Evaluation of sliding friction and contact mechanics of ... · friction behavior of elastomers regarding the two filler systems. Theoretical predictions of the stationary frictional

Evaluation of sliding friction and contact mechanics of elastomers based ondynamic-mechanical analysisAndré Le Gal, Xin Yang, and Manfred Klüppel Citation: J. Chem. Phys. 123, 014704 (2005); doi: 10.1063/1.1943410 View online: http://dx.doi.org/10.1063/1.1943410 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v123/i1 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Evaluation of sliding friction and contact mechanics of elastomers basedon dynamic-mechanical analysis

André Le Gal,a� Xin Yang, and Manfred Klüppelb�

Deutsches Institut für Kautschuktechnologie e.V., Eupener Strasse 33, D-30519 Hanover, Germany

�Received 2 March 2005; accepted 4 May 2005; published online 13 July 2005�

The paper presents a combined experimental and theoretical approach to the understanding ofhysteresis and adhesion contributions to rubber friction on dry and lubricated rough surfaces. Basedon a proper analysis of the temperature- and frequency-dependent behaviors of nonlinearviscoelastic materials such as filler reinforced elastomer materials, master curves for the viscoelasticmoduli are constructed. It is shown that the classical williams–Landel–Ferry equation cannot beapplied in its simple form, but needs the introduction of an energy term describing the temperaturedependency of glassy polymer bridges, which transmit the forces within flocculated filler clusters.The activation energy for carbon black and silica-filled elastomers is compared based on twodifferent evaluation methods. The obtained dynamic data are shown to be related to a differentfriction behavior of elastomers regarding the two filler systems. Theoretical predictions of thestationary frictional behavior of the systems are in fair agreement with the experimental friction dataat low sliding velocities. It is found that the formulated adhesion plays a dominant role on rough drysurfaces within this range of velocities. © 2005 American Institute of Physics.�DOI: 10.1063/1.1943410�

I. INTRODUCTION

In the last decade, much progress has been obtained inmodeling the friction behavior of elastomer materials onrough substrates, which commonly can be described by frac-tal or self-affine roughness spectra.1–11 The study of roadsurface texture by fractal geometry and its relationship tofriction of slipping tires was highlighted by Rado1 in theearly 1990s, representing a starting point for the develop-ment of a first elementary hysteresis friction model.2 Thebasic idea was the consideration of multiple frequency exci-tations governing the damping behavior of the dynamicallystrained rubber while sliding over rough interfaces. It wasrealized by referring to the spectral power density of theroughness spectra, which could be well described by threefractal descriptors. This concept was extended to apply for amore general description of rubber viscoelasticity on a mo-lecular level3 and a theory of dynamic contact on self-affineinterfaces was developed,4 allowing, e.g., for the predictionof the true contact area or the frequency interval of contact,which is of high relevance for the prediction of tire tractionon wet roads. Later on, an even more fundamental approachto hysteresis friction on rough substrates was developed,which was based on a three-dimensional analysis of dynamiccontact by referring to a two-dimensional spectral powerdensity,5 and other topics, such as adhesion, lubrication, andfriction on wet road tracks, have been addressed.6,7

A comparison of the two models developed in Refs. 4and 5 has shown that a quite similar behavior of the hyster-esis friction coefficient and the true contact area with varying

sliding velocity and temperature is predicted, which stronglydepends on the track roughness and the viscoelastic proper-ties of the rubber.8 However, there are also general differ-ences, e.g., for the predicted load dependency of the frictioncoefficient, which depends on the height distribution of thetrack profile in one case,4 but is independent in the othercase.5 First experimental validations of the two models haveshown a fair agreement between theory and measurements ofthe stationary friction coefficient of carbon-black-filled rub-bers on asphalt9 and corundum10,11 in the low sliding velocityrange.

