I'm Professor Filleys, and this is course 597. What we're going to do today is to start discussing dielectric relaxation, which is actually a wonderfully useful experimental technique because it measures a substantial number of

rather different variables, all at the same time in the same measurement. The general notion is that, once again the chalk has gone for a walk, the general

notion is we have a polymer chain, and the polymer chain has a number of different features which we can actually measure using dielectric relaxation. So let us look at a few of the features, and then I'll describe how dielectric relaxation does it, and we'll finally get to results.

The first issue, we have this polymer chain here, and one thing you might ask is, well, how big is it? There are several different ways of saying how big it is. One of them, which is usually written little r, is the end-to-end vector, and the notion

of the end-to-end vector is simply, it starts at one end of the polymer chain, and points at the other end of the polymer chain. So we actually have a vector that tells us the distance between the two ends.

In a certain sense, this tells you what is the orientation of the polymer chain, which way does the chain point. Now of course, if you imagine little vectors running along the polymer chain, at different places, the vectors point in different directions, but what the end-to-end vector does is tell

you the total of all of those little pointing directions. You can also imagine measuring the mean square length of the end-to-end vector, and that gives you a size figure.

It doesn't give you the only size figure, because there is an alternative size figure, the radius gyration, r sub g.

To compute r sub g, what we say is, here is a polymer chain, and someplace in there it has a center of mass. It has a point where it's uniformly balanced in each direction. This is the average of the position of all of the little beads of the polymer chain.

Then we construct vectors out from the center of mass to each piece of the chain, and we calculate for these little vectors, and I will call the little vectors r so you don't

confuse it with that r, we calculate the average value of r square, and that gives us the square of the radius of gyration. You should realize that the radius of gyration is not like the radius of a sphere.

You have a sphere, and I ask you what its radius is, there is a sphere. Any sane person will say that is the radius of the sphere. It's the distance of the outer edge. First of all, for a polymer, it's a coil, it's flexible, there's a lot of empty

space, it doesn't have a sharp outer surface. So what is done is to calculate the mean square distance, the average of the square of the distance from this to each of the beads, and you get some number that's sort of like, that is, it's a distance that is much of the way out, but it's not all

the way out. And since you may have some little bit of the chain, it managed to stick way, way out. If you tried to draw a bounding surface around a polymer, it might be fairly large.

So there are two estimates of how big the chain is. Now, the chain, these chain sizes have two features. One is, both of these depend on the polymer molecular weight.

The more of these, the longer the random coil is, or more or less random coil is, the bigger the chain is going to be. And there is an entire field involved in studying how big the chain is relative to its molecular weight. We get to that a bit later in the chapter.

I don't do it in vast detail, but we get to it a bit. The second issue though, is that both of these are also functions of the polymer concentration. You might ask, why are they functions of the polymer concentration at all?

And the answer approximately is, here's a polymer coil. And if you look at all of the random paths that the polymer coil could describe in space, well there are a lot of them, but some of them are discouraged because the two polymer

pieces of polymer chain get close to each other. And you can imagine, in fact, sketch hypothetical paths that do not exist, in which the coil would have to intersect itself and go on through. That's of course impossible. So there are paths that are discouraged, and those paths, since they can't happen, have

the effect, or rather their absence has the effect of expanding the polymer coil. But suppose you put another polymer coil out here. Each polymer coil bumps into its neighbors, and it behaves like, so to speak, a polite

person at a cocktail party. It tends to pull in a bit. Furthermore, at high concentrations, the whole solution is filled with uniform polymer everywhere, and the tendency of a polymer to repel itself, that is to not to have

the states in which the polymer occupies the same place twice. This tends to be canceled by the other chains, and the net result is the chain shrinks.

So if we plot the size of the polymer versus polymer concentration, we find that there's some downhill trend. I am not yet telling you what the trend is.

In fact, there was considerable theoretical work done on this 20 or 30 years ago, and what people said at the time was that at low concentrations, nothing much would happen. This is theory.

And then at elevated concentrations, the radius of gyration would shrink, and RG square would be proportional to C to the minus one quarter. And I should emphasize that as a particular theory, and it was not based on experiment.

But it is a particular theory. It really does exist, and there was considerable work done to test it. There was also considerable work that had also been done already that did test it, and people didn't realize that there were other sorts of measurements out there that

would answer the question. If you want to ask, well, how do you measure the size of a polymer chain, there are three sorts of answers, and one is neutron scattering, which requires a big atomic reactor and some extremely expensive instrumentation.

