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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]