Liquid-Liquid Criticality in the WAIL Water Model

Unveiling a common phase transition pathway of high-density amorphous ices through time-resolved x-ray scattering

Here, we investigate the hypothesis that despite the existence of at least two high-density amorphous ices, only one high-density liquid state exists in water. We prepared a very-high-density amorphous ice (VHDA) sample and rapidly increased its temperature to around 205 ± 10 K using laser-induced isochoric heating. This temperature falls within the so-called "no-man's land" well above the glass-liquid transition, wherein the IR laser pulse creates a metastable liquid state. Subsequently, this high-density liquid (HDL) state of water decompresses over time, and we examined the time-dependent structural changes using short x-ray pulses from a free electron laser. We observed a liquid-liquid transition to low-density liquid water (LDL) over time scales ranging from 20 ns to 3 μs, consistent with previous experimental results using expanded high-density amorphous ice (eHDA) as the initial state. In addition, the resulting LDL derived both from VHDA and eHDA displays similar density and degree of inhomogeneity. Our observation supports the idea that regardless of the initial annealing states of the high-density amorphous ices, the same HDL and final LDL states are reached at temperatures around 205 K.

Liquid-liquid transition and ice crystallization in a machine-learned coarse-grained water model

Mounting experimental evidence supports the existence of a liquid-liquid transition (LLT) in high-pressure supercooled water. However, fast crystallization of supercooled water has impeded identification of the LLT line TLL(p) in experiments. While the most accurate all-atom (AA) water models display a LLT, their computational cost limits investigations of its interplay with ice formation. Coarse-grained (CG) models provide over 100-fold computational efficiency gain over AA models, enabling the study of water crystallization, but have not yet shown to have a LLT. Here, we demonstrate that the CG machine-learned water model Machine-Learned Bond-Order Potential (ML-BOP) has a LLT that ends in a critical point at pc = 170 ± 10 MPa and Tc = 181 ± 3 K. The TLL(p) of ML-BOP is almost identical to the one of TIP4P/2005, adding to the similarity in the equation of state of liquid water in both models. Cooling simulations reveal that ice crystallization is fastest at the LLT and its supercritical continuation of maximum heat capacity, supporting a mechanistic relationship between the structural transformation of water to a low-density liquid (LDL) and ice formation. We find no signature of liquid-liquid criticality in the ice crystallization temperatures. ML-BOP replicates the competition between formation of LDL and ice observed in ultrafast experiments of decompression of the high-density liquid (HDL) into the region of stability of LDL. The simulations reveal that crystallization occurs prior to the coarsening of the HDL and LDL domains, obscuring the distinction between the highly metastable first-order LLT and pronounced structural fluctuations along its supercritical continuation.

Potential energy landscape of a flexible water model: Equation of state, configurational entropy, and Adam-Gibbs relationship

The potential energy landscape (PEL) formalism is a tool within statistical mechanics that has been used in the past to calculate the equation of states (EOS) of classical rigid model liquids at low temperatures, where computer simulations may be challenging. In this work, we use classical molecular dynamics (MD) simulations and the PEL formalism to calculate the EOS of the flexible q-TIP4P/F water model. This model exhibits a liquid-liquid critical point (LLCP) in the supercooled regime, at (Pc = 150 MPa, Tc = 190 K, and ρc = 1.04 g/cm3) [using the reaction field technique]. The PEL-EOS of q-TIP4P/F water and the corresponding location of the LLCP are in very good agreement with the MD simulations. We show that the PEL of q-TIP4P/F water is Gaussian, which allows us to calculate the configurational entropy of the system, Sconf. The Sconf of q-TIP4P/F water is surprisingly similar to that reported previously for rigid water models, suggesting that intramolecular flexibility does not necessarily add roughness to the PEL. We also show that the Adam-Gibbs relation, which relates the diffusion coefficient D with Sconf, holds for the flexible q-TIP4P/F water model. Overall, our results indicate that the PEL formalism can be used to study molecular systems that include molecular flexibility, the common case in standard force fields. This is not trivial since the introduction of large bending/stretching mode frequencies is problematic in classical statistical mechanics. For example, as shown previously, we find that such high frequencies lead to unphysical (negative) entropy for q-TIP4P/F water when using classical statistical mechanics (yet, the PEL formalism can be applied successfully).

