15 Even More Probe Diffusion
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Phillies, George
Lecture 15 - more on probe diffusion. Lectures are based on my book "Phenomenology of Polymer Solution Dynamics", Cambridge University Press 2011.
 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 15, more on probe diffusion. I'm Professor Philly's, and this is lecture two of our discussion of probe diffusion, experimental studies of polymer solution dynamics, based on measuring the diffusion of rigid objects, small or large, through polymer solutions. In our last lecture, we discussed the dynamics of small and large probes, in which we have something that might be a fraction of a nanometer, up to hundreds of nanometers across. It diffuses through polymer solutions, as it diffuses, we can measure how rapidly, and we can determine how the diffusion rate depends on the concentration of the matrix polymer, the polymer and the surrounding solution, how it depends on the molecular weight of the matrix polymer, and other features. Today, we're going to push on to consider a few other things. The first thing we're going to discuss, which is section 9.5, is re-entrance. I'm not quite sure where to put re-entrance into the discussion. It could be described as, there is this extra phenomenon you see on rare occasions. It's not clear why you're seeing it. It's not clear what it is. But it's been seen enough times that we can group all of the observations together. The starting point, we measured diffusion versus concentration, and we are doing this for polystyrene spheres, and polyethylene oxide water. This is my own work with the Oman's. We are in polyethylene oxide water, and the simple behavior, which is what is usually seen for the diffusion coefficient, gives us a curve that looks like this on a log-log plot. That's a stretched exponential. However, and we didn't actually quite discover it in the natural order of things on how we worked things out, if you take particular size of spheres and particular molasses, and particular molecular weights of polyethylene oxide, and since it's not the small ones or the large ones, I think it must be some specific chemical property of the particular polymers. At least that was our best guess. The right size sphere is what you see is a phenomenon that looks like that. That is the diffusion coefficient first increases with increasing polymer concentration. And then goes down again, and gets back to about where you would have expected D to be at that concentration. Gee, what is this? Well, that's a good question. You can see the same phenomenon or something somewhat similar. There is nice work by Wann at all, and what Wann at all looked at was a cross-linked polystyrene sphere and polyvinyl methyl ether toluene. And what they found was something that looked, if you look at their diffusion coefficients, a bit different. Namely, here is the smooth curve you find if you fit most of their measurements to a stretched exponential. But in the middle, what you suddenly see is this region where the diffusion coefficient heads off for whatever reason, and then comes back again. We're talking about polystyrene spheres that are seemingly of a reasonable size. The concentration in what are called natural units, where the intrinsic viscosity has dimensions one over concentration, C eta gets up to about 36. And there is this narrow regime where, gee, things don't work the way you would have expected. The question is, what are you looking at? Well, there are other cases that look like the first of these two found for polyelectrolytes. I've just shown two examples for neutral polymers. The question is, how are you seeing this? What are you seeing? One general explanation has to do with the question, well, what sort of a diffusion coefficient are you looking at if you are measuring things with quasi-elastic light scattering? And if you simply have a single set of diffusing species, everything is completely uniform, your light scattering spectrum or the dynamic structure factor, that's all discussed in chapter 4, on a semi-log plot is a nice straight line. You can measure its slope anywhere, you get the same answer. Life is very simple. And that would correspond, as a general statement, to Brownian particles in water, some system with no memory. And polymer solutions, though, life becomes more complicated. And if you look at the relaxation spectrum that you get with light scattering, you can get several relaxation modes. What information does light scattering give about those modes? Well, let us redraw this, and I will now redraw the light scattering spectrum on a log-log plot. And if the modes are separated, you get a first relaxation like that. That's sort of what an exponential looks like on a log-log plot. And then you may get another exponential or stretched exponential or something. And you have different things happening on different time scales. The reasonable presumption, which is there, then there's a way to test it, we'll get to in a second, is that both of these processes are going on at all times. And therefore, if you measure the initial slope of the relaxation, it's the sum of the rate determined by this faster mode plus the rate determined by this slower mode. And light, if you do what we call cumulant analysis, that is, if you ask, what does the initial slope of this curve look like, and it doesn't have to be approximated as a straight line, it could be approximated as something that is fancier than a straight line, what does that initial slope give us? Well, it gives us the average of all of the relaxation rates weighted for how much amplitude they each have. On the other hand, if you sit around to long times, if you look out here and measure the relaxation rate, you only see the slowest mode. The fast mode has relaxed away. Now, that doesn't mean that if you wait a bit, the particles are no longer moving in the rapid mode. What it means is the particles have moved far enough that their positions with respect to the rapid mode are not correlated with their initial positions, and therefore, even though the rapid process is going on, the rapid process quits contributing to the relaxation spectrum because this has gone off to zero. The new process relates to particles starting at positions that are uncorrelated with their initial positions. It is sometimes said, if you see these fast and slow modes, there are people who look purely at the time dependence of the diffusion process and say, oh, you're seeing caging. And the notion in caging, which is perfectly legitimate possibility, is that we have a pro-particle. It is somehow confined into a region of space where it can move rather rapidly, but it can't get very far. And if it wants to move a large distance, it must, for example, hop to a new region where it's happy to sit, and the hopping process is slow. Now, if you do have caging, if you have regions that are in some sense low potential energy or dynamically restricted but free to move within, then over short times, you'll see motion within a cage and that long times, you will see the motion, the slower process in which things hop from cage to cage to cage. Now, once the object gets here, it still does have the fast motion within its new cage, but these positions and these positions are uncorrelated, and therefore this motion does not contribute to the spectrum. Having described caging, I would like to emphasize that the time dependence measurement gives you absolutely no evidence as to whether the caging interpretation is correct. Question? What happens to the processes when it's moving from the confined region and moving slowly when it's hopping? What happens between here and there? How does that show up in the time scale like that? The slower process, how does that show up in the time scale? The slower process when it's doing this is the slow mode. That is, there is a part of the position of the object that stays correlated with its initial position as long as the object is trapped here. And only when the object has done a bunch of hops does the last piece of its initial position get forgotten. Thank you. Okay? So that is the question. The reason you don't see anything is that light scattering spectroscopy, a single spectrum, only gives you information about motion on one distance scale. Now it's a Fourier scale, not a linear distance scale, so you're looking at the relaxation of a spatial Fourier component of the concentration. That's why you can see both modes in the light scattering spectrum. However, if you just look at a single queue and measure the spectrum as a function of time, which is how the experiment is usually done, you are not directly generating any evidence of hopping. Indeed, if you go back a few chapters, there is this paper by, I believe it was Nemoto, the collaborators, where they look at light scattering and sedimentation. Sedimentation is intrinsically a long distance process, light scattering, light wavelength distance. And what they demonstrate is that the long distance motions are faster than the slow distance motions, which is sort of the negative of caging in some sense. Okay, so we have said, at short time, you are seeing all of the relaxations put together in a light scattering spectrum, and at long time, the light scattering spectrum only shows you the slowest relaxation of modes. You will sometimes encounter a literature error, and the literature error claims that if you look at the light scattering spectrum at short times and measure the initial slope or whatever, you are purely and exclusively seeing the fast modes. That is complete nonsense. It is simply not true. Okay, so if there are several modes, can we just measure them directly? And that is answered in section 9.6, and 9.6 discusses the spectrum with multiple modes. Now, we will come back to discuss multiple mode spectra again when we discuss optical probe studies of hydroxy-probe, cellulose solutions. However, there are so many studies of HPC solutions that they are grouped all in their own subchapter, their own section, and we will get to those separately. And we start out with studies by Bremmel. And what was done was to say, well, we will take solutions of sodium polyacrylamide, and it is a polyelectrolyte, and we will put in probes, and the probes that we will put in are polystyrene latex or hematite particles, and we look at the spectrum. And what was found was that there were two spectral modes. The two spectral modes were both fairly close to exponential. They could be characterized separately in terms of a diffusion coefficient. But what was observed under various conditions is that if you have several modes, life can become much more complicated. And you can have modes whose relaxation rate does this and modes whose rate does that. That starts to look familiar. That's the polyethylene oxide thing. And modes for which you only see this, and well, you could always think it might do that. And so you have two modes visible in the spectrum, and the behavior of the two modes is different in different cases. Now, having said that, there is now a possible explanation, or partial explanation, for some of the polyethylene oxide data. Namely, if you have a spec system that's giving you two spectral modes, it's possible that under some conditions one mode is dominant, under some conditions the other mode is dominant. With more limited anyhow technology that was available in the early 1980s, which is when the polyethylene oxide data was done, well, sometimes you basically see this as the dominant mode in the spectrum, and sometimes you see this. And if you examined the same system with modern technology, you'd find that two modes were present, and one or the other was more prominent. Okay, let us consider to advance to some additional work by Bremel and Dunstan, and we are looking at polystyrene sulfonate spheres. What is the point we say polystyrene, and you will notice there, there are carboxylate modified polystyrene spheres, or this modified or that modified, and in the case in question, they're in a copolymer. Okay, why do I mention the modification? Well, it's like this, you have a sphere, it's made of polystyrene. Polystyrene is not at all water soluble. If it were water soluble, you'd pour coffee into your cup in the morning, and the cup would