Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

CHAPTER ONE

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy Lucia Comez*,†,1, Claudio Masciovecchio‡, Giulio Monaco}, Daniele Fioretto† *IOM-CNR, c/o Universita` di Perugia, Perugia, Italy † Dipartimento di Fisica, Universita` di Perugia, Perugia, Italy ‡ Elettra-Sincrotrone Trieste, Area Science Park, Basovizza Trieste, Italy } European Syncrotron Radiation Facility, Grenoble, France 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 2. Background 2.1 The spectrum of density fluctuations 2.2 Brillouin scattering cross-section 3. Experimental 3.1 BLS: The tandem multipass Fabry–Perot interferometer 3.2 Inelastic ultraviolet scattering 3.3 Inelastic X-ray scattering 4. Liquids and Glass Transition 4.1 Introduction 4.2 Structural and secondary relaxations approaching the glass transition 4.3 Brillouin scattering and relaxation processes through the liquid–glass transition 4.4 Acoustic analysis: Sensitivity of Brillouin scattering to secondary relaxations 4.5 Full spectrum analysis: Divergence of a relaxation time and a–b splitting 4.6 Experimental determination of the nonergodicity factor 5. Glasses 5.1 Introduction 5.2 More on QES: Correlation between collective and tagged particle dynamics 5.3 More on BP: Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths in glasses 6. Future Developments Acknowledgments References

Solid State Physics, Volume 63 ISSN 0081-1947 http://dx.doi.org/10.1016/B978-0-12-397028-2.00001-1

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2012 Elsevier Inc. All rights reserved.

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1. INTRODUCTION Liquid and fluid systems in general are characterized by long-range translational invariance, and by the interplay between collective properties and the degrees of freedom associated with the intra- and intermolecular structure. These features are at the basis of many macroscopic properties peculiar to the fluid state and distinguish it from the solid and, in particular, from the crystalline phases. If long-range and long-time properties of a f luid are analyzed, simple hydrodynamics is sufficient to account for the average properties of the system. Moving toward shorter times and smaller distances, a more complex and fascinating landscape shows up where spatial and temporal discontinuities play an important role. Here, more cumbersome theoretical tools are needed to give a rationale for the results of experiments, such as the generalized molecular hydrodynamics [1]. In particular, at times shorter than those of molecular motions, one measures the unrelaxed (solid-like) response of the liquid, while at longer times, the relaxed (liquid-like) condition is generally observed. Time- (or frequency-) dependent response is the realm of generalized hydrodynamics that, through the determination of the relaxation times, gives access to the characteristic dynamics of molecular motions. In addition, if the explored length scale (q range) matches the spatial discontinuities of the system, a q-dependent response is expected that gives information on possible intermolecular correlations and aggregations. If “ordinary” liquids show these exotic properties at picosecond time scales and subnanometer length scales, far enough from our common sense, special attention is due to glass-forming liquids, which demonstrate a spectacular increase in structural relaxation time of more than 14 orders of magnitude, so that molecular arrangements develop in the scale of minutes, or hours, or more. In this condition, the solid-like nature of a liquid becomes experimentally accessible in human-lifetime scales, giving origin to viscoelastic and, eventually, glassy materials, such as window glasses, solid plastics, and hardened glues. Due to the complexity of atomic and molecular processes driving a liquid toward solidification into an amorphous structure, a number of questions arise: “What is the nature of the glass transition between a fluid or regular solid and a glassy phase? What are the physical processes that determine the general properties of glasses?” [2–6]. In glass-forming systems, different diffusion processes in the liquid phase, or hopping and tunneling processes in glasses, naturally introduce different

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time scales t, which are strongly dependent on the specific thermodynamic state. These time scales affect the collective dynamical properties differently, depending on their value as compared to the time scale, tD, the time for the particles’ vibrations around their quasi-equilibrium positions. This is on the same order as the inverse Debye frequency of a crystal with similar density and sound velocity. Moreover, the topological disorder inherent in the amorphous phases introduces other length scales, x, beside the interparticle distance, d. As a matter of fact, the rich phenomenology observed in the dynamics of disordered systems can be ascribed to the interplay between these different structural (d,x) and dynamical (t,tD) scales. This complex scenario has left open several relevant questions that have not been cast in an exhaustive and commonly accepted framework so far. Among them are the following: What is the molecular origin of the dynamic arrest at the basis of the glass transition? Are there any collective excitations with wavelengths approaching x and d? And, if so, to what extent do their eigenvectors deviate from plane waves as in crystals? How does the topological disorder affect the vibrational spectrum and the thermal behavior of glasses? Moreover, is the elastic continuum theory appropriate at the mesoscopic length scales, where the disorder of amorphous systems becomes relevant? What is the origin of sound attenuation in glasses? A large amount of information, which would be indispensable if we were to try to answer the above open questions, can be deduced by the experimental determination of the density–density correlation function, F(q,t), or, equivalently, of its Fourier transform, the dynamic structure factor, S(q,o), in the largest momentum (q) and time (t) or energy transfer (E ¼ ℏo) region. Special attention should be given to the portion of the (q,o) or (q,t) plane corresponding to the aforementioned characteristic length scales (d,x) and time scales (t,tD). In most materials, d and x are about a few tenths and tens of nanometers, respectively, while tD is usually in the sub-picosecond range. Concerning the relaxation time scales, they can assume rather disparate values, as they strongly depend on both the specific nature of the relaxation process under consideration and the thermodynamic conditions. However, the experience accumulated so far in this field points out that the relaxation processes that are the primary influence on the physical behavior of disordered systems are those of the structural relaxation in the early supercooling process and those on the same order of x/cS (cS being the sound velocity), both falling in the 0.1–100 picosecond range. It is then natural to infer that, for studying the physics of disordered systems, it is important to have access to the (q,E ) region included in the 0.01–10 nm1 and 0.01–10 meV ranges.

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In bulk materials, the S(q,o) can be directly measured by means of inelastic photon or thermal neutron-scattering experiments [1,5–8]. However, a single technique cannot completely cover, by itself, the entire range, from interatomic distances to the continuum scale. In Fig. 1.1, we show in a log–log plot the E and t versus q space where a relevant part of the condensed matter dynamics takes place. The regions where different experimental methods are currently available are also displayed. The leading energy-resolved techniques among them are inelastic light scattering (ILS), inelastic ultraviolet scattering (IUVS), inelastic X-ray scattering (IXS), and inelastic neutron scattering (INS). ILS, including Raman and Brillouin light scattering (BLS), is a useful technique for studying a large class of materials; however, it has the limitation that only very low-momentum transfers, no higher than 0.03 nm1, can be studied because of the small momentum carried by the photons at visible light wavelengths. Recently, a UV synchrotron source has been used to push the light-scattering technique up to 0.1 nm1. INS typically probes much larger q values. In principle, it is possible to investigate an extremely wide q region ( 0.3–300 nm1) with INS. However, the kinematics of the neutron-scattering process strongly limits the accessible E range at a given q value. This unavoidable constraint is particularly effective at low q. Conversely, IXS does not suffer from such a restriction; however, the energy resolution of current IXS spectrometers is limited to 1.5 meV.

102

10-14

IXS

100

10-12

10-1

10-11

10-2

/s

00

INS

m

10-10

70

10-3

10-9

/s

0

Time = t = v -1 (sec)

10-13

IUVS

E = h v = h / t (meV)

ILS 101

m

50

10-4 10-3

10-2

10-1

100

101

10-8 102

q (nm-1)

Figure 1.1 Schematic cartoon representing the dynamical windows covered by different scattering techniques.

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In the following section, we discuss how BLS, IUVS, and IXS can provide access to the collective part of density fluctuations by means of Brillouin scattering experiments, present a brief overview of the experimental setups recently developed to access these techniques, and report on a selection of results recently obtained in the study of liquids and glass transition.

2. BACKGROUND 2.1. The spectrum of density fluctuations A few general ideas are recalled of a simple formalism that is often used to describe the atomic dynamics at high frequencies and wavenumbers, that is, the so-called molecular hydrodynamics. For a complete discussion of this topic, we refer to well-known monographs [1,9]. We are here essentially concerned with the normalized correlation function, Fq(t), of the density fluctuations: D E drq ð0Þdrq ðtÞ E; ½1:1 Fq ðt Þ ¼ D drq ð0Þdrq ð0Þ where drq(t) is the q component of the f luctuation of the microscopic number density, r(r,t). The dynamic structure factor, S(q,o), is then defined as ð1 dteiot Fq ðtÞ; ½1:2 Sðq; oÞ ¼ SðqÞ 1

where we have introduced the static structure factor D E 1 ð1  do Sðq; oÞ: SðqÞ ¼ drq ð0Þdrq ð0Þ ¼ 2p 1

½1:3

This corresponds to the zeroth moment of the dynamic structure factor, In the low-q limit, appropriate for light-scattering experiments, the following relation holds for liquids:

M(0) S .

ð0Þ

MS ¼ Sðq ! 0Þ ¼

rKB T wT ; Mmol

½1:4

where Mmol is the molecular mass; kB, the Boltzmann constant; r, the mass density; and wT, the isothermal compressibility. We also recall the second (nonzero) moment of S(q,o), which is given by the relation

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ð2Þ MS

1 ¼ 2p

ð1 1

doo2 Sðq; oÞ ¼

q2 KB T : Mmol

½1:5

An equation of motion for Fq(t) can be written in the form of a generalized Langevin equation: ðt 0 @ 2 Fq ðtÞ 2 0 0 @Fq ðt Þ þ o ð q ÞF ð t Þ þ dt m ð t  t Þ ¼ 0; ½1:6 q q 0 @t 2 @t 0 0 Analogously, in the frequency space, the dynamic structure factor can be written as Sðq;oÞ ¼

1 2u20 q2  2 I o  o0 ðqÞ2  iomq ðoÞ ; o

½1:7

where I denotes the imaginary part. The parameter o0(q) introduced above is completely determined once the second sum rule for S(q,o) is fulfilled, and it turns out to be o20 ðqÞ ¼

ðqu0 Þ2 ; SðqÞ

½1:8

where u0 is the classical thermal speed, which is u20 ¼ KBT/Mmol. In Eqs. (2.6) and (2.7), we have introduced the second memory function, mq(t) (mq(o)), of the so-called Zwanzig–Mori expansion of Fq(t) (S(q,o)). Alternatively, these equations can be simply considered as relations that define mq(t). As a matter of fact, the real advantage of these equations is that the introduction of simple models for mq(t)—rather than for Fq(t)— guarantees that at least the first two nonzero spectral moments of S(q,o) are always respected. Clearly, at this stage, these equations, being formal expressions, correctly describe the atomic dynamics at any time (frequency) and wavenumber. According to the different kind of dynamics which is probed, the (q,o) space is usually divided into different regions. Two quantities can be introduced as reference points in such a space: the average intermolecular distance, d, and the characteristic time, t, of the so-called structural relaxation. Among all the relaxations that can be active in a liquid system, and which mirror its different available dissipative processes, the structural relaxation plays a particularly important role as it is related to the cooperative processes by which the local structure, after being perturbed by an external disturbance or by a spontaneous fluctuation, rearranges toward a new equilibrium position. This relaxation is intimately related to the many-body effects that

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differentiate liquids from, for example, diluted gases, and can be considered as a sort of fingerprint of the liquid state. With reference to d and t, we can introduce two regions of the (q,o) space that are relevant for the discussion that follows. i. The conditions qd  1 and ot  1 define the region of the (q,o) space where simple hydrodynamics holds. In this region, the fluctuations are collision-dominated. Here, the changes in the liquid structure induced by the density fluctuation are supposed to take place sufficiently slowly for the system to be considered in a state of local thermodynamic equilibrium. Under this condition and in the continuum limit, it is possible to obtain a closed set of equations describing the space–time variations of the conserved variables, namely the particle number and the current and energy densities. This description becomes explicit when the values of appropriate thermodynamic derivatives and transport coefficients are specified. The results of such calculations are available in analytical form and can be expressed by the Langevin equation with the following memory function: mq!0 ðtÞ ¼ o20 ðqÞ½g  1eDT q t þ 2nl q2 dðt Þ; 2

½1:9

where g ¼ CP/CV is the constant pressure to constant volume-specific heat ratio; DT ¼ k/(rCV), where k is the thermal conductivity and nl is the kinematic longitudinal viscosity. ii. If condition ot  1 fails and qd  1, one enters the region of the so-called molecular hydrodynamics [1]. In this region, the atomic dynamics is influenced both by structural and relaxational effects. For what concerns the former, they come directly into play through the structure factor, S(q). For what concerns the latter, on the other hand, an appropriate viscoelastic model has to be introduced. In fact, if an external disturbance is applied to a liquid system, the observed response depends on the relative duration of the perturbation o1 as compared to the relaxation time t. If ot  1, the system can respond to the perturbation, and quickly takes up a new configuration; this corresponds to the simple hydrodynamic region previously referred to. If, conversely, ot  1, the system has no time to respond before the perturbation is removed and the preexisting equilibrium state is unchanged; in this situation, the system behaves as a solid, and actually the condition ot  1 can be used to define the purely elastic state. Measurements made in between these two time scales, that is, in the relaxation region, enable one to determine the relaxation time. The smooth transition from the simple hydrodynamic regime to the molecular one is the main argument used to extend the hydrodynamic

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description by retaining the formal structure of the equations, but replacing the thermodynamic derivatives and the transport coefficients with functions which can vary both in space (or wavenumber) and time (or frequency). Thus, Eq. (1.9) can be generalized in the following way [1]: mq ðtÞ ¼ o20 ðqÞ½gðqÞ  1eDT ðqÞq t þ Kl ðq;tÞ; 2

½1:10

where g(q) and DT(q) are the q-dependent generalizations of the corresponding thermodynamic quantities, and Kl(q,t) is the corresponding q- and t-dependent generalization of the longitudinal kinematic viscosity. In fact, the requirement that Eq. (1.10) joins Eq. (1.9) in the simple hydrodynamic regime imposes the following condition: ð1 lim dtKl ðq; tÞ ¼ q2 nl : ½1:11 q!0 0

In addition to Eq. (1.11), further constraints can be imposed on mq(t) by the knowledge of higher spectral moments. It has to be emphasized that the formulation presented here is a generalization of the formalism used at low q and, thus, does not take into account the transverse contributions that are known to enter the density–density dynamics of liquids at high q [3,10]. This latter effect can, in fact, be expected at q values sufficiently high such that the concept of pure transverse or longitudinal character of the modes in a liquid system begins to lose significance. Equations (2.7) and (2.10) establish a formal connection between the S(q,o) and the q- and o-dependent generalization of the longitudinal kinematic viscosity. A similar connection can be established between the S(q,o) and the generalized elastic moduli. In a macroscopic formulation, the elastic moduli are defined as the coupling constants between the components of the (macroscopic) stress tensor and those of the (macroscopic) strain tensor. For an isotropic system, there are two independent elastic moduli: the shear modulus G and the bulk modulus K. The longitudinal modulus M is related to them by the expression M ¼ K þ 4G/3. The concept of elastic properties can be applied to a fluid system only in the limit of high frequencies. In fact, at low frequencies, the response of a liquid is viscous in nature, and the stress tensor is instead coupled to the rate of strain (Newton’s law), where in this case, the shear  and the bulk b viscosity coefficients play the role of coupling constants. A particularly useful way of describing both the elastic and viscous properties of a fluid is introducing a frequency-dependent stress-to-strain relation, which reduces to the standard equations for a solid and a fluid in the appropriate limits [11]. This can be accomplished by introducing a frequency

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dependence in the viscosities and in the elastic moduli, the connection between them being GðoÞ ¼ ioðoÞ

½1:12

K ðoÞ ¼ K0 þ iob ðoÞ

½1:13

and

where K0 is the zero frequency limit of the bulk modulus and is given by the inverse of the adiabatic compressibility ws. The simplest ansatz for the frequency dependence of the viscosity is the one corresponding, in time domain, to a single exponential decay. This is the basis of the Maxwell theory of viscoelasticity. A more complicated ansatz will be introduced in the following sections to take into account the complex phenomenology of the glass transition. With the formal connection between generalized viscosity and elastic moduli contained in Eqs. (2.12) and (2.13), we can write 00

Sq M0 M ðoÞ Iq ðoÞ / Sq ðoÞ ¼ ; 0 po ½M ðoÞ  ro2 =q2 2 þ ½M 00 ðoÞ2

½1:14

where S(q) is the static structure factor, r the mass density, M0 the relaxed (lowfrequency) longitudinal acoustic modulus, M*(o) ¼ M0 (o) þ iM00 (o) the generalized longitudinal acoustic modulus, and q the exchanged momentum. It is worth noting that S(q,o) written in terms of generalized viscosity is typical of liquid-state theories, while the generalized modulus is conventionally used for solid-like approaches. Studying the structural relaxation, that is, the dynamic transition from solid- to liquid-like behavior, both formalisms can be found in the literature. In the following section, the relaxation of density fluctuations is described in terms of the generalized longitudinal modulus.

