Hard superball model: Difference between revisions
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(simplified definition of superball and volume formula) |
Carl McBride (talk | contribs) m (Corrected reference errors) |
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:<math> | :<math> | ||
\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(1/2q\right) \right]^3}{3q^2\Gamma\left(3/2q\right)}, | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
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==Phase diagram== | ==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>. | 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== | ||
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)