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A critical pressure along any isotherm is defined by the intersection of the two percolation transitions. Above Tc, the liquid percolation has the higher pressure, below Tc 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 Entropy 2020, 22, 437 5 of 26 the critical point Tc-pc 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 > Tc. 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 square-well liq-vap coexistence curves LIQUID STATES red 1.005 orange 1.25 lime 1.375 green 1.5 blue 1.75 indigo 2.0 violet 3.0 black 5.0 ρpb(λ) VAPOR STATES PERCOLATION TRANSITIONS PA PE 0.1 1 10 100 temperature (kBT/ε) Figure 3. Coexistence curves of square-well fluids from references [6,7,19–24] with empirical limiting Figure 3. Coexistence curves of square-well fluids from references [6,7,19–24] 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 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 densities of Elliott and Hu [21]; densities at l = 3.0 from Benavides et al. [22]; densities and curves for λ λ = 1.005 and λ = 5 [6,7]; the upper dashed line is the hard-sphere (HS) available volume percolation = 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 transition loci and the lower horizontal dashed lines are the extended volume percolation transition loci, of the hard-sphere reference fluid [6,7]; ρPB(T) is the high-T limit, λ-dependent, bonded-cluster loci, of the hard-sphere reference fluid [6,7]; ρPB(T) is the high-T limit, λ-dependent, bonded-cluster percolation density at Tc of square-well fluids. percolation density at Tc of square-well fluids. A critical pressure along any isotherm is defined by the intersection of the two percolation Since the original discovery [6,7] of the mesophase in the square-well fluids, equally compelling transitions. Above Tc, the liquid percolation has the higher pressure, below Tc the (metastable) empirical evidence for the supercritical divide and a supercritical mesophase have been reported for liquid percolation loci, previously known as spinodal, have the lower pressure. At the intersection, other model molecular Hamiltonians, with essentially the same conclusions. An extensive computer the two states, gas and liquid, have the same T and p, and hence also the same μ (chemical potential), simulation study of the Lennard–Jones (L-J) fluid [25] reported a critical density hiatus and a but different densities. Two coexisting states with constant T,p,μ at the percolation loci intersection supercritical mesophase with linear isothermal state functions of density. Note that there are no define the critical point Tc-pc in the T-p plane, and a line of critical transitions from 2-phase coexistence impurities or gravitational potential in computer experiments! The pressure-density data points were (P = 2, F =1) to a single phase (P =1, F = 2), for T > Tc. computed for systems of 10,000 and 32,000 L-J atoms for 2000 state points along 6 near-critical Since the original discovery [6,7] of the mesophase in the square-well fluids, equally compelling isotherms with the results shown in Figure 4. The critical isotherm shows a perfectly horizontal empirical evidence for the supercritical divide and a supercritical mesophase have been reported straight line between coexisting maximum gas and minimum liquid densities. Significantly, there was for other model molecular Hamiltonians, with essentially the same conclusions. An extensive no observable particle number dependence that one might expect to see in the vicinity of any critical computer simulation study of the Lennard–Jones (L-J) fluid [25] reported a critical density hiatus singularity with divergent properties. and a supercritical mesophase with linear isothermal state functions of density. Note that there are no impurities or gravitational potential in computer experiments! The pressure-density data points were computed for systems of 10,000 and 32,000 L-J atoms for 2000 state points along 6 near-critical isotherms with the results shown in Figure 4. The critical isotherm shows a perfectly horizontal straight line between coexisting maximum gas and minimum liquid densities. Significantly, there was no observable particle number dependence that one might expect to see in the vicinity of any critical singularity with divergent properties. Another advantage of computer simulation is that detailed structural data can be obtained that helps to explain the phenomenological behaviour at the molecular level. The Lennard–Jones fluid was found to have a critical isotherm at kTc/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 density (Νσ 3/V)PDF Image | Supercritical Fluid Gaseous and Liquid States
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