^{1}, C. Vega

^{1}, J. P. K. Doye

^{2}and A. A. Louis

^{3}

### Abstract

The phase diagram for a system of model anisotropic particles with six attractive patches in an octahedral arrangement has been computed. This model for a relatively narrow value of the patch width where the lowest-energy configuration of the system is a simple cubic crystal. At this value of the patch width, there is no stable vapor-liquid phase separation, and there are three other crystalline phases in addition to the simple cubic crystal that is most stable at low pressure. First, at moderate pressures, it is more favorable to form a body-centered-cubic crystal, which can be viewed as two interpenetrating, and almost noninteracting, simple cubic lattices. Second, at high pressures and low temperatures, an orientationally ordered face-centered-cubic structure becomes favorable. Finally, at high temperatures a face-centered-cubic plastic crystal is the most stable solid phase.

This work was funded by Dirección General de Investigación (Grant No. FIS2004-06227-C02-02) and Comunidad Autónoma de Madrid (Grant No. S-0505/ESP/0229). The authors would also like to acknowledge financial help from the European Union Project No. MTKD-CT-2004-509249 and from the Universidad Computense de Madrid Project No. 910570. E.G.N. wishes to thank the Ministerio de Educación y Ciencia and the Universidad Complutense de Madrid for a Juan de la Cierva fellowship and J.P.K.D. and A.A.L. are grateful to the Royal Society for financial support. We would like to thank David J. Wales and Mark A. Miller for useful discussions, and Emanuela Zaccarelli and Francesco Sciortino for organizing the interesting workshop entitled “Patchy Colloids, Proteins and Network Forming Liquids: Analogies and New Insights from Computer Simulations,” held at the CECAM. They are also grateful to F. Sciortino for sending them a copy of Ref. 40 prior to publication.

I. INTRODUCTION

II. METHOD

A. Model

B. Solid structures

C. Equation of state for the fluid and solid phases

D. Free energy calculations

E. Coexistence lines

F. Direct coexistence simulations

III. RESULTS

IV. CONCLUSIONS

### Key Topics

- Crystal structure
- 27.0
- Anisotropy
- 20.0
- Free energy
- 20.0
- Melting
- 17.0
- Crystallization
- 14.0

## Figures

A schematic representation of the geometry of the interaction between two particles. For clarity, we depict a two-dimensional analog of the three-dimensional model used in this work. In this two-dimensional model, the particles have four patches arranged regularly with their directions described by the patch vectors, . In the particular case shown in the figure, patch 4 on particle interacts with patch 2 on particle because they are the closest to the interparticle vector.

A schematic representation of the geometry of the interaction between two particles. For clarity, we depict a two-dimensional analog of the three-dimensional model used in this work. In this two-dimensional model, the particles have four patches arranged regularly with their directions described by the patch vectors, . In the particular case shown in the figure, patch 4 on particle interacts with patch 2 on particle because they are the closest to the interparticle vector.

(Color online) Unit cells of the (a) bcc and (b) orientationally ordered fcc-o structures. In both cases the patches are aligned with the second neighbors.

(Color online) Unit cells of the (a) bcc and (b) orientationally ordered fcc-o structures. In both cases the patches are aligned with the second neighbors.

Determination of the coexistence point between the fluid (solid line) and the bcc (dashed line) phases at .

Determination of the coexistence point between the fluid (solid line) and the bcc (dashed line) phases at .

(Color) The variation of the internal energy per particle during simulations of a box containing the bcc solid in contact with the fluid at . A few trajectories at different temperatures are shown.

(Color) The variation of the internal energy per particle during simulations of a box containing the bcc solid in contact with the fluid at . A few trajectories at different temperatures are shown.

(Color) (a) Snapshot of the initial configuration of the simulation box containing the bcc and the fluid phases in contact. (b) and (c) Snapshots of the final configurations for and , respectively. The pressure was set to .

(Color) (a) Snapshot of the initial configuration of the simulation box containing the bcc and the fluid phases in contact. (b) and (c) Snapshots of the final configurations for and , respectively. The pressure was set to .

Dependence of the vapor-liquid coexistence curve on the patch width . At lower we were unable to find the (metastable) coexistence curves.

Dependence of the vapor-liquid coexistence curve on the patch width . At lower we were unable to find the (metastable) coexistence curves.

phase diagram of our octahedral six-patch particle system (with ). Labels show the region of stability of each phase. The points at which the reentrant behavior occurs are indicated with a cross.

phase diagram of our octahedral six-patch particle system (with ). Labels show the region of stability of each phase. The points at which the reentrant behavior occurs are indicated with a cross.

phase diagram of the octahedral six-patch particle system (with ). Labels show the region of stability of each phase. The reentrant behavior occurs at the points at which there is a change of sign in the slope of the phase boundaries. These points are indicated by a cross. The black circle shows the point where inverse melting occurs.

phase diagram of the octahedral six-patch particle system (with ). Labels show the region of stability of each phase. The reentrant behavior occurs at the points at which there is a change of sign in the slope of the phase boundaries. These points are indicated by a cross. The black circle shows the point where inverse melting occurs.

(Color online) Detailed view of the phase diagram in the region of the sc-liquid-bcc triple point. The triple point is shown with a dashed line. Labels indicate the region of stability of each phase. The point at which reentrant behavior occurs is indicated by a cross.

(Color online) Detailed view of the phase diagram in the region of the sc-liquid-bcc triple point. The triple point is shown with a dashed line. Labels indicate the region of stability of each phase. The point at which reentrant behavior occurs is indicated by a cross.

Equation of state for the liquid phase (circles) and the sc crystal (asterisks) at .

Equation of state for the liquid phase (circles) and the sc crystal (asterisks) at .

## Tables

Coefficients obtained by fitting the integrand of Eq. (5) to a polynomial of degree six [Eq. (14)].

Coefficients obtained by fitting the integrand of Eq. (5) to a polynomial of degree six [Eq. (14)].

Free energy of the solid phases, as obtained by thermodynamic integration from the Einstein crystal. The free energies (, , and ) and the lattice energy are given in units of .

Free energy of the solid phases, as obtained by thermodynamic integration from the Einstein crystal. The free energies (, , and ) and the lattice energy are given in units of .

Coefficients of the polynomial fit to the equation of state of the solid phases. The points were fitted to a third-degree polynomial, except for the bcc structure at , for which a fourth-degree polynomial significantly improves the fit.

Coefficients of the polynomial fit to the equation of state of the solid phases. The points were fitted to a third-degree polynomial, except for the bcc structure at , for which a fourth-degree polynomial significantly improves the fit.

Coexistence points obtained using the thermodynamic integration method. The points marked with an asterisk were used as the starting points for the Gibbs–Duhem approach. The other points served to test our calculations.

Coexistence points obtained using the thermodynamic integration method. The points marked with an asterisk were used as the starting points for the Gibbs–Duhem approach. The other points served to test our calculations.

Melting points obtained using the direct coexistence method. For comparison, the coexistence points obtained from the free energy calculations (see Table IV) are also shown. Note that the sc-fluid coexistence point was obtained using the Gibbs-Duhem method (see Table VI).

Melting points obtained using the direct coexistence method. For comparison, the coexistence points obtained from the free energy calculations (see Table IV) are also shown. Note that the sc-fluid coexistence point was obtained using the Gibbs-Duhem method (see Table VI).

Some of the coexistence points obtained with the Gibbs–Duhem method.

Some of the coexistence points obtained with the Gibbs–Duhem method.

Thermodynamic states for the two triple points in the phase diagram.

Thermodynamic states for the two triple points in the phase diagram.

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