I'm Professor Filley's, and today we're going to finish off our discussion of viscosity and push on to discuss viscoelasticity. Let me recall where we left off, roughly speaking.

We left off in chapter 12, discussing figures 12.22 to 12.24, which are treatments of the molecular weight dependence of the viscosity of the solution.

Now, there are actually a fairly considerable number of measurements of these, and they're actually fairly consistent. Namely, what is found with considerable consistency in many systems is that the viscosity depends on the molecular weight

as a stretched exponential and molecular weight. And that's brought out quite emphatically by, for example, figure 12.24. 12.24 are measurements that actually appear in Doy and Edwards, who report them as measurements done by Onogi et al.

The measurements are a little more complete than the measurements in the original paper, actually. Having said that, an interesting feature here is that if you go back into the literature and ask what people said about these measurements, you can find people who say, we have measured the molecular weight dependence of the viscosity,

and as predicted, we see a power law dependence of the molecular weight. There actually are measurements that find power law molecular weight dependencies. For example, as we discussed earlier, tau et al.'s measurements, which push out to the melt,

find close to the melt a power law dependence of viscosity on molecular weight, and volume fraction one, log log plot,

what are rather clearly straight lines, viscosity changing as a power law on M. And if you actually say, we'll stay fairly close to the melt, and we will exclude points that seem to be deviating from the original straight line, you actually could find in tau et al. M to the three behavior.

However, most of the measurements you find, which are done at considerably lower concentrations, do not find power law behavior, they find stretched exponential behavior.

An interesting example of this is provided, and this is also in essence stretching over to the next section, which is a discussion of topology, are the measurements by Gracely et al.

And Gracely look at polymers, and they look at linear chains, and they look at four arm stars, and they look at six arm stars. And if you look in at the results they show, as a function of concentration for the linear chain,

and their linear chain is a molecular weight of 1.66 mega dolton, as a function of concentration, a log log plot,

they find what is very clearly a straight line, you look at it, it's clearly a straight line, and it's quite clear that you are seeing power law behavior. On the other, this is for the two arm star, the linear chain, two arm star.

On the other hand, for F equals four, and for F equals six, very clearly on the same plots, these are chains of about the same molecular weight,

this is about 1.95 mega dolton polymer, and the other is about a, what was it, it was about 1.44 mega dolton polymer. For the other two, you very clearly see stretched exponential molecular weight dependencies,

concentration dependencies, I mean. These are fixed molecular weight, you're looking at concentration dependencies. If you look at the molecular weight dependence, where they had a considerable number of samples, they were looking at concentrations of about 300 gram per liter of polymer,

extremely concentrated systems, and the molecular weight dependence was a straight line for the linear polymer, and was a stretched exponential, two stretched exponentials actually,

for the star polymers. Taken as a function of concentration, the upper figure, the linear chain showed the power law behavior,

the stars showed a stretched exponential behavior, the viscosity of the star was for the most part less than the viscosity of the linear polymer. However, if you look carefully at the measurements against concentration,

you can see a point up here, and so if the concentration was taken to be high enough, eventually some of the star polymers, as they're obliged to do given the functional behavior, did show a viscosity that was higher than the viscosity of the stars, of the linear chain I mean.

However, for the most part, star polymers, comparable molecular weights, are up to fairly high concentrations, less viscous than linear chains, and at fixed concentration as a function of molecular weight again,

the stars are less viscous than the linear chain is. Okay, you can also look at results of, let this erase for a second.

You can also look at results of Kajura and collaborators. They looked at three arm stars.

Their stars were polyalpha, methyl, styrene. They look at a series of concentrations and molecular weights. You can see the results in figure 12.26.

And what they found was that again, if you're at about the same concentration and about the same molecular weight, the viscosity of the star polymer, this is the F equal three star, is less than the viscosity of the linear chain.

Now why is the statement that the star polymers have a lower viscosity than the linear chain of some interest? Well, there are a number of models of polymer dynamics which assert that linear chains have extra modes of motion that are in principle not accessible to stars.

