Introduction to the Elastic Continuum Theory of Liquid Crystals

In the preceding chapters of this book we have seen that in terms of molecular theories1,2 one can calculate arid successfully explain various properties of meosphase transitions. However, there exists a class of liquid-crystal phenomena involving the response of bulk liquid-crystal samples to external disturbances, with respect to which the usefulness of a molecular theory is not immediately obvious. These phenomena are usually distinguished by two characteristics: (1) the energy involved, per molecule, in producing these effects is small compared to the strength of intermolecular interaction; and (2) the characteristic distances involved in these phenomena are large compared to molecular dimensions. In describing these large-scale phenomena, it is more convenient to regard the liquid crystal as a continuous medium with a set of elastic constants than to treat it on a molecular basis. Based on this viewpoint, Zocher,3 Oseen,4 and Frank5 developed a phenomenological continuum theory of liquid crystals that is very successful in explaining various magnetic (electric) field-induced effects. It is the purpose of the present chapter to develop this elastic continuum theory for nematic and cholesteric liquid crystals and to discuss and illustrate its use. In this paper, the derivation of the fundamental equation of the elastic continuum theory is followed by the application of the theory to four effects: (1) the twisted nematic cell, (2) the magnetic (electric) coherence length, (3) the Freedericksz transition, and (4) the magnetic (electric) field-induced cholestericnematic transition.