Thermodynamic metric geometry of the two-state ST2 model for supercooled water

Phase boundaries and the Widom line from the Ruppeiner geometry of fluids

The Ruppeiner geometry has been shown to provide novel ways for constructing the phase boundaries and the Widom line of certain fluids. This paper examines the applicability of these geometric constructions to more general fluids. We develop a general equation-of-state expansion for fluids near a critical point that mainly assumes analyticity with respect to the number density. Based on this general parametrization of fluids, we prove the equivalence of the Ruppeiner geometric construction and the standard Maxwell construction of phase boundaries near the critical point. In contrast, we find that the usual prescription based on the Ruppeiner geometry for the Widom line does not produce the expected Widom line for arbitrary cases of our general fluid equation of state. This usual prescription relies on the Ruppeiner metric induced on a particular hypersurface of the thermodynamic manifold. We show that by choosing a different hypersurface, which we call the Ruppeiner-N surface, and using its associated induced metric, the Ruppeiner construction generates the entire Widom line of the van der Waals fluid exactly, even away from the critical point. Interestingly, this alternative hypersurface yields another benefit. It improves the classification scheme originally proposed by Diósi et al. for partitioning the van der Waals state space into its different phases using geodesics of a thermodynamic metric. We argue that, whereas the original Diósi boundaries did not correspond to any clear thermodynamic lines, the corresponding boundaries based on the Ruppeiner-N metric become sensitive to the presence of the van der Waals Widom line and provide the correct classification of all van der Waals states. These results suggest that the Ruppeiner-N surface may be the more appropriate hypersurface to use when studying phase diagrams with thermodynamic geometry.

Thermodynamic metric geometry and the Fisher-Widom line of simple fluids

Two boundary lines are frequently discussed in the literature, separating state regions dominated by repulsion or attraction. The Fisher-Widom line indicates where the longest-range decay of the total pair correlation function crosses from monotonic to exponentially damped oscillatory. In the context of thermodynamic metric geometry, such a transition exists where the Ricci curvature scalar vanishes, R=0. To establish a possible relation between these two lines, R is worked out for four simple fluids. The truncated and shifted Lennard-Jones, a colloid-like and the square-well potential are analyzed to investigate the influence of the repulsive nature on the location of the R=0 line. For the longer-ranged Lennard-Jones potential, the influence of the cutoff radius on the R=0 line is studied. The results are compared with literature data on the Fisher-Widom line. Since such data are rare for the longer-ranged Lennard-Jones potential, dedicated simulations are carried out to determine the number of zeros of the total correlation function, which may provide approximate information about the position of the Fisher-Widom line. An increase of the repulsive strength toward hard sphere interaction leads to the disappearance of the R=0 line in the fluid phase. A rising attraction range results in the nonexistence of the Fisher-Widom line, while it has little effect on the R=0 line as long as it is present in the fluid state.