Behavior of Supercritical Fluids across the “Frenkel Line” - The Journal

Letter pubs.acs.org/JPCL

Behavior of Supercritical Fluids across the “Frenkel Line” T. Bryk,†,‡ F. A. Gorelli,¶,§ I. Mryglod,† G. Ruocco,∥,⊥ M. Santoro,¶,§ and T. Scopigno*,∥,⊥ †

Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Street, UA-79011 Lviv, Ukraine ‡ Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University, UA-79013 Lviv, Ukraine ¶ Istituto Nazionale di Ottica INO-CNR, I-50019 Sesto Fiorentino, Italy § European Laboratory for Non Linear Spectroscopy, LENS, I-50019 Sesto Fiorentino, Italy ∥ Dipartimento di Fisica, Universita di Roma La Sapienza, I-00185 Roma, Italy ⊥ Center for Life Nano Science @Sapienza, Istituto Italiano di Tecnologia, 295 Viale Regina Elena, I-00161 Roma, Italy ABSTRACT: The “Frenkel line” (FL), the thermodynamic locus where the time for a particle to move by its size equals the shortest transverse oscillation period, has been proposed as a boundary between recently discovered liquid-like and gas-like regions in supercritical fluids. We report a simulation study of isothermal supercritical neon in a range of densities intersecting the FL. Specifically, structural properties and single-particle and collective dynamics are scrutinized to unveil the onset of any anomalous behavior at the FL. We find that (i) the pair distribution function smoothly evolves across the FL displaying medium-range order, (ii) low-frequency transverse excitations are observed below the “Frenkel frequency”, and (iii) the high-frequency shear modulus does not vanish even for low-density fluids, indicating that positive sound dispersion characterizing the liquid-like region of the supercritical state is unrelated to transverse dynamics. These facts critically undermine the definition of the FL and its significance for any relevant partition of the supercritical phase.

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such a separation line and determined as the locus of specific heat maxima. Alternative representations of the Widom line were later proposed in order to mark the separation line toward higher temperatures.6−13 Notably, the pivotal role of the Widom line was anticipated within the context of dynamical crossovers in systems displaying liquid−liquid phase transitions, such as water.14−17 From the theoretical side, PSD in liquids was originally rationalized by mode-coupling theory18 and by the memory function formalism.19,20 The explicit account for nonlocal coupling of fluctuations of conserved quantities within the mode-coupling approach allowed one to obtain a nonanalytical correction to the linear hydrodynamic dispersion law for acoustic excitations

pon increasing the pressure and temperature of a substance beyond its critical point, any thermodynamic discontinuity between the liquid and gas phase disappears and the system is said to be in a fluid state. Accordingly, conventional wisdom would suggest that structural and dynamical properties of supercritical fluids smoothly depend on temperature and pressure. Albeit the very nature of supercritical fluids has been debated for over 2 centuries,1 only recently the single-phase scenario has been challenged by Gorelli et al.,2,3 when it was suggested to analyze the propagating density fluctuations in supercritical fluids as a function of pressure/density, based on the idea that dynamical properties could be more sensitive to phase changes than the structural properties. As an example, despite strikingly similar pair distribution functions, in liquid and glass systems the density−density time correlation functions behave in an absolutely different way because of the dynamic arrest of particle motion and the emergence of nonergodicity in the glass state. Using a combination of inelastic X-ray scattering experiments and molecular dynamics (MD) simulations, it was discovered that the onset of deviation from the hydrodynamic dispersion of sound (the so-called “positive sound dispersion” (PSD)) could be used as a boundary between liquid-like and gas-like regions existing in a fluid as reminiscent of the subcritical behavior.4 Obviously, any separation line of liquid-like and gas-like types of dynamics of supercritical fluids must emanate from the critical point. Accordingly, in ref 4, the Widom line,5 a prolongation of the coexistence line, was suggested as a suitable candidate for © XXXX American Chemical Society

ωMCT(k) = csk + αk5/2 + ...