Water Potential from Adaptive Force Matching for Ice and Liquid with Revised Dispersion Predicts Supercooled Liquid Anomalies in Good Agreement with Two Independent Experimental Fits

A revised version of the Water potential from Adaptive force matching for Ice and Liquid (WAIL) was developed by using the previous data set for fitting the WAIL model but with a dispersion term calculated using symmetry adapted perturbation theory (SAPT). The model has no adjustable parameters and relies solely on fitting first-principles information. The new model, named revised WAIL (rWAIL), shows improved predictions of most properties of water when compared to the previously published WAIL model. The rWAIL model also compares favorably to other first-principles-derived water models, such as MB-Pol, at only a fraction of the computational cost. The rWAIL model is used to study the properties of supercooled water. The model shows evidence of a liquid-liquid phase transition (LLPT) in the supercooled regimes with the liquid-liquid critical point (LLCP) at 203 K and 90 MPa. This estimate is in good agreement with a recent polynomial fit to the experimental density of water. Also, the fit to the surface tension of supercooled water based on the rWAIL model shows excellent agreement with the corresponding fit to the experimental data. Consistent with previously published molecular dynamics and experimental data, the surface tension of water exhibits exponential growth in the supercooled regime, which is likely a result of the emergence of a low-density liquid form of water. The simulation thus unites two separate experimental fits with one first-principles-based model, lending strong evidence of an LLPT in real water.

Free-energy landscape and spinodals for the liquid-liquid transition of the TIP4P/2005 and TIP4P/Ice models of water

Continued increases in computational power now make it possible to evaluate the free-energy landscape associated with the first-order liquid-liquid transition in realistic models of water for which an accurate estimate of the liquid-liquid critical point exists, and to explore its change with pressure near the coexistence line. We report the results of 50 μs-long NPT umbrella sampling simulations for two realistic models for water, TIP4P/2005 and TIP4P/ice, 3-9 K below their critical temperatures. The free energy profile at different pressures clearly shows the presence of two well-defined free energy basins and makes it possible to identify the liquid-liquid spinodal points, the limits of stability that define the (temperature dependent) pressure range within which two distinct free energy basins exist. The results show that for temperatures less than 10 K below the critical temperature, metastable states are possible across a very limited pressure interval, information that is relevant to the interpretation of experiments probing the metastable phase behavior of deeply supercooled water in the so-called no-man's land.

Deciphering Density Fluctuations in the Hydration Water of Brownian Nanoparticles via Upconversion Thermometry

We investigate the intricate relationship among temperature, pH, and Brownian velocity in a range of differently sized upconversion nanoparticles (UCNPs) dispersed in water. These UCNPs, acting as nanorulers, offer insights into assessing the relative proportion of high-density and low-density liquid in the surrounding hydration water. The study reveals a size-dependent reduction in the onset temperature of liquid-water fluctuations, indicating an augmented presence of high-density liquid domains at the nanoparticle surfaces. The observed upper-temperature threshold is consistent with a hypothetical phase diagram of water, validating the two-state model. Moreover, an increase in pH disrupts the organization of water molecules, similar to external pressure effects, allowing simulation of the effects of temperature and pressure on hydrogen bonding networks. The findings underscore the significance of the surface of suspended nanoparticles for understanding high- to low-density liquid fluctuations and water behavior at charged interfaces.

Identification of local structures in water from supercooled to ambient conditions

Studies of water thermodynamics have long been tied to the identification of two distinct families of local structures, whose competition could explain the origin of the many thermodynamic anomalies and the hypothesized liquid-liquid critical point in water. Despite the many successes and insights gained, the structural indicators proposed throughout the years were not able to unequivocally identify these two families over a wide range of conditions. We show that a recently introduced indicator, Ψ, which exploits information on the hydrogen bond network connectivity, can reliably identify these two distinct local environments over a wide range of thermodynamic conditions (188-300 K and 0-13 kbar) and that close to the liquid-liquid critical point, the spatial correlations of density fluctuations are identical to those of the Ψ indicator. Our results strongly support the idea that water thermodynamic properties arise from the competition between two distinct and identifiable local environments.