dissolve. And it would be full of polystyrene, which is not necessarily something you want to drink huge quantities of, not if it's water soluble suddenly. And the answer is in order to get this stuff into water and persuade it to stay there, what you have to do is to charge up the surface, and the charge up the surface is usually done by chemical surface modification. So for example, you have, there's a carboxylate group, it's an organic acid group, and it's been fully ionized, so you add a little base, or that was done by the manufacturer, really. And now you have these little charged spheres, and since the spheres are charged, they repel each other, and they're happy to stay in water, and they're a real solution. Okay, so what Bremel and Dunstan did was to look, and they see their modes again, and they see things that look like this, except, well, perhaps it doesn't go on quite as far. And then they could say something, they do two things. The first thing they do is to say, gee, the diffusion coefficient can be used to calculate a microdiscosity, a data micro, this is the Stokes Einstein equation, but it's being written where the variables I measure are the particle size and the diffusion coefficient, and the quantity I calculate from them, and K and T and 6 and pi, the quantity I calculate from them is the microdiscosity, the viscosity that would explain the diffusion coefficient. And what was found for these experiments was that the microdiscosity for the slow mode appeared to be larger than the micro, the orthodox viscosity of the solution, and the microdiscosity that corresponded to the fast mode was clearly less than the viscosity of the solution. That is, if you have two modes, you have two diffusion coefficients that you can infer from them, and because you have two diffusion coefficients that you can infer from them, you can calculate a microdiscosity for each one, and since the two curves are not parallel at all, you get two different behaviors for the microdiscosity. The next thing that they did was to say, well, we will look at several scattering angles, and because we are looking at several scattering angles, the wave vector Q, the scattering vector in the light scattering experiment, changes. Now, the reason that's significant is the thing that you're measuring in light scattering, the dynamic structure factor depends on the scattering vector. What does the dynamic structure factor tell you? Well, we wait a while, and eventually there is a fluctuation in the concentration, a cosine-useoidal fluctuation, that has a wave vector Q, and the wave vector Q corresponds to the scattering vector of the scattered light. Now, actually, these fluctuations go on all the time. They occur simultaneously at all wave vectors, not just one, but if you wait for a moment when the scattered light is really bright, you can correctly infer that the fluctuation of the correct scattering vector to get light out of the laser beam and to your detector, that fluctuation must be fairly large. And now you wait and ask what happens at later times. And on the average, you have to do this many times. That's what the correlation function measures. The size of the fluctuation on the average relaxes back to zero. It doesn't do it all the time, relaxing back to zero, or it could never get big. It fluctuates, but on the average, if you wait for moments when it's large, the fluctuation relaxes back to zero. And the relaxation is the scattering vector. If you change the scattering vector, you're changing the distance scale over which you're watching emotions. Now, since you're looking at a concentration fluctuation that looks like this, particle emotions over distances that are significantly less than this wavelength contribute to relaxing the fluctuation. And the particles can move various different distances, but if they start at a maximum and head off towards the minimum, they reduce the size of the fluctuation. So you're not looking at exactly emotions on a single distance, but there's a distance scale in there. So what happens? What happens when they did this experiment? There are two modes, a fast mode and a slow mode. And what they found was that the diffusion coefficient of the slow mode is independent of scattering angle. And that is the behavior you get if you have a process that is purely diffusive. A single diffusion process will give you a relaxation that goes as e to the minus the diffusion coefficient q squared t. That's a single exponential relaxation. The relaxation rate goes as q squared. But if you pull the q square out and ask what is the diffusion coefficient, the diffusion coefficient is the same on all distance scales. And so what they said is that their slow mode process goes as q to the zero, and that implies that their slow mode process is in fact diffusive. The fast mode process, however, was quite different. They measured the apparent diffusion coefficient at different scattering vectors. A larger scattering vector corresponds to a smaller typical displacement. Large q is short distances. And what they found is that as they made the distance scale shorter and shorter, the particle motion was faster and faster. The important issue is that since they're saying at shorter distances, the particles move more rapidly, we are not discussing a diffusive process here. We're discussing something rather different. The other issue is, going back, the slow process is q independent. And so, gee, that actually speaks to Cajun models. The reason it speaks to Cajun models is that if you have a particle that does Brownian motion by parking in a cage, hopping to another cage, parking in a cage, hopping to another cage, and this random hop, hop, hop is the random walk that gives you the slow diffusion. If you look at very short distances, something should happen to that slow process. Because at short distances, the slow process hop, hop, hop starts to become visibly discrete. Well, that's not what you see. What they see is that the slow process, whatever it is, is q independent. Okay. Let us chug ahead. That's Bremel and Dunston. And we'll look at another set of experiments. And the new set of experiments are due to delphenol that we collaborate with. And the notion in the new set of experiments is we will look at a polymer. And the polymer they looked at was carboxy-methylvalyloids. And the material they looked at was fairly high molecular weight. And as a result, the radius of gyration of the polymer happened to be about 50 nanometers. Question? Is carboxy-methylylvalyloids used in chromatographic columns for adsorptions? Um, you might... I actually don't know. The short-form answers know this stuff is water-soluble, so it's not used in column. But a lot of these should remember, this is now a synthetic issue, a lot of polymers, yeah, they're water-soluble, but then you do cross-linking, and instead of having a linear chain you form blob that is big, big, big, and it ceases to be soluble in a practical way, and now you have a column. But this is not cross-linked. And what they did was to take spheres of three sizes, 17, 47, and 102 nanometers. And guess what? They have here spheres that are considerably smaller than Rg, and they have at the other end spheres that are considerably larger than, well, twice Rg, and they have spheres that are about the same as Rg, so they have three sizes of spheres, and they compare the diffusion of the three sizes of sphere in the same system. And you are now comparing the diffusion of objects that are smaller than the polymer molecule the same size as or larger than. Now that's a perfectly reasonable comparison viewed abstractly to make. You should, however, notice that there are theoretical models that say you have a polymer solution and it's fairly concentrated. You have these long chains and they are entangled. And once they're entangled, the model says, well, it doesn't matter how long they are because the ends don't do very much except from forward or backwards. And the interesting distance scale becomes this distance scale, so it's now xi, and xi is sort of the size of a hole in the polymer mesh. And if you say that's true, then the interesting comparison is between objects of different sizes and the size of the hole, and the hole is much smaller than a polymer chain. The sort of limit, why do we know the hole is much smaller than a polymer chain? Well, suppose we have a dilute solution that is just on the edge of entangling. Okay? In that case, the distance between entanglement points can't be much larger than a polymer chain because if it was, this polymer chain would only be entangled at one point. There wouldn't be a mesh work. So the entanglement notion is the lowest concentration at which you can possibly have entanglement is approximately when the radius of the chain is the same as the distance between entanglement points. And at all higher concentrations, xi becomes smaller and smaller, the radius of gyration shrinks, but much less. And in that case, the interesting length scale and solution is much smaller than the polymer chain. Well, having said that, they did the experiment. And the reason the experiments appear in this chapter is the spectra they found were bimodal. Their spectra were adequately fit by a single exponential, or two exponentials. That is, each mode was close to being a pure exponential. That says that each mode, separately, could perhaps be viewed as a diffusive process though there must be some complications we are not quite seeing. Nonetheless, they do see two modes. Now, the reason you have to worry about saying it's pure exponential is as follows. Suppose semi-log plot. Here's the dynamic structure factor. Here's time. This is a semi-log plot. Suppose the actual spectrum looked like that on a semi-log plot, meaning the relaxation is not a pure exponential. It occurs more rapidly at fast times and on the semi-log plot less rapidly at long times. Suppose you had a spectrum like that. Suppose you charge in, and not paying much attention to the data, fit this to a single exponential. Well, you could say it's an approximation, but there's a problem with the approximation. If I just fit to the measurements up here, I get one slope. If I just fit to the measurements down here, I get a quite different slope. And the pure exponential fit, unless you're careful, and I'm not saying the paper we're talking about was not careful, I'm just showing where there's a hazard here. The hazard is that if you fit on different time scales, you get different slopes for the exponential. And that's because the exponential is somewhat cooperative. You can always draw a straight line through a set of points. It may be a foolish straight line because the points don't resemble a straight line, but you can always draw that straight line, and you will get a result. And so what happens when they do this? Well, first of all, what they found was that there are two modes, and for the small spheres, the fast mode was dominant. For the slow spheres, I didn't say that right, sorry. At low concentrations, they found the fast mode was dominant. At high concentrations, the slow mode was dominant. And so far, we've only talked about the intermediate-sized spheres. I said that backwards the first time. We can draw a picture showing this. The other thing they found, so we'll start with the small spheres, and there are two modes, and the two modes both slow down as you increase. This is the diffusion coefficient corresponding to the mode. This is the concentration. And the first thing they find is that for the small spheres, the slow down is over the observed range of polymer concentrations. It's, oh, a factor of three or six. I'm being approximate. For the large spheres, the figure is in the book, and you can look at this, and for the large spheres, the slow down is, oh, maybe a factor of 10. And for the slow mode, it's so number like 300 or 400. It's more than two orders of magnitude. And so what you're saying is, for the small spheres, the two modes are affected only modestly. Well, actually, it's factor of three. It's plenty to see by the polymer. For the large spheres, the effect of the polymer is vastly more dramatic. And there is a crossover region when the spheres are about the same size as the polymer molecule. And it really does appear, though, of course, they