 Let's wait for everybody to join in on this colloquium. I see the number of participants is increasing as the capable Zoom servers allow the connections. Three people per second. So maybe we can start and people will keep joining as the first minutes go. It's a great pleasure for us to have today Professor Pablo de Benedetti from Princeton University. He will be giving this Zoom ICTP colloquium today. And just a quick bio sketch. Professor de Benedetti is one of the world leaders in the study of metastable liquids, which will tell us about today. He's class of 1950 professor in engineering and applied science and professor of chemical and biological engineering and dean of research. He joined the faculty of Princeton University more than 30 years ago and among his many professional owners is a member of the National Academy of Engineering, the American Academy of Arts and Science, the National Academy of Science, and his fellow of the American Institute of Chemical Engineers, the American Association for the Mountains of Science and the American Physical Society. And he wrote a very nice book called Metastable Liquids. And oh, she's here. Okay, very good. So we also have the director. So I was doing the honors. Thank you. Thank you. Actually, there was some issue with my zoom link. We got used to this now. Okay, so I was just telling the audience that Professor de Benedetti is a great expert in metastable liquids. And I was just making a list of the many, many honors that that he has achieved in his career. I joined you in the beginning, but I know so I'm very pleased actually, let me say on behalf of ICTP that we are very pleased to have Professor Pablo de Benedetti here. My colleague, Antonio has already given introduction to his many achievements. As I understand his last visit to ICTP was in 2004. Once an activity in glass physics as a speaker. So it will have a nice to welcome him here in person, but maybe hopefully sometime in the future. We can bring him back to ICTP. Thank you. So, please go ahead and yours and and you can share the screen and start the presentation. One notice to the audience. There is a Q&A window where you can ask questions that will be moderated and if you know relevant to ask to Professor de Benedetti during his talk. I suppose you don't mind being interrupted every now and then. No, no, not a problem. Not a problem. As you said, I cannot, I cannot see the questions when I'm sharing my screen so you can forward them to me, hopefully that will work. But anyway, it's a it's a real honor and a pleasure to join you this morning or this afternoon for you. I'm going to be talking about the thermodynamics of liquid water below the melting point or below the freezing point. This is a subject that has caught the interest of physicists, chemists, atmospheric scientists and biophysicists for more than half a century. For reasons that I will explain and it is an area where computer simulations have played a very important role. And I will explain why that is the case. So I will begin with presenting some experimental facts about super cool water and introduce the so called liquid liquid phase transition hypothesis and then I'll talk about three works very recent. One in which we find clear evidence of a second critical point into classical but very realistic models of water. One in which we obtain evidence consistent with again a second critical point in an ab initio neural network model of water. And the third more recent work in which we find an intriguing connection between criticality and the long range structure of glassy phases of water and I will end with conclusions. So just to put everyone on the same playing field. We all know that atmospheric pressure liquid water is stable between the melting and the boiling point. But if you're a good experimental is you can keep water in the liquid state above the boiling point that is called superheated water. And the world's record to which this has been achieved is 180 degrees above the boiling point. On the other hand you can cool water below the freezing point that's what I'm going to be talking about today and maintain it in the liquid state. And that can be done up to now up to minus 44 degrees C you see that these lines that I'm talking about are dotted. They are fixed because they are dictated not by thermodynamics but by kinetics and therefore it's interaction of thermodynamics and kinetics and your own experimental ingenuity that dictates how metastable you can be water like any liquid cool sufficiently fast come form a glass. If you heat that glass there is a glass transition. Above that, you have spontaneous crystallization to cubic ice and I'm going to be talking a lot about the properties of water in this region where water is cold enough that it tends to spontaneously freeze upon cooling but also hot enough that it tends to freeze upon heating from below the vitrifying. So it's very challenging to observe metastable liquid under these conditions. Cool water is not just laboratory curiosity the greatest inventory existing clouds. So here I show you schematically the temperature profiles in the atmosphere and I'm going to be concentrating on one on on the troposphere which is the lower 11 to 12 kilometers where weather happens. This is much better at absorbing solar radiation than the gases in the atmosphere. The troposphere is heated from below. So we probably it's very easy as sufficient altitude to reach sub freezing conditions. And because of the relative positive the of efficient ice nucleating particles is very common to see droplets of water super cool in clouds. And temperatures below minus 30 degrees C. The freezing when it happens it happens homogeneously without the need of solid particles at relatively higher temperatures. It happens to terror geniously catalyze by solid particles. And there is a lot of super cool water. Here I show you the major inventories of water on earth. If you look at the left hand column, I've translated everything to the radius of an equivalent sphere if we condense all the water in liquid form. You can see that we would have a sphere of radius of nine miles. And all the all the water that is in the atmosphere. Not all of it in the form of liquid droplets of course a lot of it in the form of water vapor. The properties of super cool water are highly unusual. The pioneering work here was done by the late Austin Angel that passed away recently. And the thermal compressibility that is the response of density to changes in pressure. And whereas most liquids, when you cool them the compressibility does nothing interesting in water, it really shoots up but continues increasing as far as one can measure. And a few years later they did similar measurements for the heat capacity which is the response function for the entropy or for the enthalpy. Again, highly unusual behavior you can see that the more you super cool, the higher the heat capacity becomes. And if you're interested in thermodynamics and statistical mechanics, this really gets your juices flowing because these are response functions that cannot keep increasing by themselves, something needs to happen. And the key question in this field is what is about to happen is what is water tending to the properties of super cool water. All of them are unusual. The most well known is probably the density maximum the fact that below four degrees C the density decreases instead of increasing upon cooling and sharp contrast to the behavior of simple liquid. Here I show the thermal expansion coefficient which not only becomes negative but the rate at which its magnitude increases continues to increase sharply upon super cooling. The heat capacity increases dramatically and I just told you that and the compressibility also increases all in sharp contrast to the behavior of a simple liquid. And microscopically, each of these response functions is associated with a fluctuation. So, super cool water is unusual because there is a negative correlation between entropy and volume fluctuations. The heat capacity increase indicates an increase in enthalpy fluctuations, the compressibility increase indicates an increase in density fluctuations. So what is water fluctuating between that is a question that one seeks to answer. The next step in the, at least the formulation of a governing hypothesis was taken in 1992 this pioneering work in the group of jeans Stanley, done by then graduate student Peter pool, working with then postdoc Francesco shorty now. If I could concentrate your attention on the curse denoted by liquid, they were studying the equation of state of super cool water. The equation is that as you lower the volume as you compress the liquid, the pressure will start increasing sharper and sharper. But as you can see, as they decrease the temperature, all of a sudden they observe exactly the opposite, the liquid became highly compressible, as you can see by an almost horizontal portion of the curse. If you take this with a grain of salt but look carefully this looks almost like the beginnings of a van der Waals loop. Notice that this is not a typographical mistake. The pressure is decreasing at constant density as we increase the temperature that's one of the anomalies of water. It's related to the fact that the thermal expansion coefficient in this region is negative. And if you very rapidly quench and form a glass on the computer and then compress the glass, instead of going continuously from a low density to high density, you see almost a sharp discontinuity. And so this led these authors to propose the bold hypothesis that they were approaching a liquid liquid transition a metastable liquid liquid transition, and perhaps a critical point, as you would have if you had a van der Waals loop in this region. This experiment this work is remarkable because it's one of the few instances where a computer simulation preceded the experimental result. Two years later Osama Mishima did a remarkable set of experiments in which he compressed so called low density amorphous ice this is the glass that you form by cooling liquid water very fast. When you compress you see that the volume decreases, almost through a first order phase transition of course these are metastable phases they're out of equilibrium. When you do a cycle, there's here low density amorphous ice high density amorphous ice, you see hysteresis, I see here a question, let me see. What I was asked for me is how does water remain liquid when the temperatures increase more than 100 degrees. Well, water is metastable. That's the answer and the way to do that experiment is for example by putting liquid liquid water in in an immiscible fluid in which it rises and continue to heat the immiscible fluid and in which water rises. The absence of a sharp interface with respect to a solid allows water to be superheated, and that has been done in the lab and you can do it up to 180 degrees C so I hope I answered that question, but returning to this slide. Mishima did the experiment exactly what I showed in the computer simulation except that here it's rotated by 90 degrees because he manipulated the pressure not the volume and simulations they manipulate the volume not the pressure. And so this led to the formulation of the so called liquid liquid transition hypothesis for water which I show here. That says that deep in the super cool region below the homogeneous nucleation temperature but above the glass transition temperature, there is a metastable liquid liquid transition terminating at a critical point. And that critical point is the origin of all the anomalies that you saw experimentally measured that is the increase in heat capacity, the increase in compressibility. Observing this has been a challenge and I'm going to show you a list of experimental papers. The preponderance of the evidence is certainly consistent with the existence of the critical point, but we don't have what we would call in the US a smoking gun that is a completely definitive and part of it is the difficulty in observing liquid water without crystallizing in this region. On the other hand in computer simulations, it's very easy to super cool and avoid crystallization so computer simulations provide a complimentary set of characteristics that make this problem very challenging to study but very appropriate to study by computer simulation, because the characteristic times for relaxation and freezing are very well separate. And so computer simulations have played a crucial role in exploring the possibility of the existence of this liquid liquid transition and a critical point again the main idea. As we know from the theory of critical phenomena, the presence of a critical point is always felt not just in the immediate vicinity of the critical point but very far away. And it is hypothesized that it is this critical point that give rise to this unusual behavior. Here I