Viscoelastic Response of Elastohydrodynamically Lubricated Compliant Contacts below Glass-Transition Temperature

Citation: Krupka, J.; Dockal, K.; Sedlacek, T.; Rebenda, D.; Krupka, I.; Hartl, M. Viscoelastic Response of Elastohydrodynamically Lubricated Compliant Contacts below Glass-Transition Temperature. Polymers 2023,15, 2528. https:// doi.org/10.3390/polym15112528 Academic Editor: Pavlos Stephanou Received: 31 March 2023 Revised: 19 May 2023 Accepted: 27 May 2023 Published: 30 May 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). polymers Article Viscoelastic Response of Elastohydrodynamically Lubricated Compliant Contacts below Glass-Transition Temperature Jiri Krupka 1,* , Krystof Dockal 2, Tomas Sedlacek 3, David Rebenda 1,4 , Ivan Krupka 1and Martin Hartl 1 1Faculty of Mechanical Engineering, Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic; [email protected] or r[email protected] (D.R.); [email protected].cz (I.K.); [email protected].cz (M.H.) 2HVM Plasma spol. s.r.o., Na Hutmance 2, 158 00 Praha 5, Czech Republic; [email protected] 3Centre of Polymer Systems, Tomas Bata University in Zlin, Trida Tomase Bati 5678, 760 01 Zlin, Czech Republic; [email protected] 4Footwear Research Centre, University Institute, Tomas Bata University in Zlin, Nad Ovcirnou IV 3685, 760 01 Zlin, Czech Republic *Correspondence: jiri.kr[email protected] Abstract: The widespread use of polymers in the high-performance engineering applications brings challenges in the field of liquid lubrication in order to separate the rubbing surfaces by the coherent fluid-film thickness relative to not only the inelastic material response of the polymers. The determination of the mechanical properties by the nanoindentation and the dynamic mechanical analysis represents the key methodology to identify the viscoelastic behavior with respect to the intense frequency and temperature dependance exhibited by polymers. The fluid-film thickness was examined by the optical chromatic interferometry on the rotational tribometer in the ball-on-disc configuration. Based on the experiments performed, first, the complex modulus and the damping factor for the PMMA polymer describing the frequency and temperature dependence were obtained. Afterwards, the central as well as minimum fluid-film thickness were investigated. The results revealed the operation of the compliant circular contact in the transition region very close to the boundary between the Piezoviscous-elastic and Isoviscous-elastic modes of the elastohydrodynamic lubrication regime, and a significant deviation of the fluid-film thickness from the prediction models for both modes in dependence on the inlet temperature. Keywords: compliant contact; elastohydrodynamic lubrication; transition region; fluid-film thickness; optical chromatic interferometry; viscoelastic behavior 1. Introduction After the WW2, the first phase of the widespread use of polymers to many areas of industry was recorded [ 1 , 2 ]. Originally, the polymer machine elements, such as gears, were mainly operated under dry conditions or lubricated by greases and used to transfer the motion rather than to transmit torque. Conversely, the current research in the field of tribology of polymer materials is focused on the operation of machine elements under liquid lubrication conditions, especially, in the elastohydrodynamic lubrication (EHL) regime [ 3 – 8 ], where the rubbing surfaces are fully separated by a coherent film thickness of the lubricant. Nowadays, the polymers in connection with the EHL regime make available the implementation into the high-performance applications where dynamics, tribology and extreme operating conditions interact. Transmissions, differentials and mechanisms with polymer gears are examples of the use in automotive, aviation, space industries, etc. One of the constrains of polymers is a significant dependence of mechanical properties on the temperature, which leads to the manifestation of viscoelastic behavior. In terms of tribology, the question arises, how the viscoelastic response of the polymer affects the formation of Polymers 2023,15, 2528. https://doi.org/10.3390/polym15112528 https://www.mdpi.com/journal/polymers Polymers 2023,15, 2528 2 of 20 film thickness in compliant contacts operated in the EHL regime under operating conditions far from the glass-transition temperature of the polymer selected. The current paper is concerned with the basic oriented research. The main aim of this paper is to investigate the formation of fluid-film thickness in the compliant circular contact operating in the EHL regime with respect to the temperature and time (or frequency), which characterize the viscoelastic response of the polymer. For validation, the amorphous transparent polymer Polymethyl-methacrylate (PMMA) and the reference synthetic lubricant 5P4E were employed. The results are presented in part A and part B where the former deals with the analysis of the material properties of PMMA by nanoindentation (nano-DMA) and the dynamic mechanical analysis (DMA). The latter is concerned with the analysis of the fluid-film thickness in compliant circular contact using the method of optical chromatic interferometry. The subject of this paper is further described in the Supplementary Information (SI)—A. 2. Materials and Methods 2.1. Selection of Materials In part A, a plate-shaped PMMA polymer specimens of 10 (w) × 10 (l) × 0.5 (t) mm were used for determination of mechanical properties based on the nano-DMA. Moreover, the cylinder-shaped PMMA specimens of 3 mm in diameter and 8 mm in length for DMA were used. In Part B, a compliant contact was simulated between the flat PMMA polymer disc with a diameter of 120 mm, and the 100Cr6 bearing steel ball of 25.4 mm in external diameter. The purchased XT PMMA sheets (NUDEC, S. A (ES)) were