nanoHUB-U Principles of Nanobiosensors/Lecture 3.2: Potentiometric Sensors Charge Screening for a Planar Sensor
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[Slide 1] Welcome back. We are talking about different types of biosensors in this segment two of the lectures. As I mentioned the first lecture, first set, had about 12, we talked about how long it takes for the biomolecules to arrive on the sensor surface. And in this set of nine lectures in part two, we are discussing different types of biosensors. And we'll be focusing on three, you may remember, potentiometric biosensors which is a sort of, "camera" for the charge. So, it says the charge. It has to be charged. The biomolecule must be charged. And then like a DNA or like a protein, and once the charge comes, it changes the potential of a MOSFET or some other types of potentiometric sensors and new sensor. There are of course amperometric and can deliver with sensors which we'll discuss later on. Now, if you remember towards the end of the last lecture, I mentioned this dilemma that the sensitivity is actually logarithmic for a MOSFET, as a function of annelid density biomolecule density where as we expected a linear dependence. My goal in today's lecture is to explain where that logarithmic dependent came from. We already gave you a clue that it has to do something with the salt, but I'll explain exactly where-- what does the salt do and why you have to sort of have salt everyday if you want to survive.
[Slide 2] So, I'll explain this difference of a MOSFET and biosensor. They look about the same but they are not the same. There's this-- gate has been shifted as a reference electrode now and the fluid has filled. The water-containing salt is filled within the-- on the gate surface. And so therefore, there is an important distinction how a MOSFET works and how a potentiometric biosensor work. We'll talk about charges, salt and screening. I'll start with a simple theory that will get us part way into the-- towards the direction we want to go regarding explanation and role of salt, but we'll see that that's not really good enough, and I'll explain why not. And a little bit more sophisticated theory the results are actually derived in the appendix, but I will give you a feel about how the theory works. So, it's sort of relatively simple, few lines of algebra, and then I'll conclude.
[Slide 3] Now, you may remember just very quickly that we have two configurations before the analytes land on the sensor surface and after the analyte has landed on the sensor surface on top of the gate insulator which is shown here in light blue in both cases. And the change in the current related to the original current before any molecule have landed, that defines the sensitivity, and we are after trying to calculate, the sensitivity classical theory says that this should be linearly proportional to the analyte density, and we found that that's not really the case experimentally. Optically, we found that this is indeed the case. The molecules are arriving on the sensor surface with the correct number and correct rates. So, there's nothing wrong in the capture equation itself. There must be something wrong the way these charges are being reflected on the channel. There is where we have to focus a little bit more on.
[Slide 4] Very quickly, I really want to remind you that the necessity of the salt essentially came by-- the reason was that a DNA string is negatively charged, the target DNA is also negatively charged, they will simply rip each others apart if you didn't put salt. We would-- our self would self explore in some way if we didn't have salt because the force is so strong, 4 billion charges on a line with another 4 billion charges on another line and separated by 1 nanometer distance. That is how close they are and how much are there is, huge amount of force. So therefore, without the salt screening this interaction, we would not be able to sort of keep them together. And an important point was that in many cases, we will-- a quick note is that in many cases, we will be drawing this DNA as a blue charge. And from now on, I will be drawing this salt on top of it. But you get the idea that the salt is actually in between the DNA strand, sort of keeping them apart and keeping their interactions apart and reducing the repulsion. So, although the picture is like this, you realize the real scenario is slightly more complicated.
