Supercritical Mesophase Research Articles

Global transformation of fluid structure and corresponding phase behavior

The laws governing the fluid structure transformation from a homogeneous state to the special state in the critical region remain intriguing. In addition, there is a drastic change in the supercritical fluid from a liquid-like type to gas-like when crossing the Widom line, which is still highly debatable. Such challenges require new insights to understand the fluid nature in its entirety. The authors have developed a Two-Phase Hierarchical Model (TPHM), which represents the fluid by a system of various size blocks nested in each other. The TPHM shows that supercritical fluids within a certain region of thermodynamic parameters exist in a form of liquid-like and gas-like clusters (supercritical mesophase) and demonstrates the formation of a hierarchical structure, most pronounced near the critical point. Based on the TPHM, a method for correcting the phase behavior in the critical region is presented, applicable both to pure substances and to multi-component mixtures.

Hypotheses in phase transition theories: “What is ‘liquid’?”

Theories predicting thermodynamic properties that describe liquid phase transitions and critical phenomena have resulted in the award of three Nobel prizes in physics: (i) “Continuity of Gaseous and Liquid States” hypothesis of van der Waals [1910], (ii) “Critical Point Universality” hypothesis embodied in the renormalization group (RG) theory of Wilson [1982], and (iii) “Topological Defect Melting” hypothesis that 2D-crystal-liquid states exhibit ‘hexatic’ phases in KTHNY theory [Kosterlitz et al. 2016]. All three hypotheses are invalidated by the reality of experimental results and raise a fundamental question first posed by Barker and Henderson in 1976: “What is liquid”. A single Gibbs phase, that includes triple-point (Tt) liquid, extends over the whole fluid density range to temperatures above the Boyle temperature (TB). Below TB, above the critical temperature Tc, predominantly gas- and liquid-like states are bounded by a narrow colloidal ‘supercritical mesophase’ with constant rigidity (ω = (dp/dρ)T). The liquid phase also becomes colloidal at the onset of pre-freezing growth and percolation of crystallites in a narrow density range below freezing density for all T > Tt. Whereas the Boyle line (RT = p/ρ) defines a crystalline ground state, a rigidity line, RT = ω, interpolates to an amorphous ground-state akin to random close packing (RCP) at T = 0. All states of gas, liquid, and crystals, are present in the stable ‘liquid phase’ and, are represented in thermodynamic p-T states all along the rigidity line. For 2D liquid–crystal coexistence in constrained computer models, the KTHNY theory describes a non-equilibrium fracture process. Hetero-phase fluctuations, leading to percolation transitions, have been misconstrued as “hexatic” in 2D, as also have 2-phase coexistence states, that are homogeneous in the absence of gravity.

Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results.

We review the experimental evidence, from both historic and modern literature of thermodynamic properties, for the non-existence of a critical-point singularity on Gibbs density surface, for the existence of a critical density hiatus line between 2-phase coexistence, for a supercritical mesophase with the colloidal characteristics of a one-component 2-state phase, and for the percolation loci that bound the existence of gaseous and liquid states. An absence of any critical-point singularity is supported by an overwhelming body of experimental evidence dating back to the original pressure-volume-temperature (p-V-T) equation-of-state measurements of CO2 by Andrews in 1863, and extending to the present NIST-2019 Thermo-physical Properties data bank of more than 200 fluids. Historic heat capacity measurements in the 1960s that gave rise to the concept of “universality” are revisited. The only experimental evidence cited by the original protagonists of the van der Waals hypothesis, and universality theorists, is a misinterpretation of the isochoric heat capacity Cv. We conclude that the body of extensive scientific experimental evidence has never supported the Andrews–van der Waals theory of continuity of liquid and gas, or the existence of a singular critical point with universal scaling properties. All available thermodynamic experimental data, including modern computer experiments, are compatible with a critical divide at Tc, defined by the intersection of two percolation loci at gaseous and liquid phase bounds, and the existence of a colloid-like supercritical mesophase comprising both gaseous and liquid states.

