nanoHUB-U Principles of Nanobiosensors/Lecture 2.7: Beating the Diffusion Limit: Enhanced Diffusion and Fluid Flow
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[Slide 1] Welcome back. As you may remember that we had been discussing how biomolecules diffuse around and the time it takes for the biomolecules to land on the sensor surface, and we have discussed that how diffusion fundamentally limits this process. Of course, in nanobiosensors makes the best out of this very bad situation, the diffusion limited situation. The nano structure itself through its fractal dimension allows the diffusion process to allow it to collect more biomolecules on the sensor surface per unit time, compared to a old or classical sensor. Now in the last class, we discussed ways to beat the diffusion limit, and I mentioned two specific approaches. One was to do with throw lots of sensors at the analyte. It is something like putting a lot of policemen on the street to look for the suspect rather than waiting for the suspect to randomly walk around and give him or herself up in the police station itself. That will take, of course, a very long time. The second approach, that was based on droplet evaporation. As we saw that a droplet can essentially reduce by a million fold, and, thereby, concentrate the biomolecule, whatever rare biomolecule we had in the analyte on the sensory cell. We will discuss a third technique today, and then we'll talk about a technique that you think that would work very well but actually doesn't. So those are the two topics that I want to discuss today.
[Slide 2] I want to begin by talking about diffusion in multiple dimensions and how it allows biomolecules to be captured faster in the target, and you see this is actually a very simple idea that we actually often do as we are searching for something. I'll then talk about the sensor if you put it on a flow channel and allow the analyte to come to you, and you may think that actually this will make things better. After all, the molecules are directly coming to you by transport or by drift rather than by diffusion. Wouldn't it make things better? So that is the question we'll address, and then finally I will conclude.
[Slide 3] Consider a analyte volume, and on the bottom, we have a spherical sensor. Now these red analytes are immersed in an analyte density rho three, and we already know that the steady state flux can be calculated by the diffusion, equivalent diffusion co-efficient, rho three minus rho naught. rho naught is the surface density on the spherical sensor. Now rho naught would be small because it will assume that everything that is coming into the sensor are being captured. Kf is infinity. And in that case, we already know the value of CD,ss. If you put it in, which is the diffusion equivalent capacitance and corresponding with lb1 over A, which is the radius, and 1 over r. Now r in three dimension after a certain period of time you may remember the diffusion forms are gradually expanding away from the sensor surface, and, therefore, r would be infinity, r would be, 1 over r would be 0 within a very short period of time. We are talking about compare to 1 over a, and, therefore, its spherical sensors as you have already seen is captures particles which is flux. to these specialF sensors is directly proportional to the analyte density and the total number of particles captured, well, it increases linearly with time t. This is one value if you didn't do anything special, then its sphere actually captures this many particle per second on its own. Can we do better? Here is an idea taken from biology. An idea is something like this. We still have our sensor surface, but, instead, we now, in addition, we now insert a nanowire like structure. Remember the nanowire itself is not a sensor in this particular configuration, but, rather, nanowire will catch the particles, funnel it down to the sensor surface. So it's like an antenna. Now, of course, the molecules will get captured, can diffuse down a little bit, and can be re-emited back to the solution. Of course, then, there's no benefit for the nanosphere sensor. On the other hand, if the molecules are captured within a diffusion length, then before it gets captured, released, re-released, it is actually captured by the sensor, and in that case, we have gotten a signal. So the question is of these two, which one would perform better? Now we can easily calculate what the corresponding flux of this thing would be. I3 would be the flux that is coming from the three dimensional volume to the cylindrical surface. So cylindrical surface is, like, a collection surface, and then they are getting funneling, funnel down like raindrops on a window, windowsill and towards the sensor surface. And, of course, in steady state, I3 and I1 will balance each other, and let's calculate those things to see what type of benefit we might get. Now I3 would be proportional to the density difference between rho 3, which is the above volume, and rho 1, which is the one dimensional density on the cylindrical nanowire itself, and I1 would be the difference between rho 1 to rho naught. rho naught is the density on the spherical sensor. Let's calculate these two fluxes.
