Attenuation of shear sound waves in jammed solids

Abstract

We study the attenuation of long-wavelength shear sound waves propagating through model jammed packings of frictionless soft spheres interacting with repulsive springs. The elastic attenuation coefficient, α(ω), of transverse phonons of low frequency, ω, exhibits power law scaling as the packing fraction φ is lowered towards φc, the critical packing fraction below which rigidity is lost. The elastic attenuation coefficient is inversely proportional to the scattering mean free path and follows Rayleigh law with α(ω) ∼ ω 4 (φ − φc) −5/2 for ω much less than ω * ∼ (φ − φc) 1/2 , the characteristic frequency scale above which the energy diffusivity and density of states plateau. This scaling of the attenuation coefficient, consistent with numerics, is obtained by assuming that a jammed packing can be viewed as a mosaic composed of domains whose characteristic size ℓ * ∼ (φ − φc) −1/2 diverges at the transition.

Sound attenuation in stable glasses

“…In several theories, these inhomogeneities have been modeled as local fluctuations of elastic constants [3][4][5][6][7][8] . These theories predict that the sound attenuation scales with the fourth power of the wavevector, Γ λ (k) ∼ k 4 (λ = L denotes longitudinal waves and λ = T denotes transverse waves) for small wavevector k. Mean-field theories [9][10][11][12] arrive at the same prediction, albeit in a different way. Yet another theoretical treatment, the soft-potential model, predicts that a quartic scaling regime exists due to phonons interacting with soft modes 13 .…”

Phonon transport and vibrational excitations in amorphous solids

“…Furthermore, we study two additional length scales, D T and D L , for transverse and longitudinal waves. At Ω ω ex0 , we can define a length D that characterizes the structural disorder responsible for the Rayleigh scattering [51]. To do this, we solve the scattering problem for elastic waves [90].…”

“…Let us consider a situation in which an elastic wave propagates in an elastic medium with scattering sources. If we assume that its wavelength is much longer than the length D of the scattering sources, we can derive the Rayleigh scattering law and formulate the attenuation rate as follows [51,90,91] δγ L = δ(K + 4G/3)/(K + 4G/3) are for transverse and longitudinal waves, respectively (see Appendix D). We remark that transverse and longitudinal waves are scattered by different elastic inhomogeneities, namely, shear (G) and longitudinal (K+4G/3) moduli inhomogeneities, respectively [92].…”

“…In this approach, two-level systems [34][35][36][37][38] and soft localized vibrations [39][40][41][42][43][44] are essential ingredients in producing the vibrational and thermal anomalies. A third approach is based on the mean-field theory of jamming [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60], in which isostaticity and marginal stability are crucial. It is not easy to judge the validity of these theories because critical tests are difficult to conduct due to the presence of phenomenological fitting parameters.…”

“…Although these results are consistent with experimental observations [9][10][11][12][13][14], the precise location of the crossover frequency has not yet been settled. Moreover, the length scale of the structural disorder responsible for the Rayleigh scattering remains unclear [51]. Furthermore, and more critically, even the validity of the Rayleigh scattering law has been seriously questioned in recent work [22].…”