9 Dielectric Relaxation, Part 2
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Phillies, George
Lecture 9 - dielectric relaxation, part 2. George Phillies lectures on polymer dynamics, based on his book "Phenomenology of Polymer Solution Dynamics".
 Classes in Polymer Dynamic. Based on George Philly's book, Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011. And today, this lecture is lecture 9, More Dielectric Relaxation. I'm Professor Philly's, and this is part 2 of my classes on dielectric increment in polymer solutions, based on my book, Phenomenology of Polymer Dynamics. Okay, so what we are going to do today is to continue with our reading of chapter 7. I'll have homework again next time. The issue is as follows, dielectric spectroscopy is an enormously effective tool that gives us a wide variety of different measurements about a polymer solution. In particular, if we have a polymer chain in the solution, it has a few features. It has an end-to-end vector, that is, a vector that starts at one end of the chain and points straight to the other end. And that's a measurement of how big the polymer is in the solution. It's not the only measure, but it is a measure. Furthermore, we can use dielectric relaxation spectroscopy to characterize the relaxation of this vector. That is, as time goes on, the little segments along the chain all perform Brownian motion. They diffuse. Now, they have to stay attached to each other as they diffuse, so they can't just move randomly completely. But if the chain starts in this configuration, after a while, two sorts of, or three sorts of things will have happened. Number one, the center of the mass of the chain, which is originally here, will move, say, over there. Number two, the two-chain ends will have moved, I'm marking the two-chain ends relative to each other. And therefore, the end-to-end vector will change length and direction. And number three, the parts of the chain between the end will change which way they're pointed. Now, the details of how a chain moves, how it translates and rotates and changes shape, are quite complicated and not shown in that sketch. Furthermore, there are a number of entirely contradictory theoretical models for how polymer chains move in solution. The key feature here is from the standpoint of dielectric spectroscopy, if we have a chain whose pieces are all asymmetric, so each piece has a little dipole lined up along the polymer chain, the result of all of these little dipoles lined up is that the polymer chain has a dipole moment which points from one end of the polymer chain to the other end of the polymer chain. And what dielectric relaxation spectroscopy and the measurement of the dielectric increment can do is to determine number one, the average length of these things, after all, average, yes, it's different for each chain in solution. The relaxation with time in the length and the direction, that is, as time goes on, the direction fluctuates, the length fluctuates, it's fluctuate one value at one time, another value at another time. And this relaxation and the direction gives us, after a piece of theoretical work, the frequency dependence of the dielectric response. Oh, that reminds me, I said dielectric constant is 81 and you asked me the dimensions and I went off mentally and way the wrong direction. The answer you were looking for is what units are you in? And the answer is that if we write the potential energy between two charges in that form, we are in CGS ESU units. Epsilon is one in a vacuum and oh, maybe about four in the oil and at low, this is low frequency, it's standard, around 80 or so in water and Epsilon is dimensionless. However, most of you are probably more used to SI units, in which case you have a four pi Epsilon not down here in vacuum and you pile up an extra constant or you change Epsilon, not Epsilon, and that's SI units of electrostatics in which the charges in Coulombs and in that case Epsilon is no longer 81, it's a number which we could go into but doesn't matter. The important issue is that the dielectric constant in water is much higher than it is in the vacuum because in water if you apply an electric field you can line up all of the little water molecules at least a bit. If you think for a moment you may ask, well, gee, if you apply a big enough electric field, don't you line them up completely and then don't they, don't you stop getting an increase in the electric increment and that's correct. Now if you actually try to apply an electric field like that externally, you will have an entertaining time with your experimental efforts. On the other hand, if you take water and you drop into, say, a sodium ion, presumably chloride someplace, this is a lot of charge and it's very tiny and there is a big electrical field here and the water molecules very near a sodium ion are very much lined up. So surrounding ions in water there is a shell where the water molecules are pretty well lined up and at that point, since they're basically already completely aligned, they behave basically like an oil in their dielectric increment. That is you get some increase in the dielectric constant because you can move the electron shells around a little bit. But basically you can't move the water molecules, they're already lined up by the local ion. Okay, let us go back to here and what I was saying is you can measure different sorts of things about a polymer chain. The next thing you can do is to say, well, suppose we have more than one chain at the same time. So there is a chain and there is another chain. What happens? 50 years ago this was a major literature dispute and there were no very easy ways to do the measurements. You knew roughly how big a polymer chain was but the question is, if you increase the polymer concentration what happens? And one answer is, if you really run up the concentration, the chains shrink in on each other but don't overlap. It's now quite clear that if you run up the polymer concentration, the chains do shrink a bit. But they also spend a lot of time interpenetrating each other. I suppose that leads to an obvious question, why do they interpenetrate? How can they interpenetrate? Well, there is a trick. We have a polymer chain and it has some typical size r. It matters a bit which method you use to characterize the typical size because it's a random structure and it's not a solid body. But r goes as molecular weight to a power around a half or a good solvent, not quite 0.6. Suppose you have an object whose radius grows as the half or a bit more power of the molecular weight. That means the molecular weight of the object is proportional to r squared or maybe a bit less. And so you have an object whose molecular weight grows as the square of the radius. Have you seen an object like that before? Yes, it's a spherical shell. If you have a spherical shell, the mass is proportional to the surface area. Yes? And there it is, mass proportional to surface area. Of course, polymers are not spherical shells but they have the same properties as spherical shells. If you increase the mass, the object has to get bigger fairly large, bigger fairly quickly I should say. And furthermore, a spherical shell is mostly empty space inside. Now a polymer chain, the polymer strand runs around all inside this blob where the polymer is. However, as I increase the molecular weight of the polymer, this polymer gets bigger and bigger and it still has an important property of the spherical shell, namely if I imagine running a line through it, on the average I hit the polymer coil twice. So if I have a really big polymer and it's really big radius and I send a line through it, and if I only hit the tiny polymer, you can cross tiny in cross section, polymer twice, most of this is empty space and polymers can then interpenetrate and that's what they do. Now if you will take out your books in advance to figure 7.4, I'm fairly sure, I grabbed the wrong thing this morning, but I'm fairly sure 7.4 is heptane solutions of 140 and 743 kilodalt and polyisoprene. Yes? Okay, very good. And what is the point of this? Well, the first issue is, the vertical axis is delta epsilon over C. Delta epsilon is the change in the dielectric constant, there are some constants involved, of the solution. And as you add more polymer, well if you add more polymer, each molecule makes its contribution to the dielectric constant. But if you divide out, hang on a second, this is getting noisy. If you divide out the concentration, what is left is the dielectric increment per molecule. And the dielectric increment per molecule is approximately proportional, there are bunches of constants involved, to the mean square size of the molecule. So let us start with figure A and you have a comparison there, and as you make the molecule larger, R square gets larger because the chain is longer, and therefore, hmm, R square gets larger, and therefore, the dielectric increment gets larger, and therefore, if you stay at the concentration equals 0, the left hand axis, the larger molecule gives you a larger dielectric increment. Now the next thing you can do is to increase the concentration of the molecules. And if you do that, there is one thing you might need to worry about, that is, suppose I put a lot of molecules into solution. So here is a polymer molecule, and here is another polymer molecule, and they are somewhat wrapped around each other. And one thing you might ask is, well, are those chains independent of each other? And the reason you would worry about this is that if you pack the chains, and the chains line up, for example, parallel to each other, the dielectric increment of a lot of chains is not linear in how many there are, it is some larger number. But the answer is, here is our first chain, and if I do an average over the solution, the second chain could run through this way, or it could run through in the opposite direction and have its end-to-end factor pointed oppositely. And these two cases are essentially equally likely to occur. The two states have about the same free energy. And because the two states have the same free energy, the direction of this factor and the direction of that factor are completely independent from each other. And so when we look at the dielectric increment of a concentrated solution, we are just seeing this very good approximation, the single chain behavior. Now there is an exception to that. There is an exception to that, and the exception to that is if you have molecules that you run up the concentration and they do some sort of a phase transition to a liquid crystal, in which case the molecules are straightened out, at least approximately. They tend to, they perhaps pull all point parallel rather than parallel and anti-parallel. There is a bunch of interesting phenomena here that are outside this course. In this case, if you run up the concentration enough, suddenly there is a very dramatic change in the dielectric increment, which we aren't going to talk about. So what happens if we increase the concentration of the polymers? The point of this discussion I just gave you is that this relationship between the dielectric increment and the mean square size of the polymer chain does not care about the polymer concentration. And so if I plot delta epsilon over C versus polymer concentration, I am essentially seeing the single chain behavior. I am seeing that to good approximation. And what do I see happen? Well, if you look at 7.4a, as I run up the polymer concentration, the dielectric increment of the single chain falls. And if I do that to a bigger chain, the dielectric increment falls, and it falls faster. So at low concentrations, with increasing molecular weight, the dielectric increment increases, but at high concentration, these curves can cross, and with increasing polymer molecular weight, the dielectric increment of the system is less. What does this mean? It means that a very large chain at high concentration has contracted down, and now its size has gone decreased considerably, and its ability to increase the dielectric constant of the solution has been substantially reduced. So that is figure 7.4a. Figures 7.4b and c, b is the big one at the bottom, show a slightly different effect, which is also important, but it shows an effect that is of separate physical, class of experiment, I should say, that is of separate physical importance. And what is shown in those figures is that we take a polymer chain, and the general experiment is as follows. We have a polymer chain which we are interested in. We put it in a solution of other chains, this is the probe, out here is the matrix, and we choose the matrix polymer so that our experimental technique doesn't see it. So from the standpoint of our experimental technique, we are looking at a probe polymer, there is a solution out here, the solution, well actually it does contribute at very high frequencies to the dielectric relaxation, but the solution is basically inert, dielectricly or optically or whatever, and we only see the probe chains. Now you might ask how can you do this with a polymer and dielectric relaxation? Go back to what we were saying about last lecture about how a polymer contributes at low frequencies to the dielectric constant. A polymer contributes at low frequencies to the dielectric constant, and if it has as a result of its molecular structure, a bunch of little dipoles inside and the dipoles are arranged so they are fixed to lie along the backbone of the polymer. For example, this is a polyester, and the important feature here is this is the repeat unit, the two ends of the repeat unit are different from each other, and furthermore the structure in the middle does not give me any way to get mirror symmetry, so this has a dipole moment which lies along the bond axis one