Frenkel Line” † ‡ ¶ § † ∥ ⊥ ¶ § ∥ ⊥ T

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.

pon increasing the pressure and temperature of a substance such a separation line and determined as the locus of specific heat U beyond its critical point, any thermodynamic discontinuity maxima. Alternative representations of the Widom line were later proposed in order to mark the separation line toward higher between the liquid and gas phase disappears and the system is − said to be in a fluid state. Accordingly, conventional wisdom temperatures.6 13 Notably, the pivotal role of the Widom line would suggest that structural and dynamical properties of was anticipated within the context of dynamical crossovers in supercritical fluids smoothly depend on temperature and systems displaying liquid−liquid phase transitions, such as 14−17 pressure. water. Albeit the very nature of supercritical fluids has been debated From the theoretical side, PSD in liquids was originally 18 for over 2 centuries,1 only recently the single-phase scenario has rationalized by mode-coupling theory and by the memory 19,20 been challenged by Gorelli et al.,2,3 when it was suggested to function formalism. The explicit account for nonlocal fl analyze the propagating density fluctuations in supercritical fluids coupling of uctuations of conserved quantities within the as a function of pressure/density, based on the idea that mode-coupling approach allowed one to obtain a nonanalytical dynamical properties could be more sensitive to phase changes correction to the linear hydrodynamic dispersion law for acoustic than the structural properties. As an example, despite strikingly excitations similar pair distribution functions, in liquid and glass systems the ωα()kckk=+5/2 + ... density−density time correlation functions behave in an MCT s absolutely different way because of the dynamic arrest of particle where k is the wavenumber and cs is the adiabatic speed of sound. motion and the emergence of nonergodicity in the glass state. Critically, it appeared that the pressure dependence of the Using a combination of inelastic X-ray scattering experiments positive coefficient α for liquid Ar21 was not the one observed for 4 and molecular dynamics (MD) simulations, it was discovered PSD in inelastic X-ray scattering experiments. The memory that the onset of deviation from the hydrodynamic dispersion of function formalism, on the other hand, allowed one to connect sound (the so-called “positive sound dispersion” (PSD)) could different channels of correlation decay in the second-order 20,22 be used as a boundary between liquid-like and gas-like regions memory function with the PSD. More recently, another existing in a fluid as reminiscent of the subcritical behavior.4 theoretical approach of generalized collective modes Obviously, any separation line of liquid-like and gas-like types of dynamics of supercritical fluids must emanate from the critical Received: August 17, 2017 point. Accordingly, in ref 4, the Widom line,5 a prolongation of Accepted: September 25, 2017 the coexistence line, was suggested as a suitable candidate for Published: September 25, 2017

⟨ 2⟩ ⟨ 2⟩ 2 ﬁ Figure 1. Left: Mean square displacements R for supercritical Ne at 295 K. The line R = Rmax , where Rmax is the rst peak position of g(r), allows ﬁ τ estimation of the speci c Frenkel time F. 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 sensitive to some single-particle specific time. This was for the dispersion law of acoustic excitations on the boundary of highlighted taking as an illustrative example the supercritical the hydrodynamic regime.26 Two dynamic models, viscoelastic argon, where the pressure dependence of the minimum of the and thermoviscoelastic ones, were analytically solved in ref 26 in adiabatic speed of sound as a function of temperature cs(T) order to understand different contributions to the PSD in fluids. (taken from the NIST database31) shows a clear mismatch with It was found within the GCM approach that the first correction the location of the Frenkel line.32 to the hydrodynamic dispersion law is proportional to k3 with a A third important issue with the Frenkel line approach coefficient β concerns the behavior of nonhydrodynamic shear waves. On the τ “ ” ω π basis of the Frenkel time scale F,a Frenkel frequency F =2 / ωβ()kckk=+3 + ... τ fi ff GCM s F can be de ned, which would identify a low-frequency cuto bound for transverse excitations in fluids.29 This is against which in viscoelastic approximation decays