Sound attenuation in stable glasses

Abstract

The temperature dependence of the thermal conductivity of amorphous solids is markedly different from that of their crystalline counterparts, but exhibits universal behaviour. Sound attenuation is believed to be related to this universal behaviour. Recent computer simulations demonstrated that in the harmonic approximation sound attenuation Γ obeys quartic, Rayleigh scattering scaling for small wavevectors k and quadratic scaling for wavevectors above the Ioffe-Regel limit. However, simulations and experiments do not provide a clear picture of what to expect at finite temperatures where anharmonic effects become relevant. Here we study sound attenuation at finite temperatures for model glasses of various stability, from unstable glasses that exhibit rapid aging to glasses whose stability is equal to those created in laboratory experiments. We find several scaling laws depending on the temperature and stability of the glass. First, we find the large wavevector quadratic scaling to be unchanged at all temperatures. Second, we find that at small wavectors Γ ∼ k 1.5 for an aging glass, but Γ ∼ k 2 when the glass does not age on the timescale of the calculation. For our most stable glass, we find that Γ ∼ k 2 at small wavevectors, then a crossover to Rayleigh scattering scaling Γ ∼ k 4 , followed by another crossover to the quadratic scaling at large wavevectors. Our computational observation of this quadratic behavior reconciles simulation, theory and experiment, and will advance the understanding of the temperature dependence of thermal conductivity of glasses.

Energy transport in glasses

“…We found that the frequency dependence of the thermal diffusivity in glasses, as calculated within the harmonic approximation, can be divided into four main regions. For low ω energy transport is dominated by transverse sound waves whose attenuation Γ obeys a Rayleigh scattering law 24,33 . Therefore, at low frequencies d(ω) = A low ω −4 where A low can be predicted from the attenuation of transverse sound waves.…”

“…We now show that the frequency dependence of d(ω) is consistent with d(ω) = v L (ω)/3 where the mean free path (ω) = v L /Γ L (ω) and Γ L (ω) is the sound attenuation coefficient. In earlier work we found that transverse sound attenuation Γ T (ω) could be rescaled by a constant factor so that it overlaps with longitudinal sound attenuation Γ L (ω) 33 . In the inset to Fig.…”

“…The mean free path of the transverse excitation (ω) is given by (ω) = 3d(ω)/v T (ω), and the relationship between d(ω) and sound attenuation Γ(ω) was discussed in Section V. In previous work we demonstrated that Γ(k) = B T k 4 for small k, and thus (ω) = 2v 6 T /(B T ω 4 ) assuming a linear dispersion relation ω = v T k. For ω = 0.5 we use our fit to Γ(k) from the previous work 33 and obtain (0.5) ≈ 2011. Shown as a dashed line in Fig.…”

“…Once the phonons are the main carriers of the energy, then the energy transport is ballistic and δr 2 (ω, t) = (v L (ω)t) 2 , which is shown as the blue dashed line. The longitudinal speed of sound v L (ω) obtained from previous work on sound attenuation 33 . We observe that the calculated δr 2 (ω, t) nearly matches this prediction for ω = 0.3.…”

“…Due to the small size of the simulation box, it is not possible to obtain diffusivity at small ω using the method of Allen and Feldman. Additionally, it has been recently observed that there are finite size effects in calculations of vibrational modes from the Hessian matrix 33,34,38 , which may result in calculated thermal diffusivity d AF (ω) that does not correspond to the thermodynamic limit.…”