Introduction

Fast crystal growth and the underlying molecular mechanisms have gained considerable attention in condensed matter physics and materials science1,2. Water crystals display remarkable polymorphism, with experimental evidence supporting the presence of twenty unique ice forms thus far3,4,5,6. Moreover, simulations anticipate the discovery of additional distinct ice structures7,8. Despite the large variety in hydrogen-bond networks observed in different ice forms9, little attention has been paid to the potential variations in growth mode and rate.

The growth rate of ordinary ices, ice Ih and ice Ic, is relatively slow, with a maximum rate of 0.1 m/s10,11,12. By contrast, numerous dynamic compression experiments have identified rapid formation of ice VII on a sub-microsecond time scale, implying that either the nucleation, the melt growth, or both processes are fast13,14,15,16. Ice VII has a body-centred cubic oxygen arrangement with two interpenetrating hydrogen-bond networks. One of the sub-networks is identical to that of ice Ic, indicating that both ice VII and ice I (Ih and Ic) share robust hydrogen-bond networks. The question of whether ice VII can form significantly faster than ice I, along with the molecular mechanisms associated with this difference, remains a topic of great interest. However, the growth kinetics of ice VII are still not well understood, primarily due to the challenges of directly accessing the melt growth process via an interface between two condensed phases in experiments, especially under high-pressure conditions.

In general, when the temperature T is controlled to be uniform in space, the intrinsic normal growth rate Vn is expressed as the product of a kinetic factor K(T) and a thermodynamic term:

$${V}_{{{{{{{{\rm{n}}}}}}}}}(T)=K(T)\times \left[1-\exp (-\Delta \mu /{k}_{{{{{{{{\rm{B}}}}}}}}}T)\right],$$
(1)

where Δμ = μliq − μcry is the difference between the chemical potentials of the crystal and liquid, and kB is the Boltzmann constant. The thermodynamic term arises from the gain of free energy during crystallisation, whereas the kinetic factor represents the rate of addition of atoms to the crystal surface. Unlike the thermodynamic term, the kinetic factor has been the subject of intense discussion1,2,17,18,19.

The Wilson–Frenkel (WF) model predicts that the kinetic factor is governed by the self-diffusion coefficient D in the melt17,20,21:

$${K}^{{{{{{{{\rm{wf}}}}}}}}}(T)=\frac{6dDf}{{\lambda }^{2}}{e}^{-\Delta s/{k}_{{{{{{{{\rm{B}}}}}}}}}},$$
(2)

where d is the interplanar distance, λ is the displacement length associated with D, Δs is the entropy of the liquid with respect to that of the crystal, and f is the fraction of the growth sites. \(\ln D\) is proportional to the activation energy for diffusion (Ed), as \(\ln D\propto -{E}_{{{{{{{{\rm{d}}}}}}}}}/{k}_{{{{{{{{\rm{B}}}}}}}}}T\). The consideration of interfacial energy owing to preordering may refine the prediction19. Although the WF model may fail to predict the absolute value, the overall temperature dependence of the growth rate for organic and inorganic molecules22, metallic alloys19, silicon23, and hexagonal ice24 (except faceted surfaces under low supercooling25) is reasonably well described by the diffusion-controlled scenario, in which K(T) is strongly depressed upon cooling owing to reduced diffusivity.

By contrast, the growth rate of some materials does not drop even under high supercooling, suggesting a barrierless ordering1,2. An apparent turnover of the growth rate was predicted for pure metals at ~ 0.7Tm (where Tm is the melting temperature)26, but it was later confirmed to be a consequence of the instability limit of the supercooled liquid, below which a number of homogeneous nucleations occur simultaneously throughout the liquid2. The collision-controlled model treats crystallisation as wave propagation, in which the ordering simultaneously proceeds in many surface layers rather than at a sharp crystal–liquid interface. In this case, \(K(T)\propto \sqrt{3{k}_{{{{{{{{\rm{B}}}}}}}}}T/m}\), where m is the particle mass18,27. Harrowell et al. recently demonstrated that the WF model can describe a barrierless growth when the activation energy for crystallisation (Ea) is applied instead of Ed, and the ratio of Ea/Ed is smaller than 0.252,28. The molecular mechanism of barrierless growth has been attributed to a less frustrated solid–liquid interface, which enables liquid to crystallise through a small adjustment via preordering1 or an interfacial liquid with a crystalline ground state2,28. In addition to the unique temperature dependence, the barrierless ordering, which involves crossing little to no activation barrier, results in an anomalous growth rate (>102 m/s for monoatomic metals)2,29.

Globular molecules, which have isotropic interactions, such as monoatomic metals and colloids, tend to exhibit faster growth rates than orientationally ordered crystals1,2,19. Although ice polymorphs, including ice VII (proton disordered, but possible orientations are limited), are categorised into the latter group, the only exception may be plastic ice crystals, in which water molecules nearly freely rotate at their lattice sites and behave like globular molecules30,31,32.

