# Statistical physics: Landau theory liquid crystal (2D)

1. May 4, 2015

### rolotomassi

1. The problem statement, all variables and given/known data

A simple model for liquid crystals confined to 2 dimensions is o assume each molecule can only align in one of 2 perpendicular directions. To construct a landau model its convenient to define order parameter, s :

s = 2 ( Np - 0.5Nt ) / Nt
Np - number molecules aligned parallel to some director
Nt - Np - number of molecules aligned perpendicular to some director
Nt - total number of molecules

The free energy is F = a + bs + cs^2 + ds^3 + es^4 , determine appropriate values of expressions for a, b, c, d and e for liquid crystal confined to 2 dimensions if a transition is observed from randomly aligned to oriented at some temperature T_critical.

2. Relevant equations

3. The attempt at a solution

I work out the order parameter for 3 configurations : s=1 (all parallel wrt director), s= -1 (all perpendicular wrt director) and s=0 (randomly aligned)

The free energy should be the same regardless of whether they are aligned parallel or perpendicular w.r.t some arbitrary direction so there should be symmetry for s = +/- 1. This means the coefficients of odd powers of 's' are zero.
F(s) is a quartic so will have 2 stable equilibria and 1 unstable equilibrium. I differentiate and get
s = 0 or s^2 = -c/2e

From what i've seen of other similar problems im thinking that the phase transition occurs when we go to a stable equilibrium point i.e - s^2 ---> + s^2 when 'c' changes sign. This happens at T_critical so let c(T) = c(T-Tc).
Not even sure if this is right or where I go from here

2. May 9, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 10, 2015

### rolotomassi

Reading it back now it is abit wordy isnt it.
Yes I have solved it its okay. Was half way there.

4. May 14, 2017

### James_12345

Hi, I'm stuck trying to solve a very similar question to this. Any chance you remember your solution?