SklogWiki - User contributions [en]

Hard superball model

2012-09-16T20:08:58Z

Ranni: /* Phase diagram */

[[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].]]

A superball is defined by the inequality

:<math>\left|\frac{x}{a}\right|^{2q} + \left|\frac{y}{a}\right|^{2q} +\left|\frac{z}{a}\right|^{2q} \le 1</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 left figure.

== Particle Volume ==
The volume of a superball with the shape parameter ''q'' and radius ''a'' is given by

:<math>
\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}
</math>
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>.

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

==References==
<references/>

Hard superball model

2012-09-16T20:08:18Z

Ranni:

File:Phase diagram superball.png

2012-09-16T20:07:30Z

Ranni:

Hard superball model

2012-09-16T20:07:11Z

Ranni:

[[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].]]

A superball is defined by the inequality

:<math>\left|\frac{x}{a}\right|^{2q} + \left|\frac{y}{a}\right|^{2q} +\left|\frac{z}{a}\right|^{2q} \le 1</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 left figure.

== Particle Volume ==
The volume of a superball with the shape parameter ''q'' and radius ''a'' is given by

:<math>
\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}
</math>
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>.

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

==References==
<references/>

Hard superball model

2012-09-16T20:06:36Z

Ranni:

[[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].]]

A superball is defined by the inequality

:<math>\left|\frac{x}{a}\right|^{2q} + \left|\frac{y}{a}\right|^{2q} +\left|\frac{z}{a}\right|^{2q} \le 1</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 left figure.

== Particle Volume ==
The volume of a superball with the shape parameter ''q'' and radius ''a'' is given by

:<math>
\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}
</math>
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>.

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

==References==
<references/>

Hard superball model

2012-09-16T19:19:42Z

Ranni: /* Phase diagram */

[[Image:shape.png|thumb|right|The shape of superballs interpolates between octahedra (''q'' = 0.5) and cubes (''q'' = ∞) via spheres (''q'' = 1).]]

A superball is defined by the inequality

:<math>\left|\frac{x}{a}\right|^{2q} + \left|\frac{y}{a}\right|^{2q} +\left|\frac{z}{a}\right|^{2q} \le 1</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 left figure.

== Particle Volume ==
The volume of a superball with the shape parameter ''q'' and radius ''a'' is given by

:<math>
\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}
</math>
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>.

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

==References==
<references/>

Hard superball model

2012-09-16T19:18:07Z

Ranni:

Hard superball model

2012-09-16T19:11:38Z

Ranni:

Hard superball model

2012-09-16T19:10:45Z

Ranni:

Hard superball model

2012-09-16T19:06:24Z

Ranni: /* Particle Volume */

Hard superball model

2012-09-16T19:05:50Z

Ranni: /* Particle Volume */

Hard superball model

2012-09-16T19:04:52Z

Ranni: /* Particle Volume */

Hard superball model

2012-09-16T19:03:56Z

Ranni:

Hard superball model

2012-09-16T18:59:26Z

Ranni:

Hard superball model

2012-09-16T18:58:32Z

Ranni:

Hard superball model

2012-09-16T18:57:30Z

Ranni:

[[Image:shape.png|thumb|right|The shape of superballs interpolates between octahedra (''q'' = 0.5) and cubes (''q'' = ∞) via spheres (''q'' = 1).]]

A superball is defined by the inequality
:<math>\frac{x^2}{a^{2q}} + \frac{y^2}{a^{2q}} + \frac{z^2}{a^{2q}} \le 1</math>

where ''x'', ''y'' and ''z'' are scaled Cartesian coordinates with ''q'' the deformation parameter, and we use radius a of the particle as our unit of length. 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.

File:Shape.png

2012-09-16T18:52:03Z

Ranni: