Thermodynamic equations of state (EoS) are fundamental tools in thermal and chemical engineering (1−3) They enable calculating properties of multicomponent systems as a function of experimentally accessible quantities such as temperature, pressure, and composition. Equations of state differ in the breadth of their applicability and complexity, and depending on the field of application, there are requirements on functionality, robustness, precision, and computational speed. A milestone in the description of fluid phases was provided by van der Waals’ equation of state, (4) which is based on a (coarse) molecular model and first led to description of vapor/liquid coexistence. Prominent modifications of the model by van der Waals that preserve the cubic volume-dependence are Redlich–Kwong, (5) Soave–Redlich–Kwong, (6) and Peng–Robinson. (7) Due to their fast evaluation times, cubic EoS are still widely used in technical applications. However, they lack a physical basis when describing nonspherical, polar, or hydrogen-bonding substances and mixtures. Modern EoS are developed based on a molecular model, i.e., a description of (pairwise) intra- and intermolecular interactions. Although theories for fluids with simple spherical intermolecular interactions were developed in the 1970s, (8) the field made a leap with a development of M.S. Wertheim, who derived a description for highly directional interactions. (9−12) Wertheim’s theory led to the statistical associating fluid theory (SAFT), (13) which regards molecules as chains of spherical segments (thus accounting for the nonspherical shape of real molecules due to covalent bonds) and allows for short-ranged attractive (hydrogen) bonds. The success of SAFT led to the development of a plethora of derived models, of which the most used ones are PC-SAFT, (14−17) SAFT-VR-Mie, (18) and soft-SAFT. (19,20) A combination of a cubic equation of state with the association contribution from TPT1 was published as cubic + association (CPA). (21−23) Finally, TPT1 and SAFT can resolve individual segments on a molecule, leading to the development of heterosegmented EoS likeSAFT-γ-Mie (24) and gc-PC-SAFT. (25)

Cubic and SAFT-type EoS aim to describe fluids based on a few either macroscopic (cubic) or microscopic (SAFT) properties. The small number of parameters ensures a robust extrapolation to state points for which no experimental data is available and the molecular model underlying these EoS ensure a meaningful description of mixtures with few binary interaction parameters or even predictions of mixture properties. However, experimental data is abundant for some fluids, and reference equations of state that use a large number of adjustable parameters to represent the experimental data of those substances have been developed. Reference EoS were published for pure components like water, (26) CO2, (27) and nitrogen, (28) but also for mixtures like natural gas or related systems. (29)
While EoS are used to compute properties of homogeneous fluid phases and to model phase equilibria and phase stability, they cannot model properties of microscopically inhomogeneous systems like fluids at interfaces or in porous media. Being able to model these phenomena is essential for dynamic processes such as droplet coalescence or formation of micelles or engineering applications such as adsorption. Classical density functional theory (30) (DFT) is a framework that extends fluid theories (i.e., equations of state based on a molecular model) to inhomogeneous systems. In DFT, the system is described by the grand potential, which can be expressed as a functional of the (partial) density profiles. Despite the fact that DFT is a formalism that has been used and studied in research for decades, it is not a commonly used method in industry as of today, although it can be used to predict properties that are difficult or expensive to measure, such as surface tensions of mixtures, (31) contact angles, (32) and adsorption isotherms. (33,34) In addition, it can provide insights into phenomena that are experimentally difficult to assess, such as the accumulation of light-boiling or amphiphilic molecules at interfaces. (35)

Within the last years, multiple open-source packages in the field of thermodynamics and equations of state were published, each with its own focus. CoolProp (36) and thermo (37) provide databases, correlations for properties such as vapor pressures and activity coefficients, and equations of state. Thermopack (38,39) (written in Fortran) focuses on equations of state with an emphasis on robust algorithms for phase equilibria and critical points, including multiphase flashes to be used within computational fluid dynamics simulations. phasepy (40,41) provides Python implementations of equations of state and algorithms for phase equilibria, including square gradient theory which can be used to describe density profiles across vapor liquid interfaces. teqp (42) (written in C++) and Clapeyron.jl (43) (written in Julia)─similar to FeOs─both utilize automatic differentiation, which circumvents the need to implement analytic derivatives of the Helmholtz energy. All of these projects have a common feature: they provide an interface to a dynamic, high-level programming language. Clapeyron.jl is implemented in the Julia programming language, while the others are either fully implemented in Python (phasepy, thermo) or provide Python bindings (CoolProp, Thermopack, teqp). Clearly, for a modern toolkit, this is a necessity as it enables access to a broader range of users and allows using tools such as Jupyter notebooks that make research more transparent and reproducible.

A recent survey of the Working Party of Thermodynamics and Transport Properties of the European Federation of Chemical Engineering summarizes the most important gaps and concerns raised by industry in the field of applied thermodynamics and outlines specific requirements for model frameworks. (44) In a subsequent publication, the authors outline the “main directions that [they] believe the applied thermodynamics community should adopt in the coming decade”: (3)

There is a need for methods to assess the properties of fluids under confinement, at interfaces, and in the presence of external fields.
Users must be able to parametrize the model and assess its uncertainties and range of applicability

In the same vein, access to these models and parameters has to be transparent, ideally in a standardized form and including the data used for the parametrization.

Finally, there is increasing demand for ongoing education, training, and collaboration.

FeOs is well suited to address these important topics. The implementation of DFT in FeOs is designed with respect to the modeling of industrially relevant problems. It provides ad hoc functionalities to describe confined media and interfaces or to specify external potentials. Furthermore, FeOs provides utilities to optimize model parameters within a couple of lines of code. All of this is possible from within Jupyter notebooks without sacrificing performance. With the simple installation and availability on all major platforms, results can easily be shared and reproduced.

FeOs is developed with two use-cases in mind. First, it provides implementations for equations of state and Helmholtz energy functionals for DFT together with algorithms for critical point and phase equilibrium calculations as well as solvers for density profiles in multiple dimensions and coordinate systems. It can therefore be used as a toolkit to compute thermodynamic properties of homogeneous and inhomogeneous systems. And second, it provides interfaces and data types that can be used to extend the code, e.g., to implement new models and algorithms...