CHAPTER II

PHASE DIAGRAMS IN BINARY ALLOYS

1. INTRODUCTION

The thermodynamics of alloys and their phase diagrams have been studied for

a long time due to their obvious technological importance. Most of these studies

use a combination of basic thermodynamical principles and empirically determined

thermodynamic data like enthalpy. activities etc [1.2]. The use of empirical

information is unavoidable. particularly in those situations where the phase

diagrams are complex. involving a number of intermediate phases. However. from

the examination of these diagrams. it is clear that there are certain features

common to most alloy systems. Superposed on these are other features that are to

be understood in terms of individual chemical properties of the components of the

concerned alloy. These common features have been understood in a schematic way

from general thermodynamic principles and the simplest atomic properties. Here

one feels the need for a simple statistical model from which generic phase

diagrams can be derived by variation of a few microscopic parameters. This will provide a qualitative insight of the kind the Ising model provides for one

component systems. Indeeq. a considerable amount of work has been done to develop

statistical models for multicomponent systems [3,4.5]. However. such models

involve multidimensional parameter spaces. Hence they have not been explored as

thoroughly as the corresponding models for one component systems. In particular.

the statistical mechanics that underlies the typical binary mixture phase

diagrams in the concentration - temperature plane (pressure being held fixed) and

the freezing of binary liquids has not been explicated - although it is implicit

in many earlier works. This is possibly because these works have not studied the

thermodynamics in terms of constraints and variables that are commonly employed

in laboratory studies. Thus the purpose of the present study is to provide a

detailed but model analysis of a well-known statistical model under appropriate constraints.

18

Another motivation comes from the DSC studies that have been done on a

series of binary mixtures [6]. These investigations reveal an interesting pattern

of evolution of enthalpy with temperature as the relative component concentration

of tl\~ mixture is varied. Though one can understand such features in these and

other related experiments in a general qualitative way from thermodynamic

principles, not much attention seems to have been given to quantitative aspects.

Accordingly, the emphasis here is on the calculation of enthalpy changes that

occur as the temperature of a binary mixture is varied through two phase regions.

The model under consideration is a spin-one Ising model. This model has been

considered from various points of view in the literature [3,4,5], but has not

been explored to study the features mentioned above. Here there are two order

parameters. One of them is density and allows for coexistence of phases with

different densities. The second order parameter is the relative concentration of

the two components and allows for coexistence of phases with different

compositions. The model contains two independent system dependent microscopic

parameters which can be varied to generate all the familiar types of phase

diagrams and allows comparison with a number of properties that would prove

useful for developing more detailed understanding. The model has been studied in

the mean-field approximation. This is the simplest nontrivial approximation and

is quite adequate for the present purposes. In contrast to the thermodynamic

models in which different expressions of free energy for different phases that

can occur are assumed, this model has one unified expression for the relevant

free energy. On minimising this free energy one can produce different types of phase diagrams by variation of the interaction parameters and the operating conditions.

The chapter is organised as follows: In section 2 the basic model and

calculation of the free energy in the mean-field approximation is presented. In

section 3 some other thermodynamic potentials are derived which are more

appropriate for the generation of phase diagrams under different conditions as

mentioned above. In particular, the thermodynamic potential for generating phase

diagrams under the condition of fixed ambient pressure is considered. Section 4 discusses the detailed numerical results for various types of fixed pressure phase diagrams. The calculated and the representative organic

enthalpy and heat evolution in typical cases

results compared with the experimental data binary mixture. Section 5 contains a summary

19

are also

for a of the

results.

2. MOD~ AND FREE ENERGIES

In the lattice model of a binary mixture of particles (atoms or molecules)

of types A and B, a ftxed volume of space is divided into No identical cells each

having volume v 0 just big enough to contain one particle of either type. The

basic simplification consists in assuming that the cell is either fully occupied

or empty. Thus partial occupation of a cell is disallowed. To describe the binary

mixture one introduces occupancy variables p~ and p?, such that P~ ( B ) =1 if the

ith cell is occupied by particle A(B) and zero otherwise. Assuming that only

particles occupying nearest neighbour cells interact, one writes the energy H of

any conftguration as

-2 ~. P~ + ~B p~l 1

(11.2.1)

where J AA ' J AD and J DD represent the strength of interaction between particle

pairs AA, AB and BB respectively. To ensure the existence of phase transitions in

pure A and pure B systems both J and J DD are taken to be positive. JAB is also AA

taken to be positive. The last term is introduced as the description will be in

terms of a grand canonical ensemble. ", A and "'B represent chemical potentials for

the two species. Eq. (11.2.1) can be rewritten in the language of the spin-one

Ising model by defining a new variable (J i in terms of which

and 2

(l-(J.)o . pB = 1 1

i --2-- (11.2.2)

