SklogWiki - User contributions [en]

Janus particles

2013-11-02T20:17:24Z

FSmallenburg: /* Phase diagram */

[[Image:Janus.png|thumb|right| Artists impression of a Janus particle]]
'''Janus particles''' are particles consisting of (at least) two parts with different interactions. For example, the typical Janus particle is a sphere which has a short-range attraction on one half of the particle, but is purely repulsive on the other side. However, the term has also been used for non-spherical particles. The name derives from the two-faced Roman god Janus. Janus particles can be considered as a one-patch [[Patchy particles|patchy particle]].

Experimentally, the different interactions can be achieved by (for example) making the two parts of the surface hydrophobic and hydrophilic, positively and negatively charged, or smooth and rough (leading to different interactions in the presence of [[Depletion force|depletants]]). In simulations, these particles are often modeled using the [[Kern and Frenkel patchy model|Kern-Frenkel]] interaction potential.

==Phase diagram==
<ref>[http://dx.doi.org/10.1103/PhysRevLett.103.237801 Francesco Sciortino, Achille Giacometti, and Giorgio Pastore "Phase Diagram of Janus Particles", Physical Review Letters '''103''' 237801 (2009)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4801438 Teun Vissers, Zdeněk Preisler, Frank Smallenburg, Marjolein Dijkstra, and Francesco Sciortino "Predicting crystals of Janus colloids", Journal of Chemical Physics '''138''', 164505 (2013)]</ref>

==See also==
*[[Dipolar Janus particles]]
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3177238 Steve Granick, Shan Jiang, and Qian Chen "Janus particles", Physics Today '''62''' pp. 68-69 (July 2009)]
*[http://dx.doi.org/10.1063/1.4707954 Gerald Rosenthal, Keith E. Gubbins, and Sabine H. L. Klapp "Self-assembly of model amphiphilic Janus particles", Journal of Chemical Physics '''136''' 174901 (2012)]
*[http://dx.doi.org/10.1063/1.4793626 Miguel Ángel G. Maestre, Riccardo Fantoni, Achille Giacometti, and Andrés Santos "Janus fluid with fixed patch orientations: Theory and simulations", Journal of Chemical Physics '''138''' 094904 (2013)]

[[category: phase diagrams]]

Janus particles

2013-11-02T20:16:45Z

FSmallenburg: /* Phase diagram */

[[Image:Janus.png|thumb|right| Artists impression of a Janus particle]]
'''Janus particles''' are particles consisting of (at least) two parts with different interactions. For example, the typical Janus particle is a sphere which has a short-range attraction on one half of the particle, but is purely repulsive on the other side. However, the term has also been used for non-spherical particles. The name derives from the two-faced Roman god Janus. Janus particles can be considered as a one-patch [[Patchy particles|patchy particle]].

Experimentally, the different interactions can be achieved by (for example) making the two parts of the surface hydrophobic and hydrophilic, positively and negatively charged, or smooth and rough (leading to different interactions in the presence of [[Depletion force|depletants]]). In simulations, these particles are often modeled using the [[Kern and Frenkel patchy model|Kern-Frenkel]] interaction potential.

==Phase diagram==
<ref>[http://dx.doi.org/10.1103/PhysRevLett.103.237801 Francesco Sciortino, Achille Giacometti, and Giorgio Pastore "Phase Diagram of Janus Particles", Physical Review Letters '''103''' 237801 (2009)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4801438 Teun Vissers, Zdeněk Preisler, Frank Smallenburg, Marjolein Dijkstra, and Francesco Sciortino "Predicting crystals of Janus colloids", Journal of Chemical Physics '''138''', 164505 (2013)</ref>

==See also==
*[[Dipolar Janus particles]]
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3177238 Steve Granick, Shan Jiang, and Qian Chen "Janus particles", Physics Today '''62''' pp. 68-69 (July 2009)]
*[http://dx.doi.org/10.1063/1.4707954 Gerald Rosenthal, Keith E. Gubbins, and Sabine H. L. Klapp "Self-assembly of model amphiphilic Janus particles", Journal of Chemical Physics '''136''' 174901 (2012)]
*[http://dx.doi.org/10.1063/1.4793626 Miguel Ángel G. Maestre, Riccardo Fantoni, Achille Giacometti, and Andrés Santos "Janus fluid with fixed patch orientations: Theory and simulations", Journal of Chemical Physics '''138''' 094904 (2013)]

[[category: phase diagrams]]

