Good morning. I'm Professor Filleys, and this is 597D, Introduction to Polymer Solution Dynamics. Today, I'm going to finish off the discussion on polymer self diffusion. That brings us approximately to the end

of the first half of the course. Last time, we were discussing tracer diffusion. We were discussing what happened if you measured the diffusion coefficient of a polymeric probe through a polymer of a different molecular weight. And what was found was that if the pro-molecular weight, P,

and the matrix molecular weight, M, were not too different, so that P over M and M over P were both less than two about three, you consistently observe a single behavior,

namely the diffusion coefficient is proportional to E to the minus alpha C to the nu, M to the gamma, P to the delta. If you move outside of that range, the phenomenology becomes more complicated.

For example, if you look at a probe of fixed molecular weight and we plot the diffusion coefficient versus the molecular weight of the matrix, that's figure 8-33. What you see is a rollover, a stretched exponential.

And that continues until roughly M over P about three. And for larger matrices, things roll over. That is, very large matrices and smaller probes are not behaving quite the same way

that you see when the matrix and the probe are the same size. OK, that's what we're going to say about self-diffusion and tracer diffusion. Let us advance to an entirely different set of experiments. And the entirely different set of experiments are due to Chang and a truly illustrious group

of collaborators. And what was done was to look at the diffusion of polystyrene through polyvinyl methyl ether, once again, in orthofluoro toluene. Or was it toluene?

You'll have to look it up in the book. So they did the measurements, but they did something different. They did the same measurements, both with quasi-elastic light scattering spectroscopy and with one of the fluorescence techniques.

What is the point of doing two sets of measurements? Well, the point of doing two sets of measurements is that two sets of measurements measure diffusion on two very different distance scales.

They also each measure motion on distance scales, where, gee, if you're clever, you can manage to go in and you can manage to vary the distance scale by changing the experimental apparatus.

FRS gives measurements on distance scales of O one to two micrometers. Quasi-elastic light scattering gives measurements on the distance scales of O maybe 100 nanometers,

a fraction of a wavelength. You can get down to a moderate fraction of a light wavelength. You can get out to much larger distances if you want. And the main result was that the relaxation rate,

the relaxation rate which for diffusion should go is d, the diffusion coefficient, times the square of the characteristic length vector. If you're looking at relaxation of a cosine wave,

you're looking at a concentration fluctuation or a concentration that looks like a cosine Qx. For light scattering, Q is determined by the scattering vector. For FRS, Q is determined by how you set up the gradings.

But in either case, you're looking at diffusion over some range of distances. And the major result, this is what is expected. And that is precisely what is found. And as Tim Lodge has described to many of us,

you have this beautiful graph of gamma versus Q square. And the important thing on the graph, which does continue eventually to the appropriate zero intercept, the nice feature of the graph is that you

have this nice straight line in the intercept down here at gamma equals zero. And you have it for Q square in the range. Gee, it's a substantial range. It's 10 to the 7 out to 10 to the 11 per centimeter

square. That is over four orders of magnitude in Q square. You are seeing a single diffusive process. Now, why is that interesting? Well, that's interesting because the polymer molecule

is something like 40 to 50 nanometers across. And therefore, you are seeing diffusion not only over distances large compared to the size of the probe, but over distances that are moderate, not really tiny, but moderate compared to the size of the probe.

And you see the same diffusive process all the way along. Now, you aren't yet looking at distances that are very small compared to the size of the probe yet. Nonetheless, there are people who propose, I have a polymer. I have what is sometimes called a correlation hole.

That is because my probe is here. There's less matrix polymer in the same region of space. They do tend to exclude each other a bit. And therefore, the diffusion process over short distances inside the correlation hole and large distances outside the correlation hole

might not be the same. Well, the experiment's been done. The one experiment that has been done sees one diffusion process. You could always wish, though, because after all, this was not precisely what the experiment was focused on, you could always wish to look at a larger set of experiments

and see if this result continued to be true. However, I can tell you what's in the literature, not what's not yet in the literature. We now push ahead to a very clever set of experiments.

And the very clever set of experiments are due to Hsu and, in large part, his collaborators. And the experiments, as described, say the following.

We are going to take DNA molecules. Here's a DNA molecule. And we are going to do clever things to them. In particular, we're going to fluorescently tag them so we could actually see the molecule. Now, if we haven't done anything to the molecule, it's a fairly compact ball.

