Lecture 23: Diffusion

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One announcement, there will be a weekly quiz on Tuesday based on homework nine, which will cover glasses and chemical kinetics. Speaking of which, that's something we started talking about last day, I found this cute quote, that kinetics is nature's way of making sure that everything doesn't happen all at once.

And, we looked at the rate laws, the most general rate law is shown here, the rate of consumption of a particular reactant goes has some function of its instant concentration raised to the power, , wnhich is the order of reaction.

The specific chemical rate constant, k, is the proportionality. And, k can be influenced by temperature through this Arrhenius type relationship, which compares the exponential of the activation energy with the available thermal energy.

And, we saw that we can integrate that rate equation, and put the data to the test. If we have a first-order reaction, we should get a semi-log dependence. And, I think I've got that shown here. Here's the data that I showed you last day.

This is just a normal decrease in concentration with time, typical attenuation curve. If we put it to the first-order test, according to this relationship we should get a straight line with a slope of minus k.

And, we saw that in the special case of first-order actions, it makes sense sometimes to talk about half-life in particular with radioactive decay. We also looked at second-order reactions, which have a slightly different functional dependence shown here.

And, then we went back and looked at the underlying physical chemistry of the activation energy. And, we came up with this model, which we presented in terms of mechanical analog. And, we saw that we could look at the decrease in chemical potential as not being monotonic but as requiring some initial activation.

And so, we have here to energy state of the reactants at one value, the energy state of the products at a lower value, and the difference between those values is the driving force for the reaction, the delta E of the reaction.

But, in order to get from the reactant state to the product state, we first have to activate. And physically, there are a number of things that are going on, and at the end of the lecture I'll show you an example with reference to an automobile exhaust catalyst, what is happening in terms of forming some kind of activated complex, which allows the reactants to reorganize themselves so that they can recognize that there is an energy to be gained by their reacting with one another, and how a catalyst facilitates this.

And specifically, we see that there is a tiny, tiny fraction of the population distribution that has enough energy to surmount the activation barrier and the inhibitor and the catalyst respectively either impede the reaction by raising the effective activation energy, thereby lopping off an even greater amount of the distribution, or facilitate the reaction by lowering the activation energy, thereby making a greater fraction of the thermal distribution available for participation in the reaction.

And, I think that's where we left it last day. And so, today what I'd like to do is to continue the whole discussion of what is broadly termed rate phenomena. So, chemical kinetics is part of a unit here in 3.091 that I'm going to call rate processes.

And, rate processes are basically processes that involve rate of change with respect to time. And, so one of course is chemical kinetics. And we just spent some time looking at chemical kinetics. But, there's another area that we want to touch upon, and it's an area called transport phenomena.

And, there's a reason for talking about this. When we think about how we're going to process materials in order to achieve solids with desired properties, we have to think about the two basics that are required to sustain a reaction.

And those are matter and energy. So, let's look at what's involved in each of these, matter and energy. We have to furnish these to the reaction. And so, if we're going to think about energy in terms of transport phenomena, that gets us into heat transfer.

But, we're not going to discuss heat transfer in 3.091. I can't do everything. Instead, what we're going to do is we're going to look at something that's representative of transport phenomena and we're going to look at mass transport.

And mathematically there are many parallels between heat transfer and mass transport. When we talk about mass transport, if we're trying to drive a reaction, we have to do two things. We had to deliver reactants to the site, and we have to remove products from the site.

If we don't do both, the reaction could either starve or choke. And, we saw a good example of that early in the semester when we talked about the Hindenburg. The Hindenburg was a good example of a reaction that had a huge driving force of hydrogen explosion.

But, the Hindenburg did not explode because we had 7,000,000 cubic feet of hydrogen. And, in order for that to react explosively, we would have had to deliver 7,000,000 cubic feet of oxygen instantaneously.

And we couldn't do that. We could not do that, so because the reaction was starved thanks to a transport phenomena problem, instead of an explosion, what we got was a burn. And, as a result, two thirds of the occupants of the vessel were able to walk off it.

And, not so many perished as otherwise would have. So, transport phenomena, clearly important. And, the dominant form of mass transport in solids is diffusion. So, when we're talking about mass transport in solids, we have to talk about diffusion.

