Analytic representation of critical equations of state

Abstract

We propose a new form for equations of state (EOS) of thermodynamic systems in the Ising universality class. The new EOS guarantees the correct universality and scaling behavior close to critical points and is formulated in terms of the scaling fields only -- unlike the traditional Schofield representation, which uses a parametric form. Close to a critical point, the new EOS expresses the square of the strong scaling field $\Sigma$ as an explicit function $\Sigma^2=D^{2e_{-1}}\Gamma(D^{-e_0}\Theta)$ of the thermal scaling field $\Theta$ and the dependent scaling field $D>0$, with a smooth, universal function $\Gamma$ and the universal exponents $e_{-1}=\delta/(\delta+1)$, $e_0=1/(2-\alpha)$. A numerical expression for $\Gamma$ is derived, valid close to critical points. As a consequence of the construction it is shown that the dependent scaling field can be written as an explicit function of the relevant scaling fields without causing strongly singular behavior of the thermodynamic potential in the one-phase region. Augmented by additional scaling correction fields, the new EOS also describes the state space further away from critical points. It is indicated how to use the new EOS to model multiphase fluid mixtures, in particular for vapor-liquid-liquid equilibrium (VLLE) where the traditional revised scaling approach fails.