Abstract
An accurate and computationally efficient molecular level description of mesoscopic behavior of icewater systems remains a major challenge. Here, we introduce a set of machinelearned coarsegrained (CG) models (MLBOP, MLBOP_{dih}, and MLmW) that accurately describe the structure and thermodynamic anomalies of both water and ice at mesoscopic scales, all at two orders of magnitude cheaper computational cost than existing atomistic models. In a significant departure from conventional forcefield fitting, we use a multilevel evolutionary strategy that trains CG models against not just energetics from firstprinciples and experiments but also temperaturedependent properties inferred from onthefly molecular dynamics (~ 10’s of milliseconds of overall trajectories). Our ML BOP models predict both the correct experimental melting point of ice and the temperature of maximum density of liquid water that remained elusive todate. Our ML workflow navigates efficiently through the highdimensional parameter space to even improve upon existing highquality CG models (e.g. mW model).
Introduction
Ice nucleation and grain growth are ubiquitous phenomena. Ice nuclei, when formed, are nanoscopic^{1}—critical sizes start from tens of molecules—and subsequently consolidate into larger grains at the mesoscopic scale^{2}. A molecular level picture of phase transformations in water, especially at mesoscopic scales, is most desirable but remains inaccessible to fully atomistic simulations^{3}. The underlying phase transitions and dynamical processes in supercooled mesoscale systems are often inaccessible due to system size and timescale limitations, which are further compounded by their sluggish kinetics. While exascale computers may cope with such mesoscopic length scales, time scale challenges will remain (Supplementary Figure 1). It is important to have a water model that accurately captures the melting point, liquid and solid densities, as well as other thermodynamic and transport properties at modest computational cost. Numerous atomistic^{4,5,6} and coarsegrained^{7} (CG) water models exist. They differ in terms of predictive power and computational cost/efficiency. The best performing nonpolarizable atomistic model is TIP4P/2005^{5}. However, it underpredicts the melting point by 20 K and is too computationally expensive for largescale molecular dynamics (MD) studies involving multimillion molecule icewater systems. Polarizable models such as MBpol^{8} and AMOEBA^{9} have comparable accuracy and can treat charged species, but are computationally expensive (Tables 1 and 2). CG models are computationally efficient, but often less accurate. The monoatomic water (mW) model^{10} remains the best performing CG model^{11}, predicting the correct melting point and several thermodynamic properties, but does not quantitatively capture the density anomaly and over predicts density of ice (Supplementary Figure 2).
A correct description of water’s complex properties with a potential model, especially in CG form, is challenging. Here, we introduce a machinelearning (ML) workflow (Fig. 1) that can be used to train models that accurately describe the behavior of ice and liquid water at mesoscopic scales. We develop a set of bondorder CG models (MLBOP and MLBOP_{dih}) that are up to two orders of magnitude cheaper (Table 2, Supplementary Figure 3) than the most accurate nonpolarizable atomistic models (TIP4P models and TIP5P) of comparable accuracy. As with the mW model^{10}, our models treat each water molecule as one bead; the interactions between the beads are treated using a bondorder potential (BOP) both with and without explicit fourbody term, i.e., onthefly dihedrals to describe tetrahedral solids. We use a multilevel hierarchical global optimization strategy to navigate the highdimensional parameter space and train the ML models. We introduce ML models that adequately describe the thermodynamic and dynamical properties of water. Moreover, we also demonstrate that our ML strategy can be used to reoptimize existing highquality water models, such as mW, and improve their overall performance.
Results
Machine learning workflow for training CG water models
The ML workflow to train CG models involves three main aspects: Model selection, Training data generation, and Multilevel hierarchical objective optimization to parameterize models against target training data. The various stages involved in the ML workflow are discussed below.
CG model of water
Water molecules are modeled using a 1:1 CG model. The mapping of atomistic water molecules into CG water beads is done via the removal of hydrogen atoms, such that the CG beads are positioned at the positions of the oxygen atoms. This representation of water molecules as monoatomic beads and the use of the ML models can lead to a much more significant speedup in MD simulations than the naive factor of three from the spatially reduced number of atoms. This is a result of larger simulation time steps being possible due to the absence of fast O–H vibrations, a significantly reduced number of pairwise interactions due to the reduced number of atoms, and the very simple CG potential form. A 1:1 CG model of water achieves both simplicity and computational efficiency.
Machine learned bondorder potential for CG water
The MLBOP model is based on the TersoffBrenner formalism^{12} (Pauling bondorder concept), which is used here to describe the shortrange directional interactions between CG water beads. The pair potential function V_{pair} is given by
where f_{C}(r_{ij}), f_{R}(r_{ij}), and f_{A}(r_{ij}) are the cutoff, repulsive, and attractive pair interactions, respectively, between bead i and j separated by a distance r_{ij}, and b_{ij} is a bondorder parameter which modifies the pair interaction strength between bead i and j depending on their local chemical environment.