[Slide 4] Now we can easily calculate the first flux. That is because we have previously done the nanowire sensor. Do you remember when the flux came directly to the nanowire sensor? All we had to do was to calculate from three dimension to two dimension, this capacitance, which is essentially a cylindrical capacitance of r divided by a. A is essentially the radius of the cylinder, and LA is the length over which the flux when the particles once captured are directly transported to the spherical sensor. LA, of course, is not the physical length of the whole thing because, of course, particles which are captured here are released. So therefore-- as if they didn't count or didn't exist. There's side three. What about I1? I1 we have to think about it a little bit. Think about some particles which are coming in here at a distance x from the sensor surface. Now it is going to diffuse in one dimension from here to the sensor surface, and we'll call that Jx. Of course, the injection could occur at any x. So, therefore, we'll integrate it over, and we'll have to multiply with 2 pi a so that it accounts for the surface of the cylinder itself. Once you evaluate these two quantities, this is the diffusion equivalent capacitance in 1d. rho 1 minus rho naught is the density difference between these two points, and you integrate it out, and you can get an expression for I1. And so once you have calculated I1 by integrating over from 0 to LA, and you already know I3. So, therefore, we can equate them and find out what the value of I1 is, what the flux to the spherical sensor is.
[Slide 5] Now this is relatively simple. We have the known values for, from the previous slide, C3,2 and C2,1 from the previous slide. You equate them, eliminate rho 1, and so, therefore, I can directly calculate the value for I1. Now here is the trick. If the diffusion in the cylindrical surface is faster than diffusing around in the volume, then D1 is significantly larger than D3. You substitute that approximation, and then it immediately gives you an expression where C2,1 sort of drops out. Because actually this diffusion is so much faster that it is no longer limited by the diffusion of this process. It essentially gets limited by how long it takes for the molecules to swim in 3D and get collected by the nanowire itself. So now once you put this expression for C3,2,-- see once we put the value for C3,2 in, simplify a little bit, and out comes a beautiful result. It says that compared to the spherical sensor alone, when you put this nanowire on top of it, it's total amplification, the total number of particle captured will be amplified by the length, by the capture length to the radius. And if it turns out to be a factor of 100 bigger, than we will have a factor of hundred amplification in the flux rate. It catches like a cylinder, but it integrates to a point, and, hence, the amplification.
[Slide 6] Now this actually turns, appears to have deep roots in biophysics. This strategy of finding something or diffusing to a target faster than you'd think otherwise possible. For example, you may remember in the second lecture I told about-- we discussed about a DNA being a long polymer. Here is my DNA. It's a long green polymer, and on one segment of the DNA, lets say there is a target, and this particular protein is looking for that target operated. Now, of course, the protein could, once it could search for this place on its own by looking for that target in three dimension, never landing on the sensor itself. So interrogating it point by point, but it turns out there is much better, much better strategy is to first land on the DNA, then slide along, and after a little bit get detached, jump to a different segment of the DNA, and then keep doing this, and so long this diffusion process in 1D along the DNA is fasted, then diffusion in 3D, it turns out that it finds the target much, much faster. Sounds complicated? Actually not because you can see if you lost a child in a city. 00:12:13,866 --> 00:12:17,176 Lets say somewhere here, a city that has subways, then you wouldn't really go around randomly walking in the city looking for the child. It is much easier to look for the child a little bit somewhere, hop on the subway, randomly get off at some other point, look for the child. So this interaction between three dimensional search and one dimensional fast transport is the essence of the amplification that we got to beat the classical diffusion limit that we have talked about in the previous lectures. That said, three techniques I told you about. There are others, but for the time being, I think these three approaches will give you an idea about research direction that people are taking.
[Slide 7] Let's go onto the final topic, and that has to do with how bringing the molecules to the sensor.