show a list of experimental papers that are consistent with liquid liquid transition. You can all see this. I don't expect you to memorize all this this talk is being recorded. The point that I want to make. These are not the titles of the papers these are the techniques that were used in any case, and there is a wide variety of techniques, all of which if you read these papers show very clear evidence that is very consistent with the existence of critical point. So let me now begin with the first piece of work from our lab that I want to talk about. We're going to be talking about simulations of a classical model to models called to poor P 2005 tip for P eyes. They belong to a family of rigid body models of water with localized charges. But these have the characteristics that they are very realistic in fact they do a fairly good job at representing the very complex solid phase diagram of water which is a highly non trivial thing to do. And so I'm going to show you results of calculations in which we measure density and energy fluctuations and analyze the results in light of the theory of critical phenomena. So this is the raw data. I have the simulations of tip five p and tip for P 2005 tip for p eyes close to what will turn out to be their meta stable critic second critical point. And you notice that the density is doing something very curious, instead of oscillating around an equilibrium value. It's fluctuating between two equilibrium values. This of course is the signature of a vicinity to a phase transition. For those of you who do simulations I want you to notice the extremely long times 40 microseconds so simulations on the order of tens of microseconds are necessary in order to collect good statistics. And using the techniques that I'm going to explain the next slide, we're able to convert this to smooth two dimensional probability distribution diagrams for density and energy fluctuations at the critical points and these are again metastable critical points because these are highly super cool conditions. So how do we obtain the values of the critical point. We use a technique called histogram reweighting I'll explain in words and then I'll take you through these equations. In words the idea is if you're interested in the statistics of the distribution of an order parameter and in isobaric simulation, an order parameter could be the fluctuating density or the fluctuating energy. Then what you do is you collect histograms of the value of these fluctuating order parameters, and you approximate the true distribution by the collection of histograms. You weigh the histograms with statistical weights, and then you require that you minimize the deviation between the histograms that you collected and the true distribution of the order parameter. And when you do that, you end up with a set of couple equations, where I going from I to R are the thermodynamic conditions under which you do simulations. F sub I is the histogram in the I thermodynamic condition that has values of volume equal to be an energy equal to E. This gives you the probability distribution of observing the order parameter at conditions P and T. F is simply the total number of observations or configurations in simulation I and C is an effective free energy defined here. So these two equations need to be solved in a coupled numerical way. When you do that, you will see here I show you a collection of extrapolated histograms, essentially they allow it to extrapolate the histograms of fluctuating quantity away from the conditions at which you simulate. In this series of curves, the symbols denote simulations, the curves are calculated histograms and you see that the technique works extremely well. These are a series of isothermal simulations at a series of pressures going from 1.5 kilobars to 2 kilobars. What you see what the system is doing, it's evolving from a low density to a high density as you increase the pressure that's expected. But instead of doing it continuously, it goes through a clear region of a phase transition because you have a very broad distribution and in fact you have by modality. And so this again indicates the existence of a phase transition and the proximity to a critical point. We wanted to really qualify and understand what type of criticality we had. And of course, when you have short range interactions and a scalar order parameter, the default thing that you look for is the icing universality class. As shown by wilding the relevant order parameter is a linear combination of density and energy here S is a field mixing parameters not the entropy and T is simply the combined linearly combined order parameter. So, this is because of the lack of particle hole symmetry in a real fluid manifested by the breakdown for example of the law of linear diameters. And we use the very useful parametrization very accurate but empirical parametrization of the distribution of magnetization fluctuations due to SIPPEN and BLUTE who fitted this to the universalizing distribution. And we compare it with our own observed fluctuation distributions for the rescaled value of this mixed order parameter defined such that it has zero mean and unit variance. And so really what you do is you compare the observed fluctuations you regress the temperature and the pressure that gives you the best fit to the icing universality class. And from which you obtain the critical parameters. So the green curve here is the universalizing distribution and the dots are the observed distributions and the simulations and you can see that you get a very good approximation to the icing model. And you can regress values of the critical parameter using this technique. So this gives you clear proof that indeed the fluctuations correspond to a critical point of the icing universality class. In this same work, we did a second calculation. So we simulated a scattering experiment we calculated the structure factor. The structure factor is related in the K equals zero. Can I interrupt you a second. Yes, please. Yes, so regarding this icing. Yes. Symmetry class. So since you said there is no particle or symmetry. It's like in the icing model, when you have an external magnetic field, right. Yes. So, and, and so you are following that you are looking at the transition the first of the transition along that line and getting closer to the critical point. Actually, the line is at the critical point and the fit of the curves are the best fit that