with a decrease of 33 8,34−36 density in agreement with observations for PSD in inelastic X-ray textbook paradigms and theoretical studies of transverse scattering experiments and for low-density states is practically dispersions in liquids, clearly demonstrating the existence of a zero. The effect of coupling to heat fluctuations within the wavelength rather than frequency gap. Moreover, a thermody- thermoviscoelastic model leads to possible negative values of β namic basis for the Frenkel line was suggested in ref 37, where (i.e., emergence of “negative” sound dispersion) for fluids with a based on the solid-state approach of nondamped phonons the large ratio of specific heats γ, like hard-disk27 and hard-sphere28 authors obtained an expression for a contribution from fluids, or supercritical ones in the vicinity of the critical point.26 transverse excitations to the total energy, which supposedly In 2012, an alternative approach to dynamics of supercritical drops to zero at the Frenkel line. Recently, the solid-state fluids was proposed, connected with the so-called “Frenkel approach to the thermodynamics of liquids was promoted by line”,29,30 which is defined on the basis of single-particles’ claiming that there exist collective excitations in condensed fi fi matter systems (either ordered or disordered) that do not decay properties. Speci cally, its de nition is given by the equivalence “ ” of the characterictic single-particle time scale of “Frenkel jumps” because of the energy conservation law ( damping myth in refs τ 38 and 39), clearly challenging the fluctuation−dissipation F (i.e., time needed for a particle to reach its nearest-neighbor 40 shell) to a shortest oscillation time of transverse excitations in theorem and the role of dissipation in creating new fl τ fluctuations. uids D (which is connected with an analogy of the Debye τ τ All of these facts call for a detailed benchmark of the Frenkel frequency): F = D. This line supposedly discriminates between rigid and nonrigid fluids and was named in ref 30 as a line of line concept and possible implications, which we present here for “ − ” the case of supercritical neon. The choice of this specific system is liquid gas transition in the supercritical region . Because the 41 Frenkel line crosses on the phase diagram the standard liquid− partly motivated by a recent report, in which the disappearance 12 of medium-range order in X-ray diffraction experiments of gas coexistence (at TF) below the critical point (TF < Tc), it immediately raises a question on the potential connection of the supercritical Ne was claimed when the derived third peak of the Frenkel line with liquid-like and gas-like features: a contradiction pair distribution function g(r) was not practically detectable after crossing a pressure value corresponding to the intersection with stands between the general ability to sustain the liquid phase for 41,42 T < T and the identification of a gas-like region at the liquid-side the Frenkel line. Additionally, we will scrutinize other c predictions of the Frenkel line approach, related to dispersion of of the binodal line for TF < T < Tc. Nevertheless, the Frenkel line was claimed to have impact on several structural and dynamic longitudinal and transverse excitations. Definitions of the Frenkel Time Based on Single-Particle quantities such as the pair distribution function, speed of sound, fi τ ff 29 Dynamics. According to its original de nition, F is the average di usion, and spectra of collective excitations because of the “ supposedly fundamental role of a single-particle time needed for time needed for a particle to move the average interparticle distance a” or “to move a distance comparable to its own size” particles to reach the nearest-neighbor distance. 29 In this respect, a second contradiction arises for an example and is given as with the behavior of the macroscopic adiabatic speed of sound in 2 fluids, c , which is governed by the local conservation laws and in a s τF = a continuum system without any atomistic structure cannot be 6D (1)

4996 DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001 The Journal of Physical Chemistry Letters Letter

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 L/T ω spectral functions C (k, ). Straight blue lines correspond to hydrodynamic dispersion law with the adiabatic speed of sound cs. The dotted horizontal ﬀ ω ω ω lines correspond to the cuto Frenkel frequency F; according to ref 29, the transverse excitations with T(k)< F should not exist.

