# Role of length-polydispersity on the phase behavior of freely-rotating hard-rectangle fluid

## TL;DR

This study uses density functional theory to analyze how length-polydispersity influences phase transitions and stability in a fluid of freely-rotating hard rectangles, revealing effects on transition order and phase stability.

## Contribution

It provides a detailed theoretical analysis of the impact of polydispersity on phase behavior and phase transitions in hard-rectangle fluids, including stability of the tetratic phase.

## Key findings

- Polydispersity increases the first-order nature of the isotropic-nematic transition.
- The tricritical point shifts to higher aspect ratios with increased polydispersity.
- The tetratic phase is slightly destabilized as polydispersity increases.

## Abstract

We used the Density Functional formalism, in particular the Scaled Particle Theory, applied to a length-polydisperse hard-rectangular fluid to study its phase behavior as a function of the mean particle aspect ratio ($\kappa_0$) and polydispersity ($\Delta_0$). The numerical solutions of the coexistence equations were calculated by transforming the original problem with infinite degrees of freedoms to a finite set of equations for the amplitudes of the Fourier expansion of the moments of the density profiles. We divided the study into two parts: The first one is devoted to the calculation of the phase diagrams in the packing fraction ($\eta_0$)- $\kappa_0$ plane for a fixed $\Delta_0$ and selecting parent distribution functions with exponential (the Schulz distribution) or Gaussian decays. In the second part we study the phase behavior in the $\eta_0$-$\Delta_0$ plane for fixed $\kappa_0$ while $\Delta_0$ is changed. We characterize in detail the orientational ordering of particles and the fractionation of different species between the coexisting phases. Also we study the character (second vs. first order) of the Isotropic-Nematic phase transition as a function of polydispersity. We particularly focused on the stability of the Tetratic phase as a function of $\kappa_0$ and $\Delta_0$. The Isotropic-Nematic transition becomes strongly of first order when polydispersity is increased: the coexisting gap widens and the location of the tricritical point moves to higher values of $\kappa_0$ while the Tetratic phase is slightly destabilized with respect to the Nematic one. The results obtained here can be tested in experiments on shaken monolayers of granular rods.