# Ordering of hard rectangles in strong confinement

## TL;DR

This study investigates the structural phases of hard rectangles confined between parallel walls, revealing three distinct ordering states influenced by density, shape, and confinement dimensions using theoretical approaches.

## Contribution

It introduces a combined transfer operator and fundamental measure theory approach to analyze phase behavior of hard rectangles under confinement.

## Key findings

- Identified three main structural phases: fluid with parallel alignment, layered planar order, and homeotropic order.
- Demonstrated phase transitions depend on density, shape, and pore width.
- Observed complex structural behavior, including multiple phases with increasing density.

## Abstract

Using transfer operator and fundamental measure theories, we examine the structural and thermodynamic properties of hard rectangles confined between two parallel hard walls. The side lengths of the rectangle ($L$ and $D$, $L>D$) and the pore width ($H$) are chosen such that maximum two layers are allowed to form in planar order ($L$ is parallel to the wall), while only one in homeotropic order ($D$ is parallel to the wall). We observe three different structures: (i) a low density fluid phase with parallel alignment to the wall, (ii) an intermediate and high density fluid phase with two layers and planar ordering and (iii) a dense single fluid layer with homeotropic ordering. The appearance of these phases and the change in the ordering direction with density is a consequence of the varying close packing structures with $L$ and $H$. Interestingly, even three different structures can be observed with increasing density if $L$ is close to $H$.