The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy

Alessandro Cunsolo

doi:10.3390/app6030064

The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy

Cunsolo, Alessandro 2016-02-26 00:00:00 applied sciences Review The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy Alessandro Cunsolo National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA; acunsolo@bnl.gov; Tel.: +1-631-344-5564; Fax: +1-631-344-8189 Academic Editor: Christoph Hauri Received: 24 November 2015; Accepted: 25 January 2016; Published: 26 February 2016 Abstract: Approximately 50 years of inelastic scattering studies of noble gases are reviewed to illustrate the main advances achieved in the understanding of the THz dynamics of simple systems. The gradual departure of the spectral shape from the hydrodynamic regime is discussed with an emphasis on the phenomenology of fast (sub-ps) relaxation processes. This review shows that relaxation phenomena in noble gases have an essentially collisional origin, which is also revealed by the parallelism between their characteristic timescale and the interatomic collision time. Additionally, recent THz spectroscopy results on noble gases at extreme thermodynamic conditions are discussed to illustrate the need for a revision of our current understanding of the supercritical phase. Keywords: inelastic X-ray scattering; inelastic neutron scattering; noble gases; relaxation phenomena; liquid and supercritical systems 1. Introduction The study of the collective molecular dynamics of liquids and glassy materials has been a vibrant ﬁeld of research since the dawn of modern science. However, in spite of an intensive theoretical, experimental and computational scrutiny, the dynamic response of these systems still presents many unsettled aspects, mainly owing to the inherent disorder and complex movements of their microscopic constituents. Among various variables unveiling the dynamic behavior of a ﬂuid, the spectrum of density ﬂuctuations, S(Q,!), is of particular interest, since it is directly accessible by both spectroscopic techniques and computer simulation methods. In spectroscopy experiments, S(Q,!) is probed by generating density ﬂuctuations on a target sample through the collision with a beam of particle-waves, i.e., photons or neutrons. During this collision, an energy h ¯! and a momentum h ¯ Q (here h ¯ = h/2 with h being the Plank constant) are transferred from the probe particles to the sample and the former are scattered over the entire solid angle. It was demonstrated that for both neutron [1] and X-ray [2] probes, the scattering intensity is proportional to the dynamic structure factor, S(Q,!), which provides a snapshot of the dynamic response of the sample through an average over timescales 9 ! and distances 9 Q . Therefore, the (Q,!) region explored in a spectroscopy measurement can be expanded enough to cover the whole transition of the dynamics from the macroscopic regime to the microscopic one. The shape of S(Q,!) is exactly predicted within these two limits and essentially unknown in between, i.e. in the so-called mesoscopic region, corresponding to Q’s and !’s matching the inverse of intermolecular separations and cage oscillation frequencies, respectively. Investigations of this region may unravel fundamental aspects of molecular motions and mutual interactions in disordered systems and can be ideally performed by X-ray (IXS) and neutron (INS) scattering techniques, as well Appl. Sci. 2016, 6, 64; doi:10.3390/app6030064 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 64 2 of 31 as by computational methods. Due to the lack of rigorous theoretical predictions, our understanding of the dynamics of disordered systems at mesoscopic scales is often limited to a phenomenological analysis of experimental and computational results. 1.1. The Dynamics of Molecules at Mesoscopic Scales In the macroscopic limit, a liquid appears as a continuum, whose dynamic response is averaged over a large amount of intermolecular interactions. For such a system, the hydrodynamic theory can be consistently used to describe S(Q,!) and ultimately leads to predict a sharp triplet proﬁle [3]. This is composed by the heat diffusion (Rayleigh) central peak, relating to entropy ﬂuctuations, which diffuse at constant pressure (P), and the two Brillouin acoustic side peaks, connected to P ﬂuctuations, propagating at constant entropy. In the opposite regime, customarily referred to as single particle or impulse approximation limit, the probed dynamic event is essentially the free recoil of the single struck atom after the collision with the probe and before any successive interaction with ﬁrst neighboring atoms. In this limiting region, the dynamic response of the ﬂuid reﬂects the momentum distribution of the struck atom, which is Gaussian in shape for both classical and quantum systems. Several theoretical models have been developed in an attempt to account for the gradual departure of the spectral shape from the hydrodynamic regime. Perhaps the most successful approach is based upon the framework of the Zwanzig-Mori [4,5], or memory function, formalisms. Its core assumption is that the departure from the hydrodynamic regime is gradual enough to be accurately described by leaving the formal structure of hydrodynamic equations unchanged, yet generalizing some thermodynamic and transport parameters as local variables of space and time. From a physical point of view, this local character reﬂects the non-homogeneous, non-stationary nature of matter at mesoscopic scales. A similar approach was successfully proposed by a large number of IXS works on samples of increasing complexity, including: (1) noble gases [6,7]; (2) diatomic liquids [8]; (3) liquid metals [9–11]; (4) hydrogen-bonded systems [12–14]; (5) glass formers [15–18]. 1.2. The Departure from Hydrodynamic Regime and the Onset of Relaxation Processes Among various phenomena occurring at the departure from the hydrodynamic regime, the coupling of density ﬂuctuations with relaxation phenomena is perhaps the most solidly understood [6,7,12,13,19]. To gain a basic insight on the physics of a relaxation process, one may start from considering that scattering-generated density waves bring the sample away from its local equilibrium. In response, energy redistributions take place from the density perturbation toward internal degrees of freedom, ultimately driving the sample to a new local equilibrium. The timescale needed by the sample to relax to the equilibrium, , depends both on the wavelength = 2/Q of the considered density wave and on the sample characteristics. For instance, if increases so do both the number of rearranging molecules and the relaxation time. Furthermore, the timescale of structural relaxations has a temperature (T-)dependence roughly as steep as the one of viscosity, due to the circumstance that viscous resistance hampers internal rearrangements. These last considerations only apply the so-called structural relaxations, involving collective rearrangements of the structure, which are dominating in highly interconnected many body systems. Conversely, for monatomic ﬂuids relaxation phenomena mostly have a collisional nature [7,20], especially when matches ﬁrst neighboring molecules’ separations. Collisional relaxations relate Appl. Sci. 2016, 6, 64 3 of 31 to energy transfers between density ﬂuctuations and mutual inter-particle collisions. Compared to structural relaxations, they are faster (usually spanning the 10 s range) and weakly dependent on thermodynamic conditions [7,20]. In general, collisional and structural relaxations coexist in a given ﬂuid and their relative relevance depends on the nature of intermolecular interactions, thermodynamic conditions and probed Q value. Experimental studies of the Q or T-transition between structural and collisional relaxation regimes often present major challenges, as the required access to large Q (or T) intervals and the high statistical accuracy in the spectral shape measurement. Furthermore, advances in the ﬁeld are currently held back by instrumental limitations, especially concerning resolution width and spectral contrast. Nonetheless, important advances in the understanding of these phenomena have been made and, most importantly, next generation instruments with the required performance will be soon available for the scientiﬁc community. 1.3. The Central Role of Noble Gases Noble gases are ideal samples to investigate relaxation phenomena, due to their extremely simple microscopic constituents and mutual interactions, which make them ideal prototypes to test theoretical models. In contrast to molecular ﬂuids, their microscopic components lack internal degrees of freedom and, contrary to, e.g., metallic liquids, they have simple and short-ranged inter-particle interactions. For instance, the intrinsic simplicity of these samples is crucial to reliably approximate the interatomic potential in molecular dynamics (MD) computer simulations [21]. Another advantage of noble gases is the large compressibility, which is a key factor to achieve substantial variations of density or, equivalently, of the strength of atomic interactions, even with moderate thermodynamic changes. The purpose of this review is to provide a short overview of THz spectroscopy studies of S(Q,!) in noble gases and in particular, to discuss the insight they shed on fast relaxation phenomena. Following an essentially chronological order, the review covers a period of approximately 50 years, from the mid-1960s to the present. Beginning from pioneering INS studies mostly seeking for reminiscences of an hydrodynamic S(Q,!) shape beyond the continuous limit, it discusses the new opportunities offered by the development of IXS toward the end of the past millennium, while illustrating the insight they provide onto the viscoelastic character of the THz dynamics of noble gases. The last sections of this review illustrate recent measurements in the deeply supercritical region calling for a global reconsideration of our understanding of the phase diagram of a ﬂuid. 2. Measurements of the THz Spectrum of Noble Gases up to the 1990s 2.1. The Spectral Line-Shape in the Continuous Limit As mentioned in the introductory section, the variable of interest for this review is the spectrum of density ﬂuctuations, or dynamic structure factor: » » 8 A E Ñ Ñ Ñ Ñ SpQ, wq d r drp r , tqdr p r , 0q exp ipQ r wtq dt (1) V 8 Ñ Ñ with drp r , tq being the space p r q and time (t) dependent density ﬂuctuation of the target sample having volume V. In the quasi-macroscopic regime, S(Q,!) can be described using the hydrodynamic theory of continuous media [3]. This approach stems from the formal expression of the conservation laws of the (density of) mass, momentum and energy of the ﬂuid plus two constitutive laws: the Navier-Stokes’ and heat transport’s equations. As extensively discussed in several monographies [3,22], the derived spectral shape is dominated by few long-lived collective modes, customarily referred to as hydrodynamic modes. These modes form a triplet well approximated by the following expression: Appl. Sci. 2016, 6, 64 4 of 31 SpQq A z z z h h S S SpQ, wq A 2 2 2 2 2 2 p w z pw w q z pw w q z S S S S (2) pw w q pw w q S S A b 2 2 2 2 pw w q z pw w q z S S S S with b rA z {p1 A q z s{w and I p1 I q{2. s s s h h h h One readily recognizes the three main components of the spectral shape (from left): (1) The heat diffusion (Rayleigh) central peak, which relates to entropy ﬂuctuations diffusing at constant pressure (P). (2) The two Brillouin acoustic side peaks, connected to P-ﬂuctuations propagating at constant entropy. (3) An additional contribution (term 9 b) asymmetric around the Brillouin peaks position having negative tails. This term distorts the Lorentzian terms 1) and 2) ultimately enabling the convergence of the spectral moments w SpQ, wqdw, with n ¤ 2. The hydrodynamic proﬁle in Equation (2) contains shape parameters whose Q-dependence is not explicitly considered here. However, at low Qs the latter can be expressed through a polynomial Q-expansion (see, e.g., Equations (42)–(45) of [23]), which at the lowest order reads: w c Q (3a) ' s s z rpg 1q D n s Q {2 (3b) T L z D Q (3c) Here c , D and u are, respectively, the adiabatic sound velocity the thermal diffusivity and the s T L longitudinal kinematic viscosity. 2 2 It should be recognized that Equation (3a–c) are valid only if D Q ,GQ c Q, that is if the T s lifetime of hydrodynamic modes is much longer than the acoustic period. Under these conditions, the hydrodynamic modes are long-lived excitations appearing in the spectrum as well-resolved and sharp peaks, customarily referred to as Rayleigh-Brillouin triplet. Experimental methods such as Brillouin light scattering (BLS) provide a measurement of Rayleigh-Brillouin triplet by typically spanning the 3 2 1 Appl. Sci. 2016, 6, 64 5 of 34 10 –10 nm Q-range. An example of the simpliﬁed hydrodynamic spectrum is presented in Figure 1 along with the three separate components discussed above. Rayleigh-Brillouin triplet Brillouin peaks Rayleigh peak Asymmetric factor 0.2 0.0 -15 -10 -5 0 5 10 15 (GHz) Figure 1. Typical shape of the Rayleigh‐Brillouin triplet measured by Brillouin light scattering. The separate Figure 1. Typical shape of the Rayleigh-Brillouin triplet measured by Brillouin light scattering. contributions to the total shape are represented by line of different color, as indicated in the legend. The separate contributions to the total shape are represented by line of different color, as indicated in the legend. The Rayleigh‐Brillouin triplet in Equation (2) will be hereafter quoted as either generalized or simplified hydrodynamic spectrum, respectively with or without the lowest order Q approximation in Equations (3a–c). 2.2. 1922–1969: From Theoretical Prediction to Actual Measurements The hydrodynamic shape of the spectrum was theoretically predicted since the seminal work of Brillouin [24] and the successive theoretical work of Landau and Plazeck during the 1930s [25] (see also the review by Mountain). However, the first experimental measurements of the Brillouin spectrum of liquids −7 could only be performed much later [26], mainly owing to the extremely fine resolving power (ΔE/E < 10 ) required for similar measurements in typical liquid samples. This performance imposes severe technical challenges that can only be addressed by complex interferometric techniques. In regards to noble gases, the first Brillouin study dates back to the measurement on Ar and Ne along the coexistence line [27] performed with a single‐pass Fabry‐Perot interferometer. Figure 2 provides an example of the Brillouin spectrum of dense Ar measured in this work. The hydrodynamic spectrum reconstructed using thermophysical and transport properties of the sample is also reported for comparison. It clearly appears that the latter profile essentially consists of three extremely narrow peaks, in full consistency with the simple hydrodynamic condition . Indeed the width of triplet components measured in [27] was either coincident DQQ ,Γ cQ Ts with the resolution function (for the Raileigh peak) or only slightly larger (for Brillouin peaks). Intensity (arbit. units) Appl. Sci. 2016, 6, 64 5 of 31 The Rayleigh-Brillouin triplet in Equation (2) will be hereafter quoted as either generalized or simpliﬁed hydrodynamic spectrum, respectively with or without the lowest order Q approximation in Equation (3a–c). 2.2. 1922–1969: From Theoretical Prediction to Actual Measurements The hydrodynamic shape of the spectrum was theoretically predicted since the seminal work of Brillouin [24] and the successive theoretical work of Landau and Plazeck during the 1930s [25] (see also the review by Mountain). However, the ﬁrst experimental measurements of the Brillouin spectrum of liquids could only be performed much later [26], mainly owing to the extremely ﬁne resolving power (DE/E < 10 ) required for similar measurements in typical liquid samples. This performance imposes severe technical challenges that can only be addressed by complex interferometric techniques. In regards to noble gases, the ﬁrst Brillouin study dates back to the measurement on Ar and Ne along the coexistence line [27] performed with a single-pass Fabry-Perot interferometer. Figure 2 provides an example of the Brillouin spectrum of dense Ar measured in this work. The hydrodynamic spectrum reconstructed using thermophysical and transport properties of the sample is also reported for comparison. It clearly appears that the latter proﬁle essentially consists of three extremely narrow 2 2 peaks, in full consistency with the simple hydrodynamic condition D Q ,GQ c Q. Indeed the width of triplet components measured in [27] was either coincident with the resolution function (for Appl. Sci. 2016, 6, 64 6 of 34 the Raileigh peak) or only slightly larger (for Brillouin peaks). 1.0 0.8 0.6 0.4 0.2 0.0 -4 -3 -2 -1 0123 4  (GHz)  Figure 2. The Rayleigh Brillouin spectrum of Ar measured along the coexistence line by Brillouin light Figure 2. The Rayleigh Brillouin spectrum of Ar measured along the coexistence line by Brillouin light scattering. The data are redrawn with permission from [27], copyrighted by the American Physical Society. scattering. The data are redrawn with permission from [27], copyrighted by the American Physical The hydrodynamic spectrum reconstructed using tabulated thermophysical properties of the sample and Society. The hydrodynamic spectrum reconstructed using tabulated thermophysical properties of the bulk viscosity measurements [28] are reported for comparison as a black line. sample and bulk viscosity measurements [28] are reported for comparison as a black line. From the position of the peak in the Rayleigh‐Brillouin spectrum, Fleury and Boon derived a value of sound velocity slightly lower than the one previously measured by ultrasound (US) absorption techniques From the position of the peak in the Rayleigh-Brillouin spectrum, Fleury and Boon derived at ≈1 MHz. Although such an apparent frequency‐decrease of sound velocity, also observed in water [29], a value of sound velocity slightly lower than the one previously measured by ultrasound (US) initially raised some interest, it lacks a particular physical significance [30] (see also [22], page 263). Rather, absorption techniques at 1 MHz. Although such an apparent frequency-decrease of sound velocity, it follows from the circumstance that the sound velocity measured by BLS (hypersonic velocity) is also observed in water [29], initially raised some interest, it lacks a particular physical signiﬁcance [30] expectedly lower by few percent than the counterpart measured by US (ultrasonic velocity), unless (see also [22], page 263). Rather, it follows from the circumstance that the sound velocity measured by competing viscoelastic effects become relevant. BLS (hypersonic Overall, the velocity) shape of is expectedly the spectrum lower predic by tedfew by per the cent hydrthan odynami theccounterpart theory was experi measur mentally ed by US confirmed by Brillouin measurement in [27], even though a slight ω‐dependence of viscosity was inferred. (ultrasonic velocity), unless competing viscoelastic effects become relevant. At this stage the occurrence of a hydrodynamic triplet at macroscopic scales seemed well‐assessed Overall, the shape of the spectrum predicted by the hydrodynamic theory was experimentally experimentally, while a remaining question concerned the persistence of a suitably generalized conﬁrmed by Brillouin measurement in [27], even though a slight !-dependence of viscosity was hydrodynamic profile down to mesoscopic scales. Most experimental studies described in the following inferred. At this stage the occurrence of a hydrodynamic triplet at macroscopic scales seemed paragraphs attempted to answer to this question. well-assessed experimentally, while a remaining question concerned the persistence of a suitably generalized hydrodynamic proﬁle down to mesoscopic scales. Most experimental studies described in 2.3. 1964–1967: Seeking for a Triplet Shape at Mesoscopic Scales the following paragraphs attempted to answer to this question. The use of a bare hydrodynamic description of the spectrum becomes questionable when approaching the mesoscopic regime, as therein the matter is no longer continuous, nor stationary. However, there are solid reasons to assume a suitably generalized hydrodynamic theory to hold validity even in this time‐ space domain. To illustrate this point, it is useful to recognize that for a system at densities typical of the −10 liquid phase, the mean free path spans the 10 m window, potentially becoming much smaller than the atomic size. Under these conditions, the movements permitted to the atoms are mainly short, vibration‐ −13 −1 like, cage oscillations (≈10 s). Consequently, even in the (Q ≈ nm , ω ≈ THz) mesoscopic range, the response of the system is still “averaged” over a large number of elementary dynamic interactions, which is a necessary pre‐requisite for a suitably generalized hydrodynamic description. In this perspective, the persistence of spectral features in the THz spectrum reminiscent of Brillouin hydrodynamic peaks did not appear to be a groundless expectation. Indeed, several INS experiments performed during the mid‐1960s aimed at validating this hypothesis by examining inelastic peaks in the spectrum of simple fluids. For instance, this is the case of an INS measurements on dense Ne, Ar and D2 [31], which clearly documented Intensity (arbit. units) Appl. Sci. 2016, 6, 64 6 of 31 2.3. 1964–1967: Seeking for a Triplet Shape at Mesoscopic Scales The use of a bare hydrodynamic description of the spectrum becomes questionable when approaching the mesoscopic regime, as therein the matter is no longer continuous, nor stationary. However, there are solid reasons to assume a suitably generalized hydrodynamic theory to hold validity even in this time-space domain. To illustrate this point, it is useful to recognize that for a system at densities typical of the liquid phase, the mean free path spans the 10 m window, potentially becoming much smaller than the atomic size. Under these conditions, the movements permitted to the atoms are mainly short, vibration-like, cage oscillations (10 s). Consequently, even in the (Q nm , ! THz) mesoscopic range, the response of the system is still “averaged” over a large number of elementary dynamic interactions, which is a necessary pre-requisite for a suitably generalized hydrodynamic description. In this perspective, the persistence of spectral features in the THz spectrum reminiscent of Brillouin hydrodynamic peaks did not appear to be a groundless expectation. Indeed, several INS experiments performed during the mid-1960s aimed at validating this hypothesis by examining inelastic peaks in the spectrum of simple ﬂuids. For instance, this is the case of an INS measurements on dense Ne, Ar and D [31], which clearly documented the presence of “extended hydrodynamic” modes in the spectrum. Due to the largely coherent cross section of these samples, both inelastic and quasi-elastic spectral features appeared in the measured spectrum. In particular, it was observed that at low Q values the inelastic shift of the side peaks approached from above the linear hydrodynamic law c Q predicted by Equation (3a). This urged the authors to interpret these peaks as the ﬁnite-Q extensions of the acoustic Brillouin peaks to mesoscopic scales. Conversely, Kroo et al. [32] reported no direct evidence of inelastic structures, however they inferred the existence of phonon like excitations indirectly from a comparison between liquid and solid phase spectra. The discrepancy between the results of Chen and those of Kroo is a likely consequence of the different incident wavelength used in the two experiments (4.1 Å and 5.3 Å respectively), which made the two sets of data hardly comparable. In a successive INS work, Sköld et al. [33] measured the spectrum of Ar in liquid (at T = 94 K and 102 K) and solid (at T = 68 K and 78 K) phases. The authors observed a linear Q-dependence of the inelastic peak at the lowest Qs, with a slope consistent to the adiabatic sound velocity. Furthermore, they showed that, at higher Qs, the sound dispersion curve of the liquid sample vaguely resembles the phonon dispersion curve of the solid, thus suggesting that the local pseudo-periodicity of the liquid structure gives rise to quasi-periodic zones reminiscent of the Brillouin zones of a crystal. Since clear signatures of acoustic excitations in the liquid were found in a Q range equivalent to the second pseudo-Brillouin zone, the authors concluded that at least two somehow “loose” pseudo-Brillouin zones could be identiﬁed in liquid argon. A similar conclusion was reached by an earlier INS investigation on liquid Pb [34], as well as a computer simulation on Rb [35]. The latter work used an inter-particle potential model characterized by an oscillatory decay to zero [36] instead of the Lennard-Jones potential routinely used to simulate noble gases. These differences in the interatomic interaction explain why in liquid metals inelastic modes are well-resolved up to almost the position of ﬁrst diffraction peak (Q ), whereas in noble gases they become overdamped for Q > Q /2. m m 2.4. 1971: First Evidences of a Rayleigh-Brillouin triplet beyond the Continuous Limit The ﬁrst convincing evidence of the persistence of well-deﬁned inelastic peaks beyond the hydrodynamic regime was reported in an INS measurement of Bell and collaborators on supercritical 1 1 neon [37]. This work spanned unusually low Q values (0.6 nm ¤ Q ¤ 1.4 nm ), which substantially reduced the dynamic gap existing between the quasi-continuous regime probed by BLS (with Q 2 1 spanning the 10 nm window) and the range covered by “standard” INS measurements (typically Q ¥ 2–3 nm ). A best ﬁt of measured line-shape with a simpliﬁed hydrodynamic similar to the one deﬁned by Equations (2) and (3a–c) enabled the authors to conclude that the simple hydrodynamic theory consistently describes the spectral shapes measured well beyond the continuous approximation. Figure 3 provides an example of the spectral proﬁles measured by Bell et al., clearly showing the persistence of neat side peaks reminiscent of the hydrodynamic acoustic excitations. Appl. Sci. 2016, 6, 64 7 of 31 Again, the inelastic shift of side peaks reportedly depends linearly on Q, with a slope consistent with the known value of adiabatic sound velocity. Interestingly, the linear dispersion seemed to persist over the whole Q-range explored by Bell et al., even when the side peaks transform to 2 2 broad shoulders, for which the simple hydrodynamic approximation (D Q ,GQ c Q) is clearly unfulﬁlled. Additionally, the widths of both inelastic (Brillouin) and quasi-elastic (Rayleigh) peaks were found to depend on Q nearly consistently with the predictions of Equations (3a–c). These aspects will be discussed in greater detail below. It is worth acknowledging that comparably well-resolved THz inelastic peaks were previously observed in a largely quantum system as superﬂuid He at 1.1 K [38] and in the mentioned work on liquid metals by Dorner et al. [39]. 2.5. 1973–1975: First Signatures of A Viscoelastic Behavior in Real and Simulated Noble Gases A successive experiment performed on neon by Bell et al. [37] covered Q values largely exceeding Appl. Sci. 2016, 6, 64 8 of 34 1 1 the simple hydrodynamic limit (2.7 nm ¤ Q ¤ 15 nm ), strongly overlapping the range covered by more “standard” INS measurements. Not surprisingly, the modeling of the line-shape at these higher Qs A successi was less ve exp straightforwar eriment perfod, rmed requiring on neon aby generalization Bell et al. [37] coof vered transport Q values variables, largely excee asd pr inescribed g the −1 −1 simple hydrodynamic limit (2.7 nm ≤ Q ≤ 15 nm ), strongly overlapping the range covered by more by the “molecular hydrodynamic” approach. Speciﬁcally, the used line-shape model consisted in “standard” INS measurements. Not surprisingly, the modeling of the line‐shape at these higher Qs was less a generalization of the simple Brillouin triplet deﬁned by Equations (2) and (3a–c) in which allowance straightforward, requiring a generalization of transport variables, as prescribed by the “molecular was made for a Q and/or !