way or the other. On the other hand, suppose I took, those are hydrogens, the repeat unit here is actually that, that's polyethylene oxide. Well, polyethylene oxide, yeah, it looks like the repeat unit repeats, but if I drew the repeat unit like that, it's the same repeat unit, and so there's a center of reflection and there's no net dipole along the bond axis. So this material does not contribute to the dielectric constant, and if we stir it into the solution, we get no effect on the dielectric behavior. Now you have to be a little careful of that statement, because this is replacing your solvent, and just as the solvent has a dielectric increment, which is the same at most frequencies, a pretty good approximation, so does this. And so you have to pay a little attention to the fact, but even though this doesn't have a big dipole moment, you're doing a little bit of something and you should think about that on occasion. Nonetheless, here's our probe, here's our invisible matrix, and we are now looking at what a ternary, three components, ternary system. So with the system you have a probe, polymer, in this case, you have a matrix, which in this case is also a polymer, and you have a solvent. And you are looking at, G, the behavior of the probe as you put a polymer into the solution. Now why would you want to do a probe matrix experiment, as opposed to just stirring the polymer in? After all, you are going to see in this experimental technique, no matter whether you have a tracer probe or a high concentration of probe, the experimental method gives you something close to single molecule behavior. The reason you do this is that the matrix has a polymer molecular weight, M, the probe has a polymer molecular weight, P, and if these are different species, I can vary P and M separate from each other. That's very useful. If you look back at 7.4a, you notice that, G, the small polymer contracts slowly with increasing concentration. The big polymer starts out large, but contracts very fast with increasing concentration. The solid lines, by the way, you notice they are data points in the figure in solid lines. The solid lines are stretched exponentials, I didn't call it alpha, I called it rho. The solid lines are stretched exponentials, they are smooth curves. If you plot them on semi log paper, you don't quite get straight lines out. The issue is, yeah, there's okay, so you mix chains together and they shrink. And big chains shrink more than little chains. But we come to the key question. Is the statement, big chains shrink more than little chains, occurring because this guy is big and shrinks more, or is it occurring because a big neighbor is more affected by causing a probe to shrink, no matter how big the probe is? Question? For instance, like, if data percentage of a string is shrink, the large chain may shrink to the size of the larger down the small chain. But what about the percentage? If you, 7.4a is delta epsilon over c, delta epsilon is proportional to the mean square radius. If you consider that, you find that for the smaller chain, which is a 140 kilo-deltin polyisoprene, we drop the delta epsilon over c in these units from about 0.18 to about 0.14, which is 20% sort of, or a little more. For the large chain, the 743 kilo-deltin, the shrinkage in r square is from 0.222 at a much lower concentration, 0.14, which is a much larger fractional change. I'm looking at 7.4a, and therefore the larger chain fractionally is actually contracting more. Now, you'd have to do, you have to do, sit there and do some arithmetic if you want to calculate what the change in the length as opposed to the fractional change in the length is. But the answer, I think, is that, yeah, the big chain really, but is it that the big chain is just more sensitive and compresses more, or is it that the surrounding chains are bigger and are more effective at contracting? And the first step to understanding that is a ternary probe matrix solvent experiment, such as the experiments you see in figures B and C. Figure B shows the 140 kilo-daltin polymer, and you notice, well, I will point it out, there are a series of curves, this is B. And as you sweep this way, what is happening is you are increasing the size of the matrix polymer. And therefore, a larger matrix polymer is more effective at, that's C. A larger matrix polymer is more effective at compressing neighboring chains than a small matrix polymer is. In contrast, well, it's not really contrast, it's the same effect. If you go to C, which is a large chain, and you ask what happens as you increase the matrix size, you see the same behavior. That is, as you increase the matrix size, I'll explain the significance of that in a second, there is more compression of the chain that's being observed. Now, I will point out an issue. Suppose I have polymers at a certain concentration, very low. And suppose I increase the molecular weight. As I increase the molecular weight, the M goes up, the radius goes up, the radius goes from M to the one-half, so what happens at low concentration to the density of polymer inside a polymer coil as I increase the molecular weight? The density of polymer inside a polymer coil, not in the solution as a whole, that's determined by the concentration, but inside a single coil, the density of that chain inside the coil is the ratio of the molecular weight to the volume of the chain, which is the molecular weight to a radius cube, which is the molecular weight. Oh yes, R is proportional to M to the half or M to the 0.6. So this is M to the 1.5 or 1.8, and therefore the density of a polymer inside its own self is proportional to, well, take the ratio, it's M to the minus 0.5 or M to the minus 0.8, or some number in between. This is the behavior you would get in a theta solvent. This, or very close, is what you get in a good solvent. 0.6, R proportional to M to the 0.6 in a good solvent is a slight overstatement, but you can also get solvents that are in between these two. And the important issue I want to make is that as you make the molecular weight larger and larger, the density inside the coil gets smaller and smaller, and therefore the same number of grams of polymer at low concentration occupies more and more space as the molecular weight is increased. And thus, if we have a solution at some concentration, the bigger the molecular weight, the closer the chains are to rubbing shoulders with each other. And rubbing shoulders is about right because we're talking about the radius of the chain, sort of like this. Okay, so that's figure 7.4. The other thing we can talk about, if you want to ask how effective are matrix chains at compressing a solvent, at compressing a polymer chain, let us look at the next figure. The next figure is 7.5, and it gives us a plot of rho, I'll say what this rho is in a second, against matrix molecular weight. And I remind you the notion is that R squared, which is the same, up to units is like delta epsilon over C, is like a constant e to the minus rho c to the nu. And rho is the constant that describes how steep this stretched exponential is. Well, I've plotted this as a function of matrix molecular weight, and there are two things you should notice. And the first is rho does this. I would not put too, too much emphasis on the last hollow point, which indicates that it might curve back at very large molecular weight. It's possible, but there's only one data point there, you really don't want to extrapolate based on one data point. That's a little dangerous. The other thing that happens though, there are actually two curves there, and the two curves show the 140 and the 743 kilodalton probes. And so the answer is, the large probe has a larger rho, and thus contracts faster in some sense than the smaller probe does. And furthermore, rho stays larger for the large probe than for the small probe at all molecular weights. And therefore, we can say, yes, there's a contraction, and part of it is the difference between these two lines, so part of it is that large chains are easier to compress. After all, they're in a certain sense more and more hollow inside. And it's also the case that if you increase the matrix molecular weight, there is a dramatic change in rho, meaning big chains are more effective at compressing each other than small chains are. So that is how that works. Okay, there's something else that can be done, and the analysis in figure 7.6 could be carried out in another step. And what 7.6 does is to say, well, we have polymer properties, different properties, and we could compare them. And these are results from Urukawa at all, and what they did is to measure two things. You will observe, there are footnotes, and I don't always read all of them out loud, but they are there. And the first thing they did is to measure the viscosity. And they measured the viscosity relative to the solvent viscosity. After all, if you change the solvent, the viscosity of the solution does change, but it's not changing because of polymer properties. And what they observe is a series of curves, viscosity versus concentration. And with increasing polymer molecular weight, there's only one polymer in solution. This is a binary system we're looking at. With increasing polymer molecular weight, what happens is the viscosity of the solution goes up. So if you have a solution, the bigger the chain is, the more viscous the solution is. And that is a nice, uniform rule. However, they then take this stuff, and they do the experiment on a binary solution and a ternary solution, and they determine how the relaxation time, and what the relaxation time tells you is how long it takes for the polymer coil to the end-to-end vector to forget which way it was pointing. So if we have a typical polymer, it has an end-to-end vector by arm. As time goes on, the vector gets longer and shorter, and it points in different directions. And eventually, the initial direction of the polymer end-to-end vector has nothing to do with the final direction, and the two directions are independent from each other. That is, if you wait long enough, yes, originally the end-to-end vector was pointing this way, the direction my arm is pointing, but if I wait long enough, the end-to-end vector is equally likely to be pointing in every direction. So first, are good? Well, if I increase polymer concentration, it takes longer for the end-to-end vector to reorient and to change length. I'm not telling you how it reorients. I'm just saying it does reorient. That's proven by this experiment. And therefore, as I increase the polymer concentration, that's the horizontal axis, the orientation time goes up. And you can now do a binary experiment or a ternary experiment. You can look at the polymer chain in solution with itself, or you can look at the polymer chain in a solution with a matrix. And what we observe is that as we increase the matrix molecular weight, you notice there are two families of curves in 7.6b, and as I increase the matrix molecular weight, even at very high concentrations, the reorientation slows down. So the molecular weight of the matrix has a direct effect on how fast the polymer coils can reorient. This result is possibly not equally consistent with all theories of polymer dynamics, all theories of how polymers move in solution. However, you can always say, well, the concentration isn't high enough, the molecular weight of the matrix isn't high enough, and therefore it isn't that the theory, some particular theory is wrong, is that it's really not applicable to these solutions. Okay, we now push ahead, and we are going to reach section 7.3. Chain dimensions. Chain dimensions, we're actually going to talk about all of the techniques that can be used to measure chain dimensions. This is one of these sections that was, well, it's a little too short for a chapter by itself, so you ask the sensible question, where do you put it? The answer is there were lots of sensible places, it could have been placed. Many people would have put it in an early chapter discussing dilute solution behavior. The basic question we are talking about is, we have a polymer molecule, it has a molecular weight, M, it has some characteristic size, and lots of ways to characterize the size are, and are, however you define it, is proportional to M to some power. Now, the reason why you're interested in this question is actually an analytical chemistry question. That is, you have people sitting there, and you have the chemical plant going, ka-chunk, ka-chunk, ka-chunk, or whatever, and you're looking out polymer, and you would like to characterize the polymer so you know what the molecular weight of the material you're making is. The reason for this is there are large numbers of industrial applications of polymers. Many of them are sensitive to some particular property of the polymer, such as its molecular weight, and you would like to know how big the polymer is and how it would be paying for a particular molecular weight and would be annoyed if it doesn't show up, or something like this. And so what you do, what you do historically is you take the polymer coming out of the plant, and the polymer coming out of the plant may be a solid or a melt, or a very concentrated solution, or God knows what, and you take this, and you almost certainly are not producing a dilute, you could be, a dilute solution that tends to be inconvenient on an industrial scale because you have to pay for the concentration step. And you take this material to the lab, and what you do is you take a bit of it, and you dilute it. The reason you dilute it is that once it's dilute, there's a polymer chain here, and there's a polymer chain there, and there's a polymer chain here, and on the average the chains are quite separated from each other. And because they're separated