Acoustic excitations and elastic heterogeneities in disordered solids

In the recent years, much attention has been devoted to the inhomogeneous nature of the mechanical response at the nanoscale in disordered solids. Clearly, the elastic heterogeneities that have been characterized in this context are expected to strongly affect the nature of the sound waves which, in contrast to the case of perfect crystals, cannot be completely rationalized in terms of phonons. Building on previous work on a toy model showing an amorphization transition, we investigate the relationship between sound waves and elastic heterogeneities in a unified framework by continuously interpolating from the perfect crystal, through increasingly defective phases, to fully developed glasses. We provide strong evidence of a direct correlation between sound wave features and the extent of the heterogeneous mechanical response at the nanoscale.In crystals, molecules thermally oscillate around the periodic lattice sites and vibrational excitations are well understood in terms of quantized plane waves, the phonons (1). The vibrational density of states (vDOS) in the low-frequency regime is well described by the Debye model, where the vibrational modes are the acoustic phonons. In contrast, disordered solids, including structural glasses and disordered crystals, exhibit specific vibrational properties compared with the corresponding pure crystalline phases. It is not possible here to give a fair review of the extensive theoretical and experimental work generated by these issues; we therefore mention below a few facts that we consider the most relevant in the present context. The origin of the vDOS modes in excess over the Debye prediction around ω ∼1 THz, the so-called Boson peak (BP), is still debated (see, among many others, refs. 2 and 3). At the BP frequency, Ω^BP, localized modes have also been observed (4). Acoustic plane waves, which are exact normal modes in crystals, can still propagate in disordered solids. Indeed, at low frequencies, Ω, and long wavelengths, Λ, acoustic sound waves do not interact with disorder and can propagate conforming to the expected macroscopic limit. However, as Ω is increased beyond the Ioffe–Regel (IR) limit, Ω^IR, acoustic excitations interact with the disorder and are significantly scattered (5–7). Interestingly, this strong scattering regime occurs around the BP position, Ω^IR ∼ Ω^BP (8, 9). The exact origin of this phenomenon and its connection to the BP remain elusive.A possible rationalization of the above issues is based on the existence of elastic heterogeneities (10), which can originate from structural disorder, as in structural glasses (2), or disordered interparticle potentials, even in lattice structures such as disordered colloidal crystals (11). In the heterogeneous-elasticity theory of refs. 7 and 12 this amounts to consider spatial statistical fluctuations of the shear modulus. Within the framework of jamming approaches and using effective medium theories, elastic heterogeneities are related to the proximity of local elastic instabilities (13). Recent simulation work (14–16) has clearly demonstrated their existence in disordered solids. This is at variance with the case of simple crystals, which are characterized by a fully affine response and homogeneous moduli distributions (17). More specifically, in the large length scale limit, macroscopic moduli are observed. In contrast, as the length scale is reduced, moduli heterogeneities are detected, at a typical length scale ξ ≃ 10−15σ (15), where σ is the typical atomic diameter. Breakdown of both continuum mechanics (18) and Debye approximation (5, 6) has been demonstrated at the same mesoscopic length-scale ξ, where they are still valid for crystals. Remarkably, the wave frequency corresponding to the wavelength Λ ∼ ξ is very close to Ω^IR ∼ Ω^BP (19). Altogether these results indicate that a close connection must exist between elastic heterogeneities and acoustic excitations. In this paper we precisely address this point.In ref. 20 we considered a numerical model featuring an amorphization transition (21). We showed how to systematically deform the local moduli distributions, evaluated by coarse-graining the system in small domains of linear length scale w. We characterized the degree of elastic heterogeneity in terms of SD of those distributions and studied the effect on normal modes (eigenvalues of the Hessian matrix) and thermal conductivity. Building on that work, we are now in the position to investigate the relation between elastic heterogeneities and acoustic excitations, unifying in a single framework ordered and disordered solid states and considering quantities directly probed by experiments. By interpolating in a controlled way from perfect crystals, through increasingly defective phases, to fully developed amorphous structures, we (i) calculate the dynamical structure factors, extracting the relevant spectroscopic parameters; (ii) characterize the wave vector dependence of sound velocity and broadening of the acoustic excitations and clarify their nature in terms of the IR limit; and (iii) provide, for the first time to our knowledge, direct evidence of the correlation of the excitations lifetimes and Ω^IR with the magnitude of the elastic heterogeneities.