I'm Professor Filleys, and this is the continuing series of lectures on polymer dynamics and their phenomenology. Today we're going to be continuing and probably concluding our discussion of Chapter 11 on dynamic light scattering and the relaxation of concentration fluctuations in polymer solutions.

We go back first to the discussion of the neutral polymer slow mode, and I'm going to bring out a few particular features that we didn't entirely discuss last time.

The first issue is that if you look at polymers in solution, and there is, for example, work of own, there are a couple of things you can do. And one thing you can do is to look at the first cumulant of the spectrum, which

is a light scattering intensity weighted average of all of the modes. And another thing you can do is to do a mode decomposition. And what is found, if we look in, is that at low concentration there is a mode,

and its relaxation rate, this is gamma, the relaxation rate increases with increasing concentration, and then at some concentration you start to see a slow mode, we're not on the same scale.

The slow, important issue with the drawing is the gamma of the slow mode decreases as you increase polymer concentration. So you have two modes, they both show this behavior. The other important feature of this, we're saying there are modes, and the modes are

both found to be Q square dependent. The significance of Q square dependence is that it corresponds to diffusion.

That is, you have something where there is a diffusion current, which is proportional to a diffusion coefficient and a concentration gradient. And then you have a continuity equation, DC DT equals del dot J, divergence of current.

And if you stick the diffusion current in here, you get DC DT goes as del dot D grad C. Now if you approach this by saying we will put in, take a spatial Fourier transform,

equivalently we will look at the relaxation of a spatial cosine wave, each of the grads gets replaced by an IQ, and you have DC DT proportional to minus D Q square T. And

that is a mark of normal conventional diffusion, namely that the modes have a Q square dependence in. If you however go into systems that show the slow mode, there are a variety of ways

of breaking down how many modes there are and what you're seeing. There's some question of how many parameters you can really pull out of a light scattering spectrum, and I've presented simulation evidence that a reasonable number is six or

eight rather than twenty. Nonetheless there are alternative approaches for doing this, and for example I note the work of Wynn Brown, who of course was looking at light scattering spectra that had very wide ranges of decay times, meaning it's easier to pull out more parameters,

and who found in the slow mode a mixture of Q square dependent modes and Q to the zero, that is wave vector in dependent modes. How can you get something that's Q to the zero?

You're looking at some internal relaxation and the particles do not have to move to change how much light they're scattered, so there's no particle motion involved, instead the particles for example change their ability to scatter light through some manner, which

is not at all specified, and they're just sitting there, so this seems the same at all scattering vectors. There's another way to push beyond this, which was discussed when we talked about probe diffusion and hydroxyprobe cellulose, and that is to take spectra and if you

don't think that the modes are well characterized, well you try to fit the modes to something. One choice is to fit the modes to sums of a few cumulant series, which as far as I know is perfectly legitimate, but has never been explored very much.

Another alternative though is to fit the dynamic structure factor to something of the form A, E to the minus theta, E to the beta, plus A, slow, E to the minus theta, T,

these P slow, beta slow, printers hate you if you give them something like a form like that, and so you fit to a sum of stretched exponentials in time. The point of fitting to stretched exponentials in time is that stretched exponentials in time

appear to describe well the time dependence, accommodate well to things that really aren't simple exponentials, and only use a few parameters. However, based on the comparison with concentration stretched exponentials, where you can pull

out a derivation based on renormalization group arguments, there's no reason not to suppose that those aren't the fundamental forms. So that is the sort of thing you have. Now if you then go in and ask what sort of behavior you have, there are a few slightly

odd things that you sometimes find. For example, there have been found slow modes in which the slow mode goes as Q square up

to a point, then rolls over and is basically independent of Q. So at short distances you're seeing something that's Q independent, some sort of structural whatever, and at small Q, small Q corresponds to large distances, at large distances you're

seeing something diffusive. There is also, one perhaps should note, an odd note due to Nicolau, they look at mode relaxation times and they find the slowest mode, and it has a relaxation rate.

And then they go and look at mechanical relaxations, that is you take the liquid, you apply an oscillating shear and vary the frequency, and there are internal relaxations that contribute one way or another to what you're seeing, but those have a characteristic time.

And if you measure the characteristic mechanical time, it corresponds exactly, or at least approximately to the slowest time you see in the optical spectrum.

