nanoHUB-U Principles of Nanobiosensors/Lecture 3.11:Cantilever Sensors Static Response
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[Slide 1] Welcome back. So in the last lecture we had been talking about cantilever-based biosensors. And I focused primarily on dynamic-- bio sensing. That once the biomolecule lands, the natural resonant frequency of the cantilever changes. And today I'll say a few more things about the cantilever dynamic biosensors. But then move onto static bio sensing where the deflection occurs but you are not relying on the oscillation but rather the degree of deflection associated with the cantilever. And we'll see in the next lecture that how the static deflection can have very large sensitivity, much larger than you might expect from potential metric or amperometric sensors. So this is sort of in essence the preparatory lecture for the next one.
[Slide 2] Let me tell you one or two more things about dynamic biosensing before we leave this topic. And then I will move into static biosensing. Immediately first emphasize that static biosensing on its own doesn't work. You cannot measure true static biosensing alone. The mass of the bacteria, you always need dynamic biosensing and I will explain why. But under specific circumstances if you design the device in a particular way, you will be able to measure it under static biosensing also. And that has to do with nonlinear biosensing. I'll explain how that works before I conclude.
[Slide 3] Okay, so you remember the special feature of the reversal of frequency at the nanoscale. This is a very important thing to remember, that generally we always have expected that as biomolecules sort of gradually build up, then the resonant frequency should go down, should gradually go down. And you see that happens, especially at very low thicknesses as the biomolecules are building up and other part does go down as one would expect. But what happens as more and more biomolecules land on the sensor surface, it makes it stiff. Just like starching your shirt makes things stiff. And so therefor what happens is it begins to compensate the mass effect. And at some point what you will see is that because this composition is so perfectly balanced that you will not see any shift in the resonant frequency, even after the biomolecule has landed on the sensor surface. And of course if you allow a longer period of time the shift in the frequency will actually become positive. And I explain to you the critical frequency and how to calculate it. So this is an important consideration at nanoscale biosensing that does not happen at millimeter scale or even at micrometers thickness membranes. All right.
[Slide 4] Now one thing I wanted to emphasize, that although I have focused on the peak, the resonant frequency, you must have noticed by now that this is not a delta function at the omega naught. But rather, the red curve is broadened, and so is the blue curve. And this broadening and the peak position is actually not given by the simple formula that I told you about. This simple formula where the omega naught is simply proportional to k divided by m and the square root. That's really not everything in this story. Remember, I neglected the-- response or the damping associated with this. And the damping fluid is of course very different from the damping in air. And that can have an important effect. That is the reason why this broadening occurs.
[Slide 5] And this is an important thing, especially when you are looking at the experimental data. That when the damping is present, the gamma is present, even if you don't have any constant external forces. You give it a ping, allow it to oscillate. But depending on whether it is oscillating in water, which is high gamma, or in air which is low gamma, the response would be considerably different. And it turns out that you can solve this equation. Instead of just having e to the power i omega naught t, by a more damped oscillation with the sine omega prime t. This is the new resonant frequency, the omega prime. And it turns out that the solution, which will be given in the appendix that you can check in the appendix, it turns out the omega naught, the new resonant frequency is k divided by m square root. That's fine. But shifted by this gamma squared, by 4m squared. So if your damping is significant, you can see the resonant frequency would be significantly affected and it will be shifted at a lower frequency. So one must account for this damping when you want to know what is your original omega naught and how far it has shifted from the original position. By the way, the gamma with respect to the omega naught, original frequency omega naught, undamped, is related to the quality factor Q. therefore you will often hear that high quality factor is very important in order for accurate cantilever-based biosensing. All right, so that's all I have to say about dynamic biosensing. Look at the appendix in order to see how this variation works. But that's essentially the basic story about nanoscale dynamic sensing at the nanoscale cantilever.
[Slide 6] Now let's talk about mass-based sensing, challenges of mass-based static sensing. Now dynamic biosensing requires that you monitor the frequency and monitor the deflection of the laser beam on a ray. That's generally the question in AC circuit and others. So therefore it would have been much better if we could just bend it just like a person standing on a diving board at the very end. If the person is heavy, then the bending would be more. If the person is light the bending will be less. If we could just measure the bending in the static, life would have been much, much, much nicer. But it turns out that is difficult. Let me explain why.
