film thickness 化学信息学

ORIGINAL PAPERThe Effect of Load (Pressure)for Quantitative EHL Film ThicknessIvan Krupka •Punit Kumar •Scott Bair •M.M.Khonsari •Martin HartlReceived:2October 2009/Accepted:4December 2009/Published online:19December 2009ÓSpringer Science+Business Media,LLC 2009Abstract New quantitative numerical simulations of the elastohydrodynamic lubrication (EHL)film thickness using realistic pressure and shear-dependent rheology and real-istic compressibility have indicated that the dependence of central film thickness upon Hertz pressure (or load)for the classic Newtonian,slightly compressible solution is merely a lower limit with magnitudes three times as great being possible.Experimental measurements of central film thickness employing Hertz pressures from 1.0to 2.6GPa confirm that for a neat mineral oil,the classical pressure dependence is accurate,while for two gear oils the experimental pressure dependence is much larger.Shear-dependent viscosity is a major factor and compressibility plays a lesser role,while there is evidence that mechanical degradation is also important.New experimental evidence of the enhanced scale sensitivity resulting from shear-thinning has also been obtained.These results for the pressure and scale dependence have dire implicationsfor the usual practice of extrapolation of film thickness from experimental measurements at large scale and low pressure using effective pressure–viscosity coefficients.For machines of small scale and high pressure,the extrapola-tion will sometimes result in substantially overestimated film thicknesses.Keywords Elastohydrodynamic ÁFilm thickness ÁEHL ÁRheology ÁCompressibilityList of Symbols B Doolittle parameter b Hertzian contact radius,m a *Redness-greenness coordinate a V Thermal expansivity,K -1E 0Reduced Young’s modulus of two surfaces,Pa f i i th areal fraction F Contact load,N ^G Dimensionless material parameter G i i th shear modulus,Pa h Film thickness,m h c Central film thickness,m H Dimensionless film thickness H =hR x /b 2K Isothermal bulk modulus,Pa K 0Isothermal bulk modulus at p =0,Pa K 00Isothermal bulk modulus at p =0,T =0K,Pa K 00Pressure rate of change of isothermal bulkmodulus at p =0L *Psychometric lightness coordinate M Molecular weight,kg/kmol N Number of flow units n i i th power-law exponent P Pressure,Pa P H Hertz (maximum)pressure,PaI.Krupka ÁM.HartlInstitute of Machine and Industrial Design,Faculty of Mechanical Engineering,Brno University of Technology,Brno 61669,Czech RepublicP.KumarNational Institute of Technology Kurukshetra,Kurukshetra,Haryana 136119,IndiaS.Bair (&)Georgia Institute of Technology,Center for High-Pressure Rheology,George W.Woodruff School of Mechanical Engineering,Atlanta,GA 30332-0405,USA e-mail:scott.bair@M.M.KhonsariDepartment of Mechanical Engineering,Louisiana StateUniversity,2508Patrick Taylor Hall,Baton Rouge,LA 70803,USATribol Lett (2010)37:613–622DOI 10.1007/s11249-009-9559-7R Reduced radius in the rolling direction,ball radius for a ball-on-disk geometry,mR g Universal gas constant=8314.34Pa m3/kmol/K T Temperature,KT R Reference temperature,Ku1,u2Velocities of lower and upper surfaces,respectively;m/sU Dimensionless speed parameter"u Average surface velocity or rolling velocity,m/s V Volume at T and p,m3V R Volume at reference state,T R,p=0,m3V0Volume at p=0,m3V Occupied volume,m3V R Occupied volume at reference state,T R,p=0,m3 ^W Dimensionless load parameterx,X Coordinate in the direction of surface velocity,m, X=x/by,Y Coordinate in the transverse direction,m,Y=y/b z,Z Coordinate along the surface normal to the surface velocity,m,Z=z/ha Reciprocal asymptotic isoviscous pressurecoefficient,Pa-1b K Temperature coefficient of K0,K-1e Occupied volume thermal expansivity,K-1g Rate-dependent shear viscosity,Pa sl Limiting low-shear viscosity,Pa sl0Low-shear viscosity at p=0,Pa sl R Low-shear viscosity at reference state,T R,p=0,Pa sl r Low-shear viscosity at T,p for shear-dependent viscosity measurement,Pa sq Mass density,kg/m3q0Inlet mass density,kg/m3s Shear stress,Pa1IntroductionA new‘‘quantitative approach’’to elastohydrodynamic lubrication(EHL)has emerged recently[1,2]wherein the behavior of the lubricated concentrated contact is calcu-lated using accurate,measurable rheological properties of the liquid.For example,a full generalized Newtonian thermal EHL analysis,including the temperature and pressure dependence of