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)

${\mathrm{\Phi }}_{ij}({\mathbf{r}}_{ij};{\tilde{\mathrm{\Omega }}}_i,{\tilde{\mathrm{\Omega }}}_j)=\underset{ij}{\overset{\mathrm{H}\mathrm{S}\mathrm{S}\mathrm{W}}{\mathrm{\Phi }}}({\mathbf{r}}_{ij})\cdot f({\tilde{\mathrm{\Omega }}}_i,{\tilde{\mathrm{\Omega }}}_j)$

where the radial component is given by the square well model (Eq. 2)

$\underset{ij}{\overset{\mathrm{H}\mathrm{S}\mathrm{S}\mathrm{W}}{\mathrm{\Phi }}}({\mathbf{r}}_{ij})=\{\begin{array}{ccc}\mathrm{\infty }& ;& r<\sigma \\ -ϵ& ;& \sigma \le r<\lambda \sigma \\ 0& ;& r\ge \lambda \sigma \end{array}$

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

$f_{ij}({\hat{\mathbf{r}}}_{ij};{\tilde{\mathrm{\Omega }}}_i,{\tilde{\mathrm{\Omega }}}_j)=\{\begin{array}{ccc}1& \mathrm{i}\mathrm{f}& \{\begin{array}{ccc}& ({\hat{e}}_{\alpha }\cdot {\hat{r}}_{ij}\le \cos \delta )& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}{\alpha }\mathrm{o}\mathrm{n}i\\ \mathrm{a}\mathrm{n}\mathrm{d}& ({\hat{e}}_{\beta }\cdot {\hat{r}}_{ji}\le \cos \delta )& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}{\beta }\mathrm{o}\mathrm{n}j\end{array}\\ 0& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}& \end{array}$