And there's less visible. But if you tag one in, you can imagine, for example, grabbing the tag with optical tweezers or any of several other methods and drawing the tag out. And you are now going to ask, well,

what happens if you do this? What sort of experiments can you make? Well, one thing you can do is to stretch the molecule and ask how it relaxes. Another thing you can do is to say there's a molecule here. And I can measure the location of its center. And since I can see lots of the molecule and do averaging,

I can measure the location of the center with extremely high precision, better than the nominal optical resolution of the microscope. OK, so what happens if we do that? Well, we can measure x square, the mean square displacement, versus time.

And we see these nice, pretty linear graphs. And we can then infer from x square what the molecular weight is like. What Chu and group claimed to do is that they had validated the reputation picture. And they did this on the basis of three observations.

The first observation was that they denied to stretch out a polymer molecule. And having stretched it out, they released the end they'd stretched. And what they found was that the molecule on a short time scale moved by pulling in its end. And they proposed they were seeing motion

down the polymeric tube. And therefore, they were seeing the presence of the tube itself, reputation. They also claimed to have found that ds was proportional to m to the minus 2. And they proposed to see scaling.

Now, these are very interesting results. However, there are certain technical obstacles to the interpretation. The first obstacle to the interpretation is that if you take a chain and stretch it a great deal

on a short time scale, it's usually proposed that it relaxes by what are called the higher order Rouse modes. That is, the forces that tend to relax the chain point along the chain at every point.

And because the net force points along the chain, pulling back from the direction of the stretch, the forces pull the chain in. You've not, however, yet seen diffusion of the center of mass within the tube.

That takes a different time scale. But you can certainly show that if you stretch the chain, it wants to pull back in again. The second proposal was that they found that the self-diffusion coefficient was inverse as the square of the molecular weight.

This was alleged to be the signature of reputation. Now it is indubitably true that the reputation model, the simple reputation model, does make this prediction. Unfortunately for the argument, there are a significant number of other models that make the same prediction.

In particular, there are some nice models due to Skolnik and collaborators in which you see this particular dependence of the self-diffusion coefficient on polymer molecular weight. But the polymers in question are indubitably not reputating. For example, they are not reputating

because one of their two ends is anchored to a plane, and the polymers rise above the plane like seaweed above a pond bottom. So here is a result. The last claim was that concentration scaling was seen. There are two issues here.

The one that is more a matter of taste, but you should look at the measurements yourself. The one that is more a matter of taste is that the visible evidence for concentration scaling was not very strong. You didn't see large numbers of points. These are not easy experiments. Don't fault people. But the evidence for concentration scaling

could have been better. The other point, though, having done the prior two lectures, is that if one did find concentration scaling here, there is an important issue, namely the result would contradict almost every other measurement of the same quantity.

And therefore, you would have to say, allowing that all the measurements are good, that what you were seeing here was some special chemical effect, and the specific chemical effect was leading to this behavior as opposed to it being a generic behavior of all polymers.

The last issue, which we will come back to perhaps later on, is that M to the minus two, if it's correct, leads to significant difficulties with observational results on viscosity, which instead of finding an expected M to the minus three based on this being M to the minus two,

I'm sorry, it's M to the plus three, find more like M to the plus 3.4. That leads to some issues. Chu, however, and collaborators did find, they don't emphasize it as much, but it's there in their papers if you read carefully,

another very important finding for understanding polymer motion. What they did, as I said, was to measure the mean square displacement as a function of time.

And they get this nice linear curve going down to a few seconds. It looks like it goes straight into zero. And that's very nice. That's what you would expect of diffusive behavior. However, the reputation model says something different. The reputation model introduces a time scale,

tau sub D. And on times, short by comparison with tau sub D, up to tau sub D, the polymer chain travels by moving back and forth within its initial tube.

So it is doing a random walk along a random path. And because it is doing a random walk along a random path, x squared is proportional to about t to the half. There's then a crossover, and these models generally do not predict crossovers,

they're formed too precisely. And at larger times, the chain at a large time can be described as doing a random walk from one point to the next. But because during time tau D, the chain escapes from its tube

and finds itself in a new one, it's an orthodox random walk. And x squared is proportional to t to the first. First. It's very nice, except there's an issue. And the issue is that the experimentalists here made several estimates of what tau D is.

