R. M. R. Fick's Laws

FICK'S
LAWS
I once had an algebra text book in which the author devoted several
paragraphs to a discussion of the nature of mathematical proof and he cited
several subjects around which one could still stir up a heated debate among the
experts. A prime ingredient in
mathematical proof, according to this author, was our intuition regarding
countable sets and their members as often comes up with regard to questions in
probability theory. Consider the
toss of a coin, for example. If we discard the cases when the coin comes to rest on its
edge, there are two possible outcomes, namely heads and tails.
We require the probability of heads or tails to be certainty, P{h
or t} = 1, and since we can find no logical argument to favor one outcome over
the other, our intuition regarding this set of outcomes is that P{h} =
P{t} = 1/2. Moreover, our
intuition tells us that P{h} = P{t} = 1/2 on the next toss yet to be
performed is completely independent of the set of outcomes at all previous
tosses. What, then, is the
probability P{hh} = P{tt} in 2 tosses? We
can enumerate all of the possible outcomes: hh, ht, th, tt and use our intuition
regarding countable sets and their members to find P{hh} = P{tt} = 1/4, since we
can find no logical argument to favor any one of these outcomes over any other.
In general we can say that the probability of a sequence of independent
outcomes is just the product of the individual probabilities.
Thus the probability of 5 heads in five tosses one after the other is
(1/2)^5 = 1/32.
Suppose, now, that we construct a machine to do the tossing and record
the results and then find, as a matter of fact, that in one sequence of 100
tosses heads came up 100 times in a row. The
question arises: is the machine and/or the coin biased in some way, or has an
extremely unlikely event occurred and what is the probability of a head or a
tail on the next toss? Our
intuition regarding sets and their members is very strong, but so is our
intuition regarding the relative frequency of events.
One may argue that the facts speak for themselves and the probability of
a head on the next toss is all but certain.
Others will argue that a careful examination of the machine and the coin
has turned up no evidence of a bias and the probabilities are still 50/50.
I have heard people I consider to be far more intelligent than I am argue
the matter off and on for weeks without coming to any agreement.
The people who operate the games in Nevada and elsewhere strongly promote
the idea that their customers should bet the farm when they find themselves on a
"winning streak", but my intuition regarding sets and their members is
stronger than my intuition regarding relative frequencies, so I have a problem
with the concept of a "winning streak", particularly when independent
events are at issue.[1]
My former office mate, the historian, was deeply concerned with such
dilemmas. He once chided me for
what he found to be an unwarranted reverence on my part for certain fundamental
laws of physics as absolute truths. He
said that his study of the history of science showed a painfully turbulent and
slow acceptance of many concepts which were routinely taught as proven facts in
college at the time. He was of the
strong opinion that the laws of physics as we know them had come to be accepted
only because they had been proved out on the battlefield.
Warring nations, he submitted, had been hiring technical consultants
since time immemorial to design weapons and strategies and the winners were
simply more likely to be those who hired people immersed in the lore of modern
physics while the losers tended to hire astrologers and mystics and those who
specialized in reading the entrails of birds.
I will now consider the problem of estimating the residence time for an
atomic particle trapped by its neighbors and in thermal equilibrium with them.
Extensive experience in the laboratory shows that there is usually a well
defined threshold energy for escape in this situation and a finite probability,
however small, that the particle will acquire the required energy given
sufficient time. In this model, the
trapped particle oscillates back and forth between its neighbors rebounding each
time it gets too close to any of them. The
particle may gain or lose energy at each rebound, with the average energy
directed along each of three spatial coordinates being kT, roughly .026 electron
volts at room temperature. The
probability of a sequence of favorable rebounds mostly increasing the energy of
the particle is extremely small, but not zero, while the frequency of rebounds
is extremely high. The combined
effect is that some particles trapped in solid solution can be expected to
remain in a fixed location at room temperature for times comparable to the age
of the universe or longer while other particles can be expected to migrate
perceptibly as we watch with every possibility in between.
As the temperature is increased the residence times decrease, often
dramatically, and all of the elements we know can be transformed into a liquid
or a vapor at temperatures attainable in the laboratory.
Let p represent the probability of escape at a trial.
The probability of failure to escape is thus (1p).
Since the probability of escape, or failure, at a trial is independent of
all previous trials, we see that the probability of a sequence of n failures is
simply (1p)^n = (1np/n)^n. This
form is studied extensively in advanced high school algebra and beginning
calculus courses. As n gets
indefinitely large the expression converges to the form EXP(x) where x = np, a
function found on most, if not all, pocket calculators.
When I enter x = np = 1, for example, my calculator gives the value
.367879441. If the frequency of
escape attempts is f per second, then n = ft where t is the elapsed time since
we last looked and found the particle in residence.[2]
The probability that the particle is still in residence after n trials (t
seconds) is just EXP(np) = EXP(ftp) = EXP(t/to) where to = 1/fp seconds.