In fact, this is an approximation of the exact average time for region. Star symbols indicate when the exact condition (2), with particles to move some distance a, which may be obtained via the a being the position of the first g(r) peak (see the next section), is expression for the mean square displacement ⟨R2⟩(t) fi τ full lled. The values F calculated from exact eq 2 will be used 2 2 later for estimation of the cutoff Frenkel frequencies for ⟨Ra⟩=()τ (2) F transverse excitations in supercritical Ne. The impact of the In Figure 1a, we show the time dependence of the Ne mean approximation in eq 1 is illustrated in Figure 1b, where Frenkel square displacements for different densities in the supercritical time is reported as estimated from approximated eq 1 and from

4997 DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001 The Journal of Physical Chemistry Letters Letter the exact eq 2. The discrepancy increases from 20% up to a factor contains the expected propagation gap in the long-wavelength − of 2 by moving from the highest to the lowest density explored in region.33 35 For lower densities, though, the transverse current this study. We note that the failure of the approximation in eq 1 spectral functions become more of a Gaussian-like shape, and can be traced back to the assumption of a linear time dependence therefore, we were unable to resolve peaks of CT(k,ω), a similar of the mean square displacement. While this is certainly observation as was reported recently for hard-sphere fluids.28 asymptotically valid, it is clearly untenable on the nearest- Our MD results shown in Figure 3 give evidence that the neighbor transit time scale implied in the FL definition (see Frenkel frequency does not play the role of the low-frequency Figure 1a). Accordingly, use of eq 1 undermines any reliable cutoff, as was stated in ref 29, and the dispersion of transverse conclusions on the Frenkel line scenario. excitations starts from zero frequency but from nonzero ff Structural Properties at the Frenkel Line. Here we aim to verify wavenumber kc predicted by di erent methodologies in refs the recently reported41 disappearance of medium-range order in 33−35. Interestingly, similar results showing existing low- X-ray diffraction experiments of supercritical Ne. To this frequency transverse excitations were reported in ref 43, although purpose, MD simulations are an ideal tool to enable one to the inconsistency with the Frenkel line approach was not ω converge the pair distribution functions in classical simulations emphasized. Jointly with our results, they clearly show that F with the necessary precision required to track the behavior of the cannot be used as (one of) the definition of the Frenkel line given height of the third maximum of g(r) upon crossing the Frenkel in ref 29 line. T In Figure 2a, the density dependence of pair distribution ωωFD= (4) functions is reported over the whole explored range. A gradual fi ωT reduction of the rst and second peaks of g(r) is observed with a with D being the highest possible Debye frequency for well-defined minimum between them. In Figure 2b,c, we zoom in transverse excitations. the region of distances where the third peak of g(r) is located. Consistency of the Definition of the Frenkel Line from a Solid- Data do not show any drastic change in the height of this feature State Approach to the Specific Heat Cv of Fluids. The inconsistency at the Frenkel line (which is shown by the dotted vertical line in between the absence of a low-frequency cutoff of transverse the bottom frame) but rather a slow decrease upon density waves with the definition of the Frenkel line is critically related to reduction. Critically, the height of the third peak of g(r) always another important claim: the value of specific heat at constant remains larger than unity for densities well below the FL. volume Cv =2kB, which coincides with the Frenkel line, separates fl It has to be pointed out, however, that