On the Empirical Determination of a Gas–Liquid Supercritical Mesophase and its Phenomenological Definition

We respond to recent articles (Int. J. Thermophysics 39 139 2018, ibid.40 21 2019) that seek to deny the veracity of the empirical discovery and thermodynamic description of a gas–liquid supercritical mesophase. These IJT articles are misleading because they are based upon a false premise that the mesophase is a hypothetical concept. There is no “mesophase hypothesis” as wrongly stated in the title of article IJT, 40 21, 2019. Unlike the critical point continuity hypothesis of van der Waals (1873), or the universal critical point scaling hypothesis advocated by Anisimov and Sengers, and others since 1965, the supercritical mesophase is an empirical entity. If a hypothetical van der Waals fluid with the properties of a critical-state point with divergent isochoric heat capacity, and with the scaling properties of “universality” theory, were to exist, it could be used to vitiate the second law of thermodynamics. By contrast, the supercritical mesophase, discovered originally from computer experiments, is an empirically established equilibrium fluid region, of two-state systems of gas plus liquid, within a single Gibbs phase; the properties of which are determined by the laws of thermodynamics. Here, we provide a phenomenological definition of the gas–liquid supercritical mesophase that accords with the body of compelling experimental results over a 150-year period since van der Waals including modern computer experiments.

Thermodynamics of Gas–Liquid Colloidal Equilibrium States: Hetero-Phase Fluctuations

Following on from two previous JETC (Joint European Thermodynamics Conference) presentations, we present a preliminary report of further advances towards the thermodynamic description of critical behavior and a supercritical gas-liquid coexistence with a supercritical fluid mesophase defined by percolation loci. The experimental data along supercritical constant temperature isotherms (T ≥ Tc) are consistent with the existence of a two-state mesophase, with constant change in pressure with density, rigidity, (dp/dρ) T, and linear thermodynamic state-functions of density. The supercritical mesophase is bounded by 3rd-order phase transitions at percolation thresholds. Here we present the evidence that these percolation transitions of both gaseous and liquid states along any isotherm are preceded by pre-percolation hetero-phase fluctuations that can explain the thermodynamic properties in the mesophase and its vicinity. Hetero-phase fluctuations give rise to one-component colloidal-dispersion states; a single Gibbs phase retaining 2 degrees of freedom in which both gas and liquid states with different densities percolate the phase volume. In order to describe the thermodynamic properties of two-state critical and supercritical coexistence, we introduce the concept of a hypothetical homo-phase of both gas and liquid, defined as extrapolated equilibrium states in the pre-percolation vicinity, with the hetero-phase fractions subtracted. We observe that there can be no difference in chemical potential between homo-phase liquid and gaseous states along the critical isotherm in mid-critical isochoric experiments when the meniscus disappears at T = Tc. For T > Tc, thermodynamic states comprise equal mole fractions of the homo-phase gas and liquid, both percolating the total phase volume, at the same temperature, pressure, and with a uniform chemical potential, stabilised by a positive finite interfacial surface tension.

Physical-Constant Equations-of-State for Argon Isotherms

Using argon as the exemplary fluid, we report a representation of equations-of-state in fluid regions bounded by percolation loci with mainly physical constants that define state bounds and characterize the fluid phase diagram. Both gas- and liquid-state pressures can be represented by 3- or 4-term virial expansions. Gaseous states require only known virial coefficients and physical constants belonging to the fluid, i.e., Boyle temperature (TB), Tc, pc and coexisting densities of gas (ρcG) and liquid (ρcL) at Tc. A notable finding is that for isotherms below TB, the contribution of the fourth virial term is negligibly small within experimental uncertainty. In the supercritical mesophase, both gaseous and liquid states percolate the configurational phase volume, whereupon state functions of the density obey a linear combination equation-of-state in this region. We also find evidence for the percolation line that bounds the low-temperature existence of the pure liquid argon state giving rise to an equilibrium prefreezing mesophase. In this region, which probably exists for all liquids, there is a colloidal dispersion of the metastable crystalline structure with an energy density closest to that of the liquid, wherein coexisting amorphous and crystalline states both percolate the phase volume. We report equations-of-state for isotherms of argon, over the whole equilibrium fluid range; i.e., for four regions, gas, supercritical mesophase, liquid, and a liquid + bcc prefreezing and metastable fluid mesophase up to a glass transition pressure. We compare with experiment using the Tegeler–Span–Wagner equation via the NIST fluid thermophysical property database.

Probabilistic characterization of the Widom delta in supercritical region.

We present a probabilistic classification algorithm to understand the structural transition of supercritical Lennard-Jones (LJ) fluid. The classification algorithm is designed based on the exploratory data analysis on the nearest Voronoi neighbors of subcritical vapor and liquid. The algorithm is tested and applied to LJ type fluids modeled with the truncated and shifted potential and the Weeks-Chandler-Andersen potential. The algorithm makes it available to locate the Widom delta, which encloses the supercritical gas-liquid boundary and the percolation transition loci in a geometrical manner, and to conjecture the role of attractive interactions on the structural transition of supercritical fluids. Thus, the designed algorithm offers an efficient and comprehensible method to understand the phase behavior of a supercritical mesophase.

