Master crossover functions for one-component fluids.

By introducing three well-defined dimensionless numbers, we establish the link between the scale dilatation method able to estimate master (i.e., unique) singular behaviors of the one-component fluid subclass and the universal crossover functions recently estimated [Garrabos and Bervillier, Phys. Rev. E 74, 021113 (2006)] from the bounded results of the massive renormalization scheme applied to the Phi(d)(4)(n) model of scalar order parameter (n=1) and three dimensions (d=3), representative of the Ising-like universality class. The master (i.e., rescaled) crossover functions are then able to fit the singular behaviors of any one-component fluid without adjustable parameter, using only one critical energy scale factor, one critical length scale factor, and two dimensionless asymptotic scale factors, which characterize the fluid critical interaction cell at its liquid-gas critical point. An additional adjustable parameter accounts for quantum effects in light fluids at the critical temperature. The effective extension of the thermal field range along the critical isochore where the master crossover functions seems to be valid corresponds to a correlation length greater than three times the effective range of the microscopic short-range molecular interaction.

Master singular behavior for the Sugden factor of one-component fluids near their gas-liquid critical point.

We present the master (i.e., unique) behavior of the squared capillary length-the so-called Sugden factor-as a function of the temperaturelike field along the critical isochore, asymptotically close to the gas-liquid critical point of about twenty (one-component) fluids. This master behavior is obtained using the scale dilatation of the relevant physical fields of the one-component fluids. The scale dilatation method introduces the fluid-dependent scale factors in a manner analog to the linear relations between physical fields and scaling fields needed by the renormalization theory applied to any physical system belonging to the Ising-like universality class. The master behavior for the Sugden factor satisfies hyperscaling. It can be asymptotically fitted by the leading terms of the theoretical crossover functions for the correlation length and the susceptibility in the homogeneous domain, recently obtained from massive renormalization in field theory. In the absence of corresponding estimation of the theoretical crossover functions for the interfacial tension, we define the range of the temperaturelike field where the master leading power law can be practically used to predict the singular behavior of the Sugden factor, in conformity with the theoretical description provided by the massive renormalization scheme within the extended asymptotic domain of the one-component fluid "subclass."

Master crossover behavior of parachor correlations for one-component fluids.

The master asymptotic behavior of the usual parachor correlations, expressing surface tension sigma as a power law of the density difference rho(L)-rho(V) 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 (beta(c))-1 , alpha(c) , Z(c) , and Y(c) , for each fluid. This is accomplished by the scale dilatation method of the fluid variables where, in addition to the energy unit (beta(c))-1 and the length unit alpha(c) , the dimensionless numbers Z(c) and Y(c) 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 delta-1/delta+1=eta-2/2d , we show that the Ising-like asymptotic value pi(a) of the parachor exponent is unequivocally linked to the critical exponents eta or delta by pi(a)/d-1=2/d-(2-eta)=delta+1/d (here d=3 is the space dimension). Such mixed hyperscaling laws combine either the exponent eta or the exponent delta , which characterizes bulk critical properties of d dimension along the critical isotherm or exactly at the critical point, with the parachor exponent pi(a) which characterizes interfacial properties of d-1 dimension in the nonhomogeneous domain. Then we show that the asymptotic (symmetric) power law [abstract; see text] is the two-dimensional critical equation of state of the liquid-gas interface between the two-phase system at constant total (critical) density rho(c) . This power law complements the asymptotic (antisymmetric) form [abstract; see text] of the three-dimensional critical equation of state for a fluid of density rho not equal to rho_(c) and pressure p not equal to p_(c) , maintained at constant (critical) temperature T=T_(c)} [mu_(rho)(mu_(rho,c)) is the specific (critical) chemical potential; p_(c) is the critical pressure; and T_(c) is the critical temperature]. We demonstrate the existence of the related universal amplitude combination [abstract; see text] = universal constant, constructed with the amplitudes D_(rho)(sigma) and D_(rho)(c) , separating then the respective contributions of each scale factor Y_(c) and Z_(c) , 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.

Master singular behavior from correlation length measurements for seven one-component fluids near their gas-liquid critical point.

We present the master (i.e., unique) behavior of the correlation length, as a function of the thermal field along the critical isochore, asymptotically close to the gas-liquid critical point of xenon, krypton, argon, helium-3, sulfur hexafluoride, carbon dioxide, and heavy water. It is remarkable that this unicity extends to the correction-to-scaling terms. The critical parameter set, which contains all the needed information to reveal the master behavior, is composed of four thermodynamic coordinates of the critical point and one adjustable parameter which accounts for quantum effects in the helium-3 case. We use a scale dilatation method applied to the relevant physical variables of the one-component fluid subclass, in analogy with the basic hypothesis of the renormalization theory. This master behavior for the correlation length satisfies hyperscaling. We finally estimate the thermal field extent where the critical crossover of the singular thermodynamic and correlation functions deviates from the theoretical crossover function obtained from field theory.