Sound attenuation in stable glasses

Lijin Wang,1, 2 Ludovic Berthier,3 Elijah Flenner,2 Pengfei Guan,1 and Grzegorz Szamel2

1Beijing Computational Science Research Center, Beijing 100193, P. R. China2Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, USA

3Laboratoire Charles Coulomb (L2C), University of Montpellier, CNRS, 34095 Montpellier, France(Dated: today)

Understanding the difference between universal low-temperature properties of amorphous andcrystalline solids requires an explanation of the stronger damping of long-wavelength phonons inamorphous solids. A longstanding sound attenuation scenario, resulting from a combination ofexperiments, theories, and simulations, leads to a quartic scaling of sound attenuation with thewavevector, which is commonly attributed to Rayleigh scattering of the sound. Modern computersimulations offer conflicting conclusions regarding the validity of this picture. We simulate glasseswith an unprecedentedly broad range of stabilities to perform the first microscopic analysis of sounddamping in model glass formers across a range of experimentally relevant preparation protocols.We present a convincing evidence that quartic scaling is recovered for small wavevectors irrespectiveof the glass’s stability. With increasing stability, the wavevector where the quartic scaling beginsincreases by approximately a factor of three and the sound attenuation decreases by over an order ofmagnitude. Our results uncover an intimate connection between glass stability and sound damping.

I. INTRODUCTION

Many theoretical descriptions of sound attenuationin low temperature (athermal) amorphous solids pre-dict a quartic scaling of the sound attenuation withthe wavevector. Early arguments, used to explain theplateau in the temperature dependence of the thermalconductivity1,2, invoked the picture of scattering of soundwaves by uncorrelated inhomogeneities that are muchsmaller than the wavelength, which is the physical sce-nario known as the Rayleigh scattering. In several the-ories, these inhomogeneities have been modeled as localfluctuations of elastic constants3–8. These theories pre-dict that the sound attenuation scales with the fourthpower of the wavevector, Γλ(k) ∼ k4 (λ = L denotes lon-gitudinal waves and λ = T denotes transverse waves) forsmall wavevector k. Mean-field theories9–12 arrive at thesame prediction, albeit in a different way. Yet anothertheoretical treatment, the soft-potential model, predictsthat a quartic scaling regime exists due to phonons in-teracting with soft modes13.

Longitudinal sound attenuation can be directly ob-tained from X-ray and light scattering experiments. Acompilation of many experimental results14–28 shows thatthe wavevector dependence of the longitudinal sound at-tenuation parameter, ΓL(k), can be divided into threeregimes: (1) ΓL(k) ∼ k2 for low k; (2) ΓL(k) ∼ k4 foran intermediate k regime; and (3) ΓL(k) ∼ k2 for largek. While the intermediate wavevector quartic and thelarge wavevector quadratic scalings of the sound attenu-ation parameter are well-documented, the small wavevec-tor quadratic dependence was only seen in a few exper-iments17–20. Because the experiments are performed atfinite temperature, the small wavevector quadratic scal-ing can be ascribed to thermal and anharmonic effects.

Computer simulations offer a conflicting view of theseresults. Most computer studies investigate sound atten-uation in the limit of zero temperature, in order to re-

move anharmonic effects. To our knowledge, no simu-lation reproduced the ΓL(k) ∼ k2 scaling observed atsmall wavevectors in experiments17–20. Regarding thequartic Rayleigh scattering regime, no firm conclusioncan be drawn either. By simulating large glasses createdby quenching configurations from a mildly supercooledliquid, Gelin et al.29 found a logarithmic correction tothe quartic scaling, Γλ(k) ∼ k4 ln(k). They invoked theexistence of correlated inhomogeneities of the elastic con-stants30 to rationalize this observation. However, a morerecent, larger-scale study31 of harmonic spheres close totheir unjamming transition confirmed the Rayleigh scat-tering scenario in 2D glasses and conjectured its validityin 3D glasses. Finally, a very recent preprint32 (whichappeared when the present paper was being finalized forsubmission) presented the first convincing evidence of thesmall wavevector quartic scaling of the transverse soundattenuation in a 3D glass created by quenching from amildly supercooled liquid. However, the status of thelongitudinal sound attenuation, even for simple glass-formers in the zero-temperature harmonic limit, remainsunsettled.

