Vibration Modes and Characteristic Length Scales in Amorphous Materials
The numerical study of the mechanical responses of amorphous materials at the nanometer scale shows characteristic length scales that are larger than the intrinsic length of the microstructure. In this article, we review the different scales appearing upon athermal elastoplastic mechanical load and we relate it to a detailed study of the vibrational response. We compare different materials with different microstructures and different bond directionality (from Lennard–Jones model materials to amorphous silicon and silicate glasses). This work suggests experimental measurements that could help to understand and, if possible, to predict plastic deformation in glasses.
Amorphous materials are widely present in our environment, but their mechanical properties, although very interesting, are still poorly understood because of the lack of periodicity in their atomic structure. These materials have no intrinsic structural length scale despite the interatomic distance, as can be probed by measuring its static structure factor1 or, equivalently, pair correlation function. It includes amorphous semiconductors like silicon,2,3 oxide glasses like silica and silicate glasses,4 metallic glasses,5 and other disordered assemblies like colloidal glasses.6 These materials are brittle at large scale but ductile at small scale.7 Their mechanical response is characterized by a very high strength with localized deformation.8 It is thus very important to understand the origin and the organization of plastic rearrangements, and if possible, to prevent plastic damage. One unsolved question concerns the connection between plastic rearrangements and structural defects. Also important is the search of visualization tools to measure local rearrangements and to prevent further plastic damage. Nanoindentation experiments and direct visualization through transmission electron microscopy, for example, need relevant interpretation tools; the former is sensitive to the loading geometry9 and the latter is distorted by the scattering due to structural disorder.10 In this context, vibrational spectroscopy and anelastic neutron and x-ray experiments are interesting and well-controlled methods to infer the mechanical properties of glasses. Indeed, vibrational spectroscopy was already successful to identify vibrational anomalies like the Boson peak in glasses11 and was proposed to quantify plastic deformation at the micrometer scale.12 Brillouin spectroscopy was used to measure the low-frequency sound velocities and thermally activated processes.13 Anelastic neutron and x-ray scattering was used to measure dynamical structure factors, Boson peak, and mean-free paths,14 as well as relaxational processes with the help of time-resolved photon correlation measurements.15 In situ deformations give access to the strain tensor at the micrometer scale.16 However, nanometric heterogeneities make the interpretation of the measurements always difficult,17 and it is necessary first to obtain an insight into the elementary processes responsible for vibrational dynamics and local plastic rearrangements in glasses. Elementary processes responsible for plastic deformation in glasses were identified 30 years ago as shear transformation zones18 and were compared with free volume theory.19 Evidence of local nanometric rearrangements was found recently in different systems with the help of atomistic simulations.20,21 However, the location of the center of the rearrangements and its composition dependence is still a matter of debate.22 At the same time, it was shown that in the elastic reversible regime, displacement field already shows large-scale correlations that have a signature in the vibrational response.23
In this article, we review first the different length scales identified numerically in glasses upon elastoplastic loading and discuss their composition dependence. We then look more accurately on the vibrational response and its sensitivity to mechanical deformation and structural changes resulting from plastic deformation. Finally, we show examples of the possible signature of these characteristic lengths and associated structural rearrangements in spectroscopic measurements.
Elasticity Versus Plasticity in Amorphous Materials
Depending on the composition (and on the pressure applied), the core of the plastic events can be either shear like or compression like, thus changing the corresponding constitutive laws. We have shown in Stillinger–Weber samples that the size W depends strongly on the bonds directionality and can be used as the analog of the size of dislocations core to infer the value of the tensile stress (Peierls stress).3 The general dependence is a decay of W with bonds directionality. In Fig. 3, we see that bond directionality affects the shape of the vortices in the elastic response: Higher bond directionality yields to a screening of the vortices that appear chopped and the corresponding increased fluctuations in the displacement field affect the acoustic scattering processes (see the “Vibration Modes” section).
