Note that this is a short lecture. The remainder of the lecture on the hydrodynamic scaling model has been transferred to lecture 30. I'm Professor Filley's, and this is the next lecture on Polymer Solution Dynamics.

What I'm going to do today is to discuss more generalizations and inferences that can be made on the basis of the phenomenology we presented in most of the earlier parts of the course. I'll then advance to treat a theoretical model that perhaps explains how polymer solution dynamics works.

We've reached the point in our discussion of inferences where we're going to get to discussing hydrodynamic interactions and the allegations that hydrodynamic interactions might be screened.

The beginning point of this is a very simple issue. We have moving objects in the solution. It's a frictional point. It's moving very slowly. It sets up a wake in the surrounding solution, and the wake acts on remote objects.

And given that we have a velocity here, an induced velocity here, we can ask how these are related, and the relationship, after all, we want something that maps a vector onto a vector is the Oscene tensor and up to constants which aren't critical at this point.

The form of the Oscene tensor is one over the distance from the particle creating the wake to the object on which the wake is acting.

And there is the tensor piece. So there is the Oscene tensor. The notion that you will sometimes find in texts on polymer dynamics is that if we go from a couple of particles in water to a polymer solution,

the assertion is made that the Oscene tensor is screened, that is the distance dependence goes as e to the minus kappa r over r or something similar. Now, screening, the phrasing is borrowed from an understanding

of electrostatics in ionic solutions. That is, if you're in an ionic solution, we have here a positive ion, large positive ion, and surrounding it, if we've got salt,

or if we're just in water, which autoionizes, there will be a preference for the negative ions in the solution to come in closer to the positive ion. And for the positive ions on the average, but not every time, to be repelled. And if you ask what the electrostatic potential around this positive ion looks

like, well, let us plot, starting at the center, the amount of charge encased within a Gauss's law sphere. So we have charges inside a sphere, and if it's atypical protein types here, the charges are all at the surface.

And so you have almost no charge until you hit the surface. And within a small number of atoms of the surface, there is where this charge lies, it's all at the surface. And then as we make the sphere larger and larger, well,

we're encompassing, you notice, more minus ions than positive ions. And therefore, the total charge inside this volume falls. And the net charge eventually falls, and if you go out far enough, the solution is electro neutral and there's no net charge.

Now, if there hadn't been the surrounding negative ions, if you just had the lump of the shell of positive ions at the surface, the potential would have gone as 1 over R, a Coulomb potential. However, because we do have these ions on the surface,

the monopole component is screened as e to the minus kappa R over R. Note I said monopole component. The charge distribution may have dipole, quadrupole, etc moments. And if you look up the paper by Kirkwood and

Scatchard, which is about three-quarters of the century old at this point, you discover that in general, the lth moment goes as e to the minus kappa R. Times a polynomial in kappa R over R to the l plus 1.

You might sensibly ask, that's very interesting, but what is kappa? And kappa is the Debye length, Debye screening length. It's actually an inverse length.

Kappa is determined by how much salt you have in the surrounding solution. The less salt there is in the solution, the longer the range of the interaction. So that is how Debye screening works.

We can also draw a freshman physics picture of screening in terms of electrical lines coming out of positive and negative charges. This is the Gauss's law picture as shown to freshmen in two dimensions.

But the idea is all that matters. Here's the positive charge. And if there's no other ions around, the lines of electrical, the electrical field lines all come out of the positive ion and keep on going. And since the surface of a spherical shell out here gets larger as R square,

g, the density of field lines falls off as 1 over R square, which is what you expect, yes? That's what the field lines and hence the force falls off as.

However, if you have some negative ions out here, the line flows out from the big positive charge in the center, reaches a negative line, a negative ion, and comes to a stop. Because it's swallowed up, cuz that's what negative ions do.

They act as sinks for the field lines. And therefore, you started off coming out of here with eight lines. And as you get out further and further, the number of surviving field lines is smaller because the negative charges, the counter ions, act as a sink.

So that is a picture of how electrostatic screening works. You will occasionally find people who make the amusing claim that all long range forces are screened. And of course, that's absurd nonsense because

there's this other long range force in nature called gravity. And except on Star Trek television shows and the like, gravity is not screened. So you have two long range forces in nature which you can easily point at, one of which is screened, one of which is not screened.

And therefore, you are under no particular obligation to believe whether long range forces should be screened or not. Well, having said that, you then ask, well, how would screening of hydrodynamics work?