you can get to the distribution where you are regressing. Of course, the fit will be different at different temperatures and pressures. So you're regressing the best fit for the critical temperature and the critical pressure. It's a numerical fit to the icing distribution. Okay, I was just asking, I mean, at the critical point, the icing, the icing model as critical exponents and everything. I will show you that's where I'm going. I will show you the exponents. Okay, very good. So this fit is just for the distribution of the magnetization and then you will show for the correlations. Okay. Exactly. So we calculate the structure factor, which is related to the compressibility in the K equals zero limit. We use the fact that close to the critical point the structure factor can be decomposing to a normal and anomalous component. The anomalous component has a Lorentzian form where Chi is the correlation length. Chi is the wave vector. And we do simulations with a very large system. So we have in this case, these are very expensive simulations on the order of 50,000 molecules exactly 36,000. Why so many, because we want to measure the correlation length and we want to make sure that our computational box contains several correlation lens. So we fix the density and we converge to the expected critical point and you can see very clearly the enhanced fluctuations given by the growth in the structure in the in the low key value of the structure factor. And so here's the question that you were asking. So what are the predictions from if you have a system that satisfies the icing universality class. These are the icing critical exponents. So we took our numerical data for the correlation length and the compressibility. And here we show a line with a slope fixed by the predictions of the critical exponents of the Isaac universality class. The critical temperature as the regress parameter and you can see that within numerical accuracy. This is also very consistent with the icing universality class. It's as good as you can do in the simulations these are extremely demanding simulations, both because of the size of the computational cell, and because of critical slowing down so you need to run extremely long simulation but very satisfying me. So the critical temperature that we regress is very consistent with what we obtained by doing the icing fit. So this first piece of work shows the existence of a metastable critical point in two models that do a very good job at representing the properties of water, but these models are empirical. The question is, can we do better than this. There's no question that simulations have helped us gather insights into the behavior of water and there are many, many examples of very active field. Can we do the same. Can we improve and say something about the behavior of water in this region that I've been talking about. I've been a type models or initial type calculations until very recently, the answer to this would have been known because of the very long times required for relaxation and comparatively large sample sizes. So I'm going to show you in this work how using a combination of machine learning computational chemistry and statistical mechanics. We were able to develop an ab initio quality model without adjustable parameters. And this model shows behavior consistent with the existence of a metastable critical point. So the development of model involves generation of a deep neural network model of water with training data prepared with scan at the density functional theory level with a scan exchange correlation functional. And so as to reproduce ab initio energies and forces, and we use deep potential molecular dynamics in the multi thermal multi barric ensemble to generate the equation of state I'll go into those details in a second. So initial training doing density functional theory calculations on a small number of configurations involving both the liquid and 15 nice phases and add new new training data by iterative exploration of configurational space. And here are the details. We ended up having roughly 20,000 structures in the final training set. And so this model was then obtained in this iterative fashion fashion. So how does this work so I'm going to show you the predictions of the model. I want you to concentrate on the top left figure. Here I'm showing the height of the second peak of the oxygen oxygen pair correlation function with the diamonds are experimental data and the circles are simulations. We see very good agreement in order to obtain this good agreement. We had to shift the simulation data by 47 Kelvin but once we do that. We obtain a good fairly good representation of the properties of super cool water, and these are highly non trivial predictions remember there are no adjustable parameters here. And in fact, we're able to reproduce both the density maximum again, diamonds are simulations circles are experiments. So this is one of water's anomalies, the diffusion anomaly if you notice the diffusion goes up with pressure when water is cold that's another of water's anomaly, both increasing diffusivity and the decrease in viscosity upon pressurization are intensive anomalies of liquid water. And here we show the anomalous increase in compressibility. Once we shift the temperatures, the model does a very good job at one kilobar. It does a good job at moderate super cooling but not as good a job at deep super cooling at one bar. Meaning sources of discrepancy and this is an active area of investigation could involve nuclear quantum effects is a nuclear treated classically, or limitations of the density functional theory but this is the model that we're going to use. Even with this, even though we were it runs much faster than an ab initium molecular dynamics. It is still too slow for the problem that we're interested in, because you need, you can only sample if you use straightforward molecular dynamics, the immediate vicinity of the point at which you're simulating. We'd like to be able to enhance the fluctuations of volume and energy and sample a broader range of configurations. So we use the technique of multi thermal multi barric multi thermal multi barric ensemble developed in the killer Pyrenees group. Pablo Piagis a co author of this paper that again enhances by adding a bias to the potential enhances the fluctuations in volume and energy. And essentially what you're doing is you're reweighting what you observe to other