I'm Professor Filleys. This is the last of the lectures on dielectric relaxation

as a method of studying polymer dynamics. Next time, we'll be advancing to the next chapter of the book, which is polymer self-diffusion. And that will get us through single-chain motion. So where we are, we've discussed chain size, which

our technique measures rather nicely. We've discussed single relaxation times. And now we're going to advance to discuss relaxation spectrum. The general notion is that you can measure the two

quadrants of the dielectric relaxation function, that is, an in-phase and an out-of-phase. And we will almost entirely talk about the dielectric loss function. And it's a function of frequency.

And if you look at a typical graph like 7-11, what you see is there's a low-frequency behavior. There's a roll-off. There's something that looks very convincingly like a power law.

And then there may be some interesting behavior at high frequencies. One of you asked, well, if you push out to sufficiently high frequency, don't you reach something resembling a high-frequency limit because the solvent has electron shells, for starters? And the answer is yes.

If you could take this out far enough, eventually you'd see something else which has its own relaxation at, gee, optical frequencies. But we do not get out there. You should, however, notice, I mentioned it looks like a power law. A standard comment on fitting things to power laws

is that if your function, your range, and your change in the dependent function are not very large, almost anything looks like a power law over short distances on a log-log plot. That is a very important basic result. Namely, if you just have small changes

in your independent variable and small changes in your dependent variable, you can be pretty sure that you can at least draw a power law curve that is adequately tangent to resemble the measurements. Here, however, for example, in figure 7-11, you notice the nominal frequency covers 10 orders of magnitude.

The strength of the dielectric response covers, well, it's actually really nine orders of magnitude or eight because there are no data points quite up to the top and the bottom. And the net result is you have very large sweeps of measurement in which you can very clearly and unambiguously see power laws.

And up to the, maybe it's only a power law at the 1% level and not at the 0.1% level. Those really are power laws. What you, however, get is a curve that looks like this. Now, why do we get this curve? The answer is that the polymer can do something

that changes its dielectric moment, meaning it changes the end-to-end vector. And there is a time scale on which the polymer reorients or does whatever else it's doing to change the value of the end-to-end vector, which is what we're measuring in most of these chain systems.

And if we look in at a high frequency, a frequency that acts on time scales short by comparison with the time on which the polymer can move, the dielectric increment of the polymer tends to fade out. That is, if you want to say,

I am going to apply an electric field and I am going to line these dipoles up, the electric field has to point in the same way, long enough for the dipoles to get turned around and point in the needed direction. If before the dipole can move significantly,

you flip the direction of the electric field, the dipole now tries to start moving back the other way. Nothing much happens and it does not contribute to the dielectric constant of the material. Well, this is the loss function, that's frequency,

and you see as you run up the frequency, things fade. Now, what I've actually chosen to plot for the dielectric function is epsilon double prime of omega divided by omega.

If we had information on the dynamic response function, and there's just not a lot of it out there, you would be looking at epsilon prime of omega and the reasonable division is by omega square. The reason for these choices is that dielectric response,

frequency-dependent response in the phenomenological analysis I give in my book is subject to the same issues as mechanical response to shear is, and therefore by direct analogy, these are the functions we look at.

We'll get to dynamic response, mechanical response, viscoelasticity, that is, late in the course, so you'll sort of have to wait on this one. This was one of the issues. I could have dropped in viscoelasticity, which is a collective property of all of the chains in the solution before I did this,

and then it would be very easy to see why this was correct, but that would sort of destroy most of the rest of the book, so you have this little thing. If you want to understand what's going on, you have to skip ahead a number of chapters. Nonetheless, we have this function,

and what is proposed by direct analogy with the discussion of viscoelastic response is that this function has two forms. It has a form e to the minus a omega to the delta, and it has a form omega to the minus x,

and these two functions are separated by some crossover frequency omega c, so you have one behavior at low frequencies

and another behavior at high frequencies. This description is partly incomplete. The reason it's incomplete is you have a polymer chain. It's this big complicated object, and it can have several physically entirely different ways of changing

which way it's end-to-end vector points, okay? And each of these will have a dielectric response function. That is, as you increase the frequency of your applied field, eventually each of these processes for moving the polymer around can't respond fast enough to contribute

to the dielectric increment of the solution, and so you will actually have several of these things going on at the same time, additively. Let us look at figure 7-11, and we will actually see an illustration, and so at very low frequency,

you see several curves. The reason you see several curves is that you are looking at several samples at different molecular weights. With increasing molecular weight,

