 OK, I think we can get started. So when we left last time, I had introduced you to this Tafel relationship, which I'll remind you dates back to 1905, which I'm taking as the beginning of modern electrochemistry. Big breath of history here. And what I was starting to do was develop some sort of theoretical concept that would get us from simple ideas about rates of reaction to this equation. That is, why did Tafel see this? Excuse me. And it starts off with a simple realization that electrochemical processes are first order processes. Like every other chemical reaction that's a first order process, it'll follow a simple differential equation. And it will be controlled eventually by an energy of activation or, more correctly, a free energy of activation. We're all familiar with that idea. We need to throw in a slight new wrinkle I was mentioning at the end of the hour last time. And that is, if we take our total energy of activation as delta G double dagger bar, and I'll write this specifically for the forward going reaction, recall that the reaction we're talking about now is the oxidized species being reduced to the reduced species, which is our standard reaction. If we're looking at the species in for the forward going reaction, oxidized going to reduced, then we're going to say that that is equal to a standard energy of activation for the forward reaction plus a second term, which has to do with the fact that we're moving electrons around and therefore we have an electrical potential and electrical work that we're going to deal with. So we're adding on this new term. This new term will be related to the number of electrons that flow in the circuit and the potential that they have to flow through. So in other words, we're going to have to change a sign here if you recall from last time. We're looking at a change in potential times fair days constant to get us converted from moles to rather Coulomb's to moles times a parameter here alpha, which I'll leave for one second. I'll tell you yes. So this is just a regular chemical one. OK, in fact, let's do it this way. All right, there's up here, this one. Thank you, yeah, there's a bar up there, yeah. So in other words, this is some of these two things. I'm sorry. Yeah, so if we are going to look at this delta G bar, now it's an activation energy, but in terms of reaction coordinate, we want to end up with a picture that looks something like this, where we're going to start off with our oxidized species and we're going to end up with our reduced species. And we're going to go over this barrier in the forward direction, which is just that delta G double dagger forward bar. And what I'm suggesting is you could break that down then into two components. One that is the normal delta G double dagger that you would have for a reaction. And that particular reaction in this case may not even have a negative delta G associated with it. So this is our delta G. And so we have a forward reaction here, when again from ox to red. And it may turn out that the free energy of the oxidized species and the free energy of the reduced species are about the same. It may even turn out that the free energy of the reduced species is higher than the oxidized species. That is that you have a positive value for this delta G, because we have a second term here that we can use to drive this reaction. And so in the case where this doesn't turn out to be net negative, and I'm just drawing sort of something close to 0 there to make the point, we can drive the reaction and get a negative delta G over here by having this term kick in and help us out. So there's going to be a second term over here, which we need to draw a picture of, which is going to be this delta G for the electron part of the process. And let me leave that picture blank for a second. I'll come back and fill that in. But how are we going to handle that picture and how are we going to handle it? The reason I want to leave it blank is without some idea of where that comes from, I can't draw anything. In classical theory, that is just the symmetry factor of the barrier. That is, you get to the top of this barrier, and it may not be exactly symmetric, and there's a possibility of it falling to the right or the left. And normally, experimental chemists don't have any idea what that number should be, so we like 0.5 is a good number since that has to go between 0 and 1. So 0.5 can't be too far off, totally symmetric barrier. But another way of looking at that in this case is we can say that this free energy for the forward reaction of activation of the electron part is going to be proportional to the total free energy change for the electron part. And we'll use alpha as our constant proportionality. So again, alpha is somewhere between 0 and 1. And then by that same argument, if that's the case, and I have to go over some kind of a symmetric barrier, then I can also state for the back reaction that that must be minus 1 minus alpha delta g of the charge transfer part of the process. So now I can finish my picture anyway by doing that. You'll notice if I do that, and I'm planning this now versus the total free energy, then I'm going to have some sort of statement like this for the activation barrier, where that delta g double dagger back is this energy, and the delta g double dagger forward is that energy for the electron. So I'm saying to add those two together, you're going to get this. So I can take a reaction that doesn't have any chemical potential to drive it. And I can couple in the potential of the electrode to give me a reaction that goes in the desired direction. OK, that's the picture. Now how do we actually do this? We're going to start off, and I think I'm going to do just to make sure I don't run out of board space. I'm going to take a second here and give myself a clean board. We're going to start off with our oxidized species, ox. And we're going to point out that it might start off with a charge on it of z. And we're going to obviously add n electrons to that. And we're going to end up with our reduced species. And that reduced species will have a charge of z prime. And the other comment to make about this, that's sort of explicit in this, is those electrons, before they've gone into the reduced species, are sitting in the electrode. So what I'm really talking about here, if I'm talking about this in terms of free energy or chemical potential, is a free energy of the electrons in the electrode. So there are n electrons sitting in that electrode that are going to come out at some point and change free energy when they do that. So if I look at the absolute free energy then for this process, that is for the oxidized species, g bar, ox, then that is equal to a chemical potential plus the potential of the electrons. So I can define a chemical potential, mu sub e, for electrons. Chemical potential for this species. And overall, that gives me the free energy for this system. I could then relate that back to some sort of a standard state. And you know for this species then that there's some standard state mu ox knot. Plus a term that takes into account the fact that this molecule with some charge on it, z, is going to come out from a solution where it's not experiencing a field and move up to an electrode. So it moves into an electric field. And so I have a term when this molecule is near the electrode that is the charge on the molecule times Faraday's constant times the potential out in solution. That's for a solution. So that's just this first term. And then I have a second term over here for the electrons, which would be the standard chemical potential of the electrons in the metal, in the electrode, minus a term that takes into account the fact that they are experiencing a potential on the metal side of the interface. So that's my oxidized species, my reactants on the product side. Of course, I have a total free energy for the product, the reduced species, which is a little simpler. That's just the standard free energy for the reduced species plus the electric field effect on whatever charge it has. Let's keep my phi as constant here, and it's out in solution. There's no charge transfer on that side of the equation, so all I have is the fact that I have some molecule with some charge, z prime on it, sitting in an electric field. And then, of course, I have a delta g for this overall reaction, which is just products minus reactants. So I have a delta g not for the products we have for the electrons, let's see, which is just product electron minus reaction or electron. So that is for the potential parts, the difference between z prime and z times Faraday's constant times this potential plus n Faraday's constant potential of electrons in the metal. And I'll point out to you that the difference between the charge after I carry out the redox reaction and the charge before I carry out the redox reaction is just exactly the number of electrons n that had to flow to carry out that