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)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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)

$f_{ij}\left({\hat {\mathbf {r} }}_{ij};{\tilde {\mathbf {\Omega } }}_{i},{\tilde {\mathbf {\Omega } }}_{j}\right)=\left\{{\begin{array}{clc}1&\mathrm {if} &\left\{{\begin{array}{ccc}&({\hat {e}}_{\alpha }\cdot {\hat {r}}_{ij}\leq \cos \delta )&\mathrm {for~some~patch~\alpha ~on~} i\\\mathrm {and} &({\hat {e}}_{\beta }\cdot {\hat {r}}_{ji}\leq \cos \delta )&\mathrm {for~some~patch~\beta ~on~} j\end{array}}\right.\\0&\mathrm {otherwise} &\end{array}}\right.$

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 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 \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