Attila R. Imre and Thomas Kraska- Stability limits in binary fluids mixtures

8/3/2019 Attila R. Imre and Thomas Kraska- Stability limits in binary fluids mixtures

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Stability limits in binary fluids mixtures

Attila R. ImreKFKI Atomic Energy Research Institute, H-1525 Budapest, P.O. Box 49, Hungary

Thomas Kraskaa

Department of Physical Chemistry, University Cologne, Luxemburger Strasse 116, D-50939 Kln, Germany

Received 7 September 2004; accepted 17 November 2004; published online 2 February 2005

The stability limits in binary fluid mixtures are investigated on the basis of the global phase diagram

approach employing a model for the attracting hard-sphere fluid. In addition to the diffusion

spinodals the mechanical spinodals are included. As a result one finds topologically different types

of the diffusion spinodals while only one shape exists for the mechanical spinodals which are

present in the region of liquid-vapor equilibria only. The diffusion spinodals represent the

underlying properties of the phase behavior. The types of stable phase behavior therefore resemble

that of the spinodal behavior. The different shapes of the spinodals can be important for

nonequilibrium processes in nature and technology. 2005 American Institute of Physics.

DOI: 10.1063/1.1847651

I. INTRODUCTION

Besides the stable homogeneous liquid and gas phases

fluids can also exist under metastable condition.1

The border-

line between stable and metastable states at positive pressure

is the coexistence or saturation curve. A part of the meta-

stable region of the liquid phase is located at negative

pressures14 Fig. 1. The metastable region is limited by

another curve called spinodal separating the metastable states

from the unstable region in which states cannot exist. As a

limit of the metastable region the spinodal also reaches nega-

tive pressure on the liquid side. Hence, if a liquid is isotro-

pically stretched it is transferred into a metastable state either

at positive or negative pressure. By overheating only the

metastable region at positive pressure can be reached. In ex-periments one can usually not reach the spinodal but a so-

called attainable stability limits, also called homogeneous

nucleation limit.1,5

The presence of a heterogeneous nucleus

such as an impurity lowers the critical work of the formation

of a vapor nucleus and therefore leads to a phase separation

at lower supersaturation. A vapor can be transferred into a

metastable state by undercooling or pressurizing. The meta-

stable region of the vapor phase also has a limiting spinodal

which is entirely located at positive pressures. The vapor and

the liquid spinodal meet continuously at the critical point in

the pand Tprojections. This is the only point in the phase

diagram where the spinodal touches the stable region. Hence,

it is the only point at which the diverging density fluctuations

at the spinodal reach the stable region of the phase diagram.

All other points of the spinodal are screened from the stable

gas- or liquid-phase regions by metastable regions as shown

in Fig. 1. In the pressure-temperature pT projection the two

branches of the spinodal meet in a cusp at the critical point.

Some similarities exist for the behavior close to a critical

point and close to a spinodal such as the diverging compress-

ibility and isobaric heat capacity. However, other properties

such as the exponents of the powers laws differ when ap-proaching the critical point and the spinodal.

5

For pure substances two topologically different liquid-

vapor spinodal types are discussed which are marked as type

A and B in Fig. 1a.4

The difference between types A and B

is the existence of a minimum in type B. Since the experi-

mental determination of the spinodals is difficult and only

possible by extrapolation,6

the spinodals of real materials are

not well known. The three best known systems are water and

the two isotopes of the helium. The spinodals of the helium 3

and helium 4 are type B and type A, respectively, while for

water it is still under discussion whether it is type A or

B.1,4,7,8

In binary liquid mixtures not only liquid-vapor LV butalso liquid-liquid LL phase transition can take place.

Therefore, it is also possible to reach a metastable liquid

state by crossing the LL coexistence curve. The limit of such

metastable liquid is a LL spinodal. When reaching the LL

spinodal the system immediately splits into two liquid phases

with different compositions. Although experimentally the

spinodal cannot be reached or jumped over, due to the slow

kinetics of phase separation, in some systems one can ob-

serve the so-called spinodal decomposition. Such jump can

be done by changing the temperature in a very fast manner or

changing the pressure which can be faster and easier for LL

phase separation than for LV phase separations.

9

A jump into the nonstable region can lead to two differ-

ent structures which are nucleation-and-growth type or spin-

odal type. The nucleation-and-growth type of phase separa-

tion happens in the metastable region and forms islands of

the new phase in the mother phase. At the spinodal the phase

separation is spontaneous and happens immediately within

the whole body of liquid leading to two bicontinuous, spon-

gelike phase.1

Although these structures disappear quickly, it

is possible to follow LV phase separations using very fast

photographic technique.10

In case of LL phase separationsaElectronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 122, 064507 2005

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the two different topologies, nucleation-and-growth type, and

spinodal decomposition type can be long lasting, especially

when very high molecular weight components are involved.

