Direct Numerical Simulation of Electric Double Layer Formation

We will present an innovative method to simulate the formation of electric double layers on complex electrode geometries. Electric double layers are omnipresent and affect the ionic diffusion in many natural systems. It plays the most crucial role in the energy storage of supercapacitors and electric-field desalination. Upon imposing an electrostatic potential, the cations and anions near the electrode surface separate to form a localized polarization counterbalancing the potential. The charge-separated region usually spans 20 to 50 nanometers, whereas other significant length scales, e.g., particle size or inter-particle space in electrodes, are in tens to hundreds of microns. A direct numerical simulation in the continuum scale requires to resolve the ionic concentration and potential gradients across 3~4 orders of magnitude in spatial scales and is incredibly challenging. Therefore, continuum-scale numerical studies are limited to one-dimensional or simple symmetric geometries, or in nanoscales where double-layer thickness and particle size are comparable. In this work, we use the smoothed boundary method that defines complex microstructures with a phase-field-order-parameter-like continuous function to reformulate the Nernst-Planck-Poisson equations and solve the equations on adaptively refined meshes. Thus, the method allows for accurately simulating the charge separation and potential distribution in arbitrarily complex geometries with resolutions spanning from nanometers in the double layer to micron/millimeters in the particle/electrode scales. Simulations of the dynamics during double layer formation and the resultant capacitance on complex electrode microstructures will be presented. The coupling of charge separation and fluid mechanics in simulating the electric-field desalination process through porous electrodes will also be discussed.