the average microscale distance between points in each component is small e.g., in thin films or for highly dis-
persed phases, interfaces and common lines. In addition, the functional dependences obtained by Assumption III
must still lead to thermodynamic relations involving com- ponent properties consistent with the thermodynamic
analysis of the system as a whole.
4.2 Constitutive postulates for phase energy functions
The dependence of energy of the fluid phases on their prop- erties is postulated as:
E
w
¼ E
w
S
w
, V
w
, M
w
, A
wn
, A
ws
ÿ 14a
E
n
¼ E
n
S
n
, V
n
, M
n
, A
wn
, A
ns
ÿ 14b
The solid phase energy depends on the state of strain of the solid
9
such that it is expressed as: E
s
¼ E
s
S
s
, V
s
E
s
, M
s
, A
ws
, A
ns
ÿ 14c
The inclusion of a dependence on the interfacial areas in these expressions is a departure from the type of postulate
made when a system is to be modeled at the microscale. This is to account for changes in energy that may occur
when the amount of surface area per volume of phase is large. Additionally, note that the nature of a solid accounts
for its energy being postulated as depending on the state of deformation rather than its volume. From these equations,
because energy is a homogeneous first order function,
9,12
the Euler forms of the energy are: E
w
¼ v
w
S
w
¹ p
w
V
w
þ m
w
M
w
þ c
w wn
A
wn
þ c
w ws
A
ws
15a E
n
¼ v
n
S
n
¹ p
n
V
n
þ m
n
M
n
þ c
n wn
A
wn
þ c
n ns
A
ns
15b and
E
s
¼ v
s
S
s
¹ j
s
: V
s
E
s
þ m
s
M
s
þ c
s ws
A
ws
þ c
s ns
A
ns
15c As an example, note that the partial derivative of E
w
with respect to one of its independent variables, as listed in eqn
14a, is simply equal to the coefficient of that variable in eqn 15a. Similar observations apply for all the phase
energies as well as the interface and common line energies to be discussed subsequently.
Now convert eqns 15a, 15b and 15c such that they are on a per unit system volume basis:
ˆ E
w
ˆh
w
, e
w
, e
w
r
w
, a
wn
, a
ws
¼ v
w
ˆh
w
¹ p
w
e
w
þ m
w
e
w
r
w
þ c
w wn
a
wn
þ c
w ws
a
ws
16a ˆ
E
n
ˆh
n
, e
n
, e
n
r
n
, a
wn
, a
ns
¼ v
n
ˆh
n
¹ p
n
e
n
þ m
n
e
n
r
n
þ c
n wn
a
wn
þ c
n ns
a
ns
16b ˆ
E
s
ˆh
s
, e
s
E
s
j ,
e
s
r
s
, a
ws
, a
ns
¼ v
s
ˆh
s
¹ j
s
: e
s
E
s
= j þ m
s
e
s
r
s
þ c
s ws
a
ws
þ c
s ns
a
ns
16c Make use of the definition of the grand canonical potential:
ˆ Q
a
¼ ˆ E
a
¹ v
a
ˆh
a
¹ m
a
e
a
r
a
: 17
and employ Legendre transformations on ˆh
a
and e
a
r
a
to obtain:
ˆ Q
w
v
w
, e
w
, m
w
, a
wn
, a
ws
ÿ ¼ ¹ p
w
e
w
þ c
w wn
a
wn
þ c
w ws
a
ws
18a ˆ
Q
n
v
n
, e
n
, m
n
, a
wn
, a
ns
ÿ ¼ ¹ p
n
e
n
þ c
n wn
a
wn
þ c
n ns
a
ns
18b ˆ
Q
s
ˆv
s
, e
s
E
s
j ,
m
s
, a
ws
, a
ns
¼ ¹ j
s
:e
s
E
s
= j þ c
s ws
a
ws
þ c
s ns
a
ns
18c where
] ˆ Q
a
] v
a
¼ ¹ ˆh
a
18d ] ˆ
Q
a
] e
a
¼ ¹ p
a
a ¼ w ,
n 18e
] ˆ Q
s
] e
s
E
s
= j
¼ ¹ j
s
18f ] ˆ
Q
a
] m
a
¼ ¹ e
a
r
a
18g ] ˆ
Q
a
] a
ab
¼ c
a ab
: 18h
For eqns 16a, 16b and 16c, it is also worth noting that their respective Gibbs–Duhem equations are:
0 ¼ ˆh
w
dv
w
¹ e
w
dp
w
þ e
w
r
w
dm
w
þ a
wn
dc
w wn
þ a
ws
dc
w ws
19a 0 ¼ ˆh
n
dv
n
¹ e
n
dp
n
þ e
n
r
n
dm
n
þ a
wn
dc
n wn
þ a
ns
dc
n ns
19b and
0 ¼ ˆh
s
dv
s
¹ e
s
E
s
j : dj
s
þ e
s
r
s
dm
s
þ a
ws
dc
s ws
þ a
ns
dc
s ns
: 19c
4.3 Constitutive postulates for interfacial energy functions
