Hard ellipsoid model: Difference between revisions

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[[Image:ellipsoid_red.png|thumb|right|A prolate ellipsoid.]]
[[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 ==  
== Interaction Potential ==  
The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by
The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by
Line 8: Line 10:
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)]
<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==
==Geometric properties==
The mean radius of curvature is given by (Ref. 2)
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>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],
Line 43: Line 48:
:<math>V = \frac{4 \pi}{3}abc.</math>
:<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>.


==See also==
==Hard ellipse model==
*[[Hard ellipsoid equation of state]]
:''Main article: [[Hard ellipse model]]'' (2-dimensional ellipsoids)
==References==
==References==
#[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)]
<references/>
#[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)]
'''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]]
[[Category: Models]]

Revision as of 13:49, 14 November 2013

A uniaxial prolate ellipsoid, a>b, b=c.
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

where , and define the lengths of the axis.

Overlap algorithm

The most widely used overlap algorithm is that of Perram and Wertheim [1].

Geometric properties

The mean radius of curvature is given by [2] [3]

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 [4] [5]

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 [6]). Camp and Allen later studied biaxial ellipsoids [7]. It has recently been shown [8] [9] that the SM2 structure is more stable than the face centered cubic structure for aspect ratios and densities .

Hard ellipse model

Main article: Hard ellipse model (2-dimensional ellipsoids)

References

Related reading