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

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}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 C

_{v}. 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 T

_{c}, 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.

## 1. Introduction

_{c}, there is deemed to be a merging of the properties of the gas and liquid states in coexistence. At a critical temperature T

_{c}, Andrews proposed the hypothesis of “continuity of liquid and gas” [2]. He suggested that there is no distinction between gas and liquid at or above T

_{c}and hence definitive state bounds along any supercritical isotherm. The experimental pressure measurements for CO

_{2}, as a function of density and temperature, in the original paper by Andrews are shown in Figure 2a.

_{c}whereupon the first two derivatives of pressure with respect to changes in density or volume go to zero at a singular node on the Gibbs density surface. At this hypothetical point, the difference in density, and hence all other distinguishing properties, between gas and liquid are deemed to disappear. For nigh 150 years, Andrews–van der Waals theory of critical and supercritical continuity of gas and liquid has been the accepted and taught physics of liquid-gas criticality. The critical point envisaged by van der Waals, however, has never been defined thermodynamically, and has always remained an experimentally unsubstantiated hypothesis.

_{2}, we show in Figure 2b the same 6 isotherms taken from a modern experimental p-ρ-T thermodynamic fluid property databank [4]. Also shown are the coexistence curves and the law of rectilinear diameters (LRD) extended into the supercritical region [5]. A foremost observation is that, when the density is on a linear scale, there is a linear region to the supercritical isotherms centred upon the extended LRD. The slope ${\omega}_{T}={\left(\partial p/\partial \rho \right)}_{T}$, i.e., the thermodynamic state function rigidity, decreases with T at this midpoint, approaching zero at the critical temperature (T

_{c}= 29.9 °C).

_{c}), nobody has reported a direct observation of a liquid-gas critical density. This hypothetical property has only ever been defined and obtained by an a priori assumption of its existence by application of LRD [5], which seems to be illogical cyclic reasoning.

_{c}, T

_{c}), on Gibbs p-T surface, is the intersection of the two percolation loci, which bound the existence of pure gas and liquid states [6,7,8]. At this juncture, the two states, liquid and gas, have the same T, p and chemical potential, but slightly different densities, by about 20%.

## 2. Experimental Gas-Liquid Equilibria

#### 2.1. Computer Experiments

_{A}+ V

_{E}= V) percolate the phase volume. Although the HS fluid has a well-defined mesophase, it cannot account for gas-liquid condensation and 2-phase coexistence without an attractive potential term. The SW-model molecular Hamiltonian is defined by adding a constant energy of attraction (ε) of finite width (λ) to a HS (diameter σ) pair potential. The range must be finite, and not infinite, as implicit in the attractive term of van der Waals equation and similar mean-field theories, to give a fluid that complies with thermodynamic laws. The SW attraction introduces another percolation transition of cohesive gaseous clusters, the density of which depends upon the range of the attraction λ, as also shown in Figure 3. Properties of square-well fluids of various values of well-width λ have been reported from computer experiments. The simulation data shown in Figure 3 are taken from references [6,7,20,21,22,23,24,25]. These results from computer experiments are reproduced from the original reports of the existence of the supercritical mesophase [6,7].

_{c}) the two percolation transitions intersect, the pressures and hence the chemical potentials become equal. At the critical percolation intersection state point, T

_{c}-p

_{c}, there is a maximum coexisting gas-phase density, and a minimum coexisting liquid-phase density.

_{B}T/ε → infinity) for any value of well-width λ, defines the existence of a supercritical mesophase.

_{c}, the liquid percolation has the higher pressure, below T

_{c}the (metastable) liquid percolation loci, previously known as spinodal, have the lower pressure. At the intersection, the two states, gas and liquid, have the same T and p, and hence also the same μ (chemical potential), but different densities. Two coexisting states with constant T,p,μ at the percolation loci intersection define the critical point T

_{c}-p

_{c}in the T-p plane, and a line of critical transitions from 2-phase coexistence (P = 2, F =1) to a single phase (P =1, F = 2), for T > T

_{c}.

_{c}/ε = 1.3365 ± 0.0005 (where k is Boltzmann’s constant and ε is the attractive minimum energy of the L-J pair potential). Several supercritical isotherms were studied in more detail with a very high precision, including the kT/ε = 1.5 isotherm over the whole density range. Accurate structural information was obtained, including the radial distribution functions (RDFs) over the whole density range. Although individual state point RDFs are of very high statistical precision, they do not reveal any information about subtle structural changes that must inevitably accompany the onset of hetero-phase fluctuations, and the higher-order order percolation transitions. When the normalized pair probabilities g(r) at specific interatomic distance (r) are viewed as a function of density, however, a picture of the three supercritical regions, with the mesophase, emerges. The detailed density-dependent structural properties (Figure 5) show the unequivocal evidence that there exist three distinct structural state regions, liquid, meso and gas. The g(r,ρ) also reveals the evidence of the hetero-phase fluctuations that are the precursor states to the percolation transitions. The structural data reproduced in Figure 5 show that at all intermolecular distances from highly repulsive overlap to long-range distances of four times the most probable pair distance, there are extremely slight, but statistically significant, structural differences, between gas, liquid and meso, as evidenced by the subtle deviations from uniform probabilities as a function of density along a supercritical isotherm.

#### 2.2. Surface Tension

_{c}, when the percolation lines PA and PB intersect, there is no barrier to nucleation; hence the surface tension must go to zero at the different coexisting respective densities of gas and liquid. Evidence that this indeed happens can be found in a Monte Carlo computer calculation of the surface tension of Lennard–Jones fluids by Potoff and Panagiotopoulos [27]. In the original interpretation of their MC results, these authors overlooked the fact that the surface tension becomes zero at a finite density difference, having a priori assumed the van der Waals and universal singularity hypotheses to be established science.

_{c}. It varies with the cube of density difference between gas and liquid, and becomes zero, not at a van der Waals hypothetical critical point at which the density difference is zero, but at a finite density difference.

