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A large and mostly unexplored part of the phase diagram lies above the critical point. The supercritical matter was traditionally believed to be physically homogeneous with no discernible differences between liquidlike and gaslike states. More recently, several proposals have been put forward challenging this view, and here we review the history of this research. About a decade ago, it was proposed that the Frenkel line (FL), corresponding to the dynamical transition of particle motion and related thermodynamic and structural transitions, gives a unique and path-independent way to separate the supercritical states into two qualitatively different states and extends to arbitrarily high pressure and temperature on the phase diagram. Here, we review several lines of enquiry that followed. We focus on the experimental evidence of transitions in deeply supercritical Ne, N 2 , CH 4 , C 2 H 6 , CO 2 and H 2 O at the FL detected by a number of techniques including X-ray, neutron and Raman scattering experiments. We subsequently summarise other developments in the field: recent extensions of analysis of dynamics at the FL, quantum simulations, topological and geometrical approaches, the universality of properties at the FL including transport properties, their fundamental bounds and the implications of the supercritical crossover for astrophysics and planetary science. Finally, we review current theoretical understanding of the supercritical state including its thermodynamic theory and list open problems in the field.

Recent studies on supercritical nitrogen revealed clear changes in structural markers and dynamical properties when the coordination number approaches its maximum value. The line in P and T space where these changes occur is referred to as the Frenkel line. Here, we qualitatively reproduce such changes in the supercritical regime using the popular “optimized potential for liquid simulation” (OPLS) classical force field for molecular dynamics. Unfortunately, at 160 K, OPLS nitrogen predicts sublimation rather than producing a liquid phase; therefore, we developed our own force field to achieve quantitative agreement with experimental data. We confirm the asymptotic behavior of the coordination number on crossing the Frenkel line and note an associated change in the diffusion constant, consistent with the non-rigid to rigid liquid-like character of the “transition”. The simulations allow us to track the Frenkel line to subcritical temperatures and demonstrate that it terminates at the triple point. This establishes the experimentally measurable changes, which could unequivocally determine the Frenkel line in other systems.

We have performed a neutron scattering experiment on supercritical \ud uid nitrogen\ud at 160 K (1.27 TC) over a wide pressure range (7.8 MPa / 0.260 g/ml to 125 MPa /\ud 0.805 g/ml). This has enabled us to study the process by which nitrogen changes from a fluid that exhibits gas-like behaviour to one that exhibits rigid liquid-like behaviour at\ud a temperature close to, but above, the critical temperature by crossing the Widom lines followed by the Frenkel line on pressure (density) increase. We find that the Frenkel\ud line transition is indicated by a transition to a regime of rigid liquid-like behaviour in\ud which the coordination number remains constant within error, in agreement with our\ud previous work at 300 K. The Frenkel line transition takes place at approximately the\ud same density at 160 K and 300 K. The data do not conclusively show an additional\ud transition at the location of the known Widom lines. We find that behaviour remains\ud gas-like until the Frenkel line is crossed and our data support the hypothesis that the\ud Widom line transitions are density increase-driven.

We have studied in detail the transition from gas-like to rigid liquid-like behavior in supercritical N2 at 300 K (2.4 TC). Our study combines neutron diffraction and Raman spectroscopy with ab initio molecular dynamics simulations. We observe a narrow transition from gas-like to rigid liquid-like behavior at ca. 150 MPa, which we associate with the Frenkel line. Our findings allow us to reliably characterize the Frenkel line using both diffraction and spectroscopy methods, backed up by simulation, for the same substance. We clearly lay out what parameters change, and what parameters do not change, when the Frenkel line is crossed.

We investigate the temperature evolution of dynamics and structure of partially confined Lennard Jones (LJ) fluids in supercritical phase along an isobaric line in the P-T phase diagram using molecular dynamics simulations. We compare the Frenkel line (FL) crossover features of partially confined LJ fluids to that of the bulk LJ fluids in supercritical phase. Five different spacings have been chosen in this study and the FL crossover characteristics have been monitored for each of these spacings for temperatures ranging from 240 K to 1500 K keeping the pressure fixed at 5000 bar. We characterize the FL crossover using density of states (DoS) function and find that partially confined supercritical fluids (SCF) exhibit a progressive shift of FL crossover point to higher temperatures for smaller spacings. While the DoS perpendicular to the walls shows persistent oscillatory modes, the parallel component exhibits a smooth crossover from an oscillatory to non-oscillatory characteristics representative of FL crossover. We find that the vanishing of peaks in DoS parallel to the walls indicates that the SCF no longer supports shear mode excitations and could serve as an identifier of the FL crossover for confined systems just as is done for the bulk. Layer heights of density profiles, self-diffusivity and the peak heights of radial distribution function parallel to the walls also feature the FL crossover consistent with the DoS criteria. Surprisingly, self-diffusivity undergoes an Arrhenius to super-Arrhenius crossover at low temperatures for smaller spacings as a result of enhanced structural order evidenced via pair-excess entropy. This feature, typical of glass-forming liquids and binary supercooled liquids, is found to develop from the glass-like characteristic slowdown and strong caging in confined supercritical fluid, evidenced via mean squared displacement and velocity autocorrelation function respectively, over intermediate timescales.