2.2. Brillouin scattering cross-section The main characteristics of the Brillouin light and X-ray scattering crosssections are recalled here. The reader is encouraged to see the well-known monographs [12,13] for a comprehensive presentation. An ideal photon-scattering experiment can be schematized as follows: a monochromatic beam of radiation (the probe) with frequency (wavenumber, polarization) oi(ki,ei) impinges on a sample (the target) and the radiation with

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frequency of  Do/2 scattered at an angle 2y with respect to the incident beam is detected within a solid angle DO. The total Hamiltonian reads ^ tot ¼ H ^p þ H ^ t þ V^ int H

½1:15

where Hˆp (Hˆt) is the unperturbed probe (target) Hamiltonian and V^ is the interaction potential. For the sake of simplicity, the target can be schematized as a nonabsorbing system made up of N distinct molecules and Ntot particles with charge qi, momentum p^i , and mass Mi. The probe is an electromagnetic ˆ . Then, neglecting the magnetic field specified by the potential vector A terms, the interaction potential is [14] Ntot 2 Ntot 1 X qi ^ 2 1X qi ^ V^ int ¼ 2 p^ Aðri ;t Þ; A ðri ; t Þ  2c i¼1 Mi c i¼1 Mi i

½1:16

where c is the speed of light. It is possible to calculate the cross-section for the scattering process in which the target goes from the initial state |tii to the final state |tfi while the probe goes from the initial state |pii, specified by its frequency oi and wavenumber ki to a final state (of,kf) within the frequency and solid angle intervals Do and DO. One photon scattering, which is the process of interest here, is described ˆ 2. Therefore, the first term of the interaction potential gives a by terms in A contribution in a perturbation treatment to first order, while the second one shall be considered to second order. The calculation is usually carried out using the following assumptions: (i) the electromagnetic field is approximated as a plane wave both when it gets on the target and when it is scattered within the (small) solid angle DO and (ii) the N molecules that constitute the system, each with center of mass in Rl, are treated within the usual Born–Oppenheimer approximation, which (parametrically) separates the nuclear degrees of freedom from the electronic ones. The calculation is then performed assuming further approximations appropriate to the wavelength range of interest, and different results are obtained depending on whether radiation is in the visible or in the X-ray range. If the X-ray range is considered, it is possible to exploit the fact that the probe energy is much higher than the typical energy differences between electronic levels; as a consequence, only the first term of the interaction potential makes a significant contribution to the cross-section, which reduces to the classical Thomson expression. Explicitly, the scattering cross-section in the X-ray range reads

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy



 2 @2s of  ¼ r02 ei ef  jf ðqÞj2 Sðq;oÞ; oi @o@O X

11

½1:17

where f(q) is the molecular form factor given by the (spatial) Fourier transform of the molecular electron charge density and r0 is the classical radius of the electron. Thus, the X-ray scattering cross-section is directly proportional to the dynamic structure factor, S(q,o). The presence of the molecular form factor plays an important role in the exploitation of X-ray scattering as a tool to measure the S(q,o). In fact, f(q) decreases monotonically with increasing q: at q ¼ 0, it is equal to Z, the total number of electrons in the molecule, and then it drops to zero on a scale fixed by the inverse of the average molecular size. Thus, IXS at high q is strongly suppressed, making the determination of S(q,o) difficult at q roughly higher than the position of the main peak in the structure factor. If the visible range is considered, in turn, the following assumptions are usually made, namely (i) the dipole approximation, thus neglecting the phase differences for the light scattered from different parts of the same molecule, and (ii) the probe energy is much lower than the typical energy differences between electronic levels. In this case, the two terms of the interaction potential Eq. (1.16) make contributions of similar amplitude, and combine together to give  2  ð D E @ s oi o3f X a b g   g  1 iot e egd ðq; 0Þ ; P ¼ e e ð e Þ e dt e ð q;t Þ P ab i f f i @o@O L 2Npc 4 abgd 1 ½1:18 where the statistical average has to be performed, in the spirit of the Born–Oppenheimer approximation, first on the electronic states, which depend parametrically on the nuclear states, and then on these latter states. In this equation, Peab ðq; tÞ is the space Fourier transform of the total polarizability tensor: ! ð N X l e Peab ðq;t Þ ¼ dreiqr aab ðfRl ðtÞgÞdðr  Rl ðtÞÞ ¼

N X

l¼1

e alab eiq Rl ðtÞ ;

½1:19

l¼1

where e al ab ðfRl ðtÞgÞ is the effective polarizability tensor of the lth molecule. Basically, this is a true many-body property that depends on the position of

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all molecules in the target. Thus, the light-scattering cross-section is proportional to the (many particles) polarizability correlation function and not to the (two particles) density correlation function. In Eq. (1.19), e alab ðfRl ðtÞgÞ is not the bare polarizability that characterizes a molecule in vacuum, but a polarizability renormalized by the many-body interactions with all the other entities in the target. The bare polarizability, e alab , can be expressed as e alab ¼ a þ blab , where a is its isotropic component and blab its traceless component. Moreover, if we call ðDe aÞlab the renormalization contribution, we obtain e aÞlab : alab ¼ a þ blab þ ðDe

½1:20

The first, isotropic term, a, in Eq. (1.20) does not depend on the molecular coordinates and its self-correlation function contributes to the cross-section through  2  oi o3f 2   2 @ s ¼ a ei ef  Sðq; oÞ; ½1:21 @o@O L;iso 2N pc 4 and thus, as in the X-ray scattering case, through a term proportional to the dynamic structure factor. This term appears only in the polarized configuration, that is, when ei||ef. The second term blab in Eq. (1.20) depends only on the orientational degrees of freedom, while the third term ðDe aÞlab arises mainly, in simple molecular systems, from the renormalization effects on the bare polarizability of the lth molecule coming from the local electromagnetic fields produced by the multipoles of the other molecules of the system. These multipoles, in turn, are induced both by the external field (probe) and by the local electromagnetic field. To first order in the deviation of the local field to the external one, and in the further hypothesis of an almost symmetrical bare polarizability (||b||  a), the result for the effective polarizability is [15] X ð2Þ De alab ðDIDÞ ¼ a2 ½1:22 Tab ðl; l0 Þ; l6¼l0

which is known as the pure dipole-induced dipole (DID) contribution to 0 the renormalized polarizability. Here, T(2) ab (l,l ) represents the usual dipole 12 propagator. The relative spectral contribution of the two tensors blab and l De aab ðDIDÞ, both of which are traceless, second rank tensors, depends on the details of the molecular structure. In general, however, they give rise

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to two contributions in the spectrum: one q-dependent and the other q-independent. The q-independent contribution appears in the spectra, on a qualitative ground, as a smooth band with a temperature-dependent width and with tails that extend up to at least a few THz, that is, up to the frequency range where the optical bands appear in most organic liquids. This contribution is fully depolarized, that is, the ratio r (depolarization ratio) of its intensity in the depolarized configuration (ei? ef) to the intensity in the polarized one (eikef) is fixed and equal to 0.75. The q-dependent spectral feature reflects, most prominently, the transverse dynamics in supercooled liquids and glassy states [16]. This contribution is forbidden by a selection rule in the polarized spectrum and its intensity in the depolarized spectrum depends on the scattering angle, which is zero in

the backscattering (2y ¼ 180 ) geometry. In addition, the q-dependent spectral feature gives rise, in supercooled liquids and glassy states, to a contribution that reflects the longitudinal dynamics and that adds up in the isotropic spectrum [16]. The intensity of this contribution, however, depends on the degree of rotation–translation coupling in the considered liquid and, usually, small, dedicated experiments are required to dig it out of the dominating isotropic contribution. In principle, then, by measuring light-scattering spectra in the depolarized and polarized configurations, by multiplying the former by 0.75, and then, by subtracting the former from the latter, the dynamic structure factor can be obtained. It has to be pointed out that among all the possible induction mechanisms, the (pure) DID (which is, however, by far the dominant one in nonionic systems) gives rise to a pure, second-rank, renormalized polarizability tensor. It is possible, however, that other induced effects, each with its peculiar spectral shape, affect the polarized and depolarized spectra in a different fashion: r can no longer be 0.75 and, moreover, it can also be frequency-dependent. If this is the case, obviously, the previously discussed procedure to determine the S(q,o) spectrum from the polarized and depolarized ones becomes unreliable. Only a careful experimental analysis of the polarized and depolarized spectra can establish whether the otherthan-DID-induced effects, with their eventual isotropic component, are effectively active in the system under examination. Summarizing, in molecular liquids with weak rotation–translation coupling strength, the polarized Brillouin spectrum in the backscattering configuration, Ik, is, to a good approximation, the sum of a q-dependent isotropic term, IISO, proportional to the S(q,o) and of a q-independent anisotropic contribution, I?, while the fully depolarized spectrum consists only

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of the latter contribution. It is always possible to derive the isotropic spectrum as IISO ¼ I  r 1 I? ;

½1:23

where r is the depolarization ratio. The S(q,o) contribution to Ik usually goes to zero at frequencies higher than about 40 GHz so that one can check whether, above this limit, the Ik and I? spectra have the same shape and whether the direct measurement of r actually gives 0.75. If this is the case, one can reasonably assume that the isotropic-induced effects have a negligible weight and that the IISO spectrum is actually proportional to the S(q,o) one. In both cases, that is, X-ray and light scattering, the kinematics of the process imposes, with high accuracy due to the small value of (oi  of)/oi, the following relation:



4pn y q¼ sin ; ½1:24 l 2 where n is the refractive index, l the wavelength of the incident beam, and 2y the scattering angle between the incident and scattered light directions.

3. EXPERIMENTAL 3.1. BLS: The tandem multipass Fabry–Perot interferometer The tandem multipass Fabry–Perot interferometer (T-FPI) was developed in its actual configuration more than 30 years ago by John Sandercock [17], and the detailed description of the setup and of its parts can be found elsewhere [18,19]. This interferometer, due to its high contrast (>1010), ensured by the multipass operation, and to the wide frequency range accessible without annoying replicas from higher orders of the Fabry–Perot, thanks to the tandem configuration, made it possible to measure surface acoustic and magnetic excitations in solids. BLS from bulk systems, although developed many years before the advent of T-FPI, has benefited greatly from the development of this new interferometer as it allows considerable enlargement of the accessible frequency range, so that full spectrum analysis methods can now be applied to study the spectrum of collective density fluctuations (Section 2.1). The accessible frequency range can be estimated from the values of the instrumental resolution and of the scattering geometry.

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15

In fact, as reported in Section 2.1, BLS deals with light inelastically scattered by density fluctuations related to thermally activated acoustic modes. Brillouin peaks can be revealed at frequency shifts approximately given by o ¼ vq, where v is the velocity of the acoustic modes, q ¼ 2nkisin(y/2) is the momentum exchanged (see Eq. 1.24) in the scattering process, n is the refractive index of the sample, and ki is the wave vector of incident light. Samples with longitudinal acoustic modes propagating at 2–3 km/s give rise to Brillouin peaks at about 10–20 GHz in the backscattering configuration of Fig. 1.2. As the intensity of the spectrum quickly drops to zero after the Brillouin peak, the backscattering configuration is the most convenient because it corresponds to the highest possible value for the peak frequency position. In addition to the use of a single lens for focusing and collecting light from the sample, it maximizes the scattering volume and minimizes the alignment efforts and the error in the exchanged q, which, at first order, is given by Dq  nki cos(y/2)Dy. Going to a higher

Figure 1.2 Schematic diagram of a BLS setup in the backscattering configuration (y ¼ 180 ) available at the GHOST laboratory in Perugia (http://ghost.fisica.unipg.it). In this geometry, the same lens is used to focus polarized light from a single-mode 514.5 nm Arþ laser onto the sample and to collect the backscattered photons for analysis through the tandem-multipass Fabry–Perot interferometer. The polarized spectrum, Ik, is obtained by setting parallel directions for both polarizer and analyzer, while the depolarized spectrum, I?, is obtained by using orthogonal polarizations. The light passing through the analyzer and filtered by the interferometer is detected by a photomultiplier working as a photon counter and the resulting signals are stored and processed by a dedicated data acquisition system. [20].

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Lucia Comez et al.

Intensity (arb. units)

106

105

104

I//

T = 333 K

IISO I^

103

102 0.1

1

10

100

Frequency (GHz)

Figure 1.3 Typical log–log plot of Ik, I?, and IISO Brillouin spectra for an epoxy resin at T ¼ 333 K [21]. The IISO is obtained by subtracting the appropriate anisotropic scattering contribution from the Ik spectrum (Eq. 2.23).

frequency is possible only by increasing ki, that is, by IUVS and IXS described in the following sections. Conversely, the lowest accessible frequency is fixed by the frequency resolution of the interferometer that, for the T-FPI, is fixed at about 100–200 MHz. Typical spectra acquired in polarized and depolarized configurations (see Section 2.2) are shown in Fig. 1.3, in a log–log plot, which emphasizes the wide frequency and intensity range accessible to the T-FPI.

3.2. Inelastic ultraviolet scattering Stringent requirements have to be fulfilled in order to make IUVS experiments in the mesoscopic kinematic region feasible. In particular, it is mandatory to have (i) an incident photon energy in the 5–11 eV region, (ii) an incident photon flux on the sample larger than 1011 photons/s, and (iii) a resolving power larger than 105. Because of the high flux needed, the radiation source for the IUVS beamline at ELETTRA (see Fig. 1.4) has to give at least 1015 photons/s/0.1% BW in the desired energy range. This calls for an undulator of the maximum length compatible with available length in the straight sections of the storage ring (namely 4.5 m). This automatically implies a very high emitted power and power density, which can be harmful to the optical elements of the beamline. For this reason, an exotic insertion device, the figure-of-8 undulator [22], has been constructed [23] as an alternative to the standard vertical field devices. The main advantage is a much reduced on-axis power density that is obtained with no penalty on the useful photon flux.

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

Sync.

Heat load + focusing

Figure-8 undulator

NIM monochromator

Sample

2q = 172°

Focusing mirror

Collection mirror

NIM analyzer

Figure 1.4 Schematic diagram of the instrument available at the IUVS beamline at the Elettra synchrotron in Trieste (http://www.elettra.trieste.it/elettra-beamlines/iuvs.html).

Using a 32-mm period figure-of-8 undulator with maximum deflection parameters Kx ¼ 3.4 and Ky ¼ 9.4, at the exit of a 600 600 mrad2 pinhole, the total power of the synchrotron radiation is reduced to about 20 W while the first harmonic delivers 2 1015 photons/s/0.1% BW. The beam coming from the undulator must be cleaned from the high-order harmonics and, for this reason, three reflections have been used, also allowing the transfer of the radiation into the monochromator stage. More specifically the beam impinges on a gold-coated GLIDCOP mirror internally water-cooled, which deviates the photons in the vertical plane with an angle of 6 . A second externally watercooled silicon mirror is used to bring back the beam parallel to the floor and to filter high-energy harmonics. The beam is then focused by a spherical silicon mirror onto the entrance slits of the monochromator with a demagnification M ¼ 20:1 and an incident angle of 85 . This allows cutting all harmonics above 20 eV photon energy. With the source size roughly measuring 1 1 mm2, a spot of 50 200 mm2 (vertically, the astigmatism makes the focus larger) is obtained at the entrance of the monochromator. The Czerny–Turner normal incidence monochromator (NIM) optical design has been chosen for the monochromator [24]. This design is based on an entrance slit, a spherical mirror, which collimates and sends the beam to an echelle plane grating, and a second spherical mirror

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Lucia Comez et al.

that collects the diffracted beam and focuses it on the exit slit. The relative energy resolution, assuming that the intrinsic contribution coming from the grating is negligible, is given by the formula (DE/E) ¼ dcot’/2F, where d is the slit opening, F is the focal length of the spherical mirror, and ’ is the blaze angle. Using d ¼ 50 mm, F ¼ 8 m, and ’ ¼ 70 , we get a relative resolution of 1.1 106. We decided to build an 8-m-focal length monochromator to match the best compromise between the required resolving power and the mechanical feasibility. The grating used has 52 lines/mm and works at a blaze angle of 69 (DE/E ¼ 1.2 106). At the exit of the monochromator, the beam impinges on a spherical mirror, which focuses the radiation on the sample on a spot size of about 30 100 mm2. A second spherical mirror is used to collect the radiation scattered from the sample and send it to the entrance slit of the analyzer unit. The analyzer has the same design as the NIM. The energy scan can be carried out by rotating the diffraction grating and collecting the number of photons scattered at a given energy. All optics after the heat load units were coated by aluminum with an MgF2 protective coating, allowing very good reflectivity up to 11 eV. The photons can be detected by a photomultiplier or, if the energy range to be covered is relatively small (<0.1 meV), a position-sensitive detector can be used. The quantum efficiency of commercial detectors for incident energies in the 5–11 eV range is larger than 10%. The momentum transfer can be varied by changing the scattering angle, y, according to Eq. (1.24). The instrument energy resolution has been measured by collecting the isotropic scattered intensity from a high-roughness copper surface tilted with respect to the beam by about 40 . The energy resolution measurements obtained at 7.5 eV incident photon energy and a 50-mm opening for the monochromator and analyzer slits gave a total relative energy resolution of 2.4 106, indicating that the obtained performance is very close to the theoretical expectation 1.7 106 given by the convolution of the energy resolution of the analyzer and that of the monochromator.