In particular, for example, the reputation type models say that if you have a linear polymer, it can creep through holes in a transient lattice like the snake in a bamboo grove. But if I am a star polymer, in order to make forward progress,

what I have to do is retract by fluctuation one of my arms. That's difficult to do. Push it out in a new direction and now I can take a very modest step forwards. The net result is that one would expect

that it is harder for star polymers to move with respect to each other than, at least if they're very concentrated, than it is for linear chains. That does not appear to be quite what we are observing. Question? Can it also be possible that since linear molecules are a linear structure,

they have better packing mechanisms, therefore they adhere closer to each other? That could be an issue in the melt. However, in the solution, the amount of packing is determined by the concentration. The answer is that we make comparison at equal concentration for linear and star.

The other thing you should recall with respect to let us pack chains into a melt is that I am a star polymer.

I have one point where there are three chains coming together. I'm an F equals three star. There is a narrow region around this where there are three of my chains close together. However, if I'm a large star polymer over most of my length, I look exactly like a linear chain.

Therefore, this is the sort of effect you might find interesting effects with small chains, but with large ones you would not expect that to arise. Okay, we advanced to figure O discussion around figure 12.28.

And the issue around figure 12.28 and figure 12.27 is suppose we take a polymer chain and we change the solvent quality.

What happens if we change the solvent quality? Well, the expectation is that if we are in a theta solvent or close thereto, which means we also have to be at the right temperature, we have a polymer chain in the solution.

It would rather see itself as its neighbor than it would see something else. And therefore, if we increase the concentration, nothing terribly interesting will happen to the chain radius.

I didn't say nothing at all because theta, you can't get perfect cancellation of effects. It's an approximation. On the other hand, if you were in a good solvent, we see what we saw in the chapter on dielectric relaxation, which is what has been used for most of these measurements.

Namely, as we increase the concentration and look at the chain radius, the chain radius falls with increasing concentration. So these, however, are static properties. How big is the chain if we take a single snapshot?

We're talking about viscosity here. Viscosity is a dynamic property. And we ask what happens to the behavior of the system if we change solvent quality in the suction oven? The first answer is, if we look at a series of chains of different molecular weight,

the viscosity, as has been the case in most other systems, not all other systems, and not at sufficiently high molecular weight, concentration rather, in those systems, the concentration dependence is a stretched exponential.

And what we happen, find, as we increase the polymer molecular weight and do our measurements, we find that the parameter alpha increases and the parameter nu falls.

So there is alpha and there is nu. And both parameters are molecular weight dependent. The second thing we can do is to compare these parameters in a good solvent and in a theta solvent.

And what we find is that in a theta solvent, measurements discussed here, the parameter nu is from 0.8 to 0.94. And if we go, however, take the same material over to a good solvent,

the parameter nu is 0.6 to 0.73. That is, we are going in and when we change the solvent quality, when we go into a better solvent, the parameter nu falls.

The model that derives this equation, in fact, predicts this behavior. We will eventually get to it in later lectures. If you look at the figures, especially 12.28, this change of nu is physically quite visible. If you look at a semi-log plot,

log viscosity versus concentration, for a theta solvent, the concentration dependence of eta is fairly close to a straight line because this is fairly close to a stretched exponential. But if you look at the measurements in the good solvents, you typically see a more pronounced curvature

because you are looking at a stretched exponential rather than a pure exponential.

Now, having said this, let us push ahead and consider the general features of viscosity. We've looked at lots of specific measurements

and we did, of course, find that the viscosity depends on the polymer concentration and the polymer molecular weight, not to mention one could put in the topology and one could put in the solvent quality

With respect to topology, there was a very long dispute, this goes back a considerable piece, about the viscosity of ring polymers. That was mostly melts, which the book does not cover.

The issue there was, well you can try to make melts but you have to ask, you can try to make rings and put them into melts. Here is a ring, is it a nice open loop or during your synthesis process did it get itself all tied in knots so that it is a ball?

Under modern conditions, if you want to answer that question, you can resort to using DNAs since you can synthesize DNAs with absolute molecular weight control. Absolutely mean we have something that is 180,000 base pair

meaning its molecular weight is tens of millions and the molecular weight is well there are fluctuations because there are several isotopes of carbon and deuterium in nature. Carbon and deuterium, hydrogen I should say in nature.

But the molecular weights are totally exactly controlled, no fluctuation. You can also synthesize things, synthetic DNA, not stuff that occurs in nature, that is a star and completely mono disperse. You can also synthesize, it happens naturally,

some viral DNAs do this. DNAs that are ring polymers that are again completely mono disperse electron micrographs of those rings indicate that the rings are completely open, they aren't tied up in knots at all.