manufactured by an extrusion process. The amorphous PMMA disposes of excellent light transmittance, over 90%, necessary for implementation of optical methods. To achieve a sufficient interference of light, and to avoid the presence of parasitic light, the PMMA disc was coated with semi-reflective chromium and antireflective layers vapordeposited on the bottom side (in contact with ball) and on the top side of disc, respectively. The mechanical properties of PMMA [ 9 , 10 ] are close in values to the engineering [ 11 ] and high-performance [ 12 ] polymers (PA66, POM, PEEK) frequently used to produce machine elements. Selected properties of specimens, such as RMS roughness (R q ) or E, are stated in Table 1. Table 1. Selected properties of specimens. Specimen Material Rq(µm) E (GPa) υ(−) Tg(◦C) Disc, plate and cylinder PMMA Rq1 < 0.005 3.3 0.39 105–110 Ball 100Cr6 Rq2 < 0.01 206 0.30 − 2.2. Selection of Lubricants In part B, the compliant contact was lubricated with Santovac ® 5—Polyphenyl Ether (5P4E), a synthetic diffusion pump oil produced by SANTOLUBES LLC company (Spartanburg, SC, USA). The selected lubricant belongs to the API (American Petroleum Institute, Washington, DC, USA) Group V according to the base oil categories including synthetic lubricants such as polyglycols, silicones and esters. One of the main characteristics of 5P4E lubricant is its extreme viscosity, approximately η0≈3 Pa s at 24 ◦C. Hence, it is usually used in aviation and aerospace applications for extreme environmental and operating conditions. In addition, the 5P4E is frequently used as a reference lubricant in the field of EHL [ 7 , 13 ]. The rheological properties of 5P4E, such as the dynamic viscosity at atmospheric pressure ( η0 ) and pressure–viscosity coefficient ( α ) at 40 and 100 ◦C, are given in Table 2. Polymers 2023,15, 2528 3 of 20 Table 2. Properties of reference lubricant—5P4E. Lubricant—5P4E T (40 ◦C) T (100 ◦C) Dynamic viscosity, η0(Pa s) 0.490 0.015 Pressure–viscosity coefficient, α(GPa−1)39.0 16.1 2.3. Experimental Apparatus Four commercial and one non-commercial experimental apparatuses were employed to determine the properties of PMMA, to evaluate the surface topography of selected specimens and the film thickness of the lubricant in compliant contact. In part A, Hysitron TI Premier Nanoindenter (Bruker) was used to analyze mechanical properties. For the chemical analysis, we employed the Raman spectroscopy using the Renishaw inVia Reflex Raman spectroscope and differential scanning colorimetry (DSC) by TA Instruments. DMA describing the viscoelastic properties of studied PMMA specimen was carried out in laboratories of the Centre of Polymer Systems (CPS) using a dynamic mechanical analyzer DMA1 (Mettler Toledo, Switzerland) equipped with clamps for compression testing mode. In part B, the 3D optical profilometer Contour GT-X by Bruker was utilized for the surface texture analysis. For the analysis of the fluid-film thickness in compliant circular contact, the rotary optical tribometer in ball-on-disc configuration, developed by the tribology group of Brno University of Technology (BUT) in the Czech Republic, was employed. This experimental apparatus has recently been described by the authors in greater detail in [14,15]. 2.4. Experimental Conditions and Methods In part A, the nano-DMA experiments of PMMA were performed to determine the E 0 and E 00 moduli and the tan δ , see Equations (SA8) and (SA9). The experimental conditions for part A are stated in Table 3. The nanoindentation technique nanoDMA III on the Bruker Hysitron Premier Ti nano-indenter was used to measure the dynamic mechanical properties of PMMA. Two types of experiments were conducted with a maximum load (by normal force) of 13 mN at a temperature interval from 15 to 80 ◦ C for the six discrete temperatures. Table 3. Experimental conditions for nano-DMA and DMA measurements—part A. Parameter Nano-DMA DMA Specimen plate cylinder DMA loading mode indentation compression Dimensions of specimen 10 (w) ×10 (l) ×0.5 (t) mm Ø 3 ×8 mm Frequency, f 1–250 Hz 10−3–102Hz Temperature, T 15, 20, 25, 30, 60, 80 ◦C 24, 40, 70, 80 ◦C The first experiment consisted of a frequency sweep from 1 Hz to 250 Hz, and the second involved CMX to obtain a modulus at different indentation depths. The load rate was 1.3 mN/s, while an ideal Berkovich indenter with a tip radius of 150 nm was used for all tests. During each experiment, the sample was kept in the xSol temperature stage. Secondly, PMMA cylinder specimens were employed for the DMA frequency sweep analysis in the frequency range between 10 −3 and 10 2 with five measuring points per decade. Based on the preliminary results of the amplitude dependent viscoelastic characterization, the linear viscoelastic region (deformation amplitude of 3 µ m) was defined and applied for further experiments at specified temperatures. Based on frequency sweep experiments, the TTSP principle was applied to obtain an MC that extends the frequency range beyond the measured frequency (f). Then, the MC was regressed according to the WLF equation (Equation (SA11)), and model parameters were derived. Besides this, the T g region was measured by DSC at heating rate of 10 ◦ C/min. Raman spectra of PMMA were collected before and after the tribological experiments as well. To obtain these spectra, the laser 785 nm was used. Polymers 2023,15, 2528 4 of 20 In part B, prior to the film thickness experiments, the 3D optical profilometer was employed for the surface structure analysis. The PMMA disc has an optically smooth surface, see Table 1; however, the steel ball was repeatably polished to decrease the R q and to get rid of parasitic optical reflectivity. After the preparation steps, the optical chromatic interferometry was implemented for evaluation of the h c and h m in the compliant