23 Viscosity, viscoelasticity
Phillies, George
Lecture 23 - Viscosity, viscoelasticity. George Phillies lectures from his book "Phenomenology of Polymer Solution Dynamics".
 Classes in Polymer Dynamic. Based on George Philly's book, Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011. And today, this lecture is lecture 23, Viscosity, Introduction to Viscoelasticity, the Temporal Scaling Onsots. I'm Professor Philly's, and today we're going to finish off our discussion of viscosity and push on to discuss viscoelasticity. Let me recall where we left off roughly speaking. We left off in chapter 12 discussing figures 12.22 to 12.24, which are treatments of the molecular weight dependence of the viscosity of the solution. Now, there's actually a fairly considerable number of measurements of these, and they're actually fairly consistent, namely what they find, what is found with considerable consistency in many systems, is that the viscosity depends on the molecular weight as a stretch exponential and molecular weight. And that's brought out quite emphatically by, for example, figure 12.24. 12.24 are measurements that actually appear in Doyle and Edwards, who report them as measurements done by Onogi et al. The measurements are a little more complete than the measurements in the original paper, actually. Having said that, an interesting feature here is that if you go back into the literature and ask what people said about these measurements, you can find people who say, we have measured the molecular weight dependence of the viscosity, and as predicted, we see a power law dependence of the molecular weight. There actually are measurements that find power law molecular weight dependence. For example, as we discussed earlier, POW at all measurements, which push out to the melt, find close to the melt a power law dependence of viscosity on molecular weight, and volume fraction one, log log plot, what are rather clearly straight lines, viscosity changing as a power law on M, and if you actually say, we'll stay fairly close to the melt and we will exclude points that seem to be deviating from the original straight line, you actually could find in POW at all M to the 3B8. However, most of the measurements you find, which are done at considerably lower concentrations, do not find power law behavior, they find stretched exponential behavior. An interesting example of this is provided, and this is also, in essence, stretching over to the next section, which is a discussion of topology, are the measurements like Grace Lee and L, and Grace Lee look at polymers, and they look at linear chains, and they look at four-armed stars, and they look at six-armed stars. And if you look in at the results they show as a function of concentration, for the linear chain, and their linear chain is a molecular weight of 1.66 megadolptin, as a function of concentration, a log-log plot, they find what is very clearly a straight line, you look at it, it's clearly a straight line, and it's quite clear that you are seeing power law behavior. This is for the two-armed star, the linear chain, two-armed star. On the other hand, for F equals 4, and for F equals 6, very clearly on the same plots, these are chains of about the same molecular weight. This is about 1.95 megadolptin polymer, and the other is about a, what was it, it was about 1.44 megadolptin polymer. For the other two, you very clearly see stretched exponential molecular weight dependencies, these are fixed molecular weight, you're looking at concentration dependencies. If you look at the molecular weight dependence, where they had a considerable number of samples, they were looking at concentrations of about 300 gram per liter of polymer, extremely concentrated systems, and the molecular weight dependence was a straight line for the linear polymer, and was a stretched exponential, two stretched exponentials actually, for the star polymers. As a function, taken as a function of concentration, the upper figure, the linear chain showed the power law behavior, the stars showed a stretched exponential behavior, the viscosity of the star was, for the most part, less than the viscosity of the linear polymer. However, if you look carefully at the measurements against concentration, you can see a point up here, and so if the concentration was taken to be high enough, eventually some of the star polymers, as they're obliged to do given the functional behavior, did show a viscosity that was higher than the viscosity of the stars, of the linear chain, I mean. However, for the most part, star polymers, comparable molecular weights, are up to fairly high concentrations, less viscous than linear chains, and at fixed concentration as a function of molecular weight, again, the stars are less viscous than the linear chain is. Okay. You can also look at results of... We'll get this erased in a second. You can also look at results of Kajura and collaborators. They looked at three-arm stars. Their stars were a poly-alpha methyl styrene. They look at a series of concentrations and molecular weights. You can see the results in figure 12.26. And what they found was that, again, if you're at about the same concentration and about the same molecular weight, the viscosity of the star polymer, this is the F equal 3 star, is less than the viscosity of the linear chain. Now, why is the statement that the star polymers have a lower viscosity than the linear chain of some interest? Well, there are a number of models of polymer dynamics which assert that linear chains have extra modes of motion that are, in principle, not accessible to stars. In particular, for example, the reputation-type models say that if you have a linear polymer, it can creep through holes in a transient lattice like the snake in a bamboo grove. But if I am a star polymer, in order to make forward progress, what I have to do is retract by fluctuation one of my arms. That's difficult to do. I can push it out in a new direction, and now I can take a very modest step forwards. The net result is that one would expect that it is harder for star polymers to move with respect to each other than, at least if they're very concentrated, than it is for linear chains. That does not appear to be quite what we are observing. Question? Can it also be possible that since linear molecules are a linear instruction, they have better packing methods, therefore, they adhere closer to each other? That could be an issue in the melt. However, in the solution, the amount of packing is determined by the concentration. And the answer is that the concept, we make comparison and equal concentration for linear and star. The other thing you should recall with respect to, let us pack chains into a melt, is that I am a star polymer. I have one point where there are three chains coming together. There, if I'm an F equals three star, there is a narrow region around this where there are three of my chains close together. However, if I'm a large star polymer over most of my length, I look exactly like a linear chain. And therefore, this is the sort of effect. You might find interesting effects with small chains, but with large ones, you would not expect that to arise. Thank you. Okay. So we advanced figure, oh, discussion around figure 12.25. And the issue around figure 12.28 and figure 12.27 is suppose we take a polymer chain and we change the solvent quality. What happens if we change the solvent quality? Well, the expectation is that if we are in a theta solvent or close there too, which means we also have to be at the right temperature, we have a polymer chain in the solution. It would rather see itself as its neighbor than it would see something else. And therefore, if we increase the concentration, nothing terribly interesting will happen to the chain radius. I didn't say nothing at all. Because theta, you can't get perfect cancellation of effects. It's an approximation. On the other hand, if you were in a good solvent, we see what we saw in the chapter on dielectric relaxation, which is what has been used for most of these measurements. Namely, as we increase the concentration and look at the chain radius, the chain radius falls with increasing concentration. So these, however, are static properties. They're how big is the chain if we take a single snapshot? We're talking about viscosity here. Viscosity is a dynamic property. And we ask what happens to the behavior of the system if we change solvent quality and such and on. The first answer is, if we look at a series of chains of different molecular weight, the viscosity, as has been the case in most other systems, not all other systems and not at sufficiently high molecular weight, but concentration rather in those systems, the concentration dependence is a stretched exponential. And what we happen, fine, as we increase the polymer molecular weight and do our measurements, we find that the parameter alpha increases and the parameter nu falls. So there is alpha and there is nu. And both parameters are molecular weight dependent. The second thing we can do is to compare these parameters in a good solvent and in a theta solvent. And what we find is that in a theta solvent as discussed here, the parameter nu is from 0.8 to 0.954. And if we go, however, take the same material over to a good solvent, the parameter nu is 0.6 to 0.73. That is, we are going in and when we change the solvent quality, when we go into a better solvent, the parameter nu falls. The model that derives this equation, in fact, predicts this behavior, we will eventually get to it in later lectures. If you look at the figures, especially 12.28, this change of nu is physically quite visible. If you look at a semi-log plot, log viscosity versus concentration, for a theta solvent, the concentration dependence of eta is fairly close to a straight line because this is fairly close to a stretched exponential. But if you look at the measurements in the good solvents, you typically see a more pronounced curvature because you are looking at a stretched exponential rather than a pure exponential. Now, having said this, let us push ahead and consider the general features of viscosity. We have looked at lots of specific measurements and we did, of course, find that the viscosity depends on the polymer concentration and the polymer molecular weight, not to mention the one could put in the topology and one could put in the solvent quality. With respect to topology, there was a very long dispute, this goes back to a considerable piece, about the viscosity of ring polymers, but that was mostly melts, which the book does not cover. The issue there was, well, you can try to make melts, you have to ask, are the, you can try to make rings and put them into melts. Here is a ring, is it a nice open loop or during your synthesis process, did it get itself all tied in knots so that it's a ball? Under modern conditions, if you want to answer that question, you can resort to using DNAs since you can synthesize DNAs with absolute molecular weight control. But absolutely mean we have something that's 180,000 base pair, meaning its molecular weight is tens of millions, and the molecular weight is, well, there are fluctuations because there are several isotopes of carbon and deuterium in nature, a carbon and deuterium hydrogen, I should say in nature, but the molecular weights are totally exactly controlled, no fluctuation. You can also synthesize things, this is synthetic DNA, not stuff that occurs in nature, that is a star and completely mono-dispersed. You can also synthesize, it happens naturally, some viral DNAs do this. DNAs that are ring polymers that are again completely mono-dispersed. Electron micrographs of those rings indicate that the rings are completely open, they are tied up in knots at all. However, having said, the ring polymers are mono-dispersed, the star polymers are truly mono-dispersed, synthetic DNA comes that way, there is the significant practical issue that people have not done studies of DNA viscosity that are as extensive as the viscosity studies that have been done on synthetic polymers. Okay, having said this, let us look at the question of what we find. And the first point, which I am going to emphasize significantly more than was emphasized in the book, is that eta is not a universal function of C and M. That is, if you think that you can take your viscosity measurements and yes, there will be some chemical dependent constants and there will be things that depend on the concentration in the molecular weight or maybe the length of strands in the molecular weight. Some representation of amount of material and size of the chain. If you think that all of the chains are the same because of that and therefore there should be a universal function that gives you viscosity as a function of concentration, you are going to be disappointed. Instead, it is quite clear from the literature that viscosity is not a universal function of C and M. Instead, there are two distinct phenomenologies and in one of the phenomenologies, you see simply a stretched exponential which in measurements of Taeger goes up, Taeger and a few other people go out more or less to melt. And in another set of systems, you see a stretched exponential up to a point and then beyond the point, you see a power wall. It is worthwhile to recall that the same phenomenology this class is occupied by spherical colloids. Spherical colloids show the same behavior. The slope of the power law depends on the exact material. For spherical colloids, the slope is appreciably steeper than for linear chains. And if you look at many arm star polymers, the slope creeps up and makes a transition from linear chain behavior up to very nearly heart-sphere behavior. It should be clear that any model that explains this phenomenon is subject to the constraint that it must be of work for spherical colloids. Spheres can do a number of things, but form entanglement lattices is not really one of them. And form and reptate is certainly not one of them. And so we have the issue that the viscosity behavior might not be quite what you would have expected from some of your readings from some theorists. Okay. Furthermore, if you merge up to viscosities above about 10 to the 8, that's eta over eta-zero above about 10 to the 8. And most of those measurements are literature studies due to Drival, though there's some nice work due to Colby. However, if you look at that, it does appear that if you get above here, instead of continuing along on the power law, there is an upwards deviation. The nature of the upwards deviation is not quite certain. One might, however, for example, propose since we're at extremely high concentrations that it's some sort of a jamming transition in which things actually mechanically get in each other's way, and the chains can't slide across each other because they can't move after all their rigid overshort distances. They can't move sideways as easily enough to get through each other. However, that is a fairly specific, that's a fairly rare exception because there are not a lot of studies that have looked at this. In particular, if you look at predictions that eta should be some c to the x, m to the y, that prediction is very generally not true in polymer solutions. There are some exceptions. There is this family, but predictions that you can simply assume scaling are not sustained. If you want to say we have scaling, you need to explain why sometimes you have scaling and sometimes you don't. Otherwise, you can't just assume. Okay. So that is the sort of behavior we get. We might then ask, well, what other generalizations can we find? And one answer is, for a very large family of parents, we have eta over eta zero is e to the some constant alpha c to the nu, and now I'm pulling the molecular weight of dependence of alpha. We'll actually see graphs of that dependence in a few moments. But if we actually look at the measurements and ask, okay, we get this thing to fit, what do we see? Well, gamma is usually in a range around a half. And I can be more precise than that, but that's a reasonable approximation. In point of fact, if we say, oh, I should add something. We go from theta to good solvents. We go from theta solvents to good solvents. The parameter nu goes down, and what I will call alpha m, because it's alpha with m pulled out, what I will call alpha m, m to the gamma, what's left, what's not c to the nu, increases as we go from theta to good solvents. Generalization. Okay, so what behavior do we see for gamma? And the answer is seen in figure 12.30, where we plot alpha, which in terms of this equation is alpha m, m to the gamma. It's the coefficient of c. And we plot alpha against polymer molecular weight. Now, if I just showed you the points, your reaction would be, gee, that was a nice piece of graph paper, Professor Philly's. At least it was very nice before you stood way back and started firing the shotgun at it, and the points are all over the place. However, if you look carefully at the points, I have labeled the points by the chemical family of the polymer. So we have identified separately each set of homologous polymers, polymers that are chemically identical with different lengths. And if you do that, you see that for each single family of homologous polymers, alpha lies very much on a straight line. The straight line has a slope. So this is a log-log plot, so we say alpha goes as m to the power gamma. And what can we say about gamma if we look at all of the measurements? That is, all of the measurements were the same way. I've looked at a large number of different polymer molecular weights. And other than that, things are consistent. And the answer is that gamma is typically in the range 0.6 to 0.67. One can find a case where it's 0.5, if I recall correctly, 0.94, where it's quite large, and you can find a case where it's 0.51. If you look at this, you notice there are different slopes for different materials, and therefore it's not quite a universal constant. But for a given set of samples, the power law dependence is actually fairly sharp. This result is also implicit in the systematic review of Dervol and many collaborators, and all of the wonderful Russian data he compiled in one place for us, because what Dervol says is that C eta is a good reducing