 OK, but today we're going to look at AC techniques, which I'm going to define kind of broadly. And it seems to me that although we think of the sort of history of electrochemistry as starting with DC techniques and then making its way very recently, actually, the AC techniques, that in fact that it started with AC techniques, what I'm not starting with though is a pointer I noticed. OK, we start with AC techniques in that very early in the history of modern of electrochemistry, we have a troll maker today, very early in the history of modern electrochemistry. Thank you. The technicopolarography was developed and it was developed for the dropping mercury electro, the DME. And although that is typically considered a DC technique, in fact it's an AC technique. And that involves a time-dependent change in the current on a rather fast time scale. And so I thought I would throw that in with AC techniques for completeness and we'll start there with dropping mercury electro and polarography. Go over that quickly because you really already know that, even though you don't know that you know that. And then move on to solid electrodes and the actual application of a sinusoidal potential to the electrode. So here, this is considered a DC technique because we are applying a DC potential to the electrode. When we do polarography, we talked about doing it at a solid electrode. But of course when we do polarography, the waveform that we are talking about is just a simple linear ramp with time. And what makes this different from the things we've been talking about recently is this is a slow time change. So we're talking about scanning somewhere between 1 and 10 millivolts, let's say, per second. So very slow compared to what we would normally do for something like cyclic voltammetry. And we're going to attempt to develop a steady state situation and use that to understand the dynamics primarily, but also kinetics in the case where the kinetic times fit in with what we're looking at. So this idea of polarography and the dropping mercury electrode that goes with that as a start comes out of the Czech Republic or Czechoslovakia at that time by a gentleman by the name of Petrowski who received the Nobel Prize for his work in the late 50s. And I was just looking at his Nobel address this morning. And Tom doesn't realize this, but he's in the process of mounting that address on the website. As soon as he checks his email and discovers that it was sent to him in the last 30 seconds. And so what's interesting about this address is it's amazing how this guy did this. This is work done in the 40s, right? And his quote unquote potential stat is nothing like you would conceive of today. So his recording medium is sort of a drum all geared up the right way with large gears and whatnot. So he's got a time dependent baseline, much like when you go and you see the seismic recordings that are done on the drum, you guys have all seen. So he has something like that going. And then how do you make a dropping mercury electrode? The way he made it and the way it was made for a long time, and still is really today, I guess, is you simply go and you take a reservoir of mercury, pass it down a tube into a capillary. And then you just have a little wire going into that capillary over here, which connects to your pretentious stats. We have a working electrode. And there'll be a natural formation of drops at the tip of the capillary. And the rate that these drops drop out of that capillary and the size of those drops depends on several features. One is the diameter of the capillary, but the controllable one, or the variable one, I should say, the one that you control during the experiment, is the height of this reservoir. And as you raise that up and lower it, you change the drop time. And that's, of course, how Heterovsky did that. And based on that, you can generate a voltamogram, such as the one that I'm showing. This is one of Heterovsky's over here. So this happens to be a chromium triscipyperidine, which I took out of his address over here. He called it something different, but that's what we would call it today. And these are, he wasn't a real big fan of labelling his axes, but there it is. This is potential, and this is current. And this is going more reducing. So this is long before the IUPAC conventions were in place. So these are reductions. And you can see three distinct waves here, which are the reductions of the bipyridine ligands on that chromium system. And this is done, I believe, in water. And what you notice here, in addition to these nice steps, is that there is this jiggle on here. And that's the drops forming and falling off. So if you were to go and blow up one of these on the plateau over here, what you actually see out of this thing, when you look now, say, at current versus time, just looking at one of those little jiggles, is what's happening is you see the current going up, stabilizing, and then dropping quickly as the drop falls off. So you have drop formation. And then loss, it falls off. And you start over again, and you get another one of these little jiggly things coming up. So that's really what all that is. Now, when I have done this myself, and there are reasons today to do this, that jiggliness fills the page. I'm going, this guy, Hedorovsky, wait for you to get the Nobel Prize. They did design the technique. It's really good because that's not bad looking. The little drop thing is pretty good. It's not filling the page here, where you can barely make out the waveform. And I couldn't figure out initially why he was so good, other than he was obviously good. And that's part of the answer. But it occurs to me that what's happening here is he had a very slow, we'll call it, strip chart recorder, amazingly slow. And it probably could not keep up with the fall off and current in particular when the drop fell off. As that drop was dislodging, his recording mechanism couldn't keep up with that. So his lines probably shoot way down on the page here, these vertical lines, but you just can't see it. So this is a good example of where poor equipment can actually help you out. Now what you want to do here again is you would like to work out an analytical solution to this and you could think about situations, again, where you're mass transport limited, reversible reaction, et cetera, et cetera, the normal condition. And that was worked out by a gentleman by the name of Ilkovic, who Hederovsky refers to in his address. And thus we have a name equation here, which I'm not going to derive through the Ilkovic equation. And what it describes very specifically is one of these little transients when you are sitting on this plateau. See, when you're on the plateau, you're mass transport limited. Anytime you get a plateau, whether using this technique or any technique, right, you are mass transport limited. It levels out because you can't bring stuff up any faster than you are. In this case, you have a certain kind of intrinsic mixing in the system, right? Because as that drop forms, it stirs things up around it. So it's not a diffusion limit, a simple diffusion limit. But it is a mass transport limit. And what Ilkovic did was recognize, in fact, how that mixing works. They have two things actually going on here. As the drop