And that's why we're going to do it. So, let's think of an example. Suppose I've got this wafer of silicon. And we've talked about doping. We've talked about doping. Suppose I want to render it p type.

So, what I want to do is I want to infuse this with a subvalent dopant such as boron. So, how do we do that? Do we smear this with boron and just wait? What's the process? Well, let's take a look more closely.

What we can do is we can take our wafer, and I'm going to exaggerate the dimensions. So, it's going to look more like a hockey puck than a silicon wafer. So, here's the silicon, and we're going to put this into a furnace.

And, we want to get boron into the wafer. And, we are not going to do this by melting. We're going to do this while the wafer is the solid state. That's why we're over here talking about diffusion.

It's a solid-state process. So, what we do is pass a form of boron over the wafer as the gas diborane. And, the diborane passes over, and on contact with the silicon is decomposes to form boron, which goes into the silicon, plus hydrogen.

And that hydrogen, of course, remains in the gas phase. And, we could even write a rate constant because this is a chemical reaction. This is the composition of diborane into hydrogen and boron, and we could even take a guess that the rate of this, I haven't measured it, but the rate would go as some relationship involving the concentration of diborane to some power, n.

So, we've seen that, but now I want to do is I want to ask, what is the process by which boron enters the silicon? So, now I want to zoom in here to the near surface. And here's the free surface of silicon.

This is the surface. This is inside the wafer. And, I've got boron that's decomposing and dissolving. Remember, it goes up substitutionally, sitting on silicon sites. So, now I'm asking, how does the boron enter the silicon? What are the kinetics of the ingress of boron? So, that's what we want to ask: rate of ingress of boron through silicon.

That's what we want to look at. And, this is not some kind of a cooperative process because the silicon is solid. So, individual atoms have to move step by step through because it's a substitutional solution that's forming here.

So, we can take a guess at it. If I now make this the coordinate system, call the free surface x equals zero, and this is moving down into greater depth. I can make a plot of ingress of boron into silicon.

So, this is x which is depth from surface. This is the free surface at x equals zero. And, I'm going to fix the concentration of diborane in the reactor. And so, the concentration of boron at the surface will be fixed.

As long as I keep the concentration of diborane fixed, concentration of boron at the surface will be fixed. So, what I'm going to do is plot the concentration of boron in the silicon as a function of depth from the surface.

And, we know deep inside the silicon, there's no boron. So, here it should be zero. We know at the surface it's constant. The question is, what is the shape of the curve? Just as I showed you last day, we talked about first order, second order.

What it boils down to is, what is the shape of the curve? And, we know it's going to look something like this. Doesn't that look a lot like the attenuation curve for the loss of matter in a chemical reaction? It starts at some constant value and trails off asymptotically.

So, really the question is, what is the shape? And, how do we answer that question? We can't look at this and eyeball it. Instead, what we do is we do a new mapping. Map concentration into some function of concentration.

Map depth into some function of depth such that I will get a straight line. And then I can, with the naked eye, say, I've got it. I know what's going on here. And so, therefore, the model that transformed c into f of c, x into g of x, then underlies the whole representation.

And so, the mathematical formulation of diffusion we owe to someone by the name of Adolf Fick. He was a physician. But he loved mathematics. And, he did to diffusion what Balmer did to spectroscopy.

Recall, Balmer studied the measurements that were made by Angstrom. And it was Balmer who fit Angstrom's data to that equation that ultimately we learned to be in concert with the Rydberg Equation.

What Fick did: he looked at Thomas Graham's. Thomas Graham had published some diffusion data for the rate of ingress of some materials into aqueous media. And, Fick was the one who was able to fit Thomas Graham's data and give us a model.

And, here is, let's jump through here. This is the cover page. This is the first page of the paper. This was published in Annalen der Physik und Chemie in Leipzig, 1855. And, here we are on page 59, "On Diffusion" by Dr.

Adolf Fick from Zurich. And it starts off as you've seen before. There's a little bit of background. There is references to the antecedent literature. And away we go. 1855: so, what did he tell us in this important paper? Well, let's take a look at his representation of what's going on in this process.

He was able to quantify this and give us a measure that has predictive capacity. And what he said was that the rate of ingress expressed by a term called flux, flux measures the rate of infusion. It's a mass flow rate.