Another answer is light scattering, which is a whole lot easier and cheaper to do. And the third answer, which gives you a slightly different number, because it measures a slightly different variable, is dielectric relaxation.

However, I am starting out by commenting and simply saying that there is this interesting question, how big a chain is. But there are other things you can ask about polymer chain, an intact chain, and one thing

you can ask is, if I have a polymer, here's a polymer chain, it has an end-to-end factor. And the question we can ask is, what happens to that end-to-end factor as time goes on?

Well, the polymer is sitting in solution. It's floppy like an overboiled noodle. And therefore, as time goes on, the ends of the chain move with respect to each other, and there are two sorts of things you can imagine happening. One is that you can imagine that as time goes on, the end-to-end vector gets longer

or shorter. And it's certainly allowed to do that. The second thing that can happen, though, is that the end-to-end vector can change the direction in which it's pointing.

Because one end of the polymer has rotated with respect to the other. Now when I say rotation, you have to be a little careful in your thinking. If I tell you this is an eraser and it's rotating, it's rotating as a rigid body. And all of the parts of the eraser stay in fixed positions with respect to each other.

Yes? They all stay, that is, rigid body, if you think back to freshman physics, it's a rigid body, and you can write the rotational velocity of the parts as, if you think way,

way back in many years, as v is r cross omega. That may look familiar from freshman physics, but it's a long time ago for some of you. Okay, so that's rigid body rotation. A polymer is not a rigid body.

However, there is a rotational motion which you can actually measure. And so as time goes on, if we had, this is supposed to be the initial end-to-end vector at time zero, and we take the dot product of that with the end-to-end vector

at time t. As time goes on, the vector forgets which way it was pointing, its length fluctuates, and this quantity on the average, if you look at lots of polymer chains, is going to change.

In particular, while it could be the vector gets longer and it could be the vector gets shorter, there is a nice mathematical result, which we will skip over, which guarantees that this object, once we average it, decays, and at very long times, the vector was originally

like this, at long times it could be pointing in any direction with equal probability, and this function decays out to zero. Okay.

Well, it doesn't have to be, I mean, if it's pointed like this at time one, and like that at time two, this number becomes negative, dot product of two anti-parallel vectors. But it could be the dot product of two perpendicular vectors, and there will be some likelihood that it will come back and is parallel with each other.

The main thing is, all of the possible directions and lengths at long times are equally likely, and therefore when you average this, it averages to zero. Yes? Okay.

So, gee, what could we say about this? Well, one thing we could do is to say this thing vaguely resembles an exponential, it doesn't have to be exactly an exponential. If it were an exponential, we could define a typical time on which this function relaxes.

We've plotted this function versus time, and there's some typical time on which the thing relaxes. Now, if it relaxes simply as an exponential, it's very easy to say what the sensible

definition of the relaxation time is. If the relaxation function is more complicated, you have alternatives, it isn't quite critical which alternative you follow. So far so good? The third thing you could do is to say, well this is a function of time, and therefore

since it's a function of time, I can characterize the shape of the function. I can actually measure it. Now we are going to take a math step, and this is a math step that comes up if you

have ever looked at rheology or response, that is if I say I have something that is a function of time, I can equally say I will do the measurements at a series of frequencies, and I will have something that is a function of frequency. And so for example, I will take the viscoelastic mechanical case, I have a block, and I compress

the block, and this is some soft object, not a block of aluminum.

If I compress it, the first thing that happens is that I get a force pushing back. However if I sit, the force disappears, it fades, because the object is soft and

gradually changes shape. And so if I compress the object and I measure the force of the object pushing back as a function of time, I get some sort of a relaxation. However I could also say, what I will do is I will take the object, it's between

two plates, and I will take the upper plate and I will shake the upper plate up and down, like this, yes? Well if I do this at a very, very low frequency, the object always changes its shape as I

move, and there's basically no, there may be frictional resistance or not. But if I change things very slowly, this pushback has decayed. On the other hand, if I do the same motion over the same distance at high frequency,

there will be a response force. And the response force will have two parts. There will be an in phase part, and there will be an out of phase, yes?

So far so good? And because there is an in phase part and an out of phase part, the in phase part basically behaves, the part, the object behaves like a spring. Like the senesal waves, right?

