Confined Crystals on Substrates : Order and Fluctuations in Between One and Two Dimensions

Abstract

The effect of lateral confinement on a crystal of point particles in d = 2 dimensions in a strip geometry is studied by Monte Carlo simulations and using phenomenological theoretical concepts. Physically, such systems confined in long strips of width D can be realized via colloidal particles at the air–water interface, or by adsorbed monolayers at suitably nanopatterned substrates, etc. As a generic model, we choose a repulsive interparticle potential decaying with the twelfth inverse power of distance. This system has been well studied in the bulk as a model for two-dimensional melting. The state of the system is found to depend very sensitively on the boundary conditions providing the confinement. For corrugated boundaries commensurate with the order of the bulk (i.e., a triangular crystalline lattice structure), both orientational and positional orders are enhanced, and near the boundaries surface-induced order persists also at temperatures where the system is fluid in the bulk. For incommensurate corrugated boundaries, however, soliton staircases near the boundaries form, causing a pattern of standing strain waves in the strip. However, smooth unstructured repulsive boundaries enhance orientational order only, positional long-range order is destroyed. The strip then exhibits a vanishing shear modulus. A comparison with surface effects on phase transitions in simple Ising and XY-models is also made. Finally, possible applications are discussed.