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.

Advances in the study of supercooled water.

In this review, we report recent progress in the field of supercooled water. Due to its uniqueness, water presents numerous anomalies with respect to most simple liquids, showing polyamorphism both in the liquid and in the glassy state. We first describe the thermodynamic scenarios hypothesized for the supercooled region and in particular among them the liquid-liquid critical point scenario that has so far received more experimental evidence. We then review the most recent structural indicators, the two-state model picture of water, and the importance of cooperative effects related to the fact that water is a hydrogen-bonded network liquid. We show throughout the review that water's peculiar properties come into play also when water is in solution, confined, and close to biological molecules. Concerning dynamics, upon mild supercooling water behaves as a fragile glass former following the mode coupling theory, and it turns into a strong glass former upon further cooling. Connections between the slow dynamics and the thermodynamics are discussed. The translational relaxation times of density fluctuations show in fact the fragile-to-strong crossover connected to the thermodynamics arising from the existence of two liquids. When considering also rotations, additional crossovers come to play. Mobility-viscosity decoupling is also discussed in supercooled water and aqueous solutions. Finally, the polyamorphism of glassy water is considered through experimental and simulation results both in bulk and in salty aqueous solutions. Grains and grain boundaries are also discussed.

Free energy surface of two-step nucleation.

We test the theoretical free energy surface (FES) for two-step nucleation (TSN) proposed by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)] by comparing the predictions of the theory to numerical results for the FES recently reported from Monte Carlo simulations of TSN in a simple lattice system [James et al., J. Chem. Phys. 150, 074501 (2019)]. No adjustable parameters are used to make this comparison. That is, all the parameters of the theory are evaluated directly for the model system, yielding a predicted FES, which we then compare to the FES obtained from simulations. We find that the theoretical FES successfully predicts the numerically evaluated FES over a range of thermodynamic conditions that spans distinct regimes of behavior associated with TSN. All the qualitative features of the FES are captured by the theory, and the quantitative comparison is also very good. Our results demonstrate that Iwamatsu's extension of classical nucleation theory provides an excellent framework for understanding the thermodynamics of TSN.

Experimental observation of the liquid-liquid transition in bulk supercooled water under pressure.

We prepared bulk samples of supercooled liquid water under pressure by isochoric heating of high-density amorphous ice to temperatures of 205 ± 10 kelvin, using an infrared femtosecond laser. Because the sample density is preserved during the ultrafast heating, we could estimate an initial internal pressure of 2.5 to 3.5 kilobar in the high-density liquid phase. After heating, the sample expanded rapidly, and we captured the resulting decompression process with femtosecond x-ray laser pulses at different pump-probe delay times. A discontinuous structural change occurred in which low-density liquid domains appeared and grew on time scales between 20 nanoseconds to 3 microseconds, whereas crystallization occurs on time scales of 3 to 50 microseconds. The dynamics of the two processes being separated by more than one order of magnitude provides support for a liquid-liquid transition in bulk supercooled water.

Surface tension of supercooled water nanodroplets from computer simulations.

We estimate the liquid-vapor surface tension from simulations of TIP4P/2005 water nanodroplets of size N = 100 to 2880 molecules over a temperature T range of 180 K-300 K. We compute the planar surface tension γp, the curvature-dependent surface tension γs, and the Tolman length δ, via two approaches, one based on the pressure tensor (the "mechanical route") and the other on the Laplace pressure (the "thermodynamic route"). We find that these two routes give different results for γp, γs, and δ although in all cases, we find that δ ≥ 0 and is independent of T. Nonetheless, the T dependence of γp is consistent between the two routes and with that of Vega and de Miguel [J. Chem. Phys. 126, 154707 (2007)] down to the crossing of the Widom line at 230 K for ambient pressure. Below 230 K, γp rises more rapidly on cooling than predicted from behavior for T ≥ 300 K. We show that the increase in γp at low T is correlated with the emergence of a well-structured random tetrahedral network in our nanodroplet cores and thus that the surface tension can be used as a probe to detect behavior associated with the proposed liquid-liquid phase transition in supercooled water.

