Relaxation and vibrational properties in metal alloys and other disordered systems

Alessio Zaccone1,2,3 1Department of Physics ”A. Pontremoli”, University of Milan, via Celoria 16, 20133 Milano, Italy. 2Statistical Physics Group, Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site, CB2 3RA Cambridge, U.K. and 3Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K.

The relaxation dynamics and the vibrational spectra of amorphous solids, such as metal alloys, have been intensely investigated as well separated topics in the past. The aim of this review is to summarize recent results in both these areas in an attempt to establish, or unveil, deeper connections between the two phenomena of relaxation and vibration. Theoretical progress in the area of slow relaxation dynamics of liquid and glassy systems and in the area of vibrational spectra of glasses and liquids is reviewed. After laying down a generic modelling framework to connect vibration and relaxation, the physics of metal alloys is considered where the emergence of power-law exponents has been identified both in the vibrational density of states (VDOS) as well as in density correlations. Also, theoretical frameworks which connect the VDOS to the relaxation behaviour and mechanical viscoelastic response in metallic glasses are reviewed. The same generic interpretative framework is then applied to the case of molecular glass formers where the emergence of stretched-exponential relaxation in dielectric relaxation can be put in quantitative relation with the VDOS by means of memory-function approaches. Further connections between relaxation and vibration are provided by the study of phonon linewidths in liquids and glasses, where a natural starting point is given by hydrodynamic theories. Finally, an agenda of outstanding issues including the appearance of compressed exponential relaxation in the intermediate scattering function of experimental and sim- ulated systems (metal alloys, colloidal gels, jammed packings) is presented in light of available (or yet to be developed) mathematical models, and compared to non-exponential behaviour measured with macroscopic means such as mechanical spectroscopy/rheology.

I. INTRODUCTION Pioneering ideas by J. Frenkel in the 1930s [2, 3] estab- lished a link between the α relaxation time and Maxwell A. Relaxation viscoelasticity, by identifying the α relaxation time τ with the time necessary for an atom to escape the nearest- neighbour cage, τ = τ0 exp(U/kBT ). Here τ0 is an at- In high-temperature liquids, where ”high- tempt frequency, while U is an activation energy barrier temperature” here refers to temperatures significantly which separates two minima (basins) in the energy land- higher than the freezing transition and the glass transi- scape. Clearly, in the supercooled regime this escape- tion, the intermediate scattering function F (q, t), defined from-the-cage process is highly cooperative and collective as the Fourier transform of density fluctuations in the because the dynamics of many atoms is nonlinearly cou- fluid, exhibits a simple exponential decay ∼ exp(−t/τ), pled through a feedback mechanism that is quantitatively which reflects an underlying simple diffusive dynamics of captured by the framework of Mode-Coupling Theory [4]. the building blocks (atoms, molecules). Upon lowering A second decay of F (q, t) may happen on a shorter the temperature close to the freezing (crystallization) or time-scale in certain systems, which is known as β or the glass transition, liquids become very viscous, with secondary relaxation. While α relaxation represents a a spectacular increase of viscosity in the supercooled global or collective process, the β relaxation is localized regime by up to 13 orders of magnitude. In the su- and has been variously associated with either molecular percooled regime, the atomic motions become highly degrees of freedom or with string-like motion of a small correlated and each atom becomes effectively ”caged” on number of atoms/molecules. a time-scale which becomes larger and larger. Mean-field More recently, non-exponential relaxation models such as the Mode-Coupling Theory predict that ∼ exp(−t/τ)β with β > 1 (or compressed expo- this timescale diverges at a critical temperature, the nential relaxation) has been observed in several systems Mode-Coupling temperature Tc [1], however there is no including, notably, metal alloys both in experiments [5] evidence of diverging quantities in experiments. and in simulations [6]. This form of the decay of F (q, t) In the supercooled state, the decay of density cor- can be accompanied by ballistic dynamical signatures relations on long time scales is collective and is no such as a relaxation time that scales as τ ∼ q−1, in longer described by a simple exponential, but rather by contrast with the scaling τ ∼ q−2 typical of diffusion. a stretched-exponential function, ∼ exp(−t/τ)β, with a In general, as shown by molecular simulations [7], stretching exponent β < 1 which can vary depending on the contributions from the inherent structures (i.e. the temperature, system composition and many other fac- potential energy basins) to the equilibrium free en- tors. This process is known as α-relaxation. ergy of supercooled liquids completely decouples from 2 the vibrational contribution. Hence, the relation be- tween vibration and relaxation requires conceptual tools from nonequilibrium statistical mechanics. In the fol- (a) lowing, various phenomenological facts and interpreta- tive/modelling frameworks are reviewed, starting from the vibrational properties.

B. Vibrational spectra of liquids and glassy solids

On the other hand, much of previous research in con- densed matter structure and dynamics has focused on the vibrational properties of amorphous systems, includ- ing amorphous materials, viscous supercooled liquids and normal liquids. The key quantity here is the vibrational density of states (VDOS), which is defined as the nor- FIG. 1. (a) The vibrational density of states (VDOS) of a nor- mal liquid, Argon, computed numerically and with theoreti- malized distribution of eigenfrequencies of a material. cal approximations, the imaginary frequencies (instantaneous For a solid, the VDOS is directly related to the distri- normal modes) corresponding to negative (unstable) eigenval- bution of eigenvalues of the Hessian (dynamical) matrix, ues of the Hessian are plotted on the negative axis. Reprinted and it is generally semi-positive defined being the ma- (adapted) with permission from Ref. [8]. Copyright (2019) terial rigid. For liquids, which are not rigid but flow American Chemical Society. (b) The VDOS of a glassy solid under an applied stress, negative eigenvalues occur, as below Tg, specifically the case of metal alloy Cu50Zr50, show- is well known, which result in purely imaginary frequen- ing experimental data points (symbols) and MD simulations cies in the VDOS, also known as instantaneous normal (solid line). Adapted from Ref. [14]. modes [8]. Although these eigenfrequencies cannot be ac- cessed experimentally, they are important as they repre- sent non-propagating exponentially-decaying, hence re- disordered structure which causes phonon scattering. laxing modes. These imaginary modes physically de- The combination of disorder-induced and anharmonic scribe the local relaxation of unstable saddle points which scattering leads through a Ioffe-Regel crossover into a are continuously generated by thermal fluctuations in the regime where vibrational modes propagate in a diffusive- liquid. like manner (diffusons). At this crossover, the VDOS 2 The VDOS of solids can be measured experimentally normalized by ω features a peak which is known as the using inelastic scattering techniques, such as X-ray and boson peak. Historically, it was shown early on in Ra- neutron scattering, where the dynamic structure factor man scattering experiments that this peak seems to obey S(q, ω) is directly measured. Upon summing over all the same frequency and temperature dependence as the −1 wavevectors q, the VDOS is obtained. In particular, for Bose-Einstein function n(ω)+1 = [1−exp(−~ω/kBT )] metallic glasses and metal alloys inelastic neutron scat- [11], which might suggest that mainly harmonic processes tering and inelastic X-ray scattering are the most used are involved. Nevertheless, recent experimental and the- techniques thanks to the favourable contrast for, respec- oretical work on perfectly ordered crystals [12, 13] has tively, isotopic and electronic scattering length. For oxide shown that the boson peak is a more general phenomenon and for organic glasses formed by either small molecules which in certain circumstances may be generated by con- or polymers, Raman and Brillouin scattering techniques tributions of anharmonicity and optical modes. have also been used extensively. A comparison between the VDOS of liquids (includ- The accessible part of the vibrational spectrum of ing INMs, from simulations) and that of glassy solids is liquids measured in inelastic scattering experiments shown in Fig. 1 to illustrate the main differences dis- presents a symmetric structure for positive and negative cussed above. energy transfer, with a diffusive Rayleigh peak (quasi- elastic contribution) due to spontaneous thermal fluctua- tions, and then two symmetric Brillouin doublets related C. Linking relaxation and vibration to acoustic contributions. The structure of the liquid spectrum has been derived using hydrodynamics (conser- Very few approaches have attempted to link relaxation vation equations) by Landau and Placzek [9] and later and vibration in glassy systems. Historically, the first ap- on in the frame of generalized or molecular hydrodynam- proach is rooted in Mode-Coupling theory (MCT), and ics [10]. provides a way to build the VDOS from the same dynam- In glassy solids, the low-frequency part of the VDOS ics of density fluctuations which controls the dynamical behaves in agreement with the Debye law ∼ ω2 from ω = slow-down in MCT. This approach, as shown by Goetze 0 up to a crossover frequency at which phonons no longer and Mayr [15], and also by Das [16], is able to recover propagate ballistically but start to feel the mesoscopic the boson peak feature in the VDOS. 3

More recently, a different approach has been proposed works which are able to explain and recover stretched- for glasses below the Tg, which is based on lattice dy- exponential relaxation starting from very different as- namics generalized to disordered solids. This approach, sumptions, which are reviewed below. starting from the Newton equations for a disordered lat- tice of atoms/molecules in presence of viscous damping, arrives at a Generalized Langevin Equation in the par- A. The trap model ticle coordinates which can be used also to describe the action of an external field and hence either dielectric or This is a model of electronic relaxation via non- mechanical relaxation. Upon summing all contributions radiative exciton-hole recombination where holes are ran- in eigenfrequency space, the VDOS naturally enters the domly distributed traps that ”eat up” the diffusing ex- description thus providing a direct link between relax- citons, originally proposed by Ilya M. Lifshitz and rep- ation and vibration. This approach will be described resented in later works in particular by Procaccia and more in detail in Section VI.C. Grassberger, and by J.C. Phillips [21–23]. As shown in Both the above approaches provide a link between the Ref. [23], according to this trap model, under the assump- low-frequency features of the VDOS (in particular, the tions of diffusive dynamics with an annihilation term and boson peak) and the α-relaxation. However, there are at Poisson statistics, one finds that the density of not-yet least three main open problems that have emerged over trapped (or survival probability) of excitons decays at the years in the field of amorphous metal alloys where long times as ∼ exp(−td/d+2), where d is the space di- the relation between vibration and relaxation is crucial, mensionality, which gives a Kohlrausch exponent β = 0.6 yet it cannot be easily handled with current models, thus in d = 3. In spite of the elegance of this model, it is calling for new theoretical efforts. not straightforward to apply it to elucidate Kohlrausch The two main problems can be summarized as follows: relaxation in supercooled liquids and structural glasses, (i) the observation of compressed exponential relaxation due to the difficulty of identifying proper excitations and with β > 1 in glassy metal alloys below Tg; traps in those systems. Nevertheless, the trap model pre- (ii) the emergence of power-laws in both the vibrational dicts ”magic values” of the stretching exponent β = 3/5, spectrum as well as in the density correlations in metal from putting d = 3 in ∼ exp(−td/d+2), and β = 3/7 for alloys; systems with long-range relaxation where long-range re- (iii) the microscopic nature of secondary (β) relaxation, laxation channels are characterized by an effective fractal in connection with local vibrational modes. dimensionality d = 3/2 leading to β = 3/7. These val- In the following sections, models of relaxation and ues found experimental confirmation in some systems [24] vibration will be reviewed separately, and then models and may be interpreted in terms of effective dimension- which attempt to establish a link between the two phe- alities in molecular configuration space. In supercooled nomena will be discussed more in detail. Finally, the liquids, however, it rather appears that there is a con- open problems (i), (ii) and (iii) will be given detailed con- tinuous spectrum of β values which continuously evolves sideration along with ideas for future modelling strate- (increases) with temperature, instead of discrete ”magic” gies. values, as suggested by experimental dielectric measure- ments [25].