Clearly (J i = + 1 (-1) corresponds to the occupation of the ith cell by particle

A(B) and (J i = 0 corresponds to the cell being empty. Denoting the number of A(B)

20

particles in the system by Nit. ( B) and the total number of particles by N,

(11.2.3)

Expression (11.2.1) for H can now be rewritten as

(11.2.4)

where

(II.2.5a)

(II.2.5b)

(11.2.5c)

(II.2.5d)

Mean field approximation is used to calculate the partition function. For this

purpose the following order parameters are defined

and (11:2.6)

where < > denotes ensemble average. <(1~> and «] i > are independent of cell index

because of translation symmetry. Note that p and M correspond to the average

particle density and the concentration difference between the two species respectively. Now vniting

(1~ = p + ~ i and (1 i = M + &J i (II.2.7)

and substituting these values in Eq.(11.2.4) gives

21 7Jj-5832-

K J H = -"2 l (p + 6p i) ( P + 6p j) - 2 l (M + 00 i) (M + 00 j )

i , j 1 , j

~ l [<M + 00 1 ) (p + 6pj) + (p + 6Pi) (M +OOj)] i,J

- ~ 2 (1 i - D 2 (17 (11.2.8)

Mean field approximation consists of neglecting tenns quadratic in 6p 1 and 00 i .

This gives

K J - - - -H = - NoZp2 - - NozM2 - LNozMp - (Kp + Ut) \' 6p - (JM + Lp) \'Mi

2 2 L 1 L ~

(11.2.9)

Substituting values of 6p 1 and 00 1 from Eq.(II.2.7) gives

N ow the partition function can be written as

Z = Tr e -IlH

-IlNoZ(Kp2+n~1:2+2LMP)/2[ - _ ]No = e 1 +2eIl(D+KpZ+LMZ)coshll(~JMz+Lpz) (II.2.U)

Then the free energy per site at the temperature T is given by

-;: ..

22

tJ(zKp+zLM+D) - } - k8 T log{I+2 e cosh(zJM+zLp+J.l) (II.2.12)

Here tJ = (kB T) -1 and z is the coordination number of the lattice. Further the

self-consistent equations detennining p and M are obtained by minimising Q and

are given by

p=-----l+2e~Xcosh tJY

(II.2.13)

- 2e~Xsinh pY M=-----

l+2etJXcosh tJY (II.2.14)

where

x = zKp + zLM + D (1I.2.15a)

Y =. zJM + zLp + IJ (II.2.15b)

For given values of T, D and IJ. the self-consistent equations (11.2.13) and

(II.2.14) are solved for p and M. These solutions are then substituted in

Eq.(II.2.12) to obtain the free energy Q from which all the thermodynamic

properties can be obtained. This free energy has been examined in some detail in

a series of papers of Lajzerowicz and Sivardiere in various physical contexts

[4,51. Depending on the values of T, D, and IJ, one can get single or multiple

solutions for (P,M). However, the free energy Q with independent :'ariables D and

IJ is not the most convenient one to use for deriving the commonly used phase

diagrams in temperature-concentration (T-M) plane at a fixed pressure. Since the

goal here is to derive such phase diagrams and also exhibit the variation of

enthalpy in the T-M plane, Legendre transformations are used to obtain the

appropriate thermodynamic potential with the desired set of independent

variables.