There are processes such as membrane production or phase

separation in polymer blends in which the partly phase-

separated liquid is further cooled down until one or both

phases solidify. In this way the initial structure can be frozen,

leading to different mechanical behavior of the solid. Hence,

the morphology and properties of the composite material

strongly depends on the type of the phase transition. There-

fore, the knowledge of the location of the spinodal is crucial

for the understanding and prediction of the properties of new

materials.

Furthermore, there are a lot of important liquid mixtures

including binary ones which can be temporarily in meta-

stable condition at which the physical and chemical proper-

ties can change very fast. Stretched binary liquids at absolute

negative pressure can easily split into two liquid phases

which can cause changes in the viscosity and several other

properties abruptly. Prominent examples for multicompo-

nents fluid are biological fluids such as blood or sap, crude

oil or aqueous solutions in soil. Blood or other body fluids

can be in metastable state locally when they are exposed to

medical ultrasound see Ref. 4 and references therein. Sap is

under moderate negative pressure down to 4 bar in the

xylem,11

crude oil can experience negative pressure during

flow or sudden decompression,12

while aqueous solutions

can be under deep negative pressure 100 MPa in the cap-

illaries of the soil.13,14

Miscibility measurements as well as

other measurements of other properties under metastable

conditions are very difficult.1520

Therefore, methods have

been developed in which properties measured at positive

pressure are extrapolated into the region of negative

pressure.14,2125

For this kind of extrapolation the knowledge

of the spinodals is crucial, because extrapolating beyond the

spinodal would yield artificial results.25

While there are several investigations of the stability

limits in pure substances experimentally and theoretically5,26

there is yet no thorough investigation on the stability limits

in binary fluid mixtures. For such study the systematic be-

havior of the global phase diagram approach is very useful.

The global phase behavior of binary fluid mixtures has beeninvestigated over the last decades in a general fashion

2732as

well as applied to specific problems and phenomena in phase

behavior.3336

In such investigations usually only the stable

parts of the phase diagram are considered. In some cases

nonstable parts are included such as metastable and unstable

parts of a critical curve.31,36,37

Although there are studies about the LV spinodal of pure

liquids and some more about the LL spinodal of binary liq-

uids, the relative position and topology of LL and LV spin-

odals in binary mixtures has been rarely studied.1,38,39

In this

work we study systematically the spinodals exemplary for

attracting hard-sphere fluid binary mixtures.

II. METHOD

The conditions for the stability of phases can be obtained

from the second law of thermodynamics saying that a closed

system is stable if the entropy is at its maximum value. From

the second-order variation of the entropy with respect to the

volume, the condition for the mechanical stability limit, the

mechanical spinodal, can be obtained:1,38,40

pV

T

pV= 0 . 1

A phase is mechanically stable for pV0 and mechanically

unstable for pV0. This is the case for pure substances and,

in principle, also for mixtures. Integration of Eq. 1 gives

2A/V2TA2V= 0. The mechanical spinodal is therefore

given on the liquid side by the minima and on the gas side by

the maxima of an isothermal van der Waals loop Fig. 1b.

As the maximum and the minimum approach each other with

increasing temperature and eventually coincide, the spinodal

goes through the critical point Fig. 1b.

In binary fluid mixture phase separation usually leads to

two phases with different mole fraction with exception of

azeotropy. The second-order variation of the entropy with

respect to the mole fraction in a closed system gives the

thermodynamic condition of the diffusion spinodal inmixtures:1,38,40

2Gmx2

p,T

G2x = 0 . 2

In analogy to the extrema of the van der Waals loop for the

mechanical spinodal the diffusion spinodal is given by the

extrema of the chemical potentials of the two compounds as

function of the mole fraction. At the diffusion spinodal the

mole fraction exhibits diverging fluctuations. In pure sub-

stances only LV equilibria exist. In mixtures there is the pos-

sibility either for LL or LV phase separations. Although ther-

modynamically there is no difference between these two

FIG. 1. Schematic sketch of the vapor-liquid phase diagram of a pure sub-

stance in a pT projection and b pprojection. a Dotted, vapor pressure

curves; dashed, different types of spinodals; and filled circle, liquid-vapor

critical point. b Solid, coexistence curve; long dashed, mechanical spin-

odal; and short dashed, isotherm.

064507-2 A. R. Imre and T. Kraska J. Chem. Phys. 122, 064507 2005



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cases, i.e., Eq. 2 always holds, experimentally the differ-

ence can be crucial. While in LL phase transitions the vol-

ume of the new phases is in the same order of magnitude as

that of the original supersaturated phase, in case of LV tran-

sition the volume of the vapor phase can be orders of mag-

nitudes larger.

