 Prv je Marko Baezzi z Padua, in nekaj se prišličajo mekanizacija v pojenju včivarii vseh vseh vseh, zelo prej, hibrič na vseh. Dobro. Svečo prednejo, nekaj vseh mekanizam v vzpešenih vljubnih vljubnih, zelo vseh vseh vseh. Vseh je dobroha v Trefano, Eubinii in Dendoor, Landinii. Stefano is now in Florence, while Enzo is in Padova. And so basically what I talk to today is about the micro rheology in the first place, because it's the problem that motivated our curiosity about this. And then I will talk about a little bit of a coarse-grain model of this world like missile networks that we have developed for simulations, and then it turned out that we could also do the mean field model for that, that works quite well. So the kind of micro rheology experiments that we are interested in deal with these micro bits that are dragged by actively, let's say, if you want, by optical tweezers. And this morning I took this picture of the, because it's not easy to picture the scale easily. The grid that you see, the mosquito net, is more or less the mesh size that this kind of micro bit sees when it meets this complex fluid. So it's really a very small set of holes that are in front of the thing. So when it's pulled, it's pretty difficult for the fluid to rearrange easily after the passage of this huge object that is distorting the motion. So there is a constant velocity v pulling the trap there, and so the bid. And in the end, one can quantify how fast this is going by this dimensional Weissenberg number, which is the ratio of two timescales. One is the relaxation timescale of the fluid to be found, and the other one is given by this ratio of dimension of the sphere divided by the velocity of the trap. So as long as the Weissenberg number is small, it's meaning that one is essentially in equilibrium and the fluid has enough time to rearrange. If the Weissenberg number is large instead, one is far from equilibrium. So there were numbers available for these experiments, and the one that caught our attention is this one by Berner et al., so backingers groups, essentially, in which they find these non-monotonic relaxations to equilibrium. So they have a situation in which, for example, for a finite Weissenberg number, I wanted to do something like this one. Anyway, maybe it's better for the people at home if I do something like this. So this is a situation with finite Weissenberg number, in which you see that there are these sort of stick-slip oscillations that are looking not so much close to equilibrium. But still, the Weissenberg number is small, and one is believing that the situation is in equilibrium. It's becoming even weirder if one looks at the description of the Langeven equation, the generalized Langeven equation provided by this publication, let's say. So in the generalized Langeven equation, you have a memory kernel gamma, and if you are close to equilibrium, you expect the fluctuation dissipation theorem to hold, and so to keep the same structure for gamma, also for small velocities, let's say. But this is not what happens. If one wants to fit the data from these experiments, one needs to have some negative terms and things going on, which are essentially perturbing to match the situation. It's not anymore the same memory kernel one as for passive diffusion of the bead. So this motivated our interest in the topic, because we wanted to understand what was going on at the level of the microscopic relaxations in this complex fluid. So specifically, because eventually there is a chance that the time scales are not currently evaluated, maybe there is some very long time scale, which is not understood. So specifically, in this case, we are talking about micelles, which are these surfactants forming micelles, eventually forming structures more complex. In a given regime where you, for example, your shampoo and lubricants are examples of this kind, in which you have a network of tubes formed by these surfactants, which are very complex. So they are mostly linear polymers joining in some cross links somewhere. And they are called leaving polymers, because they are all the time attaching and detaching. So they are not really a linear polymer where the linearity is kept forever. These kind of tubes are joining and detaching all the time with what's called the schisium. Or we have distinguished the breaking from schisium at different angles to talk about another process in which the cross link is broken to have a free end and linear strength. So we had a coarse grain model for this kind of dynamics. And we consider one of these units, including a lot of these surfactants. And the blue spots are those that are sticking with each other. So this is a model for simulations in which we get this rigid unit in which you can form easily a chain of these red balls, because the blue sticky points are sticking to each other. But if they are exposed enough outside of the red hardcore, let's say, there will be also the chance of having this kind of triple contacts. So the model can tune a little bit the propensity to do that by increasing this parameter lambda, which is the distance between the sticky spots. And so with this model tuned accordingly also to what we see in experiments, because it turns out that the schisium, so the breaking of this living polymer is very costly at the level of KBT. 30 KBT is a lot, so it's not happening all the time, this kind of process. We can tune our parameter to meet these results. And I can skip the details of the model. And what we find in practice is that in a temperature quench, so we start with all the balls by themselves in the fluid. And then we lower the temperature suddenly, so that they want to start to form the linear structures, essentially. So it's a very strong quench, let's say. After some time, we start to see the formation of this network that resembles those of myself. So we have the structures with haps, in which we have these cross links, let's say. And the color of the balls is according to the number of contacts. So the grayish, greenish ones, let's say, are those with three contacts, and the linear ones you see in reddish. While there are these end points, which have a yellow color, to see that there are also present these kind of motifs within the