Aus Wissenschaft und Forschung 5 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 Measurement, Modelling, and Application of Lubricant Properties at Extreme Pressures Patrick Wingertszahn, Sebastian Schmitt, Stefan Thielen, Manuel Oehler, Balázs Magyar † , Oliver Koch, Hans Hasse, Simon Stephan* Eingereicht: 23.05.2023 Nach Begutachtung angenommen: 18.09.2023 Messung, Modellierung und Anwendung von Schmierstoffeigenschaften bei extremen Drücken Schmierstoffe spielen in vielen technischen Anwendungen eine zentrale Rolle, z. B. in Lagern und Getrieben sowie in der Fertigungstechnik. In solchen Anwendungen sind Schmierstoffe im Kontaktbereich extremen Bedingungen ausgesetzt. In Schmierspalten kann der Druck Werte bis zu 5 GPa erreichen. Das Stoffverhalten des Schmierstoffs, insbesondere die Viskosität, hat einen großen Einfluss auf das Reibungs- und Verschleißverhalten eines Tribosystems. Dementsprechend sind zuverlässige Stoffdatenmodelle eine Voraussetzung für prädiktive tribologische Simulationen, z. B. elastohydrodynamische Simulationen. Die meisten derzeit verfügbaren experimentellen Stoffdaten liegen jedoch nur bis ca. 1 GPa vor. Daher werden zuverlässige und robuste Modelle mit gutem Extrapolationsverhalten bei extremen Drücken benötigt. In dieser Arbeit wurden Viskositätsmessungen von Squalan in einem Temperaturbereich zwischen 20 °C und 100 °C bei Drücken bis zu 1 GPa durchgeführt. Auf der Grundlage dieser Daten wurde ein physikalisches Modell für die Viskosität von Squalan entwickelt. Das Modell basiert auf einer molekular-basierten Zustandsgleichung in Kombination mit der Entropieskalierung. Abschließend wird gezeigt, wie dieses Modell für die Beschreibung des Stoffverhaltens vorteilhaft in eine EHL-Simulation über eine API integriert werden kann. Der neuartige hybride Simulationsansatz ist vielversprechend für zukünftige Anwendungen. Schlüsselwörter Schmierstoffe, Viskosität, Modellierung von Stoffeigenschaften, Entropieskalierung, Elastohydrodynamik Lubricants play a central role in many technical applications, e.g. in bearings and gears as well as in machining processes. In such applications, lubricants are exposed to extreme conditions in the contact area. In lubrication gaps, the pressure can reach values up to 5 GPa. The thermophysical properties of lubricants, and in particular the viscosity, at such extreme conditions have an important influence on the friction and wear behavior of a tribosystem. Accordingly, reliable lubricant property models are a prerequisite for accurate tribological simulations, e.g. elastohydrodynamic lubrication (EHL) simulations. Presently, the vast majority of experimental thermophysical property data are only available up to 1 GPa. Thus, reliable and robust models with strong extrapolation capabilities to higher pressure are required. In this work, viscosity measurements of squalane in a temperature range between 20 °C and 100 °C and pressures up to 1 GPa were carried out. Based on that data, a physical model for the viscosity was developed. The model is built by combining a molecular-based equation of state with the so-called entropy scaling approach. Finally, we demonstrate how this fluid property model can be favorably integrated in an EHL simulation by an application programming interface (API). The novel hybrid modeling approach is promising for future applications. Keywords Lubricants properties, viscosity, modeling of material properties, entropy scaling, elastohydrodynamic simulation Kurzfassung Abstract * Patrick Wingertszahn 1 , Sebastian Schmitt 2 , Stefan Thielen 1 , Manuel Oehler 1 , Balázs Magyar 1 † , Oliver Koch 1 , Hans Hasse 2 , Simon Stephan 2 (corresponding Author) 1 Institute of Machine Elements, Gears and Tribology (MEGT), RPTU Kaiserslautern, Germany 2 Laboratory of Engineering Thermodynamics (LTD), RPTU Kaiserslautern, Germany † Presently at University Paderborn; Design and Drive Technology TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 5 solid base. Such models, if carefully designed, often enable an extrapolation to states that were not considered during the parametrization [Sch23, Ste20, Urs23, Ewe16, Sta23]. In this work, we demonstrate the feasibility of this route in three steps by studying a model lubricant, namely squalene. In a first step, experiments were carried our using state-of-the-art