Like the senesal waves. Yes. The out of phase part, the 90 degree phase part, out of phase part, says that there's frictional resistance and therefore there is loss. And for the dielectric response, these are called epsilon prime and epsilon double prime.

And this is the loss part, and this is the dynamic part, and I could go into more detail. But the important issue is that instead of looking at something in time domain, I could

look in frequency domain. Now you might ask, is it useful to do both? And if you have perfect instruments, the answer is no. All of the information that you have in time domain you also have in frequency domain and vice versa.

Furthermore, called Cramer's Chronic Relation, and Cramer's Chronic Relation tells us

that these two parts, I said there is an in phase response and there is an out of phase response of the system, these two are not independent. If you know one at all frequencies, you know the other.

The reason this is true, except it's a little hard to see why it's true, is that if I say at time zero I squunched the block down, the force looks like this and at negative times there is no response. The statement that the time response has a functional form at positive times and is

zero at negative times constrains what these have to look like. I have not really told you how, but it does. Okay, so the main point I wanted to make was that you can have a response which you

can actually measure. Good so far? Now we come to dielectric measurements. And the first question is, what is going on?

Well, imagine we have a pair of parallel plates, yes, and we have put a charge on the two plates, yes, and we ask what voltage we had to put on the two plates to get

the charge. There are several ways of asking the same question. You could also apply a voltage and ask what current flowed. If you think way back, this is roughly a freshman physics question.

However, suppose I take the two plates and put molecules in the space between the plates as opposed to vacuum. What happens? Well, it may be the case that the molecules somehow have parts with charge on them.

We'll get into detail what I mean by parts in a bit. And as a result, the charges are oriented as indicated, yes? Well, they're oriented as indicated and inside in the middle we have minus, plus,

minus, plus, and nothing has happened. This is electrically neutral because they're minus and plus groups right next to each other. It's electrically neutral on a reasonable distance scale. However, at the two surfaces, you run out of molecules and there will be a line of molecules here with their minus groups up against the surface.

And there will be a line of molecules here with their plus groups against the surface. And in order to get the voltage difference to be the same, yes? Well, the minuses and pluses are canceling and in order to get the same voltage

between the two plates, I have to put a larger charge on the two plates. Have you actually seen this? Yes. If you look inside the integrated circuit or a circuit board, you see things called capacitors. And the point of a capacitor is that it's something that stores charge.

The fact that you have a material that polarizes like this between the two plates means that the capacitor can store much more charge than used to be the case if that material were not there.

I have a question. Did that use an electric field to generate a dipole in the molecules? We're going to come to that in a moment. The question is, where do the dipoles come from? And there are at least three major sources and I'll talk about that in a few seconds.

Okay, now the question is, is this a large effect? Yes. Suppose you have a hypothetical 1 farad capacitor. When I was an undergraduate, a 1 farad capacitor would have filled a significant part of this room, like the whole rear wall.

Thanks to modern materials, it polarized much better and 1 farad capacitor is now something that you can actually pick up and walk off with. It's a huge capacitance. The typical capacitors you see in electronic circuits are microfarads or picofarads in size.

They're really tiny, but if you wanted a nominal 1 unit capacitor, thanks to materials, you can now get them at reasonable sizes. Okay, so how do you get those dipoles?

Okay, how do you get the dipoles? There are three sorts of things that can happen. The first issue, which happens in almost any material,

is that you have material and there are atoms and the atom has a nucleus which is positive charge and an electron which is negatively charged and it may also be the case that you have covalent bonding and the electrons overlap with neighboring atoms

and if you apply an electric field, you move the electrons in one direction to some very tiny extent and you move the nuclei in the other direction to some very tiny extent and you have done something with the electronic states

and the position of the electrons inside the matter and you polarize the matter. And this, if you have a solid, what I have just shown you is the most common sort of polarization effect. This polarization effect does not change very much

until you get up to very, very high frequencies because if you say, imagine you apply an oscillating electric field, the electrons can very easily move back and forth with respect to the nuclei and that continues to be true until you start to approach the infrared and optical frequencies

in which the life does change a bit. However, the important issue is, one thing you can do is to move the electrons around. We're going to talk, however, about liquids and in liquids there are two other effects.

Here is a biopolymer and it's a typical biopolymer like a protein and biopolymers have the feature that they fasten charged groups to their surfaces. For example, we have here carboxylic acid

and it may be like this with a bound hydrogen. This is an organic acid or it may be like that and that's what it looks like if it is let go of its proton.