grows, of course, its mass gets bigger. As its mass gets bigger, its volume gets bigger. And as its volume gets bigger, its area gets bigger. So you don't have a constant area of the electrode. The area is changing with time. And so that's going to cause, in part, the current to go up, obviously. And that should go up as r. That is the ratio where r is the ratio of the volume of the drop to the area of the drop. The second thing that's happening though is the double layer. The diffusion layer, bringing these chemicals in, is expanding also as you do this, right? As that electrode expands, it forces out both the double layer and the diffusion layer. And so you have a fairly interesting not-for-day current that goes with this. But in addition to that, you have this change in the diffusion layer. And so you have a condition here where that's constantly changing on you. So what you can do is you can simply take the Cotrell equation, and then you can add into it the fact that you have this ever-growing electrode, and that you have a spherical geometry and all that. And if you do that, you come up with the Ilkovic equation. And if you're interested in the details of that Bard runs through that, I think that's in chapter 5. But really, this is just the Cotrell equation for expanding electrode. And what you see is there's some constants that come into this. But once more, you have n to the 1 power. You have the diffusion coefficient, just like Cotrell tells us, the 1 half power. Concentration is linear here. This is the flow of mercury in the system. That m is in milligrams per second of mercury flowing through your capillary. So that's where your growth is controlled. And then we have a time dependence, obviously, as that drop goes up. And you have two time dependencies that are coming into play here. One is the Cotrellian t to the 1 half time dependence for diffusion, as modified by this change in the diffusion layer. And the second is the time dependence of the growth of the drop of mercury. It's growing with time. So it turns out when you convolve that all together, it's a t to the 1 sixth power. And that gives you something like this. You can see you can get very good data out of this. Now one way you might refine this, it turns out, because I've tried this. If you do things this way, probably Hederovsky was good at it, but I can't come in on two separate days and get my reservoir to exactly the same height. Now this is just on a ring stand, which is more or less what Hederovsky was using also, and at exactly the same height so I get exactly the same drops per unit time and size drop and whatnot and reproduce my data from day to day. So what is that about? The machine's getting tired, huh? OK, stay awake machine. So you would like a better way of doing that. In addition, if these shakes are really as big as I'm suggesting they can be, do these drops, because you have equipment now that's going to follow the whole drop, then it gets very hard to try and understand the data. It looks artistically very beautiful, but it's not all that useful. So here's one, in fact, this one's very good also. But this gives you a little bit more of a feel for what's happening, and this is taking out of a text here. But you see, as the current gets bigger, you get a little bit of jiggliness as the current rises. But once you get into this plateau region, there's a lot of jiggling going on here. And again, this is an old study. So the jiggling is not as big as it would be if you use modern electronics to capture the whole Ilkovic transient there. But it gets hard. Where do you draw your line now? If you want the diffusion current, they've drawn a line right down the middle here where the ink is a little thicker. That's how you do it. Where do you draw that? So one way you might improve this is you might use a sampled current technique, where every time the drop, say gets to this point right here, you're not going to measure the current continuously, but just measure it right at one point, maybe right before the drop falls off. And you can turn then this jiggly line into a nice, solid line by sampling that. So we have the DME, and then we have the sampled dropping mercury electrode. That just makes your life a little bit simpler. But how are you going to do that if you're using just the natural drop frequency here? That is your electronics don't know when to sample, because they don't know when the drop is going to fall off. And there are subtle things in that, like exact heights and whatnot, and whether there's any vibrations in the building, things like that, that will change the natural drop rates. It's not exactly a perfect reproducible phenomenon. So what you want to add to your system, which was done, I believe, around in the 60s, maybe even in the early 70s, is a drop knocker. And a drop knocker is just a little piston-like thing, a little tapping device right here that goes and taps the end of your capillary, or shakes it, and you want to think about that. And every time it shakes it, a drop falls off. And of course, since you're controlling this now, you can knock a drop off once every second or whatever. And as long as you pick a time frame that is faster than the natural falling off rate of the drop, you're in great shape, and now you get a very good reproducibility. And you can do things like sampled polarography. That now opens up another possibility. There is, you can see, on the one hand, although this jiggly current is painful, on the other hand, it's useful in that the current here is larger than you would get if you didn't have that mercury drop going in there and stirring up things and changing the diffusion layer. So you're actually enhancing the current. That is, the current at the top here is a much larger current than you would see at a solid electrode under the same circumstances. So if your goal in life was looking at a very low concentration of something in solution, this would be an enhancement if you wanted to do analytical chemistry. Now another way you can get that enhancement is instead of changing the mercury drop, you could change the potential. So instead of just using a simple linear potential wave form, let's go and do something like this where we keep on taking our potential back to some zero, arbitrary zero, with time, and then we just map out in a very reproducible way that wave form. This would be called pulse polarography. And if I have a drop knocker, I can do this now because I can sink this wave form to my knocker. And so on a given drop, I can go up, have the drop form, I can jump the potential, and then I can go and do a sampled, if I try and look at the current coming out of that thing continuously, with the drops falling off and this happening, forget it, I've got a total mess. But if I go and say, I'm going to sample my current now right at the end of these square waves, then I can generate a very nice polarogram. And now I'm no longer limited by the diffusion limited current because I'm breaking down the mercury drop when it falls off, stirs up the electrolyte. I'm breaking down the diffusion layer as the drop falls off. And I don't have a potential there because I have an off potential to re-establish it. So I see the initial current, which is a much larger current, than