Good morning. Today will be our last lecture on probe diffusion. We're going to consider a few particular special cases and sorts of experiments that also look at probes, mesoscopic particles moving through polymer solutions.

And we'll see what information we get out of those measurements. We'll also consider, at least at short length, a summary of the sorts of experiments that we have looked at and what we get out of them. So first thing we're going to consider is rotational diffusion. That is, we have a polymer solution.

We put it into a probe. Even if the probe is a sphere, we can arrange its innards to be ordered in some sense. And if the innards are ordered, when light impinges on the sphere,

the scattered light is depolarized. The intensity of the depolarized light depends on the scattering angle and the orientation of some internal orientation of the sphere with respect to the incident light and the scattered light.

Hiding in between, there's a description of the sphere. If the sphere rotates, the intensity of the depolarized light changes. And therefore, if we have polarized light in, polarized light out, and

we monitor the intensity-intensity correlation function of the scattered light, we can actually measure rotational diffusion times for nominally spherical particles in a polymer solution. This has been done by several work authors.

I will note here, for example, work of Condorink, who looked at spheres and 4-megadalt and Xanthan, and who did a whole bunch of different experiments.

The bunch of different experiments is useful because you have a system, you look at a series of different parameters, and by comparing the behavior of the parameters, you get out considerably more information than you would out of the same four parameters being measured in four different systems. What was found, looking at the experiment, and

what is being varied is the concentration of the polymer. Varying the concentration of the polymer, what was found was that the probe diffusion coefficient and the rotational diffusion coefficient fall,

but not very much. You can follow along in the book in figure 9.37. What also measured was the sedimentation coefficient, which was affected considerably more dramatically than the probe or

rotational diffusion coefficients were. And also measured was the viscosity, I plot here the inverse of the viscosity, the fluidity. And the fluidity of the solution was much more affected than the polymer

than were any of the probe transport coefficients. What we can infer from this, first of all, you're seeing here very transparently non-Stokes-Einsteinian motion. And that is the viscosity of the solution, the macroscopic viscosity, is not determining the transport coefficients.

Furthermore, the sedimentation measurement is what can be called a micro-rheology measurement, that is you have an object. You are applying a true external force to the object. You're watching the object fall through solution.

And the apparent viscosity that you can infer from the sedimentation coefficient is not the same as the macroscopic viscosity of the fluid by direct measurement. Similar measurements have been done by any number of other people. For example, we have figure, let's erase this.

We have figure 9.38, which looks at tobacco mosaic virus. The virtue of tobacco mosaic virus is that it's a rod.

It's a very uniform rod. All of the TMV molecules are the same size. The disadvantage of TMV, and many other biological probes, is that they're charged and you have to worry a little bit about electrostatic interactions between the probe and the environment.

Well, you can't always have everything. Nonetheless, what was done was to measure the R again, and once again, on an appropriate plot,

the R versus concentration, what was found was a stretched exponential concentration dependence. The interaction between the polymers and the probe, giving us a decay of the decrease of the rotational diffusion coefficient.

This polymer concentration was increased. An interesting variation on this, see the text again, was to look at polybenazole L-glutamic acid as the probe.

It's a nice, another rod-like polymer. It's like TMV. The background material, however, in this case, were little spheres. And so you were looking at a rod diffusing through spherical particles. That's a very clever additional experiment.

We'll come back to spheres diffusing when we get along to look at, gee, colloids. In fact, there's a whole chapter on colloids. And we're coming to it quite soon. What sort of other experiments can we do?

An extremely important whole set of experiments is provided by particle tracking. The notion of particle tracking is that instead of looking at

light scattering spectroscopy, which measures the relaxation of the spatial foyer component, a fluctuating component in the concentration. Or looking at fluorescence relaxation after photobleaching or some similar method, which gives you a diffusion measurement, but

only over fairly long distances. We will actually look at the particles through a microscope. We'll use modern video technology to measure position versus time of the particle. And we can actually get a detailed statistical distribution

of the particle position as a series of times. And therefore, we can get detailed statistics for displacement factors r of tau minus r of tau plus t.

And from the particle tracking, what we can do is to actually talk about in somewhat more detail how particles move in solution. Now, the limitation at the time I speak, I suspect this is going to be obsolete in a few decades or sooner.

The limitation on the technique is that if you want to measure how the particles move, you actually have to record the positions. This is done on a large scale in many cases using video technology, using video cameras. And if you do this, you hit a limitation, namely the frame rate of the video camera.

