18 Colloid Dynamics
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Phillies, George
Lecture 18 - colloid dynamics, a topic rarely included in discussions of polymer dynamics. The forces are the same; the particle shapes are different. George Phillies lectures on polymer dynamics, based on his book "Phenomenology of Polymer Solution Dynamics".
 Lecture 18, Colloid Dynamics, based on George Philly's book, Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011. And today, this lecture is lecture 18, Colloid Dynamics. I'm Professor Philly's, and this is the next lecture in my series on polymer dynamics. Our last lecture finished off with a discussion of optical probe diffusion. What I'm going to do today is to push on to an entirely new topic in chapter. The new chapter is 10, and the new topic of chapter 10 are colloid suspensions and colloid solutions. If you go through the reviews found in the early part of the book, the reviews by people like Tyrell and Lodge, and a whole variety of other books on polymer dynamics, you'll make an interesting observation. There are lots of discussions of polymer diffusion, polymer viscosity. Almost none of the other sources you look at discuss probe diffusion at any length, and there's absolutely no discussion of colloid dynamics at all. Now, why might you be interested in looking at how colloid particles move? Well, let's remember what a colloid particle is. The simplest case, we have objects in solution, and the objects in solution are spheres that are quite small, like a half a nanometer or tens or hundreds of nanometers. They've been surface treated in some way in most cases, and as a result, well, you say this is a suspension of particles, they're big particles, but they can be density matched, and even if they're not, they can be charged or coated, given polymer coats on the surface, so that they sit in solution. Why would we be interested in these round hard objects? They're quite rigid. I've drawn a sphere, but some of them aren't spheres, they're ellipsoids or whatever. Why would we be interested in these in a discussion of colloid? Why would we be interested in these in a discussion of polymer dynamics? Why are we talking about colloid dynamics? The answer is, these are sitting in solution, and there are three sorts of forces on them. There are random thermal forces due to fluctuations in the solvent. That's what drives Brownian motion. There are hydrodynamic interactions. I'll talk about those in a bit. Namely, as one particle moves, it sets up a wake in the surrounding solution, like the wake of a motorboat. Well, not really exactly like, but somewhat similar. And the wake drags neighboring particles along with it. And then there are direct interactions of which the most important is volume exclusion. Because these objects are solids, they cannot pass through each other. I could also discuss polymer solution. Here is a non-dilute polymer solution. The important feature of the non-dilute polymer solution is that the forces between the polymer coils, thermal interactions, which are the polymer coils of the solvent, are dynamic interactions between polymer coils, and volume exclusion interactions between polymer coils. Those forces are precisely the same list of forces that act between colloids. So we have two systems. We have the same sets of forces acting. We're looking on the same time scales, so the same basic dynamic equations apply. However, there is one substantial difference between polymer dynamics and colloid dynamics. And the substantial difference is shape, or as some people would call it, topology. My apologies to real topologists who will point out that this, if linear, and that, both simply connected volumes end of discussion. They're the same, aren't they? Well, no, not really. However, the core difference is, this is a sphere. That's something that's long and stringy. And you can certainly imagine this object having modes of motion and interactions that are not stunningly the same as spheres. For example, if you have two polymer coils, you can certainly imagine one of the polymer coils wrapping itself around the other, like two rattlesnakes in a rattlesnake ball. Sears, however, just sit there. They can't wrap around each other. And if you believe in the arguments that propose there are entanglement constraints, though the entanglement models don't say what an entanglement is, then you would say that a polymer coil plausibly entangles, at least for many pictures of what entanglement means, but spheres certainly do not do so. Okay, so there is a fundamental difference in shape and in possible dynamics, but the basic forces are the same, and the basic, gee, this is over-damped motion, f equals ma equations are the same. There's some similarities, some differences. One thing one might, however, propose is that phenomena that are common to colloid dynamics and to polymer dynamics cannot be explained on the grounds that polymers can wrap up in knots, and polymers can perform repetitions, snake-like motion like the boa constrictor going through the bamboo grove. Anything that is common to these two systems very certainly cannot be assigned to topology. So that is our perspective here, and what we are going to do is to look at a variety of discussions of how sears-interim move as opposed to how polymers move, and we are going to look at colloid phenomenology, and that will tell us something about, gee, the importance of topology. The short discussion on how does this compare with other works is, there is no comparison, and the reason there is no comparison is that if you look at other treatments of polymer dynamics, you will be very hard pressed to find another that does a substantial discussion of colloid motions. So let us start out, and we will say a bit about hydrodynamic interactions. The front-end piece is as follows. We have here is a colloid sphere. It is small enough that we can approximate it as a point in terms of the other distance scales. It moves through a polymer-polymer solution. As it moves through a polymer solution, it is exerting a force, there is the force, f being exerted on the solution, and this causes the liquid to move, and so you get a flow field. And if out here the flow field has a velocity v, the coupling at low frequencies between v and f is v equals t. T is a tensor. It is a 3 by 3 matrix. The reason it is a tensor is that we are starting off with a vector here. We are ending up with a vector here, and the two vectors are not parallel. The mathematical structure, this is linear, that we can use to transform one vector into the other, is a 3 by 3 tensor, the O-sine tensor, whose I will strip out constants. The O-sine tensor folds off as 1 over r, where r is the distance from the point particle to the point of interest in the fluid. And then the tensor dependence is the identity tensor, plus this is an outer product, r hat, r hat. r hat is the unit vector pointing from the sphere to the place in the fluid. There are some constants which you can put in differently depending on whether you describe here the force or the velocity of this point. What are the important features of the O-sine tensor? First of all, the flow field is not simply parallel to the velocity of the original object. It has this motion that I have sort of sketched. Second, the O-sine tensor falls off as 1 over r. That is not a potential energy falling off as 1 over r. If I have an object which I try to hold still at this point, the flow field creates a force on the object, f prime. And the force, we have a force here creating a force there. The force falls off as 1 over r. This is the longest range force in nature. Now you may worry a bit, oh gee, is it really infinitely long-ranged? And I will now point out the constraints because O-sine is an approximation. The first constraint is that if I apply a force here for a brief period of time, you have to set up the O-sine flow field and that takes a while. And until you have set up the flow field completely, there are a more complicated set of equations due to Bowson-esque to describe how a force here for some short period of time creates a flow there. The flow field has to propagate outwards and momentum propagation in a fluid occurs at the speed of sound. So if you are looking at very long ranges or very short times, this is not perfect. But we aren't at very long ranges, we aren't at very short times. There is another constraint that is sometimes lost, which is as follows. We are actually doing this experiment inside a container. And the container is filled with a solvent, and the solvents we are talking about are mostly pretty incompressible. As a result, if I draw a line, an imaginary mathematical surface, across the container at some point, the total volume flow of fluid across that line has to be zero. If it wasn't, we'd be compressing the fluid and this can't be done to any significant extent. And therefore, as we approach the walls of the container, somehow there has to be a backflow, which one could in principle calculate, and the backflow has the effect that the total volume flow across the system is zero. Now you have to put one qualification in. Suppose I drew the mathematical line here. In that case, the mathematical line is intercepting the solid object. And if we have a flow delta V of object into this volume, we must have a net flow minus delta V of solvent out of it. This is incompressibility when you remember this object has a volume. This is a significant effect in real experiments. At a certain reasonable level of approximation, the whole thing is known as reference frame corrections. The core issue in reference frame corrections is that the reference frame that is stationary with respect to the walls of the container, that's an obvious inertial reference frame, and the reference frame that is stationary with respect to the solvent, because the solvent is moving this way, are not the same. And reference frame corrections are treated in great detail in a paper by John Gamble Kirkwood and collaborators, 1960 paper. The paper actually appears after Kirkwood's death in the Kirkwood Memorial volume, and his four co-authors insisted he'd done the work so he should be the lead author. So there is something called a reference frame correction. However, the Oceane tensor says there are these interactions, and one moving particle puts a force on its neighbors, unless the neighbors simply are bobbing along with the solvent. And if the neighbors are bobbing along with the solvent moving at this speed V, they're moving, and this velocity and that velocity are correlated. Okay, so that was the Oceane tensor, and now we're going to put down something else. And the next piece comes out of an arc, what is known as a fluctuation dissipation argument. Fluctuation dissipation arguments are potentially tricky and hazardous. That is, it is possible to charge in, make a set of statements, all of which seem reasonable, and you can get surprised by what you predict, and by the way, you were correct to be surprised because the prediction somehow went a bit astray. However, in this case, it's fairly clear what you're saying, and it's been tested experimentally. The statement is, if I have a particle here that does this under the influence of an external force, say there are little magnetic bits inside the particle and I apply a magnetic field gradient, a nearby particle will be dragged along, and the correlation between the displacement of particle one and the displacement of particle two, the correlation is given by the Oceane tensor. That is, it's something that falls off as one over R, and it has this tensor nature rather than everything simply being parallel. So that's the fluctuation dissipation argument. And what we say is that if you see this behavior for driven motion, and if you're in the low speed linear regime, which we are, you will also see it for diffusive motion, and therefore, if you look at the random walk of two particles in a liquid, their motions are correlated as described by the Oceane tensor. Let me say, if you think back to what you may have read about Brownian motion and undergraduate thermal physics, you may have thought, well, gee, aren't the motions of Brownian particles uncorrelated? And the answer is, if they're way far apart, or if you don't know what questions to ask, they appear to be uncorrelated. And the whole study of Brownian motion for a very long time was that you could do all of these statistical correlations, but until Langevon did his very clever equation, until Langevon did his work on Brownian motion, it wasn't clear what questions to ask about Brownian motion, and therefore, you got lots of answers that weren't stunningly helpful. And afterwards, everything worked. So there is the statement, well, at the front end, we have driven motion, and then if you look at diffusion, you get the same