 Great, so here we are They're gonna be inspired This is a fantastic that we have these these lectures that are Giving us an extra dimension of inspiration and today In the the president's lecture series. We have Paul check in shaking taken. Sorry Paul check in and you're a physicist that you will You will love we will also hear about a little bit about your background and story Then from your host here, but but you're a physicist and now based in New York University and The it's it's it's really Great that we can have these type of activities here Then and also combine it with that you will meet with in different constellations with people here at all During your time and and how many days have you stayed here now? Last week. Oh, it's okay It's so it's it's it's really good to be able to to stay for a week You can meet with the other different type of Disciplines with students and then maybe see a little bit of the fantastic inspirational island here so this the speakers that that we are inviting for the presidential lectures come of course from many different scientific Backgrounds and so I just encourage you to think also about who else we could invite going forward and and so that we can continue and Also broaden up and more inspiration for for everybody here at at OIST so so with that I think yeah, it's Professor Bandy that has the role of introducing our guest here today, so please Thank you President marketers. So good afternoon everyone and thanks for coming. My name is Mahesh. I'm on the faculty here at OIST And it's my distinct pleasure and honor to introduce Professor Paul Chakin Paul received his PhD from University of Pennsylvania with Alan Hager Nobel Laureate in physics I believe his PhD thesis topic was on many many body problems in superconductivity and following his PhD he started his academic career at UCLA as assistant professor and continued working in superconductivity but at some point the lure of actually seeing the microscope Microscopics of the systems led him to jump to soft matter physics And he was one of the founding fathers of an entire sub branch of soft matter physics We know as colloidal physics and that has blown up into a Big field. We have colloidal glasses crystals We have all sorts of various phenomena, but Paul has also worked on topology nanotechnology polymers all sorts all kinds of problems, but the Particular problem he's chosen to work on is a very old one the question itself took a long time to formally Define I think but as if you have read his abstract he refers to Luke from the holy Bible, but there are references to this problem in pockets of Japanese history as well So those of you who have heard of sangaku these were wooden tablets with mathematical puzzles that were hung At Shinto and Buddhist temples one of them refers to a circle packing problem It's quite famous and that is related to the topic Paul will discuss today. Another one is found in the Kojiki records Kojiki literally translates to the book of the ancients in Japan Which discusses the efficient packing of soybeans Similar references are found in Desfera Mundi by Johannes desacro Bosco From the 14th century where he discusses twisted bead moroccan purses that came from the Maghreb By traders moroccan traders into Europe. So there are references to this problem throughout history and Paul is going to enlighten a definitive aspect of it from objective Physicist standpoint today. So over to you Paul. Thank you. Thank you So I'd like to thank the president for inviting me and Mahesh for showing me around and Seeing what a really innovative fascinating place for research voice this I'm going to talk to you to give give the talk today. Oops if I can find my Activator here About an ancient problem as Mahesh said I'm going to relate it however to a model that I invented for a completely different problem Which has to do with the irreversibility of lower the Reversibility of low Reynolds number of flow, which we'll get to and talk about but the model is really simple I mean it's simple as simple simple enough that I could actually understand it and Write it down. So the idea is you throw down particles and the particles can overlap one another or not. Okay, and if they overlap they're considered active and if they're active they take a random step irrespective to the step that the the other pair of the of the overlapping pair is and This thing progresses. Okay until as you see It goes to a state where there are no overlapping particles Okay, and so this is a form of organization that wasn't sort of known before it's simply taking things where there's a problem and Only moving things to solve that problem only moving the particles that are causing the problem if you like And it turns out that will give us some insights new insights into this as we just heard ancient problem a Particle packing and a typical example of particle packing is shown over here This is a crystalline packing this over here is a disordered packing both of them sort of coexist in this just heap of particles and What we're gonna find out is that this model over here actually leads The highest density you can get with this Oh, I should have mentioned something of course if I increase the number of particles here You can all imagine it will get to a stage where I cannot never satisfy everything and so it will keep on moving Okay forever it will reach an active steady state and there's a transition between the absorbing states where everything stops and where you get this activity and that's a critical state and it's actually a Dynamical phase transition and the critical point of that phase transition Actually will be something that defines this disordered state over here and gives you with nothing else than saying I'm looking for the highest density that I can get from this model gives you randomness Jamming which Sid could have talked about but didn't really isotropy Which says that all directions things look the same independent of the direction you look Isostaticity which I'll define for you later, but essentially means it's marginally stable Hyper uniformity, which I'll also define for you later, which means that long-range density fluctuations are killed and It has an upper critical dimension of four, which means anything higher than four dimensions is the same as infinity Okay, all of them have the same behavior Okay, so the packing problem the densest packing of spheres were solved Millennia ago four or five six millennia ago as soon as people had fruit and In backward countries like the United States our grocers still Pack the fruit like this and so every child knows that the densest packing of spheres is this structure over here Which is called face-centered cubic and I'll show you why in a minute or so, okay? And it fills space to 74 percent leaving 26 percent void Okay, now the reason I say that's what we learn in the United States is because if you go to a market here Okay, you see the grocers don't pack their fruit that way anymore Technology is taking over everything here Okay, and so you don't learn as a child that the densest packing is FCC so it turns out this FCC packing was As I say, it's an ancient problem and people solved it grocers knew how to do it a long time ago It became the Kepler conjecture in 1611 Kepler said what's the densest packing I can have for cannonballs on a ship for the Royal Navy? Gauss showed that if you look only at lattices FCC is the densest lattice and the problem after millennia was actually Solved meaning not only did they have a packing that was the densest But they could prove it was the densest and that was by Thomas Hales and it took refereeing of the paper took from 1994 to 2018 just the refereeing of the paper because part of it was a computer proof and they had several thousand examples Maybe hundreds of thousands of examples that they had to check individually aside from the proof Okay Okay So those problem that problem is solved. We now know precisely what the densest packing is now in this Heap of particles that you see here from a paper by Bernal over there You've got this other phase over here, which is disordered Okay, and it fills space It's known Bernal named it random close packing rather than just close packing close packing is the densest packing That's this but the random close packing fills space to sixty four percent Okay, and its properties are not that well known. That's sort of what we're going to investigate here So why are people interested in these problems? well Originally is because back when when Grocers were stacking their spheres up instead of selling fruit or beans or whatever By Weight they sold it by volume So if you could figure out a way to pack less densely Right, you could sell more bushels. You could make more money. Okay, you could buy your neighbors farm You could become rich you could do all sorts of things and that supposedly wasn't good and that's where this quote comes from about packing of grain including was for it to get the commentary and that my Mahesh Told you on random packing which st. Luke attributes to Jesus give it shall be given unto you good measure Press down and shaken together and running over shall men given to your bosom for the same measure that you mote with all Shall be measured to you again in other words Don't cheat. Okay. You have to pack it down and make sure it's as dense as you can get it And of course, nobody really knows how dense that is Now it's also important for other reasons one of them is the structure of liquids and solids and that's what I'm going to talk about here When People started doing thermodynamics. Okay, everybody learned in Public school, I think here PV is NKT the ideal gas law, right and Vandervalls when he started saying oh, well not everything's an ideal gas. We have gases which have interactions between them He said the first thing you should consider before Attractions between the particles is the particles take up space So instead of having a volume down here or up there He says what you really want is the free volume so it should subtract from that volume the volume of all the spheres Which is just the number of spheres times the volume per sphere and I can write this in a fashion where it's Nkt over v times 1 minus 5 over 5c where that's the critical volume for that's the critical volume fraction So 74 percent for the FCC packing and 64 percent for the random closed packing okay, and Of course when you're at that there's no volume you can't the particles can't move around they're all touching one another the jammed Okay, and so the pressure diverges there and that's what I've shown over here At that Volume fraction the entropy actually goes to zero and what we want to do is maximize the entropy or lower the pressure Depending on how you want to look at it and what that tells you is sometime before you get To random a close packing where the pressure diverges you will be pushed from the disordered phase Into the ordered phase into the crystalline phase over here So it tells you that the stable state for the system is the crystal rather than the disordered arrangement and this is a very similar problem and and what that says is whatever has the highest density Okay, either the crystal phase over here or the disordered phase if I switch position Then the system would be pushed into the disordered phase so it's a question now of is the stable phase Disordered or ordered okay, and everybody's intuition Okay, is that the ground states of system of systems are ordered rather than disordered Okay, and the everybody's intuition as well is that the densest packing is going to be ordered rather than disordered And in fact nobody has a proof for that that's very fundamental But there's no proof for it and as one of my colleagues said is the only reason that you accept that is Because your mother told you when you were a kid that if you fold it up You know your clothing you could fit it into your draw and get more stuff into your draw You know when you were packing it, okay, but aside from that which is just an intuition nobody knows, okay? Well, we can look at this Transition which is seeing the pressure looking like it's diverging then Hopping over from a disordered phase to an ordered phase by doing a an experiment Which comes directly from one of Einstein's two nine two thousand five 1905 papers, okay in this paper. He's looking at colloids. He says that PV is nkt over here He's using particles that are not interacting so far, but here's the ideal gas law again. He says colloidal particles are just like Particles they're gonna obey the ideal gas law, okay, or something like it, and he tells us something else He tells us if we want to know what the pressure is we can integrate the weight of the particles that sit Above the level I'm at this that's essentially the same thing is saying is the pressure right here in this room Okay, corresponds to the weight of the of the weight of all the gas from here Up to the end of the atmosphere because the atmosphere just tails off exponentially But that's what gives us the pressure in this room is the weight of the particles above us So you can do an experiment and this is an experiment with colloids and what you find is Here is P over nkt so And this is the density or the volume fraction and what you see is something that looks like it will diverge Somewhere over here, although it's hard to really extrapolate that but what you do see is at this point You jump from a phase Which you can look at is? Liquid like because it looks milky These are the colloids and it jumps to a phase which is colorful and Those colors are a crystal is forming and the crystal has planes in it and the planes Diffract light and the wavelength they diffract is like the size of the colloid which is a fraction of a micron And that's the the color the wavelength of visible light. So they're Bragg scattering This is what if you looked at a piece of metal I don't see any aluminum around although that must be If you see it if you look at a metal and your eyes saw in the X-rays, this is what you would see You would see Bragg scattering looks like that. So this phase over here, which is right below that line is crystal and poly balls is what we call Colloids in my lab, okay, so this shows that that that exists and all of this comes directly from the fact That the densest packing is crystalline not random Okay, now You can use this some other ways, okay that Form that I showed you which is the pressure goes like one over one minus phi c is useful in other contexts when I was thinking about this I happened to go to a Bruce Springsteen concert and There's it was it is it an auditorium a basketball court And so here's the basketball kind of court over here And here's the the stage where he was and people are attracted To Bruce Springsteen, okay, and they're dancing around over here to the music So this like pressure like diffusion, okay, and in fact sitting back because we were way in the back You know with cheap seats, okay, you could see all this motion and you could see what this profile looked like okay, and You could integrate it up and find the pressure that and what you find is that people pack to 82% Okay, from the same way that we found from where the divergence of the colloids was okay, so The As of 1995 there was this