And their numbers are in the range of one and a third to two minutes. You should not be concerned that the numbers don't agree sharply. This is a somewhat diffuse concept,

and there is no particular reason to expect that different experimental methods of the same approach will give exactly the same answer. Nonetheless, they give us this nice plot of x squared versus t, going down to a modest number of seconds,

and certainly for t down to about tau D over seven, times in which the polymer chain is supposed to be confined to the tube, and showing this behavior, what is instead found is this experimental behavior. Now, this is a significant issue if you believe

in the reputation and entanglement tube picture. It is a complete non-issue. It's unsurprising if you believe in the hydrodynamic scaling model. And at some point I'm going to give an introduction to those models. So those are the experiments of Chu

and the interpretation that that group put on the work. And on the other hand, it's possible to put another interpretation on the same very interesting measurements.

Okay, we are now going to advance to yet another experimental technique. It's an experimental technique that was viewed with considerable interest in the 70s. It then died out because there were basically synthetic chemistry issues. It has recently come back.

The technique was originally called fluorescence correlation spectroscopy, or FCS,

because the original detection technique was fluorescence. There are alternatives. In fact, the original experiments that used what is basically the same method, which were done before World War I,

used an ultra-microscope, very bright beam of light coming in from the side, black background, gold salt particles, they're very good at scattering light, even if they're only a few nanometers across, and a graduate student, instead of electronics,

and the time base was not a digital clock, it was a ticking metronome. What is the experiment that can be done with these very disparate experimental instruments? The answer is, we look in at a small volume of solution, and we count the number of particles

within that small volume. We're counting the number of tagged molecules if we're talking about fluorescence, or the number of scattering molecules if we're talking about gold salt particles, but in any event, we count the number of particles,

and we find a count, N of T versus T, and the number fluctuates up and down. The larger the number is, the less dramatic the fluctuations appear to be on something like, if you use the same scale.

And then you ask, how does N of zero, N of T, viewed as a correlation function, decay? That is, you look at the number of particles at an initial time, you ask what happens to the number of particles at a later time,

and what, in essence, you measure experimentally is that if there is a fluctuation, if the number of particles is unusually large, or unusually small at time zero, well, you sit around and wait, and if you wait a piece, the number relaxes back

to the average, or toward the average. The relaxation is, if you were looking at a cosine concentration fluctuation, the relaxation would be an exponential. Here, you're looking at a particular shape, there are microscope optical issues, and the functional form here may be a bit different.

Nonetheless, you can pull out of the functional form, if you are clever, a diffusion coefficient. Because in order for the number to relax, a particle here has to diffuse out, or other particles have to diffuse in.

And you are looking at diffusion over a characteristic distance, which is sort of the size of the region you're looking in at. And now we go back to one of my papers in the mid-1970s. And what I said is that if you were doing the experiment,

and you have tagged a small percentage of the particles in the field of view, the diffusion coefficient you measure is the self-diffusion coefficient. And if you have tagged all of the particles in the field of view, the diffusion coefficient you are measuring

is the mutual diffusion coefficient. The self-diffusion coefficient describes the diffusion of one particle through a uniform background. The mutual diffusion coefficient describes the relaxation of a concentration gradient. It describes the relationship,

not only of self-diffusion, but also issues that arise because if one particle moves, the motion of other particles can be correlated with it. So that is what the theory predicts, and the nice experiments of Scalitar and collaborators,

and this is actually quite recent now, actually sit there and confirm the theoretical model that has been sitting there looking for a test for 30 years. It's a beautiful confirmation of theory, and it makes it very clear why you have two translational diffusion coefficients and why they are not equal in non-dilute solution.

That's it for interesting experiments. Undoubtedly, in the future, there will be more interesting experiments, but they weren't still in time to be included in my book.

Let us advance then from section 8.4, these interesting other experiments, and we shall advance to section 8.5. The point of section 8.5 is to take the measurements

that we've discussed one set at a time in earlier parts of the chapter, bring them all together, and see what systematic findings we make. I have been giving hints as to what the systematic findings were as we push ahead, but now what I'm going to say is we have all of these systematic findings.

Let's assemble them coherently and see what we see. Okay, number one, self-diffusion coefficient, and we have at least two theoretical models.

We have at least two theoretical models for how diffusion would depend on experimental variables. The first thing we can say is that almost without exception, we do not find scaling laws.

There is almost no experimental evidence at all for systems in which the concentration, the self-diffusion coefficient, falls off as a power law in concentration. There aren't, it's not quite zero, but it's very close.