redox reaction. So you have a term n there. And so this is just n Faraday's constant times the difference in potential between the metal and the solution. Now we can think about the electrode potential in general. So I have an electrode sitting at some potential, e. And that potential, if I'm going to carry out this reaction, must just be, obviously, the difference between the potential of the metal part of the electrode and the potential on the solution side of the interface. So that difference in potential. Plus I have to throw in a constant here because we all have a different idea about what the zero of energy ought to be. So in other words, this is a relative difference. If I want to say I will define the zero of energy as the vacuum level, then k is going to be the energy that takes us from our redox potential up to the vacuum level. If I want to use a reference electrode and set that as my zero of energy, then k will be the difference between the potential of the reference electrode and the reaction of interest. So all I'm simply saying is I'm dropping electrons through a potential gradient at an interface there. I'm going to change potential when I go through the interface. And that defines the potential, if you will, of the electrode with respect to the electrolyte that it's sitting in. So then given that, we have that therefore the change in free energy for the electrons is equal to just n, there it is, constant times this value of E minus k. And therefore, going back to my original distribution of the potential here, we're out of boards. Time to take a break and erase something. Yeah. Thank you, yeah. Now you can tell somebody was really on top of this. We'll leave Taffel's relationship up there since we're trying to get there eventually. Very good. OK, so going back to my original distribution, then what I'm simply saying is that if I have a free energy of activation, this bar now, let's get rid of that bar. Make it into a box there. If I have a free energy of activation for the forward reaction, just the electron part, that is alpha nf E minus k. That is, that's the total difference in free energy for the electrons. And I'm saying it's going to be distributed so that some fraction of it is associated with the forward reaction. And some fraction of it, the remaining fraction, is associated with the back reaction. And so then taking that idea and putting that into a context of absolute rate law, then we're saying that we have a forward rate constant, k sub f, which has to do with a frequency factor, kt over h. Those two k's are not the same k's, right? This is a rate constant. That is the energy term, Boltzmann's constant. E to the minus delta g double dagger bar forward over RT. So this is just the equation I wrote out for you at the very end of the time last class. And substituting in now our understanding of delta g double dagger bar, we have this pre-exponential factor times the exponential, which has two terms in it. And I'm doing this per mole now. So we'll point out this term is per mole, so it has an r in it. This term is per molecule, so it's based on k delta g double dagger forward for the chemical component plus delta g double dagger forward for the electrical component. And that is equal to kt over h bar times an exponential that takes into account the chemical free energy change of the reaction going from the starting material up to the activated complex times a term substituting in now that takes into account whatever zero of energy we want to use to say what the energy electrons are when they're sitting in the metal times this term, which is going to take into account the fact that the electrons are moving from the metallic interface out to the electrolyte. So if I fix, well, even if I don't fix the chemical reaction, if I take a random chemical reaction, don't tell you what it is. In fact, I'm going to look at five or six or seven different chemical reactions. As long as I tell you that we're going to operate under a certain condition of temperature, we're going to be in the lab or something like that, you'll notice all I have here is a bunch of constants. So this is just a constant. It's an adjustable factor. It only depends on temperature. And it's just setting what my zero is going to be. So we can ignore this term, basically. It comes along for the ride. If you want to take this term and move it out in the front, essentially, and make it part of the pre-exponential stuff, that would be fine. It won't affect anything at all. This term is going to depend on exactly what I want to oxidize and reduce. That is, that term is either positive or negative, depending on what I choose to oxidize or reduce. That's molecule-specific. Of course, once I've picked a molecule, that term is fixed. Or I should say, a redox couple, really, two molecules. And then finally, I have this term over here. And this term is the term that depends on the electrode potential. So this is where this whole concept comes from, that the electrode potential, when you start off in sort of zero-thorough thinking, is a thermodynamic parameter. It has to do with just the change in free energy. But in fact, when you start to work out the kinetic scheme here, it comes right out of the energy of activation. So it's also a kinetic parameter. That is, I can change the potential barrier. That's what my little picture was before of an electrochemical reaction simply by changing the potential of the electrode. It's a flexibility that you don't have in any other sort of chemistry, really. You're not allowed to play with that barrier. I could write this in a slightly more compact form and just say that there's a forward reaction. Ray constant has to do with the heterogeneous charge transfer that is moving an electron from the metal through the interface out to the electrolyte. So that's what that s has to do, heterogeneous charge transfer, which is equal to a rate constant. Let's get the right rate constant here. I did it right. It's equal to a rate constant for that charge transfer that is potential independent. It just has to do with the difference in free energy between the oxidant and the reduced species, times any constant, such as these guys that we have to throw in there. So a standardized rate constant that has no potential dependence to it, times an exponential term, e to the minus n F e divided by RT. So there's my explicit potential dependence now on the rate constant that I have an electrochemical reaction. And if you realize that that rate constant is very closely related to the current, remember, the current is the rate of the reaction, and the rate constant obviously scales with that, or I should say the current scales, the rate scales with the rate constant, you can now see how we end up with an interaction that looks like that. So you can see where Tafel ended up with his relationship. Now what we really have to do to get all the way to Tafel's relationship is not just consider that there's a forward reaction, but I've ignored the back reaction here obviously. I worked it out up to this point, and I can write another equation obviously, which is a rate constant for the back reaction. I'm not going to write it out because the flum's going to be very similar, but just let me write down here that it exists. Not like that it doesn't exist. Got all the letters wrong there, I think, except for the k. Let's see. Put the b down there for back reaction, and the s up there for the fact that it's a heterogeneous charge transfer. And the only difference is going to have a term of 1 minus alpha up here obviously to take that into account. So then if we want to write down, actually take another break here and erase the board first before we write anything else down. If we want to write down then a current expression for this whole thing, we could say that, yes, sure. Yes, that's right, because the delta g double dagger will be a different number for the back reaction. Absolutely, yeah. Maybe I should have written out another equation. But yeah, we're changing. We have two differences between these two equations, the one I wrote and the one I didn't write. There is a possibility that alpha is not 0.5, and therefore going over the barrier and going in one direction and going back over the barrier in the other direction are not the same process. That is alpha and 1 minus alpha are not the same. So that would change this exponential term. Although to be honest, even though there sometimes is an asymmetry, and in most cases we can't measure it, it's not very big asymmetry. So maybe alpha is 0.4 in one case and 0.6, and no one could ever tell you whether that's right or wrong. Anyway, so we're talking about too much. The big term actually is to turn that Bruce pointed out that this rate constant, remember, goes back to that free energy term for the chemical reaction. And that could be very different. In fact, one would guess it'd be very different for the forward and back reaction. OK, so