Anomalous phonon scattering and elastic correlations in amorphous solids

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1 216MacmilanPublishersLimited,partofSpringerNature.Alrightsreserved.SUPPLEMENTARY INFORMATION DOI: 1.138/NMAT4736 Anomalous phonon scattering and elastic correlations in amorphous solids Simon Gelin 1,2, Hajime Tanaka 2, and Anaël Lemaître 1 1 NAVIER, UMR 825, École des Ponts, IFSTTAR, CNRS, UPE, Champs-sur-Marne, France 2 Institute of Industrial Science, University of Tokyo, Komaba, Meguro-ku, Tokyo , Japan Linear scattering analysis As detailed in the main text, we study sound damping by exciting standing waves in inherent states (ISs) obtained after equilibration of a numerical model of glass. In each IS, we compute the dynamic matrix D ij = 1 H, with m the particle mass and H the Hessian, and then m ij ij solve the discrete wave equation: ü i (t) =D ij u j (t)+a sin (k r i + φ) δ(t) (S1) starting from rest ( u i = u i =, for t<). Here k is the probed wavevector. By linearity, the phase φ only needs to take the values φ = or π/2, and a can be restricted to unit vectors, which are either parallel or perpendicular to k. Since the delta term vanishes at any t>, this problem is equivalent to the elastic response without any source term, for the initial conditions u i = and u i = a sin (k r i + φ) u i at t = +. The generalized position and velocity vectors are denoted u {u i } and u { u i } respectively. The dynamic matrix, as an operator acting on these fields, is denoted D. Introducing its (normalized) eigenmodes ψ m and eigenvalues ωm 2, the formal solution of this problem is: u(t) = m u ψ m cos (ω m t) ψ m, (S2) where the sum runs over the relevant modes (i.e. after eliminating the translations associated with periodicity). It follows that the velocity auto-correlation function reads: C p (t) = u(t) u / u u = m ξ 2 m cos (ω mt) (S3) NATURE MATERIALS 1

2 MacmsLitetoprire.Alrigwith the hybridation coefficients : SUPPLEMENTARY INFORMATION DOI: 1.138/NMAT4736 ξ m = ψ m u u u. (S4) Equation (S1) can be solved numerically either via a full diagonalization of the Hessian, or by numerical Verlet integration of the source-free problem starting from the initial condition u i =, u i = u i a sin (k r i + φ) at t =. We have checked using L 84 systems, that both methods produce identical results. Since a complete diagonalisation is out of reach for large systems, we rely on the Verlet integration in all the data presented here. A Supplementary Movie is provided. It shows the scattering of a transverse standing wave of wavelength λ = L/12 28 excited in a system of size L 336. The left panel displays the time evolution of the instantaneous velocity field (only the atoms lying in the [, 2λ] 2 region of the system are represented). The graph on the right shows the corresponding evolution of the autocorrelation C T (t) = u (t) u / u u. The velocity autocorrelation C T (t) is reported on the top row of Fig. S1, using two systems (black and red) of size L 84, for two different wavevectors, k = 2π (n, ), with n = 2 (left) L and 6 (right). In both cases, C T (t) presents fast oscillations which are modulated in amplitude over time. The initial decay of the correlation is clearly visible for n = 6, but not so for n = 2. In either case, C T (t) does not fully decay, but presents complex beatings at long times. To understand this phenomenon, let us consider equation (S3): for any polarization p, C p (t) is a sum of cosines with weights ξm 2. If the initial excitation were a single eigenmode of the system, only one of these weights would be non-zero, and C p (t) would beat at all times without any loss. In any finite-sized disorder system, however, the plane wave u decomposes in general over many eigenmodes of the system. The weights ξ 2 m are plotted on the bottom row of Fig. S1, as a function of the associated frequencies ω n. For the smaller wavevector, n = 2 [Fig. S1c], only a very few of the ξ 2 m take large values, and they are peaked around a unique frequency. The sum of cosines in equation (S3), hence, comprises only a small number of non-zero terms, which is why the response presents very large oscillations at later times. For n = 6, as shown in Fig. S1d, many more modes present non-vanishing ξm 2 values. The dominant ones are peaked around a typical frequency around which the system beats initially. Since the excited eigenmodes have slightly different frequencies, their contributions [the cosines in the sum (S3)] progressively loose phase coherence as time increases. This is the cause of the initial decay of C p (t). At later times the excited modes beat with completely uncorrelated phases. In such a finite-sized system, scattering is hence the consequence of the progressive loss of phase coherence for the excited modes. Returning to the top row of Fig. S1 (panels a and b), it is clear that the responses of our two systems (black and red curves) are initially in phase (the curves superpose), and that phase coherence is lost at later times. As a consequence, C p (t) should vanish at long times, 2 NATURE MATERIALS 216ilanPublisherimd,parfSngerNatuhtsreserved.

3 MacmsLitetoprire.AlrigSUPPLEMENTARY INFORMATION when averaged over many configurations. This is confirmed by examining the averaged data for both C T (t) and the weights ξ 2 m shown by blue lines in the corresponding plots of Fig. S1. Eighteen samples, with both phases φ = and π/2, were used in the averaging procedure. The initial decay of the correlation is clearly visible even for n = 2 and the beatings are essentially suppressed for n = 6. In the following, we will systematically study the velocity autocorrelation C p (t) for various k = 2π (n, ) and polarization p = L or T, and using different system sizes. In all cases, the L data are obtained by averaging over ensembles of configurations, as listed in Table S1, and using both phases φ = and π/2. a 1 n = 2 b 1 n = C T (k,t) C T (k,t) t t c.5 n = 2 d.6 n = 6.4 ξ m 2.3 ξ m DOI: 1.138/NMAT ω m.1 1 ω m Figure S1: Transverse velocity autocorrelation. Top: C T (t) vs. t for two systems (black and red) of size L 84 (N = 1135), and two wavevectors k = 2π (n, ), with n =2(a) and L 6(b). The blue lines show the same data when averaged on 18 independent systems and on phases φ = and π/2. Bottom: the corresponding hybridization coefficients for n =2(c) and 6(d). NATURE MATERIALS ilanPublisherimd,parfSngerNatuhtsreserved.

4 MacmsLitetoprire.AlrigN N S L Number of samples n δn Table S1: Systems used in this study and associated parameters. See text for definitions. SUPPLEMENTARY INFORMATION DOI: 1.138/NMAT4736 As shown in the main text (Fig. 1), the ensemble-averaged C p (t) fits quite nicely the form exp( Γ p t/2) cos(ω p t) for large enough n. It should be noted, however, that the fit is rather poor at small n s (not shown). We attribute this observation to a finite size effect. Indeed, in the low-frequency region, the eigen-frequencies ( 2πc T /L for transverse modes) are small compared with the gaps between them. As a result, for a small n excitation, as seen in Fig. S1c, a very small number of the ξ m are large while others are essentially vanishing; C p (t) adds a small number of cosines and does not fit well the exponential damping, in particular at short times. The fit improves fast with increasing n, however, as the set of non-vanishing ξ m s progressively densifies [see Fig. S1d], and is already quite satisfactory as soon as n lies beyond, say 5 or 6. A finite-size scaling analysis will be used to assess the conditions in n under which damping data are reliable. We will extract for various L s and n s the sound speed and damping coefficient using the best fit of the ensemble-averaged velocity autocorrelation to the form exp( Γ p t/2) cos(ω p t), knowing it should become poor at low n. To avoid any bias of the measurements with k, the fits are performed over a time interval [,t max (k)], where t max (k) is determined self-consistently so that exp( Γ p t max /2) < 1 4. Table S1 provides, for each L, the size of the ensemble, the starting value n of n, and the interval δn between the successive n values for which damping is measured. The resulting values for the k-dependent damping coefficient are presented on Fig. S2 via plots of Γ p /k 3 vs. k. A lack of collapse between the data obtained at different L s is clearly visible for the transverse damping coefficient (right). But this corresponds only to data points obtained with n values that are smaller than, say 6, i.e. in conditions where the velocity autocorrelation departs from the damped cosine form, exp( Γ p t/2) cos(ω p t). Excluding these small n values, all the data form with increasing L a master curve which very clearly does not saturate at low k. This finite size analysis supports our use of the L 1348 data (filled symbols) for n 1 in the main text. Note that, for a given k, transverse phonons 4 NATURE MATERIALS 216ilanPublisherimd,parfSngerNatuhtsreserved.

5 MacmsLitetoprire.AlrigSUPPLEMENTARY INFORMATION a b L 168 L 336 L 674 L L 168 L 336 L 674 L 1348 Γ L /k 3 15 Γ T /k k k Figure S2: Finite-size effects on the k-dependent damping coefficient. Plots of Γ p /k 3 vs. k for a, longitudinal (p = L) and b, transverse (p = T) excitations. Four different system sizes are used as described in Table S1. Filled symbols (L 1348) are the data used in the main text. more severely suffer from finite size effects than longitudinal ones: this is most likely because transverse excitations beat at a smaller frequency Ω T (k) < Ω L (k): they hence hybridize with lower-frequency modes and are thus more affected by the discreteness of the spectrum. Analysis of published data In this section, we revisit previously published data for the damping coefficient in 3D systems experiments and simulations and will show that they are consistent with the k d+1 ln k scaling rather than with the Rayleigh scaling k d+1 as previously claimed for these data. Experimental data DOI: 1.138/NMAT4736 All the experimental data we examine were obtained using Inelastic Xray Scattering (IXS), to the exception of [1], which was obtained with a tunneling junction technique. Available data cover: (i) sorbitol at 8 K (T g = 266 K) [2], (ii) glycerol at 15 K (T g = 189 K) [3], (iii) densified vitreous silica at 57 K [4], (iv) sodium silicate at 1 K (T g = 694 K) [5], (v) vitreous silica (T g = 145 K) at 162 K [6], 3 K [7], and 1 K [1]. NATURE MATERIALS 5 blir216ilanpusherimd,pafsngernatuhtsreserved.

6 MacmsLitetoprire.AlrigSUPPLEMENTARY INFORMATION DOI: 1.138/NMAT4736 The data of systems (i) to (iv) are plotted as Γ L /k 4 vs. k (in log-lin) in Fig. S3a. Note that in Refs. [2, 3, 4] the damping data were obtained at a single temperature; only in Ref. [5] was temperature varied to assess the conditions of emergence of anharmonic effects, thus evidencing their absence at 1 K for this system. In any case, the authors of these works expect to capture the linear elastic response and fit their damping data with the Rayleigh prediction. Quite clearly, there is no hint, however, in these plots, of a low-k plateau which would mark the Rayleigh scaling. Instead, all these data scale quasi-linearly in this log-lin representation, which corresponds to the proposed k d+1 ln k scaling. The case of vitreous silica deserves special attention. The data of [1, 6, 7] are plotted as Γ L /k 4 vs. k (in log-lin) in Fig. S3b. Baldi et al. [6, 7] brought evidence that their data capture the large-wavelength harmonic scattering regime in the range below k 1.5 nm 1. Dietsche and Kinder [1] measured the mean free path of longitudinal phonons propagating in silica thin films of thicknesses of order 1 µm. They showed that their results depend on neither film thickness nor temperature, which supports that they capture the bulk elastic response of the material. On this ground, Baldi et al. compared their results with Dietsche and Kinder s. They also attempted [7] to interpolate between these two data sets using the Rayleigh law, but found such an interpolation did not work. In view of this, it is quite striking that both data sets match the same k d+1 ln(k/k ) scaling (red line). This provides overall a decade of scaling in k for the relation we propose. Numerical data The numerical data we examine come from: (i) a model of glycerol at 15 K [8], (ii) a mono-atomic LJ model at T = 1 3 (LJ units) [9], (iii) a binary soft sphere system at T =5.1 5 (LJ units) [1]. In these three studies, sound properties were accessed via the analysis of thermal fluctuations in molecular dynamics (MD) simulations; the original data were presented as a function of frequency or pulsation. In order to plot Γ p /k 4 vs. k, we have reconstructed the relation between frequencies and wavevectors point by point from the respective dispersion relations. The data from Refs. [8] and [9] are shown in Fig. S3c and d respectively; the data from Ref. [1] is used in the main text (see Fig. 3a). It is worth noting that the MD data of Γ L for glycerol (Fig. S3c; black filled circles) coincide quantitatively with the corresponding experimental data (Fig. S3a; black filled circles) for k>2 nm 1. In these works, Γ p data were provided down to very small wavenumbers, namely corresponding to values of n kl/(2π) where we observe important finite size effects and deviations from the k d+1 ln k scaling. From these data, we have only removed a very few points at the 6 NATURE MATERIALS blir216ilanpusherimd,pafsngernatuhtsreserved.