II. RELAXATION MODELS B. Trachenko’s cage rearrangement model for A typical feature of glassy systems and not just of stretched-exponential relaxation α-relaxation, since its first observation by Kohlrausch in 1847 [17], has been observed in the time-dependent Trachenko [26] proposed a model based on the concept relaxation of various (viscoeelastic, dielectric, electri- of Local Relaxation Events (LREs), which may be de- cal) macroscopic observables in nearly all structurally scribed as rearrangements of the nearest-neighbour cage disordered solids and in crystalline solids with defects. whereby an atom escapes from the cage. The escape Stretched-exponential relaxation has recently been ob- process is associated with a mechanical distortion of the served also in the resistivity of thermally-annealed in- cage, hence with an additional stress. This relaxes the dium oxide films [18]. Over the last century, stretched- local stress, resulting in more local stresses experienced exponentials have been used in countless experimental by other cages in the system. Within a scheme similar contexts to provide empirical fittings to experimental to the elastic or shoving model [27], this additional stress data. Although it is common knowledge that stretched- times the local activation volume (roughly, the volume of exponential relaxation relates somehow to spatially het- the cage) causes an increases of the energy barrier for the erogeneous many-body interactions or to heterogeneous relaxation events that are yet to occur. As time goes by, distribution of activation energy barriers [19, 20], only more relaxation events take place (ageing), such that the very few models or theories are able to predict stretched- energy barrier keeps increasing, through a feedforward exponential relaxation from first-principle dynamics [21]. mechanism. Upon setting a kinetic equation for the time There are at least three main theoretical frame- derivative of the number of LREs as proportional to the 4 number of LREs that still need to take place (assuming D. Random energy landscape (dynamical there is a fixed total number of LREs at time equal infin- heterogeneity) ity) and also proportional to an Arrhenius factor of the energy barrier (the relaxation event probability), the re- Yet a different way of recovering stretched-exponential sulting solutions were studied. If the energy barrier in the relaxation has been studied in the field of spin glasses. Arrhenius factor is constant with time, simple exponen- Such systems can have a number of quasi-degenerate en- tial relaxation is recovered. If, instead, the energy bar- ergy states or minima in the energy landscape. These rier increases with the number of LREs, the feedforward energies can be treated as random independent variables, mechanism leads to stretched-exponential relaxation. In- as is done in the analytically-solvable random energy terestingly, the value of stretching exponent β in this model (REM) developed by Derrida [31]. The relaxation model decreases upon decreasing T in a predictable way, to equilibrium of these models has been studied within consistent with most experimental measurements. the frame of a master kinetic equation by De Dominicis, Orland and Lainee [32]. These authors found that the self-correlation for the probability of finding the system in a state with a certain energy behaves like a stretched- C. Mode-Coupling Theory exponential at long time. Also this result is derived under equilibrium-like assumptions such as imposing detailed- The other widespread theoretical framework is the balance in order to solve the master kinetic equation, Mode-Coupling Theory (MCT) of supercooled liquids. and thus hardly applicable to frozen-in glassy systems This is a radically different approach based on hydrody- that are out of equilibrium. namics, where atomic dynamics on the Liouvillian level More generically, and also more simplistically, one can is systematically coarse-grained by means of projection- describe the energy landscape of a glassy system as a operator techniques. By means of this coarse-graining rugged function with many minima separated from each process in phase space, the slow variables (density fluc- other by barriers (saddles) of variable height E. This nat- tuations) can be isolated and shown to satisfy a self- urally introduces a distribution of barrier heights. If the consistent Generalized Langevin Equation (GLE) for main transport mechanism is diffusive, relaxation from a the dynamics, where, typically, the dependent variable well into another across the barrier is a simple exponen- is given by the intermediate scattering function F (q, t) tial process ∼ exp(−t/τ). Since there is a distribution where q is the wavevector (note that in the following we of relaxation times ρ(τ) (induced by the distribution of will use q and k for the wavevector interchangeably). The barrier heights), the global correlation function will be a damping or frictional term in the GLE is represented by weighted average of the form [33] a non-Markovian memory kernel, which itself is a func- tion of F (q, t), thus making the GLE self-consistent and Z ∞ to be solved numerically [1, 4, 28]. φ(t) ∼ ρ(τ) exp(−t/τ)dτ (1) Much research has been devoted to determining the 0 memory kernel for the damping term in the GLE-like dynamical equation of MCT, which normally requires a if ρ(τ) is Gaussian, it is very easy to see that φ(t) starts number of assumptions. The theory can then be shown off as a simple exponential decay and then crosses over to produce a non-ergodicity transition at a dynamical into a stretched-exponential with β ≈ 2/3. This is a crossover temperature, below which the F (q, t) does not typical value of the stretching exponent that is observed decay to zero at long time, but reaches a plateau (lo- in the α-relaxation of structural glasses. A similar result calization or caging). This caging transition occurs at can be obtained assuming that the quenched disorder is temperatures that are somewhat higher than the actual in the rate of jumping of an atom outside of the nearest- glass transition temperature of the system. In the α- neighbour cage, and has been numerically demonstrated relaxation regime, the decay of F (q, t) at wavevectors in Ref. [34]. An assumption underlying Eq. (1) is that comparable to the size of a nearest-neighbour cage can the relaxation spectrum ρ(τ) exists a priori and does not be fitted with a stretched-exponential function [1, 29]. In evolve with time on the observation time scale. general, as emphasized by G¨otzeand co-workers, no spe- In general, a stretched-exponential relaxation will re- cial meaning can be attributed to the value of β since this sult from the integral in Eq. (1) if the distribution ρ(τ) is essentially the result of a fitting procedure. However, satisfies certain (merely mathematical) properties. The there is an asymptotic limit of MCT where the stretched- properties of the distribution (including its moments) can exponential relaxation becomes exact, namely at q → ∞, be linked to the value of β as discussed in Ref. [35]. as was pointed out in Ref. [30]. On time-scales shorter In general, while stretched-exponential relaxation is than the α-relaxation process, MCT successfully predicts ubiquitous in the solid-state, it has not been possible to power-law decay in time including the von Schweidler re- trace it back to a single well-defined mechanism in the laxation which also shows up as a power-law in frequency many-body dynamics, or to a well-defined microscopic behaviour in the high-frequency spectrum of dielectric re- descriptor of the dynamics so that this remains a very laxation [4]. active field of research. 5

E. β relaxation ondary relaxation process by elucidating the connection between relaxation dynamics/kinetics and the localized Secondary or β relaxation typically manifests itself as force-network and vibrational modes associated with sec- a secondary (smaller) peak in the dielectric loss 00 as ondary relaxation. a function of electric field frequency [36], or in the vis- Finally, orientationally-disordered crystals (or plastic coelastic loss modulus as a function of the mechanical crystals) have provided a new playground with new op- oscillation frequency. It can also be inferred from the vis- portunities to precisely isolate molecular motions asso- coelastic loss modulus measured as a function of temper- ciated with α and β relaxation. In particular, it has ature, as was done for metallic glasses in Ref. [37] where been shown using Nuclear Magnetic Quadrupole reso- it shows up as a low-T wing attached to the α-peak. Ini- nance techniques [48] that α and β relaxation may arise tially discovered by Johari and Goldstein in molecular from molecular re-orientational motions with different glass formers, it was demonstrated to be a feature not dynamics that can be ascribed, separately, to molecules only of molecules with excitable internal degrees of free- in the crystalline lattice that belong to non-equivalent dom, but also present in liquids/glasses formed by rigid molecular environments from the point of view of sym- molecules [38]. From a theoretical point of view, thanks metry. These findings highlight the need to reconsider to work done within the frame of MCT, β relaxation α and β relaxation phenomena in the perspective of the can be understood as a faster relaxation decay which lack of a universal microscopic mechanism. precedes α-relaxation [1] and is described, within MCT and in agreement with simulations and experiments, as a power-law relaxation in time. Furthermore, in metallic III. MODELS OF VIBRATIONAL SPECTRA alloys, an analysis based on the so-called coupling model, which establishes a link between the characteristic time Early research into the vibrational spectrum and the scale of the Johari-Goldstein relaxation and that of the VDOS of glassy solids has been motivated by the dis- α-relaxation, has shown that the experimentally [37] es- covery, by Zeller and Pohl in the early 1970s, of certain timated time scale of β relaxation does indeed coincide anomalies in the specific heat and in the thermal con- with the time scale of a genuine Johari-Goldstein pro- ductivity of glasses at low temperatures. In particular, cess [39]. it was realized that the specific heat of glasses at low T Other approaches, in particular simulations, have high- deviates from the T 3 predicted by the Debye model, and lighted the prominence of mesoscopic cluster motions, in exhibits instead a linear in T behaviour. Similarly, the particular string-like motions, in the β relaxation pro- thermal conductivity behaves like T 2, instead of T 3 pre- cess [40], a phenomenon that has been observed also in dicted again from the Debye model together with Peierls- metal alloys [41–43], and from a theoretical point of view Boltzmann transport assumptions. Clearly, since the can be recovered from the Random First Order Theory VDOS enters as a factor inside the integrals over eigen- (RFOT) of the glass transition upon considering the con- frequency space which define the specific heat and the figurational entropy of the clusters [44]. thermal conductivity, any deviations from the ∼ ω2 law In the current state of the art, we therefore have, on are expected to lead to deviations from the T 3 scaling, one hand, theoretical models, such as MCT, which can which represented a motivation to study and understand provide a good description of the relaxation dynamics the vibrational spectra of glasses. For example, for the for β relaxation but fail to capture the microscopic na- specific heat, the relationship to the VDOS is given by ture of thermally-activated jumps and locally cooperative the following expression: motions involved in the process. Instead, simulations, on Z ∞  2  −2 the other hand, have pinned down the microscopic nature ~ω ~ω C(T ) = kB sinh D(ω) dω of the motions involved in the relaxation process but do 0 2 kB T 2 kB T not provide analytical insights into the time dependence (2) of correlations functions. where D(ω) is the VDOS. A further promising line of research is based on the As a model system, the silica glass has been used in energy landscape concept [45], and has recently brought early experimental studies of the VDOS of glasses [49] new insights. In particular, it has been recognized that using INS which provided early evidence of the excess of the β relaxation process kinetics is related to the rate vibrational states above the ∼ ω2 prediction, known as of jumps from one local minimum to an adjacent one, the boson peak. within the same meta-basin of the energy landscape. In Since then, the boson peak feature has been observed higher dimensions, it has been recently recognized that in practically all glassy materials: polymer glasses [50], the structure of the local minima becomes self-similar or amorphous metal alloys [51, 52], colloidal glasses [53], fractal beyond a certain critical temperature (or a criti- jammed packings [54], proteins [55], orientationally dis- cal value of some other control parameter), a transition ordered organic crystals [56], and the already mentioned called the Gardner transition in analogy with similar phe- oxides [57], and several theoretical approaches have been nomena in spin glasses [46, 47]. This approach may, in developed to describe this phenomenon. Below we briefly future work, bring a more complete picture of the sec- review some of the theoretical models proposed for the 6 description of the VDOS of glasses. Debye phonons thus resulting in the boson peak excess of vibrational states [61–63]. The key assumption of anhar- monicity has been shown in recent work to be inessen- A. Fluctuating elasticity (FE) model tial to capture the essence of the boson peak. However, the soft potential model provided the first prediction One of the most popular approaches is based on of the existence of quasi-localized modes at frequencies evaluating the Green’s function within a self-consistent lower than the boson peak, the distribution of which lo- Born approximation assuming that the shear modulus cally goes like ω4. These modes have been discovered is spatially heterogeneous, i.e. the fluctuating elasticity numerically in recent molecular simulations where care- (FE) model by Schirmacher, Ruocco and co-workers [58]. ful tuning of the system sizes allows one to disentangle The starting point is typically the assumption that the these ω4 modes from the acoustic phonons [64, 65]. An shear elastic modulus fluctuates throughout the mate- independent 1D derivation of these modes based on a rial around an average value, with Gaussian statistics simple localization model was provided by Gurarie and (although the latter specification is not necessary, other Chalker [66]. non-Gaussian distributions lead to similar results). An elastic Lagrangian is then formulated and the Gaussian disorder assumption leads straightforward to the evalua- C. Shifted/smeared van Hove singularity tion of the free energy with quenched disorder and hence to the evaluation of the Green’s function. In the simplest An important model to explain the occurrence of the version of the theory, the Green’s function resembles the boson peak in glasses and semi-ordered system has been Green’s function of a damped harmonic oscillator [59], the one based on adding disordered perturbatively to the X X 1 lattice dynamics in order to study the shifting and broad- G(z) = G(k, z) = (3) −z2 + k2[c2 − Σ(z)] ening of van Hove (vH) singularities. Within this model |k|

