ST2 Model Research Articles

Spatial calibration of WAVEWATCH III model against satellite observations using different input and dissipation parameterizations in the Black Sea

The quality of the third generation wave model WAVEWATCH III, referred WW3, using the different source term packages is investigated in the context of wave growth and dissipation in a semi-closed sea, Black Sea. Accuracy in wave height hindcast is tested for five different source term parameterizations ST1, ST2, ST3, ST4, and ST6 in WW3 against the 2020-year multi-mission satellite observations over the entire sea. The aims are to determine the most appropriate formulation and optimum tuneable coefficients for wind waves modelling in the Black Sea. For this purpose, significant wave heights (Hm0) are produced using five different source term packages implemented in the WW3 model. A sensitivity test was applied to define the tuneable parameters that contributes to the accuracy of the models, and subsequently the coefficient values of each parameters was turned to calibrate the WW3 models. The tuneable parameter allowing to obtain the most accurate result in each package are defined for each source term package. Comparing between model results and satellite observations, we found that the default WW3 model with all source terms as well as the calibrated ST2 model underestimate Hm0 in the entire domain and mainly in the eastern Black Sea region and are not suitable physical source term for the Black Sea. In the Black sea, the calibrated ST1, ST3 and ST4 forced with ERA5 wind are recommended for long term wave climate simulation, and the calibrated ST6 is recommended for extreme wave simulation.

Signatures of sluggish dynamics and local structural ordering during ice nucleation.

We investigate the microscopic pathway of spontaneous crystallization in the ST2 model of water under deeply supercooled conditions via unbiased classical molecular dynamics simulations. After quenching below the liquid-liquid critical point, the ST2 model spontaneously separates into low-density liquid (LDL) and high-density liquid phases, respectively. The LDL phase, which is characterized by lower molecular mobility and enhanced structural order, fosters the formation of a sub-critical ice nucleus that, after a stabilization time, develops into the critical nucleus and grows. Polymorphic selection coincides with the development of the sub-critical nucleus and favors the formation of cubic (Ic) over hexagonal (Ih) ice. We rationalize polymorphic selection in terms of geometric arguments based on differences in the symmetry of second neighbor shells of ice Ic and Ih, which are posited to favor formation of the former. The rapidly growing critical nucleus absorbs both Ic and Ih crystallites dispersed in the liquid phase, a crystal with stacking faults. Our results are consistent with, and expand upon, recent observations of non-classical nucleation pathways in several systems.

Thermodynamic metric geometry of the two-state ST2 model for supercooled water

Liquid water has anomalous liquid properties, such as its density maximum at 4 °C. An attempt at theoretical explanation proposes a liquid-liquid phase transition line in the supercooled liquid state, with coexisting low-density liquid (LDL) and high-density liquid (HDL) states. This line terminates at a critical point. It is assumed that the LDL state possesses mesoscopic tetrahedral structures that give it solidlike properties, while the HDL is a regular random liquid. But the short-lived nature of these solidlike structures makes them difficult to detect directly. We take a thermodynamic approach instead and calculate the thermodynamic Ricci curvature scalar R in the metastable liquid regime. It is believed that solidlike structures signal their presence thermodynamically by a positive sign for R, with a negative sign typically present in less organized fluid states. Using thermodynamic data from ST2 computer simulations fit to a mean field (MF) two state equation of state, we find significant regimes of positive R in the LDL state, supporting the proposal of solidlike structures in liquid water. In addition, we review the theory, compute critical exponents, demonstrate the large reach of the MF critical regime, and calculate the Widom line using R.

Structure of supercooled water: Polarizable BK3 model versus non-polarizable models

The polarizable BK3 model of water was simulated, using both Monte Carlo and molecular dynamics methods, along isobars p = 1 and 1000 bar for temperatures ranging from supercritical down to 228.15 K with the focus on details of its structure and changes with temperature. Since overwhelming majority of results for supercooled model water have been obtained for non-polarizable models, in most cases the ST2 one, we also simulated at corresponding conditions this model along with another, and qualitatively different non-polarizable model, TIP4P/Ice for comparison. In addition to commonly used various order parameters to characterize the structure, and a detailed analysis of the spatial distribution function over the first three coordination shells, we also employ a geometric construction based on the Delaunay tessellation, the method independent of any a priori defined quantity. All these analyses aim at an examination to what extent qualitative differences in the models (force fields) affect the observed structure. It is found that in most cases the polarizable BK3 model and TIP4P/Ice water yield very similar structural properties. On the contrary, ST2 water exhibits, in majority cases, even qualitatively different properties and these findings are discussed in detail. The obtained results suggest that any conclusion on the behavior of supercooled water based on the ST2 model should be taken, at least, with caution.

