2.1 Fick's equations

Diffusion of atoms in solids can be described by the Fick's equations. The first equation relates the flux ($\vec{j}$ : number of atoms crossing a unit area per unit time) to the gradient of the concentration ($\rho$ : number of atoms per unit volume) via the diffusion coefficient tensor $\hat{D}$ :

$$\vec{j}=-\hat{D}\mathrm{grad}\rho.$$ (2.1)

This equation permits to determine the diffusion coefficient in cases, where the concentration gradient is time independent (steady state regime).

In non steady state regime, the diffusion flux and concentration are function of time and position. In order to be able to determine the diffusion coefficient, it is necessary to take into account the conservation of matter. For not interacting particles (no chemical reaction, no reactions between different types of sites in a crystal, etc.), this is the continuity equation:

$$\frac{\partial\rho}{\partial t}+\mathrm{div}\vec{j}=0.$$ (2.2)

Combining equations (2.1) and (2.2), one obtains the second Fick's law:

$$\frac{\partial\rho}{\partial t}=\mathrm{div}\left(\hat{D}\mathrm{grad}\rho\right).$$ (2.3)

For cubic crystals and isotropic media, the diffusion coefficient tensor reduces to a scalar $D$ , thus the first Fick's law is:

$$\vec{j}=-D\mathrm{grad}\rho.$$ (2.4)

Moreover, if the concentration varies only in the $x$ direction:

$$j=-D\frac{\partial\rho}{\partial x},$$ (2.5)

and equation 2.3 reduces to:

$$\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial x}\left(D \frac{\partial\rho}{\partial x}\right).$$ (2.6)

If, additionally, the diffusion coefficient is independent of the concentration, equation (2.6) can be written in the following form:

$$\frac{\partial\rho}{\partial t}=D\frac{\partial^2\rho}{\partial x^2}.$$ (2.7)

From mathematical point of view, equation 2.7 is a second order, linear partial differential equation. Initial and boundary conditions are necessary to solve it.