the thermal properties of the liquid, has been obtained[3]for thefilm thickness and friction showing excellent agreement with experiment.This quan-titative approach is superior to the technique of adjusting the viscosity to match simple but inaccurate models to EHL contact behavior.With the quantitative approach,the contribution of each property of the liquid tofilm formation and friction may now be assessed individually;the differ-ent contributions of thermal softening and shear-thinning may be discriminated;differences in performance among liquids of different chemical structure may be understood; and extrapolation of experimental results to very severe contact conditions may be made with improved confidence.A quantitative EHL analysis recently led to an exciting prediction of new EHL behavior from calculation before it was observed experimentally.This involved the sensitivity of the centralfilm thickness,h c,to scale of contact, q(log h c)/q(log R),characterized by the reduced radius, R.The sensitivity of thefilm thickness to scale of contact was shown to be dependent upon the shear dependence of viscosity in the inlet zone[4,5]before this effect was validated by direct measurement[6].New results showing the scale effect are shown in Fig.1a and b for a linear perfluorinated polyalkylether,Z25[7],and a polyglycol gear oil,PG460[8],respectively,for a steel ball rolling against a glassflat.The ball diameter,equal to2R,is given in the legend and other conditions are given in[6].The curves in Fig.1represent the centralfilm thickness cal-culated from the Hamrock and Dowson[9]equation for a slightly compressible Newtonian liquid in circular contact, h c¼1:55a0:53ðl0"uÞ0:67E00:061R0:33pÀ0:201Hð1Þwhich predicts that q(log h c)/q(log R)=0.33.Figure1 clearly reveals that the discrepancy between the Newtonian prediction and the measuredfilm thickness increases as the scale decreases.Specifically,q(log h c)/q(log R)=0.45 and0.48for Z25and PG460,respectively.Similarly,in1996,a simple inlet zone calculation[4] showed that,for an increase in Hertz pressure,p H,from0.5 to1.0GPa,the sensitivity of thefilm thickness to pressure, expressed as D(log h c)/D(log p H)was-0.22for a Newto-nian liquid and-0.46for a Carreau-like liquid with Newtonian limit,G=0.1MPa,and power-law exponent, n=0.5at10m/s and R=1cm.Kudish[5]used a per-turbation analysis to arrive at afilm thickness formula for an Ostwald-deWaele liquid which yieldsh c/pÀ13nþ1Hð2ÞThe prediction is thus h cµp H-0.25for Newtonian and h cµp H-0.4for n=0.5.An Ostwald-deWaele liquid has no Newtonian plateau.Also,a line-contact simulation was performed in preparation for the experimental work pre-sented here.For a shear-thinningfluid in a glass/steel contact with G=0.01MPa and n=0.4,D(log h c)/ D(log p H)was found to be-0.63,much greater than the -0.23of the Newtonian result.Of course,the Hamrock and Dowson Equation1for circular contact gives q(log h c)/q(log p H)=-0.201.This article presents an experimental and analytical investigation of the effect of load on EHLfilm thickness.In a significant departure from previous study,viscosity has not been adjusted to ensure agreement between the two approaches;the viscosities have been measured in visc-ometers known to be accurate.2Experimental LiquidsFour liquids are the subject of thefilm thickness mea-surements and computations reported here.They are a linear perfluorinated polyalkylether,Z25[7];a polyglycol gear oil,PG460[8];a mineral-based,polybutene-thickened gear oil,LF5346,formulated about40years ago for which the viscosity is quite shear-dependent and an abundance of rheological data exists[10];and a mineral oil,SR600, which has been previously employed to assess thefilm dependence on contact load[11].New measurements of low-shear viscosity,l,are reported in this study for SR600 at shear stress of less than70Pa in a falling-body vis-cometer[12],and the results are listed in Table1.Similar data for the Z25and PG460can be found in[12]and for LF5346in[6].Low-shear viscosity data were used to obtain the parameters of the Tait-Doolittle free volume model given in Table2.The equation of state most widely used in high-pressure physics is the empirical Tait equation[12]often held to be the most accurate for extrapolation of volume,V, to very high pressures.VV0¼1À11þK00ln1þpK0ð1þK00Þ!ð3ÞThe ambient pressure bulk modulus may be represented as exponential in temperature.K0¼K00expðÀb K TÞð4Þand a relation between ambient pressure volume and temperature is required.V0V R¼1þa VðTÀT RÞð5ÞThe most widely used model in high-pressure physics for the variation of the low-shear viscosity with temperatureTable1New low-shear viscosity measurementsSR600Temperature/C:40Pressure/MPa0.10.099500.3091000.885200 5.86030031.64001775001250550ThixotropicViscosity in Pa s for SR600Table2The Parameters of the Tait Equation of State(3–5)and the Doolittle Viscosity Equation6 Fluid Doolittle equation parameters Tait equation parametersl R(Pa s)T R(C)B V1RV R K0K00(GPa)b K(K-1)a V(C-1)Z250.36025 5.6800.597611.0150 5.127 6.119910-3 1.0384910-3 PG4600.43740 2.6000.6938119 6.5910-38910-4 LF5346 3.4550 4.3250.666910.8597.428 6.48910-38910-4 SR6000.09940 3.1750.7191119 6.5910-38910-4and pressure has been the free-volume model,using the Doolittle equation.l¼l R exp B V1RV R1VV RÀV1RV RÀ11ÀV1RV R!"#ð6ÞThe viscosity data do not encompass the pressure–viscosity inflection[13]for all materials and,therefore,the Doolittle parameters in Table2should not be unique.In addition, there was no compressibility data available for PG460and SR600to regress the Tait parameters;the‘‘universal’’parameters suggested in[12]were employed instead.The shear stress,s,dependent viscosity,g of Z25was measured in[14],the shear-dependent viscosity of PG460 was measured in[15],and the shear-dependent viscosity of LF5346was measured in[10].Here,the mixture model of reference[16]is employed as the viscosity function.g¼lX Ni¼1f i1þsG i2"#1À1n iðÞ2;X Ni¼1f i¼1ð7ÞThe parameters are listed in Table3.The average molecular weight,M,of the mineral oil, SR600,was estimated from data for a similar paraffinic mineral oil to be520kg/kmol resulting in a Newtonian limit in terms of shear stress using Einstein-Debye[17]of about G=q R g T/M=5MPa.Among EHL lubricants,the shear-thinning response in the inlet zone displayed by SR600and LF5346should represent the low and high extremes,respectively.3EHL Experimental ProcedureFilm thickness measurements were performed using an optical test rig[18].In this apparatus,a circular EHD contact is formed between a transparent disk and a ball. The lower surface of the disk is coated with semi-reflective chromium layer to obtain optimal contrast of chromatic interferograms forfilm thickness evaluation.The contact-ing bodies are enclosed in a thermally isolated,heated chamberfilled with lubricant to ensure fullyflooded con-ditions at constant test temperature,which is checked by a small thermocouple located close to the contact inlet.A halogen lamp is used as a light source,and chromatic interferograms are recorded with a color digital video camera attached to a reflected light microscope.Film thickness values are evaluated from interferograms by thin film colorimetric interferometry[19],which provides lubricantfilm thickness measurement through the colori-metric analysis of chromatic interferograms using an appropriate color matching algorithm and color/film thickness calibration curves.These calibration curves are obtained before each measurement for the specific lubri-cant and contacting bodies.It is believed that thefilm thickness resolution is approximately1nm.The configu-ration of the optical test rig was slightly different for the study of the scale effect and the effect of load(pressure).Results shown in Fig.1were obtained with balls having diameter of5.5,9,25.4,and35mm[6].The root-mean-square surface roughnesses of the balls and the glass disk are about0.005and0.002l m,respectively.Balls having 35and25.4mm in diameter were attached to the shaft driven by servomotor through the planetary gearbox.In this case,the disk was driven by the ball in nominally pure rolling.Balls having9and5.5mm in diameter were sup-ported by four small rolling bearings and were driven by the disk in nominally pure rolling conditions.Film thick-ness experiments were carried out at the loads of1.4,3.8, 30.2,and57N for 5.5-,9-,25.4-and35-mm ball, respectively.The loads correspond to the same maximum contact pressure of518MPa,and the lubricated contact diameters of72,118,334,and458l m for5.5-,9-,25.4-, and35-mm ball,respectively.Lubricant bath temperature was set to25°C for Z25and40°C for PG460.The effect of load(pressure)on centralfilm thickness was studied in a circular EHD contact formed between a sapphire disk and a tungsten carbide(WC)ball of25.4mm in diameter.The disk is115mm in