On the other hand, the stretched exponential with, if we write this for self-diffusion where the probe and matrix have the same molecular weight, we have one more parameter here. Well, having said we have one more parameter,

life is a little easier, but this form works very well indeed. The next thing we could do though is to ask, well, suppose we actually say e to the minus alpha c to the nu, the simple stretched exponential,

one stretched exponential per polymer molecular weight. Does alpha behave in a reasonable manner? Alternatively, does it just behave in a random manner? After all, if you just find some empirical fitting function, there's no particular reason for your parameters to be related at all closely

to properties of the solution. On the other hand, if the parameters follow fairly tightly the behavior of the solution, you might reasonably say that, gee, we have something interesting. And the answer on that is found in the first few figures of section eight.

And we tried two experiments. One is to relate alpha to the polymer molecular weight. And the other is to relate alpha to the polymer radius of gyration. Now, there is the practical difficulty that you have a bunch of different polymers.

Most authors did not measure the radius of the polymer directly. And the net result is that if you try comparing these two, you discover that when you translate polymer molecular weight to polymer radius, you introduce large amounts of noise and life is not too interesting.

But in figure 8.34, we have a plot of alpha versus M. And you notice for a bunch of different polymers, it's fairly noisy. But you can draw a line like this.

You can also reach in and say, gee, I'm going to do something a little more emphatic. I'm going to be very systematic about including very high and low molecular weights when I can do that. I'm going to insist that I look at measurements in which if the people plot D versus C,

not only do they give me measurements out here, but they give me measurements in here so the entire shape of the curve is well defined. For scaling laws, all you need is the measurements out here in semi-dilute solution.

If you believe in scaling laws, the results in dilute solution are independent of these and there's not much point of measuring them. And therefore, there are bunches of sensible experimentalists who only give us results out here. However, there were people who were more systematic and they did give us nice measurements of alpha versus M.

And you can find the nice, good measurements in 8-36. And what you see in 8-36 is, yeah, the measurements spread out a bit, alpha spreads out a bit at low molecular weight,

but converges towards a straight line at high molecular weight. And in fact, I give you a straight line. And the straight line does a nice job of conforming with the upper points all the way out. Okay, so we can put a line through points. But I didn't put that line through the points.

That line, this line, is a prediction of the hydrodynamic scaling model. The important feature of this prediction is that it has no free parameters. Parameters, none at all.

The closest you get to a free parameter is how you relate the molecular weight to the polymer radius, but that's a well understood result. Furthermore, that line is here.

It's not up here. It's not down there. It is where it is drawn on the graph. And therefore, the hydrodynamic scaling model actually predicts alpha quantitatively. Better yet, if we go back to this form,

and we say our interesting physical quantity is the logarithm of the self-diffusion coefficient. The form is log of ds, which is perfectly well what you could say the measurable is,

goes as alpha c to the nu, well there's a minus sign, plus a constant. In that case, nu is a scaling exponent, and a alpha, which we calculated, is a scaling pre-factor.

There are lots of theoretical models which let you calculate scaling exponents. There's a big industry calculating them, and those calculations are very good. If you hunt about in the literature though, looking for people who calculate the scaling pre-factors,

you will discover there are rather fewer of them. But that's what we have here, a calculation of alpha. Okay, that's alpha. However, look up there, there's alpha. There's another parameter. The other parameter is nu.

And the question is, how does nu depend on variables? Well, if the stretched exponential form is accurate, nu should be the independent of concentration. You just see a value of nu, and it gives you the shape of the D versus C curve.

But, I can also imagine plotting nu against polymer molecular weight, and I have done that for most of the measurements. You have the graph and the text, and the question is what you see

when you plot nu versus M. Well, what happens is, at large molecular weight, nu is pretty close to a half. And that's true for molecular weights above a quarter or a half a million.

Half a million anyhow. At smaller molecular weight, this spreads out. You could propose, and there are actually a lot of measurements that are consistent, with nu being about one at low molecular weight,

but this is clearly not true in every system. And then a crossover function, like that, from nu of one to nu and a half. But if you look hard, you realize there are a lot of measurements that do not fall on this curve, and sorting out why has not yet been done. You also notice, if you look hard,

there are points up here with nu greater than one. That is, there are cases where the stretched exponential has been stretched in the opposite direction. It's really been compressed. And so, if you have nu greater than one, and you plot log D versus C,

well, if you had a simple exponential, nu equals one, you'd see a straight line. With nu greater than one, you see a rollover. Nu greater than one has been with this sort of study from very early times.

It's a fairly clearly seen effect. So there is the behavior, nu versus polymer molecular weight. Okay, let's think for a second. Why is nu not one?

As we will eventually see, nu is not one because chains expand and contract. In the cases where we could measure chain expansion and contraction directly, that's done with dielectric relaxation. It's an earlier chapter that we've already discussed. What was found was that the extent of chain contraction,

as measured by the change in the length of the end-to-end vector as you change concentration, the contraction of the chain with increasing concentration as measured directly, that effect accounts quantitatively

for the deviation of the diffusion coefficient from a simple exponential in concentration. That's a quantitative explanation. It's not an approximate explanation. Okay, that's nu.

Can we measure anything else? Oh yes. Remember we were talking about tracer diffusion? Well, for tracer diffusion, the probe and the matrix, I should say,

have independent molecular weights. And if these two molecular weights are not too different, say if both of these numbers are less than about three, you have a fairly consistent behavior.

You have the stretched exponential. You have something of the form d zero, probe to the minus a. By the way, that's a typo in the book. In the book, at least in the draft manuscript, that shows up as an m. And you have an exponential e to the minus a,

c to the nu, m to the gamma, t to the delta. And then you encounter a little puzzle. And the little puzzle you encounter is that in the cases where these quantities

have been measured, gamma is about 0.25 or 0.3. T to the delta, delta is about 0.3. And therefore you have gamma plus delta

is about 0.6 or maybe a bit less. Well that's very nice, except if you look in systems where you were looking at self-diffusion, you found not 0.6, but a significantly larger number close to one.

And it is not quite clear, at least I didn't find a way to analyze the data to explain why it is that if you're looking at tracer diffusion, you get a weaker dependence on the molecular weights than you do if you look at self-diffusion. I suspect it's one of these, you are looking

at certain ranges of molecular weights, you're looking at this, you're looking at that. And the two numbers are perfectly consistent and don't refer quite the same thing. But I didn't solve that problem in the book.

One of the more powerful ways to study an unknown issue in polymer dynamics is to bring to bear on the same system simultaneously several different experimental techniques.

We saw that effectively with dielectric relaxation because dielectric relaxation can measure at the same time in the same experiment the mean square end to end vector of the chain, at least if it's the right one, the characteristic relaxation time,

and more sophisticated than a characteristic relaxation time, a full relaxation function, including secondary relaxations encountered at high frequency. You can do all of this at once, but those of us who are less fortunate have to do several different types of experiments.

People have done this. In particular, we know work of Martin and work of Numesawa.

And what both of these experimental studies did was to measure at more or less one swoop the polymer self-diffusion coefficient and the polymer viscosity, the solution viscosity. And so they measure two rather different things about the same system.

For Martin, the system was, yes, the usual polystyrene, polyvinyl methyl ether, but the important issue was that the matrix was about 110 kilodaltons. The probe molecular weights went from 50 kilodaltons

and then there was 100 and there was a 420. And at the top end, there was a 900. And so you put in some very different size probes into the same matrix and you ask what you see.

Okay, how do we compare D and eta? You could put them on the same graph. You'd have to use different units. You would want to plot D, which decreases with polymer increasing matrix concentration, and you'd plot eta inverse the fluidity of the solution,

the ability to pour, which also falls as you make the matrix more concentrated. That would be a very legitimate way to go. What was instead done equally legitimate was to say, okay, I have the measurements.

I will plot the product ds eta. Well-defined product. You multiply the diffusion coefficient of the probe by the viscosity of the solution, which is basically the viscosity of the matrix. And you do this as a function of matrix concentration.

If you do it as a function of matrix concentration, what you find is that for a very large probe, ds eta, this is a very large polymer probe, ds eta is approximately constant. But as you look at smaller and smaller and smaller probes,

and remember, the molecular weights of these probes are 50, 100, 420, and 900 kilodalton. And the matrix molecular weight is slightly larger than that. It's about 110.

Well, if you look at that, what you find is that ds eta increases with increasing matrix concentration and it continues to do so until the probe is much larger than the matrix. If the probe is much larger than the matrix,

p much greater than m, you find that d eta is approximately a constant. Now there's another experiment you can do at the same time which I shall simply point out, which is to say I have the self-diffusion coefficient

of the probe and I will plot the diffusion coefficient of the probe versus the probe concentration. Why would you do that? Why, if it's a probe, are you caring about the probe concentration? Well, in general, the diffusion coefficient of the probe

could depend on how much probe was present. And therefore, if you want to say, I'm looking at true self-diffusion, what you want to do is to extrapolate in from the probe concentration you're using to zero probe concentration.

