 to everyone for joining in today on the pre-Christmas webinar. It's a real pleasure for me to introduce Goul Zerze. Goul is a chemical engineer by training. And Goul has been working now. She's a postdoc with Pablo de Benedetti at Princeton. And I'm very jealous of Goul because she has managed to work and have conversations with Frank Stilinger, which is a dream of anyone who's worked in the water community. So I can only dream to be able to do that one day. And so some of you may already know this. Goul has published a very recent interesting paper in science on the second critical point of water. And this is going to be the subject of today's talk. So Goul, the floor is yours. Thank you very much for this kind introduction, Ali. And I have to admit that Frank Stilinger, I mean, he is an amazing guy. Although we don't work on water with him, he has an amazing insight into very random subjects, like very random stuff. But yeah, I'm definitely blessed to get to meet with Frank every single week. Yeah, I'm so happy. I'm so very happy. OK, hello again, everyone. Well, first of all, thanks again, Ali, for this kind invitation to give a virtual talk here. Yeah, I was saying that I probably should also be thankful for the pandemic as it made our lives all the way virtual. I probably wouldn't be able to give this talk otherwise. Anyway, pandemic or not, I'm so glad to be here today. And as Ali introduced, I'm a postdoc at Princeton working with Pablo de Benedetti. For people who may not know, we are a computational group in chemical engineering department. We mostly work in the field of statistical mechanics. And one topic that's been a particular focus in our group is low temperature phase behavior of water. That is something that we do care a lot about for the reasons that I will explain in a minute. But first, I thought most of you guys probably don't actually know me. So I want to take this opportunity to introduce my research a little bit, just to give you some ideas about the other things that I like to do, things other than cold water. So I just squeezed in this one slide. So I am mostly a classical modeling and simulation person, atomistic modeling and simulations of intrinsically disordered proteins, or I'll call them IDPs in short. That was the major topic of my PhD thesis. IDPs are special kinds of proteins. They live on a time scale that is far smaller than the time scale that normally foldable proteins are living. They are very dynamic. They rapidly interconvert. They rapidly reorient. They are like a dream subject for people studying statistical mechanics. And single molecule properties of IDPs is the area that I have contributed the most. I had thermodynamic characterization like free energy landscapes of IDPs. I had structural characterization, like determination of partial structural order in IDPs and their relevance to the diseases. I had kinetic characterization, like determination of time scales of contact formation and its relevance to the partial structural order. And in addition to the single molecule properties, I also have some studies about phase transitions of IDPs where we push the time and length scale limits of atomistic simulations. These are from collaborations with an experimental group, but we had some unique contributions there. For example, elucidation of structural ensembles that will give rise to particular NMR signals as well as determination of molecular and atomistic level interactions that possibly drive the phase separations of IDPs. Interfacial phenomena have also been a particular interest for me related to that I have some important contributions, especially for pinpointing the role of water mediated interactions and direct surface interactions of proteins and nucleic acids on a variety of different types of surfaces ranging from self-assembled monolayers to carbon nanotubes to even lipid bilayers. I guess you start to notice that water has a special place in my research. None of the phenomena that I just talked about would happen in absence of water. And all the modeling involved in my IDP research was about improved protein-water interactions. I have religiously used a model that improves protein-water interactions and that model provides a far more accurate representation of unfolded and disordered proteins. So my special interest in water in the context of biological phenomena is I think what brought Pablo and me together. Since I joined in Pablo's group in late 2017, we obviously did some very fundamental research on pure water, which is the topic that you'll hear today. But in addition to that, we also worked on the curious case of biomolecular homo-chirality. And this is one of the most mysteries and interesting thing that I know about biomolecules. All biomolecules in our bodies are exclusively homo-chiral, but no one really knows where it originated from. Now with this motivation in mind, we took a somewhat twisted approach and asked the question, what would happen if biomolecules were heterochiral? So we have some studies related to chiral perturbations to proteins where we looked at the folding transitions and we found that even the simplest possible chiral perturbation modifies the folding equilibrium substantially, it's even possible to create a library of alternative protein bolts by chiral modifications. Okay, I guess I took enough time to walk you through my past experience. I will be more than happy to talk more about any of these subjects later if anyone is interested in. But for now, let's go back to our actual business, pure water. Like I said, none of the biological phenomena that I found very interesting would happen in the way that we know without water. So there must be something that makes water very special. And I'm pretty sure you already know that there are a lot of things that make water very special. When we look at the physical properties of water, we see that they are fundamentally different from those of most other substances. I guess the most familiar one is the volumetric anomalies of water, by which what I mean is water shows a maximal density at four degrees Celsius. This is obviously at ambient pressure, but it is not just that. Thermodynamic response functions of water such as isothermal compressibility and heat capacity also show abnormal behavior. As opposed to the response functions of a simple liquid which would just decrease as temperature decreases, those of water increase as temperature decreases. So this is very curious and obviously abnormal behavior because each response function is associated with fluctuations in a thermodynamic quantity. And even the common sense tells us that they will just decrease as temperature decreases. And thermodynamic consistency demands that they cannot keep increasing indefinitely together. They either have to pass through a maximum and then decrease or they will diverge. And this divergence argument tells us that there must be a critical point, a point of singularity presumably located at deeply super cool conditions like this. According to the critical point hypothesis, there is a pressure where the response functions diverge as temperature decreases. And this pressure would be the critical pressure and the temperature at which the response functions diverge would be the critical temperature. TC and PC together as a state point, they would be the point of singularity. In other words, the critical point. Now, only little problem is this argument is impossible to justify or falsify to the current experimental technology because ice nucleates too fast before any measurements can be done in the super cool liquid. So this is the reason why, not one of the reasons why molecular simulations are extremely important in this field because simulations don't necessarily suffer from the same problem. I can keep water as a super cool liquid in my simulations for a long enough period of time to make reasonable measurements. Now, before going any further, I wanna talk a little bit about what would be the significance of this critical point if it exists. To understand that, I like to take a look at this phase diagram of water that's for non-crystalline water. It means it excludes the solid states, but it includes meta stable liquid states. Now, critical points mark the termination of phase coexistence lines. This one, for example, this is the well-known first critical point in water that terminates the vapor liquid equilibrium. And this is where the hypothesized second critical point is. Now, according to the computational findings, this critical point terminates the coexistence of a low-nensity liquid and a high-nensity liquid. Now, I'm telling hypothesized critical point and I'm also telling according to the computational findings because there is no rigorous experimental proof that shows the coexistence of these two liquids. Well, I guess I had to say, there wasn't until very recently because this past month, Nielsen and co-workers created another magic and prepared samples of supercooled high-density liquid, high-density liquid water. And they have observed that it discontinuously transitions from high-density liquid to low-density liquid. And this transition happens on a time scale that is orders of magnitude smaller than the time scale of crystallization. Now, I'm not an experimentalist to reveal all the tricks that they have done, but I can imagine that it must be an ultra-fast heating because they obtained the high-density liquid from high-density amorphous ice in a pressurized tiny container. And then as they allow the expansion, isothermal expansion, they have observed a discontinuous shift from high-density liquid to low-density liquid. So this was a major, major breakthrough implying the coexistence of these liquids. And therefore the presence of critical point as well. But it is still not a rigorous proof of criticality. Proving criticality is still very important because it would be the criticality that explains the anomalies of water, like the increase in response functions as temperature decreases. So experimental illustration of coexistence of these two liquids was very recent, but the coexistence of the glassy versions of these liquids have been shown for a while ago. In mid-1980s, before I was born, Mishima and coworkers obtained the amorphous ice by pressurizing the ice crystal. And then they observed a first order like phase transition from low-density amorphous ice to high-density amorphous ice. And in 1992, Gene Stanley and coworkers formulated the second critical point hypothesis based on the series of MD simulations that they have with the ST2 model of water, where they found an evidence of first order like phase transition from low-density liquid to high-density liquid. And in this scenario, the liquid-liquid equilibrium line has been obtained simply by extending the LDA-HDA coexistence line into the lower pressures and higher temperatures, exactly as outlined here. So later on, of course, ST2 became a favorite model to study this interesting phenomenon. Like many people work further on ST2 and eventually unambiguous proofs of metastable criticality have been shown for this water model. This was the