the loss function becomes larger. This is at low frequency, and then as you increase the frequency, we get a behavior, and the first behavior you get, I'll just sketch this, you get a roll-off, and that's this function,

and then there is a long period on the graph where what you see is a straight line. The straight line you are looking at is a power law, because log-log plot, on a log-log plot, you see a straight line. The problem with this is

that if you take almost anything and put it on a log-log plot for not too large a distance, it looks kind of straight, and therefore it's very easy to fool yourself into thinking you are seeing a power law behavior when there is no power law there. Here, however, if you try, the points are very tiny

and you can barely see them. There are clearly a huge number of points. This is a beautiful piece of work by Adachi et al., and you very clearly get a straight line here. This behavior. Now, I will point out something that may not be instantly obvious.

If you go up to the upper left corner of the graph, you see the straight line and a curved line that are not adorned by points, and what those show is, here's the power law behavior, and I continue the line up here

so you can see it clearly, and here's the, it's actually often a pure exponential behavior, and you can see its curve, it doesn't really quite curve back, so you can actually see the two curves. As you head down, something interesting happens, namely, as you get out to sufficiently high frequency,

instead of the highest molecular weight of a material having the largest value of this function, it drops off and there are curves crossing each other, and if you look carefully, curves cross and they head out here,

and what happens at large frequency? Well, let's stay with the highest molecular weight material, which is rolled off and it's in a power law, and it stays in a power law down to about 10 to the six hertz, and then there is something that looks like this,

and the something that looks like this is a second relaxation process, that is, there is a second one of these, it's actually quite close to exponential usually, curves, and then there is a second power law, the slope of the second power law and the first

look quite similar but are not necessarily the same. The other thing you will notice is there is this region where the curve is concave upwards, and if you look just where we are going over from one function to the other,

the curve is concave upwards. Why is the curve concave upwards? These processes are additive. If I could use some sort of magic and measure them separately, the first process would be doing this, the second process would be doing this,

except in this narrow region, the higher frequency process is too weak to be the change the curve significantly, but right in here both processes have about the same intensity and are both contributing to the dielectric loss to noticeable extents,

and you have to be careful and do addition. There is nothing in the measurements that would suggest that this process that goes out to high frequencies does not also extend into zero frequency, and there's no sensible reason to suppose it doesn't,

but you would have to be a very good experimentalist indeed to convince yourself that that's what it was doing. So what I've now shown you is here is a dielectric response function, there are two physical processes here, and you sort of ask what the two physical processes are.

Sensible question. Well, here is a polymer chain, and it has an end-to-end vector, the polymer we're looking at, this is I believe cis polyisoprene, yes,

is composed of little molecular subunits which have a dipole component along the backbone, and they're all linked up more or less head to tail, and therefore this vector in space from one end to the other is, up to a multiplicative constant,

is the dielectric dipole vector of the molecule, the part we're seeing. So how do we change this? Well, the slowest process you can imagine, at least I can imagine, you might be clever and find something better,

the slowest obvious process is that you rotate things. Now, when I say rotate, I am actually being very imprecise. What is being rotated is the direction of this vector, and there are at least two ways you can imagine rotating the vector, and the first is you pick up the whole molecule,

and the whole molecule does something resembling rigid body rotation. The second way you can imagine that this vector rotates is that one end moves this way, the other end moves, say, that way,

and now look, the vector is pointed a different direction because the two ends have moved. Of course, at the same time, when the two ends have moved, everything else has moved too, and maybe the chain is off like this, or maybe the chain is off like that,

where there's a centerpiece that hasn't moved much, and the chain has moved along its own length. There are people who believe that, but the only issue is the whole chain has rotated, and this happens rather slowly.

And therefore, at fairly low frequencies, this whole chain rotation process cannot keep up with the dielectric response, excuse me, the applied frequency, and the contribution to the dielectric loss disappears. Then what is the second process?

And Adashi and people say entirely convincingly that the second process is segmental motion, but there is a piece which is not necessarily quite obvious until you think about it. Here is our polymer chain.

There is the end-to-end vector. Which segmental motions count? Well, you could have a purely internal mode, and a purely internal mode, for example,

takes this and turns it into this. That is, part of the inside of the polymer has moved. But if you look at this purely internal mode, you realize the one I drew does not move the end-to-end vector, and therefore does not contribute to dielectric relaxation.

How then can segmental motion be contributing to dielectric relaxation? And the answer is, these two chain ends have some distance over which the monomers are sort of lined up, resistance-length.

