nanoHUB-U Principles of Nanobiosensors/Lecture 2.6: Settling time Beating the Limits – Droplet Evaporation
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[Slide 1] Thank you for joining me again. This is week two and we're discussing how to beat the diffusion limit. In the first week and even for the first lecture in this week, I said that diffusion is a fundamental process, not only for biosensing but for organism of all sort, that the molecules must diffuse to the sensor's surface and on the organism's surface. And in that case, it is often the geometry of the structure that determines how efficient the capture is. But no matter what you do with the geometry, at the end, there is a certain limit beyond which you cannot go. And so, of course, that's the concern. And we are finding ways to go around that "fundamental limit." I have already discussed one approach based on bio-barcode, essentially throwing a lot of policemen, putting a lot of policemen in search of suspects. And I explained to you how that dramatically reduces the capture time. Here's a second approach that we'll be talking about.
[Slide 2] And the second approach is based on the notion of droplet evaporation. I will explain the theory and I will tell you a little bit about how this is actually made, how the device that does this droplet-based measurements or sensing how they are made because I think you'll find it very interesting.
[Slide 3] Remember, where we are in terms of this big "Mendeleev's table," we have understood the diffusion limit associated with these types of-- various types of sensors and organize them according to how they respond. And in fact, if you take any modern sensors and don't do anything special, you can immediately analyze it in terms of, in terms of their geometry. In fact, this is the first thing we should do anytime we have a new sensor, understand its fundamental limits, diffusion limits. Now, I haven't explained to you how the bio-barcodes sensor work and today, I want to explain to you how the droplet-based sensor works is that as I said, very elegant and beautiful.
[Slide 4] So, the three approaches that we have discussed, the one that we focused today, right now, is this approach based on the droplet. The key idea is this. That we'll be beating the diffusion limit or making the tau, diffusion limited tau smaller by making the L, the length or the size of the box essentially gradually progressively smaller. This is what happens when you concentrate something. So the molecule might be originally sitting anywhere within this box, which is the blue molecule. And then, if you didn't do anything special, the molecule will walk around and take a long time to get to the sensor's surface. What can happen instead, if you allow the water to evaporate, of course, the molecule cannot evaporate, the molecule is sort of trapped. It's a big molecule. So, water evaporates, the molecule doesn't. So, it is increasingly confined to smaller and smaller boxes until it is sort of captured by the sensor itself. And this time dependent L, increasingly smaller L makes tau small. So, let's discuss that. This would describe that approach.
[Slide 5] Now, before I get there, this is a very nice example of how sort of ideas from biology or biomimetics essentially inspires this type of droplet based sensing. In some way, you could say, even the diffusion limited approach that I discussed previously in the last week and also in the beginning of this week also has the same biomimetic approach, the analogy to biology, diffusion limit and all allow that to understand the sensors. And this time, the analogy to biology, this biomimetics will also essentially also tell us how to beat the diffusion limit. So, nature teaches us both how to analyze and how to overcome. So here, what I wanted to tell you. So, the analogy that one should-- droplet and everyone can think of is this lotus leaf. You can think-- we all know that a drop of water on this lotus leaf is very spherical, essentially it sits on the surface and any particle that you put in here, let's say, you put a biomolecule or a bacteria or something, it will always be confined by the trap, by the cage of the droplet. Now, of course, one problem of this droplet is it moves around. So, that's the problem. But there's another thing that we also see, also sort of goes with this general idea about trapping is the coffee ring. You know, in the morning, when you pick the coffee, coffee cup, you will often see that after a while, an unwashed coffee cup may actually develop a ring like this. What happened? You see, what happened was there is a fluid like this and essentially, because the surface was not superhydrophobic, as the fluid evaporated, this essentially deposited the molecules of the coffee around the ring. So somehow, if we could marry this concept of superhydrophobic surface or hydrophobic surface which keeps the molecule together and once it shrinks, it sort of drops the biomolecule in the center. And this concept of concentration of the analyte, if we could combine them, then we would be able to beat the diffusion limit. So, let me explain.
