Kern and Frenkel patchy model

The Kern and Frenkel patchy model published in 2003 is an amalgamation of the hard sphere model with attractive square well patches (HSSW). The model was originally developed by Bol (1982), and later Chapman(1988) reinvented the model as the basis for numerous articles describing properties of associating particles from molecular simulation and theory. The computational advantage of Bol's model is that only a simple dot product is required to determine if a particle is in the bonding orientation.

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)

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

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.

Multiple patches

The "two-patch" and "four-patch" Bol (Chapman or Kern and Frenkel) model was extensively studied by Chapman and co-workers for bulk and interfacial systems using hard sphere and Lennard-Jones reference systems. Later other groups, including Sciortino and co-workers, considered stronger association energies for the "two-patch" hard sphere reference .

Four patches

Main article: Anisotropic particles with tetrahedral symmetry

Single-bond-per-patch-condition

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

$\sin \delta \le \frac{1}{2(1+\lambda \sigma )}$

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

Hard ellipsoid model

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

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)
• Z. Preisler, T. Vissers, F. Smallenburg and F. Sciortino "Crystals of Janus colloids at various interaction ranges", Journal of Chemical Physics 145 064513 (2016)