-dependence for some thermodynamic and transport parameters appearing hydrodynamic” approach. Specifically, the used line‐shape model consisted in a generalization of the in the hydrodynamic equations. simple Brillouin triplet defined by Equations (2) and (3a–c) in which allowance was made for a Q and/or In particular, it was observed that the ﬁnite Q generalization of both D and had the form of ω‐dependence for some thermodynamic and transport parameters appearing in the hydrodynamic equations. a sharp, nearly Lorentzian, decay, indicating a decreasing weight of thermal ﬂuctuations at mesoscopic In particular, it was observed that the finite Q generalization of both DT and γ had the form of a sharp, distances. nearly Lorentzian, Furthermor dee, cathe y, indicating longitudinal a decr viscosity easing weneeded ight of th to ermal be generalized fluctuations at as meso a frequency scopic distances. dependent Furthermore, the longitudinal viscosity needed to be generalized as a frequency dependent variable as a variable as a typical manifestation of viscoelasticity. Although viscoelastic effects on the line-shape typical manifestation of viscoelasticity. Although viscoelastic effects on the line‐shape will be discussed in will be discussed in greater detail in a successive section, it’s worth emphasizing here that the core greater detail in a successive section, it’s worth emphasizing here that the core feature of viscoelastic fluids feature of viscoelastic ﬂuids is their sharply frequency dependent dynamic response, resembling either is their sharply frequency dependent dynamic response, resembling either the one of a liquid or the one of the one of a liquid or the one of a solid at low or high frequency, respectively. The crossover between a solid at low or high frequency, respectively. The crossover between the liquid‐like (or viscous) and solid‐ the liquid-like (or viscous) and solid-like (or elastic) regimes is marked by the relaxation frequency like (or elastic) regimes is marked by the relaxation frequency 1/τ. In a scattering experiment the viscous‐ 1/. In a scattering experiment the viscous-to-elastic transition can be observed when the applied to‐elastic transition can be observed when the applied perturbation (e.g., a scattering generated acoustic perturbation waves) attai (e.g., ns a fr ae scattering quency (csQ generated ) much larg acoustic er than 1/waves) τ. In practice attains , this a fr ca equency n be obta(inceQ d )by much increasing larger Qthan and/or τ (by lowering T). 1/. In practice, this can be obtained by increasing Q and/or (by lowering T). -1 Q = 1.4 nm -1 Q = 1.2 nm -1 Q = 1.0 nm -1 Q = 0.8 nm -1 Q = 0.6 nm -0.4 -0.2 0.0 0.2 0.4   (meV) Figure 3. Spectral lineshapes of neon measured by inelastic neutron scattering INS in neon at the low Figure 3. Spectral lineshapes of neon measured by inelastic neutron scattering INS in neon at the exchanged momenta indicated in the plot, the red line through data roughly connects the tops of inelastic low exchanged momenta indicated in the plot, the red line through data roughly connects the tops of peaks. Data are redrawn with permission from [40]. inelastic peaks. Data are redrawn with permission from [40]. On this ground, it is reasonable to expect that even a simple fluid at some Q, T values can exhibit solid‐ like properties, such as, for instance, the ability of supporting transverse acoustic waves. This possibility could only be tested using computer MD simulation, which can determine the correlation function between transverse components of atomic velocities, or its Fourier transform, customarily referred to as the transverse current spectrum. The transverse plane is defined as orthogonal to the momentum exchanged in the scattering event, Q . Indeed, the first indication of a high Q transverse acoustic propagation in a Appl. Sci. 2016, 6, 64 8 of 31 On this ground, it is reasonable to expect that even a simple ﬂuid at some Q, T values can exhibit solid-like properties, such as, for instance, the ability of supporting transverse acoustic waves. This possibility could only be tested using computer MD simulation, which can determine the correlation function between transverse components of atomic velocities, or its Fourier transform, customarily referred to as the transverse current spectrum. The transverse plane is deﬁned as orthogonal to the momentum exchanged in the scattering event, }Q. Indeed, the ﬁrst indication of a high Q transverse acoustic propagation in a monatomic system was inferred from the presence of a clear inelastic peak in the transverse current spectrum of a Lennard-Jones model representing Ar [41]. In this work, viscoelastic effects on the spectrum of density ﬂuctuations were described assuming a Gaussian time decay of memory function. A successive MD simulation performed on a Lennard-Jones model of Ar at the triple point [42] used instead a “more standard” exponential ansatz for the memory function within the assumptions of either a single or a double timescale. It was observed that the two alternative hypotheses did not yield appreciably different best-ﬁt results. Furthermore, this study conﬁrmed the previous observation of a Q-decay of “the thermal” parameters D and g and indicated a relaxation time ranging in the 10 ps interval with smooth variations. 2.6. 1978–1987: The Failure of the Three Modes Description High Q in Liquid Noble Gases and the Test of Kinetic and Mode Coupling Theories Successive INS works on Ne [43] and Ar [43–47] proposed a line-shape modeling consisting on the sum of generalized Lorentzian terms. Different versions of such a sum were used by superimposing the convergence of spectral moments of increasing order. Overall, it was observed that a spectral shape similar to the one in Equation (2) provides an accurate description of the spectrum of both liquid and supercritical noble gases. However, the use of such a model led to somehow controversial results for liquid sample at high Qs, as the presence of a propagation gap, that is a Q-region where ! = 0. A similar gap was also reported by INS measurements in molten salts [48] and liquid He [49,50] and its presence was also predicted by the kinetic theory [51]. Its physical origin was connected to the prevalence of dissipative forces over elastic ones, which prevent the sound from propagating at some Qs. In this regard, it is interesting that the gap seemingly disappeared upon increasing the density and approaching solidiﬁcation [45]. An example of the pressure dependence of this effect is proposed by Figure 4. There, the various dispersion curves are divided for the corresponding sound velocity, thus expectedly joining the same hydrodynamic linear law, upon approaching the Q = 0 limit. From the plot it can be readily noticed that the Q-window for which ! = 0 reduces upon increasing the pressure. Although the presence of a region of forbidden sound propagation raised some initial interest among researchers, its physical signiﬁcance seemed also controversial. In this regard, it should be acknowledged that the observed gap is centered on the position of the ﬁrst diffraction peak, where the so-called de Gennes narrowing occurs [52], and the spectral shape is reduced to a narrow featureless peak. For this reason, the use of a “three pole” line-shape modeling may give questionable results in this Q-window, also considering that inelastic features in the spectrum become dramatically overdamped [53,54], and their real position somehow ill-determined. Apart from those based on Equation (2), no other line-shape modeling used in the literature ever indicate the occurrence of such a gap and, furthermore, its presence leads to a clear physical inconsistency. In fact from a simple ﬁnite Q extension of the compressibility theorem ([55], see also Equation (13a)) one has w ¥ k T{SpQq, from which it can be deduced that a vanishing ! implies s B s a diverging S(Q), which clearly has no physical foundation for a disordered system. Aside from these high Qs ﬂaws, the “three pole” proﬁle in Equation (2) has, in principle, a solid physical ground. In fact, from one side, it can be derived using a macroscopic description of density ﬂuctuations as the hydrodynamic theory; on the other, it is consistent with the result of a truly microscopic approach as the kinetic theory [56,57]. This can be demonstrated by developing the S(Q,!) Appl. Sci. 2016, 6, 64 9 of 31 as a sum of generalized Lorentzian terms [58], upon retaining only the three dominant ones and Appl. Sci. 2016, 6, 64 10 of 34 superimposing on them the fulﬁlment of the ﬁrst three sum rules. Argon at T = 120 K P = 400 bar P = 115 bar P = 20 bar 0 5 10 15 20 -1 Q(nm ) Figure 4. The sound dispersion curves measured in Ar at increasing pressure [47] and normalized to the Figure 4. The sound dispersion curves measured in Ar at increasing pressure [47] and normalized to respective sound velocities, as derived from thermodynamic data [59]. The arrow indicates the presence of the respective sound velocities, as derived from thermodynamic data [59]. The arrow indicates the a propagation gap, i.e. of a Q region where the sound frequency vanishes. The solid line represents the presence of a propagation gap, i.e., of a Q region where the sound frequency vanishes. The solid line hydrodynamic limiting dispersion expectedly joined by all curves in the Q = 0 limit. The dashed curves are represents the hydrodynamic limiting dispersion expectedly joined by all curves in the Q = 0 limit. the low Q prediction of the Mode Coupling Theory as derived for the curves of corresponding color. The dashed curves are the low Q prediction of the Mode Coupling Theory as derived for the curves of corresponding color. It is worth noting that the kinetic approach provides, in principle, a microscopically rigorous theory of density fluctuation. This stems from the assumption that the dynamic variables of the system depend on It is worth noting that the kinetic approach provides, in principle, a microscopically rigorous both atomic velocities and spatial coordinates, as required for a reliable account of microscopic interactions. theory of density ﬂuctuation. This stems from the assumption that the dynamic variables of the Unfortunately, the predictions performed using the kinetic theory usually give poor results at high density system depend on both atomic velocities and spatial coordinates, as required for a reliable account of due to the increasing weight of correlated collisions. The INS measurements of Postol and Pelizzari on microscopic interactions. Unfortunately, the predictions performed using the kinetic theory usually supercritical Ar [60] proposed an experimental test of the kinetic theory predictions based on the give poor results at high density due to the increasing weight of correlated collisions. The INS generalized Enskog approach. As a result, no quantitative agreement was demonstrated between measurements of Postol and Pelizzari on supercritical Ar [60] proposed an experimental test of the theoretical and experimental results, although at the low density (10.5 atoms/nm ) and high temperature kinetic theory predictions based on the generalized Enskog approach. As a result, no quantitative (295 K) probed in the experiment it was reasonable to assume the kinetic theory description to hold validity. agreement was demonstrated between theoretical and experimental results, although at the low density A successive work on liquid Ar at different pressures by de Schepper et al., whose results are reported (10.5 atoms/nm ) and high temperature (295 K) probed in the experiment it was reasonable to assume in Figure 4, attempted a general test of the mentioned Lorentzian sum development of S(Q,ω) [47]. The the kinetic theory description to hold validity. effect of spectral rule fulfillment was therein analyzed by comparing the results obtained by imposing the A successive work on liquid Ar at different pressures by de Schepper et al., whose results are fulfillment of sum rules of increasing order ( 2). Furthermore, in such a work the authors attempted to reported in Figure 4, attempted a general test of the mentioned Lorentzian sum development of describe the measured dispersion curves in terms of the predictions of the mode‐coupling theory (MCT) S(Q,!) [47]. The effect of spectral rule fulﬁllment was therein analyzed by comparing the results (see e.g. [61]). The non‐analytic dispersion relations predicted by the MCT, having the form of truncated Q‐ obtained by imposing the fulﬁllment of sum rules of increasing order (¤ 2). Furthermore, in such expansions with fractional exponents, were compared with experimental dispersions. The accuracy of the MCT a work pred the ictio authors n can be attempted judged by to comparing describe the themeasur dashed ed cu disper rves in sion Figur curves e 4 wi inthterms the loof w the Q po prrtio edictions n of the dispersio of the mode-coupling n curves. The cotheory mparison (MCT clearly ) (see demo e.g.nstrates [61]). The that non-analytic at the lowest dispersion Qs the sound relations disperpr sion edicted curve −1 is well described by the MCT for all explored pressures, although for Q > 4 nm this theory does not provide by the MCT, having the form of truncated Q-expansions with fractional exponents, were compared a co with nsistent experimental account of dispersions. the observeThe d beh accur avior acy of of quasi the‐ela MCT sticpr and ediction inelastcan ic linewidths be judged of by th comparing e spectrum.the dashed curves in Figure 4 with the low Q portion of the dispersion curves. The comparison clearly −1 2.7. 1990: “Extended Hydrodynamics Modes” up to Q ≈ 1 nm demonstrates that at the lowest Qs the sound dispersion curve is well described by the MCT for all explored pressures, although for Q > 4 nm this theory does not provide a consistent account of the observed behavior of quasi-elastic and inelastic linewidths of the spectrum. -1  /c (nm ) s s Propagation gap ( 0) s Appl. Sci. 2016, 6, 64 10 of 31 Appl. Sci. 2016, 6, 64 11 of 34 2.7. 1990: “Extended Hydrodynamics Modes” up to Q 1 nm After the ﬁrst low Q measurement on Ne by Bell et al., a second INS experiment was performed After the first low Q measurement on Ne by Bell et al., a second INS experiment was performed by 1 1 by Baﬁle et al. [62] on supercritical Ar at extremely low Q’s (0.35 nm ¤ Q ¤ 1.25 nm ). −1 −1 Bafile et al. [62] on supercritical Ar at extremely low Q’s (0.35 nm ≤ Q ≤ 1.25 nm ). This measurement This measurement further extended the probed dynamic region to lower Qs, thus reducing the further extended the probed dynamic region to lower Qs, thus reducing the gap with light scattering gap with light scattering measurements. Figure 5 displays two typical spectral shapes measured in measurements. Figure 5 displays two typical spectral shapes measured in this INS work, which clearly this INS work, which clearly conﬁrm the persistence of extended Brillouin peaks at mesoscopic scales. confirm the persistence of extended Brillouin peaks at mesoscopic scales. -1 Q = 1 nm -1 Q = 0.5 nm 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9  (meV) Figure 5. Two representative INS spectral lineshapes of Ar measured by Bafile et al. [62] at low Figure 5. Two representative INS spectral lineshapes of Ar measured by Baﬁle et al. [62] at low exchanged momenta. exchanged momenta. Most importantly, in this measurement the superior accuracy in the count statistics coupled with the Most importantly, in this measurement the superior accuracy in the count statistics coupled unprecedented dense Q‐grid, enabled a very detailed analysis of the Q‐dependence of line‐shape with the unprecedented dense Q-grid, enabled a very detailed analysis of the Q-dependence of parameters of Equation (2). Best fit values of such parameters are reported in Figure 6 and therein compared line-shape parameters of Equation (2). Best ﬁt values of such parameters are reported in Figure 6 and with either the simple hydrodynamic prediction (Equations (3a–c)) or a higher‐order polynomial expansion therein compared with either the simple hydrodynamic prediction (Equation (3a–c)) or a higher-order (see, e.g., Equations (42)–(45) of [23]). In the same Figure, the result previously obtained by Bell and polynomial expansion (see, e.g., Equations (42)–(45) of [23]). In the same Figure, the result previously collaborator on neon is also reported for comparison along with the corresponding simple hydrodynamic obtained by Bell and collaborator on neon is also reported for comparison along with the corresponding prediction of Equations (3a–c). simple hydrodynamic prediction of Equation (3a–c). All hydrodynamic curves included in the figure were obtained using thermodynamic and transport All hydrodynamic curves included in the ﬁgure were obtained using thermodynamic and parameters reported in the original works. transport Overalparameters l, data in Figru eported re 6 confirm in the that original the Q‐works. dependence of line‐shape parameters is roughly consistent with the low Q approximation in Equations (3a–c). This clearly indicates that “extended hydrodynamic” Overall, data in Figure 6 conﬁrm that the Q-dependence of line-shape parameters is roughly modes survive well above the hydrodynamic limit, up to Q values exceeding the light scattering domain consistent with the low Q approximation in Equation (3a–c). This clearly indicates that “extended by nearly two orders of magnitude. hydrodynamic” modes survive well above the hydrodynamic limit, up to Q values exceeding the light scattering domain by nearly two orders of magnitude. 2.8. 1998: First IXS Measurements of the THz Spectrum of Noble Gases 2.8. 1998: First IXS Measurements of the THz Spectrum of Noble Gases The only THz spectroscopic technique available until the mid‐90s was INS, a method intrinsically hampered by kinematic constraints (see, e.g., [63] pp. 63–101) limiting the accessible portion of dynamic The only THz spectroscopic technique available until the mid-90s was INS, a method intrinsically plane (Q,ω). These limitations are particularly penalizing at low Qs, where the collective nature of hampered by kinematic constraints (see, e.g., [63] pp. 63–101) limiting the accessible portion of dynamic structural rearrangements become dominating, as well as at large ω values, where instead short‐time, plane (Q,!). These limitations are particularly penalizing at low Qs, where the collective nature of collisional events have a visible influence on density fluctuations. structural rearrangements become dominating, as well as at large ! values, where instead short-time, These problems were successfully overcome after the development of the first high resolution IXS, a collisional events have a visible inﬂuence on density ﬂuctuations. spectroscopic technique virtually free from kinematic limitations [2,64]. These problems were successfully overcome after the development of the ﬁrst high resolution IXS, a spectroscopic technique virtually free from kinematic limitations [2,64]. Intensity (arbit. units) Appl. Sci. 2016, 6, 64 12 of 34 Appl. Sci. 2016, 6, 64 11 of 31 0.4 Ne at T= 70 K, n =14.45 atoms/nm Ar at T = 301 K, n = 5.04 atoms/nm 0.3 0.2 0.1 0.0 0.12 0.09 0.06 0.03 0.00 0.08 0.06 0.04 0.02 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 -1 Q(nm ) Figure 6. Relevant shape parameters of the S(Q,ω) of supercritical Ne and Ar as measured, at the indicated Figure 6. Relevant shape parameters of the S(Q,!) of supercritical Ne and Ar as measured, at the thermodynamic conditions, in [40,62], respectively. The reported solid and dashed lines are the indicated thermodynamic conditions, in [40,62], respectively. The reported solid and dashed lines are hydrodynamic predictions referring to symbols of corresponding color as derived from the lowest Q the hydrodynamic predictions referring to symbols of corresponding color as derived from the lowest expansion in Equations (3a–c). Q expansion in Equation (3a–c). The first IXS measurement on a dense noble gas was performed on deeply supercritical neon (T = 295 The ﬁrst IXS measurement on a dense noble gas was performed on deeply supercritical neon K, n = 29.1 atoms/nm ) and discussed in combination of a MD simulation on a Lennard‐Jones model (T = 295 K, n = 29.1 atoms/nm ) and discussed in combination of a MD simulation on a Lennard-Jones representative of the same sample [65]. In Figure 7, two representative IXS spectra (upper panels) and model representative of the same sample [65]. In Figure 7, two representative IXS spectra (upper panels) corresponding MD simulations (lower panels) are compared (middle panels) after the former have been and corresponding MD simulations (lower panels) are compared (middle panels) after the former multiplied with the detailed balance factor and convoluted with the instrumental energy resolution have been multiplied with the detailed balance factor and convoluted with the instrumental energy function. The good agreement between measured and computed line‐shapes suggests that the Lennard‐ resolution function. The good agreement between measured and computed line-shapes suggests that Jones model provides a consistent description of the spectrum. Furthermore, it is apparent that even at Q the Lennard-Jones model provides −1 a consistent description of the spectrum. Furthermore, it is apparent values extending well above 1 nm , the inelastic peaks are visible in the MD simulated line‐shapes and, in that even at Q values extending well above 1 nm , the inelastic peaks are visible in the MD simulated spite of some resolution limitations, in the experimental spectra as well. line-shapes and, in spite of some resolution limitations, in the experimental spectra as well.  z (meV)  z (meV)  (meV) s Appl. Sci. 2016, 6, 64 13 of 34 Appl. Sci. 2016, 6, 64 12 of 31 -1 -1 Q = 2 nm Q = 5 nm 0.8 1.2 1.0 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0.0 0.0 0.8 1.2 1.0 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0.0 0.0 0.95 0.95 0.76 0.76 0.57 0.57 0.38 0.38 0.19 0.19 0.00 0.00 -6 -3 0 3 6 -10 -5 0 5 10   (meV) Fi Figure gure 7.7. Up Upper per pan panels: els: lolow w Q Q inelast inelastic ic X‐X-ray ray scatte scattering ring IXS IXS spe spectra ctra memeasur asured ed on on dee deeply ply sup super ercriti critical cal neon (T neon =295 ( TK, =295 n = 29.1 K, n atoms/n = 29.1 atoms/nm m ). The ra).wThe data raw (opdata en circles) (open ar cirecles) comar pa ere compar d with ed corwith respo corr ndin esponding g resolution fu resolution nctions (dfunctions ashed lines) (dashed . Lowe lines). r paneLower ls: corrpanels: espondin corr g MD esponding spectraMD com spectra puted on computed a Lennaon rd aJo Lennar nes mo ddel representative of the same sample (blue lines). Middle panels: Comparison between IXS spectra (open Jones model representative of the same sample (blue lines). Middle panels: Comparison between circles) and the corresponding molecular dynamics (MD) ones convoluted with the resolution functions (red IXS spectra (open circles) and the corresponding molecular dynamics (MD) ones convoluted with the lines). Data are redrawn with permission from [65], which is copyrighted bt the American Physical Society. resolution functions (red lines). Data are redrawn with permission from [65], which is copyrighted bt the American Physical Society. At this stage, a pending question relates to the Q‐threshold, Qh, defining the limit of validity for the “extended hydrodynamic” regime. Based on the fit of MD spectra reported in Figure 8 and discussed in Refs. At this stage, a pending question relates to the Q-threshold, Q , deﬁning the limit of validity −1 [65–67], one could conclude that, at least for dense supercritical neon, Qh is as high as 7.5 nm . According for the “extended hydrodynamic” regime. Based on the ﬁt of MD spectra reported in Figure 8 and to the plot, the transport parameters derived from best‐fit results through Equations (3a–c) are essentially discussed in Refs. [65–67], one could conclude that, at least for dense supercritical neon, Q is as high consistent with the macroscopic, or hydrodynamic, limit at least for Q lower than the values included in as 7.5 nm . According to the plot, the transport parameters derived from best-ﬁt results through the shadowed area. Thanks to the high statistical accuracy of Q‐dependent data in Figure 8, one may infer Equation (3a–c) are essentially consistent with the macroscopic, or hydrodynamic, limit at least for that the simple hydrodynamic approximation is not accurate for the thermal diffusivity. In fact, low‐Q Q lower than the values included in the shadowed area. Thanks to the high statistical accuracy values of DT seem slightly lower than the macroscopic value. of Q-dependent data in Figure 8, one may infer that the simple hydrodynamic approximation is not accurate for the thermal diffusivity. In fact, low-Q values of D seem slightly lower than the macroscopic value. Intensity (arbit. units) ApAppl. pl. ScSci. i. 2016 2016 , 6,, 64 6, 64 13 14 of of 31 34 Mesoscopic regime Extended Hydrodynamics 0.25 b) 0.20 0.15 0.10 0.05 0.00 c) 0.14 0.07 0.00 0 5 10 15 20 25 -1 Q(nm ) Figure 8. The relevant transport parameters deduced from the fits of of IXS spectra of deeply supercritical Figure 8. The relevant transport parameters deduced from the ﬁts of of IXS spectra of deeply neon [67] and those of MD spectra from a Lennard Jones model representative of the same sample [65]. The supercritical neon [67] and those of MD spectra from a Lennard Jones model representative of the same shadowed area roughly locates the transition from the extended hydrodynamics and the mesoscopic regime. sample [65]. The shadowed area roughly locates the transition from the extended hydrodynamics and Horizontal dashed lines represent the macroscopic (hydrodynamic) values deduced from thermodynamic the mesoscopic regime. Horizontal dashed lines represent the macroscopic (hydrodynamic) values [59] and US spectroscopy data [28]. deduced from thermodynamic [59] and US spectroscopy data [28]. 2.9. The Onset of a Positive Sound Dispersion in Liquid Noble Gases 2.9. The Onset of a Positive Sound Dispersion in Liquid Noble Gases In the various works on noble gases reported in literature, it was observed that transport parameters In the various works on noble gases reported in literature, it was observed that transport drastically change upon relatively moderate P and T variations. This may not be surprising, given that the parameters drastically change upon relatively moderate P and T variations. This may not be surprising, large compressibility of these systems leads to substantial variations of inter‐particle distances and, given that the large compressibility of these systems leads to substantial variations of inter-particle consequently, of interaction strengths. distances and, consequently, of interaction strengths. For instance, clear transformations were observed in the Q‐dependence of the acoustic frequency and For instance, clear transformations were observed in the Q-dependence of the acoustic frequency damping when measured either in the liquid or in the supercritical phase. In particular, it was generally and damping when measured either in the liquid or in the supercritical phase. In particular, it was found that, in liquid Ne and Ar [43,44,58,68] ωs systematically exceeds the hydrodynamic prediction at generally found that, in liquid Ne and Ar [43,44,58,68] ! systematically exceeds the hydrodynamic low/intermediate Qs. A similar effect was also observed ins liquid He [48,69,70], although the quantum prediction at low/intermediate Qs. A similar effect was also observed in liquid He [48,69,70], although character of this sample makes this finding less straightforward to interpret. Furthermore, in liquid phase sample the quantum s an “anom charaacter lous”of Qthis ‐depend sample ence makes was al this so re ﬁnding portedless for th straightforwar e sound absodrp totion int erpr coeffici et. ent, Furthermor as derive,ed frin om liquid the wid phase th ofsamples inelastican pe“anomalous” aks, zs. ConverQ sely, -dependence all these di was spersi also ve ref eported fects see for m to the dsound isappear absorption at extreme supercritical conditions [40,62,65,66]. coefﬁcient, as derived from the width of inelastic peaks, z . Conversely, all these dispersive effects These trends are clearly exemplified by Figure 9, which compares spectral parameters obtained in seem to disappear at extreme supercritical conditions [40,62,65,66]. supercritical [65] and liquid [71] neon. Overall, data reported therein indicate that, while in liquid Ne the These trends are clearly exempliﬁed by Figure 9, which compares spectral parameters obtained sound frequency (ωs) and absorption (zs) at low Qs are, respectively, higher and lower than predicted by in supercritical [65] and liquid [71] neon. Overall, data reported therein indicate that, while in liquid Equations (3a–c), no substantial discrepancy is instead observed in supercritical Ne. In Figure 9 it can be Ne the sound frequency (! ) and absorption (z ) at low Qs are, respectively, higher and lower than s s predicted by Equations (3a–c), no substantial discrepancy is instead observed in supercritical Ne. In Figure 9 it can be also noticed that for the latter sample the agreement with the simple hydrodynamic prediction is excellent up to Q 7.5 nm . 