The stability-limit conjecture revisited.

The stability-limit conjecture (SLC) proposes that the liquid spinodal of water returns to positive pressure in the supercooled region and that the apparent divergence of water's thermodynamic response functions as temperature decreases are explained by the approach to this re-entrant spinodal. Subsequently, it has been argued that the predictions of the SLC are inconsistent with general thermodynamic principles. Here, we reconsider the thermodynamic viability of the SLC by examining a model equation of state for water which was first studied to clarify the relationship of the SLC to the proposed liquid-liquid phase transition in supercooled water. By demonstrating that a binodal may terminate on a spinodal at a point that is not a critical point, we show that the SLC is thermodynamically permissible in a system that has both a liquid-gas and a liquid-liquid phase transition. We also describe and clarify other unusual thermodynamic behavior that may arise in such a system, particularly that associated with the so-called "critical-point-free" scenario for a liquid-liquid phase transition, which may apply to the case of liquid Si.

State variables for glasses: The case of amorphous ice.

Glasses are out-of-equilibrium systems whose state cannot be uniquely defined by the usual set of equilibrium state variables. Here, we seek to identify an expanded set of variables that uniquely define the state of a glass. The potential energy landscape (PEL) formalism is a useful approach within statistical mechanics to describe supercooled liquids and glasses. We use the PEL formalism and computer simulations to study the transformations between low-density amorphous ice (LDA) and high-density amorphous ice (HDA). We employ the ST2 water model, which exhibits an abrupt first-order-like phase transition from LDA to HDA, similar to that observed in experiments. We prepare a number of distinct samples of both LDA and HDA that have completely different preparation histories. We then study the evolution of these LDA and HDA samples during compression and decompression at temperatures sufficiently low that annealing is absent and also during heating. We find that the evolution of each glass sample, during compression/decompression or heating, is uniquely determined by six macroscopic properties of the initial glass sample. These six quantities consist of three conventional thermodynamic state variables, the number of molecules N, the system volume V, and the temperature T, as well as three properties of the PEL, the inherent structure (IS) energy EIS, the IS pressure PIS, and the average curvature of the PEL at the IS SIS. In other words, (N,V,T,EIS,PIS,SIS) are state variables that define the glass state in the case of amorphous ice. An interpretation of our results in terms of the PEL formalism is provided. Since the behavior of water in the glassy state is more complex than for most substances, our results suggest that these six state variables may be applicable to amorphous solids in general and that there may be situations in which fewer than six variables would be sufficient to define the state of a glass.

Phase transitions in fluctuations and their role in two-step nucleation.

We consider the thermodynamic behavior of local fluctuations occurring in a stable or metastable bulk phase. For a system with three or more phases, we present a simple analysis based on classical nucleation theory that predicts thermodynamic conditions at which small fluctuations resemble the phase having the lowest surface tension with the surrounding bulk phase, even if this phase does not have a lower chemical potential. We also identify the conditions at which a fluctuation may convert to a different phase as its size increases, referred to here as a "fluctuation phase transition" (FPT). We demonstrate these phenomena in simulations of a two dimensional lattice model by evaluating the free energy surface that describes the thermodynamic properties of a fluctuation as a function of its size and phase composition. We show that a FPT can occur in the fluctuations of either a stable or metastable bulk phase and that the transition is first-order. We also find that the FPT is bracketed by well-defined spinodals, which place limits on the size of fluctuations of distinct phases. Furthermore, when the FPT occurs in a metastable bulk phase, we show that the superposition of the FPT on the nucleation process results in two-step nucleation (TSN). We identify distinct regimes of TSN based on the nucleation pathway in the free energy surface and correlate these regimes to the phase diagram of the bulk system. Our results clarify the origin of TSN and elucidate a wide variety of phenomena associated with TSN, including the Ostwald step rule.