result. Notice, however, we've said the effect falls off as inverse as distance, and the basic unit is the size of the particle over the distance, so we have a dimensionless number here. So if the particles are tiny and way far apart, that's what far apart means, the interaction falls off a lot. Nonetheless, it doesn't fall off to zero, and there have been a series of experiments due to O. Crocker, a very clever fellow, and Miner, and Quake. And what they do is to say, well, let us look into a system, let us actually, one way or another, track the motion of parallel particles, pairs of particles, we get the particles close together, and we can then actually look at diffusive motion, and we can ask, how does diffusive motion work? And the short-format answer is, you can actually do the experiment, and you find that displacement one, displacement two are correlated. This defines what is known as a cross-diffusion tensor. The cross-diffusion tensor describes the relative motion of pairs of particles, and the cross-diffusion tensor, as is predicted by the fluctuation, dissipation arguments, is proportional to the O-seam tensor, and falls off as one over R at long distances. So those are hydrodynamic interactions. Hydrodynamic interactions were inserted into polymer dynamics very early on by the original treatment of Kirkwood and Reisman. Same Kirkwood as the reference frame fell, a very clever man, leading American statistical mechanization of the first half of the century, of the last century, Kirkwood and Reisman. And what they did is to say, here is a polymer chain, and we can model the polymer chain as a string of frictional beads, and the frictional beads are attached to each other by little springs, or perhaps the frictional beads are monomers and they're bonded together. And if the chain moves in solution, well, the chain has some center of mass velocity, B. If I apply an external flow field, the polymer flips head over tail, and has, this is coming out perpendicular to the board, it has a rotational velocity omega, and then the individual beads have fluctuation velocities relative to the general drift motion. But you know, these things are all attached to each other. It's like a very long snake. If it moves a large distance, the head and the tail have to stay attached. And therefore, this object will sort of like a bag of beads with internal fluctuations. What they put in though is the key issue, is that these beads, or at least some of them, are obliged to be moving with respect to the flow field. Why? Well, let us draw a polymer coil again. It has a center of mass velocity this way. I've applied a velocity shear to the fluid, so the fluid is going in opposite directions. The polymer is tumbling head over tail as it does this. And so up here, the beads try to move with the fluid. And down here, the beads try to move with the fluid. But if you think of the recall, this is circular motion, that means that here, and not so probable, let's extend this a bit, here, the beads have to be moving sideways. So at least some of the beads, to some extent, have to be moving with respect to the fluid. If they weren't, the shear would tear the polymer apart. And these motions with respect to the fluid, not resolved. We have a bead, it's moving with respect to the fluid, it is exerting a force on the fluid. And so all of these beads, moving with respect to the fluid, each create, and there are lots of beads, but they each create their own flow fields, which act on all of the other beads. And then you have to be a little clever in the math to re-sum things to get a self-consistent solution. And you do, and this is the Kirkwood-Ricewood model for polymer dynamics. If you are clever and careful, you can take this model, and you can use it to treat pairs of interacting polymer chains. And when you put it in to treat pairs of interacting polymer chains, which can be done, you get out the concentration dependence of all the self-diffusion coefficient or the viscosity. You have to work a bit harder to do that, but it's possible. Okay, I've talked about hydrodynamic interactions. For most of the objects we're talking about, the direct interaction is basically excluded volume. That is, we have a sphere here, and we have a sphere there, and they can not actually, they can touch, but they can't interpenetrate because they're solids. Now, there is a bit of a cheat at this point. The bit of a cheat at this point is that the real spheres you're talking about, well, they might be charged because you put them into water and you suspend them by charging them. They might have very short polymer molecules adhered to their surface to keep them from sticking. They may have van der Waals forces between them, and the net result of all of those is that at very short distances there's some extra interactions that are not exactly hard sphere. Now, approximately speaking, if you're just talking about hydrodynamics, this is not serious, but the real hydrodynamic interaction between two spheres is not exactly O-sene, that is, the real interaction tensor, real, is the O-sene tensor, which is 1 over r, and then there are 1 over r to the n, and there are a bunch of higher order corrections. Higher order corrections are short range. Their importance depends on how likely it is who find two of these objects real close together, and that likelihood of being real close together is modulated by these short range potentials, which are a little harder to determine in great detail. So life gets a little trickier than we would like, but approximately speaking, it can be made to work. Suppose, however, we had real hard spheres, and I get real hard spheres, sure I can. I can do a computer simulation. Real hard spheres have thermodynamic properties. The thermodynamic property is described by phi, their volume fraction, and what is phi? Phi is equal to the number of spheres in some volume times 4 pi r sphere cube over 3, the volume of a single sphere, this is the total volume of all of the spheres, divided by the total volume of the system. So we can plot from the 0 out to, I suppose you could say 1, but you can't get to 1. I'll point out why not in a second. 0 to, you actually don't get out here. Because the spheres are hard, their distribution functions, their statistical mechanical distribution functions, don't depend on temperature. That is, either a configuration of spheres is allowed, because no sphere is overlapped, or it's forbidden, because you've got two spheres trying to be in the same place at the same time. And guess what, no matter what you do to the temperature, it's allowed or forbidden. The highest volume fraction you can actually get is close packed, and that's about 0.74 volume fraction. You can be a little more precise, and there are actually several close packed configurations, but they all have the key feature, you can't get more than 0.74. As a historical aside, this result was believed, but had not actually been proven when I was a graduate student. The issue, which is a math issue, is that you couldn't find any configurations that were closer packed than this, but what the mathematicians had to do was to prove there was no way to pack a sphere. Here is a sphere. So the sphere had 13 other spheres touching it. If you could do that, which you can't, you could create that configurations that were regular in all these other things, and had densities higher than hexagonal close packed. Well, at some point, fairly recently, the mathematicians were able to prove this result, which most people believe. And we can now say, this is the upper limit, that's how many spheres you can fit, because that's all the packing. On the other hand, you can imagine, current spheres are like ball bearings or toy marbles. So you take the toy marbles or the ball bearings, you put them in a cloth sack, and you shake them. And you really want a cloth sack, not a box, because the ordering from the walls of a nice rectangular box propagates into the side. So what happens if you put ball bearings or old flour or salt, anything that's nice and regular, in a bag, and you shake it? Well, the first thing that happens if you shake it, it starts to settle. And if you shake it longer, it settles more. And after a while, you shake more, and you just don't make more progress. And it is fairly hard to pack things so that you get a volume fraction that is much above 0.64. Now, that's not an exact result, because if you keep shaking, you can find better packing. But there's an upper limit about here to what an irregular packing will look like. There are two more pieces. The two more pieces are due to a set of computer experiments by Ray and Hoover. The computer experiments go back almost half a century at this point. And what they did was to study hard spheres and the behavior of hard spheres at different densities. And what they demonstrated is that there is a boundary at about 0.49, people will quote, four. I think the four, that terminal four is a little enthusiastic, given the size of the system that we studied. And coming up here to about 0.55. The issue with hard spheres is as follows. Suppose I have less dense hard spheres here and more dense hard spheres there. Well, if these were small molecules that could form a liquid, we would have, for example, a gas up here, a liquid down there. And there would be a free surface that would be supported by the attractive interactions between the molecules. So you have a well-defined liquid and you have a well-defined gas. Hard spheres have no attractive interactions. That's why they're called hard. And therefore, there is nothing to keep, to maintain this boundary between two regions of different densities. Nonetheless, if you do the computer experiments, what you discover is that you have one phase out to about this density. You then have a region in which you have two phases, you have a biphase at region. And in the biphase at region, there are less densest zones and more densest zones. And then up above here, going out to here anyhow, you have a single phase. And what is said is the lower density phase is a fluid gas, basically. And the upper density phase is claimed to be an expanded solid because its structure is more or less solid-like, but the density is too low. But if you think about things, the argument I gave as to why hard spheres cannot have a gas-liquid boundary works just as well for a gas-solid boundary. And therefore, what you presumably actually have here, this must be a gas because it can't be anything else. And this is actually really also a gas because there's no attractive potential to hold the things into the solid. And therefore, we have here what would appear to be an example of a gas-gas phase transition. But one of the two gases is really very much solid-like. And the solid crystal structure, if you really get up, where the closely packed amorphous structures, and gee, there are more gaps in it. My search of the literature says this issue has not been studied very recently. And if you studied things with much larger systems, you might get a more interesting result or a clearer result anyhow. The important issue is, if you are packing hard spheres at about half by volume, there is expected to be a thermodynamic phase transition leading up at a density of 0.55 to what is a distinct phase. And in between, you have two phases present. And that's worth keeping in mind as we chug through this. Okay, so what have people done? Well, people have done on colloids more or less all of the same experiments that have been done on polymer solutions. We haven't talked about all of the polymer measurements yet. We'll get to them. What sort of things can you do? Well, the only thing you can do for a colloid solution is to measure the single particle, self or tracer, diffusion coefficient. And if you do that, this is for example figure 10.1, you measure the diffusion coefficient using any variety of methods. And it's some function of sphere concentration and the diffusion coefficient falls. You have one interesting little bit which has sort of been brought up also with polymers. That is, you can measure the diffusion coefficient over fairly short time periods. You can measure it using light scattering spectroscopy. And you can measure the diffusion rate over very long time periods using, for example, fluorescence recovery after photobleaching. Or you can measure the mutual diffusion coefficient, concentrated system, and use a very large scattering vector. This works too. That's due to PUSI and co-workers. And the net result is you can measure two diffusion coefficients more or less. You can measure a short time diffusion coefficient which does roughly this. You can measure also a long time diffusion coefficient. And the long time diffusion coefficient drops off significantly more rapidly than the short time diffusion coefficient that stays down there. For