2Mean field theory
III Theoretical Physics of Soft Condensed Matter
2.1 Binary fluids
Consider a binary fluid, consisting of a mixture of two fluids
A
and
B
. For
simplicity, we assume we are in the symmetric case, where
A
and
B
are the same
“type” of fluids. In other words, the potentials between the fluids are such that
U
AA
(r) = U
BB
(r) 6= U
AB
(r).
We consider the case where
A
and
B
repulse each other (or rather, repulse each
other more than the
A

A
and
B

B
repulsions). Thus, we expect that at high
temperatures, entropy dominates, and the two fluids are mixed together well. At
low temperatures, energy dominates, and the two fluids would be wellseparated.
We let
ρ
A
(
r
) and
ρ
B
(
r
) be the coarsegrained particle density of each fluid,
and we set our order parameter to be
φ(r) =
ρ
A
(r) − ρ
B
(r)
(N
A
+ N
B
)/V
,
with
N
A
and
N
B
, the total amount of fluids
A
and
B
, and
V
the volume. This
is normalized so that φ(r) ∈ [−1, 1].
We model our system with Landau–Ginzburg theory, with free energy given
by
βF =
Z
a
2
φ
2
+
b
4
φ
4
 {z }
f(φ)
+
κ
2
(∇φ)
2
dr,
where a, b, κ are functions of temperature.
Why did we pick such a model? Symmetry suggests the free energy should
be even, and if we Taylor expand any even free energy functional, the first few
terms will be of this form. For small
φ
and certain values of
a, b, κ
, we shall see
there is no need to look further into higher order terms.
Observe that even without symmetry, we can always assume we do not have
a linear term, since a
cφ
term will integrate out to give
cV
¯
φ
, and
¯
φ
, the average
composition of the fluid, is a fixed number. So this just leads to a constant shift.
The role of the gradient term
R
κ
2
(
∇φ
)
2
d
r
captures at order
∇
(2)
the non
locality of E
int
,
E
int
=
X
i,j∈{A,B}
Z
ρ
i
(r)ρ
j
(r
0
)U
ij
(r − r
0
) dr dr
0
,
If we assume
φ
(
r
) is slowly varying on the scale of interactions, then we can
Taylor expand this E
int
and obtain a (∇φ)
2
term.
Now what are the coefficients
a, b, κ
? For the model to make sense, we want
the free energy to be suppressed for large fluctuating
φ
. Thus, we want
b, κ >
0,
while
a
can take either sign. In general, the sign of
a
is what determines the
behaviour of the system, so for simplicity, we suppose
b
and
κ
are fixed, and let
a vary with temperature.
To do mean field theory, we find a single
φ
that minimizes
F
. Since the
gradient term
R
κ
2
(
∇φ
)
2
d
x ≥
0, a naive guess would be that we should pick a
uniform φ,
φ(r) =
¯
φ.
Note that
¯
φ
is fixed by the constraint of the system, namely how much fluid of
each type we have. So we do not have any choice. In this configuration, the free
energy per unit volume is
F
V
= f(
¯
φ) =
a
2
¯
φ
2
+
b
4
¯
φ
4
.
The global of this function depends only on the sign of
a
. For
a >
0 and
a <
0
respectively, the plots look like this:
¯
φ
f
a > 0
a < 0
a > 0
a < 0
We first think about the a > 0 part. The key point is that the function f (φ) is
a convex function. Thus, for a fixed average value of
φ
, the way to minimize
f(φ) is to take φ to be constant. Thus, since
βF =
Z
f(φ(r)) +
κ
2
(∇φ)
2
dr,
even considering the first term alone tells us we must take
φ
to be constant, and
the gradient term reinforces this further.
The
a <
0 case is more interesting. The function
f
(
φ
) has two minima,
φ
1,2
= ±φ
B
, where
φ
B
=
r
−a
b
.
Now suppose
¯
φ
lies between
±φ
B
. Then it might be advantageous to have some
parts of the fluid being at
−φ
B
and the others at
φ
B
, and join them smoothly
in between to control the gradient term. Mathematically, this is due to the
concavity of the function f in the region [−φ
B
, φ
B
].
Suppose there is
V
1
many fluid with
φ
=
φ
1
, and
V
2
many fluid with
φ
=
φ
2
.
Then these quantities must obey
V
1
φ
1
+ V
2
φ
2
= V
¯
φ,
V
1
+ V
2
= V.
Concavity tells us we must have
V
1
f(φ
1
) + V
2
f(φ
2
) < (V
1
+ V
2
)f(
¯
φ).
Thus, if we only consider the
f
part of the free energy, it is advantageous to
have this phase separated state. If our system is very large in size, since the
interface between the two regions is concentrated in a surface of finite thickness,
the gradient cost will be small compared to the gain due to phase separation.
We can be a bit more precise about the effects of the interface. In the first
example sheet, we will explicitly solve for the actual minimizer of the free energy
subject to the boundary condition
φ
(
x
)
→ ±φ
B
as
x → ±∞
, as in our above
scenario. We then find that the thickness of the interface is (of the order)
ξ
0
=
−2κ
a
,
and the cost per unit area of this interface is
σ =
−8κa
3
9b
2
1/2
.
This is known as the interfacial tension. When calculating the free energy of a
phase separated state, we will just multiply the interfacial tension by the area,
instead of going back to explicit free energy calculations.
In general the meanfield phase diagram looks like
a
−1 1
¯
φ
a(T ) = 0
Within the solid lines, we have phase separation, where the ground state of the
system for the given
a
and
¯
φ
is given by the state described above. The inner
curve denotes spinodal instability, where we in fact have local instability, as
opposed to global instability. This is given by the condition
f
00
(
¯
φ
)
<
0, which
we solve to be
φ
S
=
r
−a
3b
.
What happens if our fluid is no longer symmetric? In this case, we should
add odd terms as well. As we previously discussed, a linear term has no effect.
How about a cubic term
R
c
3
φ
(
r
)
3
d
r
to our
βF
? It turns out we can remove
the
φ
(
r
) term by a linear shift of
φ
and
a
, which is a simple algebraic maneuver.
So we have a shift of axes on the phase diagram, and nothing interesting really
happens.