Editing Kern and Frenkel patchy model
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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]] | 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). | 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> | :<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> | ||
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:<math> | :<math> | ||
f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{ {\mathbf \Omega}}_i, \tilde{ {\mathbf \Omega}}_j \right) = | f_{ij} \left(\hat{ {\mathbf r}}_{ij}; \tilde{{\mathbf \Omega}}_i, \tilde{{\mathbf \Omega}}_j \right) = | ||
\left\{ \begin{array}{clc} | \left\{ \begin{array}{clc} | ||
1 & \mathrm{if} & \left\{ \begin{array}{ccc} & (\hat{e}_\alpha\cdot\hat{r}_{ij} \ | 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} \ | \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. | 0 & \mathrm{otherwise} & \end{array} \right. | ||
</math> | </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. | 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" | 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== | ==Four patches== | ||
:''Main article: [[Anisotropic particles with tetrahedral symmetry]]'' | :''Main article: [[Anisotropic particles with tetrahedral symmetry]]'' | ||
==References== | ==References== | ||
<references/> | <references/> | ||
[[category: models]] | [[category: models]] |