Hard superball model: Difference between revisions

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</math>
</math>
where <math>\Gamma</math> is the Gamma function.
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>[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>.

Revision as of 20:10, 16 September 2012

The shape of superballs interpolates between octahedra (q = 0.5) and cubes (q = ∞) via spheres (q = 1).

A superball 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 left figure.

Particle Volume

The volume of a superball with the shape parameter q and radius a is given by

Failed to parse (unknown function "\begin{eqnarray}"): {\displaystyle \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} }

where is the Gamma function.

Overlap algorithm

The most widely used overlap algorithm is on the basis of Perram and Wertheim method[1] [2].