the short time diffusion coefficient, we can write dS is d01 plus some constant short volume fraction. And you can actually measure k1S and you get O minus, oh, what's the magic number, 1.8 plus or minus 0.06. And there is a theoretical result which is about minus 1.73. Those two are in quite satisfactory agreement because part of this k1S depends on fairly short range interactions. And any deviation of the sphere is from perfectly hard sphere would give you a slightly different number here. So you can actually predict this concentration dependence. You can also do a completely different set of experiments. And the completely different set of experiments are due to D Georgiou. And the notion here is we will take spheres. And because we have a wonderful supplier, we have spheres that are optically anisotropic inside. And as a result, if I send in polarized light and look at the scattered light, some of the scattered light is polarized the same way as the incident light. And that basically just says the light sees a sphere. And some of the light is depolarized, so I put in vertically polarized light, light polarized perpendicular to the scattering plane. I look at, I'm putting in prisms here and here, horizontally polarized light. And that light, the amount of that light is determined by the shear orientation. Now there are two features of the VH scattering. First of all, if here's a depolarizing sphere, if it rotates, the sphere rotates, the amount of VH scattering changes. The sphere is doing Brownian random rotation, so the intensity of the VH scattering fluctuates in time. And the time scale of the VH intensity fluctuations exactly tracks the time scale on which the sphere reorients. So after a great deal of clever math, I can use VH scattering to measure rotational diffusion. Question? This is an average measure of the light. What you do is to measure the time correlation function of the scattered light. So we have the intensity of the VH scattered light at time t. We now measure it at a later time, t plus tau. We do this, repeat this for lots and lots of times t to get an average behavior. And what is left is a function of tau, and tau is the time separation. And so what it means is if the light is bright at one instant in time, so IVH of t is large, because this is pointing exactly the right way, after a while the sphere forgets which way it's pointing, and IVH relaxes back to some average value. The correlation function, however, this is why you have staff mech classes discuss correlation functions, does not average to zero, instead if the light is initially bright, it remembers that it's bright and stays bright for a while. If the light is initially dim, it remembers it's dim, it's dim, it stays dim for a while, and these products do not cancel out if you form them properly. So the first thing is you can see rotational motion. The second issue is if I have two spheres, the directions in which they point, the static correlation averages to zero, that is the spheres will bump into each other, they won't get closer than a certain distance, but these optical and nice tropies are hiding inside the spheres, and so one sphere does not know which way the other sphere is pointing. As a result, the VH scattering from this sphere and the VH scattering from that sphere are uncorrelated, and when we do quasi-elastic light scattering, we see the single particle correlation function, we measure how fast single particles are moving, and we don't see any of the effects due to the position of this particle and that particle being correlated. This is not the same as regular quasi-elastic light scattering in which the relative displacements of these two particles contribute to the light scattering spectrum. So in any event, D'Giorgio was able to do this, and he was able in one experiment to measure both how ds depends on concentration, and he found a slope of O minus 1.83, and he was also able to measure how the rotational diffusion coefficient depends on concentration, and he found a slope of minus, oh what was it, about minus 0.55. So sphere-sphere interactions are less effective at affecting rotation than they are at affecting self-diffusion. He also was able to measure the curvatures of these curves. They both curve upwards, that is, you have something that's headed down, but since it can't go negative, it sort of has to pull away, gets to larger volumes, and it does. Okay. What sort of other experiments can you do? Well, one thing you could do is say I can measure the self-diffusion coefficient of the sphere. I can measure the viscosity of the sphere system. This was done by Van Bladeren. And what Van Bladeren did was to show that you have a long-time diffusion coefficient, and that large concentration, but not small concentration, just large concentration, d long times the viscosity was approximately constant. So you had a pseudo-stokes Einstein behavior. It's pseudo because the product isn't necessarily what it was at dilute solution, but there's a region in which self-diffusion, the slow self-diffusion coefficient times the viscosity, is more or less independent of concentration. Okay. So that's those experiments. Another set of experiments you can do are to measure mutual diffusion. The mutual diffusion coefficient, d sub m, describes the relaxation of a concentration gradient. Now you would correctly infer that the single particle diffusion, particles each individually moving around at random, also contributes in part to the relaxation of a concentration gradient. If we have interacting objects in a concentration gradient, more particles here, fewer particles there, the interactions between the particles contribute to d sub m. And you can then say that d sub m should depend on concentration as 1 plus k1m5. So you can measure the slope. There are a couple of challenges here. The first challenge is the slope isn't very big. Big issue. The second issue is you can't go out to very large phi. Remember phi is always less than 1, but you can't go out to phi of a half or a tenth or whatever because if you do, approximating this as a straight line isn't adequate. Furthermore, if you have concentrated spheres in a solvent, unless you're very careful, the spheres and the solvent are not perfectly, or nearly perfectly index mashed, and you get multiple scattering in which the photon bounces off several spheres before it gets to your detector. You can describe the diffusion measurement as a Doppler shift, but it's not straight line motion at constant speed. And these multiple Doppler shifts make the particles appear to diffuse faster than they're really moving. Nonetheless, MOS and collaborators using homodyne coincidence spectroscopy, if you want to know what it is, you read the two papers where I first invent the technique and then demonstrate that it works. You can read my two papers doing that. We're able to measure case of 1M. There is, however, another complication which is not trivial, which is you would like to know what the volume fraction of spheres is. And the problem with, there are several problems with this, namely, if you actually have spheres in solvent, they may imbibe a bit of solvent. If you drop the spheres into vacuum and measure their radius with an electron microscope, gee, you measure it with an electron microscope, there may be artifacts that change the shape a bit. And so you know the volume fraction, but you don't quite know it perfectly. Nonetheless, case of 1M was something like minus 0.3 or minus 0.8 approximately. Is that a large error bar? Well, you should realize that if you look at the plot of d sub m versus phi, over the observed range of phi, d sub m only changes by a modest number of percent. The slope then, to be accurate, requires that you have a measurement accuracy, which is small relative to a modest number of percent. You know, that's pretty difficult. These are wonderful measurements, and this is in a reasonable agreement with the theory. At least if you read Phillies and Carter, was one of my Michigan undergraduates, very smart woman, and the predicted value was a little smaller than that, maybe minus 0.9, but the number is certainly in the right ballpark. Okay. If you increase the concentration of spheres, you can do light scattering spectroscopy from a uniform system, or you can do light scattering spectroscopy from a system in which a small number of the spheres are different from the rest, so you only get light scattering from a few spheres. The first experiment gives you the mutual diffusion coefficient. The second experiment actually gives you self-diffusion. The important issue is that both of these measurements, though not at the same concentration, find a multimodal relaxation. Multimodal, well, if we plot intensity at one time, intensity at the other, if we plot the correlation function of the scattered light versus time, for simple Brownian motion, you get a straight line of constant slope. For these experiments, what you find is an initial decay, and then you get a second decay, which is also more or less exponential. And you can then pull out, approximately speaking, two diffusion coefficients. One diffusion coefficient that describes short-term motion, and one diffusion coefficient that describes long-term motion. So at long times, you have a long-time diffusion coefficient. At short times, both processes are presumably active, and you see an average relaxation, which is typically pretty close to the short-time relaxation rate, or you do clever high-accuracy measurements and curve fitting, and you can pull out the two diffusion coefficients. Now we come to an odd result of sagray and Pusie. And what sagray and Pusie did is to do this experiment, and they did this experiment at a series of different scattering angles. So you have an incident light beam. The scattered light goes out to the side and is measured. This is where you measure I of T. And there is a scattering angle theta that by that convention is defined the way I have described it. As you change theta, you change the distance over which you are measuring particle motion. That is, at low theta, you are looking at very long-distance motion. At shorter theta, the distance becomes shorter. Have you seen this before? Well, if you have sat through a quantum course, you first probably saw the first order, born approximation, yes, or you had a description of scattering in solids. In all of these, you had scattering something with wave vector coming in K as 2 pi over lambda. And the scattered light comes out at some K or x-rays or whatever, which is very close to the same but not quite. K initial, K final. And K final minus K initial defined a scattering vector Q. Well, Q does not measure particle motions over a single distance and see how long it takes. You are looking at spatial Fourier components of the density. And in this case, because it is a liquid, the spatial Fourier components relax as time goes on. The relaxation that Segre and Pusey found was bimodal. The other thing they found was that if you plot the intensity of scattering versus Q, you find the curve has bumps. And in particular, they found at some scattering vector a maximum in Q. And the maximum in Q corresponds to a typical case. We have spheres. There are packing constraints between the spheres. The concentration here, after all, is pretty high. And therefore, there is a typical distance between the spheres. And if you use Q that corresponds to 1 over this distance, you get a lot of scattering because there are a lot of pairs of spheres separated by that distance. Boy, did I just oversimplify, but that's the general idea. In any event, the first thing they found was that if you compare D-short and D-long, yes, both of these diffusion coefficients depend on Q, at least for some Q, but their ratio goes as Q to the 0. And the ratio is approximately Q independent. The other thing they found, here is a wave vector Q sub m at which S of Q is a maximum. Now, if I say S of Q is a maximum, that means that it's relatively easy to make fluctuations of that wave vector, and therefore, that wave vector, the fluid in some sense, is relatively soft because it's easier to make fluctuations of that wave vector. And what they found is if you compare D-zero with the diffusion coefficient you measure at Q max, of course, it's going to be inverses, but it scales as 8 of 0 inverse over 8 of 0 concentration. This is at the concentration here. The viscosity does determine the diffusion coefficient, but it does it at a specific wave vector, namely the wave vector, which is easy to make fluctuations. How can you explain that? Well, S of Q is a maximum where it's relatively easy to make fluctuations, so the fluid is soft. And if you apply a shear to the fluid, the fluid tends to yield by yielding where it's softest, namely here, in some sense. Of course, this is a microscopic result. This