symmetry. As a consequence, as atoms move towards (a)  (b) their prescribed positions (imposed by the applied strain  fi = 0 tensor) which are called affine positions, they are sub- ject to forces from their nearest-neighbours (also moving incoming towards their respective positions). In a centrosymmet- f  0 vibration  i ric crystal lattice all these forces acting on atom i would cancel each other out by inversion symmetry. Instead, in scattered an amorphous solid (or in a non-centrosymmetric lattice vibration such as α-quartz) these forces cannot identically sum up to zero, and a residual force acts on atom i in the affine position. The only way to relax this force is that atom i undergoes an additional displacement away from the affine position. This additional displacement is called nonaffine displacement, and plays the key role in the physics of deformation of glassy materials, in particular FIG. 2. (a) The top cartoon shows a comparison of the force- the nonaffine displacement times the force that acts on balance in the affine position (prescribed by the macroscopic the atom provides a negative contribution to the internal strain tensor) in a centrosymmetric crystal (top) and in a energy of deformation of the solid, which results in the glass (bottom). In the affine position in the centrosymmetric mechanical weaking. The nonaffine deformation mecha- crystal all forces from nearest neighbours cancel by symmetry nism is depicted in the cartoon in Fig. 2(a). and the atom i is automatically in mechanical equilibrium, no The local breaking of inversion symmetry is not only further displacement is needed. In the glass, the forces do not responsible for the mechanical softening of the material, cancel due to the lack of inversion symmetry, hence there is a but also affects phonon transport, as schematically de- net non-zero force in the affine position that has to be relaxed picted in Fig.2(b). In recent work Ref. [69], it has been through an additional nonaffine displacement. The bottom cartoon depicts the nonaffine displacements for two bonded shown on the paradigmatic example of harmonic disor- atoms: if the displacements were purely affine (like in a cen- dered lattices that the boson peak height strongly cor- trosymmetric crystal) the atoms would still lie on the dashed relates with an order parameter that quantifies the local lines also in the deformed (sheared) configuration. Instead, degree of inversion-symmetry breaking. It does not cor- in glasses, they are to be found away from the dashed line, relate at all, instead, with the usual bond-orientational and the nonaffine displacements are defined as the distance order parameter. In other words, the stronger the sta- between the actual end position (away from the dashed line) tistical lack of inversion-symmetry in the atomic envi- and the affine position (on the dashed line). Rij indicates ronment, the stronger the boson peak. This picture A the interatomic distance at rest, whereas rij labels the inter- is thus consistent with identifying the local degree of atomic distance if the atoms were both in the affine positions. non-centrosymmetry as a key atomistic structural feature (b) Schematic representation of what happens in a defective that promotes the scattering of phonons on mesoscopic (disordered) crystal lattice e.g. when bonds are randomly re- moved, or, which is the same, in the presence of vacancies. length-scales, as depicted schematically for the case of a An incoming phonon cannot continue along the original di- defective crystal lattice in Fig. 2(b). This also provides a rection due to the missing bond and it is therefore deviated unifying framework for both glasses and disordered crys- (scattered) into a different direction. This illustrates the role tals. of non-centrosymmetry (in either atomic positions or atomic bonding) in promoting scattering phenomena on mesoscopic length scales which in turn cause a crossover from ballistic E. The fracton model phonon propagation into diffusive-like transport due to scat- tering where vibrational excitations become quasi-localized Alexander and Orbach [76] famously proposed a simple (Ioffe-Regel crossover). The same phenomenology is likely to be at play in glasses and defective crystals [69]. model for the density of states on a fractal, called the fracton model or the Alexander-Orbach model. Their starting point was a relationship between the DOS and peak frequency [72]. Furthermore, the height of the bo- the single site Green’s function, similar to Eq. 4 above, son peak has been shown, in 2D simulations, to scale as written for the spectral eigenvalue λ, ∼ 1/G, where G is the shear modulus. 1 ρ(λ) = − ImhP˜ (−λ + i0+)i (5) In parallel, much work has been devoted to reformu- π 0 lating lattice dynamics in order to account for disorder. These efforts, which started with early molecular dynam- where P˜0 is the Laplace transform of the time-correlation ics work aiming to account for atomic relaxations in the function P0(t) which tells the position of a particle (ini- deformation of solids [73], have led to an extended lat- tially at the origin) after a time t, and is clearly re- tice dynamics formalism called nonaffine lattice dynam- lated to the Green’s function G(z) introduced above. ics (NALD) [74, 75]. The basic idea is that atoms in Assuming anomalous or fractional diffusion as appropri- an amorphous solid do not occupy centers of inversion ate for transport on a fractal object, the mean square 8 displacement reads as hr2(t)i ∼ t2/(2+δ¯), where δ¯ is the frequency at which phonons start to feel the random- anomalous diffusion index which depends on the geom- matrix structure of the eigenvalues of the Hessian (dy- −1 etry. Assuming furthermore that hP0(t)i ∼ [V (t)] , namical) matrix. ¯ where V (t) ∼ hr2(t)id/2 denotes the available volume The eigenvalue distribution of a sparse random matrix for the diffusion process with d¯ the fractal dimension of with stochastic entries tends to follow a simple analytical −d/¯ (2+δ¯) the object, they obtained hP0(t)i ∼ t . Upon re- law known as the Marchenko-Pastur (MP) distribution, placing the latter result in Eq. (3) they arrived at the which is a generalization of the Wigner semi-circle law. following result for the VDOS: In particular, random matrices drawn from the Wishart ensemble of matrices M have a distribution of eigenvalues ¯ ¯ N(λ) ∼ λ[d/(2+δ)]−1 (6) given by the MP distribution. The Wishart ensemble is created by starting from a m×n Gaussian random matrix and upon changing to eigenfrequency: 1 T A using M = n AA . This matrix has the eigenvalue ¯ ¯ distribution (M v = λv): D(ω) ∼ ω[2d/(2+δ)]−1 (7) √ √ p((1 + ρ)2 − λ)(λ − (1 − ρ)2) This model thus produces a VDOS which also departs p(λ) = (8) from the Debye prediction ∼ ω2, on account of the frac- 2πρλ tal structure of the material. Furthermore, it has been shown to produce a Ioffe-Regel crossover from propa- where we introduced the parameter ρ = m/n. Since we gating phonons to (quasi)localized excitations (having a are interested in the vibrational√ density of states D(ω) mean-free path lower than their wavelength) called frac- of the eigenfrequencies ω = λ we transform p(λ) to the tons [77], where the localization crossover is related to frequency space: p(λ)dλ = D(ω)dω the existence of a correlation length in the self-similar structure. This model will be used in the following to p √ 2 2 2 √ 2 interpret emerging power-laws in the vibrational and re- ((1 + ρ) − ω )(ω − (1 − ρ) ) Drandom(ω) = laxation properties of metal alloys. πρ ω (9) This contribution to the VDOS represents the effect of IV. EMERGING PICTURE FOR THE VDOS OF randomness in the Hessian, and still does not contain DISORDERED SOLIDS features related to the existence of Debye phonons and of short-range breaking of translation invariance reponsible Growing consensus is pointing towards a Ioffe-Regel (also in glasses) for van Hove-like peaks. These effects crossover from propagating (Debye) phonons to quasi- can be implemented in the following way. localized ”diffusons”, controlled by structural disorder, In order to accommodate phonons, one can shift this as the origin of the boson peak [69, 72, 78–86]. In partic- distribution in the frequency space by δ and introduce ular, the frequency of the crossover, which controls the the width of the spectrum b by the transformation: ω → 2 frequency of the boson peak, can be identified with the b (ω − δ), which gives:

q √ 2 4 2 4 2 √ 2 ((1 + ρ) − b2 (ω − δ) )( b2 (ω − δ) − (1 − ρ) ) D(ω) = 2 . (10) πρ b |ω − δ|

This shifted spectrum belongs to a matrix M 0 that can be trix M 0: first we need to correct the lowest edge of the derived by the original Wishart matrix M in the following spectrum by a factor that behaves like ∼ ω−1/2 for ω → 0 0 1/2 b 1/2 way [72, 87]: (M ) = 2 M +δ1, where δ and b both and like ∼ 1 for ω  0, and second we need to add peak depend on the minimal and maximal eigenfrequencies of functions to model the relics of the van Hove singularity b √ b √ the system, ω− = 2 (1 − ρ) + δ and ω+ = 2 (1 + ρ) + peaks which become more prominent for systems with δ, respectively, which define the support of the random high values of Z due to the topology of the random net- matrix spectrum. work becoming influenced by the limiting FCC lattice to In particular, the value of δ controls the shift of the which any lattice will converge for Z = 12. This second lower extremum of the support of the random matrix correction is achieved by modelling the two relics of the distribution, and thus its value controls the frequency van Hove peaks with two Gaussian functions. The final ∗ ω ≈ ω− which is associated with the boson peak. We still need to make two additional modifications that cannot be induced by a corresponding change in the ma- 9 result for the fitting formulae of the VDOS reads [86]:

0.6 q Z = 6 (a) Z = 7 (b) √ 2 4 2 4 2 √ 2 0.5 [(1 + ρ) − b2 (ω − δ) ][ b2 (ω − δ) − (1 − ρ) ] 0.4 D(ω) =1.2 πρ 2 |ω − δ| 0.3 b D ( ω ) 0.2 !1/4 0.652 0.1 0.60.0 × + 0.25 + G1(Z, ω) + G2(Z, ω) Z = 8 (c) Z = 9 (d) ω 0.5 (11) 0.4 0.3