Thermodynamic Fluid Equations-of-State.

As experimental measurements of thermodynamic properties have improved in accuracy, to five or six figures, over the decades, cubic equations that are widely used for modern thermodynamic fluid property data banks require ever-increasing numbers of terms with more fitted parameters. Functional forms with continuity for Gibbs density surface ρ(p,T) which accommodate a critical-point singularity are fundamentally inappropriate in the vicinity of the critical temperature (Tc) and pressure (pc) and in the supercritical density mid-range between gas- and liquid-like states. A mesophase, confined within percolation transition loci that bound the gas- and liquid-state by third-order discontinuities in derivatives of the Gibbs energy, has been identified. There is no critical-point singularity at Tc on Gibbs density surface and no continuity of gas and liquid. When appropriate functional forms are used for each state separately, we find that the mesophase pressure functions are linear. The negative and positive deviations, for both gas and liquid states, on either side of the mesophase, are accurately represented by three or four-term virial expansions. All gaseous states require only known virial coefficients, and physical constants belonging to the fluid, i.e., Boyle temperature (TB), critical temperature (Tc), critical pressure (pc) and coexisting densities of gas (ρcG) and liquid (ρcL) along the critical isotherm. A notable finding for simple fluids is that for all gaseous states below TB, the contribution of the fourth virial term is negligible within experimental uncertainty. Use may be made of a symmetry between gas and liquid states in the state function rigidity (dp/dρ)T to specify lower-order liquid-state coefficients. Preliminary results for selected isotherms and isochores are presented for the exemplary fluids, CO2, argon, water and SF6, with focus on the supercritical mesophase and critical region.

Water dispersions of natural graphene based carbon nanoparticles: ESR spin probe study

Dynamic and phase behavior of an aqueous dispersion of shungite carbon (ShC) nanoparticles, one of natural graphene-like nanomaterials structured in a special way in an aqueous medium, is of considerable interest, as in the case of other carbon nanomaterials, due to its effects and presumable technological and biomedical applications. ShC nanoparticles in aqueous dispersions of different composition were studied in the temperature range 7–60°C using electron spin resonance (ESR) technique of stable spin probes. Hydrophobic probe 5-DOXYL stearic acid was introduced into the structure of hydrated ShC nanoparticles. Hydrophilic probe 4-oxo-TEMPO was dissolved in the surrounding bulk solvent. It is assumed that an increase in the immobilization of the hydrophobic probe in the presence of NaCl, urea, HCl, NaOH, sucrose is associated with a change in the state of hydration of ShC nanoparticles. At the same time, an increase in the carbon concentration during the evaporation of water did not cause changes in the dynamics of the probe until the nanoparticles were dehydrated. Changes in hydration of the hydrophobic probe on the ShC nanoparticles at a temperature of about 17–20°C can be associated with a change in the state of nanoparticle dispersion near the critical mixing temperature of the main and metastable phases. The sigmoid type Arrhenius dependence of the mobility of both the hydrophilic and hydrophobic probes has maximum steepness in the region of the supercritical mesophase near 37°C. The structural and dynamic variations of the dispersion associated with the change in the dynamic regime in the supercritical mesophase near the Frenkel line are discussed as a probable explanation. The data obtained are interpreted on the basis of a hypothetical phase diagram of the dispersion of fullerenes and ShC nanostructures.

Percolation Transitions of the Ideal Gas and Supercritical Mesophase

High-temperature and pressure boundaries of the liquid and gaseous states have not been defined thermodynamically. Standard liquid-state physics texts use either critical isotherms or isobars as ad hoc boundaries in phase diagrams. Here we report that percolation transition loci can define liquid and gas states, extending from super-critical temperatures or pressures to “ideal gas” states. Using computational methodology described previously we present results for the thermodynamic states at which clusters of excluded volume (VE) and pockets of available volume (VA), for a spherical molecule diameter σ, percolate the whole volume (V = VE VA) of the ideal gas. The molecular-reduced temperature (T)/pressure (p) ratios (T* = kBT/pσ3) for the percolation transitions are T*PE = 1.495 ± 0.01 and T*PA = 1.100 ± 0.01. Further MD computations of percolation loci for the Widom-Rowlinson (W-R) model of a partially miscible binary liquid (A-B) show the connection between the ideal gas percolation transitions and the 1st-order phase-separation transition. A phase diagram for the penetrable cohesive sphere (PCS) model of a one-component liquid-gas is then obtained by analytic transcription of the W-R model thermodynamic properties. The PCS percolation loci extend from a critical coexistence of gas plus liquid to the low-density limit ideal gas. Extended percolation loci for argon, determined from literature equation-of-state measurements exhibit similar phenomena. When percolation loci define phase bounds, the liquid phase spans the whole density range, whereas the gas phase is confined by its percolation boundary within an area of low T and p on the density surface. This is contrary to a general perception, and reopens a debate of “what is liquid”. We append this contribution to the science of liquid-gas criticality and liquid-state bounds with further open debate.