And you can draw a picture of this, I'll give myself more space. So here is the moving object, and it's moving at some speed, v.

It exerts a force on the solvent because it's moving through it, a drag force, and that means it's exerting a force on the solvent in this direction. So what happens, it's dumping momentum into the solvent, and the viscosity of the fluid ensures that the momentum

is transported away from the moving object. In fact, it moves out in all directions. That's what the Oseen tensor is, it's describing the transport of momentum, which is a vector this way, but it's being transported sideways.

So the moving object creates a wake out here, it dumps momentum into the fluid out here, so the momentum had to move sideways. Now the notion is, if there is another particle here, here's another particle, the moving fluid encounters this particle,

and puts a force, Fv, on the particle. And since there is a force, Fv, on the particle, the usual third law argument says, there must be a force minus Fv on the solvent.

And therefore, because the solvent is putting a force on the particles out here, there's a backwards force the other way, and momentum is sucked out of the fluid. This is a remarkably deceptive and totally wrong argument. The reason it's totally wrong is that we're talking about a solution,

and these particles are free-floating. And because they are free-floating, over any significant period of time, their motion is completely damped. The fact they have inertia is basically irrelevant, except on the very shortest of time scales.

And therefore, at almost all time scales, V particle minus V solvent is equal to zero. And therefore, the particle bobs along with the fluid and exerts no force on it.

Furthermore, these particles out here are at thermal equilibrium. And because they are at thermal equilibrium, we know that V square at time t equals V square at time zero. So the weight spends some time accelerating the particle.

By and by, it must be the case that the particle will notice the solvent friction and will slow down again. And when it slows down, the momentum that has gone into this particle has not been destroyed. Momentum is conserved.

That's the opposite of this situation, where the positive field line flows into the negative charge, yes, and comes to a stop. I'll give an amusing electrostatic comparison with this. This field line is swallowed up and destroyed. The momentum that went into this particle,

unless it has some magic momentum storage mechanism, by and by on the femtosecond time scale, has to come out again. And therefore, the momentum comes out again. And it may be scattered. That is, the momentum that was swallowed up here may head out in different directions.

It's scattered, but it is not absorbed. Let me give the correct electrostatic analogy. We will put out here a piece of metal, okay?

And the piece of metal has acting on it a positive field line. And what happens? Well, this is physics two also. You pile up negative charges on this side. But because you piled up negative charges on this side, those charges had to come from someplace.

They are not enthusiastic about lurking inside the metal, although of course there's a skin depth. They are matched by positive charges out here. And therefore, the field line comes to here, and it comes to a stop when it reaches the metal. But metals do not screen, except in their own interior.

Instead, the positive field line appears out here and goes on its way, as any competent freshman understands. Similarly, the momentum is absorbed here briefly, but it goes on its way again. Well, that leads to the amusing question, why would anyone think that there could be screening?

And the answer is, there's a related but different problem in which there is screening. And the related problem is the sand bed, by which I name this big bed of very finely crushed rock, beach sand.

And you exert a force on the fluid in the sand. It's water packed, and ask what happens. And the answer is, if this is not a free particle but a sand grain, there's a sand grain, the fluid comes along.

It exerts a force on the sand grain, and the sand grain is jammed, it's up against its neighbors. So the momentum dumps into the sand grain. And now the momentum can transfer through sand grains at the speed of sound, and disappears, and is not there to be dumped back into the liquid at a later date.

Thus, you do reasonably expect hydrodynamic screening in sand beds. But if you understand the laws of mechanics, it's perfectly clear that at low frequencies, you cannot have high screening hydrodynamically. I'm going to emphasize I said low frequencies.

At very high frequencies, where you have to worry about the inertial mass of the solvent, and the inertial mass of the particle, life can become more complicated. Okay, so that is a picture of hydrodynamic screening, and

why maybe you shouldn't have expected it. And let us now look at experiment. I will say facts can be your friends, except in politics, where they're always your friends, because you get to make them up.

Okay, so let us consider experiments that are done that speak to hydrodynamic screening. Okay, so what experiments can we do? Well, number one, we have for spheres and polymer coils, a self-diffusion coefficient,

a first cumulant of the light scattering spectrum, a rotational diffusion coefficient. And these all go as the zero value times one plus some constant times the concentration of the first power, plus other good stuff.

