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Hard tetrahedron model

2012-12-04T03:00:02Z

141.213.172.221:

[[Image:tetrahedron.png|thumb|right]]
The '''hard tetrahedron model''' is a subset of [[hard polyhedra model]] that has been put forward as a potential model for [[water]]<ref>[http://dx.doi.org/10.1080/00268979500100281 Jiri Kolafa and Ivo Nezbeda "The hard tetrahedron fluid: a model for the structure of water?", Molecular Physics '''84''' pp. 421-434 (1995)]</ref>.
==Maximum packing fraction==
It has recently been shown that regular tetrahedra are able to achieve packing fractions as high as <math>\phi=0.8503</math><ref>[http://dx.doi.org/10.1038/nature08641 Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray and Sharon C. Glotzer "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra", Nature '''462''' pp. 773-777 (2009)]</ref> (the [[hard sphere model |hard sphere]] packing fraction is <math>\pi/(3 \sqrt{2}) \approx 74.048%</math> <ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>). This is in stark contrast to work as recent as in 2006, where it was suggested that the "...regular tetrahedron might even be the convex body having the smallest possible packing density"<ref>[http://dx.doi.org/10.1073/pnas.0601389103 J. H. Conway and S. Torquato "Packing, tiling, and covering with tetrahedra", Proceedings of the National Academy of Sciences of the United States of America '''103''' 10612-10617 (2006)]</ref>.
==Phase diagram==
<ref>[http://dx.doi.org/10.1063/1.3651370 Amir Haji-Akbari, Michael Engel, and Sharon C. Glotzer "Phase diagram of hard tetrahedra", Journal of Chemical Physics '''135''' 194101 (2011)]</ref>
==Truncated tetrahedra==
Dimers composed of Archimedean truncated tetrahedra are able to achieve packing fractions as high as <math>\phi= 207/208 \approx 0.9951923</math>
<ref>[http://dx.doi.org/10.1103/PhysRevLett.107.155501 Joost de Graaf, René van Roij, and Marjolein Dijkstra "Dense Regular Packings of Irregular Nonconvex Particles", Physical Review Letters '''107''' 155501 (2011)]</ref><ref>[http://dx.doi.org/10.1063/1.3653938 Yang Jiao and Salvatore Torquato "A packing of truncated tetrahedra that nearly fills all of space and its melting properties", Journal of Chemical Physics '''135''' 151101 (2011)]</ref> while a non-regular truncated tetrahedra can completely tile space <ref>[http://dx.doi.org/10.1021/nn204012y Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces", ACS Nano '''6''' pp. 609-614 (2012)]</ref>.

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1103/Physics.3.37 Daan Frenkel "The tetrahedral dice are cast … and pack densely", Physics '''3''' 37 (2010)]

[[category: models]]

Hard polyhedra model

2012-12-04T02:56:59Z

141.213.172.221:

The '''hard polyhedra model''' is an approximation to describe the behavior of anisotropic colloidal particles<ref>[http://dx.doi.org/10.1038/nmat1949 Sharon C. Glotzer and Michael Solomon "Anisotropy of building blocks and their assembly into complex structures", Nature Materials '''6''' pp. 557 - 562 (2007)]</ref> with screened interaction.

It was first pointed out by Agarwal and Escobedo <ref>[http://dx.doi.org/10.1038/nmat2959 Umang Agarwal and Fernando A. Escobedo "Mesophase behaviour of polyhedral particles", Nature Materials '''10''' pp. 230–235 (2011)]</ref> the possibility that measures of shape could lead to a roadmap of the structures to be self-assembled: very anisotropic particles would lead to the formation of liquid crystals while very spherical ones would form plastic crystals at intermediate packing fractions.

Later, Damasceno, Engel and Glotzer <ref>[http://dx.doi.org/10.1126/science.1220869 Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Predictive Self-Assembly of Polyhedra into Complex Structures", Science '''337''' pp. 453-457 (2012)]</ref>. showed that, in addition to the sphericity of the particle (measured by calculating its isoperimetric quotient), by knowing the types of "bonds" that a given polyhedron is able to make already in the dense liquid the class of crystalline structure could be predicted. That was made possible due to the observation that facetted particles tend to maximize their face-to-face contacts at intermediate packing fractions. <ref>[http://dx.doi.org/10.1021/nn204012y Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces", ACS Nano '''6''' pp. 609-614 (2012)]</ref>.