3.3. Inelastic X-ray scattering Brillouin IXS experiments aim at probing the collective excitations in solids and liquids (energy on the order of 1 meV) by means of hard X-rays (energy on the order of 10 keV), and therefore need a resolving power DE/E of at least 107. It is then not surprising that these experiments require the use of intense X-ray beams produced by high-energy storage rings such as those in the ESRF in Europe, the APS in the United States, and SPring-8 in Japan. At the ESRF, beamline ID28 is currently optimized for this kind of experiment.

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

The basic principle of the setup used for IXS experiments resembles that of triple-axis neutron spectrometers, where the first axis selects the incident photon energy, the second one selects the scattering wavenumber, and the third axis selects the scattered photon energy. The schematic layout of this setup is shown in Fig. 1.5. The radiation source is composed of three 1.6-m-long undulators with a magnetic period of 32 mm. The undulator beam is characterized by a broad energy spectrum (DE/E  102) and thus has to be monochromatized before impinging on the sample. The monochromatization is achieved in two steps. The first step is obtained by means of an Si(1,1,1) double-crystal monochromator kept at cryogenic temperature. In fact, the thermal expansion coefficient of silicon is zero at 150 K, and therefore the operation of the crystals at cryogenic temperature minimizes spoiling the beam properties via the thermal deformation over the beam footprint caused by the absorption of part of the power coming from the undulator. This first monochromator provides DE/E  104, and therefore reduces the FWHM of the incident beam energy spectrum to 1 eV. The second step of monochromatization is achieved by means of a flat Si crystal monochromator working at a high reflection order and in the near-backscattering geometry, which reduces the previously referred FWHM to 1 meV. The combination of high reflection order and backscattering geometry allows one to obtain small-energy bandwidths and angular acceptances larger than the divergence of the X-ray beam, thus optimizing the throughput of the monochromator. Most of the IXS experiments on disordered systems have been carried out using the Si(11,11,11) or Si(12,12,12) reflections. Detectors Multilayer mirror Monochromatic beam

Backscattering monochromator

Mirror

75 m

70 m

Spherical analyzers Sample Collim. Be lens

Premonochromatic beam

65 m

60 m

55 m

High heat load pre-monch.

50 m

45 m

Undulator source

White beam

40 m

35 m

30 m

0m

Figure 1.5 Layout of the high-resolution IXS setup used at beamline ID16/ID28 at the ESRF (until the year 2011, both ID16 and ID28 having the same characteristics were used). The abscissa scale gives an idea of the global dimensions of the beamline (http://www.esrf.eu/UsersAndScience/Experiments/Beamlines/).

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Lucia Comez et al.

The monochromatized beam is then focused by a toroidal mirror at the sample stage, where it arrives with 100 300 mm2 transversal dimensions and 60 120 mrad2 angular divergence. The photons scattered by the sample are selected in energy by means of a set of nine crystal analyzers that work, schematically, like the monochromator (i.e., in the near-backscattering configuration and at the same reflection order as the monochromator) and reflect the selected photons onto the detector (Si diode). These crystal analyzers are mounted at the extremity of a 6.5-m long arm, which is used to select the momentum transfer q rotating in the horizontal scattering plane, with the center at the sample stage. The crystal analyzers are constructed gluing 104 independent Si crystal cubes of size c on a spherical Si substrate with radius R. The aim of this design is set by the necessity of collecting the scattered radiation over a sufficiently large solid angle (on the order of an mrad, compatible with the required q resolution) and with the highest possible energy resolution provided by the use of flat perfect crystals. Moreover, in the 1:1 Rowland geometry, the contribution to the energy resolution due to the finite dimensions of the cubes (Bragg angle variation) scales with c/R. With c  0.6 mm and R ¼ 6.5 m at a Bragg angle 89.98 , this contribution is smaller than the Darwin width of the Bragg reflections mentioned earlier, and the crystal analyzers reflect the scattered X-rays with almost the intrinsic energy bandwidth of the corresponding Bragg reflection. The energy scans are performed by varying the temperature of the monochromator with respect to that of the analyzer (which is kept constant), thus changing their relative d-spacings. The relative change in the d-spacings between monochromator and analyzer is given by Dd/d ¼ aTDT, where the coefficient of thermal expansion, aT, in silicon at 294 K is 2.52 106 K1. It follows that to obtain sufficiently small step sizes ( one-fifth of the energy resolution), the temperature of the two crystals has to be controlled with a precision of 0.5 mK. The instrumental functions obtained using the Si(11,11,11) and Si(12,12,12) Bragg reflections are characterized by an FWHM of 1.5 and 1.3 meV, respectively. This matches very closely the expected calculated values [29].

4. LIQUIDS AND GLASS TRANSITION 4.1. Introduction When a liquid is cooled below its melting temperature, it usually crystallizes. However, if the quenching rate is sufficiently fast and/or the system at molecular level is complex enough to hinder crystallization, it continues

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

21

to stay in a disordered state—the supercooled one—gradually becoming less fluid upon further cooling. When the time needed for the rearrangement of the local atomic structure reaches approximately one hundredth of a second, the system becomes “solid” for any practical intent: this condition conventionally identifies the glass transition temperature, Tg Fig. 1.6. Although some general aspects in the dynamics of different systems suggest that, behind material-specific aspects, some common basic interpretation should be found for the liquid to glass transition, a single unified view of the whole process does not yet exist that gives a rationale of the divergence of the structural relaxation time and the connected singularities starting from few basic and (possibly) microscopic properties of the liquid. One can find useful reviews [27,30,31] where different models developed over the past years are described, which are used to explain the temperature dependence of the structural relaxation time. At present, at least two main regimes are recognized in the supercooling process of a liquid: the first one corresponds to the early stage of supercooling, while the second is particular to the temperatures close to the arrested state. The final arrest of the system is described by potential energy landscape and configurational entropy concepts, relating the slowing down of diffusional dynamics to a progressive decrease of the number of equilibrium states for the system [27,32–36]. Conversely, in the early stage of supercooling, the energy landscape slightly influences the dynamics, which is dominated by kinetic effects. In this regime, approaching structural arrest from the liquid side, the motion of a molecule becomes more and more difficult due to the collisions with its nearest neighbors. In simple liquids, the cage effect arises from packing constraints, because any particle is entrapped and oscillates between the walls made of the surrounding particles until a transitory structural breakdown permits it to escape [37]. This cage effect (see Fig. 1.6) and its influence on density fluctuations and related dynamical variables have been successfully described by the modecoupling theory (MCT), introduced almost 20 years ago [38]. Light, UV, and X-ray Brillouin scattering, measuring the spectrum of density fluctuations Simple liquids

Covalent or associate liquids

Caging

Bonding

Figure 1.6 Schematic representation of caging effects.

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Lucia Comez et al.

in the dynamical region described by the theory, have been widely used to test MCT predictions. The most common way to immobilize the dynamics of a system and see this cage effect at work is by cooling down a liquid. However, a similar effect can also be obtained via alternative paths such as increasing pressure (physical vitrification) or through a chemical process (chemical vitrification) involving the formation of covalent bonds between adjacent molecules. The polymerization process of liquid monomers, in which the arrested state is obtained via the slowing down of diffusive motions caused by the spontaneous growth of macromolecules, represents an example of this last category. Testing the predictions of the theory through these alternative thermodynamic paths has been an important workbench to understand its potential and limits, to test its generality, and to suggest new views and challenges. In the following sections, after a brief overview of the phenomenology connected to the glass transition, results of Brillouin scattering investigations of glass-forming liquids are reported, with the emphasis on some peculiar behaviors, such as the divergence of the structural relaxation time, the occurrence of secondary relaxations and their interplay with the structural one, and the nonergodicity parameter. Different thermodynamic paths have been used as well as diverse possible ways to elaborate Brillouin spectra. In particular, from the simple determination of frequency position and width of Brillouin peaks, related to single-frequency elastic wave velocity and attenuation, up to the full spectrum analysis of BLS, IUVS, and IXS spectra, these methods make Brillouin scattering an elastic spectroscopy technique, capable of measuring the real and imaginary part of the elastic modulus in almost the entire range in Fig. 1.1.

4.2. Structural and secondary relaxations approaching the glass transition Condensed matter usually responds with some delay to an external perturbation. After the perturbation, the system relaxes toward the new equilibrium with one or more characteristic times t, given by the characteristic times of molecular motions [39]. Simple exponential relaxations are rare in condensed matter; more frequently, the complex dynamics taking place at the molecular level is better described by stretched exponential, or Kohlrausch–Williams–Watts (KWW), relaxation functions F ðtÞ ¼ F0 exp½ðt=tK ÞbK , with bK < 1 and tk and bk being the characteristic time and the stretching of the relaxation process, respectively [40]. If the perturbation is periodic, as for longitudinal acoustic modes, the presence

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

23

of a relaxation process causes absorption and dispersion in acoustic propagation, which can be described by a frequency-dependent elastic modulus. The most used phenomenological function that describes the effect of a stretched relaxation on the longitudinal elastic modulus is the Cole–Davidson (CD) function [41]: M  ðoÞ ¼ M1 

rD ð1 þ iotÞb

;

½1:25

where r is the mass density, t the relaxation time, b < 1 the stretching parameter, D ¼ (c21  c20) is the relaxation strength with c1 the unrelaxed and c0 the relaxed sound velocity, and M1 ¼ rc21 is the unrelaxed modulus. The average relaxation time of the CD function is hti ¼ bt. In low damping conditions, the apparent longitudinal sound velocity c and attenuation a can be obtained at a given frequency o from the real M0 and imaginary M00 parts of the longitudinal modulus through the relationships c ¼ (M0 /r)1/2 and a ¼ oM00 /(2rc3). Though CD is not the exact Fourier transform of the KWW, an approximate numerical conversion can be defined between relaxation times and stretching parameters of the two functions [42]. If different motions at molecular level couple with the density fluctuations associated with acoustic modes, two or more relaxation functions must be added to reconstruct the frequency dependence of M *, giving rise to rather complex relaxation patterns. Due to the different coupling mechanisms with microscopic motions, it is common that different applied perturbations (electric field rather than magnetic field or elastic stress) show different relaxation maps. Among the variety of relaxation processes, one exhibits a certain character of generality. This is the structural, or a, relaxation process, existing in all liquids and dense fluids in a frequency region that is almost independent from the nature of the applied perturbation, and typically associated to the collective structural rearrangement of groups of atoms or molecules. This relaxation manifests strong temperature dependence, being generally proportional to the shear viscosity, thus becoming slower and slower and growing by several orders of magnitude in characteristic time, driving the system toward the glassy state. The condition in which the structural relaxation time equals a given value, arbitrarily fixed to 100 s, conventionally defines the glass transition. 4.2.1 Divergence of the structural relaxation The huge increase in the structural relaxation time ta(T ) and viscosity (T ), both rising by  14 orders of magnitude between the melting temperatures Tm and the glass transition temperature Tg, represents the most salient and

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Lucia Comez et al.

best known signature of the approach to the liquid–glass transition. These two quantities are usually proportional to each other, as suggested by the Maxwell theory of viscoelasticity in which ta ¼ /G1, where G1 is the high-frequency shear modulus. During the transition from normal liquid to glass, the microscopic molecular motion of particles changes in nature, reflecting on the changes of the temperature behavior of ta[26]: it generally passes from an Arrhenius, thermally activated behavior: t ¼ t0 eEA =KB T

½1:26

where EA is the activation energy, to a steeper (super-Arrhenius) temperature dependence, dominated by collisions and cage effects. According to MCT, in this regime, a power-law [34] t / ðT  Tc Þg

½1:27

better describes the divergence of the relaxation time. In this regime, an a-scale universality is predicted by MCT, that is, the same temperature dependence of the relaxation times relative to all the physical observables that couples with density fluctuations. On decreasing the temperature further, the collisional regime progressively gives way to more cooperative rearrangements. In this landscape-dominated regime, the structural relaxation is frequently rationalized in terms of the Vogel–Fulcher–Tammann (VFT) equation [43]: t ¼ t0 eBT0 =ðT T0 Þ

½1:28

where t0, B, and T0 (usually located several degrees below Tg) are constants Fig. 1.7. Depending on the liquid, glass transitions may be observed occurring over an enormous range in temperature, from below 50 K to above 1500 K. Moreover, some processes appear as “sharp” (indicating a narrow glass transformation range) and others are instead wide-ranging in temperature. One reason for this difference may be ascribed to the “fragility” of the glass former, a quality related to the deviation of the relaxation time temperature dependence from the Arrhenius behavior [25,26]. This deviation determines the steepness of the Arrhenius plot near Tg, and therefore the “sharpness” of the glass transition. The parameter B in the VFT equation is somehow related to the fragility of the system: the higher the values of B, the sharper is the transition.

25

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

log[h (poise)]

14 12

SiO2

10

Glycerol oTP

8 ng

ro

6

St

4 2 ile

0

ag Fr

-2 -4 0.0

0.2

0.4

0.6

0.8

1.0

Tg / T

Figure 1.7 The liquid fragility. The viscosity (proportional to the relaxation time) behavior of different glass-forming liquids is plotted as a function of temperature scaled to the glass transition (Tg). Strong network-forming liquids have an Arrhenius-like (linear) viscosity dependence, while fragile liquids show a dramatic change in viscosity as a function of temperature. [25–27].

Accordingly, glass-forming systems have been classified into two big families—“strong” and “fragile” glass formers, the “intermediate” systems being in between—and are represented on the well-known “Angell-plot” [25,26], where the viscosity of several glass-forming liquids is plotted as a function of temperature scaled to the glass transition temperature (Tg), and where strong liquids exhibit an approximately linear behavior, while fragile liquids exhibit super-Arrhenius behavior (see Fig. 1.7). Strong liquids, such as the networkforming SiO2 and germanium dioxide (GeO2), have tetrahedrally coordinated structures, while fragile liquids have nondirectional forces among molecules. A further phenomenological parameter introduced to differentiate “long and short” glasses is the kinetic “fragility” m measuring the slope of log [(1/T)] across the glass transition [26,27]: d logðÞ  m ¼ lim  T !Tg d Tg =T

½1:29

The lowest fragility value is found around m ¼ 17, corresponding to “strong” liquids. Conversely, for the most “fragile” systems, it is empirically found that m 150. A nontrivial correlation has been recently proposed between the elastic properties of glasses in the high-frequency regime detected

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Lucia Comez et al.

by Brillouin scattering techniques and the kinetic fragility m, which is, conversely, related to the very low-frequency (fractions of hertz) viscous properties of a liquid close to glass formation [44]. This correlation has been the starting point of intense research, which has, however, led to different and sometimes controversial findings [45–50]. Some recent results are discussed in Section 4.6. 4.2.2 Secondary relaxations The main a process may be accompanied by the occurrence of other processes, of inter- or intramolecular nature, which depends on the nature of the specific system, and are different in simple or molecular liquids, polymers, and mixtures Fig. 1.8. These are the secondary relaxations usually referred to as b, g, . . . , following the tradition of polymer physics. They have a weaker temperature dependence compared to the a relaxation. Approaching the glass transition, the increase in dynamically different regimes of fast relaxations, deriving from local fluctuations, and a-relaxation, involving cooperative rearranging regions, results in a bifurcation of relaxation time scales [51–53]. The small-scale motions generally associated with secondary relaxations exhibit a simple Arrhenius temperature dependence (see Eq. 1.26) that survives through the glass transition (see Fig. 1.8). Moreover, in relatively simple glass-forming systems, at least one secondary

12 10

Epoxy resin

log(1/t)

8 6

a

g

4 b

2 0 Tg = 254 K

-2 2

3

4

5 6 1000 (T)

7

8

Figure 1.8 Arrhenius plot of relaxation times for an epoxy resin [28]. Structural (a) and secondary (b and g) processes measured by dielectric spectroscopy are visualized.