Here, we perform molecular dynamics (MD) simulations of the crystal–liquid coexistence of ice Ih and two bcc-forming ice crystals (ice VII and plastic ice) and bring evidence that ice VII and plastic ice grow much faster than ice Ih. An unsupervised machine-learning (UML) technique suggests the presence of an interfacial plastic layer between ice VII and liquid, which does not appear on the ice Ih surface. We conclude that the decoupling of translational and rotational ordering, combined with the change in surface roughness and interface width, is associated with the unique growth mode of ice VII.

Results

Growth rate

Figure 1a–c present the schematic descriptions of ice Ih, ice VII, and plastic ice. A comparison of Fig. 1b, c reveals that oxygen atoms in ice VII and plastic ice have bcc lattice arrangements (see molecules in lime). A water molecule in ice VII forms four hydrogen bonds with its neighbours and the possible orientation of water dipole vector is limited, whereas water molecules in plastic ice rotate freely. We first compute the growth rate of those ice crystals in the classical WF model (\({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}\)) from Eqs. (1) and (2) combined with Δμ = (Tm − T)Δhm/Tm, where Δhm is the enthalpy of fusion at the melting temperature Tm. Table 1 summarises the physical quantities constituting Eq. (2). The fraction of favourable growth sites f is not precisely determined; it is likely within the range of 1/4 and 117,23,29. We here suppose that f=1.0. Δs is approximated by Δhm/Tm. The displacement length λ is taken from the effective molecular radius33, although λ can be an actual displacement that is slightly shorter than the radius of a molecule28. As shown in the inset of Fig. 1d, the results of the WF model suggest that the growth rate of ice Ih is the lowest, whereas plastic ice exhibits the highest growth rate. The difference predominantly arises from the entropy change during crystallisation.

Fig. 1: Various melt growth modes observed for three different ice forms.
figure 1

a–c Schematic descriptions of crystal structures. In (a), the primary prismatic surface of ice Ih is oriented toward the top. In (b) and (c), the side, top, and front are all {100} surfaces of a bcc-lattice. Nine lime molecules represent a bcc-lattice. d–f The normal growth rate measured in molecular dynamics simulations (Vn), the Wilson-Frenkel prediction (\({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}\)), and the scaled \({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}\) for the best fit to Vn. Tm is the melting temperature of each ice crystal. The solid orange line is a fitting to Vn using the activation energy for crystallisation (Ea). The inset of (d) compares \({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}\) s for the three ice crystals. g–i The Arrhenius plot of the kinetic factor K(T) obtained from Vn and Eq. (1). The solid orange line represents a linear fitting (2.5 < 1000/T < 2.9 is used for ice VII).

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Table 1 Quantities used in the Wilson–Frenkel prediction for ice Ih, ice VII, and plastic ice.
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Figure 1d–f plot the growth rate Vn measured in MD simulations for an elongated crystal–liquid system. The melt growth dynamics are seen in Supplementary Movie 1. Vn is computed from Eq. (4) and a linear fitting to the time evolution of ice-like molecules (Supplementary Fig. 2). Upon cooling from Tm, liquid water exhibits a glass transition at 4.8 GPa, and plastic ice transitions to ice VII at 10.5 GPa (Supplementary Fig. 3). In the current research, although the crystallisation of high-pressure ices from glassy water represents a captivating subject matter34, we have chosen not to delve into this particular topic. As a consequence, the coexistence between ice VII and liquid water persists in 0.80 < T/Tm at 4.8 GPa, and that between plastic ice and liquid water persists in 0.75 < T/Tm at 10.5 GPa.

In Fig. 1d, the diffusion-controlled scenario is apparently identified for ice Ih, as previously reported10,24, in which Vn first increases and then decays toward zero upon cooling. As the best fit to Vn is achieved by a slight scaling (\({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}/1.{1}^{2}\)), the absolute values and temperature dependence of the measured Vn are reasonably well predicted by the WF model.

Figure 1e demonstrates that Vn for ice VII also follows the diffusion-controlled scenario. However, the WF prediction seriously fails in describing the measured Vn, except for the maximum Vn temperature. In particular, ice VII grows 220 times faster than ice Ih, and reaches a maximum Vn of 31 m/s at T/Tm = 0.86, which is comparable to the fast growth rate observed for Pb and Ag (at T/Tm ~ 0.7) with a barrierless mechanism2.

Figure 1f shows that Vn of plastic ice rises monotonically with cooling, unlike ice Ih and ice VII. The Vn eventually reaches 116 m/s at T/Tm = 0.75, which positions plastic ice in the category of materials with the highest growth rate29. Water molecules in plastic ice maintain bcc-oxygen arrangements, similar to ice VII, but they lose their distinct hydrogen-bond network and rotate nearly freely at lattice sites. It is not surprising that the nearly spherical nature of water molecules in the plastic phase facilitates the propagation of crystal front. However, unlike previous studies that compared growth patterns between different molecular species, this study presents the finding that the growth kinetics of an identical molecule is significantly accelerated by the presence of orientational disorder within the crystal structure.