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1126/science.1226162 de Graaf and Mana "A Roadmap for the Assembly of Polyhedral Particles", Science '''337''' pp. 417-418 (2012)]

[[category: models]]

Hard polyhedra model

2012-12-04T02:56:28Z

141.213.172.221:

[[Image:polyhedron.png|thumb|right]]
The '''hard polyhedra model''' is an approximation to describe the behavior of anisotropic colloidal particles<ref>[http://dx.doi.org/10.1038/nmat1949 Sharon C. Glotzer and Michael Solomon "Anisotropy of building blocks and their assembly into complex structures", Nature Materials '''6''' pp. 557 - 562 (2007)]</ref> with screened interaction.

It was first pointed out by Agarwal and Escobedo <ref>[http://dx.doi.org/10.1038/nmat2959 Umang Agarwal and Fernando A. Escobedo "Mesophase behaviour of polyhedral particles", Nature Materials '''10''' pp. 230–235 (2011)]</ref> the possibility that measures of shape could lead to a roadmap of the structures to be self-assembled: very anisotropic particles would lead to the formation of liquid crystals while very spherical ones would form plastic crystals at intermediate packing fractions.

Later, Damasceno, Engel and Glotzer <ref>[http://dx.doi.org/10.1126/science.1220869 Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Predictive Self-Assembly of Polyhedra into Complex Structures", Science '''337''' pp. 453-457 (2012)]</ref>. showed that, in addition to the sphericity of the particle (measured by calculating its isoperimetric quotient), by knowing the types of "bonds" that a given polyhedron is able to make already in the dense liquid the class of crystalline structure could be predicted. That was made possible due to the observation that facetted particles tend to maximize their face-to-face contacts at intermediate packing fractions. <ref>[http://dx.doi.org/10.1021/nn204012y Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces", ACS Nano '''6''' pp. 609-614 (2012)]</ref>.

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1126/science.1226162 de Graaf and Mana "A Roadmap for the Assembly of Polyhedral Particles", Science '''337''' pp. 417-418 (2012)]

[[category: models]]

Hard polyhedra model

2012-12-04T02:54:48Z

141.213.172.221:

[[Image:polyhedron.png|thumb|right]]
The '''hard polyhedra model''' is an approximation to describe the behavior of anisotropic colloidal particles<ref>[http://dx.doi.org/10.1038/nmat1949 Sharon C. Glotzer and Michael Solomon "Anisotropy of building blocks and their assembly into complex structures", Nature Materials '''6''' pp. 557 - 562 (2007)]</ref> with screened interaction.

It was first pointed out by Agarwal and Escobedo <ref>[http://dx.doi.org/10.1038/nmat2959 Umang Agarwal and Fernando A. Escobedo "Mesophase behaviour of polyhedral particles", Nature Materials '''10''' pp. 230–235 (2011)]</ref> the possibility that measures of shape could lead to a roadmap of the structures to be self-assembled: very anisotropic particles would lead to the formation of liquid crystals while very spherical ones would form plastic crystals at intermediate packing fractions.

Later, Damasceno, Engel and Glotzer <ref>[http://dx.doi.org/10.1126/science.1220869 Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Predictive Self-Assembly of Polyhedra into Complex Structures", Science '''337''' pp. 453-457 (2012)]</ref>. showed that, in addition to the sphericity of the particle - measured by calculating its isoperimetric quotient, by knowing the types of "bonds" that a given polyhedron is able to make already in the dense liquid the class of crystalline structure could be predicted. That was made possible due to the observation that facetted particles tend to maximize their face-to-face contacts at intermediate packing fractions.[http://dx.doi.org/10.1021/nn204012y Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces", ACS Nano '''6''' pp. 609-614 (2012)]</ref>.

