Role of stacking disorder in ice nucleation

Abstract

The freezing of water affects the processes that determine Earth’s climate. Therefore, accurate weather and climate forecasts hinge on good predictions of ice nucleation rates1. Such rate predictions are based on extrapolations using classical nucleation theory1,2, which assumes that the structure of nanometre-sized ice crystallites corresponds to that of hexagonal ice, the thermodynamically stable form of bulk ice. However, simulations with various water models find that ice nucleated and grown under atmospheric temperatures is at all sizes stacking-disordered, consisting of random sequences of cubic and hexagonal ice layers3,4,5,6,7,8. This implies that stacking-disordered ice crystallites either are more stable than hexagonal ice crystallites or form because of non-equilibrium dynamical effects. Both scenarios challenge central tenets of classical nucleation theory. Here we use rare-event sampling9,10,11 and free energy calculations12 with the mW water model13 to show that the entropy of mixing cubic and hexagonal layers makes stacking-disordered ice the stable phase for crystallites up to a size of at least 100,000 molecules. We find that stacking-disordered critical crystallites at 230 kelvin are about 14 kilojoules per mole of crystallite more stable than hexagonal crystallites, making their ice nucleation rates more than three orders of magnitude higher than predicted by classical nucleation theory. This effect on nucleation rates is temperature dependent, being the most pronounced at the warmest conditions, and should affect the modelling of cloud formation and ice particle numbers, which are very sensitive to the temperature dependence of ice nucleation rates1. We conclude that classical nucleation theory needs to be corrected to include the dependence of the crystallization driving force on the size of the ice crystallite when interpreting and extrapolating ice nucleation rates from experimental laboratory conditions to the temperatures that occur in clouds.

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Figure 1: Reaction coordinate and free energy landscape for ice nucleation at 230 K.
Figure 2: Randomly selected critical ice nuclei with cubicity and size corresponding to the saddle point of the free energy landscape (white circle in Fig. 1c).
Figure 3: Stability and cubicity of ice crystallites along the pathway of nucleation and growth.

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Acknowledgements

We thank A. Haji-Akbari for discussions and for sharing configurations from nucleation trajectories of the TIP4P/ice model reported in ref. 6. This work was supported by the National Science Foundation through the Center of Chemical Innovation award CHE-1305427 “Center for Aerosol Impacts on Climate and the Environment” and the Environmental Chemical Sciences award CHE-1309601. We thank the Center for High Performance Computing at the University of Utah for an award of computing time and technical support.

Author information

V.M. and L.L. conceived the work. L.L. performed and analysed all molecular simulations and calculations with the 1D stacking model. M.G. contributed the 2D lattice model and analysis. A.H. computed thermodynamic parameters for the stacking models. R.G.M. provided codes for maximum likelihood optimization. A.H.N. provided codes for clustering analysis. L.L., B.P., M.G., A.H. and V.M. interpreted the results. L.L. and V.M. wrote the paper with input from B.P. and M.G.

Correspondence to Valeria Molinero.

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Extended data figures and tables

Extended Data Figure 1 Evolution of the cubicity of crystallites along the 28,510 shooting points sampled with aimless shooting.

Different colours correspond to distinct sets of shooting points obtained through aimless shooting starting from different seeds (Methods section ‘Aimless shooting’). The first four sets from the left are seeded with configurations from two spontaneous crystallization trajectories at 205 K, in which ice crystallites are stacking-disordered. The last eight sets originate from two configurations of liquid water seeded with a sphere of hexagonal ice of critical size at 230 K. Configurations containing critical-sized crystallites (highlighted with small circles) span all values of cubicities. We note that all critical-sized crystallites are considered in the committor analysis, irrespective of their cubicity. In the last eight sets, ice crystallites in the seeded configurations start with C = 0 (hexagonal crystallites) and evolve through the aimless shooting procedure towards stacking-disordered crystallites. These results are consistent with the existence of a thermodynamic driving force towards stacking-disordered ice crystallites.

Extended Data Figure 2 Free energy difference between stacking-disordered and hexagonal ice computed with the 1D stacking model.