[Slide 6] So, this is how it works. This is cartoon, let's say, if it were a teaspoonful full of seawater, there'd be millions of biomolecules, millions of bacteria in here. But even if it's a small quantity of blood, you may have a-- within this system, you may have at a-- femtomolar concentration, quite a few copies of the biomolecule. Now, if you waited for them to diffuse down and get to the bottom of the sensor's surface as you know, you'd have to wait a long time. We have discussed that already. Instead, you could do the following. For example, if you put this droplet on a surface and these dots are special, I'll come back to that in a little bit. After 10 minutes, let's say the diameter is something like 500 micron or the radius is about 500 micron, now the water evaporates. It becomes little smaller, smaller and by 35 minutes, which is half an hour, essentially it has, the size has reduced to five micron, let's say. How many force have the concentration increased do you think? Well, if this was 500 micron and if this is 5 micron, then 4 by 3 pi r cube. So, 500 micron divided by q divided by 5 micron q, almost as I say, million fold. So, any concentration that you had that was let's say, an at attomolar concentration here, at this point, the concentration has become picomolar. Now, classical sensors can subtly detect a picomolar. That's not a problem and although the original concentration was an attomolar. So, this ability to concentrate has given us a way to beat the diffusion limit. And in fact, once the molecule is deposited, and this picture is taken from this particular paper, you can image it and detect where the molecule itself was.
[Slide 7] So, how does it work? How long should I wait? You know, we want to be-- I know the droplet will evaporate quickly. But, you know, droplet of all sizes will not work. If you make a humongous droplet, that will also take a long time before it evaporates, right? If you have a swimming pool, it doesn't simply evaporate overnight and gives you the-- all that residues on the bottom of the pool the next day. So, we have to sort of have a theory of how long it-- how long it takes to evaporate-- given size of droplet to see whether we'll have any advantage or not. So, the idea is this. Consider this droplet of radius R, and assume that the concentration on the surface is Cs, concentration of the vapor. And at infinity, the concentration is zero. As the droplet is evaporating, the diffusion of the water molecules is going away and it's going away through the diffusion equation because the particles are living. They are sort of randomly walking around and getting lost to infinity. So, inverse problem of the captured by a nanosphere. Here, the particles are going away because the source is here and sink is at the infinity. Now, you could immediately get the solution of this problem. The amount of-- the amount of water lost is dm, dt, and that must be given by the difference in the analyte-- the difference of waters vapor concentration on the surface and at infinity. And the CD,ss, because we are solving the diffusion equation, here comes the CD,ss again but in a different context. And the capacitance, equivalent-- diffusion equivalent capacitance is 2 pi Dr because it's a sphere. So therefore, if you go back to the formula of the capacitance for a sphere, you will see this formula. Now the m, the amount of mass lost is essentially lost a shell of a shell of the droplet at a given time, delta t. So, you will write it as 4 pi r squared Dr and that's the amount of mass lost multiplied by the density of water, right? That's how much you lost and the only unknown variable in this particular case is r and t. And so, we can connect r with t and be done with it. And once you have done it, it tells you how long the droplet-- it will take for the droplet to go from the initial radius of Ri, let's say 500 micron to the final radius, which would be five micron or even smaller. And if you will sort of neglect this quantity with respect to this, assume the concentration of vapor at infinite distance, infinite, meaning let's say 100 micron, is zero. So, then it immediately tells you that your-- in order to make it go very fast, then your initial radius has to be relatively small. It helps if you have a lower density because it will go faster. It helps if the water molecules, the vapor can diffuse fast in the environment. And it helps to have a higher-- lower vaporization. But the bottom line is this r squared. Is the reason why a pool doesn't evaporate overnight but a little droplet on the order of micron squared or 100 micron q essentially will evaporate very, very fast. And this is where the nanotechnology of biosensing helps you to beat the diffusion limit. Now, the cS, the called surface concentration depends on the pressure, the temperature and the universal gas constant and so on and so forth. But for the time being, just let's focus on t and r that the r has to be small in order for the diffusion time, the concentration time to be faster than the diffusion limit. All right.