So, clearly it must have some kind of units of mass per unit time, so I'm going to use SI units, kilograms per second, but diffusion is quantified on the basis of unit cross sectional area. So it's a mass flow rate of species, i, in, I'm going to make it in one direction, in the x direction, the way I've set things up in a previous sketch.

So, it's kilograms per second on a unit area a basis. So, flux is mass flow rate per unit area. So, the flux of species i, what he observed is it's related to the concentration gradient in species i.

It is the concentration gradient that dictates the rate of ingress. Gradient in concentration of i in the x direction. Later on, you can generalize this in three directions. You can replace the derivative with the del, and do it with three dimensions simultaneously.

The constant of proportionality is D, the diffusion coefficient or diffusivity. And, finally, there is a minus sign. Why? Because species move from high concentration to low concentration. So, if I look at the curve above, I expect that since the concentration is high at the free surface, that the flux is moving from left to right.

We would expect J sub i to move from high concentration to low concentration. But, look at the slope. The slope here is negative. So, in order to have a positive flux with a negative slope, we fix it, no problem.

Math is easy. You just put a minus sign. So, that's the genesis of the minus sign to make the coordinates come out properly. And, units, well, we know the units on concentration. Concentration must be in kilograms per cubic meter.

So, we've got that. D by dx, the dx, this x actually calls for a unit. So, that's one over meter. And, if I'm going to have kilograms per square meter per second on the left, and these are the units of dc by dx, then it follows that diffusivity must have units of length squared per time.

So, diffusivity is expressed in meters squared per second, concentration, kilograms per cubic meter, the derivative one over m, and the flux in the following units. But, there's more. Fick, in this paper, was so visionary.

He said, you know, this describes diffusion of matter into matter. But he says, this also bears a resemblance, are you ready for this? I said there are some parallels between heat transfer and mass transport.

Fick recognized in his paper that, you know, this is similar to Fourier's first law for heat conduction. He said, look at the parallel. This is the rate. This is the heat flux. Instead of a mass flux, this is heat flux.

So, this is joules per meter squared per second, goes as minus some thermal conductivity times the gradient in temperature. So, temperature is the potential acting on heat conduction the way concentration is the potential acting on diffusion.

And, this is thermal diffusivity. And then he says, why stop at two? Let's go for the hat trick. Let's go for three. He says, you know what? This is just like Ohm's law. This is just like Ohm's law.

Well, you know Ohm's law as V equals IR. But, you can write V as minus the gradient in electrical potential, minus the gradient in electrical potential gives you voltage. And then, I can write instead of resistance, I can write conductance.

So, I can map that into, since I know V, V equals minus that. I can then write Ohm's law as the current. See, this is a current. It's heat flux. This is a current. It is a mass flux. So, let's write Ohm's law not with potential as a function of current.

Let's write current density. This is amps, which is Coulombs per second per square meter goes as minus the conductivity G times d phi by dx. So, in one paper, he unifies electron drift, heat conduction, and mass transport by diffusion.

Not bad, not bad. And this guy was a medical doctor. So, it shows you that knowledge knows no boundaries. So, what is diffusion? Let's define diffusion as mass transport by random atomic motion. And, I'll make that point by the end of the lecture so you'll understand what it is.

So, what does it tell us so far? It tells us if we want to increase the rate of ingress, remember, we are still running this doping operation, I hope you haven't forgotten about our silicon fab line here.

We're still trying to dope this with boron. So, what it's telling us is that if I want to get a higher flux, I need to get a higher gradient, which means I need to have a higher surface concentration.

So, if I have a higher surface concentration, I will increase the flux by increasing c sub s, which means increase the pressure of B2H6. So, that's good. But, there's another way we can increase the rate of ingress.

We can increase the rate of ingress, we know from experiment, increase J by increasing temperature. We saw that with the chemical kinetics. Well, where's temperature? I don't see any temperature effect here unless you're going to say, well, concentration is density, and density is a mild function of temperature.

No, here's where all the chemistry is. It's inside the diffusivity because what's diffusivity going to be related to? If it's boron moving through silicon, diffusivity must be related to atom mobility.

