I'm George Filleys, and this is lecture 12 of Phenomenology of Polymer Solution Dynamics. Today I'm going to continue my discussion of the experimental studies of polymer self diffusion. The basic notion here is, we have a polymer solution,

and we are somehow able to track the diffusion, the random walk of a single polymer, a star polymer or a linear polymer. Now, there are experiments where you actually are tracking

one single polymer at a time. More typically, you have a dilute solution of labeled polymers, and you are tracking the dilute labeled polymers as they move through the matrix polymer, an unseen polymer which may be diluted or concentrated.

There are also a few techniques, pulsed field gradient NMR, for example, in which you are in fact able to track the displacement of each polymer with respect to its own location and still be measuring single probe, single molecule self diffusion.

So, let us start on the discussion from where we left off last time, and we'll begin by looking at figure 8-12. Figure 8-12 shows experimental studies of Xu Xin and

collaborators using pulsed field gradient NMR. They're looking at polyisoprene dissolved in carbon tetrachloride. Now, the point of carbon tetrachloride is that it's inert relative to the NMR measurements, and therefore you're actually seeing the polymer.

What they are able to do here is to look at 18-arm star polymers, meaning there is a central core, there are arms reaching out, and there aren't three or four of them, there are 18 of them.

And they look as a function of polymer concentration for C up to about 70 grams per liter. They look at star polymers of molecular weights. Now these are the total molecular weights, 340, 800, and

in the end, 6,300, I think that's the right number, kilodaltons. And they do a comparison with a linear chain, which is a 300 kilodaltons round number linear chain.

So what do they see when they measure diffusion versus concentration? Well, the first answer on what they see is that as you increase the concentration, the diffusion coefficient falls.

The dependence of the diffusion coefficient on polymer concentration is in fact a stretched exponential in concentration. They also, however, manage to find, it's the center of the three lines in the graph, a linear chain whose diffusion coefficient at low

concentration is the same as that of the 800 kilodaltons star. Now you should realize, these are the molecular weights of the complete stars. If you have an 800 kilodaltons material, and

it has 18 arms coming out, the molecular weight of one of those arms is not 800 kilodaltons. It's more like round number 45 or a bit less kilodaltons. However, they find a linear chain, they find a star.

The diffusion coefficients at low concentration are about the same. And as they take up the concentration, the two curves track each other with a reasonable accuracy. Why is this result significant? Why should we care that we can measure a linear chain and

a star chain that start out the same size? And then we find that as we run up the concentration, the diffusion coefficients track. And the answer is that if you believe that star chains and linear chains diffuse through completely different processes, there'd be no reason to suppose the two curves would track each other.

Clearly, you can always find some linear chain of some molecular weight whose diffusion coefficient matches the diffusion coefficient of the star chain. But there's no reason for the two curves to lie on top of each other if you happen to believe they use completely different diffusive mechanisms.

Well, rather clearly, that's not what we are seeing. We see one curve. Let's jump ahead. The jump ahead is to figure 8-14.

8-14 is a composite. There are actually four pieces, A, B, C, D. And in addition to there being composite pieces, A, B, C, D, I've used tricks with displacing one vertical axis relative to the other.

So that, in fact, there are more than four chemical systems represented there. So there is a huge amount of measurement, a great deal of work by a bunch of different people, all on the same graph. And in fact, in particular, you see measurements of polystyrene in

carbon tetrachloride and perduterobenzene. You see measurements on polydimethylsiloxane. You see measurements on polyethylene oxide in water. And in every case, plot diffusion coefficient versus concentration.

These are, again, I will draw the sketch as a log-log plot. You see something that looks like this. Each of these studies looked at a whole series of concentrations,

and they looked at a series of polymers of different molecular weights. Now, the first observation is that if you put in a polymer, if you put in and look at one polymer, one molecular weight, a series of concentrations, the measurements fall on a stretched exponential in concentration.

You could certainly wish that the measurements had been conducted down to very low concentration in all cases, and had been conducted out as far as possible in high concentrations. But in every case, there is a stretched exponential fit. However, there's something more.

For a series of these measurements, the work was done using polymers of several different molecular weights. And so for each polymer molecular weight, you see, there's a stretched exponential. But in fact, for all of those different molecular weights, there is only one stretched exponential.

The stretched exponential is d proportional to e to the minus alpha, sum alpha, concentration to a power, polymer molecular weight to a power. And so, to describe the measurements at three, four,

whatever molecular weights, you have a scaling pre-factor, the concentration scaling exponent. The molecular weight scaling exponent. You have a very small number of parameters used to reduce a lot of different measurements.