only water for which the criticality has been proven, including all other computational waters and real water as well. But there was a fundamental reason why we were not 100% satisfied with these findings to say, yes, this must be the scenario in real water. And the reason lies in the model. This model ST2 is known to over-structure water. What I mean by over-structuring is being overly tetrahedral. And problem with that is that over-structuring, like perfectly tetrahedral particles are already known to have this kind of phase separation. Water, on the other hand, is tetrahedral, but it's much more subtle as opposed to being a three-dimensional tetrad structure like ST2. So once again, we set out the look for an evidence of criticality, now in the most realistic classical models of water, like T4P 2005. And the challenge for these models is that the expected critical point is located at much colder temperatures. So you suffer from a severe slowdown in the kinetics. And the slowdown in the kinetics makes the free energy calculations of a phase separation impossibly challenging for these models, at least for today's computational power. So what we did here instead is we just approached to the critical point from higher temperatures. And for the first time, we unambiguously showed that meta-stable criticality exists not only in the model that artificially favors the phase separation at elevated temperatures, but also for the most realistic classical models of water. We used two very independent techniques to prove our point. In the first one, we simulated the scattering experiments. And in the second one, we used the modern theory of critical phenomena to drive the order parameter that shows the phase symmetry at the critical point. So at this point in my talks, especially when I give a talk in engineering departments, I often briefly lecture the audience about the concept of criticality and more importantly, the universal behavior of criticality. But that would be an insult to do at ICTP. So I'm not gonna do that. I will only note that there's one key assumption that we made in the beginning of this work, which we later validated. And the assumption is that if a second critical point exists in water, it will also belong to the icing universality class. This assumption will come handy in the analysis of my calculations. Okay, this is the first analysis where I mimic the scattering experiments. Here I did simulations at constant volume on the critical isochor. These are pretty large size simulations as they need to accommodate for diverging correlation length. And I calculated the structure factor exactly as in scattering experiments. For people who may not know, structure factor would simply be the Fourier transform of radial distribution function or G of R. And in the reciprocal space or in the K domain, the increased correlations would reflect them as at the low K region, S of K at low K. And that's the reason why I'm zooming in here so that we can clearly see that correlations are growing substantially stronger as temperature decreases. And then I used Ornstein-Zernike formalism, which is a relatively old theory commonly used for this purpose. The purpose is to calculate the correlation length and the limiting value of S or S of zero. Limiting value of S or S of zero is linearly proportional with isothermal compressibility. One of the response functions that would diverge as we approach to the critical point. And correlation length is correlation length. It will of course diverge as we approach to the critical point. And they are supposed to diverge following a power-low relation as a function of reduced temperature. TC here is the temperature of singularity. In other words, that's the critical temperature. And you can see that there is an order of magnitude increase in the isothermal compressibility as a function of reduced temperature. Now, we know that critical behavior is universal and I wanna remind you that we assume that second critical point would also follow icing universality. So based on that, I use the critical exponents expected of icing universality both for the compressibility and the correlation length. And I simultaneously regress them to obtain a TC that would satisfy the exponents expected of icing universality. And you can see that I obtained pretty high quality of the fits. The lines are fits here and symbols are simulation data. And I got a TC of 173.3 Kelvin. Now, what does it mean? The fact that it is a very good fit over an order of magnitude change, it is a very strong support that we are approaching to a critical point. And that critical point belongs to icing universality class. Go, can I ask you a question, Zary? Please. Yeah. So just how does... So was this done before for ST2? Like the same plot? For ST2, yes. And how different is it? Qualitatively, quantitatively. So quantitatively, they use the same exponents. They did it for icing exponents. But it is the temperature, the TC, the value of TC that is shifted for ST2. I think it's around 245. That is the TC. But once you pull out it in reduced temperature, they are the same. Like I haven't overlaid them. I haven't overlaid them. Okay, okay, thanks. Sure. So this was a very convincing and macroscopically related way of studying critical phenomena. But if you are a simulation person, there are other rigorous statistical mechanical techniques that would allow you to use the modern theory of critical phenomena. For example, finite size scaling. So we studied systems by systematically varying the system sizes where all of them are finite. Of