And, atom mobility must be related to atom arrangements. And, what dictates atom arrangements? Bonding. And what dictates bonding? Electronic structure. You see, chemistry is not just a litany of silly little laws and rules and facts.

There's a unity here. So, the chemistry is all contained in D, and that's also where the temperature effect is contained. It should come as no surprise to you that the temperature dependence of diffusivity obeys an Arrhenius type relationship where we have some pre-exponential, which instead of A in honor of Arrhenius is D naught, they don't honor Arrhenius when they talk about diffusion.

They just put D naught, exponential minus Q, not to be confused with the heat flux. This is the term that they use: Q over RT. And, Q is an activation energy. It's an activation energy for diffusion, which means it's an activation energy for atomic motion.

And, you are normally going to see it RT. R is simply, not to be confused with the Rydberg constant, this is the gas constant, which is the product of the Avogadro number and the Boltzmann constant.

You're going to say, well have I know whether I'm supposed to put the Boltzmann constant here or the universal gas constant? Simple: the exponential must be dimensionless. So, if the units of Q are joules per atom, then you're going to use the Boltzmann constant.

If the units on Q are joules per mole, then you're going to use the gas constant. Otherwise, you're going to be off by a mere factor of six times ten to the 23rd. And, that should tip you off that something's wrong because you're going to get diffusion coefficients that are either blindingly fast, you would have atoms moving at a gazillion times the speed of light, or you will have five times the universe to dope the silicon with the boron.

So, that should tip you off that something's wrong. And, as far as typical values go, Q is on the order of about 2 electron volts or 200 kilojoules per mole when we're talking about diffusion in solids.

And, I do want to draw attention, I mean, clearly when we're talking about matter moving through matter, solid, it's got to be diffusion. In liquids, we can have cooperative motion, convection, stirring, but, you can have in capillaries and narrow fissures, you can have diffusion in the liquid state.

And, in the case of the liquid, the value is on the order of about a tenth of that, about 0.2 eV, which is about 20 kilojoules per mole in liquids. So, that will give you some sense. So that means, if we plot the natural law of the diffusion coefficient as a function of the reciprocal of the absolute temperature as we saw before with the Arrhenius plot for the specific chemical rate constant, you get a straight line with a negative slope because high temperature is to the left.

One over T increases to the right. And, the slope of this is minus the value of the activation energy for diffusion divided by either the Boltzmann constant or the universal gas constant. So, that's the story.

Now, I want to try to give some insight into what the activation process must be. What's going on at the atomistic level that we're trying to activate? And, for that, I go to this cartoon. We see the cartoon showing that this is now diffusion of metal in a close packed system.

So, we see an atom with a vacancy next to it. That's the first thing. The atom must diffuse into a vacancy. If there's no vacancy, the atom can't move. But, we know there have to be vacancies. We've already seen from earlier that it's impossible above zero Kelvin to have a vacancy free metal, or solid crystal.

And so, this cartoon shows the atom moving from the site to the left jumping into the vacancy, and eventually occupying the site to the right. And, there's a few things good and bad about this sketch.

Let's look at it from the top. Forget the bottom. The bottom is showing you interstitial diffusion. Let's just focus on the top. First of all, you've got to have a vacancy. This is important. You know that if you watch traffic, you ever been stuck in traffic? You are, say, the fourth car back from the traffic light.

The light turns green. Then you don't move. The light turned green. What's the matter with everybody? Well, it takes time for the first car to move to create a vacancy behind it. And then, the second car moves, creates a vacancy behind it.

The third car moves to create a vacancy in front of you. So, you drive into car vacancies. And, you can stand on a bridge sometime and look at cars pulling away from a light. And instead of watching the cars move, watch the vacancy move backwards.

That's what you have to have here. If you cease to wait for the vacancy, then you end up with what's shown here. And you see the atoms deforming. OK, now the atoms don't deform. This is nuts. This picture is incorrect.

So, what's really going on? What's really going on? This is the lattice at zero Kelvin. This is the lattice at zero Kelvin. What do we know is going on? Those atoms are vibrating. So, we have to consider a pulsating lattice.

And, we have to understand what the range of pulsation is. What's the amplitude of the pulsation, and what's the frequency of the pulsation? OK, so I need frequency, and I need amplitude. Because the squishy, these are not Nerf balls.