contact. A detailed description of this method could be found in [ 16 ]. The fluid-film thickness was measured, calibrated, and evaluated in software Achilles (ver. 4.0.117, Radek Poliscuk, Brno University of Technology, Brno, Czech Republic). The h c and h m formed in the compliant contact between the PMMA disc and the steel ball were evaluated under a single constant load, W = 35 N. Hence, the contact pair formed a circular contact with respect to the Hertzian theory corresponding to the ellipticity, k= 1. The temperature was controlled by thermocouples in the oil reservoir and thermocouples placed in the vicinity of inlet of the lubricant to the contact, which corresponds to the oil temperature and the inlet temperature (T), respectively. Four discrete inlet temperatures T from 24 to 80 ◦C below the widely used Tginterval (105–110 ◦C) [17] of PMMA were selected. The experiments were carried out on the rotary optical tribometer under the pure rolling conditions where U 1 = U 2 (1 − disc, 2 − ball) corresponding to the sliding-rolling ratio SRR = 0. Therefore, the entrainment speed U E is equal to the surface speed of the PMMA disc U 1 as well as of the steel ball U 2 . Based on this, the frequency of loading f L could be expressed by the ratio of U E and the Hertzian contact radius a H , see Equation (1), which corresponds to the state of loading and unloading of the contact during the one cycle. fL=UE aH (1) Utilizing the 5P4E lubricant, the influence of U E and T on h c and h m was determined under the condition of isothermal full-film separation of contact surfaces. The film thickness was measured in the ascending order relative to the U E . Four discrete values of h c (100, 150, 200 and 250 nm) were selected for which the profiles of film thickness in transverse (x) and longitudinal (y) direction to UEwere assessed. To detect a possible transition of h m [ 18 ], the minimum film thickness was evaluated from interferograms at the exit of contact h mr as well as at the side lobes h ms of horseshoe. After collecting the experimental data, h c and h m were compared with the soft [ 18 – 20 ] and hard [ 5 ] EHL prediction models including Equations (SA4)–(SA7), and differences were discussed. Moreover, the operation region of the contact was identified relative to the EHL modes via a hydrodynamic map [21,22] by implementation of Equations (SA1)–(SA3). The material response of the PMMA polymer was determined from the interferograms of initially circular contact at the individual T. Thus, the variations in R c , ellipticity kand contact area Afor static (U E = 0) as well as running (U E6= 0) contacts were evaluated. For the comparison, the maximal Hertzian contact pressures, p H and a H , were calculated according to Equations (2) and (3) as 56 MPa and 443 µm at 24 ◦C, respectively. pH=3W 2πaH2(2) aH=3 s3WR 2ER (3) 2 ER =1−υ2 1 E1 +1−υ2 2 E2 (4) Subsequently, the reduced elastic modulus ( ER ) and the elastic modulus of PMMA (E 1 ) were reversibly calculated from Equations (3) and (4) for individual T after the a H was substituted by the experimentally obtained R c . Nevertheless, these values were determined for the static contact (U E = 0) assuming a pure elastic response of the PMMA disc, which Polymers 2023,15, 2528 5 of 20 did not manifest the effects of viscoelastic behavior under dynamic cyclic loading. The experimental conditions for part B are summarized in Table 4. Table 4. Experimental conditions for fluid-film thickness measurements—part B. Parameter Value Entrainment speed, UE0.00025–0.8 m/s Loading frequency, fL~0.3–900 Hz Normal load, W 35 N Inlet temperature, T 24, 40, 70, 80 ◦C Sliding-rolling ratio, SRR 0 Ellipticity of contact, k1 3. Results 3.1. Part A. Analysis of Material Properties of PMMA From the part A, the frequency and temperature dependence of PMMA was determined employing the nanoindentation and the dynamic mechanical analysis considering the experimental conditions, see Table 3. To characterize the response of the material, five parameters—elastic E, storage E 0 , loss E 00 and complex E * (see Equation (SA8)) moduli and damping factor tan ( δ ) (see Equation (SA9))—were measured below T g , which reflects the experimental conditions of tribological experiments in part B, see Table 4. The glasstransition for the PMMA specimens was determined by DSC corresponding to the interval Tg∈ h 104, 115 i◦ C. This procedure was repeated before and after tribological experiments in part B together with acquisition of the Raman spectra to detect the possible changes in the PMMA structure. However, none of the measurements showed any significant variations during the tribological experiments. The temperature dependence of E 0 and E 00 moduli and tan ( δ ) was determined at the constant reference frequency of 220 Hz and 1 Hz for nano-DMA and DMA, respectively, see Figure 1a,b. For the former and latter, frequency sweep and frequency as well as temperature sweep tests were performed. As expected, the values obtained from DMA are significantly lower compared to those from nano-DMA due to a different scale of the experiment. The E 0 exhibited in both analyses a gradual linear decrease in the whole measured temperature range. The E 00 obtained by nano-DMA demonstrated only a slight but gradual increase with temperature where for the DMA data, such increase begins only at 80 ◦ C. The tan ( δ ) develops in similar manner relative to the E 00 with minimal value of approx. 