_{c}= 1.312, Δρ* = zero, and γ* is reported to be very slightly negative. Zero or negative surface tension would be consistent with the critical dividing line above which there is a mesophase of a colloidal nature. On the liquidus side, above T

_{c}, if the surface tension of a real fluid were to be zero or negative, there will be no barrier to the spontaneous fluctuations to gaseous states, which percolate the phase volume below ρ

_{PA}. Likewise, on the gas side, for densities above ρ

_{PB}, there will be no barrier to the spontaneous formation of percolating liquid-like droplets. Secondly, the data from reference [27] shows surface tension γ* varying linearly with T. We obtain from Table 1 in reference [27] T* = 2.5975γ* + 1.3093. It follows that the density difference ρ

^{*}

_{liq}– ρ

^{*}

_{gas}is a linear function of (T – T

_{c})

^{1/3}in the immediate vicinity of T

_{c}as seen in Figure 7.

_{s}= Aγ. For all subcritical gas-liquid coexisting states in the 2-phase region both G

_{s}and γ are positive, whence G

_{s}and therefore G

_{total}for the whole system is minimized by phase separation. For the equilibrium liquid in coexistence however, from the triple point to the critical point, the surface tension goes to zero. The difference between the rigidities of coexisting liquid and gaseous states (Δω = ω

_{liq}− ω

_{gas}) correlates with the surface tension (Figure 6). This can be explained: the difference in work required to increase the density of the coexisting states by adding molecules at constant volume is physically equivalent to the work required to create the interface by reducing the density of the liquid and simultaneously increasing the density of the vapour on either side of the Gibbs dividing surface; a line defined within the interface such that the total mass within an arbitrary volume containing the interface remains constant.

_{c}(Figure 7), the density difference of the coexisting homo-phases, both at T

_{c}and in the supercritical range T > T

_{c}, must remain finite. This suggests that the critical state of two coexisting homo-states within a single Gibbs phase extends well into the supercritical region, and probably all the way to the Boyle temperature as evidenced by the correlation of equation-of-state thermodynamic properties [28]. In the next section, we will re-assess the experimental evidence from historic laboratory measurements of near-critical and supercritical thermodynamic properties in the new light of the supercritical mesophase discovered in computer experiments.

#### 2.3. Mesophase Dispersion

_{c}can be found in the papers by Bradley et al. in 1908 [29]. Notwithstanding these results, in 1910 van der Waals gave his Nobel lecture [30] in which he ignored the Bradley et al. research with the declaration “at the critical volume the densities of gas and liquid become equal”. After exploring the effect of mechanical vibrations on the properties of near-critical CO

_{2}, however, Bradley et al. were able to state with some certainty that their experiments “proved” the existence of two phases far above the critical point. Moreover, their experimental observation that the meniscus disappears over a range of mean densities has never been properly explained. This reproducible experimental result, reviewed by Mills [31], was dismissed as an artefact of gravity, impurities or non-equilibrium states by the Dutch school in their argument in favour of verification of van der Waals hypothesis, and generally ignored by many of his disciples, but with the notable exception of Traube [32] who cited 10 experimental papers published prior to 1908 that vitiated the van der Waals hypothesis.

_{c}. Both liquid and vapour coexisting densities at T

_{c}are bounded by the respective percolation loci. At the temperature of intersection of the percolation loci (T

_{c}), the meniscus disappears within the tube over a certain range of mean densities, even when the intensive thermodynamic state functions, including the densities, of coexisting liquid and gas phases, are still unequal.

_{c}, Maass et al. measured large density differences and density gradients at fixed temperature and pressure. The description of their findings and even the titles of some of their publications, such as “Persistence of the Liquid State of Aggregation above the Critical Temperature” are inconsistent with the van der Waals hypothesis. Maass and co-workers acknowledged [35,36,37] that some of these effects had been seen before and gave due recognition to Traube and his co-researchers who had reached essentially the same conclusion [32].

_{c}.” These conclusions, from nearly a decade of painstaking research by Maass and his collaborators, are compelling evidence for the existence of a supercritical mesophase. In their last paper in the series Maass and Noldrett [39] write, quote: “Further evidence of the existence of a two-phase system above the temperature of meniscus disappearance as normally determined is presented. The existence of a latent heat of vaporization above this temperature is pointed out”. Finally, Maass et al. in the last articles [40,41] conclude “this is considered to be evidence for the formation of a dispersion of liquid and vapour before the critical temperature is reached”. Maass et al. suggested the nomenclature "critical dispersion temperature" instead of critical point [41].

#### 2.4. Density Hiatus

_{c}, T

_{c}and a mid-critical density ρ

_{c}defined by the law of rectilinear diameters. Likewise, any molecular-reduced corresponding states law also still applies, using molecular size, molecular mass and a characteristic interaction energy to define reduced time. The revised science of criticality does not change scaling effects of size or mass of a molecule, on thermodynamic state functions.

_{T}isotherms above T

_{c}were found to have a finite slope, but it was not possible to say whether the critical isotherm has a finite horizontal portion or "flat top". Following further experiments on xenon, Weinburger and Schneider [44] found a coexisting density gap but argued that the van der Waals theory can explain a large portion of the flat top if the effect of gravity is taken into account. Initially, Schneider and his co-authors had reported a flat top but also inferred, erroneously as we shall see below, that it was the consequence of gravity.

_{c}, to below the T

_{c}, and at densities starting well above the critical density. They concluded: “the critical isotherm is considerably flatter and broader over a range of densities than that corresponding to a van der Waals equation, and at the critical point, the third and fourth derivatives of pressure with respect to volume also appear to be zero”. This had been the conclusion of two independent theoretical predictions, just two years before Hapgood and Schneider’s paper, by Zimm [46] and by Harrison and Mayer [47]. Zimm found that all of the derivatives at T

_{c}are zero. Any state function for which all the derivatives go to zero is a horizontal line [45]. There followed theoretical papers by O.K. Rice [48,49] supporting the conclusions of Zimm, Mayer and Harrison of the existence of a density hiatus at T

_{c}.

#### 2.5. Percolation State Bounds

_{c}.