The present study reports a systematic analysis of a wide variety of structural, thermodynamic, and dynamic properties of supercritical water along the near-critical isotherm of T = 1.03Tc and up to extreme pressures, using molecular dynamics and Monte Carlo simulations. The methodology employed provides solid evidence about the existence of a structural transition from a liquidlike fluid to a compressed, tightly packed liquid, in the density and pressure region around 3.4ρc and 1.17 GPa, introducing an alternative approach to locate the crossing of the Frenkel line. Around 8.5 GPa another transition to a face-centered-cubic plastic crystal polymorph with density 5.178ρc is also observed, further confirmed by Gibbs free energy calculations using the two-phase thermodynamic model. The isobaric heat capacity maximum, closely related to the crossing of the Widom line, has also been observed around 0.8ρc, where the local density augmentation is also maximized. Another structural transition has been observed at 0.2ρc, related to the transformation of the fluid to a dilute gas at lower densities. These findings indicate that a near-critical isotherm can be divided into different domains where supercritical water exhibits distinct behavior, ranging from a gaslike one to a plastic crystal one.

This paper presents an application of mean force kinetic theory (MFT) to the calculation of the self-diffusivity of CO2 in the supercritical fluid regime. Two modifications to the typical application of MFT are employed to allow its application to a system of molecular species. The first is the assumption that the inter-particle potential of mean force can be obtained from the molecule center-of-mass pair correlation function, which in the case of CO2 is the C-C pair correlation function. The second is a new definition of the Enskog factor that describes the effect of correlations at the surface of the collision volume. The new definition retains the physical picture that this quantity represents a local density increase, resulting from particle correlations, relative to that in the zero density homogeneous fluid limit. These calculations are facilitated by the calculation of pair correlation functions from molecular dynamics (MD) simulations using the FEPM2 molecular CO2 model. The self-diffusivity calculated from theory is in good agreement with that from MD simulations up to and slightly beyond the density at the location of the Frenkel line. The calculation is compared with and is found to perform similarly well to other commonly used models but has a greater potential for application to systems of mixed species and to systems of particles with long range interatomic potentials due to electrostatic interactions.

Recently there have been significant theoretical advances in our understanding of liquids and dense supercritical fluids based on their ability to support high frequency transverse (shear) waves. Here, we have constructed a new computer model using these recent theoretical findings (the phonon theory of liquid thermodynamics), to model liquid internal energy across a wide pressure-temperature range. We have applied it to a number of real liquids in both the subcritical regime and the supercritical regime, in which the liquid state is demarcated by the Frenkel line. Our fitting to experimental data in a wide pressure-temperature range has allowed us to test the new theoretical model with hitherto unprecedented rigour. We have quantified the degree to which the prediction of internal energy and heat capacity is constrained by the different input parameters: The liquid relaxation time (initially obtained from the viscosity), the Debye wavenumber and the infinite-frequency shear modulus. The model is successfully applied to output the internal energy and heat capacity data for several different fluids (Ar, Ne, $N_2$, Kr) over a range of densities and temperatures. We find that the predicted heat capacities are extremely sensitive to the values used for the liquid relaxation time. If these are calculated directly from the viscosity data then, in some cases, changes within the margins of experimental error in the viscosity data can cause the heat capacity to exhibit a completely different trend as a function of temperature. Our code is computationally inexpensive, and it is available for other researchers to use., Comment: The following article has been accepted by Physics of Fluids, and published at https://doi.org/10.1063/5.0025871

We calculate the maximal Lyapunov exponent for a bulk system of 256 Lennard-Jones particles in constant energy molecular dynamics simulations deep into the supercritical state. We find that the maximal Lyapunov exponent undergoes a crossover, and that this crossover coincides with the dynamical crossover at the Frenkel line from liquid physics. We explain this crossover in terms of two different contributions to dynamical instability: diffusion in the liquid-like state below the Frenkel line, and collisions in the gas-like state above. These results provide insight into the phase space dynamics far from the melting line and densities where rare-gas approximation are inapplicable., 8 pages, 6 figures

Molecular dynamics (MD) simulations to understand the thermodynamic, dynamic, and structural changes in supercritical water across the Frenkel line and the melting line have been performed. The two-phase thermodynamic model [J. Phys. Chem. B, 2010, 114(24), 8191–8198] and the velocity autocorrelation functions are used to locate the Frenkel line and to calculate the thermodynamic and dynamic properties. The Frenkel lines obtained from the two-phase thermodynamic model and the velocity autocorrelation criterion do not agree with each other. Structural characteristics and the translational diffusion dynamics of water suggest that this inconsistency could arise from the two oscillatory modes in water, which are associated with the bending of hydrogen bonds and intermolecular collisions inside the first coordination shell. The overall results lead us to conclude that the universality of the Frenkel line as a dynamic crossover line from rigid to nonrigid fluids is preserved in water.