Yes? Well, if you believe that you have something whose shape you know, and you know what the hydrodynamic interactions are, you can calculate k. And you can calculate it separately for each of these. You can also do it for the probe diffusion coefficient,

a probe sphere going through polymer coils. And you do all these calculations, and if you do this carefully, you discover that yes, there's the 1 over r part, the oscene part. But there are a whole series of parts that were first developed by Kinch and

then by Binacher and Mazur, and in addition to the 1 over r term, there's a 1 over r cubed term, a 1 over r fourth term, etc, etc, etc. These objects each depend on the different parts of the full

hydrodynamic interaction tensor, of which the oscene tensor is very important, but a lowest order dominant contribution. These quantities and their case depend on all of the tensors in different ways.

And therefore, when you look at different of these, you're looking at different parts of the hydrodynamic interaction tensor, at least the weightings are different. And you can calculate all of these for spheres, for a probe going through polymer coils, or if you go back to the chapter on dynamic quasi-elastic light scattering for polymer coils diffusing.

You can calculate all of these, and you always get reasonable answers for K. Now, reasonable means plus or minus one if the concentration is in volume fraction units.

In some cases, the agreement is considerably better than that. You should realize that these things, if you're mostly sensitive to short range interactions, you really have to know accurately what the interaction potential is to know how often the particles are close together.

And therefore, those calculations intrinsically are going to be harder to do well, or harder to match with experiment well, maybe, I should say. And those all work. Well, that's very nice. Okay, so what else can you do? Well, there are a series of experiments due to Crocker,

and I actually have a list of other authors in the book. There are a series of people who've done this, who do two point hydrodynamics. And they ask, what are we going to do? We have two particles. We can see both of them. They are diffusing.

And we can calculate, measure the displacement of particle one, the displacement of particle two. And on the average, these two things are correlated. There is a cross diffusion tensor which tells you

that the motion of particle one and the motion of particle two are not independent. Now, if you were brought up on the Langevin equation, maybe you didn't realize this. Langevin is the utterly bottom order approximation for extremely dilute part monodisperse particles.

In fact, there is a correlation, it's measurable. There was someone who looked for it a century ago, but didn't quite know what to calculate, so we didn't find it. Perrin was a very good experimentalist after all. So there is the correlation. And D12 is found to be proportional to the Ocine tensor.

And since the Ocine tensor also includes, it's actually one over eta r i plus r hat r hat. I'm still leaving out some constants.

You could infer a viscosity for the solution from the cross diffusion tensor. However, if you're doing that, you're also incidentally saying the hydrodynamic interactions fall off as one over r.

Well, the experiment's been done. This form does seem to work. Though the people who did the experiments were looking for eta, not asking if one over r was correct. At least they don't mention this in the paper. They may have known it was an issue and it was so obvious. There was nothing to say. And what they find is a one over r dependence even for

pairs of probes in concentrated polymer solutions. That is, people have looked directly for hydrodynamic screening, and it is not at home. Okay, we now come to a third experiment that tests for hydrodynamic screening.

And this is an experiment due to Martin. This is not the Martin of the Martin equation. This is another Martin, who is now out at Los Alamos. And what we have is a 48 mega-dolton polymer.

And there's a tiny number of them. And it's in a bath formed by 110 kilodolton polymers. Another species at a concentration of 40%. And we have chosen the solvent, so it index matches the tiny polymer.

So we know it's there, you can't see it. And we can look, this is a huge polymer. You're doing light scattering, we can look at the internal modes. This is something we talked about back in about chapter eight. And what we find is that for reasonably large Q,

you can see the motion of one part of the polymer with respect to the next. And the relaxation rate goes as Q cubed. That is, the larger the Q is, the smaller the scaling distance is.

If you had a diffusive process, you would say rigid object moving, you'd see Q square. Here you see Q cubed. And there's two theoretical calculations here. And the two calculations are due to Rouse and the Zimm.

The Rouse model treats beads on a chain. And the beads have no hydrodynamic interactions. The Zimm model treats beads on the same chain. It's actually exactly the same chain, except remote beads interact by the Oscene tensor.

