Entropy, information and order in soft matter
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Glotzer, Sharon
Entropy, information, and order are important concepts in many fields, relevant for materials to machines, for biology to econophysics. Entropy is typically associated with disorder; yet, the counterintuitive notion that a thermodynamic system of hard particles (colloids) might - due solely to entropy - spontaneously assemble from a fluid phase into an ordered crystal was first predicted in the mid-20th century. First demonstrated for rods, and then spheres, the ordering of colloids by entropy maximization upon crowding is now well established. In recent years, surprising discoveries of ordered entropic colloidal crystals of extraordinary structural complexity have been predicted by computer simulation and observed in the laboratory. These findings, presented in this talk, demonstrate that entropy alone can produce order and complexity beyond that previously imagined, and that, in situations where other interactions are also present, the role of entropy in producing order may be greatly underestimated. Glotzer discusses how new statistical mechanical principles learned from recent findings can be used to design shapes that promote long-range entropic order.
 Welcome, everyone. I know there will be some people streaming in, but I think we are nearly complete and I want to start off this last part of the whole meeting, last part of the day, last part of the meeting. Stay tuned. We are going to have some prizes coming up. First, it gives me great pleasure to introduce Sharon Glotzer, today's closing plenary speaker, former colleague of mine from the University of Michigan. Actually we were remembering Monday night over drinks, she actually reminded me, just how embarrassingly short a period we did overlap there. I think it was literally days, well we met briefly, but I took off days after she officially started. Just a coincidence, of course, because I'm a big fan of her work. She's made a number of key advances in soft matter science. I'll just mention two. Very well, rather recently she's really done beautiful pioneering work on so-called patchy particles, colloids. If that's unfamiliar to you, it's emerging as sort of a field of its own. It's a beautiful way of controlling interactions and self-assembly of colloids. In addition to that, though, she also did some very fundamental work in glass formation. I want to emphasize that word fundamental because I would like to point out that her official affiliation is actually in an engineering department at Michigan, chemical engineering and material science, both actually. Sharon is a great example of something near and dear to my heart, also a topic of current discussion within foam, namely bridges between fundamental science, say physics and engineering. Sharon's a living example of that kind of a bridge. Her background, of course, is maybe not too surprisingly coming from a physics background in both undergrad and graduate UCLA and Boston University, respectively. She started her independent career at NIST, National Institutes of Standard and Technology, and after a few years there, then joined engineering at Michigan. She also has appointments in physics and applied physics. I think she represents a very nice mix of the fundamental and the applied, and not only taking fundamental insights and finding or seeking applications, but the other way around, which I think is equally important to emphasize, very good fundamental problems can arise from practical considerations. I think we're going to be hearing today about entropy and what is more fundamental, for instance, than the second law of thermodynamics. You'll know that that also has its origins and very practical problems. Sharon has a number of honors. I won't go through the list, but just point out that she's been very recently elected to the National Academy of Sciences, also a member of the Academy of Arts and Sciences, and very recently won the MRS prize of the Materials Research Society, and that was just this last fall, I believe. Without further ado, please, Sharon, we're happy to have you give the closing lecture. Thank you very much. Good afternoon, everyone. I am really just thrilled and honored to be here, to be able to attend the meeting this week and to be able to give this closing lecture. I've listened to a number of really wonderful talks and learned quite a bit. As Fred said, I'm going to talk about entropy. There's nothing more fundamental than that. Behind me, you see a molecular dynamics or Monte Carlo computer simulation of a suspension of nanoparticles in solution that when the colors are blue indicates that the system is in a fluid state, then you start to see yellow regions emerging. The yellow is, we color the particles yellow when the particles start to organize locally and start to nucleate a crystal that grows and takes over the whole system. This is playing on a loop, so you can watch it a few times. This simulation is meant to model a typical situation of nanoparticles that the majority of nanoparticles today that are grown in solution from metals, metal ions like silver, will grow in a faceted way to form little polyhedral crystals. Here, we have little octahedrally-shaped crystals. In normal circumstances, this might be a very nice simulation, but it wouldn't be so fascinating except for the fact that there are no forces, no intrinsic forces between the particles in the simulation whatsoever. The only thing that there is in this system is excluded volume. The particles are hard particles. They're moving around by brownie in motion, rotating, translating around as though they're in suspension, and they simply cannot overlap one another. Yet, they are going from a disordered fluid state spontaneously to a crystal state. This is happening solely due to entropy. We're going to talk about entropy. I know it's the last talk of the day, so I don't want to get too heavy with equations, but I want to start out with a few relationships that I know will be familiar to you. When we think about entropy, we think about the Gibbs entropy, and the way that Gibbs formulated it was that the entropy can be thought of as the product of the probability of a given microstate or instantiation of a system. An example of a microstate in the previous example might be one snapshot of this collection of nanoparticles at one instant of time. That's one possible arrangement of the particles, one possible thermodynamically-allowed microstate of the system. The product of the microstate, of the probability of that microstate here, p sub i, times the logarithm of that probability, summed over all of the microstates in the system. Generally speaking, in any system, the probability depends on the energy of that microstate, and so it depends on the temperature of the system. If you think in terms of quantum field theory, quantum mechanical applications, that we think in terms of the von Neumann entropy, where we think in terms of a density matrix where you can write the entropy as the trace of this density matrix times the logarithm of the density matrix. When this density matrix can be written in terms of its eigenvectors, then we can write the entropy in a form that's very similar to the form in which Boltzmann wrote the form of the entropy. We understand what entropy means in that case. When entropy is zero, you have a pure state, and then the measure of the entropy away from zero is a measure of the departure of your quantum system from a pure state. When we go to now information theory, then we have Shannon's expression for entropy. The way to think about the entropy in this context is that entropy is basically the equivalent to the minimum number of binary or yes-no questions that you have to ask to fully determine a system. You guys are familiar with the game of 20 questions, this game where I say, I'm thinking of a famous person. You have 20 questions that have to be only yes or no questions to guess who I'm thinking of. But now let's imagine that we're playing a bazillion questions instead of 20 questions. I say I'm thinking of a famous person. Well, Fred here, with his finely honed inquisitive skills, might start asking to try to figure out who I'm thinking of. He might say, is it Gandhi? Is it Barack Obama? Is it Lady Gaga? Is it Einstein? He could keep going, and eventually he might guess who I'm thinking of. In particular, if he doesn't know anything about me, and so all choices of famous people are equally likely, on average it would take him a very long time to guess the right person. But someone else, one of the more advanced students in the audience, for example, who's for whom statistical mechanics and thermodynamics is very fresh in their minds, might say a livor or dead, male or female, under 40, well, you wouldn't ask it that way because you have to be yes or no. So you'd ask male. So really quickly, you would design your questions to eliminate huge parts of, huge numbers of possibilities or huge parts of phase space so that you could hone in on the correct answer as quickly as possible. And if you could optimize the number of questions to minimize the number of questions that you have to ask to get to the answer, that's how to think about what entropy means. Now, if we think about a closed thermodynamic system, then all microstates are equally likely. In general, in any system, all microstates with the same energy are equally likely. But if we have a closed system, they all have the same energy, and all microstates are equally likely. And so if we have omega microstates, then the probability of anyone is just one over the total number of microstates. And then very quickly, the Gibbs expression for the entropy reduces to the Boltzmann entropy, the famous expression that we know, K, with a constant in front, K Boltzmann for purposes of units times the log of the number of microstates in the system. OK, so why is that important? That is a situation that we are thinking of when we look at that simulation from the beginning where hard particles spontaneously organized under no intrinsic interparticle forces but solely due to entropy in the system. And so the key point that I want you to take away from the lecture today is that entropy can order stuff if there are more ways for the system to be