Interaction of Potential Sources in Infinite 2D Arrays: Diffusion through Composite Membranes, Micro‐Electrochemistry, Entrance Resistance, and Other Examples

Abstract

Regular 2D arrays of potential sources on impervious “screens” are a mathematical idealization for the description of a number of natural and/or technological processes. At steady state, they are described by Laplace equation with suitable boundary conditions. This study explains the evolution of boundary conditions from a given potential to a given potential gradient at infinity with increasing size of arrays and provides a criterion for micro‐ and macro‐arrays in terms of distance to potential‐defining surfaces. For regular macro‐arrays, the problem is formulated and solved numerically by using cell models. The use of rigorous cell models requires 3D numerical simulations, but a simplified (cylindrical, 2D) cell model is shown to have an excellent accuracy. Numerical as well as theoretical analyses reveal a simple far‐field behavior described by an asymptotic expression for an additional potential drop (applicable for sources whose size does not exceed about 40% of the intersource distance) dependent on just one numerical constant. This expression is used for the derivation of useful approximate formulae for several applications (diffusion resistance of composite membranes, limiting current in arrays of microelectrodes, entrance diffusion resistance in arrays of scarce and short nanopores) and compared with relevant interpolation formulae available in the literature.

All-Atom Molecular Dynamics Simulations of Communication Between Nanochannel Arrays

“…Understanding the transport phenomena of a single or isolated nanochannel is of fundamental interest of many researchers. In experiments, the transport characteristics of the nanochannel array are treated as a collective behavior of individual or single channel characteristics. − To elucidate the ion transport phenomena, a single or isolated nanochannel is studied theoretically and is extrapolated to the nanochannel array. − However, it has been demonstrated that the physics observed using a single or isolated nanochannel is quite different as compared to nanochannel arrays and nano slots. − Furthermore, ion transport through the nanochannel array is shown as a function of interchannel spacing and communication among the channels. − ,− The scalability of the nanoscale transport phenomena from a single channel to a large-scale system (i.e., nanochannel array and nanoporous membrane) is very challenging and is meagerly explored.…”

“…Communication between the channels is established through the access region effects, and overlapping of the diffusion zones due to the ion concentration polarization (ICP) phenomenon. − ,− ICP is a nonequilibrium phenomenon indicating the depletion and enrichment of ions across the ion-selective devices under the action of an external electric field. ,,− This ICP phenomenon results in nonlinear I – V characteristics (including ohmic and nonohmic regions). In literature, it is demonstrated that the independent diffusion transport through each pore can be achieved if the interpore distance is greater than 56 times of pore diameter .…”

“…In literature, it is demonstrated that the independent diffusion transport through each pore can be achieved if the interpore distance is greater than 56 times of pore diameter . Also, the conductance of the array depends on the interchannel spacing, and the number of channels presents in the array. ,,− Only a few computational and experimental studies show a dependence on ionic conductance of individual channels in the nanochannel array, and the communication among the channels. ,,− ,− However, the pattern of communication among the channels in the nanochannel array (i.e., the current carried by the individual channel and its dependence) in both the ohmic and nonohmic regions, and the mechanism responsible for the communication in the nonohmic region are not explored and remain unclear so far.…”

Evaporation-driven electrokinetic energy conversion: Critical review, parametric analysis and perspectives

“…A recent study [36] analyzed transport phenomena described by Laplace equations (in particular, vapor diffusion in air within stagnant layers) in systems involving infinite regular arrays of "sources" (these can be, in particular, nanopore outlets in evaporation from porous materials). For infinite regular flat arrays with stagnant layers, the diffusion resistance was demonstrated to be well approximated by an in-series connection of the diffusion resistance of the stagnant layer and of an effective parallel connection of multiple nanopores.…”