These are hard spheres. So, this cartoon is only partly correct. So, first of all, what's the pulsation frequency? Well, this is the vibrational frequency of a crystal is called the Debye frequency denoted nu sub D, the Debye frequency.

And, the Debye frequency at room temperature is on the order of about 10 trillion per second, ten to the 13th hertz, ten to the 13th times a second the atoms in all of us are vibrating. But, diffusion, if we work backwards from established data, we learned that the jump frequency, in other words, the jump rate, the jump frequency which is denoted capital gamma, is on the order of about 100 million per second.

So, that's a big number, but it still 100,000 times smaller than the Debye frequency. So, most of the atomic motion is oscillatory. Most of it is oscillatory. So, this is one attempt in ten to the fifth is successful.

So, this means that even if you have a vacancy next door, even if you have a vacancy next door, you will try 100,000 times to jump into the vacancy before you get in. And what has to happen? What has to happen is the two atoms on either side, the ones that are shown here as squashed, those two atoms, at least one of them has to be moving out at the moment you're moving in.

And, when that atom moves out at the moment you try to move it, you can squeeze through without having to result to the Nerf ball model. OK, let's not have any deformable atoms. So, that's what we are learning here, that we have this to wait for.

So, what does this mean? This means, then, that the activation energy can be broken into two components. First of all, we have to form the vacancy. So, we've got that. We learned that earlier when we studied defects.

So, there is an enthalpy of vacancy formation. You have to pay for that, pay to open up a vacancy next door, and then there's the energy associated with this fluctuation. And that's called the enthalpy of migration, atom migration.

And the first one is related to bonding because this is really, in essence, the inverse of the bonding energy. And, the second one must be related to structure. It must be related to free volume. And, here's a cute graph that illustrates this.

These are all FCC metals: lead, aluminum, silver, gold, copper, and iron. Iron is both FCC and BCC. But, we'll put it on the graph for completeness. And so, what's plotted here is activation energy.

And, it's a function of melting temperature, activation energy as a function of melting temperature. Activation energy for diffusion as a function of the melting point: well, we know that the melting point must be related to bond energy.

And, bond energy is related to the energy of vacancy formation. So, a low melting metal has a low activation energy for diffusion. A high melting metal has a high bond energy, therefore, a high vacancy formation energy.

There is such beauty in this graph. There is completeness here. It ties everything together. It's really so elegant. And, we can also look at diffusion by an interstitial mechanism. That's what's shown underneath here, same idea.

The only difference is with interstitials, there are so many interstitials, they exist. We don't have to pay to form the interstitials. The interstitials are there by virtue of the fact that we don't have 100% packing.

The closest pack structure we have is FCC. And, that is 74% atomic. So therefore, it's 26% free volume. So, we've got 26% to play with, and it's unlikely we've got interstitial loading to consume all of that.

So, in the case of interstitials, the Q is simply equal to delta H of migration in the case of interstitials. You don't have to pay for formation of vacancies. So now, let's get to the question of random atomic motion.

This is what will convince you that we have random atomic motion. Let's look at a really clean experiment. Let's look at this one. We talked about radioactivity last day. Suppose I make a diffusion sandwich that consists of the following: I'm going to put cobalt, cobalt, and cobalt.

It's all pure cobalt. The only difference is the center of the sandwich is going to consist of radioactive cobalt. And, the two layers of bread are cool cobalt. So, there's no concentration gradient here because you know that radio isotopes behave identically, chemically.

The fact that there's radioactivity does not affect the chemical bonding capabilities of cobalt because it doesn't affect the electronic structure. And as far as the mass effect goes, it's so tiny that it can be disregarded.

So, what happens over time? If there's no concentration gradient, according to Fick, there should be no diffusion But what happens, as this cartoon shows, if we have a center band here that consists of a mix of radio cobalt and regular cobalt, over time we find that the cobalt is uniformly distributed throughout the specimen.

And, if we stop action and measure the rate of dispersion of cobalt through the specimen, we find that the rate is described by Fick's law. So, we have random jumping at the atomic level in accordance with Fick's first law.

