Nanobattery Diffusion Front: The Radial Concentration Profile

The 3D "Nanobattery Charging" sim evaluates Crank's closed-form spherical-diffusion series directly to colour a field of particles. This 2D companion instead solves the underlying partial differential equation itself: Fick's second law in spherical coordinates is marched forward on a 241-point radial grid with an explicit finite-difference scheme, so the concentration profile inside an electrode particle is built up numerically, one timestep at a time, rather than read off a formula. The left disc renders that live profile as a diffusion front sweeping inward from the particle's surface; the right panel reuses the same universal, nondimensionalized solve to colour an entire population of particles spanning nano-to-micron radii, each looked up at its own physical time θᵢ = D t / Rᵢ². A live readout compares the finite-difference solution against the analytic Crank series frame by frame, so you can watch two independent methods — one numerical, one closed-form — agree on the same physics.