However, having said, the ring polymers are mono disperse, the star polymers are truly mono disperse, synthetic DNA comes that way. There is the significant practical issue that people have not done

of DNA viscosity that are as extensive as the viscosity studies that have been done on synthetic polymers. Okay, having said this, let us look at the question of what we find. And the first point which I am going to emphasize

significantly more than was emphasized in the book is that eta is not a universal function of C and M. That is, if you think that you can take

your viscosity measurements and yes, there will be some chemical dependent constants and there will be things that depend on the concentration and the molecular weight or maybe the length of strands and the molecular weight. Some representation of amount of material and size of a chain.

If you think that all of the chains are the same because of that, and therefore there should be a universal function that gives you viscosity as a function of concentration, you are going to be disappointed. Instead, it is quite clear from the literature that viscosity is not a universal function

of C and M. Instead, there are two distinct phenomenologies and in one of the phenomenologies you see simply a stretched exponential which in measurements of Tager and a few other people

go out more or less to melt. And in another set of systems you see a stretched exponential up to a point and then beyond the point you see a power law.

It is worthwhile to recall that the same phenomenology this class is occupied by spherical colloids. Spherical colloids show the same behavior. The slope of the power law

depends on the exact material. For spherical colloids the slope is appreciably steeper than for linear chains and if you look at many arms star polymers the slope creeps up

and makes a transition from linear chain behavior up to very nearly hard sphere behavior. It should be clear that any model that explains this phenomenon is subject to the constraint that it must be a work for spherical colloids.

Spheres can do a number of things but form entanglement lattices is not really one of them and form and reptate is certainly not one of them and so we have the issue that the viscosity behavior might not be quite what you would have expected

from some of your reading from some theorists. Okay. Furthermore, if you merge up to viscosities above about 10 to the 8 that's eta over eta 0 above about 10 to the 8

and most of those measurements are literature studies due to Dravol though there's some nice work due to Colby. However, if you look at that it does appear that if you get above here instead of continuing along on the power law there is upwards deviation.

The nature of the upwards deviation is not quite certain. One might however, for example, propose since we're at extremely high concentrations that it's some sort of a jamming transition in which things actually mechanically get in each other's way

and the chains can't slide across each other because they can't move after all they're rigid over short distances they can't move sideways as easily enough to get through each other. However, that is a fairly specific

that's a fairly rare exception because there are not a lot of studies that have looked at this. In particular, if you look at predictions that eta should be some C to the X, M to the Y that prediction is very generally

not true in polymer solutions. There are some exceptions. There is this family but predictions that you can simply assume scaling are not sustained. If you want to say we have scaling you need to explain why sometimes you have scaling

and sometimes you don't. Otherwise you can't just assume it. Okay. So that is the sort of behavior we get. We might then ask well what other generalizations can we find?

One answer is for a very large family we have eta over eta zero is e to the some constant alpha C to the nu

and now I'm pulling the molecular weight of dependence of alpha out. We'll actually see graphs of that dependence in a few moments. But if we actually look at the measurements and ask okay we get this thing to fit

what do we see? Well gamma is usually in a range around a half. I can be more precise than that but that's a reasonable approximation. In point of fact if we

oh I should add something. We go from theta to good solvents. We go from theta solvents to good solvents. The parameter nu goes down and what I will call alpha m

because it's alpha with m pulled out what I will call alpha m m to the gamma what's left that's not C to the nu increases as we go from theta to solvents to good solvents. Generalization.

Okay so what behavior do we see for gamma? And the answer is seen in figure 12.30 where we plot alpha which in terms of this equation

is alpha m m to the gamma. It's the coefficient of C and we plot alpha against polymer molecular weight. Now if I just showed you the points your reaction would be and gee that was a nice piece of graph paper Professor Phileas at least it was very nice before you stood way back

and started firing the shotgun at it and the points are all over the place. However if you look carefully at the points I have labeled the points by the chemical family of the polymers. So I have identified separately each set of homologous polymers.

