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Entropy 2020, 22, 437 4 of 26 This mathematical tour-de-force also, eventually, gave rise to the concept of ‘universality’ first mooted by Uhlenbeck [10], yet accepted as established science by Rowlinson [11], Fisher [12] and Sengers [13], and further extended by Kadanoff and his collaborators [14]. The concept transcending dimensions 1D to 3D was later supported in the three papers on the Kac mean-field model in 1, 2 and 3D [15–17] and renormalization group theory of Wilson [18]. However, universality is inconsistent with our knowledge of percolation transition loci, and their relationships to criticality and supercritical properties of the liquid and gaseous states. In fact, percolation phenomena are non-existent in 1D, and very different between 2D and 3D, thereby invalidating the generalizations of the universality hypothesis. We review below the experimental evidence, spanning 150 years of research, which indicates the existence of a supercritical mesophase, neither liquid nor gas, but a colloidal dispersion in which both gaseous and liquid states can percolate the phase volume [19]. This could not happen in 2D as percolations must coincide. This colloidal mesophase is a new macroscopically homogeneous equilibrium state of matter besides crystal, liquid and gas. It is a single homogeneous Gibbs phase, with two degrees of freedom, but which is inhomogeneous on the nanoscale. The experimental evidence for its existence has been in the literature since the early 1900s, i.e., two years before van der Waals was awarded his Nobel Prize in 1910 for his thesis “On the continuity of gas and liquid states” [3]. 2. Experimental Gas-Liquid Equilibria 2.1. Computer Experiments The supercritical mesophase was discovered originally from computer experiments on model square-well (SW) fluids, which can be understood as a perturbation of the hard-sphere (HS) fluid. The hard-sphere fluid exhibits two percolation transitions of the excluded volume (PE) and accessible volume (PA), between which both complementary volumes (VA + VE = 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 [6,7,20–25]. These results from computer experiments are reproduced from the original reports of the existence of the supercritical mesophase [6,7]. The effect of the square-well perturbation on the HS fluid properties is to reduce the pressure difference between the two percolation transitions PE (or bonded-cluster percolation PB for λ < 2σ) and PA, because there is a much higher cohesion with decreasing temperature at the higher liquid density. The temperature dependence of the percolation transitions is only slight [11]. At a critical temperature (Tc) the two percolation transitions intersect, the pressures and hence the chemical potentials become equal. At the critical percolation intersection state point, Tc-pc, there is a maximum coexisting gas-phase density, and a minimum coexisting liquid-phase density. The hard-sphere mesophase at high temperature or large well-width can be seen to narrow to around 20% in the sticky-sphere limit λ → 0. Computer experimental results show maximum observable coexisting vapour phase densities and minimum coexisting liquid phase densities. Viewed alongside the HS percolation transitions, these results are inconsistent with the theoretical concept of a singular critical density with divergent thermodynamic properties. The observation of a critical divide, and its empirical percolation intercept definition origins, with PB and PA boundaries, extending all the way to the HS limit (kBT/ε → infinity) for any value of well-width λ, defines the existence of a supercritical mesophase.PDF Image | Supercritical Fluid Gaseous and Liquid States
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