# A continuum model reproducing the multiple frequency crossovers in acoustic
attenuation in glasses

H. Luo  V. M. Giordano  A. Gravouil  A. Tanguy  Univ Lyon, INSA-Lyon, CNRS
UMR5259, LaMCoS, F-69621, France  ONERA, University Paris-Saclay, Chemin de la
Hunière, BP 80100, 92123 Palaiseau, France  Institut Lumière Matière, UMR 5306
Université Lyon 1-CNRS, F-69622 Villeurbanne Cedex, France

###### Abstract

Structured metamaterials are at the core of extensive research, promising for
thermal and acoustic engineering. However, the computational cost required for
correctly simulating large systems imposes to use continuous modeling, able to
grasp the physics at play without entering in the atomistic details.
Crucially, a correct description needs to describe both the extrinsic
interface-induced and the intrinsic atomic scale-originated phonon scattering.
This latter becomes considerably important when the metamaterial is made out
of a glass, which is intrinsically highly dissipative and with a wave
attenuation strongly dependent on frequency and temperature. In amorphous
systems, the effective acoustic attenuation triggered by multiple mechanisms
is now well characterized and exhibits a nontrivial frequency dependence with
a double crossover of power laws. Here we propose a continuum mechanical model
for a viscoelastic medium, able to bridge atomic and macroscopic scales in
amorphous materials and reproduce well the phonon attenuation from GHz to THz
with a $\omega^{2}-\omega^{4}-\omega^{2}$ dependency, including the influence
of temperature.

###### keywords:

Acoustic attenuation, Constitutive modeling , Dynamic mechanical analysis ,
Finite-element simulation , Nanomaterials , Viscoelasticity

##  1 Introduction

Heterogeneous architectured materials have attracted considerable attention on
their potential applications [1]. They are manmade structural materials
designed for obtaining ad hoc  properties, that cannot be generally found in
nature. They are generally obtained by engineering the interplay between the
shape, the properties and possible association of materials at different
scales, either in a random spacial distribution (composite materials) either
with the repetition of regular patterns (metamaterials). Depending on their
application, the lengthscale of such patterns can span from nanometer to
macroscopic range. For example, metamaterials have been initially investigated
for applications in acoustics, for realizing acoustic guides, filters, lenses
[2, 3, 4, 5]. And recently, they have also been introduced into the field of
thermal science, for realizing tunable multi-functional thermal metamaterials
[6, 7], thermal cloaks and camouflage [8], etc. In both applications, it is
matter of guiding, filtering, hindering, the propagation of acoustic waves
(phonons), responsible for the sound propagation when their wavelength is
macroscopic, and for thermal transport at room temperature when it is
nanometric.

A heterogeneous architectured material affects the propagation of the acoustic
waves mainly by introducing an extrinsic source of scattering, represented by
the interfaces. In the specific case of metamaterials featuring periodicity,
novel coherent effects arise due to the Bragg scattering. As a result, a new
Brillouin Zone can be defined, which strongly modifies the phonon band
structure by folding the dispersions at its boundaries, thus introducing new
optic modes which can scatter the originally acoustic ones. Moreover, the
presence of elemental bricks in the micro-structure introduces local
resonances with phonons, leading to the aperture of forbidden gaps in the
phonon dispersion [2, 3], useful for the realization of acoustic filters and
for impacting thermal transport [9]. Recently, it has been proposed to use
amorphous materials as building blocks for metamaterials for thermal transport
engineering, due to their renowned low thermal conductivity [10, 11, 12, 13,
14, 15, 9]. Nanocomposites made of amorphous and crystalline components
promise to strongly reduce thermal transport.

To correctly describe complex structured materials, it is required to model
phonon propagation on large scale systems, more apt at representing real
materials, where the effect of heterogeneities and interfaces can be properly
investigated. Such large systems are out of reach for atomistic simulations
and can be realistically modeled only through continuous modeling. However, a
correct description of phonon attenuation in heterogeneous architectured
materials requires to take into account not only the extrinsic scattering
sources introduced by the architectured micro-structure, but also the
intrinsic attenuation in the component materials, especially when these latter
are highly dissipative, as it is the case for amorphous materials.

A multiscale approach is therefore needed, able to connect the atomistic
scale, where the intrinsic phonon attenuation takes its origin, and the
macroscopic scale, where the metamaterial assembly introduces the extrinsic
scattering sources. A particularly restrictive requirement is, that such model
should be able to reproduce the frequency- and temperature-dependence of the
intrinsic acoustic attenuation, which is specific to the microscopic
mechanisms at play.

In glasses, apparent acoustic attenuation can be related to a viscous
consumption of mechanical energy induced by anharmonic effects [16], or it can
result from acoustic scattering in disordered harmonic systems [17, 18, 19,
20]. It can be temperature-dependent, or not[21, 22, 23, 24, 25, 26]. Most
interestingly, it is frequency-dependent[22, 27]. Numerous experiments and
numerical works (atomistic simulations) have investigated the acoustic
attenuation [22, 23, 24, 25, 28, 29, 21, 30, 31, 32, 33, 34, 35, 26, 36, 18,
37, 38, 39]. A non-exhaustive summary of experimental and theoretical
literature results on acoustic attenuation in glasses is reported in Tab.1 and
SM Tab. S1. Many attenuation channels are involved, whose importance depends
on phonon frequency and temperature: tunneling due to two level systems [40,
41], soft modes [42, 43, 44], thermally activated relaxation processes [27],
anharmonicity [16], and scattering resulting from structural disorder [17, 18,
20]. This last contribution is temperature independent, and dominates acoustic
attenuation at frequencies in the GHz-THz range [19]. For lower frequencies,
anharmonicity gives rise to a phonon attenuation, or damping, $\Gamma$,
inversely proportional to phonon lifetime, which changes with frequency as
$\omega^{a}$ with $a\approx 1.5-2$ [26]. Approaching the THz range,
corresponding to phonon wavelengths in the nanometer scale, scattering on
structural atomistic disorder starts to dominate, characterized by a Rayleigh-
like dependence $\Gamma\propto\omega^{4}$ [18, 45], with a progressive
transition to a strong scattering regime, finally leading to a new high
frequency $\Gamma\propto\omega^{2}$ regime, as reported in a number of
experimental and theoretical studies [23, 24, 46, 47, 25, 22, 48, 27, 49],
eventually followed by a sudden drop [19]. The transition to strong scattering
takes place at frequencies comparable with the Boson Peak, i.e. the excess of
modes in the phonon density of states with respect to the Debye prediction at
low frequency, and has been explained as due to phonon scattering by
nanometric elastic heterogeneities [50, 51, 45, 48, 49, 52, 53, 54]. At low
temperature, molecular-dynamics simulations have been able to reproduce this
attenuation crossover, with some different scaling rules [29, 52, 17, 18, 36].
The addition of anharmonicity at finite temperatures has finally provided a
complete picture of the combination of the different attenuation channels [46,
26].

On the whole, three regimes of acoustic attenuation versus frequency can be
classified : (1) $\Gamma\propto\omega^{a}$ due to the anharmonicity at low
frequencies, with $a\approx 1.5-2$ and temperature dependent strength; (2)
$\Gamma\propto\omega^{4}$ due to Rayleigh-like scattering induced by disorder
and leading to a dramatic phonon lifetime reduction, so that the collective
vibrational modes lose progressively their plane wave character; (3)
$\Gamma\propto\omega^{2}$ above Ioffe-Regel frequency
($\omega_{IR}=\pi\Gamma$), where phonons cannot be considered as propagative
plane waves anymore. As such, they do not propagate anymore but are diffusive,
and, at higher frequency, even may localize (in the Anderson’s definition of
localization, but for acoustic waves) [55, 17, 20, 56, 57].

Table 1: Experimental results of sound attenuation of glasses in the
literature. (TJ: tunneling junction; BLS: Brillouin light scattering; BUVS :
Brillouin ultraviolet light scattering; POT: picosecond optical technique.)

Materials |  Method |  T (K) |  Power |  Frequency range and/or $\nu_{c}$ |  LA velocity (m/s) |  Ref
---|---|---|---|---|---|---
v-SiO2 | TJ | 1 | 4 | 0.1-0.4 THz |  | [34]
BUVS | 300 | 2 |  5-70 GHz* |  | [31]
IUVS | 300 | 2 | 75-95 GHz* |  | [30]
IUVS | 300 | 2-4 |  75-150 GHz $\nu_{c}=100\pm 10$ GHz | 5950 | [32]
POT | 300 | 2 |  near 250 GHz | 5940 | [33]
IXS | 1050 | 2 |  1-3 THz | 5800 | [39]
IXS | 1620 | 4-2 |  1- 4THz $\nu_{c}=$1.5 THz | 6500 | [22]
d-SiO2 | IXS | 565 | 4.21 | 1.2-1.9 THz |  | [23]
Li2O-²2B2O3 | BLS |  300 573 | 1 | 20-40 GHz | 7600 | [24]
IXS | 573 | 4 | 1.2-2.4 THz |  | [24]
Glycerol | IXS |  16 167 | 2 | 0.6-3.5 THz $\dagger$ |  | [21]
IXS | 150.1 | 4-2 |  1-2.5 THz $\nu_{c}=$1.2 THz | 3630 | [25]
BeF2 | IXS | 297 | 2 |  0.6-6 THz | 5500 |  [37] [35]
Poly butadiene | IXS | 140 | 2 |  0.6-3.6 THz | 2770 |  [38] [35]

  * *

Use the sound speed value in Ref.[32].

