I'm Professor Filleys, and this is our next lecture on the topic of polymer solution dynamics. Today, we began our discussion of chapter 16, inferences about polymer dynamics based on the known and observed phenomenology.

We're going to start with the simplest sort of discussion, and then we'll work up to look at things that are a bit more complicated. So the first issue is, there actually are a very large number of theoretical models out there that make different, sometimes quite contradictory, predictions about how polymers move in solution.

If I had done, if there were suggestions that I should do this, a review of theoretical models, I would have another book as thick as the first, with one minor difference. Almost all of those models are certainly at least somewhat wrong, because they don't all agree with each other. And therefore, most of the book would be filled with nonsense.

This didn't seem like a very useful thing to write, I can assure you, having spent four years writing the phenomenology. So however, there is a nice feature of most of those models. I'm not going to say absolutely every one of them. And the nice feature of most of them is that most of those models fall into two general classes.

And the two general classes are scaling and exponential. The issue that we look at first

is, well, we may not be able to cram every single model to those classes, but we can certainly get most of them into those classes. And that's good enough for some first steps. So what we're going to do is to say, we have these two groups of models that predict either scaling laws or stretched exponentials,

or in some cases, both. And we ask, can we use the measurements to determine what we're looking at? And so the first question is, can you tell?

So if we have a series of measurements of, I'm not saying which parameter, as a function of which other parameter, the question is, if you look at a linear or a log-log plot or whatever, can you decide which set of models

is consistent with the data? Or is it that the data aren't really up to telling the difference? And the answer is clarified by several sets of measurements that we've talked about already. For example, the electrophoretic mobility is a function of probe size. The storage and loss moduli, which

are functions of frequency. And in both of these cases, we had, using the same set of measurements on the same plot, we had for this, we had a stretched exponential region, which was very clearly stretched exponential. And we had a power law region, which was pretty clearly power law, though the variation

vertically there was a bit smaller. And for the moduli, as normalized, we had a region which was quite clearly, on a log-log plot, a stretched exponential. And as a consequent region, which

was very clearly a power law. And this would go over a fair number of orders of magnitude. So there was no question you really were on a log-log plot, looking at a straight line. Very great experience. You can see these in figures such as O3.2, 12.12, or O13.2.

You can actually very clearly tell when you have a stretched exponential, when you have a power law. And you have measurements which on a log-log plot, like these dots, very clearly lie on a smooth curve.

Now, in many cases, it's a mathematical requirement. It's the statement that you have a function that has derivatives. You can say there is some region of the curve which is at least tangential to a straight line.

And so you can say you see scaling law someplace. However, the model doesn't predict where you see that with any precision or at all. Furthermore, no matter what slope you've seen within some fairly wide ranges you've predicted, you would have found it someplace.

And so simply saying, you have a tangential scaling law when you cannot predict where you should have seen the scaling law. That is, if you say you saw this slope, were you supposed to see it here? Or are you supposed to see it up there where it didn't work as well? Well, that's not a scientific theory yet,

because it's not falsifiable. In addition, you will find people in the literature who will take the smooth data and they will fit it to not one,

but several power law regimes separated by crossovers. And you should look carefully and ask how many data points are being covered by some of the power law regimes. I found cases where the number is two or three. And this doesn't really work. So having said that, we can tell the difference.

We can do this numerically. We then ask, well, what do we see? And what we see is that the stretched exponential models, the models that predict log-log plots

that are smooth curves are at least possibly correct. And models that predict scaling behavior, that is extended regions in which log variable, log c plots give you straight lines are not consistent with the measures.

There are, however, localized exceptions. And in fact, the localized exceptions we have already used because the localized exceptions, particular regions of particular types of measurements

are how we validate the claim that we can tell the difference between a stretched law and a power law, a stretched exponential and a power law, how we can tell them apart experimentally. If we couldn't do that, there'd be no real point of going any further.

Now, once upon a time, there were critiques of the stretched exponential model and the phrasing of the critique was unusually flexible.

However, let us look at something here. So here are a set of measurements that are fit by a stretched exponential.

And the fit actually has three parameters. On the other hand, here is a fit in which we have a set of measurements that in some region really are fit by a power law. But if you look at the power law, the power law has, say, an omega zero

and then a power law for omega bar to the x. Oh, omega bar times omega to the x and there is a parameter, the place where you start. And here is omega bar,

which is the height of the function and omega equals one. The function may not actually go through at that point. It's the extrapolation to omega one and there is x, the slope. And guess what? That's three parameters also.