However, there is a complication. The complication is that if you do this in a theta solvent and you take up the temperature, if you take up the temperature, the optical mode disappears. The mechanical mode is still there. So apparently the mechanical mode was coupled to something that scattered light, and whatever

the coupling was, either the coupling may not have disappeared but the light scattering did, however the slow mode is still there. Now one thing you might do is to ask, well you have a fast mode, which gets faster

and faster as you increase the polymer concentration, you have a slow mode, which when it appears, slows down as you are increasing the polymer concentration. And you reasonably ask, how is this to be interpreted?

What are we seeing? An indication of this is done by crossing over and looking at polyelectrolytes. It has been known for a rather long time that polyelectrolytes show a slow mode, and

the issue with the polyelectrolyte slow mode was asking what it was or was not. Some of the same debates that also took place with the neutral polymer slow mode

were present, and the issue is that if you go back to 1980, digital correlator technology was really at or beyond the edge of what it could handle. I do recall when we were working at Michigan with polyacrylic acid and probe diffusion

and polyacrylic acid, you hit a point at which there was a viscosity transition of the sort we'll be discussing. The diffusion coefficient of the probes decoupled more and more from the viscosity of the solution, and there were issues with the scattering spectra of the probes, and since

what we had at the time was a fairly limited 128 channel linear correlator, it really appeared as though you were getting additional slow modes, but we knew we couldn't get at it with the technology of the time, so we sort of cut off where we were going about the time the slow mode cut in.

Most of this text does not treat polyelectrolytes, which are a very complicated problem. The reason we are discussing polyelectrolytes here is that we have a specific set of experiments due to Sedlak, which clarify what the polyelectrolyte slow mode is, and do so in a way which makes

reasonably clear that similar sorts of things are reasonably interpreted as happening for neutral polymer slow modes. Sedlak worked with 50 and 710 kilogoltons of sodium polystyrene sulfonate.

A polystyrene sulfonate is a polystyrene that has been chemically modified, and out at the edge you have a group that can be neutralized, and at this point the polystyrene

sulfonate is a polyelectrolyte and is quite water soluble. So what Sedlak did was to work with these, and the first thing he did was to say we'll dissolve the powder, it typically comes as a powder, and we will look at the spectrum,

and since there's some question of whether we're at equilibrium or not, we will simply keep looking at the same spectrum. And he did this out to a one year time scale, which requires patience, and what he observed was that S of Q, which is the light scattering intensity versus angle, which

gives you the distribution of sizes of whatever they are in solution, was independent of time, proportional to time to the zero, and therefore the distribution of clusters

sizes was not changing, however as time went on, the scattering intensity fell, and the diffusion coefficient increased. Now that's a little peculiar, and the question is how can the diffusion coefficient

be increasing and the intensity falling if the size distribution isn't changing. The following picture appears, which is consistent with everything else you're going to hear, appears to explain this. We start out, and there are some number of fairly dense clusters, and the fairly dense

clusters will occasionally have a single polymer strand sticking out. This is not yet an equilibrium arrangement. Over a year what happens is, these very exposed chains, which are not very well attached

to whatever it is, fall off, the number of chains in a cluster decreases because chains get out of the cluster and not back into them. Because the cluster has less matter in it, it scatters less light.

Because the cluster is more porous, because you've taken parts out, it has a higher diffusion coefficient, and this picture from the initial cluster to the final cluster explains all of the experimental measures.

Now that doesn't prove it's right, and one of the questions you ask is, well, is it possible you're just looking at the fact you didn't really dissolve things completely, and if you had waited to a ten year or a century time scale, is it possible that you would have found better data?

And indeed, on glass problems, there are several Dutch groups that have set samples. We need to do long time scale, and the working time scale of the experiment, which will not be done by the current people completely, is out to a century. Of course, they have to be very careful with their samples.

So having asked, is this equilibrium or not, Sedlak tried another set of experiments. Now let me back off a step in a historical note. The polyelectrolyte slow mode probably should, the early studies should all be credited to

making sure who very recently retired, Washington and his collaborators, and the feature that is observed is that if you have a polyelectrolyte at high salt, there is no slow mode.

But if you take the polyelectrolyte down to low salt, very little dissolved salt, you see the slow mode. And the fact that this is salt sensitive shows that it's a polyelectrolyte effect

in some sense. The fact that you see odd effects for polyelectrolytes in solution in the absence of background ions, this is a general issue and low ionic strength polyelectrolytes are an even more complicated

problem. Nonetheless, what Sedlak did, which was in line with this, was to say, we will take sodium polystyrene sulfonate and high salt and we will pass it through an 0.05 micron filter, 0.05 micron is about the finest filter you can find commercially, to filter water

soluble whatever, getting anything through it is a feat of great patience, but he did

it and having done all this, he found high salt very well filtered, there was absolutely no slow mode. Now, when Sedlak said no slow mode, his experiments did one thing that a lot of others do not.