if we're going to look at the current now then, we have a current that we might consider under conditions first, where there is no applied field. That is, let's get rid of this e term for a moment, and then we'll have a current I0, which is just going to be the number of electrons that transfers. Fair day's constant. The area of the electrode, this rate constant under zero field conditions that we've been talking about for the forward reaction, times the concentration of the solution species. And I'm going to add a new terminology there that Bard uses, which is this star. That refers to the concentration of the solution species in the bulk of the solution. I mentioned to you that since this reaction only takes place at the electrode surface, you're going to get a concentration gradient going from the bulk concentration up to whatever it is at the electrode surface. So we're going to normalize everything to the bulk concentration. And then in the case where we are going to apply an electric field now, we're going to have an exponential term, either minus n alpha n, fair day's constant, over RT, times this potential, which I'm going to write a little differently now, as the difference between whatever potential the electrode is at at equilibrium in the solution minus a standard redox potential at equilibrium. And for the specific case where there's no potential dependence, what I'm simply saying is I'm sitting at a 0 there. That is, if I take my electrode potential and I move it so that it's the potential that the Nernst equation tells me is the potential of the solution species under those conditions of concentration, temperature, the parameters of the Nernst equation, then I'll get this i not out. It just depends on the concentration solution. I will not have a gradient, in other words, under those conditions. There will be no charge transfer taking place on that equilibrium or very close to it. Now that equilibrium, you can't see it in this equation, by the way, because I'm only writing the forward reaction. And of course, in the back reaction, that equilibrium I'm balancing out. So although I'm talking about a current here, this is not a real current that I can observe, because there's an equal current at equilibrium that makes this total current 0. Question? This is yes. This is all in the exponential. Let's see. I should show that a little more carefully, right? Thank you. OK. I'm going to use this now. There's a lot of little constants that are running around there. So I'm going to use this as a normalizing term. That is, I don't want to have to write all this NFA, RT stuff all the time. So when I look at a current now, some other current, let me just normalize that by dividing through by this term. And then in general, for any current, I could tell you that what I'm going to be looking at is a concentration of oxidized species. And again, I have to define that concentration of oxidized species in terms of two parameters now. One is a positional parameter, x, distance from the electrode. So what I'm saying is I need the concentration right at the electrode surface, and the second is time. So anytime I write a kinetic statement, again, if I was going to write a simple first order rate law, I would just throw in the concentration, and I would assume it's homogeneous throughout my whole chemical reactor. But I have the problem here that I'm only interested in it at the surface, and I know it's not homogeneous there. So I have that divided by this term, which I'm curing through, the bulk concentration of the oxidized species. By the way, I'm being a little sloppy here, I notice that I probably will do this all the time, in that I tend to write either 0 or, not 0, I should say, oh, or OX for oxidized. I'll try and write OX, because that way you can keep it separate from some sort of a standard state. But hopefully, you'll bear with me on that one. So we have the bulk concentration. And then we have this potential dependent term, that 0 in this case, but won't be for some random case. And I'm going to make my life simple here. I'm going to find a little f, which is equal to the big Faraday's constant f divided by RT, just so I don't have to keep writing that. And I believe this is a little nomenclature trick that Bard uses also, times the over potential for the system. That is the difference between the actual electrode potential in a non-equilibrium situation. This is for equilibrium, non-equilibrium situation, and the redox potential for the chemical species. How far beyond the potential that thermodynamics tells us is going to be required to make this total delta G negative. Do I have to go in order to get whatever current I'm specifying over here? And then I have the back reaction, which I can't ignore any longer. So there's a reduced species. And it has some concentration at the electrode surface that varies as a function of time. And I'm going to normalize that to an equivalent bulk term for the reduced species. And I unfortunately put my little definition of f. I'm going to have to move that in a very bad place. Let's move that into the corner over here, which is there to totally confuse anybody who's trying to follow this. So minus the change in the concentration of the reduced species, and then another potential term. And again, that's 1 minus alpha now. And little f times the same over potential. An electrode can only be at one potential, so there's one over potential that we're at. That term disappears. That's nice. Well, no, if this assumes there are some, of course, as soon as you start the react, even if you start with no reduced species, and in which case I have a problem definition, as soon as I start the reaction, the first molecule makes a molecule reduced species, so this term exists. Now, if you're starting with that situation, where it's just going to be in the oxidized state, then the amount of actual conversion in the bulk that you're going to get is going to be trivially small. And so you can just throw out this term. Well, except that you're going to be balanced out by this exponential term, because you're going to be running out of over potential that enhances this rate. And therefore, by its very definition, is retarding the second process. So yeah, that pre-exponential factor might be large, but the exponential will dominate it. You can come up with a situation where it could be large, but it would still be dominated by the over potential. But you're right, Bruce. Most times there's nothing in this term that you have to worry about if you're going to start off just by dumping some oxidized molecules in your electrochemical cell and go from there. Now, again, what I've done in deriving this is I have totally ignored any other issue except for the rate of charge transfer to the interface. That is that activation energy that we've been talking about, that Delta G double dagger, only has to do with the rate that electrons go through the interface. It doesn't have to do, for example, with the fact that it may require some energy to move the molecules from out in solution up to the electron surface. That is, there may be a mass transport component here that becomes a dominant component under certain circumstances. So this assumes we are under a set of conditions, which are very, very similar to the set of conditions that Tafel apparently inadvertently stumbled upon, where we have a huge activation barrier and a lot of oxidized or reduced species around so that there never is a mass transport issue. The only thing that is rate limiting for this derivation is the activation barrier. And I can clean this up a little bit then and say, if there is no mass transport issue, if I'm in this Tafel extreme, this is a real life if. This isn't like an if that it'll always be true. This is in the special case, if there's no mass transport, and there usually is, then you'll notice that my concentration at the surface is equal to my bulk concentration. So for my ith species, and I can throw away the pre-exponential terms in this special case that I have just a ton of whatever I want to oxidize or reduce around, which was exactly the situation Tafel was in, and I have i equals i naught e to the minus alpha and little f times the overpotential minus e to the 1 minus alpha and little f again times the overpotential. So I get to this nice little equation now based on some fundamental understanding of molecular interactions. This is the Butler-Volmer equation. And you'll notice that if I go back and I simply identify a prime like I did very early on with this term i naught and identify now b with these things like alpha n times this little f, that is the temperature and symmetry factors in this, then the Butler-Volmer equation and the Tafel equation are the same thing. And in particular, in this one, I'm allowing both a forward reaction and backward reaction to occur. In this one, I only have the forward reaction that's occurring. And again, that's