Polymers that are chemically identical but different lengths. And if you do that you see that for each single family of homologous polymers alpha lies very much on a straight line. The straight line

has a slope so there's a log-log plot so we say alpha goes as m to the power gamma. And what can we say about gamma if we look at all of the measurements? All of the measurements where the same lab looked at a large number of different polymer molecular weights

and other than that things are consistent and the answer is gamma is typically in the range of 0.6 to 0.67. One can find a case where it's 0.94

where it's quite large and you can find a case where it's 0.51. If you look at this you notice there are different slopes for different materials and therefore the slope is not quite a universal constant

but for a given set of samples the power law dependence is actually fairly sharp. This result is also this result is also implicit in the systematic review of De Waal and many collaborators

and all of the wonderful Russian data he compiled in one place for us because what De Waal says is that C eta is a good reducing variable. That is if you take all of these measurements

and you plot them as functions of C times the intrinsic viscosity rather than C. So we have eta log log again C eta here. You find a whole series of different polymer molecular weights

but the measurements lie on the same curve. Now in order for the measurements to lie on the same curve when you are doing this plot it must be the case that eta is some function

of C eta which means that all of the molecular weight dependence of the viscosity has to be built into the molecular weight dependence of the intrinsic viscosity. You can't have any other molecular weight dependence out here someplace

because if you did when you plotted viscosity for a series of polymers they lie on the same line. They lie on the same line because viscosity is a function of C times the intrinsic viscosity.

That statement universal function goes all the way up to the melt. However it is an important fact that viscosity the intrinsic viscosity depends on M as a power and therefore this equation

is a function of what is it a function of C times and there are some constants M to the A. An example of a function that has the property that viscosity has this

dependence is given by this function. Now you would have to say this is E to the alpha M C M to the A to the power of nu and therefore gamma

this gamma is equal to A times nu the exponent of an exponent you obtain by multiplying the exponents and therefore this equation and that equation are totally consistent so long as gamma A and nu are related like this. Nonetheless

your vols results are entirely consistent with everything we said. I would be delighted to say that I can draw a pretty picture how nu depends on molecular weight or solvent quality but that really isn't quite true.

The first obstacle is that if you look at this formula for the viscosity eta proportional to E to the alpha C to the nu nu is the exponent of an exponent

or nu doesn't change a whole lot if we said nu is almost always in the range one half to one that's correct and therefore there's not a lot of range

and if there's any difficulty at all measuring nu accurately you run into a problem that the graphs are a bit scattered. There are some other parameters that we can talk about we now advance to figure 12.33

is the transition concentration C sub T transition concentration yes we are now limiting ourselves to systems that show the solution-like melt-like transition that's the transition we actually found and we charge up like this

and there is a concentration I'll use it C sub T in which the transition occurs and correspondingly there is a viscosity eta sub T when we discuss this for colloids well for spheres

there's always there's really only one curve because the volume fraction is the only variable and phi T was about 0.41 or a bit higher and the viscosity of the solvent was about 10 plus or minus 5

that's very clearly not the concentration and viscosity at which you start getting this phase effect and the question is now for linear polymers how does the transition concentration behave and the answer given in 12.33 is there a bunch of points

a bunch of points the transition concentration goes as M to the minus O 1.1 or so that is as you increase the polymer molecular weight the concentration at the transition falls

of course you need a set of measurements where the same material that shows the transition has been studied for a lot of different polymer molecular weights and the results that do this

are on poly- and hexyl isocyanate which we've discussed before and you look at those and gee you get roughly this behavior so you might okay so that's the transition concentration

you might also say well gee that's the transition concentration and maybe we should re-express the transition concentration in natural units when I say a natural concentration unit for polymer solutions what people generally mean

is c times the intrinsic viscosity now there are people who will say c over c star where c star is the overlap concentration but in fact the overlap concentration is usually inferred from the intrinsic

viscosity namely c star equals some number in the range I've seen things as small as one or as large as four in the literature over eta so this is a distinction but not a difference okay so we look at c eta

and we will ask at what transition concentration is till we see the transition and if you go through all of the accumulated measurements you can find transitions as high as 80 you can find things that occurred at about 35

and that is a more typical value you can also find transition concentrations as low for example hydroxypropyl cellulose which Carol Streletsky and I and others and so if you look at the transition concentration

in natural units it covers a very wide range of values you should realize this is a significant part of the way to the melt the melt concentrations in Dravol's-Medel's analysis were not more than about 300

the transition concentration for hydroxypropyl cellulose the intrinsic viscosity was one over one or two grams per liter and the transition occurred at around

