The mean field theory for liquid crystals was developed by the German physicist Wilhelm Maier and his student Alfred Saupe in the 1960s. While still employing the hard-rod molecule model, it included electrical effects by postulating induced dipole–dipole interactions between the molecules, which were treated as perturbations that engender attractive forces among the molecules that are responsible for the stability of the liquid crystal mesophase.
The Maier–Saupe theory successfully describes the observed mesophase to isotropic liquid phase transition at the clearing temperature as a discontinuous phase transition, and also provides an expression for the dielectric anisotropy as a function of the orientational order parameter, as will be seen below.
The Maier–Saupe theory first assumes that each hard-rod liquid crystal molecule is subject to a mean potential field arising from long-range induced dipole–dipole interaction forces; the Maier–Saupe potential field does not consider any short-range forces. From the cylindrical symmetry about the long axes of the liquid crystal molecules (often called the preferred axis) and the non-polarity of the liquid crystal as a whole, and based on the general physical truth that energy is the square of a field quantity, the orientational potential energy of the ith molecule is postulated to be proportional to the second order Legendre polynomial multiplied by the orientational order parameter variable S, as follows*