3. TRANSFORMED THERMODYNAMIC POTENTIALS

First a thermodynamic potential f(T .p,M) is defined in which the

23

independent variables are T, p, M. This IS done by a Legendre transformation

keeping in mind Euler's relation. Thus

f(T,p,M) = O(T,D,,..Ll + Dp + J.JM (11.3.1)

where D and J1 are replaced by their expressions in terms of M and p (obtained by

inverting Eqs.(II.2.13) and (1I.2.14) ). D and J.J can also be expressed in terms

of the equations

= D and - =J.J (11.3.2) ap aM

Defining a new variable M = M/p = <NA-Na>/<NA+Na> corresponding to the relative

concentration difference, the expression for f, with M rather than M as an

independent variable, becomes

f(T,p,M) Z 2 2

= - - p [ K + 2LM + JM ] + kaT ( (l-p)log(l-p) 2

1 1 1 1 + - p(1-M) log - p(1-M) + - p(1+M) log - p(l+M)]

2 2 2 2 (11.3.3 )

For a single phase situation to be thermodynamically stable f should be a convex

function of its arguments. When this global convexity is lost the system

minimises its free energy by decomposing into two coexisting homogeneous phases.

Densities and concentrations for these two phases are found by making a tangent

construction on the free energy surface. The procedure used is as follows:

Suppose (pl,M 1) and (p2,M 2) are the densities and the concentration

differences in these two phases. If the system is constrained to have an average

density p and average concentration difference M then the volume fraction (r) of

the first phase will have .to satisfy the following equations:

p = rPt + (l-r)p2 (II.3.4)

M = rM 1 + (l-r)M 2 (II.3.5)

24

These equations simply reflect the conservation of the number of particles of

each type. Notice that equations (11.3.4) and (11.3.5) implY that the point (p,M)

lies on the straight line connecting the points (pl,M 1 ) and (p2,M 2 ). Now the free

energy is minimised with respect to decomposition into all possible pairs of

points Qn a straight line of a given slope (t) passing through the point (p,M).

In this search procedure there is only one independent variable. Denote this

minimum free energy by S(t,T,p,M). Finally S is minimised with respect to t. This df df

procedure includes but goes beyond the situation when and - are identical in op oM

the two phases.

In most of the cases phase diagrams of binary mixtures 'are studied under

conditions of fixed pressure P and fixed relative concentration M. This means

that one should work with a free energy for which the pressure is an independent

variable. For a fixed value of the elementary cell volume Vo the variable

conjugate to pressure is No:

(11.3.6)

where F = N of. Equation (II.3.6) can also be written as

P = - [f - p D - M 11] (11.3.7)

Under these conditions, the thermodynamic potential G needed is generated by the

following Legendre transformation:

(II.3.S)

or

g= (f + P) / p = (pD + ~)/ p (11.3.9)

where g=G/<N>. From Eqs.(II.3.6) and (II.3.9) the expressions for pressure and

25

free energy can be written as

122 P = - - Z P ( K + 2LM + JM ) - kBT 10g(1-p) 2

(II.3.1O)

g(T.M.P) = -z p( K + 2LM + JM2)

[ p 1-M 1-M 1+M 1+M]

+ kB T log- + - log - + - log--1-p 2 2 2 2

(11.3.11)

To get the phase diagram in the T -M plane for a given pressure. first of all p is

eliminated from the equation (11.3.11) by using equation (11.3.10). Thus- the fr"L"

energy is obtained explicitly as a function of T. M and P. For some ranges of M this function may be multiple valued with one or three branches, Thermodynamic properties are governed by the function g(T,M,P) which is obtained by combining the lowest branches in the whole range of M. If g is a globally convex function a homogeneously mixed single phase is stable at that temperature and pressure for any relative concentration. Otherwise the single phase system is

thermodynamically unstable and it decomposes into a combination of two coexisting but distinct phases. The values of M for these two coexisting phases are found through the procedure of common tangent construction. Denoting these solutions by M 1 and M 2 the number fraction (<X) of the particles in the first phase is given by

the lever rule

Enthalpy (H) can be calculated from G (Eq.II.3.8) by use of the relation

where S = . Then the enthalpy per particle (h) is aT

koT h = -Zp( K +- 2LM + 1M2) - 10g(1-p)

p

(11.3.12)

(11.3.13)

(II.3.14)

When the single phase situation is thermodynamically unstable the enthalpy per

26

particle of the system can be written as

(11.3.15)

where h 1 and h 2 are the entllalpies per particle of the two phases that the system

decomposes into.