course, I don't want you to get me wrong. All atomistic simulations are finite. What I mean by finite here is that the characteristic size of the system is smaller than the characteristic size of the correlation length. And then I initially intended to walk around my critical point. So I picked my state points to do the simulations. These are the blue points accordingly. But I don't precisely know where my critical point is. When we started this work, I didn't even know whether there's a critical point or not. I only have phenomenological guesses where that could be if it exists. So this is a problem because I'm trying to obtain properties of a point that I don't exactly know where that is. So how do I deal with that problem? A rigorous statistical mechanical technique, histogram, reweighting, I will explain the technique a little better in the future slides. But first I wanna show you one of the simulations that I have for the smallest particle size at the coldest temperature, which showed us convincing enough fluctuations between low density and high density. But I want you to note that this is only near the critical point. At the critical point or very, very close to the critical point, these fluctuations will be more perfectly two-state kind. But this information is still useful. The fact that I have simulations at many other state points like this, I will be able to use the histogram reweighting to recover the density of states at other state points nearby, which will include the critical point as well. Now, briefly with the histogram reweighting, I obtained the histograms of my thermodynamic quantities of interests by extrapolating or interpolating. These are, for example, the density histograms, including extrapolations. I have actual data on 179 and 177 Kelvin, but all colder temperatures are actually extrapolations, where only the coldest one is showing a little bit of noise, the black one. So I'm pretty confident with the quality of my data. But what I want you to do is notice the splitting of these two peaks as temperature gets colder. As these peaks between low density and high density as these peaks split deeper and deeper, I'm actually passing through the critical point without showing a singularity because this is the signature of finite size effects. Singularities will be rounded and shifted as you already know, but still the question is, where exactly is my critical point? Now, given that my system's behavior is consistent with icing universality, I can compare my system once again with the perfect icing. For an icing system, there is an expected distribution of the order parameter at the critical point. Here, we are looking for an analogy to the icing system for our liquid system. And what we need to do is compare the spectrum of our density fluctuations with the known asymptotic distribution of the order parameter M for icing. So parameter M for icing is systems magnetization, spin up or down. So in analogy, we thought that our order parameter could be just density, but with a big difference. The two of the states, low density liquid and high density liquid are not symmetric of each other unlike the spin ups and downs in icing, which are perfectly symmetric. And this asymmetry between these two states yields a lopsided distribution of the order parameter. But there's a solution brought into that problem which is field mixing. The rigorously defined mixing of two fields, energy and density in this case, yields an order parameter, which I will also call M. And that is going to be symmetric for the two phases. I used my histogram revading data where I fit the distribution of the order parameter on the expected icing distribution. And this fit will eventually tell me the state point at which the distribution of the order parameter is closest to the icing. And that point I will call the critical point that belongs to icing universality class. So I did the same calculations for a few other system sizes as well, not just for 300 particles. For each size, I got a measure of TC and PC. And then I looked up the scaling of those. I am exemplifying the TC here as a function of system size, but I couldn't find any meaningful system size dependence within my numerical accuracy. I note that this doesn't necessarily mean that there is no size dependence. I would say it is most likely because scaling analysis is a very sensitive analysis and it requires very precise measurements. So with this size of errors, it is impossible to show a conclusive scaling. But at least we have this fit on the icing so we can estimate the TC and PC. So, okay, we are actually very close to the end, but I just wanted to show the computational cost of studying a realistic classical model near its critical point. In total, I think we approximately dedicated five million CPU hours and 0.3 million GPU hours only for this project that's spent within a little longer than a year, but it totally worth it for the results that I obtained. I hope you like them too. But before moving to conclusions, I just wanna show that we repeated all the same calculations for another model, tip 4PIs, first finite size scaling. So we studied bunch of state points again near the hypothesized critical point. So this is the fluctuations in one of them and I want you to notice the fluctuations here. In this case, it's a much better example of two state because this particular state point is much closer to the critical point. And then I fit all of them to the icing model for all the system, all different system sizes. Got an estimate of TC and PC. And you can see