A pinch of salt in a pinch of salt: Diffusiophoresis in concentration gradients of three ionic species

Diffusiophoresis under electrolyte gradients of binary electrolytes is a well-established field in the literature. However, mixtures of electrolytes containing more than two ions are significantly more complex to handle as the interactions between them are modeled using the nonlinear multi-ion Poisson-Nernst–Planck equations. The nonlinearity in this system leads to numerical difficulties and results in a lack of intuitive understanding. Here we focus on the case where three ions are present, with two electrolytes that share either a cation or an anion. By using perturbation analysis, we develop a linearized model for ionic and diffusiophoretic transport in the limit that one electrolyte is dilute relative to the other, or a pinch of salt in a pinch of salt. We do not place any restriction on the valency and diffusivity of each ion. We apply our perturbation analysis to a steady-state micro-channel and a transient dead-end pore and demonstrate that the perturbation model is able to reproduce trends from direct numerical simulations for different parameters and boundary conditions. We highlight that a naive superposition of electrolyte gradients is insufficient to capture this interaction, even when one of the electrolytes is dilute, and provide key mathematical and intuitive corrections to account for the complexities of a multi-ion system. • Diffusiophoretic mobility for a sphere is presented for a three-ion system. • If one electrolyte is dilute, it still plays an important role in the final mobility. • Perturbation solutions for concentration and velocity are presented.