the diffusion limited current at each of these pulses. And so I gain sensitivity by doing this. Don't learn any more information. But you'll notice, I'm getting thermodynamic parameters out of this. You can see you've got a halfway potential. And you can relate that, as you already know, to the redox potential. And the slope here, if that was not exactly what you expected from the Nernst equation, then you could get some kinetic taffle type information out of this, et cetera. If there was multiple charge transfers, we saw in the prior picture over here, you can pick that up. So there's a lot of information in there. Now let's make things more complicated. Now, by the way, is this just a historical interest or would you actually do this today? The answer is you do it today. There are plenty of papers published right today using dropping mercury electrodes. And why is that? Either because the mercury drop gives you an advantage because of its renewability, or simply because that surface is a surface that is catalytic for the process you want. For example, if you want to look at a process in water that's fairly reducing, it would be great to have a surface that can't reduce water, but could reduce your molecule of interest. And mercury, of course, has a very high overpotential for hydrogen evolution. So you can see things in the aqueous electrolyte with mercury you simply can't see on a platinum electrode because of the potential range. The renewal of the electrode is very important. In particular, there's an absorption phenomena that is hurting you. If you have something that's absorbing onto the electrode that is maybe insulating, then being able to make a new electrode once a second or whatever is an extremely helpful phenomenon. And finally, there are just processes that are catalytic at mercury and none other surfaces. So several years ago, I was asked to look at the possibility of looking for a certain organic molecule, a derivative of a carboxylic acid, that a certain company was concerned that they might be dumping into a certain Canadian river. And this was the company being conscientious, actually, and trying to figure out if they were doing this. They wanted to monitor the water in the river. And they decided that probably electrochemistry was the best way to look for this. And they asked me to do a little project to see how to do this. And I started off with nice clean platinum electrodes and doing cyclic voltammetry and got nowhere really quickly and ended up doing a dropping mercury electrode experiment. And it worked perfectly. And from the height of those peaks, you could easily pick out exactly how much of the substance was around. And there wasn't a lot around. Yes. You can, the earliest cyclic voltammograms, in fact, they're in that heteroski paper, were done on dropping mercury electrodes. The whole CV is done on one drop. So you set up the drop time, so you have 10 seconds or whatever, and you get the whole, yeah. So yeah. So that brings up yet another technique, which is the hanging mercury drop electrode. In other words, you have a drop there, but you don't let it drop. And that allows you to CV or bulk electrolysis or things like that. If you want to do a electrolysis where you want to make a lot of material, then probably a little mercury drop isn't the way to do it, because the area is so small. And so then you go to a mercury pool electrode. You could just pour some mercury in the bottom of your electrochemical cell, stick a wire down in there. You've got a great electrode. Large area electrode doesn't renew, but the mercury is the electrode of choice. You do that. Now another derivation on this, which is useful. We have pulse polarography is now differential pulse polarography, which is getting more AC again. Differential pulse polarography might be done on a mercury electrode, but need not be done on a mercury electrode. It might be done on a solid electrode now also. So the idea is going to be, in this case, this is differential pulse, superimpose on our classic linear ramp and potential, we're going to put a little interrogating pulse. They're supposed to be evenly spaced, like this. These are parallel to the line, actually, is the way it's done, yeah. So in other words, what you're doing is you have some waveform here, e, that is equal to an initial potential plus a scan rate, which is very slow. And then on top of that, we're going to add a faster change, which is just a small delta v here, which would be somewhere between 10 and 100 at the most, millivolts. We're going to pop up on this. So we're superimposing that on top of this. And this is going to be for a fairly short time, say a second would be the maximum. So you could use a dropping mercury electrode for this, but you need not use it. And the idea is going to be, we'll measure the current, the beginning of this flat part, and at the end of the flat part, and report then a delta i for each of these pulses divided by the time length of the pulse. And so we have a pseudo derivative here. And of course, the fast way pulse, the closer it is to a real derivative. And so we'll now make a plot of di dt versus potential. And this does two things for you. First of all, by doing the subtraction, if you have a underlying baseline that's getting in the way, it's automatically subtracts it out. And second of all, it just enhances your sensitivity, both to actual signal and to noise, but it enhances your sensitivity, because now you're looking at this differential quantity, or something approaching a differential quantity here. Do the peak potentials that you get out correlate to the peak potential that you get from simple CV? Yeah, so I'm getting a good question. We carried out a study, which you have up here. Several years ago with Andy Hamilton, now the provost of Yale, I guess. But he was an assistant professor of chemistry once upon a time, and he's a wonderful synthetic chemist. And he could build these porphyrin systems and do wonderful things and strap things over them. And so he built for us this bipyridine strapped over this nice porphyrin, and we decorate it with a ruthenium, obviously. So we have a ruthenium tris bipyridine on top of a porphyrin. And we were actually interested in this for reasons of photochemistry and photophysics. We wanted to look at photo induced charge transfer between these two groups, or energy transfer, things like that. And they both emit nicely, and it was fun to do the photophysics. But we needed to figure out the thermodynamics of the system before we could work out the photophysics. And so we needed to know the redox potentials of these systems. Well, this thing is not very water soluble. Actually, it's not very soluble in anything. And so we tried cyclic voltammetry so we could get the redox potentials. And we could barely see anything above the baseline. It was there were little peaks in there, but we couldn't resolve them enough that we could see them and get reasonable values for the halfway potentials and hence the redox potentials. So we went over to differential post-polarography. And so what happens when you do this is you end up with a peak. And if I was comparing that peak to a polarographic peak, then this maximum of the peak falls at the inflection point in the