_{1}

^{*}

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 (
V_{E}
) and pockets of available volume (
V_{A}
), for a spherical molecule diameter
σ
, percolate the whole volume (
V
=
V_{E}
V_{A}
) of the ideal gas. The molecular-reduced temperature (
T
)/pressure (
p
) ratios (
T
^{*}
=
k_{B}T
/
pσ
^{3}
) 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 1
^{st}
-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.

Almost 40 years ago, in their classic review on the status of liquid state theory [

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-density limit, is simply

where p^{*} is a molecular-reduced pressure (_{B} is Boltzmann’s constant ρ^{*} is a reduced density (^{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 [_{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 _{i}) of species i

Equation (3), with Equation (2), defines

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 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.

We designate the percolation transition reduced temperatures as

d = 1 no percolation

d = 2 PE and PA coincide

d = 3 there is an inequality

There is a fundamental difference between 2 and 3 dimensions. For d = 2, there are two regions, “gas-like” ^{*} > T_{PA}, liquid-like T^{*} < T_{PA} and a mesophase,

PE both for d = 2 and 3 has been investigated by a number of authors for the present and related systems [_{PE} ~ 1.2 and for d = 3Φ_{PE} ~ 0.35, where Φ is the excluded volume fraction (=πρσ^{3}/6). Heyes and coworkers [_{PE} = 0.346 (d = 3). An extensive study of ideal-gas PE states for exclusion squares, cubes, and other geometric shapes has been reported [

Estimates of

There are no reports of PA (d = 3) having been previously investigated or determined for the ideal gas, although the transition density

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. [^{1/3} (

Our method for determining

Rowlinson (W-R) model fluid [

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 _{B} in the same MD simulation run. All the values in _{i}(ρ, N) ® 0, against N (N_{A} in MD run in

We have also determined ^{*} is defined as T^{*} = 1/p^{*} and p^{*} = pσ^{3}/k_{B}T. MD simulations have some advantages over Grand Canonical Monte Carlo [_{B} by integrating the solute cluster probability distribution P(n) which decreases monotonically, from a maximum at n = 1, to zero for clusters of B in solution of A, or vice-verser. Accurate MD pressures are calculated from A-B collision frequencies.^{ }

What is the effect on the percolation transitions of increasing the mole fraction of B from the ideal gas limit (X_{B} = 0)? For the isopleth at X_{B} = 0.1, and for N = 10,000, the reduced pressures at which the two transitions occur, i.e. _{B} = 0.1 and beyond, both percolation pressures increase with X_{B}, PE more so than PA, roughly according to

where_{A} and V_{E} increasing, but V_{E} increases more than V_{A}; adding B causes A-sites to cluster more, whilst creating more spherical B-pockets. The pressures of percolation transitions for finite X_{B} appear to be coincident with higher-order discontinuities (

_{B} = 0.1 has four distinct regions. At high density, in the two-phase region, the MD pressures averaged over 100 million A-B collisions still show fairly large uncertainties. The maximum pressure along any isopleth coincides with the first-order mixing-demixing transition. This reflects the thermodynamic equilibrium condition of minimal Gibbs energy (G) (since_{B} = 0.1 in the mesophase region pressure increases linearly with density. In the one-phase region, the MD data is sufficient to observe that the percolation loci appear to be associated with changes in slope that could reflect higher-order thermodynamic phase transitions, but presently not sufficiently accurate to establish the order or strength of discontinuities.

The vertical dashed lines in

the methods described in the text and referenced [_{B} = 0.5 show three regions; there is no PA, just the PE transition at the density 0.65. The change in the slope of^{*}, is more pronounced. The rigidity function (dp/dρ)_{T} is again constant in the mesophase, with a very slight slope.

From the coexistence pressures, and direct computations of PA and PE, we are able to construct a phase diagram for the W-R binary fluid. The temperature loci of the percolation transitions fit the trendlines (dashed lines in

showing the connection between percolation and the demixing phase transition. Note the perfect symmetry about the isopleth X_{B} = 0.5.

