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Hard superball model

2018-04-02T21:56:55Z

143.232.125.22:

[[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>|x|^{2q} + |y|^{2q} +|z|^{2q} \le a^{2q}</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{align}
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 = \frac{2a^3\left[ \Gamma\left(1/2q\right) \right]^3}{3q^2\Gamma\left(3/2q\right)},
\end{align}
</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

2018-04-02T21:55:14Z

143.232.125.22:

[[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>|x|^{2q} + |y|^{2q} +|z|^{2q} \le a^{2q}</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{align}
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 \\
& = & \frac{2a^3\left[ \Gamma\left(1/2q\right) \right]^3}{3q^2\Gamma\left(3/2q\right)},
\end{align}
</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

2018-04-02T21:53:55Z

143.232.125.22: simplified definition of superball and volume formula

[[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>|x|^{2q} + |y|^{2q} +|z|^{2q} \le a^{2q}</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{align}
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 \\
& = & \frac{2a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{3q^2\Gamma\left(1+ 3/2q\right)},
\end{align}
</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]]