variable. That is, if you take all of these measurements and you plot them as functions of C times the intrinsic viscosity rather than C, so we have eta log log again, C eta here, and we find a whole series of different polymer molecular weights that the measurements lie on the same curve. Now, in order for the measurements to lie on the same curve, when you are doing this plot, it must be the case that eta is some function of C eta, which means that all of the molecular weight dependence of the viscosity has to be built into the molecular weight dependence of the intrinsic viscosity. You can't have any other molecular weight dependence out here someplace, because if you did, when you plotted viscosity for a series of polymers of different molecular weights, they wouldn't lie on the same line. They lie on the same line because viscosity is a function of C times the intrinsic viscosity. That statement, universal function, goes all the way up to the melt. However, it is an important fact that the intrinsic viscosity depends on M as a power, and therefore this equation says that the viscosity is a function of, what is it a function of? C times, and there's some constants, M to the A. An example of a function that has the property that viscosity has this dependence is given by this function. Now, you would have to say this is e to the alpha M, Cm to the A to the power of nu, and therefore gamma, this gamma is equal to A times nu, the exponent of an exponent you obtained by multiplying the exponents. And therefore this equation and that equation are totally consistent so long as gamma, A, and nu are related like this. Nonetheless, your Vol's results are entirely consistent with everything we've said. I would be delighted to say that I can draw a pretty picture of how nu depends on molecular weight and quality, but that really isn't quite true. The first obstacle is that if you look at this formula for the viscosity, A to the proportional to e to the alpha C to the nu, nu is the exponent of an exponent, which means it's a bit hard to measure. Nu doesn't change a whole lot if we said nu is almost always in the range 1 half to 1, that's correct. And therefore there's not a lot of range, and if there's any difficulty at all measuring nu accurately, you run into a problem that graphs are a bit scattered. There are some other parameters, however, that have been sort of hidden, and we can point at a few of them. One we can talk about, we now advance to figure 12.33, is the transition concentration C sub t. Transition concentration, yes, we are now limiting ourselves to systems that show the solution-like, melt-like transition, that's the transition we actually found. And we charge up like this, and there is a concentration, I'll use it as a subscript, C sub t, which the transition occurs, and correspondingly there is a viscosity, A to sub t. When we discuss this for colloids, well, for spheres, there's really only one curve, because the volume fraction is the only variable, and phi t was about 0.41 or a bit higher, and the viscosity of the transition divided by the viscosity of the solvent was about 10 plus or minus 5. That's very clearly not the concentration and viscosity at which you start getting this phase effect. And the question is now for linear polymers, how does the transition concentration behave? And the answer given in 12.33 is there are a bunch of points, a bunch of points are more or less on a straight line, and the transition concentration goes as m to the minus o, 1.1 or so. And the answer is as you increase the polymer molecular weight, the concentration at the transition falls. Of course, you need a set of measurements where the same material that shows the transition has been studied for a lot of different polymer molecular weights, otherwise you should be hesitant about believing the fit. The results that do this are on poly and hexalisocyanate, which we've discussed before, and you look at those and g, you get roughly this behavior. So you might, okay, so that's the transition concentration. You might also say, well, g, that's the transition concentration, and maybe we should re-express the transition concentration in natural units. When I say a natural concentration unit for polymer solutions, what people generally mean is c times the intrinsic viscosity. Now, there are people who will say c over c star, where c star is the overlap concentration, but in fact the overlap concentration is usually inferred from the intrinsic viscosity, namely c star equals the sum of number in the range. I've seen things as small as one of as large as four in the literature over eta. So this is a distinction but not a difference. Okay, so we look at c eta and we will ask at what transition concentration in natural units do we see the transition? And if you go through all of the accumulated measurements, you can find transitions as high as 80. You can find things that occurred at about 35, and that is a more typical value. You can also find transition concentrations as low, for example, hydroxypropylcellulose, which Carol Streletsky and I and others studied for. And so if you look at the transition concentration in natural units, it covers a very wide range of values. You should realize this is a significant part of the way to melt. The melt concentrations in D'Rivolve's Metals analysis were not more than about 300. The transition concentration for hydroxypropylcellulose, the intrinsic viscosity was one over, oh, one or two grams per liter, and the transition occurred at around six grams per liter. So if we say we'll put the concentration in natural units, you don't see any tendency of measurements to collapse onto each other. There is, however, something you can do which gets the measurements to match. Not perfectly, but actually pretty well. And the answer is if you look at all of those curves where you see something like this for linear polymers, not for spheres, for the measurements where people looked at a lot of different molecular weights and you look at their graph, you realize that the crossovers occur at about the same viscosity. And in fact, it is approximately but not exactly universally true that eta t over eta is some number like two or three hundred. That is, there is not a concentration at which you see the crossover, but there does appear to be a universal viscosity at which the crossover sets in except in hard sphere systems which may be different. And the crossover at viscosity occurs when eta over eta is zero, the reduced viscosity is a few hundred. So you produce something that's fairly viscous but not completely impossible to work with. Oh yes, you remember we said there was power law behavior out here? You might legitimately ask, well that's very nice, what is the exponent? And the answer for a fair number of systems is that x is about three and a half. However, you can find systems in which x is much larger, well I don't know, much larger, but somewhat larger. For example, in hydroxypropylcellulose, x is about four point four. And therefore there is a power law exponent and I can even tell you what the power law exponent is. Let us skip back from the systems that do show the transition which, well it is universal, a universal transition is a function of the transition viscosity. It's not a universal transition viewed as a function of the concentration. And it doesn't even happen in some systems. So let us go back and let us look at the systems in which we do not see a transition which is a large number of them. Going out in some cases to very high viscosities, I seem to recall reduced viscosities up to about ten to the seven in a few cases in which no transition is seen. Okay, so there is the curve and one of the things you might legitimately ask is, well, G, couldn't we manage to describe this measurement perhaps with a couple of power laws? And if you go through the literature, you find people who draw a power law curve up here, which is sort of tangential or goes through the points over some distance. And you find down here typically a linear curve. And in between there is some transition which is sort of talked over. As G, there is a transition that is broad. And therefore you might propose, yes, the measurements actually show a crossover from one type of behavior to the other, but the crossover is a little hard to see because there are other things going on at the same time. So there is not a clear transition. The way there would be if there was a phase transition, which I don't think anyone has proposed for, well, that isn't quite true. Almost no one has proposed for these systems. So there is a transition, but it's clearly not a phase transition. And the question is, is this a reasonable description? And the answer, I think, is that if we look at e to the alpha c to the nu behavior, the stretched exponential behavior marches out, alpha and nu are independent of concentration. They're proportional to c to the zero, independent of concentration. And that's true along the whole curve. And therefore, viewed from this perspective, there is no transition. There is a single smooth curve that just chugs ahead. So far so good. Okay. I have just run us out of chapter 12. We have now come to the end of the discussion of viscosity. I am now going to advance to discuss viscoelasticity. And in order to do that, I'm going to have to draw a picture. The idealized picture is here are two infinitely long plates. Now, that's a little dangerous because plates aren't infinitely long. And there are several ways around this, for example, bend plates in circles. And you have two concentric cylinders, or you have a flat plate, and you have a rotating cone. You may say, why would you want a rotating cone? And the answer is that the velocity up here, v, is omega r. That's velocity, oh, parallel to the plate. This distance, distance out from the center, increases as r. And therefore, this distance, because it's a cone, this distance is proportional to r. And therefore, if I look at the velocity of the plate over the distance from the plate to the bottom, and there is a velocity gradient here, and the velocity gradient is uniform over the distance. Now, you have to be a little careful with this, because there's going to be some question whether things are uniform as you move this way. There are potential complications. Nonetheless, we shall persevere, and I shall draw this pretty picture. And the bottom plate is stationary, and the top plate is being moved back and forth. And the top plate has a displacement, which I'll call d for the moment, and the displacement is some amplitude A cosine omega t. That is, we have a displacement, and the extent of the displacement oscillates as a harmonic in time. So you can do that equally with either that or that. The displacements are fairly small. Now, you might say, well, gee, could you get information out if you made the displacements large? And the answer is yes, and there is an object, an experimental method called large angle oscillatory shear. And we shall reach Laos in the next chapter, chapter 14. However, we have just reached chapter 13, which talks about linear viscoelastic behavior. OK, so we have this displacement, and in order to get this displacement, we have to apply a force on the upper plate to move it back and forth. Now, if you think about this for a moment, you realize that if the plate has inertia, you're also accelerating the plate at the same time, and you have to deal with this, but that's basically an experimental question. And the inertia of the plate has to be made small enough, in some sense, that you can measure the part of the force that is due to the fact there's a liquid in here. I'm going to call this A0, the amplitude of oscillation. We'll come back to other notation for it in a moment. And the reason is that if we look at this system, if you make the plate bigger and bigger, the amount of force you need goes up. But it's basically a linear process, and therefore the interesting thing is the force per unit area. And the force per unit area is known as the stress. And the first force per unit area, which by the way is going to be oscillating as a function of time, the stress is determined by the amount of oscillation, the displacement divided by the distance L between the plates. That is, if I take this whole apparatus and make it twice as big vertically, the amount of oscillation seen locally by a molecule, here's a molecule, and it starts out and has some neighbors along this line, and when we increase D to its maximum, this is oscillating, the neighbors are displaced sideways one way or the other in an back-and-forth manner, and the amount of local displacement is determined by the ratio of the linear displacement of the plate to the distance between the plates. And this is known as the strain. So far so good. Now, suppose we actually do this experiment. There are a couple of different ways we could do it. One thing we could do is to apply an oscillatory force up here and measure the motion. And so we have a force which is very precisely controlled and shifts things back and forth, and we ask how the plate moves. We could also have a driver and feedback electronics, and we move the plate back and forth through a fixed distance, and we instrumentally determine how much force we're having to exert so that the plate actually does this. Those two experiments are equivalent. That is, they measure a series of curves where a certain amount of force per unit area, a certain stress, and a certain strain are matched with each other one for one. But whether you are actually experimentally creating the stress and measuring the strain or vice versa, doesn't matter. There's one curve, and the distinction there is purely how you built your machine. Okay? Now, having said all this, we now go in and we measure this question. This is assuming that it's linear response. When you're basically saying that the stress, when you measure the stress, it's equivalent to measuring the strain. It's basically assuming that it's linear response. It seems to me the answer is no. That statement is more general than its linear response. That is, you have a stress, you have a strain. If I double the stress, what happens to the strain? Well, the strain may or may not double in the linear response, it does. But however, the statement is there's a stress-strain relationship would be true whether it was linear or not. The only place where you would get into trouble with this is that you could have a hysteresis issue where if the... you apply different, for example, different strains and the stress changes some place or vice versa, you might not be able to reach every point on the curve instrumentally very easily by doing one controller or the other. Now, this is linear response in the sense that the theoretical analysis that is done of linear viscoelasticity assumes that if I apply a series of forces at different times, the fluid has a response at later times, and I can just do linear response that is add up all of the responses of the fluid. Of course, if I've applied the force at several times, there's several time delays, and I will get the answer. And we'll get to linear in a bit. So what is the net result, though? The net result is that if we actually look at the strain, the stress, the force bringing in that area, and we compare it with the strain, we find something. And what we can divide up, because this is linear, we can get force, we can divide out the A0, and we have force per unit area per unit of strain. And that force turns out to have two pieces. There is one piece that is determined by a function g' of omega, and g' of omega is now multiplying the displacement. Traditionally, it's sine rather than cosine. Oh, I'll write it as cosine to avoid confusion. This is cosine omega t. There is one component where the response has a response that is in phase with the displacement. So you have a displacement, and there is a restoring force. That's what it is. It grows linearly in the displacement. That is the oscillating displacement. And then we have another component. Another component is 90 degrees out of phase, and so there is a response that is 90 degrees out of phase to the displacement, meaning this is a force that is largest when the displacement is at zero. Now that may look very odd, but you have in phase and out of phase responses. And you might ask, what does this all mean? Why is there a force that's in or out of phase with the displacement? This is a liquid, after all. If you made the displacement stop and sat there after a while, the force would disappear because the liquid would just flow. That's a little more complicated than it sounds. And the answer is that the low shear viscosity is......is actually a low frequency limit of G double prime of omega over omega. These two objects have names, by the way. G prime is the storage modulus, and G double prime is the loss modulus. And I might legitimately wonder, gee, that seems like a peculiar way to represent things. And I will now show something that makes much clearer from a physicist's point of view what you are looking at. I am going to multiply each of these objects by one. And of course, whenever a physicist says, I am going to multiply by one, you know it's going to be some huge object that is in fact one, but not if you don't look hard. And I am going to multiply this by one in the form omega square over omega square. And I am going to apply this one by one in the form omega over omega. That's perfectly legitimate. You notice this now looks exactly like the viscosity. But now I ask you, okay, I have something that goes as omega times the displacement, and it's 90 degrees out of phase with the displacement. And I have something that goes as omega square times the displacement. What am I looking at here? Well, if you think of a harmonic oscillator, you realize that this object is the velocity, and I have a term in the force that is determined by the velocity. Gee, that's a lost term, isn't it? And here I have a term that goes as omega square times the displacement. That's the acceleration. And the thing that goes as the acceleration is a restoring force. The system looks like it has little springs in it. And in fact, there are a set of math models blamed on, if