friction in elastohydrodynamic lubrication990483/fulltext01.pdf · friction in elastohydrodynamic...

DOCTORA L T H E S I S

Department of Engineering Science and MathematicsDivision of Machine Elements

Friction in Elastohydrodynamic Lubrication

Marcus Björling

ISSN 1402-1544ISBN 978-91-7439-999-8 (print)ISBN 978-91-7583-000-1 (pdf)

Luleå University of Technology 2014

Marcus B

jörling Friction in Elastohydrodynam

ic Lubrication

ISSN: 1402-1544 ISBN 978-91-7583-XXX-X Se i listan och fyll i siffror där kryssen är


Marcus Björling

Luleå University of TechnologyDepartment of Engineering Science and Mathematics,

Division of Machine Elements

Cover figure: Top left: Friction map from ball-on-disc experiment.Top right: Elastohydrodynamic contact.Bottom left: Results from ball-on-disc measurements.Bottom right: Ternary phase diagram of DLC.

Title page figure: Friction map from ball-on-disc experiment.


Copyright © Marcus Björling (2014). This document is freelyavailable at

www.ltu.se

or by contacting Marcus Björling,

[email protected]

This document may be freely distributed in its original formincluding the cur-rent author’s name. None of the content may be changed or excluded withoutpermissions from the author.

Printed by Luleå University of Technology, Graphic Production 2014

ISSN: 1402-1544ISBN: 978-91-7439-999-8 (print)ISBN: 978-91-7583-000-1 (pdf)

Luleå 2014

www.ltu.se

This document was typeset in LATEX 2ε

“Nothing has such power to broaden the mind as the ability to investigatesystematically and truly all that comes under thy observation in life.”

- Marcus Aurelius

Preface

The work presented in this doctoral thesis has primarily been carried out atLuleå University of Technology at the Division of Machine elements. I wouldlike to thank my supervisors Professor Roland Larsson and Associate Profes-sor Pär Marklund for their help, support, guidance and valuable discussionsover the course of this research. I would also like to thank Professor ElisabetKassfeldt at Luleå University of Technology for helpful discussions regardingthe friction maps presented in Paper A. I am also grateful forthe collaborationwith Dr. Scott Bair at Georgia Institute of Technology and Dr. Wassim Habchiat Lebanese American University, which led to Papers C and D in this thesis.

I would also like to thank Professor Arto Lehtovaara and JuhaMiettinen forthe collaboration that led to Paper E, including an interesting time at TampereUniversity of Technology doing experimental work.

Furthermore, I would also like to express my gratitude to allmy friendsand colleagues at Luleå University of Technology for providing an enjoyableplace to work. Special thanks go to Kim Berglund and Patrik Isaksson fortaking their time for discussions that have aided me in my work.

The support and assistance from my industrial partners Volvo ConstructionEquipment, Scania and Vicura (former SAAB Powertrain), including manyhelpful people working there is gratefully acknowledged. Acknowledgmentsshould also be made to Statoil Lubricants for providing testlubricants and toIonBond for providing DLC coatings.

I would especially like to express my gratitude to the Swedish Foundationfor Strategic Research (SSF), ProViking and VR (Swedish Research Council)for financial support. Without this funding my research would not have beenpossible.

Finally, I would like to thank my family for their support andencourage-ment, much leading to the person I am and what I have achieved today.

Marcus Björling, Luleå, September 2014

7

Abstract

Today, with increasing demands on industry to reduce energyconsumptionand emissions, the strive to increase the efficiency of machine components ismaybe bigger than ever. This PhD thesis focus on friction in elastohydrody-namic lubrication (EHL), found in, among others, gears, bearings and camfollowers. Friction in such contacts is governed by a complex interaction ofmaterial, surface and lubricant parameters as well as operating conditions. Inthis work, experimental studies have been conducted that show how frictionvaries over a wide range of running conditions when changingparameters likelubricant viscosity, base oil type, surface roughness and lubricant temperature.These measurements have also been used to predict the friction behaviour in areal gear application.

Numerical modeling of elastohydrodynamic (EHD) friction and film thick-ness are important for increased understanding of the field of EHL. Due tothe high pressure and shear normally found in EHD contacts itis crucial thatappropriate rheological models are used. An investigationhas been carriedout in order to assess the friction prediction capabilitiesof some of the mostwell founded rheological models. A numerical model was usedto predict fric-tion coefficients through the use of lubricant transport properties. Experimentswere then performed that matches the predicted results rather well, and thedeviations are discussed. The numerical model in combination with experi-mental measurements were used to investigate the friction reducing effect ofdiamond like carbon (DLC) coatings in EHL. A new mechanism offriction re-duction through thermal insulation is proposed as an alternative to the currenthypothesis of solid-liquid slip. These findings opens up fornew families ofcoatings where thermal properties are in focus that may be both cheaper, andmore effective in reducing friction in certain applications than DLC coatingsof today.

9

Contents

I Comprehensive Summary 31

1 Introduction 33

2 Elastohydrodynamic lubrication 352.1 Conformal and non-conformal contacts . . . . . . . . . . . . 352.2 Hydrodynamic and Elastohydrodynamic Lubrication . . . .. 372.3 The EHD contact . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Lubrication regimes . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 The film parameter . . . . . . . . . . . . . . . . . . . 422.4.2 Amplitude reduction . . . . . . . . . . . . . . . . . . 432.4.3 The Stribeck curve . . . . . . . . . . . . . . . . . . . 45

2.5 Lubricant rheology . . . . . . . . . . . . . . . . . . . . . . . 472.5.1 Lubricant transport properties . . . . . . . . . . . . . 472.5.2 Temperature-Viscosity relationship . . . . . . . . . . 482.5.3 Pressure-Viscosity relationship . . . . . . . . . . . . . 492.5.4 Equation of state . . . . . . . . . . . . . . . . . . . . 532.5.5 Non-Newtonian behaviour . . . . . . . . . . . . . . . 552.5.6 Limiting stress . . . . . . . . . . . . . . . . . . . . . 592.5.7 Models for shear thinning . . . . . . . . . . . . . . . 612.5.8 Glass transition . . . . . . . . . . . . . . . . . . . . . 652.5.9 Time-Temperature-Pressure superposition . . . . . . . 672.5.10 Temperature, pressure and thermal conductivity . . .. 68

2.6 Machine components . . . . . . . . . . . . . . . . . . . . . . 692.7 Friction reduction in machine components . . . . . . . . . . . 702.8 DLC coatings in EHL . . . . . . . . . . . . . . . . . . . . . . 722.9 Solid-liquid slip . . . . . . . . . . . . . . . . . . . . . . . . . 752.10 Thermal conductivity of DLC coatings . . . . . . . . . . . . . 782.11 Ball-on-disc tribotester . . . . . . . . . . . . . . . . . . . . . 81

11

3 Scope and objectives 85

4 Friction mapping 874.1 Concept of friction mapping . . . . . . . . . . . . . . . . . . 894.2 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 Predicting friction in EHL 975.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3 Inclusion of thin, thermally insulating layers . . . . . . .. . . 1045.4 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 The influence of DLC coatings on EHD friction 1096.1 Friction reduction with DLC coatings . . . . . . . . . . . . . 109

6.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . 1106.1.2 Friction measurements . . . . . . . . . . . . . . . . . 110

6.2 An alternative hypothesis . . . . . . . . . . . . . . . . . . . . 1126.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . 1126.2.2 Lubricant temperature increase with DLC coatings . . 113

6.3 Thin layer thermal insulation . . . . . . . . . . . . . . . . . . 1156.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3.2 Thermal insulation and friction . . . . . . . . . . . . . 116

6.4 Coating thickness and friction in EHL . . . . . . . . . . . . . 1206.4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4.2 Results and discussion . . . . . . . . . . . . . . . . . 122

6.5 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Friction mapping and gear contacts 1277.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.2 Ball-on-disc and gear contact friction . . . . . . . . . . . . . .1307.3 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8 Conclusions 133

9 Future Work 135

II Appended Papers 137

A EHL Friction mapping 139

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

A.2.1 Ball on disc tribotester . . . . . . . . . . . . . . . . . 144A.2.2 Test specimens and lubricants . . . . . . . . . . . . . 144A.2.3 Test procedure . . . . . . . . . . . . . . . . . . . . . 147

A.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . 147A.3.1 Surface roughness . . . . . . . . . . . . . . . . . . . 152A.3.2 Temperature and viscosity . . . . . . . . . . . . . . . 154A.3.3 Additives . . . . . . . . . . . . . . . . . . . . . . . . 156A.3.4 Base oil type . . . . . . . . . . . . . . . . . . . . . . 157

A.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

B The influence of DLC coating on EHL friction 159B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

B.2.1 Ball on disc tribotester . . . . . . . . . . . . . . . . . 166B.2.2 Test specimens and lubricants . . . . . . . . . . . . . 166B.2.3 Test procedure . . . . . . . . . . . . . . . . . . . . . 168B.2.4 Simulation model . . . . . . . . . . . . . . . . . . . . 169

B.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . 172B.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

C Towards the true prediction of EHL friction 181C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 184C.2 Overall Methodology . . . . . . . . . . . . . . . . . . . . . . 186

C.2.1 Investigation Procedure . . . . . . . . . . . . . . . . . 186C.2.2 Lubricant transport properties . . . . . . . . . . . . . 187C.2.3 Numerical model . . . . . . . . . . . . . . . . . . . . 189C.2.4 Ball on disc tribotester . . . . . . . . . . . . . . . . . 189

C.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . 191C.3.1 Film thickness and roughness effects . . . . . . . . . 195C.3.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . 196

C.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

D Thin layer thermal insulation in EHL 203D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 209D.2 Overall Methodology . . . . . . . . . . . . . . . . . . . . . . 210

D.2.1 Investigation Procedure . . . . . . . . . . . . . . . . . 210D.2.2 Lubricant transport properties . . . . . . . . . . . . . 211

D.2.3 Numerical model . . . . . . . . . . . . . . . . . . . . 213D.2.4 Ball on disc tribotester . . . . . . . . . . . . . . . . . 214

D.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . 218D.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

E The effect of DLC coating thickness on EHD friction 227E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 230E.2 Overall Methodology . . . . . . . . . . . . . . . . . . . . . . 231

E.2.1 Test specimens and lubricant . . . . . . . . . . . . . . 232E.2.2 Ball on disc tribotester . . . . . . . . . . . . . . . . . 233E.2.3 Test procedure . . . . . . . . . . . . . . . . . . . . . 234E.2.4 Surface energy and wetting . . . . . . . . . . . . . . . 235E.2.5 Surface tension . . . . . . . . . . . . . . . . . . . . . 236E.2.6 Spreading parameter . . . . . . . . . . . . . . . . . . 237

E.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237E.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242E.5 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 245

F Correlation between gears and ball-on-disc friction 247F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 251F.2 Overall Methodology . . . . . . . . . . . . . . . . . . . . . . 252

F.2.1 Ball-on-disc tribotester . . . . . . . . . . . . . . . . . 252F.2.2 Gear test rig . . . . . . . . . . . . . . . . . . . . . . . 253F.2.3 Test specimens and lubricant . . . . . . . . . . . . . . 253F.2.4 Gear test procedure . . . . . . . . . . . . . . . . . . . 255F.2.5 Correlation methodology . . . . . . . . . . . . . . . . 257F.2.6 Ball on disc test procedure . . . . . . . . . . . . . . . 258

F.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . 259F.4 In conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 263

References 265

Appended Papers

A M. Björling, R. Larsson, P. Marklund and E. Kassfeldt."EHL friction mapping - The influence of lubricant, roughnes s, speedand slide to roll ratio."Proceedings of the Institution of Mechanical Engineers, Part J: Jour-nal of Engineering Tribology, 2011 May; vol. 225, Issue 7, p. 671-681.In this paper, a ball-on-disc test device has been used to investigate howthe friction behaviour of a system changes with surface roughness, baseoil type, EP additive content and operating temperature under a widerange of entrainment speeds and slide to roll ratios. Furthermore, analternative way of presenting the results called friction mapping is intro-duced. The development of the test method and all experimental workwas carried out by Marcus Björling who also wrote the paper. RolandLarsson, Pär Marklund and Elisabet Kassfeldt were involvedin the dis-cussion of the method and the results from the experiments.

B M. Björling, P. Isaksson, P. Marklund and R. Larsson."The influence of DLC coating on EHL friction coefficient."Tribology Letters, 2012 August; vol. 47, Issue 2, p. 285-294.Several experiments were carried out in a ball-on-disc testdevice to in-vestigate how a DLC coating applied on either the ball, the disc, or bothwould influence the coefficient of friction in a wide range of entrainmentspeeds and slide to roll ratios. A numerical simulation model was de-veloped to investigate how the different thermal properties of the DLCcoating compared to the substrate would effect the lubricant film temper-ature. All experimental work was carried out by Marcus Björling, whoalso wrote the paper. Patrik Isaksson aided in the work of developing thenumerical model, and discussing the simulation results. Roland Larssonand Pär Marklund were involved in the discussion of the results.

15

C M. Björling, W. Habchi, S. Bair, R. Larsson and P. Marklund."Towards the true prediction of EHL friction."Tribology International, 2013 April; vol. 66, p. 19-26.In this work, primary measurements of lubricant transport propertiesof Squalane were performed, and used in a numerical model to predictEHD friction. Afterwards, friction was measured in a ball-on-disc tri-botester. No tuning of the lubricant properties, model or test setup wereapplied. All experimental work was carried out by Marcus Björling,who also wrote most of the paper. Wassim Habchi performed thefric-tion predictions, aided by the work in lubricant transport properties byScott Bair. All authors were involved in the discussion of the results.

D M. Björling, W. Habchi, S. Bair, R. Larsson and P. Marklund."Friction reduction in elastohydrodynamic contacts by thin layerthermal insulation."Tribology Letters, 2014 January; vol. 54, Issue 2, p. 477-486.In this article a numerical friction model was used to predict the frictioncoefficient in an EHD contact by the addition of a thin thermally insu-lating layer. The predicted friction results were comparedto experimen-tally obtained values from a ball-on-disc test device. Marcus Björlingand Roland Larsson conceived the project. Marcus Björling designedthe experiments and the input of running conditions for the numericalmodel. Wassim Habchi developed the numerical model and performedthe numerical analysis. Scott Bair performed most of the measurementsfor the lubricant transport properties. Marcus Björling wrote the paper.All authors analyzed the data, discussed the results and commented onthe manuscript.

E M. Björling, R. Larsson and P. Marklund."The effect of DLC coating thickness on elastohydrodynamicfric-tion."Tribology Letters, 2014 June; vol. 55, Issue 2, p. 353-362.Friction measurements were performed in a ball-on-disc test rig withspecimens coated with the same DLC coating, but with different coat-ing thicknesses. Contact angle measurements were performed both withreference fluids, and the lubricant used in the friction measurements tobe able to calculate the surface energy of the specimens, andto get thecontact angles with the test fluid. All experimental work wascarried outby Marcus Björling who also wrote the paper. Roland Larsson and PärMarklund commented on the manuscript.

F M. Björling, J. Miettinen, P. Marklund, A. Lehtovaara and R. Lars-son."The correlation between gear contact friction and ball-on-disc fric-tion measurements."To be submitted to journal.A series of measurements were conducted in a FZG gear test rigwiththree different oils. The measurements were performed in such a waythat the contact friction losses could be separated from thetotal losses,and a mean friction coefficient was computed for each specifictest case.The same oils were also used in measurements in a ball-on-disc machinewith conditions to simulate the gear contact. The correlation between thetwo test rigs were discussed, and a method to assess gear friction for aspur gear pair with arbitrary geometry was presented. The FZG geartests were carried out by Marcus Björling and Juha Miettinen. The ball-on-disc tests were performed by Marcus Björling who also wrote the pa-per. All authors discussed the results and commented on the manuscript.

Additional publications notincluded in the thesis

• T. Cousseau, M. Björling, B. Garca, A. Campos, J. Seabra and R. Lars-son."Film thickness in a ball-on-disc contact lubricated with greases,bleed oils and base oils."Tribology International, 2012 April; vol. 53, p. 53-60.In this paper, a ball-on-disc rig with attached optical interferometry wasused to measure the film thickness in a circular EHD contact for threelubricant greases and their base and bleed oils. The measurements werecompared to calculations using Hamrock film thickness equations. Theball-on-disc experiments were conducted by Tiago Cousseauand Mar-cus Björling, who also did most parts of the results analysisand wrotethe paper. All authors were involved in commenting on the manuscript.

• C.H. Zhang, Y.C. Zhai, M. Björling, Y. Wang, J.B. Luo and B. Prakash."EHL Properties of Polyalkylene Glycols and Their Aqueous Solu-tions."Tribology Letters, 2012 October; vol. 45, issue 3, p. 379-385.In this work, four types of polyalkylene glycols (PAGs) withdifferentmolecular weight and their aqueous solutions with different concentra-tions were studied in a ball-on-disc test device with optical interferom-etry. The purpose was to investigate the suitability to us PAG’s as basestock for water based lubricants. The ball-on-disc experiments werepartly conducted by Marcus Björling, who was also involved in ana-lyzing the results and commenting on the manuscript.

19

• Y. Shi, I. Minami, M. Grahn, M. Björling and R. Larsson."Boundary and elastohydrodynamic lubrication studies of glycerolaqueous solutions as green lubricants."Tribology International, 2014 January; vol. 69, p. 39-45In this paper, the boundary and elastohydrodynamic lubricating behaviourof glycerol and its aqueous solutions are discussed in both rolling andsliding contacts with the goal of assessing the use of glycerol as a greenlubricant. The glycerol and its aqueous solutions were subjected to testsin both boundary, mixed and boundary lubrication in both rolling, slidingand reciprocating motion. A ball on test rig with optical interferometrywas also used to measure the film thickness of the glycerol aqueous so-lutions. The ball-on-disc experiments (both optical and friction) werepartly conducted by Marcus Björling, who was also involved in analyz-ing the results. All authors were involved in the process of writing themanuscript.

Conference presentations

• M. Björling, R. Larsson, P. Marklund and E. Kassfeldt."EHL friction mapping - The influence of lubricant, roughnes s, speedand slide to roll ratio."14th Nordic Symposium on Tribology: NORDRIB 2010. Storforsen,Sweden, June 8-10, 2010.

• M. Björling, W. Habchi, S. Bair, R. Larsson and P. Marklund."Towards the true prediction of EHL friction."International Tribology Symposium of IFToMM. Luleå, Sweden, March19-21, 2013.

• M. Björling, W. Habchi, S. Bair, R. Larsson and P. Marklund."On the effect of DLC coating on full film EHL friction."World Tribology Congress, Torino, Italy, September 8-13, 2013.

21

Nomenclature

α Pressure-viscosity coefficient [Pa−1]

β Viscosity-temperature gradient at operating temperatureβ = ∂η/∂T

βK Temperature coefficient ofK0 [K−1]

βV Specific volume of the molecules without the space between them

χ Dimensionless heat capacity scaling parameter

χi Area fraction in the Ree-Eyring equation

δts Time scaling coefficient

ε Shear strain rate [s−1]

γ Shear rate [s−1]

ε Thermal expansion coefficient [◦C−1]

εg Addendum contact ratio of gear

εp Addendum contact ratio of pinion

εt Contact ratio

η Generalized (shear dependent) viscosity [Pa s]

η0 Viscosity at atmospheric pressure and operating temperature [Pa s]

γDl Dispersive component of surface tension [N/m]

γDs Dispersive component of surface energy [J/m2]

23

γPl Polar component of surface tension [N/m]

γPs Polar component of surface energy [J/m2]

γl Total surface tension [N/m]

κ Dimensionless conductivity scaling parameter

Λ Limiting stress pressure coefficient

λ Relaxation or characteristic time [s]

Λ f Film parameter

λi Characteristic or relaxation time in the Ree-Eyring equation [s]

λm Maxwell relaxation time [s]

λR Relaxation time atTR and ambient pressure [s]

λx,λy Wavelength or autocorrelation length in direction of x or y [m]

λEB Einstein-Debye relaxation time [s]

µ Limiting low-shear viscosity [Pa s]

µf Coefficient of friction

µm Approximate gear friction coefficient

µR Low shear viscosity atTR and ambient pressure [Pa s]

µ1 Low shear viscosity [Pa s]

µ2 High shear viscosity [Pa s]

µ∞ Viscosity extrapolated to infinite temperature [Pa s]

µbl Boundary lubrication friction coefficient

µEHL Sliding friction coefficient in full film conditions

µsl Sliding friction coefficient

∇1 Dimensionless wavelength parameter∇1 = (λx/b)(M3/4/L1/2)

ν Kinematic viscosity at operating temperature of oil [mm2/s]

ω Rotational speed [rpm]

ρ Mass density [kg/m3]

ρR Mass density at reference state,TR, p = 0 [kg/m3]

τ Shear stress [Pa]

τe Eyring stress [Pa]

τi Characteristic stress in the Ree-Eyring equation [Pa]

τL Limiting shear stress [Pa]

θ Contact angle [deg]

ϕ Dimensionless viscosity scaling parameter

ϕ∞ Viscosity scaling parameter for unbounded viscosity

ϕbl Weighting factor for the sliding friction coefficient equation

ϕish Inlet shear heating reduction factor

ϕrs Kinematic replenishment/starvation reduction factor

A Coefficient in the dimensionless conductivity scaling parameter

a Yasuda parameter

Ad Dimensionless deformed amplitude

Ai Dimensionless undeformed amplitude

At Heat transfer area [m2]

av Thermal expansivity defined for volume linear with temperature [K−1]

b Half width of Hertzian contactb =√

(8w1Rx)/(πE′) [m]

B Hertzian contact width [m]

BF Fragility parameter in the new viscosity equation

C Parameter in the conductivity function [W/m K]

c Coating thickness [m]

C0 Parameter in the heat capacity function [J/m3 K]

Ck Parameter in the conductivity function [W/m K]

cp Specific heat capacity [J/kg K]

Cv Lubricant volumetric heat capacity [J/Km3]

D Bearing outside diameter [mm]

d Bearing bore diameter [mm]

DF Fragility parameter in the VTF equation

dm Bearing pitch diameter [mm]

dT Temperature difference across coating [K]

E′ Effective elastic modulus,E′ = 2[(1−v21)/E1+(1−v2

2)/E2]−1

Ei Youngs modulus [GPa]

F Load [N]

Fr Radial bearing load [N]

G Effective shear modulus [Pa]

g Thermodynamic interaction parameter

GR Material modulus associated withλ at the reference state,TR, p = 0[Pa]

Grr Geometric and load dependent variable for rolling frictional moment

Gsl Geometric and load dependent variable for sliding frictional moment

H Hersey number

Hv Gear loss factor

hcen Hamrock and Dowson central film thickness [m]

hmin Hamrock and Dowson minimum film thickness [m]

i Gear ratio

K Consistency [Pans]

k Thermal conductivity [W/mK]

K′

0 Pressure rate of change of isothermal bulk modulus atp = 0

kB Boltzmann constant = 1.380622 x 10−23 [J/K]

K00 K0 at zero absolute temperature [Pa]

K0 Isothermal bulk modulus atp = 0 [Pa]

Krs Replenishment/starvation constant

Kz Bearing type related geometric constant

L Dimensionless material parameterL = G(2U)0.25

M Molecular weight [kg/kmol]

m Parameter in the heat capacity function [J/m3 K]

M1 Dimensionless (One dimensional) load parameterM1 =W1(2U)−0.5

Mrr Rolling frictional moment [Nmm]

Msl Sliding frictional moment [Nmm]

N Number of flow units in the Ree-Eyring equation

n Power law exponent

p Pressure [Pa]

Pg Glass transition pressure [Pa]

Ph Hertzian pressure [Pa]

Pl Total gear power loss [W]

Pm Gear mesh power loss [W]

Pt Total transmitted power [W]

Pbl Load dependent bearing power loss [W]

Pnl Load independent gear power loss [W]

Q Invard heat flux [W/m2]

q Coefficient in the dimensionless conductivity scaling parameter

qt Heat transfer [W]

R Ball radius [m]

R1 Geometric constant for rolling frictional moment

Rg Universal gas constant = 8314.34 [m3/kmol/K]

Rx Effective radius [m]

S Slide to roll ratio

s Exponent in the conductivity scaling model

S1 Geometric constant for sliding frictional moment

Sa Arithmetic average of absolute roughness [m]

Sn Slip numberSn = Ur/Ue

Sq Root mean square roughness [m]

Sqc Root mean square combined roughness [m]

SRR Slide to roll ratio

T Temperature [K]

t Time [s]

T0 Initial temperature [◦C]

tc Contact time [s]

Tg Glass transition temperature [K]

TR Reference temperature [K]

T∞ Divergence temperature [K]

U Dimensionless speed parameterU = Ueη0/E′Rx

Ub Surface velocity of ball [m/s]

Ud Surface velocity of disc [m/s]

Ue Mean entrainment speed [m/s]

Ui Surface velocity [m/s]

Ur Surface velocity of rough surface [m/s]

V Volume [m3]

V0 Volume atp = 0 [m3]

vi Poisson’s ratio

Vm Molecular volume [m3]

VR Volume at reference state,TR, p = 0 [m3]

w′ Contact load, 1d-geometry [Nm]

W1 Dimensionless load parameter 1d-geometry (line contact),W1 =w′/(E′Rx)

Wf Friction power [W]

Wi Weissenberg number

Wq Heat source [W/m3]

x Cross film position [m]

z Number of gear teeth

Z1 Viscosity-pressure index

Acronyms

AISI American Iron & Steel Institute

AW Anti Wear

COF Coefficient Of Friction

DIN Deutsches Institut für Normung

DLC Diamond Like Carbon

EHD Elasto Hydrodynamic

EHL Elasto Hydrodynamic Lubrication

EOS Equation Of State

EP Extreme Pressure

FOV Field Of View

FZG Forschungsstelle für Zahnräder und Getriebebau

HL Hydrodynamic Lubrication

HRC Rockwell Hardness

MoDTC Molybdenum DiThio Carbamates

MTM Mini Traction Machine

OWRK Owens-Wend-Rabel-Kaelbe

PACVD Plasma Assisted Chemical Vapour Deposition

PAO Poly Alpha Olefin

PECVD Plasma Enhanced Chemical Vapour Deposition

PVD Physical Vapour Deposition

RMS Root Mean Square

SRR Slide to Roll Ratio

SSF Swedish Foundation for Strategic Research

TEHL Thermal Elasto Hydrodynamic Lubrication

VI Viscosity Index

VTF Vogel, Tammann and Fulcher

WAM Wedeven Associates Machine

WLF William-Landel-Ferry

ZDDP Zinc DialkylDithioPhosphate

Part I

Comprehensive Summary

31

Chapter 1

Introduction

Lubrication is vital for most modern machine components to work properly,be efficient and have a satisfactory service life. The use of lubricants is how-ever not a new phenomenon connected to machines of today and the industrialrevolution. In ancient times, people found that animal fat,olive oil or othermaterials placed between two objects rubbing against each other often wouldreduce the force required to keep them in motion. In principle, the lubricant,be it a solid or a liquid, creates a film of easily sheared material between thesurfaces that reduces friction, and often also wear.

An important finding in the field of lubrication was made in 1883 byBeuchamp Tower [1] where he found a pressure build up in the lubricant inhis test rig for journal bearings. The explanation came onlya few years afterthis discovery. Osborne Reynolds [2] managed to use a reduced form of theNavier-Stokes equations and the continuity equation to generate a second orderdifferential equation for the pressure in a narrow converging gap between twosurfaces. A pressurized lubricant film could partially or totally carry a loadbetween two surfaces and thereby significantly reduce the wear and frictionbetween the surfaces in motion. This is the fundamental lubrication principleof many of todays most important machine components such as gears, bearingsand cam followers.

Today, with increasing demands on industry to reduce energyconsumptionand emissions, the strive to increase the efficiency of machine components ismaybe bigger than ever. This PhD thesis focus on friction in elastohydrody-namic lubrication (EHL), found in, among others, gears, bearings and camfollowers. Friction in such contacts is governed by a complex interaction ofmaterial, surface and lubricant parameters as well as operating conditions. Inthis work, it is shown how friction varies over a wide range ofrunning condi-

33

34 Chapter 1. Introduction

tions when changing parameters like lubricant viscosity, base oil type, surfaceroughness and lubricant temperature. Different friction regimes is used to ex-plain the frictional behaviour in a system when varying running conditions.Furthermore, a methodology is introduced where measurements in a boll-on-disc machine are used to predict friction in an actual spur gear contact.

Numerical modeling of elastohydrodynamic (EHD) friction and film thick-ness are important for increased understanding of the field of EHL. Due to thehigh pressure and shear normally found in EHD contacts, it iscrucial that ap-propriate rheological models are used. In this work, a numerical model wasused to predict friction coefficients through the use of lubricant transport prop-erties, and compared to experimental measurements. This numerical modeluse some of the most well founded rheological models and no tuning of thelubricant properties, model and test setup was applied.

Moreover, this thesis show the effect of DLC (diamond like carbon) coat-ings on EHD friction in a series of experiments. The effect ofcoating both,or only one of the contacting bodies is investigated as well as the effect of thecoating thickness. The experimental results together withnumerical simula-tion are used to investigate the friction reducing mechanism of DLC coatings.A new mechanism of friction reduction through thermal insulation is proposedas an alternative to the current hypothesis of solid-liquidslip.

This thesis is divided into two parts. Part I, the comprehensive summary,contains an introduction to the field of EHL including the EHDcontact, lu-brication regimes, lubricant rheology, EHL in machine components, coatingtechnology etc as a foundation for the reader to better understand the researchpresented in subsequent chapters. Part I also contains a summary of the re-search performed that has led to the writing of this thesis. Part II, appendedpapers, contains the articles that have been the result of this research project.The articles hold detailed information that is not always discussed at length inpart I.

Chapter 2

Elastohydrodynamiclubrication

Lubrication is vital in most modern machinery. Without lubrication, most oftodays inventions involving moving parts in relative contact would simply notwork, or only be able to operate at limited times at very low efficiency due tohigh friction coefficients and wear rates. Lubricants, either solids or liquids(in some cases even gases) are used to create a layer of easilyshared mate-rial between the surfaces in relative motion in machine components to reducefriction and wear. In addition to the friction and wear reducing capabilities ofliquid lubricants they are also often used to cool components, and to removedebris and contaminants from the contact. In this work mainly liquid lubri-cation is considered, and the following sections explain the fundamentals ofelastohydrodynamic lubrication (EHL).

2.1 Conformal and non-conformal contacts

In a simplistic way of viewing two surfaces in contact, a distinction can bemade between conformal and non-conformal contacts. Figure2.1 pictures oneconformal, and one non-conformal contact. In case of a conformal contactthe surfaces of the two contacting bodies fit well into each other geometricallyin the way that the apparent area of contact is large, and thatthe load is dis-tributed over a large contact area compared to the thicknessof the lubricantfilm. Journal bearings and slider bearings are examples of conformal contactswhere the load is carried over a relatively large area and theload carrying arearemains almost constant when the load is increased. In journal bearings the

35

36 Chapter 2. Elastohydrodynamic lubrication

radial clearance between journal and sleeve is typically around one thousandthof the journal diameter [3]. In case of non-conformal contacts the two con-tacting bodies do not fit well into each other and the load between the bodiesis carried by a relatively small area. Furthermore, the lubrication area in anon-conformal contact generally increases considerably with increasing load.Even so, since the contact area is relatively small the contact pressure increasessubstantially with increased loads. Many common machine components havenon-conformal contacts, such as gears, roller bearings andcam followers.

Figure 2.1: Conformal and non-conformal contacts.

Johnson [4] divided lubrication of non-conformal contactsinto regimes bythe influence of pressure on the elastic deformation of the solid bodies, and theinfluence of pressure on the viscosity of the lubricant. Fourdifferent regimeswere identified:

1. Iso-viscous rigid: In this regime the pressure generated in the film istoo low to substantially alter the viscosity of the lubricant or change thegeometry of the solid bodies relative to the thickness of thefilm.

2. Piezo-viscous rigid:The piezo-viscous rigid regime is reached whenthe pressure in the fluid film is high enough to substantially increase theviscosity of the lubricant, but at the same time not high enough to changethe shape of the solid bodies.

3. Iso-viscous elastic:The pressure in the fluid film is not affecting theviscosity of the lubricant, but is changing the shape of the solid bodies.This regime is often found in systems where the solid bodies are made of

2.2. Hydrodynamic and Elastohydrodynamic Lubrication 37

materials with low elastic modulus, and possibly also when the viscos-ity of the lubricant is very insensitive to pressure. This regime is oftenreferred to as soft EHL, and is for instance encountered in the hip jointin the human body, and in an elastomeric seal.

4. Piezo-viscous elastic:Many common machine components such as gearsand rolling bearings operate in the piezo-viscous elastic regime wherethe pressure in the fluid film is high enough to significantly affect boththe viscosity of the lubricant, and the shape of the solid bodies. This isoften referred to as hard EHL.

Johnson also suggested parameters that would show what kindof lubricationregime that was encountered in a particular case, a work thatwas later contin-ued by Hooke [5]. Although non-conformal contacts can have many differentkinds of geometries there are three specific cases that are mostly used in re-search for more simplistic type of experimental or numerical investigations.Two rigid cylinders with parallel axes, or a cylinder on a plane will form acontact patch known as a line contact, which will expand to a rectangle whenload is applied on the elastic bodies. A sphere on a plane, a sphere on a sphereor two cylinders with identical radii and perpendicular axes will form whatis known as a point contact, and will expand to a circular contact when loadis applied. Finally, an elliptical point contact that will expand to an ellipticalcontact patch is formed by a sphere on a cylinder or by crownedrollers wherethe radius around the symmetry axis differs from the crown radius.

2.2 Hydrodynamic and Elastohydrodynamic Lubrica-tion

Hydrodynamic lubrication generally occurs in conformal contacts through apositive pressure build-up in the fluid film due to a converging gap, wherefluid is dragged in and pressurized. The pressurized film creates the possibilityto apply a load which is carried by the fluid film, a technique that is utilizedin journal and thrust bearings. The pressure range of this type of bearingsis usually between 1 and 5 MPa, and is thus not generally enough to causesignificant elastic deformation of the surface materials.

Non-conformal contacts are often associated with elastohydrodynamic lu-brication where elastic deformation of the surfaces becomes significant. Themaximum pressure in machine components operating in EHL is typically be-tween 0.5 and 4 GPa, and the minimum film thickness is normallyless than 1


µm [3]. The reason that machine components operating in EHL can manageto carry the load in such a small lubricated area without suffering catastrophicwear is mainly contributed to two effects. Many lubricants express a nearexponential increase in viscosity with pressure that increases load carrying ca-pacity and keeps the lubricant from flowing out of the contact. For organicliquids the viscosity is roughly doubled for every increaseof 0.05 GPa in pres-sure [6]. Secondly, the elastic deformation in EHL machine components isusually several orders of magnitude greater than the minimum film thickness.This elastic deformation leads to an increased contact area, and creates a gapfor the lubricant to pass through.

2.3 The EHD contact

In Fig. 2.2 a principal (piezo-viscous elastic) EHD contactis depicted. In thiscase two cylinders are loaded against each other in the presence of a lubricantforming a flat and narrow contact. The actual film pressure is marked in blue incomparison to the Hertz analytical dry contact pressure in black. The pressuresare similar except in the outlet zone where the actual film pressure has a spikein connection to the restriction in the outlet of the contact. Since the pressureprofiles are so similar the Hertzian theory is often used to calculate pressuredistributions in EHD contacts.

The flow profiles at three positions of the contact is also shown in thebottom of Fig. 2.2. The flow has two components, Couette flow, or surfacedriven flow, and Poiseuille flow, or pressure driven flow. The Poiseuille flow issignificantly larger near the inlet and the outlet of the contact due to the highpressure gradients. The Poiseuille flow is also counteracting the Couette flowin the inlet of the contact while being in the same direction in the outlet. Due tocontinuity requirements, the total flow must be the same at all positions in thecontact. To keep flow continuity, a closing gap and an abrupt rise in pressureis a must to counteract the increase in flow that would otherwise occur in theoutlet with the steep pressure gradient [3]. As a consequence, the entrainedamount of lubricant controls the film thickness both in the central part of thecontact, and in the closing gap.

U1 andU2 are the surface velocities of the two bodies. It is this motionthat drags, or entrains the lubricant into the contact, creating the surface drivenflow, and is thus one of the most important parameters for the film thicknessinside the contact. The combined motion that entrains the lubricant into thecontact is usually defined as the entrainment speed:

2.3. The EHD contact 39

Figure 2.2: Principal sketch of EHD contact.

Ue =U1+U2

2(2.1)

The film thickness is in most cases defined in the centre of the contact,hc andin the constriction where the minimum film thickness occurs,hmin. Severalsimple equations for calculations of film thickness in EHD contacts have beenproposed during the years of research on EHL. One of the most famous isthe Hamrock and Dowson equation for elliptical contacts. For minimum filmthickness, this equation can be written as:

Hmin = 3.63R0.47x U0.68

e η0.490 E′−0.12α0.49w−0.073(1−e−0,068k) (2.2)

whereRx is the effective radius,Ue the entrainment speed,η0 the viscosity,α the pressure-viscosity coefficient,w the load andk the ellipticity parameter.While this equation will only give an approximation of the film thickness undercertain fairly limited conditions, the equations gives some clues about which


parameters that are most important for the film thickness. Entrainment speedand viscosity are by far the most influential parameters on film thickness, whilethe load has much less influence. Since temperature has a great impact onlubricant viscosity it will thus have a large influence on thefilm thickness.The pressure-viscosity coefficientα does also have a big influence on filmthickness, and will be discussed in more detail in chapter 2.5.3.

If the surface velocitiesU1 andU2 are the same, the system is said to bein a state of pure rolling. In this state, only the Poiseuilleflow will causeshear forces to act on the lubricant, and the generated friction forces will bevery small. However, as soon as there is a difference in velocity between thesurfaces, additional shear forces will occur which may giverise to substantiallyhigher friction. The amount of rolling to sliding in an EHD contact is usuallydescribed as the slide to roll ratio (SRR), commonly defined as:

SRR=U1−U2

Ue(2.3)

where the output is a number between 0 and 2. Sometimes, sliding is also ex-pressed as slip, and in this case, 200% of slip equals 2.0 in slide to roll ratio.In the case of SRR = 2.0 or 200% slip, one of the surfaces will bestation-ary and there will be a pure sliding condition. Note that the film thicknessformula above, eq. 2.2, does not take slide to roll ratio intoaccount whileit certainly has an effect on film thickness due to shear heating, and possiblyalso shear thinning of the lubricant. Correction formulas for shear heating andshear thinning are discussed in section 2.4.1 and shear thinning is disussed inmore detail in section 2.5.5.

The viscous friction generated in an EHD contact is governedby the ef-fective viscosity in the Hertz zone of the contact. The viscosity is stronglyrelated to pressure, temperature and shear rate in a rather complicated manner.Lubricant rheology is therefore of utmost importance for EHD friction, and isdiscussed in section 2.5.

Another important aspect of the EHD contact is the relative independencebetween the inlet zone, and the central region of the contactthat would be incontact if the liquid was not present, usually called the Hertz zone. The con-ditions of the lubricant in the inlet zone, the region of the conjunction beforethe Hertz zone, is mostly determining the film thickness. In the inlet zone, thepressure is slowly increasing from atmospheric to approximately 150 MPa [7],and the film thickness is governed by the viscosity of the lubricant over thispressure range, which will determine the quantity of lubricant entrained. Aswill be discussed in more detail in section 2.5.5 the relatively low strain rate in

2.4. Lubrication regimes 41

the inlet will lead to a Newtonian behaviour of most base fluids. Once the lu-bricant is entrained, the film is so thin that it is forced through the contact withnegligible side-flow, and the properties of the lubricant inthis high pressurestate has no effect on film thickness.

In the Hertz zone, the friction forces are transferred across the film, anddependent on the high pressure properties of the lubricant and the shear forcesinduced by the amount of sliding in the contact. The high pressure in combina-tion with sliding gives rise to higher strain rates, and non-Newtonian behaviourof the fluid.

For instance, the decoupling of the inlet zone, and the Hertzzone meansthat viscous heating inside of the Hertz zone will influence friction, but noteffect the film thickness unless the temperature is also increased in the inletzone.

2.4 Lubrication regimes

Ideally, both hydrodynamic and elastohydrodynamic systems are running witha fluid film that is so thick that there is no contact at all between the solids ofthe mating surfaces. A system operating under these conditions experiencesvirtually no wear at all, and the friction coefficients are generally low, andonly attributed to shearing of the lubricant. However, if lubricated machinecomponents are running at too low speeds or having too high contact pressure,the lubricant film will be penetrated by the asperities of thecontacting bodies.Surface finishing plays an important role here since smoother surfaces willallow for higher loads or lower speeds without the asperities breaking the fluidfilm. A common approach is to divide a lubricated system into three regimes;boundary lubrication, mixed lubrication and full film lubrication.

1. Boundary lubrication:Boundary lubrication is characterized by asperityinteractions carrying all the contact load. In this regime the effect oflubricant bulk properties is almost negligible. The contact friction andwear performance are governed by physical and chemical properties ofthin lubricating films formed on the surfaces in combinationwith theproperties of the bulk material or surface coatings. Lubricants usuallycontain additives that are engineered to react with the surface physicallyand/or chemically to produce boundary films to reduce friction and wearin a situation of fluid film breakdown.

2. Mixed lubrication: In mixed lubrication, or partial (elasto) hydrody-namic lubrication, the contact load is carried by both asperity interac-


tion, and hydrodynamic or elastohydrodynamic effects. Depending onrunning conditions the coefficient of friction in the mixed lubricationregime can vary over a wide span depending on how much of the loadthat is carried by asperity interactions and hydrodynamic action respec-tively. Since there are varying degrees of asperity interaction, chemicaleffects of the lubricant and properties of the bulk materialor coating arestill important.

3. Full film lubrication: When there is no asperity interaction between thesurfaces, the full film regime has been entered, and the contact load istotally carried by hydrodynamic effects. Friction in this regime is mainlygoverned by lubricant properties, but also on the thermal properties ofthe solids and/or surface coatings, as will be discussed in Chapter 6. Thefull film lubrication regime is the main area of interest in this thesis.

2.4.1 The film parameter

Historically, a common way to judge which lubrication regime a system isrunning in is by use of the film parameter,Λ f . This dimensionless parameteris the ratio of the film thickness to the surface roughness.

Λ f =hmin

√

S2q1+S2

q2

(2.4)

wherehmin is the minimum film thickness, andSq is the composite surfaceroughness. A general rule of thumb is that whenΛ f < 1 the system is run-ning in boundary lubrication, and 1< λ f < 3 is the mixed lubrication regime,whereasΛ f > 3 is the full film regime [8].

Furthermore, another use of the film parameter can be found ina paperfrom 1982, where Bair and Winer [9] present experimental evidence of thepresence of three regimes of traction in a concentrated contact. Tests were con-ducted in a rolling contact simulator by varying lubricant,temperature, rollingspeed, load and surface roughness while holding a constant slide to roll ratio.They proposed that whenΛ f < 1.0 the traction coefficient is mainly attributedto the shear properties of the lubricant adsorbed on the mating surfaces. Theasperity interactions are quite severe at these low lambda values, and thus fitthe classification of boundary lubrication. Furthermore, if 1.0< Λ f < 10, Bairand Winer propose that the film thickness and the combined roughness are ofcomparable magnitude, the traction will be determined by the bulk propertiesof the lubricant, and by the shear properties of the lubricant adsorbed on the


surfaces in local asperity contacts. This is an example of mixed lubricationor micro-EHL. Finally, whenΛ f > 10, the film thickness is greater than thesurface roughness, the traction coefficient will be governed only by the bulkrheological properties and running conditions.

In addition to the somewhat confusing use of different lambda parametersin the literature, there are some limitations in the film parameter theory as awhole. Firstly, the film parameter depends on the correct calculation of thefilm thickness, and most quick methods to calculate film thickness, for exam-ple the empirical expression for isothermal film thickness derived by Hamrockand Dowson [3] only gives an approximation of the film thickness. The ac-curacy of the prediction depends on, among other things, running conditionsand lubricant properties. For instance, in case of higher amounts of sliding,thermal heating is influencing the lubricant, which becomesthinner than theisothermal prediction. To compensate for this effect, there are thermal cor-rection factors proposed, by among others Gupta [10] and Hsuand Lee [11],that multiplied with the isothermal film thickness gives thethermally correctedfilm thickness. Furthermore, shear thinning, that is discussed in section 2.5.5may also take place in the inlet of the contact, even in cases of low sliding,thus reducing the film thickness. Several authors have proposed correctionfactors for shear thinning [12–16]. However, all of these correction factors areapproximations, only valid within certain boundaries.

Secondly, another source of error is the surface roughness measurements.When a surface is measured with an optical profilometer or a stylus profilome-ter, the results may not perfectly reproduce the original surface depending onthe software, limitations of the optics, and size of the stylus. Therefore, theresults given by one profilometer can differ quite significantly from what youget from another profilometer measuring the exact same sample [17].

2.4.2 Amplitude reduction

Another objection against the film parameter is the fact thatmany modernelastohydrodynamically lubricated machine components are operating with alambda ratio less than 1 without showing any signs of problems [18]. At firstit seems strange that machines operating without problems have film thick-nesses lower than the composite roughness. The explanationis that the actualroughness inside the EHD contact is lower than the roughnessmeasured out-side of the contact, and therefore full film lubrication can be maintained evenif the lambda ratio is less than 1. The surface roughness is flattened insidethe contact due to surface deformation and flow of the pressurized and there-


fore highly viscous lubricant. Morales-Espejel and Greenwood [19] showed inan analytical investigation that film formation in line contacts with transversalroughness can be attributed to two effects. The first effect is dominating inrolling contacts where the asperities are largely deformedwhen they enter theEHD contact region. The other effect becomes more importantwhen sliding isintroduced in the contact. The film formation is dependent onthe entrainmentof lubricant in the contact, which means the mean velocity ofthe surfaces. Incase of sliding, the surfaces will move with different velocities, and due tothe roughness of the surfaces, the entrainment of lubricantwill be fluctuating.When a valley enters the contact, much more lubricant is entrained comparedto when an asperity enters the contact, and the result is a fluctuating entrain-ment that causes a film thickness variation moving with the mean velocity. Asa consequence, the asperities move with a different speed than the film thick-ness fluctuations it causes, a fact that makes rough surface EHL analyses morecomplex requiring transient solutions of the coupled EHD equations.

The effect of asperity flattening in rough surface EHL has been studied byLubrecht and Venner [20–22]. They studied the behaviour of asimple casewith a sinusoidal waviness represented by two parameters, the wavelengthλx,and an undeformed amplitudeAi. Ad represents the deformed amplitude. Forrolling/sliding line contacts the expression is:

Ad

Ai=

1

1+0.125∇1+0.04∇12 (2.5)

where∇1 = ∇1/

√

Sn = (λx/b)(M3/41 /L1/2)/

√

Sn (2.6)

and∇1 is the dimensionless (one dimensional) wavelength parameter, Sn theslip number,λx the waviness wavelength,b the half-width of Hertzian con-tact,M1 the dimensionless materials parameter andL the dimensionless loadparameter.

The slip number is defined as the ratio between the velocity ofthe roughsurface and the mean entrainment velocity:

Sn =Ur/Ue (2.7)

This implies that one of the surfaces is expected to be smooth, whereas theother one is rough. Figure 2.3 shows the ratioAd/Ai versus∇1. One conclu-sion is that short wavelengths are hardly deformed, while long wavelengths arealmost completely deformed. However, what are short and long wavelengths,and thus also the degree of deformation is also dependent on operating condi-


tions. If the roughness of a real surface is dominated by one wavelength thismethod could be used to calculate the in contact roughness and thus be used tocalculate a more accurate film parameter.

In essence, this means that it is difficult to accurately predict the transi-tion from full film to mixed lubrication due to several factors: film thicknesscalculations are approximative, as are surface roughness measurements, andthe topography inside of the contact are compressed differently depending onroughness wavelength and operating conditions. For this reason, many exper-iments have been conducted with resistive techniques to detect the transitionfrom mixed to full film lubrication [23–25]. The equipment needed for the re-sistive measurements is not fitted to all test rigs and may be difficult to installfor different machines, but offers a more precise indication than calculations.

Figure 2.3: Ratio between the amplitudes of deformed and initial roughnessversus wavelength parameter [22].

2.4.3 The Stribeck curve

A popular way of showing the regimes of lubrication is the Stribeck curve, firstpresented by Stribeck in 1902 [26]. The y-axis is the coefficient of friction andthe x-axis is a dimensionless number, often referred to as the Hersey numbergiven by:


H =ηωp

(2.8)

whereη is the absolute viscosity,ω the rotational speed andp the pressure.

Figure 2.4: Stribeck curve, the effect on Hersey number (ηω/p) on coefficientof friction.

Figure 2.4 shows a typical Stribeck curve with the differentregimes oflubrication. Generally a small Hersey number indicates a thin lubricant film,and consequently a high Hersey number indicates a thick lubricant film. Forthat reason, it is possible to find many different Stribeck like curves in liter-ature with a variety of parameters on the x-axis, such as entrainment speed,rotational speed and film thickness. The smallest Hersey numbers representthe boundary lubrication regime, usually representing a coefficient of frictionof about 0.1. This is just a general value, and depending on the materials ofthe mating surfaces and the properties of the lubricant thisvalue can be bothhigher and lower. When the Hersey number increases a rapid decrease in fric-tion coefficient can be observed when a transition is made from the boundary

2.5. Lubricant rheology 47

lubrication regime to the mixed lubrication regime. The explanation for thisrapid decrease in friction coefficient is that a larger part of the load is carried byhydrodynamic action with increasing film thickness. The lowest value of thefriction coefficient usually marks the transition from mixed to full film lubri-cation where the fluid film is just thick enough to avoid asperity collisions. Aminor increase in friction coefficient with respect to increased Hersey numberin the full film region is usually found in the Stribeck curves. This effect is at-tributed to increased viscous losses in certain bearing types, especially journalbearings which was presented in the work of Stribeck. For machine compo-nents operating in EHL the curve may look a bit different witha coefficientof friction not increasing in the full film regime due to higher pressures anddifferent thermal conditions.

