2007/07/13 by Yves Garrabos, Fabien Palencia, Carole Lecoutre +4
Engineering · Materials Science · Physics and Astronomy · #Material Dynamics and Properties #Phase Equilibria and Thermodynamics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.76.061109
published as Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 76, 6 (2007) 061109 (22 p.)
arxiv created 2007/07/13 · openalex publication_date 2007/12/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The master asymptotic behavior of the usual parachor correlations, expressing surface tension \ensuremathσ as a power law of the density difference \ensuremathρL\ensuremath-\ensuremathρV between coexisting liquid and vapor, is analyzed for a series of pure compounds close to their liquid-vapor critical point, using only four critical parameters (\ensuremathβc)^\ensuremath-1, \ensuremathαc, Zc, and Yc, for each fluid. This is accomplished by the scale dilatation method of the fluid variables where, in addition to the energy unit (\ensuremathβc)^\ensuremath-1 and the length unit \ensuremathαc, the dimensionless numbers Zc and Yc are the characteristic scale factors of the ordering field along the critical isotherm and of the temperature field along the critical isochore, respectively. The scale dilatation method is then formally analogous to the basic system-dependent formulation of the renormalization theory. Accounting for the hyperscaling law \frac\ensuremathδ\ensuremath-1\ensuremathδ+1=\frac\ensuremathη\ensuremath-22d, we show that the Ising-like asymptotic value \ensuremathπa of the parachor exponent is unequivocally linked to the critical exponents \ensuremathη or \ensuremathδ by \frac\ensuremathπad\ensuremath-1=\frac2d\ensuremath-(2\ensuremath-\ensuremathη)=\frac\ensuremathδ+1d (here d=3 is the space dimension). Such mixed hyperscaling laws combine either the exponent \ensuremathη or the exponent \ensuremathδ, which characterizes bulk critical properties of d dimension along the critical isotherm or exactly at the critical point, with the parachor exponent \ensuremathπa which characterizes interfacial properties of d\ensuremath-1 dimension in the nonhomogeneous domain. Then we show that the asymptotic (symmetric) power law (\ensuremathαc)^d\ensuremath-1\ensuremathβc\ensuremathσ=D_\ensuremathρ^\ensuremathσ(\frac\ensuremathρL\ensuremath-\ensuremathρV2\ensuremathρc)^\ensuremathπa is the two-dimensional critical equation of state of the liquid-gas interface between the two-phase system at constant total (critical) density \ensuremathρc. This power law complements the asymptotic (antisymmetric) form (\ensuremathμ_\ensuremathρ\ensuremath-\ensuremathμ_\ensuremathρ,c)\frac\ensuremathρcpc=\ifmmode±\else\textpm\fiD_\ensuremathρc\ensuremath|\frac\ensuremathρ\ensuremath-\ensuremathρc\ensuremathρc\ensuremath|^\ensuremathδ of the three-dimensional critical equation of state for a fluid of density \ensuremathρ\ensuremath≠\ensuremathρc and pressure p\ensuremath≠pc, maintained at constant (critical) temperature T=Tc [\ensuremathμ_\ensuremathρ\phantom\rule0.2em0ex(\ensuremathμ_\ensuremathρ,c) is the specific (critical) chemical potential; pc is the critical pressure; and Tc is the critical temperature]. We demonstrate the existence of the related universal amplitude combination D_\ensuremathρc(D_\ensuremathρ^\ensuremathσ)^d∕(1\ensuremath-d)=R_D\ensuremathσ=universal constant, constructed with the amplitudes D_\ensuremathρ^\ensuremathσ and D_\ensuremathρc, separating then the respective contributions of each scale factor Yc and Zc, characteristic of each thermodynamic path, i.e., the critical isochore and the critical isotherm (or the critical point), respectively. The main consequences of these theoretical estimations are discussed in light of engineering applications and process simulations where parachor correlations constitute one of the most practical methods for estimating surface tension from density and capillary rise measurements.