The cutoff function limits the range of interaction mainly to improve computational efficiency. The function is given by
where R and D are free parameters that are chosen to include only the first nearest neighbors, such that their pair interactions are smoothly reduced starting from the distance R−D and are completely turned off beyond the distance R+D.
The repulsive and attractive pair interactions between CG water beads are modeled using exponential decay functions given by
where A, B, λ_{1}, and λ_{2} are free parameters that control the overall strength and length scale of the repulsive and attractive potentials. Furthermore, the strength of f_{A}(r) between beads i and j is scaled by a bondorder term b_{ij} which is given by
where β, n, c, d, and cosθ_{0} are free parameters. ξ_{ij} defines the effective coordination of bead i, taking into account the relative distances r_{ik} and interatomic angles θ_{ijk} of its neighboring beads. The threebody angular dependence is described by the function g(θ), which has minima defined by cosθ_{0} and the strength and sharpness of its effect controlled by c and d.
Machine learned bondorder potential with onthefly dihedrals
Typical potentials (e.g., StillingerWeber or bondorder based such as Tersoff as described above) are based on first nearest neighbor interactions and hence the functional forms do not explicitly distinguish (energetically) between cubic and hexagonal ice structures. To address this limitation, we extend the Tersoff bondorder potentials to include onthefly dihedral calculations similar to that implemented in AIREBO type models^{13}. The dihedral potential function is described by
where k_{dih} is the minimum welldepth, p controls the steepness of the well, and φ is the dihedral angle. In contrast to the dihedral potential functions typically used in rigidbond models, the welldepth (and number of minima) of this potential changes depending on the number and local coordination of water beads. To improve computational efficiency and to handle any discontinuities due to this reactive characteristic, the Tersoff cutoff function, f_{C}(r), is applied to every pair of water beads constituting a dihedral angle and an angular cutoff function, f_{D}(θ), is applied to every triplet of those water beads.
where cosθ_{1a} and cosθ_{2b} define the lower and upper bounds of the angular cutoff analogous to R+D in f_{C}(r), cosθ_{1b} and cosθ_{2a} define the switching angle for the lower and upper bound angular cutoffs analogous to R−D in f_{C}(r).
Model parameterization
The parameterization of MLBOP for water requires simultaneous optimization of 11 free parameters (R, D, A, B, λ_{1}, λ_{2}, β, n, c, d, cosθ_{0}). Likewise, the parameterization of MLBOP_{dih} for water requires optimization of 17 free parameters (R, D, A, B, λ_{1}, λ_{2}, β, n, c, d, cosθ_{0}, cosθ_{1a}, cosθ_{1b}, cosθ_{2a}, cosθ_{2b}, k, p), which makes independent fitting of the parameters infeasible. Most of these parameters do not correspond to physical properties of the system, so they cannot be chosen based on intuition. In this work, we employ global and local optimization techniques and stateoftheart machine learning principles to search for an optimized parameter set for water as described below.
Multilevel hierarchical objective machine learning workflow
Our machine learning workflow to train the CG models is illustrated in Fig. 1. In our training scheme (Fig. 1a), we introduce a multilevel evolutionary strategy (hierarchical objective genetic algorithm—HOGA) to train the ML models against an extensive training data set of energies and structural properties of ice and liquid water derived from the best available atomistic model (TIP4P/2005), supplemented by experimental data. The training data in the case of MLBOP_{dih} also includes first principles energetic differences reported for cubic and hexagonal ice phases^{14}. This elaborate training data set ensures an adequate representation of the diverse configurational space of ice and liquid water while amply sampling the energy landscape. We use HOGA to perform a global search followed by local optimization to find the optimized model parameters (Fig. 1b). This circumvents problems encountered with the local minimizers often used in force field fitting that rely on good starting guesses. An important new aspect of our scheme is that the iterations involve not just static evaluations of potential properties but also temperaturedependent properties sampled dynamically from several MD trajectories during the evolutionary process (10’s of milliseconds of overall MD trajectories). HOGA aids in an accelerated evolutionary search by efficiently sampling the parameter landscape within a given GA generation, and overcoming the limitation of assigning arbitrary weights within a single objective thereby ensuring that all the properties (static or dynamic) are equally well described.