You are taking an image of the system, but you can take the image of the system, what? 30 times a second, 60, 100 times a second. On the other hand, light scattering spectroscopy, you get particle position information going down to the, certainly,

50 or 10 nanosecond time scale and routinely down to the microsecond time scale. So there's still some technical limitations, but particle tracking is really useful because you actually measure directly how the particles move. Now, there is a historical theoretical basis for this,

which is the Langevin equation, which describes particle motion. The Langevin equation says, we'll keep this very simple. We have a particle in solution. It is moving. It experiences a drag force minus FV,

which in the original model was the Stokes law drag coefficient. We'll come to the problem with that in a second. The particle is also subject to what is called a random force F. Now the force F is not random in the sense there's no physical determination.

It's due to the solvent molecules around the particle we're interested in. We can't see the solvent molecules directly, so we don't know what that random force is going to be. Nor, since we can't see all the solvent molecules and track all of them and all this other things, can we predict what F of t is in the sense we can

predict what the sun's gravitational force on the Earth will be for the next year. So there's a force we can't measure directly or predict. However, it is a force, and we can therefore use the random force and

the drag force to write F equals MA, F equals MA for a Brownian particle. The random force in the Langevin model has the important feature that if I tell you the value of the force at one time,

that gives me no information whatsoever about the value of the force at any other time. The force fluctuates, but the correlation time in the force is approximately zero.

From this, and a great deal of work described in my other book, Elementary Lectures in Statistical Mechanics, and there's an accompanying set of video lectures on that now being produced too. From this and this, we can say, gee,

we can calculate how the particle moves, and we can calculate what the particle's displacements look like. And for a one-dimensional component of random displacement, say displacement along the x-axis, the probability of some

displacement x during time t is proportional to e to the minus delta x square over, and in one dimension, it's 4dt, isn't it?

Yes. Well, any event, the key point is, it is a Gaussian in delta x. Now, the proof that the form is a Gaussian relies on two pieces. And one piece is called the central limit theorem.

And the central limit theorem says that if you add up a large number of random events, the distribution of the sum of the random events, the statistical distribution, is a Gaussian.

So if I roll one die, it'd be a d6, right? You get a one, two, three, four, five, or six equally likely. If I take 100 dice and roll their roll and add up the numbers I get, the average I get for 100 dice is 35,

as the sum of all of the points showing. However, 35, 350, 100 times 3.5. However, if I look at the distribution of die rolls, the distribution of the sum of 100 rolls around this average,

which is 350 for 10, 100 dice, I get a Gaussian of some width. The other piece of this, which is just as important as the central limit, is that this is what is known as a Markov process.

A Markov process is a process with no memory. So that the velocity of the particle, or displacement of the particle, over one piece of time, and the displacement over the next piece of time, and the displacement over the third piece of time are all uncorrelated.

Now obviously at very short times, this is not a Markov process. But over reasonable time periods, it's in my other book, you can show that the behavior from the Langevin equation is a Markov process. And therefore, you get this Gaussian distribution of displacements versus time.

Okay, now having said that, there are then people who say, well, we have this nice result. You can find the result in the beautiful book of Burnham-Pecora.

And they actually show you a considerable number of details. And Burnham-Pecora, book on light scattering, then shows that,

well, if you have a Langevin vertical, the displacement distribution is a Gaussian.

And therefore, the light scattering spectrum, and I will write the field correlation function, is e to the minus some constant e to the minus q square average displacement during the time t over two.

This result is perfectly true for the system in which it was derived, namely a system in which the particle motions are a Markov process. There's no memory here. And in which the drag force on the particle is minus fv,

where f is the Stokes Law drag coefficient if you have spheres. Now we come to the minor technical difficulties with this result. The first technical difficulty, which goes back to the 1970s,

is that if you have a Brownian particle in a liquid, the Brownian particle in its motion does not move at a constant speed in a constant direction. Its velocity is not constant, that's what Stokes Law is. Instead, Brownian particles move like drunkards.

I will not attempt to fake drunkardness too much, but drunk can walk. And if you have a particle moving with an irregular speed through a fluid, and it's simply driven motion, Stokes Law is not the correct form.

And the correct form is the Boussinesq equation. A significant effort in statistical mechanics back in the 1970s, there's several very nice papers about it. You can look up, for example, Herman's and

Shao, and there's a bunch of others, but that's the first that comes to my memory, demonstrate that if you have a particle that obeys the Boussinesq equation, this is spheres in water doing Brownian motion, the calculated diffusion coefficient is the same as the diffusion coefficient

you get assuming Stokes Law. However, we now hit the other two, some more technical obstacles. The first technical obstacle is, of course, we're talking about complex fluids. Polymer solutions, colloid solutions, concentrated micelle systems.

You can come up with a very long list of these. And all of the very long list of these have in common, why are we interested in them as complex fluids? Well, rather consistently, the fluids are viscoelastic.

