 Dwi'n edrych i'r ddefnyddio amser wrth gweithio am y cyfnodd. Oeddwn i amser o'r rhan wych yn gwneud ddweud ar yr unrhyw yng Nghymru. Mae'r rhan o bobl sydd wedi byddig yn gwybod ar y cyfnodd. Ond i amser o'r rhan o'r ddefnyddio ar y ddweud, mae'r rhan o'r hwn yn gwneud i'r ddweud, ond o'r ddweud sut oedd yna'n ddweud. O'r rhan o'r ddweud o'r rhan o'r pan pwynt o'r ddweud. o pobl wrth pwrthwyd, mae'r tre'w argyn大家都 hynny i'n gwneud ffordd. Mae'r gwasll lleadau Pwysigol hynny yn fawr o'rités sydd, ond mae eich gwneud o'r ffordd gyda rhyngw oligol am gyfllud. Mae'r gwneud i'r gwเขll? Mae'r gwneud i'r innas gyda Llywodraeth, mae'r cymstech, mae'r gwneud o'r magwyr o lwythkeld o'r fforddol gyda hynny. Felly mae'r rhan o'r casmyd eich Llywodraeth yn delsydd. dyma cymd高u ym mhotoedd o'r leidau. maen nhw, maen nhw. Rwy'k nodi'r cymdeithas. Mae'n symyn i hyn, yn polymu'r cynnwys, fel mae'r cyllidau ymlaen, mae'n mynd i chi gweithio'r cynnwys i fynd i chi'n gweld fyddoeddwnnod o fwy o gyllidau a heddiw, торwch o'r hollbwy arweithio, fel mae'n gofynu sydd bywys. Mae'n gwneud i'r cwmwy o'r hollbwy o'r bobl wedi bod, is that in the glass transition as you cool them, you end up with stress relaxation or dynamics that becomes exponentially or even stretched exponentially slow. So the idea is if you plotted a time scale, some kind of relaxation time or the log of some relaxation time against temperature, once you get down to below the glass transition temperature, that's in this cool region here, everything starts to slow down, the relaxation times get long. And the best you can probably say about this is that the glass transition temperature TG characterises some apparent divergence in the relaxation time. So what I'm going to tell you about today, everything we're going to do is well above TG. So any kind of glass that I might refer to is absolutely not to do with a loss of microscopic degrees of freedom as you would find at low temperatures, rather it's something that occurs at high temperatures. And if you needed any more motivation, this is a quote from Phil Anderson who's really just saying that, you know, the glass transition is something we really need a bit more understanding on. So basically I'm going to sort of come to this at the end. I'm going to join up the idea that there might be some kind of slowing down in these ring polymer systems that looks like a glass. And because polymers, at least to a physicist look very universal, we can treat them as strings, featureless strings in space. There's some universality associated with the dynamics of polymers. And then, therefore, maybe we can port that universality into some understanding of jamming and glassiness. OK, so the system I'm going to talk about today has nothing to do with jammed colloids or frozen polymers. It's a system that's like these rubber bands. So I'm going to ask you what do you think the nature of this system is? So is this what, reologically or dynamically, is the nature of the system? I had, if we were in Warwick, I'd get a big bucket full of rubber bands that I keep by my desk and I'd put it on the table and I'd say reach into this big bucket full of rubber bands and grab one and try and pull it out. And what you'll find if the rubber bands are long enough and I've shaken the box well, as you try to pull one rubber band out, all of the other rubber bands come out somehow entangled with the first rubber band. Now we know that these rubber bands are topologically unlinked. They're not prepared like Randy's chain mail that he showed us this canetaplast. They're unlinked. So in principle there's no true topological interactions here. The nature of this entanglement is something that's not truly topological. And of course the physics would be rather different if first I set about cutting all of these rubber bands into linear strands and then I asked you to do the same experiment. You could reach in, you could take one of these linear strands and it would just snake out of the bucket without any problem. So there's something about the topology of these rings, these polymer rings or in this case rubber bands, that leads to this kind of strange entanglement. And really this talk is going to be about what is the nature of this kind of entanglement. So if you want to do any kind of calculations for ring polymers, you've got one advantage which you can write down a standard polymer description that's universal. And so the chemical details, as I've mentioned, aren't important, provided you're well above the glass transition temperature. But if you try and actually calculate anything, doing ring statistical mechanics is very hard because you need to typically calculate things like partition functions at fixed topology. So I need to look through all my microstates and I need to not just ask is this particular microstate of a ring, this is a ring, acceptable or what's the thermodynamic weight associated with that microstate. I have to also ask are there any other microstates that are linked with that microstate and therefore violate the condition of global fixed topology, unlinked? So that makes doing sums rather difficult. So I'm going to tell you today about simulations. And I guess this was all motivated by an interest some time ago where if you look in the literature back in the 80s and 90s, you start to find some sort of rather I'm going to call speculative statements by some authors and I'll let you read this short piece. But essentially these authors are referring to the possibility that you could get two rings, these are my two rings, and I could take one ring and I thread it through the first ring. And once I've threaded it through the first ring, the motion of the threaded ring, the one in the middle, is severely limited. It can't go very far, it can't diffuse very far until the first ring has moved out of the way. So that's essentially what these authors were referring to. I think it's a really neat idea. But the problem is, I mean, how can you actually quantify that? Can you further quantitative statement about whether these threadings either exist or if they exist, are they important? Okay, so today I'm going to tell you about two different ensembles, two different problems. One is ring polymers in the melt and there's extensive literature of people who've done simulations on ring polymers in the melt. We've done some too. This is a rather poor picture. I've just picked out a red ring polymer in a background where I've taken all the other monomers and I've grained them out. First, actually, I'm going to tell you about ring polymers in another ensemble. This is ring polymer plus gel. So the idea here is you take rings, lots of rings in the concentrated limit and you force them into a gel. So the gel here is these kind of grey struts that are periodic and cubic and the different colours in here are the different ring polymers. So I've pushed them all into this box and I'm going to ask how do they move. So this ensemble also has been looked at before some years ago. It's got some advantages and I'm going to try and convince you that it's an interesting system to look at. So the first advantage if you like is that it's a system that's probably the most widely performed experiment on the planet, which is you take a gel if you're a biologist and you try and separate polymers by length by forcing them into a gel just like I sketched on the last slide. So this is a very common experiment. Here are some ring polymers. They're actually bacterial plasmids. So in principle we could put bacterial plasmids into these gels and look at the dynamics. So this is not a crazy ensemble. It's something that's very accessible. And if you don't like the kinds of gels that biologists use if you consider them a bit disordered, although I think that's in itself interesting, you can also perhaps look at micro patent surfaces that are being made these days and think about putting ring polymers into these more ordered systems. So Davidea and the rest of us spent some time thinking about ring polymers in gels, thinking about the self-interactions of ring polymers in gels. So if you put a field on these ring polymers, something that can happen if you look in the bottom slide here, if you can just see it, is this ring polymers threaded through itself and the field's pulling it this way, but it's tied itself in something like a knot around the lattice of the gel and it can't move anywhere. So we spent some time thinking about the dynamics of ring polymers, single ring polymers in gels. I'm not going to tell you about that today because I'd like to spend the time on something else. So here's the first of the two things I'm going to tell you about. This is ring polymers embedded in a gel and just to emphasise the system, we take unknotted, unlinked ring polymers. We take them in a concentrated solution so they're well above the overlap concentration. Many ring polymers are in contact with one another and we embed them in a rigid gel and we require that they're not linked with the gel. So first we prepare a perfect cubic lattice, then we put the rings in. We treat it as a cubic lattice for simplicity and again because we don't want to burn computer time simulating the vibrational dynamics of this lattice, we completely neglect the gel dynamics, we treat it as a rigid object, I don't think this is important. And I'm going to tell you today about two things, some molecular dynamic simulations and also some Monte Carlo simulations. So here's the idea, here's our system, you can just about see the grey ends of the lattice rods sticking out. Here are our different ring polymers and I'm going to zoom in in a minute and I'm going to look at what's happening in the middle of that box. So first let me just set up the simulation background and if there are any technical questions I'm going to look at Davide. But these simulations were performed using lamps and basically this is a bead and spring type simulation, it's a large van dynamics simulation in which you've got a Lennar Jones potential between the beads, you've got some backbone bending potential so you can introduce some kind of stiffness and there's a finite extensibility potential between the beads that means they can never move apart sufficiently far that you get a strand crossing. So we can preserve topology, we check that. And the parameters that we're going to use here if any of you are interested is we've got a persistence length for our chains which we set to be the same as the mesh size for our gel and that's 10 beads. And we work at densities that are typically 10% volume fractions that's quite concentrated and we look in boxes that are between 20 and 90 beads across each edge and that's enough to make sure we don't have any periodic self-interactions between the rings. If any of you are interested you can read the paper here. So this is a movie, it's a short snapshot or short sequence from the simulation in which just two chains in fact there's a grey one in the back but you should just focus on the red, sorry the yellow and the green one, there are lots of other chains that have