D ( ω ) 0.2 0.1 2 q 2 2 where G1 = (0.011(Z −6) +0.175) exp(−2(ω−1.6) ) 0.0 π 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 q 8 ω ω and G2 = (0.011(Z − 6) + 0.045) π exp(−8(ω − 2.3 − 0.07(Z − 6))2) are the two Gaussian functions used to model the van Hove peaks. The parameters of these FIG. 3. Numerical VDOS (symbols) of harmonic random net- Gaussian functions are chosen empirically to match the wors and random-matrix model analytical fitting provided by numerical data, although the centers of the Gaussians Eq. (11) (solid lines) [86]. Debye phonons are visible (as must be related to the pseudo-Brillouin zone boundaries quadratically growing from ω = 0) in the numerical data, but are not implemented in the analytical model. of the systems. The latter must coincide with those of the FCC crystal which one recovers in the limit Z = 12. Equation (11) with ρ = 1.6 can now be compared matrix for a harmonic random network reads as: with numerical data obtained for random networks of harmonic springs which have a narrow distribution of αβ X α β α β spring length and all have the same spring constant. Hij = δij κcisnisnis − (1 − δij)κcijnijnij (12) These networks were obtained from a precursor Lennard- s Jones glass upon replacing nearest-neigibhour interac- tions with harmonic springs (and removing further than where α, β are Cartesian indices, cij is a random (scalar) nearest neighbour interactions) [69]. The disorder in the coefficient matrix with cij = 1 if atoms i and j are near- numerically generated networks can be tuned upon ran- est neighbors and cij = 0 otherwise. cij is a matrix domly removing springs, all the way from the FCC limit where each row and each column have on average z el- Z = 12 (ordered crystal) down to the marginally-rigid ements equal to 1 distributed randomly with the con- system with Z = 6 (the ”epitome” of disorder). The straint that the matrix be symmetric. Hence, each ij comparison is shown in Fig. 3. In particular, the expres- entry of the Hessian is a 3 × 3 block given by the factor α β sions of all the parameters δ, b, ρ, and those inside G1, nijnij. This is not taken into account by the MP distri- G2, either remain fixed upon changing Z or evolve with bution, which has been shown in Ref. [88] to converge to Z. Hence, Eq. (11) captures the variation of the VDOS the actual distribution of the Hessian only in the limit spectrum upon varying the coordination number Z of the of d → ∞. This is a result consistent with independent network. results obtained within the context of the replica the- The model parametrization given by Eq. (11) is excel- ory of glasses in Ref. [82]. The eigenvalue spectrum of lent and provides an accurate description of the data for the actual 3d Hessian can only be obtained numerically, all Z values considered in the broad range from Z = 9 but it has been shown that the differences with the MP- down to the unjamming transition at Z = 6. In par- derived spectrum, although noticeable, are not essential, ticular, the expressions of all the parameters δ, b, ρ, and especially in the regime of the boson peak. those inside G1, G2, either remain fixed upon changing Z Hence, the above comparison provides a strong indica- or evolve with Z. Hence, Eq. (11) captures the variation tion of how the VDOS in the vicinity of the boson peak, of the VDOS spectrum upon varying the coordination i.e. in the region where one sees the crossover from Debye number Z of the network. phonons to random-matrix behavior, is effectively dom- A word of caution here is in order. The random-matrix inated by the random-matrix statistics of the Hessian’s model presented above (and originally developed in [86]), eigenvalues. is based on extrapolating the MP distribution which is Future work should be addressed to provide further exact for an infinite-dimensional space (see e.g. the dis- insights into the deeper relation between random-matrix cussion in [82] and in [88]) to a 3d lattice. This means behaviour of the VDOS and the quasi-localized diffusons that, in the reality of a 3d atomic lattice in a solid, the nature of vibrational excitations. Hessian matrix is a block random matrix, with 3 × 3 The emerging picture for the randomness-controlled blocks due to the fact that each interatomic connection Ioffe-Regel crossover which lies at the origin of the boson is described by a unit vector nij, in 3d. The Hessian peak is schematically illustrated in Fig. 4.

10 )

λ coordinates φ one obtains:

( boson peak p ρ ~  −1/2 ¨ ˙ random-matrix spectrum mφp + νφp + kpφp = 0 (14) Eigenvalue distribution Debye phonons of the Hessian matrix ~ λ with p = 1, 2, ..., N, and N the total number of atoms in the lattice, and where kp is the spring constant asso- λ p

) 2 ciated with eigenfrequency ωp = kp/m. With initial ω

(  D relics of van Hove conditions φ (0) = 0 and φ˙ (0) = 0 the above equation boson peak singularities p p is readily solved and the solution presents a combination ~ Aω+B VDOS of exponentials which contain the damping parameter ν. 2 p ~ ω Upon taking the strong damping limit, ν  4mkp, the following relation for the autocorrelation function of Debye phonons random matrix ω normal mode displacement is obtained:

2 FIG. 4. Schematic illustration of the fundamental structure hφp(0)φp(t)i ≈ hφp(0)i exp(−t/τp) (15) of the VDOS of disordered solids. The boson peak arises from the (Ioffe-Regel type) crossover from Debye phonons at low with the following fundamental relation between relax- frequency into quasi-localized excitations the distribution of ation time and eigenfrequency valid for an arbitrary which is described by random-matrix statistics of the eigen- glassy (or simply anharmonic) solid in the strong damp- values of the Hessian matrix [86]. ing limit:

ν V. THE RELATION BETWEEN RELAXATION τp = 2 (16) TIME AND VIBRATION FREQUENCY IN mωp GLASSY SYSTEMS It is important to note that, unlike τα for example, τp A. Relation between eigenfrequency and relaxation is not a special time scale but simply a time scale associ- time ated with one of the p = 1...N normal modes (it would be p = 1...3N in 3d). The above relation is important as it A simple analytical relation between relaxation time establishes a crucial and direct connection between the and vibration eigenfrequency valid in an arbitrary solid relaxation spectrum or distribution of relaxation times with damping has been derived by Tobolsky [89, 90]. The τn in a glassy material and its VDOS. In spite of its im- starting point is the usual Newton’s equation for the dy- portance, this relation has been substantially overlooked namics of an atomic lattice, in presence of Markovian in previous work, aside from few lines of investigation. damping: These are the fluctuating elasticity (FE) model intro- duced earlier, the Frenkel description of liquids [92–94], X mr¨ + νr˙ + H r = 0 (13) which establishes a direct link between the frequency of j j ij i k-gapped transverse modes and the Maxwellian relax- i ation time, and the nonaffine lattice dynamics (NALD) where for simplicity we restrict ourselves to a 1d lattice, approach to glasses [74, 95, 96], which provides a direct but the final result holds for arbitrary dimensions [90]. link between VDOS and α-relaxation [97]. In the above equation, ν is a viscous damping coeffi- Finally it is worth mentioning that in the study of cient, Hij is the Hessian or dynamical matrix defined the phonon Boltzmann transport equation using DFT, as usual (here for a 1d lattice), and rj(t) is the displace- collective vibrational excitations, ”relaxons”, have been ment from equilibrium of atom j. At the atomic level identified recently, which have a well defined relaxation there is of course no frictional damping term, this is an time. This feature is important as it is directly related to emergent feature that arises from the dynamical coupling heat-flux dissipation and relaxons can be used to provide of atom i with many atoms in the material, mediated by an exact description of thermal transport in insulating the long-range (anharmonic) part of the interaction po- crystals [98]. tential. Such a frictional term can be recovered within particle-bath Hamiltonians [91], as will be done below in Sec. VI.C with more details, leading to a Generalized Langevin Equation which contains a frictional term. Un- B. Relation between vibrational linewidth and der the assumption that the dynamic coupling between relaxation time an atom and all the other atoms is chosen to be a con- stant, the Markovian damping of Eq. (13) above is re- A natural question to ask is also about the relation trieved. between vibration lifetime (or linewidth, its inverse) and Upon rewriting the above equation in normal mode relaxation time. 11

1. Supercooled liquids and melts question the identification of τα with τ(kmax) where kmax is the position of the first peak in S(k). In the hydrodynamic theory of liquids, such a rela- tion emerges from the analysis of the coupled momen- tum transfer (Navier-Stokes) and heat transfer equations. 2. Glasses below Tg Upon taking a double Fourier (in space) and Laplace (in time) transformation, the determinant of the hydrody- In glasses well below Tg, the situation is quite differ- namic matrix of coefficients of the coupled Navier-Stokes ent. Here the linewidth is controlled by the disorder and and heat-transfer equations leads to factorization of lon- is a purely harmonic phenomenon, at least at low enough gitudinal and transverse modes, with poles that contain temperatures. In this ”harmonic disorder” regime, the 4 imaginary parts that are related to heat diffusion and to linewidth goes as Γ ∼ ω [58], i.e. in agreement with the sound attenuation. For longitudinal modes, it is straight- Rayleigh scattering law. Recently, numerical studies have forward to show that the k-component of density relaxes shown that long-range power-law correlations in elastic as a function of time according to [99]: constant may produce a logarithmic enhancement to this dependence [103]. This effect has been theoretically pre-  2 2  dicted by extending FE theory to the case of long-range ρk(t) ∼ exp(−DT k t) + exp(−Γk t) power-law correlations of the elastic constants [104]. The 4 where we omitted prefactors, and k denotes the wavevec- relaxation process associated with the ∼ ω harmonic tor. The first exponential is controlled by the thermal dif- disorder-induced scattering of vibrations is rather pecu- liar and has been elucidated using MD simulations of LJ fusivity DT , whereas the second exponential is controlled by imaginary part of the acoustic dispersion, Γ, which is glasses in [105]. In this process, the energy of an incom- related to the viscosity of the material. Γ coincides with ing plane wave gets redistributed to other eigenmodes the acoustic linewidth which sets the lifetime of phonons, of the glass without any viscous dissipation. Interest- ingly, using a MCT-type Generalized Langevin Equation τph. Hence, in the limit of large thermal diffusivity DL, the decay of density fluctuations is controlled by Γ and approach to describe density correlations, Ruocco and the above equation establishes a simple relation between co-workers arrive at the following relation between re- laxation time of the harmonic wave-energy relaxation in- the vibrational lifetime τph and the relaxation time τ of −1 duced by disorder, τ, and the linewidth, Γ: density fluctuations, τph ∼ Γ ∼ τ. This relationship always holds in liquids as well as in amorphous solids in Γ(k) the hydrodynamic limit k → 0. τ(k) ≈ (v2 − v2)k2 In supercooled liquids at finite wave- ∞ 0 lengths/frequencies, however, things are certainly where Γ is quadratic in k and quartic (Rayleigh) in ω, not so simple [99]. A simple, practical, but essentially while τ is thus k-independent at low k, a result supported empirical way of describing relaxation is by means by recent simulations in the slow-dynamics regime [106]. of the viscoelastic model, where a memory function In the above expression, v∞ and v0 denote the speed of ML ∼ exp −(t/τ(k)) is introduced within the hydro- sound in the infinite and zero-frequency limit, respec- dynamic treatment. Typically τ(k) has a peak that tively. matches the first peak in the static structure factor Other relaxation channels involve quantum tunneling S(k), which denotes a slowing down of the dynamics around T ∼ 1K, and a relationship between ultrasonic at- due to cooperativity, a phenomenon known as de tenuation and the time-correlation function dependent on Gennes-narrowing. In general, Γ(k) and τ(k) are not the relaxation time has been presented within the frame- correlated for supercooled liquids, as demonstrated on work of Two-Level State theory [107, 108]. At higher the example of molten potassium in Refs. [100, 101] T , in several systems also other relaxation channels be- where the viscoelastic relaxation time and the linewidth come important which include thermally-activated hop- from either X-ray or neutron scattering are clearly ping, an effect that can be measured with Brillouin scat- uncorrelated as a function of k. In the case of liquid tering or ultrasound techniques, and controls the atten- Argon, the linewidth displays a non-trivial dependence uation around T = 100K [109]. Anharmonic damping on k as evidenced by Mode-Coupling theory and MD has been observed also in network glasses in the THz simulations [102], and actually exhibits a minimum regime where it exhibits a viscoelastic-modified Akhiezer in correspondence of the first peak of S(k) around behaviour, with a Maxwell dependence on frequency: 0.2nm where the relaxation time τ(k) is expected to be largest. Recent MD studies have clarified that the de ω2τ Γ ∼ Gennes narrowing might actually be caused by geometric 1 + ω2τ 2 factors, in particular by the long-range oscillations in g(r), as demonstrated through the analysis of van Hove with a single viscoelastic relaxation time τ and a vis- correlation function, rather than by collective dynamics. coelastic crossover at ωτ ∼ 1. At low frequency ω, this This is further supported by the fact that in real space recovers the hydrodynamic dependence Γ ∼ τ ∼ η where τ(r) increases with r linearly. These results call into η is the viscosity of the material. 12