To our knowledge, all prior simulations investigatedsound attenuation in glasses with stabilities dramaticallydifferent from the ones of typical laboratory glasses, pre-venting direct comparison between numerics and real ma-terials. This constraint is imposed by the large prepara-tion times required to equilibrate systems close to the ex-perimental glass transition, which, therefore, cannot besimulated using conventional techniques. In this work,we use an efficient swap Monte-Carlo algorithm33 thatwas recently developed34,35 to prepare glasses with sta-bilities comparable to, or even exceeding, the stability ofexperimental glasses. If we quantify the glass stabilityin terms of a cooling rate, the improvement due to theswap algorithm is equivalent to decreasing the coolingrate by more than 10 orders of magnitude, thus closingthe gap between previous computer investigations and

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realistic materials. In previous studies, it was demon-strated that both the low-frequency vibrational proper-ties36 and the mechanical properties37 of computer gener-ated glasses dramatically evolve with increasing the glassstability over such a broad range.

We find that changing the glass stability over a broadrange fully clarifies the elusive picture of sound attenu-ation. Generally, sound attenuation decreases with in-creasing stability, implying that more stable glasses arealso less dissipative solids (classical zero temperaturecrystalline solids are non-dissipative). More importantly,we find the wavevector dependence of sound attenua-tion at low wavevectors exhibits a quartic scaling, forboth transverse and longitudinal modes and in glasseswith very different stabilities. Thus, we unambiguouslydemonstrate the universality of the Rayleigh scatteringscaling in 3D glasses. The quartic scaling of the sound at-tenuation with the wavevector is more prominent in morestable glasses, which adds to the conjectured connectionbetween glass stability and sound damping.

II. METHODS

A. Simulation details

We perform computer simulations using a three-dimensional cubic system composed of polydisperse par-ticles with equal mass m = 1. The distribution of parti-cle diameters σ follows P (σ) = A

σ3 , where σ ∈ [0.73, 1.63]and A is a normalization factor. The cross-diameter σijis determined according to a non-additive mixing rule,σij =

σi+σj

2 (1 − ε|σi − σj |) with ε = 0.2. The interac-tion between two particles i and j is given by the in-

verse power law potential, V (rij) =(σij

rij

)12+ Vcut(rij),

when the separation rij is smaller than the potential cut-off rcij = 1.25σij , and zero otherwise. Here, Vcut(rij) =

c0 + c2

(rijσij

)2+ c4

(rijσij

)4, and the coefficients c0, c2 and

c4 are set to guarantee the continuity of V (rij) at rcij upto the second derivative.

We produce zero-temperature glasses by instan-taneously quenching supercooled liquids equilibratedthrough the swap Monte Carlo algorithm at differ-ent parent temperatures Tp, which uniquely controltheir stability36, to their local potential minima us-ing the fast inertia relaxation engine minimization38.We calculate the normal modes by diagonalizing thedynamic matrix using Intel Math Kernel Library(https://software.intel.com/en-us/mkl/) and ARPACK(http://www.caam.rice.edu/software/ARPACK/). Westudy glasses with Tp ranging from well above the onset ofsupercooling, denoted as Tp = ∞, down to Tp = 0.062,which is about 60% of the mode-coupling temperatureTc ≈ 0.108 35. The onset of slow dynamics in an equi-librated fluid occurs around To = 0.2. The parent tem-perature Tp = 0.062 is lower than the estimated exper-

CT(t)

−1.0

−0.5

0

0.5

1.0

t0 20 40 60 80 100

FIG. 1: Decay of CT (t) for a transverse excitation with awavevector k = (0, 4π/L, 0) for our most stable glass, Tp =0.062. The red curve is a fit to CT (t) = exp(−Γλt/2) cos(Ωλt).The velocity field for the whole system is shown in the upperleft corner and for a section at representative times corre-sponding to the peaks in CT (t) indicated by the arrows.

imental glass temperature Tg ≈ 0.072 for this model35,and thus the glass with Tp < 0.072 qualifies as ultra-stable. The particle number N varies between 48000 and1000000 for glasses at Tp = ∞, and between 48000 and192000 for glasses with 0.062 ≤ Tp ≤ 0.120. For allglasses studied the number density ρ = 1.0.

B. Sound attenuation

We use two different methods to obtain sound attenu-ation:1) we calculate the T = 0 dynamic structure factorutilizing the eigenvalues and eigenvectors of the dynamicmatrix39; 2) we study the decay of an excited sound wavein the harmonic approximation29.

We calculate the T = 0 dynamic structure factors usingthe eigenvalues and eigenvectors of the dynamic matrix39,

Sλ(k, ω) =

(k2

Nω2

) 3N−3∑n=1

Fn,λ(k)δ(ω − ωn), (1)

where λ is T for transverse or L for longitudinal structurefactor, ωn is the frequency (square root of the eigenvalue)associated with the n-th eigenvector. The sum is takenover all but the three modes corresponding to a universaltranslation. In eqn 1

Fn,T (k) =

∣∣∣∣∣∣N∑j=1

(en,j × k)eik·r0j

∣∣∣∣∣∣2

, (2)

and

Fn,L(k) =

∣∣∣∣∣∣N∑j=1

(en,j · k)eik·r0j

∣∣∣∣∣∣2

, (3)

3

Tp = ∞Tp = 0.200Tp = 0.100Tp = 0.085Tp = 0.075Tp = 0.062k4

k4

k2

a

Γ T

10−4

10−3

10−2

10−1

1

k0.1 1

k4

k4k2

b

Γ L

10−4

10−3

10−2

10−1

1

k0.1 1

Tp = ∞Tp = 0.200Tp = 0.100Tp = 0.085Tp = 0.075Tp = 0.062

c

Γ T/k

4

0

10

20

30

k0.05 0.1 0.2 0.5 1

d

Γ L/k

4

0

50

100

150

200

k0.05 0.1 0.2 0.5 1

FIG. 2: Wavevector k dependence of sound attenuation (a) ΓT (k) and (b) ΓL(k) in from poorly annealed glasses (Tp =∞) to stable glasses (Tp = 0.062). The different symbols denote different system sizes: star=1000K, plus=600K, x=450K,triangle=192K, square=96K, circle=48K. The k2 dependence is evident at large wavevectors and the crossover to k4 scalingcan be seen for Tp =∞ and Tp = 0.062. The reduced sound attenuation (c) ΓT /k

4 and (d) ΓL/k4. A straight line with negative

slope would indicate a logarithmic correction, which is valid only for a small range of wavevectors.

where en,j is the polarization vector of particle j in then-th eigenvector, r0j is the position of particle j in the in-herent structure, k is the wavevector satisfying periodic

boundary conditions, k ≡ |k| and k = k/|k|. We ex-tract the damping coefficients Γλ and the characteristicfrequencies Ωλ by fitting Sλ(k, ω) to a damped harmonicoscillator model40,

Sλ(k, ω) ∝ Ω2λ(k)Γλ(k)

[ω2 − Ω2λ(k)]2 + ω2Γ2

λ(k). (4)

Another method to determine Γλ and Ωλ is to studythe decay of excited sound waves in the harmonic ap-proximation, and most of our results shown in this workare from this method (unless specified). Specifically, fol-lowing Ref.29, we excite a sound wave at t = 0 by giving

each particle a velocity u0i = aλ sin(k · r0i ), where aL ∝ k

and aT · k = 0. We then numerically solve the equationsof motion,

ui(t) = −N∑j=1

Dij · uj(t) + u0i δ(t). (5)

Here, Dij is dynamic matrix and ui(t) denotes the dis-placement of particle i at t from its inherent structureposition. We calculate the velocity correlation function,

Cλ(t) =

∑Ni=1 ui(0) · ui(t)∑Ni=1 ui(0) · ui(0)

, (6)

and fit it to

Cλ(t) = exp (−Γλ(k)t/2) cos(Ωλ(k)t), (7)

to determine the frequency Ωλ and the sound attenuationΓλ. Since the calculation obtains Ωλ through a fit for afixed k, the wavevector is precisely known but there isuncertainty in Ωλ.

Shown in Fig. 1 is an example of the excited soundwave method29. The snapshots in Fig. 1 show the veloc-ity field for k = (0, 4π/L, 0) in a 48000 particle systemfor times at the peak values of CT (t) indicated in thefigure. As expected, the sound wave is scattered and theinitial velocity profile decays.

