nanoHUB-U Principles of Nanobiosensors/Lecture 2.3: Settling Time - Classical Sensors
========================================
[SLide 1] Welcome back from the break. Now, let's continue with the discussion we had -- we are having about bio sensors and the response time.
[Slide 2] And I told you before the break, we discussed that, actually, a very simple argument about the geometry of diffusion -- of how molecules get captured by a planar sensor and by a nanowire sensor -- essentially can explain why the nanowire sensors are far more sensitive. And we use a very simple rule that we'll just use square root of D t to see the number of molecules captured within a given time, and that led us to this remarkable result. But we want to do a little bit more using this.
[Slide 3] For example, we want to explain that if going from the planar to the nanowires gave us this six orders of magnitude improvement -- should we just go ahead and make a nanosphere sensor? And if we did that, will it make it -- make us give us another four or five orders of magnitude enhancement in sensitivity? So, that is what we want to do. The simple approach we took to calculate the response of a planar and a nanowire sensors require a little bit more generalization. Again, something very simple and elementally -- elemental you'll see in a second how this is done.
[Slide 4] So, the topic -- that approach I am going to tell you about now is a slightly generalized approach based on diffusion of a triangle. Diffusion triangle is called diffusion capacitance limit. First, let me tell you, diffusion capacitance is not a capacitance. It is an analog of capacitance and I will explain how it works out.
[Slide 5] Now, remember maybe you may remember from your high school days that if you had two electrodes with the charges, let's say if you a had steady state sensing problem to begin with. Let's say you have a sensor and then there's the outer perimeter on which the density of the analyte is kept fixed. We are not solving the transient problem anymore. I am just thinking about steady state diffusion. Then we know that the solution of the diffusion equation can be given by various types of functions but since -- first of all, it is steady state, therefore, I'd not have a D rho D t term anymore, it's said to have been set to equal to 0. In general, I do not know how to solve this problem because this type of diffusion equation may have variety of solutions depending on the configuration of the sensor, and the configuration of the surrounding media which is held at a constant density. But you see, although I'd not know the solution of that problem, I actually know the solution and all of you know the solution of this problem. This is electrostatic problem. Assume that you have a cylinder -- two concentric cylinders -- the outer surface -- the outer electrodes is held at a potential sin naught and the inner electrode at psi of S. There is no charge anywhere itself and so, therefore, we have simply Gauss' Law. If epsilon secondary relative of phi is equal to 0. The Poisson equation. And we know the solution of this equation. If you wanted to know the charge Q of the electrode, which simply says that this is equal to the capacitance between these two electrodes and difference of the potential psi naught and psi S. Of course, I have just shown you a concentric cylinder. Had it been a parallel plate capacitor you all know the capacitance. Capacitance is given by epsilon multiplied by W. Of course, you have to multiply it by a in order to get the total capacitance. If it were concentric -- two concentric cylinders. One in the middle of the other then, again, we know 2 pi epsilon log of W -- W being the radius, outer radius. Plus, a naught is the inner radius. And so log -- ratio -- and then log is taken. It's a concentric cylinder. You know this answer also. And finally, if it were two concentric spheres, one big sphere surrounding a middle one once again, you can look it up from any text book and you will get 4 pi epsilon 1 over a naught of the smaller sphere. And the radius of the bigger sphere, W plus a naught. Now, you see immediately, that once you know this solution, you should be able to say for the solution of this equation the diffusion problem is steady state diffusion problem. Because rho, if you replace phi by rho, and epsilon by D, then these two problems are exactly the same. And if that is the case, then I can immediately write that the current flux into the system is given by "a diffusion capacitance." Rho 0 minus rho S -- in analogy of phi 0 minus phi s. and the diffusion equivalent capacitance everywhere you have epsilon, replace with W. and then you are done. In that case, if you had two nanowires, one sort of sinking in carriers -- molecules from the other one, or two nanospheres, right? In that case, the results are immediately given. You can find out where the steady state flux of molecules are from this surface to the sensor surface. Now, of course, we do not have a steady state problem per se but this analogy up to this point is exact and closely met.
[Slide 6] Now, in general, of course, these biomolecules may not -- may have -- the sensor may have finite capacity. It's easy to account for the finite capacity of this sensor. Because remember, there's not only a diffusion problem, but there is also a capture problem. The problem how many molecules can be captured per unit time. And so you can simply integrate the diffusion flux as a function of time. This is the number of molecules captured per unit time is given by rho naught, capacitance multiplied by rho naught. We are assuming that rho s the concentration close to the sensor surface is very small. So we can neglect it. And because this is, sort of, catching things and removing things from the system very quickly. And if you integrate this system the number of molecules captured for this structure it is simply given by the capacitance -- diffusion equivalent capacitance the density of analyte and lineally proportional to time. Because say continues steady state flux. And this capacitance, of course, depends on the geometry of the sensor and the surrounding media itself.
