17 More Probe Diffusion
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Phillies, George
Lecture 17 - probe diffusion, part the last. George Phillies lectures from his book "Phenomenology of Polymer Solution Dynamics".
 Classes in Polymer Dynamic. Based on George Philly's book, Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011. And today, this lecture is lecture 17, Final Discussion of Probe Diffusion. 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 the 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. And in the writing in between, there is 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 authors. I will note here, for example, worth pondering who looked at spheres in the 4-megadolton Zanthan 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. 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-reology 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 a 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 dr again. And once again, on an appropriate plot of dr 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 as polymer concentration was increased. An interesting variation on this, see the text again, was to look at polybenzyl-L-lutamic 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 are 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, um, G, colloids. In fact, there's a whole chapter on colloids and we are 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 a spatial Fourier 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 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 are 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 Langevon equation, which describes particle motion. The Langevon 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 Langevon 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 and Statistical Mechanics, and there's an accompanying set of video lectures on that now being produced too, from this and this we can say, G, 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 the 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 4 dt. 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 would be a d6, right? You get a 1, 2, 3, 4, 5, or 6, 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 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 over, 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 Langevant 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-Pakora. And they actually show you a considerable number of details, and Burnham-Pakora book on light scattering then shows that, well, if you have a Langevant particle, the displacement distribution is a Gaussian, and therefore the light scattering spectrum, and I will write the field correlation function, is 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, oh, 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 Bosnian equation. A significant effort in statistical mechanics back in the 1970s, there are several very nice papers about it, you can look up for example Hermann's and Schau, 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 Bosnian equations, this is spheres in water, doing Brownian motion, the calculated diffusion coefficient is the same as the diffusion coefficient that you get assuming Stokes law. However, we now have the other two, some more technical obstacles. The first technical obstacle is of course we're talking about complex fluids, polymer solutions, colloid solutions, my concentrated, my cell systems, you can come up with a very long list of these, 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 Bosnian 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 the 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 long javan description does not apply. Now you might start to worry, gee, how can I tell if long javan 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, is Dubes 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 random motion, the 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 long javan equation. The 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 long javan equation? And the answer is there is an exact result due to the Zimni-Morrie and Bob Swanseig, known as the Morrie-Swanseig 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 it's 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 mind field hot hiding here, and people who say, oh Gaussian long javan 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 the list of other authors. The paper as an aside is noteworthy, is 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 analysis, 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 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 of... I mean, I've seen several other co-authors looking at the same sort of thing. And at all, you also look at delta X squared at fairly large times. And what they found is that if you go to large times, delta X squared grows as a power in time, and the power A is less than 1. That is, you have what could be described as a sober diffusive behavior. Same and weird. Well, having said that, it's not a Gaussian, there's an important aside on this. If you go through the literature, you can find bunches of experiments under the title Microbiology, 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 Sen. There are a series of other people who have done Microbiology, and let's see Crocker and Flaborators, and who else, Shen, and there are a series of other references in the text, and the other is the same data in a sense, but two experiments. And one is to say, here is a particle, and it has a displacement delta 1, and we can look at, for example, delta 1 square of T, and its average behavior. And the other is to say, we have here two particles, delta 1 and delta 2, and if the two particles are moving, we can look at the cross correlation. The cross correlation is the displacement of particle 1 during time T, and this object describes a cross diffusion object. These are both vectors, but composition is legitimately an outer product, a tensor product, so D1 and 2, in the general case, is a cross diffusion tensor, and it depends on the vector displacement between the two particles, and the viscoelastic system, you might discover there was in principle time dependence. Now, we've discussed at some length, we have a mean square displacement here, and therefore, if we pretend this is a diffusion coefficient, we can extract from it 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 microviscocity, D is 1 over the viscosity, and then there are a bunch of constants and numbers, and we can extract from each of these a microviscocity. 