Almost 40 years ago, in their classic review on the status of liquid state theory [1] , Barker and Henderson began with the words “Liquids exist in a relatively small part of the enormous range of temperatures and pressures existing in the universe”. The tiny liquid area, in the T-p projection of Gibbs density surface, was defined within either a critical isotherm, or isobar, and a triple point. Above a critical temperature (or pressure) and below the triple-point, the liquid state did not exist. Not everyone agreed with these ad hoc bounds, however. For example, recent research on percolation transition loci on Gibbs thermodynamic surfaces [2] shows that J. D. Bernal may have been closer to the truth. Besides noting that the liquid state, albeit metastable, should extend down to absolute zero using random close packing as a starting point, Bernal also argued that the liquid state should extend to supercritical temperatures and pressures, where it is bounded from the gas phase by a “hypercritical” line of discontinuity.

Here, we report results for ideal gas properties, which, alongside real experimental p-V-T properties of a typical real fluid (argon), comprise compelling evidence that the liquid state is not bounded, by either the critical isotherm or isobar. Liquid and gas phases are terminated by percolation loci along any isotherm. Moreover, we find that percolation loci extend all the way from critical coexistence to low density states with ideal gas properties.

The equation-of-state of a real gas with finite molecular size (diameter σ) behaving ideally within a low-den- sity limit, is simply

where p^* is a molecular-reduced pressure (pσ³/k_BT), T is temperature (K), k_B is Boltzmann’s constant, ρ^* is a reduced density (ρ^* = Nσ^d/L^d), where L^d is length (L, d = 1), area (A, d = 2) or volume (V, d = 3). Equation (1) has an abiding role in the description of thermodynamic properties of real molecular fluids. Pressure is everywhere continuous; second and all higher derivatives of p(ρ) are zero. Because of this simplicity, all state functions are exactly known for any d. Equation (1) is a universal scaling law that spans the dimensions.

Within the ideal gas limit of obedience to Equation (1), real fluids with finite size, i.e. σ > 0, however, exhibit various properties that cannot scale with d, linear transport coefficients, for example. Percolation transitions, not unrelated to the transport coefficients, are also strongly dimension dependent in form, and are known to determine thermodynamic phase changes in model lattice gases [3] . Percolation transitions of the available volume (V_A) and excluded volume (V_E) for the insertion of one more molecule of a finite diameter are properties relating to Gibbs energies that effect phase transitions.

For hard-core fluids,

then, the ensemble averages and equate with chemical potential (μ_i) of species i

Equation (3), with Equation (2), defines and for real fluids.

For the ideal gas, percolation of V_E is defined as a density above, or temperature below which, the overlapping exclusion spheres of radius σ/2 from a point in a uniformly random distribution of N points, form clusters that can span the whole of V. V_A comprises a distribution in configuration space of accessible pockets in which there are no ideal gas point molecules within one sphere diameter anywhere in the pocket. The percolation transition for V_A is the density above, or temperature below which, the empty pockets coalesce to span the system. For temperatures above percolation, V_A comprises a network of connecting pathways to the whole system accessible to a diffusing sphere in the static ideal gas equilibrium configuration.

For an ideal gas the exclusion sphere diameter defines. Experimental coexistence data on binary liquid phase diagrams is generally obtained at constant pressure (1 atm.) and presented with temperature (T) the dependent variable as a function of mole fraction (X_B), hence at this stage, with binary liquids in mind, we choose T^* as the state variable. (note: for an ideal gas) We designate the percolation transition reduced temperatures as and respectively. Relationships between dimensionality and percolation transitions can be summarized:

d = 1 no percolation

d = 2 PE and PA coincide and

d = 3 there is an inequality and

There is a fundamental difference between 2 and 3 dimensions. For d = 2, there are two regions, “gas-like” and “liquid-like”, whereas for d = 3 there are three regions, gas-like T^* > T_PA, liquid-like T^* < T_PA and a mesophase,. In the mesophase, both the pockets of availability and clusters of exclusion sites percolate the system. The mesophase is both gas-like and liquid-like.

PE both for d = 2 and 3 has been investigated by a number of authors for the present and related systems [4] - [11] Rough estimates of PE for d = 2 and 3 ideal gases can be gleaned from Figure 3 and Figure 4 of the paper by Bug et al. [4] . Extrapolating their data points, when their attraction parameter ε = 0, to zero density one obtains for d = 2 Φ_PE ~ 1.2 and for d = 3 Φ_PE ~ 0.35, where Φ is the excluded volume fraction (=πρσ³/6). Heyes and coworkers [5] - [8] report related investigations of PE for various models by MC and MD simulations. From an interpolation to zero density of the hard-sphere fluid variable-exclusion shell percolation threshold Heyes et al. [7] obtain Φ_PE = 0.346 (d = 3). The most accurate values for both the d = 2 (PE and PA) percolation transition are probably those of Ziff and coworkers: the values for ρ^* are 1.275 (d = 2) and 0.653 (d = 3). An extensive study of ideal-gas PE transitions for exclusion squares, cubes, and other geometric shapes has also been reported [11] .

Educational insight, and estimates of for d = 2 can easily be obtained pictorially in just a few minutes using an EXCEL spreadsheet. Figure 1 shows a typical configuration, 2000 random numbers from a uniform distribution 0 - 1, i.e. an ideal gas (N = 1000) in the vicinity of the percolation transition. Disc diameters do not vary when the square area is expanded or contracted on the display. Fixing the diameter at 5 mm, percolation

occurs at L ~ 14.0 cm hence or, i.e. in close proximity to reference [9] . Computation of PA for the ideal gas d = 3 has not previously been reported.

There are no reports of PA (d = 3) having been previously investigated or determined for the ideal gas, although the transition density is known to be 0.537 ± 0.05 for the hard-sphere fluid [12] . Here, we use the same methods, and criteria for percolation, as described previously for the hard-sphere fluid. Both and have been computed for a range of finite size systems; the results are summarized in Figure 2. Thermodynamic limiting values (N à ∞) are obtained from the linear trendlines.

Every configuration either has a percolating cluster or it does not. Clearly, for small finite systems, there will be configurations that percolate, and some that do not, in the vicinity of PE. The percolation threshold in the computations of Heyes et al. [5] - [8] was defined when 50% of configurations have a percolating cluster, with details described by Seaton et al. [13] . Here, we define PE using an ensemble average definition of a percolation

density [12] ; i.e. is the saddle-point density above which the cluster size probability distribution P(n) is bimodal. This is the normalized probability of a site belonging to a cluster of size n. Above, P(n) is a monotonic gas-like distribution, for all PE below it is bimodal. Plotting the saddle-point definition of against 1/N^1/3 (Figure 2(a)) gives a linear trendline that interpolates to the result ().

Our method for determining is essentially that described previously for hard spheres [12] , except that it is easier for the ideal gas. Here we use N-V-T MD for non-additive binary spheres that can simulate the Widom- Rowlinson (W-R) model fluid [14] [15] . This belongs to the general class of symmetric binary non-additive hard-sphere fluid mixtures defined by collision diameters

where δ is a dimensionless non-additivity, that varies from −1, for the W-R penetrable-sphere model binary fluid, via zero for one-component hard spheres, to infinity. Positive δ relates to ionic liquids and ionic crystal structures when mole fraction X_B = 0.5.

The MD program solves equations of motion of a binary mixture N_A + N_B. The results for the PA values in Figure 2(b) are obtained by the mean-squared displacements of B average over many frozen random configurations of ideal gas A. As the B particles do not interact with themselves, we average over all N_B in the same MD simulation run. All the values in Figure 2(b) were obtained for equimolar systems. Plotting the point of zero dif-

fusivity, D_i(ρ, N) à 0, against N (N_A in MD run) gives a linear trendline with the result ().

Probably the simplest 3D model Hamiltonian of a molecular fluid, which is continuous in phase space, and exhibits liquid-gas criticality and two-phase gas-liquid coexistence, is the penetrable cohesive sphere (PCS) fluid [14] [15] . The internal energy (U) is simply

where k_B is Boltzmann’s constant and T is temperature (K); the angular brackets denote a configurational average. Equation (5) defines an attractive molecular energy (ε) complementary to the volume of overlapping clusters, i.e. V_E as defined above for an ideal gas, of a configuration of N penetrable spheres, and ν₀ (=4πσ³/3) is the volume of a sphere. At low temperatures, this model exhibits the exact properties of an ideal gas in both the low- density (gas phase) and high-density (liquid-phase) limits. Here again, there is a liquid-like state with the properties of the ideal gas. Both W-R and PCS models, therefore raise a curious conundrum. Could a supercritical fluid with the properties of an ideal gas be described as “liquid”?

Experimental thermodynamic liquid-gas coexistence phase diagrams [21] [23] have traditionally been obtained along isotherms by measurements of coexistence pressures. Hence, we prefer to maintain the connection with experimental data of gas-liquid thermodynamic equilibrium measurements established over a period of 150 years by plotting p(ρ)_T isotherms (note the contrast with [15] ). A more detailed analysis of the connection of percolation loci with experimental isotherms is recently published [18] .