retrieval of the g(r) via liquid-like (Cv >2kB) and gas-like (Cv <2kB) uids with and the experimental structure factors S(k) requires an inversion without transverse excitations, correspondingly.30 However, very procedure that is subject to systematic errors. This could explain recently, it was shown28 that these arguments do not apply to the different conclusions reported in ref 41. hard-sphere fluids, where for dense fluids the transverse fi fi Frenkel Frequency and Transverse Dynamics. From the obtained excitations are well-de ned, although for the speci c heat, Cv is τ values of the Frenkel time F the corresponding Frenkel equal to 1.5kB at any density value. frequencies can be easily calculated using the definition in ref 29 In an effort to overcome such inconsistency, it has been 37 fi fl 2π proposed to arti cially decompose the energy of a uid into ωF = contributions from relaxing and propagating processes, although τ (3) F such decomposition and relative contributions to Cv naturally − The Frenkel frequency is the low-frequency cutoff for transverse originate from the Landau Placzek-like ratio for the dynamics of fl fl 44,45 37 excitations in the fluid, a cornerstone of the whole Frenkel line heat uctuations in uids. Nevertheless, it was shown that 29 ∼ approach, and in a combination with the highest possible each branch of collective excitations contributes 0.5kB to Cv, “Debye frequency”, it has been proposed as a way to discriminate and reduction of Cv to 2kB signals the disappearance of transverse between fluids with and without transverse excitations. The collective modes. The whole proposal is built on a solid-state fi Frenkel frequency calculated from eqs 3 and 2 drops monotoni- picture with nondamped phonon-like modes, and in the speci c − − ff cally from 10.28 ps 1 for the lowest studied density to 4.81 ps 1 case of transverse modes, the low-frequency cuto (whose for the highest one. absence has been discussed above) was taken as the lower In Figure 3, we show dispersions of longitudinal (L) and integration bound of transverse (T) collective excitations calculated from peak ω T L/T ω D positions of L/T current spectral functions C (k, ). The ECT = ETg(,ωωω ) ()d L/T ω ∫ T C (k, ) were obtained from time-Fourier transformation of ωF (5) the MD-derived L/T current−current time correlation func- ω tions. Note, because the transverse current spectral function at where gT( ) is the density of transverse phonon-like states and C fi zero frequency is de ned by the inverse of k-dependent shear is a normalization constant. The reduction of the contribution ET T ω ≈ η−1 fl ω ωT viscosity, C (k, =0) (k), for low-density uids with low to the total energy, ultimately to zero in the case of F = D (the viscosity, it is problematic to observe any peak structure of condition for the Frenkel line), is here the increase of the lower T ω ω C (k, ). Although we report here the spectra of collective integration bound F with density. However, as already excitations estimated from peak positions of CL/T(k,ω), we stress discussed, there is no low-frequency cutoff for the transverse that only a reasonable theory that takes into account excitations, and the dispersion law for transverse excitations contributions from different relaxing and propagating modes to starts from zero frequency. Correspondingly, the density of the shape of CL/T(k,ω) can reveal the issue of existence/absence transverse phonon-like states starts from zero frequency as well, of short-wavelength shear waves in fluids. In Figure 3, the low- and the integration in eq 5 must have zero as the lower ff ω fi fi frequency cuto s F are shown by pink dashed lines. One can integration bound. Hence, even the speci c heat-based de nition easily see that for densities 1400 kg/m3 and higher the transverse of the Frenkel line (eq 4) is not consistent with the observed excitations propagate in supercritical Ne, and their dispersion dispersion of transverse excitations in fluids.