Advances in Computational Studies of the Liquid-Liquid Transition in Water and Water-Like Models.

There has been uninterrupted interest in supercooled water ever since the pioneering experiments of Speedy and Angell revealed sharp increases in this substance's response functions upon supercooling. One intriguing hypothesis that was formulated to explain this behavior involves the existence of a metastable liquid-liquid transition (LLT) at deeply supercooled conditions. The preponderance of experimental evidence is consistent with this hypothesis, although no definitive proof exists to date. Computational studies have played an important role in this area, because ice nucleation can in principle be controlled in silico. It has been claimed, controversially, that the LLT is a misinterpreted liquid-solid transition in all models of water. Recent studies disprove this viewpoint by providing unambiguous counter-examples of distinct liquid-liquid and liquid-crystal transitions in tetrahedral models. In one, state-of-the-art sampling methods were used to compute the free energy surface of a molecular model of water and revealed the existence of two liquid phases in metastable equilibrium with each other and a stable crystal phase, at the same, deeply supercooled thermodynamic conditions. Further studies showed that, by tuning the potential parameters of a model tetrahedral system, it is possible to make the LLT evolve continuously from metastability to being thermodynamically stable with respect to crystallization. Most recently, it has been shown that the simulation code used to challenge the hypothesis of an LLT contains conceptual errors that invalidate the results on which the challenge was based, definitively resolving the controversy. The debate has vastly expanded the range of fundamental questions being pursued about phase transitions in metastable systems and ushered the use of increasingly sophisticated computational methods to explore the possible existence of LLTs in model systems.

Thermodynamic and structural anomalies of water nanodroplets.

Liquid water nanodroplets are important in earth's climate, and are valuable for studying supercooled water because they resist crystallisation well below the bulk freezing temperature. Bulk liquid water has well-known thermodynamic anomalies, such as a density maximum, and when supercooled is hypothesised to exhibit a liquid-liquid phase transition (LLPT) at elevated pressure. However, it is not known how these bulk anomalies might manifest themselves in nanodroplets. Here we show, using simulations of the TIP4P/2005 water model, that bulk anomalies occur in nanodroplets as small as 360 molecules. We also show that the Laplace pressure inside small droplets reaches 220 MPa at 180 K, conditions close to the LLPT of TIP4P/2005. While the density and pressure inside nanodroplets coincide with bulk values at moderate supercooling, we show that deviations emerge at lower temperature, as well as significant radial density gradients, which arise from and signal the approach to the LLPT.

Evaluating the Laplace pressure of water nanodroplets from simulations.

We calculate the components of the microscopic pressure tensor as a function of radial distance r from the centre of a spherical water droplet, modelled using the TIP4P/2005 potential. To do so, we modify a coarse-graining method for calculating the microscopic pressure (Ikeshoji et al 2003 Mol. Simul. 29 101) in order to apply it to a rigid molecular model of water. As test cases, we study nanodroplets ranging in size from 776 to 2880 molecules at 220 K. Beneath a surface region comprising approximately two molecular layers, the pressure tensor becomes approximately isotropic and constant with r. We find that the dependence of the pressure on droplet radius is that expected from the Young-Laplace equation, despite the small size of the droplets.

"Swarm relaxation": Equilibrating a large ensemble of computer simulations⋆.

It is common practice in molecular dynamics and Monte Carlo computer simulations to run multiple, separately-initialized simulations in order to improve the sampling of independent microstates. Here we examine the utility of an extreme case of this strategy, in which we run a large ensemble of M independent simulations (a "swarm"), each of which is relaxed to equilibrium. We show that if M is of order [Formula: see text], we can monitor the swarm's relaxation to equilibrium, and confirm its attainment, within [Formula: see text], where [Formula: see text] is the equilibrium relaxation time. As soon as a swarm of this size attains equilibrium, the ensemble of M final microstates from each run is sufficient for the evaluation of most equilibrium properties without further sampling. This approach dramatically reduces the wall-clock time required, compared to a single long simulation, by a factor of several hundred, at the cost of an increase in the total computational effort by a small factor. It is also well suited to modern computing systems having thousands of processors, and is a viable strategy for simulation studies that need to produce high-precision results in a minimum of wall-clock time. We present results obtained by applying this approach to several test cases.

