Steric model for the smectic-Cphase

A steric model of the smectic-$C$ phase is presented, with molecules assumed to have a symmetric zig-zag shape (this kind of molecular gross shape may arise from obliquely placed end chains). The interaction between such molecules depends on mixed tensors ${\overrightarrow{\nu }}^3{\overrightarrow{\nu }}^2$, etc., as well as on the usual tensors ${\overrightarrow{\nu }}^3{\overrightarrow{\nu }}^3$, ${\overrightarrow{\nu }}^2{\overrightarrow{\nu }}^2$, and similar higher-order tensors, where ${\overrightarrow{\nu }}^3$ and ${\overrightarrow{\nu }}^2$ are unit vectors denoting the directions of the long molecular axis and a transverse axis, respectively. A simple interaction, which includes a term that mimics the effect of the zig-zag shape, is constructed in terms of these tensors. Using quite plausible values for the magnitude of the zig-zag part of the interaction, a second-order phase transition to a smectic-$C$ phase is found, characterized by nonvanishing values of biaxial order parameters $〈{\overrightarrow{\nu }}^3{\overrightarrow{\nu }}^2〉$, etc. In the limit of perfect nematic order, the biaxial order parameters of the model are essentially equivalent to the vector order parameters used by de Gennes and by McMillan. The smectic planar structure is accounted for in the simple density-wave approximation of Kobayashi and McMillan. Only one phase with more orientational order than the smectic-$A$ phase is obtained; this is in contrast to McMillan's dipole model, which shows three different ordered phases (one of which corresponds to the smectic-$C$ phase). The present model predicts that the characteristic biaxialities in the smectic-$C$ phase may easily grow up to values of $O\left({10}^{-1\mathrm{}}\right)$.