laboratory equipment for determining the viscosity in a temperature range of 20 °C to 100 °C and pressures up to 1 GPa. Second, a physicallybased model was developed and used for describing the lubricant properties, especially (but not limited to) the viscosity. The extrapolation behavior of the new model was evaluated and compared to an empirical model. Third, a hybrid simulation model is proposed in which the physically-based lubricant property model is coupled with an EHL simulation. This is done using a simple benchmark test case for demonstration purposes. Accordingly, this paper is structured as follows: introducing the experimental method, introducing the modeling approach, presenting the viscosity measurement and model results, and presenting the hybrid simulation model setup and first preliminary results. Finally, conclusions are drawn and an outlook is given. 2 Experimental methods Different types of experimental instruments exist for viscosity measurements [Alu21, Mas49]. Most of them use the falling body principle. Hence, they are based on a relatively simple measurement principle. Also the highpressure viscosimeter used in this work is based on that measurement principle. Aus Wissenschaft und Forschung 6 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 1 Introduction Lubricated contacts between machine elements are subject to very high pressures - up to 5 GPa [Sta17]. The thermophysical properties of the lubricant in the contact zone play a central role for the performance of the tribosystem [Bhu00, Kuw17, He15, Ste18, Ste23]. For the modelling and simulation of lubricated tribocontacts by continuum mechanic models such as elastohydrodynamic lubrication (EHL) models, reliable fluid property models - especially for the viscosity - are required [Kie17, Spi14, Win23, Bai22, Thi20, Kam18, Kam22]. Usually, in such simulation models, empirical correlations for thermophysical property data are employed. Evidently, their accuracy is limited by the accuracy of the underlying measurement data and, more importantly, limited by the thermodynamic state range considered (and accessible) in the experiments. Extrapolating experimental data using empirical mathematical models is obviously risky. In general, empirical models are only applicable in the state range considered for the parametrization. At extreme pressures up to about 5 GPa, thermophysical properties of lubricants are challenging to determine experimentally. Experiments for determining the viscosity are presently limited to about 1 GPa - using specialized instruments [Bai04]. Moreover, even for pressures ≈ 1 GPa, thermophysical property data for lubricants is sparse. Accordingly, empirical viscosity models are, in the best cases, valid up to only about 1 GPa. Physically-based models are attractive candidates for providing a reliable extrapolation behavior beyond the range of data used for the parametrization. Especially models from molecular thermodynamics are interesting as they are constructed from the molecular architecture and molecular interactions, which provides a profound Figure 1: Structure of the high-pressure viscometer. Left part represents the actual measuring device. The right part induces the pressure. TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 6 The viscometer used here was developed by Bair [Bai04]. It was purchased under the auspices of the Collaborative Research Centre 926, modified, and extended at the RPTU Kaiserslautern. The basic structure of the viscometer is shown in Figure 1. It is suitable for measurements up to a pressure of 1 GPa and temperature up to 110 °C. In the viscosimeter (cf. Figure 1), the pressure is applied by a combination of devices, which includes a hydraulic press, a low-pressure cylinder, a high-pressure cylinder, and a pressure regulator connected to a high-pressure test chamber. As the pressure-transmitting fluid, di(2-ethylhexyl) is used. The applied pressure is measured directly in the pressurized fluid using a pressure transducer. On the high-pressure side, there is a pressure vessel closed by the viscometer plug. It contains a Bridgman seal. The Bridgman seal consists of a plug piston with threaded stem, two unhardened tool steel back-up rings, and glass fiber reinforced TFE and rubber packing. The packing is compressed via a thread, which creates the sealing