# Diffusion-driven self-assembly of rod-like particles: Monte Carlo simulation on a square lattice

## TL;DR

This study uses Monte Carlo simulations to explore how rod-like particles self-assemble on a square lattice, revealing stages of pattern formation and the influence of particle length and system size.

## Contribution

It introduces a detailed simulation framework for diffusion-driven self-assembly of rod-like particles, identifying key stages and pattern attractors in the process.

## Key findings

- Long rods ($k \\geq 6$) form diagonal stripe domains.
- Self-assembly involves three stages: destruction, coarsening, and stripe formation.
- Pattern formation time increases with system size.

## Abstract

The diffusion-driven self-assembly of rod-like particles was studied by means of Monte Carlo simulation. The rods were represented as linear $k$-mers (i.e., particles occupying $k$ adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size $L\times L$ up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, $L$, was varied from $128$ to $2048$, and periodic boundary conditions were applied along both $x$ and $y$ axes, while the length of the $k$-mers (determining the aspect ratio) was varied from $2$ to $12$. The $k$-mers oriented along the $x$ and $y$ directions ($k_x$-mers and $k_y$-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of $k$-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the $k$-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long $k$-mers. For long $k$-mers ($k\geq 6$), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state, the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of $k$-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth.