The Van der Waals interaction approach to the liquid-liquid phase-transition

Since the pioneering work of Stell and Hemmer [1], proposing the existence of a new critical point inside liquids, we have seen experiments, computational simulations and theory working together to understand the nature of the liquid-liquid transition [2]. However no definitive model of how it happens has been yet established. Some theoretical models, mainly in one-dimension, and some computational works, indicate that an attractive potential combined with an strongly repulsive hard core are essential ingredients for this transition [3]. We will present here a three-dimensional derivation of this transition. Our model is, essentially, a extension of the approach that let to the Van der Waals state equation and the gas-liquid transition. We will analytically show that, with a small improvement, this longstanding approach also predicts the existence of a further critical point, at a high pressure and a high temperature. Namely, we will consider the two particles Mayer cluster expansion [4], and show that when we take into account not only the first, but also its second order terms in the inverse of the temperature, two critical points are obtained; the first corresponding to the usual gas-liquid transition, and the second corresponding to a liquid-liquid transition.
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[2] M. Aur?lio, A. Barbosa, E. Salcedo and M. C. Barbosa. Pry. Rev. E 87, 032303 (2013).
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[4] R. K. Pathria, Statistical Mechanics (Pergamon Press, Oxford, 1977).