Negative Thermal Expansion and Some Elastic properties of a Class of Solids

TL;DR

This paper develops a theoretical model for solids with rigid units that predicts negative thermal expansion and decreasing elastic constants with temperature and pressure, aligning with experimental observations.

Contribution

It introduces a new invariant-based elasticity theory for solids with rigid polyhedral units, explaining negative thermal expansion and elastic property variations.

Findings

Negative thermal expansion coefficient predicted

Elastic constants decrease with temperature and pressure

Model aligns with experimental observations

Abstract

We consider the thermal expansion, change of sound velocity with pressure and temperature, and the Poisson ratio of lattices which have rigid units (polyhedra very large stiffness to change in bond-length and to bond-angle variations) connected to other such units through relatively compressible polyhedra. We show that in such a model, the potential energy for rotations of the rigid units can occur only as a function of the combination ${\boldsymbol \Theta}_i \equiv ({\boldsymbol \theta}_i - (\nabla \times {\bf u}_i)/2)$, where ${\boldsymbol \theta_i} $ are the orthogonal rotation angles of the rigid unit $i$ and ${\bf u}_i$ is its displacement. Given such new invariants in the theory of elasticity and the hierarchy of force constants of the model, a negative thermal expansion coefficient as well as a decrease in the elastic constants of the solid with temperature and pressure is shown to follow. These are consistent with the observations.