Hard superball model: Difference between revisions
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[[Image:shape.png|thumb|right|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).]] | ||
[[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> | :<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 cube (''q'' = ∞) via the sphere (''q'' = 1) as shown in the | 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 == | == Particle Volume == | ||
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:<math> | :<math> | ||
\begin{ | \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 | 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} | |||
\end{ | |||
</math> | </math> | ||
where <math>\Gamma</math> is the Gamma function. | where <math>\Gamma</math> is the [[Gamma function]]. | ||
==Overlap algorithm== | ==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>[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>. | 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== | ==Phase diagram== | ||
The full phase diagram of hard superballs whose shape interpolates from cubes to octahedra was in Ref | 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== | ||
<references/> | <references/> | ||
[[Category: Models]] | |||
Latest revision as of 12:51, 18 June 2018
The hard superball model is defined by the inequality
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 cube (q = ∞) via the sphere (q = 1) as shown in the right figure.
Particle Volume[edit]
The volume of a superball with the shape parameter q and radius a is given by
![{\displaystyle {\begin{aligned}v(q,a)&=&8a^{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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8d8d5827a28817c4bae8213009e5465970de86)
where is the Gamma function.
Overlap algorithm[edit]
The most widely used overlap algorithm is on the basis of Perram and Wertheim method [1] [2].
Phase diagram[edit]
The full phase diagram of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref [2].
References[edit]
- ↑ 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)
- ↑ 2.0 2.1 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)