{{Author_|


first_name = Eduardo|
middle_initial = |
last_name = Sanz|

image_name = eduardo.png|
image_caption = |

position = Assistant professor|

institution = [http://www.ucm.es Universidad Complutense de Madrid] |
department = [http://www.ucm.es/info/quifi/ Departamento de Quimica Fisica I] |

groups = [http://www.ucm.es/info/molecsim/ Grupo de Termodinámica Estadística de Fluidos Moleculares]|

homepage_link = [http://marie.quim.ucm.es Eduardo Sanz] |

email = [http://marie.quim.ucm.es See my home page] |

}}

File:Eduardo.png

2011-09-19T15:06:53Z

Esanz:

2011-09-15T16:30:47Z

Esanz:

2011-09-15T16:25:31Z

Esanz:

Structure factor

2011-09-15T16:25:17Z

Esanz:

Structure factor

2011-09-15T16:23:06Z

Esanz:

Kern and Frenkel patchy model

2011-08-08T14:43:30Z

Esanz: Undo revision 11683 by Esanz (talk)

The '''Kern and Frenkel''' <ref>[http://dx.doi.org/10.1063/1.1569473 Norbert Kern and Daan Frenkel "Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction", Journal of Chemical Physics 118, 9882 (2003)]</ref> [[Patchy particles |patchy model]] is an amalgamation of the [[hard sphere model]] with
attractive [[Square well model | square well]] patches (HSSW). The potential has an angular aspect, given by (Eq. 1)

:<math>\Phi_{ij}({\mathbf r}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j) =\Phi_{ij}^{ \mathrm{HSSW}}({\mathbf r}_{ij}) \cdot f(\tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j) </math>

where the radial component is given by the square well model (Eq. 2)

:<math>
\Phi_{ij}^{ \mathrm{HSSW}} \left({\mathbf r}_{ij} \right) =
\left\{ \begin{array}{ccc}
\infty & ; & r < \sigma \\
- \epsilon & ; &\sigma \le r < \lambda \sigma \\
0 & ; & r \ge \lambda \sigma \end{array} \right.
</math>

and the orientational component is given by (Eq. 3)

:<math>
f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j \right) =
\left\{ \begin{array}{clc}
1 & \mathrm{if} & \left\{ \begin{array}{ccc} & (\hat{e}_\alpha\cdot\hat{r}_{ij} \leq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i \\
\mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \leq \cos \delta) & \mathrm{for~some~patch~\beta~on~}j \end{array} \right. \\
0 & \mathrm{otherwise} & \end{array} \right.
</math>

where <math>\delta</math> is the solid angle of a patch (<math>\alpha, \beta, ...</math>) whose axis is <math>\hat{e}</math> (see Fig. 1 of Ref. 1), forming a conical segment.
==Two patches==
The "two-patch" Kern and Frenkel model has been extensively studied by Giacometti et al. <ref>[http://dx.doi.org/10.1063/1.3415490 Achille Giacometti, Fred Lado, Julio Largo, Giorgio Pastore, and Francesco Sciortino "Effects of patch size and number within a simple model of patchy colloids", Journal of Chemical Physics 132, 174110 (2010)]</ref>.
==Four patches==
:''Main article: [[Phase diagram of anisotropic particles with tetrahedral symmetry]]''
==References==
<references/>

[[category: models]]

Kern and Frenkel patchy model

2011-08-08T14:40:01Z

Esanz: /* Four patches */

The '''Kern and Frenkel''' <ref>[http://dx.doi.org/10.1063/1.1569473 Norbert Kern and Daan Frenkel "Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction", Journal of Chemical Physics 118, 9882 (2003)]</ref> [[Patchy particles |patchy model]] is an amalgamation of the [[hard sphere model]] with
attractive [[Square well model | square well]] patches (HSSW). The potential has an angular aspect, given by (Eq. 1)

:<math>\Phi_{ij}({\mathbf r}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j) =\Phi_{ij}^{ \mathrm{HSSW}}({\mathbf r}_{ij}) \cdot f(\tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j) </math>

where the radial component is given by the square well model (Eq. 2)

:<math>
\Phi_{ij}^{ \mathrm{HSSW}} \left({\mathbf r}_{ij} \right) =
\left\{ \begin{array}{ccc}
\infty & ; & r < \sigma \\
- \epsilon & ; &\sigma \le r < \lambda \sigma \\
0 & ; & r \ge \lambda \sigma \end{array} \right.
</math>

and the orientational component is given by (Eq. 3)