7 MacmsLitetoprire.AlrigSUPPLEMENTARY INFORMATION a b 3.16 monaco29 ruta212 baldi213 baldi214 TJ (1K) IXS (162 K) IXS (3 K) h _ Γ L / k 4 (mev.nm 4 ).12.8 h _ Γ L / k 4 (mev.nm 4 ) k (nm -1 ) k (nm -1 ) c d h _ Γ / k 4 (mev.nm 4 ).12.8 Γ / k 4 (LJ units) DOI: 1.138/NMAT k (nm -1 ) k (LJ units) Figure S3: Damping of sound measured in various glasses. Top panels a and b present experimental data. a: IXS data for glycerol [3] (black filled circles), sorbitol [2] (red crosses), densified vitreous silica [4] (green filled squares), and sodium silicate [5] (blue filled triangles). b: vitreous silica probed via IXS at 162 K [6] (black open circles) and 3 K [6] (blue filled triangles), and using a tunneling junction method at 1 K [1] (black crosses). The sound velocity c L = 6 m s 1 [7] was used to convert the mean-free path vs. frequency data of [1] intoaγ L = c L /l L vs. k relation. The red line is a k d+1 ln(k/k ) fit, with k =3.4 nm 1. Bottom panels c and d show MD simulation data, with circles and triangles representing longitudinal and transverse damping respectively. c: a model of glycerol [8]. d: a monoatomic LJ system [9]. NATURE MATERIALS ilanPublisherimd,parfSngerNatuhtsreserved.

8 MacmsLitetoprire.Alriglowest k s when they were outside the plotting range, but report all available data starting at n = 2 from [8], n = 3 from [9] and n = 3 from [1]. Hence, the stronger-than-linear upturn of Γ p /k 4 shown by these data at the lowest k is likely due to finite-size effects (see the discussion of Fig. S2). The numerical data we gather here were previously plotted as Γ vs. k in log-log plots and argued to obey the Rayleigh scaling. Our Γ p /k 4 vs. k plots quite clearly contradict this idea as there is no hint of the expected low-k plateau. Moreover, on the available k range, these numerical data agree strikingly well with our scaling. Supplementary Movie A transverse standing wave of wavelength λ = L/12 28 is excited in a system of size L 336. Left: Evolution of the velocity field along time (only the atoms lying in the [, 2λ] 2 region of the system are represented). Right: autocorrelation C T (t) = u (t) u / u u as a function of time, for this excitation. DOI: SUPPLEMENTARY INFORMATION References (21) /NMAT4736 [1] Dietsche, W. & Kinder, H. Spectroscopy of Phonon-scattering In Glass. Physical Review Letters 43, (1979). [2] Ruta, B. et al. Acoustic excitations in glassy sorbitol and their relation with the fragility and the boson peak. Journal of Chemical Physics 137, (212). [3] Monaco, G. & Giordano, V. M. Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths in glasses. Proceedings of the National Academy of Sciences of the United States of America 16, (29). [4] Baldi, G. et al. Emergence of crystal-like atomic dynamics in glasses at the nanometer scale. Physical Review Letters 11, (213). [5] Baldi, G. et al. Anharmonic damping of terahertz acoustic waves in a network glass and its effect on the density of vibrational states. Physical Review Letters 112, (214). [6] Baldi, G., Giordano, V. M., Monaco, G. & Ruta, B. Sound attenuation at terahertz frequencies and the boson peak of vitreous silica. Physical Review Letters 14, [7] Baldi, G., Giordano, V. M. & Monaco, G. Elastic anomalies at terahertz frequencies and excess density of vibrational states in silica glass. Physical Review B 83, (211). 8 NATURE MATERIALS 216ilanPublisherimd,parfSngerNatuhtsreserved.

9 [8] Busselez, R., Pezeril, T. & Gusev, V. E. Structural heterogeneities at the origin of acoustic and transport anomalies in glycerol glass-former. Journal of Chemical Physics 14, (214). [9] Monaco, G. & Mossa, S. Anomalous properties of the acoustic excitations in glasses on the mesoscopic length scale. Proceedings of the National Academy of Sciences of the United States of America 16, (29). [1] Marruzzo, A., Schirmacher, W., Fratalocchi, A. & Ruocco, G. Heterogeneous shear elasticity of glasses: the origin of the boson peak. Scientific Reports 3, 147 (213). NATURE MATERIALS 9 SUPPLEMENTARY INFORMATION DOI: 1.138/NMAT MacmilanPublishersLimited,partofSpringerNature.Alrightsreserved.