And if the chain, there is an internal mode that changes the chain ends, it's an internal, it's a local mode, it's not a whole-chain rotation. But look, if you move the two ends around, you change the direction of the end-to-end vector. And therefore, the two ends of the polymer,

like my two hands, can wiggle, and those wiggles contribute not very strongly to the dielectric relaxation. It's not very strong because it's only a small part of the polymer making the contribution. If you believe that argument,

you would also tend to say, there's a feature, oh, I didn't point this out in the book, did I? But I will point it out now. Here's a big, long chain. And the motion of the part near the end can contribute to the dielectric relaxation.

However, the amount of chain whose motion near the end is correlated with the location of the end point, there's some distance in here where if you move monomers here, the end has to come along with them.

That distance is 10 or 20 or 40 monomers. This behavior doesn't care how long the polymer is. The high frequency relaxation that is due to local modes is looking at the behavior of the two ends of the chain.

Well, guess what? If you're only looking at something of fixed molecular weight, no matter what the molecular weight of the whole chain is, this behavior should be independent of polymer molecular weight. Approximately speaking, you've got to be very careful with arguments like this and not take them too completely seriously.

And that would say that if you look at the rollover and power law drop of the fast relaxation, it should occur at sort of the same point at all molecular weights of polymer, as opposed to the chain rotation,

which occurs very differently depending on the molecular weight. And that's exactly what you see. And that tends to explain why at very high frequency, these curves all come together. They may not come together perfectly exactly, but they tend to come together.

Yes? Is that possible with the frequency increase, end-to-end vector check? You are applying weak fields. You are not applying fields that are strong enough,

for example, to stretch the polymer out. And therefore, the first approximation, the confirmation of the polymer, the shapes it adopts in the solution, as opposed to the direction, are not greatly perturbed by the electric field.

They are perturbed somewhat because you're seeing these local segments rotate. But the polymer shape is not greatly distorted from equilibrium. If you go to high frequency, the effect of the electric field,

as you can actually see that curve dropping, drops a great deal. If you were looking for an effect sort of like that, you might propose, if you are at very low frequencies, and apply a sufficiently large electric field, which may not be practical experimentally,

you might, in addition to orienting things, you might discover that you were getting local field gradients that would tend to stretch the polymer out. However, I wouldn't count on that being easy to measure.

Okay, let us push that ahead to 7-14. And 7-14 is two figures. Each figure refers to a polymer having a single molecular weight.

So we have a polymer, it has a single molecular weight, but we've changed the concentration. And as we change, we ask what happens to the polymer behavior as we change the concentration.

And the thing that happens, well, there are two things that happen. And one is the dielectric loss function increases. You notice this is not normalized by polymer concentration.

The second thing that happens is that if you look at these things, you're gonna roll over from a low frequency behavior, and there is a region which is e to the minus a omega

to some power which is actually close to one, if not exactly one. And then at higher frequencies, the curves almost but not quite blend with each other. And they almost but not quite all relax as power laws of about the same slope.

Now, the virtue of what you are seeing here is that when I say there's a power law, I am not saying we have a big sheet of graph paper and I can draw a line through the points. I am saying we have an area that I've identified as showing power law behavior,

and I will do non-linear least squares. I will do a computer fit that extracts an accurate parameter for the slope. And the virtue of computer fits is that you can get numerical results that are much more precise than you might have gotten out of graphical analysis. The measurements are quite good enough to support this,

and you can actually do it. What else you notice is that, since that way is increasing concentration, at larger and larger concentrations,

the frequency at which you start to see the rollover gets smaller and smaller. Small frequency corresponds to a large time. That is what you're saying is that you increase the concentration, the time required that is needed

for a polymer to reorient gets longer and longer. And so, you are actually in the concentration dependence of this phenomenon, seeing something about molecular dynamics. Not necessarily something very surprising,

namely that if you're in concentrated solution, a polymer has much more difficulty reorienting. Okay? So that is 7.14. Another figure that shows the same sort of behavior is 7-12, if you go back a few pages.

And in 7-12, you can see two very clear power law regions. One function is not being replaced by another. The two functions are additive. And you can actually see very cleanly

how good or not good, I'd call it fairly good, the functions I'm describing are as giving fits to the data. And the answer is, if you look at those, you see you get lines. The lines go straight through the measured points. And it is fairly clear, if you compare those, that the functional description I'm proposing

is quite accurate. The other piece of saying it's quite accurate, on those curves, there is a low frequency behavior, which is the quasi-exponential. There is a high frequency behavior, which is a power law. And if you look very hard at the intersection,

you can't see some region in here where things don't quite work right. Instead, you see you have one behavior, you have the other behavior, and you have a very sharp transition from one to the other. There is no reason why the transition shouldn't be sharp.