_{c}, both PA and PB (Figure 10) intersect in the p-T plane and cross the critical coexistence line, to become subcritical limits of existence of the metastable saturated compressed gas (p > p

_{sat}) and expanded saturated liquid phases (p < p

_{sat}), respectively. These percolation loci define the limits of existence of gas and liquid phases, not only above T

_{c}, i.e., in the supercritical region, but also in the subcritical metastable regions of existence of gas and liquid states.

^{3}. The mean of the two extreme recorded liquid and vapour densities is 536 kg/m

^{3}. The lowest co-existing liquid mass density they report is 602 kg/m

^{3}. The highest vapour mass density they can observe near T

_{c}is 470 kg/m

^{3}. The line of critical states connects these two points (Figure 9).

## 3. Thermophysical Property Compilations

#### 3.1. Multiparameter Equations-of-State

_{c}have also been incorrectly employed. Thus, even though no one has ever measured a vanishing density difference up to T

_{c}directly, all the NIST thermodynamic state functions have mistakenly presumed a continuity of gas and liquid to be the underlying science from the outset.

_{c}and in the supercritical mid-range between gas and liquid phases. The mesophase, confined within percolation loci that bound the gas and liquid phases by higher-order discontinuities, can readily be identified. A simple numerical differentiation of NIST equations-of-state, for example, can demonstrate the existence of the supercritical mesophase and locate the phase bounds, along any isotherm, of any fluid (e.g., CO

_{2}at T/T

_{c}= 1.25) for any of the 200 fluids in the NIST Thermophysical Property data bank [4]. These boundaries are smoothed over by the equations-of-state used to parameterize the original experimental data.

#### 3.2. Rigidity Symmetry at State Bounds

_{B}) rigidity decreases with density:

#### 3.3. Physical-Constant Equations-of-State

_{c}= 1.15, of CO

_{2}. The critical and Boyle temperatures for CO

_{2}according to NIST databank [4] are T

_{c}= 305 K and T

_{B}= 725 K. Using the same method described previously for argon [8], the coexisting densities at T

_{c}are found to be ρ

_{c}(gas) = 7.771 mol/L and ρ

_{c}(liq) = 13.46 mol/L. Substituting these physical constant values into equations (5 to 7) for T = 350 K, we obtain the gas and liquid state density bounds ρ

_{PB}(gas) = 7.305 mol/L and ρ

_{PA}(liq) = 12.02 mol/L, and the rigidity w(350 K) = 0.318kT (0.925 kJ/mol).

^{6}+ 0.000457ρ

^{5}– 0.009373ρ

^{4}+ 0.096245ρ

^{3}– 0.596317ρ

^{2}+ 3.459264ρ– 0.189239 with mean square regression R² = 0.999125.

^{2}) in the trendline polynomial coefficients as given. In addition, shown in Figure 12 are redefined equations to give the virial coefficients as shown on the plots. The coefficient a

_{1}in the liquid-state expansion is taken to be equal to the value of w

_{T}in the mesophase to describe the third-order discontinuity between the mesophase and the liquid state. There is a symmetry between the rigidity of gas and liquid on either side of the mesophase. This observation is empirical though it has a molecular origin in the fluctuations of the available volume and its relationship to the chemical potential of both gas and liquid states [52].

_{2}(Figure 13) show a clear symmetry on either side of the mesophase along an isotherm [53], which further suggests that the mesophase bounds narrow with increasing T and merge at or close to the Boyle temperature T

_{B}. This is the temperature above which the second virial coefficient is positive and below which it is negative but the analytic form as it passes through zero remains unknown. The same behaviour is seen in the rigidity plots for other liquids, for example water [54].

_{T}. The literature critical-point universality theory predicts that the temperature or pressure scales as Δρ

^{d}along the critical isotherm, which could therefore appear to be very flat anyway within the hypothesis. Many p-V-T experimental results for real molecular fluids have required exponents d = 3 to 4, or even higher [54] in order to parameterize an apparent flat top within the experimental uncertainty. The presentation of near-critical experimental results, however, has been adversely affected by hypotheses, which are here seen to be incorrect in the light of experimental data, and hence misrepresent the critical divide at T

_{c}and the supercritical mesophase.

## 4. Near-Critical Heat Capacities

#### 4.1. Universality and C_{v}

_{6}) in the late 1990s [59].

_{rev}and Q

_{rev}/T, enthalpy and entropy, respectively, are state functions”; where Q

_{rev}is the reversible heat absorbed. It follows that one cannot reversibly add heat to a classical Gibbs thermodynamic fluid without either doing work of expansion or increasing the temperature. The apparent divergence of C

_{v}in these experiments is based upon a misinterpretation of experimental heat capacities in the two-phase regions [60].

#### 4.2. Heat Capacity Definitions

_{v}and C

_{p}, are defined for the two-phase region, i.e., at T < T

_{c}, according to the lever rule:

_{v}does not diverge either below or above T

_{c}, the heat capacity at constant pressure C

_{p}diverges both in the two-phase region and in the supercritical region, i.e., as T → T

_{c}both above and below T

_{c}, we obtain:

_{rev}/ΔT, therefore, requires the definition of a heat capacity at saturation of gas or liquid, which is usually designated C

_{σ}. This can be calculated from the heat capacity C

_{p}if the variation in thermal pressure γ

_{σ}= (dp/dT)

_{σ}along the coexistence line is known. C

_{σ}(liq) and C

_{σ}(gas) are defined as the heat to reversibly increase the temperature of that phase in coexistence. C

_{σ}for gas or liquid can be expressed in terms of available properties C

_{p}, α

_{p}and γ

_{σ}:

_{p}is the thermal expansivity, defined by:

_{λ},

_{v}is the latent heat of evaporation. C

_{σ}for both liquid and gas, and enthalpies of coexisting phases, and hence C

_{λ}, can be obtained for most pure atomic and molecular fluids from the NIST fluid property data bank [4]. Thermal expansivities in Equation (8) are calculated from Joule–Thompson coefficients:

#### 4.3. Near Critical C_{v} Measurements

_{c}with universal scaling properties [55,62], are reproduced in Figure 14. The values of C

_{p}, C

_{v}and C

_{λ}for argon at the hypothetical critical density referred to in reference [49] are also shown in Figure 11. The values of C

_{λ}calculated using Equations (7) to (10) can be compared to “C

_{v}” data reproduced from the first divergent C

_{v}results published in 1963 [62]. Figure 13 explains this apparent divergence of C

_{v}. This evident measurement of C

_{λ}, which looks the same as a lambda-like transition for the near critical data, is then re-used in reference [48]. This observation of a lambda-like dependence was first made by Uhlenbeck [10], half a century ago, who described it as “surprising” but encouraging with regard to the embryonic concept of universality.