phase diagram that people had come up with which said okay Sometime before you get to 63 64% you'd have to go from disordered Which is liquid to crystal was solid the actual Equilibrium point is you are liquid up to 49% and crystal above 54% And then you should be crystal from there all the way to 74% But somehow people found that when they got to this range 58% Things didn't move around so freely and they said they had a glass and that was results from computer simulations Let's see what I what else I have here. So There's this beautiful experiment Okay, that was done by Pusy and Van Megan in the mid 80s where they took colloids And they let let them settle under gravity similar to the experiment. I showed you before but looking at this phase diagram Okay, and when they had less than 50 percent over here You can just see it's cloudy over here have coexistence You have crystal down there and liquid up here more crystal less liquid more crystal less liquid over here It's completely crystalline right and then the funny thing is over here You've got a crystal floating which means it's less dense above a liquid Okay, and this is what they identified as a glass. Okay, this is a great experiment a really cool experiment and You know you learn a lot from it and gave you Essentially pretty much what was what was predicted by the computer simulations by the best that people knew at the time The only problem with this experiment is it costs ten dollars Okay, and if you're in a physics department and all of your colleagues are doing, you know particle physics and going to CERN spending billions of dollars You don't get any respect if you got a ten dollar experiment So the question is okay, how do I make this experiment more expensive? Comparable to what these guys are paying right and there's an easy way Send it into space Okay, this guy costs a half a billion dollars each time you launch it This was in the days of the space shuttle. Okay, so we sent up some experiments on the space shuttle Okay, and what we found is this was our sort of phase diagram liquid over here You see crystals forming you see a dendritic growth over here So you they grow like snowflakes which wasn't known before and then as you go up you go all the way up from 50% to almost 64 percent over here No glass So the one thing we found from this well, we found several things from this But one thing we found is that glass phase doesn't exist the sample was returned to the earth You don't want to if you want to make sure you had the right concentration and stuff So you want to mix it up. Let's say it returns as a crystal You want to mix it up and see whether it's still a glass and you mix it up on earth And this is a stir bore and the bottom of this tube is down here and a year later This is sitting here, which means it's rigid which means it's not crystalline because it's clear essentially here Here's the crystal that formed in space Okay, and the crystalline space grows into the glass which means it's the more stable phase here and so after that this Phase disappeared from the phase diagram and even from the simulation phase diagram Not because of the experiment because they had better computers in the late 90s than they had in the 80s when they when they were first doing these experiments, okay now What I didn't tell you is that Saying that FCC is the densest phase Okay, it's a little bit of a cheat because there are many many phases Which have exactly the same density as FCC The way you make a dense phase in one dimension is you make a line of particles in two dimensions You make two lines of particles, but you don't put one directly on top of the other You put one in the inter interstices of the Layer below it and that gives you this hexagonal lattice over here, which is the densest packing in two dimensions Now what you do is you say, okay? I'm gonna stack these guys on top of one another But I can do that by putting the next layer in that interstice or in this Well in that one or in that one, okay? And these where you place it are given names like this layer is called a This layer is called C. This layer is B. The FCC structure is ABC ABC ABC You can also do a B a B a B. That's another well-known structure That's hexagonal close packed instead of face-centered cubic face-centered cubic is basically a cubic structure Okay, the difference between these two Is only two parts of a thousand so the actual Thermodynamically stable phase wasn't known at the time so we went back to space. This was last year and we got this We did an experiment with the confocal microscope Which just lets you look into the sample unless you scan the sample and also look into the sample and get 3d information We took a strip like this That's the whole strip here. I'm showing you images from this strip. They're stitched together Overlaid in these regions. It's a single crystal. You can see these lines going straight through like that and when you unravel it by looking at many layers so you can see the three-dimensional structure you see That when you piece it together you see this one one one face over here It corresponds to this hexagonal face Okay, and when you rotate it and take another cut what you see is this image over here Which clearly has square symmetry and that corresponds to this face of a cube or this face of the pile And so we know from this experiment. This is sort of the biggest single single crystal of Colloids, okay, that's ever been made. That's why I called it a monster Well, I wasn't the only one that called it a monster and it says that even though there's only point two percent difference The actual ground state for the system is the FCC crystal and this is the first time that's shown Let me get back to random packing Because this problem is an old problem and Crystal packing is now completely solved theoretically and we know what the ground state is and everything else It's still not really understood what this random phase is and people Smart people have done really cool experiments with this one of them is Stephen Hales Okay, who's a botanist. Okay, who studied how Peas swelled for example and he did two experiments one I could think of one for the life Me I'd never think of as much too clever for me Okay, one of them is you find out the packing fraction over here by filling your jar with peas Putting in water and draining out the water you measure how much water there is with him without the peas You know how much room the peas took up, okay? Then what he did is he put this heavy lid on top of it and he cooked it and When you cook it the peas swell and when the peas swell They make a border with the particles they touch and that allows you to find the configuration of the cage around each particle and the number nearest neighbors and So I assigned this was when I was in Princeton I assigned an undergraduate to do this for his project for his thesis project undergraduate thesis project And he tried it and he couldn't get the peas to swell Okay, so I talked with Sir Sam Edwards who's You know one of the top guys in the field of soft and dense matter one of the founders of it I said what gives the peas don't swell because he's English and these this was done in England He said oh, don't you know that for the past couple hundred years peas have been bred not to swell Okay, so he said I'm gonna send you peas that you can do the experiment with it So he sent us the peas and they didn't swell either, but the student Wasn't stopped by that. He just uses really couscous and here's what happens when you cook there It's really couscous and you drain out the water and then you put ink back in and what you find is What the