only have three sphere sizes. It does appear there's a transition between the sphere is much smaller than the polymer and the sphere is much larger than the polymer. Now, if you really want to say that statement and say there's a transition, you would really want more than three sphere sizes for the simple reason that you ask, well, is it a continuous rollover or is there a real change from behavior class A to behavior class B? Does the rollover actually occur when r is equal to rg or is it someplace to one side or another? And if you want to settle that question, you'd have to use plenty of sphere sizes, and then, if you're lucky, life becomes transparent. Well, that's what Karel Stroletsky did. He was my PhD student for his doctoral thesis. He did this in HPC, not carboxy-methylcellulose, but he used a whole bunch of sphere sizes and he, in fact, found exactly what is being implied by the very nice experiments of Delfino. Namely, there is a behavioral class for spheres that are smaller than rg. There is a different behavioral class for spheres that are sort of larger than rg. There are several ways of characterizing the size of a polymer molecule, so let's say it's a bit imprecise as to exactly what you call the size of the transition. But, in fact, the transition is fairly large. It occurs approximately at the size of the polymer coil, and therefore, the solution, in my opinion, unsurprisingly, has a longest characteristic length scale, which is the size of a polymer chain. Well, that's fine unless you believe entanglement models because entanglement models say that the characteristic longest length scale is this much shorter length scale psi. Yeah, that's not my problem. Okay. We now advance to section 9.7. And 9.7 talks about probes in polyelectrolyte solutions. Now probes in polyelectrolyte solutions are something that is not covered very heavily by my book. At some point I decided I have to stop someplace, or the book will get bigger and bigger, and the publisher only gave me 500 pages. And the time required to complete, instead of being a fixed date within my lifetime, will be divergent, and I'll never finish the book. So I had to stop at some point. And I remind you, things that we stopped before getting to include rods, polyelectrolytes, liquid crystal polymers, block copolymers, salts, oh, thread like micelles, there are solutions, small molecules, but they form dynamic structures in solutions that are long and thready, that look a lot like polymer chains, except they're not covalently cross-linked, and they can go straight through each other. Now, having said that, there are a bunch of things we did not cover. Nonetheless, I do briefly mention some of the results that do exist on polyelectrolyte systems, and I bring up some of the polyelectrolyte results so that you know they're there. So what did we say about polyelectrolytes? Well, the first issue is there's some extra variables in a polyelectrolyte system. That is, if you have a polyelectrolyte, the first material we worked on, polyacrylic acid, could be a polyelectrolyte, but we worked with the non-neutralized material. These are results of tyholin and I, in which here are the carboxylic acid groups, and they're mostly non-neutralized. Well, some of them spontaneously ionize a bit. But what you could do is add base to the system, and now you have ions on the chain, and you have sodium ions, and you added sodium hydroxide, and guess what? You now have neutralized polyelectrolytes. Well, how much base did you add? There is a percent neutralization that determines what fraction of those groups are charged. Also, you could have a salt concentration determined by its ionic strength, I, which for monovalent ions is simply the molarity of the solution. We can go on. There's more complication, but we aren't doing polyelectrolytes, and there are extra variables. And having said, there are extra variables, there are all sorts of experiments you can measure. Okay. For example, we will take, I believe it's figure 9-25, and we can measure the diffusion coefficient. We're looking at polystyrene spheres in a partly neutralized polyacrylic acid. This one happens to be 60% neutralized. It's a nice round number. It is, if I recall correctly, a 600 or a slightly smaller polymer. And we measure the diffusion coefficient of spheres as we add salt to the solution. And so we start with no added salt. That's, now you may say, gee, that's simple, isn't it? No, that's actually quite hazardous. The reason it's quite hazardous is, A, water ionizes a bit, and so you really can never get less than in water, about 10 to the minus 7 molar ions, because the water is ionized. Furthermore, water absorbs carbon dioxide from the air, unless you're really rigorous about dealing with issues. And therefore, the pH of water, if you actually look at it under these conditions, is more like 5, meaning you have something like 10 to the minus 5 molar hydrogen ions and other good stuff floating about. And so, unless you're extremely careful in your work, it's very hard to avoid introducing at least traces of small ions. And therefore, if you want to know what's going on, the experimental process is to be sure to add at least tiny amounts of salt, sodium chloride, potassium chloride, whatever, 10 to the minus 3 molar, 10 to the minus 4 molar, and then you are probably closer to knowing what's going on in the solution. Nonetheless, that's I equals 0. We didn't add anything. And as we increase the amount of salt, well, at 0 polymer concentration, this is D versus polymer concentration, changing the salt concentration does almost nothing. But at higher salt concentrations, with increasing ionic strength, those curves flatten out. And as you increase the salt concentration, the polymer is less and less able to retard the motion of the spheres. Why? Well, that's a good question. And you can come up with all sorts of explanations. Gee, you added salt, the polymer is less rigid. You added salt to the electrostatic interactions between the polymer and the sphere are weakened. You can come up with all sorts of explanations. You notice it started to get very complicated. You can also look at Stokes-Einstein behavior, and you can compare the product diffusion coefficient times measured viscosity of the solution with what you would have in pure water. If the Stokes-Einstein equation worked, this number would stay 1 as you run up the polymer concentrations, increase the salt concentration. That's not what happens. In fact, what happens is you have non-Stokes-Einsteinian behavior, and the viscosity goes up, the diffusion coefficient goes down, but the matrix polymer is more effective at increasing the viscosity than it is at retarding the diffusion, and this number is larger than 1. There are systems where it's less than 1. The statement, it's larger than 1, has the attractive feature that you simply cannot claim, oh, it's just the spheres are aggregating, there's stuff sticking to the spheres. That would give an effect of the wrong sign. So as we proceed, we see non-Stokes-Einsteinian behavior, and we ask, well, are there ways to make the non-Stokes-Einsteinian behavior less? And the answer is that this number tends towards 1, it doesn't get there. If you make the spheres larger, if you make the ionic strength larger, or for that matter, if you weaken the polyelectrolyte effect by reducing the percent neutralization, if you do any of these things, or all of them, the degree of non-Stokes-Einsteinian behavior goes down. Okay, there is another theoretical piece in here, and the theoretical piece is there are people who will claim, this is a claim, you can find it in the literature for yourself, that you get non-Stokes-Einsteinian behavior if the probe size is less than, or maybe much less than, all of the characteristic lengths and solution. However, if the probes are big, the claim is that you must be getting Stokes-Einsteinian behavior. Well, that's very nice, but if you believe that, gee, if you see spheres and they're showing non-Stokes-Einsteinian behavior, the spheres must be smaller than the length scale. How big were the spheres we used in that experiment? Well, there were a series of sized spheres from 21 up to 655 nanometers, yes, diameter, and based on our measurements, since we were seeing non-Stokes-Einsteinian behavior, except perhaps for the largest spheres, the longest length scale in these solutions must be greater than 300 nanometers. Certainly, it must be greater than 50 nanometers, and it appeared it had to be greater than 300 nanometers. Well, that's a very long length scale. That's not what you would necessarily have expected, but that's in fact what's there. Okay, there's not a figure I drop in, and the figure is, again, the work of myself and its work with LeCroy and Yamburt, and its figure, if I recall correctly, 9-27. And if you look at the figure, yeah, it's B versus C again, so if you look at the figure, yes, it's a semi-log, it's a linear semi-log plot, if you look at it, you realize D doesn't change very much, and we have a whole lot of points, and the points are concentrated on finding what is very clearly the initial slope. And in some cases, a simple exponential slope is adequate, in some cases, a linear slope fit was adequate, and we worked very hard. We used spheres of size 734, and if I recall correctly, 95 nanometers. The measurements you see are for two sizes of polystyrene sulfonate, but we had a fair amount of sulfonate, and we look at several molecular weights of polystyrene sulfonate, and the question is, why did we work so hard to determine the initial slope, which is what we did? We determined the leading linear effect of adding polymer, and how it slows down, a probe diffusion. The reason we did this is that there was a theoretical model that was mycoutic, it's based on the Kirkwood-Reisman picture of single chains, and what it says is, here's a probe sphere. We are in dilute solution, we want to find the leading slope, which is the dilute slope, so here is a polymer chain. The sphere moves, and as the sphere moves, the polymer chain gets to respond, but it's somewhat constrained. It can move, it can translate, so it has some translation. It can notice this side of the chain is closer to the sphere than this side is, so it can rotate, and it has some rotation vector, omega, which is perpendicular to the blackboard, and we can describe the polymer chain, lowest approximation, as a sort of bag of frictional beads that translates and rotates. But we can now calculate the hydrodynamic interaction back and forth between the sphere and the beads, and we can therefore calculate quantitatively how effective the chain is at slowing down the spheres. And the point of this was to see, well, can we do the calculation and does the calculation work? And the direct calculation ends up with a single free parameter, there's a way of beating that limit, which we did eventually, but the point is we could actually calculate the slopes and we get the right answer. So that particular paper is a theoretical model test. It's an extremely important theoretical model test because it tests the hydrodynamic interaction theory, and it works. Okay, I'm doing fine on time since I started a bit late. Okay, let us step ahead again, and the step ahead again is to say there is something called solvent quality. And you can sort of look blindly and wonder, what is the solvent quality? Suppose we have a polymer chain. Here's a polymer coil. We have states of the system in which here comes a polymer coil, here comes the polymer, very crudely drawn, and here comes another piece, and the two are in more or less direct contact. Yes. And you might ask, does this polymer prefer to have another piece of polymer coil as neighbor, or does it prefer that the two polymers stay well apart from each other, and it has solvent molecules as neighbors? It isn't an absolute, we must have one or the other. It's a thermodynamic preference in contact. And the net result is, if the polymer chains prefer to be next to each other, the polymer coil stays fairly compact. If the polymer prefers to be in contact with solvent, well, you have different pieces of the polymer in this picture have to push apart from each other so they don't bump into each other as much, and suddenly the polymer