So, this is a demonstration that we have things occurring at the atomistic level involving random jumping. And, to make the case more potent, the actual curve here obeys a Gaussian, which is exactly the random walk problem, you know, the drunken sailor walks out the door, takes one step or two steps, and so the chances that he gets five steps away are vanishingly small.

The chances are that he falls after one or two steps are higher. So, you will get a Gaussian curve here for the distance at the cobalt has gone as a function of time. So, what's wrong with this cartoon, by the way? We've talked about this is cobalt jumping through cobalt.

Where does cobalt have to jump to? It has to jump into a vacancy. I don't see any vacancies here. So, the chances of this cobalt jumping and exchanging with the adjacent cobalt, now we're not talking about a tiny fluctuation to allow cobalt to squirt into a vacancy.

We are talking about a fluctuation like this to allow two cobalts to crisscross. And, I mean, that's just not going to happen, maybe once in the life of the universe. But I don't think it's going to happen.

So, we really need to have vacancies in here. And, ultimately we have to recognize that diffusion is heavily influenced by defects: no vacancies, poor diffusion. Many vacancies, better diffusion. So, let's take a look at some evidence of this.

This is a plot of diffusion coefficient is a function of temperature. It's an Arrhenius plot in accordance with this curve here. And what we have is for a variety of impurities in solid iron. So, let's look up at the top here.

This is hydrogen. And you can see that the diffusivity of hydrogen increases with temperature up to the temperature at which iron converts from BCC to FCC, and then there's an abrupt drop in the diffusivity of hydrogen.

What's happening? What's happening? What's the void fraction of BCC versus the void fraction of FCC? FCC is more tightly packed, and so we see an abrupt fall. Here's carbon; same thing. The diffusivity rises up to the transformation temperature, falls, and then continues to rise.

And, note the difference in the slope. The slope of carbon in BCC iron is gentler than the slope of carbon in FCC iron. Well, what's that mean? It means that the slope in FCC iron is this upper value.

The slope in BCC iron is the lower value. You don't have to pay for the formation of vacancies. Here's the self diffusion of iron in iron. Look, it drops two orders of magnitude when it moves from BCC to FCC.

And, just to make the point, this is carbon in graphite. This is self diffusion of carbon in graphite. What's the problem there? Where do we see that as operating? Well, we said that delta H vacancy formation is related to bonding.

So, here you see the comparison between metallic bonding, which is what you have in iron, so self diffusion of iron if I were to project this curve up to the temperature, so we can compare at the same temperature, we see seven, eight orders of magnitude difference between self diffusion of iron in iron.

That's a metal in a metallic crystal, versus carbon in graphite, which is carbon in a covalent crystal. So: big difference in performance. Now, there are other ways of looking at it. Let's look at other defects.

How do other defects affect? So, here's a cartoon showing the diffusion of some solute into a material, only this is a real material. It's a polycrystalline material. So, there is a grain boundary.

And, as you know, grain boundaries have a little more openness. The atoms at the grain boundary have fewer nearest neighbors. Well, if what you are trying to do is to jump to that little narrow saddle point waiting for those atoms to pulsate, in a grain boundary, you don't have to wait.

The atoms are farther apart. So, it's like the difference between having to drive down an expressway, or drive down a narrow city street. You've got a much easier time moving quickly down the expressway.

All right, so we can see what the cartoon is trying to show is that up at the top, we see the rate of ingress of this solute through the bulk, or we say it is lattice diffusion, classical lattice diffusion.

Down the grain boundary, at the same time, we see a much deeper penetration because the material is able to move with less constraint. And, along the surface, it can move very readily. Surface diffusion is very, very rapid.

And, here's some data. These are three different investigations of diffusion of silver, self diffusion. OK, this is silver diffusing through silver. So, this is classical. What they call volume diffusion is what we would call lattice diffusion or bulk diffusion, single crystal, silver moving through silver.

How would you make these measurements? What's the only way you could identify one silver from another? You have to use radio tracer. OK, so using radio tracer, they're able to get self diffusion coefficients, and you see, what they've done is they've flipped this around.

This is very wimpy. This is so wimpy; look at what they've done. These people, the people that plotted this, they're such sissies. Look, do you have any trouble figuring out that temperature is increasing in this direction? I mean, you can think in one over T space.

