Editing Hard ellipsoid model
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[[Image:ellipsoid_red.png|thumb|right|A | [[Image:ellipsoid_red.png|thumb|right|A prolate ellipsoid.]] | ||
'''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]]. | '''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 == | == Interaction Potential == | ||
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axis. | axis. | ||
==Overlap algorithm== | ==Overlap algorithm== | ||
The most widely used overlap algorithm is that of Perram and Wertheim | The most widely used overlap algorithm is that of Perram and Wertheim: | ||
*[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)] | |||
==Geometric properties== | ==Geometric properties== | ||
The mean radius of curvature is given by | The mean radius of curvature is given by (Refs. 5 and 6) | ||
:<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>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], | ||
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[[Mathematica]] notebook file for | [[Mathematica]] notebook file for | ||
[http://www. | [http://www.qft.iqfr.csic.es/personal/carl/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid] | ||
==Maximum packing fraction== | ==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 | 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> | models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>. | ||
#[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)] | |||
#[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)] | |||
==Equation of state== | ==Equation of state== | ||
:''Main article: [[Hard ellipsoid equation of state]]'' | :''Main article: [[Hard ellipsoid equation of state]]'' | ||
==Virial coefficients== | ==Virial coefficients== | ||
:''Main article: [[Hard ellipsoids: virial coefficients]] | :''Main article: [[Hard ellipsoids: virial coefficients]] | ||
==References== | ==References== | ||
#[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)] | |||
''' | #[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/10.1063/1.473665 Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics '''106''' pp. 6681- (1997)] | |||
#[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/10.1063/1.472110 G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics '''105''' pp. 2429-2435 (1996)] | |||
#[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)] | |||
[[Category: Models]] | [[Category: Models]] |