[Slide 7] So assume that in this top figure the biomolecules are yet to arrive on the sensor's surface and captured by this magenta probe molecules. And this is a simple silicon nitride micro cantilever and could be really thin. And once the biomolecule arrives, because of the mass, also because of the electrostatic push or repulsion or because of the hydration or simply because the biomolecules are large, pushing against each other. The whole cantilever might bend. It's just like biometallic strips. When you increase the temperature the whole thing bends because the surface stress. The physics is no different. Now it could be because of the mass, surface stress or the change in the spring constant. The whole thing may deflect in steady state. The question is, can you measure it? Let's see.
[Slide 8] So let's calculate. We are thinking about steady state, so life is good. I don't have any acceleration. I don't have any damping. One of the only thing that I have is the force is being balanced by the spring. What is the force? Well force is mg. g is a gravitational constant. All right, so life is simple. So you can take this quantity, this balanced force. Now you can take log on both sides, so log k plus log y is log m plus log g, and take a differential. So that makes it delta k over k naught, delta y over y naught and so on so forth. You remember you see there is no term called delta g over g naught because of course hopefully the gravitational constant doesn't change significantly when you change things by a nanometer. So that is not there. And so therefore the net change is equal to how the mass change due to mass and change due to spring. Mass is trying to make it heavier. Mass is trying to make it bigger, landing on the total spring and the spring is sort of pushing back. And the net difference gives you the net shift in delta y. All right, so that's simple. By the way, you may remember that in the resonant frequency also, we have this spring competition between spring constant and the mass. And so essentially the reason is that the essential difference comes from the same physics. That will be explained in a second. All right, so this looks like a good strategy.
[Slide 9] So let's take an example. Let's say you have a silicon beam again. Length is 5 micron, width 1.5 micron. very thin, density is silicon, so therefore 2,300 grams. And let's say you have some protein molecules, this prostate specific antigen marker for prostate cancer. That comes in, lands on the sensor surface, squirts it on the cantilever. Again I am assuming it about coats the whole thing uniformly, length 5 micron, width about 1.5 micron. I'm sorry, this would be micron. Both of them are micron. H is about 50 nanometer and the density is somewhat less dense. So about 1,000 kilograms per meter cubed. Let's put it in the previous equation. And you have the delta over m. And once you calculate it you see the deflection is huge. Whatever was the original deflection is about 100 percent increase in the deflection. So you would be very happy that this is going to be a very good sensor. Except that once you calculate the y itself, how much deflection has occurred, you can see about 40 femtometer which is very small. There's no technique that allows you to measure a distance that small. And since it is unmeasurable, therefore even with great sensitivity this is a technique that we cannot use. So somehow if you could reduce the k, if you could somehow reduce the k. Spring constant, of course we could increase it. Now we want to reduce it by quite a bit, right? Because you see .0152 which is already small in the nanoscale. That didn't do it. So we may have to reduce it by a factor of hundred or thousand before we can actually measure something. And so therefore somehow we have to soften the spring. Now how are we going to soften the spring? This is already small silicon cantilever hanging there. How do you soften this spring? This is something that I'm going to discuss a little bit later.
[Slide 10] Before I do so, let me tell you why it is that we could measure things in dynamic oscillation, measure mass. There is no problem. But when you try to do it in static, we said that oh it is too small. Why is that? It turns out that for the dynamic case, which was the last lecture, you remember y was Ae to the power i omega t in the real function A equals sine omega t. So if you get that acceleration, you will have omega squared multiplied by A is the amplitude of the oscillation. Now do you remember what the omega was? The omega that we calculated was a couple of gigahertz. And so by the time and oscillation was about a nanometer. When you multiply these things, you can see that we are oscillating at about 20,000 g. So actually you have artificially sort of making the acceleration significantly larger. That allowed us to make the measurement there successfully. But with the g alone static deflection just with a single g. We cannot measure anything. So that's why this dynamic sensor worked but the static sensor does not work. We have to do something about it.