4. NUMERICAL RESULTS

In all the numerical calculations the unit of energy and temperature has

been choosen to be zK and zK/kB• respectively. Thus the only independent

interaction parameters are J/K and L/K. The numerical values of both of these

dimensionless parameters lie in the range of -1 to 1. LI K is always taken to be

positive since a simultaneous reversal of signs of M and L/K corresponds to

exchanging the roles of particles of types A and B. Thus the phase diagram for

(J/K. -L/K) is obtained from that for (J/K. L/K) through a simple reflection

around the line M = 0 in the T-M plane. The pressure P (=0.01 in the unit of zK)

is kept constant for all the calculations. In all cases the stable phase at the

highest temperatures has the lowest density when three solutions for p exist. By

convention it is called the liquid phase. At the lowest temperatures there is

only one solution for p - its numerical value being very close to one. This phase

will be referred to as the solid phase. According to this description there is

always a solid-liquid phase transition for pure A and pure B systems. In the

intermediate temperature range labelling a phase as solid or liquid is done in a

way so· that continuity of desmption is maintained. When there are three

possible values of free energy the highest one corresponds to v.llat will be

referred to here as the metastable state.

4.1 Phase diagrams

In order to classify the different types of phase diagrams possible for

various values of J IK and L/K it is important to remember the following points:

When J IK is positive demixing of particles of types A and B is energetically

favourable in general. For negative values of 11K homogeneous mixing is preferred. However. entropic considerations favour mixing. At the highest

temperatures, when the role of entropy is dominant, the stable phase is always a

27

liquid phase in which particles of types A and B can be mixed in any proportion

(even when J/K is positive). At the lowest temperatures, when energy rather than

entropy is the deciding factor and the system is in the solid phase, whether or

not species A and B can be mixed in any proportion is decided by the sign of J/K.

For J/K<O this is indeed possible whereas for J/K>O it is not. Since the overall

structure of a phase diagram is constrained by these limiting situations it is

clear that J I K is the appropriate parameter in terms of which all possible phase

diagrams should be categorised. The numerical data presented below correspond to

a set of values of J/K and L/K which have been selected such that they generate

all the distinct types of phase diagrams possible for this model Pl.

(a). JlK=O

For J=L=O J AA' J BB and JAB have the same value. Since energywise there is no

distinction among the different kinds of pairs possible, particles of types A and

B are equivalent and thermodynamic properties do not depend on relative

concentration M. The two species mix in any proportion at any temperature and

there is 8. single liquid-solid phase transition at a temperature which is

independent of M. A liquid-solid coexistence region appears as L/K becomes

nonzero. The size of this thermodynamically unstable region in the T - M plane

increases with the value of L/K. For L/K = 0.2 the free energy at three different

temperatures is shown in Fig.(II.l). At temperature T=0.222 the lowest branch of

the free energy function corresponds to the liquid phase. At the lower

temperature T=0.155 two phases (liquid and solid) coexist and the values of M for

these coexisting phases are the abscissa of the points where a common tangent

touches the two branches of the free energy function (Fig.(ll.lb». At a still

lower temperature T=0.110 the solid branch of the free energy function becomes

the lowest, showing the existence of a single solid phase at this temperature.

The corresponding phase diagram is shown in Fig.(lI.2a). Phase diagram for L/K = 0.05 is shown in Fig.(II.2b) to illustrate the point that the coexistence region

shrinks as L/K decreases.

(b). JlK<O

This is the situation in which there is complete miscibility in both solid

and liquid phases. The free energy at different temperatures for 11K = -0.15 and

28

>-01 .... " c CI.I -0.80 CI.I CI.I ~

-O~--~~~~--~~ -10 -0.5 1.0

- 0.10 (C)

t

" £ - 0.60

-0.80 - ,.0 -O.S

CI.I ~ - 0.60

Lo..

1.0

0.0 1.0 M

Figure II.I: Free energy at temperatures (a) T=O.222. (b) T=O.155 and (c) T=O.110 for J I K =0 and LI K =0.2.

