^{1,a)}, Matthew Dennison

^{2}and Andrew Masters

^{2}

### Abstract

Monte Carlo simulations are used to map out the complete phase diagram of hard body UFO systems, in which the particles are composed of a concentric sphere and thin disk. The equation of state and phase behavior are determined for a range of relative sizes of the sphere and disk. We show that for relatively large disks, nematic and solid phases are observed in addition to the isotropic fluid. For small disks, two different solid phases exist. For intermediate sizes, only a disordered fluid phase is observed. The positional and orientational structure of the various phases are examined. We also compare the equations of state and the nematic-isotropic coexistence densities with those predicted by an extended Onsager theory using virial coefficients up to .

M.A.B. is grateful to the Royal Society for financial support and for funding the computer cluster used for these simulations. A.J.M. and M.D. gratefully acknowledge support from EPSRC.

I. INTRODUCTION

II. SUMMARY OF THE PHASE BEHAVIOR OF THE HARD BODY UFO

III. SIMULATIONS AND THEORY

A. Monte Carlo simulations

B. Extended Onsager theory for the nematic-isotropic transition

IV. EQUATIONS OF STATE AND PHASE CHARACTERIZATION

A. Region 1:

B. Region 2:

C. Region 3:

V. CONCLUSIONS

### Key Topics

- Equations of state
- 38.0
- Crystal structure
- 19.0
- Phase diagrams
- 11.0
- Particle distribution functions
- 9.0
- Crystalline solids
- 8.0

## Figures

The phase diagram for hard body UFOs as a function of the relative size of the disk compared to the sphere, , as determined by constant pressure Monte Carlo simulations. The solid lines are a guide to the eye. The dotted vertical lines indicate the boundaries between the regions where different types of behaviors are observed. The density is shown as both and for comparison to the hard sphere and hard thin disk limit, respectively. The phases are labeled as follows: is the isotropic phase; is the nematic phase; is a cubic crystal and is a tetragonal crystal observed only for small ring systems; and is a hexagonal close packed crystal observed for large ring systems.

The phase diagram for hard body UFOs as a function of the relative size of the disk compared to the sphere, , as determined by constant pressure Monte Carlo simulations. The solid lines are a guide to the eye. The dotted vertical lines indicate the boundaries between the regions where different types of behaviors are observed. The density is shown as both and for comparison to the hard sphere and hard thin disk limit, respectively. The phases are labeled as follows: is the isotropic phase; is the nematic phase; is a cubic crystal and is a tetragonal crystal observed only for small ring systems; and is a hexagonal close packed crystal observed for large ring systems.

Equation of state for the UFO model with . (White points) simulation data for compression from the isotropic phase. (Gray points) compression/expansion from the fcc solid phase. (Black point) expansion from the solid phase. Dashed lines indicate the virial expansion of the pressure for the fluid phase, truncated at (large dashes, bottom) to (small dashes, top).

Equation of state for the UFO model with . (White points) simulation data for compression from the isotropic phase. (Gray points) compression/expansion from the fcc solid phase. (Black point) expansion from the solid phase. Dashed lines indicate the virial expansion of the pressure for the fluid phase, truncated at (large dashes, bottom) to (small dashes, top).

The radial distribution function for the hard body UFO model with in (a) the isotropic phase at , (b) the phase at , and (c) the phase at . The solid lines show the structure for the UFO model; the dashed lines (where shown) indicate the structure for the hard sphere model at the equivalent density in the same phase. The distance is shown as .

The radial distribution function for the hard body UFO model with in (a) the isotropic phase at , (b) the phase at , and (c) the phase at . The solid lines show the structure for the UFO model; the dashed lines (where shown) indicate the structure for the hard sphere model at the equivalent density in the same phase. The distance is shown as .

Snapshots of the UFO system in (a) the isotropic phase , (b) the phase , and [(c) and (d)] two different views of the phase .

Snapshots of the UFO system in (a) the isotropic phase , (b) the phase , and [(c) and (d)] two different views of the phase .

The even rank orientational order parameters for the (gray points) and (black points) branches of the equation of state: (circles) , (triangles) , and (squares) . The orientational distribution function in the (b) phase and (c) phase; here the (100), (010), and (001) axes are along , , and , respectively. The dark regions show high probability, and the light regions indicate low probability.