While Eq. 1 can be easily applied to an equation of

state, Eq. 2 has to be rewritten in terms of the Helmholtz

energy A by Jacobi transformation:

G2x = A2x AVx

2

A2V= 0 . 3

The abbreviations of the derivatives in general are given41

by

n+kA/Vn xkTAnVkx. All other natural variables of the

functions are kept constant at differentiation. The Helmholtz

energy can be calculated from the equation of state by inte-

gration as for the residual part:

Ares = presdV. 4After adding the ideal mixing term one can compute the

mechanical and the diffusion spinodal for a given equation of

state. The calculations here are performed for the attracting

hard-sphere fluid modeled by the Carnahan-Starling-van der

Waals equation of state:42

p =RT

Vm1 + y + y2 y3

1 y3 a

Vm2

. 5

Here a is the attraction parameter and y = b/ 4Vm the packing

fraction with the covolume parameter b. The choice of this

equation is twofold: first it is a simple and good description

the attracting hard-sphere model fluid, second the global

phase behavior of this equation is very wellinvestigated.

32,43,44The latter point makes it possible to lo-

cate easily the different types of stable phase behavior in

terms of the global parameters , , and . These global

parameters are reduced differences of the equation of state

parameters a and b,

=b22 b11

b22 + b11, 6

=

a22

b222

a11

b112

a22

b222

+ a11

b112

, 7

=

a22

b222

2a12

b11b22+

a11

b112

a22

b222

+a11

b112

. 8

The parameters of the mixture are calculated from the corre-

sponding parameters of the pure substances aii, bii and the

cross-interaction parameters a12, b12 by the one-fluid mixing

theory with quadratic mixing rule

a =i

j

xixjaij , 9

b =i

j

xixjbij . 10

III. RESULTS

The diffusion and the mechanical spinodals are calcu-

lated by the computer algebra program MAPLE Ref. 45 with

a code developed for this investigation. Although the me-

chanical spinodal is hidden behind the diffusion spinodal inmixtures, its location in phase space can be important be-

cause it influences the magnitude of the density fluctuation in

the system at a metastable state point. A third type of spin-

odal, the isentropic spinodal, is not considered here. In an

recent investigation of pure substances26

the isentropic spin-

odal has been located far below the mechanical spinodal and

therefore its influence on the metastable region is expected to

be negligible.

A. Type I phase behavior

In order to analyze the spinodal behavior we start with

the simplest possible binary system, the type I system in theclassification of van Konynenburg and Scott.

27,28In such sys-

tems the critical points of the pure substances are connected

by a continuous binary liquid-gas critical curve. Besides this

critical curve there are no further critical curves. Such system

can be obtained with the parameter set =0.2, =0.1, and

=0.0 as shown in Fig. 2. One can see in Fig. 2b that in the

limits of the pure substances the mechanical and the diffu-

sion spinodal approach each other. In case of equal covol-

umes =0.0 the cusps of the mechanical spinodal in the pT

projection form a straight line connecting the critical points

of the pure substances Fig. 2a, dotted line. In the px pro-

jection shown in Fig. 2b it appears as a continuous curve.

FIG. 2. a Type I phase behavior in a pressure-temperature projection for

= 0, =0.2, and =0.1. Solid curve, binary vapor-liquid critical curve;

long-dashed curves, diffusion spinodal isopleths; short-dashed curves, me-

chanical spinodal isopleths; dotted line, pseudocritical curve; and dot-

dashed, pure substance vapor pressure curves. b The same phase diagram

in pressure-mole fraction diagram. The numbers mark different T/Tc1values.

064507-3 Stability limits in binary mixtures J. Chem. Phys. 122, 064507 2005



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Along this curve also A3V=0 holds, which corresponds to the

pure substance critical condition. Therefore it is occasionally

called pseudocritical curve. It is a straight line in the pTprojection because the critical conditions for pure substances

A2V=A3V=0 give Tc =0.3773 a/Rb and pc =0.070669 a/b2

for the CarnahanStarlingvan der Waals equation. Here Tcand pc both depend linearly on the attraction parameter ax

which is the only mole fraction dependence in case of equal

sized molecules. Therefore the pseudocritical curve is a lin-

ear interpolation of the pure critical points in the pT projec-

tion. It should be emphasized that the pseudocritical curve is

by no means a real critical curve. The binary critical curve is

rather located on the surface to the diffusion spinodal surface

in pTx space.

In Fig. 2 one can see that for a jump from the stable gas

phase into the nonstable region one first reaches the diffusion

spinodal. The mechanical spinodal is located behind the dif-

fusion spinodal. It follows that the vapor-liquid phase transi-

tion in binary mixtures is dominated by concentration fluc-

tuations and the system is decomposed already before at the

diffusion spinodal. The same can be observed for a jump

from the liquid phase into the two-phase region. This is in

agreement with inspection of Eq. 3 which shows that first

G2x and than A2V vanishes.46

For a mechanically stable phase

A2V is positive and because of the square the complete sec-

ond term of Eq. 3 AVx2 /A2V is positive. It follows that for

vanishing G2x the first of term Eq. 3 A2x has to be positive

as well. At the mechanical spinodal A2V vanishes and henceG2x diverges to minus infinity. Therefore, vanishing G2x re-