been removed and the yellows and the green ones are interacting. In fact if you look you can see the green ring here is pushed in a protrusion through the yellow ring. So this is something like one of these kind of threading events that I just described in which the green chain has threaded itself through the yellow chain. Okay so by eye it looks like these kinds of things might be happening in our simulation. Let me just come back. Right so there's many things we can do now. The first thing we can do is we can look at the mean squared displacement so averaged over all chain segments and we can ask how does that scale with time. So what you've got is you've got a sub-diffusive regime and then for the short rings you've got a clear linear and T-diffusive regime at late times. For the longer rings the sub-diffusive regime extends much further and in particular it extends into a regime where the mean squared displacement of the rings is actually bigger than their radius of duration. So this is a little bit strange, but the rings have got some kind of memory that lasts longer than the time it takes for them to move outside of their immediate environment. So what's the sort of meaning behind this? Well maybe this is some sort of signature of the fact that in this sub-diffusive regime you've got some memory of the threadings whilst in the fully linear and T-regime you've gone to times where you don't have any memory of the threadings. So that's a little bit speculative. We can look in more detail over the next few slides about that proposition. So I've told you a little bit about these threadings. Maybe one ring polymer is threading through another as I've shown you a couple of times now like this. So is this important? Before you can ask whether it's important you have to ask how you identify them and that's actually not trivial. So here's two rings that most people in the audience would freely admit are interpenetrating. But the problem is if you look on a knot table you see that actually this is topologically equivalent of course to the two unlinked state because these two rings are not in any way linked. So any kind of topological measure that you might apply to this problem will reveal simply that these two rings are unlinked. They will say nothing about whether or not they're threading because that's not fundamentally a topological quantity. So we were in the cafe yesterday Gareth and Mark and I and we've seen quite a lot of these knot tables in the talks so far but there was a fantastic picture in the cafe upstairs you should go and take a look at it a fan of classification table of coffees in Italy I'm sorry it's not a very good picture so obviously the Italians take their coffee as seriously as we take our knots. So okay let's let's work a bit with this threading so let's try and come up with some sort of definition and the definition that's extremely convenient in the gel is that we can exploit the gel architecture. So here's a snapshot where I've taken my gel and I've removed all of the other many chains in the system and I've just kept the yellow ones and the green ones and I'm going to look at a lattice volume in which both the yellow and green chains happen to be present. Here it is and you can see by eye that the green chain is clearly threading through the yellow chain but again how do we make that statement firm? So what we do is we start some closure operation so we take the yellow chain and we take the yellow chain where it leaves the lattice volume that's here and we close the yellow chain using just a simple straight line construction between the two points that the yellow chain leaves the lattice volume and then with the green chain we identify whether the green chain leaves the lattice volume and we close those not on the face but at infinity or if you like behind the cell. So what we've made now is we've made topologically we've identified these two double strands of the two different rings with this kind of system in which the yellow ring, I've now constructed this kind of pseudo loop for this piece of the yellow ring polymer is linked with the two little pseudo pieces of the green ring polymer. So now we've got something we can identify topologically and it's actually we can run with that so here's what we do so first of all we see that how do we un-enticiw whether there's a threading or not. So what we do is we say that the number of threadings between ring I and ring J at time t is constructed as follows. You do a sum over all of the lattice volumes, the cells C and for each lattice volume you sum over all of the segments of the jth chain, that's in this case the green one, that are present in that cell and then you compute the linking number between the closed loop constructed for the ith chain in that cell with the closed loops constructed for each of the jth segments in that cell and you divide it by two because here you can see we've got one penetration, one threading but we're going to divide by two for the linking number so this dividing by two just gives us a weight of one for this if you like single duplex threading and what's interesting is there's a kind of non commutative symmetry here if ring I is threaded by ring J and there's both of these rings are threaded in some sense but there's an active threading in this case that's the green one because the green one has if you like moved through the yellow one and the yellow one is passively constrained the green one moves away and then the yellow one can move again so there's a different character to the