. For machining processes, such as drilling, grinding, and cutting, ﬂuids play a crucial role for lubrication and cooling. For adequately describing such pro-cesses, robust models for the thermophysical properties of the ﬂuids are a prereq- uisite. In the contact zone, extreme conditions prevail, e.g. regarding temperature and pressure. As thermophysical property data at such conditions are presentlyoften not available, predictive and physical models are required. Molecular-based equations of state (EOS) are attractive candidates as they provide a favorable trade-off between computational speed and predictive capabilities. Yet, withoutexperimental data, it is not trivial to assess the physical reliability of a given EOS model. In this work, Brown’s characteristic curves are used to assess molecular- based ﬂuid models. Brown’s characteristic curves provide general limits that are to be satisﬁed such that a given model is thermodynamically consistent. Moreover, a novel approach was developed, which uses pseudo-experimental data obtainedfrom molecular simulations using high-accurate force ﬁelds. The method is gen- eralized in a way that it can be applied to different force ﬁeld types, e.g. model potentials and complex real substances. The method was validated based on the(scarcely) available data in the literature. Based on this pseudo-experimental data, different thermodynamic EOS models were assessed. Only the SAFT-VR Mie EOS is found to yield thermodynamically consistent results in all cases. Thereby,robust EOS models were identiﬁed that can be used for reliably modeling cutting ﬂuids at extreme conditions, e.g. in machining processes. 1 Introduction Cutting ﬂuids experience extreme conditions in a contact zone of machining processes (cf. Fig. 1), i.e. large temperature and pressure as well as extreme gradients in these properties [ 8,47,49]. The pressure in a tribological contact zone can be up to several GPa and the temperature up to 1 000 K [ 6,8]. For modeling such processes, reliable and robust models for describing the thermophysical properties of the ﬂuids, e.g. the heat capacity, compressibility, and density at a given temperature and pressure are required. Yet, classical laboratory experiments for determining thermophysical properties at suchconditions are practically not feasible. Also, empirical models ﬁtted to the available data © The Author(s) 2023 J. C. Aurich et al. (Eds.): IRTG 2023, Proceedings of the 3rd Conference on Physical Modeling for Virtual Manufacturing Systems and Processes , pp. 170–188, 2023. https://doi.org/10.1007/978-3-031-35779-4 _10 Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 171 at moderate conditions often exhibit an unrealistic extrapolation behavior to extreme temperature and pressure. Hence, physical predictive and reliable models are required. Models based on molecular thermodynamics can be favorably used for predict- ing thermophysical properties due to their physical kernel, i.e. mathematical structure.In particular, the strong physical background of these models often allows success- ful extrapolations to conditions that were not considered for the model development such as extreme pressure and temperature. Both molecular dynamics (MD) simulationsand molecular-based equation of state (EOS) models are attractive candidates for such applications. Fig. 1. Lubricated contact zone of a machining process between a workpiece (light gray) and a tool (dark gray). In the contact zone, extreme temperature and pressure prevail in the cutting ﬂuid (blue). The most popular modeling approach based on molecular thermodynamics is molec- ular simulation, i.e. molecular dynamics (MD) or Monte Carlo (MC) simulations based on classical molecular force ﬁelds. In molecular simulations, matter is modeled on theatomistic scale using a particle-based scheme. These particles interact with intermolec- ular potentials, i.e. the force ﬁelds. The accuracy of a molecular simulation primarily depends on the quality of the force ﬁeld, which are available today for a large numberof substances [ 46]. Molecular simulation has been extensively used for predicting ther- mophysical properties at extreme conditions, e.g. Refs. [ 32,34,35]. Due to the strong physical basis and reliability of the predictions, molecular simulations data is at timesused as pseudo-experimental data where no ‘real’ experimental data is available [ 32,34]. Yet, molecular simulations (MD or MC) are computationally expensive. For predicting thermophysical properties at a single state point, on the order of magnitude of 10 2CPUh are required. An alternative modeling approach are molecular-based equation of state (EOS) models. They are algebraic models formulated in the Helmholtz energy as a func-tion of the temperature and density, i.e. a=a(T,ρ). Thereby, ﬂuid properties at a given state point can be evaluated in milliseconds. Also, since the Helmholtz energy ansatz is a thermodynamic