what Tafel was doing. He was taking water, and he was evolving hydrogen in his experiment. And so he's really driving this reaction. And so there's not much of a back reaction going. We can ignore this. That gets back to this concept of saying a minute ago. If you poise your potential so that your overpotential for the forward reaction is large, then your overpotential for the back reaction will be very small. So one of these terms is going to dominate. Now at this point, you're all supposed to have riders cramped. And I could say something like, well, we could keep going here. But let's not do that. Yeah. The eta is the same. Well, no, excuse me. The eta is, this has got a minus sign in front of it, right? And this one does not. So yeah. So the statement I'm trying to make, I'm probably beating on this too much. But it's very simple. At any given time, your electrode is only at one potential. And that potential is either favorable for running one reaction or the other reaction. You can't find, you can't have it at two potentials. And you can't find it at one potential where both reactions are going to go at very fast rates. You can find a potential where the forward reaction goes at a moderate or slow rate. And likewise, the back reaction does. And then you're very close to the equilibrium situation. But that's the best you can do. Now, this is sort of a sad statement I'm going to make now. This 1905, when Tafel sees this, it took a little bit of time beyond that to realize that there was a molecular basis for this. Butler and Volmer did that. But this is fundamentally old stuff. This is about the oldest modern electrochemical experiment that you can do. That is the oldest experiment you can do where you measure current. And it's an important parameter, which is my definition of modern. And yet, one of the hottest fields in electrochemistry, I'll stipulate today, is fuel cell research. And about the only major tool that you have on the electrochemical side to analyze what a fuel cell does is the Tafel relationship. Take that for what it's worth. Now, why is that? The answer is pretty simple. You can make models of electrochemical fuel cells. You might say, I'm interested in evaluating new catalysts for taking hydrogen to protons or something like that. And you can go into some kind of electrochemical cell and say, I'm going to evaluate this catalyst as a model, not in a fuel cell, but in some electrochemical cell and learn something about the catalyst. That's fine. But once you've done that now, you say, OK, I want to see how this catalyst actually behaves in a real life fuel cell. You are building a device that is going to generate a tremendous amount of current. It is easy for us in our labs to generate between 10 and 20 amps. That's not an exaggeration of current in a fuel cell that has a 1 square centimeter of active area, 1 square inch of active area. And our absolute best cells will get up to 20 amps out of that cell, and on our OK cells, it's still at 10 amps. So we actually have battery cables coming off of our fuel cell to take care of the current, these huge currents that are coming out. This equation, we can handle any current. But all of the other electrochemical equations assume that the perturbation you're making on the system is fairly small, and hence, the currents that you're going to generate in the other potential stuff experiment. The currents that you're going to generate are going to be small, because the perturbation is small. And there's all kinds of assumptions in the analysis that assumes small currents. And so you can't use any of the wonderful stuff we're going to develop as we go through this turn, cyclovoltametry, chronoamperometry, under conditions that you find in a fuel cell. It's just too much current to deal with. The one exception that's just coming online now is AC voltametry, for the first time, is being done in fuel cell environments, which is a real big challenge. Because there, of course, you're oscillating the potential. You're looking at an AC current that comes out. And we're talking about oscillating this system at these very high current densities. So there's all kinds of control electronics that become an issue when you try to do that. So that's the bad news. The good news is, is I don't have to teach you anything else to allow you now to understand what's happening in fuel cell research. You know everything there is. So I thought we could turn our attention for the remainder of the time into the type of fuel cell research that we're doing in our labs at Princeton. And I could show you how you take this sort of approach, this Tafel approach, and apply it to the analysis of a fuel cell. So we'll do that on the PowerPoint, I think. There we go. OK. So when one is talking about a fuel cell, I'll just point out very quickly here that's going to run something like this car over there. That, by the way, that's a Daimler Chrysler minivan that is running on this fuel cell stack that Ballard has put together and uses a Millennium cell hydrogen storage system based on sodium borohydride. It apparently goes down the road. At least it looks like it might be moving there. This fuel cell has a little bit, it's hard to tell. They always show pictures like this. They're wonderful pictures. And let's look at that car just moving right down the highway there or whatever, maybe a city street. They don't show you the three or four cars behind it, or the engineers that are making sure it keeps running so they can take this picture. This one works a lot better, actually. This is United Technologies fuel cell stack. That's the stack that's used in the space shuttle. So you can build things that work. And this one works also. I didn't mean to imply that this one doesn't work. That car really does move down the road. Yes? In terms of kilowatts or size. This dimension here is about a foot and a half. It's not very big. If you take about a cubic foot of volume, you can build a stack that will move a decent sized car. Not quite a minivan, but a little smaller car down the road at normal speeds. This is not going to be your 0 to 60 in three second car. With the size I just spec for you, it may not even be a California freeway driving car, but it handle regular roads OK, or freeways on the East Coast are a lot slower. The 405 would do great on. Except for that uphill part there as you're going out towards the coast. So in this case, the fuel tank is not an effect. The fuel tank on this car. I'm going to get all the way past my introductory slide which says the title of it. The fuel tank in this car is sitting right back here with a regular gasoline tank. So it takes no more space than that does. And it's just a polyethylene container in this case because it's using sodium borohydride aqueous solution. So density of that is the density of water essentially. A little more than the density of water. So it's very comparable to the gasoline that you're carrying around. A little heavier because of the density difference. And this car is getting an effective storage if you use kind of the super-grade of sodium borohydride solution that is getting as close to saturation as you can deal with. That's equivalent to having a gas cylinder which is by mass giving you 7% hydrogen. So your typical steel gas cylinder that we have in our labs. It's about 2,000 psi typically. And the storage there is somewhat under 1% to give you a feeling for this. And you don't get very much range, obviously, if that's what you're going to deal with. You can take those gas cylinders up to higher pressures, say 5,000 psi. But of course to do that you have to make the steel thicker. And we've probably all lugged a tank down the hull once. We know it's already heavy enough. When you essentially get to 5,000 psi, double the mass of the tank, then you're paying an awful lot of energy costs for carrying around that tank. There are, however, now, and they could be going to use these composite tanks, carbon composite tanks. They're very light. And they will get up to 7% storage. They are approved right now for 7,000 psi storage of hydrogen. And it is claimed by the Department of Energy that soon they'll be rated for 10,000 psi. They tend to give out somewhere around 12,000 psi. So I know this trick. I just gave this lecture in my class at Princeton and that class was going for 50 minutes and I have 50 minutes, approximately 45 minutes of transparencies here. And it took me two hours to get through it. And the reason was that the same thing happened there that's happening now and that is I kind of turned on the first transparency and we didn't get past it in the first hour. So let me move along and then you can ask questions as we go. And if this takes more than today to get through, that's fine. But I understand it's an interesting point. Now, when one talks about a fuel cell, you see in the public literature this concept of fuel cell and you're supposed to automatically realize that it means that it's using hydrogen and oxygen, particularly in air, as the fueling system. There are other fuel cells. The direct methanol fuel cell is now coming along pretty nicely. There are some other systems that have a military application. But by and large, when one says fuel cell, one means this hydrogen air system because it's the one that works. Now, even if I go that far, I really haven't specified for you what I'm really talking about in terms of fuel cell. And then there's got to be at least a half dozen different varieties of hydrogen, oxygen, or hydrogen air fuel cells. The ones I had on the front page were proton exchange membrane fuel cells. It's a small footprint fuel cell that has an obvious application as an automotive component or perhaps empowering a residence or a small office building. However, another fuel cell that people are very interested in that does the same chemistry, but works very differently, is the high temperature fuel cell based on a solid oxide electrolyte. So we can take materials like zirconia and variations on that theme and they will conduct oxygen as O2 minus nomally as an electrolyte at much more elevated temperature. So about the lowest temperature you can run those at is somewhere around 700 degrees C and you'll quite often find those running at 1,000 degrees C. Now on the one hand, that is a bit of a drawback for say something like a car because the question was asked well how big a cell do you need and obviously you want it to be a fit under your hood so what I was telling you was good but if you have to take something that's a cubic foot and keep it at 1,000 degrees even with the world's best insulation you're going to spend a lot of energy keeping that thing hot. So you really, if you're going to run a solid oxide cell you would like a lot of thermal mass and so you want a big cell and so if you want to run an office building or a university or something like that where you're going to be generating hundreds of kilowatts of power and you have a reasonable volume to surface ratio that sort of cell makes a lot of sense. It has a big advantage if you get into that application in that once you get up to somewhere around 1,000 degrees then the same kind of catalysts that you're going to use to promote your electrochemistry are good reforming catalysts so you no longer have to use hydrogen in that cell. You can just pour propane, methane, your favorite hydrocarbon into that cell and it will automatically be reformed to hydrogen in the way you go so you avoid some of the hydrogen problem by doing that. But for an automotive application probably not going to work. There are also phosphoric acid fuel cells where the phosphoric acid we're talking about is not the stuff you buy in your stock room here which is probably 12 molar phosphoric acid which you think is concentrated but real phosphoric acid that is no water around and that runs about 200 degrees C so a more moderate temperature and cells, UTC cells a cell that's 200 kilowatts that's based on a foz acid process you can go out and buy one of those today if you have a lot of money. That is a cell that by the way it's actually it's sold as a reformer you take natural gas in it's reformed into hydrogen CO2 and then those fuel cells next to it and if it's in a trailer that's about half the size of this room we're in right now just pull up your trailer hook it up to your natural gas line turn it on according to UTC there are other electrolytes there are alkaline electrolytes there are carbonate electrolytes there's a whole variety all of them are at higher temperature the solid oxide being the highest temperature so on the one hand we have the proton exchange membrane operating at low temperature and then all the way up and depending what you want to do different cells make sense so our interest at Princeton is in these proton exchange membrane systems these so-called low temperature systems and the cells that we deal with are single cells so the pictures I was showing you a minute ago are stacks where we put together a whole bunch of single cells and why does one do that well I already mentioned you're getting a tremendous current out but what I cleverly didn't mention is the voltage of these cells is pretty unimpressive you're talking about a half a volt per cell so in order to get up to something that you're going to drive a car with and people argue whether it should be 50 volts or 80 volts or 100 volts you're going to stack a lot of these little cells together to get both the current and the voltage that you need so if we're looking at a single cell we're going to have a device that looks something like this it involves a membrane electrolyte in the center and the typical material that's used now because it's the one material that everybody agrees works is a commercial material made by DuPont called napheon it's a persulfinated polymer that has persulfinated persulfinated polymer too bad it isn't persulfinated that has sulfonate groups as acidic sites in it and one way of looking at it is in terms of this little cartoon that we've drawn here that is it has areas in red that look pretty much like Teflon and then it self-assembles so the sulfonic acid groups are adjacent to each other and so you have other areas these green areas where you have hydrogen bonding between sulfonic acid groups if you expose this material to water then it readily absorbs the water absorbs it in these areas that are acidic not in the red areas and so you end up with these little pools of water in the material and little channels aligned with the sulfonic acids that interconnect those pools that material I don't know if you've seen it but if you've seen a transparency that you use on overhead projector it looks about the same it's not much different in terms of just visually looking at it that's what it looks like the thickness here has been going down about ten years ago now the standard thickness used was it's done in thousands of an inch in mills so it's five mills thick about 125 microns across and today people build cells that are typically about 40 microns or two mills across and they're trying to get down to one mill dimensions so one of the important aspects of this material besides the fact that it obviously allows protons to transport is that it's fairly mechanically robust not ideal but you can make these fairly thin films of this material and it doesn't fall apart on you the second attribute of this material is that teflonic regions are okay not perfect again but okay and making sure that the hydrogen you're going to put on the cell stays on one side of the cell and the oxygen on the other side of the cell there's a fair amount of crossover especially as the material gets thinner but it does a decent job with regard to that so you take this material and you're attached to electrodes to it the electrodes are a carbon cloth material it's pretty much like the clock that we make our clothes out of it's a fairly open weave and the idea is that we're going to bring in our reactants either our hydrogen in the case of the anode or our oxygen in the case of the cathode from the back of the weave so that's the gray thing right there is the carbon cloth conducting carbon so there's our hydrogen molecules that come in they're going to transport through that weave when they do that they come up to this front section which is our catalyst bed the catalyst bed consists of micron sized carbon particles and sitting on them are nanometer size sort of three to five nanometer size platinum particles standard catalyst for the hydrogen side of the fuel cell and we're going to carry out this redox reaction obviously at that surface now you have to watch very carefully we'll see how many people are paying attention okay now when that redox reaction happens you see what you end up with is protons those are those nice little purple things right there and they get injected into the electrolyte probably saw that happening right and then of course they have electrons those are the yellow things over there although professor Lewis claims that electrons are blue and he has a case to make for that today they're yellow you like the platinum being black and the carbon being kind of you know I had a wonderful visiting researcher in my lab that did this graphic and I don't complain because I can't do this this is just wonderful now the electrons end up in the carbon cloth and then we have a wire out here and they end up in the external circuit and of course you're all paying a lot of attention and you notice that the light bulb was not on until the electrons got up there how many people noticed that? saw that okay you got to keep an eye on the light bulb that's the important thing in electric chemistry cathode side we have oxygen coming in it doesn't sink okay Bruce on the cathode side we have oxygen coming in it's going to also come through this carbon cloth support it hits the catalyst bed again typically