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?
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.