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)

$\Phi _{ij}^{\mathrm {HSSW} }\left({\mathbf {r} }_{ij}\right)=\left\{{\begin{array}{ccc}\infty &;&r<\sigma \\-\epsilon &;&\sigma \leq r<\lambda \sigma \\0&;&r\geq \lambda \sigma \end{array}}\right.$

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta } is the solid angle of a patch (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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: Anisotropic particles with tetrahedral symmetry