Continuum theory of electrostatic-elastic coupling interactions in colloidal crystals

A mobile Coulomb gas permeating a fixed background crystalline lattice of charged colloidal crystals is subject to an electrostatic-elastic coupling, which we study on the continuum level by introducing a minimal coupling between electrostatic and displacement fields. We derive linearized, Debye-H\"uckel-like mean-field equations that can be analytically solved, incorporating the minimal coupling between electrostatic and displacement fields leading to an additional effective attractive interaction between mobile charges that depends in general on the strength of the coupling between the electrostatic and displacement fields. By analyzing the Gaussian fluctuations around the mean-field solution we also identify and quantify the region of its stability in terms of the electrostatic-elastic screening length. This detailed continuum theory incorporating the standard lattice elasticity and electrostatics of mobile charges provides a baseline to investigate the electrostatic-elastic coupling for microscopic models in colloid science and materials science.