Every state of the PCS fluid corresponds to a transcribed state of the W-R binary model fluid. The equations for the transcription from the W-R binary percolation and coexistence pressures (Figure 7(a)) to the PCS one-component gas-liquid pressure can be derived, for example, from an analysis of the respective grand partition functions as described by de Miguel et al. [15] . The transcription equations we need here are

pressure (6)

density (7)

where is the thermodynamic activity of a component defined as lnZ^* = μ/k_BT and μ is Gibbs chemical po-

tential relative to the ideal gas at the same T, p. Gibbs energy change, hence Z^*, can be obtained by integrating the excess pressure loci at constant T, with respect to density.

For the one-component isomorphism of W-R fluid, pressure is the natural variable so the p^* vs. ρ^* representation along isotherms relates directly to experimental results and phase diagrams. We plot the values of the pressure at coexistence vs. ρ^* which we plot in Figure 7(b). Itshows that the simplest imaginable continuous Hamiltonian model to exhibit liquid-gas coexistence and a critical temperature has a coexistence line at the intersection percolation loci as observed for square-well and Lennard-Jones model fluids and many real fluids [24] - [28] . There is a liquid-gas critical dividing line between 2-phase coexistence and a supercritical mesophase. The liquid state extends to an ideal gas zero density and pressure limit. This raises the question: could there exist a high temperature limit of the percolation transition loci in real fluids at very low density in the region of obedience to the ideal gas law?

The evidence suggests this is indeed the case, provided we re-interpret the experimental thermodynamic properties of real fluids in the light of our knowledge of percolation transitions. For 80 important gases or liquids, including the simplest real liquid argon, in the NIST “Thermophysical Properties of Fluid Systems” data bank [23] , equation-of-state p-V-T data with 6-figure accuracies are obtainable. These p(ρ)_T supercritical isotherms have been formulated, however, using a large number of parameters and an assumption of a supercritical continuity of liquid and gas. If there were to be no continuity of liquid and gas, there would need to exist three different equations-of-state to describe the gas and liquid phases, bounded by percolation loci, and the mesophase region in between. Present findings suggest theory-based equations-of-state with far fewer parameters, and with scientifically correct functional forms should eventually replace the NIST equations. Present observations indicate a virial expansion for the gas phase, perhaps a free-volume expansion for the liquid, and a linear combination for the mesophase.

Although lacking a molecular-level definition, for any real fluid, for which an exact Hamiltonian is generally unknown, the percolation loci can be defined and obtained phenomenologically along any thermodynamic equilibrium isotherm by the rigidity inequalities [18] . In the case of real fluids with attractive potentials, the percolation transition bounding the gas phase, i.e. the counterpart of PE for impenetrable spheres, has been designated PB, as it is a percolation of bonded clusters.

Rigidity (ω_T) is the work required to isothermally increase the density of a fluid; with dimensions of molar energy and is plotted for a range of isotherms for CO₂ in Figure 8. The symmetry of the rigidity in both the subcritical and supercritical regions is the subject of a recent paper [18] . We note from Figure 8 that the percolation loci PB and PA appear to be converging at the Boyle rigidity in the case of CO₂.

The rigidity state function relates directly to the change in Gibbs energy (G) with density at constant T according to

The following inequalities (10)-(12) are presently empirical, but do have a molecular-level explanation in terms of the number density fluctuations of gas and liquid respectively [18]

gas (10)

meso (11)

liquid (12)

It is clear from Equation (9) that ω > 0, i.e. rigidity must always be positive: Gibbs energy cannot decrease with pressure when T is constant. By these definitions, moreover, not only can there be no “continuity” of gas and liquid, but the gas and liquid states are fundamentally different in their thermodynamic description. Rigidity is determined by number density fluctuations at the molecular level, which have different density dependencies in either phase. The gas phase comprises one large void with many small clusters, which determine fluctuations, whereas the liquid phase comprises one large cluster and with many vacant pockets, which determine density fluctuations. Since the properties of a hole or an occupied site are statistically equivalent, this give rise to a symmetry of supercritical properties between gas and liquid phases [18] .