That's a feature of scaling laws that only apply in some region. The cutoffs at the two end are parameters. And if you set the cutoff with someplace else, your RMS error, if you said the cutoff was down here

and the data did this, your RMS error would go through the ceiling. So, having said that, you have two sets of functions, each of which are characterized by three parameters. G, the functions are not singular

for the accessed values of parameter space and therefore one set of functions is exactly as flexible as the other. They're each three parameter functions. They each cover some limited range of function space, different ranges, but the limits must be exactly the same

in terms of what measure of function space they're able to probe because they each have three free parameters. Now, you might then ask, well, gee, where did the line about unusually flexible come from? And I shall now repeat an anecdote which I did not put into the book. And the anecdote I happen to know

because some years ago we had a visitor. And the visitor actually talked about something else except she or he, I'm not saying, was at a research group meeting at a French research group centered in Paris, a theoretical group, at which they had to confront the fact

I'd started publishing the stretched exponential papers. And the question was what was to be said about it. And finally, the supreme leader of the group said, well, what we will say is that the function is unusually flexible and that's why it works. And there were a couple of graduate students who said,

you know, the fits are very, very good. The agreement is excellent and maybe we should take this seriously. And when you have research groups based on leadership from the top, or so the description to me went, eventually we came up with,

no, we are going to say it's very flexible. And it's a very clever throwaway line but it's not really very solid scientific criticism. But now you know where the unusually flexible came from and why the line started cropping up in so many places at exactly the same time. I mean, you could imagine it's the same idea

and 10 or 15 people think of it once but no, that's not quite what happened. So now you've heard a little scientific gossip. Okay, now we have a few limitations on this. What sort of limitations do we have?

Let's start with a significant one. My book does not treat polymer melts. Nonetheless, if we look at discussions of polymer melts, there is fairly good agreement that the viscosity scales as M to a power

usually said experimentally to be 3.4 and the self-diffusion coefficient scale to the power usually said to be, oh, well, let's see, viscosity goes up with increasing M,

diffusion goes down with increasing M but there are several papers which find something closer to that. And we'll see why there is some issue as to what these two numbers are. We'll get to that later. The important issue is that if you look at several transport parameters

and these are the ones that were traditionally examined. If you look at transport parameters, you find power laws in M. However, in solutions, what I have been saying is, for example, dS goes as d0 e to the a.

It's not going to be alpha this time because we're going to pull out the molecular weight dependence. This is self-diffusion, not tracer. There's only one molecular weight there. And having pulled this out, we then ask, gee, these don't attach

to each other very nicely. Now, what this would mean is, here's the self-diffusion coefficient in the melt. Here we are in solution and we would typically say

that diffusion and solution might be faster but this function is going to fall, I'll make a positive number, as an exponential and therefore, at some molecular weight, you would say that dS in solution,

if you just extrapolate the function, would be less than d in the melt. Well, that's an unreasonable outcome. At least most people would agree it's unreasonable. If you start with the melt and start sticking solvent molecules in, and there are systems where you can do this, you expect the small molecules

of most reasonable solvents to function as plasticizers and to soften the melt. And in fact, anyone who is hearing this has seen plasticizers in action. There is this wonderful stuff called Teflon tape, which is a substitute for other ways of attaching plumbing fixtures.

And Teflon tape is this nice, flexible, soft stuff. Well, pure Teflon is not flexible and soft. It's rigid and brittle, but you add stuff to it, and the Teflon tape, G, is now soft and does all of the wonderful things it does. So we expect adding a solvent should not lead to this effect,

and therefore, take the polymer solution, and we go to very, very high molecular weight. Eventually, something has to happen in here.

And what exactly happens is not clear. However, the statement is that something does happen, and since I was in a nice conversation with Tim Watt, who has made wonderful contributions to polymer science of all sorts, this sort of leads to the very nice paper by Tao and all

where they do measurements that go out to the melt but also come down to reasonably low concentrations, and you can see in the book what happens. Okay, so that is the melt, that is the solution to melt transition.

You can see cases where there are crossovers that show what happens here, perhaps. For example, mu versus probe size electrophoretic mobility, you in fact see a crossover from one probe size dependence to the other.

That probe size transition is probably not the same as this one, but the important issue is you can get transitions. Okay, so there is an issue which is not completely resolved

because there aren't enough measurements yet. The important issue is that there is a melt behavior, there is a solution behavior, and presumably, you somehow have to get from one to the other. Okay, so we also said, last bit,

e to the alpha, c to the mu, and alpha we find experimentally is proportional to m to the gamma, which we find for all sorts of different experiments, but there's one little issue. We could do a gamma as corresponding to viscosity or gamma as corresponding to probe diffusion

or gamma as corresponding to polymer self-diffusion. We can pull out a gamma for each of these, and for viscosity, gamma is some number like two-thirds. However, I'm going to note for schizo-phylans,

and the reason I'm noting it is that it is a really superb set of measurements, and for those, gamma is pretty close to five-sixths. It may not be a perfect number, but it's pretty good. Oh, I'm sorry, not five-sixths, 0.92.

For probe diffusion, gamma is about 0.84, and gee, that is about five-sixths, isn't it? I'm not saying it is five-sixths, but it's an interesting coincidence. And then finally, for self-diffusion,

it's some number close to one. The statement that gamma for self-diffusion is some number close to one, gee, we have something related here.