The Rouse model finds Q to the four behavior. The Zimm model finds Q cubed behavior, as you see here. And therefore, for this object in a solution where the hydrodynamic interactions ought to be screened, this is a,

the matrix is very concentrated, very substantially entangled. If you believed in the screening, you'd say, you should see non-interacting beads, Rouse behavior. In fact, you see Q cubed behavior. And therefore, as Martin says, the interaction,

the hydrodynamic interaction between the beads, must be falling off as 1 over R, the unscreened behavior. Let me comment that all of these results are in the literature. Some of them have been around for a long time.

However, what I just showed you, putting them all in the same place, and saying we have all these tests for the existence of the alleged hydrodynamic screening phenomenon. Well, no one has ever done that, at least you, other than me.

And therefore, there are people who believe in hydrodynamic screening, even think they're experimental tests that show it, though they never cite it. Okay, let us then chug ahead and ask, well, what about theories?

What are the actual detailed theories, as opposed to the hand-waving treatments? And there are several important theoretical treatments, and one is due to Altenberger. And Altenberger looked both at low frequency and also at higher frequencies.

And what Altenberger said, well, here's the moving object creating a wake. And if this thing is fixed, it easily can be shown to contribute to screening. This is Darcy's Law for sand beds. It's a 19th century calculation.

On the other hand, if these things are entirely mobile, there's no screening at all. Beenacher did a calculus, put his name up, good guy. Beenacher did a really clever calculation.

What he said is, well, the way hydrodynamics works is we have a moving particle, it creates a wake that heads out to here. And the electrostatic analogy of screening is that the wake is scattered. We'll talk more about why they're scattering,

either later in this lecture or the next lecture. And so the wake can be scattered off and hit another particle, and another particle, and eventually it gets back to here, or eventually it gets over to there. Well, let us suppose we take these, now of course there's a lot of scattering

patterns, some of which, like that, get complicated. But let us take what are called the ring diagrams.

Yes, these are the longest range diagrams. The ring diagrams, if you do the electrostatic calculation, are the diagrams that cause screening. They're the longest range part. Well, he resums the ring diagrams, very clever work.

And what he finds is that the Oscene tensor is renormalized. That is the strength of the hydrodynamic interaction is changed by all of these scattering effects. However, the range remains as 1 over R.

Okay then, so where did hydrodynamic screening come from? Did it just appear by magic? And the answer is no. There is a paper by DeJean, early 70s paper, and

he says that the range of the hydrodynamic interaction goes as some R over psi where psi is a screening length. And how does he calculate this? Well, he doesn't calculate it at all. In fact, what he does is to cite a paper by Freed and Edwards.

And the nice paper by Freed and Edwards purports to derive hydrodynamic screening. Well, having said it, purports to derive hydrodynamic screening, if you hunt through the literature, this is a very hard

calculation, you have to do a lot of approximations. At a later date, Freed and Perico did a better calculation. And that is the normal process of doing theory. You do things better and better. And if you're lucky, you eventually get closer and closer to the right answer.

And the Freed and Perico calculation shows no, there's no screening, it's 1 over R. And so this paper on being improved gets to that paper. And this paper cites a paper that, well, has been improved upon.

And the improvement says there is no screening. However, lots of people reference this source without bothering to trace back in the literature and see what's going on. Yes, that is the, that is, there is only one Dejean.

There are several Martins, but there is only one Dejean. Okay, so much for the myth of hydrodynamic screening. And now we push ahead to length scales.

And I first know why the presence of length scales is theoretically sensitive. And they're actually, there is a core issue, which is that a lot of these calculations use scaling arguments.

And the scaling arguments say that we look at the rate and ratio of, say, a particle radius to some length in the solution. That ratio is, of course, dimensionless, it's length over length.

And we can then do very clever theoretical arguments to say this is going to be raised to some power delta. And if I am very smart indeed, I can actually derive values of delta. For melts, for diffusion,

there is actually a rational derivation of a scaling law. In solutions, scaling, at least in all of the papers I've found, had, I'm sure there are more out there, has been assumed. And the core issue is that you derive delta. And there are some nice papers, for example, by Dale Shaffer,

who shows how to do this and get delta. It's a very nice calculation. Critical to the statement that I can use scaling laws to derive delta is that there is one dominant length.

And whatever the dominant length is, I know what it is. The reason you need one dominant length is if there were two of these things, like psi and that thing,

that's dimensionless, that's dimensionless. And now you don't know what power of psi enters the calculation, because you have two unknown exponents in basically one equation.

So if you want this to work, you really want one length scale, and it has to be dominant. If you have several length scales, simple scaling arguments cannot be made to work. Not because there might be a scaling law, but you have to be more clever to derive it.

