Manifestations of metastable criticality in the long-range structure of model water glasses

Much attention has been devoted to water’s metastable phase behavior, including polyamorphism (multiple amorphous solid phases), and the hypothesized liquid-liquid transition and associated critical point. However, the possible relationship between these phenomena remains incompletely understood. Using molecular dynamics simulations of the realistic TIP4P/2005 model, we found a striking signature of the liquid-liquid critical point in the structure of water glasses, manifested as a pronounced increase in long-range density fluctuations at pressures proximate to the critical pressure. By contrast, these signatures were absent in glasses of two model systems that lack a critical point. We also characterized the departure from equilibrium upon vitrification via the non-equilibrium index; water-like systems exhibited a strong pressure dependence in this metric, whereas simple liquids did not. These results reflect a surprising relationship between the metastable equilibrium phenomenon of liquid-liquid criticality and the non-equilibrium structure of glassy water, with implications for our understanding of water phase behavior and glass physics. Our calculations suggest a possible experimental route to probing the existence of the liquid-liquid transition in water and other fluids.

I find the work to be solid, of high quality, to address a subject of pivotal current concern and to provide results of great relevance of interest for a wide community of researchers across the physical sciences. Thus, I am glad to recommend publication. Nontheless, there is only one point I would like to raise for the authors to comment: In Fig. 1 a, a very notable extremely sharp peak develops in S(0) around the critical pressure as temperature is lowered. This behavior stems from the increase in the long-range density fluctuations and it would be expected to grow as we approach the critical temperature, T_C, where fluctuations should be maximal (diverge if at the critical point). However, when going down in temperature from around T_C to the lower temperatures studied, the curves clearly superimpose each other. Is this a sign of the presence of finite size effects? Does the system size become small in comparison with the range of the fluctuations?
Reviewer #3 (Remarks to the Author): In this paper, the authors report an interesting numerical simulation observation that the amorphous state formed by temperature cooling under various pressures remembers the critical fluctuations that water experiences during the cooling process. The authors used a model water, TIP4P/2005, clearly proven to have a second critical point. As a result, long-range density correlations were observed in the amorphous state obtained by temperature cooling at pressures near the critical pressure. These results of the TIP4P/2005 model were compared with those of the mW water and Kob-Andersen binary mixtures. The results show that the peculiar behavior observed in the TIP4P/2005 model is not observed at all in the KA model, which does not have a second critical point. On the other hand, the mW model results show a weak signal indicating the enhancement of density fluctuations at a pressure near the Tg(P)'s minimum.
The enhancement of long-range density correlations in the glassy state results from an intriguing combination of criticality and slow glassy dynamics. This stems from the special relationship between the critical-point location and the glass-transition line in the water's T-P phase diagram. Although it may be technically challenging, it would be possible to apply this strategy to experiments, as suggested by the authors. This unique strategy may provide a new way to detect the second critical point of water, which has been difficult to prove experimentally. This is the first systematic study of the effect of critical fluctuations on water's glassy state to the best of my knowledge.
Given that the criticality of water associated with the liquid-liquid transition has received considerable attention in the community, this manuscript will significantly impact the field and also stimulate experimental research. Thus, I warmly support its publication in Nature Communications. However, before it is accepted, I recommend that the authors consider the following points.
(1) First of all, S(0) is proportional to the isothermal compressibility. Then, the isothermal compressibility is composed of the non-critical background part and the critical part. The high-temperature value of S(0) reflects the background part. For water, it is known that even the non-critical part originating from the twostate feature can have a maximum with decreasing temperature (see, e.g., Fig. 12 of Ref. A). Thus, there is no one-to-one correspondence between the correlation length of critical fluctuations and S(0), as long as the critical contribution is not dominant. In principle, these two contributions can be separated by the detailed analysis of S(k), but which seems difficult in the present case (see Fig. S4). I recommend the authors mention that there can be these two contributions to S(0).
(2) Since it has been proven to have a second critical point for TIP4P/2005 water, the observed S(0) peak as a function of P probably comes from critical fluctuations, as the authors claimed. In contrast, the small peak observed for mW water may not come from the criticality but from the non-critical part's increase due to the two-state feature. A simple two-state model without criticality predicts that the non-critical compressibility peak height as a function of T should monotonically increase with an increase in P (see, e.g., Fig. 12 of Ref. A). On the other hand, Tg decreases with P, which may induce S(0)'s quicker decay for a higher P side. The competition between these two tendencies may explain a small S(0) peak around 6 kbar observed for the mW model. However, to draw a definite conclusion, the relationship between the cooling rate and the structural relaxation rate is necessary. There is also a possibility that the second critical point is hidden in the glass state. I recommend the authors discuss these issues briefly.
(3) For density fluctuations to be frozen in glass, they need to grow as they approach the critical point and freeze by glassiness upon further cooling. Therefore, the P-dependence of S(0) may reflect the relationship of the T, P-dependence of the correlation length and lifetime of the density fluctuations with the T, Pdependence of the glass transition. The systematic shift of the S(0) peak to the lower pressure side with increasing cooling rate, i.e., the asymmetric pressure dependence of the S(0) peak, may also reflect the Tg's pressure dependence. The dynamic aspect of the origin of the asymmetry deserves some more discussion.
(4) As shown in Fig. 4, Tg decreases as the cooling rate decreases. At the lowest cooling rate qT=-0.1 K/ns, Tg(P) lies below Tc. Thus, one would expect a sharp increase of S(0) peak at the critical pressure at this cooling rate. In Fig. 2, however, we can see only a slight increase of the S(0) peak as the cooling rate decreases. This seems to indicate that critical fluctuations cannot fully develop due to the rapid increase in the lifetime of critical fluctuations in the critical point vicinity. It suggests an interesting competition among the critical slowing down towards Tc, the slowing down of the glassy dynamics towards Tg, and the cooling rate. I recommend the authors discuss this point in more detail.
(5) It would be useful for readers if the authors could provide a rough estimate of the correlation length of density fluctuations. According to Fig. S4, it looks like about 5A if we use the Ornstein-Zernike-like analysis (although the lengthscale might be too short for the OZ analysis to be valid).
(6) On page 11, the authors mention the relationship between the minimum of Tg and the maximum diffusivity around 2 kbar. I want to point out that there is another possibility in this regard. Recent studies have shown that the diffusivity maximization around 2 kbar, which is observed even at high temperatures well above Tg, is not a consequence of glassiness but due to the dependence of the activation energy of the liquid on the degree of tetrahedral ordering (see Ref. B). A similar conclusion was also derived recently (see Ref. C). I agree that there is a relationship between the maximum diffusivity and the minimum Tg, but the former may not be a direct consequence of the glassiness. It seems natural to assume that both are due to structural crossover from LDL-like to HDL-like liquids. This scenario may also explain the similar behavior observed for the mW water.
(7) The analysis of the nonequilibrium index X is quite interesting. The increase of dX/dT near the critical pressure clearly indicates the importance of the critical slowing down. I recommend the authors add plots of dX/dT near the onset of the X increase to Fig. 5. (8) Recently, it has been shown that the first sharp diffraction peak in the structure factor of water may serve as a good descriptor for characterizing the LDL-like and HDL-like structures (see Ref. D; see also Ref. E on the systematic change of S(k)). So, plotting S(k) together with g(r) in Figs. S1 and S2 would be useful for readers.

RESPONSE TO REVIEWS
Manuscript ID: NCOMMS-21-01760-T Title: "Manifestations of metastable criticality in the long-range structure of model water glasses" Authors: Thomas E. Gartner III, Salvatore Torquato, Roberto Car, and Pablo G. Debenedetti We thank you for your time in considering our manuscript, your thorough comments and helpful suggestions, and your overall positive evaluation of the manuscript. We have edited the manuscript to address your comments/suggestions, which are summarized in our responses below. We have also prepared a version of the manuscript with our changes highlighted in yellow, which is included with our submission of the revision.

In the manuscript "Manifestations of metastable criticality in the long-range structure of model water glasses", Gartner et al simulate, using classical molecular dynamics, the cooling of liquid water (TIP4P/2005 and mW) and of KA at different pressures and cooling rates. At each thermodynamic point, the authors compute the value of S(0) and the 'non-equilibrium index' X. Interestingly, the authors find a pronounced increase in the long-range density fluctuations in the vicinity of the pressure associated with the liquid-liquid transition in TIP4P/2005, while no such increase is present in other glasses that, in the liquid phase, do not show a LLT. The authors also find that water-like systems exhibit a strong pressure dependence on the non-equilibrium index, that instead does not show such dependence in simple liquids. Based on these results, the authors claim the existence of a link between metastable equilibrium phenomena of the LLCP and non-equilibrium long-range structures in glassy water. Moreover, the authors claim that the protocol described in this manuscript can potentially be implemented in experiments as another route to study supercooled liquids and locate a LLCP.
Overall, the paper is well written, and the results are interesting. The introduction of the reported protocol to study supercooled and glassy water is novel and the signatures of the LLT in the S(0) in the glassy states adds new understanding on the field of water. In my opinion, the manuscript deserves to be published in Nature Communications after the author address the following points.

RESPONSE:
We thank the reviewer for the summary of our findings and the recommendation to publish.