There have been ample studies of the many phases of H2O in both its solid and low pressure liquid states, and the transitions between them. Using molecular dynamics simulations we address the hitherto unexplored deeply supercritical pressures, where no qualitative transitions are thought to take place and where all properties are expected to vary smoothly. On the basis of these simulations we predict that water at supercritical pressures undergoes a structural crossover across the Frenkel line at pressures as high as 45 times the critical pressure. This provides a new insight into the water phase diagram and establishes a link between the structural and dynamical properties of supercritical water. Specifically, the crossover is demonstrated by a sharp and pronounced at low pressures, and smooth at high pressures, signified by changes in the pair distribution functions and local coordination which coincide with the dynamical transition (the loss of all oscillatory molecular motion) at the Frenkel line on the phase diagram.

The physics of supercritical states is understood to a much lesser degree compared to subcritical liquids. Carbon dioxide, in particular, has been intensely studied, yet little is known about the supercritical part of its phase diagram. Here, we combine neutron scattering experiments and molecular dynamics simulations and demonstrate the structural crossover at the Frenkel line. The crossover is seen at pressures as high as 14 times the critical pressure and is evidenced by changes of the main features of the structure factor and pair distribution functions.

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.

A fluid particle changes its dynamics from diffusive to oscillatory as the system density increases up to the melting density. Hence, the notion of the Frenkel line was introduced to demarcate the fluid region into rigid and nonrigid liquid subregions based on the collective particle dynamics. In this work, we apply a topological framework to locate the Frenkel lines of the soft-sphere and the hard-sphere models relying on the system configurations. The topological characteristics of the ideal gas and the maximally random jammed state are first analyzed, then the classification scheme designed in our earlier work is applied. The classification result shows that the fraction of solid-like atoms increases from zero to one in the rigid liquid region. The dependence of the solid-like fraction on the bulk density is understood based on the theory of fluid polyamorphism. The percolation behavior of solid-like clusters is described based on the fraction of solid-like molecules in an integrated manner. The crossover densities are obtained by examining the percolation of solid-like clusters. The resultant crossover densities of soft-sphere fluids converge to that of hard-sphere fluid. Hence, the topological method successfully highlights the generality of the Frenkel line., 9 pages, 5 figures, 2 tables

Experimental, computational, and theoretical results have shown that the notion of a homogeneous supercritical state space has to be replaced with distinct liquid-like and gas-like regions, divided by a cross-over line. Several such cross-over lines have been proposed, such as the Frenkel line, the Fisher-Widom line, and the Widom line. We use thermodynamics arguments, molecular dynamics (MD) simulations, and experimental data to investigate the relation between these lines. This work proposes a new interpretation of the Widom line based on the curvature of the Gibbs free energy, showing that the supercritical cross-over can be evaluated as a projection of the subcritical phase transition from a liquid to an ideal gas state to supercritical conditions. We show that the cross-over across the Widom line should not be regarded as instantaneous, but instead as spreading over a finite temperature interval. The thermodynamic character of the Widom line becomes negligible at reduced pressures p/pcr > 3 and disappears completely at higher pressures p/pcr > 10. We find evidence that measured transitions attributed to the Widom line do in fact match the Frenkel line better. In this way, we suggest a consistent view of the supercritical state space and resolve apparent contradictions.

A hallmark of a thermodynamic phase transition is the qualitative change of system thermodynamic properties such as energy and heat capacity. On the other hand, no phase transition is thought to operate in the supercritical state of matter and, for this reason, it was believed that supercritical thermodynamic properties vary smoothly and without any qualitative changes. Here, we perform extensive molecular dynamics simulations in a wide temperature range and find that a deeply supercritical state is thermodynamically heterogeneous, as witnessed by different temperature dependence of energy, heat capacity and its derivatives at low and high temperature. The evidence comes from three different methods of analysis, two of which are model-independent. We propose a new definition of the relative width of the thermodynamic crossover and calculate it to be in the fairly narrow relative range of 13%-20%. On the basis of our results, we relate the crossover to the supercritical Frenkel line.