ordered than there are for it to be disordered. That's counterintuitive to many people. Oftentimes, we think of entropy as being synonymous with disorder. And oftentimes, that is the case. And strictly speaking, you can think of it as disorder in the microstates in the phase space. But in terms of thinking in spatial arrangements and thinking about the order as we conceive of it in a typical way, entropy isn't usually thought to order stuff, but indeed it can. And Ansager knew this in 1949, he showed us that if you have a system of hard rods and you crowd them enough and you have hard rods that are solely excluded volume interactions, they can't overlap, nothing else. And if the aspect ratio of the rods is long enough, then at some volume fraction, those rods will go from a completely disordered fluid state where they're pointed in any direction to an aligned state where on average the rods are aligned with one another. And here we show the aligned or ordered state by green and the disordered state by red. And it's in this way that a system of hard rods will undergo an isotropic to pneumatic transition. It occurs through entropy maximization. The system gives up rotational degrees of freedom to gain translational degrees of freedom because there are more ways to arrange the particles if they line up, there's more excluded volume available per particle. So there are more ways to range the system, more entropy, and that's what stabilizes the state. The same is true, and this is a little less obvious, for hard spheres. And this was first predicted by Kirkwood in the 50s and shown in really the very first molecular dynamic simulation of particles that was done by Alder and Wainwright in the late 50s, and at the same time one of the first Monte Carlo simulations of particles done by Wood and Jacobson, they both set out using computer simulation to test Kirkwood's prediction that even hard spheres, if crowded to a high enough concentration, might spontaneously crystallize just due to entropy. And in fact, it does at around 50% packing fraction of a completely thermodynamic equilibrium fluid of hard particles, hard colloidal spheres, will spontaneously form an FCC crystal because there are more microstates that are recognizable as crystal ordered in a regular periodic arrangement than there are ways for the particles to remain disordered. So it's a purely, it's an entropically driven phase transition towards ordering. And nowhere are entropy driven phase transitions described more eloquently and more clearly than in this very classic paper by Don Frakel on entropy-driven phase transitions published in 1999 in Physica A. This is a special picture of Don, he doesn't like to show it because the crown is tilted, he likes it when it's straight, but Don is, he really is the king of entropy-driven phase transitions and those of us like myself who work in this field learned everything we know from Don Frakel who of course made his career here and now is the chair of chemistry at Cambridge University. So we got started on this problem because, not because I started off being so obsessed with entropy like I am now, but we were trying to design cat-teliuride, cat-cellanide and other type of semi-conducting nanoparticles that a colleague of mine, Nick Kotoff, makes in his lab across the hall from me. Their little nanometer sized particles that happen to grow is little tetrahedra and they're coated with ligands and they have all sorts of interactions. They're Van der Waals interactions between these particles and solutions. The ligands can be hydrogen bonding with one another, the particles can be charged, they can have intrinsic dipole moments in them and so we were trying to understand all these different kinds of structures that he was seeing and at some point we said, well what if we stripped away, because the simulators we can do this, what if we stripped away all of the interactions and left only the shape of the particle and asked what would entropy do? And we were really surprised to find that entropy would do a lot. And in fact entropy produced for us just in the same way as hard spheres spontaneously crystallize, a crystal of hard tetrahedrally shaped particles that was actually a dodecagonal 12-fold rotationally symmetric quasi crystal, which is a crystal that has no long range translational order. There's no repeat unit like there is in a face centered cubic crystal. You can't find a part of the crystal and then repeat it and get your crystal back. But instead it has long range rotational ordering to it like in penrose tiles. And this was and still is to my knowledge the single most complex structure known or in least in the computer simulation do solely to entropy, just coming from hard colloidal type particles. It's so complicated that you could see here that there's yellow motifs and blue motifs and the so five particles will come together in a pentagonal dipyramid and then there's 12 particles that will come together in a 12-member ring where the tetrahedra go up down, up down, up down, all the way around. And then they stack up, pentagonal dipyramid, 12-member ring, dipyramid, 12-member ring. And then those things form logs and those logs stand up next to each other and tilt a little bit. And all of this spontaneously occurs out of the disordered fluid just sitting there in thermodynamic equilibrium and then just slowly compressing, compressing until you get around 50% packing fraction and then you wait for it and this just comes out robustly every time. If you were to find the nearest periodic crystal approximate, it's called, it would have 82 particles in the unit cell. So I just found that amazing. This is what the bond order diagram along the top and along the bottom, the diffraction pattern looks like on the sides, you see the diffraction pattern in the periodic direction. So dodecagonal quasicrystals are an example of an axiol or planar quasicrystals. So it's quasicrystal in two dimensions when you look down the 12-fold axis but it's periodic in the third direction and so that's what you see there. But if you look along the 12-fold axis, you see these beautiful high-resolution diffraction patterns from the simulations that look as nice as a crystallographer would ever want to get in an experiment. You could clearly see all these 12-fold rings going around reflecting this long-range rotational symmetry. So quasicrystals were discovered back in the early 80s by Dan Scheckman and his colleagues at NIST, my former home, but that was way before my time. Although I did work on the hallway where Dan Scheckman and John Kahn worked, you could feel the excitement. And so Dan Scheckman won the Nobel Prize for discovering quasicrystals in the lab in 2011 because it was not necessarily because quasicrystals were so technologically important yet, but because it represents an entire paradigm shift in the way that we think about crystals to think that you could have structures that violated typical symmetry conditions and could have this kind of long-range quasic periodicity. Now our particular dodecaiton quasicrystal of tetrahedra is actually isostructural to another chalcogenide quasicrystal comprised of tantalum and tellurium. And when we take the tantalum-tallurium diffraction pattern and superimpose it on our diffraction pattern, we see that it matches up peak for peak. So it's as though the centers of our little hard tetrahedra are sitting at the atomic positions of the tantalum and tellurium atoms in the chalcogenide quasicrystal. Now there are other soft matter quasicrystals, but they make up a tiny fraction of all of the quasicrystals that are known. Since the discovery of quasicrystals in the 80s, there have been more than 10,000 publications on quasicrystals. But only about a couple dozen in the field of soft matter. And those only started in like 2003, 2004 with reports of soft matter quasicrystals formed from dendrimor molecules, which are dendrimor molecules, are branch molecules that have a starting point at the core, that come out with branches. And then from each branch tip come more branches and more branches, like little fractal objects. You can make them more and more and more generations and you have this big kind of squishy thing with a hard rigid core in the middle. And those things take the place, if you will, of atoms. They're like giant atoms. And although micelles, such as dendrimors or blockopolymer micelles or surfactant micelles are known to self-assemble into face-centered cubic or body-centered cubic, and even A15 and other type phases, it was never seen before that they could form quasicrystalline-like phases. And so these are just some examples where 12-fold and even 18-fold rotationally symmetric quasicrystals were found. For the most part, they're all 12-fold symmetric. And what they all have in common is that they're all formed of these micelles. And when micelles form in blockopolymers, in surfactant systems, even in systems of little nanoparticles with little polymer tethers on them, once those micelles form, they're sort of like closed objects. They self-assemble and they're sort of terminal objects. And those objects have a much weaker interaction with each other than the molecules within them making up the micelle do. So you could, and they're typically spherical, and you could almost think of the self-assembly of these micelles like the self-assembly of hard spheres or soft spheres, squishy spheres. And in fact, it's possible to understand the self-assembly of these as being analogous to what happens in metallic systems. So what is typically described in metals via pairwise interactions between metal atoms as being responsible for long-range quasicrystal in order is achieved here in micellar systems due solely to entropy. Okay, so how far can we go with entropy? Was this quasicrystal finding of hard tetrahedra just one example with many more to come? Was it the tip of the iceberg or did we just get lucky? Well, we definitely got lucky, but was that it? Was there not much more to be found? And so we embarked upon this study that we've been doing now for several years of every shape under the sun and trying to understand what is the role of shape and the entropy due to shape in organizing, in self-assembling systems. And in particular, for hard particle systems where you could actually imagine