peas look like is this they have flattened faces with their neighbors over here and some of them not all of them But some of them have faces which are pentagons Okay, there's one which is a pentagon. There's one this pentagon. This one's a hexagon But you have some pentagons thrown in there and pentagons Okay Have five-fold symmetry and five-fold symmetry. Okay is not allowed Crystallographically why well for one thing if you have a lattice, okay You have to have a point at the inverse all of them have inversion symmetry And you'll note if I do this in two dimensions for example If I take that point and I find the inversion inversion point It should be over there and it doesn't exist for a pentagon for a hexagon it will but not for a pentagon and Seemingly if I made something with only pentagonal faces I would make an icosahedron and this point should be on the other side and the back face over there And that doesn't exist either. Okay So you can't have crystallographically a Crystal with five-fold symmetry or a cosahedral symmetry in three dimensions that led to Quasi crystals which I'm not going to talk about because one of the inventors Dove Levine is sitting in the back row there was the inventor of quasi crystals. Okay in any event What what does that mean? So the idea is if you start randomly packing things and you get things in icosahedral Configuration you're going to frustrate the formation of any crystal and that may be what frustrates in three dimensions Okay Forming a crystal all the time unless you prepare it specially you have to prepare it specially if you just pour the spheres in They're going to go in randomly because they're frustrated. They can't crystallize if you have some of these icosahedron in there okay now so This random close packing crystal close packing we've decided okay thousands of years ago it was solved and Finally mathematically we solved a couple years ago random close packing isn't solved yet Okay, and there's this paper that which brought a lot of attention is random close packing of spheres will define Cell torque harder will come up in a little while so I was at a Faculty party at Princeton and I happened to run into John Conway John Conway is one of the Best mathematicians in the latter half of the 20th century he invented the game of life and stuff like that I said he's an expert on sphere packing in any number of dimensions I asked them how come you don't work on random close packing because that's Unsolved okay, and what he said mathematician is Mathematician which won't touch the problem and the reason is that some things that are random are well-defined a random walk Is well-defined a random you know Set random steps or this random picking of particles out is well-defined what it means is they're uncorrelated The system is completely uncorrelated that has no memory of what goes before and what he said is Mathematicians won't touch the problem random is uncorrelated Excluded volume that is the fact that the spheres can't in pen interpenetrate one another Gives you correlations You know you can't have two particles over lacking distance between them has to be set as the diameter of the particles You can't get them close together that correlates the system anything else you want to say Doesn't matter because it's no longer random I can't define as random and that's what's held the the field up for for a while now One question that I've been aiming for one of the big questions I've always been after and as I explained to you isn't known yet is this question of Whether you can ever find anything which which Which that stacks? more densely Randomly than it does in a crystal okay, and so the idea is well If I want to see whether this is true that everything Crystallizes everything is more Ordered in its ground state. Okay, I should try some examples and the easiest example if you want to deviate from spheres is ellipsoids okay, and The best place to get monodisperse ellipsoids is from the Morris Company in terms of M&Ms So M&Ms look like this They're identical to one another to better than half a percent Okay, they're all the same shape. Okay. They're all the same size Here is an ellipso and ellipses around this projection this cut of the M&M for instance If you choose skittles, okay, they deviate by three percent or more And the reason I want to choose something that's ellipsoids because ellipsoids mathematicians have actually worked on and they can say something about it so I Gave this problem to another undergraduate Okay, I said why don't we find out the random close packing of M&Ms? Okay, so he did it. Okay. He went and he measured and he said Look when I take regular M&Ms or mini M&Ms. Okay, I get point six seven point six seven six whatever and For ball bearings point six four So this is what spheres do this was just to check that you'd get the well-known answer and these guys are higher So he says okay. Well M&Ms pack more densely that don't know we don't know yet about That's random. We think it's random. We'd have to prove that it's random, but in any event they pack denser than spheres and I said You're crazy. You're nuts. This isn't true mathematicians have proved it and what mathematicians had actually proven Oops, here we go is the following is that if you take this dense packing this FCC packing over here Okay, and what you do is you take if this is a program and I just put my pointer over there and pull this down by the aspect ratio 1.91 to 1 which is the aspect ratio for M&Ms So I just pull this side down like that. I get the densest packing of ellipsoids and it's exactly the same as the densest packing of the spheres Because I haven't changed the config I haven't changed the number of particles and the volume of an ellipsoid is four-thirds well for a Sphere it's four-thirds pi r cubed for an ellipsoids. It's four-thirds pi a b c where those are the axes the principal axes Okay, so if I change one of them. There's an oblate ellipsoid. I only change one of them I have r squared times r over 1.91 almost the factor of 2 and I've changed my container by exactly the same amount Volume fractions the same proof that you know ellipsoids that are this is a proof by mathematicians in the early 1900s that ellipsoids pack identically to spheres Okay, so I went off to fire the student right because he screwed up Okay, and then I said wait before I do that Maybe I should look at it. Maybe it's different if it's random Okay, and so I tried the same thing with my in my what my imagination Told me was a random packing which looks like this and when I pull the cursor down I get something looks like that and you look at it for a little bit and you say well There's nothing wrong with this this sort of looks okay, but this doesn't look right and You look at it for a while and finally you figure out what doesn't look right What doesn't look right is for example if I look at this particle over here Which was This particle over here, which is this particle over here Okay, before I pull it down. This guy is clearly stable, right? It's held in place by those neighbors But when I look at this guy With a some weight pushing down over here, I'm held in by these three But there's some weight pushing down in here, so I have a torque on this particle Okay, which is unresolved Okay, that is if these are frictionless. This is just going to push the particle down into this hole And so it's really different Having spheres and having ellipsoids and the difference is I have a degree of freedom here that I didn't have before I can rotate the particle. I rotate the particle here. Nothing happens. I wrote the particle rotate the particle I change the geometry