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Electrically driven transport of molecules and ions within aqueous electrolytes is of long-standing interest, with direct relevance to applications that include the delivery/release of biologically active solutes to/from cells and tissues. Examples include iontophoretic and electroporation-mediated drug delivery. Here, we describe a robust method for characterizing electrodiffusive transport in physiologic aqueous media. Specifically, we treat the case of solute present in sufficiently low concentration as to negligibly contribute to the total ionic current within the system. In this limiting case, which applies to many systems of interest, the predominant electrical behavior due to small ions is decoupled from solute transport. Thus, electrical behavior may be characterized using existing methods and treated as known in characterizing electrodiffusive molecular transport. First, we present traditional continuum equations governing electrodiffusion of charged solutes within aqueous electrolytes and then adapt them to discretized systems. Second, we examine the time-dependent and steady-state interfacial concentration gradients that result from the combination of diffusion and electrical drift. Third, we show how interfacial concentration gradients are related to electric field strength and duration. Finally, we examine how discretization size affects the accuracy of these methods. Overall these methods are motivated by and well suited to addressing an outstanding goal: estimation of the net ionic and molecular transport facilitated by electroporation in biological systems.