Modulated phase formation vs macroscopic phase separation in electric double layers with mobile surface charges

We investigate whether an electric double layer with mobile surface charges becomes unstable toward the formation of a finite-wavelength lateral modulation of the surface charge density before undergoing macroscopic phase separation into large domains with different surface charge densities. To address this question, we consider an isolated planar macroion in contact with a symmetric 1:1 electrolyte and perform a perturbation analysis at the onset of instability. The mobile surface charges are described as a regular lattice solution on the mean-field level, while the electrolyte is treated within nonlinear Poisson-Boltzmann theory. We determine the minimum degree of nonideality required to destabilize a homogeneous electric double layer and identify whether the resulting instability leads to macroscopic phase separation or to the emergence of a thermodynamically stable modulated phase. The nature of the instability is governed by the Bjerrum length lB, the Debye length lD, and the area a per lattice site of the underlying regular solution. We present an instability diagram that predicts the necessary conditions a>35.54lB2 and lD > 16.92lB for the appearance of a modulated phase. When a modulated phase exists, its characteristic wavelength is larger than 52.6lB. Our model suggests that weakly charged macroions at low salt are promising candidates for the formation of laterally periodic surface-charge patterns.