nanoHUB-U Principles of Nanobiosensors/Lecture 3.10: Cantilever-based Sensors Basic Operation
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[Slide 1] Welcome. Today we'll be talking a third type of biosensors. One of the simplest and I find it most interesting. And this is called a cantilever-based sensors. Essentially a mass sensor, and we'll understand or try to interpret the physics associated with this cantilever-based mass sensors today.
[Slide 2] So I'll beginning by-- with the basic introduction that how does this particular sensor compare with the two previous types of sensors that we have talked about. And then I'll explain briefly how the sensor actually works. But this will be qualitative-- qualitative discussion. The real discussion, quantitative discussion, would be on the physics of linear biosensing. I'll explain the framework, theoretical framework which uses this in order to measure the mass of biosensing. And today we'll focus on dynamic business, that if you have a mass based sensor then this particular technique will use oscillation of the-- of the cantilever beam in order to measure the mass. In the next lecture, we'll talk about static deflection. So let's get started before-- And then of course we will conclude.
[Slide 3] Now let me briefly remind you that we had been talking about three types of sensors. Potentiometric, essentially a camera for the charge, so therefore this one the charge of the biomolecule, yellow biomolecule, changes the channel potential. And as a result, the current flow from the source to the drain changes, reflecting that the biomolecule has arrived. Similarly, an amperometric sensors which is a like a glucose sensors that we have discussed in the last three lectures. Essentially identifies when a molecule has landed on the working electrode by changing the current flow through the circuit. So this would be the working electrode, that could be the counter electrode, and we have seen that because the redox reaction associated with this arrival of the biomolecule. This current changes, and that allows us to detect the presence of the biomolecule. So in-- in some way it is a camera for electron affinity. The mechanical biosensors that we'll be discussing today, measures the mass of a biosensor when the biomolecule arrives on a cantilever, let's say, or this spring mass system. The mass of the molecule changes the mass of the suspended beam changes, as we'll explain in a second. And this particular sensor can measure the mass of the arriving biomolecules, thereby essentially identifying the molecule itself. Now what type of mass are we really talking about? How small are these things? Especially because in nano-biosensing we are used to dealing with, you have seen so far, very, very small masses.
[Slide 4] So the masses we're talking about, remember a single molecule of glucose, real small molecules, is about 180 Daltons biomolecule or about 0.3 zepto-gram. Bonded base pair of DNA is 300 Dalton or about 0.5 zepto-- zepto-gram. And if you have a DNA which is 100 base pair long, then you can see that this would be essentially 50 zepto-gram. So we are talking about extremely small masses. As we go to a larger molecule, per amino acid, that's the AA, is 125 Daltons. You may have 200, 300 amino acid in a particular protein. So you can see again a zepto-gram, on the 100s of zepto-gram level masses. Now once you get to viruses or bacteria, things become much larger. Mega-Daltons, 10s of attograms, pico-grams, and a human cell can be as large as a nanogram. So therefore this mass base cantilever base sensors generally can detect these nanogram to pico-gram level-- materials relatively easily. But once you get to this very low level, very low concentration it requires some other amplification before the molecule can be detected. But it turns out theoretically at least in vacuum, one should be able to measure 100s of zepto-gram level-- level masses using this sensor technology. So therefore, is an extremely sensitive nanosense, nano-biosensors.
[Slide 5] Now why do we care about nano-biosensors? Remember there are cantilever-based nano-biosensors. Remember I told you that there are many biosensors. Right? Optical, biosensors, optics-based biosensors, various level based biosensor technology. One of the key things that makes this cantilever-based-- based sensors very attractive is that we don't need any reference electrode. Remember we are measuring mass. And so therefore there's no current flow involved. And so we don't need these electrodes, the extra electrodes which are necessary for potential metric or amperometric sensors. And therefore, we can minerate-- miniaturize them far more effectively. Also, there's no salt necessary directly and so therefore the charge screening. Salt is necessary, of course, in order for the binding to occur. However, since we are not measuring the charge associated with the DNA molecule the screening doesn't matter. Screening is there, but screen doesn't matter. Because we're just measuring the added mass associated with the biosensor. And therefore and also it goes very well for various nano-fabrication technology. And so therefore you can miniaturize them, put hundreds of them or thousands of them in parallel so, therefore, this is an ideal technology for nano-biosensing. Of course, the other problem generally-- general problems associated with settling time. That how long the molecules will take before it lands on the sensor surface. Well that is independent of what type of sensor you use, and so therefore cantilever sense-- sensors will have the same the limitations as we discussed other sensors have. Now selectivity is an issue because, you see, if you have a mass based sensor, and if two proteins have exactly the same masses, and they then land on the sensor's surface then you will not be able to differentiate them. And so just like potentiometric sensors had this selectivity problem with something that we'll discuss in the next set of lectures, this one has similar selectivity problem.