  * $\dagger$

Use the sound speed value in Ref.[25].

Nevertheless, little consideration has been given so far to the use of those
results on acoustic attenuation at the macroscale due to the lack of adapted
continuum constitutive laws. A continuous model is expected to describe the
effective mechanical behavior without knowing the atomistic details. However,
the peculiar frequency and temperature dependence of attenuation in glasses
needs to be somehow reproduced in order to have realistic macroscale dynamical
simulations of acoustic properties for large structures or composites. In this
context, we thus need to connect atomistic and macroscopic scale, keeping the
information on the active intrinsic attenuation channels and the microscopic
frequency and temperature dependence of phonon attenuation.

To do so, a model is urgently needed to homogenize the effective attenuation
triggered by multiple mechanisms, and characterized by a defined frequency
dependence. Within Stoke’s theory [58], the amplitude of a plane wave
decreases exponentially with the propagation distance in viscous media, with a
decay rate given, in the low frequency limit $\omega\tau\ll 1$, by
$\gamma=\frac{\omega^{2}\tau}{2c}$ , with $\tau=\frac{4\eta}{3\rho c^{2}}$
where $\eta$ is the dynamic viscosity coefficient, $\rho$ is the fluid density
and $c$ is the speed of sound in the absence of viscosity. This behavior leads
to a typical attenuation distance $l=1/\gamma$ scaling as
$\propto\omega^{-2}$. A similar exponential attenuation of acoustic wave
packets, eventually with another frequency dependence, was also evidenced in
amorphous materials up to the Ioffe-Regel frequency [20], allowing defining a
mean-free path from the corresponding Beer-Lambert law for acoustic transport
attenuation. This behavior is thus similar in amorphous materials and in
Newtonian liquids. Glasses could be then characterized by an effective
viscoelastic behaviour for their acoustic properties, even below the glass
transition temperature (that is, in the solid state). This corresponds to the
requirement that the internal friction in the material can be characterized by
the quality factor $Q^{-1}=G^{\prime\prime}/G^{\prime}$ where
$G=G^{\prime}+iG^{\prime\prime}$ is the complex elastic modulus in the linear
regime [40]. This is indeed the case in a glass [19] for frequencies up to the
Ioffe-Regel crossover, where acoustic phonons (named propagons [55]) still
maintain well-defined wave vectors and whose amplitude exhibits an exponential
decay. A relation has been proposed between the microscopic quantity (phonon
attenuation) and the macroscopic one (quality factor): $\Gamma/\omega=Q^{-1}$
[59, 19, 60, 53, 61, 62] able to interpret experimental neutron-scattering
data for example, within the Damped Harmonic Oscillator Model for phonons
[63].

Recently, we have proved that the above concept can describe acoustic
attenuation in an amorphous material at THz frequencies, in a limited range
below the Ioffe-Regel criterium[64]. In that work, we have assumed a constant
viscosity, thus leading to a $\omega^{2}$ behaviour in the acoustic
attenuation. However, at such energies, the dominant scattering source is the
structural disorder at the atomic scale [49, 19] thus leading to a
$\omega^{4}$ behaviour, as found in [24, 25, 22, 27, 49, 26]. The
corresponding low scattering is potentially responsible for the unusual
temperature dependence $\kappa\propto T$ observed in the low temperature
regime [15], while this low temperature sensitivity was also related to
anharmonic effects in the 2-level model [40, 42]. The progressive transition
to a stronger scattering and to the resulting diffusive motion of initial
plane waves will be responsible for the progressive saturation which may be
observed as a plateau in the glassy thermal conductivity at around 10 K
followed by an increase of $\kappa$ with the temperature, and is also
responsible for the peak in the specific heat at the same temperature [65, 66,
15]. Modeling amorphous materials for thermal applications clearly needs
including different frequency dependences, with at least three successive
regimes including $\Gamma\propto\omega^{a}$ with $a\approx 2$ to take account
of possible anharmonic effects in the low frequency range (low temperatures
[26]) , $\Gamma\propto\omega^{4}$ in the low scattering regime and then
$\Gamma\propto\omega^{2}$ again at very high frequencies in the strong
scattering (diffusive) regime.

In this work, we develop a new continuum mechanical model for a viscoelastic
medium, where no disorder is introduced, by taking into account the different
attenuation regimes acting in parallel in the rheological responses of
glasses. We will show that our model is able to reproduce the attenuation at a
macroscopic level, through a single continuum model, involving the combined
effect of parallel sources of acoustic attenuation and their crossovers. By
tuning the parameters of the model, our constitutive law can reproduce
qualitatively and quantitatively the three regimes of acoustic attenuation
versus frequency: successively $\Gamma\propto\omega^{a},\omega^{4},\omega^{2}$
with $a\approx$ 2.

##  2 Model

Let’s assume an isotropic, homogeneous and viscous solid (amorphous materials
being homogeneous and isotropic above the nanometer scales [67] ): we can
express the elastic constitutive laws describing the stress-strain relation
using the Hooke’s law and separating the hydrostatic from the deviatoric
components

$\sigma_{ij}=3K\epsilon^{sph}_{ij}+2G\epsilon^{dev}_{ij}$ (1)

where $K$ is the generalized bulk modulus and $G$ is the generalized shear
modulus. The hydrostatic (or spherical) part of strain is given by
$\epsilon^{sph}_{ij}=\frac{1}{3}\delta_{ij}\epsilon_{kk}$ and the deviatoric
part, which is isochoric, is
$\epsilon^{dev}_{ij}=\epsilon_{ij}-\frac{1}{3}\delta_{ij}\epsilon_{kk}$, where
$\delta_{ij}$ is Dirac function.

Figure 1: Illustration of the viscoelastic model: two Maxwell-like models in
parallel.

In Ref.[64], we proposed to use two different simple rheological models for
the spherical and deviatoric parts separately, determining the longitudinal
and transverse quality factor. Here, we propose a new model for the transverse
modes. As shown in Fig. 1, we assume two Maxwell-like models in parallel, with
$\mu_{1}$ and $\mu_{2}$ the rigidity and $\eta_{1}\omega^{-4}$ and
$\eta_{2}\omega^{-2}$ the frequency-dependent viscosity, respectively. The
physical idea behind this model is the asynchronous activation of the
$\omega^{4}$ and the $\omega^{2}$ dependence of the attenuation, resulting
from two processes contributing additionally to the global stress in the
system. The frequency-dependent elastic modulus
$G^{*}(\omega)=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)$ for the model
illustrated in Fig. 1 can be expressed as:

$\begin{split}G^{*}(\omega)&=\frac{i\mu_{1}\eta_{1}\omega^{-3}}{i\eta_{1}\omega^{-3}+\mu_{1}}+\frac{i\mu_{2}\eta_{2}\omega^{-1}}{i\eta_{2}\omega^{-1}+\mu_{2}}\end{split}$
(2)

We can write down the full complex constitutive elastic tensor
$\mathbb{G}^{*}$ [64], which is reduced to a $6\times 6$ symmetric matrix due
to isotropy, imposing the conditions
$G^{*}_{11}=G^{*}_{22}=G^{*}_{33}=K^{*}(\omega)+\frac{2}{3}G^{*}(\omega)$ and
$G^{*}_{44}=G^{*}_{55}=G^{*}_{66}=G^{*}(\omega)$, being directly related to
sound propagation and attenuation. By definition, isotropic longitudinal (L)
and transverse (T) sound speeds are defined as

$\displaystyle\begin{cases}v_{L}(\omega)^{2}&=\frac{G^{\prime}_{11}(\omega)}{\rho}\\\
v_{T}(\omega)^{2}&=\frac{G^{\prime}_{44}(\omega)}{\rho}\end{cases}$ (3)

similarly, the inverse quality factors are obtained as:

$\displaystyle
Q^{-1}_{L}(\omega)=\frac{G^{\prime\prime}_{11}(\omega)}{G^{\prime}_{11}(\omega)}\
\text{and}\
Q^{-1}_{T}(\omega)=\frac{G^{\prime\prime}_{44}(\omega)}{G^{\prime}_{44}(\omega)}$
(4)

where $G^{\prime}_{ij}=\operatorname{Re}(G^{*}_{ij})$ and
$G^{\prime\prime}_{ij}=\operatorname{Im}(G^{*}_{ij})$.

We will derive the complex elastic modulus in order to obtain the analytic
expression of the quality factor $Q^{-1}=G^{\prime\prime}/G^{\prime}$. Holding
the relation $Q^{-1}=\Gamma/\omega$, ideally, we expect to get a
$\omega^{3}$-$\omega^{1}$-$\omega^{3}$ behavior for $Q^{-1}$. At the same
time, the amplitude of $\Gamma/\omega$ and the crossover position should be
clearly reproduced.

