Analysis and extension of perturbation theories for modelling Helmholtz energies

Abstract

Equations of state expressed in the Helmholtz energy enable accurate predictions of a broad range of fluid properties, i.e., PV T -behavior, densities, phase equilibria, or heat capacities, at low computational cost. Perturbation theories provide a powerful physical basis for developing equations of state. In this work, an equation of state for Mie ν-6 fluids is developed within the framework of the uv-theory. The equation of state is based on interpolation between two first-order perturbation theories, and reproduces the virial expansion up to third order at low densities. The new model is not only comparable in accuracy to state-of-the-art empirical models, but also offers several advantages due to its physical basis. These include a better description of the meta-stable and unstable regions, and a more systematic extension to non-spherical fluids and mixtures. As a first step towards extending the equation of state to chain fluids, an algebraic model for the second virial coefficient of Lennard-Jones chains is developed. This model combines a third-order perturbation theory with empirical terms adjusted to newly generated molecular simulation data. Although perturbation theories can be formulated rigorously for mixtures, this is rarely done in practice, as deriving algebraic models is challenging, and numerical evaluation is computationally demanding. Instead, one-fluid approximations are typically used, treating mixtures as effective pure fluids at the cost of accuracy. An algebraic correction term is developed for size-asymmetric mixtures of square-well and Lennard-Jones fluids. This correction can be applied to various perturbation-theory-based equations of state using a one-fluid approximation. The correction term improves the prediction of mixture properties without introducing (significant) additional complexity. The functional form of the correction was obtained using symbolic regression. More complex interactions, specifically Coulomb interactions are additionally addressed. Formulating analytic models for electrolytes based on perturbation theories is challenging due to non-converging integrals. A common approach is to replace terms associated with Coulomb interactions with simple Helmholtz energy expressions based on the Debye-Hückel theory, the mean spherical approximation, or the Born theory of solvation. Using molecular simulations to generate reference data for ion-related Helmholtz energy contributions, it is shown that these substitutions are not sufficiently accurate without empirical adjustments.