fundamental expression, all other thermodynamic properties can bederived from such a model – including phase equilibria, which are for example important for describing cavitation in a contact zone. Moreover, equation of state models can be favorably combined with further physical theories for modeling for example transportand interfacial properties, e.g. entropy scaling [ 20] and density gradient theory [ 40,44]. In particular, molecular-based EOS models can be directly integrated for scale bridging in macroscopic models, e.g. in phase ﬁeld or CFD simulations [ 15]. Molecular-based EOS come along with the drawback that approximations and assumptions are made 172 J. Staubach and S. Stephan within the model formulation. Hence, compared to molecular simulation, molecular- based EOS have a less strong physical backbone. A good extrapolation behavior can therefore not be presumed a priori. Different strategies have been proposed for the assessment of EOS models [ 1,2,17, 39,48]. The thermodynamic consistency of a pure component model can for example be assessed using ab initio virial coefﬁcients in the low-density limit and the Nezbeda compressibility or Clausius-Clapeyron test for the vapor-liquid equilibrium [ 30,48]. An interesting strategy for testing the extrapolation behavior at extreme pressure and temperature liquid states is based on Brown’s characteristic curves. Brown’s charac- teristic curves deﬁne lines on the thermodynamic equilibrium surface and are namedZeno, Amagat, Boyle, and Charles curve. Each of the four characteristic curves has – for a given thermophysical property – the same behavior as the ideal gas [ 9]. These curves are located within a large pressure and temperature range. For a given molec-ular ﬂuid, Brown’s characteristic curves are known to exhibit certain thermodynamic features [ 9]. Therefore, Brown’s characteristic curves have become an important tool for the assessment of the extrapolation behavior of new EOS [ 1,39,43]. Nevertheless, due to the extreme conditions, there is practically no experimental data available on the characteristic curves. Therefore, Brown’s characteristic curves are usually only used for a qualitative assessment of the behavior of the model, i.e. evaluating the limits and thegeneral shape. It should moreover be noted that providing thermodynamically consistent Brown’s characteristic curves is a necessary, but not sufﬁcient condition for a thermo- dynamic model being physically reasonable, which is also shown in an example in this work. In this work, a novel approach was developed for testing the extrapolation behavior of EOS models based on Brown’s characteristic curves using pseudo-experimental data obtained from molecular simulation. This approach conveniently combines the strong physical backbone and predictive capabilities of molecular force ﬁeld models [ 32,34, 35] with the computational advantages of molecular-based EOS. By using molecular simulation pseudo-experimental data, not only the thermodynamic consistency of the EOS model regarding Brown’s characteristic curves can be assessed, but also the accu-racy of the model can be simultaneously evaluated. The computational procedure for determining Brown’s characteristic curves for a given force ﬁeld model is adopted from [51]. The new approach for assessing EOS models with pseudo-experimental charac- teristic curve data was tested on both model ﬂuids and real substances. This paper is outlined as follows: First, the methods used for determining the characteristic curves are described. Then, the results for the different substances are presented and discussed. 2 Methods 2.1 Brown’s Characteristic Curves The characteristic curves of a molecular ﬂuid were postulated by E.H. Brown [ 9]a s curves on the thermodynamic pvT surface, along which the compressibility factor Zor its derivatives are identical to the values of an ideal gas. Brown’s characteristic curves include state points at extreme conditions regarding temperature and pressure and cantherefore be used as a tool for testing the extrapolation behavior of EOS. Moreover, on the Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 173 thermodynamically consistent surface, Brown deduced from physical arguments that the curves exhibit certain features and limits (details are given below) that can be favorably used for the assessment of models at extreme conditions. The four characteristic curves are: •Zeno curve Z (also called ideal curve) •Amagat curve A (also called Joule inversion curve) •Boyle curve B •Charles curve C (also called Joule-Thomson inversion curve) Figure 2shows a schematic representation of Brown’s characteristic curves. The char- acteristic curves are shown in a double-logarithmic temperature-pressure ( pT) diagram. Based on rational thermodynamic arguments, Brown derived several requirements forthe characteristic curves to be thermodynamically consistent: Each characteristic curve