4998 DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001 The Journal of Physical Chemistry Letters Letter

Consistency of the Definition of Positive Sound Dispersion within to the ratio of transverse to longitudinal sound velocity as per eq the Frenkel Line Approach. The viscoelastic increase of the 6. Such a ratio has very little system dependence with respect to frequency of acoustic excitations outside of the hydrodynamic the wide range of positive dispersion observed in fluids, from a − regime observed in dense fluids by scattering experiments2 4,6 few pecent in simple liquids (such as liquid metals) to a factor of and MD simulations7,26,36,46 is another process for which the two in water. “Frenkel mechanism” has been proposed as a possible MD simulations for supercritical Ne for different pressures at demarcation line: below the Frenkel line, the PSD exists, but 295 K shows the following facts 29 not above the Frenkel line. This was based on the empirical fact (i) There are no sudden changes in the density dependence of that the propagation speed increases from the macroscopic g(r) by crossing the Frenkel line. More specifically, the adiabatic value (for which the low-frequency shear modulus G0 is height of the third peak of g(r) slowly reduces to unity for ρ 1/2 zero) [Bs/ ] to some high-frequency value [(B∞ + (4/3)G∞)/ ρ 1/2 29 densities far below the value at the Frenkel line. ] with a nonzero high-frequency shear modulus. (ii) Low-frequency transverse excitations are observed for ω < One can immediately argue on these intuitive speculations that ω F, as expected from the well-established theory of shear the high-frequency shear modulus, G∞, is nonzero for all 33−35 wave propagation in liquids. densities (see Figure 4), which would cause nonzero PSD in the (iii) The high-frequency shear modulus does not drop to zero even for low-density fluids, indicating that any inter- pretation 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 Figure 4. High-frequency B∞, adiabatic Bs, low-frequency B0 bulk moduli, and high-frequency G∞ shear module as functions of the density supercritical region is critically undermined by the present study. of supercritical Ne at 295 K. The location of the Frenkel line from ref 41 is shown by the vertical blue dashed line. ■ METHODS We performed MD simulations of supercritical Ne using a system 4 whole range of densities that contradicts experimental and of 4000 particles in a microcanonical ensemble with Lennard- 41 simulation results. Moreover, in the hydrodynamic region (on Jones two-body potentials. The parameters of the Lennard-Jones the macroscopic scale) and in the high-frequency regime (on the potential were the same as those in ref 41. Several studies atomistic scale), the bulk modulus B (as it is given erroneously in investigated the correction of three-body contributions to two- ref 29) is not the same quantity: on macroscopic scales, it is the body interactions of noble gases.47,48 In the specific case of the fi adiabatic bulk modulus Bs that is solely de ned by the velocity Lennard-Jones interatomic potential, however, thermodynamical field and heat fluctuations in the fluids, while at high frequencies, properties are very well reproduced, as demonstrated by the ∞ fi fi it is B that is de ned by the microscopic forces acting on agreement (5%) of the calculated speci c heat Cv and adiabatic ff 31 particles and is essentially an isothermal quantity. In Figure 4,we speed of sound cs for di erent densities with NIST values. The show the density dependence of the adiabatic Bs bulk modulus temperature in simulations was kept at 295 K, that is, T/Tc = (which is γ times larger than