The theoretical and experimental developments of slid-ing friction on self-affine rough interfaces provide a funda-mental physical background for the understanding of hyster-esis and adhesion contributions for rubber friction on dry andlubricated rough interfaces. Technical relevance of this ques-tion is given, e.g., for dynamic contact problems of tires withroad tracks during cornering and breaking, especially underantiblocking system �ABS� conditions.11–14 A deeper insightinto the friction mechanism of tires can serve as a valuabletool in the development of tread compounds for specific ap-plications, e.g., for dry, wet, or ice traction. In particular, aquestion of high interest is the microscopic mechanism gov-erning the well-known improvement of wet traction of tiretreads, when carbon black is replaced by silica. Recently, ithas been argued that the improvement is due to differences inthe morphology of the filler network, as quantified by thelower activation energy in the case of silica.15 This was re-lated to “dynamically softer” filler-filler bonds of the silicanetwork, which can be assumed to be responsible for adeeper penetration of the rubber into the track roughness.

The aim of this paper is to study the relations betweenviscoelastic material properties of variously filled rubbers

a�Electronic mail: [email protected]�Author to whom correspondence should be addressed. Electronic mail:

[email protected]

THE JOURNAL OF CHEMICAL PHYSICS 123, 014704 �2005�

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and friction properties on dry and wet rough surfaces byreferring to the predictions of friction theory as well as ex-perimental investigations of stationary friction at low slidingvelocities. In the first part, a generalized master procedurerelated to the time-temperature superposition principle forfiller reinforced rubbers is presented. In this context, verticalshift factors have to be introduced. They are interpreted viathe thermal activation of glassy polymer layers between floc-culated filler particles, which alters the stiffness of the fillernetwork. The associated activation energy values are shownto be related to the filler type and polymer chain architecture.In the second part, the obtained master curves of the complexmodulus serve as input functions for the prediction of fric-tion coefficients on a broad frequency and velocity scale,which are compared to experimental friction data. The roleof the dynamic-mechanical properties for the evaluation ofrubber friction on rough surfaces is addressed, providing adeeper comprehension on how tread material properties in-fluence tire traction. Finally, the impact of hysteresis andadhesion contributions to rubber friction under dry and wetconditions is demonstrated.

II. EXPERIMENT

The samples employed in this study include a solution-styrene-butadiene rubber �S-SBR� with 50 wt % vinyl and 25wt % styrene �BUNA VSL 5025-0 HM� often used for treadcompounds and an amorphous EPDM �Keltan 512�. Thepolymers were mixed with two different filler systems, car-bon black �N339� and silica �Coupsil 8113�. The Coupsil8113 consists of silica particles coated with a bifunctionalsilane, ensuring a reasonable dispersability during mixingand formation of covalent bonds with the polymer duringvulcanization. The formulation of compounds is completedwith antiaging �2-phr IPPD� as well as processing agents�1-phr stearic acid and 2.5-phr zinc oxide�. In order toachieve a sufficient level of elasticity, both polymers werecross-linked in a steam press with 1.7-phr sulfur in combi-nation with 2.5-phr accelerator �CBS�. The full formulationsof the composites are listed in Table I.

The unfilled polymers were mixed on a roller mill for 5min while the filled systems had a two-step mixing stagewith first 5 min in an internal mixer followed by 5 min on theroller mill. The curing time was measured on a Monsantorheometer at a temperature of 160 °C and corresponds to t95,

i.e., the required time to reach 95% of the maximum torquededuced from the vulcameter curves of each compound. Testsamples were finally vulcanized in a heating press under200-bar pressure at a curing temperature of 160 °C.

The dynamic-mechanical measurements were performedin the torsion mode with 2-mm-thick clamped strip specimenon an ARES rheometer �Rheometrix�. On the one side, thedynamic moduli were measured over a wide range of tem-perature at a frequency of 1 Hz and 0.5% strain amplitude.On the other side, in order for the master procedure to beapplied, isothermal frequency sweeps between 0.1 and 100Hz were applied at 0.5% and 3.5% strain amplitudes.

