Fick's Laws of Diffusion: Concentration Gradients and Flux

Diffusion is a random walk in disguise

Drop a band of dye into still water and it spreads even though no current is pushing it. Every molecule is being kicked in a random direction by thermal collisions - a random walk - and while any single molecule's path is unpredictable, the statistics of billions of them are not. A sharp band inevitably flattens into a broad, smooth bell-shaped profile, and it always flattens in the same direction: from high concentration to low. That directionality does not come from any force pulling molecules downhill; it comes from simple counting. More molecules are near the peak than the edges, so more random steps carry molecules away from the peak than toward it.

Fick's First Law: flux from a gradient

Adolf Fick, in 1855, borrowed the mathematical form of Fourier's heat-conduction law and applied it to matter. Fick's First Law states that the flux J - the amount of substance crossing a unit area per unit time - is proportional to the local concentration gradient:

J = -D · (dC/dx)

J : flux (mol per m² per s)
D : diffusion coefficient (m²/s) - depends on medium, temperature, molecule size
dC/dx : concentration gradient
minus sign: flux runs down the gradient, from high C to low C

D is not a universal constant - it is a property of the pair (solute, medium) at a given temperature. In a gas D is largest, of order 10⁻⁵ m²/s, because molecules travel far between collisions. In a liquid it drops to roughly 10⁻⁹ m²/s, since the solute is constantly jostled by a dense neighbour shell. In a solid or a gel it can fall to 10⁻¹² m²/s or lower, because the surrounding lattice or polymer mesh blocks most paths outright. The Stokes-Einstein relation ties D in a liquid to temperature, viscosity and particle radius, which is why a smaller ion or a hotter, thinner solvent diffuses faster.

Fick's Second Law: how the profile evolves in time

The first law gives flux at an instant; it does not say how concentration changes over time. Combine it with mass conservation - what flows into a thin slice minus what flows out must equal the change of amount stored in the slice - and you get Fick's Second Law, a partial differential equation for C(x,t):

∂C/∂t = D · ∂²C/∂x²

// same mathematical form as the 1-D heat equation
// concentration diffuses exactly like temperature diffuses

For an instantaneous point source this equation has an exact solution: a Gaussian that widens over time while its peak height falls, always conserving the total amount of substance. The width of that Gaussian - and more generally the rough distance a diffusion front has travelled - scales not linearly with time but as the square root of time:

x_rms ≈ √(2 D t)     // 1-D case

// doubling the distance takes 4x the time, not 2x
// this is why diffusion alone is a hopeless transport
// mechanism over macroscopic distances - the reason
// blood has a circulatory system instead of relying
// on oxygen diffusing from your lungs to your toes

Steady state and Fick's Law in practice

When a system is fed at one boundary and drained at the other for long enough, dC/dt goes to zero everywhere and the concentration profile stops changing shape even though matter keeps flowing through it - a steady state. Across a slab of thickness L with fixed concentrations C1 and C2 at the two faces, the profile becomes exactly linear and the flux is constant everywhere:

J_steady = D · (C1 - C2) / L

// same functional form as Ohm's Law (I = V/R) and
// as steady-state heat conduction - flux is a
// concentration difference divided by a resistance L/D

This steady-state form is what shows up in gas-exchange membranes, dialysis units, transdermal drug patches and semiconductor doping profiles: engineers pick the membrane thickness and material specifically to hit a target flux for a given concentration difference.

Where the simple picture breaks down

Fick's laws assume D is constant, but in concentrated solutions, in swelling gels, or when diffusion is coupled to an electric field (ions) or to a chemical reaction consuming the diffusing species, D itself becomes a function of concentration, position or time, and the simple Gaussian solution no longer applies exactly. Reaction-diffusion systems - where a diffusing species is also being created or destroyed - need an extra source/sink term added to the second law, and that small addition is enough to produce Turing patterns, the same mechanism behind animal coat markings.

Frequently asked questions

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