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1126/science.1226162 de Graaf and Mana "A Roadmap for the Assembly of Polyhedral Particles", Science '''337''' pp. 417-418 (2012)]

[[category: models]]

Hard polyhedra model

2012-12-04T02:54:15Z

141.213.172.221: Created page with "right The '''hard polyhedra model''' is an approximation to describe the behavior of anisotropic colloidal particles<ref>[http://dx.doi.org/10.1..."

[[Image:polyhedron.png|thumb|right]]
The '''hard polyhedra model''' is an approximation to describe the behavior of anisotropic colloidal particles<ref>[http://dx.doi.org/10.1038/nmat1949 Sharon C. Glotzer and Michael Solomon "Anisotropy of building blocks and their assembly into complex structures", Nature Materials '''6''' pp. 557 - 562 (2007)]</ref> with screened interaction.

It was first pointed out by Agarwal and Escobedo <ref>[http://dx.doi.org/10.1038/nmat2959 Umang Agarwal and Fernando A. Escobedo "Mesophase behaviour of polyhedral particles", Nature Materials '''10''' pp. 230–235 (2011)]</ref> the possibility that measures of shape could lead to a roadmap of the structures to be self-assembled: very anisotropic particles would lead to the formation of liquid crystals while very spherical ones would form plastic crystals at intermediate packing fractions.

Later, Damasceno, Engel and Glotzer[http://dx.doi.org/10.1126/science.1220869 Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Predictive Self-Assembly of Polyhedra into Complex Structures", Science '''337''' pp. 453-457 (2012)]</ref>. showed that, in addition to the sphericity of the particle - measured by calculating its isoperimetric quotient, by knowing the types of "bonds" that a given polyhedron is able to make already in the dense liquid the class of crystalline structure could be predicted. That was made possible due to the observation that facetted particles tend to maximize their face-to-face contacts at intermediate packing fractions.[http://dx.doi.org/10.1021/nn204012y Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces", ACS Nano '''6''' pp. 609-614 (2012)]</ref>.

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1126/science.1226162 de Graaf and Mana "A Roadmap for the Assembly of Polyhedral Particles", Science '''337''' pp. 417-418 (2012)]

[[category: models]]

Idealised models

2012-12-04T02:34:13Z

141.213.172.221: /* 'Hard' models */

'''Idealised models''' usually consist of a simple [[Intermolecular pair potential|intermolecular pair potential]], whose purpose is often to study underlying physical phenomena, such as generalised [[phase diagrams]] and the study of [[phase transitions]]. It is entirely possible that a number of the models bear little or no resemblance to [[Realistic models | real molecular fluids]].
==Lattice models==
*[[Barker-Fock model]]
*[[Blume-Emery-Griffiths model]] (including the Blume-Capel model)
*[[Bond fluctuation model]]
*[[Hard hexagon lattice model]]
*[[Hard square lattice model]]
*[[Henriques and Barbosa model]]
*[[Kagomé-lattice eight-vertex model]]
*[[Kratky-Porod model]] (also known as ''semiflexible worm-like chains'')
*[[Lattice gas]]
*[[Lattice hard spheres]]
*[[Lebwohl-Lasher model]]
*[[Potts model]]
**[[Ashkin-Teller model]]
**[[Kac model]]
*[[Roberts and Debenedetti model]]
*[[RP(n-1) model]]
*N-vector model:
**[[Self-avoiding walk model]] (n=0)
**[[Ising Models]] (n=1)
**[[XY model]] (n=2)
**[[Heisenberg model]] (n=3)
*[[Toda lattice]]
*[[Triangular lattice ramp model]]

