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Ion transport in porous media is present in a wealth of technologies, e.g., energy storage devices such as batteries and supercapacitors, and environmental technologies such as electrochemical carbon capture and capacitive deionization. Recent studies on flat-plate electrodes have demonstrated that asymmetries in ion properties, such as valences and diffusivities, lead to intriguing and counter-intuitive physical phenomena. Yet, the consequences of such asymmetries to ion transport have seldom been explored in porous geometries. To bridge this knowledge gap, we conduct a perturbation expansion of the Poisson-Nernst-Planck equations in a cylindrical pore in the limit of small potentials for a binary electrolyte with arbitrary valences and diffusivities. We obtain good agreement between the perturbation analysis and direct numerical simulations. Our analysis reveals that the charge and the salt transport are coupled with each other. Further, the coupling between the charge and salt transport processes is enhanced with an increase in valence and diffusivity asymmetries of ions. We observe that the mismatch of the ionic diffusivities induces a non-trivial salt dynamics, producing either transient depletion or enhancement of salt in the pore. In the regime of high static diffusion layer conductance, we obtain an analytical solution to our perturbation model. The solution elucidates how electrolyte asymmetry induces two charging timescales that are set by the relative pore size. In the overlapping-double-layer regime, these timescales reduce to the diffusion times of each ion such that the transport of the two ions appears to be decoupled. Overall, our work underscores that the asymmetry in cation and anion diffusivities fundamentally alters the behavior of ionic transport inside a charged cylindrical pore and opens up new avenues of research on electrolyte transport in porous materials.