Mathematical Modeling of Ion Transport Dynamics in Asymmetric Electrolytes

We use perturbation methods to analyze mathematical models for the ion transport dynamics of asymmetric electrolytes, specifically, with unequal ionic diffusion coefficients. We adopt the Poisson-Nernst-Planck (PNP) framework to describe ion transport. We consider a model electrochemical cell, with an asymmetric, binary electrolyte between planar, parallel, inert electrodes subject to a time-dependent voltage. We relate the mesoscale ion transport to measurable quantities, particularly: the current in the circuit, surface force between electrodes, motion of particles in the electrolyte, or the impedance response of the cell. The primary consequence of unequal diffusion coefficients is the generation of a transient ionic strength gradient in the electrolyte, whose dynamics follow a (slow) diffusive time scale. The ionic strength gradient further drives a potential gradient, thus the current in the circuit reflects this diffusive time scale, in addition to the typical (faster, “RC”) charging time scales of these systems. We investigate the effect of non-dilute concentration of ions by incorporating Stefan-Maxwell fluxes, that inherently account for ion asymmetry, into the PNP equations. We calculate the long-time asymptotic behavior of the current in the circuit when a model cell has been subject to a step voltage. The diffusion time scale due to ion asymmetry is apparent in the current evolution at long times. Next, we consider the dynamic force between two charged surfaces in a dilute electrolyte that are subject to ac, dc, and pulsed voltages. The surface force is inherently nonlinear in the applied voltage, thus a sinusoidal input voltage with a zero average can result in a steady force. Thereafter, we develop an asymptotic approximation for the asymmetric rectified electric field (AREF) that arises when an asymmetric electrolyte is subject to an ac voltage. We also discuss how particle motion in these systems is affected by both the electric field and the transient ionic strength gradient. Finally, we derive asymptotic approximations for theimpedance of a model cell having an asymmetric electrolyte with unequal valences and diffusion coefficients. We consider two frequency regimes, corresponding to the typical inverse RC charging time scale, and the inverse bulk diffusion time scale that is relevant to asymmetric electrolytes.