_{λ}(T) near T

_{c}in references [56,57,58,63,64], and Figure 13, occurs because C

_{σ}(T) for both liquid and gas diverges when the rigidity (dp/dρ)

_{T}goes to zero on either side of the critical divide at T

_{c}. Any experimental measurement of C

_{λ}(T) at density ρ

_{exp}within the critical divide, i.e., at T

_{c}, ρ

_{gas}< ρ

_{exp}< ρ

_{liq}, should thus show a divergence. If the hypothetical coexistence parabola were to exist, however, with a van der Waals singular point, the divergence would only occur at the hypothetical critical density ρ

_{c}. The isochoric measurements on argon [63,64] were performed at the densities of 521.0 kg/m

^{3}and 531.0 kg/m

^{3}[60], whereas the critical density given in NIST 2017 is ρ

_{c}= 535.6 kg/m

^{3}[4]. The results for a heat capacity divergence are supportive of the critical divide picture [7,8,9] and are inconsistent with the concept of a scaling singular point on the density, where a distinction between liquid and gas is deemed to disappear.

_{v}. Whilst stirring may reduce undesirable effects of gravitational phase separation, it creates an inhomogeneous shear field throughout the sample. For otherwise homogeneous Newtonian fluids with low viscosities this is normally acceptable for a single phase, e.g., in supercritical range. The thermodynamic states of these samples below T

_{c}, however, are in the 2-phase region. The stirred steady state is effectively a micro-emulsion. The surface energy of the emulsification contributes to the heat capacity measurements but is difficult to quantify. Below T

_{c}the liquid-vapour surface tension is positive; it becomes zero at T

_{c}. The is no thermodynamic definition of a surface tension other than for two phases in coexistence; we can only speculate regarding the contribution of interfacial effects in the supercritical 2-state mesophases.

_{v}in Equation (5) below T

_{c}. The surface work is of opposite sign, so the stirred heat capacities in the vicinity of T

_{c}are in-between C

_{λ}and C

_{v}with a weak divergence at T

_{c}as seen in Figure 14. The pointed peak at around 40 J/K·mol in the isochoric heat capacity at T

_{c}is explained by an increase in surface tension and heterophase fluctuations starting around 140 K in the subcritical 2-phase range as Tc is approached. For T > T

_{c}, C

_{v}returns to the lower liquid-state value on exiting the mesophase at higher temperatures around 175K (see Figure 11).

#### 4.4. Space Shuttle Experiments

_{6}along a near critical isochore aboard the NASA space shuttle [59]. The publicly available data points of the NASA Space shuttle experiment are shown in Figure 15 alongside the heat capacities C

_{p}, C

_{v}and C

_{λ}[4]. A comparison shows that what was actually measured in the space-shuttle experiment (labelled C

_{ss}) relates to C

_{λ}a few degrees below T

_{c}, and to C

_{v}a few degrees above T

_{c}, but is possibly made to look like a lambda-transition in the vicinity of T

_{c}by the stirring or steady-state emulsification.

_{c}(gas) and ρ

_{c}(liq) at T

_{c}, with an intermediate singularity, is widespread and misleading. It leads to unreal values in the vicinity of T

_{c}. An example in the present context is the misinterpretation of SF

_{6}space shuttle data. The real thermodynamic C

_{v}data for SF

_{6}are the NIST values. An equation-of-state proposed by Kostrowicka-Wyczalkowska and Sengers [65] assumes a scaling singularity at the outset, and uses a “cross-over” mathematical device in order to accommodate the spurious divergent form of the experimental space shuttle measurements, although these data are not the thermodynamic equilibrium C

_{v}, defined by the lever rule as illustrated here in Figure 15.

#### 4.5. Heat Capacities from V(p,T)

_{v}) derived from p-V-T equation-of-state measurements by the Michels group of the van der Waals Laboratory in Amsterdam, published in 1958 [61,62], are also widely used as evidence of continuity with a scaling singularity. The data in reference [54] (Table XXIV, p.793) suggest just the opposite conclusion: it shows no evidence for any critical point. The highest coexisting temperature reported [61,62] at which liquid and gas coexist of −122.5 °C (= 150.7 K) is right on the critical temperature (150.87 ± 0.015K [4]). At this temperature Michels et al. find a maximum coexisting gas density of 258.13 amagat (= 461.0 kg/m

^{3}) and a minimum coexisting liquid density of 343.25 amagat (= 613.2 kg/m

^{3}). These limiting coexistence densities compare well with the argon gas and liquid densities (475.6 and 598.6 kg/m

^{3}respectively) obtained from the intersection of percolation loci used to define T

_{c}and the coexisting state bounds ρ

_{c}(gas) and ρ

_{c}(liq) at T

_{c}thermodynamically [8]. In addition, the isochoric heat capacities of these two state points at T

_{c}, C

_{v}(gas) and C

_{v}(liquid), give a mean C

_{v}(lever-rule) value, which is near the NIST subcritical and supercritical values to within ± 0.1 K, as shown in Figure 14.

## 5. Conclusions

_{c}.

^{rd}order phase transitions [67]. Unfortunately, by the time the 2nd Edition of Mayer and Mayer’s “Statistical Mechanics” was published, 20 years later, Joseph Mayer was no longer active in research, and the concept of universality had been acclaimed as established physics [10,11,12,13]. The original chapter on “Condensation and the Critical Region” is missing from the 2nd Edition of Mayer and Mayer.