We have the observation for viscosity of Dravol that c times the intrinsic viscosity is a good reducing variable, and that is equivalent to saying that c times m to the power is what we have for the viscosity, what the basic underlying variable is.

Oh, you will sometimes find in the literature statements that c times m to the first is a good reducing variable for the viscosity. That doesn't appear to be quite true based on our careful numerical analysis. There were theoretical reasons

for supposing these were reasonable arrangements. Okay. So that is what we find, and there are exponents, and the exponents aren't the same. That is, alpha does depend on polymer molecular weight, and it depends on polymer molecular weight

for slightly different extents for the viscosity for probe diffusion and for self-diffusion. This last bit already has an implication. Namely, there are people who will model probe diffusion, and here is the probe, and here is a very large polymer,

and there are models that propose that Dp is proportional for very large chains and high concentrations to m to the zero. And the reason is the probe is almost never close to a chain end. It's just close to the chain middle, and therefore it can't see

how big the polymer molecular weight is, and thus it must be, the D must be independent of polymer molecular weight. Well, those models are conclusively rejected by experiment. There's no doubt about that.

So that is roughly what we can say about the molecular weight dependence. Next thing I want to talk about a bit is the diffusion coefficient, self-diffusion coefficient,

and the viscosity. We can measure the self-diffusion coefficient. We can measure the viscosity. We can also do this for probe diffusion. The important issue I want to make in the cases where the experiment has been done

is that the product D eta is not independent of concentration. Instead, at large concentration, D eta is much larger than D zero eta zero.

And that is true both for self-diffusion and also for probes, for probe diffusion. And this is rather unmistakably the case. Now what you might, though, ask is, gee, why is this interesting? And the reason it's interesting is there are a number of theoretical models that propose that we can relate the diffusion coefficient

and the viscosity of the solution, namely the diffusion coefficient is, say, the time required for the polymer to move a distance. I'm going to skip,

I'm going to insert an intervening step, divided by some critical time. And this is, as a proportionality, and I should make this a proportionality too, this is Rg squared, the size of the polymer chain, which is, of course, not quite concentration independent, divided by the solution viscosity.

That is, the proposal is that D and eta are related as indicated. Now the point of this is, is that Rg squared, the mean square size of the polymer chain, if it's in concentrated solution,

goes approximately as M to the first. That is also the behavior the polymer chain would have in a theta solvent. In a good solvent, we would say Rg is proportional to M

to the, it's a little less than 0.6. However, this is the size of the chain, and it does not care very much what the polymer concentration is. It cares a little bit.

And therefore, D and eta ought to be related as indicated and therefore, D should go as M over the viscosity. But if you have a model that says eta is proportional to M to the X,

then D goes as M to the minus X plus one. And therefore, the diffusion coefficient and the viscosity are proportional to each other. Our diffusion coefficient is like this,

viscosity is like that, and D over eta is a ratio determined by a polymer molecular weight and very little else. I could also write this as D eta proportional to Rg squared. Yes? Yes.

Well, that says that the product D eta cannot depend very much on concentration. It can depend a bit, a factor of two, because polymer chains change size between dilute and concentrated solution. However, experiment says something very different

and the variation here can be by several orders of magnitude. And therefore, we can say that models that say D goes as radius squared over viscosity are rejected by experiment.

And if members of the audience who are not familiar with polymer theory may be saying, oh yeah, of course, sure, and members who are who may be saying something closer to oh, dear me. Okay, there has been a historical puzzle

in the melt literature. And the historical puzzle in the melt literature is that it was supposed D went as M to the minus two. Viscosity seemed experimentally to go as M to the 3.4. And you will notice that the product D eta,

if you believe these two was proportional to M to the 1.4, and 1.4 is not one, and the measurements are perfectly good enough to tell the difference. So the question is, what is going on? And you worked very hard to say, gee, the measurements have this issue or that issue,

or we can do better measurements. Or you say, we need to tweak the theory a little bit here or there. The other outcome, of course, is what you're seeing is a basic failure of the conceptual model that leads to this expectation. I have said much less about the molecular weight dependence of nu.

For example, self-diffusion or probe diffusion tends to be scattered at low molecular weight, and at large molecular weight, the measurements are much better nailed down.

Okay, there are, however, several systematizations that do link various static and dynamic parameters, and these are the ones that are actually working. And one is to go in and look at dielectric relaxation.

And dielectric relaxation is this wonderful technique that gives us the mean square end-to-end vector. It gives us a critical time. It actually gives us full spectra and multiple modes. It gives you a huge amount of information about neutral polymers and solution.

And what we find, and this is in chapter five or so, six, is that the characteristic time we measure is sum tau zero e to the, and then there's the stuff in the exponential. And there is an a constant, and there is a c to the first

and then there is an r square. R, if you prefer, over r zero square, so this little factor is one at concentration zero to the power of psi, and psi is equal to three-halves.

