Materials Science & Metallurgy
Master of Philosophy, Materials Modelling,
Course MP4, Thermodynamics and Phase Diagrams, H. K. D. H. Bhadeshia
Lecture 3: Models of Solutions
List of Symbols
Symbol
Meaning
∆
G
M
Molar Gibbs free energy of mixing
∆
H
M
Molar enthalpy of mixing
∆
S
M
Molar entropy of mixing
∆
e
G
Excess Gibbs free energy per mole of solution
∆
e
H
Excess enthalpy per mole of solution
∆
e
S
Excess enthalpy per mole of solution
Mechanical Mixtures and Solutions
Consider the pure components
A
and
B
with molar free energies
µ
o
A
and
µ
o
B
respectively. If the components are initially in the form of powders
then the average free energy of such a mixture of powders is simply:
G
{
mixture
}
= (1
−
x
)
µ
o
A
+
xµ
o
B
(1)
where
x
is the mole fraction of
B
. It is assumed that the powder particles
are so large that the
A
and
B
atoms do not “feel” each other’s presence
via interatomic forces between unlike atoms. It is also assumed that the
number of ways in which the mixture of powder particles can be arranged
is not sufficiently different from unity to give a significant contribution
to a configurational entropy of mixing. Thus, a blend of powders which
obeys equation 1 is called a
mechanical mixture
. It has a free energy that
is simply a weighted mean of the components, as illustrated in Fig. 1a
for a mean composition
x
.
Fig. 1:
(a) The free energy of a mechanical mixture,
where the mean free energy is simply the weighted
mean of the components. (b) The free energy of an
ideal atomic solution is always lower than that of a
mechanical mixture due to configurational entropy.
In contrast to a mechanical mixture, a
solution
is conventionally
taken to describe a mixture of atoms or molecules. There will in general
be an enthalpy change associated with the change in near neighbour
bonds. We shall show later that much more probable arrangements
of atoms become possible with intimate mixtures; this enters thermo-
dynamics via the configurational entropy, ensuring a reduction in free
energy on mixing even when there are no enthalpy changes. The free
energy of the solution is therefore different from that of the mechanical
mixture, as illustrated in Fig. 1b. The difference in the free energy be-
tween these two states of the components is the free energy of mixing
∆
G
M
, the essential term in all thermodynamic models for solutions.
Ideal Solution
An ideal solution is one in which the atoms are, at equilibrium, dis-
tributed randomly; the interchange of atoms within the solution causes
no change in the potential energy of the system. For a binary (
A
–
B
)
solution the numbers of the different kinds of bonds can therefore be
calculated using simple probability theory:
N
AA
=
z
1
2
N
(1
−
x
)
2
N
BB
=
z
1
2
N x
2
N
AB
+
N
BA
=
zN
(1
−
x
)
x
(2)
where
N
AB
represents both
A
–
B
and
B
–
A
bonds which cannot be dis-
tinguished.
N
is the total number of atoms,
z
is a coordination number
and
x
the fraction of
B
atoms.
It is assumed that there is a random distribution of atoms in an
ideal solution. There is no enthalpy of mixing since there is no change
in energy when bonds between like atoms are broken to create those
between unlike atoms. This is why the atoms are randomly distributed
in the solution.
Configurational Entropy
Thus, the preparation of a binary alloy by this route would involve
taking the two elemental powders (
A
and
B
) and mixing them together
in a proportion whereby the mole fraction of
B
is
x
. The pure powders
have the molar free energies
µ
o
A
and
µ
o
B
respectively, as illustrated on
Fig. 1. The free energy of this mechanical mixture of powders is given
by:
G
{
mixture
}
= (1
−
x
)
µ
o
A
+
xµ
o
B
−
T
∆
S
M
(3)
where ∆
S
M
is the change in configurational entropy as a consequence of
the mixing of the powders. We have assumed here, and shall continue
to assume, that there is no change in enthalpy in the process since the
atoms are indifferent to the their neighbours whatever they might be.
The change in configurational entropy as a consequence of mixing
can be obtained using the Boltzmann equation
S
=
k
ln
{
w
}
where
w
is
the number of configurations.
Suppose that there are
m
A
atoms per particle of
A
, and
m
B
atoms
per particle of
B
; the powders are then mixed in a proportion which
gives an average concentration of
B
which is the mole fraction
x
.
There is only one configuration when the heaps of powders are sep-
arate. When the powders are randomly mixed, the number of possible
configurations for a mole of atoms becomes (see Appendix):
¡
N
a
([1
−
x
]
/m
A
+
x/m
B
)
¢
!
(
N
a
[1
−
x
]
/m
A
)! (
N
a
x/m
B
)!