The two methods introduced above encode the samedynamical information, but there exists a finite size effectthat is impossible to correct for using the normal modeanalysis. See Appendix section for details on how weaccount for this finite size effect and for details on howwe obtain Γλ.

III. SOUND ATTENUATION IN STABLEGLASSES

Shown in Fig. 2 are Γλ(k) for a range of stabilities for(a) transverse sound waves and (b) longitudinal soundwaves. For large wavevectors we observe quadratic scal-ing, which is consistent with previous results. There is

4

no difference in the attenuation for Tp = 0.2 and Tp =∞suggesting that zero-temperature glasses quenched fromparent temperatures above the onset temperature To =0.2 have identical attenuation. There is a crossover toquartic scaling, Rayleigh scaling, for our least stableglasses Tp = ∞ and our most stable glasses Tp = 0.062.Therefore, Γλ(k) = Bλk

4 for small wavevectors irrespec-tive of the glass’s stability.

To examine the stability dependence of Bλ and thepossibility of a logarithmic correction, in Fig. 2 weplot Γλ(k)/k4 for Tp = ∞, 0.1, 0.085, 0.075, and 0.062for transverse sound (c) and the longitudinal sound (d).There is a factor of 15 decrease in Bλ from our least sta-ble glass to the most stable glass. We note that in therepresentation of Fig. 2 a straight line with a negativeslope would indicate the −k4 ln(k) scaling suggested byGelin et al. 29. We can identify a range of wavevectorsthat is described by ΓT (k) ∼ −k4 ln(k) for our least sta-ble glasses, but this does not provide a good descriptionfor a wide range of wavevectors. Instead we observe adistinct plateau at low wavevectors, indicating a purelyquartic scaling without a logarithmic correction.

As noted by Monaco and Mossa40 when studyingglasses created by quenching from mildly supercooledliquids, the transverse and longitudinal sound attenu-ation differ by a constant factor when examined as afunction of frequency ω = vλk, where vT =

√G/ρ,

vL =√

(K + 4G/3)/ρ, G is the shear modulus, and K isthe bulk modulus, Fig. 3. We find ΓL(ω) = ΓT (ω)/n irre-spective of the glass’s stability, but the scaling factor n isstability dependent with n ≈ 5 for our poorly annealedglass, Tp = ∞, and n ≈ 3 for our most stable glass,Tp = 0.062, indicating a decreasing difference betweenΓT (ω) and ΓL(ω) with increasing stability. This scalingsuggests that the sound attenuation is governed by a sta-bility dependent frequency (time) scale and possibly nota characteristic length scale. However, a changing lengthscale cannot be ruled out.

With increasing stability, the glass becomes less dissi-pative and quartic scalings of ΓT and ΓL start at largerwavevectors. The wavevector at which the quartic scalingbegins depends on the polarization, transverse or longi-tudinal, of the sound wave. In contrast, if we plot thesound attenunation as a function of frequency, the fre-quency where the quartic scaling begins does not dependon the transverse or longitudinal sound wave. Again,this crossover frequency increases with increasing stabil-ity. The glass is becoming more uniform, resulting in adecrease in the dissipation 29,30 with an increase in thestability.

For small and intermediate wavevectors thewavevector-dependent speed of sound vT (k) = ΩT /k isa well defined quantity. In particular, for every parenttemperature the k → 0 limit is given by

√G/ρ, which

is shown as horizontal lines in Fig. 4. However, withincreasing wavevector different methods lead to slightlybut systematically different results for the wavevector-dependent speed of sound. If we determine the speed

ω4ω4

blue = transversen

red = longitudinal

Tp = ∞

Tp = 0.062

Γ T/n

, ΓL

10−4

10−3

10−2

10−1

1

ω= vλk0.1 1

FIG. 3: Frequency ω = vλk dependence of sound attenuationfor our least stable glass Tp =∞ (filled symbols) and our moststable glass Tp = 0.062 (open symbols). The different sym-bols denote different system sizes: star=1000K, plus=600K,x=450K, triangle=192K, square=96K, circle=48K. The redsymbols are the results for the longitudinal attenuation andthe blue symbols are results for the transverse attenuation.The transverse attenuation is scaled by a Tp dependent factorn, where n = 5 for Tp =∞ and n = 3 for Tp = 0.062.

of sound from the fit to the frequency-dependentdynamic structure factor (filled circles), the resultingquantity exhibits a minimum, which has been reportedin previous simulations8,29,31,40 and experiments23,24,42.This minimum is replaced by a plateau for our sta-ble glasses. However, if we rely upon the fit to thetime-dependent function Cλ(t) (open symbols), thewavevector-dependent speed of sound exhibits a morepronounced minimum, which is also present for thestable glasses. The difference between the two methodsis small but systematic.

