Normal mode approach for predicting the mechanical properties of solids from first principles: Application to compressibility and thermal expansion of zeolites

Abstract

We present a method for analyzing the mechanical properties of solids, based on normal modes and their coupling to lattice strains. This method was used to study elastic compression and thermal expansion of zeolites, with parameters calculated from density functional theory. We find in general that the bulk modulus can be divided into two contributions: a positive term arising from compression without internal relaxation, and a negative term from coupling between compression and internal vibrational modes. For silica polymorphs, the former term varies little among the phases studied, reflecting the intrinsic rigidity of $\mathrm{Si}{\mathrm{O}}_{4}$ tetrahedra. In contrast, the latter term varies strongly from one polymorph to the next, because each polymorph exhibits different symmetry constraints on internal vibrations and their couplings to lattice strains. Typically only a few normal modes contribute to the bulk modulus. To facilitate parametrization of this normal mode model, we constructed a simplified classical spring-tetrahedron model for silica. After fitting to properties of silica sodalite, this model reproduces cell volumes and predicts bulk moduli of $\ensuremath{\alpha}$-cristobalite and silica zeolites CHA, LTA, and MFI. We incorporated anharmonic effects into the theory, allowing the calculation of the thermal expansion coefficient. The resulting expression provides a generalization of classical Gr\"uneisen theory, taking into account additional anharmonicities. This method was used to study thermal expansion of fcc aluminum and an aluminosilica sodalite, yielding good agreement with experiment.