Anisotropic particles with tetrahedral symmetry

2013-11-02T20:12:50Z

FSmallenburg: /* Crystallization */

[[Image:patchy_4.png|thumb|right| Artists impression of a tetrahedral patchy particle]]
'''Anisotropic particles with tetrahedral symmetry'''
==Kern and Frenkel model==
===Phase diagram===
The [[Phase diagrams |phase diagram]] of the tetrahedral [[Kern and Frenkel patchy model | Kern and Frenkel ]] [[patchy particles | patchy model]] exhibits the following solid phases<ref>[http://dx.doi.org/10.1021/jp9081905 Flavio Romano, Eduardo Sanz and Francesco Sciortino "Role of the Range in the Fluid−Crystal Coexistence for a Patchy Particle Model", Journal of Physical Chemistry B '''113''' pp. 15133–15136 (2009)]</ref><ref>[http://dx.doi.org/10.1063/1.3393777 Flavio Romano, Eduardo Sanz and Francesco Sciortino "Phase diagram of a tetrahedral patchy particle model for different interaction ranges", Journal of Chemical Physics '''132''' 184501 (2010)]</ref>:
[[Building up a diamond lattice |diamond crystal]] (DC),
[[Building up a body centered cubic lattice | body centred cubic]] (BCC) and [[Building up a face centered cubic lattice |face centred cubic]] (FCC). The gas-liquid [[critical points | critical point]] becomes metastable with respect
to the diamond crystal when the range of the interaction becomes short (roughly less than 15% of the
diameter).

:[[Image:romanojpcb09.gif]]

In contrast to isotropic models, the critical point becomes only weakly metastable with respect to the solid as the interaction range
narrows (from left to right in the figure).

===Crystallization===

Tetrahedral Kern-Frenkel patchy particles crystallise spontaneously into open tetrahedral networks for narrow patches (solid angle < 30). The interaction range does not play an important role in crystallisation <ref>[http://dx.doi.org/10.1063/1.3578182 Flavio Romano, Eduardo Sanz, and Francesco Sciortino "Crystallization of tetrahedral patchy particles in silico", Journal of Chemical Physics '''134''' 174502 (2011)]</ref>

[[Image:fig5.jpg]]

Interaction range, <math>\delta</math>, versus patch angular width.
Diamonds correspond to crystallising and circles to glass–forming models.
The point studied in Ref. <ref>[http://dx.doi.org/10.1063/1.3578182 Zhenli Zhang, Aaron S. Keys, Ting Chen, and Sharon C. Glotzer "Self-Assembly of Patchy Particles into Diamond Structures through Molecular Mimicry", Langmuir '''21''' 11547 (2005)]</ref> is included.

When the patches in this model are made even wider (while still enforcing the limit of a single bond per patch), the diamond phase becomes metastable with respect to a liquid phase, which is stable even in the zero-temperature limit <ref>[http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys2693.html Frank Smallenburg and Francesco Sciortino "Liquids more stable than crystals in particles with limited valence and flexible bonds", Nature Physics '''9''' 554 (2013)]</ref>.

==Modulated patchy Lennard-Jones model==
The solid phases of the [[modulated patchy Lennard-Jones model]] has also been studied <ref>[http://dx.doi.org/10.1063/1.3454907 Eva G. Noya, Carlos Vega, Jonathan P. K. Doye, and Ard A. Louis "The stability of a crystal with diamond structure for patchy particles with tetrahedral symmetry", Journal of Chemical Physics '''132''' 234511 (2010)]</ref>
==Lattice model==
<ref>[http://dx.doi.org/10.1080/00268976.2010.523521 N. G. Almarza and E. G. Noya "Phase transitions of a lattice model for patchy particles with tetrahedral symmetry", Molecular Physics '''109''' pp. 65-74 (2011)]</ref>

==See also==
*[[PMW]] (primitive model for [[water]])

== References ==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3582904 G. Munaó, D. Costa, F. Sciortino, and C. Caccamo "Simulation and theory of a model for tetrahedral colloidal particles", Journal of Chemical Physics '''134''' 194502 (2011)]
[[category: models]]

