Fick's Laws of Diffusion - Fick's Second Law

Fick's second law predicts how diffusion causes the concentration to change with time:

where

• is the concentration in dimensions of, example
• is time
• is the diffusion coefficient in dimensions of, example
• is the position, example

It can be derived from Fick's First law and the mass conservation in absence of any chemical reactions:

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiation and multiply by the constant:

and, thus, receive the form of the Fick's equations as was stated above.

For the case of diffusion in two or more dimensions Fick's Second Law becomes

,

which is analogous to the heat equation.

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, Fick's Second Law yields

An important example is the case where is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant, the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.