For some experimental methods and probes, this is somewhat pointless because you're already at such extreme probe dilution that the diffusion coefficient of the probe doesn't care what the probe concentration is, at least not in your experiments. However, you can do this experiment and if you do the experiment,

what you find is you get a series of slopes. That is, the diffusion coefficient of the probe is sensitive to probe concentration and the slope here is initially weakly positive,

but as you drop, excuse me, as you increase the matrix concentration, as you increase the matrix concentration, the dependence of DP on probe on probe concentration becomes more and more negative. So at first, with very dilute matrix,

you add probe polymers and their diffusion coefficient increases. As you increase the matrix concentration, though, and go to elevated matrix concentration, as you increase the concentration of the probe, the diffusion coefficient of the probe falls.

Okay, and those are results of a Martin. We now advance to a very ingenious paper

and an ingenious paper due to Rumasawa and collaborators. What they did was to look at a ternary system and they have a series of probes

and they have a series of matrix polymers and they can combine them and they look at a product, namely the product of the diffusion coefficient of the probe the radius of the probe, those large probes diffuse more slowly, but this product might be independent

of probe concentration and the viscosity of the solution. There wasn't, and you look at this as you vary the matrix concentration and the matrix molecular weight. It's a well-defined set of experiments. There's some very nice chemistry getting it all to work.

Someone has to make all those polymers for you after all and what was done was to propose that you were going to see two sorts of regimes. If you had a small matrix and a large probe,

the proposal was that you would see Stokes-Einstein behavior and if you see Stokes-Einstein behavior, this quantity would be fairly constant

because in Stokes-Einstein diffusion, the diffusion coefficient goes inversely as the viscosity. However, as you head this way, as you head to looking at larger probes,

maybe I should have said that differently, larger matrices and still large probes, if you make the matrix polymer larger

or you make the concentration larger, what is going to happen? Well, if you believe in the reputation model, as you increase the concentration and you use large matrix polymers, the matrix chains can entangle and once they have entangled,

the expectation is that Stokes-Einstein behavior goes, you go over to replication behavior, this product, DSR eta over its value

when you're not yet entangled, up here, this quantity is some number like one and down here, the reputation expectation is that DSR eta can become large because the polymer is more effective

at increasing the viscosity of the solution, the matrix increases the viscosity of the solution but the probes are able to thread their way through the matrix and are not slowed down as much so this product becomes large. Experimentally, that's what you find, you find a regime where DSR eta is about constant

and when you move to large matrices and elevated concentrations, when you increase the concentration of the matrix polymer and the concentration of the matrix polymer, you discover that this product increases quite markedly.

Well, that's very good. The question though is, is this a proof of the reputation model? And the answer is that it is not. The result is consistent with the reputation ideas but it does not prove it.

Why not? Understand why not. We would have to advance to a chapter we have not yet reached, namely the chapter on probe diffusion and in probe diffusion we look at spheres, large spheres or small spheres,

diffusing through a polymer matrix as we change the concentration and we, as you would infer from the name probe diffusion, measure the diffusion coefficient of these spherical probes and while there's several things you could plot, one of them is indubitably D eta.

Now, a large sphere is very different from polymer coil in one sense. If you believe in reputation and the probe is reasonably large, larger than the holes between the entanglement points as drawn here, big probe, narrow tube,

then you can't predict that the sphere reputates, it can't fit, it can't move like that. Instead, the sphere has to move through the solution of which it is exposed to a fair amount

and in that case, the prediction or expectation, I'm not sure it's quite counts as a prediction, was that Dp eta would be about constant, that is, large spheres would show Stokes, Einstein behavior.

Well, at low matrix concentrations and with small matrix polymers, that's not a bad prediction. If you have a big sphere and a very low molecular weight polymer, you tend to see behavior like this. However, if you increase the concentration of the matrix

and you make the matrix chains quite large, so they are moving towards being entangled, D eta experimentally probe eta does this. Actually, that's sort of what Numosawa saw, namely for probes that were small

and matrix polymers that were small. There was Stokes, Einstein behavior, but as you made the matrix polymers large and the matrix concentration high, so that you got up like that, gee, what do you find?

You find for chains, the product D eta increases quite markedly, but that's exactly what you find for spheres. That is, the observation that D eta increases for large probes and large concentrated

high molecular weight matrices is consistent with the reputation picture, but it does not in any sense prove the reputation picture. It does not prove the reputation picture because you see exactly the same phenomenology for probes that cannot possibly be rectated.