polarogram. So I've increased my sensitivity. And in terms of finding a halfway potential, it's actually easier because if you don't have a lot of signal here, finding the inflection point here can be a challenge. But even without a lot of signal, of course, finding a peak is a lot easier. So this is your half. That was supposed to go right down the center, by the way, and there's a wonderful artwork here. And there's your halfway potential, which, again, assuming diffusion coefficients aren't too wacko, is the redox potential. So that's the one big piece of information you get out of this. And likewise, if this was a cyclical tamagram, since that's what we were trying to start with, then it turns out that would have ended up being the halfway potential more or less the cyclical tamagram. But again, if you have a lot of baseline and a little signal, this is much. The black curve there is much easier to pull out. Now, the downside on this is great for finding halfway potentials, but not very good for doing anything else. That is, there is not a nice mathematical framework that would let you look at mechanistic information doing this. Things are just too complicated here. And you're not going to be able to distinguish subtle changes in the width and height of this peak that's going to let you know what the mechanism is. When you do this, I should point out, you typically have control over how big this little perturbing potential is and how long you dwell here. And obviously, the bigger the potential is and the longer you dwell here, then the more gain you're going to have and the bigger your signal is going to be. But also, the more digital it's going to be. That is, you're going to start missing points. That is, this potential gets too big. It perturbs your ramp too much. And you have made a change that you don't want to make. So there's something about looking at the forest or the trees or something here, right? So what happens here? So we got this solid curve out, which is a kind of complicated looking thing, you'll notice, when we ran this on this complicated looking molecule. And then we simply threw in a little bit of rutheniatrists by Puridine, spiked the solution with that. And you observe, when you do that, that these three waves right here, there's a little bit of movement bounce up. But nothing's happening over here. This one you can see there's a little change in the baseline and similar for these, although we didn't show it. So these three ways are the reductions. The three reductions are the rutheniatrists by Puridine, one electron in each of the rings, if you will. And then out here, we're picking up the reductions of the porphyrin system. So we get the halfway potentials for all the different redox active species out of something like that. And with good resolution, compared to what we were able to observe in solution just by cyclic voltammetry, where we could barely see anything and now we have these nice strong peaks. One closing comment to make on dropping mercury electron, maybe two. The first is, you need to use amazingly pure mercury if you decide to do this. Don't borrow somebody's dropping mercury electrode. Go to your stock room and get reagent-grade mercury and pour it in, because you will lose a friend if you do that. The problem is, there are enough impurities in reagent-grade mercury that they will clog up very quickly, this capillary. And once that happens, it's time to buy a new capillary. And that's the only expensive part in this whole thing. So typically, you use triply distilled mercury in there, or it won't pass through the capillary. The X-mercury, still. Anything less than that will not work. And I've tried to be cheap about it at times, and it's not worth it, I discovered. Because the capillaries cost a lot more than the amount of money you're saving on not getting the good mercury. So that's number one. Number two is, today, really it's only come out in the last 20 years or so, there is a more sophisticated way of generating these mercury drops. And what's done is you can buy a system which pressurizes the mercury reservoir. So there's no more of this lifting or drop knocking business or whatnot. But it pressurizes it, and it does it with a little relay piston system that generates a pulse. And so you can pressurize mercury through the system. And you can very precisely regulate that, both in terms of the time of the drop and in terms of the size of the mercury bead that you generate. That is by applying a bigger pressure pulse, you can make a bigger bead, and of course the frequency of the pulse gives you the drop time. So you have very precise ways of measuring this now. And you get out a very good result. It's very easy to monitor using either the sampled mercury electrode or one of these pulse polarography techniques. So those are a few practical hints. Now, what about real AC techniques? So really when people talk about AC techniques, they're not talking about a time dependence like this, but they are talking about a time dependence where you apply a classical sinusoidal signal to your electrode. So the idea is going to be you have a potential, you're going to put perturbation, a small potential perturbation there. No DC potential on this, let's say, to start with. And then you're going to monitor the current as a function of time. And you may get out a sine wave there that is shifted, the phase shift in respect to the potential shift. And why might that happen? Well, if this is just a classic, this by the way is just a picture right out of a bar, if this is a classic electrical circuit, then we know that if we put a sine wave in and we have a pure resistive load, that Ohm's Law tells us that all that's going to happen is we're going to get a current out that goes as the amplitude of the potential wave divided by the resistance. And we don't see any shift in the sinusoidal dependence. On the other hand, if we have a capacitor, right, just using q equals cv, then we have to take the derivative, assuming the capacitance is not time dependent. We come up with a cosine function and a frequency dependence in the linear part of the curve also. Or just rewriting that, since this is a sine wave, we have a 90 degree shift in our sine wave plus this additional frequency dependence out here. And so if we have some kind of an RC circuit like we would draw for our typical electrochemical cell, we're going to have both of these things going on. So in general, we expect when we put in some sort of sinusoidal dependence here to come up with a frequency dependent current in terms of the amplitude of the current that is phase shifted with regard to the potential that we are applying there. Now, given that, there's a whole variety of techniques that one can use. For example, I might put that sinusoidal dependence on top of some sort of a linear ramp. And that's called AC voltammetry. If I do that, and it would have, of course, similar advantages to doing something like this. And since I would be using a much higher frequency signal, I'd get more of a derivative over here. So there are things like that. That one might do. You can get fancy, for example, instead of looking at the primary harmonic coming out of this, you might look at the second harmonic coming out of the current