Okay, having said that, we then sensibly ask, okay, so there has to be a length scale, what is the length scale? And a standard answer is we have a transient lattice model.

And the transient lattice model says, here are polymer coils. And every so often, the polymer coils move close to each other and form what is called an entanglement point.

Now, if we were talking about covalently cross-linked gels, that is, true chemical gels, these would not be entanglement points. They would be covalent bonds linking the large chains. The covalent bonds are permanent on a thermodynamic basis.

Well, almost permanent, anyhow. And therefore, you say there is a real physical gel here. What DeJane said is, you get these temporary entanglement points so you have a transient lattice. And there are timescales on which the polymer solution looks like a cross-linked gel.

His original development of this, he was quite clear on this, is that this is a guess. And we explore the consequences of the guess. And if we're lucky, the guess works. And if it doesn't, we're no worse off than we were before.

Now, there is one question, there is one critical issue you want to be careful of. It is very easy to draw an entanglement point. There is an entanglement point. There is absolutely nothing in the theory that describes the physical nature of entanglement points.

In fact, well, this is a point, it's a real point. You could actually see it if you had infinitely good vision. You could also say this is some sort of a mean field approximation.

And there aren't really entanglement points at all. It's just a mean field description of the system. The reason the transient lattice is important is that if I have another polymer coil coming through here like this, and it tries to move sideways. The claim of the degenerate model is, the chain moves sideways until

it encounters a neighbor, and it is then obliged to stop. Now if you're in a true gel, we sort of know that's true. Because if you watch a very long chain polymer doing electrophoresis through a true gel, like a huge DNA, you can actually watch the DNA move.

There's some interesting complications on how DNAs move through gels. In fact, they do have to do sort of a snake-like motion. They have to move this way, because motion that way over long distance is blocked by the neighbors.

The related polymer motion is called reputation, from repterre to crawl. So, that is the picture, and associated with this is a width of the hole,

which is claimed to be about the same as the distance between entanglement points. If you think about this, I waved my hands very fast, and this is called psi. And so the reputation picture says the gel looks like a fishnet, and it has holes of size psi.

Well, we can't actually see the gel, the transient lattice, and its entanglement points. In fact, we wouldn't necessarily know. We had an entanglement point if we were looking at it, because we don't know what they look like.

So the question is, how do you find how big psi is? And the following, I believe I should credit Ben Law for this. It's in Dejean's book. The notion is, here is an invisible fishnet, and you cannot see the holes in the net.

You cannot see the mesh. How do I determine the mesh size of an invisible fishnet? And the answer is, I go to the aquarium, and I buy a bunch of fish,

different sizes, and I confront them all with the mesh. And the small fish will be able to swim through the mesh.

And if I get the fish to be big enough, they can't get through the mesh. Size exclusion. Yes, size exclusion. Now, you can actually get size exclusion in real cross-linked gels. It works very well. That's the basis of a 40-year-old chromatography method, size exclusion.

The small objects get into the gel, and therefore don't move as fast through the column as the bigger objects do. Well, having said that, people have sort of done the experiment. You can do the experiment with optical probes and light scattering.

And the impression you would get from this picture is that if I plot diffusion coefficient versus R, there is some size and over some range.

After all, this is a random lattice. The large particles are blocked and cannot advance, other than very slowly. And the small ones can swim right through. Well, that is what this model predicts, and it's really not true. Yes, polymer solutions do a certain amount of seething,

and therefore in some systems, the small particles move faster. In some systems, the large particles seem to move faster. Small is what is usually the case. However, you certainly do not find a blockage for particles above some size.

And therefore, this picture is simply not sustained by studies of the diffusion of probes through polymer solutions. The experiment has been done. Well, you can, however, take a polymer solution

and look at the light scattering spectrum for a whole bunch of different size of probes. And that was done by Kirill Streletsky in my laboratory, who used something like 10 or a dozen different size probes from quite small to quite large.

It's also been done by Paul Russo, who used smaller probes than we did. And what Kirill discovered is that if you look at the probes, the light scattering spectra are bimodal. That means for motion on some fixed distance scale,

as you're looking at a single cube, if you compare at a fixed distance scale, there is some mode of motion which allows some of the particles to move quickly and others to move only slowly. And you can now, since you have these two relaxations, they each have a relaxation rate,

and you can look at the behavior of the relaxation rate as a function of probe size. And for example, you find for, here's R increasing, and if you plot the relaxation rate versus probe size,

you discover, for example, for one of the relaxation rates, the diffusion rate does not depend very much on polymer concentration, it does a little bit, until you get up to a critical size, and above that size, there's a much more dramatic dependence of the diffusion rate,

or rather, that mode relaxation rate on particle size. The spectra you find are sums of stretched exponentials, so there are four or five, counting the amplitude ratio, parameters you can look at.

And what you discover is there are a characteristic set of small probe behaviors, there are a characteristic set of large probe behaviors, you can get into a discussion as you move from here to here, as to which mode you are supposed to associate with which mode,

because you have two modes with relaxation rates that aren't incredibly far apart, so that it would be sort of obvious. And so there is a critical length scale in solution that you can actually see using the probe method, using Benoit's idea. Question?

And this critical concentration, it's not dependent on the probe size at all, is it? Oh, this is a critical probe size. Probe size, okay. Yes, and you see the probe size in the concentration dependence of some of the spectral parameters, which are different for small and large probes.

And the critical probe size is approximately, or maybe twice, RG, the radius of the polymer. That is, the visible typical length scale is something like that,

the size of the entire polymer chain. That's very different than having a length scale that is a small part of the size of a polymer chain. So there is a visible length scale, but it's not the ones that are being predicted by these models.

There are a number of other experiments you can do that speak to the same question. And when we say the things speak to the same question,

well, let us consider a set of experiments. Let us look at diffusion of a polymer, and we can do the experiment using light scattering spectroscopy. And light scattering spectroscopy is sensitive to some distance like lambda over four pi.

That's an approximation, and it depends on the scattering angle. But this is distances of, for example, 500 angstrom. And you can get smaller than that and larger than that somewhat by changing your scattering angle.

You can also study the same experiment doing forced Rayleigh scattering. The idea in forced Rayleigh scattering, there are several other methods, such as fluorescence recovery after photobleaching, where the key step is the same.

You send in a laser beam that has been beam split and is being brought in at two angles. This is equivalent to looking at a two slit interference pattern. And if you look in the liquid, there are regions that are bright,

and there are regions that are dim, and there is here a space in L that's determined by the light wavelength and this angle, and it now becomes an undergraduate physics calculation. But you can create a grating in the solution.

And the next step is you've created a grating, and for example, you have photobleached particles at some distance, or you have created zones that are lower or higher concentration with some separation. You've done something, and you can then observe the relaxation of the grating as,

for example, the particles that were originally here diffuse to fill in the gaps. I am leaving out a great deal of detail on the experiment, except the core issue, which is that you are observing particle motion over much longer length scales.

And there are several people who have done this, and the several people who have done this, for example, Chang, but if you plot gamma, the relaxation rate versus Q square, you get out this nice straight line, which covers orders of magnitude and particle distance.

And therefore, diffusion over small distances, fairly small distances, and diffusion over very large distances, proceeds at about the same rate. Now, in particular, Chang et al.'s measurements appear to get to sufficiently short length scales

so that they are looking at the motion of this particle over distances that are smaller than the full particle size, meaning that they are looking, here is particle or polymer coil, it is inside what is known as a correlation hole,

that is, other chains are less likely to be found here, because we've already said this chain is here, and the motion of the part of the polymer inside the correlation hole, though over fairly large distances, we're not looking at really short distances yet, the motion over moderate distances is the same as the long range distance.

There are theoretical explanations for these bimodal spectra that say, here is a particle, it's in some sort of a hole in the solution, a hole created by the fact that the particle is here, and therefore the particle can move rapidly over short distances,

but only move much more slowly over long distances, and thus you see a fast mode and a slow mode. Well, if you understand light scattering spectroscopy, you just realize that I have just confuted two completely different issues, and the two issues I have conflated are, well, number one, fast and slow motion,

that's a time scale issue, and number two, short and long range motion, that's an issue that you would see by looking at a series of different values of Q. And experiments done at a single scattering angle

only look at particle motions over a fixed distance. Well, it's of course a spatial Fourier component relaxing, so fixed distance is a little different than what you see on your car odometer, but the core issue is you're looking at a fixed distance,

and therefore the slow and fast modes both correspond to particles moving, say, yay far, not to particles moving small distances and large distances. Okay, and what this experiment says is that we don't seem to see small and large distance effects.