The even rank orientational order parameters for the (gray points) and (black points) branches of the equation of state: (circles) , (triangles) , and (squares) . The orientational distribution function in the (b) phase and (c) phase; here the (100), (010), and (001) axes are along , , and , respectively. The dark regions show high probability, and the light regions indicate low probability.

The equation of state for the UFO model. (White points) simulation data for compression from the isotropic phase. (Black points) expansion from the solid phase. The vertical lines indicate the location of the NI transition. Dashed lines indicate the virial expansion of the pressure in the isotropic and nematic phases, and the location of the transition between them, truncated at (large dashes, bottom) through to (smaller dashes, top).

The equation of state for the UFO model. (White points) simulation data for compression from the isotropic phase. (Black points) expansion from the solid phase. The vertical lines indicate the location of the NI transition. Dashed lines indicate the virial expansion of the pressure in the isotropic and nematic phases, and the location of the transition between them, truncated at (large dashes, bottom) through to (smaller dashes, top).

Snapshots of [(a) and (b)] the nematic phase of the UFO model at and [(c) and (d)] the hexagonal phase at .

Snapshots of [(a) and (b)] the nematic phase of the UFO model at and [(c) and (d)] the hexagonal phase at .

The radial distribution function for the hard body UFO model with in (a) the isotropic phase at ,(b) the nematic phase at , and (c) the phase at . The solid lines show the structure for the UFO model, and the dashed and dotted lines (where shown) indicate the structure for the hard sphere model and the infinitely thin disk model, respectively, at the equivalent density. The distance is shown as .

The radial distribution function for the hard body UFO model with in (a) the isotropic phase at ,(b) the nematic phase at , and (c) the phase at . The solid lines show the structure for the UFO model, and the dashed and dotted lines (where shown) indicate the structure for the hard sphere model and the infinitely thin disk model, respectively, at the equivalent density. The distance is shown as .

The equation of state for the UFO model. (Whitepoints) simulation data for compression from the isotropic phase. (Black points) expansion from the solid phase. The vertical lines indicate the location of the NI transition. The inset shows the equation of state at higher pressures. Dashed lines indicate the virial expansion of the pressure in the isotropic and nematic phases, and the location of the transition between them, truncated at (larger dashes, bottom) through to (smaller dashes, top). Note that the results with truncation at are not shown, since no transition is found at this level of the theory.

The equation of state for the UFO model. (Whitepoints) simulation data for compression from the isotropic phase. (Black points) expansion from the solid phase. The vertical lines indicate the location of the NI transition. The inset shows the equation of state at higher pressures. Dashed lines indicate the virial expansion of the pressure in the isotropic and nematic phases, and the location of the transition between them, truncated at (larger dashes, bottom) through to (smaller dashes, top). Note that the results with truncation at are not shown, since no transition is found at this level of the theory.

The structure of the UFO model at high pressure .

The structure of the UFO model at high pressure .

The radial distribution function for the hard body UFO model with in (a) the isotropic phase at and (b) the nematic phase at . The solid lines show the structure for the UFO model, and the dashed and dotted lines indicate the structure for the hard sphere model and the infinitely thin disk model, respectively, at the equivalent density. (c) The radial distribution function for the solidlike phase at (thin line) and (thick line). The distance is shown as .

The radial distribution function for the hard body UFO model with in (a) the isotropic phase at and (b) the nematic phase at . The solid lines show the structure for the UFO model, and the dashed and dotted lines indicate the structure for the hard sphere model and the infinitely thin disk model, respectively, at the equivalent density. (c) The radial distribution function for the solidlike phase at (thin line) and (thick line). The distance is shown as .

The coexistence densities for region 3 as a function of ring size. The black points indicate the NI coexistence densities as determined by computer simulation. Lines indicate the coexistence densities determined by Onsager theory. The virial expansion is truncated at (large dashes) to (small dashes); truncation at is shown as the solid lines.

The coexistence densities for region 3 as a function of ring size. The black points indicate the NI coexistence densities as determined by computer simulation. Lines indicate the coexistence densities determined by Onsager theory. The virial expansion is truncated at (large dashes) to (small dashes); truncation at is shown as the solid lines.

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