diameter.The root-mean-square surface roughnesses of the ball and disk are about0.006and0.002l m,respectively.The ball is attached to a shaft that is driven by servomotor through a planetary gearbox.The disk is driven by the ball in nom-inally pure rolling.Material properties of the contacting bodies are summarized in Table4.Experimental condi-tions in terms of dimensionless parameters are given in Table5.Lubricant bath temperature was set to40°C for PG460and SR600and50°C for LF5346.Film thickness experiments were carried out at the loads of12,85,and 195N that correspond to the maximum contact pressure of 1029,1977,and2607MPa,respectively.It should be pointed out that material properties of WC ball and sapphire disk given in Table4were obtained from the manufacturers.However,these parameters are used inTable3Parameters for the Mixture Model(7)Liquid Z25PG460LF5346l r(Pa s)102102103N112f1110.96f20.04n10.820.660.6n20.013G1(Pa)7910449105 5.49103 G2(Pa) 1.29107the numerical solution,and it will be shown later that the pressure sensitivity is dependent upon these elastic prop-erties so that the error in their estimation can influence calculations.Therefore,static contact experiments were carried out to check the elastic modulus of the sapphire disk and the WC ball.The constant load of65N was used. Atfirst,the static contact between the glass disk and the WC ball was evaluated.Figure2shows a chromatic interferogram and corresponding cross section providing L*and a*color coordinates across the contact.L*is a psychometric lightness coordinate and a*is a redness-greenness coordinate of the CIELAB uniform color space that is recommended by the Commission Internationale d’Eclairage laboratory for the use in the colorimetric applications.The colorimetric analysis of these color val-ues within chromatic interferogram captured under static contact conditions can be used to evaluate the contact diameter because the Hertzian contact can be recognized as aflat region on the L*and a*curves plotted across the contact.Then,the contact diameter can be used to check the material properties of the WC ball as the modulus of elasticity and Poisson’s ratio of the glass disk is well known.The contact diameter obtained from chromatic interferogram was found to be402.3l m which compares well with value(403.5l m)calculated from data in Table4.Similar comparison was done using the sapphire disk and WC ball(Fig.3)and the measured and calculated contact diameter was263.3l m and262.1l m,respec-tively.Therefore the values given in Table3can be used in this study with confidence.Centralfilm thickness values were obtained from3Dfilm thickness maps evaluated from EHD chromatic interferograms.Figure4shows the com-parison of EHD chromatic interferograms andfilm thick-ness profiles for LF5346captured at rolling speed of 0.0585m/s.4Theoretical AnalysisThe governing equations and the numerical solution scheme are reported in[20].Briefly,the pressure distri-bution is obtained by solving the generalized Reynolds Equation which can be expressed in dimensionless form as follows:Table4Properties of contacting bodiesGlass DiscModulus of elasticity(GPa)81 Poisson’s ratio0.208 Sapphire DiscModulus of elasticity(GPa)420 Poisson’s ratio0.29 Steel ballModulus of elasticity(GPa)210Poisson’s ratio0.3 WC ballModulus of elasticity(GPa)641 Poisson’s ratio0.26Table5Experimental conditionsp H/GPa^U^W^GSR6001.0297.57E-13to4.56E-12 1.35E-07126541.9779.51E-13to4.57E-129.58E-07126542.6079.68E-13to4.95E-12 2.97E-0612654 PG4601.0299.65E-12to1.25E-12 1.35E-0772071.9779.65E-12to1.25E-129.58E-0772072.6079.65E-12to1.25E-12 2.97E-067207 LF53461.0297.62E-11to9.85E-12 1.35E-07131491.9777.62E-11to9.85E-129.58E-07131492.6077.62E-11to9.85E-12 2.97E-0613149(a) L*o o X "q "F 2o P o X þo o Y "q "F 2o P o Y ¼k o o X ð"q H Þþk S 2oo X "q H 1À2"F 1"F 0!