:<math>
f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j \right) =
\left\{ \begin{array}{clc}
1 & \mathrm{if} & \left\{ \begin{array}{ccc} & (\hat{e}_\alpha\cdot\hat{r}_{ij} \leq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i \\
\mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \leq \cos \delta) & \mathrm{for~some~patch~\beta~on~}j \end{array} \right. \\
0 & \mathrm{otherwise} & \end{array} \right.
</math>

where <math>\delta</math> is the solid angle of a patch (<math>\alpha, \beta, ...</math>) whose axis is <math>\hat{e}</math> (see Fig. 1 of Ref. 1), forming a conical segment.
==Two patches==
The "two-patch" Kern and Frenkel model has been extensively studied by Giacometti et al. <ref>[http://dx.doi.org/10.1063/1.3415490 Achille Giacometti, Fred Lado, Julio Largo, Giorgio Pastore, and Francesco Sciortino "Effects of patch size and number within a simple model of patchy colloids", Journal of Chemical Physics 132, 174110 (2010)]</ref>.
==Four patches==
:''Main article: [[Anisotropic particles with tetrahedral symmetry]]''

==References==
<references/>

[[category: models]]

Anisotropic particles with tetrahedral symmetry

2011-08-08T14:37:34Z

Esanz: /* Crystallization */

[[Image:patchy_4.png|thumb|right| Artists impression of a tetrahedral patchy particle]]
==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.

==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]]

Anisotropic particles with tetrahedral symmetry

2011-08-08T14:36:59Z

Esanz: /* Crystallization */

[[Image:patchy_4.png|thumb|right| Artists impression of a tetrahedral patchy particle]]
==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.

==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]]

2009-11-28T16:24:10Z

Esanz:

The phase diagram of tetrahedral, patchy particles <ref>[http://dx.doi.org/10.1021/jp9081905 F. Romano, E. Sanz and F. Sciortino "Role of the Range in the Fluid−Crystal Coexistence for a
Patchy Particle Model", J. Phys. Chem. B '''113''' pp. 15133–15136 (2009)]</ref>
exhibits the following solid phases: Diamond Crystal (DC),
Body Centered Cubic (BCC) and Face Centered Cubic (FCC). The gas-liquid 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]]

By contrast to isotropic models, the critical point
has only a weak metastability
with respect to the solid as the interaction range
narrows (from left to right in the figure).

----

=== References ===
<references/>

File:Romanojpcb09.gif

2009-11-28T16:08:16Z

Esanz:

Anisotropic particles with tetrahedral symmetry

2009-11-28T16:07:50Z

Esanz:

The phase diagram of tetrahedral, patchy particles <ref>[http://dx.doi.org/10.1021/jp9081905 F. Romano, E. Sanz and F. Sciortino "Role of the Range in the Fluid−Crystal Coexistence for a
Patchy Particle Model", J. Phys. Chem. B '''113''' pp. 15133–15136 (2009)]</ref>
exhibits the following solid phases: Diamond Crystal,
Body Centered Cubic and Face Centered Cubic. The gas-liquid 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). Interestingly, and differently from the isotropic case, the supersaturation of the fluid at the critical point does not significantly increase upon going toward the adhesive (vanishing interaction range) limit.

[[Image:romanojpcb09.gif]]

----

=== References ===
<references/>

File:Jp-2009-081905 0005.gif

2009-11-28T16:02:32Z

Esanz: Phase diagram of a tetrahedral patchy model for two ranges interaction ranges. For short ranges (on the right) the critical point becomes metastable with respect the DC (Diamond Crystal) solid.

Phase diagram of a tetrahedral patchy model for two ranges interaction ranges. For short ranges (on the right) the critical point becomes metastable with respect the DC (Diamond Crystal) solid.

Anisotropic particles with tetrahedral symmetry

2009-11-28T15:59:42Z

Esanz: /* References */

The phase diagram of tetrahedral, patchy particles <ref>[http://dx.doi.org/10.1021/jp9081905 F. Romano, E. Sanz and F. Sciortino "Role of the Range in the Fluid−Crystal Coexistence for a
Patchy Particle Model", J. Phys. Chem. B '''113''' pp. 15133–15136 (2009)]</ref>
exhibits the following solid phases: Diamond Crystal,
Body Centered Cubic and Face Centered Cubic. The gas-liquid 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). Interestingly, and differently from the isotropic case, the supersaturation of the fluid at the critical point does not significantly increase upon going toward the adhesive (vanishing interaction range) limit.

----

=== References ===
<references/>