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 × 108 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) 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.
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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