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

27

relaxation is typically found, the so-called b Johari–Goldstain (bJG) process [54–57]. The bifurcation of a and b relaxation times usually intervene in a narrow frequency/temperature range defining a distinct region in the Arrhenius diagram above Tg, typically called crossover region, where a change in the temperature behavior of ta may also be found. More complex situations, with two or more crossover regions can be observed, depending on the specific systems under study [28,58]. The sensitivity of Brillouin scattering to secondary relaxation in supercooled liquids and glasses is shown in Section 4.4 and a significant case of ab bifurcation in water-supercooled solutions is reported in Section 4.5. It is to be noticed that the existence of these intramolecular secondary relaxations, which may appear in the spectrum of density fluctuations of simple and complex molecular liquids, is not explicitly taken into account by MCT. MCT indeed relies only on the behavior of the a relaxation and of a b region [38] that is the short time precursor of the structural relaxation, and which is frequently masked by strong secondary relaxations, as shown in Section 4.6. This is the reason why it has become indispensable to optimize a fitting procedure of the spectrum of density fluctuations measured by Brillouin techniques using phenomenological relaxation functions including structural and secondary relaxations, rather than by an MCT-only density correlation function, as in early BLS tests of the theory [30]. In fact, as exposed in Section 4.6, the prediction of the theory is robust enough to correctly describe the structural relaxation parameters, once the effects of the secondary ones are correctly taken into account. 4.2.3 The nonergodicity factor Within standard MCT models [59,60], the ideal liquid–glass transition is also signaled by a square-root singularity of the nonergodicity factor fq, a parameter that measures the strength of the structural relaxation. MCT associates the glass transition to a transition from ergodic (liquid) states, where the correlation functions of density fluctuations decay to 0 for long times ’q(t ! 1) ¼ 0 to nonergodic (glassy) states, where density fluctuations are frozen to a nonequilibrium value, ’q(t ! 1) ¼ fq, 0 < fq < 1, the nonergodicity factor. The arrest of density fluctuations can be induced by keeping all external parameters fixed except for one, called z, which can be temperature T, pressure P, chemical conversion a, etc. Therefore, according to the MCT theory, a nonergodicity transition occurs if z passes through a critical value zc. Leading-order solutions of MCT equations, near the critical point, can be expressed in terms of the small parameter e ¼ (zc  z)/zc. As a central result of the theory, the nonergodicity factor at any wave vector q is predicted to

28

Lucia Comez et al.

changep from to a square-root z-dependence fq ðzÞ ¼ ffiffiffiffiffi a constant for liquid state c c fq þ hq jej for glass states. Here, fq is the critical nonergodicity parameter and hq the critical amplitude at fixed wave vector q, which are expected to be in phase and in antiphase with the static structure factor, S(q), respectively. It is important to notice that the critical point for the nonergodicity transition is the same where the power law divergence of the relaxation time is expected (see previous paragraph) and that close to this critical point, the relaxation parameters, like power law exponents and stretching parameters, are related to one another and, in principle, obtainable from the static structure factor only. In this sense, MCT is a “microscopic” theory of glass-forming systems [38]. In the ideal case of a relaxation-only systems, the value of this factor is given by the limiting sound velocities, fq ¼ 1  c20/c21, obtained by ultrasonic measurements and by the frequency position of Brillouin lines taken in the appropriate relaxed and unrelaxed limits. Equivalently, if the spectrum of density fluctuations is taken in the IXS regime, where the width of the central line is larger than the characteristic rate of the structural relaxation, the nonergodicity parameter can be obtained by the ratio of the intensity of the central line over the total integrated intensity of the spectrum, and the q dependence of fq can also be explored by changing the scattering geometry [44]. In fact, in this condition, the “frozen” component of the density fluctuations contributes to the quasi-elastic scattering only. The earliest MCT experimental works from the early 1990s concern the comparison between the leading-order asymptotic formulae derived for the dynamics near glass transition singularities and the results of neutron scattering, depolarized light scattering, impulsive stimulated light scattering, and dielectric-loss spectroscopy for conventional liquids. Although MCT was originally conceived for rather simple systems, experimental verifications have been progressively directed toward more complex systems to test the universality of the theory. In the following section, this aspect is emphasized, one reason being that the progress and development of highfrequency and high-q Brillouin techniques (IXS and IUVS) have proved to be a fundamental tool for the experimental tests of MCT in real systems.

4.3. Brillouin scattering and relaxation processes through the liquid–glass transition The study of the whole relaxation dynamics and the experimental determination of the relaxation times, strengths, and stretching parameters during the glass transformation necessarily require broadband spectra and/or the

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

29

use of complementary techniques, such as BLS, IUVS, and IXS, to be addressed. The isotropic spectrum, proportional to the dynamic structure factor, S (q,o), was derived in Section 2.1 within the framework of generalized hydrodynamics, and can be written as 00

I0 M ðoÞ Iq ðoÞ ¼ : 0 o ½M ðoÞ  ro2 =q2 2 þ ½M 00 ðoÞ2

½1:30

The occurrence of relaxation processes can be taken into account by the o-dependent generalized modulus, which can be written as M∗(o) ¼ M1  DM∗(o)þio1, where M1 is the high-frequency (unrelaxed) longitudinal modulus; DM∗(o), the relaxing part of the generalized modulus; M0 (o) and M00 ’(o), the real and the imaginary part of M∗(o), respectively; and 1/r, the high-frequency (unrelaxed) longitudinal kinematic viscosity [61,62]. This latter term contains the effect of relaxation processes with characteristic rates far above the experimental frequency window, including those of nondynamic origin [63], and this is usually referred to as microscopic term or instantaneous contribution. DM*(o) can be expressed by one or more phenomenological functions, depending on the complexity of the relaxation pattern affecting the investigated Brillouin frequency window. The simple case of an a relaxation-only scenario can be described by a single CD relaxation function, as reported in Eq. (1.25). If in the frequency windows covered by BS experiments more relaxation processes affect the spectra of density fluctuations, a different phenomenological function must be added to properly build M∗(o); for instance, a minimal way to express the frequency dependence of the complex modulus considering two relaxation processes and an “instantaneous” contribution is given by M  ðoÞ ¼ M1 

rD1 ð1 þ iot1 Þ

b1



rD2 ð1 þ iot2 Þb2

þ io1 :

½1:31

In Fig. 1.9, real and imaginary parts of the complex longitudinal moduli are represented at a given temperature for a glass former characterized by two relaxation processes, where the limiting relaxed and unrelaxed areas and the intermediate relaxing region can be recognized. A schematic remanding to the location of frequencies probed by BLS, IUVS, and IXS helps to visualize the potentiality of a combined Brillouin investigation.

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Lucia Comez et al.

US 10-3

10-5

10-1

BLS IUVS IXS 103 101

Instantaneous contribution

M0 a

10-5

10-3

b

10-1 101 Frequency (GHz)

M¢¢

M¢

M¥

103

Figure 1.9 Schematic representation of real and imaginary parts of the longitudinal acoustic moduli determined using Eq. (1.31) and the parameters of Ref. [62] for polybutadiene at 270 K.

Though a full spectrum analysis of BLS, IUVS, and IXS spectra obtained on the same system at a given temperature is the best way to gain a rather complete relaxation map of a given system, some useful information can also be obtained by a cheaper and faster single-frequency procedure, which is the most ancient way used to elaborate Brillouin spectra. It consists in fitting the spectra around Brillouin peaks and deducing the sound velocity and attenuation from the frequency position and linewidth of these peaks. To this aim, it can be seen that in the frequency region around the peak, if the width of the peak is considerably narrower than the dispersion region, M0 and M00 can be considered constant and the formula for the scattered light can be approximated by that of a damped harmonic oscillator (DHO): Iq ðoÞ ¼

I0 Go2LA : p ½o2LA  o2  þ ½oG2

½1:32

In Eq. (1.32), oLA/2p and G/2p approximately correspond to the frequency position and to the linewidth (FWHM) of the Brillouin peaks [21, 61] and are related to the real and imaginary parts of the modulus at the frequency of the Brillouin peak through the relationships o2LA ¼ q2M0 (oLA)/r and G ¼ q2M00 (oLA)/roLA. Moreover, they give the apparent longitudinal sound velocity cL and the apparent longitudinal kinematic viscosity D

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

through cL ¼ oLA/q and D ¼ L/r ¼ G/q2. Making measurements as a function of, for instance, temperature, the relaxation processes move in frequency following the VFT (Eq. 1.28) or the Arrhenius (Eq. 1.26) laws, and the effect of velocity dispersion and absorption can be revealed by the temperature dependence of c and D. We call this investigation scheme “acoustic analysis,” a heritage of the ultrasonic investigation of liquids and solids. Conversely, looking at the low-frequency limit of Eq. (1.30), the so-called Mountain region [21], one can easily see that, in this condition, 00

oI /

M ðoÞ 00 00 2 2 ¼ J ðoÞ 0 M ðoÞ þ M ðoÞ

½1:33

which gives direct access to the imaginary part of the acoustic compliance, that is, to the shape of the relaxation function affecting density fluctuations. This region is accessible only by very high-resolution and contrast techniques, such as BLS, that have recently made it possible to detect the presence of secondary relaxation processes in the spectrum of density fluctuations, as shown in the following section.

4.4. Acoustic analysis: Sensitivity of Brillouin scattering to secondary relaxations Brillouin techniques have proved to be capable of obtaining information about the number and typology of relaxation processes that, being active in the GHz–THz frequency region, couple with density fluctuations [64–67]. As introduced in the previous sections, by changing the temperature, these relaxations progressively vary their characteristic time and thus their position in frequency: the structural process suffers an enormous change, while secondary relaxations, if present, experience a less drastic variation. A first acoustic analysis, rather minimal but instructive, can be obtained by fitting the narrow-frequency region around the Brillouin peak, shown in Fig. 1.10, by the DHO equation (Eq. 1.32). Typical behaviors of cL and D, obtained by this procedure applied to the intermediate glass former glycerol, are reported in Fig. 1.11 as a function of temperature [61]. Looking at these pictures, the data reveal the well-known features expected in the presence of the structural relaxation process: a marked dispersion of the sound velocity, accompanied by a maximum of the absorption. In particular, the hypersonic velocity shows the typical S-shape dispersion curve that is bounded by the limiting high-frequency (unrelaxed)

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Lucia Comez et al.

1

10

105

Intensity (a.u.)

104 103

BLS IXS

T = 295 K

200

0 -2000

0

2000

Frequency (GHz)

Figure 1.10 Typical Brillouin spectra of glycerol at ambient temperature collected in the visible (upper panel: only the anti-Stokes side is shown) and in the X-ray frequency domain (lower panel). Green and blue areas indicate the region around the Brillouin peak. (a)

(b) 4

1

3

c¥

Tg

BLS IXS

0.1

D (cm2/s)

cL (km/s)

BLS IXS US

2

a

0.01 1 ⫻ 10-3

Glycerol

1 0

Tg

c0

100 200 300 400 500 T (K)

1 ⫻ 10-4

100

200 300 T (K)

400

Figure 1.11 Apparent longitudinal sound velocity (a) and kinematic viscosity (b) determined by Brillouin techniques for glycerol [61]. Reproduced with permission from Ref. [61]. Copyright (2003) American Chemical Society.

and low-frequency (relaxed) values of the velocity, c1 and c0, respectively. The c1 values can be measured by IXS as, especially for low temperatures, the structural relaxation is definitively out of its spectral window (unrelaxed condition). Conversely, the c0 values can be obtained at much lower frequencies (KHz–MHz range) by ultrasonic measurements. The change in

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

the slope of c(T) at about 187 K indicates the occurrence of the liquid–glass transition. The longitudinal kinematic viscosity obtained by BLS displays a well-defined maximum close to 350 K that corresponds to the matching between the relaxation rate and the characteristic frequency of the longitudinal acoustic excitations. Particularly relevant is the comparison between BLS and IXS, which typically measures the unrelaxed values of G/q2, that is, 1/r: the two sets of data are consistent, within their error bars, in a range of about 80 K below Tg, although they refer to q values that differ by a factor of 100. Below Tg, the structural relaxation is located at frequencies lower than 1 Hz, and it is plausible that there is no significant contribution from it in the GHz region. The absence of an excess damping in BLS data over the IXS unrelaxed ones at Tg points in favor of an a relaxation-only scenario for the dynamics of glycerol. This is a peculiar behavior of glycerol, probably connected with its associated nature [68]. However, such behavior is an exception rather than the rule and the signature of more than one relaxation is more frequently found in glassforming liquids. To this respect, Fig. 1.12 reports the temperature dependence 10–1

BLS q » 0.04nm–1 IXS q = 2.5nm–1

10–1

a

10–2

G/q 2 (cm2/s)

–1 BLS q » 0.02nm BLS fit

a

10–2 T g = 244K

10–3 100

200

T g = 257K

oTP 300

400

200

DGEBA 300

250

350

–1

BLS q » 0.03nm–1 IXS q = 2nm–1 IXS q = 4nm–1

10–1

BLS q » 0.03nm IXS q = 2nm–1

10–1

a

a

10–2

10–2

T g = 170K

10–3 0

100

200

PB

T g = 187K

10–3 0

300

100

200

m-tol 300

T (K)

Figure 1.12 Apparent longitudinal kinematic viscosity, D, determined by Brillouin techniques from polybutadiene (PB), diglycidyl etherof bisphenol-A (DGEBA), o-terphenyl (oTP), and m-toluidine [64,66]. The glass transition temperatures for each system are indicated by a dashed line.

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Lucia Comez et al.

of D for a polymer, an epoxy resin, and two molecular liquids, namely polybutadiene (PB), diglycidyl ether of bisphenol-A (DGEBA), o-terphenyl (oTP), and m-toluidine. In analogy to glycerol, all these systems show an excess of attenuation in the high-temperature region in large part due to the presence of the structural relaxation. Besides this similarity, however, a difference with respect to glycerol is evident: for temperatures lower than Tg (indicated with the dashed line), the linewidth continues to decrease as the temperature is decreased although it still remains at a finite value. Because, at these temperatures, the structural relaxation has reached a characteristic rate of fraction of hertz, the excess attenuation in the gigahertz range must necessarily be attributed to secondary relaxations. Studying the temperature behavior of the sound attenuation in greater depth can give additional information about the nature of the secondary relaxations. In particular, for all the systems described earlier, the excess found at low temperatures seems to reflect the coupling of the longitudinal acoustic modes with internal, molecular degrees of freedom, of either vibrational or diffusional nature. The possible influence of fast intramolecular relaxations in the damping of the longitudinal acoustic modes was identified in the 1950s by ultrasonic studies of the dynamics of normal liquids, as described in the extensive work of Herzfeld and Litovitz [68]. In this work, liquids were classified on a general scheme according to their acoustic properties. Fast relaxations were explained in terms of coupling among the acoustic modes and one or more intramolecular vibrations, the liquids affected by intramolecular relaxations being identified as “Kneser.” Reexamining the list of Kneser liquids under the perspective of the glass transition, it is possible to recognize most of them as fragile glass formers [26,31]. In contrast, going on with the comparison with glycerol, for which the longitudinal kinematic viscosity does not show any excess over the unrelaxed plateau, we observe that it does not belong to the class of Kneser liquids, but rather to that of “associated liquids,” for which also ultracoustic works give no evidence of the presence of intramolecular relaxations [68]. It is interesting to note that in the modern Angell’s classification as well, glycerol actually differs from fragile liquids, falling in the intermediate class of glass formers having intermediate features with respect to fragile and strong liquids (Fig. 1.7). The interpretation of the secondary relaxations revealed by the acoustic analysis in fragile systems as intramolecular in nature is supported by independent observations obtained by different measurements on the same systems.

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

(i) In the case of oTP, this idea is corroborated by the presence of the same relaxation process in the glassy, the liquid, and the single-crystal phases [69]; by the absence of its effect on the transverse dynamics [69]; and by a molecular dynamics study of a flexible molecule model of oTP [70]. (ii) In DGEBA, the intramolecular nature of this relaxation was proven by a direct comparison of dielectric and BLS spectra in the Mountain region [21]. (iii) In the case of PB, m-toluidine, and other systems, the analysis of the temperature dependence of these processes allows the association of the coupling among the translational and intramolecular degrees of freedom to specific movements of parts of the molecule [62,64,66]. i. For oTP, the comparison between the room temperature spectrum of the single crystal and the spectrum of the supercooled liquid at T ¼ 245 K, reported in Fig. 1.13, proves to be very informative as just at first glance they show similar linewidths for the Brillouin peaks and a similar central shape. The complete analysis as a function of temperature of the Brillouin linewidth of the single crystal, Gc, and of the liquid–glass, Glg, is also shown in Fig. 1.13. At temperatures above Tg, in the undercooled and normal liquid phase, Glg is governed by the structural relaxation, which, on the other hand, is not expected to affect the acoustic damping in the gigahertz region below Tg. On further cooling, Glg still manifests a temperature dependence, remaining above the unrelaxed G1 value for at least 200 , thus suggesting 0.6 Supercooled liquid 245 K

104

2

0.4 103 L

102

L Crystal (001) 296 K

101

T

Tg

3

Tg

1

G (GHz)

Intensity (arb. units)

10

L

L

5

0 200

400

600

0.2

T

100 10–1

0.0 –20

0 Frequency (GHz)

20

0

100

200

300

T (K)

Figure 1.13 (Left panel) BLS spectra in backscattering geometry of the single crystal [q along the (001) direction] and of the strongly supercooled oTP at the indicated temperatures. (Right panel) Temperature dependence of the measured linewidths in the oTP single crystal Gc (hexagons), and glass–liquid, Glg (circles). The inset shows the same data on a larger temperature range [69]. The cross open circles represent the only contribution of the secondary, fast relaxation to the linewidth in the glass. Adapted from Ref. [69].