screening out the interactions, then you really want to know what entropy is going to do. But even in systems where you still have other kinds of van der Waals, electrostatic, and other interactions between your particles, knowing the role of entropy that's coming just from the shape alone is very important in the design of nanoparticle systems for particular targeted structures. There are a lot of groups in the community working on these kinds of problems, and one of the most prolific groups is right here in the Netherlands, Maryland. Dykstra's group, of course, is one of the leading groups in the world in simulating entropically driven phase transitions as a function of shape and comparing those with dense packings of shapes and trying to understand how these kinds of phases are stabilized in all sorts of different shape systems. Now what we did is we started with our tetrahedron. We said, well, how robust is it? What happens if the tetrahedron aren't perfect? When we synthesized them in the lab, the tetrahedron are really perfect. The tips might be high energy, apices, they might get blunted, and they could be slightly polydipers in size, et cetera, et cetera. So we said, let's do a study where we take our tetrahedron, we start snipping the corners. Now if you snip a perfect tetrahedron enough, you get to a perfect octahedron. And so we said, all right, let's study all those shapes in between. And what I'm showing you here is the tetrahedron on your left, which is what gives us the quasicrystal. And you can actually snip it a lot and still get that quasicrystal to form very robustly. And then at some point, you snip it too much. And now instead of the quasicrystal self-assembling out of the fluid phase, a diamond crystal self-assembles out of the fluid phase. So now the particles, the tetrahedra particles, flip in a staggered way, every other one, in much the same way that you would see carbon atoms in diamond in a staggered ethane configuration. If we keep snipping even more, not quite to the octahedron, then we see this very complicated bipartite structure with 16 particles in the unit cell that's isostructural to a high pressure lithium phase, where the centers of our particles would be like the lithium atoms under high pressure. And then if you snip even more, eventually you get to a BCC phase that's known for octahedra. And so in looking at that, we try to understand, well, what is governing what structure forms as a function of the shape of the particle? And the thing that really screamed out to us was that the particles, I mean, we know that the particles are trying to maximize entropy to minimize free energy because there is nothing else. To maximize entropy means you want to maximize the amount of free volume in the system. And what we could see here is that the particles all want to maximize the alignment of the biggest faces of the particles. Because by doing so, it's like on-sug or on-steroids. So instead of rods coming together and having to align to get more translate, give up rotation for translation, these tetrahedra are coming together or the truncated tetra are coming together and they're aligning on their faces to create more wiggle room, more free volume in the system. That's governing what's going on. So then we got really eager. And we embarked upon a much bigger study. My student Pablo Shotgun Damacino decided to take a shotgun approach at this. He went to Mathematica and he said, what's the first 145 shapes in Mathematica? So he studied all of the platonic solids, all of the Archimedian solids and their duels, the Catalan solids, there are 92 Johnson solids, and then he picked a bunch of Zona-hedra prisms and antiprisms for good measure. And each one of those, we did a study like we did of the tetrahedron and said, okay, as we increase the density, what do we get? Do we get a phase transition? If we do, what kind of structure do we see? Et cetera. So just to give you an example, so out of those 145 particles, 101 of them self-assembled into a recognizable structure, these are just a few of them. We get things that are like your typical FCC, BCC, simple cubic structures. We also get at the bottom, you see liquid crystal type structures where the particles align in either in some overall nomadic phase. They're kind of pointing in one direction or some smectic layered phase or even discotic type phases. But there's some of them I want to highlight those that I'll just blow them up here. From dodecahedra and truncated augmented dodecahedra and this barrel-like structure that form beta-manganese structure with 20 particles in the unit cell or gamma brass with 52 particles in the unit cell. I mean, I'm just amazed that these things figure out to form crystals with such large unit cells due solely to entropy. It's not something that I would have anticipated years ago before starting this project. And in particular, structures that typically you need multiple types of atoms like gamma brass. To see, we even got a decagonal quasicrystal, but the decagonal quasicrystal was always not a stable to face-centered cubic. So we'd have a liquid, out would come this beautiful structure in the lower right of a decagonal quasicrystal. We watched that for a while and it would just spontaneously crystallize to face-centered cubic because that's the lower free energy or higher entropy state of the system. Now some of the crystal structures that we found have been validated, have been found by various groups. Chad Merkin's group has found a number of the simpler structures are trying to get some of the more complicated structures even without Chad's DNA programmable assembly just throwing hard particles in solution. Also Paidong Yang's group has reported this high pressure lithium phase in octahedra but using depletion interactions, which is another entropic force that we'll talk about in just a couple minutes. And so the search is on for some of these more complicated structures. And we're trying to see how complex can we go with this. Having all this data allowed us to start to see trends. And so we could group things together. So we grouped them together into all the liquid crystals and all of the proper crystals that we saw. And then in the upper right-hand corner you see 66 different crystals that we call plastic crystals. These are rotator phases, meaning that the particles self-assemble into crystals where the centers of the particles are sitting very in an organized way on the lattice sites of some crystal symmetry but the particles are free to rotate around. And that rotational entropy appears to stabilize these crystals. In other words, if you were to stop them from rotating, in some cases the crystals just fall apart. And so that rotational degeneracy is critical in stabilizing these kinds of crystal structures. So one of the things that I found most surprising about all this is that when we looked at, we took all of our crystals, so 101 of those formed crystals, but that means that 44 of them, they just did not form crystals no matter what we did. It doesn't mean that they can't, but just kinetically they just didn't. But if we took all the ones that form crystals and we looked at in the crystal phase, we said, okay, what's the coordination number of the lattice that you're in? That's like asking what's the valence of the particle. And then we plotted that and then we said, okay, now if we go back into the fluid where any diffraction pattern would show you nothing, bond order diagram shows you nothing, no correlations, it's just a fluid, and you ask what's the coordination number or valence there? We plotted it, we found a nearly one-to-one correlation between the valence, the effect of valence in the fluid and the crystal, meaning that already in the fluid, there's a type of emergent valence due to shape that's already dictating what crystal is going to form. And so if we look further, we could see that there are different classes of shapes. There are shapes where they act a lot like metallic bonding systems. These are the ones that make the really complex, big, large unit cells like beta manganese. And all of those are rotation, are rotator crystals. We see ones where they're like covalently bonded systems. And instead of being, instead of having, you know, many, many facets, they have some really distinct large facets that control the coordination number. And then we see others that are like molecular liquid crystals and form these kind of molecular liquid crystal, smectic, pneumatic, and columnar phases. And so what we learned from this is that shape entropy creates a valence that morphometrically mimics their atomic and molecular counterparts upon crowding. So how do we quantify this? The right way to think about what's going on here is that if you were to grab any nearest neighbor pair of particles in the system, they're all moving around by Brownian motion. But there's a sea of other particles around them. And that sea of other particles around them is imposing an osmotic pressure on those two particles. And there's a competition. These two particles on their own want to have as much wiggle room, as much free volume as they can. So if we look at like a pair of the truncated purple tetrahedra or the pair of the orange cubes, they want to have as much wiggle room as possible. On the other hand, the closer they get together, the more free volume there is for everybody else in the system. And those two competing terms can be quantified by writing down basically a potential of mean force and torque. This is a standard kind of construct that's used in colloidal systems where typically you'd write it down. You have some bare interaction term in which all your intrinsic interactions between the particles would be put. But here we don't have any. We're saying these are just hard particles. We've gotten rid of all the bare intrinsic interactions. We just have steric interactions. And then you have this other, the competition between these other two terms that counts the number of ways that a pair of particles can be arranged relative to each other. And for each one of those, all the ways in which you can arrange all the other particles in the system. And the reason why by writing this down is useful is it gives us, for the first time, a tool to quantify just how strong is one of these entropic bonds in the