from each other, these polymer chains change the behavior of the solution, but they do so in a way that is not affected by their interactions, it's just determined, if you're lucky, by how big the chain is. And so you measure dilute solution properties, and the dilute solution properties can be used to tell you how big the polymer is. Now you might sensibly ask, well, what dilute solution properties are there? And one dilute solution property is the viscosity, which is proportional to the viscosity of the solvent, one plus a constant, times the concentration plus. But if you just add a little chain, bit of chain to the solution, the linear term is quite completely accurate. And the feature here is that K... Is that a power long? Power long. Well, it is a power long. This is a power series. However, you can measure K1 quite accurately, and K1 is approximately proportional to some power, like the Q, of the size of the polymer. Now I put a subscript on R. This is the viscometric radius. It's the radius inferred by asking, how effective are these chains at changing the viscosity of the solution? The reason I bring up often say, this is the viscometric radius, is that, well, there are several other techniques for measuring polymer size. For example, you could take the solution and shine a light beam. We now use a laser, but they didn't, when this technique was first developed, through here, they didn't have lasers back then. The solution scatters light, it becomes cloudy, opalescent. And the reason the solution becomes cloudy is the little polymer chains in solution do not have the same index of refraction that water does, so they scatter light. And if you measure the intensity of the scattered light as a function of angle, why is a function of angle? Well, if the polymer chain is very small, it scatters light effectively in all directions. I am skipping a light polarization issue, which is experimentally very important. But if vertical polarization in, vertical polarization out, scattering plane horizontal, so if we're looking down from above, we look at the light scattering like this, and the polarization vector of the light is perpendicular to the paper. I can make the statement, if the molecule is very small, it scatters light equally in all directions. However, if we make the molecule bigger, eventually the molecule starts to approach being comparable to the wavelength of light. This is a light wave, you see, it oscillates. And as a result, this part of the molecule is being illuminated by this part of the wave and has one phase. This part of the molecule is being illuminated by a different part of the light wave, and scatters light with a different phase, and the light scattered from these two points can interfere. I am omitting details. And as a result, the light scattering depends on the scattering angle. How does it depend on the scattering angle? As you make the polymer bigger and bigger, you make the polymer bigger and bigger, the light scattering is focused more and more in the forward direction. The extreme case of this, this is shown by window glass. If you make the polymer enormous so that you have this block, and it's only one molecule, the laser beam goes straight through. If you want to say there's scattering, it's only in exactly the forward direction, and if the block is homogeneous, there's almost no scattering to the side at all. The object is transparent. A similar mathematical effect, it's not a polymer, is seen in window glass. Window glass is transparent because to the extent you get scattering, it's forward. So, you say the scattering versus angle depends on the polymer molecular weight, and therefore there is another radius we can infer, and the other radius we can infer is determined by the scattering experiment. I can run down a series of these, and there are a series of different ways we can physically measure an apparent radius of the molecule. For example, we could measure the diffusion coefficient, how fast an individual chain moves, and the diffusion coefficient, d, is proportional to r to the minus one power average. We could do an osmotic pressure measurement, I won't get it, go into that. So, we have a bunch of different ways we can do the measurement, and for each of them we can ask r is proportional to m to some power in dilute solution. The power depends on the solvent. It's a little less, a shade under 0.6 in a good solvent. It's 0.5 in a theta solvent. Those are the same numbers I put up a few moments ago. However, most experimental techniques give the same number out. That is, most experimental techniques, if you ask how does the scattering power, how does the experiment depend on molecular weight, you turn out to get the same solvent. The one exception is that if you do light scattering measurements under some conditions, you get a slightly different exponent. Now, as long as you know what you're doing and are using the same technique, you need to know what the exponent is. Of course, it's very nice to say, oh yes, it depends on the power law. How do we determine what that power is? We take some property like that constant K1 I mentioned, and we measure it for a series of different, of polymers of different molecular weight. There are a bunch of different ways to determine polymer molecular weight. And having done this, we plot log K versus log M, and we get, if it's power law behavior, a straight line of some slope. Well, if K is proportional to M to the nu, log K is proportional to nu log M, and the slope here, the slope of this line, gives us the exponent. There's one modest challenge. The modest challenge is that if you want to determine this slope, this number and this number both have to change by a significant amount. And therefore, if you say, I would like to get a slope accurately, well, of course, it depends how accurately you can measure K. If you can measure K and M incredibly accurately, you can determine the slope with a fairly short piece of line. The problem is the data points are doing this. And if you just look in a little box, yeah, the slope could be this, or look at where those points are. Maybe the slope is more like that, or maybe it's more like that. And if you don't cover much range this way or that way, relative to the scatter in your points, you can't tell accurately what the slope is. So what you need to do is have something that covers ideally orders of magnitude here and fewer orders of magnitude there, and then you can make things work. If you look at the read-through chapter, you will observe that people looked at synthetic polymers with molecular weights up to about 50 million. 50 million is a heroic molecular weight to synthesize, except for biopolymers. DNAs do not view 50 million as extremely large. And they are extremely high molecular weight of the polyethylene. Yes? It's more than 15 million. Yes, you can get extremely high molecular weights. However, if you go to a standard industrial catalog and look at the polymers they're selling, you will find they run in the tens of thousands and the hundreds of thousands, and maybe a shade over a million. If you want something that is, say, 48 million Daltons and that is highly mono-dispersed, you have a significant synthetic chemical challenge ahead of you which people have risen to salt. The reason they rose to solve it is that there are theories for dilute polymers that predict this slope, and there are people who wanted to test those theories and were willing to go to a great deal of work to test them. Okay. Now we will skip ahead to figure 7.8, but I won't talk about it for a few moments. The issue here is, now what happens if you increase the polymer concentration? And what turns out to happen if you increase the polymer concentration is the polymer coils shrink. We've already talked about dielectric relaxation data. And in the dielectric relaxation data, um, G, dielectric relaxation, if we increase the concentration, the dielectric increment per unit concentration, the increment per molecule, falls the chains contract. However, there are two other ways of doing the same experiment. And the first is light scattering. And the second is neutron scattering. The original light scattering experiments, there's a historical note here. Go to a paper by, I am pulling the names from memory, but it's in the book Bouchot and Benoit. I have met Benoit. And they looked at this interesting problem. Suppose we do experiments on dilute solution. It's, the technique was very well established to get polymer molecular weight of polyethylene, for example, in dilute solution by looking at how much light it's got. However, if you looked at a random copolymer, you didn't get very consistent or reasonable results. A copolymer, well, there are block copolymers. This is a block copolymer. This is a die block. This is a block copolymer. There are copolymers that are absolutely patterned. And then there are random copolymers, where the other A's and their B's. But the synthetic process doesn't impose a great deal of ordering here. And so there are polymers like, you try to measure the molecular weight of these and you don't get very sensible results. And the question was why? And they had this very bright idea, which they tested. And the bright idea was that the reason you weren't getting quite sensible results is that each chain was not the same as all of its neighbors in the ordering of the A's and the B's. And therefore you got excess scattering because there was this difference between one chain and the next. I'm not explaining the details of how that worked. So they asked, how can we test if the excess scattering is arising because the chains are heterogenous? That is, they don't all have the same pattern. Well, it's a little hard to control the synthesis. So what we'll do is to make a polymer that is as a die block that is as heterogenous as possible. And finally we will look at a mixture of these and these. You can't get much and you can't get any more dramatically different in composition chain by chain than what you see here. And what they confirmed was that yes, the fact that these aren't the same contributes to the light scattering. And this proceeded along, the development proceeds along to the point where if you have these chains the index match, have the same index of refraction as the solvent people realize, you can't see these anymore and you just see the behavior of the B chains. And therefore if you have large B chains where you can use light scattering to measure the radius and you stick in, start substituting in A chains for solvent, the A chains will cause the B chains to contract. Exactly the behavior we talked about for dielectric relaxation, but now we are using light scattering to see the contraction. And there were a series of experiments and memory serves the core author in that list will be Kuhn. And the observation is that if we have a chain of some size and we increase the concentration of invisible matrix chains, R square of the radius is what you measure, you discover that the chains shrink. That experiment is hard to push out to high-polymer concentration. At least it wasn't. However, there is a third way to do the experiment. And the third way to do the experiment is neutron scattering. You can't get much more scientifically impressive than an experiment where you need a large nuclear reactor in your lab in order to carry things out. Now actually you do not have a large nuclear reactor in your lab. Instead you go to a national facility, there are a very small number of these in the world, which has a big reactor which produces a very large, very stable, very well characterized neutron flux. It has little holes in the side. Neutron beams are filtered and come out and you can now use them for scattering. And you are very careful to do bunches of things for safety. But the net result is you can scatter neutrons. You might ask, why do you want to scatter neutrons? And the first answer is neutrons will go through things rather cleanly, at least if they're low atomic weight. You might get absorption eventually. If you walk around with silver coins in your pocket and you were careless with where the neutrons are going, after a while you notice that the neutrons are becoming radioactive, the silver is becoming radioactive rather, because it's absorbing neutrons. And there are some interesting pre-World War II cyclotron descriptions where people should have been more careful and eventually were. However, having said that, the virtue of neutron scattering is they scatter very effectively from hydrogen and from deuterium. But they do not scatter the same way. Again, there is a large theoretical development which I am skipping over. But the net result is that if I take perduterated, this is perduterated polyethylene, and I put it into a sea of hydrogenated polyethylene, I mostly see only one of the, or vice versa. I really only see one of these. And if I am very clever, instead of I take a material that is some amount h and some amount x, and if I get the ratio just right, the neutron scattering almost does not see this at all. And I can now produce invisible polymer chains. Neutrons scattering invisible polymer chains. And I then put in them the chains that are going to scatter. And so I have, for example, trace quantities of the visible chain and large amounts of the invisible chain. And I can use neutron scattering on a polymer that is very well characterized. The material you are looking at is polystyrene and toluene. And also polystyrene and carbon disulfide. Why carbon disulfide? It doesn't contain any hydrogen. It is neutron invisible. Well, it is not quite neutron invisible, but it is good enough. And we can plot in figure 7.8 the chain radius versus the polymer concentration. And what you see if you don't look