And here you can fairly clearly see it is. If you expand the figure, it remains obvious it's a sharp transition. Okay, what else can you do with this technique? Well, there's something else that's clever you can do.

You can say, here's a polymer molecule. And so far we have been talking entirely about polymer molecules composed entirely

of monomers having type A dipoles. That is a dipole that simply points along the backbone. However, if you are a clever synthetic chemist, and there are plenty of those out there, you can say, I will generate a polymer that is this long,

but it will be a block copolymer. That is, there will be a piece of one species, a piece of a second species, and a piece of, well, it could be a third species. And in particular, I will choose things so, for example, only this section

of the polymer has type A dipoles in it. And the rest does not. And now I am looking not at how the whole chain moves, I am actually looking in at the motion of this small segment. And you can actually make this experiment work.

You can also, and that is what you see in figure 716, this is a di-block polymer, there's the division point.

The two chains are of the same length. And G, we've hooked them up together head to head. And so, we have a half end vector like that, and a half end vector like that. And the relaxation includes a term that tells us

the correlation between this direction and that direction and how they relax. And you might correctly assume that since this is basically a polymer, half as long as the full polymer, it relaxes more quickly. And the alignment of these two vectors

relaxes more quickly. However, the relaxation function, if you look at 7-16, the relaxation function is the stretched exponential in frequency at short times. The power law at long times. And this is a fairly large figure,

so if you look near 10-cube or 10-four kilohertz, there is a point in the middle where both curves are very close to the measurements. You see where both curves are close to the measurements? And the two curves are tangent to each other.

One curve just switches over to the other in a fairly sharp manner. Now if you ask, is the crossover really sharp, you always have the following problem. Here's the first curve. Here's the second curve tangent to the first. Here are the data points which,

no matter how good the experimentalist is, have a certain amount of noise to them, not quite as much as I'm drawing. And there is a region in here in which the accuracy of the measurement does not really let you distinguish between the two curves. And while I am saying crosses over at the intersection,

if you wanted to say something else happened and the correct function did something like this and then hopped over, I can't prove it's not true, but you can't see any hopover in the data. You just see a smooth pair of functions that are tangent.

However, experimentally the statement there is nothing else there is always subject to the fact that your experiment has a certain limit. There is a more complicated version of this picture

and I published a paper on it this summer. No one's tried doing it. It's a very hard experiment. Synthetic chemistry. Here's a polymer. And we will take two identifiable segments

and we will generate one dipole pointing that way. Two, one of them flipped. Three, the other one flipped. You see this one is now backwards. And both of them flipped. And I will measure the four dielectric response functions.

That is, I produce four polymers and in one of them the type A dipoles are pointing this way. And in the second I flipped this piece around so the type As are pointing the other way. And there's a third where instead of flipping this one

I flipped that one. And there is a fourth in which I flipped both of them. Okay, I now have four different compounds each of whom has a dielectric response that I can in principle measure with sufficient accuracy.

I didn't say this was an easy experiment but you know with digital electronics high accuracy measurements get easier. And now I add and subtract the dielectric response functions in a correct pattern. It's all in the paper. I'm just going to give the result. In a conventional experiment

with just one type A dipole all along you're measuring the direction of the vector at one time dot the direction of the vector at the other time and then there's some average behavior. So the end to end vector at one time is like this

and a later time it's like that. And the dipole eventually forgets which way it was pointing relative to which way it was pointing originally. And that leads to the dielectric response. If you just measure these two, this one compound

your dielectric response is a combination of things. And one is each dipole has to remember which way it was pointed relative to itself. And there's another part I'm going to call the two pieces A and B.

The orientation of one piece of the polymer at time zero and the orientation of the other piece of the polymer at time T. Yes? So far so good? Well, this is what is called cross correlation.

That is we're asking how is the orientation of this piece of the chain at one time related to the orientation of this piece of the chain at other times? And that tells you a great deal about the dynamic behavior of the polymer. It probably tells you more

if the two pieces are right next to each other than if they're way off in the distance. If the two pieces are way off in the distance relative to each other you may discover there's nothing very interesting happening other than there's bulk rotation which you can actually see. Well, in principle you can do this measurement.

Question? There is a question down there. For polymer chain it can rotate or orientate in three dimensional not just in one plane like showing in that platform. That's correct. This function or this one we start like this

and the end to end vector could have done this rather than that. It does both of them. It's a three dimensional behavior. However, it's kind of hard to draw this on the blackboard so I don't. So, but you're perfectly correct. The relaxation does not occur in a plane.