system. So here's an example taken from this truncated tetrahedron system that's truncated, not enough to form diamond, still wants to form a quasicrystal. So it's one of these very dense fluid states. And in this dense fluid state, we just pick a pair, one pair, that's next to each other. We say what's the difference in free energy of the system between that pair being aligned along their faces or pointing like this? And that's what's shown here. The difference in free energy is 3KT. That's big. That puts this emergent and tropic interaction, and this is still in the fluid phase. This puts these entropic bonds, if you will, of the same importance numerically as hydrogen bonds or Van der Waals interactions or even some screened electrostatic interactions in some of these systems. So even in the presence of other interactions, it's important to be able to quantify the role of entropy in contributing to order in your system rather than disorder in your system. Now when we look at our systems, we're typically thinking in terms of we have a bunch of particles and they're in some solution which we just imagine is imparting Brownian motion to the particles and we're just confining them more and more to make it denser, denser, denser. There's another entropic force that is closely related which is typically thought of as a depletion interaction and that's where you have colloidal particles in solution. And in addition, you have a third component which is a much, much smaller particle or a little polymer molecule, something that maybe for a one micron sized colloidal particle might be 15 nanometers in something in diameter. And the more of these little depletant particles or depletant molecules you throw in, then the more likely you are to get an effective attraction between the big colloidal particles. And the way to think about it is in this way, every one of these big colloidal particles has an excluded volume region to it and that takes up some volume where other particles can't go. And then you have to think about all the micro states available to all the depletant molecules in the system. There's a gazillion of them. But if the colloidal particles come together, then all of a sudden it frees up a little bit of local free volume that is now available to these much, much tinier particles, giving you many more micro states for the particles, making it very entropically favorable. And so that is what's behind the depletion interaction and it produces an effective osmotic pressure of the depletant molecules on the particles, on the big particles holding them together. So that's another example of entropy ordering things. The systems that we're talking about are related to that except that we don't have depletants, but imagine that the little depletants are now the original particles themselves. Of course, you have many, many fewer of them. The actual depletion interactions can be much, much stronger, but still the ones that we're seeing are very, very strong and can produce ordering in these systems. And you can find out everything you ever wanted to know, but we're afraid to ask about colloids and the depletion interaction in this textbook, of course, from Hank Lecker-Kerker and Ramco Tonier. I'm sure I'm not saying that properly. Which is a beautiful treatise on colloids and the depletion interaction. And so what we've tried to do is pull together what is known about entropic forces from depletion interactions and entropic forces from these purely hard particle systems from what Anzagernov, from Asakura, Osawa, and put it all within this one generalized framework that makes it possible to think of these things in a similar way and allows us to connect a lot of the work from all of these wonderful Dutch physicists, some of which are here with us today, like René van Roisweyve mentioned yet and Alphonse van Blottern, who have studied various examples of entropic forces in colloidal systems due to depletions and other types of problems. And so Fred mentioned that my group has worked a lot on patchy particles and the idea behind that is trying to understand how can we design nanoparticles and colloids with interactions such that we could get them to self-assemble into structures that we have decided that we want a priori. What we can do now with shape is that we can now understand how, even in the absence of any sticky patches, explicit sticky patches, we can put shape into particles and get effective sticky patches that can help to get the kind of order that you might want in your system. So I want to close with just a couple of examples of how far we think we can go with entropy just to show you some very recent things. So you might think, well, if you want a target crystal structure, why not take the crystal structure and find out what its Voronoi cell is, right? So you can take any crystal structure and find out what is the Voronoi cell of that. And some crystal structures will have multiple Voronoi cells. There's multiple Wiccoph positions to consider. And so then imagine that you know what the Voronoi cell is, so you know already by construction and the limit of infinite pressure, they have to fit that together like a puzzle to give you the space filling structure and that's your lattice that you started with. So what if you could make particles in the shape of the Voronoi cells, then throw those in water, confine them and see if they self-assemble. And what we find is that sometimes that works. What we find also is sometimes that doesn't work. And sometimes you get a totally different phase and it seems to be related to the symmetry, like the highest symmetry particles work and the lowest symmetry particles don't work. But one of the things that jumped out at me that I just was amazed by was that in some of the systems where you have two types of Voronoi particles and you know that they have to be ordered in the final structure, you take those two types of Voronoi particles, you throw them in a box and out comes, like in this case, the target structure that you started with. But there's occupational degeneracy in the positions of the Voronoi particles. So in other words, somehow entropy maximization and the shapes of the particles are enough to give you the structure you started with but without everybody in their right positions. So that's something interesting that we're still looking to figure out. We were really excited to see this recent paper from Alphonse von Blottern's group where they figured out this really clever way to start with spherical particles and grow them in a crystal in such a way that they form the Voronoi, they adopt the Voronoi shape of the crystal structure that they started with that they want to have. And so we think that this Voronoi particle shape approach might be something that's viable in the future. My friend and colleague Nick Kotoff at Michigan and I have done a lot of work where we study his little cattailuride cattcellonide particles, sometimes we mix them together with proteins. And over the past couple years, we've been studying a specific system of highly polydispersed nanoparticles that self-assemble and they're charged, all like-charged particles, also with Van der Waals interactions. And instead of forming big crystals or large elongated structures, they come together and form micelles, many, many, many micelles. And each one of the micelles is basically exactly the same size with several hundred nanoparticles in them. They adopt a core shell structure by size in the nanoparticles and once they grow up to a certain size, they stop. And if you put another nanoparticle near it, it doesn't want to attach. The force is completely balanced because of a charge renormalization that goes on during self-assembly due to the screening of the interactions by salt in the solution. And what's amazing is that the structure of those superparticles that form is so good that those big spheres now crystallize into an FCC cubic crystal. And that's what's shown in the TEM micrograph behind the cartoon images from the simulations is this nice-ordered crystal. So we can-and at that point, those big spheres are organizing by entropy, even though just like micelles from surfactant systems originally was other interactions that pulled those spheres together into spheres. Just in this month's issue of nature materials, there's a beautiful series of papers. There's a-especially beautiful paper by Alphonse von Blottern and Marilyn Dykstra showing what happens if you-you know, if you take hard spheres, they're going to crystallize into FCC like we saw. So what they showed is that if you confine it so that it-you inhibit the spheres from adopting the FCC crystal and promote the natural tendency towards local polytetrahedra, local icosahedral ordering of the spheres, that you can actually grow an icosahedral crystal-a periodic crystal with icosahedral symmetry and fastening out to about a thousand particles, which is much bigger than anyone thought you could do before, and that's do solely-do solely to entropy. And I'm going to close with an example of something that we haven't gotten yet from entropy, but which we're hunting for, and I leave it as a challenge to the students and postdocs in the audience to think about how you might do this is I mentioned that-so most of the-the known quasi crystals are-are icosahedral full three-dimensional quasi crystals. So they're quasi periodic in all three directions, yet all the soft matter ones so far are planar. So in the same issue of nature materials, we reported on the first computer simulation of an icosahedral quasi crystal, which we got not by shapes and entropy, but by point particles interacting with a short range, totally isotropic, pair potential, and one component. So that's already weird and crazy and unexpected, that you don't have to have multiple components and you don't need long-range interactions and that you don't need multi-body-many body terms which might give you angular correlations to angular interactions. But what we-we-but so far that-that is a quasi crystal that is energetically stabilized and requires explicit intrinsic interaction. And so we're thinking about what kind of shapes you might be able to make that could self-assemble into structure of this incredible level of complexity where in this structure, the nearest quasi-the nearest crystal periodic approximate, we can rule out already things with five to ten thousand particles in the unit cell, so something would have to be bigger than that. So I leave that as a puzzle to you and I'll end there. And thank you very much for the opportunity and for sticking around so late in the day. Thank you. Thank you.