Gibbs Density Surface of Water and Steam: 2nd Debate on the Absence of Van Der Waals’ “Critical Point”

A revised phase diagram for water shows three distinct fluid phases. There is no continuity of liquid and gas, and no “critical point” on Gibbs’ density surface as hypothesized by van der Waals. A supercritical colloidal mesophase bounded by percolation transition loci separates supercritical liquid water and gas-phase steam. The water phase is bounded by a percolation transition (PA) of available volume, whereas steam is bounded by the loci of a percolation transition (PB) at a density whereupon a bonded molecular cluster suddenly percolates large distances. At the respective percolation densities, there is no barrier to nucleation of water to steam (PA) or steam to water (PB). Below the critical temperature, the percolation loci become the metastable spinodals in the two-phase coexistence region. A critical divide is defined by the interception of PA and PB the p-T plane. Critical parameters are obtainable from slopes and intercepts of pressure-density supercritical isotherms within the mesophase. The supercritical mesophase is a fourth equilibrium state besides ice, water and steam. A thermodynamic state function rigidity (dp/dρ)T defines a distinction between liquid and gas, and shows a remarkable symmetry due to an equivalence in number density fluctuations, arising from available volume and molecular clusters, in liquid and gas respectively. Following an earlier debate in these pages [“Fluid phases of argon: A debate on the absence of van der Waals’ critical point” Natural Science 5 (2) 194-206 (2013)], we here report further debate on a science of criticality applied to water and steam (APPENDIX 1).

Nature of the Supercritical Mesophase

It has been reported that at temperatures above the critical there is no “continuity of liquid and gas”, as originally hypothesized by van der Waals [1]. Rather, both gas and liquid phases, with characteristic properties as such, extend to supercritical temperatures [2]-[4]. Each phase is bounded by the locus of a percolation transition, i.e. a higher-order thermodynamic phase change associated with percolation of gas clusters in a large void, or liquid interstitial vacancies in a large cluster. Between these two-phase bounds, it is reported there exists a mesophase that resembles an otherwise homogeneous dispersion of gas micro-bubbles in liquid (foam) and a dispersion of liquid micro-droplets in gas (mist). Such a colloidal-like state of a pure one-component fluid represents a hitherto unchartered equilibrium state of matter besides pure solid, liquid or gas. Here we provide compelling evidence, from molecular dynamics (MD) simulations, for the existence of this supercritical mesophase and its colloidal nature. We report preliminary results of computer simulations for a model fluid using a simplistic representation of atoms or molecules, i.e. a hard-core repulsion with an attraction so short that the atoms are referred to as “adhesive spheres”. Molecular clusters, and hence percolation transitions, are unambiguously defined. Graphics of color-coded clusters show colloidal characteristics of the supercritical mesophase. We append this Letter to Natural Science with a debate on the scientific merits of its content courtesy of correspondence with Nature (Appendix).

Critical and supercritical properties of Lennard–Jones fluids

Critical properties for Lennard–Jones fluids are seen to be consistent with an alternative description of liquid–gas criticality to the van der Waals hypothesis. At the critical temperature (Tc) there is a critical dividing line on the Gibbs density surface rather than a critical point singularity. We report high-precision thermodynamic pressures from MD simulations for more than 2000 state points along 7 near-critical isotherms, for system sizes N=4096 and 10,976 for a Lennard–Jones fluid. We obtain kBTc/ɛ=1.3365±0.0005 and critical pressure pcσ3/ɛ=0.1405±0.0002 which remains constant between two coexisting densities ρc(gas) σ3=0.266±0.01 and ρc(liquid) σ3=0.376±0.01 determined by supercritical percolation transition loci. A revised plot of the liquid–vapor surface tension shows that it goes to zero at this density difference, which, along with a direct evaluation of Gibbs chemical potential along the critical isotherm, reaffirms a coexisting dividing line of critical states. Analysis of the distribution of clusters from MD simulations along a supercritical isotherm (T*=1.5) gives new insight into the supercritical boundaries of gas and liquid phases, and the nature of the supercritical mesophase.