exhibits a single pressure maximum. Furthermore, the Zeno curve crosses each of the other three curves in one point (cf. Fig. 2), whereas the Boyle, Charles, and Amagat curve do not intersect each other. Moreover, the Charles curve terminates on the vapor- liquid binodal and the Boyle curve terminates on the spinodal. The characteristic curves end at low pressure (corresponds to low densities) at speciﬁc temperatures, which are directly linked to the second virial coefﬁcient B[43], which is often known with very high accuracy. In the virial expansion up to second order, the compressibility factor Z can be written as Z=1+Bρ. (1) Hence, there is a direct link between the zero-pressure limit of the characteris- tic curves and the second virial coefﬁcient [ 29,43]. In the following, the deﬁnition and general features of the four characteristic curves are introduced. A comprehensiveintroduction to Brown’s characteristic curves is given in Refs. [ 4,5,29,43]. State points on the Zeno curve satisfy the relation Z=vp RT=1, (2) where Zis the compressibility factor, vthe molar volume, pthe pressure, Rthe molar gas constant, and Tthe temperature. The deﬁnition can be rewritten in terms of the Helmholtz energy as ρ/parenleftbigg∂˜a ∂ρ/parenrightbigg T=0, (3) where ρis the molar density and ãis the molar Helmholtz energy deﬁned as ˜a=A NAkBT with the Boltzmann constant kBand the Avogadro constant NA. The Zeno curve ends at the so-called Boyle temperature in the zero-pressure (and therefore zero-density) limit, cf. Fig. 2. At the Boyle temperature, the second virial coefﬁcient is zero B(TBoyle )=0, which directly follows from comparing the deﬁnition of the Zeno curve Eq. ( 2) with the virial expansion Eq. ( 1). The Zeno curve intersects all other characteristic curves. The intersection of the Zeno and the Boyle curve islocated in the zero-pressure limit at the Boyle temperature. The temperature of the 174 J. Staubach and S. Stephan Z B CA Fig. 2. Schematic representation of Brown’s characteristic curves in the temperature-pressure (pT) projection. The Zeno curve Z ( red), Amagat curve A ( orange ), Boyle curve B ( blue ), and C h a r l e sc u r v eC( pink ) are shown. The gray-shaded area indicates the solid phase region. The VLE, spinodals, and critical point of the ﬂuid are indicated by the solid black line, the dashedblack line, and the star, respectively. intersection with the Charles curve is approximately the critical temperature of the ﬂuid, cf. Fig. 2. The intersection of the Zeno curve with the Amagat curve is located at very low temperatures – for many substances in the solid phase region. The Boyle curve is deﬁned by the following relations: /parenleftbigg∂Z ∂1/ρ/parenrightbigg T=0,/parenleftbigg∂Z ∂p/parenrightbigg T=0. (4) These deﬁnitions can be rewritten in terms of the Helmholtz energy as ρ/parenleftbigg∂˜a ∂ρ/parenrightbigg T+ρ2/parenleftbigg∂2˜a ∂ρ2/parenrightbigg T=0. (5) The Boyle curve starts at the Boyle temperature in the zero-pressure limit, cf. Fig. 2. The Boyle curve has a bell shape and ends at the spinodal in the metastable region of the ﬂuid, where the ﬁrst and second density derivative of the Helmholtz energy are zero. The Charles curve is deﬁned by one of the following relations: /parenleftbigg∂Z ∂1/ρ/parenrightbigg p=0,or/parenleftbigg∂Z ∂T/parenrightbigg p=0,or/parenleftbigg∂H ∂p/parenrightbigg T=0, (6) where His the enthalpy. These relations can be rewritten in terms of the Helmholtz energy as ρ/parenleftbigg∂˜a ∂ρ/parenrightbigg T+ρ2/parenleftbigg∂2˜a ∂ρ2/parenrightbigg T+ρ T∂˜a ∂ρ ∂ 1/T=0. (7) The Charles curve starts in the zero-pressure limit at high temperatures where the tangent to the second virial coefﬁcient curve passes through the origin, i.e. the equation Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 175 dB/dT=B/Tholds. At low temperatures, the Charles curve ends at the vapor-pressure curve. The Charles curve has a bell shape and encloses the Boyle curve. The Charles curve is also known as the Joule-Thomson inversion curve, which is of fundamental importance for refrigeration technologies. There are several studies available in theliterature investigating the Joule-Thomson inversion curve for refrigerants [ 10,12,13, 22,52,53]. The Amagat curve is deﬁned by one of the following relations: /parenleftbigg∂Z ∂T/parenrightbigg ρ=0,or/parenleftbigg∂U ∂1/ρ/parenrightbigg T=0, (8) where Uindicates the internal energy. These relations can be rewritten in terms of the Helmholtz energy as ρ T∂˜a ∂ρ ∂ 1/T=0. (9) The Amagat curve starts in the zero-pressure limit at high temperatures, where the second virial coefﬁcient exhibits a maximum. At low temperatures, the Amagat curveends at the vapor-pressure curve. The Amagat curve has a bell shape and encloses the Boyle curve and the Charles curve, cf. Fig. 2. To be thermodynamically consistent, all four curves are required to exhibit a concave shape throughout in the double logarithmic pTprojection. Moreover, for model ﬂuids, the zero-pressure limit state point can be computed exactly using the virial route [ 51]. Hence, the zero-pressure limit is known exactly for model ﬂuids. 