conditions noted here by T prime and P prime. So these are the results. In the left panel I show you the equation of state this is density versus temperature and various pressures. And though it's very difficult to see I'm actually showing two sets of data, the data that are reweighted by this multi thermal multi barric ensemble and the data obtained by MPT simulations you see that they're almost indistinguishable. And the model is showing a low temperatures and enhance vast enhanced fluctuations which you can see by an increase in the slope of row versus P. Here we take the derivatives and we calculate the compressibility. And you notice lines through the points. And so what we use to get the lines through the points is a phenomenological thermodynamic model called two state thermodynamics. What happens in super cool water is that molecules can be very easily distinguished by the local configurations in a computer simulation. And these are intend to be by model, either a molecule has a local environment that is tetrahedral the coordinated and low density, or the tetrahedral is highly distorted with the invasion of a fifth neighbor for example, and high density so you can easily classify the water molecules in one or two environments. So you develop a so called mixture model where X is the fraction of water that has a low density environment. You postulate an ideal entropy of mixing with some interaction that reflects the non ideality. But what I want to stress is that when I use the word mixture here is in quotation marks, because this is not a true mixer is a single component. The difference of molecules that have a low density environment is obtained by minimization of the Gibbs free energy is not imposed a priori. So you postulate an expansion between the difference in free energy between water molecules that have low density and high density environments. In general this to if I mean you can easily see by taking two derivatives since this model is symmetric by the exchange X to one minus X that at Omega equals to you have a critical point, and you make this more general you make interaction energy temperature dependent. You have a series of adjustable parameters which you obtained by fitting the model to the simulation data for example for the fraction of water molecules that have a low density environment. I see a question. So the question is what do GA and GB pertain to in the first equation of two state thermodynamics GA and GB are the chemical potential of free energy per molecule of a water molecule that has a low density local environment and a high density local environment. So they are the precursors of the two different liquid phases. And when you do that, you obtain very good fit to simulation data this is earlier work, but for our model these are the fits. You can see that you obtain excellent fit to the simulation data. We had to stop at this temperature because of the low temperature long relaxation times, but you can see that the fit through the model predicts again the mixing and the existence of a liquid liquid critical point. In this case at 224 Kelvin and you subtract remember we to the 47 degree shift. You get a pretty decent agreement with our previous predictions the important thing here is that this is informed, whereas it before this was a rigorous profile by numerical. Here the simulation stop at a temperature below the critical point exists below the temperature that we simulated in this work, but nonetheless, the fit is very satisfactory and is strong evidence consistent with again the existence of metastable critical point in this model. Can I stop you for a second so just to, to understand a little bit so GA and GB are essentially fitting parameters for for for the day they're not found microscopically you're using your two state model to to find this delta P delta T which are essentially GA minus GB from the data. So I postulate that GA minus GB has an expansion in temperature and pressure and what I obtained are the coefficients. Okay, and what I'm measuring my simulations is X, that is to say at any given point, I do simulations I can by using a certain type of when a molecule has a low density local environment and a high density local environment at any given TMP, I know what X is. So that gives me a parameter to fit, and I obtained with this the parameters of model, and that then gives me the lines, but also T C is a is a fitting parameter. Exactly, exactly. So, from from that, what shows when GB and GA are equal for the temperature which they are equal and pressure for which they are equal. That's the point of the face transition that's the red circle there. That's the locus of the face transition in fact that condition defines the critical point and the continuation which is the weird online when GA is equal to GB. Okay. Okay. Just just to make clear to Thank you. And once you have that then you can do excellent fit to the simulation so here we have the predicted critical point. This is a line of maximum density, the symbols are the simulations, the line is the predicted fit, the two state fit. The same thing with the maximum of heat capacity the maximum of compressibility. And then they merge at the critical point as they should so this gives you essentially control of the whole thermodynamics. But as I said, our simulations, numerically stop at 250 and the critical point is predicted to be below that. The last topic that I want to talk takes us back to the glasses so I want to remind you the basic phenomenology that I showed a low density glass and a high density glass. These clearly are non equilibrium phases, and we became very interested in the connection between glasses and criticality by a very interesting paper by salt or quattro falso martelli. Nicholas Johan Batista and Roberto car of a few years ago. We did numerical calculations on water glasses, and show that the glasses were what cell has come to call hyper uniform. That is to say that, rather than having the structure factor go to infinity as I showed before the knowing the noting enhance long range fluctuations. The structure factor that goes to zero, or that is very small denoting an anomalous suppression density fluctuations. So then we asked the question something really interesting is happening here because you go from s equals zero to a diverging as of zero. And so the change in the structure of the glass, perhaps could reflect passage close to the critical