Scaling with a fractional power: Overcoming the shortage of scaled-particle variables for the hard-disk fluid

Within scaled-particle theory, we construct an equation of state (EOS) for hard-disk mixtures by making use of an additional scaled-particle variable which weighs the densities of the different components by its radii to the power χ. This allows us to simultaneously respect exact results pertaining to the cases of a large particle or a point particle being added to the mixture. In the limit χ → 2, the mixture EOS of Santos et al. [Mol. Phys. 96, 1 (1999)] is recovered, while the limit χ → 0 yields the accurate expression for the interfacial free energy of Martin et al. [J. Chem. Phys. 149, 084701 (2018)]. From the low-density limit of the EOS, the value χ ≈ 0.8 is extracted, which is shown to yield a mixture EOS that is significantly more accurate than the expressions due to Santos et al. and Martin et al. In particular, the systematic deviation inherent to these prior results is remedied.