Research ArticleGibbs Thermodynamics of the Renninger-Wilemski Problem

Victor Kurasov

Faculty of Physics Saint Petersburg University Kamennoostrovskiy Prospekt 5431 Office 60 Saint Petersburg 197022 Russia

Correspondence should be addressed to Victor Kurasov victor kurasovyahoocom

Received 5 August 2014 Accepted 2 January 2015

Academic Editor Y Peles

Copyright copy 2015 Victor Kurasov This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Renninger-Wilemski problem in nucleation is analyzed The Gibbs dividing surfaces method with external parameters is usedto enrich the initial model It is shown that both the traditional (Doyle) model and the Renninger-Wilemski model are not completeones and namely the Gibbs dividing surface approach can solve this problem It is shown that the application of the Gibbs approachalso requires some model constructionsThe simplified Gibbs model is proposed It is shown that the simplified Gibbs model givesfor the height of activation barrier the same numerical results as the Renninger-Wilemski model

1 Introduction

Although the principles of thermodynamics are wellgrounded there exist many narrow questions in applicationsOne of such questions belongs to the new phase embryosformation Despite the fact that the free energy of the newphase embryo can be calculated in frames of equilibriumthermodynamics there remain many questions of accountof specific conditions for the critical embryo appearance

There is no need to demonstrate the actuality of nucle-ation taking place practically overall The importance ofnucleation initiates investigations fromWilsonrsquos experiments[1ndash3] up to modern investigations Thermodynamic aspectsof investigations occupy an important place in this fieldconsidering not only the fundamental question of stationarity(see to clarify it eg [4 5]) but also the problems of smallnessof characteristic embryos and corrections appeared becauseof its sizes and shapes [6] The results given by thermo-dynamics immediately lead to quantitative description ofkinetics [7] and can be included into consideration of somerather complex aggregates with very specific properties [8]This consideration is based mainly on thermodynamics ofsmall embryos Here we will discuss namely some importantspecific features of the small embryos thermodynamics

The bright demonstration of specific features of ther-modynamic aspects is the Renninger-Wilemski problemwhich occupies an important place in the nucleation of

multicomponent liquids An abbreviated version of thisproblem sounds as follows

(i) Is it necessary to take the derivative of the surface ten-sion on concentration to determine the characteristicsof a critical droplet

From the first glance this problem sounds so simple that theanswer seems to be evident But it took a dozens of years toclarify this problem

This problem arises from the very beginning of investi-gations of the multicomponent nucleation since the pioneerpaper written by Reiss [9] but until the paper of Renninger etal [10] this problem was somehow in a shadow The classicaltheory of nucleation [11 12] is standing somehow aside ofthis question The reason of this shadow is the rather weakdependence of the surface tension on concentration in theintermediate region of concentrations for the dominatingmajority of mixtures The physical nature of this weaknessis the enrichment of the surface layer with the surface-active molecules When it is easy to find in the bulk of anembryo a molecule of the surface-active substance then itis profitable to catch it and to put in the surface Roughlyspeaking only an entropy of mixing has to be paid here Theenergy profit in this situation is a surface with a surface-activesubstance So the surface tension appears to be a relativelysmooth function of concentration Only when it is difficult tofind such a molecule in a bulk liquid one has to pay some

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2 Journal of Thermodynamics

relatively essential work to find it and the surface tensionbegins to depend on concentration essentially But the lastsituation occurs at small concentrations and this case shouldbe preferably described as a one-component nucleation withsome impurities

The mentioned weak dependence allows many authorsin 1950ndash80s to ignore the fact of dependence of the sur-face tension on concentration although there exist someinvestigations like that by Mirabel and Katz [13] In [13] thecorrection for the Young-Laplace formula with the derivativeof the surface tension on concentration was presented Thiscorrection is rather doubtful

The statement that it is necessary to forbid to differentiatethe surface tension was announced by Renninger and coau-thors in [10] criticizing mainly Doyle [14] and Mirabel andKatz [13] Before [10] the situation was the following thereexists the expression for the free energy it is clear that the freeenergy of the critical embryo is the free energy at the saddlepoint and it is clear that to get the coordinates of the saddlepoint it is necessary to take the free energy partial derivativesand to put them to zero but no special attention was paidfor the question what to do with the derivative of the surfacetension on the solution concentration

It was Doyle [15] who presented contrary to [10] thearguments for the traditional version with the differentiationof the free energy on concentration having justified his initialapproach in [14] Later Wilemski clarified the problem in [1617] and one could say after [16 17] that the phenomenologicalrecipe required that it was necessary to forbid to differentiatethe surface tension This seemed to be the final conclusionin this question Since that time the problem is called theRenninger-Wilemski (RW) problem and seemed to be theclosed problem in thermodynamics

This problemwas revitalized in 2003 byReiss andReguera[18 19] who showed by a refinement of some connections forderivatives of chemical potentials in droplet the alternativevariant of derivation of equations which are similar to thoseused by Dole One can say that technically the publication[18 19] was the top of analysis of this problem But theGibbs approach has not been considered in [18 19] Itwas announced in [18 19] that the Gibbs approach [20]ldquois of course the most sophisticated and general but itsapplication requires more information than is available inmacroscopic thermodynamic observablesrdquo Meanwhile therecent investigations on thermodynamics of complex systemssuccessfully operate with the quantities of excesses which istypical for the Gibbs dividing surfaces conception

One has to mention that already after Wilemski in thefield of multicomponent nucleation many models use theelements of the Gibbs dividing surface method but in theirproper constructions Instead of application of thermody-namicswith some parameters of themodel these papers try tobuild their ldquotruerdquo constructions without any choice of param-etersHere one canmention the approach ofDebenedetti [21]the model of Nishioka and Kusaka [22] and the approachof Laaksonen et al [23] which was analyzed also in [24] Allthese models have the features of the Gibbs dividing surfacemethod but all of them propose some specific relations toclose the system of equations All of them announced the

recipe for determination of the critical embryo characteristicsas the only ldquotruerdquo one although the Gibbsmodel is principallya model with external parameters

Here we will analyze both traditional models (the Doylemodel and the Renninger-Wilemski model) and show thatboth of them are not self-consistent The alternative modelas realization of the Gibbs dividing surface method will bepresented In the last part it will be shown how to simplifythis approach

2 Initial Remarks

To determine the coordinates of the critical embryo one canfollow at least two ways

(i) The first way is to construct some expression (namelythe expression in the capillary approximation) forthe total free energy of the critical embryo formationand then to determine the coordinates of the criticalembryo as the coordinates of the free energy saddlepoint by differentiating and putting the derivatives tozero

(ii) The second way is to use the general relations comingfrom the differential analysis such as the Laplace-Young formula and the Gibbs-Thomson formula forthe equilibrium (critical) droplet and to put into theseformulas some concrete models for necessary values

When the expression for the free energy is precise thenboth approaches have to give one and the same result Thetrouble is that everything we can suggest is no more thanan approximation In fact these approaches historically led totwo different results and this discrepancy has now the nameof the Renninger-Wilemski problem (for such a terminologysee [18 19]) Now it is clear that both results can be derived inboth approaches but historically Renninger et al [10] used thesecond approach and their opponent Doyle [15] used the firstone Following the history of the questionwewill speak aboutRW-model and about D-model correspondingly Certainlythis classification is rather artificial

The question of thermodynamic derivations of the sur-face characteristics was the subject of numerous investiga-tions The thermodynamic definition of the surface tensionis discussed in [25] Thermodynamics of a surface layer isanalyzed in [26]

At first it is necessary to present the formal thermody-namic derivation in order to see where the hidden supposi-tions about the structure of the system under considerationhave been done Here we follow the manner of [18 19]because namely that derivation opened the possibility to seedifferent approaches in frames of the differential formalismCertainly some essential modifications will be introduced

Since in the systemwhere the embryo is born the variables119879 (temperature) and119881 (volume) are supposed to be fixed onehas to use the Helmholtz free energy

119865 = 119880 minus 119879119878 (1)

Really according to principles of thermodynamics

119889119865 = minus119878119889119879 minus 119901119889119881 (2)

Journal of Thermodynamics 3

We suppose that the terms sum119894120583119894119873119894(120583119894are the chemical

potentials 119873119894are the numbers of particles and index 119894

numerates the components) are included into thermody-namic potentials1

One has to split119880 and 119878 into contributions from the liquidphase (index 119897) and the vapor phase (index V)2 Then

119880 = 119880119897+ 119880V 119878 = 119878

119897+ 119878V 119881 = 119881

119897+ 119881V

119889119880V = 119879119889119878V minus 119901V119889119881V +sum

119894

120583V119894119889119873V119894

119889119880119897= 119879119889119878

119897minus 119901119897119889119881119897+sum

119894

120583119897119894119889119873119897119894

(3)

Since

119889119881119897= minus119889119881V 119889119873V119894 = minus119889119873

119897119894 (4)

we come to

119889119880 = 119879119889119878 minus (119901119897minus 119901V) 119889119881119897 +sum

119894

(120583119897119894minus 120583V119894) 119889119873119897119894 (5)

Here the symbol sdot sdot sdot indicates that the pressure is the pressureunder the curved surface that is the Young-Laplace correc-tion 2120574119903 has to be added (120574 is a surface tension 119903 is a radiusof a droplet) Then

119889119865 = minus(119901119897minus (119901V +

2120574

119903)) 119889119881

119897+sum

119894

(120583119897119894minus 120583V119894) 119889119873119897119894 (6)

In the state of equilibrium we have 119889119865 = 0 which gives

119901119897= 119901V +

2120574

119903 (7)

This is the Young-Laplace equation and

120583119897119894[119901V +

2120574

119903] = 120583V119894 [119901V] (8)

Here and later the square brackets show the functionaldependence to avoid misreading as multiplication If theindex of variable is absent it means that the dependence takesplace over the whole set of variables

Now it is necessary to go in chemical potentials from 119901

to 119901 + 2120574119903 explicitly To do this it is necessary to calculate120597120583119897119894120597119901 To fulfil this transformation we take the Maxwell

relations for the Gibbs free energy 119866119897= 119880119897minus119879119878119897+119901119897119881119897with a

differential

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894 (9)

Then

120597120583119894119897

120597119901119897

= V119897119894 (10)

where V119897119894is the volume per one molecule of component 119894 in a

liquid phase We suppose the liquid to be incompressible Sothe rhs of the previous equation is constant and can be easily

integrated Equation (10) is of great importance since it willlead to the Gibbs-Thomson equation Then

120583119897119894[119901 +

2120574

119903] = 120583119897119894[119901] +

V1198971198942120574

119903(11)

and (8) looks like

Δ120583119894=V1198971198942120574

119903 (12)

whereΔ120583119894is the difference of chemical potentials at the exter-

nal pressure in the system The last relation is the so-calledGibbs-Thomson equation well known in thermodynamics

Equation (12) leads to invariant

Δ120583119897119894

V119897119894

=

Δ120583119897119895

V119897119895

(13)

and allows a clear physical interpretation Namely (12) is thebasis for RW-model

Here it is necessary to make one very essential commentHaving written equation (9) for the differential of 119866

119897= 119880119897minus

119879119878119897+ 119901119897119881119897we miss the term 120574119860 that is the surface tension

120574 multiplied on the surface area 119860 We can include this termand write

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (14)

This does not change the derivation It is another momentwhich is important here Beside this we have to note thatordinary it is implied that 120574119860 is independent of othercharacteristics presented in (14) But in the droplet there isa connection between 119860 and119873

119897119894 This puts a question on the

validity of this derivation It will be discussed belowEquation (6) for 119889119865 can be integrated from119873

119897119894= 0 up to

the current size which gives 119865[119873119894] To fulfil this procedure

it is convenient to present the dependence on 119873119897119894as the

dependence on concentrations defined3 as

120585119894=

119873119897119894

1198730

(15)

where

1198730= sum

119894

119873119897119894 (16)

is the total number of molecules in a dropAs an extensive variable4 we can take119873

0 but some other

variants of choice are possible and will be presented belowInstead of integration one can suggest to present the

model for 119865 and then to check whether it leads to (7) (12)by differentiation We are interested here namely in the lastrelations

One can easily suggest the following expression for 119865

119865 [119873119897119894] = 119865 [0] minussum

119894

Δ120583119894119873119897119894+ 120574119860 (17)

Here and below119860 is the surface area of the embryoThe term119881119897(119901119897minus119901) ismissed here since it is very small It corresponds to

4 Journal of Thermodynamics

the difference between the situation when the volume of thesystem is fixed or the pressure in the system is fixed Concreteconditions ordinary are not well posed

The condition for the equilibrium is

120597119865

120597119873119897119894

= 0 (18)

The explicit differentiation has to be done with account of theGibbs-Duhem equation for the bulk liquid

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 = 0 (19)

which is actually

sum119873119897119894119889120583119897119894= 0 (20)

The last relation will be satisfied automatically and inevitablywhen we use some concrete dependencies for chemicalpotentials in liquid

The result of differentiation (we take for simplicity a two-component case) on account of

119860 = 119862(sum

119894

V119897119894119873119897119894)

23

119862 = (36120587)13 (21)

is

120597119865 [119873119897119894]

120597119873119897119894

= minusΔ120583119894+

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (22)

where

V119898= sum120585

119894V119897119894 (23)

We see that the equilibrium condition

Δ120583119894=

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (24)

differs from the Gibbs-Thomson equation by the last termThis equation is the basis in D-model

In RW-model one can also explain how to come to (12)starting from (17) In the differentiation of 119865 instead of (19) itis necessary to take the surface variant of the Gibbs-Duhemequation

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 + 119860119889120574 = 0 (25)

which can be approximately reduced to

sum119873119897119894119889120583119897119894+ 119860119889120574 = 0 (26)

The last term in the previous equation cancels the derivativeof the surface tension on concentration and we come toGibbs-Thomson equations (12)

For a long time it seemed that (12) is preferable becausethere exists a differential method to see (12) But Reiss andReguera showed that it is possible to get (24) in modificationof the way we came here to (12) In integration 119889120583

119897119894119889119901 it is

necessary to take into account not only the direct dependence

120597120583119897119894120597119901 = V

119897119894but also the dependence on size of the embryo

through the dependence

120597120583119897119894

120597119860=

120597120574

120597119873119897119894

(27)

The last relation can be found if we start with the Gibbspotential taking into account that in119880

119897there already exits the

surface term 120574119860 Then

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (28)