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 microviscocity depends on the time step we 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 1, this is mu 2, and eta mu 1 over eta mu 2, the two viscosities are not equal to each other, their ratio can have a difference, which is a factor of 2 or a factor of 3. So the viscosity viscoelastic properties that cover single particle motion and cover two particle correlations are not the same. You also could find, at least in the system that was studied, that the two particle microviscocity 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 macrophysiocytic viscosity, I mean the viscosity you measure if you actually get a reometer, 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 the series of others, and the core issue is they do what is called two particle microreology. And having done two particle microreology, 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 tracks T, the O-Sine tensor, which the O-Sine tensor describes hydrodynamic interactions between pairs of spheres. There's a power series in one over the distance to various powers, the O-Sine 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 microreology. Okay, very pretty experiment. We're going to look at microreology again, but we're going to look at the segmental scale, and this is a wonderful experiment by Dichtel and Sackman. 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 at 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 coarse 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 they 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 timescales on which these observations are being made are much shorter than the timescale on which this polymer molecule, in terms of the repetition model, has escaped from its tube and moved to a new location site over here. So we are looking at measurements on still fairly short timescales. However, what we find is the parallel motion is faster than the perpendicular, and this is actually a prediction of models that on short timescales you have local chain motions which can get fairly fast. There is, 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 microviscocity, and the microviscocity is something like an order of magnitude larger than the solvent viscosity. There is a microviscocity in there, but the polymer segments do not think they are moving through playing the solvent. They think they are moving through something that is quite resistive to their motion. Okay, well this is a very pretty experiment. It is all done microscopically. It is an experiment, it would be wonderful. Gee, this is a computer technology issue. If it could be carried out and you observe the motion out for very long times, and perhaps by the time this lecture is over, I will discover that it is in fact now been done, but it did not 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 is all this biotechnology stuff. But from the polymer physics standpoint, there is 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. They were essentially totally mono-dispersed. And thanks to modern biotechnology, you can go into the ring and cut the ring, and be sure you are cutting the ring exactly once. So you get a linear polymer whose molecular weight is identical to 1 part and 10 to the 5, or 1 part and 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 microbiology in these two systems. They got up to concentrations in natural units only about 6, unfortunately. However, what they found was that the microviscocity of the ring system, yeah, that is related to the microviscocity of the linear system, but in 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 ramp, and so the thing not only is biting its own tail, but it's doing loop-to-loop around itself. There could be concatenated rings, and there were all sorts of complaints if you discuss 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 do 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 might 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 are 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 going to be shown directly, there are result regions which are relatively fluid. There are regions where particle motion is more slow down, which are in some sense vitrified. Those of you who are familiar with the Kibblesen glass model and other recent work on glasses may realize that I am describing a solution. It should be heterogenious, 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 varied that 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 Kibblesen glass model, which proposes that glass efficaciation occurs because you start forming heterogeneous regions which are in some sense vitrified and resist motion. Well, here's the experiment and you're seeing it directly. Okay, so much for the pretty work on wheat glyadins. Let us chug ahead and we have another section. In the next section of the paper, we are going to talk about true microbiology. What do I mean by true microbiology as opposed to diffusion and a microfascosity from diffusion measurements? I mean, we're going to do an experiment in which we take a mesoscopic particle. We apply to the mesoscopic particle of force F. We discover the particle has an associated velocity B. 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 that we're actually applying an external force to the system. Okay, a whole bunch of experiments are studying this. And so, for example, we got our amplard. What is done is to apply a force to the system, and this can be done, 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 square 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 number around three-quarters. And so, what they did 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 in this patient argument. The diffusion experiment measurement showed the same behavior, namely, we have T to the A for A being some power. 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? 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-wavars, 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-fascosity agreeing with macro-fascosity. Okay. There are also a series of other experiments. Let's put a few names on it. Schmidt, a Ho, a Ho, a Ho, and so there are a series of experiments in which real microbiology is done. We have a mesoscopic particle, the force is applied, a viscosity is inferred. And what is found is that the micro-fascosity 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 a single particle microbiology and two particle microbiology, and people said that the micro-fascosity inferred by looking at the 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-fascosity. 