=='Hard' models==
*[[Hard core Yukawa]]
*[[Hard cube model]]
*[[Hard ellipsoid model]]
**[[Hard ellipse model]]
*[[Hard superball model]]
*[[1-dimensional hard rods]]
*[[3-dimensional hard rods]]
*[[Hard pentagon model]]
*[[Hard polyhedra model]]
*[[Hard sphere model | Hard sphere]]
**[[Hard disks]] (in a two dimensional space)
**[[Hard disks in a three dimensional space]] (including hard-cut spheres)
**[[Hard hyperspheres]]
**[[Dipolar hard spheres]]
*[[Hard spherocylinders]]
**[[Charged hard spherocylinders]]
**[[Oblate hard spherocylinders]]
*[[Hard tetrahedron model]]
*[[Parallel hard cubes]]
*[[Rough hard sphere model]]
*[[Sutherland potential]]
*[[Widom-Rowlinson model]]
*[[Zwanzig model]]
====Multi-site models====
*[[Branched hard sphere chains]]
*[[Flexible hard sphere chains]] (also known as the ''pearl-necklace model'')
*[[Fused hard sphere chains]]
*[[Hard dumbbell model]]
**[[Asymmetric hard dumbbell model]]
*[[Rigid fully flexible fused hard sphere model]]
*[[Tangent linear hard sphere chains]]
*[[Tetrahedral hard sphere model]]

==Piecewise continuous models==
*[[Buldyrev and Stanley model]]
*[[Harmonic repulsion potential]]
*[[Hemmer and Stell model]]
*[[Hertzian sphere model]]
*[[modified Lennard-Jones model]]
*[[Penetrable sphere model]]
*[[Penetrable square well model]]
*[[Pseudo hard sphere potential]]
*[[Ramp model]] (also known as the '''Jagla model''')
*[[Square well model]]
*[[Square well lines potential]]
*[[Square well spherocylinders]]
*[[Soft-core square well model]]
*[[Square shoulder model]]
*[[Square shoulder + square well model]]
**[[Double square well model]]
*[[Sticky hard sphere model]]
*[[Triangular well model]]
*[[Soft sphere potential]]

=='Soft' models==
*[[Born-Huggins-Meyer potential]]
*[[Buckingham potential]]
*[[Continuous shouldered well model]]
*[[Durian foam bubble model]]
*[[Exp-6 potential]]
*[[Flexible molecules|Flexible molecules (intramolecular interactions)]]
*[[Fomin potential]]
*[[Gaussian overlap model]] (including the Gaussian core model)
*[[Gay-Berne model]]
*[[Harmonic repulsion potential]]
*[[Intermolecular Interactions]]
*[[Kihara potential]]
*[[Lennard-Jones model | Lennard-Jones model (3D)]]
**[[Lennard-Jones model in 1-dimension]] (rods)
**[[Lennard-Jones disks | Lennard-Jones model in 2-dimensions]] (disks)
**[[Lennard-Jones model in 4-dimensions]]
**[[Lennard-Jones sticks]]
**[[modified Lennard-Jones model]]
**[[n-6 Lennard-Jones potential]]
**[[8-6 Lennard-Jones potential]]
**[[9-3 Lennard-Jones potential]]
**[[9-6 Lennard-Jones potential]]
**[[200-100 Lennard-Jones potential]]
**[[10-4-3 Lennard-Jones potential]]
**[[Soft-core Lennard-Jones model]]
**[[Stockmayer potential]]
**[[Two center Lennard-Jones model]]
*[[m-6-8 potential function]]
*[[Manning and Rosen potential]]
*[[Mie potential]]
*[[Morse potential]]
*[[Repulsive shoulder system with attractive well potential]]
*[[Rosen and Morse potential]]
*[[United-atom model]]
*[[Single site anisotropic soft-core potential]]
*[[Snub hexagonal model]]
*[[Soft-core square well model]]
*[[Soft sphere potential]]
*[[Soft sphere attractive Yukawa model]]
*[[Tietz potential]]
*[[Wei potential]]
*[[Yoshida and Kamakura model]]
====Multi-site models====
*[[Rigid linear Lennard-Jones chains]]

