The Kern–Frenkel Model

Overview

Questions

• What are patchy particles?

• What is the Kern-Frenkel potential?

• How can I evaluate the Kern-Frenkel potential between two particles?

Objectives

• Describe the Kern–Frenkel model mathematically and graphically.

Patchy Particles

In the Introducing Molecular Dynamics tutorial, you simulated a system of particles interacting through the Lennard-Jones pair potential. The Lennard-Jones pair potential is spherically symmetric. In other words, the pair potential does not depend on the relative orientation of the interacting particles.

Some systems are better represented by anisotropic pair potentials that depend on relative particle orientations. For example, patchy particles interact via anisotropic pair potentials where some relative orientations are more attractive than others.

The Kern–Frenkel Model

The Kern–Frenkel model, introduced in a 2003 Journal of Chemical Physics paper, models particles with a single patch. It consists of a directional pairwise energetic interaction in addition to hard sphere volume exclusion. The pair potential $U_{ij}(\vec{r}_{ij}, \mathbf{q}_i, \mathbf{q}_j)$ between particles $i$ and $j$ at a center-to-center separation $\vec{r}_{ij}$ and orientations $\mathbf{q}_i$ and $\mathbf{q}_j$ is of the form

$$U_{ij} = \begin{cases} \infty & r_{ij} < \sigma_{ij} \\ -\varepsilon\cdot f(\vec{r}_{ij}, \mathbf{q}_i, \mathbf{q}_j) & \sigma_{ij} \leq r_{ij} < \lambda_{ij}\sigma_{ij} \\ 0 & r_{ij} \geq \lambda_{ij}\sigma_{ij} \end{cases}$$

where $\varepsilon$ is the strength of the patchy interaction, $\sigma_{ij}$ is sum of the radii of particles $i$ and $j$, $\lambda_{ij}$ is the range of the square well attraction, and $f$ is an orientational masking function given by

$$f(\vec{r}_{ij}, \mathbf{q}_i, \mathbf{q}_j) = \begin{cases} 1 & \hat{e}_i \cdot \hat{r}_{ij} \ge \cos \delta \land \hat{e}_j \cdot \hat{r}_{ji} \ge \cos \delta \\ 0 & \mathrm{otherwise} \end{cases}$$

where $\hat{e}_i$ is the director of the patch on particle $i$ and $\delta$ is the half-opening angle of the patch.

Graphically, this pair potential corresponds to the following: two particles interact with energy $\infty$ if the gray shaded regions on the two particles overlap at all, $-\varepsilon$ if the blue shaded regions on the two particles overlap, and zero otherwise.