Mechanical instability at finite temperature.

Xiaoming Mao,T C Lubensky,Anton Souslov,Carlos I Mendoza

doi:10.1038/ncomms6968

Abstract

Many physical systems including lattices near structural phase transitions, glasses, jammed solids and biopolymer gels have coordination numbers placing them at the edge of mechanical instability. Their properties are determined by an interplay between soft mechanical modes and thermal fluctuations. Here we report our investigation of the mechanical instability in a lattice model at finite temperature T. The model we used is a square lattice with a φ(4) potential between next-nearest-neighbour sites, whose quadratic coefficient κ can be tuned from positive to negative. Using analytical techniques and simulations, we obtain a phase diagram characterizing a first-order transition between the square and the rhombic phase and different regimes of elasticity, as well as an 'order-by-disorder' effect that favours the rhombic over other zigzagging configurations. We expect our study to provide a framework for the investigation of finite-T mechanical and phase behaviour of other systems with a large number of floppy modes.

Highlights

Many physical systems including lattices near structural phase transitions, glasses, jammed solids and biopolymer gels have coordination numbers placing them at the edge of mechanical instability
Lattices with coordination number z 1⁄4 zc 1⁄4 2d in d spatial dimensions, which we will call Maxwell lattices[5], exist at the edge of mechanical instability, and they are critical to the understanding of systems as diverse as engineering structures[6,7], diluted lattices near the rigidity threshold[8,9,10], jammed systems[11,12,13], biopolymer networks[14,15,16,17,18] and network glasses[19,20]
Hypercubic lattices in d dimensions and the kagome lattice and its generalization to higher dimensions with nearest-neighbour (NN) Hookean springs of spring constant k are a special type of Maxwell lattice whose phonon spectra in a system of N unit cells have harmonic-level zero modes at all N(d À 1)/d points on (d À 1)-dimensional hyperplanes oriented along symmetry directions and passing through the origin[21] of the Brillouin zone

Summary

Introduction

Many physical systems including lattices near structural phase transitions, glasses, jammed solids and biopolymer gels have coordination numbers placing them at the edge of mechanical instability. We use these simulations to investigate the phase diagram[38] corresponding to the Hamiltonian (3) and to investigate the properties of the phases we encounter, such as ground-state degeneracy, order-by-disorder and negative thermal expansion.

Results

Conclusion