Isomorph theory is one of the promising theories to understand the quasi-universal relationship between thermodynamic, dynamic and structural characteristics. Based on the hidden scale invariance of the inverse power law potentials, it rationalizes the excess entropy scaling law of dynamic properties. This work aims to show that this basic idea of isomorph theory can be extended by examining the microstructural features of the system. Using the topological framework in conjunction with the entropy calculation algorithm, we demonstrate that Voronoi entropy, a measure of the topological diversity of single atoms, provides a scaling law for the transport properties of soft-sphere fluids, which is comparable to the frequently used excess entropy scaling. By examining the relationship between the Voronoi entropy and the solid-like fraction of simple fluids, we suggest that the Frenkel line, a rigid-nonrigid crossover line, {be} a topological isomorphic line where the scaling relation qualitatively changes., 21 pages, 7 figures

Isothermal-isobaric path integral centroid molecular dynamics simulations were conducted for fluid 4He at more than 600 state points in the proximity of the critical point to reveal the detailed states and relevant quantumness underlying the supercritical state. Through intensive analyses of the thermodynamic, dynamic, and quantum properties, we revealed the hidden state boundaries that separate the liquid-like and gas-like states in the supercritical region of this fluid. The Widom line, defined as the locus of the maxima of isobaric heat capacity CP, is also the quantum boundary at which there are changes in the isobaric temperature-dependence of the quantum wavelength, λquantum, i.e., maximum amplitude of the Feynman imaginary-time paths (necklaces) of individual atoms. The Frenkel line, the famous dynamic state boundary, was observed to start from nearly the same point, 0.73-0.76 Tc, on the P-T plane as observed for classical fluids. Several state boundaries based on the new criteria were found to emanate from the critical point or its vicinity on the P-T plane and are discussed in comparison with these boundaries. The quantumness of this fluid was expressed as (a) non-classical significant depression of CP observed in the liquid-like state; (b) the depression of the slopes dP/dT of the Widom line and the liquid-gas coexistence line near the critical point; and (c) the depression of the heat of pseudo-boiling across the Widom line. This is explained in terms of the decreasing kinetic energy with temperature observed in the liquid-like state below the Widom temperature TWidom, or alternatively in terms of the lattice model heat capacity, including the λquantum.

The Frenkel line, a crossover line between rigid and nonrigid dynamics of fluid particles, has recently been the subject of intense debate regarding its relevance as a partitioning line of supercritical phase, where the main criticism comes from the theoretical treatment of collective particle dynamics. From an independent point of view, this Letter suggests that the two-phase thermodynamics model may alleviate this contentious situation. The model offers new criteria for defining the Frenkel line in the supercritical region and builds a robust connection among the preexisting, seemingly inconsistent definitions. In addition, one of the dynamic criteria locates the rigid-nonrigid transition of the soft-sphere and the hard-sphere models. Hence, we suggest the Frenkel line be considered as a dynamic rigid-nonrigid fluid boundary, without any relation to gas-liquid transition. These findings provide an integrative viewpoint combining fragmentized definitions of the Frenkel line, allowing future studies to be carried out in a more reliable manner., Comment: 15 pages, 4 figures, 1 Supporting Information (PDF file)

It has been recently proposed that supercritical fluids experience a crossover between two different regimes of the particle dynamics. The crossover takes place at the Frenkel line which can be defined from vanishing of oscillations of the velocity autocorrelation functions (vacfs). Here, we present a novel detailed analysis of vacfs of binary isotopic mixtures of Lennard-Jones and Soft Spheres particles and consider temperature dependence of vacfs of individual components as well as mixtures. We show that partial vacfs can be very different: for the mass ratio of about 2, the temperature at which the oscillatory behavior of vacf vanishes differs eightfold. Notably, oscillatory behavior of vacfs of the whole mixture vanishes at about the same temperature as in the one-component fluid. We conclude that the criterion to locate the Frenkel line on the basis of vacfs is general and applies to mixtures, reflecting more fundamental changes of system dynamics such as the disappearance of transverse mode from the excitation spectrum.

We introduce a unified approach to states of matter (solid, liquid and gas) and describe the thermodynamics of the pressure-temperature phase diagram in terms of phonon excitations. We derive the effective Hamiltonian with low-energy cutoff in two transverse phonon polarizations (phononic band gaps) by breaking the symmetry in phonon interactions. Further, we construct the statistical mechanics of states of aggregation employing the Debye approximation. The introduced formalism covers the Debye theory of solids, the phonon theory of liquids, and thermodynamic limits such as the Dulong-Petit thermodynamic limit, the ideal gas limit and the new thermodynamic limit, dubbed here the Frenkel line thermodynamic limit. We discuss the phonon propagation and localization effects in liquids above and below the Frenkel line, and explain the "fast sound" phenomenon. As a test for our theory we calculate velocity-velocity autocorrelation and pair distribution functions within the Green-Kubo formalism. We show the consistency between dynamics of phonons and pair correlations in the framework of the unified approach. New directions towards advancements in phononic band gaps engineering, hypersound manipulation technologies and exploration of exotic behaviour of fluids relevant to geo- and planetary sciences are discussed. The presented results are equally important both for practical implications and for fundamental research., Comment: 21 pages, 7 figures, 3 tables

