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Hard superball model

2012-10-02T06:31:53Z

137.224.242.104:

[[Image:shape.png|thumb|right|The shape of superballs interpolates between octahedra (''q'' = 0.5) and cubes (''q'' = ∞) via spheres (''q'' = 1).]]
[[Image:phase_diagram_superball.png|thumb|right|Phase diagram for hard superballs in the <math>\phi</math> (packing fraction) versus 1/''q'' (bottom axis) and ''q'' (top axis) representation where ''q'' is the deformation parameter [2].]]

The '''hard superball model''' is defined by the inequality

:<math>\left|\frac{x}{a}\right|^{2q} + \left|\frac{y}{a}\right|^{2q} +\left|\frac{z}{a}\right|^{2q} \le 1</math>

where ''x'', ''y'' and ''z'' are scaled Cartesian coordinates with ''q'' the deformation parameter and radius ''a''. The shape of the superball interpolates smoothly between two Platonic solids, namely the octahedron (''q'' = 0.5) and the [[Hard cube model |cube]] (''q'' = ∞) via the [[Hard sphere model |sphere]] (''q'' = 1) as shown in the right figure.

== Particle Volume ==
The volume of a superball with the shape parameter ''q'' and radius ''a'' is given by

:<math>
\begin{eqnarray}
v(q,a) & = & 8 a^3 \int_{0}^1 \int_{0}^{(1-x^{2q})^{1/2q}} (1-x^{2q}-y^{2q})^{1/2q} \mathrm{d}\, y \, \mathrm{d}\, x \nonumber\\
& = & \frac{8a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{\Gamma\left(1+ 3/2q\right)},
\end{eqnarray}
</math>
where <math>\Gamma</math> is the [[Gamma function]].

==Overlap algorithm==
The most widely used overlap algorithm is on the basis of Perram and Wertheim method <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> <ref name="superballs">[http://dx.doi.org/10.1039/C2SM25813G R. Ni, A.P. Gantapara, J. de Graaf, R. van Roij, and M. Dijkstra "Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra", Soft Matter '''8''' pp. 8826-8834 (2012)]</ref>.

==Phase diagram==
The full [[phase diagrams |phase diagram]] of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref <ref name="superballs"> </ref>.

==References==
<references/>

[[Category: Models]]

Hard superball model

2012-09-16T18:51:06Z

137.224.242.104: Created page with "The shape of superballs interpolates between octahedra (''q'' = 0.5) and cubes (''q'' = ∞) via spheres (''q'' = 1)."

[[Image:shape.png|thumb|right|The shape of superballs interpolates between octahedra (''q'' = 0.5) and cubes (''q'' = ∞) via spheres (''q'' = 1).]]

Idealised models

2012-09-16T18:38:01Z

137.224.242.104: /* 'Hard' models */

'''Idealised models''' usually consist of a simple [[Intermolecular pair potential|intermolecular pair potential]], whose purpose is often to study underlying physical phenomena, such as generalised [[phase diagrams]] and the study of [[phase transitions]]. It is entirely possible that a number of the models bear little or no resemblance to [[Realistic models | real molecular fluids]].
==Lattice models==
*[[Barker-Fock model]]
*[[Blume-Emery-Griffiths model]] (including the Blume-Capel model)
*[[Bond fluctuation model]]
*[[Hard hexagon lattice model]]
*[[Hard square lattice model]]
*[[Henriques and Barbosa model]]
*[[Kagomé-lattice eight-vertex model]]
*[[Lattice gas]]
*[[Lattice hard spheres]]
*[[Lebwohl-Lasher model]]
*[[Potts model]]
**[[Ashkin-Teller model]]
**[[Kac model]]
*[[Roberts and Debenedetti model]]
*[[RP(n-1) model]]
*N-vector model:
**[[Self-avoiding walk model]] (n=0)
**[[Ising Models]] (n=1)
**[[XY model]] (n=2)
**[[Heisenberg model]] (n=3)
*[[Toda lattice]]
*[[Triangular lattice ramp model]]

=='Hard' models==
*[[Hard core Yukawa]]
*[[Hard cube model]]
*[[Hard ellipsoid model]]
**[[Hard ellipse model]]
*[[Hard superball model]]
*[[1-dimensional hard rods]]
*[[3-dimensional hard rods]]
*[[Hard pentagon model]]
*[[Hard sphere model | Hard sphere]]
**[[Hard disks]] (in a two dimensional space)
**[[Hard disks in a three dimensional space]] (including hard-cut spheres)
**[[Hard hyperspheres]]
**[[Dipolar hard spheres]]
*[[Hard spherocylinders]]
**[[Charged hard spherocylinders]]
**[[Oblate hard spherocylinders]]
*[[Hard tetrahedron model]]
*[[Parallel hard cubes]]
*[[Rough hard sphere model]]
*[[Sutherland potential]]
*[[Widom-Rowlinson model]]
====Multi-site models====
*[[Hard dumbbell model]]
*[[Branched hard sphere chains]]
*[[Flexible hard sphere chains]] (also known as the ''pearl-necklace model'')
*[[Fused hard sphere chains]]
*[[Tangent linear hard sphere chains]]
*[[Tetrahedral hard sphere model]]