that I again, have no clear system size dependence. Then I repeated the structure factor calculations as well for tip 4PIs, the substantial growth in the correlation length as temperature decreases. And I got the correlation length and the compressibility from Orstein-Zernike and I again simultaneously regressed them using the exponents expected of icing. And I got high quality of fits and I got a TC of 188.6 Kelvin. This estimate is not very different than the estimate that I have from finite size systems. So bottom line is we were able to unambiguously show from two very different ways that these two classical models of water have a liquid-liquid critical point that belongs to icing universality class. The significance of this work is in the reliability of these models. With this work, we essentially showed that it is not just ST2 that has a meta stable criticality. Instead, these two classical models of water that are widely regarded as to be among the most realistic ones, they also have a liquid-liquid critical point. Still, this doesn't necessarily conclude about real water, but it definitely brings us one step closer to the real water. This work is published in Science, as Ali said, in the summer. I hope you enjoyed as much as I did. I'm blessed to work with this couple of people in this work. Pablo is my postdoc advisor and Francesco Shortino of Sapienza University of Rome, Pablo's long-term collaborator. And I again, thank all of you for paying attention. It was great to connect with ICTP and I will be more than happy to take any questions you might have. Thank you very much, Gold, for that great presentation. I have a virtual clap for you. Okay, so we have lots of time for questions. I have many, but I will wait for the audience to start. So if you have a question, go ahead and unmute yourself and ask away. Yes, I take the opportunity. I have questions related with the densities, profiles that you show. I guess that they are global densities, yes. You are just measuring the density of the box that you are simulating, right? That is correct, yes. Okay, is there any correlation with some kind of microscopic measures of the density, like the five or something like that? That's a good question. Let me stop sharing for a moment and I'll share back in an instant. Let me start sharing again. I am looking for some additional slides that I have because I'm gonna show you something you may have found very interesting. So this is the density, the global density that we are measuring. So what I did is I have extracted the parts that I consider high density and low density. So the pink and red exemplifies high density, green and blue exemplifies low density. And when I measure the G of R and S of K for those sub-ensemble, let me say that way, I can see the structural differences. You see for the low density one, S of K indicates more structured configuration. And from larger simulations, what I have is, this is LSI index colored simulation box and the red one was I think the low density, the blue one was the high density. I see the formation of those large local domains of low density liquid and high density liquid in the case of big box. In case of finite size system, it's a flip. The system, the whole system flips back and forth between low density and high density. But in the case of more realistic system size, what happens is I see the formation of these large local domains of low density liquid and high density liquid. Okay, this is at the critical temperature, right? Not exactly. We are close, but not exactly at critical. It's a little bit about the critical temperature. Just a little bit, like a few Kelvin about the critical temperature. Okay, thank you. Yeah, sure. Can I ask a question? Go ahead, Uriel. Regarding this figure that you have on the screen, so this means that these two faces, they don't like to mix so much. I mean, because they seem to be kind of segregated, one from each other, from one another. You mean this one? No, the previous one. You have two faces, no? Yeah. No, yes, sorry. What happens is, you know, as I approach to the critical point, like, okay, I have another slide, let me see. Oops, sorry. Where was, where was that? Okay, so this is very similar to what happens in ferromagnetic particles, you know, the formation of the large local domains of two different phases. So of course, you know, well, about the critical temperature, they would be very well mixed. But the fact that I am getting close to the critical point, I see the segregation. Well, I'm not sure whether I should say segregation or I should rather say formation of large local domains of, yeah. So I wonder why, like, why is that? I mean, why is not, why it tends to be separated and not mixed, like the interaction between the two types of liquids is somehow special or it has a preferential interaction with particles of the same, in the same, you know, phase. I see, so this is more of a philosophical question, I guess, because this is kind of a well-known behavior, like, as we approach to the critical point. Of course, yeah, of course, well-about to the critical point, they would be very well mixed and it actually is, you know, I don't have LSI colored example here, but they are actually very well mixed if you're well-about the critical temperature. But as we approach to the critical temperature, what happens is the system gets, the correlations get longer and longer length scales, length and time scales. As correlations become stronger, the system gets more and more correlated, that correlation is the reflection of the formation