coil gets larger. And if we choose our solvent, we can have what is called a good solvent, in which the polymer prefers to see neighboring molecules be solvent. We can have what is called a theta solvent, in which there's sort of indifference, and we can also have a poor solvent. An extreme example of poor solvent behavior are protein molecules. Protein molecules are nice and neatly folded and have particular groups on the outside and are lumps because proteins view water as a very poor solvent. And so they fold up tightly. And they fold up tightly so that all of the aliphatic groups, the hydrocarbons, try to hide on the inside of the polymer chain. Of course, they can't all manage to do this. And the groups on the polymer that are charged, the carboxylic acid groups, the amine groups, the charge, sit on the outside and be charged. And so the polymer has charges on the outside or tries to, and things that look more like oil on the inside, and it doesn't expand at all. And this is why polymer molecules are stable, because water is a very poor solvent. Now, you might think, aren't proteins soluble in water? Yeah, they're still soluble because they do this, but it's still a poor solvent effect. Okay, so we ask what happens if we measure the diffusion coefficient versus polymer concentration? And we take the same polymer and we put it in two different solvents. Well, the answer, there are two sets of experiments that are both in the book. And one approach to this is we will change the chemical identity of the solvent. And if we do that, we see a theta solvent behavior, and we see a good solvent behavior. And the diffusion coefficient in the theta solvent just falls stretched as a pure exponential, e to the minus alpha c to something close to one. And then the good solvent is e to the minus alpha c to the nu. And g at high concentrations, we're out here in concentration, the theta solvent slows down the polymer more effectively than the good solvent does. Rather, the polymer and the theta solvent slows things down more effectively. Now, there's another way you can change solvent conditions without having to change the solvent molecule. And the alternative way to do things instead of changing the solvent is to change the temperature. Because there are systems that approach being good solvents at one temperature. But if you change the temperature enough, you approach a theta point, and at the theta temperature, you get a different behavior. And those were experiments that were done by Delphine, Coleman, Neal and I. And once again, in the theta system, you see something that's sort of like a straight line. And in the good system, you see something that's sort of like this. And oh my, what's occurring? Well, the answer is you have different good and theta behavior. And from a bit it appeared, we were at the first bit of enthusiastic, g we found a solution, we found a prediction that is actually made by my model of polymer dynamics. And you see exactly what you expected to see. And then we realized, but if you look at the alternative models, all the models we found actually show the same prediction. So this very pretty experiment doesn't really tell you anything, except that in some sense the models agree with each other. And the last thing I will discuss. There was a period in the late 80s, early 90s, when I would go around giving speeches on probe diffusion and what you saw and what it appeared to imply about polymer dynamics. And so I would go here or there or someplace else and I would give my remarks. And someone would stand up and complain that I had not reduced my measurements relative to the glass temperature. Now hiding behind this not at all innocent question is a very complicated piece of theoretical issue. But a piece of the notion is as follows. Here is a polymer solution, a polymer and solution. And I am representing the polymers looking like a pearl necklace. The rate at which the polymer can move is determined by a drag coefficient, a resistance to motion of the individual beads. That's the same as the Stokes law drag coefficient, but it refers not to the whole chain, but to the little piece. Now if you work in polymer melts, the issue is that G, as you change the temperature up or down, the drag coefficient of the individual polymer beads slows down or increases. The drag coefficient goes down or up. And as you change the temperature there for the whole dynamics of the system, changes on its time scale, not because something particular is happening to the whole chain motion, but because the drag coefficient of the individual beads is changed. In fact there is a whole experimental approach called time temperature reduction. And the notion in time temperature reduction is that you can change the time scale on which things are happening by changing the temperature and therefore you can do what appears to be a measurement over a very wide range of frequencies, response frequencies, by doing a measurement over a much narrower range and doing it over a series of temperatures and at each temperature the same behavior occurs at different frequencies and therefore by changing the frequency of the temperature you can compress all of these behaviors so you actually look at them experimentally in one relatively narrow frequency regime. Well that's time temperature dependence. Now how does time temperature dependence come into our probe diffusion work? And the answer is that this drag coefficient was said to depend and it was a very vague question. It wasn't a detailed analysis, exponentially on the concentration and therefore as I changed the concentration I was changing the drag coefficient. Now what does this have to do with the glass? Well I'm going to draw another picture. Two pictures. And so we will plot a property that depends on how easily the beads can move, namely the viscosity, the resistance to pouring. And I will plot this as a function of temperature. And if I plot this as a function of temperature, what happens is I cool the system off, the viscosity increases and then it increases very rapidly indeed and eventually I get to something called glass. Now the picture I've just drawn is for a polymer mept. And the notion is that as T approaches this temperature Tg, the glass temperature, the viscosity sort of diverges. Now it doesn't really diverge, but if the viscosity is 10 to 14 times the viscosity that you had when you had a simple fluid melt, it doesn't pour. It's practically but not quite a solid. That's the glass temperature. If you look at large numbers of behaviors, you can say the interesting variable is T minus Tg, the distance from the glass temperature. And if you want to compare how different polymeric melts are behaving, the sensible comparison is not to compare them at the same temperature, but at the same temperature distance away from the glass temperature. Now you could also be a little fancier and say maybe you ought to normalize like this because there are systems that have very low glass temperatures and systems that have very high glass temperatures, but that is the glass temperature idea. The notion for polymer solutions is that if we look at 1 over D, which is sort of like a viscosity, and we look versus polymer concentration, 1 over D is doing this, there is some glass in some sense that is eventually things don't move or something happens, and therefore I ought to reduce relative to this glass temperature. So Carol Quinlan and I did this experiment, and what we said is, well, how do we reduce relative to the glass temperature? And the answer is we will do measurements of D at a series of temperatures, and the reason we will do this at a series of temperatures is as follows. This picture here actually leads to something called the Vovogl-Folker-Tamann equation. It says D should go as sum D0, and then there is a temperature T because the basic diffusion coefficient scales as temperature, and then there is an e to the minus sum constant, there is a constant, over T minus Tg. And you notice this thing has the feature as we approach, oh, I should put a plus sign there, the sign is of course not meaningful because A is a signed number. But the issue is that as the temperature approaches the glass temperature, this factor becomes extremely large, and therefore I will put the minus sign in its diffusion, and therefore this number becomes large, and therefore the temperature goes to zero, as you approach the glass temperature. Well, if I am doing this in polymer solutions, the claim is at each concentration, because Tg is a function of polymer concentration, as I change the polymer concentration I am changing the glass temperature, and therefore instead of quoting everything at the same temperature, I should quote all of my measurements at the same T minus Tg. Okay? So some place I should establish Tg for these solutions, and if I quote measurements at different concentrations, I should also be at the same time be comparing measurements at different temperatures rather than at the same temperature, which is what I have been doing. Okay, well this equation has a feature. It makes a prediction as to how D should depend on temperature, doesn't it? There is however another prediction as to how D should depend on temperature, namely D should, if we compare with the D with T over eta of the Solmon, we should get a straight line. Yes? Okay, so we did all the experiments. And the first thing that happened is that if you measure D in a series of temperatures and fit to this form, you discover that Tg is extremely low, like minus 100 centigrade. That's not a physical temperature, of course, for water because it freezes. However, what we found, you do the temperature dependence at a series of different polymer concentrations, and you ask how does the apparent Tg, whether you believe this makes any sense or not, depend on temperature, and the answer is the apparent Tg is independent of polymer concentration. And therefore, if you believe that you were supposed to reduce relative to Tg, well, we did the experiment, and the answer is Tg is the same at all concentrations contrary to what was being said, and therefore the reduction relative to Tg wouldn't do anything. The second thing we said is, well, we can look at D versus T over eta sub s, and Gd is linear in T over eta. This is eta of the solution, or eta of the solvent. It doesn't really matter which because they just track each other. And if you look for curvature here, well, the curvature is slightly larger than the scatter of the points, and so you could say there's a little curvature, which, by the way, is independent of polymer molecular weight. And the best we can say is, quite good approximation, D just scales as temperature over solution viscosity linearly, and there's no sign of any deviation from linear behavior. Maybe there's a slight sign, it's sort of a one or a two percent effect, and your measurements are accurate to a bit better than one percent. So you're sort of at the point where you are leaning hard on your data claim that you're seeing a deviation from what Stokes Einstein would tell you to expect for the temperature dependence. So that's temperature dependence. However, we were challenged to do a reduction relative to the glass temperature, and what we showed was that the relevant glass temperature for these experimental measurements is independent from polymer concentration. And since it's independent from polymer concentration, the reduction relative to the glass temperature doesn't do anything. And we could also say, ask, well, what is the dependence on solvent viscosity, which is temperature dependent, and the answer is D is linear and T over A is a solvent, and therefore any hydrodynamic picture or any picture that says the solve viscosity controls things, works just fine. So that's it for temperature dependence. And we are now at the end of today's lecture.