==Piecewise continuous models==
*[[Buldyrev and Stanley model]]
*[[Harmonic repulsion potential]]
*[[Hemmer and Stell model]]
*[[Hertzian sphere model]]
*[[Penetrable sphere model]]
*[[Penetrable square well model]]
*[[Ramp model]] (also known as the '''Jagla model''')
*[[Square well model]]
*[[Square well lines potential]]
*[[Square well spherocylinders]]
*[[Soft-core square well model]]
*[[Square shoulder model]]
*[[Square shoulder + square well model]]
**[[Double square well model]]
*[[Sticky hard sphere model]]
*[[Triangular well model]]
*[[Soft sphere potential]]

=='Soft' models==
*[[Born-Huggins-Meyer potential]]
*[[Buckingham potential]]
*[[Continuous shouldered well model]]
*[[Durian foam bubble model]]
*[[Exp-6 potential]]
*[[Flexible molecules|Flexible molecules (intramolecular interactions)]]
*[[Fomin potential]]
*[[Gaussian overlap model]] (including the Gaussian core model)
*[[Gay-Berne model]]
*[[Harmonic repulsion potential]]
*[[Intermolecular Interactions]]
*[[Kihara potential]]
*[[Lennard-Jones model | Lennard-Jones model (3D)]]
**[[Lennard-Jones model in 1-dimension]] (rods)
**[[Lennard-Jones disks | Lennard-Jones model in 2-dimensions]] (disks)
**[[Lennard-Jones model in 4-dimensions]]
**[[Lennard-Jones sticks]]
**[[n-6 Lennard-Jones potential]]
**[[8-6 Lennard-Jones potential]]
**[[9-3 Lennard-Jones potential]]
**[[9-6 Lennard-Jones potential]]
**[[200-100 Lennard-Jones potential]]
**[[10-4-3 Lennard-Jones potential]]
**[[Soft-core Lennard-Jones model]]
**[[Stockmayer potential]]
**[[Two center Lennard-Jones model]]
*[[m-6-8 potential function]]
*[[Manning and Rosen potential]]
*[[Mie potential]]
*[[Morse potential]]
*[[Repulsive shoulder system with attractive well potential]]
*[[Rosen and Morse potential]]
*[[United-atom model]]
*[[Single site anisotropic soft-core potential]]
*[[Snub hexagonal model]]
*[[Soft-core square well model]]
*[[Soft sphere potential]]
*[[Soft sphere attractive Yukawa model]]
*[[Tietz potential]]
*[[Wei potential]]

==Patchy models==
*[[Patchy particles]]
**[[Kern and Frenkel patchy model]]
**[[Modulated patchy Lennard-Jones model]]
**[[Smith and Nezbeda associated fluid model]]
==Charged or polar models==
*[[Coulomb's law]]
*[[Charged hard dumbbells]]
*[[Charged hard spherocylinders]]
*[[Dipolar hard spheres]]
*[[Dipolar square wells]]
*[[Dipolar square wells | Quadrupolar square wells]]
*[[Drude oscillators]]
*[[Keesom potential]]
*[[Quadrupolar hard spheres]]
*[[Restricted primitive model]]
*[[Shell model]]
*[[Stockmayer potential]]

==Three-body and many-body potentials==
*[[Many-body interactions]] - a general discussion page.
*[[Axilrod-Teller interaction]]
*[[Keating potential]]
*[[Tersoff potential]]
*[[Stillinger-Weber potential]]
== Metals ==
*[[Dzugutov potential]]
*[[Embedded atom model]]
*[[Finnis-Sinclair]]
*[[Gupta potential]]
*[[Sutton-Chen]]
*[[Z1 and Z2 potentials]]
[[category:models]]
[[category:Computer simulation techniques]]

Phase diagrams

2012-09-16T16:51:07Z

137.224.242.104: /* Recommended reading */

==Generic phase diagrams==
*[[Pressure-temperature]]
*[[Density-temperature]]
*[[Binary phase diagrams]]
*[[Eutectic mixtures]]
==Phase diagrams for idealised models==
The following are partial or complete phase diagrams for various [[idealised models]]:
*[[Phase diagram of anisotropic particles with octahedral symmetry |Anisotropic particles with octahedral symmetry]]
*[[Phase diagram of anisotropic particles with tetrahedral symmetry|Anisotropic particles with tetrahedral symmetry]]
*[[Phase diagram of the Gay-Berne model | Gay-Berne model]]
*[[Phase diagram of the hard spherocylinder model | Hard spherocylinder model]]
*[[Phase diagram of the Lennard-Jones model |Lennard-Jones model]]
*[[Phase diagram of the two center Lennard-Jones model |Two center Lennard-Jones model]]
*[[Phase diagram of the Yukawa potential |Yukawa potential]]

==Recommended reading==
*Paul Ehrenfest "Phase conversions in a general and enhanced sense, classified according to the specific singularities of the thermodynamic potential", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen '''36''' pp. 153-157 (1933)
*[http://dx.doi.org/10.1088/0953-8984/20/15/153101 C. Vega, E. Sanz, J. L. F. Abascal and E. G. Noya "Determination of phase diagrams via computer simulation: methodology and applications to water, electrolytes and proteins", Journal of Physics: Condensed Matter '''20''' 153101 (2008)]
*[http://dx.doi.org/10.1039/C2SM25813G R. Ni, A.P. Gantapara, J. de Graaf, R. van Roij, and M. Dijkstra "Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra", Soft Matter '''8''' 8826 (2012)]
Experimental phase diagrams:
*[http://www.ucpress.edu/books/pages/2708.html David A. Young "Phase Diagrams of the Elements", University of California Press (1991)]
[[category: phase diagrams]]