six grams per liter so if we say we'll put the concentration in natural units you don't see any tendency of the measurements to collapse onto each other there is however something you can do which gets the measurements to match

not perfectly but actually pretty well and the answer is if you look at all of those curves where you see something like this for linear polymers not for spheres for the measurements

where people looked at a lot of different molecular weights and you look at their graph you realize that the crossovers occur at about the same viscosity and in fact it is approximately but not exactly universally true

that eta T over eta is some number like two or three hundred that is there is not a concentration but there does appear to be a viscosity a universal viscosity at which the crossover

sets in except in hard sphere systems which may be different and the crossover at viscosity occurs when eta over eta zero the reduced viscosity is a few hundred so you produce something that is fairly viscous but not completely impossible to work with

you might legitimately ask well that is very nice what is the exponent and the answer for a fair number of systems is that x is about three and a half however you can find systems

in which x is much larger well I don't know much larger but somewhat larger for example in hydroxypropyl cellulose x is about four point four power law exponent and I can even tell you what the power law exponent is

let us skip back from the systems that do show the transition which well it is universal a universal transition is a function of the transition viscosity it's not a universal transition viewed as a function of the concentration

and it doesn't even happen in some systems so let's go back and let us look at the systems in which we do not see a transition which is a large number going out in some cases to very high viscosities I seem to recall

reduced viscosities up to about ten to the seven in a few cases in which no transition is seen okay so there is the curve and one of the things you might legitimately ask is G, couldn't we manage

to describe this measurement perhaps with a couple of power laws and if you go through the literature you find people who draw a power law curve up here which is sort of tangential or goes through the points over some distance and you find down here

typically a linear curve and in between there is some transition as G there is a transition that is broad and therefore you might propose yes the measurements actually show a crossover from one type of behavior

to the other but the crossover is a little hard to see because there are other things going on at the same time there is not a clear transition the way there would be if there was a phase transition which I don't think everyone has proposed

for these systems so there is a transition but it is clearly not a phase transition and the question is is this a reasonable description and the answer I think is that if we look at E to the alpha C to the nu behavior the stretched exponential behavior marches out alpha and nu

are independent of concentration they are proportional to C and that is true along the whole curve and therefore viewed from this perspective there is no transition

there is a single smooth curve that just chugs ahead so far so good ok I have just run us out of chapter 12 we have now come to the end

of the discussion of viscosity I am now going to advance to discuss viscoelasticity and in order to do that I am going to have to draw a picture

the idealized picture is here are two infinitely long plates now that is a little dangerous because plates are not infinitely long and there are several ways around this for example you bend plates in circles and you have two concentric cylinders

or you have a flat plate or you have a rotating cone you may say why would you want a rotating cone and the answer is that the velocity up here v is omega r that is velocity o

parallel to the plate this distance distance out from the center increases as r and therefore this distance because it is a cone this distance is parallel it is proportional to r and therefore if I look

at the velocity of the plate over the distance from the plate to the bottom and there is a velocity gradient here and the velocity gradient is uniform over the distance now you have to be a little careful with this

because there is going to be some question of whether things are uniform as you move this way there are potential complications we shall persevere and I shall draw this pretty picture and the bottom plate

is stationary and the top plate is being moved back and forth and the top plate has a displacement which I'll call d for the moment and the displacement is some amplitude a

cosine omega t that is we have a displacement the extent of the displacement oscillates as a harmonic in time you can do that equally with either that or that

the displacements are fairly small now you might say well gee could you get information out if you made the displacements large and the answer is yes and there is an object an experimental method called

large angle oscillatory shear and we shall reach this in the next chapter chapter 14 however we have just reached chapter 13 which talks about linear viscoelastic behavior

okay so we have this displacement and the result in order to get this displacement we have to apply a force on the upper plate to move it back and forth now if you think about this for a moment you realize

that we are accelerating the plate at the same time and you have to deal with this but that is basically an experimental question and the inertia of the plate has to be made small enough in some sense that you can measure the part of the force that is due to the fact there is a liquid in here