Then theMaxwell relation on variables119873119897119894119860 gives (27)Then

the integration will lead precisely to (24) This derivationrevitalizes D-model

To complete the overview it is necessary to add theapproach of Mirabel and Katz [13] who follow the differentialway of consideration which led to (12) In order to get (24)Mirabel and Katz modified Young-Laplace equation (7) as

119901119897minus 119901V minus

2120574

119903sim (

119889120574

119889120585119894

) (29)

In [10] this approach was classified as an inappropriateone We see that this approach is one of explanations how toreconcile the requirement to get the Gibbs-Kelvin equationand the requirement to have the free energy in the capillarymodel as described above So under the mentioned form ofthe free energy the true alternative is to modify the Gibbs-Kelvin equation or to modify the Young-Laplace equation

3 Physical Model

Ideas to find a tool outside the nucleation theory to solvethe problem formulated above seem to be very attractiveBut unfortunately there is no such tool Speaking about thepossibility of the direct measurement of the free energy ofembryos one has to confess that except for some specificcases (see [27]) there is no direct mechanism to realize thispossibility

Certainly practically every reliable model can be ex-pressed in terms of statistic-mechanical approach (see [28])but this can not be a tool to decide whether one can cancel thederivative of the surface tension Here also both approachescan be justified on statistic-mechanical basis the matter iswhich reference state of equilibrium is chosen

One has to stress that the density functional theorycannot be the instrument to decide to cancel the derivatives[29 30] because here themodel formof the free energy is sim-ply postulated at the level of the elementary hydrodynamicsubsystems

Unfortunately the comparisonwith the 2D Isingmodel asit is proposed in [31] to solve the question of the necessity ofcancelation cannot be fulfilled because one cannot generalizethis model to the multicomponent case

So we have to formulate some physicalmodel and discussit

Journal of Thermodynamics 5

31 Different Meanings of the Surface Tension One cannotargue that some excesses initiated by the interface have to beput into the theoryThemost common one is the excess of thefree energy that is the surface tension

The answer to the question what approach is the mostappropriate one lies in the method of construction of thecapillary approximation and the expression for a free energyin this approximation We have to reformulate a questionwhat is the surface tension which we will put in the surfaceterm for the free energy of the embryo Several answers arepossible

(i) This is the surface tension of an embryoThen the nucleation theory is the absolutely self-closed theory and it cannot be checked At themodern level of experimental devices there is no wayto measure the surface tension of droplets otherwisethan in the nucleation rate experiments Everythingwhich is measured in experiments is attributed tothe value of surface tension according to theoreticaldependencies derived in frames of an approach tocalculate the nucleation rate

(ii) This is the surface tension 120574119905observed in a capillary

tube (index 119905) of the radius equal to the radius of theembryo (with corresponding wetting)Here one can perform direct experiments to measurethe surface tension This is the way to constructa theory with the predicting force contrast to theprevious case

(iii) The surface tension is the plane surface tension 120574119901

Certainly this is a rather rough approximation butordinary namely the flat surface tension is knownBy application of the methods of thermodynamicsof small objects it is possible to construct someapproximation for the surface tension in a tube and toreduce this case to the previous one Also one has tomention that in some sense this case can be preferablebecause the surface excesses are mainly known in thecase of the flat surface

(iv) One can propose the conventional variant 120574 is thesurface tension of a flat surface but at the surpluspressure (plus the Laplace-Young addition) this valuecan be found by interpolation from 119875 119879 coexistencecurve to surplus temperature or pressureThere are no principal differences between this vari-ant and the previous one So no special attention willbe paid to this variant

To go further one has to note that the capillary tube andthe flat liquid are not a closed object as the droplet is This isthe inevitable difference and this difference will be importantThe question is now the following

(i) is it enough to account the compactness of the embryoonly at the level of expression for 119860 as a function of thenumber ofmolecules inside the embryo or is it necessaryto rewrite the analog of the Gibbs-Duhem equationwhich will be different from (25)

Our conclusion is that there is no direct answer on thisquestion in frames of macroscopic thermodynamics butthe principle of correspondence formulated below is ratherreliable and solves this question

32Mean Chemical Potentials in DifferentModels Theobjectof further investigation will be the cluster of a liquid phasewhich is dense and compact contrast to a vapor phase Sowe will omit the index 119897 use ]

119894instead of 119873

119897119894 and speak

about chemical potentials instead of differences in chemicalpotentials5

We will formulate the principle of correspondence

(i) It is necessary to use the Gibbs-Duhem equation fromthe system which provides the surface tension usedin the model If we use the surface tension from aflat surface we write the Gibbs-Duhem equation fora flat surface If we use the surface tension from acapillary tube we write the Gibbs-Duhem equationfor a capillary tube

Certainly the most similar system with the explicitly mea-sured surface tension for the situation of the embryo is thecapillary tube We take the surface tension from the capillarytube and have to write the Gibbs-Duhem equation for thecapillary tube This leads (as it will be shown later) to theRenninger-Wilemski recipe But one has to stress that theprinciple of correspondence is no more than a suppositionAlso it is necessary to add that one can act in a manner of theexplicit attribution of the surface excesses which is typical forthe Gibbs dividing surface approach

In capillary models one has to use the surface tensionfrom the situation with a capillary tube It leads to thefollowing consequence discussed below

(i) In the droplet there is a strong connection betweenthe radius 119903 and the number of molecules ]

119894 this

connection is absent in the capillary tube

The last connection can be written in a most simple versionof the theory (without surface excesses taken into account) as

41205871199033

3= sum

119894

]119894V119897119894 (30)

If we use the Gibbs formalism there are surface excessesΨ119894of

the substance (index 119904marks the surface) and then

41205871199033

3= sum

119894

(]119894minus Ψ119894) V119897119894 (31)

We attribute in frames of conception of a phase in the Gibbsmethod all other molecules to the bulk phase

]119887119894= ]119894minus Ψ119894 (32)

(index 119887marks the bulk region)Approximately Ψ

119894are proportional to the surface area

41205871199032

Ψ119894= 4120587119903

2120588119894 (33)

where 120588119894is the density of excess6

6 Journal of Thermodynamics

As the main dividing surface it is preferable to take thesurface of tension where the Young-Laplace equation is pre-cise

Then

41205871199033

3= sum

119894

(]119894minus 4120587119903

2120588119894) V119897119894 (34)

is an equation on 119903 as a function of ]119894

The concentrations are defined as

120585119894=

]119887119894

sum119895]119887119895

(35)

It is a principal requirement that in the definition of concen-tration there stand the bulk numbers of molecules

Now we will present constructions on the base of acapillary tube as a corresponding system for the free energyin capillary approximation Until the place where the Gibbsmodel is formulatedwe forget about the excesses at the dividingsurface In the capillary tube there is no such balance relationas ]119887119894= ]119894minusΨ119894 If one constructs the free energy of a capillary

tube one gets

119865119905= sum

119894

119873119894120583119894+ 120574119905119860 (36)

(index 119905 marks that we deal with a tube 119873119894are the numbers

of molecules in a tube) Since the numbers 119873119894are not fixed

in principle (the tube is connected with a bath) one has toexpress the last relation through concentrations This gives

119865119905= sum119873

119894119861119887[120585] + 120574119860 (37)

where 119861119887is

119861119887= sum120585

119894120583119894[120585] (38)

The value of 119861119887has the sense of the averaged chemical poten-

tial Consider

119861119887= ⟨120583⟩ (39)

The condition for the equilibrium will be

119889119861119887

119889120585= 0 (40)

Since Gibbs-Duhem equation (19) takes place the last condi-tion means

120583119894= 120583119895 (41)

with a clear physical senseThe same picture will be in a situation with a plane

surface The difference is that the surplus chemical potentialsare counted here from the coexistence line while in the caseof a tube they are counted from the potentials in vapor phaseunder the Laplace pressure

The term 120574119860 does not influence the concentration equi-librium when we suppose that the contact angle in a capillary

tube is constant (namely this situation is considered) Cer-tainly there exists a dependence on concentration throughthe increase of pressure but we suppose that the dependenceof activity coefficients that is the functions 119891

119894staying in

standard expressions

120583119894= ln(

119901

119901eq) minus ln 120585

119894minus ln119891

119894[120585119894] (42)

on pressure is small Here 119901eq is the pressure over a pureplane liquid phase and ln(119901119901eq) is a logarithm of thesupersaturation plus one

The situation in a droplet is another oneThe surface area119860 depends on 119873

119894and we have to take this dependence into

account There are several ways how to do itTo see the preferable way we will analyze the structure of

the free energy in the embryo We present 119865 from (17) in theform like that in the one-dimensional case

119865 = minus119861119889Ω + Ω

23 (43)

Here Ω is defined in order that Ω23 is the surface surplusinput into the droplet free energy

Ω23

= 119860120574 (44)

For 119861119889one can get on the base of (17)

119861119889= 119862minus32

120574minus32

(119861119887

V119898

) (45)

where V119898is the mean volume of a molecule in a droplet

The value of 119861119889allows the following interpretation

119861119889sim

⟨120583⟩

12057432 ⟨V⟩(46)

and one can consider the denominator in the last expressionas the mean surface tension for one molecule in a power 32(certainly one molecule has no surface tension)

Condition for an equilibrium concentration 120585119888will be

119889119861119889

119889120585119894

= 0 (47)

Condition for an equilibrium when the derivative of 120574 is putto zero will be

119889 (119861119887V119898)

119889120585119894

= 0 (48)

Here one can also get the interpretation for 119861119887V119898as the ratio

of two mean values ⟨120583⟩ = 120583119898and ⟨V⟩ = V

119898

Both conditions are independent of the value of Ω andit is important When we choose as a variable which issupplementary to 120585 any other variable we can come to thedependence of 120585

119888on Ω but at the saddle point the result will

be the sameWewill call119861

119889themean chemical potential in the droplet

in D-model and 119861RW sim 119861119887V119898the mean chemical potential

in the droplet in RW-modelThe property of independence on 120574 of the minimum of

119861119887V119898is the serious argument for RW-model and against D-

model

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

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2 Journal of Thermodynamics

relatively essential work to find it and the surface tensionbegins to depend on concentration essentially But the lastsituation occurs at small concentrations and this case shouldbe preferably described as a one-component nucleation withsome impurities

The mentioned weak dependence allows many authorsin 1950ndash80s to ignore the fact of dependence of the sur-face tension on concentration although there exist someinvestigations like that by Mirabel and Katz [13] In [13] thecorrection for the Young-Laplace formula with the derivativeof the surface tension on concentration was presented Thiscorrection is rather doubtful

The statement that it is necessary to forbid to differentiatethe surface tension was announced by Renninger and coau-thors in [10] criticizing mainly Doyle [14] and Mirabel andKatz [13] Before [10] the situation was the following thereexists the expression for the free energy it is clear that the freeenergy of the critical embryo is the free energy at the saddlepoint and it is clear that to get the coordinates of the saddlepoint it is necessary to take the free energy partial derivativesand to put them to zero but no special attention was paidfor the question what to do with the derivative of the surfacetension on the solution concentration

It was Doyle [15] who presented contrary to [10] thearguments for the traditional version with the differentiationof the free energy on concentration having justified his initialapproach in [14] Later Wilemski clarified the problem in [1617] and one could say after [16 17] that the phenomenologicalrecipe required that it was necessary to forbid to differentiatethe surface tension This seemed to be the final conclusionin this question Since that time the problem is called theRenninger-Wilemski (RW) problem and seemed to be theclosed problem in thermodynamics

This problemwas revitalized in 2003 byReiss andReguera[18 19] who showed by a refinement of some connections forderivatives of chemical potentials in droplet the alternativevariant of derivation of equations which are similar to thoseused by Dole One can say that technically the publication[18 19] was the top of analysis of this problem But theGibbs approach has not been considered in [18 19] Itwas announced in [18 19] that the Gibbs approach [20]ldquois of course the most sophisticated and general but itsapplication requires more information than is available inmacroscopic thermodynamic observablesrdquo Meanwhile therecent investigations on thermodynamics of complex systemssuccessfully operate with the quantities of excesses which istypical for the Gibbs dividing surfaces conception

One has to mention that already after Wilemski in thefield of multicomponent nucleation many models use theelements of the Gibbs dividing surface method but in theirproper constructions Instead of application of thermody-namicswith some parameters of themodel these papers try tobuild their ldquotruerdquo constructions without any choice of param-etersHere one canmention the approach ofDebenedetti [21]the model of Nishioka and Kusaka [22] and the approachof Laaksonen et al [23] which was analyzed also in [24] Allthese models have the features of the Gibbs dividing surfacemethod but all of them propose some specific relations toclose the system of equations All of them announced the

recipe for determination of the critical embryo characteristicsas the only ldquotruerdquo one although the Gibbsmodel is principallya model with external parameters

Here we will analyze both traditional models (the Doylemodel and the Renninger-Wilemski model) and show thatboth of them are not self-consistent The alternative modelas realization of the Gibbs dividing surface method will bepresented In the last part it will be shown how to simplifythis approach

2 Initial Remarks

To determine the coordinates of the critical embryo one canfollow at least two ways

(i) The first way is to construct some expression (namelythe expression in the capillary approximation) forthe total free energy of the critical embryo formationand then to determine the coordinates of the criticalembryo as the coordinates of the free energy saddlepoint by differentiating and putting the derivatives tozero

(ii) The second way is to use the general relations comingfrom the differential analysis such as the Laplace-Young formula and the Gibbs-Thomson formula forthe equilibrium (critical) droplet and to put into theseformulas some concrete models for necessary values

When the expression for the free energy is precise thenboth approaches have to give one and the same result Thetrouble is that everything we can suggest is no more thanan approximation In fact these approaches historically led totwo different results and this discrepancy has now the nameof the Renninger-Wilemski problem (for such a terminologysee [18 19]) Now it is clear that both results can be derived inboth approaches but historically Renninger et al [10] used thesecond approach and their opponent Doyle [15] used the firstone Following the history of the questionwewill speak aboutRW-model and about D-model correspondingly Certainlythis classification is rather artificial

The question of thermodynamic derivations of the sur-face characteristics was the subject of numerous investiga-tions The thermodynamic definition of the surface tensionis discussed in [25] Thermodynamics of a surface layer isanalyzed in [26]

At first it is necessary to present the formal thermody-namic derivation in order to see where the hidden supposi-tions about the structure of the system under considerationhave been done Here we follow the manner of [18 19]because namely that derivation opened the possibility to seedifferent approaches in frames of the differential formalismCertainly some essential modifications will be introduced

