# Fick's law

#### Prerequisite

In our discussion of diffusion, we modeled what was happening in terms of a simple mathematical model of random walkers taking a step back and forth randomly with equal probabilities. The mathematics of probabilities, finding the number of different ways that lead to the same final displacement (sort of like figuring out the odds of throwing a 7 or 11 with a pair of dice) led to a normal (Gaussian) distribution with a height that fell like one over the square root of the number of steps, and a width that grew like the square root of the number of steps. We interpreted the number of steps as time.

So our analysis said that if we released a blob of chemical at a point in space that it would spread out; the value at the starting point would decrease by a factor of the square root of time, and the width of the distribution would increase like the square root of time.

We might or might not find this mathematical model convincing or appropriate. What we really want to know is:

If we have a concentration of molecules in one region, how does it spread to a region with a lower concentration?

And for biological mechanisms we are interested in how this works at cellular scales — scales of microns.  Since each cubic micron typically contains trillions of molecules, we can use reasoning that averages over many molecules (like kinetic theory) to figure out how concentrations change by diffusion.

## A molecular model of diffusion

We'll make a more sophisticated mathematical model of molecular motions than throwing dice, but it will still be simple enough to get some useful results.

• We have a uniform fluid of molecules (colored black) containing a few interesting molecules (colored red) that has a concentration that varies as a function of position. For simplicity, let's assume that it varies in just one dimension, say the "x" direction.
• Rather than working with a continuously varying concentration, $n(x)$, let's break the x-direction up into tiny steps of width $Δx$ and treat the concentration as if it were a constant in each bin.  (This is like the way we think about an integral.)

We are interested in how the concentration changes as a result of the random molecular motions. Imagine an area, $A$, perpendicular to the x-axis. It lies in the y-z plane that is perpendicular to the screen at the point we have marked $x = 0$ and shown by a slightly thicker blue line.

The molecules we are interested in (the red ones) will cross $A$ from both directions. What is the net number of particles will pass through the area $A$ in a given amount of time?  That number is called the particle flux, and given the symbol $J$. So

net number of molecules crossing $A$ in a time $Δt = JAΔt$.

Now let's simplify our model.

• We only care about the red molecules, so let's ignore all the (black) molecules of the fluid (even though there are lots more of them than there are red ones).
• Molecules are moving in the $y$ and $z$ directions, but that doesn't affect whether they cross our area, A, so let's ignore motion in those directions.
• Molecules are also moving at different speeds, but let's simplify that by using an average speed for all our molecules, $\langle v_x \rangle$.
• Let's choose our bin width, $Δx$, to equal the average distance our molecule will travel between collisions: its mean free path, $λ$.

The resulting model looks like the figure at the right. More red molecules will hit our area $A$ from the left than from the right so there will be a net flux from negative to positive $x$.

We should consider a time interval that will take all the molecules in our bin to the left of $A$ that are going to the right through $A$. If they don't undergo a collision, this will make $Δx$ equal to

$$λ = \langle v_x \rangle Δt$$

Notice two things:

• The number of red molecules in each x-bin decreases as we move to the right.
• In each x-bin there are the SAME number of molecules going in each direction!

This means a surprising and somewhat counter-intuitive result:

The net flow of red molecules from left to right is a result of the change in concentration, not because the molecules are moving more preferentially in one direction than in another.

### Finding the flux: Fick's law

Now we can work out what $J$ must be. The number of red molecules that pass through $A$ from the left in a time $Δt$  is the volume of the cylinder based on A times the number density of red molecules in that volume on the left.

1/2  x density of red molecules in the bin to the left x volume of cylinder
=
$\frac{1}{2} n_L (Aλ)$

where $n_L$ is the density of red molecules in the bin on the left of $x = 0$. The half comes from the fact that only half of the red molecules in the bin, on the average, are moving to the right.

The number of red molecules that pass through $A$ from the right in a time $Δt$ is (the volume of the cylinder based on $A$) times (the number density of red molecules in that volume on the right).

1/2  x density of red molecules in the bin to the right x volume of cylinder
=
$\frac{1}{2} n_R (Aλ)$

where $n_R$ is the density of red molecules in the bin on the right of $x = 0$. The half comes from the fact that only half of the red molecules in the bin, on the average, are moving to the left.

The resulting net flow to the right (assuming the density is greater on the left) is our $J$ (number per unit area per unit time) times the area, $A$, times the time, $\Delta t$:

net flow through area $A$ in time $\Delta t = JA \Delta t = \frac{1}{2} n_L A \lambda - \frac{1}{2} n_R A \lambdaJA \Delta t = -\frac{1}{2} (n_L - n_R) A \lambda$$For a small distance \Delta x, this becomes$$JA \Delta t = -\frac{1}{2} \big(\frac{dn}{dx} \Delta x \big) A \lambda$$Dividing by A and Δwe get$$J = -\frac{1}{2} \frac{dn}{dx} \frac{\Delta x}{\Delta t} \lambda$$Since in this model, the width of the bin divided by our time interval is the average x velocity, \langle v_x \rangle, we get that J is proportional to the concentration gradient times a combination of constants: half the average velocity times the mean free path.$$J = -(\frac{1}{2} \langle v_x \rangle \lambda) \frac{dn}{dx}$$so we define the diffusion constant$$D = \frac{1}{2} \langle v_x \rangle \lambda$$and get:$$J = -D \frac{dn}{dx}$\$

The above equation is called Fick's Law, and is simply the statement of how the changing concentration results in a flow due to diffusion.

## Making sense of the result

In order to "find the dog" in the equation that is Fick's law, note what this analysis tells us:

1. The flow of particles is driven by the concentration gradient — how it changes in space.
2. The minus sign tells us that the flow is opposite the derivative -- in the direction of decreasing concentration. So the flow is from high concentration to low.
3. 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.
4. The diffusion coefficient, D, that tells how fast the flow happens, is 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.

Ben Geller and Joe Redish 12/3/11

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