0.04. In Figure 1b, using the DMA analysis, the E 00 and tan ( δ ) data exhibit a significant fluctuation between 40–60 ◦ C; however, E 0 qualitatively corresponds to the nano-DMA data. Quantitatively, E 0 and E 00 moduli from nano-DMA overrated the data from the DMA temperature sweep tests several times. Conversely to E 00 and E 0 , the data of tan ( δ ) were in accordance with the results in Figure 1a. The nano-DMA and DMA frequency sweep experiments were also performed in the frequency interval of f∈ h 1, 250 i Hz and f∈ h 10 −3 , 10 2i Hz for several discrete temperatures. This provided necessary input data to compose the MC by implementation of the TTSP. The MC were derived for E 0 , see Figure 2a,b, at TR= 60 ◦C on account of the supplementation of this parameter into the tribological prediction models of h c and h m in part B. After implementation of TTSP, the reduced frequency (f red ) extended the operating frequency range from 10 0 to 10 7 for the nano-DMA and from 10 −7 to 10 4 for the DMA analysis. However, the significant difference of nano-DMA and DMA data relative frequency (or temperature) is quite evident. The former corresponds to the increase in E 0 from 3 to 7 GPa, for the latter, E 0 was noticeable lower, ranging only between 0.5 and 1.5 GPa with increase in fred. The superposition of the data highlighted a difference between the operating frequency of nano-DMA as well as DMA experiments in part A (see Table 3) and tribological Polymers 2023,15, 2528 6 of 20 experiments in part B (see Table 4, evaluated according to Equation (1)). Hence, this justified using of the TTSP together with derivation of a T that were regressed according to the WLF equation, see (Equation (SA11)). Then, the constants C 1 and C 2 were obtained, see Figure 2a,b. Polymers2023,15,xFORPEERREVIEW6of22    (a)(b) Figure1.TemperaturedependenceofstorageE′andlossE″moduli,anddampingfactortan(δ)for purePMMAatconstantreferencefrequency:(a)nano-DMA,(b)DMA. InFigure1b,usingtheDMAanalysis,theE″andtan(δ)dataexhibitasignificant fluctuationbetween40–60°C;however,E′qualitativelycorrespondstothenano-DMA data.Quantitatively,E′andE″modulifromnano-DMAoverratedthedatafromtheDMA temperaturesweeptestsseveraltimes.ConverselytoE″andE′,thedataoftan(δ)werein accordancewiththeresultsinFigure1a. Thenano-DMAandDMAfrequencysweepexperimentswerealsoperformedinthe frequencyintervalof𝑓∈ 〈1, 250〉Hzand𝑓∈ 〈10,10 〉Hzforseveraldiscretetemperatures.ThisprovidednecessaryinputdatatocomposetheMCbyimplementationof theTTSP.TheMCwerederivedforE′,seeFigure2a,b,at T 60 °Conaccountofthe supplementationofthisparameterintothetribologicalpredictionmodelsofhcandhmin partB. AfterimplementationofTTSP,thereducedfrequency(fred)extendedtheoperating frequencyrangefrom10to10forthenano-DMAandfrom10to10fortheDMA analysis.However,thesignificantdifferenceofnano-DMAandDMAdatarelativefrequency(ortemperature)isquiteevident.TheformercorrespondstotheincreaseinE′ from3to7GPa,forthelatter,E′wasnoticeablelower,rangingonlybetween0.5and1.5 GPawithincreaseinfred. Figure 1. Temperature dependence of storage E 0 and loss E 00 moduli, and damping factor tan ( δ ) for pure PMMA at constant reference frequency: (a) nano-DMA, (b) DMA. Polymers2023,15,xFORPEERREVIEW7of22    (a)(b) Figure2.Master-curveofstoragemodulusE′atreferencetemperatureTR=60°C.(a)nano-DMA, (b)DMA. Thesuperpositionofthedatahighlightedadifferencebetweentheoperatingfrequencyofnano-DMAaswellasDMAexperimentsinpartA(seeTable3)andtribological experimentsinpartB(seeTable4,evaluatedaccordingtoEquation(1)).Hence,thisjustifiedusingoftheTTSPtogetherwithderivationofaTthatwereregressedaccordingtothe WLFequation,see(Equation(SA11)).Then,theconstantsC1andC2wereobtained,see Figure2a,b. 3.2.PartB.AnalysisofFluid‐FilmThicknessinCompliantContact FromthepartB,dataofthehcandhmwereevaluatedforexperimentalconditions, seeTable4.Atfirst,interferogramswerecenteredinx-ycoordinatestodeterminethecenterofthecontactattheintersectionofhorizontalandverticalpathsaccordingtotheNewton’sfringes.Then,hcwasevaluatedrelativetotheproductofentrainmentspeedUEand dynamicviscosityη0,asFigure3a(24and40°C)andFigure3b(70and80°C)demonstrate.  (a)(b) Figure 2. Master-curve of storage modulus E 0 at reference temperature T R = 60 ◦ C. ( a ) nano-DMA, (b) DMA. 3.2. Part B. Analysis of Fluid-Film Thickness in Compliant Contact From the part B, data of the h c and h m were evaluated for experimental conditions, see Table 4. At first, interferograms were centered in x-y coordinates to determine the center of the contact at the intersection of horizontal and vertical paths according to the Newton’s fringes. Then, h c was evaluated relative to the product of entrainment speed U E and dynamic viscosity η0 , as Figure 3a (24 and 40 ◦ C) and Figure 3b (70 and 80 ◦ C) demonstrate. From Figure 3a, it is evident that h c qualitatively corresponds well, but it quantitatively differs with increase in U Eη0 . On the other hand, in Figure 3b, h c differs only slightly in the values, and the development of h c at 70 and 80 ◦ C in dependence on U Eη0 is practically identical. In both figures, the product of U Eη0 spans a similar range from 10 −3 to 10 −1 Pa · m in the entire temperature interval corresponding to h c < 500 nm. Nevertheless, the absolute Polymers 2023,15, 2528 7 of 20 difference of h c develops in a dissimilar manner between 24 and 40 ◦ C as well as between 70 and 80 ◦C relative to the product of UEη0. The minimum film thickness was identified at the side lobes of the contact h ms without no signs of transition of the minimum to the exit h mr in all experiments, see Figure 4a,b. The difference between hmr and hms gradually decreases with the increasing temperature. The increase in temperature (T ≥ 40 ◦ C) at lower values of U Eη0 (up to 10 −2 ) demonstrated a tendency of h mr and h ms to unify into a single curve; however, at higher values of U Eη0 , this led to the increase in the variation between hmr and hms. Polymers2023,15,xFORPEERREVIEW7of22    (a)(b) Figure2.Master-curveofstoragemodulusE′atreferencetemperatureTR=60°C.