And that actually brings us, approximately speaking, to the end of the first half of this course.

In the first half of the course, we've discussed things that correlate with driven motion of polymers and we have discussed motion of small molecules, sections of polymer, and single polymer chains.

In the rest of the course, we are going to advance, we're going to talk about probe diffusion, rigid objects moving through polymer solutions. We're going to talk about collective behaviors, the mutual diffusion coefficient, the viscosity, and viscoelasticity.

I'm also going to drop into there one completely non-conventional topic, namely we're going to talk about colloid dynamics. Why would we talk about colloid dynamics? Well, colloidal particles are spheres. They don't have all of the inner structure and motion that polymer coils do.

However, their interactions with other spheres are hydrodynamic and excluded volume. The forces in colloid systems are exactly the same as the forces you would encounter in polymer coil systems.

The one difference is that spheres indubitably cannot reptate. They can't deform to get through small holes in the solution. They just sit there and therefore, anything that is common to sphere dynamics and coil dynamics can't be attributed

to the ability of the coils to change shape. We'll see what those common features are when we get to that chapter. Let us, however, spend a few minutes recalling what we've done so far in the course. We began with chapter two,

and chapter two talks about sedimentation. That is, we take a centrifuge, we spin it up, we put a polymer solution in the centrifuge before we turn it on, and what we observe is that the polymer sediments up or down relative to the solvent. Up or down depending on its density,

larger or smaller than the solvent density. We can also take a polymer solution in the centrifuge and add to it probe particles, colloids, other polymer chains, and we can look at the ternary mixture and ask how a probe component is moving

with respect to the polymeric matrix. We then talked about, well, we did talk about viscosity a bit for the first time, and what was observed was that the ability

of a polymer matrix to increase the viscosity of the solution was significantly larger than the ability of the same matrix to retard the sedimentation of the probe macromolecules. Okay, we did sedimentation, and then we advanced,

and we talked about, gee, electrophoresis. We take charged probes, we apply an electrical field, and we measure the diffusion of the probes, actually the migration of the probes as well,

mostly the migration, though there is diffusion going on, through the matrix, usually of neutral polymers. And what we found was, what we'd also seen with sedimentation, namely you plot sedimentation, or you plot electrophoretic mobility

versus concentration of the matrix, and there is a prominent reduction in the sedimentation rate, or the mobility, as you increase the concentration of the polymers. That's what you do find experimentally. There were several models presented

for how these quantities might depend on matrix concentration, or matrix molecular weight, but what we saw for the first time was a stretched exponential in polymer concentration. For the dependence of mobility on probe molecular weight,

there was a more complicated behavior, namely there was a stretched exponential in probe molecular weight until we hit a crossover. And above the crossover, which in this system is pretty sharp,

there is a decrease, a very weak decrease, of mobility with increasing probe size. The decrease turns out to be sufficiently weak that it's at least tricky to use the dependence of mobility on probe size

to do separations out here. There is some evidence, it's not as well sorted out as one might wish, that this change is actually a linear to non-linear transport change, namely the dynamics change if the force of the probe polymer matrix is large enough.

I'm not sure I view that as settled. We did take an intermission to discuss light scattering spectroscopy and its ability to measure the mutual diffusion coefficient. So we talked about quasi-elastic light scattering,

which is sensitive to fluctuations in a single spatial Fourier component of the polymer concentration. For colloids, it's possible to calculate how the diffusion coefficient depends on the size of the diffuse.

spheres and their concentration and the concentration calculation in that style was displayed. Okay so that's light scattering spectroscopy. I did at the end for once toss in a few comments on experimental constraints. The

comments on experimental constraints were, well it'd be very nice to take our light scattering spectrum and fit it to a huge number of exponentials and pull out the diffusion coefficients separately of everything present. You can't do that. You can pull out a modest number of parameters from a light

scattering spectrum and when I say modest number I mean six to eight, not the mobilities of 2,000 species. It just doesn't work that way. Okay that was chapter four and now we push ahead. And what did we push ahead to do? First

we discussed small molecule motions and we began with ions and similar such

things doing either diffusion or electrical mobility through viscous and non-viscous solvents. And for not very viscous solvents we found that the diffusion coefficient and the lamp and the electrical mobility, this goes back a very long way in time for some of us, were inversely inverse to