of these large local domains. When you talk about correlation, you're always meaning this geophile, like the Fourier transform of the geophile, right? Yeah, that would reflect itself at the low K. So this is not clear what happens in the low K because this is small box, the small simulations, the finite size ones. So K is not small enough to show the correlations, but in the case, you know, when the K is small enough, actually even this one looks better. If the K is small enough, that means your box is large enough, you can see that correlations at low K, yeah. I see. Okay. Yeah, low K means large R. Yes. Yeah. One more question. Sorry, can I, Ali, or? Please, please, please, go ahead. One little question. So have you tried to do, these are kind of correlations like structural, know, based on the G of R. Yes. Have you tried to do a dynamical correlation, like time correlation functions to see, like, what? Yeah, good question. Yes, I did. Let me see whether I copied, pasted. Okay, I guess no. Okay, I haven't copied them here. But yes, I did intermediate scattering function, which is SKT. It's not just S of K, but it's SKT as a function of T as well. And the growth in time scale correlations is pretty much the same as in length scale correlations. So they become substantially slower as you, so there is a real critical slowdown that happens here. I see, I see, I see. Very nice, thank you. Sure. Okay, maybe I can, is someone wants to? Yes, hi. Yeah, go ahead, I do. Yeah, so I'm a bit curious about this histogram reweighting technique. How does the error propagate when you try to infer the histogram of the lower temperatures from that of the higher temperatures? Yeah, I think this is a great question. So the way that I checked the errors, the quality of my histogram reweighting data, I looked mostly up, I looked up the noise because as you extrapolate further and further, of course you cannot cover the whole, the entire range, the every single temperature, the every single pressure you cannot cover, there's a limit. And that limit I thought is, you're gonna understand that you approach that, you're getting to that limit as you start seeing the noise in your extrapolations. Data starts to be very noisy. So this was my sense of the error. Otherwise, the other way that I have estimated the error is I split it, my whole histogram data into two equal non-overlapping parts. And I looked up the difference between the histograms that I obtained from two independent samples and the difference between them are not very large. I don't have the data posted here, but that was one way that I have checked the statistical errors in my data. Thanks. Sure. Okay, Gull, I had a couple of questions for you. So the first one is related to the question that Alex asked, which is about local descriptors of what is going on. And it's related also to the point that, so ST2, which is an overstructured liquid, also predicts, let's say the right physics of, well, supposedly the right physics of the second critical point. Okay, it's a model that misses a lot of the molecular interactions, et cetera, et cetera. So, I mean, would you say that it's right for the wrong reasons and if it is, what else is going on in the more realistic models that's causing the shift in the temperatures and, yeah. It's actually not right for the wrong reasons. It's actually right for exactly the right reasons because the problem is that, the problem with ST2 is that it is way too tetrahedral. So perfect tetrahedral particles, they actually do have this kind of phase separation. They do split into two. They do have high-density versions and low-density versions. Problem with water is that it's not that tetrahedral. So that was, I think, the whole reason why people keep arguing about whether critical point is real or not, because the counter argument is that, if there's no critical point, the counter argument is that you are crossing through a spinodal, which essentially means that you cannot keep water in its metastable form anymore. And this is because of the subtlety in the tetrahedrality. It's so subtle, the real water is so subtle, whether you have that much of a tetrahedrality or not. But the thing is, these models, let me see whether I have a picture of it. I think I do. These models, they are not tetrad structures. They are not defined in three dimensions. This is only defined in two dimensions. It's on a plane, like the liquid water. So it's described on a plane. They just have one additional interaction site. So they are four sites because of that reason. So the problem in ST2 is it's too perfectly tetrahedral. So it gives correct answers for exactly correct reasons. Okay, okay, okay, okay, I see. But I mean, I guess related to that is, you mentioned this in your abstract, where you didn't say anything about it, which is, I know there is, I guess some recent work by Roberto Carr. Yes. On, let's say, ab initio neural network stuff. That's good. I mean, let's say philosophically, what's your perspective on this? I mean, it seems like the classical model gets it right. So what else do you gain from the ab initio? The first time I presented my stuff to Frank Stalinger, this water stuff to Frank Stalinger, the first question he asked was, how do you think the nuclear quantum effects will impact your findings? So I think those are kind of the stuff, important information that you would probably get from those kinds of models. Well, the thing is, well, Francesco Shortino has a work on how bond flexibility, bond GDT in S2 impacts the liquid-liquid-face separation, which is