2.2 Substances In this work, both model ﬂuids and real substances were studied to demonstrate and test the novel approach. As model ﬂuids, the Lennard-Jones (LJ) ﬂuid, the Lennard-Jonestruncated and shifted (LJTS) ﬂuid, and ﬁve Mie ﬂuids were studied. For the studied model ﬂuids, spherical molecules were considered that interact with the respective interaction potential. The model potentials used here can be employed for describing a large num-ber of real substances, for example available in the MolMod force ﬁeld database [ 46]. The underlying interaction potential are moreover important, as they are often used as building block in complex molecular force ﬁelds of real substances [ 28,46]. Three real substances were studied in this work: toluene, ethanol, and dimethyl ether. The three sub- stances strongly differ regarding the molecular structure and intermolecular interactions such that the novel approach is tested here for different situations. The (full) Lennard-Jones potential u LJis deﬁned as uLJ(r)=4ε/bracketleftbigg/parenleftBigσ r/parenrightBig12 −/parenleftBigσ r/parenrightBig6/bracketrightbigg , (10) with the energy parameter ε, the size parameter σ, and the distance between two par- ticles r. The LJ potential models repulsive interactions between particles at very small distances and attractive (dispersive) interactions at intermediate distances. The exponent 176 J. Staubach and S. Stephan 12 characterizes the repulsive interactions and the exponent 6 the attractive interactions between particles. The LJ potential is relatively simple, but a good approximation for intermolecular interactions. It has been extensively used since the early days of computer simulations and, accordingly, high quality benchmark data is available for the LJ ﬂuid[48]. The Lennard-Jones truncated and shifted (LJTS) potential u LJTS is computationally much cheaper, but simpliﬁes the dispersive long-range interactions. The LJTS potential is deﬁned as uLJTS(r)=/braceleftbigguLJ(r)−uLJ(rc)r≤rc 0 r>rc,(11) with the cutoff radius rc=2.5σ. The Mie potential is a generalization of the LJ potential deﬁned as uMie(r)=Cε/bracketleftbigg/parenleftBigσ r/parenrightBigλn−/parenleftBigσ r/parenrightBigλm/bracketrightbigg , (12) with C=λn λn−λm/parenleftbiggλn λm/parenrightbiggλm λn−λm. (13) Hence, the Mie λn,λmpotential has two additional parameters: the exponent for the repulsive interactions λnand the exponent for the attractive interactions λm. Often, λm =6 is chosen, while λnis often used as an additional adjustable parameter. In this work, ﬁve Mie ﬂuids were considered with ( λn,λm)=(8,6) (12,6) (20,6) (12,4) (12,8). For the real substances, toluene, ethanol, and dimethyl ether were chosen as high- accurate force ﬁelds for these substances are available in the literature [ 18,25,38,46]. The force ﬁelds for toluene, ethanol, and dimethyl ether are based on a united-atom approach and modeled as rigid bodies, where the hydrogen atoms are not explicitlymodeled. The toluene force ﬁeld [ 25] has seven Lennard-Jones interactions sites and six point quadrupoles modeling the πorbital. The ethanol force ﬁeld [ 38] has three Lennard-Jones interactions sites and three point charges modelling the alcohol groupand enabling h-bonding. The dimethyl ether force ﬁeld [ 18] has three Lennard-Jones interactions sites and a single point dipole modeling the overall polarity of the molecule. Also, equation of state models for these substances are available in the literature. All three substances are base components in the chemical industry. Moreover, the three sub- stances have strongly different molecular architecture and interactions: ethanol stronglyforms hydrogen bonds, toluene is an aromatic compound with strong quadrupolar inter- actions caused by π-orbitals, and dimethyl ether has strong dipolar interactions. Hence, the three substances challenge the predictive capabilities of the molecular-based EOSmodels regarding different aspects. 2.3 Molecular Simulation In this work, reference data for the assessment of the EOS models was generated using molecular dynamics simulations. Hence, the MD data is considered as pseudo- experimental data for the characteristic curves. In particular, high accurate molecu- lar force ﬁeld models were used that are known to provide excellent agreement withexperimental data [ 18,25,38]. Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 177 For determining the characteristic curve state points from a given force ﬁeld, a new simulation method was developed by our group [ 51]. For the development, a large number of sampling routes were compared. Based on this systematic approach, for each characteristic curve, the most favorable simulation route was determined [ 51]. The new method has several advantages, e.g. a rigorous method for determining the statistical uncertainties and evaluating the thermodynamic self-consistency of the results [ 51]. Nevertheless, it should be noted that also systematical uncertainties – as well-knownfrom laboratory experiments – may affect the reproducibility of the simulation results [36,42], e.g. the simulation engines and implemented algorithms may have some minor inﬂuence on the results. Molecular dynamics simulations were performed with the molecular simulation tool ms2[21,33] using at least N=2,000 particles. For each characteristic curve, at least 14 state points were determined. For each characteristic curve state point, ﬁve state pointsin the vicinity of the initial guess characteristic curve were considered as a set. Each simulation of a simulation set was carried out at the same temperature. The simulations were carried out in the NVT ensemble for the Zeno, Boyle, and Charles curve. The NPTensemble was used for the simulations of the Amagat curve. From the simulation results of a set, the characteristic curve state point was determined using the respective relations. In Table 1, the relations used for determining the characteristic curve state points are listed. For the zero-pressure limit, cluster integrals were evaluated for determining the characteristic curve state point via the virial route with high accuracy [ 51]. For the studied model ﬂuids, these integrals can be evaluated exactly. For the real substance ﬂuids, the integrals were evaluated using Monte Carlo simulations for sampling different orientations and distances between two molecules. For both, the model ﬂuids and thereal substances, the second virial coefﬁcient was computed in a wide temperature range. From these results, the zero-pressure limit state points of the characteristic curves were obtained. Determining Brown’s characteristic curves from molecular simulation for agiven substance requires approximately 10 4CPU hours. However, the force ﬁeld type has a signiﬁcant inﬂuence on that. Details on the molecular simulation methodology are given in Ref. [ 51]. Table 1. Conditions used for determining characteristic curve state points using MD simulations based on the method proposed in Ref. [ 51]. Zeno Amagat Boyle Charles Z−1=0/parenleftBig ∂U ∂1/ρ/parenrightBig T=0/parenleftBig ∂Z ∂1/ρ/parenrightBig T=0/parenleftBig ∂H ∂p/parenrightBig T=0 2.4 Molecular-Based Equation of States Molecular-based EOS have been primarily developed within the chemical engineering community for modelling in particular phase equilibria of mixtures [ 11,14]. Yet, due to their physical general mathematical backbone, they can be favorably applied also in 178 J. Staubach and S. Stephan other ﬁelds for modeling properties of ﬂuids. The most popular molecular-based EOS is PC-SAFT [ 24]. Molecular-based EOS are formulated as ˜a=˜arep+˜aatt+˜achain+˜aassoc+˜aD+˜aQ, (14) where ˜aindicates the conﬁgurational Helmholtz energy, ˜arepthe contribution due to repulsive interactions between particles, ˜aattthe contribution due to dispersive attractive interactions, ˜achain the contribution due to the chain length of molecules, ˜aassoc the contri- bution due to h-bonding interactions [ 19],˜aDthe contribution due to dipole interactions [23], and ˜aQthe contribution due to quadrupole interactions. Each contribution term has one or two substance-speciﬁc parameters that characterize features of the molecule or intermolecular interactions, e.g. the chain term ˜achain comprises chain length parameter mand the attractive contribution ˜aattthe dispersion energy ε. Details on the molecular parameters of this model class are given in the review of Economou [ 19]. Moreover, ˜a is a function of the temperature Tand the density ρ(and for mixtures the composition vector x), which also holds for the individual terms on the RHS of Eq. ( 14). According to Eq. ( 14), the Helmholtz energy contributions from the different molecular interac- tions and features are independent. This is an approximation made in the model, whichmay seem crude, but has proven reliable and ﬂexible. Thermodynamic properties can be straightforwardly computed from Eq. ( 14) using well-known relations between the derivatives of the Helmholtz energy with respect to the density and inverse temperature,e.g. the pressure can be computed as p=ρTR(1+∂˜a/∂ρ)and the heat capacity can be computed as c v=−/parenleftbig ∂2˜a/∂(1/T)2/parenrightbig . Only derivatives of the Helmholtz energy with respect to the density and inverse temperature (and eventually composition) are requiredfor the calculation of thermodynamic properties of interest from a given EOS. In this work, the characteristic curves were computed from different EOS models using the thermodynamic deﬁnitions given by Eqs. ( 3), (5), (7), and ( 9). Two different molecular-based EOS frameworks were considered in this work: PC- SAFT [ 24] and SAFT-VR Mie [ 27]. The latter was directly