the catalyst bed is a platinum bed that is a standard material used same scheme here other than the protons end up here the oxygen ends up here the electrons end up here and we're going to make water on this side of the cell so this side of the cell is dry this side of the cell we're generating water on now one of the big tricks in this whole PEM thing is that you need to keep water in the electrolyte because in fact the protons here are moving through water you can see so no water no proton motion high resistance no fuel cell on the other hand it turns out that if you get water in the weaves of this cloth it blocks the pores and you can't get the gases through so that's called flooding so you have an interesting situation here you have something that looks like a sandwich but remember this is a couple of mills thick it looks like a transparency and it's got these two pieces of cloth on either side so this whole device here is a rather thin sort of thing coming up maybe on a millimeter in thickness on the most and you've got to keep the outside dry and the inside wet and you're making water over here and of course you don't want water just over here you want it everywhere and it turns out in order to do that you don't make enough water because you're not making it in the right place really and so you have to humidify your gases in particular you really have to humidify this hydrogen side but typically you humidify both sides but you don't over humidify because then you'll get flooding so this is a nice engineering trick and we've been talking about redox reactions so just remind you what's going on here at our anode we are generating protons at our cathode we are generating water we have a delta E for that of 1.23 volts assuming no current is flowing that's our standard redox potential for this so we could expect to get maybe on a good day something like 1 volt out of this cell because of course there's some resistances and things that are going to have to happen but as I already mentioned we're not even going to get that we're going to get about half of that out of this cell under operational conditions so what do we expect? here's where Mr. Taffel comes in the standard experiment that one does in this fuel cell is simply takes a resistive load typically a resistor and places it between the cathode and the anode and measures the current and voltage that are coming under the cell as you vary that resistance so you could actually take a set of resistors and physically put them in there and you can be fancy and have an electronic load that does this under computer control but it's the same situation so our adjustable parameter in these cells is the external resistance we're not potentiostat there's no referencing that's it it's real life Taffel conditions and we're simply going to plot that cell potential that we get and the current density that we observe here now this is a cartoon it has some kind of random numbers down here but what you see what you'd like to see of course ideally is that you get 1.29 volts out of this thing and no matter what current you're drawing you'd like to see that 1.29 volts that of course won't happen by the way it says 1.29 there on the last transparency it said 1.23 volts this is not because I've done my normal error this is a real difference in that these cells are going to run in the 60 to 80 degree range and so there's an adjustment to the standard potentials we have to make so under the conditions of operating the cell it should be about 1.29 volts this difference of a couple of millivolts doesn't matter given that we're going to throw away hundreds and hundreds of millivolts in a second here but just to keep the books balanced we have a little bit of a gain here because of this temperature we're operating that temperature because under those conditions we have favorable rates of reaction at the two electrodes and we have a favorable rate of reaction to the plant that allows you to make sure that the bacteria that's running in the plant is not on a very high level you can actually see that we're operating at a low level in terms of the lights and we're also working on the results of the process and how we're going to be able to predict the plant's病 and the total effect that's going to be significant because we're also working on that looks like this curve that we're seeing here. We see initially some sort of an open circuit voltage, which is less than the 1.29 or less than the 1.23, and more typically more like 1.0 on a good day, or maybe even a little less than that. And what's the reason for that? Well, as I mentioned, the napheon is not perfect in terms of its ability to keep hydrogen on one side and oxygen on the other side. So you get some crossover, and that attracts from your voltage. That's like a short circuit. That is, if I get some hydrogen moving over to my oxygen electrode, then I get the chemical reaction. Hydrogen puts oxygen just happening at the platinum sites there, and it doesn't get counted in my external circuit here. So I have some crossover loss to start with. I also may have some intrinsic resistances that show up at very, very low current densities, and that gives rise to a loss. So I start to pull current here. I lower my resistance in others, my external resistance, and I see a very steep drop-off in my potential as I do it. This is the taffle part of that curve. This is exactly this relationship right here. That drop-off in potential is due to the over-potential for carrying out this reaction. Now, it turns out on a well-prepared platinum surface, the over-potential for hydrogen protons is really small. In fact, negligible compared to the over-potential for the oxygen reduction. So we're seeing about a 400 millivolt drop-off in this region, and just about all of that can be attributed to the oxygen electrode. If there's one thing you want to do to improve fuel cells, find a new oxygen electrode. You can have more impact there than you'll have anywhere else. Also, a new oxygen electrode that runs on some venous and platinum would be nice also, because there's a little bit of a cost issue there. This next region, at this point, we have all the over-potential we need, and the reaction is just going as fast as it wants. And we want to run into a region now where we're limited by the resistance of the cell. That is primarily the resistance of the napheon, but there's also a fairly important resistive component from the napheon electrode interface. I mentioned to you before that the charge transfer itself could be thought of as a resistance, but that resistance is up in this region here, in the Tafel region. I'm talking about the fact that, unlike most electrochemical cells, I'm taking an electron and a proton in this case and moving it from a metal system into another solid state, not into a solution, but into this piece of napheon. And so I have a resistance for that interface, and an additional resistance for that interface. That is, it is easy to make an electrode-electrolyte interface for most electrochemists. You take your electrode, you stick it in a beaker of water, or a cedar nitrile, or whatever your favorite electrochemical solvent is, and you've made the interface. You don't think too much about it. However, the way you make a polymer-electrolyte interface is very problematic. It's more of an art than a science. And there's resistance associated with that, even on the best of days. So we have two key resistances, that interface resistance and the resistance of the napheon itself. And so we get something that looks pretty much like Ohm's Law in the center here. We see a drop-off in current with voltage. That's the linear. And we can back out of resistance, overall resistance for the cell, by just looking at the slope over here. And then finally, we get up to currents that are absolutely ridiculous. And we cannot support that level of current density. That is, you cannot pump the gases into the cell fast enough to keep up with the redox processes. And so we end up with a mass transport limitation over here. So from a pragmatic point of view, the solution there is really easy. Just don't run your cell out there. So all the data I'm going to show you actually has to do with cells that we're going to cut it before we get to that point. Sometimes that point can move way up here. And that's problematic. And that's this flooding situation. You'll really slow down mass transport if you flood your electrodes. And this will swing in. So you have to find engineering situations where that won't be the case. Now one of the things that you should take away from everything I've said so far is that a poor electrochemist by himself doesn't stand a chance of building a fuel cell, probably. That is, there's a lot of electrochemistry going on here. Charges are moving around. But there's a lot of engineering issues going on here. Transport, both of charge and of mass, is critical in heat actually in the cells in