 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 15, more on probe diffusion.  I'm Professor Philly's, and this is lecture two of our discussion of probe diffusion,  experimental studies of polymer solution dynamics, based on measuring the diffusion  of rigid objects, small or large, through polymer solutions.  In our last lecture, we discussed the dynamics of small and large probes,  in which we have something that might be a fraction of a nanometer,  up to hundreds of nanometers across.  It diffuses through polymer solutions, as it diffuses, we can measure how rapidly,  and we can determine how the diffusion rate depends on the concentration of the matrix  polymer, the polymer and the surrounding solution, how it depends on the molecular  weight of the matrix polymer, and other features.  Today, we're going to push on to consider a few other things.  The first thing we're going to discuss, which is section 9.5, is re-entrance.  I'm not quite sure where to put re-entrance into the discussion.  It could be described as, there is this extra phenomenon you see on rare occasions.  It's not clear why you're seeing it.  It's not clear what it is.  But it's been seen enough times that we can group all of the observations together.  The starting point, we measured diffusion versus concentration,  and we are doing this for polystyrene spheres, and polyethylene oxide water.  This is my own work with the Oman's.  We are in polyethylene oxide water, and the simple behavior, which is what is usually  seen for the diffusion coefficient, gives us a curve that looks like this on a log-log  plot.  That's a stretched exponential.  However, and we didn't actually quite discover it in the natural order of things on how we  worked things out, if you take particular size of spheres and particular molasses,  and particular molecular weights of polyethylene oxide, and since it's not the small ones or  the large ones, I think it must be some specific chemical property of the particular polymers.  At least that was our best guess.  The right size sphere is what you see is a phenomenon that looks like that.  That is the diffusion coefficient first increases with increasing polymer concentration.  And then goes down again, and gets back to about where you would have expected D to be  at that concentration.  Gee, what is this?  Well, that's a good question.  You can see the same phenomenon or something somewhat similar.  There is nice work by Wann at all, and what Wann at all looked at was a cross-linked polystyrene  sphere and polyvinyl methyl ether toluene.  And what they found was something that looked, if you look at their diffusion coefficients,  a bit different.  Namely, here is the smooth curve you find if you fit most of their measurements to a  stretched exponential.  But in the middle, what you suddenly see is this region where the diffusion coefficient  heads off for whatever reason, and then comes back again.  We're talking about polystyrene spheres that are seemingly of a reasonable size.  The concentration in what are called natural units, where the intrinsic viscosity has dimensions  one over concentration, C eta gets up to about 36.  And there is this narrow regime where, gee, things don't work the way you would have expected.  The question is, what are you looking at?  Well, there are other cases that look like the first of these two found for polyelectrolytes.  I've just shown two examples for neutral polymers.  The question is, how are you seeing this?  What are you seeing?  One general explanation has to do with the question, well, what sort of a diffusion coefficient  are you looking at if you are measuring things with quasi-elastic light scattering?  And if you simply have a single set of diffusing species, everything is completely uniform,  your light scattering spectrum or the dynamic structure factor, that's all discussed in chapter 4,  on a semi-log plot is a nice straight line.  You can measure its slope anywhere, you get the same answer.  Life is very simple.  And that would correspond, as a general statement, to Brownian particles in water, some system with no memory.  And polymer solutions, though, life becomes more complicated.  And if you look at the relaxation spectrum that you get with light scattering,  you can get several relaxation modes.  What information does light scattering give about those modes?  Well, let us redraw this, and I will now redraw the light scattering spectrum on a log-log plot.  And if the modes are separated, you get a first relaxation like that.  That's sort of what an exponential looks like on a log-log plot.  And then you may get another exponential or stretched exponential or something.  And you have different things happening on different time scales.  The reasonable presumption, which is there, then there's a way to test it, we'll get to in a second,  is that both of these processes are going on at all times.  And therefore, if you measure the initial slope of the relaxation,  it's the sum of the rate determined by this faster mode plus the rate determined by this slower mode.  And light, if you do what we call cumulant analysis, that is, if you ask,  what does the initial slope of this curve look like,  and it doesn't have to be approximated as a straight line,  it could be approximated as something that is fancier than a straight line,  what does that initial slope give us?  Well, it gives us the average of all of the relaxation rates  weighted for how much amplitude they each have.  On the other hand, if you sit around to long times,  if you look out here and measure the relaxation rate,  you only see the slowest mode.  The fast mode has relaxed away.  Now, that doesn't mean that if you wait a bit,  the particles are no longer moving in the rapid mode.  What it means is the particles have moved far enough  that their positions with respect to the rapid mode  are not correlated with their initial positions,  and therefore, even though the rapid process is going on,  the rapid process quits contributing to the relaxation spectrum  because this has gone off to zero.  The new process relates to particles starting at positions  that are uncorrelated with their initial positions.  It is sometimes said, if you see these fast and slow modes,  there are people who look purely at the time dependence  of the diffusion process and say, oh, you're seeing caging.  And the notion in caging, which is perfectly legitimate possibility,  is that we have a pro-particle.  It is somehow confined into a region of space  where it can move rather rapidly, but it can't get very far.  And if it wants to move a large distance,  it must, for example, hop to a new region  where it's happy to sit, and the hopping process is slow.  Now, if you do have caging, if you have regions  that are in some sense low potential energy  or dynamically restricted but free to move within,  then over short times, you'll see motion within a cage  and that long times, you will see the motion,  the slower process in which things hop from cage to cage to cage.  Now, once the object gets here,  it still does have the fast motion within its new cage,  but these positions and these positions are uncorrelated,  and therefore this motion does not contribute to the spectrum.  Having described caging, I would like to emphasize  that the time dependence measurement  gives you absolutely no evidence  as to whether the caging interpretation is correct.  Question?  What happens to the processes when it's moving  from the confined region and moving slowly when it's hopping?  What happens between here and there?  How does that show up in the time scale like that?  The slower process, how does that show up in the time scale?  The slower process when it's doing this is the slow mode.  That is, there is a part of the position of the object  that stays correlated with its initial position  as long as the object is trapped here.  And only when the object has done a bunch of hops  does the last piece of its initial position get forgotten.  Thank you.  Okay?  So that is the question.  The reason you don't see anything  is that light scattering spectroscopy, a single spectrum,  only gives you information about motion on one distance scale.  Now it's a Fourier scale, not a linear distance scale,  so you're looking at the relaxation of a spatial Fourier component  of the concentration.  That's why you can see both modes in the light scattering spectrum.  However, if you just look at a single queue  and measure the spectrum as a function of time,  which is how the experiment is usually done,  you are not directly generating any evidence of hopping.  Indeed, if you go back a few chapters,  there is this paper by, I believe it was Nemoto,  the collaborators, where they look at light scattering  and sedimentation.  Sedimentation is intrinsically a long distance process,  light scattering, light wavelength distance.  And what they demonstrate is that the long distance motions  are faster than the slow distance motions,  which is sort of the negative of caging in some sense.  Okay, so we have said, at short time,  you are seeing all of the relaxations put together  in a light scattering spectrum,  and at long time, the light scattering spectrum  only shows you the slowest relaxation of modes.  You will sometimes encounter a literature error,  and the literature error claims that if you look at the light  scattering spectrum at short times and measure the initial slope  or whatever, you are purely and exclusively seeing the fast modes.  That is complete nonsense. It is simply not true.  Okay, so if there are several modes,  can we just measure them directly?  And that is answered in section 9.6,  and 9.6 discusses the spectrum with multiple modes.  Now, we will come back to discuss multiple mode spectra again  when we discuss optical probe studies of hydroxy-probe,  cellulose solutions.  However, there are so many studies of HPC solutions  that they are grouped all in their own subchapter,  their own section, and we will get to those separately.  And we start out with studies by Bremmel.  And what was done was to say,  well, we will take solutions of sodium polyacrylamide,  and it is a polyelectrolyte,  and we will put in probes,  and the probes that we will put in  are polystyrene latex or hematite particles,  and we look at the spectrum.  And what was found was that there were two spectral modes.  The two spectral modes were both fairly close to exponential.  They could be characterized separately  in terms of a diffusion coefficient.  But what was observed under various conditions  is that if you have several modes,  life can become much more complicated.  And you can have modes whose relaxation rate does this  and modes whose rate does that.  That starts to look familiar.  That's the polyethylene oxide thing.  And modes for which you only see this,  and well, you could always think it might do that.  And so you have two modes visible in the spectrum,  and the behavior of the two modes is different in different cases.  Now, having said that,  there is now a possible explanation,  or partial explanation,  for some of the polyethylene oxide data.  Namely, if you have a spec system  that's giving you two spectral modes,  it's possible that under some conditions  one mode is dominant,  under some conditions the other mode is dominant.  With more limited anyhow technology  that was available in the early 1980s,  which is when the polyethylene oxide data was done,  well, sometimes you basically see this  as the dominant mode in the spectrum,  and sometimes you see this.  And if you examined the same system with modern technology,  you'd find that two modes were present,  and one or the other was more prominent.  Okay, let us consider to advance  to some additional work by Bremel and Dunstan,  and we are looking at polystyrene sulfonate spheres.  What is the point we say polystyrene,  and you will notice there,  there are carboxylate modified polystyrene spheres,  or this modified or that modified,  and in the case in question, they're in a copolymer.  Okay, why do I mention the modification?  Well, it's like this, you have a sphere,  it's made of polystyrene.  Polystyrene is not at all water soluble.  If it were water soluble,  you'd pour coffee into your cup in the morning,  and the cup would dissolve.  And it would be full of polystyrene,  which is not necessarily something  you want to drink huge quantities of,  not if it's water soluble suddenly.  And the answer is in order to get this stuff into water  and persuade it to stay there,  what you have to do is to charge up the surface,  and the charge up the surface is usually done  by chemical surface modification.  