network, let's say. So these are the protagonists of what I'm going to tell you about during the last part of the talk, which is the mean field model. And the other ones, of course, are the branching points. Okay? So at the level of this temperature quench, a very easy quantity, a very easily measurable quantity is the length of the strands within the network, which are essentially the lengths of these polymers within to end, which are branches or end points, let's say. And this grows over time, so it starts from zero, because everything isolated, but from the beginning, then you have these linear strands growing and merging, and starting to detach from each other, and so on. So you see that at the beginning, there is some sort of dynamical scaling, because it's going up, let's say, linearly with time, then there is a plateau, and then it starts again to grow linearly in time, finally meeting an equilibrium level at the end. And here we are able to map everything to the level of seconds. So for this particular setup of the simulation, in any case, we can go to very long times, because it's a very cost-grain simulation. So now we want to understand what is this time scale, which is at the end of a power load, by the way, so it's not so easy to distinguish normally. Dynamical scaling in the end, so this is the name of the game, if you want. At the end, we don't have so much branching anymore, because the average length is growing and growing, for the reason being that the polymers become longer and longer with fewer cross-links, let's say. Eventually, in the end, in this system, you have up to a few units of cross-links. So the mean field model tries to model these kind of things, so, of course, it's a mean field model, so it's very simple and it doesn't have a spatial structure, but it's just a mean field equation. So it turns out that these are the few reactions that we can take into account that are sufficient for describing the evolution of the simulations, and I'm going to show you a comparison with the simulations. So, for example, there is the merging of two free units in a double unit, let's say, like that, or if there is a situation like this in which you have a free end joined by another isolated unit, and then it becomes a free end attached to the rest of the polymer. So this can be read. These dashed lines means usually there is the rest of the polymer going on after the structure. So this, the flux that we will see for this kind of transition number C, let's say, is the crucial one, so it's the annihilation of the free ends. Because without this, in the end, you have a lot of free ends, which are energetically unstable and they cannot result in an equilibrated situation. You want long polymers in the end, and this is the only way of getting rid of the free ends and generating long polymers. There is an alternate solution, which is getting a free end and sticking it to the middle of a polymer, creating a link, a cross link of three. So a hub, if you want. But this is not energetically as good as this one. So the bottleneck is going to be this transition C. But this is a temporary solution, and it's present. It is explaining the results that we see in the simulations. So we term ni, the fraction, the numerical fraction of the unit, so the number, the fraction, let's say, of units of kind 0, of kind 1, 2, 3. And there is also the probability of having this unit, which is this ratio. So we get rid of m0 in this probability. So these are the kind of transitional rates that we can imagine being useful for this system. There is a transitional rate W for the forward transition and the transitional rate U for the backward transition, which is a detachment transition, in which you need to pay some KBT of the order of 30 for detaching, let's say. And, OK, it took some time to sort out of the over 2 times 2 and so on, but in the end, we got them correctly. And with this transitional rate, now we see what happens. We can have these fluxes for these transitions. So one uses those transition rates to generate the fluxes. And these are put inside just a simple linear equation, in which we have a sort of master equation, in which we have the variation in time of these densities given by this stoichiometric matrix multiplying these fluxes. OK. N's and p's. So the N's are directly the densities. So it's the number of units. The p's are slightly different, because we don't consider the five minutes, OK? We don't consider the isolated units. So there was a little bit of tuning of these kind of things, because sometimes you need to have the things attached. So it's assumed that they are attached, and so you don't want to use N1, but you use p1, because they are supposed to be attached. OK, and using these kind of things in the end by putting it some inside the thermodynamics, so you see that the detachment rates are proportional to the attachment rates, but there is exponential of minus of free energy for detaching. So they are smaller, including entropic terms and so on. We can do that. And for the beginning, for example, for short times, we have this nice symmetric situation, which is just the solution of the previous equations, in which it's a simple set of equations. I don't know if it's just an academic set of a couple differential equations, but in the end, you get these very simple solutions for short times, ignoring the presence of the abs. If you start to include everything, in the end, in any case, the numerical solution we get fits the data that are the points very well. So the mean field model, in the end, gets also correctly the right asymptotic values, which are these dot lines, and it's satisfactory because it's at least explaining the simulations. So in this mean field model, the bottleneck, as I was anticipating, is this transition. When you create these abs, they need to be broken at some point, and they need to somewhat be luck enough to generate a free end, meeting another free end somewhere else that annihilate each other. The annihilation, in any case, is present already at the beginning. I mean, in this initial power law, there is already an annihilation of free ends. This is explained by the annihilation equation, let's see. So, trying scaling, so one sees that, for