water. It's all critical. And so all of the work that we've done at Princeton has been a collaborative effort with a professor, Jay Benzinger, in our chemical engineering department. So he thinks long and hard about cell design, reactor design, about how you make those electrode-electrolyte interfaces so that they'll work efficiently. And we think about the materials that we're going to have in terms of the catalyst and the polymer. And that all comes together. And we hopefully end up with a fuel cell after that. So that curve I just showed you comes from an equation now that hopefully is going to look really familiar to you. We've kind of thrown the terms together in a slightly different way to make life simple. But the first part of the curve is over here. Now instead of just having a redox potential over there, I have what's called a formal potential, which takes into account a couple of things. It takes into account the fact that I'm not going to run my cell under standard conditions. I don't want to have to write up the whole Nernst equation and point out that my hydrogen and my oxygen are not at one atmosphere. For example, my temperature is not room temperature. So I'm going to scoop all of that stuff into this term over here. It also takes into account the fact that there may be some diffusion coefficients, which Tafel didn't know about. And I don't want to have to worry about involved here. And so those diffusion processes that are not all that important for this discussion are scooped into that term. And then we have our standard Tafel term there using his nomenclature, over potential, and log of the current. We add on now something that Tafel was not concerned about because he had lots of water around in his cell. We don't, and that is this resistive loss in the cell. This is for all resistances in the cell, whether it's the naphthion, the interface, whatever. And then finally out here, we add on a term that takes care of this mass transport limitation. If we get to the situation, we can't get our reactants into the cell fast enough. And there's just some adjustable parameters here, a pre-exponential and an exponential factor that adjusts. And it simply says that it has to do something with the current over here. So everything here now looks very familiar, hopefully. And oh, I should point out, although I write the symmetry factor explicitly here, we assume it's 0.5. So that's the background. What are we going to try and do here? Why aren't we already running around with fuel cells doing everything for us? In particular, proton exchange membrane fuel cells. The big problem with the proton exchange membrane fuel cell today, actually the big problem isn't listed here. It's cost, the cost of fortune. But the big scientific problem, engineering problem, with these cells is the question of where we get our hydrogen from. The Lewis group has been less than successful at generating hydrogen from water cheaply. And as a result, all of our hydrogen today comes from oil companies. It comes from either partial oxidation or steam reforming of natural gas, things like that. And when you do that, you make a lot of hydrogen. So all the hydrogen in those tanks in your lab, that's what that's coming from, an oil company, stripped out of hydrocarbon. When you do that, you generate other things. You're supposed to only generate CO2, which is an issue in and of itself. If you think that fuel cells are the answer to greenhouse gases. But you could think about things like, you're going to do this in a big factory, and you're going to collect the CO2 and stick it in California because you live in New Jersey or something like that. The real problem is you generate CO, quite a bit of CO. In a typical one-step reforming reaction, you generate 1% CO in your output stream. And we all know CO is bad news for us. It's really bad news for a platinum catalyst. So we have 1% coming out in a simple reforming stream. These systems I'm showing, you can't really tolerate more than about 10 parts per million of CO in the hydrogen stream. They poison at 10 parts per million over a matter of a few hours. You can get this number down by doing things like doing two-stage reforming and adding in a water gas shift reaction. And you probably want to do some of that anyway, for example, the water gas shift, because that CO is a fairly energy-rich molecule. And you would like to get more hydrogen out of it. So 1% is on the high side. And you're probably not going to deal with that. But even having said that, you're not getting down to 10 parts per million by doing all those things. So you do all those things, and then you polish off your final feed stream through a palladium, or I guess people like a palladium silver filter now. Palladium will let the hydrogen through, keeps all that CO and CO2 back where it belongs. And you get a tank of hydrogen, like the tanks of hydrogen that we all use in our labs. And those have less than 10 parts per million of CO in them. You'll be happy to know. And your fuel cell will run fine. But if I now want to think about I'm going to stop at the local gas station and tell them to fill it up with real-life gas with hydrogen, I don't really want to pay for reagent-grade hydrogen that has less than 10 parts per million of CO in it. I'll never be able to afford that. So we need to generate a fuel cell that maybe doesn't tolerate 1% CO, but does better than 10 parts per million. To the extent I can step away from this, the hydrogen just gets cheaper. So that is really the big ticket issue. Can I build a proton exchange membrane fuel cell that will tolerate some CO, at least? And then while you're coming up with your wish list, you might as well also point out that thermal management is an issue in these cells. That is, I've already told you, it's better not go above 80 degrees. And it has to be above 60 degrees. And so if you're going to do this, for example, in a car, then you're going to have to have a radiator that is about the excessive heat that's just going to be generated from the resistance of the cell so that we keep it below 80 degrees. And the way we would normally do that in a car is with a radiator, of course. And we do that already with our engines. But our engines are running more in the order of 150 degrees C. And it's really easy to remove 150 degree heat down to room temperature, right? It's not so easy, the so-called low-grade heat, to remove 80 degree heat down to room temperature. So that means you need a much larger radiator than you currently have. So in fact, the size of the fuel cell is an issue. But where you put this radiator is an issue. And if you decide you're going to drive your car in a part of the world where the temperature gets close to 80 degrees C, and there are a few limited places where that happens, then it's not going to work really well. No thermal gradient. So if you could do something about the thermal management, that would be nice. And while you're at it, I've already mentioned this water management issue, electrodes dry, membrane wet. If you could do something that would make that easier to manage, go ahead. So you solve all three of those problems and do it cheaply, and you can sell it to me. So, boy, this guy, he goes and he steals your punch line. The answer is simple. All three of those are fixed by going up in temperature. Yeah. If you can get up above 120 degrees C, then the CO starts to compete with the Platinum for sites on the hydro, let's try again. The CO starts to compete with the hydrogen for sites on the Platinum catalyst. This works because aluminum and CO and a lot of hydrogen. So even though CO binds much more tightly to Platinum than hydrogen does, you have a concentration effect going in your favor. It also turns out that the free energy for binding of hydrogen to Platinum is pretty temperature invariant in between room temperature and up to actually 200 degrees C, whereas the binding of CO to Platinum, that free energy of binding is falling off very nicely above 100 degrees C. So right around here, you start to win, sort of. And if you could run a little hotter, maybe 130, 140 degrees, you can win even more. That's sort of a magic number. And also, obviously, if you're running your cell at 120 to 150 degrees, your thermal management is automatically taken care of, because now you have a nice thermal gradient between room temperature and your cell, so you can dump your heat very nicely. And finally, it turns out that your water management is helped, because you can set your cell up. If you're going to run this way, you have to run under pressure. There's not a lot of liquid water around at 120 degrees C. So you're going to pressurize your cell a little bit. And you can set it up so you have sufficient pressure in the center of the cell to keep the water liquid, but sort of towards the edges of the cell, it's going to evaporate a lot more easily. And of course, that's where your electrodes are. So you