That is, there are a lot of experiments that will report effectively a ratio of the intensities of the fast and slow modes. What Sedlak did was to use light scattering standards to calibrate and he was therefore able to measure the absolute intensity of the fast mode and the slow mode separately.

So, he could actually say how much fast mode there was. The reason this is of interest is revealed by what happens when you take these samples and dialyze to remove salt or equivalently if you just make samples in different salt

concentrations, namely you do dialysis which extracts the salt from solution and after you have done this you find the slow mode.

Now the question is why are you seeing the slow mode? And there are in fact two sorts of explanations, one of which is ruled out by Sedlak's experiments. One notion relies on the statement I have a polymer solution, a polyelectrolyte solution.

I pull the salt out, I measure the intensity of the scattering or the intensity of the observed fast mode as I take the salt out and what I find is that when I pull the

salt out the intensity plummets. Once the particles repel each other it's much harder for them to form concentration fluctuations and so the concentration fluctuations are small and therefore the amount of light scattering is small.

Now one thing that could happen if the intensity of the fast mode is falling rapidly is there actually was a slow mode here all along however at high salt the slow mode was so weak that it could not be seen due to the scattering by the fast mode.

As you pull the salt out though the intensity of the slow mode is perhaps independent from concentration, salt concentration and therefore the slow mode rises up out of the deeps

like a rock left behind on a beach as the tide recedes. Well that's very clever however you can rule that out if you do absolute intensity measurements because if you know the absolute intensity of the light here you can work out how intense

the slow mode would have been here, you know how intense the fast mode is here and you can simply ask if there was actually this much slow mode present would it have been hidden or not and the answer is no it would not have been hidden it would have been quite visible. And so what Sedlak showed by measuring scattering intensity is that quite clearly the slow mode

actually does appear when you pull the salt out. Now the other thing that Sedlak did was to sit there and look at the scattering due to

the slow mode as time went on and what he found was that s of q scattering versus angle was fixed that is you had some distribution of objects and solutions that were contributing to the slow mode and they were large enough to be comparable in size with a light wavelength

and their size distribution did not change if you let the sample sit for very long times. On the other hand if you let things sit the intensity of the slow mode rose, this

is this year experiment again, and the diffusion coefficient fell and if you looked at intensity of the slow mode versus time you could start with let's dissolve things in pure water and the intensity of the slow mode fell as time went on. You could start with no

clusters let's dissolve things in pure water, let's pull the salt out and the intensity changed the other way and the two attempted to converge. There is a reasonable interpretation of this observation which is that you are looking

at equilibrium clusters, if you have them and their innards are too concentrated and the innards empty out, if you have them and their innards are too dilute they pull in more chains and they tend towards an equilibrium size and therefore the reasonable interpretation

is that you are looking at equilibrium domains and solution and you can approach the domains either from a side where you have them initially too concentrated or a side where

they are initially too dilute and they converge to the same point. We can also note two other sets of experiments, one set is due to Kong et al and this is the Russo group in Louisiana and what they did was to say well let us look at this accusation

that the domains are due to problems with dissolving the powder. We will synthesize our polymers from scratch and we will never take them out of solution. They were made in solution, they have always been in solution, there is no dissolution step

in their history to cause any difficulty. The second experiment, quite different, is due to Tanotou and Kriol and they did in essence optical microscopy and they looked

at systems among other things, systems that show domains. Well they look at the systems that show domains microscopically and they can actually visualize things that are about the right size. They are not very distinct because after all these are solution structures,

they aren't solid bodies and the solution structures, whatever they are, are about, if I said they were about a half a micron or a bit less, that's about it. So they are quite large, they are much larger than an individual polymer chain and you can actually

see them. Maybe you can't see them very well but you can actually see them. Oh, last experiment, Sedlak. Sedlak started out with, again, and this time he started out with

polystyrene sulfonate that was non neutralized so that, well non neutralized is not precisely correct because if you have any acid group in water, organic acid, whatever, it auto ionizes to some extent so it has some charge on it but mostly it's not charged.

