Isothermal binodal curves near a critical endpoint

Thermodynamics in the vicinity of a critical endpoint with nonclassical exponents α, β, γ, δ, … , is analyzed in terms of density variables (mole fractions, magnetizations, etc.). The shapes of the isothermal binodals or two-phase coexistence curves are found at and near the endpoint for symmetric and nonsymmetric situations. The spectator- (or noncritical-) phase binodal at T=Te is characterized by an exponent (δ+1)/δ (≃1.21) with leading corrections of relative order 1/δ (≃0.21), θ4/βδ (≃0.34) and 1−(βδ)−1 (≃0.36); in contrast to classical (van der Waals, mean field, etc.) theory, the critical endpoint binodal is singular with a leading exponent (1−α)/β (≃2.73) and corrections which are elucidated; the remaining, λ-line binodals also display the “renormalized exponent,” (1−α)/β but with more singular corrections. [The numerical values quoted here pertain to (d=3)-dimensional-fluid or Ising-type systems.]