Since in the systemwhere the embryo is born the variables119879 (temperature) and119881 (volume) are supposed to be fixed onehas to use the Helmholtz free energy

119865 = 119880 minus 119879119878 (1)

Really according to principles of thermodynamics

119889119865 = minus119878119889119879 minus 119901119889119881 (2)

Journal of Thermodynamics 3

We suppose that the terms sum119894120583119894119873119894(120583119894are the chemical

potentials 119873119894are the numbers of particles and index 119894

numerates the components) are included into thermody-namic potentials1

One has to split119880 and 119878 into contributions from the liquidphase (index 119897) and the vapor phase (index V)2 Then

119880 = 119880119897+ 119880V 119878 = 119878

119897+ 119878V 119881 = 119881

119897+ 119881V

119889119880V = 119879119889119878V minus 119901V119889119881V +sum

119894

120583V119894119889119873V119894

119889119880119897= 119879119889119878

119897minus 119901119897119889119881119897+sum

119894

120583119897119894119889119873119897119894

(3)

Since

119889119881119897= minus119889119881V 119889119873V119894 = minus119889119873

119897119894 (4)

we come to

119889119880 = 119879119889119878 minus (119901119897minus 119901V) 119889119881119897 +sum

119894

(120583119897119894minus 120583V119894) 119889119873119897119894 (5)

Here the symbol sdot sdot sdot indicates that the pressure is the pressureunder the curved surface that is the Young-Laplace correc-tion 2120574119903 has to be added (120574 is a surface tension 119903 is a radiusof a droplet) Then

119889119865 = minus(119901119897minus (119901V +

2120574

119903)) 119889119881

119897+sum

119894

(120583119897119894minus 120583V119894) 119889119873119897119894 (6)

In the state of equilibrium we have 119889119865 = 0 which gives

119901119897= 119901V +

2120574

119903 (7)

This is the Young-Laplace equation and

120583119897119894[119901V +

2120574

119903] = 120583V119894 [119901V] (8)

Here and later the square brackets show the functionaldependence to avoid misreading as multiplication If theindex of variable is absent it means that the dependence takesplace over the whole set of variables

Now it is necessary to go in chemical potentials from 119901

to 119901 + 2120574119903 explicitly To do this it is necessary to calculate120597120583119897119894120597119901 To fulfil this transformation we take the Maxwell

relations for the Gibbs free energy 119866119897= 119880119897minus119879119878119897+119901119897119881119897with a

differential

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894 (9)

Then

120597120583119894119897

120597119901119897

= V119897119894 (10)

where V119897119894is the volume per one molecule of component 119894 in a

liquid phase We suppose the liquid to be incompressible Sothe rhs of the previous equation is constant and can be easily

integrated Equation (10) is of great importance since it willlead to the Gibbs-Thomson equation Then

120583119897119894[119901 +

2120574

119903] = 120583119897119894[119901] +

V1198971198942120574

119903(11)

and (8) looks like

Δ120583119894=V1198971198942120574

119903 (12)

whereΔ120583119894is the difference of chemical potentials at the exter-

nal pressure in the system The last relation is the so-calledGibbs-Thomson equation well known in thermodynamics

Equation (12) leads to invariant

Δ120583119897119894

V119897119894

=

Δ120583119897119895

V119897119895

(13)

and allows a clear physical interpretation Namely (12) is thebasis for RW-model

Here it is necessary to make one very essential commentHaving written equation (9) for the differential of 119866

119897= 119880119897minus

119879119878119897+ 119901119897119881119897we miss the term 120574119860 that is the surface tension

120574 multiplied on the surface area 119860 We can include this termand write

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (14)

This does not change the derivation It is another momentwhich is important here Beside this we have to note thatordinary it is implied that 120574119860 is independent of othercharacteristics presented in (14) But in the droplet there isa connection between 119860 and119873

119897119894 This puts a question on the

validity of this derivation It will be discussed belowEquation (6) for 119889119865 can be integrated from119873

119897119894= 0 up to

the current size which gives 119865[119873119894] To fulfil this procedure

it is convenient to present the dependence on 119873119897119894as the

dependence on concentrations defined3 as

120585119894=

119873119897119894

1198730

(15)

where

1198730= sum

119894

119873119897119894 (16)

is the total number of molecules in a dropAs an extensive variable4 we can take119873

0 but some other

variants of choice are possible and will be presented belowInstead of integration one can suggest to present the

model for 119865 and then to check whether it leads to (7) (12)by differentiation We are interested here namely in the lastrelations

One can easily suggest the following expression for 119865

119865 [119873119897119894] = 119865 [0] minussum

119894

Δ120583119894119873119897119894+ 120574119860 (17)

Here and below119860 is the surface area of the embryoThe term119881119897(119901119897minus119901) ismissed here since it is very small It corresponds to

4 Journal of Thermodynamics

the difference between the situation when the volume of thesystem is fixed or the pressure in the system is fixed Concreteconditions ordinary are not well posed

The condition for the equilibrium is

120597119865

120597119873119897119894

= 0 (18)

The explicit differentiation has to be done with account of theGibbs-Duhem equation for the bulk liquid

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 = 0 (19)

which is actually

sum119873119897119894119889120583119897119894= 0 (20)

The last relation will be satisfied automatically and inevitablywhen we use some concrete dependencies for chemicalpotentials in liquid

The result of differentiation (we take for simplicity a two-component case) on account of

119860 = 119862(sum

119894

V119897119894119873119897119894)

23

119862 = (36120587)13 (21)

is

120597119865 [119873119897119894]

120597119873119897119894

= minusΔ120583119894+

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (22)

where

V119898= sum120585

119894V119897119894 (23)

We see that the equilibrium condition

Δ120583119894=

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (24)

differs from the Gibbs-Thomson equation by the last termThis equation is the basis in D-model

In RW-model one can also explain how to come to (12)starting from (17) In the differentiation of 119865 instead of (19) itis necessary to take the surface variant of the Gibbs-Duhemequation

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 + 119860119889120574 = 0 (25)

which can be approximately reduced to

sum119873119897119894119889120583119897119894+ 119860119889120574 = 0 (26)

The last term in the previous equation cancels the derivativeof the surface tension on concentration and we come toGibbs-Thomson equations (12)

For a long time it seemed that (12) is preferable becausethere exists a differential method to see (12) But Reiss andReguera showed that it is possible to get (24) in modificationof the way we came here to (12) In integration 119889120583

119897119894119889119901 it is

necessary to take into account not only the direct dependence

120597120583119897119894120597119901 = V

119897119894but also the dependence on size of the embryo

through the dependence

120597120583119897119894

120597119860=

120597120574

120597119873119897119894

(27)

The last relation can be found if we start with the Gibbspotential taking into account that in119880

119897there already exits the

surface term 120574119860 Then

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (28)

Then theMaxwell relation on variables119873119897119894119860 gives (27)Then

the integration will lead precisely to (24) This derivationrevitalizes D-model

To complete the overview it is necessary to add theapproach of Mirabel and Katz [13] who follow the differentialway of consideration which led to (12) In order to get (24)Mirabel and Katz modified Young-Laplace equation (7) as

119901119897minus 119901V minus

2120574

119903sim (

119889120574

119889120585119894

) (29)

In [10] this approach was classified as an inappropriateone We see that this approach is one of explanations how toreconcile the requirement to get the Gibbs-Kelvin equationand the requirement to have the free energy in the capillarymodel as described above So under the mentioned form ofthe free energy the true alternative is to modify the Gibbs-Kelvin equation or to modify the Young-Laplace equation

3 Physical Model

Ideas to find a tool outside the nucleation theory to solvethe problem formulated above seem to be very attractiveBut unfortunately there is no such tool Speaking about thepossibility of the direct measurement of the free energy ofembryos one has to confess that except for some specificcases (see [27]) there is no direct mechanism to realize thispossibility

Certainly practically every reliable model can be ex-pressed in terms of statistic-mechanical approach (see [28])but this can not be a tool to decide whether one can cancel thederivative of the surface tension Here also both approachescan be justified on statistic-mechanical basis the matter iswhich reference state of equilibrium is chosen

One has to stress that the density functional theorycannot be the instrument to decide to cancel the derivatives[29 30] because here themodel formof the free energy is sim-ply postulated at the level of the elementary hydrodynamicsubsystems

Unfortunately the comparisonwith the 2D Isingmodel asit is proposed in [31] to solve the question of the necessity ofcancelation cannot be fulfilled because one cannot generalizethis model to the multicomponent case

So we have to formulate some physicalmodel and discussit

Journal of Thermodynamics 5

31 Different Meanings of the Surface Tension One cannotargue that some excesses initiated by the interface have to beput into the theoryThemost common one is the excess of thefree energy that is the surface tension

The answer to the question what approach is the mostappropriate one lies in the method of construction of thecapillary approximation and the expression for a free energyin this approximation We have to reformulate a questionwhat is the surface tension which we will put in the surfaceterm for the free energy of the embryo Several answers arepossible

(i) This is the surface tension of an embryoThen the nucleation theory is the absolutely self-closed theory and it cannot be checked At themodern level of experimental devices there is no wayto measure the surface tension of droplets otherwisethan in the nucleation rate experiments Everythingwhich is measured in experiments is attributed tothe value of surface tension according to theoreticaldependencies derived in frames of an approach tocalculate the nucleation rate

(ii) This is the surface tension 120574119905observed in a capillary

tube (index 119905) of the radius equal to the radius of theembryo (with corresponding wetting)Here one can perform direct experiments to measurethe surface tension This is the way to constructa theory with the predicting force contrast to theprevious case

(iii) The surface tension is the plane surface tension 120574119901

Certainly this is a rather rough approximation butordinary namely the flat surface tension is knownBy application of the methods of thermodynamicsof small objects it is possible to construct someapproximation for the surface tension in a tube and toreduce this case to the previous one Also one has tomention that in some sense this case can be preferablebecause the surface excesses are mainly known in thecase of the flat surface

(iv) One can propose the conventional variant 120574 is thesurface tension of a flat surface but at the surpluspressure (plus the Laplace-Young addition) this valuecan be found by interpolation from 119875 119879 coexistencecurve to surplus temperature or pressureThere are no principal differences between this vari-ant and the previous one So no special attention willbe paid to this variant

To go further one has to note that the capillary tube andthe flat liquid are not a closed object as the droplet is This isthe inevitable difference and this difference will be importantThe question is now the following

(i) is it enough to account the compactness of the embryoonly at the level of expression for 119860 as a function of thenumber ofmolecules inside the embryo or is it necessaryto rewrite the analog of the Gibbs-Duhem equationwhich will be different from (25)

Our conclusion is that there is no direct answer on thisquestion in frames of macroscopic thermodynamics butthe principle of correspondence formulated below is ratherreliable and solves this question

32Mean Chemical Potentials in DifferentModels Theobjectof further investigation will be the cluster of a liquid phasewhich is dense and compact contrast to a vapor phase Sowe will omit the index 119897 use ]

119894instead of 119873

119897119894 and speak

about chemical potentials instead of differences in chemicalpotentials5

We will formulate the principle of correspondence

(i) It is necessary to use the Gibbs-Duhem equation fromthe system which provides the surface tension usedin the model If we use the surface tension from aflat surface we write the Gibbs-Duhem equation fora flat surface If we use the surface tension from acapillary tube we write the Gibbs-Duhem equationfor a capillary tube

Certainly the most similar system with the explicitly mea-sured surface tension for the situation of the embryo is thecapillary tube We take the surface tension from the capillarytube and have to write the Gibbs-Duhem equation for thecapillary tube This leads (as it will be shown later) to theRenninger-Wilemski recipe But one has to stress that theprinciple of correspondence is no more than a suppositionAlso it is necessary to add that one can act in a manner of theexplicit attribution of the surface excesses which is typical forthe Gibbs dividing surface approach

In capillary models one has to use the surface tensionfrom the situation with a capillary tube It leads to thefollowing consequence discussed below

(i) In the droplet there is a strong connection betweenthe radius 119903 and the number of molecules ]

119894 this

connection is absent in the capillary tube

The last connection can be written in a most simple versionof the theory (without surface excesses taken into account) as

41205871199033

3= sum

119894

]119894V119897119894 (30)

If we use the Gibbs formalism there are surface excessesΨ119894of

the substance (index 119904marks the surface) and then

41205871199033

3= sum

119894

(]119894minus Ψ119894) V119897119894 (31)

We attribute in frames of conception of a phase in the Gibbsmethod all other molecules to the bulk phase

]119887119894= ]119894minus Ψ119894 (32)

(index 119887marks the bulk region)Approximately Ψ

119894are proportional to the surface area

41205871199032

Ψ119894= 4120587119903

2120588119894 (33)

where 120588119894is the density of excess6

6 Journal of Thermodynamics

As the main dividing surface it is preferable to take thesurface of tension where the Young-Laplace equation is pre-cise

Then

41205871199033

3= sum

119894

(]119894minus 4120587119903

2120588119894) V119897119894 (34)

is an equation on 119903 as a function of ]119894

The concentrations are defined as

120585119894=

]119887119894

sum119895]119887119895

(35)

It is a principal requirement that in the definition of concen-tration there stand the bulk numbers of molecules

Now we will present constructions on the base of acapillary tube as a corresponding system for the free energyin capillary approximation Until the place where the Gibbsmodel is formulatedwe forget about the excesses at the dividingsurface In the capillary tube there is no such balance relationas ]119887119894= ]119894minusΨ119894 If one constructs the free energy of a capillary

tube one gets

119865119905= sum

119894

119873119894120583119894+ 120574119905119860 (36)

(index 119905 marks that we deal with a tube 119873119894are the numbers

of molecules in a tube) Since the numbers 119873119894are not fixed

in principle (the tube is connected with a bath) one has toexpress the last relation through concentrations This gives

119865119905= sum119873

119894119861119887[120585] + 120574119860 (37)

where 119861119887is

119861119887= sum120585

119894120583119894[120585] (38)

The value of 119861119887has the sense of the averaged chemical poten-

tial Consider

119861119887= ⟨120583⟩ (39)

The condition for the equilibrium will be

119889119861119887

119889120585= 0 (40)

Since Gibbs-Duhem equation (19) takes place the last condi-tion means

120583119894= 120583119895 (41)

with a clear physical senseThe same picture will be in a situation with a plane

surface The difference is that the surplus chemical potentialsare counted here from the coexistence line while in the caseof a tube they are counted from the potentials in vapor phaseunder the Laplace pressure

