The nature of the rectilinear diameter singularity

TL;DR

This paper provides a rigorous explanation for the singularity in the rectilinear diameter equation, constructs an optimal order parameter, and demonstrates the universal ratio of amplitudes near critical points, supported by analysis of argon and water.

Contribution

It offers a rigorous theoretical explanation for the rectilinear diameter singularity and constructs an optimal symmetric order parameter within the canonical formalism.

Findings

The ratio of amplitudes is universal across substances.

The critical exponent β is approximately 0.33 for argon and water.

The singularity term |t|^{2β} is rigorously justified.

Abstract

The rigorous explanation for the term $| t |^{2\beta}$ in the rectilinear diameter equation is given ($t = (T_c-T)/T_c$, $\beta$ is the critical exponent for the asymptotic form of the equation of state). The optimal order parameter, for which the branches of binodal are symmetric is constructed within the canonical formalism. It is shown that the ratio of the amplitudes $\f{D_{2\beta}}{D_{1-\alpha}}$ before $|t|^{2\beta}$ and $|t|^{1-\alpha}$ where $\alpha$ determines the behavior of the heat capacity, takes the universal character. The analysis of entropy for argon and water leads to $\beta = 0.33$ and $\f{D_{2-\beta}}{{D_{1-\alpha}}}\approx - 3.5$.