That is, the concentration dependence of the relaxation time is determined by an overall exponential effect,

and the deviation from exponentiality is determined quantitatively to good precision by the chain contraction. We have another one of these. We can measure the storage modulus or the loss modulus,

or we can look at shear thinning. And the lower frequency rate behavior is sum gi zero e to the minus a omega. We apply that to the delta.

And if we look at this, what we discover is that a, this constant in front, is proportional to gi zero to some power x. And the exact power x depends on the parameter.

For g prime, x is about a quarter. For g double prime, x is about 0.4 or 0.5. Too good approximation.

And for eta, shear thinning, x is about g, two-thirds in most cases, but we did find one case where it was 0.9. However, if you look at the graph for each chemical system, the correlation for shear thinning is superb.

And therefore, this object, which is basically a dynamic parameter in some sense, is being entirely determined by what in some sense are the low frequency properties,

which are pseudo-static properties. Okay, so we actually have this, okay, that's it for, you can relate dynamic parameters to static parameters and actually see interesting things.

Let's shove ahead a bit. We hit another issue.

And the new issue we are going to hit are transitions. Now when I say transitions, or when transitions arise in this discussion, you have to be a little careful. Large numbers of people see the word transition, and think of phase transitions.

Phase transitions have the feature that they are sharp. That is, you chug along heating ice and heating ice, and you bring it up towards zero centigrade. And when you hit zero centigrade,

you suddenly notice you are putting in more and more heat and the temperature does not change at all, at least within limits of experimental measurement accuracy. And then the last of the ice melts, and you can start heating it again, but the specific heat is not the same as it was before. And phase transitions, when you talk about phase transitions,

you are talking about something that is usually sharp, as presented along most reasonable variable axes. Of course, you can always do things to confuse the issue. When people hear transitions, they think phase transitions, and they must think sharp.

And that is not necessarily what we are talking about here. That is, there are a number of cases of theoretical models which say we have a dynamic domain A, we have a dynamic domain B, we have some sort of a boundary between them, and at the boundary, we have a region which is a crossover.

And in the crossover, the two dynamic sets of dynamic effects are competing. And because they are competing, we do not see a sharp change from point A to point B. Thus, for example, you might say,

here is a domain, here is a domain, and in between, we have the crossover regime in which neither domain model is good enough by itself to apply. We don't see a merged model. And therefore, there is a region

in which you don't see other models. There is very little work done on modeling what you would expect the width of these transitions to be. The only discussion I have ever seen of this referring to, as you make a solution more concentrated so the chains overlap,

you get what is called an entanglement transition, is the observation that an entanglement transition in which you form this transient lattice is a percolation transition. Percolation means you can walk along the polymers, jumping from one polymer to the next

where there are entanglements, and when you have the entanglement transition, you can walk across the whole bottle of polymer, simply skipping from chain to chain. Percolation transitions are usually extremely sharp. Sharpness, however, is not the desired effect

for a lot of the polymer discussions because if the transition were sharp, it would not resemble experiment. And therefore, you will see in the literature the assertion that hydrodynamic crossovers are broad. There is, however, and I have read fair parts of the literature, absolutely no theoretical basis for this claim

other than if it weren't true, the theory wouldn't work. So what transitions did we see, or could we see? Well, we did not talk a great deal about G of T. G of T is the time domain equivalent

of the storage modulus and the loss modulus. The notion is we have two parallel plates. We displace one of them quite quickly relative to all of the relaxations in the solution. There is then a restoring force on the plates and a friction force on the lower plate.

And that plate dissipates, and that force dissipates, and the dissipation is described by G of T. And looks sort of like this, except if you go out to, and we better make this log time,

if you start making the polymer solution concentrated, you eventually end up with, that's supposed to be flat, one, a relaxation which is extremely long-lived and which has a plateau, it's not really exactly a plateau, it just doesn't change very much.

And then down here at long times, corresponding to short frequencies, you have what is called the terminal regime. And so that is, in a sense, a transition, you start seeing this plateau. Okay, so did we actually see transitions? Well, yes, we saw several of them.

For example, for small molecule diffusion or conductivity, I was able to point out that if we look at these as a function of viscosity, we see something that depends as eta to the minus one, this is in the literature, up to some boundary,

and above that, we see eta for the diffusion coefficient to the minus two-thirds or so. And that transition, well, it's clearly going to depend on what the solvent is. But if I, in the aqueous systems that were looked at, it was some place near five centipoise.

And that's meant to be a somewhat imprecise statement. You can also look at NMRT one times for polymers, and there is, I didn't go into it in any detail in the lecture, but there is something interesting here

about maybe two centipoise. Okay, on the other hand, if you look at polystyrene spheres, which are hundreds of angstroms across, in water glycerol, D of the polystyrene latex spheres goes as T over eta

of the solution over at least three orders of magnitude. You can take that out to pure glycerol, or nearly pure glycerol. You can go over a wide range of temperatures, and this is what you see. Eventually, if you super cool the glycerol, something else happens. Here's another transition we saw.

