Kern and Frenkel patchy model: Difference between revisions
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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]] is an amalgamation of the [[hard sphere model]] with | 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: | 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> | |||
==References== | ==References== | ||
<references/> | <references/> | ||
[[category: models]] | [[category: models]] | ||
Revision as of 12:11, 10 May 2010
The Kern and Frenkel [1] patchy model is an amalgamation of the hard sphere model with attractive square well patches (HSSW). The potential has an angular aspect, given by (Eq. 1)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) }
where the radial component is given by the square well model (Eq. 2)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. }
and the orientational component is given by (Eq. 3)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. }