The statement, the fluids are viscoelastic means, Stokes Law does not apply. It is possible you could rejigger Boussinesq's calculation to handle the statement that the viscosity is frequency dependent.

There's several other ways of saying the same thing. However, the important result is the system has memory. If you have a viscoelastic fluid, thanks to the fluctuation dissipation theorem, we can say that if we look at what we were calling a random force.

Thanks to the fluctuation dissipation theorem, we can say that the random force is some function, has some dependence on tau. So the random force at one time and the random force at another time

are partly correlated with each other, and thus the Langevin description does not apply. You might start to worry, gee, how can I tell if Langevin is applicable? How can I tell if I'm in a system in which I can use this result and

that result, or as opposed to a system in which I can't? Fortunately, there is a wonderful answer to this, and a wonderful answer to this is Doom's theorem. Let's see, it's a nice mid-1940s result.

It's an analysis of random motion. It's an analysis of random walks that show this sort of behavior. And the important answer is that if you have a Brownian Markov process, if you have a process for which the whole development leads to this result,

then we can say with absolute mathematical certainty that the light scattering spectrum, gamma is a number, the light scattering spectrum is a single pure exponential.

If the light scattering spectrum is not a pure exponential, then you aren't looking at something described by the simple Langevin equation. This displacement distribution function is not a simple Gaussian.

In fact, successive displacements are correlated. G is not determined by the mean square particle displacement. And you actually have to know what you're doing somewhat to get any further. You might ask, well, what do we do to replace the Langevin equation?

And the answer is there is an exact result due to Hosseini-Mori and Bob Swanzig, known as the Mori-Swanzig theorem. That is also covered in great detail in my book, Elementary Lectures in Statistical Mechanics.

This is an exact result, and it tells you what happens next. Of course, evaluating it, you may correctly infer, is a little more tricky. Okay, so I have now said, gee, there's this question of what is going on.

And you have to realize that there is something of an intellectual minefield hiding here. And people who say, Gaussian, Langevin equation, well, that's unlikely to be correct. In fact, as I'll show you in a sec, we know it's wrong. And polymer solutions.

And now we come to several extremely important works. The first of which is the nice paper by Apgar, by Sen, and a list of other authors.

The paper, as an aside, is noteworthy as a superb illustration of professional ethics. Namely, I have listed two names because the paper has two first authors. And there is even a footnote telling people there are two first authors, and they equally deserve credit for their incredibly important piece of work.

And what they did was to do particle tracking of 0.43 to 0.6 micron spheres. And they looked in a series of solutions. And they started out, okay, does the experiment work? Well, let's look in something like water or water glycerol.

They also looked in solutions of actin, and solutions of actin fashion, these are proteins. Actin polymerizes and makes long threads.

Indeed, if you're trying to work with it, one of the problems is that it will do so under all sorts of different conditions. And controlling the polymerization is a bit tricky. And what they actually did was to measure, not talk about,

but measure the displacement distribution at different times. Now they were doing this with particle tracking. And the limitation of we're doing it with particle tracking and we're doing computer, you can do computer analysis of images these days, is that the number of measurements you get out is a bit less than you would get out

if you were doing quasi-elastic light scattering. And so there's a certain amount of signal to noise question, which they've obviously worked very hard on. The important result is that p of delta xt for these spheres and these protein solutions,

which have been very heavily studied by variations on this technique. P of delta xt is not a Gaussian in delta x. Well, since we believe that polymer solutions are viscoelastic,

that's not a big surprise. If you understood the underlying theory, that would be an immediately obvious result. There's absolutely no reason to expect a Gaussian for p of delta x. The core issue is that in order to have Gaussian behavior, the central limit theorem won't do it for you.

You also need that the system be a Markov process in its motions, which in viscoelastic systems you do not have. Okay, so we do not have a Gaussian. This was not the only other experiment to look at this. There is also another nice pair by Tseng and

several other co-authors looking at the same sort of thing. Tseng and et al, you also look at delta x square at fairly large times.

And what they found is that if you go to large times, delta x square grows as a power in time. And the power a is less than one. That is, you have what could be described as a soda diffusive behavior.

Tseng and Wirth. Well, having said that, it's not a Gaussian. There's an important side on this. If you go through the literature, you can find bunches of experiments

under the title Micro-Rheology, where people actually assume that p of delta x is a Gaussian. All of the techniques which simply assume Gaussian behavior in a fundamental way, as opposed to saying they're going to assume it and

then doing something quite different that doesn't really rely on that. But all of the techniques that really rely on the assumption that p of delta x t is a Gaussian and delta x, the outcomes of those methods are invalid. They're invalid because the underlying ground assumption is incorrect.

Can they be patched up? Well, if you're clever, you can do a lot of patching. Now that is Apgar and Tseng. There are a series of other people who have done micro-rheology.