(4)
The numerator in equation 4 is the factorial of the total number of
particles and the denominator the product of the factorials of the
A
and
B
particles respectively. Assuming large numbers of particles, we may
use Stirling’s approximation (ln
y
! =
y
ln
y
−
y
) to obtain the molar
entropy of mixing as
∆
S
M
kN
a
=
(1
−
x
)
m
B
+
xm
A
m
A
m
B
ln
½
N
a
(1
−
x
)
m
B
+
xm
A
m
A
m
B
¾
−
1
−
x
m
A
ln
½
N
a
(1
−
x
)
m
A
¾
−
x
m
B
ln
½
N
a
x
m
B
¾
(5)
subject to the condition that the number of particles remains integral
and non–zero. This equation reduces to the familiar
∆
S
M
=
−
kN
a
[(1
−
x
) ln
{
1
−
x
}
+
x
ln
{
x
}
]
when
m
A
=
m
B
= 1.
Molar Free Energy of Mixing
The molar free energy of mixing is therefore:
∆
G
M
=
N
a
kT
[(1
−
x
) ln
{
1
−
x
}
+
x
ln
{
x
}
]
(6)
Fig. 2 shows how the configurational entropy and the free energy of
mixing vary as a function of the concentration. ∆
G
M
is at a minimum
for the equiatomic alloy because that is when the entropy of mixing
is at its largest; the curves naturally are symmetrical about
x
= 0
.
5.
The form of the curve does not change with temperature though the
magnitude at any concentration scales with the temperature. It follows
that at 0 K there is no difference between a mechanical mixture and an
ideal solution.
The chemical potential per mole for a component in an ideal solu-
tion is given by:
µ
A
=
µ
o
A
+
N
a
kT
ln
{
1
−
x
}
(7)
and there is a similar equation for
B
. Since
µ
A
=
µ
o
A
+
RT
ln
a
A
, it
follows that the activity coefficient is unity.
Fig. 2:
The entropy of mixing and the free energy of
mixing as a function of concentration in an ideal bi-
nary solution where the atoms are distributed at ran-
dom. The free energy is for a temperature of 1000 K.
Regular Solutions
There are no solutions of iron which are ideal. The iron–manganese
liquid phase is close to ideal, though even that has an enthalpy of mix-
ing which is about
−
860 J mol
−
1
for an equiatomic solution at 1000 K,
which compares with the contribution from the configurational entropy
of about
−
5800 J mol
−
1
. The ideal solution model is nevertheless useful
because it provides reference. The free energy of mixing for a non–ideal
solution is often written as equation 7 but with an additional excess free
energy term (∆
e
G
= ∆
e
H
−
T
∆
e
S
) which represents the deviation from
ideality:
∆
G
M
= ∆
e
G
+
N
a
kT
[(1
−
x
) ln
{
1
−
x
}
+
x
ln
{
x
}
]
= ∆
e
H
−
T
∆
e
S
+
N
a
kT
[(1
−
x
) ln
{
1
−
x
}
+
x
ln
{
x
}
]
(8)
One of the components of the excess enthalpy of mixing comes from
the change in the energy when new kinds of bonds are created during the
formation of a solution. This enthalpy is, in the
regular solution
model,
estimated from the
pairwise
interactions. The term “regular solution”
was proposed to describe mixtures whose properties when plotted var-
ied in an aesthetically regular manner; a regular solution, although not
ideal, would still contain a random distribution of the constituents. Fol-
lowing Guggenheim, the term regular solution is now restricted to cover
mixtures that show an ideal entropy of mixing but have a non–zero
interchange energy.
In the regular solution model, the enthalpy of mixing is obtained by
counting the different kinds of near neighbour bonds when the atoms are
mixed at random; this information together with the binding energies
gives the required change in enthalpy on mixing. The binding energy
may be defined by considering the change in energy as the distance
between a pair of atoms is decreased from infinity to an equilibrium
separation (Fig. 3). The change in energy during this process is the
binding energy, which for a pair of
A
atoms is written
−
2
²
AA
. It follows
that when
²
AA
+
²
BB
<
2
²
AB
, the solution will have a larger than
random probability of bonds between unlike atoms. The converse is
true when
²
AA
+
²
BB
>
2
²
AB
since atoms then prefer to be neighbours
to their own kind. Notice that for an ideal solution it is only necessary
for
²
AA
+
²
BB
= 2
²
AB
, and not
²
AA
=
²
BB
=
²
AB
.
Fig. 3:
Curve showing schematically the change in
energy as a function of the distance between a pair of
A
atoms.
−
2
²
AA
is the binding energy for the pair of
atoms. There is a strong repulsion at close–range.