Anomalous scattering in supercooled ST2 water

ABSTRACTLarge-scale molecular dynamics (MD) simulations of systems containing up to 256,000 molecules were performed to investigate the scattering behaviour of the ST2 water model at deeply supercooled conditions. The simulations reveal that ST2 exhibits anomalous scattering, reminiscent of that observed in experiment, which is characterised by an increase in the static structure factor at low wavenumbers. This unusual behaviour in ST2 is linked with coupled fluctuations in density and local tetrahedral order in the liquid. The Ornstein–Zernike correlation length estimated from the anomalous scattering component exhibits power-law growth upon cooling, consistent with the existence of a liquid–liquid critical point (LLCP) in the ST2 model at ca. 245 K. Further, spontaneous liquid–liquid phase separation is observed upon thermally quenching a large system with 256,000 water molecules below the predicted critical temperature into the two-phase region. The large-scale MD simulations therefore confirm the existence of a metastable liquid–liquid phase transition in ST2 and support findings from previous computational studies performed using smaller systems containing only a few hundred molecules. We anticipate that our analysis may prove useful in interpreting recent scattering experiments that have been performed to search for an LLCP in deeply supercooled water.

Anomalous Features in the Potential Energy Landscape of a Waterlike Monatomic Model with Liquid and Glass Polymorphism.

We study the potential energy landscape (PEL) of a waterlike monatomic liquid that exhibits a liquid-liquid phase transition (LLPT) and glass-glass transformation (GGT). We identify two anomalous features of the PEL that give origin to both phenomena. Specifically, during the pressure-induced LLPT and GGT, (i)the inherent structures (IS) energy becomes a concave function of volume, and (ii)the IS pressure exhibits a van der Waals-like loop. We argue that features (i) and (ii)imply that the GGT is a (nonequilibrium) first-order phase transition, analogous to the LLPT. Interestingly, contrary to the case of the classical ST2 model for water, (a)we do not find two separate PEL megabasins (one for the low-density glass and liquid, and another for the high-density glass and liquid), and (b) features (i)-(ii) persist at temperatures well above the LLPT.

A free energy study of the liquid-liquid phase transition of the Jagla two-scale potential

A fundamental understanding of pure-component liquid-liquid phase separation in network-forming fluids remains an open challenge. While considerable progress has been recently made in demonstrating the existence of such a phase transition in some models via rigorous free energy calculations, it remains unclear what aspects of a model are sufficient, necessary, and/or prohibited in order for it to exhibit a liquid-liquid phase transition (LLPT). Among the simplest models capable of producing water-like anomalies is the spherically-symmetry two-scale Jagla potential, for which an LLPT has been identified via equation of state calculations. In this work, we perform rigorous free energy calculations to demonstrate the existence of an LLPT in the Jagla model. We also utilize finite-size scaling analysis to calculate the surface tension associated with the LLPT. In addition to the thermodynamics of the model, we investigate the relaxation times for density and bond-orientational order in both liquid phases and show that, contrary to assertions in the literature, the characteristic relaxation time of bond-orientational order is not orders of magnitude slower than that of density. To the contrary, we actually identify conditions for which density is the slowly relaxing order parameter. In addition to the original parameterization of the Jagla model, we provide in the “Appendix” preliminary free energy surface calculations for select parameterizations of the generalized family of Jagla potentials spanning from the original (anomalous, water-like) Jagla model to the Lennard-Jones model. These calculations indicate that, as the parameterization moves towards the Lennard-Jones limit, the LLPT disappears within the range of parameters explored. Throughout the paper, we compare our results for the Jagla model with those found in the literature for the ST2 model of water in order to emphasize key similarities and differences between two models that exhibit pure-component liquid-liquid phase separation. Synopsis: We perform rigorous free energy calculations to demonstrate the existence of a liquid-liquid phase transition (LLPT) in the spherically-symmetry two-scale Jagla potential. We also calculate the surface tension associated with the LLPT and investigate the relaxation times of density and bond-orientational order to gain further insight into the LLPT phenomenon.

Pressure-induced transformations in glassy water: A computer simulation study using the TIP4P/2005 model.

We study the pressure-induced transformations between low-density amorphous (LDA) and high-density amorphous (HDA) ice by performing out-of-equilibrium molecular dynamics (MD) simulations. We employ the TIP4P/2005 water model and show that this model reproduces qualitatively the LDA-HDA transformations observed experimentally. Specifically, the TIP4P/2005 model reproduces remarkably well the (i) structure (OO, OH, and HH radial distribution functions) and (ii) densities of LDA and HDA at P = 0.1 MPa and T = 80 K, as well as (iii) the qualitative behavior of ρ(P) during compression-induced LDA-to-HDA and decompression-induced HDA-to-LDA transformations. At the rates explored, the HDA-to-LDA transformation is less pronounced than in experiments. By studying the LDA-HDA transformations for a broad range of compression/decompression temperatures, we construct a "P-T phase diagram" for glassy water that is consistent with experiments and remarkably similar to that reported previously for ST2 water. This phase diagram is not inconsistent with the possibility of TIP4P/2005 water exhibiting a liquid-liquid phase transition at low temperatures. A comparison with previous MD simulation studies of SPC/E and ST2 water as well as experiments indicates that, overall, the TIP4P/2005 model performs better than the SPC/E and ST2 models. The effects of cooling and compression rates as well as aging on our MD simulations results are also discussed. The MD results are qualitatively robust under variations of cooling/compression rates (accessible in simulations) and are not affected by aging the hyperquenched glass for at least 1 μs. A byproduct of this work is the calculation of TIP4P/2005 water's diffusion coefficient D(T) at P = 0.1 MPa. It is found that, for T ≥ 210 K, D(T) ≈ (T - T(MCT))(-γ) as predicted by mode coupling theory and in agreement with experiments. For TIP4P/2005 water, T(MCT) = 209 K and γ = 2.14, very close to the corresponding experimental values T(MCT) = 221 K and γ = 2.2.

Tuning the Liquid-Liquid Transition by Modulating the Hydrogen-Bond Angular Flexibility in a Model for Water.

We propose a simple extension of the well known ST2 model for water [F. H. Stillinger and A. Rahman, J. Chem. Phys. 60, 1545 (1974)] that allows for a continuous modification of the hydrogen-bond angular flexibility. We show that the bond flexibility affects the relative thermodynamic stability of the liquid and of the hexagonal (or cubic) ice. On increasing the flexibility, the liquid-liquid critical point, which in the original ST2 model is located in the no-man's land (i.e., the region where ice is the thermodynamically stable phase) progressively moves to a temperature where the liquid is more stable than ice. Our study definitively proves that the liquid-liquid transition in the ST2 model is a genuine phenomenon, of high relevance in all tetrahedral network-forming liquids, including water.

Phase diagram of the ST2 model of water

We evaluate the free energy of the fluid and crystal phases for the ST2 potential with reaction field corrections for the long-range interactions. We estimate the phase coexistence boundaries in the temperature–pressure plane, as well as the gas–liquid critical point and gas–liquid coexistence conditions. Our study frames the location of the previously identified liquid–liquid critical point relative to the crystalline phase boundaries, and opens the way for exploring crystal nucleation in a model where the metastable liquid–liquid critical point is computationally accessible.