quires positive A2V and hence first the diffusion spinodal and

then the mechanical spinodal is reached. In Fig. 3 a sche-

matic sketch of an isothermal vapor-liquid equilibrium in-

cluding the spinodals is shown. The chosen temperature is

between the critical temperatures of the pure substances. The

diffusion spinodal long dashed touches the binodal solid

at the binary critical point. The mechanical spinodal short

dashed is in this case present only in the lower part of the

phase equilibrium. Half of the diffusion spinodal belongs to

the liquid phase LV spinodal; from the bottom-left corner to

the critical point; the other half belongs to the vapor phase

VL spinodal. The liquid part can be approached only com-

ing for the liquid side and vice versa as indicated by the

arrows in Fig. 3. When jumping into the two-phase region by

pressure increase from the vapor phase or pressure decrease

from the liquid phase one eventually reaches the diffusion

spinodal. Figure 3 also shows that a spinodal can be located

in the one-phase region in such diagram, for example, the

gas-phase spinodal in the liquid phase region. This may ap-

pear unusual but makes sense because the spinodals belongto the mother phase on the other side of the corresponding

branch of the coexistence curve. The limit of both, the dif-

fusion and the mechanical spinodal for vanishing mole frac-

tion is the pressure of mechanical spinodal of the pure sub-

stance at given temperature. This is shown in the left

diagram in Fig. 3. The fact that the spinodals can be located

outside the vapor-liquid coexistence region is inseparably re-

lated to this limiting behavior.

B. Type II phase behavior

After having analyzed a type I liquid-vapor system the

next step is to turn to a system with a liquid-liquid immisci-

bility. Type II is such system. In Fig. 4 different projections

of a type II system as calculated with Eq. 5 for the global

parameters = 0, =0.396, and =0.005 are plotted. In Fig.

4a the pT projection of the binary critical curve is shown.

One can see a type I VL critical curve and in addition a LL

critical curve at low temperatures. At high pressure the LL

critical curve goes to infinite pressure within a fluid model,

however, it is terminated by solidification in real systems. In

some cases this LL branch of the critical curve behaves mo-

notonously in the pT projection, in other cases it can exhibit

a temperature minimum.36

At low pressure the LL-critical

curve goes trough a minimum to a cusp in the pT projection.This cusp is the stability limit of the critical curve G4x = 0

and hence the remaining branch of the LL-critical curve go-

ing to negative pressure is an unstable critical curve. While a

normal phase becomes unstable at G2x =0 a critical phase

becomes unstable at G4x = 0. Somewhere, typically below

and close to the vapor pressure curve of the more volatile

substance a critical endpoint separates the stable and the

metastable branch of the LL-critical curve. In the px projec-

tions in Figs. 4a and 4c as well as in the pTx diagram in

Fig. 4e one can recognize that the cusp of the LL-critical

curve in the pT projection is actually a continuous curve.

Figure 4e also shows that in case of type II the diffusion

spinodals have an additional branch going to high pressure.Together with the spinodal branches known from type I the

diffusion spinodal isopleths form a &-like shape in the con-

stant mole fraction section. Further analysis shows that the

high pressure spinodal is also present in type I systems but

located at negative absolute temperature which is of course

outside the physical meaningful range in these systems.

When varying the molecular parameters the high pressure

branch of the spinodals can move to positive temperature

which results in the appearance of the LL-critical curve. This

appearance corresponds to the hypothetical transition from

type I to type II which is called zero-Kelvin transition.47

At

this transition the system has a LL critical curve at zero

FIG. 3. Schematic vapor-liquid phase diagram with the coexistence curve

solid, the mechanical spinodal short dashed, and the diffusion spinodallong dashed. The arrows symbolize a jump into the two-phase region from

the vapor or the liquid phase. In all cases first the diffusion spinodal is

reached. The pressure-temperature diagram on the left side corresponds to

one of the pure substances with x =0. Here the vapor pressure curve solid

curve of the pure substance and the mechanical spinodals short dashed are

plotted.




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Kelvin, an approach which neglects of course the solidifica-

tion. In the isothermal sections at high temperature such as

T/Tc1 =2.0 the diffusion spinodals behave as in type I sys-

tems shown in Fig. 4a. With decreasing temperature the

spinodals in the isothermal section undergo a topological

transition.

