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 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. as well as others .

Four patches

Main article: Anisotropic particles with tetrahedral symmetry

References

Related reading

• Christoph Gögelein, Flavio Romano, Francesco Sciortino, and Achille Giacometti "Fluid-fluid and fluid-solid transitions in the Kern-Frenkel model from Barker-Henderson thermodynamic perturbation theory", Journal of Chemical Physics 136 094512 (2012)
• Emanuela Bianchi, Günther Doppelbauer, Laura Filion, Marjolein Dijkstra, and Gerhard Kahl "Predicting patchy particle crystals: Variable box shape simulations and evolutionary algorithms", Journal of Chemical Physics 136 214102 (2012)