threading and threader the threadee and the threader here which we refer to as an active and passive chain and that comes out of the order in which so if I change the identity I change the order of which chain is threaded with which okay so now we can identify these threadings we've got a way of defining them in the gel we can count them so we can look at the number of threadings per ring polymer per chain and we can vary the chain length so m in all of these slides will be the number of beads on the ring polymer and this is the number of threadings n will be the number of polymers so the total number of threadings divided by the number of ring polymers is the number of threadings per polymer and what we find is that this trend is almost perfectly linear it shows no sign of breaking down so that says that the number of threadings between these rings seems to be extensive in their mass and although we can only access a regime where we've got perhaps two or three threadings per ring our MD breaks down at this point there's no obvious limit this trend suggests there's no obvious limit on the number of threadings we could generate so if we took ring polymers and we were able to have a computer that could go to 10,000 or 100,000 this trend suggests that we could generate a very large number of threadings between these rings and count them we can do other things with them we can measure their temporal correlations so we can construct a correlation function at time t which tells us if there's a threading between ring i and j at t0 is there a threading between ring i and j at some later time t appropriately normalised so here's the correlation function for the threadings as a function of time and you can see that first it has a very slow relaxation and then later it drops off it comes to a constant value because this has a non-zero infinite time asymptote an interesting inset is that we've here compared the stress relaxation due to the stress carried by the ring polymers themselves with the correlation function for the threading the loss of threadings and you can see that stress is lost before the threadings are lost and that might seem a bit odd but if you think about it if you've got a chain threading through another one the chain that's passively threaded the one that I'm holding with my thumb and finger here this one can move around quite freely it can't freely diffuse because it's still threaded but the stress that's associated with most of the tube segments on this chain can still relax so it's not inconsistent actually that the stress relaxation can die faster than the threadings so we can define a correlation time for the threadings we say we wait until a tenth of the threadings that were initially present have died away have been lost and then we can drop down where the number of threadings is the tenth of the initial number and we can pull off a correlation time which I call T0.1 so this is the correlation time for our threadings right that's one thing we can do there's another thing we can do which is once we've got this equilibrated system of rings and we can count threadings we can associate threadings together in this kind of network so the idea here is let me break this down this is a busy slide so we're looking at different lengths of ring polymers this is the polymer bead number we'll start with the smallest ones and for the smaller systems these are 256 length polymers there are 50 of them in the box and this is the linking network or the threading network that you get so if you look on the top here this is telling you that ring number 26 there's an arrow associated with this ring number 19 so this is ring 26 and this is ring 19 so there is in our box, in our simulation box a threading event like that between ring 26 and 19 they've all got indices because of course we're running this on a computer we can trace their identity and what we see for these small rings is that we see a few small networks of inter threadings actually one or two rings there's a slightly larger one here but as we go to systems of longer rings so here's for 512, 1024 and 15 whatever it is 16 or something this, if we just jump straight to the largest one this is the network that we find so we find an extremely large inter penetrating network that contains about half in fact I think it's more than half of the rings in the system many of which are both threading and threaded by each other so these kinds of large highly inter penetrating networks are generically difficult to undo so what you have to do is if you think about the rings the rings on the edge of this network that are threaded you have to remove one and then you can release the next one that can then go on and release the one that follows and so on so undoing these kinds of networks you would expect generically would take quite a long time and the networks again we see no necessarily plateauing of this kind of trend if we go to longer rings we should expect an extremely inter penetrating network so the main body of the text here is showing a betty number which is some measure of the loops the number of little loops in these graphs and the number in the most strongly connected component which in this case is actually it's almost all the rings there are 50 rings I think in this system so they're all rings so the the network are just it's a perfect lattice yes you can well so I'm not allowing the rings must be topologically unlinked from the network that's that's required because I construct I mean I'm a chemist say and I construct this lattice and then