Furthermore, this form, as is seen from these curves, is successful at simultaneously reducing both the concentration and the molecular weight dependence to what is in fact one single curve. You might say curve, looks like a bunch of curves.

Well, d here is really a function of two variables, concentration and molecular weight. And therefore, you could imagine doing a three dimensional graph, or in some science fictional future in 15 years, having the holo-blackboard

display, concentration, molecular weight. Those are two perpendicular axes seen in perspective. Diffusion coefficient this way, third axis seen in perspective.

You notice these are all perpendicular to each other. This is a perspective drawing. And this d is a function of concentration and molecular weight. Gives, here are the two, the concentration axis, molecular weight axis. Diffusion that way. This function gives me a smooth surface.

When we do measurements for a set of fixed polymer molecular weights, and a whole bunch of concentrations. Remember, it's much easier to do measurements in a series of concentrations than in a series of polymer molecular weights. That's just the way chemistry works out.

When we do this, what we are doing is looking at this smooth surface and taking cuts at fixed constant polymer molecular weight. Each cut gives me a line which is the shape of the surface at that value of m.

So if the molecular weight axis is this way, I have a cut of the surface here. I have a cut of the surface here. I have a cut of the surface here. That's three lines. And I put the lines down on the graph. But in fact, I am doing something much more complicated than just saying,

I can represent the concentration dependence as a curve. Or I can represent the polymer molecular weight dependence as a curve. What is in fact being done is to say, I can represent simultaneously the concentration and molecular weight dependencies as a smooth sheet.

And this function does it. Okay, let us push ahead. And we will now push ahead, and we will reach section 8.3.

And section 8.3 of the book, by book Phenomenology of Polymer Solution Dynamics, discusses tracer diffusion.

The tracer diffusion measurements are substantially the same, in some respects, as the polymer self diffusion measurements. But in a core respect, they're very different. What is the same notion? The same respect is that we are measuring displacement,

or perhaps mean square displacement, versus time. Or we are measuring e to the i, q. That's the scattering vector in light scattering, dot displacement in time.

And we are doing this for one chain at a time. So we may actually do an average by looking at a bunch of chains simultaneously. But the physical parameter we're pulling out, describes only how a single chain moves.

Gives you no information on how one chain moves relative to its neighbors. It just tells you how one chain moves. That's the sense in which tracer diffusion is the same as self diffusion. There's also, however, an aspect by which tracer and self diffusion are very different.

And the aspect arises from a question. And tracer diffusion is how you get some hook on trying to answer the question. The question is as follows. If we look at self diffusion as a function of concentration and

molecular weight of the diffusing species, as we increase the polymer molecular weight, the solutions become more and more effective at slowing down the particle that's trying to move. Well, that's very interesting. Why does the solution become more effective at slowing down the polymer

that's trying to move? And one could imagine the answer is, I'm the diffusing polymer. I'm getting bigger and bigger and bigger. And therefore, in some sense, I get attached to more and more things. And the resistance increases because the moving object is getting bigger.

That's incidentally beyond the statement that the diffusion coefficient in concentrated solution is dependent proportional to the diffusion coefficient at high dilution.

If you look at ds over d0, the form we just talked about, e to the minus alpha, c to the mu, m to the gamma. There is a molecular weight dependence of how effective the solution is on a fractional basis of slowing down diffusion.

So one possible explanation for this term is that it arises because the diffusing object is getting bigger. However, it could also be the case that the core issue is not that the diffusing object is getting bigger, but that here is an object that is trying to move.

And as you surround it with chains that are longer and longer and longer, the chains out here, because they're longer, are more effective at slowing down the same moving object. A little, let's be reasonable about this, suggests that you shouldn't be

surprised if both of these effects are acting at the same time. Well, that's very nice. So we have two different effects acting at the same time. They both contribute to the molecular weight dependence of the self-diffusion coefficient. How can we sort them out?

The answer on how we sort them out is tracer diffusion. The core recognition behind tracer diffusion, which is an experiment we have seen before when we talked about dielectric relaxation, we have seen before when we talked about centrifugation,

we have seen before when we talked about electrophoresis, is that you have the pro particle, you have the polymer matrix. And if you have some way of identifying your pro particles and seeing them and making the matrix polymers invisible,

you can separate out the dependence of D on the molecular weight of the probe, and the dependence of D on the molecular weight of the matrix. And you can study these separately, or you can study these in tandem.

There are bunches of other things you could also study. You could study the topology of the probe in the matrix. You could study the solvent quality. The core issue in tracer diffusion is we can sort out the probe and matrix polymer molecular weight effects.

Word of warning to repeat what I said last time. There are a number of authors who cleverly use M to stand for the molecular weight of the probe, and P to stand for the molecular weight of the matrix. All I can say is there are reasons why Leibniz notation

overran Newtonian notation for derivatives. And the reason is that Leibniz thought about his notation to make it useful. And Newton was so bright that he didn't have to think about it. Leibniz was actually the smarter man of the two in this respect. So there is the calculus issue, and there is the notation issue.

And here is the experiment. And the question is, well, that's very nice. So how are we actually going to do this? Well, the main answer is you have to make the matrix invisible. One choice is to use a light scattering spectroscopy,

which gives us a measurement of the diffusion coefficient. And we use the trick that if we have a polymer, for example, and we'll start off with figure 8-15.

And we have a polymer, polymethyl methacrylate, and we are in a solvent, to be precise, benzene, which is pretty nasty to work with, and you should treat it with great respect. If you're looking at that mixture, PMMA and benzene are relatively close to being iso-refractive.

And therefore, if you do light scattering from them, you don't see very much. But if you add to this polystyrene, which is also soluble in benzene, you can now watch the diffusion of very dilute polystyrene matrix,

excuse me, polystyrene pro-polymers through a PMMA matrix. Well, this works fine except for one minor technical detail. And the one minor technical detail is that these two polymers are incompatible with each other.

That is, if you attempt to mix them, there's an unfavorable thermodynamic interaction. And the net result is that if you go in and you have a polystyrene chain,

and you add more and more PMMA to the solution, you expect the polystyrene chain will tend to get smaller and smaller because it will, on a molecular scale, it will be being repelled by the PMMA, and therefore you have a pro-particle that is doing something inconvenient, namely it's shrinking.

If you actually look at the measurements though, in this case, so we plot the self-diffusion coefficient of the polystyrene, the species we can see, against the concentration of the matrix polymer,

the polymer whose concentration we're varying, because the pro-polymer is always left dilute. What we discover is, there's a drop-off in D if the pro-polymer is bigger than about 25 kilodaltons.

But if we're down at a pro-polymer of about 2 kilodaltons, which is a very low molecular weight, if we're down at a very low molecular weight, what happens is that the fusion coefficient just sits there. The polystyrene chains don't shrink up on themselves,

because it's a very short chain, and also they apparently are retarded very effectively in their motions by the presence of the matrix. As a practical matter, there are a fairly limited number of sets of measurements,

where people have studied the motion of fairly low molecular weight polymeric probes. You can find studies, it was in the earlier chapter, on solvent diffusion or small molecule probe diffusion, fluorescein for example, or any of a variety of dyes.

But if you look for the behavior of small probes, small polymers, you don't find quite as much information as you do on larger polymers. That's just the way it is. Okay, we now chug ahead, and we advance to figure 8-16.

And 8-16 is a series of papers associated with names including Leger and

Herve, and the graph is noteworthy in particular that this is more or less one of the first systematic studies of polymer self-diffusion. And the authors also did tracer diffusion.

And what you observe, if you don't look hard at the graph, yes, as you increase the concentration of the polymer, the self-diffusion coefficient slows down. As you increase the concentration of the matrix polymer,

the diffusion coefficient of the probe slows down. You also notice a slight anomaly. If you look hard at the self-diffusion measurements on the graph, your data actually looks like that. That is, there is this hump, which is, to my eye,

25%, but I've heard it described as a bit larger, it depends where you measure it. Where the diffusion coefficient of the probe first increases with increasing matrix concentration and then falls again.

This is for D-self. The tracer measurements don't show this. Now you could say, this is an example of re-entrance. Re-entrance is a phenomenon we will be encountering once and again as we go through the course. And in this case, what you're saying is that something odd happens

at low concentrations of polymer. However, it could also be noted that there are a whole series of people who studied at least more or less the same chemical system, and no one else sees this effect. The answer, the magic phrase we use, is specific chemical effect.

The point of specific chemical effect is that you're actually dealing with the product of a complicated synthetic process. And even though you've been very careful and you've done all sorts of

good things, every so often something odd happens with a particular sample of material, and you may not even realize it's happening. And it may not be worth chasing down exactly what happened. However, the other point of specific chemical effect is that you are seeing

something that is not the generic behavior of polymer systems. And therefore, you have to sort out which measurements you look at, the bulk of them, and which measurements you decide something exotic happened, which is not the fault of the experimenters, of course. It should be noted, however, that one of the consequences of this specific

chemical effect is that you appear to see scaling law behavior. And these are more or less the only measurements. They're not quite the only measurements, but they're pretty close, in which you actually can lay down a ruler on the log log plot and say,

look, clear scaling law behavior, not stretched exponentials at all. Now we advance to a longish series of experiments, and

the longish series of experiments are due to Tim Lodge and his wonderful and huge research group. The other names on these papers include Wheeler and Marklund, and this is not, I'm not sure I'm giving you an exhaustive list, but noted authors.