I am going to call this a zero the amplitude of oscillation and as you make the plate bigger and bigger the amount of force you need goes up but it is basically a linear process

and therefore the interesting thing is the force per unit area and the force per unit area is known as the stress and the first force per unit area which by the way is going to be oscillating

as a function of time the stress the amount of oscillation the displacement divided by the distance L between the plates that is if I take this whole apparatus

and make it twice as big vertically the amount of oscillation is seen locally by a molecule here is a molecule and it starts out it has some neighbors along this line and when we increase D to its maximum this is oscillating the neighbors are displaced

sideways one way or the other in a back and forth manner and the amount of local displacement is determined by the ratio of the linear displacement of the plate to the distance between the plates and this is known as the spring so far so good

now how can we actually do this experiment there are a couple of different ways we could do it one thing we could do is to apply an oscillatory force

up here and measure the motion and so we have a force which is very precisely controlled and shifts things back and forth and we ask how the plate moves we can also have a driver and feedback electronics and we move the plate back and forth

to the distance and we instrumentally determine how much force we're having to exert so that the plate actually does this those two experiments are equivalent that is they measure a series of curves

where a certain amount of force per unit area a certain stress and a certain strain are matched with each other so whether you are actually experimentally creating the stress and measuring the strain or vice versa doesn't matter there's one curve and the distinction there

is purely how you build your machine okay now having said all this we now go in and we measure this question

this is assuming that it's linear when you're basically saying that the stress when you measure the stress is equivalent to measuring the strain it's basically assuming it seems to me

the answer is no that statement is more general than its linear response that is you have a stress you have a strain if I double the stress what happens to the strain well the strain does not double in the linear response it does but however the statement is

there's a stress-strain relationship would be true whether it was linear or not the only place where you would get into trouble with this is that you could have a hysteresis issue where if you apply for example different strains

and the stress changes someplace or vice versa you can reach every point on the curve instrumentally very easily by doing one control or the other now this is linear response in the sense that the theoretical analysis

that is done of linear viscoelasticity assumes that if I apply a series of forces at different times the fluid has a response at later times and I can just do all of the responses of the fluid of course if I've applied

the force at several times there are several time delays and I will get the answer and we'll get to linear in a bit so what is the net result though the net result is that if we actually look at the strain the stress the force and we compare it

with the strain we find something we can divide out because this is linear we can get force we can divide out the a zero we have force per unit area

per unit of strain and that force turns out to have two pieces there is one piece that is determined by a function g prime of omega and g prime of omega

is now multiplying the displacement traditionally sine rather than cosine oh I'll write it as cosine to avoid confusion this is cosine omega t

there is one component where the response has a response that is in phase with the displacement so you have a displacement

and there is a restoring force that grows linearly in the displacement here is the oscillating displacement and then we have another component and the other component

is 90 degrees out of phase and so there is a response that is 90 degrees

out of phase to the displacement meaning this is a force that is largest when the displacement is at zero

you have in phase and out of phase responses and you might ask what does this all mean why is there a force that is in or out of phase with the displacement this is a liquid after all if you made the displacement stopped and sat there after a while the force

would disappear because the liquid would just float that's a little more complicated and the answer is that the low shear viscosity is

well it's really a low frequency limit of G double prime of omega over omega these two objects have names by the way G prime is the storage modulus

and G double prime is the loss modulus and you might legitimately wonder G that seems like a peculiar way to represent things and I will now show something

that makes much clearer from a physicist's point of view what you are looking at I am going to multiply each of these objects by one and of course whenever a physicist says I am going to multiply by one you know it's going to be some huge object that is in fact one

and I multiply this by one in the form omega square over omega square and I'm going to multiply this one by one in the form omega over omega okay that's perfectly legitimate you notice

this now looks exactly like the viscosity but now I ask you okay I have something that goes as omega times the displacement that goes out of phase with the displacement and I have something that goes as omega square

times the displacement what am I looking at here well if you think of a harmonic oscillator you realize that this object is the velocity and I have a term in the force that is determined by the velocity G that's a loss term

of displacement that's the acceleration and the thing that goes as the acceleration is a restoring force the system looks like it has little springs in it and in fact there are a set of math models

blamed on if I recall correctly Maxwell and this whole picture can be modeled as here is a system and it has a spring is something known as a dashpot

dashpots were very important back when people had teletypes the dashpot the platen of the or the moving object in the teletype comes back very fast and you don't want it to whack the far end hard because if it does

the vibration will damage things so you have this little piston like object and the moving object coming this way has an arm coming out there's a rubber disc here that's the traditional image and the rubber disc hits this thing and there is a little hole here

and this is a piston but it's a leaky piston and so you compress the air and it slows things down and then when it's compressing the air the air blows out the end and you can tune the opening because there's a little sliding arm and so eventually

after a very small fraction of a second the moving object on the teletype is brought to an end here it's brought to a stop at exactly the right location and with no unpleasant sharp acceleration that would damage anything

