💧 Fick's Laws of Diffusion — Concentration Gradients & Flux

Fick's First Law states that the diffusive flux J is proportional to the negative of the local concentration gradient: J = -D·(dC/dx). Particles move, on average, from regions of high concentration toward regions of low concentration, and the steeper the gradient, the larger the flux.

Fick's Second Law, ∂C/∂t = D·∂²C/∂x², describes how a concentration profile evolves over time. It follows from combining Fick's First Law with conservation of mass (a continuity equation): the local rate of concentration change equals minus the divergence of the flux, and substituting J = -D·dC/dx gives the diffusion equation.

D is a proportionality constant with units of length² / time that measures how quickly a species spreads through a medium. It depends on the diffusing particle's size, the temperature, and the viscosity or structure of the surrounding medium — gases have the largest D, liquids much smaller, and solids smaller still.

Diffusion is a random walk: each step is uncorrelated with the last, so displacements partially cancel out. The mean squared displacement grows linearly with time (⟨x²⟩ = 2Dt), and taking the square root to get a typical distance gives L ≈ √(Dt) — a hallmark signature that distinguishes diffusion from directed motion, which grows linearly with time.

A concentration gradient is the rate of change of concentration with position, dC/dx. It is not a physical force; instead, diffusion arises because random thermal motion is more likely to carry particles from a crowded region into a less crowded one than vice versa, simply because there are more particles available to leave the crowded side.

Diffusion is driven by thermal motion, which never stops as long as temperature is above absolute zero. Once the concentration is uniform, particles still jitter around randomly, but because there is no net gradient left, the number moving in each direction balances out and the net flux drops to zero — this is dynamic, not static, equilibrium.

Start with conservation of mass for a thin slice of the medium: the concentration inside the slice changes only because flux enters one face and leaves the other, so ∂C/∂t = -∂J/∂x. Substituting Fick's First Law, J = -D·∂C/∂x, gives ∂C/∂t = D·∂²C/∂x² — the diffusion equation, valid whenever D is constant in space.

Each arrow shows the local diffusive flux at that point along the concentration profile: its direction points from higher to lower concentration (down the gradient), and its length is proportional to how steep the local gradient is, exactly as required by J = -D·dC/dx.

Diffusion has no built-in direction of travel — every particle moves independently and randomly. What look like a spreading and flattening curve is simply the accumulated statistics of many random walks: material moves down whatever local gradient exists, so peaks erode and valleys fill in until the profile is uniform, rather than the whole curve translating.

Fick's laws describe oxygen and carbon dioxide exchange in the lungs and across cell membranes, the spread of a perfume or gas leak through still air, drug release from a transdermal patch or tablet coating, dopant diffusion during semiconductor chip manufacturing, and salt or heat spreading through soil, water, or metal.