UPHILL DIFFUSION

Fick's 1st law gives the flux of a solute down the concentration gradient:

But, multicomponent diffusion can be upgradient, as illustrated in the following example.
We fill a column with 0.1 mM NaCl and 0.1 mM HNO3 solution, the pH is 4. We make another solution with the same NaCl concentration, but with 0.001 mM HNO3, the pH is 6. This is the constant boundary solution shown in the figure.

After 1 hour diffusion, the concentrations in the column become (PHREEQC input file uphill.phr):

Note that H+ has decreased more in the column than NO3-.
Note also that the Na+ concentration bulges upward, although the initial concentration gradient was zero everywhere. On the other hand, the Cl- concentration has decreased, despite the initially zero concentration gradient which, according to Fick's 1st law, gives zero flux.

How to explain these results?
The diffusion coefficient of H+ is approximately 5 times higher than of NO3-. Consequently, H+ diffuses quicker from the column than NO3-. After some time, the column contains more NO3- than H+. The negative charge that would develop is balanced by Na+ which enters, and Cl- which leaves the column. Diffusion of Na+ and of Cl- starts although the initial concentration gradient is 0, and continues even against the concentration gradients shown in the figure. This is an effect of the zero charge flux condition used by PHREEQC when calculating multicomponent diffusion.

Alternatively, the example can be calculated with a zero-current condition in the Nernst-Planck equation, showing that a small, negative potential develops in the column because of the rapid out-diffusion of H+, see electro-migration and -remediation.

This example is from
Lichtner, P.C., 1995. Principles and practice of reactive transport modeling, Mat. Res. Soc. Symp. Proc. Vol. 353, 117-130.

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