Small-size plastic rearrangements locate at a place and a scale where mechanical strain is ill defined.26 When looking at the mechanical functions coarse-grained at a slightly larger scale, we saw that they can be predicted by the statistical analysis of the local elastic moduli.26,28 Heterogeneous elasticity and plastic rearrangements are thus closely linked, and the latter could be revealed by an appropriate analysis of the vibration modes. We will now describe the resonant vibrations of amorphous materials.
At low frequencies (below 2πc/ξ where c is the velocity of transverse waves), plane waves coexist with soft modes that are the precursors of plastic instabilities. The number of soft modes depends on the proximity to a plastic rearrangement,28 on the quenching rate, and on the composition of the glass. For example, contrary to Lennard–Jones glasses, silica glasses contain many soft modes at rest and this number increases with the pressure applied. Plane waves are not purely plane waves but contain a small amount of nonaffine displacements whose amplitude is increasing when approaching the frequency 2πc/ξ. In this case, these modes are also called quasi-localized modes.32
At intermediate frequencies, between 2πc/ξ and the frequency characterizing the transition between propagating (or diffusing33) acoustic modes and localized optic modes, vibration modes show a rotational structure (rotons) analog to the nonaffine displacement field observed in the athermal elastic regime (Fig. 3).
Finally, at the transition between acoustic and optic modes (which can be identified either by the saturation of the dispersion relation measured by the dynamical structure factor or by the decay in the vibrational density of states (VDOS) possibly followed by a deep increase), the vibration modes show a multifractal structure similar to the one observed in Anderson localization.34
The boson peak can be measured in glasses with different spectroscopic techniques (Raman and Brillouin spectroscopy, x-ray and neutron scattering). We will now look at a different signature of the mechanical deformation in vibrational spectroscopy.
Scattering measurements of the mechanical response can be regrouped into two different groups: scattering on the atomic positions (light, x-ray, and neutron scattering) versus scattering due to temporal variation of the local optical polarizability resulting from atomic motion (Raman and Brillouin scattering). The former will allow to get the dispersion relation and the acoustic mean-free paths in the boson peak frequency range,14 whereas the latter will be more sensitive to local vibrations and structural changes.38
Quasi-static athermal atomistic simulations helped to identify two characteristic length scales in the nonaffine displacement field of amorphous materials submitted to a mechanical load. These length scales are related respectively to the elastic and to the plastic deformation of athermally driven amorphous materials and can be inferred from the analysis of elastic constants at the nanometer scale. Consequently, they have a signature in the vibrational response and vibrational spectroscopy measurements. Elastic deformation acts as a scatterer and contributes to the boson peak. Irreversible plastic structural changes can be probed with Raman spectroscopy. Among other perspectives, the detailed analysis of the vibration modes and of their sensitivity to mechanical load strongly encourages researchers to evaluate a spectroscopic signature of the soft modes that are the precursors for the plastic instability and could be used to predict plastic deformation.
This research benefited strongly from interactions with T. Albaret, Y. Beltukov, N. Cuny, C. Fusco, C. Goldenberg, B. Mantisi, C. Martinet, D. Parshin, N. Shcheblanov, M. Tsamados, P. Umari, and J.P. Wittmer. This work was supported by the French Research National Agency program ANR MECASIL, Labex IMUST, and ANR Initiative d’Excellence.
- 1.G. Cuello, J. Phys. 20, 2441091 (2008).Google Scholar
- 31.P. Sheng, Introduction to Waves Scattering, Localization and Mesoscopic Phenomena (Boston: Academic Press, 1995).Google Scholar
- 35.C. Kittel, Introduction to Solid State Physics (New-York: Wiley, 1995).Google Scholar
- 37.Y. Beltukov, C. Fusco, D. Parshin, A. Tanguy (unpublished research, University Claude Bernard Lyon, Lyon, France, 2015).Google Scholar
- 39.P. Umari (Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2003).Google Scholar
- 40.N. Shcheblanov, B. Mantisi, P. Umari, A. Tanguy (unpublished research, University Claude Bernard Lyon, Lyon, France, 2015).Google Scholar