Okay, and one of the ways that this shows up, so this just emphasizes that point Okay, you don't see that this doesn't look quite kosher. It doesn't look quite right. Okay, it's not stable Okay, so it turns out I said people don't know a lot about random packing one of the things they know about Packings like this which is actually due to Maxwell same Maxwell as the DNM Okay, is something that you learned in public school Okay, what you learned in public school is if I have a certain number of variables that I have to solve for I Have to have the same number of equations as I have variables, right? That will determine the problem then I can determine those variables, right? Well, where are you going to get equations in a pile of m and m's or a pile of spheres? The only way you get equations is from contacts between them So the contact I can say well the position this position and that position are related by this line that joins them together And that can determine where what are the where the variables are so the number of variables must equal the number of contacts Okay, the number of contacts is the number of neighbors Okay, this is for the whole system the number of variables is the dimensionality times the number of particles So for instance, it's D is three in three dimensions two in two dimensions But Z is times n over two because a contact involves two neighbors one with the other Okay, and so when I solve this the number of neighbors is supposed to be 2d Okay, so for spheres in 1d you need two neighbors, which is obvious In 2d you need four neighbors on average and 3d you need six, but for ellipsoids for m and m's Okay, they're ellipsoids of revolution So there's a vector and that vector has two angles that determine its orientation, right? So I have two more degrees of freedom. I have five degrees of freedom I have to multiply that by two I have to get ten and when the experiment was done Okay, we calculated and the average number of contacts for the m and m's was ten 9.87 which is close as we can get to it now We want to see whether this is really disordered or not so how do you see whether it's disordered or not? Well, you can look at and you say it looks disordered, but that's not good enough Okay, you have to find the 3d structure of this and it's hard You can't use a confocal microscope to look through a packet of m and m's so what you do oops I forgot to say something so this can this Where you satisfy this condition you have enough contacts to determine the variables, but just enough Okay, so that's called isostatic. That's the minimum number you need to define it. Okay And to understand what isostatic means the easiest example is a chair which can move Down Right tip right and left and tip forward and backwards. So that's three degrees of freedom You need three legs two legs won't do four legs is too many. Okay, so isostatic would be three here Okay, how do you know that it's not ordered you want a 3d image of it? You go to the Princeton hospital. He said my head hurts. I need an MRI You stick your head in guy goes out of room. He's put back in a bottle You know your packing of m&m's and now you can see what the structure is and you can look whether it's ordered And it's not ordered and you can calculate even whether everything is aligned. Okay much less crystalline Okay, so that works. Let's see. How am I doing? I better speed up. Okay, so You can now say well if And with if five degrees of freedom is good six degrees is even better So if I make a shape where the three axes the ellipsoids are Are different then maybe I can get even a denser Packet and here's sort of what's predicted Okay for the packing fraction. It's getting up point near point seven four for that particular geometry So we made these guys Okay, ellipsoids these are no longer by buying them from M&Ms. We used Essentially our own version because it wasn't easy. They weren't really good 3d printers at a time for making this We made a whole bunch of them and we find that found the packing of them and they packed to 70 74.7 percent better than Better than you know the the spheres better than the ellipses of revolution and better than FCC crystals Now remember I told you that it was proven by mathematicians Okay, that ellipsoids the densest packing of ellipsoids is the same as the densest packing of spheres So what happens it looks like we got something here and looks like we got something that packs better randomly Okay, then it does crystalline Well, you shouldn't really trust Mathematicians, okay, because when mathematicians tell you they have a crystal what they really mean is they have a lattice Okay, and crystals as we know them are not just lattice sometimes in each unit cell I can have more than one object where the lattice has exactly the same object in every unit cell and in fact if you take an FCC structure like that and Instead of putting your ellipsoids down like that in this plane and the plane below it You put them like this in the top plane and you put them I don't know you can see it like that in the next plane So you have these and you put them perpendicular to one another Then what you find is you get something that packs denser than spheres do and it actually packs up to 77 percent Okay, so now the question is is the ratio that we're using how does it fit relative to this? Relative to what you get for different aspect ratios here And the answer is the random packing which we found at 77 4.7 percent is beaten by an by an ellipsoid with three different axes at 7.5 at 75.7 okay, so as Of now there is nothing that packs better randomly Than crystalline in actually at any dimension that I know of but Certainly for 3d. Okay. Now. I want to go. I still have 15 minutes left. Okay, so let's see where I can do this Now I want to show you how I we got this algorithm and what it does so here is a here is a demonstration by gi Taylor master of hydrodynamics Who's going to demonstrate to you something which? Is really remarkable That flow at low rentals number that is flow That's very viscous and slow in a viscous medium with a slower with a slow motion is time reversible now Everybody knows that the laws of physics Are time reversible that means something else that means if you play a movie one way and play a Movie the other way backwards when you look at what goes on it looks like the laws of physics are baited in both of them But this is something different. This actually tells you that What happens for low rentals number the time reversibility is that the movie plays backward Played backward is just the reverse of the movie played forward and that means that the fluid elements motion is Locked is completely determined by the motion of the boundary Okay, so what what he does is in this video is He's got two concentric cylinders is shown over here. Okay, the inner one Rotates the outer one is static. He puts some ink in over here, right? And now what he's going to do is he's going to rotate it like that Okay And it looks like it's dissolving right, okay, and he's not Taking really any care to go exactly the same speed. He's doing his finger, which doesn't go at the right at the same speed To show that he's not just playing the movie backward. He moves his finger to the other side He doesn't you know again be very careful about the speed He's going as long as it's slow and you see what happens is that it comes back and That's really remarkable Okay, you will remember that movie the rest of your life Okay Okay, so Low Reynolds number flow is reversible. It's time reversible. Okay, so the question that some of my colleagues had Dave Pine and Jerry Gallup is Well, what happens if you've