[Slide 6] So let's-- let me explain how this biosensor look like. If you look at this particular cantilevers. It's a hanging cantilever connected to a substrate, and when a biomolecule lands on the sensor surface just like a springboard or diving board when somebody stands on the end, the cantilever bends. So you can see this diving board like configuration. And once the biomolecule, and here it's a virus. Once the biomolecule sort of here lands on the sensor surface, then it will change the deflection, change the oscillation, resonant frequency, something that we'll discuss in a few minutes. And as a result, this added mass is deflected in the shift in the frequency of this cantilever, the oscillation frequency of these cantilever. And that allows us to detect its mass. Remember, this is a virus particle, it's very, very small, small mass and still the sensor is sensitive enough to detect a single virus molecule. Now the size of the sensor, well it's about-- Width is about, in this case, about a micron, 2 micron. Couple of microns in length. So the micron size, but the real nano-scale features comes in the thickness. The thickness is only about 25 to 50 nanometer, and so that gives us, as you will see, huge sensitivity of this slice of sensors. Now how do you measure the deflection? Well what you do is essentially once the biomolecules have landed, you bounce off a laser beam, and use a photo detector to measure-- measure the location. Remember when the cantilever was flat, in that case, after bouncing light it will, sort of, it will land somewhere on the screen of the photo detector, and after it has bent or once it begins oscillating then, of course, this one will move back and forth on the detector array, and from there you can essentially calculate what the resonance frequency is, or how much the deflection has, how much deflection has taken place. Now, of course, this cantilever need not be one. As I said, that you can miniaturize them and put many of them in parallel, so here I show you an array. It's a beautiful example of an array, cantilever array. Where you can see there's all these springboards are in parallel. So if you use different receptors for each one of these "diving board," quote unquote, in that case you'll be able to detect different molecules in parallel in solution using this cantilever-based sensors.
[Slide 7] Now very simply, the deflection-- measurement of deflection is a very important thing in here. And this measurement of deflection could be static deflection or it could be the resonant frequency, change in the resonant frequency. And so you can do it optically, as I just explained by bouncing off the laser, and so there by this deflection is change in the position on the photo detector is proportional to the amount of deflection you have. That's one way. But integration of a laser and the corresponding photo detector could be bulky. Right. Could be difficult to miniaturize. Other methods are also possible. For example, this is a cantilever. You can see a u-shaped cantilever hanging just like a springboard. But now if you use a piece of electric material which with resistance changes when it's lengthened or shortened, in that case after the biomolecule has landed on the cantilever, it will bend a little bit and that will change along the green line the resistance associated with it. And thereby you can also detect whether a biomolecule has landed or not by simply looking at the resistance. And finally, you can also look at the capacitance. So if you have a bottom plate and the top cantilever, so you know, it's epsilon naught A over D is the gap. And once the molecule has landed, the D will become smaller, the capacitance will change, and by reading of the capacitor you can also see whether a biomolecule has landed or not. Now in many cases, you will set up the cantilevers in oscillation simply by applying a voltage, and generally you can always measure the capacitor between the suspended beam, which is shown here, and the bottom electrode of a capacitor. And by looking at the difference in the cap-- difference in the capacitance you can infer the change in delta y that must have occurred to give this response.
[Slide 8] Now how should you bring the biomolecule to the sensor? Right. You can-- One thing you can always do is to put the sensor and in the solution containing the analyte molecule, proteins or DNA, this type of thing. Other viruses, that's possible. In that case it will be in the fluid. The biosensor will be within the fluid. The one problem with that and that's why you-- it's difficult to detect less than a pico-gram of molecule, is because this is in the fluid. So as it is trying to go back and forth, oscillate, then the oscillation would dampened by the fluidic motion. And so, there is-- This is, by the way, most common putting it in the fluid, in the fluidic channel measuring-- measuring the change in the capacitors or changing piece's electric response or looking at the optical deflection. But another one it's very ingenious is to-- This is the cantilever, once again, you can see this is hanging within the-- within the-- as a springboard. But now the fluidic channel is embedded within the cantilever itself, cantilever itself. And so the biomolecules come from 1, 2, 3, 4, 5. So it flows through it, and once this yellow biomolecule is there it changes the mass of the cantilever, and thereby changes the resonant frequency, but now this is oscillating in air so the dampening is very good, very small. And therefore you can measure smaller amount of mass using this technique.