From Eq.2, $G^{\prime}$ and $G^{\prime\prime}$ can be extracted:

$\begin{split}G^{\prime}(\omega)&=\frac{\mu_{1}\eta_{1}^{2}}{\mu_{1}^{2}\omega^{6}+\eta_{1}^{2}}+\frac{\mu_{2}\eta_{2}^{2}}{\mu_{2}^{2}\omega^{2}+\eta_{2}^{2}}\\\
&=\mu_{1}(\frac{\tau_{1}^{2}}{\omega^{6}+\tau_{1}^{2}}+\alpha\frac{\tau_{2}^{2}}{\omega^{2}+\tau_{2}^{2}})\end{split}$
(5)

$\begin{split}G^{\prime\prime}(\omega)&=\frac{\mu_{1}^{2}\eta_{1}\omega^{3}}{\mu_{1}^{2}\omega^{6}+\eta_{1}^{2}}+\frac{\mu_{2}^{2}\eta_{2}\omega}{\mu_{2}^{2}\omega^{2}+\eta_{2}^{2}}\\\
&=\mu_{1}(\frac{\tau_{1}\omega^{3}}{\omega^{6}+\tau_{1}^{2}}+\alpha\frac{\tau_{2}\omega}{\omega^{2}+\tau_{2}^{2}})\end{split}$
(6)

where $\tau_{1}=\frac{\eta_{1}}{\mu_{1}}(\mathrm{s}^{-3})$,
$\tau_{2}=\frac{\eta_{2}}{\mu_{2}}(\mathrm{s}^{-1})$ and
$\alpha=\frac{\mu_{2}}{\mu_{1}}$. As such, there are only three parameters
left to characterize the attenuation. Finally, the quality factor
$Q^{-1}=G^{\prime\prime}/G^{\prime}$ is given by:

$\begin{split}Q^{-1}(\omega)&=\frac{\frac{\tau_{1}\omega^{3}}{\omega^{6}+\tau_{1}^{2}}+\alpha\frac{\tau_{2}\omega}{\omega^{2}+\tau_{2}^{2}}}{\frac{\tau_{1}^{2}}{\omega^{6}+\tau_{1}^{2}}+\alpha\frac{\tau_{2}^{2}}{\omega^{2}+\tau_{2}^{2}}}\\\
&=\frac{\alpha\tau_{2}\omega^{7}+\tau_{1}\omega^{5}+\tau_{1}\tau_{2}^{2}\omega^{3}+\alpha\tau_{1}^{2}\tau_{2}\omega}{\alpha\tau_{2}^{2}\omega^{6}+\tau_{1}^{2}\omega^{2}+(1+\alpha)\tau_{1}^{2}\tau_{2}^{2}}\end{split}$
(7)

As discussed in the SM Appendix, we show the existence of
$\omega$-$\omega^{3}$-$\omega$ behavior given by the quality factor $Q^{-1}$
of our model, indicating that the model is capable to describe
$\Gamma\propto\omega^{2}$-$\omega^{4}$-$\omega^{2}$ energy-dependent
attenuation.

##  3 Results

In order to use our model, we need to calibrate it against attenuation results
from microscopic experimental or theoretical results. Similarly to what we
have done in Ref.[68], we calibrate our model on literature data on the
prototype glass SiO2. Reported attenuations are for longitudinal modes only,
but it has been demonstrated that the same phenomenology is present in
transverse modes at higher values of $q$ and $\omega$, making the computations
more comfortable [26]. The experimental results of SiO2 are reported in Fig. 2
in which sound attenuation $\Gamma$ is measured at 300K in the subterahertz
range ($\nu<0.3$ THz with $\omega=2\pi\nu$) and at 1620K at terahertz
frequencies ( $\nu>1$ THz). The three regimes mentioned before can be clearly
found in the figures. At high frequency, a $\omega^{4}$-$\omega^{2}$ crossover
of $\Gamma$ appears around $\omega=1\times 10^{13}\ \rad\cdot\mathrm{s}^{-1}$
($\nu=$ 1.5 THz) as shown in Fig. 2. In addition, in the subterahertz range,
there should be another $\omega^{2}$-$\omega^{4}$ crossover which is not
explicitly given by the experimental data but shown as the intersection of the
auxiliary lines ($\omega^{2}$ and $\omega^{4}$). By adjusting the parameters
of the model ($\tau_{1}$, $\tau_{2}$ and $\alpha$), we give our best fit in
Fig. 2 for high (red line) and low (green line) temperature. The fitting
parameters for high temperature are reported in Tab. 2. The results of the fit
at high temperature is excellent. As for the low temperature data, it would be
extremely interesting to compare with data in the 0.1-10 THz range.
Unfortunately, no data in this range at low temperature are available in
literature.