We are looking at the self-diffusion coefficient of the solvent as a function of polymer concentration. And what we find is an exponential out to some place near here, and then beyond here,

a stretched exponential, the two behaviors are very clearly very different. At the crossover, the two curves are tangent to each other. But there really is a crossover, and it's something like 350 or 500 grams per liter.

I proposed, I didn't put it into the book. I hadn't thought of it at the time. I proposed an explanation, which is that roughly in here, the polymer coils have gotten fairly concentrated, and roughly in here, if we have a solvent molecule,

the gaps between the polymer coils start approaching the size of a polymer, of a solvent molecule, and the statement, the solvent as a continuum really stops working. Now how exactly small do the gaps have to be relative to the size of the solvent?

That's quite imprecise, so doing a more accurate geometric calculation is pointless because there's this constant you don't have predicted. Oh yes, so there's a real transition.

If we look at the zero share of viscosity, the low share of viscosity, we found that some systems, but not all of them, do have a transition. That is, there are some systems in which the viscosity

is just a stretched exponential in concentration all the way up, and there are other systems in which we have a stretched exponential up to a point, and then we have something that is very clearly a power law. This is the solution-like, melt-like transition.

It is quite sharp. Now I have said, gee, there are these theoretical models that say you get scaling, and here we see scaling and viscosity. However, you should realize it's only scaling and viscosity in some systems.

Furthermore, there's a transition concentration at which this occurs, and the transition concentration in natural units as a function of the intrinsic viscosity, or rather CT times the intrinsic viscosity,

is four or maybe 35, that's more typical, but maybe as large as 80, and therefore, the transition occurs at nothing like the concentration at which the chains start to overlap, at least not in a systematic way. However, there is also a transition viscosity,

and if we look at eta T divided by the solvent viscosity, we discover this is rather consistently 100 to 300. Question? I have a question.

Theory predicts either viscosity or self-diffusion. Suppose you're an application engineer or something, or working on it. Typically, you'd be looking at the experimental data, and suppose somebody has given you an order to make something different based on what you know.

In that situation, would you use the theory to plot both the stressed explanation as well as the power law? Let's just consider viscosity, and then give your client back that information, or you make an educated guess, what exactly do you do? Well, the answer is, if you want to know

what you're going to do because you'd like to interpolate, the sort of answer is you have a number of points, and fortunately, it is quite easy to tell the difference between a stretched exponential or a power law. And what you go do is you simply go in

and generate your log-log plot of the measurements, and it will be painfully visible whether as far out as you've gone, there's nothing but stretched exponential behavior, or whether there is a crossover in power law behavior. And once you've done that, you'll know whether, if you are doing industrial-type stuff, you want to use a single stretched exponential,

or you need two functions and can specify a crossover. This crossover, by the way, in viscosity, is sharp.

So the answer is there is a sharp concentration. It typically, in the cases we've looked at, it's always found at about the same relative viscosity. That was an about, not an exact. And therefore, you know roughly where to look whether something interesting is going to happen or not.

Thank you. Of course, if you actually have a system and you can't get a lot of data points, you may be less happy with this statement. You do need some decent number of measurements to see something. Okay, so that is an actual crossover in the viscosity.

We also talked, however, let us stop for a second. You brought up self-diffusion, and the self-diffusion coefficient is just a stretched exponential through a wide range of code of, I drew it upside down, that was one over ds.

This is ds. You increase the polymer concentration, ds falls as a stretched exponential. And therefore, if you think back to that model we mentioned, which predicts ds goes as RG squared over eta as a scaling approximation.

Well, that doesn't work at all because the concentration dependence of the viscosity and the concentration dependence of the self-diffusion don't use the same function in all cases. So we can rule that one out again. What else can we say about transitions?

Well, I've mentioned several times for the electrophoretic mobility as a function of probe size. And I sort of made a guess at what you might be looking at. That is, you're looking at a linear to non-linear dynamics transition. And the non-linearity is shown

because the mobility depends on the applied field to a power. One last transition. And the last transition has been seen at several points. And it has not been seen very clearly because there are not a lot of measurements,

but you appear to see it. And this is due to the work of the late Sesquier-Kronyak. And it is a transition near about 800 grams per liter. And that is very approximate because the data in cases that I've seen are spaced 100 grams per liter apart.

And it's not clear that it's sharp or anything else. And the core issue is that if you look at the light scattering, and there's also experiments that look both at VV and VH scattering, you see a slow mode. And the slow mode slows down

as you increase the polymer concentration. However, there is another feature which is that the slow mode depends on scattering vector squared. It depends on scattering vector squared if you are at smaller concentration or larger temperature.