can minimize flooding by running at the higher temperature and thinking about the pressure gradients in your cell. So this is a great thing to do. There's only one problem. It doesn't work. It doesn't work in that you take yourself to 120 degrees, and you evaporate the water out of your naphthion and no fuel cell anymore. And at 120 degrees, the vapor pressure of water is a very impressive number. We all think of these vapor pressure curves from room temperature up to 100 degrees. You've all seen the phase diagram for water. We know at 100 degrees, it gets up to one atmosphere. But although engineers look at it above 100 degrees, chemists usually don't. I've never seen it until I started to do this. And that thing takes off exponentially above 100 degrees. So by the time you're up at 150 degrees, you're talking about numbers like five atmospheres of water in your cell. And you go a little bit above that. And it's 10 atmospheres of water. These are tremendous numbers. So to operate in the 120 to 130 degree range, we're going to pressurize our cells to three atmospheres. That'll give us a half atmosphere of reacting gases and 2 and 1 half atmospheres of water in there. So we have a water issue. And under those conditions, even saying, well, I'll just feed in more water external to the cell to keep it wet, doesn't work. You cannot feed water in fast enough to overcome the evaporation rate. What's the original thought? It turns out it's not evaporation at all. That's affecting this. So if you look at the original thinking on this, the reason this doesn't work is evaporation. You just evaporate the water out of your cell. That is not the issue at all, as I'll show you. If that was the issue, then I just build a pressurized cell, and I can stop all evaporation in this story. It turns out what you have to do to make this thing work is not only use a modest amount of pressure, but you have to change your membrane somewhat. And what we have discovered worked was that if we put a small particle phase, a metal oxide phase, into our membrane, our naphyan membrane, then we can get a very good result. We have two ways of doing this. One is a sol-gel process where we take something like this tetrothoxycyline. We put it into the naphyan membrane. The naphyan membrane is a strong acid environment. It's got those sulfonic acids in it. Strong acid is a great catalyst for making siloxanes. And so you make a siloxane matrix of some ill-defined nature inside the naphyan membrane. It's ill-defined because it's inside the naphyan membrane. So you're not exactly sure what you have there, but it's good stuff, whatever it is. And we're not the first people that suggested that. That was initially suggested to improve the low temperature operation of these naphyan base cells by Professor Maroutes, who had a big background in this. And we simply took his synthesis and said we'll try it at high temperature. Something that we found, because this doesn't involve some chemistry, is that you could get a very similar result by just taking small particles, nanoparticles, if you will, of your favorite metal oxide and mixing them in to the naphyan. That is, take the naphyan, solubilize it, get it in an alcohol solvent, throw in your particles, stir everything up, sonicate a little bit to get a really nice distribution of things, and recast your membrane. This, when we started, was supposed to be a really bad idea. It sounds like a bad idea on paper, because, well, this thing sounds nice and homogeneous, and you'll have some kind of interesting material. This sounds like a big mess. And why would it give you an interesting material? And there were some preliminary experiments that would have suggested it was a big mess and didn't give you nice materials. But if you go down to, say, tens of nanometer particles and you do this, in fact, you get a very homogeneous material. And it works as well as this material. And the big advantage is I can put any metal oxide in I want here, because I can buy nanoparticles of anything I want. And over here, I'm stuck with a fairly limited chemistry. So we do all of our work now this way. Just to show you that, I'm being honest about this, this electron micro probe data that we obtained on one of these matrices, these polymer electrolytes, in which we put silica into the electrolyte, not via this sol-gel business, but just by mixing it in. And so what we've done is we've taken a naphyl membrane. We've recast it one piece with no silica in it, one piece with the silica in it, about 3% by weight. Taken the membrane, we have freeze fractured it. And in this picture, which is a map of the oxygen in the electron micro probe, there's one side of the membrane, there's the other side. You can see we're looking right across the membrane. You can see there's oxygen in there, which is good news, because the naphyl membrane, the backbone of the naphyl is Teflon, really. It's just a flow of carbon. But there are side chains, the sulfonic acid groups, which are one source of oxygen, are off these side chains. In addition to that, the side chains have ether linkages in them. So we expect oxygen in them, and we're picking it up. On the other hand, if we look for silicon in this pure recast membrane, you can see you can't make out the membrane, because there's no silicon in there. We take our composite membrane now. We look for oxygen. And you notice the green indicates there's more oxygen in the blue there in this map. That there's a lot more oxygen. And it's fairly homogeneously distributed in it. Not perfect, but fairly homogeneous. And we look for silicon. And we see a very homogeneous distribution of silicon in that membrane. So we can make the membrane just by mixing in particles. And it's extremely homogeneous. Now, more important, if I take a regular napheon membrane, this is a napheon. The way DuPont does its numbering, you have to kind of not only, if you want to become a fuel cell person, I'm not even sure it's a chemist, you kind of have to become a sales person also, as well as an engineer. You have to understand how they do their cataloging and everything else. So the way napheon does this, these numbers over here, the last digit tells you the thickness of the membrane. So that's a 5 mil thick membrane. And the first two digits tell you the ratio in some really wacko units of the sulfonic acid groups to the rest of the polymer. So it turns out the way they've ratioed this, a smaller number is better. That is, it has more acid in it. And it's done in grams. It's not done in moles. And so it doesn't have any immediate significance other than this is a fairly high acid loading in this polymer of 115. So you take that polymer, you slap your electrons on, very carefully, and you run one of these current voltage curves, and you get the black curve that I am showing you right there. And that is not a bad curve. It starts a little bit, you can't see it, because the red that will recovers it, about a 1 volt versus SCE. And you'll notice that we're going out to about 1.5 amps per square centimeter in this cell. And then we get into mass transport out here. We've cut it. The resistance isn't bad over here. A good rule of thumb is look at the cell performance at 0.6 volts. And to the extent that that's somewhere near 3 quarters of an amp, half of an amp is probably good enough, but 3 quarters of an amp is better, you have a pretty decent cell. So that is a very nice cell. Take that cell, heat it up to 130 degrees. We want to operate from 80 degrees where we were operating. And you get this response. Activation part of the taffle slope looks just about the same. It's actually it's a little higher over here. It turns out you can't tell from the data I'm showing you. That makes sense. We heated the temperature up. We have a temperature component here. So things are going a little better at open circuit. Not very useful. And then we get into resistance regime over here where we have a dead cell. That is a high-resistance cell. We've taken our R from that value to this value. And so where we're getting 3 quarters of an amp out before at 0.6 volts here, we're getting a couple hundred milliamps out. That's a dead cell. What this data doesn't show you is if I now go and cool the cell back down and run the curve again, it looks just like the red curve. It does not recover. Now, what you do is you take one of these composite films and do the same thing, run it at 130 degrees, and you get the orange curve here. And you'll notice that looks really good. Not only is it better than the red curve, it's better than the black curve, the 80 degree curve. So we are rapidly running out of time here. And I'm going to stop at that first result that we heat that cell up with the composite membrane and we get an improved response. I just want to tell you one other thing. Besides learning about how to make a good cell, you just learned what Princeton's school colors are. OK, we will continue next hour.