The primary prismatic surface for ice Ih and {100} surface for the two bcc-forming ice crystals are chosen, but a similar temperature dependence is observed for the other surfaces (see literature for ice Ih10,24 and Supplementary Fig. 4 for ice VII and plastic ice). By examining systems with a surface area four times larger, our results suggest that the growth rate is independent of the size of the interface area (N=49152); Vn of ice Ih, ice VII, and plastic ice is 0.14 ± 0.02 m/s at 0.97 Tm, 31 ± 2 m/s at 0.85 Tm, and 116 ± 7 m/s at 0.78 Tm, respectively, for the current system (see Fig. 1d–f); 0.10 ± 0.01, 30 ± 1, and 111 ± 2 m/s for the quadruple system.

We also compute the kinetic factor K from Vn combined with Eq. (1). In Fig. 1g–i, the Arrhenius plot of K provides the activation energy for crystallisation (Ea) within the generalised WF equation (Table 1). The resulting Ea values for ice Ih and ice VII are, respectively, comparable to the activation energy for self-diffusion (Ed) in the melt, which agrees with the fact that the temperature dependence of the measured Vn overlaps well with the scaled \({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}\). We later discuss why \(\ln K\) deviates from a linear function of 1/T at high temperatures in the case of ice VII. Unlike ice Ih and ice VII, Ea for plastic ice is certainly lower than Ed, which shifts the Vn peak to a lower temperature than that of \({V}_{{{{{{{{\rm{n}}}}}}}}}^{{{{{{{{\rm{wf}}}}}}}}}\). The resulting Ea/Ed of 0.75 is still much larger than the threshold of 0.25 for a (nearly) barrierless growth mode2,28. Thus, if the plastic phase can be cooled below 0.75Tm without the transition to ice VII, the Vn peak would be detectable, as suggested by the solid orange line in Fig. 1f. More importantly, we find that Ea for ice VII is almost equal to that of plastic ice, implying that the incorporation of liquid molecules into ice VII and plastic ice occurs via a similar ordering mechanism.

Classification of local environments

An intriguing phenomenon we encountered is that ice VII can grow much faster than ice Ih, even though in both cases the two ice crystals construct robust hydrogen-bond networks, unlike plastic ice. To autonomously characterise their local molecular environments on a single particle level, we employ the UML scheme that uses a neural network-based autoencoder combined with the Gaussian mixture model (GMM)35,36. Our descriptor for the j-th molecule is encoded into a 6-dimensional vector; \({{{{{{{\bf{Q}}}}}}}}(j)=(\{{\bar{q}}_{l}(j)\})\), where l is the positional symmetry of neighbours which can take the value 2, 4, 6, 8, 10, and 1237. A neural network is trained using two independent data sets; (i) the vectors Q(j) from all the particles in crystal–liquid configurations at T/Tm=0.8, 0.9, and 1.0 for ice Ih; and (ii) those for the bcc-forming ice crystals (i.e., ice VII and plastic ice). To find the optimal dimensionality of the bottleneck layer, we evaluate the reconstruction of mean square errors (MSE) of the autoencoder with varying dimensionality38. In Fig. 2a, we detect the presence of an elbow when the number of bottleneck nodes is equal to 3 (see L-method39) for both training sets. These results suggest that a three-dimensional nonlinear projection of the original input vectors is sufficient to preserve the relevant information. The relative importance (RI) of the input variables in neural network models refers to the degree to which each predictor contributes to the network output. To reduce the risk of bias in determining relative importance, we compare the results of two approaches, including the improved stepwise40 and the input perturbation41 methods. Both methods yielded results that concur in identifying that \({\bar{q}}_{l}\) exhibits the largest variation in the training data (Supplementary Fig. 5). For ice Ih, the most relevant feature is \({\bar{q}}_{6}\) with a RI of about 46% and the second is \({\bar{q}}_{4}\) with RI of about 27%. For bcc-forming ice crystals, the most relevant features are \({\bar{q}}_{6}\), with a RI of about 54%, and \({\bar{q}}_{8}\) with a RI of about 27%.

Fig. 2: Classification of local molecular environments into relevant clusters, using the unsupervised machine learning approach for crystal-liquid coexistence systems.
figure 2

a Mean square errors (MSE) of the autoencoder as a function of the number of bottleneck nodes. The inset describes the architecture of a neural-network-based autoencoder, which finds a low-dimensional representation of the input vectors Q, from which the decoder reconstructs an approximation of the input as output \({{{\hat{{{{{\bf{Q}}}}}}}}}\) while preserving the relevant information. b Bayesian information criterion (BIC) as a function of the number of Gaussian components in the Gaussian mixture model. c Probability distributions of three clusters found for the coexistence between liquid and bcc-forming ice crystals, projected onto each axis in the determined χ1-χ2-χ3 subspace.

