Ionic conductivity and hydrodynamic permeability of inhomogeneous (cavity doped) polyelectrolyte hydrogels

Abstract

Ion transport in polyelectrolyte membranes (charged hydrogels) is of significant technological (and biological) importance, but little is known of how micro-structural inhomogeneity affects ionic conductivity. Whereas a uniform electric field drives uni-directional electro-migrative and electro-osmotic ion fluxes in perfectly uniform microstructures, this study considers the influence of spherical inclusions/cavities on the hydrodynamic and ion permeability of charged hydrogels. Such cavities have a high permeability, but they can bear a much lower conductivity due to the partitioning of counter-ions between the cavity and bulk hydrogel phases, also inducing micro-scale electro-osmotic flow. To understand these, perturbations from a nonlinear Poisson–Boltzmann equilibrium state are used to compute the velocity disturbances, and electrostatic and ion-concentration polarization. These furnish three independent Onsager coefficients: one of which is the effective hydrodynamic permeability, and all of which contribute to the two principal electrical conductivities (distinguished by electrode configuration). Cavities with diameters in the range $10$ – $1000$  nm are found to be readily polarized, decreasing the effective conductivity of an otherwise uniform polyelectrolyte. In highly permeable hydrogels, however, electro-osmosis may enhance the electrical conductivity when flow is blocked by impenetrable electrodes. Explicit formulas for the hydrodynamic permeability are provided, complementing a simplified (Maxwell–Donnan) analysis of the conductivity, which neglects diffuse double-layer effects and ion-concentration perturbations.