And what they did is to find a polymer system, ternary system, with the following properties. Property one, you have a polymer, polyvinyl methyl ether, which for

some reason is usually referred to as PBME, and it goes into a solvent, orthofluoro toluene, or OFT.

And the important feature of OFT and PBME is that they are quite thoroughly isorefractive. Now if you want better isorefractive, there is an approach due to Ben Chu.

He looked at a mixed solvent, that is you have two small molecule solvents instead of one. You adjust the composition ratio, and then you tune the temperature because the index of refraction changes with temperature, and you can get really incredibly good matches.

Of course, there's a bit of work involved. Lodge was lucky enough to find PBME-OFT, and then he had a useful probe, the S, polystyrene.

Polystyrene as a probe for this system has two enormous advantages. The first advantage is that you can make polystyrene synthetically that are very pure in their molecular weight distribution, so

you know exactly what the molecular weight of the polymer is. The second statement about polystyrene and PBME is that they're compatible. So that OFT is a usable solvent with both of these materials. And the materials, the two polymers mix together rather than having some

very negative thermodynamic interaction which causes them to fold up. And that's very useful because it means you know that the probe behavior is reasonably well behaved. Okay, so having said that, what did he do? Well, for the PBME, there were three matrix polymers,

140, 630, and a bit over 1,300. What was the magic number? If I recall correctly, it was 1,340 kilodaltons. So you have a fairly low molecular weight polymer,

a medium molecular weight polymer, a quite respectively large molecular weight polymer, and since these are readily soluble in PBME, it was possible, though an experimental chore, to take the concentrations of these polymers out to some large value.

In particular, if you go through the full series of papers, you find that they took the concentration out to 300 gram per liter of polymer.

They also did viscosity measurements. We'll talk about viscosity more later in the course. But the core answer was that the intrinsic viscosity, or rather its inverse, was for these three materials,

11, 6, or 3 gram per liter. Eta intrinsic is in liters per gram, of course. And these are overlap concentrations, C star. You can have different numbers up here and different authors do that. And therefore, they estimated the entanglement concentrations,

the concentrations at which the polymer changes intertwined with each other. So you have what the model would call a semi-dilute polymer solution. They estimated C as 50 for the small polymer. 11 for the 12 I mean, for the intermediate polymer.

And about 6 for the large polymer. Meaning they got up to C over CE for the largest polymer up to something around 50. Now, the transitions we are talking about here

are not thermodynamic phase transitions. It's not that you chug up to some concentration, you have hit the solubility limit, and now you have a solid phase appearing right at that point. Instead, you have chains that are intertwining with each other. And at some point, in terms of this model,

the intertwining has become rheologically effective. And they become rheologically effective over a range which most versions of the theory do not calculate. That is, it starts becoming effective. You run up the concentration some imprecise amount, and now it's really effective.

But they got up to quite high concentrations. Oh, I should pause. I am aware of one model analysis which does predict the sharpness of this transition. The prediction is based, in essence, on the assertion this entanglement transition, chains all getting entangled with each other,

is a percolation transition, that is you move from chains that are unattached to chains that have entanglements, so you could walk along a chain, hop at an entanglement, walk along a chain, and cover the system. Percolation transitions are extremely sharp, at least usually.

However, there are people who don't agree that's how you should analyze it. I just bring it up, that there is this range of opinions on what the transition should look like, and at least some of them say sharp and some say otherwise. It doesn't matter here, we clearly got way above the transition.

Okay, so that was what was done. And then Lodge worried about probes. This, after all, is all information on the matrix. However, there are probes, and what Lodge did was to realize

we can do synthetic chemistry, and we can make star polymers. And so there is a three arm star, and we can also make, etc., f equals 12 arm stars.

And of course, we have a linear polymer, but you can view a linear polymer as being a two arm star. And the linear polymers, and each of these were made with molecular weights. Well, let's see, linear polymers have molecular weights from 65 to 10 to 50 kilodaltons, meaning quite small to quite large.

The three arm stars were the same. The 12 arm stars went from 55 to 1640 kilodaltons. You should realize 1640 kilodaltons spread over 12 arms is something like,

Or one of the arms is something like, 130 kilodaltons. Very round number down in my head. And therefore, if you look from here across there, it's nowhere near as big as a 1640 kilodalton linear chain would be.

So here is this beautiful experimental system. And the beautiful experimental system, we have a bunch of different molecular weight probes. We have three different molecular weight matrix polymers.

We have three probe topologies. There are a lot of different combinations there. And what Lodge did was to realize we should systematically study these combinations and see what we would find.