the zero-frequency isothermal 6.63, and the drift of energy over the production runs of 300 000 macroscopic bulk modulus, B0) and the high-frequency bulk B∞ time steps (Δt = 0.5 fs) was not larger than 0.02%. We studied 10 and shear G∞ moduli; one can immediately see the essential different densities along the isothermal line in the range from 691 ff ∞ ∞ 3 ρ 3 ρ di erence between Bs and B . Hence, replacing B with Bs in kg/m (1.428 c) to 1600 kg/m (3.306 c), calculating both order to obtain the expression for the macroscopic propagation single-particle and collective dynamic properties. Additionally, in 43 speed the density range of 1000−1600 kg/m3, we performed longer 2 2 2 simulations evaluating pair distribution functions every 10 steps v =+cv(4/3) (6) l s t in order to converge the g(r) and clarify the issue about the where vl and vt are respectively the speeds of longitudinal and possible disappearence of the third peak of g(r) upon crossing the transverse excitations, is clearly unjustified. In fact, Figure 4 Frenkel line. unambiguously shows that there is no qualitative change in the For each density, we calculated the longitudinal (L) and density dependence of different bulk and high-frequency shear transverse (T) current−current time correlation functions L/T moduli upon crossing the Frenkel line, which for the Ne isotherm FJJ (k,t), which upon time-Fourier transformation resulted in ρ L/T ω in the focus of this study is located close to density F = 1330 kg/ L and T current spectral functions C (k, ). In order to 3 41 m . Moreover, the high-frequency shear modulus is nonzero in estimate the macroscopic adiabatic speed of sound cs from MD the whole studied density range, which means nonzero positive simulations, we calculated energy−energy and energy−density sound dispersion according to eq 6, even above the Frenkel line static correlators, obtained the wavenumber-dependent thermo- (for densities smaller than 1330 kg/m3). In passing, we note that, dynamic quantities, and, from a smooth k dependence of the beside the above-mentioned issues, the Frenkel line-based ratio γ(k)/S(k)(γ(k) and S(k) are k-dependent ratios of the quantification of positive dispersion would be entirely related specific heats and structure factor, respectively) together with the

4999 DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001 The Journal of Physical Chemistry Letters Letter thermal speed, straightforwardly obtained cs, as was shown A. Z.; Russo, J.; et al. Water: A Tale of Two Liquids. Chem. Rev. 2016, before.6,26,49 Note that the k-dependent static correlators and 116, 7463−7500. PMID: 27380438. time correlation functions were additionally averaged over all (18) Ernst, M. H.; Dorfman, J. R. Nonanalytic dispersion relations for possible directions of wave vectors having the same absolute classical fluids. J. Stat. Phys. 1975, 12, 311. value. (19) Balucani, U.; Zoppi, M. Dynamics of the liquid state; Clarendon Press: Oxford, U.K., 1994. ■ AUTHOR INFORMATION (20) Scopigno, T.; Balucani, U.; Ruocco, G.; Sette, F. Density fluctuations in molten lithium: inelastic x-ray scattering study. J. Phys.: Corresponding Author Condens. Matter 2000, 12, 8009. *E-mail: [email protected]. (21) de Schepper, I.; Verkerk, P.; van Well, A.; de Graaf, L. Non- ORCID analytic dispersion relations in liquid argon. Phys. Lett. A 1984, 104,29− M. Santoro: 0000-0001-5693-4636 32. (22) Scopigno, T.; Ruocco, G.; Sette, F. Microscopic dynamics in T. Scopigno: 0000-0002-7437-4262 liquid metals: The experimental point of view. Rev. Mod. Phys. 2005, 77, Notes 881. The authors declare no competing ﬁnancial interest. (23) deSchepper, I. M.; Cohen, E. G. D.; Bruin, C.; van Rijs, J. C.; Montfrooij, W.; de Graaf, L. Hydrodynamic time correlation functions ■ REFERENCES for a Lennard-Jones fluid. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 271. (1) McMillan, P.; Stanley, H. Fluid phases: Going supercritical. Nat. (24) Mryglod, I. M.; Omelyan, I. P.; Tokarchuk, M. V. Generalized Phys. 2010, 6, 479. collective modes for the Lennard-Jones fluid. Mol. Phys. 1995, 84, 235. (2) Gorelli, F. A.; Santoro, M.; Scopigno, T.; Krisch, M.; Ruocco, G. (25) Bryk, T.; Mryglod, I.; Kahl, G. Generalized collective modes in a Liquidlike Behavior of Supercritical Fluids. Phys. Rev. Lett. 2006, 97, binary He0.65 - Ne0.35 mixture. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, 245702. Relat. Interdiscip. Top. 1997, 56, 2903. (3) Gorelli, F. A.; Santoro, M.; Scopigno, T.; Krisch, M.; Bryk, T.; (26) Bryk, T.; Mryglod, I.; Scopigno, T.; Ruocco, G.; Gorelli, F.; Ruocco, G.; Ballerini, R. Inelastic x-ray scattering from high pressure Santoro, M. Collective excitations in supercritical fluids: Analytical and fluids in a diamond anvil cell. Appl. Phys. Lett. 2009, 94, 074102. molecular dynamics study of “positive” and “negative” dispersion. J. (4) Simeoni, G.; Bryk, T.; Gorelli, F. A.; Krisch, M.; Ruocco, G.; Chem. Phys. 2010, 133, 024502. Santoro, M.; Scopigno, T. The Widom line as the crossover between (27) Huerta, A.; Bryk, T.; Trokhymchuk, A. Collective excitations in liquid-like and gas-like behaviour in supercritical fluids. Nat. Phys. 2010, 2D hard-disc fluid. J. Colloid Interface Sci. 2015, 449, 357. 6, 503. (28) Bryk, T.; Huerta, A.; Hordiichuk, V.; Trokhymchuk, A. Non- (5) Widom, B. In Phase Transitions and Critical Phenomena; Domb, C., hydrodynamic transverse collective excitations in hard-sphere fluids. J. Green, M. S., Eds.; Academic, 1972; Vol. 2; p 79. Chem. Phys. 2017, 147, 064509. (6) Gorelli, F. A.; Bryk, T.; Krisch, M.; Ruocco, G.; Santoro, M.; (29) Brazhkin, V. V.; Fomin, Y. D.; Lyapin, A. G.; Ryzhov, V. N.; Scopigno, T. Dynamics and Thermodynamics beyond the critical point. Trachenko, K. Two liquid states of matter: A dynamic line on a phase Sci. Rep. 2013, 3, 1203. diagram. Phys. Rev. E 2012, 85, 031203. (7) Bryk, T.; Gorelli, F.; Ruocco, G.; Santoro, M.; Scopigno, T. (30) Brazhkin, V. V.; Fomin, Y.; Lyapin, A. G.; Ryzhov, V. N.; Tsiok, E. Collective excitations in soft-sphere fluids. Phys. Rev. E 2014, 90, N.; Trachenko, K. "Liquid-Gas" Transition in the Supercritical Region: 042301. Fundamental Changes in the Particle Dynamics. Phys. Rev. Lett. 2013, (8) Bryk, T.; Gorelli, F.; Ruocco, G.; Santoro, M.; Scopigno, T. In 111, 145901. Physics of Liquid Matter: Modern Problems; Bulavin, L., Lebovka, N., Eds.; − (31) Thermophysical Properties of Fluid Systems. http://webbook.nist. 2014; pp 77 102. gov/chemistry/ﬂuid/ (2017). (9) Banuti, D. T.; Raju, M.; Ihme, M. Similarity law for Widom lines (32) Bryk, T. Comment on ”Dynamic Transition of Supercritical and coexistence lines. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Hydrogen: Defining the Boundary between Interior and Atmosphere in Interdiscip. Top. 2017, DOI: 10.1103/PhysRevE.95.052120. Gas Giants. Phys. Rev. E 2015, 91, 036101. (10) Raju, M.; Banuti, D. T.; Ma, P. C.; Ihme, M. Widom Lines in (33) Hansen, J.