Motivation

Consider an atomistic picture of diffusion in a binary alloy as shown in Fig. 1. Assuming random jumps for the atoms (and that the jump frequency is independent of composition), as can be seem from the schematic, purely because of the difference in the number of atoms of a particular type, there is a net flux. And this flux is opposite to the concentration gradient.

Thus, Fick's first law, which states that the flux of atoms is proportional to concentration gradient and is in such a way as to reduce concentration gradients is a very reasonable one:

$$J = - D \nabla c$$ (1)

Figure 1: An atomistic picture in which if we assume random jumps for atoms and that the jump frequency is independent of composition, the atomic flux is proportional to composition gradient.

[scale=0.6,angle=0]Figures/atomisticpicture.pdf

As noted in Part I of these course notes, the proportionality constant D is known as diffusivity and is positive. Again as noted in Part I, coupling this constitutive law with the conservation law for mass one obtains the classical diffusion equation, namely

$$\frac{\partial c}{\partial t} = D \nabla^2 c$$ (2)

Even though the classical diffusion equation is based on a reasonable constitutive law and a conservation law it is known to fail in certain cases. Such failure occurs in what is known as phase separating systems. Why? And, how does one fix this failure?