[Slide 9] All right, so this is the general way by which a sensor works. Now let me explain the physics of it in a little bit more quantitative terms.
[Slide 10] Now although the cantilever where originally it was undeflected, and the green layer is the selective layer which recognizes the molecule, just like the DNA layer was present and target DNA came and bound to it. Or an antibody might be present in order to catch that target bacteria or protein associated with it. That's the green layer which is the receptor layer. And then once the red biomolecules have come in, the mass has changed, and the whole thing is deflected by amount delta S. Now the analysis of this cantilever is sort of difficult. Right. You have to take a few classes in mechanical engineering in order to-- in order to understand how much deflection takes place. So I'll tell you a simpler way of doing it. In the simpler way, we'll think about this original system as is physics spring mass system. It's just like a scale that you use in the supermarket to weigh the produce, for example. Tomatoes and other things, you put it in and from the scale you see the weight of this-- of this produce that you are going to buy. So it's a simple spring mass system, doesn't have the complication associated with it. And then once the biomolecule comes in, let's say it's-- the deflection gets larger. Initially it was y naught. It has sort of deflected a little bit more. Now these are two equivalent systems of the cantilever. So therefore this equivalence will show the equivalence in a sec-- in a second, but in order to know the dynamics of this cantilever, you know, it's very easy. The first term is here the acceleration, the mass multiplied by the acceleration. That's the force. And then second term is gamma dy, dt . This is the velocity multiplied by the dampening coefficient, the faster its trying to move, dampening will try to prevent it. And the spring force, which is trying to pull things back is k, y naught-- y minus y naught. And this is the spring force which is trying to push things back. So when you add these three terms in together that will be equal to the external force associated. In this case, for example, there's no external force. Just the mass and, therefore, the sum of these three things must equal to 0, approximately. Or if you have a spring for external force, you can simply add to it. This is a general equation.
[Slide 11] Now how does this equivalent circuit compare with the original cantilever? Assume that you have a cantilever of width W, length L, and height H. Now remember this was about a couple of micron, width was couple of micron, 3 - 4 microns in the previous example you saw. And H was how much? Like 25, 30 nanometer, very, very thin. Now, once you have this basic cantilever it turns out-- this equivalent mass that will allow you to go from the cantilever to the spring mass system equivalent is something like this. Rho b is the mass of the original cantilever. b stands for before, before the molecule has arrived. LWH is a volume. And so this is essentially the weight of the-- weight of the cantilever. And .24 really comes from how the cantilever is suspended. We'll not worry too much about it. And so when you're solving this simple equation, rather than this complicated equation, instead of m we'll just put mb in that equation. Now what about this k? Well k is also simple. It is the Young's modulus that for every material, if it's made of silicon we'll have a Young's modulus associated. We can just Google-- Google it up. I is the moment of inertia, I'll explain. For the cross-section, depends on the cross-section of the beam. And L is the length, length cubed. And I before any molecule has arrived is given by, for this rectangular thing, W is the width and H cubed, the thickness cubed. That's why if you make it very thin-- If you make it very thin then it really reduces the moment of inertia significantly. Now this factor of 12 comes from, again, a calculation that is in given in a standard strength of materials-- textbook. But we'll not get into that for the time being. Let's accept these equations. Now, once the biomolecule has sort of landed on the sensor surface, then afterwards the height of the cantilever has sort of gotten a little bigger. In that case that will be equal to plus delta H. So the mass will have increased after the biomolecule has landed. Correspondingly, the spring will also change. Spring constant will also change. How? Because the height is no longer H but H plus delta H. You know half of the delta H is on the top, half is on the bottom because the cantilever is sitting in the fluid. And biomolecule will come and attach both on the top surface as well as on the bottom surface. I'm telling-- I'm saying this delta H, over 2 on either side. And so, therefore, I will also change. And, therefore, the spring constant will also change. Now I'm not talking about gamma yet. We'll see on that gamma is relatively small. These things are happening. What are happening in the air specifically, I'll talk about gamma a little bit later. So let's think about the ideal case where the gamma is relatively small.