somewhat related to the nuclear economy, somewhat, not exactly, but on the same page. And what he found is the presence of such effects and bond flexibility and stuff, they favor the liquid-liquid-face separation over ice crystallization. But I guess, bottom line answer to your question is, more insights that you can obtain from things like nuclear quantum effects and things like that. Okay, I see, okay, thanks. Yeah, sure. More questions before I ask my next one. I have a question. I think, wait, hold on, Victor, sorry. I think Mehdi had a question and I stopped. Hello to everybody. Thank you for presentation. Is there any advantage for having larger system size for simulation? I mean, when you go to ice phase, there might be several polycrystallized structure in the system and it might be needed to have larger system size rather than the system size you have. Do you have any comments? I think, yes, you're absolutely right in that, but thank God I don't work on ice crystallization and it would be a much bigger of a trouble to try to nucleate the ice in a bigger system size. So in my case, the liquid system, the benefit of having more realistic, you know, when I say more realistic, a little more away from the finite size is that you can actually show the growth in the correlation length. So to get something like this, you need to have a large, very large system size because in the finite size systems, exactly as I showed here, what happens is the correlations are cut here. So you cannot get any useful information about the correlations in your system. And also as a result of the finite size effects, what happens is your system flips back and forth between the high density and low density. And as opposed to that, what happens in a large system is a real phase separation, like formation of an interface instead of phase flip. What happens is there would be a formation of an interface and the box will split into high density liquid and the low density liquid, which I didn't observe because I didn't do simulations beyond the critical point. Well, I can't, it's just so slow. I can't do that. But I hope this kind of answers your question why, you know, what would be the benefits of having a larger simulation box? Okay, thank you very much for your answer. But when we are talking about interface, it might be a closed shape for a given interface and it might be needed to consider kind of a large system. So it might be, I don't know. No, we have to. We actually have to. Like I said, in a finite size system where the size of the system is smaller than the size of the correlation line, what happens is system flips back and forth. There's no interface formation. For interface formation, we absolutely need a much bigger simulation box. Yeah. Okay, thank you very much. Sure. Victor. Victor. Hi, so I have a question. You keep talking about high density and low density liquids, but you have actually seen ice formation. You say you didn't study the nucleation, but does it ever occur and do you get any time resolved properties that can tell you how solid like you are? Yes. Let me see whether I have those here. Man, I have all of them here somewhere. Let me find it. Okay. I have to pull it up from. I didn't want to do this, but okay. I guess we have to. Let me see the supporting information. I looked up some properties. There's the supplementary. Let me click on PDF. Because in the supplementary, I have measured some quantities, supplementary materials. There you go. I measured some quantities that are related to ice formation. There you go. So we checked for cubic ice, hexagonal ice, collaterate interfacial ice. And we found very small fraction of molecules that forms ice in my box. And this is kind of good news for me because if it forms once, you cannot stop ice growing. So I'm glad that they didn't nucleate any ice during the time that I performed the simulations. Yeah. Okay, thank you. Sure. More questions from the audience. We've had quite a few already. Go, there's been all this discussion, of course, on the dynamical anomalies, breakdown of Stokes-Einstein. And its interpretation in terms of the LDHD business. I guess, now you have the trajectories. Is this something that you guys are thinking about, or are you able to go in, for example, and like for example, when you have these extended connected regions of low density and high density, would you be able to see my computing some dynamical quantities, like rotational diffusion, that that is reflected there as well? Yeah, that would be actually interesting. And I think it would be reflected, but there's not something actually I did calculate. I probably should have done that. But yeah, you're right that it would be reflected. It's just, I didn't... Yeah, no, no, it's really a monster of a project anyway. But so I guess, yeah, the challenge is that, yeah, I guess converging dynamical quantities would be even harder. Yeah, actually, so those, speaking of those dynamical stuff, now that I've pulled up the supporting information already, I can show that the SKT stuff, and you can see the time scale of relaxation. So this one, the smallest, well, smallest K vector, of course, is the slowest one because that's the largest, longest length scale correlation. And it takes, it's the slowest one. Let me see which temperature was this. So this was at 183, which is fairly above the critical point I need to say, because critical point is 173 Kelvin. So this is 10 Kelvin about the critical point. And even at the 10 Kelvin about, you see that the time scale of relaxation is on the order of