0.16

0.12

0.08L--~L----'---~---...J -1.0 -0.5 0.0

M 0.5 1.0

(b)

0.18

0.17

0.16

0.15 L..-__ --L ____ -'--__ ----L __ ----'

-1.D -0.5 0.0 M

0.5 1.0

Figure 11.2: Phase diagrams for (a) JlK=O and L/K=O.2 and (b) JlK=O and L/K=O.05.

29

L/K = 0 is shown in Fig.(II.3). At T 1 :: 0.188 the lowest branch of the free

energy is of liquid phase indicating existence

temperature. At lower temperature T 2 =0.156,

of single liquid phase at this

two phases (liquid and solid)

coexist and their coexisting solutions are the points where a common tangent

touches the two branches of free energy. At still lower temperature (T 3 =0.128)

the solid branch of the free energy becomes the lowest, shov/ing the existence of

single solid phase at this temperature. The corresponding phase diagram is shown

in Fig.(II.4a). Here there are two coexistence regions symmetrically situated

about M = O. There are three values of relative concentration (M = -1, 0 and +1)

for which the system undergoes a simple liquid-solid phase transition as

temperature changes. For any ,other proportion of mixing there is 8 two phase

coexistence region. As L/K becomes nonzero the phase diagram becomes asymmetric.

The melting point of pure A system becomes higher than the melting point of pure

B system and the coexistence region on the right shrinks. Beyond 8 critical value

of L/K, which depends on J/K, there is only one coexistence region (as in

Fig.(lI.2). The phase diagram for such a situation, with J/K = -0.075 and L/K =

0.1, is shown in Fig.(II.4b).

(c). J/K >0

As stated earlier, for J = L = 0 there is a simple liquid-solid phase

transition at a temperature which is independent of relative concentration. When

J I K increases in the positive direction ty.ro features immediately develop in the

phase diagram: (i) At the lowest temperatures, where a single solid phase is

stable for J=O, a coexistence region between two solid phases de\:elops, its upper

boundary being convex upwards. This is shown in Fig.(Il.Sa). Oi) As shown in

Fig.(II.3a) two coexistence regions, symmetric about M=O and meeting at a point,

develop. However, in contrast to Fig.(l1.3a), the lower boundary is concave

upwards. In between there is a range of temperatures where the pha..c:e is solid

with mixing possible in any proportion. Immediately above this temperature range

coexistence between liquid and solid phases takes place wherea...~ below this the

system is either in a pure solid phase or in a state of coexistence between two

solid phases. The free energy at four different temperatures for this situation

is shown in Fig.(II.5b).

As J/K increases beyond this temperature range, the region where a single

30

-0.35 .-------------, (0 )

>--0.45 ~ c ., ., QI ~ -0.55

-0.65 -lD -0.5 1.0

-0.40 t c )

~-0.50 .. ., c QI

QI QI ~ -OJ)()

-0.70 -lD - 0.5

S

0.0 ,..

-O.I/J (b)

>-0.50 ~ ., c ., ., ., ~ -oro

-0.70 -1.0

O.S

L

1.0

1D

Figure 11.3: Free energy at temperatures (a) T l' (b) T 2 and (c) T 3 for J/K=-Q.15 and L/K=O.

0.17 .-----------1

0.16

.... ....

O.,L.L[--..l------I.I---'I-----1DC - O.SO 0.00 O.SO 100 1.0

M M

Figure n.4: Phase diagram for (a) JlK=-Q.15 and L/K=O and (b) JlK=-o.075 and L/K=O.1.

31

....

020 'I

L

0.16 5

T)

Tc ------ c 'I. 0.12 oc.~

0.0& L--_--'-__ --'-_--..J'--_..J -1.0 -0.5 0.0

M

O.S 1.0

Figure 11.58: Phase diagram for J/K=O.15 and L/K=O.

(0) ( b)

-0.55

>-01 -0.65 >-~

~ '" c '" '" c

'" "'-0.65 '" '" ~ -0.75 '" ~

-0.85 -0.75 -1.0 - 0.5 0.0 0.5 1.0 -to - 0.5 0.0 0.5

M M

(d)

-0.57

~ -0.52 >-...

'" 01 C

'" II II t -0.60 II

II ~

-0.68 0.59 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 OD O.S

M tot

Figure II.5b: Free energy at temperatures (a) T1' (b) T 2 • (c) T3

corresponding to Fig.II.5a.