Parallel hard cubes

2013-04-28T13:34:09Z

FSmallenburg: Added information and references on phase transition

'''Parallel hard cubes''' are a simple particle [[Models |model]] used in [[statistical mechanics]]. They were introduced by B. T. Geilikman <ref>B. T. Geilikman "", Proceedings of the Academy of Science of the USSR '''70''' pp. 25- (1950)</ref> in 1950. The [[virial equation of state]] ([[pressure]] as a power series in the density) was studied by Zwanzig, Temperley, Hoover, and De Rocco <ref>[http://dx.doi.org/10.1063/1.1742621 Robert W. Zwanzig "Virial Coefficients of "Parallel Square" and "Parallel Cube" Gases", Journal of Chemical Physics '''24''' pp. 855-856 (1956)]</ref><ref name="6and7">[http://dx.doi.org/10.1063/1.1732443 William G. Hoover and Andrew G. De Rocco, "Sixth and Seventh Virial Coefficients for the Parallel Hard-Cube Model", Journal of Chemical Physics '''36''' pp. 3141- (1962)]</ref>. The latter two authors computed seven-term series for the models <ref name="6and7"> </ref>. Both the sixth and seventh terms in the hard-cube series are negative, a counter-intuitive result for repulsive interactions. In 1998 E. A. Jagla <ref>[http://dx.doi.org/10.1103/PhysRevE.58.4701 E. A. Jagla "Melting of hard cubes", Physical Review E '''58''' pp. 4701-4705 (1998)]</ref> investigated the melting transition for both parallel and rotating cube models, finding a qualitative difference in the nature of the transition for the two models. In that same year Martinez-Raton and Cuesta described cubes and mixtures of cubes (See [[Parallel hard cubes#Mixtures | Mixtures]] <ref name="6and7"> </ref>).
====Usefulness of the Model====
Parallel hard cubes have another use, beyond providing a simple model for which seven terms in the Mayers' virial series can be evaluated. In 2009 the Hoovers pointed out <ref>[http://dx.doi.org/10.1103/PhysRevE.79.046705 Wm. G. Hoover and C. G. Hoover "Nonlinear stresses and temperatures in transient adiabatic and shear flows via nonequilibrium molecular dynamics: Three definitions of temperature", Physical Review E '''79''' 046705 (2009)]</ref> that these models can be used as "ideal gas thermometers" capable of measuring the tensor [[temperature]] components <math>\{ T_{xx},T_{yy},T_{zz}\}</math>. Kinetic theory shows that particles colliding with a hard-cube [[Maxwell velocity distribution |Maxwell-Boltzmann]] [[ideal gas]] at temperature <math>T</math> will lose or gain energy according to whether the particle kinetic temperature exceeds <math>T</math> or not. The independence of the temperature components for the hard parallel cubes (or squares in two dimensions) allows them to serve as gedanken-experiment thermometers for all three temperature components.

==Phase behavior==
The phase diagram of parallel hard cubes shows a second-order phase transition from a fluid to a simple cubic crystal
<ref>[http://dx.doi.org/10.1063/1.1342816 B. Groh and B. Mulder, "A closer look at crystallization of parallel hard cubes", J. Chem. Phys. '''114''' pp. 3653 (2001)]</ref>,
which contains a large number of vacancies
<ref>[http://dx.doi.org/10.1063/1.3699086 M. Marechal, U. Zimmermann and H. Loewen, "Freezing of parallel hard cubes with rounded edges", J. Chem. Phys. '''136''' pp. 144506-144506 (2012)]</ref>.

==Mixtures==
<ref>[http://dx.doi.org/10.1103/PhysRevLett.76.3742 José A. Cuesta "Fluid Mixtures of Parallel Hard Cubes", Physical Review Letters '''76''' pp. 3742-3745 (1996)]</ref>
<ref>[http://dx.doi.org/10.1063/1.474298 José A. Cuesta and Yuri Martínez-Ratón "Fundamental measure theory for mixtures of parallel hard cubes. I. General formalism", Journal of Chemical Physics '''107''' pp. 6379- (1997)]</ref>
<ref>[http://dx.doi.org/10.1063/1.479273 Yuri Martínez-Ratón and José A. Cuesta "Fundamental measure theory for mixtures of parallel hard cubes. II. Phase behavior of the one-component fluid and of the binary mixture", Journal of Chemical Physics '''111''' pp. 317- (1999)]</ref>
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.1725285 William G. Hoover, "High-Density Equation of State of Hard Parallel Squares and Cubes", Journal of Chemical Physics '''40''' pp. 937- (1964)]
*[http://dx.doi.org/10.1063/1.450974 T. R. Kirkpatrick "Ordering in the parallel hard hypercube gas", Journal of Chemical Physics '''85''' pp. 3515-3519 (1986)]
*[http://dx.doi.org/10.1103/PhysRevLett.78.3681 José A. Cuesta and Yuri Martínez-Ratón "Dimensional Crossover of the Fundamental-Measure Functional for Parallel Hard Cubes", Physical Review Letters '''78''' pp. 3681-3684 (1997)]
*[http://dx.doi.org/10.1063/1.4754836 S. Belli, M. Dijkstra, and R. van Roij "Free minimization of the fundamental measure theory functional: Freezing of parallel hard squares and cubes", Journal of Chemical Physics '''137''' 124506 (2012)]