_{c}[25,27]. Surface tension, nevertheless, is the clue to understanding the thermodynamic states both above and below the critical divide. It is positive and well-defined below T

_{c}, and we have two-phase coexistence with liquid condensation that minimizes the interfacial area and hence also total Gibbs energy. At T

_{c}, the surface tension is zero, but it cannot be thermodynamically well-defined above T

_{c}as the two states are not coexisting phases, but colloidally dispersed states that do not exist independently of each other, as the precursor states to the percolation transitions are dispersed hetero-phase fluctuations [69].

_{c}within the mesophase, the pressure difference between liquid and gas states along an isotherm at the percolation bounds increases. Equilibrium is obtained if both the liquid and gas pure phases, both inter-disperse to percolate the phase volume, with an interfacial surface tension that balances any Gibbs energy difference between gas and liquid at the same T,p state point. Thus, both pure gas and pure liquid states coexist at the same T,p states with uniform chemical potential throughout at equilibrium in the mesophase. Looking at the experimental data, (Figure 9, Figure 10 and Figure 11) it appears that the mesophase may extend all the way to low density at the Boyle temperature, narrowing the density gap as it does so.

_{c}. In the region of the critical divide and the immediate supercritical region, data points are either fabricated by hypothetical scaling equations, or distorted by the continuity hypothesis implicit in the equations-of-state used. These multiparameter equations tell us nothing about the underlying science of critical and supercritical state bounds; they produce hypothetical near-critical and supercritical mesophase data that have never been experimentally measured. Figure 16 shows how the scientific malpractice of assuming the van der Waals and universal scaling hypotheses to be established scientific truth, leads to the spurious experimental p-V-T data [4,19] in the vicinity of T

_{c}.

_{c}” as discovered by the historic measurements of the 1960s and confirmed by the space shuttle microgravity experiment. It seems rather unusual, if these measurements were such an important fundamental cornerstone of the theory of critical-point universality [10,11,53], that for several decades, nobody has attempted to reproduce a divergent C

_{v}. It is also rather curious that the space-shuttle measurements in microgravity, designed to eliminate gravitational phase separation effects, found almost identical behaviour to the existing mundane laboratory measurements, i.e., a quasi-vertical increase in C

_{v}(T) for T ≥ T

_{c}. We conclude that there is a need for further direct accurate measurements of T > T