Table 2: Parameters of the new viscoelastic continuum model. Calibration on silica glass at high temperature [27, 22]. $\tau_{1}(\mathrm{s}^{-3})$ | $\tau_{2}(\mathrm{s}^{-1})$ | $\alpha$
---|---|---
$3\times 10^{39}$ | $2.7\times 10^{13}$ | 1.4
Figure 2: Comparison between $\Gamma/\omega$ and $Q^{-1}$ with
$\tau_{1}=3\times 10^{39}\mathrm{s}^{-3}$, $\tau_{2}=2.7\times
10^{13}\mathrm{s}^{-1}$ and $\alpha=1.4$ (red line) or $\alpha=0.05$ (green
line) . The red (green) line is the $Q^{-1}$ of the present viscoelastic
model. Experimental data are from: IXS [22], POT [27], BUVS [32, 31],BLS [69]
and TJ [34]. The blue dashed lines are two power-law fits representing
$\Gamma/\omega\propto\omega$ and $\Gamma/\omega\propto\omega^{3}$,
respectively.

It is worth going more in the detail of the fitting procedure to highlight the
role of each parameter. A good fit should give a $Q^{-1}$ reproducing not only
the amplitude of the acoustic attenuation, but also at least the crossover
from $\omega^{4}$ to $\omega^{2}$ around $1\times
10^{13}~{}\rad\times\mathrm{s}^{-1}$ ($\nu=\omega/2\pi=$ 1.5 THz). It is not
easy to reach these two conditions through a simple optimization algorithm,
because most of the time it only meets the amplitude condition but the
crossover condition is ignored. To meet the two conditions simultaneously, we
start with a know-how to illustrate the evolution of the amplitude and the
crossover frequencies versus the three parameters $\tau_{1}$, $\tau_{2}$ and
$\alpha$. This approach is necessary as the first step of the parameters
identification before finding an alternative optimization algorithm in future
work.

The roles of the three parameters are displayed one by one in Fig. 3. By
default, we set
$\tau_{1}=10^{30}~{}\mathrm{s}^{-3},\tau_{2}=10^{10}~{}\mathrm{s}^{-1}$ and
$\alpha=1$. The chosen order of magnitude for different parameters can be
estimated based on the crossover positions.

Figure 3: Evolution of $Q^{-1}$ as a function of $\tau_{2}$, $\tau_{1}$ and
$\alpha$, respectively. The red arrow indicates the growth direction of the
parameter.

Let’s first consider $\tau_{2}$, as shown in the left panel of Fig. 3. We can
find that the global amplitude of the $Q^{-1}$ decreases with $\tau_{2}$.
Therefore, in the first step, we change $\tau_{2}$ in order to get the same
order of magnitude for $Q^{-1}$ obtained in the numerical model, and the
experimental values of $\Gamma/\omega$. During this process, one peak appears
at $\omega=1\times 10^{10}~{}\rad\cdot\mathrm{s}^{-1}$ whose position and
amplitude are determined by the two other parameters. This peak corresponds to
a transition between two different dominant frequency powers in the
constitutive model and depends on the relation among three parameters of the
model. In the current configuration, that is for $\tau_{1}=1\times
10^{30}~{}\mathrm{s}^{-3}$ and $\alpha=1$, it shows a $\omega^{3}-\omega^{-3}$
crossover.

Let’s now consider the variations of $\tau_{1}$, after having optimized
$\tau_{2}$. The influence of $\tau_{1}$ is shown in the middle panel of Fig.
3. Increasing $\tau_{1}$ pushes to the right the above
$\omega^{3}$-$\omega^{-3}$ peak. When this peak moves to the right, it becomes
smaller and the $\omega^{-3}$ part disappears first due to the collapse with
the increasing $\omega$ background. In this step, only the peak moves and the
amplitude of left or right wing is unchanged.

Finally, let’s consider the role played by the ratio $\alpha=\mu_{1}/\mu_{2}$,
which characterizes the rigidity ratio between the two parallel processes of
dissipation contributing additionally to the stress state. As shown in the
right panel of Fig. 3, changing $\alpha$ will raise or lower the left wing and
modify the power law near the crossover between left and right wing. When
$\alpha\ll 1$, the left wing will drop significantly. However, when $\alpha\gg
1$, the crossover $\omega^{3}$-$\omega^{-3}$ will progressively disappear and
the whole line, especially the part $\propto\omega^{3}$, tends to a single
$\omega$ power law. Between them, the $\omega^{3}$ tends to $\omega$.
Therefore, an appropriate $\alpha$ should raise the left wing to fit the
experiments data, while keeping the $\omega^{3}$ apparent power law as much as
possible.

