# Kern and Frenkel patchy model

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The Kern and Frenkel [1] 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),[2] and later Chapman (1988) [3] [4] 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)


${\displaystyle \Phi _{ij}({\mathbf {r} }_{ij};{\tilde {\mathbf {\Omega } }}_{i},{\tilde {\mathbf {\Omega } }}_{j})=\Phi _{ij}^{\mathrm {HSSW} }({\mathbf {r} }_{ij})\cdot f({\tilde {\mathbf {\Omega } }}_{i},{\tilde {\mathbf {\Omega } }}_{j})}$

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

${\displaystyle \Phi _{ij}^{\mathrm {HSSW} }\left({\mathbf {r} }_{ij}\right)=\left\{{\begin{array}{ccc}\infty &;&r<\sigma \\-\epsilon &;&\sigma \leq r<\lambda \sigma \\0&;&r\geq \lambda \sigma \end{array}}\right.}$

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

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

where ${\displaystyle \delta }$ is the solid angle of a patch (${\displaystyle \alpha ,\beta ,...}$) whose axis is ${\displaystyle {\hat {e}}}$ (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 [5][6][7].

## Four patches

Main article: Anisotropic particles with tetrahedral symmetry

## Single-bond-per-patch-condition

If the two parameters ${\displaystyle \delta }$ and ${\displaystyle \lambda }$ fullfil the condition

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

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 [3][5][7]

## Hard ellipsoid model

The hard ellipsoid model has also been used as the 'nucleus' of the Kern and Frenkel patchy model [8].