Okay, more experiments done by Streletsky on hydroxypropyl cellulose came to the conclusion that at higher concentrations, you see a third very slow mode, and the very slow mode

corresponds to the motion of objects that are quite large. How do we know it corresponds to the motion of objects quite large? Well, we took the same polymer solution he did, we put in it spheres of different sizes,

and we discover how fast the sphere of each different size diffused. And therefore, we can infer what size of polystyrene sphere would give us a relaxation rate about as fast as the relaxation rate we see is the ultra slow hydro HPC mode.

And that corresponds to an object which is, oh, half a nanometer across. It's much bigger than a single polymer chain, which is only half a nanometer across. Excuse me. This is about 0.5 mu.

And the single polymer chains are more like 0.05 mu, sort of, 50 nanometers. And therefore, the moving objects are much larger than single polymer coils.

We may describe these as vitrified regions. What is the nature of the vitrified regions? Well, we actually don't know. One thing we can say is that they're not local regions of high or low concentration. Why?

Because Rob O'Connell had measured the intensity. He thought it was intensity over concentration versus concentration. If you simply had objects that didn't interfere or anything, the intensity would be constant. If the objects repel each other, which is typically the case, because they can't move

through each other, as you make them more concentrated, it gets harder to form a concentration fluctuation, and the intensity therefore falls. Here's the concentration at which you start seeing the ultra slow mode, and there's no

bump up of the concentration as you cross that line. That is, the vitrified regions are regions that are dynamically paralyzed, but they aren't dynamically paralyzed because they're only semi-dissolved.

By topology, we mean the connectivity of the polymer chain.

And so, we have linear chains, we have ring polymers, we have star polymers, linear, ring, star.

We can make things that look like, this is called a pom-pom, it vaguely resembles an object used by cheerleaders in football games.

Well, once upon a time, anyhow. You can also make something that looks like this, this is an H. You can also make something that looks like this. You will be astonished to learn this is called a cone.

There's a lot of information on linear polymers. There's a considerable amount of information on star polymers. There is some information on ring polymers. Now, the early studies of rings have a problem, but synthesizing ring polymers, say ring

polystyrene, is actually tricky, because you have to start out with linear polystyrene, activate the ends, and persuade the ends to attach to each other, not the other chains. And you want to do this in such a way that you didn't accidentally form a concatenate,

that is rings that are, yes, rings that are not actually chemically bonded, but are attached. And when people did measurements on rings and didn't quite get the expected result, there was a lot of controversy as to what was being seen.

Under modern conditions, there's a much better solution. You take any of a number of viral DNAs, and there are viral DNAs that are rings, and they are clean rings, and if you put them down on an electron microscope, there's no sign that they are knotted up, at least they don't normally appear to be knotted up.

And therefore, they're all exactly the same size, too, they're rings. And if you go in with a restriction enzyme, you can cut things once, and you get a linear chain that is exactly the same size within a part in 10 to the 6 as the original ring.

Um, so you can actually study topology. Now having said, you can study topology. If you go through the text, there are lots of bits and pieces here and there on one effect

or another effect, and here's what happens when we change the topology. I could have arranged this where I had a chapter, effect of topology, like I had a chapter on colloids and all dynamic studies of colloids. I could have had a chapter on star polymers and all of the dynamic stuff involving star

polymers, but then I wouldn't have had all the star polymers readily available to be compared with linear polymers for each technique. So let us consider a few bits. And for example, we have electrophoretic mobility.

And if you hunt through the appropriate chapter, you will find this nice picture of electrophoretic mobility, which is of probes that are about 2 times 10 to the 5 base pair, except some of them are linear, and some of them are extremely synthetic stars.

You cannot have a star DNA with the normal synthetic process in cells, but you can make this as a synthetic chemical. It's very clever. It's truly clever to think of doing that. And then you have a matrix polymer, which was either a 53.5 or an 1197 kilodolton

polymer at various concentrations. So you then look at the mobility of the linear chain, the mobility of the star as a function of concentration. And what you discover is two lines are pretty much on top of each other.

That is, if you look at electrophoretic mobility, linear chains and stars of quite large size move at the same speed, pretty much. You can also look at polymer self-diffusion, where the concentration dependence of the self-diffusion coefficient

is characterized by our parameters alpha and nu. And you can compare alpha and nu for linear chains and star polymers. And especially if you say we will do the comparison for, we will keep the size of the probes constant.