The statement that you see Q squared dependence in the relaxation means that the relaxation is diffusive. That is, if you stick in combined Fick's law

and the continuity equation, and you ask how does a cosine and concentration relax, it relaxes, there's a decaying exponential in time, E to the minus D Q squared. There's a Q from Fick's law and a Q from the continuity equation. You get a relaxation that goes as Q squared.

However, if you take this and you go to larger concentration or colder, the relaxation goes as Q to the zero.

That is, you see a relaxation, but if you study the relaxation on different distance scales, nothing interesting changes. The relaxation rate does not change. And that is characteristic of a relaxation for which the word structural is applied.

I should emphasize that when I say structural, that is the name attached to it. And if you try to be picky and ask, what do you mean structure? Well, you're seeing something that isn't related to stuff moving over a distance. That would be diffusion. But something is happening inside,

like local packing was changing or something, or that or something covers an awful lot of ground. And the net result is, you see a change to Q to the zero behavior. And this occurs someplace near here and it is more prominent in colder systems.

And if I appear to not be very precise as to what is going on, that's because there are a limited number of measurements that have seen it. And that is about what I can say.

So let us push on. And now we come to the transition we do not see.

Suppose we look at more or less all of the parameters we've talked about as a function of concentration. And what we see are smooth curves. You might have to plot the inverse of the parameter

to get this curve, say, or solve the future. We see smooth curves, and the smooth curves have fixed constant parameters. And so the same parameters were here and here and here and you get a single curve that describes

the whole range of concentration behavior. Well, in particular, the smooth curves chug ahead. And someplace here are a couple of concentrations. And one of them is C star, the overlap concentration.

And the overlap concentration is some small integer like one or four divided by the intrinsic viscosity. This is the concentration at which the polymer chains in solution are sort of shoulder to shoulder. And if you put another chain in,

it has to go through the others because there's no place else for it to fit. Well, there is an overlap concentration and there is an entanglement concentration which is some multiple of C star. And there are models that predict dynamics changes dramatically as you cross these transitions.

It may not change instantaneously, sharply, but you have different dynamics at higher concentrations and lower ones, and therefore things should happen. Well, pointing at very large numbers of measurements using very different physical parameters,

we can say quite definitively, except maybe for the solution-like, melt-like viscosity transition, that these expectations are incorrect. It is not true that you have one set of dynamics in dilute solution and a different set of dynamics in semi-dilute

or entangled solutions. The experiment are fairly thorough in rejecting this. The only way you could avoid this is to say, well, the dynamics are completely different, the modes of motion are completely different, but everything is so set up to conspire that a smooth curve with constant parameters,

and thus parameters you could calculate up here using one set of dynamics, and the same parameters must come out of the other set of dynamics down there. This could happen, but I don't think the smart money is betting on it. The only exception to this

is the solution-like, melt-like transition, but there we have an important issue. You might say the solution-like, melt-like transition is an example of the crossover here, except of course that it's only seen in some systems.

Not universal. The viscosity is not a universal function of the polymer concentration. That isn't very good for some of these models either.

However, the important issue is that if you look at most sets of measurements, like self-diffusion, and you look someplace near the overlap or so-called entanglement concentration, nothing odd happens there, and there's not really a crossover or anything like that.

Okay, I inserted into the book a discussion of colloid dynamics.

The reason I inserted colloid dynamics into the book is that colloids and polymers have two similarities and one differences. One of the similarities is that the forces between them are the same. The other similarity is that the dynamic equations,

meaning a Morrie-Swanzig or whatever you prefer equation, are the same. That is, you are looking at heavily damped motions of many interacting objects. The difference between colloids and polymers

is that it's topology. That is, colloids, spherical colloids, cannot tie each other in knots. Polymers can certainly do so. Whether this is dynamically important is a separate question, but certainly the two topologies are the same.

Colloids can jam, that is, if you have something that's like 60% by volume of spheres, it doesn't flow very smoothly. But chains, presumably in cross-section, could also jam.

In fact, that might be the 800 gram per liter transition I mentioned. However, the net result is, if we compare the dynamics of colloids to polymer chains, we can sort out which features of the system are due to topology and which are not.

And so, for example, we can look at viscosity versus sphere volume fraction or versus polymer concentration. And for colloids, we see a stretched exponential that comes up, and then there is a completely sharp crossover to a very steep power line.

This is solution-like, melt-like behavior. It's seen for spheres, so it has nothing, it has not a lot to do with topology, with one minor exception. For polymers, the curve just keeps on going.

For spheres, there is a sharp upwards turn. You should realize that the concentration and viscosity axes may be very different for these different systems. If, however, you look at many arm stars,

the arm stars start out down here, and as you increase the number of arms, you get all the way up to here. That is, as you look at things that look less and less like spheres, or excuse me, less and less like a single random coil and more and more like spheres,

like 200-armed stars, which are fairly spherical. As you go from here to there, what you find is there's a transition from a polymer behavior to a spherical behavior, and therefore this piece could be assigned to topology.