6 2  2 10 D (m /s)   (m /s) c (m/s) L s T Appl. Sci. 2016, 6, 64 15 of 34 also noticed that for the latter sample the agreement with the simple hydrodynamic prediction is excellent Appl. Sci. 2016, 6, 64 14 of 31 −1 up to Q ≈ 7.5 nm . supercritical Ne liquid Ne viscoelastic A) B) consistency 8 0 effects with simple hydrodynamics D) C) 0 0 0 5 10 15 20 0 5 10 15 20 25 30 -1 Q(nm ) Figure 9. Left column: shift (panel a) and half width (panel c) of the inelastic peak in the spectrum of Figure 9. Left column: shift (panel a) and half width (panel c) of the inelastic peak in the spectrum supercritical Ne at T = 294 K and n = 29 atoms/nm as derived from a best fit line‐shape with Equation (2). Red of supercritical Ne at T = 294 K and n = 29 atoms/nm as derived from a best ﬁt line-shape with and black dots are IXS and MD data from [65] and [67] respectively. Data from [65] are redrawn with Equation (2). Red and black dots are IXS and MD data from [65] and [67] respectively. Data from [65] permission from the original publication, under the copyright of the American Physical Society. The solid are redrawn with permission from the original publication, under the copyright of the American lines represent the corresponding simplified hydrodynamic prediction in Equations (3a–c). In which Physical Society. The solid lines represent the corresponding simpliﬁed hydrodynamic prediction in literature values of thermodynamic properties [72] and bulk viscosity [28] coefficients were inserted. Right Equation (3a–c) In which literature values of thermodynamic properties [72] and bulk viscosity [28] column: the corresponding quantities are reported for liquid neon at T = 35 K and n = 36.65 atoms/nm . These coefﬁcients were inserted. Right column: the corresponding quantities are reported for liquid neon data are adapted with permission from [71] , which is copyrighted by the American Physical Society. at T = 35 K and n = 36.65 atoms/nm . These data are adapted with permission from [71] , which is copyrighted by the American Physical Society. As mentioned, “anomalous” dispersive effects observed for the liquid sample can be ascribed to the onset of a viscoelastic response induced by the coupling with a relaxation process. As mentioned, “anomalous” dispersive effects observed for the liquid sample can be ascribed to To illustrate this point, it is useful to recognize that, in a spectroscopy experiment, the scattering‐ the onset of a viscoelastic response induced by the coupling with a relaxation process. excited acoustic waves cause a time‐dependent perturbation of the local equilibrium of the target sample. To illustrate this point, it is useful to recognize that, in a spectroscopy experiment, the As a response, decay channels redistribute the energy carried by the acoustic wave toward some internal scattering-excited acoustic waves cause a time-dependent perturbation of the local equilibrium of the degrees of freedom of the fluid. These energy rearrangements ultimately drive the sample to relax in a new target sample. As a response, decay channels redistribute the energy carried by the acoustic wave local equilibrium within a timescale τ. toward some internal degrees of freedom of the ﬂuid. These energy rearrangements ultimately drive Two limiting scenarios can thus occur: (1) the time‐dependent acoustic perturbation has period, 2π/ωs, the sample to relax in a new local equilibrium within a timescale . much longer than any internal degrees of freedom of the system. Under these conditions, the latter relaxes Two limiting scenarios can thus occur: (1) the time-dependent acoustic perturbation has period, to equilibrium “instantaneously” (i.e., within a timescale τ << 1/ωs) and the acoustic propagation essentially ta2kes p/! place , much over longer success than ive any equiinternal librium st degr ates ees (viof sco fru eedom s limit)of ; (2 the ) Conv system. erselUnder y, if the these acouconditions, stic wave has the an ex latter tremel relaxes y short to pe equilibrium riod, it “perceives” “instantaneously” internal rearra (i.e. ng ,ement within s as a timescale frozen‐like (<< τ >>1 1/ /!ωs)) an and d does the acoustic not couple with them. Consequently, the acoustic propagation is elastic, i.e., it occurs with virtually no energy loss propagation essentially takes place over successive equilibrium states (viscous limit); (2) Conversely, if (elastic limit). the acoustic wave has an extremely short period, it “perceives” internal rearrangements as frozen-like Therefore, when a relaxation is active, the response of a system to the acoustic propagation depends ( >> 1/! ) and does not couple with them. Consequently, the acoustic propagation is elastic, i.e., it on how the acoustic frequency, ωs, compares to the relaxation time. When ωs increases from the viscous occurs with virtually no energy loss (elastic limit). Therefore, when a relaxation is active, the response of a system to the acoustic propagation depends on how the acoustic frequency, ! , compares to the relaxation time. When ! increases s s from the viscous (! << 1) to the elastic limit (! >> 1), the viscoelastic transition manifests itself s s through a systematic decrease of acoustic dissipation and a corresponding increase of sound velocity. Since in scattering experiments ! is typically varied by changing Q (through the dispersion relation ! = ! (Q)), in these measurements viscoelastic behavior is reﬂected by a Q-increase of ! and s s s a Q-decrease of z .  z (meV)  ( meV) S s Appl. Sci. 2016, 6, 64 15 of 31 This explains why in panels 9b and 9d these shape parameters are, at the lowest Qs, respectively larger and lower than their hydrodynamic predictions. Once the phenomenological signature of a viscoelastic behavior is experimentally assessed, a more quantitative understanding of this phenomenon could only come from the modeling of measured line-shape with a viscoelastic model. The next section is devoted to the derivation of such a model within the framework of the memory function formalism. 3. A Model for the Measured Spectral Shape 3.1. The Memory Function Formalism Let Aptq rA ptq..........A ptqs be a vector whose components are stochastic variables describing 1 n the properties of a system of N interacting particles (with N > u). It can be shown that under very general conditions, the equation of motion of Aptq has the following form: Ñ Ñ Ñ dAptq iW Aptq Kptq Apt tqdt f ptq (4) dt In the formula above the following quantities have been introduced: B F B F Ñ Ñ Ñ Ñ The antisymmetric matrix iW Ap0q, iLAp0q Ap0q, Ap0q is the proper frequency of the system, which describes the oscillatory behavior of Aptq. This matrix depends on equilibrium (static) properties of the system. Here the symbol x.......y indicates a statistical (thermal) average. Ñ Ñ ip1PqLt The variable f ptq e ip1 PqLAp0q is the random ﬂuctuating force. Noticeably, the B F Ñ Ñ presence of the p1 Pq term implies that Ap0q, f ptq 0, namely that the ﬂuctuating force is orthogonal (statistically uncorrelated) to Ap0q. B F B F Ñ Ñ Ñ Ñ The matrix Kptq f , f ptq Ap0q, Ap0q is memory matrix, or memory function if = 1, as in the cases of interest here. It is worth noticing that the integrand in Equation (4) includes all Aptq values between 0 and t, “weighted” by the memory, which thus deﬁnes the ability of the system to keep memory of the past. B F Ñ Ñ The equation of motion for the correlation matrix Cptq AptqAp0q can be readily derived from Ñ Ñ Equation (4) through a scalar product, while using the orthogonality between Ap0q and f ptq. Explicitly: dCptq iW Cptq Kptq Cpt tqdt (5) dt Equations (4) and (5) are referred to as memory equation or generalized Langevin equation either for dynamic variables or for their correlations, respectively. When dealing with a single dynamic variable, time reversal requirements impose W = 0. Using a procedure described in various monographies [22,63], the Langevin equation can be solved following an iterative approach, which eventually leads to the following recursive formula: Cpsq K p0q 1 1 rs s (6) Cp0q K p0q K p0q s . . . customarily referred to as the continued fraction expansion. Here K (t) is the i-th (iteration) order memory function associated to the variable of interest, Appl. Sci. 2016, 6, 64 16 of 31 zt Cpsq dte Cptq while is the Laplace transform of the correlation function. The t = 0 values of the i-th order memory function K (t) can be determined by superimposing the sum rule fulﬁllment, which for the few lowest orders yields: A E K p0q w (7a) .. @ D @ D 4 2 w w K p0q K p0q @ D @ D (7b) 2 0 2K p0q w w .. @ D @ D 6 4 w w K p0q 1 K p0q @ D @ D (7c) 2 2 2K p0q D w w 2 2 d Cptq 1 8 1 n n n Herexw y i r s rCp0qs w CpwqdwrCp0qs is the n-order spectral moment dt t0 of the correlation function C(!). Among all possible ﬂuctuating variables, we are here interested in density ﬂuctuations since the Fourier-Laplace transform of their correlation function, S(Q,!) is directly measured by both IXS and INS. S(Q,!) can be derived as the Fourier anti-transform of the intermediate scattering function A E Ñ Ñ Ñ Ñ FpQ, tq drp r , tqdr p r , 0q exp iQ r d r . The latter can be expressed through the continued fractions expansion of Equation (4), whose 2-nd order truncation yields: @ D FpQ, sq SpQq s m pQ, sq Where m pQ, sq is the second order memory function for density ﬂuctuations while S(Q) = F(Q,0) is the static structure factor. The S(Q,!) can be derived from Equation (7) as follows: @ D SpQ, wq 1 FpQ, s iwq 1 Re Re iw SpQq p SpQq p iw m pQ, s iwq The last term of the above formula can be calculated explicitly to eventually obtain: @ D 2 1 w m pQ, wq SpQ, wq 1 (8) @ D SpQq p 2 2 2 2 1 rw w wm pQ, wqs w rm pQ, wqs L L where m pQ, wq and m pQ, wq are, respectively, real and imaginary parts of the Fourier transform L L of m (Q,t). At this stage, the problem of choosing the most appropriate model is shifted from S(Q,!) to m (Q,t), or m (Q,!). In many literature works on liquid systems attempting a memory function L L based modeling of the spectrum (see e.g. [13,73]), it can be shown that a sensible ansatz for the time dependent memory function is given by: 2 2 m pQ, tq D exppt{t q D exppt{t q 2G dptq (9) L T a m T a The three terms in the right hand side of Equation (9) are (from left): (1) The thermal contribution. The ﬁrst term describes diffusive thermal motions triggered by 2 1 spontaneous temperature gradients. Its timescale and amplitude are t pgD Q q and T T 2 2 D pg 1qpc Qq respectively, with D , c , and being the generalized diffusivity, the T T T isothermal velocity and the constant pressure to constant volume speciﬁc heats ratio. Appl. Sci. 2016, 6, 64 17 of 31 (2) The viscous relaxation contribution. The second term accounts for relaxation processes affecting 2 2 2 2 the viscosity and having a timescale t and amplitude D pc gc qQ , where c is the a 8 a 8 generalized inﬁnite frequency or “elastic” sound velocity of the sample. (3) The instantaneous contribution. The last term accounts for the coupling of density ﬂuctuations with the ultra-fast vibrational dynamics here accounted by a (t) proﬁle. It was often found that the amplitude 2G has a quadratic Q-dependence and is essentially insensitive to thermodynamic changes [20]. It is immediate to recognize that real and imaginary parts of Fourier transform of the memory function in Equation (9) read as: t t T a 1 2 2 m pQ, wq D D G (10a) L T a 2 2 1 pwt q 1 pwt q 2 2 pwt q pwt q 2 T a 2 2 wm pQ, wq D D (10b) L T a 2 2 1 pwt q 1 pwt q T a respectively. These two equations can be inserted in Equation (8) to obtain a model for the classical part of the line-shape. It should be emphasized that at mesoscopic scales all transport and thermodynamic coefﬁcients introduced above and in the following, unless otherwise speciﬁed, are Q-dependent generalizations of their macroscopic counterparts, although such a dependence in not made explicit in the notation. The model used for the memory function often results from suitable approximation of Equation (9) as appropriate to the sample, thermodynamic conditions and dynamic range probed. For simplest systems as noble gases, the single exponential decay or viscoelastic model has shown to provide a reasonable approximation [42]. Regardless on the approximation adopted, the exponential ansatz for the viscoelastic term, lends itself two at least these two fundamental objections: (1) In principle, structural relaxation phenomena in highly viscous systems can be better described by assuming a stretched exponential rather than a simple exponential time-decay of the memory function [17,74,75]. However, the use of this model would introduce an unwanted additional parameter (the stretching coefﬁcient) and, perhaps more importantly, its Fourier transform cannot be cast on an analytical form. Fortunately, for simplest systems as noble gases, of interest in this paper, it is reasonable to assume that the simple exponential decay is a rather accurate approximation. (2) A second remark concerns the possible presence in the spectrum of a second inelastic excitation as observed in systems as diverse as liquid water [73,76–78], tetrahedrally arranged glasses [79], glass formers [80] liquid metals [81–85], complex biophysical samples [86–88] and mixtures [89]. The presence of this low frequency mode in the S(Q,!) is customarily ascribed to the onset of shear mode propagation, although this assignment could seem suspicious as S(Q,!) couples primarily with longitudinal movements only. In a liquid, the onset of shear waves in the spectrum of density fluctuations can only occur via the so-called longitudinal-transverse coupling [73,76,90], that is a mixing between acoustic modes having orthogonal polarization. The presence of this mode-mixing is not properly accounted for by any known analytic model for the memory function. However, such a coupling has never been observed in noble gases, likely due to the highly isotropic character of their interatomic interaction, and will not be discussed in the remainder of this paper. 3.2. The Departure from the Hydrodynamic Shape of the Spectrum In the hydrodynamic regime, density ﬂuctuations are probed for long time lapses (t >> ) over which the viscous relaxation is perceived as a very rapid decay, well approximated by a Markovian Appl. Sci. 2016, 6, 64 18 of 31 term (9 (t)) in the time-dependence of memory function. Overall, it can be shown that the following memory function: 2 2 2 m pQ, tq pg 1qpc Qq exp gD Q t 2n Q dptq (11) T T L L,hyd is fully consistent with the Rayleigh-Brillouin triplet expressed by Equation (2) under the approximation deﬁned by Equation (3a–c) As mentioned, at mesoscopic scales all parameter entering in Equation (11) should be generalized as Q dependent variables. Due to the lack of ﬁrm theories describing analytically the departure of the spectrum from the simple hydrodynamic regime, this Q-dependence is usually determined empirically through the best-ﬁt modeling of spectra measured at different Qs. Albeit phenomenological in character, this approach has proven to be successful in shedding a deep light on the fast dynamics of various disordered systems, including noble gases. In particular, it has elucidated many phenomenological aspects associated to the coupling of density ﬂuctuations with THz relaxation processes. 3.3. The Simple Viscoelastic Model The simplest method to account for THz viscoelastic effects on the spectrum of density ﬂuctuation is to assume a single exponential decay of the memory function: m pQ, tq D exppt{tq (12) L 2 where the amplitude of the memory function is ﬁxed by the superimposition of sum rules to the spectrum, which eventually leads to: @ D k T 2 B 2 w @ D Q (13a) MSpQq @ D @ D w (13b) @ D @ D 4 2 w w 2 2 2 @ D D c c Q (13c) 8 0 w SpQq @ D here w SpQq, while k and M are the Boltzmann constant and atomic mass, respectively; Equations (12) and (13a–c) deﬁne the so called Debye [91] or simple viscoelastic model [92]. Clearly, this model can hold validity only if: (1) thermal relaxations have a negligible weight on the time decay of memory function, i.e., if 1 and (2) the contribution from fast relaxation (term 9 (t) in Equation (9)) phenomena is also negligible. Conceptually, the simple viscoelastic description stems from the idea that, upon crossing the viscoelastic transition, the sound velocity c , changes from c w {Q B{r with B being the bulk 0 0 modulus—as appropriate for a liquid—to c w {Q M{r with M B 4{3G and G being, 8 8 respectively, the elastic and the shear modulus—as appropriate for solids. The superimposition of sum rules links the finite Q generalization of B and M to the second and fourth normalized moments, respectively. In a way the viscoelastic model through the sum rules’ fulfillment interpolates between the low Q, viscous, response and the high Q elastic one. Specifically, one side, Equation (13a) extends to intermediate Qs the macroscopic compressibility result [55] c K T{MSp0q on the other side, the T B high Q link between M and normalized fourth moment, [1] is here extended down to the same Qs values. 4. 2001–2007: The Memory Function-Based Modeling of IXS Spectra of Noble Gases The success of the memory function based modeling of experimental spectra was ultimately enabled by the dramatic improvement of the incident ﬂux and, consequently, of the count statistics Appl. Sci. 2016, 6, 64 19 of 31 prompted by the development of IXS. Furthermore, these high ﬂuxes coupled with the reduced beam cross section enabled spectroscopy measurements on substantially smaller samples, thus opening the access to extreme thermodynamic conditions. The ﬁrst IXS work investigating viscoelastic effects on liquid and supercritical neon [6] took advantage of these opportunities there by exploring an unprecedented broad thermodynamic range. Measurements were performed on dense neon from liquid to supercritical conditions following both an isochoric (n = 29 atoms/nm ) and an isothermal (T = 32 K) path. The sound dispersion curves of neon determined in such a work are reported in Figure 10. Best ﬁt values of w c Q and w c Q 0 0 8 8 derived from the best ﬁt modeling of the line-shape are compared with the acoustic frequency evaluated from the position of the maxima of current spectra C (Q,!), W . The current spectrum is deﬁned as the L l Fourier transform of the correlation function between longitudinal components of atomic velocities. The position of its maxima is often identiﬁed as the dominant acoustic frequency, this identiﬁcation being rigorous in the hydrodynamic limit only. Within the viscoelastic approach, W is assumed to undergo a transition between the viscous and the elastic limit respectively represented by ! and ! . 0 8 Figure 10 displays the values of W , ! and ! of liquid (upper panel) and supercritical (bottom 0 8 panel) neon reported in Ref. [6] and compares them with the hydrodynamic straight line c Q. From the inspection of the two plots the following trends are readily noticed: (1) the liquid phase datum bears evidence for the more-than-linear behavior of W (purple dots) for Q ¤ 10 nm , i.e., the systematic bending of W upwards the hydrodynamic linear law and toward ! (the mentioned PSD effect) . Conversely, the liquid phase point is characterized by a sizable PSD. (2) At higher Qs one can observe the “backward transition” of W from ! to ! , which is actually l 8 0 joined for Q larger than the position of the ﬁrst di Q (22 nm ). (3) Again, viscoelastic effects disappear in deeply supercritical conditions (lower panel) since ! , W and ! merge into each other (at least at the lowest Qs), thus suggesting that the viscous Appl. Sci. 2016, 6, 64 21 of 34 relaxation term in Equation (9) has a vanishing strength. Liquid, T = 40 K Supercritical, T = 294 K 0 3 6 9 12 15 18 21 24 27 30 -1 Q(nm ) Figure 10. The acoustic frequency Ωl derived from the maxima of current spectra (magenta dots) are Figure 10. The acoustic frequency W derived from the maxima of current spectra (magenta dots) are reported against the zero (black dots) and infinite frequency (blue dots) limiting dispersions extracted from reported against the zero (black dots) and inﬁnite frequency (blue dots) limiting dispersions extracted best fits the IXS spectra of neon. Fits were obtained with a single exponential viscoelastic model for the memory function decay (Equation (12)). The linear hydrodynamic sound dispersion (Equation (3a)) is also from best ﬁts the IXS spectra of neon. Fits were obtained with a single exponential viscoelastic model reported for reference as derived using tabulated values of the adiabatic sound speed [71]. Data are adapted for the memory function decay (Equations (12)). The linear hydrodynamic sound dispersion (see with permission from [6] copyrighted by American Physical Society. Equations (3a)) is also reported for reference as derived using tabulated values of the adiabatic sound As previously mentioned, the validity of the simple viscoelastic model demands that the thermal speed [71]. Data are adapted with permission from [6] copyrighted by American Physical Society. contribution to the memory function decay has negligible amplitude (γ ≈ 1). The soundness of this assumption is supported by the bottom panel of Figure 11, where the value of γ computed for supercritical Ne [65,66] is reported.  (Q),  (Q),  (Q) (meV) 0  l Appl. Sci. 2016, 6, 64 20 of 31 As previously mentioned, the validity of the simple viscoelastic model demands that the thermal contribution to the memory function decay has negligible amplitude ( 1). The soundness of this assumption is supported by the bottom panel of Figure 11, where the value of computed for Appl. Sci. 2016, 6, 64 22 of 34 supercritical Ne [65,66] is reported. 0.18 0.09 0.00 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 0 5 10 15 20 25 30 -1 Q( nm ) Figure 11. Lower panel: the generalized specific heat ratio as derived from MD computations for Lennard Figure 11. Lower panel: the generalized speciﬁc heat ratio as derived from MD computations for Jones models simulating deeply supercritical neon ([66], red dots). The unit value is reported as a dashed Lennard Jones models simulating deeply supercritical neon ([66], red dots). The unit value is reported line for reference. Lower panel: the thermal diffusivity, as derived from MD simulations in [66], while using as a dashed line for reference. Lower panel: the thermal diffusivity, as derived from MD simulations the simplified hydrodynamic expression in Equation (3c). In all plots horizontal dashed line indicate the in [66], while using the simpliﬁed hydrodynamic expression in Equations (3c). In all plots horizontal macroscopic values (from National Institute of Standards and Technology NIST database [72] of the symbols dashed line indicate the macroscopic values (from National Institute of Standards and Technology having corresponding color. NIST database [72] of the symbols having corresponding color. The curve is consistent with the macroscopic limit at the lowest Qs. Overall, it can be inferred that the The curve is consistent with the macroscopic limit at the lowest Qs; however both curves suggest −1 weight of the thermal contribution (proportional to γ − 1) is drastically reduced beyond Q ≈ 7 nm since that the weight of the thermal contribution (proportional to 1) is drastically reduced beyond therein γ differs from 1 by less than 10%. Q 7 nm since therein differs from 1 by less than 10%. The generalized thermal diffusivity obtained from the same work is reported in the upper plot. One The generalized thermal diffusivity obtained from the same work is reported in the upper plot. can confidently conclude that thermal diffusivity drastically decreases with Q consistently with the INS One can conﬁdently conclude that thermal diffusivity drastically decreases with Q consistently with result of Bell et al. [40]. Furthermore, it appears that in the supercritical sample the departure from the the INS result of Bell et al. [40]. Furthermore, it appears that in the supercritical sample the departure hydrodynamic value happens at higher Q values and is much smoother. This seems consistent with the −1 from the hydrodynamic value happens at higher Q values and is much smoother. This seems consistent extended hydrodynamic behavior (up to ≈8 nm ) of acoustic parameters already discussed in reference to with the extended hydrodynamic behavior (up to 8 nm ) of acoustic parameters already discussed Figure 10. It has to be noticed that trends similar to the ones in Figure 11 were also reported in a previous MD in rwork eference on ato LeFigur nnard e Jones 10. It sy has steto m re bepresenta noticedtive that oftr Ar ends at th similar e triple to po the int.ones in Figure 11 were also reported in a previous MD work on a Lennard Jones system representative of Ar at the triple point. A final remark concerns the instantaneous term in the memory function decay (last term in Equation (9)). It A is ﬁnal reason remark able to concerns expect that the thinstantaneous is contribution term is rather in the wememory ak for low function ‐viscosity decay and non (last‐associ termate ind fluids Equation as noble (9)). gases. It is Th reasonable is conclusion to expect is support that the this so contrib undness ution of ais simple rather viweak scoelafor stic low-viscosity assumption. On and the other hand in Ref. [42], using a double exponential model allows one to extract essentially the same value non-associated ﬂuids as noble gases. This conclusion is support the soundness of a simple viscoelastic −1 for the two timescales involved, at least for Q > 4–5 nm . assumption. On the other hand in Ref. [42], using a double exponential model allows one to extract essentially the same value for the two timescales involved, at least for Q > 4–5 nm . 