The term 120574119860 does not influence the concentration equi-librium when we suppose that the contact angle in a capillary

tube is constant (namely this situation is considered) Cer-tainly there exists a dependence on concentration throughthe increase of pressure but we suppose that the dependenceof activity coefficients that is the functions 119891

119894staying in

standard expressions

120583119894= ln(

119901

119901eq) minus ln 120585

119894minus ln119891

119894[120585119894] (42)

on pressure is small Here 119901eq is the pressure over a pureplane liquid phase and ln(119901119901eq) is a logarithm of thesupersaturation plus one

The situation in a droplet is another oneThe surface area119860 depends on 119873

119894and we have to take this dependence into

account There are several ways how to do itTo see the preferable way we will analyze the structure of

the free energy in the embryo We present 119865 from (17) in theform like that in the one-dimensional case

119865 = minus119861119889Ω + Ω

23 (43)

Here Ω is defined in order that Ω23 is the surface surplusinput into the droplet free energy

Ω23

= 119860120574 (44)

For 119861119889one can get on the base of (17)

119861119889= 119862minus32

120574minus32

(119861119887

V119898

) (45)

where V119898is the mean volume of a molecule in a droplet

The value of 119861119889allows the following interpretation

119861119889sim

⟨120583⟩

12057432 ⟨V⟩(46)

and one can consider the denominator in the last expressionas the mean surface tension for one molecule in a power 32(certainly one molecule has no surface tension)

Condition for an equilibrium concentration 120585119888will be

119889119861119889

119889120585119894

= 0 (47)

Condition for an equilibrium when the derivative of 120574 is putto zero will be

119889 (119861119887V119898)

119889120585119894

= 0 (48)

Here one can also get the interpretation for 119861119887V119898as the ratio

of two mean values ⟨120583⟩ = 120583119898and ⟨V⟩ = V

119898

Both conditions are independent of the value of Ω andit is important When we choose as a variable which issupplementary to 120585 any other variable we can come to thedependence of 120585

119888on Ω but at the saddle point the result will

be the sameWewill call119861

119889themean chemical potential in the droplet

in D-model and 119861RW sim 119861119887V119898the mean chemical potential

in the droplet in RW-modelThe property of independence on 120574 of the minimum of

119861119887V119898is the serious argument for RW-model and against D-

model

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

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ThermodynamicsJournal of

Journal of Thermodynamics 3

We suppose that the terms sum119894120583119894119873119894(120583119894are the chemical

potentials 119873119894are the numbers of particles and index 119894

numerates the components) are included into thermody-namic potentials1

One has to split119880 and 119878 into contributions from the liquidphase (index 119897) and the vapor phase (index V)2 Then

119880 = 119880119897+ 119880V 119878 = 119878

119897+ 119878V 119881 = 119881

119897+ 119881V

119889119880V = 119879119889119878V minus 119901V119889119881V +sum

119894

120583V119894119889119873V119894

119889119880119897= 119879119889119878

119897minus 119901119897119889119881119897+sum

119894

120583119897119894119889119873119897119894

(3)

Since

119889119881119897= minus119889119881V 119889119873V119894 = minus119889119873

119897119894 (4)

we come to

119889119880 = 119879119889119878 minus (119901119897minus 119901V) 119889119881119897 +sum

119894

(120583119897119894minus 120583V119894) 119889119873119897119894 (5)

Here the symbol sdot sdot sdot indicates that the pressure is the pressureunder the curved surface that is the Young-Laplace correc-tion 2120574119903 has to be added (120574 is a surface tension 119903 is a radiusof a droplet) Then

119889119865 = minus(119901119897minus (119901V +

2120574

119903)) 119889119881

119897+sum

119894

(120583119897119894minus 120583V119894) 119889119873119897119894 (6)

In the state of equilibrium we have 119889119865 = 0 which gives

119901119897= 119901V +

2120574

119903 (7)

This is the Young-Laplace equation and

120583119897119894[119901V +

2120574

119903] = 120583V119894 [119901V] (8)

Here and later the square brackets show the functionaldependence to avoid misreading as multiplication If theindex of variable is absent it means that the dependence takesplace over the whole set of variables

Now it is necessary to go in chemical potentials from 119901

to 119901 + 2120574119903 explicitly To do this it is necessary to calculate120597120583119897119894120597119901 To fulfil this transformation we take the Maxwell

relations for the Gibbs free energy 119866119897= 119880119897minus119879119878119897+119901119897119881119897with a

differential

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894 (9)

Then

120597120583119894119897

120597119901119897

= V119897119894 (10)

where V119897119894is the volume per one molecule of component 119894 in a

liquid phase We suppose the liquid to be incompressible Sothe rhs of the previous equation is constant and can be easily

integrated Equation (10) is of great importance since it willlead to the Gibbs-Thomson equation Then

120583119897119894[119901 +

2120574

119903] = 120583119897119894[119901] +

V1198971198942120574

119903(11)

and (8) looks like

Δ120583119894=V1198971198942120574

119903 (12)

whereΔ120583119894is the difference of chemical potentials at the exter-

nal pressure in the system The last relation is the so-calledGibbs-Thomson equation well known in thermodynamics

Equation (12) leads to invariant

Δ120583119897119894

V119897119894

=

Δ120583119897119895

V119897119895

(13)

and allows a clear physical interpretation Namely (12) is thebasis for RW-model

Here it is necessary to make one very essential commentHaving written equation (9) for the differential of 119866

119897= 119880119897minus

119879119878119897+ 119901119897119881119897we miss the term 120574119860 that is the surface tension

120574 multiplied on the surface area 119860 We can include this termand write

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (14)

This does not change the derivation It is another momentwhich is important here Beside this we have to note thatordinary it is implied that 120574119860 is independent of othercharacteristics presented in (14) But in the droplet there isa connection between 119860 and119873

119897119894 This puts a question on the

validity of this derivation It will be discussed belowEquation (6) for 119889119865 can be integrated from119873

119897119894= 0 up to

the current size which gives 119865[119873119894] To fulfil this procedure

it is convenient to present the dependence on 119873119897119894as the

dependence on concentrations defined3 as

120585119894=

119873119897119894

1198730

(15)

where

1198730= sum

119894

119873119897119894 (16)

is the total number of molecules in a dropAs an extensive variable4 we can take119873

0 but some other

variants of choice are possible and will be presented belowInstead of integration one can suggest to present the

model for 119865 and then to check whether it leads to (7) (12)by differentiation We are interested here namely in the lastrelations

One can easily suggest the following expression for 119865

119865 [119873119897119894] = 119865 [0] minussum

119894

Δ120583119894119873119897119894+ 120574119860 (17)

Here and below119860 is the surface area of the embryoThe term119881119897(119901119897minus119901) ismissed here since it is very small It corresponds to

4 Journal of Thermodynamics

the difference between the situation when the volume of thesystem is fixed or the pressure in the system is fixed Concreteconditions ordinary are not well posed

The condition for the equilibrium is

120597119865

120597119873119897119894

= 0 (18)

The explicit differentiation has to be done with account of theGibbs-Duhem equation for the bulk liquid

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 = 0 (19)

which is actually

sum119873119897119894119889120583119897119894= 0 (20)

The last relation will be satisfied automatically and inevitablywhen we use some concrete dependencies for chemicalpotentials in liquid

The result of differentiation (we take for simplicity a two-component case) on account of

119860 = 119862(sum

119894

V119897119894119873119897119894)

23

119862 = (36120587)13 (21)

is

120597119865 [119873119897119894]

120597119873119897119894

= minusΔ120583119894+

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (22)

where

V119898= sum120585

119894V119897119894 (23)

We see that the equilibrium condition

Δ120583119894=

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (24)

differs from the Gibbs-Thomson equation by the last termThis equation is the basis in D-model

In RW-model one can also explain how to come to (12)starting from (17) In the differentiation of 119865 instead of (19) itis necessary to take the surface variant of the Gibbs-Duhemequation

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 + 119860119889120574 = 0 (25)

which can be approximately reduced to

sum119873119897119894119889120583119897119894+ 119860119889120574 = 0 (26)

The last term in the previous equation cancels the derivativeof the surface tension on concentration and we come toGibbs-Thomson equations (12)

For a long time it seemed that (12) is preferable becausethere exists a differential method to see (12) But Reiss andReguera showed that it is possible to get (24) in modificationof the way we came here to (12) In integration 119889120583

119897119894119889119901 it is

necessary to take into account not only the direct dependence

120597120583119897119894120597119901 = V

119897119894but also the dependence on size of the embryo

through the dependence

120597120583119897119894

120597119860=

120597120574

120597119873119897119894

(27)

The last relation can be found if we start with the Gibbspotential taking into account that in119880

119897there already exits the

surface term 120574119860 Then

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (28)

Then theMaxwell relation on variables119873119897119894119860 gives (27)Then

the integration will lead precisely to (24) This derivationrevitalizes D-model

To complete the overview it is necessary to add theapproach of Mirabel and Katz [13] who follow the differentialway of consideration which led to (12) In order to get (24)Mirabel and Katz modified Young-Laplace equation (7) as

119901119897minus 119901V minus

2120574

119903sim (

119889120574

119889120585119894

) (29)

In [10] this approach was classified as an inappropriateone We see that this approach is one of explanations how toreconcile the requirement to get the Gibbs-Kelvin equationand the requirement to have the free energy in the capillarymodel as described above So under the mentioned form ofthe free energy the true alternative is to modify the Gibbs-Kelvin equation or to modify the Young-Laplace equation

3 Physical Model

Ideas to find a tool outside the nucleation theory to solvethe problem formulated above seem to be very attractiveBut unfortunately there is no such tool Speaking about thepossibility of the direct measurement of the free energy ofembryos one has to confess that except for some specificcases (see [27]) there is no direct mechanism to realize thispossibility

Certainly practically every reliable model can be ex-pressed in terms of statistic-mechanical approach (see [28])but this can not be a tool to decide whether one can cancel thederivative of the surface tension Here also both approachescan be justified on statistic-mechanical basis the matter iswhich reference state of equilibrium is chosen

One has to stress that the density functional theorycannot be the instrument to decide to cancel the derivatives[29 30] because here themodel formof the free energy is sim-ply postulated at the level of the elementary hydrodynamicsubsystems

Unfortunately the comparisonwith the 2D Isingmodel asit is proposed in [31] to solve the question of the necessity ofcancelation cannot be fulfilled because one cannot generalizethis model to the multicomponent case

So we have to formulate some physicalmodel and discussit

Journal of Thermodynamics 5

31 Different Meanings of the Surface Tension One cannotargue that some excesses initiated by the interface have to beput into the theoryThemost common one is the excess of thefree energy that is the surface tension

The answer to the question what approach is the mostappropriate one lies in the method of construction of thecapillary approximation and the expression for a free energyin this approximation We have to reformulate a questionwhat is the surface tension which we will put in the surfaceterm for the free energy of the embryo Several answers arepossible

(i) This is the surface tension of an embryoThen the nucleation theory is the absolutely self-closed theory and it cannot be checked At themodern level of experimental devices there is no wayto measure the surface tension of droplets otherwisethan in the nucleation rate experiments Everythingwhich is measured in experiments is attributed tothe value of surface tension according to theoreticaldependencies derived in frames of an approach tocalculate the nucleation rate

(ii) This is the surface tension 120574119905observed in a capillary

tube (index 119905) of the radius equal to the radius of theembryo (with corresponding wetting)Here one can perform direct experiments to measurethe surface tension This is the way to constructa theory with the predicting force contrast to theprevious case

(iii) The surface tension is the plane surface tension 120574119901

Certainly this is a rather rough approximation butordinary namely the flat surface tension is knownBy application of the methods of thermodynamicsof small objects it is possible to construct someapproximation for the surface tension in a tube and toreduce this case to the previous one Also one has tomention that in some sense this case can be preferablebecause the surface excesses are mainly known in thecase of the flat surface

(iv) One can propose the conventional variant 120574 is thesurface tension of a flat surface but at the surpluspressure (plus the Laplace-Young addition) this valuecan be found by interpolation from 119875 119879 coexistencecurve to surplus temperature or pressureThere are no principal differences between this vari-ant and the previous one So no special attention willbe paid to this variant

To go further one has to note that the capillary tube andthe flat liquid are not a closed object as the droplet is This isthe inevitable difference and this difference will be importantThe question is now the following

(i) is it enough to account the compactness of the embryoonly at the level of expression for 119860 as a function of thenumber ofmolecules inside the embryo or is it necessaryto rewrite the analog of the Gibbs-Duhem equationwhich will be different from (25)

Our conclusion is that there is no direct answer on thisquestion in frames of macroscopic thermodynamics butthe principle of correspondence formulated below is ratherreliable and solves this question

32Mean Chemical Potentials in DifferentModels Theobjectof further investigation will be the cluster of a liquid phasewhich is dense and compact contrast to a vapor phase Sowe will omit the index 119897 use ]

119894instead of 119873

119897119894 and speak

about chemical potentials instead of differences in chemicalpotentials5

We will formulate the principle of correspondence

(i) It is necessary to use the Gibbs-Duhem equation fromthe system which provides the surface tension usedin the model If we use the surface tension from aflat surface we write the Gibbs-Duhem equation fora flat surface If we use the surface tension from acapillary tube we write the Gibbs-Duhem equationfor a capillary tube

Certainly the most similar system with the explicitly mea-sured surface tension for the situation of the embryo is thecapillary tube We take the surface tension from the capillarytube and have to write the Gibbs-Duhem equation for thecapillary tube This leads (as it will be shown later) to theRenninger-Wilemski recipe But one has to stress that theprinciple of correspondence is no more than a suppositionAlso it is necessary to add that one can act in a manner of theexplicit attribution of the surface excesses which is typical forthe Gibbs dividing surface approach

In capillary models one has to use the surface tensionfrom the situation with a capillary tube It leads to thefollowing consequence discussed below

(i) In the droplet there is a strong connection betweenthe radius 119903 and the number of molecules ]

119894 this

connection is absent in the capillary tube

The last connection can be written in a most simple versionof the theory (without surface excesses taken into account) as

41205871199033

3= sum

119894

]119894V119897119894 (30)

If we use the Gibbs formalism there are surface excessesΨ119894of

the substance (index 119904marks the surface) and then

41205871199033

3= sum

119894

(]119894minus Ψ119894) V119897119894 (31)

We attribute in frames of conception of a phase in the Gibbsmethod all other molecules to the bulk phase

]119887119894= ]119894minus Ψ119894 (32)