==Patchy models==
*[[Patchy particles]]
**[[Kern and Frenkel patchy model]]
**[[Modulated patchy Lennard-Jones model]]
**[[Smith and Nezbeda associated fluid model]]
==Charged or polar models==
*[[Coulomb's law]]
*[[Charged hard dumbbells]]
*[[Charged hard spherocylinders]]
*[[Dipolar hard spheres]]
*[[Dipolar Janus particles]]
*[[Dipolar square wells]]
*[[Dipolar square wells | Quadrupolar square wells]]
*[[Drude oscillators]]
*[[Keesom potential]]
*[[Quadrupolar hard spheres]]
*[[Quadrupolar Lennard-Jones model]]
*[[Restricted primitive model]]
*[[Shell model]]
*[[Stockmayer potential]]

==Three-body and many-body potentials==
*[[Many-body interactions]] - a general discussion page.
*[[Axilrod-Teller interaction]]
*[[Keating potential]]
*[[Tersoff potential]]
*[[Stillinger-Weber potential]]
== Metals ==
*[[Dzugutov potential]]
*[[Embedded atom model]]
*[[Finnis-Sinclair]]
*[[Gupta potential]]
*[[Sutton-Chen]]
*[[Z1 and Z2 potentials]]
[[category:models]]
[[category:Computer simulation techniques]]

Hard tetrahedron model

2012-12-04T02:32:00Z

141.213.172.221: /* Truncated tetrahedra */

[[Image:tetrahedron.png|thumb|right]]
The '''hard tetrahedron model'''. Such a structure has been put forward as a potential model for [[water]]<ref>[http://dx.doi.org/10.1080/00268979500100281 Jiri Kolafa and Ivo Nezbeda "The hard tetrahedron fluid: a model for the structure of water?", Molecular Physics '''84''' pp. 421-434 (1995)]</ref>.
==Maximum packing fraction==
It has recently been shown that regular tetrahedra are able to achieve packing fractions as high as <math>\phi=0.8503</math><ref>[http://dx.doi.org/10.1038/nature08641 Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe G. Petschek, Peter Palffy-Muhoray and Sharon C. Glotzer "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra", Nature '''462''' pp. 773-777 (2009)]</ref> (the [[hard sphere model |hard sphere]] packing fraction is <math>\pi/(3 \sqrt{2}) \approx 74.048%</math> <ref>[http://dx.doi.org/10.1038/26609 Neil J. A. Sloane "Kepler's conjecture confirmed", Nature '''395''' pp. 435-436 (1998)]</ref>). This is in stark contrast to work as recent as in 2006, where it was suggested that the "...regular tetrahedron might even be the convex body having the smallest possible packing density"<ref>[http://dx.doi.org/10.1073/pnas.0601389103 J. H. Conway and S. Torquato "Packing, tiling, and covering with tetrahedra", Proceedings of the National Academy of Sciences of the United States of America '''103''' 10612-10617 (2006)]</ref>.
==Phase diagram==
<ref>[http://dx.doi.org/10.1063/1.3651370 Amir Haji-Akbari, Michael Engel, and Sharon C. Glotzer "Phase diagram of hard tetrahedra", Journal of Chemical Physics '''135''' 194101 (2011)]</ref>
==Truncated tetrahedra==
Dimers composed of Archimedean truncated tetrahedra are able to achieve packing fractions as high as <math>\phi= 207/208 \approx 0.9951923</math>
<ref>[http://dx.doi.org/10.1103/PhysRevLett.107.155501 Joost de Graaf, René van Roij, and Marjolein Dijkstra "Dense Regular Packings of Irregular Nonconvex Particles", Physical Review Letters '''107''' 155501 (2011)]</ref><ref>[http://dx.doi.org/10.1063/1.3653938 Yang Jiao and Salvatore Torquato "A packing of truncated tetrahedra that nearly fills all of space and its melting properties", Journal of Chemical Physics '''135''' 151101 (2011)]</ref> while a non-regular truncated tetrahedra can completely tile space <ref>[http://dx.doi.org/10.1021/nn204012y Pablo F. Damasceno, Michael Engel and Sharon C. Glotzer "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces", ACS Nano '''6''' pp. 609-614 (2012)]</ref>.

==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1103/Physics.3.37 Daan Frenkel "The tetrahedral dice are cast … and pack densely", Physics '''3''' 37 (2010)]

[[category: models]]