Screened Coulomb interaction and melting in two dimensions

Abstract

The electrostatic potential of a two dimensional lattice formed by a colloidal suspension of spheres within a thin film of water is derived as a solution to the linear Debye-Hückel equation. The applicability of this solution to the full nonlinear Boltzmann-Poisson equation (with a renormalized charge) is then discussed. The elastic constants of a triangular lattice are calculated within the harmonic approximation and used to predict the density at which the lattice melts via dislocation unbinding according to the Kosterlitz-Thouless (KT) theory. The results do not agree very well with those of a recent experiment. We suggest that geometry-dependent charge renormalization, anharmonic effects, screening by dislocation pairs, as well as finite dielectric contrast, εr, and finite particle size effects, all can affect our theoretical predictions substantially. Although after these corrections the KT theory may account for the behaviour of the colloidal system, we note that in general the potential of a two dimensional system governed by screened Coulomb interactions can, in some regions of parameter space, be very insensitive to lattice spacing (particle density). For such a system at fixed temperature, the KT theory or any other purely thermodynamic mechanism may not be appropriate for describing melting as a function of particle density.