Metastable liquid–liquid transition in a molecular model of water

Liquid water's isothermal compressibility and isobaric heat capacity, and the magnitude of its thermal expansion coefficient, increase sharply on cooling below the equilibrium freezing point. Many experimental, theoretical and computational studies have sought to understand the molecular origin and implications of this anomalous behaviour. Of the different theoretical scenarios put forward, one posits the existence of a first-order phase transition that involves two forms of liquid water and terminates at a critical point located at deeply supercooled conditions. Some experimental evidence is consistent with this hypothesis, but no definitive proof of a liquid-liquid transition in water has been obtained to date: rapid ice crystallization has so far prevented decisive measurements on deeply supercooled water, although this challenge has been overcome recently. Computer simulations are therefore crucial for exploring water's structure and behaviour in this regime, and have shown that some water models exhibit liquid-liquid transitions and others do not. However, recent work has argued that the liquid-liquid transition has been mistakenly interpreted, and is in fact a liquid-crystal transition in all atomistic models of water. Here we show, by studying the liquid-liquid transition in the ST2 model of water with the use of six advanced sampling methods to compute the free-energy surface, that two metastable liquid phases and a stable crystal phase exist at the same deeply supercooled thermodynamic condition, and that the transition between the two liquids satisfies the thermodynamic criteria of a first-order transition. We follow the rearrangement of water's coordination shell and topological ring structure along a thermodynamically reversible path from the low-density liquid to cubic ice. We also show that the system fluctuates freely between the two liquid phases rather than crystallizing. These findings provide unambiguous evidence for a liquid-liquid transition in the ST2 model of water, and point to the separation of time scales between crystallization and relaxation as being crucial for enabling it.

Heating-induced glass-glass and glass-liquid transformations in computer simulations of water

Water exists in at least two families of glassy states, broadly categorized as the low-density (LDA) and high-density amorphous ice (HDA). Remarkably, LDA and HDA can be reversibly interconverted via appropriate thermodynamic paths, such as isothermal compression and isobaric heating, exhibiting first-order-like phase transitions. We perform out-of-equilibrium molecular dynamics simulations of glassy water using the ST2 model to study the evolution of LDA and HDA upon isobaric heating. Depending on pressure, glass-to-glass, glass-to-crystal, glass-to-vapor, as well as glass-to-liquid transformations are found. Specifically, heating LDA results in the following transformations, with increasing heating pressures: (i) LDA-to-vapor (sublimation), (ii) LDA-to-liquid (glass transition), (iii) LDA-to-HDA-to-liquid, (iv) LDA-to-HDA-to-liquid-to-crystal, and (v) LDA-to-HDA-to-crystal. Similarly, heating HDA results in the following transformations, with decreasing heating pressures: (a) HDA-to-crystal, (b) HDA-to-liquid-to-crystal, (c) HDA-to-liquid (glass transition), (d) HDA-to-LDA-to-liquid, and (e) HDA-to-LDA-to-vapor. A more complex sequence may be possible using lower heating rates. For each of these transformations, we determine the corresponding transformation temperature as function of pressure, and provide a P-T "phase diagram" for glassy water based on isobaric heating. Our results for isobaric heating dovetail with the LDA-HDA transformations reported for ST2 glassy water based on isothermal compression/decompression processes [Chiu et al., J. Chem. Phys. 139, 184504 (2013)]. The resulting phase diagram is consistent with the liquid-liquid phase transition hypothesis. At the same time, the glass phase diagram is sensitive to sample preparation, such as heating or compression rates. Interestingly, at least for the rates explored, our results suggest that the LDA-to-liquid (HDA-to-liquid) and LDA-to-HDA (HDA-to-LDA) transformation lines on heating are related, both being associated with the limit of kinetic stability of LDA (HDA).

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.

Pressure-induced transformations in computer simulations of glassy water

Glassy water occurs in at least two broad categories: low-density amorphous (LDA) and high-density amorphous (HDA) solid water. We perform out-of-equilibrium molecular dynamics simulations to study the transformations of glassy water using the ST2 model. Specifically, we study the known (i) compression-induced LDA-to-HDA, (ii) decompression-induced HDA-to-LDA, and (iii) compression-induced hexagonal ice-to-HDA transformations. We study each transformation for a broad range of compression/decompression temperatures, enabling us to construct a "P-T phase diagram" for glassy water. The resulting phase diagram shows the same qualitative features reported from experiments. While many simulations have probed the liquid-state phase behavior, comparatively little work has examined the transitions of glassy water. We examine how the glass transformations relate to the (first-order) liquid-liquid phase transition previously reported for this model. Specifically, our results support the hypothesis that the liquid-liquid spinodal lines, between a low-density and high-density liquid, are extensions of the LDA-HDA transformation lines in the limit of slow compression. Extending decompression runs to negative pressures, we locate the sublimation lines for both LDA and hyperquenched glassy water (HGW), and find that HGW is relatively more stable to the vapor. Additionally, we observe spontaneous crystallization of HDA at high pressure to ice VII. Experiments have also seen crystallization of HDA, but to ice XII. Finally, we contrast the structure of LDA and HDA for the ST2 model with experiments. We find that while the radial distribution functions (RDFs) of LDA are similar to those observed in experiments, considerable differences exist between the HDA RDFs of ST2 water and experiment. The differences in HDA structure, as well as the formation of ice VII (a tetrahedral crystal), are a consequence of ST2 overemphasizing the tetrahedral character of water.