[Slide 12] Okay, so let's start with a simple example. Let's say I have no dampening. I throw away gamma. And I don't have any external force, continuous external force. So I take this spring on which the biomolecules have landed. Give it a little push and let it oscillate, and look at the resonant frequency. Or look at the frequency of oscillation, natural frequency of oscillation. How does it depend on the mass? Well it's relatively simple, you see. Once you-- if you write y minus y naught as delta y, the change in the y. Then you can see the first term, y, you can write it as y plus delta y. y naught is a constant, so once you take a derivative that will go away. And the solution of this equation you can write it in the form of Ae to the power I naught omega t. Omega naught is the natural frequency of oscillation. So once you insert this delta y, expression for delta y into the-- into the equation of motion, then you can immediately see once you solve it that the resonant frequency is given by k, divided the spring constant, divided by the mass of the spring mass system under this square root. Now you may have seen this equation many times in your high school physics-- high school or college physics courses. And so, therefore, there is nothing surprising in this-- in this particular formulation. So if you make the mass bigger-- If you make the mass bigger or the spring constant-- If you-- the resonant frequency will decrease, and even also if you make it very, very stiff the resonant frequency will increase. And so, therefore, these are well-known quantities. So before the biomolecules have landed, we'll have kb, the spring constant before it comes in, and mb is the mass of the cantilever before the biomolecule have landed. And afterwards, it will be a changed spring constant and changed ma. Now by the way, what does it mean to have a-- Change in the mass is easy to understand. But what does it mean to have a change in the spring constant? Well change in the spring constant simply means that once the biomolecule has landed that it has gotten sort of stiff, more stiff. Difficult to oscillate as much as it used to be. Right. That will be the change in the spring constant. Any time you take a shirt and launder it and then in that case if you put starch on it, that changes the, sort of, the spring constant because, sort of, things have gotten more difficult to bend. And so that would be the corresponding change in that k is spring constant. ka-- after the biomolecule has landed. All right.
[Slide 13] The second quick example, let's say I have a silicon beam. 3 micron, width about 1.5 micron, height 25 nanometer. It is silicon so the Young's modulus is about 70 gigapascal, density 2330 kilogram per meter cubed. Put these values in, the alpha 1 is 0.24, remember that constant that depends on the support of the cantilever. You calculate it. You see this is a very small mass, this tiny springboard of-- is essentially about 63 femtogram. That's very, very small, this cantilever. That's why it's so sensitive. And correspondingly, you can calculate the spring constant by putting the values in. And you can see the 25 nanometer cubed because H cubed, remember. And this is the width, and this is the corresponding E. E is the Young's modulus. And once you put it in, you get about 0.01 Newton per meter. That's the spring constant associated with it again. Relatively small. And that's primarily because this H is very small. So very, very thin. So easy to bend compared with thick cantilever. And if you put-- put the values in you will see you'll have a resonant frequency before the biomolecules come in, on the order a few megahertz.
[Slide 14] Now you can do a couple of things. You can, for example, you do-- can do a mass only dynamic biosensing. Where you assume that the spring constant doesn't change. Lets assume that and in that case only the mass changes. So the mass after is equal to mass before plus the delta m. Delta m is equal to the mass of the biomolecule. And once you put it in, and assume that spring constant has not changed. It's just an assumption. In that case change in the resonant frequency is directly proportional to the change in the mass of the biomolecule. By the way, how do you get it? Get this expression, delta omega divided by omega b. You take a log-- You take a log of this quantity and then take a differential. That will give you this particular relationship. And once you put these things in, you can immediately show, it takes a few lines of algebra, is that the shift in the resonant frequency after the biomolecule of mass delta m has landed depends on a bunch of constant associated with the-- with the cantilever itself, cantilever properties, but it's directly proportional. Remember, It's no logarithm independence, as was the case for the potentiometric sensors. Right. It's directly, linearly proportional. And how can you make this change significant? Well you can make the width smaller. Or you can make the length smaller and thereby you essentially can have a significant increase in the differential change. And so there are many ways you can essentially make a cantilever base biosensor very, very sensitive.
[Slide 15] You could go to the other extreme. You could say, well, mass is so small. The biomolecule mass is so small that I can essentially ignore. And only the spring constant, you know after you have coordinated the layer of bimolecules it has gotten steeper-- stiffer and that is the more dominant quantity. In that case again, you would neglect the delta m, put all the constants in, assume this time the delta k change in the k is coming because of change in H. So it is H plus delta H to the cubed. And if you know delta k, that will be the differential, which will be 3 times delta H. You know for a small delta H. And the delta H will be proportional to the number of molecules you have, ms, area of individual biomolecules right. Do you remember for the DNA or for the protein we have certain specific sizes. And Ht is the height of individual biomolecules. You put those things in, that will give you the delta H, and once you put the delta H in this expression you will know the delta k. And once you know the delta k, you correspondingly will find the shift in the resonant frequency due to change in the stiffness alone. And so correspondingly, if you take some, you know, typical values. 20 nanometer H and the biomolecules is approximately 10 nanometer high, let's say. And correspondingly calculate these numbers, then you will see spring constant can change by several percent. Easily detectable by-- by modern nano-scale cantilever-based sensors.
[Slide 16] In fact, experimentally, people have seen, in fact, even 20% to 40%-- 10 to 40% change in k once the biomolecules have landed. And so there are-- This is a significant effect. This is not a significant effect in large scale sensors. This is a significant effect in nano-scale sensors.
[Slide 17] The remarkable thing about nano-scale biosensors is that the frequency may increase or decrease depending on how many biomolecules have landed. I will explain that quantity-- I explain that issue as a defining feature of nano-scale biosensors and then conclude. You see, if you have-- There are certain experiments to go and measure in the lab, certain experiments. Let's say the amplitude versus frequency. This is the resonant frequency, means the natural frequency of oscillation, omega naught. That's what the peak is. And when the biomolecule has not arrived, is yet to arrive, then let's say you have this red curve. After the biomolecule has arrived, then the curve may have shifted to the left. This is one experimental observation. But it turns out that a second student, who has gone in the lab, and waited for a little bit longer or just a slightly different sensor can see the opposite effect. That in the beginning the peak was somewhere, and afterwards the peak has actually increased. The resonant frequency actually has increased. What is going on? This would never happen in this type of reversal would never happen in a larger scale cantilever-based sensors, but it happens at nano-scale. Well, what you can show, and I will explain in a second, that actually the frequency goes down like this. If the biomolecule that has arrived on the sensor surface is not too high. So the delta H is small, let's say you have 20 nanometer or so. A thin coating, in that case the frequency goes down. But if you wait a little bit longer and the delta H gradually builds up, then the frequency will reverse direction and will actually increase. So this is for a 5 micron long cantilever. If you have a somewhat smaller cantilever, in fact these effects will be even more accentuated. Remember delta omega is inversely proportional to q of the length of the cantilever. So once you make it actually smaller, these effects get even more accentuated. So you can have negative or positive depending on the types of cantilever that you have used. And you have to sort of deconvolve it in order to get the mass of the biomolecule that you have captured. Now it turns out that this frequency at which, or this thickness at which this reversal occurs is given by a very simple formula. Delta H critical thickness is given by this simple formula, and derive it in the-- in the next slide. But the bottom line is frequency goes down because the of mass effect. Frequency goes up because of the spring effect. Spring constant effect, increasing the spring. And so, therefore, these two things are sort of balancing against each other at the nano-scale doesn't happen at the micro- or millimeter length scales.
[Slide 18] How would you get the green point, the critical point? Very easy. You see, you know the resonant frequency before the molecule is captured, omega naught b, and after the molecule has been captured, omega naught a. Equate them, because at that point there'll be no change in the resonant frequency. And if you equate them, then, correspondingly, you know all the expressions for mass before and mass after. Which has increased by delta is increased by delta Hc. And the spring constant before and spring constant after, put them in together. And that will immediately give you the critical frequency with respect to the original-- original thickness of the cantilever expressed in terms of the spring constant as density of the biomolecule and compared to the original density of the silicon cantilever, as well as the spring constant. Before-- I'm sorry, the Young's modulus before and after. Put some values in and you will see about 20 or 30 nanometer. By the time molecules have, sort of, collected 20 or 30 nanometer, very, very small, actually the cantilever will behave as if very differently from, compared to a scale that you use, let's say, in supermarket.
[Slide 19] So let me conclude then. In this first lecture, we have talked about cantilever sensors, how it measures mass, and stiffness following the capture of biomolecules. Right. k and m after the capture of biomolecules. And this generally we always expect the frequency to decrease in a traditional cantilever-based sensors, but at the nano-scale it can go either way. And I explained to you how this spring constant and the mass essential are balanced against each other. And one weird thing about these sensors is that we don't need reference electrode. That's really a very important thing because it allows us to densely pack them. That's one of the features why cantilever sensors are so-- so helpful. And-- But the final thing you should always have to remember is that the diffusion limit and selectivity problems are still there. It's a great sensor, very sensitive, can measure a virus even people have been attempting to measure proteins, individual proteins. But you see it is very difficult, these problems of selectivity and diffusion limit are still there, and so, therefore, will still be detected by this settling time limits. So let me stop here. In the next lecture, we'll talk a little bit more about static deflection associated with this cantilever. Until next time, take care.