instead of the fundamental and things like that. And that can give you some sensitivity and remove some baseline and things like that. So from a purely analytical point of view, if I can't get a signal out and I'd like to get one, there are some advantages to doing AC voltammetry. But again, you're more or less limiting yourself to finding a redox potential. You can get something with a peak, hopefully, and a redox potential. You're not going to get any mechanistic information. Another experiment, and the one I want to focus on that you can do, essentially do not have a DC component. Just have an AC component. And so our parameter now, our variable, is going to become the frequency. So we're not going to scan our potential at all. We're going to fix our potential. We pick a potential that from some other experiment we know something about. And we just are going to change our frequency and look at this phase shift as a function of frequency. And then once you do that, you have lots of different ways that you might portray your data. So for example, again, out of Bard, you might go and this is the log of the frequency. If you can't see that down there, and this is the log of the amplitude, you might look at that, amplitude of the output current as a function of frequency. You might look at how the phase angle changes as a function of the applied frequency. One nice way of combining all of that is in the so-called Nyquist plot. And in that plot, we're going to plot the real part of the current signal versus the imaginary part. And this real part, remember, from that we're deducing the resistance of the system. And from this, we're deducing the capacitance of the system. Actually, it's the partial to 1 over the frequency times the capacitance of the system. And so if we have a pure RC circuit, nothing electrochemical, but a real live resistor and a capacitor in a circuit, then you'll get a plot just like shown on the PowerPoint. You get this nice semicircle here, where the total resistance of the circuit can be determined by where this circuit or semicircle impinges on the resistive axis here. The way this is working is we're moving from low frequency to high frequency. We're generating that semicircle. Third way that, of course, you might try and analyze this is just in terms of the standard vector notation that one uses with real and imaginary vectors. OK, what are we going to do? So you end up with a semicircle. So what? What do you do with it? That is on a good day you end up with a semicircle. What are you going to do with it? The way this is analyzed, and this is the plus and the minus of doing this, is you need to draw out an equivalent circuit. And you analyze the response of that equivalent circuit to this system. Now, the data you're getting out of this is very sensitive to the equivalent circuit. So it will be easy to determine whether the equivalent circuit you draw does not fit the data. So that's the good news. The bad news is we're going to have to take these real life resistive and capacitive components that we're drawing in our equivalent circuit and try and equate them to chemistry. And to the extent that we do that correctly, then we get a useful answer, and to the extent that we don't get a useful answer. The other problem you have is that we're going to be drawing circuits with several circuit elements in them. And that means that there's probably more than one solution to the problem. There may not be a unique solution. And therefore, again, there's a little bit of a guess going on here. So the circuit that you saw on the prior PowerPoint and what I've drawn up here that goes with this is just a system like we've been talking about all term, where we have a resistor and a capacitor in series in parallel. And you can see without doing any math what's happening here. Normally, of course, what you're going to need to do is you have to develop a mathematical relationship between that circuit that I just drew on the board and this set of data over here. But in fact, it's very easy to see what's happening. I come in here and I come in here at different frequencies. And when the perturbing wave gets to this point, it's got to decide whether it goes up or down. And it's going to make that decision simply based on the lowest resistance path. You know that if this frequency is very, very low, let's say it's DC at so low, so frequency of zero, then this path up here is blocked. You can't get through a capacitor with a DC current. And therefore, this is a high impedance path and all of the current will flow through the resistor as a result. On the other hand, as I change my resistance, excuse my frequency, then at some point, since this thing goes as 1 over omega c, the impedance here becomes lower than the resistance over here. And there's some magic frequency where I start passing everything through this. And in that case, nothing will go through here. And of course, there's a set of frequencies that are involved here that are where c or 1 over omega c and r are pretty comparable. And I have current going through both of those. So at this end, right here, zero of frequency. Everything's going through the resistor. And then as I go to higher and higher frequency, I get to a point where everything's going through the capacitor. All the currents here. So when I measure this point, I'm measuring the resistance of this circuit. And when I'm measuring this point over here, there is no resistance from the resistor because everything's going through this upper branch. And I'm measuring the capacitance of the system. And you can show that when you're right in the middle of the semicircle here, that's 1 over the rc time constant for this. Now the resistor I'm talking about is this resistor down here, not this r sub omega, because everything obviously passes through that. That will be an offset to this curve. To the extent that r omega exists, that will just push this curve down the axis. So without doing any math, it's pretty easy to see what's happening here. And now all I have to do is say, well, it's not a circuit or an electrical circuit with components in it that I'm interested in. But what I'm really interested in is a double layer here and a charge transfer resistance right here. And now you can see, if I have a way of measuring the value of these components, which I obviously do over here, then I'm going to back out the charge transfer kinetics and the double layer properties of this system. So typically what I would do is I would figure out this point right here by extrapolating to zero frequency and then use the maxim over here to come up with the capacitance once I have the resistance. Now life isn't that simple though. Typically when one goes and does the experiment, it doesn't look like a semicircle like that. Actually on a good day it looks like this. And sometimes there's more of that line than there is a semicircle and sometimes there's more of a semicircle than there's a line. You'll see in a moment we're going to want to both have the line and the semicircle so to the extent that one kind of wipes out the other one. That's a bad day. But things ideally look like this. So where does this tail come from? And we'll have to modify our circuit a little bit if we want to get a little