And what he then did was to say we will now add sodium hydroxide and we will neutralize the polymer, meaning we will pull off the protons from the acidic groups and the polymer

will now become extremely heavily charged. And what was found was, as you add NaOH, the intensity of the fast mode drops dramatically and the intensity of the slow mode, you sit

around and the intensity of the slow mode keeps climbing. That is, you can approach this along one more axis where the solvent basically doesn't change. The polymer is dissolved in the solution and all you are doing is changing the charge on the polymer

molecule and you see the same behavior this other way. Okay, so we have seen all of these alternatives. Question? Yes, the experiments all converge to agree that you are looking

at an equilibrium structure and solution, a structure that is considerably larger than a single polymer molecule. And the various properties, some of which I've skipped over of this picture, agree completely with the polymer slow mode seen for neutral polymers,

namely for neutral polymers, both modes are Q-square dependent. The fast mode, which corresponds to chains interacting with each other as single chains, D increases as you

increase the concentration because the chains repel each other. The slow mode, the vitrified region, whatever it is, acts like a diffusing large object that behaves as though you were doing probe diffusion and therefore as you increase the polymer concentration, the probes

are slowed down. Okay, so you might reasonably ask, how does this compare with what we know about glasses or things where you might see this phenomenon? And there is an interesting analogy which I will pursue for a few minutes. An interesting analogy is with the Kibbleson

glass model. This is Dan Kibbleson mostly. He was one of my post-doctoral supervisors. He's since passed away. The issue is as follows. Suppose I will plot this versus temperature. We look at the behavior of a liquid versus temperature and we look at

the viscosity. And if we look at the viscosity versus temperature as we cool something off, the simple behavior is you cool it off, you cool it off, you get to the melting point, and at the melting point you go from a liquid to a solid, a crystal, and gee,

there's no more viscosity. This thing just sits there. That's simple freezing. However, many substances, if you take them and cool them off, you cross the melting

point and you can just keep on going. And the viscosity goes up and up. Now there are some substances in which you cool and cool and you eventually hit a lower limit

below which you cannot have a liquid. And the lower limit, what happens is, if you start, the likelihood of forming crystallites in solution goes up very fast with decreasing temperature and the material turns into a crystal and solid. But there are other

liquids where you cool things off and cool things off and cool things off and in the end you get an amorphous solid. But it really does seem to be solid. It's glass.

The question of the nature of the glass transition is very controversial and very complicated and it's mostly beyond what we're going to talk about. There are, however, a few other peculiar features of the glass transition, one of which is you pull the liquid off,

you now heat it up again, you heat it up to out here someplace, and you see excess light scattering relative to the amount of light scattering you expected in the solution. And the excess light scattering stays around for very long times. It wasn't here originally,

but once you've run the liquid down and back up again, you get this excess light scattering. And the question is, how are we to interpret all of these different phenomena? What Kibbleson proposed was, you cool the liquid off and at some temperature it starts

forming clusters. He talks about, and his collaborators talk about, clusters that are icosahedral packings of atoms. And the clusters are thermodynamically stable.

However, the clusters cannot lead to crystallization. Why not? Because icosahedra aren't space-filling. If you get 20-sided dice and try packing them together, you can pack cubes and make nice crystals. You cannot persuade icosahedra to pack because it's not a space-filling

geometry in three-dimensional space. And as a result, the icosahedral crystals objects try to grow, but as they grow, because it's not space-filling, they have to distort

and there's strain energy coming in. And at some size, you have a frustration limit and the clusters cannot get any bigger. That's the front part of the model.

Now the fact you make more and more of these clusters down here sort of explains why the viscosity goes up. The viscosity is going up for the same reason that the viscosity of ice slush is higher than the viscosity of liquid water, namely there are these little unbending things in solution and they get in each other's way.

Now we come to the truly brilliantly creative contribution. And the creatively brilliant contribution is the statement, these structures are not space-filling, they can't give you the crystal, but they're thermodynamically stable. That is, the clusters have a melting

temperature and the melting temperature of the clusters is higher than the melting temperature of the pure crystal. As a result, they're still stable in liquid out here. Once you've

made them, you heat up beyond the melting temperature and the clusters stay around to some significantly higher temperature. And they then contribute to the viscosity increase,

they then increase, and the large increase and the extra light scattering, they explain why you form a glass. And there's one other thing. Suppose you've made these and you would like to crystallize, you would like to rearrange the atoms in a nice crystal lattice. In order to get from here to here, you have to break up this structure

and form the preferred structure. Well, the potential energy, the free energy barrier in between can be quite large. So even though this is the preferred structure, the stable