It would be expected that the two methods could dis-agree when the excitation is no longer well describedas a propagating sound wave, which is generally asso-ciated to when the mean free path is equal to half thewavelength, i.e. the Ioffe-Regel limit. Shown in the in-set to Fig. 4 is the Ioffe-Regel limit obtained from whenΩT (kIR) = πΓT (kIR) as a function of the parent temper-ature. For this calculation we used ΩT determined fromthe fits to the dynamic structure factor. The result is notsensitive to which method is used to determine ΩT . ForTp = 0.2, kIR ≈ 0.5 and for Tp = 0.062, kIR ≈ 0.87. Bothof these quantities lie slightly above where the two meth-ods to obtain the wavevector-dependent speed of soundbegin to diverge. Thus, the classification of these exci-tations as propagating sound waves is breaking down forwavevectors slightly smaller than kIR.

Nevertheless, we find that increasing the stability ofthe glass allows for propagating sound waves at smallerwave lengths, and this can be quantified by the change inkIR. For decreasing Tp, kIR increases by a factor of 1.8over our range of stability. For wavevectors above kIRit is expected that the vibrations are more localized andthere is a change in the energy transport from a prop-agating regime below kIR to a diffusive regime above

5

k IR

0.5

0.6

0.7

0.8

0.9

Tp

0.05 0.10 0.15 0.20

Tp = 0.062

Tp = 0.085

Tp = 0.200

v T(k

)

1.8

1.9

2.0

2.1

2.2

2.3

k0 0.2 0.4 0.6 0.8 1.0

FIG. 4: The wavevector dependence of sound speed for dif-ferent parent temperatures Tp. The horizontal lines indicatethe corresponding macroscopic values in the long-wavelengthlimit. The open symbols are obtained through fits of CT (t)and the closed symbols are obtained through fits to ST (k;ω).(Inset) Ioffe-Regel wavevector kIR as a function of Tp.

kIR43–45. Therefore, the decreased dissipation and in-

crease in kIR should have significant effects on the ther-mal conductivity and the stability dependence of thermalenergy transport.

IV. CONNECTION BETWEEN SOUNDATTENUATION, VIBRATIONAL MODES, AND

THE BOSON PEAK

A recurring idea is that sound attenuation and the ex-cess in vibrational modes over the Debye theory are in-timately connected. Recall that in the Debye theory thedensity of states increases with a decrease in the speedof sound. Using this idea, the minimum in vT (ω) hasbeen associated with an increase of the density of statesD(ω) and the boson peak using a generalized plane waveapproach40. However, we find that the description ofthe vibrational modes as well defined sound waves breaksdown for wavevectors below the boson peak whose posi-tion approximates kIR

46.A generalized Debye model of Mizuno and Ikeda31 and

the theoretical treatment of Schirmacher et al.4 both re-late the excess number of low frequency modes abovethe Debye model, Dex(ω), to sound attenuation. Bothof these treatments predict that Dex ≈ 4BT /(πk

2Dv

6T )ω4,

where kD = (6πρ)1/3, for small wavevectors. In previousstudies36,47 it was found that the low frequency modescould be divided into extended and localized modes.The density of the low frequency extended modes obeysDebye theory and the density of the localized modesDloc = A4ω

4. Therefore, these localized modes are themodes in excess of the Debye theory, and we can associatethem with Dex(ω). We find that A4 is 20% larger than4BT /(πk

2Dv

6T ) for our poorly annealed glass and 150%

larger for our most stable glass. Therefore, these modelsare currently not quantitatively predictive and get worsewith increasing stability.