used for describing the LJ and Mie model ﬂuid as well as for the real substances toluene, ethanol, and dimethyl ether. For describing the LJ and LJTS model ﬂuid within the PC-SAFT framework, the models from Refs. [ 26,45] were used, which are re-parametrized PC-SAFT monomer models. For comparison, also empirical EOS models were used in some cases [ 37,50, 54]. For the LJ ﬂuid, the molecular-based EOS of Stephan et al. [ 45] and that of Laﬁtte et al. [ 27] were used. For the LJTS ﬂuid, the molecular-based EOS of Heier et al. [ 26] and the empirical EOS of Thol et al. [ 50] were used. For the Mie ﬂuid, the EOS of Laﬁtte et al. [ 27] was used. For modelling the three real substances, toluene, ethanol, and dimethyl ether, both the SAFT-VR Mie EOS and the PC-SAFT EOS were used.Moreover, for ethanol, the empirical EOS from Schroeder et al. [ 37] and, for dimethyl ether, the empirical EOS from Wu et al. [ 54] were used. The PC-SAFT and SAFT- VR Mie model parameters for all three real substances are summarized in Table 2.F o r dimethyl ether, a new SAFT-VR Mie model was parametrized within this work. The other model parameters were taken from the literature [ 3,23,24,27]. Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 179 Table 2. Model parameters used for the SAFT-VR Mie and PC-SAFT EOS. The model parameters were taken from Refs. [ 3,23,24,27]. The model parameters for dimethyl ether for the SAFT- VR Mie EOS were obtained within this work. Columns indicate the dispersion energy ε, particle size parameter σ, attractive λnand repulsive λmexponent, chain length m, association volume κ, association energy /epsilon1, and dipole moment μ. Substance ε/kB/Kσ/Å λn λm m κ /epsilon1/kB/Kμ/D PC-SAFT Toluene 285.69 3.7169 - - 2.8149 - - - Ethanol 191.31 3.1477 - - 2.4382 0.03481 2599.8 1.7 Dimethyl ether 210.29 3.2723 - - 2.2634 - - 1.3 SAFT-VR Mie Toluene 409.73 4.2770 16.334 6 1.9977 - - - Ethanol 168.15 3.4914 7.6134 6 1.9600 0.34558 2833.7 - Dimethyl ether 216.74 3.306 12 6 2.2340 - - - 3 Results For the real substance ﬂuids, classical SI units are used for presenting the results. For the model ﬂuids, the Lennard-Jones units are used, i.e. using the potential well depth ε, the particle size parameter σ(cf. Eq. ( 10)), and the mass of the particle Mas well as the Boltzmann constant k Bconstitute the base unit system. Details on the LJ unit system are given in Ref. [ 48]. 3.1 Lennard-Jones Fluids Figures 3and 4show the results for the studied Lennard-Jones ﬂuids: Fig. 3for the (full) LJ ﬂuid and Fig. 4for the LJTS ﬂuid. In both cases, results from different EOS and molecular simulation are shown. The latter is taken as reference. For the LJ ﬂuid (cf. Fig. 3), two molecular simulation data sets are compared. Over- all, the data from Deiters and Neumaier [ 16] and the results using the new simulation method from our group [ 51] are in excellent agreement. In particular, both data sets are in excellent agreement with the zero-density limit data point obtained from the exact virial route. Moreover, the simulation results for all four characteristic curves exhibit a smooth trend. For the LJ ﬂuid, two molecular-based EOS were used for predictingthe characteristic curves. The EOS from Stephan et al. [ 45] (re-parametrized PC-SAFT monomer) yields a realistic Zeno, Boyle, and Charles curve. For the Amagat curve, however, the re-parametrized PC-SAFT monomer yields an erratic behavior, cf. Fig. 3. The EOS from Laﬁtte et al. [ 27], on the other hand, is in excellent agreement with the reference data. Moreover, this EOS yields physically realistic predictions for all four characteristic curves. Interestingly, the reference data and the EOS from Laﬁtte et al. agree very well even in the VLE metastable region. For the LJTS ﬂuid (cf. Fig. 4), only the data set determined by our group is available [51]. The simulation results show a very smooth trend and are also consistent with the 180 J. Staubach and S. Stephan 0.1 1 100.010.1110p/ εσ-3 T/εk-1 B Fig. 3. Characteristic curves of the Lennard-Jones 12,6 ﬂuid. Symbols indicate MD simulation results from Deiters and Neumaier ( )[16] and from this work ( ). Lines correspond to predictions from the SAFT-VR Mie EOS (solid lines) and the EOS from Stephan et al. (dashed lines). Results for the Boyle curve ( blue ), Zeno curve ( red), Charles curve ( pink ), and Amagat curve ( orange ). The vapor pressure curve ( —) and the spinodal curves (---) as well as the critical point ( )a r e shown – computed from the EOS from Stephan et al. exact zero-pressure limit results obtained from the virial route. Based on the molecular simulation reference data, two EOS models are assessed: the EOS of Thol et al. [ 50] and the EOS of Heier et al. [ 26]. The latter is also a re-parametrized PC-SAFT monomer, which is the reason that it yields a spurious Amagat curve. It was shown by Boshkhova and Deiters [ 7] that these defects are caused by a problematic formulation of the repulsive term of the underlying EOS framework. In the range of the Zeno, Boyle, and Charles curve, the EOS of Heier et al. [ 26] yields a realistic behavior. The EOS of Thol et al. [50] also exhibits an erroneous Amagat curve, but in the low-temperature regime. Brown showed that the Amagat curve is to intersect the Zeno curve in that regime, which is not the case for the EOS of Thol et al. Nevertheless, the EOS of Thol et al. is in reasonable agreement with the molecular simulation reference data in most cases. Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 181 0.1 1 100.010.1110p / εσ-3 T / εk-1 B Fig. 4. Characteristic curves of the Lennard-Jones truncated and shifted (LJTS) ﬂuid. Symbols indicate MD simulation results [ 51]. Lines correspond to predictions from the LJTS EOS from Heier et al. (solid lines) and the EOS from Thol et al. (dashed lines). Results for the Boyle curve ( blue ), Zeno curve ( red), Charles curve ( pink ), and Amagat curve ( orange ). The vapor pressure curve ( —) and the spinodal curves (---) as well as the critical point ( )a r es h o w n . 3.2 Mie Fluids Figure 5and 6show the results for the studied Mie ﬂuids. Figure 5shows the results for Mie ﬂuids with different repulsive exponent λn;F i g . 6the results for Mie ﬂuids with different attractive exponent λm. While a large number of equations of state are available in the literature for the classical (full) LJ ﬂuid, only some Mie EOS are presently available. Here, the Mie EOS from Laﬁtte et al. [ 27] was considered. Overall, the predictions from the Mie EOS are in excellent agreement with the molecular simulation reference data – for all studied λn, λmcombinations. This is impressive considering the fact that only data for moderate conditions was used for the model development [ 27]. This highlights the advantage of the physically-based model to reliable extrapolate to extreme temperatures and pressures.Only, some small deviations to the reference data are observed for the Amagat curve, cf. Fig. 5and 6. Overall, the EOS of Laﬁtte et al. captures the effect of both the repulsive and the attrac- tive exponent well and is in excellent agreement with the pseudo-experimental data – in a very wide temperature and pressure range. Hence, the EOS of Laﬁtte et al. is an excel- lent candidate for modeling real substance components and testing their extrapolationbehavior. 182 J. Staubach and S. Stephan 0.1 1 100.010.1110p / εσ-3 T / εk-1 B Fig. 5. Characteristic curves of the Mie λn,6 ﬂuid. Symbols indicate MD simulation results [ 41]. Lines correspond to predictions from the SAFT-VR Mie EOS. Results for three λn,6 Mie ﬂuids: 8,6 ( and dashed lines), 12,6 ( and solid lines), and 20,6 ( and dotted lines). Results for the Boyle curve ( blue ), Zeno curve ( red), Charles curve ( pink ), and Amagat curve ( orange ). 0.1 1 10 1000.010.1110100p / εσ-3 T / εk-1 B Fig. 6. Characteristic curves of the Mie 12, λmﬂuid. Symbols indicate MD simulation results [41]. Lines correspond to predictions from the SAFT-VR Mie EOS. Results for three 12, λmMie ﬂuids: 12,4 ( and dashed lines), 12,6 ( and solid lines), and 12,8 ( and dotted lines). Results for the Boyle curve ( blue ), Zeno curve ( red), Charles curve ( pink ), and Amagat curve ( orange ). 3.3 Toluene, Ethanol, and Dimethyl Ether For the model ﬂuids (cf. Sect. 3.2), the molecular-based EOS only used a repulsive and an attractive term for modeling the simple spherical particles. For the real substances discussed here, also the chain contribution and the association contribution were used. This signiﬁcantly increases the complexity of the models. Figure 7shows the results for toluene. Again, molecular simulation data is taken as reference. These data are used for the assessment of the two EOS models: the PC-SAFT Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 183 EOS and the SAFT-VR Mie EOS. For the PC-SAFT EOS, a spurious Amagat curve is obtained. This is not surprising as this defect is inherited from the repulsive term (cf. Sect. 3.2). The SAFT-VR Mie EOS predictions, on the other hand, are physically reason- able and in good agreement with the pseudo-experimental data. Only, the temperatureof the characteristic curves in the vicinity of the zero-pressure limit is overestimated by the SAFT-VR Mie EOS. 100 1000 100001101001000p / MPa T / K Fig. 7. Characteristic curves of toluene. Symbols ( ) indicate MD simulation results [ 51]. Lines are predictions from the SAFT-VR Mie EOS (solid lines) and the PC-SAFT EOS (dashed lines). Results for the Boyle curve ( blue ), Zeno curve ( red), Charles curve ( pink ), and Amagat curve ( orange ). The vapor pressure curve ( —) predicted from the SAFT-VR Mie EOS is shown. For ethanol and dimethyl ether, only the Charles curve was studied, cf. Figures 8and 9, respectively. The molecular simulation reference data is compared with the predictions from three EOS. Both the PC-SAFT and SAFT-VR Mie EOS were used for both substances. Moreover, the empirical EOS from Schroeder et al. was used for modeling ethanol andthe empirical EOS from Wu et al. was used for modeling dimethyl ether. Both empirical EOS yield an unphysical behavior, i.e. a convex shape at high pressure. The PC-SAFT EOS yields physically reasonable Charles curve for both components. For both the PC-SAFT and SAFT-VR Mie EOS, the results are qualitatively in accordance with Brown’s postulates. For ethanol, the PC-SAFT results show best quantitative agreement with the pseudo-reference data. For dimethyl ether, the SAFT-VR Mie results show the best agreement with the reference data. This is probably due to the underlying substance model parameters, cf. Table 2. 