A hydrogen ion is a proton, there's nothing else there and the proton has floated off into water. Now suppose I apply an electrical field. Well one of the things, these protons let go and go off into water

and are bound again with great frequency and one of the things that can happen is the proton here could wander over there and now this is neutral because there's a hydrogen atom attached and this is negatively charged because it lost an H+, a hydrogen ion, a proton

and we suddenly have the case that we can move the dipole moment of the molecule, the charges on its surface around by moving protons around, hydrogen ions. So the hydrogen ions can move and migrate.

Furthermore, if this is a typical biopolymer, it has surface charge, it's either positive or negative and the surface charges like this one attract ions of the countervailing charge out of solution.

Now which ion it attracts depends on what's available in the solution. The important issue is that if you have a molecule and you attract counter-ions, there's a cloud of charges surrounding each molecule

and this charge leads to something called Debye screening. The essence of Debye screening is here's something in the solution that's negatively charged and it's got lots of negative charges on it.

It tends to attract positive ions to the surface and if I sit out here and ask how big is the electric field, I can do a Gauss's law argument and I ask how much charge is in there and the answer is that the negative charges are partly cancelled by the positive charges

and out here this object looks as though it's much more weakly charged. Well if I apply an electric field, I tend to hold positive charges one way and the negative charge the other way with respect to its positive charges

and the net result is I have created a dipole. You see positive charges here, negative charges there. That is I get a dipole moment out of this thing because I have stretched the ion cloud. Now if there's an ion cloud, there's something else unfortunate going on at the same time,

namely these things are free to move. There's an induced electrical current and heating and this creates all sorts of interesting experimental issues which we aren't going to get into. The major issue though is I have for biopolymers and polyelectrolyte polymers,

whether they're biological or not, in solution there are two additional mechanisms for getting a dipole moment. One is that you move around the bound protons if we're in water solution and the other is you distort the ion cloud. Okay, now this process will not continue to as high frequencies

as the processes involved in the electronic polarization. Why not? Because if I flip the electronic field back and forth, these are fairly large objects. They're not tiny like electrons

and they take a while to move back and forth. And if I flip the electric field fast enough, they sit here and they don't have time to move. So they don't, so at high enough frequencies this effect goes away.

Okay, now we'll go to the last sort of dielectric moment issue and we're now going to the one that matters for studying polymers. Suppose I have a molecule, and I'm going to phrase this in terms of organic molecules,

but it doesn't have to be that way. If I have an organic molecule, for example, I won't tell you what R is. It's some organic piece. That repeats.

This is what is called a polyester. That's an ester group. Polyesters mean you have a whole bunch of these linked together. The way you synthesize them, most typically you would form this bond to synthesize them.

And if you drop it into water and heat it up, you can break the ester bonds and you will break it there. Okay, so that's a polyester. And the important issue though is, this is oxygen, this is carbon, this is hydrogen, this is whatever. And each atom to some extent is electropositive or electronegative

and tries to attract electrons from the neighboring atoms. Now it doesn't attract them completely. It's not like sodium chloride. The chlorine atom strips an electron off the sodium.

You have a positive charge there and you have a negative charge there. Okay? Well, however, it does to an extent. And therefore, the electron, the oxygen tends to be negatively charged with respect to the carbon.

That's also true of this oxygen, though not as much. The hydrogen and the carbon are much less significant than the oxygen. But there is some tendency for each molecule and each bond to have a dipole moment.

The dipole moments are fixed. They're fixed by quantum mechanics. If I apply an electric field of any reasonable size, the electrical field has very, very little effect on these dipoles.

It may create its own dipole moment, but it doesn't change very much how big this dipole moment is. Nonetheless, in a solution you can create polarization. How do you create polarization? You take the molecule, this is a little hard to do on a blackboard,

and you rotate the molecule so it is now facing like this. And if you physically rotate the molecule one way or the other,

I may have rotated it the wrong way, you can actually align the dipole moments and you polarize the liquid by aligning the dipoles. Is this a large effect? Well, yes. Consider water.

Water at low frequency where you can align the dipoles has a dielectric constant around 80. On the other hand, if you ask what is the... Unit. The unit of the...

I'm now trying to remember which units I just quoted things in. And I'm now pulling a blank and it will undoubtedly come to me as soon as class is over. However, if you repeat the same experiment in the visible, and ask what is the index of refraction of water, it's about 1.3.