I construct the ring oh yes there are certainly loops I'm sorry in this network there are absolutely loops and in fact this is the measure that there are in this network so here you can see that there are there are loops in this network well that's just a crap picture but take my word for it there are loops here I'm sorry so these represent so they're coloured if they're a strongly connecting component which means that they're they're both threaded and threading another ring is that right Davide can reach any node from any other node on one of these colours right so we can also of course do some more kind of typical if you like physics computer experiments we can look at relaxation times so we take our ring system and we construct a measure of relaxation times we ask how long does it take for us to lose most of our threadings how long does it take us to relax stress and there's another measure here which I actually won't talk about and what we find is that there's a clear power law behaviour but for the largest rings and only for the largest rings we start to see a break from the power law it's pretty clear for the time taken for the threadings to relax for the stress relaxation we're actually not sure this is fully equilibrated from a stress relaxation perspective so this is a lower bound on the stress relaxation so we think maybe there are some hints that in the standard dynamical quantities we're starting to see a slowing down so what else can we do well we can exploit the fact that these ring polymers form a duplex structure what do I mean by that I mean that if you look in a cell the ring has to come in with one branch do something in the rest of the system and then come out with another branch and that's a condition of it being unlinked with the gel and that means I can kind of sketch the problem as a duplex system in which I need to have these kind of two stranded excursions everywhere in the gel and that means I'm not allowed these kinds of excursions because they link the polymer with the gel and that was forbidden so of course as soon as I see a picture like this I start thinking of lattice animals I start thinking of linear polymers maybe with branches and I'm going to even simplify beyond that and I'm going to think about what if I treated these ring polymers as purely linear objects quasi linear objects so I forget about the existence of this branch and I just treat it like a polymer the polymer happens to have two strands one going one way and the other returning back again so if I do that I've got this kind of wormy picture in which I've got a ring this is ring one if you trace it you can see it's continuous it comes back on itself but it looks like a polymer or a sausage because I haven't allowed it to have any branches so this is a massive oversimplification but it's somewhere to start and in this particular sketch I've shown two other rings one of which is providing this ring one with a passive penetration so ring one is kind of stuck by the fact that ring two is threading through it until ring two gets out of the way and over here ring one is providing an active penetration to ring three so ring three is kind of stuck until ring one gets out of the way and we're going to treat this in a kind of standard reptative picture in which all of these sort of linear like objects can reptate like linear polymers but subject to the existence of these threading constraints so we're going to take the curvilinear diffusion and we're just going to map it in the contour length space so this is contour length I'm going to treat all of the polymers as if they're linear sausages and I'm going to sorry it hasn't come out very well I'm going to idealise this kind of fully three-dimensional picture I showed by the existence of these passive active threading pairs so everywhere I've got an active threading by one polymer obviously I need a passive threading on another polymer the one that it's threading through so this is an attempt to these red dots are passive threadings so I've drawn them inside the contour of this kind of sausage to give you the idea that it's like a pin it's like a nail that's penetrating the sausage the sausage can't get past this nail until the nail gets out of the way and the nail gets out of the way when this sausage moves through the corresponding active threading and annihilates the active passive pair so this sketch is the representation of the sketch I showed you on the previous slide this one where we've just got these two active passive threading pairs both involving chain one like this and we'll discretise space so now we can diffusively move these polymers and we'll just look at a later time when the system's moved a little bit what's happened well this particular ring has curvilineally diffused a little bit to the left this ring has curvilineally diffused a little bit to the right the bottom ring has also curvilineally diffused a little bit to the right but what's happened is that with some external probability that we can control within this model it's generated a passive active threading pair ring two at this point and generated this new pair of of threadings and then at a later time what's happened is well ring two's moved a little bit to the right it's actually annihilated this pair which is now no longer present and ring three has moved to the right was attempting to move to the right but it's pinned by the presence of this passive threading that belongs to chain one so this move in our Monte Carlo code is not allowed because it's jammed by the