is a microscopic result. More precise we cannot be. I should, however, also note the results of Del Santy. Very nice person, better ones. And if we put in D sub S, there's a gamma fast, there's a gamma slow, there are two relaxation rates, and as we increase the polymer concentration, the fast mode relaxes more rapidly, and the slow mode relaxes more slowly. Okay, why is this interesting? Well, if you do the same experiment on a polymer solution, you see it, you see the same thing. You see a fast mode and a slow mode, and their concentration dependence is crack each other. If you put probes into the polymer solution, you get other results, and we'll consider this in much more detail later on in the course. So that is, you see multiple modes and their concentration dependence. Okay. And now we will shove ahead, and we've discussed light scattering, and I have skipped some details here, but they're useful to talk about after we've talked about polymer systems. We will now skip ahead and talk about particle tracking. And the issue is that you can look in with a video camera, like the video camera that's recording, only fancier, and you can actually in a microscope see particles and you can watch them move. And therefore you can pull out full details of the statistics of the motion. Now there are a couple of disadvantages to this experiment that aren't quite as obvious. First is you're limited by the video scan rate, and you cannot do video scan rate with current technology on a microsecond time scale. The second thing is you're only looking at a small number of particles at once, so you have to spend a lot of time and do a lot of computation if you want to pull out considerable results. A time scale problem is not as serious as it sounds. That is, instead of speeding the camera up, what you can do is to say we will increase the viscosity of the liquid, say replace water with water, glycerol, and presumably all of the same things will happen, but they will happen on a time scale described by the viscosity, which is one or two or three orders of magnitude longer. And therefore instead of speeding up the camera, we slow down the system. And the nice set of experiments are due to CASPER. And what was done was to watch this motion as you change the polymer, increase the polymer concentration, and the first thing that was found was the particle motion slows down. And the next thing you do is you find that the polymer motion becomes saltatory. Saltation is moving by jumping. A cricket spends much of its time moving by saltation. So what happens is the sphere sort of sit here and move, and sit here and then move. This is not an actual picture. And that's saltatory motion. And if you run up the concentration enough, eventually you get to the point where there is relatively little motion. The spheres sort of are trapped at their current locations. Okay. Now you might like to interpret this. You could do this motion, this measurement, and that measures diffusion. You can also do quasi-elastic light scattering spectroscopy. And that measures a diffusion rate, and you can compare the two. And what was done was to interpret the light scattering spectra. This result is not always correct. I've learned about this before as e to the minus q square x square. Before you could get out a mean square motion that depends on how far apart the two time intervals are. That's according to this equation, which is only true under certain conditions. You could also measure mean square displacements microscopically. Yes. At a volume fraction of about 0.6, these two agreed. At smaller concentrations, they didn't. However, at the volume fraction of 0.6, the light scattering spectra was a single exponential. And as we know from Dubb's theorem, this result is not true if the spectrum is not a single exponential. If the spectrum is a single exponential, Dubb's theorem says this result is true. And so when Dubb's theorem said you were in good shape with this equation, which is only true at one volume fraction, the indirect and direct measurement studies of particle displacements agreed with each other. You can, however, because you're doing video tracking, do measurements that you cannot pull out of quasi-elastic light scattering. For example, you could measure the particle displacement between times t and t plus tau. And the particle displacement between the joining time interval between t plus tau and t plus theta. You do both of those. And this was actually done by Gao and Kilfoil. And what they demonstrated, approximately speaking, was that you had, first of all, you could, if you had a substantial motion this way in one time interval, in the next time interval you tended to have some recoil as particle more often than not would drift back. You could also, and a series of people have done this, look at how neighboring particles move. And what happens if you look in at spheres and look at their motion? Well, what you find is that there are groups of spheres that form somewhat ordered clusters. I'm being imprecise about that. And that move more or less as a group, and this is sort of a ball, in a rather slow manner. You find other groups of particles, and the other groups of particles lie on ribbons, typical, that move rapidly. You may say, okay, why is that happening? What is going on? And I can give one piece of an explanation. Namely, there are also large-scale computer simulations of glass formation. And there are a number of people, large group is at Michigan, that have done this sort of work. And they find experiment, they find simulationally, that as you approach glass, you get groups of more mobile particles and groups of less mobile particles. The less mobile particles tend to lie in lumps, the more mobile particles tend to lie on lines and ribbons. So we have now seen experiments that validate the computer simulation work. Okay. What else can we do? Well, I was saying we could observe single particle confusion. And so here is a probe radius rp. Here is a matrix particle, radius rm. rm may be larger than or less than rp. And you can measure the self-diffusion coefficient of the probe as you change the concentration of the probe. And you can do this for tracer particles that are small or large relative to the matrix particle. And here are small probes, small relative to the matrix. And here are middle size. And here are large probes, meaning small matrix particles. And you can actually do these experiments and you can measure how the probe diffusion coefficient depends on the relative size of the probe particle and the matrix particle. Okay. We are almost out of time, but I have enough time to discuss viscosity and viscosity by estheticity. For herd spheres, viscosity as a function of concentration has a fairly consistent experimental form that everyone has found. And here with hypothesized data points on it is the lower concentration region e to the alpha phi nu. And we chug along and we get to some point and at some point we have a power law behavior phi dx. You can continue the mathematical curve of the stretched exponential, but the data is doing this. In at least a few experiments people have gotten points that are very close to here and it does not appear that there is any crossover regime in most systems. We talked about a similar behavior for polymer solution viscosity, but for polymer solution viscosity at the crossover, the slope of the power law line and the slope of the stretched exponential line were the same. For herd spheres the crossover is not analytic, the power law line just takes off. There has been a substantial amount of work to study this or rather to study viscosity of herd spheres. Most people didn't say, well the viscosity goes up very quickly and they didn't have the functional form description that would tell you sharply what you were seeing. You can measure approximately where this is and this crossover occurs in a volume fraction of about 0.4 to 0.45. It's a little hard to measure more exactly than that and it's slightly different in different systems. And the crossover occurs at a viscosity 8 over 8 of 0 where 8 of 0 is the solvent viscosity of 05 to 15. Also slightly different in different systems. Now most of these studies were interested in something completely different. Namely if you continue this curve up to 0.494, give or take, there is a viscosity at the point that thermodynamic phase transition sets in. And that viscosity is about f50 plus or minus. It's a little hard to be more precise because this is an extremely steep curve. A small error in determining or deciding what the concentration is leads to a significant difference in viscosity. The core point I would make is that this crossover is down here. So what happens at the phase transition is up here and very, very certainly the stretched exponential to power law transition is not vaguely at the same location as is the thermodynamic phase transition. We are out of time. We're pretty close. And that's it for this lecture.
 Lecture 18, Colloid Dynamics, based on George Philly's book,  Phenomenology of Polymer Solution Dynamics, Cambridge University Press, 2011.  And today, this lecture is lecture 18, Colloid Dynamics.  I'm Professor Philly's, and this is the next lecture in my series on polymer dynamics.  Our last lecture finished off with a discussion of optical probe diffusion.  What I'm going to do today is to push on to an entirely new topic in chapter.  The new chapter is 10, and the new topic of chapter 10 are colloid suspensions and colloid solutions.  If you go through the reviews found in the early part of the book,  the reviews by people like Tyrell and Lodge, and a whole variety of other books on polymer dynamics,  you'll make an interesting observation.  There are lots of discussions of polymer diffusion, polymer viscosity.  Almost none of the other sources you look at discuss probe diffusion at any length,  and there's absolutely no discussion of colloid dynamics at all.  Now, why might you be interested in looking at how colloid particles move?  Well, let's remember what a colloid particle is.  The simplest case, we have objects in solution,  and the objects in solution are spheres that are quite small, like a half a nanometer or tens or hundreds of nanometers.  They've been surface treated in some way in most cases,  and as a result, well, you say this is a suspension of particles, they're big particles,  but they can be density matched, and even if they're not, they can be charged or coated,  given polymer coats on the surface, so that they sit in solution.  Why would we be interested in these round hard objects?  They're quite rigid.  I've drawn a sphere, but some of them aren't spheres, they're ellipsoids or whatever.  Why would we be interested in these in a discussion of colloid?  Why would we be interested in these in a discussion of polymer dynamics?  Why are we talking about colloid dynamics?  The answer is, these are sitting in solution, and there are three sorts of forces on them.  There are random thermal forces due to fluctuations in the solvent.  That's what drives Brownian motion.  There are hydrodynamic interactions.  I'll talk about those in a bit. Namely, as one particle moves,  it sets up a wake in the surrounding solution, like the wake of a motorboat.  Well, not really exactly like, but somewhat similar.  And the wake drags neighboring particles along with it.  And then there are direct interactions of which the most important is volume exclusion.  Because these objects are solids, they cannot pass through each other.  I could also discuss polymer solution.  Here is a non-dilute polymer solution.  The important feature of the non-dilute polymer solution is that the forces between the polymer coils,  thermal interactions, which are the polymer coils of the solvent,  are dynamic interactions between polymer coils,  and volume exclusion interactions between polymer coils.  Those forces are precisely the same list of forces that act between colloids.  So we have two systems. We have the same sets of forces acting.  We're looking on the same time scales, so the same basic dynamic equations apply.  However, there is one substantial difference between polymer dynamics and colloid dynamics.  And the substantial difference is shape, or as some people would call it, topology.  My apologies to real topologists who will point out that this, if linear, and that,  both simply connected volumes end of discussion.  They're the same, aren't they? Well, no, not really.  However, the core difference is, this is a sphere.  That's something that's long and stringy.  And you can certainly imagine this object having modes of motion and interactions  that are not stunningly the same as spheres.  For example, if you have two polymer coils,  you can certainly imagine one of the polymer coils wrapping itself around the other,  like two rattlesnakes in a rattlesnake ball.  