_{c}near-critical heat capacities, not least to characterise the properties of the supercritical mesophase, and to investigate and characterize the effect of surface tension terms that must be present but difficult to define and quantify.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gibbs, J.W. A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans. Conn. Acad. Arts Sci.
**1873**, 2, 382. [Google Scholar] - Andrews, T. On the continuity of gas and liquid states of matter. Proc. R. Soc. Lond.
**1869**, 159, 575–590. [Google Scholar] - Van der Waals, J.D. Over de Continuiteit van den Gas-en Vloeistoftoestand. Ph.D. Thesis, University of Leiden, Leiden, The Netherlands, 1873. [Google Scholar]
- NIST Thermo-physical Properties of Fluid Systems. Available online: http://webbook.nist.gov/chemistry/fluid/ (accessed on 4 March 2020).
- Reif-Acherman, S. History of the Law of Rectilinear Diameters. Quim. Nova
**2010**, 33, 2003–2013. [Google Scholar] [CrossRef] [Green Version] - Woodcock, L.V. Thermodynamic description of liquid-state limits. J. Phys. Chem. B
**2012**, 116, 3734–3744. [Google Scholar] [CrossRef] - Woodcock, L.V. Observations of a thermodynamic liquid–gas critical coexistence line and supercritical phase bounds from percolation loci. Fluid Phase Equilibria
**2013**, 351, 25–33. [Google Scholar] [CrossRef] - Woodcock, L.V. Gibbs density surface of fluid argon: Revised critical parameters. Int. J. Thermophys.
**2014**, 35, 1770–1784. [Google Scholar] [CrossRef] - Onsager, L. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev.
**1944**, 65, 117. [Google Scholar] [CrossRef] - Uhlenbeck, G.E. The classical theories of critical phenomena. In Critical Phenomena: Proceedings of a Conference in Washington DC; Green, M.S., Sengers, J.V., Eds.; National Bureau of Standards: Washington, DC, USA, 1966. [Google Scholar]
- Rowlinson, J.S. Critical states of fluids and fluid mixtures: A review of the experimental position. In Critical Phenomena: Proceedings of a Conference in Washington DC; Green, M.S., Sengers, J.V., Eds.; National Bureau of Standards: Washington, DC, USA, 1966. [Google Scholar]
- Fisher, M.E. Notes, definitions and formula for critical-point singularities. In Critical Phenomena: Proceedings of a Conference in Washington DC; Green, M.S., Sengers, J.V., Eds.; National Bureau of Standards: Washington, DC, USA, 1966; pp. 21–25. [Google Scholar]
- Sengers, J.V. Behaviour of the viscosity and thermal conductivity of fluids near the critical point. In Critical Phenomena: Proceedings of a Conference in Washington DC; Green, M.S., Sengers, J.V., Eds.; National Bureau of Standards: Washington, DC, USA, 1966. [Google Scholar]
- Kadanoff, L.P.; Gotze, W.; Hamblen, D.; Hecht, R.; Lewis, E.A.S.; Palciauskas, V.V.; Rayl, M.; Swift, J.; Aspnes, D.; Kane, J.W. Static phenomena near critical points: Theory and experiment. Rev. Mod. Phys.
**1967**, 39, 395. [Google Scholar] [CrossRef] - Kac, M.; Uhlenbeck, G.E.; Hammer, J. On the van der Waals theory of the vapor-liquid equilibrium I. J. Math. Phys.
**1963**, 4, 216. [Google Scholar] [CrossRef] - Kac, M.; Uhlenbeck, G.E.; Hammer, J. On the van der Waals Theory of the vapor-liquid equilibrium II. J. Math. Phys.
**1963**, 4, 2239. [Google Scholar] - Kac, M.; Uhlenbeck, G.E.; Hammer, J. On the van der Waals theory of the vapor-liquid equilibrium III Discussion of the Critical Region. J. Math. Phys.
**1963**, 5, 60–74. [Google Scholar] - Wilson, K.G. Renormalization Group and Critical Phenomena. Phys. Rev.
**1971**, B4, 3174–3183. [Google Scholar] [CrossRef] [Green Version] - Woodcock, L.V. On the Empirical Determination of a Gas–Liquid Supercritical Mesophase and its Phenomenological Definition. Int. J. Thermophysics.
**2020**, 41, 70. [Google Scholar] [CrossRef] - Vega, L.; de Miguel, E.; Rull, L.F.; Jackson, G.; McLure, I.A. Phase equilibria and critical behavior of square-well fluids of variable width by Gibbs ensemble Monte Carlo computation. J. Chem. Phys.
**1992**, 96, 2296–2305. [Google Scholar] [CrossRef] [Green Version] - Elliott, J.R.; Hu, L. Vapor-liquid equilibria of square-well fluids. J. Chem. Phys.
**1999**, 110, 3043–3048. [Google Scholar] [CrossRef] - Benavides, A.; Alejendre, J.; del Rio, F. Properties of square-well fluids of variable width IV molecular dynamics test of the van der Waals and long-range approximation. Mol. Phys.
**1991**, 74, 321–331. [Google Scholar] [CrossRef] - Magnier, H.J.; Curtis, R.; Woodcock, L.V. Nature of the supercritical mesophase. Nat. Sci.
**2014**, 6, 797–807. [Google Scholar] [CrossRef] [Green Version] - Magnier, H.J. Understanding Biopharmaceutical Aggregation Using Minimalist Models Based on Square-Well Potential. Ph.D. Thesis, University of Manchester, Manchester, UK, 2016. [Google Scholar]
- Heyes, D.M.; Woodcock, L.V. Critical and supercritical properties of Lennard-Jones fluids. Fluid Phase Equilibria
**2013**, 356, 301–308. [Google Scholar] [CrossRef] - He, S.; Attard, P. Surface tension of the Lennard-Jones liquid under supersaturation. Phys. Chem. Chem. Phys.
**2005**, 7, 2928–2935. [Google Scholar] [CrossRef] - Potoff, J.T.; Panagiotopoulos, A.Z. Surface tension of the three-dimensional Lennard-Jones fluid from histogram reweighting Monte Carlo simulations. J. Chem. Phys.
**2000**, 112, 6411–6415. [Google Scholar] [CrossRef] - Woodcock, L.V. Thermodynamic fluid equations-of-state. Entropy
**2018**, 20, 22–37. [Google Scholar] [CrossRef] [Green Version] - Bradley, W.P.; Browne, A.W.; Hale, C.F. Liquid above the critical temperature. Phys. Rev.
**1908**, 27, 90–95. [Google Scholar] [CrossRef] [Green Version] - Van der Waals, J.D. Nobel Lecture “The Equation of State of Gases and Liquids”. Available online: https://www.nobelprize.org/prizes/physics/1910/waals/lecture/ (accessed on 4 March 2020).
- Mills, A.A. The critical transition between the liquid and gaseous conditions of matter. Endeavour
