# The excluded area of two-dimensional hard particles

## TL;DR

This paper investigates how the excluded area between two-dimensional hard particles varies with orientation, using principal component analysis to classify shapes and Monte Carlo simulations to analyze their phase behavior.

## Contribution

It introduces a PCA-based method to classify excluded areas of 2D particles and links shape features to excluded area properties and phase behavior.

## Key findings

- Three principal components suffice to represent excluded area variations.
- Minimum excluded area occurs when particles are antiparallel.
- Elongation significantly influences the excluded area.

## Abstract

The excluded area between a pair of two-dimensional hard particles with given relative orientation is the region in which one particle cannot be located due to the presence of the other particle. The magnitude of the excluded area as a function of the relative particle orientation plays a major role in the determination of the bulk phase behaviour of hard particles. We use principal component analysis to identify the different types of excluded area corresponding to randomly generated two-dimensional hard particles modeled as non-self-intersecting polygons and star lines (line segments radiating from a common origin). Only three principal components are required to have an excellent representation of the value of the excluded area as a function of the relative particle orientation. Independently of the particle shape, the minimum value of the excluded area is always achieved when the particles are antiparallel to each other. The property that affects the value of the excluded area most strongly is the elongation of the particle shape. Principal component analysis identifies four limiting cases of excluded areas with one to four global minima at equispaced relative orientations. We study selected particle shapes using Monte Carlo simulations.