Well, yeah, it does that and it does it via high quality Victorian engineering there are essentially no moving parts in this other than the object you're trying to stop there's no active feedback system no electronics it's all done mechanically and that is a dash pod

And you can fill the dashpot in this picture with some liquid so you have this hanging object in the liquid and the hanging object in the liquid is just a liquid with a viscosity and no frequency dependence. Or it has a frequency dependence and this object represents the storage modulus

and this piece represents the loss modulus. So that is a picture of the viscoelastic parameters, the linear, in terms of a storage modulus and a loss modulus and you notice that I have re-parameterized them starting here

and then doing this for consistency basically. And once I've done this, the loss modulus looks exactly like the viscosity. There is however, and I'm just going to show the qualitative picture first,

there is however a consequence of re-parameterizing things. If I plot G prime or G double prime themselves, you remember they contain something,

I divide out an omega which I can do safely. And therefore G prime and G double prime both go to zero. This is a little hard to do on a log-log plot but they both go down and at low frequencies, this is more typically a log plot so you never really get to zero.

There is a linear behavior which is omega to a power and then there are things that happen out here and perhaps the curves cross. And if I keep making omega larger and larger, eventually the curves turn up again.

I am not going to draw this in any detail, you can find it in standard text. The important issue is that if you talk about G prime or G double prime themselves, you find that they have curves that look sort of like this

and there is a region where things are reasonably flat and there is a low frequency region and a high frequency region, I could introduce terminology, we'll get to that next time. The main issue is though there is a shape for the curves. If I instead say I am going to go in and I am going to plot G prime over omega square

or G double prime over omega, if I do that, I have divided out the leading slope. And if I divide out the leading slope, that says that at low frequency,

the behavior that I am looking at is frequency independent, which is sort of what you would naturally expect. And these curves all look, as you will see from the book, about the same, namely there is a flat region, there is a rollover,

and then down here there is at least at first a power law. And then maybe something interesting happens at larger frequencies. And the question we ask given that I have drawn these pictures is,

well, how do I derive or rationalize a curve that explains this? And so I will give an ansatz based on a theoretical model we have not yet discussed in detail

and a renormalization group argument. And so we are going to actually invoke, which we have not actually done before, the hydrodynamic scaling model, which is my model, and we are going to invoke a renormalization group argument.

The renormalization group argument being invoked is much more general and qualitative than many renormalization group arguments, but it will be clear what it is.

So we are going to start out, and here is a straight line representing concentration. And we will imagine that the viscosity is plotted perpendicular to the blackboard out towards you.

And what we find is, the viscosity increases as I go to higher concentration, and eventually, at least in some systems, we reach the transition concentration C sub T. And down here is a solution-like regime, and in the solution-like regime,

you see stretched exponential in concentration behavior. And up here is the melt-like regime M, where you see C to the X behavior. That's the pure phenomenology.

Now what I say is, I have used something called the Altenberger-Dollar positive function renormalization group method,

and I applied this in the context of the hydrodynamic scaling model. And I said, I can calculate a certain lead behavior at lower concentrations, and the Altenberger-Dollar renormalization group method

lets me turn the lower concentration calculation into the stretched exponential. And we're going to assume that's true. Now I'm going to put another little bit on top of it. The PFRG, the Altenberger-Dollar positive function renormalization group method,

actually predicts two sorts of functional behaviors. The functional behaviors it predicts depend on where what are called the fixed points of the renormalization group are located.

In particular, if you have a fixed point at zero, then near zero, you predict stretched exponential behavior as is found. On the other hand, out here someplace, if you have a fixed point that is way out there, and that is the dominant fixed point.

There can be several fixed points, but as we go along here, there's always a dominant fixed point. If the dominant fixed point is out here, you would get power law behavior as is observed.

Next trick, the viscosity is the low frequency limit of G double prime omega over omega. The viscosity is a function of concentration, and so is the dynamic loss modulus.