The percolation loci on the T^{*}-X_{B} surface (_{B} an intersection occurs when _{B} when _{B} = 0.339 for the critical coexistence mole fraction.

The present results show a coexistence line at

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 [

where k_{B} is Boltzmann’s constant and T is temperature (K); the angular brackets denote a configurational aver-

age. Equation (5) defines an attractive molecular energy 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 v_{0} (=4πσ^{3}/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 dense fluid with the properties of an ideal gas be described as “liquid”?

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 (

pressure [PCS]

density [PCS]

where ^{*} =μ/k_{B}T and μ is Gibbs chemical potential relative to the ideal gas at the same T, p. G change, hence Z^{*}, can be obtained by integrating the excess pressure loci at constant T, with respect to density._{ }

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 [_{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 [

Rigidity is the work required to increase the density of a fluid; with dimensions of molar energy, it relates directly to the change in Gibbs energy (G) with density at constant T according to

The inequalities that distinguish gas from liquid are:

gas

meso

liquid

It is clear from Equation (9) that ω ³ 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 fundamentally different in each phase. There are many small clusters in a gas with one large void, and many vacant pockets in the liquid with one large cluster.

The isotherms [_{c} = 151 K) to 500 K are plotted in

gression between 0.999999 and 1.0 for all the supercritical isotherms in

where ω_{ }along any isotherm T(K) and p_{0} 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 results presented here for the percolation transition loci comparing both real and model fluids reaffirm previous observations [

Returning to the question about universality and dimensionality dependence of the description of criticality, we observe that for d = 2, since PE = PA for all densities (or concentration X_{B}), the phase behavior and criticality will be quite different; there can be no mesophase in 2-dimensions. We conjecture, therefore, that the percolation locus intersects the equimolar isopleth with a critical singularity at X_{B} = 0.5 for the d = 2 W-R model. Another consequence of the absence of a d = 2 mesophase would be no metastability beyond the first-order phase boundaries, and, unlike d = 3, no metastability and hence no spinodals within the subcritical bimodals. The existence of a mesophase is a property only of d = 3 systems. This difference in the description of liquid-gas criticality between 2 and 3 dimensions vitiates the hypothetical “universality” concept as applied to liquid-gas and binary-liquid, criticality.

Finally, we conjecture an answer to the question of Barker and Henderson “What is liquid?” [

sity. On the basis of these observations, it is the “gas phase”, i.e. defined by the inequality Equation (10) that exists in a limited area of the universal T-p plane. We note, however, that there can be no zero of density/pressure for a real fluid, as they become logarithmic to high vacuum levels. The truly ideal gas is a ficticious concept; the sign of the second virial coefficient determines the designation gas or liquid in this limit. The liquid area of existence extends to infinite pressure and temperature; whereas the gas phase extends to infinite vacuum, but only below a certain temperature. It appears that the liquid phase will extend upwards in temperature, for all pressures to perhaps continuously become plasma (

We acknowledge correspondence regarding the scientific content of this paper courtesy of Prof. G. Jackson Editor of Molecular Physics (Appendix 1) and Prof. R. Piazza Dep. Editor IoP J. Physics (C) (Appendix 2). These discussions add to the previous debates on a new science of liquid-gas criticality published in Natural Science [

Leslie V. Woodcock, (2016) Percolation Transitions of the Ideal Gas and Supercritical Mesophase. Open Access Library Journal,03,1-19. doi: 10.4236/oalib.1102499

The author studies the vapour/liquid and liquid/liquid transitions, and their relation to what looks like two rather artificial percolation thresholds: for clusters with specified bond lengths, and for holes in these clusters. The paper is a theoretical one, using highly approximate theories. The author then plots rather odd looking diagrams, e.g.