structure at low temperature, in order to get from here to here, you'd need to supply very large amounts of thermal energy to the transition state. And guess what? That thermal energy is not available. And therefore, even though at low temperatures, the frustrated

crystal may be less stable than the real crystal, once you've made them, you can't make them go away easily, not just by sitting and waiting. So that is the Kivelson glass model. I have done molecular dynamics with Paul Whitford, molecular dynamics simulations which

appear to show, reveal the presence of Kivelson clusters that have exactly the properties that Kivelson would ascribe to them, including a few that I haven't gone into. The only difference between the clusters we found and the ones he described is that the clusters Whitford

and I found show septahedral sevenfold ordering, not icosahedral twentyfold ordering. Septahedral ordering is extremely unusual in nature. If you didn't think there was a reason to look for it, you might not have done so, but that is what we found.

Okay, what does this have to do with the slow-mo? And the answer is, we have the polymer molecules, and they form these objects which in polyelectrolytes have a higher

density than the surrounding solution, and in neutral polymers apparently do not. And these objects are frustration limited, meaning they can't grow more than they do, clusters,

they form at higher concentration and contribute to the viscosity because you've got clustering. And so these are the glassy objects of the Kivelson glass model, except we have a working experimental case where they actually exist. Okay, we are now approximately done with the discussion of the slow-mo. And the question

is where we push on to next, and one answer is Rayleigh Brillouin scattering.

We will talk for a piece about Rayleigh Brillouin scattering. This is in some sense the scattering that answers the question, why does the sky glow bluish in daylight? In any event, the answer is that we have set this thing up,

we have a liquid, we scatter light from it, and the light to some extent changes frequency when it's scattered. We can do this with a simple liquid that does not

contain diffusing macromolecules. The scattering light changes in frequency and scattering are quite large, like 10 to the 9 hertz or more, meaning you don't use a digital correlator to study them, you use a Fabry-Perot interferometer, or one of several relatives.

And if we put in monochromatic laser light, here's frequency, I will put in laser light of one frequency, Nu0, and I will ask what frequencies of light are scattered out. And the answer is, you see a spectrum sort of like this, and there is a central peak,

and there are two shifted peaks, one shifted up in frequency, one shifted down in frequency.

And how do we interpret this? Well this peak is due to heat diffusion. You have fluctuations in the local energy density and solution. The energy fluctuations create mass density fluctuations, so they scatter light. The way the fluctuations go away is

the energy diffuses out of them, which is heat diffusion, and so the width here is determined by the diffusion coefficient for heat. The two side peaks are out at

C, and the frequency C is the local speed of sound waves in the solution. Why are there sound waves in the solution? They're thermally excited just the way diffusive motion is thermally excited. And the sound wave peaks also have a width, and the width is determined

by dissipation that kills off sound waves. So you have three peaks there, and one of the things you can imagine doing is to add a polymer to the solution and ask what

happens. And the short form answer is that polymer molecules occur on a very long time scale. This material is at the gigahertz range, a very short time scale, and therefore

the two don't couple to each other a lot. However, if you go through from low polymer concentration out to the melt, what you find is that the frequency shift here changes with polymer concentration, the widths change, and therefore there is some sort of weak

sensitivity of the Rayleigh-Brill 1 spectrum to the fact that you're replacing the solvent with polymer. That shouldn't be very surprising. After all, there's no reason for the polymer

to have the same thermal diffusion coefficient and speed of sound as the solvent, and as you move from pure solvent to pure polymer, something ought to happen, and it does. Another thing you can study is the Soret coefficient. The notion in the Soret coefficient

is revealed by the experiment used to study it. You use laser interferometry to create an interference grading in the solution. That is, you send in two beams of light,

same source, so they're coherent. They come in at two different angles and they interfere. And because they interfere, they appear to produce a brightness grading. Well, if you do this with a high-power source, you don't just have a brightness grading, you have locally

heated the material. And now there are several things that happen. And the first thing that happens is you produce a fluctuation in the local energy density, and that diffuses out due to thermal diffusion. The second thing that happens is that you have the

Soret diffusion, and the notion here is that you have a matter current which is proportional to the Soret coefficient and the temperature gradient. That is, if you create a temperature gradient in the solution, or in a gas, objects diffuse parallel to the temperature

gradient. There are a couple of minor complications here. First of all, there are two major differences. First of all, the effect is also known in gases, and the mechanism in gases would appear to be quite different from that in liquids. Second, unlike some other

diffusion coefficients, the Soret diffusion coefficient can have either sign. That is, if I produce a temperature gradient like this, the temperature gradient will just drive the motion of the macromolecules, but it may drive them one way, or it can drive them

the other way, and each is an allowed outcome. That's the Soret diffusion coefficient. People have actually observed that in polymer solutions, and you can actually see things.