 Welcome, everyone.  I know there will be some people streaming in, but I think we are nearly complete and  I want to start off this last part of the whole meeting, last part of the day, last part of  the meeting.  Stay tuned.  We are going to have some prizes coming up.  First, it gives me great pleasure to introduce Sharon Glotzer, today's closing plenary speaker,  former colleague of mine from the University of Michigan.  Actually we were remembering Monday night over drinks, she actually reminded me, just  how embarrassingly short a period we did overlap there.  I think it was literally days, well we met briefly, but I took off days after she officially  started.  Just a coincidence, of course, because I'm a big fan of her work.  She's made a number of key advances in soft matter science.  I'll just mention two.  Very well, rather recently she's really done beautiful pioneering work on so-called patchy  particles, colloids.  If that's unfamiliar to you, it's emerging as sort of a field of its own.  It's a beautiful way of controlling interactions and self-assembly of colloids.  In addition to that, though, she also did some very fundamental work in glass formation.  I want to emphasize that word fundamental because I would like to point out that her  official affiliation is actually in an engineering department at Michigan, chemical engineering  and material science, both actually.  Sharon is a great example of something near and dear to my heart, also a topic of current  discussion within foam, namely bridges between fundamental science, say physics and engineering.  Sharon's a living example of that kind of a bridge.  Her background, of course, is maybe not too surprisingly coming from a physics background  in both undergrad and graduate UCLA and Boston University, respectively.  She started her independent career at NIST, National Institutes of Standard and Technology,  and after a few years there, then joined engineering at Michigan.  She also has appointments in physics and applied physics.  I think she represents a very nice mix of the fundamental and the applied, and not only  taking fundamental insights and finding or seeking applications, but the other way around,  which I think is equally important to emphasize, very good fundamental problems can arise from  practical considerations.  I think we're going to be hearing today about entropy and what is more fundamental, for  instance, than the second law of thermodynamics.  You'll know that that also has its origins and very practical problems.  Sharon has a number of honors.  I won't go through the list, but just point out that she's been very recently elected to  the National Academy of Sciences, also a member of the Academy of Arts and Sciences, and very  recently won the MRS prize of the Materials Research Society, and that was just this last  fall, I believe.  Without further ado, please, Sharon, we're happy to have you give the closing lecture.  Thank you very much.  Good afternoon, everyone.  I am really just thrilled and honored to be here, to be able to attend the meeting this  week and to be able to give this closing lecture.  I've listened to a number of really wonderful talks and learned quite a bit.  As Fred said, I'm going to talk about entropy.  There's nothing more fundamental than that.  Behind me, you see a molecular dynamics or Monte Carlo computer simulation of a suspension  of nanoparticles in solution that when the colors are blue indicates that the system  is in a fluid state, then you start to see yellow regions emerging.  The yellow is, we color the particles yellow when the particles start to organize locally  and start to nucleate a crystal that grows and takes over the whole system.  This is playing on a loop, so you can watch it a few times.  This simulation is meant to model a typical situation of nanoparticles that the majority  of nanoparticles today that are grown in solution from metals, metal ions like silver, will  grow in a faceted way to form little polyhedral crystals.  Here, we have little octahedrally-shaped crystals.  In normal circumstances, this might be a very nice simulation, but it wouldn't be so fascinating  except for the fact that there are no forces, no intrinsic forces between the particles  in the simulation whatsoever.  The only thing that there is in this system is excluded volume.  The particles are hard particles.  They're moving around by brownie in motion, rotating, translating around as though they're  in suspension, and they simply cannot overlap one another.  Yet, they are going from a disordered fluid state spontaneously to a crystal state.  This is happening solely due to entropy.  We're going to talk about entropy.  I know it's the last talk of the day, so I don't want to get too heavy with equations,  but I want to start out with a few relationships that I know will be familiar to you.  When we think about entropy, we think about the Gibbs entropy, and the way that Gibbs  formulated it was that the entropy can be thought of as the product of the probability  of a given microstate or instantiation of a system.  An example of a microstate in the previous example might be one snapshot of this collection  of nanoparticles at one instant of time.  That's one possible arrangement of the particles, one possible thermodynamically-allowed microstate  of the system.  The product of the microstate, of the probability of that microstate here, p sub i, times the  logarithm of that probability, summed over all of the microstates in the system.  Generally speaking, in any system, the probability depends on the energy of that microstate,  and so it depends on the temperature of the system.  If you think in terms of quantum field theory, quantum mechanical applications, that we think  in terms of the von Neumann entropy, where we think in terms of a density matrix where  you can write the entropy as the trace of this density matrix times the logarithm of  the density matrix.  When this density matrix can be written in terms of its eigenvectors, then we can write  the entropy in a form that's very similar to the form in which Boltzmann wrote the  form of the entropy.  We understand what entropy means in that case.  When entropy is zero, you have a pure state, and then the measure of the entropy away from  zero is a measure of the departure of your quantum system from a pure state.  When we go to now information theory, then we have Shannon's expression for entropy.  The way to think about the entropy in this context is that entropy is basically the equivalent  to the minimum number of binary or yes-no questions that you have to ask to fully determine  a system.  You guys are familiar with the game of 20 questions, this game where I say, I'm thinking  of a famous person.  You have 20 questions that have to be only yes or no questions to guess who I'm thinking  of.  But now let's imagine that we're playing a bazillion questions instead of 20 questions.  I say I'm thinking of a famous person.  Well, Fred here, with his finely honed inquisitive skills, might start asking to try to figure  out who I'm thinking of.  He might say, is it Gandhi?  Is it Barack Obama?  Is it Lady Gaga?  Is it Einstein?  He could keep going, and eventually he might guess who I'm thinking of.  In particular, if he doesn't know anything about me, and so all choices of famous people  are equally likely, on average it would take him a very long time to guess the right person.  But someone else, one of the more advanced students in the audience, for example, who's  for whom statistical mechanics and thermodynamics is very fresh in their minds, might say a  livor or dead, male or female, under 40, well, you wouldn't ask it that way because you have  to be yes or no.  So you'd ask male.  So really quickly, you would design your questions to eliminate huge parts of, huge numbers of  possibilities or huge parts of phase space so that you could hone in on the correct answer  as quickly as possible.  And if you could optimize the number of questions to minimize the number of questions that you  have to ask to get to the answer, that's how to think about what entropy means.  Now, if we think about a closed thermodynamic system, then all microstates are equally likely.  In general, in any system, all microstates with the same energy are equally likely.  But if we have a closed system, they all have the same energy, and all microstates are equally  likely.  And so if we have omega microstates, then the probability of anyone is just one over  the total number of microstates.  And then very quickly, the Gibbs expression for the entropy reduces to the Boltzmann entropy,  the famous expression that we know, K, with a constant in front, K Boltzmann for purposes  of units times the log of the number of microstates in the system.  OK, so why is that important?  That is a situation that we are thinking of when we look at that simulation from the beginning  where hard particles spontaneously organized under no intrinsic interparticle forces but  solely due to entropy in the system.  And so the key point that I want you to take away from the lecture today is that entropy  can order stuff if there are more ways for the system to be ordered than there are for  it to be disordered.  That's counterintuitive to many people.  Oftentimes, we think of entropy as being synonymous with disorder.  And oftentimes, that is the case.  And strictly speaking, you can think of it as disorder in the microstates in the phase  space.  But in terms of thinking in spatial arrangements and thinking about the order as we conceive  of it in a typical way, entropy isn't usually thought to order stuff, but indeed it can.  And Ansager knew this in 1949, he showed us that if you have a system of hard rods and  you crowd them enough and you have hard rods that are solely excluded volume interactions,  they can't overlap, nothing else.  And if the aspect ratio of the rods is long enough, then at some volume fraction, those  rods will go from a completely disordered fluid state where they're pointed in any direction  to an aligned state where on average the rods are aligned with one another.  