put particles in does the same thing hold so they said well, let's just do the GIT Taylor Taylor experiment again. We'll put particles in here Okay, we'll look at a few of them some of them will index match so you can't see them They're invisible in the fluid and some of them will be able to see and we will just do the experiment where we go Back and forth and change the amplitude of the back and forth motion We can go all the way around and come all the way back or move it a small amount Okay, and now they're going to look at it from the side And what they find is This and this are both movies. These are as a function of volume fraction and Strain amplitude that is how much you rotated the amplitude of the motion and This guy clearly is not reversible. That is it's a this is a strobe movie So you're taking a picture every cycle and here they're not coming back Okay, it would be like that ink spot was spreading out Or a cord in different places where it's here. You can see this some motion here, but pretty much every cycle They come back so it looks like it's reversible here But if you either make the density too high or the amplitude too high it becomes non Non-reversible like that not reversible okay, so When they measured it they said okay, what we find is a threshold here and there's no diffusion Effectively everything comes back to the same place if you're below this strain amplitude for a particular volume fraction and The diffusion is different in one direction the other along the radial direction and tangential direction But there's a real threshold that happens okay, so I Made this model to try and understand it and the model Comes in because the model is very simple because I don't really know how to do hydrodynamics What I imagined is I have particles and suspension and I'm gonna shear them right and they're gonna come close to one another It turns out they never really you're gonna go like that because they can't penetrate one another They're gonna go around one another, but I'm gonna say if I do an affine deformation, which means I just Tilt everything like as shown over here displace them according to these vectors. Okay, this one will go Move further than that one. They'll collide over there. I don't know how to handle a collision. That's really hard Okay, but what I'll do is I'll say I'll suppose that it moves and then when I come back to the original ones When I go back to here Okay, so I've strained it and now I bring it back. That's my cycle I'll bring it back to the original spot in other words those two But then I'll give each of them a random displacement because I don't know where they're gonna end up Okay, and that's what the model is and then I just repeat and repeat and so what's shown here is What happens in the simulation which is similar to the first one? I've shown you only here I'm showing you it's strobed so in between what you're seeing the particles are shearing and touching one another or not And you see what happens over here? Well, these are particles the blue ones in that cycle haven't touched anything else. Okay, they eventually get infected Over here the activity which is the number of red particles the one that have had collisions, okay, sort of stays Active here and over here you see what happens is They die out Okay So something funny is going on. Okay, these guys continue. Okay, these guys die out So what I did is I went to my computer and smacked it because I thought something was wrong And they didn't start up again And then I ran it again and as soon as I ran at the second time they said oh, this is remarkable What's happening is these guys because they're moving around have found positions Such that when I share them they don't collide with anything. They've organized themselves It's a problem of self-organization. Okay, and you might say well, how do I know that this thing? Okay, isn't going to find the same, you know Absorbing state as I have over here where everything stops and the answer is because I can look Oops, this thing isn't working now Hmm Well, it sometimes works. I can look at the activity as a function of the number of cycles Okay, and here if I'm above the threshold It sort of goes down and then goes to some average steady state Whereas below the threshold goes down to zero and it turns out I can measure how long it takes to get to steady state Or how long it takes to get to zero and from either side of that transition I find the divergence and that's like critical slowing down for a thermodynamic phase transition But this is a dynamical phase transition rather than a thermodynamic phase transition, right? But nonetheless its second order, okay, because I can see it coming by the divergences That's what defines a second order from first order first order You don't see it coming until you have it until you're there until you have it second order. You see it coming Okay, okay, now I'm going to tell you something else So I've we've established now that for shearing the part that for shearing these particles You have a transition between absorbing state and Active state, okay, and I'm going to introduce a new concept This was done actually by Sal Torcado was one of the guys that I did the M&M's experiment with and here's what Hyper-uniformity is about Sal says suppose I throw particles down in space in a d-dimensional space two three nine whatever I throw them down randomly a Poisson distribution as a random distribution. I ask first I now put a volume around them I make a volume which in cases these guys an imaginary volume, okay And I want to know the average number of particles inside and that's easy It's the density times the volume Okay Now and that means it goes like r to the d as shown over here, right r to the d Okay The coefficient depends on what dimension you're in now. He says suppose I move it around Okay, the number inside is going to change Okay, and so if I move this guy from here and I move it over here as I move it around I'm going to get fluctuations now If it's random this Poisson distribution, we know the answer the randomness is just the square root of n Right delta n the randomness I'll get is the square root of n That means delta n squared which is the variance rather than the fluctuations is just n which goes like v Which goes like r to the d so there are big fluctuations in the number inside as they move things around Now the question is can I do better than that and everybody's immediately say sure you can do better than that You make a crystal, okay, and a crystal you're gonna say well, it doesn't matter where I draw my circle Okay, I'm gonna have the same number of particles inside and that is everybody's intuition and it's wrong Because as I move this circle around sometimes this point is in the circle and sometimes it's out So the variation in the get goes with the perimeter rather than the area or it goes like r to the d minus 1 Okay, and that means that this thing the variation is less than is there But there's still some variation is always gonna be some variation if you have points instead of just something that's uniform What that corresponds to actually is that the structure factor Okay Goes to zero as q goes to zero as the wave vector goes to zero It means that density fluctuations at large scale are reduced a lot, okay Okay, and here's what one of those things looks like this is disordered Right, it's clearly not completely disordered You can sort of see that the spacing there between the particles is so it's certainly not crystalline though And it's also certainly not random, right? So this is sort of what a hyper uniform structure looks like Okay, now. What does this have