Diffusion 
3. Phenomenological Modelling of Diffusion 

Preliminary Remarks  
"Phenomenological" is a
phenomenal word with 7 syllables! A journalist I know was almost fired for
using a 5syllable word in the Ithaca Journal. What does it mean? According to a dictionary it means "relating to phenomenology", and phenomenology is the "science" of phenomena. That really helps 

"Phenomenological Modelling" in plain words means that you model what you "see", what you know from experiments. However, you do not just describe some phenomena one by one, you try to find a common underlying pattern. As far as possible, you let yourself be guided by the facts and not by hypotheses or theories about how things should be. If you are lucky, the underlying pattern can be couched in the form of a rule that can be expressed in equations (often and wrongly called a "law") that contains the outcome of all experiments already performed and allows to predict the outcome of experiments that haven't been done yet.  
Ohm's "law", for example, is probably the best know phenomenological model for certain phenomena concerning the flow of electrical current through some materials. I put the word "law" in quotation marks since Ohm's law was not a law of nature, it just described the essence of many experiments. Today Ohm's law is recognized as a law of nature  but only under certain, precisely defined circumstances. That is so because modern physics can derive Ohm's law from it's basic laws or axioms, and thus knows exactly what is behind it, under what kind of circumstances it is valid, and when and why it might be completely wrong.  
What we are looking at her is Adolf Fick's first and second "law" of diffusion. You might never have heard about those laws but they are still in the very center of material processing. My students most certainly hear about Fick's laws in detail, and that always involves a certain amount of smirking or giggling because not only is "Adolf" a first name that has not been used for more than 65 year in Germany; Adolf's last name in German means exactly what it would mean in English if you exchange the "i" for another vowel.  
Adolf Fick, by the way, was not a physicist but a medical Doctor. In 1855 young Fick, employed by the University of Zürich in Switzerland, published a seminal paper in "Poggendorf's Annalen der Physik" (Poggendorf's annals (yearly records) of physics). The title was simply: "Über Diffusion" (About dffusion).  
How come a medical guy put down some
of the more important basic equations of Materials Science? Simple. The
diffusion of water through biological
membranes like cell walls is a basic process of life. However, it did not
attract much attention from the physicists in those times so Fick addressed the
matter himself. For about 50 years Fick's "laws" were empirical and not justified by deeper insights into the mechanisms of diffusion. It was Albert Einstein who derived Fick's law from the atomistic theory of diffusion. In a perfect world that would have been his second Nobel price (with about four more to come). In the world we live in, he only got one. 

So in what follows I will first work through Ficks's two laws and then tie them to the atomistic modelling of diffusion.  
Fick's First and Second Law  
First we need to define the term particle current density. That is actually quite easy. It is just another word for how much of something expressed as number of particles flows through some area in a given time. Since everything that exists can be broken down to particles, we can express any current, anything that flows from here to there, in terms of particles flowing from here to there.  
Let's look at some
examples:


Imagine yourself (or a person of your choice) to be a very tiny little devil who can sit inside a redhot steel rod, watching carbon atoms coming by. He or she imagines some area A(z) or reference plane at some depth z, measured from the surface of the iron, which is the source of the carbon, and starts counting.  
How many carbon atoms, running around at random
like the famous
drunken
sailor, happen to move through the area A(z) into the
depth of the iron, and how many happen to move back in the opposite direction?
The figure below gives some idea of what our little devil will see. 



The red spheres symbolize carbon atoms in iron
(or phosphorous atoms in silicon, or...); the arrows indicate the next jump
they are going to make. Only up/down and left/right jumps are indicated since I
can't do better in drawing this. The surface of the iron is on the left, and we
assume that there is always plenty of carbon available on the surface but don't
show that for clarity's sake. The little devil looks at a circular reference plane at some depth z. It's circular but only because that makes drawing easier. You are free to imagine any shape you like. The cylinder is meaningless and only there "to guide the eye". The way I have drawn this figure, we have a decrease in the carbon concentration c(z) going from left to right or with increasing z. That is the reason why on average more carbon atoms will jump through the reference plane from lefttoright (lr) than from righttoleft (rl). The little devil would measure a larger partial current j_{lr} than in the opposite direction. The net current is j = j_{lr}  j_{rl} 

Now that we know what particle
currents are, we need to wonder about the reasons for their existence. First
thing to note is that there are no particle currents in systems that have
achieved 

It is thus the local
particle concentration
that changes if there are (net) particle currents. Let's put that in a rule:
Particle currents cause concentrations changes. You and I have no problem seeing this. 