2.5 Lubricant rheology

This section is written as an introduction to the rheological phenomena actingas a prerequisite for the function of machine components working in EHL,and secondly governs viscous friction generation. The section covers both theprinciples of rheology, and some examples of how this has been modeled overthe years. This section should not be seen as a complete guideof rheology, andfor a more thorough review of high pressure rheology the reader is referredto other sources such as Bair’s book on high pressure rheology [6] and laterreview papers [27, 28]. One should keep in mind when reading this section,that friction in an EHD contact is mainly governed by the viscosity in theHertz zone of the contact. The viscosity is strongly dependent on pressure,temperature and shear rate. The understanding of rheology,and the use ofproper models to model rheological behaviour is therefore crucial for accuratepredictions of EHD friction.

2.5.1 Lubricant transport properties

Transport processes concerns the transport of mass, energyor momentum fromone region of a material to another under the influence of composition, tem-perature and/or velocity gradients [29]. If we consider a material volume iso-lated from its surroundings where chemical composition, temperature or veloc-ity vary within the volume, transport processes acts to render these quantitiesuniform throughout the material. The non-uniform state required to generatethese transport processes causes them to be known as non-equilibrium pro-cesses. They do not reflect what happens at equilibrium, but the rate at which


equilibrium is reached. For any given gradient of a quantitysuch as temper-ature or velocity, the equilibrium after isolation from thesurroundings occursthrough transport processes and its rate depends on the properties of the ma-terial known as transport properties. There are several transport properties,but the most important ones are viscosity, thermal conductivity and diffusivity,that are connected to transport of momentum, energy and massrespectively.As will be described throughout the following sections, especially viscosity,but also the thermal conductivity of lubricants are very important for the func-tion and performance in EHL. Diffusivity has a minor influence in EHL andwill therefore not be discussed at all.

2.5.2 Temperature-Viscosity relationship

The fact that viscosity tends to decrease with increasing temperature has beenobserved first-hand by most people, for instance when cooking oils tend tofloat more freely at higher temperatures or that the effort toshift gears witha manual transmission is increased at lower temperature. The viscosity is ofhigh importance for both friction and film thickness in EHD applications. Theviscosity index (VI), is a measure of how much the viscosity of a particularlubricant is changed with temperature. A high VI means a small change inviscosity with temperature, while a low VI means a large change in viscositywith temperature. It is generally desirable to have a lubricant with a high VI.The temperature-viscosity coefficient,β, is a measure of how much viscositychanges with temperature and is dependent on both temperature and pressure.In general,β will increase with pressure, with the exception of low pressureand high temperature [6]. A common model for the temperature-viscosity re-lationship in the field of EHL is the Vogel, Tammann and Fulcher (VTF) equa-tion [30]:

µ= µ∞expDFT∞

T −T∞(2.9)

whereµ is the viscosity,µ∞ is the viscosity extrapolated to infinite tempera-ture,DF the fragility parameter andT∞ the Vogel temperature, the temperatureat which the viscosity diverges. Some of these parameters are discussed inmore detail in section 2.5.8. The temperature-viscosity coefficient for the VTFequation is then given by:

β =DFT∞

(T −T∞)2 (2.10)


2.5.3 Pressure-Viscosity relationship

Many fluids exhibits piezo-viscous properties, which meansthat the viscosityis coupled to the pressure the fluid is subjected to. Without the increase inviscosity with pressure many machine components like gears, roller bearingsand cam followers would not work properly since the lubricant film wouldnot be able to partly or totally carry the load. It is found that the increase inviscosity with pressure of the lubricant is nearly exponential. However, notall liquids possess this property. While most organic liquids doubles in vis-cosity for every 0.05 GPa increase in pressure, liquid metals have much lesssensitivity to pressure and doubles about every 3 GPa. For other liquids thatexpands on freezing, such as water and gallium, the viscosity could actuallyreduce with pressure for conditions close to the melting temperature [6]. Thepressure-viscosity relationship is of great importance for EHL and has there-fore been the topic of many authors. In 1882-1893 Barus published a seriesof papers [31] where he reported a viscosity increase with pressure of solidmarine glue that he described with a linear model:

η = η0(1+αp) (2.11)

whereη0 is the absolute viscosity,α the pressure-viscosity coefficient andpthe pressure. In 1916 Hersey [32] used Barus linear equationto describe theincrease of viscosity for two different oils up to 20 MPa. However, in con-secutive measurements up to 50 MPa he found a faster then linear increase inviscosity.

In 1926, Bridgman [33] was able to perform measurements of viscosityup to 1.2 GPa of pressure and could observe a close to exponential behaviour.The exponential form of the pressure-viscosity relationship below has beenused extensively in the literature discussing EHL and is often referred to as theBarus equation, even though Barus as explained above proposed a linear modelin his work. It is not clear when the exponential model was first proposed:

η = η0eαp (2.12)

Another commonly used relationship for the pressure-viscosity relationshipwere proposed by Roelands. Between 1963 to 1966 he presentedthree differ-ent models [34, 35]. It is mainly the third model that has beenused in EHLliterature. This model for isothermal cases can be expressed as:


Table 2.1: Lubricant properties.

Synthetic SuperrefinedFluid designation paraffinic naphthenicAbsolute viscosity @ p=0, 38◦C, η0 [Pas] 4.14 0.0681Pressure viscosity coefficient @ 38◦C, α 1.77e−8 m2/N 2.51e−8 m2/NDimensionless viscosity pressure index @ 38◦C 0.43 0.67

η = η0

(

η∞

η0

)1−(1+p/cp)Z1

(2.13)

whereZ1 is the viscosity-pressure index, and constantsη∞ = -6.3110−5 Pa sandcp = 0.1961 GPa. The pressure-viscosity indexZ1 needs to be obtainedfor each specific lubricant whereasη∞ and cp are general parameters foundby Roelands to generate reasonably accurate results. It wasalso shown byRoelands that for most fluids,Z1 is usually independent of temperature up toabout 100 degrees C. Figure 2.5 show pressure-viscosity responses for twodifferent lubricants expressed with both the pure exponential and Roelandsformulas. The values used as input for the calculation were obtained by Joneset al. [36] and presented in Table 2.1. Blok [37] found that it is possible torelate the pressure-viscosity coefficient to pressure-viscosity index by the useof the following equation:

Z1 =α

(1/cp)(lnη0− lnη∞)(2.14)

While it is possible to find specific lubricants where the pureexponentialmodel or Roelands model are able to predict accurate resultseven at pressuresabove 1.0 GPa this is generally not the case. Roelands statedhimself that theupper limit of his model is roughly 0.5 GPa, that has more recently been dis-cussed in a paper by Bair [38]. As a consequence, these modelsmay be usefulfor predicting the pressure-viscosity behaviour at lower pressure, for instancein the inlet zone of an EHD contact where the film thickness is mainly gov-erned. However, these models are probably not suitable for accurate predictionof the pressure-viscosity behaviour at the high pressure central regions of theEHD contact where friction is governed. Several other attempts can be foundin literature to describe the pressure-viscosity relationship at higher pressures.A modified two slope model was utilized by Cioc [39] to avoid overestimationof the viscosity under heavy loads (high pressures), which was influenced bythe work of Allen [40] in 1973.


0 2 4 6 8 10 12 14

x 108

10−2

100

102

104

106

108

1010

1012

1014

Pressure [Pa]

Abs

olut

e vi

scos

ity [P

as]

Oil 1 − BarusOil 1 − RoelandsOil 2 − BarusOil 2 − Roelands

Figure 2.5: Pressure viscosity response for two different lubricants obtainedfrom Barus and Roelands formulas.

Several attempts have been made to derive empirical based models for thepressure-viscosity relationship that is generally valid for pressures up to themaximum Hertzian pressures often found in EHD contacts. Irving and Bar-low [41], and Bair and Kottke [42] proposed a four respectivefive parameterempirical model applicable to more realistic EHD pressures. Except requiringmore parameters than equations 2.12 and 2.13, it is not generally clear howto accurately implement the temperature dependence of viscosity into theseformulations [6].

Another alternative to this are models building on the free volume theoryof viscous flow. The free volume model was used for the pressure-viscosityrelationship at almost the same time as Roelands proposed his model [43] andis most likely the most widely used model outside the field of EHL [44, 45].The basic equation of the free volume model was proposed by Doolittle [46] inan attempt to correlate viscosity of alkanes with temperature at ambient pres-sure. A few years later, Williamset al. [47] realized that the Doolittle equationwith its abrupt increase in viscosity when the free volume becomes small coulddescribe the behaviour of lubricants at high viscosities, near the glass transi-tion. Cohen and Turnbull [48] gave the free volume concept physical meaning


with their hypothesis that molecular transport occurs whena fluctuation in thefree volume opens up a void greater than the critical size to allow movementof a molecule into that void. Since the pure Doolittle equation is a volumet-ric relationship on viscosity it must be used with an appropriate equation ofstate such as Tait or Murnaghan and/or a model for the temperature-viscosityrelationship to yield results relevant for EHD contacts. Yasutomiet al. [49]proposed a correlation derived from the Doolittle free volume equation thatdoes not need an equation of state, based on a pressure modified version of theWilliam-Landel-Ferry (WLF) equation [47]. Very recently an improvementhas been proposed with a modified Yasutomi-WLF model that better handlesfluids that present an inflection, and gives a more accurate representation ofviscosity at low pressures [50]. Ref [51] contains Yasutomi-WLF parametersfor a range of different lubricants.

It has been shown in non-equilibrium molecular dynamics simulations[52, 53] that the repulsive intermolecular potential dominates the transport ofmomentum and that if a power law behaviourr−3g is assumed for repulsivepotential, then viscosity must scale asTVg [54]. Whereg is a material con-stant or a thermodynamic interaction parameter. This parameter is in principlereflecting the magnitude of the intermolecular forces and thusg will decreasegoing from van der Waals fluids to ionic liquids. This Ashurst-Hoover scal-ing rule was more recently validated for several liquids [54,55]. A convenientscaling parameter can be written as [56]:

ϕ =

(

TTR

)(

VVR

)g

(2.15)

whereϕ is the dimensionless viscosity scaling parameter,TR a reference tem-perature andVR a reference volume defined atT = TR and in the limit ofp =0. It is now possible to find a value forg where viscosity data for several dif-ferent temperatures plotted against the scaling parameterwill fall onto a singlemaster curve. An accurate scaling function can be obtained from a Vogel likeform [56]:

µ= µ∞exp

(

BFϕ∞

ϕ−ϕ∞

)

(2.16)

whereϕ∞ is the viscosity scaling parameter for unbounded viscosityandBF isthe fragility parameter. It has been shown that this scalingrule is more accu-rate than the free volume model, without needing more parameters [56]. Veryrecently an attempt to relate the normalization ofϕ back to the molecular levelby using molecular volumesVm from theoretical models as scaling parame-


ters rather thanVR was made [57]. The new formulation with the variableβV

includes both thermodynamic and molecular scaling:

βV =

(

1T

)(

Vm

V

)g

(2.17)

whereVm is the specific volume of the molecules without the space betweenthem.

2.5.4 Equation of state

While liquids in many cases can be treated as incompressiblefor calculationsin a variety of engineering applications, this should not beassumed for EHL,especially due to the high pressures found in these kinds of contacts. With theassumption of an incompressible liquid, the increase in viscosity with pres-sure would be lost. A liquid pressurized to 1 GPa at 0◦C may be relativelycompressed by 20% where at the same pressure, the same liquidcould becompressed by 30% at 200◦C. Furthermore, the volume is compressed at anon-linear rate. The first half of the total compression at 1 GPa is reachedat a pressure of 1/3 GPa for a temperature of 0◦C and already at 1/4 GPa at200◦C [6]. The effect of compressibility of the lubricant have been shown togenerally only have minor effects on minimum film thickness [58–60], whilethe effect on central film thickness is significant [60, 61]. However, the ef-fect of compressibility on central film thickness is still minor in comparisonto the effect on film thickness by entrainment speed or viscosity. The centralfilm thickness is approximately reduced by the amount of the relative compres-sion [61].

To include compressibility in EHD calculations an equationof state (EOS)is necessary. This equation relates the volume (or density)to temperature andpressure. One of the most commonly used EOS for the EHD problem withcompressible lubricants is the Dowson and Higginson isothermal equation ofstate [62]. Their equation was obtained from curve fitting data from a mineraloil at one temperature up to about 350 MPa. The increase of density withpressure will rapidly level off with increasing pressure, and reach a limit sothat at sufficiently high pressure, the liquid will in principle be incompressibleagain. For these reasons, and the fact that it is isothermal,this equation may notbe the most suitable for high pressure EHD, as demonstrated by comparisonswith high pressure measurements in a publication by Bair [28]. Several otherequations of state has been proposed, among others by Jacobson and Vinet [63]and Ramesh [64].


The most used EOS in high pressure physics is the isothermal Tait equa-tion, often considered to be the most accurate for extrapolation to high pres-sures [65,66]. The Tait equation written for the volume relative to the volumeat ambient pressure:

VV0

= 1−1

1+K′

0

ln

[

1+p

K0(1+K

′

0)

]

(2.18)

whereV is the volume,V0 is the volume atp = 0, K0 the bulk modulus,K, at p= 0 andK

′

0 is the pressure rate of change ofK at p = 0 . The bulk modulus forthe Tait EOS depends on pressure as:

K =

{

1−1

1+K′

0

ln

[

1+p

K0(1+K

′

0)

]}

[

K0+ p(1+K′

0)]

(2.19)

The Tait EOS has been used up to 3 GPa of pressure with good accuracy[67].

Another isothermal EOS with slightly lower compressibility at high pres-sures is the Murnaghan equation [68], derived from a linear theory of finitestrain [69].

For an EOS to be most useful for EHD calculations it must include tem-perature, at least for a limited, but relevant range of temperatures. As mosttemperature modifications has been done to the Tait EOS, the descriptions herewill be limited to that equation. It is generally found thatK

′

0 is independentof temperature [6, 69] and has therefore been assigned an universal value,K

′

0= 10.2 [70]. However, it should be noted that the universal value were basedon measurements performed by Cutleret al. [70] on a variety of hydrocarbonliquids, and they found values as low as 9 for light hydrocarbons, and up to 12for high weight lubricants.

SinceK′

0 is considered independent of temperature, the effect of tempera-ture on compressibility for the Tait EOS is therefore entirely governed by thetemperature dependence ofK0. There have been several forms proposed bydifferent authors [70–72], where the latter, Fakhreddine and Zoller proposed:

K0 = K00exp(−βKT) (2.20)

whereK00 andβK are approximately 9 GPa and 0.006K−1 respectively. Thisequation seem to work well for all temperatures [6]. Finally, a relationship


between the ambient pressure volume and temperature is needed:

V0

VR= 1+av(T −TR) (2.21)

whereav is the thermal expansivity defined for volume linear with temperature.Equations 2.18, 2.20 and 2.21 now gives a complete EOS for thecompressibil-ity of lubricants including temperature dependence.

2.5.5 Non-Newtonian behaviour

In a Newtonian fluid, the relationship between shear stress,τ and shear strainrate,γ, hereafter only called shear rate, is linear:

τ = ηγ (2.22)

However, EHD contacts often experience transient conditions with high pres-sure variations and high shear rates. During these conditions it is questionablethat the lubricant in general acts in a Newtonian way. One indication of thisis the non-linear friction response seen in experimental testing when friction isplotted against SRR, as can be seen in Fig. 4.2. A generalizedexample withspecified friction regimes is given in Fig. 2.6.

These figures indicate that the lubricant shear stress is still related to theshear rate, but not always following a linear relationship.At low slide to rollratios (or shear stress), friction is increasing linearly with SRR indicating thatfriction is governed by the Newtonian behaviour of the lubricant. At some rateof shear, friction is starting to increase in a non-linear manner and will at a cer-tain point reach a limiting value, or stress. In connection to this limiting value,friction asymptotically reaches a plateau showing little variation with shearrate before finally starting to decrease with increasing shear rate. The limitingstress will be discussed in more detail in section 2.5.6 while this section willcover the non-linear and thermoviscous part of the frictioncurve.

It should be mentioned that at very high pressures and reallylow SRRs,the friction behaviour is not governed by the linear Newtonian behaviour, butby another linear effect. It has been found in experimental investigations thatat really low SRRs in a high pressure contact between two rollers, the frictionfollowed the same linear slope both with and without the lubricant present.This phenomena has been attributed to the elastic creep of the rollers [73].

The non-linear decrease in viscosity with shear rate is usually called shearthinning, and is one of the main reasons for the behaviour in the non-linearfriction regime shown in Fig. 2.6. Shear thinning means thatthe viscosity


Slide to roll ratio

Fric

tio

n

Linear

Non-LinearPlateau

Thermoviscous

Figure 2.6: Friction response and regimes with varying SRR.


µ2

µ1

Shear stress

Vis

cosity

1st Newtonian plateau Transition 2nd Newtonian plateau

Figure 2.7: Typical behaviour of shear thinning lubricant [76].

will decrease with an increase in shear stress. Shear thinning is constitutivebehaviour, which means that the apparent viscosity measured at a particularshear rate has a specific value independent of history. For example, if theviscosity is measured following a measurement at higher shear rates the resultswill be the same as if the measurement were done after a measurement at alower shear rate. Figure 2.7 shows a flow curve for a shear thinning fluid.Typically the fluid is behaving in a Newtonian manner with a viscosity,µ1,independent on shear rate. At a certain shear stress, the lubricants viscositystarts to reduce with increased shear stress. Some lubricants enters a secondNewtonian,µ2 at really high shear rates [16, 74, 75]. It is quite unusual toobserve the first and second Newtonian plateaus in a single experimental flowcurve, and often only an inflection in the curve is seen opposed to the secondplateau.

Shear thinning can be said to occur for a specific liquid when the Weis-senberg number becomes greater than unity:

Wi = λγ (2.23)

whereγ is the shear rate andλ a relaxation or characteristic time. It has beenshown with non-equilibrium molecular dynamics simulations [77] that this re-


laxation timeλ, is close to the rotational relaxation time for a molecule. TheEinstein-Debye relation [78] for the rotational relaxation time of a moleculeis:

λEB =µVm

kBT(2.24)

whereVm is a molecular volume andkB is the Boltzmann constant. Assumingthat the molecular volume is given by the total liquid volumeper moleculegives:

Vm =MKb

ρRg(2.25)

whereM is the molecular weight,ρ is the mass density andRg is the universalgas constant. Combining eq. 2.24 and eq. 2.25 gives:

λEB =µM

ρRgT(2.26)

This means that the molecular structure of a liquid is strongly connected tothe tendency for shear thinning at a particular shear stress. Low molecularweight liquids are much less likely to shear thin, especially in the inlet of thecontact, while high molecular weight liquids may shear thinat the inlet evenfor contacts in pure rolling [79–82]. A molecular weight of around 2000 g/molhas been reported as an approximate limit for inlet shear thinning [83].

This poses an interesting problem in multi grade oils where VI improversare used to reduce the effect of temperature on viscosity. The VI improversare usually high molecular weight polymers that will lead toshear thinning atmuch lower shear stress than would be the case for the pure base oil. Shearthinning in the inlet region may lead to substantially lowerfilm thickness thancalculated using the low shear viscosity.

There are several other events that may give similar effect on viscosityat increased shear rate, such as slip within the liquid, or atthe solid-liquidinterface, thermal softening, thixotropy, molecular degradation and cavitation.

A thixotropic material has a viscosity dependent on shear rate, but alsotime of shear. If such a material is subjected to shear stressthe viscosity isreduced to a certain degree. If the shear rate is then loweredto the originalvalue again, it takes a certain time for the viscosity to return to its originalstate. Depending on the applied shear rate, going back to theoriginal viscositymay take hours [84].

In the case of molecular degradation, a permanent loss of viscosity with


shear is caused by a chemical change in the material [85]. Usually the reduc-tion in viscosity is due to chemical bonds being broken by theshear forces,also leading to a lower molecular weight [86]. In this case the viscosity willnot go back to its original state, no matter the time without shear. However,recent research has shown that the permanent viscosity reduction caused byshear may only influence the viscosity at low shear stress, while the viscosityat high shear stress remains unchanged [87].

It has also been shown in experiments in a Couette viscometerwith dimethyl-silicone oils that cavitation occurs when the shear stress is larger than the pres-sure [88], thus resulting in a behaviour which is similar to shear thinning.

Thermal softening is a result from viscous heating of the lubricant, wherethe temperature increase will lower the viscosity of the lubricant, which in-evitably will happen at sufficiently high pressures and shear rates. It is onlyat very low shear rates that thermal softening can be ignoredin comparison toshear thinning effects. While this may not pose a problem in EHD contacts inmachine components, it could be problematic for measurements conducted toinvestigate the shear dependent viscosity of a lubricant. This also means thatthermal softening is not only present in the thermoviscous regime in Fig. 2.6.While being the dominant part for the friction behaviour in the thermoviscousregime, thermal softening is, in many cases also present, tosome extent, in theother regimes as well. In fact, this can be said to be true for all regimes in Fig.2.6. Each regime is dominated by either Newtonian, shear thinning, limitingshear stress or thermal softening behaviour, but many effects could be at playat the same time. Recently, Habchiet al. [89] published a paper where threedimensionless numbers were used to identify the friction regimes shown inFig. 2.6. This model also makes it possible to see which friction mechanisms,or regimes that are at play at the same time for a specific set ofconditions.

Models for shear thinning are discussed in section 2.5.7, but it is appropri-ate to introduce the concept of the limiting shear stress before going into thesemodels.

2.5.6 Limiting stress

As mentioned in section 2.5.5, it is often observed in traction curves a limitwhere the shear stress is no longer increasing with shear rate. It is commonlybelieved that this behaviour is due to a limiting shear stress, when the ma-terial/lubricant is assumed to deform plastically, a concept originating fromSmith in 1960 [90]. This behaviour is most likely the most important part offriction in EHL, but also one of the least understood. There is controversy


whether the limiting shear stress exist in the way of plasticyield, or if there isanother phenomena behind. In 1983 Evans and Johnson associated the limitingshear stress behaviour with shear bands [91] and it has been shown later thatlocalizations of deformation form bands where the shear stress is much higherthan in the bulk of the liquid that can be looked upon as slip planes [92, 93].As summarized by Bair: "The observation of shear bands operating at highshear stress suggests that the rate independent response inEHL traction andhigh pressure viscometer measurements is a result of intermittent slip alongbands inclined to the flow direction in a manner similar to theyield behaviourof some solid polymers" [6]. The shear bands do, maybe surprisingly so, notoccur in line with the flow direction. There are two types of shear bands, onethat is rotated approximately 20 degrees from the flow direction, and the otherone at an angle of approximately 15 degrees from the cross filmdirection. Theexplanation for these directions could be found in the Mohr-Coulomb failurecriterion [6]. The orientation of the bands causes an intersection of the shearband with a solid boundary which means that the shear bands cannot operatecontinuously. Therefore, bands form, slip, subsides, and another band forms.The limiting shear stress is often modeled as dependent on pressure as:

τL = Λp (2.27)

whereτL is the limiting shear stress andΛ is the limiting stress pressure coef-ficient. Λ can be calculated through the use of internal friction and the Mohr-Coulomb failure criterion [6] or as much more common, from EHD frictionmeasurements conducted at high pressures [73, 94, 95]. However, the lattermethod has recently been shown to be partly inaccurate sincethe plateau of-ten observed in high pressure friction curves, and used to deduce the limitingstress coefficient is not solely determined by the limiting shear stress, but alsoshear thinning at the same time [89]. For that reason, limiting shear stress ismost likely better experimentally assessed in a rheometer showing rate inde-pendent shear stress or limiting shear stress [96]. Furthermore, the assumptionthat limiting shear stress is independent of temperature isincorrect since it hasbeen shown that the limiting shear stress in general decreases with an increasein temperature [97–99]. The limiting shear stress problem is therefore still notcompletely resolved, and is one of the greatest challenges in the field of EHLin order to make accurate friction predictions.

In addition to the hypothesis of limiting shear stress, another theory thatcould explain the phenomena is wall slip, which is also knownas boundaryslip or solid-liquid slip. In this case, the assumption thatthe velocity of the


liquid at an infinitesimal distance from the solid boundary is equal to the ve-locity of the solid boundary is violated for shear stress above a certain value.Boundary slip has been experimentally observed [100–102],also at typicalEHD pressures using various lubricants and solid materials[103–106] wherethe authors propose that boundary slip could be an explanation for the limitingshear stress behaviour, or at least a complement to the limiting shear stress the-ory. However, all of these studies were made with very smoothsurfaces, below10 nm, and it has been shown and discussed by several authors that boundaryslip requires very smooth surfaces [101,107–109]. It is therefore questionableif boundary slip is involved in the rate-independent behaviour found in frictiontest with surfaces with engineering roughness of more than 10 nm. Solid-liquidslip is discussed in more detail in section 2.9.

2.5.7 Models for shear thinning

Several authors have addressed the non-linear behaviour ofviscosity with sheardescribed in section 2.5.5. One that has had a large impact onthe field of EHLwas Eyring, who 1936 applied his theory of thermally activated reaction ratesto viscous flow, plasticity and diffusion [110]. The result were a number ofempirical equations of which one was the sinh law:

µ=τe

γsinh

(

ττe

)

(2.28)

whereγ is the shear rate andτe is the so called "Eyring stress" which essentiallyis the stress level where a specific lubricant changes from Newtonian to nonNewtonian behaviour. Bair [84] summarized the molecular justification of thesinh law as: “Both diffusion and viscous shear flow were assumed to resultfrom a molecule overcoming a potential barrier to jump to an adjacent hole.For viscous shear, the frequency of jumps in one direction was assisted byshear stress, and jumps in the opposite direction were retarded by the stressaccording to an exponential relationship. The net frequency of the jumps inthe assisted direction was then the difference between exponentials with equalarguments of opposite sign, hence a hyperbolic sinh law for the shear rate as afunction of shear stress.”

However, Eyring himself explained in subsequent work [111]that he foundequation 2.28 not to be an accurate description of shear-thinning, but insteadof limited use for the description of thixotropy, as discussed shortly in section2.5.5, which he addressed in a separate paper [112]. Empirical tests performeda couple of years after Eyring presented equation 2.28 also showed that shear


thinning follows a power law rather than a sinh law [113]. In fact, it seems thatthat the sinh law in practice describes the effect of thermalsoftening ratherthan shear thinning [28]. As an answer to the criticism, Ree and Eyring [114,115] presented a modification to the sinh law. Here, they elaborated that themolecular motion was the result of jumps fromN individual heterogeneousflow units of the molecule, each occupyingχi areal fraction and each having arelaxation time,λi . The Ree-Eyring shear-thinning equation:

τ =N

∑i=1

χiτisinh−1(λi γ) (2.29)

whereτi = µ/γi . The inclusion of several Eyring stresses,τi made it possibleto use the Ree-Eyring equation to approximate power law behaviour and hasled to successful predictions of both film thickness and friction in EHL [116].However, Eyring explicitly excluded the case ofN=1 for shear thinning [111]and for some cases it requires up to five flow units and 10 parameters [117] toaccurately reproduce a power law behaviour for a specific fluid over a limitedrange of shear rate.

The molecular mechanics of the Eyring theory have been criticized for be-ing unsound [118, 119] and it was later shown experimentally[120] and bymolecular dynamics simulations [121] that shear thinning is not the effect ofmolecular jumps as described by Eyring, but by molecular alignment. Basi-cally, shear thinning is the onset of orientational order and the alignment of themolecules with the flow direction. The alignment angle for alkane for exam-ple, is typically 45 degrees at very low strain rates but becomes significantlyreduced at higher shear rates. Alkane molecule chains will now align to theflow direction and the dominant motion will be sliding of chains parallel to theflow.

In 1965 Tanner [122] presented a generalized Maxwell model for lubrica-tion:

τ+λmDτ

Dt= η(τ)γ (2.30)

where D/Dt is the convected time derivative andλm is a relaxation time [83].Equation 2.30 is however time dependent and not a generalized Newtonianmodel with the exception of theη(τ) expression. Tanner used a power-lawform for this expression:

η(τ) = K(τ)1−(1/n) (2.31)


whereK is the consistency andn is the power law exponent. Equation 2.31,known as the Ostwald-de Waele equation [123] can not describe the linearNewtonian behaviour for liquids at low shear rate (stress),but it is proven tobe realistic for larger values. Tanner’s model has been usedin several tractionstudies, and is usually attributed to a paper by Johnson and Tevaarwerk [124]in 1977. However, in their paper, the nonlinear viscous function is based onthe sinh law instead of a power law.

The power law behaviour described by the Ostwald de Waele equation isa limiting high shear behaviour, and has been used in numerous viscosity-stress functions. One of the most famous is the (double Newtonian) Carreauequation [125]:

η = µ2+µ−µ2

[1+(λγ)2](1−n)/2(2.32)

whereµ2 is the second Newtonian viscosity. The second Newtonian viscosityis sometimes seen as a plateau in the viscosity, measured as afunction of rateor stress. In most cases, the plateau is not fully developed in the measurementrange, but an inflection can be seen that to some extent definesthe secondNewtonian. In case that no second Newtonian is present, the Carreau equationis reduced to the single Newtonian form:

η =µ

[1+(λγ)2](1−n)/2(2.33)

A more general form of the Carreau equation is the Carreau-Yasuda form:

η = µ2+µ−µ2

[1+(λγ)a](1−n)/a(2.34)

wherea is the Yasuda parameter, and controls the breadth of the transition fromNewtonian to power-law behaviour. A small value ofa gives a wide transitionand indicates a wide distribution of molecular weight. Setting a = 2 givesthe original Carreau form, and is typical for monodisperse liquids. Settinga = 1− n yields the Cross equation [126]. For additional power law modelsthe reader is referred elsewhere [6]. Carreau parameters for numerous liquidsof different kinds can be found in [127].

At the present, the power law exponent,n can only be obtained throughexperiments or molecular dynamics simulations [128] sinceno theory exists toaccurately relaten to structure [83]. The characteristic time,λ may be writtenas a Maxwell relaxation time [28]:


Table 2.2: Parameters used in Fig. 2.8.

TR = 40 ◦ C ϕ∞ = 0.1743K

′

0 = 11.74 BF = 24.50av = 8.36×10−4K−1 µ∞ = 0.9506×10−4 Pa·sK00 = 8.658 GPa µR = 15.6 mPa·sβK = 6.332×10−3K−1 λR = 2.26×10−9 sg = 3.921 n = 0.463

λ =µG

(2.35)

whereG is a modulus associated with the longest relaxation time. Asdescribedin section 2.5.5 the longest relaxation time,λ, has been shown in molecular dy-namics simulations to be nearly the rotational relaxation time for the molecule:

λ =µM

ρRgT(2.36)

And hence:

G≈ρRgT

M(2.37)

The critical shear stress for the limit of Newtonian behaviour varies inverselywith molecular weight.

Figure 2.8 shows viscosity versus pressure for several different shear ratesfor Squalane at 40◦C. The viscosities were calculated using Tait EOS withequations: 2.18, 2.20 and 2.21. The pressure dependence wascalculated withthe thermodynamic scaling rule, equations: 2.15 and 2.16. For the shear de-pendence of viscosity a t-T-p shifted Carreau equation was used:

η(γ,T, p) = µ

[

1+

(

γλRµµR

TR

TVVR

)2](n−1)/2

(2.38)

Time-Temperature-Pressure superpositioning is discussed in section 2.5.9. Theparameters used in the equations are summarized in Table 2.2, obtained fromrefs [129,130].


0 0.5 1 1.5 2

x 109

10−2

100

102

104

106

108

Pressure [Pa]

Effe

ctiv

e V

isco

sity

[Pa

s]

Shear rate = 0Shear rate = 10Shear rate = 1000Shear rate = 1e6Shear rate = 1e8

Figure 2.8: Viscosity versus pressure for several different shear rates forSqualane @ 40◦C.

2.5.8 Glass transition

In sections 2.5.5 and 2.5.7 we have seen that a transition to shear thinning isa result from the molecule having insufficient time to attainits equilibriumorientation for an applied rate of shear. Another transition that occurs as aresult from the molecule being unable to respond to changes in its environmentis the glass transition. The glass transition for a liquid happens with eithercooling or compression to a point where the liquid enters a solid like statebelow its freezing point, a process called supercooling. The temperature atwhich this happens is the glass transition temperature,Tg. The glass transitiontemperature is strongly dependent on pressure, and a sufficiently high pressuremay induce a transition from liquid to glass without requiring a decrease intemperature [131]. The glassy state should not be confused with freezing orcrystallization which is a phase transition, whereas the glassy state is reachedwithout any pronounced change in material structure. It is widely believedthat a true equilibrium state is always crystalline, and therefore the glass is notat equilibrium. As a direct result of the non equilibrium state, the history ofthe glass forming has an influence on both the structure and properties of theglass, but also on the point of transition. For instance, by lowering the rate of


cooling, the glass transition temperature will decrease. In the same way, theglass transition pressure,Pg, is decreased when the rate of pressure change isincreased. If the pressure is increased slowly, the molecules of the lubricantwill have more time to rearrange and the glass transition will take place at ahigher pressure. Furthermore, a glass that is formed under pressure will havegreater density than a glass formed at ambient pressure, anda glass formed at ahigh rate of change in pressure or temperature will have a lower density than aglass formed with a lower rate of change [132]. This means that all propertiesthat depend upon density will be dependent on the conditionswhen the glasswas formed [6,133].

Because of the high pressures usually found in EHD contacts,the glasstransition temperature is increased to such extent that many lubricants will bein the glassy state within the EHD contact. This is especially the case for highpressure contacts with low SRR where a minimal amount of thermal soften-ing may increase the temperature of the lubricant above the glass transitiontemperature. In practice, the glass formation has no effecton film thicknesssince the film thickness is governed in the relatively low pressure inlet region,and most liquids will not reach the glass transition there. When the glass stateis present in the Hertz zone of the contact, friction is believed to be mainlygoverned by the limiting shear stress discussed in section 2.5.6.

Liquids that show a large change in properties as the glass transition isapproached from low viscosity are called strong liquids, opposed to fragileliquids that experience low changes of properties as the glass transition is ap-proached. The term fragility was made popular by Angell [30]to describe thedifference in behaviour of supercooled glass forming liquids. TheD f parame-ter in the VTF equation (2.9) is used to describe the fragility of a liquid. Thefragility of the liquid has been shown to be important for entrapment in EHDcontacts [134].

When a liquid is supercooled, the difference in entropy between the liquidand solid phase decreases. By extrapolation of the heat capacity of the super-cooled liquid below its glass transition temperature, the temperature of wherethe differences in entropy is zero could be found. This temperature is calledthe Kauzmann temperature, and it has been found [135] to correlate with theVogel temperature, also called the idealized glass transition temperature,T∞,also used in the VTF equation. The idealized glass transition temperature isusually 15-40◦C less than the glass transition temperature [6].


2.5.9 Time-Temperature-Pressure superposition

Flow curves of lubricants with true shear thinning behaviour, when plotted aseither log viscosity versus log shear rate (or stress), or log shear stress versuslog shear rate have an useful property. Any curve can be shifted vertically andhorizontally without rotation to superimpose over anothercurve, often withexperimental accuracy. An exception being portions of a flowcurve influ-enced by a second Newtonian plateau. This graphical shifting of flow curvesto superimpose has been called time-temperature-pressuresuperposition, orthe method of reduced variables [6]. The time-temperature-pressure superpo-sition principle is very important in EHL since it gives the opportunity to useflow curves obtained at convenient pressures and temperatures to be used todescribe the shear dependence of viscosity at more extreme pressures and tem-peratures. The superposition principle fixesn and sets up a shifting rule forG(T, p) [28]. If performing a viscosity measurement at a reference tempera-ture, and pressure gives a value ofG(TR, pR) = GR, thenG is found from oneof the following shifting rules:

G= GR (2.39)

G= GR

(

µµR

)m

(2.40)

G= GRTρ

TRρR(2.41)

G= GRTVR

TRV(2.42)

whereGR is the material modulus at a reference state ofT = TR andp = 0, andρR is the mass density at a reference state ofTr , p = 0.

Consider the single Newtonian Carreau equation written with λ = µ/G:

η = µ

[

1+

(

µγG

)2](n−1)/2

(2.43)

If the Ferry rule (eq. 2.42) for systems with constant mass, which has beenderived from the Einstein-Debye equation (2.24) [136] is used for shifting to-gether with the fact that the maxwell relaxation time for thereference state is:λR = µR/GR, the result is a t-T-p shifted single Newtonian Carreau equation:


η(γ,T, p) = µ

[

1+

(

γλRµµR

TR

TVVR

)2](n−1)/2

(2.44)

2.5.10 Temperature and pressure dependence on thermal conduc-tivity

Although the thermal properties of a lubricant such as heat capacity and ther-mal conductivity are usually not considered in rheology, they may have a sig-nificant effect on friction in EHL. Since film thickness in EHLis usually rel-atively thin, heat storage within the liquid is small, and the heat capacity ofthe lubricant will therefore not significantly change the performance. Ther-mal conductivity on the other hand, has been shown to have significant effecton EHD friction. Kumar and Khonsari [137] have shown in numerical cal-culations that by doubling the thermal conductivity, the predicted friction isincreased by 15 percent.

In 1931, Bridgman [138] found that the thermal conductivityincreasedunder a pressure of 1.2 GPa by a varying amount for different liquids. Hemeasured an increase of 150 percent for water, and 270 percent for normalpentane. More recently Larsson and Andersson [139] presented measurementsfor the pressure and temperature dependence on thermal conductivity and heatcapacity for several lubricants. They performed measurements at 22 and 107◦Cwith pressures up to 1.1 GPa using the hot-wire method. They concluded thatthe thermal conductivity roughly doubled at 1 GPa while it had only a 5 percentdecrease when temperature was increased from 22 to 107◦C.

Bair [28] proposed a scaling parameter for the thermal conductivity as afunction of volume and temperature:

κ =

(

VVR

)[

1+A

(

TTR

)(

VVR

)q]

(2.45)

whereA and q are coefficients obtained from measurements on the specificfluid. The thermal conductivity is then expressed as:

k=Ckκ−s (2.46)

whereCk ands are coefficients obtained from measurements on the specificfluid. It is still not known if this scaling parameter and function is universallyaccurate.

2.6. Machine components 69

2.6 Machine components

Many of our most common machine components such as gears, cams and rollerbearings are working in EHL. However, the working conditions for these ma-chine components can vary greatly in terms of load, entrainment speed andSRR, and also in terms of environmental conditions such as temperature andcontamination. Gears may be the machine component with the widest op-erating conditions and is therefore exemplified here. The pressure generatedbetween the mating teeth in a gear transmission can reach several GPa and thefrictional losses in a gearbox are therefore, depending on running conditionsto a large extent governed by EHD behaviour. It has been assessed that 33% of the fuel energy in a car is used to overcome friction, and that 7-18 %of these losses originates from the transmission [140]. In heavy road vehiclesand buses, 33.5 % of the fuel energy is used to overcome friction, and 13 %of these losses originates from the transmission [141]. Thelosses in a geartransmission can be devided into two categories, load dependent, and load in-dependent losses. The load independent losses is typicallyviscous losses dueto oil churning and are mostly governed by lubricant visosity, density and thegeometrical design of gears and housing. The load dependentlosses are due tofriction in the rolling and sliding interfaces between the mating gear teeth, andare influenced by a large numbers of parameters.

Figure 2.9 shows how several parameters are continuously changing alongthe line of action in a spur gear at a constant rotational speed. The load,and consequently the pressure in the contact are governed bythe transmit-ted torque, number of teeth in contact, and the geometry of the gear teeth. Theentrainment speed is governed by the geometry of the teeth and increased withrotational speed while SRR is only governed by teeth geometry and is inde-pendent of rotational speed. In practice, the conditions ina gear contact arehighly transient due to adjustments in load and speed on the transmission anddynamic behaviour in the driveline. A common way to model theentrainmentspeed, SRR and load conditions for a specific spur gear pair along the line ofaction is to use an analytical approach where the contact is represented by twocylinders whose radii vary along the line of action [142–146]. This type of so-lution does usually not take elastic deformations of the gears into account andis one of the reasons that finite element or finite difference [147] numericalmodels [148, 149] are used for similar calculations including the elasticity ofthe gears.


Figure 2.9: Changing conditions along the line of action in a meshing spur gear.

2.7 Friction reduction in machine components

There have been continuous research efforts to improve the efficiency and ser-vice life of machine components working in EHL by looking at several differ-ent aspects of the lubricated system. Much research have been conducted tounderstand the influence of lubricant properties such as viscosity and limitingshear stress on friction and wear. One example of progress isthe viscosity in-dex of lubricants that have improved over the years by VI modifiers and morerefined base oils with a result that machine components can work more effec-tively over a wider range of temperatures. The choice of lubricant in a gearboxhas been shown to have a substantial effect on the efficiency [150–154] ofthe gearbox. The lubricant in a gearbox influences friction power losses inseveral areas. First of all does the viscosity and limiting shear stress governthe friction losses in full film lubrication. Moreover do viscosity, pressure-viscosity and VI have an influence on when the system is operating in full filmlubrication by their effect on lubricant film thickness. Thesecond part of thelubricants influence on contact friction is the behaviour when asperity inter-actions occur, which is partly connected to the properties of the base oil, andprimarily governed by lubricant additives. These additives control friction and

2.7. Friction reduction in machine components 71

wear in boundary and mixed lubrication. Especially gear oils developed forhigh torque transfer applications usually contain anti-wear (AW) and extreme-pressure (EP) additives that reacts with the steel surface on the gears to reducewear and risk of scuffing. These reactions also influences thefriction coeffi-cients although the friction coefficients is reported both to increase, decrease,or be unaffected depending on several factors such as lubricant temperature,surface roughness, additive type etc [155–157]. There is however, no ques-tion that under beneficial circumstances that additive formed films can reducecontact friction in boundary and mixed lubrication.

The geometrical design of gears has been investigated and gears constructedfor lower amounts of sliding has been shown to have increasedefficiency[158–161]. Generally this is done by increasing the number of teeth dramati-cally while reducing teeth size, reportedly still allowingthe same load carryingcapacity. However, due to the lower module of the low loss gear, more precisetolerances has to be used to maintain proper centre distanceof the gears.

Investigations have also shown that using gears made of different mate-rials compared to classical gear steels may lower the contact friction losses.Examples are high pressure nitrided steel [162] and austempered ductile iron[161, 163]. However, the mechanism behind this difference in contact frictiondue to different gear materials has not really been explained. One possibil-ity is that the additives in the lubricant interacts in different ways dependingon the kind of material in the gear, thus affecting friction between contactingasperities.

Studies also show that gears with lower surface roughness due to finergrinding techniques or polishing has lower contact friction power losses [164,165]. This reduction of contact friction is mainly attributed to the fact thatgears with smoother surfaces will run in full film lubrication at lower speedscompared to gears whose surfaces have higher roughness.

In recent years, tribological surface coatings have been more frequentlyused to improve performance of machine components. Severalstudies havebeen performed using gear test rigs with a positive result onefficiency. Manydifferent types of coatings have been investigated, such asmolybdenum disul-phide/titanium, MoS2/Ti [166–168], carbon/chromium, C/Cr [167, 169] andtungsten carbon carbide, WC/C [170, 171]. The increased efficiency of thecoated gears in these investigations are attributed to lower friction coefficientfor asperity interactions, and therefore a reduction of friction in both boundaryand mixed lubrication. The WC/C coating is part of a group called diamondlike carbon (DLC) coatings, and can also be denoted as a-C:H:W or W-DLC.The use of DLC coatings in EHL are considered in the next section, and the


friction reducing properties are discussed, not only in boundary and mixedlubrication, but also in full film lubrication.

2.8 DLC coatings in EHL

DLC coatings are relatively thin, hard coatings typically deposited using phys-ical vapour deposition (PVD) [172], plasma-enhanced chemical vapour de-position (PECVD) or plasma assisted chemical vapour deposition (PACVD).DLC coatings consist of amorphous carbon materials, and their structures pri-marily rely on the ratio of sp3 to sp2 bonds. Sp3 bonded carbon atoms have adiamond like, or tetrahedral structure, while sp2 bonded carbon atoms have agraphitic, or trigonal structure. A DLC coating with a high ratio of sp3 bonds istherefore harder than a coating with a lower ratio of sp3 bonds. DLC coatingsexists as amorphous carbon (a-C) or hydrogenated amorphouscarbon (a-C:H),depending on if they contain hydrogen or not. DLC coatings having a highsp3 content are often called tetrahedral, ta-C or ta-C:H depending on if theycontain hydrogen or not. DLC coatings also often contain other fillers suchas silicon, fluorine, boron and nitrogen. For instance a nitrogen doped hydro-genated amorphous carbon is usually denoted a-C:H:N or N-DLC. Figure 2.10show a ternary phase diagram with some different forms of DLC.

Except the hardness, the ratio of sp3 to sp2 bonds, hydrogen content andother fillers influences several different properties of thecoating such as; sur-face energy, electrical- and thermal conductivity. Beforeapplying the DLCcoating to the substrate it is common to use a base layer of forinstance chromiumor silicon to improve the adhesion of the coating.

High hardness, high elastic modulus, low friction characteristics, high wearand corrosion resistance, chemical inertness and thermal stability are factorsthat make DLC coatings the subject of many studies, and seem suitable foruse in, amongst others, machine components and cutting tools. DLC coat-ings show low dry friction (friction coefficients are typically in the range of0.08-0.2) and wear rates compared to steel on steel in dry conditions, an effectmainly attributed to the formation of easily sheared graphitic layers on one orboth of the mating surfaces [173, 174]. However, most machine componentsare operating under lubricated conditions, and several studies have thereforebeen conducted on the topic of DLC performance under lubricated conditions.

In general, the friction in boundary lubrication is lower for a steel-DLCsystem compared to a steel-steel system and especially during the running inprocesses a system containing at least one DLC coated surface shows much

2.8. DLC coatings in EHL 73

Figure 2.10: Ternary phase diagram showing various forms of DLC.

lower initial friction coefficients, and a faster and more efficient running in[173, 175]. A system where both surfaces are DLC coated also shows a lowerinitial friction value but a longer running in period, as expected since bothsurfaces have high hardness and therefore need a longer timeto smoothendown.

While most studies in the literature focus on the influence ofcoatings onwear and friction in boundary lubrication and pure sliding contacts, few stud-ies can be found concerning rolling and sliding EHD friction, especially inthe mixed and full film regime. Renodeauet al. [176] used a rolling slidingmini traction machine (MTM) to test discs coated with four different com-mercial DLC coatings against a steel ball with four different lubricants. Theentrainment speed in the test device was chosen to simulate mixed lubrication.Friction and wear of the different configurations were measured and showeddissimilar results. Some material/lubricant combinations showed lower fric-tion and wear compared to the steel-steel system whereas some combinationsshowed higher friction and wear. Topolovec-Miklozicet al. [177] also per-formed tests in a MTM where two types of commercial DLC coatings werestudied, one hydrogenated diamond like and the other Cr-doped, non hydro-genated and graphitic. The study focused on the effect of boundary friction


with several types of additives. Bobzinet al. [178] performed tests in a twindisc tribometer with two different PVD coatings, WC/C and CrAlN, comparedto uncoated discs. Both of the latter studies report lower friction coefficientsfor the coated specimens in both boundary and mixed lubrication. It is howevernot clear if the friction reduction achieved in the mixed lubrication regime wassolely attributed to lower friction in the asperity interactions or if there also wasa beneficial effect in the hydrodynamic part of the mixed lubrication regime.

Few studies can be found in literature focusing on DLC coatings in fullfilm EHL. Xu [179] used a Wedeven Associates Machine (WAM) ball-on-disctest rig with both uncoated and DLC coated steel specimens. Results fromthe tests showed substantially lower friction with DLC coated specimens com-pared to the uncoated ones, even in the full film regime. The friction reduc-tion with the DLC coatings were highest when both ball and disc specimenswere coated. There was, however, no given explanation for the observation.Evanset al. [180] also performed tests in a WAM tribotester where discswith three different coatings were compared to an uncoated disc. All discswere run against an uncoated steel ball. The tested coatingswere chromiumnitride (CrxN), tungsten carbide-reinforced amorphous hydrocarbon (WC/a-C:H) and silicon incorporated diamond like carbon (Si-DLC). At lower en-trainment speeds where the asperity interaction between the mating surfaceswere large the CrxN coating had the highest friction whereas the WC/a-C:Hand especially Si-DLC showed lower friction than the uncoated surface. Athigher entrainment speeds, where full film lubrication was assumed, the Si-DLC and WC/a-C:H showed lower friction coefficients than theuncoated sur-face. The authors correlated the difference in full film friction to the differentsurface energies of the investigated specimens. The mechanism of the frictionreduction was proposed to be a phenomena described as boundary slip, interfacial slip or slid-liquid slip. A more recent study was doneby Kalin and Po-lajnar [181] where three different DLC coatings, A-C:H, N-DLC and F-DLCwere tested in full film lubrication and compared to uncoatedsteel referencespecimens. All investigated coatings provided lower friction coefficients infull film compared to the uncoated specimens. Even in this case, the authorssuggested that the friction reduction achieved with the coatings were due tosolid-liquid slip.

2.9. Solid-liquid slip 75

2.9 Solid-liquid slip

One of the most important assumption often made when modeling or dis-cussing fluid flow is the no-slip boundary condition. This dictates that a liq-uid element adjacent to a moving surface assumes the velocity of the surface.While this assumption may be reasonable in many situations,there are caseswhere this assumption leads to unrealistic behaviour such as the spreading ofa liquid on a solid substrate [182–185], corner flow [186] or extrusion of poly-mer melts [187]. As a consequence, several boundary conditions that allowsfor finite slip at the solid-liquid interface have been proposed [183, 188, 189].While these investigations show mathematically that the no-slip boundary con-dition is invalid in certain cases there have also been experimental observationsof solid-liquid slip. Légeret al. showed that polymer melts can slip againsta solid surface [190]. Vinogradova observed flow propertiesof water to besignificantly different between hydrophobic surfaces compared to hydrophilicsurfaces. These changes were interpreted as slippage [191]. Pit et al. uti-lized fluorescence recovery after photobleaching (FRAP) technique to assessthe velocity of a liquid close to a solid surface under simpleCouette shear.They used both a sapphire surface, and an identical sapphiresurface with amonolayer of octadecyltrichlorosilane, thus making it lyophobic to the fluidhexadecane used in the tests. They observed higher fluid flow adjacent to thesapphire surface with the applied monolayer suggesting slip at the solid-liquidinterface [192, 193]. Several other evidence for solid-liquid slip can be foundin literature [100,102,194–196].

