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
The dependence of the internal energy of the interfaces on their properties are postulated as:
E
wn
¼ E
wn
S
wn
, A
wn
, M
wn
, V
w
, V
n
, L
wns
20a E
ws
¼ E
ws
S
ws
, A
ws
, M
ws
, V
w
, V
s
E
s
, L
wns
20b E
ns
¼ E
ns
S
ns
, A
ns
, M
ns
, V
n
, V
s
E
s
, L
wns
20c Multiphase porous-media flow
529
Here the common line length is included as an indicator of the length of the boundary of the interface, a measure of
whether the microscale areas are small and distributed or large. The inclusion of the volumes of the adjacent fluid
phases and the strain tensor of the adjacent solid phase adds generality that may be important when the amount of
volume per area is small. The Euler forms of the energy equations are:
E
wn
¼ v
wn
S
wn
þ g
wn
A
wn
þ m
wn
M
wn
¹ c
wn w
V
w
¹ c
wn n
V
n
¹ c
wn wns
L
wns
21a E
ws
¼ v
ws
S
ws
þ g
ws
A
ws
þ m
ws
M
ws
¹ c
ws w
V
w
¹ j
ws
: V
s
E
s
¹ c
ws wns
L
wns
21b E
ns
¼ v
ns
S
ns
þ g
ns
A
ns
þ m
ns
M
ns
¹ c
ns n
V
n
¹ j
ns
: V
s
E
s
¹ c
ns wns
L
wns
21c Conversion of these expressions to a per-unit-volume basis
where ˆ E
ab
is the energy per unit volume of medium gives: ˆ
E
wn
¼ ˆ E
wn
ˆh
wn
, a
wn
, a
wn
r
wn
, e
w
, e
n
, l
wns
ÿ ¼
v
wn
ˆh
wn
þ g
wn
a
wn
þ m
wn
a
wn
r
wn
¹ c
wn w
e
w
¹ c
wn n
e
n
¹ c
wn wns
l
wns
22a ˆ
E
ws
¼ ˆ E
ws
ˆh
ws
, a
ws
, a
ws
r
ws
, e
w
, e
s
E
s
= j
, l
wns
ÿ ¼
v
ws
ˆh
ws
þ g
ws
a
ws
þ m
ws
a
ws
r
ws
¹ c
ws w
e
w
¹ j
ws
:e
s
E
s
= j ¹ c
ws wns
l
wns
22b ˆ
E
ns
¼ ˆ E
ns
ˆh
ns
, a
ns
, a
ns
r
ns
, e
n
, e
s
E
s
= j
, l
wns
ÿ ¼
v
ns
ˆh
ns
þ g
ns
a
ns
þ m
ns
a
ns
r
ns
¹ c
ns n
e
n
¹ j
ns
:e
s
E
s
= j ¹ c
ns wns
l
wns
22c Make use of the definition of the grand canonical potential
of the form: ˆ
Q
ab
¼ ˆ E
ab
¹ v
ab
ˆh
ab
¹ m
ab
a
ab
r
ab
23 and Legendre transformation of the independent variables
ˆh
ab
and a
ab
r
ab
to obtain: ˆ
Q
wn
v
wn
, a
wn
, m
wn
, e
w
, e
n
, l
wns
¼ g
wn
a
wn
¹ c
wn w
e
w
¹ c
wn n
e
n
¹ c
wn wns
l
wns
ð24aÞ ˆ
Q
ws
v
ws
, a
ws
, m
ws
, e
w
, e
s
E
s
= j
, l
wns
¼ g
ws
a
ws
¹ c
ws w
e
w
¹ j
ws
: e
s
E
s
= j ¹ c
ws wns
l
wns
24b ˆ
Q
ns
v
ns
, a
ns
, m
ns
, e
n
, e
s
E
s
= j
, l
wns
¼ g
ns
a
ns
¹ c
ns n
e
n
¹ j
ns
: e
s
E
s
= j ¹ c
ns wns
l
wns
24c where
] ˆ Q
ab
] v
ab
¼ ¹ ˆh
ab
24d ] ˆ
Q
ab
] a
ab
¼ g
ab
24e ] ˆ
Q
ab
] m
ab
¼ ¹ a
ab
r
ab
24f ] ˆ
Q
ab
] e
a
¼ ¹ c
ab a
24g ] ˆ
Q
as
] e
s
E
s
= j
¼ ¹ j
as
24h ] ˆ
Q
ab
] l
wns
¼ ¹ c
ab wns
: 24i
From eqns 22a, 22b and 22c, the Gibbs–Duhem equa- tions for the interfacial energies are obtained, respectively, as:
0 ¼ ˆh
wn
dv
wn
þ a
wn
dg
wn
þ a
wn
r
wn
dm
wn
¹ e
w
dc
wn w
¹ e
n
dc
wn n
¹ l
wns
dc
wn wns
25a 0 ¼ ˆh
ws
dv
ws
þ a
ws
dg
ws
þ a
ws
r
ws
dm
ws
¹ e
w
dc
ws w
¹ e
s
E
s
j : dj
ws
¹ l
wns
dc
ws wns
25b and
0 ¼ ˆh
ns
dv
ns
þ a
ns
dg
ns
þ a
ns
r
ns
dm
ns
¹ e
n
dc
ns n
¹ e
s
E
s
j : dj
ns
¹ l
wns
dc
ns wns
: 25c
4.4 Constitutive postulate for the common line energy function