And let's see, Crocker and collaborators, and who else? Tseng, and there are a series of other references in the text.

And what they do is to do two sorts of, the same data in a sense, but two experiments. And one is to say, here is a particle, and it has a displacement delta one. And we can look at, for example, delta one square of t.

And it's average behavior. And the other is to say, we have here two particles, delta one and delta two. And if the two particles are moving, we can look at the cross correlation.

The cross correlation is the displacement of particle one during time t, the displacement of particle two during time t. And this object describes a cross diffusion object.

These are both vectors. The composition is legitimately an outer product, a tensor product. So d1,2 in the general case can be described as a cross diffusion tensor.

And it depends on the vector displacement between the two particles. And the viscoelastic system, you might discover there was a principle of time dependence.

Now we've discussed at some length, we have a mean squared displacement here. And therefore, if we pretend this is a diffusion coefficient, we can extract from some sort of a microscopic viscosity. We can also go in here, this is a different diffusion coefficient. It's a cross diffusion tensor.

However, the same general analysis says we can extract from here another micro viscosity. D is one over the viscosity, and then there are a bunch of constants. And numbers, and we can extract from each of these a micro viscosity.

The interest in doing this is that if you go in and you do this twice, well, there are two key features. One is there is a time or a frequency dependence here. That is the apparent micro viscosity depends on

the time step you use to measure the displacement. And this corresponds to the system having, in some sense, a dynamic storage modulus and a G double prime, a dynamic loss modulus.

And so you have two moduli, and you can look at both of them. And what comes out of this? Well, the first thing that comes out, you can say these are in essence a frequency dependent viscosity.

We will be considerably more precise in a later chapter. And the important issue is that this is sort of mu one, this is mu two. And eta mu one over eta mu two. The two viscosities are not equal to each other.

Their ratio can have a difference which is a factor of two or a factor of three. So the viscosity viscoelastic properties that cover single particle motion and that cover two particle correlations are not the same.

You also could find, at least in the system that was studied, that the two particle micro viscosity is something like, or quite close to, the macroscopic viscosity of the liquid. Well, that's very interesting. Of course, there's some complications.

When I said macroscopic viscosity, I mean the viscosity measure if you actually get a rheometer, which is a box of some size, and measure fluid flow over distances you can see with the naked eye. Okay, these are the nice experiments of Crocker and a series of others.

And the core issue is they do what is called two particle micro rheology.

And having done two particle micro rheology, they infer viscosity. Now they actually did something else. So it's sort of, you read the paper very carefully and you notice it. This is a cross diffusion tensor. It's cross diffusion tensor. And the cross diffusion tensor tracked t, the Oscene tensor,

which in the Oscene tensor describes hydrodynamic interactions between pairs of spheres. There's a power series in one over the distance to various powers. The Oscene tensor is the lead term.

There's an interesting implication of this, which we come back to very late in the course. So these are the studies of Crocker and many other people on two particle micro rheology.

Okay, very pretty experiment. We're going to look at micro rheology again. But we're going to look at it the segmental scale.

And this is a wonderful experiment by Bichel and Sackmann. And what they did was take a long polymer, so there is our long polymer. And what they do is to attach to the long polymer a bunch of, if I recall correctly, little gold particles.

And these little particles are attached to various points along the chain. And therefore, you can see the chain and where it is. And you're taking pictures of it, and you can watch the chain move. And one thing you can do beyond,

we can measure the center of mass motion as we can say. I can look at this piece of the chain. And this piece of the chain has a motional component parallel to the apparent backbone and perpendicular. So there's parallel and there's perpendicular. And this piece of the chain and every other piece, by the way,

has a parallel component and a perpendicular component. Of course, parallel and perpendicular are now pointing completely different ways at different places along the chain. But because they very cleverly attached all these beads, I can see a course outline of where the chain is.

And I can tell where parallel and perpendicular points at each place along the chain. Isn't that clever? And I can now measure a parallel diffusion coefficient and a perpendicular diffusion coefficient. They actually did the experiment.

So having done the experiment, what did they find? Well, the first thing they found was that at very short times, typical displacements, parallel and perpendicular, are about the same.

But if you wait a while and look at significantly longer times, the parallel displacements can be rather rapid. That's parallel to the chain. The perpendicular displacements are now much slower than the parallel displacements.

I should stress the time scales on which these observations are being made are much shorter than the time scale on which this polymer molecule, in terms of the reptation model, has escaped from its tube and moved to a new location, say over here.

So we are looking at measurements on still fairly short time scales. However, what we find is the parallel motion is faster than the perpendicular. And this is actually a prediction of models that on short time scales, you have local chain motions which can get fairly fast.