Sears, however, just sit there. They can't wrap around each other.  And if you believe in the arguments that propose there are entanglement constraints,  though the entanglement models don't say what an entanglement is,  then you would say that a polymer coil plausibly entangles,  at least for many pictures of what entanglement means,  but spheres certainly do not do so.  Okay, so there is a fundamental difference in shape and in possible dynamics,  but the basic forces are the same,  and the basic, gee, this is over-damped motion, f equals ma equations are the same.  There's some similarities, some differences.  One thing one might, however, propose is that phenomena that are common to colloid dynamics  and to polymer dynamics cannot be explained on the grounds that polymers can wrap up in knots,  and polymers can perform repetitions, snake-like motion like the boa constrictor going through the bamboo grove.  Anything that is common to these two systems very certainly cannot be assigned to topology.  So that is our perspective here,  and what we are going to do is to look at a variety of discussions  of how sears-interim move as opposed to how polymers move,  and we are going to look at colloid phenomenology,  and that will tell us something about, gee, the importance of topology.  The short discussion on how does this compare with other works is,  there is no comparison, and the reason there is no comparison is that if you look at other treatments of polymer dynamics,  you will be very hard pressed to find another that does a substantial discussion of colloid motions.  So let us start out, and we will say a bit about hydrodynamic interactions.  The front-end piece is as follows.  We have here is a colloid sphere.  It is small enough that we can approximate it as a point in terms of the other distance scales.  It moves through a polymer-polymer solution.  As it moves through a polymer solution, it is exerting a force,  there is the force, f being exerted on the solution,  and this causes the liquid to move,  and so you get a flow field.  And if out here the flow field has a velocity v,  the coupling at low frequencies between v and f is v equals t.  T is a tensor. It is a 3 by 3 matrix.  The reason it is a tensor is that we are starting off with a vector here.  We are ending up with a vector here, and the two vectors are not parallel.  The mathematical structure, this is linear, that we can use to transform one vector into the other,  is a 3 by 3 tensor, the O-sine tensor, whose I will strip out constants.  The O-sine tensor folds off as 1 over r, where r is the distance from the point particle  to the point of interest in the fluid.  And then the tensor dependence is the identity tensor, plus this is an outer product, r hat, r hat.  r hat is the unit vector pointing from the sphere to the place in the fluid.  There are some constants which you can put in differently depending on whether you describe here  the force or the velocity of this point.  What are the important features of the O-sine tensor?  First of all, the flow field is not simply parallel to the velocity of the original object.  It has this motion that I have sort of sketched.  Second, the O-sine tensor falls off as 1 over r.  That is not a potential energy falling off as 1 over r.  If I have an object which I try to hold still at this point,  the flow field creates a force on the object, f prime.  And the force, we have a force here creating a force there.  The force falls off as 1 over r.  This is the longest range force in nature.  Now you may worry a bit, oh gee, is it really infinitely long-ranged?  And I will now point out the constraints because O-sine is an approximation.  The first constraint is that if I apply a force here for a brief period of time,  you have to set up the O-sine flow field and that takes a while.  And until you have set up the flow field completely,  there are a more complicated set of equations due to Bowson-esque  to describe how a force here for some short period of time creates a flow there.  The flow field has to propagate outwards and momentum propagation in a fluid occurs at the speed of sound.  So if you are looking at very long ranges or very short times, this is not perfect.  But we aren't at very long ranges, we aren't at very short times.  There is another constraint that is sometimes lost, which is as follows.  We are actually doing this experiment inside a container.  And the container is filled with a solvent,  and the solvents we are talking about are mostly pretty incompressible.  As a result, if I draw a line, an imaginary mathematical surface, across the container at some point,  the total volume flow of fluid across that line has to be zero.  If it wasn't, we'd be compressing the fluid and this can't be done to any significant extent.  And therefore, as we approach the walls of the container, somehow there has to be a backflow,  which one could in principle calculate,  and the backflow has the effect that the total volume flow across the system is zero.  Now you have to put one qualification in.  Suppose I drew the mathematical line here.  In that case, the mathematical line is intercepting the solid object.  And if we have a flow delta V of object into this volume,  we must have a net flow minus delta V of solvent out of it.  This is incompressibility when you remember this object has a volume.  This is a significant effect in real experiments.  At a certain reasonable level of approximation,  the whole thing is known as reference frame corrections.  The core issue in reference frame corrections is that the reference frame that is stationary  with respect to the walls of the container, that's an obvious inertial reference frame,  and the reference frame that is stationary with respect to the solvent,  because the solvent is moving this way, are not the same.  And reference frame corrections are treated in great detail in a paper by John Gamble Kirkwood  and collaborators, 1960 paper.  The paper actually appears after Kirkwood's death in the Kirkwood Memorial volume,  and his four co-authors insisted he'd done the work so he should be the lead author.  So there is something called a reference frame correction.  However, the Oceane tensor says there are these interactions,  and one moving particle puts a force on its neighbors,  unless the neighbors simply are bobbing along with the solvent.  And if the neighbors are bobbing along with the solvent moving at this speed V,  they're moving, and this velocity and that velocity are correlated.  Okay, so that was the Oceane tensor, and now we're going to put down something else.  And the next piece comes out of an arc, what is known as a fluctuation dissipation argument.  Fluctuation dissipation arguments are potentially tricky and hazardous.  That is, it is possible to charge in, make a set of statements, all of which seem reasonable,  and you can get surprised by what you predict,  and by the way, you were correct to be surprised because the prediction somehow went a bit astray.  However, in this case, it's fairly clear what you're saying, and it's been tested experimentally.  The statement is, if I have a particle here that does this under the influence of an external force,  say there are little magnetic bits inside the particle and I apply a magnetic field gradient,  a nearby particle will be dragged along,  and the correlation between the displacement of particle one and the displacement of particle two,  the correlation is given by the Oceane tensor.  That is, it's something that falls off as one over R, and it has this tensor nature rather than everything simply being parallel.  So that's the fluctuation dissipation argument.  And what we say is that if you see this behavior for driven motion,  and if you're in the low speed linear regime, which we are, you will also see it for diffusive motion,  and therefore, if you look at the random walk of two particles in a liquid,  their motions are correlated as described by the Oceane tensor.  Let me say, if you think back to what you may have read about Brownian motion and undergraduate thermal physics,  you may have thought, well, gee, aren't the motions of Brownian particles uncorrelated?  And the answer is, if they're way far apart, or if you don't know what questions to ask, they appear to be uncorrelated.  And the whole study of Brownian motion for a very long time was that you could do all of these statistical correlations,  but until Langevon did his very clever equation, until Langevon did his work on Brownian motion,  it wasn't clear what questions to ask about Brownian motion, and therefore, you got lots of answers that weren't stunningly helpful.  And afterwards, everything worked.  So there is the statement, well, at the front end, we have driven motion, and then if you look at diffusion, you get the same result.  Notice, however, we've said the effect falls off as inverse as distance,  and the basic unit is the size of the particle over the distance, so we have a dimensionless number here.  So if the particles are tiny and way far apart, that's what far apart means, the interaction falls off a lot.  Nonetheless, it doesn't fall off to zero, and there have been a series of experiments due to O.  Crocker, a very clever fellow, and Miner, and Quake.  And what they do is to say, well, let us look into a system, let us actually, one way or another,  track the motion of parallel particles, pairs of particles, we get the particles close together,  and we can then actually look at diffusive motion, and we can ask, how does diffusive motion work?  And the short-format answer is, you can actually do the experiment, and you find that  displacement one, displacement two are correlated.  This defines what is known as a cross-diffusion tensor.  The cross-diffusion tensor describes the relative motion of pairs of particles,  and the cross-diffusion tensor, as is predicted by the fluctuation, dissipation arguments,  is proportional to the O-seam tensor, and falls off as one over R at long distances.  So those are hydrodynamic interactions.  Hydrodynamic interactions were inserted into polymer dynamics very early on by the original treatment  of Kirkwood and Reisman.  Same Kirkwood as the reference frame fell, a very clever man, leading American statistical  mechanization of the first half of the century, of the last century, Kirkwood and Reisman.  And what they did is to say, here is a polymer chain, and we can model the polymer chain as  a string of frictional beads, and the frictional beads are attached to each other by little  springs, or perhaps the frictional beads are monomers and they're bonded together.  And if the chain moves in solution, well, the chain has some center of mass velocity, B.  If I apply an external flow field, the polymer flips head over tail, and has, this is coming  out perpendicular to the board, it has a rotational velocity omega, and then the individual beads  have fluctuation velocities relative to the general drift motion.  But you know, these things are all attached to each other.  It's like a very long snake.  If it moves a large distance, the head and the tail have to stay attached.  And therefore, this object will sort of like a bag of beads with internal fluctuations.  What they put in though is the key issue, is that these beads, or at least some of them,  are obliged to be moving with respect to the flow field.  Why?  Well, let us draw a polymer coil again.  It has a center of mass velocity this way.  I've applied a velocity shear to the fluid, so the fluid is going in opposite directions.  The polymer is tumbling head over tail as it does this.  And so up here, the beads try to move with the fluid.  And down here, the beads try to move with the fluid.  But if you think of the recall, this is circular motion, that means that here, and not so probable,  let's extend this a bit, here, the beads have to be moving sideways.  So at least some of the beads, to some extent, have to be moving with respect to the fluid.  If they weren't, the shear would tear the polymer apart.  And these motions with respect to the fluid, not resolved.  We have a bead, it's moving with respect to the fluid, it is exerting a force on the fluid.  And so all of these beads, moving with respect to the fluid, each create, and there are lots of beads,  but they each create their own flow fields, which act on all of the other beads.  And then you have to be a little clever in the math to re-sum things to get a self-consistent solution.  And you do, and this is the Kirkwood-Ricewood model for polymer dynamics.  If you are clever and careful, you can take this model, and you can use it to treat pairs of interacting polymer chains.  And when you put it in to treat pairs of interacting polymer chains, which can be done,  you get out the concentration dependence of all the self-diffusion coefficient or the viscosity.  