Hard superball model

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The shape of superballs interpolates between octahedra (q = 0.5) and cubes (q = ∞) via spheres (q = 1).
Phase diagram for hard superballs in the \phi (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

|x|^{2q} + |y|^{2q} +|z|^{2q}  \le a^{2q}

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

         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)},

where \Gamma 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].