Thermodynamics of Liquid Polyamorphism: Equation of State of a Fluid with Thermodynamic Equilibrium Between Two Structures

“Liquid polyamorphism” is the existence of two alternative amorphous structures in a single-component liquid. Liquid polyamorphism is found in a broad group of very different materials, such as triphenyl phosphite, silicon, silicon dioxide, cerium, and hydrogen, usually at extreme conditions. In particular, this phenomenon is hypothesized in metastable, deeply supercooled water. Cold and supercooled water is assumed to exist in two forms, a low-density or high-density liquid. The existence of these two alternative structures could, at certain conditions, result in a metastable liquid-liquid separation in pure water. The hypothesized liquid-liquid metastable coexistence is not directly accessible in bulk-water experiments because it is presumably located a few degrees below the empirical limit of homogeneous ice formation. We consider a phenomenological approach to describe liquid polyamorphism in a single-component fluid. The first fundamental question to be addressed is the separation of time scales: a system with two inter-convertible fluid structures can be thermodynamically treated as a single-component fluid if the time of observation is longer than the time of conversion (fast conversion). In the opposite limit (slow conversion) the system thermodynamically behaves as a two-component mixture. We propose a simple equation of state which can generically describe liquid-gas and liquid-liquid transitions in the same single-component fluid. The lattice-gas model and the van der Waals equation of state are adopted for one of the alternative structures. In order to predict liquid polyamorphism, the lattice-gas equation of state and the van der Waals equation of state are supplemented by a thermodynamic constraint associated with the “chemical-reaction” equilibrium between the two inter-convertible liquid structures. The equations provide a closed formulation for the thermodynamics of the system. This system of equations is nonlinear and is solved numerically to obtain both the liquid-gas and liquid-liquid equilibrium and the corresponding stability limits. The results are tested on realistic water-like models for which the properties obtained by MD simulations are available.