effect. The pressure vessel contains a thin-walled tube, which is closed on one side with a plug and on the other side with a movable cylinder. The cylinder serves for equalizing the pressure such that the fluid sample in the tube is pressurized to the same pressure as the pressure vessel. The sample fluid is filled into the tube and a sinker is inserted. The metallic sinker is magnetic, which allows its motion to be detected with a variable differential transformer (LVDT). Two different sinkers can be used in the viscometer. One of the two sinkers has been optimized for low viscosity measurements. The second sinker is suitable for measuring high viscosity values. The movement of the sinker through the LVDT causes a voltage signal, which is recorded by a digital multimeter and sent to a measuring computer. There, the measurement signal is recorded and further post-processed. For prescribing the temperature, heating air is supplied via a pressure regulator and heated by a 600 W heating cartridge in a large tube in front of the regulator. The temperature is adjusted by changing the supply voltage of the heating element. The temperature is measured with a type J thermocouple. The accuracy of the temperature measurement is approximately ± 0.5 °C. The viscosity η is determined from the falling speed v of the sinker and the density of the sinker ρ sinker and the density of the fluid ρ fluid . The viscosity is evaluated as (1) where ϑ is the temperature, p is the pressure, and C is a calibration variable that is a function of the temperature. The latter characterizes the geometry of the experimental setup, i.e. the influence of the geometry of the drop body and the sample tube on the falling velocity and the flow field. The calibration was performed using tri(2-ethylhexyl) trimellitate [Bai16] and di(2-ethylhexyl) sebacate [Kle53, Sad23] data. Based on Eq. (1), the viscosity of the lubricant is determined for a given pressure p and temperature ϑ. The velocity is evaluated in a stationary range of the fall time of the sinker as exemplarily shown in Figure 2. It shows the measurement signal, which indicates the position of the magnetic sinker, caused from the motion of the magnetic sinker in the tube. Figure 2 shows two sequential movements. For the evaluation of the fall curve, only the stationary part of the curve is used, cf. fitted linear function depicted in Figure 2. The shown example is for a relatively low viscosity. For high viscosities, the falling time can be up to several hours. In this work, measurements were carried out at three temperatures, namely 20 °C, 60 °C, and 100 °C. The pressure was varied between ambient pressure 0.1 MPa and 1000 MPa. For each temperature, 13-19 pressure values were studied. In total, 47 state points were studied. Each measurement was conducted at least three times. The final value was then calculated as the average of the individual measurement values. The uncertainty of the data was estimated from the standard deviation of the repeated measurements. 3 Physically-based lubricant property model In this work, methods from molecular thermodynamics were applied for modelling the properties of the lubricant. The approach used here is a combination of a molecular-based equation of state (EOS) and entropy scaling. Molecular-based equations of state [Nez20, Ste20, Mue01, Kon21] have been primarily developed by the chemical engineering community and are therein today well-established. Molecular-based equations of state are known to yield an excellent extrapolation behavior in ( , ) = ( ) ∙ ∙ − ( , ) , Aus Wissenschaft und Forschung 7 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 Figure 2: Example of a recorded fall curve of the sinker during viscosity measurements: Measurement signal (indicating the sinker position) as a function of the measurement time. TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 7 the reduced configurational entropy s ̃ conf = s conf / k B / m as (2) where ɑ 2 and ɑ 3 are component-specific adjustable parameters and g 1 and g 2 are global parameters of the entropy scaling model fitted to molecular simulation data of the Lennard-Jones fluid [Sch23b]. The parameters ɑ 2 and ɑ 3 were fitted to the viscosity data for squalane determined in this work and are given in Table 2. ( ̃ ) = ̃ + ̃ 1 + ( ̃ − 1 ) + ̃ , Aus Wissenschaft