Okay, and we start with figure 8-17. And we continue through figure, and 8-17 shows linear chain.

And we continue through the other figures, and we find three arm stars, and we find 12 arm stars. And therefore, we have a range of different polymers with

different numbers of arms. And we can plot ds versus concentration. And we can do this for a series of different probes. And of course, the larger probes diffuse more slowly.

And we get measurements of d of the probe versus concentration of the matrix. For four different sizes of probe. Well, that's very nice. And if you look at the graph, you'll see these nice lines that really do a fine job of going through the data points.

They really do, with a couple of exceptions I will get to. And what are those lines? Those lines are stretched exponentials e to the minus a, c to the nu, matrix molecular weight to a power p,

probe molecular weight to a power gamma. And if you work through, there are in fact three different matrix polymers. There are four different probe polymers, yes. And there's concentration.

There are 12 lines generated. The graphs show you one set of matrices at a time for a series of different probes. However, the three graphs in one figure correspond to all three matrices, and all of the concentrations that were used, and all of the probes that were used.

And then there are the lines through those graphs. We showed, in at least some cases, two sets of lines. The first are the single stretched exponential fits. One stretched exponential per matrix probe combination. And then we do the hard thing.

Namely, we say we will take all of the measurements at the same time, and we will fit all of the measurements at the same time to a stretched exponential. And if we fit all of the measurements at the same time to a stretched exponential, we get a family of 12 curves, all defined by the same one, two, three, four parameters.

And the curves do a very nice job of representing all of the measurements simultaneously. There is one point, if you look hard, where clearly things aren't nearly as good.

Namely, if you look at the smallest molecular wave probe, the measurements do this, the fitted curve does that, and there is quite visible discrepancy. You might ask, gee, what's going on?

Well, what is going on, if you compare with the single fits, is that the concentration exponent depends on the molecular weight of the probe, and the matrix in some combination. And the concentration exponent starts very near one, for small polymers, and heads down towards about half for large polymers.

The reason for this one proposes is that small polymers do not change their radius of gyration very much as you increase the concentration of the matrix, but large polymers contract.

And we saw that with dielectric relaxation, where small polymers, by direct measurement, did not contract as you ran up the matrix concentration. But large polymers very definitely did go to smaller radius, and we could actually describe that quantitatively. That was in an earlier lecture.

Well, the fitting form that was used just took nu to be a constant. And saying nu is a constant rather than some well-defined function of probe size is not quite good enough an approximation for very small probes. If you wanted to go to the next step, you would have to put in that nu was

some function of m to the, I got the exponents wrong, I usually say gamma and delta, m to the gamma, t to the delta. And nu in the graph is being treated as a constant.

Well, it's an approximation. Okay, well that's very nice for linear polymers. However, there were also three-arm star polymers. There were also 12-arm star polymers. And Lodge did, Lodge and his very wonderful group, did not only

the linear chains, but the three-arm stars and the 12-arm stars, and giving us three families of graphs. One for the linear chains, one for the three-arm chains, and one for the 12-arm chains. Gee, three families of graphs.

For each family of graph, this expression was applied to the number of sets of measurements. It would be six or eight, it would be, let me get the numbers right. It's not the same number of sets of measurements because there were a certain number of linear chains and a certain number of three-arm chains and

a certain number of 12-arm chains. But for each set of chain, each type of chain, three or 12, we do the same fitting process and we see the lines which describe very nicely not only the molecular weight dependence on the matrix,

but the molecular weight dependence on the probe, and a reasonable but not perfect job for reasons I've explained with the concentration dependence. So, that's very nice. These are absolutely beautiful measurements. They're taken to matrix concentrations way above the entanglement concentration.

And there is one market feature, which perhaps needs to be stressed. And the market feature is that if you look at one of those sets of measurements, here's d sub s versus concentration. You see a smooth, on a log-log plot,

you see a smooth curve of constantly increasing the slope. We start out quite close to flat, we roll over, we go downhill, there is absolutely no indication in the measurements of a well-defined region in which you see a transition to scaling behavior.

That is, if you look at the large measurements and ask, where is scaling? It's quite clear that there isn't any. There's no region on a log-log plot where you see a straight line. Instead, you see a smooth curve of constantly increasing slope.

And that smooth curve of constantly increasing slope is the extremely uniform behavior you find for polymer self-diffusion. Well, we can push ahead, and we can also look at figure 8-20.

And the figure in question shows us comparison of linear and star chains, d versus c. And for small chains, the linear and star chains, if you get two that match,

have about the same concentration dependences. And for the larger chains, for the big chains, there's some difference.