In addition, the relaxation times $\tau_{1}\omega^{-4}$ and
$\tau_{2}\omega^{-2}$ should make sense physically, which requires them being
longer than $1\times 10^{-14}-1\times 10^{-15}~{}\mathrm{s}$. Using the orders
of magnitude of $\tau_{1}$ and $\tau_{2}$, we estimate the corresponding
relaxation time in Tab.3. Relaxation times shorten as the frequency increases,
indicating as upper frequency limit for this model $3\times 10^{13}$ Hz, that
is satisfactory.

Table 3: Estimation of the relaxation time for $\tau_{1}$ = $3\times 10^{39}$ $\mathrm{s}^{-3}$ and $\tau_{2}$ = $2.7\times 10^{13}$ $\mathrm{s}^{-1}$. $\omega$ ($\rad\cdot\mathrm{s}^{-1}$) |  $\tau_{1}\omega^{-4}$ (s) |  $\tau_{2}\omega^{-2}$ (s)
---|---|---
$1\times 10^{11}$ | $3\times 10^{-5}$ | $2.7\times 10^{-9}$
$1\times 10^{12}$ | $3\times 10^{-9}$ | $2.7\times 10^{-11}$
$1\times 10^{13}$ | $3\times 10^{-13}$ | $2.7\times 10^{-13}$
$3\times 10^{13}$ | $3.7\times 10^{-15}$ | $3\times 10^{-14}$

In SM Appendix, we discuss in more details the conditions giving rise to a
$\omega-\omega^{3}-\omega$ behavior. If these conditions are strictly met,
then the crossover frequencies can be expressed in terms of $\alpha$,
$\tau_{1}$ and $\tau_{2}$. We recall the two crossover frequencies evidenced
in SM Appendix:

  * 1.

Position of the crossover $\omega$ \- $\omega^{3}$:

$\omega_{1-3}=\frac{1}{\alpha+1}\tau_{2}$

  * 2.

Position of the crossover $\omega^{3}$ \- $\omega$:

$\omega_{3-1}=(\frac{1}{\alpha})^{1/2}(\frac{1}{\alpha+1})^{1/2}\tau_{2}$

Note that in the present model, the ratio $\omega_{3-1}/\omega_{1-3}$ depends
only on $\alpha$. It is given by :

$\frac{\omega_{3-1}}{\omega_{1-3}}=\sqrt{\frac{\alpha+1}{\alpha}}$ (8)

where $\alpha>0.76$ is a necessary condition to have a perfect
$\omega-\omega^{3}-\omega$ behavior without exhibiting other power laws (this
condition disappears if other crossovers are allowed). This gives an upper
limit of the ratio $\omega_{3-1}/\omega_{1-3}$ less than 1.52. In practice,
these conditions may not be strictly met but were sufficient in our case.
Moreover, the results about the crossover frequencies are still very useful
because they determine the magnitude of the parameters. For example, knowing
that $\omega_{3-1}\approx 2\pi\times 1.5$ THz $\approx 9.4\times
10^{12}~{}\rad\cdot\mathrm{s}^{-1}$ and $\alpha$ is normally supposed as 1 at
first, then we have $\tau_{2}=\omega_{3-1}/\sqrt{0.5}\approx 1.3\times
10^{13}~{}\mathrm{s}^{-1}$ and $\tau_{1}=\frac{\tau_{2}^{3}}{4}\approx 5\times
10^{38}~{}\mathrm{s}^{-3}$.

##  4 Discussion and Conclusion

Provided the parameters $\tau_{1}$, $\tau_{2}$ and $\alpha$, the interesting
case with three principal frequency dependencies
$\omega$-$\omega^{3}$-$\omega$ is only a special case offered by the visco-
elastic model. For this special case, we can write down the asymptotic
approximation for the three frequency dependencies : (1) Low frequency:
$Q^{-1}_{(1)}\sim\frac{\alpha}{1+\alpha}\frac{1}{\tau_{2}}\omega$; (2)
Intermediate frequency:
$Q^{-1}_{(2)}\sim\frac{1}{1+\alpha}\frac{1}{\tau_{1}}\omega^{3}$; (3) High
frequency: $Q^{-1}_{(3)}\sim\frac{1}{\tau_{2}}\omega$. It is noticed that the
process with viscosity $\eta_{2}\omega^{-2}$ dominates the low and high
frequency regimes, while the $\eta_{1}\omega^{-4}$ process activates mainly in
the intermediate frequencies corresponding to the Rayleigh-like damping below
the Ioffe-Regel frequency. Especially, it can be seen that, when $\alpha\gg
1$, $Q^{-1}_{(1)}$ tends to $\frac{1}{\tau_{2}}\omega$, thus $Q^{-1}_{(3)}$,
and the intermediate regime disappears $Q^{-1}_{(2)}\sim 0$ as illustrated in
Fig.3. In contrast, when $\alpha\ll 1$, low-frequency $\omega$ process gives a
smaller value of the quality factor compared to the high-frequency ones. This
$\alpha$-dependent $\omega$ process is impressively adapted to describe the
temperature-dependent acoustic attenuation $\Gamma/\omega\propto\omega$ at low
frequencies (see Fig.2). This low frequency behavior is known to be due to
anharmonicity, thermal activation, often described with the help of two level
systems and is also shown in the MD simulations [26]. As such, small $\alpha$
corresponds to low temperature and large $\alpha$ corresponds to a relatively
high temperature, meaning that the effective role of the temperature would be
to soften the $\eta_{2}\omega^{-2}$ process by decreasing its related rigidity
$\mu_{2}$. Moreover, our model predicts an extension of the $\omega^{3}$
regime at high frequencies in the low temperature case as can be seen in Fig.2
and possibly additional frequency dependencies at higher frequencies difficult
to reach experimentally.

