Coexistence curves at liquid-liquid critical points: Ising exponents and extended scaling

Abstract

New measurements have been made of coexisting mass densities of isobutyric acid + water with a precision of 20 ppm and within 3.5\ifmmode^\circ\else\textdegree\fi{}C of the critical temperature. The measurements were made using a single sample of composition very close to critical. It is found that the coexistence curve is more symmetric in terms of the difference in volume fraction ($\ensuremath{\Delta}\ensuremath{\varphi}$) of coexisting phases than in terms of the difference in mass density. The difference $\ensuremath{\Delta}\ensuremath{\varphi}$ is well fitted for $\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}T)}{{T}_{c}}<0.006$ by the expression $\ensuremath{\Delta}\ensuremath{\varphi}=B{\ensuremath{\epsilon}}^{\ensuremath{\beta}}$, where $B=1.071\ifmmode\pm\else\textpm\fi{}0.023$ and $\ensuremath{\beta}=0.328\ifmmode\pm\else\textpm\fi{}0.004$. (Uncertainties are given as 3 times the standard deviation.) A new analysis has been made of the recent data of Gopal et al. on the difference in volume fraction of coexisting densities of carbon disulfide + nitromethane. For this system, $\ensuremath{\Delta}\ensuremath{\varphi}$ can be fitted for $\ensuremath{\epsilon}<0.2$ by an extended scaling expression suggested by Wegner's work, $\ensuremath{\Delta}\ensuremath{\varphi}=B{\ensuremath{\epsilon}}^{\ensuremath{\beta}}+{B}_{1}{\ensuremath{\epsilon}}^{\ensuremath{\beta}+{\ensuremath{\Delta}}_{2}}+{B}_{2}{\ensuremath{\epsilon}}^{\ensuremath{\beta}+2{\ensuremath{\Delta}}_{2}}$. The exponent ${\ensuremath{\Delta}}_{2}$ is fixed at 0.5; the fit gives $\ensuremath{\beta}=0.316\ifmmode\pm\else\textpm\fi{}0.008$, $B=1.63\ifmmode\pm\else\textpm\fi{}0.09$, ${B}_{1}=0.77\ifmmode\pm\else\textpm\fi{}0.31$, and ${B}_{2}=\ensuremath{-}2.43\ifmmode\pm\else\textpm\fi{}0.40$. This work suggests that liquid-liquid critical phenomena are consistent with the functional forms obtained from renormalization-group calculations and with asymptotic exponents which are like those presently calculated for the Ising model. The range of asymptotic behavior seems to be larger for a liquid-liquid critical point than for a liquid-gas critical point.