In this work we present numerical methods that are suitable for studying a variety of electrochemical systems, such as charging kinetics of porous electrodes used in energy storage devices (e.g. supercapacitors, batteries, or fuel cells), calculation of binding energies in biomolecules, and fluid flow problems in micro-fluidic devices. A central feature in these applications is the formation of a thin (~ 10 nm) and charged layer known as the Electric Double Layer (EDL) near charged surfaces. Numerical resolution of EDL is challenging and computationally expensive since physical quantities, such as electric potential and ion concentration, vary exponentially across this layer.

To address this challenge, we present numerical methods based on adaptive Quadtree grids (two spatial dimensions) and Octree grids (three spatial dimensions) that enable accurate, yet efficient, calculation of quantities of interest inside the EDL. Specifically, we present algorithms for the solution of Poisson-Boltzmann (PB) and Poisson-Nernst-Planck (PNP) Partial Differential Equations (PDEs). These governing equations are widely used to study these various electrochemical systems as mentioned above.

Finally, we present several examples which illustrate the convergence of presented algorithms and provide two applications, namely in studying the potential distribution around charged biomolecules and in studying the charging dynamics of supercapacitors. Interestingly, in the second application, our detailed calculations provide unreported insights into the role of pore micro-structure on the charging kinetics of porous electrode. Specifically, we find that certain micro-structures that provide continuous conducting pathways (e.g. regular arrays of carbon nanotubes (CNTs) or graphene sheets) allow for high-density surface currents which decrease the charging times and hence can potentially improve the power density of porous electrodes.