microseconds or tens of micro, between microseconds or half microseconds. Very good. Okay. Yeah, so can you go back to the slide where you show the structure factor in GOVR? GOVR of the small systems or the, first let me find this one. Yes. And do you have an idea of qualitatively, like when you visualize the, I mean the structures in the high density or the low density, do you have an inkling about what is different between the two? Yeah, so these are actually the snapshots that I pulled up from my simulation boxes. So the low density one is the one that is beautifully coordinated, tetrahedrally coordinated one. The high density one is four molecules in the first solvation shell, four molecules plus one more squeezed itself in, breaking the hydrogen bonding network. So, you know, it overcrowded here, but so that's the reason why this is called entropically stable, this is entropically stable. This is more, yeah. So this is, so it's defective, like there's a lack of a hydrogen bond. There's a lack of a hydrogen bond, but at the cost of introducing one more molecule to the first solvation shell, a first neighboring shell, whichever solvation shell is not a correct, solvation of water is not very good term probably, but first neighbor shell, the first neighbor, yeah. But in terms of the geometric definition, you would still find, would you find that it's over coordinated or would you, or that it's under coordinated here? Well, this is what happens. Well, I wouldn't say over coordinated because this is, so this is actually what I mean when I said it's subtle. It is coordinated, but it is in terms of individual molecules, it is described by a single plane, not three-dimensional, you know, the individual molecule itself is not three-dimensional tetrad. Individual molecule is a planar molecule, it's a flat molecule, but it still forms these coordinated structures. So I wouldn't say it is over-coordination. Okay, thanks. Sure. Did you look at the lifetime of these hydrogen bonds? There will be actually quite interesting to do lifetime of the hydrogen bonds, but I didn't, but it would be quite interesting to do. And actually there are some folks in the lab right now who are doing it for other purposes, not for proving the criticality, but for other purposes, they are looking at the lifetime of hydrogen bonds. Do you have the diffusion constant between the high-density and low-density liquids? I don't have this either, but let me think. I did calculate the diffusion constant at some point, and I have seen that beautiful ballistic part, the diffusive part, but I didn't have any rigorous analysis of diffusion between high-density to low-density, but I'm not sure whether it relates to the criticality, though. But yeah, I don't have it. Okay. More questions? If there are no more questions, join me in thanking Gull again for a wonderful talk and for keeping a lot of time for discussion. I think this is very good for virtual talks. Actually, very short, but more time for discussion. Yeah, I'm so very happy to receive a lot of questions from you guys. I guess this is the first time that I'm receiving that many questions, and I'm very happy with that. I think it stirs a lot of beautiful discussions and further thought about the time. I actually have something else came to mind. So going back to the beginning of your presentation about your other problems that you've been interested in, water interfaces, proteins. What are the implications of, let's say these findings on water? I know it has implications, for example, water under confinement, right? Yeah, yeah. But can you say something about that? So the thing that we broadly call hydrophobic effect is technically arising from this. I mean, arising from the same reasons. The reason is water being tetrahedral. So I think this is the most important implication that I can immediately think of. The thing that's been broadly described as hydrophobic effect. It's very similar to what you said about confinement and evaporation of water between the plates. So, yeah, those arise for the same reasons. For the reasons that water is tetrahedral. And the argument about the critical point, whether it exists or not, like I said, the counter argument is spinodal. You're crossing through a spinodal. But the thing is, we wouldn't see, well, in the case of, but okay. If you're approaching the spinodal, you would also see those increase in the response functions as temperature decreases. You would see more or less the same thing. But what you wouldn't see is an icing like scaling. Icing like scaling only happens if you are approaching to a true critical point. And you need to get close enough to the critical point to be able to see an icing like scaling unless you are very, very close to the critical point, you would see more like a mean field scaling instead of an icing one. So, yeah, I think this was, this was, I think, very helpful for me in general. Phase transitions, I'm so happy to be able to develop analysis techniques for phase transitions because they happen pretty often in biological systems too. Yeah, sure, sure, sure. Yeah, yeah. I know, very nice, very cool. Okay, very good. Well, thank you very much, Gull. Thank you all for joining in today. And I guess it's happy holidays to everyone. It was my pleasure to be here and happy holidays, everyone. I thank all of you for your attention and for the beautiful questions that you have asked. Thank you for joining us, Gull. Thank you for closing the tough year. Bye. All right, take care, guys. Thanks, happy holidays. Happy holidays. Bye, thanks a lot.