und Forschung 8 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 many cases [Ste20, Urs23, Ste23b] (yet, it cannot be taken for granted). For modeling transport properties of fluids, equations of state need an extension. A possible transport property extension is the so-called entropy scaling [Dyr18], which can take favorably advantage of the physical framework of molecular-based equations of state. Entropy scaling exploits the phenomenon, that transport properties follow (within certain limits) a monovariate function with respect to the configurational entropy - when properly made dimensionless by the temperature and the density [Ros99]. Thereby, transport properties can be described as a simple function of the entropy, which accordingly acts as a transducer between the state point (e.g. T, ρ) and the transport property. Entropy scaling was originally proposed by Rosenfeld [Ros77, Ros99] and is today a popular tool in chemical engineering [Dyr18, Loe18]. In this work, a molecular-based EOS in combination with entropy scaling was used for modeling the viscosity of squalane. Molecular-based EOS provide an expression for the Helmholtz energy A = A(T,ρ) based on the physical properties of a molecule, i.e. the chain length, the dispersion energy, and the segment diameter (cf. Table 1). From the Helmholtz energy model A = A(T,ρ), all static thermodynamic properties such as the pressure, compressibility, and entropy can be computed using basic thermodynamic relations. Dynamic, i.e. transport properties, like the viscosity cannot be derived from the Helmholtz energy without further ado. Entropy scaling utilizes the observation that transport property data like the viscosity at different temperature and pressure exhibit an approximately monovariate dependency with respect to the configurational entropy s conf [Ros77, Ros99]. The latter can be straightforwardly modeled by a molecular-based EOS. Hence, the entropy s conf = s conf (T,ρ) computed from the equation of state brings in the temperature and density dependency of the model. In the entropy scaling model, the transport properties have to be scaled properly; details are given for example in Refs. [Ros77, Ros99, Dyr18, Loe18, Sch23]. To describe the scaled viscosity as function of the entropy, i.e. η s = η s (s conf ), only few experimental data are required. This reflects the fact that entropy scaling models have good extrapolation capabilities. In this work, squalane (C 30 H 62 ) was considered. Here, the PC-SAFT equation of state [Gro01] was used for modeling the configurational entropy s conf . The PC-SAFT parameters for squalane were taken from Bamgbade et al. [Bam15]. The parameters are given in Table 1. The chain length parameter reflects the elongation of the C30 molecule in the model, cf. Ref. [Gro01] for details. The entropy scaling approach proposed by Schmitt et al. [Sch23b] was used in this work. Therein, the scaled viscosity η s = η s (s ̃ conf ) is described by a rational function of Symbol Unit value m - 16.6709 σ Å 3.536 ε / k B K 227.53 Table: 1: PC-SAFT parameters for squalane [Bam15]: chain length parameter m, dispersion energy ε, and segment diameter σ. 4 Results and discussion for the viscosity of squalane Figure 3 shows the results for the viscosity of squalane. Both, the experimental data and the entropy scaling model are shown. Additionally, experimental viscosity data from the literature as well as the viscosity obtained from an empirical model are depicted. The measurement data from this work is in good agreement with the data from the literature [Bai06]. The mean relative statistical uncertainty of the experimental data from this work is approximately 5 %. The numeric values of the experimental viscosity data obtained in this work is reported in Table 3. The entropy scaling model for squalane was fitted to the experimental data obtained in this work. The entropy scaling model yields a relative mean deviation to the measurement data from this work of 23 % - with only two adjustable parameters. The empirical Doolittle model (taken from Ref. [Bai06]) yields a relative mean deviation of 10 %. Yet, in the Doolittle model, five parameters were adjusted. The extrapolation behavior of the entropy scaling model and the Doolittle model are shown in Figure 3-right. The entropy scaling model shows a trustworthy behavior compared to the empirical Doolittle model. The entropy scaling model yields a more meaningful behavior, which is due to the strong physical base of the entropy scaling approach and the molecularbased equation of state. Yet, the entropy scaling model could probably be improved by using more sophisticated Symbol Value ɑ 2 3.5561 ɑ 3 0.4868 Table: 2: Entropy scaling parameters for squalane. TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 8 molecular-based equation of state models, e.g. the SAFT-VR Mie equation of state [Laf23, Urs23]. 