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)

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

where $\delta$ is the solid angle of a patch ($\alpha ,\beta ,...$) 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. .