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We then apply GMM in the respective three-dimension subspace and compute the Bayesian information criterion (BIC) for various numbers of Gaussian components in the mixture model. Figure 2b shows the presence of an elbow when the number of Gaussian components is equal to two for ice Ih. For the bcc-forming ice crystals, an elbow is observed when the number of Gaussian components is equal to three. Accordingly, the local molecular environment is classified into two relevant clusters for ice Ih and three for the bcc-forming ice crystals. Figure 2c shows how the original probability distributions for the bcc-forming ice crystals are decomposed into three Gaussian distributions on each axis in the determined subspace (named as χ1, χ2, and χ3).

In Fig. 3a–c, water molecules are coloured according to the type of cluster to which they belong. Figure 3a suggests that the coexistence between ice Ih and liquid is well-segregated into the ice domain (grey) and the liquid domain (blue). Their direct contact indicates that liquid water molecules directly incorporate into the ice Ih surface. By contrast, Fig. 3b shows that an interfacial domain (green) is present between ice VII (orange) and liquid (blue). Figure 3d illustrates that the density profile of the interfacial domain retains its layered structures, which are somewhat wider than those observed in ice VII. The thin interfacial layer is invariably detected between ice VII and liquid for all investigated temperatures. The thickness of the interfacial layer, although it fluctuates, is constant during the freezing of ice VII (see Supplementary Fig. 6 for details). Therefore, liquid water molecules necessarily transition to ice VII through the intermediate green layer.

Fig. 3: Crystal-liquid coexistence structures coloured according to the clustering using the unsupervised machine learning approach.
figure 3

a–c The crystal--liquid coexistence structures for ice Ih, ice VII, and plastic ice at varying temperatures. d The z-dependent density profile in the vicinity of the ice VII-liquid interface at Tm and 4.8 GPa. Dynamical features, e reorientational correlation function (RCF) and f mean square displacement (MSD), for water molecules in each cluster of ice VII–liquid coexistence at the melting temperature (Tm).

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Our approach identifies an intermediate layer even when we only provide the ice VII-liquid coexistence structures as input for UML, excluding the plastic ice-liquid coexistence structures (Supplementary Fig. 7).

Figure 3c shows that the coexistence of plastic ice and liquid is represented by two domains at T/Tm=0.8, 0.9, and 1.0, similar to that for ice Ih. At these temperatures, liquid water molecules (blue) directly incorporate into plastic ice (green). When the temperature is further cooled to T/Tm=0.7—the phase boundary between ice VII and plastic ice is crossed—almost half of the crystal domain is identified as ice VII (orange). The crystal region is not fully covered by orange dots, because the small free energy barrier between ice VII and plastic ice at this condition allows their incessant alternations42. Our results demonstrate that although oxygen atoms in ice VII and plastic ice both adopt a bcc arrangement, the distortion caused by the lack of specific hydrogen-bond networks is sufficiently large to distinguish plastic ice from ice VII if we choose an appropriate order parameter.

Characterisation of the interfacial layer

The clustering analysis suggests that the interfacial domain existing between ice VII and liquid possesses the same structural features as plastic ice (both coloured in green). In addition, we compare their dynamic features. Figure 3e, f show the reorientational correlation function (RCF) 〈ui(Δt + t0) ⋅ ui(t0)〉, and the mean square displacement (MSD) 〈∣ri(Δt + t0) − ri(t0)∣2〉 for water molecules belonging to three different domains in the ice VII–liquid configuration at Tm. Here, ui(t) represents the unit vector along the dipole direction, and ri(t) represents the oxygen position for the i-th molecule at time t; the parentheses 〈 ⋯  〉 denote the ensemble average over the particles and over the reference time t0. The particles are labelled based on their configuration at t0. The results show that the RCF of the interfacial domain decays similarly to that of the liquid. However, the MSD for the interfacial domain resembles that of ice VII, rather than liquid. The non-zero slope in the MSD for the interfacial domain is attributed to the diffusion into the liquid region. Supplementary Fig. 8 shows the z-dependence of the diffusion coefficient and the relaxation time. The observed facile rotational but less diffusional characteristics indicate a plastic phase. Therefore, the interfacial domain between ice VII and liquid is classified as a plastic crystal from both structural and dynamic perspectives.

The presence of such an intermediate wetting layer suggests that the surface tension (free energy) between ice VII and liquid is greater than the sum of the surface tensions between ice VII and plastic ice and between plastic ice and liquid, analogous to a quasi-liquid layer at the ice–vapour interface43. The appearance of a metastable plastic phase (not bcc-plastic) is also observed during the homogeneous nucleation of ice VII, which is in line with the Ostwald step rule8. However, because the arrangement of oxygen atoms in the metastable plastic ice differs from that in the final state (ice VII), both oxygen and hydrogen atoms must be simultaneously rearranged in the second step of homogeneous nucleation. In contrast, we underline that almost complete decoupling of translational and rotational ordering during crystallisation occurs in the melt growth of ice VII.