Okay, that's the Soret. Oh, I was describing mode. So we have an energy mode that relaxes. We have a concentration mode, which relaxes with the diffusion coefficient for mass concentration. And then in some systems, there's also an intermediate mode, which is sometimes described

as an alpha mode, and sometimes described as a structural mode. However, the evidence that it's, if I asked you what is the structure that is doing its structurating, there's

no real answer. It's just described as a structural mode. It is a reasonable interpretation by comparison with viscoelasticity, but if I asked you what is the structure, well, that's, we're still working on that one. Okay, another set of experiments, a different

set of experiments. And the set of experiments are, here is a solution, and here are some

A polymers, and here are some B polymers, and we take the As and the Bs, and we dissolve them in a solvent, and we say that the mixture, that is, the polymers are such that

there was a matrix polymer that was allowed to be concentrated or not, and a tracer polymer, which was always dilute, and we looked at the single particle diffusion of the tracer

polymers through the matrix solution. Here, matrix and tracer are both potentially concentrated. There are an extensive series of experiments to test theory by Ben-Munah. Agreement is

reasonable. There are a number of tricks you can pull in this system. First of all, if I am clever, I can arrange things to do index matching, and if I index match, perhaps

I can only see one of the two polymers, and not the other. Second, I can arrange things to what is called zero average contrast, so that if I increase the concentration of

A, and I increase the concentration of B, but I keep the ratio constant, that is if I have a concentration fluctuation that moves more mixture into the solution, that's a fluctuation,

I can arrange the solvent so that there is no change in the index of refraction, and therefore, fluctuations in the total concentration of polymer, assuming it's uniform in composition,

do not scatter light. Under this condition, what scatters light is something that changes the concentration of A relative to the concentration of B. So you do a theory for this and get

out answers. What does the theory look like? The core issue is as follows. There is a diffusion coefficient for the motion of A as driven by a concentration gradient

in A, and that causes the concentration of A to change. Also, there is a diffusion coefficient that couples to the concentration gradient of B, and that causes the concentration of

B to change. I should actually be consistent and write these as the currents, not the concentration of B time. However, a non-dilute solution, these are cross-toupled. There is a DBA,

which is the diffusion of B as driven by the concentration of A, and there is a DAB, which is the diffusion of A driven by the concentration of B. So there are cross diffusion

coefficients. Furthermore, there are reference frame effects. The basic issue in reference frame effects is if I say there is a current of A, there are A particles moving that way. There are two sensible ways to measure the current of A. One is to look at the motion

of A relative to the solvent. The other is to say we are in a closed container. If there is an A particle moving that way, because it is a closed container, it must be displacing

the solvent. So the solvent moves that way, and it just displaces everything in a non-preferential way, and therefore there are B particles pushed the other direction because A is displacing them. This is the reference frame effect. The reference frame description is due to

the wood, and the reference frame motions are independent of these cross diffusion coefficients.

That is, the two effects add independently. The consequence of the reference frame effects is that if I write the diffusion equation in a reference frame fixed on the scattering cell rather than the solvent, these various D's get mixed up in different ways to some

extent. So DAA, concentration gradient of A driving a motion of A, will also give us directly due to reference frames contribute to the motion of B, and vice versa. So there

are some interactions. The problem was solved for polymers by Ben Muna. There have been a series of experiments to test his theoretical models, and the models, well, work pretty well. They really do. They are somewhat coarse-grained models in the sense

their description of polymers is not we are looking at single polymer chains and polymer hydrodynamics in the Kirkwood-Reisman sense. It's a very coarse-grained description of what's going on, but at that level it works quite well. I shall very briefly

note alternative to light scattering. Neutron scattering. Now the core issue in neutron

scattering is in particular inelastic neutron scattering. We send in neutrons of one energy. They are scattered by the sample in its first one-order approximation scattering.

They come out in some direction, and they come out with some change in energy. The change in energy is equivalent to a change of frequency for visible light, and it corresponds to the fact that they are scattering off something that is moving in the system. Now

the one difficulty is that in order to use this to study, say, diffusion, the energy changes are extremely small, and therefore you have to do something very clever in order to measure them, and the clever thing is a result due to Mezzei, known as neutron spin

echo, and it's an experiment which is very clever. It's been done in a small number of laboratories, and it in fact gives you the energy shifts to very high precision.

