Acoustic excitations and elastic heterogeneities indisordered solidsHideyuki Mizunoa,b,1, Stefano Mossac,d,e,2, and Jean-Louis Barrata,b,f

aUniversité Grenoble Alpes, Laboratoire Interdisciplinaire de Physique, F-38000 Grenoble, France; bCentre National de la Recherche Scientifique, LaboratoireInterdisciplinaire de Physique, F-38000 Grenoble, France; cUniversité Grenoble Alpes, Institut Nanoscience et Cryogénie, Structures et Propriétés d’ArchitecturesMoléculaires, F-38000 Grenoble, France; dCommissariat à l’Énergie Atomique et aux Énergies Alternatives, Institut Nanoscience et Cryogénie, Structures etPropriétés d’Architectures Moléculaires, F-38000 Grenoble, France; eCentre National de la Recherche Scientifique, Institut Nanoscience et Cryogénie, F-38000Grenoble, France; and fInstitut Laue-Langevin, F-38042 Grenoble, France

Edited by Peter Harrowell, University of Sydney, Sydney, Australia, and accepted by the Editorial Board July 9, 2014 (received for review May 22, 2014)

In the recent years, much attention has been devoted to the in-homogeneous nature of the mechanical response at the nanoscalein disordered solids. Clearly, the elastic heterogeneities that havebeen characterized in this context are expected to strongly affect thenature of the sound waves which, in contrast to the case of perfectcrystals, cannot be completely rationalized in terms of phonons.Building on previous work on a toy model showing an amorphiza-tion transition, we investigate the relationship between soundwaves and elastic heterogeneities in a unified framework by con-tinuously interpolating from the perfect crystal, through increas-ingly defective phases, to fully developed glasses. We providestrong evidence of a direct correlation between sound wave fea-tures and the extent of the heterogeneous mechanical response atthe nanoscale.

elasticity | amorphous materials | molecular dynamics simulation |vibrational properties | sound transport

In crystals, molecules thermally oscillate around the periodiclattice sites and vibrational excitations are well understood in

terms of quantized plane waves, the phonons (1). The vibrationaldensity of states (vDOS) in the low-frequency regime is welldescribed by the Debye model, where the vibrational modes arethe acoustic phonons. In contrast, disordered solids, includingstructural glasses and disordered crystals, exhibit specific vibra-tional properties compared with the corresponding pure crys-talline phases. It is not possible here to give a fair review of theextensive theoretical and experimental work generated by theseissues; we therefore mention below a few facts that we considerthe most relevant in the present context. The origin of the vDOSmodes in excess over the Debye prediction around ω ∼1 THz,the so-called Boson peak (BP), is still debated (see, among manyothers, refs. 2 and 3). At the BP frequency, ΩBP, localizedmodes have also been observed (4). Acoustic plane waves, whichare exact normal modes in crystals, can still propagate in dis-ordered solids. Indeed, at low frequencies, Ω, and long wave-lengths, Λ, acoustic sound waves do not interact with disorderand can propagate conforming to the expected macroscopiclimit. However, as Ω is increased beyond the Ioffe–Regel (IR)limit, ΩIR, acoustic excitations interact with the disorder andare significantly scattered (5–7). Interestingly, this strong scat-tering regime occurs around the BP position, ΩIR ∼ ΩBP (8, 9).The exact origin of this phenomenon and its connection to the BPremain elusive.A possible rationalization of the above issues is based on the

existence of elastic heterogeneities (10), which can originatefrom structural disorder, as in structural glasses (2), or disor-dered interparticle potentials, even in lattice structures such asdisordered colloidal crystals (11). In the heterogeneous-elasticitytheory of refs. 7 and 12 this amounts to consider spatial statisticalfluctuations of the shear modulus. Within the framework ofjamming approaches and using effective medium theories, elasticheterogeneities are related to the proximity of local elasticinstabilities (13). Recent simulation work (14–16) has clearly

