Year: 2007
Type: article
Abstract: The master asymptotic behavior of the usual parachor correlations, expressing surface tension $\ensuremath{\sigma}$ as a power law of the density difference ${\ensuremath{\rho}}_{L}\ensuremath{-}{\ensuremath{\rho}}_{V}$ 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 ${({\ensuremath{\beta}}_{c})}^{\ensuremath{-}1}$, ${\ensuremath{\alpha}}_{c}$, ${Z}_{c}$, 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 ${({\ensuremath{\beta}}_{c})}^{\ensuremath{-}1}$ and the length unit ${\ensuremath{\alpha}}_{c}$, 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 $\frac{\ensuremath{\delta}\ensuremath{-}1}{\ensuremath{\delta}+1}=\frac{\ensuremath{\eta}\ensuremath{-}2}{2d}$, we show that the Ising-like asymptotic value ${\ensuremath{\pi}}_{a}$ of the parachor exponent is unequivocally linked to the critical exponents $\ensuremath{\eta}$ or $\ensuremath{\delta}$ by $\frac{{\ensuremath{\pi}}_{a}}{d\ensuremath{-}1}=\frac{2}{d\ensuremath{-}(2\ensuremath{-}\ensuremath{\eta})}=\frac{\ensuremath{\delta}+1}{d}$ (here $d=3$ is the space dimension). Such mixed hyperscaling laws combine either the exponent $\ensuremath{\eta}$ or the exponent $\ensuremath{\delta}$, which characterizes bulk critical properties of $d$ dimension along the critical isotherm or exactly at the critical point, with the parachor exponent ${\ensuremath{\pi}}_{a}$ which characterizes interfacial properties of $d\ensuremath{-}1$ dimension in the nonhomogeneous domain. Then we show that the asymptotic (symmetric) power law ${({\ensuremath{\alpha}}_{c})}^{d\ensuremath{-}1}{\ensuremath{\beta}}_{c}\ensuremath{\sigma}={D}_{\ensuremath{\rho}}^{\ensuremath{\sigma}}{(\frac{{\ensuremath{\rho}}_{L}\ensuremath{-}{\ensuremath{\rho}}_{V}}{2{\ensuremath{\rho}}_{c}})}^{{\ensuremath{\pi}}_{a}}$ is the two-dimensional critical equation of state of the liquid-gas interface between the two-phase system at constant total (critical) density ${\ensuremath{\rho}}_{c}$. This power law complements the asymptotic (antisymmetric) form $({\ensuremath{\mu}}_{\ensuremath{\rho}}\ensuremath{-}{\ensuremath{\mu}}_{\ensuremath{\rho},c})\frac{{\ensuremath{\rho}}_{c}}{{p}_{c}}=\ifmmode\pm\else\textpm\fi{}{D}_{\ensuremath{\rho}}^{c}{\ensuremath{\mid}\frac{\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{c}}{{\ensuremath{\rho}}_{c}}\ensuremath{\mid}}^{\ensuremath{\delta}}$ of the three-dimensional critical equation of state for a fluid of density $\ensuremath{\rho}\ensuremath{\ne}{\ensuremath{\rho}}_{c}$ and pressure $p\ensuremath{\ne}{p}_{c}$, maintained at constant (critical) temperature $T={T}_{c}$ [${\ensuremath{\mu}}_{\ensuremath{\rho}}\phantom{\rule{0.2em}{0ex}}({\ensuremath{\mu}}_{\ensuremath{\rho},c})$ 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 ${D}_{\ensuremath{\rho}}^{c}{({D}_{\ensuremath{\rho}}^{\ensuremath{\sigma}})}^{d∕(1\ensuremath{-}d)}={R}_{D\ensuremath{\sigma}}=\text{universal}$ constant, constructed with the amplitudes ${D}_{\ensuremath{\rho}}^{\ensuremath{\sigma}}$ and ${D}_{\ensuremath{\rho}}^{c}$, 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.
Source: Physical Review E
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1980740847