Training data set
The machine learning workflow begins with the preparation of an extensive data set, which is necessary for a supervised training method. We build the training set from atomistic MD trajectories of 1600 TIP4P/2005 water molecules, simulated at pressure P = 1 bar over a wide range of temperatures using the LAMMPS simulator^{15} with a 13 Å interaction cutoff, the particleparticle particlemesh method for longrange electrostatic interactions, and a 1 fs time step. The training set consists of various ice and liquid water configurations, which includes hexagonal ice at 123 K < T < 273 K, supercooled liquid water at 253 K < T < 273 K, normal condensed phase liquid water at 273 K < T < 373 K, and icewater interfaces. Unlike most typical force field fitting procedures, we fit to the structure and energetics of TIP4P/2005 water configurations in the training set but also go beyond that by using the TIP4P/2005 training set as good starting configurations for running MD simulations with ML models during the fitting process. Properties including dynamical properties can be sampled from these simulations and be used to fit directly to experimental values of thermodynamic properties. Note that the main limitation of the TIP4P/2005 model is its inability to get the correct melting point (T_{m}) and the relative difference between temperature of maximum density (TMD) and T_{m}. In such cases, we use known experimental values as targets.
All MD simulations performed during our force field fitting workflow are run in an isobaricisothermal ensemble at pressure P = 1 bar and different target temperatures using the LAMMPS simulator^{15}. The equilibration time of these simulations varies from 150 ps to 4 ns depending on the configuration and temperature (e.g., shortest for ice and longest for supercooled water). Note that the training set contains configurations of ice and liquid water over a wide range of temperatures, static properties as well as timeaveraged properties such as ΔH_{m} and ρ_{i} sampled from MD simulations.
Hierarchical objective genetic algorithm (HOGA)
The quality of a proposed parameter set is evaluated based on a hierarchical objective function (see pseudo code in Supplementary Note 1). In the HOGA evolutionary scheme, we truncate the evaluation of a parameter set which leads to large errors in hierarchical property classes and assign it a penalty depending on which class it fails at. The selection of hierarchical classes is at the discretion of the user. In this case, the hierarchy of the property classes is as listed in Supplementary Table 2. Note that a higher preference is given to the temperaturedependent densities of ice, water and the melting point to ensure that the models reproduce the density anomaly and the relative locations of melting point and TMD. The hierarchical approach aids in an accelerated evolutionary search by efficiently sampling the parameter landscape within a given generation, and overcoming the limitation of assigning arbitrary weights within a single objective thereby ensuring that all the properties (static or dynamic) are equally well described.
Given the objective function definition as described above, we proceed to apply a twostage optimization technique to search for a suitable parameter set for water in the multidimensional parameter space. The goal is to locate the global minimum in the objective value landscape. We strategically start with a broad survey of the landscape using global optimization methods followed by a deeper refinement search using local optimization methods. In principle, any combination of global and local optimization methods should work for such a workflow. Here, we choose to use the genetic algorithm^{16} (GA) for global optimization and the NelderMead simplex algorithm^{17} for local optimization.
Using HOGA, the global optimization process begins with the initialization of a population of N_{p} random parameter sets. The objective value Δ^{(i)} for each of these parameter sets is evaluated and their convergence is checked. If the convergence criteria are not met, then a new list of N_{p} parameter sets is derived using genetic operations (selection, crossover, mutation, etc.) from the m old parameter sets having the lowest objective values. The selection operation creates a list of best parameter sets based on their objective values, which mimics the principle of “survival of the fittest” in evolution. The crossover operation intermixes these parameter sets to generate new potential good candidates, analogous to how good traits are passed from biological parents to their offspring. The mutation operation introduces sufficient diversity into the population to avoid premature convergence of the GA run, which also provides the population the opportunity to improve beyond those possible via inheriting traits from parent structures (crossover). In this work, we used tournament selection without replacement as the selection operation, the simulated binary method as the crossover operation with an operation probability of 0.9, and a polynomial of order 20 for the mutation operation with an operation probability of 0.1. The objective value is evaluated for the new parameter sets followed by convergence test. This routine is iteratively performed until convergence.
To effectively survey the objective landscape, we typically perform at least 20 GA runs simultaneously (up to a total of 100 runs), where each GA run has a population size of 200 and run for about 100 generations. The global optimization stage typically returns a list of closetooptimal parameter sets which we further refine using local optimization techniques. In this work, we use the NelderMead simplex algorithm^{17} for local optimization, and the final parameter set is chosen based on the performance in validation tests. The best parameter sets for MLBOP and MLBOP_{dih} optimized through HOGA are provided in Table 3.