to do with what I was anything I was telling you before? Dove Levine who's sitting in the back of the room here who is part of Mahesh's conference last week showed That act the critical point of this phase transition. So here. This is for high sheer amplitude Okay, you are active. This is the fraction of active particles Okay, you go to a steady state over here, which is active on the other hand if the amplitude is low So if I'm on a line over here then as I go through several many cycles These guys arrange themselves until they're no longer colliding with one another and I get an absorbing state and Exactly at the separatrix between here, which is the critical state for the system what? Dove found is that Your hyper uniform and this is not hyper uniform This is what you expected a phase transition any regular thermodynamic phase transition has what's called critical opalescence Like the formation of clouds, right? Which scatter light a lot because they have a whole range of different lens scales in them just when you go from the gas phase to the liquid phase that's called critical opalescence and It's it's you know Disordered at all lens scales here. Okay, and what we found what what dove found for these absorbing state phase transitions is that The long-range correlations the long-range fluctuations die out it becomes uniform So we decided to test that and our experiment unlike the coettes system was a system where we wanted to be able to see What goes on with the colloids? So here are the colloids they're in what's called a cone and plate geometry like that that makes sure that the strain rate That the strain actually is the same over here as it over here because the amount that I'm going in Angle into the blackboard over here is lower than they are But the distance I'm doing that is also lower So the strain here and the strain here are the same Cone and plate geometry this guy rotates if I can get it to rotate Oops, I can get it to rotate like that It's shaking because he took the movie with his cameras. It was handheld, but So you can do this experiment now and rotate back and forth and also look at with a microscope from the top and He wants to find a way Okay, to find the critical point of this transition. That's hard to do Okay, you have to usually take many samples around it to find out where's but he had a much clever idea He said suppose I start over here at a high strain rate. So I'm moving it a lot It will be an active state. I will go to some steady state out here I will do 400 cycles like that. So it's gonna end up somewhere over here Okay, and then what I'll do is I'll lower the strain To here and then I'll follow this curve and after 400 cycles I'll be somewhere over there and then I do that again and Then I do it again and as soon as I cross the separatrix over here The first curve I hit on this side would be just below critical and they are they organize themselves Okay, and they don't move right they've organized themselves and now they don't move anymore And if I do anything less than that they remain in that configuration It's really a real insight into what's going on. Okay, and this movie is just really cool This shows you what this idea of the organization that you have by these collisions does this is going to be This is a picture of confocal microscope picture of one plane in the sheared solution Okay, and again, it's going to be strobed So it's going to go 400 cycles back and forth Okay, and then we take a picture and then 400 cycles again with a slightly a slightly lower Strain a slightly lower shear. Okay, and what you'll see is this thing will move around Because everything is active, but as soon as it crosses this line, which is the critical Strain Okay, it will stop So watch here. It is every 400 cycles. There's a picture and now Foon it just stops right and mind you when it stops over there It's still being sheared back and forth like that. It's just it's organized themselves be below that threshold And so now each time it does that it repeats precisely. Okay, that's kind of neat. Okay now One of the things so one of the things that dove predicted is that the system is hyper uniform And that means the structure factor s of q goes like up goes to zero is q goes to zero in Dove's prediction and the prediction of that model the value of q Okay, and s the S of q goes like q to the alpha. I should have written that there it is S of q goes like q to the alpha and alpha for three dimensions in this model is supposed to have a slope of point two five And you can see if you're not at critical It doesn't have that if you're critical you have that if you're above critical It's also doesn't have that slope the critical value here is the yellow curve And you can do it for different values of Concentration and when you get to critical all of them collapse to that power law So it looks like you have a system, which is hyper uniform, okay, which is cool. So we found something out now Sam the student that did this experiment said well Let's see it whether I can fit it with the random organization model And what he found out is his volume fractions that he was using were up to 40 percent and in fact You never get in random organization a volume fraction that's higher than 20 percent Okay, so he said well, maybe we should modify the model to be able to go to a higher density And so what he did is instead of any time particles overlap Okay, they move randomly. So this one takes a random step This one takes a random step not away from the other it could be toward it. It could be orthogonal to it, but Random he said let's replace that by a step where they actually separate from one another That should allow us to organize better and go to a higher state so he did that and this is a function of How much when they overlap I give them a kick I'm each giving them a kick it could be a very big kick It could be a very small kick. This is actually the size of the kick relative to the radius Okay, and what he finds is if I don't if the kicks are random I get this curve if the kicks are directed away so the particles are separating I get this and this guy over here Extrapolates to point six four Point six four woke us up point six four is what I've been telling you about that's Random close packing it looks like So it looks like maybe we found something goes there now You might say okay, is it only when I do purely repulsive interaction? It turns out all you need is one of the steps out of a Thousand is repulsive and the others are random and it still goes to point six four Okay, and now you look at what other people have looked at they've looked at structure factors They've looked at the distance between Particles that are not touching the distribution of those all of those fit With what people associate with random close packing including the fact that is epsilon goes to zero the number of contacts Goes to six and six is isostatic and that's like jamming Okay, that's like what we found when when we looked at you know The average coordination you need to be barely Stable okay Well, I won't tell you much about this This just says that when I look at other properties like critical slowing down and like how many active particles Are there when I go past jamming it falls in a particularly universality class That is it fits in the class that random org and mana does so BRO is in the same Universality class as random org was Here is what happens when you now look at s of q As a function for this bias random org as we call it For different kick size and as the kick size goes to zero Okay Actually for the critical value for any of these kick sizes you find that the slope here