Okay, so you scatter from neutrons. It's like scattering from light. There's one minor difference. If I have a polymer, let's say polyethylene, it might be regular ethylene.

We could also replace the protons with the deuterons, and now we have perduteroethylene in which all of the hydrogen has been replaced with D, and we could also do this so there are different amounts of H and D. The scattering properties for neutrons of

hydrogen 1 and deuterium are radically different, and as a result, by changing the isotopic substitution in what is otherwise exactly the same system, you can see some

fairly interesting results. Of particular note, if you adjust the HD ratio just right, you have say a polymer and it's randomly substituted with H and D. If you get the ratio just right, the polymer becomes very hard to see with neutron scattering, and

you can focus your attention on other things, and this allows you to do what is in essence tracer diffusion, self-diffusion measurements in polymer solutions. What has been done is also to do experiments on dilute chains. Dilute chains have been studied. The feature

here is, and this goes back to what we said a lecture or two ago, Kora made fairly specific predictions, namely there is a mode that is diffusive. There is a second mode

that includes diffusion and also a relaxation object whose relaxation goes roughly as QQ.

The issue there is you can actually see the two modes. This has also been done before using light scattering. However, with neutron scattering, a representative wavelength on which you might be working, say eight angstroms, so you are looking at motion on a much shorter

distance scale with neutrons than with visible light. That brings us to the end

of our discussion of light scattering, and I'm going to put a break in the tape at this point. And greetings to the second part of today's lecture. We're now going to advance from a discussion of quasi-elastic light scattering, neutron scattering, and such-not

to a discussion of viscosity. The physical notion of viscosity is represented by a single sketch. We have two extremely large, flat plates. One of the plates is stationary.

One of the plates is moving with some speed VX. The two plates are separated by some distance L in the Z direction, and therefore, if I plot V sub X versus position, the traditional

assumption was that V sub X versus position is linear in position, and therefore, DVX equals this V zero X, the actual velocity at the top, over L. You see, it's a velocity

gradient, velocity that way, gradient that way. The assertion that the velocity gradient is the same everywhere across has recently been shown to run into some severe

problems if you push hard on the system. Now, of course, there's an obvious alternative assumption if I take two parts and move one with respect to the other. If I have a solid block and I start moving the solid block top with respect to the bottom, I

get crazing and sheer planes and the thing severs. Well, polymers are somewhat in between simple liquids, which do this, and solids, which break if you twist them hard enough. Come back to that later in the course.

Having said that, in order to keep the upper plate moving you have to apply a force per unit area and the force per unit area is determined by the velocity gradient, assumed to be the same everywhere across, and the constant eta, eta is the Greek letter, and it stands

for viscosity, it's the resistance to pouring. Now that is, this is what is called shear, and this is the shear viscosity. There are two other viscosities that arise somewhat.

One is, if we imagine a little volume of material here, the viscosity that comes in, if we imagine compressing or decompressing the fluid, for example at some frequency, and changing its volume, and this is what is called bulk viscosity.

The polymer solutions we're talking about are mostly dissolved in substantially incompressible solvents and therefore you can't do this a great deal. There is however a third viscosity, which is extremely important in polymer engineering,

and the third viscosity arises if we take a thin long piece of polymer, this is usually done with a melt, and you grab the two ends of it and pull them in opposite directions, and as you stretch there is a resistance to stretching.

The resistance to stretching is known as extensional viscosity, and the point on extensional viscosity, there is a resistance to stretching, well this is how you make a lot of polymer

threads in some part. You push the polymer through holes and then if you want a really fine thread, you may for example stretch it as shown here. Well stretching is not quite trivial, and I've sort of indicated how it's done.

Okay so there are several different viscosities that you can imagine measuring, and what is found is that the viscosity of a polymer solution depends on concentration.

So as you add polymer, the viscosity of the solution changes, and if there is no polymer there at all, you have eta zero, the viscosity of the solvent, and then you have something of the form one plus k one c plus k two c squared plus dot dot dot.

K one has a name. K one, which is usually given the symbol, eta embraces, is the intrinsic viscosity.

It's the low concentration leading linear slope. Yes? Low concentration leading linear slope. We can rewrite this equation, traditional rewrite, one plus c eta plus k h c eta squared.