ð8Þwhere X ¼x b ,Y ¼y b ,P ¼p p h ,"q ¼q q o,k ¼U E 0p h R b ÀÁ3and S ¼ðu 2Àu 1Þu o is the slide-to-roll ratio with u o representing the average rolling velocity.The present simulations pertain to pure rolling,i.e.,S =0.The integral functions in dimensionless form are:"F0¼Z 11"gd Z ;"F1¼Z 1Z "g d Z and "F 2¼Z 10Z "gZ À"F 1"F 0 d Zð9Þwhere Z =z /h and "g ¼g =l o The pressure distributionobtained numerically satisfies the load balance condition given below in dimensionless form:ZZP d X d Y ¼2p3ð10ÞThe solution domain in the present simulations ranges from X =-4to 1.5,Y =-3to 3discretized using a uniform mesh of 5519601points.It has been verified that thecontacts are fully flooded and any further mesh refinement cause negligible change in the results.5ResultsFilm thickness measurements and calculations are pre-sented for comparison in Figs.5,6,7for LF5346,PG460,and SR600,respectively.The Newtonian solution is seen to be accurate for the neat mineral oil,SR600,at all pressures in Fig.7and for the polyglycol,PG460,at the lowest pressure in Fig.6.As load increases in Fig.6,the mea-sured film thickness becomes progressively less than the Newtonian prediction.That is to say,the Newtonian pre-diction understates the load sensitivity.In Fig.5,the Newtonian prediction is extremely inaccurate for LF5346at all loads,and the load sensitivity is understated.In both Figs.5and 6,the generalized Newtonian solutions offer improvements in the accuracy of prediction of film thickness.It is difficult to compare the pressure sensitivity,q (log h c )/q (log p H ),of experiment and simulation from the plots in Figs.5,6,7.This comparison is made in Table 6where the sensitivities were obtained from a power-law regression of results at the indicated velocities.Clearly,the generalized Newtonian calculation is a major improvement over the classical Newtonian,q (log h c )/q (log p H )&-0.20.However,the magnitude of the pressure sensitivity is understated by the generalized Newtonian calculation as well.6DiscussionThe simulations presented here are quantitative in the sensethat the viscosities used in the calculations were not adjusted to fit experiment or to improve numerical stability;the viscosities were measured in viscometers.Therefore,conclusions generated from the comparisons with experi-ment should provide insight into the real behavior of liquids in the operating contacts.6.1Shear-Dependent ViscosityIn the previous study of the scale effect [6],q (log h c )/q (log R ),the generalized Newtonian calculation also understated the increase of scale effect as a result of shear-thinning for LF5346.In that study,mechanical degradation was advanced as a likely possibility for the increased effect.The same reduction in molecular weight as a result of the imposed shear stress may be responsible here.However,in the course of the viscosity measurements,(a)L*shear stresses of up to 2and 8MPa were applied to PG460and LF5346,respectively,without loss of repeatability.Some insight may be gained from the computed values of shear stress in the inlet.Figure 8shows the evolution of pressure and shear stress along the inlet sweep for two loads.For comparison,x =0has been shifted to the edge of the Hertz boundary.Increasing the load results in increased pressure gradient and shear stress in theimportant region near the Hertz boundary.Since viscosity is reduced by an increase in shear stress when the stress exceeds G 1,the increase in load (contact pressure)must result in a greater decrease in film thickness for the shear-thinning case than for the Newtonian case where viscosity is unaffected by shear.(a)35012 N85 N 195 N 12NN 591N58An additional interesting effect was discovered in the course of the calculations.The load(pressure)effect is dependent upon the elastic modulus of the contacting bodies.Table7summarizes the calculatedfilm thicknesses for different values of E0where conditions have been adjusted to have similar centralfilm thickness.Small val-ues of E0lead to greater load(pressure)sensitivity.This result means that the usual simple power-law form of Eq.1 will not be useful for generalized Newtonian response.The individual exponents may not be independent of the other terms.Clearly,there must be some combination of small scale, R,and high contact pressure,which will result in shear-thinning of even an unformulated low-molecular-weight mineral oil in the inlet zone with the attending departures from classical Newtonianfilm behavior.Table6Pressure sensitivities,q(log h c)/q(log p H),for experiment and analysis for rolling velocity of0.1m/s for PG460and LF5346 and0.2m/s for SR600LF5346PG460SR600Experimental-0.49-0.39-0.21 Analytical-0.31-0.28-0.19The classical Newtonian result is q(log h c)/q(log p H)&-0.20 Table7The pressure effect for variable E0Lubricant E0(GPa)h c/nm(p H=1GPa)h c/nm(p H=2GPa)q(log h c)/q(log p H)Shear-thinning,l o=1Pa.s(G=10kPa,n=0.6)123580433-0.42 Newtonian,l o=0.0978Pa.s123580469-0.30 Shear-thinning,l o=0.79Pa.s(G=10kPa,n=0.6)550580482-0.27 