3. Organic liquids particular, MCT fittings have been proved able to de- scribe the α peak and the α relaxation wing of organic A comprehensive study of structural relaxation in the molecular glass formers such as propylene carbonate in normal liquid phase of glycerol has been reported in Ref. [112], with few fitting parameters. Fittings of the Ref. [110], where different techniques (Brillouin spectra loss viscoelastic modulus of colloidal glasses in compar- in the visible, UV, inelastic X-ray scattering and hyper- ison with rheological measurements data have also been sonic measurements) were combined in an attempt to reported in recent work [113]. cover the broadest range of frequency. Also in this case the frequency dependence of the longitudinal linewidth, Γ(ω), displays a viscoelastic crossover but this time it B. The fluctuating elasticity (FE) model had to be fitted with a (more cooperative) Cole-Davidson model: As mentioned in Sec. IIIA, the FE model assumes, as its starting point, quenched Gaussian disorder in the elas- 1 − h(iω) Γ(ω) = Γ + (v2 − v2)Re tic shear modulus. Referring to the simplest version of ∞ ∞ 0 iω the theory and focusing on longitudinal excitations, the dynamical susceptibility is found within this approach to where be  β 2 1 k 2 h(iω) = χL(k, z) = 2 2 2 = k GL(k, z) (17) 1 + iωτ −z + k cL(z)

+ is the Cole-Davidson function, with τ the viscoelastic re- where, again, z = ω + i0 , while cL(z) is the complex laxation time and β ≤ 1 the stretch parameter. The longitudinal speed of sound, and GL(k, z) is the longitu- data taken over such a broad range of frequencies dis- dinal disorder-averaged Green’s function obtained from played a two-step viscoelastic relaxation. The above the FE approach as outlined in Sec. IIIA. The complex Cole-Davidson model had to be supplemented with an viscoelastic modulus is thus obtained as additional simple Maxwell (β = 1) process which de- 2 0 00 scribes the first viscoelastic crossover at low frequency. E(z) = ρcL(z) = E (ω) − iE (ω) (18) This suggests that the first low-frequency process is a purely hydrodynamic one, similar to the one mentioned This is actually a generalization of the FE to fluctuating viscoelastic moduli, also called Heterogeneous Viscoelas- above for glasses below Tg since in the hydrodynamic limit there ought to be no difference between liquid and ticity [60]. glass. The second viscoelastic crossover in the normal This theory provides a direct link between the VDOS liquid is probably related to caging and dynamical slow- and the α-relaxation process because the Green’s func- down as described by MCT and related approaches. tion G(z) is the same input used to compute the VDOS as mentioned in Sec. IIIA as well as to compute the viscoelastic loss modulus E00(ω) where the α-relaxation process is visible as a wing to the high-frequency side of VI. LINKING VIBRATIONAL AND RELAXATION PROCESSES IN METAL ALLOYS the loss peak. In its more sophisticated version where the Green’s function is evaluated using Coherent Potential Approxi- In this section we will review the results obtained with mation (CPA), the theory yields the following description theoretical approaches which have been proposed in an of the α-relaxation wing in the loss modulus, shown in attempt to link the vibrational spectrum, i.e. the VDOS, Fig. 5. with the relaxation features observed in material be- In this comparison there are a few fitting or adjustable haviour. We will focus on models which build a more parameters, which coincide with the parameters that de- direct connection between the VDOS and the stretched- fine the Gaussian function (see inset of Fig. 5) used exponential in α relaxation wing, since the latter process to describe the spatially-fluctuating distribution of the can be accessed experimentally in metal alloys [111]. high-frequency shear modulus, in turn related to the spa- tial distribution of energy barriers (dynamical hetero- geneity) through the shoving model of the glass tran- A. Mode-Coupling Theory sition [27, 114]. Thus the same ”disorder parameters” which define the Gaussian distribution for the fluctuat- Among the models which can describe the α relax- ing elasticity control both the VDOS of the system and ation in real materials, the more first principles, perhaps the α-relaxation wing visible in Fig. 5. Upon Fourier- remains MCT although it does not provide a direct link transforming the latter into the time-domain, a function between vibration and relaxation (though the VDOS can of time similar to a stretched-exponential can be ob- also be recovered from the MCT description of density tained. Also, the same model also predicts a boson peak fluctuations, as shown in [15], as mentioned above). In in the specific heat, according to Eq. 2 [60]. 13

by [91]

2 N " 2  2# P 1 X p Fα(Q) H = +V (Q)+ α + m ω2 X − 2m 2 m α α α m ω2 α=1 α α α (19) where the first two terms are the Hamiltonian of tagged particle with (effective) mass m and coordinate Q, while 2 1 PN pα 2 2 ( +mαω X ) is the Hamiltonian of the bath of 2 α=1 mα α α harmonic oscillators coupled to the tagged particle with linear coupling function Fα(Q) = cαQ where cα are the coupling strength coefficients which are different for all the different atoms the tagged atom is interacting with (e.g. cα is expected to be large for nearby atoms and small for atoms far away in the material). FIG. 5. Comparison between the Heterogeneous Viscoelastic- This configuration gives rise to a second-order inho- ity theory description (based on the FE model) and various mogeneous differential equation for the position of the sets of experimental data (including metallic glasses) for the α-th oscillator of the bath, whose solution leads to the shear viscoelastic loss modulus G00(ω) as a function of me- following GLE [91]: chanical oscillation frequency ω. Adapted with permission Z t 0 from Ref. [60], Copyrights American Physical Society (2019). ¨ 0 ν(t − t ) dQ 0 mQ = −V (Q) − 0 dt + Fp(t). (20) −∞ m dt

Finally, at high frequencies this theory recovers the FE where Fp(t) is the thermal noise, with the usual model, which indicates that the FE model works within fluctuation-dissipation statistics (which, however, ac- the frame of affine elasticity. It is known, indeed, that the quires new terms when the system is subjected to an frequency-dependent elastic modulus of an amorphous external oscillatory field [115]). material reaches a high-frequency plateau that coincides As is standard for normal mode analysis, one can intro-√ with the affine elasticity approximation [95]. Implemen- duce the rescaled tagged-particle displacement q = Q m tation of nonaffine elasticity within the FE and the Het- in the Hamiltonian, such that the resulting equation of erogeneous Viscoelasticity frameworks remains a task for motion, using mass-rescaled coordinates, becomes ongoing and future work. Z t 0 0 dq 0 q¨ = −V (q) − ν(t − t ) 0 dt + Fp(t). (21) −∞ dt C. Linking vibration and relaxation in metal alloys The noise term can be shown to drop out upon taking an with the NALD approach ensemble average, or equivalently, upon averaging over cycles. 1. The viscoelastic NALD approach As shown in Ref. [91], the friction coefficient ν arises from the long-range coupling between atoms in the The starting point of the Nonaffine Lattice Dynamics particle-bath model, which effectively takes care of long- (NALD) approach is to consider a tagged atom i and range and many-body anharmonic tails of interatomic to model it as interacting with its nearest-neighbours interaction. The friction, in general, arises due to the (through the Hessian, in the same way as in Eq. (13) coarse-graining of a number of degrees of freedom (basi- above) and then also with all other particles (atoms) cally all the coordinates and momenta of the bath parti- in the material to which it is coupled dynamically via cles, as well as the momentum of the tagged particle). a bilinear term in the Hamiltonian. Effectively the The friction is typically non-Markovian (i.e. time or tagged atom behaves like a tagged particle that lives in history-dependent) and given by a memory kernel, in the a local environment (a force-field) given by its nearest- most general case, as is also the case in the MCT equa- neighbours, and in addition its dynamics is coupled tions. to all other atoms in the system (the bath oscilla- Upon applying a deformation described by the strain tors), which is an effective way of modelling long-range tensor η, the nonaffine dynamics of a tagged particle i many-body interactions. This class of Hamiltonians are interacting with other atoms satisfies the following equa- tion for the displacement {x (t) = ˚q (t) − ˚q } around a known as particle-bath or Caldeira-Leggett Hamiltonians i i i and their classical (non-quantum) formulation is due to known rest frame ˚q (see Ref. [97] for details of deriva- i Zwanzig [91]. tion): The Hamiltonian of the tagged atom coupled to its nearest-neighbours as well as all the many other atoms Z t x¨ + ν(t − t0)x ˙ dt0 + H x = Ξ η , (22) i i ij j i,xx xx in the material (treated as harmonic oscillators) is given −∞ 14 which can be solved by performing Fourier transfor- mation followed by normal mode decomposition that decomposes the 3N-vectorx ˜, that contains positions of p all atoms, into normal modesx ˜ = x˜ˆp(ω)φ (p is the index labeling normal modes). Note that we specialize on time-dependent uniaxial strain ηxx(t). For this case, the vector Ξi,xx represents the force per unit strain acting on ● atom i due to the motion of its nearest-neighbors which are moving towards their respective affine positions (see e.g.Refs. [74, 75] for the analytical expression and derivation) and in the case of metal alloys it also includes electronic effects empirically via the Embedded Atom Model (EAM) potential [116].