32

lD

1.0

and (d) T4

solid phase exists shrinks steadily to zero. This happens at a finite value of

J I K beyond which there exists a temperature when a single liquid phase can be in

equilibrium with two distinct solid phases (Fig.(II.6a)). The corresponding free

energy at four different temperatures is shown in Fig.(I1.6b). The relative

concentration 'domain where a single solid phase is stable shrinks as J I K

increases (Fig.(II.7). The asymmetry in figures (II.4b) and (II. Sa) is due to

tile nonzero value of L/K. The evolution of the phase diagram as UK increases is

illustrated here by using the phase diagrams in Fig.(1I.8) and Fig.(Il.9). Figure

(11.8) can be understood from the phase diagram in Fig.(lI.Sa) where L = O. As

L/K increases the melting point of pure A increa.~es relative to the melting point

of pure B. Simultaneously the solid-liquid coexistence region on the left shrinks

to zero. The solid-solid coexistence region survives. It does not appear in

Fig.(1l.8) because the temperature is never low enough but the existence of this

region can be demonstrated analytically. The phase diagram for J/K=O.lS, L/K=O.2

is shown in Fig.(II.9). At T 1 the phase is liquid in and two components can mix

in any proportion. At T 2 two phases (liquid and ~-solid) can be in coexistence.

At a still lower temperature T 3 a-solid phase of a specific composition can be in

equilibrium with the liquid phase or with the ~-solid phase. This temperature is

known as the peritectic temperature. At T 4 the free energy function developes two

convex hulls with no common points. One gives the coexistence of a liquid (L) and

a solid phase (a), and the other one gives coexistence of two solid phases ( a

and ~ ). Two solid phases are in coexistence at T 5' This phase diagram can be

understood as an evolution from Fig.(II.6a) as the value of L/K is increased. In

Fig.(II.6a) JAB < J BB < J AA because both J and L are positive and J > L. This

explains the relative . positions of the following three temperatures: the melting

point of pure A, the melting point of pure B and the eutectic temperature.

However, in the case of Fig.(II.9), J < L. Thus J AA > JAB > J BB, leading to the

consequence that the peritectic temperature lies between the melting points of

pure A and pure B. This constraint forces the upper part of Fi8.(II.6a) to

transform into the corresponding part of Fig.(II.9). The lower sections of the two figures are of the same type.

4.2 Enthalpy changes

This section present some representative data on the evolution of enthalpy

with temperature at fixed values of the relative concentration difference M. The

33

0.2\ 1,

0.19 L 11

t- 1)

14

QI3L---~----~----~--~

- 10 - 0.5 0.0 05 1.0 M

Figure II.6&: Phase diagram for J/K=O.2 and L/K=O.03.

~.85 - 0.7 - 1.0 1.0 -1.0 1.0

M M

( b) (d)

-0.50 -O.S

>-01 ... " >-c C7I

" ... " c " -0.6 tI

" ... ....

-0.7 -Q7 - 1.0 -0.5 0.0 0.5 1.0 -1.0 -o.S 0.0 .0.5 1.0 I

M M

Figure II.6b: Free energy at temperatures (a) T l' (b) T z' (c) T3 and (d) T4 corresponding to Fig.II.6a.

34

0.28 r------------,

0.18

I-

0.20 0+/3 0.16

0.16 L.-_---A __ ..I....__---A __ ~ 0.14~---L--..I....---'-...I. __ ....J -1.00 -0.50 0.00 0.50 1.00 -to -0.5

tot 0.0 tot

0.5 to

Figure 11.7: Phase diagram corresponding to parameters JlK=O.7 and L/K=O.OS.

Figure 11.8: Phase diagram corresponding to parameters JlK=O.075 and L/K=O.1.

O. 2i. r------------------, 1,

0.22

0.20 12

0.18

115

0.12

l

0.1 0 L-------L.-----'L-----'--::---~, 00 - 1.00 - 0.50 .

Figure 11.9: Phase diagram corresponding to parameters J/K=O.15 and L/K=O.16.

