In this work, we present an open-source software package, referred to as FeOsFramework for Equations of State and Classical Density Functional Theory. FeOs is a collection of interfaces and data types that can be used (1) to implement thermodynamic equations of state and Helmholtz energy functionals for classical density functional theory, and (2) to compute thermodynamic properties of pure substances and mixtures, phase equilibria, and interfacial properties such as surface tensions and adsorption isotherms. The framework is written in the Rust programming language with a complete Python interface and is designed with a focus on usability and extensibility. It is openly available on GitHub (). Equations of state can be implemented in Rust, yielding performant code, or as a Python class, which is useful for prototyping and with less emphasis on execution speed. In both cases, the user has to implement a single function: the Helmholtz energy. FeOs then uses generalized (hyper-) dual numbers to evaluate the Helmholtz energy as well as the required exact partial (higher-order) derivatives. Using this type of automatic differentiation delivers performance without the need for implementing any analytical derivatives. The performance is further enhanced by a caching mechanism that avoids duplicate model evaluations. Together with the core interfaces and functionalities for equations of state and classical density functional theory, we provide implementations for multiple models such as the PC-SAFT equation of state (with homo- and heterosegmented group contribution methods) and Helmholtz energy functionals (including segment-based functionals). To showcase a selection of FeOs’ features, an example study of the adsorption of biogas in porous media using the PC-SAFT functional is provided.

The curvature dependence of interfacial properties has been discussed extensively over the last decades. After Tolman published his work on the effect of droplet size on surface tension, where he introduced the interfacial property now known as Tolman length, several studies were performed with varying results. In recent years, however, some consensus has been reached about the sign and magnitude of the Tolman length of simple model fluids. In this work, we re-examine Tolman's equation and how it relates the Tolman length to the surface tension and we apply non-local classical density functional theory (DFT) based on the perturbed chain statistical associating fluid theory (PC-SAFT) to characterize the curvature dependence of the surface tension of real fluids as well as mixtures. In order to obtain a simple expression for the surface tension, we use a first-order expansion of the Tolman length as a function of droplet radius R, as δ(R) = δ + δ/R, and subsequently expand Tolman's integral equation for the surface tension, whereby a second-order expansion is found to give excellent agreement with the DFT result. The radius-dependence of the surface tension of increasingly non-spherical substances is studied for n-alkanes, up to icosane. The infinite diameter Tolman length is approximately δ = -0.38 Å at low temperatures. For more strongly non-spherical substances and for temperatures approaching the critical point, however, the infinite diameter Tolman lengths δ turn positive. For mixtures, even if they contain similar molecules, the extrapolated Tolman length behaves strongly non-ideal, implying a qualitative change of the curvature behavior of the surface tension of the mixture.

The leading order terms in a curvature expansion of the surface tension, the Tolman length (first order), and rigidities (second order) have been shown to play an important role in the description of nucleation processes. This work presents general and rigorous expressions to compute these quantities for any nonlocal density functional theory (DFT). The expressions hold for pure fluids and mixtures, and reduce to the known expressions from density gradient theory (DGT). The framework is applied to a Helmholtz energy functional based on the perturbed chain polar statistical associating fluid theory (PCP-SAFT) and is used for an extensive investigation of curvature corrections for pure fluids and mixtures. Predictions from the full DFT are compared to two simpler theories: predictive density gradient theory (pDGT), that has a density and temperature dependent influence matrix derived from DFT, and DGT, where the influence parameter reproduces the surface tension as predicted from DFT. All models are based on the same equation of state and predict similar Tolman lengths and spherical rigidities for small molecules, but the deviations between DFT and DGT increase with chain length for the alkanes. For all components except water, we find that DGT underpredicts the value of the Tolman length, but overpredicts the value of the spherical rigidity. An important basis for the calculation is an accurate prediction of the planar surface tension. Therefore, further work is required to accurately extract Tolman lengths and rigidities of alkanols, because DFT with PCP-SAFT does not accurately predict surface tensions of these fluids.

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Models for surfactants need to incorporate the amphiphilic character of the molecules to describe key properties such as the adsorption at interfaces and the reduction of interfacial tensions. One possibility is to model the surfactant molecules as heteronuclear chains. Therefore, we revisit the heterosegmented density functional theory and present a theory consistent with the group contribution perturbed-chain statistical associating fluid theory equation of state. The model is used to study water/surfactant and water/surfactant/octane systems with surfactants from the group of polyethylene glycol alkyl ethers, a commonly used group of nonionic surfactants. The model parameters are obtained by fitting to pure component data of small surfactants. Binary interaction parameters are required to model the water/alkane subsystem and to account for the polarity of the head groups of the surfactant. The model is able to reproduce the significant enrichment of surfactant molecules at both vapor–liquid surfaces and liquid–liquid interfaces and the corresponding reduction of interfacial tensions. For liquid–liquid interfaces, the competing solubility of the surfactant in both phases has to be taken into account when searching for an optimal surfactant molecule.