Mathematical Modeling of Ion Selective Membrane Systems Subject to Electrical Polarization
Authors: Flavin Matthew Thomas , Han Jongyoon , Freeman Daniel 
Published: February 2020

Ion selective membrane (ISM) electrodes are widely used for selective ion sensing applications. We have, in our work, applied the ISM to modulate chemical concentrations utilizing the physical process of ion concentration polarization (ICP) for applications in neuroscience and neurotechnology.1 Here, we discuss physicochemical modeling of the ISM and adjacent phases under electrical polarization, focusing on key phenomena relating to ISM-based electrochemical modulation. We have developed several multidimensional models using Nernst-Planck-Poisson (NPP) with bulk reaction for multi-ion solute transport and full Navier-Stokes for solvent transport, making minimal simplifying assumptions. These models were solved using the finite element method (FEM) in COMSOL Multiphysics, and analytical solutions were derived for certain key situations. Previous approaches to modeling this system have relied on assumptions such as electroneutrality,2 excess supporting electrolyte,3 Butler-Volmer interfacial charge transfer,4 and equilibrium reactions.5 By avoiding these, we have been able to observe unique phenomena arising from ISM polarization, expanding the applicability of modeling predictions.
Crucial transport properties of the membrane such as conductivity and primary ion transference can be evaluated using a one-dimensional model of chemical transport based on NPP with bulk reaction (shown in Figure 1A). We were able to show the process by which the membrane loses its selectivity for the primary ion upon significant depletion in the adjacent aqueous phase, upon which the membrane starts to transport interfering ions. In addition, we can show how limiting behavior and loss of selectivity occurs when the concentration of ions within the membrane are sufficiently polarized, above which point over-limiting current is carried by leaked counter ions such as chloride. Finally, the solutions to this model allows us to obtain critical information on the time-scale of the relevant processes that govern concentration changes.
The dimensions of the ICP region are determined by three different processes, each with different characteristic length-scales: spontaneous convection, forced convection, and three-dimensional diffusion. To evaluate this, two-dimensional NPP with Navier-Stokes (Figure 1B-C) and three-dimensional axisymmetric NPP (Figure 1D-E) were solved for the cases of forced convection and three-dimensional diffusion respectively. Using these models, we were able to determine explicit relationships between variables such as flow rate and electrode dimensions, and the concentration profile of ions throughout the ICP region. In addition, we performed an experimental visualization of the forced convection model using fluorescence imaging of calcium-sensitive dye in a microfluidic channel (Figure 1F).
These results are important not only for chemical modulation devices, but also for dynamic ISM sensors based on chronopotentiometry,6 coulometry,7 and amperometry.8

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Figure 1