You have to work a bit harder to do that, but it's possible.  Okay, I've talked about hydrodynamic interactions.  For most of the objects we're talking about, the direct interaction is basically excluded volume.  That is, we have a sphere here, and we have a sphere there, and they can not actually, they can touch,  but they can't interpenetrate because they're solids.  Now, there is a bit of a cheat at this point.  The bit of a cheat at this point is that the real spheres you're talking about,  well, they might be charged because you put them into water and you suspend them by charging them.  They might have very short polymer molecules adhered to their surface to keep them from sticking.  They may have van der Waals forces between them, and the net result of all of those is that at very short distances  there's some extra interactions that are not exactly hard sphere.  Now, approximately speaking, if you're just talking about hydrodynamics, this is not serious,  but the real hydrodynamic interaction between two spheres is not exactly O-sene,  that is, the real interaction tensor, real, is the O-sene tensor, which is 1 over r,  and then there are 1 over r to the n, and there are a bunch of higher order corrections.  Higher order corrections are short range.  Their importance depends on how likely it is who find two of these objects real close together,  and that likelihood of being real close together is modulated by these short range potentials,  which are a little harder to determine in great detail.  So life gets a little trickier than we would like, but approximately speaking, it can be made to work.  Suppose, however, we had real hard spheres, and I get real hard spheres, sure I can.  I can do a computer simulation.  Real hard spheres have thermodynamic properties.  The thermodynamic property is described by phi, their volume fraction,  and what is phi?  Phi is equal to the number of spheres in some volume times 4 pi r sphere cube over 3,  the volume of a single sphere, this is the total volume of all of the spheres,  divided by the total volume of the system.  So we can plot from the 0 out to, I suppose you could say 1, but you can't get to 1.  I'll point out why not in a second.  0 to, you actually don't get out here.  Because the spheres are hard, their distribution functions, their statistical mechanical distribution functions,  don't depend on temperature. That is, either a configuration of spheres is allowed,  because no sphere is overlapped, or it's forbidden, because you've got two spheres trying to be in the same place at the same time.  And guess what, no matter what you do to the temperature, it's allowed or forbidden.  The highest volume fraction you can actually get is close packed,  and that's about 0.74 volume fraction.  You can be a little more precise, and there are actually several close packed configurations,  but they all have the key feature, you can't get more than 0.74.  As a historical aside, this result was believed, but had not actually been proven when I was a graduate student.  The issue, which is a math issue, is that you couldn't find any configurations that were closer packed than this,  but what the mathematicians had to do was to prove there was no way to pack a sphere.  Here is a sphere. So the sphere had 13 other spheres touching it.  If you could do that, which you can't, you could create that configurations that were regular in all these other things,  and had densities higher than hexagonal close packed.  Well, at some point, fairly recently, the mathematicians were able to prove this result, which most people believe.  And we can now say, this is the upper limit, that's how many spheres you can fit, because that's all the packing.  On the other hand, you can imagine, current spheres are like ball bearings or toy marbles.  So you take the toy marbles or the ball bearings, you put them in a cloth sack, and you shake them.  And you really want a cloth sack, not a box, because the ordering from the walls of a nice rectangular box propagates into the side.  So what happens if you put ball bearings or old flour or salt, anything that's nice and regular, in a bag, and you shake it?  Well, the first thing that happens if you shake it, it starts to settle.  And if you shake it longer, it settles more.  And after a while, you shake more, and you just don't make more progress.  And it is fairly hard to pack things so that you get a volume fraction that is much above 0.64.  Now, that's not an exact result, because if you keep shaking, you can find better packing.  But there's an upper limit about here to what an irregular packing will look like.  There are two more pieces.  The two more pieces are due to a set of computer experiments by Ray and Hoover.  The computer experiments go back almost half a century at this point.  And what they did was to study hard spheres and the behavior of hard spheres at different densities.  And what they demonstrated is that there is a boundary at about 0.49, people will quote, four.  I think the four, that terminal four is a little enthusiastic, given the size of the system that we studied.  And coming up here to about 0.55.  The issue with hard spheres is as follows.  Suppose I have less dense hard spheres here and more dense hard spheres there.  Well, if these were small molecules that could form a liquid, we would have, for example, a gas up here, a liquid down there.  And there would be a free surface that would be supported by the attractive interactions between the molecules.  So you have a well-defined liquid and you have a well-defined gas.  Hard spheres have no attractive interactions. That's why they're called hard.  And therefore, there is nothing to keep, to maintain this boundary between two regions of different densities.  Nonetheless, if you do the computer experiments, what you discover is that you have one phase out to about this density.  You then have a region in which you have two phases, you have a biphase at region.  And in the biphase at region, there are less densest zones and more densest zones.  And then up above here, going out to here anyhow, you have a single phase.  And what is said is the lower density phase is a fluid gas, basically.  And the upper density phase is claimed to be an expanded solid because its structure is more or less solid-like, but the density is too low.  But if you think about things, the argument I gave as to why hard spheres cannot have a gas-liquid boundary works just as well for a gas-solid boundary.  And therefore, what you presumably actually have here, this must be a gas because it can't be anything else.  And this is actually really also a gas because there's no attractive potential to hold the things into the solid.  And therefore, we have here what would appear to be an example of a gas-gas phase transition.  But one of the two gases is really very much solid-like.  And the solid crystal structure, if you really get up, where the closely packed amorphous structures, and gee, there are more gaps in it.  My search of the literature says this issue has not been studied very recently.  And if you studied things with much larger systems, you might get a more interesting result or a clearer result anyhow.  The important issue is, if you are packing hard spheres at about half by volume, there is expected to be a thermodynamic phase transition leading up at a density of 0.55 to what is a distinct phase.  And in between, you have two phases present.  And that's worth keeping in mind as we chug through this.  Okay, so what have people done?  Well, people have done on colloids more or less all of the same experiments that have been done on polymer solutions.  We haven't talked about all of the polymer measurements yet. We'll get to them.  What sort of things can you do?  Well, the only thing you can do for a colloid solution is to measure the single particle, self or tracer, diffusion coefficient.  And if you do that, this is for example figure 10.1, you measure the diffusion coefficient using any variety of methods.  And it's some function of sphere concentration and the diffusion coefficient falls.  You have one interesting little bit which has sort of been brought up also with polymers.  That is, you can measure the diffusion coefficient over fairly short time periods.  You can measure it using light scattering spectroscopy. And you can measure the diffusion rate over very long time periods using, for example, fluorescence recovery after photobleaching.  Or you can measure the mutual diffusion coefficient, concentrated system, and use a very large scattering vector.  This works too. That's due to PUSI and co-workers.  And the net result is you can measure two diffusion coefficients more or less.  You can measure a short time diffusion coefficient which does roughly this.  You can measure also a long time diffusion coefficient.  And the long time diffusion coefficient drops off significantly more rapidly than the short time diffusion coefficient that stays down there.  For the short time diffusion coefficient, we can write dS is d01 plus some constant short volume fraction.  And you can actually measure k1S and you get O minus, oh, what's the magic number, 1.8 plus or minus 0.06.  And there is a theoretical result which is about minus 1.73. Those two are in quite satisfactory agreement because part of this k1S depends on fairly short range interactions.  And any deviation of the sphere is from perfectly hard sphere would give you a slightly different number here.  So you can actually predict this concentration dependence.  You can also do a completely different set of experiments.  And the completely different set of experiments are due to D Georgiou.  And the notion here is we will take spheres.  And because we have a wonderful supplier, we have spheres that are optically anisotropic inside.  And as a result, if I send in polarized light and look at the scattered light, some of the scattered light is polarized the same way as the incident light.  And that basically just says the light sees a sphere.  And some of the light is depolarized, so I put in vertically polarized light, light polarized perpendicular to the scattering plane.  I look at, I'm putting in prisms here and here, horizontally polarized light.  And that light, the amount of that light is determined by the shear orientation.  Now there are two features of the VH scattering.  First of all, if here's a depolarizing sphere, if it rotates, the sphere rotates, the amount of VH scattering changes.  The sphere is doing Brownian random rotation, so the intensity of the VH scattering fluctuates in time.  And the time scale of the VH intensity fluctuations exactly tracks the time scale on which the sphere reorients.  So after a great deal of clever math, I can use VH scattering to measure rotational diffusion.  Question?  This is an average measure of the light.  What you do is to measure the time correlation function of the scattered light.  So we have the intensity of the VH scattered light at time t.  We now measure it at a later time, t plus tau.  We do this, repeat this for lots and lots of times t to get an average behavior.  And what is left is a function of tau, and tau is the time separation.  And so what it means is if the light is bright at one instant in time, so IVH of t is large, because this is pointing exactly the right way,  after a while the sphere forgets which way it's pointing, and IVH relaxes back to some average value.  The correlation function, however, this is why you have staff mech classes discuss correlation functions,  does not average to zero, instead if the light is initially bright, it remembers that it's bright and stays bright for a while.  If the light is initially dim, it remembers it's dim, it's dim, it stays dim for a while, and these products do not cancel out if you form them properly.  So the first thing is you can see rotational motion.  The second issue is if I have two spheres, the directions in which they point, the static correlation averages to zero,  that is the spheres will bump into each other, they won't get closer than a certain distance,  but these optical and nice tropies are hiding inside the spheres, and so one sphere does not know which way the other sphere is pointing.  As a result, the VH scattering from this sphere and the VH scattering from that sphere are uncorrelated,  and when we do quasi-elastic light scattering, we see the single particle correlation function,  we measure how fast single particles are moving, and we don't see any of the effects due to the position of this particle and that particle being correlated.  This is not the same as regular quasi-elastic light scattering in which the relative displacements of these two particles contribute to the light scattering spectrum.  