nanoHUB-U Principles of Nanobiosensors/Lecture 3.9: Amperometric Sensors - Beating the diffusion limit by Nanogap Amperometry
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[Slide 1] Welcome back. In the last two lectures we have talked about amperometric sensors. You see, when the biomolecules land on a electrode surface, it initiates the reaction that changes the current flow and therefore, and then you can detect the presence of biomolecule through that approach. Remember, this is really, we talked about really glucose sensors, and if you think about all among the sensor that are daily used, it's about eighty percent of the market, billions of dollars worth of sensors; it's a very important sensor, but it has an important limitations and I'll discuss how a Nanogap amperometry, especially at low concentration, how a Nanogap amperometry can overcome that limit, especially the diffusion limits.
[Slide 2] So we'll start by discussing what the diffusion limit of a amperometric sensor is; explain how to improve our go around this limits by Nanogap ameprometry; explain how to sense DNA. We talked about glucose sensor, which is very good, but if you wanted to do DNA sequencing, especially at very low DNA concentrations, then we'll see how to use that; it's a very interesting use of amperometric sensors; then we'll conclude.
[Slide 3] Let's quickly remind ourselves that how the glucose sensor actually work. So the glucose sensor essentially had an amperometric sensors electrochemical cells in the back, and then on the front end there's a coating on this platinum electrode, and what happened the glucose came and the oxidize, glucose oxidize essentially allowed the enzyme, the red enzymes, allowed reaction to occur, hydrogen peroxide came out, and then the correspondingly that gave up the electrons and the current flow was detected. But if you think about the big picture, in the big picture glucose came and the current, there was a current in response. Now, what happened to the glucose concentration in, let's say, in my lab report, for example, generally that I showed in the beginning of the course, generally very high, so you don't have to worry about diffusion limits and other things. So in that case, life was simple; the only thing we focused on was that how many glucose molecule is there and how much current, current flow; then we saw that there is a one to one relationship.
[Slide 4] So this is sort of a equivalent picture then. We have an analyte come in, a bunch of enzymes, which facilitates the reaction, recognizes the specific and analyte we are after, and the product diffuses off, and the current flows. This is the basic picture of all amperometric detection, all amperometric sensors. So now we can think about these basic pictures and see how it would be modified when the analyte concentration is extremely low, there's a picomolar a femtomolar. How would things change in that case?
[Slide 5] So the way things would change is that, remember, there was this Redox reaction going on; reaction oxidation reaction going on, on a particular electrode; this is, let's say, the platinum working electrode; the other electrode I'm not showing here, and then as a result there was a net current flow. Now, of course, if the concentration is very low, then what will happen is that the molecules will essentially diffuse towards the electrode, and this is the bulk concentration if you wait longer and longer, then this initial wait would be certain value, then it would be longer, and it would go as square root of Dt. You may remember from the first set of lectures that any time you have low analyte concentration, the depletion goes as square root of Dt, so this would go as square root of DT; and the total flux will be proportional to this proportional to this slope of this concentration profile. Now, this time around what is special is that now you have also a product. Initially we didn't have a product; molecules came in and essentially was captured by the sensors surface; that's how we assume what we assumed. But here you see there'll be this oxidized product, and the product would also have to diffuse away because if it doesn't diffuse away fast enough, then there will be a reverse reaction, and you may remember that that was the essence of thus Butler Volmer equation, so you see rho sR; this is the surface concentration of the reduced species, kf is the reaction rate. So this part, without the second term, is just fine; it says that whatever flux is coming in, current is proportional to that, no problem. The problem is that there is also a negative term associated with the oxidized product, and so therefore, what happens that if you don't move it fast enough, if the concentration builds up, the pink concentration builds up, it would drive a reverse reaction and the reverse reaction would subtract from the current, and your current sensitivity would go down. So we want this to be fast, R to diffuse fast, and O to diffuse fast as well. So there is a diffusion limit here that sort of controls the total current I. So we will do the math; you'll see in a second how it works out, but this is the basic physics picture.