at the figure too hard is that as you change the chain rate, as you change the matrix concentration, the chains shrink. Now, that figure is a little more complicated than you might think if you didn't look at it hard. First of all, the two polymer chains are very nearly the same size. The two chains we are looking at, Daoud et al., the solid points, are looking at 110 kilodalton polystyrene. A king et al. are looking at 111 kilodalton polystyrene in different solvent. So what I did to generate this figure, and this is one of these things that is worth talking about, to generate a figure, is if you look hard, you will notice there is a size scale on the left and right axis, and they are not the same. The reason is that the polymer in these two solvents has a different, even though it is almost the same molecular weight, it is different by 1%. And you should realize that saying these polymers differ in molecular weight by 1% means you have an unreasonable confidence in your experimental accuracy, or you are working with DNA where you know the molecular weight exactly. So what I did was say, okay, we will take two vertical scales, one for each set of experiments, and they will have the same top end, so that at zero concentration we have the same starting point. And the two axes will then cover the same fractional change in radius. And that is what we do, and you notice the axes are quite different, but they do have the same starting point, and they do both cover approximately a factor of two change in radius. Exactly a factor of two. The other problem I had, which is a little more also significant, is that one of the research groups perfectly reasonably reported the polymer concentration in grams per liter, grams of polymer per liter of solution. That is a very reasonable concentration unit. The other group reported the mole fraction, and you then have the little difficulty that those two are of mole fraction of polymer, except at the two end points, those two scales aren't quite the same, fundamentally. Why aren't they the same? The reason they're not the same is something called volume of mixing. The traditional freshman chemistry example of this is to take a liter of water and a liter of non-denatured ethanol, that's tax stamped ethanol, and you mix them, and you now have, for all practical purposes, vodka. However, do you end up with two liters? No. No, you end up with, does anyone happen to remember the number? Oh, 1.7 over 8. Yes, it's about that. I was going to say 1.9, but you're right. You end up with, you have a liter and a liter, and you mix them together, and as you mix them together, the system contracts. And the details of that are quite complicated. The reason this is important is that if you want to convert how many moles of this and moles of that we have, or weight fraction, we have so much weight, I think that may actually be weight fraction, I think I misremembered, we have 50% by weight of polystyrene. That doesn't mean you have, given the density is 0.9 roughly, that doesn't mean you have 450 grams of polystyrene in a liter of solution. It means you have 450 grams of polystyrene in 900 grams of solution, and you cannot convert unless you do some auxiliary measurements, which were not done, or at least which I did not find. And so there are two sets of concentration units, bottom and top. They are arranged so infinite dilution, and melt, all polymer, line up, because in the melt, 100% by weight, and whatever the, it's around, oh, 943, if I recall, gram per liter, the melt numbers are the same, and so we have composed this graph to show two sets of data. Now why are, what is the issue here? Well, those two sets of data, they're both very hard to measure. They were done by two slightly different methods. Daoud used a tracer amount of polystyrene, labeled polystyrene. King used concentrated polystyrene and used a statistical mechanics trick to pull out the radius of the labeled polystyrene from the unlabeled polystyrene. We're very controversial for a very long time because they do not agree with each other. The core issue, why are we even interested in this graph? Well, there is a scaling theory of polymers. It's in the Daoud paper, and the references are there. And what is claimed, if we look at R squared versus concentration, at low concentration it's claimed that nothing happens. Well, that's what it says. And then there's some sort of crossover, which means there is a region where several things are happening simultaneously, and the theory doesn't make a clear prediction. And then out here, the theoretical prediction is that R squared is proportional to C to the minus one and four. Chains contract as you increase the concentration, and the prediction was that you get a power log. What is the prediction? And so there was an effort made to measure experimentally what the power was. And there are these two sets of data, and if you plot the two sets of data on log-log paper, you get two somewhat different exponents. And the black dots agree sort of with the exponent that was predicted, and the open dots get a different number. And since some people were very attached to the theory, there was a great deal of back and forth as to what was going on. Let me, however, point out a couple of features of this graph that aren't emphasized quite as much, except by people who aren't quite as involved in the dispute. And the first is that if you look at that graph, I said there was a crossover, and the crossover occurs roughly at a concentration, a C star, the overlap concentration, at which the polymer chains, approximately speaking, are shoulder to shoulder, but haven't started to inter-penetrate too much. Well, that's what theory says. But if you look at that data, it does this. And in fact, something like, oh, a third of the contraction has occurred in this dilute solution region where there's not supposed to be any contraction. And then the contraction continues. You can see the contraction. And if you ask what happens in solution, well, the contraction continues and continues. And I'll draw a line here, which is something like 500 grams per liter, sort of half a polymer. And beyond that, there's, perhaps it continues a bit, that's the solid dots, or perhaps it runs out of steam. The open dots sort of show that the polymer doesn't contract much at all, above about 50% by weight. And there are people who will be happy to explain why you should be seeing that. But the net result is the curve does not look a great deal like the theory qualitatively. If you only look at the high concentration dots, the constant d, if you only look at the dots that are, oh, above, say, 100 grams per liter, you can put them on a power law. This is a log, log plot, radius squared versus concentration. You can put them on a power law, but the variation this way isn't exactly very big. The variation this way isn't exactly huge. So there's some uncertainty in the power law. And there was then a great deal of dispute as to what was happening. I put in something different from a power law. I put in a, those two solid curves are stretched exponentials again. And you see the solid curves do quite nicely at going through all of the points, including the low concentration points that the power law does not predict. And the only difficulty occurs if you go out to the melt concentration. Okay, that's it for chain size. We will now push ahead, and I will at least slightly talk about relaxations. The first point is experimentally, the easy way to do the experiment is to take the sample, put it between metal plates, and apply an oscillating electric field that oscillates at a given frequency. Okay, so we have a capacitor. Yes? So we can apply an oscillating electric field at some frequency, and we can then measure the capacitance of the capacitor. Now how you measure the capacitance, well, if you want to do it accurately, you have to work a bit. The important issue, and this is actually physics too, is the capacitance of the capacitor is determined by the dielectric constant of the material between the plates. Okay? Yeah. Dielectric constant. And if we are talking to freshmen, we tell them there's a material that has a dielectric constant which can be small or large. In the real world, epsilon depends on omega. Dielectric constant depends on frequency. You've actually seen this in practice. That is, if you ask, what is the dielectric constant of water, it's very large. If you go up to optical frequency, the index of refraction is directly related to the dielectric constant. Yes, water has a dielectric constant which is not one, and an index of refraction which is not one, but it's not very big. And in fact, if you plot the dielectric constant versus frequency, you discover, we'll use log omega because we want to cover lots of frequencies, and we will use log of the dielectric constant. It rolls off. It rolls off quite dramatically. We'll discuss the details of the roll off next time. Now, what I want to do first is to simply get to the question, well, why does it roll off? Why does the dielectric constant drop at very high frequency? Well, let's go back and think. Here is a molecule, and it has two charged ends. This is a simple case in water. Say, in amino acid. It's actually got charges on it that it's fixed. We apply an electric field. We get a dielectric constant because the molecule can line up with the field. However, if I have to, if I flip the field around, it's an oscillating field, the molecule has to turn around through 180 degrees to match. Okay? That takes a while. If we are at very low frequency, the molecule has no difficulty flipping and flopping each time we change the sign of the electric field in the capacitor. However, if I crank up the frequency enough, the molecule starts to flip. It can't keep up. And before it's rotated significantly, the field is pointing back the way it was originally. And as a result, at very high frequency, the molecules simply aren't capable of keeping up with the field, and they do not, and they simply sit there. Yes? Okay. But, um... Gee, that's the rule. Now, there's something else that can happen that does not come up in dielectric constant studies but comes up in certain related studies. Namely, um... As you keep changing the field around, the flip and flop back and forth may get out of phase with the charge you're applying here. You could describe this either as I am applying a voltage across the plates, or you could say I am applying a charge on the two plates. You can do several different things, and there are some complications that we will not get into in dielectric response. So there is something I have over somehow time for the dipoles to relax. That would happen at very high frequency. The dipoles just sit there, and they do not contribute to the dielectric relaxation. However, the dipoles still... the molecules still have electrons, and the electrons shuttle back and forth even though the molecules do not have time to rotate. And so at very high frequencies, way, way out in the optical almost, you will still have a dielectric response, but it will be much smaller because it will just correspond to the electrons moving. However, the molecular rotation or local rotation, part of the dipole behavior, doesn't contribute anything.
 Classes in Polymer Dynamic. Based on George Philly's book,  Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011.  And today, this lecture is lecture 9, More Dielectric Relaxation.  I'm Professor Philly's, and this is part 2 of my classes on dielectric increment in polymer solutions,  based on my book, Phenomenology of Polymer Dynamics.  Okay, so what we are going to do today is to continue with our reading of chapter 7.  I'll have homework again next time.  The issue is as follows, dielectric spectroscopy is an enormously effective tool  that gives us a wide variety of different measurements about a polymer solution.  In particular, if we have a polymer chain in the solution, it has a few features.  It has an end-to-end vector, that is, a vector that starts at one end of the chain  and points straight to the other end.  And that's a measurement of how big the polymer is in the solution.  It's not the only measure, but it is a measure.  Furthermore, we can use dielectric relaxation spectroscopy to characterize the relaxation of this vector.  That is, as time goes on, the little segments along the chain all perform Brownian motion.  They diffuse.  Now, they have to stay attached to each other as they diffuse, so they can't just move randomly completely.  But if the chain starts in this configuration, after a while, two sorts of, or three sorts of things will have happened.  Number one, the center of the mass of the chain, which is originally here, will move, say, over there.  Number two, the two-chain ends will have moved, I'm marking the two-chain ends relative to each other.  And therefore, the end-to-end vector will change length and direction.  And number three, the parts of the chain between the end will change which way they're pointed.  Now, the details of how a chain moves, how it translates and rotates and changes shape,  are quite complicated and not shown in that sketch.  Furthermore, there are a number of entirely contradictory theoretical models for how polymer chains move in solution.  The key feature here is from the standpoint of dielectric spectroscopy, if we have a chain whose pieces are all asymmetric,  so each piece has a little dipole lined up along the polymer chain,  the result of all of these little dipoles lined up is that the polymer chain has a dipole moment  which points from one end of the polymer chain to the other end of the polymer chain.  