[Slide 8] Assume that you have a sensor, and in this sensor, if it was sitting in a middle of the flow channel, and the fluid is coming from one reservoir and going to the other. The molecules which are in the reservoir, let's say it's very rare, and, obviously, the fluid will bring the molecules to the sensor surface, and you might think that this would be better. Because instead of sitting around like a jellyfish, it is actually the flow is sort of bringing the food to the jellyfish directly. Wouldn't it be better, and wouldn't a nanosensor benefit from it? You might think the answer is yes, but you can immediately a little bit closer thought immediately will tell you that there is, the answer isn't so obvious. Because, after all, when the molecule is coming to, its fast, that's true, but the molecule is also going away from it fast. That's also sort of counterbalancing it. Do I win at the end of the day? In order to answer that question, or in the next three or four slides, I'd like to do a simple calculation with the following idea. Assume that I have rotated the sensor sort of, you'd say, rectangular cross sectional sensor, flow channel, I've rotated it sidewise, and so I'm looking at it from the side. The sensor is on the bottom, which was the circular disc here. On the side, this is laying flat like a little rectangle, that
[inaudible] rectangle. The fluid flow goes from 0 on one edge of the plate to the other edge of the plate, and there is a parabolic flow profile. And the idea of that gives this enhancement is that instead of this depletion from gradually going away, as it would happen in normal diffusion limited system, this flow will continuously replenish the density here. So the density will essentially at the distance delta is called the stagment layer, the density will stay clamped, and, therefore, there may be faster diffusion to this system because if it were not clamped, the diffusion front will move away, and in that case, the diffusion slow down might have caused me trouble. But how big is delta, and how much enhancement do I get, and here is a relatively simple calculation.
[Slide 9] In order to understand this calculation, we fairly quickly, let's quickly define a number. It's called the Peclet number. In this case, it is essentially a ratio of if you didn't do anything versus if you did something, this ratio of these two things, if you did something means that if you had a certain flow rate q divided by if you just allowed it to diffuse, and there was no flow. And the related enhancement of the flow rate is the flaunt with respect to the field diffusion is given by this number. Now, a is the radius of the disc, h is the height, and w is the width of the channel. Now, after a, some specific calculations are done, your result will be something like this. It will be rho naught aD. If the velocity or flow rate was extremely small, then the result is essentially the same as that of a diffusion case, and you can immediately see that. Forget about when the rho of p is small. I can drop these two terms. You can see 4 rho naught aD is essentially the diffusion, diffusion result that we have seen before for a disc sensor. However, if the flow rate is significant, where q is large, P is large, in that case, you can get a significant gain with a cube root of Pe. Let me explain how that comes about.
[Slide 10] In order to understand how that comes about, you can compare two situations. Situation one, a disc sitting on the bottom of that flow channel. What is the flux that it will catch? By now, we are experts in doing this. So we will simply do the CD,ss of the disc. You see, there is no 4 pi. 4 pi was there when it was this sphere or hemisphere. Here is a disc, and you can look it up in any book. The capacitance of a disc in infinite medium, and then it will be 4 pi rho naught. One over a and 1 over r. And the maximum value that this disc can get, purely under diffusion will essentially be the diffusion and the diffusion front is relatively large. It can drop it, and the maximum value would simply given by 4D rho naught a. Let's say that's my value. Will this be significantly better? The answer is yes. The answer is yes because this r will no longer go to infinity. Now the r will be picked at the thickness of the stagnant layer. So instead of simply saying a plus r with r setting to infinity, I will simply have a plus delta being the extent of the r. And once you put it in, we'll get a ratio of these two quantities and relative enhancement of the flux. If you take the ratio of these two quantities, it will simply be a over delta. Essentially of that order. So if your a is relative size, a's actually larger than, significantly larger than delta. Then you have achieved something because there will be a significant enhancement of flux capture.
[Slide 11] Let's calculate very quickly. These are relatively simple calculations. We want to calculate what the value of a and delta is. My goal is I already know a, the size of my sensor. Delta is what we are after. If the velocity profile is parabolic, then it's given by y1 minus y. That type of relationship normalized to h. H is the height of the channel, and then I can calculate the average velocity. U over 6, and the average velocity must be equal to the flow rate. Q is the total flow rate, volume, and the blue is the cross sectional area of the sensor. I can equate this to find the expression for u, and once I know u, I can also calculate the value of the change in the velocity or-- if at the bottom point at y equal 0, and if I calculate that, we'll get an expression for q.