Others, however, another piece of this. Namely, if you look at the parallel motion, you can extract from the parallel motion, which is, after all, diffusive, a micro viscosity. And the micro viscosity is something like an order of

magnitude larger than the solvent viscosity. As there is a micro viscosity in there, but the polymer segments do not think they are moving through plain solvent. They think they're moving through something that is quite resistive to their motion.

Okay, well this is a very pretty experiment. It's all done microscopically. And as an experiment, it would be wonderful, gee, this is a computer technology issue, if it could be carried out and you observe the motion after very long times. And perhaps by the time this lecture is over,

I will discover that has in fact now been done. But it didn't seem to have been done at the time I wrote the book. But it was very nice. Okay, let's look at another experiment. And the other experiment we will look at is due to Goodman.

And what was done was to look at diffusion and motion in DNA solutions. Now why are we interested in DNA solutions? Well, there's all this biotechnology stuff. But from the polymer physics standpoint, there's something very important here.

There are a number of DNAs you can procure that are rings. And they can be quite large. The largest they looked at was greater than 1 times 10 to the 4 base pair.

The rings are essentially totally mono-dispersed. And thanks to modern biotechnology, you can go into the ring and cut the ring and be sure you're cutting the ring exactly once. So you get a linear polymer whose molecular weight is identical to,

oh, 1 part in 10 to the 5 or 1 part in 10 to the 6. After all, you probably did insert a hydrogen atom or an OH or something in here. And you have split the ring. And these two are exactly the same molecular weight,

except this is now a linear polymer. And having said that, okay, well, we will now do spheres and micro-rheology in these two systems. They got up to concentrations in natural units only about six, unfortunately.

However, what they found was that the micro-viscosity of the ring system, yeah, that's related to the micro-viscosity of the linear system. But at elevated concentrations, the ring system was considerably less viscous by several fold.

Now, the reason this is of some interest is that there are theoretical treatments of rings. It is always possible to claim that those theoretical treatments are so

brilliant that they do not yet apply to this system, because the system is not concentrated enough. But what the sit predictions say is that linear chains have modes of motion, for example, translation along their whole length, that are not usefully available to ring polymers.

And therefore, the viscosity of a ring solution ought to be considerably larger than the viscosity of a solution of linear chains. Such a behavior has never been observed. Historically, when you talked about ring polymers though, there were always objections that well, the rings,

instead of being the pretty picture I've drawn, could be tangled all in knots and so the thing not only is biting its own tail, but it's doing loop-the-loop around itself. There could be concatenated rings, and there were all sorts of complaints if you discussed synthetic ring polymers.

With biopolymers, these objections are invalid. With biopolymers, you can say, as a result of the way they're synthesized, there are essentially zero concatenated chains.

Furthermore, the chains are all wonderfully mono-dispersed. There are no linear chains mixed in there until you've done the cutting. Furthermore, you can do electron microscopy and you can see rings as they come out of various objects.

If you see rings, you don't see balls. And therefore, with great certainty, we can say, we actually have a rings polymer system. It's mono-dispersed, it's not concatenated, it doesn't have all of those nasty criticisms that have been made of it. And this is a legitimate test of the system, up to the limitation

that you wish the viscosity had been taken to be some higher number. Last experiment, and you may say, well, this system, we're about to talk about very pretty work of Shu et al.

And we are looking at wheat gliadin, which is a natural product. And we are looking at the diffusion of objects through wheat gliadin.

And gee, there's some very odd results that come out. They're quite solid. And one result is that we have spatial heterogeneity. That is, if you measure the motion of probes through the solution,

in different parts of solution, you can sort out, there are regions where the properties of the liquid are a bit different. And one can imagine saying, well, as a qualitative description,

this is not in the shoe directly, there are regions which are relatively fluid. There are regions where particle motion is more slowed down, which are in some sense vitrified. Those of you who are familiar with the Kivelson glass model, and

other recent work on glasses, may realize that I am describing a solution. It should be homogeneous, shouldn't it? Well, no, it's not. And the implication is, you might, well, let's look at the last piece of the experiment, and then I will say what you might.

They looked at a series of concentrations, such as 250 gram per liter, and 400 gram per liter. And the 250 gram per liter p of delta x, the displacement distribution function was a Gaussian.

And the 400 gram per liter of this material p of delta was very definitely not a Gaussian. Gee, the system, it's a liquid, but it's not homogeneous. It's thickening up.

We are looking at the Kivelson glass model, which proposes that glassification occurs because you start forming heterogeneous regions, which are in some sense vitrified resist motion. Well, here's the experiment, and you're seeing it directly.