[Slide 5] Now, the first thing you have to understand is salt is ionized in water. Now, that may sound strange. I mean, why wouldn't it be? But if you think about it, your table salt is solid. It is not ionized in water. If you look at the crystal, solid crystal, the sodium and chlorine are nicely arranged in a nice three dimensional square lattice. And therefore, our cubic lattice, and therefore, is very surprising that the salt, as soon as you put it in water, then they essentially dissolve each other, dissolve away from each other. And the reason is actually the mutual interaction. Because you see, if you look at the potential between the sodium and chlorine, q squared divided by 4 pi of sin and r, in water, the dielectric constant is close to 100, like 78 or so. So what happens that although you had, let's say, 3 to 4 eV of-- or about-- for sodium chloride, it's about 1.5 eV. In water, as soon as you divide by 100, essentially it turns out that the interaction is so weak that the water molecules just walking around, diffusing around, can bomb them apart because their interaction has been reduced considerably by the dielectric constant of water itself. As a result, what happens, that the sodium and chlorine, essentially they move in a fall of-- in a fall. So, the chlorine might be just 2 nanometers away but the sodium wouldn't know because the interaction has been screened by the dielectric constant of the water. Now, this distance where beyond which it cannot see, for water, it is about 0.7 nanometer or so and it's called a Bjerrum length, essentially explaining that how long does it have to be before, what is the distance, beyond which the charge is essentially-- will be separated. They will not be sort of attracted and form a crystal anymore. Now, this is very important because if you put solid salt in DNA, to DNA, that wouldn't have really worked because the positive and negative charges will be together, and essentially as if you haven't put any charge at all. What happens in reality, that as soon as you put these blue charges, the DNA is negative just on salt to be. And therefore, all the sodium ions, they are all free now, remember. They will get close. And all the chlorine atoms, negative, repelled, and they will be pushed back. Therefore, this association is very important for this scheme to work. And as a result, what will happen? That the sodium ions will essentially, this magenta line will pile up, trying to sort of screen the original biomolecules in blue. The chlorine, whatever is the equilibrium concentration was close to the surface, it will be less now because the molecules would be pushed away. And we have to sort of account for these extra charges because now, they will be sinking some thin lines. This salt will eat away some of the charges, "therefore less will be available for the MOSFET". And you see, that is where we will see the logarithm independence gradually are coming from. That when you increase the biomolecule concentration by a factor of 100, the MOSFET has a smaller and smaller share as the density goes up, and hence, the log dependence. Now, this concentration is easily understood. It is given by essentially the number of articles you have in equilibrium which is the concentration, bulk concentration and how to get those are numbered. Bulk concentration is expressed in molar. Otherwise, you can just simply count the whole thing and say it is numbered by centimeter cube. I guess density would be fine. This is just one way of writing it. And it depends by the Boltzmann equation exponentially on the potential. So, as the potential goes up, it goes to the charged surface. Remember, if you try to bring in biomolecule close to a charge, gradually the potential goes up because the charges are being repelled. And in that case, it will go exponential. The carrier density goes exponentially with the potential, called the Boltzmann relationship. So, we have an expression for N and we have an expression for the sodium and then chlorine, we are ready to go.
[Slide 6] Remind-- just quickly to remind ourselves how this is different. The calculation that you are going to do is different from the calculation I did in the last lecture is that in last lecture, we didn't account for the salt. And so therefore, whatever was the biomolecule concentration in blue, the charges on the channel was, the green, was exactly the same. And therefore, we said that the Q biomolecule is equal to the Q channel charge, and therefore we expected this current to be directly proportional to the Q biomolecule, number of biomolecules, mu VD over L is simply the velocity. We will not worry about it because it will always be the exact same number. And when you take a ratio for the sensitivity, remember this will drop out. So, we will not worry about-- worry about these constants too much, focus on Q bio.
[Slide 7] Now, with salt and electrolyte which are dissociated and charged, we now have a second capacitor on the other side because this charge, blue charge, some of it, this charge can sink at the sodium sides or the chlorine sides, depending on whether it's a positive charge or a negative charge. And whatever doesn't get sunk on this side, on this yellow will come to the MOSFET side. And some of these two yellow must be equal to the blue, which is the biomolecule charge. Total amount of charge must be conserved. So, let's do this calculation, two capacitors essentially in parallel. It is simply high school algebra. We can find out that the Q charge is equal to the charge in the left and charge in the right, two yellow charges. We'll call it double layer because-- and double layer capacitors multiplied by sin because Q is CV, and C oxide multiplied by V, that gives you the charge in the MOSFET. Solve for phi simply by solving this particular equation. And once you know phi, we can find out how much charge is on the MOSFET which is C naught multiplied by phi. This is phi, this is C naught, gives you the total current. And now you can see that because of the CDL, the salt itself, the response would be a little bit lower. Previously of course, the whole biomolecule charge was coming in. Now, only the ratio of C naught divided by CDL plus C naught, only a fraction is now available for the MOSFET to operate with.