[Slide 12] So I want to know what the velocity at the top of this stagnant layer is. How do I do that? If I know the rate of change of velocity, I can multiply with delta, and once I calculate it, I will get an expression that relates the flow rate to the velocity and other geometrical factors. Then you have to make a special argument, and the argument is the following. That the time it takes for a molecule to go from one side of the sensor after arriving it before it leaves the sensor and goes away forever versus the time it takes to diffuse down and get captured by the sensor surface. These two times must be equal. It's only then there will be a capture. Because if the diffusion time takes too long to get to the sensor surface, it will simply disappear forever, and it will never be captured. So, therefore, that argument simply is reflected by this. If delta is the distance, then the diffusion time is delta square over 2D, and the transport time tau C is the size of the sensor divided by the velocity of the fluid, and velocity of the fluid I already know. I put it in, solve for delta, and there is my expression for delta over a, and which goes as the Peclet number to the cube root.
[Slide 13] So to wrap it up, wrap this discussion up, all you have to do now is to put this expression in here. I max, I already know from the previous calculation. So once I put it in, then I will get an expression that has the flux in terms of the flow parameters. And you can take a log, and you can see if your diffusion co-efficient is faster, you have larger flux. If you have higher flow rate, larger flax. Smaller cross sectional area, obviously, larger flux, and so on, so forth. But relative improvement is what we are after. In a biosensor, does the flow help or not? Once we have the expression, we can answer this question, and there will be a big surprise for us.
[Slide 14] So let's see what we mean by that.
[Slide 15] Let's say I look at the total flux and the volume metric flow rate, which is q. I plot it as a function of volume metric flow rate. The diffusion co-efficient is 150, and let's say we are thinking about one femtomolar. Very, rare analyte density. Very low analyte density. If you evaluate the previous expression that we just arrived, it will say that if you have a sphere of 100 micron, then you get a tenfold increase in the capture rate by flow. Tremendous. That's significant, but as you go to nanoscale, what you begin to see is that as your radius is going from 100 to 1 micron, right. Many of the nanoscale sensors are of that order. The relative gain even at very high flow velocity is inconsequential, very, very small. What's going on here? In fact, you will see that this happens almost in every place. And the reason is, and I'll explain that a little bit later, is this delta relative to a. You see these two things for nanoscale sensors, a is very small. So in order to get considerable enhancement, you have to make delta also very, very small in comparison. That means impractically high velocity, and, therefore, you do not get as much enhancement in the practical case.
[Slide 16] You could plot the relative enhancement to check it out. How significant the gains are, and the relative enhancement is for one micron, as you can see, is essentially negligible. Yes, for larger sensors, significant gain. For smaller sensors, the blue line, not very much at all.
[Slide 17] So if you put, for example, a cylindrical nanowire sensors in a flow channel, you will see that as you go from ten micron to one micron down to one nanometer, the gain is very small, and, therefore, the main conclusion that I want to draw from this analysis is that the large scale sensors, yes, flow helps a lot, but for small nanoscale sensors, putting it in fluid channel to the 0th order doesn't give you very much.
[Slide 18] Let me conclude, then. The first topic today I discussed was this dimensional modification allowing from 3D capture it in a cylinder, let it slide down on the cylinder, and then captured by the nanosphere. That gives considerable and considerable-- enhancement, and this is biomimetic in the sense that biology, in biology, DNA proteins detect targets in the DNA exactly the same way. Now sensors in a flow, a flow channel enhances detection. There's no doubt about it because you are bringing things faster, but the counter problem, point is that it also is getting away faster, and so, therefore, there is a balance, and what we saw that for larger flow rate, of course, for better detection, you know larger flow rate. That makes things better because it brings the molecules faster, and you want smaller cross section. That makes sense, right, because then the fluid will be hugging the sensor better, and the chances of the biomolecule actually getting caught by the sensor surface is improved. But the point I really wanted to emphasize that this nanobiosensors do not benefited as much as nano, as much as larger classical sensor. So this is another twist of nanobiosensing. Just like the diffusion limit was one type of twist, that larger sensor don't generally worry about. Similarly, this counterintuitive idea that flow doesn't help nanobiosensors is another aspect that we should be thinking about. OK. That's it. In the next class, I will continue with this discussion of diffusion limits and how to build it, and until that time,