Model validation and performance of machine learned CG water models
Figure 1c, d compares structural and dynamicsinferred properties with experimental data. Our MLBOP models successfully capture the bestknown thermodynamic anomaly, the existence of a density maximum at 277 K (Fig. 1c); they correctly describe the freezing/melting transition at 273 ± 1 K, and densities of ice (140 K–273 K) and water (243 K–373 K) within 1.4% of experiments. Capturing the correct value of the TMD relative to the melting point has remained a challenge for all water models^{10,18,19}. TIP4P/2005 is the best atomistic model to depict TMD but underestimates the melting point by 20 K. Regarding transport properties (Fig. 1d), the room temperature diffusivity, MLBOP models is ~3 × 10^{−5} cm^{2} s^{−1} in close agreement with experiment^{20} (2.3 × 10^{−5} cm^{2} s^{−1}). Both ML models slightly overestimate diffusivities in the supercooled range but outperform other existing water models (Supplementary Figure 2b).
Figure 1e compares the O–O radial distribution function (RDF) for ice I_{h} at 77 K and (supercooled) liquid water at 254 K derived from experiments. The location and intensities of the peaks corresponding to first, second and third coordination shells are in good agreement. MLBOP models, however, overstructure water, and underestimate the exchange of water molecules between first and second coordination shell^{21,22} (deeper minimum in the radial distribution function or RDF at ~3.4 Å). Our model is suitable for mesoscopic phenomena, such as ice nucleation and grain growth as well as applications involving polycrystalline ice, e.g., friction, mechanics of ice, melting of ice crystals, or pollutant effects on nucleation and ice grain growth (for example, see Supplementary Figure 4). The model captures the temperature and pressure dependent (Supplementary Figure 6) trends of these peaks. The MLBOP calculated number of water neighbors in the first solvation shell, integrated out to the predicted temperature independent isosbestic point (r = 3.25 Å), is 4.7 in accordance with the experimental range of 4.3–4.7^{23,24}. Also, the angular distribution function at 298 K agrees well with TIP4P/2005 (Supplementary Figure 5b). The MLBOP heat capacities for liquid water, with respect to their values at 309 K, reproduce the thermodynamic anomaly indicated by the sharp increase in C_{p} of supercooled water (Fig. 1f). We also introduce an MLBOP_{dih} which represents a modification of MLBOP model to include onthefly dihedrals. MLBOP_{dih} performs on par with MLBOP and additionally was trained using HOGA to capture the DFT predicted free energy difference (~1.4 meV/atom per water molecule) between ice polymorphs. The performances of both MLBOP and MLBOP_{dih} are detailed in Tables 4–6 and Supplementary Figure 26. Overall, the trained ML models perform better or on par with the best available water models in several of the properties listed, but at a fraction of the computational cost.
HOGA to retrain existing best performing CG models
Our machine learning strategy is quite general and can be used to improve a variety of existing material models. To demonstrate this capability, we retrain the best available coarse grained model, for water, i.e. the mW model^{10}, against our training dataset using the HOGA ML workflow. The new mW model trained using the machine learning workflow (termed MLmW and given in Table 3) correctly captures the TMD, the density and structure of ice in the supercooled regime (140–270 K) as well as improves several thermodynamic and transport properties compared to original mW while retaining the structure (e.g., RDF) of liquid mW water. The limitation of the MLmW is that the melting point is slightly overpredicted (~289 K) and, in contrast with the MLBOP and MLBOP_{dih} models, is unable to get the relative difference between T_{m} and TMD. Nevertheless, the overall predictions of MLmW are better than the original mW in several properties (see Fig. 2 and Tables 1 and 5).
Briefly, the ML trained mW improves upon several of the properties compared to the original mW. For example, the diffusion coefficient at 300 K is 4.8 × 10^{−5} cm^{2} s^{−1}, which is closer to the experimental value of 2.4 × 10^{−5} cm^{2} s^{−1}. Likewise, the density of ice at the melting point improves from 0.978 to 0.930 g cm^{−3}, which is closer to the experimental ice density (0.917 g cm^{−3}). The volume change upon melting, the TMD as well as the enthalpy of melting show an improvement over the original mW model. Other properties such as (dp/dT)_{melt} also show a significant improvement as detailed in Table 5. Note that these improved predictions come by sacrificing the melting point; the MLmW predicts the melting point to be 289 K which is 16 K higher than the original mW and experimental melting point. The HOGA algorithm is able to efficiently sample the highdimensional parameter space and arrive at an optimal set of mW parameters with an improved overall score for the properties listed in Table 1.