(a)nano-DMA, (b)DMA. Thesuperpositionofthedatahighlightedadifferencebetweentheoperatingfrequencyofnano-DMAaswellasDMAexperimentsinpartA(seeTable3)andtribological experimentsinpartB(seeTable4,evaluatedaccordingtoEquation(1)).Hence,thisjustifiedusingoftheTTSPtogetherwithderivationofaTthatwereregressedaccordingtothe WLFequation,see(Equation(SA11)).Then,theconstantsC1andC2wereobtained,see Figure2a,b. 3.2.PartB.AnalysisofFluid‐FilmThicknessinCompliantContact FromthepartB,dataofthehcandhmwereevaluatedforexperimentalconditions, seeTable4.Atfirst,interferogramswerecenteredinx-ycoordinatestodeterminethecenterofthecontactattheintersectionofhorizontalandverticalpathsaccordingtotheNewton’sfringes.Then,hcwasevaluatedrelativetotheproductofentrainmentspeedUEand dynamicviscosityη0,asFigure3a(24and40°C)andFigure3b(70and80°C)demonstrate.  (a)(b) Figure 3. Central film thickness h c versus product of U Eη0 at different inlet temperatures. Temperatures (a) 24 and 40 ◦C, (b) 70 and 80 ◦C. Polymers2023,15,xFORPEERREVIEW8of22   Figure3.CentralfilmthicknesshcversusproductofUEη0atdifferentinlettemperatures.Temperatures(a)24and40°C,(b)70and80°C. FromFigure3a,itisevidentthathcqualitativelycorrespondswell,butitquantitativelydifferswithincreaseinUEη0.Ontheotherhand,inFigure3b,hcdiffersonlyslightly inthevalues,andthedevelopmentofhcat70and80°CindependenceonUEη0ispracticallyidentical.Inbothfigures,theproductofUEη0spansasimilarrangefrom10to 10Pa·mintheentiretemperatureintervalcorrespondingtohc<500nm.Nevertheless, theabsolutedifferenceofhcdevelopsinadissimilarmannerbetween24and40°Caswell asbetween70and80°CrelativetotheproductofUEη0. Theminimumfilmthicknesswasidentifiedatthesidelobesofthecontacthmswithoutnosignsoftransitionoftheminimumtotheexithmrinallexperiments,seeFigure 4a,b.Thedifferencebetweenhmrandhmsgraduallydecreaseswiththeincreasingtemperature.Theincreaseintemperature(T≥40°C)atlowervaluesofUEη0(upto10−2)demonstratedatendencyofhmrandhmstounifyintoasinglecurve;however,athighervaluesof UEη0,thisledtotheincreaseinthevariationbetweenhmrandhms.  (a)(b) Figure4.MinimumfilmthicknessatcontactexithmrandonsidelobeshmsversusproductofUEη0 atdifferentinlettemperatures.(a)24°C,(b)80°C. Inthenextstep,thefilmthicknessprofileswereinvestigatedintheperpendicular andparalleldirectionrelativetoUEalongthecenterlineofthecontactforfourdiscretehc (100,150,200and250nm),seeFigures5and6showingdimensionlesscoordinatesXand YversusfilmthicknessH.At24°C,thecoherentfilmthicknessisformedandthehorseshoeshapewithconstrictionatthesidelobesisevidentfromthetransverseprofilein Figure5a.AnincreaseinUEledtoamorepronouncedhorseshoefilmshape,sothedifferencebetweenhcandhmsismoresignificant.IntheparalleldirectiontoUE,thefilm thicknessprofilealongthecenterlinewasalmostconstantexceptforthecontactinlet(on theleft)andfortheexitfromthecontact(ontheright)where,forthelatter,significant variationsinthewedgeshapewereobserved,seeFigure5b. Figure 4. Minimum film thickness at contact exit h mr and on side lobes h ms versus product of U Eη0 at different inlet temperatures. (a) 24 ◦C, (b) 80 ◦C. In the next step, the film thickness profiles were investigated in the perpendicular and parallel direction relative to U E along the center line of the contact for four discrete h c (100, 150, 200 and 250 nm), see Figures 5and 6showing dimensionless coordinates X and Y versus film thickness H. At 24 ◦ C, the coherent film thickness is formed and the horseshoe shape with constriction at the side lobes is evident from the transverse profile in Figure 5a. An increase in U E led to a more pronounced horseshoe film shape, so the difference between h c and h ms is more significant. In the parallel direction to U E , the film thickness profile along the center line was almost constant except for the contact inlet (on Polymers 2023,15, 2528 8 of 20 the left) and for the exit from the contact (on the right) where, for the latter, significant variations in the wedge shape were observed, see Figure 5b. Polymers2023,15,xFORPEERREVIEW9of22    (a)(b) Figure5.Filmthicknessprofilesalongthecontactcenterlineat24°C.(a)Transverseprofile,(b) longitudinalprofile.  (a)(b) Figure6.Filmthicknessprofilesalongthecontactcenterlineat80°C.(a)Transverse,(b)longitudinal. Contrarytothicknessprofilesobtainedat24°C,anincreaseininlettemperatureto 80°C,seeFigure6aresultsinareductioninthesidelobesofhorseshoe.Moreover,the gradualchangeinthefilmthicknessinthetransverseprofilewasobservedinthedirection fromthesidelobestothecentralregionalongthecenterline,whichcorrespondstothe increaseintheratiohms/hc. ThefilmthicknessprofilesparalleltoUE,seeFigure6b,demonstrateasimilarbehaviorsuchasat24°C,withoutinclinationoffilmthicknessprofilewithincreasingUE.In addition,theincreaseinTprobablyinitializesthepolymerconstitutivevariationinmaterialstiffnessconnectedwithachangeinthepressuredistributionandthustheasymmetricalchangeintheradiusofrunningcontactRcinthetransverseandlongitudinalprofiles. However,fortheformer,thisdeviationismorenoticeable.Inthelongitudinalprofilesat thecontactinlet,thesecondarylocalminimumoffilmthicknesswasnotrecorded. Figure 5. Film thickness profiles along the contact center line at 24 ◦ C. ( a ) Transverse profile, (b) longitudinal profile. Polymers2023,15,xFORPEERREVIEW9of22    (a)(b) Figure5.Filmthicknessprofilesalongthecontactcenterlineat24°C.(a)Transverseprofile,(b) longitudinalprofile.  (a)(b) Figure6.Filmthicknessprofilesalongthecontactcenterlineat80°C.