It took the genius of Adolf Fick to realize that the deeper truth is to express this the other way around. Cause and effect in the sentence above are related upside down. Turn it around and you have  


So what is a gradient? The difference of something like a
concentration c(x) here at the position x
and a little bit off at c(x + Dx). Divide by Dx and you have the gradient. In other words, a
gradient in one dimension is simply the slope or derivative
If you don't get this you must go back to start. I can't teach you elementary calculus on the side just so. Of course, if you should decide to send me lots of money, I might reconsider. 

In three dimensions the gradient is
written "grad" or
(read "napla")
and if applied to a function like 



The ¶/¶x,
y, z are simply the partial derivatives of the function, done
in the usual way for the particular variable, assuming the others to be
constants. Nothing to it. You also see that "havancd math" is to some
extent just the invention of smart abbreviations. The napla operator "" is an extremely powerful abbreviation. It makes equations short and easy to manipulate, saving a lot of writing 

Good old Adolf Fick figured that not only do concentration gradients drive the particle currents we call diffusion currents, but that these currents are simply proportional to the concentration gradients.  
Bingo, we have Fick's first law:  


D is the required proportionality coefficient and called "diffusion coefficient. The "–" (minus) sign is convention, it simply makes sure that the mathematical particle current flows in the same direction as the particles. That is not really required. Electrons, for example, flow in opposite direction of the technical electrical current because when the electrical sign conventions were made, nobody had the faintest notion of what was really flowing.  
Fick's first law states, for example,
that there is always a net flow of diffusing particles from the higher to the
lower concentration. Big deal! You would know that without fancy equations! Yes  but now we can calculate what happens! For simple situations and for complicated ones. For carbon diffusing into iron and diffusing out of it. And so on. 

Before I move on to the more
important second law, let me make one thing
perfectly clear: A concentration gradient is always what we call a driving force for particle currents, but it is not the only driving force there is. Electrical potential differences (vulgo called "voltage") drive currents of charged particles, pressure differences drive the flow of gas particles, gravitation drives the flow of water and pretty much everything else, and so on. Semiconductor electronics, for example, results to a large extent from the competition of the diving forces "potential differences" and "concentration gradients". Both are driving forces that drive currents of charged particles and thus electrical currents inside semiconductor devices, and they tend to drive the charged particle in different directions. The resulting balance determines electrical current flow as a function of voltage and particle concentration and thus the behavior of the device 

Fick's first law, in the money
analogy, determines how much money flows into or out of your account as a
function of money that's somewhere else. If your landlords account, for
example, contains less money than there should be, we have a kind of
concentration gradient that drives some money from your account into your
landlords account. But now let's move to Fick's second law. It makes a statement on how the balance of your account develops with time or in more general terms how local concentrations change with time. It is easy to derive this. All we need to do is to balance the account, subtracting everything that flows out in a given time from everything that flows in. Let's do that in one dimension for starters: 



Once more, drawing a cylinder is just for convenience. We are looking at the (differentially) small volume between z and z + dz. The change of the volume concentration with time, i.e. dc(z)/dt is simply given by the difference of what flows in,j(z)/dz, minus what flows out, j(z + dz)/dz.  
Why do I divide the currents by
dz? Because we have to convert the areal density of the current (per
cm^{2}) into a volume
density (per cm^{3}). If 5 people per minute keep flowing into a
room with 10 m length, the volume density in there is only half of what it
would be in a room with 5 m length. Ponder it, or believe me. If we write that down we get: 



We know j at any position from Fick's first law. All we have to do is to substitute j in the equation above by Fick's first law. If we do that (and go to three dimensions right away), we obtain Fick's second law:  


D = ^{2} is the delta operator, showing once more that smart abbreviations save a lot of work.  
Fick's second law simply states that the change in concentration somewhere in your specimen (¶c(x,y,z/¶t) in math lingo) that is caused by diffusion is proportional to the second derivative of the concentration there. The proportionality constant once more is the diffusion coefficient D.  
Of course, whenever we process a
piece of material by employing diffusion of something, we need to know how the
concentration at some point given by the coordinates x, y,
z changes with time t. The concentration is thus a function of four variables, Solving partial differential equations for some starting and boundary conditions given by the particular problem you are looking at, is considered an art. Fick's second law is a kind of partial differential equation that is notoriously difficult to solve. I'll give you a short taste treat below. 

Using Fick's Second Law for a Standard Diffusion  
Let's do a simple onedimensional standard diffusion. We have a piece of iron, put a lot of carbon on the surface with a concentration c_{0} that is always the same, heat up the whole contraption and watch the carbon diffuse into the iron. In other words, we case harden our blade.  
Or we do it with phosphorous and a
silicon crystal. In this case we need to know what happens with extreme
precision because we are now starting to make tiny electronic circuits. Or we do it with .... It's easy to come up with any number of examples because this is essentially the standard "experiment" concerning diffusion of something into something. 