And here we show the aligned or ordered state by green and the disordered state by red.  And it's in this way that a system of hard rods will undergo an isotropic to pneumatic  transition.  It occurs through entropy maximization.  The system gives up rotational degrees of freedom to gain translational degrees of freedom  because there are more ways to arrange the particles if they line up, there's more excluded  volume available per particle.  So there are more ways to range the system, more entropy, and that's what stabilizes the  state.  The same is true, and this is a little less obvious, for hard spheres.  And this was first predicted by Kirkwood in the 50s and shown in really the very first  molecular dynamic simulation of particles that was done by Alder and Wainwright in the  late 50s, and at the same time one of the first Monte Carlo simulations of particles  done by Wood and Jacobson, they both set out using computer simulation to test Kirkwood's  prediction that even hard spheres, if crowded to a high enough concentration, might spontaneously  crystallize just due to entropy.  And in fact, it does at around 50% packing fraction of a completely thermodynamic equilibrium  fluid of hard particles, hard colloidal spheres, will spontaneously form an FCC crystal because  there are more microstates that are recognizable as crystal ordered in a regular periodic arrangement  than there are ways for the particles to remain disordered.  So it's a purely, it's an entropically driven phase transition towards ordering.  And nowhere are entropy driven phase transitions described more eloquently and more clearly  than in this very classic paper by Don Frakel on entropy-driven phase transitions published  in 1999 in Physica A. This is a special picture of Don, he doesn't like to show it because  the crown is tilted, he likes it when it's straight, but Don is, he really is the king  of entropy-driven phase transitions and those of us like myself who work in this field learned  everything we know from Don Frakel who of course made his career here and now is the  chair of chemistry at Cambridge University.  So we got started on this problem because, not because I started off being so obsessed  with entropy like I am now, but we were trying to design cat-teliuride, cat-cellanide and  other type of semi-conducting nanoparticles that a colleague of mine, Nick Kotoff, makes  in his lab across the hall from me.  Their little nanometer sized particles that happen to grow is little tetrahedra and they're  coated with ligands and they have all sorts of interactions.  They're Van der Waals interactions between these particles and solutions.  The ligands can be hydrogen bonding with one another, the particles can be charged, they  can have intrinsic dipole moments in them and so we were trying to understand all these  different kinds of structures that he was seeing and at some point we said, well what  if we stripped away, because the simulators we can do this, what if we stripped away all  of the interactions and left only the shape of the particle and asked what would entropy  do?  And we were really surprised to find that entropy would do a lot.  And in fact entropy produced for us just in the same way as hard spheres spontaneously  crystallize, a crystal of hard tetrahedrally shaped particles that was actually a dodecagonal  12-fold rotationally symmetric quasi crystal, which is a crystal that has no long range  translational order.  There's no repeat unit like there is in a face centered cubic crystal.  You can't find a part of the crystal and then repeat it and get your crystal back.  But instead it has long range rotational ordering to it like in penrose tiles.  And this was and still is to my knowledge the single most complex structure known or  in least in the computer simulation do solely to entropy, just coming from hard colloidal  type particles.  It's so complicated that you could see here that there's yellow motifs and blue motifs  and the so five particles will come together in a pentagonal dipyramid and then there's  12 particles that will come together in a 12-member ring where the tetrahedra go up  down, up down, up down, all the way around.  And then they stack up, pentagonal dipyramid, 12-member ring, dipyramid, 12-member ring.  And then those things form logs and those logs stand up next to each other and tilt  a little bit.  And all of this spontaneously occurs out of the disordered fluid just sitting there in  thermodynamic equilibrium and then just slowly compressing, compressing until you get around  50% packing fraction and then you wait for it and this just comes out robustly every  time.  If you were to find the nearest periodic crystal approximate, it's called, it would have 82  particles in the unit cell.  So I just found that amazing.  This is what the bond order diagram along the top and along the bottom, the diffraction  pattern looks like on the sides, you see the diffraction pattern in the periodic direction.  So dodecagonal quasicrystals are an example of an axiol or planar quasicrystals.  So it's quasicrystal in two dimensions when you look down the 12-fold axis but it's periodic  in the third direction and so that's what you see there.  But if you look along the 12-fold axis, you see these beautiful high-resolution diffraction  patterns from the simulations that look as nice as a crystallographer would ever want  to get in an experiment.  You could clearly see all these 12-fold rings going around reflecting this long-range rotational  symmetry.  So quasicrystals were discovered back in the early 80s by Dan Scheckman and his colleagues  at NIST, my former home, but that was way before my time.  Although I did work on the hallway where Dan Scheckman and John Kahn worked, you could  feel the excitement.  And so Dan Scheckman won the Nobel Prize for discovering quasicrystals in the lab in 2011  because it was not necessarily because quasicrystals were so technologically important yet, but  because it represents an entire paradigm shift in the way that we think about crystals to  think that you could have structures that violated typical symmetry conditions and could  have this kind of long-range quasic periodicity.  Now our particular dodecaiton quasicrystal of tetrahedra is actually isostructural to  another chalcogenide quasicrystal comprised of tantalum and tellurium.  And when we take the tantalum-tallurium diffraction pattern and superimpose it on our diffraction  pattern, we see that it matches up peak for peak.  So it's as though the centers of our little hard tetrahedra are sitting at the atomic  positions of the tantalum and tellurium atoms in the chalcogenide quasicrystal.  Now there are other soft matter quasicrystals, but they make up a tiny fraction of all of  the quasicrystals that are known.  Since the discovery of quasicrystals in the 80s, there have been more than 10,000 publications  on quasicrystals.  But only about a couple dozen in the field of soft matter.  And those only started in like 2003, 2004 with reports of soft matter quasicrystals  formed from dendrimor molecules, which are dendrimor molecules, are branch molecules  that have a starting point at the core, that come out with branches.  And then from each branch tip come more branches and more branches, like little fractal objects.  You can make them more and more and more generations and you have this big kind of squishy thing  with a hard rigid core in the middle.  And those things take the place, if you will, of atoms.  They're like giant atoms.  And although micelles, such as dendrimors or blockopolymer micelles or surfactant micelles  are known to self-assemble into face-centered cubic or body-centered cubic, and even A15  and other type phases, it was never seen before that they could form quasicrystalline-like  phases.  And so these are just some examples where 12-fold and even 18-fold rotationally symmetric  quasicrystals were found.  For the most part, they're all 12-fold symmetric.  And what they all have in common is that they're all formed of these micelles.  And when micelles form in blockopolymers, in surfactant systems, even in systems of little  nanoparticles with little polymer tethers on them, once those micelles form, they're sort  of like closed objects.  They self-assemble and they're sort of terminal objects.  And those objects have a much weaker interaction with each other than the molecules within  them making up the micelle do.  So you could, and they're typically spherical, and you could almost think of the self-assembly  of these micelles like the self-assembly of hard spheres or soft spheres, squishy spheres.  And in fact, it's possible to understand the self-assembly of these as being analogous  to what happens in metallic systems.  So what is typically described in metals via pairwise interactions between metal atoms  as being responsible for long-range quasicrystal in order is achieved here in micellar systems  due solely to entropy.  Okay, so how far can we go with entropy?  Was this quasicrystal finding of hard tetrahedra just one example with many more to come?  Was it the tip of the iceberg or did we just get lucky?  Well, we definitely got lucky, but was that it?  Was there not much more to be found?  And so we embarked upon this study that we've been doing now for several years of every  shape under the sun and trying to understand what is the role of shape and the entropy  due to shape in organizing, in self-assembling systems.  And in particular, for hard particle systems where you could actually imagine screening  out the interactions, then you really want to know what entropy is going to do.  But even in systems where you still have other kinds of van der Waals, electrostatic, and  other interactions between your particles, knowing the role of entropy that's coming  just from the shape alone is very important in the design of nanoparticle systems for  particular targeted structures.  There are a lot of groups in the community working on these kinds of problems, and one  of the most prolific groups is right here in the Netherlands, Maryland.  