where k is the wavenumber and cs is the adiabatic speed of sound. Critically, it appeared that the pressure dependence of the positive coefficient α for liquid Ar21 was not the one observed for PSD in inelastic X-ray scattering experiments.4 The memory function formalism, on the other hand, allowed one to connect different channels of correlation decay in the second-order memory function with the PSD.20,22 More recently, another theoretical approach of generalized collective modes Received: August 17, 2017 Accepted: September 25, 2017 Published: September 25, 2017 4995

DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001

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The Journal of Physical Chemistry Letters

Figure 1. Left: Mean square displacements ⟨R2⟩ for supercritical Ne at 295 K. The line ⟨R2⟩ = Rmax2, where Rmax is the first peak position of g(r), allows estimation of the specific Frenkel time τF. Right: Frenkel time for supercritical Ne at 295 K calculated from approximate eq 1 and exact eq 2 expressions for the time needed for particles to move the average nearest-neighbor distance.

(GCM)23−25 was applied in order to obtain analytical expression for the dispersion law of acoustic excitations on the boundary of the hydrodynamic regime.26 Two dynamic models, viscoelastic and thermoviscoelastic ones, were analytically solved in ref 26 in order to understand different contributions to the PSD in fluids. It was found within the GCM approach that the first correction to the hydrodynamic dispersion law is proportional to k3 with a coefficient β

sensitive to some single-particle specific time. This was highlighted taking as an illustrative example the supercritical argon, where the pressure dependence of the minimum of the adiabatic speed of sound as a function of temperature cs(T) (taken from the NIST database31) shows a clear mismatch with the location of the Frenkel line.32 A third important issue with the Frenkel line approach concerns the behavior of nonhydrodynamic shear waves. On the basis of the Frenkel time scale τF, a “Frenkel frequency” ωF = 2π/ τF can be defined, which would identify a low-frequency cutoff bound for transverse excitations in fluids.29 This is against textbook paradigms33 and theoretical studies8,34−36 of transverse dispersions in liquids, clearly demonstrating the existence of a wavelength rather than frequency gap. Moreover, a thermodynamic basis for the Frenkel line was suggested in ref 37, where based on the solid-state approach of nondamped phonons the authors obtained an expression for a contribution from transverse excitations to the total energy, which supposedly drops to zero at the Frenkel line. Recently, the solid-state approach to the thermodynamics of liquids was promoted by claiming that there exist collective excitations in condensed matter systems (either ordered or disordered) that do not decay because of the energy conservation law (“damping myth” in refs 38 and 39), clearly challenging the fluctuation−dissipation theorem40 and the role of dissipation in creating new fluctuations. All of these facts call for a detailed benchmark of the Frenkel line concept and possible implications, which we present here for the case of supercritical neon. The choice of this specific system is partly motivated by a recent report,41 in which the disappearance of medium-range order in X-ray diffraction experiments of supercritical Ne was claimed when the derived third peak of the pair distribution function g(r) was not practically detectable after crossing a pressure value corresponding to the intersection with the Frenkel line.41,42 Additionally, we will scrutinize other predictions of the Frenkel line approach, related to dispersion of longitudinal and transverse excitations. Def initions of the Frenkel Time Based on Single-Particle Dynamics. According to its original definition, τF is the average time needed for a particle “to move the average interparticle distance a” or “to move a distance comparable to its own size” and is given as29

ωGCM(k) = csk + βk3 + ...