At T/Tc1 =1.03 close to the critical temperature of the

more volatile substance Fig. 4b, the looplike topology of

the LV diffusion spinodals has changed to a curve with two

maxima and one minimum. These extrema of the isothermal

diffusion spinodal are points on the critical curves: the maxi-

mum at lower pressure is a critical point on the metastable

branch of the LL-critical curve, the maximum at higher pres-

sure is a stable VL critical point, and the minimum is a point

on the unstable branch of the LL-critical curve. These three

points are marked by symbols in Fig. 4b and by a dot-

dashed line in Fig. 4d. At slightly higher temperature below

T/Tc1 = 1.07 the maximum and minimum corresponding to

the unstable and metastable LL-critical points meet and the

diffusion spinodal exhibits a saddle point. This saddle point

corresponds to the cusp of the LL-critical curve in the pTprojection with G4x =0 connecting the metastable and un-

stable branch of the critical curve. In Fig. 4c spinodals at

lower temperature are shown. The spinodals are similar as

for T/Tc1 = 1 Fig. 4a but exhibit a pronounced maximum

at high pressure corresponding to a stable LL-critical point.

For T/Tc1 =0.5 the extrema are marked by symbols in Fig.

4c and as well as by a dot-dashed line in Fig. 4 d.

C. Type V phase behavior

Type V phase behavior consists of a LV-critical curve

which is interrupted by a LLV three-phase curve as shown

schematically in Fig. 5d. The calculations for = 0, =0.5,and =0.2 shown in Fig. 5a indicate that the spinodals at

high temperature are similar to the ones of type I. At T/Tc1=1 the diffusion spinodal has a topologically similar shape as

at higher temperature but the maximum is more pronounced.

Therefore, the mechanical spinodal is at much lower pressure

than the diffusion spinodal compared to the type I phase

diagram described above. This distance between mechanical

and diffusion spinodal, which is also present for the type II

phase diagram, appears to be typical for LL equilibria. The

appearance of such LL type diffusion spinodal corresponds

to the LL type critical curve of type V at T/Tc11. The

maximum is maintained with decreasing temperature. Fur-

thermore, an additional minimum at high mole fraction be-comes visible, for example, at T/Tc1 =0.8 in Fig. 5c. These

two extrema coincide in a saddle point at negative pressure

corresponding to a cusp in the critical curve G4x = 0 in the

pT projection at which the metastable and unstable branches

meet. In Fig. 5d the situation corresponding to T/Tc1 =0.8

is shown schematically. In Fig. 5b the region close to the

critical point of the more volatile substance is enlarged. At

T/Tc1 =1.01 one can recognize one minimum corresponding

to the unstable branch of the critical curve and a maximum

corresponding to the very short LV-critical branch connected

to the critical point of the more volatile substance Fig. 5d.

At T/Tc1 =1.03 these two extrema have vanished and hence

the cusp of the critical curve is located between T/Tc1=1.01 marked by a dot-dashed line in Fig. 5d and

T/Tc1 =1.03.

D. Type III phase behavior

In type III systems the LV-critical curve starting at the

critical point of the less volatile substance is continuously

connected to the LL-critical curve going to high pressure.

The critical curve is monotonously going to high pressure,

i.e., there is no pressure minimum or maximum.32

Figure

6a shows the spinodals of such system = 0, =0.45, and

=0.1 which are at high temperature as T/Tc1 =2.0 also

FIG. 4. Type II spinodal isotherms calculated for =0 , =0.396, and

=0.005. a High temperature isotherms; b enlargement of the region near

the critical point of the more volatile substance; c low temperature iso-

therms; and d pT projection. Legend as for Fig. 3. The numbers markdifferent T/Tc1 values. The dot-dashed lines at T/Tc1 =0.5 and 1.03 in d

correspond to the diffusion spinodal isotherms plotted as bold dashed curves

in b and c. e Three dimensiona1 plot: bold solid curve, critical curve;

light solid curves, diffusion spinodal isopleths; and dashed curves, mechani-

cal spinodal isopleths.




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similar to those of type I. With decreasing temperature, but

still above the critical temperature of the less volatile sub-

stance, the diffusion spinodal forms a pronounced maximum

at high pressure corresponding to a LL-critical point. The

continuous appearance of the LL type spinodal represents the

continuous transition from LV to LL phase equilibria. At

T/Tc1 = 1 the spinodals reach the critical point of the less

volatile substance in a cusp as for the other types of phase

behavior mentioned above. This diffusion spinodal is not

continuously connected to the less volatile substance. It

rather diverges to infinite pressure coming from both pure

substances. Figure 6b shows how the spinodals behave at

high pressure. At T/Tc1 =1.28 the spinodals go to high pres-

sure as for T/Tc1 =1, while at T/Tc1 =1.29 they consist of

two parts, one at high pressure with a minimum and one

below with a maximum. Since these extrema correspond toLL-critical points the LL-critical curve exhibits a critical

temperature minimum between T/Tc1 =1.28 and T/Tc1=1.29. The topology of this transition resembles that of the

binodals at a critical temperature minimum.48

E. Type IIIm phase behavior

Type IIIm phase behavior is similar to that of type III but

the critical curve exhibits a pressure minimum and a maxi-

mum in Fig. 7b. In Fig. 7a some diffusion spinodal iso-

therms are shown for a type IIIm phase diagram near the

region of the critical point of the more volatile substance

= 0, =0.4, and =0.05. At T/Tc1 =1.05 and T/Tc1 =1.1one can recognize two maxima and one minimum which are