On-site observations describe how each molecule is absorbed onto the crystal surface. Even in cases of the first-order solidification, the molecular-level incorporation process can differ between materials due to their surface structure, and hence, affects the growth rate17,18,20,21,27. Figure 4a tracks the Voronoi volume of a typical water molecule at the instance when it incorporates into ice VII from liquid, together with the cluster identification. A stepwise decrease in the Voronoi volume can be seen, although it largely fluctuates, consistent with the change of cluster type. In both transitions, from liquid to interfacial plastic and from interfacial plastic to ice VII, the variation appears not gradual but distinct. Accordingly, although the growth rate of ice VII largely deviates from the WF prediction, the one-by-one freezing supposed in the WF model describes the molecular picture more accurately than the gradual density variation expected in the wave-propagation scenario of the collision-controlled model.

Fig. 4: Properties of the interfacial plastic domain.
figure 4

a Voronoi volume of a typical water molecule at the instant when it is incorporated into ice VII (T/Tm=0.9 and 4.8 GPa). Tm is the melting temperature. White and black lines, respectively, show the running average over 0.4 ps (40 points) and sigmoidal-curve fitting. The colour slabs represent the clusters to which the molecule belongs. b Number of ice VII molecules vs. time for the coexistence between ice VII and plastic ice (T/Tm=0.9 and 4.8 GPa). The red line is a linear fitting of the period between 0 and 10 ps. The insets depict a part of the coexistence system, in which orange and green particles represent ice VII and plastic ice crystals. c z dependence of the fraction of clusters for a typical snapshot of ice VII--liquid coexistence, with sigmodal functions (solid lines) fitting the ice VII and liquid fractions (dots). The black solid lines and arrows indicate the width of interfacial plastic region. Temperature dependence of d the width of interfacial plastic domain at 4.8 GPa and e the roughness \({k}_{{{{{{{{\rm{B}}}}}}}}}T/\pi \tilde{\alpha }\) for the surface between (interfacial) plastic ice and liquid at 4.8 GPa and 10.5 GPa.

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Which process dominates the growth of ice VII from the liquid? To answer this question, we build a coexistence between ice VII and plastic ice and relax it at 383 K (0.9Tm for ice VII) and 4.8 GPa. (see Methods section for details). Figure 4b shows the time evolution of the number of ice VII molecules. As can be seen in the insets of Fig. 4b, spinodal decomposition occurs in many places at 20 ps because plastic ice under this thermodynamic condition is extremely unstable. However, the sole advance of ice VII crystal front is observed, albeit briefly, during the period from 0 to 10 ps (Supplementary Fig. 9). A linear fitting to that period, combined with Eq. (4), estimates the growth rate as 130 m/s, which is much faster than that of ice VII from liquid (at most 31 m/s). Thus, the transition from liquid to interfacial plastic layer, rather than from interfacial plastic layer to ice VII, primarily controls the overall growth rate of ice VII from liquid.

The temperature dependence of the interfacial domain width is examined. Figure 4c shows the fraction of each cluster as a function of z for a crystal–liquid configuration at Tm of 4.8 GPa. As expected, the plastic ice fraction exhibits a pronounced peak at the interface between ice VII and liquid. A small portion of the plastic particles is also observed in bulk ice VII (13 % at Tm), which tends to disappear with cooling. The curves for ice VII and liquid fractions are fitted using a sigmoid function, and their 90% drop is used to define the width of the interfacial domain. Figure 4d shows that the width of the interfacial domain is reduced upon cooling. As demonstrated for colloids and metals, pre-ordering plays a key role in fast crystal growth, where only a small adjustment is required for the transition to the crystal1,2. If we define the activation energy for crystallisation as \({E}_{{{{{{{{\rm{a}}}}}}}}}=-{k}_{{{{{{{{\rm{B}}}}}}}}}\partial \ln K/\partial (1/T)\), Ea (computed from Fig. 1h) increases upon cooling, indicating that the transition from liquid to ice VII becomes more difficult. Thus, regarding the interfacial plastic region as the pre-ordering of ice VII, the increases in Ea may arise from the reduction of the pre-ordering layer.

We also characterise the surface roughness by the height correlation function25,44,45,46.

$$G(r)=\left\langle {[h({{{{{{{\bf{r}}}}}}}})-h({{{{{{{\bf{0}}}}}}}})]}^{2}\right\rangle ,$$
(3)