Origin of the improvement in the ML model performance
To elucidate the improvements of the MLmW and MLBOP models, we compare the pairwise interaction energy curves of these two models with the original mW model (Fig. 3a). As seen in the energy curves representing only the 2body interactions (solid line style), there are two notable differences as we go from mW to MLmW to MLBOP. There is a progressive steepening of the repulsive wall at r < 2.7 Å, and the interaction cutoffs become shorter (4.3 Å to 4.0 Å to 3.6 Å). A large increase in repulsive interaction can also be inferred from the ~3.3 times larger value of parameter B (coefficient of the repulsive term) in the functional form of MLmW vs. mW (Supplementary Table 1). Furthermore, in contrast to a previous study that mW has the shortest optimal interaction cutoff necessary for capturing the anomalous properties of water^{25}, HOGA is able to find a model with a shorter cutoff that improves the original model. The shorter cutoff of MLmW also contributes to its improved efficiency over mW (Table 2). MLmW has a 3° deviation (left shift of minimum in Fig. 3b) from the ideal tetrahedral angle of 109.47° (in mW), and has a larger 3body energy penalty for interatomic angles θ >130° but a smaller penalty for θ < 70°. The dashed and dotted cross marks in Fig. 3b mark the interatomic angles (~37°, ~72°, ~155°) at which the dashed and dotted energy curves in (a) are evaluated. The overall effect of bond order in the two models appears similar. We note that there have been prior efforts by Molinero and coworkers at improving the parameterization of the mW model using relative entropy minimization^{26} (REM) as well as using uncertainty quantification^{25} (UQ). Both of these studies provide useful insights into the effect of model parameters on system properties. Note that while the search spaces in UQ were localized around the already optimized mW set, the parameter search in the REM procedure was global. In both the cases, the overall performance of those reparameterized models were found to be poorer when compared to the original mW. In the present case, the performance improvements in MLmW arise from a drastic deviation of potential parameters from the already optimized mW parameter set. This signifies the effectiveness of HOGA in navigating the highdimensional parameter space and arriving at a set of optimal parameters that outperforms other optimization techniques such as REM and UQ.
Although both MLmW and MLBOP have quantitively improved the description of liquid density anomaly as well as the density of ice in mW, only MLBOP (in fact out of all currently existing water models) is able to capture the correct ordering and the relative temperature difference between the melting point and TMD of water. A major difference is the use of explicit cutoff by the Tersoff functional form (MLBOP models) as against the implicit cutoff functions employed by StillingerWeber form (mW models). An explicit cutoff function provides the flexibility to modify the tail portion of the pair interaction energy curve independent of the rest of features such as the repulsive wall, location and depth of minimum, etc. (see inset of Fig. 3c). As the MLBOP cutoff (R+D) becomes smaller, while keeping the switching distance (R−D) fixed, the relative separation between the melting point (cross marks) and TMD (vertical dotted lines) reduces and their ordering eventually flips (Fig. 3c). We further note that the tail portion of the interaction energy curve also has a strong influence on other properties including the liquid densities, ice densities close to the melting point, enthalpy of melting, diffusion coefficients, etc., which exemplifies the challenge in simultaneously optimizing many properties (i.e., multiobjective) and the need of a ML workflow in place of the local optimization based fitting procedures and/or driven by human intuition.
Simulations of ice nucleation in supercooled water
As a representative test case, we perform MD simulations on multimillion water molecules using MLBOP models to understand at the molecular level homogeneous nucleation of supercooled water leading up to the formation and growth of grains of ice. Figure 4 summarizes the initial stages of nucleation leading up to the formation of polycrystalline ice for one such trajectory when water is slowly cooled from 275 to 210 K over 130.4 ns (cooling rate ~0.5 K ns^{−1}). Following the appearance of the first stable nuclei at ~210 K, the temperature was held at 210 K for a further 100 ns to study the nucleation and growth processes in this homogeneously nucleated water. Figure 4a shows the potential energy variation as a function of time during the cooling phase and constant temperature phase. We identify four distinct stages during the freezing process: a long quiescent time period of ~130 nanoseconds before the first nucleation events; a period of slow transformation with a limited number of nuclei (13 at t = 150 ns, Fig. 4d); accelerated transformation driven by growth of a greater number of nuclei (~185 at 200 ns); and completion of grain growth to form a polycrystalline box of ice. Figure 4b shows the corresponding snapshots during the initial quiescent period when the system explores the relatively flat energy landscape before entering the nucleation and growth period. The molecular level illustration is consistent with classical nucleation theory; the quiescent period is marked by pronounced fluctuations of many subcritical nuclei which rapidly form, break and reform in the supercooled liquid as shown in Fig. 4d. The postquiescent period shown by MD snapshots in Fig. 4c is marked by formation of multiple stable nuclei which grow slowly followed by a rapid growth phase when the grains begin to percolate through the entire threedimensional space. The completion of the growth phase is characterized by the formation of a polycrystalline ice with the nanoscopic grains separated by boundaries comprised of amorphous ice. A local structure analysis (see Methods) of the growing structure reveals that the grains are comprised of stacking disordered ice (I_{sd}) i.e., randomly mixed alternating sheets of hexagonal and cubic ice (see Supplementary Figure 7 for the local structure). Figure 4e shows that the evolving ice structure becomes increasingly rich in I_{c} phase compared to the more stable I_{h} phase with the ratio of cubic to hexagonal to be ~1.85 at the end of t = 350 ns. The observed preference for cubic ice formation is consistent with multiple experimental results in the past including a recent Xray diffraction study^{27} and CG simulations^{28} as well as atomistic simulations using forwardflux sampling technique^{29}.