marked by symbols for T/Tc1 =1.1. The maximum at low

pressure and the minimum has vanished for T/Tc1 =1.2

FIG. 5. Type V phase behavior calculated for =0 , =0.5, and

=0.2. aHigh temperature diffusion and mechanical spinodal isotherms. b Enlarge-

ment of the region near the critical point of the more volatile substance. The

bold dashed curve calculated for T/Tc1 =1.01 shows the existence of three

extrema of the diffusion spinodal isotherm together with a maximum athigher pressure not shown in this scale. c Low temperature diffusion and

mechanical spinodal isotherms. The minimum of the diffusion spinodal iso-

therms close to x =1 corresponds to the unstable branch of the critical curve.

Legend as for Fig. 3. The numbers mark different T/Tc1 values. d Sche-

matic sketch of the pT diagram of this system. The dot-dashed lines mark

the isothermal sections shown in b and c.

FIG. 6. Type III phase behavior calculated for =0 , =0.45, and =0.1. a

Diffusion and mechanical spinodal isotherms in the low pressure region. b

Diffusion spinodal isotherms in the high pressure region. Legend as for Fig.

3. The numbers mark different T/Tc1 values.

FIG. 7. a Type IIIm phase behavior calculated for = 0, =0.4, and

=0.05. The three squares mark the extrema of the diffusion spinodal iso-

therm at T/Tc1 =1.1. Legend as for Fig. 3. The numbers mark different T/Tc1values. b Corresponding pT diagram. Solid curves, critical curves; and

dashed curves, vapor pressure curves of the pure substances. The isotherm at

T/Tc1 =1.1 marks a section shown in a.




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which means that the cusp formed by the unstable critical

curve and the short critical curve connected to the critical

point of the more volatile substance is passed. The maximum

at high pressure corresponds to the stable LL-critical curve.

F. Types VI and VII phase behavior

The high temperature part of the type VII hase behavior

shown in Fig. 8a for = 0, =0.42, and =0.019 re-

sembles that of type V discussed above. In addition type VII

systems have a LL closed loop critical curve. Type VI phase

behavior is similar to type VII but with a continuous LV-

critical curve as in type I. Therefore, it is not discussed here

separately. It has been shown32

and confirmed several

times44,49 that Eq. 5 is able to generate such closed loopphase behavior, however, at very low temperature. Experi-

mentally LL closed loop phase behavior have been found for

water containing systems.50,51

Figure 8b shows the spin-

odals corresponding to a LL equilibria at low temperature

calculated with Eq. 5. One can see a sequence of LL diffu-

sion spinodals forming the closed-loop dome. At these tem-

peratures the mechanical spinodal is at very large negative

pressure not shown here which indicates that this diffusion

spinodal is LL type, being far away from the mechanical

spinodal.

G. Phase behavior in the shield region

The shield region is a small area in the global phase

diagram which exhibits several complex phase diagrams in-

cluding four-phase equilibria. Here, we have investigated the

diffusion spinodals for a system which very close to the sym-

metric systems at =0.0. The chosen parameters are = 0,

= 0.02, and =0.35. The diffusion spinodals and the pT

projection of this system are shown in Fig. 9a. Each re-

duced temperature, for which a diffusion spinodal is calcu-

lated in Fig. 9a is marked by a thin line in Fig. 9b. The

square symbols in Fig. 9a correspond to the same symbols

in Fig. 9b. At T/Tc1 =0.9 we find two spinodals correspond-

ing to LV equilibria ending at the pure substance spinodals at

x =0 and x =1. At higher pressure a spinodal related to a LL

equilibria with a LL-critical point at the minimum exists.

With decreasing temperature these three spinodals intersect

in two steps. First the LV spinodal at low mole fraction in-

tersects with the LL spinodal as one can see in Fig. 9 a for

T/Tc1 = 0.85. For lower temperatures the VL spinodal at high

mole fractions joins the remaining part of the LV spinodal at

low mole fraction. These two branches do not form a cusp at

T/Tc1 =0.85 as Fig. 9a suggests. Actually both have a con-

tinuous maximum being LV-critical points. At T/Tc1 =0.8 the

system has already passed the second transition. One diffu-

sion spinodal connects the pure substance spinodals continu-

ously while the diffusion spinodals starting at the pure sub-

stance spinodal at low pressure go to very high pressure at

about x =0.18 and x =0.82. The remaining island of the dif-

fusion spinodal between x =0.4 and 0.6 around p/pc1 =0.5 is

related to partly unstable critical curves G4x0 as one can

see in Fig. 9b.

