Bistability in a self-assembling system confined by elastic walls. Exact results in a one-dimensional lattice model

TL;DR

This paper provides exact and asymptotic analysis of a one-dimensional lattice model to understand how confinement influences self-assembly, revealing bistability and structural deformations related to phase transition-like behavior.

Contribution

It introduces exact and asymptotic formulas for confined SALR particles, highlighting the effects of incommensurability and elastic walls on self-assembly and bistability.

Findings

Exact formulas agree with simulations for at least 5 layers.

Incommensurability causes structural deformations and defect changes.

Soft elastic walls induce bistability with nearly equal probabilities for two system sizes.

Abstract

The impact of confinement on self-assembly of particles interacting with short-range attraction and long-range repulsion (SALR) potential is studied for thermodynamic states corresponding to local ordering of clusters or layers in the bulk. Exact and asymptotic expressions for the local density and for the effective potential between the confining surfaces are obtained for a one-dimensional lattice model introduced in [J. P\k{e}kalski et al. $J. Chem. Phys.$ ${\bf 140}$, 144903 (2013)].The simple asymptotic formulas are shown to be in good quantitative agreement with exact results for slits containing at least 5 layers. We observe that the incommensurability of the system size and the average distance between the clusters or layers in the bulk leads to structural deformations that are different for different values of the chemical potential $\mu$. The change of the type of defects is reflected in the dependence of density on $\mu$ that has a shape characteristic for phase transitions. Our results may help to avoid misinterpretation of the change of the type of defects as a phase transition in simulations of inhomogeneous systems. Finally, we show that a system confined by soft elastic walls may exhibit bistability such that two system sizes that differ approximately by the average distance between the clusters or layers are almost equally probable. This may happen when the equilibrium separation between the soft boundaries of an empty slit corresponds to the largest stress in the confined self-assembling system.