This is how I feel comfortable. Why look in an Arrhenius plot, I want a negative slope and a straight line. And, what they did is look at the numbers. See the numbers? They're ascending from right to left because they're so chicken that they're afraid to have temperature rising from right to left.

So, what they did is they plotted the diffusion coefficient, natural log of the diffusion coefficient. But, they've got one over T rising from right to left so that they'll get positive looking slopes on their lines.

It's very girly. I don't like this. I don't like this. OK, so look at this. Now, here we have, first of all, a very steep slope. So, that indicates a higher activation energy because we have a higher lattice energy with which we have to deal.

Here's the grain boundary. The grain boundary, first of all, it's several orders of magnitude higher, and the slope is gentler, indicating that we have less constraint. And, finally up at the top, we have the surface diffusion which has, and I draw your attention to the values.

Here, we are down to around ten to the minus eight, ten to the minus nine, ten to the minus ten in units of centimeters squared per second. Look at the surface diffusion. Ten to the minus five, ten to the minus four, here's diffusion in molten ferrous alloys, ten to the minus five, ten to the minus four.

So, what we've got here is diffusion across the surface is on a par with diffusion in a liquid. So, it gives you a very vivid demonstration of how there's a relationship between the physical constraints of the lattice and diffusion coefficients.

So, we can capture that by saying that the activation energy for what I've been calling bulk diffusion or lattice diffusion, this is classical vacancy stuff, OK? Vacancy mechanism, OK, via vacancy, this activation energy is much greater than the activation energy for diffusion along grain boundaries, which is much greater than the activation energy across a free surface.

So, what do we see? The trend in moving from left to right is decreasing coordination number, which means more freedom. And, as Q goes down, the value of D goes up. So, all other things being equal, this is for, let's compare apples to apples.

So, this is for self-diffusion, silver in silver, iron in iron, what have you, the diffusion coefficient for lattice diffusion is going to be less than the diffusion coefficient for grain boundary diffusion which is less than the diffusion coefficient for surface diffusion.

And so, things are making sense. And finally, to make the point, it even applies in amorphous solids. So, I want to show you some data. This is helium entering different glasses. We just finished the unit on glasses.

So, let's put it all together. So, what people have done here is measure, they're using a permeation velocity. For all intents and purposes, the ordinate is proportional to the diffusion coefficient.

So: high permeation velocity, a high value of permittivity, K, is the same as high value of D, diffusivity. And, what do you find? First of all: Arrhenius type behavior, and secondly you see the most rapid infusion of helium is with fused silica, Vycor.

As we come down here, we see more and more modification. Down here we have soda lime glass. So, let's pick a constant temperature, let's say, 200°C. Fused silica is strictly network former. Borosilicate has network former plus some borate intermediate.

Down here, we have soda lime, which consists of network former plus a fair bit of sodium oxide. So, the chain lengths are getting shorter. We have tighter packing. The void space is smaller. And, down here, we have lead borate.

Lead borate has a huge amount of modification. It's a euphemism for crystal. So, we have lead crystal down here. We have amorphous silica here. At constant temperature, we can see this is ten to the minus eight.

This is ten to the minus 14, a million times difference in the coefficient for helium entering the glasses at the same temperature. So, again, the message is, as the channel is less and less constrained, you have a higher and higher rate of diffusion.

And so, what they did here is they plotted, as a function of the amount of, these are all network formers. So, as the fraction of network formers rises, you get a more open structure. As the fraction of network formers falls, that's to say the chain lengths are getting shorter.

You've got tighter packing. So, this is beautiful. It's an isotherm showing the rate of helium ingress as a function of amount of network formation. And, here's the last one I'll show you. I couldn't resist.

These are so beautiful. Look at this. This is now the same glass, but instead of asking, what's the rate of helium? Let's compare: same glass, same structure; what's the rate of helium ingress, neon ingress, nitrogen ingress, and what do you find? The larger the species.

And, this is H, but in fact it's H2. So, what do you find? The larger the species, the greater the activation energy for ingress of the species into the same glass. So, this is changing the size of the diffuser and keeping the size of the channel constant.

This one here was keeping the size of the diffuser constant, and changing the size of the channel. So, everything is making sense. So, I wanted to give you an example showing how the catalyst works.