Within NALD, the stress can be derived upon taking the total derivative of the internal energy associated with FIG. 6. Comparison between the NALD theoretical descrip- the above equation of motion, by imposing mechanical tion and experimental data for the Young viscoelastic loss 00 equilibrium at all steps in the deformation. Upon further modulus G (ω) of Cu50Zr50 alloys as a function of mechan- performing a normal mode analysis and upon computing ical oscillation frequency ω. The temperature range is from the stress, one arrives at the following expressions for the slightly above to slightly below the glass transition temper- stress (in the case of longitudinal deformation along the ature. Adapted with permission from Ref. [97], Copyrights American Physical Society (2019). x-axis): 1 X σ˜xx(ω) = EAη˜(ω) − Ξˆp,xxx˜ˆp(ω) between the memory kernel and the intermediate scatter- V p ing function was derived from first-principles by Sjoegren ˆ ˆ and Sjoelander [118] 1 X Ξp,xxΞp,xx = E η˜(ω) + η˜(ω) A V ω2 − ω2 − iν˜(ω)ω Z ∞ p p ρkBT 4 2 ν(t) = 2 dkk Fs(k, t)[c(k)] F (k, t) (25) 6π m 0 = Exxxx(ω)˜η(ω). (23) where c(k) is the direct correlation function of liquid- from which the viscoelastic Young modulus is readily ob- state theory, Fs(k, t) is the self-part of the intermedi- tained: ate scattering function and F (k, t) is the intermediate Z ωD D(ω )Γ(ω ) scattering function [118]. All of these quantities are E∗(ω) = E − 3ρ p p dω (24) A 2 2 p functions of the wave-vector k and the integral over k 0 ωp − ω + iν˜(ω)ω leaves a time-dependence of ν(t) which is exclusively where we have dropped the Cartesian indices for conve- given by the product Fs(k, t)S(k, t). Similar relations nience, since we are specializing on uni-axial extension, between time-dependent friction and intermediate scat- and ρ = N/V s the atomic density of the solid. tering function have been derived within the frame of the Self-Consistent Generalized Langevin Equation (SC- Here we have used ωp to denote the eigenfrequency to GLE) theory by Medina-Noyola and co-workers and ap- avoid ambiguity with ω which denotes the frequency of plied to colloidal systems [119, 120] and monoatomic liq- the applied mechanical oscillation. uids [121]. Using the EAM method to describe interatomic inter- From approximate fittings to the solution to MCT actions in metal alloys, it has been possible to evaluate equations, and also from scattering experiments and sim- the above expression for the case of binary CuZr alloys. ulations, we know that in supercooled liquids F (k, t) ∼ The comparison (taken from Ref. [97]) is shown in Fig. exp(−t/τ)β, with values of the stretching exponent that 6 below. normally lie in the range β = 0.6 − 0.8 [5]. In turn, this argument gives ν(t) ∼ exp[−(t/τ)b], with stretch- ing exponent b for the memory kernel. For Cu50Zr50 2. Memory function to bridge from microscopic to near Tg in the comparison in Fig. 6, it was found macroscopic relaxation b = 0.58 − 0.75 for the memory kernel ν(t), correspond- ing to β = 0.76 − 0.85 in F (k, t) ∼ exp(−t/τ)β using Eq. In the above comparison few fitting parameters are (25), which is in good agreement with experimental de- used in the modelling of the memory kernel, which is terminations via X-ray photon-correlation spectroscopy taken to be a stretched-exponential function. The lat- (XPCS) for supercooled metallic melts [5], where β ≈ 0.8 ter choice is motivated by the fact that, as within MCT in the supercooled regime near the glass transition tem- schemes, the memory kernel dictates the shape of the in- perature. Also,β = 0.76 was found in the normal liq- termediate scattering function [117]. A beautiful relation uid state with quasielastic scattering [122]. From direct 15 stretched-exponential fittings of E00(ω) in DMA data, This fact introduces a separation of time scales in the however, typically one finds stretch exponents which are dynamics that shows up in distinct peak contributions systematically lower, around β ≈ 0.4 − 0.5 [123–125]. in the VDOS. In order to reproduce both α and β re- A possible resolution to this discrepancy is indicated laxation processes in the fitting of the viscoelastic loss by Eq. (25) above and by the model fitting of Ref. [97], modulus G00(ω) it is necessary to implement two terms where the stretch exponent of the memory kernel for the in the memory kernel, ν(t) = ν1(t) + ν2(t), both mod- fitting of E00(ω) (which is closer to the stretch exponent elled as stretched-exponentials, with different character- that one gets from a direct stretched-exponential fitting istic times. It was found that the β relaxation process re- of macroscopic data) is basically the square of the stretch ceives a dominant contribution from the dynamics of the exponent of F (k, t). Al atoms, which are the lightest in terms of atomic mass. The advantage of this approach is the transparent and This finding is consistent with the picture of activat- direct link between the VDOS and the macroscopic ma- ing units of atoms which are able to perform thermally- terial response, e.g. the loss modulus, hence the relax- activated jumps outside the cage, where, in a system with ation behaviour. Upon taking the memory kernel as a wide separation of atomic masses, the lightest atoms are stretched-exponential function, the only fitting parame- going to be the majority components of the activating ters are the prefactor, the characteristic relaxation time units. Overall, these findings point toward the existence and the stretching-exponent b. As mentioned already, of degrees of freedom with well separated time-scales as however, b is found to agree with experimental measure- the possible origin of β-relaxation processes and this idea ments [5, 126] in the supercooled regime, which effec- should be explored further in future studies possibly in tively reduces the number of fitting parameters to two combination with the so-called coupling model [127]. non-trivial adjustable parameters. It will be shown later on in the context of dielectric relaxation, that the α wing and hence the stretched- VII. VIBRATION AND RELAXATION IN exponential in the material response are recovered by the METALLIC GLASSES: POWER LAWS GLE model even for constant (Markovian) friction, and a strong correlation has been shown (on toy models) be- The VDOS of amorphous metal alloys (metallic tween the boson peak and the extent of the α wing and glasses) can be accessed via inelastic neutron scattering the stretching exponent. (INS) as well as from MD simulations. Among the ear- liest measurements are those carried out by Suck and co-workers [128, 129]. Interestingly, these results (later 3. Separation of timescales in XPCS and DMA confirmed by many other investigators) evidenced the ex- istence of a power-law regime in the VDOS in an energy We have noted that the characteristic time-scale window near the boson peak, and in particular between probed in DMA is significantly larger than the time- 4.7 meV and 8 meV, where the VDOS exhibits the fol- scale measured in XPCS and quasielastic scattering ex- lowing power-law behaviour in glassy CuZr alloys: periments. The only systematic study available on the D(ω) ∼ ω4/3. (26) same system, Mg65Cu25Y10, has evidenced that the re- laxation timescales probed by DMA are on the order of In spite of the large amount of experimental and numer- −1 0 τ = 10 − 10 s, while the timescales probed in XPCS ical studies that followed (and substantially confirmed) 2 4 are in the range τ = 10 − 10 . This wide separation of the data of Suck et al., the emergence of this power-law timescales (at least two orders of magnitude in τ) may behaviour has remained largely unexplained to date. also suggest that, in fact, widely different relaxation pro- It is tempting to interpret the power-law in the VDOS cesses altogether might be at play. This is an impor- of CuZr alloys measured experimentally [128] in light tant point which deserves more extensive and systematic of the fracton model of Alexander and Orbach [76] (re- studies across multiple time and length-scales possibly viewed in Sec. III.E) and of recent structural results combining numerical simulations, experiments and theo- which unveiled fractal patterns in the atomic packing retical approaches. and density correlations of metallic glasses. In particular, both neutron and X-ray diffraction results [130] as well as numerical MD simulations [131], suggest the existence of 4. Modelling of α and β relaxation fractal correlations with a fractal exponent d¯= 2.3 in the relation between the number and size of activating units. Finally, the NALD approach has been used also to de- The activating units are defined as clusters of mobile scribe both α and β relaxation in metal alloys. Also in atoms which display thermally activated jumps beyond this case, the VDOS extracted from experiments (or al- nearest-neighbour distances. Similar clusters have been ternatively, from MD simulations) was implemented in detected also in entropically-stabilized BCC (Wigner) Eq. (24), for the paradigmatic case of a ternary alloy, crystals of charged colloidal particles [132]. ¯ La60Ni15Al25. In this system there is one atomic com- Hence, we can set d = 2.3 for the fractal dimension ponent, La, which is much heavier than the other two. of the activating-clusters network, motivated by experi- 16 ments and simulations [130, 131]. Furthermore, we can to the glassy state [36, 137] as well as in disordered set δ¯ ≈ 0 which corresponds to normal diffusion, because solids, such as orientationally-disordered (plasic) crys- the atoms in the activating units effectively diffuse on the talline phases [138], in general. length scale of the cluster due to the thermally activated ¯ ¯ In spite of this large body of experimental work, theo- jumps. Hence, we obtain [2d/(2 + δ)] − 1 ≈ 1.3 which is retical work has been comparatively much more limited, very close to the exponent 4/3 = 1.33 measured by Suck partly due to the difficulty of modelling such strongly in- et al. [128]. teracting molecular systems. Modelling has been mostly This new interpretation of the 4/3 law in the VDOS of limited to empirical models such as Havriliak-Negami or amorphous metal alloys based on the fracton model, pro- Cole-Cole relations, which consist of empirical ”decora- posed here, may need to be further supported in future tions” of the classic Debye relaxation model. The Debye work. This could be done, for example, by comparing the relaxation assumes that molecules can be described as characteristic energy of the activating units which form independently reorienting dipoles in the alternating elec- the fractal network with the energy window where the tric field, and solves the Smoluchowski diffusion equa- 4/3 law is observed in the VDOS for specific systems. tion for the angular coordinates of the orienting dipole. If this picture will be confirmed, it may indicate that The time-dependent solution, as for any diffusive process, within this energy interval the dominant contribution to is of course just a simple decaying exponential in time. the VDOS comes from the fractal network of activating Therefore, the time-dependent dielectric response goes units, within which vibrational excitations propagate in a like  ∼ exp(−t/τ) with a single characteristic relaxation diffusive-like manner. This may then imply that the bo- time τ which is related to the specifics of the molecular son peak in metallic glasses coincides with a Ioffe-Regel dipole. Upon Fourier transforming, the dielectric com- crossover [77] from ballistic Debye phonons into quasi- plex modulus is localized excitations that move diffusively along the frac- tal network of activating units. Also, the identification of fracton excitations may be ∗ S − ∞  (ω) = ∞ + (27) important to provide a more solid theoretical framework 1 + iωτ for the description of β relaxation in amorphous metal alloys, which is due to stringy and cluster-like motions of where S is the static electric constant, while ∞ is the mobile atoms [41]. This could be a possible route towards infinite frequency value. The imaginary part of the above linking vibrational (fracton) properties and β relaxation expression gives the loss modulus 00(ω) which features a in metal alloys. symmetric absorption peak centered on a characteristic Finally, power-laws in amorphous metal alloys have frequency ω ∼ 1/τ. been observed also in avalanches, i.e. intermittent stress drop events associated with plastic rearrangements un- Since the Debye relaxation assumes independently re- der external strain [133, 134]. Numerous studies have orienting dipoles, it neglects the cooperativeness of col- shown that avalanches in amorphous solids follow power- lective motions which is of course crucial for the descrip- law statistics P (x) ∼ x−s where x is a variable that char- tion of supercooled liquids and glasses in the α relaxation acterizes the size of the avalanche, and recently these regime, and which shows up as an asymmetric wing to phenomena have been observed experimentally in metal- the absorption peak, as already discussed above. The lic glasses [135]. A further step could be to link mean- Havriliak-Negami, Cole-Cole, Cole-Davidson and other field analytical models of avalanche statistics [136] with similar models empirically modify the Eq. (27) by ele- fracton excitations in an attempt to produce a unifying vating the denominator and the term (iωτ) within it, to picture of power-laws emerging in amorphous metals. power-law exponents which are then fitted to the experi- mental data. In general, all these models can provide fit- tings of data but do not provide insights into the physics of relaxation. VIII. RELAXATION AND VIBRATION IN MOLECULAR GLASS FORMERS For β relaxation, typically a Cole-Cole empirical func- tion is used for the modelling of this relaxation process Dynamical relaxation in the liquid and glass state has in BDS experiments. A more physical model is the one been studied intensely by means of Broadband Dielectric proposed by Colmenero and co-workers which is based on Spectroscopy (BDS) over the last decades, using small modelling the secondary β relaxation process by taking organic molecules (glycerol, propylene carbonate, etc) as a Gaussian distribution of activation energies together model systems. BDS allows one to access the loss di- with the Arrhenius thermal activation law [139]. electric modulus 00(ω) over many orders of magnitude in Other than empirical models, more physical and first- electric field frequency ω. Such a breadth of frequency principles descriptions are very rare, and basically re- range is only possible with dielectric spectroscopy tech- duce to two approaches, that have already been men- niques, and it has provided deep insights into the vari- tioned above: the MCT approach, and the NALD-GLE ous relaxation phenomena occurring in supercooled liq- approach. We will briefly review some results from these uids as temperature evolves upon going from the liquid two approaches. 17