Dykstra's group, of course, is one of the leading groups in the world in simulating  entropically driven phase transitions as a function of shape and comparing those with  dense packings of shapes and trying to understand how these kinds of phases are stabilized in  all sorts of different shape systems.  Now what we did is we started with our tetrahedron.  We said, well, how robust is it?  What happens if the tetrahedron aren't perfect?  When we synthesized them in the lab, the tetrahedron are really perfect.  The tips might be high energy, apices, they might get blunted, and they could be slightly  polydipers in size, et cetera, et cetera.  So we said, let's do a study where we take our tetrahedron, we start snipping the corners.  Now if you snip a perfect tetrahedron enough, you get to a perfect octahedron.  And so we said, all right, let's study all those shapes in between.  And what I'm showing you here is the tetrahedron on your left, which is what gives us the  quasicrystal.  And you can actually snip it a lot and still get that quasicrystal to form very robustly.  And then at some point, you snip it too much.  And now instead of the quasicrystal self-assembling out of the fluid phase, a diamond crystal  self-assembles out of the fluid phase.  So now the particles, the tetrahedra particles, flip in a staggered way, every other one,  in much the same way that you would see carbon atoms in diamond in a staggered ethane configuration.  If we keep snipping even more, not quite to the octahedron, then we see this very complicated  bipartite structure with 16 particles in the unit cell that's isostructural to a high pressure  lithium phase, where the centers of our particles would be like the lithium atoms under high  pressure.  And then if you snip even more, eventually you get to a BCC phase that's known for octahedra.  And so in looking at that, we try to understand, well, what is governing what structure forms  as a function of the shape of the particle?  And the thing that really screamed out to us was that the particles, I mean, we know  that the particles are trying to maximize entropy to minimize free energy because there  is nothing else.  To maximize entropy means you want to maximize the amount of free volume in the system.  And what we could see here is that the particles all want to maximize the alignment of the  biggest faces of the particles.  Because by doing so, it's like on-sug or on-steroids.  So instead of rods coming together and having to align to get more translate, give up rotation  for translation, these tetrahedra are coming together or the truncated tetra are coming  together and they're aligning on their faces to create more wiggle room, more free volume  in the system.  That's governing what's going on.  So then we got really eager.  And we embarked upon a much bigger study.  My student Pablo Shotgun Damacino decided to take a shotgun approach at this.  He went to Mathematica and he said, what's the first 145 shapes in Mathematica?  So he studied all of the platonic solids, all of the Archimedian solids and their duels,  the Catalan solids, there are 92 Johnson solids, and then he picked a bunch of Zona-hedra prisms  and antiprisms for good measure.  And each one of those, we did a study like we did of the tetrahedron and said, okay,  as we increase the density, what do we get?  Do we get a phase transition?  If we do, what kind of structure do we see?  Et cetera.  So just to give you an example, so out of those 145 particles, 101 of them self-assembled  into a recognizable structure, these are just a few of them.  We get things that are like your typical FCC, BCC, simple cubic structures.  We also get at the bottom, you see liquid crystal type structures where the particles  align in either in some overall nomadic phase.  They're kind of pointing in one direction or some smectic layered phase or even discotic  type phases.  But there's some of them I want to highlight those that I'll just blow them up here.  From dodecahedra and truncated augmented dodecahedra and this barrel-like structure that form  beta-manganese structure with 20 particles in the unit cell or gamma brass with 52 particles  in the unit cell.  I mean, I'm just amazed that these things figure out to form crystals with such large  unit cells due solely to entropy.  It's not something that I would have anticipated years ago before starting this project.  And in particular, structures that typically you need multiple types of atoms like gamma  brass.  To see, we even got a decagonal quasicrystal, but the decagonal quasicrystal was always  not a stable to face-centered cubic.  So we'd have a liquid, out would come this beautiful structure in the lower right of  a decagonal quasicrystal.  We watched that for a while and it would just spontaneously crystallize to face-centered  cubic because that's the lower free energy or higher entropy state of the system.  Now some of the crystal structures that we found have been validated, have been found  by various groups.  Chad Merkin's group has found a number of the simpler structures are trying to get some  of the more complicated structures even without Chad's DNA programmable assembly just throwing  hard particles in solution.  Also Paidong Yang's group has reported this high pressure lithium phase in octahedra but  using depletion interactions, which is another entropic force that we'll talk about in just  a couple minutes.  And so the search is on for some of these more complicated structures.  And we're trying to see how complex can we go with this.  Having all this data allowed us to start to see trends.  And so we could group things together.  So we grouped them together into all the liquid crystals and all of the proper crystals that  we saw.  And then in the upper right-hand corner you see 66 different crystals that we call plastic  crystals.  These are rotator phases, meaning that the particles self-assemble into crystals where  the centers of the particles are sitting very in an organized way on the lattice sites of  some crystal symmetry but the particles are free to rotate around.  And that rotational entropy appears to stabilize these crystals.  In other words, if you were to stop them from rotating, in some cases the crystals just  fall apart.  And so that rotational degeneracy is critical in stabilizing these kinds of crystal structures.  So one of the things that I found most surprising about all this is that when we looked at,  we took all of our crystals, so 101 of those formed crystals, but that means that 44 of  them, they just did not form crystals no matter what we did.  It doesn't mean that they can't, but just kinetically they just didn't.  But if we took all the ones that form crystals and we looked at in the crystal phase, we  said, okay, what's the coordination number of the lattice that you're in?  That's like asking what's the valence of the particle.  And then we plotted that and then we said, okay, now if we go back into the fluid where  any diffraction pattern would show you nothing, bond order diagram shows you nothing, no correlations,  it's just a fluid, and you ask what's the coordination number or valence there?  We plotted it, we found a nearly one-to-one correlation between the valence, the effect  of valence in the fluid and the crystal, meaning that already in the fluid, there's  a type of emergent valence due to shape that's already dictating what crystal is going to  form.  And so if we look further, we could see that there are different classes of shapes.  There are shapes where they act a lot like metallic bonding systems.  These are the ones that make the really complex, big, large unit cells like beta manganese.  And all of those are rotation, are rotator crystals.  We see ones where they're like covalently bonded systems.  And instead of being, instead of having, you know, many, many facets, they have some really  distinct large facets that control the coordination number.  And then we see others that are like molecular liquid crystals and form these kind of molecular  liquid crystal, smectic, pneumatic, and columnar phases.  And so what we learned from this is that shape entropy creates a valence that morphometrically  mimics their atomic and molecular counterparts upon crowding.  So how do we quantify this?  The right way to think about what's going on here is that if you were to grab any nearest  neighbor pair of particles in the system, they're all moving around by Brownian motion.  But there's a sea of other particles around them.  And that sea of other particles around them is imposing an osmotic pressure on those two  particles.  And there's a competition.  These two particles on their own want to have as much wiggle room, as much free volume as  they can.  So if we look at like a pair of the truncated purple tetrahedra or the pair of the orange  cubes, they want to have as much wiggle room as possible.  On the other hand, the closer they get together, the more free volume there is for everybody  else in the system.  And those two competing terms can be quantified by writing down basically a potential of mean  force and torque.  This is a standard kind of construct that's used in colloidal systems where typically  you'd write it down.  You have some bare interaction term in which all your intrinsic interactions between the  particles would be put.  But here we don't have any.  We're saying these are just hard particles.  We've gotten rid of all the bare intrinsic interactions.  We just have steric interactions.  And then you have this other, the competition between these other two terms that counts  the number of ways that a pair of particles can be arranged relative to each other.  And for each one of those, all the ways in which you can arrange all the other particles  in the system.  And the reason why by writing this down is useful is it gives us, for the first time,  a tool to quantify just how strong is one of these entropic bonds in the system.  So here's an example taken from this truncated tetrahedron system that's truncated, not enough  to form diamond, still wants to form a quasicrystal.  So it's one of these very dense fluid states.  And in this dense fluid state, we just pick a pair, one pair, that's next to each other.  We say what's the difference in free energy of the system between that pair being aligned  along their faces or pointing like this?  And that's what's shown here.  The difference in free energy is 3KT.  That's big.  