element. So we decided to investigate this numerically. By doing the following computational experiments that is quenching water glasses and we're going to be using the tip for P 2005 at different pressures and see if something unusual happens to glasses formed at pressures that cross the critical element that we previously calculated. And obviously the answer is yes otherwise I wouldn't be talking about this and so the rest of the talk I want to show you some interesting results. Now we wanted to have a comparison as to the expected behavior. So we use three models one is this tip for P 2005. The other one is a very popular course green model of water that doesn't have a liquid liquid transition. It's a single side model developed and use very successfully by valeria mollinero that enforces to tragedy reality by three body term. So this is intermediate in the sense that it has all of water's anomalies but doesn't have a metastable critical point. And then something that is a canonical glass former. It's a mixture of Leonard Jones with different energy and size parameters is called the Cobb Anderson mixture. It's a canonical glass former. It's very difficult to crystallize and very easy to form glasses, but has no reason to form to display any of water's anomalies. So we quench these at different pressures and measure the structure factor. So this is the Cobb Anderson binary Leonard Jones. So what you see on the on the on the right hand side is the numerical calculation of something called the spectral density, which denotes volume fraction fluctuations it's analogous to the structure factor. So, again, we're quenching isobarically the red is the initial temperature, blue denotes the final temperature, and nothing unusual happens in the structure factor. As we change the pressure, you see reduced fluctuations as you increase the pressure the behavior that you would expect for MW. You see the same although there's an interesting non monotonicity, but basically you see the same qualitative behavior as in the Cobb Anderson mixture. But for water, or at least for the tip for P model of water you see very interesting and very different behavior. So the structure factor at the K equals zero limit of the structure factor shows pronounced enhancements in the glass phase. Very close to the pressure where we calculated that there is a critical point. So this was very intriguing to us because it's telling us that the glass shows remnants of long range density fluctuations that originate by being close to the critical point. So we decided to look at this more carefully. So here I show you the same information, but I'm showing you the structure factor as it evolves during the quench of the liquid to the glassy phase. You see that both at low pressures and at very high pressures the trend is monotonic. You see a decrease in the structure factor, but an intermediate pressures close to the critical point as you super cool the liquid, the structure factor starts to increase, and then it's frozen into the glass. So then the question that drove this investigation is, is what we see a sign of fluctuations being frozen upon crossing the widow line now the widow line. I alluded to it before is the locus of maximum correlation lens every critical point has a widow line emanating from it. And the line of maximum compressibilities and maximum heat capacities asymptotically become the widow line asymptotically close to the critical point. So we decided to look at that. And the question we are asking is how does the glass transition influence these phenomena. So we computed here the glass transition of this tip for P 2005 water model for different cooling rates as you lower the cooling rate. The glass transition dips lower that's basic phenomenology, but you see one more of water's anomalies, you see two types, or two families of glass transition at lower pressures, you have water like anomalous behavior in that the glass transition decreases as you pressurize the fluid, whereas at high pressure the glass transition increases as you pressurize the fluid. This is the normal behavior that you see in any liquid. This is telling you that as you compress the liquid, the molecules gain mobility if you want to speak loosely, and that is because at low density water forms a very open low density to tradeally coordinated network. This is the critical point that we calculated and you see that the glass transition line computed at the lowest cooling rate that we attained in the simulations is close to but below the critical temperature. So even though today I talk very little about dynamics, I want you to notice that this point has a combination of critical slowing down. It is also metastable with respect to a crystal so you have to worry about nucleation, and it is also close to the glass transition so dynamics here is extremely rich and very interesting. So here I'm showing you the structure factor as a function of pressure for these points, and you see that as you at different cooling rates, you see a maximum as you lower the cooling rates, the maximum becomes sharper. And the lowest cooling rate that we could see this maximum coincides with the critical pressure and within numerical accuracy. These maxima correspond to points along the widow line emanating with the critical point. So the physical picture is extremely rich and attractive and is summarized here as you cool a liquid. At constant pressure, you're quenching and then you're going to freeze the fluctuations, the maximum fluctuations that you see when you cool and the glass transition intersects the widow line. Those are the fluctuations that are being frozen. And that's why you see signatures of criticality because if you're able to cool at a rate, excuse me, such as the glass transition lies below the critical point. You will preserve the fluctuations that you see in passing through the critical point. This is extremely intriguing because it suggests an experimental access to the possible existence of critical point by studying the long range structure of glasses as a function of pressure and I'm collaborating with Thomas Lerting at Innsbruck, where he was doing experiments, trying to see if we see this effect. So summarizing the behavior of the glass transition that we saw for the covenants of mixture normal liquid behavior for MW you see water like behavior. But you don't see an in an anomalous increase in the structure factor correspond to passage