Nature of polycrystalline ice and transformation of stacking disordered ice to hexagonal ice
The final microstructure at 230 ns (Fig. 4c) is fine grained (average grain size ~9300 water molecules) and is expected to anneal over long times (microsecond to seconds) to naturally observed larger grains. We slowly anneal the nanocrystalline sample by heating from 210 to 260 K over 100 ns and then hold the sample at 260 K until the grains coarsen. The polycrystalline sample evolves into a single grain at the end of 1 microsecond of simulation (Fig. 5a). The internal structure of the grains is ice I_{sd}, i.e., randomly mixed alternating sheets of hexagonal and cubic ice, comprised of stacking faults that evolve over time (Fig. 5b). The ice I_{sd} structure observed in our simulations is rich in I_{c} phase compared to the more stable I_{h} phase with the ratio of cubic to hexagonal being ~2 by 1200 ns (Fig. 4e).
The preference for ice I_{sd} formation during nucleation is consistent with recent experiments and simulations^{27,28,29,30} but the stacking disorder in polycrystalline ice has been much debated^{27}. Kuhs et al.^{31} have analyzed neutron diffraction data and electron microscopy images to study the extent of stacking disorder in ice in the 170–190 K range. They tracked the evolution of cubicity as a function of time and note that the fraction of cubic stacking sequences is ~0.5. At temperatures > 180 K, cubicity decreases slowly to approach pure I_{h} after annealing over 10–12 h. Molinero and coworkers^{28} have analyzed the structure of ice that crystallizes at 180 K and shown that the ratio of cubic to hexagonal stacking sequences is ~2:1, which is similar to those found in our work. Indeed, more recent studies by Amaya et al.^{32} using femtosecond wideangle xray scattering confirm that ice formed by nanodroplets that freeze rapidly at timescales of the order of 1 microsecond indeed have much higher cubicity values ~0.78 ± 0.05. This value is much higher than that reported by Kuhs et al.^{31}. Nonetheless, these studies suggest that there can be a range of stacking disordered ice with different cubicity. The differences in the extent of stacking disorder were attributed to the differences in freezing temperatures^{33}, the size of droplets (nanosized vs. micron sized) and the freezing rates^{34} (microseconds vs. seconds) to name a few.
Capturing the energetic ordering and the subtle energetic differences between ice phases within a molecular model remains a major challenge. While the stable phase at weak undercooling is I_{h}, the ice phase that nucleates from supercooled water is, however, the stacking disordered ice. Free energy calculations performed by Molinero and coworkers^{30}, using the mW model, show that the entropy of mixing of cubic and hexagonal layers makes stackingdisordered ice the stable phase for crystallites sizes up to 100,000 molecules. We note that the free energy cost of producing a growth fault in ice I_{h} for the mW model is ~15.3 ± 2.3 J mol^{−1} (0.159 ± 0.024 meV), which is consistent with the experimental value of 16.5 ± 1.7 J mol^{−1} (0.171 ± 0.018 meV) reported by Hondoh et al.^{35}. Depending on the experimental conditions and the method of sample preparation, there is a range of free energy or enthalpy reported for the transformation of stacking disordered ice to pure hexagonal ice. For example, Ghormley et al.^{36} report transformation of cubic to hexagonal crystals to be ~22 J mol^{−1} (0.228 meV) in heating from 223 to 268 K. Differential scanning calorimetry of transformation of cubic ice (prepared by rapid quenching of liquid water at 190 K) to hexagonal ice report a slightly higher value ~56 J mol^{−1} (0.580 meV). On the other hand, McMillian et al.^{37} used calorimetry measurements and report a heat of transformation ΔH = 160 J mol^{−1} (1.658 meV) between ‘cubic’ and hexagonal ice. Likewise, Shilling et al.^{38} prepared amorphous ices by vapor deposition at 90 K and transformed them to stacking disordered ice by heating up to about 160 K. They report a free energy change of 155 ± 30 J mol^{−1} (1.606 ± 0.31 meV) for transforming I_{sd} to I_{h}. This value is much higher than that reported by Ghormley et al.^{36}. Differential thermal analysis also suggests heat release of similar order when I_{c} transforms to more thermodynamically stable I_{h}^{39}. Note that the mW predicted free energy is lower than the experimental values of Shilling et al.^{38} and McMillian et al.^{37}. On the other hand, recent ab initio studies also suggest a thermodynamic preference for I_{h} compared to I_{c} (~1.4 meV per H_{2}O arising from the difference in anharmonicity between cubic and hexagonal ice)^{14}. While DFTPBE may not be the best method for estimating the energetics of ice, this high free energy difference between I_{c} and I_{h} cannot be captured by nearest neighbor interactions as is the case in MLBOP, MLmW and mW. We, therefore, introduce an additional fourbody term to the MLBOP in the form of onthefly dihedrals model and retrained this MLBOP_{dih} model by including the average energy difference between cubic and hexagonal ice (reported in ref. ^{14} using DFTPBE) in the training data set. This 4body term essentially captures the energetics difference provided by the PBE input data in ref. ^{14}. One can retrain the 4body term (Eqs. 8–10) if new improved ab initio data becomes available.