This will come to mind in a moment. Don't worry about it. In any event, the main issue is at low frequencies you can line up those water dipoles. At high frequencies you can't. And if you ask to what extent, if you measure the dielectric constant,

whose units of course depend on which system of units you're using, if you measure the dielectric constant at low frequencies you get one answer, and as the frequency goes up, the dielectric constant falls towards what you find in the optical, which is quite modest.

Okay. So having said that, having said we can line up molecules, molecules have dipole moments, we now go back to Stockmeyer, and you remember I discussed three sorts of motion.

Now why are those sorts of motion interesting? Well suppose I have some organic molecule whose structure I am not giving in detail,

and it has a side group, a pendant group, and the pendant group is even free to rotate. Yes? Well, if this object is perfectly symmetric, if it has mirror symmetry, it will not have a dipole moment.

But there are many organic groups that do not have mirror symmetry around every axis, and therefore there are three sorts of dipole moments we can imagine. First of all, this pendant group to some extent has a dipole moment, which I'll call C, which is perpendicular to this axis.

This doesn't mean the whole dipole moment points this way, it means the dipole moment has a component this way. It may also have a component that way. This component relaxes by reorientation around this bond,

which is a very fast process. So this process occurs at very high frequencies. Then this whole pendant group may have a piece of its dipole moment facing that way.

And that dipole moment can only be relaxed, where it's not the same as being in a liquid, by rotation like this. You may say, can't it rotate like that? The answer is no, it can't rotate like that. It's attached to the polymer backbone here. The pendant group can rotate around this bond, around this axis only, like this.

This whole thing can only rotate that way. And finally, if I look at a monomer, the monomer may have a net dipole moment along the bond, along the chain backbone.

In order for there to be a net moment along the chain backbone, there is a symmetry requirement. I will start with polyethylene. There we go. As simple as you can get, yes.

Okay, and which way does the dipole moment point? Well the answer is, this and this are the same,

so there's no reason to prefer one direction to the other, yes. And therefore there can be no net dipole moment. Suppose I go in though, and I replace these two side groups with R and R prime.

Well now this direction and that direction are obviously different, yes. And this bond can have a dipole moment that way. Unfortunately there's a little problem.

This is the mirror image of that. And so this and that are going to cancel along the backbone. They may have a component perpendicular to the backbone. Remember these are carbon atoms, so they're tetrahedra. They aren't squished flat in the plane. How do we beat that?

There is a bond angle between the carbon atoms, so what I mean is it's now to 180 degrees, so that. Let me draw a picture of that. I am now drawing it. So there is a vector like this and a vector like that.

The component along the backbone sums to zero. There will however be a component perpendicular to the backbone,

and that contributes to the type B relaxation. Yes, see that. So having said that, the bond angle does matter, but it contributes to,

that's why there's a contribution to type B, not to the type A. So how do I get something that's non-zero? And the answer is I need some polymer whose structural sequence looks like that.

And a nice example is, here's a polyester. The repeat is here. So there's some dipole here, which has some size, and a dipole there, and a dipole there, and this is the same as that.

And if you add them up, their sum along the backbone does not have to be zero.

Okay, so if you select the right polymer, and unfortunately the world list of right polymers is not as large as you might like, it is the case that if you select the correct polymer,

the dipole moment along the backbone has some value. There are cases where it would be interesting that the dipole moment is close to zero. For example, DNA, double-stranded DNA, the dipole moment is unsurprisingly zero because the two add to zero.

Well, so what can we do with this? Here's a polymer chain. And each group of monomers gives us a dipole moment that points along the chain.

And if you add up, so here's D1 and D2, these are the dipoles of the little pieces, D3. And if you add up the dipoles of all of the little pieces, and they're all vectors,

and they all lie parallel to the chain, when you add them all up, you get a vector that has to be parallel to the end-to-end vector. It does not have to be equal to the end-to-end vector. For starters, the units are different because this is a distance times a charge.

It's a dipole moment. This is just a distance. But the sum of all the little dipole moments has to add to a vector that is parallel to the end-to-end vector. Furthermore, unless the molecule has some floppy bits so that, for example,

part of the molecule has a dipole moment, then there's a little connector that has no dipole moment. Then there's a repeat unit with a dipole moment and another little floppy bit. So long as the whole molecule has dipole moments, it must be the case that this length

is rigidly proportional to the sum of the dipole vectors.