(index 119887marks the bulk region)Approximately Ψ

119894are proportional to the surface area

41205871199032

Ψ119894= 4120587119903

2120588119894 (33)

where 120588119894is the density of excess6

6 Journal of Thermodynamics

As the main dividing surface it is preferable to take thesurface of tension where the Young-Laplace equation is pre-cise

Then

41205871199033

3= sum

119894

(]119894minus 4120587119903

2120588119894) V119897119894 (34)

is an equation on 119903 as a function of ]119894

The concentrations are defined as

120585119894=

]119887119894

sum119895]119887119895

(35)

It is a principal requirement that in the definition of concen-tration there stand the bulk numbers of molecules

Now we will present constructions on the base of acapillary tube as a corresponding system for the free energyin capillary approximation Until the place where the Gibbsmodel is formulatedwe forget about the excesses at the dividingsurface In the capillary tube there is no such balance relationas ]119887119894= ]119894minusΨ119894 If one constructs the free energy of a capillary

tube one gets

119865119905= sum

119894

119873119894120583119894+ 120574119905119860 (36)

(index 119905 marks that we deal with a tube 119873119894are the numbers

of molecules in a tube) Since the numbers 119873119894are not fixed

in principle (the tube is connected with a bath) one has toexpress the last relation through concentrations This gives

119865119905= sum119873

119894119861119887[120585] + 120574119860 (37)

where 119861119887is

119861119887= sum120585

119894120583119894[120585] (38)

The value of 119861119887has the sense of the averaged chemical poten-

tial Consider

119861119887= ⟨120583⟩ (39)

The condition for the equilibrium will be

119889119861119887

119889120585= 0 (40)

Since Gibbs-Duhem equation (19) takes place the last condi-tion means

120583119894= 120583119895 (41)

with a clear physical senseThe same picture will be in a situation with a plane

surface The difference is that the surplus chemical potentialsare counted here from the coexistence line while in the caseof a tube they are counted from the potentials in vapor phaseunder the Laplace pressure

The term 120574119860 does not influence the concentration equi-librium when we suppose that the contact angle in a capillary

tube is constant (namely this situation is considered) Cer-tainly there exists a dependence on concentration throughthe increase of pressure but we suppose that the dependenceof activity coefficients that is the functions 119891

119894staying in

standard expressions

120583119894= ln(

119901

119901eq) minus ln 120585

119894minus ln119891

119894[120585119894] (42)

on pressure is small Here 119901eq is the pressure over a pureplane liquid phase and ln(119901119901eq) is a logarithm of thesupersaturation plus one

The situation in a droplet is another oneThe surface area119860 depends on 119873

119894and we have to take this dependence into

account There are several ways how to do itTo see the preferable way we will analyze the structure of

the free energy in the embryo We present 119865 from (17) in theform like that in the one-dimensional case

119865 = minus119861119889Ω + Ω

23 (43)

Here Ω is defined in order that Ω23 is the surface surplusinput into the droplet free energy

Ω23

= 119860120574 (44)

For 119861119889one can get on the base of (17)

119861119889= 119862minus32

120574minus32

(119861119887

V119898

) (45)

where V119898is the mean volume of a molecule in a droplet

The value of 119861119889allows the following interpretation

119861119889sim

⟨120583⟩

12057432 ⟨V⟩(46)

and one can consider the denominator in the last expressionas the mean surface tension for one molecule in a power 32(certainly one molecule has no surface tension)

Condition for an equilibrium concentration 120585119888will be

119889119861119889

119889120585119894

= 0 (47)

Condition for an equilibrium when the derivative of 120574 is putto zero will be

119889 (119861119887V119898)

119889120585119894

= 0 (48)

Here one can also get the interpretation for 119861119887V119898as the ratio

of two mean values ⟨120583⟩ = 120583119898and ⟨V⟩ = V

119898

Both conditions are independent of the value of Ω andit is important When we choose as a variable which issupplementary to 120585 any other variable we can come to thedependence of 120585

119888on Ω but at the saddle point the result will

be the sameWewill call119861

119889themean chemical potential in the droplet

in D-model and 119861RW sim 119861119887V119898the mean chemical potential

in the droplet in RW-modelThe property of independence on 120574 of the minimum of

119861119887V119898is the serious argument for RW-model and against D-

model

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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ThermodynamicsJournal of

4 Journal of Thermodynamics

the difference between the situation when the volume of thesystem is fixed or the pressure in the system is fixed Concreteconditions ordinary are not well posed

The condition for the equilibrium is

120597119865

120597119873119897119894

= 0 (18)

The explicit differentiation has to be done with account of theGibbs-Duhem equation for the bulk liquid

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 = 0 (19)

which is actually

sum119873119897119894119889120583119897119894= 0 (20)

The last relation will be satisfied automatically and inevitablywhen we use some concrete dependencies for chemicalpotentials in liquid

The result of differentiation (we take for simplicity a two-component case) on account of

119860 = 119862(sum

119894

V119897119894119873119897119894)

23

119862 = (36120587)13 (21)

is

120597119865 [119873119897119894]

120597119873119897119894

= minusΔ120583119894+

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (22)

where

V119898= sum120585

119894V119897119894 (23)

We see that the equilibrium condition

Δ120583119894=

2120574

119903+ (

3V119898(1 minus 120585

119894)

119903)(

119889120574

119889120585119894

) (24)

differs from the Gibbs-Thomson equation by the last termThis equation is the basis in D-model

In RW-model one can also explain how to come to (12)starting from (17) In the differentiation of 119865 instead of (19) itis necessary to take the surface variant of the Gibbs-Duhemequation

sum119873119897119894119889120583119897119894+ 119878119897119889119879 minus 119881

119897119889119901 + 119860119889120574 = 0 (25)

which can be approximately reduced to

sum119873119897119894119889120583119897119894+ 119860119889120574 = 0 (26)

The last term in the previous equation cancels the derivativeof the surface tension on concentration and we come toGibbs-Thomson equations (12)

For a long time it seemed that (12) is preferable becausethere exists a differential method to see (12) But Reiss andReguera showed that it is possible to get (24) in modificationof the way we came here to (12) In integration 119889120583

119897119894119889119901 it is

necessary to take into account not only the direct dependence

120597120583119897119894120597119901 = V

119897119894but also the dependence on size of the embryo

through the dependence

120597120583119897119894

120597119860=

120597120574

120597119873119897119894

(27)

The last relation can be found if we start with the Gibbspotential taking into account that in119880

119897there already exits the

surface term 120574119860 Then

119889119866119897= minus119878119897119889119879 + 119881

119897119889119901119897+sum

119894

120583119897119894119889119873119897119894+ 120574119889119860 (28)

Then theMaxwell relation on variables119873119897119894119860 gives (27)Then

the integration will lead precisely to (24) This derivationrevitalizes D-model

To complete the overview it is necessary to add theapproach of Mirabel and Katz [13] who follow the differentialway of consideration which led to (12) In order to get (24)Mirabel and Katz modified Young-Laplace equation (7) as

119901119897minus 119901V minus

2120574

119903sim (

119889120574

119889120585119894

) (29)

In [10] this approach was classified as an inappropriateone We see that this approach is one of explanations how toreconcile the requirement to get the Gibbs-Kelvin equationand the requirement to have the free energy in the capillarymodel as described above So under the mentioned form ofthe free energy the true alternative is to modify the Gibbs-Kelvin equation or to modify the Young-Laplace equation

3 Physical Model

Ideas to find a tool outside the nucleation theory to solvethe problem formulated above seem to be very attractiveBut unfortunately there is no such tool Speaking about thepossibility of the direct measurement of the free energy ofembryos one has to confess that except for some specificcases (see [27]) there is no direct mechanism to realize thispossibility

Certainly practically every reliable model can be ex-pressed in terms of statistic-mechanical approach (see [28])but this can not be a tool to decide whether one can cancel thederivative of the surface tension Here also both approachescan be justified on statistic-mechanical basis the matter iswhich reference state of equilibrium is chosen

One has to stress that the density functional theorycannot be the instrument to decide to cancel the derivatives[29 30] because here themodel formof the free energy is sim-ply postulated at the level of the elementary hydrodynamicsubsystems

Unfortunately the comparisonwith the 2D Isingmodel asit is proposed in [31] to solve the question of the necessity ofcancelation cannot be fulfilled because one cannot generalizethis model to the multicomponent case

So we have to formulate some physicalmodel and discussit

Journal of Thermodynamics 5

31 Different Meanings of the Surface Tension One cannotargue that some excesses initiated by the interface have to beput into the theoryThemost common one is the excess of thefree energy that is the surface tension

The answer to the question what approach is the mostappropriate one lies in the method of construction of thecapillary approximation and the expression for a free energyin this approximation We have to reformulate a questionwhat is the surface tension which we will put in the surfaceterm for the free energy of the embryo Several answers arepossible

(i) This is the surface tension of an embryoThen the nucleation theory is the absolutely self-closed theory and it cannot be checked At themodern level of experimental devices there is no wayto measure the surface tension of droplets otherwisethan in the nucleation rate experiments Everythingwhich is measured in experiments is attributed tothe value of surface tension according to theoreticaldependencies derived in frames of an approach tocalculate the nucleation rate

(ii) This is the surface tension 120574119905observed in a capillary

tube (index 119905) of the radius equal to the radius of theembryo (with corresponding wetting)Here one can perform direct experiments to measurethe surface tension This is the way to constructa theory with the predicting force contrast to theprevious case

(iii) The surface tension is the plane surface tension 120574119901

Certainly this is a rather rough approximation butordinary namely the flat surface tension is knownBy application of the methods of thermodynamicsof small objects it is possible to construct someapproximation for the surface tension in a tube and toreduce this case to the previous one Also one has tomention that in some sense this case can be preferablebecause the surface excesses are mainly known in thecase of the flat surface

(iv) One can propose the conventional variant 120574 is thesurface tension of a flat surface but at the surpluspressure (plus the Laplace-Young addition) this valuecan be found by interpolation from 119875 119879 coexistencecurve to surplus temperature or pressureThere are no principal differences between this vari-ant and the previous one So no special attention willbe paid to this variant

To go further one has to note that the capillary tube andthe flat liquid are not a closed object as the droplet is This isthe inevitable difference and this difference will be importantThe question is now the following

(i) is it enough to account the compactness of the embryoonly at the level of expression for 119860 as a function of thenumber ofmolecules inside the embryo or is it necessaryto rewrite the analog of the Gibbs-Duhem equationwhich will be different from (25)

Our conclusion is that there is no direct answer on thisquestion in frames of macroscopic thermodynamics butthe principle of correspondence formulated below is ratherreliable and solves this question

32Mean Chemical Potentials in DifferentModels Theobjectof further investigation will be the cluster of a liquid phasewhich is dense and compact contrast to a vapor phase Sowe will omit the index 119897 use ]

119894instead of 119873

119897119894 and speak

about chemical potentials instead of differences in chemicalpotentials5

We will formulate the principle of correspondence

(i) It is necessary to use the Gibbs-Duhem equation fromthe system which provides the surface tension usedin the model If we use the surface tension from aflat surface we write the Gibbs-Duhem equation fora flat surface If we use the surface tension from acapillary tube we write the Gibbs-Duhem equationfor a capillary tube

Certainly the most similar system with the explicitly mea-sured surface tension for the situation of the embryo is thecapillary tube We take the surface tension from the capillarytube and have to write the Gibbs-Duhem equation for thecapillary tube This leads (as it will be shown later) to theRenninger-Wilemski recipe But one has to stress that theprinciple of correspondence is no more than a suppositionAlso it is necessary to add that one can act in a manner of theexplicit attribution of the surface excesses which is typical forthe Gibbs dividing surface approach

In capillary models one has to use the surface tensionfrom the situation with a capillary tube It leads to thefollowing consequence discussed below

(i) In the droplet there is a strong connection betweenthe radius 119903 and the number of molecules ]

119894 this

connection is absent in the capillary tube

The last connection can be written in a most simple versionof the theory (without surface excesses taken into account) as

41205871199033

3= sum

119894

]119894V119897119894 (30)

If we use the Gibbs formalism there are surface excessesΨ119894of

the substance (index 119904marks the surface) and then

41205871199033

3= sum

119894

(]119894minus Ψ119894) V119897119894 (31)

We attribute in frames of conception of a phase in the Gibbsmethod all other molecules to the bulk phase

]119887119894= ]119894minus Ψ119894 (32)

(index 119887marks the bulk region)Approximately Ψ

119894are proportional to the surface area

41205871199032

Ψ119894= 4120587119903

2120588119894 (33)

where 120588119894is the density of excess6

6 Journal of Thermodynamics

As the main dividing surface it is preferable to take thesurface of tension where the Young-Laplace equation is pre-cise

Then

41205871199033

3= sum

119894

(]119894minus 4120587119903

2120588119894) V119897119894 (34)

is an equation on 119903 as a function of ]119894

The concentrations are defined as

120585119894=

]119887119894

sum119895]119887119895

(35)

It is a principal requirement that in the definition of concen-tration there stand the bulk numbers of molecules

Now we will present constructions on the base of acapillary tube as a corresponding system for the free energyin capillary approximation Until the place where the Gibbsmodel is formulatedwe forget about the excesses at the dividingsurface In the capillary tube there is no such balance relationas ]119887119894= ]119894minusΨ119894 If one constructs the free energy of a capillary

tube one gets

119865119905= sum

119894

119873119894120583119894+ 120574119905119860 (36)

(index 119905 marks that we deal with a tube 119873119894are the numbers

of molecules in a tube) Since the numbers 119873119894are not fixed

in principle (the tube is connected with a bath) one has toexpress the last relation through concentrations This gives

119865119905= sum119873

119894119861119887[120585] + 120574119860 (37)

where 119861119887is

119861119887= sum120585

119894120583119894[120585] (38)

The value of 119861119887has the sense of the averaged chemical poten-

tial Consider

119861119887= ⟨120583⟩ (39)

The condition for the equilibrium will be

119889119861119887

119889120585= 0 (40)

Since Gibbs-Duhem equation (19) takes place the last condi-tion means

120583119894= 120583119895 (41)

with a clear physical senseThe same picture will be in a situation with a plane

surface The difference is that the surplus chemical potentialsare counted here from the coexistence line while in the caseof a tube they are counted from the potentials in vapor phaseunder the Laplace pressure

The term 120574119860 does not influence the concentration equi-librium when we suppose that the contact angle in a capillary

tube is constant (namely this situation is considered) Cer-tainly there exists a dependence on concentration throughthe increase of pressure but we suppose that the dependenceof activity coefficients that is the functions 119891