If we do probe diffusion, we'll use linear chains and stars. You find for 3, 8, and 18-arm stars, that alpha and nu are certainly not very different from linear chains. I didn't say they're not different at all,

but they're not enormously different as if the star polymers had different methods of propagating than linear chains did. Now this has been done with various sizes of polymers. Now these, actually these are synthetic polymers. The self-diffusion studies are almost all synthetic polymers.

Self-diffusion in concentrated solution, you would face the issue DNA is charged, and you are now looking at a charged concentrated polyelectrolyte solution. That is one of the topics I did not consider in my book. We have to stop some of this. Okay.

So what else can you do to do the same experiments? well, one other thing you can do in this is to look at viscosity and This was actually done 20 years ago and People compared ring and linear chains and found that the viscosity of the ring Was considered was less than the viscosity of the linear chain

And since this was the opposite of what the reputation model predicted there were lots of let's attack the experiment discussion Attempting to explain how this could be true in terms of for example trace amounts of linear

Chains that cause some weird effect and this discussion goes on for some time Recently however Goodman Did the experiment using DNAs ring DNA and cut DNA

Because of the manner of synthesis of ring DNA there is no possibility of concatenation There are good reasons to suppose that the rings are not routinely tied knots The ring and linear DNAs are really really the same size and Goodman found the viscosity

Of ring polymers is significantly factor of three or five if I recall less than the viscosity of linear polymers at different concentrations So that is the topology effect Mm-hmm. I drop in a few theoretical

Simulational issues

So what can we pull out of theory? Well, there is a computer simulation due to Skolnik Which says here is a polymer here polymer chains and the polymer chains are bonded to a surface at one end

Now the bonding in order because of the way the computer model works allows The bond to move up or down about one one or so bond lengths So Bob's up and down a bit, but it stays basically at the surface and what is done it was

Computationally determined the diffusion coefficient of these chains and the diffusion coefficient was found To Now you will find people who claim that the proportional to M to the minus 2 Is the signature that proves you have reputation and this?

simulation conclusively disproves that claim for the excellent reason that these chains cannot Possibly rep tate because they're attached at one end they can only move sideways and nonetheless you see this behavior One may propose that what reputation is saying is that you actually have all

some general feature of constrained motion there is what appears to be a significant calculation by Brereton and Bruce Lee

This is actually in the review by lodge, which is an excellent review On the trip on the traditional style beautifully thorough covers melts covers mostly melts a little bit of solution though and Dropped in there is this interesting paper and what they do is a fluctuation dissipation

That's equivalent to saying more responsive Calculation On how chains move and what they said Is here is polymer chain. It's subject to random thermal forces the same way that a

Brownian particle is but there are a whole lot of random forces acting along the length and what they say Is that if the random forces are highly correlated in space and they propose to be

to The space that the moving polymer occupies in the transient lattice The tube model is an example of this they are able to extract scaling laws

However the conditions required to get scaling laws are actually Considerably more relaxed than the assertion that you have this transient lattice and one Polymer that can only move back and forth along its length through the tube

the issue with the stress tensor Is if we want to talk about viscoelasticity We ask the question how does one part of the fluid put forces on a different part of the fluid and

The general answer if you simply have a simple fluid like this is the particles attract or repel each other And there are intermolecular forces that act to create

This part of the fluid pushing or pulling on that part of the fluid viscosity for example What the reputation type models say is that for a polymer The important issue is here is a polymer chain stretched across the boundary and the piece of the chain here

Can pull on the stuff on the other side and therefore? Intramolecular forces are dominant in creating stress cancer here Intermolecular forces don't do anything only intermolecular forces act However there have been a number of calculations done on this

They're cited in the book and the conclusion is the net contribution of this array piece of the Interactions to the stress tensor is small and it is really all the intermolecular Polymer forces that are creating the stress tensor. That's a direct calculation

Finally, returning to something I said for very early on in this summary lecture, but very important Viscosity is not Universal it's not a universal function of concentration

And that's very important because any model that says well we have these general Considerations about polymer chains which we can represent as long strings and the general

Considerations work equally for everything and they lead to particular predictions like scaling laws well those models have the problem that eta of C if you plot eta versus C do not have a universal function of concentration and That is it for this part of the lecture so we shall stop for a moment