Fick's second law
/ FIKS /
Most real diffusion problems are not steady — they are stories that unfold in time. Push carbon into the surface of a cold steel gear and, hour by hour, the carbon-rich zone creeps deeper and the profile keeps changing. Fick's first law cannot handle this, because it assumes a fixed gradient. Fick's second law is the tool for the changing case: it tells you how the concentration at every point evolves as the diffusion proceeds.
The equation is dC/dt = D times (d2C/dx2). In words: the rate at which concentration changes with time at a point (the left side) is set by how curved the concentration profile is there (the second derivative on the right). Where the profile is bowed — a peak losing atoms, a valley gaining them — concentration changes fast; where it is a straight line, it holds steady. It is the same mathematical form as the equation for heat spreading through a bar, which is why heat and mass diffusion look so alike. For the common case of a constant surface concentration (Cs) diffusing into a semi-infinite solid that started uniform (C0), the solution uses the error function: (Cx - C0)/(Cs - C0) = 1 - erf(x / (2 times sqrt(D times t))).
That solution hides a profoundly useful rule of thumb. The depth to which diffusion reaches grows not with time, but with the square root of time — because x sits inside the combination x/sqrt(Dt). To make a carburized case twice as deep, you must diffuse four times as long, not twice. Equivalently, if you know one recipe (a time and temperature that give a certain depth) you can scale to another: keeping Dt constant keeps the profile the same. This one relationship lets engineers design heat treatments — how hot, how long — to hit a target case depth.
If a carburizing run at fixed temperature gives a 1 mm case in 4 hours, reaching a 2 mm case needs Dt to quadruple — so 16 hours, not 8, because depth scales as sqrt(Dt). To go faster instead, raise the temperature: that boosts D exponentially and gets the same depth in far less time.
Depth grows as the square root of time — double the depth costs four times the time.
The square-root-of-time rule (constant Dt gives the same profile) is the single most useful takeaway, but it assumes D stays constant — true only at fixed temperature, since D itself climbs steeply with heat.