119894staying in

standard expressions

120583119894= ln(

119901

119901eq) minus ln 120585

119894minus ln119891

119894[120585119894] (42)

on pressure is small Here 119901eq is the pressure over a pureplane liquid phase and ln(119901119901eq) is a logarithm of thesupersaturation plus one

The situation in a droplet is another oneThe surface area119860 depends on 119873

119894and we have to take this dependence into

account There are several ways how to do itTo see the preferable way we will analyze the structure of

the free energy in the embryo We present 119865 from (17) in theform like that in the one-dimensional case

119865 = minus119861119889Ω + Ω

23 (43)

Here Ω is defined in order that Ω23 is the surface surplusinput into the droplet free energy

Ω23

= 119860120574 (44)

For 119861119889one can get on the base of (17)

119861119889= 119862minus32

120574minus32

(119861119887

V119898

) (45)

where V119898is the mean volume of a molecule in a droplet

The value of 119861119889allows the following interpretation

119861119889sim

⟨120583⟩

12057432 ⟨V⟩(46)

and one can consider the denominator in the last expressionas the mean surface tension for one molecule in a power 32(certainly one molecule has no surface tension)

Condition for an equilibrium concentration 120585119888will be

119889119861119889

119889120585119894

= 0 (47)

Condition for an equilibrium when the derivative of 120574 is putto zero will be

119889 (119861119887V119898)

119889120585119894

= 0 (48)

Here one can also get the interpretation for 119861119887V119898as the ratio

of two mean values ⟨120583⟩ = 120583119898and ⟨V⟩ = V

119898

Both conditions are independent of the value of Ω andit is important When we choose as a variable which issupplementary to 120585 any other variable we can come to thedependence of 120585

119888on Ω but at the saddle point the result will

be the sameWewill call119861

119889themean chemical potential in the droplet

in D-model and 119861RW sim 119861119887V119898the mean chemical potential

in the droplet in RW-modelThe property of independence on 120574 of the minimum of

119861119887V119898is the serious argument for RW-model and against D-

model

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Superconductivity

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Statistical MechanicsInternational Journal of

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Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

Journal of Thermodynamics 5

31 Different Meanings of the Surface Tension One cannotargue that some excesses initiated by the interface have to beput into the theoryThemost common one is the excess of thefree energy that is the surface tension

The answer to the question what approach is the mostappropriate one lies in the method of construction of thecapillary approximation and the expression for a free energyin this approximation We have to reformulate a questionwhat is the surface tension which we will put in the surfaceterm for the free energy of the embryo Several answers arepossible

(i) This is the surface tension of an embryoThen the nucleation theory is the absolutely self-closed theory and it cannot be checked At themodern level of experimental devices there is no wayto measure the surface tension of droplets otherwisethan in the nucleation rate experiments Everythingwhich is measured in experiments is attributed tothe value of surface tension according to theoreticaldependencies derived in frames of an approach tocalculate the nucleation rate

(ii) This is the surface tension 120574119905observed in a capillary

tube (index 119905) of the radius equal to the radius of theembryo (with corresponding wetting)Here one can perform direct experiments to measurethe surface tension This is the way to constructa theory with the predicting force contrast to theprevious case

(iii) The surface tension is the plane surface tension 120574119901

Certainly this is a rather rough approximation butordinary namely the flat surface tension is knownBy application of the methods of thermodynamicsof small objects it is possible to construct someapproximation for the surface tension in a tube and toreduce this case to the previous one Also one has tomention that in some sense this case can be preferablebecause the surface excesses are mainly known in thecase of the flat surface

(iv) One can propose the conventional variant 120574 is thesurface tension of a flat surface but at the surpluspressure (plus the Laplace-Young addition) this valuecan be found by interpolation from 119875 119879 coexistencecurve to surplus temperature or pressureThere are no principal differences between this vari-ant and the previous one So no special attention willbe paid to this variant

To go further one has to note that the capillary tube andthe flat liquid are not a closed object as the droplet is This isthe inevitable difference and this difference will be importantThe question is now the following

(i) is it enough to account the compactness of the embryoonly at the level of expression for 119860 as a function of thenumber ofmolecules inside the embryo or is it necessaryto rewrite the analog of the Gibbs-Duhem equationwhich will be different from (25)

Our conclusion is that there is no direct answer on thisquestion in frames of macroscopic thermodynamics butthe principle of correspondence formulated below is ratherreliable and solves this question

32Mean Chemical Potentials in DifferentModels Theobjectof further investigation will be the cluster of a liquid phasewhich is dense and compact contrast to a vapor phase Sowe will omit the index 119897 use ]

119894instead of 119873

119897119894 and speak

about chemical potentials instead of differences in chemicalpotentials5

We will formulate the principle of correspondence

(i) It is necessary to use the Gibbs-Duhem equation fromthe system which provides the surface tension usedin the model If we use the surface tension from aflat surface we write the Gibbs-Duhem equation fora flat surface If we use the surface tension from acapillary tube we write the Gibbs-Duhem equationfor a capillary tube

Certainly the most similar system with the explicitly mea-sured surface tension for the situation of the embryo is thecapillary tube We take the surface tension from the capillarytube and have to write the Gibbs-Duhem equation for thecapillary tube This leads (as it will be shown later) to theRenninger-Wilemski recipe But one has to stress that theprinciple of correspondence is no more than a suppositionAlso it is necessary to add that one can act in a manner of theexplicit attribution of the surface excesses which is typical forthe Gibbs dividing surface approach

In capillary models one has to use the surface tensionfrom the situation with a capillary tube It leads to thefollowing consequence discussed below

(i) In the droplet there is a strong connection betweenthe radius 119903 and the number of molecules ]

119894 this

connection is absent in the capillary tube

The last connection can be written in a most simple versionof the theory (without surface excesses taken into account) as

41205871199033

3= sum

119894

]119894V119897119894 (30)

If we use the Gibbs formalism there are surface excessesΨ119894of

the substance (index 119904marks the surface) and then

41205871199033

3= sum

119894

(]119894minus Ψ119894) V119897119894 (31)

We attribute in frames of conception of a phase in the Gibbsmethod all other molecules to the bulk phase

]119887119894= ]119894minus Ψ119894 (32)

(index 119887marks the bulk region)Approximately Ψ

119894are proportional to the surface area

41205871199032

Ψ119894= 4120587119903

2120588119894 (33)

where 120588119894is the density of excess6

6 Journal of Thermodynamics

As the main dividing surface it is preferable to take thesurface of tension where the Young-Laplace equation is pre-cise

Then

41205871199033

3= sum

119894

(]119894minus 4120587119903

2120588119894) V119897119894 (34)

is an equation on 119903 as a function of ]119894

The concentrations are defined as

120585119894=

]119887119894

sum119895]119887119895

(35)

It is a principal requirement that in the definition of concen-tration there stand the bulk numbers of molecules

Now we will present constructions on the base of acapillary tube as a corresponding system for the free energyin capillary approximation Until the place where the Gibbsmodel is formulatedwe forget about the excesses at the dividingsurface In the capillary tube there is no such balance relationas ]119887119894= ]119894minusΨ119894 If one constructs the free energy of a capillary

tube one gets

119865119905= sum

119894

119873119894120583119894+ 120574119905119860 (36)

(index 119905 marks that we deal with a tube 119873119894are the numbers

of molecules in a tube) Since the numbers 119873119894are not fixed

in principle (the tube is connected with a bath) one has toexpress the last relation through concentrations This gives

119865119905= sum119873

119894119861119887[120585] + 120574119860 (37)

where 119861119887is

119861119887= sum120585

119894120583119894[120585] (38)

The value of 119861119887has the sense of the averaged chemical poten-

tial Consider

119861119887= ⟨120583⟩ (39)

The condition for the equilibrium will be

119889119861119887

119889120585= 0 (40)

Since Gibbs-Duhem equation (19) takes place the last condi-tion means

120583119894= 120583119895 (41)

with a clear physical senseThe same picture will be in a situation with a plane

surface The difference is that the surplus chemical potentialsare counted here from the coexistence line while in the caseof a tube they are counted from the potentials in vapor phaseunder the Laplace pressure

The term 120574119860 does not influence the concentration equi-librium when we suppose that the contact angle in a capillary

tube is constant (namely this situation is considered) Cer-tainly there exists a dependence on concentration throughthe increase of pressure but we suppose that the dependenceof activity coefficients that is the functions 119891

119894staying in

standard expressions

120583119894= ln(

119901

119901eq) minus ln 120585

119894minus ln119891

119894[120585119894] (42)

on pressure is small Here 119901eq is the pressure over a pureplane liquid phase and ln(119901119901eq) is a logarithm of thesupersaturation plus one

The situation in a droplet is another oneThe surface area119860 depends on 119873

119894and we have to take this dependence into

account There are several ways how to do itTo see the preferable way we will analyze the structure of

the free energy in the embryo We present 119865 from (17) in theform like that in the one-dimensional case

119865 = minus119861119889Ω + Ω

23 (43)

Here Ω is defined in order that Ω23 is the surface surplusinput into the droplet free energy

Ω23

= 119860120574 (44)

For 119861119889one can get on the base of (17)

119861119889= 119862minus32

120574minus32

(119861119887

V119898

) (45)

where V119898is the mean volume of a molecule in a droplet

The value of 119861119889allows the following interpretation

119861119889sim

⟨120583⟩

12057432 ⟨V⟩(46)

and one can consider the denominator in the last expressionas the mean surface tension for one molecule in a power 32(certainly one molecule has no surface tension)

Condition for an equilibrium concentration 120585119888will be

119889119861119889

119889120585119894

= 0 (47)

Condition for an equilibrium when the derivative of 120574 is putto zero will be

119889 (119861119887V119898)

119889120585119894

= 0 (48)

Here one can also get the interpretation for 119861119887V119898as the ratio

of two mean values ⟨120583⟩ = 120583119898and ⟨V⟩ = V

119898

Both conditions are independent of the value of Ω andit is important When we choose as a variable which issupplementary to 120585 any other variable we can come to thedependence of 120585

119888on Ω but at the saddle point the result will

be the sameWewill call119861

119889themean chemical potential in the droplet

in D-model and 119861RW sim 119861119887V119898the mean chemical potential

in the droplet in RW-modelThe property of independence on 120574 of the minimum of

119861119887V119898is the serious argument for RW-model and against D-

model

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Soft MatterJournal of

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ThermodynamicsJournal of

6 Journal of Thermodynamics

As the main dividing surface it is preferable to take thesurface of tension where the Young-Laplace equation is pre-cise

Then

41205871199033

3= sum

119894

(]119894minus 4120587119903

2120588119894) V119897119894 (34)

is an equation on 119903 as a function of ]119894

The concentrations are defined as

120585119894=

]119887119894

sum119895]119887119895

(35)

It is a principal requirement that in the definition of concen-tration there stand the bulk numbers of molecules

Now we will present constructions on the base of acapillary tube as a corresponding system for the free energyin capillary approximation Until the place where the Gibbsmodel is formulatedwe forget about the excesses at the dividingsurface In the capillary tube there is no such balance relationas ]119887119894= ]119894minusΨ119894 If one constructs the free energy of a capillary

tube one gets

119865119905= sum

119894

119873119894120583119894+ 120574119905119860 (36)

(index 119905 marks that we deal with a tube 119873119894are the numbers

of molecules in a tube) Since the numbers 119873119894are not fixed

in principle (the tube is connected with a bath) one has toexpress the last relation through concentrations This gives

119865119905= sum119873

119894119861119887[120585] + 120574119860 (37)

where 119861119887is

119861119887= sum120585

119894120583119894[120585] (38)

The value of 119861119887has the sense of the averaged chemical poten-

tial Consider

119861119887= ⟨120583⟩ (39)

The condition for the equilibrium will be

119889119861119887

119889120585= 0 (40)

Since Gibbs-Duhem equation (19) takes place the last condi-tion means

120583119894= 120583119895 (41)

with a clear physical senseThe same picture will be in a situation with a plane

surface The difference is that the surplus chemical potentialsare counted here from the coexistence line while in the caseof a tube they are counted from the potentials in vapor phaseunder the Laplace pressure

The term 120574119860 does not influence the concentration equi-librium when we suppose that the contact angle in a capillary

tube is constant (namely this situation is considered) Cer-tainly there exists a dependence on concentration throughthe increase of pressure but we suppose that the dependenceof activity coefficients that is the functions 119891

119894staying in

standard expressions

120583119894= ln(

119901

119901eq) minus ln 120585

119894minus ln119891

119894[120585119894] (42)

on pressure is small Here 119901eq is the pressure over a pureplane liquid phase and ln(119901119901eq) is a logarithm of thesupersaturation plus one

The situation in a droplet is another oneThe surface area119860 depends on 119873

119894and we have to take this dependence into

account There are several ways how to do itTo see the preferable way we will analyze the structure of

the free energy in the embryo We present 119865 from (17) in theform like that in the one-dimensional case

119865 = minus119861119889Ω + Ω

23 (43)

Here Ω is defined in order that Ω23 is the surface surplusinput into the droplet free energy

Ω23

= 119860120574 (44)

For 119861119889one can get on the base of (17)

119861119889= 119862minus32

120574minus32

(119861119887

V119898

) (45)

where V119898is the mean volume of a molecule in a droplet

The value of 119861119889allows the following interpretation

119861119889sim

⟨120583⟩

12057432 ⟨V⟩(46)

and one can consider the denominator in the last expressionas the mean surface tension for one molecule in a power 32(certainly one molecule has no surface tension)

Condition for an equilibrium concentration 120585119888will be

119889119861119889

119889120585119894

= 0 (47)

Condition for an equilibrium when the derivative of 120574 is putto zero will be

119889 (119861119887V119898)

119889120585119894

= 0 (48)

Here one can also get the interpretation for 119861119887V119898as the ratio

of two mean values ⟨120583⟩ = 120583119898and ⟨V⟩ = V

119898

Both conditions are independent of the value of Ω andit is important When we choose as a variable which issupplementary to 120585 any other variable we can come to thedependence of 120585

119888on Ω but at the saddle point the result will

be the sameWewill call119861

119889themean chemical potential in the droplet

in D-model and 119861RW sim 119861119887V119898the mean chemical potential

in the droplet in RW-modelThe property of independence on 120574 of the minimum of

119861119887V119898is the serious argument for RW-model and against D-

model

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

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ThermodynamicsJournal of

Journal of Thermodynamics 7

(A1) In the capillary tube which gives the tension usedin the capillary approximation for 119865 there exists thesame independence the concentration is the argu-ment which provides the minimum of 119861