However, this and the fact that you see a power law at all is clearly not due to topology because you see it both for spheres and for polymer chains. Okay, what else do you see? You could look at the storage modulus

or the loss modulus as functions of frequency. We could certainly do that. And what we observe for colloids is exactly the same as the behavior we saw for polymer coils.

The general form of the dynamic modulus, the dynamic moduli, both of them, have the same dependence on frequency for spheres as for random coils, and therefore we can say there's no topology. I will put in the question mark on this now.

If you do this for polymer coils and you go out far enough in frequency, you see a rollover or you see bumps that do various things. That's not seen for hard spheres. However, if you look at the numerics,

polymers have been carried out to much higher frequencies using time temperature superposition. And it might be the case that if you took spheres, hard spheres, and you did the studies over a much wider range of frequencies,

you would start to see exactly the same effect you see for polymer coils. That is, the sphere measurements were not carried out as far in frequency, and therefore you don't want to say there's a topology difference here because it's not clear that you've actually looked at it. Okay, what else can you do?

Well, you can look at the dynamic structure factor. This is the light scattering spectroscopy, which has relaxation. And if you go up to elevated concentration, for sphere as always, but in a certain sense all sphere systems are the same, they have no real ability to be different from

each other. And for many polymer systems, you see two modes. You see a fast mode, a relaxation that is fast. Here's the relaxation rate, here's the concentration. And it gets faster as you increase the concentration.

And at elevated concentration, you see a slow mode that gets slower as you increase the concentration. And both of these modes, except for that interesting issue that I asked for a cognac uncovered, go as Q square. They're both diffusive modes.

And so we see this, it's seen both for spheres and for polymer coils. So that effect cannot be a topological effect either. Okay, there are significant studies that have been done using video microscopy.

Video microscopy lets us actually observe the motions of objects in solution.

Now, you have to work hard, and you may have to tag a small number of them so you can watch the small number moving through the background. But you can do this. And if you do video microscopy of spheres, you find that there are lumps that move slowly,

and you find that there are what I will describe as ribbons, where particles move fairly rapidly. Could this be a microscopic artifact?

Well, you can also do computer simulations on, for example, super-cooled Lennard-Jones fluids. And there are very nice sets of experiments, for example, from the Michigan groups that show rather clearly, you have semi-stationary lumps, rapidly moving ribbons.

And therefore, what is seen in the microscopy agrees with the computer simulations. And this becomes, you don't see this at high temperature, but at low temperature, meaning for spheres, high concentration, you do see this dynamic effect.

Oh, aside, we have hard spheres. The Boltzmann or Gibbs factor, e to the minus beta u, is either one or zero, because the potential energy is either zero or infinity. Therefore, e to the minus beta u is temperature independent.

And while in a Lennard-Jones fluid, you can see things that happen as you heat it or cool it, for spheres, the heating or cooling does nothing. It is an a thermal system, and therefore, you can only change things by changing the concentration.

Well, that's the visualization for spheres. But in fact, there have been experiments done for polymers. There's a nice coat of micrographs in the literature. And you look in, and there are systems which show the slow mode in which the experiments of Sedlak say the slow mode is lumps that are equilibrium objects.

And you can actually see irregularities in the liquid with the naked eye, or at least with a good microscope. You can actually see the slow moving vitrified regions. And therefore, the video appears to show both for spheres and for polymers,

you get vitrified, glassified regions. The nature of the vitrification is unclear.

Okay, that's vitrified regions. What else can we do experimentally? Well, g, colloids have a self-diffusion coefficient.

They have a mutual diffusion coefficient. They have a rotational diffusion coefficient. And these things have low concentration behavior.

And so if we plot a D versus concentration, we get for a self-diffusion coefficient, we get a downhill slope. And for a mutual diffusion coefficient, the calculations I've done show a downhill slope. And you can do the calculation for rotation.

The rotational diffusion coefficient is the one that is most sensitive to short range forces at the sphere surface because the hydrodynamic interaction falls off very sharply, like one over r to the sixth with distance.

So dr is the hardest to calculate. But we can write, for example, ds equals d0, 1 plus some constant k1 volume fraction. And if we calculate k1, and we calculate k1 on the assumption we have

hard spheres, and we have the ocine hydrodynamic interaction and its extensions, which historically are actually kinch. We have hard spheres and the ocine pinch hydrodynamic interaction.

We can calculate k1, and it's at least approximately accurate, with a plus or minus one or better. I should point out for mutual diffusion, the initial slope has a contribution from direct interactions.

It has a contribution from hydrodynamic interactions. They're additive, they're each some number around, say, eight. And the addition happens to give us a number close to zero. But the terms separately are quite large.

And therefore, we're actually doing a measurement and calculation that's decently accurate. And we get the right answer. The statement that we are seeing and able to calculate the slope reasonably accurately says, there's an implication here. And the implication is that we understand the forces.

There has been a very long time going back, say, to the Kirkwood-Reisman model where people say, we will treat the hydrodynamics with the ocine tensor. We will treat volume exclusion by saying these things are hard spheres.