I recall correctly, Maxwell. And this whole picture can be modeled as, here is a system, and it has a spring. And coupled to the spring is something known as a dashpot. Dashpots were very important back when people had teletypes. The dashpot, the plating of the moving object in the teletype comes back very fast, and you don't want it to whack the far end hard because if it does, the vibration will damage things. So you have this little piston-like object, and the moving object coming this way has an arm coming out. There's a rubber disk here, that's the traditional image. And the rubber disk hits this thing, and there is a little hole here. And this is a piston, but it's a leaky piston, and so you compress the air and it slows things down. And then when it's done compressing the air, the air blows out the end, and you contume the opening because there's a little sliding arm. And so eventually, after a very small fraction of a second, the moving object on the teletype is brought to an end here. It's brought to a stop at exactly the right location, and with no unpleasant, sharp acceleration that would damage anything. Well, yeah, it does that, and it does it via high-quality Victorian engineering. There are essentially no moving parts in this other than the object you're trying to stop. There's no active feedback system, no electronics, it's all done mechanically. And that is a dashpot. And you can fill the dashpot in this picture with some liquid. So you have this hanging object in the liquid, and the hanging object in the liquid, it's just a liquid with a viscosity and no frequency dependence. And so you, or it has a frequency dependence, and this object represents the storage modulus, and this piece represents the loss modulus. So that is a picture of the viscoelastic parameters, the linear, in terms of a storage modulus and a loss modulus. And you notice that I have reparameterized them starting here and then doing this for consistency, basically. And once I've done this, the loss modulus looks exactly like the viscosity. There is, however, and I'm just going to show the qualitative picture first. Here's, however, a consequence of reparameterizing things. If I plot G prime or G double prime themselves, you remember they contain something, I divide out an omega, which I can do safely. And therefore G prime and G double prime both go to zero. So it's a little hard to do on a log-log plot, but they both go down. And at low frequencies, this is more typically a log plot, so you never really get to zero, there is a linear behavior, which is omega to the power. And then there are things that happen out here, and perhaps the curves cross. So if you're making omega larger and larger, eventually the curves turn up again. I am not going to draw this in any detail, you can find it in standard texts. The important issue is that if you talk about G prime or G double prime itself, themselves, you find that they have curves that look sort of like this. And there's a region where things are reasonably flat, and there's a low frequency region and a high frequency region I could introduce terminology. We'll get to that next time. The main issue is, though, there's a shape for the curves. If I instead say I am going to go in, and I am going to plot G prime over omega square, or G double prime over omega, if I do that, I have divided out the leading slope. And if I divide out the leading slope, that says that at low frequency, the behavior that I am looking at is frequency independent, which is sort of what you would naturally expect. And these curves all look, as you will see from the book, about the same, namely there's a flat region, there's a rollover, and then down here there is at least at first a power wall. And then maybe something interesting happens at larger frequencies. The question we ask, given that I have drawn these pictures, is, well, how do I derive or rationalize a curve that explains this? And so I will give an Anzatz based on the theoretical model we have not yet discussed in detail, and a renormalization group argument. We are going to actually invoke, which we have not actually done before, the hydrodynamic scaling model, which is my model, and we are going to invoke a renormalization group argument. The renormalization group argument being invoked is much more general and qualitative than many renormalization group arguments, but it will be clear what it is. So we are going to start out, and here is a straight line representing concentration. And we will imagine that the viscosity is plotted perpendicular to the blackboard out towards you. And what we find is the viscosity increases as I go to higher concentration, and eventually, at least in some systems, we reach the transition concentration C sub t. And down here is a solution-like regime, and in the solution-like regime, you see stretched exponential in concentration behavior. And up here is the melt-like regime M, where you see C to the X behavior. That is the pure phenomenology. Now what I say is I have used something called the Altenberger dollar positive function renormalization group method, and I applied this in the context of the hydrodynamic scaling model. And I said I can calculate a certain lead behavior at lower concentrations, and the Altenberger dollar renormalization group method lets me turn the lower concentration calculation into the stretched exponential, and we are going to assume that is true. Now I am going to put another little bit on top of it. The PFRG, the Altenberger dollar positive function renormalization group method, actually predicts two sorts of functional behaviors. The functional behaviors it predicts depend on where what are called the fixed points of the renormalization group are located. In particular, if you have a fixed point at zero, then near zero you predict stretched exponential behavior as is found. On the other hand, out here someplace, if you have a fixed point that is way out there, and that is the dominant fixed point. There can be several fixed points, but as we go along here, there is always a dominant fixed point. If the dominant fixed point is out here, you would get power law behavior as is observed. Next trick. The viscosity is the low frequency limit of double prime omega over omega. Viscosity is a function of concentration, and so is the dynamic loss modulus. Therefore, when we say we are looking along this curve at viscosity, I could also say, I am going to put another axis on here. I am going to insert the vertical axis I have skipped over. I am going to call the vertical axis omega frequency. I am going to say viscosity is what we measure along the omega more or less, maybe not quite, zero one. We measure at low frequencies, and we get the viscosity, zero frequency limit. However, it is the zero frequency limit, and while I wrote it as eta, I could just as well have written it as Gw prime of omega, and by the way, concentration, over omega. And therefore, I now have a two variable function, and I have used the renormalization group argument to work along at omega is zero. Okay? Now comes the somewhat bold part. I will start here. Yes. And I will march sideways. And at first I am in a regime where this fixed point at zero is presumably still there and dominant. And then at some point, I move sideways enough, and these other fixed points, about which I know very little, become dominant. And therefore, I have moved from a solution-like regime to a melt-like regime. However, I am marching along at C fixed with omega as the variable, because I am moving sideways along this graph. And I am looking at something that is really only a function of omega. If this picture is correct, and there is a great deal of extrapolation in this, it's an anzotz, not a derivation. At lower frequencies, this fixed point is dominant, and therefore I should have a stretched exponential in frequency. However, eventually I cross this line, and once I am across this line, I am in a region where the melt-like fixed points out there are dominant, and I should have omega to a power behavior. This doesn't really calculate what the power is at all. And in order to do the extrapolation even at low frequency, I would need something that gives me viscosity is 8 to 0 plus presumably, some intrinsic viscosity, 8 and a lot C, plus some beta omega. That is, I would need some way to generate a model that gives me the low frequency, so I could move off 0 at all. And after I have moved off 0 at all, I can then use the Altman-Burger-Duller renormalization group to say what happens with frequency. However, I have not found, I have been working on other things truthfully, I have not yet found the frequency, yes, and therefore the assertion is that you find this linear step and you can then fill in the low frequency behavior. That has not yet been done, and the crossover and the fixed points out there have not been done even for concentration. Okay? So that is the rationale, and you notice what it predicts. It predicts stretched exponential in frequencies and lower frequencies, and it predicts power-long frequencies and higher frequencies. However, there is some interesting, if you look carefully, some little bits I have skipped around. And one bit I have skipped around is what it says about frequency dependence way out here. That is, if you are way out there, you're in the mount-like regime all the way, and therefore you see simply power law and frequency behavior. Well, you can actually point at data that looks like this, and the question is, are you really going to see that, or is it the case that there is a very low frequency, stretched exponential in frequency regime? And corresponding to that, is it possible that this curve actually bends over and very close to zero frequency if you get down to sufficiently low frequencies, the seed of the X behavior would disappear? I don't have an answer to that. The other point is that if we go up to very high frequencies, is it going to be the case that we have gotten into the omega to the X regime at very low concentrations, or does this curve bend over so that at very, very low concentrations where we can just barely see particle interactions at all, we are still in the concentration to a stretched exponential regime. That is, the model does not handle this very, does not discuss this, the model sort of dodges around that, it covers this large area. Now, you could say, of course, gee, this picture explains why some systems show the solution like, melt like transition, and some do not. Namely, depending on exactly where the fixed points out here are, it might be the case that in some of these systems, you chug along the concentration axis all the way to the melt. This curve does not go to infinity, it stops at the melt. And even at the melt, the detailed numerical parameter is going into the calculation or such, that you can get all the way out here, and the fixed point at zero is still dominant. But there are other systems, and all you need are differences in chemically dependent parameters, such that these fixed points become dominant at some concentration, and out here you see a power lock. And therefore, one universal physical model plus chemically dependent parameters that determine where which fixed point is dominant predicts both behaviors. Well, that's fine for the hydrodynamic scaling model, which actually allows you to have two behaviors in one general model. It is not at all so good for models that assume scaling, because they assume that you always get scaling out here, which it seems that you do not. Okay, that is the Anzatz. And having given you the Anzatz, we have now reached experiment. We have also reached approximately the end of the hour, and therefore we have reached a natural point to stop. Therefore, I am going to stop. So this has been a lecture finishing off my treatment of viscosity and advancing to my treatment of the hydrodynamic scaling model and its Anzatz for predicting linear viscoelasticity. I'm George Phillies. This is the end of the 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 23,  Viscosity, Introduction to Viscoelasticity, the Temporal Scaling Onsots.  I'm Professor Philly's, and today we're going to finish off  our discussion of viscosity and push on to discuss viscoelasticity.  Let me recall where we left off roughly speaking.  We left off in chapter 12 discussing figures 12.22 to 12.24,  which are treatments of the molecular weight dependence of the viscosity of the solution.  Now, there's actually a fairly considerable number of measurements of these,  and they're actually fairly consistent, namely what they find,  what is found with considerable consistency in many systems,  is that the viscosity depends on the molecular weight as a stretch exponential and molecular weight.  And that's brought out quite emphatically by, for example, figure 12.24.  12.24 are measurements that actually appear in Doyle and Edwards,  who report them as measurements done by Onogi et al.  The measurements are a little more complete than the measurements in the original paper, actually.  Having said that, an interesting feature here is that if you go back into the literature  and ask what people said about these measurements,  you can find people who say,  we have measured the molecular weight dependence of the viscosity,  and as predicted, we see a power law dependence of the molecular weight.  There actually are measurements that find power law molecular weight dependence.  For example, as we discussed earlier,  POW at all measurements, which push out to the melt,  find close to the melt a power law dependence of viscosity on molecular weight,  and volume fraction one, log log plot,  what are rather clearly straight lines,  viscosity changing as a power law on M,  and if you actually say, we'll stay fairly close to the melt  and we will exclude points that seem to be deviating from the original straight line,  you actually could find in POW at all M to the 3B8.  However, most of the measurements you find,  which are done at considerably lower concentrations,  do not find power law behavior, they find stretched exponential behavior.  An interesting example of this is provided,  and this is also, in essence, stretching over to the next section,  which is a discussion of topology,  are the measurements like Grace Lee and L,  and Grace Lee look at polymers,  and they look at linear chains,  and they look at four-armed stars,  and they look at six-armed stars.  And if you look in at the results they show as a function of concentration,  for the linear chain,  and their linear chain is a molecular weight of 1.66 megadolptin,  as a function of concentration, a log-log plot,  they find what is very clearly a straight line,  you look at it, it's clearly a straight line,  and it's quite clear that you are seeing power law behavior.  This is for the two-armed star, the linear chain, two-armed star.  On the other hand, for F equals 4,  and for F equals 6,  very clearly on the same plots,  these are chains of about the same molecular weight.  This is about 1.95 megadolptin polymer,  and the other is about a, what was it,  it was about 1.44 megadolptin polymer.  For the other two, you very clearly see  stretched exponential molecular weight dependencies,  these are fixed molecular weight,  you're looking at concentration dependencies.  If you look at the molecular weight dependence,  where they had a considerable number of samples,  they were looking at concentrations of about 300 gram per liter of polymer,  extremely concentrated systems,  and the molecular weight dependence was a straight line  for the linear polymer,  and was a stretched exponential,  two stretched exponentials actually,  for the star polymers.  As a function, taken as a function of concentration,  the upper figure,  the linear chain showed the power law behavior,  the stars showed a stretched exponential behavior,  the viscosity of the star was, for the most part,  less than the viscosity of the linear polymer.  However, if you look carefully at the measurements  against concentration,  you can see a point up here,  and so if the concentration was taken to be high enough,  eventually some of the star polymers,  as they're obliged to do given the functional behavior,  did show a viscosity that was higher than the viscosity of the stars,  of the linear chain, I mean.  However, for the most part,  star polymers, comparable molecular weights,  are up to fairly high concentrations,  less viscous than linear chains,  and at fixed concentration as a function of molecular weight,  again, the stars are less viscous than the linear chain is.  Okay.  You can also look at results of...  We'll get this erased in a second.  You can also look at results of Kajura and collaborators.  They looked at three-arm stars.  Their stars were a poly-alpha methyl styrene.  They look at a series of concentrations and molecular weights.  You can see the results in figure 12.26.  And what they found was that, again,  if you're at about the same concentration  and about the same molecular weight,  the viscosity of the star polymer,  this is the F equal 3 star,  is less than the viscosity of the linear chain.  Now, why is the statement that the star polymers  have a lower viscosity than the linear chain of some interest?  Well, there are a number of models of polymer dynamics  which assert that linear chains have extra modes of motion  that are, in principle, not accessible to stars.  In particular, for example, the reputation-type models  say that if you have a linear polymer,  it can creep through holes in a transient lattice  like the snake in a bamboo grove.  But if I am a star polymer,  in order to make forward progress,  what I have to do is retract by fluctuation one of my arms.  That's difficult to do.  I can push it out in a new direction,  and now I can take a very modest step forwards.  The net result is that one would expect  that it is harder for star polymers  to move with respect to each other  than, at least if they're very concentrated,  than it is for linear chains.  That does not appear to be quite what we are observing.  Question?  Can it also be possible that since linear molecules  are a linear instruction,  they have better packing methods,  therefore, they adhere closer to each other?  