Editing Kern and Frenkel patchy model

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]] published in 2003 is an amalgamation of the [[hard sphere model]] with
attractive [[Square well model | square well]] patches (HSSW). The model was originally developed by Bol (1982),<ref>[http://dx.doi.org/10.1080/00268978200100461 W. Bol "Monte Carlo simulations of fluid systems of waterlike molecules", Molecular Physics '''45''' pp. 605-616 (1982)]</ref> and later Chapman (1988) <ref name="Chapman">[W.G. Chapman, Doctoral Thesis, Cornell University (1988)]</ref> <ref>[G. Jackson, W.G. Chapman, K.E. Gubbins, Molecular Physics 65, 1-31 (1988)]</ref> reinvented the model as the basis for numerous articles describing properties of associating particles from molecular simulation and theory. The computational advantage of Bol's model is that only a simple dot product is required to determine if a particle is in the bonding orientation. 
 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} \geq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i  \\ 
                                                            \mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \geq \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.
==Multiple patches==
The "two-patch" and "four-patch" Bol (Chapman or Kern and Frenkel) model was extensively studied by Chapman and co-workers for bulk and interfacial systems using hard sphere and Lennard-Jones reference systems.  Later other groups, including Sciortino and co-workers, considered stronger association energies for the "two-patch" hard sphere reference <ref name="bianchi">[http://dx.doi.org/10.1063/1.2730797  F. Sciortino, E. Bianchi, J. Douglas and P. Tartaglia "Self-assembly of patchy particles into polymer chains: A parameter-free comparison between Wertheim theory and Monte Carlo simulation", Journal of Chemical Physics '''126''' 194903 (2007)]</ref><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><ref name="rovigatti">[http://dx.doi.org/10.1063/1.4737930  José Maria Tavares, Lorenzo Rovigatti, and Francesco Sciortino "Quantitative description of the self-assembly of patchy particles into chains and rings", Journal of Chemical Physics '''137''' 044901 (2012)]</ref>.

==Four patches==
:''Main article: [[Anisotropic particles with tetrahedral symmetry]]''

==Single-bond-per-patch-condition==
If the two parameters <math>\delta</math> and <math>\lambda</math> fullfil the condition

:<math>
\sin{\delta} \leq \dfrac{1}{2(1+\lambda\sigma)}
</math>

then the patch cannot be involved in more than one bond. Enforcing this condition makes it possible to compare the simulations results with [[Wertheim's first order thermodynamic perturbation theory (TPT1)| Wertheim theory]] <ref name="Chapman"/><ref name="bianchi"/><ref name="rovigatti"/>

==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1063/1.3689308 Christoph Gögelein, Flavio Romano, Francesco Sciortino, and Achille Giacometti "Fluid-fluid and fluid-solid transitions in the Kern-Frenkel model from Barker-Henderson thermodynamic perturbation theory", Journal of Chemical Physics '''136''' 094512 (2012)]
*[http://dx.doi.org/10.1063/1.4722477 Emanuela Bianchi, Günther Doppelbauer, Laura Filion, Marjolein Dijkstra, and Gerhard Kahl "Predicting patchy particle crystals: Variable box shape simulations and evolutionary algorithms", Journal of Chemical Physics '''136''' 214102 (2012)]
*[http://dx.doi.org/10.1063/1.4960423  Z. Preisler, T. Vissers, F. Smallenburg and F. Sciortino "Crystals of Janus colloids at various interaction ranges", Journal of Chemical Physics '''145''' 064513 (2016)]


[[category: models]]