[[category: models]]

Hard cube model

2013-04-28T13:23:53Z

FSmallenburg: fixed reference

{{stub-general}}
[[Image:cube.png|thumb|right]]
The '''Hard cube model''' models cube-shaped particles interacting purely through excluded-volume interactions. The phase behavior has been studied extensively
<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>
<ref>[http://dx.doi.org/10.1073/pnas.1211784109 F. Smallenburg, L. Filion, M. Marechal, and M. Dijkstra "Vacancy-stabilized crystalline order in hard cubes", Proc. Natl. Acad. Sci. USA '''109''' pp. 17886-17891 (2012)]</ref>. The simple cubic crystal phase has been shown to contain a high number of mobile, delocalized vacancies, similar to those seen in [[Parallel hard cubes]].

==Implementation==
Overlaps between cubes can be checked based on the [http://en.wikipedia.org/wiki/Hyperplane_separation_theorem separating axis theorem], which says that if two convex objects are not interpenetrating, there must be an axis for which the projections of the two objects will not overlap. In the case of two
convex polyhedral particles, only a finite number of possible separating axes need to be
checked: in that case, the possible separating axes are either parallel to a normal of one
of the faces of either of the two particles, or perpendicular to the plane spanned by one
of the edges of the first particle and one of the edges of the second particle.

For two cubes, the overlap check therefore corresponds to the projection of both shapes onto 15 axes: the three edge vectors of both particles (6 in total), and the axes given by the 9 possible cross products of an edge vector of one cube with an edge vector of the other cube. The particles overlap if and only if the projections of the two particles on all 15 axes overlap as well.

==References==
<references/>

'''Related reading'''
*[http://dx.doi.org/10.1063/1.4734021 Umang Agarwal and Fernando A. Escobedo "Effect of quenched size polydispersity on the ordering transitions of hard polyhedral particles", Journal of Chemical Physics '''137''' 024905 (2012)]
[[category: models]]

Hard cube model

2013-04-28T13:16:26Z

FSmallenburg: Added some information and a reference

{{stub-general}}
[[Image:cube.png|thumb|right]]
The '''Hard cube model''' models cube-shaped particles interacting purely through excluded-volume interactions. The phase behavior has been studied extensively <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>
<ref>[http://dx.doi.org/10.1073/pnas.1211784109 F. Smallenburg, L. Filion, M. Marechal, and M. Dijkstra, "Vacancy-stabilized crystalline order in hard cubes",
Proc. Natl. Acad. Sci. USA '''109''' 17886 (2012)]</ref>. The simple cubic crystal phase has been shown to contain a high number of mobile, delocalized vacancies, similar to those seen in [[Parallel hard cubes]].

==Implementation==
Overlaps between cubes can be checked based on the [http://en.wikipedia.org/wiki/Hyperplane_separation_theorem separating axis theorem], which says that if two convex objects are not interpenetrating, there must be an axis for which the projections of the two objects will not overlap. In the case of two
convex polyhedral particles, only a finite number of possible separating axes need to be
checked: in that case, the possible separating axes are either parallel to a normal of one
of the faces of either of the two particles, or perpendicular to the plane spanned by one
of the edges of the first particle and one of the edges of the second particle.

For two cubes, the overlap check therefore corresponds to the projection of both shapes onto 15 axes: the three edge vectors of both particles (6 in total), and the axes given by the 9 possible cross products of an edge vector of one cube with an edge vector of the other cube. The particles overlap if and only if the projections of the two particles on all 15 axes overlap as well.

==References==
<references/>

'''Related reading'''
*[http://dx.doi.org/10.1063/1.4734021 Umang Agarwal and Fernando A. Escobedo "Effect of quenched size polydispersity on the ordering transitions of hard polyhedral particles", Journal of Chemical Physics '''137''' 024905 (2012)]
[[category: models]]