In this work, we report on inelastic X-ray scattering experiments combined with the molecular dynamics simulations on deeply supercritical Ar. The presented results unveil the mechanism and regimes of sound propagation in the liquid matter and provide compelling evidence for the adiabatic-to-isothermal longitudinal sound propagation transition. We introduce a Hamiltonian predicting low-frequency transverse sound propagation gaps, which is confirmed by experimental findings and molecular dynamics calculations. As a result, a universal link is established between the positive sound dispersion (PSD) phenomenon and the origin of transverse sound propagation revealing the viscous-to-elastic crossover in liquids. The PSD and transverse phononic excitations evolve consistently with theoretical predictions. Both can be considered as a universal fingerprint of the dynamic response of a liquid, which is also observable in a subdomain of supercritical phase. The simultaneous disappearance of both these effects at elevated temperatures is a manifestation of the Frenkel line. We expect that these findings will advance the current understanding of fluids under extreme thermodynamic conditions., Comment: 18 pages, 5 figures

We review recent work aimed at understanding dynamical and thermodynamic properties of liquids and supercritical fluids. The focus of our discussion is on solid-like transverse collective modes, whose evolution in the supercritical fluids enables one to discuss the main properties of the Frenkel line separating rigid liquid-like and non-rigid gas-like supercritical states. We subsequently present recent experimental evidence of the Frenkel line showing that structural and dynamical crossovers are seen at a pressure and temperature corresponding to the line as predicted by theory and modelling. Finally, we link dynamical and thermodynamic properties of liquids and supercritical fluids by the new calculation of liquid energy governed by the evolution of solid-like transverse modes. The disappearance of those modes at high temperature results in the observed decrease of heat capacity.

Thermal density fluctuations of supercritical CO_{2} were explored using small-angle neutron scattering (SANS), whose amplitude (susceptibility) and correlation length show the expected maximum at the Widom line. At low pressure, the susceptibility is in excellent agreement with the evaluated values on the basis of mass density measurements. At about 20 bar beyond the Widom line, SANS shows the formation of droplets accompanied by an enhanced number density of the supercritical fluid. The corresponding borderline is interpreted as a Frenkel line separating gas- and liquidlike regimes.

We have conducted Raman spectroscopy\ud experiments on liquid ethane (C2H6) at 300 K, obtaining a\ud large amount of data at very high resolution. This has enabled\ud the observation of Raman peaks expected but not previously\ud observed in liquid ethane and a detailed experimental study of\ud the liquid that was not previously possible. We have observed a\ud transition between rigid and nonrigid liquid states in liquid\ud ethane at ca. 250 MPa corresponding to the recently proposed\ud Frenkel line, a dynamic transition between rigid liquid\ud (liquidlike) and nonrigid liquid (gaslike) states beginning in\ud the subcritical region and extending to arbitrarily high pressure\ud and temperature. The observation of this transition in liquid\ud (subcritical) ethane allows a clear differentiation to be made\ud between the Frenkel line (beginning in the subcritical region at\ud higher density than the boiling line) and the Widom lines (emanating from the critical point and not existing in the subcritical\ud region). Furthermore, we observe a narrow transition at ca. 1000 MPa to a second rigid liquid state. We propose that this\ud corresponds to a state in which orientational order must exist to achieve the expected density and can view the transition in\ud analogy to the transition in the solid state away from the orientationally disordered phase I to the orientationally ordered phases\ud II and III.

In a previous letter and other papers, we have interpreted the supercritical region of fluids as being parted into a gas-like and a liquid-like regimes. The boundary between these two distinct regions of the P-T phase diagram of substances is represented by the Widom line, extending the subcritical liquid-gas 1st order transition line above the critical point. Out authors have been commenting on our interpretation, claiming that the two regions are parted by the so called Frenkel line, which does not reconnect to the subcritical liquid-gas phase boundary. In this work, we show on a solid theoretical basis that the Frenkel line is incorrect and that indeed the Widom line is the relevant boundary between to two supercritical regimes.

The dynamics of supercritical fluids, a state of matter beyond the gas-liquid critical point, changes from diffusive to oscillatory motions at high pressure. This transition is believed to occur across a locus of thermodynamic states called the Frenkel line. The Frenkel line has been extensively investigated from the viewpoint of the dynamics, but its structural meaning is not still well understood. This letter interprets the mesoscopic picture of the Frenkel line entirely based on a topological and geometrical framework. This discovery makes it possible to understand the mechanism of rigid/non-rigid transition based not on the dynamics of individual atoms, but on their instantaneous configurations. The topological classification method reveals that the percolation of solid-like structures occurs above the rigid-nonrigid crossover densities., Comment: 15 pages, 4 figures, and the Supporting Information

The phase diagram (pressure versus temperature) of the pure fluid is typically envisioned as being featureless apart from the presence of the liquid-vapor coexistence curve terminating at the critical point. However, a number of recent authors have proposed that this simple picture misses important features, such as the Widom line, the Fisher-Widom line, and the Frenkel line. In our paper we discuss another way of augmenting the pure fluid phase diagram, lines of zero thermodynamic curvature $R=0$ separating regimes of fluid solid-like behavior ($R>0$) from gas-like or liquid-like behavior ($R, Comment: 38 pages, 22 figures, 58 references