bit more realistic. That is we have our overall cell resistance. We have our double layer capacitance. We have our charge transfer resistance. But now we have to add in a component here, which is not a component you can get at Radio Shack. And this is the so-called Warburg impedance. Because this is a variable impedance. It's not something that we can model with a fixed go out and buy a resistor. And it is in there to handle diffusion. We have mass transport limitations. So in other words, when I'm at very, very high frequency in this area over here, I don't have to worry about mass transport because I'm oscillating my potential so fast that the molecules, if you will, don't have time to move. And so there is no mass transport over the period of the wave. When I get to very low frequencies, then there is molecular motion on the time scale that I'm oscillating the potential. And so I expect that to come into play. And it turns out that comes into play as a frequency-dependent impedance. So it's another element that falls on the lower part of this arm in the circuit. And when I put that in and model that as a diffusive process, then I get this linear tail for the diffusion. So it's only at low frequency. And as Bard chose you right here, again, you can extrapolate here is your offset frequency right there due to your r sub omega. Here is your total resistance, which would be r omega plus r charge transfer if you extrapolate the semi-circle down to the axis. And then here is the Warburg impedance, which comes in and changes that resistive axis and extrapolates back, if you can get enough data points there, to over the charge transfer resistance. So you can back all the terms out by doing this. And you can determine, if you have that shape, where you are in the system by where you sit on the shape. That is, over here you are charge transfer limited, and over here you are mass transport limited. Yes? So should that intersect then? The intercept is not the center of the semi-circle. This intercept right here of the linear part, the center of the semi-circle is still 1 over r times the capacitance up here. So this line here does not extrapolate to the center of the semi-circle. You do not have a diffusive component at this frequency. You're going too fast already. Wouldn't the center also be? The center is r, yeah. The center would be r omega plus half of r Ct. No, this is just extrapolating back. I believe this is just wonderfully played there. That's just 1 half of r Ct. All right, so what are we going to do with this? One of the big applications for this, by the way, is in corrosion studies. You have a piece of iron or whatever that is corroding away, and you would like to monitor the thickness of the corrosion layer on that piece of iron. And so since the thickness will change, the resistance of the interface as well as the capacitance where the semi-circle falls will tell you about the thickness of the corrosion layer. And so you can do some calibration curves. You don't even have to work out specific details. It's not important what the resistance in the capacitance is, but you just need a calibration curve that shows you how this semicircle shifts as the corroding layer gets thicker. And so you can monitor a corrosion rate by doing this, by monitoring this over a period of time. Yeah? Sorry, again. Let's go back there, yeah. But starting with strength and wind, that intercept happens in the left intercept. This intercept here? Yeah, it would have to intercept the x-axis left of the center of the semicircle. Yeah, you got a good point. In addition, it does work. Right, so I know it doesn't. It does do it to the right. And I'm going to have to go back and look at the equations. That I can't intuit. It's just not clear enough. Yeah, I'll have to go back and look at the equations. Apologize for that. So I'll get back to you on that. OK, so let's take this to a, you can get really exciting now with these circuits you see. So the problem that I've been trying to model all of these techniques on has been this nickel fairy cyanide electrode you will call. And so let's go back to that electrode system. And you remember that in that system we have an ion current, essentially. That is, we're pumping ions in and out of that thin nickel fairy cyanide nickel layer. And I had argued that there was an effective viscosity for the nickel fairy cyanide layer. And therefore, we had this time-dependent diffusion coefficient when we do a chrono-amperometry experiment. Because not only are we pumping ions in or out of the surface when we do a potential step, but we're changing the dimensions of the crystal structure simultaneously. And that changes the flow ability of the ions through that structure. That gave us this time-dependence to the diffusion coefficient. So the question is, yes, there's a time-dependence to diffusion coefficient, but is the physical meaning that I have given you for that correct? That it's actually the lattice parameter changing that's giving this time-dependence. And so we went back and said, well, if that's the case, let's go to a potential where a small perturbation will not dramatically change the number of ions in the layer, and hence will not dramatically change the structure of the layer. And let's see then what happens in terms of the diffusion coefficient. Because if our model's right, then suddenly we won't have a time-dependent diffusion coefficient anymore under these circumstances. So we pick a potential based on the cyclical tamagram for nickel-fairy cyanide, which you will recall. It's the nickel-fairy cyanide, where we're perturbing the 2-3-oxidation state of the ion. We're going to pick a potential here. Here's our cyclical tamagram. Nope, that's just not going to make it. That's a little better. Here's our E1-1-2 at surface confined, so it's symmetric, believe it or not. And we're going to want to run our AC experiment either over here or way over here. And we picked over there because we were concerned about corrosion out there. Why? Because if we do it somewhere around here, then we're getting massive changes in the number of ions as we change the potential there. Even if we made a small potential change here, we're making a big change in the ratio of oxidized to reduced, whereas if we do it over here, then the change is pretty small. And so we'd expect we're not making much of a structural change. So we use a small, say, a 10-millivolt AC perturbation sitting at the potential over here, around zero volts versus SCE. And we're going to look and see what happens then to that. And we'll do that as a function of the cations that are present to see what's happening as we'll change to different alkali cations. But before we do that, of course, we need an equivalent circuit. And so we found this equivalent circuit in the literature for another chemically modified electrode surface that we decided that looked pretty good. So again, we have our standard cell resistance. We have our double layer. Nothing too surprising here. Here is the charge transfer resistance. They call r sub r in this particular thing. There's your Warburg impedance. And now they have added in two more circuit elements to model their circuit. The first one is a resistance that's associated with the ions going