[Slide 7] Now, of course, we are not really interested in steady state sensing -- steady state capture problem. But you see, you could think about it a transient problem as a series of snapshot in steady state. For example, when you see a movie, all the images sort of are -- each can be static, but when they are run at a quick succession as a function of time they look like a moving image. Similarly, you can think about the outer electrode not really as a real electrode, but sort of a transient position where the -- beyond which the bulk concentration persist. And within here, it has, sort of, diffused, and, sort of, has been captured by the sensor itself. As a function of time, this sensor this depleted region will gradually get bigger and bigger as the square root of D t, t is the time. And so, therefore, it looks like the electrode spacing will now be time dependent "Electrode spacing" will be time dependent and it will go as square root of D t. All you have to is that where ever you saw this W which was a fixed thing in the previous case -- fixed number for the steady state problem will now become a square root of D t. You have 2Dt in one dimension, 4Dt in two dimension, 6Dt in three dimensions. When you have spherical sensing problem where the molecules are being captured by the small sphere within the -- in the center. Now these 2, 4, and 6, we needn't worry much about it because the number these prefactors themselves are not important. But you immediately will see that these results give you pretty remarkable answers. We already know about these -- these two because these was planar sensors, this is nanowire sensor we already know about them. We'll check it out whether we have gotten the right result using this approach. But this is completely new. We did not know how a nanosphere sensor would behave. So let's check it out.
[Slide 8] It turns out that the transient response is a -- this I told you before -- is given by Nt. The transient diffusion equivalent capacitance rho naught Nt from two slides before. I know about CDt for each type of system. A different fractal dimension 2, 1, and 0. Now if you put it in -- this value in here the square root of t will cancel and will give square root of t dependence for one dimensional sensor. Do you remember this is exactly the result we had when we analyzed the system by the simple approach of lost capacit-, lost triangle. Similarly, for this nanowire sensors once you put this value in you see this logarithm independence is very weak. So once you put it in, it will be a state of constance. This will be very weak dependence, and essentially the number of particles captured will increase linearly in time. That is also a result that we have seen before. However, for the nanosphere sensors with the fractal dimension of 0, you'll see that as time progresses, 1 over 6 square root of delta t, this will become smaller and smaller. And over a time, then therefore, this will simply flip over. And that will become a constant. Rho naught t will get multiplied. And so, therefore, both in nanowire sensor and in nanosphere sensor have same dependencies. Both appear to be independent, sort of, beaten the diffusion limit in the sense that the molecules do not show -- the time exponent do not show diffusion slow down unlike the planar case. But these nanosphere sensor is not significantly different from a nanowire sensor in that respect. So, therefore, there's no point in actually making this sensor compare to a nanowire sensor for example.
[Slide 9] Now the result we just got using this simple capacitor approach something that you learned in high school perhaps, does it, sort of, is it real? Does it compare well with exact solution? It turns out that the nanosphere sensors also can be solved exactly and analytically. And the solution is given by this flux density D rho naught divided by a naught multiplied by some complicated factor, you can integrate the whole thing to get the total number of particles. That has been captured by the nanosphere sensor. And once you have integrated it out, there'll be a complicated expression but the bottom line is that this linear dependent -- this a naught is a small quantity so this will drop out, and you will see that the linear dependence with time is exactly the produced. So, therefore, the simple approach we took in terms of-- diffusion equivalent capacitance, in fact, gives us the correct result.
[Slide 10] So here is the summary of the types of response time that we have captured. The number of molecules that we captured by sensors of different nanostructures or surfaces. If you have a planar sensor, or if you have a nanowire sensors, then both cases, you can use a very simple relationships where the Df is a fractal dimension. Rho naught is the analyte density in order to calculate the number of particles captured for a given amount of time. On the other hand, for a nanosphere sensors, which is the fractal dimension between 0 and 1, the response time doesn't go like this, but rather simply as rho naught t to the power of 1. And, therefore, this doesn't have significant difference compared to how a nanowire sensor captures molecule.
[Slide 11] So let's summarize then. So the summary would be something like this. We know the planar sensor now. 1 over rho squared. The nanowire 1 over rho that you have seen already. And the nanodot sensor well, this is a little better but not significantly. And, therefore, and the experiments if you put it in the literature -- from the literature you'll see it halts the line. And, therefore, this gives you fundamental limits to the nanobiosensor problem.
[Slide 12] Now let me conclude by suggesting a few things. The settling time is the time or the response time is the time needed before a sensor captures a certain number of molecules. And this minimum number of molecules that's got to be captured depends on two things. If the density if very high, then the time will be low, but at the same time, if the sensor has a fractal dimension which is favorable there's one dimensional, zero dimensional capture, that will also make things much, much faster. Now this is a fundamental limit in the sense that it really doesn't matter what type of sensor you use to detect the biomolecule. This result would be -- could be correct. And, in fact, this is very similar to Heisenberg uncertainty principle that it relays fundamental quantities. In their case, energy and time. In this case, density and time giving a constant. Of course, it's not as fundamental as Heisenberg's principle. But it has -- because it can be overcome using technological approaches. But for the time being, the analogy is very close. And the important point is it doesn't matter what type of sensors you have used. And the size of the biomolecule if you use a DNA versus a protein, of course the response would be different. And the size of the biomolecule is hidden in the diffusion coefficient. A larger molecule has smaller diffusion coefficient compared to a smaller molecule. And at the end of the day, the geometry of diffusion, as encapsulated in the fractal dimension of the sensor's surface, that determines the response of the sensors. And the next time around, we'll talk a little bit more about more complex surfaces because we now know how to do planar sensors, nanowire sensors, nanodot sensors. But you realize there are sensors which may have a random collection of nanowires are more complicated version of surfaces -- nanostructure surfaces. How do you get the response of those sensors we'll discuss it in the next lecture. And until that time, take care.