4 Conclusions In this work, the extrapolation behavior of physically-based thermodynamic models was studied. Therefore, a novel approach was developed that uses molecular dynamics simu-lations for generating pseudo-experimental data. Using that data as a reference, different 184 J. Staubach and S. Stephan 100 1000110100p / MPa T / K Fig. 8. Charles curve of ethanol. Symbols indicate MD results [ 31]. Lines indicate predictions from different EOS: SAFT-VR Mie (solid line), PC-SAFT (dashed line), and the empirical multi-parameter EOS from Schroeder et al. [ 37] (dotted line). The vapor pressure curve ( —) and the spinodal curves (---) as well as the critical point ( ) were computed from the EOS of Schroeder et al. 100 1000110100p / MPa T / K Fig. 9. Charles curve of dimethyl ether. Symbols indicate MD results [ 31]. Lines indicate pre- dictions from different EOS: SAFT-VR Mie (solid line), PC-SAFT (dashed line), and the empiricalmulti-parameter EOS from Wu et al. [ 54] (dotted line). The vapor pressure curve ( —) and the spinodal curves (---) as well as the critical point ( ) were computed from the EOS of Wu et al. molecular-based equation of state models were assessed regarding their applicability for modeling ﬂuid properties at extreme temperature and pressure. In a ﬁrst step, a new simulation method was developed for determining Brown’s characteristic curves from a given molecular force ﬁeld model [ 51]. This method com- bines statistical mechanics for (exactly) computing the zero-pressure limit state point of Prediction of Thermodynamic Properties of Fluids at Extreme Conditions 185 the characteristic curve via the virial route with classical MD simulations in the high- pressure regime. The method can be applied to both simple model ﬂuids as well as complex real molecular substances. It has not yet been tested on electrolyte and polymer systems, which would be interesting for future work. Moreover, we have only addressedpure substances here. In technical applications, however, ﬂuid mixtures are in practi- cally all cases present. An extension of the computational approach to mixtures would therefore also be interesting for a future work. The performance of two molecular-based EOS frameworks was compared, i.e. the PC-SAFT EOS and the SAFT-VR Mie EOS. The PC-SAFT model comprises more strongly simplifying approximations compared to the SAFT-VR Mie EOS. This is prob-ably the reason that the SAFT-VR Mie EOS is found to extrapolate signiﬁcantly better to extreme conditions regarding temperature and pressure. The PC-SAFT EOS, which is very frequently used in chemical engineering, on the other hand exhibits an artiﬁcialAmagat curve and should therefore not be applied at extreme temperatures and pressures. Hence, the SAFT-VR Mie EOS is found to be an excellent candidate for modeling ﬂuid properties at extreme conditions, e.g. in tribological contact processes. Impressively,the SAFT-VR Mie EOS does not only satisfy the requirements of Brown’s character- istic curves, the predictions of the EOS are in most cases in excellent agreement with the pseudo-experimental reference data. This is surprising considering the fact that theSAFT-VR Mie EOS was parametrized using data at moderate conditions alone. This supports the fact that models with a strong physical backbone can provide an excellent extrapolation behavior. In some cases, also empirical EOS were used for comparison and found to exhibit an unphysical behavior in some state region. Based on the novel approach using MD pseudo-experimental reference data, for the ﬁrst time, Brown’s characteristic curves were used for a quantitative assessment of thermodynamic equation of state models. For future work, it would be interesting to study long chain alkane molecules, which are important substances used as lubricants.Moreover, it would be interesting to study the reproducibility of Brown’s characteristic curves across different MD codes as well as the inﬂuence of the force ﬁeld used for modeling a given substance. Acknowledgement. The authors gratefully acknowledge access to the ELWE supercomputer at Regional University Computing Center Kaiserslautern (RHRK) under the grant TUKL-MTD. 186 J. Staubach and S. Stephan