in and out of a membrane. So there's resistance associated with that. So that's this r sub a for absorption of ions resistance. And it turns out there's also a capacitance associated with that absorption process. And so you get this nice complicated circuit now, which you can analyze, though. And again, what you expect is you expect a semicircle and a linear part. That is, we have the Warburg impedance and then a semicircle. And we're doing that now under two different conditions. So we have the little triangles, whatever that shape is down there, or what shape that is. But those open things down there are not triangles. Whatever that shape is, circles or whatever those are, that is with only sodium nitrate present in the supporting electrolyte. And then we have this solid line over here. Well, we have a mixture of sodium and cesium ions present in the electrolyte, and hence in the layer. And the first comment to make is you'll notice, independent of what ions are around, this part of the curve does not change. And that part of the curve, remember, has no diffusive component in it. That is, we're wriggling things so fast that there's not time for the ions to move appreciably, or certainly for the last to expand or contract. And so it doesn't matter what's around. Nothing's really happening over here. And then as we get into this Warburg part, you can see there is a big difference between whether we have cesium ions around or do not have cesium ions around. Now, there's another reason to show you this data, and that this is pretty typical data. This is pretty commonly what you get. You don't get things that look like that. That would be nice, but I think I've seen that once in my life. You get, on a good day, things that look like this, where you can say, well, I see enough of this semicircle that I certainly can extend it down to the axis there. And I certainly see enough of this that I could draw a straight line through it if I was forced to. And this line, by the way, is a little more complicated circuit. So the fitting is, this is a fit to the circuit. It's a little more complicated than a simple semicircle and a straight line. But this is not typical. And sometimes this line, you can see it could, totally dominates the spectrum. And you can't pull this out very well at all. That's not good. Anyhow, that line, like I said, is a fit to this circuit that I just showed you. And you can see we've got a very good fit. But then how could we miss, in a sense? That is, this is a 1, 2, 3, 4, 5, 6-parameter fit. And I couldn't go wrong, right? What do you do to go for the Wolverine heat? It's just fit. In other words, it's just a least squares minimization. And it changes, and you allow that to change the frequency. Yeah, right. The way you approximate this is, over a limited frequency range, you don't let it change too much. So you hold it constant. So on the one hand, it's a great fit. Both those lines have wonderful fit to the data. On the other hand, as I said, you couldn't miss. And so there's going to be now two big questions here. One is, is that fit physically realistic? That is, there are, presumably, with six unknowns. There are other numbers we could have popped in for those unknowns and gotten just as good a fit. So do we have the right set of numbers that we come up with with our fit? Number one, and number two is, is the physical significance that we associate with these various components reasonable? That is, is there, in fact, a charge transfer resistance here, for example, and is it, in fact, in series with this absorption resistance and whatnot? So the first thing that one gets is you notice that the resistance for the electrolyte is small and constant. That's good, because we certainly expect that. So that looks pretty good to us. We get a charge transfer resistance that is changing the cation, whether we're sodium nitrate or the mixture. That seems physically compelling, also, in that recall that the cyclic voltamogram looks different for the two cases. And remember, in the case of pure sodium nitrate, it's ideal, so it should be very fast charge transfer, lower resistance. And in the case of pure cesium ions, remember, it's a very non-ideal and broadened out suggesting charge transfer limitations. So expect the resistance to go up, so it has gone up. So this is not pure cesium. You notice this is 8 millimolar cesium ions in the presence of 1 molar sodium. Remember, we have to do that because of the selectivity of the surface. It's very high for the cesium ions. But so a little bit of cesium does a little bit of damage there on the charge transfer rate constant. So that, that seems to physically fit well. And now we have this resistance to associate with the ions moving in and out of the surface layer. And we would, of course, want that to change if we have some physical significance between sodium and cesium. And in fact, you see that that is changing significantly from 33 to 1.6 as we do that. We expect potentially some change in our double layer capacitance as we do this, but not a big change. That is, anything that changes on the surface can change the double layer. We're making a change on the surface, but we're not making a very big change. This double layer capacitance is a very sensitive parameter. And again, there is a little change there, but not significant. And in fact, if someone said, you know, you fit this wrong and these two numbers are actually the same, I'm sure there's another fit to this data that looks just as good as the one I just showed you that has these two numbers the same, I'd buy that also. We don't need this number for anything. This is just a check. If that number was fluctuating wildly, we'd just throw out the fit. And then we have the capacitance associated with the ions moving in and out of this double layer. We expect that to have a big dependence. It does. Now, so the model seems to meet reality pretty well. That's a check. The real question, though, is if we take these parameters now, in particular these parameters, and we generate the diffusion coefficients, what do we see? So we do that, right? And remember, the way we got the diffusion coefficients before was by chronocoolometry. So how does that compare with what we saw before by chronocoolometry? So first of all, how are we going to compare it to our chronocoolometry diffusion coefficients? Because remember, they're time dependent. Remember, they went as something we call d0 over 1 plus t, the quantity squared. So we're simply saying we will use this parameter up here as a one point parameter for this system, because we don't know what time to look at it otherwise. The argument is, again, that since we're making a small perturbation over here, it should be like a big step at early time. At early time, of course, the bottom collapses to 1 here. And so presumably this is what we want to look at for early time. And so we have the numbers over here, and you'll notice I need to make a big point of this before. But all these numbers are in the order of 10 to the minus 10, centimeters squared per second. That is a typical number for a solid state diffusion process, as opposed to something like 10 to the minus 5, which is a typical number for diffusion through a solution. So we know that we're looking at the ions moving out of the layer, because that order meant to this number can only be associated with the ions moving through the layer. So it's a solid state process. And we notice now that when we take this at early time and we compare it to these numbers here, that the comparison appears pretty good. There is some difference. But remember, we are doing a slight perturbation here, and we are making an approximation here. And so we conclude that this looks all very consistent, and that the model I presented to you last hour, where we have this change in lattice, the lattice expanding or contracting as we do the redox process, and hence the ability of the ions to move through the lattice changing, is consistent. And when we freeze out that lattice motion, if you will, which we're doing in this AC experiment, and we lose the time dependence, we're able to pull out a diffusion coefficient in the normal way. And that diffusion coefficient seems to match up pretty well with what we get for the early time chronogram. Yes. The top, this one here? Yeah, this one's 10 to the minus 11. Yeah. That's 50 times. And I think that's pretty close for diffusion coefficients. Yeah. I'm curious how the other ones are really close. Yeah, I mean, to some extent, I think we got lucky. This is for the oxidation. So we're expelling cations from the surface as we do that. I would have guessed if we were going to be off on one, it would have been one of these down here, because this is the pure case where we have a more ideal electrode. But getting within a couple hours of magnitude on a diffusion coefficient is typically good enough. That is, it would be nice if there was really good ways of measuring diffusion coefficients that got you the right answer within a factor of two or whatever, but we don't really have those. Now, it is quite possible that we haven't actually published this data. I'll tell you why in a second. That maybe we do have the wrong circuit here. Maybe that's it. The reason we haven't published this data is that it is a six-parameter fit. I'm a little leery to believe a six-parameter fit, even though I think it's right, based on just two sets of conditions. So really, one needs to go through and look at a wide variety of different sodium and cesium concentrations and make sure it always at least fits at least this good, if not better as we do that. The problem with doing that is once you get much higher in concentration than this, the cesium is going to start to swamp the sodium effect. So you need to go to very low concentrations of cesium, and therefore you introduce a new error, and that is your ability to know how much cesium you have in the system. That is, sure, you can go down to one million more cesium. We believe we can make a solution that was one million more cesium, but we've got to go a lot lower than that. And you start to run these problems with cesium, can be absorbing onto your glassware and your electrodes and things like that once you get to low concentrations. Remember, we can see nanomolar cesium. And so you introduce more errors, and so that's the issue that goes with that. But it looks like that we have something that is consistent with the model that I presented to you. And yes, a factor of 50 in a diffusion coefficient is within the error of the measurement, unfortunately. OK, questions about this? In the differential pulse, you only get a self-potential ground base. Yeah. Now, there are a whole variety of differential pulse techniques. Is there any advantage to one another, or any real? There used to be a great advantage of one over the other. And by the way, I suppose I should add to this. There's another technique, which is perhaps the latest differential pulse techniques that I totally ignored, which is so-called square wave voltammetry. Same idea, I use a square wave here. The big advantage used to be in the day before computers is you bought a machine, and this was hardwired into it, the various pulse sequences. And so you used what you had built into your machine. Now, generating a wave form is not so complicated. Even though there's some nice concepts about getting rid of baseline and low end and sometimes noise and things like that, it doesn't always work. And that's typically why you're doing this. So usually, you will just run through these techniques from an analytical point of view to use the technique that gives you the best signal. And you cannot sit here and say, oh, this one will always give you the best signal. This one will always give you the best signal. It doesn't work that way. You would think it should, based on what I said, but it looks like a nice sophisticated technique, but it doesn't. It's also not totally a true statement that all you can use this for is figuring halfway potentials. And I guess it goes without saying, because I guess I didn't say it, but the amount of material that you have there, that's typically what this is used for. You want to correlate some height here with a concentration. That's the primary use for this for pure analytical chemistry. I have x-millimoles or micromoles or whatever around. There has been some work. Some work has been done looking at various mechanisms and seeing how the waves either move in position or broaden with the mechanism. But you run into, even if you were going to do that, you run into the standard problem that you run into, say, with chrono-amperometry. And that is the shape of the curve that comes out does not have any significance to the eye. There is no pattern recognition going on there where you could look at it and say, oh, look, it shifted over to an EC mechanism based on the scan rate dependence or something like that. Because you're talking about subtle changes in the width of the curve. And of course, there are a variety of reasons why a curve might broaden out, changes in resistance, changes in mechanism, et cetera, et cetera. So there is one technique where you scan forward using this differential pulse and you scan back. And of course, you could easily imagine in the perfectly reversible system, everything lines up. And then as you get kinetic complications in those waves split a bit. But assuming you can see these sorts of things by cyclical tammetry, it would be infinitely easier and faster to do it that way. So you're going to do it that way. The only reason you would try and do a mechanistic fit to this data is that this is the only technique that would give you enough signal that you could see. And you were stuck with it, therefore. And it might be this technique, or as I said, I should write down, it might be square wave tammetry. And which one will give you a better result? Basically, you try it. Other questions? And I should point out, Bard goes through these techniques, I believe it's in chapter 5 again in a fair amount of detail if you do want to look at the equations, things like that. But they get very specialized. Their primary use again is for pure analytical chemistry. Very good then. OK, so we will do a quick jump into photoelectric chemistry next time. And then you get a little vacation. And Professor Lewis is going to talk about ultramicroelectrodes and some more information about the double layer. Develop that to a little further extent than I have. Very good.