35

pattem of enthalpy change cOITesponding to the phase diagrams of Figs.(1I.5a) and (II.8) is shown in Fig.(ll.lO). Enthalpy variation with temperature at different concentrations for the eutectic phase diagrams of Figs.(1l.6a) and (II. 7) is shown in Fig.(lU1). Fig.(l1.12) shows enthalpy variation corresponding to the phase diagranl of Fig.(lI.2). The interesting points to note here are the following: The enthalpy versus temperature curve is continuous everywhere except at those temperatures where there is phase transition from solid to liquid or from one type of solid to another. The latent heat of fusion is given by the magnitude of discontinuity. At those temperatures where the

system goes from a mixture of two phases P 1 and P 2 to either pure P 1 or pure P 2

phase the enthalpy is continuous but the slope changes discontinuously. One empirical fact worth pointing out here is that in all the cases of solid-liquid

transition, which in general involve passing through a two-phase coexistence region, the total change of enthalpy for complete melting increa."es with the temperature at which melting starts.

In Differetial Scanning Calorimetry (DSC) studies of phase transitions what one measures directly is the heat flow when a mixture is either cooled or heated at a constant (heatig or cooling) rate. This is a quantity proportional to specific heat. Thus for the purpose of comparison with DSC data the appropriate quantity to calculate is the derivative of enthalpy with temperature. This is identical with specific heat except at those points where the derivative is not defined. Specific heat at three compositions (M = -0.62, -0.25, 0.72, ) corresponding to the eutectic phase diagram of Fig.(II. 7) is shown in Fig.(l1.14). For the sake of direct comparison with experimental data the negative of specific heat versus temperature has been plotted. Heat flow (equivalent to specific heat here) data for a binary mixture of two organic compounds (Cis-decalin and Bromobenzene) at three different concentrations (18.8%, 38.3%, 72% bromobenzene) is shown in Fig.(1l.15). The phase diagram of this binary system (Fig.3 of ref.6 which is reproduced here as Fig.(11.16» is of the same type as tilOse presented in Figs.(II.6a) or (11.7). Figs.(1I.14) and (II. IS) clearly bring out the extent of agreement between the nvmerically calculated specific heat data and the experimental results. The singular looking peaks in Fig.(II.14) and the corresponding points in Fig.(ll.lS) reflect the fact

dH that at these points the derivative of enthalpy ie. - is not defined.

dT

36

~ a. o

.s=

(a)

0.1

-0.1

c- 0.3 w M=-1

~ a. -

- 0.5 M=l

- 0.7 L-_--'-__ .L-_---I... __ ..a.....-.----01

Ol3 Q15 Q17 Q19 Temperature

0.2

(b) r-r 1. M =0 1 2 3 4

o 2. M:O.4 3. M=0.8 4. M:1.0

,g -0.2 ... c LLJ

- 0.4 - .J

0.19 0.21 Temperature

Figure ILIO: Enthalpy variation with temperature at indicated concentrations (M) corresponding to the phase diagrams of (a) Fig.n.S and (b) Fig.n.S(a).

37

(0)

0.1

M:-0.17

-0.1 ~ a. 0

.J:. E -0.3 LaJ

-0.5 M = 1

-0.7 O.u. 0.16 0.18 0.20

0.2

~

.9- -0.2 o

.J:. -C LaJ

-0.6

Temperature

(b)

M=-o'25

M=l

0.22

-to ~~~_-'-_...L-----,....L._-L.._.....J 0.18

Temperature

Figure n.ll: Enthalpy variation with temperature at indicated concentrations (M) corresponding to the phase diagrams of (a) FigJI.6(a) and (b) Fig.II.7.

38

>-a. o

.s= ... c:

0.15

0.05

- Q 15

UJ - 0.35

-0.55

- 0.75 0.08

M=_O_

M=0.3

M='

0.13 0.18 0.23

Temperature

Figure 11.12: Enthalpy vanaUon with temperature at indicated concentrations (M) corresponding to the phase diagram of Fig.II.2a.

39

~ o

1001 so 1,0 (a) (b) I Cel

0

0. C - SO .. 0 ... a ~ a 0.,72 ~:- 0..62 0 .. .. .. -100 L L L

,~ - 100 u

M-:;"I I ~ -40. ..