Suppose now that we retain the approximation that the atoms are
randomly distributed, but assume that the enthalpy of mixing is not
zero. The number of
A
–
A
bonds in a mole of solution is
1
2
zN
a
(1
−
x
)
2
,
B
–
B
bonds
1
2
zN
a
x
2
and
A
–
B
+
B
–
A
bonds
zN
a
(1
−
x
)
x
where
z
is the
co–ordination number. It follows that the molar enthalpy of mixing is
given by:
∆
H
M
'
N
a
z
(1
−
x
)
xω
(9)
where
ω
=
²
AA
+
²
BB
−
2
²
AB
(10)
The product
zN
a
ω
is often called the regular solution parameter, which
in practice will be temperature and composition dependent. A com-
position dependence also leads to an asymmetry in the enthalpy of
mixing as a function of composition about
x
= 0
.
5. For the nearly
ideal Fe–Mn liquid phase solution, the regular solution parameter is
−
3950 + 0
.
489
T
J mol
−
1
if a slight composition dependence is neglected.
A positive
ω
favours the clustering of like atoms whereas when it
is negative there is a tendency for the atoms to order. This second case
is illustrated in Fig. 4, where an ideal solution curve is presented for
comparison. Like the ideal solution, the form of the curve for the case
where ∆
H
M
<
0 does not change with the temperature, but unlike the
ideal solution, there is a free energy of mixing even at 0 K where the
entropy term ceases to make a contribution.
Fig. 4:
The free energy of mixing as a function of con-
centration in a binary solution where there is a pref-
erence for unlike atoms to be near neighbours. The
free energy curve for the ideal solution (
∆
H
M
= 0
)
is included for comparison.
The corresponding case for ∆
H
M
>
0 is illustrated in Fig. 5, where
it is evident that the form of the curve changes with the temperature.
The contribution from the enthalpy term can largely be neglected at very
high temperatures where the atoms become randomly mixed by thermal
agitation so that the free energy curve has a single minimum. However,
as the temperature is reduced, the opposing contribution to the free en-
ergy from the enthalpy term introduces two minima at the solute–rich
and solute–poor concentrations. This is because like–neighbours are
preferred. On the other hand, there is a maximum at the equiatomic
composition because that gives a large number of unfavoured unlike
atom bonds. Between the minima and the maximum lie points of in-
flexion which are of importance in spinodal decomposition, which will
be discussed later. Some of the properties of different kinds of solutions
are summarised in Table 1.
Type
∆
S
M
∆
H
M
Ideal
Random
0
Regular
Random
6
= 0
Quasichemical
Not random
6
= 0
Table 1:
Elementary thermodynamic properties of
solutions
Appendix: Derivation of equation 9
Energy, defined relative to infinitely separated atoms, before mixing:
1
2
zN
a
·
(1
−
x
)(
−
2
²
AA
) +
x
(
−
2
²
BB
)
¸
Fig. 5:
The free energy of mixing as a function of
concentration and temperature in a binary solution
where there is a tendency for like atoms to cluster.
The free energy curve for the ideal solution (
∆
H
M
=
0
) is included for reference.
since the binding energy per pair of atoms is
−
2
²
and
1
2
zN
a
is the
number of bonds. After mixing, the corresponding energy is given by:
1
2
zN
a
·
(1
−
x
)
2
(
−
2
²
AA
) +
x
2
(
−
2
²
BB
) + 2
x
(1
−
x
)(
−
2
²
AB
)
¸
where the factor of two in the last term is to count
AB
and
BA
bonds.
Therefore, the change due to mixing is the latter minus the former,
i.e.
=
−
zN
a
·
(1
−
x
)
2
(
²
AA
) +
x
2
(
²
BB
) +
x
(1
−
x
)(2
²
AB
)
−
(1
−
x
)(
²
AA
)
−
x
(
²
BB
)
¸
=
−
zN
a
·
−
x
(1
−
x
)(
²
AA
) +
−
x
(1
−
x
)(
²
BB
) +
x
(1
−
x
)(2
²
AB
)
¸
=
zN
a
(
x
)(1
−
x
)
ω
given that
ω
=
²
AA
+
²
BB
−
2
²
AB
.
Appendix: Configurations
Suppose there are
N
sites amongst which we distribute
n
atoms of
type
A
and
N
−
n
of type
B
(Fig. 6). The first
A
atom can be placed
in
N
different ways and the second in
N
−
1 different ways. These two
atoms cannot be distinguished so the number of different ways of placing
the first two
A
atoms is
N
(
N
−
1)
/
2. Similarly, the number of different
ways of placing the first three
A
atoms is
N
(
N
−
1)(
N
−
2)
/
3!
Therefore, the number of distinguishable ways of placing all the
A
atoms is
N
(
N
−
1)
. . .
(
N
−
n
+ 2)(
N
−
n
+ 1)
n
!
=
N
!
n
!(
N
−
n
)!
Fig. 6:
Configurations