On the basis of the break down of the no-slip boundary conditions a mod-ified Reynolds equation has been proposed that assumes a no-slip boundarycondition at the moving surface, but allows slip at a critical shear stress at thestationary solid-liquid interface [108,109,197]. This “half-wetted” bearing de-sign was theoretically shown to combine good load support with lower frictionthan a conventional fully wetted bearing. Other work concludes that differ-ent adhesion properties of the two surfaces in a journal bearing may cause thebearing to operate in an instable manner [198]. Solid-liquid slip has also exper-imentally been seen to reduce both film thickness and friction in hydrodynamiclubrication [107, 199–201], although at least one author reports that the loadsupport is higher with the fully-wetted bearing compared tothe half-wettedcounterpart [201].

It is believed that solid-liquid slip occurs when the shear forces at the in-terface is higher than the solid-liquid interactions, thusleading to some de-gree of slip at the interface. This principle is illustratedin Fig. 2.11. The


Figure 2.11: Velocity profile of a liquid with no slip, and with slip at the uppersolid-liquid interface.

slip alters the flow profile of the lubricant thus reducing both film thicknessand friction. However, it is still not entirely clear how themechanism ofslip works, and several hypotheses can be found in the literature. In manycases poor wetting, or high contact angle is proposed as the main feature topromote slip [100, 108, 109, 196, 200–202], and in other cases low surfaceenergy [180, 181]. However, using only surface energy as a means to deter-mine the potential of solid-liquid slip may not be suitable since surface en-ergy is a material property and tells nothing about the interaction of a spe-cific material with a specific fluid. On the other hand, contactangle measure-ments represent a property of the specific surface and lubricant combinationand could intuitively seem to be more suitable. However, Kalin and Polajnarhave recently published studies including many different lubricants and coat-ings where surface and liquid properties like surface energy and contact anglesare discussed [203, 204]. They conclude that contact angle cannot be used inisolation to predict the wetting behaviour, and instead propose the use of aspreading parameter which correlates well with the surfaceenergy. They alsoshowed the correlation between previously discussed interface properties withfriction coefficients in full film EHL, where especially the spreading parameterand the polar part of the surface energy correlates very wellwith the frictionmeasurements [181].

Aside from non-fully wetted surfaces, several authors [100,107–109,205]have reported that the surfaces must be very smooth, typically below 6-12nm RMS for solid-liquid slip to occur. The critical roughness is most likelydepending on the size of the molecules of the liquid used [193,206,207]. Pear-

2.9. Solid-liquid slip 77

son and Petrie [207] means that no slip can occur when the molecular size issmaller than the wall roughness scale. However, it is not necessarily so that assmooth a surface as possible will lead to the greatest amountof slip. At rela-tively low roughness (below 15 nm) it has been shown that RMS roughnessesof 0.7 and 4.0 nm produced less slip than a 12.2 nm surface [195]. That anoptimized roughness is superior to very smooth surfaces in some cases may beapparent by considering solutions from nature such as lotusleaves [208, 209].In nature, for instance low wettability is created by cleveruse of surface tex-turing.

The work on solid-liquid slip in EHL is much more limited. Theinvestiga-tions performed at atmospheric pressure or hydrodynamic pressure discussedabove may not be applicable on EHL. Solid-liquid slip has been reported tobe reduced with pressure in the hydrodynamic pressure range[123, 210], butalso observed to increase with pressure in the EHD pressure region [211,212].In favor for solid-liquid slip in EHL is amplitude reduction, described in sec-tion 2.4.2, that describes elastic deformation of the roughness inside the highpressure contact. Studies show that the roughness reduction is dependent onseveral factors, such as entrainment speed, slide to roll ratio, and pressure,and that long wavelengths are compressed more than short wavelengths. Thismeans that the actual roughness inside the high pressure region may, depend-ing on the roughness parameters and running conditions, be smoother thanoutside of the contact. This would, at least according to Pearson and Petrie bebeneficial for solid-liquid slip. The effect of amplitude reduction is howeverreduced with the increase of SRR.

While solid-liquid slip has been hypothesized to be the reason for the fullfilm friction reduction observed in EHD contacts by some authors [180, 181,202, 212], the present author is not aware of any validated numerical simula-tions that supports the hypothesis, although numerical work have been doneto incorporate interfacial wall slip in EHL [213]. To the author’s knowledge,only a few studies have observed solid-liquid slip in EHD contacts [103, 106,211, 212], and they have all used relatively smooth surfaces(below 12 nm).The liquids they used are either polyphenyl ether (5P4E) which has very highsurface tension and is known to show poor wetting at metal surfaces, or highmolecular weight polybutene. It is still to be elucidated ifsolid-liquid slip canbe observed with the use of surfaces and lubricants more often encountered inmachine elements like roller element bearings and gears.


2.10 Thermal conductivity of DLC coatings

As mentioned in section 2.8, DLC coatings have several properties that may beof interest in tribological applications. However, a property that may not havedrawn much attention, at least not in the field of EHL is the thermal conduc-tivity. As will be explained in sections 6.2, 6.3 and 6.4, thermal conductivitymay in fact be a very important property of a coating in EHL.

Carbon structures have a wide range of potential thermal conductivitiesdue to difference in structure. Highly ordered carbons suchas crystalline dia-mond (mainly sp3 bonds) and carbon nanotubes (sp2 bonds) have high thermalconductivities, in the range of thousands of Wm−1K−1 [214]. In these ma-terials, impurities, grain boundaries and isotropic effects controls the thermalproperties. Amorphous carbons on the other hand, as described in section 2.8and often used in tribological applications have much lowerthermal conduc-tivities, often in the range of a few Wm−1K−1 [215–222].

Morath et al. [216] used a picosecond thermal reflectance technique tomeasure the thermal conductivity of several 70-300 nm thickDLC films atroom temperature. They got values ranging from 3 to 10 Wm−1K−1 dependingon the hydrogen content of the coating. The higher the hydrogen content, thelower the thermal conductivity.

Bullenet al.[218] used the 3ω method to measure the thermal conductivityof several DLC films ranging between 20 and 280 nm, and one film at 3800nm at temperatures between 50 and 400 K. They got values ranging from 0.1to 3 Wm−1K−1 depending on the testing temperature and the density of thecoating. For all cases, the thermal conductivity increasedwith temperatureand density of the coating.

Chenet al. [219] utilized photothermal reflectance technique to measurethe thermal conductivities of ta-C films with thicknesses varying from 20-100nm. They got an average thermal conductivity of 4.7 Wm−1K−1, relativelyindependent of film thickness.

Shamsaet al. [220] measured thermal conductivities of a wide range ofa-C:H films with different hydrogen content with the 3ω method. The filmthicknesses ranged between 18.5 and 100 nm, and the thermal conductivitiesfrom 0.277 to 3.5 Wm−1K−1. Their results coincides with earlier findings, butfor a much greater number of different types of films. For all films, the thermalconductivity increase with temperature in the range from 50to 400 K. Theyalso found a correlation between the amount of sp3 bonds and thermal conduc-tivity. The higher the amount of sp3 bonds, the higher the thermal conductivity.At the the same time, the sp3 content scales with density in such a way as pre-

2.10. Thermal conductivity of DLC coatings 79

viously reported that a higher density equals higher thermal conductivity.Balandinet al. [221] used the 3ω method to measure the thermal conduc-

tivity of ta-C films in the range between 0.9 and 20 nm. They found ther-mal conductivities as low as 0.09 Wm−1K−1 for the thinnest coating and 1.4Wm−1K−1 for the thickest coating. They attribute the thickness effect on ther-mal conductivity to a reduction in sp3 as the film gets thinner, thus reducingthe thermal conductivity. Furthermore, they claim that theeffect of thermalboundary resistance on the interface layer between the substrate and the filminduces an even faster reduction of thermal conductivity asthe film thicknessdecreases.

Kim et al. [223] prepared a-C:H films of 200, 600, 1200 and 1800 nm, andperformed thermal conductivity measurements with the 3ω method. The mea-surements showed a strong correlation between thermal conductivity and filmthickness. The authors made a curve fit to the measurements and the resultsare shown in Fig. 2.12, where the stars represent the actual measurements.The dependence of film thickness on thermal conductivity wasattributed tothe interfacial thermal resistance between the a-c:H film, and the substrate.

Interfacial thermal resistance, thermal boundary resistance, or kapitza ther-mal resistance are discussed in refs [221, 224, 225] and are essentially an ex-tra thermal resistance caused by the differences in electronic and vibrationalproperties of the two materials forming an interface. It maybe possible, as inthe work by Chenet al. [219] to compensate for the effect of thermal bound-ary resistance and calculate the bulk thermal conductivityof the film. In thiscase the results should be a thermal conductivity much less dependent on filmthickness. However, in practice, the effective thermal conductivity will be in-fluenced by interface effects, and to a greater extent the thinner the film is. Athicker film will have a thermal conductivity more close to bulk behaviour ofthe material.

A few concluding remarks on the thermal conductivity of DLC coatingsare in place. It is clear that there is a rather big span in measured conductivi-ties for different kinds, and thicknesses of DLC coatings. However, evidently,most DLC coatings have thermal conductivities more than 10 times lower thansteel. The thermal conductivity of carbon structures is mainly related to theamount and structural disorder of sp3 phase. If the sp3 phase is amorphous,as is the case with DLC coatings discussed in this thesis, thermal conductivityscales, above all, with density and sp3 content. Since doping of DLC coat-ings with for instance nitrogen or fluorine changes the structure and amountof sp3 in the coating [226–228], it is reasonable to believe that the doping willalso change the thermal conductivity of the coating, although the author is not


0 500 1000 1500 20000

0.5

1

1.5

2

2.5

Coating thickness [nm]

The

rmal

con

duct

ivity

[W/m

K]

Figure 2.12: Thermal conductivity versus film thickness for a-C:H films.

2.11. Ball-on-disc tribotester 81

aware of any such studies. The interfacial thermal resistance is likely to have asignificant influence on the thermal transport across interfaces. In many casesa base layer of chromium or silicon is applied to the substrate before the actualDLC coating to improve the adhesion of the coating. Such a layer, no matterthe thickness, will add a second interface for thermal transport, and thus makeit even more challenging to predict the effective thermal conductivity of a spe-cific combination, and thickness of a substrate, base layer and DLC coating.

2.11 Ball-on-disc tribotester

The ball-on-disc friction measurements for the investigations in this thesiswere conducted in a Wedeven Associates Machine (WAM) ball-on-disc testdevice, model 11, whose principle setup is shown in detail inFig. 2.13. A ballis loaded against a solid disc resulting in a circular EHD contact. The lubri-cant is supplied at the centre of the disc in an oil dispenser that distributes thelubricant across the disc surface. Lubricant is circulatedin a closed loop fromthe oil bath, through a peristaltic pump to the oil dispenserat the centre of thedisc. The rotation of the ball and disc surfaces drags lubricant into the contactand a lubricant film is formed. The ball and the disc are drivenby separateelectric motors, the former to a speed up to 25000 rpm and the latter up to12000 rpm. From the rotational speed of the spindle connected to the ball, thespindle angle and the radius of the ball, the surface speed ofthe ball,Ub canbe calculated. In the same way by knowing the track radius on the disc, andthe rotational speed, the surface speed of the disc,Ud can be calculated. Theentrainment speed,Ue is then defined as:

Ue =Ub+Ud

2(2.47)

By adjusting the rotational speeds of the motors separately, different surfacespeeds of the ball and the disc can be achieved, resulting in apartly slidingcontact. The slide to roll ratio (SRR) is defined as:

SRR=Ub−Ud

Ue(2.48)

According to this definition, a positive SRR means that the ball surface speedis higher than the disc’s, while a negative SRR gives the opposite, with thedisc surface speed that is higher than the ball’s. The machine is calibratedfor pure rolling, SRR = 0, by adjusting spindle angle and positioning of theball to ensure a condition of no spinning. In the experimentsconducted for


Figure 2.13: Principle of WAM ball-on-disc test device.

2.11. Ball-on-disc tribotester 83

the research in this thesis, SRRs as low as 0.02 has been used with acceptableaccuracy.

Load cells are used to measure the force on the three principal axes wherethe machine operates, and are used to calculate the frictioncoefficient in thecontact. The accuracy of the measurements are depending on several factors.One of the most prominent is the applied load. The contact load between theball and disc are governing the contact pressure for the tests where a higherload leads to a higher contact pressure. A higher contact load will also giverise to a higher friction force, which in turn will give more consistent valuesfor the friction measurements due to the sensitivity of the load cells. ConsiderFigs. 2.14 and 2.15. Figure 2.14 contains data from 7 friction measurementsperformed with the fluid squalane at 40◦C at an entrainment speed of 1.6 m/sand a SRR varying from 0.02 to 1.05. The results for the same test proce-dure, but at a 300 N load is shown in Fig. 2.15. It is clearly seen in thiscomparison that the standard deviation is generally lower for the case with thehigher load. Furthermore, the standard deviation is generally larger at lowerSRRs where a small change in SRR has a large impact on frictioncoefficient.This effect is amplified at higher loads where the friction coefficient increasewith increasing SRR is even larger, which can also be seen in Figs. 2.14 and2.15. These experiments have been performed in full film lubrication with alubricant without additives, and no asperity interactionsor tribofilm formationcould therefore influence the results. A larger standard deviation is to be ex-pected when running tests in mixed or boundary lubrication where wear willcontinuously change the contact conditions, and thereforeinfluence the fric-tion measurements. Using lubricants with different additives will also in somecases cause larger standard deviations due to tribofilm formation.


0 0.2 0.4 0.6 0.8 10.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt

Figure 2.14: Mean value and standard deviations from seven repeated test withconstant entrainment speed of 1.6 m/s, 80 N load and varying SRR.

0 0.2 0.4 0.6 0.8 10.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt

Figure 2.15: Mean value and standard deviations from seven repeated test withconstant entrainment speed of 1.6 m/s, 300 N load and varying SRR.

Chapter 3

Scope and objectives

Although much work has been done in the field of EHL over the years, thereare still areas where more research is needed for a better understanding of thefriction mechanisms. Friction in an EHD contact should be viewed as a systemproperty, which is influenced by many parameters such as; lubricant, materialand surface properties, as well as running conditions like load, entrainmentspeed, slide to roll ratio (SRR) and lubricant temperature.When changing oneor more of these parameters, the effect on friction may both increase or de-crease depending on which friction regime that is considered. At the present,the literature is lacking investigations where these parameters have been stud-ied in a wide range of entrainment speeds and SRRs. Understanding these in-teractions are important when optimizing machine components for increasedefficiency, and therefore important objectives of this workincludes:

• Find a suitable method to experimentally investigate how changes in sev-eral parameters influences the friction coefficient in EHL using a widerange of entrainment speeds and SRRs.

• Correlate the contact friction measurements conducted ina simplifiedexperimental system, a ball-on-disc machine, to contact friction mea-sured in a gear test rig.

85

86 Chapter 3. Scope and objectives

The use of surface coatings in EHL have been shown to reduce friction inboth boundary, mixed and full film regimes. The mechanism of the frictionreduction in full film with surface coatings is not fully understood, and thefollowing research topics were included in this work to elucidate the frictionmechanism:

• Investigate under which working conditions the friction reduction oc-curs.

• Explore possibilities of other explanations for the friction reduction be-sides the solid-liquid slip theory.

At the same time, there has been advances in the numerical modeling of EHLduring the last decades as new knowledge has been gained in the field. Pre-diction of friction in EHL is especially demanding due to thenature of thehigh pressure rheology found in EHD contacts. Several different models areproposed in the literature for the prediction of EHD friction. It is howevernot entirely clear which models are most suitable to use and their capabilitiesfor true prediction of EHD friction. This leads to additional objectives of thiswork:

• Investigate the current state of the art in numerical EHD friction pre-diction, and identify potential weaknesses in these modelsfor furtherinvestigation.

• Use numerical modeling of EHL as an aid in understanding EHDfric-tion mechanisms to propose ways to reduce EHD friction as a means toincrease the efficiency of machine components.

Chapter 4

Friction mapping

Friction coefficients in experimental or numerical investigations in EHL areoften presented as Stribeck curves, as introduced in section 2.4.3, or as µ-slipcurves, as discussed in section 2.5.5. Figure 4.1 presents friction coefficientsmeasured in a ball-on-disc machine, described in chapter 2.11, against entrain-ment speed at three different SRRs in the form of a stribeck curve. The mostprominent feature of Fig. 4.1 being the high friction coefficients at low en-trainment speeds that gradually reduces when the speed is increased. Initiallythe decrease of the friction coefficients are due to a reduction of asperity inter-actions between the surfaces and in other words a transitionfrom mixed to fullfilm lubrication. However, contrary to the classical description of the Stribeckcurve, the friction coefficients continues to reduce with speed even after thefull film regime is reached. It is obvious that the friction reduction is highlycorrelated to the amount of sliding in the contact, and not only the thicknessof the lubricant film which is mainly governed by the entrainment speed. Forthe same entrainment speed the difference in film thickness between the threedifferent SRRs will be almost negligible even though the friction coefficientsin some cases differ greatly. The explanation for this is thedecoupling of filmthickness and friction due to their different origins. The film thickness is gov-erned by the conditions in the inlet of the contact, whereas friction is governedby the conditions in the high pressure zone of the contact. Different SRRs willinduce different amounts of non-newtonian effects and thermal softening onthe lubricant which greatly affects friction. The inlet conditions are to a muchlesser extent affected by shearing, and the film thickness istherefore largelyunchanged.

Figure 4.2 shows friction coefficients against SRR from similar experi-ments at three different entrainment speeds in the form of a aµ-slip curve.

87

88 Chapter 4. Friction mapping

0 2 4 6 8 100.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Entrainment speed [m/s]

Fric

tion

coef

ficci

ent

SRR=9 %SRR=14 %SRR=49 %

Figure 4.1: Example of Stribeck curve.

This clearly shows the non-linear response on friction withincreased shearrate as described in section 2.5.5. Initially as the slidingincreases from zero,friction rapidly increases with shear (or stress) in a linear manner, where atsome shear rate the friction response will become non-linear, reach a maxi-mum value and then start to decrease. Here, it is also noticeable that the lowestfriction coefficients is measured for the highest entrainment speeds where bothfilm thickness and sliding speed is the highest.

While in many cases both of these ways of presenting frictiondata arevery useful to interpret and understand friction behaviourin various systems,they may have shortcomings when observing systems in a wide range of en-trainment speeds and SRRs. In Paper A, experiments were conducted usinga wide range of entrainment speeds and SRRs where several parameters wereinvestigated; base oil type, lubricant viscosity, EP additive content, lubricanttemperature and surface roughness. As a complement to µ-slip and Stribeckcurves the authors used what they call friction mapping as a way to presentand analyze the results from the experiments. Friction mapping gives the pos-sibility to compare two different systems in terms of friction in a wide rangeof entrainment speeds and SRRs that is more accessible to some extent thanseveral stribeck or/and µ-slip curves. This chapter gives an overview of the

4.1. Concept of friction mapping 89

0 0.1 0.2 0.3 0.4 0.50.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Slide to roll ratio

Fric

tion

coef

ficie

nt

U_e=9.6 m/sU_e=7.68 m/sU_e=6.144 m/s

Figure 4.2: Example of µ-slip curve.

concept of friction mapping and discusses some of the key findings of PaperA. For more details about the test specimens, lubricants andthe test procedure,see sections A.2.2 and A.2.3.

4.1 Concept of friction mapping

As a complement to the µ-slip curves and Stribeck curves the friction map-ping technique is proposed since it contains both SRR, entrainment speed andcoefficient of friction and can thus replace several µ-slip-and Stribeck curvesallowing for a better overview of the friction characteristics of the studied sys-tem. A 3D friction map is presented in Fig. 4.3 where frictioncoefficientis displayed versus entrainment speed and SRR. Included in the figure arealso friction regimes inspired by the work of Johnson and Tevaarwerk [124].The friction regimes generally corresponds to the featuresdiscussed for theStribeck and µ-slip curves in Figs. 4.1 and 4.2.

In the linear region, shear stress is proportional to shear rate, and will thuslead to a linear increase in friction. This linear relationship is only seen atlow SRRs, and the upper limit (in SRR) becomes even lower withincreasingpressures as the shear stress increases faster with shear rate at higher pressures


Mixed

Linear

Non-Linear

Thermal

Figure 4.3: 3D friction map with regimes.

and at lower entrainment speeds.The non-linear region is influenced by shear thinning and also by the growth

of a limiting stress region. The shear stress will no longer increase proportion-ally to shear rate. It will rather increase at a reduced rate until limiting shearstress is attained over much of the contact. Then, shear stress becomes inde-pendent of shear rate. However, if the sliding speed is further increased, ther-mal effects eventually dominate the frictional response overwhelming othereffects including limiting shear stress which is no longer reached. As a conse-quence, friction starts decreasing with increasing SRR dueto combined ther-mal and shear-thinning effects. This regime has been identified as the thermo-viscous, or thermal regime.

The last region is the mixed lubrication region where asperity contacts oc-cur between the surfaces, and the coefficient of friction is therefore a combi-nation of hydrodynamic effects and asperity interaction.

The boundaries of these regions are exclusive for each studied system, de-pending on lubricant and material properties as well as load, surface roughnessand operating temperature. The location of the mixed lubrication boundary is



Slid

e to

rol

l rat

io

0.07

0.065

0.06

0.055

0.05

0.045

0.04

0.035

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 4.4: 2D friction map - Low viscosity oil @ 40◦C.

assumed to be where the coefficient of friction is no longer decreasing withincreased SRR for a certain entrainment speed, which would imply incipientasperity interactions. However, this is a floating boundarycontrolled by severalparameters, among others the balance between increasing asperity interactionsdue to a thinner lubricant film, and the decrease of limiting shear stress andthermal softening of the lubricant. The location of the thermal boundary isassumed to be where the limiting shear stress has been reached, and above all,where thermal softening of the lubricant is dominating the coefficient of fric-tion. In essence, the friction map can be seen as a fingerprintof the frictionalbehaviour in a specific system.

While the 3D friction map is useful for getting an overview ofthe frictionalbehaviour of a studied system, a 2D friction map is in many cases more usefulfor more detailed comparisons of friction between two systems. Figure 4.4shows a 2D friction map corresponding to the 3D friction map in Fig. 4.3.The test results shown in Figs. 4.3 and 4.4 are both from PaperA, whereexperiments have been conducted at a temperature of 40◦C with a lubricantwhose kinematic viscosity is 30.8 cSt at the test temperature. As a comparison,a test under the same conditions has been performed with another lubricantwith a higher kinematic viscosity of 94.9 cSt at the same temperature. Theresult from this test is presented in Fig. 4.5 as a 2D frictionmap. Both oils are



Slid

e to

rol

l rat

io

0.07

0.065

0.060.055

0.05

0.045

0.04

0.035

0.03

0.025

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 4.5: 2D friction map - High viscosity oil @ 40◦C.

mineral base oils without additives. The difference in friction between the twooils is therefore most likely almost entirely an effect of the viscosity difference.More lubricant properties can be found in Table A.2.

Comparing lubricants with different viscosities are relevant since there isa current trend in industry, especially in the automotive industry, to go towardsthinner oils in transmissions. It is also often believed that reducing the viscos-ity also reduces friction, which is only partly true. To makeit more convenientto investigate how these two cases with different viscosities differ from eachother, another map is presented where the absolute frictioncoefficients fromthe high viscosity oil is subtracted from the low viscosity case, as presented inFig. 4.6.

In this figure, it is possible to see the difference between the two cases atdifferent entrainment speeds and SRRs. In this example, a negative numberindicates a higher friction coefficient for the high viscosity case, and a positivenumber indicates a lower friction coefficient for the high viscosity case. Asseen in the figure, the higher viscosity oil gives lower contact friction over amajority of the tested range of entrainment speeds and SRRs.This indicatesthat in many situations, a higher viscosity oil will reduce the contact frictioncoefficient, even in the full film region, compared to a lower viscosity oil.However, it is to be pointed out that a lower viscosity oil, used in for instance



Slid

e to

rol

l rat

io 0.008

0.004

0

−0.004

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 4.6: 2D friction map - difference between low viscosity oil and high vis-cosity oil @ 40◦C. Positive values indicate lower friction for the high viscosityoil.



Slid

e to

rol

l rat

io

−50−40−30

−20−10

0

10

20

30

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 4.7: 2D friction map - difference in % between 40◦C and 90◦C. Positivevalues indicate lower friction for the higher oil temperature.

a gearbox, may give lower total losses than a higher viscosity oil when othersources of losses aside from contact friction are considered, such as viscouslosses.

Another example is given in Fig. 4.7 where the thicker oil in Fig. 4.5 iscompared at 40 and 90◦C, the latter in Fig. A.8(b). In this case, the frictioncoefficients from the 90◦C test is subtracted from the 40◦C test and presentedas the difference in percent when going from an oil temperature of 40◦C to90◦C. Here it is very clear that the effect of increasing the oil temperaturefrom 40 to 90◦C has a very different effect depending on entrainment speedandSRR. At low entrainment speeds and SRRs an increase in oil temperature from40 to 90◦C is substantially decreasing the friction coefficients, while at higherentrainment speeds and SRRs the friction coefficients are instead increasing.This result also indicate that the film thickness generated is thick enough tokeep the surface mostly or totally separated even for the 90◦C case.

An approximation of the friction power loss in a specific system can bedone by multiplying the measured friction coefficient with entrainment speed,SRR and contact pressure. Figure 4.8 shows the friction power correspondingto the case in Fig. 4.3. It is evident that reducing the coefficient of friction



Slid

e to

rol

l rat

io220

200180

160140

120100

8060

40

20

1 2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 4.8: Friction power map [MW/m2].

in the high speed, high sliding region has a bigger influence on friction powerloss compared to a reduction of the friction coefficient in the low speed, lowsliding region. With the friction power map in mind, the casein Fig. 4.6 canbe studied again. Depending on the operating conditions of aspecific machinecomponent, a change from a high viscosity oil to a low viscosity oil could ei-ther increase, or reduce contact friction. If the machine component is workingin a region with very low SRR, a change of oil would reduce the efficiency ofthe machine component in terms of contact friction, but if the machine com-ponent is working with high SRR a change of oil from low viscosity to highviscosity seems feasible. Finally, if the machine component operates in bothareas, the concept of friction power can be used to give an insight of if thenet effect would be positive or negative. In this case it would almost certainlybe positive since the high viscosity oil has better performance in the regionswhere most power is entered to the system.


4.2 In conclusion

So called friction maps have been used to evaluate the results from a series offriction measurements performed in a ball-on-disc machineas a complementto the more commonly used stribeck and µ-slip plots. Tests have been per-formed with different lubricant viscosities, temperatures, surface roughness,base oil type and additive content. The measurements were conducted under awide range of entrainment speeds and SRRs. The results show that changingfor instance lubricant temperature or viscosity may eitherreduce, or increasefriction depending on the operating conditions. It is therefore important toconsider the normal operating conditions for a specific machine component toassess if such a change will have a beneficial effect on power loss.

In Chapter 7, ball-on-disc measurements in combination with friction map-ping is used to predict the contact friction in a gear test rig. In the followingchapter, friction mapping is used in a comparison of friction coefficients be-tween experimental measurements and predictions from a numerical model.

Chapter 5

Predicting friction in EHL

The capability to predict film thickness and friction in EHD contacts is crucialfor better understanding of the field, and the means of a scientific way to de-velop more efficient machine components. EHL is a complex field, and to beable to predict film thickness and friction with good accuracy, realistic modelsof the lubricant behaviour have to be used. In section 2.5, many of the fun-damental parts of lubricant behaviour have been discussed,such as the effecton viscosity by temperature, pressure and shear. A short review of some ofthe most commonly used rheological models in EHL shows that many of thesemodels lacks a rigorous physical foundation, and that some of them are usedfor modeling of relationships they were not intended for. Many comparisonsbetween predictions and experiments involve some kind of tuning of the modelin order to match the experimental results. In real, true prediction, this cannotbe done since there would be no experimental results to compare to. Instead,all three components occur independently from each other:

• Primary measurement of lubricant transport properties.

• Prediction using measured transport properties as input data togetherwith operation and geometric data.

• Experimental validation

Such an investigation was conducted in Paper C. The three components listedabove have been carried out at three different laboratories. Initially the trans-port properties of squalane (a low molecular weight branched alkane) werecharacterized at Georgia Tech. These properties were laterused in a numericalmodel to predict friction at the Lebanese American University. Finally, friction

97

98 Chapter 5. Predicting friction in EHL

Table 5.1: Investigated conditions.

Temperature 40◦ CContact load 50 and 300 NMaximum hertzian pressure 1.07 and 1.94 GPaEntrainment speed,Ue 0.34 to 9.6 m/sSlide to Roll Ratio, SRR 0.0002 to 0.49

was measured in a ball-on-disc test device at Luleå University of Technology.No tuning of the lubricant properties, model or test setup have been applied.The presented results on EHD friction are therefore a true representation of thecurrent level of EHD friction prediction.

5.1 Method

The numerical prediction and experimental measurements were both performedusing the same conditions reported in Table 5.1. The investigation includesa combination of high entrainment speeds and contact pressures that to theauthors knowledge have not been considered in any similar previous studies.Both numerical predictions and experiments were performedwith the samelubricant, squalane, a commercially available low molecular weight branchedalkane (2,6,10,15,19,23-hexamethyltetracosane). An oilwithout additives waschosen to minimize the effect of tribochemical reactions onthe friction coeffi-cient. At the test temperature of 40◦C, the ambient viscosity of squalane is 15mPas and the pressure-viscosity coefficient is 18 GPa−1 [129].

The transport properties used for the numerical model in this study are ob-tained from several earlier studies on squalane [94,129,130]. The formulationsused in this investigation are found in section C.2.2 and arefurther discussedin section 2.5.

The numerical model employed in this work was described in detail in[229–231]. The model is based on a finite element, fully coupled solutionof the EHD equations: Reynolds, linear elasticity and load balance equations.The latter are solved simultaneously providing robust and fast converging solu-tions. The generalized Reynolds equation [232] is used to account for the sheardependence of the lubricant. Special formulations are introduced in order tostabilize the solution of Reynolds equation at high loads. The temperature dis-tribution in the contact is obtained by solving the 3D energyequation in thelubricant film and solid bodies. The model incorporates the variations of thelubricant’s thermal properties with pressure and temperature throughout the

5.2. Results 99

contact. Then an iterative procedure is applied between therespective solu-tions of the EHD and thermal problems as described in [229,230]. During theiterative procedure, every time the shear stressτ is evaluated (using viscositydata provided by a combination of the Carreau and Vogel-likemodels) it iseither truncated toτL if it exceedsτL or, otherwise, it is kept unchanged. Thenumerical model does not take asperity interactions into account and shouldtherefore only be seen as a full-film EHL model.

The experiments were carried out in a WAM 11, ball-on-disc test devicedescribed in section 2.11. Since the numerical model used inthis work does nottake asperity interactions into account, the tests were performed using grade 20balls and polished discs giving a combined surface roughness of 43 nm RMSto reduce the occurrence of asperity interactions during testing. More detailsof the specimens are given in section C.2.4.1. The test procedure following theconditions presented in Table 5.1 is given in section C.2.4.2.

5.2 Results

The results from both the numerical predictions and the ball-on-disc frictionmeasurements are presented as friction maps, as introducedin Chapter 4. Re-cent work on friction regimes in quantitative EHL [89] has led to more de-tailed knowledge about the regimes, and one more regime is included in thiswork, see Fig. C.11 compared to Fig. 4.3 in Chapter 4. The added regime isthe Plateau regime, which lies in the boundary between the mixed lubricationregime, and the thermoviscous regime. Here, friction reaches an asymptoticvalue and shows little variation indicating that the frictional response of thecontact is governed by the limiting shear stress behaviour of the lubricant.

Figures 5.1 and 5.2 corresponds to the numerical and experimental resultsat 1.07 GPa while Figs. 5.3 and 5.4 corresponds to the resultsat 1.94 GPa. Fig-ure 5.5 shows the difference between the experiment and calculation in percentfor the test performed at 1.07 GPa, and the difference for the1.94 GPa test isshown in Fig. 5.6. A positive number in those figures indicates that the pre-dicted coefficient of friction is higher than the experimentally obtained value,whereas a negative number indicates that the predicted friction coefficient islower than the experimentally obtained value. Figs. C.9 andC.10 show thegenerated power from friction in the experiments for the 1.07 and 1.94 GPacases respectively. The friction power is calculated as:Wf = µf ∗Ue∗SRR∗L.Whereµf is the coefficient of friction,Ue is the mean entrainment speed, andL is the applied load.


Figure 5.1: Predicted friction map at 1.07 GPa and 40◦C.

Figure 5.2: Measured friction map at 1.07 GPa and 40◦C.

5.2. Results 101

Figure 5.3: Predicted friction map at 1.94 GPa and 40◦C.

Figure 5.4: Measured friction map at 1.94 GPa and 40◦C.



SR

R

5

10

15

20 15

20

2020

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 5.5: Difference in friction coefficient; Calculation-experiment [%], 40◦ C,1.07 GPa.


SR

R

20

25

25

20

15

10

5

0−5

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 5.6: Difference in friction coefficient; Calculation-experiment [%], 40◦ C,1.94 GPa.

5.2. Results 103

It is assumed that most parts of the measurements are performed in the fullfilm regime, with the exception of the lowest entrainment speeds in the higherpressure case. A more thorough discussion of the roughness effects and filmthickness is given in section C.3.1.

By looking at the overview of the four cases in Figs. 5.1 to 5.4it is clearthat the numerical model manages to capture the overall frictional behavioursuch as the linear increase in friction with shear rate at lowSRR, the non linearincrease in friction leading to the limiting shear stress, and the reduction offriction at higher entrainment speeds and SRRs due to thermal effects. Mixedlubrication effects are however not included in the numerical model, and couldtherefore be part of the difference between the numerical prediction and theexperimental results at the lowest entrainment speeds.

At lower entrainment speeds, up to about 2 m/s, the predictions are ingeneral not far off; the deviation is typically below 10 % andin line withprevious comparisons at these entrainment speeds [231]. Athigher speeds thecoefficient of friction is constantly overestimated by the numerical model, andgenerally more so at higher SRRs. When looking at absolute values, and thedifference between the numerical calculations and the experiment as presentedin Figs. 5.5 and 5.6, it is clear that the model does not accurately predictall parts of the tested region, or that some of the boundary conditions in thenumerical model does not fit the experiment.

One of the most striking differences between the predictions and the ex-periments are the plateau regime in the predictions, Figs. 5.1 and 5.3 wherethe coefficient of friction is almost constant at a high levelover a wide rangeof entrainment speeds and SRRs. Such behaviour is not as distinct in the ex-periments, Figs. 5.2 and 5.4 where instead the friction coefficients drops fasterwith an increase in entrainment speeds and SRRs.

By looking at the difference in percent between the predictions and the ex-periments in Figs. 5.5 and 5.6 it is clear that the levels of difference are situatedin the same manner as the levels of friction power in Figs. C.9and C.10. Itis therefore likely that one of the main reasons for the deviation between theprediction and the experiments are found in how the model handles thermal ef-fects and/or some thermal behaviour in the test rig and specimens that are notmatched in the boundary conditions in the numerical model. It should how-ever be pointed out that the biggest discrepancies does not occur at the highestentrainment speeds and SRRs, but rather in connection to theplateau regime.

The numerical model is using bulk temperature as a boundary condition forthe inlet temperature of the lubricant and a steady state condition is calculatedfor the specific running conditions (entrainment speed and SRR.) Any global


effects on the specimen temperature are therefore not takeninto account, whilein the experiment it is possible that especially the ball surface temperatureis increasing during the test. As a specific running condition is measured inthe test rig, it takes a few seconds for the machine to adjust rotational speedsof both specimens to achieve the correct entrainment speed and SRR. Someadditional time is also required for the data capture process. It is possiblethat during this time, the specimens are heated due to the friction generated inthe contact, and therefore the lubricant temperature will be slightly higher inthe inlet and central region of the contact compared to the numerical model.Looking at the friction power for the low pressure, 1.07 GPa case in Fig. C.9,there is a maximum of 7 W. For the high pressure, 1.94 GPa case in Fig. C.10,with a maximum friction power of 55 W, this is more likely to have an effecton the results, and may be the reason why the deviation in general is slightlylarger for the higher pressure case at higher sliding speeds. If the reasoningabove is the cause for the largest part of the deviations, theeffect of the globalgeometry on intake and surface temperature must be taken into account.

However, it is also possible that the deviations between theexperimentalresults and the predictions are due to shortcomings in the rheology modelsused in the study.

Furthermore, recent research [89] suggests that it is not possible to accu-rately estimate the limiting shear stress coefficient from EHD friction measure-ments, since shear-thinning was found to affect the friction coefficient evenwhen the limiting shear stress is reached in parts of the contact.

Moreover is the limiting shear stress in the numerical modelused in thiswork assumed to be only depending on pressure. It has been shown earlierin measurements that temperature has an influence on the limiting shear stress[97–99]. It is reasonable to believe that this is one of the reasons why theplateau regime is more pronounced in the numerical predictions. With theinclusion of the temperature dependence of limiting shear stress the plateauwould probably level off with increased sliding due to a reduction in limitingshear stress before thermal effects starts to dominate the friction behaviour.

5.3 Inclusion of thin, thermally insulating layers

The same numerical model as described in section 5.1 was used, in paper D,to investigate the effect of thermally insulating layers onEHD friction whoseresults are presented in section 6.3. The inclusion of coatings into the numeri-cal model consists simply in adding a thin layer to the surface of the 3D solid

5.3. Inclusion of thin, thermally insulating layers 105

body representing the contacting solids. This layer possesses different elasticmaterial and thermo-dynamic properties than those of the substrate. The in-fluence of hard coatings on EHD film formation and pressure distribution hasbeen studied in several publications [233–235]. It has beenshown that a coat-ing that is harder than the substrate reduces the contact width, and increasesthe central pressure and the pressure spike, as is seen in Figs. 5.7 and 5.8. Acoating that is softer than the substrate would have the opposite effect. Botheffects increase with the thickness of the coating. However, the simulation ofa typical isothermal Newtonian EHD line contact with insulating coatings ofdifferent thickness shows that for coatings of 2-5 µm thickness (such as theones considered in section 6.3), the effect of the coating ondimensionless filmthickness and pressure is negligible, as suggested by Figs.5.7 and 5.8. [236].Therefore, for the sake of simplicity, in the EHD part of the model the surfacesare assumed to be uncoated and the coatings are only considered in the thermalpart. This leads to a significant reduction in the size of the discrete problemas the thin layer coating requires a very large number of elements to discretizeit. The temperature distribution in the contact is obtainedby solving the 3Denergy equation in the lubricant film and solid bodies (substrates and coat-ings). The inclusion of the coatings consists in inserting ageometrical layerbetween the lubricant and each substrate and applying the thermal propertiesof the coating to it. Obviously, the continuity of heat flux needs to be imposedbetween the lubricant film and the coatings as well as betweenthe coatings andthe substrates. For a complete description of the numericalmodel, see sectionD.2.3.


Figure 5.7: Effect of DLC coating thickness on the dimensionless film thickness(top) and pressure (bottom) distribution of a typical EHL line contact.

5.3. Inclusion of thin, thermally insulating layers 107

Figure 5.8: Effect of DLC coating thickness on the dimensionless film thickness(top) and pressure (bottom) distribution of a typical EHL line contact (Zoom onFigure 5.7).


5.4 In conclusion

A friction prediction investigation has been performed where lubricant trans-port properties have been measured at one laboratory, and later used at a secondlaboratory to predict friction in a numerical model for circular EHD contacts.Finally, the same parameters were used for friction measurements in a ball-on-disc machine in a third laboratory.

The predictions were found to capture the main features of the frictionbehaviour at different entrainment speeds and SRRs. In absolute values thenumerical results were found to be remarkably good considering the true pre-diction approach. At low entrainment speeds the deviationsbetween the nu-merical predictions and the experimental results were in general less than 10%, while increasing at higher entrainment speeds and SRRs. To achieve evenbetter precision in the predictions there is a need to refine the assessment ofthe limiting shear stress of the lubricant, including the effect of temperature.In addition, it may be necessary to refine the boundary conditions in the modelto account for global effects on temperature.

The model was also extended to include thin thermally insulating layers tobe able to predict the effect of coatings with low thermal conductivity on EHDfriction.

The use of primary measurements or “true” measurements of the liquidproperties in EHL has distinct advantages over the previoustechniques of ad-justing properties to satisfy perhaps inappropriate assumptions. If friction is tobe controlled by lubricant selection and formulation, there must be an under-standing of the contribution of each of the transport properties to the energydissipation. However, the use of the true prediction approach investigated herewill generally require more effort from a measurement and modeling point ofview compared to some of the previous techniques.

Chapter 6

The influence of DLC coatingson EHD friction

As explained in section 2.8, DLC coatings have been found to reduce frictionin EHL, not only in mixed and boundary lubrication, but also in the full filmlubrication regime. The mechanism for the friction reduction in full film hasbeen hypothesized to be solid-liquid slip, as detailed in section 2.9. However,it cannot be said for certain that solid-liquid slip is the only plausible frictionreducing mechanism of the DLC coating. On the contrary, the following sec-tions presents results that indicate that it is not possibleto explain the frictionreduction achieved in full film EHL with DLC coatings by only consideringsolid-liquid interactions. In connection to these findings, an alternative hy-pothesis to the solid-liquid slip theory is also presented along with convincingevidence of its relevance.

6.1 Friction reduction with DLC coatings

One of the questions raised when reviewing the literature onsolid-liquid slipis whether slip can be achieved also with surfaces that are not atomisticallysmooth, or below a limit of approximately 10-15 nm RMS. To investigate this,ball-on-disc experiments were conducted in Paper B, with both uncoated steeland DLC coated specimens to characterize the friction behaviour over a widerange of entrainment speeds and SRRs according to the concept of frictionmapping presented in Chapter 4. The test specimens had a surface roughnesssubstantially higher than what is generally considered thelimit for solid-liquidslip to occur.

109

110 Chapter 6. The influence of DLC coatings on EHD friction

6.1.1 Method

The friction measurements were conducted in a WAM ball-on-disc machine asdescribed in section 2.11. All tests were performed with thesame lubricant,a pure mineral oil without additives, with details in Table B.1. An oil with-out additives was chosen to minimize the effect of tribochemical reactions onthe friction coefficient. Tests were performed both with uncoated steel spec-imens as a reference, but also the combinations of DLC coatedball and disc,uncoated ball with DLC disc, and DLC coated ball with uncoated disc. Thespecimens had a combined roughness ranging from 155 to 355 nmRa (TablesB.3 and B.4) which is considerably more than the limit for solid-liquid slip tooccur that is usually believed to be below 15 nm. The DLC coating used wasthe commercially available Tribobond 44, an a-C:Cr coatingdeposited using aPVD process. The coating thickness was 2 µm measured with calotest. Moreinformation about the specimens and the coating is found in section B.2.2. Thefriction measurements were performed at a pressure of 1.7 GPa with entrain-ment speeds between 0.34 and 9.6 m/s and SRRs from 0.0002 to 0.49 and twooil temperatures of 40 and 90◦C. The test procedure is described in detail insection B.2.3.

6.1.2 Friction measurements with coated and uncoated specimens

Figures 6.1 to 6.3 show the results from the ball-on-disc friction measurementsperformed at 40◦C. Figure 6.1 shows the friction reduction in percent achievedwith coating only the disc with DLC compared to the uncoated reference case,while Fig. 6.2 shows the friction reduction when coating only the ball withDLC. Finally, Fig. 6.3 shows the friction reduction achieved when both balland disc were coated with DLC. The absolute values for the friction reductioncan bee seen in Figs. B.5 to B.7. It is evident that the DLC coating leads toa reduction in friction coefficient in all cases, with the exception of the lowestentrainment speeds where the rougher surface of the DLC coated disc leads toa transition from full film to mixed lubrication at higher speeds. The biggestfriction reduction is achieved when both specimens are coated followed byonly coating the ball, and then only coating the disc. Interestingly, this frictionreduction was achieved with surface roughness values one ortwo orders ofmagnitude larger than the ones usually considered to be the limit for solid-liquid slip, as discussed in section 2.9.

6.1. Friction reduction with DLC coatings 111


SR

R

10

15

50

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 6.1: Friction reduction in percent achieved with DLC coated disc in com-parison with uncoated specimens at 40◦C.


SR

R

15

10

5

0

5 −51 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 6.2: Friction reduction in percent achieved with DLC coated ball in com-parison with uncoated specimens at 40◦C.



SR

R

20

15

10

5

0

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure 6.3: Friction reduction in percent achieved with DLC coated specimensin comparison with uncoated specimens at 40◦C.

6.2 An alternative hypothesis

Since the friction measurements whose results are presented in section 6.1.2showed significant friction reductions with the DLC coatings in full film, evenwith surfaces with considerably higher roughness than whatis often seen asas the limit for solid-liquid slip, an alternative hypothesis was developed. Asdiscussed in detail in section 2.10, there is compelling evidence that most DLCcoatings have thermal conductivities that are much lower than steel. As a com-plement to the friction measurements presented in section 6.1.2, a numericalmodel was developed in Paper B, to investigate if there is a possibility that thelow thermal conductivity of the DLC coatings can affect the friction coefficientin an EHD contact.

6.2.1 Method

A one-dimensional numerical model was developed that features a thin lubri-cant film between two infinitely wide and flat, full film lubricated surfaces,with or without coatings subjected to a heat source. A schematic view is givenin Fig. 6.4. The solution is time dependent where the temperature rise in a

6.2. An alternative hypothesis 113

Figure 6.4: Schematic view of thermal model, not to scale.

line directed in the cross section of the film thickness passing the centre of thecontact from leading edge to trailing edge is calculated. Inthat way the tem-perature distribution in a plane crossing the centre of the contact is obtained.The time it takes for the line to move from the leading edge to the trailing edgeis calculated by dividing the Hertzian contact zone with thesurface velocity ofthe faster moving surface. The dimensions of the contact zone is approximatedby the Hertzian theory using the 52100 steel values for elastic modulus andPoisson’s ratio. Since the coatings are thin, the properties of the substrate areassumed to predominate the EHD deformation in the contact. The authors didnot have the possibility to perform thermal measurements onthe DLC coatingused in this study, and therefore the thermal properties were approximated toa thermal conductivity of 2.0 Wm−1K−1 and a heat capacity of 1.0 J kg−1K−1

from work performed by other authors [223,237–239]. Properties of the DLCcoating and substrate used in the model are given in Table B.2, and the oilproperties in Table B.1. A complete description of the numerical model andboundary conditions are given in section B.2.4. The purposeof the model wasto obtain an approximation of the temperature rise in a lubricant element pass-ing through the contact at different entrainment speeds andSRRs, and mostimportantly, the effect of a low thermal conductivity coating on the increase inlubricant temperature.

6.2.2 Lubricant temperature increase with DLC coatings

Figure 6.5 shows a graph with values obtained from the numerical simulationmodel. The presented temperatures represents the maximum value of the meancross film temperature increase in the lubricant as the computational domainpasses the contact zone. Maximum temperatures are presented for four dif-ferent combinations of entrainment speeds and SRRs. The four cases werechosen so that they represent widely varying running conditions. In two ofthe cases the entrainment speed is high, 7.7 m/s while the SRRis either low orhigh, 0.04 or 0.49 respectively. In the other two cases, the entrainment speed is


Oil

film

tem

per

atu

rein

crea

se[

◦C

]

DLC disc - DLC ballDLC disc - Steel ballSteel disc - DLC ballSteel disc - Steel ball

Ue:7.7,S:0.49 Ue:7.7,S:0.04 Ue:2.0,S:0.49 Ue:2.0,S:0.040

20

40

60

80

100

120

140

160

180

Figure 6.5: Simulated temperature increase in lubricant.

relatively low, 2.0 m/s, and again, the SRR is either 0.04 or 0.49. The simula-tion has been run for both the reference case with uncoated surfaces, and alsowith the other cases with one or both surfaces coated. All simulations havebeen performed with an initial temperature of 40◦C.

There is a significant increase in temperature compared to the uncoatedcase for all cases where one or both surfaces are coated with DLC. The high-est temperature increase is found for the case when both surfaces are coatedwith DLC, followed by the case where the ball is coated with DLC and thedisc is uncoated, and ultimately the case where only the discis coated withDLC. These findings are in line with the experimental resultsthat showed thebiggest decrease in friction for the cases where both surfaces were coated withDLC, followed by coating only the ball, and then coating onlythe disc. The oiltemperature increase is largest at the high sliding, high entrainment speed caseand drops primarily with SRR, but also with entrainment speed. At the sametime, the measured friction reduction is also larger at higher sliding speeds.Although the absolute difference in oil film temperature increase between thedifferent surface combinations vary with entrainment speed and SRR, the rel-ative difference seem to be more or less the same in the four cases. The nu-merical model clearly show that a 2 µm thin thermally insulating DLC coating

6.3. Thin layer thermal insulation 115

can substantially change the lubricant temperature in the EHD contact, whichis likely to have a noticeable effect on the friction coefficient. An increasedlubricant temperature will lower the viscosity and limiting shear stress of thelubricant, thus leading to lower friction. Furthermore, the increase in lubricanttemperature will only have a marginal effect on film thickness since the inletis not subjected to much heating compared to the hertz region. It is thereforelikely that the friction reduction measured for the combinations with one orboth specimens coated with DLC to some extent is due to the thermal insu-lation of the coating. If there is no solid-liquid slip in thecontact due to theroughness of the surfaces, or too strong solid-liquid interactions, the increaseof lubricant temperature in the contact may in fact be the only reason to thedecrease in friction. It is also possible that the friction reduction is an effect ofboth solid-liquid slip, and thermal insulation of the DLC coating.