is point two five But even at the hot at the lowest kick size which gives you the highest density is it It's the slope is point two five. It's hyper uniform Okay, and then you go over and you compare with what other people have measured for what they consider random close packing Okay, and these are two other models and this is our model and they fit right on top of one mother okay, and so We can now say, okay, let's propose That BRO is the thing that defines RCP and can we learn anything from that and one of the things we learn from it is that You get the answer that people have seen before For three dimensions for four dimensions for five dimensions. You find something different From what lots of people find you find that there is no such thing as random close packing in 2d or in 1d 1d is sort of trivial 2d it turns out when you do this you get this hexagonal pattern Which is the densest packing okay in two dimensions, right and it always goes there It doesn't go random and the interesting thing is we didn't tell this model That the ground state has to be ordered and in fact in three dimensions is not ordered We didn't tell it it has to be disordered either and in two dimensions. It's ordered right so there's something going on here Which is really interesting now it turns out the in In in phase transition theory there's something that's called the upper critical The upper critical dimension right and what that means is Above the fluctuations in the system when you get above that dimension the fluctuations are irrelevant Okay, it's you go to what's a mean field Okay, and that's what you expect when the number of neighbors you have when you can average Things over just your neighbors and you can average things over your neighbors very well if the dimension is infinite Because then you have infinite number of neighbors Okay, and so the physics that you get from the upper critical dimension on is the same as the physics you get at infinite dimension Okay, and it turns out for this universality class Okay, the upper critical dimension which gives you mean field here is four Two is different than one is different than two is different than three but four out to infinity is the same Okay, and here is the fraction of active the exponent for the fraction of active particles This is what we get in this bias random organization model in two dimensions We get point six four what's predicted is point six four Three dimensions and we get point eight four Four dimensions and above four and five dimensions we get one The same thing is true for the hyper uniformity the hyper uniformity, okay is For one dimension point four one with that we don't have for in two dimensions point four five three dimensions point two five Four dimensions is not hyper uniform anymore. It's random Okay, so one thing we found is if you do if you look at random close packing Okay, if you go above four dimensions and above all the way to infinity. It's random Okay, it's not hyper uniform. It's not density fluctuations going away and That's sort of interesting because and this is one thing I skipped over which I shouldn't Well, I'll get to it in a second I guess since I'm running out of time anyway, okay, and this also shows that the Number of the number of touching neighbors goes like Twice the dimension right in this model Okay, I want to do that now the one thing I forgot to tell you is why people are no no longer care About selling fruit by the bushel, right? So why do they care about any of this now? Okay, and people do care about this now and they particularly care about what packing is like in high dimensions And the reason is in if you want to do Information theory or if you want to transmit information or you want to store information For communications for computing for whatever, okay, you want to store your information Information, okay as far away Each piece of information as far away from each other piece of information as you can So that noise doesn't confuse them so that noise doesn't ruin the information you're trying to send Okay, and typically the way that's done Is by saying okay, I'm in a right Let's say an 8-bit word and I want that 8-bit word to represent Yes or no Okay, and the way I'm going to do it is yes is going to be eight ones And no is going to be eight zeros and why what I do with eight when I do with one because if things are corrupt in the Transmission, okay, I may instead of having eight ones only have seven ones or only six ones And what I'm going to do is I'm going to say the majority rules Okay, so the larger I make my word there the more certain I am that I'm protected against mistakes against errors It's error-correcting and then I want to make it that One eight ones is over here and eight zeros is as far away from that as I can get it Okay, and that's why people nowadays are interested in packing problems Okay, and they're interested in particular densest packing because the densest packing gets your units as far away from them as you can get Okay, so Well, I guess so I want to do is close this up if I can By showing you what's known about packing or one of the things is I should mention one other thing So if we shear the system in three if we don't share the system in three dimensions We get random close packing if we share the system in three dimensions. We actually get an FCC crystal Okay So our dream which almost certainly won't be true is if we go to high dimensions And we do random and we do random organization BRO We're going to end up with is random close packing And if we shear it we're going to end up with a densest phase And one of the problems with that is we don't know what the densest phases are In if you look in higher dimensions, so this is from Conway's book One packing so this is dimension and This is what people suspect are the structures Okay, they're densest, but the only thing that's known for sure is One two three Four is not this should have been one two three. It's just stopped there eight and twenty four those guys are known exactly from work that That's she did with within the past two years or so Okay, and all the rest is unknown so all the rest is open for us to find out whether whether we can get there or not Okay, so Where I've shown you through a lot whoops and way over is that it looks like This self-organization algorithm that we have random Organization can explain can give you RCP can give you the properties of RCP in One two three and now four and five dimensions at least and maybe more and that these other things which people Put in in order to define the problem So people sometimes to define random close packing they put in that the system must be disordered They put in that the system should be isotropic They put in that the system should be isostatic that has had z is equal to twice the dimension You're in that it's jammed and then it's hyper uniform and these guys Just spill out the only thing we do is we say we've got a model We're looking for the highest density critical point in that dimension for this model and the answer These things are a merchant properties then that come from that Okay, and the real hero of all of this is Sam who's over here Who sort of did the experiment to find it was hyper uniform and then suggested that we should modify random Organization to get that and other people that are involved were Sal for the Paul and and waning and Sal for looking at ellipsoids Random organization was an experiment that was done by Dave Pine and Jerry Gallup and later by the role and John Royer dove is as I say back there. He's showed that The systems were hyper uniform And here's Sam again. Okay. Thanks for your attention