The stationary friction measurements were made at vari-able load with quadratic rubber sheets of 50�50-mm2 sizeand 2-mm thickness on a rough granite surface. The charac-terization of surface roughness was performed by stylus mea-surements with a Form Talysurf Intra �Taylor Hobson� with ahorizontal resolution of 0.5 �m and a vertical resolution of0.1 �m.

III. VISCOELASTICITY OF ELASTOMER MATERIALS

Relaxation spectroscopy has been a useful techniqueover the last decades to investigate the relationship betweenthe microstructure and some macroscopic properties of poly-meric materials. The dynamic-mechanical analysis and thedielectric spectroscopy are probably the two most used tech-niques to provide some information on the frequency- andtemperature-dependent relaxation processes occurring in therubber matrix. In both cases, specific fluctuations of the ma-terial response are detected using small perturbations of theexternal force field. This leads for instance to the definitionof the shear modulus G*�� ,T� when a dynamic-mechanicaldeformation is applied to a rubber sample.

The introduction of the free volume and chain mobilityconcepts put the emphasis on the ability for a polymer matrixto rearrange within a shorter time with increasing tempera-ture. The time-temperature superposition principle was thenderived from this consideration, giving a unique picture onthe viscoelastic behavior of polymers over a wide range offrequency.16 The addition of a filler system, such as carbonblack, improves the rubber mechanical properties by a rein-forcing mechanism but at the same time introduces a nonlin-ear dependency upon the dynamic strain amplitude.17,18 Theconsiderable change in the damping properties of the rubbermust anyway be controlled, as it is directly connected withthe phenomenon of heat generation resulting in an alterationof the product performance.

From a morphological point of view, the insertion of afiller structure in the polymer matrix strongly modifies thepolymer dynamics in the vicinity of the particles, where theinteraction with the filler surface hinders the chain mobility.It results in a slowed-down dynamics of the so-called “boundrubber” around filler particles, implying that the local glasstransition of the polymer nanolayers close to the surface isoccurring at significantly higher temperatures than the onefrom the bulk polymer matrix. This effect can typically beobserved in polymer thin films and in the confining areabetween flocculated filler particles, which are separated by

TABLE I. Formulation of rubber compounds with weight units in phr.

Sample 1 2 3 4 5 6

S-SBR 100 100 100 / / /EPDM / / / 100 100 100N339 / 60 / / 60 /

Coupsil 8113 / / 60 / / 60ZnO 2.5

Stearic acid 1IPPD 2CBS 2.5

Sulfur 1.7DPG / / / 1.5 1.5 1.5

014704-2 Le Gal, Yang, and Klüppel J. Chem. Phys. 123, 014704 �2005�

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nanoscopic bridges of immobilized polymer.19–22 At moder-ate temperatures the polymer layers between flocculatedfiller particles are considered to be in a glassy state. Sincethese glassy polymer bridges are well attached to the fillersurface, they can transmit large stresses between adjacentparticles, leading to a high stiffness and strength of filler-filler bonds. In turn, the stiffness and strength of the bondsand its thermal activation govern that of the filler networkand the whole rubber sample.15,22 The associated structure offiller networks in elastomers is depicted in Fig. 1, where thebound rubber and the glassy polymer bridges are indicated.The mechanism related to the thermal activation of theglassy polymer bridges is associated with an activation en-ergy directly evaluated via an Arrhenius plot of thetemperature-dependent elastic modulus. As an alternative so-lution, the activation energy of the glassy layers can also bedetermined during the master procedure through the differentisothermal curve segments and the introduction of a verticalshift factor taking into account the temperature dependenceof the filler reinforcement. Below, we will give a numericalcomparison of both methods using different polymer andfiller systems.