FIG. 8. Schematic illustration of the modelling strategy used within the NALD-GLE approach to describe the motions of FIG. 7. Comparison between MCT theoretical calculations partial negative and positive charges of polarized molecules in (solid line) of dielectric loss 00(ν) as a function of electric an electric field in dielectric spectroscopy experiments. Each field frequency ν at T = 157K and experimental BDS data for positive or negative partial charge interacts with its neigh- propylene carbonate (symbols) The dashed line represents the bours with an effective interaction. Furthermore, it is also dy- calculation with just one contribution to the memory kernel, namically coupled to all other particles (via long-range inter- namely the collective density fluctuations, which are shown actions) which is taken into account via the bi-linear particle- to capture the α relaxation peak. The dashed-dotted line bath oscillator coupling, the last term on the r.h.s. in Eq. represents the calculation with just the other contribution to (19). the memory kernel, which is due to coupling between probe- molecule and collective fluctuations, shown to capture the sec- ondary β relaxation process. Adapted with permission from VDOS by implementing the same physics as encoded in Ref. [112], Copyrights American Physical Society (2019). the memory functions inside the calculation scheme for the VDOS based on MCT description of density fluctua- tions [15]. A. MCT description of dielectric relaxation of glasses B. GLE-NALD description of dielectric relaxation Within the schematic MCT framework (i.e. the one of glasses where the wavevector in F (q, t) is fixed to be equal to the cage size, corresponding to the first peak of the An approach similar to the one described in Sec. VI.C static structure factor S(q)), α relaxation and stretched- for the viscoelastic loss modulus can be also applied to exponential behaviour can be reproduced upon suitably describe the dielectric loss modulus of supercooled liquids accounting for thermally-activated intermittent jumps and glasses. using ad hoc memory kernels. The main shortcoming Within this picture, positive and negative partial of the Mori-Zwanzig projection operator formalism on charges in the polarized molecules form a network or lat- which MCT relies is that it is not trivial to accommo- tice, similar to an ionic lattice, as schematically depicted date thermally-activated processes within projection op- in Fig. 8. erators. However, certain forms of the correlators which Each of these partial charges, that we may call ”par- make up the memory kernel of the MCT equation of mo- ticles”, interacts with some neighbours via some effec- tion have been shown to be able to reproduce, to some tive interactions which define an effective spring constant. extent, the intermittent jumps [112]. Furthermore, long range (Coulombic) interactions estab- Using two distinct contributions to the memory kernel, lish dynamical couplings between each of these partial one stemming from collective density fluctuations and the charges and all the others in the system, in the spirit of other from the correlations between a probe-particle and particle-bath dynamical coupling. Hence, one can use the the rest of the system, Voigtmann and co-workers [112] Hamiltonian Eq. (19) where there is a term describing have managed to provide an excellent description of both the local interaction with neighbours and a term (the last α and secondary β relaxation in propylene carbonate, as one on the r.h.s. of Eq. (19)) which instead implements shown in Fig. 7. the long-range coupling between the tagged particle and Clearly, future work in this area will be addressed to- all the other particles in the system. wards strengthening the connection between the analyt- Also in this case, upon eliminating fast variables (mo- ical forms of the contributions to the memory kernel and menta) from the Hamiltonian Eq. (19), one arrives at the underlying microscopic dynamical processes. This the GLE, Eq. (21). Now, in the case of dielectric re- approach also has the potential of providing a way of laxation one does not have to worry about nonaffine connecting the material response, in terms of 00 with the displacements, since due to the electric field being uni- 18 form in the x direction and only the x-component of the response matters, the microscopic displacements are all affine because the nonaffine components vanish by sym- 10 metry. Since in this case the external forcing term is not the applied strain, but the applied (oscillatory) electric field, the term on the r.h.s. side will be different from the one in Eq.(22) and will be given, instead, by the electric 1 force acting on the tagged particle:

t ϵ''(ω) [arb. units] Z 0 b m¨r + ν e−[(t−t )/τ] r˙ dt0 + H r = q E. (28) 0.1 i 0 i ij j e −∞ 10-5 10-4 10-3 10-2 0.1 1 where we have already explicity implemented a stretched- ω[Hz] exponential expression for the memory kernel. Upon performing the usual normal mode decomposi- tion and upon Fourier transforming, we finally obtain the FIG. 9. Comparison between the NALD-GLE theoretical following expression for the complex dielectric modulus: calculations (solid line) of dielectric loss 00(ω) as a func- tion of electric field frequency ω at T = 184K and ex- Z ωD ∗ D(ωp) perimental BDS data for glycerol (symbols) measured in  (ω) = 1 − A 2 2 2 dωp (29) 0 ω − i(˜ν(ω)/m)ω − C ωp Ref. [141]. The dashed line represents an empirical stretched- exponential (Kohlrausch) fitting. Adapted with permission where ωp denotes the normal mode frequency, whereas from Ref. [140], Copyrights American Physical Society (2019). ω denotes the frequency of the external electric field. In the above equation, A is a rescaling constant, since ex- (a) perimental data are normally given in arbitrary units, 0.20 whereas C = pk/m is related to the average effective spring constant k of nearest-neighbour interactions and is 0.10 treated as a constant parameter, while ν(ω) is the Fourier −[(t−t0)/τ]b ϵ'' 0.05 transform of the memory kernel ν(t) = ν0e in Eq. (28). Upon taking the imaginary part of Eq. (29), the dielec- 0.02 tric loss modulus 00 is readily obtained. This has been used in Ref. [140] to fit experimental BDS data of glycerol 10-2 10-2 1 100 104 106 108 1010 obtained by the Augsburg group of Loidl, Lunkenheimer and co-workers [141]. The comparison is shown in Fig. ω[Hz] 9. The calculation implemented a model VDOS for a ran- 0.20 dom network of harmonically interacting particles [140] (b) since experimental data for the VDOS of the glycerol sys- tem were not available. The model VDOS however also features a strong boson peak near the glass transition, 0.10

which modulates the α wing asymmetry. ϵ'' In later work on orientationally-disordered (plastic) crystals of the fluorinated molecules Freon112 and Freon113, the VDOS measured experimentally via Inelas- 0.05 tic Neutron Scattering for the same molecular systems was used in Eq. (29) which led to a significant improve- 10-3 0.1 10 1000 105 ment in the quality of the fitting, results are shown in ω[Hz] Fig. (10). Furthermore, the continuous eigefrequency spectrum of dynamical coupling constant cα in the Hamiltonian Eq. (19) could be extracted from the fitting. This spec- FIG. 10. Fitting of experimental data using Eq. (29) (solid trum provides information on the strength of dynamical lines) for Freon 112 (a) at 91 K (red circles), 115 K (brown coupling (anharmonic interactions) in a certain energy squares) and 131 K (blue diamonds) and for Freon 113 (b) at window. The analysis gave evidence for a bump in the 72 K (red circles), 74 K (brown squares) and 76 K (blue dia- energy window 2 − 5 THz, which may correspond to sec- monds). A rescaling constant was used to adjust the height of ondary β relaxation. The bump is significantly larger (by the curves since the data are in arbitrary units. Experimental a factor 2) for Freon112 than for Freon113, which is con- data for Freon 112 were taken from Ref. [142], while data for sistent with the fact that Freon112 has more long-range Freon 113 were taken from Ref. [143]. 19 interactions and anharmonicity than Freon113. This could lead to medium-range dynamical coupling associ- ated with clustered motions responsible for the secondary 60 β relaxation. Hence, this approach, by allowing the iden- tification of the vibrational energy window where the sec- ondary relaxation takes place, opens up the possibility of 40 isolating the vibrational modes associated with β relax- ation. This is a new and active line of research that is 20 being tested further on different systems. ϵ(t)[arb. units]

0 C. The link between boson peak and 0.1 10 1000 105 107 stretched-exponential relaxation t[s]

The above relationship Eq. (29) establishes a direct link between the VDOS and the relaxation through the FIG. 11. Comparison between Eq. (30) (solid line) and a loss dielectric modulus. Through this equation it is there- pure stretched-exponential dielectric relaxation in the time fore possible to study how the boson peak in the VDOS domain Eq. (33) (dashed line) with β = 0.56, where the correlates with stretched-exponential relaxation. In or- VDOS is the one of a harmonic random network featuring a der to do this it is best to work with a uniform constant prominent boson peak [69]. Adapted with permission from (Markovian) friction ν = const. Ref. [140], Copyrights American Physical Society (2019). In the time domain, the relationship reads as: (t) = B + which recovers Eq. (16) derived by Tobolsky and co- workers within a constant factor 1/C2. Z ωD D(ω ) e(K−ν/2m)t e−(K+ν/2m)t  A p + dω , (30) However, due to the distribution of relaxation 2K K − ν/2m K + ν/2m p 0 times/eigenfrequencies provided by D(ωp) in Eq. (30), q 2 the resulting behaviour for (t) will be stretched expo- where K ≡ ν − (Cω )2 (please note that there is 4m2 p nential: a difference of a minus sign in the definition of K in Ref. [140] due to a typo). The relationship is plotted (t) ≈ B − A0 exp(−t/τ)β (33) in Fig. 11 below in comparison with the dielectric relax- ation given by a pure stretched-exponential (Kohlrausch) as demonstrated in Fig. 11. function. It is clear that Eq. (30) is able to reproduce This result establishes an important point: the stretched-exponential relaxation (even without non- Markovianity), as the dielectric relaxation is given by stretched-exponential relaxation can be recovered within an integral where simple exponentials are weighted by Eq. (1) from simple exponential decay with a distribution a distribution, the latter being nothing else than the of relaxation times that is provided by the VDOS. VDOS. One can compare, indeed, Eq. (30) with the generic expression Eq. (1) for the emergence of stretched- exponential relaxation from the superposition of simple In previous work, Eq. (29) evaluated using VDOS for exponentials with a suitable distribution of relaxation a model system with variable boson peak has been shown times (provided by the VDOS as explained below). to provide a direct correlation between boson peak height In Eq. (30), the simple exponentials can give rise to and the extent of α wing asymmetry, in that a larger either decaying or increasing exponential behaviour in boson peak translates into a wider or more asymmetric time, or oscillatory behaviour, depending on the choice α relaxation wing [144]. of the parameters. For strong damping, decaying ex- Finally, in terms of future work it will be interesting ponential behaviour in time dominates the other be- to explore whether a similar mechanism like Eqs. (30)- haviours, and the leading term can be written in the form (33) leading to stretched-exponential relaxation starting exp((K −ν/2m)t). Upon Taylor expanding to second or- from a VDOS with a boson peak, might be active in crys- der in 1/ν, we obtain the following approximate equality tals as well. On one hand, recent work has highlighted, (neglecting higher order terms in 1/ν): both experimentally [12, 145] and theoretically [13], that r 2 2 a VDOS with a boson peak does arise also in perfect crys- ν 2 2 mωp 2 − (Cωp) − ν/2m ≈ −C (31) tals thanks to anharmonic diffusive-like damping. On the 4m ν other hand, experimental results were presented in the from which, upon replacing back in the exponential, we past which demonstrate stretched-exponential behaviour obtain in the dielectric relaxation of perfect crystals, see the ex- (K−ν/2m)t −t/τ 1 ν ample of TiBO3 crystals [146]. Hence, future work could e ≈ e , with τ = 2 2 (32) C mωp be directed towards understanding whether a boson peak 20