If we think of the residence time as a random variable, it works out that
the expected residence time is just to = 1/fp seconds. The probability of escape at a trial, from Boltzmann
Statistics, is just p = EXP(Qd/kT) while the frequency of escape attempts per
second is on the order of 10^12 to 10^13 times per second.
Figure 41 is a plot of LOG(Expected Residence Time) vs Temperature for
various activation energies from 0.25 ev to 2 ev assuming that f = 10^13
trials/second. By way of
stimulating the intuition, note that Qd for the hydrogen nucleus in soft iron is
about .24 ev while Qd for sulphur in stainless steel is in excess of 2 ev.
There are very few, if any, values of Qd less than .24 ev in the
literature, but a large number in excess of 2 ev.
We can estimate the frequency of escape attempts in terms of the mass of
the particle in question, the physical size of the cell in which the particle is
trapped between its neighbors, and its average kinetic energy as determined by
the temperature. It turns out,
however, that this quantity appears in our final calculations along with other
factors so that precise knowledge of this factor alone is not important. We can measure directly in the laboratory the overall
combined effect of all of these factors without knowing the individual factors
precisely.
Suppose we have, in a 3 dimensional solid, a distribution of atomic
particles (AP) subject to thermally activated random walk.
We can define a concentration, C, AP/Cm^3, at each point in the solid.
This concentration will, in general, be a function of the three spatial
coordinates, say x, y, and z and the time, t and we may find it convenient to
use the notation C = C(x,y,z,t) from time to time. The population of diffusing AP in a neighborhood will
decrease in proportion to the population present due to emigration and increase
in proportion to the population of neighboring points due to immigration.
Suppose we have an incremental increase in concentration, ¶C, corresponding to an incremental
displacement, ¶x, in the x direction.
We can define a concentration gradient,
¶C/¶x,
in this direction and postulate a net flow of AP in the direction corresponding
to a decrease in concentration. We
designate the flow of AP as J, AP/Sec/Cm^2, and argue, from intuition at this
point, that J is proportional to the negative of the concentration gradient.
This is Fick's First Law of diffusion which may be stated formally as
follows: Jx
= D ¶C/¶x
AP/Second/Cm^2 Jx being
the x component of the total flow. There
are, of course, diffusion currents in the y and z directions as well provided
that there are also concentration gradients in those directions.
The 3 dimensional vector equation may be expressed in vector notation as
J = D ÑC,
but we shall be concerned mostly with the simpler one dimensional version.
The constant of proportionality, D, has the dimensions Cm^2/Second, and
it can be shown that D takes the form D = N*d^2*f*EXP(Qd/kT)/2 where f is the
frequency of escape attempts per second, mentioned earlier, d is the average
distance between traps where the AP may reside between migrations, and N is the
average number of d spacings the AP travels between traps.
In general, an AP may not fall into the nearest trap as soon as possible
after a successful escape, but it may wander through the solid passing up a
number of opportunities before falling into one.
The uncertainty in the value of N is covered up by a further uncertainty
in the value of f, although d is generally knowable from Xray studies of the
solid crystal lattice under consideration.
f is, in general, a slowly varying function of temperature, but the
temperature dependence expressed in the factor EXP(Qd/kT) dominates the whole
process so overwhelmingly that any experimental determination of f as a function
of temperature is not possible. The
diffusion parameter, D, is usually expressed in the form D = Do EXP(Qd/kT)
where Do is a constant which can be determined experimentally in a number of
ways, for example, as Lewis Hall and I did in the case of the diffusion of
sulphur in stainless steel. We also
measured the heat of diffusion, Qd, at the same time.
The derivation of Fick's Second Law of Diffusion is based on the idea
that the total number of AP in the extended neighborhood of any interior point
is a constant. At a surface, of course, we will allow AP to escape the solid
as a whole. Consider the simple
case where there are no concentration gradients in the y or z directions and
thus no net migration of AP in those directions.
If we find that Jx is constant with variation in x then there are as many
AP entering a small volume in the neighborhood of x as there are leaving it and
the concentration within the volume is constant in time.
If, on the other hand, Jx is not constant with a variation in x, the
concentration will not be constant in time.
If there are more AP entering a small volume than there are leaving it,
the concentration will increase with time, and visa versa.
We can express this idea in mathematical notation as follows: ¶J/¶x
=  ¶C/¶t,
AP/cm^3/Sec. We can take a partial
derivative of both sides of Fick's First Law with respect to x and find ¶J/¶x
= D ¶^{2}C/¶x^{2}
=  ¶C/¶t.
This is Fick's Second Law of Diffusion, presented again in the form here: ¶^{2}C/¶x^{2}
= (1/D) ¶C/¶t
The left hand side of this expression is roughly identified with the
upward curvature of the concentration profile while the right hand side is
proportional to an increase, with respect to time, of the local concentration.
If the local concentration is decreasing with time, the concentration
profile will be curved downward. 
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