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| [[Image:ellipsoid_red.png|thumb|right|A uniaxial prolate ellipsoid, a>b, b=c.]] | | [[Image:ellipsoid_red.png|thumb|right|A prolate ellipsoid.]] |
| [[Image:oblate_1.png|thumb|right|A uniaxial oblate ellipsoid, a>c, a=b.]]
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| '''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]].
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| == 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 |
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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: |
| <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>.
| | *[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)] |
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| ==Geometric properties== | | ==Geometric properties== |
| The mean radius of curvature is given by | | The mean radius of curvature is given by (Refs. 2 and 3) |
| <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>
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| <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>
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| :<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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| :<math>V = \frac{4 \pi}{3}abc.</math> | | :<math>V = \frac{4 \pi}{3}abc.</math> |
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| [[Mathematica]] notebook file for
| | [http://www.qft.iqfr.csic.es/personal/carl/SR_B2_B3_ellipsoids.nb Mathematica notebook file for calculating the surface area and mean radius of curvature of an ellipsoid] |
| [http://www.sklogwiki.org/SR_B2_B3_ellipsoids.nb calculating the surface area and mean radius of curvature of an ellipsoid] | | ==See also== |
| | | *[[Hard ellipsoid equation of state]] |
| ==Maximum packing fraction==
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| 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
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| models whose maximal aspect ratio is greater than <math>\sqrt{3}</math>
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| <ref name="Donev1"> [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>
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| <ref name="Donev2">[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>
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| ==Equation of state==
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| :''Main article: [[Hard ellipsoid equation of state]]''
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| ==Virial coefficients==
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| :''Main article: [[Hard ellipsoids: virial coefficients]]
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| ==Phase diagram==
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| One of the first [[phase diagrams]] of the hard ellipsoid model was that of Frenkel and Mulder (Figure 6 in
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| <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>).
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| Camp and Allen later studied biaxial ellipsoids
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| <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
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| <ref name="Radu1"> [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>
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| <ref name="Radu2">[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>
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| 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 <ref name="Radu1"/> <ref name="Radu2"/> and the maximal packing fraction <ref name="Donev1"/> <ref name="Donev2"/>, 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>.
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| ==Hard ellipse model==
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| :''Main article: [[Hard ellipse model]]'' (2-dimensional ellipsoids)
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| ==References== | | ==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)] |
| '''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/10.1063/1.473665 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/10.1080/02678299008047365 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/10.1016/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/10.1063/1.4812361 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)]
| | #[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)] |
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| [[Category: Models]] | | [[Category: Models]] |