4.4. More about diffusion[[Student version, December 8, 2002]] 117
is (that is, onX,Y,orZ); the important thing is not really the number crossing the boundary
“a,” but rather the numberperunit areaof “a.” This is such a useful notion that the average rate
of crossing a surface per unit area has a special name, theflux,denoted by the letterj.Thus, flux
has dimensionsT−^1 L−^2.
Wecan restate the result of the two preceding paragraphs in terms of the number density
c=N/(LY Z), finding that
j=1
YZ×∆t×
1
2
×L×
(
−
d
dx
LY Zc(x))
=−
1
∆tL^2
2
×
dc
dx.
Also, we have already given a name to the combinationL^2 /2∆t,namely, the diffusion constantD
(see Equation 4.5b). Thus, we have
j=−D
dc
dx. Fick’s law (4.18)
jis the net flux of particles moving from left to right. If there are more on the left than on the
right, thencis decreasing, its derivative is negative, so the right-hand side ispositive.That makes
sense intuitively: A net drift to the right ensues, tending to even out the distribution, or make
it more uniform. If there’s structure (or order) in the original distribution, Fick’s law says that
diffusion will tend to erase it. The diffusion constantDenters the formula, because more-rapidly
diffusing particles will erase their order faster.
What “drives” the flux? It’snotthat the particles in the crowded region push against each other,
driving each other out. Indeed, we assumed that each particle is moving totally independently of
the others; we’ve neglected any possible interactions among the particles, which is appropriate if
they’re greatly outnumbered by the surrounding solution molecules. The only thing causing the net
flow is simply that if there aremoreparticles in one slot than in the neighboring one, and if each
particle is equally likely to hop in either direction, then more will hop out of the slot with the higher
initial population.Mere probability seems to be “pushing” the particles.This simple observation is
the conceptual rock upon which we will build the notion ofentropic forcesin later chapters.
Fick’s law is still not as useful as we’d like, though. We began this subsection with a very
practical question: If all the particles are initially concentrated at a point, so that the concentration
c(r,0) is sharply peaked at one point, what will we measure forc(r,t)atalater timet?We’d like
an equation we could solve, but all Equation 4.18 does is tell usjgivenc.That is, we’ve found one
equation intwounknowns, namely,candj. But to find a solution, we need one equation inone
unknown, or equivalently a second independent equation incandj.
Looking again at Figure 4.10, we see that the average numberN(x)changes in one time step
for two reasons: Particles can cross the imaginary wall “a,” and they can cross “b.” Recalling that
jrefers to the net flux from left to right, we find the net change
d
dt
N(x)=(
YZj(x−L 2 )−YZj(x+L 2 ))
.
Once again, we may take the bins to be narrow, whereupon the right-hand side of this formula
becomes (−L)times a derivative. Dividing byLY Zthen gives thatddct=−ddjx,aresult known as
thecontinuity equation.That’s the second equation we were seeking. We can now combine it with
Fick’s law to eliminatejaltogether. Simply take the derivative of Equation 4.18 and substitute to