where h is the height at r = (x, y) on the lateral plane. 〈 ⋯  〉 is an ensemble average that runs over the sampled snapshots and the position of the origin 0. G diverges logarithmically as \({k}_{{{{{{{{\rm{B}}}}}}}}}T/\pi \tilde{\alpha }\ln (r/d)\) for a rough surface, where \(\tilde{\alpha }\) is a surface stiffness, whereas G saturates for a faceted surface45. The surface is defined as the oxygen atoms in the plastic domain that are accessible from the liquid domain using a spherical probe with a radius of 0.2 nm47. We perform MD simulations for a more extensive system than that of Vn and find that G(r) diverges with r for all the configurations examined, thus indicating a rough surface (Supplementary Fig. 10). \({k}_{{{{{{{{\rm{B}}}}}}}}}T/\pi \tilde{\alpha }\) is then obtained from a linear fitting to the semi-log plot of G(r). Analogous to the correlation between the step stiffness and the density of kinks on a step45, \({k}_{{{{{{{{\rm{B}}}}}}}}}T/\pi \tilde{\alpha }\) may correlate positively with the density of kinks on a surface and thus also with the fraction of favourable growth sites (f) in the WF model. Figure 4e shows that \({k}_{{{{{{{{\rm{B}}}}}}}}}T/\pi \tilde{\alpha }\) increases upon cooling in the case of the plastic–liquid surface on ice VII at 4.8 GPa, whereas it remains relatively stable for the plastic–liquid surface at 10.5 GPa. In other words, the plastic–liquid surface becomes rougher; thus, f increases upon cooling at 4.8 GPa, but not at 10.5 GPa. Assuming Ea is a constant, the increase in f results in the deviation of \(\ln K\) from a linear function of 1/T, as observed in Fig. 1h. Further studies are required to clarify whether Ea, f, or a combination of the two, is associated with the non-linear dependence of \(\ln K\) on 1/T for the melt growth of ice VII.

Difference between ice VII and plastic ice

Using a simple model, we establish that the greatly distorted bcc-lattice structure (representative of plastic ice) is distinguishable from the moderately distorted bcc-lattice structure (representative of ice VII) within our methodology. We assume that the model bcc-lattice contains a large number of independent three-dimensional harmonic oscillators of the same spring constant c. The particle position around a lattice site (Δr) follows the Gaussian distribution (\(\propto \exp (-\Delta {r}^{2}/2{\sigma }^{2})\)) with the standard deviation \(\sigma =\sqrt{{k}_{{{{{{{{\rm{B}}}}}}}}}T/c}\). The distorted bcc-lattice of 16000 particles is generated by the pyscal package, and the resulting local environment is projected on the χ1 − χ2 − χ3 subspace. In Fig. 5a, the distributions with σ/a = 0.04, 0.08, and 0.15 (a is the lattice constant and the cut-off for neighbours is 1.14a) reproduce reasonably well the distributions for ice VII, plastic ice, and liquid in Fig. 2c, respectively. Figure 5b provides a schematic representation of these deviations. These findings bring evidence that the degree of distortion present in the bcc lattice differs between ice VII and plastic ice.

Fig. 5: Influence of the lattice distortion in the model bcc crystal composed of independent harmonic oscillators on identifying the local structural environments.
figure 5

a Probability distribution of the distorted bcc-lattice with σ/a=0.04, 0.08, and 0.15 on the χ1-χ2-χ3 subspace. σ is the standard deviation of particle position around a lattice site; a is the lattice constant of bcc-lattice; χ1,χ2,χ3 are the three-dimensional nonlinear projection of the original input vectors. b Schematic description for those distortions. The dashed lines in grey are for the eye guide.

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Conclusions

In summary, the results of our study have suggested that ice VII and plastic ice grow outstandingly faster than ice Ih, which significantly exceeds the growth rate expected from the classical WF model. The discovered fast growth rate of ice VII, which maintains a specific hydrogen bond network, was attributed to the decoupling of translational and orientational ordering occurring through an interfacial plastic layer. The accelerated growth kinetics associated with the globular molecular features in the plastic phase is consistent with the computational observation that the monoatomic water model, due to the absence of hydrogen, overestimates the growth rate of ice Ih by order of magnitude of 2 to 3 compared to the all-atom models10. The bcc-like local structure of high-density water may also impact the fast local ordering of the plastic ice (layer)48,49.

The classification of each molecule into relevant local molecular environments provided further details on how liquid molecule freezes into ice VII. More specifically, we found that (i) the crystal front advances via distinct transitions of each molecule, (ii) the incorporation from liquid to interfacial plastic layer dominates the overall growth rate, and (iii) the width and roughness of the interfacial domain correlates with the growth kinetics. Furthermore, although many studies have focused on the growth front and nearby liquid in order to rationalise the molecular mechanism of fast growth, the observed remarkably high growth rate for plastic ice suggests that the nature of bulk crystal itself—in our case, it is orientational disordering—also plays an essential role in determining the growth mode.

Our numerical approach of the fine combination of MD simulations with UML is directly applicable to other high-pressure ice crystals, e.g., to study the role of high-density liquid layer found on ice III and ice VI50, as well as the structural changes in hydration shell which support the functions of macromolecules51,52,53. Because UML does not require supervision—this differs from supervised machine learning, where one would need to assume that the structures are known, i.e., labelled by the expert—it can detect unknown ice structures as long as the data is of sufficient amount and quality and that the descriptors are relevant and informative about those structures.

Ice VII occurs under extreme environments, such as the Earth’s deep mantle and extraterrestrial space54,55. However, recent studies predicted ice VII-like structures even at ambient conditions in nanospaces between metal nanoparticles56 or polymers57. More generally, a plastic (or rotator) phase is not limited to water58,59,60. Thus, our findings would provide a better understanding of the melt growth mechanism of various materials and offer a guide for producing high-quality crystallines. The potential formation of an orientationally disordered wetting layer would be important information when we examine the binding mechanism of additives to crystals (e.g., antifreeze proteins)61 or chemical reactions on crystals62.