to a critical point. But for tip for P you see both water like behavior in the sense of having a low density branch and a high density branch with different pressure dependence of the glass transition line. But also along this branch and anomalous increase in the long range density fluctuations in the glass. So let me summarize the implications of the third part, the third calculation that I talked about the long range structure of water like tip for P 2005 glasses shows a surprising signature of metastable criticality. In W model exhibits water like anomalies in TG shows two branches, but no evidence of liquid liquid transition. And we hypothesize that the low that the locate at the low way vector limit of the structure factor in his pressure dependence may distinguish between systems that have a liquid liquid phase transition and those that don't. And so it could be maybe an experimentally accessible accessible metric that does not rely on having to observe liquid water without without crystallizing in in this region. So let me summarize the conclusions of what I told you both tip for P 2005 and tip for P is possess a second critical point. We see signatures of liquid liquid phase transition and that I've been issued deep neural network model of water. I've shown you long range structure of tip for P 2005 glasses showing signatures of criticality. And the intriguing concept that the intersection of the widow line and the glass transition line freezes long range structure in glasses. I don't want you to leave this talk with impression that I'm saying that real water has a critical point a second critical point. This will need to be determined by experiments, but some of the experimental evidence the papers that I listed in that single slide is very compelling and consistent with the existence of criticality. But again, the definitive proof will have to come from experiments not from simulations. We don't have a systematic understanding of how force fields affect and shift the liquid liquid transition. I alluded to the fact that dynamics is extremely rich we're starting to explore it there is a confluence of critical slowing down, nucleation and vitrification. I didn't talk today about scaling and universality but there are several features associated with the second critical point that allow considerable collapse of very different models provided you do the right scaling. And I do want to end by really acknowledging the people who did this work on the metastable criticality this is my postdoc good sir say and I've had the pleasure of collaborating with Francesco shorty know in Rome. On the deep neural network model postdoc Thomas Gardner and Pablo Piaggi, former postdoc Lin Feng Zhang, and this was a collaboration with Thanos Panajatopoulos and Roberto Carr. And the work on water glasses again the work of Thomas Gardner collaboration with Roberto Carr and Salto Quattro funding of National Science Foundation and Department of Energy. And we made abundant use of the Princeton super computing resources so I really want to thank you for your attention and I'm very happy to try to answer any questions you may have thank you very much. Thank you. Thanks, Pablo for for the beautiful talk so we do have one question which was asked by Stefano Baroni and but to be asked at the at the end of the talk, as instructed, can you read it, or should I read it to you. Yes. Okay, so you can read it you can you can also answer. It says, in one of your first slides you showed us the time evolution of density or energy fluctuations featuring solid on light jumps I presume these data were from isothermal isobaric simulations which are designed to you correctly were properties, but not necessarily dynamical ones. Have you got any idea on how the details of these results depend on the specific thermostat embarrass that employed in the simulation. So we did very thermostat embarrassed at the outset of the of the project. The equilibrium results as long as you use physically meaningful parameters for the thermostat embarrassed that do not depend on the values. Of course the dynamics of the fluctuations do depend on that. But the slide, the calculation that we were doing is trying to obtain the equilibrium distribution two dimensional histogram of energy and density fluctuations. And so those are provided you use physical values for the parameters independent of the bare set and thermostat constants. So we say we answer this question. Yeah. Ali, you have a question. Yes, if I can take steal this opportunity. So, so thanks very much Pablo. So I had a curiosity about, so you know comparing the classical potentials versus the deep neural network model. Let's say from the deep neural network, of course, it's trained with all the full electronic polarization, etc, etc. Do you have a sense on, you know, when you talk about the high density low density fluctuations. Is this an interesting coupling going on with dielectric fluctuations as well. And is that something that could be teased out from. We did not look at that. It certainly is something that could be teased out. We were interested in equation of state calculations. It could. Let me, I do want to clarify something if you hope you don't mind I'll leave taking your question. I don't want to sound as though I'm overselling the deep neural network model when I said there's no adjustable parameters. Because of course, you know what points you use for the training said how many layers you use in your neural network. Those are in a sense parameters. You know, but there's no assumption as to the model you know you're collecting data, doing a neural network fit of the model and this is what you get so it is very nice to see that you're able to reproduce the water anomalies already. I would imagine and we're doing work in that direction that training neural network with more points and more ice phases will read will give you more accurate model and hopefully lead to the elimination or the narrowing of this 47 Kelvin gap between theory and experiment. Okay, thanks. I do have a question about that 47 Kelvin gap. So that that's, that's a few million electron volt in energy right so so you said you were expecting that to come from, you know, nuclear fact quantum effects, which kind of can you elaborate a little bit more on what these simulations you treat the nuclei I mean you're doing electronic structure calculations and the nuclear treated classically in