nanoHUB-U Principles of Nanobiosensors/Lecture 2.4: Settling time Sensors with Complex Geometry
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[Slide 1] Welcome to Week 2 of the course. In the first week, we discussed why nanobiosensors are so interesting for a wide variety of applications. And we essentially focused on 3 things, 3 essential things. One was the settling time, how long does it take for the biomolecule to diffuse around and, eventually, land on the sensor surface? And once it lands on the sensor surface, then sensitivity becomes very important because every sensor, some sensors respond at low concentration, some at a slightly higher concentration so that's very important. And, finally, there is this question about selectivity that we want the sensor to respond only to the analyte that it's looking for, not to the parasitic-- other molecules that may be floating around. That sensitivity-- selectivity is fundamental to this question about false positive or false negative type of analysis. This week, we will continue with this discussion about settling time. We're thinking about how long it takes for the biomolecule to come and land on the sensor surface.
[Slide 2] Now, let me remind you the essence of last three lectures, and essentially what we saw that there is a fundamental limit to detection, this rho naught. How low an analytic-- analyte concentration, or a biomolecule concentration can you detect and that depends on 2 things. One is the settling time, how long are you willing to wait? But, equally important, it also depends on the shape of a sensor. That's surprising, that's a little bit of a surprise. Now, that it depends on the how long you are willing to wait is not surprising. Let's say, my child is running a high fever. I put a thermometer in her mouth, 10 seconds later, the temperature is right there because she's running a high fever. It's very quick. On the other hand, if she has a low-grade fever, then I have to keep requesting her, keep a little in your mouth a little bit longer, a little bit longer because, if I don't wait too long, or if I not wait long enough, then I might even miss that she has a fever. So, therefore, this reciprocal relationship between the quantity that is being measured and the time that we allow to do the measurement is fundamental. There's nothing really fundamentally new about it. What is new is this shape dependence, the dramatic dependence on shape. And what we did, in the last class, 2, 3 classes, was to show that it works very well, describes the response very well for the nanobiosensor, whereas the fractal dimension is 1 or a planar sensor. And it explains why nanowire sensor are so much more sensitive. What I'd like to do today is to show that, even when the sensor is far more complicated, for example, this surface has a random network of nanotubes. Even in that case, this seemingly simple, but elegant, formula continues to hold. So that will be the purpose of the lecture today.
[Slide 3] So, the types of complex network we'll be talking about are, essentially, 4 to begin with. One will be an array of one dimensional nanowires or nanotubes. Another could be, essentially, two dimensional grid. The third could be a set of the cantilevers of different length. And the fourth could be a random network. I'm going to discuss just these two because that essentially captures the essence of the problem. Now one thing I want to highlight from the very beginning is that you see the problem we are solving is that how long it takes for the magenta molecule to come and land on the sensor surface. How the sensors actually detect it, whether it changes the current from the source the one green source to the out drain or whether use an amperometric method or whether these could be a set of cantilevers. We are not really asking that question right now. We'll be thinking about that down the road. We are asking a more fundamental question. Given the geometry, what is the response time? How long does it take before a certain number of molecules have been captured?
[Slide 4] Now let me then, in order to do this, let's go and recap the procedure, the algorithms of how this problem should be solved. And this has to do with diffusion capacitance. Given the approach I showed you last time, it looked simple. But you will soon find out that this is a very powerful approach. So here is the summary. I told you that, well, that N(t) is directly proportional to the capacitance. This is the diffusion equivalent capacitance, density of analyte and the time you waited, the time t. And the trick is to calculate this capacitance by analogy to electrostatics. So whatever system we have, we can have a 2 dimensional. I assume that this in parallel plate configuration. In that case, the answer is epsilon naught over W. And we change the epsilon to the diffusion coefficient D and, thereby, calculate the value C. So that was step 1. Find an equivalent for whatever sensor problem you're looking into. Find an equivalent electrostatic problem and simply look up the corresponding capacitance from any book or from any table. So, for example, this is what we have done for a cylindrical nanowire or this is for a spherical, spherical nanobiosensors. The second step, after this is done, is to replace this W. This W, distance between the two electrodes and make it time dependent. That is, the second electrode is actually a hypothetical or virtual electrode. Because, as the diffusion front is going away from the sensor surface, because their sensor is capturing the molecule. Then this W would keep expanding with square root of Dt so you do this replacement. And once you have done the replacement, then we are done.
