Editing Liquid crystal (section)

===McMillan's model===
McMillan's model, proposed by William McMillan,<ref>{{cite journal|title= Simple Molecular Model for the Smectic A Phase of Liquid Crystals|journal=Phys. Rev. A|volume= 4 |date=1971|issue=3|page=1238|doi=10.1103/PhysRevA.4.1238|bibcode = 1971PhRvA...4.1238M| vauthors = McMillan W }}</ref> is an extension of the Maier–Saupe mean field theory used to describe the phase transition of a liquid crystal from a nematic to a smectic A phase. It predicts that the phase transition can be either continuous or discontinuous depending on the strength of the short-range interaction between the molecules. As a result, it allows for a triple critical point where the nematic, isotropic, and smectic A phase meet. Although it predicts the existence of a triple critical point, it does not successfully predict its value. The model utilizes two order parameters that describe the orientational and positional order of the liquid crystal. The first is simply the average of the second [[Legendre polynomials|Legendre polynomial]] and the second order parameter is given by:

: <math>\sigma = \left\langle\cos\left(\frac{2\pi z_i}{d}\right)\left(\frac{3}{2}\cos^2\left(\theta_i\right) - \frac{1}{2}\right)\right\rangle</math>

The values ''z<sub>i</sub>, θ<sub>i</sub>'', and ''d'' are the position of the molecule, the angle between the molecular axis and director, and the layer spacing. The postulated potential energy of a single molecule is given by:

:<math>U_i(\theta_i, z_i) = -U_0\left(S + \alpha\sigma\cos\left(\frac{2\pi z_i}{d}\right)\right)\left(\frac{3}{2}\cos^2\left(\theta_i\right) - \frac{1}{2}\right)</math>

Here constant α quantifies the strength of the interaction between adjacent molecules. The potential is then used to derive the thermodynamic properties of the system assuming thermal equilibrium. It results in two self-consistency equations that must be solved numerically, the solutions of which are the three stable phases of the liquid crystal.<ref name=b5/>