Fick's second law
/ Fick = FIK /
Fick's first law tells you how fast atoms flow right now, but it does not tell you how a concentration pattern reshapes itself as the hours pass. Watch a drop of dye spread in gel: the sharp blob softens into a fuzzy cloud, the peak sinks, the edges creep outward. Fick's second law is the bookkeeping that predicts exactly that — how the concentration at every point rises or falls over time as diffusion smears sharp features into smooth ones.
It comes from combining Fick's first law with the plain accounting rule that atoms are not created or destroyed: whatever flows into a thin slab and does not flow out must pile up inside it. The result is dC/dt = D times (d^2C/dx^2). In words: the concentration at a point changes in time at a rate set by the curvature of the profile. Where the profile is bowl-shaped (curving up, a valley), material accumulates and C rises; where it is dome-shaped (a peak), material drains away and C falls. Diffusion always erodes peaks and fills valleys. A classic solution is the error-function profile you get when two solids are welded face to face: the sharp step in composition blurs into an S-curve whose width grows as sqrt(D times t) — double the width needs four times the time.
That sqrt(D times t) is the single most useful number in all of diffusion: the distance an atom typically wanders in time t. It is why doubling a diffusion depth costs four times the time, and why firing schedules trade temperature against time. In ceramics it governs how far a dopant penetrates on firing, how quickly a concentration difference in a powder compact evens out, and how deep a colour or a reacted layer grows. The honest caveat: solving it cleanly assumes D is constant, but in real ceramics D can depend on concentration, on oxygen pressure, and steeply on temperature, so the tidy error-function answers are a first approximation, not gospel.
Weld a block of pure MgO to a block of NiO and hold them hot: nickel and magnesium ions interdiffuse across the join, and Fick's second law predicts the smooth S-shaped composition curve that widens as sqrt(D times t) — after four times the time, the blurred zone is twice as deep.
The diffusion equation: concentration changes fastest where the profile is most curved, so peaks erode and valleys fill; features spread as sqrt(D times t).
A common trap is to expect diffusion depth to grow in proportion to time. It does not — it grows as the square root of time, so the second hour adds far less penetration than the first. The rule of thumb sqrt(D times t) is worth memorising.