III Theoretical Physics of Soft Condensed Matter

4The variational method

4.2 Smectic liquid crystals

We use this to talk about the isotropic to smectic transition in liquid crystals.

The molecules involved often have two distinct segments. For example, we may

have soap molecules that look like this:

The key property of soap molecules is that the tail hates water while the head

likes water. So we expect these molecules to group together like

In general, we can imagine our molecules look like

and like attracts like. As in the binary fluid, we shall assume the two heads are

symmetric, so

. If we simply want the different parts to stay

away from each other, then we can have a configuration that looks like

In general, we expect that there is such an order along the

direction, as

indicated, while there is no restriction on the alignments in the other directions.

So the system is a lattice in the

direction, and a fluid in the remaining two

directions. This is known as a smectic liquid crystal, and is also known as the

lamellar phase. This is an example of microphase separation.

As before, we let

be a coarse grained relative density. The above ordered

phase would then look like

φ(x) = cos q

for some

that comes from the molecular length. If our system is not perfectly

ordered, then we may expect it to look roughly like A cos q

z for some A.

We again use Landau–Ginzburg model, which, in our old notation, has

βF =



(∇φ)

(∇

φ)



dr.

If we write this in Fourier space, then we get

βF =

(a + κq

+ γq

)φ

−q

−(q

)

Notice that the quartic term results in the rather messy sum at the end. For the

iso-smectic transition, we choose κ < 0, γ > 0.

Again for simplicity, we first consider the case where

= 0. Then this is a

Gaussian model with

G(q) = a + κq

+ γq

= τ + α(q −q

)

Varying

gives a linear shift in

(

). As we change

, we get multiple different

curves.

G(q)

a = a

Thus, as a decreases, S(q) = h|φ

i =

G(q)

blows up at some finite

q = q

−κ

2γ

, a

4γ

We should take this blowing up as saying that the

|q|

states are highly

desired, and this results in an ordered phase. Note that any

with

|q|

highly desired. When the system actually settles to an ordered state, it has to

pick one such

and let the system align in that direction. This is spontaneous

symmetry breaking.

It is convenient to complete to square, and expand

about

and

a = a

. Then we have

G(q) = τ + α(q −q

)

where

τ = a − a

, α =

) = −2κ.

Then the transition we saw above happens when τ = 0.

We now put back the quartic term. We first do this with mean field theory,

and then later return to the variational method.

Mean Field Theory

In mean field theory, it is easier to work in real space. We look for a single field

configuration that minimizes

. As suggested before, we try a solution of the

form

φ = A cos q

which is smectic along

. We can then evaluate the free energy (per unit volume)

to be

βF

= βF[φ] =

(∇φ)

(∇

φ)

where the bar means we average over one period of the periodic structure. It is

an exercise to directly compute

, (∇φ)

, (∇

φ)

, φ

This gives

βF



+ κA

+ γA







(a − a

)

| {z }





Note that

is fixed by the system as we originally had, while

is the amplitude

of the fluctuation which we get to choose. Plotting this, we get a familiar graph

βF/V

τ > 0

τ < 0

τ > 0

τ < 0

τ >

0, then the optimum solution is given by

= 0. Otherwise, we should

pick

A 6

= 0. Observe that as we slowly reduce

across 0, the minimum varies

continuously with τ:

|A|

Variational method

We now consider the variational theory. In the notation we were using previously,

we have

H =

−q

G(q) +

−(q

)

Our trial H

−q

J(q).

Since this is a Gaussian model, we know that

log

J(q)

To use our inequality, we need to evaluate our other two bits. We have

hφ

−q

J(q).

We already calculated

hφ

−q

J(q)

Thus, we have

Here it is clear that we must impose a cutoff on

. We can think of this 1 as the

equipartition theorem.

We can also compute

hHi

J(q)

G(q) +

hφ

−(q

)

| {z }

In the Gaussian model, each

is a zero mean Gaussian random variables, which

have certain nice properties. Wick’s theorem tells us we have

habcdi

= habi

hcdi

+ haci

hbdi

+ hadi

hbci

Applying this, together with the result

hφ

= h|φ

,−q

we obtain

U = 3

h|φ

= 12

J(q)

Thus, we have

G(q)

J(q)

We can then compute

F =



log

J(q)

− 1 +

G(q)

J(q)



J(q)

We minimize over J(q) by solving

∂

∂J(q)

= 0

for all q. Differentiating, we obtain

J(q)

−

G(q)

J(q)

−

V J(q)

J(q

)

= 0.

Multiplying through by J

, we get

J(q) = G(q) +

J(q

)

For large V , we can replace the sum by an integral, and we have

J(q) = G(q) +

(2π)

J(q

)

It is very important that once we have fixed

(

), the second term is a constant.

