Fick’s Law

The transport of material by diffusion is due to the random thermal movement of molecules and is described by Fick’s law:

(4.1)

where J_i is the molar flux [mol m⁻² s⁻¹] of species i. The flux represents the rate at which material passes through a plane of unit area. It is a vector quantity with both direction and magnitude. In one dimension, the gradient is simply the derivative, and the flux is

The driving force for transport by diffusion is the concentration gradient, and material is transported in the x-direction from a region of high concentration to low concentration. D_i is the proportionality constant called the diffusion coefficient or diffusivity [m²·s⁻¹]. These gradients in concentration, shown in Figure 4.2, drive diffusion of O and R to and from the surface.

The flux defined here, J, is the flux relative to the molar average velocity. It is generally more convenient to work with a flux relative to a fixed frame of reference. This quantity is also a flux, but will be designated with N. For our purposes, we can relate the two quantities by a combination of molecular diffusion and a contribution from bulk flow

(4.2)

where v is the molar average velocity of the fluid.