Nanobattery Charging: The R² Diffusion Law

Electrode Particle Field

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

Average SOC

τ₈₀ smallest particle

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τ₈₀ largest particle

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How it works

Charging an intercalation electrode particle is a diffusion problem: lithium ions must random-walk from the particle surface to its centre before the whole particle reaches its new state of charge (SOC). For a sphere of radius R held at a fixed surface concentration, Fick's second law in spherical coordinates gives the classic Crank solution for the volume-averaged fractional uptake:

SOC(t) = 1 − (6/π²) Σ_{n=1}^∞ (1/n²) · exp(−n²t/τ)
   where   τ = R² / (π²D)

Everything else being equal, the characteristic charging time τ scales with the square of the particle radius. Shrink a particle from a 1 µm (1000 nm) grain down to a 100 nm nanoparticle — a 10× smaller radius — and τ drops by a factor of 100. That single scaling law is the entire reason nanostructured "nanobattery" electrodes (nanoparticles, nanowires, thin films) can absorb or release charge far faster than the same active material in coarse, micron-scale grains, without changing the chemistry at all.

Particle radius slider — sets the mean radius of the electrode particle field; each visible sphere is drawn at its own randomly sampled radius around that mean, log-spread across roughly two orders of magnitude so the R² effect is visible at a glance.
Diffusion coefficient slider — solid-state Li⁺ diffusivities span roughly 10⁻¹⁶ m²/s (dense oxide cathodes) to 10⁻¹⁰ m²/s (fast conductors), set here directly.
τ₈₀ readouts — the time for the smallest and largest particle currently on screen to reach 80% SOC, computed from the formula above, not just read off the animation.

This is the working principle behind real fast-charge electrode engineering: LiFePO₄ nanoparticles, silicon and graphite nanowire anodes, and thin-film "nanobattery" architectures all trade particle/feature size for charge-rate capability.

Low state of charge High state of charge

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