Statistical physics: Landau theory liquid crystal (2D)

[rolotomassi]

Homework Statement

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.

Homework Equations

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 I am 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[/B]

 

[rolotomassi]

Reading it back now it is abit wordy isn't it.
Yes I have solved it its okay. Was half way there.

 

[James_12345]

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