So in any event, D'Giorgio was able to do this, and he was able in one experiment to measure both how ds depends on concentration,  and he found a slope of O minus 1.83, and he was also able to measure how the rotational diffusion coefficient depends on concentration,  and he found a slope of minus, oh what was it, about minus 0.55.  So sphere-sphere interactions are less effective at affecting rotation than they are at affecting self-diffusion.  He also was able to measure the curvatures of these curves.  They both curve upwards, that is, you have something that's headed down,  but since it can't go negative, it sort of has to pull away, gets to larger volumes, and it does.  Okay. What sort of other experiments can you do?  Well, one thing you could do is say I can measure the self-diffusion coefficient of the sphere.  I can measure the viscosity of the sphere system.  This was done by Van Bladeren.  And what Van Bladeren did was to show that you have a long-time diffusion coefficient,  and that large concentration, but not small concentration, just large concentration, d long times the viscosity was approximately constant.  So you had a pseudo-stokes Einstein behavior.  It's pseudo because the product isn't necessarily what it was at dilute solution,  but there's a region in which self-diffusion, the slow self-diffusion coefficient times the viscosity,  is more or less independent of concentration.  Okay. So that's those experiments.  Another set of experiments you can do are to measure mutual diffusion.  The mutual diffusion coefficient, d sub m, describes the relaxation of a concentration gradient.  Now you would correctly infer that the single particle diffusion, particles each individually moving around at random,  also contributes in part to the relaxation of a concentration gradient.  If we have interacting objects in a concentration gradient, more particles here, fewer particles there,  the interactions between the particles contribute to d sub m.  And you can then say that d sub m should depend on concentration as 1 plus k1m5.  So you can measure the slope.  There are a couple of challenges here.  The first challenge is the slope isn't very big.  Big issue.  The second issue is you can't go out to very large phi.  Remember phi is always less than 1,  but you can't go out to phi of a half or a tenth or whatever  because if you do, approximating this as a straight line isn't adequate.  Furthermore, if you have concentrated spheres in a solvent, unless you're very careful,  the spheres and the solvent are not perfectly, or nearly perfectly index mashed,  and you get multiple scattering in which the photon bounces off several spheres before it gets to your detector.  You can describe the diffusion measurement as a Doppler shift,  but it's not straight line motion at constant speed.  And these multiple Doppler shifts make the particles appear to diffuse faster than they're really moving.  Nonetheless, MOS and collaborators using homodyne coincidence spectroscopy,  if you want to know what it is, you read the two papers where I first invent the technique  and then demonstrate that it works.  You can read my two papers doing that.  We're able to measure case of 1M.  There is, however, another complication which is not trivial,  which is you would like to know what the volume fraction of spheres is.  And the problem with, there are several problems with this, namely,  if you actually have spheres in solvent, they may imbibe a bit of solvent.  If you drop the spheres into vacuum and measure their radius with an electron microscope,  gee, you measure it with an electron microscope,  there may be artifacts that change the shape a bit.  And so you know the volume fraction, but you don't quite know it perfectly.  Nonetheless, case of 1M was something like minus 0.3 or minus 0.8 approximately.  Is that a large error bar?  Well, you should realize that if you look at the plot of d sub m versus phi,  over the observed range of phi, d sub m only changes by a modest number of percent.  The slope then, to be accurate, requires that you have a measurement accuracy,  which is small relative to a modest number of percent.  You know, that's pretty difficult.  These are wonderful measurements, and this is in a reasonable agreement with the theory.  At least if you read Phillies and Carter,  was one of my Michigan undergraduates, very smart woman,  and the predicted value was a little smaller than that, maybe minus 0.9,  but the number is certainly in the right ballpark.  Okay.  If you increase the concentration of spheres,  you can do light scattering spectroscopy from a uniform system,  or you can do light scattering spectroscopy from a system  in which a small number of the spheres are different from the rest,  so you only get light scattering from a few spheres.  The first experiment gives you the mutual diffusion coefficient.  The second experiment actually gives you self-diffusion.  The important issue is that both of these measurements,  though not at the same concentration,  find a multimodal relaxation.  Multimodal, well, if we plot intensity at one time, intensity at the other,  if we plot the correlation function of the scattered light versus time,  for simple Brownian motion, you get a straight line of constant slope.  For these experiments, what you find is an initial decay,  and then you get a second decay, which is also more or less exponential.  And you can then pull out, approximately speaking, two diffusion coefficients.  One diffusion coefficient that describes short-term motion,  and one diffusion coefficient that describes long-term motion.  So at long times, you have a long-time diffusion coefficient.  At short times, both processes are presumably active,  and you see an average relaxation, which is typically pretty close  to the short-time relaxation rate,  or you do clever high-accuracy measurements and curve fitting,  and you can pull out the two diffusion coefficients.  Now we come to an odd result of sagray and Pusie.  And what sagray and Pusie did is to do this experiment,  and they did this experiment at a series of different scattering angles.  So you have an incident light beam.  The scattered light goes out to the side and is measured.  This is where you measure I of T.  And there is a scattering angle theta that by that convention is defined  the way I have described it.  As you change theta, you change the distance over which you are measuring particle motion.  That is, at low theta, you are looking at very long-distance motion.  At shorter theta, the distance becomes shorter.  Have you seen this before?  Well, if you have sat through a quantum course,  you first probably saw the first order, born approximation, yes,  or you had a description of scattering in solids.  In all of these, you had scattering something with wave vector coming in K as 2 pi over lambda.  And the scattered light comes out at some K or x-rays or whatever,  which is very close to the same but not quite.  K initial, K final.  And K final minus K initial defined a scattering vector Q.  Well, Q does not measure particle motions over a single distance  and see how long it takes.  You are looking at spatial Fourier components of the density.  And in this case, because it is a liquid, the spatial Fourier components relax as time goes on.  The relaxation that Segre and Pusey found was bimodal.  The other thing they found was that if you plot the intensity of scattering versus Q,  you find the curve has bumps.  And in particular, they found at some scattering vector a maximum in Q.  And the maximum in Q corresponds to a typical case.  We have spheres.  There are packing constraints between the spheres.  The concentration here, after all, is pretty high.  And therefore, there is a typical distance between the spheres.  And if you use Q that corresponds to 1 over this distance, you get a lot of scattering  because there are a lot of pairs of spheres separated by that distance.  Boy, did I just oversimplify, but that's the general idea.  In any event, the first thing they found was that if you compare D-short and D-long,  yes, both of these diffusion coefficients depend on Q, at least for some Q,  but their ratio goes as Q to the 0.  And the ratio is approximately Q independent.  The other thing they found, here is a wave vector Q sub m at which S of Q is a maximum.  Now, if I say S of Q is a maximum, that means that it's relatively easy to make fluctuations  of that wave vector, and therefore, that wave vector, the fluid in some sense, is relatively soft  because it's easier to make fluctuations of that wave vector.  And what they found is if you compare D-zero with the diffusion coefficient you measure at Q max,  of course, it's going to be inverses, but it scales as 8 of 0 inverse over 8 of 0 concentration.  This is at the concentration here.  The viscosity does determine the diffusion coefficient, but it does it at a specific wave vector,  namely the wave vector, which is easy to make fluctuations.  How can you explain that?  Well, S of Q is a maximum where it's relatively easy to make fluctuations, so the fluid is soft.  And if you apply a shear to the fluid, the fluid tends to yield by yielding where it's softest,  namely here, in some sense.  Of course, this is a microscopic result.  This is a microscopic result.  More precise we cannot be.  I should, however, also note the results of Del Santy.  Very nice person, better ones.  And if we put in D sub S, there's a gamma fast, there's a gamma slow, there are two relaxation rates,  and as we increase the polymer concentration, the fast mode relaxes more rapidly,  and the slow mode relaxes more slowly.  Okay, why is this interesting?  Well, if you do the same experiment on a polymer solution, you see it, you see the same thing.  You see a fast mode and a slow mode, and their concentration dependence is crack each other.  If you put probes into the polymer solution, you get other results,  and we'll consider this in much more detail later on in the course.  So that is, you see multiple modes and their concentration dependence.  Okay.  And now we will shove ahead, and we've discussed light scattering,  and I have skipped some details here, but they're useful to talk about after we've talked about polymer systems.  We will now skip ahead and talk about particle tracking.  And the issue is that you can look in with a video camera, like the video camera that's recording, only fancier,  and you can actually in a microscope see particles and you can watch them move.  And therefore you can pull out full details of the statistics of the motion.  Now there are a couple of disadvantages to this experiment that aren't quite as obvious.  First is you're limited by the video scan rate, and you cannot do video scan rate with current technology on a microsecond time scale.  The second thing is you're only looking at a small number of particles at once,  so you have to spend a lot of time and do a lot of computation if you want to pull out considerable results.  A time scale problem is not as serious as it sounds.  That is, instead of speeding the camera up, what you can do is to say we will increase the viscosity of the liquid,  say replace water with water, glycerol, and presumably all of the same things will happen,  but they will happen on a time scale described by the viscosity, which is one or two or three orders of magnitude longer.  And therefore instead of speeding up the camera, we slow down the system.  And the nice set of experiments are due to CASPER.  And what was done was to watch this motion as you change the polymer, increase the polymer concentration,  and the first thing that was found was the particle motion slows down.  And the next thing you do is you find that the polymer motion becomes saltatory.  Saltation is moving by jumping.  A cricket spends much of its time moving by saltation.  So what happens is the sphere sort of sit here and move, and sit here and then move.  This is not an actual picture.  And that's saltatory motion.  And if you run up the concentration enough, eventually you get to the point where there is relatively little motion.  The spheres sort of are trapped at their current locations.  Okay.  Now you might like to interpret this.  You could do this motion, this measurement, and that measures diffusion.  You can also do quasi-elastic light scattering spectroscopy.  And that measures a diffusion rate, and you can compare the two.  And what was done was to interpret the light scattering spectra.  This result is not always correct.  I've learned about this before as e to the minus q square x square.  Before you could get out a mean square motion that depends on how far apart the two time intervals are.  That's according to this equation, which is only true under certain conditions.  You could also measure mean square displacements microscopically.  Yes.  At a volume fraction of about 0.6, these two agreed.  At smaller concentrations, they didn't.  