However, the question is, is this right or are we completely off the track, given that we're on a very atomic scale? And the answer, based on these calculations, it appears we are very much on the right track in what we did, okay? Okay, so that is what we can say about colloids.

Oh, one last bit on colloids. For colloids, over an extended concentration range, the diffusion coefficient follows a stretched exponential.

That's also true for polymers. Now, the numerical coefficients may be different. After all, there is some difference between the systems. But the functional form is there, and therefore we can say that the statement we have a stretched exponential is not topology.

You may correctly conclude, since I kept saying not topology, not topology, not topology, there is not a lot of room left for places in which these things are different in shape, and therefore,

their behavior is qualitatively different. There are quantitative effects, but the possibility of qualitative differences between colloid dynamics and chain dynamics are substantially suppressed. Not completely. For example, spheres have this phase transition,

starting at volume fraction a half, going up to volume fraction 0.58. And then at volume fraction 0.72, slightly less than three-quarters by volume. They're close-packed, you can't fit anymore. Polymers will form melt, in which more or less all of space is filled by polymers.

And that actually is a geometric packing issue. Okay, let's go back to video, and we will look at electrophoresis.

And there are very nice experiments done on synthetic DNA stars. And what you find is that if you have a star polymer,

this is somewhat artistic, headed that way, the star polymer moves by stretching its arms out, and the body stands behind it. We pointed this out before, way in an earlier lecture. And this category of motion is two-fitted, squid-like.

The chain moves like this. We appear to be in the low-field regime. That is, if you go lower and lower electric fields, things move more and more slowly, the extension becomes less and less. But we seem to be in the low-field regime.

If we are in the low-field regime, and that is a very big if, because there's plenty of room for going to very low fields, and gee, it does change. But if you are in the low-field regime, then the fluctuation dissipation

theorem and linear response theory says that this class of motion is how star polymers diffuse through polymer solutions. And I should stress this is solutions, not melts. If you have a star chain diffusing through a solution, and

here is a cross-linked gel, when the star comes up to the gel, it can't penetrate, it just flattens out on the surface. Well, if that's correct, then star polymers diffuse through solution via teuthetic motion. The arms move, the body is dragged behind.

If there are other polymer coils that are entrained by the polymer chain, if there are other entrained coils, then it must be the case that gee,

it must be the case that the chains are being dragged along. Well, that says a great deal about how polymers move through solution, because it speaks to a very fundamental model issue. And I'm going to stop here, and

I'm going to point out the fundamental model question. Here is a polymer coil chain and solution. And here comes another one, and it's diffusing that way. And at some point, it's done that.

And I'm trying to draw the overlap so you can see which chain is in front and which chain is behind. And gee, at this point, the two chains are like this, yes? Two chains are like this. Well, the question is,

what happens next if this chain tries to move that way some more? And one answer is, if I plot v versus time of the two chains, we have the moving chain, which is chugging along, and it reaches the chain that was sitting there.

And the two chains then come to a stop, because one chain keeps the other from moving due to the excluded volume force. The other, which is the interaction scene, say, in a rugby scrum or an American football game, is this chain is moving, and it encounters the second chain, and they get dragged along.

And therefore, the second chain is dragged along by the first. Now, of course, the two chains, if they're the same size, are co-equal. So there'll be some compromise. But what they do, though when they're very close,

they will have very much the same random thermal forces on them. The assertion that both chains stop when they collide is the fundamental basis of the tube model of all of the tube reputation models.

The assertion that they drag along is the fundamental basis of most of the hydrodynamic models. This experiment is the critical test for determining which model is correct. Because if I am a star polymer, here I am, I'm a star polymer.

I am headed this way through a concentrated solution. If I encounter another polymer here, what I have to do is diffusively retract my arm in, stick it out on the far side. And now I've freed up this arm, and I can move ahead again until my arm encounters

something, and then I have to retract my arm to disentangle it. And after I've retracted my arm, I can stick it out again on the far side of the obstacle. You notice, if you try walking this way through any sort of set of obstacles, this is a very clumsy way to move, and I would be greatly slowed down.

The hydrodynamic models say, I'm the star polymer, I'm moving this way. If I get near another polymer chain here, I am as privileged as it is as to whether I move or not. We are both subject to somewhat similar external random forces,

because the random forces are spatially correlated due to the Oscene tensor. And therefore, we will continue to move. And I will not have to pull my arms in in order to move. And there is the critical experiment that tests these two models and rejects the notion that polymers function as tubes.

That is, it is the critical experiment if we are in the low field regime, a statement of which we cannot be certain.

I started a bit late, so I will do one last short topic, and we are done. And the last somewhat short topic, no, we won't do the short topic, because it's actually a long topic. That is the end of today's lecture. Next time, I will finish off the current chapter, 16,

inferences that can be made from observations on polymer dynamics. Until then, class is over.