^{1,a)}, Jérôme Delhommelle

^{1,b)}and Claude Millot

^{1,c)}

### Abstract

We report on a molecular simulation study of the homogeneous nucleation of in the supercooled liquid at low pressure and for degrees of supercooling ranging from 32% to 60%. In all cases, regardless of the degree of supercooling, the structure of the crystal nuclei is that of the Pa3 phase, the thermodynamically stable phase. For the more moderate degree of supercooling of 32%, the nucleation is an activated process and requires a method to sample states of high free energy. In this work, we apply a series of bias potentials, which promote the ordering of the centers of mass of the molecules and allow us to gradually grow crystal nuclei. The reliability of the results so obtained is assessed by studying the evolution of the nuclei in the absence of any bias potential, and by determining their probability of growth. We estimate that the size of the critical nucleus, for which the probability of growth is 0.5, is molecules. Throughout the nucleation process, the crystal nuclei clearly exhibit a Pa3 structure, in apparent contradiction with Ostwald’s rule of stages. The other polymorphs have a much larger free energy. This makes their formation highly unlikely and accounts for the fact that the nucleation of proceeds directly in the stable Pa3 structure.

Financial support from Sanofi-Aventis is gratefully acknowledged. The authors also acknowledge the Centre Informatique National de l’Enseignement Supérieur for a substantial allocation of computer time.

I. INTRODUCTION

II. SIMULATION METHODS

A. Structural features of liquid, Pa3 and Cmca

B. Determination of the melting temperature

C. Simulation of the homogeneous nucleation process

III. NUCLEATION FROM STRONGLY SUPERCOOLED LIQUIDS

IV. NUCLEATION FROM A MODERATELY SUPERCOOLED LIQUID

V. DISCUSSION AND CONCLUSION

### Key Topics

- Carbon dioxide
- 40.0
- Nucleation
- 31.0
- Crystallization
- 25.0
- Crystal structure
- 23.0
- Free energy
- 17.0

## Figures

Distributions of the local order parameters , , , and for liquid (straight line), Pa3 (dashed line), and Cmca (dotted line) at .

Distributions of the local order parameters , , , and for liquid (straight line), Pa3 (dashed line), and Cmca (dotted line) at .

Distributions of the orientational order parameters and for liquid (straight line), Pa3 (dashed line), and Cmca (dotted line) at .

Distributions of the orientational order parameters and for liquid (straight line), Pa3 (dashed line), and Cmca (dotted line) at .

Time evolution of the total potential energy, density, and global order parameter at (circles), 250 K (squares), and 260 K (triangles). Each symbol stands for an average taken over 4 ps.

Time evolution of the total potential energy, density, and global order parameter at (circles), 250 K (squares), and 260 K (triangles). Each symbol stands for an average taken over 4 ps.

Time evolution of the total potential energy, density, global order parameter , and number of solidlike molecules for a system of 864 molecules at (circles), 120 K (squares), and 125 K (triangles). Each symbol stands for an average taken over 50 ps.

Time evolution of the total potential energy, density, global order parameter , and number of solidlike molecules for a system of 864 molecules at (circles), 120 K (squares), and 125 K (triangles). Each symbol stands for an average taken over 50 ps.

Time evolution of the number of solidlike molecules for a system of 864 molecules at (straight line), 120 K (dashed line), and 125 K (dotted line).

Time evolution of the number of solidlike molecules for a system of 864 molecules at (straight line), 120 K (dashed line), and 125 K (dotted line).

Distributions of local translational (, , , and ) and orientational ( and ) order parameters for crystals obtained at (straight line), 120 K (dashed line), and 125 K (dotted line).

Distributions of local translational (, , , and ) and orientational ( and ) order parameters for crystals obtained at (straight line), 120 K (dashed line), and 125 K (dotted line).

Time evolution of the total potential energy, density, global order parameter , and number of solidlike molecules for a system of 864 molecules at (circles) and 130 K (squares). Each symbol stands for an average taken over 100 ps.

Time evolution of the total potential energy, density, global order parameter , and number of solidlike molecules for a system of 864 molecules at (circles) and 130 K (squares). Each symbol stands for an average taken over 100 ps.

Distributions of the size of the largest nucleus obtained in a system of 4000 molecules obtained from twelve umbrella sampling simulations. refer to the bias potential applied to the system.

Distributions of the size of the largest nucleus obtained in a system of 4000 molecules obtained from twelve umbrella sampling simulations. refer to the bias potential applied to the system.

Growth probability for a crystal nucleus during an unbiased molecular dynamics simulation.

Growth probability for a crystal nucleus during an unbiased molecular dynamics simulation.

Distributions of local translational (, , , and ) and orientational ( and ) order parameters for pre-critical (bias —straight line), critical (bias —dashed line), and postcritical (bias —dotted line) nuclei.

Distributions of local translational (, , , and ) and orientational ( and ) order parameters for pre-critical (bias —straight line), critical (bias —dashed line), and postcritical (bias —dotted line) nuclei.

Snapshots of configurations containing, from left to right, a precritical nucleus (bias ), a critical nucleus (bias ), or a postcritical nucleus (bias ). The larger sticks represent solidlike molecules while the smaller sticks stand for liquidlike molecules.

Snapshots of configurations containing, from left to right, a precritical nucleus (bias ), a critical nucleus (bias ), or a postcritical nucleus (bias ). The larger sticks represent solidlike molecules while the smaller sticks stand for liquidlike molecules.

## Tables

Parameters for the umbrella sampling simulations. is the radius of the sphere on which the bias is applied, the value for the global order parameter imposed in the sphere, time the overall duration of the molecular dynamics simulations, the average number of molecules belonging to the subsystem, the average size of the largest nucleus, the average value measured for the order parameter in the sphere, the average value measured for the order parameter in the whole system. Standard deviations are given in parentheses. We use a force constant in all these simulations.

Parameters for the umbrella sampling simulations. is the radius of the sphere on which the bias is applied, the value for the global order parameter imposed in the sphere, time the overall duration of the molecular dynamics simulations, the average number of molecules belonging to the subsystem, the average size of the largest nucleus, the average value measured for the order parameter in the sphere, the average value measured for the order parameter in the whole system. Standard deviations are given in parentheses. We use a force constant in all these simulations.

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