5 Application: Hybrid physically-based fluid property + EHL model The developed molecular thermodynamics model provides a robust tool for modeling thermophysical properties of a fluid lubricant - here squalane. In particular, different properties are captured in a thermodynamically consistent approach since the different properties are derived from a fundamental Helmholtz energy expression. Thereby, not only the density and the viscosity, but also the heat capacity, the compressibility, the vapor pressure etc. can be modeled (without any further parameter adjustment). This class of material model (molecular-based equation of state combined with entropy scaling) can be beneficially used in tribological continuum simulations. This is demonstrated here using an elastohydrodynamic lubrication (EHL) simulation model. Hence, the physically-based fluid property model is combined with an Aus Wissenschaft und Forschung 9 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 °C P MPa η Pa s °C p MPa η Pa s °C P MPa η Pa s 20 0.1 0.044 60 0.1 0.008 100 0.1 0.003 42.1 0.109 40.0 0.016 42.1 0.006 82.1 0.237 42.1 0.015 82.1 0.010 122.1 0.475 80.0 0.027 122.1 0.015 162.1 0.823 82.1 0.026 162.1 0.020 233.5 2.233 120.0 0.047 283.3 0.080 283.3 4.301 122.1 0.045 333.0 0.109 382.5 15.95 160.0 0.082 382.5 0.140 481.1 52.88 162.1 0.070 481.1 0.270 579.1 172.57 200.0 0.125 579.1 0.528 676.6 553.03 283.3 0.325 676.6 0.986 773.5 1 444.9 333.0 0.573 773.5 1.795 869.9 6 010.3 382.5 0.906 869.9 3.293 481.1 2.330 965.7 5.338 579.1 5.349 982.9 4.918 676.6 11.68 773.5 26.28 869.9 61.26 982.9 75.50 Table: 3: Experimental viscosity data for squalane obtained from the high-pressure viscosimeter in this work. Figure 3: Viscosity of squalane as function of the pressure for three temperatures ϑ = 20 °C, 60 °C, and 100 °C (color coded): Results from the entropy scaling model, the empirical Doolittle model (taken from Ref. [Bai06]), experimental data from this work, and experimental data from Ref. [Bai06]. Left: Pressure range 0 ≤ p / MPa ≤ 1000 (experimentally accessible). Right: Extrapolation behavior of the models up to 5000 MPa. TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 9 the materials need to be specified. At the end of the simulations, the results are provided, e.g. pressure and temperature profiles (as a function of the direction of motion coordinate x, cf. Figure 4). For demonstration purposes and as a proof-of-concept, we used a simple geometry for describing a tribological contact. The geometry proposed by Woloszynski et al. [Wol15] was used. For the fluid properties, namely the density and the viscosity at a given temperature and pressure, the squalane model introduced in Section 4 was used. The results for the lubrication gap and pressure profiles are depicted in Figure 5. The resulting pressure profile shows the expected shape. Elastic deformations were not included and the simulation was isothermal. The density as well as the viscosity of the lubricant were on-the-fly calculated via Mic- Therm, i.e. the molecular-based EOS and entropy scaling. The results shown in Figure 5 demonstrate the general applicability of the hybrid simulation scheme. Aus Wissenschaft und Forschung 10 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 EHL model to a hybrid simulation model - as schematically depicted in Figure 4. The coupling is carried out internally and on-the-fly via an application programming interface (API). Thus, the state points relevant within the EHL model are handed over to the molecular thermodynamics software, which then provides the requested material property data. In this work, we used the open-access EHL code from Hansen et al. [Han22]. For the molecular thermodynamics part, an in-house developed software tool box was used, which is called