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Lucia Comez et al.

the presence of secondary relaxations. As a result, the total Glg can be described as an additive combination of three different contributions: Glg ¼ Ga(T ) þ Gf(T ) þ G1, where Ga(T) is the contribution of the structural relaxation and becomes negligible below Tg, Gf(T) is the contribution of the secondary, fast relaxation, and G1, the unrelaxed term, is assigned to the contribution of the topological disorder [71]. Within this background, it is possible to discern the contribution of the secondary relaxation by simply subtracting Ga(T) and G1 from the total linewidth as shown in Fig. 1.13. Furthermore, these results are compared with data obtained on the oTP single crystal, also shown in Fig. 1.13. The linewidth of the single crystal, Gc(T), displays below Tg a temperature dependence similar to that of Glg; in addition, Gc(T) closely mimics Gf(T). The conclusion is that the anharmonic process detected in the single crystal and the fast process detected in the liquid and glass [65,71] do share a common origin. In order to remove any reasonable doubt about the intramolecular nature of the fast process in oTP, the absorption of the transverse acoustic waves has also been measured by BLS [71]. Indeed, if the anharmonicity is of intramolecular origin, one does not expect any appreciable contribution from it to the linewidth of transverse acoustic peaks, as the vibrational relaxations strongly couple only to density fluctuations [72]. Conversely, if the fast relaxation is of intermolecular origin, one expects a transverse Brillouin linewidth comparable to the longitudinal one around Tg or, at least, with a comparable T-dependence. The result of this experiment is shown in Fig. 1.14, where the apparent longitudinal and transverse kinematic viscosities, DL and DT respectively, are represented as a function of temperature. It is at once evident that, above Tg, DT is dominated by the influence of the structural process and that, proceeding toward lower temperatures, it does not follow DL, going to gradually lower and lower values. This is an unequivocal indication that the fast process does not couple to the transverse sound waves and, therefore, that this process may be ascribed to a vibrational relaxation. A molecular dynamics (MD) simulation investigation of the relaxation processes active in oTP performed in a wide temperature range, spanning the normal and supercooled liquid region, confirmed this view [70]. Indeed, MD results have shown a strong coupling between the density fluctuations and the vibrational excitations, this coupling giving rise to a vibrational relaxation in the 1011 s time scale, comparable with that determined via the analysis of Brillouin spectra (Section 4.5.2). The phenyl–phenyl stretching has been recognized as the internal vibration primarily responsible for such a relaxation.

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

G/q 2 (cm2 s–1)

10–1

OTP LA OTP TA

10–2

10–3 0.0

0.5

1.0

1.5

2.0

2.5

T/Tg

Figure 1.14 Temperature dependence of Brillouin linewidths normalized by q2: longitudinal data (LA) from Ref. [65], transverse data (TA) from Ref. [69]. The dashed line indicates the low temperature limiting value of the longitudinal apparent kinematic viscosity, 1/r. Figure adapted from Ref. 70.

ii. Brillouin and dielectric profiles collected on the epoxy system, DGEBA, at the same temperatures are shown in Fig. 1.15. The intensity of Brillouin spectra has been multiplied by o in order to obtain the imaginary part of acoustic compliance in the low-frequency side of the spectra, as previously described by Eq. (1.33). The rapid increase of BLS spectra at high frequency is given by the resonance, the Brillouin peak, which “disturbs” the spectral shape, fixing the upper limit for the Mountain region. In the frequency range reported in the picture, dielectric measurements reveal the presence of two relaxations [28]: the structural relaxation, perceptible in the low-frequency part of the spectra for temperatures higher than 298 K, that, on cooling, progresses rapidly toward lower frequencies; and a strong secondary relaxation, associated with the rotational diffusion of the epoxy rings of the molecules, positioned at about hundreds of megahertz, with a relaxation time that is weakly temperature-dependent. This secondary relaxation is still discernible in the spectra of both the supercooled and the glassy states and it is the unique relaxation detectable in the reported frequency window for temperatures lower than 283 K. The presence of this feature even in the Mountain region of Brillouin spectra, which scale surprisingly well over dielectric ones, is a clear indication of the sensitivity of BLS to secondary processes, and provide a rationale for the excess absorption revealed for temperatures below Tg in Fig. 1.12.

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Lucia Comez et al.

1.6

313 K

1.2

298 K

0.8

e²,wIISO

283 K

0.4 263 K

0.3 0.2

243 K 223 K

0.1 0.0 107

108

109

1010

Frequency (Hz)

Figure 1.15 Comparison between dielectric spectra (full lines) and longitudinal compliance (solid circles) obtained from the low-frequency part of the Brillouin spectra from Ref. [21].

iii. The analysis of the temperature dependence of D obtained by the width of Brillouin lines can give an estimation of the activation energy of the secondary relaxation, which provides an important hint for recognizing its molecular origin [64–67]. In fact, assuming that the intramolecular relaxations are thermally activated, that is, their characteristic time follows the Arrhenius law of Eq. (1.26), and that the relaxation function can be written as in Eq. (1.31), it can be easily demonstrated [67] that, if the strength of the fast relaxation process is negligibly temperature-dependent, the major contribution to the temperature dependence of the width of Brillouin lines comes from the temperature dependence of the relaxation time through the relationship

GBR 1 D p ð1þbÞ b bEA lim 2 ¼ þ sin b oLA : ½1:34 t0 exp  r r KB T 2 ot1 q Therefore, analyzing the temperature behavior of D for T < Tg through Eq. (1.34) (see fitting curve in Fig. 1.16) and given the value of b (typical b values for polymers are around 0.3) for the stretching of the relaxation function, one can determine the value of the activation energy. The solid (red) line is the result of the fitting procedure with Eq. (1.34) to the experimental data [67]. This calculation was performed for different systems and it was possible to compare the values of EA, obtained by means of this procedure (typically

39

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

PB

G/q 2 (cm2/s)

10–1

BLS IXS IXS

10–2

0

100

200

300

400

T (K)

Figure 1.16 Apparent longitudinal kinematic viscosity of polybutadiene as obtained from Brillouin light scattering.

some units of KJ/mol) with the energy required for specific rearrangements of part of the studied molecule, obtained by complementary techniques, thus making achievable an assignment of molecular motion involved in the relaxation process. Case by case, fast relaxations have been attributed to torsional dynamics of segments of the polymer backbone in PB [67]; rotation of the lateral methyl group around its bond with the polymer main chain in PMA [67]; and rearrangements of the NH2 and CH3 groups in m-toluidine [66]. It should be emphasized that the presence of similar relaxation processes has also been observed by BLS in systems characterized by the presence of phenyl rings, as oTP [73]. Overall, these results show that a traditional acoustic analysis can provide important information on the high-frequency relaxation pattern of a system. Such information may also be used as a starting point for a Brillouin full spectrum analysis, which remains the most complete and informative way to analyze Brillouin spectra.

4.5. Full spectrum analysis: Divergence of a relaxation time and a–b splitting In order to obtain a complete characterization of relaxation times and strengths of relaxation processes coupled with density fluctuations and on their temperature dependence across the liquid to glass transition, the simultaneous analysis of Brillouin spectra collected in different spectral domains is indispensable. In fact, the relaxation functions being rather complex and there being a large number of parameters, the joint analysis allows one to constrain some of them, so as to provide unambiguous information from the fitting procedure.

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Lucia Comez et al.

In particular, IXS typically provides information about the highfrequency, unrelaxed parameters (M1 and 1), while ultrasonic measurements consent to measure the low-frequency, relaxed modulus (M0). The intermediate regime between the relaxed and the unrelaxed limits, actually covered by BLS and IUVS, is also highly informative because the characteristic time and shape of the relaxation processes can be better revealed within this spectral region (see, for instance, Fig. 1.9). 4.5.1 a-Only relaxation scenario: The case of glycerol and water As already observed from the traditional acoustic analysis, there are different classes of systems with different relaxation patterns. For associated liquids, that is, those in which hydrogen bonding plays an important role, a relaxation scenario dominated by the structural relaxation seems to emerge. This class includes prototypic liquids, like glycerol and water. Water, ubiquitous in nature, is interesting per se because it is known to exhibit a number of anomalies compared to many other liquids, such as a negative melting volume, a density maximum in the normal liquid range [4,74,75], and the possible (and much debated) existence of a second critical point of water in the liquid supercooled metastable phase [76,77]. Glycerol and water also have the peculiarity of being transparent to UV radiation, so in recent times, a number of specific experiments have been performed to study these materials by means of BLS, IUVS, and IXS techniques in the same thermodynamic conditions. Representative spectra as a function of temperature are reported in Fig. 1.17. Due to the fact that in these systems there is no evidence, below Tg, of an excess absorption of acoustic waves, indicating the absence of strong b-type relaxations coupled to density fluctuation (Section 4.4), a model given by an a relaxation plus an instantaneous contribution can be applied to properly represent Brillouin spectra so that Eq. (1.31) becomes M  ðoÞ ¼ M1 

rD ð1 þ iotÞb

þ io1 ;

½1:35

At each temperature, the data collected in different frequency regions and exchanged momentum have been simultaneously analyzed using this approach. The procedure has been equally adopted in the case of both water and glycerol, obtaining the temperature dependence of the structural relaxation, as reported in the Arrhenius plot of Fig. 1.18. What is immediately evident is the large variation experienced by the structural relaxation process toward the glass transition (for glycerol, Tg is located at about 187 K [61] and for water it is presumed to be at about

41

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

BLS IUVS

S(q,w) (a.u.)

T=333 K

IXS Glycerol

BLS IUVS T=297 K

T=295 K

T=277 K

T=243 K

T=268 K

T=213 K

BLS 180⬚

T=253 K

10 1000 Frequency (GHz)

BLS 90⬚ BLS 180⬚ IUVS

IUVS IXS

1

Water

1

10 Frequency (GHz)

100

Figure 1.17 Selected Brillouin spectra as a function of temperature collected in different frequency regions and using different scattering geometries. (Left panel) Isotropic spectra of glycerol: symbols refer to experimental data, dashed lines to the analysis reported in Ref. [61]. (Right panel) Polarized (solid lines) and depolarized (dotted lines) spectra of water [78,79].

136 K, but this putative location is still a highly debated topic [57]), from picoseconds to thousands of seconds, and the similarity with the behavior of shear viscosity by lowering the temperature (red lines). Another relevant aspect concerns the comparison between ta determined from the Brillouin full spectrum analysis with that obtained by other techniques: ultrasonic, photon correlation, depolarized light scattering, and impulsive stimulated thermal scattering (ISTS) data (see Refs. [61,78,80] for details), all of them rescaled over the Brillouin ones (see Fig. 1.18). The collapse of all the data on a single master curve indicates the validity of the a-scale universality predicted by MCT over a wide time–temperature region.

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Lucia Comez et al.

1 0

10

Relaxation time

Relaxation time

10

10

7

10

–2

1

4

10

10

1

0

0

10 10

–2 –1

10 200

–6

300 T(K)

300

400

400 T(K)

–1 –8

log(l/t [ps])

log(l/t [ps])

–4

–10 –2 –12 Glycerol

Water

–14 –3 3

4

5 1000/T (K)

6

7

3

4

5

1000/T (K)

Figure 1.18 Arrhenius plot of the structural relaxation times of glycerol (left panel) and water (right panel). The relaxations times are obtained from a Brillouin full spectrum analysis (violet symbols) and compared with various literature data obtained by other spectroscopic techniques [61,78,80]. The red lines represent rescaled viscosity data showing the same temperature dependence of structural relaxation time. The insets show the master plot of the structural relaxation data as a function of temperature.

Studying the dynamics of density fluctuations, one can learn that, despite these peculiarities and its “poor glass-forming ability” [81], when crystallization is avoided, water manifests features generally observed in glassforming systems during the viscoelastic changeover, such as the divergence of the structural relaxation passing from the liquid to the deep supercooled state [79,80,82–85]. The divergence of the structural relaxation was fitted by the power law behavior predicted by MCT (dashed line in the right panel of Fig. 1.18), with a critical temperature of about 220 K, an exponent g ¼ 2.3, and a stretching parameter b ¼ 0.6, in very good agreement with the MCT theory and simulations [82–85] and with results of independent optical Kerr effect measurements [86]. 4.5.2 a–b Splitting scenario: The case of oTP and LiCl aqueous solutions In the case of more complex systems such as molecular liquids, resins, polymers, and mixtures, we have learned that the scenario becomes more complicated and, because of the presence of secondary relaxations, phenomenological functions with at least two relaxations, for example, Eq. (1.31),

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

43

are required to proceed in the analysis of the full spectrum. In the case of these materials, besides the temperature dependence of the relaxation times, it is also interesting to analyze the variation of their strength. This allows one to understand when a particular process begins to play an important role in the dynamics of the density fluctuations, to locate the splitting region of a and b relaxation times, and to gain information on possible anomalies associated with this splitting phenomenon. We have selected two interesting cases studied by Brillouin scattering: for the class of molecular liquids, (i) the fragile glass former oTP, and for the class of aqueous solutions, (ii) the LiCl–6H2O water mixture. i. The fragile glass former orthoterphenyl was one of the first systems in which, making use of high-resolution BLS, it was possible to detect the presence of secondary relaxations of intramolecular nature in the glassy state (Section 4.4). Furthermore, taking advantage of measurements of the dynamic structure factor in a wide temperature range, from below the glass transition region up to the boiling point (boiling point, Tb  610 K; melting point, Tm ¼ 329 K; glass transition temperature, Tg ¼ 244 K), it was also possible to outline a specific mapping of the relaxation dynamics with evidence of the a–b splitting region. Three distinct dynamical regions, associated with the different states of matter, were identified in the gigahertz frequency domain: (1) the glass, (2) the supercooled liquid, and (3) the normal liquid regions. Each of them has required a specific data analysis treatment (according to Eqs. (4.6) and (4.7)), which is extensively explained in Refs. [65,71]. The fitting procedure of BLS spectra has been made using complementary ultrasonic (megahertz frequency range) and Brillouin X-ray scattering (terahertz frequency range) data, thereby limiting the number of free parameters. 1. In the glassy phase, the spectra (see Fig. 1.19) indicate, in a direct way and independently from any theoretical guess, the occurrence of a relaxational process revealed by the presence of a Mountain-like feature at frequencies below the Brillouin peak. More quantitatively, this fast relaxation, described by a simple Debye function (b ¼ 1 in Eq. (1.25)), results in a characteristic time on the order of 1011 s and a rather small strength, both reported in Fig. 1.20 as a function of temperature. However, in spite of this small strength, the fast process gives rise to a significant contribution to the Brillouin linewidth mainly because it is located precisely in the Brillouin spectral window. It was attributed to the coupling between the density modes and intramolecular degrees of freedom (vibrational relaxation) on the basis of arguments reported in Section 4.4.

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Lucia Comez et al.

10–3 oTP

S (q, n) (GHz–1)

10–4 10–5 10–4 245 K

10–5 10–6 10–5

226 K

10–6 10–7

186 K 1

10 n (GHz)

Figure 1.19 Examples of BLS spectra in the o-terphenyl glass taken at the indicated temperatures. The solid and dashed lines are the fits obtained using either a two-relaxation-based memory function or a simple unrelaxed ansatz, respectively [71]. Adapted from Ref. [71].

3.5 1010 3.0 108

t (ns)

bJG

104 102

D2 (GPa)

a

106

Tm

a

2.5 2.0 1.5 1.0

100

Fast

Fast

0.5

10-2 2

3 1000/T

4 (K-1)

5

0.0

200

300

400 T (K)

500

600

Figure 1.20 (Left panel) Activation plot of o-terphenyl. The results of the Brillouin data analysis are represented by full symbols: squares for the fast, activated relaxation time; circles for the structural relaxation time, which has been imposed by the temperature dependence of the shear viscosity (dotted line); and diamonds for the characteristic times of the average relaxation in the normal liquid phase. Further values of the average relaxation time (open symbols) refer to other complementary techniques (details in Ref. [65]). (Right panel) Temperature dependence of the corresponding strengths of the relaxation processes determined by Brillouin data analysis. Adapted from Ref. [65].

