^{1}and Jonathan P. K. Doye

^{1,a)}

### Abstract

We simulate the homogeneous nucleation of ice from supercooled liquid water at 220 K in the isobaric-isothermal ensemble using the MW monatomic water potential. Monte Carlo simulations using umbrella sampling are performed in order to determine the nucleationfree energy barrier. We find the Gibbs energy profile to be relatively consistent with that predicted by classical nucleation theory; the free energy barrier to nucleation was determined to be ∼18 *k* _{B} *T* and the critical nucleus comprised ∼85 ice particles. Growth from the supercooled liquid gives clusters that are predominantly cubic, whilst starting with a pre-formed subcritical nucleus of cubic or hexagonal ice results in the growth of predominantly that phase of ice only.

We would like to thank the Engineering and Physical Sciences Research Council for financial support and A. J. Williamson and F. Romano for helpful discussions.

I. INTRODUCTION

II. SIMULATION METHODS

III. RESULTS

A. Free energy profile

B. Nucleation pathway

C. Nucleation rate

IV. CONCLUSIONS

### Key Topics

- Ice
- 112.0
- Nucleation
- 88.0
- Free energy
- 68.0
- Crystal growth
- 23.0
- Homogeneous nucleation
- 18.0

## Figures

A typical probability distribution for all pairs of , where particles *i* and *j* are within 3.6 Å of each other. The three states depicted have all been equilibrated at 220 K and the ice structures are not, therefore, “perfect.” This figure is analogous to those in Refs. 64 and 68; we reproduce it here for convenience.

A typical probability distribution for all pairs of , where particles *i* and *j* are within 3.6 Å of each other. The three states depicted have all been equilibrated at 220 K and the ice structures are not, therefore, “perfect.” This figure is analogous to those in Refs. 64 and 68; we reproduce it here for convenience.

The free energy profile of MW nucleation as a function of the size of the largest crystalline cluster in the system. Simulation results from different windows are depicted in alternating colours to show their overlap. The dashed line corresponds to the classical nucleation theory prediction; dotted lines depict fits to the simulation data. The free energy profiles for ice nucleation seeded with hexagonal crystal clusters (I_{h}) and ice nucleation directly from the supercooled liquid (FG) are shown.

The free energy profile of MW nucleation as a function of the size of the largest crystalline cluster in the system. Simulation results from different windows are depicted in alternating colours to show their overlap. The dashed line corresponds to the classical nucleation theory prediction; dotted lines depict fits to the simulation data. The free energy profiles for ice nucleation seeded with hexagonal crystal clusters (I_{h}) and ice nucleation directly from the supercooled liquid (FG) are shown.

Representative nucleation snapshots from umbrella sampling simulations. Particles within 3.6 Å are connected with lines; the lines are yellow within the largest ice cluster and cyan elsewhere (in the top row only). Particles are not connected across the periodic boundary. Spheres represent particles classified as ice: red spheres correspond to cubic ice, orange spheres correspond to hexagonal ice, and pink spheres (in the top row only) correspond to ice particles not within the largest crystalline cluster. In (a), a 30-particle cluster as nucleated from the supercooled liquid is shown. In (b), an 83-particle cluster as grown in a simulation initially seeded with an equilibrated cubic ice cluster is shown. In (c), a 165-particle cluster as grown in a simulation initially seeded with an equilibrated hexagonal ice cluster is shown. In each case, the top and bottom pictures depict the same cluster from different perspectives; one within the liquid framework and one showing solely the largest crystalline cluster.

Representative nucleation snapshots from umbrella sampling simulations. Particles within 3.6 Å are connected with lines; the lines are yellow within the largest ice cluster and cyan elsewhere (in the top row only). Particles are not connected across the periodic boundary. Spheres represent particles classified as ice: red spheres correspond to cubic ice, orange spheres correspond to hexagonal ice, and pink spheres (in the top row only) correspond to ice particles not within the largest crystalline cluster. In (a), a 30-particle cluster as nucleated from the supercooled liquid is shown. In (b), an 83-particle cluster as grown in a simulation initially seeded with an equilibrated cubic ice cluster is shown. In (c), a 165-particle cluster as grown in a simulation initially seeded with an equilibrated hexagonal ice cluster is shown. In each case, the top and bottom pictures depict the same cluster from different perspectives; one within the liquid framework and one showing solely the largest crystalline cluster.

The proportion of core ice particles classified as hexagonal for the set of simulations in which the crystalline cluster was grown directly from the supercooled liquid. Error bars show the standard deviation for the population of configurations at each ice cluster size.

The proportion of core ice particles classified as hexagonal for the set of simulations in which the crystalline cluster was grown directly from the supercooled liquid. Error bars show the standard deviation for the population of configurations at each ice cluster size.

A plot of two sphericity parameters against the size of the largest crystalline cluster calculated for snapshots of the system along the local order parameter used to drive nucleation in our system. Error bars show the standard deviation for the population of configurations at each cluster size. The diagram shows results for nucleation from a supercooled liquid; the other two systems behave analogously.

A plot of two sphericity parameters against the size of the largest crystalline cluster calculated for snapshots of the system along the local order parameter used to drive nucleation in our system. Error bars show the standard deviation for the population of configurations at each cluster size. The diagram shows results for nucleation from a supercooled liquid; the other two systems behave analogously.

(a) The global order parameters *Q* _{6} and ζ calculated along the local order parameter used to drive nucleation in our system. Error bars show the standard deviation for the population of configurations at each cluster size. The cyan solid line indicates the approximate location of the nucleation barrier as calculated by local order parameters. The results depicted here refer to the 576 particles around the centre of mass of the ice nucleus for comparison with Ref. 42, and not to the full 1400-particle system. (b) An analogous diagram for a single brute-force trajectory at 210 K.

(a) The global order parameters *Q* _{6} and ζ calculated along the local order parameter used to drive nucleation in our system. Error bars show the standard deviation for the population of configurations at each cluster size. The cyan solid line indicates the approximate location of the nucleation barrier as calculated by local order parameters. The results depicted here refer to the 576 particles around the centre of mass of the ice nucleus for comparison with Ref. 42, and not to the full 1400-particle system. (b) An analogous diagram for a single brute-force trajectory at 210 K.

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