Diffusion and Fick's Laws

Core Idea

Fick's first law states that the diffusion flux J = −D(∂c/∂x) is proportional to the concentration gradient, with diffusion coefficient D. Fick's second law ∂c/∂t = D(∂²c/∂x²) describes how concentration profiles evolve in time, with Gaussian spreading: ⟨x²⟩ = 2Dt for one-dimensional diffusion. The diffusion coefficient for a gas scales as D ∝ T^3/2/p from kinetic theory; for a sphere in a liquid, the Stokes-Einstein equation gives D = kT/(6πηr), connecting diffusion to viscosity η and solute radius r. Self-diffusion, mutual diffusion, and tracer diffusion coefficients are distinct but related through the Onsager reciprocal relations.

How It's Best Learned

Solve the diffusion equation analytically for a point source and verify ⟨x²⟩ = 2Dt. Use the Stokes-Einstein equation to estimate the size of proteins from measured diffusion coefficients (a technique used in DLS).

Common Misconceptions

• Confusing Fick's first law (steady-state flux) with Fick's second law (time-dependent concentration change); the first applies to steady-state, the second to transient diffusion.
• Thinking diffusion only applies to gases; the Stokes-Einstein equation and Fick's laws describe diffusion in liquids and solutions equally well.

Explainer

Imagine dropping a crystal of food dye into still water and watching it spread. The dye does not move in any preferred direction — every molecule executes a random walk, jostling in all directions due to thermal motion. Yet the net result is clear: dye moves from regions of high concentration toward regions of low concentration. Fick's laws describe this macroscopic consequence of microscopic randomness.

Fick's first law captures the steady-state picture: the diffusion flux J (moles crossing unit area per unit time) is proportional to the local concentration gradient, with the diffusion coefficient D as the proportionality constant: J = −D(∂c/∂x). The negative sign is essential — flux flows in the direction of decreasing concentration. If the gradient is steep, flux is large; if concentration is uniform, flux is zero regardless of how high the concentration is. This law applies when the concentration profile has settled into a time-independent shape (steady state), as in a membrane separating two well-mixed reservoirs.

Fick's second law extends the analysis to transient situations, where concentration profiles evolve over time: ∂c/∂t = D(∂²c/∂x²). This is a partial differential equation whose solutions describe how an initial concentration distribution spreads and flattens over time. For a point source of material released at x = 0 and t = 0, the solution is a Gaussian profile whose width grows in time. The key result is ⟨x²⟩ = 2Dt: mean squared displacement is proportional to t, not t². Diffusion is slower than ballistic motion — a consequence of the random-walk mechanism where forward and backward steps partially cancel.

The diffusion coefficient D is not a universal constant — it depends on what is diffusing and in what medium. For a gas, kinetic theory gives D ∝ T^3/2/p: faster molecules (higher T) and fewer collisions (lower p) mean faster diffusion. For a spherical particle in a liquid, the Stokes-Einstein equation gives D = kT/(6πηr), where η is viscosity and r is the particle radius. This equation has a remarkable practical use: measuring D by tracking particle motion (e.g., in dynamic light scattering) lets you infer the particle's radius — a standard technique for sizing proteins and nanoparticles.

Finally, do not fall into the trap of thinking Fick's laws are only for gases. They describe diffusion in any medium — liquids, solids, and biological membranes — and are foundational to understanding everything from drug delivery kinetics to oxygen transport in tissue to heat conduction (which obeys an identical mathematical form).