~ III

I

.. - I!:IO ... u u .. ~ Q

III - 200 III - 200 I I - 80.

- 250

- JOo. I I -30'b -120 0.,19 0.,21 Q,n 0.,25 19 0.,20. 0.,21 0.12 0.,23 024 0.,18 0,20 0..22 0.,24 0.26

T.mp.roture r.mp.,.otur. T.mp'.r~t,':Ir.

Figure n.13: Variation of specific heat with temperature at compositions (a) M=-0.62. (b) M= -0.25 and (c) M=O.72 corresponding 'to the phase diagram of Fig.I1.?

(a) 0..0. b) I (e)

t 01 0.5 0. 0..5 ! ! "i -0.,5

3 3 0 0..0. 0 .2 0.,0. .... .... .,1.0. ... .... .... .... 0 ~ -0. 5 tI 0

I ' I-I. 5 ~ -QS

-2,0. - 1,5 -75

Figure 11.14: Experimental data for enthalpy change in the mixture of cis-decalin and bromobenzene at concentrations of (a) 18.8%, (b) 38.3% and (c) 72% ~romobenzene.

0.,28

245

24lJ 21.0

235

x 23 L 230 - !I::

t- -t-l"'~

220 220

215 x x

21 210 1.0

X

Figure,. ILlS: Experimentally determined phase diagram of the binary mixture of cis-decalin and bromobenzene (from ref.6).

41

5. SUMMARY

The results described in the previous section show that the calculations

based on a !pin-one Ising model provide a unified and comprehensive description

of most of the basic features that occur in phase diagrams of binary mixtures. By

considering an appropriate free energy function which is a Legendre transform of

the free energy calculated from the statistical model. it is shown that all the

typical binary mixture phase diagrams in temperature-concentration plane can be

derived by varying just two interaction parameters ( 11K. L/K ). Furthermore,

this treatment allows calculation of enthalpy - which enables us to understand

quantitatively the nature of heat absorption as the temperature of a binary

mixture is increased. The results obtained are in good qualitative agreement with

the data obtained in DSC measurements.

42

(III.4.5)

where

G. j = . 1, ij = A,B (III.4.6)

The expressions for these derivatives are given in the Appendix. Introducing new • •

variables ~A = N A -N A and ~o = N o-N B' and substituting Eq.(III.4.5) into

Eq.(IlI.4.3). one obtains

(IIl.4.7)

The relevant solution of Eq. (III.4.7) is of the fonn:

(IIl.4.8)

where the variable u is a linear combination of ~A and ~B'

(IlI.4.9)

where the coefficients "A and "B' yet unknown, are determined in the following.

Substituting Eq. (IlI.4.8) into Eq. (1I1.4.7), 2 * Z ,. 2 aF 1 ,.

(DA "A + Do "0) - - -.- lDA (~GAA ~A + ~GAO ~B) "A &12 kBT

* aF + DB(~GBA ~A + ~GBB ~B) "B] - = 0 (IIl.4.10) au

aF The validity of this equation requires that the coefficient of is proportional

au to u itself, that is,

• • DA IlGAA "A + DB IlG OA "0 = KvA • •

-DA IlGAO "A + Do ~GAO "0 = K"o (111.4.11)

54

- ----------------------------

REFERENCES

(1) Y.K.Rao, 'Pha..~ diagrtJ111s: MateriBl Science & Technology, Vol.1, (edited by Alper M. Allen), p.1, Academic Press, New York and London (1970).

(2) A.H.Cottrell, 'TheoreticBl Stmctural Metallurgy, chapter X, The Camelot Press, London (1955).

(3) M.Blume, V.J.Emery and R.B.Griffiths, Phys. Rev. A 4, 1071 (1971). [4] J.Lajzerowicz and J.Sivardiere, paper II, Phys. Rev. A 11, 2090 (1975). [5] J.Lajzerowicz and J.Sivardiere, paper III, Phys. Rev. A 11, 2101 (1975). (6) S.S.N.Murthy and Deepak Kumar, J.Chem.Soc. Faraday Trans., 89, 2423 (1993). (7) L.D.Landau and E.M.Lifshitz, 'Statistical Physics' I (3rd edition), p.295,

Pergamon Press, New York (1980).

43