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 microbiology experiment using mesoscopic distance scales, does not equal the macroscopic viscosity. And therefore, the diffusion viscosity inferred from two particle tracking, the fact that it's equal to the macroscopic viscosity, gee, what does that mean? Shouldn't the micro-fascosity agree that the diffusion micro-fascosity agree with the real-reology 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 return 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 1, and it has a size exponent, which is for the probe, which is about 3-fifths, but you should realize this is all for probes that are small enough to get through the holes. If you present a large object, it gets trapped. It does not get through. Okay. One more set of experiments. These are all duquets, ruby-felps, and collaborators, and I am not going to sort out exactly who leaves. We can talk about k-buby-felps and work, and the core issue is let us attempt to model how objects move within a cell. There's two sorts of issues, and one is, 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, sort of what you might have expected, except this is the actual experiment showing it, is that you have the interior of a cell, and there's sort of long, thready things that actually do trap and solve, but not all large particles, and mixed in with the long, thready things are smaller macromolecules. Here they were there, and the smaller macromolecules also act to retard diffusion, but not in the same way that the matrix meshwork does. And so this very pretty set of experiment 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. Okay. Okay. 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 micro-reology, 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, let's use 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. What'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 Popeshko. It's a light scattering experiment, except what was to do, Don, 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 in 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 do 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 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-light 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 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 at all. You're a little limited to 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-reology literature based on our friend here. So in two literatures, they don't talk to each other much. However, we're 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, yeah, there's 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're 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 hydroxy-prol-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's 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 1 times 10 to the 4, up to something like 5 times 10 to the 5, 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 bound to about a lot of 6 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 James Yambert and Mickey McCroy and I, they did all the work. We will look at alpha against polymer molecular weight, these are for probes and 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 of 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 emphasized, if you can infer from the diffusion coefficient of microviscocity, there is a very complicated story here, but we in general find that the microviscocity is not the same as the viscosity, except for polymer solutions in a very small molecular weight. If the polymers in the solution are very small, the microviscocity and the macroviscocity are pretty much close to each other. There are a variety of tests of the so-called Langevon-Rondelais picture, which was derived for centrifugation, and in defense of the original authors, there are people who apply this to 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 dextrans 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 viscosity. So, 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 is twice as big by cutting the 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 is 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 the 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 9. We'll discuss something else in the next lecture.
 Classes in Polymer Dynamic. Based on George Philly's book,  Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011.  And today, this lecture is lecture 17, Final Discussion of Probe Diffusion.  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 the 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.  And in the writing in between, there is 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 authors.  I will note here, for example, worth pondering who looked at spheres in the 4-megadolton Zanthan  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.  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-reology 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 a 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 dr again.  And once again, on an appropriate plot of dr 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 as polymer concentration was increased.  An interesting variation on this, see the text again, was to look at polybenzyl-L-lutamic 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 are 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, um, G, colloids.  In fact, there's a whole chapter on colloids and we are 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 a spatial Fourier 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 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 are 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 Langevon equation,  which describes particle motion.  The Langevon 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 Langevon 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 and Statistical Mechanics,  and there's an accompanying set of video lectures on that now being produced too,  from this and this we can say, G, 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 the 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 4 dt.  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 would be a d6, right?  You get a 1, 2, 3, 4, 5, or 6, 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 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 over, 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 Langevant 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-Pakora.  And they actually show you a considerable number of details,  and Burnham-Pakora book on light scattering then shows that,  well, if you have a Langevant particle, the displacement distribution is a Gaussian,  and therefore the light scattering spectrum, and I will write the field correlation function,  is 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, oh, 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 Bosnian equation.  A significant effort in statistical mechanics back in the 1970s,  there are several very nice papers about it,  you can look up for example Hermann's and Schau,  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 Bosnian equations,  this is spheres in water, doing Brownian motion,  the calculated diffusion coefficient is the same as the diffusion coefficient  that you get assuming Stokes law.  However, we now have the other two, some more technical obstacles.  The first technical obstacle is of course we're talking about complex fluids,  polymer solutions, colloid solutions, my concentrated, my cell systems,  you can come up with a very long list of these,  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 Bosnian 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 the 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 long javan description does not apply.  