demonstrated their existence in disordered solids. This is atvariance with the case of simple crystals, which are character-ized by a fully affine response and homogeneous moduli dis-tributions (17). More specifically, in the large length scale limit,macroscopic moduli are observed. In contrast, as the lengthscale is reduced, moduli heterogeneities are detected, at a typ-ical length scale ξ ’ 10−15σ (15), where σ is the typical atomicdiameter. Breakdown of both continuum mechanics (18) andDebye approximation (5, 6) has been demonstrated at the samemesoscopic length-scale ξ, where they are still valid for crystals.Remarkably, the wave frequency corresponding to the wave-length Λ ∼ ξ is very close to ΩIR ∼ ΩBP (19). Altogether theseresults indicate that a close connection must exist betweenelastic heterogeneities and acoustic excitations. In this paperwe precisely address this point.In ref. 20 we considered a numerical model featuring an

amorphization transition (21). We showed how to systematicallydeform the local moduli distributions, evaluated by coarse-graining the system in small domains of linear length scale w. Wecharacterized the degree of elastic heterogeneity in terms of SDof 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 investi-gate the relation between elastic heterogeneities and acousticexcitations, unifying in a single framework ordered and disor-dered solid states and considering quantities directly probed byexperiments. By interpolating in a controlled way from perfectcrystals, through increasingly defective phases, to fully de-veloped amorphous structures, we (i) calculate the dynamical

Significance

What makes the difference when sound propagation is in-vestigated in a crystal or in a glass? One hundred years ago,Debye rationalized the former case in terms of phonons. Incontrast, years of effort have failed to provide a convincing pic-ture for vibrations in disordered solids. We provide a contributionto this issue by reporting clear evidence that a mechanical fea-ture, elastic heterogeneity at the nanoscale, profoundly affectsthe main properties and even the very nature of sound waves.Our picklock is the numerical study of a toy model that, at fixedmacroscopic thermodynamical conditions, allows to investigatein a unified framework the perfect crystal, increasingly defectiveordered phases, and the fully developed amorphous state.

Author contributions: H.M., S.M., and J.-L.B. designed research, performed research,analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. P.H. is a guest editor invited by the Editorial Board.1Present address: Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- undRaumfahrt, 51170 Köln, Germany.

2To whom correspondence should be addressed. Email: stefano.mossa@cea.fr.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1409490111/-/DCSupplemental.

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structure factors, extracting the relevant spectroscopic param-eters; (ii) characterize the wave vector dependence of soundvelocity and broadening of the acoustic excitations and clarifytheir nature in terms of the IR limit; and (iii) provide, forthe first time to our knowledge, direct evidence of the corre-lation of the excitations lifetimes and ΩIR with the magnitudeof the elastic heterogeneities.

ResultsWe study by molecular dynamics simulation in the NVT en-semble, at constant temperature T = 0.01 and number densityρ̂=N=V = 1:015 (V being the system volume), a 50: 50 mixture,composed by N atoms with different diameters, σ1 and σ2, andsame mass, m = 1. We consider two different system sizes N =108,000 and 256,000, to improve statistics and wave vector rangeand confirm that results are not affected by finite-size effects.Particles interact via a soft-sphere potential, vαβ = e(σαβ/r)

12, withσαβ = (σα + σβ)/2 and α, β ∈ 1,2. The potential is cut off andshifted at r = 2.5σαβ. In a one component approximation, wedefine an “effective” diameter σ3eff =

Pα;β=1;2σ

3αβ=4 (22). Starting

from a perfect face-centered cubic crystal, defects are added inthe form of size disorder, by simultaneously decreasing σ1 belowthe initial value σ1 = 1 and increasing σ2, keeping a constant σeff ≡ 1(21). The size ratio, λ = σ1/σ2 ≤ 1, quantifies the size disorderand is our control parameter. λ = 1 corresponds to the perfectcrystal case, whereas for λ = 0.7 a completely developed amor-phous structure is observed. An amorphization transition occursat λ = λ* ’ 0.81 (20, 21). Additional details can be found in ref.20. Simulations have been realized by using the large-scale,massively parallel molecular dynamics computer simulation codeLAMMPS (23).We first focus on the acoustic sound velocities in the macro-

scopic limit. In crystals, sound propagation depends on the di-rection of the wave vector, ~q (1). This is at variance with theisotropic amorphous phases, where only the wave vector modu-lus is relevant. In the macroscopic limit, the q-independentsound velocity is c=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiMeff=ρ