Liquid/vapor and liquid/liquid separation has been studied in experiment extensively for well over 100 years. There are clearly critical points, and UCSTs and LCSTs. For example, a number of liquid mixtures separate into two coexisting phases over some T range. On heating, the compositions of the two phases become more and more similar and at a point they become identical. This is the UCST. At higher Ts there is just one liquid phase, and as far as we know thermodynamic variables such as the energy are continuous and have continuous derivatives.

There is a huge amount of data on this. It is not in doubt, as far as I know. It may well be possible to define various percolation thresholds in the single liquid phase but the experimental evidence is that they don’t affect the thermodynamics (or dynamics so far as I know) and they are in effect irrelevant. Science is not about things we cannot measure as they have no effect. I do not see how to try and locate the suggested percolation thresholds. Critical points are well established, http://www.kayelaby.npl.co.uk/chemistry/3_5/3_5.html for a large number of measured liquid/vapour critical points.

Author: in the 150 year history of experimental measurements of critical parameters, as tabulated in in “Kaye and Laby”, and all other literature compilations e.g. NIST Thermophysical Properties (reference [_{c.}, usually in conjunction with a cubic equation-of-state, or similar {see e.g. S. Reif-Acherman “History of the law of rectilinear diameters”, Quimica Nova, 33, 2003-2013 (2010)}. This also applies to the critical densities of computer models using Gibbs ensemble Monte Carlo methods e.g. the various papers of Jackson and coworkers: see references [_{c} are flat on top. The hypothetical “critical point” has been obtained using a priori hypothesis of existence, and defined only by numerical parameterization.

(Henderson continued) In conclusion, as the predictions made using highly approximate theories disagree with experiments, unlike a large body of existing theory that agrees with experiment, I cannot recommend publication. Theory should help us understand experimental data, and make predictions that are testable and may be right, not disagree with experimental data when that data is established beyond reasonable doubt.

Author: What “highly approximate theories”? There is no theory in this paper, just results. The simulation results we present for the W-R mixture and PCS fluid should stimulate further laboratory experiment to explore the percolation loci. That’s my understanding of the way science works!

I do not think any revision will make this manuscript acceptable for publication.

Review 2: Prof. D. Frenkel (Cambridge) and Prof. G. Jackson (IC London)This paper is a further contribution in the author’s series on the theme of percolation lines, mesophases and the absence of a critical point. The present paper is very much in the same spirit as these earlier papers. Indeed Sec.5 tells much the same story about the interpretation of the argon phase diagram as that in several earlier Woodcock contributions. Although the Introduction reads somewhat differently from the earlier papers, it becomes abundantly clear, by pgs. 8, 9, that Woodcock continues to deny the existence of a liquid-gas or a liquid-liquid critical point. Rather he maintains that liquid-gas coexistence terminates in a “critical” coexistence line where lines of percolation transitions meet the coexistence curve. These lines bound what Woodcock terms a “supercritical mesophase”. An example is given, for a square-well model, in figure 11 of [

In Sec.1 the author claims “percolation transitions are known to determine thermodynamic phase transitions in model lattice gases”. He then says “percolation transitions of the … are properties relating to Gibbs energies that effect phase transitions”. Both statements are staggering. Both are erroneous.

Author: There are various references to the connection between percolation and changes in thermodynamic state functions can be found in reference [_{A} or V_{E} to chemical potentials, which determine equilibrium between phases. Both statements are essentially correct.

These two statements set the flavour of the remainder of the paper. Section 2 describes the determination of what the author designates ideal gas percolation transitions. As far as I can glean this refers to percolation in interpenetrating (ideal) objects in the continuum. He appears to argue that the percolation threshold had not been determined for disks in d = 2 or for spheres in d = 3 and proceeds to estimate these. A glance at Wikipedia or, indeed, at the paper of Baker et al. [

Author: I was not aware of these papers and apologise to the authors for their omission: these are indeed highly accurate determinations of V_{E} percolation thresholds. For d = 2, the correct reference is J. Quintanilla, S. Torquato and R. M. Ziff, J. Phys. (A) Math. Gen. 399-407 (2000) and for d = 3, C. D. Lorentz and R. M. Ziff, J. Chem. Phys. 114(8), 3659-61 (2001).