[Slide 5] We can essentially get all the results that we need. For example, I showed you, for a planar sensor, once you substitute this diffusion equivalent capacitance into this formula, we got the right dependence as would be expected from the exact solution. That was the same case for nanowire sensor and as well as for a nanosphere sensor. So in that way, essentially, we saw that this approach works when the fractal dimensions are integers. Now what about when the fractal dimensions are not integers? The complex shaped objects.
[Slide 6] Let's find out. Here, we have an array of sensors and I have taken a cross-section through those, through a sensor array. And you will see that each element, the elements are separated by a distance P. So there is a periodicity, P, that separates the electrodes. Now I can look up from a book that how the capacitance, if what is the capacitance between a given electrode and a set of wires. In fact, I got this capacitance from old vacuum tube literature. For those of you who may have studied it, you may remember that there is a cathode which essentially injects electrons. And there is a metal grade essentially which modulates the current flow. So I just looked up from there and we'll be doing this looking up business quite a bit because that gives the-- that is essentially gives the power of this technique. So here is the final formula. We can see there's an epsilon which is the dielectric constant. W is telling us how far apart this grade is with respect to the other electrode. And P, of course, is the periodicity of the grade. Remember the rules, epsilon gets replaced by D, the diffusion coefficient. W gets replaced by square root of Dt. And once that is done, we have just solved a problem that computers will find very difficult to solve, and will take a lot of time for it to solve it properly.
[Slide 7] And here is the answer. Calculate the total number N(t). And in this case, we will just substitute the value of CD(t). See this W has been replaced by 2 DT and the diffusion coefficient D has replaced epsilon. Put it in, plot it out, log of N(t) and log of time, and you will get a curve like this, which the circles are numerical simulation. Just to check it out that we didn't do any mistake and the blue line is the analytical solution. Now you immediately see there's something funny about this curve that, unlike the integer sensors, you know, the nanowire, 1 dimensional nanowire, 2 dimensional planar sensors. In this case, this log N to log t is not a constant line. But rather there's a knee in between. What's going on here? This is something we didn't expect before. Now you can immediately see where this knee is coming from if you pay a little bit of attention to the diffusion front as a function of time. These three white dots, or white squares, are essentially the cross- section of the nanowire. It's a poor man's version. I replaced the circle with a square, but that's it. And you can see the very early stage of the time the diffusion fronts are hugging each one of the sensors, each one of the nanowires. And, therefore, as they are depleting, they do not really know that there are other others present. They are just behaving like isolated nanowires. And, therefore, in the early stage, the diffusion is 1 dimensional, that is, almost like a nanowire. And that's why, in the beginning, the slope we see for log t as a function of log time is approximately 1 because, it is 1 dimensional diffusion is going on. A little bit later, however, what happens that these diffusion fronts begin to merge. And once they do, as you go farther and farther out into the solution, then these array of sensors look as if they are a given plane. And, in that case, this looks like a 2 dimensional diffusion front. And, correspondingly, the number of particle captured responds to that, or reflects that fact. Now the slope is half. And so, therefore, this knee, the surprising knee we talked about, is essentially nothing but 1D to 2D translation. Now does this look surprising to you, what I just mentioned? It shouldn't be. When you are in a plane far up in the sky, all the buildings essentially may look almost like a continuous thing, especially in a crowded city. Once you come down and are about to land, all the buildings in the airport they essentially look apart. This is essentially the reverse problem as if a plane is taking off from here and gradually going up, each individual buildings were distinct when the plane was at a low altitude. As you are going further out, then everything looks more continuous. There is nothing more complicated than that in this particular physics. Now remember one thing, this picture is really like a composite picture especially at low density. There will be nothing like the state of diffusion fronts. What will happen is that individual particles will essentially diffuse down and be captured by one or the other at low densities. So this picture, you should view it as if there are thousands of snapshots you have taken, thousands of sensors working in parallel. And you have taken that statistical or ensemble average and this essentially represents the ensemble average of lots of sensors. At low density, this is not a single sensor we are talking about, but lots of sensors. The picture is essentially a composite.