That is, if I move the end-to-end vector, move the two ends of the polymer apart, I have changed the end-to-end vector, I've stretched it, and at the same time, I must have lined up these dipoles so the molecular dipole moment is larger.

So, what can we do with this? Well, first, we can take our polymer solution, put it in the dielectric apparatus, and we can ask, how much does it contribute to the dielectric increment

of the material between the capacitor? What happens to the capacitance if I just measure the capacitance of the capacitor at low frequency? And the answer is, here are two plates. The capacitance tells me how much charge I can store on the plates at a given voltage.

And the larger the dipoles, if I go from small dipoles between the plates to really big dipoles between the plates, as I increase the dipole moment of the molecules between the plates and the extent to which the molecules are lined up, guess what?

I increase the capacitance of the capacitor, yes? And there is a rigid mathematical relationship such that I look at the dielectric increment, the change in the dielectric constant of the material between the plates

as I add the polymer, and that is linearly proportional to this quantity squared, which is linearly proportional to the mean square end-to-end vector. Why do I have to say mean? Well, you know, one molecule has one configuration,

another molecule has another configuration, a third molecule is way stretched out. So at a given time, each molecule has a different dipole moment, and the molecules all move, so the dipole moments change. But on the average, which is all we see at low frequency,

they're contributing to the capacitance of the capacitor, and corresponding to that, they're contributing a quantity that's proportional to the average mean square end-to-end vector. So that is the very first thing we can do, we can see with dielectric constant measurements.

The second thing we can do though, is we can apply an electric field between the plates, an oscillating field, and if we apply an oscillating field, here is a molecule that is a type that has an end-to-end vector.

If I flip the electric field, in order for the molecule to contribute to the dielectric constant, it has to get itself turned around, and that takes a while.

And if you ask how long does it take, well, there's a force on the molecule, a torque, a force that induces rotation of torque. There's also a frictional force, because this thing is trying to move through the solution, and the ratio of the driving force to the frictional force tells you how fast things move.

That's sort of freshman physics. In freshman physics though, we usually talk about a person jumping out of an airplane and opening a parachute. And there is a force of gravity down, and there is a drag force up,

and the drag force up is proportional to how fast the person is falling. That's true. If you actually are doing this with real parachute dynamics, it's squared.

In freshman physics, we usually talk about very low speed motion, which you do not get with conventional parachutes, and it's only linear. However, whether it is linear or squared, there is a force up that is determined by the velocity, a force down that is gravity, and the force all put in your squares.

We'll do a real parachute. And there is a force of gravity. This is the forces, and this is equal to m times the acceleration. However, if you jump from any reasonable height, you hit terminal velocity. Your velocity is a constant, and therefore your acceleration is zero.

And therefore, F V square equals m g, and V equals the square root of m g over F. Actually, I'm not sure I knew that one about parachutes. It's not something I'm practically interested in.

I prefer to stay on the ground, thank you. Having said that, the important issue is you apply a driving force, and there is a relaxation rate, which is determined by the ratio of your driving force to a resistance.

And the larger the resistance, the slower the motion is, and the longer it takes for things to get around. And if I apply a driving force of fixed size, and I change the frequency,

well, if I am at very low frequency, even a very slow motion lets me rotate the dipole. If I am at sufficiently high frequency, these things notice they're supposed to start switching places, but they have hardly moved at all before the electric field switches sign.

So far, so good? Another question is that the solution also has some time, a relaxation time, so does that affect the polymer's relaxation time?

Well, let's see. The solution will have several relaxation times. Some of them correspond to motions of the polymer. We are measuring those times directly, so it's the same time. There may also be times that are not related to what the polymer is doing,

that are related to, say, what the solvent is doing, and the things that the solvent is doing occur at much higher frequencies. And therefore, those things happen very quickly and are over with, and you don't really see them. Okay, so we actually can do these experiments,

and we can actually measure the size of the polymer as we change its concentration. And that leads us to figure 7.1, which is a few pages into the chapter.

And what figure 7.1 shows you, first of all, is the size of the polymer, the mean square length of the end-to-end vector, as you change the concentration of polymers around it.

And so what you find is that there's a mean square size, or a contribution to the dielectric constant. That's the same thing, effectively, for polymers, given what are giving us the dipole moment.

