Diffusion to a cube: A 3D implicit finite difference method

Abstract

We report the diffusional mass-transport to an isolated cube and numerically determine both the steady-state and the chronoamperometric flux, the latter at both short and long timescales. It is found that across a wide range of timescales the total flux is well approximated (to within 5% error) by the response expected for a sphere of equivalent surface area where the equivalent radius is equal to 1.38 times the half side length of the cube. Under steady-state conditions the size of the equivalent sphere is marginally smaller (equivalent radius 1.34). However, the prime difference between the flux to a cube and a sphere is that in the case of a cube the steady-state flux density across the particle is non-uniform. Towards the cubic particle edges the flux density increases asymptotically. This numerical study was undertaken using a graphics processor unit solving the diffusion equation using a 3D finite difference fully backward implicit scheme. The results are of interest to current-time responses seen in single nanoparticle experiments where non spherical shapes are frequently encountered.