A. Time-temperature superposition principle

While the measurement of the modulus as a function ofthe temperature over a broad range at a given frequency canbe realized on a broad temperature scale, the same mechani-cal characterization carried out at one temperature by varyingthe frequency can easily be applied only up to about 102 Hz.The estimation of the high-frequency moduli is, however,accessible via the time-temperature superposition principle.It states that the effect of changing the temperature is the

same as applying a multiplication factor aT to the time scale.Reducing the relaxation data to the reference temperature Tref

leads to the following expression:

log aT =− C1�T − Tref�C2 + T − Tref

. �1�

This principle is often referred to as the Williams–Landel–Ferry �WLF� equation.16 The constants C1 and C2 vary withthe chosen reference temperature and in general also frompolymer to polymer.

The application of the WLF equation is demonstrated inFig. 2 for the case of the unfilled S-SBR. Since the systemsmust be in the thermodynamic equilibrium to be describewith the WLF formulation, sufficiently high chain mobility isrequired. This can be seen in the right diagram of Fig. 2,where discrepancies between Eq. �1� and the experimentalshift factors occur when the glass transition temperature ofthe polymer is approached. Details on the dynamic glasstransition temperature of the unfilled rubbers and their WLFcoefficients are listed in Table II.

B. Nonlinear viscoelasticity of filled rubbers

With the introduction of fillers as reinforcing agent, thecomplex interactions between the filler network and thepolymer matrix are at the origin of strong nonlinear effectswith respect to the amplitude of deformation.17 The influenceof the filler system during the master procedure is consideredvia the introduction of a temperature-dependent vertical shiftfactor physically interpreted as the activation energy of theglassy polymer bridges in the vicinity of nanoscopic gapsbetween adjacent filler particles. Figure 3 illustrates the ne-cessity to introduce a vertical shift factor for low-frequencydata, corresponding to high temperatures according to theWLF equation. Indeed, once the isothermal moduli segmentsare horizontally shifted with the same parameters as for the

FIG. 1. Schematic view of a filler cluster with bound rubber shell �lightgray� and glassy polymer bridges �dark gray�. In the case of carbon black orsilica, every black disk represents a structured primary filler aggregate.

FIG. 2. Isothermal dynamic storage moduli at differenttemperatures with strain amplitude of 0.5% �left� andadaptation of the experimental shift factors to Eq. �9�for the unfilled S-SBR �right�. The filled symbols arenot used for the fit.

TABLE II. Viscoelastic material parameters evaluated from temperature andfrequency sweeps of the unfilled S-SBR and EPDM samples. The WLFparameters are obtained from adaptations of the horizontal shift factors withEq. �9�.

Dynamic Tg �°C� WLF Parameters

Polymer tan �max G�max Tref �°C� C1 C2 �°C�

S-SBR −6 −14 21 7.7 82.7EPDM −42 −50 21 3.6 99.1

014704-3 Friction and contact mechanics of elastomers J. Chem. Phys. 123, 014704 �2005�

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unfilled compound, the poor overlapping in the low-frequency region indicates that the filler system is influenc-ing the dynamic behavior within this experimental range andshould be associated with a thermally activated mechanism.The physical origin of the observed shifting behavior is thesuperposition of two relaxation processes in filled elas-tomers, the one of the polymer matrix and that of the fillernetwork. The applied horizontal and vertical shifting proce-dures allow for a separation of both relaxation processes.

The evaluation procedure of the activation energies isdemonstrated in Fig. 4 for the case of the carbon-black-filledS-SBR sample by plotting the logarithm of the vertical shiftfactors against inverse temperature. As expected, an Arrhen-ius dependence is found within a range of sufficiently hightemperatures physically related to thermally activated pro-cesses with a probability of occurrence equal toexp�−Ea /RT�, where Ea is the activation energy and R is thegas constant. For a comparison of the different elastomersystems, the evaluation of activation energies was carried outat strain amplitudes of 0.5% and 3.5%, which are already inthe range of nonlinear viscoelastic response. Therefore, theKramers–Kronig relations are not fulfilled and both compo-nents of the dynamic modulus transform independently. Ac-cordingly, G� and G� are considered separately during theapplication of the vertical shifting procedure. This differentbehavior at low frequency can clearly be seen in Fig. 3,

