Dislocation-mediated melting of anisotropic layers

Abstract

Using the ideas of Kosterlitz and Thouless to describe dislocation-mediated melting of two-dimensional (2D) crystals, we consider the melting of anisotropic layers of molecules. Depending on the symmetry of the Burgers vector of the dislocation most prone to unbind, new types of melting behavior occur. In the most interesting case, the properties of the melted phase are described by three characteristic lengths. There are crossovers between regimes of 2D solidlike, 2D smecticlike, 2D nematiclike, and quasi-isotropic behavior. At the melting temperature, there are divergences in the anisotropic properties of the crystal due to one type of dislocation being free, but the other type being effectively bound. In the presence of an incommensurate crystalline substrate, the 2D smectic properties may be stabilized at large distances giving rise to a distinct smectic phase. Similarly, 2D smectic order may be stabilized by the interactions between the layers of a 3D smectic giving rise to a distinct "bismectic" phase, intermediate between the smectic-$C$ and -$H$ phases. Consequences of these results for various scattering experiments have been calculated. The general theory of dislocations in an anisotropic 2D solid is worked out in detail in Appendix A with explicit calculations for the interactions between dislocations and the stress and displacement fields associated with them.