Now you might start to worry, gee, how can I tell if long javan 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,  is Dubes 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 random motion,  the 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 long javan equation.  The 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 long javan equation?  And the answer is there is an exact result due to the Zimni-Morrie and Bob Swanseig,  known as the Morrie-Swanseig 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 it's 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 mind field hot hiding here,  and people who say, oh Gaussian long javan 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 the list of other authors.  The paper as an aside is noteworthy,  is 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 analysis,  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 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 of...  I mean, I've seen several other co-authors looking at the same sort of thing.  And at all, you also look at delta X squared at fairly large times.  And what they found is that if you go to large times,  delta X squared grows as a power in time,  and the power A is less than 1.  That is, you have what could be described as a sober diffusive behavior.  Same and weird.  Well, having said that, it's not a Gaussian,  there's an important aside on this.  If you go through the literature, you can find bunches of experiments  under the title Microbiology, 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 Sen.  There are a series of other people who have done Microbiology,  and let's see Crocker and Flaborators,  and who else, Shen,  and there are a series of other references in the text,  and the other is the same data in a sense, but two experiments.  And one is to say, here is a particle, and it has a displacement delta 1,  and we can look at, for example, delta 1 square of T, and its average behavior.  And the other is to say, we have here two particles, delta 1 and delta 2,  and if the two particles are moving, we can look at the cross correlation.  The cross correlation is the displacement of particle 1 during time T,  and this object describes a cross diffusion object.  These are both vectors, but composition is legitimately an outer product,  a tensor product, so D1 and 2, in the general case,  is a cross diffusion tensor, and it depends on the vector displacement  between the two particles, and the viscoelastic system,  you might discover there was in principle time dependence.  Now, we've discussed at some length, we have a mean square displacement here,  and therefore, if we pretend this is a diffusion coefficient,  we can extract from it 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 microviscocity,  D is 1 over the viscosity, and then there are a bunch of constants and numbers,  and we can extract from each of these a microviscocity.  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 microviscocity depends on the time step we 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 1,  this is mu 2, and eta mu 1 over eta mu 2,  the two viscosities are not equal to each other,  their ratio can have a difference, which is a factor of 2 or a factor of 3.  So the viscosity viscoelastic properties that cover single particle motion  and cover two particle correlations are not the same.  You also could find, at least in the system that was studied,  that the two particle microviscocity 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 macrophysiocytic viscosity,  I mean the viscosity you measure if you actually get a reometer,  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 the series of others,  and the core issue is they do what is called two particle microreology.  And having done two particle microreology, 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 tracks T, the O-Sine tensor,  which the O-Sine tensor describes hydrodynamic interactions between pairs of spheres.  There's a power series in one over the distance to various powers,  the O-Sine 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 microreology.  Okay, very pretty experiment.  We're going to look at microreology again,  but we're going to look at the segmental scale,  and this is a wonderful experiment by Dichtel and Sackman.  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 at 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 coarse 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 they 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 timescales on which these observations are being made  are much shorter than the timescale on which this polymer molecule,  in terms of the repetition model, has escaped from its tube and moved to a new location site over here.  So we are looking at measurements on still fairly short timescales.  However, what we find is the parallel motion is faster than the perpendicular,  and this is actually a prediction of models that on short timescales  you have local chain motions which can get fairly fast.  There is, 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 microviscocity,  and the microviscocity is something like an order of magnitude larger than the solvent viscosity.  There is a microviscocity in there, but the polymer segments do not think they are moving through playing the solvent.  They think they are moving through something that is quite resistive to their motion.  Okay, well this is a very pretty experiment. It is all done microscopically.  It is an experiment, it would be wonderful.  Gee, this is a computer technology issue.  If it could be carried out and you observe the motion out for very long times,  and perhaps by the time this lecture is over,  I will discover that it is in fact now been done,  but it did not 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 is all this biotechnology stuff.  But from the polymer physics standpoint, there is 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.  They were essentially totally mono-dispersed.  And thanks to modern biotechnology, you can go into the ring and cut the ring,  and be sure you are cutting the ring exactly once.  So you get a linear polymer whose molecular weight is identical to 1 part and 10 to the 5,  or 1 part and 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 microbiology in these two systems.  They got up to concentrations in natural units only about 6, unfortunately.  However, what they found was that the microviscocity of the ring system,  yeah, that is related to the microviscocity of the linear system,  but in 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 ramp,  and so the thing not only is biting its own tail,  but it's doing loop-to-loop around itself.  There could be concatenated rings,  and there were all sorts of complaints if you discuss 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 do 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 might 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 are 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 going to be shown directly,  there are result regions which are relatively fluid.  