And what dielectric relaxation spectroscopy and the measurement of the dielectric increment can do is to determine number one,  the average length of these things, after all, average, yes, it's different for each chain in solution.  The relaxation with time in the length and the direction, that is, as time goes on, the direction fluctuates,  the length fluctuates, it's fluctuate one value at one time, another value at another time.  And this relaxation and the direction gives us, after a piece of theoretical work,  the frequency dependence of the dielectric response.  Oh, that reminds me, I said dielectric constant is 81 and you asked me the dimensions  and I went off mentally and way the wrong direction.  The answer you were looking for is what units are you in?  And the answer is that if we write the potential energy between two charges in that form,  we are in CGS ESU units.  Epsilon is one in a vacuum and oh, maybe about four in the oil and at low, this is low frequency,  it's standard, around 80 or so in water and Epsilon is dimensionless.  However, most of you are probably more used to SI units,  in which case you have a four pi Epsilon not down here in vacuum  and you pile up an extra constant or you change Epsilon, not Epsilon,  and that's SI units of electrostatics in which the charges in Coulombs  and in that case Epsilon is no longer 81, it's a number which we could go into but doesn't matter.  The important issue is that the dielectric constant in water is much higher than it is in the vacuum  because in water if you apply an electric field you can line up all of the little water molecules  at least a bit.  If you think for a moment you may ask, well, gee, if you apply a big enough electric field,  don't you line them up completely and then don't they, don't you stop getting an increase  in the electric increment and that's correct.  Now if you actually try to apply an electric field like that externally,  you will have an entertaining time with your experimental efforts.  On the other hand, if you take water and you drop into, say, a sodium ion,  presumably chloride someplace, this is a lot of charge and it's very tiny  and there is a big electrical field here and the water molecules very near a sodium ion  are very much lined up.  So surrounding ions in water there is a shell where the water molecules are pretty well lined up  and at that point, since they're basically already completely aligned,  they behave basically like an oil in their dielectric increment.  That is you get some increase in the dielectric constant because you can move the electron shells around a little bit.  But basically you can't move the water molecules, they're already lined up by the local ion.  Okay, let us go back to here and what I was saying is you can measure different sorts of things about a polymer chain.  The next thing you can do is to say, well, suppose we have more than one chain at the same time.  So there is a chain and there is another chain.  What happens?  50 years ago this was a major literature dispute and there were no very easy ways to do the measurements.  You knew roughly how big a polymer chain was but the question is,  if you increase the polymer concentration what happens?  And one answer is, if you really run up the concentration, the chains shrink in on each other but don't overlap.  It's now quite clear that if you run up the polymer concentration, the chains do shrink a bit.  But they also spend a lot of time interpenetrating each other.  I suppose that leads to an obvious question, why do they interpenetrate?  How can they interpenetrate?  Well, there is a trick.  We have a polymer chain and it has some typical size r.  It matters a bit which method you use to characterize the typical size because it's a random structure and it's not a solid body.  But r goes as molecular weight to a power around a half or a good solvent, not quite 0.6.  Suppose you have an object whose radius grows as the half or a bit more power of the molecular weight.  That means the molecular weight of the object is proportional to r squared or maybe a bit less.  And so you have an object whose molecular weight grows as the square of the radius.  Have you seen an object like that before?  Yes, it's a spherical shell.  If you have a spherical shell, the mass is proportional to the surface area.  Yes?  And there it is, mass proportional to surface area.  Of course, polymers are not spherical shells but they have the same properties as spherical shells.  If you increase the mass, the object has to get bigger fairly large, bigger fairly quickly I should say.  And furthermore, a spherical shell is mostly empty space inside.  Now a polymer chain, the polymer strand runs around all inside this blob where the polymer is.  However, as I increase the molecular weight of the polymer, this polymer gets bigger and bigger  and it still has an important property of the spherical shell, namely if I imagine running a line through it,  on the average I hit the polymer coil twice.  So if I have a really big polymer and it's really big radius and I send a line through it,  and if I only hit the tiny polymer, you can cross tiny in cross section, polymer twice,  most of this is empty space and polymers can then interpenetrate and that's what they do.  Now if you will take out your books in advance to figure 7.4, I'm fairly sure,  I grabbed the wrong thing this morning, but I'm fairly sure 7.4 is heptane solutions  of 140 and 743 kilodalt and polyisoprene.  Yes? Okay, very good.  And what is the point of this?  Well, the first issue is, the vertical axis is delta epsilon over C.  Delta epsilon is the change in the dielectric constant, there are some constants involved, of the solution.  And as you add more polymer, well if you add more polymer, each molecule makes its contribution to the dielectric constant.  But if you divide out, hang on a second, this is getting noisy.  If you divide out the concentration, what is left is the dielectric increment per molecule.  And the dielectric increment per molecule is approximately proportional, there are bunches of constants involved,  to the mean square size of the molecule.  So let us start with figure A and you have a comparison there, and as you make the molecule larger,  R square gets larger because the chain is longer, and therefore, hmm, R square gets larger,  and therefore, the dielectric increment gets larger, and therefore, if you stay at the concentration equals 0,  the left hand axis, the larger molecule gives you a larger dielectric increment.  Now the next thing you can do is to increase the concentration of the molecules.  And if you do that, there is one thing you might need to worry about, that is,  suppose I put a lot of molecules into solution.  So here is a polymer molecule, and here is another polymer molecule,  and they are somewhat wrapped around each other.  And one thing you might ask is, well, are those chains independent of each other?  And the reason you would worry about this is that if you pack the chains,  and the chains line up, for example, parallel to each other,  the dielectric increment of a lot of chains is not linear in how many there are,  it is some larger number.  But the answer is, here is our first chain,  and if I do an average over the solution, the second chain could run through this way,  or it could run through in the opposite direction and have its end-to-end factor pointed oppositely.  And these two cases are essentially equally likely to occur.  The two states have about the same free energy.  And because the two states have the same free energy,  the direction of this factor and the direction of that factor are completely independent from each other.  And so when we look at the dielectric increment of a concentrated solution,  we are just seeing this very good approximation, the single chain behavior.  Now there is an exception to that.  There is an exception to that, and the exception to that is if you have molecules  that you run up the concentration and they do some sort of a phase transition to a liquid crystal,  in which case the molecules are straightened out, at least approximately.  They tend to, they perhaps pull all point parallel rather than parallel and anti-parallel.  There is a bunch of interesting phenomena here that are outside this course.  In this case, if you run up the concentration enough, suddenly there is a very dramatic change in the dielectric increment,  which we aren't going to talk about.  So what happens if we increase the concentration of the polymers?  The point of this discussion I just gave you is that this relationship between the dielectric increment  and the mean square size of the polymer chain does not care about the polymer concentration.  And so if I plot delta epsilon over C versus polymer concentration,  I am essentially seeing the single chain behavior.  I am seeing that to good approximation.  And what do I see happen?  Well, if you look at 7.4a, as I run up the polymer concentration,  the dielectric increment of the single chain falls.  And if I do that to a bigger chain, the dielectric increment falls, and it falls faster.  So at low concentrations, with increasing molecular weight, the dielectric increment increases,  but at high concentration, these curves can cross,  and with increasing polymer molecular weight, the dielectric increment of the system is less.  What does this mean?  It means that a very large chain at high concentration has contracted down,  and now its size has gone decreased considerably,  and its ability to increase the dielectric constant of the solution has been substantially reduced.  So that is figure 7.4a.  Figures 7.4b and c, b is the big one at the bottom,  show a slightly different effect, which is also important,  but it shows an effect that is of separate physical,  class of experiment, I should say, that is of separate physical importance.  And what is shown in those figures is that we take a polymer chain,  and the general experiment is as follows.  We have a polymer chain which we are interested in.  We put it in a solution of other chains, this is the probe,  out here is the matrix,  and we choose the matrix polymer so that our experimental technique doesn't see it.  So from the standpoint of our experimental technique,  we are looking at a probe polymer, there is a solution out here,  the solution, well actually it does contribute at very high frequencies to the dielectric relaxation,  but the solution is basically inert, dielectricly or optically or whatever,  and we only see the probe chains.  Now you might ask how can you do this with a polymer and dielectric relaxation?  Go back to what we were saying about last lecture about how a polymer contributes  at low frequencies to the dielectric constant.  A polymer contributes at low frequencies to the dielectric constant,  and if it has as a result of its molecular structure,  a bunch of little dipoles inside and the dipoles are arranged  so they are fixed to lie along the backbone of the polymer.  For example, this is a polyester,  and the important feature here is this is the repeat unit,  the two ends of the repeat unit are different from each other,  and furthermore the structure in the middle does not give me any way to get mirror symmetry,  so this has a dipole moment which lies along the bond axis one way or the other.  On the other hand, suppose I took, those are hydrogens,  the repeat unit here is actually that, that's polyethylene oxide.  Well, polyethylene oxide, yeah, it looks like the repeat unit repeats,  but if I drew the repeat unit like that, it's the same repeat unit,  and so there's a center of reflection and there's no net dipole along the bond axis.  So this material does not contribute to the dielectric constant,  and if we stir it into the solution, we get no effect on the dielectric behavior.  Now you have to be a little careful of that statement,  because this is replacing your solvent,  and just as the solvent has a dielectric increment, which is the same at most frequencies,  a pretty good approximation, so does this.  And so you have to pay a little attention to the fact,  but even though this doesn't have a big dipole moment,  you're doing a little bit of something and you should think about that on occasion.  Nonetheless, here's our probe, here's our invisible matrix,  and we are now looking at what a ternary, three components, ternary system.  So with the system you have a probe, polymer, in this case,  you have a matrix, which in this case is also a polymer,  and you have a solvent.  And you are looking at, G, the behavior of the probe  as you put a polymer into the solution.  Now why would you want to do a probe matrix experiment,  as opposed to just stirring the polymer in?  After all, you are going to see in this experimental technique,  no matter whether you have a tracer probe or a high concentration of probe,  the experimental method gives you something close to single molecule behavior.  The reason you do this is that the matrix has a polymer molecular weight, M,  the probe has a polymer molecular weight, P,  and if these are different species, I can vary P and M separate from each other.  That's very useful.  If you look back at 7.4a, you notice that, G,  the small polymer contracts slowly with increasing concentration.  The big polymer starts out large, but contracts very fast with increasing concentration.  The solid lines, by the way, you notice they are data points in the figure in solid lines.  The solid lines are stretched exponentials,  I didn't call it alpha, I called it rho.  The solid lines are stretched exponentials, they are smooth curves.  If you plot them on semi log paper, you don't quite get straight lines out.  The issue is, yeah, there's okay, so you mix chains together and they shrink.  And big chains shrink more than little chains.  But we come to the key question.  