Bizarrely the author chooses to present values in terms of a reduced temperature T^{*}. This follows from an assumption that T^{*} = 1/ρ^{*} for this ideal system.

Author: 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 the choice of T^{*}.

There are no new results in Section 2.

Author: In this statement Frenkel and Jackson miss the central point of the paper. The essential new result is for the percolation of V_{A} i.e. T*_{PA} in 3d (_{PE}. Between these transition temperatures, both V_{E} and V_{A} percolate, hence the mesophase exists in 3 dimensions, not 2. It is an important new result because as solute concentration (X_{B}) increases in the W-R fluid, two percolation temperatures come together and coincide. This intersection triggers the first-order phase transition, with the two different phases having the same T, p and chemical potential. It is this fundamental property of percolation in 3d that does not exist in 2d that vitiates the hypothetical concept of universality, and confirms the new science of criticality for both liquid-gas, and now also liquid-liquid coexistence.

In Section 3 the author turns attention to the W-R model in the binary mixture representation. In this version the pair interactions are either zero (like-like) or hard (unlike). Thus the system is strictly athermal; temperature plays no role in determining phase behaviour. Yet Woodcock introduces a T^{*}. It is not clear to me how this is defined for binary W-R mixture and what its relevance might be.

Author: I do not consider a reversible isobaric thermal expansion or contraction process to be “athermal”. The work in expanding or compressing the equilibrium W-R fluid along an isopleth is associated with a reversible heat dq_{rev} that is non-zero; this enthalpy change increases the temperature and entropy on expansion. Thus, T^{*} is here the natural reduced state variable for comparison with experiment (see ^{*} for W-R model is defined T^{*} = 1/p^{*} and p^{*} = pσ^{3}/k_{B}T (I thought it was obvious). The definition is now included in the text.

In _{B}. These display PE and PA transitions as well as a sharp 2-phase boundary at large ρ^{*}. How was this phase boundary determined? No indication is provided other than some mention of a maximum pressure.

Author: Along any isopleth, the pressure is a maximum at the two-phase boundary to comply with the thermodynamic requirement of minimum Gibbs energy as shown in _{B} = 0.5 equimolar isopleth (probably more accurately than [

By contrast [_{B}. For X_{B} = 0.1 rho^{*} ~0.86. Because of the symmetry of the W-R model the fluid-fluid critical point must be at X_{B} = 0.5.

Author: This statement is the original hypothesis of Widom and Rowlinson [_{B} = 0.5 for the 3D WR fluid!

Thus in

Author: Yes, this line is the calculated critical density.

At this stage it is useful to consider ^{*} re-appears and

emerges from the mist. And there is a splendid sentence “…gives X_{B} = 0.339 for the critical coexistence mole fraction.” This marks the onset of Woodcock’s critical coexistence line referred to above. _{B} phase diagram.

Author: It appears that the reviewers have misunderstood the definition of reduced temperature T^{*}. (I have dealt with this above) The phase diagram could just as easily be presented as p^{*}, ρ^{*} or V^{*}, it would not change the science. I choose T^{*} to identify directly with real binary-liquid experimental phase diagrams that exhibit a similar upper critical consolute temperature (see

And then to speak of “liquid states” of the ideal gas is curious, to say the least. I think the author is plotting in this strange transcription what he thinks is equivalent to the density vs. X_{B} phase diagram, now with the accompanying MESO phase, that should replace figure 1 of [

Author: I expect it is going to take a while for the scientific community to adjust to a novel language of liquid-state criticality. p(ρ) isotherms are the “natural representation” of liquid-gas coexistence and criticality.