Okay, so much for the pretty work on wheat gliadin. Let us chug ahead, and we have another section. In the next section of the paper, book,

we are going to talk about true microrheology. What do I mean by true microrheology, as opposed to

diffusion and a micro viscosity from diffusion materials? I mean, we're going to do an experiment in which we take a mesoscopic particle. We apply to the mesoscopic particle the force F. We discover the particle has an associated velocity v.

We can do such things as applying an oscillating force, and then we get a velocity omega. This is a real viscosity measurement in the sense we are actually applying an external force to the system.

Okay, whole bunch of experiments of studying this. And so, for example, read author Amblard. What is done is to apply a force to the system, and they did something careful.

They both applied a force, and they looked at diffusive motion. And they asked, okay, what happens to, for example, the mean squared displacement at long times? Or what happens to the displacement if I apply a force at long times?

And both of these were found to depend on p to the a for a sum over around three quarters. That is, at long times, at least as long out as they went, they applied a force and the response, the displacement increased as a power long time, but not linearly.

And the diffusion measurement did what you would sort of expect from a fluctuation dissipation argument. The diffusion experiment showed the same behavior. Namely, we have t to the a for a being some power. That was very clever.

Okay, one can also, and there are a series of papers in the book, look at such things as let's look at spheres that are bigger and bigger and bigger. There is sometimes a hand-waving argument, hand-waving is very dangerous. There is sometimes a hand-waving argument that for small spheres,

you might expect that the Stokes-Einstein equation could fail. What does small mean? It means small relative to some length scale unspecified in the liquid. But if you have really big spheres, to the really big spheres, the fluid outside looks like a continuum.

And therefore, say the hand-wavers, if you make the sphere big enough, you will see Stokes-Einstein type diffusive and driven behavior. Well, that doesn't appear to be the case. The experiment has been done. And if you go to large particles, you rather definitely do not move over to the micro-viscosity agreement, the macro-viscosity.

Okay, there are also a series of other experiments. Let's put a few names on it. Schmid, Ho, Ho-Yang.

And so there are a series of experiments in which real micro-rheology is done.

We have a mesoscopic particle, the force is applied, viscosity is inferred. And what is found is that the micro-viscosity is not at all the same as the viscosity measured experimentally. And if you look at frequency dependence, well, that doesn't help you.

Okay, so what is the importance of that result? Let's step back a bit. Do you remember I mentioned single particle micro-rheology and two particle micro-rheology? And people said that the micro-viscosity inferred by looking at correlation in the motion of two particles was the same as the viscosity

measured macroscopically, the true viscosity measured macroscopically. Well, yeah, the true viscosity measured macroscopically. Let's just call it eta, may match the two particle micro-viscosity. But there's an interesting issue here.

That is, if you do a real viscosity measurement on mesoscopic particles, the real viscosity measurement, this is a real micro, real rheology experiment using mesoscopic distance scales, does not equal the macroscopic viscosity.

And therefore, the diffusion viscosity inferred from two particle tracking, you can't, the fact that it's equal to the macroscopic viscosity, gee, what does that mean? Shouldn't the micro viscosity agree, the diffusion micro viscosity

agree with the real rheology experiments made on the same distance scales? There are some interesting issues here that are still under investigation. Okay, we shall advance a piece, and we shall advance two gels.

For the most part, this book does not discuss true gels. When I say a true gel, I mean we have long pieces,

and the long pieces are covalently cross-linked so that for essentially all purposes, those are permanent bonds and you have mesh work. The book does not cover true gels a great deal. Nonetheless, we do look at it a little bit, and

we can make a few observations about it. The first observation is that a true gel, not a concentrated solution, but a true gel, is a size filter. That is, a true gel really does let small particles through,

and really does stop large particles so they cannot enter. This physical result has been the basis of gel permeation chromatography since I was an undergraduate.

More years ago than I care to emphasize in detail, but there is the result. True gels are size filters. Real polymer solutions are not size filters. They may retard large and small particles to different extents, at least modestly.

However, particles of all sizes can pass through polymer solutions. They can't pass through gels. For small particles, there are some experiments by Park et al. On how the diffusion coefficient depends on solution properties.

And you do still get a stretched exponential. The stretched exponential has a concentration exponent, which is slightly less than one. And it has a size exponent which is, for the probe, which is about three-fifths.

But you should realize this is all from probes that are small enough to get through the holes. If you present a large object, it's trapped, it does not get through. Okay, one more set of experiments.

These are all new, Kate Luby Phelps and collaborators. And I am not going to sort out, I can't, exactly, leaves.

We can talk about Kate Luby Phelps and the work. And the core issue is let us attempt to model how objects move within a cell. And there's two sorts of issues. And one is, we're looking at a diffusion coefficient.

