The 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, · · ·) theory, the critical endpoint binodal is singular with 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.) a)Current address: Center for Polymer Studies, Center for Computational Science, and Department of Physics, Boston University, Boston, MA 02215 USA 1