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.

Liquid-liquid transition in ST2 water

We use the weighted histogram analysis method [S. Kumar, D. Bouzida, R. H. Swendsen, P. A. Kollman, and J. M. Rosenberg, J. Comput. Chem. 13, 1011 (1992)] to calculate the free energy surface of the ST2 model of water as a function of density and bond-orientational order. We perform our calculations at deeply supercooled conditions (T = 228.6 K, P = 2.2 kbar; T = 235 K, P = 2.2 kbar) and focus our attention on the region of bond-orientational order that is relevant to disordered phases. We find a first-order transition between a low-density liquid (LDL, ρ ≈ 0.9 g/cc) and a high-density liquid (HDL, ρ ≈ 1.15 g/cc), confirming our earlier sampling of the free energy surface of this model as a function of density [Y. Liu, A. Z. Panagiotopoulos, and P. G. Debenedetti, J. Chem. Phys. 131, 104508 (2009)]. We demonstrate the disappearance of the LDL basin at high pressure and of the HDL basin at low pressure, in agreement with independent simulations of the system's equation of state. Consistency between directly computed and reweighted free energies, as well as between free energy surfaces computed using different thermodynamic starting conditions, confirms proper equilibrium sampling. Diffusion and structural relaxation calculations demonstrate that equilibration of the LDL phase, which exhibits slow dynamics, is attained in the course of the simulations. Repeated flipping between the LDL and HDL phases in the course of long molecular dynamics runs provides further evidence of a phase transition. We use the Ewald summation with vacuum boundary conditions to calculate long-ranged Coulombic interactions and show that conducting boundary conditions lead to unphysical behavior at low temperatures.

Nanoscale Dynamics of Phase Flipping in Water near its Hypothesized Liquid-Liquid Critical Point

One hypothesized explanation for water's anomalies imagines the existence of a liquid-liquid (LL) phase transition line separating two liquid phases and terminating at a LL critical point. We simulate the classic ST2 model of water for times up to 1000 ns and system size up to N = 729. We find that for state points near the LL transition line, the entire system flips rapidly between liquid states of high and low density. Our finite-size scaling analysis accurately locates both the LL transition line and its associated LL critical point. We test the stability of the two liquids with respect to the crystal and find that of the 350 systems simulated, only 3 of them crystallize and these 3 for the relatively small system size N = 343 while for all other simulations the incipient crystallites vanish on a time scales smaller than ≈ 100 ns.

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.

ATTRACTOR FLOWS IN st2 BLACK HOLES

Following the same treatment of Bellucci et al. we obtain, the hitherto unknown general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing and vanishing central charge Z for the so-called st2 model, the minimal rank-two [Formula: see text] symmetric supergravity in d = 4 space-time dimensions. We also make useful comparisons with results that already exist in literature, and introduce the fake supergravity (first-order) formalism to be used in our analysis. An analysis of the BPS bound all along the non-BPS attractor flows and of the marginal stability of corresponding D-brane charge configurations has also been presented.

Two-center black holes duality-invariants for stu model and its lower-rank descendants

We classify 2-center extremal black hole charge configurations through duality-invariant homogeneous polynomials, which are the generalization of the unique invariant quartic polynomial for single-center black holes based on homogeneous symmetric cubic special Kä hler geometries. A crucial role is played by a horizontal $SL(p,\mathbb {R})$SL(p,R) symmetry group, which classifies invariants for p-center black holes. For p = 2, a (spin 2) quintet of quartic invariants emerge. We provide the minimal set of independent invariants for the rank-3 $\mathcal {N}=2$N=2, d = 4 stu model, and for its lower-rank descendants, namely, the rank-2 st2 and rank-1 t3 models; these models, respectively, exhibit seven, six, and five independent invariants. We also derive the polynomial relations among these and other duality invariants. In particular, the symplectic product of two charge vectors is not independent from the quartic quintet in the t3 model, but rather it satisfies a degree-16 relation, corresponding to a quartic equation for the square of the symplectic product itself.