4.1. Microscopic vs. Structural Relaxations 4.1. Microscopic vs. Structural Relaxations The results discussed in [6] indicate that the relaxation phenomenon causing the viscoelastic response of neon has a microscopic origin as opposite to the structural character of relaxations routinely observed in The results discussed in [6] indicate that the relaxation phenomenon causing the viscoelastic associated and/or highly viscous fluids. It should be emphasized that structural and microscopic response of neon has a microscopic origin as opposite to the structural character of relaxations relaxations have different physical origin and phenomenology: the former are cooperative in character and routinely observed in associated and/or highly viscous ﬂuids. It should be emphasized that structural involve readjustments of the structure in response to a mechanical or a scattering‐induced perturbation. and microscopic relaxations have different physical origin and phenomenology: the former are Consequently, they are prevalent in glass‐forming systems or associated fluids, i.e., fluids with extended cooperative in character and involve readjustments of the structure in response to a mechanical or network of bonds, this is, e.g., liquid water [12,13,93–95], hydrogen fluoride [14], glycerol [17]. Since a scattering-induced perturbation. Consequently, they are prevalent in glass-forming systems or collective rearrangements of the structure are hampered by viscous processes, the timescale of structural associated ﬂuids, i.e., ﬂuids with extended network of bonds, this is, e.g., liquid water [12,13,93–95], relaxation follows a temperature dependence roughly as steep as the one of viscosity. hydrogen ﬂuoride [14], glycerol [17]. Since collective rearrangements of the structure are hampered by 6 2 10 D (m /s) Appl. Sci. 2016, 6, 64 21 of 31 viscous processes, the timescale of structural relaxation follows a temperature dependence roughly as steep as the one of viscosity. Conversely, microscopic relaxations are induced by single molecules’ motions in the 10 s window and are dominating in simple systems, such as dense gases and non-associated, weakly viscous, ﬂuids. These fast motions can be naturally identiﬁed with vibration-like cage oscillations. Due Appl. Sci. 2016, 6, 64 23 of 34 to their non-cooperative nature, microscopic relaxations are not signiﬁcantly slowed down by viscosity and their timescale exhibits only a weak dependence on thermodynamic conditions. To quantitatively assess the hypothesized microscopic nature of viscoelastic pr−oce 13 sses in neon, Conversely, microscopic relaxations are induced by single molecules’ motions in the 10 s window and are dominating in simple systems, such as dense gases and non‐associated, weakly viscous, fluids. sensible variable to take as a reference is the mean free time between interatomic collisions, These fast motions can be naturally identified with vibration‐like cage oscillations. Due to their non‐ . This depends on atomic size, mass and shape, as well as on the interaction potential and cooperative nature, microscopic relaxations are not significantly slowed down by viscosity and their thermodynamic state of the system. Without sophisticated calculations, a reasonable estimate can be timescale exhibits only a weak dependence on thermodynamic conditions. obtained assuming as model a hard sphere gas following the Maxwell-Boltzmann statistics. For this To quantitatively assess the hypothesized microscopic nature of viscoelastic processes in neon, system the inter-collision time can be easily calculated as: sensible variable to take as a reference is the mean free time between interatomic collisions,τc. This depends on atomic size, mass and shape, as well as on the interaction potential and thermodynamic state of the system. Without sophisticated calculations, a reasonable estimate can be obtained assuming as model a t (14) hard sphere gas following the Maxwell‐Boltzmann statistics. For this system the inter‐collision time can be 16pd k T easily calculated as: where d is the hard sphere diameter. In this context can be identiﬁed as the characteristic timescale of τ  (14) 16πdk T interatomic interactions and for microscopic, or collisional, relaxations is reasonable that the relaxation time and have comparable values. In Figure 12 the Q-dependence of the reduced relaxation time where d is the hard sphere diameter. In this context τc can be identified as the characteristic timescale of interatomic interactions and for microscopic, or collisional, relaxations is reasonable that the relaxation time Y = / is reported as obtained using values measured in neon in Ref. In spite of the important τα and τc have comparable values. In Figure 12 the Q‐dependence of the reduced relaxation time Ψ = τα/τc scattering of data, it is clear that for Q larger than about 8 nm Y approaches the unit value, as is reported as obtained using τα values measured in neon in Ref. In spite of the important scattering of data, expected in the merely collisional limit. In this perspective, data in Figure 12 lead to the conclusion −1 it is clear that for Q larger than about 8 nm Ψ approaches the unit value, as expected in the merely that for Q > 8 nm THz relaxation processes of neon are intimately related to interatomic−1collisions. collisional limit. In this perspective, data in Figure 12 lead to the conclusion that for Q > 8 nm THz This relamay xationbe proce ansses expected of neon conclusion, are intimatelyparticularly related to intersince atomicr co elaxation llisions. processes are activated by the This may be an expected conclusion, particularly since relaxation processes are activated by the coupling of density ﬂuctuations with some internal degree of freedom of the system. Given that coupling of density fluctuations with some internal degree of freedom of the system. Given that microscopic components of a monatomic ﬂuid lack internal degrees of freedom, it seems natural to microscopic components of a monatomic fluid lack internal degrees of freedom, it seems natural to identify identify local cage oscillations as the internal degree of freedom that density ﬂuctuations couple to. local cage oscillations as the internal degree of freedom that density fluctuations couple to. -3 n = 29 nm T = 32 K -3 T = 40 K n = 34.7 nm -3 T = 50 K n = 36.7 nm 10 -3 n = 38.5 nm 5 10 152025 30 -1 Q (nm ) Figure 12. The Q‐dependence of reduced relaxation time Ψ = τα/τc as obtained from Ne data at the Figure 12. The Q-dependence of reduced relaxation time Y = / as obtained from Ne data at thermodynamic conditions indicated in the plot. Data are adapted with permission from [6] copyrighted by the thermodynamic conditions indicated in the plot. Data are adapted with permission from [6] American Physical Society. The horizontal line marks the unit value characteristic of the collisional regime. copyrighted by American Physical Society. The horizontal line marks the unit value characteristic of the collisional regime. At this stage, THz viscoelastic data available in literature can be used to compare the case of Ne to the one even more studied of water. This comparison seems particularly meaningful due to the disparate nature of microscopic structure and, in particular, to the presence of a large-scale connectivity in water. Appl. Sci. 2016, 6, 64 22 of 31 A meaningful reference to compare viscoelastic effects in different materials is provided by the generalized longitudinal viscosity, whose link the viscoelastic spectral parameters is given by the formula: 2 2 rpw w qt h (15) Figure 7 displays the Q-dependence of extracted (through Equation (15)) from the viscoelastic modeling of neon spectra in [6] at the thermodynamic conditions considered in Figure 4. Results on neon are compared with those obtained in liquid water from an IXS [13] and a joint INS and INS result on heavy water [78]. Some clear features readily emerge from the comparison: (1) Viscosity curves of Ne are only weakly dependent on thermodynamic conditions, although the macroscopic viscosity in the spanned thermodynamic interval undergoes important variations. Combined thermodynamic [72] and bulk viscosity data [28] of neon lead to estimate a more than 300% variation of within the probed thermodynamic range, clearly non reproduced by mesoscopic measurements. Conversely, viscosity data of water in the same plot have a T dependence as sharp as the one of macroscopic viscosity [96]. This discrepancy is likely due to the large weight of structural processes in water, especially at the low Qs covered by data of [13]. Structural relaxations, owing to their collective nature can only be observed over long distances, or, equivalently, at high Qs and are expectedly dominating in associate liquids. (2) High T values of approach from above low-Q neon data. This seems consistent with the inferred link between structural relaxation and local readjustment of the hydrogen bond (HB) network in water [12,13]. At the highest Ts the average number of HBs per molecule is sensibly lower which weakens the strength of structural relaxations [13,19]. (3) The strong Q-dependence of in water suggests that at Q higher than the values covered by Ref. [13], viscosity curves of water and neon tend to get closer to each other also considering that the latter have an essentially ﬂat Q-dependence at high Qs. This impression is conﬁrmed by data derived from extremely high resolution INS measurements of on deuterated water (line + open squares symbols) at T = 288 K, which covered a Q-range larger than those of other water data reported in the same plot. Overall, data in Figure 13 lead to the conclusion that the merely collisional component of relaxation in water becomes more relevant either at high T, i.e., upon approaching supercritical condition, or at Appl. Sci. 2016, 6, 64 25 of 34 high Qs, where structural relaxation become less prevalent. T= 277 K T= 288 K T= 333 K H O 1E-3 T= 433 K 1E-4 1E-5 Ne 1E-6 10 20 30 -1 Q (nm ) Figure 13. Longitudinal viscosity data derived using Equation (15) either from IXS measurements [13], on Figure 13. Longitudinal viscosity data derived using Equation (15) either from IXS measurements [13], normal water (line + squares symbols), or from joint IXS and INS measurements [78] on heavy water (line + on normal water (line + squares symbols), or from joint IXS and INS measurements [78] on heavy open squares). Data are redrawn with permission from [13] and [78] copyrighted by the American Physical water Soci (line ety +. open The temperatu squares). res Data are ind aricated e redrawn in the plo with t. Ne permission on data correspo fromn[d13 to ] and the sa [m 78e] ther copyrighted modynamicby the conditions and symbols as in Figure 4 and are redrawn with permission from [6] copyrighted by American American Physical Society. The temperatures are indicated in the plot. Neon data correspond to the Physical Society. same thermodynamic conditions and symbols as in Figure 4 and are redrawn with permission from [6] copyrighted by American Physical Society. 4.2. 2001–2003: Quantum Effects on the Line‐Shape The treatment discussed so far is fully classical in character, being based on the assumption that all involved operators are commuting and the correlation functions are even function of time or, equivalently, that the spectral shape is symmetric in ω. It can be assumed that the onset of quantum effects primarily induces an asymmetry on the shape of S(Q,ω) due to the different statistical population of states having distinct ω values. In all experimental and computational works mentioned in this review, the spectrum is described within a quasi‐classical approximation. According to such approximation, quantum effects only influence the statistical populations of states having different ωs, through the so‐called detailed balance principle [97]. This can be fulfilled in infinite possible ways; one of the most commonly used consisting in the multiplication of the classical S(Q,ω) profile by the factor n(ω,) T = ω/[ kT1exp( ω/) kT], i.e.: BB  ω 1 SQ (,ω)= S (Q,ω) (16)  C kT 1e xp( ω/) kT BB Here is the “classical”, or symmetric, part of the spectrum which is essentially the Fourier SQ (,ω) transform of the intermediate scattering F(Q,t). While deriving a model for it is assumed that the SQ (,ω) operator defining the time evolution of density are classical commuting variables. However, in real fluids quantum deviations may be important. In this respect, it’s worth distinguishing between two possible quantum effects on S(Q,ω): i) “diffraction” or “delocalization” effects, which arise from the non‐commutative nature of Hamiltonian operators; and ii) exchange effects, reflect the symmetry restriction to be fulfilled in a many body system of identical particles. 1/ 2 Both effects depend on the de Broglie wavelengh , however: =2  π mk T  Diffraction effects are relevant only when  matches the length‐scales over wich the interparticle potential U(rij) varies apppreciably.  (Pa s) } Appl. Sci. 2016, 6, 64 23 of 31 4.2. 2001–2003: Quantum Effects on the Line-Shape The treatment discussed so far is fully classical in character, being based on the assumption that all involved operators are commuting and the correlation functions are even function of time or, equivalently, that the spectral shape is symmetric in !. It can be assumed that the onset of quantum effects primarily induces an asymmetry on the shape of S(Q,!) due to the different statistical population of states having distinct ! values. In all experimental and computational works mentioned in this review, the spectrum is described within a quasi-classical approximation. According to such approximation, quantum effects only inﬂuence the statistical populations of states having different !s, through the so-called detailed balance principle [97]. This can be fulﬁlled in inﬁnite possible ways; one of the most commonly used consisting in the multiplication of the classical S(Q,!) proﬁle by the factor npw, Tq }w{k Tr1 expp}w{k Tqs, i.e.: B B }w 1 SpQ, wq S pQ, wq (16) k T 1 expp}w{k Tq B B Here S pQ, wq is the “classical”, or symmetric, part of the spectrum which is essentially the Fourier transform of the intermediate scattering F(Q,t). While deriving a model for S pQ, wq it is assumed that the operator deﬁning the time evolution of density are classical commuting variables. However, in real ﬂuids quantum deviations may be important. In this respect, it’s worth distinguishing between two possible quantum effects on S(Q,!): i) “diffraction” or “delocalization” effects, which arise from the non-commutative nature of Hamiltonian operators; and ii) exchange effects, reﬂect the symmetry restriction to be fulﬁlled in a many body system of identical particles. 1{2 Both effects depend on the de Broglie wavelengh L }p2p{mk Tq , however: Diffraction effects are relevant only when L matches the length-scales over wich the interparticle potential U(r ) varies apppreciably. ij Exchange effects are instead relevant when Lbecomes comparable with interatominc centers of mass distance and are thus observable only for small mass atoms, for which such a distance can be relatively small. These effects only emerge at very low temperature and responsible for few “spectacular” quanto-mechanical manifestations, such as, e.g., Bose condensation and superfluidity. One can reasonably assume that all sampple considerd in this review are either fully classical or moderately quantum ﬂuids, for which exchange effects can be discarded, while quantum deviation can be achieved through h ¯ -order corrections of classical results. For moderately quantum ﬂuids, one can still derive F(Q,t) using a memory function formalism. However, the operators deﬁning the Hamiltonian of the system are generally non-commuting. For these systems, the time dependence of density ﬂuctuation is derived as the solution of the Heisenberg equation of motion: . i drpQ, tq irdr, Hs rexpriQ r ptqs , Hs (17) where the Hamiltonian, H, for a N particle system reads as: ¸ ¸ H U r (18) ij 2M i1,...N ij where r is the Laplacian operator and U r is a pair potential acting between the i-th and j-th atoms. i ij Three IXS works performed at the beginning of the new millennium were focused on the onset of quantum effects on the spectrum of supercritical He [98] and liquid Ne [99,100]. Whilst in the ﬁrst work , quantum effects were investigated for moderate Qs as a function of temperature and density; in the two IXS works on Ne, due to the unprecedented wide Q range the whole transition in Q form the classical (continuous) to the quantum (single particle) regimes was crossed. Appl. Sci. 2016, 6, 64 24 of 31 In all these IXS works, quantum effects were sought for in the coefﬁcients governing the short time dynamics, i.e. the spectral moments, which for a quantum ﬂuid read as: n n n n xw y 1{i pB {Bt q Fpq, tq| xrrrr, Hs Hs .......Hsy| (19) t0 t0 In principle thexw y can be computed from direct integration of IXS intensity. However, a rigorous computation is non-trivial, since one must cope with all spurious intensity effects and the contribution to the integral from the ﬁnite instrumental resolution. One can easily eliminate spurious intensity effect—as long as these are !-independent—by dealing with spectral moments’ ratios. Conversely, eliminating the resolution contribution would require a numerical deconvolution of the measured line-shape, which is ill-determined unless the spectral shapes has the form of a single featureless peak. In the IXS works mentioned above spectral moments were instead computed from direct integration of best ﬁt (non-convoluted) model line-shape. This procedure led to estimate that, for instance, quantum effects in the neon measurement of Figures 12 and 13 although clearly sizable, are < 15%–20%, even at the lowest Ts and/or the highest densities. This estimate is roughly consistent with the one reported by Bell et al. [40]. One may wonder what inﬂuence these effects can have on the relaxation process. It can be shown that the so-called quantum viscoelastic [101], an exponential ansatz, can still be used. The direct consequence of quantum effects is a global softening of the interaction potential. This stems from the delocalized nature of the tagged particle, which enables a deeper penetration in the region where the interatomic potential is repulsive. 5. 2006-Present: Toward a New Vision of the Supercritical Phase Based on the results previously discussed, it may be natural to conclude that the viscoelastic behavior is a dynamic ﬁngerprint of the liquid phase, which disappears when supercritical conditions are reached. Indeed, the PSD has often been considered an almost universal feature of liquid aggregates, since almost ubiquitously observed in liquid systems as disparate as, for instance hard sphere [102] and Lennard-Jones models [103], noble gases [43,46,71], as well as diatomic [8], associated [12,13] and glass forming [15,17] systems. Its disappearance at deeply supercritical conditions was considered as an indication that supercritical ﬂuids do not exhibit a high frequency viscoelasticity. The actual test of this conjecture requires a thorough investigation of the THz spectral shape of supercritical ﬂuid at extreme P and T values. For many decades, this has been an almost prohibitive task, owing to a number of experimental difﬁculties. As mentioned, the development of IXS brought about a substantial narrowing of the beam cross section, while the improved performance of undulators and crystal optics have dramatically enhanced the ﬂux achievable in ordinary THz spectroscopy measurements. This paved the way to new class of IXS experiments at extreme pressures based upon the use of diamond anvil cells (DACs). Taking advantage of this opportunity, an IXS DACs measurement on deeply supercritical oxygen demonstrated that a PSD of about 20% in amplitude is visible at temperatures as high as nearly twice the critical one [104]. The results of this measurement imposed a global revision of the previous idea that THz viscoelasticity phenomena could represent a dynamical ﬁngerprint of the liquid phase. Another breakthrough came from a successive joint IXS and MD work on Ar at extreme pressures [105], which showed the occurrence of a dynamic transition upon crossing a boundary in the supercritical domain. A dramatic reduction in the amplitude of PSD was, in fact, observed upon crossing the Widom line [106] and was interpreted as a crossover between a “liquid like” and a “gas like” sub-regions of the supercritical phase. As a reminder, the Widom line is deﬁned as the locus of speciﬁc heat maxima, which emanates from the critical point toward the supercritical domain with essentially the same slope as the coexistence line. For this reason, the Widom line is often considered a “prosecution” of the coexistence line beyond the critical point. Perhaps the most intriguing aspect of these results is that the mere existence of a boundary in the thermodynamic plane challenges the long-standing vision of the supercritical phase as intrinsically uniform [107], thus opening up unexplored theoretical scenarios and novel investigation opportunities. Appl. Sci. 2016, 6, 64 25 of 31 Another important result came from the recent conclusion that PSD effects are intimately related to the onset of shear propagation. The occurrence of a shear mode propagation in argon was ﬁrst demonstrated by a MD simulation on a Lennard Jones Ar [41,42]. As a common result, it was observed that at some wavelengths transverse current spectra bore evidence for a well-deﬁned maximum, indicating the occurrence of a shear mode propagation. More recently, a joint IXS and MD simulation work on deeply supercritical argon [108] evidenced a close connection between the ability of a ﬂuid system to support shear mode propagation and the presence of a sizeable PSD. In this work IXS spectra were measured along a nearly isobaric path (with P 1 GPa) with T spanning the 300 K–436 K temperature range, while MD data spanned the same isobar within a larger T-interval (up to 800 K). As a result, it was there shown that the disappearance of the PSD in deep supercritical conditions is accompanied by a parallel shear modes overdamping (see Figure 14). Overall, the fundamental character of the PSD and its link to thermodynamic transformations (previously disputed by [109]), are convincingly demonstrated in the work of Bolmatov et al. There, in fact, the PSD and the shear mode propagation are linked together and interpreted as different manifestations of the same universal property of ﬂuids: the onset of a high frequency viscoelasticity. Appl. Sci. 2016, 6, 64 28 of 34 The results of this work suggest that the dynamic response of supercritical ﬂuids is characterized by different regimes characteristic of distinct thermodynamic subdomains: (1) A gas‐like domain, featured by the absence of PSD effects and the related inability of the fluid to (1) A gas-like domain, featured by the absence of PSD effects and the related inability of the ﬂuid to support transverse wave propagation and support transverse wave propagation and (2) A liquid‐like domain in which the system exhibits merely viscoelastic features such as a sizable PSD and the onset of shear mode propagation. (2) A liquid-like domain in which the system exhibits merely viscoelastic features such as a sizable PSD and the onset of shear mode propagation. Furthermore, in the work of Bolmatov and collaborators [108] the Frenkel line [110] was identified as the crossover line between the two thermodynamic regions. The interpretation of the Frenkel line as a Furthermore, in the work of Bolmatov and collaborators [108] the Frenkel line [110] was identiﬁed crossover line demarcating the presence/absence of transverse acoustic propagation was originally as the crossover line between the two thermodynamic regions. The interpretation of the Frenkel line as discussed in [111]. It was also previously predicted that the occurrence, across the Frenkel line, of both a crossover line demarcating the presence/absence of transverse acoustic propagation was originally dynamic and structural crossovers accompanied by changes in phonon states [112–115]. Finally, it is worth mentioning a more recent work on deeply supercritical Ar [116], which provides compelling evidence for discussed in [111]. It was also previously predicted that the occurrence, across the Frenkel line, of both important dynamical changes occurring upon crossing the Frenkel line and having the form of strong dynamic and structural crossovers accompanied by changes in phonon states [112–115]. Finally, it is localization of the longitudinal sound mode. worth mentioning a more recent work on deeply supercritical Ar [116], which provides compelling Most importantly, the assessed parallelism between the disappearance of the PSD and the evidence for important dynamical changes occurring upon crossing the Frenkel line and having the overdamping of shear mode finally indicates that the observed viscoelastic behavior in noble gases is form of strong localization of the longitudinal sound mode. related relaxation process primarily affecting the shear viscosity. T = 600 K 12 T = 300 K 0 5 10 15 20 -1 Q (nm ) Figure 14. The dispersion curve of deeply supercritical Ar at P = 1 GPa, as evaluated from the maxima Figure 14. The dispersion curve of deeply supercritical Ar at P = 1 GPa, as evaluated from the position of MD simulated longitudinal (red dots) and transverse current spectra (open circles). The maxima position of MD simulated longitudinal (red dots) and transverse current spectra (open circles). corresponding hydrodynamic dispersion is also reported as a dashed line for comparison. Adapted with The corresponding hydrodynamic dispersion is also reported as a dashed line for comparison. Adapted permission from [108]. Copyright (2016) of American Chemical Society. with permission from [108]. Copyright (2016) of American Chemical Society. 6. Conclusions and Future Perspectives In summary, this brief review shows that THz studies on simplest systems as noble gases have, across the years, evidenced a rather complex behavior. The study of relaxation processes in these systems and, in particular, the comparison with those observed in more complex, associated fluids led the scientific community to identify a new class of relaxation phenomena completely different from structural (or α‐ )relaxations routinely observed glass formers and highly viscous materials. These phenomena are faster (in the sub‐ps) and only weakly dependent on thermodynamic conditions. Furthermore, their timescale is weekly affected by first neighbors’ arrangement, as suggested to its flat Q‐dependence, which presents no reminiscence of S(Q) oscillations. These collisional relaxations become visible also in associate fluids at temperature approaching critical conditions—due to the decrease in network connectivity—and/or over   ,  (meV) L t Appl. Sci. 2016, 6, 64 26 of 31 Most importantly, the assessed parallelism between the disappearance of the PSD and the overdamping of shear mode ﬁnally indicates that the observed viscoelastic behavior in noble gases is related relaxation process primarily affecting the shear viscosity. 6. Conclusions and Future Perspectives This brief review results THz studies on the viscoelastic response of simplest systems as noble gases have, across the years evidenced a rather complex behavior. The study of relaxation processes in noble gases and, in particular, the comparison with those observed in more complex, associated ﬂuids led the scientiﬁc community to identify a new class of relaxation phenomena completely different from structural (or -)relaxations routinely observed glass formers and highly viscous materials. These new relaxation phenomena are faster (in the sub-ps) and only weakly dependent on thermodynamic conditions. Furthermore, their timescale is weekly affected by ﬁrst neighbors’ arrangement, as suggested to its ﬂat Q-dependence which presents no reminiscence of S(Q) oscillations. These collisional relaxations become visible also in associate ﬂuids at temperature approaching critical conditions—due to the decrease in network connectivity—and/or over short distances, over which collective rearrangements cannot take place. These phenomena have recently observed to drastically affect the shear components of viscosity and to be intimately related to the presence of transverse acoustic modes propagation. Most importantly, the study of viscoelastic phenomena at extreme supercritical conditions challenges the long-standing vision of the supercritical phase as inherently uniform state of matter. Nowadays these results call for a new and more complex understanding of the supercritical domain, based on the existence of liquid-like (viscoelastic) and gas-like (merely viscous) sub-domains separated by a thermodynamic boundary. Shedding further light onto this revolutionary hypothesis is certainly one of the main motivations for the scientiﬁc community working in the ﬁeld. Looking ahead, further advances in the study of relaxation phenomena in noble gases and more speciﬁcally in noble gases are still held back by inherent limitation in the used techniques. If from one side INS is still hampered by both incident ﬂux and kinematic limitation, the main drawback of current IXS is the broad and slowly decaying (essentially Lorentzian) resolution wings. When focusing on relaxation processes the relevant spectral information is gathered in the extremely low frequency or quasi-elastic spectral window, often on the side of a dominating central peak. A narrow resolution width is thus required to properly resolve the spectral features of interest, while a superior contrast is necessary to discern them from the broad wings of the elastic line. Fortunately, a new concept optics for both monochromatization and energy analysis holds the promise for new generation spectrometers meting the required performance [117]. This will enable to perform IXS experiments with a resolution function <1 meV broad and essentially Gaussian in shape. The feasibility of such high resolution optical schemes has been demonstrated in a recent work [118], while a beamline based on this working principle has recently become operative at the new synchrotron NSLS II of Brookhaven National Laboratory [119]. This new opportunity is certainly quite encouraging and, in the near future, deemed to further advance our knowledge of the complex THz dynamics of simple liquids. 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Publisher: Multidisciplinary Digital Publishing Institute
Copyright: © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
ISSN: 2076-3417
DOI: 10.3390/app6030064
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Abstract