[Slide 8] Now this knee is very important. And I will come to that in a little bit later. But, for the time being, let's calculate how will this sensor respond? Very simple. You can ask the question that if there is NS particle captured, the sensor is ready to respond, lets say for this given technology. And I already know what CD(t) is. And when I put this 2 information together, I can calculate that in order to reach a certain analyte concentration NS, how long should I wait? You should notice the periodicity P of the sensors and the diffusion coefficient all around. The size of the molecule, the environment in which they are swimming, these are all hidden in D. And, of course, this one looks like a 2 dimensional sensors if the P is very close, because if they are very close, it's almost like a plane. And if it is very far apart, then it's like a 1 dimensional sensor. And that's what this knee transitions, the knee reflects that. For high P, this transition will occur at lower point. For low P, when they are very sparse, this will occur at a longer, at a slightly longer time.
[Slide 9] So here is the summary then. It tells us how this type of array sensor should behave. You remember the planar sensor plotted in terms of the settling time as a function of the analyte density. Planar sensor is actually very bad because it goes as 1 over rho squared. And so, therefore, for a given analyte density, it takes a long time for the molecule to be detected by a planar sensor. Nanowire sensors is far better because, for a given analyte density, it goes as 1 over rho and, therefore, this blue line gives you a much shorter capture time. Because it's capturing from surrounds, all side, rather than from one side alone. Where does this array sensor stand? Well, it stands somewhere in between because you see its dimension is somewhere in between so it stands somewhere in between. And if you plot the corresponding value of TS from the previous slide, you can see in the beginning it looks like a 1 dimensional sensor. And then, there beyond the knee point, it essentially turns around and looks almost like a planar sensor. So at low densities, you don't get as much advantage. At high density, of course, then you get almost it looks like it's a nanowire sensor. And it behaves in between the responses in between and experiments have repeatedly confirmed this assertion.
[Slide 10] All right, we are done with the planar sensor. We know how a nanowire and a nanosphere sensors work. We now know how we an array works, 1 dimensional array or 2 dimensional array behaves about the same. So we'll not worry about that anymore, just remember that it sits somewhere in between the 2 extremes. Now what about this one, this system? This looks like something that would be very difficult to simulate and very difficult to know what type of response this type of sensor might get us into. So let's see how we can handle this.
[Slide 11] Recall, very briefly, that in order to calculate the fractal dimension of a surface, we have to repeatedly divide the surface and the number of box occupied as a function of the division. And whatever power is there, that essentially expressed the fractal dimension of the surface. Surface is 2 dimensional, line is 1 dimensional and a dot is 0 dimensional. And I also told you that a random network has fractal dimension in between because it is neither a planar sensor nor a single line so it will be somewhere in between. And we calculated what that number is for a particular network last time.
[Slide 12] Now if you recall, that once you have determined the fractal dimension of this network by repeatedly dividing it and counting how many pixels are occupied, then there is a simple algorithm to essentially get this. Once you get this number, there's a simple algorithm to convert it to a regularized sensor. And do all the analysis here because this would be very difficult to solve numerically. And I don't know whether anyone has solved this numerically exactly to this point. And the idea is very simple. That once you know the fractal dimension, you can choose a particular value of m. That gives you the value of n and, once you know the m for this fractal dimension, you just keep dividing into n pieces and keeping m pieces. M, this pieces and generate the corresponding structure. These two have exactly the same fractal dimension.
[Slide 13] Now this is how it works then. You see, first we start on the very right corner and very left corner this composite network. Create an equivalent sensor which has a regularized fractal surface on the bottom. And simply ask the questions that how these red biomolecules are captured by those sensors. And if you did that, you will get the following simulation. The diffusion fronts will move as a function of time and essentially, in the beginning, whatever the diffusion front of each one is, a little bit later, they will merge. So it will start from 1D just like the previous case. Then the neighbors will merge. Look like 2D. Then a little bit later, the 2D will behave as a unit. Then, subsequently, the pairs will merge and, since it is a self-similar work, remember that we repeatedly did the chopping. And so therefore-- and threw things out-- and therefore, this diffusion front is self-similar. And so, as a result, you will get the following capture profile as a function of time. In the very beginning, it will be 1 dimensional when the surface, the diffusion surface are very close by. Then it will be a little bit later. It will be 2 dimensional when the neighboring diffusion fronts have moved, merged, and it has gradually moved up together. Then a little bit later, the whole thing will behave as 1 unit. It will return to the 1 dimensional diffusion front. And then eventually the 2 dimensional diffusion will return. And this will keep oscillating. So that's why I'm calling dimensionally frustrated diffusion because, depending on the height you look at, it may look, the structure may look 1 dimensional or 2 dimensional. One-dimensional and 2 dimensional and, as a result, this thing will oscillate, never settling down in 1D or 2D. And look at this knee. This knee, we have discussed before in the case of array sensors. Only that this knee repeatedly comes back for this type of structure. And, therefore, you can see the net response is going to be somewhere in between, and that will be dictated by the fractal dimension which control the spacing between the sensors themselves, sensor element themselves. Now if the sensor was very dense, if the fractal dimension is very high, you see closer to 2D, let's say 1.89 for example. In that case, what will happen that the elements will be far closer so the 1 dimensional segment will get suppressed and the 2 dimensional segment will get elongated. One-dimensional, 2 dimensional will repeat but the knee has shifted to the left and, overall, this is going to behave almost like a 2 dimensional surface. And, therefore, once you have done this calculation, you can always show that the total number of particle captured is given by exactly this formula. The slope of this line is indeed 3 minus DF over 2 and, depending on the value of DF, the slope will change as reflected from this diagram.