However, at the volume fraction of 0.6, the light scattering spectra was a single exponential.  And as we know from Dubb's theorem, this result is not true if the spectrum is not a single exponential.  If the spectrum is a single exponential, Dubb's theorem says this result is true.  And so when Dubb's theorem said you were in good shape with this equation,  which is only true at one volume fraction, the indirect and direct measurement studies of particle displacements agreed with each other.  You can, however, because you're doing video tracking, do measurements that you cannot pull out of quasi-elastic light scattering.  For example, you could measure the particle displacement between times t and t plus tau.  And the particle displacement between the joining time interval between t plus tau and t plus theta.  You do both of those.  And this was actually done by Gao and Kilfoil.  And what they demonstrated, approximately speaking, was that you had, first of all,  you could, if you had a substantial motion this way in one time interval,  in the next time interval you tended to have some recoil as particle more often than not would drift back.  You could also, and a series of people have done this, look at how neighboring particles move.  And what happens if you look in at spheres and look at their motion?  Well, what you find is that there are groups of spheres that form somewhat ordered clusters.  I'm being imprecise about that.  And that move more or less as a group, and this is sort of a ball, in a rather slow manner.  You find other groups of particles, and the other groups of particles lie on ribbons, typical, that move rapidly.  You may say, okay, why is that happening? What is going on?  And I can give one piece of an explanation.  Namely, there are also large-scale computer simulations of glass formation.  And there are a number of people, large group is at Michigan, that have done this sort of work.  And they find experiment, they find simulationally, that as you approach glass,  you get groups of more mobile particles and groups of less mobile particles.  The less mobile particles tend to lie in lumps, the more mobile particles tend to lie on lines and ribbons.  So we have now seen experiments that validate the computer simulation work.  Okay.  What else can we do?  Well, I was saying we could observe single particle confusion.  And so here is a probe radius rp.  Here is a matrix particle, radius rm.  rm may be larger than or less than rp.  And you can measure the self-diffusion coefficient of the probe as you change the concentration of the probe.  And you can do this for tracer particles that are small or large relative to the matrix particle.  And here are small probes, small relative to the matrix.  And here are middle size.  And here are large probes, meaning small matrix particles.  And you can actually do these experiments and you can measure how the probe diffusion coefficient  depends on the relative size of the probe particle and the matrix particle.  Okay.  We are almost out of time, but I have enough time to discuss viscosity and viscosity by estheticity.  For herd spheres, viscosity as a function of concentration has a fairly consistent experimental form that everyone has found.  And here with hypothesized data points on it is the lower concentration region e to the alpha phi nu.  And we chug along and we get to some point and at some point we have a power law behavior phi dx.  You can continue the mathematical curve of the stretched exponential, but the data is doing this.  In at least a few experiments people have gotten points that are very close to here and it does not appear that there is any crossover regime in most systems.  We talked about a similar behavior for polymer solution viscosity, but for polymer solution viscosity at the crossover,  the slope of the power law line and the slope of the stretched exponential line were the same.  For herd spheres the crossover is not analytic, the power law line just takes off.  There has been a substantial amount of work to study this or rather to study viscosity of herd spheres.  Most people didn't say, well the viscosity goes up very quickly and they didn't have the functional form description that would tell you sharply what you were seeing.  You can measure approximately where this is and this crossover occurs in a volume fraction of about 0.4 to 0.45.  It's a little hard to measure more exactly than that and it's slightly different in different systems.  And the crossover occurs at a viscosity 8 over 8 of 0 where 8 of 0 is the solvent viscosity of 05 to 15.  Also slightly different in different systems.  Now most of these studies were interested in something completely different.  Namely if you continue this curve up to 0.494, give or take, there is a viscosity at the point that thermodynamic phase transition sets in.  And that viscosity is about f50 plus or minus.  It's a little hard to be more precise because this is an extremely steep curve.  A small error in determining or deciding what the concentration is leads to a significant difference in viscosity.  The core point I would make is that this crossover is down here.  So what happens at the phase transition is up here and very, very certainly the stretched exponential to power law transition is not vaguely at the same location as is the thermodynamic phase transition.  We are out of time. We're pretty close.  And that's it for this lecture.
Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_0_1067.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_1068_3635.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_3636_4595.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_4596_4643.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_4644_6275.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_6276_8579.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_8580_8963.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_8964_9455.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_9456_9791.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_9792_11123.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_11124_11267.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_11268_12191.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_12192_12587.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_12588_15263.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_15264_16235.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_16236_18251.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_18252_18875.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_18876_21431.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_21432_23555.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_23556_23831.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_23832_24287.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_24288_24491.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_24492_26999.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_27000_29867.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_29868_31919.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_31920_31991.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_31992_32627.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_32628_33923.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_33924_36623.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_36624_37067.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_37068_38099.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_38100_40067.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_40068_41183.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_41184_42275.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_42276_42503.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_42504_43355.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_43356_44147.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_44148_45371.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_45372_46439.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_46440_46775.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_46776_47675.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_47676_48035.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_48036_48659.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_48660_48803.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_48804_50219.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_50220_50507.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_50508_50963.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_50964_53303.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_53304_54887.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_54888_55535.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_55536_55991.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_55992_57143.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_57144_57695.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_57696_58919.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_58920_59123.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_59124_59423.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_59424_59855.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_59856_60359.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_60360_63947.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_63948_64535.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_64536_65603.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_65604_65843.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_65844_66671.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_66672_67703.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_67704_68111.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_68112_69431.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_69432_69767.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_69768_71423.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_71424_72503.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_72504_73571.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_73572_73751.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_73752_76223.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_76224_77651.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_77652_77987.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_77988_78707.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_78708_79187.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_79188_79943.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_79944_80531.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_80532_80987.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_80988_81371.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_81372_81875.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_81876_82979.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_82980_84779.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_84780_85079.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_85080_85223.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_85224_86159.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_86160_87443.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_87444_88343.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_88344_89915.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_89916_90107.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_90108_90275.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_90276_90443.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_90444_90611.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_90612_93239.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_93240_93779.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_93780_94127.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_94128_94787.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_94788_97739.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_97740_99107.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_99108_100331.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_100332_100427.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_100428_101267.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_101268_101903.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_101904_102023.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_102024_102347.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_102348_103451.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_103452_103535.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_103536_105455.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_105456_106739.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_106740_107303.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_107304_107543.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_107544_107663.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_107664_107771.png Classes_in_Polymer_Dynamics_-_18_Colloid_Dynamics_107772_109080.png