chapter ii phase diagrams in binary...
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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.
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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)
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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