**1993**, 17, 203–209. [Google Scholar] [CrossRef] - Traube, I. On the critical temperature. Trans. Faraday Soc.
**1938**, 34, 1234–1235. [Google Scholar] [CrossRef] - Tapp, J.S.; Steacie, E.W.R.; Maass, O. An investigation into the density of a vapor in equilibrium with a liquid near its critical temperature. Can. J. Res.
**1933**, 9, 217–239. [Google Scholar] [CrossRef] - Winkler, C.A.; Maass, O. Density differences at the critical temperature. Can. J. Res.
**1933**, 9, 613–629. [Google Scholar] [CrossRef] - Maass, O.; Geddes, A.L. The persistence of the liquid state of aggregation above the critical temperature: The system ethylene. Phil. Trans. R. Soc.
**1937**, A236, 303–332. [Google Scholar] - Maass, O. Changes in the liquid state in the critical temperature region. Chem. Rev.
**1938**, 23, 17–27. [Google Scholar] [CrossRef] - McIntosh, R.L.; Maass, O. Persistence of the liquid state of aggregation above the critical temperature. Can. J. Res.
**1938**, 16b, 289–302. [Google Scholar] [CrossRef] - Dacey, J.; McIntosh, R.L.; Maass, O. Pressure, volume, temperature relations of ethylene in the critical region I. Can. J. Res.
**1939**, 17b, 206–213. [Google Scholar] [CrossRef] - McIntosh, R.L.; Dacey, J.R.; Maass, O. Pressure, volume, temperature relations of ethylene in the critical region II. Can. J. Res.
**1939**, 17b, 241–250. [Google Scholar] [CrossRef] - Mason, S.G.; Naldrett, S.N.; Maass, O. A study of the coexistence of the gaseous and liquid states in the critical temperature region: Ethane. Can. J. Res.
**1940**, 18b, 103–117. [Google Scholar] [CrossRef] - Naldrett, S.N.; Maass, O. A study of the coexistence of the gaseous and liquid states in the critical temperature region: Ethylene. Can. J. Res.
**1940**, 18b, 118–121. [Google Scholar] [CrossRef] - Guggenheim, E.A. The Principle of Corresponding states. J. Chem. Phys.
**1945**, 13, 253–261. [Google Scholar] [CrossRef] [Green Version] - McCormack, E.; Schneider, W.G. Isotherms of Sulphur Hexafluoride in the critical temperature region. Can. J. Chem.
**1951**, 29, 699–714. [Google Scholar] [CrossRef] - Weinberger, M.A.; Schneider, W.G. On the liquid vapor coexistence curve of Xenon in the region of the critical temperature. Can. J. Chem.
**1952**, 30, 422–437. [Google Scholar] [CrossRef] [Green Version] - Habgood, H.W.; Schneider, W.G. PVT Measurements in the critical region of Xenon. Can. J. Chem.
**1954**, 32, 98–112. [Google Scholar] [CrossRef] - Zimm, B. Contribution to the Theory of Critical Phenomena. J. Chem. Phys.
**1951**, 19, 1019–1023. [Google Scholar] [CrossRef] - Mayer, J.E.; Harrison, S.F. Statistical mechanics of condensing systems. III. J. Chem. Phys.
**1938**, 6, 87. [Google Scholar] [CrossRef] - Rice, O.K. On the behavior of pure substances near the critical point. J. Chem. Phys.
**1947**, 15, 314–332. [Google Scholar] [CrossRef] - Rossini, F.D. Thermodynamics and Physics of Matter, 1st ed.; Oxford University Press: London, UK, 1955; pp. 419–500. [Google Scholar]
- Gilgen, R.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (pressure, density, temperature) relation of argon. I. The homogeneous gas and liquid regions in the temperature range from 90 to 300 K at pressures up to 12 MPa. J. Chem. Thermodyn.
**1994**, 26, 383–398. [Google Scholar] [CrossRef] - Gilgen, R.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (pressure, density, temperature) relation of argon. II Saturated-liquid and saturated vapour densities and vapour pressures along the entire coexistence curve. J. Chem. Thermodyn.
**1994**, 26, 399–413. [Google Scholar] [CrossRef] - Hoover, W.G.; Poirier, J.C. Determination of virial coefficients from potential of mean force. J. Chem. Phys.
**1962**, 37, 1041–1042. [Google Scholar] [CrossRef] - Woodcock, L.V. Thermodynamics of gas-liquid criticality: Rigidity symmetry on Gibbs density surface. Int. J. Thermophys.
**2016**, 37, 24–33. [Google Scholar] [CrossRef] - Woodcock, L.V. Gibbs Density Surface of Water and Steam: 2
^{nd}Debate on the Absence of Van Der Waals’ “Critical Point”. Nat. Sci.**2014**, 6, 411–432. [Google Scholar] [CrossRef] [Green Version] - Sengers, J.V.; Anisimov, M.A. Comment on Gibbs Density Surface of Fluid Argon, L.V.; Woodcock, Int. J. Thermophys. (2014) 35:1770–1784. Int. J. Thermophys.
**2015**, 36, 3001. [Google Scholar] [CrossRef] - Voronel, A.V.; Smirnov, V.A.; Chashkin, R.Y. Measurements of isochoric heat capacities of near-critical argon. JETP Lett.
**1969**, 9, 229. [Google Scholar] - Anisimov, M.A.; Berestov, A.T.; Veksler, L.S.; Koval’chuk, B.A.; Smirnov, V.A. Measurements of isochoric heat capacities of near-critical argon. Sov. Phys. JETP
**1972**, 39, 359. [Google Scholar] - Anisimov, M.A.; Koval’chuk, B.A.; Smirnov, V.A. Experimental study of the isochore heat capacity of argon in a broad range of parameters of state, including the critical point. In Thermophysical Properties of Substances and Materials; Izd-vo Standartov: Moscow, Russia, 1975; pp. 237–245. (In Russian) [Google Scholar]
- Haupt, A.; Straub, J. Evaluation of the isochoric heat capacity measurements at the critical isochore of SF6 performed during the German Spacelab Mission D-2. Phys. Rev.
**1999**, E59, 1975–1986. [Google Scholar] - Woodcock, L.V. On the interpretation of near-critical heat capacities. Int. J. Thermophys.
**2017**, 38, 139–144. [Google Scholar] [CrossRef] - Michels, A.; Levelt, J.M.; de Graaff, W. Compressibility isotherms of argon at temperatures between −25 °C and −155 °C and at densities up to 60 amagat. Physica
**1958**, 24, 659–671. [Google Scholar] [CrossRef] - Michels, A.; Levelt, J.M.; Wolkers, G.J. Compressibility isotherms of argon at temperatures between 0 °C and −155 °C and at densities up to 640 amagat. Physica
**1958**, 24, 769–794. [Google Scholar] [CrossRef] - Anisimov, M.A. 50-Years of Breakthrough Discoveries in Fluid Criticality. Int. J. Thermophys.
**2011**, 32, 2003. [Google Scholar] [CrossRef] [Green Version] - Bagatskiĭ, M.I.; Voronel’, A.V.; Gusak, V.G. Measurements of isochoric heat capacities of near-critical argon. Sov. Phys. JETP
**1963**, 16, 517. [Google Scholar] - Kostrowicka-Wyczalkowska, A.; Sengers, J.V. Thermodynamic properties of sulphur hexafluoride in the critical region. J. Chem. Phys.
**1999**, 111, 1551. [Google Scholar] [CrossRef] - Mayer, J.E.; Mayer, M.G. Statistical Mechanics, 1st ed.; Wiley: New York, NY, USA, 1940; Chapter 14. [Google Scholar]
- Woodcock, L.V. Percolation transitions and fluid state boundaries. CMST
**2017**, 23, 281–294. [Google Scholar] [CrossRef] [Green Version] - Tegeler, C.; Span, R.; Wagner, W. New equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressures up to 1000 MPa. J. Phys. Chem. Ref. Data
**1999**, 28, 779–850. [Google Scholar] [CrossRef] [Green Version] - Woodcock, L.V. Thermodynamics of supercritical colloidal equilibrium states: Hetero-phase fluctuations. Entropy
**2019**, 21, 1189. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Illustration of a Gibbs (1873) surface: p(V,T) for a typical one-component pure fluid showing a hypothetical critical volume postulated by van der Waals (1873).

