Vibrational states and disorder in continuously compressed model glasses

Abstract

We present in this paper a numerical study of the vibrational eigenvectors of a two-dimensional amorphous material, previously deeply studied from the point of view of mechanical properties and vibrational eigen-frequencies [7], [8], [9], [10]. Attention is paid here to the connection between the mechanical properties of this material in term of elastic heterogeneities (EH), and how these inherent heterogeneous structures affect the vibrational eigenvectors and their plane waves decomposition. The systems are analysed for different hydrostatic pressures, and using results from previous studies, a deeper understanding of the boson peak scenario is obtained. The vibrational spectrum of a continuously densified silica glass is also studied, from which it appears that the pulsation associated with the boson peak follows the same pressure dependence trend than that of transverse waves with pulsation associated with the EH characteristic size.

Introduction

Glasses develop a resistance to a macroscopic deformation, like solids, but depict a lack of structural order, like liquids. In solids, the response to such a deformation is theoretically approached by the use of the continuum elasticity, which involve phonons to model the vibrational excitations. Due to the loss of translational invariance in glasses in comparison to the corresponding crystal, propagating phonons are difficult to describe, but have already been observed down to the atomic level [1]. It's also admitted that a common characteristic of disordered solids, when compared to crystalline ones, lies in the existence of an excess of low-frequency vibrational states, the well-known boson peak. This excess produces the also well-known specific heat anomaly of glasses at temperatures T ∼ 10 K, and seems to be a signature of the disorder in glasses beyond the nanometer scale.

Many attempts have been proposed to theoretically interpret the boson peak anomaly [2], [3], [4], [5], [6], [24], [25], [28], [38], and among them, Shintani and Tanaka [11] have recently proposed an interpretation in term of involved atomic arrangements with transverse vibrations that tend to be particularly strong, in two-dimensional systems, around favoured structures of five-fold symmetry. The framework in which the boson peak is more related with transverse vibrations than longitudinal ones has been already approached in a series of papers dealing with the vibrational and mechanical properties of two- and three-dimensional soft (Lennard–Jones) and strong (silica) glasses [7], [8], [9], [10]. In those systems, transversal motions of the particles give rise to large vortices of characteristic size ξ_naff, when they respond to a macroscopic deformation. This noisy displacement field has been called non-affine displacement field, and involve correlated displacement of atoms from ξ_naff ∼ 20 inter-atomic distances in three dimensions, to ξ_naff ∼ 30 in two dimensions. In both soft and strong glasses, the application of the classical continuum elasticity theory is subject to strong limitations below such length scales ξ_naff. This effect is particularly important when the wavelength of the vibrational excitation is less than this characteristic size ξ_naff. The origin of the departure from classical behaviour is very likely related to the disorder in inter-atomic interactions (local stresses, inhomogeneities in elastic constants, local anisotropy). This leads to an important decrease of the average shear modulus in amorphous systems compared with the corresponding crystals. Note that the typical size ξ_naff depends on the pressure: it decreases and saturates at high pressures for both soft and strong glasses. Finally, it has been also previously noticed that the estimate of the frequency associated with ξ_naff is in good agreement with the boson peak position [10], and thus should encourage to consider the boson peak as a length marking a crossover between (i) a regime where vibrations with wavelengths larger than ξ_naff may be well described by a classical continuum theory, and (ii) a small wavelength regime where vibrations are strongly affected by the elastic heterogeneities (EH).

In the following, such a picture will be confirmed in the case of a two-dimensional glass. After a summary of the theoretical concepts related with the continuum elasticity and an overview of the simulated system, the pressure dependence of the vibrational density of states (vDOS) and the boson peak will be approached for both strong and soft glasses. We then expect a significant change of the vDOS upon compression, namely a decrease of boson peak height and its shift toward higher frequencies [12]. While this kind of pressure dependence study is merely used, in the literature, to improve the theoretical interpretations of the boson peak [13], [14], [15], [16], [21], [32], most of the present work will be devoted in the effect of increasing pressure, on the interplay between elastic heterogeneities (EH) and elastic wave propagation in a two-dimensional glass.