You are here: Material Science/Chapter 1/Diffusion

Simply put, diffusion is the movement of a substance (atoms/ions/molecules is what we’re mostly talking about here) from an area of high concentration to an area of low concentration.  More concisely, we can call it material transport by atomic motion.  Things in the natural world desire equilibrium, and will always try to correct anything that’s not in equilibrium.  That’s the way it is, and the way it will always be.

The motivation for this topic: diffusion plays a role in the mechanical properties of metals, so these concepts will be used later on.

Introduction to Diffusion

Say you press two different pieces of metals together with a high amount of force.  For example, copper and nickel.  You wouldn’t expect much to change right?  And that’s mostly correct, at least at room temperature. But let’s try heating the metals now.  And remember what heating really is: when you heat the metals, you are raising the temperature, and the temperature is really just a measure of how much the atoms are vibrating on average, or how much energy they have.  If you increase the temperature of the metal by heating, the vibrational energy of the atoms will increase.  How quickly do these atoms vibrate?   At room temperature, the vibrational frequency of a typical atom – how often they move back and forth –  is about 10^13.  And the magnitude, or how far they move, is about a few thousandths of a nanometer.  Don’t bother trying to visualize either the speed or the distance. Just remember that each atom vibrates very quickly and move extremely small distances (at least from our perspective). A metal melts when you raise the temperature such that the average vibrational energy of each atom is so large that the bonds to neighbouring atoms no longer hold the atom in place – they are ruptured – and the atom is free to move wherever it likes.

If we begin heating our pieces of metals past room temperature, the diffusion between the two metals will increase.  Some of the copper atoms will diffuse into the nickel bar, and some of the nickel atoms will diffuse into the copper bar.

Why does diffusion even take place?  As in, why do atoms move from higher concentrations to lower concentrations?  They don’t really do this intentionally.  It’s not like the atom knows where it is and makes a conscious decision to move somewhere else.  All atoms are vibrating and moving around in the material.  It is this “random movement” or random walk of the atoms that results in atoms travelling from areas where there are many (high concentration) to areas where there are few (low concentration).  If you let this happen for a long enough period of time, the concentrations will eventually balance out due to this random walk.  (At the risk of being pedantic, some clarification: the motion of single atoms/ions/molecules is not truly random – it may seem random – but really it is due to “collisions” with other atoms/ions/molecules.  For our purposes, you can imagine it as random).

Back to our two pieces of metal bars.  At room temperature, not much will happen, because the atoms don’t have enough energy to escape their bonds.  But if we heat these bars while pressing them together, then the atoms start vibrating with more energy – and a few of the atoms will be able to break their bonds.  Note that although the average vibrational energy of the atoms isn’t enough to cause the metal to flow (melt) there will be some atoms who do indeed have enough energy – more than the average – to break their bonds and move around.  And the atoms will move from areas of high concentration to areas of low concentration.  On one side, we have a large concentration of copper.  On the other side, we have a high concentration of nickel.  What we’ll end up with, then, is an alloyed region in the middle – equal amounts copper and nickel.  As we move towards either end, we’ll see more and more of the original atoms and less diffused atoms, since it is further away for the atoms to travel.  This type of diffusion, where one type of atom diffuses into another,  is called interdiffusion.  Or impurity diffusion (one atom being the “impure” atom).  Diffusion still occurs even if there is only one type of atom.  For example, if we take our copper bar and heat it, there will be copper atoms that are moving around and switching places.  Of course, nothing really results from this – we don’t end up with any type of alloy, just the same old copper.

If we drop down to the atomic level again for a second, we can examine diffusion more closely.  Diffusion is what?  Is atoms moving around.  Atoms aren’t just anywhere in metals, if you recall – they are located at specific lattice locations within the crystal structure.  Diffusion is atoms moving from lattice site to lattice site, and this is happening all the time within a solid.  The atom needs a couple of things before it can switch lattice sites.  First, it needs a place to go.  A lattice site next to it needs to be open.  Then, it needs enough energy to break the bonds that is currently holding it in it’s lattice position.  It also needs enough energy to squeeze through to that new position, because the movement of an atom will cause a minor distortion in the nearby crystal structure.  Conceivably, it could have enough vibrational energy that it’s bonds are ruptured, but doesn’t have enough energy to actually move to the new site because of the distortion that it will cause like trying to move through a crowd of people.

It’s also worth noting that at room temperature, a very small number of atoms will have enough energy to break their bonds and move around – just the lucky atoms that have ended up with above average energy for whatever reason.  Of course, as you heat the metal and raise the average vibrational energy, then more and more atoms will have enough energy to break bonds and diffuse.

Vacancy & Interstitial Diffusion

Let’s recall a couple of terms from the defect section: vacancy, and interstitial.  Vacancy diffusion is when the atom moves from a lattice site to a vacant lattice site – what we just talked about above.  As this method of diffusion requires empty lattice sites, the diffusion is limited by the overall number of vacant sites.  It’s interesting to note that the motion of the atom and the vacancy site is opposite.  The atom may move into an empty site to the right, and that empty site “moves” to where the atom used to be, to the left.  Of course the empty site isn’t so much moving as just switching spots, whereas the atom does physically move.  Both self-diffusion (another name is interdiffusion) and impurity diffusion generally move by this vacancy diffusion method.  In the impurity diffusion case, there is an impurity atom that occupies a previously empty lattice site.

Interstitial diffusion is when an atom moves from an interstitial site (if you recall, an interstitial site is the small space between the regular lattice positions in the crystal structure – so there isn’t really much room there, and so the interstitial impurity atoms are generally smaller than the “host” atoms).  Some regular users of this type of diffusion include hydrogen, carbon, nitrogen and oxygen, all of which are pretty small and can fit in these interstitial sites.  Host atoms rarely occupy interstitial sites; they are too big.  This type of diffusion is usually pretty rapid, since there are a lot of these interstitial spaces available if you’re small enough.  And since the atoms are small, they’re more mobile and can whizz around.  Usually the bonds are weaker too, so they are easier to break – less vibrational energy require to rip them apart.  So interstitial diffusion is quick and abundant, but usually occurs with impurity atoms – small atoms that aren’t the same type as the ones forming the bulk of the material.

Diffusion Flux

Generally, how quickly does diffusion occur?  This section in textbooks is usually pretty math heavy, but we’re just going over the concepts here.  You’ll agree that diffusion is a time dependant process, meaning that it takes time for these atoms to move around.  The amount of diffusion that occurs is a function of how much time has past.  If we took those two bars in our earlier example, pressed them together and heated them for just 1 second before pulling them apart, not much will have changed.  On the other hand, leave the heated bars pressed together for a few hours and significant diffusion will have occurred. We can measure this, and we call the amount of diffusion occurring the diffusion flux (flux is a fancy word that means flow – you could describe the amount of water flowing past a certain point on a river as flux).  Diffusion flux is represented with the letter J, usually.  In our river example, what exactly are we measuring?  We could measure the mass of the water flowing, or the volume – those are reasonable things to measure. With diffusion flux, we measure the mass or the number of atoms that are diffusing through a particular area per unit of time.  We need to know the cross-sectional area; a measure of mass per time would be useless.  Imagine if I told you that 100 litres was flowing down the river per second.  You’d need to know how big the river was in order to get a sense of how rapidly the water was flowing.  If the river is the size of the Nile, then the current would be very slow to only be moving 100 litres of water across any one point of the river per second.  Usually, the units of diffusion flux is the number of kilograms that diffuse over an area of 1 square meter every second.  If you prefer, you can switch out the number of kilograms for the number of atoms with some simple math.

Fick’s First Law

If diffusion does not change over time (i.e. the exact same number of atoms is diffusing for hours and hours without changing) then it is termed steady-state (literally, the state of the diffusion is steady).  How can we calculate the diffusion flux?  We need to introduce a couple more concepts – the concentration profile and the diffusion coefficient.

The concentration profile is simply the concentration of atoms versus the position within the solid.  It makes sense that if you are closer to the source of the diffusing atoms, or not very far into the solid, then the concentration of diffusing atoms will be higher – they haven’t had to travel as far.  If you move further into the solid and further away from the source of the diffusing atoms, then the concentration of the diffusing atoms will be lower. Simple.  We can actually take the slope at any point on this concentration profile to give us the concentration gradient for that particular point.  If the profile is linear (a straight line) then it is simply a matter of taking the rise over the run to get the concentration gradient. This concentration gradient basically tells us how quickly the concentration changes as we move further into the solid.

The diffusion coefficient, D, is a number with rather un-intuitive units that ultimately depends on the velocity of the moving particles (which in turn depends on temperature, the viscosity of the fluid, and the size of the particles).  We’ll leave that here for now.

What matters is that we can find the amount of substance diffusing per unit of area per unit of time with D and the concentration gradient.  This is known as Fick’s First Law. Fick’s First Law also says that the diffusing particles move from a region of high concentration to a region of low concentration.  That’s what’s important here.

If Fick’s First Law dealt with steady-state diffusion, then it is a good guess that Fick’s Second Law dealt with nonsteady-state diffusion.  And you would be correct with that guess.  Of course, nonsteady-state means that the diffusion flux, or the amount of particles diffusing, varies with time. For example, initially maybe there are many, many atoms diffusing – a high J.  Wait an hour, and maybe now less particles are diffusing – a lower J.  J, the diffusion flux, has changed with time.  You’d see this in the real world more often than steady-state diffusion, which is unfortunate, because steady-state generally means that it will be simpler, and the math will be easier (but you can safely read on because you won’t see any math on these pages).

Diffusion Summary

We haven’t talked about mechanical properties yet (that’s next) but here I can give you a practical example of why we just suffered through something as boring as diffusion.  Say you have something made out of steel, and you want this steel to be ductile – you want it to bend easily without it snapping.  Say that it’s a piece of cutlery, like a spoon.  A strong but somewhat bendable spoon is desirable, so that it doesn’t shatter when you try to scoop out ice cream that’s too frozen.  But ductility often comes at the price of hardness.  In other words, the spoon will be soft, which we want, but it will also be easy to scratch, which is bad. We want our spoons looking nice, not all scratched up.  What can we do?  Armed with the knowledge that carbon makes steel harder (to be discussed), we can put the spoon in a carbon rich environment – a controlled atmosphere rich in carbon. This is known as carburizing. Some of the carbon will diffuse into the spoon, but we’ll take the spoon out before the carbon reaches very far – just so that the carbon diffuses onto the surface, but not much deeper.  We can control this by understanding how diffusion works.  What we’re left with is a spoon that has a hard steel exterior, but maintains that softer interior, making it the perfect cutlery.

You are here: Material Science/Chapter 1/Diffusion
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