Thermodynamic status of random close packing

An amorphous ground state reminiscent of random close packing (RCP) of spheres terminates a liquid phase that spans all temperatures. On the Gibbs density surface, the liquid phase has bounded by a supercritical percolation line, two-phase liquid–gas coexistence line, and below the triple point, the metastable liquid branch terminating at T = 0 K with the RCP state. There is no “continuity of gas and liquid phases”; they are separated by a supercritical mesophase bounded by percolation transitions. As a consequence, the gas phase cannot be a starting point for liquid-state theory. RCP has a thermodynamic status. For square-well model fluids, evidence from computer simulations shows that the amorphous ground state is the same RCP state of the hard-sphere model. The RCP limiting density state of the hard-sphere fluid, with its reproducible and well-characterized structure, can be obtained by well-defined irreversible and reversible processes which establish a thermodynamic status. Empirical results, within margins of numerical uncertainty, are that the amorphous ground state density corresponds to a packing fraction = 0.6366 ± 0.0005 (Buffon’s constant 2/π) and a residual entropy per sphere ΔS(0) ∼ kB (Boltzmann constant). We conclude that RCP of spheres is a well-defined state with a thermodynamic status, and a central role in the description of liquid phase equilibria, as originally suggested, 50 years ago, by J.D. Bernal. We postulate that the metastable RCP state is the starting point for a modern theory of the liquid state.

Gibbs Density Surface of Fluid Argon: Revised Critical Parameters

A phase diagram of simple fluids, based upon percolation transition loci determined from literature experimental density-pressure isotherms of argon, shows three fluid phases. The liquid phase spans all temperatures from a metastable amorphous ground state to supercritical temperatures. There is a supercritical mesophase of a colloidal nature between two percolation transition loci, PA and PB, that bound the existences of liquid and gas phases, respectively. There is no ‘critical point’ on the Gibbs density surface. When two percolation transitions have the same pressure, i.e., on intersection of the loci in the \(p\)–\(T\) plane, an equilibrium ‘line of critical states’ is thermodynamically defined. This description of criticality yields slightly different values of critical parameters than those hitherto obtained by interpolation of experimental \(p\)–\(\rho \)–\(T\) data from the coexistence region. Revised values of the critical temperature (\(T_\mathrm{c})\) and critical pressure (\(p_\mathrm{c})\) of argon can be obtained from the original high precision thermodynamic measurements for fluid argon, using only data from one-phase supercritical isotherms. Values of coexisting densities at \(T_\mathrm{c}\) and the density loci of the percolation transitions that bound the existence of supercritical gas and liquid phases for \(T>T_\mathrm{c}\) are also reported. Percolation loci, for \(T<T_\mathrm{c}\), extend into the gas–liquid coexistence region to become the spinodal lines that limit the existence of metastable gas and liquid phases.

Fluid phases of argon: A debate on the absence of van der Waals’ “critical point”

A phase diagram of argon based upon percolation transition loci determined from literature experimental p-V isotherms, and simulation values using a Lennard-Jones model shows three fluid phases. The liquid phase spans all temperatures, from a metastable amorphous ground state at 0K, to ultra-high T. There is a supercritical mesophase bounded by percolation transition loci, and a gas phase. Intersection of two percolation loci in the p-T plane thermodynamically defines a critical line between two coexisting gas and liquid critical states at T = Tc, and the single mesophase for T > Tc. A debate on the absence of a van der Waals critical point in the Gibbs p-T density surface is appended.

Observations of a thermodynamic liquid–gas critical coexistence line and supercritical fluid phase bounds from percolation transition loci

We extend previous investigations into the thermodynamics of liquid state boundaries by focusing on the origins of liquid–gas criticality. The singular point hypothesis of van der Waals is re-examined in the light of recent knowledge of the hard-sphere percolation transitions and further analysis of simulation results for the supercritical properties of the square-well fluids. We find a thermodynamic description of gas–liquid criticality that is quite different from both van der Waals hypothesis and modern mean-field theory. At the critical temperature (Tc) and critical pressure (pc), in the density surface ρ(p,T), there is no critical point. Using tabulations of experimental ρ(p,T) data, for supercritical argon, and also water, as examples, at Tc a liquid phase coexists with a vapor phase determined by percolation transition densities. In the ρ(p,T) surface, there is a line of critical coexistence states of constant chemical potential at the intersection of two percolation loci in the p–T plane. For temperatures above this line, there exists a supercritical mesophase bounded by percolation transition loci. Below the line of critical states there is the familiar subcritical liquid–vapor two-phase coexistence region. Unlike the hypothetical van der Waals critical point, all thermodynamic state points on the line of critical states are consistent with Gibbs phase rule.