(a)Transverse,(b)longitudinal. Contrarytothicknessprofilesobtainedat24°C,anincreaseininlettemperatureto 80°C,seeFigure6aresultsinareductioninthesidelobesofhorseshoe.Moreover,the gradualchangeinthefilmthicknessinthetransverseprofilewasobservedinthedirection fromthesidelobestothecentralregionalongthecenterline,whichcorrespondstothe increaseintheratiohms/hc. ThefilmthicknessprofilesparalleltoUE,seeFigure6b,demonstrateasimilarbehaviorsuchasat24°C,withoutinclinationoffilmthicknessprofilewithincreasingUE.In addition,theincreaseinTprobablyinitializesthepolymerconstitutivevariationinmaterialstiffnessconnectedwithachangeinthepressuredistributionandthustheasymmetricalchangeintheradiusofrunningcontactRcinthetransverseandlongitudinalprofiles. However,fortheformer,thisdeviationismorenoticeable.Inthelongitudinalprofilesat thecontactinlet,thesecondarylocalminimumoffilmthicknesswasnotrecorded. Figure 6. Film thickness profiles along the contact center line at 80 ◦ C. ( a ) Transverse, ( b ) longitudinal. Contrary to thickness profiles obtained at 24 ◦ C, an increase in inlet temperature to 80 ◦ C, see Figure 6a results in a reduction in the side lobes of horseshoe. Moreover, the gradual change in the film thickness in the transverse profile was observed in the direction from the side lobes to the central region along the center line, which corresponds to the increase in the ratio hms/hc. The film thickness profiles parallel to U E , see Figure 6b, demonstrate a similar behavior such as at 24 ◦ C, without inclination of film thickness profile with increasing U E . In addition, the increase in T probably initializes the polymer constitutive variation in material stiffness connected with a change in the pressure distribution and thus the asymmetrical change in the radius of running contact R c in the transverse and longitudinal profiles. However, for the former, this deviation is more noticeable. In the longitudinal profiles at the contact inlet, the secondary local minimum of film thickness was not recorded. Polymers 2023,15, 2528 9 of 20 Further results represent the evaluation of interferograms of static contacts (U E = 0) at different temperatures. The results summarized in Table 5showed a linear increase in R c from 24 ◦ C to 80 ◦ C where the absolute difference represents the increase in R c , approximately by 17%. Moreover, the difference of R cx and R cy in the transverse and longitudinal directions, respectively, was almost negligible. Table 5. Dimensional parameters of static contacts at different inlet temperatures. Inlet Temperature T, ◦C 24 40 70 80 Contact radius Rcx, mm 0.439 0.466 0.503 0.528 Contact radius Rcy, mm 0.440 0.458 0.501 0.527 Average Rc, mm 0.439 0.462 0.502 0.527 Ellipticity k,−0.997 1.015 1.002 1.001 Area A, mm20.606 0.671 0.788 0.874 Apart from R c , the area of contact Ademonstrated a significant enlargement (about 30%) when the inlet temperature increased from 24 ◦ C to 80 ◦ C. R cx and R cy were averaged and then substituted into Equations (3) and (4), from which the E R and E 1 were calculated, see Table 6. Table 6. Reduced elastic modulus E R and elastic modulus E 1 of PMMA calculated for static contacts at different inlet temperatures. Inlet Temperature T, ◦C 24 40 70 80 Reduced elastic modulus ER, GPa 7.853 6.746 5.301 4.538 Elastic modulus of PMMA E1, GPa 3.388 2.903 2.274 1.943 On the other hand, the running contact (U E6= 0) was always dimensionally smaller exhibiting a proportional difference of R c between the transverse and longitudinal directions in the entire temperature interval relative to the static contact (U E = 0) see Figure 7. This results in the ellipticity variation in the interval k∈(1.03, 1.10) where higher values of kcorrespond with enhancement in inlet temperature. Only a slight transition from the circular (k= 1) to the wide elliptical contact (k> 1) was recorded. Despite the increase in ellipticity, the contact area of the running contact Awas reduced in size by up to 12% in dependence on U E for individual inlet temperatures. Contrary to this, the effect of temperature is more significant than U E throughout the selected h c (100, 150, 200 and 250 nm), which leads to the enlargement of contact area Aof running contact in the interval from 0.5 to 20% relative to the contact area at 24 ◦ C. However, this difference gradually decreases with increasing hc, see Figure 7a,b. Polymers2023,15,xFORPEERREVIEW10of22   Furtherresultsrepresenttheevaluationofinterferogramsofstaticcontacts(UE=0) atdifferenttemperatures.TheresultssummarizedinTable5showedalinearincreasein Rcfrom24°Cto80°CwheretheabsolutedifferencerepresentstheincreaseinRc,approximatelyby17%.Moreover,thedifferenceofRcxandRcyinthetransverseandlongitudinal directions,respectively,wasalmostnegligible. Table5.Dimensionalparametersofstaticcontactsatdifferentinlettemperatures. InletTemperatureT,°C24407080 ContactradiusRcx,mm0.4390.4660.5030.528 ContactradiusRcy , mm0.4400.4580.5010.527 AverageRc,mm0.4390.4620.5020.527 Ellipticityk,− 0.9971.0151.0021.001 AreaA , mm20.6060.6710.7880.874 ApartfromRc,theareaofcontactAdemonstratedasignificantenlargement(about 30%)whentheinlettemperatureincreasedfrom24°Cto80°C.RcxandRcywereaveraged andthensubstitutedintoEquations(3)and(4),fromwhichtheERandE1werecalculated, seeTable6. Table6.ReducedelasticmodulusERandelasticmodulusE1ofPMMAcalculatedforstaticcontactsatdifferentinlettemperatures. InletTemperatureT,°C24407080 ReducedelasticmodulusER , GPa7.8536.7465.3014.538 ElasticmodulusofPMMAE1,GPa3.3882.9032.2741.943 Ontheotherhand,therunningcontact(UE≠0)wasalwaysdimensionallysmaller exhibitingaproportionaldifferenceofRcbetweenthetransverseandlongitudinaldirectionsintheentiretemperatureintervalrelativetothestaticcontact(UE=0)seeFigure7. Thisresultsintheellipticityvariationintheinterval𝑘 ∈ 󰇛1.03, 1.10󰇜wherehighervaluesofkcorrespondwithenhancementininlettemperature.Onlyaslighttransitionfrom thecircular(k=1)tothewideellipticalcontact(k>1)wasrecorded.Despitetheincrease inellipticity,thecontactareaoftherunningcontactAwasreducedinsizebyupto12% independenceonUEforindividualinlettemperatures.Contrarytothis,theeffectoftemperatureismoresignificantthanUEthroughouttheselectedhc(100,150,200and250nm), whichleadstotheenlargementofcontactareaAofrunningcontactintheintervalfrom 0.5to20%relativetothecontactareaat24°C.However,thisdifferencegraduallydecreaseswithincreasinghc,seeFigure7a,b. (a)  Figure 7. Cont. Polymers 2023,15, 2528 16 of 20 to