which in viscoelastic approximation decays with a decrease of density in agreement with observations for PSD in inelastic X-ray scattering experiments and for low-density states is practically zero. The effect of coupling to heat fluctuations within the thermoviscoelastic model leads to possible negative values of β (i.e., emergence of “negative” sound dispersion) for fluids with a large ratio of specific heats γ, like hard-disk27 and hard-sphere28 fluids, or supercritical ones in the vicinity of the critical point.26 In 2012, an alternative approach to dynamics of supercritical fluids was proposed, connected with the so-called “Frenkel line”,29,30 which is defined on the basis of single-particles’ properties. Specifically, its definition is given by the equivalence of the characterictic single-particle time scale of “Frenkel jumps” τF (i.e., time needed for a particle to reach its nearest-neighbor shell) to a shortest oscillation time of transverse excitations in fluids τD (which is connected with an analogy of the Debye frequency): τF = τD. This line supposedly discriminates between rigid and nonrigid fluids and was named in ref 30 as a line of “liquid−gas transition in the supercritical region”. Because the Frenkel line crosses on the phase diagram the standard liquid− gas coexistence (at TF) below the critical point (TF < Tc),12 it immediately raises a question on the potential connection of the Frenkel line with liquid-like and gas-like features: a contradiction stands between the general ability to sustain the liquid phase for T < Tc and the identification of a gas-like region at the liquid-side of the binodal line for TF < T < Tc. Nevertheless, the Frenkel line was claimed to have impact on several structural and dynamic quantities such as the pair distribution function, speed of sound, diffusion, and spectra of collective excitations29 because of the supposedly fundamental role of a single-particle time needed for particles to reach the nearest-neighbor distance. In this respect, a second contradiction arises for an example with the behavior of the macroscopic adiabatic speed of sound in fluids, cs, which is governed by the local conservation laws and in a continuum system without any atomistic structure cannot be

τF = 4996

a2 6D

(1) DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001

Letter

The Journal of Physical Chemistry Letters

Figure 2. (a) Pair distribution functions below and above the Frenkel line. (b,c) Same as (a), in density region 1000 kg/m3−1350 kg/m3, zoomed in to emphasize the third peak of g(r). (d) Comparison of experimental results41 with our MD data. The location of the Frenkel line41 is shown as a dashed line.

Figure 3. Dispersion of longitudinal (L) and transverse (T) collective excitations for supercritical Ne at 295 K obtained from peak positions of current spectral functions CL/T(k,ω). Straight blue lines correspond to hydrodynamic dispersion law with the adiabatic speed of sound cs. The dotted horizontal lines correspond to the cutoff Frenkel frequency ωF; according to ref 29, the transverse excitations with ωT(k) < ωF should not exist.

region. Star symbols indicate when the exact condition (2), with a being the position of the first g(r) peak (see the next section), is fullfilled. The values τF calculated from exact eq 2 will be used later for estimation of the cutoff Frenkel frequencies for transverse excitations in supercritical Ne. The impact of the approximation in eq 1 is illustrated in Figure 1b, where Frenkel time is reported as estimated from approximated eq 1 and from

In fact, this is an approximation of the exact average time for particles to move some distance a, which may be obtained via the expression for the mean square displacement ⟨R2⟩(t)

⟨R2⟩(τF) = a 2

(2)

In Figure 1a, we show the time dependence of the Ne mean square displacements for different densities in the supercritical 4997

DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001

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contains the expected propagation gap in the long-wavelength region.33−35 For lower densities, though, the transverse current spectral functions become more of a Gaussian-like shape, and therefore, we were unable to resolve peaks of CT(k,ω), a similar observation as was reported recently for hard-sphere fluids.28 Our MD results shown in Figure 3 give evidence that the Frenkel frequency does not play the role of the low-frequency cutoff, as was stated in ref 29, and the dispersion of transverse excitations starts from zero frequency but from nonzero wavenumber kc predicted by different methodologies in refs 33−35. Interestingly, similar results showing existing lowfrequency transverse excitations were reported in ref 43, although the inconsistency with the Frenkel line approach was not emphasized. Jointly with our results, they clearly show that ωF cannot be used as (one of) the definition of the Frenkel line given in ref 29

