Hard superball model: Difference between revisions

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(simplified definition of superball and volume formula)
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\begin{align}
\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+1/2q\right) \right]^3}{3q^2\Gamma\left(1+ 3/2q\right)},
         & = & \frac{2a^3\left[ \Gamma\left(1/2q\right) \right]^3}{3q^2\Gamma\left(3/2q\right)},
\end{align}
\end{align}
</math>
</math>

Revision as of 22:55, 2 April 2018

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

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

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

where is the Gamma function.

Overlap algorithm

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

Phase diagram

The full phase diagram of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref [2].

References