applied sciences Review The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy Alessandro Cunsolo National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA; acunsolo@bnl.gov; Tel.: +1-631-344-5564; Fax: +1-631-344-8189 Academic Editor: Christoph Hauri Received: 24 November 2015; Accepted: 25 January 2016; Published: 26 February 2016 Abstract: Approximately 50 years of inelastic scattering studies of noble gases are reviewed to illustrate the main advances achieved in the understanding of the THz dynamics of simple systems. The gradual departure of the spectral shape from the hydrodynamic regime is discussed with an emphasis on the phenomenology of fast (sub-ps) relaxation processes. This review shows that relaxation phenomena in noble gases have an essentially collisional origin, which is also revealed by the parallelism between their characteristic timescale and the interatomic collision time. Additionally, recent THz spectroscopy results on noble gases at extreme thermodynamic conditions are discussed to illustrate the need for a revision of our current understanding of the supercritical phase. Keywords: inelastic X-ray scattering; inelastic neutron scattering; noble gases; relaxation phenomena; liquid and supercritical systems 1. Introduction The study of the collective molecular dynamics of liquids and glassy materials has been a vibrant ﬁeld of research since the dawn of modern science. However, in spite of an intensive theoretical, experimental and computational scrutiny, the dynamic response of these systems still presents many unsettled aspects, mainly owing to the inherent disorder and complex movements of their microscopic constituents. Among various variables unveiling the dynamic behavior of a ﬂuid, the spectrum of density ﬂuctuations, S(Q,!), is of particular interest, since it is directly accessible by both spectroscopic techniques and computer simulation methods. In spectroscopy experiments, S(Q,!) is probed by generating density ﬂuctuations on a target sample through the collision with a beam of particle-waves, i.e., photons or neutrons. During this collision, an energy h ¯! and a momentum h ¯ Q (here h ¯ = h/2 with h being the Plank constant) are transferred from the probe particles to the sample and the former are scattered over the entire solid angle. It was demonstrated that for both neutron [1] and X-ray [2] probes, the scattering intensity is proportional to the dynamic structure factor, S(Q,!), which provides a snapshot of the dynamic response of the sample through an average over timescales 9 ! and distances 9 Q . Therefore, the (Q,!) region explored in a spectroscopy measurement can be expanded enough to cover the whole transition of the dynamics from the macroscopic regime to the microscopic one. The shape of S(Q,!) is exactly predicted within these two limits and essentially unknown in between, i.e. in the so-called mesoscopic region, corresponding to Q’s and !’s matching the inverse of intermolecular separations and cage oscillation frequencies, respectively. Investigations of this region may unravel fundamental aspects of molecular motions and mutual interactions in disordered systems and can be ideally performed by X-ray (IXS) and neutron (INS) scattering techniques, as well Appl. Sci. 2016, 6, 64; doi:10.3390/app6030064 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 64 2 of 31 as by computational methods. Due to the lack of rigorous theoretical predictions, our understanding of the dynamics of disordered systems at mesoscopic scales is often limited to a phenomenological analysis of experimental and computational results. 1.1. The Dynamics of Molecules at Mesoscopic Scales In the macroscopic limit, a liquid appears as a continuum, whose dynamic response is averaged over a large amount of intermolecular interactions. For such a system, the hydrodynamic theory can be consistently used to describe S(Q,!) and ultimately leads to predict a sharp triplet proﬁle [3]. This is composed by the heat diffusion (Rayleigh) central peak, relating to entropy ﬂuctuations, which diffuse at constant pressure (P), and the two Brillouin acoustic side peaks, connected to P ﬂuctuations, propagating at constant entropy. In the opposite regime, customarily referred to as single particle or impulse approximation limit, the probed dynamic event is essentially the free recoil of the single struck atom after the collision with the probe and before any successive interaction with ﬁrst neighboring atoms. In this limiting region, the dynamic response of the ﬂuid reﬂects the momentum distribution of the struck atom, which is Gaussian in shape for both classical and quantum systems. Several theoretical models have been developed in an attempt to account for the gradual departure of the spectral shape from the hydrodynamic regime. Perhaps the most successful approach is based upon the framework of the Zwanzig-Mori [4,5], or memory function, formalisms. Its core assumption is that the departure from the hydrodynamic regime is gradual enough to be accurately described by leaving the formal structure of hydrodynamic equations unchanged, yet generalizing some thermodynamic and transport parameters as local variables of space and time. From a physical point of view, this local character reﬂects the non-homogeneous, non-stationary nature of matter at mesoscopic scales. A similar approach was successfully proposed by a large number of IXS works on samples of increasing complexity, including: (1) noble gases [6,7]; (2) diatomic liquids [8]; (3) liquid metals [9–11]; (4) hydrogen-bonded systems [12–14]; (5) glass formers [15–18]. 1.2. The Departure from Hydrodynamic Regime and the Onset of Relaxation Processes Among various phenomena occurring at the departure from the hydrodynamic regime, the coupling of density ﬂuctuations with relaxation phenomena is perhaps the most solidly understood [6,7,12,13,19]. To gain a basic insight on the physics of a relaxation process, one may start from considering that scattering-generated density waves bring the sample away from its local equilibrium. In response, energy redistributions take place from the density perturbation toward internal degrees of freedom, ultimately driving the sample to a new local equilibrium. The timescale needed by the sample to relax to the equilibrium, , depends both on the wavelength = 2/Q of the considered density wave and on the sample characteristics. For instance, if increases so do both the number of rearranging molecules and the relaxation time. Furthermore, the timescale of structural relaxations has a temperature (T-)dependence roughly as steep as the one of viscosity, due to the circumstance that viscous resistance hampers internal rearrangements. These last considerations only apply the so-called structural relaxations, involving collective rearrangements of the structure, which are dominating in highly interconnected many body systems. Conversely, for monatomic ﬂuids relaxation phenomena mostly have a collisional nature [7,20], especially when matches ﬁrst neighboring molecules’ separations. Collisional relaxations relate Appl. Sci. 2016, 6, 64 3 of 31 to energy transfers between density ﬂuctuations and mutual inter-particle collisions. Compared to structural relaxations, they are faster (usually spanning the 10 s range) and weakly dependent on thermodynamic conditions [7,20]. In general, collisional and structural relaxations coexist in a given ﬂuid and their relative relevance depends on the nature of intermolecular interactions, thermodynamic conditions and probed Q value. Experimental studies of the Q or T-transition between structural and collisional relaxation regimes often present major challenges, as the required access to large Q (or T) intervals and the high statistical accuracy in the spectral shape measurement. Furthermore, advances in the ﬁeld are currently held back by instrumental limitations, especially concerning resolution width and spectral contrast. Nonetheless, important advances in the understanding of these phenomena have been made and, most importantly, next generation instruments with the required performance will be soon available for the scientiﬁc community. 1.3. The Central Role of Noble Gases Noble gases are ideal samples to investigate relaxation phenomena, due to their extremely simple microscopic constituents and mutual interactions, which make them ideal prototypes to test theoretical models. In contrast to molecular ﬂuids, their microscopic components lack internal degrees of freedom and, contrary to, e.g., metallic liquids, they have simple and short-ranged inter-particle interactions. For instance, the intrinsic simplicity of these samples is crucial to reliably approximate the interatomic potential in molecular dynamics (MD) computer simulations [21]. Another advantage of noble gases is the large compressibility, which is a key factor to achieve substantial variations of density or, equivalently, of the strength of atomic interactions, even with moderate thermodynamic changes. The purpose of this review is to provide a short overview of THz spectroscopy studies of S(Q,!) in noble gases and in particular, to discuss the insight they shed on fast relaxation phenomena. Following an essentially chronological order, the review covers a period of approximately 50 years, from the mid-1960s to the present. Beginning from pioneering INS studies mostly seeking for reminiscences of an hydrodynamic S(Q,!) shape beyond the continuous limit, it discusses the new opportunities offered by the development of IXS toward the end of the past millennium, while illustrating the insight they provide onto the viscoelastic character of the THz dynamics of noble gases. The last sections of this review illustrate recent measurements in the deeply supercritical region calling for a global reconsideration of our understanding of the phase diagram of a ﬂuid. 2. Measurements of the THz Spectrum of Noble Gases up to the 1990s 2.1. The Spectral Line-Shape in the Continuous Limit As mentioned in the introductory section, the variable of interest for this review is the spectrum of density ﬂuctuations, or dynamic structure factor: » » 8 A E Ñ Ñ Ñ Ñ SpQ, wq d r drp r , tqdr p r , 0q exp ipQ r wtq dt (1) V 8 Ñ Ñ with drp r , tq being the space p r q and time (t) dependent density ﬂuctuation of the target sample having volume V. In the quasi-macroscopic regime, S(Q,!) can be described using the hydrodynamic theory of continuous media [3]. This approach stems from the formal expression of the conservation laws of the (density of) mass, momentum and energy of the ﬂuid plus two constitutive laws: the Navier-Stokes’ and heat transport’s equations. As extensively discussed in several monographies [3,22], the derived spectral shape is dominated by few long-lived collective modes, customarily referred to as hydrodynamic modes. These modes form a triplet well approximated by the following expression: Appl. Sci. 2016, 6, 64 4 of 31 SpQq A z z z h h S S SpQ, wq A 2 2 2 2 2 2 p w z pw w q z pw w q z S S S S (2) pw w q pw w q S S A b 2 2 2 2 pw w q z pw w q z S S S S with b rA z {p1 A q z s{w and I p1 I q{2. s s s h h h h One readily recognizes the three main components of the spectral shape (from left): (1) The heat diffusion (Rayleigh) central peak, which relates to entropy ﬂuctuations diffusing at constant pressure (P). (2) The two Brillouin acoustic side peaks, connected to P-ﬂuctuations propagating at constant entropy. (3) An additional contribution (term 9 b) asymmetric around the Brillouin peaks position having negative tails. This term distorts the Lorentzian terms 1) and 2) ultimately enabling the convergence of the spectral moments w SpQ, wqdw, with n ¤ 2. The hydrodynamic proﬁle in Equation (2) contains shape parameters whose Q-dependence is not explicitly considered here. However, at low Qs the latter can be expressed through a polynomial Q-expansion (see, e.g., Equations (42)–(45) of [23]), which at the lowest order reads: w c Q (3a) ' s s z rpg 1q D n s Q {2 (3b) T L z D Q (3c) Here c , D and u are, respectively, the adiabatic sound velocity the thermal diffusivity and the s T L longitudinal kinematic viscosity. 2 2 It should be recognized that Equation (3a–c) are valid only if D Q ,GQ c Q, that is if the T s lifetime of hydrodynamic modes is much longer than the acoustic period. Under these conditions, the hydrodynamic modes are long-lived excitations appearing in the spectrum as well-resolved and sharp peaks, customarily referred to as Rayleigh-Brillouin triplet. Experimental methods such as Brillouin light scattering (BLS) provide a measurement of Rayleigh-Brillouin triplet by typically spanning the 3 2 1 Appl. Sci. 2016, 6, 64 5 of 34 10 –10 nm Q-range. An example of the simpliﬁed hydrodynamic spectrum is presented in Figure 1 along with the three separate components discussed above. Rayleigh-Brillouin triplet Brillouin peaks Rayleigh peak Asymmetric factor 0.2 0.0 -15 -10 -5 0 5 10 15 (GHz) Figure 1. Typical shape of the Rayleigh‐Brillouin triplet measured by Brillouin light scattering. The separate Figure 1. Typical shape of the Rayleigh-Brillouin triplet measured by Brillouin light scattering. contributions to the total shape are represented by line of different color, as indicated in the legend. The separate contributions to the total shape are represented by line of different color, as indicated in the legend. The Rayleigh‐Brillouin triplet in Equation (2) will be hereafter quoted as either generalized or simplified hydrodynamic spectrum, respectively with or without the lowest order Q approximation in Equations (3a–c). 2.2. 1922–1969: From Theoretical Prediction to Actual Measurements The hydrodynamic shape of the spectrum was theoretically predicted since the seminal work of Brillouin [24] and the successive theoretical work of Landau and Plazeck during the 1930s [25] (see also the review by Mountain). However, the first experimental measurements of the Brillouin spectrum of liquids −7 could only be performed much later [26], mainly owing to the extremely fine resolving power (ΔE/E < 10 ) required for similar measurements in typical liquid samples. This performance imposes severe technical challenges that can only be addressed by complex interferometric techniques. In regards to noble gases, the first Brillouin study dates back to the measurement on Ar and Ne along the coexistence line [27] performed with a single‐pass Fabry‐Perot interferometer. Figure 2 provides an example of the Brillouin spectrum of dense Ar measured in this work. The hydrodynamic spectrum reconstructed using thermophysical and transport properties of the sample is also reported for comparison. It clearly appears that the latter profile essentially consists of three extremely narrow peaks, in full consistency with the simple hydrodynamic condition . Indeed the width of triplet components measured in [27] was either coincident DQQ ,Γ cQ Ts with the resolution function (for the Raileigh peak) or only slightly larger (for Brillouin peaks). Intensity (arbit. units) Appl. Sci. 2016, 6, 64 5 of 31 The Rayleigh-Brillouin triplet in Equation (2) will be hereafter quoted as either generalized or simpliﬁed hydrodynamic spectrum, respectively with or without the lowest order Q approximation in Equation (3a–c). 2.2. 1922–1969: From Theoretical Prediction to Actual Measurements The hydrodynamic shape of the spectrum was theoretically predicted since the seminal work of Brillouin [24] and the successive theoretical work of Landau and Plazeck during the 1930s [25] (see also the review by Mountain). However, the ﬁrst experimental measurements of the Brillouin spectrum of liquids could only be performed much later [26], mainly owing to the extremely ﬁne resolving power (DE/E < 10 ) required for similar measurements in typical liquid samples. This performance imposes severe technical challenges that can only be addressed by complex interferometric techniques. In regards to noble gases, the ﬁrst Brillouin study dates back to the measurement on Ar and Ne along the coexistence line [27] performed with a single-pass Fabry-Perot interferometer. Figure 2 provides an example of the Brillouin spectrum of dense Ar measured in this work. The hydrodynamic spectrum reconstructed using thermophysical and transport properties of the sample is also reported for comparison. It clearly appears that the latter proﬁle essentially consists of three extremely narrow 2 2 peaks, in full consistency with the simple hydrodynamic condition D Q ,GQ c Q. Indeed the width of triplet components measured in [27] was either coincident with the resolution function (for Appl. Sci. 2016, 6, 64 6 of 34 the Raileigh peak) or only slightly larger (for Brillouin peaks). 1.0 0.8 0.6 0.4 0.2 0.0 -4 -3 -2 -1 0123 4  (GHz)  Figure 2. The Rayleigh Brillouin spectrum of Ar measured along the coexistence line by Brillouin light Figure 2. The Rayleigh Brillouin spectrum of Ar measured along the coexistence line by Brillouin light scattering. The data are redrawn with permission from [27], copyrighted by the American Physical Society. scattering. The data are redrawn with permission from [27], copyrighted by the American Physical The hydrodynamic spectrum reconstructed using tabulated thermophysical properties of the sample and Society. The hydrodynamic spectrum reconstructed using tabulated thermophysical properties of the bulk viscosity measurements [28] are reported for comparison as a black line. sample and bulk viscosity measurements [28] are reported for comparison as a black line. From the position of the peak in the Rayleigh‐Brillouin spectrum, Fleury and Boon derived a value of sound velocity slightly lower than the one previously measured by ultrasound (US) absorption techniques From the position of the peak in the Rayleigh-Brillouin spectrum, Fleury and Boon derived at ≈1 MHz. Although such an apparent frequency‐decrease of sound velocity, also observed in water [29], a value of sound velocity slightly lower than the one previously measured by ultrasound (US) initially raised some interest, it lacks a particular physical significance [30] (see also [22], page 263). Rather, absorption techniques at 1 MHz. Although such an apparent frequency-decrease of sound velocity, it follows from the circumstance that the sound velocity measured by BLS (hypersonic velocity) is also observed in water [29], initially raised some interest, it lacks a particular physical signiﬁcance [30] expectedly lower by few percent than the counterpart measured by US (ultrasonic velocity), unless (see also [22], page 263). Rather, it follows from the circumstance that the sound velocity measured by competing viscoelastic effects become relevant. BLS (hypersonic Overall, the velocity) shape of is expectedly the spectrum lower predic by tedfew by per the cent hydrthan odynami theccounterpart theory was experi measur mentally ed by US confirmed by Brillouin measurement in [27], even though a slight ω‐dependence of viscosity was inferred. (ultrasonic velocity), unless competing viscoelastic effects become relevant. At this stage the occurrence of a hydrodynamic triplet at macroscopic scales seemed well‐assessed Overall, the shape of the spectrum predicted by the hydrodynamic theory was experimentally experimentally, while a remaining question concerned the persistence of a suitably generalized conﬁrmed by Brillouin measurement in [27], even though a slight !-dependence of viscosity was hydrodynamic profile down to mesoscopic scales. Most experimental studies described in the following inferred. At this stage the occurrence of a hydrodynamic triplet at macroscopic scales seemed paragraphs attempted to answer to this question. well-assessed experimentally, while a remaining question concerned the persistence of a suitably generalized hydrodynamic proﬁle down to mesoscopic scales. Most experimental studies described in 2.3. 1964–1967: Seeking for a Triplet Shape at Mesoscopic Scales the following paragraphs attempted to answer to this question. The use of a bare hydrodynamic description of the spectrum becomes questionable when approaching the mesoscopic regime, as therein the matter is no longer continuous, nor stationary. However, there are solid reasons to assume a suitably generalized hydrodynamic theory to hold validity even in this time‐ space domain. To illustrate this point, it is useful to recognize that for a system at densities typical of the −10 liquid phase, the mean free path spans the 10 m window, potentially becoming much smaller than the atomic size. Under these conditions, the movements permitted to the atoms are mainly short, vibration‐ −13 −1 like, cage oscillations (≈10 s). Consequently, even in the (Q ≈ nm , ω ≈ THz) mesoscopic range, the response of the system is still “averaged” over a large number of elementary dynamic interactions, which is a necessary pre‐requisite for a suitably generalized hydrodynamic description. In this perspective, the persistence of spectral features in the THz spectrum reminiscent of Brillouin hydrodynamic peaks did not appear to be a groundless expectation. Indeed, several INS experiments performed during the mid‐1960s aimed at validating this hypothesis by examining inelastic peaks in the spectrum of simple fluids. For instance, this is the case of an INS measurements on dense Ne, Ar and D2 [31], which clearly documented Intensity (arbit. units) Appl. Sci. 2016, 6, 64 6 of 31 2.3. 1964–1967: Seeking for a Triplet Shape at Mesoscopic Scales The use of a bare hydrodynamic description of the spectrum becomes questionable when approaching the mesoscopic regime, as therein the matter is no longer continuous, nor stationary. However, there are solid reasons to assume a suitably generalized hydrodynamic theory to hold validity even in this time-space domain. To illustrate this point, it is useful to recognize that for a system at densities typical of the liquid phase, the mean free path spans the 10 m window, potentially becoming much smaller than the atomic size. Under these conditions, the movements permitted to the atoms are mainly short, vibration-like, cage oscillations (10 s). Consequently, even in the (Q nm , ! THz) mesoscopic range, the response of the system is still “averaged” over a large number of elementary dynamic interactions, which is a necessary pre-requisite for a suitably generalized hydrodynamic description. In this perspective, the persistence of spectral features in the THz spectrum reminiscent of Brillouin hydrodynamic peaks did not appear to be a groundless expectation. Indeed, several INS experiments performed during the mid-1960s aimed at validating this hypothesis by examining inelastic peaks in the spectrum of simple ﬂuids. For instance, this is the case of an INS measurements on dense Ne, Ar and D [31], which clearly documented the presence of “extended hydrodynamic” modes in the spectrum. Due to the largely coherent cross section of these samples, both inelastic and quasi-elastic spectral features appeared in the measured spectrum. In particular, it was observed that at low Q values the inelastic shift of the side peaks approached from above the linear hydrodynamic law c Q predicted by Equation (3a). This urged the authors to interpret these peaks as the ﬁnite-Q extensions of the acoustic Brillouin peaks to mesoscopic scales. Conversely, Kroo et al. [32] reported no direct evidence of inelastic structures, however they inferred the existence of phonon like excitations indirectly from a comparison between liquid and solid phase spectra. The discrepancy between the results of Chen and those of Kroo is a likely consequence of the different incident wavelength used in the two experiments (4.1 Å and 5.3 Å respectively), which made the two sets of data hardly comparable. In a successive INS work, Sköld et al. [33] measured the spectrum of Ar in liquid (at T = 94 K and 102 K) and solid (at T = 68 K and 78 K) phases. The authors observed a linear Q-dependence of the inelastic peak at the lowest Qs, with a slope consistent to the adiabatic sound velocity. Furthermore, they showed that, at higher Qs, the sound dispersion curve of the liquid sample vaguely resembles the phonon dispersion curve of the solid, thus suggesting that the local pseudo-periodicity of the liquid structure gives rise to quasi-periodic zones reminiscent of the Brillouin zones of a crystal. Since clear signatures of acoustic excitations in the liquid were found in a Q range equivalent to the second pseudo-Brillouin zone, the authors concluded that at least two somehow “loose” pseudo-Brillouin zones could be identiﬁed in liquid argon. A similar conclusion was reached by an earlier INS investigation on liquid Pb [34], as well as a computer simulation on Rb [35]. The latter work used an inter-particle potential model characterized by an oscillatory decay to zero [36] instead of the Lennard-Jones potential routinely used to simulate noble gases. These differences in the interatomic interaction explain why in liquid metals inelastic modes are well-resolved up to almost the position of ﬁrst diffraction peak (Q ), whereas in noble gases they become overdamped for Q > Q /2. m m 2.4. 1971: First Evidences of a Rayleigh-Brillouin triplet beyond the Continuous Limit The ﬁrst convincing evidence of the persistence of well-deﬁned inelastic peaks beyond the hydrodynamic regime was reported in an INS measurement of Bell and collaborators on supercritical 1 1 neon [37]. This work spanned unusually low Q values (0.6 nm ¤ Q ¤ 1.4 nm ), which substantially reduced the dynamic gap existing between the quasi-continuous regime probed by BLS (with Q 2 1 spanning the 10 nm window) and the range covered by “standard” INS measurements (typically Q ¥ 2–3 nm ). A best ﬁt of measured line-shape with a simpliﬁed hydrodynamic similar to the one deﬁned by Equations (2) and (3a–c) enabled the authors to conclude that the simple hydrodynamic theory consistently describes the spectral shapes measured well beyond the continuous approximation. Figure 3 provides an example of the spectral proﬁles measured by Bell et al., clearly showing the persistence of neat side peaks reminiscent of the hydrodynamic acoustic excitations. Appl. Sci. 2016, 6, 64 7 of 31 Again, the inelastic shift of side peaks reportedly depends linearly on Q, with a slope consistent with the known value of adiabatic sound velocity. Interestingly, the linear dispersion seemed to persist over the whole Q-range explored by Bell et al., even when the side peaks transform to 2 2 broad shoulders, for which the simple hydrodynamic approximation (D Q ,GQ c Q) is clearly unfulﬁlled. Additionally, the widths of both inelastic (Brillouin) and quasi-elastic (Rayleigh) peaks were found to depend on Q nearly consistently with the predictions of Equations (3a–c). These aspects will be discussed in greater detail below. It is worth acknowledging that comparably well-resolved THz inelastic peaks were previously observed in a largely quantum system as superﬂuid He at 1.1 K [38] and in the mentioned work on liquid metals by Dorner et al. [39]. 2.5. 1973–1975: First Signatures of A Viscoelastic Behavior in Real and Simulated Noble Gases A successive experiment performed on neon by Bell et al. [37] covered Q values largely exceeding Appl. Sci. 2016, 6, 64 8 of 34 1 1 the simple hydrodynamic limit (2.7 nm ¤ Q ¤ 15 nm ), strongly overlapping the range covered by more “standard” INS measurements. Not surprisingly, the modeling of the line-shape at these higher Qs A successi was less ve exp straightforwar eriment perfod, rmed requiring on neon aby generalization Bell et al. [37] coof vered transport Q values variables, largely excee asd pr inescribed g the −1 −1 simple hydrodynamic limit (2.7 nm ≤ Q ≤ 15 nm ), strongly overlapping the range covered by more by the “molecular hydrodynamic” approach. Speciﬁcally, the used line-shape model consisted in “standard” INS measurements. Not surprisingly, the modeling of the line‐shape at these higher Qs was less a generalization of the simple Brillouin triplet deﬁned by Equations (2) and (3a–c) in which allowance straightforward, requiring a generalization of transport variables, as prescribed by the “molecular was made for a Q and/or !