[Slide 8] So, how do you do this? How do you make such a device? You can make it relatively easily. You can have an array of electrodes that will create a sensor's surface. You see, we don't have lotus leaf and how would-- even if you had a lotus leaf, how would you actually put contact within the lotus leaf. So, we have to make our own lotus leaf to beat the diffusion limit. And this is a way of making the lotus leaf, I will-- as I will explain in a little bit. So, what is done is that you make a series of pillars, certain electrodes and you can electrically connect them in order to measure the analyte concentration. And from a top view, it looks like a coffin. So, you can see as if somebody is lying around there but essentially, this is a droplet, elongated droplet because the pillars are along in a particular direction, so that elongates it. In a lotus leaf, it would be-- the structure would be symmetric making it spherical. But essentially, the physics will essentially, generally be the same. So then therefore, if you start with such structure like this, within a few minutes essentially, you can evaporate the sensor's surface, evaporate the droplet. And while you're evaporating the droplet, you can measure for the analyte concentration. And that will give you the-- allow you to beat the diffusion limit. And it's very similar to the picture that you had seen before. Now, let me tell you a little bit about how to create this superhydrophobic or hydrophobic surface because that is sort of essential element to this strategy.
[Slide 9] So, if you think about a water droplet sitting on a surface, most of the time or often, depending on the standard surface, it may look slightly elongated. It's not hydrophobic enough to sort of roll up or ball up. Why not? Think about these three systems. So you have a vapor, you have liquid and a solid, and for the time being, forget about this evaporation. So, we have a time independent droplet sitting somewhere. So, how-- what determines its shape? Well, what determines its shape is essentially at this point, at the corner, you can essentially look what the forces are. If the forces are very strong, then it will sort of get elongated, forces between the solid and the liquid. If they like each other very much, it will get elongated. If they repel each other, then it will be sort of pushed in and it does-- the droplet will become spherical. And so therefore, you should be looking at-- we should be looking at three surfaces. One is the top half is the air, bottom half is the liquid, that sort of gives you the force on this side of the interphase. In here, bottom side is solid, top side is liquid. So in this corner, you will have essentially balance of these two forces and on this side, when going away, you have solid on the bottom, vapor on the top and the balance of these two forces will determine the net force in this particular case. So, balance of these three forces will determine what type of shape do you actually have for such a system.
[Slide 10] Now, this is something Young has developed a long time ago. Again, the sphere, the hemisphere is-- a sort of cut sphere is representing the droplet. The radius r is the radius of curvature and the angle theta, you can convince yourself that this angle theta also defines the angle theta of the droplet with respect to the sensor's surface. And as you can see, bigger the theta, more spherical the shape. The volume of a truncated sphere is given by, in terms of theta and r, by this formula. The surface area is given by R sin theta which gives you the radius and pi r squared gives you the area. And the top surface area is given by the following formula. And so, you can see the total energy of the system. LV stands for liquid to vapor. So on the top surface, we have liquid on the bottom and vapor on the top, so you multiply with S, that gives you the total surface energy. This is surface tension. And then on the bottom, we have solid and liquid, that's the interphase and their spread over A. And finally, solid to vapor is everything that is not A. So, all these areas outside, A infinity minus A, is essentially the energy associated with the solid to, solid to the vapor. And therefore, if you calculate, you know, this S depends on R and theta and you can replace everything therefore, after you do the substitutions, you can replace R and express G, the energy G, in terms of theta alone. Once you know theta and once you know the volume, right, once you know these two things, you know the total energy of the system. Now, of course, you may say that the theta could be variable. For a given V, I can have all sorts oftheta. Yes, but this energy must be minimized for the shape to be stable. And once you minimize this energy with respect to theta, you try out different theta in order to see where the energy is minimized, you correspondingly get Young's equation which relates the theta associated with the fluid equate to the surface tension between solid to liquid, solid to, you know, liquid to vapor and solid to vapor using this simple relationship. Now, in most surfaces, it will not be superhydrophobic or hydrophobic. So therefore, our original strategy of beating the diffusion limit will not work. So, we need to work a little bit harder as I'll show you in the next slide.