That puts this emergent and tropic interaction, and this is still in the fluid phase.  This puts these entropic bonds, if you will, of the same importance numerically as hydrogen  bonds or Van der Waals interactions or even some screened electrostatic interactions in  some of these systems.  So even in the presence of other interactions, it's important to be able to quantify the  role of entropy in contributing to order in your system rather than disorder in your system.  Now when we look at our systems, we're typically thinking in terms of we have a bunch of particles  and they're in some solution which we just imagine is imparting Brownian motion to the  particles and we're just confining them more and more to make it denser, denser, denser.  There's another entropic force that is closely related which is typically thought of as a  depletion interaction and that's where you have colloidal particles in solution.  And in addition, you have a third component which is a much, much smaller particle or  a little polymer molecule, something that maybe for a one micron sized colloidal particle  might be 15 nanometers in something in diameter.  And the more of these little depletant particles or depletant molecules you throw in, then  the more likely you are to get an effective attraction between the big colloidal particles.  And the way to think about it is in this way, every one of these big colloidal particles  has an excluded volume region to it and that takes up some volume where other particles  can't go.  And then you have to think about all the micro states available to all the depletant molecules  in the system.  There's a gazillion of them.  But if the colloidal particles come together, then all of a sudden it frees up a little  bit of local free volume that is now available to these much, much tinier particles, giving  you many more micro states for the particles, making it very entropically favorable.  And so that is what's behind the depletion interaction and it produces an effective  osmotic pressure of the depletant molecules on the particles, on the big particles holding  them together.  So that's another example of entropy ordering things.  The systems that we're talking about are related to that except that we don't have  depletants, but imagine that the little depletants are now the original particles themselves.  Of course, you have many, many fewer of them.  The actual depletion interactions can be much, much stronger, but still the ones that we're  seeing are very, very strong and can produce ordering in these systems.  And you can find out everything you ever wanted to know, but we're afraid to ask about colloids  and the depletion interaction in this textbook, of course, from Hank Lecker-Kerker and Ramco  Tonier.  I'm sure I'm not saying that properly.  Which is a beautiful treatise on colloids and the depletion interaction.  And so what we've tried to do is pull together what is known about entropic forces from depletion  interactions and entropic forces from these purely hard particle systems from what Anzagernov,  from Asakura, Osawa, and put it all within this one generalized framework that makes  it possible to think of these things in a similar way and allows us to connect a lot  of the work from all of these wonderful Dutch physicists, some of which are here with us  today, like René van Roisweyve mentioned yet and Alphonse van Blottern, who have studied  various examples of entropic forces in colloidal systems due to depletions and other types  of problems.  And so Fred mentioned that my group has worked a lot on patchy particles and the idea behind  that is trying to understand how can we design nanoparticles and colloids with interactions  such that we could get them to self-assemble into structures that we have decided that  we want a priori.  What we can do now with shape is that we can now understand how, even in the absence of  any sticky patches, explicit sticky patches, we can put shape into particles and get effective  sticky patches that can help to get the kind of order that you might want in your system.  So I want to close with just a couple of examples of how far we think we can go with entropy  just to show you some very recent things.  So you might think, well, if you want a target crystal structure, why not take the crystal  structure and find out what its Voronoi cell is, right?  So you can take any crystal structure and find out what is the Voronoi cell of that.  And some crystal structures will have multiple Voronoi cells.  There's multiple Wiccoph positions to consider.  And so then imagine that you know what the Voronoi cell is, so you know already by construction  and the limit of infinite pressure, they have to fit that together like a puzzle to give  you the space filling structure and that's your lattice that you started with.  So what if you could make particles in the shape of the Voronoi cells, then throw those  in water, confine them and see if they self-assemble.  And what we find is that sometimes that works.  What we find also is sometimes that doesn't work.  And sometimes you get a totally different phase and it seems to be related to the symmetry,  like the highest symmetry particles work and the lowest symmetry particles don't work.  But one of the things that jumped out at me that I just was amazed by was that in some  of the systems where you have two types of Voronoi particles and you know that they have  to be ordered in the final structure, you take those two types of Voronoi particles,  you throw them in a box and out comes, like in this case, the target structure that you  started with.  But there's occupational degeneracy in the positions of the Voronoi particles.  So in other words, somehow entropy maximization and the shapes of the particles are enough  to give you the structure you started with but without everybody in their right positions.  So that's something interesting that we're still looking to figure out.  We were really excited to see this recent paper from Alphonse von Blottern's group  where they figured out this really clever way to start with spherical particles and  grow them in a crystal in such a way that they form the Voronoi, they adopt the Voronoi  shape of the crystal structure that they started with that they want to have.  And so we think that this Voronoi particle shape approach might be something that's viable  in the future.  My friend and colleague Nick Kotoff at Michigan and I have done a lot of work where we study  his little cattailuride cattcellonide particles, sometimes we mix them together with proteins.  And over the past couple years, we've been studying a specific system of highly polydispersed  nanoparticles that self-assemble and they're charged, all like-charged particles, also  with Van der Waals interactions.  And instead of forming big crystals or large elongated structures, they come together and  form micelles, many, many, many micelles.  And each one of the micelles is basically exactly the same size with several hundred  nanoparticles in them.  They adopt a core shell structure by size in the nanoparticles and once they grow up  to a certain size, they stop.  And if you put another nanoparticle near it, it doesn't want to attach.  The force is completely balanced because of a charge renormalization that goes on during  self-assembly due to the screening of the interactions by salt in the solution.  And what's amazing is that the structure of those superparticles that form is so good  that those big spheres now crystallize into an FCC cubic crystal.  And that's what's shown in the TEM micrograph behind the cartoon images from the simulations  is this nice-ordered crystal.  So we can-and at that point, those big spheres are organizing by entropy, even though just  like micelles from surfactant systems originally was other interactions that pulled those spheres  together into spheres.  Just in this month's issue of nature materials, there's a beautiful series of papers.  There's a-especially beautiful paper by Alphonse von Blottern and Marilyn Dykstra showing  what happens if you-you know, if you take hard spheres, they're going to crystallize  into FCC like we saw.  So what they showed is that if you confine it so that it-you inhibit the spheres from  adopting the FCC crystal and promote the natural tendency towards local polytetrahedra, local  icosahedral ordering of the spheres, that you can actually grow an icosahedral crystal-a  periodic crystal with icosahedral symmetry and fastening out to about a thousand particles,  which is much bigger than anyone thought you could do before, and that's do solely-do solely  to entropy.  And I'm going to close with an example of something that we haven't gotten yet from  entropy, but which we're hunting for, and I leave it as a challenge to the students  and postdocs in the audience to think about how you might do this is I mentioned that-so  most of the-the known quasi crystals are-are icosahedral full three-dimensional quasi crystals.  So they're quasi periodic in all three directions, yet all the soft matter ones so far are planar.  So in the same issue of nature materials, we reported on the first computer simulation  of an icosahedral quasi crystal, which we got not by shapes and entropy, but by point  particles interacting with a short range, totally isotropic, pair potential, and one  component.  So that's already weird and crazy and unexpected, that you don't have to have multiple components  and you don't need long-range interactions and that you don't need multi-body-many body  terms which might give you angular correlations to angular interactions.  But what we-we-but so far that-that is a quasi crystal that is energetically stabilized and  requires explicit intrinsic interaction.  And so we're thinking about what kind of shapes you might be able to make that could self-assemble  into structure of this incredible level of complexity where in this structure, the nearest  quasi-the nearest crystal periodic approximate, we can rule out already things with five  to ten thousand particles in the unit cell, so something would have to be bigger than  that.  So I leave that as a puzzle to you and I'll end there.  And thank you very much for the opportunity and for sticking around so late in the day.  Thank you.  Thank you.
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