-P.; McDonald, I. R. Theory of simple liquids; Academic: Binary Mixtures of Supercritical Fluids. Sci. Rep. 2017, DOI: 10.1038/ London, 1986. s41598-017-03334-3. (34) MacPhail, R. A.; Kivelson, D. Generalized hydrodynamic theory (11) Tareyeva, E. E.; Ryzhov, V. N. Supercritical fluid of particles with a of viscoelasticity. J. Chem. Phys. 1984, 80, 2102. Yukawa potential: A new approximation for the direct correlation (35) Bryk, T.; Mryglod, I. Optic-like excitations in binary liquids: function and the Widom line. Theor. Math. Phys. 2016, 189, 1806−1817. (12) Ruppeiner, G.; Dyjack, N.; McAloon, A.; Stoops, J. Solid-like transverse dynamics. J. Phys.: Condens. Matter 2000, 12, 6063. features in dense vapors near the fluid critical point. J. Chem. Phys. 2017, (36) Bryk, T. Non-hydrodynamic collective modes in liquid metals and 146, 224501. alloys. Eur. Phys. J.: Spec. Top. 2011, 196, 65. (13) Artemenko, S.; Krijgsman, P.; Mazur, V. The Widom line for (37) Bolmatov, D.; Brazhkin, V. V.; Trachenko, K. Thermodynamic supercritical fluids. J. Mol. Liq. 2017, 238, 122−128. behaviour of supercritical matter. Nat. Commun. 2013, 4, 2331. (14) Xu, L.; Kumar, P.; Buldyrev, S. V.; Chen, S.-H.; Poole, P. H.; (38) Brazhkin, V. V.; Trachenko, K. Collective Excitations and Sciortino, F.; Stanley, H. E. Relation between the Widom line and the Thermodynamics of Disordered State: New Insights into an Old dynamic crossover in systems with a liquid−liquid phase transition. Proc. Problem. J. Phys. Chem. B 2014, 118, 11417. Natl. Acad. Sci. U. S. A. 2005, 102, 16558−16562. (39)Brazhkin,V.V.;Trachenko,K.Betweenglassandgas: (15) Gallo, P.; Corradini, D.; Rovere, M. Widom line and dynamical Thermodynamics of liquid matter. J. Non-Cryst. Solids 2015, 407, 149. crossovers as routes to understand supercritical water. Nat. Commun. (40) Boon, J.-P.; Yip, S. Molecular Hydrodynamics; McGraw-Hill: New 2014, 5, 5806. York, 1980. (16) Franzese, G.; Stanley, H. E. The Widom line of supercooled water. (41) Prescher, C.; Fomin, Y.; Prakapenka, V. B.; Stefanski, J.; J. Phys.: Condens. Matter 2007, 19, 4th Workshop on Nonequilibrium Trachenko, K.; Brazhkin, V. V. Experimental evidence of the Frenkel Phenomena in Supercooled Fluids, Glasses and Amorphous Materials, line in supercritical neon. Phys. Rev. B: Condens. Matter Mater. Phys. Pisa, ITALY, SEP 17−22, 2006; p 205126, 10.1088/0953-8984/19/20/ 2017, 95, 134114. 205126 (42) Prescher, C.; Fomin, Y.; Prakapenka, V. B.; Stefanski, J.; (17) Gallo, P.; Amann-Winkel, K.; Angell, C. A.; Anisimov, M. A.; Trachenko, K.; Brazhkin, V. V. Experimental evidence of the Frenkel line Caupin, F.; Chakravarty, C.; Lascaris, E.; Loerting, T.; Panagiotopoulos, in supercritical neon; Arxiv e-print, 2016.

5000 DOI: 10.1021/acs.jpclett.7b02176 J. Phys. Chem. Lett. 2017, 8, 4995−5001 The Journal of Physical Chemistry Letters Letter

(43) Fomin, Y.; Ryzhov, V. N.; Tsiok, E. N.; Brazhkin, V. V.; Trachenko, K. Crossover of collective modes and positive sound dispersion in supercritical state. J. Phys.: Condens. Matter 2016, 28, 43LT01. (44) Bryk, T.; Ruocco, G.; Scopigno, T. Landau-Placzek ratio for heat density dynamics and its application to heat capacity of liquids. J. Chem. Phys. 2013, 138, 034502. (45) Bryk, T.; Scopigno, T.; Ruocco, G. Heat capacity of liquids: A hydrodynamic approach. Condens. Matter Phys. 2015, 18, 13606. (46) Bryk, T.; Ruocco, G. Generalized collective excitations in supercritical argon. Mol. Phys. 2011, 109, 2929. (47) Jakse, N.; Bomont, J. M.; Charpentier, I.; Bretonnet, J. L. Effects of dispersion forces on the structure and thermodynamics of fluid krypton. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 62, 3671. (48) Jakse, N.; Bretonnet, J. L. Use of state-dependent pair potentials in describing the structural and thermodynamic properties of noble gases. J. Phys.: Condens. Matter 2003, 15, S3455. (49) Bryk, T.; Ruocco, G. Generalised hydrodynamic description of the time correlation functions of liquid metals: Ab initio molecular dynamics study. Mol. Phys. 2013, 111, 3457.

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