K one is eta. In this form, I factored k two into a key k h and a c eta squared. K h is the Huggins coefficient.

The Huggins coefficient is the lowest concentration term that reflects the interaction of polymer coils with each other. The linear term here appears simply because there are polymer coils, and you can calculate it to some level of precision by calculating how a single polymer chain moves and then

saying, well, there are a lot of them, so there's lots times as many effects. Huggins coefficient describes interaction between polymer chains. So that's k sub h.

Okay, so having said that, that's k sub h, and we have an intrinsic viscosity. What do we do next? Well, one thing we could do is to ask what happens at higher concentrations, and the classical answer to that is the Martin equation.

Martin presented a paper in 1943. My footnotes give the paper's title thanks to our very hard working library. I have not found another source that supplies the title other than the original conference

meeting report. And what Martin proposed was that eta would be eta zero e to the constant times the concentration. And he had data that supported this. It was a nice approximate formula, and that is the Martin equation.

Now, one can do better than that, but I am going to pause for a second and point out something about the intrinsic viscosity. This has dimension one, and this has dimension one, therefore this must have dimension one, and in units, the intrinsic viscosity in units must be one over a concentration.

So, C eta is dimensionless.

C eta can be described as the natural units for concentration units for discussing viscosity. How big, where is C eta of one? Well, for a largish industrial type polymer, meaning molecular weight is a hundred thousand

a million, something in there, C eta is one when the concentration is one or ten grams per liter, something like that. It's a fairly low concentration. It's not like volume fraction where phi of one is unattainable because you can't

pack it better than 0.72. C eta you can get up to, you'll be approaching the melt, a hundred or a couple hundred. Now, there are two other sets of predictions of viscosity at higher concentration,

and one, there are some nice papers due to Dale Schaefer who put this together, and the prediction is that viscosity at elevated concentrations goes as C to the X, M to the Y, where X and Y are powers. This is a reputation of scaling prediction.

Now, for polymer melts, and I emphasize for melts, for polymer melts, the theory derives not only the Y, but also the fact that it's a power loss.

That is, for melts, the model actually predicts the functional form. For solutions, that's not the case, but in solutions, you can predict the X and the Y.

Can we put in some numbers here? Sure we can. Let's get some more board space. You can actually measure viscosity of polymer melts for different molecular weights. It's not quite as easy as it sounds, and you have the serious issue that if you have a polymer,

how do you determine exactly what its molecular weight is? And the answer is there are a bunch of methods that give you information, but they're several to 10% methods, not 0.01% methods. In any event, the reputation prediction, looking at C to the X, M to the Y,

is that Y is some number, the traditional number is 3. Experimentally, Y is about 3.4.

This is from melts. And there are some theoretical treatments that endeavor to resolve the difference. For solutions, you can also predict X, and X is some number in the range of 3.75 to 5,

depending on solvent predictions, that's the predicted. And so you can actually predict some exponents. The alternative to this is eta proportional to eta zero, E to the alpha C to the nu,

where alpha is proportional to M to the power. And that's the stretched exponential form for viscosity based on the Kirkwood-Reisman hydrodynamics

and based also on the positive function renormalization group method, you can actually predict this form, you can predict nu, you can predict numerical values with no pre-factors for alpha,

and the agreement is, well, the cup is half empty or half full, it's actually pretty full. So, those are predictions.

Let me just start on measurements since we are almost out of time. And we will start with figure 12.1. 12.1 shows measurements of Jameson and Telford. The measurements are on a 7.8 megadaltons polystyrene in, if I recall correctly, tetrahydrofuran.

Now, 7.8 megadaltons is an extremely large polymer. They looked at viscosity versus concentration.

They did some other things which are in the book. However, if you look at viscosity, this is a log plot, you see a nice stretched exponential form. The stretched exponential form works well at all concentrations.

You can also look, for example, at figure 12.3. 12.3 shows measurements of Animoto, and these are on a schizophyllan.

And again, if you plot viscosity versus concentration, you see these nice curves, and the lines are at least decently close to them. If you look very carefully at the curves for the highest molecular weight polymer and dilute solution, the measurements actually show a viscosity which is a bit lower than you'd expect from the curve,

which describes everything else quite well. But if you look at those curves, you notice the viscosities up here are extremely large indeed. We're looking at increases in the viscosity of multiple orders of magnitude,

in some cases up to five or seven. And the stretched exponential describes nicely over the full range of concentrations what the viscosity is. I see, however, we are out of time, and therefore we will continue this in the next lecture.