119887 The posi-

tion of this minimum is independent of 120574

Immediately there arise two arguments for D-modelagainst RW-model

(A2) The number of molecules and the surface term120574119860 (120574 is the surface tension 119860 is the surface area) arenot independent in droplet

In the tube the situation is opposite They are indepen-dent

When the molecule comes to the droplet it enlarges thevolume and thus the surface term 120574119860 In the tube such effectis absent

So here appears a question why we will use the model ofa tube7

Even when the molecule does not change the volume ofan embryo (eg this molecule is changed by the molecule ofanother type but with the same volume) the surface tensionwill be changed We come to the following conclusion

(A3) The composition of the surface layer has to bechosen not only to minimize the surface tension asit is in the tube but to minimize the whole surfaceterm 120574119860 that is the composition of the bulk regionalso acts on 119860 and thus on the surface term in a waydifferent from the case of a tubeThis argument strengthens the conviction not to usethe tube as the source of approximation for 119865 but touse directly D-model

But the most important argument comes from the analy-sis of the Gibbs-Duhem relation It is formulated below Theproblem staying there is that the Gibbs-Duhem relation hasto be used inevitably in any model but to write it one has toadopt also some model This model has the direct influenceon the result

33 Interpretation of the Gibbs-Duhem Relation The princi-ple of correspondence states that since we use the surfacetension from the measurements with a capillary tube we haveto use the Gibbs-Duhem relation for the surface of a tubeThe Gibbs-Duhem relation for the surface in a tube has thefollowing form8

119889120574 +sum120588119894119889120583119894[120585119887119894] = 0 (49)

One can easily note that if

120588119894

sum119895120588119895

= 120585119887119894 (50)

then

sum

119894

120588119894119889120583119894[120585119887119894] equiv 0 (51)

and the derivative of the surface tension cannot be compen-sated9

So the physical reason of the compensation of thederivative of the surface tension on the concentration is thedifference of the concentration in the surface layer from thebulk concentration Surface enrichment as a complicationfor consideration was noticed in [32] but there the differentpossibilities to formulate the reference system were not evenmentioned

Really in any surface layer (let it be the layer of animaginary thickness 119897) we see that the volume of the layer isapproximately 119897119860 and there are

120585119887119894119897119860

V119898

+ 120588119894119860 (52)

molecules of a sort 119894 So the concentration 120585119904119894in a surface layer

will be

120585119904119894=

120585119887119894119897V119898+ 120588119894

sum119895120585119887119895119897V119898+ 120588119895

(53)

and it does not coincide with 120585119887119894 The difference between 120585

119887119894

and 120585119904119894explains why segregation is size dependent in diluted

solutions as in droplets of small size there is no sufficientnumber of atoms Certainly 119897 is a parameter and 120585

119904119894cannot

be calculated explicitly But under any 119897 the value 120585119904119894does not

coincide with 120585119887119894

We see that D-model is the model with uniform liquidand RW-model is the model with nonuniform liquid Theaccount of nonuniformity of liquid in frames of RW-model isthe simplest one but we see that this account is very importantand makes the model more complete It is rather difficult torealize that RW-model is the model with nonuniform liquidbecause the region of enrichment is not described explicitlyAlso there are no explicit values of excesses which also canshow the nonuniformity The Gibbs-Duhem equation is theonly trace of such nonuniformity It is simply a necessarycondition of enrichment in the model

We come to the following argument

(A4) Having used 120574 from the situation with capillarytube of flat surface where the surface tension regionis enriched with the surface-active component itis reasonable to take the model where the surfaceregion is enriched according to the same rule that isaccording to Gibbs-Duhem equation (49)

This argument is the decisive argument in a choice favorablefor RW-model

Despite the choice of RW-model as a more adequatemodel we see that both models are not free from objectionsFor the RW-model the arguments A2 and A3 are the featuresignored by thismodel For theD-model the argumentsA1 andA4 are the unresolved problems Below the Gibbs dividingsurfacemethod (G-model) will be presented and thismethodis more sophisticated than RW- and D-models But still theGibbs dividing surface method (at least at the level of theseveral first terms of decomposition) is nomore than amodelbecause the arguments A2 and A3 are not fully resolved But

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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Journal of

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ThermodynamicsJournal of

8 Journal of Thermodynamics

it is the best model one can propose at the relatively simplelevel

The terminology ldquoGibbs modelrdquo or ldquoGibbs dividing sur-face methodrdquo means nothing without any concrete recipe ofapproximations Certainly if we can get the characteristicsof the curved surface of the closed confined droplet thenthe description becomes precise The problem is that we donot know these characteristics the optimal thing we can dois to use the characteristics of the curved surface over theunbounded system like the capillary tube isThe transmissionof these characteristics to the situation of a droplet means theusage of the model Evidently the capillary model is ratheraccurate and natural But it is still a model Here lies thesource of discrepancy between theory and experiments onnucleation which tortures this field more than half a centuryCertainly this difficulty takes place in one-component casealso

4 Gibbs Method

The more adequate model than D- and RW-models can beconstructed on the base of surface excesses and it will be donein this section

First of all it is necessary to stress that we speak aboutthe simplest vulgar version of the Gibbs dividing surfacesmethod Namely such a version is necessary to be installedinstead of D-model or RW-model because the informationabout excesses even in a flat case is rather poor

The structure of the Gibbs dividing surface method inthe simplest (zero) approximation requires to consider thesurface (it is more convenient to consider the surface of ten-sion ie the surface where the Young-Laplace formula takesplace precisely) and the surface excesses of all components119904119894which will be marked by Ψ

119894 Then the surface area 119860 will

be calculated on the base of

119860 = 119862(sum

119894

V119897119894(]119894minus Ψ119894))

23

(54)

which forms an equation on 119860

119860 = 119862(sum

119894

V119897119894(]119894minus 119860120588119894))

23

(55)

The first iteration is sufficient

119860 = 119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

(56)

We again recall that here and later we suppose the liquidto be incompressible and the volumes V

119897119894to be independent

of concentration

From the first glance it seems that the free energy 119865 hasto be approximately transformed from 119865 = sum

119894120583119894]119894+ 120574119860 to

119865 = sum

119894

120583119894]119894

+ 120574(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

(57)

But actually this transformation can be avoided by use ofspecial variables as it will be clear below

One can fulfil the same analysis as above but instead ofΩone has to choose

Ω 997888rarr 12057432

(119862(sum

119894

V119897119894(]119894minus 119862(sum

119894

V119897119894]119894)

23

120588119894))

23

)

32

(58)

Then

119865 = minussum

119895

120582119895120583119895minussum

119895

984858119895120583119895

Ω23

120574 (120585)+ Ω23 (59)

with

120582119894= ]119894minus Ψ119894 (60)

Certainly

120582119894

120582119895

=120585119894

120585119895

(61)

One can choose instead of Ω the variable 120581 by thefollowing formula

120581 = 11986032

(120574 minussum

119894

984858119894120583119894)

32

(62)

In these variables the free energy 119865 has the form

119865 = minus120581119887119892 (120585) + 120581

23 (63)

with the generalized chemical potential

119887119892=

sum119894120582119894120583119894

120581(64)

or

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

120581 (65)

One has to show that 119887119892does not depend on 120581 To fulfill this

derivation one can come to

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

11986032 (120574 minus sum119896984858119896120583119896)32

(66)

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

Journal of Thermodynamics 9

or10

119887119892= sum

119894

120585119894120583119894

sum119895120582119895

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120582119897

(67)

It can be also presented as

119887119892= sum

119894

120585119894120583119894

1

(120574 minus sum119896984858119896120583119896)32

sum119897V119897120585119897

(68)

The last relation evidently shows that 119887119892is really a function of

120585 The dependence on 120581 is absentFormula (63) is valid for the arbitrary Gibbs model

This form will be called the canonic representation Now wesee that it takes place for the arbitrary model in frames ofthe Gibbs dividing surface method This result is of greatimportance Directly from this result follows theGibbs rule Itmeans that the free energy of the critical embryo is one-thirdof the surface energy Now this rule is shown for the arbitrarymodel from the Gibbs dividing surface method

One can use expression (68) to clarify the Renninger-Wilemski problem According to the Gibbs absorptionrelation11

119889120574 = sum

119895

984858119894119889120583119894 (69)

the derivative of the surface tension on concentration iscanceled by the corresponding derivatives of 984858

119894on 120585 So if

we write 119887119892without surface excesses as

119887119892= sum

119894

120585119894

1

12057432sum119895V119895120585119895

(70)

we have to forbid the differentiation of 120574 on concentrationWe see that in frames of the Gibbs approach (G-model) theRenninger-Wilemski problem is explained Here there are noartificial requirements not to differentiate the surface tensionon concentration It is necessary to stress that the reason is notthe formal Gibbs absorption equation but the difference ofconcentrations in the bulk solution from the integral values

One has also to recall that in frames of G-model there isone undefined parameter because in the 119871 component casethere are only 119871 minus 1 independent equations

119889120574

119889120585119887119894

=

sum119895120588119895119889120583119895

119889120585119887119894

(71)

One additional parameter in G-model is inevitable Ordi-nary it is the sum of excesses with some weights

5 The Reduced Gibbs Model

The G-model is wider than the D-model and the RW-model It is important that G-model is the self-consistent oneBut there exists one additional parameter which cannot becalculated Namely for two excesses 120588

1and 120588

2there exists

only one equation

12058811198891205831

119889120585119887

+12058821198891205832

119889120585119887

=119889120574

119889120585119887

(72)

For 119871-component case there exist 119871 minus 1 equations

sum119895120588119895119889120583119895[120585119887]

119889120585119887119894

=119889120574

119889120585119887119894

(73)

and the last 119871th equation

sum119895120588119895119889120583119895[120585119887]

119889120585119887119871

=119889120574

119889120585119887119871

(74)

is the superposition of the previous 119871 minus 1 equations So it isnecessary to add one equationWhen one knows the distance119897 between the surface of tension and the total equimolecularsurface determined as

sum

119895

]119895V119897119895= (

4120587

3) 1199033

(75)

(certainly it is possible to account the density in a vaporphase) then (119897 ≪ 119903)

sum119860120588119894V119897119894= 119860119897 (76)

Themost simple variant is to say that the total equimolec-ular surface coincides with the surface of tension Thiscorresponds to initial relation (30) It gives

sum120588119894V119897119894= 0 (77)

Certainly this is artificial requirement and generally speakingit is wrong The only justification is that this requirement isthe simplest one and that it corresponds to the recipe (30)of the simplest and the oldest capillary approximation whichis used since [9] In [23 24] this connection was announcedto be the only true one One has to stress that according tothe Gibbs dividing surface method this connection has nopreferences

The main conclusion for the SG-model (simplified Gibbsmodel) is that the results for the free energy of the criticalembryo for concentration 120585

119887and for the variable Ω (here 120581)

coincide with results of RW-modelThe last statement can be easily checked by comparison

of corresponding formulasThe coincidence of the free energy of the critical embryo

the concentration and the surface of the critical embryoin both (RW and SG) models does not mean that allcharacteristics coincide In RW-model one can get

]119894=

120585119887119894Ω

V1198981205743211986232

= 119873119887119894 (78)

Certainly to see the critical numbers ]119888119894of the molecules

of 119894th component in SG-model one has to solve the system ofequations and add 120588

11989441205871199032 to

]119887119894=

120585119887119894Ω

V1198981205743211986232

(79)

Then

]119894=

120585119887119894Ω

V1198981205743211986232

+ 12058811989441205871199032 (80)

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Thermodynamics

or

]119894=

120585119887119894Ω

(V1198981205743211986232)

+ 120588119894Ω23

120574minus1 (81)

and the result slightly differs from RW-modelBut this difference means absolutely nothing for the

height of activation barrier For the rate of nucleation sincethe change of the Zeldovich factor is negligible the error isnegligible also

We come to the same functional dependencies but withslightly changed values of coefficients Such simple depen-dencies are probably the source for the simple scaling factorslike it is observed in [33] One can also mention that theapplicability of the scaling approach to the free energy likethe Fisher droplet model and analogous ideas by Stauffer[34] and later by Bauchspiess and Stauffer [35] can be alsointerpreted as the result of the simple scaling observed hereThey can be also included in this picture by some moderatetransformations

6 Self-Consistency ofThermodynamic Approaches

From the first glance the situation with the Renninger-Wilemski problem is strange there are no evident errors inbothD- andRW-models but at least one result is wrongDoesit mean that in construction of thermodynamic theories it ispossible to make an ldquoinvisible errorrdquo Fortunately it will beshown below that the difference in results for the free energyof the critical embryo obtained in different models has theorder of correction

To see this smallness one can redefine 120581 as11986032 and forgetabout excesses The value of 120581 is the big parameter of thetheory Really the thermodynamic description is valid onlyat great number of molecules in the embryo which requires abig value of 120581

Then 119865 = minus119861119860120581 minus 120574120581

23 and it is important that 119861119860

sim

119861119887V119898does not contain 120574 Really from equation

120597119865

120597120585

10038161003816100381610038161003816100381610038161003816120581=fixed=

120597120574

12059712058512058123

minus120597119861119860

120597120585120581 (82)

it is seen that the first term (120597120574120597120585)12058123 with the derivative

120597120574120597120585 has the correction order 12058123

120597120574

12059712058512058123

sim 12058123 (83)

in comparison with the second term (120597119861119860120597120585)120581 having the

order 120581

120597119861119860

120597120585120581 sim 120581 (84)

It means that the correction will be smallWe extract this result because of its importance for the

reconstruction of the logical self-consistency of thermody-namics Only the correcting order of the term with thederivative of the surface tension allows to ignore it in themain

order and to return the leading role of the ordinary capillaryapproximation

Certainly in situations of practical realization in exper-iments the difference between the order 120581 and 120581

23 can behardly seen

Since the formal recipe to resolve the Renninger-Wilemski problem is to forbid the differentiation of 120574 onconcentration then the equation on concentration will bedifferent It would cause the impression that there is a shiftin a leading term The true answer is that this result causesthe shift in 119865

119888which has a correction order as it follows from

the last equationOne can treat the surface tension as a coefficient in

the first correction term proportional to the surface of theembryo The coefficients at 120581