And so, I'd like to talk about catalytic converters in automobiles. This is the number one market for industrial catalysts: catalytic converters. A little bit of background. Gasoline is a mix of hydrocarbons, but the dominant one is octane here: C8H18, which we burn with air.

And air, of course, is only 20% oxygen. It's roughly 80% nitrogen. So, when we burn gasoline, we make not just water vapor, carbon monoxide, carbon dioxide, but the air at those temperatures forms various nitrous oxides.

And these are, in fact, the precursor to smog. So, back in the early 70's, it was decided that in order to improve urban air quality, we need to convert unburned hydrocarbons. Some of the gasoline actually doesn't end up being burned.

And, some of it's completely burned to carbon monoxide. And, of course, there's nitrous oxide. So, here's what the mandate was: to convert the hydrocarbons to water vapor and carbon dioxide. I know carbon dioxide is implicated in greenhouse gases, but believe me, if you've got a choice of carbon monoxide or carbon dioxide, I recommend you go with carbon dioxide.

So, we'll solve the small problems first. And then, nitrous oxides convert then to nitrogen, oxygen, and CO2. Now, look at the nature of these three reactions. The first two involve oxidation reactions.

The last one involves a reduction reaction. Now, if I told you I want to make a single reactor in which you should simultaneously conduct three different chemical conversions, two of which are reductions, and one of which is an oxidation, in the same place at the same time, you would say, you're nuts.

You can't do that. Well, that's the challenge that was presented to the auto industry. And, on top of that, some pinhead in the government decided to invoke the Sherman Antitrust Act, which was designed to prevent people from colluding on price-fixing.

But they invoked it in colluding on technological developments. So, all of the auto companies were obliged to work independently. So, GM, and Ford, and Chrysler, couldn't pool their resources. GM alone tried 1,500 formulations.

Why did they try 1,500? Because people don't understand the basis science of catalysis. And, when you don't understand, there's only one way to proceed. It's called by inspection, cook and look. It's very costly.

And, finally, this is what they settled on: platinum palladium for the first two reactions, and rhodium for the last one. Basically, you go to the most expensive part of the periodic table. And, that's the formulation.

Tom, if we could go to the document camera, this catalyst is put on a substrate. So, we want to use as little of the precious metals as possible because it's a surface reaction. What's happening is the catalyst is sitting here.

So, this is platinum, rhodium, and palladium. And, what happens is it provides a site for selectivity, absorption of CO's. So, CO's can absorb as follows as opposed to in the gas phase, two of these molecules are zooming along at high speed.

And, it's not good enough for them to just collide. They actually have to be close enough together for a long enough period of time for bonds to decay and form. If we adsorb, they can sit side by side long enough for something to do this.

And then they have to desorb. You have to adsorb the reactants and desorb the products. So, this is now a ceramic catalyst. You can see it's been extruded like pasta and then fired. So, all up and down these honeycomb channels, we put a dilute wash of acids containing platinum, rhodium, palladium, fire because we want the thinnest, thinnest layer of the precious metal all up and down this ceramic catalyst.

So, that's what's inside. If you look up underneath the car, you'll see the exhaust tubing, and it will flare out to something about this size. And then, it goes on to the muffler. So, this is the three-way catalytic converter.

And then, Tom, could we go back to the computer, please? And then, hand and glove with this came unleaded gasoline. We couldn't have catalytic converters with leaded gasoline because leaded gasoline contains, as the name implies, lead as tetraethyl lead shown here with this compound.

And, one of the combustion products is lead oxide. Well, lead oxide you've met before. It's a modifier. But, it's also volatile. It's volatile. So, it will go down through the exhaust stream to the catalytic converter.

And, it is converted as NOX is to elemental lead. And then, it decomposes. So, now you have elemental lead, and elemental lead alloys with platinum, palladium, and rhodium. Now, what do you think the catalytic value of lead is? Well, let me tell you.

Given the difference in price between these three and this, if this had any catalytic value, we wouldn't be using these. So, what this does is it poisons the catalyst, and after several tank-fulls of gas, renders the catalyst useless.

So, the advent of the three-way catalytic converter was accompanied by the advent of unleaded gasoline. Have a nice weekend.

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