Newtonian,l o=0.0797Pa.s550580494-0.23 Shear-thinning,l o=1Pa.s(G=10kPa,n=0.4)12313189-0.56 Newtonian,l o=0.0118Pa.s123131105-0.32 Shear-thinning,l o=0.715Pa.s(G=10kPa,n=0.4)55013196-0.45 Newtonian,l o=0.0093Pa.s550131108-0.286.2CompressibilityThe compressibility of the liquid further complicates any discussion of the sensitivity of centralfilm thickness to load.Equation19of Venner and Bos[21]can be written ash c¼h incompc Vðp¼p HÞV0ð11Þwhere h c incomp is the centralfilm thickness of an incompressible liquid.Differentiating yieldso h c o p H ¼o h incompco p HVV0p¼p Hþh incompc1V0o Vo pp¼p Hð12Þwhich can be recast aso log h c o log p H ¼h incompch cVV0p¼p Ho log h incompco log p Hþp HVo Vo pp¼p H"#ð13ÞApplication of Eq.11and the definition of bulk modulus K(p)=-V d p/d V giveso log h c o log p H ¼o log h incompco log p HÀp HKðp¼p HÞIn other words,q(log h c)/q(log p H)is the superposition of a component that results from the effect of pressure on geometry,stress and rheology,q(log h c incomp)/q(log p H) with a component resulting from effect of pressure on liquid volume,-p H/K.Differentiation of Eq.3yieldsK¼1À11þK0ln1þpK0ð1þK00Þ!&'½K0þpð1þK00Þð14ÞThe component of q(log h c)/q(log p H)that results from compressibility alone,-p H/K,is listed in Table8for the experimental liquids for which compressibility data are available and for the Tait equation of state with the uni-versal parameters and for the Dowson and Higginson (D&H)equation of state[9]which was used in the simu-lations which resulted in the Hamrock and Dowson Equa-tion1.Clearly,from Table8,a large component(about one-third)of[q(log h c)/q(log p H)]E0,R=const=-0.201for the Hamrock and Dowson Equation1comes from com-pressibility,which is understated by the D&H equation of state except for low pressure and press-ibility increases exponentially with temperature[22].A portion of the discrepancy between the classical prediction of load sensitivity and experiment is therefore a result of compressibility.The glass transition has not been considered above where the equation of state describes liquid-like behavior only. When the viscosity reaches to some very large value of107to 1012Pa s,further liquid-like compression is restricted by the viscosity and a solid-like compressibility governs.7Conclusions1.Quantitative calculations of EHLfilm thickness usingrealistic pressure and shear-dependent rheology and realistic compressibility have indicated that the classical pressure sensitivity,[q(log h c)/q(log p H)]E0,R=const= -0.2,is merely a lower limit with values as great as-0.6 being possible.2.Experimental measurements employing Hertz pressuresfrom1.03to2.6GPa reveal that the pressure sensitivity may be as great as[q(log h c)/q(log p H)]E0,R=const= -0.5.3.The distribution of shear stress along the inlet sweepdepends upon the contact load(pressure).If the viscosity is shear-dependent for the shear stress existing along the inlet,then the viscosity will have a dependence on the load,which differs from the Newtonian case.4.Realistic compressibility accounts for a portion of theenhanced pressure dependence in some cases.5.Mechanical degradation of large lubricant moleculesmay play a part in the pressure sensitivity.5.New experimental evidence of the enhanced scale sensi-tivity resulting from shear-thinning has been obtained. 6.Most published pressure–viscosity coefficients haveresulted from thefitting of a classicalfilm thickness formula to a measurement offilm thickness rather than from a direct measurement of viscosity under pressure.These results have dire implications for the usual practice of extrapolation offilm thickness from experimental measurements at large scale and low pressure using the adjusted pressure–viscosity coeffi-cient.For machines of small scale and high pressure, the extrapolation will sometimes result in substantially overestimatedfilm thickness.Acknowledgments Film thickness measurements were carried out with the support of the Ministry of Education,Youth and Sports of the Czech Republic—Project No.ME09025.Bair was supported by the National Science Foundation under grant number EEC#0540834.Table8The component of q(log h c)/q(log p H)that results from compressibility alone,-p H/KT=20°C T=150°Cp H(GPa)=0.5120.512LF5346-0.084-0.097-0.109-0.100-0.112-0.124 Z25-0.088-0.100-0.112-0.102-0.113-0.125 Universal Tait-0.079-0.093-0.105-0.095-0.107-0.119 D&H-0.075-0.067-0.049-0.075-0.067-0.049。