## Diffusion

### Molecular Motion

We can safely ignore attractive forces if the thermal energy of the particles is sufficient to ensure that all collisions are elastic. This is the billiard ball model, and we shall make use of it to investigate the properties of molecular events on a microscopic scale. We will further simplify our discussion to one dimensional space, as none of the essential physics are lost by doing this. All particles have thermal energy which is given by the equipartition theorem as kT/2 for each axis. T is the absolute temperature and k is Boltzmann’s constant.

(5.1.1)

If we consider the thermal energy as kinetic energy, then we can write the equation

(5.1.2)

where the mean square velocity is proportional to the temperature and inversely proportional to the mass of the particle. When we imagine a bunch of these particles whizzing around, it is clear that they will be colliding with other particles quite regularly. With each collision, the velocity and direction of motion will change. With 1023 particles the situation is complicated enough that we can essentially consider each change in motion of a single particle to be uncorrelated with all previous collisions. This situation can be modelled as a random walk. A single particle is started at some point and has equal probability of moving one step to the left or one step to the right.

Fig. 5.1.1

We’ll call the step length d and the time interval t (these will be treated as constants, although they will differ for different particles and different temperatures). So if we let our particle "walk" for 5 steps, its path might be RRLRL which would leave it one step to the right of its starting point. This procedure is isomorphic to tossing a coin, so we can learn something about it by looking at that situation. For N tosses of a fair coin, the probability of x heads appearing is given by the binomial distribution.

Fig. 5.1.2

Naturally the highest probability is given to the situation in which there are an equal number of heads and tails, but there is a spread in the distribution.

Going back to the random walk of a particle, we can set up an equation that tells us the displacement of the particle after n time intervals

(5.1.3)

If we were to perform a random walk with N particles, then the average displacement would be given by the sum of each individual displacement divided by N

(5.1.4)

Substituting our earlier expression, we get

(5.1.5)

The plus/minus term becomes zero, so on average the particle stays where it is. This just shows the obvious fact that the mean displacement is equal to zero (due to the symmetry of equal probability for movement to the right and left). To see how the particles will spread out with time, we can look at the root-mean-square displacement (since the square of a negative number is positive, this will show us how the particles move away from their starting point). First we square the displacement

(5.1.6)

Then we take the mean

(5.1.7)

Again the plus/minus term goes to zero, so the mean square displacement is given by

(5.1.8)

Since xi(0) = 0, it follows that <x2(1)> = 1d2, <x2(2)> = 2d2, etc. so that <x2(n)> = nd2, and since each step occurs in time t, t = n t.

(5.1.9)

Since d and t are considered constants, we can replace them with a single constant

(5.1.10) so that

(5.1.11)

This is the diffusion equation, where D is the diffusion constant (the reason for the 2 will become apparent). If we look at the root-mean-square displacement, we get

(5.1.12)

So we can see that the spreading of the particle is proportional to the square root of time. This means that to cover twice as much distance, four times as much time will be required.

If we go back to the velocity due to thermal motion

(5.1.13)

We can substitute this into

(5.1.14)

to get

(5.1.15)

Now using the equipartition theorem again, we can see that

(5.1.16)

Thus allowing one to determine the mean time between collisions by measuring macroscopic quantities. This is essentially what Einstein did to prove the atomic hypothesis and show how we could measure the size of atomic particles. Although we rarely see past the celebrity of relativity and quantum theory, this was surely Einstein’s most fruitful contribution to the physics that people actually do.

### Diffusion

Now let us suppose that we know the numbers of particles at two points on a line (x and x + d) and want to see what the net flux is across an imaginary boundary separating these points.

Fig. 5.1.3

At time t + t, half of the particles at x will move one step to the right, and half of the particles at x + d will move to the left. The net number of particles crossing the boundary will be

(5.1.17)

The net flux is the number of particles moving divided by the cross-sectional area per unit time

(5.1.18)

We can multiply the right side by d2/d2

(5.1.19)

We saw before that d2/2t is the diffusion constant. The number of particles at a point divided by the area times the length of each point is simply the number of particles per unit volume. We usually call this the concentration.

(5.1.20)

Now, as d becomes very small, the limit of the above equation as d approaches zero is simply the definition of a partial derivative, so we can write it as

(5.1.21)

This is Fick’s first law and it simply states that the net flux at a point is proportional to the concentration gradient. It is now clear why the factor of 2 was introduced into the definition of the diffusion constant, to make it compatible with Fick’s equation. We can generalize this result to include time as well by assuming that particles are conserved. If we consider a thin box, the concentration change in some time period will be a function of the fluxes at either side of the box.

Fig. 5.1.4

So the number of particles accumulating in the box is given by

(5.1.22)

On the right side the t’s and the A’s cancel and in the limit in which both t and d go to zero we’re left with

(5.1.23)

Substituting Fick’s first law leaves us with Fick’s second law

(5.1.24)

This is often referred to as the diffusion equation and is perhaps the most significant equation in all of biology. Although it can be written out with just a few symbols, when it is generalized to three dimensions, solutions can be very hard to come by (although the equation itself looks quite elegant--it won't hurt to take a quick peek).

(5.1.25)

In fact, the best way to obtain solutions is to look in a book called "The Mathematics of Diffusion" by John Crank and hope that he solved it for the situation which you are studying.

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Document last updated Oct. 14, 1998.