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Maier-Saupe mean field theory
This statistical theory, proposed by Alfred Saupe and Wilhelm Maier , includes contributions from an attractive intermolecular potential from an induced dipole moment between adjacent liquid crystal molecules. The anisotropic attraction stabilizes parallel alignment of neighboring molecules, and the theory then considers a mean-field average of the interaction. The M-S theory is amazingly successful in accounting for the basic features of the nematic-isotropic (N-I) phase transition and of the temperature dependence of the anisotropy exhibited by the dielectric, diamagnetic and optical properties in the nematic phase.

The mean-field character of the M-S theory is due to the neglect of correlations between the orientations of neighboring molecules in the nematic liquid. Thus, a molecule is assumed to orient under the action of a mean-field, independently of the orientations of its neighbors.

The orientation of molecules can be defined as long axis a which has a distribution function $$g_a(\theta,\phi)$$ which has a constraint that the sum of it over all solid angle is the total concentration:

$$\int g_a\mathrm{d}\Omega = 1$$

It's convenient to introduce free enthalpy per molecule $$F(p.T)$$. $$G$$ depends on the angular distribution function $$G_a(\theta,\phi)$$.

$$F(p,T) = F_{iso}(p,T) + k_BT\int g_a\log(4\pi g_a)\mathrm{d}\Omega + F_1(p,T,S)$$

The first term $$F_i$$ is the free enthalpy of the isotropic phase. The second term reflects the decrease due to anisotropic angular distribution. The last term $$F_1$$ describe the interaction of molecules which can be assumed to have a quadratic form of order parameter $$S$$, :

$$F_1 = -\frac{1}{2}u(p,T)S^2$$

where $$u(>0)$$ is the mean field potential energy which describe attractive intermolecular potential.

Since $$G$$ has a constraint, it can be minimized by using the Lagrangian multipliers. We expand the definition of nematic scalar order parameter in $$\delta G_1$$:

$$\delta F-\lambda \int \delta g(\theta,\phi)\mathrm{d}\Omega = \int k_BT\ \delta g[1+\log (4\pi g)]\mathrm{d}\Omega+\delta F_1 - \lambda \int \delta g\mathrm{d}\Omega= 0$$

$$\delta F_1 = -uS\delta S = -uS \int \frac{1}{2}(3\cos^2\theta-1)\delta g\ \mathrm{d}\Omega$$

where the quadruple expression of order parameter is used. Then the normalized distribution function is:$$G(\theta) = \frac{1}{Z}\exp[uS\ P_2(\cos \theta)/k_B]= \frac{1}{Z}\exp(-\beta U(\theta))$$

$$U(\theta) = -uS\ P_2(\cos \theta)$$

Here $$Z$$ is the normalization constant. $$U(\theta)$$is the potential of mean torque which describes the orientational intermolecular interaction potential.

It is convenient to introduce a new variable $$m$$to do further analysis. $$m$$is a dimensionless parameter describing the strength of the nematic interaction relative to the thermal energy.

$$m = \frac{3uS}{2k_BT}$$

We can calculate the

$$S = -\frac{1}{2} + \frac{3}{2}<\cos^2\theta> = -\frac{1}{2}+\frac{3}{2}\frac{\partial Z}{Z\partial m}$$

We can solve $$S(T)$$ and $$m(T)$$ graphically.

By using the M-P mean field theory, we can calculate the Helmholtz energy per molecule.

$$f\equiv\frac{F-F_{iso}}{Nu} = \frac{1}{2}(\tau m)^2-\frac{3}{2}\tau \ln [\frac{e^{2m/3}D(\sqrt m)}{\sqrt m}]$$

$$\tau= \frac{S}{m} = \frac{3\sqrt m -(3+2m)D(\sqrt m)}{4m^2D(\sqrt m)}$$

$$\tau$$is the dimensionless scaled temperature. $$D(x) = e^{-x^2}\int ^{\sqrt m}_0e^{t^2}\mathrm{d}t$$is the Dawson integral.

Using these expressions we can compare the result of M-S mean field theory and LdG theory. And these two reveal qualitatively similar behaviors.

Although this theory is quite simple and not quiet agree with what is observed in experiment. However the deviation is not large for thermotropic nematics. Improvement has been obtained either by improving on the mean-field approximation or by using more elaborate forms for the mean-field potential.