Influence of sample preparation on the transformation of low-density to high-density amorphous ice: An explanation based on the potential energy landscape.

Experiments and computer simulations of the transformations of amorphous ices display different behaviors depending on sample preparation methods and on the rates of change of temperature and pressure to which samples are subjected. In addition to these factors, simulation results also depend strongly on the chosen water model. Using computer simulations of the ST2 water model, we study how the sharpness of the compression-induced transition from low-density amorphous ice (LDA) to high-density amorphous ice (HDA) is influenced by the preparation of LDA. By studying LDA samples prepared using widely different procedures, we find that the sharpness of the LDA-to-HDA transformation is correlated with the depth of the initial LDA sample in the potential energy landscape (PEL), as characterized by the inherent structure energy. Our results show that the complex phenomenology of the amorphous ices reported in experiments and computer simulations can be understood and predicted in a unified way from knowledge of the PEL of the system.

Potential energy landscape of the apparent first-order phase transition between low-density and high-density amorphous ice.

The potential energy landscape (PEL) formalism is a valuable approach within statistical mechanics to describe supercooled liquids and glasses. Here we use the PEL formalism and computer simulations to study the pressure-induced transformations between low-density amorphous ice (LDA) and high-density amorphous ice (HDA) at different temperatures. We employ the ST2 water model for which the LDA-HDA transformations are remarkably sharp, similar to what is observed in experiments, and reminiscent of a first-order phase transition. Our results are consistent with the view that LDA and HDA configurations are associated with two distinct regions (megabasins) of the PEL that are separated by a potential energy barrier. At higher temperature, we find that low-density liquid (LDL) configurations are located in the same megabasin as LDA, and that high-density liquid (HDL) configurations are located in the same megabasin as HDA. We show that the pressure-induced LDL-HDL and LDA-HDA transformations occur along paths that interconnect these two megabasins, but that the path followed by the liquid is different from the path followed by the amorphous solid. At higher pressure, we also study the liquid-to-ice-VII first-order phase transition, and find that the behavior of the PEL properties across this transition is qualitatively similar to the changes found during the LDA-HDA transformation. This similarity supports the interpretation that the LDA-HDA transformation is a first-order phase transition between out-of-equilibrium states. Finally, we compare the PEL properties explored during the LDA-HDA transformations in ST2 water with those reported previously for SPC/E water, for which the LDA-HDA transformations are rather smooth. This comparison illuminates the previous work showing that, at accessible computer times scales, a liquid-liquid phase transition occurs in the case of ST2 water, but not for SPC/E water.

Free energy of formation of small ice nuclei near the Widom line in simulations of supercooled water.

The ST2 interaction potential has been used in a large number of simulation studies to explore the possibility of a liquid-liquid phase transition (LLPT) in supercooled water. Using umbrella sampling Monte Carlo simulations of ST2 water, we evaluate the free energy of formation of small ice nuclei in the supercooled liquid in the vicinity of the Widom line, the region above the critical temperature of the LLPT where a number of thermodynamic anomalies occur. Our results show that in this region there is a substantial free-energy cost for the formation of small ice nuclei, demonstrating that the thermodynamic anomalies associated with the Widom line in ST2 water occur in a well-defined metastable liquid phase. On passing through the Widom line, we identify changes in the free energy to form small ice nuclei that illustrate how the thermodynamic anomalies associated with the LLPT may influence the ice nucleation process.

Two-state thermodynamics of the ST2 model for supercooled water.