6.3 Thin layer thermal insulation

To further investigate the hypothesis presented in section6.2 where the thermalinsulating properties of DLC coatings were proposed to leadto a reduction infull film EHD friction, a more advanced 3D numerical model wasused inPaper D. The numerical model which uses lubricant transportproperties andwell founded rheological models is described in detail in Chapter 5. A series offriction experiments were also conducted in a ball-on-discmachine where anuncoated pair of specimens was compared to a pair of DLC-coated specimensto validate the numerical results.

6.3.1 Method

The numerical predictions and experimental measurements were both per-formed using the same conditions as reported in Table 6.1. Both were per-formed with uncoated specimens, and DLC coated specimens. The coatingused was Tribobond 43, a hydrogenated amorphous carbon, (Cr+) a-C:H, ap-plied through PACVD. The coating thickness was measured to 2.8 µm usingCalotest. A chromium based interlayer with a thickness of 0.1-0.3 µm de-posited by magnetron sputtering were used to improve the adhesion. Moreinformation regarding the specimens is found in section D.2.4.1.

Both numerical predictions and experiments were performedwith the samelubricant, squalane, a commercially available low molecular weight branchedalkane (2,6,10,15,19,23-hexamethyltetracosane). An oilwithout additives was



Temperature 40◦ CContact load 80 and 300 NMaximum hertzian pressure 1.25 and 1.94 GPaEntrainment speed,Ue 1, 1.6 ,3.145, 6.144 m/sSlide to Roll Ratio, SRR 0.0002 to 1.05Coating none or Tribobond 43

chosen to minimize the effect of tribochemical reactions onthe friction coeffi-cient. At the test temperature of 40◦C, the ambient viscosity of squalane is 15mPas and the pressure-viscosity coefficient is 18 GPa−1 [129].

The experiments were carried out in a Wedeven Associates Machine (WAM)11, ball-on-disc test device described in section 2.11. Since the numericalmodel used in this work does not take asperity interactions into account, thetests were performed using grade 20 balls and polished discsgiving a com-bined surface roughness of 43 nm RMS to reduce the occurrenceof asperityinteractions during testing. The test procedure followingthe conditions pre-sented in Table 6.1 is given in section D.2.4.2.

6.3.2 Thermal insulation and friction

The results from the measurements and predictions are showntogether in Figs.6.6 and 6.7. The figures show how the friction coefficient is changing withSRR both in the measurements, and in the predictions. Results from othercombinations of loads and entrainment speeds are found in section D.3. It isclear that the DLC coating offers a substantial friction reduction, in the caseof Fig. 6.7, around 40%. Although the numerically predictedfriction coef-ficients deviate from the experimental ones in absolute value, the shape andtrend of the friction curves are qualitatively similar for both the coated and un-coated cases. In view of the results and discussions of the numerical model inChapter 5, a perfect correlation between the predictions and the measurementswas not expected. Nevertheless, the results from the numerical predictionsshow that the addition of a thin (2.8 µm) thermal insulating coating leads to asubstantial reduction of the friction coefficient in an EHD contact. Since thenumerical model does not incorporate any chemical/surfaceinteraction effects,such as asperity interactions or solid-liquid slip, the observed reduction in thenumerical results can only be attributed to thermal effects.

Figure 6.8 shows the temperature distributions in the Z direction acrossthe film and solids (substrate and coatings (for the case of coated surfaces))


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt

Experiment − UncoatedExperiment − CoatedSimulation − UncoatedSimulation − Coated

Figure 6.6: Friction results at 1.25 GPa, 40◦C at 3.145 m/s entrainment speedcomparing uncoated and coated specimens in both simulation and experiment.

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt


Figure 6.7: Friction results at 1.94 GPa, 40◦C at 6.144 m/s entrainment speedcomparing uncoated and coated specimens in both simulation and experiment.


from the inlet (X = -1.5) to the outlet (X = 1.5) at the same conditions as inFig. 6.7. X represents the position in the circular contact in the entrainmentdirection. In an EHD contact the film thickness, and thus the ability to separatethe surfaces from contacting each other is governed by the viscous behaviour ofthe lubricant in the inlet of the contact. On the other hand, friction is governedby the viscous behaviour of the lubricant in the central areaof the contact.Therefore a sufficiently viscous lubricant is required in the inlet to generate anadequate film thickness, while low viscosity is desirable atthe contact centreto reduce friction.

The difference in temperature in the inlet is very small, andwill thus lead tonegligible differences in film thickness between the coatedand uncoated case.In the central parts (X = -0.5 to X = 0.5) of the contact there isa substantialdifference in lubricant temperature. The higher temperature in the lubricantwith the coated specimens will therefore lead to lower viscosities and conse-quently lower friction. However, in the outlet of the contact (X = 1.0 and X= 1.5) the lubricant temperatures are actually lower with the coating. At thisstage, the lubricant film is ruptured, and would therefore not contribute to thefriction. The higher lubricant temperature at the contact outlet for the uncoatedcase results from heat transfer by conduction and advectionwithin the solidsfrom the centre of the contact (where most of the heat is generated) towards theoutlet. This leads to high solid surface temperatures at theexit of the contact,which maintains the lubricant film at a higher temperature. In the coated case,less heat is transferred to the liquid at the exit of the contact since DLC actslike an insulating material. Figure 6.9 shows the flow profile(the x-velocitycomponent of the lubricant flow across the film) at the centre of the contactfor both coated and uncoated surfaces at the same conditionsas in Fig. 6.7.The difference in velocity is large around the midlayer of the lubricant filmand near the faster moving surface. A lower velocity variation across the lubri-cant film around the midlayer, and thus lower shear rates, in combination withlower viscosities due to the higher lubricant temperaturesin the coated caseleads to the friction reduction compared to the uncoated case.

As seen in Figs. 6.6 and 6.7, the greatest reduction in friction with thecoating happens at the highest SRR, as expected since more heat is generatedat higher sliding speeds. Substantial friction reductionsare however achievedalready at much lower SRRs. Table 6.2 summarizes the friction reductionachieved with the coating for all entrainment speed and loadcases at the high-est SRR value of 1.05. It is clear that the relative reductionin friction is great-est at the highest entrainment speed and reduced when entrainment speed de-crease. Note that the relative reduction in friction is not much smaller at the


Lubricant CoatingCoatingPlane Ball

Lubricant BallPlane

Figure 6.8: Numerical simulation of lubricant film temperature in different loca-tions of the lubricant film for uncoated specimens and coated specimens.


Table 6.2: Friction reduction with DLC coating at SRR = 1.05.

Load [N] Entrainment Measured Predictedspeed [m/s] reduction [%] reduction [%]

300 6.144 41 48300 3.145 35 4680 6.144 39 4680 3.145 37 4680 1.611 30 3980 1.0 24 28

3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Liquid velocity (m/s)

Z

CoatedUncoated

Figure 6.9: Lubricant x-velocity profile at the centre of the contact across filmthickness. Z=1 is the faster moving surface (ball) and Z=0 is the slower movingsurface (disc).

low load compared to the high load. Even at the lowest load andthe lowestentrainment speed, Fig. D.7, the friction reduction is significant even at a SRRof 0.2. This indicates that excessive heat generation is notneeded to benefitfrom a thermally insulating coating, and that SRR has greater influence on thefriction reduction than contact pressure.

6.4 Coating thickness and friction in EHL

The findings presented in sections 6.1 to 6.3 strongly suggests that thermal in-sulation is of great importance for the observed friction reductions with DLCcoatings in full film EHL. To further increase the understanding of the fric-

6.4. Coating thickness and friction in EHL 121

tion reducing capabilities of DLC coatings in full film EHL, the solid-liquidslip theory discussed in section 2.9 is related to the hypothesis of thermal in-sulation. In Paper E, specimens coated with the same DLC coating, but withdifferent coating thickness are investigated in terms of friction reduction in fullfilm EHL, and compared to measurements of contact angle and surface energyof the coatings. By investigating two coatings with supposedly the same sur-face energy and contact angle, but with different thicknessand thus thermalproperties, the goal was to provide further information about the mechanismsbehind the full film EHD friction reducing capabilities of DLC coatings.

6.4.1 Method

A series of friction measurements were conducted in the WAM ball-on-disctest rig described in section 2.11 with the conditions listed in Table 6.3. Thetests were performed with both uncoated specimens, and specimens coatedwith Tribobond 43, a hydrogenated amorphous carbon, (Cr+) a-C:H. The spec-imens were prepared with two different coating thicknesses, 0.8 and 2.8 µm,measured using Calotest. The coatings were applied by PACVDand a chromiumbased interlayer with a thickness of 0.1-0.3 µm deposited bymagnetron sput-tering were used to improve the adhesion. The thermal conductivities of thecoatings were approximated to 1.75 W/mK for the 0.8 µm coating and 2.23W/mK for the 2.8 µm coating. An analysis using Fourier’s law of thermalconductive heat transfer indicated that the thinner coating, despite its lowerthermal conductivity, would conduct 2.75 times as much heatgiven identicalboundary conditions as the thicker coating. More details regarding the speci-mens are found in section E.2.1 and the test procedure is described in detail insection E.2.3.

The lubricant used for the tests was squalane, a commercially availablelow molecular weight (422.81 g/mol) branched alkane (2,6,10,15,19,23- hex-amethyltetracosane). A lubricant without additives was chosen to minimizethe effect of tribochemical reactions on the friction coefficient. At the testtemperature of 40◦C, the ambient viscosity of squalane is 15 mPas and thepressure-viscosity coefficient is 18 GPa−1 [129].

The surface energies of the specimens were evaluated with the Owens-Wend-Rabel-Kaelbe (OWRK) method using both demineralisedwater and di-iodomethane. Contact angle measurements were also performed with squalaneon all specimens to determine the wetting performance of thelubricant usedin the friction tests. Surface tension measurements were also performed onsqualane using the pendant drop method. Finally, a spreading parameter pro-



Temperature 40◦ CContact load 80 and 300 NMaximum hertzian pressure 1.25 and 1.94 GPaEntrainment speed,Ue 1.611, 3.145 and 6.144 m/sSlide to Roll Ratio, SRR 0.0002 to 1.05Coating thickness 0.8 and 2.8 µm

posed by Kalin and Polajnar [181] was used to evaluate the wetting perfor-mance between squalane and the different specimens. More information aboutthe surface energy measurements and calculations can be found in sectionE.2.4 while the surface tension measurements and spreadingparameter cal-culations are described in sections E.2.5 and E.2.6.

6.4.2 Results and discussion

The results from the surface energy calculations are shown in Fig. 6.10. Itshows that uncoated steel has the highest surface energy of 50 mJ/m2 whilethe DLC coatings have 48.1 mJ/m2 and 48.5 mJ/m2 for the 0.8 µm and 2.8 µmcoatings respectively. When looking at the distribution indispersive and polarcomponents of the surface energies for the investigated surfaces they are allvery similar in the sense that the dispersive component of the surface energy isdominant and only a small part is coming from the polar component.

Figure 6.11 shows the results for the calculated spreading parameters forthe combination of squalane and the different surfaces. In all cases the spread-ing parameter is positive, suggesting that the lubricant spreads over the sur-faces with time and does not obtain a constant contact angle immediately. Ac-cording to the definition, the lubricant will spread most easily on the uncoatedsteel surface, and less on the coated surfaces. In this case,the thicker coat-ing has a slightly higher spreading parameter suggesting that it would providelower wetting of the surface with squalane compared to the thinner coating.

Figure 6.12 shows the results from the contact angle measurements forsqualane on the specimens. The values were taken after about16 seconds andrepresents a steady state contact angle. The low contact angles indicate goodwetting on all specimens.

The results from one of the ball-on-disc friction measurements are shownin Fig. 6.13. More results can be found in section E.3 and theyall show thesame trend, although the absolute values are different. Theuncoated specimenshave the highest friction coefficients for all tested combinations of pressures

6.4. Coating thickness and friction in EHL 123

Steel DLC 0.8 DLC 2.80

10

20

30

40

50

60

70

Sur

face

ene

rgy

[mJ/

m2 ]

Total surface energyDispersive partPolar part

Figure 6.10: Surface energies and the corresponding dispersive and polar partsof investigated surfaces determined using the OWRK method.

Steel DLC 0.8 µm DLC 2.8 µm0

2

4

6

8

10

12

14

16

18

Spr

eadi

ng p

aram

eter

[mJ/

m2 ]

Figure 6.11: Spreading parameter values for squalane and the investigatedsurfaces.



1

2

3

4

5

6

7

8

Con

tact

ang

le [°

]

Figure 6.12: Contact angle for squalane on the investigated surfaces.

Table 6.4: Friction reduction with DLC coatings at SRR = 1.05.

Load [N] Entrainment Reduction Reductionspeed [m/s] 0.8 µm [%] 2.8 µm [%]

300 6.144 26 41300 3.145 20 3580 6.144 29 3980 3.145 23 3780 1.611 16 30

and entrainment speeds. The thinner DLC coating leads to a significant re-duction in friction coefficient, and friction is further reduced for the specimenswith the thicker coating. For all entrainment speeds the percental friction re-duction is increased with the increase of SRR. Table 6.4 shows the reductionin friction for all investigated cases at the highest SRR of 1.05.

If the friction reduction can be explained by the interaction between thesurface and the liquid as suggested by the solid-liquid sliptheory, it must bepossible to be determined by utilizing appropriate measurements. Such mea-surements have included contact angle, surface energy and spreading param-eter. However, in this work, the two investigated DLC coatings have almostidentical values for contact angle, surface energy, and spreading parameter,but still generate significantly different friction reductions compared to the un-coated specimens. These results indicates that measurements of contact angle,

6.5. In conclusion 125

0 0.2 0.4 0.6 0.8 1 1.20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt

UncoatedTB43 0.8 µmTB43 2.8 µm

Figure 6.13: Friction measurements for squalane at 6.144 m/s entrainmentspeed and 1.94 GPa of maximum hertzian pressure for uncoated steel andtwo different thicknesses of the same DLC coating.

surface energy, or spreading parameter are not sufficient todescribe the fullfilm EHD friction reduction capability of a surface coating.Studies that indi-rectly indicate solid-liquid slip as a consequence of reduced friction in EHLmay in fact, as indicated by this study, be an effect of thermal insulation, orpossibly, a combination of thermal insulation and solid-liquid slip.

6.5 In conclusion

In this chapter, several investigations have been presented with the goal ofelucidating the mechanisms behind the friction reduction achieved with DLCcoatings in full film EHL. From the empirical studies, the following importantfacts can be highlighted:

• DLC coatings provides a reduction in EHD friction even in the full filmregime where there is no asperity interactions between the two matingsurfaces.

• The full film friction reduction can be achieved by coating only one ofthe surfaces in contact, but the largest friction reductionis achieved bycoating both surfaces.


• The friction reduction is achieved even though the surfaceroughness ofthe specimens are one to two orders of magnitude rougher thanwhat isgenerally considered the limit of solid-liquid slip.

• When applying the same kind of DLC coating, but with different thick-ness to two pair of specimens, the thicker coating provides alarger fric-tion reduction than the thinner coating. The difference in friction reduc-tion is observed even if measurements showed that the two coatings hadalmost the same surface energy, contact angle and spreadingparameter.

The fact that two coatings with the same surface energy, contact angle andspreading parameters have significantly different friction reducing capabilitiesleads to an important conclusion. It is not possible to explain the friction re-duction observed in full film EHL with surface coatings by only consideringsolid-liquid interactions.

As an alternative to the solid-liquid slip theory another hypothesis wereproposed where the low thermal conductivities of DLC coatings was shown tohave an effect on the EHD friction coefficient. In a first step,a 1-dimensionalnumerical simulation model was developed to calculate the temperature in-crease in a finite lubricant element that passes the EHD contact. The resultsfrom the numerical model showed substantially higher temperature increasesof the lubricant when the surfaces were coated with DLC compared to un-coated surfaces. The highest temperature increases calculated with both sur-faces coated coincides with the largest friction reductionthat was measuredwhen both specimens were coated.

In a second step, a thoroughly validated thermal elastohydrodynamic lubri-cation (TEHL) numerical model was used to predict the friction coefficient forthe tested cases. A series of friction measurements were also made to validatethe numerical results. Although the numerically predictedvalues, which ig-nore accumulated heat, deviate from the experimental ones in absolute values,the shape and trend of the friction curves are qualitativelysimilar. The nu-merical model does not incorporate any chemical/surface interaction effects,and thus the observed reduction in friction for the predicted results can onlybe attributed to thermal effects. These findings presents strong evidence thatthermal insulation has an important role in full film EHL.

It is possible that some part of the friction reduction observed in the exper-imental results presented in this chapter is caused by a combination of solid-liquid slip and thermal insulation. It is however believed that the majority, orpossibly all of the friction reduction in these experimentsare due to thermalinsulation as an effect of the low thermal conductivities ofthe DLC coatings.

Chapter 7

Friction mapping and gearcontacts

As a means of reducing energy consumption and emissions in several fieldslike the automotive market, research efforts are spent to increase the efficiencyof machine components. It has been assessed that 33 % of the fuel energy in acar is used to overcome friction, and that 7-18 % of these friction losses orig-inates from the transmission [140]. Running experiments with full size gear-boxes from the real application has the advantage of giving realistic results interms of power losses depending on lubricant type, load, speed and operatingtemperature. The drawback is extensive costs, and the difficulty in differenti-ating between load dependent and load independent losses, and which part ofthe losses are coming from the gears, seals, bearings or synchronizers respec-tively. Even when a gear pair is rotating at a constant speed,several parametersare changing along the line of action between the meshing teeth, such as load,entrainment speed, and slide to roll ratio (SRR), as described in section 2.6. Byusing a ball-on-disc machine, or a twin disc test rig it is possible to measurefriction coefficients for specific entrainment speeds and SRRs. The possibilityto use friction measurements in these kinds of devices that would correlate wellwith gear tests would make friction predictions in gear transmissions cheaper,and more accessible for companies or institutes not having agear test rig.

In Paper F, an investigation is conducted where the correlation betweenfriction measurements in a ball-on-disc machine is compared to friction mea-surements in a FZG gear test rig. In addition, using the concept of frictionmapping introduced in Chapter 4, a method is proposed to predict the frictioncoefficient in an arbitrary spur gear pair from a small amountof measurements

127

128 Chapter 7. Friction mapping and gear contacts

Table 7.1: Lubricant properties.

Name Emgard Shell StatoilMTF 4250 Spirax S6 Gearway S5

Classification 75W-90 75W-90 75W-140Density @ 15◦C [kg/m3] 879 878 872Kinematic viscosity @ 40◦C [mm2/s] 140 115 190Kinematic viscosity @ 100◦C [mm2/s] 18.4 15.2 25Dynamic viscosity @ 40◦C [mPas] 123 101 166Dynamic viscosity @ 100◦C [mPas] 16 13 22

in the ball-on-disc machine.

7.1 Method

The gear tests were performed in a modified FZG rig described in section F.2.2.Tests were performed in a range of rotational speeds from 750to 2000 rpm insteps of 250 rpm. The tests were conducted at two torque levels, 302 Nm, and 0Nm. By running tests with minimal torque, the load independent losses for thesystem could be obtained for every rotating speed. An empirical formula wasused to calculate the load dependent bearing losses. By subtracting the powerlosses from the full load test, with the losses from the no-load test and thecalculated load-dependent bearing losses, the contact friction losses for eachspecific rotational speed could be assessed. Finally, an approximate averagefriction coefficient for each rotational speed were calculated. A more detaileddescription of the gear test procedure and the calculationsare given in sectionF.2.4.

The purpose of the study was to create conditions in the ball-on-disc testrig, described in section 2.11, as similar as possible to thegear tests, and com-pare the friction data between the tests. For this reason, the surface roughnessvalues were very similar for the ball-on-disc specimens, and the gear speci-mens. The combined RMS roughness was 43 nm for the ball-on-disc speci-mens, and 42 nm for the gear specimens. More information about the spec-imens can be found in section F.2.3. The experiments were performed withthree commercially available, fully formulated gearbox oils whose propertiesare shown in Table 7.1.

An analytic geometric model of the gear contact was used to calculatethe maximum Hertzian pressure with 302 Nm of torque to approximately 1.24GPa. The same model was also used to calculate SRRs and entrainment speedsalong the line of action. The SRR in a gear contact is independent of rota-

7.1. Method 129

Table 7.2: Investigated conditions in ball-on-disc rig.

Temperature 40 and 70◦CContact load 76 NMaximum hertzian pressure 1.24 GPaEntrainment speed,Ue 1 - 4 m/sSlide to Roll Ratio, SRR 0.0002 - 1.2

tional speed and was calculated to range from -1.1 to 1.1 in the gears used inthis investigation. Since both pinion and gear have the sameproperties, theentrainment speed is constant along the line of action, but dependent on therotational speed of the gears. The lowest tested rotationalspeed was 750 rpmthat corresponds to an entrainment speed of 1.37 m/s, while the highest speedof 2000 rpm gives an entrainment speed of 3.66 m/s.

The ball-on-disc tests were performed with the same three oils as the geartests and at the same temperatures, 40 and 70◦C. In the gear setup, the contactpressure is changing along the line of action due to changes in effective radius,but also since the load is sometimes carried by only one toothengagement andotherwise by two teeth. To be able to use the concept of friction mapping intro-duced in Chapter 4, all tests in the ball-on-disc machine were performed witha load of 76 N, equivalent to a maximum Hertzian pressure of 1.24 GPa whichis the same as the maximum pressure calculated for the gear contact. Frictiondata for various entrainment speeds and SRRs were measured in a range thatspans the minimum and maximum values calculated for the gearcontact atthe different rotational speeds as detailed in Table 7.2. A triangle based lin-ear interpolation method was used to create a friction map for a each lubricantat a specific pressure and temperature for a range of entrainment speeds andSRRs. An example of a 2D friction map is given in Fig. 7.1 wherefrictioncoefficient is plotted as contours with entrainment speed and SRR at the x andy axis. The figure also contains projections of the corresponding entrainmentspeeds and SRRs for three different rotational speeds, 750,1250 and 2000rpm, for the gear pair investigated in this work. Included inthe figure is alsothe corresponding entrainment speeds and SRRs for a FZG typeC gear pairat 2300 rpm. Here the entrainment speed is not constant alongthe line of ac-tion since gear and pinion are not identical. The friction map can be seen as alook-up-table for an arbitrary rotational speed of a gear pair. The line of actionwere divided into a number of finite data points in the friction map for a spe-cific rotational speed and used to calculate a mean friction coefficient that wascompared to the mean friction coefficient obtained from the same rotationalspeed in the gear test.



Slid

e to

rol

l rat

io

0.03

0.0350.04

0.0450.05

1.5 2 2.5 3 3.5 4

0.2

0.4

0.6

0.8

1

Figure 7.1: Example of friction map with corresponding entrainment speedsand SRRs along the line of action for tested gear pair at 750 rpm (solid line),tested gear pair at 1250 rpm (dashed line), tested gear pair at 2000 rpm(dashed/dotted line) and FZG type C gear pair at 2000 rpm (dotted line).

7.2 Ball-on-disc and gear contact friction

The results from the ball-on-disc friction measurements and the FZG tests areshown in Figs. 7.2 and 7.3 for the oil temperatures of 40 and 70◦C respectively.The most important observation is that the ranking of the oils are the same inboth test rigs, at both oil temperatures. It is clear that thelowest contact frictionis achieved with the Gearway oil that has the highest viscosity, while the high-est contact friction is achieved with the Spirax oil having the lowest viscosity.This is the case for both 40 and 70◦C. However, it should be mentioned thatthese differences may not solely depend on the viscosities of the lubricant, butalso other properties, such as the limiting shear stress, and different additivepackages. Furthermore, the shape of the friction curves arerelatively similarbetween the test rigs indicating that the ball-on-disc measurements are reason-ably well capturing the behaviour of the gear contact. The agreements betweenthe ball-on-disc measurements and the gear tests are well inline, or better thancomparisons between twin disc and gears found in literature[153,240,241].

However, there is a difference in absolute friction values between the ball-on-disc and gear tests. One of the reasons are most likely thesimplificationof using only one pressure in the ball-on-disc measurementsto make the use

7.2. Ball-on-disc and gear contact friction 131

800 1000 1200 1400 1600 1800 20000.01

0.02

0.03

0.04

0.05

0.06

0.07

Rotational speed [rpm]

Fric

tion

coef

ficie

nt

Gearway − WAMGearway − FZGSpirax − WAMSpirax − FZGEmgard − WAMEmgard − FZG

Figure 7.2: Friction coefficients for ball-on-disc and gear tests for three differentlubricants conducted at an oil temperature of 40◦C.

800 1000 1200 1400 1600 1800 20000.01

0.02

0.03

0.04

0.05

0.06

0.07


Fric

tion

coef

ficie

nt


Figure 7.3: Friction coefficients for ball-on-disc and gear tests for three differentlubricants conducted at an oil temperature of 70◦C.


of friction mapping possible. This pressure corresponds tothe pressure in thegear contact when only one tooth is carrying the load. In parts along the lineof action where two teeth are carrying the load, the pressurewill be lower andthe friction coefficients will drop due to the reduction in pressure. This is onereason for the ball-on-disc friction values to be slightly higher than the valuesfrom the gear test, which is also the general case in Figs. 7.2and 7.3. More-over, the deviations generally increase with rotational speed which is mostlikely caused by different thermal properties of the two systems. It is expectedthat thermal effects will increase more with rotational speed in the gear setupthan in the ball-on-disc setup due to the higher frictional power generated inthe gear rig.

Although the correlation between the gear tests and the ball-on-disc testsare not always spot on in terms of absolute friction coefficients, the shape of thefriction curves are the same in both machines. Moreover, theranking betweenthe oils are maintained for the two different rigs suggesting that the use offriction testing in a ball-on-disc machine would be a good alternative to a fullgear test for evaluating lubricants in terms of friction performance. In addition,the use of the friction mapping concepts gives a gear developer the possibilityto assess an approximate friction coefficient for a type of gear with a specificset of entrainment speeds and SRRs without having to manufacture such agear. A new gear pair, at a specific rotational speed, will cover a different setof entrainment speeds and SRRs compared to the examples given in Fig. 7.1,and if the friction map is sufficiently large, an enormous amount of differentgear geometries and rotational speeds could be assessed without having to runany more ball-on-disc tests. The exception being large differences in contactpressures that could possibly occur when changing gear geometries requiringanother ball-on-disc test to be performed at another pressure.

7.3 In conclusion

It has been shown that the friction coefficients obtained in acircular contact ina ball-on-disc machine can be used to predict the contact friction in a spur gearline contact with reasonable accuracy. This finding leads tothe possibility toperform cheaper tests in a ball-on-disc machine to evaluatedifferent lubricantsin terms of friction performance compared to full gear tests. It also gives thepossibility to assess the frictional losses in several theoretical gear pairs withdifferent geometries by performing a small amount of ball-on-disc tests withone, or several lubricants.

Chapter 8

Conclusions

The concept of friction mapping has been introduced as a complement to thecommonly used Stribeck and µ-slip curves often encounteredin literature.Friction mapping is useful to characterize the friction behaviour of a systemconsisting of lubricant properties, operating temperature, surface roughnessand possibly a surface coating in a wide range of entrainmentspeeds and slideto roll ratios. It has been shown that a change in lubricant temperature or vis-cosity can both increase or reduce friction depending on entrainment speed andslide to roll ratio.

A numerical model has been presented where lubricant transport propertieswere used to predict the friction coefficient in a circular EHD contact and com-pared to experimental measurements in a ball-on-disc machine. In absolutevalues the numerical results were found to be good considering the true pre-diction approach. The inclusion of temperature dependenceof limiting shearstress was identified as the most important area of improvement of the model.

By coating one, or both of the surfaces in an EHD contact with aDLCcoating, friction is decreased, even in the full film regime.Coating both sur-faces with DLC gives a larger friction reduction compared tocoating only oneof the surfaces. It has been shown that the friction reduction achieved withDLC coatings in full film EHL is not necessarily governed by solid-liquid slipas suggested in much of the literature, but rather due to a thermally insulatingeffect of the DLC coating. A numerical model was used to investigate the ef-fect of thin, thermally insulating layers in EHL. The results show that a thin,thermally insulating layer will increase the lubricant temperature inside of thecontact and consequently reduce the friction coefficient while only marginallyinfluencing the film thickness. The numerical model does not incorporate anychemical/surface interaction effects, and thus the observed reduction in friction

133

134 Chapter 8. Conclusions

for the predicted results can only be attributed to thermal effects.Experiments conducted in EHL with two identical DLC coatings, but with

different thickness, show that both coated cases reduces friction compared touncoated specimens, but that the thicker coating provides alarger reductionthan the thinner coating. Surface energy and contact angle measurements showonly marginal differences between the two coating thicknesses, which indicatethat it is not possible to explain the friction reduction achieved in full film EHLwith surface coatings by only considering solid-liquid interactions.

These findings open a new realm of coatings where the focus arethe ther-mal properties, resulting in cheaper and more effective coatings to reduce fric-tion.

The friction coefficients obtained in a circular contact in aball-on-discmachine can be used to predict the contact friction in a spur gear line con-tact with reasonable accuracy. This finding leads to the possibility to performcheaper tests in a ball-on-disc machine to evaluate different lubricants in termsof friction performance compared to full gear tests. It alsogives the possibil-ity to assess the frictional losses in several theoretical gear pairs with differentgeometries by performing a small amount of ball-on-disc tests with one, orseveral lubricants.

Chapter 9

Future Work

The investigations conducted in this thesis has led to a number of researchtopics that would be interesting to address in future work:

• The temperature dependence of the limiting shear stress must be in-cluded in numerical models of EHD friction in order to achieve evenbetter predictions.

• Is it possible by clever use of ball-on-disc measurements to obtain lubri-cant data to enable accurate friction predictions without the use of highpressure rheology measurements and molecular dynamics simulations?

• More studies are needed to investigate solid-liquid slip and thermal in-sulation effects of surface coatings in EHL. Is solid-liquid slip actuallytaking place in systems with engineering roughness and lubricants usedin machine components, and where are the limits in terms of contactpressure and surface roughness for solid-liquid slip to occur?

• How large friction reductions can be achieved in EHL by designing anew type of surface coating with the aim of maximizing the capabilityof thermal insulation?

• Is it possible to use experimental measurements in a ball-on-disc ma-chine to assess the friction coefficients in more complex gear geome-tries such as helical gears and hypoid gears in a similar way as it hasbeen applied to spur gears?

135

136 Chapter 9. Future Work

Part II

Appended Papers

137

Paper A

EHL friction mappingThe influence of lubricant,roughness, speed and slide to rollratio

139

141

Proceedings of the Institution of Mechanical Engineers, Part J: Journal ofEngineering Tribology, 2011 May; vol. 225, Issue 7, p. 671-681.

With kind permission of SAGE Publications Ltd

EHL friction mappingThe influence of lubricant, roughness, speed and slide to

roll ratio

M. Bjorling, R. Larsson, P. Marklund and E. Kassfeldt

Luleå University of Technology, Division of Machine Elements,Luleå, SE-971 87 Sweden

Abstract

A friction test is conducted in a WAM ball-on-disc test rig. The output fromthe test is friction coefficient versus entrainment speed and slide-to-roll ratiopresented as a 3D friction map. A number of parameters are varied whilestudying the friction coefficient; surface roughness, baseoil viscosity, base oiltype and EP additive package. Entrainment speed, slide to roll ratio and oiltemperature are also varied. The results show that the mapping is efficientin showing the different types of friction that may occur in an EHL contact.The results also show that the friction behaviour can be strongly influencedby changing surface roughness as well as base oil viscosity,base oil type, EPadditive content and operating temperature.

142 Paper A. EHL Friction mapping

A.1 Introduction

Reducing losses in transmissions has become a priority in the automotive mar-ket during the latest years, mainly due to environmental aspects leading toregulations on the automotive industry to drive the development of cars withlower fuel consumption. Rising fuel prices and increasing environmental con-cern also makes customers more prone to purchase more fuel efficient vehicles.

In addition to the fuel savings that could be done by increased efficiencyof transmissions there are other benefits as well. A more efficient transmissionwill in general generate less heat, and experience less wear. This will leadto fewer failures, longer lifetime of components, and possibly longer serviceintervals. Furthermore this implies a possibility to reduce coolant components,thus reducing the total weight of the system, leading to further decrease inconsumption and a lower impact on nature due to a reduction ofmaterial usage.A low weight design is also beneficial for vehicle dynamics and handling.

In some cases a substantial part of the losses in an automotive transmissionis attributed to gear contact friction due to sliding and rolling between the gearteeth. The total transmission losses due to gear contact friction are ranging be-tween 4.5 and 50 percent [165,242,243]. Generally gearboxes running at lowspeeds and high loads have a substantial part of gear contactlosses, whereashigh speed applications usually dominates by churning losses. Since churn-ing losses mostly are related to the oil viscosity, the best system performanceought to be obtained by optimizing the gear contact to work with as low vis-cosity as possible to minimize churning losses while still keeping a low wearof the system.

Much research has been conducted on the topics of gear contacts coveringcontact geometry, materials, surface topography, coatings as well as lubricantproperties and rheological effects. Several authors have presented papers de-scribing and modeling the gear contact behaviour. Already 1966 Dowson andHigginson [62] used their EHL theory to predict the film thickness betweentwo gear teeth at the pitch point. This work was extended by Gu[244] toinclude the whole line of action in an involute gear mesh. This study alsoincluded an approximate thermal model. The complexity of the models hasincreased over the years to consider more parameters and gives more accurateresults. Recently Akbarzadeh and Khonsari [145] presenteda model for calcu-lating the friction in spur gears considering shear thinning and surface rough-ness. Their model uses the scaling factors of Johnsonet al. [245], to predictfriction from the hydrodynamic respectively asperity contact part, and Green-wood and Tripp [246] formula for contacts between two rough surfaces. They

A.1. Introduction 143

later extended their model to incorporate thermal effects in the analysis [146].

Parallel with the theoretical research, experiments have been carried outfor different purposes. One is of course for validation of mathematical models,but it is also a way to test how altering different factors, such as lubricantproperties and surface roughness influences for instance the efficiency of thesystem. Experiments are conducted in various tests rigs, for instance the FZGback-to-back, as well as in twin disc, and ball-on-disc configurations. Allof these has their own benefits and disadvantages which must be consideredtogether with the purpose of the actual experiment. Experiments with realgears are of course closest to the real application, and in some cases the actualgear box is mounted in a test rig and the experiments are performed. Thisapproach gives very good indications on how the real system performance isinfluenced by changing certain parameters, and standardised methods like theFZG should make it easy to replicate results [153,154,162,169,247]. However,testing with real gears is generally the most expensive way of testing, and it isalso hard to draw any detailed conclusions since there are many componentsinfluencing the results and the output generally is an average friction valueeven though contact load, radii, entrainment speed and slide to roll ratio (SRR)are continuously changing along the line of action.

Many authors have used twin-disc test devices to simulate power loss andwear behaviour of gear contacts [169, 241, 247, 248]. The benefits comparedto gear testing are lower costs, and the possibility to in detail study param-eters like friction coefficient at specific entrainment speeds and slide to rollratios. Furthermore it is possible to simulate the gear contact without otherfriction losses present in gears (like churning in dip lubricated gears and bear-ing losses).

The ball-on-disc configuration shares the benefits of the twin disc, and isalso easier to control since there are not the same alignmentissues. In thepresent study a Wedeven Associates Machine (WAM) ball-on-disc configura-tion is used to study the friction behaviour in various entrainment speeds andslide to roll ratios. Additional parameters studied includes: surface roughness,base oil type, base oil viscosity, oil temperature and additive packages. Theoutput from the test is friction coefficient versus entrainment speed and slide-to-roll ratio presented as a 3D friction map. Ball on disc friction experimentshave earlier been carried out to investigate EHL film formation and frictionbehaviour during rolling and sliding [24,179].


A.2 Method

The following sections cover a description of the ball-on-disc test rig, the testspecimens and lubricants, and an overview of the test procedure.

A.2.1 Ball on disc tribotester

The experiments are conducted in a Wedeven Associates Machine (WAM)ball-on-disc test device, model 11, where the contact is shown in detail in Fig.A.1. The WAM utilizes advanced positioning technology for high precisiontesting under incipient sliding conditions. The ball and the disc are driven byseparate electric motors, the former to a speed up to 25000 rpm and the lat-ter up to 12000 rpm. Each motor is adjustable on-line to change entrainmentspeed and slide to roll ratio. The standard ball specimen hasa diameter of20.637 mm and the disc has a diameter of 101 mm. With standard sized testspecimens an entrainment speed of up to 27 m/s is possible under pure rollingconditions, and the maximum load of 1000 N which gives a maximum circularHertzian contact pressure of 2.91 GPa.

The test device contains a built in cooling and heating system allowingfor lubricant test temperatures between 5 and 100◦C. A closed loop systemsupplies the ball-on-disc contact with new lubricant.

Load cells are used to measure the force on the three principal axes wherethe machine operates, X, Y and Z. The test device also measures shaft rotatingspeeds, oil pump speed and values from up to twelve thermocouples. In thecurrent setup three thermocouples are used. One is located in the oil bath, onein the outlet of the oil supply and one measures oil film temperature very closeto the inlet region in the ball-on-disc contact.

The lubricant is supplied to the contact through the oil dispenser in themiddle of the disc in Figs. A.1 and A.2. The oil dispenser has several smallholes on the underside which distributes the lubricant on tothe disc. The sup-ply to the oil dispenser is secured by a hose pump delivering approximately 60ml/min which is connected to the oil sump.

A.2.2 Test specimens and lubricants

Two different pairs of test specimens were used in the test. The first pair,referred to as "smooth" is made from AISI 52100 bearing steel, where theballs are taken direct from factory and the discs are processed the same wayas raceway material. These specimens both have a hardness ofHRC = 60and very smooth surfaces (approximately 30 nmSa for the ball and 55 nm

A.2. Method 145

Figure A.1: WAM ball-on-disc test device.

Figure A.2: Schematic sketch of the WAM ball-on-disc test device.


Table A.1: Roughness and standard deviation values of unworn discs.

Material Diameter [mm] Sa[nm] Sq[nm] StdSa StdSq

9310 95 220 282 13 179310 81 214 274 11 169310 60 221 283 7 1052100 95 55 77 2 452100 81 54 76 3 652100 60 57 81 2 4

Table A.2: Test oil data.

Type SL324 SL211 SL212 SL326Additives None None EP None

Kin. Visc @ 40◦C 103 30.8 30.7 109.3[cSt]

Dyn. Visc @ 40◦C 94.5 27.1 27.1 94.9[mPas]

Kin. Visc @ 100◦C 15.6 5.3 5.3 11.98[cSt]

Dyn. Visc @ 100◦C 13.4 4.46 4.46 9.97[mPas]

Density @ 15◦C 928 872 872 885[kg/m3]

Viscosity Index 157 104 104 99Type Ester Mineral Mineral Mineral

Sa for the disc) whereas the second pair, referred to as "rough"is made ofAISI 9310 gear steel for both ball and disc providing a rougher grind closer togear roughness, approximately 220 nmSa for the disc and 200 nmSa for theball. The 9310 disc is case carburized to a depth of about 0.8 mm and has ahardness of HRC = 63. Both discs have a circumferential grind. The roughnessof the discs was measured with a Wyko NT1100 optical profilingsystem fromVeeco. Measurements were done using 10x magnification and 0.5x field ofview (FOV). The measurements were made at different diameters of the discs.For each diameter, mean values of seven measurements on different positionsof the discs are presented together with standard deviationin Table A.1.

Four different lubricants were used in the study. Two pure mineral base oilswith the same dynamic viscosity, 27.1 mPas at 40◦C, one of them with a twopercent EP additive content, and one pure mineral base oil ofthe same type butwith a dynamic viscosity of 94.5 mPas at 40◦C. One saturated synthetic esterwith a dynamic viscosity of 94.9 mPas at 40◦C were also tested. The lubricantdata is presented in Table A.2.

A.3. Results and discussion 147

A.2.3 Test procedure

The test cycle covers entrainment speeds between 0.34-9.6 m/s and slide to rollratios from 0.0002 to 0.49, or 0.02 to 49 % slip as used in the present paper. Inall cases of slip the ball rotates faster than the disc. SRR, or slip is defined asthe speed difference divided with the mean entrainment speed. After the test,surfaces were measured in the Wyko to observe eventual changes in surface to-pography. Before each test the device and specimens were thoroughly cleanedwith heptane and ethyl alcohol, and the test device warmed upapproximately60 minutes before starting the test with lubricant circulation to ensure temper-ature stability. During the warm up sequence the entrainment velocity is setto 2.5 m/s and there is no load applied, but the ball is positioned very close tothe disc so that lubricant is circulated over the ball to ensure warm up. Whentemperature stability is reached a 200 N load, equivalent to1.7 GPa Hertzianpressure is applied and the machine calibrated for pure rolling. The machineis run 20 minutes with these settings to ensure a mild run-in.The test cycleis then started which contains several loops where the slip is held constant foreach loop and the entrainment speed is varied from 9.6 to 0.34m/s. In the firstloop the slip is held at 0.02 % and is increased with each loop until it reaches 49%. The test cycle is then repeated in the same track for both ball and disc untilthe absolute traction coefficient does not vary more than a maximum of 0.002from the previous test cycle, excluding slip below 0.16 % where the machinescatters a bit.

When this occurs, the system is considered run in, and the data from thefinal test cycle is used for evaluation. The oil bulk temperature and the temper-ature at the disc surface is deviating less than± 1.5◦C from the target tempera-tures of 40 and 90◦C during testing. However, the actual contact temperaturesare higher than the bulk oil temperature. In the most severe cases with high en-trainment speed, SRR and coefficient of friction, the contact temperature willincrease several tens of degrees. [229]

The logged data from each test is processed separately. All measured val-ues from a specific running condition is averaged, and a triangle based linearinterpolation is used between the data points. The result iseither plotted as a3D map, or as a 2D contour map.

A.3 Results and discussion

The different test cases are shown in Table A.3, also containing surface rough-ness information after completed tests. Figure A.3 shows a 3D friction map of


Table A.3: Test cases.

Track Oil Temp[◦C] Material Sa[nm]diameter[mm] (AISI) (After test)

70 SL326 90 9310 18774 SL211 90 9310 18877 SL212 90 9310 18696 SL324 90 9310 19794 SL326 40 9310 19480 SL212 40 9310 18382 SL211 40 9310 19192 SL324 40 9310 20176 SL326 90 52100 5474 SL211 90 52100 5568 SL212 90 52100 5593 SL324 90 52100 5692 SL326 40 52100 5490 SL211 40 52100 5694 SL212 40 52100 5295 SL324 40 52100 55

one of the test cycles, where friction coefficient is displayed versus entrainmentspeed and slip. Here one can see the high gradient in frictioncoefficient whenlittle slip is induced from pure rolling, as well as the general decrease in fric-tion coefficient with increased entrainment speed. In addition to the 3D map, acontour map is usually more suitable for comparisons. Figure A.4 shows a 2Dcontour map corresponding to the friction map in figure A.3. All the results inthe present work are presented in 2D contour maps Figs. A.7(a)- A.10(d).

Figure A.5 shows a schematic 2D friction map divided into four differentregimes, with reasoning from Johnson and Tevaarwerk [124].In the linearregion "L", shear stress is proportional to shear rate, and are in the presentedmaps barely visible. The non-linear region, "NL", is dominated by shear thin-ning effects. In the thermal region, "T" the shear stress decreases with increas-ing shear rate. Finally one region is marked "M", where asperity contact occursbetween the surfaces, which is the mixed lubrication regime. The boundariesof these regions are exclusive for each system, depending onrunning condi-tions as well as on the material and lubricant parameters.

The location of the mixed lubrication boundary is assumed tobe wherethe coefficient of friction is no longer decreasing with increasing slip for acertain entrainment speed, which would imply incipient asperity interactions.However, this is a floating boundary controlled by several parameters, among


Figure A.3: 3D friction map - SL211, 40◦C, smooth.


Slip

[per

cent

]

0.0350.040.0450.050.0550.06

0.065

0.07

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

45

Figure A.4: 2D Contour map - SL211, 40◦C, smooth. Dashed lines indicatesoperating conditions for a FZG type A gear at 3200 RPM.


Figure A.5: 2D map - Regimes.


Slip

[per

cent

]

220

200180

160140

120100

80

60

40

20

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

45

Figure A.6: Power map [MW/m2].



Slip

[per

cent

]

0.0350.04

0.0450.050.0550.060.065

0.07

0.055

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

45

(a) Smooth surfaces, 40◦C


Slip

[per

cent

]

0.035

0.04

0.0450.05

0.0550.060.

065

0.04

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45

(b) Smooth surfaces, 90◦C


Slip

[per

cent

]

0.035

0.040.0450.050.055

0.06

0.065

0.07

0.0750.08

0.08

50.

09

0.0551 2 3 4 5 6 7 8 9

0

5

10

15

20

25

30

35

40

45

(c) Rough surfaces, 40◦C


Slip

[per

cent

]

0.045

0.07

50.

080.

085

0.09

0.07

0.065

0.06

0.055

0.05

0.045

0.04

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

45

(d) Rough surfaces, 90◦C

Figure A.7: 2D friction maps - SL211, low viscosity oil without EP additive.


others the balance between increasing asperity interactions, and the decreasein limiting shear strength and increased shearability withthe increase in tem-perature associated with increased slip, both affecting coefficient of friction,but in opposite ways.

The location of the thermal boundary is assumed to be where the limitingshear strength is reached for the lubricant, and thus entering the region wherethermal effects dominates the coefficient of friction.

For each combination of running conditions a mapping of the input powerof the system can be done by multiplying the measured friction coefficient withentrainment speed, SRR and contact pressure. Figure A.6 shows the powerinput corresponding to the case in figure A.3. It is evident that reducing coef-ficient of friction in the high speed, high slip region has a bigger influence onthe total frictional losses compared to a reduction of friction coefficient in thelow speed, low slip region.

The coefficient of friction is most commonly presented against SRR ina plot often referred to as a "µ-slip curve", or against entrainment speed in aStribeck like curve. The friction maps presented here provide advantages com-pared to these more conventional ways of presenting friction results. Since thefriction maps contain both SRR, entrainment speed and coefficient of friction itcan replace several µ-slip and Stribeck curves, and it is easy to see how changesin entrainment speed and SRR influences the friction coefficient. There is alsopossible to indentify the regimes discussed above and thus understand whichmechanisms that are mainy contributing to the friction in different areas of themap. By indicating the typical running condition for a certain application, forinstance a gear mesh, or a cam follower one could approximately calculatecontact friction power loss for the application, and also determine where it ismost suitable to put in resources to lower the friction coefficients, and hencethe power loss. In figure A.4 the operating conditions (entrainment speed andSRR) for a FZG type A gear at 3200 RPM is indicated as two dashedlines.

A.3.1 Surface roughness

It is evident from the test results that the surface topography has a rather biginfluence on friction characteristics. In all cases where oil type and temperatureare kept at the same level the test cycles performed with the smoother surfacesgives lower coefficients of friction. Furthermore, the mixed lubrication regimeis larger when rough surfaces are used, not surprising sincethe transition fromfull film to mixed lubrication is reached at higher entrainment speeds. Thiscould be examined in pairs, for instance, Figs. A.7(a) and A.7(c) or Figs.



Slip

[per

cent

]

0.0250.03

0.0350.040.045

0.050.055

0.06

0.065

0.07

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.035

0.040.045

0.05

0.055

0.06

0.06

50.

07

0.04

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.0250.030.0350.04

0.0450.05

0.055

0.06

0.065

0.070.0750.08

0.0551 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.035

0.040.045

0.05

0.055

0.06

0.06

50.

07

0.075

0.04

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45


Figure A.8: 2D friction maps - SL326, high viscosity oil without EP additive.

A.7(b) and A.7(d).

In case of the smooth disc and ball pair, the observations made after thetest cycles showed only minor signs of wear on the disc surface, and a verysmall wear track on the balls, suggesting a mild running in ofthe surfaces.This is also confirmed by the surface roughness measurementswhere theSa

values presented in table A.3 is very close to the measurements made outsidethe tracks presented in table A.1. The rough balls and disc onthe other handshow more pronounced wear, where theSa values, in the wear track on thedisc, generally are 30-40 nm smoother compared to the unworndisc. It is,however, difficult to draw any conclusions from the difference in wear whenusing the various oil and temperature combinations. This isbecause the result-ing surface measurements show very similar results, within20 nm inSa forall wear tracks, and that the difference could as well be attributed to differentamounts of roll-overs, since the tracks are located on different diameters on the



Slip

[per

cent

]

0.035

0.040.0450.05

0.0550.060.065

0.07

0.0551 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.035

0.040.045

0.05

0.05

5

0.06

0.04

0.035

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.035

0.04

0.0450.05

0.055

0.06

0.065

0.07

0.0750.08

0.08

5

0.09

0.051 2 3 4 5 6 7 8 9

0

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.05

0.055

0.06

0.0650.07

0.07

5

0.08

0.08

5

0.09

0.09

5

0.1

0.05

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45


Figure A.9: 2D friction maps - SL212, low viscosity oil with EP additive.

disc, and that tests have not been run the same number of timesfor all com-binations. Furthermore is the surface roughness varying atdifferent locationson the disc even before the tests are performed as shown by Table A.1. How-ever, there is an indication that tests performed with the thicker oils, SL324and SL326 were less worn.

A.3.2 Temperature and viscosity

With increasing temperature, pressure-viscosity coefficient, limiting shear stressand viscosity are decreasing [249,250]. As a consequence ofthis, the lubricant,and therefore also the friction characteristics, will havedifferent behaviours at40 and 90◦C. At the lower temperature the friction coefficients will increasemore with increased slip, and also reach higher values due toa higher lim-iting shear strength. However, after the peak value, the friction coefficients



Slip

[per

cent

]

0.020.0250.030.035

0.04

0.045

0.05

0.0550.06

0.065

0.055

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.03

0.0350.04

0.045

0.05

0.055

0.06

0.0350.03

1 2 3 4 5 6 7 8 90

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.0250.030.0350.040.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45



Slip

[per

cent

]

0.030.0350.04

0.045

0.05

0.055

0.06

0.0650.07

0.075

0.03

1 2 3 4 5 6 7 8 9

5

10

15

20

25

30

35

40

45


Figure A.10: 2D friction maps - SL324, ester oil without EP additive.


will decrease faster as well. This behaviour can be seen in Figs. A.7(a) andA.7(b) or Figs. A.8(a) and A.8(b), where the friction coefficients increases inthe thermal region, and decreases in the Linear and Non Linear regions whenthe lubricant temperature is increased from 40 and 90◦C for cases with smoothsurfaces. The same behaviour is observed for the ester as seen in Figs. A.10(a)and A.10(b).