Stars and linear chains aren't being affected in quite the same way. You can look at that for yourself. Okay, let us push ahead to the next figure, which is Studied-Newton-Martin. And Martin also looked at polystyrene in PVME solutions.

He actually looked first. This is among the early work on this. His measurements cover a range of about 10 to the 2 in self-diffusion coefficient. That is, the diffusion coefficient starts out at one value, and

it falls to about 1% of its value at the lower end of the curve. He has several different polyvinyl methyl ethers, and a joint fit reflecting the various sizes of the probe gives us the curves you see, which work reasonably well.

However, Martin did something else. In addition to measuring ds, he also measured eta, the viscosity of the solution. The issue with solution viscosity is that if you have a solvent, and

you start stirring polymer in, in almost all cases, as you add polymer, the solution becomes more and more viscous. And you might say that there is some, therefore, some approximate resistance to flow, and that approximate resistance to flow should perhaps correspond to a change in the diffusion coefficient.

If you have mesoscopic spheres, say 20 nanometers spheres, diffusing through a small molecule liquid, then you quite uniformly find ds proportional to kt over 6 pi eta r.

Where eta is the salt, this is now just the solvent viscosity, that's all you have. R is the sphere radius. This is known as the Stokes-Einstein equation.

The Stokes-Einstein equation, derived by Einstein a century ago, corresponds to an estimate of what the diffusion coefficient of the sphere should be in a simple liquid. It works all right for mesoscopic spheres. As we noted, if you go to ions or small molecules, and

you run up the viscosity above a few centipoise, Stokes-Einstein doesn't work right, because the diffusion coefficient is not proportional to one over eta. However, if you believe you're looking at a system in which the Stokes-Einstein equation is right, or is approximately right, given that the pro-polymers

really aren't spheres, you would say that you would expect ds eta to be about the same as the value of the product in pure solvent. That is, as you add polymer to the system, the viscosity goes up,

the self-diffusion coefficient goes down, and therefore you might expect, people do, some people do, that the product ought to be fairly independent of polymer matrix concentration. Well, it really isn't.

And you have non-Stokes-Einsteinian behavior. In almost all cases, the non-Stokes-Einsteinian behavior is that as you

increase the concentration of the system, the product d eta also increases. That is, the matrix polymer is more effective, sometimes orders of magnitude more effective, at increasing the viscosity of the solution, than it is at retarding the diffusion of

polymer or hard sphere probes. That's non-Stokes-Einsteinian diffusion, but it's fairly unambiguously observed by Martin for polymer coils. Unfortunately, there is not nearly as much data as one would like, where people compare

chain diffusion and viscosity, so I can't go into a great deal of discussion of this, because the measurements aren't quite there. For probe diffusion, matters are quite different. For studies of probe diffusion, spheres diffusing through polymer solution, there are lots of measurements of viscosity and probe size and probe diffusion all on the

same system, and much more extensive comparisons are possible. Let us chug ahead to the experimental study of Davis.

What Davis does is something slightly different. It's only one system, but diffusion is measured two ways. It's measured using quasi-elastic light scattering spectroscopy, and it's measured using pulsed field gradient NMR. The virtue of this measurement

is that the two techniques are sensitive to polymer motion on very different distance scales. Correspondingly, they are watching motion on very different time scales,

because in order to travel further, it takes longer. Quasi-elastic light scattering is sensitive to motions over some reasonable fraction of a light wavelength. Pulsed field gradient NMR is sensitive to motions over considerably

longer distances, and what Davis demonstrates is you look on these two distance scales, and the diffusion coefficient looks about the same. Of course, you might ask, well, how could it not be the same? I will give an example. It's a very standard example.

You should realize that it doesn't mean the example does not have to be right, but it gives one way you could get to this result. Suppose you had a system in which you had boxes, and inside the potential energy of the diffusing particle was fairly flat,

and then there were potential energy barriers that the particle had to climb. If you watch a single particle, it will do a random walk inside a box, but only rarely will it find enough kinetic energy to get over the potential energy wall and get into the next box into which it is now trapped.

If you look on short distance scales like this, you mostly only see the particle jittering back and forth in the box, which it can do quite rapidly. If, however, you do use a measurement of diffusion, which asks how long does it take the particle to get from here to here,

that's pulsed field gradient NMR, the particle has to hop repeatedly from box to box, a slow process, and in this case the diffusion found on long distance scales is slower than the diffusion found on short distance scales.

I urge you to reread the papers of Numoto where precisely the opposite circumstances is found. The long distance diffusion seems to be faster than the slow distance diffusion. I don't have a pretty picture that shows you how you get that second result.