13 ln 120581 120581minus13 and so forthdepend on intensive variables (concentration is one example)Their derivatives will be canceled by derivatives of corres-ponding excessesThe structure will resemble the Renninger-Wilemski problem But here the dimension of ldquosurfacerdquo willbe 12058113 ln 120581 120581minus13 and so forth This effect will be called theldquogeneralized cancelation of derivatives on intensive variablesrdquoThe considerations to justify the cancelation are practicallythe same as those presented here for the leading correction

The necessity to develop the theory with surface excessesis evident because the surface excesses will essentially shiftthe position of the near-critical region in comparisonwith thesize of the near-critical region In RW-model the form of thenear-critical region can not be explicitly described But thisregion is extremely important for solution of kinetic equationto get the rate of nucleation

The Gibbs method allows to get the form of the near-critical region Rigorously speaking to determine the formof the near-critical region one has to take the expression for119865 with correction terms up to the order which causes theshift of position of the near-critical region Now it is clearthat the effect results mainly in the shift of the near-criticalregion while the form of the near-critical region remainsapproximately the same

7 Results and Conclusions

Themain new results of the current analysis are the following

(i) It is shown that both D-model and RW-model are notcomplete ones

(ii) The principle of correspondence is formulated It isshown that this postulate leads to the RW-model

(iii) It is shown that canonic representation takes place inthe arbitrary Gibbs model

(iv) TheGibbs rule is shown for the arbitrarymodel in theclass of the Gibbs dividing surface method

(v) The driving physical reason of RW problem is shownIt is the difference of concentration in a surface layerfrom the concentration in a bulk liquid phase

(vi) The self-consistency of thermodynamic models isshown

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 11

Does the performance of this analysis mean that thefree energy of the critical embryo in nucleation has beenfound To comment on this question one has to note thatthe conditions of applicability of thermodynamic approachto solve the RW problem are very strong Namely ordinary itis required that

]119894≫ 1 (85)

But also the surface layer has to be the uniform media whichrequires

Ψ119894= 119860120588119894≫ 1 (86)

The last inequality is very hard to satisfyThe difficulty occursnot only because the surface layer has the small volume (wemean the real surface layer which has the finite thickness andit is only reflected in Ψ

119894in Gibbs model) but also because

there exists an effect of enrichment of the surface layer by theso-called surface active substances Then there are only fewmolecules of the surface passive substance This breaks thelast inequality for the surface-passive components for ratherbig critical embryos Otherwise we have to construct thetheory with a nonuniform surface layer which is extremelydifficult to do even approximately

The theoretical constructions presented above can begeneralized In fact there exist several model methods toimprove the construction of the free energy for examplethe Debenedetti Reiss method discussed in [36] Theserefinements can be inserted into the theory presented herewithout radical rearrangements

One can refine the theory by inserting the dependenceof the surface tension on radius (see eg [37 38]) Thespecial effects which appeared in the constrained systems (see[39]) are not taken into account here but they can be easilyintroduced into this theory One can add that the possibilityof deformation of the center of the embryo can be also takeninto accountThis effect is quite analogous to the deformationof heterogeneous centers and it was considered in [40]

We keep aside the possibility of fluctuations widely inves-tigated theoretically and observed experimentally [41] Theycan be also included into consideration without essentialtransformation of the theoryOne can insert here theTolman-like corrections as well as the scaled expansions introducedin [42] One has to note that any refinements like [43 44] ofthe classical nucleation theory can be included here directlywithout any interference with results of this considerationRefinements have to be put instead of the classical expression

However in this analysis we do not analyze the possibilitythat the bulk structure of the embryo can differ from thebulk structure of the reference bulk system in a capillary tube(for the principal possibility of such an effect see [45]) Weconsider the bulk of the embryos as the homogeneousmediafor nonuniform density one can see [46] The theory can betransformed to grasp this case also

We do not pretend to go close to the critical pointalthough the difference between concentrations in bulk andin surface layers can be interpreted in analogous manner Itseems that any extensions of the classical approach like [47]should be replaced by theoretical constructions based on therenormalized group approach

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Endnotes

1 One can also suppose that the fixed variables are thetemperature 119879 and the pressure 119901 and use the Gibbs freeenergy

119866 = 119865 + 119901119881 (lowast)

The difference in the work of the droplet formation willbe in a value of the volume of the appeared dropletmulti-plied on differences in pressure Ordinary this term isvery small and it is neglected In the closed systemsduring the process of nucleation from diluted solutionthe concentration decreases This effect is rather smallfor critical embryos and begins to be essential at bigsupercritical formations of a newphaseThe correspond-ing effect in the global kinetics of nucleation is describedin [48]

2 Already here we have to adopt some model to divide119880 and 119878 into two parts Also we suppose that thetemperature is one and the same for both parts thatis we consider the isothermal nucleation At this levelof narration we do not need the surface excesses theseexcesses will appear later

3 Here this statement used in [18 19] is not absolutelyprecise and will be corrected

4 This variable used in [18 19] as an extensive variableis only asymptotically extensive one which will bediscussed

5 Here we are not interested any more in the density ofenvironment and denote the number ofmolecules insidethe droplet by ]

119894 This notation marks that we are not

interested in the number of molecules in a vapor phase

6 Later it may be useful to write another coefficient ofproportionality as Ψ

119894= 119860120588119894

7 Namely theGibbs dividing surface for a tube leads to thecancelation of 119889120574119889120585

119894

8 For the embryo we cannot extract the surface explicitlyas an independent object since there is the connection(30) So the base for such type of the Gibbs-Duhemrelation is absent Instead of this one has to express119860 viathe numbers of molecules taking into account relation(27)

9 Since in reality it is compensated it means that thispossibility cannot be realized

10 We miss 119862 for simplicity

11 The sign is changed because of specific choice of differ-ences of chemical potentials

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Journal of Thermodynamics

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions Avol 189 no 11 pp 265ndash307 1898

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquo Philosophical Transactions of the Royal Society ofLondon vol 192 no 9 pp 403ndash453 1899

[3] C T RWilson ldquoOn the comparison effeciency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A vol 193 p 289 1900

[4] VV Ryazanov ldquoNonequilibrium thermodynamics based on thedistributions containing lifetime as a thermodynamic parame-terrdquo Journal of Thermodynamics vol 2011 Article ID 203203 10pages 2011

[5] V V Ryazanov ldquoNonequilibrium thermodynamics and distri-butions time to achieve a given level of a stochastic process forenergy of systemrdquo Journal ofThermodynamics vol 2012 ArticleID 318032 5 pages 2012

[6] R Kumar G Sharma and M Kumar ldquoEffect of size andshape on the vibrational and thermodynamic properties ofnanomaterialsrdquo Journal of Thermodynamics vol 2013 ArticleID 328051 5 pages 2013

[7] S Turmanova S Genieva and L Vlaev ldquoKinetics of non-isothermal degradation of some polymer composites changeof entropy at the formation of the activated complex fromthe reagentsrdquo Journal of Thermodynamics vol 2011 Article ID605712 10 pages 2011

[8] E Hernandez-Lemus ldquoNonequilibrium thermodynamics ofcell signalingrdquo Journal of Thermodynamics vol 2012 Article ID432143 10 pages 2012

[9] H Reiss ldquoThe kinetics of phase transitions in binary systemsrdquoThe Journal of Chemical Physics vol 18 no 6 pp 840ndash848 1950

[10] R G Renninger F C Hiller and R C Bone ldquoComment onlsquoself-nucleation in the sulfuric acid-water systemrsquordquo The Journalof Chemical Physics vol 75 no 3 pp 1584ndash1585 1981

[11] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[12] F F Abraham Homogeneous Nucleation Theory AcademicPress New York NY USA 1974

[13] P Mirabel and J L Katz ldquoBinary homogeneous nucleation asa mechanism for the formation of aerosolsrdquo The Journal ofChemical Physics vol 60 no 3 pp 1138ndash1144 1974

[14] G J Doyle ldquoSelfminusnucleation in the sulfuric acidminuswater systemrdquoThe Journal of Chemical Physics vol 35 no 3 p 795 1961

[15] G J Doyle ldquoResponse to the comment on lsquoSelf-nucleation in thesulfuric acid-water systemrsquordquoThe Journal of Chemical Physics vol75 no 3 p 1585 1981

[16] GWilemski ldquoComposition of the critical nucleus inmulticom-ponent vapor nucleationrdquoThe Journal of Chemical Physics vol80 no 3 p 1370 1984

[17] G Wilemski ldquoRevised classical binary nucleation theory foraqueous alcohol and acetone vaporsrdquo The Journal of PhysicalChemistry vol 91 no 10 pp 2492ndash2498 1987

[18] H Reiss and D Reguera ldquoNucleation in confined ideal binarymixtures the Renninger-Wilemski problem revisitedrdquo Journalof Chemical Physics vol 119 no 3 pp 1533ndash1546 2003

[19] H Reiss Methods of Thermodynamics Dover Mineola NYUSA 1996

[20] JW GibbsTheCollectedWorks of JW Gibbs Longmans GreenNew York NY USA 1928

[21] P G Debenedetti Metastable Liquids Concepts and PrinciplesPrinceton University Press Princeton NJ USA 1996

[22] K Nishioka and I Kusaka ldquoThermodynamic formulas of liquidphase nucleation from vapor in multicomponent systemsrdquoTheJournal of Chemical Physics vol 96 no 7 pp 5370ndash5376 1992

[23] A Laaksonen R McGraw and H Vehkamaki ldquoLiquid-dropformalism and free-energy surfaces in binary homogeneousnucleation theoryrdquo Journal of Chemical Physics vol 111 no 5pp 2019ndash2027 1999

[24] Y S Djikaev I Napari and A Laaksonen ldquoOn the closureconjectures for the Gibbsian approximation model of a binarydropletrdquo The Journal of Chemical Physics vol 120 no 20 pp9752ndash9762 2004

[25] F F Abraham ldquoA reexamination of homogeneous nucleationtheory thermodynamic aspectsrdquo Journal of the AtmosphericSciences vol 25 pp 47ndash53 1998

[26] R C Cammarata ldquoGeneralized surface thermodynamics withapplication to nucleationrdquo Philosophical Magazine vol 88 no6 pp 927ndash948 2008

[27] F G Shi ldquoDirect measurement of free-energy barrier to nucle-ation of crystallites in amorphous silicon thin filmsrdquo Journal ofApplied Physics vol 76 no 9 pp 5149ndash5153 1994

[28] R BMcClurg and R C Flagan ldquoCritical comparison of dropletmodels in homogeneous nucleation theoryrdquo Journal of Colloidand Interface Science vol 201 no 2 pp 194ndash199 1998

[29] A Laaksonen V Talanquer and D W Oxtoby ldquoNucleationmeasurements theory and atmospheric applicationsrdquo AnnualReview of Physical Chemistry vol 46 no 1 pp 489ndash524 1995

[30] R Evans ldquoThe nature of the liquid-vapour interface and othertopics in the statistical mechanics of non-uniform classicalfluidsrdquo Advances in Physics vol 28 no 2 pp 143ndash200 1979

[31] S Ryu and W Cai ldquoValidity of classical nucleation theory forIsingmodelsrdquoPhysical ReviewEmdashStatistical Nonlinear and SoftMatter Physics vol 81 no 3 Article ID 030601 2010

[32] D T Wu ldquoNucleation Theoryrdquo Solid State Physics vol 50 pp37ndash187 1996

[33] B N Hale ldquoScaling of nucleation ratesrdquoMetallurgical Transac-tions A vol 23 no 7 pp 1863ndash1868 1992

[34] D Stauffer ldquoScaling theory of percolation clustersrdquo PhysicsReports vol 54 no 1 pp 1ndash74 1979

[35] K R Bauchspiess and D Stauffer ldquoUse of percolation clustersin nucleation theoryrdquo Journal of Aerosol Science vol 9 no 6pp 567ndash577 1978

[36] Y S Djikaev R Bowles and H Reiss ldquoRole of constraintsin the thermodynamics of heterogeneous condensation onsolid soluble particles failure of the capillarity approximationrdquoPhysicaA StatisticalMechanics and Its Applications vol 298 no1-2 pp 155ndash176 2001

[37] F F Abraham and G M Pound ldquoA reexamination of the freeenergy of droplet formation and dependence of surface tensionon radiusrdquo Journal of Crystal Growth vol 6 no 4 pp 309ndash3101970

[38] F F Abraham and G M Pound ldquoFree energy of formationof droplets from vapor and dependence of surface tension onradiusrdquo Journal of Crystal Growth vol 2 no 3 pp 165ndash168 1968

[39] W RWilcox ldquoThe relation between classical nucleation theoryand the solubility of small particlesrdquo Journal of Crystal Growthvol 26 no 1 pp 153ndash154 1974

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Thermodynamics 13

[40] W A Cooper and C A Knight ldquoHeterogeneous nucleation bysmall liquid particlesrdquo Journal of Aerosol Science vol 6 no 3-4pp 213ndash221 1975

[41] D Schwarz R van Gastel H J W Zandvliet and B PoelsemaldquoSize fluctuations of near critical nuclei and gibbs free energyfor nucleation of BDA on Cu(001)rdquo Physical Review Letters vol109 no 1 Article ID 016101 2012

[42] I Kusaka M Talreja and D L Tomasko ldquoBeyond classicaltheory predicting the free energy barrier of bubble nucleationin polymer foamingrdquo AIChE Journal vol 59 no 8 pp 3042ndash3053 2013

[43] H Reiss andW K Kegel ldquoReplacement free energy and the 1Sfactor in nucleation theory as a consequence ofmixing entropyrdquoJournal of Physical Chemistry vol 100 no 24 pp 10428ndash104321996

[44] D Reguera and J M Rubi ldquoNonequilibrium translational-rotational effects in nucleationrdquoThe Journal of Chemical Physicsvol 115 no 15 p 7100 2001

[45] A Reinhardt and J P K Doye ldquoFree energy landscapes forhomogeneous nucleation of ice for a monatomic water modelrdquoThe Journal of Chemical Physics vol 136 no 5 Article ID054501 2012

[46] G O Berim and E Ruckenstein ldquoKinetic theory of heteroge-neous nucleation effect of nonuniform density in the nucleirdquoJournal of Colloid and Interface Science vol 355 no 1 pp 259ndash264 2011

[47] R McGraw and H Reiss ldquoNucleation near a critical tempera-turerdquo Journal of Statistical Physics vol 20 no 4 pp 385ndash4131979

[48] V B Kurasov ldquoKinetic theory for condensation ofmulticompo-nent vapor under dynamic conditionsrdquo Physica A vol 207 no4 pp 541ndash560 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of