Fick's laws of diffusion

Diffusion is the net transport of particles associated with random molecular motion. In many simple systems, a concentration gradient gives a useful macroscopic description of the resulting transport. German physiologist Adolf Fick formulated laws of diffusion in 1855, drawing an analogy with Fourier's law of heat conduction. They remain foundational models in fields including chemistry, materials science and biology.

For a one-dimensional system with constant diffusion coefficient, Fick's first law is J = −D ∂c/∂x. Here J is the diffusive flux, c is concentration, x is position, and D is the diffusion coefficient. In vector form for an isotropic medium, J = −D∇c. The minus sign indicates that the flux points down the concentration gradient, from higher concentration toward lower concentration. This relation is used for steady-state diffusion, when the concentration profile does not change with time.

The second law describes non-steady diffusion: concentration changes over time as material flows into or out of a region. For constant D in a homogeneous medium, it is ∂c/∂t = D∇²c; in one dimension, ∂c/∂t = D∂²c/∂x². This form follows by combining the first law with conservation of mass. If diffusivity varies with position or composition, the governing equation must account for that variation; the simple constant-D form cannot be used without checking its assumptions.

The simple forms of Fick's laws are most useful when a continuum description is appropriate and diffusion can be represented by a scalar, effectively constant coefficient. In mixtures with interacting components or strongly varying properties, more detailed transport models may be needed. Fickian models are applied to estimate transport through materials, describe the spread of dissolved substances, and model molecular movement in biological and chemical systems. They are idealized relations: real transport may also involve fluid flow, reactions, or other processes.