Nonetheless, the core issue is Davis did measurements on several different distance scales and he found the same results. Okay, we can now chug ahead. You note Brown, gee, this is 8 megadolton of polymer

and up to 1.43 megadolton of polymer, so the probe and matrix polymers could be very large and once again, though only the matrix molecular weight was varied, the matrix dependence is captured by the joint stretched exponential.

Let us chug ahead and we will go to the experiments of Kent et al. And Kent et al. look at three combinations of matrix polymer and probe polymer

and the three combinations are labeled by their molecular weights all in kilogalton and the combinations happen to be 23366, 233, 840, and what was the last one, 920, 840.

You may say, gee, those are weird numbers, why are they like that? The first answer on why they are like that is you actually have to purchase or synthesize

the polymer. Synthesis is a chore, purchasing, you're limited in what you can buy and you find polymers of particular molecular weights and you take as much advantage as you can of what you have either bought or made. The second point is that these combinations actually do have a pattern to them.

Here, the probe and matrix molecular weight are about equal. Here and here, we have the probe molecular weight much larger or much smaller

than the matrix molecular weight. That is, we are looking at three cases in which either the probe and matrix are about the same or the probe and matrix are very different from each other. There are models that predict that you see one behavior for p about equal to m

and very different behaviors if p and m are very different. Well, maybe you do get this behavior if they're different enough and we'll get to that a bit more in a moment. The important issue here though is there are three curves corresponding to these three

combinations of materials and the three curves are accurately described as there's concentration dependence and there is a p to the delta and an m to the gamma. There is a matrix and probe molecular weight dependence.

Now, you could say, gee, there are only three probe molecular weight. There are only three probe matrix combinations. The fact that you have the probe make three combinations and two parameters makes life easy for you. Well, that's partly true.

However, there's no particular guarantee that this sort of probe and matrix molecular weight dependence is going to work at all. Nonetheless, it does. Okay, I'll head again and we advance to a very nice figure which in the draft manuscript

is A-28 which is on polypropylene oxide and we also have measurements on g, other

polymers.

And what is interesting about this figure? The first set of measurements is a 33 kilodalton probe in a 32 kilodalton matrix except it's a mixture of 32 kilodalton and 1 kilodalton polymer. This is what is called a polymer blend.

In a blend you have, as you would expect, several molecular weights of, in this case, a given polymer. You've mixed them in combination. Now, the virtue of a blend, yes, the molecular weights are very different, but the two chains

are chemically the same. So you have long chains and short chains and there is essentially nothing that persuades the matrix polymer that it should not be soluble in what is the solvent which is more of itself except shorter pieces.

And therefore, you do not run into a solubility limit the way you do with some polymers. Instead, the combination is soluble in all proportions and now you drop a probe in. Ditto, the other groups of measurements, you have a 225 kilodalton probe, you have

matrix polymers which are molecular weights 93, 250, and something really large 20,000 kilodalton, that is 20 megadalton polymer, and you measure the diffusion of the probes

through the matrix polymers. However, because your solvent is simply the low molecular weight version of the same polymer, you can do the measurements of D versus concentration.

on some scale, and you can actually, as it was done, take the measurements out to melt. That is, you start out with dilute matrix polymer or no matrix polymer, just the probe. You add matrix polymer and you end up out here and eventually there's nothing but matrix

polymer, no solvent at all. That's what a melt is. We do not talk about melts in this course. It was simply a matter, you have to stop writing someplace or it will go on forever. And it did start almost to go on forever, but not quite. Things were brought to a stop.

Things were, we got done on schedule. However, the point of this is, we have stretched exponential concentration dependence for D. It starts here, it ends up there.

And absolutely nothing happens all the way out to the highest concentrations you can imagine, even with extremely large polymers. So we do have a fitting function description, phenomenological description, which actually, at least in this case, works over a very wide range of concentrations.

So we have now discussed blends. Are there other experiments we can do?

Well, we might want to get enthusiastic and say, we will try a much wider range of probe and matrix molecular weights. And if you chug ahead a few figures, we see a plot of diffusion coefficient.

We are, in this case, going to plot diffusion coefficient versus probe molecular weight. And we have, as seen in the figure, a curve showing self-diffusion, meaning probe and

matrix are the same molecular weight. And we have another curve which is very, very nearly the same. So we have a P equals M curve, and we have a case where the probe is much smaller than

the matrix. And we see about the same probe behavior. And finally we have some isolated points up here which sit on a power law, which correspond to the probe much larger than the matrix. As you realize, there are only a certain number of points you can fill in to a certain

extent. And now we end up finally, if I recall correctly, the next figure is 8-31. And what is done here is to say, what the authors did was to measure the diffusion

coefficient. And they measured the diffusion coefficient for probe and matrix molecular weight ratios of 0.1 and, if I recall, 3 and 4, and maybe it's 2.