example, increasing this binding free energy, there is a longer and longer time, and the scaling we find is this one, OK? Here, there is still some local equilibrium between abs and free ends, and so they go in parallel with each other. If you increase the hub free energy, so what we call g, there is also some scaling, but it is more sensible, so it's exponential of g, while, in this case, it's exponential of the free energy divided by two. So, by putting everything together, the final scaling of the relaxation times takes this form. Of course, there is a prefactor, depending on all the microscopic details, but then there is a strong exponential dependence on all these free energies. And this dense of detaching and attaching is only taking more and more time than just avoiding this kind of process. So, it's a very complex process going on for the relaxation. So, we can also try to see what happens with the gentle perturbation. We have the system in equilibrium, and we can imagine that locally, the passage of the bead is just changing a little bit the fraction of free ends by increasing them slightly, let's say, because it's, I don't know, an ideal mechanical perturbation in which one is slightly increasing the number of free ends. And this, in the end, are going to relax in the same fashion asymptotically as we had for the thermal quench. So, the final time scale is the same. It's only that the shape of these kind of things is different, but in any case, you see, again, that this blue line, which is the number of hubs, which is at the beginning in equilibrium, temporally has a maximum, because it's, again, due to the fact that temporarily the free ends find this temporary solution of meeting the middle of a strand before starting to detach and so on. So, this is a very complicated dance for relaxing to equilibrium. And so, in the end, what we see seems to point to a situation in which there is at least a time scale for the relaxation, which is not very easy to spot, which is this one due to the fact that the free ends are touching and detaching. And we are wondering whether this is something, indeed, affecting the relaxation of these micelles in experiments. So, to conclude the story about the experiments, I can tell you that, indeed, in the end, the suspects were meeting by new experiments in the sense that Matthias Kruger and co-workers did new experiments in which they went even slower with the trap. So, by reducing the velocity, they went down to Weissenberg numbers of the order of 10 to the minus three now instead of 10 to the minus one, too. And what they see is an effective viscosity, which is, indeed, the step into a different level. So, this confirms the suspect that the previous experiments were not really close to equilibrium. They were already pulling too strongly the system far from equilibrium, which means that we can slip well in the sense that the normal fluctuation dissipation TRM is restored in, I mean, is preserved close to equilibrium, we can believe. The gamma kernel is the same close to equilibrium, which is this regime, probably. And then, if we go too fast, we are in a regime which looks like in equilibrium, but it's not. So, this is what the macroscopic rheology would find as a regime. So, I close with this open problem. Now, there is the problem of understanding in this rheological and micro-reological experiments what is really close to equilibrium and whether they are rheological experiments who are measuring something which is the true viscosity or something which is actually already a non-linear effect that generates a different viscosity. Thank you. Thank you for this very nice and compact talk. Yes, questions, Andrea. We are planning to do that. There is a student already pulling a bead inside. So, we are essentially doing the micro-reology experiments in a simulation. And so, we will see what happens after that. We indeed expected to find something like the fluctuation dissipation in equilibrium and something different when we start to pull too quickly. Because you see the... Ah, yes, yes, yes. In fact, the question is not linear, then you might have effective coefficient in the linear projection, which depends on how strong you are pulling. So, probably... So, for example, if you look at the statistics of the fluctuation, whether they are Gaussian or if you develop non-Gaussian tables, so maybe this... I don't know, but if it's a situation in which the question is not linear, it should be told to everybody because I think around people are somewhat feeling that one just uses the standard, linear, generalize, generalize, generalize equation. They could be wrong, but you need to prove that. So, you are the man who can do that. Other questions? So, then I will have on... So, if I understood correctly, so what is there to be changed in the generalize, generalize equation or what is the general suspect to be changed, the time dependence or the explicit time dependence of the kernel or the linearity of the... Well, there is a possibility because in that Weissenberg number you have a ratio of timescales, but if you don't get right to the first timescale, which is the fluid timescale, you are getting a different Weissenberg number. I mean, it's a Weissenberg number based on a wrong assumption for the... So, it's possible that actually the Weissenberg numbers are not those that we read in the figures because there is a really longer timescale which is not measured in normal experiments. In normal experiments, it's possible that the bead is rotated and it's passing the same already scramble position many times, so that there was not enough time for relaxing over minutes, let's say instead of seconds, and so these kind of things are under... I don't know if there is... I talked to some experimentalists about that, but I don't know if there will be experiments clarifying this one. It's very complicated, Thomas? Yes. Yes. Yeah, I think this is deeply non-equilibrium, even if it looks like it's going slowly with the trap. It's already too fast, so it's not... I think this is also what Matias Krugel gas agrees on by now. Yeah, probably is like that. Any further questions? If someone in the chat has a question, you can email Saga. Otherwise, let's thank Marco again. Thank you.