presence of this passive threading and you can see that if I've got lots of these red dots scattered around in the system I've got a real problem because the dynamics is going to slow and almost worse than that to get rid of the red dots I've got to access these kind of blue loops and the blue loops can be hidden behind red dots so the system can get very slow this is the idea you know you could imagine some kind of jamming in which the red dots hide a bunch of green loops that themselves are connected to red dots so hierarchically you have to undo everything before you can relax the stress ok so what do we get so this is the Monte Carlo code just following that scheme that I outlined to you for a worm-like linear curvilinear diffusion I'm saying it's in zero dimensions because I'm not really correctly treating the three dimensional nature of the polymer coils I'm treating them all as if they live in the same zero dimensional system where they can all freely interpenetrate anywhere they like and this is the relaxation time sorry these haven't come out very well this is a disengagement time in units of the hot time so this is the stress relaxation time log of and this is the log of the chain length so straight line is a power law this is Doi Edwards stress relaxation p equals zero this is this external parameter that I can control the rate at which I make new threadings basically every time I make a move I've got a probability p that it generates a new threading if I never make any threadings I've just got free curvilinear stress relaxation ala Doi Edwards if I start to make threadings with some probability that I control I break away from this power law and I start to see something that looks quite exponential I see that quite clearly so the stress relaxation time is increasing by orders of magnitude here okay and I won't even talk about that in set what else can you do well actually it turns out that there's a characteristic time it's not the hop time it's not the time to move one unit if you like one tube diameter there's a time which is the time it takes to hop between penetrations so if you've got a certain number of threadings the time it takes you to diffuse between those threadings is something like this it's inverse square of the if you like the density of threadings and if you rescale the relaxation time in units of this threading hop time everything collapses onto something that looks like an exponential straight line here so this is log linear this is number of threadings against relaxation time so this is clearly indicating an exponential stress relaxation as a function of the number of threadings in the system which we can control directly by this microscopic parameter p and this is just a blow up of what's happening round zero that's important okay so we've got exponential stress relaxation here that's the end of everything I'm going to tell you about the gel where we've got a clear signature at least in this Monte Carlo code and some hints from the MD of some kind of dramatic exponential jamming ala a glass so this next section is going to be coming back to the melt actually for us that means a very concentrated solution here's a snapshot of our melt there's no longer any lattice involved here this is just a pure polymers they're still unlinked from one another so there's been we've published on this but also in the same year a couple of papers came out a bit later from groups also interested in measures of threading so this is pretty topical at the moment and right so one thing we're going to look at is contiguity so how do I define contiguity so here's a snapshot from the simulation I've just taken three or four chains and I've removed all the other ones so and I've coloured the chains so there's a cream chain here and the blue chain they touch you can see at this point so for us touch is within some measure of distance that's comparable to the bead size roughly but the cream and violet ones don't touch there's never any contact between them so something we can do in the melt is we can look at the persistence of contiguity these kinds of contacts between chains and we define a very kind of stiff contiguity function well so this is the definition of contacts as I told you basically if any two monomers are closer than the inverse cube root of the density that means the inter bead spacing then there's if they're not if there's no contact then there's no contact if there's any one bead pair that's in contact you score one and this non-contiguity correlation function is something it's the product of this which means that for all times less than t these two chains have to be contiguous they have to be in contact otherwise this records zero so this records one up until the time that the cream and blue chains are no longer in contact and then it records zero so if we average that over all chains the smooth function that tells us how the contiguity between chains is evolving so we find that contiguity is very persistent for longer chains so here's this correlation function I showed you about it turns out to be roughly stretched exponential early on with a power law tail later but we can compute that the memory time here this is the important quantity the non-contiguity time and this non-contiguity time scales there's two measures you can take you can either integrate the full correlation function or you can just do a fit to the stretched exponential and in both cases you get a decorrelation time that's exponential in the chain length so the contiguity in this system seems to be living