Is the statement, big chains shrink more than little chains,  occurring because this guy is big and shrinks more,  or is it occurring because a big neighbor is more affected by causing a probe to shrink,  no matter how big the probe is?  Question?  For instance, like, if data percentage of a string is shrink,  the large chain may shrink to the size of the larger down the small chain.  But what about the percentage?  If you, 7.4a is delta epsilon over c, delta epsilon is proportional to the mean square radius.  If you consider that, you find that for the smaller chain,  which is a 140 kilo-deltin polyisoprene,  we drop the delta epsilon over c in these units from about 0.18 to about 0.14,  which is 20% sort of, or a little more.  For the large chain, the 743 kilo-deltin,  the shrinkage in r square is from 0.222 at a much lower concentration,  0.14, which is a much larger fractional change.  I'm looking at 7.4a,  and therefore the larger chain fractionally is actually contracting more.  Now, you'd have to do, you have to do, sit there and do some arithmetic  if you want to calculate what the change in the length as opposed to the fractional change in the length is.  But the answer, I think, is that, yeah, the big chain really,  but is it that the big chain is just more sensitive and compresses more,  or is it that the surrounding chains are bigger and are more effective at contracting?  And the first step to understanding that is a ternary probe matrix solvent experiment,  such as the experiments you see in figures B and C.  Figure B shows the 140 kilo-daltin polymer,  and you notice, well, I will point it out, there are a series of curves, this is B.  And as you sweep this way, what is happening is you are increasing the size of the matrix polymer.  And therefore, a larger matrix polymer is more effective at, that's C.  A larger matrix polymer is more effective at compressing neighboring chains  than a small matrix polymer is.  In contrast, well, it's not really contrast, it's the same effect.  If you go to C, which is a large chain, and you ask what happens as you increase the matrix size,  you see the same behavior.  That is, as you increase the matrix size, I'll explain the significance of that in a second,  there is more compression of the chain that's being observed.  Now, I will point out an issue.  Suppose I have polymers at a certain concentration, very low.  And suppose I increase the molecular weight.  As I increase the molecular weight, the M goes up, the radius goes up,  the radius goes from M to the one-half, so what happens at low concentration to the density of polymer  inside a polymer coil as I increase the molecular weight?  The density of polymer inside a polymer coil, not in the solution as a whole,  that's determined by the concentration, but inside a single coil,  the density of that chain inside the coil is the ratio of the molecular weight to the volume of the chain,  which is the molecular weight to a radius cube, which is the molecular weight.  Oh yes, R is proportional to M to the half or M to the 0.6.  So this is M to the 1.5 or 1.8, and therefore the density of a polymer inside its own self  is proportional to, well, take the ratio, it's M to the minus 0.5 or M to the minus 0.8,  or some number in between.  This is the behavior you would get in a theta solvent.  This, or very close, is what you get in a good solvent.  0.6, R proportional to M to the 0.6 in a good solvent is a slight overstatement,  but you can also get solvents that are in between these two.  And the important issue I want to make is that as you make the molecular weight larger and larger,  the density inside the coil gets smaller and smaller,  and therefore the same number of grams of polymer at low concentration occupies more and more space  as the molecular weight is increased.  And thus, if we have a solution at some concentration, the bigger the molecular weight,  the closer the chains are to rubbing shoulders with each other.  And rubbing shoulders is about right because we're talking about the radius of the chain,  sort of like this.  Okay, so that's figure 7.4.  The other thing we can talk about, if you want to ask how effective are matrix chains  at compressing a solvent, at compressing a polymer chain,  let us look at the next figure.  The next figure is 7.5, and it gives us a plot of rho,  I'll say what this rho is in a second, against matrix molecular weight.  And I remind you the notion is that R squared, which is the same,  up to units is like delta epsilon over C,  is like a constant e to the minus rho c to the nu.  And rho is the constant that describes how steep this stretched exponential is.  Well, I've plotted this as a function of matrix molecular weight,  and there are two things you should notice.  And the first is rho does this.  I would not put too, too much emphasis on the last hollow point,  which indicates that it might curve back at very large molecular weight.  It's possible, but there's only one data point there,  you really don't want to extrapolate based on one data point.  That's a little dangerous.  The other thing that happens though, there are actually two curves there,  and the two curves show the 140 and the 743 kilodalton probes.  And so the answer is, the large probe has a larger rho,  and thus contracts faster in some sense than the smaller probe does.  And furthermore, rho stays larger for the large probe  than for the small probe at all molecular weights.  And therefore, we can say, yes, there's a contraction,  and part of it is the difference between these two lines,  so part of it is that large chains are easier to compress.  After all, they're in a certain sense more and more hollow inside.  And it's also the case that if you increase the matrix molecular weight,  there is a dramatic change in rho,  meaning big chains are more effective at compressing each other than small chains are.  So that is how that works.  Okay, there's something else that can be done,  and the analysis in figure 7.6 could be carried out in another step.  And what 7.6 does is to say,  well, we have polymer properties, different properties,  and we could compare them.  And these are results from Urukawa at all,  and what they did is to measure two things.  You will observe, there are footnotes,  and I don't always read all of them out loud, but they are there.  And the first thing they did is to measure the viscosity.  And they measured the viscosity relative to the solvent viscosity.  After all, if you change the solvent,  the viscosity of the solution does change,  but it's not changing because of polymer properties.  And what they observe is a series of curves,  viscosity versus concentration.  And with increasing polymer molecular weight,  there's only one polymer in solution.  This is a binary system we're looking at.  With increasing polymer molecular weight,  what happens is the viscosity of the solution goes up.  So if you have a solution,  the bigger the chain is, the more viscous the solution is.  And that is a nice, uniform rule.  However, they then take this stuff,  and they do the experiment on a binary solution  and a ternary solution,  and they determine how the relaxation time,  and what the relaxation time tells you  is how long it takes for the polymer coil  to the end-to-end vector  to forget which way it was pointing.  So if we have a typical polymer,  it has an end-to-end vector by arm.  As time goes on, the vector gets longer and shorter,  and it points in different directions.  And eventually, the initial direction of the polymer  end-to-end vector has nothing to do with the final direction,  and the two directions are independent from each other.  That is, if you wait long enough,  yes, originally the end-to-end vector was pointing this way,  the direction my arm is pointing,  but if I wait long enough,  the end-to-end vector is equally likely to be pointing  in every direction.  So first, are good?  Well, if I increase polymer concentration,  it takes longer for the end-to-end vector  to reorient and to change length.  I'm not telling you how it reorients.  I'm just saying it does reorient.  That's proven by this experiment.  And therefore, as I increase the polymer concentration,  that's the horizontal axis,  the orientation time goes up.  And you can now do a binary experiment  or a ternary experiment.  You can look at the polymer chain in solution with itself,  or you can look at the polymer chain  in a solution with a matrix.  And what we observe is that as we increase the matrix molecular weight,  you notice there are two families of curves in 7.6b,  and as I increase the matrix molecular weight,  even at very high concentrations,  the reorientation slows down.  So the molecular weight of the matrix  has a direct effect on how fast the polymer coils can reorient.  This result is possibly not equally consistent  with all theories of polymer dynamics,  all theories of how polymers move in solution.  However, you can always say,  well, the concentration isn't high enough,  the molecular weight of the matrix isn't high enough,  and therefore it isn't that the theory,  some particular theory is wrong,  is that it's really not applicable to these solutions.  Okay, we now push ahead,  and we are going to reach section 7.3.  Chain dimensions.  Chain dimensions,  we're actually going to talk about all of the techniques  that can be used to measure chain dimensions.  This is one of these sections that was,  well, it's a little too short for a chapter by itself,  so you ask the sensible question,  where do you put it?  The answer is there were lots of sensible places,  it could have been placed.  Many people would have put it in an early chapter  discussing dilute solution behavior.  The basic question we are talking about is,  we have a polymer molecule,  it has a molecular weight, M,  it has some characteristic size,  and lots of ways to characterize the size are,  and are, however you define it,  is proportional to M to some power.  Now, the reason why you're interested in this question  is actually an analytical chemistry question.  That is, you have people sitting there,  and you have the chemical plant going,  ka-chunk, ka-chunk, ka-chunk, or whatever,  and you're looking out polymer,  and you would like to characterize the polymer  so you know what the molecular weight  of the material you're making is.  The reason for this is there are large numbers  of industrial applications of polymers.  Many of them are sensitive to some particular property  of the polymer, such as its molecular weight,  and you would like to know how big the polymer is  and how it would be paying for a particular molecular weight  and would be annoyed if it doesn't show up,  or something like this.  And so what you do, what you do historically  is you take the polymer coming out of the plant,  and the polymer coming out of the plant  may be a solid or a melt,  or a very concentrated solution,  or God knows what,  and you take this,  and you almost certainly are not producing a dilute,  you could be, a dilute solution  that tends to be inconvenient on an industrial scale  because you have to pay for the concentration step.  And you take this material to the lab,  and what you do is you take a bit of it,  and you dilute it.  The reason you dilute it is that once it's dilute,  there's a polymer chain here,  and there's a polymer chain there,  and there's a polymer chain here,  and on the average the chains are quite separated from each other.  And because they're separated from each other,  these polymer chains change the behavior of the solution,  but they do so in a way that is not affected by their interactions,  it's just determined, if you're lucky,  by how big the chain is.  And so you measure dilute solution properties,  and the dilute solution properties can be used to tell you  how big the polymer is.  Now you might sensibly ask,  well, what dilute solution properties are there?  And one dilute solution property is the viscosity,  which is proportional to the viscosity of the solvent,  one plus a constant,  times the concentration plus.  But if you just add a little chain,  bit of chain to the solution,  the linear term is quite completely accurate.  And the feature here is that K...  Is that a power long?  Power long.  Well, it is a power long.  This is a power series.  However, you can measure K1 quite accurately,  and K1 is approximately proportional to some power,  like the Q, of the size of the polymer.  Now I put a subscript on R.  This is the viscometric radius.  It's the radius inferred by asking,  how effective are these chains  at changing the viscosity of the solution?  The reason I bring up often say,  this is the viscometric radius,  is that, well, there are several other techniques  for measuring polymer size.  For example, you could take the solution  and shine a light beam.  We now use a laser,  but they didn't, when this technique was first developed,  through here, they didn't have lasers back then.  The solution scatters light,  it becomes cloudy, opalescent.  And the reason the solution becomes cloudy  is the little polymer chains in solution  do not have the same index of refraction that water does,  so they scatter light.  And if you measure the intensity of the scattered light  as a function of angle,  why is a function of angle?  Well, if the polymer chain is very small,  it scatters light effectively in all directions.  I am skipping a light polarization issue,  which is experimentally very important.  But if vertical polarization in,  vertical polarization out,  scattering plane horizontal,  so if we're looking down from above,  we look at the light scattering like this,  and the polarization vector of the light  is perpendicular to the paper.  I can make the statement,  if the molecule is very small,  it scatters light equally in all directions.  