Kern and Frenkel patchy model

The Kern and Frenkel 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)

$\Phi _{ij}({\mathbf {r} }_{ij};{\tilde {\mathbf {\Omega } }}_{i},{\tilde {\mathbf {\Omega } }}_{j})=\Phi _{ij}^{\mathrm {HSSW} }({\mathbf {r} }_{ij})\cdot f({\tilde {\mathbf {\Omega } }}_{i},{\tilde {\mathbf {\Omega } }}_{j})$

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. }

where 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 \delta} is the solid angle of a patch (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 \alpha, \beta, ...} ) whose axis is ${\hat {e}}$ (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. .

Four patches

Main article: Phase diagram of anisotropic particles with tetrahedral symmetry