the exact eq 2. The discrepancy increases from 20% up to a factor of 2 by moving from the highest to the lowest density explored in this study. We note that the failure of the approximation in eq 1 can be traced back to the assumption of a linear time dependence of the mean square displacement. While this is certainly asymptotically valid, it is clearly untenable on the nearestneighbor transit time scale implied in the FL definition (see Figure 1a). Accordingly, use of eq 1 undermines any reliable conclusions on the Frenkel line scenario. Structural Properties at the Frenkel Line. Here we aim to verify the recently reported41 disappearance of medium-range order in X-ray diffraction experiments of supercritical Ne. To this purpose, MD simulations are an ideal tool to enable one to converge the pair distribution functions in classical simulations with the necessary precision required to track the behavior of the height of the third maximum of g(r) upon crossing the Frenkel line. In Figure 2a, the density dependence of pair distribution functions is reported over the whole explored range. A gradual reduction of the first and second peaks of g(r) is observed with a well-defined minimum between them. In Figure 2b,c, we zoom in the region of distances where the third peak of g(r) is located. Data do not show any drastic change in the height of this feature at the Frenkel line (which is shown by the dotted vertical line in the bottom frame) but rather a slow decrease upon density reduction. Critically, the height of the third peak of g(r) always remains larger than unity for densities well below the FL. It has to be pointed out, however, that retrieval of the g(r) via the experimental structure factors S(k) requires an inversion procedure that is subject to systematic errors. This could explain the different conclusions reported in ref 41. Frenkel Frequency and Transverse Dynamics. From the obtained values of the Frenkel time τF the corresponding Frenkel frequencies can be easily calculated using the definition in ref 29 ωF =

2π τF

ωF = ωDT

(4)

with ωTD being the highest possible Debye frequency for transverse excitations. Consistency of the Def inition of the Frenkel Line f rom a SolidState Approach to the Specif ic Heat Cv of Fluids. The inconsistency between the absence of a low-frequency cutoff of transverse waves with the definition of the Frenkel line is critically related to another important claim: the value of specific heat at constant volume Cv = 2kB, which coincides with the Frenkel line, separates liquid-like (Cv > 2kB) and gas-like (Cv < 2kB) fluids with and without transverse excitations, correspondingly.30 However, very recently, it was shown28 that these arguments do not apply to hard-sphere fluids, where for dense fluids the transverse excitations are well-defined, although for the specific heat, Cv is equal to 1.5kB at any density value. In an effort to overcome such inconsistency, it has been proposed37 to artificially decompose the energy of a fluid into contributions from relaxing and propagating processes, although such decomposition and relative contributions to Cv naturally originate from the Landau−Placzek-like ratio for the dynamics of heat fluctuations in fluids.44,45 Nevertheless, it was shown37 that each branch of collective excitations contributes ∼0.5kB to Cv, and reduction of Cv to 2kB signals the disappearance of transverse collective modes. The whole proposal is built on a solid-state picture with nondamped phonon-like modes, and in the specific case of transverse modes, the low-frequency cutoff (whose absence has been discussed above) was taken as the lower integration bound of

(3)

The Frenkel frequency is the low-frequency cutoff for transverse excitations in the fluid, a cornerstone of the whole Frenkel line approach,29 and in a combination with the highest possible “Debye frequency”, it has been proposed as a way to discriminate between fluids with and without transverse excitations. The Frenkel frequency calculated from eqs 3 and 2 drops monotonically from 10.28 ps−1 for the lowest studied density to 4.81 ps−1 for the highest one. In Figure 3, we show dispersions of longitudinal (L) and transverse (T) collective excitations calculated from peak positions of L/T current spectral functions CL/T(k,ω). The CL/T(k,ω) were obtained from time-Fourier transformation of the MD-derived L/T current−current time correlation functions. Note, because the transverse current spectral function at zero frequency is defined by the inverse of k-dependent shear viscosity, CT(k,ω = 0) ≈ η−1(k), for low-density fluids with low viscosity, it is problematic to observe any peak structure of CT(k,ω). Although we report here the spectra of collective excitations estimated from peak positions of CL/T(k,ω), we stress that only a reasonable theory that takes into account contributions from different relaxing and propagating modes to the shape of CL/T(k,ω) can reveal the issue of existence/absence of short-wavelength shear waves in fluids. In Figure 3, the lowfrequency cutoffs ωF are shown by pink dashed lines. One can easily see that for densities 1400 kg/m3 and higher the transverse excitations propagate in supercritical Ne, and their dispersion