It occurs in three dimensional space and so at one time the dipole is this way. At some later time it might be this or this or this. Yes? And therefore you have to take into, you have to remember this is a 3D problem.

There are mathematical descriptions of this called spherical harmonics that let you treat this. The fact that this is three dimensional, if you remember when we were discussing fluorescence relaxation I said there were some complications in the math.

This is the same complication. It arises because the system is three dimensional and you just have to live with it. We're two dimensional. Well, you can study 2D systems but that's not what we're talking about. Okay, what else can you do? Well, there's one interesting thing.

You can say I have a polymer and I am going to decompose the motion of the polymer into some set of normal coordinates.

For example, I will say here's a coordinate. It's the end to end vector. Here's another coordinate. It's the difference between the vector

from the middle to one end and the vector from the middle to the other end. Those two pieces, those two vectors are perpendicular to each other. This one tells you about the largest scale motion of the polymer. This one tells you about a somewhat

finer scale motion of the polymer. Now there are two general sorts of approaches to how you handle this. We go back to two theory papers. One is by Kirkwood and Reisman.

Now the experimental quantities they calculate do not include one end dielectric response. However, the math doesn't care about that. What Kirkwood and Reisman said is here is a polymer chain and it has a center

and it rotates and it also changes internal shape, fluctuation, but everything of any interest for, for example, viscosity or end to end motion

is hidden in the translation and the overall rotation. That's Kirkwood Reisman. There is another description due to Ruse and there's a second paper by Zim where he takes the Ruse model and he does something to it

which gets called Rouse-Zim because about half of it is really rouses and they're both nice models. And they decompose the polymer into a set of displacements each perpendicular to one axis.

And I am not going to do math on that. However, you get out of this theory, you say we have a shape of the polymer and the polymer, if you distort it, tends to relax back to equilibrium configurations

and the relaxation in these is a sum of exponentials in time. Well, if you have something that relaxes

and this is an exponential in time, what does its relaxation look like if you have, do this in frequency domain? And the answer is that if you have an e to the minus gamma t, that's an exponential. The frequency behavior, there's some math here I'm skipping,

is some constant divided by, in essence, omega square plus some number proportional to gamma square. I am skipping the math details and some two pis and such not that are not central because the important part here

is that at long time, at high frequency, excuse me, this should go as omega to the minus two. Yes? Well, if you look at the high frequency behavior, and by the way, if you add a bunch of things that are decaying as omega to the minus two

and you add them up, they still decay as omega to the minus two. So these models say omega to the minus two at high frequency. However, if you look at the slopes, well, actually, you can't look at them. You have to do the numerical fit and you would have to procure a copy of my book,

Complete Tables, which I don't really urge you to do for this course. It is the supplement to the Cambridge book, it is the book I published, giving all of the numerical fitting parameters for all of these curves I'm showing you.

That is, for example, in seven dash 16, there is a power law and there's something that's close to an exponential and each of those has fitting parameters that are numbers. Well, I tabulated all of the numbers and I have published them, so if you're curious what the numbers are, you can look them up.

The important issue is that if you look at the slopes of these power laws, you see something that is like omega to the minus 1.3, some number like that. You don't see the correct slope and that tends to say that these,

the simple Rouse-Zimm type models are perhaps not quite doing what you want. Now, we have to be very careful on that. It's a very simple model. Often if you take a simple model and you tweak it very gently, so it's really the same model except you fix some minor details,

it now works perfectly. So the fact you don't quite get the right prediction doesn't mean the model is totally hopeless, it means there's an opportunity for theoretical work here. But there is this modest discrepancy that you should be aware of.

We will decompose things into modes. Now, what would happen if you go into concentrated solution? These are dilute solution models. What happens in a non-dilute solution? Because the chain, like you said before, the chains may be penetrated or intended

with each other. That's absolutely correct. And so the question is, okay, they've entangled, what is this going to do to the system? And we can give a general answer as to how systems that show mode behavior

react when you start perturbing them. And the general statement is, you may still have normal modes. The new normal modes will be linear combinations of the old modes. We're going to have to take a break for a second.

