Reading the content in Fick's law


If there is a concentration of a kind of molecule immersed in a thermal bath, the random motion of the molecules of the bath (mostly water in biological examples) will cause the molecules we are looking at to undergo a random walk. However, if the concentration of those molecules is not uniform, their random motion will result in a net transport of those molecules from a region of high concentration to one of low concentration. Let's review how quantitative information about that result is coded in the equation for Fick's first law:

$$J = -D \frac{dn}{dx}$$

In order to "find the dog" in the equation that is Fick's law, note what we learned in the construction of the law (derivation) from an analysis of the random motion.

1. J — The quantity on the left is the "current density" of the kind of molecules whose concentration we are considering. (This could, for example, be a chemical signaling molecule.) This has two tricky factors: J counts the number of particles per second per unit area. So this is the number of particles crossing a unit area in each second. Since the equation is written for the x direction, we are only considering the motion of the particles in the x direction so they have to be crossing an area perpendicular to the x axis (in the y-z plane). Note that J measures the net flow of the particles we are looking at — not those of the thermal bath whose jiggling is pushing them in a random walk. 

2. dn/dx —  The flow of particles is driven by the concentration gradient — how the density it changes in space. If the concentration were uniform, its derivative would be 0. There would be as many molecules traveling on one direction as the other so the net would be 0. The flow results from the difference in concentration, not from the direction of the molecule's motion. In any small region, whatever the concentration, the molecules are going every which way.

3. The minus sign — This tells us that the flow is opposite the derivative — in the direction of decreasing concentration. So the flow is from high concentration to low.

4. D — The diffusion coefficient, D, tells how fast the flow happens. In our derivation, it came out as proportional to the product of the average speed of the molecules times their mean free path — the average distance they travel before their velocities get reoriented in a new direction. In various situations (such as in a gas or in a liquid) these parameters may be more conveniently expressed in terms of other parameters — such as the density of the thermal bath, its temperature, or its viscosity.

Note: If we have a concentration that varies in 3 dimensions, not just in x, our equation requires the gradient — the vector derivative. If we have any function of position (for example, the concentration, n), the gradient of that function at any location points in the direction in which the direction increases the fastest.  Then the result looks like this:

$$\overrightarrow{J} = -D \overrightarrow{\nabla}n$$

The "downhill" of the concentration provides the direction of the diffusive flow.

Joe Redish 8/3/15

Article 401
Last Modified: March 3, 2019