A4

0

0.01

0.02

0.03

0.04

0.05

0.06

ΓT/k40 5 10 15 20 25

FIG. 5: The coefficient A4 describing the the density oflow frequency quasi-localized modes, Dloc = A4ω

4 correlatesvery well with the plateau height of ΓT /k

4 for small wavevec-tors. They are both strongly suppressed when glass stabilityincreases.

Our findings for the transverse sound attenuation inmoderate and low stability glasses are in general agree-ment with the very recent results of Moriel et al.32.Specifically, both our study and that of Moriel et al.find quartic small wavevector scaling of the transversesound attenuation in 3D glasses. Moriel et al. also investi-gated the dependence of the sound attenuation of glasseswith different densities of low-frequency quasi-localizedmodes. They found that the decreasing density of thesemodes correlates with the decreasing extent of the in-termediate regime between the small wavevector quarticscaling and the large wavevector quadratic scaling, whichcan be fitted to the −k4 ln k form proposed by Gelin etal.29.

In contrast, we investigated a number of glasses withvery different stabilities. We find that the sound attenua-tion and the Rayleigh scattering plateau Γλ/k

4 decreaserapidly. In Fig. 5 we show that A4 and BT = ΓT /k

4

are proportional to each other, with BT ∼ A4. Thissignificantly extends the qualitative correlation found byMoriel et al.32

A recent experiment by Pogna et al.41 reported ona connection between sound attenuation and the bosonpeak. They find a decrease in the boson peak height andsound attenuation for hyperaged amber (conjectured tobe much more stable) compared to annealed amber (withordinary stability), which mirrors our results36. Pogna etal. used the fluctuating elasticity theory of Schirmacher etal.5 (which predicts the quartic scaling of sound attenua-tion with the wavevector) to fit the vibrational density ofstates. There are two main parameters in the theory, onequantifies the strength of the disorder and is related tothe width of the local elastic constant distribution, andanother quantifying the spatial range of correlations ofelasticity. They concluded that upon lowering the fictivetemperature by 9% that there was a six percent decreaseof the strength of the fluctuations and a 22% increaseof the elastic correlation length. Therefore, they conjec-tured that the change of the low-frequency vibrational

6

properties is mainly driven by an increased elastic corre-lation length. Future work should examine the change ofthe disorder strength and the elastic correlation lengthwith stability more directly to verify this conclusion.

A competing theoretical explanation for the relation-ship between sound attenuation and the boson peakis that the sound modes interact with additional softmodes13, the soft potential model. Examination andevaluation of the soft potential model requires the de-termination of several parameters, and this exercise isleft for future work.

V. DISCUSSION

The idea that a Rayleigh scattering mechanism may beresponsible for the small wavevector scaling of sound at-tenuation spans for over 60 years31,48. Mizuno and Ikedaconsidered scattering of an elastic wave. Their analy-sis determined that Γλ = δγ2λD

3λΩ4

λ/(4πv3λ), where δγ is

the strength of the elastic inhomogeneities and D is thecharacteristic size31. Since it has been suggested thatkBP = ωBP /vT is related to the length scale of elas-tic inhomogeneities3,42, and thus D, we checked to seeif this was consistent with the quartic scaling regimesfor ΓT . We used the thermodynamic approach studiedby Mizuno, Mossa, and Barrat49 to obtain the strengthδγT = δG/G, where δG is the fluctuations of the shearmodulus, of the elastic inhomogeneities. We find thatthis naive approach does not correctly predict the changein the sound attenuation for each parent temperature.Future work needs to examine the spatial correlations ofthe elastic modulus and their relationship to sound at-tenuation.

Recent experiments on amber aged for 110 millionyears suggest that the vibrational properties of amor-phous materials are controlled by the distribution of elas-tic constants and their spatial correlation41. Future nu-merical studies should examine this relationship for sim-ulated ordinary and stable glasses to clarify this relation-ship. The stability dependence of sound attenuation us-ing ultrastable glasses, experimentally available via themethod of physical vapor deposition50, has shown thatsound damping decreases with increasing stability51. Itwould be interesting to examine the wavevector depen-dence of sound damping at low temperatures where an-harmonicities may come into play52, for these ultrastableglasses.

