Scaling laws for fluid systems using generalized homogeneous functions of strong and weak variables

Abstract

We present a systematic approach to scaling at ordinary critical points with special emphasis on the critical point of a single-component fluid. Recent work on scaling in fluids has avoided the possibility of a singular coexistence surface. In particular, the consequences of satisfying the inequality $\theta \le \alpha +\beta$ as an equality have not been explored. We show that $\theta =\alpha +\beta$ is a prediction of scaling, and that, if $\theta =\alpha +\beta$, the specific heat at constant volume has a leading-order ($\alpha$-divergent) asymmetry across the coexistence surface. We further show that the asymmetric nature of the fluid critical point precludes the analyticity of the critical isochore above the critical temperature, whether the critical isochore is expressed in terms of $\mu \mathrm{}\left(T\right)\mathrm{}$ or $P\left(T\right)\mathrm{}$. A weak singularity of the form ${|T-T_c|}^{3-2(\alpha +\beta )}$ is predicted for the isochore, which may be dominated by stronger singularities.