[Slide 6] Alright, so the thing is that in this case you can immediately see that you can get into big trouble very soon because as the concentration is getting depleted with the certain square root of Dt, the gradient would gradually become smaller. Remember the one dimensional diffusion limit and therefore the flux would become smaller and smaller. Similarly, and to add to that trouble, is that the oxidized products will also not diffuse fast enough, right? It will go as square root of DT, so initially to the fast diffusion, but it would gradually slow down. So as a result what will happen that this reaction will saturate very quickly with a tiny amount of current. You know, in typical structures you can only have probably a pico amp of current, so it's not really very high. What can you do to beat it? How do you beat diffusion? Well, here's a trick that I found sort of very beautiful. So we still have the bottom electrode just like this one, where the reduced spaces gets oxidized, which is fine, the same one, but now I have added a second electrode with two little gaps here. The reference electrode is not shown; they're somewhere else, and, but that's essential really for the circuit to complete. So now assume the same molecule has come in, the same reduced space has come in, and on this bottom electrode it has become oxidized, so this pink is the oxidized one. Now, when the pink goes to the other electrode, the other electrode converts it right back, and so therefore, it goes from oxidized to reduced, so it takes the pink molecule and makes it green. Now you can see how things would build up; the green will come to the bottom electrode and it will convert it to a pink, and this osillation will go back and forth until, of course, the original molecule escapes from another escape route. So now you can see, previously I had just one reaction and corresponding to that I had one current. Generally this current is on the order of femto amp or so, for typical gaps would be like a femto amp. Now because of this constant shuttling back and forth, see how many times the green molecule has come on the bottom surface. Here I have shown three; typical is on the order of four micron sized-- sensors, and this gap is, by the way, on the order of 15 nanometer; that's what the Nanogap amperometric comes in, and only nano technology can allow this type of gaps to be fabricated. You see, now this one can be as high as 400 to 1000, so the signal has just been amplified 1000 times because you have beat the diffusion limit; you have-- and you can see this in a schematic picture like this. So this is my original electrode on the left R to O, and this is the new electrode which sort of bring things back; and you can see that there is a diffusion front going on for the green, but this is clanged to W. It cannot keep going on because as soon as the rate molecule comes here, the pink molecules, they'll be converted back to green, so this concentration never changes, and as the result the flux always remains very high. Also, for the pink one, you see, the concentration never changes because as soon as it comes here it will immediately convert it to green. So you can see that this will have a tremendous amount of amplification, compared to the other structure. So this is a recycling, of course. If you are thinking about plant and animal, we constantly do this recycling, not in the Nanogap, of course, but the oxygen and carbon dioxide, the exchange between animals and plants; essentially you can think about this cycling going on.
[Slide 7] So what would be the enhancement? Well, you can see in here, if it was sort of a parallel plate electrode, in that case the current would be set, would be reducing as the square root of D naught t, even if-- square root, and therefore, the current would essentially saturate; it would be a very small value of current. In contrast, in here, the concentration is here fixed at a distance W, and if W is significantly smaller than square root of Dt, then you can have a significant amplification. As I mentioned, that if the standard one has about the pico amp, this green one with cycling can have tens of nano amps, and therefore, this one has the ability to detect concentration, which are far lower then compared to that other case.
[Slide 8] So let's do some quick math to see because we need things quantitative a little bit, so this reaction redox current here is related to the flux that is going in to this electrode. How do I calculate it? The slope of this line is rho AR, is this concentration at this point for the green; rho BI is the concentration on the other side divided by W, that gives me the slope. D is the diffusion coefficient, A is the area of the electrode and Q is the charge. So that gives me the total amount of flux that is diffusing towards the electrode. And similarly the amount of pink molecules that are diffusing to the other electrode from A to B is given by an almost middle line formula, except the concentration positions have been exchanged. Now, once I know the concentration of the pink molecule and this electrode, electrode A, and the green molecules on electrode A, then I can use the Butler Volmer equation to know how much current will be flowing. And similarly, in B, once I know the pink and green electrode, I can correspondingly calculate how much current would be flowing through electrode B, and this would be the corresponding-- Butler Volmer equation this time for electrode B. Now, take a quick look. This f, the f is given by the faraday constant divided by RT; RT is the universal gas constant. Eta a, is the electrode, the voltage applied to the electrode A, and eta B is the voltage applied to electrode B, and E naught is the corresponding formal potential at which the reaction begins, so therefore, all these things can be found in any handbook so you can put the corresponding value in for a given electrode and solve these four equations, four and non four equations, and I'll show you what the result is.