As shown in the Tab.3, the frequency-dependent relaxation times given by the
fitted values of three parameters are reasonable up to $\omega=$ $30\times
10^{12}~{}\rad\cdot\mathrm{s}^{-1}$ which covers all spectral range of
experimental data. Therefore, we can conclude that the values of this model
are physically meaningful.

In conclusion, the model presented here which involves two parallel sources of
dissipation with a frequency dependent viscosity is shown able to reproduce
the successive attenuation regimes $\omega^{2}$ \- $\omega^{4}$ \-
$\omega^{2}$ observed in glasses. In practice, we investigated a calibration
of the model on a-SiO2 which gives a quality factor $Q^{-1}$ that fits very
well the experimental attenuation data $\Gamma/\omega$ [22, 23, 21, 30, 31,
32, 33, 34, 39]. Interestingly, the frequency dependence of the viscosities
involves two mechanisms: one is related to an attenuation time
$\tau_{1}\omega^{-4}$ (possibly related to low acoustic scattering at the
atomic scale as discussed before [18]), while another involves a relaxation
time $\tau_{2}\omega^{-2}$ (possibly related to anharmonicity or to strong
acoustic scattering at small scale, as already discussed in the introduction).
The permanent combination of these two well-known atomistic processes in
parallel allows to reproduce the two experimentally observed crossovers in the
frequency dependence of the attenuation. The comparison of our model with
experimental data on a large frequency range leads to the determination of the
3 parameters involved in the model, that is : $\tau_{1}=3\times
10^{39}~{}\mathrm{s}^{-3}$, $\tau_{2}=2.7\times 10^{13}~{}\mathrm{s}^{-1}$ and
$\alpha=1.4$ at ambient temperature (300K) but was also able to reproduce the
temperature dependence in the low frequency part, by tuning only the parameter
$\alpha$ ($\alpha$ is lowered to $0.05$ at 1 K). These numerical values are
reasonable for $\omega<3\times 10^{13}~{}\rad\cdot\mathrm{s}^{-1}$ that is the
largest frequency that can be reached at the atomic scale. Thus, our results
on the one hand give a description of the effective frequency-dependent
acoustic attenuation in SiO2 glasses ranging from GHz to THz range, and on the
other hand consolidate the possibility of homogenizing the multi-channel
intrinsic acoustic attenuation from nanometric to macroscopic scales with
appropriate viscoelastic models. As such, this model will allow for improved
and more realistic large scale simulations of metamaterials with an amorphous
component. As an example, in a recent work, we have used continuum finite
element modeling to investigate the acoustic attenuation in nanocomposites,
which has allowed us to study the effect of the interface scattering on the
real size materials, underlining the existence of propagative, diffusive and
localized regime, depending on the nanostructuration lengthscale and elastic
mismatch between the components. Still, in that work no intrinsic acoustic
attenuation was considered, the component materials being non-dissipative
[68]. The introduction of the dissipation in the amorphous matrix would allow
to evidence the competition between extrinsic and intrinsic scattering
sources, and to predict the efficiency of the nanostructuration in attenuating
phonons in a medium where attenuation is already important. Of course, we
expect to be confronted to different regimes, depending on phonon frequency: a
key role of the extrinsic source for low intrinsic scattering, and a less
important effect in the strong scattering regime.

Finally, our model paves the way to further studies, investigating other power
laws for phonon attenuation, introducing anisotropy and studying the effect of
temperature, allowing to describe thermal transport as well as sound
propagation in structured materials of arbitrary complexity.

## Acknowledgement

This work benefited form the early participation of Robin Rouzaud. Interesting
discussions with Giulio Monaco are acknowledged. H.L. is financed by the
french ministry of research. A.T., H.L. and A.G. thank ECOS chile - grant No.
C17E02 for their support.

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