So for example, you have,  there's a carboxylate group,  it's an organic acid group,  and it's been fully ionized,  so you add a little base,  or that was done by the manufacturer, really.  And now you have these little charged spheres,  and since the spheres are charged,  they repel each other,  and they're happy to stay in water,  and they're a real solution.  Okay, so what Bremel and Dunstan did  was to look, and they see their modes again,  and they see things that look like this,  except, well, perhaps it doesn't go on quite as far.  And then they could say something,  they do two things.  The first thing they do is to say,  gee, the diffusion coefficient can be used  to calculate a microdiscosity,  a data micro, this is the Stokes Einstein equation,  but it's being written where the variables I measure  are the particle size and the diffusion coefficient,  and the quantity I calculate from them,  and K and T and 6 and pi,  the quantity I calculate from them is the microdiscosity,  the viscosity that would explain the diffusion coefficient.  And what was found for these experiments  was that the microdiscosity for the slow mode  appeared to be larger than the micro,  the orthodox viscosity of the solution,  and the microdiscosity that corresponded to the fast mode  was clearly less than the viscosity of the solution.  That is, if you have two modes,  you have two diffusion coefficients  that you can infer from them,  and because you have two diffusion coefficients  that you can infer from them,  you can calculate a microdiscosity for each one,  and since the two curves are not parallel at all,  you get two different behaviors for the microdiscosity.  The next thing that they did was to say,  well, we will look at several scattering angles,  and because we are looking at several scattering angles,  the wave vector Q, the scattering vector  in the light scattering experiment, changes.  Now, the reason that's significant  is the thing that you're measuring in light scattering,  the dynamic structure factor  depends on the scattering vector.  What does the dynamic structure factor tell you?  Well, we wait a while,  and eventually there is a fluctuation in the concentration,  a cosine-useoidal fluctuation,  that has a wave vector Q,  and the wave vector Q corresponds to the scattering vector  of the scattered light.  Now, actually, these fluctuations go on all the time.  They occur simultaneously at all wave vectors,  not just one,  but if you wait for a moment when the scattered light  is really bright,  you can correctly infer that the fluctuation  of the correct scattering vector  to get light out of the laser beam  and to your detector,  that fluctuation must be fairly large.  And now you wait and ask what happens at later times.  And on the average, you have to do this many times.  That's what the correlation function measures.  The size of the fluctuation on the average  relaxes back to zero.  It doesn't do it all the time, relaxing back to zero,  or it could never get big.  It fluctuates, but on the average,  if you wait for moments when it's large,  the fluctuation relaxes back to zero.  And the relaxation is the scattering vector.  If you change the scattering vector,  you're changing the distance scale  over which you're watching emotions.  Now, since you're looking at a concentration fluctuation  that looks like this,  particle emotions over distances  that are significantly less than this wavelength  contribute to relaxing the fluctuation.  And the particles can move various different distances,  but if they start at a maximum  and head off towards the minimum,  they reduce the size of the fluctuation.  So you're not looking at exactly emotions on a single distance,  but there's a distance scale in there.  So what happens?  What happens when they did this experiment?  There are two modes, a fast mode and a slow mode.  And what they found was that the diffusion coefficient  of the slow mode is independent of scattering angle.  And that is the behavior you get  if you have a process that is purely diffusive.  A single diffusion process will give you a relaxation  that goes as e to the minus the diffusion coefficient q squared t.  That's a single exponential relaxation.  The relaxation rate goes as q squared.  But if you pull the q square out  and ask what is the diffusion coefficient,  the diffusion coefficient is the same on all distance scales.  And so what they said is that their slow mode process  goes as q to the zero,  and that implies that their slow mode process  is in fact diffusive.  The fast mode process, however, was quite different.  They measured the apparent diffusion coefficient  at different scattering vectors.  A larger scattering vector corresponds  to a smaller typical displacement.  Large q is short distances.  And what they found is that as they made the distance scale  shorter and shorter, the particle motion was faster and faster.  The important issue is that since they're saying  at shorter distances, the particles move more rapidly,  we are not discussing a diffusive process here.  We're discussing something rather different.  The other issue is, going back,  the slow process is q independent.  And so, gee, that actually speaks to Cajun models.  The reason it speaks to Cajun models  is that if you have a particle that does Brownian motion  by parking in a cage,  hopping to another cage,  parking in a cage,  hopping to another cage,  and this random hop, hop, hop is the random walk  that gives you the slow diffusion.  If you look at very short distances,  something should happen to that slow process.  Because at short distances,  the slow process hop, hop, hop  starts to become visibly discrete.  Well, that's not what you see.  What they see is that the slow process,  whatever it is, is q independent.  Okay.  Let us chug ahead.  That's Bremel and Dunston.  And we'll look at another set of experiments.  And the new set of experiments  are due to delphenol that we collaborate with.  And the notion in the new set of experiments  is we will look at a polymer.  And the polymer they looked at  was carboxy-methylvalyloids.  And the material they looked at  was fairly high molecular weight.  And as a result,  the radius of gyration of the polymer  happened to be about 50 nanometers.  Question?  Is carboxy-methylylvalyloids  used in chromatographic columns  for adsorptions?  Um, you might...  I actually don't know.  The short-form answers know  this stuff is water-soluble,  so it's not used in column.  But a lot of these should remember,  this is now a synthetic issue,  a lot of polymers,  yeah, they're water-soluble,  but then you do cross-linking,  and instead of having a linear chain  you form blob that is big, big, big,  and it ceases to be soluble  in a practical way,  and now you have a column.  But this is not cross-linked.  And what they did was to take spheres  of three sizes,  17, 47,  and 102 nanometers.  And guess what?  They have here spheres  that are considerably smaller than Rg,  and they have at the other end  spheres that are considerably larger than,  well, twice Rg,  and they have spheres that are about the same  as Rg, so they have three sizes of spheres,  and they compare the diffusion  of the three sizes of sphere  in the same system.  And you are now comparing  the diffusion of objects  that are smaller than the polymer molecule  the same size as or larger than.  Now that's a perfectly reasonable comparison  viewed abstractly to make.  You should, however, notice  that there are theoretical models that say  you have a polymer solution  and it's fairly concentrated.  You have these long chains  and they are entangled.  And once they're entangled,  the model says,  well, it doesn't matter how long they are  because the ends don't do very much  except from forward or backwards.  And the interesting distance scale  becomes this distance scale,  so it's now xi, and xi is sort of the size  of a hole in the polymer mesh.  And if you say that's true,  then the interesting comparison is  between objects of different sizes  and the size of the hole,  and the hole is much smaller than a polymer chain.  The sort of limit,  why do we know the hole is much smaller  than a polymer chain?  Well, suppose we have a dilute solution  that is just on the edge of entangling.  Okay?  In that case,  the distance between entanglement points  can't be much larger than a polymer chain  because if it was,  this polymer chain would only be entangled  at one point.  There wouldn't be a mesh work.  So the entanglement notion  is the lowest concentration  at which you can possibly have entanglement  is approximately when the radius of the chain  is the same as the distance  between entanglement points.  And at all higher concentrations,  xi becomes smaller and smaller,  the radius of gyration shrinks,  but much less.  And in that case,  the interesting length scale and solution  is much smaller than the polymer chain.  Well, having said that,  they did the experiment.  And the reason the experiments appear in this chapter  is the spectra they found were bimodal.  Their spectra were adequately fit  by a single exponential,  or two exponentials.  That is, each mode was close  to being a pure exponential.  That says that each mode, separately,  could perhaps be viewed as a diffusive process  though there must be some complications  we are not quite seeing.  Nonetheless, they do see two modes.  Now, the reason you have to worry about saying  it's pure exponential is as follows.  Suppose semi-log plot.  Here's the dynamic structure factor.  Here's time.  This is a semi-log plot.  Suppose the actual spectrum looked like that  on a semi-log plot,  meaning the relaxation is not a pure exponential.  It occurs more rapidly at fast times  and on the semi-log plot less rapidly at long times.  Suppose you had a spectrum like that.  Suppose you charge in,  and not paying much attention to the data,  fit this to a single exponential.  Well, you could say it's an approximation,  but there's a problem with the approximation.  If I just fit to the measurements up here,  I get one slope.  If I just fit to the measurements down here,  I get a quite different slope.  And the pure exponential fit,  unless you're careful,  and I'm not saying the paper we're talking about  was not careful,  I'm just showing where there's a hazard here.  The hazard is that if you fit on different time scales,  you get different slopes for the exponential.  And that's because the exponential is somewhat cooperative.  You can always draw a straight line through a set of points.  It may be a foolish straight line  because the points don't resemble a straight line,  but you can always draw that straight line,  and you will get a result.  And so what happens when they do this?  Well, first of all,  what they found was that there are two modes,  and for the small spheres,  the fast mode was dominant.  For the slow spheres,  I didn't say that right, sorry.  At low concentrations,  they found the fast mode was dominant.  At high concentrations,  the slow mode was dominant.  And so far, we've only talked about  the intermediate-sized spheres.  I said that backwards the first time.  We can draw a picture showing this.  The other thing they found,  so we'll start with the small spheres,  and there are two modes,  and the two modes both slow down as you increase.  This is the diffusion coefficient corresponding to the mode.  This is the concentration.  And the first thing they find is that for the small spheres,  the slow down is over the observed range of polymer concentrations.  It's, oh, a factor of three or six.  I'm being approximate.  For the large spheres,  the figure is in the book,  and you can look at this,  and for the large spheres,  the slow down is, oh, maybe a factor of 10.  And for the slow mode,  it's so number like 300 or 400.  It's more than two orders of magnitude.  And so what you're saying is,  for the small spheres,  the two modes are affected only modestly.  Well, actually, it's factor of three.  It's plenty to see by the polymer.  For the large spheres,  the effect of the polymer is vastly more dramatic.  And there is a crossover region  when the spheres are about the same size as the polymer molecule.  And it really does appear, though,  of course, they only have three sphere sizes.  