[Slide 11] So let's start by doing something about the static sensor. And the trick would be the following. We'll add a capacitor. Remember previously it was just a spring cantilever. Electrical things were not around sort of. Now we add a capacitor. So the capacitor has two plates. One is a bottom plate and the top plate would be this cantilever beam. And we'll apply a voltage. Let's see how the property of the sensor changes as a result. What it will do effectively is it will change the k, reduce the k significantly. That's what's going to happen. So again, you will have the force balance equation exactly the same as before. You have that acceleration term, damping term, spring term. Equal to the external force, any force that you apply. And then because I have used a capacitor, there will be an electrical force associated with the capacitor. Now in steady state of course the time-dependent terms are 0. The first and second terms are 0 so I have this spring. External force and the electrical force. How do you calculate the electrical force in a capacitor? Again, from the college-level physics courses you may have noticed this, that the energy of the capacitor is half CV squared. Everybody knows that. Now how do you get force from the energy? You take a derivative. Derivative of the energy with respect to the change in position. That's the force. So if you take a derivative, it will be half V squared dCdy. By the way, the negative sign comes from the fact that y is measured vertically down. That's why. And the V squared is a constant because here look at this battery. That battery is always connected to the constant voltage. The voltage is not changing. But once the cantilever has come down a little bit, the gap has changed and so therefore the C has changed. The capacitor has changed. And so dCdy. If it were a parallel plate capacitor, what is the formula for the parallel plate capacitor? Epsilon naught A over y. And so dCdy simply will be epsilon A naught over y squared. So you can put it in. You can immediately get the capacitors, the force associated with this additional capacitor.
[Slide 12] Now let's think about a person who does not know that you have added a capacitor. They are still thinking that the same old spring mass system shown here in B. What I want to tell you, that that person will think that the spring has the effective spring constant K effective, which is different from the physical k that you have. Because we have added this extra capacitor. This person doesn't know that there is this capacitor business going on in here. So this is the effective spring constant. And let me tell you how that spring constant becomes very, very small once you put in a capacitor like this. All right. So remember, in equilibrium the spring constant is being directly balanced by the electrostatic force associated with the capacitor. And you remember why this one is over y squared, because dCdy in a parallel plate configuration and this one over y squared. Now you see, for somebody who is in the B configuration, the person may think that he had an original spring and there is some extra force that is coming from this capacitor, which he really doesn't know about. For him, he sees the total external force. And essentially the sum of these two. And if you do that, then the corresponding spring constant would be this dF over dy. And the spring constant will be of course the k. That's fine. With a negative sign. That's fine. But there will be a y cubed dependence in here because there is a y squared here. And first of all, the spring now looks weaker, considerably weaker. This is our original k minus this quantity. The higher the V, the weaker the spring. And in fact you can make the V so large that the spring constant may disappear completely. In that case, remember I will have a huge change once the biomolecule comes in. There will be huge deflection because I have effectively made this spring constant disappear. Now this particular expression 2V squared, you can rewrite it by using the first expression here. This is how it works. So V squared epsilon A over 2 y squared. So you can essentially take this quantity, put it in here and then you will immediately see that will give you two-thirds ky naught minus y. And once you sort of put it in you will get the final expression. So what does it mean? It says that as soon as you keep increasing the bias, the voltage V, the effective spring constant will gradually go down. And eventually when it's two-thirds y naught, at that point the spring constant will essentially vanish. The spring would have softened considerably.