Cases with rough surfaces follow the same trend, however showing higherfriction coefficients in average due to a transition from full film to mixed lu-brication regime at higher speeds, thus increasing the areaof the mixed lubri-cation regime, see Figs. A.7(c) and A.7(d) or Figs. A.8(c) and A.8(d).

An interesting comparison is between the high viscosity oilSL326 at 90◦CFig. A.8(b), and the low viscosity oil SL211 at 40◦C, Fig. A.7(a) tested withsmooth surfaces. In this case the SL211 has three times as high viscositycompared to SL326 since they are run at different temperatures. Despite thisthe latter show lower coefficients of friction with the exception of a region withhigh speeds and high slip. This comparison also indicates that the film formedin both cases are sufficient, and that the lower limiting shear strength, andmore easily sheared films at higher temperature is beneficialfor the frictioncoefficients.

Changing the lubricant viscosity from low to high with smooth surfacesshow small differences, Figs. A.7(a) and A.8(a) or Figs. A.7(b) and A.8(b),suggesting that the film thickness in both cases are sufficient for full film lu-brication in most parts of the map. The friction coefficientsare lower for thehigher viscosity oil in the thermal region, but higher in thelinear and non-linear region. The former is believed to be due to higher thermal insulationin the thicker oil film for the high viscosity oil, and therefore locally highertemperatures in the film leadning to lower shear resistance.However, for thetest cases with rough surfaces, a change from the lower viscosity oil, SL211 tothe higher viscosity oil, SL326 shows a general decrease of the friction coeffi-cient and a transition from full film to mixed lubrication at lower speeds, Figs.A.7(c) and A.8(c) or Figs. A.7(d) and A.8(d). Metal to metal contact is thusmore likely to occur with a low viscosity oil.

A.3.3 Additives

When adding EP additive to the low viscosity oil, the friction coefficient in thelow temperature tests show very little difference, Figs. A.7(a) and A.9(a), orFigs. A.7(c) and A.9(c). At 90◦C and with the smoother surfaces, Figs. A.9(b)and A.7(b), the SL326 EP oil shows a slightly lower friction coefficient, but

A.4. Conclusions 157

with the rougher surfaces the friction coefficient is highercompared to thepure base oil, Figs. A.7(d) and A.9(d). Similar results werereported in [251]where higher friction coefficients were found for rough surfaces with added EP,and lower friction coefficients with added EP with smooth surfaces, comparedto the same base oil without EP additive, for measurements made at 90◦C.These results suggest that the EP additive becomes more reactive with highertemperature and an increased number of asperity collisions.

A.3.4 Base oil type

The thicker of the mineral oils, SL326 has similar dynamic viscosity comparedto the ester oil, SL324 for an operating temperature of 40◦C. The ester showlower friction coefficients for both the smooth and rough material pairs, Figs.A.10(a) and A.8(a) respectively Figs. A.10(c) and A.8(c). The main differ-ences between the ester and the mineral oil are lower limiting shear strength,pressure-viscosity coefficient and temperature-viscosity coefficient for the es-ter [99]. The difference in temperature-viscosity coefficient explains the dif-ference in viscosity index and hence the fact that the viscosities are similar forboth oils at 40◦C but lower for the mineral oil at 90◦C. The lower limiting shearstrength in combination with the lower high pressure-viscosity in the thermalregion is thought to be the reason for the lower friction coefficients for theester. When comparing the same oils at 90◦C the differences are even biggerwith lower coefficients of friction for the ester oil, Figs. A.8(b) and A.10(b)or Figs. A.8(d) and A.10(d). An effect as discussed earlier are believed to bedue to benefits of larger film thickness for the ester oil whichhas an higherviscosity at 90◦C.

A.4 Conclusions

A method for evaluating and presenting contact friction behaviour in EHL tri-bological systems with respect to surface roughness, temperature and oil pa-rameters under various running conditions are presented. The method givesa good overview, a system finger print, of the frictional behaviour in a broadoperating range, and the performed tests show the differences in friction coef-ficient with respect to different surface topographies as well as viscosity, baseoil type and temperature.

The method was applied in a study of the influence of a number ofparam-eters. The following conclusions can be drawn from this study:


1. Decreasing base oil viscosity increases contact friction in most operatingconditions, introducing a transition from full film to mixedlubricationat higher entrainment speeds due to thinner lubricating films, and alsoincreasing friction coefficients in the thermal region.

2. Increasing the operating temperature lowers the friction coefficients inthe Linear and Non Linear regions and the opposite for the high speed-high slip thermal region.

3. By using smoother surfaces the transition from full film tomixed lubri-cation occurs at lower entrainment speeds.

4. An ester oil with the same viscosity as the mineral oil shows lower fric-tion coefficients for all investigated running conditions.

Acknowledgement

The authors gratefully acknowledge industrial partners Volvo ConstructionEquipment, Scania and SAAB Powertrain for support, Statoillubricants forproviding the test lubricants, and Swedish Foundation for Strategic Research(ProViking) for financial support.

Paper B

The influence of DLC coatingon EHL friction coefficient

159

161

Tribology Letters, 2012 August; vol. 47, Issue 2, p. 285-294.With kind permission of Springer Science+Business Media

The influence of DLC coating on EHL friction coefficient

M. Bjorling, P. Isaksson, P. Marklund and R. Larsson


Best paper award 2012 in surface engineering given by the Surface EngineeringTechnical Committee for the Society of Tribologists and Lubrication Engineers

(STLE)

Abstract

High hardness, high elastic modulus, low friction characteristics, high wearand corrosion resistance, chemical inertness and thermal stability are factorsthat make diamond like carbon (DLC) coatings the subject of many studies.For the same reasons they also seem suitable for use in, amongst others, ma-chine components and cutting tools. While most studies in literature focus onthe influence of coatings on wear and friction in boundary lubrication and puresliding contacts, few studies can be found concerning rolling and sliding EHLfriction, especially in the mixed and full film regime. In thepresent paper testsare carried out in a Wedeven Associates Machine (WAM) tribotester where anuncoated ball and disc pair is compared to the case of coated ball against un-coated disc, coated disc against uncoated ball, and coated disc against coatedball. The tests are conducted at two different temperaturesand over a broadrange of slide to roll ratios and entrainment speeds. The results are presentedas friction maps as introduced in previous work [252]. Furthermore a numer-ical simulation model is developed to investigate if there is a possibility thatthe hard, thin DLC coating is affecting the friction coefficient in an EHL con-tact due to thermal effects caused by the different thermal properties of thecoating compared to the substrate. The experimental results show a reductionin friction coefficient in the full film regime when DLC coatedsurfaces areused. The biggest reduction is found when both surfaces are coated, followed

162 Paper B. The influence of DLC coating on EHL friction

by the case when either ball or disc is coated. The thermal simulation modelshows a substantial increase of the lubricant film temperature compared to un-coated surfaces when both surfaces are coated with DLC. The reduction infriction coefficient when coating either only the ball or thedisc are almost thesame, lower than when coating both surfaces but still higherthan the uncoatedcase. The findings above indicate that it is reasonable to conclude that thermaleffects are a likely cause for the decrease in coefficient of friction when operat-ing under full film conditions, and in the mixed lubrication regime when DLCcoated surfaces are used.

B.1. Introduction 163

B.1 Introduction

The elastohydrodynamic lubrication (EHL) performance of machine compo-nents is governed by many factors such as lubricant properties, load conditions,operating temperature, speed, slide to roll ratio and surface roughness. Increas-ing demands on improved efficiency, sustainability and longer service life haveled to improvements in lubrication technology and surface finishing techniquesas well as geometrical changes to components to improve the performance. Inrecent years tribological coatings have been more frequently used to improvethe performance of machine components. The use of tribological coatings havebeen shown to have several benefits, such as improved runningin performance,higher wear resistance and reduced friction. Some machine components likejournal bearings are mainly operating in the full film regime, while other com-ponents like rolling contact bearings, gears and cam followers are operatingin a range from boundary or mixed to full film EHL. It is therefore of interestto understand the influence of coatings in all regimes. The focus in this studyis on thin, hard coatings usually deposited using physical vapour deposition(PVD) or plasma enhanced chemical vapour deposition (PECVD) such as dif-ferent varieties of diamond like carbon (DLC). The following section containsa short review of the effect of DLC coatings on wear and above all friction inEHL contacts.

High hardness, high elastic modulus, low friction characteristics, high wearand corrosion resistance, chemical inertness and thermal stability are factorsthat make DLC coatings the subject of many studies, and seem suitable foruse in, amongst others, machine components and cutting tools. DLC coatingsshow low dry friction (friction coefficients are typically in the range of 0.1-0.2)and wear rates compared to steel on steel in dry conditions, an effect mainlyattributed to the formation of easily sheared graphitic layers on one or both ofthe mating surfaces [173, 174]. It is reported that in some cases the frictioncoefficient of a steel-DLC dry interface is lower than a lubricated steel-steelinterface [173]. However, most machine components are operating under lu-bricated conditions, and several studies have been conducted on the topic ofDLC performance under lubricated conditions.

In general the friction in boundary lubrication is lower fora steel-DLC sys-tem compared to a steel-steel system and especially during the running in pro-cesses a system containing at least one DLC coated surface shows much lowerinitial friction coefficients, and a faster and more efficient running in [173,175].A system where both surfaces are DLC coated also shows a lowerinitial fric-tion value but a longer running in period as expected since both surfaces have


high hardness and therefore need a longer time to smoothen down. Anotherissue that arises when both surfaces are coated are the surface additive inter-actions. Most of todays lubricants with extreme pressure (EP) and anti wear(AW) additives are formulated with steel contacts in mind, and when there isno steel surface available, the additives may not work as intended. Severalstudies indicate that EP and AW additives developed for steel-steel contactsstill give beneficial effects in systems where two, or preferably one surface iscoated with DLC [175, 253]. ZDDP has been widely used since 1960 and isproven to work well together with iron surfaces where a polyphosphate film isformed that gives good wear resistance. Podgorniket al. has found that aWS2

tribofilm is formed when sulfur is present in the lubricant ina sliding WC DLCcoating [254]. The same reaction was observed by Mistryet al. in tests runwith WC DLC together with a lubricant containing ZDDP and MoDTC whereMoS2 andWS2 tribofilms were formed resulting in promising friction and wearperformance [255]. Several other authors have also studiedthe interaction be-tween DLC coatings and the combination of ZDDP and MoDTC [256–260].However depending on lubricant composition, additive package and type ofDLC coating both friction and wear performance varies greatly so it is hard todraw any general conclusions [176,177,261,262].

While most studies in literature focus on the influence of coatings on wearand friction in boundary lubrication and pure sliding contacts, few studies canbe found concerning rolling and sliding EHL friction, especially in the mixedand full film regime. Renodeauet al. used a rolling sliding mini traction ma-chine (MTM) to test discs coated with four different commercial DLC coat-ings against a steel ball with four different lubricants [176]. The entrainmentspeed in the test device was chosen to simulate mixed lubrication. Frictionand wear of the different configurations were measured and showed dissimilarresults, some material/lubricant combinations showed lower friction and wearcompared to the steel-steel system whereas some combinations showed higherfriction and wear. Topolovec-Miklozicet al. also performed tests in a MTMwhere two types of commercial DLC coatings were studied, onehydrogenateddiamond like and the other Cr-doped, non hydrogenated and graphitic. Thestudy focused on the effect of boundary friction with several types of addi-tives [177]. Bobzinet al. performed tests in a twin disc tribometer with twodifferent PVD coatings, WC/C and CrAIN, compared to uncoated discs [178].Both of the latter studies report lower friction coefficients for the coated spec-imens in both boundary and mixed lubrication. It is however not clear if thefriction reduction achieved in the mixed lubrication regime is solely attributedto lower friction in the asperity interactions or if there isalso a beneficial effect

B.1. Introduction 165

in the hydrodynamic part of the mixed lubrication regime.A dissertation written by Hai Xu at Ohio State university caught the au-

thors interest [179]. Here tests were performed in a WedevenAssociates Ma-chine (WAM) ball-on-disc test device with both uncoated andDLC coated steelballs and discs. The tested combinations were with either both ball and discuncoated, or uncoated disc against a coated ball or both balland disc coated.Results from the tests showed significantly reduced friction coefficients evenin full film lubrication with coated surfaces, especially the case with both sur-faces coated. Other than a comment that coatings should be considered inany efficiency improvement efforts there is no further discussions regardinghow a coating can reduce friction in the full film regime. Evans et al. alsoperformed tests in a WAM tribotester where discs with three different coat-ings were compared to an uncoated disc [180]. All discs were run against anuncoated steel ball. The coatings investigated in their work were chromiumnitride (CrxN), tungsten carbide-reinforced amorphous hydrocarbon (WC/a-C:H) and silicon incorporated diamond like carbon (Si-DLC). At lower en-trainment speeds where the asperity interaction between the mating surfacesare large the CrxN coating had the highest friction whereas the WC/a-C:H andespecially Si-DLC showed lower friction than the uncoated surface. At higherentrainment speeds where full film lubrication was assumed the Si-DLC andWC/a-C:H showed lower friction coefficients than the uncoated surface. Thereason for this is hypothesized to be due to boundary slip between smooth,non wetting surfaces as addressed by Chooet al. [107]. An interfacial slipwould lead to lower shear stresses and thus lower friction inthe EHL contact.In the work presented by Evanset al. low surface energy is correlated withlow friction as the Si-DLC coating that has lower surface energy compared toWC/a-C:H showed even lower friction in the full film regime.

The authors have not found any studies in the literature focusing on theeffect of DLC coatings on the friction coefficient in the fullfilm EHL regime,especially addressing the effect of coating both surfaces or the combination ofcoating either ball or disc. Is it reasonable that boundary slip is the reason forthe reduction in friction coefficient, or is there possibly other reasons as well?

In the present paper tests are carried out in a WAM tribotester where anuncoated ball and disc pair is compared to the case of coated ball against un-coated disc, coated disc against uncoated ball, and coated disc against coatedball. The tests are conducted at two different temperaturesand over a broadrange of slide to roll ratios and entrainment speeds. The results are presentedas friction maps as introduced in previous work [252]. Furthermore a numeri-cal simulation model is developed to investigate if there isa possibility that the


hard thin DLC coating is affecting the friction coefficient in an EHL contactdue to thermal effects caused by the different thermal properties of the coatingcompared to the substrate.

B.2 Method

The following sections cover a description of the ball-on-disc test rig, the testspecimens and lubricant, and an overview of the test procedure. Furthermorethe numerical simulation model is described together with boundary conditionsand assumptions.

B.2.1 Ball on disc tribotester

The experiments were performed in a WAM 11 ball-on-disc testdevice asshown in Figure B.1. The lubricant is supplied at the centre of the disc inan oil dispenser that distributes the lubricant across the disc surface. Lubricantis circulated in a closed loop from the oil bath, through a hose pump to the oildispenser at the centre of the disc. The hose pump is delivering approximately140 ml/min. Three thermocouples are used in the test setup, one located in theoil bath, one in the outlet of the oil supply and one trailing in the oil film closeto the inlet region of the ball-on-disc contact. A more thorough description ofthe test rig and its features is presented in previous work [252].

B.2.2 Test specimens and lubricants

All tests were performed with the same lubricant, a pure mineral oil withoutany additives. An oil without additives was chosen to minimize the effect oftribochemical reactions on the friction coefficient. As explained later the testsare conducted repeatedly until a satisfactory repeatability is reached, and sincethis could take a different amount of cycles for different test cases a lubri-cant containing additives could be in different stages of tribofilm development.Properties of the test lubricant are given in Table B.1.

All specimens used in the tests, both balls and discs are madefrom AISI52100 bearing steel, where the balls are taken direct from factory and the discsare processed the same way as bearing raceway material. The balls are grade20 with a 13/16 inch (20.637 mm) outer diameter and a hardnessof about 60HRC. The discs have a 4 inch (101.6 mm) outer diameter, a circumferentialgrind and are through hardened to about 60 HRC.

B.2. Method 167

Figure B.1: WAM ball-on-disc test device.

Table B.1: Test lubricant properties.

Additives NoneKin. Visc @ 40◦C 109.3 cStDyn. Visc @ 40◦C 94.9 mPasKin. Visc @ 100◦C 11.98 cStDyn. Visc @ 100◦C 9.97 mPasDensity @ 15◦C 885 kg/m3

Viscosity Index 99Type MineralThermal conductivity 0.3 W/mKHeat capacity 1800 J/kgKPressure viscosity coefficient 19 GPa−1

Temperature viscosity coefficient 0.04 K−1

Thermal expansion coefficient 6.5e-4◦C−1 [250]


The commercially available coating Tribobond 44 was used onall coatedspecimens. Tribobond 44 is an a-C:Cr coating deposited using reactive PVDprocesses with a deposition temperature of 200◦C. The thickness of the coat-ings are 2 µm, the hardness 1500 HV0.005 and the friction coefficient againststeel in dry sliding is 0.08. Both balls and discs were measured for surfaceroughness data before and after friction testing with a WykoNT1100 opticalprofilometer system from Veeco. Measurements were performed using 10xmagnification and 1x field of view (FOV). Twenty one measurements weremade per ball, seven on the unworn surface, and seven in each wear track aftertesting. The discs were measured in several positions, and for each radius thir-teen measurements were made. For each series of measurements mean valuesand standard deviations are calculated for the root mean square roughness,Sq

and the arithmetic average of absolute roughness,Sa.

B.2.3 Test procedure

The WAM is used to generate friction data from a relatively broad range ofoperating conditions where one test cycle covers entrainment speeds between0.34 to 9.6 m/s and slide to roll ratios (SRR) from 0.0002 to 0.49. SRR isdefined as the speed difference divided by the mean entrainment speed. Alltest are performed with positive sliding only, which in thiscase means that theball is rotating faster than the disc. Both ball and disc specimens were cleanedwith heptane and ethyl alcohol before starting the experiments for each of thetest cases. Before starting the experiments for each test case the test deviceis warmed up to the desired operation temperature during approximately 60minutes with oil circulation over both ball and disc to ensure temperature sta-bility. When thermal stability is reached a 200 N load, equivalent to 1.7 GPamaximum Hertzian pressure is applied and the machine is calibrated for purerolling by adjusting spindle angle and positioning of the ball to ensure a con-dition of no spinning. These settings are then held constantfor 20 minutes toensure a mild run-in. Subsequently the test cycle is startedthat contains sev-eral loops where SRR is held constant for each loop and the entrainment speedis ramped from 9.6 to 0.34 m/s. In the first loop the SRR is held at 0.0002and is continously increased with each loop until it reaches0.49. The sametest cycle is repeated in the same track for both ball and discuntil the absolutefriction coefficient for each measured combination of entrainment speed andSRR does not differ more than 0.001 from the previous test cycle, excludingSRR below 0.0016 where the accuracy of the machine is slightly lower. Whenthis occurs, the system is considered run in, and the data from the final test

B.2. Method 169

cycle is used for evaluation. The temperature of the oil bulkand fluid film atthe disc surface is typically deviating less than± 1.5◦C from the target tem-peratures of 40 and 90◦C during testing. The actual contact temperatures arehowever higher than the bulk oil temperature. In the most severe cases withhigh entrainment speed, SRR and coefficient of friction (COF), the contacttemperature will increase several tens of degrees [229].

Data from each test is processed separately, and a triangle based linearinterpolation is used between the data points measured for specific SRRs andentrainment speeds. The results are presented as 2D contourmaps.

B.2.4 Simulation model

To investigate if there is a possibility that a hard thin DLC coating is affect-ing the friction coefficient in a EHL contact due to thermal effects caused bythe different thermal properties of the coating compared tothe substrate a nu-merical simulation model is developed. While bulk DLC can have very highthermal conductivity it is shown by several authors that thin DLC films canbecome insulating due to interfacial effects [223, 237]. A one dimensionalmodel is developed that features a thin oil film between two infinitely wideand flat, full film lubricated surfaces, with or without coatings subjected to aheat source. The solution is time dependent where the temperature rise in aline directed in the cross section of the film thickness passing the centre of thecontact from leading edge to trailing edge is calculated. Inthat way the tem-perature distribution in a plane crossing the centre of the contact is obtained.The time it takes for the line to move from the leading edge to the trailing edgeis calculated by dividing the Hertzian contact zone with thesurface velocity ofthe faster moving surface:

tc = B/Ub (B.1)

where

tc = Contact time [s]B = Hertzian contact width [m]Ub = Surface velocity of ball [m/s]

The dimensions of the contact zone is approximated by the Hertzian theoryusing the 52100 steel values for elastic modulus and Poisson’s ratio. Since thecoatings are thin the substrate is assumed to predominate the EHL deformationin the contact. Since the authors did not have the possibility to perform thermal


measurements on the DLC coating used in this study the thermal properties arecollected from work performed by other authors [223,237–239]. Properties ofthe DLC coating and substrate used in the model are given in Table B.2, theoil properties in Table B.1 and a schematic view of the model in Figure B.2.Numbers 1-6 in Figure B.2 corresponds to boundary points, and the letters A-Eare computational domains where the heat equation is being solved. The heatequation solved in the domains A-E is expressed as:

δtscpρ∂T∂t

+∇(−k∇T) =Wq (B.2)

where

δts = Time scaling coefficientρ = Density [kg/m3]cp = Heat capacity at constant pressure [J/(kgK)]k = Thermal conductivity [W/(mK)]Wq = Heat source [W/m3]T = Temperature [◦C]t = Time [s]

The energy input of the heat source that is applied uniformlythroughout thelubricant film is calculated as:

Wq = µf PhSUe

hcen+ εT

∂Ph

∂t(B.3)

where

Ph = Hertzian pressure [Pa]µf = Coefficient of frictionS = Slide to roll ratioUe = Entrainment speed [m/s]hcen = Central film thickness [m]ε = Thermal expansion coefficient [◦C−1]

The surfaces will be in contact with the oil film under different amounts oftime, since the surface speed of the ball and disc are different due to sliding.The simulation time is governed by the time it takes for the faster sliding ball topass the Hertzian contact width of the ball-on-disc contact, and a time scalingcoefficient is therefore applied on the slower moving surface of the disc, both

B.2. Method 171

in case of coated or uncoated. The time scaling coefficient for the disc is givenas:

δts =Ud

Ub(B.4)

where

Ud = surface velocity of disc [m/s]Ub = surface velocity of ball [m/s]

For the same reason a time scaling coefficient is also appliedon the lubricantfilm, but instead of being constant it is a function of the position, x across thefilm:

δts =Ud

Ub+

1− UdUb

hcenx (B.5)

For all other domains the time scaling coefficient is set to 1.The steel surfacesthat are domains A and E in Figure B.2 are assigned the thickness of the discand the diameter of the ball, and the coatings, that are assigned domains Band D have a fixed thickness set to 2 µm. The oil film is assigned domainC and the thickness is approximated by the Hamrock-Dowson equation forcentral film thickness [3] and a thermal correction model developed by Hsuand Lee [11] is used to compensate for thermal thinning of thelubricant film.The outer boundaries of the steel surfaces, points 1 and 6, are assigned thebulk oil temperature, and the interior boundaries 2-5 are assigned a continuitycondition. Therefore the boundary conditions can be described as:

Boundary 1 and 6: T=T0Internal boundariesi=2-5: ki∇Ti|− = ki∇Ti|+ andTi |− = Ti |+

where T0 is the initial temperature. All subdomains are assigned thebulk oiltemperature as an initial value (T0):

For A-E: T(t0)=T0

All solutions are checked for convergence with respect to both mesh size andtime step length.


Figure B.2: Schematic view of thermal model, not to scale.

Table B.2: Model properties.

DLC AISI 52100Density [kg/m3] 2500 7810Thermal conductivity [W/mK] 2 46.6Heat capacity [J/kgK] 1000 475Elastic modulus [GPa] 210Poisson’s ratio 0.3

B.3 Results and discussion

Eight different combinations were evaluated as presented in Table B.3. De-pending on surface roughness, coatings and operating temperature a differentamount of test cycles were required to meet the criterion of running in, as listedin Table B.3. Table B.3 also contains information about the root mean squarecombined roughness,Sqc of each test combination measured after the test hasbeen completed. Table B.4 contains information from the surface roughnessmeasurements made on the ball and disc specimens before and after testing.The measurements show that the uncoated balls wear more thanthe coatedones regardless of if the counter surface is coated or not.

Case one and two where an uncoated ball is run against an uncoated discat 40 and 90◦C are considered as reference cases, and are shown in Figs. B.3and B.4. The results are presented as 2D friction maps whose advantages andfeatures are described in detail elsewhere [252] where the friction coefficient isplotted against SRR and entrainment speed. Similar 2D maps are obtained forcase 3-8, but are not shown here. Instead the absolute coefficients of frictionfrom case 3, 5 and 7 that are conducted with a lubricant bulk temperature of40◦C are subtracted from case 1 and displayed as difference mapsin Figs. B.5-B.7. In these figures a positive number indicates that the presented case hada lower coefficient of friction (COF) compared to the reference case, and viceversa for negative numbers. The results from case 4, 6 and 8 that are conducted

B.3. Results and discussion 173

Table B.3: Test combinations.

Case Ball Disc Temp Cycles Sqc [nm]1 Uncoated Uncoated 40◦C 4 1552 Uncoated Uncoated 90◦C 11 1573 Uncoated Coated 40◦C 4 3554 Uncoated Coated 90◦C 11 2935 Coated Uncoated 40◦C 4 1706 Coated Uncoated 90◦C 11 1627 Coated Coated 40◦C 4 3158 Coated Coated 90◦C 11 259


SR

R0.0250.030.0350.040.045

0.05

0.055

0.06

0.065

0.07

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.3: Uncoated disc against uncoated ball at 40◦C.

with a bulk temperature of 90◦C are in the same way subtracted from case 2and shown in Figs. B.8-B.10.

As seen in Figs. B.5-B.7 the friction coefficients at higher entrainmentspeeds when full film lubrication is assumed to occur are generally lower whenone or both surfaces are coated with DLC, which is consistentwith earlier find-ings [179, 180]. The biggest decrease in friction coefficient is achieved whenboth surfaces are coated, as also reported elsewhere [179],followed by coatingonly the disc, and then the smallest reduction by only coating the ball. How-ever, at lower speeds where mixed lubrication is assumed to occur the frictioncoefficients are higher for all coated cases. The reason for this is probably dueto the rougher surface of the DLC coated disc that promotes a transition tomixed lubrication from full film lubrication at higher entrainment speeds.

The same trend can be seen in Figs B.8-B.10 where the increased oil bulk


Table B.4: Surface roughness measurements for ball and disc specimens.

Ball Counter Temp Sa Sq StdSa StdSq

surface [◦C] [nm] [nm] [nm] [nm]Coated* - - 25 41 3 8Coated Uncoated 40 33 54 2 6Coated Uncoated 90 31 54 1 5Coated* - - 31 53 4 11Coated Coated 40 41 72 4 9Coated Coated 90 29 51 2 13Uncoated* - - 35 53 4 8Uncoated Uncoated 40 47 78 4 13Uncoated Uncoated 90 70 124 5 6Uncoated * - - 32 47 2 6Uncoated Coated 40 88 133 10 12Uncoated Coated 90 64 90 8 13

Disc Counter Temp Sa Sq StdSa StdSq

surface [◦C] [nm] [nm] [nm] [nm]Coated* - - 215 270 22 25Coated Coated 40 244 303 32 39Coated Coated 90 203 254 31 36Coated Uncoated 40 267 329 16 18Coated Uncoated 90 224 279 39 46Uncoated * - - 110 144 7 9Uncoated Coated 40 246 309 19 27Uncoated Coated 90 206 260 17 20Uncoated Uncoated 40 201 134 4 5Uncoated Uncoated 90 108 140 5 6

*Unworn surface


SR

R

0.06

0.055

0.05

0.045

0.04

0.035

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.4: Uncoated disc against uncoated ball at 90◦C.


temperature from 40 to 90◦C leads to a thinner lubricant film and thus promotesa transition to mixed lubrication at even higher entrainment speeds comparedto the lower temperature, especially in the cases when the rougher DLC coateddiscs are used. With increasing entrainment speed and especially SRR thecombinations with one or two surfaces coated with DLC will show COF lowerthan the reference case. This is mainly believed to be due to lower hydrody-namic friction, as seen in the low temperature case, but possibly also due to alower COF in the asperity interactions compared to steel against steel. The dif-ference between figures B.8 and B.10 is therefore most likelyto be explainedby the reduction in hydrodynamic friction by coating both surfaces rather thanone, and also the slightly lower combined roughness in the DLC-DLC com-bination. In Figure B.9 where only the ball is DLC coated there seem to beno substantial differences compared to the reference case,indicating that thefriction coefficient between steel-steel asperities and steel-DLC asperities inthis case is roughly the same.

It is evident that the DLC coating in some way promotes lower frictioncoefficients compared to the uncoated references cases, in these tests most no-ticeably in the full film regime. Chooet al. [107] have discussed that smoothsurfaces are one of the key factors for the boundary slip theory. They testedcoated surfaces with RMS roughness of 0.75 and 9.5 nm and found negligi-ble difference in friction between the rough (9.5 nm) coatedsurface and anuncoated surface with roughness of the same magnitude, while the smoothersurface showed a significant reduction in friction comparedto its uncoatedcounterpart. Considering that the coated surfaces used in this study are one ortwo orders of magnitude larger than the rough surfaces used by Chooet al. itis likely that boundary slip is not the dominant reason, or possibly not part atall in the decrease in friction coefficient found with the coated surfaces in thisstudy.

Part of the explanation may be found from the thermal simulation model.Figure B.11 shows a graph of the maximum value of the mean cross film tem-perature increase in the lubricant as the computational domain passes the con-tact zone. Values are presented for four different combinations of entrain-ment speeds and SRR. The simulation has been run for both the reference casewith uncoated surfaces, but also with the other cases with one or both surfacescoated. All simulations have been performed with an initialtemperature of40◦C since the experiments conducted at 90◦C indicates mixed lubrication inlarge parts of the tested operating range.

There is a significant increase in temperature for all cases where one orboth surfaces are coated with DLC. The highest temperature increase is found



SR

R

0.004

0.004

0.002

0

−0.

002

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.5: Difference in friction coefficients between case 1 and 3, uncoatedreferences compared to uncoated ball and DLC coated disc @ 40◦C.


SR

R

0.004

0.002

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.6: Difference in friction coefficients between case 1 and 5, uncoatedreferences compared to DLC coated ball and uncoated disc @ 40◦C.



SR

R

−0.002 0

0.0020.004

0.006 0.006

0.004

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.7: Difference in friction coefficients between case 1 and 7, uncoatedreferences compared to DLC coated ball and DLC coated disc @ 40◦C.


SR

R

0

−0.005−0.01

−0.015−

0.02

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.8: Difference in friction coefficients between case 2 and 4, uncoatedreferences compared to uncoated ball and DLC coated disc @ 90◦C.



SR

R

0

0

0

0

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.9: Difference in friction coefficients between case 2 and 6, uncoatedreferences compared to DLC coated ball and uncoated disc @ 90◦C.


SR

R

0

0.005

−0.005

−0.01

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure B.10: Difference in friction coefficients between case 2 and 8, uncoatedreferences compared to DLC coated ball and DLC coated disc @ 90◦C.

B.4. Conclusions 179

for the case when both surfaces are coated with DLC followed by the casewhere the ball is coated with DLC and the disc is uncoated. These findingsare in line with the experimental results that showed the biggest decrease inCOF for the cases where both surfaces were coated with DLC, and only a neg-ligible difference between coating the ball, or the disc, but still a substantialdifference compared to having both ball and disc uncoated. An increased tem-perature in the contact leads to lower shear resistance and lower limiting shearstrength and it is therefore likely that it will result in lower friction coefficients.An increased temperature in the contact will only have a small effect on filmthickness since the lubricant film build-up is mainly governed by the viscosityin the inlet region of the contact, which is not subjected to the same temper-ature increase as the lubricant inside the contact. It is reasonable to believethat thermal effects are most likely to be one of the reasons for the decreasein coefficient of friction seen in the experiments. Especially at higher SRRthe temperature increase is substantial. However, at lowerSRR the increasein temperature is not as distinct, and even though the experiments show lessdecrease in COF at lower SRR it is possible that there are other effects as wellthat are part of the reduction in COF. One possibility is the fact that the thermalmodel does not take global effects into account, and there could also be otherfactors.

B.4 Conclusions

The experiments conducted in the ball-on-disc apparatus with both coated anduncoated specimens show that:

• There is a reduction in coefficient of friction in the full film regime whena DLC coating is applied to at least one of the surfaces.

• The biggest reduction in coefficient of friction is found when both balland disc are coated.

• There is only a very small difference in coefficient of friction comparingcoating only the ball, or only the disc.

• The decrease in coefficient of friction is generally increasing with en-trainment speed, and especially SRR.

The thermal simulation model shows that:


Oil

film

tem

per

atu

rein

crea

se[

◦C

]

DLC disc - DLC ballDLC disc - Steel ballSteel disc - DLC ballSteel disc - Steel ball

Ue:7.7,S:0.49 Ue:7.7,S:0.04Ue:2.0,S:0.49Ue:2.0,S:0.040

20

40

60

80

100

120

140

160

180

Figure B.11: Simulated temperature increase in lubricant depending on thecombination of uncoated or coated DLC specimens at different combinations ofentrainment speed and SRR.

• The increase of the lubricant film temperature is largest compared to un-coated surfaces when both surfaces are coated with DLC. The reductionin friction coefficient when coating only the ball, or only the disc arealmost the same, lower than coating both surfaces but still higher thanthe uncoated case.

• The increase of mean lubricant film temperature is larger for higher en-trainment speeds, and above all for higher SRR.

The findings above indicate that it is reasonable to concludethat thermal effectsare a likely cause for the decrease in coefficient of frictionin the full film, andmixed lubrication regime when DLC coated surfaces are used.

Acknowledgement

The authors gratefully acknowledge industrial partners Volvo ConstructionEquipment, Scania and Vicura AB for support, Ion Bond for providing thecoatings, Statoil lubricants for providing the test lubricant, and Swedish Foun-dation for Strategic Research (ProViking) for financial support.

Paper C

Towards the true prediction ofEHL friction

181

183

Tribology International, 2013 April; vol. 66, p. 19-26.With permission from Elsevier

Towards the true prediction of EHL friction

M. Björlinga, W. Habchib, S. Bairc, R. Larssona and P. Marklunda

a) Luleå University of Technology, Division of Machine Elements, Luleå,SE-971 87 Sweden

b) Department of Industrial and Mechanical Engineering, LebaneseAmerican University, Byblos, Lebanon

c) Georgia Institute of Technology, Centre for High Pressure Rheology, G.W.Woodruff School of Mechanical Engineering, Atlanta, GA 30332-0405,

USA

Abstract

The capability to predict elastohydrodynamic film-thickness and friction fromprimary measurements of transport properties of liquid hasbeen an elusivegoal for tribologists for 50 years. Most comparisons between predictions andexperiments involve some amount of tuning of the model in order to matchthe experimental results. In true prediction, this cannot be done since thereare normally no experimental results to compare to. Primarymeasurementsof lubricant transport properties of Squalane were performed, and used in anumerical friction prediction model. Afterwards, friction was measured in aball-on-disc tribotester. No tuning of the lubricant properties, model or testsetup were applied. The current work on EHL-friction is therefore a true rep-resentation of the current level of EHL-friction prediction.

184 Paper C. Towards the true prediction of EHL friction

C.1 Introduction

Elastohydrodynamic lubrication (EHL) is said to occur in lubricated, non-conformal contacts where the elastic deformation of the mating surfaces hasa significant influence on the lubricant film thickness. In many cases the elas-tic deformation occurs even for materials with a high young’s modulus dueto high contact pressures. These high pressures will in general give rise to asubstantial viscosity increase in the lubricant. Machine components like gears,rolling element bearings and cam followers operate under EHL conditions. Im-proving the performance of machine components working under EHL regimeis of great importance to reduce wear and increase efficiency.

The capability to predict elastohydrodynamic film thickness and frictionfrom primary measurements of the transport properties of the liquid has beenan elusive goal of the tribology community for more than fiftyyears. Indeed,such a capability may be considered an indication of the maturity of the EHLfield and of the understanding it has provided of the mechanics governing thelubrication process. Nearly all of the research in EHL has involved the as-sumption of simple but unrealistic rheological models and the adjustment ofthe parameters of these models for validation. Clearly, a complete understand-ing of the film forming process, free of adjustable properties, must precede aprediction of friction since the film thickness establishesthe shear rate in theHertz region where friction is mostly generated. It is important to keep in mindthat the pressure range, and thus the viscosity increase inside the contact, thatgoverns the coefficient of friction is several orders of magnitude higher thanthe pressures in the inlet that are governing film formation.

The foundation for the prediction of film thickness and friction, at least forlow pressures, has been available for some time. The pressure dependence ofdensity, viscosity and thermal conductivity was accurately described to EHLpressures (1.2 GPa) by Bridgman [138] in 1931. Precise measurements ofthe temperature and pressure dependence of lubricants to EHL pressures wereobtained in Bridgman’s laboratory and reported in the 1953 ASME viscosityreport [263]. The nature of non-Newtonian response to shearstress in lubri-cants was correctly described by Hutton in 1973 [120]. Therefore, much ofthe needed rheological descriptions have been in place for about four decadesand all that has been required has been to perform the computations with ac-curately measured properties. Some good predictions have been obtained forsqualane and a polyalphaolefin using simple isothermal calculations where thefilm thickness was assumed uniform over the Hertz region resulting in theHertz pressure distribution [81, 94]. The analysis for the polyalphaolefin was

C.1. Introduction 185

improved by a full EHL simulation [264] which calculated thereal distributionof film thickness and pressure, resulting in an extraordinarily precise agree-ment with measured contact behaviour although at low pressure and slidingvelocity.

Successful predictions in the high pressure, thermal regime have comemore recently. Using the "full system approach" [230, 265],it has been pos-sible to predict EHL friction [231] accurately at very high contact pressurewith significant sliding. The only property which was not obtained from a pri-mary measurement was the limiting-stress pressure coefficient. The ability tomeasure shear stress in a liquid when the response is rate-independent [266]apparently has been lost for about twenty years. However, when the other rhe-ological properties are well-known an approximation of this coefficient maybe extracted from a traction measurement with relative confidence. It shouldbe pointed out that recent reseach [89] has shown that an accurate value oflimiting shear stress cannot be obtained from a traction measurement unlessthe viscosity is known to be very large. This is because the magnitude of thefriction cofficient in the top parts of a friction vs slip curve is not only governedby the limiting shear stress, but also shear-thinning.

Most comparisons between numerical predictions and experiments involvesome amount of tuning of the model in order to match the experimental re-sults. In realtrue prediction this cannot be done since there are normally noexperimental results to compare to. Instead, all three components occur inde-pendently from each other:

• Primary measurement of lubricant transport properties.

• Prediction using measured transport properties as input data togetherwith operation and geometric data.

• Experimental validation

Such an investigation is presented here. The three components in this workhave been carried out at three different laboratories. Initially the transportproperties of squalane (a low molecular weight branched alkane) were char-acterized at Georgia Tech. These properties were later usedin a numericalmodel to predict friction at the Lebanese American University. Finally frictionwas measured in a ball-on-disc test device at Luleå University of Technology.No tuning of the lubricant properties, model and test setup has been applied.The presented results on EHL friction are therefore a true representation ofthe current level of EHL friction prediction. This study also includes higherentrainment speeds than previously published in this kind of investigation.


Table C.1: Investigated conditions.

Temperature 40◦ CContact load 50 and 300 NMaximum hertzian pressure 1.07 and 1.94 GPaEntrainment speed,Ue 0.34 to 9.6 m/sSlide to Roll Ratio, SRR 0.0002 to 0.49

C.2 Overall Methodology

The following sections cover the investigated cases, including running condi-tions and loads. It also contains information about the lubricant and its trans-port properties, as well as the numerical model and the underlying boundaryconditions and assumptions. Finally, the experimental equipment and speci-mens are discussed together with the test procedure.

C.2.1 Investigation Procedure

The numerical prediction and experimental measurements are performed us-ing the same conditions reported in Table C.1. SRR is defined as the speeddifference divided by the mean entrainment speed,Ue . All tests are performedwith positive sliding only, which in this case means that theball is rotatingfaster than the disc. The investigation includes a combination of high entrain-ment speeds and contact pressures that to the authors knowledge has not beenconsidered in any similar previous studies.

Both numerical predictions and experiments were performedwith the samelubricant, squalane, a commercially available low molecular weight branchedalkane (2,6,10,15,19,23-hexamethyltetracosane). An oilwithout additives waschosen to minimize the effect of tribochemical reactions onthe friction coeffi-cient. As explained later the tests are conducted repeatedly until a satisfactoryrepeatability is reached, and since this could take a different amount of cyclesfor different test cases a lubricant containing additives could be in differentstages of tribofilm development. At the test temperature of 40◦C, the ambientviscosity of squalane is 15 mPas and the pressure-viscositycoefficient is 18GPa−1 [129]. The following section gives a more in depth view of thelubri-cant parameters used in the numerical model.

C.2. Overall Methodology 187

C.2.2 Lubricant transport properties

The transport properties used for the numerical model in this study are ob-tained from several earlier studies on squalane. A brief overview of the mod-els derived from these studies is given here. Further information about themeasurements for the effect of pressure, temperature and shear on viscosity isfound in references: [94,129,130].

C.2.2.1 Equation of state

A temperature modified version of the Tait equation of state is used to modelthe temperature and pressure dependence of volume for Squalane. The Taitequation is written for the volume relative to the volume at ambient pressure,

VV0

= 1−1

1+K′

0

ln

[

1+p

K0(1+K

′

0)

]

(C.1)

with

K0 = K00exp(−βKT) (C.2)

The volume at ambient pressure relative to the ambient pressure volume at thereference temperature,TR, is assumed to vary with temperature as:

V0

VR= 1+av(T −TR) (C.3)

whereK′

0 = 11.74,av = 8.36×10−4K−1, K00 = 8.658 GPa andβK = 6.332×10−3K−1 were obtained from experimental measurements with a standard de-viation of 0.05% [129].

C.2.2.2 Viscosity

A thermodynamic scaling rule that has been found to be accurate for manyorganic liquids is:µ = f (TVg), where -3g is related to the exponent of therepulsive intermolecular potential. A useful scaling parameter can thereforebe written as:

ϕ =

(

TTR

)(

VVR

)g

(C.4)

An accurate scaling function can be obtained from a Vogel like form:


µ= µ∞exp

(

BFϕ∞

ϕ−ϕ∞

)

(C.5)

whereg = 3.921,ϕ∞ = 0.1743,BF = 24.50 andµ∞ = 0.9506×10−4 Pa·s wereobtained from experimental measurements with standard deviation of 14.9 %with respect to relative viscosity [129].

For the shear dependence of viscosity, a t-T-p shifted Carreau equation isused:

η(γ,T, p) = µ

[

1+

(

γλRµµR

TR

TVVR

)2](n−1)/2

(C.6)

whereµR = 15.6 mPa·s,λR= 2.26×10−9 s, andn = 0.463 were obtained fromNon-Equilibrium Molecular Dynamics and experimental measurements [130].The limiting shear stress was shown to depend on pressure as:

τL = Λp (C.7)

whereΛ = 0.075 was found from EHL traction experiments and is assumed tobe independent of temperature [94].

C.2.2.3 Thermal properties

The thermal conductivity and volumetric heat capacity of squalane is expressedas:

k=Ckκ−s (C.8)

with

κ =

(

VVR

)[

1+A

(

TTR

)(

VVR

)q]

(C.9)

whereq = 2 andA = -0.115, and

Cv = ρcp =C0+mχ (C.10)

with

χ =

(

TTR

)(

VVR

)−3

(C.11)

C.2. Overall Methodology 189

whereCk = 0.074 W/m·K, s = 4.5,C0 = 0.94× 106 J/m3 ·K andm = 0.62×106 J/m3 · K were obtained from experimental measurements. The thermalproperties were measured by Ove Andersson at Umeå University.

C.2.3 Numerical model

The numerical model employed in this work was described in detail in [229–231]. In this section, only the main features of this model are recalled. Themodel is based on a finite element fully coupled resolution ofthe EHD equa-tions: Reynolds, linear elasticity and load balance equations. The latter aresolved simultaneously providing robust and fast converging solutions. Thegeneralized Reynolds equation [232] is used to account for the shear depen-dence of the lubricant. Special formulations are introduced in order to stabilizethe solution of Reynolds equation at high loads. The temperature distributionin the contact is obtained by solving the 3D energy equation in the lubricantfilm and solid bodies. The model incorporates the variationsof the lubricant’sthermal properties with pressure and temperature throughout the contact. Thenan iterative procedure is applied between the respective solutions of the EHDand thermal problems as described in [229, 230]. During the iterative proce-dure, every time the shear stressτ is evaluated (using viscosity data providedby a combination of the Carreau and Vogel-like models) it is either truncatedto τL if it exceedsτL or, otherwise, it is kept unchanged.

C.2.4 Ball on disc tribotester

The experiments were carried out with a Wedeven Associates Machine (WAM)11, ball-on-disc test device. The lubricant is supplied at the centre of the disc inan oil dispenser that distributes the lubricant across the disc surface. Lubricantis circulated in a closed loop from the oil bath, through a hose pump to the oildispenser at the centre of the disc. The hose pump is delivering approximately180 ml/min. Three thermocouples are used in the test setup, one located in theoil bath, one in the outlet of the oil supply and one trailing in the oil film closeto the inlet region of the ball-on-disc contact. A more thorough description ofthe test rig and its features is presented in previous work [252].

C.2.4.1 Test specimens

All specimens used in the tests (balls and discs) are made from AISI 52100bearing steel. The balls are grade 20 with a 13/16 inch (20.637 mm) outerdiameter and a hardness of about 60 HRC. The discs have a 4 inch(101.6


Table C.2: Specimen material properties.

Material 52100 (AISI)Young’s modulus (Pa) 210x109

Poisson’s coefficient 0.3Specific heat capacity (J/kg K) 475Thermal conductivity (W/m K) 46.6Thermal diffusivity (m2/s) 13.6x10−6

Density (kg/m3) 7850

mm) outer diameter, a circumferential grind (before polish) and are throughhardened to about 60 HRC. Additional material parameters also used in thenumerical model are found in Table C.2. The surface roughness, RMS, hasbeen measured to about 25 nm for the balls, and 35 nm for the disc, whichgives a combined roughness of approximately 43 nm. The surface roughnessmeasurements have been conducted in a Wyko NT1100 optical profilometersystem from Veeco. The measurements were performed using 10x magnifica-tion and 1x field of view.

C.2.4.2 Test procedure

The ball-on-disc test device is used to generate friction data from a relativelybroad range of operating conditions where one test cycle covers entrainmentspeeds between 0.34 to 9.6 m/s and slide to roll ratios (SRR) from 0.0002 to0.49. Both ball and disc specimens were cleaned with heptaneand ethyl alco-hol before starting the experiments for each of the test cases. Before startingthe experiments for each test case, the test device is warmedup to the desiredoperating temperature during approximately 60 minutes with oil circulationover both ball and disc to ensure temperature stability. When thermal stabil-ity is reached a 50 or 300 N load, equivalent to 1.07 or 1.94 GPamaximumHertzian pressure is applied and the machine is calibrated for pure rolling byadjusting spindle angle and positioning of the ball to ensure a condition of nospinning. These settings are then held constant for 20 minutes to ensure a mildrun-in. Subsequently the test cycle is started that contains several loops whereSRR is held constant for each loop and the entrainment speed is ramped from9.6 to 0.34 m/s. In the first loop the SRR is held at 0.0002 and isthen con-tinuously increased with each loop until it reaches 0.49. The same test cycleis repeated in the same track for both ball and disc until the absolute frictioncoefficient for each measured combination of entrainment speed and SRR doesnot differ more than 0.001 from the previous test cycle, excluding SRR below

C.3. Results and discussion 191

0.0016 where the accuracy of the machine is slightly lower. When this occurs,the system is considered run in, and the data from the final test cycle is usedfor evaluation. The temperature of the oil bulk and fluid film at the disc surfaceis typically deviating less than± 1.5◦C from the target temperature of 40◦Cduring testing. The actual contact temperatures are however higher than thebulk oil temperature. In the most severe cases with high entrainment speed,SRR and coefficient of friction (COF), the contact temperature will increaseseveral tens of degrees [229].

Data from each test is processed separately, and a triangle based linearinterpolation is used between the data points measured for specific SRRs andentrainment speeds. The results are presented as 3D friction maps, that arediscussed in detail in previous work [252].

C.3 Results and discussion

In this section, the results from both experiments and numerical simulationsare presented. The results include friction coefficient measurements and cal-culations, film thickness calculations and comparisons in percentage of thedifference between experiments and numerical calculations. It also shows thecalculated power generated from friction in the simulations for both pressures.The results from the calculations and the experiments can beseen in Figs. C.1to C.4 where Figs. C.1 and C.3 are from calculations, and Figs. C.2 and C.4are from experiments.

Figure C.5 shows the difference between the experiment and calculationin percent for the test performed at 1.07 GPa, and the difference for the 1.94GPa test is shown in Fig. C.6. A positive number in those figures indicates thatthe predicted coefficient of friction is higher than the experimentally obtainedvalue, whereas a negative number indicates that the predicted friction coeffi-cient is lower than the experimentally obtained value. The calculated minimumfilm thickness for the 1.07 and 1.94 GPa cases for the entire region of the testsare shown in Figs. C.7 and C.8. Figs. C.9 and C.10 show the generated powerfrom friction in the experiments for the 1.07 and 1.94 GPa cases respectively.The friction power is calculated as:Wf = µf ∗Ue∗SRR∗L. Whereµf is thecoefficient of friction,Ue is the mean entrainment speed andL is the appliedload. The loads are 50 and 300 N respectively.