[Slide 9] So if you essentially solve the four equations, the final result would be something like this, so you can see that most of them are actually constants because these are voltage, eta A is the voltage applied to electrode A, eta B voltage applied to electrode B; so most of these things are constant. We can forget about it. One little thing I should look at is the maximum current that I can have, the limiting current, and you can see it's immediately obvious what this is; that this is inversely proportional to W. As you make things smaller and smaller, you can have more and more recycling, and so therefore, your current will be larger and larger. That's one, and there's also a diffusion limit between these two; D over W is essentially the diffusion equivalent capacitance. You may remember this from the first set of lectures, and so we'll use the same formulation here, and so that gives you a total amount of recycled current for a planar, parallel plate planar electrode, which is good. And again, as I said, three four hundred times amplification, the current is possible. Alright.
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[Slide 10] So now if your electrode were more complex; you know, these days nobody uses parallel plate electrode. You want to have tiny nano structured electrode with diameter on the order of maybe 10 nanometer to 20, 30, 40 nanometer, very, very tiny electrodes; and nano technology, nano fabrication allows you to fabricate those structures. So in that case, only thing you have to do, you see, whatever capacitance you had, if you look at the previous expressions, you have D over W here; we'll just replace that with CD SS, and CD SS you have seen before. And so, only thing that we need to do is that wherever we have the diffusion current, D over W, we'll replace it with CD SS, the Butler Volmer equation remains the same, calculate the whole thing, see there's only two places where things have changed. There was a D over W here, C D,SS has replaced it; D over W here C D,SS have replaced it. Everything else is exactly the same. By the way, the derivation will be in the appendix, so don't worry about it; just look at the basic, basic formulation.
[Slide 11] So see how powerful this can be. For the CD SS for the parallel plate electrode, you simply put this diffusion equivalent capacitance in here; you'll get the standard formula, that's not a problem. But even if you had two tiny nano electrodes like here, and you put it in the solution, very complex fluxes are going on, redox reactions are going on in the solution volume. Looks very complicated, it looks like it's very difficult to solve it by any numerical method, and yet, I know what the C D,SS is for two electrodes; I just looked up in Wikipedia and A naught is the radius of this electrode, W is the separation between the electrode, D is the diffusion coefficient, and L is the length of the nanowire, of the nano-- electrode; and all I have to do, if I wanted to know what type of current amplification I will get from this system, take the C D,SS, put it in here, done, in one shot the entire solution, I'll get one entire solution in a second.
[Slide 12] And in fact you can show how good this is by solving the numerical analytical result. For example, if you have a parallel plate electrode, you need to apply a certain voltage before the reaction begins, and you'll have a certain amount of amplification, certain current. By the way, if you didn't do this recycling, your current would be essentially unmeasurable, very small; so it has already given you amplification. However, if you do a concentric electrode, so redox reaction going on between the center electrode and the concentric surface around it, you'll get significant amplification by a factor of four or five; and if you do a two nanospheres, sort of, in that case, the reaction will be even faster. And so at a given voltage, depending on the voltage you want to operate it at, and to avoid parasitic reaction, if you start applying too high of voltage, there maybe other reactions going on, so it's always good to operate it at lower voltages and you can see this type of Nanogap ampoerometry with different nano structure geometry can give you significant advantages.
[Slide 13] So here is a numerical simulation. Under here are the two electrodes that you are seeing, here are the redox reactions that are going on, the blue is one pole; you can sort of think about, and the rate is another pole. Look at the complex flow of various diffusing and oxidizing species and if you look at the analytical result and the numerical results and symbols and lines respectively, things are very well reproduced. A simple formula based on diffusion equivalent capacitance solves a very complicated numerical problem associated with this pair of redox electrodes.
[Slide 14] Alright, so that's the first part in which we said how to beat the diffusion limit. Now, let me show you very quickly then in the next few slides how to detect DNA using this. This is a new research direction, and you should be able to understand it easily now. Remember the picture. The analyte came, product went out as a function of enzymes. If you didn't have any red enzymes here, there wouldn't be any reaction, right? So therefore, one way to do detection is to modulate the amount of enzyme rather than modulating the amount of analytes. Let me explain how it works. So assume that I have located the platinum electrode at 90 degrees and these biomolecules essentially act like an enzyme, the rate biomolecules, and initially on the sensor surface you didn't have any enzyme; so there was these analytes around, but no reaction was happening. Then you have one enzyme, red enzyme landing, then you have a bunch more landing on the sensor surface, and as they begin to land, this reaction gets initiated through the enzyme, current begins to flow, and therefore the current would now be directly proportional to the number of enzyme molecules, quote end quote, enzyme molecules that have landed on the sensor surface. Alright. Now, by the way, this land, this enzyme molecules could be DNA, could be protein, anything that initiates the reaction; you have to choose, of course, the correct molecules, reaction molecules that would recognize this process, would do.