We report experimental evidence for a crossover between a liquid-like state and a gas-like state in fluid methane (CH4). This crossover is observed in all of our experiments, up to 397 K temperature; 2.1 times the critical temperature of methane. The crossover has been characterized with both Raman spectroscopy and X-ray diffraction in a number of separate experiments, and confirmed to be reversible. We associate this crossover with the Frenkel line - a recently hypothesized crossover in dynamic properties of fluids extending to arbitrarily high pressure and temperature, dividing the phase diagram into separate regions where the fluid possesses liquid-like and gas-like properties. On the liquid-like side the Raman-active vibration increases in frequency linearly as pressure is increased, as expected due to the repulsive interaction between adjacent molecules. On the gas-like side this competes with the attractive Van der Waal’s potential leading the vibration frequency to decrease as pressure is increased.

Collective excitations in hard-sphere fluids were studied in a wide range of wave numbers and packing fractions η by means of molecular dynamics simulations. We report the observation of non-hydrodynamic transverse excitations for packing fractions η≥0.395 in the shape of transverse current spectral functions. Dispersion of longitudinal excitations in the whole range of packing fractions shows a negative deviation from the linear hydrodynamic law with increasing wave numbers even for dense hard-sphere fluids where the transverse excitations were observed. These results do not support a recent proposal within the "Frenkel line" approach that the positive sound dispersion in liquids is defined by transverse excitations. We report calculations of the cutoff "Frenkel frequencies" for transverse excitations in hard-sphere fluids and discuss their consistency with the estimated dispersions of shear waves.

We study thermo-mechanical properties of matter at extreme conditions deep in the supercritical state, at temperatures exceeding the critical one up to four orders of magnitude. We calculate the Gr\"{u}neisen parameter {\gamma} and find that it decreases with temperature from 3 to 1 on isochores depending on the density. Our results indicate that from the perspective of thermo-mechanical properties, the supercritical state is characterized by the wide range of {\gamma} which includes the solid-like values - an interesting finding in view of the common perception of the supercritical state as being an intermediate state between gases and liquids. We rationalize this result by considering the relative weights of oscillatory and diffusive components of the supercritical system below the Frenkel line. We also find that {\gamma} is nearly constant at the Frenkel line above the critical point and explain this universality in terms of pressure and temperature scaling of system properties along the lines where particle dynamics changes qualitatively.

Investigation of excitation spectra of liquids is one of the hot test topics nowadays. In particular, recent experimental works showed that liquid metals can demonstrate transverse excitations and positive sound dispersion. However, the theoretical description of these experimental observations is still missing. Here we report a molecular dynamics study of excitation spectra of liquid iron. We compare the results with available experimental data to justify the method. After that we perform calculations for high temperatures to find the location of the Frenkel line introduced in our previous works.

The existence of the Frenkel line in the supercritical regime of a Lennard-Jones (LJ) fluid shown through molecular dynamics (MD) simulations initially and later corroborated by experiments on argon opens up possibilities of understanding the structure and dynamics of supercritical fluids in general and of the Frenkel line in particular. The location of the Frenkel line, which demarcates two distinct physical states, liquidlike and gaslike within the supercritical regime, has been established through MD simulations of the velocity autocorrelation (VACF) and radial distribution function (RDF). We, in this article, explore the changes in the structural features of supercritical LJ fluid under partial confinement using atomistic walls. The study is carried out across the Frenkel line through a series of MD simulations considering a set of thermodynamics states in the supercritical regime ($P=5000$ bar, $240\phantom{\rule{0.28em}{0ex}}\mathrm{K}\ensuremath{\le}T\ensuremath{\le}1500\phantom{\rule{0.28em}{0ex}}\mathrm{K}$) of argon well above the critical point. Confinement is partial, with atomistic walls located normal to $z$ and extending to ``infinity'' along the $x$ and $y$ directions. In the ``liquidlike'' regime of the supercritical phase, particles are found to be distributed in distinct layers along the $z$ axis with layer spacing less than one atomic diameter and the lateral RDF showing amorphous-like structure for specific spacings (packing frustration) and non-amorphous-like structure for other spacings. Increasing the rigidity of the atomistic walls is found to lead to stronger layering and increased structural order. For confinement with reflective walls, layers are found to form with one atomic diameter spacing and the lateral RDF showing close-packed structure for the smaller confinements. Translational order parameter and excess entropy assessment confirms the ordering taking place for atomistic wall and reflective wall confinements. In the ``gaslike'' regime of the supercritical phase, particle distribution along the spacing and the lateral RDF exhibit features not significantly different from that due to normal gas regime. The heterogeneity across the Frenkel line, found to be present both in bulk and confined systems, might cause the breakdown of the universal scaling between structure and dynamics of fluids necessitating the determination of a unique relationship between them.