Writing

C =

J(q

)

(2π)

J(q

)

we can then say the minimum is given by

(

) =

(

) + 3

. Thus, solving for

J(q) is equivalent to finding a C such that

C =

(2π)

G(q) + 3bC

This is a self-consistency equation for C (and hence J).

There is a very concrete interpretation of

. Recall that

hφ

is the average

value of φ

at some point r (which is independent of r. So we can compute

hφ(r)

i =

hφ(r)

i dr =

h|φ

i =

J(q)

Thus, what our computation above amounts to is that we replaced

(

) by

G(q) + 3bhφ

A good way to think about this is that we are approximating the

term in

the free energy by

≈

hφ

iφ

We have to pick a value so that this approximation is consistent. We view the

hφ

above as just another constant, so that the free energy is now a Gaussian.

We can compute the expectation of

using this free energy, and then we find

that

hφ

i =

(2π)

G(q) + 3bhφ

This is the self-consistency equation as above.

We could have done the approximation above without having to through the

variational method, but the factor of

is then no longer obvious.

To solve this self-consistency equation, or at least understand it, it is conve-

nient to think in terms of τ instead. If we write our G(q) as

G(q) = τ + α(q −q

)

then we have

J(q) = ¯τ + α(q −q

)

, ¯τ = 3bhφ

The self-consistency equation now states

¯τ = τ +

(2π)

¯τ + α(q −q

)

We are interested in what happens near the critical point. In this case, we expect

¯τ

to be small, and hence the integral is highly peaked near 0. In

= 3, we can

make the approximation.

(2π)

¯τ + α(q −q

)

2π

∞

¯τ + α(q −q

)

≈

3bq

2π

∞

¯τ + α(q −q

)

While there is a nasty integral to evaluate, we can make the substitution

q 7→ ¯τq

to bring the dependency on ¯τ outside the integral, and obtain

¯τ = τ +

√

¯τ

, s =

2π

√

∞

1 + y

∼

√

The precise value of

does not matter. The point is that it is constant indepen-

dent of

¯τ

. From this, we see that

¯τ

can never reach zero! This means we can

never have a continuous transition. At small

¯τ

, we will have large fluctuations,

and the quantity

h|φ

i = S(q) =

¯τ + α(q −q

)

becomes large but finite. Note that since

¯τ

is finite, this sets some sort of length

scale where all q with |q −q

| ∼ ¯τ have large amplitudes.

We can think of the above computation as looking at neighbourhoods of the

= 0 vacuum, as that is what the variational methods see. So

¯τ

never vanishes

in this regime, we would expect a discontinuous isotropic-smectic transition that

happens “far away” from this vacuum.

Consider fluctuations about an ordered state, φ = φ

+ δφ, where

= A cos q

We can do computations similar to what we did above by varying

δφ

, and obtain

(

). Then the global minimum over all

then gives us the “true” ground

state. Instead of doing that, which is quite messy, we use the heuristic version

instead. For A = 0, we had

τ = ¯τ + 3bhφ(r)

For finite A, the quartic term now has an extra contribution, and we bet

¯τ = τ +

√

¯τ

3bA

Compare this with mean field theory, where we have

MF T

(A) =



τA



We see that for small

, the fluctuations are large, and mean field theory is

quite off. For large

, the fluctuation terms are irrelevant and mean field theory

is a good approximation. Thus, we get

MFT

We see that as

decreases, the minimum discontinuously jumps from

= 0

to a finite value of A. Mean field theory is approached at large A.

We can then plot the minimum value of A as a function of τ :

MFT

|A|

We have not calculated

, but it can be done. We shall not do this, but

we can state the result (Brazovskii, 1975):

' −(sb)

2/3

, A

' s

1/3

−1/6

It turns out the variational approach finally breaks down for

τ < τ

∼ −s

3/4

1/2

We have

 τ

b 

√

. The reason this breaks down is that at low enough

temperatures, the fluctuations from the quartic terms become significant, and

our Gaussian approximation falls apart.

To figure out what

is, we need to find leading corrections to

, as

Brazovskii did. In general, there is no reason to believe

is large or small. For

example, this method breaks down completely for the Ising model, and is correct

in no regime at all. In general, the self-consistent approach here is ad hoc, and

one has to do some explicit error analysis to see if it is actually good.

Brazovskii transition with cubic term

What happens when we have a cubic term? This would give an

term at

mean field level, which gives a discontinuous transition, but in a completely

different way. We shall just state the results here, using a phase diagram with

two parameters τ and c. In mean field theory, we get

where H is a hexagonal phase, which looks like

where each cylinder contains a high concentration of a fixed end of the molecule.

This is another liquid crystal, with two crystal directions and one fluid directions.

The self-consistent version instead looks like

˜c

Here c only matters above a threshold ˜c.