# Fick's Second Law of Diffusion

In the book "Foundations of Materials Science and Engineering" (Smith and Hashemi) the following is written:

For cases of non-steady state diffusion in which diffusivity is independent of time, Fick's second law of diffusion applies, which is

$$\frac{\partial c_{x}}{\partial t}=\frac\partial{\partial x}\left(D\frac{\partial c_{x}}{\partial x}\right)$$

In what cases are materials' diffusivity dependent on time, and in what cases are they not?

## 2 Answers

Diffusivity is not dependent on time. It is dependent on temperature (and material). Fick's second law holds because the diffusivity, $D$, is constant with respect to depth, $x$, and time, $t$, so it is treated as a constant in the equation.

$$\frac{\partial c_{x}}{\partial t}=\frac\partial{\partial x}\left(D\frac{\partial c_{x}}{\partial x}\right) = D\left(\frac{\partial^2c_x}{\partial x^2}\right)$$

Fick's second law of diffusion applies to non steady state diffusion, which concentration at a depth $x$, $c_x$ is a function of time.

In general D depends on both position x and time t, since it usually depends of the concentration c(x,t) as well as mobility factors. This is why treating D as a constant (and thus putting it outside the last partial with respect to x) is a simplification, that is mostly used for getting an approximate solution for c(x,t). In many cases, however, this will be sufficiently close to the "correct" solution.