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FOUNDATION · DIFFUSION

Diffusion in a particle

Why do smaller particles fill with lithium so much faster?

The idea

A battery electrode is a powder of tiny particles. Lithium can only get into each one through its surface, and then has to spread inward by diffusion: random hopping, with no push toward the center. How long that takes is one of the limits on how fast the material can charge.

What you’re looking at

A round particle with a quarter cut away, so you can see inside. The cut faces are colored by how much lithium is there, from dark purple for empty to blue for full, with contour lines at 20, 40, 60 and 80 percent. The beads show where the lithium is: they follow the amount, not the random path of any one ion. The film loops: lithium goes in, then comes back out.

Try it

Below the film, change the particle’s radius and compare how far lithium has spread after the same 2 seconds. Diffusion time grows with the square of the distance, so half the radius takes a quarter of the time. This is an ideal sphere with one fixed diffusivity, not a measurement of a real material.

01 — WATCH IT HAPPENBLENDER FILM

A sphere with one quarter cut away. The two cut faces are colored by lithium concentration, dark purple for empty and blue for full, with hairline contours at 20, 40, 60 and 80 percent. Beads mark where the lithium is: they gather at the surface and spread toward the center, then thin out from the surface first when the surface is emptied. The film loops every 16 seconds.

FIG. 01 SPHERICAL DIFFUSION

Lithium fills a particle from the outside in

  1. The surface fills firstLithium enters through the surface, so the outer shell is full long before the middle
  2. The center catches up slowlyDiffusion carries lithium inward. The deeper it has to go, the longer it takes
  3. Empty the surface, and lithium leavesThe outer shell drains first while the center is still full
  4. The center is the last to emptyThe same distance sets the pace in both directions
Lithiumgoing in
Time0.0 s
Average fill0%

10centerr / Rsurface
Concentration from center to surface, for the frame on screen
Lithium inLithium out
02 — TRY AN IDEAINTERACTIVE MODEL PREVIEW

Small particle
Different timescale

Same material. Same elapsed time.
Change the radius and see what happens inside.

FIG. 02 / ARADIAL CROSS SECTIONt = 2 s
Concentration through a spherical particle A circular cross section colors the concentration at each radial position. At the same diffusivity and elapsed time, a smaller particle has a more uniform profile. R
Relative concentration, c/cₛ01
FIG. 02 / BTHE SAME FIELD, AS A PROFILE
Radial concentration profileConcentration from center to surface. The horizontal axis is normalized distance r over R; the vertical axis is relative concentration from zero to one. 10centerr / Rsurface
COMPARE SIZES

Half the distance
A quarter of the time

The characteristic diffusion time scales with the square of the particle radius. A shorter path makes a big difference.

25 nm100 nm

Drag to compare the concentration profiles.

1.00×

characteristic diffusion time
Relative to a 50 nm particle, at fixed D

Fixed: D = 10⁻¹⁶ m²/s · elapsed time = 2 s

An idealized sphere. Illustrative inputs. A starting point for understanding diffusion—not a prediction of battery performance.

MODEL NOTES

A smaller distance
A different timescale

The diagram uses the analytical series solution for diffusion into a sphere (Crank, The Mathematics of Diffusion, 2nd ed., 1975, eq. 6.18). The surface concentration is held constant; concentration starts at zero throughout the interior. Diffusivity is constant and the sphere does not change shape.

The concentration profile is shown at the same illustrative elapsed time and diffusivity for each radius. The characteristic diffusion time scales as R² / D. Doubling the radius gives four times the characteristic time at fixed D.

c(x, τ) = 1 − 2/(πx) Σ [ (−1)ⁿ⁺¹ sin(nπx) / n ] e−n²π²τ

Here x = r/R and τ = Dt/R². The preview uses D = 10⁻¹⁶ m²/s, t = 2 s, and 40 series terms. These are illustrative inputs, not measured properties of a particular cathode.

The film at the top uses the same equation, solved numerically while the surface concentration switches between 1 and 0 every 8 seconds of film. It shows the repeating cycle, so the last frame joins the first. The colors and contours on the cut faces come straight from that solution. Each bead is placed so the beads on a face follow the same concentration, which makes them trace the flow of lithium rather than individual random walks. The cutaway is a viewing aid, not a change to the sphere’s boundary conditions. The film’s time readout uses the radius chosen below.

The model omits particle interactions, electrode reactions, changing diffusivity, phase transitions, and electrochemical boundary conditions. It cannot predict a real battery’s capacity or charging speed.

Background: diffusion into a sphere · UT Austin