ET = C

∫ω

ωDT F

E(ω , T )g T(ω) dω

(5)

where gT(ω) is the density of transverse phonon-like states and C is a normalization constant. The reduction of the contribution ET to the total energy, ultimately to zero in the case of ωF = ωTD (the condition for the Frenkel line), is here the increase of the lower integration bound ωF with density. However, as already discussed, there is no low-frequency cutoff for the transverse excitations, and the dispersion law for transverse excitations starts from zero frequency. Correspondingly, the density of transverse phonon-like states starts from zero frequency as well, and the integration in eq 5 must have zero as the lower integration bound. Hence, even the specific heat-based definition of the Frenkel line (eq 4) is not consistent with the observed dispersion of transverse excitations in fluids. 4998

DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001

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The Journal of Physical Chemistry Letters Consistency of the Def inition of Positive Sound Dispersion within the Frenkel Line Approach. The viscoelastic increase of the frequency of acoustic excitations outside of the hydrodynamic regime observed in dense fluids by scattering experiments2−4,6 and MD simulations7,26,36,46 is another process for which the “Frenkel mechanism” has been proposed as a possible demarcation line: below the Frenkel line, the PSD exists, but not above the Frenkel line.29 This was based on the empirical fact that the propagation speed increases from the macroscopic adiabatic value (for which the low-frequency shear modulus G0 is zero) [Bs/ρ]1/2 to some high-frequency value [(B∞ + (4/3)G∞)/ ρ]1/2 with a nonzero high-frequency shear modulus.29 One can immediately argue on these intuitive speculations that the high-frequency shear modulus, G∞, is nonzero for all densities (see Figure 4), which would cause nonzero PSD in the

to the ratio of transverse to longitudinal sound velocity as per eq 6. Such a ratio has very little system dependence with respect to the wide range of positive dispersion observed in fluids, from a few pecent in simple liquids (such as liquid metals) to a factor of two in water. MD simulations for supercritical Ne for different pressures at 295 K shows the following facts (i) There are no sudden changes in the density dependence of g(r) by crossing the Frenkel line. More specifically, the height of the third peak of g(r) slowly reduces to unity for densities far below the value at the Frenkel line. (ii) Low-frequency transverse excitations are observed for ω < ωF, as expected from the well-established theory of shear wave propagation in liquids.33−35 (iii) The high-frequency shear modulus does not drop to zero even for low-density fluids, indicating that any interpretation of positive sound dispersion based on the onset of transverse dynamics29 is untenable. (iv) The thermodynamic basis for the Frenkel line is ill-defined because the Frenkel frequency, which is the cornerstone of the Frenkel line approach, cannot be used as the lower integration bound in the integrals over the density of transverse vibrational states. Taken all together, these elucidate important issues with the Frenkel line approach. On the one hand, there are clear inconsistencies among different proposed definitions, and on the other hand, the relevance of the Frenkel line as a boundary identifying different structural and dynamical behaviors in the supercritical region is critically undermined by the present study.

Figure 4. High-frequency B∞, adiabatic Bs, low-frequency B0 bulk moduli, and high-frequency G∞ shear module as functions of the density of supercritical Ne at 295 K. The location of the Frenkel line from ref 41 is shown by the vertical blue dashed line.