It does appear there's a transition between  the sphere is much smaller than the polymer  and the sphere is much larger than the polymer.  Now, if you really want to say that statement  and say there's a transition,  you would really want more than three sphere sizes  for the simple reason that you ask,  well, is it a continuous rollover  or is there a real change from behavior class A  to behavior class B?  Does the rollover actually occur when r is equal to rg  or is it someplace to one side or another?  And if you want to settle that question,  you'd have to use plenty of sphere sizes,  and then, if you're lucky, life becomes transparent.  Well, that's what Karel Stroletsky did.  He was my PhD student for his doctoral thesis.  He did this in HPC, not carboxy-methylcellulose,  but he used a whole bunch of sphere sizes  and he, in fact, found exactly what is being implied  by the very nice experiments of Delfino.  Namely, there is a behavioral class  for spheres that are smaller than rg.  There is a different behavioral class  for spheres that are sort of larger than rg.  There are several ways of characterizing  the size of a polymer molecule,  so let's say it's a bit imprecise  as to exactly what you call the size of the transition.  But, in fact, the transition is fairly large.  It occurs approximately at the size of the polymer coil,  and therefore, the solution, in my opinion, unsurprisingly,  has a longest characteristic length scale,  which is the size of a polymer chain.  Well, that's fine unless you believe entanglement models  because entanglement models say that the characteristic  longest length scale is this much shorter length scale psi.  Yeah, that's not my problem.  Okay.  We now advance to section 9.7.  And 9.7 talks about probes in polyelectrolyte solutions.  Now probes in polyelectrolyte solutions  are something that is not covered very heavily by my book.  At some point I decided I have to stop someplace,  or the book will get bigger and bigger,  and the publisher only gave me 500 pages.  And the time required to complete,  instead of being a fixed date within my lifetime,  will be divergent, and I'll never finish the book.  So I had to stop at some point.  And I remind you, things that we stopped before getting to  include rods,  polyelectrolytes,  liquid crystal polymers,  block copolymers,  salts, oh, thread like micelles,  there are solutions, small molecules,  but they form dynamic structures in solutions  that are long and thready,  that look a lot like polymer chains,  except they're not covalently cross-linked,  and they can go straight through each other.  Now, having said that, there are a bunch of things  we did not cover.  Nonetheless, I do briefly mention some of the results  that do exist on polyelectrolyte systems,  and I bring up some of the polyelectrolyte results  so that you know they're there.  So what did we say about polyelectrolytes?  Well, the first issue is there's some extra variables  in a polyelectrolyte system.  That is, if you have a polyelectrolyte,  the first material we worked on,  polyacrylic acid, could be a polyelectrolyte,  but we worked with the non-neutralized material.  These are results of tyholin and I,  in which here are the carboxylic acid groups,  and they're mostly non-neutralized.  Well, some of them spontaneously ionize a bit.  But what you could do is add base to the system,  and now you have ions on the chain,  and you have sodium ions,  and you added sodium hydroxide,  and guess what?  You now have neutralized polyelectrolytes.  Well, how much base did you add?  There is a percent neutralization  that determines what fraction of those groups are charged.  Also, you could have a salt concentration  determined by its ionic strength, I,  which for monovalent ions is simply the molarity of the solution.  We can go on.  There's more complication,  but we aren't doing polyelectrolytes,  and there are extra variables.  And having said, there are extra variables,  there are all sorts of experiments you can measure.  Okay.  For example, we will take,  I believe it's figure 9-25,  and we can measure the diffusion coefficient.  We're looking at polystyrene spheres  in a partly neutralized polyacrylic acid.  This one happens to be 60% neutralized.  It's a nice round number.  It is, if I recall correctly,  a 600 or a slightly smaller polymer.  And we measure the diffusion coefficient of spheres  as we add salt to the solution.  And so we start with no added salt.  That's, now you may say,  gee, that's simple, isn't it?  No, that's actually quite hazardous.  The reason it's quite hazardous is, A,  water ionizes a bit,  and so you really can never get less than in water,  about 10 to the minus 7 molar ions,  because the water is ionized.  Furthermore, water absorbs carbon dioxide from the air,  unless you're really rigorous about dealing with issues.  And therefore, the pH of water,  if you actually look at it under these conditions,  is more like 5, meaning you have something like  10 to the minus 5 molar hydrogen ions  and other good stuff floating about.  And so, unless you're extremely careful in your work,  it's very hard to avoid introducing  at least traces of small ions.  And therefore, if you want to know what's going on,  the experimental process is to be sure to add  at least tiny amounts of salt,  sodium chloride, potassium chloride, whatever,  10 to the minus 3 molar,  10 to the minus 4 molar,  and then you are probably closer  to knowing what's going on in the solution.  Nonetheless, that's I equals 0.  We didn't add anything.  And as we increase the amount of salt,  well, at 0 polymer concentration,  this is D versus polymer concentration,  changing the salt concentration does almost nothing.  But at higher salt concentrations,  with increasing ionic strength,  those curves flatten out.  And as you increase the salt concentration,  the polymer is less and less able  to retard the motion of the spheres.  Why? Well, that's a good question.  And you can come up with all sorts of explanations.  Gee, you added salt, the polymer is less rigid.  You added salt to the electrostatic interactions  between the polymer and the sphere are weakened.  You can come up with all sorts of explanations.  You notice it started to get very complicated.  You can also look at Stokes-Einstein behavior,  and you can compare the product diffusion coefficient  times measured viscosity of the solution  with what you would have in pure water.  If the Stokes-Einstein equation worked,  this number would stay 1  as you run up the polymer concentrations,  increase the salt concentration.  That's not what happens.  In fact, what happens is you have  non-Stokes-Einsteinian behavior,  and the viscosity goes up,  the diffusion coefficient goes down,  but the matrix polymer is more effective  at increasing the viscosity  than it is at retarding the diffusion,  and this number is larger than 1.  There are systems where it's less than 1.  The statement, it's larger than 1,  has the attractive feature that you simply cannot claim,  oh, it's just the spheres are aggregating,  there's stuff sticking to the spheres.  That would give an effect of the wrong sign.  So as we proceed, we see non-Stokes-Einsteinian behavior,  and we ask, well, are there ways to make  the non-Stokes-Einsteinian behavior less?  And the answer is that this number tends towards 1,  it doesn't get there.  If you make the spheres larger,  if you make the ionic strength larger,  or for that matter,  if you weaken the polyelectrolyte effect  by reducing the percent neutralization,  if you do any of these things, or all of them,  the degree of non-Stokes-Einsteinian behavior goes down.  Okay, there is another theoretical piece in here,  and the theoretical piece is there are people who will claim,  this is a claim, you can find it in the literature for yourself,  that you get non-Stokes-Einsteinian behavior  if the probe size is less than, or maybe much less than,  all of the characteristic lengths and solution.  However, if the probes are big,  the claim is that you must be getting Stokes-Einsteinian behavior.  Well, that's very nice, but if you believe that,  gee, if you see spheres and they're showing  non-Stokes-Einsteinian behavior,  the spheres must be smaller than the length scale.  How big were the spheres we used in that experiment?  Well, there were a series of sized spheres from 21 up to 655 nanometers,  yes, diameter, and based on our measurements,  since we were seeing non-Stokes-Einsteinian behavior,  except perhaps for the largest spheres,  the longest length scale in these solutions  must be greater than 300 nanometers.  Certainly, it must be greater than 50 nanometers,  and it appeared it had to be greater than 300 nanometers.  Well, that's a very long length scale.  That's not what you would necessarily have expected,  but that's in fact what's there.  Okay, there's not a figure I drop in,  and the figure is, again, the work of myself  and its work with LeCroy and Yamburt,  and its figure, if I recall correctly, 9-27.  And if you look at the figure, yeah, it's B versus C again,  so if you look at the figure, yes, it's a semi-log,  it's a linear semi-log plot, if you look at it,  you realize D doesn't change very much,  and we have a whole lot of points,  and the points are concentrated on finding  what is very clearly the initial slope.  And in some cases, a simple exponential slope is adequate,  in some cases, a linear slope fit was adequate,  and we worked very hard.  We used spheres of size 734,  and if I recall correctly, 95 nanometers.  The measurements you see are for two sizes  of polystyrene sulfonate,  but we had a fair amount of sulfonate,  and we look at several molecular weights of polystyrene sulfonate,  and the question is, why did we work so hard  to determine the initial slope, which is what we did?  We determined the leading linear effect of adding polymer,  and how it slows down, a probe diffusion.  The reason we did this is that there was a theoretical model  that was mycoutic,  it's based on the Kirkwood-Reisman picture of single chains,  and what it says is, here's a probe sphere.  We are in dilute solution,  we want to find the leading slope, which is the dilute slope,  so here is a polymer chain.  The sphere moves,  and as the sphere moves, the polymer chain gets to respond,  but it's somewhat constrained.  It can move, it can translate,  so it has some translation.  It can notice this side of the chain  is closer to the sphere than this side is,  so it can rotate,  and it has some rotation vector,  omega, which is perpendicular to the blackboard,  and we can describe the polymer chain,  lowest approximation,  as a sort of bag of frictional beads  that translates and rotates.  But we can now calculate the hydrodynamic interaction  back and forth between the sphere and the beads,  and we can therefore calculate quantitatively  how effective the chain is at slowing down the spheres.  And the point of this was to see,  well, can we do the calculation  and does the calculation work?  And the direct calculation ends up with a single free parameter,  there's a way of beating that limit,  which we did eventually,  but the point is we could actually calculate the slopes  and we get the right answer.  So that particular paper is a theoretical model test.  It's an extremely important theoretical model test  because it tests the hydrodynamic interaction theory,  and it works.  Okay, I'm doing fine on time since I started a bit late.  Okay, let us step ahead again,  and the step ahead again is to say  there is something called solvent quality.  And you can sort of look blindly and wonder, what is the solvent quality?  Suppose we have a polymer chain.  Here's a polymer coil.  We have states of the system in which here comes a polymer coil,  here comes the polymer, very crudely drawn,  and here comes another piece,  and the two are in more or less direct contact.  Yes.  And you might ask, does this polymer prefer to have another piece of polymer coil as neighbor,  or does it prefer that the two polymers stay well apart from each other,  and it has solvent molecules as neighbors?  It isn't an absolute, we must have one or the other.  It's a thermodynamic preference in contact.  And the net result is, if the polymer chains prefer to be next to each other,  the polymer coil stays fairly compact.  