[Slide 13] Now what's special about this two-thirds y naught? Well in order to understand it you have to understand that this spring mass system in the presence of a capacitor is a highly nonlinear system. It is such that if you apply a voltage, the spring will come down, oscillate and then stabilize. Put a little bit more voltage, comes down, oscillates and stops. But if you put beyond the sudden voltage as you get closer to this two-thirds y naught, it can then snap shut beyond this point. It becomes highly nonlinear capacitor in this system. And if you do a sort of simulation you will see that for one voltage the black line is sort of oscillation stabilized at a given distance. y naught is about less than 3 micron. A little bit more voltage, red stabilizes. Blue stabilizes. But apply a little bit more voltage and it doesn't stabilize at all. And so we are talking about biasing the sensor in the blue curve, close to the blue curve, where the sensitivity is the greatest. So how does it work? Why is there transition like that at two-thirds y naught? In order to understand that, you have to plug the y, the displacement, as a function of the force. Remember the spring force goes as k y naught minus y, which is the red line. As you pull the spring more and more away from y naught, the force you require increases linearly. So of course you cannot go below y naught because the two places would be together. So that's the maximum point. It increases linearly with spring constant. Now what happens because of the electrostatic force goes as 1 over y squared so in the beginning the spring force is significantly larger than the electrostatic force. So nothing much happens. So this is by the way, this y square comes because of the fact as I told you before dCdy will have a y squared dependent. That's why this F electric force has a 1 over y squared dependence. So it stabilizes at a particular point. The spring constant has reduced a little bit but not significantly. Now if you increase the voltage a little bit more, then you can see the difference between the electrostatic and the spring has narrowed considerably. So the spring is getting weaker and at one point essentially they will be tangential to each other and this electrical force will be always larger. This blue line will be always larger than the red and the spring will not be able to hold it back. It will snap. And so we want to operate very close to this point because at this point the spring is the weakest. Any small vibration can cause a significant change in the displacement. So you can find out what this position is simply by noting that at this position, at this critical position we have to be a little bit below that. The forces are equal, the tangents are equal. And if you equate this to when you will do it as part of the homework, you'll find that two-third y naught the spring is the weakest. And at that point you actually have the greatest sensitivity. Now you can look at it in a slightly different way also in terms of total energy.
[Slide 14] This is the spring energy, ky naught minus y squared divided by 2. Electrical energy is half CV squared. If we don't have any voltage, then essentially the electrical force has no effect. As you increase the voltage then you have the pink and the green curve. And if you put them together, you will see that in the beginning you have the total system has equilibrium at y naught. As you apply more and more voltage, the spring gets weaker because the curvature gives you the spring constant. The curvature is the second derivative of this with respect to y. That gives me the spring constant. The curvature becomes small and at some point on the green curve essentially there is no stable point. It goes and clamps shut. So we want to operate close to the weaker spring position. And that is something I will explain in detail in the next lecture, how we can use this spring weakening, spring softening in order to have highly sensitive cantilever-based biosensor.
[Slide 15] So let me conclude then. So I told you about the importance of dynamic biosensing. In the last lecture we simplified things a little bit. I explained the frequency reversal and all, but didn't really emphasize the importance of damping, fluid damping, and broadening and quality factor. I pointed that out this time around. In fact, this is why it's difficult to go below a picogram in fluid, because of the damping issue. In fact, if you didn't have damping, if you had a vacuum, an order of 100 zeptograms is easily measurable. And so therefore this is an important thing. Damping is very important. It's not a secondary effect especially at nanoscale. Now I explained to you pure mass sensing is difficult to measure. Why? Because the gravitational constant is so weak. You know, for us it's very strong. We can walk around and we stick to the earth. That's not a problem. But think about a little virus or a bacteria. They don't care about the gravitational constant. In their lives, gravitation plays no role whatsoever. So in that case, either one has to do this dynamic biosensing, which makes the acceleration thousands of times larger. Or have to weaken the spring in such a way so that even the small mass can cause significant deflection. And one of the ways of spring-softening, you can always have a new material that you can try to find, very difficult, is to simply add a capacitor. And if you add a capacitor for the biomolecule sitting on the cantilever surface, the biomolecule doesn't know that you have added a capacitor underneath. It doesn't know that. It feels like the overall spring has weakened somehow. And when it's landing on it, the deflection is significant. And that is the physics I explained in the last three or four slides. It turns out that physics of this transition, that transition of spring weakening, is that at the heart of this phase transition, how water become solid and the solid ice becomes water. This phase transition and this physics are equivalent, and the great sensitivity that you have at the phase transition is exactly what you are using here to measure the mass of a virus or a bacteria. So that's it. Let me end here. In the next lecture I will show how the static bio deflection and transistor-based sensing can be combined together to greatly enhance sensitivity. These are nonlinear biosensing. But until that point, take care.