There are regions where particle motion is more slow down,  which are in some sense vitrified.  Those of you who are familiar with the Kibblesen glass model  and other recent work on glasses may realize that I am describing a solution.  It should be heterogenious, 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 varied that 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 Kibblesen glass model,  which proposes that glass efficaciation occurs  because you start forming heterogeneous regions  which are in some sense vitrified and resist motion.  Well, here's the experiment and you're seeing it directly.  Okay, so much for the pretty work on wheat glyadins.  Let us chug ahead and we have another section.  In the next section of the paper,  we are going to talk about true microbiology.  What do I mean by true microbiology  as opposed to diffusion and a microfascosity  from diffusion measurements?  I mean, we're going to do an experiment  in which we take a mesoscopic particle.  We apply to the mesoscopic particle of force F.  We discover the particle has an associated velocity B.  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  that we're actually applying an external force to the system.  Okay, a whole bunch of experiments are studying this.  And so, for example, we got our amplard.  What is done is to apply a force to the system,  and this can be done,  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 square 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 number around three-quarters.  And so, what they did 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 in this patient argument.  The diffusion experiment measurement showed the same behavior,  namely, we have T to the A for A being some power.  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?  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-wavars,  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-fascosity  agreeing with macro-fascosity.  Okay.  There are also a series of other experiments.  Let's put a few names on it.  Schmidt,  a Ho,  a Ho,  a Ho,  and so there are a series of experiments  in which real microbiology is done.  We have a mesoscopic particle,  the force is applied,  a viscosity is inferred.  And what is found  is that the micro-fascosity  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  a single particle microbiology  and two particle microbiology,  and people said that the micro-fascosity  inferred by looking at the 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-fascosity.  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 microbiology experiment  using mesoscopic distance scales,  does not equal the macroscopic viscosity.  And therefore, the diffusion viscosity  inferred from two particle tracking,  the fact that it's equal to the macroscopic viscosity,  gee, what does that mean?  Shouldn't the micro-fascosity  agree that the diffusion micro-fascosity  agree with the real-reology 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 return 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 1,  and it has a size exponent,  which is for the probe,  which is about 3-fifths,  but you should realize this is all for probes  that are small enough to get through the holes.  If you present a large object,  it gets trapped.  It does not get through.  Okay.  One more set of experiments.  These are all duquets,  ruby-felps, and collaborators,  and I am not going to sort out  exactly who leaves.  We can talk about k-buby-felps and work,  and the core issue is let us attempt to model  how objects move within a cell.  There's two sorts of issues,  and one is,  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,  sort of what you might have expected,  except this is the actual experiment showing it,  is that you have the interior of a cell,  and there's sort of long,  thready things that actually do  trap and solve,  but not all large particles,  and mixed in with the long,  thready things are smaller macromolecules.  Here they were there,  and the smaller macromolecules  also act to retard diffusion,  but not in the same way  that the matrix meshwork does.  And so this very pretty set of experiment 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.  Okay.  Okay.  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  micro-reology,  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,  let's use 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.  What'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 Popeshko.  It's a light scattering experiment,  except what was to do, Don,  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 in 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 do 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 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-light 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 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 at all.  You're a little limited to 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-reology literature  based on our friend here.  So in two literatures, they don't talk to each other much.  However, we're 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, yeah, there's 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're 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 hydroxy-prol-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's 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 1 times 10 to the 4,  up to something like 5 times 10 to the 5,  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 bound to about a lot of 6  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 James Yambert and Mickey McCroy and I,  they did all the work.  We will look at alpha against polymer molecular weight,  these are for probes and 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  of 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 emphasized,  if you can infer from the diffusion coefficient of microviscocity,  there is a very complicated story here,  but we in general find that the microviscocity is not the same as the viscosity,  except for polymer solutions in a very small molecular weight.  If the polymers in the solution are very small,  the microviscocity and the macroviscocity are pretty much close to each other.  There are a variety of tests of the so-called Langevon-Rondelais picture,  which was derived for centrifugation,  and in defense of the original authors,  there are people who apply this to 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 dextrans 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 viscosity.  So, 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 is twice as big by cutting the 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 is 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 the 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 9.  We'll discuss something else in the next lecture.
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