Hard ellipsoid model: Difference between revisions

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[[Mathematica]] notebook file for
[[Mathematica]] notebook file for
[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]
[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==
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>.
#[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)]
==See also==
==See also==
*[[Hard ellipsoid equation of state]]
*[[Hard ellipsoid equation of state]]

Revision as of 10:51, 19 February 2008

A prolate ellipsoid.

Interaction Potential

The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by

where , and define the lengths of the axis.

Overlap algorithm

The most widely used overlap algorithm is that of Perram and Wertheim:

Geometric properties

The mean radius of curvature is given by (Refs. 5 and 6)

and the surface area is given by

where is an elliptic integral of the first kind and is an elliptic integral of the second kind, with the amplitude being

and the moduli

and

where the anisotropy parameters, and , are

and

The volume of the ellipsoid is given by the well known

Mathematica notebook file for 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 for models whose maximal aspect ratio is greater than .

  1. 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)
  2. Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato "Unusually Dense Crystal Packings of Ellipsoids", Physical Review Letters 92 255506 (2004)

See also

References

  1. D. Frenkel and B. M. Mulder "The hard ellipsoid-of-revolution fluid I. Monte Carlo simulations", Molecular Physics 55 pp. 1171-1192 (1985)
  2. Michael P. Allen "Computer simulation of a biaxial liquid crystal", Liquid Crystals 8 pp. 499-511 (1990)
  3. Philip J. Camp and Michael P. Allen "Phase diagram of the hard biaxial ellipsoid fluid", Journal of Chemical Physics 106 pp. 6681- (1997)
  4. Carl McBride and Enrique Lomba "Hard biaxial ellipsoids revisited: Numerical results", Fluid Phase Equilibria 255 pp. 37-45 (2007)
  5. G. S. Singh and B. Kumar "Geometry of hard ellipsoidal fluids and their virial coefficients", Journal of Chemical Physics 105 pp. 2429-2435 (1996)
  6. G. S. Singh and B. Kumar "Molecular Fluids and Liquid Crystals in Convex-Body Coordinate Systems", Annals of Physics 294 pp. 24-47 (2001)