Figure C.1: Predicted friction map at 1.07 GPa and 40◦C.

Figure C.2: Measured friction map at 1.07 GPa and 40◦C.


Figure C.3: Predicted friction map at 1.94 GPa and 40◦C.

Figure C.4: Measured friction map at 1.94 GPa and 40◦C.



SR

R

5

10

15

20 15

20

2020

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure C.5: Difference in friction coefficient, Calculation-experiment [%], 40◦C,1.07 GPa.


SR

R

20

25

25

20

15

10

5

0−5

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure C.6: Difference in friction coefficient, Calculation-experiment [%], 40◦C,1.94 GPa.



SR

R 50 100

150

200

250

300

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure C.7: Calculated minimum film thickness [nm], 40◦C, 1.07 GPa.

C.3.1 Film thickness and roughness effects

The minimum film thickness values presented in Figs. C.7 and C.8 are takenfrom the numerical calculations. It is worth noting that thefilm thickness, espe-cially for the lower pressure case, Figure C.7 are almost independent of SRR.The increased temperature in the inlet of the contact with increasing entrain-ment speeds and SRRs are not substantial enough to considerably influence thefilm thickness. For the higher pressure case, Fig. C.8 the effect of increasingentrainment speed and SRRs on film thickness is larger, whichis not surprisingconsidering the difference in friction power, see Figs. C.9and C.10. However,even here the effect on film thickness is small. The effect of thermal heatingon friction is of much greater importance and will be discussed later.

Most likely the measurement performed at the lower contact pressure, 1.07GPa, Fig C.2, is in the full film regime at all combinations of speeds and SRRs,possibly with the exception of the lowest speeds. The calculated minimum filmthickness for the lowest tested speeds at 1.07 GPa is in the range of 35-40 nm,as seen in Fig. C.7, in comparison with a combined roughness (RMS) forthe contacting surfaces of about 43 nm. The possibility for slightly smoothersurfaces after running in, with the combination of surface amplitude reduc-tion [22, 267] makes full film lubrication possible. For the test performed atthe higher contact pressure, 1.94 GPa, it is more likely thatasperity interactionsoccurs at the lowest speeds, even considering running in effects and amplitude



SR

R

200

180

160

140

120

100

80604020

1 2 3 4 5 6 7 8 9

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure C.8: Calculated minimum film thickness [nm], 40◦C, 1.94 GPa.

reduction. It should however be pointed out that the surfaceroughness mea-surements performed after the tests did not show any statistical reduction inRMS roughness indicating that any running in effects were very small. Thecalculated minimum film thickness for the higher pressure case can be seen inFig. C.8.

C.3.2 Friction

As discussed in previous work [252] the friction maps could be divided intoseveral different regimes where friction is governed and influenced by differentfactors. Recent work on friction regimes in quantitative EHL [89] has ledto more detailed knowledge about the regimes, and the addition of one moreregime is also included in the present work. The regions can be seen in Fig.C.11, which results from a measurement conducted at the sameconditions asFig. C.4 but with rougher surfaces for a more pronounced mixed lubricationregion.

In the linear region, shear stress is proportional to shear rate, and will thuslead to a linear increase in friction. This linear relationship is only seen atlow SRRs, and the upper limit (in SRR) becomes even lower withincreasingpressures as the shear stress increases faster with shear rate at higher pressuresand at lower entrainment speeds.

The non-linear region is influenced by shear thinning and dominated by



SR

R

7

6

5

43

2

1

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure C.9: Friction power [W], 40◦C, 1.07 GPa.

the growth of a limiting stress region. The shear stress willno longer increaseproportional to shear rate. It will rather increase at a reduced rate until limitingshear stress is attained over much of the contact. Then, shear stress becomesindependent of shear rate. However, if one keeps on increasing the slidingspeed, thermal effects eventually dominate the frictionalresponse overwhelm-ing other effects including limiting shear stress which is no longer reached.As a consequence, friction starts decreasing with increasing SRR due to com-bined thermal and shear-thinning effects. This regime has been identified asthe Thermoviscous regime.

In the boundary between the mixed lubrication regime and theThermovis-cous regime lies the Plateau regime. Here, friction reachesan asymptotic valueand shows little variation indicating that the frictional response of the contactis governed by the limiting shear stress behaviour of the lubricant.

The last region is the mixed lubrication region where asperity contacts oc-cur between the surfaces, and the coefficient of friction is therefore a combi-nation of hydrodynamic effects and asperity interaction.

By looking at the overview of the four cases in Figs. C.1 to C.4it is clearthat the numerical model manages to capture the overall frictional behavioursuch as the linear increase in friction with shear rate at lowSRR, the non linearincrease in friction leading to the limiting shear stress and the reduction offriction at higher entrainment speeds and SRRs due to thermal effects. Mixedlubrication effects are however not included in the numerical model, and could



SR

R50

454035

30252015

10

5

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Figure C.10: Friction power [W], 40◦C, 1.94 GPa.

therefore be part of the difference between the numerical prediction and theexperimental results at the lowest entrainment speeds, as discussed in previoussection.

At lower entrainment speeds, up to about 2 m/s the predictions are in gen-eral not far off, the deviation is typically below 10 % and in line with pre-vious comparisons at these entrainment speeds [231]. At higher speeds thecoefficient of friction is constantly overestimated by the numerical model, andgenerally more so at higher SRRs. When looking at absolute values, and thedifference between the numerical calculations and the experiment as presentedin Figs. C.5 and C.6 it is clear that the model does not accurately predict allparts of the tested region, or that some of the boundary conditions in the nu-merical model does not fit the experiment.

One of the most striking differences between the predictions and the ex-periments are the plateau regime in the predictions, Figs. C.1 and C.3 wherethe coefficient of friction is almost constant at a high levelover a wide range ofentrainment speeds and SRRs. Such behaviour is not as distinct in the exper-iments, Figs. C.2 and C.4 where instead the friction coefficients drops fasterwith an increase in entrainment speeds and SRRs.

By looking at the difference in % between the predictions andthe experi-ments in Figs. C.5 and C.6 it is clear that the levels of difference are situatedin the same manner as the levels of friction power in Figs. C.9and C.10. Itis therefore likely that one of the main reasons for the deviation between the


Mixed

Linear

Non-Linear

Plateau

Thermoviscous

Figure C.11: 3D friction map with regimes.

prediction and the experiments are found in how the model handles thermal ef-fects and/or some thermal behaviour in the test rig and specimens that are notmatched in the boundary conditions in the numerical model. It should how-ever be pointed out that the biggest discrepancies does not occur at the highestentrainment speeds and SRRs, but rather in connection to theplateau regime.

The numerical model is using bulk temperature as a boundary condition forthe inlet temperature of the lubricant and a steady state condition is calculatedfor the specific running conditions (entrainment speed and SRR.) Any globaleffects on the specimen temperature are therefore not takeninto account, whilein the experiment it is possible that especially the ball surface temperature isincreasing during the test. As a specific running condition is measured in thetest rig it takes a few seconds for the machine to adjust rotational speeds of bothspecimens to achieve the correct entrainment speed and SRR.Some additionaltime is also required for the data capture process. It is possible that duringthis time, the specimens are heated due to the friction generated in the contact,and therefore the lubricant temperature will be slightly higher in the inlet andcentral region of the contact compared to the numerical model. Looking atthe friction power for the low pressure, 1.07 GPa case in Fig.C.9, there is amaximum of 7 W. For the high pressure, 1.94 GPa case in Fig. C.10, with amaximum friction power of 55 W, this is more likely to have an effect on theresults, and may be the reason why the deviation in general isslightly largerfor the higher pressure case at higher sliding speeds. If thereasoning above is


the cause for the largest part of the deviations, it is not sufficient to consider anideal circular contact. Rather must the effect of the globalgeometry on intakeand surface temperature be taken into account.

However, it is also possible that the deviations between theexperimentalresults and the predictions are due to shortcomings in the rheology modelsused in the study.

As previously discussed, recent research [89] suggests that it is not possibleto accurately estimate the limiting shear stress coefficient from EHL frictionmeasurements, since shear-thinning was found to affect thefriction coefficienteven when the limiting shear stress is reached in parts of thecontact.

Moreover is the limiting shear stress in the numerical modelused in thiswork assumed to be only depending on pressure. It has been shown earlierin measurements that temperature has an influence on the limiting shear stress[99]. It is reasonable to believe that this is one of the reasons why the plateauregime is more pronounced in the numerical predictions. With the inclusion ofthe temperature dependence of limiting shear stress the plateau would probablylevel off with increased sliding due to a reduction in limiting shear stress beforethermal effects starts to dominate the friction behaviour.

Finally, even rather small deviations in the Carreau equation and Vogel likefunction will have a substantial impact on the resulting friction coefficient.

C.4 Conclusion

In this paper the results from a true friction prediction investigation are shown.The transport properties for the fluid squalane was measuredat one laboratory,and was later used at a second laboratory to predict frictionin a numericalmodel for a circular EHL contact. Finally the same parameters were used forfriction measurements in a ball-on-disc machine in a third laboratory. Theinvestigated conditions includes a combination of higher entrainment speedsand pressures than previously reported in this kind of investigation. No tuningof the lubricant properties, model and test setup was applied.

The predictions were found to capture the main features of the frictionbehaviour at different entrainment speeds and SRRs. In absolute values thenumerical results were found to be remarkably good considering the true pre-diction approach. At low entrainment speeds the deviationsbetween the nu-merical prediction and the experimental results are in general less than 10 %,while increasing at higher entrainment speeds and SRRs. Theauthors believethat to achieve even better precision in the predictions there is a need to re-

C.4. Conclusion 201

fine the assessment of the limiting shear stress of the lubricant, including theeffect of temperature. In addition may it be necessary to refine the boundaryconditions in the model to account for global effects on temperature.

The use of primary measurements or "true" measurements of the liquidproperties in EHL has distinct advantages over the previoustechniques of ad-justing properties to satisfy perhaps inappropriate assumptions. If friction is tobe controlled by lubricant selection and formulation, there must be an under-standing of the contribution of each of the transport properties to the energydissipation. An example can be made for the liquid in this study, squalane.Johnston et al. [268] adjusted the pressure-viscosity coefficient of squalane tomake the classical formulas agree with measured film thickness. If this co-efficient, which is about one-half of the value obtained in viscometers, hadbeen used to extrapolate viscosity to the full pressure of the contact in the con-ventional manner, the importance of the low-shear viscosity would have beenseverely understated and a more severe form of shear-thinning would have tobe invoked to explain the friction. However, the use of the true predictionapproach investigated in this paper will generally requiremore effort from ameasurement and modeling point of view compared to previoustechniques.

Acknowledgement

The authors wish to thank Ove Andersson at Umeå University for performingthe thermal properties measurements. The authors from Luleå wish to thankSwedish Foundation for Strategic Research (ProViking) forfinancial support.Bair was supported by the Center for Compact and Efficient Fluid Power, aNational Science Foundation Engineering Research Center funded under co-operative agreement number EEC-0540834.

Paper D

Friction reduction inelastohydrodynamic contacts bythin layer thermal insulation

203

205

Nature, 2014 January; vol. 505, Issue 7483, p. 264

Warm carbon coat reduces friction





USA

Nature - Research highlights

A coating material made of carbon reduces friction not just by providing a slip-pery surface, but also by keeping the points of contact warm.Marcus Björlingat Luleå University of Technology in Sweden and his team coated steel ballswith “diamond-like-carbon” - a material in which carbon atoms have a bondingpattern similar to that of diamond. They rolled the balls against a metal discwith an oil lubricant in between, and showed that the carbon coating acts as aninsulator, lowering the viscosity of the lubricant and thusreducing the fricionbetween the ball and the disc. These findings could encouragethe developmentof lubricant coatings made from insulating materials.

206 Paper D. Thin layer thermal insulation in EHL

207

Tribology Letters, 2014 January; vol. 54, Issue 2, p. 477-486.With kind permission of Springer Science+Business Media

Friction reduction in elastohydrodynamic contacts by thinlayer thermal insulation





USA

Abstract

Reducing friction is of utmost importance to improve efficiency and lifetimeof many products used in our daily lives. Thin hard coatings like diamond-like-carbon (DLC) have been shown to reduce friction in fullfilm lubricatedcontacts. In this work, it is shown that, contrarily to common belief, the fric-tion reduction stems mainly from a thermal phenomenon and not only a chem-ical/surface interaction one. It is shown that a few micrometer thin diamond-like-carbon coating can significantly influence the thermalbehaviour in a lu-bricated mechanical system. The presented simulations, validated by exper-iments, show that applying a thin diamond-like-carbon coating to metal sur-faces creates an insulating effect that, due to the increased liquid lubricant filmtemperature at the centre of the contact, locally reduces lubricant viscosityand thus friction. The results of the investigation show that the addition ofthin insulating layers could lead to substantial performance increases in many


applications. On a component level the contact friction coefficient in somecommon machine components like gears, rolling element bearings and camfollowers can potentially be reduced by more than 40 %. This will most likelyopen up the way to new families of coatings with a focus on thermal propertiesthat may be both cheaper and more suitable in certain applications than DLCcoatings.

D.1. Introduction 209

D.1 Introduction

Hydrodynamic lubrication (HL) has been studied for more than a century sinceits discovery by Beauchamp Tower [1]. Osborne Reynolds derived the gov-erning equation of HL (which bears his name) from the Navier-Stokes equa-tions [2]. Elasto hydrodynamic lubrication (EHL) was discovered half a cen-tury later. Due to the non-conformal contacts found in systems working inEHL, the contact pressure is sufficient to cause elastic deformation of the con-tacting surfaces and many orders of magnitude increase of viscosity of theliquid film. Increased understanding in this field is crucialfor improvementsof machine components such as bearings, gears and cam-followers, as well asimplants for the human knee and hip joints. To reduce friction in EHL con-tacts, researchers have been focusing on the development ofnew lubricants toreduce shear forces, and surface finishing techniques to reduce asperity inter-actions between the surfaces. More recently coatings have been investigated toincrease wear resistance and reduce friction in case of solid contact. High hard-ness, high elastic modulus, low friction characteristics,high wear and corro-sion resistance, chemical inertness, and thermal stability are factors that makediamond-like-carbon (DLC) coatings the subject of much investigation. Thesefactors make the coatings promising for use in, among others, machine com-ponents and cutting tools. Of interest in this study is, however, the low thermalconductivity (in the range of 1-3 W/mK) that has been measured for these thincoatings [220,221,223].

Reports of friction reduction with DLC coated surfaces in full film EHLhave been published earlier [179,180,269]. This reductionhas been attributedby several authors to boundary slip [180, 202, 203] as a consequence of non-fully wetted surfaces [100, 106–109, 189, 192, 193, 201], where some of thework is based on atomically smooth surfaces. Slip at the boundary would leadto lower shear stresses and thus lower friction in the EHL contact. It has alsobeen observed that for boundary slip to take place, three factors must be ful-filled. First of all the surface has to be non- or partially wetted by the lubricant.Secondly, the pressure must be low, close to ambient where the lubricant wouldstill be in its liquid form, opposed to the glass state at higher pressures. Finally,the surfaces must be very smooth, generally below 6 nm RMS [101,107–109].However, the present authors have presented an investigation in which frictionreduction with DLC coatings was measured [269] in the EHL full film regimeeven when the combined RMS roughness of the surfaces was in the range of155-355 nm. Based on a simplified analytical estimation of the temperatureincrease in the lubricant film induced by DLC surface coating, the authors pro-


Lubricant

u1

F

x

y

z u2

R

Figure D.1: Schematic of the experimental and numerical model of the studiedsystem. F is the applied load, R is the radius of the ball and u1, u2 the surfacevelocities of the disc and ball. Slide to roll ratio is defined as u1-u2/(u1+u2/2).

posed that the friction reduction could be a result of thermal insulation dueto the low thermal conductivity observed for some DLC coatings. The tem-perature increase in the lubricant film would reduce the viscosity and therebyreduce the coefficient of friction. It was also shown that coating only one ofthe specimens reduced the friction coefficient, albeit not as much as when bothsurfaces were coated. In this article, a more advanced and thoroughly vali-dated [229–231] 3D numerical model was used to predict the effect of thininsulating layers on full film EHL friction. A series of friction experiments hasalso been conducted in a ball-on-disc machine where an uncoated pair of spec-imens is compared to a pair of DLC coated specimens to validate the numericalresults.

D.2 Overall Methodology

The following sections cover the investigated cases, including running condi-tions and loads. It also contains information about the lubricant and its trans-port properties, as well as the numerical model and the underlying boundaryconditions and assumptions. Finally, the experimental equipment, specimensand coating are discussed together with the test procedure.A schematic of theexperimental and numerical model setup can be seen in FigureD.1.

D.2.1 Investigation Procedure

The numerical prediction and experimental measurements are performed usingthe same conditions reported in Table D.1. SRR is defined as the velocity dif-

D.2. Overall Methodology 211

Table D.1: Investigated conditions.

Temperature 40◦ CContact load 80 and 300 NMaximum hertzian pressure 1.25 and 1.94 GPaEntrainment speed,Ue 1, 1.6 ,3.145, 6.144 m/sSlide to Roll Ratio, SRR 0.0002 to 1.05Coating none or Tribobond 43

ference divided by the mean entrainment velocity,Ue . All tests are performedwith positive sliding only, which in this case means that theball is rotatingfaster than the disc.

Both numerical predictions and experiments were performedwith the samelubricant, squalane, a commercially available low molecular weight branchedalkane (2,6,10,15,19,23-hexamethyltetracosane). An oilwithout additives waschosen to minimize the effect of tribochemical reactions onthe friction coef-ficient. At the test temperature of 40◦C, the ambient viscosity of squalane is15 mPas and the pressure-viscosity coefficient is 18 GPa−1 [129]. The follow-ing section gives a more in depth view of the lubricant parameters used in thenumerical model.

D.2.2 Lubricant transport properties

The transport properties used for the numerical model in this study are ob-tained from several earlier studies on squalane. A brief overview of the mod-els derived from these studies is given here. Further information about themeasurements for the effect of pressure, temperature and shear on viscosity isfound in references: [94,129,130].

D.2.2.1 Equation of state

A temperature modified version of the Tait equation of state is used to modelthe temperature and pressure dependence of volume for Squalane. The Taitequation is written for the volume relative to the volume at ambient pressure,

VV0

= 1−1

1+K′

0

ln

[

1+p

K0(1+K

′

0)

]

(D.1)

with

K0 = K00exp(−βKT) (D.2)


The volume at ambient pressure relative to the ambient pressure volume at thereference temperature,TR, is assumed to vary with temperature as:

V0

VR= 1+av(T −TR) (D.3)

whereK′

0 = 11.74,av = 8.36×10−4K−1, K00 = 8.658 GPa andβK = 6.332×10−3K−1 were obtained from experimental measurements with a standard de-viation of 0.05% [129].

D.2.2.2 Viscosity

A thermodynamic scaling rule that has been found to be accurate for manyorganic liquids is:µ = f (TVg), where -3g is related to the exponent of therepulsive intermolecular potential. A useful scaling parameter can thereforebe written as:

ϕ =

(

TTR

)(

VVR

)g

(D.4)

An accurate scaling function can be obtained from a Vogel like form:

µ= µ∞exp

(

BFϕ∞

ϕ−ϕ∞

)

(D.5)

whereg = 3.921,ϕ∞ = 0.1743,BF = 24.50 andµ∞ = 0.9506×10−4 Pa·s wereobtained from experimental measurements with standard deviation of 14.9 %with respect to relative viscosity [129].

For the shear dependence of viscosity, a t-T-p shifted Carreau equation isused:

η(γ,T, p) = µ

[

1+

(

γλRµµR

TR

TVVR

)2](n−1)/2

(D.6)

whereµR = 15.6 mPa·s,λR= 2.26×10−9 s, andn = 0.463 were obtained fromNon-Equilibrium Molecular Dynamics and experimental measurements [130].The limiting shear stress was shown to depend on pressure as:

τL = Λp (D.7)

whereΛ = 0.075 was found from EHL traction experiments and is assumed tobe independent of temperature [94].


D.2.2.3 Thermal properties

The thermal conductivity and volumetric heat capacity of squalane is expressedas:

k=Ckκ−s (D.8)

with

κ =

(

VVR

)[

1+A

(

TTR

)(

VVR

)q]

(D.9)

whereq = 2 andA = -0.115, and

Cv = ρcp =C0+mχ (D.10)

with

χ =

(

TTR

)(

VVR

)−3

(D.11)

whereCk = 0.074 W/m·K, s = 4.5,C0 = 0.94× 106 J/m3 ·K andm = 0.62×106 J/m3 · K were obtained from experimental measurements. The thermalproperties were measured by Ove Andersson at Umeå University.

D.2.3 Numerical model

The numerical model employed in this work is based on the full-system finiteelement approach for thermal elastohydrodynamic lubricated (TEHL) contactsdescribed in details in [229, 231, 270]. In this section, only the main featuresof this model are recalled along with the necessary amendments required forthe inclusion of surface coatings. The model is based on a finite element fullycoupled resolution of the EHL equations: Reynolds, linear elasticity and loadbalance equations. The latter are solved simultaneously providing robust andfast converging solutions. The generalized Reynolds equation [232] is usedto account for the shear-dependence of the lubricant. Special formulations areintroduced in order to stabilize the solution of Reynolds equation at high loads.The inclusion of coatings into the EHL model consists simplyin adding a thinlayer to the surface of the 3D solid body representing the contacting solids.This layer possesses different elastic material properties than those of the sub-strate. However, the simulation of a typical isothermal Newtonian EHL linecontact with DLC coatings of different thickness shows thatfor coatings of2-5 µm thickness (such as the ones considered in this work), the effect of the


coating on dimensionless film thickness and pressure is negligible as suggestedby Figs. D.2 and D.3. Therefore, for the sake of simplicity, in the EHL part ofthe model the surfaces are assumed to be uncoated and the coatings are onlyconsidered in the thermal part. This leads to a significant reduction in the sizeof the discrete problem as the thin layer coating requires a very large numberof elements to discretize it. However, the influence of hard coatings on EHLfilm formation and pressure distribution has been studied inseveral publica-tions [233–235]. It is shown (as also seen in Figs. D.2 and D.3) that a coatingthat is harder than the substrate reduces the contact width,and increases thecentral pressure and the pressure spike. A coating that is softer than the sub-strate would have the opposite effect. Both effects increase with the thicknessof the coating. The temperature distribution in the contactis obtained by solv-ing the 3D energy equation in the lubricant film and solid bodies (substratesand coatings). The inclusion of the coatings consists in inserting a geometricallayer between the lubricant and each substrate and applyingthe thermal prop-erties of DLC to it. Obviously, the continuity of heat flux needs to be imposedbetween the lubricant film and the coatings as well as betweenthe coatings andthe substrates. The model incorporates the variations of the lubricant’s thermalproperties with pressure and temperature throughout the contact. An iterativeprocedure is established between the respective solutionsof the EHL and ther-mal problems as described in [229, 270]. Throughout the iterative procedure,every time the shear stressτ is evaluated (using viscosity data provided by acombination of the Carreau and Vogel-like models), it is either truncated toτL

if it exceedsτL or, otherwise, it is kept unchanged. The numerical model doesnot take asperity interactions into account and should therefore in this designonly be seen as a full film EHL model.

D.2.4 Ball on disc tribotester

The experiments were carried out with a Wedeven Associates Machine (WAM)11, ball-on-disc test device. The lubricant is supplied at the centre of the disc inan oil dispenser that distributes the lubricant across the disc surface. Lubricantis circulated in a closed loop from the oil bath, through a peristaltic pump tothe oil dispenser at the centre of the disc. The peristaltic pump is deliveringapproximately 180 ml/min. Three thermocouples are used in the test setup, onelocated in the oil bath, one in the outlet of the oil supply andone trailing in theoil film close to the inlet region of the ball-on-disc contact. A more thoroughdescription of the test rig and its features is presented in previous work [252].


Figure D.2: Effect of DLC coating thickness on the dimensionless film thickness(top) and pressure (bottom) distribution of a typical EHL line contact.


Figure D.3: Effect of DLC coating thickness on the dimensionless film thickness(top) and pressure (bottom) distribution of a typical EHL line contact (Zoom onFigure D.2).


Table D.2: Specimen material properties.

Material 52100 (AISI) Tribobond 43Young’s modulus (Pa) 210×109 -Poisson’s coefficient 0.3 -Specific heat capacity (J/kg K) 475 1000Thermal conductivity (W/m K) 46.6 2.225Density (kg/m3) 7850 2500

D.2.4.1 Test specimens

All specimens used in the tests (balls and discs) are made from AISI 52100bearing steel. The balls are grade 20 with a 13/16 inch (20.637 mm) outerdiameter and a hardness of about 60 HRC. The discs have a 4 inch(101.6mm) outer diameter, a circumferential grind (before polish) and are throughhardened to about 60 HRC. Additional material parameters also used in thenumerical model are found in Table D.2. The tests (and predictions) were per-formed with both uncoated 52100 (AISI) specimens, and specimens coatedwith Tribobond 43 (hydrogenated amorphous carbon (Cr+) a-C:H), a com-mercially available DLC coating applied through Plasma Assisted ChemicalVapor Deposition, measured to a thickness of 2.8 µm using calotest. The sur-face roughness, RMS, has been measured to about 25 nm for the balls, and 35nm for the disc, which gives a combined roughness of approximately 43 nm.The surface roughness measurements have been conducted in aWyko NT1100optical profilometer system from Veeco. The measurements were performedusing 10x magnification and 1x field of view.

D.2.4.2 Test procedure

The ball-on-disc test device is used to generate friction data from a series oftests under different operating conditions. In each test the entrainment speedand contact pressure is held constant while the slide to rollratio (SRR) is variedfrom 0.0002 to 1.05. Both ball and disc specimens were cleaned with heptaneand ethyl alcohol before starting the experiments for each of the test cases.Before starting the experiments for each test case, the testdevice is warmed upto the desired operating temperature during approximately60 minutes with oilcirculation over both ball and disc to ensure temperature stability. When sta-bility is reached a 80 or 300 N load, equivalent to 1.25 or 1.94GPa maximumHertzian pressure is applied and the machine is calibrated for pure rolling byadjusting spindle angle and positioning of the ball to ensure a condition of nospinning. These settings are then held constant for 20 minutes to ensure a mild


run-in. Subsequently the test cycle is started wherein the entrainment speed isincreased from the lowest value to the largest value. The tests were repeatedseven times. The temperature of the oil bulk and fluid adheredat the disc sur-face is typically deviating less than± 1.5◦C from the target temperature of40◦C during testing. The actual contact temperatures are however higher thanthe bulk oil temperature. In the most severe cases with high entrainment speed,SRR and coefficient of friction (COF), the contact temperature will increaseseveral tens of degrees.

D.3 Results and discussion

The measurements conducted with coated surfaces showed significantly lowercoefficients of friction compared to the uncoated referencecase. The resultsare shown in Figs. D.4 to D.9. The figures show how the frictioncoefficient ischanging with the ratio of sliding speed to rolling speed (SRR) at four differ-ent entrainment speeds. While roller bearings usually operate with low SRR,typically below 0.05, cam followers and gears operate at much higher SRR. Athigh sliding speeds where power losses usually are the greatest, the measuredreduction in friction with the coating is more than 40% at maximum SRR forthe most prominent case, Figure D.8, while at the lowest sliding speed, thedecrease in friction at maximum SRR is at least 25 %, Figure D.7.

The results from the numerical simulations are shown together with themeasurements in Figs. D.4 to D.9. Although numerically predicted frictioncoefficients deviate from experimental ones in absolute value, the shape andtrend of the friction curves are qualitatively similar for both coated and un-coated cases. A comparison between this numerical model andsome refer-ence measurements was published recently where the discrepancy between thesimulation results, and the measurements is discussed [271]. The simulationassumes that the solid and liquid materials are entrained into the contact at theinitial bulk temperature and does not therefore account forthe accumulatedheat. In addition, a major part of the discrepancy might be attributed to the factthat the Limiting Shear Stress coefficientΛ cannot be accurately deduced fromtraction curves as suggested by [89]. In fact, in [89] the authors show that evenwhen the friction plateau is reached in EHL traction curves,shear-thinning inthe peripheral area of the contact still affects the magnitude of friction andtherefore the limiting shear stress coefficient cannot be directly deduced fromthe asymptotic friction value in the plateau region as is thecurrent practice inmost EHL studies, including the current work. Moreover is the limiting shear

D.3. Results and discussion 219

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt


Figure D.4: Friction results at 1.25 GPa, 40◦C at 6.144 m/s entrainment speedcomparing uncoated and coated specimens in both simulation and experiment.

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt




0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

SRR

Fric

tion

coef

ficie

nt



stress in the numerical model used in this work assumed to be only dependingon pressure. It has been shown earlier in measurements that temperature has aninfluence on the limiting shear stress [99]. Nevertheless, the results from thenumerical predictions show that the addition of a thin (2.8 µm) thermal insulat-ing coating leads to a substantial reduction of the frictioncoefficient in an EHLcontact. Since the numerical model does not incorporate anychemical/surfaceinteraction effects, such as asperity interactions or boundary slip, the observedreduction in the numerical results can only be attributed tothermal effects.

Figure D.10 shows the temperature distributions in the Z direction acrossthe film and solids (substrate and coatings (for the case of coated surfaces))from the inlet (X = -1.5) to the outlet (X = 1.5) at the same conditions as inFigure D.4. X represents the position in the circular contact in the entrainmentdirection. In an EHL contact the film thickness, and thus the ability to separatethe surfaces from contacting each other is governed by the viscous behaviour ofthe lubricant in the inlet of the contact. On the other hand, friction is governedby the viscous behaviour of the lubricant in the central areaof the contact.Therefore a sufficiently viscous lubricant is required in the inlet to generate anadequate film thickness, while low viscosity is desirable atthe contact centreto reduce friction. The difference in temperature in the inlet is very small, andwill thus lead to negligible differences in film thickness between the coated anduncoated case. In the central parts (X = -0.5 to X = 0.5) of the contact there is a


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

SRR

Fric

tion

coef

ficie

nt



0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt




0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt



substantial difference in temperature in the lubricant. The higher temperaturein the lubricant with the coated specimens will therefore lead to lower viscosi-ties and consequently lower friction. However, in the outlet of the contact (X =1.0 and X = 1.5) the lubricant temperatures are actually lower with the coating.At this stage, the lubricant film is ruptured, and would therefore not contributeto the friction. The higher lubricant temperature at the contact outlet for theuncoated case results from heat transfer by conduction and advection withinthe solids from the centre of the contact (where most of the heat is generated)towards the outlet. This leads to high solid surface temperatures at the exitof the contact, which maintains the lubricant film at a highertemperature. Inthe coated case, less heat is transferred to the liquid at theexit of the contactsince DLC acts like an insulating material. Figure D.11 shows the flow profile(the x-velocity component of the lubricant flow across the film) at the centreof the contact for both coated and uncoated surfaces at the same conditions asin Figure D.4. The difference in velocity is large around themidlayer of thelubricant film and near the faster moving surface. A lower velocity variationacross the lubricant film around the midlayer, and thus lowershear rates, incombination with lower viscosities due to the higher lubricant temperatures inthe coated case leads to the friction reduction compared to the uncoated case.

As seen in Figs. D.4 to D.9, the greatest reduction in friction with thecoating happens at the highest SRR, as expected since more heat is generated


Lubricant CoatingCoatingPlane all

Lubricant allPlane

Figure D.10: Numerical simulation of lubricant film temperature increase in dif-ferent locations of the lubricant film for coated and uncoated specimens.


Table D.3: Friction reduction with DLC coating at SRR=1.05.

Load [N] Entrainment Measured Predictedspeed [m/s] reduction [%] reduction [%]

300 6.144 41 48300 3.145 35 4680 6.144 39 4680 3.145 37 4680 1.611 30 3980 1.0 24 28

at higher sliding speeds. Substantial friction decreases are however achievedalready at much lower SRRs. Table D.3 summarizes the friction reductionachieved with the coating for all entrainment speed and loadcases at the high-est SRR value of 1.05. It is clear that the relative reductionin friction is great-est at the highest entrainment speed and reduced when entrainment speed de-crease. Note that the relative reduction in friction is not much smaller at thelow load compared to the high load. Even at the lowest load andthe lowest en-trainment speed, Figure D.7, the friction reduction is significant even at SRRof 0.2. This indicate that excessive heat generation is not needed to benefitfrom a thermally insulating coating, and that SRR has greater influence on thefriction reduction than contact pressure.

In this article a coating with only one specific thermal inertia has beenevaluated. The authors believes that a coating with lower thermal inertia thanthe one investigated will lead to greater reduction in friction, while a coatingwith a high thermal inertia will lead to less reduction in friction. A low thermalinertia is connected to low thermal conductivity and volumetric heat capacityand is thus insulating. It is also expected that the coating thickness will influ-ence the friction reduction in a similar way. A thicker coating will lead to agreater friction reduction, while a thinner coating will lead to a lower frictionreduction.

These findings will most likely lead to new possibilities in reducing frictionin machine components working in, or partly, in the full film EHL regime.In the experiments conducted in this study a thermal insulating DLC coatingwas used, but it is certainly possible to obtain thermal insulation using othertypes of coatings as well. These findings will probably open the way up fornew coatings (where thermal properties constitute the maindesign parameters)used for friction reduction in certain lubricated machine elements. These newcoatings may very well be cheaper and more available than DLCcoatings.The results indicate the possibility to reduce contact friction by more than 40% by using thermal insulating layers in EHL contacts. By applying thicker

D.4. Conclusions 225

3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Liquid velocity (m/s)

Z

CoatedUncoated

Figure D.11: Lubricant x-velocity profile at the centre of the contact across filmthickness. Z=1 is the faster moving surface (ball) and Z=0 is the slower movingsurface (disc).

coatings and/or coatings with even lower thermal conductivity even greaterfriction reduction could be possible.

D.4 Conclusions

This paper presents strong evidence that thermal insulation has an importantrole in friction reduction using thin coatings in full film EHL contacts. A seriesof friction tests were conducted in a ball-on-disc machine with both coated anduncoated specimens. The DLC coated specimens showed lower coefficient offriction in all tested cases. A validated TEHL numerical model was used to pre-dict the friction coefficient for the tested cases. Althoughthe numerically pre-dicted values, which ignore accumulated heat, deviate fromthe experimentalones in absolute values, the shape and trend of the friction curves are qualita-tively similar. The numerical model does not incorporate any chemical/surfaceinteraction effects, and thus the observed reduction in friction for the predictedresults can only be attributed to thermal effects. These findings open up for thedevelopment of new families of coatings where thermal properties constitutethe main design parameters. These coatings may be both cheaper to produce,more available, and provide greater friction reduction than DLC coatings in


certain applications.

Acknowledgement

The authors wish to thank Ove Andersson at Umeå University for perform-ing the thermal properties measurements, and IonBond AB forproviding theDLC coatings used in the tests. The authors from Luleå wish tothank SwedishFoundation for Strategic Research (ProViking) for financial support. Bair wassupported by the Center for Compact and Efficient Fluid Power, a NationalScience Foundation Engineering Research Center funded under cooperativeagreement number EEC-0540834.

Paper E

The effect of DLC coatingthickness on elstohydrodynamicfriction

227

229

Tribology Letters, 2014 June; vol. 55, Issue 2, p. 353-362With kind permission of Springer Science+Business Media

The effect of DLC coating thickness on elstohydrodynamicfriction

M. Björling, R. Larsson and P. Marklund


Abstract

The application of surface coatings has been shown to reducefriction in elasto-hydrodynamic lubrication (EHL), not only in the mixed and boundary regimewhen asperity interactions occur, but also in the full film regime. Several stud-ies suggest that the full film friction reduction is due to a violation of the no-slipboundary condition and thus slip is taking place between thesolid and the liq-uid. Another hypothesis proposes that the full film frictionreduction is dueto the low thermal conductivity of diamond like carbon (DLC)coatings. Inthis work, two DLC coatings with the same composition, but different thick-nesses are investigated with uncoated steel specimens as a reference, all withthe same surface roughness. Friction tests in a ball-on-disc machine show thatboth coatings reduce friction compared to the uncoated reference case in fullfilm EHL. The thicker coating is significantly more effectiveat reducing fric-tion than the thinner one at a maximum friction reduction of 41 % compared to29 % for the thinner coating. Moreover, contact angle measurements, surfaceenergy measurements, and spreading parameter calculations show no statis-tically significant differences between the two coatings, suggesting that thefriction reduction capabilities of coatings in full film EHLcannot be describedby solid-liquid interactions alone. The difference in friction reduction betweenthe specimens in this work is mainly attributed to differentthermal properties.

230 Paper E. The effect of DLC coating thickness on EHD friction

E.1 Introduction

Surface engineering has emerged as an important part in reducing friction inthe field of elastohydrodynamic lubrication (EHL). Smoother surfaces in con-tact has the advantage of pushing the transition from full film lubrication tomixed lubrication toward lower speeds, and will thus lead toreductions in bothfriction and wear. Furthermore has the use of tribological coatings grown sub-stantially in the last decade to provide various enhancements such as lower fric-tion and wear both in dry and lubricated contacts. Diamond like carbon (DLC)coatings are the subject of many studies since they possess properties such aslow friction characteristics, high wear and corrosion resistance, chemical inert-ness, thermal stability, as well as high hardness and high elastic modulus. DLCcoatings generally reduce friction in boundary and mixed lubrication regimesby lowering the contact friction between the asperities. However, the matter ofinterest in this paper is the reduction that is achieved by DLC coatings in fullfilm EHL where there is no contact between the surfaces and thelubricant car-ries all of the load. Several authors have experimentally observed a reductionin friction with DLC coated surfaces in full film EHL [179–181,269,272]. Thefriction reduction has been explained by several authors asan effect of bound-ary slip, or solid-liquid interface slip [180, 181, 202] a phenomena thoroughlydiscussed in literature [100, 106–109, 189, 192, 193, 201],where some of thework is based on atomically smooth surfaces.

However, it is still not entirely clear how the mechanism of slip works,and several hypothesis can be found in the literature. In many cases, poorwetting or high contact angle is proposed as the main featureto promote slip[100,108,109,196,200–202], and in other cases low surfaceenergy [180,181].However, using only surface energy as a means to determine the potential ofsolid-liquid slip may not be suitable since surface energy is a material propertyand tells nothing about the interaction of a specific material with a specificfluid. On the other hand, contact angle measurements represent a propertyof the specific surface and lubricant combination and could intuitively seemto be more suitable. However, Kalin and Polajnar have recently publishedstudies including many different lubricants and coatings where the influenceof surface energy and contact angle is discussed [203, 204].They concludethat contact angle cannot be used in isolation to predict thewetting behaviour,and instead propose the use of a spreading parameter which correlates wellwith the surface energy. They also showed the correlation between contactangle, surface energy and spreading parameter with friction coefficients in fullfilm EHL, where especially the spreading parameter and the polar part of the

E.2. Overall Methodology 231

surface energy correlate very well with the friction measurements [181].

Furthermore, many authors have concluded that for solid-liquid slip to takeplace, the surfaces have to be very smooth, generally below 6nm RMS [101,107–109]. However, the present authors have presented an investigation inwhich friction reduction with DLC coatings was measured [269] in full filmEHL even when the combined RMS roughness of the surfaces was in the rangeof 155-355 nm. Based on a simplified analytical estimation ofthe temperatureincrease in the lubricant film induced by DLC surface coating, the authorsproposed that the friction reduction could be a result of thermal insulation dueto the low thermal conductivity observed for some DLC coatings [220, 221,223]. The temperature increase in the lubricant film would reduce the viscosityand thereby reduce the coefficient of friction. In a more recent study [272]the present authors used a more advanced and thoroughly validated [229–231,271] 3D numerical model to predict the effect of thin insulating layers on fullfilm elastohydrodynamic (EHD) friction to be compared with experimentalmeasurements. The presented simulations, validated by experiments, showedthat applying a thin diamond-like-carbon coating to metal surfaces creates aninsulating effect that, due to the increased liquid lubricant film temperature atthe centre of the contact, locally reduces lubricant viscosity and thus friction.This model was later refined by Habchi and used to numericallyinvestigate theeffect of different coating thickness and thermal properties on EHD friction[236].

In this work, specimens coated with the same DLC coating, butwith dif-ferent coating thickness, are investigated in terms of friction reduction in fullfilm EHL and compared to measurements of contact angle and surface energyof the coatings. By investigating two coatings with supposedly the same sur-face energy and contact angle, but with different thicknesses and hence thermalproperties, the authors want to provide further information about the mecha-nisms behind the full film EHD friction reduction capabilities of DLC coatings.

E.2 Overall Methodology

The following sections cover the test specimens, lubricant, and coatings used.It also contains information about how the experimental equipment for thefriction tests are set up, and how the experiments are performed. Finally, theprocedure for the contact angle, surface energy, and surface tension measure-ments are discussed.


E.2.1 Test specimens and lubricant

The tests were performed with a commercially available DLC coating pro-duced with two different thicknesses and uncoated DIN 100Cr6 (AISI 52100)bearing steel as a reference. For the friction tests in the ball-on-disc machinepolished balls and discs were used that had been measured to asurface rough-ness, RMS of 25 nm for the balls and 35 nm for the discs, which gives a com-bined roughness of approximately 43 nm. These roughness values were alsomaintained after the specimens had been coated with DLC. Thesurface rough-ness measurements were conducted in a Wyko NT1100 optical profilometersystem from Veeco. The measurements were performed using 10x magnifica-tion and 1x field of view. The balls are grade 20 with a 13/16 inch (20.63mm) outer diameter and a hardness of about 60 HRC. The discs have a 4inch (101.6 mm) outer diameter, a circumferential grind (before polish) andare through hardened to about 60 HRC. Except the steel uncoated referencespecimens, the remaining specimens were coated with Tribobond 43, a hydro-genated amorphous carbon ((Cr+) a-C:H), through plasma-assisted chemicalvapour deposition. The specimens were prepared with two different coatingthicknesses, 0.8 and 2.8 µm, measured using calotest. A chromium-based in-terlayer with a thickness of 0.1-0.3 µm deposited by magnetron sputtering wasused to improve the adhesion. The thermal conductivities were not measuredon these specific coatings, but approximated by the formula obtained by Kimet al.[223] that is expressed in Fig. E.1. The dots represent the actual measure-ments performed by Kimet al. from which they derived the curve fit. The starsrepresent the coating thicknesses used in this work. Other work has been per-formed focusing on measuring thermal conductivities of DLCcoatings thinnerthan 20 nm where a thermal conductivity of 0.09 W/mK was obtained for acoating of approximately 3 nm [221]. The effect of thermal boundary resis-tance [224] is more influential at thinner coating thicknesses and is the mostlikely explanation why the thermal conductivity has a rapiddecrease for thin-ner coatings. It is also likely that the chromium interlayerwill further reducethe thermal conductivity due to additional thermal boundary resistance. Thethermal conductivities are expected to be around 1.75 W/mK and 2.23 W/mKfor the 0.8 µm and 2.8 µm coatings respectively. This should be compared toa value of 46.6 W/mK for the substrate material. Note that even though thethermal conductivity is higher for the thicker coating, thetotal thermal insu-lating effect is still higher due to the increase in coating thickness. ConsiderFourier’s law of thermal conductive heat transfer:


0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

Coating thickness [nm]

The

rmal

con

duct

ivity

[W/m

K]

Figure E.1: Thermal conductivity with respect to coating thickness for a-C:Hcoating. The dots represents the measured values used for the curve fit. Thestars represents the values for the coating thicknesses in this paper.

qt = kAtdT/c (E.1)

whereqt is the heat transfer,k is the thermal conductivity,At is the heattransfer area,dT is the temperature difference across the material, andc isthe coating thickness. Assuming identical values forAt anddT for the dif-ferent coating thicknesses would give approximately 2.75 times as much heatconducted through the thinner coating.

The lubricant used for the tests was squalane, a commercially availablelow molecular weight (422.81 g/mol) branched alkane (2,6,10,15,19,23- hex-amethyltetracosane). A lubricant without additives was chosen to minimizethe effect of tribochemical reactions on the friction coefficient. At the testtemperature of 40◦C, the ambient viscosity of squalane is 15 mPas, and thepressure-viscosity coefficient is 18 GPa−1 [129].

E.2.2 Ball on disc tribotester

The experiments were carried out with a Wedeven Associates Machine (WAM)11, ball-on-disc test device. The lubricant is supplied at the centre of the discin an oil dispenser that distributes the lubricant across the disc surface. Thelubricant is circulated in a closed loop from the oil bath, through a peristaltic


pump to the oil dispenser at the centre of the disc. The peristaltic pump isdelivering approximately 180 ml/min. Three thermocouplesare used in thetest setup, one located in the oil bath, one in the outlet of the oil supply, andone trailing in the oil film close to the inlet region of the ball-on-disc contact.A more thorough description of the test rig and its features is presented inprevious work [252].

E.2.3 Test procedure

In this investigation, we only tested the combination of uncoated specimens,specimens coated with 0.8 µm DLC and specimens coated with 2.8 µm DLC.Previous investigations have shown that a coating on only one of the specimensin contact still gives a reduction in friction, but not as great as if both specimensare coated [181,269]. The ball-on-disc test device was usedto generate frictiondata from a series of tests under different operating conditions. In each test, theentrainment speed and contact pressure were held constant while the slide toroll ratio (SRR) was varied from 0.0002 to 1.05. SRR is definedas the ball sur-face speed subtracted by the disc surface speed giving the sliding speed. Thesliding speed is then divided with the entrainment speed giving SRR. All testsin this investigation were hence conducted with the ball having a higher surfacespeed than the disc. Both ball and disc specimens were cleaned with heptaneand ethyl alcohol before starting the experiments for each of the test cases.Before starting the experiments for each test case, the testdevice was warmedup to the desired operating temperature during approximately 60 minutes withlubricant circulation over both ball and disc to ensure thermal stability. Whena stable temperature was reached, an 80 N or a 300 N load was applied whichis equivalent to 1.25 or 1.94 GPa maximum Hertzian pressure and the machinewas calibrated for pure rolling by adjusting spindle angle and positioning ofthe ball to ensure a condition of no spinning. These settingswere then heldconstant for 20 minutes to ensure a mild run-in. Subsequently, the test cyclewas started, wherein the entrainment speed was kept at a constant value, andthe slide to roll ratio was varied from the lowest to the highest value. The testcycle was repeated seven times for each entrainment speed. The temperatureof the oil bulk and fluid adhered at the disc surface was typically deviatingless than± 1.5◦C from the target temperature of 40◦C during testing. Fourdifferent entrainment speeds were used in the tests. The entrainment speedswere chosen such that the minimum film thickness in the most severe case(lowest entrainment speed and highest SRR with the thickestcoating) wouldstill be higher than the combined roughness of the specimensand thus still be


Table E.1: Investigated conditions.

Temperature 40◦ CContact load 80 and 300 NMaximum hertzian pressure 1.25 and 1.94 GPaEntrainment speed,Ue 1.611, 3.145 and 6.144 m/sSlide to Roll Ratio, SRR 0.0002 to 1.05Coating thickness 0.8 and 2.8 µm

in the full film regime. The film thickness for 1.25 GPa, an entrainment speedof 1.6 m/s and 1.04 in SRR give an uncoated minimum film thickness of 65nm, while the coated case would give a minimum film thickness of 63 nm.The film thickness calculations were made with the numericalmodel used in aprevious investigation including the effect of a thermallyinsulating coating onfilm thickness and friction in EHL [272]. Here, along with another numericalwork [236] it is concluded that although a thermally insulating coating couldhave a significant effect on friction, the film thickness is barely affected. Asummary of the investigated conditions can be seen in Table E.1.

E.2.4 Surface energy and wetting

The surface energies of the specimens were evaluated using the Owens-Wend-Rabel-Kaelbe (OWRK) method (Eq. E.2) [273]. This method requires contactangle measurements of the specimens with at least two liquids with knownproperties. In this investigation, demineralized water and diiodomethane wereused. The properties of these fluids needed for the OWRK method, total sur-face tension, and the dispersive and polar components of surface tension ob-tained from literature [274] are presented in Table E.2. It should be mentionedthat several theoretical models exist for the calculation of surface energy fromcontact angle measurements. Kalin and Polajnar [203, 204] recently investi-gated the surface energies of several different DLC coatings using OWRK, Ossand Wu methods. They concluded that these three models were qualitativelythe same (for the samples and fluids they used), providing thesame ranking ofthe surfaces, but with a difference in absolute values ranging between 5 and 25%. The OWRK method, eq. E.2, presented values in between the Oss methodand the Wu method, and is probably the most used model in literature whichis why it is used also in this investigation.

γl (1+cosθ) = 2

(

√

γDs γD

l +√

γPs γP

l

)

(E.2)


Table E.2: Surface tension and its polar and dispersive components for testliquids.

Liquid Total surface tension Dispersive component Polar componentγl [mN/m] γD

l [mN/m] γPl [mN/m]

Water 72.8 21.80 51.00Diodomethane 50.8 50.8 0Squalane 31.8 30.7 1.1

whereγl is the total surface tension of the fluid,θ the contact angle,γDs the

dispersive component of surface energy,γPs the polar component of surface

energy,γDl the dispersive component of surface tension andγP

l the polar com-ponent of surface tension.

The contact angle measurements were conducted with the sessile droptechnique using an optical goniometer, Fibro 1121/1122 DAT-Dynamic Ab-sorption and Contact Angle Tester. Before the measurementswere conducted,the specimens were cleaned with acetone and ethanol and dried in a stream ofhot air. The drop size was 2.5 µm, and used for all lubricants for consistency.At this small volume, the effect of the drops impact due to itsweight can beignored [275]. Each fluid and liquid combination was repeated at least 8 timesbefore the average value was calculated. The contact angle of the lubricanttypically changed with time after having been deposited on the surface. Thevalue was measured after 12 seconds for both reference fluidsand all mate-rials for consistency. In the case of squalane that showed contact angles thatchanged more with time, the measurement was done after 16 seconds.