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

45

2. Crossing the glass transition temperature (Tg ¼ 247 K) and entering into the supercooled phase, the system experiences the concomitant existence of the structural process, whose main features are found to be in agreement with the results obtained by other spectroscopies, and of the previously reported secondary process, which seems to be insensitive to the glass transition. 3. In the normal liquid phase, close to the melting temperature (Tm ¼ 349 K), the two processes join together. On inspecting Fig. 1.20, it is apparent that Brillouin spectroscopy probes some sort of average, or merged, relaxation process, which has intermediate characteristics between those of the structural and the fast processes with respect to its strength and relaxation time. In concluding, the fast relaxation detected in oTP, also present in the glassy phase, appears to be quite uncorrelated from the universal features of the structural dynamics and rather strictly linked to its peculiar attitude to exchange energy with collective density fluctuations, typical of vibrational relaxations [72]. Finally, the picture summarized by Fig. 1.20 also represents the results of the Brillouin data analysis, compared with those obtained with other correlation functions. On the whole, besides the structural relaxation and the fast vibrational relaxation, another secondary relaxation, the so-called Johari–Goldstein b relaxation (bJG), much slower in time, has been detected by dielectric spectroscopy. Different from the vibrational relaxation, bJG seems to be more connected with the structural relaxation [87,88], and the a–bJG splitting is frequently accompanied by changes in the ta behavior [28,58]. From the comparison with depolarized light scattering, and photon correlation spectroscopy and dielectric data, measuring optical and electric polarizability correlation functions, respectively, one can deduce that the a-scale universality, if present, manifests only in the deeply supercooled phase and at frequencies lower than  106 Hz, after the a–bJG splitting. ii. Electrolyte aqueous solutions have received significant scientific attention, as they are strongly connected with human life: important chemical reactions in the human body, such as membrane permeability and neuronal signaling, require ionic substances in an aqueous environment [89]. The study of these systems at temperatures below the homogeneous crystallization temperature of water, 235 K, where entropy increases and structural effects due to hydrogen bonding occur by changing the thermodynamic scenery [75], has been generating even greater interest. Among electrolyte aqueous solutions, LiCl–RH2O aqueous mixtures (where R is the molar ratio of solvent/solute) have attracted special consideration for their ability to

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supercool [90–93]. Particularly for R  6, it has been shown that the glassy state is realized by increasing the network of H-bonds among the hydration shells of ionic clusters, other than by the strengthening of ion-solvent interactions [94]. The R ¼ 6 solution possesses the peculiarity of avoiding crystallization in the whole temperature range down to the glass transition temperature Tg 135 K [91]. The wide domain of supercooling has allowed a detailed study of the structure [95] and dynamics [91] of this system. Concerning the relaxation dynamics, the structural relaxation process of LiCl–6H2O measured by different techniques was found to obey a scaling law comparable to that of viscosity. Furthermore, besides the main process, some techniques have revealed the presence of a local b relaxation in the temperature range from 130 to 220 K. [96] The coexistence of a- and b relaxations, as previously discussed, considerably complicates the analysis of density fluctuations spectra, especially if measured in narrow spectral regions. In fact, the first BLS study of aqueous LiCl solutions [97], performed only in the limited frequency region of a few gigahertz around the Brillouin peaks and using a single-relaxation guess, provided an unexpected Arrhenius behavior for the structural relaxation time. Thus, it has been necessary to organize a more comprehensive investigation of the dynamics of density fluctuations in order to gain insight into the possible interplay between the a- and b relaxations and their correlation with structural modifications occurring at the development of short-range order. This is the reason why the representative LiCl–6H2O solution has been investigated as a function of temperature by means of the three techniques for inelastic scattering of radiation that probe contiguous frequency domains, BLS–IUVS–IXS, with the purpose of proceeding with a joint analysis of the obtained spectra [98]. Spectra acquired at same temperatures by the different techniques are analyzed simultaneously using Eqs. (4.6) and (4.7). In Fig. 1.21, the experimental data together with the full lines obtained by the joint fit are reported at three different temperatures. As described earlier, IXS gives access to the unrelaxed c1 and 1 values, while the relaxed c0 values were taken from the literature [99]. Above  220 K, a single-relaxation model is sufficient to describe the spectra. Conversely, below 220 K, a progressive failure of this simple model is found. In particular, in this temperature region, in addition to the joint analysis by means of BLS, IUVS, and IXS, the help of complementary techniques such as ISTS and photon correlation spectroscopy [98] is very useful, in probing the temperature behavior of the a relaxation time, which becomes slower and slower, progressively exiting the Brillouin experimental spectral windows.

T = 292 K Q = 2 nm-1

Q = 0.074 nm-1 Q = 0.033 nm-1

1

10

100

-4

-3 -2

-1

0

1

2

3

4

T = 253 K Q = 3 nm-1

Q = 0.033 nm-1

Q = 0.074 nm-1 1

10

100

-4

-3 -2

-1

0

1

2

3

4

T = 173 K

Q = 3 nm-1

Q = 0.033 nm-1 Q = 0.075 nm-1

1

10

100 -4

-2

Frequency (GHz)

10

100 Energy (meV)

0

2

4

8

12 16

Frequency (THz)

400

-16 -12 -8

-4

0

4

Energy (meV)

Figure 1.21 Brillouin spectra acquired at the temperatures of 292, 253, and 173 K by different inelastic scattering techniques: light scattering (green diamonds), ultraviolet scattering (violet circles), and X-ray scattering (blue squares). In the left panels, only the Stokes inelastic peaks are reported [98]. Reproduced with permission from Ref. [100]. Copyright (2009) American Chemical Society.

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The characteristic times of both relaxations obtained from data analysis are displayed as a function of reversed temperature in Fig. 1.22, suggesting a splitting temperature at about 220 K that marks the change between one and two relaxation regimes. The b relaxation time is discovered to follow an Arrhenius law (see Eq. 1.26) with activation energy E  23 kJ/mol, in agreement with the literature data [96]. The corresponding relaxation strengths are also reported in Fig. 1.22, helping to clarify the nature of and the interplay between the two relaxations. In particular, it can be seen that the a relaxation manifests an onset around T ¼ 220 K, and its amplitude increases below this temperature at the expense of the amplitude of the b relaxation. A peculiarity of LiCl–6H2O that distinguishes it from systems with abJG splitting, is that the b process seems to develop as the low-temperature continuation of the single high-temperature process. The temperature behavior of its amplitude appears similar to that observed in different glass formers, such as d-Sorbitol [100], toluene, and 1-propanol [101], and in random copolymers of poly(n-butyl methacrylate-stat-styrene) [102], all characterized by a decrease in amplitude on cooling. Moreover, the onset of the a relaxation is close to the splitting and to the melting temperature. Other cases where a similar behavior was observed include ibuprofen [103], poly(phenyl glycidyl ether)-coformaldehyde (PPGE) [28], and DGEBA [58] in which two

Two relaxations

12

log (/ns)

10 8 6 a

4 2 0 -2 -4

b 3

4 5 6 1000/T (K-1)

7

8

Relaxation Strength ( m2/s2/ 106)

One relaxation

Two relaxations

One relaxation

Da

D1

10 8 6 4 2 0

Db 150

200

250 T (K)

300

350

Figure 1.22 Arrhenius plot of the relaxation times in the LiCl–6H2O solution (left panel) and corresponding relaxation strengths (right panel) obtained by joint analysis of Brillouin data following the procedure adopted in Ref. [98]. Open symbols from Refs. [96,98,99]. Reproduced with permission from Ref. [100]. Copyright (2009) American Chemical Society.

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

49

crossover regions were found, characterized by a similar splitting of relaxation processes. The high-temperature crossover was located at 1010 s, that is, close to that obtained for LiCl–6H2O. In all these cases, the fast secondary process was not a bJG relaxation. Concerning the possible interpretation of this phenomenon, it may be noticed that an onset of a from the b relaxation process qualitatively similar to the one observed in LiCl–6H2O had been predicted for the rotational relaxation in solutions of small molecules in supercooled fluids [104]. Something similar also occurs in coupling phenomena between soft lattice modes and relaxation modes of lower frequency reported on approaching structural phase transitions [105]. More intriguingly, it can be observed that an additional hypothesis about the origin of this splitting may come from recent theories and experimental results concerning the dynamics and phase diagram of water and of dilute ion solutions at low temperatures and high pressures. In this context, the dynamic splitting observed in LiCl aqueous solutions could possibly be connected with the onset of the hypothetical liquid–liquid phase transition from the homogeneous liquid phase to a mixture of a low-density liquid (LDL) pure water phase and a high-density liquid (HDL) water phase, encapsulating LiCl salt (HDL þ LiCl). LDL and HDL þ LiCl would thus be the liquid counterparts, for LiCl solutions, of the low-density amorphous [106] and high-density amorphous [107] phases found in water. Recently, Mishima and Stanley [4] proposed this two-fluid model for interpreting new calorimetric data of dilute LiCl solution [108], on the basis of previous simulation [109] and experimental work on LiCl solutions glasses [110]. Brillouin scattering measurements at lower LiCl concentrations and at higher pressures could help to further test this hypothesis.

4.6. Experimental determination of the nonergodicity factor The most significant MCT prediction that has been tested on complex glassforming systems [111–115] concerns the nonergodicity factor, fq, which, as stated earlier, should exhibit a square-root, cusp-like behavior as a function of the control parameter z close to the ideal glass transition temperature (Section 4.2.3). From the experimental point of view, it was shown that, in case of an a-only relaxation scenario, it is possible to estimate this parameter by the area of the quasi-elastic contribution to the spectrum of density fluctuations or from relaxed and unrelaxed sound velocities. However, with increasing system complexity, the experimental determination of fq becomes more and more difficult due to the presence of a variety of molecular motions corresponding to different characteristic relaxation processes, whose

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nature is connected to the specificity of the system under study. A correct estimate of fq is related to an accurate modeling of the spectrum of density fluctuations that should be able to take into account all eventual relaxations in excess with respect to the a relaxation. To this aim, a joint analysis of spectra obtained by different techniques can significantly improve the possibility of evaluating fq in real systems. In the following section, some selected examples are reported. 4.6.1 Molecular liquids Testing the MCT predictions in covalent and associated liquids, the ubiquitous class of liquids including water and silica, has not been a trivial task as, in these liquids, the local order extends over several neighboring molecules, often giving rise to a nontrivial q behavior of the static structure factor, for example, a pre-peak in the low-q region of S(q). On the one hand, computer simulations on these systems have shown that MCT quantitatively reproduces the wave vector dependence of the nonergodicity factor in the pre-peak q region [116] while, on the other hand, the few available results of experimental investigations are often mutually contradictory [112]. In addition, in this case, the use of high-resolution IXS has been proved to be a valuable tool for testing the MCT predictions. IXS having access to the spectrum of the density fluctuations in the mesoscopic regime (q values range between 1 and 20 nm1) allowed operating both the test of the usual cusp-like behavior of fq as a function of the control parameter and the verification of the link between the nonergodicity factor and the q evolution of the static structure factor. This aspect is particularly relevant for covalent and associate liquids, which are characterized by identifiable features in the static structure factor, related to the local order. In IXS spectra, it is possible to distinguish between a quasi-elastic Iel(q) and two inelastic Iinel(q) contributions, the total intensity being Itot(q) ¼ Iel (q) þ Iinel(q). The elastic contribution is modeled by a delta or a narrow Lorentian function centered at zero frequency and the inelastic by a DHO function (see Eq. 1.32). These components are highlighted in Fig. 1.23, after convolution with the experimental function. Starting from these values, it has been shown that a good estimate of the nonergodicity factor can be really provided by the ratio of the elastic to total scattered intensity Iel/Itot [44]. As an example, we report the case of m-toluidine, a molecular liquid characterized by a spatial organization of the molecules induced by hydrogen bonds extending over several molecular diameters and giving rise to

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Intensity (a.u.)

Experimental data Total contribution Iel

1000

Ianel

500

-20 -15 -10

-5 0 5 Energy (meV)

10

15

20

Figure 1.23 Typical IXS spectrum. The elastic and inelastic components are highlighted.

(a)

(b) 1.0

Tc

0.8

Tc

8

0.9

7

fq 0.6

6

fqc

0.8 q=2 nm-1

q=4 nm-1

0.6 5

1.0

fq

Tc 0.9

0.8

0

Tc

3

6

9

4

12

-1

q (nm )

3

0.9 q=7 nm-1

100

200

T (K)

100

200

T (K)

2 1

q=10 nm-1

300

Pre-peak

Static structure factor

0.9

0.8 300

0

5

10

q

15

0

(nm-1)

Figure 1.24 (a) Temperature dependence of the nonergodicity factor fq of m-toluidine for different values of the exchanged wave vector q. The solid lines are the best fits obtained using the square-root function predicted by the MCT; for each q, the corresponding value of the critical temperature Tc is indicated. (b) X-ray diffraction pattern of liquid m-toluidine taken from Ref. [120], compared with the parameter fqc (inset) obtained from IXS experimental data as explained in the text. Figure adapted from Ref. [121].

nanometer-size clusters [117–119]. From a structural point of view, m-toluidine is characterized by a pre-peak (pp) in the S(q) located at about qpp ¼ 5 nm1 (see Fig. 1.24). The IXS investigation has been performed as a function of temperature from the liquid to the glassy state [121]. The resulting nonergodicity factor is shown in Fig. 1.24 for different q values. In the figure, the solid lines are the

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best fitting curves obtained using a square-root function, giving experimental evidence for the validity of the square-root singularity predicted by the idealized MCT in the whole investigated range of q. Note that this evidence was never obtained in the mesoscopic range before the improved inelastic X-ray facilities (Section 3.3); in fact, for different glass formers, such as polybutadiene [62] and o-terphenyl [65], a discontinuous behavior close to Tc had been observed, but the statistical quality of the data at that time was not sufficient to extract a clear analytic behavior. The improved resolution and flux of the IXS experiment actually results in a better signal-to-noise ratio, essential for determining the ratio of the elastic to total scattered intensity. Another relevant aspect is that the value of the critical temperature Tc obtained by fitting the square-root function turns out to be q-independent within experimental error (average value is Tc ¼ 228  3 K in the q range between 2 and 10 nm1) in line with MCT predictions (see Fig. 1.14). Finally, the most stringent test of the predictions of MCT on m-toluidine concerns the behavior of the parameters fcq and hq in comparison with the static structure factor S(q) [120]. The values of fcq (full squares)—obtained by fitting the experimental fq(T) data using the MCT square-root function—are reported in the inset of Fig. 1.24 (right panel) together with the values of fq at 263 K (circles), which are available on a wider grid of q. These latter data points are directly calculated by the ratio Iel/Itot, and provide actual values of fcq, as the temperature T ¼ 263 K belongs to the plateau of the fq(T) curves. The main result is that the values of fcq, on the whole, follow in phase the oscillations of the static structure factor (Fig. 1.24, right panel), while hq is found to be in antiphase (data not shown). This indicates that such behavior, predicted by MCT calculations for simple liquids, also holds for a liquid with a sizable local order. 4.6.2 Epoxy resins: The chemical vitrification There are systems that progressively polymerize developing structural relaxation dynamics and evolve toward the structural arrest upon the increase of the number of covalent bonds that irreversibly form among the constituent molecules. The ability of reactive systems, such as thermoset resins [122], to vitrify during the chemical bonding process plays a role in the understanding of the extent to which different systems share a common physics [123]. The concept that, at the basis of glass formation, there may exist a basic mechanism generating universal manifestations is the focal point of the MCT explanation of glassy dynamics. Therefore, understanding if the

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MCT can interpret the bond-induced dynamical arrest as it does for the glass formation in conventional systems represents a big challenge, a severe experimental test of the theory. On this line, important evidence in favor of the universality of MCT predictions has been provided by IXS investigations during a step polymerization of selected binary mixtures [124]. These studies are similar to those carried out in liquids with a local order [121], but the uniqueness of these new samples relies upon the introduction of a new control parameter for the glass formation, very different from the usual temperature or pressure. In brief, as monomers polymerize, covalent bonds form randomly between pairs of mutually reactive groups; the fraction a of the bonds formed, called chemical conversion, is the (independently measurable) parameter that drives the system toward the structural arrest [35]. At a sufficiently low and fixed temperature, the monomers bond slowly to each other and IXS spectra can be collected at almost fixed values of the single control parameter a. The experiments were conceived as a measure of the dynamic structure factor for wave vectors q in the range between 1 and 15 nm1, while monitoring—by the total scattered intensity, I(q)—the static structure features in the same range. Linear and network polymers were subjected to the test, searching for a more convincing general result. The nonergodicity factor was calculated by the ratio Iel/Itot [44], and a representative set of results is reported in Fig. 1.25 (all the results can be visualized in the work by Corezzi et al. [124]). The main result obtained in all the investigated step polymers resides in the fact that chemical bonding, while inducing molecular ordering on a q=2 nm-1 0.84

q=4 nm-1

ac=0.35 ± 0.02

q=10 nm-1

q=7 nm-1

ac=0.34 ± 0.04

0.88

ac=0.35 ± 0.04 0.92

0.84

0.80

fq

0.84

0.76 0.88

0.80 0.80 0.72 0.0

0.2

0.4

0.6

Chemical conversion, a

0.0

0.2

0.4

0.6

Chemical conversion, a

0.0

0.2

0.4

0.6

Chemical conversion, a

0.0

0.2

0.4

0.6

Chemical conversion, a

Figure 1.25 Dependence on chemical conversion, a, of the nonergodicity factor, fq, for a step polymer at the indicated fixed q values. The solid lines are the best fit using the square-root function predicted by the MCT [124].

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mesoscopic length scale, as revealed by the development of a peak at small q in the scattered intensity, also effectively realizes the mechanism of dynamical arrest described by the MCT. This evidence is marked by a cusp singularity in the behavior of the nonergodicity factor as a function of the number of chemical bonds. Moreover, the position of the cusp singularity obtained by fitting the data with the square-root function turns out to be independent of q within experimental error, in agreement with the general MCT predictions (Fig. 1.25). In addition, structural changes occurring during the polymerization process induce strong modifications in the q dependence of the static structure factor, which reflects astonishingly well onto the q dependence of the nonergodicity factor. Figure 1.26 shows the results obtained for a linear polymerization [124], surprisingly underscoring that f(q) oscillates in phase with I(q) at any extent of reaction, in the q region of both the main and the pre-peak of I(q). In particular, the correlation holds up to a ¼ 0, where the value of f(q) corresponds to the plateau value fcq. These results, qualitatively similar to those obtained by solving numerically the MCT equations for simple liquids [59], molecular 1.2

I(q) (a.u.)