Thermodynamic properties of the ST2 model for supercooled liquid water exhibit anomalies similar to those observed in real water. A possible explanation of these anomalies is the existence of a metastable, liquid-liquid transition terminated by a critical point. This phenomenon, whose possible existence in real water is the subject of much current experimental work, has been unambiguously demonstrated for this particular model by most recent simulations. In this work, we reproduce the anomalies of two versions of the ST2 model with an equation of state describing water as a non-ideal "mixture" of two different types of local molecular order. We show that the liquid-liquid transition in the ST2 water is energy-driven. This is in contrast to another popular model, mW, in which non-ideality in mixing of two alternative local molecular orders is entropy-driven, and is not sufficiently strong to induce a liquid-liquid transition.

Heterogeneous nucleation in the low-barrier regime.

In simulations of the two-dimensional Ising model, we examine heterogeneous nucleation induced by a small impurity consisting of a line of l fixed spins. As l increases, we identify a limit of stability beyond which the metastable phase is not defined. We evaluate the free energy barrier for nucleation of the stable phase and show that, contrary to expectation, the barrier does not vanish on approach to the limit of stability. We also demonstrate that our values for the height of the barrier yield predictions for the nucleation time (from transition state theory) and the size of the critical cluster (from the nucleation theorem) that are in excellent agreement with direct measurements, even near the limit of stability.

Free energy surface of ST2 water near the liquid-liquid phase transition.

We carry out umbrella sampling Monte Carlo simulations to evaluate the free energy surface of the ST2 model of water as a function of two order parameters, the density and a bond-orientational order parameter. We approximate the long-range electrostatic interactions of the ST2 model using the reaction-field method. We focus on state points in the vicinity of the liquid-liquid critical point proposed for this model in earlier work. At temperatures below the predicted critical temperature we find two basins in the free energy surface, both of which have liquid-like bond orientational order, but differing in density. The pressure and temperature dependence of the shape of the free energy surface is consistent with the assignment of these two basins to the distinct low density and high density liquid phases previously predicted to occur in ST2 water.

Study of the ST2 model of water close to the liquid-liquid critical point.

We perform successive umbrella sampling grand canonical Monte Carlo computer simulations of the original ST2 model of water in the vicinity of the proposed liquid-liquid critical point, at temperatures above and below the critical temperature. Our results support the previous work of Y. Liu, A. Z. Panagiotopoulos and P. G. Debenedetti [J. Chem. Phys., 2009, 131, 104508], who provided evidence for the existence and location of the critical point for ST2 using the Ewald method to evaluate the long-range forces. Our results therefore demonstrate the robustness of the evidence for critical behavior with respect to the treatment of the electrostatic interactions. In addition, we verify that the liquid is equilibrated at all densities on the Monte Carlo time scale of our simulations, and also that there is no indication of crystal formation during our runs. These findings demonstrate that the processes of liquid-state relaxation and crystal nucleation are well separated in time. Therefore, the bimodal shape of the density of states, and hence the critical point itself, is a purely liquid-state phenomenon that is distinct from the crystal-liquid transition.

Dynamical behavior near a liquid-liquid phase transition in simulations of supercooled water.

We examine the behavior of the diffusion coefficient of the ST2 model of water over a broad region of the phase diagram via molecular dynamics simulations. The ST2 model has an accessible liquid-liquid transition between low-density and high-density phases, making the model an ideal candidate to explore the impacts of the liquid-liquid transition on dynamics. We locate characteristic dynamical loci in the phase diagram and compare them with the previously investigated thermodynamic loci. The low-density liquid phase shows a crossover from non-Arrhenius to Arrhenius behavior, signaling the onset of a crossover from fragile-to-strong behavior. We explain this crossover in terms of the asymptotic approach of the low-density liquid to a random tetrahedral network and show that the temperature dependence of the diffusion coefficient over a wide temperature range can be simply related to the concentration of defects in the network. Our findings thus confirm that the low-density phase of ST2 water is a well-defined metastable liquid.