Nanobattery Diffusion Front: The Radial Concentration Profile

Electrode Particle

Mean particle radius 100 nm

Li⁺ diffusion coefficient 1.0 × 10⁻¹³ m²/s

Playback speed 1.0×

Live Readouts

Empty (0% SOC) → Full (100% SOC)

Elapsed charge time

0 s

Mean-particle SOC

τ₈₀ smallest particle

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τ₈₀ largest particle

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FD vs. Crank analytic error

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Population avg. SOC

How it works

This is a genuine numerical solve, not a lookup of the closed form. Fick's second law for a diffusing species in a sphere of radius R, held at a fixed surface concentration, is

∂c/∂t = D (1/r²) ∂/∂r( r² ∂c/∂r )

Substituting x = r/R and the nondimensional time θ = D t / R² collapses this to a single universal equation with no free parameters:

∂c/∂θ = (1/x²) ∂/∂x( x² ∂c/∂x ),   c(1,θ)=1,  ∂c/∂x|₀=0

The left panel marches this equation forward on a 241-point radial grid using explicit finite differences (with the standard 6·(c₁−c₀)/Δx² stencil at the singular centre point) and draws the resulting concentration profile c(x,θ) as a false-colour disc — the diffusion front visibly eating inward from the rim. Because the equation only depends on θ, one grid solve serves every particle in the field: each particle's own θᵢ = D·t/Rᵢ² is looked up against the same precomputed table, which is what drives the population panel on the right.

Left disc — the numerically solved radial concentration profile of the mean-radius particle, redrawn every frame from the live finite-difference state.
Right field — every particle in the population, each coloured by its own θᵢ = D t / Rᵢ² read off the same universal solve — the R² law made visible: small particles saturate almost immediately, large ones lag far behind.
FD vs. Crank analytic error — this build's own finite-difference solve is checked every frame against Crank's closed-form spherical diffusion series (the same series the 3D "Nanobattery Charging" sim evaluates directly), so you can see the two independent methods agree.

Real-world relevance: this is the same R²-scaling that makes nanostructured electrodes (LiFePO₄ nanoparticles, silicon/graphite nanowires, thin-film architectures) charge orders of magnitude faster than coarse micron-scale grains of the identical active material.

Low Li⁺ concentration High Li⁺ concentration