[Slide 8] Now, let's quickly calculate CDL because I'm telling you all this, that yes, this salt will eat away certain amount of charge. I didn't really tell you anything about-- anything how to calculate it. It turns out very simple. You have to-- a few lines of algebra, for Poisson equation is something that relates charge to potential. So, we'll have to solve the Poisson equation. If you are not familiar, you can look it up, yeah, nearly most college physics textbooks have it. If not, then I will-- can also explain it in the question and answer session. Bottom line is the potential is related to the net charge, the double layer charge, and this is the dielectric constant. Notice the K, kappa w which is the dielectric constant of water. Remember, the salt is dissolved in water so it is not 1 anymore. K is not 1 anymore. And the total charge depends on how many, at a given location, how many sodium ions you have and how my many chlorine atoms do you-- chlorine ions you have. The difference of this will give you the net charge. I already know that the value of N minus and N plus from the Botlzmann relationship. And I also know that on the sensor surface, my potential will be psi naught. And then very far from the sensor of course, the charge will have no effects so the potential will go back to 0. With these two boundary conditions and these two values in, I can put it in. And once we have put it in, then you see e to the power of x minus-- e to the power minus x. That will be reflected in this psi hyperbolic. And the psi hyperbolic, when psi is relatively small with respect to kT. Let's say we are talking about very weak salt, very low concentration salt, in that case, psi hyperbolic, you can essentially expand it, write it out, and it will come out as psi e to the power of x essentially is what you're trying to, psi hyperbolic is e to the power x, sine e to the power minus x. You're trying to sort of find the difference of these two quantities. It will be directly proportional to x and the remaining constants. All these things floating around salt concentration, dielectric constant of water. All will hide in this length called LD, the Debye length, because Debye was one of the first people who came up with this type of theory. And the LD depends inversely on salt concentration, and other. There are other constants floating around. But the bottom line is the potential under these conditions when the salt is weak, you see, you know, in that case, it will simply be given by psi naught and the potential will decay as x to the power LD. LD is the Debye length. And the Debye length can be large only when the salt concentration is weak. I not is weak and so correspondingly, that solves for this potential.
[Slide 9] Now, that's a lot of calculation but let me quickly tell you that you can view the whole thing much more simply. Let's ask ourselves this question. Yeah, the charge is distributed in a complex exponential form, but what is the centroid of this charge? Effectively, where can I view the centroid of this charge? I can do that relatively quickly because I know the rho DL. Rho DL is simply n plus and n minus, the difference of it. You put the values from the previous slide. And then once you put it in, there'll be-- everything will be a constant, except there'll be an x dependence through the exponential. You put this quantity back in, in order to get the centroid. We make sure that we multiply with x with the charge density and divide by normalized by the total charge. And then that turns out to be very simple, that the centroid actually is 1 Debye length away. What does it mean? Well, what it means, that you can forget about all these dependences. Only thing you have to think about as if there is a separated second electrode, and the second electrode is only LDV. And what is the capacitance associated will all these biomolecules distributed and all those things? All you have to do is to simply remember that it is essentially a dielectric with dielectric constant of water separated by the Debye length. That's all you need. Now, once you have this, let's see how the rest of the thing works out.
[Slide 10] So now, we have two capacitors. One capacitor associated with the salt, sinking certain amount of charge, that is the CDL. And then of course I have the MOSFET which is C-OX. It's easy to calculate that what fraction of charge will be sitting on the MOSFET itself which will simply be, again, do you remember two slides before, we said that only a fraction given by this ratio of the capacitances, only a fraction of the total biomolecule will show up on the MOS side, on this capacitor. Putting the values CDL is given by 1 over LD, and remember, KW is very big, close to 100, 78 or so in water, and LD is relatively small. And once you have sort of walked down through this algebra, you'll find that this QMOS is directly proportional to the Debye length. Well, that doesn't tell me much except when you realize that it says that LD is inversely proportional to the salt concentration. So therefore, what this results immediately is telling us that the sensitivity which is proportional to the charge is actually linearly proportional to the density but inversely proportional to the square of salt concentration. This is the first time the issue of salt is coming in. Previously in the pictures, salt didn't play any role at all. But you see, we haven't really solved the original problem. The original problem was this linear dependence of rho naught. We haven't detected that one yet. Yeah, salt has come in which is good because salt is important, but our original problem of linear dependence remains. In the next three slides, I'll tell you what really-- how this log dependence really arises. Now, one thing you realize, as we increase the salt concentration, more and more charge will essentially sink on the electrolyte side and therefore your sensitivity will go down. That makes sense because you have less and less left over for the other capacitor to take care of or use.