1.0 0.8 0.6 0.4 0.2

0.95

f(q)

0.90 0.85 0.80 0.75 0

2

4

6

8

q

(nm-1)

10

12

14

Figure 1.26 q dependence of the total scattered intensity I(q) (upper panel), and of the nonergodicity factor f(q) (lower panel), at different extents of the polymerization: square (red) symbols a ¼ 0; hexagons (blue) symbols a at an intermediate reaction time; circle (green) symbols a at the end of the reaction. Figure adapted from Ref. [124].

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systems [111], and polymers [115,125], and by experiments on stable glass formers [112,121,126,127], thus give strong support to the ability of MCT to describe the early phase of structural arrest, independently of the complexity of the system and the nature of the control parameter. 4.6.3 Polymers The dynamics of polymers is typically characterized by both a relaxation and a huge number of secondary relaxations, due to their very complex molecular structure [128]. In the case of a relaxation-only systems, the low q value fq!0(z) can be determined (Section 4.2.3) from unrelaxed and relaxed sound velocities or, equivalently, unrelaxed (M1) and relaxed (M0) longitudinal moduli though the expression fq!0(z) ¼ 1  M0(z)/M1(z) [129]. The unrelaxed longitudinal modulus can be inferred by fitting Brillouin scattering or ISTS measurements and the relaxed longitudinal modulus from ultrasonic measurements. In the early tests performed by means of BLS, the spectrum of density fluctuations was reproduced assuming a single-relaxation process. This was supposed to be the structural relaxation, which is responsible for the glass transition. As a result, fq!0, determined for a variety of polymeric systems, shown in Fig. 1.27, displays a flat behavior, with no evidence of a critical behavior in the supercooled phase [130]. On the one hand, the approximation of using an a relaxation-only model is obviously too crude in the case of a coexistence of structural and thermally activated secondary relaxations, and it may profoundly affect the results of the 1.0

fq®0=1-M0 /M¥

0.8 0.6 PBA DGEBA PB PEA PMA

0.4 0.2 0.0 200

250

300

350

400

450

500

T (K)

Figure 1.27 Temperature dependence of the nonergodicity parameter of different polymers obtained from fitting Brillouin spectra with a single-relaxation function [130].

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analysis. On the other hand, the introduction of a second relaxation in the analysis of BLS spectra considerably increases the number of free parameters in the fitting procedure, and the meaning of the fit itself is likely to fail. Subsequently, it is the development of the IXS technique, operating in the terahertz frequency range, that has made it feasible to overcome the previous difficulties in the description of the rich pattern of relaxation processes of complex systems such as polymers. A careful reexamination of the case of polybutadiene [62], one of the most studied polymeric systems, has actually shown that a joint analysis of the IXS and BLS spectra, which makes it possible to considerably enlarge the accessible frequency/exchanged momentum windows, gives the opportunity of using more realistic models for the dynamic structure factor. In this case, the two-step memory function for the spectrum of density fluctuations described in Section 4.3 was successfully used to represent the dynamics of supercooled PB in the whole exchanged momentum region of 0.04–4 nm1covered by BLS and IXS, as reported in Fig. 1.28. This modeling allowed verification of the existence of a cusp-like behavior in the temperature dependence of the nonergodicity factor, which was calculated from the relaxed and unrelaxed values of the longitudinal modulus referring only to the a relaxation [62]. Moreover, from a comparison with results obtained on the same system at high q by means of neutron scattering, it can be seen that the q dependence of fq is also consistent with MCT, thus giving an important positive test to the basic predictions of the theory in the case of polymeric systems as well. 4.6.4 Fragility and nonergodicity factor Brillouin spectroscopy in the terahertz frequency region (IXS) has also shed some light on the connection between viscous flow and vibrational properties in glass-forming materials. This has been realized by exploring the fragility of a wide set of liquids and the nonergodicity factor of the corresponding glasses [44,131]. As briefly introduced in Section 4.2.1, the rapidity of the increase in the viscosity when approaching Tg from the liquid state, defined as the kinetic “fragility”, m, is a concept introduced by Angell to classify strong and fragile systems [26]. On the one hand, on moving toward the glass transition from the liquid side, such a property was found to be related to macroscopic properties (see Scopigno et al. [131] and references therein), and, on the other hand, surprisingly, a correlation has also been observed with low-temperature vibrational properties, specifically with the nonergodicity factor. Indeed, it was shown that fq(T), determined by means of IXS data as

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Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

(a)

BLS 105 10

(b)

IXS -1

q = 0.036 nm

q = 1 nm

-1

3

10

0.9

f(q)

0.8

4

0.7

2

10

Intensity (arb. units)

103 10

2

10

1

10

3

0.6 0.5 101

0.4

PB -20 -10

0

10

20

-2000

0 0

q = 2 nm-1

2000

(c)

1

-1 q (Å )

2

q = 4 nm-1 103

0.50

fq®0=1-M0 /M¥ 102

0.45

102 0.40

101

101 -2000

0

2000

-2000

0

2000

0.35 200

250

300

350

T (K)

Frequency (GHz)

Figure 1.28 (a) Density fluctuation spectra of supercooled polybutadiene obtained by BLS and IXS scattering at T ¼ 249 K and at the indicated q values. (b) q behavior of f(q) obtained in the hydrodynamic limit and by neutron spin echo measurements in the high-q region (circles) (details in Ref. [62]). The full line is proportional to the static structure factor measured by neutron scattering at T ¼ 270 K. (c) Temperature dependence of the nonergodicity parameter of PB obtained by the joint analysis of BLS and IXS spectra. Figure adapted from Ref. [62].

the ratio of the elastic to total scattered intensity [44], can be interpreted in terms of a harmonic description of the atomic vibrations and expressed as fq ðT Þ ¼

1   1 þ a T=Tg

½1:36

where a is a constant. A direct proportionality was found between the parameter a and fragility m (see Fig. 1.29), indicating a correlation between fq(T ! 0) measured in the glass, well below Tg, and the fragility, determined near Tg from the liquid side. This correspondence may indicate an effective way to apply the fragility concept to the glassy state, and, interestingly, it provides a method to determine the fragility exclusively from properties of the glass well below Tg. However, it has been shown that in some complex systems, a similar linear relationship cannot exist [45,48,132,133]. Because of these exceptions, a further effort was made to improve the theoretical support on which the correlation is based. The crucial point is that in Eq. (1.36) the experimental

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120 100

oTP

80 mtol

PB1.4

m

Se

mTCP

60 40

salol

nBB

Glycerol Silica BeF2

20 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

a

Figure 1.29 Correlation plot of the kinetic fragility, m, and the a parameter of Eq. (1.36). The dashed line is obtained by a fit of the data to a linear equation. It corresponds to m ¼ 135 a [44]. Figure adapted from Ref. [44].

BPA-PC

160

Structural relaxation

120 0.05

PET PMMA

Secondary relaxation

PS

DBP

m

S(Q,w)(ps)

0.10

80 PPG nBB

40 0.00 -30

-20

-10

0

Frequency

10

(ps-1)

20

30

0 0.0

0.5

1.0

1.5

2.0

a

Figure 1.30 (Left panel) Dynamic structure factor as measured by an IXS experiment (full line). Vibrational contribution (dashed line), and structural and secondary relaxations are also indicated. (Right panel) Correlation between the fragility, m, and the nonergodicity factor accounting for the presence of secondary relaxations. The a parameter was calculated following the indications in Ref. [131]. Reproduced with permission from Ref. [134]. Copyright (2010) American Physical Society.

determination of fq passes through the ratio Iel/Itot obtained by IXS spectra. As previously shown, this method is implicitly subject to the assumption that Iel uniquely depends on the structural relaxation. This is a genuine guess in the case the structural process is the only one or at least it represents the dominant contribution to the glass transition dynamics. If, on the other hand, strong secondary relaxations are present, making an important contribution to the quasi-elastic scattering, as schematically reported in Fig. 1.30, the determination of fq(T) from the ratio Iel/Itot fails. This problem can be solved

Progress in Liquid and Glass Physics by Brillouin Scattering Spectroscopy

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by considering the elastic contribution as an additive combination of structural and secondary relaxations, Iel ¼ Iel,a þ Iel,b, and extracting from it the proper candidate to be correlated with fragility, that is, the structural contribution [131]. A way was found to express this correction in terms of a residual excess entropy at T0 (the Vogel temperature [43]), attributed to a non-a process originating from internal degrees of freedom. The corrective contribution was applied to all those polymers that do not satisfy the m versus the a proportionality represented in Fig. 1.29 (we recall that in all these systems, secondary relaxations account for at least 20% of the total relaxation), restoring a direct relation as shown in Fig. 1.30.

5. GLASSES 5.1. Introduction The vibrational dynamics of amorphous solids and glasses has been widely studied in the last three decades [134]. There are several properties associated with vibrational dynamics that have intrigued scientists, including (i) the quasi-elastic scattering (QES), detectable in the Raman and neutron spectra of supercooled liquids and glasses; and (ii) the boson peak (BP), that is, an excess over the Debye level, present in an amorphous solid and missing in the crystalline counterpart, that emerges at energies of a few millielectronvolts in the vibrational density of states. Understanding the physical origin of these features is an important and challenging task because they are inherently related to various universal low-temperature properties of the glasses, such as low-temperature specific heat, thermal conductivity, and propagation of ultrasound [135,136], that are only weakly dependent on the specificity of the material. Light- and neutron-scattering spectra of glasses reveal the presence of a broad quasi-elastic contribution that is deeply temperature-dependent. Despite the fact that the nature of this feature has been extensively studied, a general agreement on its interpretation is still lacking. Overall, the existing literature proposes a connection between the QES intensity and the damping of the acoustic waves, hypothesizing that the temperature behavior of these two quantities might be the same [136–153]. The main limitation, in the past, to verify such a guess came from the lack of experimental data for the acoustic attenuation in a frequency range wide enough to be directly compared to the QES data.

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Similarly, the microscopic origin of the BP has been widely discussed for many decades and many theoretical assumptions have been proposed in this regard, without a general consensus. The BP was interpreted in terms of (i) soft vibrations that should be present in glasses in addition to the acoustic ones [136,154,155]; (ii) vibrational dynamics related to harmonic oscillators with disorder in the force constants [156–159]; and (iii) characteristic vibrations of nanometric clusters [160,161] that should exist in the glass with a spatially inhomogeneous cohesion and hybridize with the acoustic modes [162]. Finally, numerical studies, revisiting the idea of an inhomogeneous elastic response, showed that the classical elasticity explanation breaks down in glasses on the mesoscopic length scale [163–165] and the BP would emerge at the frequency corresponding to this length scale. In recent years, the continuous development of, and advances in, Brillouin techniques (Section 3) have made possible enormous steps forward in the interpretation of QES and BP, as they have opened up access to wavenumber (q) and energy (E)/frequency (o) regions previously unexplored (see Fig. 1.1).

5.2. More on QES: Correlation between collective and tagged particle dynamics A central problem for the study of the physical origin of QES is comparing neutron- and light-scattering spectra with acoustic attenuation data in comparable frequency/energy regions. In this regard, the combined use of Brillouin spectra (BLS–IUVS–IXS) (Section 4.5) and incoherent neutron spectra has proved to be able to demonstrate that, to all effects, a strict relation exists between acoustic dissipation and density of states of a glass-forming system, valid even beyond the region in which the quasi-elastic contribution appears in the spectra. A first interesting indication of such correspondence comes from the direct comparison between normalized isotropic Brillouin susceptibilities wLS00 (o) and taggedparticle susceptibilities wINS00 (o), obtained from neutron-scattering data. For the comparison, the latter are divided by a factor, weakly temperaturedependent, so that one obtains the overlap with the Brillouin spectra [166,167]. Figure 1.31 illustrates this comparison in the case of glycerol. The same figure also shows the acoustic compliance data (solid symbols), evaluated as J00 (oB) ¼ q2G/[roB(o2B þ G2)], scaled by a factor ℏrN J00 for a proper comparison with wLS00 (o) data. In the above equation, q is the exchanged momentum in the scattering process; r, the mass density; oB and G, the angular frequency and linewidth, respectively, of the Brillouin

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BLS IUVS

IXS T=333 K

1E-5

T=243 K

cLS²(w), J²(w), cINS²(w).s (GHz-1)

1E-6

INS

T=295 K

T=213 K

1E-4

1E-5

1E-7

T=273 K

T=150 K

1E-6

1E-7

100

101

102

w (GHz)

103

100

101

102

103

w (GHz)

Figure 1.31 Selection of susceptibility spectra in absolute units for different temperatures. Experimental BLS (open diamonds), IUVS (open triangles), and IXS (open squares) data are shown together with INS data (open circles) [166]. The INS data have been scaled by a factor s. Filled (blue) hexagons correspond to the imaginary part of the longitudinal compliance, J00 (oB/2p), at the Brillouin frequency (oB/2p), after scaling by hrN, rN being the number density. Dashed and solid (blue) lines are the Brillouin spectra joint-analysis fits and the corresponding hrN J00 (o/2p) curves, respectively. Figure adapted from Ref. [121].

peaks obtained by a damped harmonic oscillator fit of the Brillouin [61]; and rN, the number density (average number of atoms per unit volume). Finally, to obtain a global picture, Fig. 1.31 also represents the acoustic compliance curves (solid lines) corresponding to the functions J00 (o) ¼ M00 (o)/[M00 (o)2 þ M0 (o)2], reproduced using the parameters obtained from the simultaneous full spectrum analysis of the isotropic Brillouin spectra [61,166]. The good-quality superposition between the imaginary part of longitudinal compliance and the scaled neutron susceptibility suggests the existence of a remarkable correspondence between these two quantities at all temperatures and in the whole explored frequency range. This connection has

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special significance as, in mechanical experiments, oJ00 (o) is proportional to the nergy dissipation per unit time, and, in incoherent neutron-scattering experiments, owINS00 (o) is proportional to the Fourier transform of the velocity autocorrelation function g(o), which, in the solid, corresponds to the vibrational density of states. To this respect, great attention has recently been devoted to the study of the relation between g(o) and the acoustic properties in glasses, especially in the region of the BP [168,169]. The data above prove that such a relation holds—and even ore accurately—at lower frequencies as well. Moreover, the empirical observation of a good scaling of collective over single-particle scattering spectra in a wide temperature/frequency region also finds a theoretical support. In fact, a standard field-theory approach allows one to write the density of states as the superposition of longitudinal and transverse acoustic wave contributions:

2Mat o X 1 gðoÞ ¼ IfwL ðk; oÞ þ 2wT ðk;oÞg ½1:37 k
3o2 GðoÞ þ 2 3 oD ðoÞ o0

½1:38

3 where gD(o) is the Debye density of states, with cD(o) ¼ {[c3 L (o) þ 2cT (o)]/ 3}(1/3) and oD(o) ¼ kDcD(o) frequency-dependent generalizations of the Debye sound velocity and frequency, respectively, and with cL and cT the longitudinal and transverse sound velocities, respectively. grel(o) is the relaxational contribution to g(o), with o0 an interaction parameter describing the coupling between sound waves and relaxations. The application of this equation is shown in Fig. 1.32 for glycerol (see details in Comez et al. [166]). In a few words, the observed scaling of Fig. 1.31 is modeled by using a simple relation between acoustic loss and generalized density of states, proposing the use of the Cauchy-like model for deriving the shear modulus from the bulk one. This approach, independent of any theoretical consideration

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BLS IUVS IXS 10-5 10-5 10-7

T=243 K

g(w/2p) (GHz-1)

T=333 K

10-5

10-7

T=295 K

T=213 K 10-9

IXS 10-5

10-5

BP»1THz 10-7

T=150 K

T=273 K 101

102

103

101

102

103

10-10

w/2p (GHz)

Figure 1.32 Experimental g(o/2p) from INS (open down triangles) shown together with calculated g(o/2p) (full up triangles) at the Brillouin peak frequency of BLS (green), IUVS (violet), and IXS measurements (blue) [171] using Eq. (1.38). J00 '(o/2p)/s curves multiplied by o/2p are also reported as solid lines, where s has the same values as in Fig. 1.31. In the inset, the high-frequency data for the glass are emphasized, and the boson peak position is indicated by the arrow. Figure adapted from Ref. [121].

of the microscopic origin of the acoustic attenuation, holds in an astonishingly wide frequency range, comprising both structural and fast relaxation, and in a wide temperature range, from the liquid to the supercooled and the glassy regime, and establishes a clear connection between the quasi-elastic component observed in neutron-scattering experiments and the fast relaxation dynamics probed by Brillouin scattering, thus promising to provide a general link between single-particle and collective properties of disordered systems. It is worth mentioning that the scaling becomes worse below the glass transition (Tg ¼ 187 K in glycerol) in the region

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where the BP becomes prominent. In this region, more complex mechanisms may contribute to the acoustic attenuation in the glassy state and in proximity of the BP [169,171–174]. Such a case is discussed in the next section.

