Thursday, July 29, 2010

Fick's Second Law of Diffusion and Its Solution

One can find Einstein's papers on Brownian motion in Investigations on the Theory of the Brownian Movement which is still in print. In the first paper from 1905 Einstein derives Fick's Second Law of Diffusion by making a series expansion of the concentration of particles, f (φ in the Wikipedia article), as a function of time and position. Then he gives a solution for this equation which turns out to be a normal distribution with σ equal to Δ (rms), which is also known as the diffusion length. Strictly speaking, this solution is for a point source in one dimension but for a narrow column food coloring in a thin slab of jello one would expect that this would be a close approximation with x replaced by r, the distance from the column and the center of the distribution over time.

edit: For one dimension half of the particles will be within 0.674 σ of the center of the normal distribution. The fraction within 1 σ is 0.683, within 2 σ it is 0.954 and within 3 σ one will find 0.997 of all the particles. The particles slowly spread over time with σ (the diffusion length) acting as a scale factor. In two dimensions the radial distribution giving the density of particles as a function of the radius, r, only is also a normal distribution but with the one dimensional σ replaced by √2 σ.

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