-dependence for some thermodynamic and transport parameters appearing hydrodynamic” approach. Specifically, the used line‐shape model consisted in a generalization of the in the hydrodynamic equations. simple Brillouin triplet defined by Equations (2) and (3a–c) in which allowance was made for a Q and/or In particular, it was observed that the ﬁnite Q generalization of both D and had the form of ω‐dependence for some thermodynamic and transport parameters appearing in the hydrodynamic equations. a sharp, nearly Lorentzian, decay, indicating a decreasing weight of thermal ﬂuctuations at mesoscopic In particular, it was observed that the finite Q generalization of both DT and γ had the form of a sharp, distances. nearly Lorentzian, Furthermor dee, cathe y, indicating longitudinal a decr viscosity easing weneeded ight of th to ermal be generalized fluctuations at as meso a frequency scopic distances. dependent Furthermore, the longitudinal viscosity needed to be generalized as a frequency dependent variable as a variable as a typical manifestation of viscoelasticity. Although viscoelastic effects on the line-shape typical manifestation of viscoelasticity. Although viscoelastic effects on the line‐shape will be discussed in will be discussed in greater detail in a successive section, it’s worth emphasizing here that the core greater detail in a successive section, it’s worth emphasizing here that the core feature of viscoelastic fluids feature of viscoelastic ﬂuids is their sharply frequency dependent dynamic response, resembling either is their sharply frequency dependent dynamic response, resembling either the one of a liquid or the one of the one of a liquid or the one of a solid at low or high frequency, respectively. The crossover between a solid at low or high frequency, respectively. The crossover between the liquid‐like (or viscous) and solid‐ the liquid-like (or viscous) and solid-like (or elastic) regimes is marked by the relaxation frequency like (or elastic) regimes is marked by the relaxation frequency 1/τ. In a scattering experiment the viscous‐ 1/. In a scattering experiment the viscous-to-elastic transition can be observed when the applied to‐elastic transition can be observed when the applied perturbation (e.g., a scattering generated acoustic perturbation waves) attai (e.g., ns a fr ae scattering quency (csQ generated ) much larg acoustic er than 1/waves) τ. In practice attains , this a fr ca equency n be obta(inceQ d )by much increasing larger Qthan and/or τ (by lowering T). 1/. In practice, this can be obtained by increasing Q and/or (by lowering T). -1 Q = 1.4 nm -1 Q = 1.2 nm -1 Q = 1.0 nm -1 Q = 0.8 nm -1 Q = 0.6 nm -0.4 -0.2 0.0 0.2 0.4   (meV) Figure 3. Spectral lineshapes of neon measured by inelastic neutron scattering INS in neon at the low Figure 3. Spectral lineshapes of neon measured by inelastic neutron scattering INS in neon at the exchanged momenta indicated in the plot, the red line through data roughly connects the tops of inelastic low exchanged momenta indicated in the plot, the red line through data roughly connects the tops of peaks. Data are redrawn with permission from [40]. inelastic peaks. Data are redrawn with permission from [40]. On this ground, it is reasonable to expect that even a simple fluid at some Q, T values can exhibit solid‐ like properties, such as, for instance, the ability of supporting transverse acoustic waves. This possibility could only be tested using computer MD simulation, which can determine the correlation function between transverse components of atomic velocities, or its Fourier transform, customarily referred to as the transverse current spectrum. The transverse plane is defined as orthogonal to the momentum exchanged in the scattering event, Q . Indeed, the first indication of a high Q transverse acoustic propagation in a Appl. Sci. 2016, 6, 64 8 of 31 On this ground, it is reasonable to expect that even a simple ﬂuid at some Q, T values can exhibit solid-like properties, such as, for instance, the ability of supporting transverse acoustic waves. This possibility could only be tested using computer MD simulation, which can determine the correlation function between transverse components of atomic velocities, or its Fourier transform, customarily referred to as the transverse current spectrum. The transverse plane is deﬁned as orthogonal to the momentum exchanged in the scattering event, }Q. Indeed, the ﬁrst indication of a high Q transverse acoustic propagation in a monatomic system was inferred from the presence of a clear inelastic peak in the transverse current spectrum of a Lennard-Jones model representing Ar [41]. In this work, viscoelastic effects on the spectrum of density ﬂuctuations were described assuming a Gaussian time decay of memory function. A successive MD simulation performed on a Lennard-Jones model of Ar at the triple point [42] used instead a “more standard” exponential ansatz for the memory function within the assumptions of either a single or a double timescale. It was observed that the two alternative hypotheses did not yield appreciably different best-ﬁt results. Furthermore, this study conﬁrmed the previous observation of a Q-decay of “the thermal” parameters D and g and indicated a relaxation time ranging in the 10 ps interval with smooth variations. 2.6. 1978–1987: The Failure of the Three Modes Description High Q in Liquid Noble Gases and the Test of Kinetic and Mode Coupling Theories Successive INS works on Ne [43] and Ar [43–47] proposed a line-shape modeling consisting on the sum of generalized Lorentzian terms. Different versions of such a sum were used by superimposing the convergence of spectral moments of increasing order. Overall, it was observed that a spectral shape similar to the one in Equation (2) provides an accurate description of the spectrum of both liquid and supercritical noble gases. However, the use of such a model led to somehow controversial results for liquid sample at high Qs, as the presence of a propagation gap, that is a Q-region where ! = 0. A similar gap was also reported by INS measurements in molten salts [48] and liquid He [49,50] and its presence was also predicted by the kinetic theory [51]. Its physical origin was connected to the prevalence of dissipative forces over elastic ones, which prevent the sound from propagating at some Qs. In this regard, it is interesting that the gap seemingly disappeared upon increasing the density and approaching solidiﬁcation [45]. An example of the pressure dependence of this effect is proposed by Figure 4. There, the various dispersion curves are divided for the corresponding sound velocity, thus expectedly joining the same hydrodynamic linear law, upon approaching the Q = 0 limit. From the plot it can be readily noticed that the Q-window for which ! = 0 reduces upon increasing the pressure. Although the presence of a region of forbidden sound propagation raised some initial interest among researchers, its physical signiﬁcance seemed also controversial. In this regard, it should be acknowledged that the observed gap is centered on the position of the ﬁrst diffraction peak, where the so-called de Gennes narrowing occurs [52], and the spectral shape is reduced to a narrow featureless peak. For this reason, the use of a “three pole” line-shape modeling may give questionable results in this Q-window, also considering that inelastic features in the spectrum become dramatically overdamped [53,54], and their real position somehow ill-determined. Apart from those based on Equation (2), no other line-shape modeling used in the literature ever indicate the occurrence of such a gap and, furthermore, its presence leads to a clear physical inconsistency. In fact from a simple ﬁnite Q extension of the compressibility theorem ([55], see also Equation (13a)) one has w ¥ k T{SpQq, from which it can be deduced that a vanishing ! implies s B s a diverging S(Q), which clearly has no physical foundation for a disordered system. Aside from these high Qs ﬂaws, the “three pole” proﬁle in Equation (2) has, in principle, a solid physical ground. In fact, from one side, it can be derived using a macroscopic description of density ﬂuctuations as the hydrodynamic theory; on the other, it is consistent with the result of a truly microscopic approach as the kinetic theory [56,57]. This can be demonstrated by developing the S(Q,!) Appl. Sci. 2016, 6, 64 9 of 31 as a sum of generalized Lorentzian terms [58], upon retaining only the three dominant ones and Appl. Sci. 2016, 6, 64 10 of 34 superimposing on them the fulﬁlment of the ﬁrst three sum rules. Argon at T = 120 K P = 400 bar P = 115 bar P = 20 bar 0 5 10 15 20 -1 Q(nm ) Figure 4. The sound dispersion curves measured in Ar at increasing pressure [47] and normalized to the Figure 4. The sound dispersion curves measured in Ar at increasing pressure [47] and normalized to respective sound velocities, as derived from thermodynamic data [59]. The arrow indicates the presence of the respective sound velocities, as derived from thermodynamic data [59]. The arrow indicates the a propagation gap, i.e. of a Q region where the sound frequency vanishes. The solid line represents the presence of a propagation gap, i.e., of a Q region where the sound frequency vanishes. The solid line hydrodynamic limiting dispersion expectedly joined by all curves in the Q = 0 limit. The dashed curves are represents the hydrodynamic limiting dispersion expectedly joined by all curves in the Q = 0 limit. the low Q prediction of the Mode Coupling Theory as derived for the curves of corresponding color. The dashed curves are the low Q prediction of the Mode Coupling Theory as derived for the curves of corresponding color. It is worth noting that the kinetic approach provides, in principle, a microscopically rigorous theory of density fluctuation. This stems from the assumption that the dynamic variables of the system depend on It is worth noting that the kinetic approach provides, in principle, a microscopically rigorous both atomic velocities and spatial coordinates, as required for a reliable account of microscopic interactions. theory of density ﬂuctuation. This stems from the assumption that the dynamic variables of the Unfortunately, the predictions performed using the kinetic theory usually give poor results at high density system depend on both atomic velocities and spatial coordinates, as required for a reliable account of due to the increasing weight of correlated collisions. The INS measurements of Postol and Pelizzari on microscopic interactions. Unfortunately, the predictions performed using the kinetic theory usually supercritical Ar [60] proposed an experimental test of the kinetic theory predictions based on the give poor results at high density due to the increasing weight of correlated collisions. The INS generalized Enskog approach. As a result, no quantitative agreement was demonstrated between measurements of Postol and Pelizzari on supercritical Ar [60] proposed an experimental test of the theoretical and experimental results, although at the low density (10.5 atoms/nm ) and high temperature kinetic theory predictions based on the generalized Enskog approach. As a result, no quantitative (295 K) probed in the experiment it was reasonable to assume the kinetic theory description to hold validity. agreement was demonstrated between theoretical and experimental results, although at the low density A successive work on liquid Ar at different pressures by de Schepper et al., whose results are reported (10.5 atoms/nm ) and high temperature (295 K) probed in the experiment it was reasonable to assume in Figure 4, attempted a general test of the mentioned Lorentzian sum development of S(Q,ω) [47]. The the kinetic theory description to hold validity. effect of spectral rule fulfillment was therein analyzed by comparing the results obtained by imposing the A successive work on liquid Ar at different pressures by de Schepper et al., whose results are fulfillment of sum rules of increasing order ( 2). Furthermore, in such a work the authors attempted to reported in Figure 4, attempted a general test of the mentioned Lorentzian sum development of describe the measured dispersion curves in terms of the predictions of the mode‐coupling theory (MCT) S(Q,!) [47]. The effect of spectral rule fulﬁllment was therein analyzed by comparing the results (see e.g. [61]). The non‐analytic dispersion relations predicted by the MCT, having the form of truncated Q‐ obtained by imposing the fulﬁllment of sum rules of increasing order (¤ 2). Furthermore, in such expansions with fractional exponents, were compared with experimental dispersions. The accuracy of the MCT a work pred the ictio authors n can be attempted judged by to comparing describe the themeasur dashed ed cu disper rves in sion Figur curves e 4 wi inthterms the loof w the Q po prrtio edictions n of the dispersio of the mode-coupling n curves. The cotheory mparison (MCT clearly ) (see demo e.g.nstrates [61]). The that non-analytic at the lowest dispersion Qs the sound relations disperpr sion edicted curve −1 is well described by the MCT for all explored pressures, although for Q > 4 nm this theory does not provide by the MCT, having the form of truncated Q-expansions with fractional exponents, were compared a co with nsistent experimental account of dispersions. the observeThe d beh accur avior acy of of quasi the‐ela MCT sticpr and ediction inelastcan ic linewidths be judged of by th comparing e spectrum.the dashed curves in Figure 4 with the low Q portion of the dispersion curves. The comparison clearly −1 2.7. 1990: “Extended Hydrodynamics Modes” up to Q ≈ 1 nm demonstrates that at the lowest Qs the sound dispersion curve is well described by the MCT for all explored pressures, although for Q > 4 nm this theory does not provide a consistent account of the observed behavior of quasi-elastic and inelastic linewidths of the spectrum. -1  /c (nm ) s s Propagation gap ( 0) s Appl. Sci. 2016, 6, 64 10 of 31 Appl. Sci. 2016, 6, 64 11 of 34 2.7. 1990: “Extended Hydrodynamics Modes” up to Q 1 nm After the ﬁrst low Q measurement on Ne by Bell et al., a second INS experiment was performed After the first low Q measurement on Ne by Bell et al., a second INS experiment was performed by 1 1 by Baﬁle et al. [62] on supercritical Ar at extremely low Q’s (0.35 nm ¤ Q ¤ 1.25 nm ). −1 −1 Bafile et al. [62] on supercritical Ar at extremely low Q’s (0.35 nm ≤ Q ≤ 1.25 nm ). This measurement This measurement further extended the probed dynamic region to lower Qs, thus reducing the further extended the probed dynamic region to lower Qs, thus reducing the gap with light scattering gap with light scattering measurements. Figure 5 displays two typical spectral shapes measured in measurements. Figure 5 displays two typical spectral shapes measured in this INS work, which clearly this INS work, which clearly conﬁrm the persistence of extended Brillouin peaks at mesoscopic scales. confirm the persistence of extended Brillouin peaks at mesoscopic scales. -1 Q = 1 nm -1 Q = 0.5 nm 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9  (meV) Figure 5. Two representative INS spectral lineshapes of Ar measured by Bafile et al. [62] at low Figure 5. Two representative INS spectral lineshapes of Ar measured by Baﬁle et al. [62] at low exchanged momenta. exchanged momenta. Most importantly, in this measurement the superior accuracy in the count statistics coupled with the Most importantly, in this measurement the superior accuracy in the count statistics coupled unprecedented dense Q‐grid, enabled a very detailed analysis of the Q‐dependence of line‐shape with the unprecedented dense Q-grid, enabled a very detailed analysis of the Q-dependence of parameters of Equation (2). Best fit values of such parameters are reported in Figure 6 and therein compared line-shape parameters of Equation (2). Best ﬁt values of such parameters are reported in Figure 6 and with either the simple hydrodynamic prediction (Equations (3a–c)) or a higher‐order polynomial expansion therein compared with either the simple hydrodynamic prediction (Equation (3a–c)) or a higher-order (see, e.g., Equations (42)–(45) of [23]). In the same Figure, the result previously obtained by Bell and polynomial expansion (see, e.g., Equations (42)–(45) of [23]). In the same Figure, the result previously collaborator on neon is also reported for comparison along with the corresponding simple hydrodynamic obtained by Bell and collaborator on neon is also reported for comparison along with the corresponding prediction of Equations (3a–c). simple hydrodynamic prediction of Equation (3a–c). All hydrodynamic curves included in the figure were obtained using thermodynamic and transport All hydrodynamic curves included in the ﬁgure were obtained using thermodynamic and parameters reported in the original works. transport Overalparameters l, data in Figru eported re 6 confirm in the that original the Q‐works. dependence of line‐shape parameters is roughly consistent with the low Q approximation in Equations (3a–c). This clearly indicates that “extended hydrodynamic” Overall, data in Figure 6 conﬁrm that the Q-dependence of line-shape parameters is roughly modes survive well above the hydrodynamic limit, up to Q values exceeding the light scattering domain consistent with the low Q approximation in Equation (3a–c). This clearly indicates that “extended by nearly two orders of magnitude. hydrodynamic” modes survive well above the hydrodynamic limit, up to Q values exceeding the light scattering domain by nearly two orders of magnitude. 2.8. 1998: First IXS Measurements of the THz Spectrum of Noble Gases 2.8. 1998: First IXS Measurements of the THz Spectrum of Noble Gases The only THz spectroscopic technique available until the mid‐90s was INS, a method intrinsically hampered by kinematic constraints (see, e.g., [63] pp. 63–101) limiting the accessible portion of dynamic The only THz spectroscopic technique available until the mid-90s was INS, a method intrinsically plane (Q,ω). These limitations are particularly penalizing at low Qs, where the collective nature of hampered by kinematic constraints (see, e.g., [63] pp. 63–101) limiting the accessible portion of dynamic structural rearrangements become dominating, as well as at large ω values, where instead short‐time, plane (Q,!). These limitations are particularly penalizing at low Qs, where the collective nature of collisional events have a visible influence on density fluctuations. structural rearrangements become dominating, as well as at large ! values, where instead short-time, These problems were successfully overcome after the development of the first high resolution IXS, a collisional events have a visible inﬂuence on density ﬂuctuations. spectroscopic technique virtually free from kinematic limitations [2,64]. These problems were successfully overcome after the development of the ﬁrst high resolution IXS, a spectroscopic technique virtually free from kinematic limitations [2,64]. Intensity (arbit. units) Appl. Sci. 2016, 6, 64 12 of 34 Appl. Sci. 2016, 6, 64 11 of 31 0.4 Ne at T= 70 K, n =14.45 atoms/nm Ar at T = 301 K, n = 5.04 atoms/nm 0.3 0.2 0.1 0.0 0.12 0.09 0.06 0.03 0.00 0.08 0.06 0.04 0.02 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 -1 Q(nm ) Figure 6. Relevant shape parameters of the S(Q,ω) of supercritical Ne and Ar as measured, at the indicated Figure 6. Relevant shape parameters of the S(Q,!) of supercritical Ne and Ar as measured, at the thermodynamic conditions, in [40,62], respectively. The reported solid and dashed lines are the indicated thermodynamic conditions, in [40,62], respectively. The reported solid and dashed lines are hydrodynamic predictions referring to symbols of corresponding color as derived from the lowest Q the hydrodynamic predictions referring to symbols of corresponding color as derived from the lowest expansion in Equations (3a–c). Q expansion in Equation (3a–c). The first IXS measurement on a dense noble gas was performed on deeply supercritical neon (T = 295 The ﬁrst IXS measurement on a dense noble gas was performed on deeply supercritical neon K, n = 29.1 atoms/nm ) and discussed in combination of a MD simulation on a Lennard‐Jones model (T = 295 K, n = 29.1 atoms/nm ) and discussed in combination of a MD simulation on a Lennard-Jones representative of the same sample [65]. In Figure 7, two representative IXS spectra (upper panels) and model representative of the same sample [65]. In Figure 7, two representative IXS spectra (upper panels) corresponding MD simulations (lower panels) are compared (middle panels) after the former have been and corresponding MD simulations (lower panels) are compared (middle panels) after the former multiplied with the detailed balance factor and convoluted with the instrumental energy resolution have been multiplied with the detailed balance factor and convoluted with the instrumental energy function. The good agreement between measured and computed line‐shapes suggests that the Lennard‐ resolution function. The good agreement between measured and computed line-shapes suggests that Jones model provides a consistent description of the spectrum. Furthermore, it is apparent that even at Q the Lennard-Jones model provides −1 a consistent description of the spectrum. Furthermore, it is apparent values extending well above 1 nm , the inelastic peaks are visible in the MD simulated line‐shapes and, in that even at Q values extending well above 1 nm , the inelastic peaks are visible in the MD simulated spite of some resolution limitations, in the experimental spectra as well. line-shapes and, in spite of some resolution limitations, in the experimental spectra as well.  z (meV)  z (meV)  (meV) s Appl. Sci. 2016, 6, 64 13 of 34 Appl. Sci. 2016, 6, 64 12 of 31 -1 -1 Q = 2 nm Q = 5 nm 0.8 1.2 1.0 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0.0 0.0 0.8 1.2 1.0 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0.0 0.0 0.95 0.95 0.76 0.76 0.57 0.57 0.38 0.38 0.19 0.19 0.00 0.00 -6 -3 0 3 6 -10 -5 0 5 10   (meV) Fi Figure gure 7.7. Up Upper per pan panels: els: lolow w Q Q inelast inelastic ic X‐X-ray ray scatte scattering ring IXS IXS spe spectra ctra memeasur asured ed on on dee deeply ply sup super ercriti critical cal neon (T neon =295 ( TK, =295 n = 29.1 K, n atoms/n = 29.1 atoms/nm m ). The ra).wThe data raw (opdata en circles) (open ar cirecles) comar pa ere compar d with ed corwith respo corr ndin esponding g resolution fu resolution nctions (dfunctions ashed lines) (dashed . Lowe lines). r paneLower ls: corrpanels: espondin corr g MD esponding spectraMD com spectra puted on computed a Lennaon rd aJo Lennar nes mo ddel representative of the same sample (blue lines). Middle panels: Comparison between IXS spectra (open Jones model representative of the same sample (blue lines). Middle panels: Comparison between circles) and the corresponding molecular dynamics (MD) ones convoluted with the resolution functions (red IXS spectra (open circles) and the corresponding molecular dynamics (MD) ones convoluted with the lines). Data are redrawn with permission from [65], which is copyrighted bt the American Physical Society. resolution functions (red lines). Data are redrawn with permission from [65], which is copyrighted bt the American Physical Society. At this stage, a pending question relates to the Q‐threshold, Qh, defining the limit of validity for the “extended hydrodynamic” regime. Based on the fit of MD spectra reported in Figure 8 and discussed in Refs. At this stage, a pending question relates to the Q-threshold, Q , deﬁning the limit of validity −1 [65–67], one could conclude that, at least for dense supercritical neon, Qh is as high as 7.5 nm . According for the “extended hydrodynamic” regime. Based on the ﬁt of MD spectra reported in Figure 8 and to the plot, the transport parameters derived from best‐fit results through Equations (3a–c) are essentially discussed in Refs. [65–67], one could conclude that, at least for dense supercritical neon, Q is as high consistent with the macroscopic, or hydrodynamic, limit at least for Q lower than the values included in as 7.5 nm . According to the plot, the transport parameters derived from best-ﬁt results through the shadowed area. Thanks to the high statistical accuracy of Q‐dependent data in Figure 8, one may infer Equation (3a–c) are essentially consistent with the macroscopic, or hydrodynamic, limit at least for that the simple hydrodynamic approximation is not accurate for the thermal diffusivity. In fact, low‐Q Q lower than the values included in the shadowed area. Thanks to the high statistical accuracy values of DT seem slightly lower than the macroscopic value. of Q-dependent data in Figure 8, one may infer that the simple hydrodynamic approximation is not accurate for the thermal diffusivity. In fact, low-Q values of D seem slightly lower than the macroscopic value. Intensity (arbit. units) ApAppl. pl. ScSci. i. 2016 2016 , 6,, 64 6, 64 13 14 of of 31 34 Mesoscopic regime Extended Hydrodynamics 0.25 b) 0.20 0.15 0.10 0.05 0.00 c) 0.14 0.07 0.00 0 5 10 15 20 25 -1 Q(nm ) Figure 8. The relevant transport parameters deduced from the fits of of IXS spectra of deeply supercritical Figure 8. The relevant transport parameters deduced from the ﬁts of of IXS spectra of deeply neon [67] and those of MD spectra from a Lennard Jones model representative of the same sample [65]. The supercritical neon [67] and those of MD spectra from a Lennard Jones model representative of the same shadowed area roughly locates the transition from the extended hydrodynamics and the mesoscopic regime. sample [65]. The shadowed area roughly locates the transition from the extended hydrodynamics and Horizontal dashed lines represent the macroscopic (hydrodynamic) values deduced from thermodynamic the mesoscopic regime. Horizontal dashed lines represent the macroscopic (hydrodynamic) values [59] and US spectroscopy data [28]. deduced from thermodynamic [59] and US spectroscopy data [28]. 2.9. The Onset of a Positive Sound Dispersion in Liquid Noble Gases 2.9. The Onset of a Positive Sound Dispersion in Liquid Noble Gases In the various works on noble gases reported in literature, it was observed that transport parameters In the various works on noble gases reported in literature, it was observed that transport drastically change upon relatively moderate P and T variations. This may not be surprising, given that the parameters drastically change upon relatively moderate P and T variations. This may not be surprising, large compressibility of these systems leads to substantial variations of inter‐particle distances and, given that the large compressibility of these systems leads to substantial variations of inter-particle consequently, of interaction strengths. distances and, consequently, of interaction strengths. For instance, clear transformations were observed in the Q‐dependence of the acoustic frequency and For instance, clear transformations were observed in the Q-dependence of the acoustic frequency damping when measured either in the liquid or in the supercritical phase. In particular, it was generally and damping when measured either in the liquid or in the supercritical phase. In particular, it was found that, in liquid Ne and Ar [43,44,58,68] ωs systematically exceeds the hydrodynamic prediction at generally found that, in liquid Ne and Ar [43,44,58,68] ! systematically exceeds the hydrodynamic low/intermediate Qs. A similar effect was also observed ins liquid He [48,69,70], although the quantum prediction at low/intermediate Qs. A similar effect was also observed in liquid He [48,69,70], although character of this sample makes this finding less straightforward to interpret. Furthermore, in liquid phase sample the quantum s an “anom charaacter lous”of Qthis ‐depend sample ence makes was al this so re ﬁnding portedless for th straightforwar e sound absodrp totion int erpr coeffici et. ent, Furthermor as derive,ed frin om liquid the wid phase th ofsamples inelastican pe“anomalous” aks, zs. ConverQ sely, -dependence all these di was spersi also ve ref eported fects see for m to the dsound isappear absorption at extreme supercritical conditions [40,62,65,66]. coefﬁcient, as derived from the width of inelastic peaks, z . Conversely, all these dispersive effects These trends are clearly exemplified by Figure 9, which compares spectral parameters obtained in seem to disappear at extreme supercritical conditions [40,62,65,66]. supercritical [65] and liquid [71] neon. Overall, data reported therein indicate that, while in liquid Ne the These trends are clearly exempliﬁed by Figure 9, which compares spectral parameters obtained sound frequency (ωs) and absorption (zs) at low Qs are, respectively, higher and lower than predicted by in supercritical [65] and liquid [71] neon. Overall, data reported therein indicate that, while in liquid Equations (3a–c), no substantial discrepancy is instead observed in supercritical Ne. In Figure 9 it can be Ne the sound frequency (! ) and absorption (z ) at low Qs are, respectively, higher and lower than s s predicted by Equations (3a–c), no substantial discrepancy is instead observed in supercritical Ne. In Figure 9 it can be also noticed that for the latter sample the agreement with the simple hydrodynamic prediction is excellent up to Q 7.5 nm . 