That could be an issue in the melt.  However, in the solution,  the amount of packing is determined by the concentration.  And the answer is that the concept,  we make comparison and equal concentration for linear and star.  The other thing you should recall with respect to,  let us pack chains into a melt,  is that I am a star polymer.  I have one point where there are three chains coming together.  There, if I'm an F equals three star,  there is a narrow region around this  where there are three of my chains close together.  However, if I'm a large star polymer  over most of my length,  I look exactly like a linear chain.  And therefore, this is the sort of effect.  You might find interesting effects with small chains,  but with large ones, you would not expect that to arise.  Thank you.  Okay.  So we advanced figure, oh,  discussion around figure 12.25.  And the issue around figure 12.28 and figure 12.27  is suppose we take a polymer chain  and we change the solvent quality.  What happens if we change the solvent quality?  Well, the expectation is that if we are in a theta solvent  or close there too,  which means we also have to be at the right temperature,  we have a polymer chain in the solution.  It would rather see itself as its neighbor  than it would see something else.  And therefore, if we increase the concentration,  nothing terribly interesting will happen to the chain radius.  I didn't say nothing at all.  Because theta, you can't get perfect cancellation of effects.  It's an approximation.  On the other hand, if you were in a good solvent,  we see what we saw in the chapter on dielectric relaxation,  which is what has been used for most of these measurements.  Namely, as we increase the concentration  and look at the chain radius,  the chain radius falls with increasing concentration.  So these, however, are static properties.  They're how big is the chain if we take a single snapshot?  We're talking about viscosity here.  Viscosity is a dynamic property.  And we ask what happens to the behavior of the system  if we change solvent quality and such and on.  The first answer is,  if we look at a series of chains of different molecular weight,  the viscosity, as has been the case in most other systems,  not all other systems  and not at sufficiently high molecular weight,  but concentration rather in those systems,  the concentration dependence is a stretched exponential.  And what we happen, fine,  as we increase the polymer molecular weight  and do our measurements,  we find that the parameter alpha increases  and the parameter nu falls.  So there is alpha and there is nu.  And both parameters are molecular weight dependent.  The second thing we can do  is to compare these parameters  in a good solvent and in a theta solvent.  And what we find is that in a theta solvent  as discussed here, the parameter nu is from 0.8 to 0.954.  And if we go, however,  take the same material over to a good solvent,  the parameter nu is 0.6 to 0.73.  That is, we are going in  and when we change the solvent quality,  when we go into a better solvent,  the parameter nu falls.  The model that derives this equation,  in fact, predicts this behavior,  we will eventually get to it in later lectures.  If you look at the figures, especially 12.28,  this change of nu is physically quite visible.  If you look at a semi-log plot,  log viscosity versus concentration,  for a theta solvent,  the concentration dependence of eta  is fairly close to a straight line  because this is fairly close to a stretched exponential.  But if you look at the measurements in the good solvents,  you typically see a more pronounced curvature  because you are looking at a stretched exponential  rather than a pure exponential.  Now, having said this,  let us push ahead and consider  the general features of viscosity.  We have looked at lots of specific measurements  and we did, of course, find that the viscosity  depends on the polymer concentration  and the polymer molecular weight,  not to mention the one could put in  the topology and one could put in  the solvent quality.  With respect to topology,  there was a very long dispute,  this goes back to a considerable piece,  about the viscosity of ring polymers,  but that was mostly melts,  which the book does not cover.  The issue there was, well,  you can try to make melts,  you have to ask, are the,  you can try to make rings  and put them into melts.  Here is a ring, is it a nice open loop  or during your synthesis process,  did it get itself all tied in knots  so that it's a ball?  Under modern conditions,  if you want to answer that question,  you can resort to using DNAs  since you can synthesize DNAs  with absolute molecular weight control.  But absolutely mean we have something  that's 180,000 base pair,  meaning its molecular weight is  tens of millions,  and the molecular weight is,  well, there are fluctuations  because there are several isotopes  of carbon and deuterium in nature,  a carbon and deuterium hydrogen,  I should say in nature,  but the molecular weights  are totally exactly controlled,  no fluctuation.  You can also synthesize things,  this is synthetic DNA,  not stuff that occurs in nature,  that is a star  and completely mono-dispersed.  You can also synthesize,  it happens naturally, some viral DNAs do this.  DNAs that are ring polymers  that are again completely mono-dispersed.  Electron micrographs of those rings  indicate that the rings are completely open,  they are tied up in knots at all.  However, having said,  the ring polymers are  mono-dispersed,  the star polymers are truly mono-dispersed,  synthetic DNA comes that way,  there is the significant practical issue  that people have not done studies  of DNA viscosity that are as extensive  as the viscosity studies  that have been done on synthetic polymers.  Okay, having said this,  let us look at the question of what we find.  And the first point,  which I am going to emphasize  significantly more than was emphasized in the book,  is that eta is not a universal  function of C and M.  That is, if you think that you can take your viscosity measurements  and yes, there will be some chemical dependent constants  and there will be things that depend on the concentration  in the molecular weight  or maybe the length of strands in the molecular weight.  Some representation of amount of material  and size of the chain.  If you think that all of the chains are the same  because of that and therefore there should be a universal function  that gives you viscosity as a function of concentration,  you are going to be disappointed.  Instead, it is quite clear from the literature  that viscosity is not a universal function of C and M.  Instead, there are two distinct phenomenologies  and in one of the phenomenologies,  you see simply a stretched exponential  which in measurements of Taeger goes up,  Taeger and a few other people go out more or less to melt.  And in another set of systems,  you see a stretched exponential  up to a point  and then beyond the point,  you see a power wall.  It is worthwhile to recall  that the same phenomenology this class  is occupied by spherical colloids.  Spherical colloids show the same behavior.  The slope of the power law  depends on the exact material.  For spherical colloids,  the slope is appreciably steeper than for linear chains.  And if you look at many arm star polymers,  the slope creeps up and makes a transition  from linear chain behavior up to very nearly heart-sphere behavior.  It should be clear that any model  that explains this phenomenon  is subject to the constraint  that it must be of work for spherical colloids.  Spheres can do a number of things,  but form entanglement lattices  is not really one of them.  And form and reptate is certainly not one of them.  And so we have the issue that the viscosity behavior  might not be quite what you would have expected  from some of your readings from some theorists.  Okay.  Furthermore,  if you merge up to viscosities above about 10 to the 8,  that's eta over eta-zero above about 10 to the 8.  And most of those measurements are literature studies  due to Drival, though there's some nice work due to Colby.  However, if you look at that,  it does appear that if you get above here,  instead of continuing along on the power law,  there is an upwards deviation.  The nature of the upwards deviation is not quite certain.  One might, however, for example, propose  since we're at extremely high concentrations  that it's some sort of a jamming transition  in which things actually mechanically get in each other's way,  and the chains can't slide across each other  because they can't move after all their rigid overshort distances.  They can't move sideways as easily enough  to get through each other.  However, that is a fairly specific,  that's a fairly rare exception  because there are not a lot of studies that have looked at this.  In particular,  if you look at predictions that eta should be some c to the x,  m to the y,  that prediction is very generally not true in polymer solutions.  There are some exceptions.  There is this family,  but predictions that you can simply assume scaling  are not sustained.  If you want to say we have scaling,  you need to explain why sometimes you have scaling  and sometimes you don't.  Otherwise, you can't just assume.  Okay.  So that is the sort of behavior we get.  We might then ask,  well, what other generalizations can we find?  And one answer is,  for a very large family of parents,  we have eta over eta zero  is e to the some constant alpha c to the nu,  and now I'm pulling the molecular weight of dependence of alpha.  We'll actually see graphs of that dependence in a few moments.  But if we actually look at the measurements and ask,  okay, we get this thing to fit, what do we see?  Well, gamma is usually in a range around a half.  And I can be more precise than that,  but that's a reasonable approximation.  In point of fact, if we say, oh, I should add something.  We go from theta to good solvents.  We go from theta solvents to good solvents.  The parameter nu goes down,  and what I will call alpha m,  because it's alpha with m pulled out,  what I will call alpha m, m to the gamma,  what's left, what's not c to the nu,  increases as we go from theta to good solvents.  Generalization.  Okay, so what behavior do we see for gamma?  And the answer is seen in figure 12.30,  where we plot alpha, which in terms of this equation  is alpha m, m to the gamma.  It's the coefficient of c.  And we plot alpha against polymer molecular weight.  Now, if I just showed you the points,  your reaction would be, gee,  that was a nice piece of graph paper, Professor Philly's.  At least it was very nice before you stood way back  and started firing the shotgun at it,  and the points are all over the place.  However, if you look carefully at the points,  I have labeled the points by the chemical family of the polymer.  So we have identified separately each set of homologous polymers,  polymers that are chemically identical with different lengths.  And if you do that, you see that for each single family  of homologous polymers,  alpha lies very much on a straight line.  The straight line has a slope.  So this is a log-log plot,  so we say alpha goes as m to the power gamma.  And what can we say about gamma if we look at all of the measurements?  That is, all of the measurements were the same way.  I've looked at a large number of different polymer molecular weights.  And other than that, things are consistent.  And the answer is that gamma is typically in the range 0.6 to 0.67.  One can find a case where it's 0.5, if I recall correctly, 0.94,  where it's quite large,  and you can find a case where it's 0.51.  If you look at this, you notice there are different slopes  for different materials,  and therefore it's not quite a universal constant.  But for a given set of samples,  the power law dependence is actually fairly sharp.  This result is also implicit in the systematic review of Dervol  and many collaborators,  and all of the wonderful Russian data he compiled in one place for us,  because what Dervol says is that C eta is a good reducing variable.  That is, if you take all of these measurements  and you plot them as functions of C times the intrinsic viscosity  rather than C,  so we have eta log log again, C eta here,  and we find a whole series of different polymer molecular weights  that the measurements lie on the same curve.  Now, in order for the measurements to lie on the same curve,  when you are doing this plot, it must be the case that eta is some function of C eta,  which means that all of the molecular weight dependence of the viscosity  has to be built into the molecular weight dependence of the intrinsic viscosity.  You can't have any other molecular weight dependence out here someplace,  because if you did, when you plotted viscosity for a series of polymers  of different molecular weights, they wouldn't lie on the same line.  They lie on the same line because viscosity is a function of C times  the intrinsic viscosity.  That statement, universal function, goes all the way up to the melt.  However, it is an important fact that the intrinsic viscosity depends on M as a power,  and therefore this equation says that the viscosity is a function of,  what is it a function of? C times, and there's some constants, M to the A.  An example of a function that has the property that viscosity has this dependence  is given by this function.  Now, you would have to say this is e to the alpha M, Cm to the A to the power of nu,  and therefore gamma, this gamma is equal to A times nu,  the exponent of an exponent you obtained by multiplying the exponents.  And therefore this equation and that equation are totally consistent so long as gamma,  A, and nu are related like this.  Nonetheless, your Vol's results are entirely consistent with everything we've said.  I would be delighted to say that I can draw a pretty picture of how nu depends on molecular weight  and quality, but that really isn't quite true.  The first obstacle is that if you look at this formula for the viscosity,  A to the proportional to e to the alpha C to the nu, nu is the exponent of an exponent,  which means it's a bit hard to measure.  Nu doesn't change a whole lot if we said nu is almost always in the range 1 half to 1,  that's correct.  And therefore there's not a lot of range, and if there's any difficulty at all measuring nu accurately,  you run into a problem that graphs are a bit scattered.  There are some other parameters, however, that have been sort of hidden,  and we can point at a few of them.  One we can talk about, we now advance to figure 12.33, is the transition concentration C sub t.  Transition concentration, yes, we are now limiting ourselves to systems that show the solution-like,  melt-like transition, that's the transition we actually found.  And we charge up like this, and there is a concentration, I'll use it as a subscript,  C sub t, which the transition occurs, and correspondingly there is a viscosity, A to sub t.  When we discuss this for colloids, well, for spheres, there's really only one curve,  because the volume fraction is the only variable, and phi t was about 0.41 or a bit higher,  and the viscosity of the transition divided by the viscosity of the solvent was about 10 plus or minus 5.  That's very clearly not the concentration and viscosity at which you start getting this phase effect.  And the question is now for linear polymers, how does the transition concentration behave?  And the answer given in 12.33 is there are a bunch of points,  a bunch of points are more or less on a straight line,  and the transition concentration goes as m to the minus o, 1.1 or so.  And the answer is as you increase the polymer molecular weight, the concentration at the transition falls.  Of course, you need a set of measurements where the same material that shows the transition  has been studied for a lot of different polymer molecular weights,  otherwise you should be hesitant about believing the fit.  The results that do this are on poly and hexalisocyanate, which we've discussed before,  and you look at those and g, you get roughly this behavior.  So you might, okay, so that's the transition concentration.  You might also say, well, g, that's the transition concentration,  and maybe we should re-express the transition concentration in natural units.  When I say a natural concentration unit for polymer solutions,  what people generally mean is c times the intrinsic viscosity.  Now, there are people who will say c over c star, where c star is the overlap concentration,  but in fact the overlap concentration is usually inferred from the intrinsic viscosity,  namely c star equals the sum of number in the range.  I've seen things as small as one of as large as four in the literature over eta.  So this is a distinction but not a difference.  Okay, so we look at c eta and we will ask at what transition concentration in natural units do we see the transition?  And if you go through all of the accumulated measurements, you can find transitions as high as 80.  You can find things that occurred at about 35, and that is a more typical value.  You can also find transition concentrations as low, for example, hydroxypropylcellulose,  which Carol Streletsky and I and others studied for.  And so if you look at the transition concentration in natural units, it covers a very wide range of values.  You should realize this is a significant part of the way to melt.  The melt concentrations in D'Rivolve's Metals analysis were not more than about 300.  The transition concentration for hydroxypropylcellulose, the intrinsic viscosity was one over, oh, one or two grams per liter,  and the transition occurred at around six grams per liter.  So if we say we'll put the concentration in natural units, you don't see any tendency of measurements to collapse onto each other.  There is, however, something you can do which gets the measurements to match.  Not perfectly, but actually pretty well.  And the answer is if you look at all of those curves where you see something like this for linear polymers, not for spheres,  for the measurements where people looked at a lot of different molecular weights and you look at their graph,  you realize that the crossovers occur at about the same viscosity.  