Hard superball model - Revision history

Carl McBride: Corrected reference errors

2018-06-18T11:51:47Z

Corrected reference errors

← Older revision		Revision as of 13:51, 18 June 2018
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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==

143.232.125.22 at 21:56, 2 April 2018

2018-04-02T21:56:55Z

← Older revision		Revision as of 23:56, 2 April 2018
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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)},
	& = & \frac{2a^3\left[ \Gamma\left(1/2q\right) \right]^3}{3q^2\Gamma\left(3/2q\right)},
	\end{align}		\end{align}
	</math>		</math>

143.232.125.22 at 21:55, 2 April 2018

2018-04-02T21:55:14Z

← Older revision		Revision as of 23:55, 2 April 2018
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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>

143.232.125.22: simplified definition of superball and volume formula

2018-04-02T21:53:55Z

simplified definition of superball and volume formula

← Older revision		Revision as of 23:53, 2 April 2018
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	The '''hard superball model''' is defined by the inequality		The '''hard superball model''' 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>		:<math>\|x\|^{2q} + \|y\|^{2q} +\|z\|^{2q} \le a^{2q}</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 [[Hard cube model \|cube]] (''q'' = ∞) via the [[Hard sphere model \|sphere]] (''q'' = 1) as shown in the right figure.		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 [[Hard cube model \|cube]] (''q'' = ∞) via the [[Hard sphere model \|sphere]] (''q'' = 1) as shown in the right figure.
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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{8a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{\Gamma\left(1+ 3/2q\right)},		& = & \frac{2a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{3q^2\Gamma\left(1+ 3/2q\right)},
	\end{align}		\end{align}
	</math>		</math>

134.99.64.193: Fixed volume equation formatting

2015-09-14T14:14:29Z

Fixed volume equation formatting

← Older revision		Revision as of 16:14, 14 September 2015
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	:<math>		:<math>
	\begin{~~eqnarray~~}		\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 ~~\nonumber~~\\		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{8a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{\Gamma\left(1+ 3/2q\right)},		& = & \frac{8a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{\Gamma\left(1+ 3/2q\right)},
	\end{~~eqnarray~~}		\end{align}
	</math>		</math>
	where <math>\Gamma</math> is the [[Gamma function]].		where <math>\Gamma</math> is the [[Gamma function]].

137.224.242.104 at 06:31, 2 October 2012

2012-10-02T06:31:53Z

← Older revision		Revision as of 08:31, 2 October 2012
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	:<math>\left\|\frac{x}{a}\right\|^{2q} + \left\|\frac{y}{a}\right\|^{2q} +\left\|\frac{z}{a}\right\|^{2q} \le 1</math>		:<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 [[Hard cube model \|cube]] (''q'' = ∞) via the [[Hard sphere model \|sphere]] (''q'' = 1) as shown in the ~~left~~ figure.		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 [[Hard cube model \|cube]] (''q'' = ∞) via the [[Hard sphere model \|sphere]] (''q'' = 1) as shown in the right figure.

	== Particle Volume ==		== Particle Volume ==

Carl McBride: Added some internal links

2012-09-18T09:16:04Z

Added some internal links

← Older revision		Revision as of 11:16, 18 September 2012
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	[[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].]]		[[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		The '''hard superball model''' 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>		:<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.		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 [[Hard cube model \|cube]] (''q'' = ∞) via the [[Hard sphere model \|sphere]] (''q'' = 1) as shown in the left figure.

	== Particle Volume ==		== Particle Volume ==
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	\end{eqnarray}		\end{eqnarray}
	</math>		</math>
	where <math>\Gamma</math> is the Gamma function.		where <math>\Gamma</math> is the [[Gamma function]].

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

Carl McBride: Added Category (Models)

2012-09-17T14:19:40Z

Added Category (Models)

← Older revision		Revision as of 16:19, 17 September 2012
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	==References==		==References==
	<references/>		<references/>


			[[Category: Models]]

Ranni: /* Phase diagram */

2012-09-16T20:08:58Z

Phase diagram

← Older revision		Revision as of 22:08, 16 September 2012
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	==Phase diagram==		==Phase diagram==
	The full phase diagram of hard superballs whose shape interpolates from cubes to octahedra was in Ref.[2].		The full phase diagram of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref.[2].

	==References==		==References==
	<references/>		<references/>

Ranni at 20:08, 16 September 2012

2012-09-16T20:08:18Z

← Older revision		Revision as of 22:08, 16 September 2012
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	[[Image:shape.png\|thumb\|right\|The shape of superballs interpolates between octahedra (''q'' = 0.5) and cubes (''q'' = ∞) via spheres (''q'' = 1).]]		[[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].]]		[[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		A superball is defined by the inequality