p, where ρ is the mass density and

Meff is an effective macroscopic modulus that depends on boththe direction of propagation and the longitudinal or transversecharacter of the excitation. In what follows we will consider the(110) direction, with Meff = K + Gp/3 + Gs, Gs, and Gp, for thelongitudinal (L) and the two transverse (T1 and T2) branches,respectively. Here K, Gp, and Gs are the bulk, pure shear, andsimple shear moduli, respectively (16). Additional results forthe (100) and (111) directions are reported in Supporting In-formation. In Fig. 1A we show the λ dependence of c for thethree branches. For λ > λ*, cT1 > cT2 , and both slowly follow thedecrease of λ. At λ*, cT2 ’ 0, which can be associated with anelastic instability controlled by Gp (20). For λ < λ*, cT1

decreases whereas cT2 increases, and both reach the same val-ues in the fully developed amorphous state, as expected. Notethat glass and pure crystal show very similar cT2 in the macro-scopic limit. Finally, in the entire λ range the overall variationof cL is very mild.In Fig. 1B we also display the λ dependence of the SD, δM,

calculated from the probability distributions of the local moduli,M = K, Gs, and Gp, respectively. These can be evaluated bycoarse-graining the system in little cubic domains, of linear sizew = 3.16 in this case (20). Starting from a spatially homogeneousdistribution at λ = 1, δGp undergoes very important mod-ifications, strongly increasing by decreasing λ, reaching a maxi-mum at λ*, and abruptly decreasing to a stable low value onthe amorphous side. δGs follows a qualitatively similar be-havior, while quantitatively less important, and the expecteddegeneracy is recovered in the amorphous phases. Finally,longitudinal data also undergo variations similar to those ofδGs for λ ≥ 0.82, eventually staying almost unchanged acrossthe transition.

Moving from the macroscopic limit, we now investigate thewave-vector dependence of the dynamic structure factors,

Sa�~q;ω

�=

12πN

�qω

�2Z

dtDja!�

~q; t�j*a!�

~q; t�E

eiωt; [1]

where a = L, T and jL!ð~q; tÞ and jT

!ð~q; tÞ are the longitudinal andtransverse momentum currents, respectively (6, 9). It is by now con-sensual that transverse modes play the most important role in de-termining anomalies in vibrational properties (9).More specifically,the transverse branchwith the lowest elasticmodulus has been dem-onstrated to be the one that correlates most to the low-frequencyvibrational states (20). In what follows we therefore focus on the T2excitations. Additional data for the T1 and Lmodes are included inSupporting Information. In Fig. 2 we plot STð~q;ωÞ at the indicatedvalues of λ and q. For λ = 0.84 and 0.81, where the two transversesound velocities are well separated (Fig. 1A), STð~q;ωÞ features twoBrillouin peaks corresponding to T1 (high-ω) and T2 (low-ω) exci-tations, respectively. In contrast, a single Brillouin peak is visible, asexpected, in the amorphous phase at λ = 0.7, where cT1 ’ cT2 .Propagation frequency, Ωað~qÞ, and line broadening, Γað~qÞ, of

the sound excitations can be extracted from these data by fittingthe spectral region around the Brillouin peaks to a dampedharmonic oscillator model (6, 9),

Sa�~q;ω

�∼

Γa�~q�Ω2

a

�~q�

�ω2 −Ω2

a

�~q��2

+ω2Γ2a

�~q�: [2]

In Fig. 3 we show the sound velocity, cT2 =ΩT2ð~qÞ=q, and broad-ening, ΓT2 , of the T2 excitations at the indicated values of the

0

2

4

6

8

cLcT1cT2

0

1

2

3

4

5

0.7 0.75 0.8 0.85 0.9 0.95 1

δKδGsδGp

A

B

Fig. 1. Macroscopic limit of sound velocities and width of the distributionsof local elastic moduli. (A) λ dependence of the longitudinal (L) and trans-verse (T1 and T2) macroscopic sound velocities in the (110) direction. Thesedata have been calculated from the effective elastic moduli K + Gp/3 + Gs, Gs,and Gp, respectively. Here K, Gp, and Gs are the bulk, pure shear, and simpleshear moduli, respectively. The vertical dashed line indicates the transitionpoint λ = λ* ’ 0.81. (B) λ dependence of the elastic heterogeneities, δM,associated to K, Gs, and Gp. These data are the SDs of the distribution of thelocal elastic moduli for a coarse-graining length scale w = 3.16 (20).

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disorder parameter λ. For the sake of clarity, we consider firstthe isotropic amorphous case, λ = 0.7. As expected, for vanishingΩ, cT2 corresponds to the macroscopic value of Fig. 1A (horizon-tal dashed line), calculated directly from the value of Gp at thesame λ value. Next, cT2ðΩÞ decreases (softening), reaches a min-imum, and eventually undergoes positive dispersion at higherfrequencies. In the same region where cT2 shows a minimum,a cross-over from ∼Ω2 at high frequency [which can be describedby a two-mode Maxwell constitutive model (24)] to a Rayleigh-like ∼Ωα with α close to 4 at intermediate frequency is evidentfor ΓT2 around Ω ’ 1, which corresponds to ΩBP in this case(20). Both these features are consistent with previous findingsfor the Lennard-Jones glass (6, 7).As λ increases, the sound velocity at a given frequency first

decreases, goes through a minimum at λ* ’ 0.81, and eventuallyincreases steadily. We note that the maximum ratio ’3.5 be-tween the maximum and minimum value (as a function of fre-quency) is reached at λ*, whereas cT2 is essentially frequencyindependent at λ ≥ 0.9, where the Debye picture still holds.Therefore, cT2 mirrors at all frequencies the nonmonotonic be-havior of the macroscopic limit of Fig. 1A. Sound broadeningfollows a quite different pattern. As λ increases from 0.7, ΓT2 isenhanced and reaches a maximum at λ*. Next, it is strongly

suppressed for λ > λ*, converging to a very low value at λ = 1, ofanharmonic origin (damping). We remark that in this case theratio between the maximum and minimum values reached coversalmost two decades at Ω ’ 1. We will see below that this findingcan be rationalized in terms of a strong correlation with themagnitude of the elastic heterogeneity associated with the ap-propriate modulus (Fig. 1B).Next, we focus on the IR limit, ΩIR

T2, for all investigated λ’s.

In Fig. 4A we propose a different representation of the datapoints of Fig. 3, as the ratio πΓT2ðΩÞ=Ω. At the IR limit,πΓT2ðΩIR

T2Þ=ΩIR

T2= 1 (i.e., the decay time of the excitations equals

half of the corresponding vibrational period). ΩIRT2

provides anupper bound for the validity of acoustic-like descriptions of thevibrational excitations. The λ dependence of ΩIR

T2presents again

an interesting nonmonotonous pattern, which we make quanti-tative in Fig. 4B. Here we plot the ΩIR

T2extracted from the above

data, together with our results of ΩIRT1

and ΩIRL for the other

two branches T1 and L. Starting from the pure crystal, whereΩIR

a is expected to be comparable to the highest frequencycomprised in the vDOS, the IR limit decreases steadily with λin all cases, reaches a minimum at λ* for T2 and L (in the T1case ΩIR

T1continuously decreases through the transition), and

levels off to a constant value on the amorphous side. For thetwo transverse branches, this value corresponds to the ΩBP

position, whereas for the longitudinal mode ΩIRL � ΩBP, as

already shown in refs. 6 and 9. Note that a recent study (25)reported that the nature of the BP depends on the Poissonratio, ν: For fragile glasses with relatively high ν > 0.25,ΩBP ∼ΩIR

T (transverse IR limit) (6, 9), whereas for strongglasses with lower ν < 0.2, ΩBP ∼ΩIR

L (longitudinal IR limit)

10-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-1

0 0.5 1 1.5 2 2.5 3

q=0.19q=0.38q=0.56q=0.75

A

B

C

Fig. 2. Transverse dynamic structure factors, ST ð~q,ωÞ, at the indicated valuesof the wave vector ~q in the (110) direction, calculated from Eq. 1. Threevalues of λ are shown, in a defective crystal state (A), at the amorphisationtransition (B), and in the fully developed glassy phase (C). Two Brillouinpeaks, corresponding to the T1 and T2 branches, are visible for λ = 0.84 and0.81. In the glassy phase (λ = 0.7) only one degenerate excitation survives.

1

1.5

2

2.5 1.00.940.9

0.840.820.810.8

0.780.7

10-3

10-2

10-1

100

10-1 100

A

B

Fig. 3. Spectroscopic parameters calculated from the dynamic structurefactors ST ð~q,ωÞ. (A) Transverse phase velocity, cT2 ðΩÞ=ΩT2 ðqÞ=q, and (B)broadening, ΓT2 ðΩÞ, for the T2 excitations at the indicated values of λ. Thesedata have been obtained by fitting the calculated ST ð~q,ωÞ to the dampedharmonic oscillator line shape of Eq. 2. The horizontal dashed line in Acorresponds to the macroscopic limit of the sound velocity at λ = 0.7. Thedashed lines ∝ Ω2 and ∝ Ω4 in B are also guides for the eye, to emphasize theextremely complex frequency dependence of ΓT2 at different values of λ. Acomprehensive discussion of these data is included in the text.

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(8). We have checked that our soft-sphere model is a fragilesystem, with ν ’ 0.43 for λ ≤ 0.78, which is consistent withthese findings. Unfortunately, for λ > λ* we were not able todetermine reliably the value of ΩBP. However, in ref. 20 (Figs.3B and 4) we showed that ΩBP shifts to lower frequencies asλ tends to λ* from above and increases back to higher fre-quencies below the transition. This behavior clearly mirrorsthe pattern followed by ΩIR

T2in Fig. 4B, implying the lowest-

frequency T2 excitations are most related to the BP. Therefore, wecan conjecture that ΩBP ∼ΩIR

T2above the transition also.

DiscussionThe data shown in Fig. 3B acquire even more interest in light ofthe above discussion of the IR limit. Indeed, the shape of theΓT2ðΩÞ functions significantly changes with λ and follows quitecomplex patterns. These show, at frequencies close to ΩIR

T2, clear

cross-overs between regimes with different effective exponents,α, at high and intermediate frequencies (6). The expected low-frequency cross-over to the ’Ω2 behavior owing to anharmon-icity can be recognized for λ ≥ 0.94 (also see the blue dashed linein Fig. 5A), whereas it cannot be observed for lower values of λ at

the considered temperature. In Fig. 5A the red and green solidlines are the best power-law fits to the data, in the high ðΩ>ΩIR

T2Þ

and intermediate ðΩ<ΩIRT2Þ frequency ranges, respectively. The

positions of ΩIRT2

are indicated by the vertical dashed lines, for thecases λ < 0.94. For λ = 0.94, ΩIR

T2is already close to the highest

frequency comprised in the vDOS. The obtained values of theexponents α in the two regimes are shown in Fig. 5B. ForΩ<ΩIR

T2, and for the deeply amorphous state λ = 0.7, α ’ 3.7,

compatible with the expected Rayleigh scattering exponent α =4. By increasing λ, α first decreases steadily by reaching the value2 at λ* and next increases up to a value ’3.5 at λ = 0.94. ForΩ>ΩIR

T2, we recover the expected value α = 2 in the amorphous

phase (24), which decreases quite abruptly, reaching a value ’1.5 atλ*. Subsequently, α increases to a value close to 2.5 at λ = 0.86.These data, in particular those corresponding to the high- 0

0.5

1

1.5

2

2.5

3

10-1 100

λ=1.0λ=0.94λ=0.9

λ=0.84λ=0.82λ=0.81λ=0.8

λ=0.78λ=0.7

10-1

100

101

0.7 0.75 0.8 0.85 0.9

LT1T2

A

B

Fig. 4. Characterization of the IR limit. (A) Ratio πΓT2=Ω for the T2 excita-tions, calculated from the data of Fig. 3, at the indicated values of λ. Thefrequency corresponding to the intersection of each dataset with the hori-zontal line at the value 1 defines the IR limit, ΩIR

T2 . (B) λ dependence of ΩIRT2

extracted from the above data, corresponding to the T2 branch. We alsoshow ΩIR

L and ΩIRT1 for the longitudinal (L) and higher transverse (T1) branches,

respectively. The vertical line indicates the transition point λ* ’ 0.81. Thehorizontal line corresponds to the BP position, ΩBP ’ 1, for the amorphousphases at λ ≤ 0.78.