[Slide 11] The real thing is that where I trick you a little bit for the sake of giving a physical picture is that the salt concentration is not low. Salt concentration terms that were very high, 100 millimolar is what you need in order to keep the DNA together. They don't want to be together. And without a physiological condition, in our body, it's about 100 millimolar, very high salt concentration. And when the salt concentration is so high, you cannot take away the sine hyperbolic and replace it with a low density approximation. You have to keep the whole thing. And the algebra is a little bit complicated. But you see, the physics is actually transparently simple. This is how it works. In the local concentration, the charge in the double layer was proportional to the capacitance of a double layer and psi naught. psi naught is the potential at the biomolecule surface, the blue charge. And we saw that this is directly proportional to-- inversely proportional to the LD and the relationship was very simple. And the proportion between charged and potential was linear. It turns out that when the charges are very high, the salt concentration is very high, this linear dependence doesn't fall because you cannot make a small psi naught approximation and you will have to keep the whole thing. By the way, the word z, z here means the valency of the salt. In sodium chloride, one positive, one negative, z is 1. So, if you have other salt which may have two positive and two negative, one is a dissociate, in that case, z would be 2. Now while the pre-factors all have the salt concentration and other, these are all discussed nicely in the appendix so don't worry about it for the time being. Now, you remember the total biomolecule charge must be equal to the MOS charge and the double layer charge. These two must balance. Global charge must be equal to each other. And therefore, I know that MOS charge which is the oxide multiplied by psi naught. And I know the double layer charge from here in the first line, I know that. This is a horrendous nonlinear equation. You need a computer to solve it, and we don't like computers too much in here for these sort type of problems. What turns out, that when salt is very strong, the second term is so big, you can sort of drop the first term. And we'll correct for it in a second, but assume that the second term-- first term is so small that it does matter. Most of the biomolecule charges are being balanced by the double layer.
[Slide 12] Ok. That simplifies our life because in that case, the biomolecule charge is directly related to psi naught. I can get the psi naught by solving the problem, and then I can just go back and put this psi naught into the MOS equation which is small. So therefore, that one wouldn't really change the numbers very much. And I get the expression, now you see appearance of a log. It came from the strong analyte dependence of the screening. And Qbio, well, we already know the biomolecule, total number of biomolecule is proportional to Nt. This is the charge for individual biomolecules. Nt is the number of particle you captured. Remember, in the first 12 lectures, we spent a lot of time and the result was proportional to the analyte density. It had the fractal dimension and all those. Remember the formula we derived, we just put it in.
[Slide 13] And once you put it in, now you see and write it out. Remember, the sensitivity is directly proportional to the MOSFET charge. All these log dependencies, log of rho naught, lot of salt concentration, log of time, fractal dimension, they all appear. Just simply follow simple basic, simple physics. There's nothing few lines of algebra and the experiments simply comes out transparent, transparent, becomes transparently clear that what happened, where did the log dependence come from. So, this logarithim independence comes from because as the molecules are sort of landing, more and more are sort of screened by the salt and the MOSFET is getting smaller and smaller fraction of it. That's why there's a log dependence. Why is there a log dependence of time? Well, it's the same reason as the log dependence of density. there is a function of time, as they are collecting more and more biomolecules, then correspondingly the screening becomes stronger in the salt side, you are getting fewer and same is true for increasing salt concentration. They have essentially taken every-- most of the charges associated by a molecule, leaving a very little for the MOSFET to play with.
[Slide 14] So, let me conclude. So, as you remember that the biomolecules, that the potentiometric sensors work with have to be charged. If it's not charged, biomolecules will not sit. And so, therefore, the analyte doesn't exist. So, it must be charged and that means DNA, that means protein, this type of charges, it may not be virus or bacteria or glucose may not work with it. Second is that I explained that this type of many-- most biomolecules actually can only survive in the salt-- with salt because salt essentially screens them in water. Salt screens their interaction. So DNA cannot stay together without salt and protein will completely fall apart. There's all the shape gone if you don't have salt. So, salt is essentially the essence of life in this particular case that mediate this interaction, which is good for life, it's good for me that we can survive with the salt. But the tragedy is that in the case of a potentiometric sensor, this salt is essentially taking most of it and leaving a little bit left for me to measure using the potentiometric sensors. And therefore, the sensitivity doesn't go up as quickly as you might have expected. And this screening depends on complex interplay of multiple variables, right? Temperature, dielectric constant, how the salt goes, the size of the biomolecule-- size of the salt molecule themselves, I have explained the very essence of the problem. However, if you read books, there are big books explaining the details of the physics but the essence is no more difficult. So, I will end here for this lecture. In the next lecture, we will be talking about cylindrical nanobiosensors, the effect of screening on them. You see, this is planar sensors. Maybe in cylindrical sensors, things are very different. That would be next lecture. Until the next time, take care.