To test the thermodynamic preference of our new MLBOP_{dih} model, we calculate the free energies of various ice phases (Fig. 6a) within the quasiharmonic approximation (see Methods). Our model is able to capture the temperaturedependent stability of cubic, stacking disordered and hexagonal phases; hexagonal is the most stable phase and is ~1 meV per molecule lower than the metastable cubic phase at 260 K (Table 6). Despite I_{h} being energetically preferred compared to I_{sd}, we do not observe a transformation to a pure hexagonal phase even after 1.3 μs of simulations possibly due to sufficiently large activation barrier. Indeed, climbing image Nudged Elastic Band (CINEB) calculations within the framework of MLBOP_{dih} show that the energetic barrier associated with elimination of a stacking fault plane in hexagonal ice is ~170 meV (Fig. 6b). The atomicscale pathway governing the transformation of a representative I_{sd} with ABCBAB stacking to I_{h} (ABABAB) is shown in Fig. 6b. The sliding of the molecules in the Cplane to their respective Aplane positions entails a range of concerted molecular motions involving stretching/rotation of hydrogen bonds. These coordinated movements result in localized strain in the vicinity of the stacking fault plane and underlie the activation barrier (~170 meV) associated with elimination of the fault. Thermal fluctuations at 260 K (kT ~ 22 meV) are sufficiently small to preclude observation of the I_{sd} → I_{h} transformation within μs timescales. Inplane I_{sd} → I_{h} transitions have much lower barriers (~5 meV per water) and are frequently observed at MD timescales (see Fig. 5c and Supplementary Movie 3, 4). Indeed, this behavior is consistent with the partial dislocation mechanism proposed by Hondoh et al.^{40}. It is also worth noting that the hexagonal ice becomes thermodynamically more favorable as we approach the melting point; hence the free energy difference between I_{h} and I_{c} is expected to increase (as shown in Fig. 6a). The different stackingdisordered ice configurations have much smaller free energy difference (<0.2 meV per water), which can explain the presence of random stacking disorder observed in previous experiments^{41} and simulations^{27,28,31,41,42,43}.
In summary, we introduced a machine learning strategy to train CG models, namely MLBOP, MLBOP_{dih} and MLmW, for water simulations. As proofofprinciple, we use the developed ML CG models (MLBOP and MLBOP_{dih}) to elucidate the mesoscale mechanism of ice grain formation and growth from supercooled water. In light of the accuracy and speed of our ML potential models, we foresee their wide usage in problems including phase transitions, homogeneous and heterogeneous nucleation, interfacial properties, coexistent regimes, and mechanical behavior, all at system sizes and times inaccessible to current popular models such as the TIP5P and TIP4P models.
Methods
Molecular dynamics simulations
Using the final MLBOP and MLBOP_{dih} potentials, we perform massively parallel, longtime MD simulations of supercooled water to study the sequence of steps from nucleation to grain formation and coarsening. We start with a simulation cell containing 2,048,000 molecules of water at 275 K. Simulations with a larger sized cell containing ~8 million water molecules are also performed. In both the cases, the structure is minimized and equilibrated for 100 ps at 275 K in an NPT ensemble. Subsequently, the liquid water is cooled down to 200 K over 150 ns under isobaric conditions. We observe the first stable nuclei at ~210 K. Consequently, we stop the cooling at 210 K and hold the supercooled liquid at 210 K for 100 ns. Following nucleation and growth of nanocrystalline ice grains at this temperature, we anneal the structure to 260 K at a rate of 0.5 K ns^{−1} and then hold it at 260 K beyond a microsecond to study the temporal evolution of grains.