reach a given point on the disc corresponding to the distance equal to the R c ) of the E r is demonstrated in Figure 15. Polymers2023,15,xFORPEERREVIEW18of22    Figure15.RelaxationmodulusErbasedongeneralizedMaxwellmodel. TheinfluenceofErisevidentfromFigures9and10wheretheH&DmodelsforP-E andI-EmodeofEHLweremodified(H&Dm,dashed-line).At24and40°C,thedifference HcfromthemodifiedsoftandhardH&Dmmodelsremainsmainlyunchangedrelativeto H&D,andHccorrelatesbestwiththesoftandhardH&Dfortheformerandlattertemperature,respectively.However,for70and80°C,theexperimentalHcdemonstrateda verygoodagreementwithmodifiedsoftH&Dmmodelwithonlyaslightdeviationnot exceeding10%relativetosoftH&Dwhereadeviationover60%waspreviouslyrecorded. Interestingly,althoughtheviscoelasticresponsecharacterizedbyErwasincluded,theHc alsocorrespondswellwithhardH&DandH&Dmmodels,especiallyatatemperatureof 80°C. ItisevidentthattheinlettemperatureleadstoasignificantdecreaseinHc;however, theformationoffilmthicknessintheTRregionpredominantlycorrespondstotheP-E moderatherthantotheI-Emode.Thisissurprisingrelativetotheconstitutivematerial behaviorofpolymers,aswellasthepositionofexperimentsclosetotheI-Emode,see Figure13.However,thisbringsusbacktothedeterminationoftheEHLmode,whichmay differduetotheintegrationoftheErinsteadoftheE. TheintegrationoftheErinthehydrodynamicmap,seeFigure13,shiftstheoperation conditionsofthecontactslightlytowardtheP-EmodeofEHLintheTRregionwherethe magnitudeofshiftwithincreasingtemperatureisenhanced.Asimilarshiftwouldbeseen inFigure14accordingtotheArchard’sg4parameteraswell,butinadifferentmanner. Here,thedevelopmentof𝐻 ,𝐻 and𝐻 aswellasgEasymptoticallyapproachesthe boundarybetweentheTRregionandI-EmodeofEHLandwouldconvergetothevalue of0.1forg4,whichisderivedin[21]astheboundarypointofTRregion.However,the abovepointsouttoasignificantdiscrepancybetweentheexpectedoperationmodeof EHLandI-EEHLmodels.Simultaneously,thedescribedbehaviormaybeaconsequence oftheviscoelasticresponseofthepolymer,whichcausesadecreaseinfluid-filmthickness,notconsideredbytheI-EEHLmodels. Inaddition,severalreasonscanbefoundtoexplainthedescribeddeviations.Anotherpossibilitymaybestrainhardeningathighcontactpressures,whichmaynotbe evidentintheDMAanalysis.Conversely,theDMAanalysismayhaveshownsoftening duetocyclicloadingofonespotonthespecimen.Hence,thereareseveralreasonswhy thestiffnesscouldbeevenlower.Oneofthemcouldbeadecreasebyabout10%intheE measuredbynano-DMAattherollingpathafterthetribologicalexperimentwasperformedinpartB.Nevertheless,thesofteningeffectshouldonlybeparticularlynoticeable atthetemperatureclosetotheTginterval. Figure 15. Relaxation modulus Erbased on generalized Maxwell model. The influence of E r is evident from Figures 9and 10 where the H&D models for P-E and I-E mode of EHL were modified (H&D m , dashed-line). At 24 and 40 ◦ C, the difference H c from the modified soft and hard H&D m models remains mainly unchanged relative to H&D, and H c correlates best with the soft and hard H&D for the former and latter temperature, respectively. However, for 70 and 80 ◦ C, the experimental H c demonstrated a very good agreement with modified soft H&D m model with only a slight deviation not exceeding 10% relative to soft H&D where a deviation over 60% was previously recorded. Interestingly, although the viscoelastic response characterized by E r was included, the H c also corresponds well with hard H&D and H&D m models, especially at a temperature of 80 ◦C. It is evident that the inlet temperature leads to a significant decrease in H c ; however, the formation of film thickness in the TR region predominantly corresponds to the P-E mode rather than to the I-E mode. This is surprising relative to the constitutive material behavior of polymers, as well as the position of experiments close to the I-E mode, see Figure 13. However, this brings us back to the determination of the EHL mode, which may differ due to the integration of the Erinstead of the E. The integration of the E r in the hydrodynamic map, see Figure 13, shifts the operation conditions of the contact slightly toward the P-E mode of EHL in the TR region where the magnitude of shift with increasing temperature is enhanced. A similar shift would be seen in Figure 14 according to the Archard’s g 4 parameter as well, but in a different manner. Here, the development of ˆ Hc , ˆ Hmr and ˆ Hms as well as g E asymptotically approaches the boundary between the TR region and I-E mode of EHL and would converge to the value of 0.1 for g 4 , which is derived in [ 21 ] as the boundary point of TR region. However, the above points out to a significant discrepancy between the expected operation mode of EHL and I-E EHL models. Simultaneously, the described behavior may be a consequence of the viscoelastic response of the polymer, which causes a decrease in fluid-film thickness, not considered by the I-E EHL models. In addition, several reasons can be found to explain the described deviations. Another possibility may be strain hardening at high contact pressures, which may not be evident in the DMA analysis. Conversely, the DMA analysis may have shown softening due to cyclic loading of one spot on the specimen. Hence, there are several reasons why the stiffness could be even lower. One of them could be a decrease by about 10% in the E measured by nano-DMA at the rolling path after the tribological experiment was performed in part B. Nevertheless, the softening effect should only be particularly noticeable at the temperature close to the Tginterval. Polymers 2023,15, 2528 17 of 20 5. Conclusions In the present study, the constitutive viscoelastic response of PMMA was experimentally determined relative to the tribology of compliant contacts operated in the EHL regime below the glass-transition temperature of the polymer. The viscoelastic response was evaluated by the generalized