MicTherm. It comprises a large library of molecular thermodynamics models, which enable the modelling of both pure component and mixture properties. Also, a large number of substance models is included in MicTherm. Thereby, a large number of lubricant components, mixtures as well as different thermophysical properties can readily be modeled. The EHL code [Han22] was slightly adapted such that it was compatible with the MicTherm API. For carrying out simulations based on the outlined hybrid model, the geometry, the boundary conditions, and Figure 5: Results of exemplary simulation based on the Woloszynski-geometry [Wol15]: Lubrication gap height h (left) and pressure p (right) as function of the coordinate x. The temperature was ϑ = 100 °C. Input • Geometry • Boundary conditions • Substance properties of the solid • Substance model of lubricant Results material properties state point temperature profile pressure profile … API p x T x hybrid simulation Molecular thermodynamics software EHL software x Figure 4: Scheme of hybrid simulation model linking molecular thermodynamics models with EHL simulations. TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 10 6 Conclusions and Outlook In this work, we showed how physically-based models can be used for extrapolating thermophysical property data for tribological applications to extreme pressures. This was done using squalane as a model lubricant. First, experimental measurements for determining the viscosity of squalane were carried out using a specialized high-pressure viscosimeter. The measurements were carried out up to 1 GPa and 100 °C. The obtained data is in good agreement with available data from the literature. Second, a physically-based model was developed for describing the thermophysical properties of squalane - focusing on the viscosity. Therefore, a molecular-based equation of state was used in combination with the entropy scaling approach. As this model is built on a strong physical basis, there are good reasons to believe in the extrapolation behavior of the proposed model. This is supported by the fact that the empirical model (taken from the literature) shows a questionable behavior upon extrapolation. In the third part, we demonstrated that the physically-based material model can be integrated into an EHL simulation such that the fluid properties are on the fly computed. The hybrid simulation approach proposed in this work can be favorably applied for various tribological applications. Molecular-based equations of state can be used for modeling a large number of thermophysical properties in a thermodynamically consistent way, which also provides good extrapolation behavior - if used carefully. Moreover, it has been shown in the literature [Fer22] that entropy scaling is also applicable for predicting transport properties of mixtures. Hence, for future work, it would be interesting to use the hybrid simulation approach for modeling cavitation and mixtures of lubricants. Furthermore, caloric properties such as the heat capacity and the thermal conductivity could be directly used in the hybrid model for realistically considering the thermal balance of the system. Acknowledgement The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) within the SFB 926, by the BMBF within the WindHPC project, and by the KSB foundation. The simulations were carried out on the ELWE cluster at Regional University Computing Center Kaiserslautern (RHRK) under the grant TUK-MTD. Literature [Ahu21] A. Ahuja, R. Lee, Y. M. Joshi: Advances and Challenges in the High-Pressure Rheology of Complex Fluids, Advances in Colloid and Interface Science 294, 102472 (2021). https: / / doi.org/ 10.1016/ j.cis.2021.102472 [Bai04] S. Bair: A Routine High-Pressure Viscometer for Accurate Measurements to 1 GPa, Tribology Transactions 47, 356-360 (2004). https: / / doi.org/ 10.1080/ 05698190490455582 [Bai06] S. Bair: Reference Liquids for Quantitative Elastohydrodynamics: Selection and Rheological Characterization, Tribology Letters 22, 197-206 (2006). https: / / doi.org/ 10.1007/ s11249-006-9083-y [Bai16] S. Bair: The temperature and pressure dependence of viscosity and volume for two reference liquids, Lubrication Science 28, 81 (2016). https: / / doi.org/ 10.1002/ ls.1302 [Bai22] S. Bair, A Traction (Friction) Curve Is Not a Flow Curve, Lubricants 10, 221 (2022). https: / / doi.org/ 10.3390/ lubricants10090221 [Bam15] B.A Bamgbade, A. Wu, W.A. Burgess, D. Tapriyal, I.K. Gamwo, H.O. Baled, R.M. Enick, M.A. McHugh: High-Temperature, High-Pressure Volumetric Properties of Propane, Squalane, and Their Mixtures: Measurement and PC-SAFT Modeling, Industrial & Engineering Chemistry Research 54, 6804-6811 (2015). https: / / doi.org/ 10.1021/ acs.iecr.5b01173 [Bhu00] B. Bhushan (Ed.): Modern Tribology Handbook: Volume One and Two, 1 ed., CRC Press, Boca Raton, US, (2000). [Dyr18] J.C. Dyre: Perspective: Excess-Entropy Scaling, Journal of Chemical Physics 149, 210901 (2018). https: / / doi.org/ 10.1063/ 1.5055064 [Ewe16] J. Ewen, C. Gattinoni, F. Thakkar, N. Morgan, H. Spikes, D. Dini: A Comparison of Classical Force- Fields for Molecular Dynamics Simulations of Lubricants, Materials 9, 651 (2016). https: / / doi.org/ 10.3390/ ma9080651 [Fer20] D. Fertig, H. Hasse, S. Stephan: Transport properties of binary Lennard-Jones mixtures: Insights from entropy scaling and conformal solution theory, Journal of Molecular Liquids 367, 120401 (2022). https: / / doi.org/ 10.1016/ j.molliq.2022.120401 [Gro01] J. Gross, G. Sadowski: Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules, Industrial & Engineering Chemistry Research 40, 1244-1260 (2001). https: / / doi.org/ 10.1021/ ie0003887 [Han22] E. Hansen, A. Kacan, B. Frohnapfel, A. Codrignani: An EHL Extension of the Unsteady FBNS Algorithm, Tribology Letters 70, 80 (2022). https: / / doi.org/ 10.1007/ s11249-022-01615-1 [He15] Y. He, T.J. Zolper, P.Liu, Y. Zhao, X. He, X. Shen, H. Sun, Q. Duan, Q. Wang: Elastohydrodynamic lubrication properties and friction behaviors of several ester base stocks, Friction 3, 243-255 (2015). https: / / doi.org/ 10.1007/ s40544-015-0090-6 [Kam18] L. Simo Kamga, M. Oehler, B. Magyar, B. Sauer: Charakterisierung des strukturviskosen Verhaltens von Schmierstoffen durch Kombination von Experiment und EHD-Simulation mithilfe des Carreau- Modells, Forschung Ingenieurwesen 82, 301-309 (2018). https: / / doi.org/ 10.1007/ s10010-018-0283-z [Kam22] L. Simo Kamga, D. Meffert, B. Magyar, M. Oehler, B. Sauer: Simulative investigation of the influence of surface texturing on the elastohydrodynamic lubrication in chain joints, Tribology International 171, 107564 (2022). https: / / doi.org/ 10.1016/ j.triboint.2022.107564 Aus Wissenschaft und Forschung 11 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 11 Thermophysical Properties of Long Linear and Branched Alkanes, The Journal of Physical Chemistry B 127, 1789-1802 (2023). https: / / doi.org/ 10.1021/ acs.jpcb.2c07997 [Sch23b] S. Schmitt, H. Hasse, S. Stephan: Entropy scaling framework for transport properties using molecularbased equations of state, to be submitted (2023). [Spi14] H. Spikes: Basics of EHL for practical application, Lubrication Science 27, 45-67 (2014). https: / / doi.org/ 10.1002/ ls.1271 [Sta17] G. W. Stachowiak: How tribology has been helping us to advance and survive, Friction 5, 233 (2017). https: / / link.springer.com/ article/ 10.1007/ s40544- 017-0173-7 [Sta23] J. Staubach, S. Stephan: Prediction of Thermodynamic Properties of Fluids at Extreme Conditions: Assessment of the Consistency of Molecular-Based Models, In: J.C. Aurich, C. Garth, B. Linke (Eds.) Proceedings of the 3rd Conference on Physical Modeling for Virtual Manufacturing Systems and Processes (2023). https: / / doi.org/ 10.1007/ 978-3-031-35779-4_10 [Ste18] S. Stephan, M.P. Lautenschlaeger, I. Alabad Alhafez, M.T. Horsch, H.M. Urbassek, H. Hasse: Molecular Dynamics Simulation Study of Mechanical Effects of Lubrication on a Nanoscale Contact Process, Tribology Letters 66, 126 (2018). https: / / doi.org/ 10.1007/ s11249-018-1076-0 [Ste23] S. Stephan, S. Schmitt, H. Hasse, H.M. Urbassek: Molecular dynamics simulation of the Stribeck curve: Boundary lubrication, mixed lubrication, and hydrodynamic lubrication on the atomistic level, Friction 11, 2342-2366 (2023). https: / / doi.org/ 10.1007/ s40544-023-0745-y [Ste20] S Stephan, J Staubach, H Hasse: Review and comparison of equations of state for the Lennard-Jones fluid, Fluid Phase Equilibria 523, 112772 (2020). https: / / doi.org/ 10.1016/ j.fluid.2020.112772 [Ste20b] S. Stephan, U.K. Deiters: Characteristic curves of the Lennard-Jones fluid, International Journal of Thermophysics 41, 147 (2020). https: / / doi.org/ 10.1007/ s10765-020-02721-9 [Ste23b] S. Stephan, M. Urschel: Characteristic curves of the Mie fluid, Journal of Molecular Liquids 383, 122088 (2023). https: / / doi.org/ 10.1016/ j.molliq.2023.122088 [Thi20] S. Thielen, B. Magyar, B. Sauer: Thermoelastohydrodynamic Lubrication Simulation of Radial Shaft Sealing Rings, Journal of Tribology 142, 052301 (2020). https: / / doi.org/ 10.1115/ 1.4045802 [Urs23] M. Urschel, S. Stephan: Determining Brown’s Characteristic Curves Using Molecular Simulation, Journal of Chemical Theory and Computation 19, 1537- 1552 (2023). https: / / doi.org/ 10.1021/ acs.jctc.2c01102 [Win23] P. Wingertszahn, O. Koch, L. Maccioni, F. Concli, B. Sauer: Predicting Friction of Tapered Roller Bearings with Detailed Multi-Body Simulation Models. Lubricants 11, 369 (2023). https: / / doi.org/ 10.3390/ lubricants11090369 [Wol15] T. Woloszynski, P. Podsiadlo, G.W. Stachowiak: Efficient Solution to the Cavitation Problem in Hydrodynamic Lubrication, Tribology Letters 58, 18 (2015). https: / / doi.org/ 10.1007/ s11249-015-0487-4 Aus Wissenschaft und Forschung 12 Tribologie + Schmierungstechnik · 70. Jahrgang · 4-5/ 2023 DOI 10.24053/ TuS-2023-0017 [Kie17] T. Kiekbusch, B. Magyar, B. Sauer: Multi-Ebenen- Simulation zur Berechnung der lokalen Lebensdauer in hochbelasteten Wälzkontakten. 12. VDI-Fachtagung Gleit- und Wälzlagerungen 2017. Gestaltung, Berechnung, Einsatz (VDI Berichte 2308, ISBN 978-3-18-092308-6). pp. 103-112. Schweinfurt, June 27.-28. 2017. [Kle53] R. V. Kleinschmidt, D. Bradbury, M. Mark: Viscosity and Density of over Forty Lubricating Fluids of Known Composition at Pressures to 150,000 psi and Temperatures to 425 F, ASME, New York (1953). [Kon21] G. M. Kontogeorgis, R. Dohrn, I. G. Economou, J.-C. de Hemptinne, A. ten Kate, S. Kuitunen, M. Mooijer, L. F. Žilnik, V. Vesovic: Industrial Requirements for Thermodynamic and Transport Properties: 2020, Industrial & Engineering Chemistry Research 60, 4987-5013 (2021). https: / / doi.org/ 10.1021/ acs.iecr.0c05356 [Kuw17] T. Kuwahara, G. Moras, M. Moseler: Friction Regimes of Water-Lubricated Diamond (111): Role of Interfacial Ether Groups and Tribo-Induced Aromatic Surface Reconstructions, Physical Review Letters 119, 096101 (2017). https: / / doi.org/ 10.1103/ Phys- RevLett.119.096101 [Laf13] T. Lafitte, A. Apostolakou, C. Avendano, A. Galindo, C.S. Adjiman, E.A Mü ller, G. Jackson: Accurate Statistical Associating Fluid Theory for Chain Molecules formed from Mie Segments, Journal of Chemical Physics 139, 154504 (2013). https: / / doi.org/ 10.1063/ 1.4819786 [Loe18] O. Lötgering-Lin, M. Fischer, M. Hopp, J. Gross: Pure Substance and Mixture Viscosities Based on Entropy Scaling and an Analytic Equation of State, Industrial & Engineering Chemistry Research 57, 4095-4114 (2018). https: / / doi.org/ 10.1021/ acs.iecr.7b04871 [Mas49] W.P. Mason: Measurement of the Viscosity and Shear Elasticity of Liquids by Means of a Torsionally Vibrating Crystal, Transactions of the American Society of Mechanical Engineers 96, 359 (1949). https: / / doi.org/ 10.1115/ 1.4017390 [Mue01] E.A. Müller, K.E. Gubbins: Molecular-based equations of state for associating fluids: A review of SAFT and related approaches, Industrial & Engineering Chemistry Research 40, 2193-2211 (2001). https: / / doi.org/ 10.1115/ 1.4017390 [Nez20] I. Nezbeda: On Molecular-Based Equations of State: Perturbation Theories, Simple Models, and SAFT Modeling, Frontiers in Physics 8, 1 (2020). https: / / doi.org/ 10.3389/ fphy.2020.00287 [Ros77] Y. Rosenfeld: Relation between the transport coefficients and the internal entropy of simple systems, Physical Review A 15, 2545-2549 (1977). https: / / doi.org/ 10.1103/ PhysRevA.15.2545 [Ros99] Y. Rosenfeld: A Quasi-Universal Scaling Law for Atomic Transport in Simple Fluids, Journal of Physics: Condensed Matter 11, 5415 (1999). https: / / doi.org/ 10.1088/ 0953-8984/ 11/ 28/ 303 [Sad23] R. J. Sadinski, C. H. Hager, W. M. Hannon: A Proposed Pressure-Viscosity Coefficient Model Through Processing of Falling Body Viscometry Data for Di-(Ethylhexyl) Sebacate. Journal of Tribology 145, 011601 (2023). https: / / doi.org/ 10.1115/ 1.4055408 [Sch23] S. Schmitt, F. Fleckenstein, H. Hasse, S. Stephan: Comparison of Force Fields for the Prediction of TuS_4_2023.qxp_TuS_4_2023 20.09.23 09: 16 Seite 12