The new modes are linear combinations of the old modes, but at least at first, the old modes are pretty close to correct. Furthermore, you can always, since the modes are also orthogonal to each other

in a way, sine waves of different frequencies are orthogonal, you can always decompose the description of the molecular position into these modes. In low concentration, as time goes on, if you've excited only one mode,

it stays the only mode that is excited as things decay. Just as if you take any stringed instrument and pluck the string, you get a mode excited, the string oscillates, and as time goes on, you mostly see just the same oscillation sitting there,

the same frequency. However, if you have something that has gotten polymers entangled, the modes interact with each other, and if you excite one mode initially, it will decay in other modes as time goes on. Furthermore, the relaxation rates all get changed.

The relaxation rates typically are changed before the mode description starts to get more interesting. We are not going to go into that one at all, but I point out that you can do this thing, and there are results.

Okay, so let us chug ahead. 7-17 comes back to one of these very clever synthetic polymers.

Polymer, it's a tri-block copolymer, and the central piece has a dielectric response, and the two ends are dielectrically inert. That is, so far as the measurement is concerned, these are invisible, and we are just looking

at the motion of the central piece. And if you look at the motion of the central piece, you see two relaxational modes. You see something that looks like, we're plotting in the figure, 7-17, we're plotting dielectric loss versus frequency,

and you see a piece like this, and then you see another piece adding on to it at long, at high frequencies. And the question is how you interpret this. And the answer is, I'm not giving my interpretation,

the authors say the same thing, this is Adachi et al again, that at high frequencies, the chain can rearrange itself, and now the central piece changes its end-to-end vector because the chain is twitching, it's changing its shape.

And so there is a central piece, there are things where the end-to-end vector moves a bit because the chain is changing shape. That's segmental motion. However, this piece is attached to the two ends, and the only way you can completely get this around

with high probability, is if the whole chain gets turned around. Now how does it turn around? Well, it could do cartwheels, or it could tunnel along until the first end is here,

the middle piece is here pointing the other way, and the end that was up here has now moved down here. I'm not telling you how it manages to rotate, flip its ends as it does,

and you can imagine very different physical motions that give the same net result. And when this has happened, when we've gone from here to here, or when we've done a rotation, this dipole can now be pointing in any direction. And therefore, the labeled center piece shows something of how the center relaxes,

and it shows something of how the whole chain relaxes. Okay, that's 7.17. Now we're going to advance, and I am pushing on to O.

Eventually, we get to a final discussion of what is happening in the system. And I guess the first issue is that we've been sort of summarizing, we've said in the chapter, what have we seen?

Well, first of all, we can measure the mean square end-to-end length. Mean square end-to-end length is not the most commonly used characterization

of the size of a polymer chain. Most people will talk about R sub g, the radius of gyration, which is physically a different number.

However, what the Rg tells you, here's a polymer coil, here is the center of mass, here is a vector s out from the center of mass to a monomer, and if you average all of these s's and compute the mean square average,

you get Rg square. So it gives you a sort of a mean square radius of the chain. The mean square radius and the end-to-end vector are quite different. However, it's fairly clear that if the polymer expands a lot,

both of these will grow. If the polymer contracts, both of these will shrink. They're both characterizations of the size of the polymer. They're just not the same one. Okay. And what we find for R square versus polymer concentration

is that it's flat for small chains. It's flat in theta solvents. But in good solvents, chains shrink as you increase the concentration.

There are people who propose that contraction is a power law at higher concentrations and zero at low concentrations. The zero at low concentration part does not seem to be well-founded. And something that says R square is proportional to e to the minus

some constant concentration to a power that is stretched exponential again, that seems to be a reasonable picture of what is happening. Okay. So that's chain size.

I compare, there are other methods of measuring chain size, and this chapter puts all of them in one place, even though some of them use very different methods, neutron scattering or light scattering. Okay. You can also say there is a characteristic relaxation time

for the dielectric relaxation. In terms of our curves that look like this, the characteristic relaxation time is something like, is inverse in the frequency. It's not simply one over omega.

There is a frequency range in which things roll over, and the inverse of that frequency is a time, and there's a characteristic time in here. I am not putting in constants. And we can ask, what happens to the characteristic time

as we increase the polymer concentration? The answer is it goes up. And in particular, it goes up as there's some constant A, there's C because it's sort of a pure exponential,

and then there is R square to the three halves. The radius is shrinking as we increase the polymer concentration. So at low concentrations, this is fairly close to a pure exponential or a straight line,

but at higher concentrations, this number, the radius gets smaller and smaller, and we get a rollover relative to the concentration. And if you put in the static measurement, time-independent measurement, of the mean square size of the end-to-end factor

into this behavior, you describe extremely accurately what this curvature is like. Okay, what else can you do? Well, if you will find, let's see, what figure do we want?