Section 4 switches to the one-component isomorphism of W-R. Now temperature is a natural variable so the T* vs. rho* representation is the natural choice for plotting phase diagrams. This is abundantly clear in the original W-R paper [

Author: Experimental thermodynamic liquid-gas coexistence phase diagrams have always been obtained along isotherms by measurements of coexistence pressures, going all the way back to Andrews 1876 pressure v. density isotherms of CO_{2} (_{T} moreover is a fundamentally important state function that distinguishes gas from liquid as seen in

Author: Hence, unlike [

The final paragraph of Section 4 goes off on a tangent.

Author: It is relevant commentary: the new results in Section 4 show that there is no continuity of liquid and gas in supercritical equations of state equations of state which we now find can extend all the way to the ideal gas limit. The last paragraph of section 4 therefore considers the implications of this result, i.e. a requirement for three different functional forms for the equation-of-state of gas, liquid and meso.

Section 5 is concerned with the fluid phases of argon. A reasonable person might assume these were accurately established many years ago. Not so according to the current author. He re-visits his earlier analyses (see my earlier comments) and argues, yet again, for an interpretation in terms of a mesophase―see _{T}; this quantity is proportional to the inverse isothermal compressibility and must therefore be >0 for an equilibrium fluid. But on pg. 10 the discussion moves to the derivative of ω_{T} w.r.t. rho. This is the third density derivative of a free energy density. (And it happens to be something I know about.) The sign of this derivative (Equations (10)-(12)) is deemed to point to putative phase boundaries/percolation transitions. Here I lose the plot. Are the percolation transitions supposed to be related directly to the thermodynamic criteria of Equations (10)-(12)?

Author: Yes, empirically: a recent more detailed analysis is now published [

From reasonably realistic MF eqns.-of-state, one does find regions where the sign of this particular derivative changes sign. But there is no fundamental significance in this. As far as I know there is no connection with percolation transitions. Thus I fail to see what the author is arguing.

Author: On the contrary, here we extend the analysis to the high-temperature, or low-pressure, dilute ideal gas limit.

In summary, this paper appears to be an attempt by the author to bring his (curious) ideas into the realm of fluid-fluid phase separation. He has failed. This paper is not acceptable in Mol. Phys. It should be rejected.

Review 3: Prof. R. Evans (Bristol)This is the latest in a series of papers by Woodcock and co-workers extolling the conjecture of the non-existence of a fluid critical point, being subsumed instead by an extensive “mesophase” region contained between to percolation boundaries.

There is a wealth of experimental data on measurements of the divergence of thermodynamic properties close to critical points in broad ranges of systems including magnetic materials, metallic alloys, liquid crystals, plasmas and, of course, fluids and fluid mixtures. Vapor-liquid criticality in pure carbon dioxide was first recognized and studied by Andrews in the 1860s and inspired van der Waals when he published his thesis on the equation of state of gases and liquid which lead to the award of the Nobel Prize in Physics 1910. The existence of critical points in fluids and materials and is now very well established.

Though observed in very different systems, criticality is found to be universal in nature: Wilson was awarded a Nobel Prize in Physics 1982 for his theoretical work unifying critical phenomena in connection with phase transitions. There is certainly no experimental evidence supporting the view put forward by Woodcock and colleagues. I am afraid that I cannot support the premise on which this paper is based and must recommend rejection.

Author: The word “found” should read “believed” as it is unsubstantiated hypothesis. Extensive and compelling experimental evidence against van der Waals hypothesis of a liquid-gas “critical point”, and “continuity” of liquid and gas, and latterly “universality”, has been out there for 150 years, but many prominent theoreticians have overlooked it.

Evans: I take issue with Sections 9 - 12 of reference [

Author: On the contrary, Bernal’s paper shows remarkable insight. Bernal first recognised what is sometimes now referred to as the “Widom line” (10 years before Widom, and 30 years before his colleague E. Stanley called it the “Widom line”). Bernal correctly predicted implications for the non-continuity of liquid and gas in the supercritical region in contrast to van der Waals hypothesis.