And they measured ds for various combinations of probe and matrix molecular weight as a function of concentration. And because of the way things are arranged, these two curves have to come together at

an infinite dilution. And, gee, there are data, and there are smooth lines, and they also said, we will look at various probes. We'll look at ds again, and it's various probes as a function of matrix molecular weight.

And there are a lot of probes and not that many matrices, but you see another series, a family of lines, which mostly but not completely go through all of the measurements. The agreement of these curves with the measurements is not nearly as good as the

agreement of these curves. However, what you see in the two figures, parts A and B, are in fact one surface. You see eight curves, but the eight curves are not independent from each other.

Instead, we had a fitting function, ds is d0, and we have to take into account the fact that as you change the probe size, d0 changes, e to the minus a, c to the nu, m to the gamma,

e to the delta. We generate a family of curves using single values of delta, gamma, nu, and a. So they're actually only four parameters, plus d0, which you can measure separately. And all eight curves are eight separate slices through the multidimensional sheet

described by this function. The concentration slices work extremely well. The molecular weight slices do not work as well, but one function at a very small number of parameters describes an extremely wide range of probe and matrix molecular weights.

Okay, well that's very nice. The function works quite well. The final point I would call your attention to, if I recall correctly, the next figure,

which if I recall correctly, is 832. And the important feature here is that you are looking at tiny probes. Yes, they're polymers, but they're really small polymers. They're not megadaltons or hundreds of kilodaltons.

And in this case, what one finds is that d is from d0, p probe molecular weight to the minus a. That's the fact that the probe radius depends on its molecular weight. e to the minus, and we get a, a, c, and c is to the first power, and p is to some

power, delta. So we have dependence on probe molecular weight, we have a dependence on concentration. And you can judge for yourself, the agreement of fits and data do not look too bad.

There is, however, one other little bit. These are very low molecular weight probes, a few kilodaltons. And one expects for very small probes that the probe radius is more or less independent of concentration.

If you think back to the chapter on dielectric relaxation, we made the point that the concentration dependence could be written as c to the first, or radius over radius at zero concentration

cubed, and the deviation from a simple exponential was entirely covered by the contraction of the chains to the cubed power. And what we had from dielectric relaxation, this unity between chain size contraction and stretched exponential rather than simple exponential behavior.

Here we have chains that do not contract. Their radius is independent of concentration, and therefore, there should be all of the concentration dependence, and that is exactly what you find.

And the concentration dependence, as you see in the figure, are all pure exponentials in concentration. And because they are pure exponentials in concentration, it's e to the minus a c to the first. That's a coherent description of what's going on, and it's coherent in terms of the stretched

exponentiality arising from chain contraction. Alright, what have I done today? What I have done today is to go over a large number of measurements of polymer self-diffusion. There are a wide variety of variables that can be varied through studying molecular

weight, concentration, we didn't have much to say about solvent quality. You can arrange things so that the probe polymer and the matrix polymer are different. That's tracer diffusion. And in that case, the probe and matrix behaviors can be sorted out separately.

There are experimental studies due to lodge on the effect of probe topology, linear 3-arm star, 12-arm star, we explored that. There are some limited measurements where people measured the viscosity as well as

the diffusion coefficients, and there we saw clear evidence for non-Stokes-Einsteinian diffusion. We also looked at cases where there were large variations of probe or matrix or both molecular weights, and we asked how do things behave.

And throughout, in almost but not quite all cases, we found that a joint stretched exponential e to the minus a c to the nu, probe molecular weight to some power, matrix molecular

weight to some power, d0 which has to be separate for each probe, this functional form describes polymer self-diffusion. There is almost never any evidence for scaling behavior in which the self-diffusion coefficient

goes as, for example, ds proportional to m to some power like minus 2 or minus 2.3, c to some power x where you can put in such x as you want. Scaling behavior is simply not found in polymer solutions.

Well, that's very important. If you've got a theory and it predicts scaling laws, and it actually predicts scaling laws, your theory isn't in good correspondence with results. If you want a theory that treats polymer self-diffusion, it should give stretched exponentials, not power laws, in the key variables.

But we're going to continue with this theme in future lectures. We have one more lecture on polymer self-diffusion. And that one lecture will be used to unify the very different results we have. After all, we've been getting these fitting results for gamma, delta, nu, and a.

And if you are doing something sensible, you hope to find that these coefficients have some reasonable behavior on the other properties of the solution. For example, you might hope a is a constant. You might hope that gamma and delta are numbers that are about the same in different systems,

assuming you're looking at some generic property rather than at some property which is an outcome of chemically specific properties. And you can push ahead, and that's what we're going to do, and analyze these things. But that is it for now.