and Bouchaud [147], initially developed for colloidal gels. Compressed exponential relaxation has been observed many times in soft matter, for example in colloidal and jammed systems [40, 148–150], normally together with a ballistic relationship between relaxation time and wavevector, τ ∼ q−1. In this theory, the starting point is the equation of elastodynamics for the displacement field of a (visco)elastic medium in the presence of a finite concentration of point elastic dipoles, and with some vis- cous damping. The main assumption is that there exists a characteristic time-scale θ over which a single dipole relaxation event comes to completion. The intensity of the dipole decays on this time-scale. If the time inter- val which enters the time-correlation functions is much smaller than θ, the solution to the elastic field equations FIG. 12. Time-dependence of correlation functions measured for the displacement yields an intermediate scattering via X-ray photon correlation spectroscopy on Mg65Cu25Y10 function which behaves like a compressed exponential metal alloys, across the glass transition Tg. The inset reports with β = 3/2, not far from the value measured exper- the values of the stretched/exponent β across the transition, imentally [5]. which clearly shows the crossover from compressed behaviour From a physical point of view, this interpretation is (β > 1) below Tg to stretched behaviour (β < 1) above Tg. appealing because, right at Tg, and this is especially true The inset shows the values of exponent β across Tg. Adapted with permission from Ref. [5], Copyrights American Physical for metallic glasses where the quench rate is extremely Society (2019). high, internal stresses develop suddenly as the system becomes rigid. This implies that as the structure freezes in, many atoms will find themselves away from the in- due anharmonicity in perfect crystals may also lead to teraction minima with their neighbours, which means stretched-exponential relaxation in e.g. dielectric relax- that the force obtained as the derivative of the inter- ation. atomic potential is non-zero, hence there will be a lo- cal internal stress associated with every atom displaced from the potential minimum with its neighbour. Each of IX. A NEW MYSTERY IN THE ATOMIC these interatomic stresses thus behaves like a local elastic RELAXATION OF METAL ALLOYS: dipole, which lends credibility to the elastic dipole model COMPRESSED EXPONENTIAL RELAXATION to explain compressed exponential relaxation in metal- lic glasses. This approach rooted in elasticity could also One of the most puzzling phenomena that has been explain the ballistic (or acoustic-like) dependence of the −1 observed in experimental measurements on glass-forming relaxation time on wavevector, τ ∼ q . metal alloys in recent years is the crossover, right at the It should also be remarked that, as pointed out in a re- view by Bouchaud [151], superdiffusive models which are Tg from the stretched-exponential relaxation in the su- percooled liquid to a compressed exponential relaxation often invoked to explain ballistic-type compressed expo- nential relaxation, in fact predict just a simple exponen- below the Tg [5], as shown in Fig. 12. The values of the compressed exponent (close to 1.3) remain constant as a tial relaxation in time, as is the case for L´evyflights. function of temperature below Tg, while it varies with the sample aging. The characteristic relaxation time also dis- plays a marked dependence on sample age. Above Tg, the B. Superposition of collisional Gaussian functions stretching exponent decreases slightly upon approaching Tg from above, which points towards a narrowing of the A different mechanism by which compressed exponen- distribution of relaxation times, along the interpretation tial relaxation may arise could possibly be by superposi- presented above. Below Tg, however, the compressed ex- tion of Gaussian functions, as suggested in Ref. [152], in ponential behaviour cannot be interpreted within any of a way similar to how a superposition of simple exponen- the theories or frameworks illustrated above. tials yields a stretched-exponential as outlined above. In the case of metal alloys, the various Gaussian functions, with different parameters, could correspond to different A. Bouchaud-Pitard elastic dipole relaxation model atomic species or even to rigid atomic clusters. The bal- listic collisional relaxation of an atomic species is known A theoretical explanation of compressed exponential to be given by a Gaussian function of time [153], and behaviour that has been invoked to interpret these re- the presence of multiple atomic species (or even rigid sults is based on the relaxation of internal stresses as- clusters) in metal alloys could lead to a superposition of sociated with elastic dipoles, within the theory of Pitard Gaussians and hence to a compressed exponential accord- 21 ing to the mathematical model presented in [152]. Future (i) Different approaches (Fluctuating Elasticity, Mode- work using MD simulation data could be addressed to Coupling theory, and lately the Nonaffine Generalized verify this possible explanation, which, for the case of Langevin Equation framework) concur in indicating dy- metal alloys, is proposed here for the first time. namical heterogeneity as the root cause of stretched ex- ponential relaxation behaviour. Dynamical heterogene- ity provides a spectrum of relaxation processes, from C. Problems to be solved the superposition of which the stretched exponential be- haviour arises. A further connection between vibrational properties and structural relaxation is provided by the In future work, still, several questions remain to be linewidth of vibrational excitations measured with var- answered. First of all, what is the relationship be- ious techniques and its frequency dependence that in- tween compressed exponential relaxation in the interme- volves a Maxwell viscoelastic crossover with a single re- diate scattering function and the relaxation behaviour laxation time in the hydrodynamic limit [110, 159]. observed in mechanical spectroscopy/rheology (i.e. the (ii) Growing consensus in recent theoretical work from Fourier transform of G00(ω))? Experimental studies indi- a number of different authors is being gathered which cate that stretched-exponential (Kohlrausch) behaviour indicates that the vibrational density of states (VDOS) in the mechanics persists also below T [154], which g of amorphous solids is characterized by a crossover from would then call for additional modelling efforts in or- ballistic phonons at large wavelength into a regime of der to explain how compressed exponential behaviour in quasi-localized excitations, due to disorder-induced scat- the intermediate scattering function could be compatible tering becoming important at a length scale at which with Kohlrausch stretched behaviour in G00 and in G(t). the phonon wavelength is comparable with the mean free A possible explanation is suggested in Ref. [97] and Sec- path [80]. This crossover gives rise to a deviation from tion VI.C.2 of the present review, and in particular by the Debye law and gives rise to the boson peak. Re- Eq. (25) above, which suggests that the exponent of the cent work [86] has revealed a deeper link between the memory function is actually twice the exponent in the quasi-localized excitations and random matrix statistics intermediate scattering function. of the eigenvalues of the Hessian. In turn, the random An interesting question is how the compressed expo- matrix statistics can provide a natural explanation for nential behaviour connects to the VDOS. In recent work the linear-in-T anomaly in the specific heat of glasses at on polymer glasses, it has been shown that instanta- low temperature, as shown in Ref. [160]. Finally, the bo- neous normal modes (INMs), see Sec. IB and Fig. 1 son peak has been observed also in highly-ordered crys- above, are closely connected with residual or internal tals in recent experimental work on halomethanes [12], stresses [155, 156]. Hence, in future work, it could be and one cannot rule out similar results in metals. The interesting to look for signatures of the internal stresses mechanism in this case [13] is controlled by a similar Ioffe- in inelastic scattering data that could be linked to INMs Regel crossover from ballistic phonons to quasi-localized from MD simulations. excitations, with the difference that localization is in- Another important question is related to the aging de- duced by anharmonicity-induced scattering (present also pendence of both the relaxation time and the compressed in perfect crystals), as opposed to ”harmonic” disorder- exponent [126, 157, 158]. Simulation and experimental induced scattering active in glasses. work in this area should be addressed to verifying the as- (iii) The NALD-GLE approach has directly identi- sumptions on waiting times for elastic dipole relaxation fied the spectrum of relaxation processes, which controls within the Bouchaud dipole model, and the latter could stretched-exponential behaviour in mechanical and di- be extended to fully account for the aging dependence electric relaxation, with the VDOS, via Eq. (24) and of the compressed exponential behaviour and for the ob- Eq. (29) above. A similar connection, although less di- served intermittency of atomic-relaxation processes in ag- rect, emerges from the FE framework where the same ing dynamics. Green’s function with quenched disorder in the elastic modulus controls both the boson peak in the VDOS and the Kohlrausch stretched-exponential behaviour in me- X. SUMMARY chanical relaxation; (iv) An old mystery in the VDOS of metal alloys, i.e. We have reviewed results concerning the structural the observation of a 4/3 power-law regime in the VDOS in relaxation processes in supercooled liquids and glasses an energy window close to the boson peak [128], could be (with particular attention to metal alloys) and the rela- explained (as suggested in this review for the first time, to tion thereof with vibrational properties. In spite of the our knowledge) by means of the Alexander-Orbach frac- fact that the two fields have developed almost indepen- ton model of vibrational excitations on fractal structures, dently of one another for a long time, several links be- by identifying the latter with the fractal network of ac- tween relaxation and vibrational behaviour have emerged tivating units (mobile atoms undergoing medium-range in recent work. These could be summarized schematically jumps), recently observed in metal alloys [130, 131]; in as follows: future work, such power-laws could be further put in re- 22 lation with power-laws observed in avalanches and other its combination with models of nonaffine deformation of intermittent phenomena in strained alloys [135]; metal alloys [97]. (v) Secondary or β relaxation in metal alloys has been Important topics that have been left out in this review, shown in recent work to be associated with clusters of but which are also important in the current debate are mobile atoms which undergo coordinated motions [42]. the following: (i) the relationship between α and β relax- The visibility and extent of the β peak may be related to ation, in particular as a function of temperature, with the atomic size mismatch in the alloy, with the high-mobility secondary peak which moves farther apart from the pri- clusters being composed to a larger extent by the small- mary peak upon lowering the temperature, a topic that est/lightest atomic species [161]. Future work could be has been investigated with the Ngai coupling model [163] addressed to identifying which specific vibrational modes for the coupling between the two characteristic relax- are associated with atomic motions involved in β re- ation times; (ii) the complex relationship between ageing, laxation [162], using the relaxation time-eigenfrequency relaxation and vibrational properties which is hitherto equivalence relations presented in this review. This iden- completely unexplored from a theoretical point of view, tification may bring new insights into the nature of β although a large body of experimental data on ageing in relaxation at the microscopic level and new routes for its metal alloys is available [126, 164]; (iii) the physics of mathematical modelling. relaxation in the nonlinear regime, i.e. under large am- (vi) The recent observation of compressed exponen- plitudes of oscillatory strain (in metal alloys) or electric tial relaxation in the intermediate scattering function of field (in organic glass formers), which is a topic of great metal alloys below Tg and the crossover from compressed interest for current research [165]. to stretched relaxation upon crossing Tg [5] have trig- gered new fundamental questions. While compressed ex- ponential behaviour could be explained with the Pitard- Bouchaud model of relaxing elastic dipoles connected ACKNOWLEDGMENTS with internal stresses, it remains to be understood why the mechanical relaxation remains of stretched exponen- tial type below Tg, and what the connection would then Many useful discussions with W. Goetze, G. Ruocco, be between stretched exponential relaxation in the me- G. Baldi, K. Samwer, J.-L. Tamarit, Th. Voigtmann, chanics and compressed exponential relaxation in the in- E. Pineda, K. Trachenko, Z. Evenson, and with E. M. termediate scattering function. Solving this crucial issue Terentjev are gratefully acknowledged. This work was may call for a rethinking and revisiting of the Pitard- supported by contract W911NF-19-2-0055 with US Army Bouchaud model, its extension to link with the dynamics Research Office (ARO) and Army Research Laboratory of intermittent events, clustering of atomic motion, and (ARL).

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