^{1}and G. Odriozola

^{1,a)}

### Abstract

Following previous work [G. Odriozola and F. de J. Guevara-Rodríguez, J. Chem. Phys.134, 201103 (2011)]10.1063/1.3596728, the replica exchange Monte Carlo technique is used to produce the equation of state of hard 1:5 aspect-ratio oblate ellipsoids for a wide density range. Here, in addition to the analytical approximation of the overlap distance given by Berne and Pechukas (BP) and the exact numerical solution of Perram and Wertheim, we tested a simple modification of the original BP approximation (MBP) which corrects the known T-shape mismatch of BP for all aspect ratios. We found that the MBP equation of state shows a very good quantitative agreement with the exact solution. The MBP analytical expression allowed us to study size effects on the previously reported results. For the thermodynamic limit, we estimated the exact 1:5 hard ellipsoid isotropic-nematic transition at the volume fraction 0.343 ± 0.003, and the nematic-solid transition in the volume fraction interval (0.592 ± 0.006) − (0.634 ± 0.008).

The authors thank Projects Nos. Y.00116 and Y.00119 SENER-CONACyT for financial support. Authors also thank Tito González Rodríguez for reading the paper.

I. INTRODUCTION

II. HARD ELLIPSOIDAL MODELS

A. Berne and Pechukas

B. Exact numerical solution

C. Modified Berne and Pechukas

D. Comparing BP and MBP to the exact numerical solution

III. REPLICA EXCHANGE MONTE CARLO

IV. RESULTS

A. Exact vs analytical overlap distance

B. Size effects

C. Structure

V. CONCLUSIONS

### Key Topics

- Crystal structure
- 15.0
- High pressure
- 11.0
- Equations of state
- 9.0
- Exact solutions
- 9.0
- Numerical solutions
- 9.0

## Figures

Two ellipsoids at different contact configurations. The exact center-center distance at contact, σ, is given as a function of σ_{∥} and σ_{⊥}. The relations among the normal versors , and the center-center vector are also shown. The BP approximation produces the exact solution for all cases but e. MBP yields the exact solution for all shown cases.

Two ellipsoids at different contact configurations. The exact center-center distance at contact, σ, is given as a function of σ_{∥} and σ_{⊥}. The relations among the normal versors , and the center-center vector are also shown. The BP approximation produces the exact solution for all cases but e. MBP yields the exact solution for all shown cases.

(a) Appearance frequency, F, against the percentage of deviation of the analytical overlap distance with the corresponding exact distance, , for the 1:5 oblate case. The histogram is built by considering *N* _{ c } = 1 × 10^{8} random positions and orientations. (b) Average deviation, (bullets), and the optimized γ parameter, γ_{ min } (open symbols), for several aspect-ratios. The vertical dark line at 1:5 points out the case shown in panel (a). Dark (black) solid lines, light (cyan) solid lines, and (red) dashed lines correspond to the BP, MBP with γ = 1, and MBP with γ_{ min } analytical expressions, respectively.

(a) Appearance frequency, F, against the percentage of deviation of the analytical overlap distance with the corresponding exact distance, , for the 1:5 oblate case. The histogram is built by considering *N* _{ c } = 1 × 10^{8} random positions and orientations. (b) Average deviation, (bullets), and the optimized γ parameter, γ_{ min } (open symbols), for several aspect-ratios. The vertical dark line at 1:5 points out the case shown in panel (a). Dark (black) solid lines, light (cyan) solid lines, and (red) dashed lines correspond to the BP, MBP with γ = 1, and MBP with γ_{ min } analytical expressions, respectively.

(a) Equations of state of the exact and analytical hard 1:5 oblate ellipsoidal models, *Z*(φ) (φ is the volume fraction). (b) Isothermal compressibility obtained from density fluctuations, χ(φ). (c) Order parameter, *Q* _{6}(φ). For all panels, black circles, cyan squares, red crosses, and blue plus symbols correspond to the exact PW overlap distance, and the analytical solutions given by BP, MBP with γ = 1, and MBP with γ = γ_{ min }, respectively. All data correspond to *N* = 100.

(a) Equations of state of the exact and analytical hard 1:5 oblate ellipsoidal models, *Z*(φ) (φ is the volume fraction). (b) Isothermal compressibility obtained from density fluctuations, χ(φ). (c) Order parameter, *Q* _{6}(φ). For all panels, black circles, cyan squares, red crosses, and blue plus symbols correspond to the exact PW overlap distance, and the analytical solutions given by BP, MBP with γ = 1, and MBP with γ = γ_{ min }, respectively. All data correspond to *N* = 100.

(a) Probability distribution functions (PDFs) of volume fraction fluctuations for each of the *n* _{ r } = 64 pressure values and for *N* = 200. (b) Equations of state, *Z*(φ). (c) Isothermal compressibilities, χ(φ), obtained from density fluctuations. (d) Order parameters, *Q* _{6}(φ). For panels (b)–(d), dark circles and light squares correspond to *N* = 200 and *N* = 100, respectively. Vertical dotted lines highlight the *N* = 200 transitions. All data correspond to the analytical MBP case with γ = γ_{ min }.

(a) Probability distribution functions (PDFs) of volume fraction fluctuations for each of the *n* _{ r } = 64 pressure values and for *N* = 200. (b) Equations of state, *Z*(φ). (c) Isothermal compressibilities, χ(φ), obtained from density fluctuations. (d) Order parameters, *Q* _{6}(φ). For panels (b)–(d), dark circles and light squares correspond to *N* = 200 and *N* = 100, respectively. Vertical dotted lines highlight the *N* = 200 transitions. All data correspond to the analytical MBP case with γ = γ_{ min }.

Radial distribution functions (black lines), *g*(*r*), and their corresponding order parameter (cyan lines), *p*(*r*), for different pressures (increasing from left to right). Panels (a)–(d) correspond to the isotropic, nematic, crystal A, and crystal B structures. The insets show the corresponding snapshots.

Radial distribution functions (black lines), *g*(*r*), and their corresponding order parameter (cyan lines), *p*(*r*), for different pressures (increasing from left to right). Panels (a)–(d) correspond to the isotropic, nematic, crystal A, and crystal B structures. The insets show the corresponding snapshots.

Detail of the different crystal structures found in the *N* = 200 system (cases (d) and (e) of Fig. 4). The shown α angles, defined as where is the versor pointing along the stack axis, are close to 62° and 44° for crystal A and B, respectively.

Detail of the different crystal structures found in the *N* = 200 system (cases (d) and (e) of Fig. 4). The shown α angles, defined as where is the versor pointing along the stack axis, are close to 62° and 44° for crystal A and B, respectively.

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