6 2  2 10 D (m /s)   (m /s) c (m/s) L s T Appl. Sci. 2016, 6, 64 15 of 34 also noticed that for the latter sample the agreement with the simple hydrodynamic prediction is excellent Appl. Sci. 2016, 6, 64 14 of 31 −1 up to Q ≈ 7.5 nm . supercritical Ne liquid Ne viscoelastic A) B) consistency 8 0 effects with simple hydrodynamics D) C) 0 0 0 5 10 15 20 0 5 10 15 20 25 30 -1 Q(nm ) Figure 9. Left column: shift (panel a) and half width (panel c) of the inelastic peak in the spectrum of Figure 9. Left column: shift (panel a) and half width (panel c) of the inelastic peak in the spectrum supercritical Ne at T = 294 K and n = 29 atoms/nm as derived from a best fit line‐shape with Equation (2). Red of supercritical Ne at T = 294 K and n = 29 atoms/nm as derived from a best ﬁt line-shape with and black dots are IXS and MD data from [65] and [67] respectively. Data from [65] are redrawn with Equation (2). Red and black dots are IXS and MD data from [65] and [67] respectively. Data from [65] permission from the original publication, under the copyright of the American Physical Society. The solid are redrawn with permission from the original publication, under the copyright of the American lines represent the corresponding simplified hydrodynamic prediction in Equations (3a–c). In which Physical Society. The solid lines represent the corresponding simpliﬁed hydrodynamic prediction in literature values of thermodynamic properties [72] and bulk viscosity [28] coefficients were inserted. Right Equation (3a–c) In which literature values of thermodynamic properties [72] and bulk viscosity [28] column: the corresponding quantities are reported for liquid neon at T = 35 K and n = 36.65 atoms/nm . These coefﬁcients were inserted. Right column: the corresponding quantities are reported for liquid neon data are adapted with permission from [71] , which is copyrighted by the American Physical Society. at T = 35 K and n = 36.65 atoms/nm . These data are adapted with permission from [71] , which is copyrighted by the American Physical Society. As mentioned, “anomalous” dispersive effects observed for the liquid sample can be ascribed to the onset of a viscoelastic response induced by the coupling with a relaxation process. As mentioned, “anomalous” dispersive effects observed for the liquid sample can be ascribed to To illustrate this point, it is useful to recognize that, in a spectroscopy experiment, the scattering‐ the onset of a viscoelastic response induced by the coupling with a relaxation process. excited acoustic waves cause a time‐dependent perturbation of the local equilibrium of the target sample. To illustrate this point, it is useful to recognize that, in a spectroscopy experiment, the As a response, decay channels redistribute the energy carried by the acoustic wave toward some internal scattering-excited acoustic waves cause a time-dependent perturbation of the local equilibrium of the degrees of freedom of the fluid. These energy rearrangements ultimately drive the sample to relax in a new target sample. As a response, decay channels redistribute the energy carried by the acoustic wave local equilibrium within a timescale τ. toward some internal degrees of freedom of the ﬂuid. These energy rearrangements ultimately drive Two limiting scenarios can thus occur: (1) the time‐dependent acoustic perturbation has period, 2π/ωs, the sample to relax in a new local equilibrium within a timescale . much longer than any internal degrees of freedom of the system. Under these conditions, the latter relaxes Two limiting scenarios can thus occur: (1) the time-dependent acoustic perturbation has period, to equilibrium “instantaneously” (i.e., within a timescale τ << 1/ωs) and the acoustic propagation essentially ta2kes p/! place , much over longer success than ive any equiinternal librium st degr ates ees (viof sco fru eedom s limit)of ; (2 the ) Conv system. erselUnder y, if the these acouconditions, stic wave has the an ex latter tremel relaxes y short to pe equilibrium riod, it “perceives” “instantaneously” internal rearra (i.e. ng ,ement within s as a timescale frozen‐like (<< τ >>1 1/ /!ωs)) an and d does the acoustic not couple with them. Consequently, the acoustic propagation is elastic, i.e., it occurs with virtually no energy loss propagation essentially takes place over successive equilibrium states (viscous limit); (2) Conversely, if (elastic limit). the acoustic wave has an extremely short period, it “perceives” internal rearrangements as frozen-like Therefore, when a relaxation is active, the response of a system to the acoustic propagation depends ( >> 1/! ) and does not couple with them. Consequently, the acoustic propagation is elastic, i.e., it on how the acoustic frequency, ωs, compares to the relaxation time. When ωs increases from the viscous occurs with virtually no energy loss (elastic limit). Therefore, when a relaxation is active, the response of a system to the acoustic propagation depends on how the acoustic frequency, ! , compares to the relaxation time. When ! increases s s from the viscous (! << 1) to the elastic limit (! >> 1), the viscoelastic transition manifests itself s s through a systematic decrease of acoustic dissipation and a corresponding increase of sound velocity. Since in scattering experiments ! is typically varied by changing Q (through the dispersion relation ! = ! (Q)), in these measurements viscoelastic behavior is reﬂected by a Q-increase of ! and s s s a Q-decrease of z .  z (meV)  ( meV) S s Appl. Sci. 2016, 6, 64 15 of 31 This explains why in panels 9b and 9d these shape parameters are, at the lowest Qs, respectively larger and lower than their hydrodynamic predictions. Once the phenomenological signature of a viscoelastic behavior is experimentally assessed, a more quantitative understanding of this phenomenon could only come from the modeling of measured line-shape with a viscoelastic model. The next section is devoted to the derivation of such a model within the framework of the memory function formalism. 3. A Model for the Measured Spectral Shape 3.1. The Memory Function Formalism Let Aptq rA ptq..........A ptqs be a vector whose components are stochastic variables describing 1 n the properties of a system of N interacting particles (with N > u). It can be shown that under very general conditions, the equation of motion of Aptq has the following form: Ñ Ñ Ñ dAptq iW Aptq Kptq Apt tqdt f ptq (4) dt In the formula above the following quantities have been introduced: B F B F Ñ Ñ Ñ Ñ The antisymmetric matrix iW Ap0q, iLAp0q Ap0q, Ap0q is the proper frequency of the system, which describes the oscillatory behavior of Aptq. This matrix depends on equilibrium (static) properties of the system. Here the symbol x.......y indicates a statistical (thermal) average. Ñ Ñ ip1PqLt The variable f ptq e ip1 PqLAp0q is the random ﬂuctuating force. Noticeably, the B F Ñ Ñ presence of the p1 Pq term implies that Ap0q, f ptq 0, namely that the ﬂuctuating force is orthogonal (statistically uncorrelated) to Ap0q. B F B F Ñ Ñ Ñ Ñ The matrix Kptq f , f ptq Ap0q, Ap0q is memory matrix, or memory function if = 1, as in the cases of interest here. It is worth noticing that the integrand in Equation (4) includes all Aptq values between 0 and t, “weighted” by the memory, which thus deﬁnes the ability of the system to keep memory of the past. B F Ñ Ñ The equation of motion for the correlation matrix Cptq AptqAp0q can be readily derived from Ñ Ñ Equation (4) through a scalar product, while using the orthogonality between Ap0q and f ptq. Explicitly: dCptq iW Cptq Kptq Cpt tqdt (5) dt Equations (4) and (5) are referred to as memory equation or generalized Langevin equation either for dynamic variables or for their correlations, respectively. When dealing with a single dynamic variable, time reversal requirements impose W = 0. Using a procedure described in various monographies [22,63], the Langevin equation can be solved following an iterative approach, which eventually leads to the following recursive formula: Cpsq K p0q 1 1 rs s (6) Cp0q K p0q K p0q s . . . customarily referred to as the continued fraction expansion. Here K (t) is the i-th (iteration) order memory function associated to the variable of interest, Appl. Sci. 2016, 6, 64 16 of 31 zt Cpsq dte Cptq while is the Laplace transform of the correlation function. The t = 0 values of the i-th order memory function K (t) can be determined by superimposing the sum rule fulﬁllment, which for the few lowest orders yields: A E K p0q w (7a) .. @ D @ D 4 2 w w K p0q K p0q @ D @ D (7b) 2 0 2K p0q w w .. @ D @ D 6 4 w w K p0q 1 K p0q @ D @ D (7c) 2 2 2K p0q D w w 2 2 d Cptq 1 8 1 n n n Herexw y i r s rCp0qs w CpwqdwrCp0qs is the n-order spectral moment dt t0 of the correlation function C(!). Among all possible ﬂuctuating variables, we are here interested in density ﬂuctuations since the Fourier-Laplace transform of their correlation function, S(Q,!) is directly measured by both IXS and INS. S(Q,!) can be derived as the Fourier anti-transform of the intermediate scattering function A E Ñ Ñ Ñ Ñ FpQ, tq drp r , tqdr p r , 0q exp iQ r d r . The latter can be expressed through the continued fractions expansion of Equation (4), whose 2-nd order truncation yields: @ D FpQ, sq SpQq s m pQ, sq Where m pQ, sq is the second order memory function for density ﬂuctuations while S(Q) = F(Q,0) is the static structure factor. The S(Q,!) can be derived from Equation (7) as follows: @ D SpQ, wq 1 FpQ, s iwq 1 Re Re iw SpQq p SpQq p iw m pQ, s iwq The last term of the above formula can be calculated explicitly to eventually obtain: @ D 2 1 w m pQ, wq SpQ, wq 1 (8) @ D SpQq p 2 2 2 2 1 rw w wm pQ, wqs w rm pQ, wqs L L where m pQ, wq and m pQ, wq are, respectively, real and imaginary parts of the Fourier transform L L of m (Q,t). At this stage, the problem of choosing the most appropriate model is shifted from S(Q,!) to m (Q,t), or m (Q,!). In many literature works on liquid systems attempting a memory function L L based modeling of the spectrum (see e.g. [13,73]), it can be shown that a sensible ansatz for the time dependent memory function is given by: 2 2 m pQ, tq D exppt{t q D exppt{t q 2G dptq (9) L T a m T a The three terms in the right hand side of Equation (9) are (from left): (1) The thermal contribution. The ﬁrst term describes diffusive thermal motions triggered by 2 1 spontaneous temperature gradients. Its timescale and amplitude are t pgD Q q and T T 2 2 D pg 1qpc Qq respectively, with D , c , and being the generalized diffusivity, the T T T isothermal velocity and the constant pressure to constant volume speciﬁc heats ratio. Appl. Sci. 2016, 6, 64 17 of 31 (2) The viscous relaxation contribution. The second term accounts for relaxation processes affecting 2 2 2 2 the viscosity and having a timescale t and amplitude D pc gc qQ , where c is the a 8 a 8 generalized inﬁnite frequency or “elastic” sound velocity of the sample. (3) The instantaneous contribution. The last term accounts for the coupling of density ﬂuctuations with the ultra-fast vibrational dynamics here accounted by a (t) proﬁle. It was often found that the amplitude 2G has a quadratic Q-dependence and is essentially insensitive to thermodynamic changes [20]. It is immediate to recognize that real and imaginary parts of Fourier transform of the memory function in Equation (9) read as: t t T a 1 2 2 m pQ, wq D D G (10a) L T a 2 2 1 pwt q 1 pwt q 2 2 pwt q pwt q 2 T a 2 2 wm pQ, wq D D (10b) L T a 2 2 1 pwt q 1 pwt q T a respectively. These two equations can be inserted in Equation (8) to obtain a model for the classical part of the line-shape. It should be emphasized that at mesoscopic scales all transport and thermodynamic coefﬁcients introduced above and in the following, unless otherwise speciﬁed, are Q-dependent generalizations of their macroscopic counterparts, although such a dependence in not made explicit in the notation. The model used for the memory function often results from suitable approximation of Equation (9) as appropriate to the sample, thermodynamic conditions and dynamic range probed. For simplest systems as noble gases, the single exponential decay or viscoelastic model has shown to provide a reasonable approximation [42]. Regardless on the approximation adopted, the exponential ansatz for the viscoelastic term, lends itself two at least these two fundamental objections: (1) In principle, structural relaxation phenomena in highly viscous systems can be better described by assuming a stretched exponential rather than a simple exponential time-decay of the memory function [17,74,75]. However, the use of this model would introduce an unwanted additional parameter (the stretching coefﬁcient) and, perhaps more importantly, its Fourier transform cannot be cast on an analytical form. Fortunately, for simplest systems as noble gases, of interest in this paper, it is reasonable to assume that the simple exponential decay is a rather accurate approximation. (2) A second remark concerns the possible presence in the spectrum of a second inelastic excitation as observed in systems as diverse as liquid water [73,76–78], tetrahedrally arranged glasses [79], glass formers [80] liquid metals [81–85], complex biophysical samples [86–88] and mixtures [89]. The presence of this low frequency mode in the S(Q,!) is customarily ascribed to the onset of shear mode propagation, although this assignment could seem suspicious as S(Q,!) couples primarily with longitudinal movements only. In a liquid, the onset of shear waves in the spectrum of density fluctuations can only occur via the so-called longitudinal-transverse coupling [73,76,90], that is a mixing between acoustic modes having orthogonal polarization. The presence of this mode-mixing is not properly accounted for by any known analytic model for the memory function. However, such a coupling has never been observed in noble gases, likely due to the highly isotropic character of their interatomic interaction, and will not be discussed in the remainder of this paper. 3.2. The Departure from the Hydrodynamic Shape of the Spectrum In the hydrodynamic regime, density ﬂuctuations are probed for long time lapses (t >> ) over which the viscous relaxation is perceived as a very rapid decay, well approximated by a Markovian Appl. Sci. 2016, 6, 64 18 of 31 term (9 (t)) in the time-dependence of memory function. Overall, it can be shown that the following memory function: 2 2 2 m pQ, tq pg 1qpc Qq exp gD Q t 2n Q dptq (11) T T L L,hyd is fully consistent with the Rayleigh-Brillouin triplet expressed by Equation (2) under the approximation deﬁned by Equation (3a–c) As mentioned, at mesoscopic scales all parameter entering in Equation (11) should be generalized as Q dependent variables. Due to the lack of ﬁrm theories describing analytically the departure of the spectrum from the simple hydrodynamic regime, this Q-dependence is usually determined empirically through the best-ﬁt modeling of spectra measured at different Qs. Albeit phenomenological in character, this approach has proven to be successful in shedding a deep light on the fast dynamics of various disordered systems, including noble gases. In particular, it has elucidated many phenomenological aspects associated to the coupling of density ﬂuctuations with THz relaxation processes. 3.3. The Simple Viscoelastic Model The simplest method to account for THz viscoelastic effects on the spectrum of density ﬂuctuation is to assume a single exponential decay of the memory function: m pQ, tq D exppt{tq (12) L 2 where the amplitude of the memory function is ﬁxed by the superimposition of sum rules to the spectrum, which eventually leads to: @ D k T 2 B 2 w @ D Q (13a) MSpQq @ D @ D w (13b) @ D @ D 4 2 w w 2 2 2 @ D D c c Q (13c) 8 0 w SpQq @ D here w SpQq, while k and M are the Boltzmann constant and atomic mass, respectively; Equations (12) and (13a–c) deﬁne the so called Debye [91] or simple viscoelastic model [92]. Clearly, this model can hold validity only if: (1) thermal relaxations have a negligible weight on the time decay of memory function, i.e., if 1 and (2) the contribution from fast relaxation (term 9 (t) in Equation (9)) phenomena is also negligible. Conceptually, the simple viscoelastic description stems from the idea that, upon crossing the viscoelastic transition, the sound velocity c , changes from c w {Q B{r with B being the bulk 0 0 modulus—as appropriate for a liquid—to c w {Q M{r with M B 4{3G and G being, 8 8 respectively, the elastic and the shear modulus—as appropriate for solids. The superimposition of sum rules links the finite Q generalization of B and M to the second and fourth normalized moments, respectively. In a way the viscoelastic model through the sum rules’ fulfillment interpolates between the low Q, viscous, response and the high Q elastic one. Specifically, one side, Equation (13a) extends to intermediate Qs the macroscopic compressibility result [55] c K T{MSp0q on the other side, the T B high Q link between M and normalized fourth moment, [1] is here extended down to the same Qs values. 4. 2001–2007: The Memory Function-Based Modeling of IXS Spectra of Noble Gases The success of the memory function based modeling of experimental spectra was ultimately enabled by the dramatic improvement of the incident ﬂux and, consequently, of the count statistics Appl. Sci. 2016, 6, 64 19 of 31 prompted by the development of IXS. Furthermore, these high ﬂuxes coupled with the reduced beam cross section enabled spectroscopy measurements on substantially smaller samples, thus opening the access to extreme thermodynamic conditions. The ﬁrst IXS work investigating viscoelastic effects on liquid and supercritical neon [6] took advantage of these opportunities there by exploring an unprecedented broad thermodynamic range. Measurements were performed on dense neon from liquid to supercritical conditions following both an isochoric (n = 29 atoms/nm ) and an isothermal (T = 32 K) path. The sound dispersion curves of neon determined in such a work are reported in Figure 10. Best ﬁt values of w c Q and w c Q 0 0 8 8 derived from the best ﬁt modeling of the line-shape are compared with the acoustic frequency evaluated from the position of the maxima of current spectra C (Q,!), W . The current spectrum is deﬁned as the L l Fourier transform of the correlation function between longitudinal components of atomic velocities. The position of its maxima is often identiﬁed as the dominant acoustic frequency, this identiﬁcation being rigorous in the hydrodynamic limit only. Within the viscoelastic approach, W is assumed to undergo a transition between the viscous and the elastic limit respectively represented by ! and ! . 0 8 Figure 10 displays the values of W , ! and ! of liquid (upper panel) and supercritical (bottom 0 8 panel) neon reported in Ref. [6] and compares them with the hydrodynamic straight line c Q. From the inspection of the two plots the following trends are readily noticed: (1) the liquid phase datum bears evidence for the more-than-linear behavior of W (purple dots) for Q ¤ 10 nm , i.e., the systematic bending of W upwards the hydrodynamic linear law and toward ! (the mentioned PSD effect) . Conversely, the liquid phase point is characterized by a sizable PSD. (2) At higher Qs one can observe the “backward transition” of W from ! to ! , which is actually l 8 0 joined for Q larger than the position of the ﬁrst di Q (22 nm ). (3) Again, viscoelastic effects disappear in deeply supercritical conditions (lower panel) since ! , W and ! merge into each other (at least at the lowest Qs), thus suggesting that the viscous Appl. Sci. 2016, 6, 64 21 of 34 relaxation term in Equation (9) has a vanishing strength. Liquid, T = 40 K Supercritical, T = 294 K 0 3 6 9 12 15 18 21 24 27 30 -1 Q(nm ) Figure 10. The acoustic frequency Ωl derived from the maxima of current spectra (magenta dots) are Figure 10. The acoustic frequency W derived from the maxima of current spectra (magenta dots) are reported against the zero (black dots) and infinite frequency (blue dots) limiting dispersions extracted from reported against the zero (black dots) and inﬁnite frequency (blue dots) limiting dispersions extracted best fits the IXS spectra of neon. Fits were obtained with a single exponential viscoelastic model for the memory function decay (Equation (12)). The linear hydrodynamic sound dispersion (Equation (3a)) is also from best ﬁts the IXS spectra of neon. Fits were obtained with a single exponential viscoelastic model reported for reference as derived using tabulated values of the adiabatic sound speed [71]. Data are adapted for the memory function decay (Equations (12)). The linear hydrodynamic sound dispersion (see with permission from [6] copyrighted by American Physical Society. Equations (3a)) is also reported for reference as derived using tabulated values of the adiabatic sound As previously mentioned, the validity of the simple viscoelastic model demands that the thermal speed [71]. Data are adapted with permission from [6] copyrighted by American Physical Society. contribution to the memory function decay has negligible amplitude (γ ≈ 1). The soundness of this assumption is supported by the bottom panel of Figure 11, where the value of γ computed for supercritical Ne [65,66] is reported.  (Q),  (Q),  (Q) (meV) 0  l Appl. Sci. 2016, 6, 64 20 of 31 As previously mentioned, the validity of the simple viscoelastic model demands that the thermal contribution to the memory function decay has negligible amplitude ( 1). The soundness of this assumption is supported by the bottom panel of Figure 11, where the value of computed for Appl. Sci. 2016, 6, 64 22 of 34 supercritical Ne [65,66] is reported. 0.18 0.09 0.00 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0.0 0 5 10 15 20 25 30 -1 Q( nm ) Figure 11. Lower panel: the generalized specific heat ratio as derived from MD computations for Lennard Figure 11. Lower panel: the generalized speciﬁc heat ratio as derived from MD computations for Jones models simulating deeply supercritical neon ([66], red dots). The unit value is reported as a dashed Lennard Jones models simulating deeply supercritical neon ([66], red dots). The unit value is reported line for reference. Lower panel: the thermal diffusivity, as derived from MD simulations in [66], while using as a dashed line for reference. Lower panel: the thermal diffusivity, as derived from MD simulations the simplified hydrodynamic expression in Equation (3c). In all plots horizontal dashed line indicate the in [66], while using the simpliﬁed hydrodynamic expression in Equations (3c). In all plots horizontal macroscopic values (from National Institute of Standards and Technology NIST database [72] of the symbols dashed line indicate the macroscopic values (from National Institute of Standards and Technology having corresponding color. NIST database [72] of the symbols having corresponding color. The curve is consistent with the macroscopic limit at the lowest Qs. Overall, it can be inferred that the The curve is consistent with the macroscopic limit at the lowest Qs; however both curves suggest −1 weight of the thermal contribution (proportional to γ − 1) is drastically reduced beyond Q ≈ 7 nm since that the weight of the thermal contribution (proportional to 1) is drastically reduced beyond therein γ differs from 1 by less than 10%. Q 7 nm since therein differs from 1 by less than 10%. The generalized thermal diffusivity obtained from the same work is reported in the upper plot. One The generalized thermal diffusivity obtained from the same work is reported in the upper plot. can confidently conclude that thermal diffusivity drastically decreases with Q consistently with the INS One can conﬁdently conclude that thermal diffusivity drastically decreases with Q consistently with result of Bell et al. [40]. Furthermore, it appears that in the supercritical sample the departure from the the INS result of Bell et al. [40]. Furthermore, it appears that in the supercritical sample the departure hydrodynamic value happens at higher Q values and is much smoother. This seems consistent with the −1 from the hydrodynamic value happens at higher Q values and is much smoother. This seems consistent extended hydrodynamic behavior (up to ≈8 nm ) of acoustic parameters already discussed in reference to with the extended hydrodynamic behavior (up to 8 nm ) of acoustic parameters already discussed Figure 10. It has to be noticed that trends similar to the ones in Figure 11 were also reported in a previous MD in rwork eference on ato LeFigur nnard e Jones 10. It sy has steto m re bepresenta noticedtive that oftr Ar ends at th similar e triple to po the int.ones in Figure 11 were also reported in a previous MD work on a Lennard Jones system representative of Ar at the triple point. A final remark concerns the instantaneous term in the memory function decay (last term in Equation (9)). It A is ﬁnal reason remark able to concerns expect that the thinstantaneous is contribution term is rather in the wememory ak for low function ‐viscosity decay and non (last‐associ termate ind fluids Equation as noble (9)). gases. It is Th reasonable is conclusion to expect is support that the this so contrib undness ution of ais simple rather viweak scoelafor stic low-viscosity assumption. On and the other hand in Ref. [42], using a double exponential model allows one to extract essentially the same value non-associated ﬂuids as noble gases. This conclusion is support the soundness of a simple viscoelastic −1 for the two timescales involved, at least for Q > 4–5 nm . assumption. On the other hand in Ref. [42], using a double exponential model allows one to extract essentially the same value for the two timescales involved, at least for Q > 4–5 nm . 