Melting curves of ice polymorphs in the vicinity of the liquid-liquid critical point

The possible existence of a liquid-liquid critical point in deeply supercooled water has been a subject of debate due to the challenges associated with providing definitive experimental evidence. The pioneering work by Mishima and Stanley [Nature 392, 164-168 (1998)] sought to shed light on this problem by studying the melting curves of different ice polymorphs and their metastable continuation in the vicinity of the expected liquid-liquid transition and its associated critical point. Based on the continuous or discontinuous changes in the slope of the melting curves, Mishima [Phys. Rev. Lett. 85, 334 (2000)] suggested that the liquid-liquid critical point lies between the melting curves of ice III and ice V. We explore this conjecture using molecular dynamics simulations with a machine learning model based on ab initio quantum-mechanical calculations. We study the melting curves of ices III, IV, V, VI, and XIII and find that all of them are supercritical and do not intersect the liquid-liquid transition locus. We also find a pronounced, yet continuous, change in the slope of the melting lines upon crossing of the liquid locus of maximum compressibility. Finally, we analyze the literature in light of our findings and conclude that the scenario in which the melting curves are supercritical is favored by the most recent computational and experimental evidence. Although the preponderance of evidence is consistent with the existence of a second critical point in water, the behavior of ice polymorph melting lines does not provide strong evidence in support of this viewpoint, according to our calculations.

Are Neural Network Potentials Trained on Liquid States Transferable to Crystal Nucleation? A Test on Ice Nucleation in the mW Water Model

Neural network potentials (NNPs) are increasingly being used to study processes that happen on long time scales. A typical example is crystal nucleation, which rate is controlled by the occurrence of a rare fluctuation, i.e., the appearance of the critical nucleus. Because the properties of this nucleus are far from those of the bulk crystal, it is yet unclear whether NN potentials trained on equilibrium liquid states can accurately describe nucleation processes. So far, nucleation studies on NNPs have been limited to ab initio models whose nucleation properties are unknown, preventing an accurate comparison. Here we train a NN potential on the mW model of water─a classical three-body potential whose nucleation time scale is accessible in standard simulations. We show that a NNP trained only on a small number of liquid state points can reproduce with great accuracy the nucleation rates and free energy barriers of the original model, computed from both spontaneous and biased trajectories, strongly supporting the use of NNPs to study nucleation events.

Liquid-Liquid Criticality in TIP4P/2005 and Three-State Models of Water

Molecular dynamics simulations leading to the isothermal compressibility, the isobaric thermal expansivity, and the isobaric heat capacity of TIP4P/2005 water are found to be consistent with the coordinates of its second, liquid-liquid critical point reported recently by Debenedetti et al. [ Science 2020, 369, 289-292]. In accord with the theory of critical phenomena, we encounter that the rise in the magnitude of these response functions as temperature is lowered is especially marked along the critical isochore. Furthermore, response-function ratios provide a test for thermodynamic consistency at the critical point and manifest nonuniversal features sharply distinguishing liquid-liquid from standard gas-liquid criticality. The whole pattern of behavior revealed by simulations is qualitatively the same as the one of a three-state Ising model of water exhibiting a low-temperature liquid-liquid critical point. Exact solutions for the two-state components of such a three-state model are also provided.

A neural network potential with self-trained atomic fingerprints: A test with the mW water potential

We present a neural network (NN) potential based on a new set of atomic fingerprints built upon two- and three-body contributions that probe distances and local orientational order, respectively. Compared with the existing NN potentials, the atomic fingerprints depend on a small set of tunable parameters that are trained together with the NN weights. In addition to simplifying the selection of the atomic fingerprints, this strategy can also considerably increase the overall accuracy of the network representation. To tackle the simultaneous training of the atomic fingerprint parameters and NN weights, we adopt an annealing protocol that progressively cycles the learning rate, significantly improving the accuracy of the NN potential. We test the performance of the network potential against the mW model of water, which is a classical three-body potential that well captures the anomalies of the liquid phase. Trained on just three state points, the NN potential is able to reproduce the mW model in a very wide range of densities and temperatures, from negative pressures to several GPa, capturing the transition from an open random tetrahedral network to a dense interpenetrated network. The NN potential also reproduces very well properties for which it was not explicitly trained, such as dynamical properties and the structure of the stable crystalline phases of mW.

Equation of State of Liquid Water Written by Simple Experimental Polynomials and the Liquid-Liquid Critical Point

Experimental and theoretical evidence has been accumulating to support the liquid-liquid critical point (LLCP) hypothesis for water. However, no agreement has yet been reached on the pressure and temperature of LLCP. Here we made simple experimental equations that reproduced the volume of liquid water measured over a wide pressure-temperature range. They were polynomials that calculate pressure using volume and temperature as variables, and coefficients were determined by the method of least-squares. We analyzed the polynomials by changing the volume data and the number of terms in the polynomials and extrapolated them slightly to low temperatures. Consequently, the available experimental volume of liquid water indicated (but did not prove) the existence of LLCP at low temperature. Representative polynomials suggested that LLCP locates around 105 ± ∼ 9 MPa, 207 ± ∼ 5 K, and 0.993 ± ∼ 0.009 cm³/g.