5.3. More on BP: Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths in glasses Continuous progress in the IXS technique [175] has made it possible to come to novel conclusions about the interpretation of the BP. A new series of high-resolution IXS data allowed highlighting a characteristic softening in the sound velocity corresponding to longitudinal acoustic-like modes with nanometer wavelengths, which can be viewed in relation to the universal anomalies observed in the specific heat of glasses at low temperatures [135]. This softening, a sign of the breakdown of the Debye approximation, corresponds to a crossover between well-defined acoustic modes at wavelengths larger than a few nanometers and ill-defined ones at wavelengths smaller than a few nanometers. This information was primarily obtained from the q dependence of sound velocity and acoustic attenuation determined from the analysis of Brillouin spectra [171] as described in Section 4.3. The data for glycerol are reported in Fig. 1.33. By examining the acoustic dispersion curve in Fig. 1.33, it can be seen that the apparent longitudinal phase velocity, vL(q), tends at low q to the macroscopic limit given by the longitudinal speed of sound measured using low-frequency techniques [61,176,177]. In more detail, it shows a sharp decrease with q (softening) upto  2.2 nm1, and then a plateau upto 4.5 nm1. Only above this value one begins to observe the typical crystal-like decrease of the sound velocity due to the bending of the dispersion curve on approaching the first sharp diffraction peak. This means that the macroscopic Debye limit does not collapse continuously on approaching the microscopic scale, as it happens in crystals, but rather by an abrupt change: the breakdown is indicated by a sudden decrease of the sound velocity on the mesoscopic scale of a few nanometers, which can be related to the medium-range order of the glass. Moreover, looking at the q dependence of the linewidth of the Brillouin peaks, one can observe an extremely steep increase in the broadening as a function of q. For q  2.2 nm1, this increase can be effectively described by a q4 power law, which then converts into a q2 behavior for higher q values (green lines in Fig. 1.33). This phenomenology has been recognized as a

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(a)

VL (km/s)

3.6

3.4

3.2

(b)

G (meV)

10

q2

1

q4

0.1 0

1

2

3

4

5

6

q (nm-1)

Figure 1.33 Breakdown of the Debye approximation for the acoustic modes with nanometric wavelengths. (a) q dependence of the apparent longitudinal phase velocity of a glycerol glass at 150.1 K derived from IXS experiments [171] (gray hexagons) and from lower-frequency techniques: stimulated Brillouin gain spectroscopy (blue rhomb, from Ref. [176]), Brillouin light scattering (red square, from Ref. [61]), and inelastic ultraviolet scattering (green triangle, from Ref. [177]). The dashed line indicates the macroscopic sound limit from Brillouin light scattering results [61]. (b) q dependence of the broadening (FWHM) of the Brillouin peaks derived from IXS experiments (gray hexagons). The green lines correspond to the best q4 and q2 functions fitting the lowand the high-q portions of the IXS data, respectively. Figure adapted from Ref. [171].

general feature common to networks as to molecular glasses [169,171]. The other relevant aspect is that the q4 regime for the acoustic damping is found in the same energy range where the softening of the acoustic branch appears. This softening implies the existence of acoustic-like excitations in excess with respect to the Debye level on the mesoscopic scale or at energies of a few milli-electronvolts. This is exactly the energy range where the BP appears in glasses, suggesting that the softening of the sound velocity adds to the soft acoustic-like modes of the glasses that will definitely contribute to the BP. Furthermore, from a quantitative point of view, the softening phenomenon in the glass can be put in relation with the BP using a crystal-like approach in the whole q range analyzed, in the sense that q can be used to count the low-energy modes as far as they are still plane-wave-like, or as far as the Brillouin peaks are still well defined. With this background, the acoustic density of states g(o) can then be directly related to a combination

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of the longitudinal (L) and transverse (T) acoustic dispersion curves. In this regard, it has been shown that the most significant contribution to the vibrational density of states comes from the transverse acoustic branch, and a strategy has been developed to evaluate this contribution. The main assumptions are that (i) the softening observed on the longitudinal branch is simply the signature of an effect occuring in the transverse branch [165,171], and (ii) the softening in the transverse branch appears at the same energy as in the longitudinal branch, as justified by recent simulation results [178]. The dispersion of the transverse modes can thus be reconstructed starting from the longitudinal one, taking into account the relation between longitudinal and transverse moduli and assuming the bulk modulus to be essentially constant over the considered energy range [171]. The last step to calculate the acoustic contribution to the vibrational density of states is explicitly considering the fact that the acoustic-like modes that one measures through IXS are actually not plane waves but are, instead, characterized by a finite and strongly varying width. To do this, the relation given by Eq. (1.37) was used. The vibrational density of states determined using the above methodology is reported in Fig. 1.34 and compared with the measured one in the case of glycerol [171,179], showing a good agreement. It is important to

g(E)/E 2 (10-4 meV -3)

4

3

2

Debye level 1 0

2

4

6

8

E (meV)

Figure 1.34 Reduced density of vibrational states g(E)/E2 of a glass of glycerol. The hexagons are experimental data from an inelastic neutron-scattering experiment performed at 170 K [179]. The full line represents the theoretical profile calculated following the procedure described in Ref. [171]. The dashed horizontal line symbolizes the Debye level. Figure adapted from Ref. [176].

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note that, because the measured vibrational density of states is consistent with the specific heat data [179], the acoustic softening and broadening determined by IXS can then account for the well-known anomaly found in the specific heat of glasses in the 10 K temperature range. The example given here sheds light on some relevant aspects of the vibrational dynamics of glasses. First, evidence is given of the pivotal role played by the acoustic excitations to describe the origin of the BP, as assumed in many theories or models conceived to explain the low-temperature-specific heat anomalies of glasses. As a result, one observes a failure of the macroscopic Debye continuum model at energies roughly corresponding to the BP, which naturally adds soft states over the Debye level in the millielectronvolt energy range. In the case of glycerol, the observed softening of the acoustic modes is able to quantitatively account for the intensity of the BP. For other systems, it is also possible that additional modes, for example, optic-like ones, pile up in the same energy range [180]. However, ultimately the mechanism proposed is general and provides a way to explain the ubiquitous presence of the BP in glasses. This view is also supported by computational work [173] where a model monatomic glass of extremely large size was simulated. Moreover, in this way, it is indeed proved that the elastic continuum approximation for the acoustic excitations breaks down suddenly on the mesoscopic, mediumrange-order length scale of 10 interatomic spacings, where it is still valid for the corresponding crystalline state. This scale also corresponds to the region where the sound velocity shows a clear decrease compared to the macroscopic value, which is strongly related to the universal excess over the Debye model prediction found in glasses at frequencies of about a few terahertz in the vibrational density of states or at temperatures of tens of Kelvin in the specific heat. Overall, these findings are also most likely at the base of the observation that the BP intensity scales with the continuum elastic properties under different experimental conditions [181–183], providing a valid benchmark for any further modeling of their high-frequency dynamics.

6. FUTURE DEVELOPMENTS Looking at Fig. 1.1, one can readily see that the most interesting (q,E) range (i.e., 0.1–10 nm1 and 0.1–100 meV) is only partially covered by the existing spectroscopies. In particular, any available technique is able to access the q region corresponding to the disorder length scale

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(qx  2p/x  0.1–1.5 nm1) without kinematic constraints, that is, without limitations in the measurable energy/time scale. Time-resolved techniques usually have an experimental time window essentially limited by the temporal width of the source in the short-time/ high-energy side. On the other hand, very long time scales are much more easily accessible, and thus potentially allow the measurement of slow dynamics corresponding to the lower energy side of the excitation spectrum, a spectral range hardly measurable with energy-resolved techniques. Among various time-resolved techniques, the transient grating (TG) scheme—a four-wave mixing spectroscopy—permits direct measurement of F(q,t) at a given q value [184]. This capability makes the TG technique essentially equivalent to conventional energy spectroscopy, such as Brillouin scattering. To date, TG experiments are carried out using conventional laser sources. Therefore, the dynamics may be investigated in a region of q vector ( 0.0001–0.01 nm1) limited by the laser wavelength, while the shorter time scales are currently around 100 femtoseconds/40 meV. We pointed out in the introduction of this chapter (see Fig. 1.1) that today there is no instrument capable of exciting collective modes at the nanoscale (namely in the region of q  qx). The development of a VUV-transientgrating instrument at the FERMI@Elettra FEL source [185] will make it possible to fill this experimental gap. The instrument will cover the region in the q range characteristic of the topological disorder and partially overlap both the ILS/IUVS and the IXS/ INS range, thus making the FEL-TG technique the ideal bridge between macroscopic and microscopic spectroscopy. The uniqueness of the FERMI source compared to other FEL facilities consists in a well-defined (Gaussian-like) time–space profile of the photon pulses that is a mandatory requirement for TG experiments. In a TG experiment (see Fig. 1.35), two laser pulses (pump) interfere on the sample, producing a spatially periodic variation of the material optical properties. The spatial modulation of the grating defines a wave vector q ¼ 4p siny/l1, where l1 and y are the wavelength and the incidence angle of the excitation laser pulses, respectively. A second beam, typically of different wavelength, is used as a probe. It impinges on the induced grating at the Bragg angle producing a diffracted beam, whose time evolution is the signal measured in a TG experiment, and carries out the dynamic information we are looking for. Depending on the experimental setup, different response functions (such as, e.g., F(q,t)) of the investigated material) can be probed.

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Excitation pulses (pump)

Eex1, l1

Standing e.m. wave (Transient Grating)

Sample Transmitted pulse Delayed pulse (probe)

2q

Epr, l2

qB Eex2, l1

qB qB

Edif, l2

Diffracted pulse (signal) Detector

Figure 1.35 Rationale for the transient-grating experiment method.

In Fig. 1.36, we depict a sketch of a typical setup used to carry out TG experiments with laser sources. The pump pulse of wavelength l1 is split into two equal intensity beams, Eex1 and Eex2, by a beam splitter or a diffractive optical element. The two pulses are routed by a mirror and/or a lens system, and impinge into the sample at the angle y in time and space coincidence conditions. The interference pattern of Eex1 and Eex2 generates in the sample a (transient) standing electromagnetic field that periodically modulates its optical properties, that is, Eex1 and Eex2 create the transient grating. The time evolution of the induced grating can be then tracked by another pulse (Epr) of wavelength l2, properly delayed by an optical delay line. Such a probe beam impinges on the sample exactly at the Bragg angle yB ¼ asin(l2 sin (y)/l1), that is, in phase-matching conditions. The diffracted electric field intensity (|Edif|2) as a function of the time delay between the pump and probe beams at the sample position is then recorded by a detection system screened from the strong background, mainly coming from spurious scattering from the pump beams, by means of an optical filter that cuts the radiation at the pump wavelength. This idea has been implemented in numerous setups, which have proved to be extremely reliable and able to produce data of excellent statistical quality, covering over six orders of magnitude in the time scale. The most crucial aspects of a TG experiment are obtaining a reliable time resolution and performing the time scan. There are two main ways to achieve these tasks: (a) employing a pulsed source for the probe beam and, using an optical delay line, controlling the time delay, T, between pump and probe pulses. In this case, it is possible to record the TG signal at known T values and the time scan reduces to the (spatial) scan of the delay line; that is, T ¼ (L  L0)/c, where L and L0 are the actual delay line position

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Pump (l1) Probe (l 2)

Focusing lenses

50–50 beam splitter or diffractive optic

Delay line Routing mirrors

Epr

Eex2

Eex1

2q

Sample

Edif

qB Detector

Figure 1.36 Typical TG setup for laser-based instruments.

and the zero-delay position, respectively, while c is the speed of light. The advantage of this approach is twofold: (i) it is possible to perform extremely fine-grained time scans, since L steps of about 0.3 mm correspond to 1 fs steps; and (ii) the overall time resolution, DT, is essentially given by the jitter between the pump and probe pulses and by the convolution of their time widths (usually >40 fs in TG experiments), while the contribution coming

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from the spatial resolution of the scanning delay line is negligible: DTdline ¼ DL/c  0.3 fs, where DL  0.1 mm is the accuracy of a standard motorized slit. In case the pump and probe beams are generated by the same laser pulse, the contribution arising from the jitter can be neglected. The drawback of this approach is that the time scan is limited by the maximum extension, Lmax  L0, of the delay line. For instance, a 0.3-m long delay line allows measuring time delays not longer than 1 ns. (b) Alternatively, a continuous wave probe with a time-resolved detection system locked to the pump source can be used. In this case, the time resolution is limited by the associated electronic devices. Using a fast detector and a high-performance digitalization data system, it is possible to obtain sub-nanosecond time resolutions. On the other hand, the only limitation concerning the maximum measurable time delay is the repetition rate of the pump source. These two methods are truly complementary and, used together, can in principle provide the best trade-off, which will enable probing both the fast ( ps) and slow (ns) dynamics of the system. However, the characteristic time scale of the slow (acoustic) collective excitations decreases till it reaches the nanosecond range for q > 0.006 nm1, even in systems with very slow sound velocity. Therefore, at q values larger than the ones accessible by laser-based instruments, the leading dynamical features of almost all materials can be probed by using only the delay line system. The main technical challenge is adapting the optical layout depicted in Fig. 1.36 to the spectral range (l ¼ 60–10 nm), stretchable down to 3 nm in the second phase of the FERMI project characteristic of FERMI@Elettra [185]. Here, standard transmission optics cannot be used because of the very strong photon absorption. A possible solution could be to use the first and third harmonics of the FEL as the pump and probe beams, respectively. The two harmonics could be spatially separated by a half-reflecting mirror that intercepts a portion of the FEL beam. The two-excitation pulses (Eex1 and Eex2) may be generated by again splitting one of the two beams by a mirror that intercepts half of the beam. Eex1 and Eex2 have to impinge onto the sample at the appropriate angle for generating the transient grating. This task could be achieved by a set of a graphite-coated mirror for the longer wavelength and a multilayer-coated mirror for the shorter ones. These can be inserted in the beam pathways at the appropriate angle in order to obtain the desired y value. These mirrors can be mounted on motorized stages in order to provide the necessary fine-tuning for achieving both an accurate spatial and temporal coincidence at the sample position. An appropriate combination of the (y,l1) values will permit covering the

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q ¼ 0.017–4 nm1 range. The probe beam of wavelength l2 ¼ l1/3 and intensity 1% of the first harmonic can be delayed by a delay line capable of providing the difference in time on the order of nanoseconds. This is realized by using four multilayer mirrors operating at 45 angle. The use of four multilayer mirrors will guarantee a suppression of the fundamental (first harmonic) of a factor larger than 106. In this way, the probe is almost perfectly spectral-pure. The probe beam will then be routed to the sample at the Bragg angle by a set of mirrors similar to the ones used to route the pump beams. Finally, the focal spots at the sample position will be adjusted by focusing mirrors with variable focal lengths. The numerous optical elements will reduce the photon flux of both pump and probe pulses on the sample (100–1000-fold, depending on l) that will still allow obtaining a reliable TG signal. Indeed, we estimated that in several cases the diffraction efficiency has values from 104 to 107 considering pump beams with 100 nJ energy. These efficiencies are high enough to obtain a detectable diffracted signal keeping the power densities on the sample surface well below the ablation threshold and to avoid unwanted photo-induced heating of the sample. For samples that present a strong absorption in the VUV range, the standard transmission diffraction geometry, depicted in Fig. 1.36, has to be replaced by reflection geometry. In this case, the pumped transient grating will be thin and diffraction of the probe beam will also take place at angles different from the Bragg one (see, for instance, Tobey et al. [186]). A large-area CCD detector will then be used to collect the entire diffraction pattern as a function of the delay time. In summary, the main task of the TIMER project, that is, probing the dynamics in disordered systems in the q range (0.1–1 nm1) characteristic of the topological disorder over a wide time window ( 0.2–10,000 ps) never exploited before by a single instrument, can be achieved at the FERMI@Elettra FEL source. Furthermore, the possibility to create and probe sub-picosecond transient gratings with a spatial period in the nanometer range will be extremely useful in other fields of research as well, as it will provide the unique capability to measure correlations, electronic excitation lifetimes, heat transport, intramolecular dynamics, and nonlinear material responses. Finally, TG is also an extremely sensitive probe for surfaces and interfaces, with potential applications in the study of thin films and nanostructured materials. Specifically designed for film samples is instead a new experimental apparatus based on a broadband version of interferometric picosecond

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acoustics (BPA) recently developed to catch, in a single measurement, sound velocity and attenuation over a range of 30–300 GHz [187]. Besides the obvious advantage in terms of the accessible frequency–wavelength region, BPA time–domain detection enables spatial resolution, allowing detection of any spatial inhomogeneities, that is, regions of differing mobility, experienced by the phonons during their propagation. This new technique, tested on fused silica [187], is expected to be implemented in studies on a large variety of amorphous systems and over wide temperature ranges.

ACKNOWLEDGMENTS U. Buchenau, S. Corezzi, L. Lupi, A. Paciaroni, G. Ruocco, S.C. Santucci, F. Scarponi, T. Scopigno, A. Gessini, F. Bencivenga, and R. Verbeni are gratefully acknowledged for a number of discussions on the issues presented here. C. Masciovecchio acknowledges funding by the European Research Council—Contract ERC No. 202804.

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