10-2

10-1

100

101λ=0.94 λ=0.84

10-2

10-1

100

101λ=0.82 λ=0.81

10-2

10-1

100

101

10-1 100

λ=0.8

10-1 100

λ=0.7

1

1.5

2

2.5

3

3.5

4

0.7 0.75 0.8 0.85 0.9 0.95

A

B

Fig. 5. Frequency dependence of broadening for the T2 excitations at theinvestigated values of λ . (A) Ω dependence of the broadening ΓT2 at theindicated values of λ. Red and green solid lines are the best power-law fits ofthe form ’Ωα to the data in the high ðΩ>ΩIR

T2 Þ and intermediate ðΩ<ΩIRT2 Þ

frequency ranges, respectively. ΩIRT2 is indicated by the vertical dashed lines in

all cases except λ = 0.94, where ΩIRT2 is comparable to the highest frequency

comprised in the vDOS. For λ = 0.94, the low-frequency cross-over to theanharmonic ’Ω2 behavior is indicated by the blue dashed line. (B) Values ofthe exponents α extracted from the best fit to the data in A for the high(circles) and intermediate (squares) frequency ranges, respectively. A de-tailed discussion of these data is included in the text.

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frequency branch, provide information similar to that of ref. 26.There, it was shown by a quite involved analysis that frustrationseems to control the value of α. More precisely, α ’ 4 whenfrustration is absent and decreases even below the value 2 byincreasing frustration. This is consistent with our findings,which, however, provide a broader picture, including predic-tions on the low-frequency values. Note that, at variance withref. 26, here we can refer to topologically ordered and disor-dered systems described by the same family of Hamiltonians.The above complex behavior of the sound velocities certainly

has a nontrivial impact on the frequency-dependent macroscopicelastic response. Indeed, we plot in Fig. 6 the frequencydependence of the bulk (Fig. 6A), simple shear (Fig. 6B),and pure shear (Fig. 6C) moduli, at the indicated values of λ.These have been obtained by our data for the sound velocitiesas cLðΩÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKðΩÞ+GpðΩÞ=3+GsðΩÞÞ=ρp

, cT1ðΩÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGsðΩÞ=ρ

p,

and cT2ðΩÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGpðΩÞ=ρ

p, respectively. We focus on the extreme

cases of the pure crystal and the completely developed glass. Inthe first case, the shear moduli are frequency independent, asexpected, up to frequencies where they start to slightly decrease,following the bending of the respective sound velocities whenapproaching the first Brillouin zone. The data for the bulkmodulus K follow a similar scenario with a more pronounceddecrease for Ω ≥ 4. More intriguing is the result for the glass

case, λ = 0.7. Here the shear moduli mirror the behavior of therespective sound velocities, with softening followed by an in-crease at higher frequencies. In contrast, the bulk modulus Kseems to undergo a cross-over at Ω ’ ΩBP ’ 1 from the constantmacroscopic value to a clear frequency-dependent (decreasing)behavior. This result is at variance with what was reported in refs.6 and 7, where a frequency-independent bulk modulus wasproposed. This also originated a simple scaling relation betweenthe longitudinal and transverse sound velocities, which is notfulfilled here. This discrepancy could probably be reconciled byreferring to a nontrivial role played by the details of the in-teraction potential, or of the implemented polydispersity. Cer-tainly it imposes an important caveat on the universality ofapproaches based on the hypothesis of a spatially homogeneousbulk modulus (7, 12).An additional challenge for future theoretical work comes

from another very interesting feature emerging from our results.The total variation of the IR limit ΩIR

T2for the T2 branch on

approaching λ* from above is very large (an order of magni-tude), and ΩIR

T2apparently is (anti-) correlated with the elastic

heterogeneities of Fig. 1B. Above λ*, we noticed that the soundbroadening also has a quite large overall variation and seems tofollow the evolution of the elastic heterogeneities. We makequantitative these correlations in Fig. 7, which is the most rel-evant result of this work. In Fig. 7A we plot ΓT2 on both sides ofthe transition, at the low frequencies Ω ’ 0.5 and 1, and asa function of the extent of the elastic heterogeneities at the corre-sponding λ (Fig. 1B). Whereas in the nondegenerate cases λ > λ*(open symbols) the appropriate data to consider are δG ’ δGp, in

28

30

32

34

36

38

40

42λ=1.0

λ=0.94λ=0.9

λ=0.84λ=0.82λ=0.81λ=0.8

λ=0.78λ=0.7

4

6

8

10

12

14

16

0

2

4

6

8

10-1 100

A

B

C

Fig. 6. Frequency dependence of the macroscopic moduli (A) K(Ω), (B) Gs(Ω),and (C)Gp(Ω), calculated from the sound velocities, at the indicated values of λ.The horizontal dashed lines correspond to the values at Ω → 0 at λ = 0.7.