Methods

MD simulations

MD simulations are performed using the GROMACS 2020.6 package63. Water molecules are represented by the TIP5P rigid water model64. In this model, bcc-plastic ice, in which oxygen atoms adopt a bcc lattice, forms from ice VII via the first order phase transition upon heating at several gigapascals30,42,65. Although several different types of plastic ice crystals exist66,67, plastic ice in this study represents the bcc form of plastic ice.

TIP4P series have been widely used for studying phase transitions of water68,69,70. However, the TIP5P model more closely reproduces the experimental densities of ice VII when compared to the TIP4P/2005 model71. Naturally-occurring ice VII becomes the most stable ice polymorphs at pressures greater than 2 GPa. In contrast, ice VII in the TIP4P/2005 model and the SPC/E model exhibits stability at pressures exceeding 10 GPa, and hypothetical ice crystals (not existing in nature) become stable at lower pressures66. Moreover, ice VII in the TIP4P/2005 model transforms into plastic ice at higher temperatures when the pressure is greater than 4 GPa72 or 6 GPa31. Thus, ice VII in the TIP4P/2005 model would not be a stable phase at any pressure on the solid-liquid coexistence boundary in the phase diagram. Other models in the TIP4P series, such as TIP4P and TIP4P/Ice, would display comparable phase behaviours as TIP4P/2005 since they are known to primarily shift the temperature of the phase diagram73,74. Although a hypothetical ice crystal also exists between ice VI and ice VII on the phase diagram of the TIP5P model, the pressure at which ice VII becomes stable (>4 GPa) is considerably lower than that observed in the TIP4P/2005 and SPC/E models (>10 GPa)66. Furthermore, attempts to forcibly grow metastable ice VII in the TIP4P/2005 model have been observed to result in the appearance of a hypothetical ice crystal on the ice VII surface, thereby precluding measurements of ice VII growth rate75. In contrast, the TIP5P model allows for the measurement of the growth rate of both ice VII and plastic ice, as shown below.

The pair interactions are truncated at 0.9 nm. Long-range dispersion corrections are applied to energy and pressure, and long-range Coulombic interactions are evaluated using the particle-mesh Ewald algorithm. Periodic boundary conditions are applied in all directions. The equation of motion is integrated with the leapfrog algorithm using a time step of 2 fs. In the equilibrations, the temperature T and pressure p are controlled using the Berendsen algorithm with a damping constant of 20 ps (1.0 ps for 0.1 MPa), while the Nosé-Hoover thermostat with a damping constant of 2.0 ps is used for T in the production runs.

The growth rate of a planar ice surface is measured using crystal–liquid systems in an elongated box with dimensions of approximately 2.7 × 2.9 × 48.4 nm for ice Ih and 2.6 × 2.6 × 32.2 nm for ice VII and plastic ice. The total number of molecules N is 12288. The primary prismatic surface of ice Ih and the {100} surface of ice VII and plastic ice are exposed in the z-direction. The structures of ice Ih and ice VII are generated using the GenIce package76. Then, 80% of the crystal is melted at a high temperature by applying position restraints to the remaining part. Plastic ice spontaneously forms from ice VII when T and p are increased.

MD simulations are carried out under the isothermal–isobaric (NpT) ensemble. Ice Ih, ice VII, and plastic ice under 0.1 MPa, 4.8 GPa, and 10.5 GPa, respectively, are considered. Semi-isotropic pressure coupling is used for crystal–liquid systems, in which the pressure coupling is isotropic in x-and y-directions but different in z-direction. Thus, the volume change due to crystallisation results in a change in the z-direction of the simulation cell. Three independent trajectories are generated from two different initial structures; thus, the growth rate for each condition is estimated from six trajectories.

The normal growth rate Vn is computed as follows:

$${V}_{{{{{{{{\rm{n}}}}}}}}}=\frac{\Delta {N}_{{{{{{{{\rm{ice}}}}}}}}}}{2\Delta t}\times \frac{{L}_{{{{{{{{\rm{z}}}}}}}}}}{N},$$
(4)

where ΔNice/Δt is the slope of a linear fitting to the time (t) evolution of the number of ice-like molecules (Nice), and Lz is the box length along the z-axis when the entire system is frozen. For ice VII and plastic ice, ice-like molecules are defined by \({\bar{q}}_{6}\, > \,0.25\), while that for ice Ih is set as \({\bar{q}}_{6}\, > \,0.40\) (\({\bar{q}}_{6}\) is explained below). Half of the slope is associated with a single growth front because melt growth occurs at both ends of the crystal. In the same manner, we also compute the negative growth rate (i.e., for melting) and estimate the melting temperature (Tm) using the framework of the direct coexistence method (Supplementary Fig. 11)77. The obtained Tms are summarised in Table 1.