whereby the temperature dependence appears to be strongerfor the loss modulus G�. Note that in the case of very smallstrain amplitudes, in the so-called plateau regime of thesmall strain modulus typically found for strains smaller than0.2%, also filled systems show a linear response. In that casea simultaneous vertical shifting procedure of G� and G� mustbe applied, since both components form the complex modu-lus G* which is coupled to the stress relaxation modulus viaFourier transformation.

The results of the vertical shifting procedure are summa-rized in Table III. The activation energies of the differentcomposites are found in the order of 10 kJ/mol, correspond-ing to the physical range of van der Walls interactions andalso consistent with the values presented in the literature.15,23

The data in Fig. 4 and Table III make clear that the activationenergy decreases systematically with rising strain amplitudeup to 3.5%. It also becomes obvious that for all systems theactivation energy Ea found for G� is smaller than that for G�.By referring to the dynamic flocculation model of rubberreinforcement,22 the two activation processes can be tracedback to the thermal activation of two kinds of filler-fillerbonds, i.e., the glassy polymer layers between filler particles,which become softer with rising temperature �compare Fig.1�. The activation energy associated with G� is related to anArrhenius behavior of the force constant of filler-filler bondsin a virgin state, while the activation energy of G� corre-sponds to the force constant of softer filler-filler bonds in adamaged state, resulting from stress-induced breakdown andreaggregation of the bonds during cyclic deformations. Withthis concept the temperature dependency of the characteristicstress softening effect as well as filler-induced hysteresis up

TABLE III. Comparison of the activation energy values obtained at 0.5%and 3.5% strains for the differently filled composites. The results from fre-quency sweeps of G� and G�, indicated as WLF sweeps, and the directlyestimated activation energies of G� from temperature sweeps are shown.

Activation energy �kJ/mol�

T sweep WLF sweep 0.5% WLF sweep 3.5%Polymer Filler G� G� G� G� G�

S-SBR C black 11.57 8.81 15.26 5.40 7.72Silica 7.93 2.34 12.22 1.66 7.94

EPDM C black 7.29 4.89 8.81 3.62 4.91Silica 4.65 2.75 5.24 2.33 3.91

FIG. 3. Master curves at 0.5% strain amplitude of thecarbon-black-filled S-SBR before vertical shifts �left�and after vertical shifts �right�.

FIG. 4. Arrhenius plot of the vertical shift factor of G� for the carbon-black-filled S-SBR at different strain amplitudes, as indicated.

014704-4 Le Gal, Yang, and Klüppel J. Chem. Phys. 123, 014704 �2005�

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to large strain could be well explained, whereby the activa-tion energies obtained from the vertical shifting procedure at3.5% strain amplitude provided reasonable simulations forthe stress-strain cycles between 5% and 100% strains.24

These results indicate that the larger activation energiesfound for the very small strain amplitudes are possibly re-lated to a dispersion effect of bond structures, i.e., a fractionof weaker filler-filler bonds which break initially at smalldeformations.

For comparison reasons, the activation energies obtaineddirectly from temperature sweeps of G� are shown in TableIII, as well. They are found to be systematically larger thanthe one from the frequency sweeps, because for temperaturesweeps the dynamics of the filler-filler bonds is superim-posed with that of the polymer matrix. This is in contrastwith the mastering technique presented in Figs. 3 and 4,where both dynamical processes are separated via the verti-cal and horizontal shifting procedures. As an example, theevaluation of the activation energies from temperaturesweeps is shown for the EPDM systems in Fig. 5, where thestorage modulus log G��T� is plotted as a function of theinverse temperature T. Contrary to the unfilled sample, theaddition of reinforcing fillers induces an Arrhenius-type tem-perature dependence above Tg that can be described in thesame way as before as a thermally promoted phenomenon.The Arrhenius behavior for the two filled systems isobserved in the range of Tg+30 °C and Tg+100 °C, wherethe slope of the linear relationship between log G� and theinverse temperature determines the activation energy. Acomparison between the two numerical methods listed inTable III reveals that the trend exhibited by the differentcompounds is unchanged, whether the activation energyis calculated on the basis of temperature or frequencysweeps.