At present I teach a 4th year undergraduate course on Phase Transitions. One of the key messages I attempt to get across is the deep connection between Ising and fluid systems regarding the commonality of their criticality.

Author: There is of course some similarity of an Ising singularity, and a gas-liquid transition from two phases to one (the supercritical mesophase) along an isochore on crossing the critical divide. However, there is no critical point on the Gibbs density surface. Ising models cannot describe gas-liquid criticality because they are not capable of doing work like Gibbsian systems, which obey the 1^{st} and 2^{nd} laws regarding heat and work, and do not have a Gibbs chemical potential. Consequently phase transitions on a liquid-gas Gibbs surface may behave quite differently from the lattice gases, such as Onsager’s exact result for the 2D Ising model.

I trust that I do not have to explain to you that the evidence for equivalent critical behaviour is overwhelming. One can go back to Domb, Widom, Fisher, Kadanoff, K. Wilson, J. Sengers and move forward to the beautiful and compelling work of Wilding & Bruce (reviews by Wilding), Parola & Reatto. The last item has made it into the latest issue of Hansen & MacDonald. There are even recent excellent movies on U Tube by D. Ashton comparing, via simulation,near-critical configurations in Ising with those in a LJ fluid.

Author: The literature that assumes the existence of a Ising-like singularity on 3D Gibbsian density surfaces is indeed overwhelming, and that is the problem we are having. This “club of luminaries” have built their careers upon the myth of “universality” and produced an abundance of theory that is largely detached from the real experimental literature of liquid-gas thermodynamic properties of criticality. For example, take Wilson’s 30-page Nobel address in 1982. There is one brief 100-word paragraph which is an incorrect statement about the densities of water and steam near T_{c} [

All these demonstrate fluids are equivalent to Ising in the nature of their criticality. Surely you will agree that 1) Onsager proved there is a critical point in the d = 2 Ising model and 2) there must be a critical point in d = 3 Ising. A plethora of MC studies confirm the latter and a similar number of careful MC studies treating LJ and closely related model fluids confirm that the density probability functions of the fluid map precisely to those of the appropriate Ising model. Indeed this is how the critical point is located and exponents determined! (It was not for nothing that K. Wilson won the Nobel Prize.)

Author: The “plethora of MC studies” are all prejudiced by the a priori assumption of a singular point at T_{c} at the outset of the simulations. The paper with Dave Heyes [

http://dx.doi.org/10.1016/j.fluid.2013.07.056

Do you doubt the existence of a fluid critical point with the accompanying divergence of the compressibility and the Ornstein-Zernike-like treatment of critical opalescence?

Author: Of course there is a critical temperature along any isochore that spans the critical dividing line between gas and liquid, and hence a singular point on the Gibbs p, T projection surface. There is also a divergence of the compressibility at T_{c} along any isochore due to the fact that dG = pdV =0 when (dp/dV)_{T} is zero at T_{c}. The Ornstein-Zernike treatment has long been open to question (see your contribution to debate in Natural Science below). The phenomenon of critical opalescence, in the supercritical region, is probably explained by “Tyndall scattering”; a scattering effect associated with the colloidal nature of supercritical mesophases. See e.g. Natural Science (2014) 6 (6) 797-807.

http://dx.doi.org/10.4236/ns.2014.610078

(An earlier version of this paper was submitted to J. Phys. (C) for consideration as a fast-track breakthrough communication. On the following opinions of sub-editor Piazza, it was initially rejected without review.)

JPCM Editor Liquids and complex fluids: Prof. Dr. R. Piazza: Milan Polytechnic.