E.2.5 Surface tension

The surface tension of the lubricant used for the friction tests was determinedusing the same optical goniometer as for the contact angle measurements. Thependant drop method was used to establish the surface tension. Each test wasrepeated 5 times, and the average value was calculated. Since the pendant dropmethod only gives the total surface energy of the lubricant,another methodis needed to determine the polar and dispersive components of the surfacetension. By measuring the contact angle for the specific lubricant on PTFE,the dispersive component of the surface tension can be acquired. This is be-cause PTFE only has a dispersive component of surface energy, 23.53 mJ/m2

and consequently, the OWRK method could be used to calculatethe disper-sive component of the surface tension. The polar part is thencalculated usingFowkes method [276] with the following equation:

E.3. Results 237

γl = γDl + γP

l (E.3)

E.2.6 Spreading parameter

Kalin and Polajnar proposed the use of a spreading parameterfor evaluating thewetting between a surface and a lubricant [181]. They found that while contactangle measurements did not correlate with the friction reduction, the spreadingparameter did. Combining the Young equation and the OWRK model, theyderived a spreading parameter that was used to characterizethe spreading ofthe surface and liquid combinations investigated in this article:

Sp = 2

[

√

γDs γD

l +√

γPs γP

l − γl

]

(E.4)

E.3 Results

The surface energies for the investigated specimens are presented in Fig. E.2.It shows that uncoated steel has the highest surface energy of 50.07 mJ/m2

while the DLC coatings have 48.1 mJ/m2 and 48.5 mJ/m2 respectively for the0.8 µm and 2.8 µm coatings. When looking at the distribution in dispersive andpolar components of the surface energies for the investigated surfaces, they areall very similar in the sense that the dispersive component of the surface energyis dominant and only a small part is coming from the polar component.

Figure E.3 shows the results for the calculated spreading parameters for thecombination of squalane and the different surfaces. The spreading parameterdefined by Kalin and Polajnar [181] ,eq. E.4, represents the wetting behaviourof the lubricant and surface combinations. In all cases, thespreading parameteris positive, suggesting that the lubricant spreads over thesurfaces with timeand does not obtain a constant contact angle immediately. According to theequation, the lubricant will spread most easily on the uncoated steel surfaceand less on the coated surfaces. In this case, the thicker coating has a slightlyhigher spreading parameter suggesting that it would provide lower wetting ofthe surface with squalane compared to the thinner coating.

Figure E.4 shows the results from the contact angle measurements forsqualane on the specimens. The values were taken after about16 seconds andrepresent a steady state contact angle. The low contact angles indicate goodwetting on all specimens.


Steel DLC 0.8 DLC 2.80

10

20

30

40

50

60

70S

urfa

ce e

nerg

y [m

J/m

2 ]

Total surface energyDispersive partPolar part

Figure E.2: Surface energies and the corresponding dispersive and polar partsof investigated surfaces determined using the OWRK method.


2

4

6

8

10

12

14

16

18

Spr

eadi

ng p

aram

eter

[mJ/

m2 ]

Figure E.3: Spreading parameter values for squalane and the investigated sur-faces.

E.3. Results 239


1

2

3

4

5

6

7

8

Con

tact

ang

le [°

]

Figure E.4: Contact angle for squalane on the investigated surfaces.

Figs. E.5 to E.7 show the results from the ball-on-disc friction measure-ments with the uncoated specimens and the two different coating thicknesses.Although three different entrainment speeds were investigated at 1.25 GPapressure (Table E.1), only the lowest and highest entrainment speeds are shownhere since the intermediate entrainment speed showed the same trends as thetwo presented here. For 1.94 GPa, only the highest entrainment speed case isshown here as a comparison with the 1.25 GPa case at the same speed. Ta-ble E.3 shows the reduction in friction for all investigatedcases at the highestSRR of 1.05. Table E.3 also shows that in general, the percental friction re-duction is greater for the lower pressure independent of coating thickness andentrainment speed.

It is clear that the uncoated specimens have the highest friction coefficientsfor all tested combinations of pressures and entrainment speeds. The thinnerDLC coating leads to a significant reduction in friction coefficient, and thefriction is further reduced for the specimens with the thicker coating. For allentrainment speeds, the percental friction reduction is increased with the in-crease of SRR. The fact that the friction coefficients in general are higher atthe lower speed, Figure E.5, does not indicate a transition to the mixed lubri-cation regime compared to the higher speeds, but rather an effect of differentfull film lubrication conditions. A higher entrainment speed will lead to athicker film, and thus lower shear rates. Furthermore, a higher entrainmentspeed will increase thermal softening of the lubricant, reducing the viscosity


0 0.2 0.4 0.6 0.8 1 1.20

0.01

0.02

0.03

0.04

0.05

0.06

Slide to roll ratio

Fric

tion

coef

ficie

nt


Figure E.5: Friction measurements for squalane at 1.6 m/s entrainment speedand 1.25 GPa of maximum hertzian pressure for uncoated steel and two differ-ent thicknesses of the same DLC coating.

Table E.3: Friction reduction with DLC coatings at SRR=1.05.

Load [N] Entrainment Reduction Reductionspeed [m/s] 0.8 µm [%] 2.8 µm [%]

300 6.144 26 41300 3.145 20 3580 6.144 29 3980 3.145 23 3780 1.611 16 30

and also leading to a reduction in friction. When comparing the friction trendsat the same speed for both pressures, Figs. E.6 and E.7, the friction coefficientincreases faster with SRR and reaches a higher value for the higher pressure.At higher SRR when thermal softening of the lubricant is dominating the fric-tion behaviour, the high pressure case drops more rapidly due to higher heatgeneration.

E.3. Results 241

0 0.2 0.4 0.6 0.8 1 1.20

0.01

0.02

0.03

0.04

0.05

0.06

Slide to roll ratio

Fric

tion

coef

ficie

nt


Figure E.6: Friction measurements for squalane at 6.144 m/s entrainmentspeed and 1.25 GPa of maximum hertzian pressure for uncoated steel andtwo different thicknesses of the same DLC coating.

0 0.2 0.4 0.6 0.8 1 1.20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Slide to roll ratio

Fric

tion

coef

ficie

nt


Figure E.7: Friction measurements for squalane at 6.144 m/s entrainmentspeed and 1.94 GPa of maximum hertzian pressure for uncoated steel andtwo different thicknesses of the same DLC coating.


E.4 Discussion

The tribological tests performed in this investigation clearly show the frictionreducing effect of a DLC coating in full film EHL, Figs. E.5 to E.7 and TableE.3. The reduction in friction coefficient for the 2.8 µm coating at the highesttested SRRs ranges from 30 % at an entraniment speed of 1.6 m/sand 1.25 GPaof pressure to a reduction of 41 % at an entrainment speed of 6.144 m/s and1.94 GPa of pressure. For the thinner DLC coating the same conditions leadto friction reductions of 16 and 26 %. It is apparent that the thicker coating ismore effective at reducing friction.

The phenomena of reducing friction with DLC coatings in fullfilm condi-tions where there are no asperity contacts between the mating surfaces have byseveral authors been explained as solid-liquid slip [180, 201, 202]. The prin-ciple of the theory being that the often assumed no-slip boundary conditionbetween the liquid and the solid is violated. It means that the interaction be-tween the solid and the liquid is not strong enough to resist the shear forcesat the interface, and the liquid will thus have a lower velocity than the solidsurface. This will in turn reduce shear rates and friction.

If the friction reduction can be explained by the interaction between thesurface and the liquid, it must be possible to be determined by utilizing appro-priate measurements. Such measurements have included contact angle, sur-face energy, and spreading parameter. However, results in the literature havebeen non conclusive in the sense that some authors have correlated the frictionreduction to poor wetting (large contact angle) [100, 108, 109, 196, 200–202]and sometimes with surface related properties such as hydrophobicity and lowsurface energy [180, 181]. Recently, Kalin and Polajnar presented a series ofstudies [181, 203] where they conclude that there is poor correlation betweencontact angle and friction reduction with surface coatingsin full film EHL.On the other hand, they found good correlation between surface energy, andespecially the polar part of the surface energy, and friction reduction. Theyalso found good correlation between a spreading parameter,which takes intoaccount both surface and liquid properties, and friction reduction. In this pa-per, the friction reducing capabilities of two different thicknesses of the sameDLC coatings are investigated with uncoated steel specimens as a reference.As seen in Figs. E.5 and E.6 the friction reduction is more effective with thethicker coating than the thinner one. However, this difference in friction reduc-tion between these two coatings cannot be explained by contact angle, surfaceenergy, or spreading parameters. Nor is the surface roughness different be-tween the specimens. In Figure E.4 it can be seen that there isno statistical

E.4. Discussion 243

difference in contact angle between the two coated specimens. The uncoatedreference has slightly lower contact angle, but the difference may be withinthe measurement error. Furthermore, as seen in Figure E.2 there is no statisti-cally significant difference between the two DLC coatings interms of neithertotal surface energy, nor the dispersive and polar parts. The surface energies ofthe coated specimens are, however, lower than the steel reference. Moreover,no significant difference can be seen between the coatings with the calculatedspreading parameters in Figure E.3. Even here there is a distinction betweenthe coated specimens and the uncoated steel reference.

This investigation shows that contact angle and surface energy measure-ments alone are not sufficient to estimate the friction reduction observed withsurface coatings in full film EHL. The present authors have earlier [269] hy-pothesized that the friction reduction obtained in full filmEHL may be causedby the low thermal conductivity of certain DLC coatings, andthis theory waslater validated in another investigation including both experimental measure-ments and numerical simulations [272]. The mechanism at work in these casesis that applying a thin diamond-like-carbon coating to metal surfaces createsan insulating effect that, due to the increased liquid lubricant film tempera-ture at the centre of the contact, locally reduces apparent lubricant viscosityand thus friction. The same numerical model was later used toinvestigate theeffect of coating thickness and thermal properties of the coatings on frictionreduction [236] with results in line with those obtained experimentally in thispaper.

The idea that thermal properties of the specimens have an influence onfilm shape and friction is not new. The concept of a temperature-viscositywedge caused by specimens with widely different thermal conductivities wasintroduced by Cameron in the 1960’s [277, 278] and has been discussed byseveral authors [279].

It is therefore likely that some of the inconsistencies between contact an-gles and/or surface energies and friction reduction achieved in full film EHLmay be explained by the fact that solid-liquid slip alone cannot explain ordescribe the phenomena. In fact, under certain conditions,the thermally insu-lating effects of the coatings are dominating, or possibly even the only cause tothe friction reduction. In this investigation, since thereis no significant differ-ence between contact angle and surface energy of the two coating thicknesses,the difference in friction reduction may solely depend on different thermalproperties due to the different thicknesses of the coatings. However, it is moreuncertain if the friction reduction achieved with the coatings compared to theuncoated reference is only due to thermal insulation, or if asmaller part of the


friction reduction is also due to solid-liquid slip.The influence of surface roughness on solid-liquid slip has been discussed

by several authors, where many of them have concluded that for slip to oc-cur, the surfaces have to be very smooth [100, 107–109, 205],typically 6-12nm RMS. The critical roughness is most likely depending on the size of themolecules of the liquid used [193, 206, 207]. Pearson and Petrie [207] meanthat no slip can occur when the molecular size is smaller thanthe wall rough-ness scale. However, it is not necessarily so that as smooth asurface as possiblewill lead to the greatest amount of slip. At relatively low roughness (below 15nm) it has been shown that RMS roughnesses of 0.7 and 4.0 nm produced lessslip than a 12.2 nm surface [195]. It should, however, be mentioned that theroughness in this case was not only different in amplitude, but also producedwith different geometries. That an optimized roughness is superior to verysmooth surfaces in some cases may be apparent by consideringsolutions fromnature such as lotus leaves [208,209]. In nature, for instance low wettability iscreated by clever use of surface texturing.

Moreover, most of the investigations on solid-liquid slip are performedat atmospheric pressure or hydrodynamic pressure and may therefore not bedirectly applicable on EHL. Solid-liquid slip has been reported to be reducedwith pressure in the hydrodynamic pressure range [123,210], but also observedto increase with pressure in EHL [211]. In favor for solid-liquid slip in EHLis amplitude reduction [19,22] that describes elastic deformation of the rough-ness inside the high pressure contact. Studies show that theroughness reduc-tion is dependent on several factors, such as entrainment speed, slide to rollratio, and pressure, and that long wavelengths are compressed more than shortwavelengths. This means that the actual roughness inside the high pressure re-gion may, depending on the roughness parameters and runningconditions, besmoother than outside of the contact. This would at least according to Pearsonand Petrie be beneficial for solid-liquid slip.

To the author’s knowledge, only a few studies have observed solid-liquidslip in EHL [103, 106, 211], and they have all used relativelysmooth surfaces(below 12 nm). The liquids they used are either polyphenyl ether (5P4E) whichhas very high surface tension and is known to show poor wetting at metal sur-faces, or high molecular weight polybutene. It is still to beelucidated whethersolid-liquid slip can still be observed with the use of surfaces and lubricantsmore often encountered in machine elements such as ball bearings and gears.Studies that indirectly indicate solid-liquid slip as a consequence of reducedfriction in full film EHL may, in fact, as indicated by this study, be an effectof thermal insulation, or a combination between thermal insulation and solid-

E.5. In conclusion 245

liquid slip.

E.5 In conclusion

This paper shows that measurements of contact angle, surface energy, or spread-ing parameter alone are not sufficient to estimate the full film EHD frictionreduction capability of a surface coating. Friction tests were carried out ina ball-on-disc machine with specimens coated with the same DLC coating,but with different thicknesses, and compared to uncoated specimens, all withthe same surface roughness. The DLC coated surfaces showed areduction infriction in all tested cases compared to the uncoated specimens. However, thethicker DLC coating provided substantially lower frictionthan the thinner coat-ing. Contact angle measurements were carried out with reference fluids andthe test fluid to calculate surface energy and spreading parameter for the testcombinations. No significant differences were found between the two coatingthicknesses although their friction reduction effects were substantially differ-ent. The difference in friction reduction between the coatings is attributed tothe fact that the thicker coating has a greater thermal insulating effect than thethinner coating.

Acknowledgement

The authors wish to thank IonBond AB for providing the DLC coatings used inthe tests and the Swedish Foundation for Strategic Research(ProViking) andVR (Swedish Research Council) for financial support.

Paper F

The correlation between gearcontact friction and ball-on-discfriction measurements

247

249

Submitted to journal

The correlation between gear contact friction andball-on-disc friction measurements

M. Björling, J. Miettinen, P. Marklund, A. Lehtovaara and R.Larsson


Abstract

Running experiments with full-size gearboxes from the actual application hasthe advantage of giving realistic results in terms of power losses depending onlubricant type, load, speed and operating temperature. Thedrawback is ex-tensive costs, lengthy testing, and the difficulty in differentiating between loaddependent and load independent losses, and which losses arecoming from thegears, seals, bearings or synchronizers. As an alternativeto numerical pre-dictions, many authors have used twin-disc machines to simulate power loss ingear contacts. By controlling the rotational speeds of two rollers in contact, thesame entrainment speeds and slide to roll ratios can be achieved as along theline of action of the gear system that are simulated. This approach is cheaperthan running full gear tests, and gives more detailed information regarding gearcontact friction along the line of action. A ball-on-disc tribotester do not suf-fer from the same aligning problems as encountered in a twin disc machine,and may be available at research facilities not having a twindisc machine. Itis however unclear if it is possible to correlate the friction coefficient in thecircular contact in a ball-on-disc tribotester to the line contact in the spur gearcontact. The purpose of this work was to investigate the correlation betweenfriction measurements conducted in a ball-on-disc machinewith friction mea-surements conducted in a back-to-back gear test rig. The correlation betweenthe gear tests and the ball-on-disc tests were reasonably good in terms of abso-lute values, and the shape of the friction curves were similar, indicating that theball-on-disc measurements to a large extent are capturing the behaviour of thegear contact. Furthermore, the ranking of the oils showed perfect agreementat both temperatures. Using the friction mapping approach presented in this

250 Paper F. Correlation between gears and ball-on-disc friction

work also gives the possibility to assess the contact friction losses in severaltheoretical gear pairs with different geometries by performing a small amountof ball-on-disc tests with one, or several lubricants.

F.1. Introduction 251

F.1 Introduction

Reducing energy consumption and emissions have been a priority in the indus-trialized world for a long time, and even more so during the last 5-10 years.With the exception of bringing new technologies and solutions to the market,constant development is carried out to improve current technology. The auto-motive market has faced increasing restrictions in terms ofemissions and hastherefore spent large amounts of money on research and development. Risingfuel prices and increased environmental concern also make the customers moreprone to purchase more fuel efficient vehicles. It has been assessed that 33 %of the fuel energy in a car is used to overcome friction, and that 7-18 % of thesefriction losses originates from the transmission [140]. Inheavy road vehiclesand buses, 33.5 % of the fuel energy is used to overcome friction, and 13 %of these losses originates from the transmission [141]. Thelosses in a geartransmission can be devided into two categories; load dependent and load in-dependent losses. The load independent losses are typically viscous losses dueto oil churning and are mostly governed by lubricant viscosity, density and thegeometrical design of gears and housing. The load dependentlosses are dueto friction in the rolling and sliding interfaces between the mating gear teeth,and are influenced by a large numbers of parameters. The totalgear contactfriction losses are ranging between 4.5 and 55 % depending onthe design anduse of the transmission [140, 243]. Most gears are operatingin the elastohy-drodynamic lubrication (EHL) regime and the friction originating from thesekinds of contacts are the matter of interest in this paper.

Running experiments with full size gearboxes from the real application hasthe advantage of giving realistic results in terms of power losses dependingon lubricant type, load, speed and operating temperature. The drawbacks areextensive costs, lengthy testing, and the difficulty in differentiating betweenload dependent and load independent losses, and which losses are coming fromthe gears, seals, bearings or synchronizers. Even when a gear pair is rotatingat a constant speed several parameter are changing along theline of actionbetween the meshing teeth, such as load, entrainment speed and slide to rollratio (SRR). When running tests with gears and being successful in removingall other sources of losses, only an average friction coefficient can be obtained.To remedy this problem and allow more detailed studies of gear losses bothnumerical and experimental methods have been used.

Several researchers have solved the numerical EHL problem to be able topredict, and understand gear friction. Such studies include both smooth [142,280] and rough surfaces [145, 147, 281]. A reliable and accurate numerical


prediction model for gear contact friction would be the bestalternative sincethe numer of tests would be kept at a minimum, saving both timeand money.However, EHL is a complex field, and there are as far as the authors knowsno models with such true predictive capabilities to date [271]. Due to thesevere running conditions in many gearboxes, highly additivated lubricantsare often used which also puts demands on the numerical models to includetribochemical effects which is a tremendous challenge.

As an alternative to numerical predictions, many authors have used twin-disc machines to simulate power loss in gear contacts [153,169,240,241,247,248]. By controlling the rotational speeds of two rollers incontact, the sameentrainment speeds and SRRs can be achieved as in the line of action of thegear system that are simulated. This approach is cheaper andless time consum-ing than running full gear tests, and gives more detailed information regardinggear contact friction along the line of action. The twin discis seen as suit-able for mimicking a gear contact also due to the fact that both twin disc, spurand helical gears to some extent operate with line contacts.A ball-on-disc tri-botester do not suffer from the same aligning problems encountered in a twindisc machine using disc profiles creating a pure line contact, and may be avail-able at research facilities not having a twin disc machine. It is however unclearif it is possible to correlate the friction coefficient in thecircular contact in aball-on-disc tribotester to the line contact in the spur gear contact. The purposeof this work is to investigate the correlation between friction measurementsconducted in a ball-on-disc machine with friction measurements conducted ina FZG gear test rig. In addition, using the earlier presentedconcept of fric-tion mapping [252], a method is proposed to predict the friction coefficientin an arbitrary gear pair from a minimum of measurements in a ball-on-discmachine.

F.2 Overall Methodology

The following sections cover the test rigs, test specimens and lubricants usedin the experiments. It also contains information about how the experimentswere performed and how the data was processed and evaluated.

F.2.1 Ball-on-disc tribotester

The experiments were carried out with a Wedeven Associates Machine (WAM)11, ball-on-disc test device. The lubricant is supplied at the centre of the discin an oil dispenser that distributes the lubricant across the disc surface. The

F.2. Overall Methodology 253

Figure F.1: FZG test rig.

lubricant is circulated in a closed loop from the oil bath, through a peristalticpump to the oil dispenser at the centre of the disc. The peristaltic pump isdelivering approximately 180 ml/min. Three thermocouplesare used in thetest setup, one located in the oil bath, one in the outlet of the oil supply andone trailing in the oil film close to the inlet region of the ball-on-disc contact.A more thorough description of the test rig and its features is presented inprevious work [252].

F.2.2 Gear test rig

A modified FZG test rig was used for the gear tests, as depictedin Figure F.1.The test gears, described in section F.2.3 were located in two separate housingswith their own lubrication system with a capacity of 25 liters each. The gearswere spray lubricated with a flow rate of 2.0 liters per minutedirected in theentry side of the mesh. The loading of the gears were done by applying atorque on shaft 1 with the help of a rod and dead weights. The correspondingstrain caused by the twist of the shaft was measured with fullbridge straingauge system. The power circulating design of the test rig means that theelectric motor was only compensating for the energy equivalent to the lossesin the system. However, the gear friction losses were calculated by the frictionmoment measured by a torque meter on shaft 2.

F.2.3 Test specimens and lubricant

The test gears are made of case hardened steel, 21 NiCrMo2-2.The gears werecase hardened to a depth of 1.1± 0.5 mm and a hardness of 58± 2 HRC. Thetest gears were subjected to grinding and polishing, down toa surface rough-ness of around 30 nm RMS measured with a stylus mechanical profilometer.


Table F.1: Test gear geometry.

Number of teeth 20Pressure angle [◦] 20Gear Ratio 1Centre distance [mm] 91.5Normal module [mm] 4.5Profile shift 0.176Face width [mm] 20Contact ratio 1.45Addendum contact ratio 0.725

Table F.2: Lubricant properties.

Name Emgard Shell StatoilMTF 4250 Spirax S6 Gearway S5

Classification 75W-90 75W-90 75W-140Density @ 15◦C [kg/m3] 879 878 872Kinematic viscosity @ 40◦C [mm2/s] 140 115 190Kinematic viscosity @ 100◦C [mm2/s] 18.4 15.2 25Dynamic viscosity @ 40◦C [mPas] 123 101 166Dynamic viscosity @ 100◦C [mPas] 16 13 22

This gives a combined RMS roughness for the gear pair of approximately 42nm. Both pinion and gear have the same properties as shown in Table F.1,which means that the gear ratio is 1.

The ball-on-disc tests were performed with specimens made of DIN 100Cr6(AISI 52100) bearing steel. The specimens have been measured to a surfaceroughness, RMS of 25 nm for the balls and 35 nm for the discs, which gives acombined roughness of approximately 43 nm. The surface roughness measure-ments were conducted in a Wyko NT1100 optical profilometer system fromVeeco. The measurements were performed using 10x magnification and 1xfield of view. The balls are grade 20 with a 13/16 inch (20.63 mm) outer diam-eter and a hardness of about 60 HRC. The discs have a 4 inch (101.6 mm) outerdiameter, a circumferential grind (before polish) and are through hardened toabout 60 HRC.

The experiments were performed with three commercially available, fullyformulated transmission oils whose properties are shown inTable F.2.


F.2.4 Gear test procedure

The gear tests were performed at two different oil feed temperatures for eachoil, 40 and 70◦C and with a torque of 302 Nm and 0 Nm. Before starting themeasurements the lubrication systems were filled with 50 liters of the test oiland the rig was run with a 302 Nm torque at a speed of 1250 rpm for1 hourfor lubricant temperatures to stabilize and to warm up all components in thetest rig. The rig was then run with zero load at 1250 rpm for onehour forthe gear teeth to cool down. The rotational speed was now set to 750 rpm, stillwith zero load, and the measurements were started. After 15 minutes the speedwas increased by 250 rpm, and after another 15 minutes increased by another250 rpm until the maximum speed of 2000 rpm was reached. At this pointthe rig was stopped, and 302 Nm torque was applied and the procedure wasrepeated from 750 to 2000 rpm in steps of 250 each 15 minutes. The lubricanttemperature, measured with thermocouples located in the oil feed at the gearmesh, were typically deviating less than± 1◦C during the tests. Before fill-ing up with a new lubricant the lubrication system, gears andhousings werethoroughly cleaned with heptane and left to dry over night.

The total power lossPl in a gear transmission is divided into load depen-dent losses and load independent losses. The load dependentlosses are dividedinto mesh losses,Pm, and load dependent bearing losses,Pbl, while the load in-dependent losses,Pnl, includes oil churning, seal friction and bearing losses atzero load. The total power loss can thus be expressed as:

Pl = Pm+Pnl +Pbl (F.1)

By running the test rig with zero load, the load independent losses can beassessed, while running the rig with full load gives the total power loss. To beable to calculate the mesh dependent power losses, which is of main interest inthis study there is also a need to assess the load dependent bearing losses. Theload dependent power losses for the SKF 6406 deep groove ballbearings werecalculated as [282]:

Pbl = (Mrr +Msl)2πω60

(F.2)

whereMrr is the rolling frictional moment, andMsl is the sliding frictionalmoment.Mrr is calculated as:

Mrr = ϕishϕrsGrr (νω)0.6 (F.3)


with:

ϕish =1

1+1.84×10−9(ωdm)1.28ν0.64 (F.4)

ϕrs = 1/e

[

Krsνω(d+D)√

Kz2(D−d)

]

(F.5)

and:

Grr = R1d1.96m F0.54

r (F.6)

whereω is the rotational speed,ν the kinematic viscosity at operating temper-ature of the oil,D = 90 mm,d = 30 mm,dm = 60 mm,Krs = 3×10−8, Kz = 3.1,andR1 = 3.6×10−7. Msl is calculated as:

Msl = Gslµsl (F.7)

with:

Gsl = S1d−0.26m F5/3

r (F.8)

µsl = ϕblµbl +(1−ϕbl)µEHL (F.9)

and:

ϕbl =1

e2.6×10−8(ων)1.4dm(F.10)

whereµbl = 0.15,µEHL = 0.04 andS1 = 2.43×10−3. When the mesh losses arecalculated it is possible to compute an approximate averagefriction coefficient,µm, based on a method presented by Michaelis and Höhn [151]:

µm =Pm

PtHv(F.11)

wherePt is the total transmitted power andHv the gear loss factor given by:

Hv =πz1

i +1i

(1− εt + ε2p+ ε2

g) (F.12)

wherez is the number of teeth,i the gear ratio,εt the contact ratio,εp theaddendum contact ratio of the pinion, andεg the addendum contact ratio of thegear.


F.2.5 Correlation methodology

The purpose of this study was to create conditions in the ball-on-disc test rigsimilar to the gear tests and compare the friction data between the tests. Forthis reason, the surface roughness were similar for the ball-on-disc specimens,and the gears. An analytic geometric model of the gear contact was used to cal-culate the maximum hertzian pressure with 302 Nm of torque toapproximately1.24 GPa. The same model was also used to calculate SRR and entrainmentspeed along the line of action. The SRR in a gear contact is independent ofrotational speed and was calculated to range from -1.1 to 1.1in the gears usedin this investigation. Since both pinion and gear have the same geometry theentrainment speed is constant along the line of action, but dependent on therotational speed of the gears. The lowest rotational speed tested was 750 rpmwhich corresponds to an entrainment speed of 1.37 m/s while the highest speedof 2000 rpm gives an entrainment speed of 3.66 m/s.

The ball-on-disc tests were performed with the same three oils as the geartests and at the same temperatures. In the gear setup the contact pressure ischanging along the line of action due to changes in effectiveradius, but alsosince the load is sometimes carried by only one tooth engagement and other-wise by two teeth engaging. To be able to use the earlier introduced conceptof friction mapping [252] all tests in the ball-on-disc machine were performedwith a load of 76 N, equivalent to a maximum hertzian pressureof 1.24 GPawhich is the same as the maximum load calculated for the gear contact. An-other simplification was made using only positive values forSRR, where inthe actual gear contact, the SRR has opposite signs from going into contact(approach), to the pitch point, compared to from the pitch point going out ofcontact (recess). Friction data for various entrainment speeds and SRRs weremeasured in a range that spans the minimum and maximum valuescalculatedfor the gear contact at the different rotational speeds as detailed in Table F.3.A triangle based linear interpolation method were used to create a friction mapfor a specific lubricant at a specific pressure for a range of entrainment speedsand SRRs. One of the six 2D friction maps are shown in Figure F.2 wherefriction coefficient is plotted as contours of entrainment speed and SRR. Thefigure also contains projections of the corresponding entrainment speeds andSRRs for two different rotational speeds, 900 and 1300 rpm, for the gear pairinvestigated in this paper. Included in the figure is also thecorresponding en-trainment speeds and SRRs for a FZG type C gear pair at 2300 rpmwherethe entrainment speed is not constant along the line of action. The FZG typeC pattern is only included as an example, and has not been usedin any tests



Slid

e to

rol

l rat

io

0.03

0.0350.04

0.0450.05

1.5 2 2.5 3 3.5 4

0.2

0.4

0.6

0.8

1

Figure F.2: Example of friction map with corresponding entrainment speedsand SRRs along the line of action for tested gear pair at 750 rpm (solid line),tested gear pair at 1250 rpm (dashed line), tested gear pair at 2000 rpm(dashed/dotted line) and FZG type C gear pair at 2000 rpm (dotted line).

performed in this article. The friction map can be seen as a look-up-table foran arbitrary rotational speed of a gear pair. The line of action were dividedinto 200 data points in the friction map for a specific rotational speed and usedto calculate a mean friction coefficient that was compared tothe mean frictioncoefficient obtained from the same rotational speed in the gear test.

F.2.6 Ball on disc test procedure

The ball-on-disc test device was used to generate friction data from a relativelybroad range of operating conditions where one test cycle covers entrainmentspeeds between 1 and 4 m/s and SRRs from 0.0002 to 1.2. Both ball anddisc specimens were cleaned with heptane and ethyl alcohol before starting theexperiments for each of the test cases. Before starting the experiments for eachtest case, the test device was warmed up to the desired operating temperatureduring approximately 60 minutes with lubricant circulation over both ball anddisc to ensure thermal stability. When a stable temperaturewas reached a 76 Nload was applied which is equivalent to 1.24 GPa maximum Hertzian pressureand the machine was calibrated for pure rolling by adjustingspindle angle andpositioning of the ball to ensure a condition of no spinning.These settings

F.3. Results and discussion 259

Table F.3: Investigated conditions in ball-on-disc rig.

Temperature 40 and 70◦CContact load 76 NMaximum hertzian pressure 1.24 GPaEntrainment speed,Ue 1 - 4 m/sSlide to Roll Ratio, SRR 0.0002 - 1.2Oils See Table F.2

were then held constant for 20 minutes to ensure a mild run-in. Subsequentlythe test cycle was started. The test cycle contains several loops where SRRis held constant for each loop and the entrainment speed is ramped from 4 to1 m/s. In the first loop the SRR is held at 0.0002 and is then continuouslyincreased with each loop until it reaches 1.2. The temperature of the oil bulkand fluid adhered at the disc surface was typically deviatingless than± 1.5◦Cfrom the target temperatures of 40 and 70◦C during testing.

F.3 Results and discussion

Figure F.3 shows the change in SRR along the line of action going from ap-proach (gear flank position 0), to the pitch point (gear flank position 100) andrecess (gear flank position 200). Figure F.4 shows the friction coefficientsalong the line of action for the gear pair at three different rotational speedsoriginating from the ball-on-disc measurements using the Emgard oil at 40◦C.Each rotational speed corresponds to an entrainment speed,and a set of SRRsas shown in figure F.2. As the gear flanks first comes into contact, the SRRis at its highest value which leads to extensive thermal softening of the lu-bricant thus leading to low friction coefficients. The friction coefficients arethen gradually increasing as the SRR decrease following theline of action to-wards the pitch point, thus leading to reduced thermal softening and insteada behaviour dominated by shear thinning and the limiting shear stress of thelubricant. At the lowest SRRs close to the pitch point the friction coefficientis rapidly decreasing due to the low shear rates the oil is subjected to at lowsliding. When the pitch point is passed the SRR is again gradually increasedfollowing the same behaviour following the line of action towards the recess.The friction coefficients are generally lower at the higher entrainment speeds,an effect attributed to thicker oils films reducing the shearrates in the lubricantfilm. Higher entrainment speeds also leads to higher slidingspeeds causingadditional thermal softening, and possibly less asperity interactions due to the


0 50 100 150 2000

0.2

0.4

0.6

0.8

1

Gear flank position

Slid

e to

rol

l rat

io

Figure F.3: Change in SRR along the line of action using only positive values ofSRR.

thicker oil films. The minimum film thickness was calculated to 180 nm at thehighest temperature and the lowest entrainment speed for the lubricant with thelowest viscosity. 180 nm is significantly larger than the combined roughnessof the surfaces. However, the film thickness calculation didnot include shearthinning and therefore the actual film thickness may be substantially lower.Most likely, the majority of the tests were performed in fullfilm lubrication,with the possible exception of the thinnest oil at the highest temperature andlowest speeds.

The mean value of the friction coefficients measured in the ball-on-discrig along the line of action for a specific entrainment speed with a specificlubricant and temperature was used for comparison with the correspondingcase at a specific rotational speed in the FZG test rig.

The results from the ball-on-disc friction measurements and the FZG testsare shown in Figs. F.5 and F.6 for the oil temperatures of 40 and 70◦C re-spectively. The most important observation is that the ranking of the oils arethe same in both test rigs, at both oil temperatures. It is clear that the lowestcontact friction is achieved with the Gearway oil that has the highest viscosity,while the highest contact friction is achieved with the Spirax oil having thelowest viscosity. This is the case for both 40 and 70◦C. However, it shouldbe mentioned that these differences may not solely depend onthe viscositiesof the lubricant, but also other properties, such as the limiting shear stress,

F.3. Results and discussion 261

0 50 100 150 200

0.03

0.035

0.04

0.045

0.05

0.055

Gear flank position

Fric

tion

coef

ficie

nt

750 rpm − 1.37 m/s1250 rpm − 2.285 m/s2000 rpm − 3.66 m/s

Figure F.4: Friction coefficients simulated in ball-on-disc machine along the lineof action at three different rotational speeds for Emgard oil at 40◦C.

and different additive packages. Furthermore, the shape ofthe friction curvesare relatively similar between the test rigs indicating that the ball-on-disc mea-surements are reasonably well capturing the behaviour of the gear contact. Theagreements between the ball-on-disc measurements and the gear tests are wellin line with, or better than comparisons between twin disc and gears found inliterature [153,240,241].

However, there is a difference in absolute friction values between the ball-on-disc, and gear tests. One of the reasons are most likely the simplification ofusing only one pressure level in the ball-on-disc measurements to make the useof friction mapping possible. This pressure corresponds tothe pressure in thegear contact when only one tooth is carrying the load. In parts along the line ofaction the pressure is lower when two teeth are carrying the load, and the fric-tion coefficients will drop due to the reduction in pressure.This is one reasonfor the ball-on-disc friction values to be slightly higher than the values fromthe gear test, which is also the general case in Figs. F.5 and F.6. Moreover, thedeviations generally increase with rotational speed whichis most likely causedby different thermal properties of the two systems. It is expected that thermaleffects will increase more with rotational speed in the gearsetup than in theball-on-disc setup due to the higher frictional power generated in the gear rig.

Although the correlation between the gear tests and the ball-on-disc testsare not always spot on in terms of absolute friction coefficients, the shape of


800 1000 1200 1400 1600 1800 20000.01

0.02

0.03

0.04

0.05

0.06

0.07


Fric

tion

coef

ficie

nt


Figure F.5: Friction coefficients for ball-on-disc and gear tests for three differentlubricants conducted at an oil temperature of 40◦C.

800 1000 1200 1400 1600 1800 20000.01

0.02

0.03

0.04

0.05

0.06

0.07


Fric

tion

coef

ficie

nt


Figure F.6: Friction coefficients for ball-on-disc and gear tests for three differentlubricants conducted at an oil temperature of 70◦C.

F.4. In conclusion 263

the friction curves are similar in both cases. Moreover, thedifferences be-tween the oils are maintained for the two different rigs, at both temperatures,suggesting that the use of friction testing in a ball-on-disc machine could bea good alternative to a full gear test for evaluating lubricants in terms of fric-tion performance. In addition, the use of the friction mapping concepts givesa gear developer the possibility to assess an approximate friction coefficientfor a type of gear with a specific set of entrainment speeds andSRRs with-out having to manufacture such a gear. A new gear at a specific rotationalspeed will cover a different area compared to the examples given in figure F.2,and if the friction map is sufficiently large, an enormous amount of differentgear geometries and rotational speeds could be assessed without having to runany more ball-on-disc tests. The exception being large differences in contactpressures that could possibly occur when changing gear geometries requiringanother ball-on-disc test to be performed at another contact pressure level.

F.4 In conclusion

The objective of this work was to investigate if it is possible to use a ball-on-disc machine creating a circular EHD contact to assess the contact friction in aspur gear line contact. The ball-on-disc machine was used tomeasure frictionin a range of entrainment speeds and SRRs that includes the correspondingvalues calculated for the gear pair at a certain range of rotational speeds. Theball-on-disc measurements were then used to calculate an average contact fric-tion coefficient along the line of action of the corresponding gear pair at a cer-tain rotational speed. This average friction coefficient was then compared toan average friction coefficient obtained in the gear test rig. The correlation be-tween the gear tests and the ball-on-disc tests were reasonably good in terms ofabsolute values, and the shape of the friction curves were similar between thetests rigs indicating that the ball-on-disc measurements to a large extent werecapturing the behaviour of the gear contact. The ranking between the oils werethe same in the two test rigs at both oil temperatures. These findings bringsforward the possibility to perform cheaper tests in a ball-on-disc machine toevaluate different lubricants in terms of friction performance compared to fullgear tests. Furthermore, using the friction mapping approach presented in thiswork also gives the possibility to assess the contact friction losses in severaltheoretical gear pairs with different geometries by performing a small amountof ball-on-disc tests for each investigated lubricant.


Acknowledgement

The authors gratefully acknowledge industrial partners Volvo ConstructionEquipment, Scania and Vicura AB for support, and Swedish Foundation forStrategic Research (SSF), ProViking and VR (Swedish Research Council) forfinancial support. The authors would also like to thank Scania and Statoil lu-bricants for providing the test lubricants.

References

[1] B. Tower, “Second report on friction experiments (experiments on theoil pressure in a bearing),”Proc. Inst. Mech. Eng., pp. 58–70, 1885.

[2] O. Reynolds, “On the theory of lubrication and its application to mrbeuchamp tower’s experiments, including an experimental determina-tion of the viscosity of olive oil,”Philosophical Transactions of theRoyal Society, vol. 177, pp. 157–234, 1886.

[3] B. J. Hamrock, S. R. Schmid, and B. O. Jacobson,Fundamentals of fluidfilm lubrication - SE. Marcel Dekker, Inc, 2004.

[4] K. L. Johnson, “Regimes of elastohydrodynamic lubrication,” Journalof Mechanical Engineering Science, vol. 12(1), pp. 9–16, 1970.

[5] C. Hooke, “The elastohydrodynamic lubrication of heavily loaded con-tacts,”Journal of Mechanical Engineering Science, vol. 19(4), pp. 149–156, 1977.

[6] S. Bair,High Pressure Rheology for Quantitative Elastohydrodynamics.Elsevier, first edition ed., 2007.

[7] H. A. Spikes, “Sixty years of ehl,”Lubrication Science, vol. 18(4),pp. 265–291, 2006.

[8] B. J. Hamrock,Fundamentals of fluid film lubrication. McGraw-Hill,1994.

[9] S. Bair and W. O. Winer, “Regimes of traction in concentrated contactlubrication,” Transactions of the ASME - Journal of Lubrication Tech-nology, vol. 104(3), pp. 382–391, 1982.

265

266 References

[10] P. K. Gupta, H. S. Cheng, D. Zhu, N. H. Forster, and J. B. Schrand,“Viscoelastic effects in MIL-L-7808-type lubricant. part1: analyticalformulation,” Tribology transactions, vol. 35(2), pp. 269–274, 1992.

[11] C.-H. Hsu and R.-T. Lee, “An efficient algorithm for thermal elastohy-drodynamic lubrication under rolling/sliding line contacts,” Journal ofTribology, vol. 116(4), pp. 762–769, 1994.

[12] S. Bair and W. O. Winer, “A simple formula for ehd film thicknessof non-newtonian liquids,”Elastohydodynamics ’96 / D. Dowson et al.(editors) - Tribology Series, vol. 32, pp. 235–241, 1997.

[13] S. Bair, “Shear thinning correction for rolling/sliding elastohydrody-namic film thickness,”Proceedings of the Institution of Mechanical En-gineers, Part J: Journal of Engineering Tribology, vol. 219(1), pp. 67–74, 2005.

[14] J. Y. Jang, M. M. Khonsari, and S. Bair, “Correction factor formula topredict the central and minimum film thickness for shear-thinning fluidsin ehl,” Journal of Tribology, vol. 130(2), pp. 1–4, 2008.

[15] P. Anuradha and P. Kumar, “New film thickness formula forshear thin-ning fluids in thin film elastohydrodynamic lubrication linecontacts,”Proceedings of the Institution of Mechanical Engineers, Part J: Journalof Engineering Tribology, vol. 225(4), pp. 173–179, 2011.

[16] W. Habchi, S. Bair, F. Qureshi, and M. Covitch, “A film thickness cor-rection formula for double-newtonian shear-thinning in rolling ehl cir-cular contacts,”Tribology Letters, vol. 50(1), pp. 59–66, 2013.

[17] R. Ohlsson, A. Wihlborg, and H. Westberg, “The accuracyof fast 3Dtopography measurements,”International Journal of Machine Tools andManufacture, vol. 41(13-14), pp. 1899–1907, 2001.

[18] P. Cann, E. Ioannides, B. Jacobson, and A. A. Lubrecht, “The lambdaratio - a critical re-examination,”Wear, vol. 175(1-2), pp. 177–188,1994.

[19] J. A. Greenwood and G. E. Morales-Espejel, “The behaviour of trans-verse roughness in EHL contacts,”Proceedings of the Institutionof Mechanical Engineers, Part J: Journal of Engineering Tribology,vol. 208(2), pp. 121–132, 1994.

References 267

[20] C. Venner and A. Lubrecht, “Numerical simulation of a transverse ridgein a circular EHL contact under rolling/sliding,”Journal of Tribology,vol. 116(4), pp. 751–761, 1994.

[21] C. Venner and A. Lubrecht, “Numerical analysis of the influence ofwaviness on the film thickness of a circular EHL contact,”Journal ofTribology, vol. 118(1), pp. 153–161, 1996.

[22] A. A. Lubrecht and C. H. Venner, “Elastohydrodynamic lubrication ofrough surfaces,”Proceedings of the Institution of Mechanical Engi-neers, Part J: Journal of Engineering Tribology, vol. 213, pp. 397–403,1999.

[23] C. C. Chou and J. F. Lin, “Tribological effects of roughness andrunning-in on oil-lubricated line contacts,”Proceedings of the Institu-tion of Mechanical Engineers, Part J: Journal of Engineering Tribology,vol. 211(3), pp. 209–222, 1997.

[24] J. Lord and R. Larsson, “Film-forming capability in rough surfaceEHL investigated using contact resistance,”Tribology International,vol. 41(9-10), pp. 831–838, 2008.

[25] P. Lugt, R. Severt, J. Fogelström, and J. Tripp, “Influence of surfacetopography on friction, film breakdown and running-in in themixedlubrication regime,”Proceedings of the Institution of Mechanical Engi-neers, Part J: Journal of Engineering Tribology, vol. 215(6), pp. 519–533, 2001.

[26] R. Stribeck, “Characteristics of plain and roller bearings,” Zeitschrift desVereines deutscher Ingenieure, vol. 46(37), pp. 1341–1348, 1432–1438,1463–1470, 1902.

[27] S. Bair, Y. Liu, and Q. J. Wang, “The pressure-viscositycoefficientfor newtonian EHL film thickness with general piezoviscous response,”Journal of Tribology, vol. 128(3), pp. 624–631, 2006.

[28] S. Bair, “Rheology and high-pressure models for quantitative elastohy-drodynamics,”Proceedings of the Institution of Mechanical Engineers,Part J: Journal of Engineering Tribology, vol. 223, pp. 617–628, 2009.

[29] J. Millat, J. H. Dymond, and C. A. N. de Castro, eds.,Transport prop-erties of fluids: Their correlation, prediction and estimation. CambrideUniversity press, 1996.

268 References

[30] C. A. Angell, “Formation of glasses from liquids and biopolymers,”Sci-ence, vol. 267(5206), pp. 1924–1935, 1995.

[31] C. Barus, “Isothermals, isopiestics, and isometrics relative to viscosity,”Am. J. Sci., vol. 266, pp. 87–96, 1893.

[32] M. Hersey, “The theory of the torsion and the rolling ball viscometersand their use in measuring the effect of pressure on viscosity,” Journalof the Washington Academy of Sciences, vol. 6, pp. 525–531, 1916.

[33] P. Bridgman, “The effect of pressure on the viscosity offorty-three pureliquids,” Proceedings of the American Academy of Arts and Sciences,vol. 266, pp. 57–99, 1926.

[34] C. J. A. Roelands, J. C. Vlugter, and H. I. Waterman, “Theviscosity-temperature-pressure relationship of lubricating oils and its correlationwith chemical constitution,”Journal of Basic Engineering, vol. 85(4),pp. 601–607, 1963.

[35] C. J. A. Roelands,Correlational Aspects of the Viscosity-Temperature-Pressure Relationship of Lubricating Oils. PhD thesis, Delft Universityof Technology, 1966.

[36] W. R. Jones, R. L. Johnson, W. O. Winer, and D. M. Sandborn,“Pressure-viscosity measurements for several lubricantsto 5.5 x 108

newtons per square meter (8x 104 psi) and 149 degrees◦C (300 ◦F),”ASLE Trans, vol. 18 (4), pp. 249–262, 1975.

[37] H. Blok, “Inverse problems in hydrodynamic lubrication and design di-rectives for lubricated flexible surfaces,” inProceedings of the Interna-tional Symposium on Lubrication and wear(D. Muster and B. Stern-licht, eds.), pp. 1–151, 1965.

[38] S. Bair, “A note on the use of roelands equation to describe viscosityfor ehd hertzian zone calculations,”Journal of Tribology, vol. 115(2),pp. 333–334, 1993.

[39] C. Cioc, S. Cioc, L. Moraru, A. Kahraman, and T. G. K. Jr.,“A deter-ministic elastohydrodynamic lubrication model of high speed rotorcrafttransmission components,”Tribology transactions, vol. 45(4), pp. 556–562, 2002.

References 269

[40] C. W. Allen, D. P. Townsend, and E. V. Zaretsky, “New generalizedrheological model for lubrication of a ball spinning in a nonconforminggroove,” 1973.

[41] J. B. Irving and A. J. Barlow, “An automatic high pressure viscometer,”Journal of Physics E: Scientific Instruments, vol. 4(3), pp. 232–236,1971.

[42] S. Bair and P. Kottke, “Pressure viscosity relationships for elastohydro-dynamics,”Tribology Transactions, vol. 46(3), pp. 289–295, 2003.

[43] D. L. Hogenboom, W. Webb, and J. A. Dixon, “Viscosity of severalliquid hydrocarbons as a function of temperature, pressure, and freevolume,” The Journal of Chemical Physics, vol. 46(7), pp. 2586–2598,1967.

[44] C. A. Herbst, R. L. Cook, and H. E. K. Jr, “High-pressure viscos-ity of glycerol measured by centrifugal-force viscometry,” Nature,vol. 361(6412), pp. 518–520, 1993.

[45] R. L. Cook, C. A. Herbst, and H. E. K. Jr, “High-pressure viscosity ofglass forming liquids measured by the centrifugal force diamond anvilcell viscometer,”Journal of Physical Chemistry, vol. 97(10), pp. 2355–2361, 1993.

[46] A. K. Doolittle, “Studies in newtonian flow. ii. the dependence ofthe viscosity of liquids on free-space,”Journal of Applied Physics,vol. 22(12), pp. 1471–1475, 1951.

[47] M. L. Williams, R. F. Landel, and J. D. Ferry, “The temperature de-pendence of relaxation mechanisms in amorphous polymers and otherglass-forming liquids,”Journal of the American Chemical Society,vol. 77(14), pp. 3701–3707, 1955.

[48] M. H. Cohen and D. Turnbull, “Molecular transport in liquids andgases,”Journal of Chemical Physics, vol. 31(5), pp. 1164–1169, 1959.

[49] S. Yasutomi, S. Bair, and W. O. Winer, “An application ofa free volumemodel to lubricant rheology: 1-dependence of viscosity on temperatureand pressure,”Journal of Tribology, vol. 106(2), pp. 291–303, 1983.

270 References

[50] S. Bair, C. Mary, N. Bouscharain, and P. Vergne, “An improved yasu-tomi correlation for viscosity at high pressure,”Proceedings of the In-stitution of Mechanical Engineers, Part J: Journal of Engineering Tri-bology, vol. 227(9), pp. 1056–1060, 2013.

[51] P. Vergne and S. Bair, “Classical EHL versus quantitative EHL: A per-spective part I - real viscosity-pressure dependence and the viscosity-pressure coefficient for predicting film thickness,”Tribology Letters,vol. 54(1), pp. 1–12, 2014.

[52] W. T. Ashurst and W. G. Hoover, “Dense fluid shear viscosity and ther-mal conductivity - the excess,”AIChE Journal, vol. 21(2), pp. 410–411,1975.

[53] W. T. Ashurst and W. G. Hoover, “Dense-fluid shear viscosity vianonequilibrium molecular dynamics,”Physical Review A, vol. 11(2),pp. 658–678, 1975.

[54] C. M. Roland, S. Bair, and R. Casalini, “Thermodynamic scaling of theviscosity of van der waals, h-bonded, and ionic liquids,”The Journal ofChemical Physics, vol. 125(124508), pp. 1–8, 2006.

[55] A. S. Pensado, A. A. H. Pádua, M. J. P. Comuñas, and J. Fernández,“Relationship between viscosity coefficients and volumetric propertiesusing a scaling concept for molecular and ionic liquids,”Journal ofPhysical Chemistry B, vol. 112(18), pp. 5563–5574, 2008.