Experimental argon isotherms [21] - [23] from the critical temperature (T_c = 151K) to 500K are plotted in Figure 9. All the isotherms below a temperature around 400K show that there is a flat meso region, i.e. (dω/dρ)_T = 0. In this narrow range, for each supercritical isotherm, to within the uncertainty of the original experimental data, the rigidity is constant. A value can be obtained by a linear fit over a finite density range with a linear trendline regression between 0.999999 and 1.0 for all the supercritical isotherms in Figure 9. An accurate equation- of-state for the meso region is thus obtained

where ω_T is the rigidity along any isotherm T(K) and p₀ is a pressure constant. The percolation loci densities can be estimated by observing the differences p(gas) − p(meso), and p(liquid)?p(meso), both decrease quadratically with density, and interpolate to zero at the percolation loci densities ρ_PB and ρ_PA.

The initial slope of the percolation loci in the pressure plot (Figure 9(a)) corresponds to an intercept temperature obtained from percolation loci in Figure 9(b) at zero density, as ω(ρ → 0) = Nk_BT for an ideal gas. All molecular gases exhibit a characteristic temperature at which the second virial coefficient changes sign. This corresponds to a low-density limit of the percolation transition loci that may converge together at or below this temperature. The Boyle temperature of argon is 407 K [23] corresponding to a rigidity of 3.38 KJ/mol.

In Section 2 we have presented a new result for the penetrable sphere available volume percolation transition temperature () of an ideal gas and found that. We conclude that the d = 3 ideal gas has three

regions liquid-like, gas-like, and a meso-region wherein both V_E and V_A percolate. The connection to the coexistence envelope of a W-R fluid is the ultimate compelling evidence for the absence of any van der Waals singularity on the density surface of liquid-gas equilibrium or on composition surface of the partially miscible binary liquid at the critical temperature or pressure.

In Section 3 we have reported the direct determination of both percolation loci and along several

isopleths of the W-R fluid. We find that both percolation loci decrease with composition X_B (or X_A) and, around T^* ~ 1, and intersect with two phases then having the same temperature, pressure and minimal chemical potential at the known critical temperature. It is wholly consistent with previous results for the intersection of percolation loci in the p-T plane at the critical temperatures for model square-well fluids [23] , Lennard-Jones fluids [24] , and real fluids including argon [26] [26] and water [27] . To suggest that percolation transitions are unrelated to thermodynamic properties would imply that the intersection of percolation loci at p_c-T_c, in all these systems, would have to be a series of incredible coincidences. This is quite implausible.

The phase behavior (Section 4) for the model penetrable sphere fluid obtained by transcription of the W-R fluid properties, therefore, is further compelling evidence that there is no critical point on Gibbs density surface for liquid-gas equilibria. This simple model liquid-gas Hamiltonian shows a critical dividing line between a maximum coexisting gas density and a minimum coexisting liquid density, above which exists a supercritical mesophase.

We have also shown that the thermodynamic state function rigidity, (dp/dρ)_T, can define a distinction between gas and liquid. For any one-phase system rigidity is everywhere positive; in any two-phase region ω = 0. Rigidity decreases with density for a gas and increases with density for a liquid. For temperatures above critical coexistence the rigidity has a constant value in the mesophase that separates the percolation loci, which bound the limits of existence of liquid and gas phases in the supercritical region.

We have compared the results for the ideal gas percolation ratios with the high-temperature, low pressure and density, limits of argon. The results from the modern NIST thermophysical property tables [23] indicate that the supercritical mesophase, at least in the case of simple fluids, extends all the way to a dilute gas behaving ideally. This reopens the debate “What is liquid?” [1] .

Finally, we have presented results that show there cannot be “universality” in the description of criticality between 2- and 3-dimensional systems. The mesophase is a fundamental property of complementary excluded and available volume percolation loci only in 3d, which does not exist in 2d. Hence, there can be a critical point singularity on the density surface of 2d fluids, whereas it is a critical dividing line for real 3d liquid-gas thermodynamic systems.

We acknowledge this paper is written by invitation of Guest-Editor Prof. Chan Kwong-Yu of the University of Hong Kong in honor of distinguished liquid-state physicist Dr. Yiping Tang. This version has benefited from some referees comments. We remain heartened, however, that there is nothing to vitiate this new science of criticality in any previous referee reports, including those contrived by Molecular Physics editor G. Jackson to reject this new science of liquid-gas criticality. These reviews and rebuttals, together with an acknowledged list of known contributors to this debate, will be published elsewhere. https://www.researchgate.net/profile/Leslie_Woodcock

Leslie V. Woodcock, (2016) Thermodynamics of Criticality: Percolation Loci, Mesophases and a Critical Dividing Line in Binary-Liquid and Liquid-Gas Equilibria. Journal of Modern Physics,07,760-773. doi: 10.4236/jmp.2016.78071