And it's a function of concentration. We actually measure directly what is called dielectric increment, the contribution of the polymers to the dielectric constant of a liquid. But with some work, you can show that dielectric increment is linear

in the mean square size of the end-to-end vector. And what you see in figure 7.1a is that the polymer contracts as you increase the concentration. Now the contraction is not incredibly big.

We get up to 600 grams per liter, or not quite 600. That's 50% polymer, 50% solvent. That is extremely concentrated. That's not dilute solution at all. And in doing so, we get the size of the polymer

from 20 in the described units down to like 8. So we've made this very dramatic change in the polymer concentration. And the polymer size has only gone down a bit. You notice, however, if you look at the graph, the drop is steepest at small polymer concentration.

And as you get out to larger concentrations, the contraction effect slows down. Okay, well we can reasonably ask, say, that's very nice. We see that the thing is contracting. How is it contracting?

What can we say about the form of the contraction? And the answer to that smooth curve that goes through the points tells you that the mean square radius, which is some function of concentration, is equal to mean square radius at zero.

And the functional form e to the minus a constant concentration to a power. And this functional form is known as a stretched exponential. We talked about them earlier in the course.

And therefore, we have this shrinking process, which carries on all the way out to the melt, more or less. And over the observed range, out to about 600, we have a smooth curve and we have data points lying on top of the smooth curve.

Now there's something else we can measure. We can also measure, this is the rest of figure 7.1,

we can measure a relaxation time. And the relaxation time tells us how long it takes for a polymer molecule, arranged like me, to rotate through some typical angle.

We've applied, we switched the direction of the electric field, we're doing this cosine wave, and the molecules respond by rotating. Yes? So far so good? And the more resistant the solution is to motion,

the longer the time is going to be. And what we observe for time versus concentration, same system, same, actually it's the same experimental apparatus at the same time. You're measuring both of these, you can measure both of these simultaneously,

you just do a series of, you have to do a series of concentrations, is that the time increases sort of like that. And there is an initial slope, which can be measured, and there is the behavior at larger times.

And when we combine these two, we get a concentration result. And the concentration result, this result, can again be written in the form, time is sum tau zero, e to the alpha, c to the nu.

That is, you can say the time increases, and you see the smooth curves that you see in 7.1b, and the data lies on a smooth curve. You ask, gee, what is the form of that smooth curve?

And the answer is, the form of the smooth curve is, the one that I've drawn is the time is sum tau zero, and then we have the exponential, and there's a new constant a, new constant a because I'm factoring something out.

There is the size of the polymer at the concentration c, divided by the size of the polymer at concentration zero. This is a quantity that has dimension one. It's the ratio of the radii at the two concentrations, and this is raised to the three-halves power.

And then there is a c, and the c is now raised to the first power. That is, there's a concentration dependence for the relaxation time, and it's determined by the size of the whole chain, the end-to-end factors.

of the whole chain. This is approximately like end-to-end vector length cube. Now there's a low concentration slope here which is effectively a because of low

concentration you could write this as tau equals tau zero one plus a C but there's a deviation that's why this thing rolls over rather than going off straight there is a deviation of the curve from linearity and the deviation

is exactly determined by the size of the intact polymer chain. Now if you think of the polymer chain as having to rotate the way it would in water the statement that the time goes up as our cube is not very surprising because if

you have a sphere rotating in water here's a sphere I will rotate it in water and the drag force for rotation goes as the cube of the radius the size of the sphere. Yeah but what I mean is the resistance of the surface should be

the four power four now three because it's the surface area that the area of surface is a radius of power four. Three. Area, well no the area is radius squared. Yeah. Yes however the answer is the torque the resistance to force

goes as the cube and I'm trying to think of a fast hand-waving explanation the answer is yes there's this area that is going up as R square but if I

am trying to pull something a distance or out at a certain angular speed the torque it generates grows because I've moved it further out from the center and so you end up with R cube and G so there's something that depends on the

cube of the radius but however these things shrink if you increase the polymer concentration they start out big and as you increase the polymer concentration they're getting smaller and therefore their ability the their ability to contribute to rotation will resistance is going down because their

radius is going down we are able to do this measurement because well we can measure not only the dielectric increment which gives us the mean

square size we can also measure my mind we can measure the relaxation time can't we we can measure it's gone in frequency domain how long it takes for a molecule to forget which way it's end-to-end vector was initially