Acknowledgements

We thank E. Lerner and E. Bouchbinder for correspon-dence on an earlier version of this work. L.W., E. F.,and G.S. acknowledge funding from NSF DMR-1608086.This work was also supported by a grant from the Si-mons foundation (No. 454933 L. B.). L. W., and P.G. acknowledge support from the National Natural Sci-

k48K2k96K3

C(t)

−1.0

−0.5

0

0.5

1.0

t0 50 100 150 200 250 300

FIG. 6: Velocity correlation function C(t) for wavevectors ofsimilar magnitude, k ≈ 0.24, for two different system sizes.The decay rate is clearly different and does not appear expo-nential.

ence Foundation of China (No. 51571011), the MOST973 Program (No. 2015CB856800), and the NSAF jointprogram (No. U1530401). We acknowledge the com-putational support from Beijing Computational ScienceResearch Center.

Appendix

Molecular dynamics simulations can be subject to ef-fects due to small size of the simulation cell compared toexperimental systems and the use of periodic boundaryconditions. Bouchbinder and Lerner recently commentedon finite size effects in the calculation of the frequencywidth of phonon bands53, which indicates that finite sizeeffects exist for the calculation of sound attenuation inamorphous solids. We find that there are strong finitesize effects for the lowest wavevector sound waves in oursimulations, especially for our most stable glasses. Herewe describe a method to calculate sound attenuation thatis independent of system size.

One route to calculate the attenuation of sound wavesis to study the decay of an excitation in the harmonic ap-proximation as described in the Methods section. Afterexciting a sound wave, we study the decay of the velocitycorrelation function C(t), eqn (6). For small wavevectorswe expect that C(t) = exp(−Γλt/2) cos(Ωλt).

To demonstrate that a finite size effect exists we canexamine C(t) for similar wavevectors in two systems ofdifferent sizes. The magnitude of the third smallest al-lowed wavevector for the 96K system k96K3 = 0.238 andthe magnitude of the second smallest allowed wavevectoris k48K2 = 0.245. The attenuation of these sound wavesshould be similar, but we find that they are very differ-ent, Fig. 6. Specifically, at long times the peak heightsof the 96K system are much larger than for the 48K sys-tem. However, C(t) nearly overlaps at short times for

7

192K, k=0.30896K, k=0.3079238, k=0.299

Enve

lope

0.1

0.2

0.5

1

t0 50 100 150 200 250 300

FIG. 7: The envelope of of C(t) for three system sizes forthree wavevectors of nearly equal magnitude. The solid linerepresents a fit of the envelope to exp(−Γλt/2).

ST(k,ω) FitsEnvelope Fits48K96K192K9238

Γ T(k

)

10−3

10−2

10−1

k0.05 0.1 0.2 0.5

FIG. 8: Sound attenuation ΓT (k) calculated using fits toST (k, ω) (red) and the envelope fits (blue). The different sym-bols correspond to different system sizes. The inset shows anexpanded view of the results for one wavevector. There isa clear finite size effect when ΓT (k) is obtained by fittingST (k, ω), which can be removed by using the restricted enve-lope fits.

both system sizes. To study the decay of C(t) we calcu-late the envelope of C(t), which is the absolute value ofthe maximum and minimum of the oscillations.

Shown in Fig. 7 on a linear-log scale is the envelopefor three different sizes for a wavevector of similar mag-nitude. We note that the initial decay of all three en-velopes is exponential, but there are deviations from theexponential decay at a system size dependent time. Todetermine Γλ we fit the envelope to exp(−Γλt/2) up toa time when the decay is no longer exponential. Ouruncertainty in Γλ reflects the uncertainty in this fittingrange.

Another method to obtain sound attenuation isthrough the dynamic structure factor Sλ(k, ω) using theeigenvalues and eigenvectors of the dynamic matrix, asdescribed in the Methods section, or Fourier transform-ing C(t). Sound attenuation Γλ is then obtained by fit-ting with the damped harmonic oscillator model, eqn (4).

Shown in Fig. 8 as red symbols are the results of fit-ting ST (k, ω) and as blue symbols are the results of therestricted envelope fits. The different symbols indicatedifferent system sizes. The inset shows an expanded viewof a region of very similar wavevectors for four differentsystem sizes. There is a clear finite size effect when ΓT isfound through fits of ST (k, ω), which is removed by usingthe envelope fits.

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