In this paper the author makes the claim that an ideal gas has a liquid phase, on the basis of a re-definition of what it means be a “liquid”. This is in addition to his earlier claim that the liquid-vapour critical point does not exist. If true, either of these would overthrow more than a hundred years of statistical mechanics of liquids and phase transitions. Now while this is not necessarily a bad thing―indeed, science progresses by successfully challenging received wisdom―it is also true that “extraordinary claims require extraordinary evidence”. In my view, the latter is lacking, as the author’s statements appear to be underpinned mostly by his own work (refs. [

Author: In refusing to have this paper reviewed, Piazza contradicts, without any scientific basis, the general conclusions of two previously related articles, references [

[On receiving this rebuttal appeal, Dep. Editor Piazza referred it to one of his colleagues] Quote:

It is known that there is no symmetry difference between the coexisting gas and liquid phases, contrary to other systems, like ferromagnets or superfluids. This circumstance, opens the way to several alternative definitions of liquid state, which is usually regarded just as the high density portion of the fluid phase below the critical temperature. In this manuscript the author advocates one of such definitions.

Author: I know of no previous “alternative thermodynamic definition” of the liquid state.

He first examines the location of two percolation transition lines in ideal gases. These lines do not have any thermodynamic signature in the equilibrium properties of the (ideal) gas.

Author: This statement appears to be based upon a misapprehension that the “ideal gas” actually exists. In any real gas, the density cannot go to zero, it is logarithmic. The percolation transitions do indeed “have a signature” in ALL real gases behaving ideally (i.e. when p = ρkT) in the low-density limit.

The liquid phase is here identified as the region at density higher than both transition lines.

Author: The liquid phase is identified as the density range within which the excluded volume percolates but the free volume does not!

Then, in a somewhat arbitrary way, this definition is “generalized” into Equation (12), whose relationship to percolation is obscure to me. The author also claims that a “mesophase” is present and is defined by Equation (11).

Author: The experimental evidence for the existence of the colloidal mesophase is overwhelming, and can be found in references [

In systems where extensive and very accurate simulations are available (e.g. the hard sphere fluid) there is no region in the phase diagram where Equation (11) holds (the vanishing of this derivative occurs only along a line).

Author: The derivative (dp/dρ)_{T} does not vanish except at T_{c}, in the mesophase it becomes constant (

A further claim of the author, whose relationship with the previous points is not explained, regards the absence of a critical singularity in the thermodynamic properties. I cannot see why a definition of liquid state based on the existence of percolation lines should have any implication on the thermodynamics of the system.

Author: It is classical thermodynamics. When the two percolation lines that bound the existence of liquid and gas phases intersect on the Gibbs density surface the respective phases have the same temperature and pressure but different densities, and hence the same Gibbs chemical potential, and hence coexist at a first-order phase transition.

In summary: the manuscript mainly deals with an operative definition of liquid state.

Author: On the contrary, this manuscript deals mainly with new results for the percolation transitions of an ideal gas and loci for model fluids that are valuable contributions to the literature and lead to an alternative interpretation of the phase behavior of the W-R model binary liquid and the one-component liquid gas fluid. A possible alternative operative definition of “liquid state” is a corollary of these findings.

The author addresses this issue first by examining the possible occurrence of percolation transitions.

Author: The phrase “possible occurrence” of percolation transitions is inappropriate: one only needs look at references [

Then the author puts forward several statements regarding the thermodynamic behavior of the fluid, whose relationship with the previous analysis I cannot see: the equivalence between percolation transitions and “rigidity”, the presence of a mesophase with definite thermodynamic signature, the absence of a critical singularity. These statements are very provocative and require a much more convincing demonstration.

Author: The “convincing demonstration” is in references [

In conclusion I cannot recommend publication of this manuscript.

Appeal adjudicator: Editor-in-Chief: IoP Journal of Physics Condensed Matter

Dr. J. S. Gardner: NIST Center for Neutron Research Gaithersburg, Maryland USA

“I agree with the two referees that this paper should not be published in JPCM”