The Isothermal Binodal Curves Near a Critical Endpoint

Thermodynamics in the vicinity of a critical endpoint with nonclassical exponents $\alpha$, $\beta$, $\gamma$, $\delta$, $...$ 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 andnear the endpoint for symmetric and nonsymmetric situations. The spectator- (or noncritical)-phase binodal at $T=T_{e}$ is characterized by an exponent $(\delta +1)/\delta$ $(\simeq 1.21)$ with leading corrections of relative order $1/\delta$ $(\simeq 0.21)$, $\theta_{4}/\beta\delta$ $(\simeq 0.34)$ and $1 -(\beta\delta)^{-1}$$(\simeq 0.36)$; in contrast to classical (van der Waals, mean field, $...$) theory, the critical endpoint binodal is singular with leading exponent $(1-\alpha)/\beta$ $(\simeq 2.73)$ and corrections which are elucidated; the remaining, $\lambda$-line binodals also display the `renormalized exponent,' $(1-\alpha)/\beta$ butwith more singular corrections. (The numerical values quoted here pertain to $(d=3)$-dimensional-fluid or Ising-type systems.)