for a time that scales exponentially with the length of the rings it also the number of contiguous neighbors and the number of neighbors which is simply two chains that are considered neighbors if they're within rg of one another grows weekly with n but doesn't very much okay so what can we do with all of this so we've looked at contiguity we decided to look at pinning this system and I guess one motivation that you might have for that is that there's been a lot of interest recently in randomly pinned glasses so this is a picture from a recent paper by these co-workers but there's been a number of other articles in which what one does is one pins say you take a colloidal system near the glass transition you pin some fraction of the colloids externally by hand you freeze them and you ask what does that do to the nature of the glass transition so that's that's a topic at the moment in the field of glasses but we're going to do the same thing with rings so here's what we do we've got a bunch of rings in our system I'm going to call them at the moment idealize them as boxes and I take a bunch of constraints and there's a fraction C of rings that are going to be constrained and here I'm going to maybe constrain half of them so C is the fraction that are constrained and for us constraining means completely immobilizing so more recently we've been looking at some other versions of constraint but everything I'm going to show you today is completely immobilized so here we are we randomly drop these green beads into these boxes and the ones that have a bead in this case ring 2, 5, 6, 8, 9 and 11 are completely frozen so the way that the simulation works is we pre-equilibrate a melt of rings and then after it's pre-equilibrated we identify a fraction of them to completely immobilize and we ask what happens to the rest of the rings so here's a movie in which the red ring is still free to diffuse it's one of the ones that didn't have a little bead in its box but the other rings are ones that have been frozen and what you can see is that the red ring is stuck, it's stuck because it's threaded essentially by the grey, blue, green and or yellow rings so the fact that the red ring can't move is a signature, a dynamical signature of the existence of threading so the problem is here of course we're searching for a definition of threading we can't use the definition that we used in the lattice because we don't have a lattice volume to construct this nice topological closure on anymore so the idea is we look at dynamics and this signature tells us about the existence of threading so let me show you what we've got here so first let's look at this trace this trace is for ring polymers where all but one of the rings are frozen and this is the dynamical progress of the single unfrozen ring in a background where all of the other rings are frozen and this is its centre of mass diffusion against time log log so what you find is that single ring just like the red ring I showed you on the last plot is not getting anywhere its centre of mass diffusion is caged just like you saw on that movie and this dotted line is rg for the red ring so it's stuck if I take rings I unfreeze all the rings and I let them diffuse, I see this trace if I do the same thing for linear rings a linear chain so standard open linear polymers and I freeze all but one of them then the linear polymer the free one that's in the bath of n-1 frozen ones can still diffuse it doesn't feel this caging of course it doesn't feel this caging because there's no threading for linear chains so this is completely consistent with the idea that the reason why this single free ring is not moving is because it's threaded and we have frozen so this is actually the first I think good evidence that threadings actually exist in the melt so believe it or not this is such a pretty contentious question whether or not there are even threadings present but this I think unambiguously shows that they are because the dynamics of this ring is clearly massively affected by its neighbours and that is due to the fact that it's threaded okay so this is showing us that we can look at in the unfrozen fraction so we've frozen some of the rings and we look at the unfrozen fraction and we see how many of those are immobilised so basically even in the explicitly unfrozen fraction we've got some that are threaded and immobilised just like I showed you like that red ring in that movie and some that are unfredded and can diffuse so obviously if I don't if I only freeze one ring out of a thousand I'm probably going to expect most of them still to diffuse and only a few of them to be threaded but in the limit where I freeze all but one as you saw basically all of them or the only free one is threaded so you can ask how does the number of immobilised or caged rings so this is in the free fraction vary per frozen ring so the number that I freeze and how does that vary with M so for each ring that I freeze this is the number of rings that I implicitly immobilise as a function of the ring length so for every ring I freeze I get a knock on effect above this line I'm getting more than one implicitly immobilised ring for each explicitly immobilised ring and actually that number, that trend for that seems to be clearly exponential so for long rings if I just freeze one I get a massive knock on effect and you can see how this leads to a kind of topple threading susceptibility that if I get a fluctuation and there's one threading that could lead to a knock on where the immobilisation of that chain