[Slide 15] Alright. So do you remember this particular equation that enzyme kinetics? What was the equation called? This is called the Michaleis-Menton equation, and you may remember, in the last lecture we discussed it, one over i ss. Previously we focused on the number of analyte molecules coming in. For us last time one over i max was the constant, right? That was proportional to enzyme. This time around, this is our target of interest because we'll assume that there are lots of reaction molecules around; it's the enzyme molecule which is rate limiting, so we'll focus on this. Now, the i max is related to qV max. V max is the maximum velocity or maximum rate, and you may remember this proportion from the last lecture, proportional to the number enzyme molecule, and therefore, proportional to the number of DNA, so the number of red DNA that they're coming in is making the current i ss larger by proportion.
[Slide 16] So here is how it works. Assume in the beginning you have the same two thirds Nanogap sensor, two electrodes, both of them are working by the way; the reference electrode is somewhere else, and this chamber is already filled with reaction molecules, which wants to react on the electrode surface, but it cannot because the enzyme molecules is not present, the barrier is too high, so the reaction cannot happen. Here comes the DNA molecule. The DNA molecule comes and lands on the electrode surface, and then as a function of time, more and more DNA molecules come and land on the reaction surface. Once that happens, they say the biomolecule, which is there, the reaction molecule, green reaction molecule already present, but couldn't react before, now will start reacting; and because, and the reaction rate will be proportional to the number of red molecules, and so therefore by this constant cycling you will see that there'll be significant amplification of the current that can be obtained by the structure, as I said, three to four hundred times; in this particular example, only a factor of three in this cartoon. And so, this is the essence of how you can have DNA detection at ultra low concentration through this redox recycling
[Slide 17] or redox cycling All I have to do in this case, therefore, I can, again, write the full system expression; Butler Volmer equation is exactly the same. Notice the concentration of the reaction molecules at electrode A. Only thing that has changed is the pre-factor is enzyme dependent, our concentration of DNA dependent so the pre-factor has changed, but everything has remained the same. This is what allows faster reaction in the presence of a DNA, and only thing that the second electrode does is essentially recycles the product initiated by the red molecules, which in this case are DNA.
[Slide 18] So if you wanted to write out the steady state expression for DNA, then this would be number of equal to the number of receptors, DNA concentration, you know, femto molar, pico molar, rho DNA would be that concentration, you put it in. kM is already known from the reaction electrode platinum in this case, for example; dissociation, association and dissociation rates; these are all known, the kM values are known, and all you have to do is change the K naught value in the previous expression. Everything else is exactly the same. C D,SS, everything the same. The only thing that has changed, because of the DNA biomolecules have come in, the K naught has changed. And this is how you detect. So this change in the K naught, it's reflected in the change in the current, so this is how you detect DNA molecules by amperometric sensor.
[Slide 19] So let me conclude. I tried to explain in the last three lectures that amperometric sensors are broadly used as glucose sensors and they're very important, and it can be a sensitive monitor of analyte density, and even can detect DNA, especially if you have Nanogap sensors with a finite gap or you can interdigitate them, the two electrodes, so that cycling can go on in the entire interdigitated structure, and that can give you huge improvement in the signal at, even at very low analyte concentration. But remember, that still the biomolecule has to come first, that it still has a diffusion limit; once it gets trapped into the Nanogap amperometric structure, then the recycling would happen, but the molecule has to come first; and so, therefore, still subject to the diffusion limit that we talked about before. Now, this enzymatic reaction provides a built-in selectivity because you see, only that reaction can be initiated, which is enabled by this enzyme of DNA, in this particular case. So this gives it a very good selectivity, a topic of great interest that we'll discuss in the subsequent lectures, but it doesn't mean that it has solved all selectivity problem. If you start putting more and more voltage in the hope of getting more and more current, other molecules, which are just lying around, then they can begin to react with the electrode, and if that happens, then you see your signal current contaminated. So the selectivity is improved, but selectivity problem is not solved in this particular case. So these are the two important things to remember for these amperometric sensors. So we have talked about potentiometric sensors; now we just finished talking about amperometric sensors that how at nanoscale things are changing dramatically. In the next set of lectures, we'll think about cantilever sensors to small springboard essentially on which a biomolecule will come in and change the resonance frequency, and that will, and once we finish that will set-- complete set of three. So until next time. Take care.