[56] S. Bair and R. Casalini, “A scaling parameter and function for the accu-rate correlation of viscosity with temperature and pressure across eightorders of magnitude of viscosity,”Journal of Tribology, vol. 130(4),p. no. 041802, 2008.

[57] S. Bair and A. Laesecke, “Normalized ashurst-hoover scaling and acomprehensive viscosity correlation for compressed liquids,” Journalof Tribology, vol. 134(2), pp. 021801–1 – 021801–8, 2012.

[58] A. A. Lubrecht,The numerical solution of the elastohydrodynamicallylubricated line and point contact problem using multigrid techniques.PhD thesis, University of Twente, Enschede, 1987. ISBN 90-9001583-3.

References 271

[59] B. J. Hamrock, P. Pan, and R. T. Lee, “Pressure spikes in elastohydro-dynamically lubricated conjunctions,”Journal of Tribology, vol. 110(2),pp. 279–284, 1987.

[60] C. C. Kweh, H. P. Evans, and R. W. Snidle, “Elastohydrodynamic lu-brication of heavily loaded circular contacts,”Proceedings of the Insti-tution of Mechanical Engineers, Part C: Journal of Mechanical Engi-neering Science, vol. 203(2), pp. 133–148, 1989.

[61] C. H. Venner and J. Bos, “Effects of lubricant compressibility on thefilm thickness in ehl line and circular contacts,”Wear, vol. 173(1-2),pp. 151–165, 1994.

[62] D. Dowson and G. R. Higginson,Elastohydrodynamic Lubrication, theFundamentals of Roller and Gear Lubrication. Oxford: Pergamon,1966.

[63] B. O. Jacobson and P. Vinet, “A model for the influence of pressureon the bulk modulus and the influence of temperature on the solidifica-tion pressure for liquid lubricants,”Trans. ASME. Journal of Tribology,vol. 109(4), pp. 709–714, 1987.

[64] K. T. Ramesh, “The short-time compressibility of elastohydrodynamiclubricants,”Journal of Tribology, vol. 113(2), pp. 361–370, 1991.

[65] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,Molecular Theory ofGases and Liquids. New York: Wiley, 1954.

[66] J. Millat, J. H. Dymond, and C. A. N. Castro,Transport properties ofFluids Their Correlation, Prediction and Estimation. IUPAC, Cam-bridge, 1996.

[67] R. Cook, H. K. Jr., C. Herbst, and D. Herschbach, “Pressure and tem-perature dependent viscosity of two glass forming liquids:Glyceroland dibutyl phthalate,”The Journal of Chemical Physics, vol. 100(7),pp. 5178–5189, 1994.

[68] F. D. Murnaghan,Finite Deformation of an Elastic Solid. New York:Wiley, 1951.

[69] J. R. MacDonald, “Some simple isothermal equations of state,”Reviewsof Modern Physics, vol. 38(4), pp. 669–679, 1966.

272 References

[70] W. G. Cutler, R. H. Mcmickle, W. Webb, and R. W. Schiessler, “Studyof the compressions of several high molecular weight hydrocarbons,”The Journal of Chemical Physics, vol. 29(4), pp. 727–740, 1958.

[71] S. Bair, “The pressure-viscosity coefficient of a perfluorinated polyetherover a wide temperature range,”Journal of Tribology, vol. 123(1),pp. 50–53, 2001.

[72] Y. A. Fakhreddine and P. Zoller, “The equation of state of a poly-dimethylsiloxane fluid,”Journal of Applied Polymer Science, vol. 41(5-6), pp. 1087–1093, 1990.

[73] S. Bair and M. Kotzalas, “The contribution of roller compliance toelastohydrodynamic traction,”STLE Tribology Transactions, vol. 49(2),pp. 218–224, 2006.

[74] S. Bair and M. M. Khonsari, “Generalized reynolds equations forline contact with double-newtonian shear-thinning,”Tribology Letters,vol. 18(4), pp. 513–520, 2005.

[75] P. Katyal and P. Kumar, “On the role of second newtonian viscosity inehl point contacts using double newtonian shear-thinning model,” Tri-bology International, vol. 71, pp. 140–148, 2014.

[76] W. Habchi, P. Vergne, D. Eyheramendy, and G. E. Morales-Espejel,Afull-system finite element approach to elastohydrodynamiclubricationproblems: application to ultra-low-viscosity fluids. PhD thesis, Lyon,France: LaMCos, Universitete’ de Lyon, CNRS, INSA-Lyon, 2008.

[77] J. D. Moore, S. T. Cui, H. D. Cochran, and P. T. Cummings, “Rheologyof lubricant basestocks: A molecular dynamics study of C30 isomers,”Journal of Chemical Physics, vol. 113(19), pp. 8833–8840, 2000.

[78] M. Paluch, M. Sekula, S. Pawlus, S. J. Rzoska, J. Zialo, and C. M.R, “Test of the einstein-debye relation in supercooled dibutylphtha-late at pressures up to 1.4 GPa,”Physical Review Letters, vol. 90(17),pp. 175702/1–175702/4, 2003.

[79] M. Hartl, I. Krupka, R. Poliscuk, M. Liska, J. M. M. Querry, andP. Vergne, “Thin film colorimetric interferometry,”Tribology Transac-tions, vol. 44(2), pp. 270–276, 2001.

References 273

[80] S. Bair, P. Vergne, and M. Marchetti, “The effect of shear-thinning onfilm thickness for space lubricants,”Tribology Transactions, vol. 45(3),pp. 330–333, 2002.

[81] S. Bair, P. Vergne, and M. Querry, “A unified shear-thinning treatmentof both film thickness and traction in ehd,”Tribology Letters, vol. 18(2),pp. 145–152, 2005.

[82] J. P. Chaomleffel, G. Dalmaz, and P. Vergne, “Experimental results andanalytical film thickness predictions in ehd rolling point contacts,”Tri-bology International, vol. 40(10-12), pp. 1543–1552, 2007.

[83] S. Bair and F. Qureshi, “The generalized newtonian fluidmodel andelastohydrodynamic film thickness,”Journal of Tribology, vol. 125(1),pp. 70–75, 2003.

[84] S. Bair, “Actual eyring models for thixotropy and shear-thinning: Ex-perimental validation and application to ehd,”Journal of Tribology,vol. 126(4), pp. 728–732, 2004.

[85] I. Krupka, S. Bair, P. Kumar, P. Svoboda, and M. Hartl, “Mechanicaldegradation of the liquid in an operating ehl contact,”Tribology Letters,vol. 41(1), pp. 191–197, 2011.

[86] J. Holtzinger, J. Green, G. Lamb, D. Atkinson, and H. Spikes, “Newmethod of measuring permanent viscosity loss of polymer-containinglubricants,”Tribology Transactions, vol. 55(5), pp. 631–639, 2012.

[87] S. Bair, I. Krupka, P. Sperka, and M. Hartl, “Quantitative elastohydro-dynamic film thickness of mechanically degraded oil,”Tribology Inter-national, vol. 64, pp. 33–38, 2013.

[88] W. O. Winer and S. Bair, “Influence of ambient pressure onthe apparentshear thinning of liquid lubricants - an overlooked phenomena,” I MechE Conference Publications, vol. Paper C190/87, pp. 395–398, 1987.

[89] W. Habchi, S. Bair, and P. Vergne, “On friction regimes in quantitativeelastohydrodynamics,”Tribology International, vol. 58, pp. 107–117,2013.

[90] F. W. Smith, “Lubricant behavior in concentrated contact - some rheo-logical problems,”Tribology Transactions, vol. 3(1), pp. 18–25, 1960.

274 References

[91] C. Evans and K. Johnson, “The rheological properties ofelastohy-drodynamic lubricants,”Proceedings of the Institution of Mechani-cal Engineers, Part C: Journal of Mechanical Engineering Science,vol. 200(C5), pp. 303–312, 1986.

[92] S. Bair, F. Qureshi, and W. O. Winer, “Observations of shear localizationin liquid lubricants under pressure,”Journal of Tribology, vol. 115(3),pp. 507–514, 1993.

[93] S. Bair and C. McCabe, “A study of mechanical shear bandsin liquids athigh pressure,”Tribology International, vol. 37(10), pp. 783–789, 2004.

[94] S. Bair, C. McCabe, and P. T. Cummings, “Calculation of viscous ehltraction for squalane using molecular simulation and rheometry,” Tri-bology Letters, vol. 13(4), pp. 251–254, 2002.

[95] J. Kleemola and A. Lehtovaara, “An approach for determination oflubricant properties at elliptical elastohydrodynamic contacts using atwin-disc test device and a numerical traction model,”Proceedings ofthe Institution of Mechanical Engineers, Part J: Journal ofEngineeringTribology, vol. 222(7), pp. 797–806, 2008.

[96] S. Bair, “The nature of the logarithmic traction gradient,” Tribology In-ternational, vol. 35(9), pp. 591–597, 2002.

[97] S. Bair and W. O. Winer, “Shear strength measurements oflubricants athigh pressure,”Journal of lubrication technology, vol. 101(3), pp. 251–257, 1979.

[98] E. Höglund and B. Jacobson, “Experimental investigation of the shearstrength of lubricants subjected to high pressure and temperature,”Jour-nal of Tribology, vol. 108(4), pp. 571–577, 1986.

[99] E. Höglund, “Influence of lubricant properties on elastohydrodynamiclubrication,” Wear, vol. 232(2), pp. 176–184, 1999.

[100] Y. Zhu and S. Granick, “Rate-dependent slip of newtonian liquid atsmooth surfaces,”Physical Review Letters, vol. 87(9), pp. 961051–961054, 2001.

[101] Y. Zhu and S. Granick, “Limits of hydrodynamic no-slipboundary con-dition,” Physical Review Letters, vol. 88(10), pp. 1061021–1061024,2002.

References 275

[102] V. S. J. Craig, C. Neto, and D. R. M. Williams, “Shear-dependentboundary slip in an aqueous newtonian liquid,”Physical Review Let-ters, vol. 87(5), pp. 054504–1 – 054504–4, 2001.

[103] F. Guo, P. L. Wong, M. Geng, and M. Kaneta, “Occurrence of wallslip in elastohydrodynamic lubrication contacts,”Tribology Letters,vol. 34(2), pp. 103–111, 2009.

[104] X. M. Li, P. Gou, and L. Wong, “Study of boundary slippage usingmovement of a post-impact ehl dimple under conditions of pure slidingand zero entrainment velocity.,”Tribology Letters, vol. 44(2), pp. 159–165, 2011.

[105] F. Guo, X. M. Li, and P. L. Wong, “A novel approach to measure slip-length of thin lubricant films under high pressures,”Tribology Interna-tional, vol. 46(1), pp. 22–29, 2012.

[106] P. Wong, X. Li, and F. Guo, “Evidence of lubricant slip on steel surfacein ehl contact,”Tribology International, vol. 61, pp. 116–119, 2013.

[107] J. H. Choo, R. P. Glovnea, A. K. Forrest, and H. A. Spikes, “A lowfriction bearing based on liquid slip at the wall,”Journal of Tribology,vol. 129(3), pp. 611–620, 2007.

[108] H. A. Spikes, “The half-wetted bearing. part 1: Extended reynolds equa-tion,” Proceedings of the Institution of Mechanical Engineers, Part J:Journal of Engineering Tribology, vol. 217(1), pp. 1–14, 2003.

[109] H. A. Spikes, “The half-wetted bearing. part 2: Potential application inlow load contacts,”Proceedings of the Institution of Mechanical Engi-neers, Part J: Journal of Engineering Tribology, vol. 217(1), pp. 15–26,2003.

[110] H. Eyring, “Viscosity, plasticity, and diffusion as examples of absolutereaction rates,”The Journal of Chemical Physics, vol. 4(4), pp. 283–291, 1936.

[111] F. Ree, T. Ree, and H. Eyring, “Relaxation theory of transport problemsin condensed systems,”Industrial & Engineering Chemistry, vol. 50(7),pp. 1036–1040, 1958.

[112] S. J. Hahn, H. Eyring, I. Higuchi, and T. Ree, “Flow properties of lubri-cating oils under pressure,”NLGI Spokesman, vol. 21(3), pp. 123–128,1958.

276 References

[113] R. S. Spencer, “Flow of linear amorphous polymers,”Journal of Poly-mer Science, vol. 5(5), pp. 591–608, 1950.

[114] T. Ree and H. Eyring, “Theory of non-newtonian flow. I. solid plasticsystem,”Journal of Applied Physics, vol. 26(7), pp. 793–799, 1955.

[115] T. Ree and H. Eyring, “Theory of non-newtonian flow. II.solution sys-tem of high polymers,”Journal of Applied Physics, vol. 26, pp. 800–809, 1955.

[116] P. Kumar, S. Bair, and M. Khonsari, “Full EHL simulations using theactual Ree-Eyring model for shear-thinning lubricants,”Journal of Tri-bology, vol. 131(1), pp. 011802–1, 2008.

[117] W. K. Kim, N. Hirai, T. Ree, and H. Eyring, “Theory of non-newtonianflow. iii. a method for analyzing non-newtonian flow curves,”Journalof Applied Physics, vol. 31(2), pp. 358–361, 1960.

[118] T. A. Alfrey, Mechanical Behavior of High Polymers. Interscience,1948.

[119] M. Mooney, “A theory of the viscosity of a maxwellian elastic liquid,”Trans. Soc. Rheol, vol. I, pp. 63–94, 1957.

[120] J. F. Hutton, “Theory of rheology,”Interdisciplinary Approach to LiquidLubricant Technology (P.M. Ku, Ed.), vol. NASA, Washington, pp. 187–261, 1973.

[121] S. T. Cui, S. A. Gupta, P. T. Cummings, and H. D. Cochran,“Moleculardynamics simulations of the rheology of normal decane, hexadecane,and tetracosane,”Journal of Chemical Physics, vol. 105(3), pp. 1214–1220, 1996.

[122] R. I. Tanner, “Some illustrative problems in the flow ofviscoelasticnon-newtonian lubricants,”ASLE Transactions, vol. 8(2), pp. 179–183,1965.

[123] R. I. Tanner,Engineering Rheology. Clarendon Press, Oxford, 1985.

[124] K. L. Johnson and J. L. Tevaarkwerk, “Shear behaviour of elastohydro-dynamic oil films,”Proceedings of the Royal Society of London. SeriesA, Mathematical and Physical Sciences., vol. A356, pp. 215–236, 1977.

References 277

[125] P. J. Carreau, “Rheological equations from molecularnetwork theories,”Trans. Soc. Rheol, vol. 16(1), pp. 99–127, 1972.

[126] M. M. Cross, “Rheology of non-Newtonian fluids, a new flow equa-tion for pseudoplastic systems,”Journal of Colloid Science, vol. 20(5),pp. 417–437, 1965.

[127] S. Bair, “The shear rheology of thin compressed liquidfilms,” Proceed-ings of the Institution of Mechanical Engineers, Part J: Journal of En-gineering Tribology, vol. 216(1), pp. 1–18, 2002.

[128] L. I. Kioupis and E. J. Maginn, “Impact of molecular architecture onthe high-pressure rheology of hydrocarbon fluids,”Journal of PhysicalChemistry B, vol. 104(32), pp. 7774–7783, 2000.

[129] S. Bair, “Reference liquids for quantitative elastohydrodynamics,”Tri-bology Letters, vol. 22(2), pp. 197–206, 2006.

[130] S. Bair, C. McCabe, and P. T. Cummings, “Comparison of nonequi-librium molecular dynamics with experimental measurements in thenonlinear shear-thinning regime,”Physical Review Letters, vol. 5(4),pp. 583021–583024, 2002.

[131] M. Alsaad, S. Bair, D. M. Sanborn, and W. O. Winer, “Glass transitionsin lubricants: Its relation to elastohydrodynamic lubrication (ehd),”Journal of Tribology, vol. 100, pp. 404–416, 1978.

[132] D. J. Plazek and C. A. Bero, “Precise glass temperatures,” Journal ofCondensed Matter, vol. 15, pp. S798–S802, 2003.

[133] S. Bair and W. O. Winer, “Some observations in high pressure rheologyof lubricants,”Journal of Lubrication Technology, vol. 104(3), pp. 357–364, 1982.

[134] A. Martini and S. Bair, “The role of fragility in ehl entrapment,”Tribol-ogy International, vol. 43(1-2), pp. 277–282, 2010.

[135] C. A. Angell, “Relaxation in liquids, polymers and plastic crystals- strong/fragile patterns and problems,”Journal of Non-CrystallineSolids, vol. 131-133, pp. 13–31, 1991.

[136] S. Bair and W. O. Winer, “A quantitative test of the einsteinUdebyerelation for low molecular weight liquids using the shear dependence ofviscosity,” Tribology Letters, vol. 26(3), pp. 223–228, 2007.

278 References

[137] P. Kumar and M. M. Khonsari, “Traction in ehl line contacts using free-volume pressure-viscosity relationship with thermal and shear-thinningeffects,”Journal of Tribology, vol. 131(1), pp. 1–8, 2009.

[138] P. W. Bridgman,The Physics of High Pressure, pp. 98–147, 295–356.G. Bell and sons, ltd, London, 1931.

[139] R. Larsson and O. Andersson, “Lubricant thermal conductivity and heatcapacity under high pressure,”Proceedings of the Institution of Mechan-ical Engineers, Part J: Journal of Engineering Tribology, vol. 214(4),p. 337342, 2000.

[140] K. Holmberg, P. Andersson, and A. Erdemir, “Global energy consump-tion due to friction in passenger cars,”Tribology International, vol. 47,pp. 221–234, 2012.

[141] K. Holmberg, P. Andersson, N. O. Nylund, K. Mäkelä, andA. Erdemir,“Global energy consumption due to friction in trucks and buses,” Tri-bology International, vol. 78, pp. 94–114, 2014.

[142] D. Y. Hua and M. M. Khonsari, “Application of transientelastohydro-dynamic lubrication analysis for gear transmissions,”Tribology Trans-actions, vol. 38(4), pp. 905–913, 1995.

[143] R. Larsson, “Transient non-newtonian elastohydrodynamic lubricationanalysis of an involute spur gear,”Wear, vol. 207, pp. 67–73, 1997.

[144] P. Kumar, P. K. Saini, and P. Tandon, “Transient elastohydrodynamiclubrication analysis of an involute spur gear using couple-stress fluid,”Proceedings of the Institution of Mechanical Engineers, Part J: Journalof Engineering Tribology, vol. 221(6), pp. 743–754, 2007.

[145] S. Akbarzadeh and M. M. Khonsari, “Performance of spurgears con-sidering surface roughness and shear thinning lubricant,”Journal of Tri-bology, vol. 130, pp. 021503.1–021503.8, 2008.

[146] S. Akbarzadeh and M. M. Khonsari, “Thermoelastohydrodynamic anal-ysis of spur gears with consideration of surface roughness,” Tribol lett,vol. 32, pp. 129–141, 2008.

[147] H. Xu, A. Kahraman, N. E. Anderson, and D. G. Maddock, “Predictionof mechanical efficiency of parallel-axis gear pairs,”Journal of Mechan-ical Design, vol. 129, pp. 58–68, 2007.

References 279

[148] L. Bobach, R. Beilicke, D. Bartel, and L. Deters, “Thermal elastohydro-dynamic simulation of involute spur gears incorporating mixed friction,”Tribology International, vol. 48, pp. 191–206, 2012.

[149] X. Tan, C. E. Goodyer, P. K. Jimack, R. I. Taylor, and M. A. Walk-ley, “Computational approaches for modelling elastohydrodynamic lu-brication using multiphysics software,”Proceedings of the Institutionof Mechanical Engineers, Part J: Journal of Engineering Tribology,vol. 226(6), pp. 463–480, 2012.

[150] M. Yoshizaki, G. Naruse, R. Nemoto, and S. Haizuka, “Study on fric-tional loss of spur gears (concerning the influence of tooth form, load,tooth surface roughness, and lubricating oil),”Tribology Transactions,vol. 34(1), pp. 138–146, 1991.

[151] K. Michaelis and B. R. Höhn, “Influence of lubricants onpower lossof cylindrical gears,”Tribology Transactions, vol. 37(1), pp. 161–167,1994.

[152] R. Nemoto, C. Naruse, M. Yoshizaki, and N. Suyama, “Influence ofoil viscosity, chemical oil structure, and chemical additives on frictionloss of spur gears (concerning the influence of synthetic oiland mineraloil),” Tribology Transactions, vol. 37(2), pp. 358–368, 1994.

[153] B.-R. Höhn, K. Michaelis, and A. Doleschel, “Frictional behaviour ofsynthetic gear lubricants,”Tribology Series, vol. 39, pp. 759–768, 2001.

[154] R. Martins, J. Seabra, A. Brito, C. Seyfert, R. Luther,and A. Igartua,“Friction coefficient in FZG gears lubricated with industrial gear oils:Biodegradable ester vs. mineral oil,”Tribology International, vol. 39(6),pp. 512–521, 2006.

[155] K. Kubo, Y. Shimakawa, and M. Kibukawa, “The effect of gear oil vis-cosity and friction reducer type on transmission efficiency,” TribologyInternational, vol. 19(6), pp. 312–317, 1986.

[156] R. Sarin, D. K. Tuli, A. S. Verma, M. M. Rai, and A. K. Bhatna-gar, “Additive-additive interactions: Search for synergistic FM-EP-AWcomposition,”Wear, vol. 174(1-2), pp. 93–102, 1994.

[157] O. Järviö and A. Lehtovaara, “Experimental study of influence of lubri-cants on friction in spur gear contacts,”Tribologia, vol. 21(1), pp. 20–27, 2002.

280 References

[158] B. R. Höhn, K. Michaelis, and A. Wimmer, “Low loss gears,” GearTechnology, vol. 24(4), pp. 28–38, 2007.

[159] L. Magalhães, R. Martins, C. Locateli, and J. Seabra, “Influence of toothprofile and oil formulation on gear power loss,”Tribology international,vol. 43(10), pp. 1861–1871, 2009.

[160] K. Michaelis, B.-R. Höhn, and H. Michael, “Influence factors on gear-box power loss,” Industrial Lubrication and Tribology, vol. 63(1),pp. 46–55, 2011.

[161] L. Magalhães, R. Martins, and J. Seabra, “Low-loss austempered duc-tile iron gears: Experimental evaluation comparing materials and lubri-cants,”Tribology International, vol. 46(1), pp. 97–105, 2012.

[162] R. C. Martins, N. F. R. Cardoso, H. Bock, A. Igartua, andJ. H. O.Seabra, “Power loss performance of high pressure nitrided steel gears,”Tribology International, vol. 42(11-12), pp. 1807–1815, 2009.

[163] R. Martins, J. Seabra, and L. Magalhães, “Austemperedductile iron(adi) gears: Power loss, pitting and micropitting,”Wear, vol. 264(9-10),pp. 838–849, 2008.

[164] R. D. Britton, C. D. Elcoate, M. P. Alanou, H. P. Evans, and R. W.Snidle, “Effect of surface finish on gear tooth friction,”Journal of tri-bology, vol. 122(1), pp. 354–360, 2000.

[165] T. Petry-Johnson, A. Kahraman, N. E. Anderson, and D. R. Chase, “Anexperimental investigation of spur gear efficiency,”Journal of Mechan-ical Design, vol. 130, pp. 62601–1 – 62601–10, 2008.

[166] R. I. Amaro, R. C. Martins, J. O. Seabra, N. M. Renevier,and D. G.Teer, “Molybdenum disulphide/titanium low friction coating for gearsapplication,”Tribology International, vol. 38(4), pp. 423–434, 2005.

[167] R. Martins, R. Amaro, and J. Seabra, “Influence of low friction coat-ings on the scuffing load capacity and efficiency of gears,”TribologyInternational, vol. 41(4), pp. 234–243, 2008.

[168] H. B. He, H. Y. Li, Z. Z. Xu, D. Kim, and S. K. Lyu, “Effect of MoS2-based composite coatings on tribological behavior and efficiency ofgear,”International Journal of Precision Engineering and Manufactur-ing, vol. 11(6), pp. 937–943, 2010.

References 281

[169] R. I. Amaro, R. C. Martins, J. O. Seabra, S. Yang, D. G. Teer, and N. M.Renevier, “Carbon/chromium low friction surface coating for gears ap-plication,” Industrial Lubrication and Tribology, vol. 57(6), pp. 233–242, 2005.

[170] F. Joachim, N. Kurz, and B. Glatthaar, “Influence of coatings andsurface improvements on the lifetime of gears,”Gear Technology,vol. 21(4), pp. 50–56, 2004.

[171] R. Michalczewski, M. Kalbarczyk, W. Piekoszewski, M.Szczerek, andW. Tuszynski, “The rolling contact fatigue of WC/C-coated spur gears,”Proceedings of the Institution of Mechanical Engineers, Part J: Journalof Engineering Tribology, vol. 227(8), pp. 850–860, 2013.

[172] R. Anderson, J. Helmer, and K. Lai, “Physical vapor deposition employ-ing ion extraction from a plasma,” Jan. 9 1996. US Patent 5,482,611.

[173] H. Ronkainen, D. Varjus, and K. Holmberg, “Friction and wear prop-erties in dry, water- and oil-lubricated DLC against alumina and DLCagainst steel contacts,”Wear, vol. 222(2), pp. 120–128, 1998.

[174] Z. F. Zhou, K. Y. Li, I. Bello, C. S. Lee, and S. T. Lee, “Study oftribological performance of ECR-CVD diamond-like carbon coatingson steel substrates part 2. the analysis of wear mechanism,”Wear,vol. 258(10), pp. 1589–1599, 2005.

[175] B. Podgornik, S. Jacobson, and S. Hogmark, “DLC coating of bound-ary lubricated components - advantages of coating one of thecontactsurfaces rather than both or none,”Tribology International, vol. 36(11),pp. 843–849, 2003.

[176] H. Renodeau, B. Papke, M. Pozebanchukz, and P. Parthasarathy, “Tri-bological properties of diamond-like carbon coatings in lubricated auto-motive applications,”Proceedings of the Institution of Mechanical En-gineers, Part J: Journal of Engineering Tribology, vol. 223(3), pp. 405–412, 2009.

[177] K. Topolovec-Miklozic, F. Lockwood, and H. Spikes, “Behaviour ofboundary lubricating additives on DLC coatings,”Wear, vol. 265(11-12), pp. 1893–1901, 2008.

[178] K. Bobzin, N. Bagcivan, N. Goebbels, K. Yilmaz, B.-R. Hoehn,K. Michaelis, and M. Hochmann, “Lubricated PVD CrAlN and WC/C

282 References

coatings for automotive applications,”Surface and Coatings Technol-ogy, vol. 204(6-7), pp. 1097–1101, 2009.

[179] H. Xu, Development of a generalized mechanical efficiency predictionmethodology for gear pairs. PhD thesis, Graduate School of The OhioState University, 2005.

[180] R. D. Evans, J. D. Cogdell, and G. A. Richter, “Tractionof lubricatedrolling contacts between thin-film coatings and steel,”Tribology Trans-actions, vol. 52(1), pp. 106–113, 2009.

[181] M. Kalin and M. Polajnar, “The effect of wetting and surface energyon the friction and slip in oil-lubricated contacts,”Tribology Letters,vol. 52(2), pp. 185–194, 2013.

[182] C. Huh and L. E. Scriven, “Hydrodynamic model of steadymovementof a solid/liquid/fluid contact line,”Journal of Colloid and InterfaceScience, vol. 35(1), pp. 85–101, 1971.

[183] L. M. Hocking, “A moving fluid interface on a rough surface,” Journalof Fluid Mechanics, vol. 76, pp. 801–817, 1976.

[184] J. Koplik, J. R. Banavar, and J. F. Willemsen, “Molecular dynamicsof Poiseuille flow and moving contact lines,”Physical Review Letters,vol. 60(13), pp. 1282–1285, 1988.

[185] P. A. Thompson and M. O. Robbins, “Simulations of contact line mo-tion: slip and the dynamic contact angle.,”Physical Review Letters,vol. 63(7), pp. 766–769, 1989.

[186] J. Koplik and J. R. Banavar, “Corner flow in the sliding plate problem,”Physics of Fluids, vol. 7(12), pp. 3118–3125, 1995.

[187] S. Richardson, “On the no-slip boundary condition,”Journal of FluidMechanics, vol. 59(4), pp. 707–719, 1973.

[188] P. A. Durbin, “Considerations on the moving contact-line singularity,with application to frictional drag on a slender drop,”Journal of FluidMechanics, vol. 197, pp. 157–169, 1988.

[189] P. Thompson and S. Troian, “A general boundary condition for liquidflow at solid surfaces,”Nature, vol. 389(6649), pp. 360–362, 1997.

References 283

[190] L. Léger, H. Hervet, G. Massey, and E. Durliat, “Wall slip in polymermelts,” Journal of Physics Condensed Matter, vol. 9(37), pp. 7719–7740, 1997.

[191] O. Vinogradova, “Slippage of water over hydrophobic surfaces,”Inter-national Journal of Mineral Processing, vol. 56(1-4), pp. 31–60, 1999.

[192] R. Pit, H. Hervet, and L. Léger, “Friction and slip of a simple liquid ata solid surface,”Tribology letters, vol. 7(2-3), pp. 147–152, 1999.

[193] R. Pit, H. Hervet, and L. Léger, “Direct experimental evidence of slipin hexadecane: solid interfaces,”Physical Review Letters, vol. 85(5),pp. 980–983, 2000.

[194] E. Bonaccurso, M. Kappl, and H. J. Butt, “Hydrodynamicforce mea-surements: Boundary slip of water on hydrophilic surfaces and elec-trokinetic effects,”Physical Review Letters, vol. 88(7), pp. 761031–761034, 2002.

[195] E. Bonaccurso, H. J. Butt, and V. S. J. Craig, “Surface roughness andhydrodynamic newtonian fluid in a completely wetting system,” Physi-cal Review Letters, vol. 90(14), pp. 144501/1–144501/4, 2003.

[196] W. Hild, A. Opitz, J. A. Schaefer, and M. Scherge, “The effect of wet-ting on the microhydrodynamics of surfaces lubricated withwater andoil,” Wear, vol. 254(9), pp. 871–875, 2003.

[197] A. E. Fortier and R. F. Salant, “Numerical analysis of ajournal bear-ing with a heterogeneous slip/no-slip surface,”Journal of Tribology,vol. 127(4), pp. 820–825, 2005.

[198] C. W. Wu and G. J. Ma, “Abnormal behavior of a hydrodynamic lu-brication journal bearing caused by wall slip,”Tribology International,vol. 38(5), pp. 492–499, 2005.

[199] B. Ono and Y. Yamamoto, “Possibility of slip in hydrodynamic oilfilms under sliding contact conditions,”Lubrication Science, vol. 14(3),pp. 303–320, 2002.

[200] J. H. Choo, H. A. Spikes, M. Ratoi, R. Glovnea, and A. Forrest, “Fric-tion reduction in low load hydrodynamic lubrication lubrication with ahydrophobic surface,”Tribology International, vol. 40(2), pp. 154–159,2007.

284 References

[201] S. Jahanmir, A. Hunsberger, and H. Heshmat, “Load capacity and dura-bility of H-DLC coated hydrodynamic thrust bearings,”Journal of Tri-bology, vol. 133(3), p. 031301, 2011.

[202] M. Kalin, I. Velkavrh, and J. Vižintin, “The stribeck curve and lu-brication design for non-fully wetted surfaces,”Wear, vol. 267(5-8),pp. 1232–1240, 2009.

[203] M. Kalin and M. Polajnar, “The correlation between thesurface energy,the contact angle and the spreading parameter, and their relevance forthe wetting behaviour of DLC with lubricating oils,”Tribology Interna-tional, vol. 66, pp. 225–233, 2013.

[204] M. Kalin and M. Polajnar, “The wetting of steel, dlc coatings, ceramicsand polymers with oils and water: The importance and correlations ofsurface energy, surface tension, contact angle and spreading,” AppliedSurface Science, vol. 293, pp. 97–108, 2014.

[205] K. Sarkar and A. Prosperetti, “Effective boundary conditions for stokesflow over a rough surface,”Journal of Fluid Mechanics, vol. 316,pp. 223–240, 1996.

[206] A. Jabbarzadeh, J. D. Atkinson, and R. I. Tanner, “Effect of the wallroughness on slip and rheological properties of hexadecanein molecu-lar dynamics simulation of couette shear flow between two sinusoidalwalls,” Physical Review E, vol. 61(1), pp. 690–699, 2000.

[207] J. R. A. Pearson and C. J. S. Petrie,Polymer systems: deformation andflow. Macmillan, 1968.

[208] B. Bhushan, “Biomimetics: lessons from nature - an overview,” Philo-sophical Transactions of the Royal Society A: Mathematical, Physicaland Engineering Sciences, vol. 367(1893), pp. 1445–1486, 2009.

[209] B. Bhushan and Y. C. Jung, “Natural and biomimetic artificial surfacesfor superhydrophobicity, self-cleaning. low adhesion anddrag reduc-tion,” Progress in Materials Science, vol. 56(1), pp. 1–108, 2011.

[210] H. S. Tang and D. M. Kalyon, “Unsteady circular tube flowof compress-ible polymeric liquids subject to pressure-dependent wallslip,” Journalof Rheology, vol. 52(2), pp. 507–525, 2008.

References 285

[211] X. M. Li, F. Guo, and P. L. Wong, “Shear rate and pressureeffects onboundary slippage in highly stressed contacts,”Tribology International,vol. 59, pp. 147–153, 2013.

[212] A. Ponjavic and J. S. S. Wong, “The effect of boundary slip on elastohy-drodynamic lubrication,”RSC Advances, vol. 4(40), pp. 20821–20829,2014.

[213] P. Kumar and P. Anuradha, “Elastohydrodynamic lubrication modelwith limiting shear stress behavior considering realisticshear-thinningand piezo-viscous response,”Journal of Tribology, vol. 136(2),pp. 021503–1 – 021503–8, 2014.

[214] I. Suarez-Martinez and N. A. Marks, “Effect of microstructure on thethermal conductivity of disordered carbon,”Applied Physics Letters,vol. 99, p. 033101, 2011.

[215] S. Falabella, D. B. Boercker, and D. M. Sanders, “Fabrication of amor-phous diamond films,”Thin solid films, vol. 236(1-2), pp. 82–86, 1993.

[216] C. J. Morath, H. J. Maris, J. J. Cuomo, D. L. Pappas, A. Grill, V. V.Patel, J. P. Doyle, and K. L. Saenger, “Picosecond optical studies ofamorphous diamond and diamondlike carbon: thermal conductivity andlongitudinal sound velocity,”Journal of Applied Physics, vol. 76(5),pp. 2636–2640, 1994.

[217] W. Hurler, M. Pietralla, and A. Hammerschmidt, “Determination ofthermal properties of hydrogenated amorphous carbon films via mi-rage effect measurements,”Diamond and Related Materials, vol. 4(7),pp. 954–957, 1995.

[218] A. Bullen, K. O’Hara, D. Cahill, O. Monteiro, and A. V. Keudell, “Ther-mal conductivity of amorphous carbon thin films,”Journal of AppliedPhysics, vol. 88(11), pp. 6317–6320, 2000.

[219] G. Chen, P. Hui, and S. Xu, “Thermal conduction in metalized tetra-hedral amorphous carbon (ta-C) films on silicon,”Thin solid films,vol. 366(1-2), pp. 95–99, 2000.

[220] M. Shamsa, W. Liu, A. Balandin, C. Casiraghi, W. Milne,and A. Fer-rari, “Thermal conductivity of diamond-like carbon films,”AppliedPhysics Letters, vol. 89(16), p. 161921, 2006.

286 References

[221] A. Balandin, M. Shamsa, W. Liu, C. Casiraghi, and A. F, “Thermalconductivity of ultrathin tetrahedral amorphous carbon films,” AppliedPhysics Letters, vol. 93(4), p. 043115, 2008.

[222] J. L. Arlein, S. E. M. Palaich, B. C. Daly, P. Subramonium, and G. A.Antonelli, “Optical pump-probe measurements of sound velocity andthermal conductivity of hydrogenated amorphous carbon films,” Jour-nal of Applied Physics, vol. 104, p. 033508, 2008.

[223] J. W. Kim, H.-S. Yang, Y. H. Jun, and K. C. Kim, “Interfacial effecton thermal conductivity of diamond-like carbon films,”Journal of Me-chanical Science and Technology, vol. 24(7), pp. 1511–1514, 2010.

[224] E. T. Swartz and R. O. Pohl, “Thermal boundary resistance,” Reviews ofModern Physics, vol. 61 (3), pp. 605–668, 1989.

[225] S. M. Lee and D. G. Cahill, “Heat transport in thin dielectric films,”Journal of Applied Physics, vol. 81(6), pp. 2590–2595, 1997.

[226] B. Kleinsorge, A. C. Ferrari, J. Robertson, W. I. Milne, S. Waidmann,and S. Hearne, “Bonding regimes of nitrogen in amorphous carbon,”Diamond and Related Materials, vol. 9(3), pp. 643–648, 2000.

[227] N. W. Khun, E. Liu, and M. D. Krishna, “Structure, adhesive strengthand electrochemical performance of nitrogen doped diamond-like car-bon thin films deposited via dc magnetron sputtering,”Journal ofNanoscience and Nanotechnology, vol. 10(7), pp. 4752–4757, 2010.

[228] A. Bendavid, P. J. Martin, L. Randeniya, M. S. Amin, andR. Roh-anizadeh, “The properties of fluorine-containing diamond-like carbonfilms prepared by pulsed DC plasma-activated chemical vapour depo-sition,” Diamond and Related Materials, vol. 19(12), pp. 1466–1471,2010.

[229] W. Habchi, D. Eyheramendy, S. Bair, P. Vergne, and G. Morales-Espejel, “Thermal elastohydrodynamic lubrication of point contactsusing a newtonian/generalized newtonian lubricant,”Tribology letters,vol. 30(1), pp. 41–52, 2008.

[230] W. Habchi, D. Eyheramendy, P. Vergne, and G. Morales-Espejel,“A full-system approach of the elastohydrodynamic line/point contactproblem,” ASME Journal of Tribology, vol. 130,021501, pp. 1–10,2008.

References 287

[231] W. Habchi, P. Vergne, S. Bair, O. Andersson, D. Eyheramendy, andG. E. Morales-Espejel, “Influence of pressure and temperature depen-dence of thermal properties of a lubricant on the behaviour of circu-lar tehd contacts,”Tribology International, vol. 43(10), pp. 1842–1850,2010.

[232] P. Yang and S. Wen, “A generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic lubrication,”Journal of Tribol-ogy, vol. 112(4), pp. 631–636, 1990.

[233] A. A. Elsharkawy, M. J. A. Holmes, H. P. Evans, and R. W. Snidle,“Micro-elastohydrodynamic lubrication of coated cylinders using cou-pled differential deflection method,”Proceedings of the Institutionof Mechanical Engineers, Part J: Journal of Engineering Tribology,vol. 220(1), pp. 29–41, 2006.

[234] Y. Liu, W. W. Chen, D. Zhu, S. Liu, and Q. J. Wang, “An elastohydrody-namic lubrication model for coated surfaces in point contacts,” Journalof Tribology, vol. 129(3), pp. 509–516, 2007.

[235] Y. Liu, Q. J. Wang, and D. Zhu, “Effect of stiff coatingson EHLfilm thickness in point contacts,”Journal of Tribology, vol. 130(3),p. 031501, 2008.

[236] W. Habchi, “A numerical model for the solution of thermal elastohy-drodynamic lubrication in coated circular contacts,”Tribology Interna-tional, vol. 73, pp. 57–68, 2014.

[237] K. T. Wojciechowski, R. Zybala, and R. Mania, “Application ofDLC layers in 3-omega thermal conductivity method,”Journal ofAchievements in Materials and Manufacturing Engineering, vol. 37(2),pp. 512–517, 2009.

[238] M. Hakovirta, J. E. Vuorinen, X. M. He, M. Nastasi, and R. B. Schwarz,“Heat capacity of hydrogenated diamond-like carbon films,”AppliedPhysics Letters, vol. 77/15), pp. 2340–2342, 2000.

[239] Y. Oka, M. Kirinuki, T. Suzuki, M. Yatsuzuka, and K. Yatsui, “Effect ofion beam implantation on density of DLC prepared by plasma-basedion implantation and deposition,”Nuclear Instruments and Methodsin Physics Research, Section B: Beam Interactions with Materials andAtoms, vol. 242(1-2), pp. 335–357, 2006.

288 References

[240] B.-R. Höhn, K. Michaelis, and A. Doleschel, “Limitations of bench test-ing for gear lubricants,”ASTM Special Technical Publication, vol. 1404,pp. 15–32, 2001.

[241] J. Kleemola and A. Lehtovaara, “Experimental simulation of gear con-tact along the line of action,”Tribology International, vol. 42(10),pp. 1453–1459, 2009.

[242] P. Heingartner and D. Mba, “Determining power losses in the helicalgear mesh,”Gear Technology, vol. 22(5), pp. 32–37, 2005.

[243] C. Changenet, “Power loss predictions in geared transmissions usingthermal networks-applications to a six-speed manual gearbox,” Journalof Mechanical Design, Transactions of the ASME, vol. 128, pp. 618–625, 2006.

[244] A. Gu, “Elastohydrodynamic lubrication of involute gears,”Journal ofengineering for industry, vol. 95(4), pp. 1164–1170, 1973.

[245] K. L. Johnson, J. A. Greenwood, and S. Y. Poon, “A simpletheory ofasperity contact in elastohydro-dynamic lubrication,”Wear, vol. 19(1),pp. 91–108, 1972.

[246] J. A. Greenwood and J. H. Tripp, “The contact of two nominally flatrough surfaces,”Proc Inst Mech Eng, vol. 185, pp. 625–633, 1971.

[247] J. Castro and J. Seabra, “Coefficient of friction in mixed film lubrica-tion: gears versus twin-discs,”Proceedings of the Institution of Mechan-ical Engineers, Part J: Journal of Engineering Tribology, vol. 221(3),pp. 399–411, 2007.

[248] J. Kleemola and A. Lehtovaara, “Experimental evaluation of frictionbetween contacting discs for the simulation of gear contact,” Tribo test,vol. 13(1), pp. 13–20, 2007.

[249] V. Wikstrom and E. Hoglund, “Investigation of parameters affecting thelimiting shear stress-pressure coefficient: a new model incorporatingtemperature,”Journal of Tribology, vol. 116(3), pp. 612–620, 1994.

[250] R. Larsson, P. O. Larsson, E. Eriksson, M. Sjoberg, andE. Hoglund,“Lubricant properties for input to hydrodynamic and elastohydrody-namic lubrication analyses,”Proceedings of the Institution of Mechan-ical Engineers, Part J: Journal of Engineering Tribology, vol. 214(1),pp. 17–27, 2000.

References 289

[251] C. C. Chou and J. F. Lin, “A new approach to the effect of EP additiveand surface roughness on the pitting fatigue of a line contact system,”Journal of Tribology, vol. 124(2), pp. 245–258, 2002.

[252] M. Björling, R. Larsson, P. Marklund, and E. Kassfeldt, “Elastohy-drodynamic lubrication friction mapping - the influence of lubricant,roughness, speed, and slide-to-roll ratio,”Proceedings of the Institu-tion of Mechanical Engineers, Part J: Journal of Engineering Tribology,vol. 225(7), pp. 671–681, 2011.

[253] B. Podgornik, S. Jacobson, and S. Hogmark, “Influence of EP additiveconcentration on the tribological behaviour of DLC-coatedsteel sur-faces,”Surface and Coatings Technology, vol. 191(2-3), pp. 357–366,2005.

[254] B. Podgornik, D. Hren, and J. Vidintin, “Combination of DLC coat-ings and EP additives for improved tribological behaviour of boundarylubricated surfaces,”Wear, vol. 261(1), pp. 32–40, 2006.

[255] K. Mistry, A. Morina, and A. Neville, “Tribo-chemistry of lubricatedDLC coatings for engineering components under extreme-pressure con-tact conditions,” Tribology and Lubrication Technology, vol. 66(5),pp. 16–17, 2010.

[256] M. I. de Barros’Bochet, J. M. Martin, T. Le-Mogne, and B. Vacher, “Lu-brication of carbon coatings with MoS2 single sheet formed by MoDTCand ZDDP lubricants,”Lubrication Science, vol. 18(3), pp. 141–149,2006.

[257] M. I. de Barros’Bochet, J. M. Martin, T. Le-Mogne, and B. Vacher,“Boundary lubrication mechanisms of carbon coatings by MoDTC andZDDP additives,” Tribology International, vol. 38(3), pp. 257–264,2005.

[258] S. Miyake, T. Saito, Y. Yasuda, Y. Okamato, and M. Kano,“Im-provement of boundary lubrication properties of diamond-like carbon(DLC) films due to metal addition,”Tribology International, vol. 37(9), pp. 751–761, 2004.

[259] T. Haque, A. Morina, A. Neville, R. Kapadia, and S. Arrowsmith, “Ef-fect of oil additives on the durability of hydrogenated DLC coating un-der boundary lubrication conditions,”Wear, vol. 266(1-2), pp. 147–157,2009.

290 References

[260] M. Kalin, E. Roman, L. Odbolt, and J. Vidintin, “Metal-doped (Ti,WC) diamond-like-carbon coatings: Reactions with extreme-pressureoil additives under tribological and static conditions,”Thin solid films,vol. 518(15), pp. 4336–4344, 2010.

[261] K. Yoshida, T. Horiuchi, M. Kano, and M. Kumagai, “Effect of a tribo-chemical reacted film on friction and wear properties of DLC coatings,”Plasma Processes and Polymers, vol. 6(1), pp. S96–S101, 2009.

[262] M. Kalin and J. Vidintin, “Differences in the tribological mechanismswhen using non-doped, metal-doped (Ti, WC), and non-metal-doped(Si) diamond-like carbon against steel under boundary lubrication, withand without oil additives,”Thin solid films, vol. 515(4), pp. 2734–2747,2006.

[263] R. V. Kleinschmidt, D. Bradbury, and M. Mark, “Viscosity and densityof over forty lubricating fluids of known composition at pressures to150,000 psi and temperatures to 425 F,” tech. rep., ASME, NewYork,1953.

[264] Y. Liu, Q. J. Wang, S. Bair, and P. Vergne, “A quantitative solution forthe full shear-thinning EHL point contact problem including traction,”Tribology Letters, vol. 28(2), pp. 171–181, 2007.

[265] W. Habchi, I. Demici, D. Eyheramendy, G. Morales-Espejel, andP. Vergne, “A finite element approach of thin film lubricationin circularEHD contacts,”Tribology International, vol. 40(10-12), pp. 1466–1473,2007.

[266] S. Bair and W. O. Winer, “A new high-pressure, high-shear stress vis-cometer and results for lubricants,”STLE, vol. 36(4), pp. 721–725,1993.

[267] C. Venner and A. Lubrecht, “An engineering tool for thequanti-tative prediction of general roughness deformation in EHL contactsbased on harmonic waviness attenuation,”Proceedings of the Institu-tion of Mechanical Engineers, Part J: Journal of Engineering Tribology,vol. 219(5), pp. 303–312, 2005.

[268] G. J. Johnston, R. Wayte, and H. A. Spikes, “The measurement andstudy of very thin lubricant films in concentrated contacts,” TribologyTransactions, vol. 34(2), pp. 187–194, 1991.

References 291

[269] M. Björling, P. Isaksson, P. Marklund, and R. Larsson,“The influence ofDLC coating on EHL friction coefficient,”Tribology Letters, vol. 47(2),pp. 285–294, 2012.

[270] W. Habchi, D. Eyheramendy, P. Vergne, and G. Morales-Espejel, “Sta-bilized fully-coupled finite elements for elastohydrodynamic lubricationproblems,” Advances in Engineering Software, vol. 46(1), pp. 4–18,2012.

[271] M. Björling, W. Habchi, S. Bair, R. Larsson, and P. Marklund, “Towardsthe true prediction of EHL friction,”Tribology International, vol. 66,pp. 19–26, 2013.

[272] M. Björling, W. Habchi, S. Bair, R. Larsson, and P. Marklund, “Frictionreduction in elastohydrodynamic contacts by thin-layer thermal insula-tion,” Tribology Letters, vol. 54(2), pp. 477–486, 2014.

[273] D. K. Owens and R. Wendt, “Estimation of the surface free energy ofpolymers,”Journal of Applied Polymer Science, vol. 13(8), pp. 1741–1747, 1969.

[274] C. W. Fan and S. C. Lee, “Surface free energy effects in sputter-deposited WNx films,”Materials Transactions, vol. 48(9), pp. 2449–2453, 2007.

[275] E. Lugscheider and K. Bobzin, “Wettability of pvd compound materialsby lubricants,”Surface and Coatings Technology, vol. 165(1), pp. 51–57, 2003.

[276] F. M. Fowkes, “Attractive forces at interfaces,”Journal of Industrial andEngineering Chemistry, vol. 56(12), pp. 40–52, 1964.

[277] A. Cameron, “Hydrodynamic lubrication of rotating disks in pure slid-ing, a new type of oil film formation,”Inst of Petroleum, vol. 37,pp. 471–786, 1951.

[278] A. Cameron, “The viscosity wedge,”ASLE Transactions, vol. 1(2),pp. 248–243, 1958.

[279] M. Kaneta and P. Yang, “Effects of the thermal conductivity of con-tact materials on elastohydrodynamic lubrication characteristics,” Pro-ceedings of the Institution of Mechanical Engineers, Part C: Journal ofMechanical Engineering Science, vol. 224(12), pp. 2577–2587, 2010.

292 References

[280] K. F. Martin, “The efficiency of involute spur gears,”Journal of Me-chanical Design, vol. 103(1), pp. 160–169, 1981.

[281] S. Wu and H. S. Cheng, “A friction model of partial-EHL contacts andits application to power loss in spur gears,”Tribology Transactions,vol. 34(3), pp. 398–407, 1991.

[282] SKF - Rolling bearings catalogue. SKF Group, 2012.

friction in elastohydrodynamic lubrication990483/fulltext01.pdf · friction in elastohydrodynamic...

Documents