**Figure 2.**(

**a**) The original pressure vs. density experimental data for 6 isotherms of CO

_{2}in the region of the critical temperature (30.9 °C [4]) as reported by Andrews in 1869 [2]; the unlabelled abscissa scale is a logarithmic density; (

**b**) the same 6 isotherms on a linear density scale from NIST [4]; also shown are the coexistence envelope (solid black lines) and the law of rectilinear diameters (almost vertical brown straight line) extended into the supercritical region (dashed).

**Figure 3.**Coexistence curves of square-well fluids from references [6,7,19,20,21,22,23,24] with empirical limiting densities shown as closed circles and curves for λ = 1.25–2.0 from Vega et al. [20]; open circles are the densities of Elliott and Hu [21]; densities at l = 3.0 from Benavides et al. [22]; densities and curves for λ = 1.005 and λ = 5 [6,7]; the upper dashed line is the hard-sphere (HS) available volume percolation transition loci and the lower horizontal dashed lines are the extended volume percolation transition loci, of the hard-sphere reference fluid [6,7]; ρ

_{PB}(T) is the high-T limit, λ-dependent, bonded-cluster percolation density at T

_{c}of square-well fluids.

**Figure 4.**Critical and supercritical pressure isotherms of the Lennard–Jones fluid from reference [25]. (

**a**) Isotherms showing the linear mesophase regions (dashed rectangle) between gas-like and liquid-like states of decreasing and increasing rigidities. (

**b**) Expanded dashed rectangle region in (

**a**) for 50 state points along each isotherm within the mesophase.

**Figure 5.**Values of the normalized pair distribution probability function at selected intermolecular pair distances ranging r

_{0}to 4r

_{0}(

**a**–

**d**); r

_{0}(=2

^{1/6}σ) is the distance of zero force; the state parameters along the supercritical isotherm (T* = 1.35); percolation transitions at T

_{c}* = 1.336 are at reduced densities 0.266 (PB) and 0.376 (accessible volume (PA)) [25], as indicated by vertical lines.

**Figure 6.**Near critical vapour-liquid density-difference dependence upon surface tension; γ* goes to zero at a finite density difference, which coincides with the density difference between PA and PB at T

_{c}. The data points shown are taken directly from numerical results in Table 1 of the paper by Potoff and Panagiotopoulos [27].

**Figure 7.**Surface tension (γ) of saturated liquid argon (red dots) obtained from NIST tables [4] relative to the triple point value (γ

_{t}) as a function of (T/T

_{c})

^{1/2}. Also shown are the density difference (Δρ) ratio (dashed black line) and the rigidity difference (Δω) ratio (solid line) for comparison; the arrow shows that a real liquid argon shows a cubic density-difference dependence on T in the proximity of T

_{c}, consistent with the reinterpretation of the MC results in Figure 6.

**Figure 8.**Guggenheim’s classic diagram of 1945 [42]. A plot of reduced temperature (T/T

_{c}) as a function of reduced density (ρ/ρ

_{c}) of available experimental data points for the coexisting densities of 8 fluids: the solid line is an assumed parabola with a singularity of the form ρ − ρ

_{c}= (T − T

_{c})

^{1/3}to determine a critical density.

**Figure 10.**Percolation transition points along near-critical isotherms plotted directly from original measurements as reported by Gilgen et al. [50] and shown in Figure 4 in the p-T projection. The extrapolated dashed lines of available volume (PA-blue) and bonded cluster (PB-green) correspond to the experimentally observed spinodal lines from the literature. The red line and points are the coexistence data of Gilgen et al. [51].

**Figure 11.**Experimental coexistence density measurements taken directly from Table 1 of Gilgen et al. [50]; at T

_{c}the experimental density gap coincides with the linear density difference between the supercritical gas and liquid phases as defined by the sign of the rigidity state function and obtained from the NIST 2017 Thermophysical tables [4]; the percolation loci (dashed) decrease linearly with density in the supercritical range and appear to originate at the Boyle temperature; the large red data points are the LRD mean coexisting densities.

**Figure 12.**Pressure as a function of density for CO

_{2}at a supercritical temperature (350 K or T/T

_{c}= 1.15) as derived from NIST Thermophysical Properties compilation: (

**a**) gas state, (

**b**) supercritical meso-phase and (

**c**) liquid.

**Figure 13.**Supercritical isotherms of carbon dioxide rigidity: red = supercritical, purple = critical, blue = subcritical. The rigidity is obtained from NIST thermophysical tables [4]. Loci of the gas and liquid supercritical phase bounds are green and blue, respectively, and appear to merge at or near to the Boyle temperature (T

_{B}~ 700 K).

**Figure 14.**Heat capacities of argon for the isochore 13.29 mol/l. C

_{l}is calculated from the NIST thermophysical compilations [4] using Equations (7) to (10); C

_{p}and C

_{v}are the NIST values; the two experimental C

_{v}data points at T

_{c}are the values of C

_{v}at the densities of coexisting liquid (open circle) and gas (solid circle) taken from Table XX p.789 of Michels et al. [62]; the crosses are experimental data points reported and discussed in references [55,58,60]. (note that values for C

_{p}and C

_{λ}above T

_{c}diverge at T

_{c}off the scale shown).

**Figure 15.**Variation of the heat capacities of SF

_{6}along an isochore (5.0474 mol/l) within the critical divide, corresponding to the same density of the space shuttle experiment [59]. This density is about 1% lower than the mean critical density given by NIST [4] i.e., 5.0926 mol/L. C

_{ss}are the space shuttle experimental data points; the subcritical and supercritical isochoric heat capacities (C

_{v}) are from NIST thermophysical compilations [4].

**Figure 16.**Near-critical density coexistence data for argon. The blue and red dots are experimental data points up to the maximum experimentally observed coexisting gas density and minimum coexisting liquid densities reported by Gilgen et al. [51]; the blue and red experimental coexistence lines up to the critical hiatus (purple line) are NIST data; the black crosses are also NIST data points [4] fabricated using equations that accommodate the continuity and universality singular critical density hypotheses; dotted lines show the percolation lines PA and PB that bound the supercritical mesophase as defined at a T

_{c}by the coexisting densities at equal pressures; there are no known experimental p-V-T data to support the hypothetical van der Waals critical point of argon as published by NIST [4] (black spot).

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Khmelinskii, I.; Woodcock, L.V.
Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results. *Entropy* **2020**, *22*, 437.
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**AMA Style**

Khmelinskii I, Woodcock LV.
Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results. *Entropy*. 2020; 22(4):437.
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Khmelinskii, Igor, and Leslie V. Woodcock.
2020. "Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results" *Entropy* 22, no. 4: 437.
https://doi.org/10.3390/e22040437