4.1. Microscopic vs. Structural Relaxations 4.1. Microscopic vs. Structural Relaxations The results discussed in [6] indicate that the relaxation phenomenon causing the viscoelastic response of neon has a microscopic origin as opposite to the structural character of relaxations routinely observed in The results discussed in [6] indicate that the relaxation phenomenon causing the viscoelastic associated and/or highly viscous fluids. It should be emphasized that structural and microscopic response of neon has a microscopic origin as opposite to the structural character of relaxations relaxations have different physical origin and phenomenology: the former are cooperative in character and routinely observed in associated and/or highly viscous ﬂuids. It should be emphasized that structural involve readjustments of the structure in response to a mechanical or a scattering‐induced perturbation. and microscopic relaxations have different physical origin and phenomenology: the former are Consequently, they are prevalent in glass‐forming systems or associated fluids, i.e., fluids with extended cooperative in character and involve readjustments of the structure in response to a mechanical or network of bonds, this is, e.g., liquid water [12,13,93–95], hydrogen fluoride [14], glycerol [17]. Since a scattering-induced perturbation. Consequently, they are prevalent in glass-forming systems or collective rearrangements of the structure are hampered by viscous processes, the timescale of structural associated ﬂuids, i.e., ﬂuids with extended network of bonds, this is, e.g., liquid water [12,13,93–95], relaxation follows a temperature dependence roughly as steep as the one of viscosity. hydrogen ﬂuoride [14], glycerol [17]. Since collective rearrangements of the structure are hampered by 6 2 10 D (m /s) Appl. Sci. 2016, 6, 64 21 of 31 viscous processes, the timescale of structural relaxation follows a temperature dependence roughly as steep as the one of viscosity. Conversely, microscopic relaxations are induced by single molecules’ motions in the 10 s window and are dominating in simple systems, such as dense gases and non-associated, weakly viscous, ﬂuids. These fast motions can be naturally identiﬁed with vibration-like cage oscillations. Due Appl. Sci. 2016, 6, 64 23 of 34 to their non-cooperative nature, microscopic relaxations are not signiﬁcantly slowed down by viscosity and their timescale exhibits only a weak dependence on thermodynamic conditions. To quantitatively assess the hypothesized microscopic nature of viscoelastic pr−oce 13 sses in neon, Conversely, microscopic relaxations are induced by single molecules’ motions in the 10 s window and are dominating in simple systems, such as dense gases and non‐associated, weakly viscous, fluids. sensible variable to take as a reference is the mean free time between interatomic collisions, These fast motions can be naturally identified with vibration‐like cage oscillations. Due to their non‐ . This depends on atomic size, mass and shape, as well as on the interaction potential and cooperative nature, microscopic relaxations are not significantly slowed down by viscosity and their thermodynamic state of the system. Without sophisticated calculations, a reasonable estimate can be timescale exhibits only a weak dependence on thermodynamic conditions. obtained assuming as model a hard sphere gas following the Maxwell-Boltzmann statistics. For this To quantitatively assess the hypothesized microscopic nature of viscoelastic processes in neon, system the inter-collision time can be easily calculated as: sensible variable to take as a reference is the mean free time between interatomic collisions,τc. This depends on atomic size, mass and shape, as well as on the interaction potential and thermodynamic state of the system. Without sophisticated calculations, a reasonable estimate can be obtained assuming as model a t (14) hard sphere gas following the Maxwell‐Boltzmann statistics. For this system the inter‐collision time can be 16pd k T easily calculated as: where d is the hard sphere diameter. In this context can be identiﬁed as the characteristic timescale of τ  (14) 16πdk T interatomic interactions and for microscopic, or collisional, relaxations is reasonable that the relaxation time and have comparable values. In Figure 12 the Q-dependence of the reduced relaxation time where d is the hard sphere diameter. In this context τc can be identified as the characteristic timescale of interatomic interactions and for microscopic, or collisional, relaxations is reasonable that the relaxation time Y = / is reported as obtained using values measured in neon in Ref. In spite of the important τα and τc have comparable values. In Figure 12 the Q‐dependence of the reduced relaxation time Ψ = τα/τc scattering of data, it is clear that for Q larger than about 8 nm Y approaches the unit value, as is reported as obtained using τα values measured in neon in Ref. In spite of the important scattering of data, expected in the merely collisional limit. In this perspective, data in Figure 12 lead to the conclusion −1 it is clear that for Q larger than about 8 nm Ψ approaches the unit value, as expected in the merely that for Q > 8 nm THz relaxation processes of neon are intimately related to interatomic−1collisions. collisional limit. In this perspective, data in Figure 12 lead to the conclusion that for Q > 8 nm THz This relamay xationbe proce ansses expected of neon conclusion, are intimatelyparticularly related to intersince atomicr co elaxation llisions. processes are activated by the This may be an expected conclusion, particularly since relaxation processes are activated by the coupling of density ﬂuctuations with some internal degree of freedom of the system. Given that coupling of density fluctuations with some internal degree of freedom of the system. Given that microscopic components of a monatomic ﬂuid lack internal degrees of freedom, it seems natural to microscopic components of a monatomic fluid lack internal degrees of freedom, it seems natural to identify identify local cage oscillations as the internal degree of freedom that density ﬂuctuations couple to. local cage oscillations as the internal degree of freedom that density fluctuations couple to. -3 n = 29 nm T = 32 K -3 T = 40 K n = 34.7 nm -3 T = 50 K n = 36.7 nm 10 -3 n = 38.5 nm 5 10 152025 30 -1 Q (nm ) Figure 12. The Q‐dependence of reduced relaxation time Ψ = τα/τc as obtained from Ne data at the Figure 12. The Q-dependence of reduced relaxation time Y = / as obtained from Ne data at thermodynamic conditions indicated in the plot. Data are adapted with permission from [6] copyrighted by the thermodynamic conditions indicated in the plot. Data are adapted with permission from [6] American Physical Society. The horizontal line marks the unit value characteristic of the collisional regime. copyrighted by American Physical Society. The horizontal line marks the unit value characteristic of the collisional regime. At this stage, THz viscoelastic data available in literature can be used to compare the case of Ne to the one even more studied of water. This comparison seems particularly meaningful due to the disparate nature of microscopic structure and, in particular, to the presence of a large-scale connectivity in water. Appl. Sci. 2016, 6, 64 22 of 31 A meaningful reference to compare viscoelastic effects in different materials is provided by the generalized longitudinal viscosity, whose link the viscoelastic spectral parameters is given by the formula: 2 2 rpw w qt h (15) Figure 7 displays the Q-dependence of extracted (through Equation (15)) from the viscoelastic modeling of neon spectra in [6] at the thermodynamic conditions considered in Figure 4. Results on neon are compared with those obtained in liquid water from an IXS [13] and a joint INS and INS result on heavy water [78]. Some clear features readily emerge from the comparison: (1) Viscosity curves of Ne are only weakly dependent on thermodynamic conditions, although the macroscopic viscosity in the spanned thermodynamic interval undergoes important variations. Combined thermodynamic [72] and bulk viscosity data [28] of neon lead to estimate a more than 300% variation of within the probed thermodynamic range, clearly non reproduced by mesoscopic measurements. Conversely, viscosity data of water in the same plot have a T dependence as sharp as the one of macroscopic viscosity [96]. This discrepancy is likely due to the large weight of structural processes in water, especially at the low Qs covered by data of [13]. Structural relaxations, owing to their collective nature can only be observed over long distances, or, equivalently, at high Qs and are expectedly dominating in associate liquids. (2) High T values of approach from above low-Q neon data. This seems consistent with the inferred link between structural relaxation and local readjustment of the hydrogen bond (HB) network in water [12,13]. At the highest Ts the average number of HBs per molecule is sensibly lower which weakens the strength of structural relaxations [13,19]. (3) The strong Q-dependence of in water suggests that at Q higher than the values covered by Ref. [13], viscosity curves of water and neon tend to get closer to each other also considering that the latter have an essentially ﬂat Q-dependence at high Qs. This impression is conﬁrmed by data derived from extremely high resolution INS measurements of on deuterated water (line + open squares symbols) at T = 288 K, which covered a Q-range larger than those of other water data reported in the same plot. Overall, data in Figure 13 lead to the conclusion that the merely collisional component of relaxation in water becomes more relevant either at high T, i.e., upon approaching supercritical condition, or at Appl. Sci. 2016, 6, 64 25 of 34 high Qs, where structural relaxation become less prevalent. T= 277 K T= 288 K T= 333 K H O 1E-3 T= 433 K 1E-4 1E-5 Ne 1E-6 10 20 30 -1 Q (nm ) Figure 13. Longitudinal viscosity data derived using Equation (15) either from IXS measurements [13], on Figure 13. Longitudinal viscosity data derived using Equation (15) either from IXS measurements [13], normal water (line + squares symbols), or from joint IXS and INS measurements [78] on heavy water (line + on normal water (line + squares symbols), or from joint IXS and INS measurements [78] on heavy open squares). Data are redrawn with permission from [13] and [78] copyrighted by the American Physical water Soci (line ety +. open The temperatu squares). res Data are ind aricated e redrawn in the plo with t. Ne permission on data correspo fromn[d13 to ] and the sa [m 78e] ther copyrighted modynamicby the conditions and symbols as in Figure 4 and are redrawn with permission from [6] copyrighted by American American Physical Society. The temperatures are indicated in the plot. Neon data correspond to the Physical Society. same thermodynamic conditions and symbols as in Figure 4 and are redrawn with permission from [6] copyrighted by American Physical Society. 4.2. 2001–2003: Quantum Effects on the Line‐Shape The treatment discussed so far is fully classical in character, being based on the assumption that all involved operators are commuting and the correlation functions are even function of time or, equivalently, that the spectral shape is symmetric in ω. It can be assumed that the onset of quantum effects primarily induces an asymmetry on the shape of S(Q,ω) due to the different statistical population of states having distinct ω values. In all experimental and computational works mentioned in this review, the spectrum is described within a quasi‐classical approximation. According to such approximation, quantum effects only influence the statistical populations of states having different ωs, through the so‐called detailed balance principle [97]. This can be fulfilled in infinite possible ways; one of the most commonly used consisting in the multiplication of the classical S(Q,ω) profile by the factor n(ω,) T = ω/[ kT1exp( ω/) kT], i.e.: BB  ω 1 SQ (,ω)= S (Q,ω) (16)  C kT 1e xp( ω/) kT BB Here is the “classical”, or symmetric, part of the spectrum which is essentially the Fourier SQ (,ω) transform of the intermediate scattering F(Q,t). While deriving a model for it is assumed that the SQ (,ω) operator defining the time evolution of density are classical commuting variables. However, in real fluids quantum deviations may be important. In this respect, it’s worth distinguishing between two possible quantum effects on S(Q,ω): i) “diffraction” or “delocalization” effects, which arise from the non‐commutative nature of Hamiltonian operators; and ii) exchange effects, reflect the symmetry restriction to be fulfilled in a many body system of identical particles. 1/ 2 Both effects depend on the de Broglie wavelengh , however: =2  π mk T  Diffraction effects are relevant only when  matches the length‐scales over wich the interparticle potential U(rij) varies apppreciably.  (Pa s) } Appl. Sci. 2016, 6, 64 23 of 31 4.2. 2001–2003: Quantum Effects on the Line-Shape The treatment discussed so far is fully classical in character, being based on the assumption that all involved operators are commuting and the correlation functions are even function of time or, equivalently, that the spectral shape is symmetric in !. It can be assumed that the onset of quantum effects primarily induces an asymmetry on the shape of S(Q,!) due to the different statistical population of states having distinct ! values. In all experimental and computational works mentioned in this review, the spectrum is described within a quasi-classical approximation. According to such approximation, quantum effects only inﬂuence the statistical populations of states having different !s, through the so-called detailed balance principle [97]. This can be fulﬁlled in inﬁnite possible ways; one of the most commonly used consisting in the multiplication of the classical S(Q,!) proﬁle by the factor npw, Tq }w{k Tr1 expp}w{k Tqs, i.e.: B B }w 1 SpQ, wq S pQ, wq (16) k T 1 expp}w{k Tq B B Here S pQ, wq is the “classical”, or symmetric, part of the spectrum which is essentially the Fourier transform of the intermediate scattering F(Q,t). While deriving a model for S pQ, wq it is assumed that the operator deﬁning the time evolution of density are classical commuting variables. However, in real ﬂuids quantum deviations may be important. In this respect, it’s worth distinguishing between two possible quantum effects on S(Q,!): i) “diffraction” or “delocalization” effects, which arise from the non-commutative nature of Hamiltonian operators; and ii) exchange effects, reﬂect the symmetry restriction to be fulﬁlled in a many body system of identical particles. 1{2 Both effects depend on the de Broglie wavelengh L }p2p{mk Tq , however: Diffraction effects are relevant only when L matches the length-scales over wich the interparticle potential U(r ) varies apppreciably. ij Exchange effects are instead relevant when Lbecomes comparable with interatominc centers of mass distance and are thus observable only for small mass atoms, for which such a distance can be relatively small. These effects only emerge at very low temperature and responsible for few “spectacular” quanto-mechanical manifestations, such as, e.g., Bose condensation and superfluidity. One can reasonably assume that all sampple considerd in this review are either fully classical or moderately quantum ﬂuids, for which exchange effects can be discarded, while quantum deviation can be achieved through h ¯ -order corrections of classical results. For moderately quantum ﬂuids, one can still derive F(Q,t) using a memory function formalism. However, the operators deﬁning the Hamiltonian of the system are generally non-commuting. For these systems, the time dependence of density ﬂuctuation is derived as the solution of the Heisenberg equation of motion: . i drpQ, tq irdr, Hs rexpriQ r ptqs , Hs (17) where the Hamiltonian, H, for a N particle system reads as: ¸ ¸ H U r (18) ij 2M i1,...N ij where r is the Laplacian operator and U r is a pair potential acting between the i-th and j-th atoms. i ij Three IXS works performed at the beginning of the new millennium were focused on the onset of quantum effects on the spectrum of supercritical He [98] and liquid Ne [99,100]. Whilst in the ﬁrst work , quantum effects were investigated for moderate Qs as a function of temperature and density; in the two IXS works on Ne, due to the unprecedented wide Q range the whole transition in Q form the classical (continuous) to the quantum (single particle) regimes was crossed. Appl. Sci. 2016, 6, 64 24 of 31 In all these IXS works, quantum effects were sought for in the coefﬁcients governing the short time dynamics, i.e. the spectral moments, which for a quantum ﬂuid read as: n n n n xw y 1{i pB {Bt q Fpq, tq| xrrrr, Hs Hs .......Hsy| (19) t0 t0 In principle thexw y can be computed from direct integration of IXS intensity. However, a rigorous computation is non-trivial, since one must cope with all spurious intensity effects and the contribution to the integral from the ﬁnite instrumental resolution. One can easily eliminate spurious intensity effect—as long as these are !-independent—by dealing with spectral moments’ ratios. Conversely, eliminating the resolution contribution would require a numerical deconvolution of the measured line-shape, which is ill-determined unless the spectral shapes has the form of a single featureless peak. In the IXS works mentioned above spectral moments were instead computed from direct integration of best ﬁt (non-convoluted) model line-shape. This procedure led to estimate that, for instance, quantum effects in the neon measurement of Figures 12 and 13 although clearly sizable, are < 15%–20%, even at the lowest Ts and/or the highest densities. This estimate is roughly consistent with the one reported by Bell et al. [40]. One may wonder what inﬂuence these effects can have on the relaxation process. It can be shown that the so-called quantum viscoelastic [101], an exponential ansatz, can still be used. The direct consequence of quantum effects is a global softening of the interaction potential. This stems from the delocalized nature of the tagged particle, which enables a deeper penetration in the region where the interatomic potential is repulsive. 5. 2006-Present: Toward a New Vision of the Supercritical Phase Based on the results previously discussed, it may be natural to conclude that the viscoelastic behavior is a dynamic ﬁngerprint of the liquid phase, which disappears when supercritical conditions are reached. Indeed, the PSD has often been considered an almost universal feature of liquid aggregates, since almost ubiquitously observed in liquid systems as disparate as, for instance hard sphere [102] and Lennard-Jones models [103], noble gases [43,46,71], as well as diatomic [8], associated [12,13] and glass forming [15,17] systems. Its disappearance at deeply supercritical conditions was considered as an indication that supercritical ﬂuids do not exhibit a high frequency viscoelasticity. The actual test of this conjecture requires a thorough investigation of the THz spectral shape of supercritical ﬂuid at extreme P and T values. For many decades, this has been an almost prohibitive task, owing to a number of experimental difﬁculties. As mentioned, the development of IXS brought about a substantial narrowing of the beam cross section, while the improved performance of undulators and crystal optics have dramatically enhanced the ﬂux achievable in ordinary THz spectroscopy measurements. This paved the way to new class of IXS experiments at extreme pressures based upon the use of diamond anvil cells (DACs). Taking advantage of this opportunity, an IXS DACs measurement on deeply supercritical oxygen demonstrated that a PSD of about 20% in amplitude is visible at temperatures as high as nearly twice the critical one [104]. The results of this measurement imposed a global revision of the previous idea that THz viscoelasticity phenomena could represent a dynamical ﬁngerprint of the liquid phase. Another breakthrough came from a successive joint IXS and MD work on Ar at extreme pressures [105], which showed the occurrence of a dynamic transition upon crossing a boundary in the supercritical domain. A dramatic reduction in the amplitude of PSD was, in fact, observed upon crossing the Widom line [106] and was interpreted as a crossover between a “liquid like” and a “gas like” sub-regions of the supercritical phase. As a reminder, the Widom line is deﬁned as the locus of speciﬁc heat maxima, which emanates from the critical point toward the supercritical domain with essentially the same slope as the coexistence line. For this reason, the Widom line is often considered a “prosecution” of the coexistence line beyond the critical point. Perhaps the most intriguing aspect of these results is that the mere existence of a boundary in the thermodynamic plane challenges the long-standing vision of the supercritical phase as intrinsically uniform [107], thus opening up unexplored theoretical scenarios and novel investigation opportunities. Appl. Sci. 2016, 6, 64 25 of 31 Another important result came from the recent conclusion that PSD effects are intimately related to the onset of shear propagation. The occurrence of a shear mode propagation in argon was ﬁrst demonstrated by a MD simulation on a Lennard Jones Ar [41,42]. As a common result, it was observed that at some wavelengths transverse current spectra bore evidence for a well-deﬁned maximum, indicating the occurrence of a shear mode propagation. More recently, a joint IXS and MD simulation work on deeply supercritical argon [108] evidenced a close connection between the ability of a ﬂuid system to support shear mode propagation and the presence of a sizeable PSD. In this work IXS spectra were measured along a nearly isobaric path (with P 1 GPa) with T spanning the 300 K–436 K temperature range, while MD data spanned the same isobar within a larger T-interval (up to 800 K). As a result, it was there shown that the disappearance of the PSD in deep supercritical conditions is accompanied by a parallel shear modes overdamping (see Figure 14). Overall, the fundamental character of the PSD and its link to thermodynamic transformations (previously disputed by [109]), are convincingly demonstrated in the work of Bolmatov et al. There, in fact, the PSD and the shear mode propagation are linked together and interpreted as different manifestations of the same universal property of ﬂuids: the onset of a high frequency viscoelasticity. Appl. Sci. 2016, 6, 64 28 of 34 The results of this work suggest that the dynamic response of supercritical ﬂuids is characterized by different regimes characteristic of distinct thermodynamic subdomains: (1) A gas‐like domain, featured by the absence of PSD effects and the related inability of the fluid to (1) A gas-like domain, featured by the absence of PSD effects and the related inability of the ﬂuid to support transverse wave propagation and support transverse wave propagation and (2) A liquid‐like domain in which the system exhibits merely viscoelastic features such as a sizable PSD and the onset of shear mode propagation. (2) A liquid-like domain in which the system exhibits merely viscoelastic features such as a sizable PSD and the onset of shear mode propagation. Furthermore, in the work of Bolmatov and collaborators [108] the Frenkel line [110] was identified as the crossover line between the two thermodynamic regions. The interpretation of the Frenkel line as a Furthermore, in the work of Bolmatov and collaborators [108] the Frenkel line [110] was identiﬁed crossover line demarcating the presence/absence of transverse acoustic propagation was originally as the crossover line between the two thermodynamic regions. The interpretation of the Frenkel line as discussed in [111]. It was also previously predicted that the occurrence, across the Frenkel line, of both a crossover line demarcating the presence/absence of transverse acoustic propagation was originally dynamic and structural crossovers accompanied by changes in phonon states [112–115]. Finally, it is worth mentioning a more recent work on deeply supercritical Ar [116], which provides compelling evidence for discussed in [111]. It was also previously predicted that the occurrence, across the Frenkel line, of both important dynamical changes occurring upon crossing the Frenkel line and having the form of strong dynamic and structural crossovers accompanied by changes in phonon states [112–115]. Finally, it is localization of the longitudinal sound mode. worth mentioning a more recent work on deeply supercritical Ar [116], which provides compelling Most importantly, the assessed parallelism between the disappearance of the PSD and the evidence for important dynamical changes occurring upon crossing the Frenkel line and having the overdamping of shear mode finally indicates that the observed viscoelastic behavior in noble gases is form of strong localization of the longitudinal sound mode. related relaxation process primarily affecting the shear viscosity. T = 600 K 12 T = 300 K 0 5 10 15 20 -1 Q (nm ) Figure 14. The dispersion curve of deeply supercritical Ar at P = 1 GPa, as evaluated from the maxima Figure 14. The dispersion curve of deeply supercritical Ar at P = 1 GPa, as evaluated from the position of MD simulated longitudinal (red dots) and transverse current spectra (open circles). The maxima position of MD simulated longitudinal (red dots) and transverse current spectra (open circles). corresponding hydrodynamic dispersion is also reported as a dashed line for comparison. Adapted with The corresponding hydrodynamic dispersion is also reported as a dashed line for comparison. Adapted permission from [108]. Copyright (2016) of American Chemical Society. with permission from [108]. Copyright (2016) of American Chemical Society. 6. Conclusions and Future Perspectives In summary, this brief review shows that THz studies on simplest systems as noble gases have, across the years, evidenced a rather complex behavior. The study of relaxation processes in these systems and, in particular, the comparison with those observed in more complex, associated fluids led the scientific community to identify a new class of relaxation phenomena completely different from structural (or α‐ )relaxations routinely observed glass formers and highly viscous materials. These phenomena are faster (in the sub‐ps) and only weakly dependent on thermodynamic conditions. Furthermore, their timescale is weekly affected by first neighbors’ arrangement, as suggested to its flat Q‐dependence, which presents no reminiscence of S(Q) oscillations. These collisional relaxations become visible also in associate fluids at temperature approaching critical conditions—due to the decrease in network connectivity—and/or over   ,  (meV) L t Appl. Sci. 2016, 6, 64 26 of 31 Most importantly, the assessed parallelism between the disappearance of the PSD and the overdamping of shear mode ﬁnally indicates that the observed viscoelastic behavior in noble gases is related relaxation process primarily affecting the shear viscosity. 6. Conclusions and Future Perspectives This brief review results THz studies on the viscoelastic response of simplest systems as noble gases have, across the years evidenced a rather complex behavior. The study of relaxation processes in noble gases and, in particular, the comparison with those observed in more complex, associated ﬂuids led the scientiﬁc community to identify a new class of relaxation phenomena completely different from structural (or -)relaxations routinely observed glass formers and highly viscous materials. These new relaxation phenomena are faster (in the sub-ps) and only weakly dependent on thermodynamic conditions. Furthermore, their timescale is weekly affected by ﬁrst neighbors’ arrangement, as suggested to its ﬂat Q-dependence which presents no reminiscence of S(Q) oscillations. These collisional relaxations become visible also in associate ﬂuids at temperature approaching critical conditions—due to the decrease in network connectivity—and/or over short distances, over which collective rearrangements cannot take place. These phenomena have recently observed to drastically affect the shear components of viscosity and to be intimately related to the presence of transverse acoustic modes propagation. Most importantly, the study of viscoelastic phenomena at extreme supercritical conditions challenges the long-standing vision of the supercritical phase as inherently uniform state of matter. Nowadays these results call for a new and more complex understanding of the supercritical domain, based on the existence of liquid-like (viscoelastic) and gas-like (merely viscous) sub-domains separated by a thermodynamic boundary. Shedding further light onto this revolutionary hypothesis is certainly one of the main motivations for the scientiﬁc community working in the ﬁeld. Looking ahead, further advances in the study of relaxation phenomena in noble gases and more speciﬁcally in noble gases are still held back by inherent limitation in the used techniques. If from one side INS is still hampered by both incident ﬂux and kinematic limitation, the main drawback of current IXS is the broad and slowly decaying (essentially Lorentzian) resolution wings. When focusing on relaxation processes the relevant spectral information is gathered in the extremely low frequency or quasi-elastic spectral window, often on the side of a dominating central peak. A narrow resolution width is thus required to properly resolve the spectral features of interest, while a superior contrast is necessary to discern them from the broad wings of the elastic line. Fortunately, a new concept optics for both monochromatization and energy analysis holds the promise for new generation spectrometers meting the required performance [117]. This will enable to perform IXS experiments with a resolution function <1 meV broad and essentially Gaussian in shape. The feasibility of such high resolution optical schemes has been demonstrated in a recent work [118], while a beamline based on this working principle has recently become operative at the new synchrotron NSLS II of Brookhaven National Laboratory [119]. This new opportunity is certainly quite encouraging and, in the near future, deemed to further advance our knowledge of the complex THz dynamics of simple liquids. 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The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy

The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy

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The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy

The Spectrum of Density Fluctuations of Noble Gases Probed by THz Neutron and X-ray Spectroscopy

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