And in fact, it is approximately but not exactly universally true that eta t over eta is some number like two or three hundred.  That is, there is not a concentration at which you see the crossover,  but there does appear to be a universal viscosity at which the crossover sets in except in hard sphere systems which may be different.  And the crossover at viscosity occurs when eta over eta is zero, the reduced viscosity is a few hundred.  So you produce something that's fairly viscous but not completely impossible to work with.  Oh yes, you remember we said there was power law behavior out here?  You might legitimately ask, well that's very nice, what is the exponent?  And the answer for a fair number of systems is that x is about three and a half.  However, you can find systems in which x is much larger, well I don't know, much larger, but somewhat larger.  For example, in hydroxypropylcellulose, x is about four point four.  And therefore there is a power law exponent and I can even tell you what the power law exponent is.  Let us skip back from the systems that do show the transition which, well it is universal,  a universal transition is a function of the transition viscosity.  It's not a universal transition viewed as a function of the concentration.  And it doesn't even happen in some systems.  So let us go back and let us look at the systems in which we do not see a transition which is a large number of them.  Going out in some cases to very high viscosities, I seem to recall reduced viscosities up to about ten to the seven in a few cases  in which no transition is seen.  Okay, so there is the curve and one of the things you might legitimately ask is, well,  G, couldn't we manage to describe this measurement perhaps with a couple of power laws?  And if you go through the literature, you find people who draw a power law curve up here,  which is sort of tangential or goes through the points over some distance.  And you find down here typically a linear curve.  And in between there is some transition which is sort of talked over.  As G, there is a transition that is broad.  And therefore you might propose, yes, the measurements actually show a crossover from one type of behavior to the other,  but the crossover is a little hard to see because there are other things going on at the same time.  So there is not a clear transition.  The way there would be if there was a phase transition, which I don't think anyone has proposed for, well, that isn't quite true.  Almost no one has proposed for these systems.  So there is a transition, but it's clearly not a phase transition.  And the question is, is this a reasonable description?  And the answer, I think, is that if we look at e to the alpha c to the nu behavior,  the stretched exponential behavior marches out,  alpha and nu are independent of concentration.  They're proportional to c to the zero, independent of concentration.  And that's true along the whole curve.  And therefore, viewed from this perspective, there is no transition.  There is a single smooth curve that just chugs ahead.  So far so good.  Okay.  I have just run us out of chapter 12.  We have now come to the end of the discussion of viscosity.  I am now going to advance to discuss viscoelasticity.  And in order to do that, I'm going to have to draw a picture.  The idealized picture is here are two infinitely long plates.  Now, that's a little dangerous because plates aren't infinitely long.  And there are several ways around this, for example, bend plates in circles.  And you have two concentric cylinders, or you have a flat plate,  and you have a rotating cone.  You may say, why would you want a rotating cone?  And the answer is that the velocity up here, v, is omega r.  That's velocity, oh, parallel to the plate.  This distance, distance out from the center, increases as r.  And therefore, this distance, because it's a cone, this distance is proportional to r.  And therefore, if I look at the velocity of the plate over the distance from the plate to the bottom,  and there is a velocity gradient here, and the velocity gradient is uniform over the distance.  Now, you have to be a little careful with this, because there's going to be some question  whether things are uniform as you move this way.  There are potential complications.  Nonetheless, we shall persevere, and I shall draw this pretty picture.  And the bottom plate is stationary, and the top plate is being moved back and forth.  And the top plate has a displacement, which I'll call d for the moment,  and the displacement is some amplitude A cosine omega t.  That is, we have a displacement, and the extent of the displacement oscillates as a harmonic in time.  So you can do that equally with either that or that.  The displacements are fairly small.  Now, you might say, well, gee, could you get information out if you made the displacements large?  And the answer is yes, and there is an object, an experimental method called large angle oscillatory shear.  And we shall reach Laos in the next chapter, chapter 14.  However, we have just reached chapter 13, which talks about linear viscoelastic behavior.  OK, so we have this displacement, and in order to get this displacement,  we have to apply a force on the upper plate to move it back and forth.  Now, if you think about this for a moment, you realize that if the plate has inertia,  you're also accelerating the plate at the same time, and you have to deal with this,  but that's basically an experimental question.  And the inertia of the plate has to be made small enough, in some sense,  that you can measure the part of the force that is due to the fact there's a liquid in here.  I'm going to call this A0, the amplitude of oscillation.  We'll come back to other notation for it in a moment.  And the reason is that if we look at this system, if you make the plate bigger and bigger,  the amount of force you need goes up.  But it's basically a linear process, and therefore the interesting thing is the force per unit area.  And the force per unit area is known as the stress.  And the first force per unit area, which by the way is going to be oscillating as a function of time,  the stress is determined by the amount of oscillation,  the displacement divided by the distance L between the plates.  That is, if I take this whole apparatus and make it twice as big vertically,  the amount of oscillation seen locally by a molecule, here's a molecule,  and it starts out and has some neighbors along this line,  and when we increase D to its maximum, this is oscillating,  the neighbors are displaced sideways one way or the other in an back-and-forth manner,  and the amount of local displacement is determined by the ratio of the linear displacement of the plate  to the distance between the plates.  And this is known as the strain.  So far so good.  Now, suppose we actually do this experiment.  There are a couple of different ways we could do it.  One thing we could do is to apply an oscillatory force up here and measure the motion.  And so we have a force which is very precisely controlled and shifts things back and forth,  and we ask how the plate moves.  We could also have a driver and feedback electronics,  and we move the plate back and forth through a fixed distance,  and we instrumentally determine how much force we're having to exert so that the plate actually does this.  Those two experiments are equivalent.  That is, they measure a series of curves where a certain amount of force per unit area,  a certain stress, and a certain strain are matched with each other one for one.  But whether you are actually experimentally creating the stress and measuring the strain or vice versa,  doesn't matter.  There's one curve, and the distinction there is purely how you built your machine.  Okay?  Now, having said all this, we now go in and we measure this question.  This is assuming that it's linear response.  When you're basically saying that the stress, when you measure the stress, it's equivalent to measuring the strain.  It's basically assuming that it's linear response.  It seems to me the answer is no.  That statement is more general than its linear response.  That is, you have a stress, you have a strain.  If I double the stress, what happens to the strain?  Well, the strain may or may not double in the linear response, it does.  But however, the statement is there's a stress-strain relationship would be true whether it was linear or not.  The only place where you would get into trouble with this is that you could have a hysteresis issue where if the...  you apply different, for example, different strains and the stress changes some place or vice versa,  you might not be able to reach every point on the curve instrumentally very easily by doing one controller or the other.  Now, this is linear response in the sense that the theoretical analysis that is done of linear viscoelasticity  assumes that if I apply a series of forces at different times, the fluid has a response at later times,  and I can just do linear response that is add up all of the responses of the fluid.  Of course, if I've applied the force at several times, there's several time delays, and I will get the answer.  And we'll get to linear in a bit.  So what is the net result, though?  The net result is that if we actually look at the strain, the stress, the force bringing in that area,  and we compare it with the strain, we find something.  And what we can divide up, because this is linear, we can get force, we can divide out the A0,  and we have force per unit area per unit of strain.  And that force turns out to have two pieces.  There is one piece that is determined by a function g' of omega,  and g' of omega is now multiplying the displacement.  Traditionally, it's sine rather than cosine.  Oh, I'll write it as cosine to avoid confusion.  This is cosine omega t.  There is one component where the response has a response that is in phase with the displacement.  So you have a displacement, and there is a restoring force.  That's what it is.  It grows linearly in the displacement.  That is the oscillating displacement.  And then we have another component.  Another component is 90 degrees out of phase, and so there is a response that is 90 degrees out of phase to the displacement,  meaning this is a force that is largest when the displacement is at zero.  Now that may look very odd, but you have in phase and out of phase responses.  And you might ask, what does this all mean?  Why is there a force that's in or out of phase with the displacement?  This is a liquid, after all.  If you made the displacement stop and sat there after a while, the force would disappear because the liquid would just flow.  That's a little more complicated than it sounds.  And the answer is that the low shear viscosity is... ...is actually a low frequency limit of G double prime of omega over omega.  These two objects have names, by the way.  G prime is the storage modulus, and G double prime is the loss modulus.  And I might legitimately wonder, gee, that seems like a peculiar way to represent things.  And I will now show something that makes much clearer from a physicist's point of view what you are looking at.  I am going to multiply each of these objects by one.  And of course, whenever a physicist says, I am going to multiply by one, you know it's going to be some huge object that is in fact one,  but not if you don't look hard.  And I am going to multiply this by one in the form omega square over omega square.  And I am going to apply this one by one in the form omega over omega.  That's perfectly legitimate.  You notice this now looks exactly like the viscosity.  But now I ask you, okay, I have something that goes as omega times the displacement,  and it's 90 degrees out of phase with the displacement.  And I have something that goes as omega square times the displacement.  What am I looking at here?  Well, if you think of a harmonic oscillator, you realize that this object is the velocity,  and I have a term in the force that is determined by the velocity.  Gee, that's a lost term, isn't it?  And here I have a term that goes as omega square times the displacement.  That's the acceleration.  And the thing that goes as the acceleration is a restoring force.  The system looks like it has little springs in it.  And in fact, there are a set of math models blamed on, if I recall correctly, Maxwell.  And this whole picture can be modeled as, here is a system, and it has a spring.  And coupled to the spring is something known as a dashpot.  Dashpots were very important back when people had teletypes.  The dashpot, the plating of the moving object in the teletype comes back very fast,  and you don't want it to whack the far end hard because if it does, the vibration will damage things.  So you have this little piston-like object, and the moving object coming this way has an arm coming out.  There's a rubber disk here, that's the traditional image.  And the rubber disk hits this thing, and there is a little hole here.  And this is a piston, but it's a leaky piston, and so you compress the air and it slows things down.  And then when it's done compressing the air, the air blows out the end,  and you contume the opening because there's a little sliding arm.  And so eventually, after a very small fraction of a second, the moving object on the teletype is brought to an end here.  It's brought to a stop at exactly the right location, and with no unpleasant, sharp acceleration that would damage anything.  Well, yeah, it does that, and it does it via high-quality Victorian engineering.  There are essentially no moving parts in this other than the object you're trying to stop.  There's no active feedback system, no electronics, it's all done mechanically.  And that is a dashpot.  And you can fill the dashpot in this picture with some liquid.  So you have this hanging object in the liquid, and the hanging object in the liquid,  it's just a liquid with a viscosity and no frequency dependence.  And so you, or it has a frequency dependence, and this object represents the storage modulus,  and this piece represents the loss modulus.  So that is a picture of the viscoelastic parameters, the linear, in terms of a storage modulus and a loss modulus.  And you notice that I have reparameterized them starting here and then doing this for consistency, basically.  And once I've done this, the loss modulus looks exactly like the viscosity.  There is, however, and I'm just going to show the qualitative picture first.  Here's, however, a consequence of reparameterizing things.  If I plot G prime or G double prime themselves, you remember they contain something,  I divide out an omega, which I can do safely.  And therefore G prime and G double prime both go to zero.  So it's a little hard to do on a log-log plot, but they both go down.  And at low frequencies, this is more typically a log plot, so you never really get to zero,  there is a linear behavior, which is omega to the power.  And then there are things that happen out here, and perhaps the curves cross.  So if you're making omega larger and larger, eventually the curves turn up again.  I am not going to draw this in any detail, you can find it in standard texts.  The important issue is that if you talk about G prime or G double prime itself, themselves,  you find that they have curves that look sort of like this.  And there's a region where things are reasonably flat, and there's a low frequency region  and a high frequency region I could introduce terminology.  We'll get to that next time.  The main issue is, though, there's a shape for the curves.  If I instead say I am going to go in, and I am going to plot G prime over omega square,  or G double prime over omega, if I do that, I have divided out the leading slope.  And if I divide out the leading slope, that says that at low frequency,  the behavior that I am looking at is frequency independent,  which is sort of what you would naturally expect.  And these curves all look, as you will see from the book, about the same,  namely there's a flat region, there's a rollover,  and then down here there is at least at first a power wall.  And then maybe something interesting happens at larger frequencies.  The question we ask, given that I have drawn these pictures, is,  well, how do I derive or rationalize a curve that explains this?  And so I will give an Anzatz based on the theoretical model we have not yet discussed in detail,  and a renormalization group argument.  We are going to actually invoke, which we have not actually done before,  the hydrodynamic scaling model, which is my model,  and we are going to invoke a renormalization group argument.  The renormalization group argument being invoked is much more general  and qualitative than many renormalization group arguments,  but it will be clear what it is.  So we are going to start out, and here is a straight line representing concentration.  And we will imagine that the viscosity is plotted perpendicular to the blackboard out towards you.  And what we find is the viscosity increases as I go to higher concentration,  and eventually, at least in some systems, we reach the transition concentration C sub t.  And down here is a solution-like regime, and in the solution-like regime,  you see stretched exponential in concentration behavior.  And up here is the melt-like regime M, where you see C to the X behavior.  That is the pure phenomenology.  Now what I say is I have used something called the Altenberger  dollar positive function renormalization group method,  and I applied this in the context of the hydrodynamic scaling model.  And I said I can calculate a certain lead behavior at lower concentrations,  and the Altenberger dollar renormalization group method lets me turn the lower concentration calculation  into the stretched exponential, and we are going to assume that is true.  Now I am going to put another little bit on top of it.  The PFRG, the Altenberger dollar positive function renormalization group method,  actually predicts two sorts of functional behaviors.  The functional behaviors it predicts depend on where what are called the fixed points  of the renormalization group are located.  In particular, if you have a fixed point at zero, then near zero you predict  stretched exponential behavior as is found.  On the other hand, out here someplace, if you have a fixed point that is way out there,  and that is the dominant fixed point.  There can be several fixed points, but as we go along here,  there is always a dominant fixed point.  If the dominant fixed point is out here, you would get power law behavior as is observed.  Next trick.  The viscosity is the low frequency limit of double prime omega over omega.  Viscosity is a function of concentration, and so is the dynamic loss modulus.  Therefore, when we say we are looking along this curve at viscosity,  I could also say, I am going to put another axis on here.  I am going to insert the vertical axis I have skipped over.  I am going to call the vertical axis omega frequency.  I am going to say viscosity is what we measure along the omega more or less,  maybe not quite, zero one.  We measure at low frequencies, and we get the viscosity,  zero frequency limit.  However, it is the zero frequency limit,  and while I wrote it as eta, I could just as well have written it as  Gw prime of omega, and by the way, concentration, over omega.  And therefore, I now have a two variable function,  and I have used the renormalization group argument to work along at omega is zero.  Okay?  Now comes the somewhat bold part.  I will start here.  Yes.  And I will march sideways.  And at first I am in a regime where this fixed point at zero is presumably  still there and dominant.  And then at some point, I move sideways enough,  and these other fixed points, about which I know very little, become dominant.  And therefore, I have moved from a solution-like regime to a melt-like regime.  However, I am marching along at C fixed with omega as the variable,  because I am moving sideways along this graph.  And I am looking at something that is really only a function of omega.  If this picture is correct, and there is a great deal of extrapolation in this,  it's an anzotz, not a derivation.  At lower frequencies, this fixed point is dominant,  and therefore I should have a stretched exponential in frequency.  However, eventually I cross this line, and once I am across this line,  I am in a region where the melt-like fixed points out there are dominant,  and I should have omega to a power behavior.  This doesn't really calculate what the power is at all.  And in order to do the extrapolation even at low frequency,  I would need something that gives me viscosity is 8 to 0 plus presumably,  some intrinsic viscosity, 8 and a lot C, plus some beta omega.  That is, I would need some way to generate a model that gives me the low frequency,  so I could move off 0 at all.  And after I have moved off 0 at all, I can then use the Altman-Burger-Duller  renormalization group to say what happens with frequency.  However, I have not found, I have been working on other things truthfully,  I have not yet found the frequency, yes,  and therefore the assertion is that you find this linear step  and you can then fill in the low frequency behavior.  That has not yet been done, and the crossover and the fixed points out there  have not been done even for concentration.  Okay?  So that is the rationale, and you notice what it predicts.  It predicts stretched exponential in frequencies and lower frequencies,  and it predicts power-long frequencies and higher frequencies.  However, there is some interesting, if you look carefully,  some little bits I have skipped around.  And one bit I have skipped around is what it says about frequency dependence  way out here.  That is, if you are way out there, you're in the mount-like regime all the way,  and therefore you see simply power law and frequency behavior.  Well, you can actually point at data that looks like this,  and the question is, are you really going to see that,  or is it the case that there is a very low frequency,  stretched exponential in frequency regime?  And corresponding to that, is it possible that this curve actually bends over  and very close to zero frequency if you get down to sufficiently low frequencies,  the seed of the X behavior would disappear?  I don't have an answer to that.  The other point is that if we go up to very high frequencies,  is it going to be the case that we have gotten into the omega to the X regime  at very low concentrations, or does this curve bend over so that at very,  very low concentrations where we can just barely see particle interactions at all,  we are still in the concentration to a stretched exponential regime.  That is, the model does not handle this very, does not discuss this,  the model sort of dodges around that, it covers this large area.  Now, you could say, of course, gee,  this picture explains why some systems show the solution like, melt like transition,  and some do not.  Namely, depending on exactly where the fixed points out here are,  it might be the case that in some of these systems,  you chug along the concentration axis all the way to the melt.  This curve does not go to infinity, it stops at the melt.  And even at the melt, the detailed numerical parameter is going into the calculation or such,  that you can get all the way out here, and the fixed point at zero is still dominant.  But there are other systems, and all you need are differences in chemically dependent parameters,  such that these fixed points become dominant at some concentration, and out here you see a power lock.  And therefore, one universal physical model plus chemically dependent parameters  that determine where which fixed point is dominant predicts both behaviors.  Well, that's fine for the hydrodynamic scaling model,  which actually allows you to have two behaviors in one general model.  It is not at all so good for models that assume scaling,  because they assume that you always get scaling out here, which it seems that you do not.  Okay, that is the Anzatz.  And having given you the Anzatz, we have now reached experiment.  We have also reached approximately the end of the hour,  and therefore we have reached a natural point to stop.  Therefore, I am going to stop.  So this has been a lecture finishing off my treatment of viscosity  and advancing to my treatment of the hydrodynamic scaling model  and its Anzatz for predicting linear viscoelasticity.  I'm George Phillies. This is the end of the lecture.
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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 23, Viscosity, Introduction to Viscoelasticity, the Temporal Scaling Onsots. I'm Professor Philly's, and today we're going to finish off our discussion of viscosity and push on to discuss viscoelasticity. Let me recall where we left off roughly speaking. We left off in chapter 12 discussing figures 12.22 to 12.24, which are treatments of the molecular weight dependence of the viscosity of the solution. Now, there's actually a fairly considerable number of measurements of these, and they're actually fairly consistent, namely what they find, what is found with considerable consistency in many systems, is that the viscosity depends on the molecular weight as a stretch exponential and molecular weight. And that's brought out quite emphatically by, for example, figure 12.24. 12.24 are measurements that actually appear in Doyle and Edwards, who report them as measurements done by Onogi et al. The measurements are a little more complete than the measurements in the original paper, actually. Having said that, an interesting feature here is that if you go back into the literature and ask what people said about these measurements, you can find people who say, we have measured the molecular weight dependence of the viscosity, and as predicted, we see a power law dependence of the molecular weight. There actually are measurements that find power law molecular weight dependence. For example, as we discussed earlier, POW at all measurements, which push out to the melt, find close to the melt a power law dependence of viscosity on molecular weight, and volume fraction one, log log plot, what are rather clearly straight lines, viscosity changing as a power law on M, and if you actually say, we'll stay fairly close to the melt and we will exclude points that seem to be deviating from the original straight line, you actually could find in POW at all M to the 3B8. However, most of the measurements you find, which are done at considerably lower concentrations, do not find power law behavior, they find stretched exponential behavior. An interesting example of this is provided, and this is also, in essence, stretching over to the next section, which is a discussion of topology, are the measurements like Grace Lee and L, and Grace Lee look at polymers, and they look at linear chains, and they look at four-armed stars, and they look at six-armed stars. And if you look in at the results they show as a function of concentration, for the linear chain, and their linear chain is a molecular weight of 1.66 megadolptin, as a function of concentration, a log-log plot, they find what is very clearly a straight line, you look at it, it's clearly a straight line, and it's quite clear that you are seeing power law behavior. This is for the two-armed star, the linear chain, two-armed star. On the other hand, for F equals 4, and for F equals 6, very clearly on the same plots, these are chains of about the same molecular weight. This is about 1.95 megadolptin polymer, and the other is about a, what was it, it was about 1.44 megadolptin polymer. For the other two, you very clearly see stretched exponential molecular weight dependencies, these are fixed molecular weight, you're looking at concentration dependencies. If you look at the molecular weight dependence, where they had a considerable number of samples, they were looking at concentrations of about 300 gram per liter of polymer, extremely concentrated systems, and the molecular weight dependence was a straight line for the linear polymer, and was a stretched exponential, two stretched exponentials actually, for the star polymers. As a function, taken as a function of concentration, the upper figure, the linear chain showed the power law behavior, the stars showed a stretched exponential behavior, the viscosity of the star was, for the most part, less than the viscosity of the linear polymer. However, if you look carefully at the measurements against concentration, you can see a point up here, and so if the concentration was taken to be high enough, eventually some of the star polymers, as they're obliged to do given the functional behavior, did show a viscosity that was higher than the viscosity of the stars, of the linear chain, I mean. However, for the most part, star polymers, comparable molecular weights, are up to fairly high concentrations, less viscous than linear chains, and at fixed concentration as a function of molecular weight, again, the stars are less viscous than the linear chain is. Okay. You can also look at results of... We'll get this erased in a second. You can also look at results of Kajura and collaborators. They looked at three-arm stars. Their stars were a poly-alpha methyl styrene. They look at a series of concentrations and molecular weights. You can see the results in figure 12.26. And what they found was that, again, if you're at about the same concentration and about the same molecular weight, the viscosity of the star polymer, this is the F equal 3 star, is less than the viscosity of the linear chain. Now, why is the statement that the star polymers have a lower viscosity than the linear chain of some interest? Well, there are a number of models of polymer dynamics which assert that linear chains have extra modes of motion that are, in principle, not accessible to stars. In particular, for example, the reputation-type models say that if you have a linear polymer, it can creep through holes in a transient lattice like the snake in a bamboo grove. But if I am a star polymer, in order to make forward progress, what I have to do is retract by fluctuation one of my arms. That's difficult to do. I can push it out in a new direction, and now I can take a very modest step forwards. The net result is that one would expect that it is harder for star polymers to move with respect to each other than, at least if they're very concentrated, than it is for linear chains. That does not appear to be quite what we are observing. Question? Can it also be possible that since linear molecules are a linear instruction, they have better packing methods, therefore, they adhere closer to each other? That could be an issue in the melt. However, in the solution, the amount of packing is determined by the concentration. And the answer is that the concept, we make comparison and equal concentration for linear and star. The other thing you should recall with respect to, let us pack chains into a melt, is that I am a star polymer. I have one point where there are three chains coming together. There, if I'm an F equals three star, there is a narrow region around this where there are three of my chains close together. However, if I'm a large star polymer over most of my length, I look exactly like a linear chain. And therefore, this is the sort of effect. You might find interesting effects with small chains, but with large ones, you would not expect that to arise. Thank you. Okay. So we advanced figure, oh, discussion around figure 12.25. And the issue around figure 12.28 and figure 12.27 is suppose we take a polymer chain and we change the solvent quality. What happens if we change the solvent quality? Well, the expectation is that if we are in a theta solvent or close there too, which means we also have to be at the right temperature, we have a polymer chain in the solution. It would rather see itself as its neighbor than it would see something else. And therefore, if we increase the concentration, nothing terribly interesting will happen to the chain radius. I didn't say nothing at all. Because theta, you can't get perfect cancellation of effects. It's an approximation. On the other hand, if you were in a good solvent, we see what we saw in the chapter on dielectric relaxation, which is what has been used for most of these measurements. Namely, as we increase the concentration and look at the chain radius, the chain radius falls with increasing concentration. So these, however, are static properties. They're how big is the chain if we take a single snapshot? We're talking about viscosity here. Viscosity is a dynamic property. And we ask what happens to the behavior of the system if we change solvent quality and such and on. The first answer is, if we look at a series of chains of different molecular weight, the viscosity, as has been the case in most other systems, not all other systems and not at sufficiently high molecular weight, but concentration rather in those systems, the concentration dependence is a stretched exponential. And what we happen, fine, as we increase the polymer molecular weight and do our measurements, we find that the parameter alpha increases and the parameter nu falls. So there is alpha and there is nu. And both parameters are molecular weight dependent. The second thing we can do is to compare these parameters in a good solvent and in a theta solvent. And what we find is that in a theta solvent as discussed here, the parameter nu is from 0.8 to 0.954. And if we go, however, take the same material over to a good solvent, the parameter nu is 0.6 to 0.73. That is, we are going in and when we change the solvent quality, when we go into a better solvent, the parameter nu falls. The model that derives this equation, in fact, predicts this behavior, we will eventually get to it in later lectures. If you look at the figures, especially 12.28, this change of nu is physically quite visible. If you look at a semi-log plot, log viscosity versus concentration, for a theta solvent, the concentration dependence of eta is fairly close to a straight line because this is fairly close to a stretched exponential. But if you look at the measurements in the good solvents, you typically see a more pronounced curvature because you are looking at a stretched exponential rather than a pure exponential. Now, having said this, let us push ahead and consider the general features of viscosity. We have looked at lots of specific measurements and we did, of course, find that the viscosity depends on the polymer concentration and the polymer molecular weight, not to mention the one could put in the topology and one could put in the solvent quality. With respect to topology, there was a very long dispute, this goes back to a considerable piece, about the viscosity of ring polymers, but that was mostly melts, which the book does not cover. The issue there was, well, you can try to make melts, you have to ask, are the, you can try to make rings and put them into melts. Here is a ring, is it a nice open loop or during your synthesis process, did it get itself all tied in knots so that it's a ball? Under modern conditions, if you want to answer that question, you can resort to using DNAs since you can synthesize DNAs with absolute molecular weight control. But absolutely mean we have something that's 180,000 base pair, meaning its molecular weight is tens of millions, and the molecular weight is, well, there are fluctuations because there are several isotopes of carbon and deuterium in nature, a carbon and deuterium hydrogen, I should say in nature, but the molecular weights are totally exactly controlled, no fluctuation. You can also synthesize things, this is synthetic DNA, not stuff that occurs in nature, that is a star and completely mono-dispersed. You can also synthesize, it happens naturally, some viral DNAs do this. DNAs that are ring polymers that are again completely mono-dispersed. Electron micrographs of those rings indicate that the rings are completely open, they are tied up in knots at all. However, having said, the ring polymers are mono-dispersed, the star polymers are truly mono-dispersed, synthetic DNA comes that way, there is the significant practical issue that people have not done studies of DNA viscosity that are as extensive as the viscosity studies that have been done on synthetic polymers. Okay, having said this, let us look at the