this. I'm not an expert in in DFT calculation but in the method you're treating the nuclear classically and they're not. So, if you do path integral simulations and people have done that and there are people who have quantified very accurately how much of this comes from nuclear quantum effects. The answer is model specific so one would have to do calculations for the specific model that we train. And you're using a particular that functional called the scan functional that is a good functional it's not an exact functional so there are sources of error there. So what are these are the possible sources of error that we have. So but physically the idea is that part of the energy goes into degrees of freedom that you have not considered. Exactly. Exactly. Okay, so you are. And this would become particularly relevant at low temperatures which is the region that I'm interested in. You know if I were doing vapor liquid equilibrium. It would be different. Somebody has raised his hand. Can you ask to write because I cannot let the people talk but you can you can write your question in the q amp a thing please. I think he's in the panel now so you can talk. Yeah. Okay, okay. Do you hear me. Yes. Yes. Okay, great. But thanks a lot for the talk it was really really interesting. I have a very very naive question. So structurally this high and low density liquids. So I a certain point in the talk you mentioned a fifth coordinated structure with respect to the four coordinated coordinated structure. So, did you characterize them structurally and if that the case, if that's the case so how long do they live which is a lifetime of of a molecule in the surrounded by a high density. environment salvation shell and another one, so baby, low density salvation shell. Thank you that's an excellent question, because I didn't have time to go into the detail let me start the difference in structure. So the low density liquid. Every molecule is tetrahedrally coordinated, but you don't have long range orders so you don't have like hexagonal eyes so you think of tetrahedra that are orientationally disordered. Okay. So the high density liquid. If you look at the structure first of all, you have a local high dense higher density because there is like a fifth neighbor that typically intrudes into the first coordination shell. So there are many ways of distinguishing this. Some people use as an order parameter the distance to the fifth neighbor in the low density liquid that distance is higher than in the high density liquid. That's one order parameter, another order parameters to measure the height between the first and second peaks of the oxygen oxygen pair correlation function for the low density liquid that almost goes to zero so you have a very good separation between the first and second salvation shell in the high density liquid that is corrupted. Regarding the lifetime in a computer simulation. If you if you have a model that allows you to see liquid liquid coexistence and that's a non trivial aspect, because you're talking very low temperatures you need to let it equilibrate. But one such model is one that I didn't talk about today call sd2, you can see the phases coexist for a very long time when I say very long time, I mean simulation time. These phases will coexist for tens of microseconds at the right conditions. Now if you're asking for the individual molecule a given configuration in a thermalized fluid that's fluctuating all the time. It's a few picoseconds. So before the system has phase separated, you will see that a local configuration of a water molecule is either that right really coordinated or high density, and those in a property equilibrated liquid are fluctuating all the time. That's why it's important not to not to get carried away by calling this. I call it a quote unquote mixture model I don't want you to imagine that there are. I was doing a fit above the critical point. So what I'm fitting are the fraction of water molecules that have a given local environment that's in a thermalized liquid it's equilibrated that is fluctuating all the time. Okay. Yeah, thank you very much. So we have a question from area to something area. Yes, good, good morning Pablo. Hello, a question about the nucleation between the two phases when you are below the critical point. What was that nucleation look like do we know anything about it from simulations or from experiment. From simulations, we know that in a small system, which is what we've been able to simulate below the critical point. It is possible to have the system transition between low and high density phases. So it goes back and forth between low and high density phase. Of course, that's something that you can only see in a simulation it's a finite size effect. And what you see happens it happens very rapidly. All of a sudden you see a local low density local environment that forms in the high density phase, and that immediately nucleates the overall low density phase. And so the system goes from a low density to a high density and just fluctuates back and forth in experiment. It has not been seen that that that's what everyone is looking for. There is indirect evidence for example the most recent by the group of Anders Nielsen, who did a isochoric quench the glass phase and did femtosecond actually scattering and observed a discontinuity in the scattering spectrum, which he was able to assign to a transition from one phase to the other. But that's in the glassy phase or almost close to the glassy phase. So we know what it looks like in the liquid in simulations. It has not been seen in experiments unambiguously. I hope I answer your question. Thank you very much. Okay, so if there are no other questions, let me remind the public that we have a session of exclusive question answer with the students, which is a separate zoom room. If there are no other questions. Can I ask you a question? I will show that session starts at 530 for you. Yeah, it's in 20 minutes. Yes, okay. Very good. Okay, I can, you know, I can meet you there in 20 minutes. It's a different zoom like, yeah. Yeah, you have a different invitation. Okay, so if there are no other questions, then I would like to thank Pablo again for his, you know, for his, for his talk and kindly to discuss. Thank you. I'm the one who's grateful. Thank you very much. And we are really looking forward for you to visit. As soon as it will be possible. Okay, very good. Thanks Pablo. Thank you.