Oh, 7.19 is nice. So if you will find 7.19, and we're going to do decomposition of the data,

combine more and more data into a simpler and simpler form. The front end on this is, we plotted epsilon versus omega. I'll do the usual form I show it in.

Many people would plot epsilon double prime rather than epsilon double prime over omega, and you get something that is coming down. And buried in here is a characteristic time tau, and tau is the inverse of the frequency.

So we've taken this curve lots and lots of points, and we've boiled it down to one relaxation time. Now that's actually an oversimplification because the curve is more complicated than that, but it's a characteristic time, and for many purposes that's quite useful. And now we go in,

and we plot tau versus concentration, and those are the curves you see in 7-19. And what you see in 7-19 is a times increase with increasing polymer concentration. You are looking at a log-log plot.

On a log-log plot, there aren't really any straight lines there. Well, I should qualify that. If you stop with two points, or maybe three points, you can get lots of straight lines, but that sort of misses the point.

But if you just look at the entirety of the data, what you see is that tau goes as e to the some constant c, that is the relaxation time increases roughly exponentially with concentration.

So far so good. Oh, I believe I used gamma here. Did I use gamma? We'll get to that in a second. So what is happening is you have these relaxation time versus frequency. You have a huge number of measurements

of the dielectric response. You collapse that curve to a single mathematical point here, which is its characteristic time. You repeat measuring these curves for a whole bunch of concentrations. This is getting tedious, long, very hard work, very well done,

and you see how the relaxation time changes as you change the solution in which you are measuring it. Okay, and now you skip to figure seven dash 20 because in figure seven dash 19,

there were four of these curves. Each has, is an exponential with some exponential constant. And in seven dash 20, I take the next step. I plot gamma, the increase in these curves, versus M, the molecular weight of the polymer.

And seven dash 20 is again a log-log plot. There are not a lot of points left. You started here with a humongous number of dielectric response points, and you collapsed them to far fewer points,

one per sample. And you collapsed the one per sample to one per polymer molecular weight, and you're down now to four points, which I didn't quite draw too well, and a straight line which says gamma is proportional to M to the, and I actually give a number for it,

which if I recall is 0.46. However, a reasonable person would say, call it a half, you're probably not measuring it more accurately than that. And if you look at the points, they aren't perfectly on the line. In fact, you might suppose that if there were more points,

there are only four points left, it might not be a power law at all. But that's a reasonable approximation within the limits of the available measurements. And we can finally say tau is proportional to E to the A times the concentration.

And A is equal to some constant. And then it's equal, there's an exponential. And what does the exponential depend on? Yes.

an exponential and we have something of the form constant molecular weight to the one-half concentration to the first I should have I'm sorry I wrote that as a that's a mistake I was getting a step out of phase with

myself the characteristic time which I call a in the figure and I've called T elsewhere increases as an exponential and concentration and there is the molecular weight dependence and there's this extra K in front okay so that is

sort of what we find and you notice we've taken we have shown phenomenologically only that we can take a very large number of measurements and we can reduce them to very few points and finally we can reduce them to a single function and the single function tells us how the

relaxation time is determined by the polymer molecular weight in the concentration given the small number of points there you really wouldn't want to bet that this exponent is perfect the exponent this exponent on C C to the first this being a simple exponential increase actually appears to

be fairly accurate okay figure 7.21 which is the last figure in the chapter that we're going to talk about let's step back a second curves early in

the chapter the frequency dependence of the dielectric loss function and I said there was an early piece like this which we've talked about and then

there was power law decay and maybe there were some additive things down here this is omega to the minus X that's very nice how big is X you can find out how big X is by going to figure 7.21 where I have plotted X for

a substantial number of different polymers polymers on that are smaller or larger than each other and they're a bunch of different combinations of polymer and solvent there if you look at a single set of points of a single

style you'll discover they're much they're fairly smoothly behaved but if you look at all of the points at the same time well different sim different systems give you different values of X a bit but if I plot X versus

concentration what I find is things a series of things that were kind of like that and they start out at about Oh 1.4 and at first perhaps they increase a bit to 1.5 you notice there's a lot of spread in that measurement and they

then head down again and by the time we're down up to about as high as concentrations were available which is something like 100 grams per liter polymer we are back down to Oh 1.2 and so X has a concentration

dependence and if I give a series of these curves for different values of concentration the lines out here aren't quite parallel because X is weakly dependent on concentration that naturally brings us to the end of the

discussion of the chapter on dielectric relaxation pass this back if you will so we have discussed dielectric relaxation I have also now given all of you your next homework assignment which you should try to read carefully