Maxwell model and implemented in I-E and P-E EHL models where only a purely elastic response of the contact solids has been assumed so far. Hence, a significant deviation often revealed for these models, especially for I-E when the temperature increases, then fluid-film thickness is overestimated. The operating mode of EHL was identified by hydrodynamic map (g E and g V ) and ˆ H and g 4 parameters, and operating conditions of the formed circular contact were compared with research in the field of soft EHL. The main results can be summarized as follows: The operating mode of the EHL has proved to be in the TR region close to the boundary between the P-E and I-E modes of the EHL according to the hydrodynamic map and the defined interval for the TR region via the Archard’s parameter g 4 . The shift of operating conditions to the vicinity of the I-E mode is manifested by an increase in the compliance of the circular contact after the inlet temperature is enhanced. The implementation of a viscoelastic response of PMMA in the prediction models through the relaxation modulus reduced the deviation between the measured fluid-film thickness and the H&D model for the I-E and P-E modes. A decrease in deviation corresponds to an increase in contact compliance and the proximity of the operating conditions to the boundary between the TR region and the I-E mode. The minimum film thickness was always localized at the side lobes of the horseshoe of the contact without the expected transition to the exit of the contact with increasing entrainment speed. The secondary minimum of film thickness at the inlet to the contact was not manifested as a consequence of the viscoelastic response of the polymer. For the TR region, the pressure-viscosity effect of the lubricant should be included reflecting the mixed behavior at the boundary between the P-E and I-E modes of EHL, especially at lower temperatures. For T > 40 ◦ C, the I-E EHL models generally predict the film thickness which is considerably overestimated even though the pressure-viscosity effect of the lubricant is always neglected. The constitutive viscoelastic behavior should be considered in I-E EHL models; however, the variation in the material properties of individual polymers can make the generalization very difficult. Moreover, considering the lubricant rheology, this may be even more complex. The obtained results could be applicable to the design of polymer gears as well as to the optimalization of the lubrication management in the engineering applications where the rubbing surfaces of compliant and rigid solids interact under liquid lubrication conditions. For future research, the fluid-film thickness could be investigated in more detail considering the viscoelastic response of PMMA relative to the rheology of the lubricant. Supplementary Materials: The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/polym15112528/s1. Author Contributions: Conceptualization, J.K. and K.D.; Data curation, J.K., K.D. and T.S.; Methodology, J.K.; Validation, J.K. and K.D.; Formal analysis, J.K., K.D. and D.R.; Investigation, J.K., K.D. and T.S.; Visualization, J.K.; Writing—original draft preparation, J.K.; Writing—review and editing, J.K., K.D., D.R. and T.S.; Supervision, I.K.; Project administration, M.H.; Funding acquisition, M.H. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Czech Science Foundation (GACR), grant number 1826849J. The author T.S. thanks the Ministry of Education, Youth and Sports of the Czech Republic 557—DKRVO (RP/CPS/2022/003). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data concerned were obtained through the research of the author and reported exclusively in this article and Supplementary Information. Polymers 2023,15, 2528 18 of 20 Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. Nomenclature A Area of contact: mm2 aHHertzian radius, µm aT, bTHorizontal and vertical shift factor (TTSP) C1, C2Constants for WLF equation, -, ◦C E Elastic modulus, MPa E*Complex modulus, MPa EAViscoelastic activation energy, kJ mol−1 E0Storage modulus, MPa E00 Loss modulus, MPa ErRelaxation modulus, MPa ERReduced elastic modulus, MPa f Frequency, Hz fLFrequency of loading, Hz fred Reduced frequency, Hz g4Archard’s parameter of the system gEParameter of elasticity gVParameter of viscosity G Dimensionless material parameter H Dimensionless film thickness ˆ H Film thickness parameter HcDimensionless central film thickness Hmr Dimensionless minimum film thickness at the exit of contact Hms Dimensionless minimum film thickness at the side lobes hcCentral film thickness, nm hmr Minimum film thickness at the exit of contact, nm hms Minimum film thickness at the side lobes, nm i Complex number k Ellipticity of contact (Rsx/Rsy) p Contact pressure, GPa pHMaximal Hertzian pressure, MPa R Gas constant, J mol−1K−1 R’ Reduced curvature radius, mm RsCurvature radius of specimens, mm RcRadius of static and running contact, µm RqRMS roughness, µm SRR Sliding-rolling ratio t Time, s T Inlet temperature, ◦C TgGlass-transition temperature, ◦C TRReference temperature, ◦C tan δDamping/loss factor UEEntrainment speed, m/s U Surface speed of specimens, m/s U Dimensionless speed parameter W Normal load, N W Dimensionless load parameter x, y Coordinate perpendicular and parallel to UE X, Y Dimensionless x and y coordinate (x/aHand y/aH) αPressure-viscosity coefficient, GPa−1 δPhase angle between stress and strain, deg ηDynamic viscosity of lubricant, Pa s Polymers 2023,15, 2528 19 of 20 η0Dynamic viscosity of lubricant at atmospheric pressure, Pa s νPoisson’s ratio of specimens Subscripts 1, 2 Index of PMMA disc and steel ball alfa Primary relaxation of polymer beta, gamma, delta Secondary relaxations of polymer exp Experimental values mod Model values References 1. 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