[Slide 11] So, what we need to do is that in many cases, the angle may be small because this solid to liquid tension may be very, very large and therefore, it can elongate, elongate the droplet. So, this strategy is-- in the following. You can cross multiply. And then you realized on the bottom surface, essentially there's a competition between solid to vapor and solid to liquid. Somehow, we have to make the solid to vapor quantity larger in order to make the theta larger. How do you do that? Well, what you can do is that essentially cut holes, drill holes in the surface and therefore, all the points that previously had solid to liquid will now have vapor to liquid. So phi is the sort of the periodicity associated with this. So, the solid to vapor and solid to liquid, the original surface has been reduced to phi, only the top wire it's touching it, and one minus phi has been replaced with liquid to vapor because this is liquid and this hole has vapor in it. And so there correspondingly, the theta will rise dramatically. And that is how you make a superhydrophobic surface. That is the purpose of the pillars that we talked about.
[Slide 12] So very quickly then, what you do, you put down, you create the pillars with the right configuration so that it creates a droplet. Once you have the droplet, you have the anode and then the cathode. So, they not only create the superhydrophobic surface but also the sensor in between. So, you can measure the impedance between one set anode-- one set of electrode to the next and the impedance is plotted here. And as the droplet, as the droplet, and you have initially, we have one droplet that gives you a certain impedance and as the droplet is evaporating, your impedance continues to decrease. And from this value of the impedance and these values as the function of time, you can detect the analyte concentration. And in fact, if your analyte concentration is lower then, of course, your impedance would be a little bit lower. And-- But the point is, our impedance would be, if your analyte concentration is higher then your impedance would be lower. But the bottom line is, using this approach, you may be able to detect tens of attomolar approach because this has sort of allowed you to concentrate on a single point, thereby beating the diffusion limit. And this actually gives fantastic, fantastic sensitivity.
[Slide 13] So, let me summarize then, bringing us back to the old plot that the way the droplets with evaporation allows enhancement of sensitivity is that it starts at a very low concentration. Evaporation essentially pushes it-- pushes the concentration at much higher value so then you can use the planar sensor or a nanowire sensor. It doesn't matter. In that case, you'll have very high sensitivity beating the diffusion limit.
[Slide 14] So again, this is a summary which sort of tells us that these approaches essentially allows you to approach very low concentration levels that classical diffusion limit prevents these classical sensors, standard nanobiosensors from reaching.
[Slide 15] So let me conclude. I told you about superhydrophobic surface. Superhydrophobic surface simply means something is so hydrophobic that the ball is almost rolled up so that when the biomolecule is brought closer to the surface, they're all drop at a single point. Unlike the coffee ring which sort of spreads it around. Because then, you have to put sensors all around. Here is much better because it will bring everything down to the point where you really want to sense it. Now, I told you about how droplet allows pre-concentration to beat the diffusion limit. And these type of superhydrophobic surfaces are relatively easy to design and therefore, this type of sensors can be pretty interesting in terms of being able to beat the diffusion limit. So, that's it. In this lecture, I told you about one strategy, the second strategy of beating the diffusion limit. In the next lecture, I will tell you a little bit about that how there are other approaches that beats the diffusion limit. But you may have notice that I have not really told you how if you have a fluid flow, flow the fluid, bring the analyte faster to the sensor's surface whether that can give you a significant enhancement in sensitivity. The surprise is that it doesn't. And I'll explain to you in the next slide that while it beats the diffusion limit to some extent, there's not dramatic enhancement. And from that then onward, we'll be able to understand how the different techniques of beating the diffusion limit compare. So until next time. Take care.