pointing we have a characteristic timescale and we can measure it now figure 7.1 which is from results of a dachi at all is a 100 kilodalt and cis poly isoprene and benzene so that is actually only a single system you're

looking at we will talk about cis poly isoprene a great deal because cis poly isoprene has a key feature namely it is a polymer whose structure can be written as ABC ABC ABC so it has a dipole moment pointed along its

bond at its bond trail if if the molecule were just symmetric like polystyrene there would be no polystyrene at the monomer scale is a B a B a B it's symmetric there'd be no dipole moment along the length

and if you did these measurements you wouldn't see anything cis poly isoprene has the nice feature that it does have a dipole moment the points along the backbone and therefore you can see the backbone rotating and which is what we

are seeing here I shall put in the minor safety caution benzene is an extremely nasty carcinogen if you ever feel a temptation to use it you want to be quite respectful of it well that's sensible okay let us push ahead

to figure 7.2 okay and in figure 7.2 we talk about the same polymer in two solvents and so we are talking about the same polymer and the polymer is

cis poly isoprene and the solvents are benzene which is a good solvent and

dioxane which is not now why do we care about solvent quality well the notion of solvent quality is that if I this is very qualitative I have a chain and if the polymer is in a good solvent each monomer would rather

interact with the solvent than with another monomer and so there is preferentially a tendency for monomers to move apart from each other so they all have solvent next to them however as we head off from the

good solvent or it's a poor solvent we aren't going to get there yet there we first get to the center and there's what is called the fatal point theta solvent is a little misleading because most solvents really are only theta at a

specific temperature and at higher or lower temperatures they are better or worse than theta but there we come to a temperature where the polymer does not care whether it is interacting with a polymer next to it or another monomer and this effect of the poly of the polymer subunits not being able

to have two in the same place at the same time cancels that result is the chain shrinks and if we go to poor solvents it shrinks considerably more however there is a consequence of being in a theta solvent rather than a

good solvent if I run up the concentration of polymer in a good solvent each polymer chain sees its neighbors and since the monomers want to stay apart from each other I am oversimplifying vastly the polymer chain in a good solvent contracts in a theta solvent the polymer monomers

don't care whether they're touching another monomer or a solvent so what happens when I run up the concentration anyone want to guess it's

a constant correct it doesn't care and therefore in a theta solvent the size of the polymer does not change as you change the concentration that is oversimplification the reason it's an oversimplification is you eventually

get to the point where you could have three polymer beads on three chains close to each other at the same time the effect of the solvent the theta solvent is to cancel the interaction of pairs of monomers but

when you get to the point there are triples of monomers you no longer have necessarily perfect cancellation and therefore in concentrated I mean really concentrated polymer solutions things can change okay let's go back to that

expression for the time the time expression I wrote down was that the relaxation time is some constant a times radius of concentration C over radius at zero and it really is very close to the three-halves power concentration

however if it is the case that the polymer does not contract as you increase the concentration yes this is simple exponential if I plot it on a

semi log scale as a scene for example in figure 7.2 B that's the right-hand figure you know on a semi log scale if I plot this and this isn't doing anything I will see an exponential on a semi log scale I will see a straight line with the good solvent because the chain is

contracting if I plot how versus C instead of seeing a straight line I'll see something that rolls over well that's exactly what happens if you look in figure B the two upper sets of points up to fairly high concentrations I

said it breaks down eventually the measurements show that you get either pure exponential or a stretched exponential until you get up to healthy concentrations there's not there's another way to break this though I've

been talking about effect of good and theta solvents on polymer size and that's true for large chains for very short chains there is not much effect on the size of changing solvent quality because the polymer doesn't

often bend around to run into itself and so for very short chains you would expect this object to be independent of concentration and if you look at the two bottom curves in 7.2 B you see two straight lines okay everyone see that

well that's saying with the solvent quality effect is about what you expect all right what we have done today is to discuss dielectric constant measurements I have said a little bit but not very much about how they work

I have made the point that because we can do dielectric constant measurements and dielectric response measurements we can measure the mean square size of a polymer chain and we can also measure in addition to the mean square size we

can measure the typical relaxation time these two quantities are related there are a vast number of researchers who have worked on this adachi what Nabi if you go back far enough you find the papers by Stockmeyer and

Bauer yes and I have not given all of those tried to put those into my memory and drop them on to the blackboard but all of the citations in detail are in the text okay we are out of time classes dismissed