IV. CONCLUSIONS

The spinodals represent the underlying property of a

phase diagram. They form planes in pTx space which are

envelops of the critical curves. There are a limited number ofspinodal shapes such as the LV-type spinodal, the continuous

LL spinodal of types II, V, or III or the diverging spinodal of

type III at low temperature. The different types of phase

behavior as known from the classification of van Konynen-

burg and Scott further differentiate similar types of spinodal

topologies. As expected, the mechanical spinodal shape

marked as type B in pure substances Fig. 1 has not been

found with the attracting hard-sphere model fluid. In case of

a LV equilibrium a mechanical spinodal is located near the

diffusion spinodal. In case of a LL equilibrium the mechani-

cal spinodal is typically far away from the diffusion spinodal.

For type II or III phase behavior the reason for the large

FIG. 8. Type VII phase behavior calculated for = 0, =0.42, and

=0.019. a High temperature diffusion spinodal isotherms. The numbers

mark different T/Tc1 values. b Low temperature diffusion spinodal iso-

therms. Legend as for Fig. 3.

FIG. 9. a Diffusion spinodals for a phase diagram in the shield region

calculated for =0 , =0.02, and =0.35. b Corresponding pT diagram.Symbols, binary critical points forming the critical curves; and dashed

curves, vapor pressure curves of the pure substances. The isotherms at

T/Tc1 =0.8, 0.85, and 0.9 mark the sections shown in a.




8/8

distance to the mechanical spinodal can be deducted from the

pressure-volume sections. A mechanical spinodal requires a

van der Waals loop in the pressure-density diagram which is

not present at high pressure in these phase diagram types. At

low pressure the diffusion spinodal exhibits a pronounced

maximum which causes the distance to the mechanical spin-

odal.

The unusual shape of the diffusion spinodal of type II or

type IIIm systems in the vicinity of the critical temperatureof the more volatile substance is of interest for processes

including a supercritical solvent such as carbon dioxide. Car-

bon dioxide-solute systems are often type II, IV, or III and

process conditions a usually slightly above the critical tem-

perature of carbon dioxide. The spinodal isotherm at T/Tc1=1.03 in Fig. 4b showing a type II system has already two

maxima and one minimum. However, only one maximum is

related to a stable LV-critical point. In Fig. 5b showing a

type V system at T/Tc1 = 1.01 the second maximum has

emerged to high pressure and is here a stable LL-critical

point. Such type of topology can be found in mixtures of

carbon dioxide with certain organic solvents which exhibit a

LL immiscibility on top of the LV equilibria. In case of thegas antisolvent crystallization such LL decomposition leads

to undesired large agglomerated paracetamol particles rather

than fine ones as discussed recently.52

The location of the

diffusion spinodal is expected to affect the precipitation pro-

cess in such cases.

ACKNOWLEDGMENTS

This work was partially supported by the Hungarian Re-

search Fund OTKA under Contract No. T 043042, by a

bilateral program of the German Science Foundation DFG

and the Hungarian Academy of Science HAS, Grant No.

436 UNG 113/150/n-1. A.R.I. was also supported by theBolyai Research Fellowship.

1P. G. Debenedetti, Metastable Liquids: Concepts and Principles Prince-

ton University Press, Princeton, NJ, 1996.2D. H. Trevena, Cavitation and Tension in Liquids Adam Hilger, Bristol,

1987.3A. Imre, K. Martins, and L. P. N. Rebelo, J. Non-Equil. Thermodyn. 23,

351 1998.4Liquids Under Negative Pressure, NATO Science Series, Vol. II/84, edited

by A. R. Imre, H. J. Maris, and P. R. Williams Kluwer, Dordrecht, 2002.5V. G. Baidakov, Sov. Technol. Rev. B 5, 1 1994.

6R. Gomes de Azevedo, J. Szydlowski, P. F. Pires, J. M. S. S. Esperanca,

H. J. R. Guedes, and L. P. N. Rebelo, J. Chem. Thermodyn. 36, 211

2004.7Q. Zheng, D. J. Durben, G. H. Wolf, and C. A. Angell, Science 254, 829

1991.8V. P. Skripov, High Temp. 31, 448 1993.

9K. Liu and E. Kiran, Macromolecules 34, 3060 2001.

10V. Kedrinskii, A. Besov, M. Devydov, A. Makarov, and S. Stebnovsky,

Proceeding of the Fifth International Symposium on Cavitation Osaka,

Japan, 2003, http:/iridium.me.es.osaka-u.ac.jp/cav2003/index1.html, Gs-4-004

11E. Steudle, Nature London 378, 663 1995.