If the polymer prefers to be in contact with solvent,  well, you have different pieces of the polymer in this picture  have to push apart from each other so they don't bump into each other as much,  and suddenly the polymer coil gets larger.  And if we choose our solvent, we can have what is called a good solvent,  in which the polymer prefers to see neighboring molecules be solvent.  We can have what is called a theta solvent, in which there's sort of indifference,  and we can also have a poor solvent.  An extreme example of poor solvent behavior are protein molecules.  Protein molecules are nice and neatly folded and have particular groups on the outside  and are lumps because proteins view water as a very poor solvent.  And so they fold up tightly.  And they fold up tightly so that all of the aliphatic groups, the hydrocarbons,  try to hide on the inside of the polymer chain.  Of course, they can't all manage to do this.  And the groups on the polymer that are charged, the carboxylic acid groups,  the amine groups, the charge, sit on the outside and be charged.  And so the polymer has charges on the outside or tries to,  and things that look more like oil on the inside, and it doesn't expand at all.  And this is why polymer molecules are stable, because water is a very poor solvent.  Now, you might think, aren't proteins soluble in water?  Yeah, they're still soluble because they do this, but it's still a poor solvent effect.  Okay, so we ask what happens if we measure the diffusion coefficient versus polymer concentration?  And we take the same polymer and we put it in two different solvents.  Well, the answer, there are two sets of experiments that are both in the book.  And one approach to this is we will change the chemical identity of the solvent.  And if we do that, we see a theta solvent behavior, and we see a good solvent behavior.  And the diffusion coefficient in the theta solvent just falls stretched as a pure exponential,  e to the minus alpha c to something close to one.  And then the good solvent is e to the minus alpha c to the nu.  And g at high concentrations, we're out here in concentration,  the theta solvent slows down the polymer more effectively than the good solvent does.  Rather, the polymer and the theta solvent slows things down more effectively.  Now, there's another way you can change solvent conditions without having to change the solvent molecule.  And the alternative way to do things instead of changing the solvent is to change the temperature.  Because there are systems that approach being good solvents at one temperature.  But if you change the temperature enough, you approach a theta point,  and at the theta temperature, you get a different behavior.  And those were experiments that were done by Delphine, Coleman, Neal and I.  And once again, in the theta system, you see something that's sort of like a straight line.  And in the good system, you see something that's sort of like this.  And oh my, what's occurring? Well, the answer is you have different good and theta behavior.  And from a bit it appeared, we were at the first bit of enthusiastic,  g we found a solution, we found a prediction that is actually made by my model of polymer dynamics.  And you see exactly what you expected to see.  And then we realized, but if you look at the alternative models,  all the models we found actually show the same prediction.  So this very pretty experiment doesn't really tell you anything,  except that in some sense the models agree with each other.  And the last thing I will discuss.  There was a period in the late 80s, early 90s, when I would go around giving speeches on probe diffusion  and what you saw and what it appeared to imply about polymer dynamics.  And so I would go here or there or someplace else and I would give my remarks.  And someone would stand up and complain that I had not reduced my measurements relative to the glass temperature.  Now hiding behind this not at all innocent question is a very complicated piece of theoretical issue.  But a piece of the notion is as follows.  Here is a polymer solution, a polymer and solution.  And I am representing the polymers looking like a pearl necklace.  The rate at which the polymer can move is determined by a drag coefficient,  a resistance to motion of the individual beads.  That's the same as the Stokes law drag coefficient,  but it refers not to the whole chain, but to the little piece.  Now if you work in polymer melts, the issue is that G, as you change the temperature up or down,  the drag coefficient of the individual polymer beads slows down or increases.  The drag coefficient goes down or up.  And as you change the temperature there for the whole dynamics of the system,  changes on its time scale, not because something particular is happening to the whole chain motion,  but because the drag coefficient of the individual beads is changed.  In fact there is a whole experimental approach called time temperature reduction.  And the notion in time temperature reduction is that you can change the time scale on which things are happening  by changing the temperature and therefore you can do what appears to be a measurement  over a very wide range of frequencies, response frequencies,  by doing a measurement over a much narrower range and doing it over a series of temperatures  and at each temperature the same behavior occurs at different frequencies  and therefore by changing the frequency of the temperature you can compress all of these behaviors  so you actually look at them experimentally in one relatively narrow frequency regime.  Well that's time temperature dependence.  Now how does time temperature dependence come into our probe diffusion work?  And the answer is that this drag coefficient was said to depend and it was a very vague question.  It wasn't a detailed analysis, exponentially on the concentration  and therefore as I changed the concentration I was changing the drag coefficient.  Now what does this have to do with the glass?  Well I'm going to draw another picture.  Two pictures.  And so we will plot a property that depends on how easily the beads can move,  namely the viscosity, the resistance to pouring.  And I will plot this as a function of temperature.  And if I plot this as a function of temperature, what happens is I cool the system off,  the viscosity increases and then it increases very rapidly indeed  and eventually I get to something called glass.  Now the picture I've just drawn is for a polymer mept.  And the notion is that as T approaches this temperature Tg, the glass temperature,  the viscosity sort of diverges.  Now it doesn't really diverge, but if the viscosity is 10 to 14 times the viscosity  that you had when you had a simple fluid melt, it doesn't pour.  It's practically but not quite a solid.  That's the glass temperature.  If you look at large numbers of behaviors, you can say the interesting variable is T minus Tg,  the distance from the glass temperature.  And if you want to compare how different polymeric melts are behaving,  the sensible comparison is not to compare them at the same temperature,  but at the same temperature distance away from the glass temperature.  Now you could also be a little fancier and say maybe you ought to normalize like this  because there are systems that have very low glass temperatures  and systems that have very high glass temperatures,  but that is the glass temperature idea.  The notion for polymer solutions is that if we look at 1 over D, which is sort of like a viscosity,  and we look versus polymer concentration, 1 over D is doing this,  there is some glass in some sense that is eventually things don't move or something happens,  and therefore I ought to reduce relative to this glass temperature.  So Carol Quinlan and I did this experiment,  and what we said is, well, how do we reduce relative to the glass temperature?  And the answer is we will do measurements of D at a series of temperatures,  and the reason we will do this at a series of temperatures is as follows.  This picture here actually leads to something called the Vovogl-Folker-Tamann equation.  It says D should go as sum D0, and then there is a temperature T  because the basic diffusion coefficient scales as temperature,  and then there is an e to the minus sum constant,  there is a constant, over T minus Tg.  And you notice this thing has the feature as we approach,  oh, I should put a plus sign there, the sign is of course not meaningful  because A is a signed number.  But the issue is that as the temperature approaches the glass temperature,  this factor becomes extremely large, and therefore I will put the minus sign in its diffusion,  and therefore this number becomes large, and therefore the temperature goes to zero,  as you approach the glass temperature.  Well, if I am doing this in polymer solutions, the claim is at each concentration,  because Tg is a function of polymer concentration,  as I change the polymer concentration I am changing the glass temperature,  and therefore instead of quoting everything at the same temperature,  I should quote all of my measurements at the same T minus Tg.  Okay?  So some place I should establish Tg for these solutions,  and if I quote measurements at different concentrations,  I should also be at the same time be comparing measurements at different temperatures  rather than at the same temperature, which is what I have been doing.  Okay, well this equation has a feature.  It makes a prediction as to how D should depend on temperature, doesn't it?  There is however another prediction as to how D should depend on temperature,  namely D should, if we compare with the D with T over eta of the Solmon,  we should get a straight line.  Yes?  Okay, so we did all the experiments.  And the first thing that happened is that if you measure D in a series of temperatures  and fit to this form,  you discover that Tg is extremely low, like minus 100 centigrade.  That's not a physical temperature, of course, for water because it freezes.  However, what we found, you do the temperature dependence  at a series of different polymer concentrations,  and you ask how does the apparent Tg,  whether you believe this makes any sense or not, depend on temperature,  and the answer is the apparent Tg is independent of polymer concentration.  And therefore, if you believe that you were supposed to reduce relative to Tg,  well, we did the experiment,  and the answer is Tg is the same at all concentrations contrary to what was being said,  and therefore the reduction relative to Tg wouldn't do anything.  The second thing we said is, well, we can look at D versus T over eta sub s,  and Gd is linear in T over eta.  This is eta of the solution, or eta of the solvent.  It doesn't really matter which because they just track each other.  And if you look for curvature here,  well, the curvature is slightly larger than the scatter of the points,  and so you could say there's a little curvature,  which, by the way, is independent of polymer molecular weight.  And the best we can say is, quite good approximation,  D just scales as temperature over solution viscosity linearly,  and there's no sign of any deviation from linear behavior.  Maybe there's a slight sign, it's sort of a one or a two percent effect,  and your measurements are accurate to a bit better than one percent.  So you're sort of at the point where you are leaning hard on your data claim  that you're seeing a deviation from what Stokes Einstein would tell you to expect  for the temperature dependence.  So that's temperature dependence.  However, we were challenged to do a reduction relative to the glass temperature,  and what we showed was that the relevant glass temperature for these experimental measurements  is independent from polymer concentration.  And since it's independent from polymer concentration,  the reduction relative to the glass temperature doesn't do anything.  And we could also say, ask, well, what is the dependence on solvent viscosity,  which is temperature dependent, and the answer is D is linear and T over A is a solvent,  and therefore any hydrodynamic picture or any picture that says the solve viscosity  controls things, works just fine.  So that's it for temperature dependence.  And we are now at the end of today's lecture.
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