the viscosity but then there was a crossover at a viscosity of a few cent poise, not the same in every system, and above there the diffusion coefficient and the conductivity went as eta to the minus O two-thirds or so. And

that goes back to results of Heber green more than a century ago and more recent measurements since. On the other hand if you took a polystyrene latex spheres which are say 20 nanometers in radius or 10 nanometers

big objects and you look in small molecule liquids you find the diffusion coefficient of these probes goes as temperature over solvent viscosity and it does that even if the solvent viscosity is oh a thousand or a few

thousand times the viscosity of water. So large probes and small probes do not behave the same way. Well what happens if you raise the viscosity not

by using a viscous small molecule solvent but by adding a polymer. If you add a polymer and measure how rapidly the solvent moves, this is a diffusion of

the solvent versus concentration of the polymer, you discover that measurements tend to look about the same. There is a lower concentration region which is more or less exponential in polymer concentration and then someplace near oh 350 to 500 gram per liter of polymer there is a crossover and there

is a stretched exponential in polymer concentration. So the diffusion coefficient is falling with increasing matrix concentration you cross through

something and then the ability of the small molecules to move falls off much more rapidly as you run up how much polymer concentration. An interpretation of this effect which may or may not be true but it seems reasonable is that

if you have some characteristic distance psi between neighboring polymer beads not on the same chain or at least not nearby on the same chain and you have a solvent molecule which has some characteristic size R roughly speaking

when R becomes larger than psi so the solvent can't consistently find interstices through which it can pass when R becomes larger than psi which happens in about this concentration R doesn't change with concentration psi certainly does. When you cross over the mobility of the solvent changes because

it has problems finding holes. Finally you can do local measurements of solvent mobility and solvent ability to rotate and what is demonstrated

experiments of Cron, Lodge, and other people we have a polymer coil here and it has neighboring solvent molecules which are actually fairly tightly packed and roughly speaking what you discover is that the

polymer has a dramatic effect on the ability of the solvent molecules to move out to something like two solvent diameters that is an imprecise number but it's some number like that. The effect of the polymer by the way

doesn't have to be to interfere with the ability of the solvent molecules to move. If you put the right polymer, a very flexible polymer, into an arachlor solvent, arachlor is a chlorinated biphenyl, it's extremely viscous, what

you discover is that the polymer acts as a plasticizing agent and the solvent molecules are able to move or at least rotate more and more rapidly as you add more and more polymer. Okay that's it for the chapter on small

molecule motion. We then pushed ahead and talked a bit about segmental motion. We had a series of different experiments that measure motion of segments, motion

of single reorientation of single bonds, and we could do these as we change the matrix concentration and other variables. We can also put in probes that tell us the alignment of the backbone and if this was done by attaching the probe at one end, what was demonstrated at least for the

polymer in question is here's the probe, here's the polymer, the behavior of the probe changes as you make the polymer longer and longer and that effect persists until you have something like 20 to 40 monomers here. Once you have more

than about 20 to 40 monomers, once you have a long polymer chain, making the chain longer and longer has very little effect on the mobility of the chain ends. That sort of makes sense. What it says is there is a distance over which chain orientation and motion along the chain, we have different

pieces, there's a distance over which the different pieces talk to each other but if you have two pieces that are very far apart along the chain, they're really not aware of what each other is doing. Okay we finally pushed ahead

and we talked about dielectric relaxation which I discussed earlier in this class. A dielectric relaxation is wonderful because it gives us the mean

square end-to-end distance, it gives us a characteristic time tau, it gives us a relaxation function which in some cases has multiple relaxations. We get all of these very different experimental numbers out of one system. I was able to

demonstrate how tau and r-square are correlated, namely tau goes as e to the a concentration refers r-square to the 3 halves. r-square changes with

changing concentration and that drives the concentration dependence of tau away from a single exponential, however the deviation depends on chain radius. This is actually a quantitative result. The concentration dependence of r-square

explains quantitatively why tau is not a single simple exponential and polymer concentration. We were also able to look at relaxation functions and how their parameters depend on concentration. Finally, and we close that

discussion today, we discussed polymer self-diffusion which I summarized just now. And in polymer self-diffusion we find how the self-diffusion depends on the concentration of the polymer, the size of the matrix polymer, and

separately the size of the pro-polymer. So we've now done a full set of discussions of what are more or less single molecule results. In the rest of the course we will push ahead and we will discuss collective effects and other related quantities. However, for today, that's it.