Difference between revisions of "Form Factor:Superball"

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(References)
(Application to nanoscience)
 
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====Application to nanoscience====
 
====Application to nanoscience====
 
* Yugang Zhang, Fang Lu, Daniel van der Lelie, Oleg Gang "[http://prl.aps.org/abstract/PRL/v107/i13/e135701 Continuous Phase Transformation in Nanocube Assemblies]" ''Physical Review Letters'' '''2011''', 107, 135701 [http://dx.doi.org/10.1103/PhysRevLett.107.135701 doi: 10.1103/PhysRevLett.107.135701]
 
* Yugang Zhang, Fang Lu, Daniel van der Lelie, Oleg Gang "[http://prl.aps.org/abstract/PRL/v107/i13/e135701 Continuous Phase Transformation in Nanocube Assemblies]" ''Physical Review Letters'' '''2011''', 107, 135701 [http://dx.doi.org/10.1103/PhysRevLett.107.135701 doi: 10.1103/PhysRevLett.107.135701]
 +
* John Royer, George L. Burton, Daniel L. Blair and Steven Hudson [http://pubs.rsc.org/en/Content/ArticleLanding/2015/SM/C5SM00729A Rheology and Dynamics of Colloidal Superballs] ''Soft Matter'' '''2015''' [http://dx.doi.org/10.1039/C5SM00729A doi: 10.1039/C5SM00729A]
  
 
====Use in scattering====
 
====Use in scattering====
 
* [[Yager, K.G.]]; Zhang, Y.; Lu, F.; Gang, O. "[http://scripts.iucr.org/cgi-bin/paper?S160057671302832X Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems]" ''Journal of Applied Crystallography'' '''2014''', 47, 118–129. [http://dx.doi.org/0.1107/S160057671302832X doi: 10.1107/S160057671302832X]
 
* [[Yager, K.G.]]; Zhang, Y.; Lu, F.; Gang, O. "[http://scripts.iucr.org/cgi-bin/paper?S160057671302832X Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems]" ''Journal of Applied Crystallography'' '''2014''', 47, 118–129. [http://dx.doi.org/0.1107/S160057671302832X doi: 10.1107/S160057671302832X]
 
** See also [[Paper:DNA-nanoparticle superlattices formed from anisotropic building blocks|summary of paper]].
 
** See also [[Paper:DNA-nanoparticle superlattices formed from anisotropic building blocks|summary of paper]].

Latest revision as of 13:20, 10 June 2015

A superball is a general mathematical shape that can be used to describe rounded cubes. In fact, it is a general parametrization that can describe, via a parameter :

  • Empty space ()
  • Concave octahedra ()
  • Octahedra ()
  • Convex octahedra ()
  • Spheres ()
  • Rounded cubes ()
  • Cubes ()


Superball examples.png

The general equation is parametrized by the size, , and the curvature :

Obviously for , we recover the equation for a sphere. In the limit of large , we obtain a cube.

Volume

The normalized volume for a superball is:

Where and is the usual Euler gamma function.

Superball volume.png

Equations

The form factor for a superball is likely not analytic. However, it can be computed numerically.

References

Mathematical descriptions of superballs

Application to nanoscience

Use in scattering