An analysis of the results shown in Table III makes clearthat two main effects must be mentioned. On the one side,the activation energy values of the S-SBR compounds aresystematically higher than the one from the EPDM. On theother side, the samples filled with silica are associated withlower activation energy values. The concept of filler-fillerbonds depicted in Fig. 1 allows for an explanation for thistendency, because the thermal activation of the glassy poly-mer bridges is expected to be affected by the interaction

strength between polymer and filler surface.20 Then the dif-ference between S-SBR and EPDM can be explained by thehigher amount of double bonds and styrene units containedin the SBR chains, which promotes the interaction betweenpolymer and filler particles as compared to EPDM. The silicaparticles are in both cases chemically coupled with a silaneagent to achieve a strong bonding between the polymerchains and the filler particle. This implies that the polymer-filler coupling is constituted with single “contact points,”corresponding to the spatial location of the silane cross-links.On the contrary, the interaction between carbon black and thebulk polymer is dispersive and therefore materialized by anadsorbed layer of polymer chains with a modified dynamicsin the vicinity of the filler particle. The interface is in thiscase distributed over the carbon black surface, which meansthat a large amount of polymer chains is involved in thebonding process. As a consequence, the required energy toactivate thermally the immobilized polymer layers aroundand between adjacent filler particles is higher for the carbon-black-filled composites. This leads to the conclusion that dur-ing dynamic-mechanical deformations, the silica systems aredynamically softer than the carbon black systems. It can berelated to dynamically softer hingelike filler-filler bonds viathe immobilized interfacial layers between neighboring silicaparticles within the filler clusters or the filler network,respectively.15

IV. RUBBER FRICTION ON SELF-AFFINE SURFACES

The above results concerning the different viscoelasticmaterial picture are important factors for sliding friction ofrubber on rough surfaces, whereby the rubber is locally sub-jected to a deformation field due to the multiple scale asperi-ties of the substrate. The details of this process involvingsurface roughness, contact mechanics, adhesion, and hyster-esis friction are considered in the present section. The flex-ibility and hysteresis of filled rubber composites under dy-namic deformation is of high importance for tires, when thecontact between a tread element and a rigid rough road sur-face is considered. The ability of the rubber to deform andfill the cavities of the profile under dynamic excitation is adetermining factor, for instance during ABS braking on wetroads, where the braking process leads to high-frequency�kilohertz to megahertz� deformations of the tread rubbermaterial.

A. Characterization of surface roughness

Recent studies showed that the concept of self-affinitycould be applied to measure the texture of many roughsurfaces.14 Self-affine objects exhibit an invariant behaviorunder anisotropic dilations represented by the local fractaldimension D that is interpreted as a quantitative measure ofthe surface irregularity in the case of rough profiles. For anestimation of the surface descriptors, one can refer to theheight-difference correlation function Cz��� physically repre-senting the mean-square height fluctuations with respect tothe horizontal length scale �,4

FIG. 5. Arrhenius plot of the temperature-dependent storage modulus forthe filled and unfilled EPDM samples at 0.5%.