it's basically another literature search and I've tried to be a bit more precise on what you're supposed to do I imagine we will discuss it oh it's not due for quite some time but we will discuss it next Wednesday and you should try to

have made some progress by that so what is the what are we going to talk about next chapter 8 and what is the title of chapter 8 self-diffusion what

is the experimental issue suppose we have some clever technical means of reaching in and taking one of the polymer chains and painting it bright

green and suppose we have a more powerful than is actually physically possible microscope we can watch how this polymer moves as time goes up and

for starters here is the polymer coil and it has a location and if we wait a while it's doing Brownian motion diffusion that there's thermals velocity of the particles in the system and just as you get diffusion of say a perfume and air so if I were to spray perfume in the front of the room

after a while you would all smell it that's actually a bit of a cheat diffusion is very slow the turbulence the convection of the air and the room is what moves the molecules around on a room scale there's no convection here

it's diffusion it's very slow but if you wait a while the polymer chain moves and you look in for an isolated chain in dilute solution well if you repeat the experiment lots of times sometimes it moves a large distance

sometimes it doesn't move much but if you repeat the experiment lots of times you can say for displacement along one of the coordinate axes that the mean square displacement increases linearly in time you can also say that the

probability of getting some displacement Delta X during T is proportional to E to the minus X square I think I'm about to drop a constant I'm going to

it's either a two or four and I've just dropped a constant mentally but the important issue is that the probability of seeing a displacement Delta X is a Gaussian in Delta X and the width of the Gaussian increases linearly in time at large times this number is very big and therefore for P to relax to

zero Delta X has to be very large at short times T is small this function falls off rapidly and therefore at short times the scales are not quite

correct the important issue is the policy object moves out it's always more likely not to have moved far than to have moved any larger distance the

normalization on these two curves isn't the same since the total probability for all displacements has to be one has to end up someplace after all but the polymer spread if you watch the polymer it moves suppose however you go

into a non dilute solution well the things I just wrote on the border incorrect in non dilute solution it's more complicated than that nonetheless it moves that's quote Galileo actually but he was talking about the planet

here is a polymer and if you wait a while it actually moves diffuses also if you suppose that at the initial time two polymers were wrapped around each other like a pair of earthworms if you wait a while the two polymers

move they're in different places and this I'm not going to say that the genuine banglements actually look like this but the topological interlacing

that looks like that after a while they go away because the chains move away from each other I'm not saying whether those that's significant or not I'm just saying they move and there are theoretical models in which going from here to here is physically extremely important and so it's interesting to ask

how fast a polymer chain diffuses it's not doing straight line motion it's doing a random walk how fast random walk imagine a drunkard on the street someone who's as drunk as possible and still able to walk he puts his feet

down and F for the first step or two he's kind of headed the same direction but after a while yes that is a drunkards walk the steps the distance of the step is sort of fixed there's no uniform pacing but somehow he gets

away from the bar and ends up someplace else okay so that is diffusion the question is how you measure it and okay how do we measure it well one thing we can imagine doing is say we will take polymer chains and we will

tag them with a fluorescent material okay now we will send in two laser beams at slightly different angles and we will get an interference pattern the result of two laser beams crossing at slightly different angles is there will

be places that are bright and places that are dim this is a really bright pair of laser beams and where the poor defenseless fluorescent molecule is in

the really bright area it gets fried and is no longer fluorescent and so I have destroyed the tags on these molecules and not the tags on those now I illuminate the sample again fluorescence is and I measure where

there is fluorescence and where there isn't and I discovered that at short times the fluorescent intensity versus position looks like this the actual

experimental method for doing this is a little more clever than looking in at a microscope I am omitting experimental details there are nice papers by Paul Russo for example he uses this technique I'll give the name of the

technique is the last comment however what happens as time goes on well they're fluorescently labeled molecules here there are none or very few here and because of Brownian motion molecules move back and forth now all

likelihood that a polymer molecule here will move this way and the polymer molecule there will move the other direction are exactly equal but there are lots of fluorescent molecules here very few left here that we didn't bleach and therefore the net result is that there is a flux of fluorescent

molecules in that direction and as time goes on the concentration gradient the concentration profile of the fluorescent molecules decays and eventually the system becomes uniform okay well that is actually a measurement of polymer

self-diffusion the technique I have just described is called fluorescence

relaxation after photo bleaching or frappe and frappe gives you the self

diffusion coefficient there are a bunch of other techniques which I will get to next time that give you the same measurement but this class is at an end okay that's it