We can easily guess on how the concentration profile of the carbon
inside the iron, i.e. the concentration as a function of depth, must develop
with time. The concentration will be highest close to the surface, decreasing
with depth. For longer times, the penetration of the carbon into the depth of
the iron will increase. We are going to find some carbon deeper and deeper
inside the iron if we keep the experiment running. We can confidently draw some quantitative diffusion profiles as shown below. 



Now let's calculate those diffusion profiles by solving the
differential equation called Fick's second law for the peculiarities of our experiment. This peculiarities
translate into what is called boundary and
starting conditions. We must
express them in mathematical terms  here they are.


It's those conditions that make solving differential equations hard. I spare you the gruesome details and give you the standard solution straight away:  


???? What the f... is an erf? Relax. It is just the harmless
error function. There are just far more functions out there, defined by some name and an abbreviation of that name (e.g. the familiar (?) sine (sin) and cosine (cos)) than you are probably aware of. The error function (erf) belongs to the group of stochastic or statistical functions, just like the more familiar "normal distribution" or "bell curve". There is no abbreviation for the "normal distribution" because we actually can write it out. In fact, the error function is a close relative of the normal distribution, defined by 



Thanks a lot! Well, I can't help it. Solutions of
diffusion problems via Fick's laws tend to contain statistical functions  as
well they should, considering that the underlying processes are
random walks or
completely statistical events. Anyway, the solution given describes exactly the kind of simple curves we came up with by guessing. Except now we have precise numbers. 

Enough of that. Let's turn to the
last problem in the phenomenological description of diffusion. That is, of
course, the diffusion coefficient D. We need a number for that entity if we want to do calculations. What has Adolf Fick done for us in that respect? 

Nothing at all. The diffusion coefficient for a
certain problem like A diffusing in B at the
temperature T, i.e. If you, like Fick, didn't know that things are made from atoms, and that those atoms jump around in a random manner during diffusion, you simply could not take a shot at the diffusion coefficient of a given diffusion problem. As we know now, it must somehow contain what is going on on the fundamental level because that has not been considered so far. 

Enter Albert Einstein (and a few others like
Smoluchowski).
They showed (indirectly, and in a way different from what is given below) that
the 1. Fick equation can be derived from looking at all this jumping around
and that we get an expression for the
diffusion coefficient in terms of atoms making jumps. That will be the topic of the last paragraph. 

Connecting Fick's Laws and Jumping Atoms  
Let's look at a cubic primitive element crystal as shown below. We are going to look a the (1dim.) net current j_{V} of vacancies moving through some plane P_{1} at the coordinate x. That means that the same net current of atoms moves in the opposite direction.  


For doing that, we need to know how
many vacancies P_{i} per
cm^{2} are contained in the various planes
P_{i}. In other words, we need to know the areal
density and not the volume density c_{V}(x). We
also allow that this density changes with x, i.e. that we have a
density gradient. For getting areal densities out of volume densities one multiplies c_{V}(x) with the lattice constant a. We have 



I have replaced an infinitely small
dx by a small but finite
lattice constant
a. Real mathematicians will recoil in
horror from am atrocity like this and will stop speaking to you  but you don't
want to talk to these guys anyway. In the world of crystals, dimensions smaller
than about a lattice constant simply make no sense, so we are perfectly
justified in doing this. Now let's consider the jumping around of the vacancies in the the two neighboring planes P_{1} and P_{2}. We are only interested in jumps in the xdirection since all the others do not contribute to the particle movement in this direction. The jumping rate we call r once more. Looking at it in detail gives us


Since in our geometry only 1 out of
6 possible jumping directions points in the desired direction (look at
the figure to appreciate this), we take only 1/6 of the total number of
jumps per second. In different types of crystal, this lattice factor g is somewhat different. For
fcc or bcc crystals we have g = 1. It is not all that difficult
to figure that out for yourself. Otherwise just believe me. The diffusion current of vacancies in one direction is simply given by how many vacancies will jump through the area per second, and that is the jumping rate times the number of vacancies per area that we have figured out above. We have, for example: The net diffusion current j_{x} of vacancies diffusing in xdirection then is 



Expanding the expression by a/dx, once more using a or dx as essentially the same thing, gives the final result  


Bingo once more! That is Fick's second law for one direction. Except that we have (a^{2} · r / 6) instead of the diffusion coefficient D. So that expression is the diffusion coefficient:  


This is a pretty momentous equation. Let's savor it bit by bit.  
1. The
"phenomenological" diffusion coefficient in Fick's laws is now fully
explained. It is essentially the jumping rate r(T) of the
diffusing particle that is doing a random walk. For vacancy assisted diffusion,
this is the jumping rate of the
vacancy, for interstitial diffusion it is the jumping rate of the
interstitial. Of course, the direction of the diffusion current of the atom
diffusing by a vacancy mechanism is opposite to that of the vacancy current.
The jumping rate must be multiplied by the lattice constant a, to take care of dimensions, and the lattice factor g, to describes the different geometries in different types of crystals. For a simple element crystals (one kind of atoms) we have g = 1/6, 1, 1, 1/8 for cubic primitive, fcc, bcc or diamond type, respectively. There is more. The diffusion coefficient contains:




In words: The diffusion length, a kind of average penetration depth of A diffusing into B, scales with the square root of the time t and the square root of the diffusion coefficient D. If you want your carbon to penetrate 10 times as deep into your iron, you must hold it 100 times longer at some high temperature.  
2. Since "thermally activated" diffusion, or jumping of particles over an energy barrier, always contains a Boltzmann factor and thus an exponential dependence on temperature, the diffusion coefficient changes dramatically with temperature. It thus does not make sense to give a number for it. What's needed are at least two numbers (E_{D} and D_{0}) or a graph, always in the Arrhenius plot as we know it from looking at vacancy concentrations.  
3. The diffusion energy E_{D} for a jump of a vacancy is not always the same number as E_{M}, the energy that an atom of the crystal needs to overcome in order to jump into a neighboring vacancy. We must expect it to be different if some foreign atom sitting in the lattice does the jump. If an interstitial jumps, E_{M} has nothing to do with E_{M} anyway.  
4. Besides the Boltzmann factor, we only have a bunch of constants (there are a few more and rather unimportant ones if one looks at the situation a bit more closely) like the vibration frequency n and lattice parameters. We simply merge all that to the "preexponential" factor D_{0}.  
5. Not to forget, Fick's
first law is now a law of nature. It is
valid in all cases where the basic ingredients (jumping at random etc.) are
given. For other cases, we know (in principle) what we need to do. Fick's second law follows automatically from Fick's first law if we assume "continuity". That simply means that our diffusing particles can neither be destroyed nor created. They do not suddenly disappear into nothing or appear out of nothing. That is certainly true for common atoms but not, for example, for particles like neutrons that decay radioactively into something else, or photons that get absorbed and generated all the time. In this case Fick's second law must be augmented, and it is perfectly clear how to do that. 

5. Just to be on the save side. Einstein didn't really do it as shown above. His reasoning  big surprise!  was far more abstract and mathematical. In essence, he explained socalled "Brownian motion"; which leads straight to "random walk theory", which leads straight to what I have shown above.  
Let's sum up! All of the above simply
means that for any diffusion problem of
some A running around in B via a a random walk, we need to
know just two numbers:


The rest "simply" involves solving Fick's law with the numbers provided for the diffusion coefficient D. I'm not saying it is easy. But I am saying that we know how it works.  
Let's look at a few examples:  


These are measured data of the
selfdiffusion in nickel, a magnetic
material. The red line is a perfect straight line and serves only to guide the
eye Note that the measured points follow it rather closely. The two numbers for
the migration energy and the preexponential factor can be extracted easily and
are given in the table below. By the way, Helmut Mehrer, the author of the source book I used a lot here, was the teaching assistant who way back around 1972 tried to teach me about point defects and diffusion. His book is presently the "bible" for everything concerning diffusion in solids. Of course, the data I took from his book were not all measured personally by him but represent the efforts of a large number of scientists. 



You noticed that I did not include data on iron or other bcc metals. There is a reason for this, just bear with me (or jump to the iron module right away). Before I finish this module, let me show you a wealth of diffusion data in a material where it really matters:  


What you see, for example, is that
the elements of supreme importance to silicon technology (boron (B), arsenic
(As) and phosphorous (P)) are diffusing far more slowly than metals like nickel
(Ni) copper (Cu ) or iron (Fe). That is too bad because we need to diffuse the
first three into the silicon and we must keep everything else out at all costs.
You also see that selfdiffusion, the running around of silicon atoms inside the silicon crystal, is the slowest diffusion of all. And of course you realize that those curves, measured and remeasured many times, show the results of many tricky, expensive and time consuming experiments that had been made over the years. So let's now look at how you actually do the experiments. 

Back to  
Diffusion  
1. Atomic Mechanisms of Diffusion  
2. Random Walk  
On to  
4. Experimental Techniques for Measuring Diffusion Parameters  
5. Diffusion in Iron  
TTT Diagrams: 1. The Basic Idea
10.5.2 Making Steel up to 1870
Units of Length, Area, and Volume
Experimental Techniques for Measuring Diffusion Parameters
Phenomenological Modelling of Diffusion
Size and Density of Precipitates
Atomic Mechanisms of Diffusion
Segregation at Room Temperature
Global and Local Equilibrium for Point Defects
Constitutional Supercooling and Interface Stability
© H. Föll (Iron, Steel and Swords script)