10-3

10-2

10-1

100

10-1

100

0 1 2 3 4 5 6

A

B

Fig. 7. Direct correlation of features of acoustic-like excitations with localelastic heterogeneities. Here we show the dependence on the extent of theelastic heterogeneities of (A) broadening, ΓT2 , and (B) IR frequency, ΩIR

T2 , forthe T2 acoustic excitations. Data are plotted versus the SD δG of the distri-bution of the relevant local elastic modulus, calculated for a coarse-graininglength scale w = 3.16 (see Fig. 1B). δG ’ δGp for λ ≥ λ* (open symbols) andδG ’ δGp + δGs for λ < λ* in the amorphous phases (closed symbols). A dis-cussion of this point is included in the text. In A we plot values of ΓT2 cor-responding to two different fixed frequencies, Ω ’ 0.5 and Ω ’ 1. Dashedlines are guides for the eye.

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the amorphous cases (filled symbols), where the transverse moduliare degenerate, we assume additivity of the disorder sources and useδG ’ δGp + δGs. Remarkably, the data follow an exponential be-havior ΓT2 ∝ expðδG=gΓÞ for both frequencies. Similarly, we findΩIR

T2∝ expð−δG=gΩIRÞ (Fig. 7B) for both lattice and amorphous

cases. Note that no adjustable parameters are involved in theseplots. We can conservatively assert that these data are the firststrong evidence to our knowledge of a direct correlation of quan-tities related to the intrinsic nature of acoustic-like excitations inordered/defective/amorphous phases with local mechanical proper-ties at the nanoscale (i.e., heterogeneity of the elastic moduli). Weare convinced that important theoretical work will be needed in thefuture to precisely understand the origin and the possible universalcharacter of the above particular functional form.

ConclusionsIn summary, in this work we have investigated sound wavepropagation in a numerical model featuring an amorphizationtransition. By controlling the extent of a well-designed form ofsize disorder we have been able to consider a panoply of dif-ferent solid states of matter, ranging from the perfect crystal andincreasingly defective lattice structures to completely amorphousphases. This approach can be seen as a numerical analog ofexperiments that compare scattering experiments on glasses andthe corresponding (poly-) crystalline polymorphs (27, 28). Bycalculating the appropriate dynamical structure factors, we havefully characterized transverse and longitudinal vibrational excita-tions in terms of sound velocities and broadening, also providing

a very detailed analysis of the complex frequency dependence ofthe latter. The frequency behavior of the macroscopic moduli hasalso been scrutinized, demonstrating an interesting (and unex-pected) frequency-dependent bulk modulus for frequencies largerthan ΩBP. This is at variance with previous results, owing to dif-ferences of the considered interaction potentials or to hasty con-clusions, based on datasets less extended than those considered inthe present work. Most important, both the lifetime and the IRlimit of the sound-like excitations have been shown to directlycorrelate with the width of the distributions of local elastic moduli,both in the cases of lattice systems with defects and isotropicamorphous structures. The fact that elastic heterogeneities cor-relate with sound transport properties is very often referred to asevidence in the literature, but it has never actually been demon-strated. Our results therefore provide the first direct evidence toour knowledge that elastic heterogeneities crucially influence themost puzzling features in acoustic-like excitations in disorderedsystems, including strong scattering and BP. They also constitutea true challenge for important theoretical work in the near future.

ACKNOWLEDGMENTS. This work was supported by the NanosciencesFoundation of Grenoble. J.-L.B. is supported by the Institut Universitaire deFrance. Most of the computations presented in this paper were performedusing the Froggy platform of the CIMENT infrastructure (https://ciment.ujf-grenoble.fr), which is supported by the Rhône-Alpes region (GrantCPER07_13 CIRA) and the Equip@Meso project (Reference ANR-10-EQPX-29-01) of the program Investissements d’Avenir supervised by the AgenceNationale pour la Recherche.

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