Liquid-liquid phase separation in supercooled water from ultrafast heating of low-density amorphous ice

Recent experiments continue to find evidence for a liquid-liquid phase transition (LLPT) in supercooled water, which would unify our understanding of the anomalous properties of liquid water and amorphous ice. These experiments are challenging because the proposed LLPT occurs under extreme metastable conditions where the liquid freezes to a crystal on a very short time scale. Here, we analyze models for the LLPT to show that coexistence of distinct high-density and low-density liquid phases may be observed by subjecting low-density amorphous (LDA) ice to ultrafast heating. We then describe experiments in which we heat LDA ice to near the predicted critical point of the LLPT by an ultrafast infrared laser pulse, following which we measure the structure factor using femtosecond x-ray laser pulses. Consistent with our predictions, we observe a LLPT occurring on a time scale < 100 ns and widely separated from ice formation, which begins at times >1 μs.

Correlated Fluctuations of Structural Indicators Close to the Liquid-Liquid Transition in Supercooled Water

Multiple numerical studies have unambiguously shown the existence of a liquid-liquid critical point in supercooled states for different numerical models of water, and various structural indicators have been put forward to describe the transformation associated with this phase transition. Here we analyze numerical simulations of near-critical supercooled water to compare the behavior of several of such indicators with critical density fluctuations. We show that close to the critical point most indicators are strongly correlated to density, and some of them even display identical distributions of fluctuations. These indicators probe the exact same free energy landscape, therefore providing a thermodynamic description of critical supercooled water which is identical to that provided by the density order parameter. This implies that close to the critical point, there is a tight coupling between many, only apparently distinct, structural degrees of freedom.

Liquid-Liquid Transition in Water from First Principles

A long-standing question in water research is the possibility that supercooled liquid water can undergo a liquid-liquid phase transition (LLT) into high- and low-density liquids. We used several complementary molecular simulation techniques to evaluate the possibility of an LLT in an ab initio neural network model of water trained on density functional theory calculations with the SCAN exchange correlation functional. We conclusively show the existence of a first-order LLT and an associated critical point in the SCAN description of water, representing the first definitive computational evidence for an LLT in water from first principles.

Manifestations of the structural origin of supercooled water’s anomalies in the heterogeneous relaxation on the potential energy landscape

Liquid water is well-known for its intriguing thermodynamic anomalies in the supercooled state. The phenomenological two-state models—based on the assumption of the existence of two types of competing local states (or, structures) in liquid water—have been extremely successful in describing water’s thermodynamic anomalies. However, the precise structural features of these competing local states in liquid water still remain elusive. Here, we have employed a predefined structural order parameter-free approach to unambiguously identify two types of competing local states—entropically and energetically favored—with significantly different structural and energetic features in the TIP4P/2005 liquid water. This identification is based on the heterogeneous structural relaxation of the system in the potential energy landscape (PEL) during the steepest-descent energy minimization. This heterogeneous relaxation is characterized using order parameters inspired by the spin-glass transition in frustrated magnetic systems. We have further established a direct relationship between the population fluctuation of the two states and the anomalous behavior of the heat capacity in supercooled water. The composition-dependent spatial distribution of the entropically favored local states shows an interesting crossover from a spanning network-like single cluster to the spatially delocalized clusters in the close vicinity of the Widom line. Additionally, this study establishes a direct relationship between the topographic features of the PEL and the water’s thermodynamic anomalies in the supercooled state and provides alternate markers (in addition to the locus of maxima of thermodynamic response functions) for the Widom line in the phase plane.

Water's Unusual Thermodynamics in the Realm of Physical Chemistry

While it is known since the early work by Edsall, Frank and Evans, Kauzmann, and others that the thermodynamics of solvation of nonpolar solutes in water is unusual and has implications for the thermodynamics of protein folding, only recently have its connections with the unusual temperature dependence of the density of solvent water been illuminated. Such density behavior is, in turn, one of the manifestations of a nonstandard thermodynamic pattern contemplating a second, liquid-liquid critical point at conditions of temperature and pressure at which water exists as a deeply supercooled liquid. Recent experimental and computational work unambiguously points toward the existence of such a critical point, thereby providing concrete answers to the questions posed by the 1976 pioneering experiments by Speedy and Angell and the associated "liquid-liquid transition hypothesis" posited in 1992 by Stanley and co-workers. Challenges of this phenomenology to the branch of Statistical Mechanics remain.