However, if we make the molecule bigger,  eventually the molecule starts to approach  being comparable to the wavelength of light.  This is a light wave, you see, it oscillates.  And as a result,  this part of the molecule is being illuminated  by this part of the wave and has one phase.  This part of the molecule is being illuminated  by a different part of the light wave,  and scatters light with a different phase,  and the light scattered from these two points  can interfere.  I am omitting details.  And as a result,  the light scattering  depends on the scattering angle.  How does it depend on the scattering angle?  As you make the polymer bigger and bigger,  you make the polymer bigger and bigger,  the light scattering is focused  more and more in the forward direction.  The extreme case of this,  this is shown by window glass.  If you make the polymer enormous  so that you have this block,  and it's only one molecule,  the laser beam goes straight through.  If you want to say there's scattering,  it's only in exactly the forward direction,  and if the block is homogeneous,  there's almost no scattering to the side at all.  The object is transparent.  A similar mathematical effect,  it's not a polymer, is seen in window glass.  Window glass is transparent  because to the extent you get scattering,  it's forward.  So, you say the scattering versus angle  depends on the polymer molecular weight,  and therefore there is another radius we can infer,  and the other radius we can infer  is determined by the scattering experiment.  I can run down a series of these,  and there are a series of different ways  we can physically measure  an apparent radius of the molecule.  For example, we could measure the diffusion coefficient,  how fast an individual chain moves,  and the diffusion coefficient, d,  is proportional to r to the minus one power average.  We could do an osmotic pressure measurement,  I won't get it, go into that.  So, we have a bunch of different ways  we can do the measurement,  and for each of them we can ask  r is proportional to m to some power in dilute solution.  The power depends on the solvent.  It's a little less, a shade under 0.6 in a good solvent.  It's 0.5 in a theta solvent.  Those are the same numbers I put up a few moments ago.  However, most experimental techniques  give the same number out.  That is, most experimental techniques,  if you ask how does the scattering power,  how does the experiment depend on molecular weight,  you turn out to get the same solvent.  The one exception is that if you do light scattering measurements  under some conditions, you get a slightly different exponent.  Now, as long as you know what you're doing  and are using the same technique,  you need to know what the exponent is.  Of course, it's very nice to say,  oh yes, it depends on the power law.  How do we determine what that power is?  We take some property like that constant K1 I mentioned,  and we measure it for a series of different,  of polymers of different molecular weight.  There are a bunch of different ways  to determine polymer molecular weight.  And having done this,  we plot log K versus log M,  and we get, if it's power law behavior,  a straight line of some slope.  Well, if K is proportional to M to the nu,  log K is proportional to nu log M,  and the slope here, the slope of this line,  gives us the exponent.  There's one modest challenge.  The modest challenge is that if you want to determine this slope,  this number and this number both have to change  by a significant amount.  And therefore, if you say,  I would like to get a slope accurately,  well, of course, it depends how accurately you can measure K.  If you can measure K and M incredibly accurately,  you can determine the slope with a fairly short piece of line.  The problem is the data points are doing this.  And if you just look in a little box,  yeah, the slope could be this,  or look at where those points are.  Maybe the slope is more like that,  or maybe it's more like that.  And if you don't cover much range this way or that way,  relative to the scatter in your points,  you can't tell accurately what the slope is.  So what you need to do is have something  that covers ideally orders of magnitude here  and fewer orders of magnitude there,  and then you can make things work.  If you look at the read-through chapter,  you will observe that people looked at synthetic polymers  with molecular weights up to about 50 million.  50 million is a heroic molecular weight to synthesize,  except for biopolymers.  DNAs do not view 50 million as extremely large.  And they are extremely high molecular weight of the polyethylene.  Yes?  It's more than 15 million.  Yes, you can get extremely high molecular weights.  However, if you go to a standard industrial catalog  and look at the polymers they're selling,  you will find they run in the tens of thousands  and the hundreds of thousands,  and maybe a shade over a million.  If you want something that is, say,  48 million Daltons and that is highly mono-dispersed,  you have a significant synthetic chemical challenge ahead of you  which people have risen to salt.  The reason they rose to solve it  is that there are theories for dilute polymers  that predict this slope,  and there are people who wanted to test those theories  and were willing to go to a great deal of work to test them.  Okay.  Now we will skip ahead to figure 7.8,  but I won't talk about it for a few moments.  The issue here is,  now what happens if you increase the polymer concentration?  And what turns out to happen  if you increase the polymer concentration  is the polymer coils shrink.  We've already talked about dielectric relaxation data.  And in the dielectric relaxation data,  um,  G, dielectric relaxation,  if we increase the concentration,  the dielectric increment per unit concentration,  the increment per molecule,  falls the chains contract.  However, there are two other ways of doing the same experiment.  And the first is light scattering.  And the second is neutron scattering.  The original light scattering experiments,  there's a historical note here.  Go to a paper by, I am pulling the names from memory,  but it's in the book Bouchot and Benoit.  I have met Benoit.  And they looked at this interesting problem.  Suppose we do experiments on dilute solution.  It's, the technique was very well established  to get polymer molecular weight of polyethylene,  for example, in dilute solution  by looking at how much light it's got.  However, if you looked at a random copolymer,  you didn't get very consistent or reasonable results.  A copolymer, well, there are block copolymers.  This is a block copolymer.  This is a die block.  This is a block copolymer.  There are copolymers that are absolutely patterned.  And then there are random copolymers,  where the other A's and their B's.  But the synthetic process doesn't impose a great deal  of ordering here.  And so there are polymers like,  you try to measure the molecular weight of these  and you don't get very sensible results.  And the question was why?  And they had this very bright idea, which they tested.  And the bright idea was that the reason you weren't  getting quite sensible results is that each chain  was not the same as all of its neighbors  in the ordering of the A's and the B's.  And therefore you got excess scattering  because there was this difference between one chain  and the next.  I'm not explaining the details of how that worked.  So they asked, how can we test if the excess scattering  is arising because the chains are heterogenous?  That is, they don't all have the same pattern.  Well, it's a little hard to control the synthesis.  So what we'll do is to make a polymer that is as a die block  that is as heterogenous as possible.  And finally we will look at a mixture of these and these.  You can't get much and you can't get any more dramatically  different in composition chain by chain  than what you see here.  And what they confirmed was that yes,  the fact that these aren't the same contributes to the light  scattering.  And this proceeded along, the development  proceeds along to the point where if you have these chains  the index match, have the same index of refraction  as the solvent people realize, you can't see these anymore  and you just see the behavior of the B chains.  And therefore if you have large B chains where you can use  light scattering to measure the radius and you stick in,  start substituting in A chains for solvent,  the A chains will cause the B chains to contract.  Exactly the behavior we talked about for dielectric relaxation,  but now we are using light scattering to see the contraction.  And there were a series of experiments and memory serves  the core author in that list will be Kuhn.  And the observation is that if we have a chain of some size  and we increase the concentration of invisible matrix chains,  R square of the radius is what you measure,  you discover that the chains shrink.  That experiment is hard to push out to high-polymer concentration.  At least it wasn't.  However, there is a third way to do the experiment.  And the third way to do the experiment is neutron scattering.  You can't get much more scientifically impressive than an experiment  where you need a large nuclear reactor in your lab  in order to carry things out.  Now actually you do not have a large nuclear reactor in your lab.  Instead you go to a national facility,  there are a very small number of these in the world,  which has a big reactor which produces a very large,  very stable, very well characterized neutron flux.  It has little holes in the side.  Neutron beams are filtered and come out  and you can now use them for scattering.  And you are very careful to do bunches of things for safety.  But the net result is you can scatter neutrons.  You might ask, why do you want to scatter neutrons?  And the first answer is neutrons will go through things rather cleanly,  at least if they're low atomic weight.  You might get absorption eventually.  If you walk around with silver coins in your pocket  and you were careless with where the neutrons are going,  after a while you notice that the neutrons are becoming radioactive,  the silver is becoming radioactive rather,  because it's absorbing neutrons.  And there are some interesting pre-World War II cyclotron descriptions  where people should have been more careful and eventually were.  However, having said that, the virtue of neutron scattering  is they scatter very effectively from hydrogen and from deuterium.  But they do not scatter the same way.  Again, there is a large theoretical development which I am skipping over.  But the net result is that if I take  perduterated, this is perduterated polyethylene,  and I put it into a sea of hydrogenated polyethylene,  I mostly see only one of the, or vice versa.  I really only see one of these.  And if I am very clever, instead of I take a material  that is some amount h and some amount x,  and if I get the ratio just right,  the neutron scattering almost does not see this at all.  And I can now produce invisible polymer chains.  Neutrons scattering invisible polymer chains.  And I then put in them the chains that are going to scatter.  And so I have, for example, trace quantities of the visible chain  and large amounts of the invisible chain.  And I can use neutron scattering on a polymer that is very well characterized.  The material you are looking at is polystyrene and toluene.  And also polystyrene and carbon disulfide.  Why carbon disulfide?  It doesn't contain any hydrogen.  It is neutron invisible.  Well, it is not quite neutron invisible, but it is good enough.  And we can plot in figure 7.8 the chain radius  versus the polymer concentration.  And what you see if you don't look at the figure too hard  is that as you change the chain rate,  as you change the matrix concentration, the chains shrink.  Now, that figure is a little more complicated  than you might think if you didn't look at it hard.  First of all, the two polymer chains are very nearly the same size.  The two chains we are looking at,  Daoud et al., the solid points, are looking at 110 kilodalton polystyrene.  A king et al. are looking at 111 kilodalton polystyrene in different solvent.  So what I did to generate this figure,  and this is one of these things that is worth talking about,  to generate a figure,  is if you look hard, you will notice there is a size scale  on the left and right axis, and they are not the same.  The reason is that the polymer in these two solvents has a different,  even though it is almost the same molecular weight,  it is different by 1%.  And you should realize that saying these polymers  differ in molecular weight by 1%  means you have an unreasonable confidence in your experimental accuracy,  or you are working with DNA where you know the molecular weight exactly.  So what I did was say,  okay, we will take two vertical scales,  one for each set of experiments,  and they will have the same top end,  so that at zero concentration we have the same starting point.  And the two axes will then cover the same fractional change in radius.  And that is what we do, and you notice the axes are quite different,  but they do have the same starting point,  and they do both cover approximately a factor of two change in radius.  Exactly a factor of two.  The other problem I had, which is a little more also significant,  is that one of the research groups perfectly reasonably reported  the polymer concentration in grams per liter,  grams of polymer per liter of solution.  That is a very reasonable concentration unit.  The other group reported the mole fraction,  and you then have the little difficulty  that those two are of mole fraction of polymer,  except at the two end points,  those two scales aren't quite the same, fundamentally.  