^{1}, Jonathan P. K. Doye

^{1,a)}, Eva G. Noya

^{2}and Carlos Vega

^{3}

### Abstract

We present a local order parameter based on the standard Steinhardt–Ten Wolde approach that is capable both of tracking and of driving homogeneous icenucleation in simulations of all-atom models of water. We demonstrate that it is capable of forcing the growth of ice nuclei in supercooled liquid water simulated using the TIP4P/2005 model using over-biassed umbrella sampling Monte Carlo simulations. However, even with such an order parameter, the dynamics of icegrowth in deeply supercooled liquid water in all-atom models of water are shown to be very slow, and so the computation of free energy landscapes and nucleation rates remains extremely challenging.

We should like to thank the Engineering and Physical Sciences Research Council and the Dirección General de Investigación Científica y Técnica (Grant Nos. FIS2010-16159 and FIS2010-15502) for financial support.

I. INTRODUCTION

II. ORDER PARAMETERS IN NUCLEATION STUDIES

A. Global order parameters

B. Local order parameters

III. ORDER PARAMETERS FOR HOMOGENEOUS ICENUCLEATION

IV. SIMULATION DETAILS

V. NUCLEATION PATHWAYS

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Nucleation
- 89.0
- Ice
- 67.0
- Homogeneous nucleation
- 32.0
- Free energy
- 28.0
- Crystal growth
- 13.0

## Figures

A typical probability density distribution for all pairs of , where the centres of mass of molecules *i* and *j* are within 3.5 Å of each other. The three states depicted were equilibrated at 200 K (using the TIP4P/2005 water model) and the ice structures are not, therefore, “perfect”. This figure is analogous to those in Refs. 25,36,52.

A typical probability density distribution for all pairs of , where the centres of mass of molecules *i* and *j* are within 3.5 Å of each other. The three states depicted were equilibrated at 200 K (using the TIP4P/2005 water model) and the ice structures are not, therefore, “perfect”. This figure is analogous to those in Refs. 25,36,52.

An example of non-ice-like chain growth in TIP4P/2005 umbrella sampling simulations when using the order parameter without chain removal as described in the text. The system has 1000 molecules at 240 K, starting from a 24-molecule cluster. Molecules classified as being part of the largest crystalline cluster are shown in red and violet; there are 45 molecules in this cluster. Molecules whose centres of mass are within 3.5 Å are connected with lines. Molecules shown in violet would be removed from the largest crystalline cluster on application of the chain removal algorithm described in the text.

An example of non-ice-like chain growth in TIP4P/2005 umbrella sampling simulations when using the order parameter without chain removal as described in the text. The system has 1000 molecules at 240 K, starting from a 24-molecule cluster. Molecules classified as being part of the largest crystalline cluster are shown in red and violet; there are 45 molecules in this cluster. Molecules whose centres of mass are within 3.5 Å are connected with lines. Molecules shown in violet would be removed from the largest crystalline cluster on application of the chain removal algorithm described in the text.

Neighbour-averaged order parameters for systems of ices I_{h} and I_{c} and liquid water. All systems were equilibrated at 200 K using the TIP4P/2005 water model, and they contain different numbers of molecules. The neighbour cutoff distance was 3.5 Å.

Neighbour-averaged order parameters for systems of ices I_{h} and I_{c} and liquid water. All systems were equilibrated at 200 K using the TIP4P/2005 water model, and they contain different numbers of molecules. The neighbour cutoff distance was 3.5 Å.

Representative nucleation snapshots from umbrella sampling simulations of TIP4P/2005 water. In each case, two pictures depict the same cluster from different perspectives; one within the liquid framework (in cyan) and one showing solely the largest crystalline cluster. In the former, spheres represent centres of mass of molecules classified as ice: red spheres correspond to cubic ice, orange spheres correspond to hexagonal ice and pink spheres correspond to ice molecules not within the largest crystalline cluster. Pictures representing solely the largest cluster depict both the oxygen (red) and the hydrogens (white) of each molecule. In (a), an 82-molecule ice cluster grown from the supercooled liquid at 240 K is shown; in (b), a 73-molecule ice cluster grown from a small cluster of I_{h} ice at 240 K is shown; and in (c), a series of ice clusters of increasing size (comprising 23, 60, 77, 107, and 145 molecules from left to right) grown from a small cluster of I_{c} ice at 200 K is depicted. There are 1900 molecules in the system in (a) and the first three configurations of (c), and 2500 molecules in (b) and the last two configurations of (c). Simulations of nucleation from a hexagonal seed (shown in (b)) were undertaken using the hybrid Monte Carlo approach, and the rest by a standard Metropolis Monte Carlo approach. *p* = 1 bar.

Representative nucleation snapshots from umbrella sampling simulations of TIP4P/2005 water. In each case, two pictures depict the same cluster from different perspectives; one within the liquid framework (in cyan) and one showing solely the largest crystalline cluster. In the former, spheres represent centres of mass of molecules classified as ice: red spheres correspond to cubic ice, orange spheres correspond to hexagonal ice and pink spheres correspond to ice molecules not within the largest crystalline cluster. Pictures representing solely the largest cluster depict both the oxygen (red) and the hydrogens (white) of each molecule. In (a), an 82-molecule ice cluster grown from the supercooled liquid at 240 K is shown; in (b), a 73-molecule ice cluster grown from a small cluster of I_{h} ice at 240 K is shown; and in (c), a series of ice clusters of increasing size (comprising 23, 60, 77, 107, and 145 molecules from left to right) grown from a small cluster of I_{c} ice at 200 K is depicted. There are 1900 molecules in the system in (a) and the first three configurations of (c), and 2500 molecules in (b) and the last two configurations of (c). Simulations of nucleation from a hexagonal seed (shown in (b)) were undertaken using the hybrid Monte Carlo approach, and the rest by a standard Metropolis Monte Carlo approach. *p* = 1 bar.

The global order parameters *Q* _{6} and ζ calculated as a function of the size of the largest crystalline cluster, the order parameter used to drive nucleation in this work, for the system seeded with a cubic ice nucleus. Error bars show the standard deviation for the population of configurations at each cluster size. The results depicted here refer to the nearest 576 particles from the centre of mass of the ice nucleus. *T* = 200 K, *p* = 1 bar.

The global order parameters *Q* _{6} and ζ calculated as a function of the size of the largest crystalline cluster, the order parameter used to drive nucleation in this work, for the system seeded with a cubic ice nucleus. Error bars show the standard deviation for the population of configurations at each cluster size. The results depicted here refer to the nearest 576 particles from the centre of mass of the ice nucleus. *T* = 200 K, *p* = 1 bar.

MD simulations of melting. The starting point is a crystalline cluster comprising approximately 220 molecules embedded in supercooled liquid water. The curves exhibiting melting were simulated at 240 K, whilst the remaining ones were simulated at 200 K. These simulations entailed 2500 TIP4P/2005 water molecules. Note that the melting point of TIP4P/2005 ice is 252 K.^{61} *p* = 1 bar.

MD simulations of melting. The starting point is a crystalline cluster comprising approximately 220 molecules embedded in supercooled liquid water. The curves exhibiting melting were simulated at 240 K, whilst the remaining ones were simulated at 200 K. These simulations entailed 2500 TIP4P/2005 water molecules. Note that the melting point of TIP4P/2005 ice is 252 K.^{61} *p* = 1 bar.

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