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Kern and Frenkel patchy model

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The Kern and Frenkel [1] 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 \le r < \lambda \sigma \\
0         & ; & r \ge \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} \geq \cos \delta) & \mathrm{for~some~patch~\alpha~on~}i  \\ 
                                                            \mathrm{and} & (\hat{e}_\beta\cdot\hat{r}_{ji} \geq \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[edit]

The "two-patch" Kern and Frenkel model has been extensively studied by Sciortino and co-workers [2][3][4].

Four patches[edit]

Main article: Anisotropic particles with tetrahedral symmetry


If the two parameters \delta and \lambda fullfil the condition

\sin{\delta} \leq \dfrac{1}{2(1+\lambda\sigma)}

then the patch cannot be involved in more than one bond. Enforcing this condition makes it possible to compare the simulations results with Wertheim theory [2][4]

Hard ellipsoid model[edit]

The hard ellipsoid model has also been used as the 'nucleus' of the Kern and Frenkel patchy model [5].


Related reading