Identification of ice polytypes and grains
Hexagonal (I_{h}), cubic (I_{c}), and amorphous/liquid phases of ice are determined using a structure identification algorithm^{44} implemented in the visualization software OVITO^{45}. The molecules are color coded according to their local environment (see Supplementary Figure 7). A feature detection algorithm based on image processing and unsupervised machine learning (clustering) techniques^{46} is developed to identify individual grains and their size distribution. The procedure involves voxelization (5 Å bin), contrasting filters, thresholding, DBSCAN clustering^{47}, refinement, and positionbased reverse mapping. All nearest neighbor searches are performed using a periodic kd tree. This grain identification procedure accurately identifies small and large grains that are often irregularly shaped.
Energy barrier calculations of stacking faults
The activation energy associated with the elimination of stacking fault plane in hexagonal ice is computed using Climbing Image Nudged Elastic Band (CINEB) calculations within the framework of MLBOP_{dih} as implemented in LAMMPS^{48,49,50}. For these calculations, the computational supercell consists of 6 layers (432 water molecules), with 72 molecules in the stacking fault plane.
Free energy calculations
Free energies of the cubic, hexagonal, and stacking disorder ice polytypes are computed within the quasiharmonic approximation^{51} using Phonopy^{52} to account for variation of phonon modes due to thermal expansion. For each ice phase, we first optimize the geometry of the unit cell in the framework of MLBOP_{dih} until the energy difference between consecutive steps is <10^{−4} eV, and the atomic forces are within 10^{−3} eV Å^{−1}. Next, we determine the changes in volume owing to thermal expansion using 1 ns long MD simulations at zero pressure and desired temperature. At each volume, phonon frequencies and related vibrational properties are computed via finite displacement approach using sufficiently large supercells (~ 88 Å × 70 Å × 66 Å) and displacement of 0.015 Å. The computed volumedependence of phonon frequencies are then used to compute vibrational contribution to free energy at each temperature.
Iceliquid surface tension calculations
The liquidice surface tension is calculated using the mold integration method^{53}. The well depth is chosen to be 10 meV based the value chosen for the original mW model^{54}. The system consists of 1600 wells and a total of 6000 water molecules. The temperature for this calculation was chosen to be the freezing temperature for each of the respective models. The slab is 2 unit cells in thickness in the Zdirection which is the interfacial direction. To obtain the free energy vs well radius curve, a series of thermodynamic integrations are performed by incrementing the well radius by 0.1 Å increments. This covers a range from 0.4 Å up to 1.2 Å. These values were used to create a linear curve that could be used to extrapolate back to the optimal well radius to obtain the correct value for the surface tension. This is in line with the procedure outlined in ref. ^{53}. After the curve is obtained, the optimal well depth for both systems is determined by running a simulation at each well radius and monitoring the crystallinity of the system. Each simulation is run for 10 ns. Once an approximate range is identified a set of intermediate radius values is examined to find the optimal well radius. We first benchmarked our procedure for the original mW and validate that the iceliquid surface tension was 35 mJ/m^{2} consistent with that reported in refs. ^{53,54}. For the MLBOP and MLBOP_{dih} it is found to be roughly 0.65 A while for the MLMW it is found to be 0.45 Å, which yields the values in Table 4.
Code availability
Code and workflow developed in this study are available from the authors upon reasonable request.
Data availability
The data that support the findings of this study are available from the authors upon reasonable request.
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Acknowledgements
The authors thank Maria Chan, Alper Kinaci, Kiran Sasikumar, Al Wagner and Ross Harder for useful discussions. Use of the Center for Nanoscale Materials was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DEAC0206CH11357. This research also used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DEAC0206CH11357. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DEAC0205CH11231. We also acknowledge the Carbon, Fusion and LCRC computing facilities at Argonne.
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H.C., M.J.C., and B.N. contributed equally. H.C., M.J.C., B.N., and S.K.R.S. conceived and designed the project. H.C., B.N., M.J.C., and S.K.R.S. developed the machine learning framework and bondorder potential model for water with input from S.K.G. M.J.C. and H.C. performed the largescale simulations of grain formation and growth. H.C., B.N., and M.J.C. developed the feature detection algorithm for 3D analysis of grain size and distribution. T.D.L. calculated free energies of various stacked disorder ice phases. B.N. performed CINEB calculations of the energy barrier for elimination of a stacking fault plane in hexagonal ice. S.K.R.S. supervised the overall project. All the authors including C.J.B. and S.K.G. performed the data analysis and contributed to the preparation of the manuscript.
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Chan, H., Cherukara, M.J., Narayanan, B. et al. Machine learning coarse grained models for water. Nat Commun 10, 379 (2019). https://doi.org/10.1038/s41467018082226
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