Hard ellipsoid model

2023-06-21T04:25:15Z

Godriozo:

[[Image:ellipsoid_red.png|thumb|right|A uniaxial prolate ellipsoid, a>b, b=c.]]
[[Image:oblate_1.png|thumb|right|A uniaxial oblate ellipsoid, a>c, a=b.]]
'''Hard ellipsoids''' represent a natural choice for an anisotropic model. Ellipsoids can be produced from an affine transformation of the [[hard sphere model]]. However, in difference to the hard sphere model, which has fluid and solid phases, the hard ellipsoid model is also able to produce a [[nematic phase]].
== Interaction Potential ==
The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by

:<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1</math>

where <math>a</math>, <math>b </math> and <math>c</math> define the lengths of the
axis.
==Overlap algorithm==
The most widely used overlap algorithm is that of Perram and Wertheim
<ref>[http://dx.doi.org/10.1016/0021-9991(85)90171-8 John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics '''58''' pp. 409-416 (1985)]</ref>.

==Geometric properties==
The mean radius of curvature is given by
<ref>[http://dx.doi.org/10.1063/1.472110 G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics '''105''' pp. 2429-2435 (1996)]</ref>
<ref>[http://dx.doi.org/10.1006/aphy.2001.6166 G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics '''294''' pp. 24-47 (2001)]</ref>

:<math>R= \frac{a}{2} \left[ \sqrt{\frac{1+\epsilon_b}{1+\epsilon_c}} + \sqrt \epsilon_c \left\{ \frac{1}{\epsilon_c} F(\varphi , k_1) + E(\varphi,k_1) \right\}\right],
</math>
and the surface area is given by
:<math>S= 2 \pi a^2 \left[ 1+ \sqrt {\epsilon_c(1+\epsilon_b)} \left\{ \frac{1}{\epsilon_c} F(\varphi , k_2) + E(\varphi,k_2)\right\} \right],
</math>
where <math>F(\varphi,k)</math> is an [[elliptic integral]] of the first kind and <math>E(\varphi,k)</math> is an elliptic integral of the second kind,
with the amplitude being

:<math>\varphi = \tan^{-1} (\sqrt \epsilon_c),</math>

and the moduli

:<math>k_1= \sqrt{\frac{\epsilon_c-\epsilon_b}{\epsilon_c}},</math>

and

:<math>k_2= \sqrt{\frac{\epsilon_b (1+\epsilon_c)}{\epsilon_c(1+\epsilon_b)}},</math>

where the anisotropy parameters, <math>\epsilon_b</math> and <math>\epsilon_c</math>, are

:<math>\epsilon_b = \left( \frac{b}{a} \right)^2 -1,</math>

and

:<math>\epsilon_c = \left( \frac{c}{a} \right)^2 -1.</math>

The volume of the ellipsoid is given by the well known

:<math>V = \frac{4 \pi}{3}abc.</math>

[[Mathematica]] notebook file for
[http://www.sklogwiki.org/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid]

==Maximum packing fraction==
Using [[event-driven molecular dynamics]], it has been found that the maximally random jammed (MRJ) [[packing fraction]] for hard ellipsoids is <math>\phi=0.7707</math> for
models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>
<ref>[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]</ref>
==Equation of state==
:''Main article: [[Hard ellipsoid equation of state]]''
==Virial coefficients==
:''Main article: [[Hard ellipsoids: virial coefficients]]
==Phase diagram==
One of the first [[phase diagrams]] of the hard ellipsoid model was that of Frenkel and Mulder (Figure 6 in
<ref>[http://dx.doi.org/10.1080/00268978500101971 D. Frenkel and B. M. Mulder "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations", Molecular Physics '''55''' pp. 1171-1192 (1985)]</ref>).
Camp and Allen later studied biaxial ellipsoids
<ref>[http://dx.doi.org/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)]</ref>. It has recently been shown
<ref>[http://arxiv.org/abs/0908.1043 M. Radu, P. Pfleiderer, T. Schilling "Solid-solid phase transition in hard ellipsoids", arXiv:0908.1043v1 7 Aug (2009)]</ref>
<ref>[http://dx.doi.org/10.1063/1.3251054 M. Radu, P. Pfleiderer, and T. Schilling "Solid-solid phase transition in hard ellipsoids", Journal of Chemical Physics '''131''' 164513 (2009)]</ref>
that the [[SM2 structure]] is more stable than the [[Building up a face centered cubic lattice | face centered cubic]] structure for aspect ratios <math>a/b \ge 2.0</math> and densities <math>\rho \gtrsim 1.17</math>. An updated phase diagram, encompassing the [[SM2 structure]] structure and the maximal packing fraction <ref>[http://dx.doi.org/10.1126/science.1093010 Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato, and P. M. Chaikin "Improving the Density of Jammed Disordered Packings Using Ellipsoids", Science '''303''' pp. 990-993 (2004)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRevLett.92.255506 Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters '''92''' 255506 (2004)]</ref>, can be found in <ref>[https://doi.org/10.1063/1.36993314 G. Odriozola "Revisiting the phase diagram of hard ellipsoids", Journal of Chemical Physics '''136''' 134505 (2012)]</ref> <ref>[https://doi.org/10.1063/1.4789957 G. Odriozola "Further details on the phase diagram of hard ellipsoids of revolution", Journal of Chemical Physics '''138''' 064501 (2013)]</ref>.

==Hard ellipse model==
:''Main article: [[Hard ellipse model]]'' (2-dimensional ellipsoids)
==References==
<references/>
'''Related reading'''

*[http://dx.doi.org/10.1080/02678299008047365 Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals '''8''' pp. 499-511 (1990)]
*[http://dx.doi.org/10.1016/j.fluid.2007.03.026 Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria '''255''' pp. 37-45 (2007)]
*[http://dx.doi.org/10.1063/1.4812361 Wen-Sheng Xu , Yan-Wei Li , Zhao-Yan Sun and Li-Jia An "Hard ellipses: Equation of state, structure, and self-diffusion", Journal of Chemical Physics '''139''' 024501 (2013)]

[[Category: Models]]

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Hard ellipsoid model