Cholesteric twist in a model system

A highly idealized model is studied to gain some understanding of the structure of cholesteric liquid crystals. Asymmetric molecules, characterized by two perpendicular axes, the ``long'' and the ``short'' molecular axes, are considered; the centers of mass of the molecules lie on a rectangular lattice. An effective interaction that takes account of left-right asymmetry is assumed between nearest neighbors; also, an external field, correcting for a deficiency of the simple interaction, is chosen so that in the absence of a cholesteric twist the system is purely ``nematic.'' The results of the model (treated in mean field approximation) are (i) there is no cholesteric-nematic phase transition, an experimentally well-established fact; (ii) S∼ γ θ where S is a measure of asymmetry in the orientations of the ``short'' molecular axes, θ is the rotation (in radians) of the cholesteric helix from one layer of molecules to the next, and γ =O(10−2); (iii) the pitch of the helix is insensitive to temperature changes. An argument showing the reason for result (iii) in the present model is given.