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METHODS We performed MD simulations of supercritical Ne using a system of 4000 particles in a microcanonical ensemble with LennardJones two-body potentials. The parameters of the Lennard-Jones potential were the same as those in ref 41. Several studies investigated the correction of three-body contributions to twobody interactions of noble gases.47,48 In the specific case of the Lennard-Jones interatomic potential, however, thermodynamical properties are very well reproduced, as demonstrated by the agreement (5%) of the calculated specific heat Cv and adiabatic speed of sound cs for different densities with NIST values.31 The temperature in simulations was kept at 295 K, that is, T/Tc = 6.63, and the drift of energy over the production runs of 300 000 time steps (Δt = 0.5 fs) was not larger than 0.02%. We studied 10 different densities along the isothermal line in the range from 691 kg/m3 (1.428ρc) to 1600 kg/m3 (3.306ρc), calculating both single-particle and collective dynamic properties. Additionally, in the density range of 1000−1600 kg/m3, we performed longer simulations evaluating pair distribution functions every 10 steps in order to converge the g(r) and clarify the issue about the possible disappearence of the third peak of g(r) upon crossing the Frenkel line. For each density, we calculated the longitudinal (L) and transverse (T) current−current time correlation functions FL/T JJ (k,t), which upon time-Fourier transformation resulted in L and T current spectral functions CL/T(k,ω). In order to estimate the macroscopic adiabatic speed of sound cs from MD simulations, we calculated energy−energy and energy−density static correlators, obtained the wavenumber-dependent thermodynamic quantities, and, from a smooth k dependence of the ratio γ(k)/S(k) (γ(k) and S(k) are k-dependent ratios of the specific heats and structure factor, respectively) together with the

whole range of densities that contradicts experimental4 and simulation41 results. Moreover, in the hydrodynamic region (on the macroscopic scale) and in the high-frequency regime (on the atomistic scale), the bulk modulus B (as it is given erroneously in ref 29) is not the same quantity: on macroscopic scales, it is the adiabatic bulk modulus Bs that is solely defined by the velocity field and heat fluctuations in the fluids, while at high frequencies, it is B∞ that is defined by the microscopic forces acting on particles and is essentially an isothermal quantity. In Figure 4, we show the density dependence of the adiabatic Bs bulk modulus (which is γ times larger than the zero-frequency isothermal macroscopic bulk modulus, B0) and the high-frequency bulk B∞ and shear G∞ moduli; one can immediately see the essential difference between Bs and B∞. Hence, replacing B∞ with Bs in order to obtain the expression for the macroscopic propagation speed43 v l 2 = cs 2 + (4/3)vt 2

(6)

where vl and vt are respectively the speeds of longitudinal and transverse excitations, is clearly unjustified. In fact, Figure 4 unambiguously shows that there is no qualitative change in the density dependence of different bulk and high-frequency shear moduli upon crossing the Frenkel line, which for the Ne isotherm in the focus of this study is located close to density ρF = 1330 kg/ m3.41 Moreover, the high-frequency shear modulus is nonzero in the whole studied density range, which means nonzero positive sound dispersion according to eq 6, even above the Frenkel line (for densities smaller than 1330 kg/m3). In passing, we note that, beside the above-mentioned issues, the Frenkel line-based quantification of positive dispersion would be entirely related 4999

