Physics of the Jagla Model as the Liquid-Liquid Coexistence Line   Approaches Horizontal
  The slope of the coexistence line of the liquid-liquid phase transition (LLPT) can be positive, negative, or zero. All three possibilities have been found in Monte-Carlo simulations of a modified spherically symmetric two-scale Jagla model. Since the liquid-liquid critical point (LLCP) frequently lies in a region of the phase diagram that is difficult to access experimentally, it is of great interest to study critical phenomena in the supercritical region. We therefore study the properties of the Widom line, which is defined in the one-phase region above the critical point as the locus of maximum correlation length as function of the ordering field at constant thermal field. Asymptotically close to the critical point, the Widom line coincides with the loci of the response function extrema, because all response functions can be asymptotically expressed as functions of the diverging correlation length. We find that the method of identifying the Widom line as the loci of heat capacity maxima becomes unfruitful when the slope of the coexistence line approaches zero in the $T$-$P$ plane. In this case the specific heat displays no maximum in the one-phase region because for a horizontal phase coexistence line, according to the Clapeyron equation, the enthalpy difference between the coexisting phases is zero, and thus there can be no contribution to enthalpy fluctuations from the critical fluctuations. The extension of the coexistence line beyond the critical point into the one-phase region must in this case be performed using density fluctuations; the line of compressibility maxima remains well defined, though it bifurcates into a symmetrical pair of lines. These findings agree well with the linear scaling theory of the LLCP by Anisimov and collaborators. 
Jiayuan Luo, Limei Xu, C. Austen Angell, H. Eugene Stanley, and Sergey   V. Buldyrev
cond-mat.stat-mech cond-mat.soft