There's a concentration dependence. And the other is, one can look at the limit for very small particles of the diffusion coefficient and

ask what the effect of the system on very small particles is. And there is a great deal of experimentation done. And the net result of all of this, as you can say, or what you might have expected, except this is the actual experiment showing it. Is that you have the interior of a cell, and they're sort of long,

thready things that actually do trap some, but not all, large particles. And mixed in with the long, thready things are smaller macromolecules, we sort of knew they were there. And the smaller macromolecules also act to retard diffusion,

but not in the same way that the matrix mesh work does. And so this very pretty set of experiments says we can construct physical models that are made of materials we understand.

And the diffusion behavior of probes can be shown to be the same as the diffusion behavior within a cell. And we can then sort out what's going on. Very beautiful piece of basic physics.

We now hit a section of the chapter which was an original draft of the manuscript was much longer, except in the end I basically scrapped it.

And the section we are going to talk about has the name Microbe Rheology. And the reference here is to a series of experimental methods which assume,

and I really mean assume, that if we are looking at a probe in solution, delta, the light scattering spectrum, and I'll write that in terms of the field correlation function, is e to the minus.

And up here is the mean square displacement, which is of course a function of time. As we have seen, going back to the experiments of Apgar and Seng, and as we can see directly, namely if you look at the system, the spectra are not pure single exponentials.

This assumption has one basic problem, it's wrong. As a result, experimental analyses that rely on that experiment with a few.

exceptions are unreliable in their determinations of what's going on. Now there are some very clever exceptions. One of the very clever exceptions is an experiment due to Popescu. It's a light scattering experiment, except what was done was to use a diode laser

that gave us light whose wavelength, whose correlation length I could see, was extremely short. That is, if I look along the light that has come out of the laser at a series of points,

the light is more or less, but not quite, monochromatic. However, if I look at the phase at a series of points, over very tiny distances the phase of the light here tells me the phase of the light here, but if I go out any distance

at all, the phase of the light here and the phase of the light there are independent of each other. What's the experimental consequence of that? Well, suppose I use scattering off two particles. If the two particles are separated by this sort of distance, the phase of the light

here and the phase of the light there are independent from each other, and there's no interference. There's no way to tell the relative motion of the particles, with one exception.

The light is going in through a window, and we have some number of particles which are very close to the glass surface. And now we have scattering off the glass surface, and we have scattering off particles that are incredibly close to the surface, close enough that they are within a correlation

length, a light wavelength correlation length of the surface. And the light scattered by these few particles, and the light scattered by the window are coherent, and you get interference. And you can then look at the motion of a few particles very close to the window surface.

This is very useful if, for example, you have a basically opaque turbid solution, milk white solution. Regular light scattering is disappointing, not withstanding heterodyne coincidence spectroscopy and related techniques.

Light scattering spectroscopy is disappointing, but the Popescu technique, and there's a theoretical analysis, if I recall correctly, how, lets you measure the diffusion of the particles close to the window. However, Popescu and all are very careful to emphasize one thing, namely they are looking

at the diffusion of the particles over very short times, in which they are only looking at the lead term of this hypothetical exponential. The lead linear term behaves the way it's supposed to, it really does give you the

mean square displacement. All of the difficulties arise if you try to carry out over any distance. So there is this very pretty experiment, it really does these nice things, and it works. Okay, those are the experiments of Popescu et al.

You're a little limited in how far out in time you can get, but it works. And we have reached the end of the chapter. That is all the chapter there is except for a summary.

The summary is, we have looked at the diffusion of probes through polymer solutions. There is an enormous literature on probes in polymer solutions, and that enormous literature

is mostly non-communicating with the so-called micro-rheology literature based on our friend here.

So there are two literatures, they don't talk to each other much, however, we are going to be talking about the optical probe diffusion literature, optical including some other techniques like particle tracking. And the first point we can say is that if we measure diffusion using light scattering,

we find three sorts of behaviors, and one is a diffusion coefficient there and it falls as a stretched exponential in concentration. The second behavior which we see on occasion is re-entrance, where there is some regime

in which you, usually quite limited, in which you do not see this behavior, but then after you leave the regime, the behavior emerges. The third sort of thing we have seen, and this has been studied in detail in hydroxypropyl

cellulose solutions, is that on occasion you also get multiple modes, that is you see several relaxations going on at the same time in the same system.

How the system manages to do this is a little less clear, but it clearly does do it. Figure 943 turns to a study of this behavior, and what is plotted in there are alpha and nu as a function of polymer molecular weight.

For each polymer molecular weight, there is a concentration dependence, you measure D at a whole series of concentrations and you get out of that two numbers, an alpha and a nu, which correspond to one molecular weight. If you look at the figure, you see that from one times ten to the four, up to something

like five times ten to the five, over an order of magnitude and a half, there is a fairly nice line where alpha depends on molecular weight to a power. The reason I show the dextran measurements first is that historically these were the

first measurements in which on one hand the values of alpha and nu weren't very noisy, and on the other hand there were a whole bunch of different molecular weights, and you can see rather cleanly a linear behavior here.