leads to the immobilisation of other chains ok so what am I showing you here, I'm showing you the fact that if I take longer rings so this is a phase diagram and I'm looking at inverse ring length against the fraction of explicitly frozen chains and what we've done is we've looked we define a liquid to be a state in which there's some diffusion of some rings and a glass in which there's no diffusion of any of the rings most of which down here are not explicitly frozen so there seems to be a line separating a glassy state in which there's no diffusion from a liquid state and that line is coming down into this interesting corner, this universal corner which is large ring length and small number of explicitly frozen chains and I don't think we would like to take this too seriously but at least on this exponential fit there's an intersect finite chain length, I don't actually take that too seriously obviously this is the interesting corner it's also the corner that's most important or most difficult to simulate okay so basically the whole picture of this talk was the question of whether or not we ever get these hierarchical constraints whether we get a network in which in order to move one ring you have to move a bunch of other rings that also you need to move rings and this generically you'd expect to lead to exponential slowing down and it should look like a glass I mean if I take some sort of universal limit where the number of the number of threadings per ring is very large it's going to take me extremely long time to undo ring one for me that looks a bit like a glass so just to wind up on a lighter note this is entirely Davide's suggestion Davide decided to go into his kitchen and see if he could reproduce this system in pasta he's Italian what can you say so here's three tubes of pasta he's made by rolling out sheets and then sealing them along a seam and then you can turn them around and chop them to make rings like this and then you can cook them and then you can try and eat them so this is a picture of his wife picking up some of this pasta with a fork and what you can see is right at the beginning of the talk I told you what happens if you reach into a bucket of rubber bands you've got the same problem with the pasta so it doesn't come off your plate because it's tangled up with all of these other rings and how much olive oil you add so if this was spaghetti you wouldn't have a problem because these rings are not going anywhere so it's a problem and the conclusion is you need to use a bib if you want to eat this stuff because it's pretty messy and bragging rights we have a recipe published in physics world recently and there's a non-commutativity with the limit in which you add dough and eggs which Davide was quite proud about so that's it and just to round up with some conclusions so there's two systems I told you about rings embedded in a gel I think the big story here is we've actually got a way of defining threadings we can count them and track them the number of threadings per chain seems to scale extensively in the ring length which means that you can just dial up the ring length and get as many threadings as you like and it's kind of natural that you should expect jamming if you can do that so what we see is an emergence of a strongly connected interpenetrating network so this is something generically you expect to be slow to undo we see the first hints of a crossover to some sort of non-powl or maybe exponential relaxation in the MD and in this oversimplified sausage-like Monte Carlo study we see that extremely clearly then the second story I told you about rings in the melt so the intriguing bits here are we've got memory of a contiguity that seems to be exponential in M this is another hint that we've got exponential relaxation we can show for the first time that threadings exist and that's revealed by when we pin some fraction we see that there's a knock on other rings get immobilised by that and then finally this discussion point about ok so where have we got to here so we've got at least some strong hints that we might be getting to a system that gets jammed and it gets jammed purely due to the topological nature of the rings themselves not through any loss of microscopic degrees of freedom so this is a pretty unusual kind of jammed state like glasses as far as I'm aware there's always some sort of microscopic jamming so what we've got is a system that's jamming like a glass but it's got the universality of physics of polymer physics behind it so maybe this is a way of attacking the glass transition where you've really got some solid universality behind you and some acknowledgments so as I told you almost all of this work was done by Davide who's for a more than a year now been a member with Davide churning out papers as they do there which is great the Monte Carlo work was done by Weijang Loh who's now at Duke and we also benefited from a bunch of discussion so some of this work was done in collaboration with Enzo and Joe I don't think anything I told you today was done with Christian but he's been talking to us about other things Gareth was also a co-author Andrew in Oxford has been trying to make some plasmates in his lab and we've been trying to actually do some experiments and even back sort of 15 years ago or more a colleague of mine, Jan, made some plasmates and we put them in a reometer at Davide's place at Harvard and tried to look at some plasmates but without too much success so I think they deserve an acknowledgement and that's it, thanks for your attention