question of what we find. And the first point, which I am going to emphasize significantly more than was emphasized in the book, is that eta is not a universal function of C and M. That is, if you think that you can take your viscosity measurements and yes, there will be some chemical dependent constants and there will be things that depend on the concentration in the molecular weight or maybe the length of strands in the molecular weight. Some representation of amount of material and size of the chain. If you think that all of the chains are the same because of that and therefore there should be a universal function that gives you viscosity as a function of concentration, you are going to be disappointed. Instead, it is quite clear from the literature that viscosity is not a universal function of C and M. Instead, there are two distinct phenomenologies and in one of the phenomenologies, you see simply a stretched exponential which in measurements of Taeger goes up, Taeger and a few other people go out more or less to melt. And in another set of systems, you see a stretched exponential up to a point and then beyond the point, you see a power wall. It is worthwhile to recall that the same phenomenology this class is occupied by spherical colloids. Spherical colloids show the same behavior. The slope of the power law depends on the exact material. For spherical colloids, the slope is appreciably steeper than for linear chains. And if you look at many arm star polymers, the slope creeps up and makes a transition from linear chain behavior up to very nearly heart-sphere behavior. It should be clear that any model that explains this phenomenon is subject to the constraint that it must be of work for spherical colloids. Spheres can do a number of things, but form entanglement lattices is not really one of them. And form and reptate is certainly not one of them. And so we have the issue that the viscosity behavior might not be quite what you would have expected from some of your readings from some theorists. Okay. Furthermore, if you merge up to viscosities above about 10 to the 8, that's eta over eta-zero above about 10 to the 8. And most of those measurements are literature studies due to Drival, though there's some nice work due to Colby. However, if you look at that, it does appear that if you get above here, instead of continuing along on the power law, there is an upwards deviation. The nature of the upwards deviation is not quite certain. One might, however, for example, propose since we're at extremely high concentrations that it's some sort of a jamming transition in which things actually mechanically get in each other's way, and the chains can't slide across each other because they can't move after all their rigid overshort distances. They can't move sideways as easily enough to get through each other. However, that is a fairly specific, that's a fairly rare exception because there are not a lot of studies that have looked at this. In particular, if you look at predictions that eta should be some c to the x, m to the y, that prediction is very generally not true in polymer solutions. There are some exceptions. There is this family, but predictions that you can simply assume scaling are not sustained. If you want to say we have scaling, you need to explain why sometimes you have scaling and sometimes you don't. Otherwise, you can't just assume. Okay. So that is the sort of behavior we get. We might then ask, well, what other generalizations can we find? And one answer is, for a very large family of parents, we have eta over eta zero is e to the some constant alpha c to the nu, and now I'm pulling the molecular weight of dependence of alpha. We'll actually see graphs of that dependence in a few moments. But if we actually look at the measurements and ask, okay, we get this thing to fit, what do we see? Well, gamma is usually in a range around a half. And I can be more precise than that, but that's a reasonable approximation. In point of fact, if we say, oh, I should add something. We go from theta to good solvents. We go from theta solvents to good solvents. The parameter nu goes down, and what I will call alpha m, because it's alpha with m pulled out, what I will call alpha m, m to the gamma, what's left, what's not c to the nu, increases as we go from theta to good solvents. Generalization. Okay, so what behavior do we see for gamma? And the answer is seen in figure 12.30, where we plot alpha, which in terms of this equation is alpha m, m to the gamma. It's the coefficient of c. And we plot alpha against polymer molecular weight. Now, if I just showed you the points, your reaction would be, gee, that was a nice piece of graph paper, Professor Philly's. At least it was very nice before you stood way back and started firing the shotgun at it, and the points are all over the place. However, if you look carefully at the points, I have labeled the points by the chemical family of the polymer. So we have identified separately each set of homologous polymers, polymers that are chemically identical with different lengths. And if you do that, you see that for each single family of homologous polymers, alpha lies very much on a straight line. The straight line has a slope. So this is a log-log plot, so we say alpha goes as m to the power gamma. And what can we say about gamma if we look at all of the measurements? That is, all of the measurements were the same way. I've looked at a large number of different polymer molecular weights. And other than that, things are consistent. And the answer is that gamma is typically in the range 0.6 to 0.67. One can find a case where it's 0.5, if I recall correctly, 0.94, where it's quite large, and you can find a case where it's 0.51. If you look at this, you notice there are different slopes for different materials, and therefore it's not quite a universal constant. But for a given set of samples, the power law dependence is actually fairly sharp. This result is also implicit in the systematic review of Dervol and many collaborators, and all of the wonderful Russian data he compiled in one place for us, because what Dervol says is that C eta is a good reducing variable. That is, if you take all of these measurements and you plot them as functions of C times the intrinsic viscosity rather than C, so we have eta log log again, C eta here, and we find a whole series of different polymer molecular weights that the measurements lie on the same curve. Now, in order for the measurements to lie on the same curve, when you are doing this plot, it must be the case that eta is some function of C eta, which means that all of the molecular weight dependence of the viscosity has to be built into the molecular weight dependence of the intrinsic viscosity. You can't have any other molecular weight dependence out here someplace, because if you did, when you plotted viscosity for a series of polymers of different molecular weights, they wouldn't lie on the same line. They lie on the same line because viscosity is a function of C times the intrinsic viscosity. That statement, universal function, goes all the way up to the melt. However, it is an important fact that the intrinsic viscosity depends on M as a power, and therefore this equation says that the viscosity is a function of, what is it a function of? C times, and there's some constants, M to the A. An example of a function that has the property that viscosity has this dependence is given by this function. Now, you would have to say this is e to the alpha M, Cm to the A to the power of nu, and therefore gamma, this gamma is equal to A times nu, the exponent of an exponent you obtained by multiplying the exponents. And therefore this equation and that equation are totally consistent so long as gamma, A, and nu are related like this. Nonetheless, your Vol's results are entirely consistent with everything we've said. I would be delighted to say that I can draw a pretty picture of how nu depends on molecular weight and quality, but that really isn't quite true. The first obstacle is that if you look at this formula for the viscosity, A to the proportional to e to the alpha C to the nu, nu is the exponent of an exponent, which means it's a bit hard to measure. Nu doesn't change a whole lot if we said nu is almost always in the range 1 half to 1, that's correct. And therefore there's not a lot of range, and if there's any difficulty at all measuring nu accurately, you run into a problem that graphs are a bit scattered. There are some other parameters, however, that have been sort of hidden, and we can point at a few of them. One we can talk about, we now advance to figure 12.33, is the transition concentration C sub t. Transition concentration, yes, we are now limiting ourselves to systems that show the solution-like, melt-like transition, that's the transition we actually found. And we charge up like this, and there is a concentration, I'll use it as a subscript, C sub t, which the transition occurs, and correspondingly there is a viscosity, A to sub t. When we discuss this for colloids, well, for spheres, there's really only one curve, because the volume fraction is the only variable, and phi t was about 0.41 or a bit higher, and the viscosity of the transition divided by the viscosity of the solvent was about 10 plus or minus 5. That's very clearly not the concentration and viscosity at which you start getting this phase effect. And the question is now for linear polymers, how does the transition concentration behave? And the answer given in 12.33 is there are a bunch of points, a bunch of points are more or less on a straight line, and the transition concentration goes as m to the minus o, 1.1 or so. And the answer is as you increase the polymer molecular weight, the concentration at the transition falls. Of course, you need a set of measurements where the same material that shows the transition has been studied for a lot of different polymer molecular weights, otherwise you should be hesitant about believing the fit. The results that do this are on poly and hexalisocyanate, which we've discussed before, and you look at those and g, you get roughly this behavior. So you might, okay, so that's the transition concentration. You might also say, well, g, that's the transition concentration, and maybe we should re-express the transition concentration in natural units. When I say a natural concentration unit for polymer solutions, what people generally mean is c times the intrinsic viscosity. Now, there are people who will say c over c star, where c star is the overlap concentration, but in fact the overlap concentration is usually inferred from the intrinsic viscosity, namely c star equals the sum of number in the range. I've seen things as small as one of as large as four in the literature over eta. So this is a distinction but not a difference. Okay, so we look at c eta and we will ask at what transition concentration in natural units do we see the transition? And if you go through all of the accumulated measurements, you can find transitions as high as 80. You can find things that occurred at about 35, and that is a more typical value. You can also find transition concentrations as low, for example, hydroxypropylcellulose, which Carol Streletsky and I and others studied for. And so if you look at the transition concentration in natural units, it covers a very wide range of values. You should realize this is a significant part of the way to melt. The melt concentrations in D'Rivolve's Metals analysis were not more than about 300. The transition concentration for hydroxypropylcellulose, the intrinsic viscosity was one over, oh, one or two grams per liter, and the transition occurred at around six grams per liter. So if we say we'll put the concentration in natural units, you don't see any tendency of measurements to collapse onto each other. There is, however, something you can do which gets the measurements to match. Not perfectly, but actually pretty well. And the answer is if you look at all of those curves where you see something like this for linear polymers, not for spheres, for the measurements where people looked at a lot of different molecular weights and you look at their graph, you realize that the crossovers occur at about the same viscosity. And in fact, it is approximately but not exactly universally true that eta t over eta is some number like two or three hundred. That is, there is not a concentration at which you see the crossover, but there does appear to be a universal viscosity at which the crossover sets in except in hard sphere systems which may be different. And the crossover at viscosity occurs when eta over eta is zero, the reduced viscosity is a few hundred. So you produce something that's fairly viscous but not completely impossible to work with. Oh yes, you remember we said there was power law behavior out here? You might legitimately ask, well that's very nice, what is the exponent? And the answer for a fair number of systems is that x is about three and a half. However, you can find systems in which x is much larger, well I don't know, much larger, but somewhat larger. For example, in hydroxypropylcellulose, x is about four point four. And therefore there is a power law exponent and I can even tell you what the power law exponent is. Let us skip back from the systems that do show the transition which, well it is universal, a universal transition is a function of the transition viscosity. It's not a universal transition viewed as a function of the concentration. And it doesn't even happen in some systems. So let us go back and let us look at the systems in which we do not see a transition which is a large number of them. Going out in some cases to very high viscosities, I seem to recall reduced viscosities up to about ten to the seven in a few cases in which no transition is seen. Okay, so there is the curve and one of the things you might legitimately ask is, well, G, couldn't we manage to describe this measurement perhaps with a couple of power laws? And if you go through the literature, you find people who draw a power law curve up here, which is sort of tangential or goes through the points over some distance. And you find down here typically a linear curve. And in between there is some transition which is sort of talked over. As G, there is a transition that is broad. And therefore you might propose, yes, the measurements actually show a crossover from one type of behavior to the other, but the crossover is a little hard to see because there are other things going on at the same time. So there is not a clear transition. The way there would be if there was a phase transition, which I don't think anyone has proposed for, well, that isn't quite true. Almost no one has proposed for these systems. So there is a transition, but it's clearly not a phase transition. And the question is, is this a reasonable description? And the answer, I think, is that if we look at e to the alpha c to the nu behavior, the stretched exponential behavior marches out, alpha and nu are independent of concentration. They're proportional to c to the zero, independent of concentration. And that's true along the whole curve. And therefore, viewed from this perspective, there is no transition. There is a single smooth curve that just chugs ahead. So far so good. Okay. I have just run us out of chapter 12. We have now come to the end of the discussion of viscosity. I am now going to advance to discuss viscoelasticity. And in order to do that, I'm going to have to draw a picture. The idealized picture is here are two infinitely long plates. Now, that's a little dangerous because plates aren't infinitely long. And there are several ways around this, for example, bend plates in circles. And you have two concentric cylinders, or you have a flat plate, and you have a rotating cone. You may say, why would you want a rotating cone? And the answer is that the velocity up here, v, is omega r. That's velocity, oh, parallel to the plate. This distance, distance out from the center, increases as r. And therefore, this distance, because it's a cone, this distance is proportional to r. And therefore, if I look at the velocity of the plate over the distance from the plate to the bottom, and there is a velocity gradient here, and the velocity gradient is uniform over the distance. Now, you have to be a little careful with this, because there's going to be some question whether things are uniform as you move this way. There are potential complications. Nonetheless, we shall persevere, and I shall draw this pretty picture. And the bottom plate is stationary, and the top plate is being moved back and forth. And the top plate has a displacement, which I'll call d for the moment, and the displacement is some amplitude A cosine omega t. That is, we have a displacement, and the extent of the displacement oscillates as a harmonic in time. So you can do that equally with either that or that. The displacements are fairly small. Now, you might say, well, gee, could you get information out if you made the displacements large? And the answer is yes, and there is an object, an experimental method called large angle oscillatory shear. And we shall reach Laos in the next chapter, chapter 14. However, we have just reached chapter 13, which talks about linear viscoelastic behavior. OK, so we have this displacement, and in order to get this displacement, we have to apply a force on the upper plate to move it back and forth. Now, if you think about this for a moment, you realize that if the plate has inertia, you're also accelerating the plate at the same time, and you have to deal with this, but that's basically an experimental question. And the inertia of the plate has to be made small enough, in some sense, that you can measure the part of the force that is due to the fact there's a liquid in here. I'm going to call this A0, the amplitude of oscillation. We'll come back to other notation for it in a moment. And the reason is that if we look at this system, if you make the plate bigger and bigger, the amount of force you need goes up. But it's basically a linear process, and therefore the interesting thing is the force per unit area. And the force per unit area is known as the stress. And the first force per unit area, which by the way