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A. Z.; Russo, J.; et al. Water: A Tale of Two Liquids. Chem. Rev. 2016, 116, 7463−7500. PMID: 27380438. (18) Ernst, M. H.; Dorfman, J. R. Nonanalytic dispersion relations for classical fluids. J. Stat. Phys. 1975, 12, 311. (19) Balucani, U.; Zoppi, M. Dynamics of the liquid state; Clarendon Press: Oxford, U.K., 1994. (20) Scopigno, T.; Balucani, U.; Ruocco, G.; Sette, F. Density fluctuations in molten lithium: inelastic x-ray scattering study. J. Phys.: Condens. Matter 2000, 12, 8009. (21) de Schepper, I.; Verkerk, P.; van Well, A.; de Graaf, L. Nonanalytic dispersion relations in liquid argon. Phys. Lett. A 1984, 104, 29− 32. (22) Scopigno, T.; Ruocco, G.; Sette, F. Microscopic dynamics in liquid metals: The experimental point of view. Rev. Mod. Phys. 2005, 77, 881. (23) deSchepper, I. M.; Cohen, E. G. D.; Bruin, C.; van Rijs, J. C.; Montfrooij, W.; de Graaf, L. Hydrodynamic time correlation functions for a Lennard-Jones fluid. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 271. (24) Mryglod, I. M.; Omelyan, I. P.; Tokarchuk, M. V. Generalized collective modes for the Lennard-Jones fluid. Mol. Phys. 1995, 84, 235. (25) Bryk, T.; Mryglod, I.; Kahl, G. Generalized collective modes in a binary He0.65 - Ne0.35 mixture. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 2903. (26) Bryk, T.; Mryglod, I.; Scopigno, T.; Ruocco, G.; Gorelli, F.; Santoro, M. Collective excitations in supercritical fluids: Analytical and molecular dynamics study of “positive” and “negative” dispersion. J. Chem. Phys. 2010, 133, 024502. (27) Huerta, A.; Bryk, T.; Trokhymchuk, A. Collective excitations in 2D hard-disc fluid. J. Colloid Interface Sci. 2015, 449, 357. (28) Bryk, T.; Huerta, A.; Hordiichuk, V.; Trokhymchuk, A. Nonhydrodynamic transverse collective excitations in hard-sphere fluids. J. Chem. Phys. 2017, 147, 064509. (29) Brazhkin, V. V.; Fomin, Y. D.; Lyapin, A. G.; Ryzhov, V. N.; Trachenko, K. Two liquid states of matter: A dynamic line on a phase diagram. Phys. Rev. E 2012, 85, 031203. (30) Brazhkin, V. V.; Fomin, Y.; Lyapin, A. G.; Ryzhov, V. N.; Tsiok, E. N.; Trachenko, K. "Liquid-Gas" Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics. Phys. Rev. Lett. 2013, 111, 145901. (31) Thermophysical Properties of Fluid Systems. http://webbook.nist. gov/chemistry/fluid/ (2017). (32) Bryk, T. Comment on ”Dynamic Transition of Supercritical Hydrogen: Defining the Boundary between Interior and Atmosphere in Gas Giants. Phys. Rev. E 2015, 91, 036101. (33) Hansen, J.-P.; McDonald, I. R. Theory of simple liquids; Academic: London, 1986. (34) MacPhail, R. A.; Kivelson, D. Generalized hydrodynamic theory of viscoelasticity. J. Chem. Phys. 1984, 80, 2102. (35) Bryk, T.; Mryglod, I. Optic-like excitations in binary liquids: transverse dynamics. J. Phys.: Condens. Matter 2000, 12, 6063. (36) Bryk, T. Non-hydrodynamic collective modes in liquid metals and alloys. Eur. Phys. J.: Spec. Top. 2011, 196, 65. (37) Bolmatov, D.; Brazhkin, V. V.; Trachenko, K. Thermodynamic behaviour of supercritical matter. Nat. Commun. 2013, 4, 2331. (38) Brazhkin, V. V.; Trachenko, K. Collective Excitations and Thermodynamics of Disordered State: New Insights into an Old Problem. J. Phys. Chem. B 2014, 118, 11417. (39) Brazhkin, V. V.; Trachenko, K. Between glass and gas: Thermodynamics of liquid matter. J. Non-Cryst. Solids 2015, 407, 149. (40) Boon, J.-P.; Yip, S. Molecular Hydrodynamics; McGraw-Hill: New York, 1980. (41) Prescher, C.; Fomin, Y.; Prakapenka, V. B.; Stefanski, J.; Trachenko, K.; Brazhkin, V. V. Experimental evidence of the Frenkel line in supercritical neon. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 95, 134114. (42) Prescher, C.; Fomin, Y.; Prakapenka, V. B.; Stefanski, J.; Trachenko, K.; Brazhkin, V. V. Experimental evidence of the Frenkel line in supercritical neon; Arxiv e-print, 2016.

thermal speed, straightforwardly obtained cs, as was shown before.6,26,49 Note that the k-dependent static correlators and time correlation functions were additionally averaged over all possible directions of wave vectors having the same absolute value.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

M. Santoro: 0000-0001-5693-4636 T. Scopigno: 0000-0002-7437-4262 Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001

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DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001