Fick's first law: diffusion flux, sign convention and calculator

Learning objectives

By the end of this lesson you will be able to:

compute a one-dimensional diffusive flux with consistent SI units;
predict the direction of the flux from the sign of the concentration gradient;
list the assumptions behind the law and what it does not model.

Before you start: comfort with a derivative as a slope. Part 01B shows where lithium travels inside a cell.

01The law and its units

For diffusion in one dimension, the flux of a species is proportional to its concentration gradient:

J = - D \frac{d c}{d x}

Symbol	Meaning	SI unit
$J$	Molar flux through a unit area	mol m⁻² s⁻¹
$D$	Diffusion coefficient (diffusivity)	m² s⁻¹
$c$	Concentration	mol m⁻³
$x$	Spatial coordinate	m
$d c / d x$	Concentration gradient	mol m⁻⁴

Flux measures how much of the species crosses a unit area per unit time. This is a molar flux. Multiplying by the species’ molar mass converts it to a mass flux in kg m⁻² s⁻¹ without changing its direction.

02The minus sign is the physics

Passive diffusion runs down the concentration gradient. With positive $x$ as the reference direction:

a positive gradient (concentration rising toward +x) gives a negative flux — the species moves toward −x;
a negative gradient gives a positive flux — the species moves toward +x;
a zero gradient gives zero net flux.

Where concentration falls with x, the flux points toward +x. Flip the slope and the flux flips with it.

For example, with $D = 10^{- 14}$ m²/s, a gradient of +2 × 10⁶ mol/m⁴ gives $J = - 2 \times 10^{- 8}$ mol/(m²·s), and the reversed gradient gives +2 × 10⁻⁸ mol/(m²·s).

03A battery-material example

Take an illustrative solid-state diffusivity of 10⁻¹⁴ m²/s and a concentration change of −5000 mol/m³ across 5 µm of active material. The gradient is −5000 / (5 × 10⁻⁶) = −10⁹ mol/m⁴, so:

J = - (10^{- 14}) (- 10^{9}) = 10^{- 5} mol m⁻² s⁻¹

The calculator approximates the gradient as Δc / Δx, which is exact only for a linear profile. Estimating a gradient from a real profile — and choosing a mesh to do it on — is the job of a numerical model, not of the constitutive law.

04Where it shows up in a cell

Diffusion acts wherever there is a concentration gradient: inside active-material particles during intercalation (solid-state diffusion), and in the electrolyte filling the pores of the electrodes and separator described in Part 01B, where it acts alongside migration in the electric field. It also explains the shape of one aging curve: in the diffusion-limited model of Part 04, the SEI film grows like √t because the film acts as its own diffusion barrier.

Applying the law inside a porous electrode usually needs an effective transport coefficient and extra homogenisation assumptions, because the species moves through a tortuous network of pores rather than a uniform medium.

05Assumptions

A diffusion coefficient that does not depend on concentration within a single evaluation.
An isotropic medium, represented here in one spatial dimension.
A continuum description with a locally defined concentration gradient.
Diffusion driven only by the concentration gradient.
SI-unit inputs — numbers carry no unit metadata, so consistency is the user’s job.

The diffusivity may vary in space — one value per position — but not with the local concentration. Every diffusivity must be finite and non-negative, and every gradient value finite.

06What it does not model

Fick’s first law alone is not a time-evolution model, and it defines no initial or boundary conditions. It leaves out concentration- or temperature-dependent diffusivity, anisotropic diffusion tensors, migration in an electric field, convection, activity corrections for non-ideal mixtures, mechanics, reactions and interfacial transport.

Not yet in Battery Core

Fick’s second law — conservation combined with this constitutive relation, which describes how concentration changes in time — is not implemented yet. Neither are the single-particle (SPM, SPMe) or Doyle–Fuller–Newman (DFN) battery models built on it.

07Calculate it in Python

The calculator mirrors the tested ficks_first_law_flux function in the open-source battery_core package, including its input rules:

from battery_core import ficks_first_law_flux

ficks_first_law_flux(1.0e-14, 2.0e6)                 # -2e-08 mol/(m² s)
ficks_first_law_flux(1.0e-14, -5000.0 / 5.0e-6)      #  1e-05 mol/(m² s)
ficks_first_law_flux(1.0e-14, [-3e6, 0.0, 3e6])      # array: one flux per gradient

Run the transport notebook in your browser to plot flux against gradient and change the diffusivity yourself.

08Exercises

Halve the diffusivity from 10⁻¹⁴ to 5 × 10⁻¹⁵ m²/s. What happens to the flux–gradient line?

Its slope, − $D$ , halves in magnitude. Flux is proportional to $D$ , so every flux at a given gradient is half as large; the sign — the direction — does not change.

Calculate the flux for D = 2 × 10⁻¹⁴ m²/s and dc/dx = −3 × 10⁷ mol/m⁴.

$J = - (2 \times 10^{- 14}) (- 3 \times 10^{7})$ = 6 × 10⁻⁷ mol/(m²·s), positive, so toward +x.

In the battery-material example, halve the distance while keeping the concentration change fixed. What is the new flux?

The gradient doubles to −2 × 10⁹ mol/m⁴, so the flux doubles to 2 × 10⁻⁵ mol/(m²·s). Thinner layers and smaller particles shorten diffusion paths — one reason particle size matters for rate capability.

09References and further reading

These sources give historical and modern context; this page summarises the equation independently rather than reproducing textbook prose.

A. Fick, “On liquid diffusion,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10(63), 30–39 (1855). doi:10.1080/14786445508641925
J. Newman, K. E. Thomas-Alyea, Electrochemical Systems, 3rd ed., Wiley-Interscience, 2004.