[Slide 14] So here is the summary then. The summary is that in a planar sensor, we have a certain response that goes as 1 over rho squared. For a nanocomposite or for a random network, we have a response which is 3 minus DF over 2 and the DF is whatever the fractal dimension of this sensor is. For a nanowire sensor, DF is 1. And so here we have a T to the power of 1 response or it goes as 1 over rho in terms of density. And a nanodot sensor is not significantly different as I have mentioned before. And if you look at experiments from a wide variety of groups, you would see essentially these trends are followed.
[Slide 15] Now before I conclude this particular lecture, I want to emphasize a very important point. You see, when I draw a line like this, I'm really drawing an ensemble average. That this is really an average of many sensors. Assume an infinitely long 1 dimensional sensors, no sensors are infinitely long but let's assume4 a long infinite sensor. And then individual sensors are essentially we can think about individual sensors are like segments of it. So let's say I have 1000, 1000 sensing element all in parallel. Now, depending on how far the molecules are, individual molecules are, they will
[inaudible] and get captured by the sensor surface. If the molecules are close by, the response will be fast. If the molecules are far out, the response would be farther out. And this red point essentially tells you the average of all this measurements. Now 1 very important point in this particular case is that, if you do not wait long enough, in that case you may not be able to capture a certain number of sensor response. You may conclude your experiment too quickly and, therefore, you may say that you have been able to detect a femtomolar because you knew the input density. You are simply saying whether the sensor responds or not. But you see that would not be appropriate because a, slightly a factor of 10 or lower in density will also show you the same response. So, therefore, in order to get the average, we must wait until the average, so the distribution is completed. If we have 1000 devices, we have to measure them all before we decide what is the detection limit of that particular sensor technology. Now this point has caused a lot of confusion in the field and, therefore, one should be very clear on this particular point that you should not terminate the experiment too quickly. Then you have no idea what the actual detection limit for the sensor technology is. And, in fact, you can do detailed Monte Carlo experiment to establish this point. This is not exactly a Gaussian symmetric, but slightly asymmetric structure. But the point is that, once you have taken the average of all, only then you can get the detection limit.
[Slide 16] So let me conclude this lecture then by bringing together all the pictures that we had. Planar sensor we now understand. We understand how array sensors work. How a 2 dimensional grid of regular array sensors might work. We now know how nanocomposites, why the response is somewhere in between. Why nanowire sensors are so responsive, and why it doesn't pay to build a nanosphere sensors.
[Slide 17] So here are my conclusions. Geometry of diffusion is very important and it- dictates the response time. We understood that, no matter how complex the surface is, once it can be described by its fractal dimension, then its response can be approximately calculated by this very simple fundamental formula. And it is always bracketed by your planar sensor and the cylindrical shape nanosensors. Now sometimes you may want to use complex sensors, complex shape nanocomposite sensors simply because it may give you a better signal to noise ratio. There may be other reasons why you want to use it. But you wouldn't use it to get higher sensitivity because, although there are many nanowires, they are competing among themselves to capture the biomolecules. And, therefore, the overall response is actually slightly poorer compared to a single nanowire. And finally, there's a difference between first arrival time when the first sensor responds to it. And then the average arrival time of when you capture a certain number, NS number but an ensemble average is different from just the first response of the sensor. And if this distinction is not clearly understood, there are often a lot of confusion that can be caused by this inability to distinguish between these 2 fundamental concepts. So I'll end here and I will, next time, I'll tell you how to think about and beat this diffusion limit. I have showed you what the diffusion limits are, the settling time, I'll tell you how to beat them and,