Comments:
1. The authors should mention the following papers when mentioning that the Tgdependence over the cooling rate has been already investigated with another model of water: [Giovambattista et al "glass-transition temperature of water: a simulation study, PRL 93,

(2004)" and Giovambattista et al "Cooling rate, heating rate, and aging effects in glassy water, PRE 99, 050201(R) (2004)"].
RESPONSE: We have added these relevant references to our discussion of cooling rate effects on pages 9-10: "Prior simulation studies with other water models 51-53 have established qT as an important parameter controlling the structural and energetic properties of water glasses; here, we evaluate the effects of qT on TIP4P/2005's long-range structure."

Figure 3d & 4: one would expect the LLCP be close to the slowest cooling rate (0.1K/ns). Why, instead, it is closer to the second slower cooling rate (1K/ns)?
RESPONSE: We presume that the reviewer is referring to the idea that the slowest cooling rate should exhibit closest to 'equilibrium-like' behavior, hence the suggestion that the Tg obtained using the slowest cooling rate should be closest to the LLCP. However, there is no requirement that the glass transition in the limit of slow cooling be near to the LLCP, given that the two phenomena come from distinct microscopic origins. Indeed, [Giovambattista et al., Sci. Rep. 2, 390 (2012)] shows that, for the ST2 model, the Tg (achieved at a finite cooling rate) lies somewhat below the LLCP, and thus the locus of Tg(P) intersects with the liquid-liquid transition line for that model. By contrast, in the present work with TIP4P/2005, we were able to achieve Tg(P) that lie both below and above the LLCP by changing the cooling rate. This observation was the basis of our suggestion for future work to test a similar glass formation protocol for systems where the LLCP is significantly above or below the locus of Tg(P).
3. The non-equilibrium index is useful but not very handy. One is restricted to very large simulation cells to properly sample S(0). Moreover, this index does not capture the dependence on the pressure ( Figure 5). One needs to introduce info from the slope of Tg(P). The authors should comment more on this.
RESPONSE: We agree with the reviewer that any metric involving S(0) (including but not limited to the non-equilibrium index) is difficult to sample, especially for the wide parameter space explored in this work. Though challenging to calculate, the non-equilibrium index (X) was especially useful in this work, as it revealed subtle but important features such as the impact of critical slowing down in TIP4P/2005, as noted below by Reviewer #3. We also respectfully disagree with the reviewer's comment that X does not capture pressure dependence. Indeed, the key observation of Figure 5 is that the slope of X is strongly dependent on the pressure in waterlike models, whereas it is insensitive to pressure in the Kob-Andersen mixture. We also note that we account for the effect of pressure by normalizing the x-axis of Figure 5 by the (pressuredependent) Tg(P) reported in Figure 3. To clarify this point, we have added the following comment to page 13-14: "To facilitate comparison across models and state points, we normalize the temperatures in Figure 5 by the (pressure-dependent) Tg(P) reported in Figure 3." RESPONSE: We do believe that it is important to consider finite size effects when interpreting the behavior we observed in this work--the reviewer correctly notes that in the immediate vicinity of the critical point density fluctuations will grow to length scales comparable to the system size. However, in the particular case studied here, we feel that the behavior that the reviewer is noting (the S(0) not truly diverging and eventually ceasing to evolve below a certain temperature) to be more of a dynamic consequence of the non-equilibrium cooling of these systems, rather than limitations of system size. For temperatures near to or below the glass transition, large-scale structural rearrangements become increasingly infrequent, thus largely 'freezing-in' the S(0) for temperatures below a given point. If, instead of the isobaric cooling performed in this work, we were able to fully equilibrate the system at state points near to or below the LLCP, we would expect to see a much sharper increase in S(0) followed by a decrease with decreasing temperature, as this reviewer states. This point also relates to the issues raised by Reviewer 3 below in terms of underscoring the important relationship between thermodynamic and dynamic phenomena in controlling the structural evolution in these systems. We have added an additional paragraph of commentary to page 16-17, as well as a few comments throughout the manuscript to draw attention to this point--these additions are reproduced in our responses to Reviewer 3 below. The enhancement of long-range density correlations in the glassy state results from an intriguing combination of criticality and slow glassy dynamics. This stems from the special relationship between the critical-point location and the glass-transition line in the water's T-P phase diagram. Although it may be technically challenging, it would be possible to apply this strategy to experiments, as suggested by the authors. This unique strategy may provide a new way to detect the second critical point of water, which has been difficult to prove experimentally. This is the first systematic study of the effect of critical fluctuations on water's glassy state to the best of my knowledge.
Given that the criticality of water associated with the liquid-liquid transition has received considerable attention in the community, this manuscript will significantly impact the field and also stimulate experimental research. Thus, I warmly support its publication in Nature Communications. However, before it is accepted, I recommend that the authors consider the following points.