As known, the isochoric heat capacity C v of many atomic and molecular substances in liquid and supercritical fluid states decreases with increasing temperature at high temperatures. It is shown that: such a behavior of C v is related to a decrease of the contribution of interactions of atoms with each other to C v with increasing temperature; such a behavior of C v takes place for atomic substances when the fluctuations of the total kinetic energy increase with increasing temperature faster than that of the total potential energy; such a behavior of C v can be described by the density independent radial distribution function of non-ideal dilute gas consisting of atoms interacting with each other via an additive potential which is equal to the sum of a spherically symmetrical pair interaction potentials over of all pairs of particles; the pair potential can be equal to: the potential of soft spheres, bounded potential, non-positive potential, the sum of the potential of hard spheres and bounded or non-positive or attractive potential, and the sum of the repulsive potential of soft spheres and attractive London potential; the above conclusions are valid for molecular substances if the isochoric heat capacity of the molecular ideal gas does not depend on temperature; the radial distribution function of non-ideal dilute gas can describe the above behavior of C v for argon in liquid and supercritical states; the Carnahan–Starling equation of state for the hard spheres with temperature dependent diameter gives a good quantitative description of the isochoric heat capacity of argon if the diameter is defined using the Lennard-Jones potential. The equation to define the Frenkel line of molecular substances on the (temperature, density)-plane is established. The explicit expressions to define the Frenkel line are derived. It is shown that the existence of the Frenkel line established from the condition C v − C v , i g = k ∕ 2 , where C v , i g is the isochoric heat capacity of an ideal gas, does not mean that the solid-like and liquid-like states exist in liquids, and the transition from C v > 2 k to C v 2 k across the Frenkel line is the transition from the solid-like states to liquid-like ones.

At present, supercritical technologies use fluids (for example, CO2, H2O) at temperatures and pressures close to critical. It is not currently clear how important critical fluctuations are for dissolution processes and chemical reactions in supercritical fluids. At the same time, our recent work has shown that qualitative changes in the system behavior occurred also at temperatures and pressures much higher than critical. This paper briefly summarizes the latest results of the search and study of the parameters P and T of the changes of dynamic types in supercritical fluids. It turned out that this region of parameters corresponded to a narrow band, namely, the “Frenkel line.” Reaching this line with increasing temperature corresponds to the disappearance of shear excitations in the liquid at all frequencies. The Frenkel line can be considered as a boundary between the liquid and the dense gas at extremely high pressures far from the critical point. Recently, a simple and mathematically rigorous method for finding the Frenkel line was proposed based on analysis of the autocorrelation function of particle velocities. We suggest using the data of the Frenkel line parameters for actual fluids to move supercritical technologies into the region of extremely high pressures exceeding critical ones by tens and hundreds of times.

We discuss the fluid state of matter at high temperature and pressure. We review the existing ways in which the boundary between a liquid and a quasigas fluid above the critical point are discussed. We show that the proposed 'thermodynamic' continuation of the boiling line, the 'Widom line', exists as a line near the critical point only, but becomes a bunch of short lines at a higher temperature. We subsequently propose a new 'dynamic' line separating a liquid and a gas-like fluid. The dynamic line is related to different types of particle trajectories and different diffusion mechanisms in liquids and dense gases. The location of the line on the phase diagram is determined by the equality of the liquid relaxation time and the minimal period of transverse acoustic excitations. Crossing the line results in the disappearance of transverse waves at all frequencies, the diffusion coefficient acquiring a value close to that at the critical point, the speed of sound becoming twice the particle thermal speed, and the specific heat reaching 2. In the high-pressure limit, the temperature on the dynamic line depends on pressure in the same way as does the melting temperature. In contrast to the Widom line, the proposed dynamic line separates liquid and gas-like fluids above the critical point at arbitrarily high pressure and temperature. We propose calling the new dynamic line the 'Frenkel line'.

The locus of extrema (ridges) for heat capacity, thermal expansion coefficient, compressibility, and density fluctuations for model particle systems with Lennard-Jones (LJ) potential in the supercritical region have been obtained. It was found that the ridges for different thermodynamic values virtually merge into a single Widom line at T < 1.1T(c) and P < 1.5P(c) and become practically completely smeared at T < 2.5T(c) and P < 10P(c), where T(c) and P(c) are the critical temperature and pressure. The ridge for heat capacity approaches close to critical isochore, whereas the lines of extrema for other values correspond to density decrease. The lines corresponding to the supercritical maxima for argon and neon are in good agreement with the computer simulation data for LJ fluid. The behavior of the ridges for LJ fluid, in turn, is close to that for the supercritical van der Waals fluid, which is indicative of a fairly universal behavior of the Widom line for a liquid-gas transition.