12F. H. Veliyev and I. S. Guliyev, Proceedings: The Sciences of Earth 66, 5

2003.13

Y. Guan and D. G. Fredlund, Can. Geotech. J. 34, 604 1997.14

L. Mercury and Y. Tardy, Geochim. Cosmochim. Acta 65, 3391 2001.15

A. Imre and W. A. Van Hook, J. Polym. Sci., Part B: Polym. Phys. 32,

2283 1994.16

A. Imre and W. A. Van Hook, J. Polym. Sci., Part B: Polym. Phys. 35,

1251 1997.17

A. Imre and W. A. Van Hook, Chem. Soc. Rev. 27, 117 1998.18

L. P. N. Rebelo, Z. P. Visak, and J. Szydlowski, Phys. Chem. Chem. Phys.

4, 1046 2002.19Z. P. Visak, L. P. N. Rebelo, and J. Szydlowski, J. Chem. Educ. 79, 869

2000.20

Z. P. Visak, L. P. N. Rebelo, and J. Szydlowski, J. Phys. Chem. B 107,

9837 2003.21

G. M. Schneider, Ber. Bunsenges. Phys. Chem. 76, 325 1972.22

B. A. Wolf and G. Blaum, Macromol. Chem. Phys. 177, 1073 1976.23

L. Mercury, M. Azaroual, H. Zeyen, and Y. Tardy, Geochim. Cosmochim.

Acta 67, 1769 2003.24

S. Jiang, L. An, B. Jiang, and B. A. Wolf, Chem. Phys. 298, 37 2004.25

A. Drozd-Rzoska, S. J. Rzoska, and A. R. Imre, Phys. Chem. Chem. Phys.

6, 2291 2004.26

T. Kraska, Ind. Eng. Chem. Res. 43, 6213 2004.27

R. L. Scott and P. H. Konynenburg, Discuss. Faraday Soc. 49, 87 1970.28

P. H. van Konynenburg and R. L. Scott, Philos. Trans. R. Soc. London,

Ser. A 298, 495 1980.29

L. Z. Boshkov, Dokl. Bolg. Akad. Nauk 40, 901 1987.30T. Kraska and U. K. Deiters, J. Chem. Phys. 96, 539 1992.

31A. van Pelt, C. J. Peters, J. de Swaan Arons, and P. H. E. Meijer, J. Chem.

Phys. 99, 9920 1993.32

L. V. Yelash and T. Kraska, Ber. Bunsenges. Phys. Chem. 102, 213

1998.33

A. R. Imre, L. V. Yelash, and T. Kraska, Phys. Chem. Chem. Phys. 4, 992

2002.34

M. Bardas, N. Dahmen, T. Kraska, K.-D. Wagner, and L. V. Yelash, Phys.

Chem. Chem. Phys. 4, 987 2002.35

L. V. Yelash, T. Kraska, A. R. Imre, and S. J. Rzoska, J. Chem. Phys. 118,

6110 2003.36

A. R. Imre, T. Kraska, and L. V. Yelash, Phys. Chem. Chem. Phys. 2, 992

2002.37

A. R. Imre, A. Drozd-Rzoska, T. Kraska, K. Martins, L. P. N. Rebelo,

S. J. Rzoska, Z. P. Visak, and L. V. Yelash, in Nonlinear Dielectric Phe-

nomena in Complex Liquids, edited by S. J. Rzoska and V. Zhelezny,NATO Science Series Vol. 157 Kluwer, Dordrecht, 2004, pp. 177189.

38M. Modell and R. C. Reid, Thermodynamics and Its Application Prentice

Hall, Englewood Cliffs, NJ, 1983, Chap. 9.39

M. Mller, L. G. MacDowell, P. Virnau, and K. Binder, J. Chem. Phys.

117, 5480 2002.40

D. Kondepudi and I. Prigogine, Modern Thermodynamics Wiley, New

York, 1998.41

J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures Butter-

worths, London, 1982.42

N. F. Carnahan and K. E. Strarling, AIChE J. 18, 1184 1972.43

L. V. Yelash and T. Kraska, Z. Phys. Chem. Munich 211, 159 1999.44

R. J. Sadus and J.-L. Wang, Fluid Phase Equilib. 214, 67 2003.45

MAPLE 9.01 Waterloo Maple Inc, Waterloo, Canada, 1981-2003.46

J. M. H. Levelt Sengers, in Supercritical Fluid Technology: Fundamentals

for Application, edited by T. J. Bruno and J. F. Ely CRC, Boca Raton,

1991.47

U. K. Deiters and I. L. Pegg, J. Chem. Phys. 90, 6632 1989.48

G. M. Schneider, Ber. Bunsenges. Phys. Chem. 70, 497 1966.49

R. L. Scott, Phys. Chem. Chem. Phys. 1, 4225 1999.50

H. Ochel, H. Becker, K. Maag, and G. M. Schneider, J. Chem. Thermo-

dyn. 25, 667 1993.51

A. Wallbruch and G. M. Schneider, J. Chem. Thermodyn. 27, 377 1995.52

A. Weber, L. V. Yelash, and T. Kraska, J. Supercrit. Fluids 33, 107 2005.


Attila R. Imre and Thomas Kraska- Stability limits in binary fluids mixtures

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Transcript of Attila R. Imre and Thomas Kraska- Stability limits in binary fluids mixtures