Why aren't they the same?  The reason they're not the same is something called volume of mixing.  The traditional freshman chemistry example of this  is to take a liter of water  and a liter of non-denatured ethanol,  that's tax stamped ethanol,  and you mix them,  and you now have, for all practical purposes, vodka.  However, do you end up with two liters?  No.  No, you end up with, does anyone happen to remember the number?  Oh, 1.7 over 8.  Yes, it's about that.  I was going to say 1.9, but you're right.  You end up with, you have a liter and a liter,  and you mix them together,  and as you mix them together, the system contracts.  And the details of that are quite complicated.  The reason this is important is that if you want to convert  how many moles of this and moles of that we have,  or weight fraction, we have so much weight,  I think that may actually be weight fraction,  I think I misremembered,  we have 50% by weight of polystyrene.  That doesn't mean you have,  given the density is 0.9 roughly,  that doesn't mean you have 450 grams of polystyrene  in a liter of solution.  It means you have 450 grams of polystyrene  in 900 grams of solution,  and you cannot convert unless you do some auxiliary measurements,  which were not done, or at least which I did not find.  And so there are two sets of concentration units,  bottom and top.  They are arranged so infinite dilution,  and melt, all polymer, line up,  because in the melt, 100% by weight,  and whatever the, it's around, oh, 943,  if I recall, gram per liter,  the melt numbers are the same,  and so we have composed this graph to show two sets of data.  Now why are, what is the issue here?  Well, those two sets of data,  they're both very hard to measure.  They were done by two slightly different methods.  Daoud used a tracer amount of polystyrene,  labeled polystyrene.  King used concentrated polystyrene  and used a statistical mechanics trick  to pull out the radius of the labeled polystyrene  from the unlabeled polystyrene.  We're very controversial for a very long time  because they do not agree with each other.  The core issue, why are we even interested in this graph?  Well, there is a scaling theory of polymers.  It's in the Daoud paper, and the references are there.  And what is claimed, if we look at R squared  versus concentration, at low concentration  it's claimed that nothing happens.  Well, that's what it says.  And then there's some sort of crossover,  which means there is a region where  several things are happening simultaneously,  and the theory doesn't make a clear prediction.  And then out here, the theoretical prediction  is that R squared is proportional to C to the minus one and four.  Chains contract as you increase the concentration,  and the prediction was that you get a power log.  What is the prediction?  And so there was an effort made to measure  experimentally what the power was.  And there are these two sets of data,  and if you plot the two sets of data on log-log paper,  you get two somewhat different exponents.  And the black dots agree sort of with the exponent  that was predicted, and the open dots get a different number.  And since some people were very attached to the theory,  there was a great deal of back and forth as to what was going on.  Let me, however, point out a couple of features of this graph  that aren't emphasized quite as much,  except by people who aren't quite as involved in the dispute.  And the first is that if you look at that graph,  I said there was a crossover,  and the crossover occurs roughly at a concentration,  a C star, the overlap concentration,  at which the polymer chains, approximately speaking,  are shoulder to shoulder,  but haven't started to inter-penetrate too much.  Well, that's what theory says.  But if you look at that data, it does this.  And in fact, something like, oh,  a third of the contraction has occurred  in this dilute solution region  where there's not supposed to be any contraction.  And then the contraction continues.  You can see the contraction.  And if you ask what happens in solution,  well, the contraction continues and continues.  And I'll draw a line here,  which is something like 500 grams per liter,  sort of half a polymer.  And beyond that, there's, perhaps it continues a bit,  that's the solid dots,  or perhaps it runs out of steam.  The open dots sort of show that the polymer  doesn't contract much at all,  above about 50% by weight.  And there are people who will be happy  to explain why you should be seeing that.  But the net result is the curve does not look  a great deal like the theory qualitatively.  If you only look at the high concentration dots,  the constant d, if you only look at the dots  that are, oh, above, say, 100 grams per liter,  you can put them on a power law.  This is a log, log plot, radius squared versus concentration.  You can put them on a power law,  but the variation this way isn't exactly very big.  The variation this way isn't exactly huge.  So there's some uncertainty in the power law.  And there was then a great deal of dispute  as to what was happening.  I put in something different from a power law.  I put in a, those two solid curves  are stretched exponentials again.  And you see the solid curves do quite nicely  at going through all of the points,  including the low concentration points  that the power law does not predict.  And the only difficulty occurs  if you go out to the melt concentration.  Okay, that's it for chain size.  We will now push ahead,  and I will at least slightly talk about relaxations.  The first point is experimentally,  the easy way to do the experiment  is to take the sample,  put it between metal plates,  and apply an oscillating electric field  that oscillates at a given frequency.  Okay, so we have a capacitor.  Yes?  So we can apply an oscillating electric field  at some frequency,  and we can then measure the capacitance of the capacitor.  Now how you measure the capacitance,  well, if you want to do it accurately,  you have to work a bit.  The important issue,  and this is actually physics too,  is the capacitance of the capacitor  is determined by the dielectric constant  of the material between the plates.  Okay?  Yeah.  Dielectric constant.  And if we are talking to freshmen,  we tell them there's a material  that has a dielectric constant  which can be small or large.  In the real world,  epsilon depends on omega.  Dielectric constant depends on frequency.  You've actually seen this in practice.  That is, if you ask,  what is the dielectric constant of water,  it's very large.  If you go up to optical frequency,  the index of refraction is directly related  to the dielectric constant.  Yes, water has a dielectric constant  which is not one,  and an index of refraction which is not one,  but it's not very big.  And in fact, if you plot the dielectric constant  versus frequency,  you discover,  we'll use log omega  because we want to cover lots of frequencies,  and we will use log of the dielectric constant.  It rolls off.  It rolls off quite dramatically.  We'll discuss the details of the roll off next time.  Now, what I want to do first is to simply get to the question,  well, why does it roll off?  Why does the dielectric constant  drop at very high frequency?  Well, let's go back and think.  Here is a molecule,  and it has two charged ends.  This is a simple case in water.  Say, in amino acid.  It's actually got charges on it that it's fixed.  We apply an electric field.  We get a dielectric constant  because the molecule can line up with the field.  However, if I have to,  if I flip the field around,  it's an oscillating field,  the molecule has to turn around  through 180 degrees to match.  Okay?  That takes a while.  If we are at very low frequency,  the molecule has no difficulty flipping and flopping  each time we change the sign of the electric field  in the capacitor.  However, if I crank up the frequency enough,  the molecule starts to flip.  It can't keep up.  And before it's rotated significantly,  the field is pointing back the way it was originally.  And as a result, at very high frequency,  the molecules simply aren't capable  of keeping up with the field,  and they do not, and they simply sit there.  Yes?  Okay.  But, um...  Gee, that's the rule.  Now, there's something else that can happen  that does not come up in dielectric constant studies  but comes up in certain related studies.  Namely, um...  As you keep changing the field around,  the flip and flop back and forth  may get out of phase with the charge you're applying here.  You could describe this either as I am applying a voltage  across the plates,  or you could say I am applying a charge on the two plates.  You can do several different things,  and there are some complications  that we will not get into in dielectric response.  So there is something I have over somehow time  for the dipoles to relax.  That would happen at very high frequency.  The dipoles just sit there,  and they do not contribute to the dielectric relaxation.  However, the dipoles still...  the molecules still have electrons,  and the electrons shuttle back and forth  even though the molecules do not have time to rotate.  And so at very high frequencies,  way, way out in the optical almost,  you will still have a dielectric response,  but it will be much smaller  because it will just correspond to the electrons moving.  However, the molecular rotation or local rotation,  part of the dipole behavior, doesn't contribute anything.
Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_0_71.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_72_503.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_504_899.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_900_4163.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_4164_6791.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_6792_8147.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_8148_10139.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_10140_11579.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_11580_11975.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_11976_12683.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_12684_14039.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_14040_17831.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_17832_19067.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_19068_19715.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_19716_20135.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_20136_25379.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_25380_27515.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_27516_28439.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_28440_31475.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_31476_33431.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_33432_37367.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_37368_41771.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_41772_44399.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_44400_44603.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_44604_44855.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_44856_46547.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_46548_46895.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_46896_49847.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_49848_50411.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_50412_50567.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_50568_50831.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_50832_55691.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_55692_57539.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_57540_62039.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_62040_63047.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_63048_65135.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_65136_65783.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_65784_66383.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_66384_67127.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_67128_71435.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_71436_73835.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_73836_74075.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_74076_74603.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_74604_75143.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_75144_76475.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_76476_80699.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_80700_85163.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_85164_85835.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_85836_86039.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_86040_93467.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_93468_93575.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_93576_93923.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_93924_94703.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_94704_97775.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_97776_102875.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_102876_105479.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_105480_105671.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_105672_108443.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_108444_108527.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_108528_109787.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_109788_113171.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_113172_115283.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_115284_117575.png Classes_in_Polymer_Dynamics_-_9_Dielectric_Relaxation__Part_2_117576_118270.png