is going to be oscillating as a function of time, the stress is determined by the amount of oscillation, the displacement divided by the distance L between the plates. That is, if I take this whole apparatus and make it twice as big vertically, the amount of oscillation seen locally by a molecule, here's a molecule, and it starts out and has some neighbors along this line, and when we increase D to its maximum, this is oscillating, the neighbors are displaced sideways one way or the other in an back-and-forth manner, and the amount of local displacement is determined by the ratio of the linear displacement of the plate to the distance between the plates. And this is known as the strain. So far so good. Now, suppose we actually do this experiment. There are a couple of different ways we could do it. One thing we could do is to apply an oscillatory force up here and measure the motion. And so we have a force which is very precisely controlled and shifts things back and forth, and we ask how the plate moves. We could also have a driver and feedback electronics, and we move the plate back and forth through a fixed distance, and we instrumentally determine how much force we're having to exert so that the plate actually does this. Those two experiments are equivalent. That is, they measure a series of curves where a certain amount of force per unit area, a certain stress, and a certain strain are matched with each other one for one. But whether you are actually experimentally creating the stress and measuring the strain or vice versa, doesn't matter. There's one curve, and the distinction there is purely how you built your machine. Okay? Now, having said all this, we now go in and we measure this question. This is assuming that it's linear response. When you're basically saying that the stress, when you measure the stress, it's equivalent to measuring the strain. It's basically assuming that it's linear response. It seems to me the answer is no. That statement is more general than its linear response. That is, you have a stress, you have a strain. If I double the stress, what happens to the strain? Well, the strain may or may not double in the linear response, it does. But however, the statement is there's a stress-strain relationship would be true whether it was linear or not. The only place where you would get into trouble with this is that you could have a hysteresis issue where if the... you apply different, for example, different strains and the stress changes some place or vice versa, you might not be able to reach every point on the curve instrumentally very easily by doing one controller or the other. Now, this is linear response in the sense that the theoretical analysis that is done of linear viscoelasticity assumes that if I apply a series of forces at different times, the fluid has a response at later times, and I can just do linear response that is add up all of the responses of the fluid. Of course, if I've applied the force at several times, there's several time delays, and I will get the answer. And we'll get to linear in a bit. So what is the net result, though? The net result is that if we actually look at the strain, the stress, the force bringing in that area, and we compare it with the strain, we find something. And what we can divide up, because this is linear, we can get force, we can divide out the A0, and we have force per unit area per unit of strain. And that force turns out to have two pieces. There is one piece that is determined by a function g' of omega, and g' of omega is now multiplying the displacement. Traditionally, it's sine rather than cosine. Oh, I'll write it as cosine to avoid confusion. This is cosine omega t. There is one component where the response has a response that is in phase with the displacement. So you have a displacement, and there is a restoring force. That's what it is. It grows linearly in the displacement. That is the oscillating displacement. And then we have another component. Another component is 90 degrees out of phase, and so there is a response that is 90 degrees out of phase to the displacement, meaning this is a force that is largest when the displacement is at zero. Now that may look very odd, but you have in phase and out of phase responses. And you might ask, what does this all mean? Why is there a force that's in or out of phase with the displacement? This is a liquid, after all. If you made the displacement stop and sat there after a while, the force would disappear because the liquid would just flow. That's a little more complicated than it sounds. And the answer is that the low shear viscosity is......is actually a low frequency limit of G double prime of omega over omega. These two objects have names, by the way. G prime is the storage modulus, and G double prime is the loss modulus. And I might legitimately wonder, gee, that seems like a peculiar way to represent things. And I will now show something that makes much clearer from a physicist's point of view what you are looking at. I am going to multiply each of these objects by one. And of course, whenever a physicist says, I am going to multiply by one, you know it's going to be some huge object that is in fact one, but not if you don't look hard. And I am going to multiply this by one in the form omega square over omega square. And I am going to apply this one by one in the form omega over omega. That's perfectly legitimate. You notice this now looks exactly like the viscosity. But now I ask you, okay, I have something that goes as omega times the displacement, and it's 90 degrees out of phase with the displacement. And I have something that goes as omega square times the displacement. What am I looking at here? Well, if you think of a harmonic oscillator, you realize that this object is the velocity, and I have a term in the force that is determined by the velocity. Gee, that's a lost term, isn't it? And here I have a term that goes as omega square times the displacement. That's the acceleration. And the thing that goes as the acceleration is a restoring force. The system looks like it has little springs in it. And in fact, there are a set of math models blamed on, if I recall correctly, Maxwell. And this whole picture can be modeled as, here is a system, and it has a spring. And coupled to the spring is something known as a dashpot. Dashpots were very important back when people had teletypes. The dashpot, the plating of the moving object in the teletype comes back very fast, and you don't want it to whack the far end hard because if it does, the vibration will damage things. So you have this little piston-like object, and the moving object coming this way has an arm coming out. There's a rubber disk here, that's the traditional image. And the rubber disk hits this thing, and there is a little hole here. And this is a piston, but it's a leaky piston, and so you compress the air and it slows things down. And then when it's done compressing the air, the air blows out the end, and you contume the opening because there's a little sliding arm. And so eventually, after a very small fraction of a second, the moving object on the teletype is brought to an end here. It's brought to a stop at exactly the right location, and with no unpleasant, sharp acceleration that would damage anything. Well, yeah, it does that, and it does it via high-quality Victorian engineering. There are essentially no moving parts in this other than the object you're trying to stop. There's no active feedback system, no electronics, it's all done mechanically. And that is a dashpot. And you can fill the dashpot in this picture with some liquid. So you have this hanging object in the liquid, and the hanging object in the liquid, it's just a liquid with a viscosity and no frequency dependence. And so you, or it has a frequency dependence, and this object represents the storage modulus, and this piece represents the loss modulus. So that is a picture of the viscoelastic parameters, the linear, in terms of a storage modulus and a loss modulus. And you notice that I have reparameterized them starting here and then doing this for consistency, basically. And once I've done this, the loss modulus looks exactly like the viscosity. There is, however, and I'm just going to show the qualitative picture first. Here's, however, a consequence of reparameterizing things. If I plot G prime or G double prime themselves, you remember they contain something, I divide out an omega, which I can do safely. And therefore G prime and G double prime both go to zero. So it's a little hard to do on a log-log plot, but they both go down. And at low frequencies, this is more typically a log plot, so you never really get to zero, there is a linear behavior, which is omega to the power. And then there are things that happen out here, and perhaps the curves cross. So if you're making omega larger and larger, eventually the curves turn up again. I am not going to draw this in any detail, you can find it in standard texts. The important issue is that if you talk about G prime or G double prime itself, themselves, you find that they have curves that look sort of like this. And there's a region where things are reasonably flat, and there's a low frequency region and a high frequency region I could introduce terminology. We'll get to that next time. The main issue is, though, there's a shape for the curves. If I instead say I am going to go in, and I am going to plot G prime over omega square, or G double prime over omega, if I do that, I have divided out the leading slope. And if I divide out the leading slope, that says that at low frequency, the behavior that I am looking at is frequency independent, which is sort of what you would naturally expect. And these curves all look, as you will see from the book, about the same, namely there's a flat region, there's a rollover, and then down here there is at least at first a power wall. And then maybe something interesting happens at larger frequencies. The question we ask, given that I have drawn these pictures, is, well, how do I derive or rationalize a curve that explains this? And so I will give an Anzatz based on the theoretical model we have not yet discussed in detail, and a renormalization group argument. We are going to actually invoke, which we have not actually done before, the hydrodynamic scaling model, which is my model, and we are going to invoke a renormalization group argument. The renormalization group argument being invoked is much more general and qualitative than many renormalization group arguments, but it will be clear what it is. So we are going to start out, and here is a straight line representing concentration. And we will imagine that the viscosity is plotted perpendicular to the blackboard out towards you. And what we find is the viscosity increases as I go to higher concentration, and eventually, at least in some systems, we reach the transition concentration C sub t. And down here is a solution-like regime, and in the solution-like regime, you see stretched exponential in concentration behavior. And up here is the melt-like regime M, where you see C to the X behavior. That is the pure phenomenology. Now what I say is I have used something called the Altenberger dollar positive function renormalization group method, and I applied this in the context of the hydrodynamic scaling model. And I said I can calculate a certain lead behavior at lower concentrations, and the Altenberger dollar renormalization group method lets me turn the lower concentration calculation into the stretched exponential, and we are going to assume that is true. Now I am going to put another little bit on top of it. The PFRG, the Altenberger dollar positive function renormalization group method, actually predicts two sorts of functional behaviors. The functional behaviors it predicts depend on where what are called the fixed points of the renormalization group are located. In particular, if you have a fixed point at zero, then near zero you predict stretched exponential behavior as is found. On the other hand, out here someplace, if you have a fixed point that is way out there, and that is the dominant fixed point. There can be several fixed points, but as we go along here, there is always a dominant fixed point. If the dominant fixed point is out here, you would get power law behavior as is observed. Next trick. The viscosity is the low frequency limit of double prime omega over omega. Viscosity is a function of concentration, and so is the dynamic loss modulus. Therefore, when we say we are looking along this curve at viscosity, I could also say, I am going to put another axis on here. I am going to insert the vertical axis I have skipped over. I am going to call the vertical axis omega frequency. I am going to say viscosity is what we measure along the omega more or less, maybe not quite, zero one. We measure at low frequencies, and we get the viscosity, zero frequency limit. However, it is the zero frequency limit, and while I wrote it as eta, I could just as well have written it as Gw prime of omega, and by the way, concentration, over omega. And therefore, I now have a two variable function, and I have used the renormalization group argument to work along at omega is zero. Okay? Now comes the somewhat bold part. I will start here. Yes. And I will march sideways. And at first I am in a regime where this fixed point at zero is presumably still there and dominant. And then at some point, I move sideways enough, and these other fixed points, about which I know very little, become dominant. And therefore, I have moved from a solution-like regime to a melt-like regime. However, I am marching along at C fixed with omega as the variable, because I am moving sideways along this graph. And I am looking at something that is really only a function of omega. If this picture is correct, and there is a great deal of extrapolation in this, it's an anzotz, not a derivation. At lower frequencies, this fixed point is dominant, and therefore I should have a stretched exponential in frequency. However, eventually I cross this line, and once I am across this line, I am in a region where the melt-like fixed points out there are dominant, and I should have omega to a power behavior. This doesn't really calculate what the power is at all. And in order to do the extrapolation even at low frequency, I would need something that gives me viscosity is 8 to 0 plus presumably, some intrinsic viscosity, 8 and a lot C, plus some beta omega. That is, I would need some way to generate a model that gives me the low frequency, so I could move off 0 at all. And after I have moved off 0 at all, I can then use the Altman-Burger-Duller renormalization group to say what happens with frequency. However, I have not found, I have been working on other things truthfully, I have not yet found the frequency, yes, and therefore the assertion is that you find this linear step and you can then fill in the low frequency behavior. That has not yet been done, and the crossover and the fixed points out there have not been done even for concentration. Okay? So that is the rationale, and you notice what it predicts. It predicts stretched exponential in frequencies and lower frequencies, and it predicts power-long frequencies and higher frequencies. However, there is some interesting, if you look carefully, some little bits I have skipped around. And one bit I have skipped around is what it says about frequency dependence way out here. That is, if you are way out there, you're in the mount-like regime all the way, and therefore you see simply power law and frequency behavior. Well, you can actually point at data that looks like this, and the question is, are you really going to see that, or is it the case that there is a very low frequency, stretched exponential in frequency regime? And corresponding to that, is it possible that this curve actually bends over and very close to zero frequency if you get down to sufficiently low frequencies, the seed of the X behavior would disappear? I don't have an answer to that. The other point is that if we go up to very high frequencies, is it going to be the case that we have gotten into the omega to the X regime at very low concentrations, or does this curve bend over so that at very, very low concentrations where we can just barely see particle interactions at all, we are still in the concentration to a stretched exponential regime. That is, the model does not handle this very, does not discuss this, the model sort of dodges around that, it covers this large area. Now, you could say, of course, gee, this picture explains why some systems show the solution like, melt like transition, and some do not. Namely, depending on exactly where the fixed points out here are, it might be the case that in some of these systems, you chug along the concentration axis all the way to the melt. This curve does not go to infinity, it stops at the melt. And even at the melt, the detailed numerical parameter is going into the calculation or such, that you can get all the way out here, and the fixed point at zero is still dominant. But there are other systems, and all you need are differences in chemically dependent parameters, such that these fixed points become dominant at some concentration, and out here you see a power lock. And therefore, one universal physical model plus chemically dependent parameters that determine where which fixed point is dominant predicts both behaviors. Well, that's fine for the hydrodynamic scaling model, which actually allows you to have two behaviors in one general model. It is not at all so good for models that assume scaling, because they assume that you always get scaling out here, which it seems that you do not. Okay, that is the Anzatz. And having given you the Anzatz, we have now reached experiment. We have also reached approximately the end of the hour, and therefore we have reached a natural point to stop. Therefore, I am going to stop. So this has been a lecture finishing off my treatment of viscosity and advancing to my treatment of the hydrodynamic scaling model and its Anzatz for predicting linear viscoelasticity. I'm George Phillies. This is the end of the lecture.
Philly's book, Phenomenology of Polymer Solution today, this lecture is lecture 23, Viscosity,