There is also, for nu, a similar behavior in which nu goes from one down to about a lot of six over the same molecular weight range, and you see the parameters have a fairly clean polymer molecular weight dependence, which is what you should have expected if

the theory actually meant anything. Okay, another piece, these are experiments of Jenksi Amber and Vicky LaCroix and I, they did all the work, we looked at alpha against polymer molecular weight.

These are for probes in polystyrene sulfonate, and there are some fairly short polymers that really aren't random coils. And then you have a straight line and some points more or less on the straight line.

The point of the experiment was to test a theoretical calculation of alpha, and there is the line. The reason the theoretical calculation of alpha is of interest is that for big spheres and polymer coils in which the beads are quite small, there are no free parameters

in the line. The model actually gives a number out, not a number with a fitting parameter, and the line really is where it belongs, not over here or over here. So that was very satisfactory because it was a direct test of can we calculate interactions

between probe molecules and polymer coils in polymer solutions, and the answer was a rather definite, yep, we can. Figure 945 continues this, and over for polymers with a wide range of molecular

weights, if you plot alpha versus M, you get data that pretty much lie on the line. Of peculiar note, way at the bottom of the line there are a couple of points where the

polymer is in fact bovine serum albumin. Now bovine serum albumin is not a random coil, it's a ball with a well-defined structure. And so to plot bovine serum albumin on this graph, we didn't put in its actual

molecular weight. We plotted it as though it was a random coil polymer of some molecular weight, a molecular weight that would give us a random coil polymer the size of a BSA molecule. And those points sit right on the line.

Okay, what else can we do? We discussed comparison with gels, and the important point is that a real gel, a cross-linked gel is a size filter, it really does block the motion of objects that are bigger than

the holes in the gel. A polymer solution is not a size filter, its effect on moving probes may depend somewhat on the size of the probes, but even if you have really big particles, they can diffuse

through polymer solutions. We also emphasize, you can infer from d, diffusion coefficient, micro-viscosity, there is a very complicated story here, but we in general find that the micro-viscosity

is not the same as the viscosity, except for polymer solutions of very small molecular weight. If the polymers in solution are very small, the micro-viscosity and the macro-viscosity are pretty much close to each other.

There are a variety of tests of the so-called Langevin-Rondeles picture, which was derived for centrifugation, and in defense of the original authors, there are people who apply

this diffusion, perhaps this is not quite fair to the original work, nonetheless the original picture, which was for sedimentation, was that this should be some e to the minus a concentration to a power, probe size to a power, polymer molecular weight to the

power zero plus an eta zero over eta, solvent viscosity, solution viscosity. If the polymer solution is much more viscous than water, say if the viscosity of the

solution is 50 or 20 centipoise even, this term in concentrated polymer solutions disappears. Russo has given a very pretty test where they looked at fluorescein and labeled dextrins

and such not in various polymer solutions and they found including this term was useful, but they didn't get to go to very high viscosities and if eta is not much larger than eta zero, this term is indeed significant.

The important piece here as a conclusion, the one point I want to stress, is that the original theoretical model claims that there should be an M to the zero dependence, that is the resistance of the polymer solution to particle diffusion or sedimentation should

be independent of the molecular weight of the matrix. And the basis for that is the notion here is a polymer solution and the polymer coils are very large. Now if I go in and say cut the molecular weight of the polymers in half by splitting

them up, it is possible that I will go in and suddenly I have made a hole that's twice as big by cutting a polymer there. But if the polymers are really big, if I cut the molecular weight of the polymer in half, there are a very small number of cuts, there are a very large number of these

hypothesized gaps through which it's claimed the probes move. And almost none of the gaps have a polymer cut to their side. And therefore S and D in terms of this picture would be expected to be independent of polymer

molecular weight. Well the probe diffusion is quite sensitive to the matrix molecular weight. That was the pretty graph that showed alpha versus M and we saw quite strong dependence

like M to the 5 sixths or M to the first. And the reason for that must be that this picture of probes as advancing through a polymer solution by looking for gaps is incorrect.

Instead what presumably happens in a polymer solution is the probe is advancing, the polymer chains move out of its way, they're dragged with it, they move along with the probe because they're strong hydrodynamic interactions and therefore the probe particle in a polymer

solution does not look for holes between the chains because the chains aren't anchored to specific spatial coordinates, they're free to move. And because they're free to move, this mesh picture is simply incorrect. That's it for today and we have reached the end of chapter nine.

We'll discuss something else in the next lecture.