Therefore, when we say we are looking along this curve at viscosity, I could also say, I'm going to put another axis on here. I'm going to insert the vertical axis I've skipped over.

I'm going to call the vertical axis omega, frequency. I'm going to say viscosity is what we measure along the omega more or less, maybe not quite, zero one. So we measure at low frequencies, and we get the viscosity,

this is the zero frequency limit. However, it's the zero frequency limit, and while I wrote it as eta, I could just as well have written it as GW prime of omega, and by the way concentration, over omega.

And therefore I now have a two variable function, and I have used the renormalization group argument to work along at omega is zero. Now comes the somewhat bold part. I will start here, and I will march sideways.

And at first I'm in a regime where this fixed point at zero is presumably still there and dominant. And then at some point I move sideways enough and these other fixed points,

about which I know very little, become dominant. And therefore I have moved from a solution-like regime to a melt-like regime. However, I am marching along at C fixed with omega as the variable,

because I'm moving sideways along this graph. And I am looking at something that is really only a function of omega. If this picture is correct, there is a great deal of extrapolation in this.

It's an ansatz, not a derivation. At lower frequencies, this fixed point is dominant, and therefore I should have a stretched exponential in frequency.

However, eventually I cross this line, and once I am across this line, I am in a region where the melt-like fixed points out there are dominant, and I should have omega to a power behavior. This doesn't really calculate what the power is at all.

And in order to do the extrapolation even at low frequency, I would need something that gives me viscosity is eta zero, plus presumably some intrinsic viscosity eta naught C,

plus some beta omega. That is, I would need some way to generate a model that gives me the low frequency frequency dependence, so I could move off zero at all. And after I have moved off zero at all,

I can then use the Altenberger-Duller renormalization group to say what happens with frequency. However, I have not found, I've been working on other things truthfully, I have not yet found the frequency,

and therefore the assertion is that you find this linear step and you can then fill in the low frequency behavior. That has not yet been done. And the crossover and the fixed points out there have not been done even for concentration.

Okay? So that is the rationale, and you notice what it predicts. It predicts stretched exponential in frequencies at lower frequencies, and it predicts power law in frequencies at higher frequencies.

However, there are some interesting, if you look carefully, there are some little bits I have skipped around. And one bit I have skipped around is what it says about frequency dependence way out here. That is, if you are way out there,

you're in the mount-like regime all the way, and therefore you see simply power law in frequency behavior. Well, you can actually point at data that looks like this, and the question is, are you really going to see that, or is it the case that there is a very low frequency,

stretched exponential in frequency regime, and corresponding to that, is it possible that this curve actually bends over and very close to zero frequency if you get down to sufficiently low frequencies, the C to the X behavior would disappear. I don't have an answer to that.

The other point is, that if we go up to very high frequencies, is it going to be the case that we have gotten into the omega to the X regime at very low concentrations,

or does this curve bend over so that at very, very low concentrations where we can just barely see particle interactions at all, we are still in the concentration to a stretched exponential regime.

That is, the model does not handle this very, does not discuss this. The model sort of dodges around that. It covers this large area. Now, you could say, of course, that G, this picture explains

why some systems show the solution-like, melt-like transition, and some do not. Namely, depending on exactly where the fixed points out here are, it might be the case that in some of these systems, you chug along the concentration axis

all the way to the melt. This curve does not go to infinity. It stops at the melt. Even at the melt, the detailed numerical parameters going into the calculation are such that you can get all the way out here and the fixed point at zero is still dominant.

But there are other systems and all you need are differences in chemically dependent parameters such that these fixed points become dominant at some concentration and out here you see a power law. And therefore, one universal physical model

plus chemically dependent parameters that determine which fixed point is dominant predicts both behaviors. Well, that's fine for the hydrodynamic scaling model, which actually allows you to have two behaviors in one general model.

It is not at all so good for models that assume scaling and it seems that you always get scaling out here, which it seems that you do not. Okay, that is the ansatz. And having given you the ansatz, we have now reached experiment.

We have also reached approximately the end of the hour and therefore we have reached a natural point to stop. Therefore, I am going to stop. So this has been a lecture where we are finishing off my treatment of viscosity and advancing to my treatment

of the hydrodynamic scaling model and its ansatz for predicting linear viscoelasticity. I am George Filleys. This is the end of the lecture.