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Advanced Topics in Equilibrium Statistical
Mechanics
Glenn Fredrickson
7. Critical Phenomena
The subject of
critical phenomena
, i.e., the study of the equilibrium and nonequi
librium properties of systems near a critical point (e.g., a gasliquid critical
point), has a long history, but many notable recent advances (culminating in a
Nobel prize to Robert Wilson in 1982).
A. MeanField Theory: The VDW ﬂuid
The simplest theories to construct of critical phenomena are socalled
meanfeld
theories
—approaches that
average
the environment around a given molecule,
thus decoupling that molecule from its neighbors and making the statistical
mechanics calculation simple, like that of an ideal gas
.
Indeed the Van der Waals EOS for a ﬂuid can be “derived” using this sort
of reasoning:
"
p
+
a
µ
N
V
¶
2
#
[
V
−
Nb
]=
Nk
B
T
and is one of the simplest MF theories of the gasliquid critical point. Let’s
begin by rewriting the equation in terms of
ρ
=
N
V
rather than
V
:
(
p
+
aρ
2
)(1
−
bρ
)=
ρk
B
T.
Qualitatively, we have:
pρ
plane:
pT
plane:
1
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plane:
Let’s now locate the critical point in the VDW model. Clearly
∂p
∂ρ
!
T
=0
,
∂
2
p
2
!
T
,
VDW equation
give three equations that can give
p
c
,
T
c
,
ρ
c
in terms of
a, b
. Simpler is to write
out the VDW equation as
−
abρ
3
+
aρ
2
−
(
bp
+
k
B
T
)
ρ
+
p
or
ρ
3
−
1
b
ρ
2
+
µ
p
a
+
k
B
T
ab
¶
ρ
−
p
ab
.
Above
T
c
, this has 1 real, 2 imaginary roots; these merge to 3 equal real roots
at
T
c
.Thu
s
,a
t(
T
c
,p
c
):
ρ
3
−
1
b
ρ
2
+
µ
p
c
a
+
k
B
T
c
ab
¶
ρ
−
p
c
ab
=(
p
−
p
c
)
3
=
p
3
−
3
ρ
2
ρ
c
+3
ρρ
2
c
−
ρ
3
c
⇒
ρ
c
=
1
3
b
c
=
a
27
b
2
,k
B
T
c
=
8
a
27
b
,Z
c
=
β
c
p
c
ρ
c
=
3
8
Next, we introduce
reduced
variables:
ρ
r
=
ρ/ρ
c
r
=
p/p
c
,T
r
=
T/T
c
which allows the VDW equation to be rewritten as:
(
p
r
ρ
2
r
)(3
−
ρ
r
)=8
T
r
ρ
r
.
The absence of any materialdependent parameters is the basis for the familiar
“
Law of Corresponding States
”, which you have undoubtedly studied in your
thermo classes.
Next, let’s zoom in
on the critical region by further changing variables and
expanding the VDW equation:
t
≡
T
−
T
c
T
c
=
T
r
−
1
“temp diFerence”
ψ
≡
ρ
−
ρ
c
ρ
c
=
ρ
r
−
1
“order parameter”
(density diFerence)
π
≡
p
−
p
c
p
c
=
p
r
−
1
“pressure diFerence”
2
[
π
+1+3(
ψ
+1)
2
][3
−
(
ψ
+ 1)] = 8(
t
+ 1)(
ψ
This reduces to:
π
=4
t
³
1+
ψ
1
−
1
2
ψ
´
+
3
2
ψ
3
1
1
−
1
2
ψ
π
≈
4
t
(
1+
3
2
ψ
+
...
)
+
3
2
ψ
3
+
for

t
¿
1,

ψ
1.
Next, we do a thermodynamic integration to get the Helmholtz f.e.: write
A
≡
VA
V
,A
V
=
A
V
(
ρ, T
)
Helmholtz f.e.
per unit
volume (intensive)
p
=
−
∂A
∂V
´
N,T
=
−
A
V
−
V
V
´
N,T
=
−
A
V
−
V
∂ρ
´
N
V
´
T
=
−
A
V
+
ρ
V
´
T
p
c
(
π
+1)=
−
A
V
+(1+
ψ
)
V
∂ψ
´
t
or in terms of a dimensionless f.e. density
F
≡
A
V
/p
c
:
−
F
ψ
)
∂F
!
t
=1+
π
(
t, ψ
)
Integrate this ODE subject to
F
(
t,
0)
≡
F
0
(
t
)
.
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This document was uploaded on 05/18/2010.
 Spring '09
 The Land

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