ΔGsd-Ih(N,C) is plotted as a function of cubicity, for a variety of crystal sizes N and their corresponding number of layers in the crystals (see labels on the right side). The zoomed area shows the crossover to stable hexagonal crystallites for crystallites with 22 layers.

Extended Data Figure 3 Probability distributions of radius of gyration for cubic, hexagonal and stacking-disordered ice crystallites.

Radii of gyration evaluated for critical-sized ice crystallites, with N* = 356 ± 5 identified by RC7 = NCHI (that is, accounting only for the cubic, hexagonal and interfacial ice in the ice nucleus). The crystallites were collected from the equilibrium path sampling trajectories used to compute the free energy landscape. The blue line corresponds to hexagonal crystallites (C = 0 to 0.15) sampled over about 900 configurations, the green line corresponds to stacking-disordered crystallites (C = 0.6 ± 0.05) sampled over about 7,000 configurations and the red line corresponds to cubic crystallites (C = 0.9 to 1) sampled over about 400 configurations. The shape of the crystallites is independent of their cubicity, and not far from that of a perfect sphere with 350 water molecules, Rgsphere = 10.6 Å.

Extended Data Figure 4 Probability distribution of cubicity for crystallites with stacking in one or two directions in the 2D lattice model.

The black line shows that the distribution of cubicity in the equilibrium ensemble of crystallites with 33 cells (corresponding to NCH ≈ 200 water molecules) is slightly shifted towards cubic-rich crystallites. The blue line shows the distribution of cubicity for the subset of crystallites that have stacking of cubic and hexagonal layers along one direction only (the blue arrow points to a snapshot of a typical configuration with 1D stacking). The distribution of crystallites with 1D stacking is slightly shifted towards hexagonal-rich crystallites. The red line shows the distribution of cubicity for the subset of crystallites that have stacking of cubic and hexagonal layers along two directions (the red arrow points to a typical configuration with 2D stacking). This subset is defined as crystallites with hexagonal regions of at least four cells that are connected horizontally to a cubic region, and hexagonal regions of at least four cells connected vertically to a cubic region. The distribution of crystallites with 2D stacking is clearly shifted towards large values of cubicity, indicating that the occurrence of crystallites with two stacking directions causes the shift of the distribution of the entire ensemble of crystallites (black line). Stacking-disordered crystallites with an equivalent biasing mechanism towards the cubic polymorph may be expected in other materials that present cubic and hexagonal polymorphs with similar bulk free energies and for which the cubic and hexagonal layers can be seamlessly stacked.

Extended Data Figure 5 Randomly selected crystallites generated with the 2D lattice model.

All crystallites contain 33 lattice cells (equivalent to NCH = 198) as in Extended Data Figure 4 and have cubicity C = 0.60 ± 0.03. Red cells correspond to cubic ice, blue cells to hexagonal ice, and grey cells to liquid. Crystallites in snapshots a, b, d, l, m and o have stacking in two directions, while those in snapshots c, e, f, g, h, i, j, k, n and p display stacking in one direction.

Extended Data Figure 6 Cubicity of ice crystallites predicted with the 2D stacking model as a function of crystal size.

The red line corresponds to ΔGIc-Ih = 627 J mol−1, the difference in stability between cubic and hexagonal ice predicted by recent density functional theory calculations28; the black line corresponds to ΔGIc-Ih = 155 J mol−1, the upper limit47 proposed based on vapour pressures of stacking-disordered ices. Most probable values of cubicity for a given system size, that is, cubicities along the minimum free energy path, are calculated according to Methods section ‘ΔGsd-Ih using a 2D lattice stacking model’.

Extended Data Table 1 Comparison of thermodynamic properties for mW and water
Extended Data Table 2 Maximum likelihood scores for each candidate reaction coordinate RC, and predicted value of the coordinate at the transition state, RC*
Extended Data Table 3 Overlap of the transition state ensembles of four of the top size reaction coordinates of Extended Data Table 2

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Lupi, L., Hudait, A., Peters, B. et al. Role of stacking disorder in ice nucleation. Nature 551, 218–222 (2017). https://doi.org/10.1038/nature24279

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