Coupled electrokinetic transport through a nanoporous membrane: a theoretical approach

Liquid transport through nanopore is central into many applications, from water purification to biosensing or energy harvesting. Ultimately thin nanopores are of major interest in these applications to increase driving potential and reduce as much as possible dissipation sources. We investigate here theoretically the efficiency of the electrical power generation through an ultrathin nanoporous membrane by means of streaming current (electrical current induced by ionic flow in the vicinity of the liquid/solid interface) or electroosmosis (flow rate induced by an electrical potential). Upscaling from one unique pore to a nanoporous membrane is not straightforward when we consider low aspect ratio nanopore because of 3D entrance effects, which lead to interactions between the pores. Whereas these interactions have already been considered for direct transport (hydrodynamic permeability of the membrane, ionic conductance), specific effects appear when coupled transports are considered. We derive here the expression of the electroosmotic mobility for a nanoporous membrane including surface conduction, and we show that (i) it depends mainly of the distance between the pores and (ii) it is sublinear with the number of pores. Varying the pore spatial organization (square, hexagonal, disordered structure) reveals that these transport properties are only dependent on one parameter, the porosity of the membrane. Finally, when considering energy conversion yield, it is shown that increasing the number of pores is deleterious, and a non-monotonic behavior with salt concentration is reported.