The interplay between the structure and dynamics of partially confined Lennard Jones (LJ) fluids, deep into the supercritical phase, is studied over a wide range of densities in the context of the Frenkel line (FL), which separates rigid liquidlike and non-rigid gaslike regimes in the phase diagram of the supercritical fluids. Extensive molecular dynamics simulations carried out at the two ends of the FL (P = 5000 bars, T = 300 K, and T = 1500 K) reveal intriguing features in supercritical fluids as a function of stiffness of the partially confining atomistic walls. The liquidlike regime of a LJ fluid (P = 5000 bars, T = 300 K), mimicking argon, partially confined between walls separated by 10 {\AA} along the z-axis, and otherwise unconstrained, reveals amorphous and liquidlike structural signatures in the radial distribution function parallel to the walls and enhanced self-diffusion as the wall stiffness is decreased. In sharp contrast, in the gas-like regime (P = 5000 bars, T = 1500 K), soft walls lead to increasing structural order hindering self-diffusion. Furthermore, the correlations between the structure and self-diffusion are found to be well captured by excess entropy. The rich behavior shown by supercritical fluids under partial confinement, even with simple interatomic potentials, is found to be fairly independent of hydrophilicity and hydrophobicity. The study identifies persisting sub-diffusive features over intermediate time scales, emerging from the strong interplay between density and confinement, to dictate the evolution and stabilization of structures. It is anticipated that these results may help gain a better understanding of the behavior of partially confined complex fluids found in nature.

Recent research suggests that the supercritical state consists of liquidlike and gaslike states where particle dynamics and key system properties are qualitatively different. We report experimental evidence of the structural crossover in supercritical neon at pressure and temperature conditions significantly exceeding the critical point values: 250$\,P_c$ and 6.6$\,T_c$. The experimental results show a crossover of the medium-range order structure evidenced by the change of the structure factor with pressure. We also observe the crossover of the short-range order structure indicated by changes in the coordination number. The relative width of the crossover is fairly narrow and is smaller than 10-12 \% in pressure and temperature. By comparing our experimental results with molecular dynamics simulations, we suggest that the observed crossover can be attributed to the Frenkel line and discuss the relationship between the structural crossover and qualitative changes of dynamical and thermodynamic properties of supercritical matter., Comment: 7 pages, 6 figures

Supercritical fluids play a significant role in elucidating fundamental aspects of liquid matter under extreme conditions. They have been extensively studied at pressures and temperatures relevant to various industrial applications. However, much less is known about the structural behaviour of supercritical fluids and no structural crossovers have been observed in static compression experiments in any temperature and pressure ranges beyond the critical point. The structure of supercritical state is currently perceived to be uniform everywhere on the pressure-temperature phase diagram, and to change only in a monotonic way even moving around the critical point, not only along isotherms or isobars. Conversely, we observe structural crossovers for the first time in a deeply supercritical sample through diffraction measurements in a diamond anvil cell and discover a new thermodynamic boundary on the pressure-temperature diagram. We explain the existence of these crossovers in the framework of the phonon theory of liquids using molecular dynamics simulations. The obtained results are of prime importance since it implies a global reconsideration of the mere essence of the supercritical phase. Furthermore, this discovery may pave the way to new unexpected applications and to the exploration of exotic behaviour of confined fluids relevant to geo- and planetary sciences., 11 pages, 5 figures

International audience; Positively charged ions were produced in supercritical helium at temperatures from 6 to 10 K and up to 2 MPa using a corona discharge. Their mobility was measured via current–voltage curves, and the hydrodynamic radius was derived using Stokes law. An initial increase and subsequent decrease of hydrodynamic radius was observed and interpreted in terms of growth, compression and solidification of ion clusters. The mobility was modeled using a van der Waals-type thermodynamic state equation for the ion-in-helium mixed system and a temperature-dependent Millikan-Cunningham factor, describing experimental data both in the Knudsen and the Stokes flow region. Regions of maximum hydrodynamic radius and large compressibility were interpreted as boiling points. These points were modeled over a large range of pressures and found to match the Frenkel line of pure helium up to 0.7 MPa, reflecting similarity of density fluctuations in pure supercritical helium and gas–liquid phase transitions of ionic helium clusters.

In this paper we present inelastic X-ray scattering experiments in a diamond anvil cell and molecular dynamic simulations to investigate the behavior of phononic excitations in liquid Ar. The spectra calculated using molecular dynamics were found to be in a good agreement with the experimental data. Furthermore, we observe that, upon temperature increases, a low-frequency transverse phononic gap emerges while high-frequency propagating modes become evanescent at the THz scale. The effect of strong localization of a longitudinal phononic mode in the supercritical phase is observed for the first time. The evidence for the high-frequency transverse phononic gap due to the transition from an oscillatory to a ballistic dynamic regimes of motion is presented and supported by molecular dynamics simulations. This transition takes place across the Frenkel line thermodynamic limit which demarcates compressed liquid and non-compressed fluid domains on the phase diagram and is supported by calculations within the Green-Kubo phenomenological formalism. These results are crucial to advance the development of novel terahertz thermal devices, phononic lenses, mirrors, and other THz metamaterials., 8 pages, 5 figures. arXiv admin note: text overlap with arXiv:1512.07202