Phase Separation in Highly Charged Confined Ionic Solutions

We study phase separation in ionic solutions confined by solid objects carrying surface charges. Within the framework of Density Functional Theory, the Helmholtz free energy of the ionic solution is minimized under canonical constraints on the ionic densities fixing their mean value while ensuring global electroneutrality. The free energy splits into a bulk and an electrostatic contribution. The bulk contribution, which includes non-ideal terms accounting for long-range electrostatic and short-range steric correlations between ions, is evaluated with the Mean Spherical Approximation and the Local Density Approximation. The Primitive Model is considered with counterand co-ions having the same diameter. The electrostatic contribution treats the interactions between the ions and the solid object at the mean-field level through the solution of a suitable Poisson problem. The numerical methodology hinges on a regularization of the free energy and a finite element discretization of the Euler-Lagrange conditions of the constrained minimization problem on adaptively refined meshes as the regularization parameter approaches zero. Results are presented for the one-dimensional double-layer configuration and a multi-dimensional periodic network of charged circular inclusions. The main results are the formation of a condensed phase near the charge solid surface screening most of the surface charge, the stark contrast with predictions using the Poisson–Boltzmann theory, and the fact that co-ion densities are higher in the condensed phase as well. An extension of the methodology to the case where ions do not carry opposite charges is also presented.