Watch concentration profiles evolve through a slab and into a semi-infinite solid. Drag the time slider to see Fick's 2nd Law spread atoms by an error-function curve, then apply it to a real case: carburizing steel.
At steady state, the concentration profile through a slab is linear. The slope dC/dx is constant, so the flux J is the same at every depth — atoms flow downhill at a uniform rate from the high-concentration face to the low-concentration face.
A bigger gradient or higher diffusivity → bigger flux. Sign convention: J is positive when atoms move in the +x direction.
A semi-infinite solid starts at uniform C₀. The surface is held at Cₛ. Drag t to watch the profile evolve. At t = 0 the curve is a step; as time grows, atoms diffuse inward and the front advances proportional to .
The error function erf(z) rises from 0 to 1. Where ≈ 1, the local concentration is about halfway between C₀ and Cₛ. That distance — — is the practical "diffusion depth" of the process.
Low-carbon steel is held in a carbon-rich atmosphere at high temperature. Carbon diffuses in from the surface, producing a hard, wear-resistant case over a tough core. Case depth is the distance at which the carbon content reaches the target value.
Push temperature up and case depth grows fast — D is exponential in 1/T. Doubling time only grows depth by ; adding 50 °C can do far more.
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