To estimate the growth rate of ice VII from plastic ice, we transform 90% of ice VII to plastic ice by performing MD simulations under canonical ensemble at 800 K with applying position constraints to the remaining part. Using the resulting coexistence as the initial structure (Supplementary Fig. 9a), we perform 100 ps simulation at 383 K (0.9 Tm for ice VII) and 4.8 GPa. Ice VII and plastic ice are identified by the UML approach. The ΔNice/Δt is computed by a linear fitting from 0 to 10 ps, during which the sole advance of ice VII crystal front is observed (Supplementary Fig. 9). By applying Eq. (4), the growth rate of ice VII from plastic ice is determined.

Enthalpies, diffusion coefficients, rotational correlation functions, and radial distribution functions for bulk configurations are computed using 5488 water molecules in a nearly cubic box, in which the pressure is isotropically controlled. For each condition, a 1 ns equilibration run is followed by a production run for several dozens of ns. In addition, a larger system of 12.7 × 12.7 × 32.7 nm, comprising 307,200 molecules, is used to compute the height correlation function of the ice surface.

Structural characterisation

The local structure around i-th particle is characterised by the average version of bond orientational order parameter \(\bar{{q}_{l}}(i)\) with the spherical harmonics of order l78. We focus only on the coordinates of oxygen atoms.

$${\bar{q}}_{l}(i)=\sqrt{\frac{4\pi }{2l+1}\mathop{\sum }\nolimits_{m = -l}^{l}| {\bar{q}}_{lm}(i){| }^{2}},$$
(5)

where

$${\bar{q}}_{lm}(i)=\frac{1}{{N}_{{{{{{{{\rm{b}}}}}}}}}(i)+1}\left[{q}_{lm}(i)+\mathop{\sum}\limits_{k\in {\widetilde{N}}_{{{{{{{{\rm{b}}}}}}}}}(i)}{q}_{lm}(k)\right]$$

and

$${q}_{lm}(i)=\frac{1}{{N}_{{{{{{{{\rm{b}}}}}}}}}(i)}\mathop{\sum}\limits_{j\in {\widetilde{N}}_{{{{{{{{\rm{b}}}}}}}}}(i)}{Y}_{l}^{m}({{{{{{{{\bf{r}}}}}}}}}_{ij}).$$
(6)

\({\widetilde{N}}_{{{{{{{{\rm{b}}}}}}}}}(i)\) is the set of nearest neighbours of i-th particle within a cutoff of 0.35 nm for 0.1 MPa and 0.40 nm for the other pressures. \({\widetilde{N}}_{{{{{{{{\rm{b}}}}}}}}}(i)\) contains Nb(i) particles. \({Y}_{l}^{m}({{{{{{{{\bf{r}}}}}}}}}_{ij})\) are the spherical harmonics of order l with an integer m ranging from m = − l to m = + l. rij extends from i-th particle to j-th particle. Then, the local structural environment of i-th particle is represented by a six-dimensional vector:

$${{{{{{{\bf{Q}}}}}}}}(i)=({\bar{q}}_{2}(i),{\bar{q}}_{4}(i),{\bar{q}}_{6}(i),{\bar{q}}_{8}(i),{\bar{q}}_{10}(i),{\bar{q}}_{12}(i)).$$
(7)

Following the work by Boattini et al.35,36, we apply a UML approach to group similar local environments. A neural network-based autoencoder is trained to perform the input reconstruction task through a low-dimensional nonlinear projection that preserves the most relevant features of the data. The input data is the sum of the vectors Q(i) obtained from MD simulations for crystal–liquid coexistence systems. The network layers in the encoder and decoder structures are taken from PyTorch (Python package)79. We use one hidden layer of dimension 60 for both the encoder and the decoder parts of the network. A hyperbolic tangent activation function is applied between the linear layers of the autoencoder for non-linear mapping.

To prevent potential overfitting and to improve the robustness to small variations in the input data, we use a ridge regression approach80. This approach minimises the mean square errors plus a fraction of the L2 norm of the model’s weight vector38. The corresponding cost function can be expressed as follows:

$$\frac{1}{{N}_{s}}\mathop{\sum }\limits_{i=1}^{{N}_{s}}| | {{{{{{{\boldsymbol{Q}}}}}}}}(i)-{{{\hat{{{{{\boldsymbol{Q}}}}}}}}}(i)| {| }^{2}+\phi \mathop{\sum }\limits_{j=1}^{M}{w}_{j}^{2},$$
(8)

where Ns is the number of input vectors, \({{{\hat{{{{{\boldsymbol{Q}}}}}}}}}(i)\) is the output vector of i-th particle, M is the total number of weights wj, and ϕ is set to 10−535. M is automatically determined in the autoencoder. The cost function is minimised using a minibatch stochastic gradient descent with momentum81.

Similar environments in the determined low-dimensional subspace are grouped together by a GMM. The main assumption of GMM is that the data is generated from a mixture of Gaussian distributions with unknown parameters. We learn the parameters using the expectation-maximisation algorithm available in the scikit-learn library in Python82. The optimal number of clusters is determined by finding an elbow on the BIC using the L-method39.