014704-5 Friction and contact mechanics of elastomers J. Chem. Phys. 123, 014704 �2005�

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Cz��� = ��z�x + �� − z�x��2� , �2�

where the average �…� is taken over all realizations of theprofile z�x�. For self-affine surfaces, characteristic scalingproperties lead to a power-law dependence for the height-difference correlation function below a certain lengthscale,

Cz��� = ��2� �

��6−2D

for � � �� , �3�

and a plateau value ��2 is reached for ����. Both length

scales �� and �� describe the maximum roughness paralleland perpendicular to the profile, below which self-affinity isfulfilled. D is the surface fractal dimension �2�D�3�. Theevaluation of these three fractal descriptors is illustrated inFig. 6 for the profile of a granite surface as characterized bystylus measurements.

The spectral power density S�f�, required for the estima-tion of the hysteresis friction coefficient, is analytically de-rived in a similar manner as a function of the surface descrip-tor set �D ,�� ,���.

B. Evaluation of the real area of contact

The real area of contact is of considerable importance, asit determines the amount of cavities filled by the rubber andthe number and size of local contact spots involved in thefrictional process. In the case of self-affine surfaces, the in-terval of contact is found to be distributed over many lengthscales. It can be derived from an energy condition, stating

that the elastic deformation work necessary to fill up a cavityshould be larger than the elastically stored energy in thestress field of the rubber,4

����2h��� � E*���h3��� . �4�

Here, E*��� is the norm of the complex dynamic modulus atfrequency �=2v /�, v is the sliding velocity, and h��� is thedeformation of the rubber while filling up a cavity of hori-zontal size �. From Eq. �4� a minimum contact length �min

can be evaluated, characterizing the smallest length scalewhere cavities are filled up with rubber under an applied load���. The theory of dynamic contact we used here is basedon a generalized Greenwood–Williamson approach, wherethe contact parameters, such as real area of contact, penetra-tion depth, or load, are related to each other via the distanced between the rubber and the mean profile height of thesubstrate.4 It involves the Greenwood–Williamson �GW�functions,

Fn�t� = �l

�

�z − t�n��z�dz , �5�

as characterized by the height distribution ��z� of the sub-strate profile with discrete values n=0, 1, and 3/2. Here t�d / is the normalized distance with respect to the varianceof the height distribution ��z�. The real area of contact isthen derived as the sum of “external” contact spots with thesurface summits, scaling with F0, and “internal” contactspots with the surface cavities, depending on F3/2. It can beevaluated via the scaling behavior exhibited by the micro-scopic summits and cavities of the profile as the following:4,8

Ac��min� = A0 1

808

�2D − 4�2��F02� d

F3/2� d

s�E*�2v

���

s3/2�2D − 2�2���E*�2v�min

� �1/3

. �6�

The parameter s is an affine scaling factor of the order one,which relates the height distribution ��z� of the roughnessprofile to the summit height distribution �s�z� at the largestroughness scale. A good agreement was found between anumerical approach and an analytical method for the estima-

tion of the summit height distribution �s�z�. The numericalmethod calculates the local maxima within a length scaleequal to the horizontal cutoff length �� in accordance with thegeneralized Greenwood–Williamson theory. The maximadistribution is then compared with an analytical transforma-

FIG. 6. Stylus measurement of the profile �left� andcorresponding height-difference correlation function�right� of the granite surface. The fractal descriptorsD�3−H=2.17,��=0.20 mm, and �� =0.81 mm areindicated.

014704-6 Le Gal, Yang, and Klüppel J. Chem. Phys. 123, 014704 �2005�

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tion of the profile, shifting the points to the maximum profileheight in an affine way, which determines the affine scalingfactor.8 For the granite surface this procedure yields thevalue s=1.204, which is taken as a fixed parameter for thesimulations below. Note that the GW function F3/2 in Eq. �6�is related to the summit height distribution �s�z� at the larg-est roughness scale with variance s= /s. By referring tothe scaling factor s, the corresponding distance ds can also beexpressed by the distance d. Both distance parameters ds andd can be evaluated as �inverse� GW functions of the appliedload 0, implying that Eq. �6� determines the real area ofcontact in dependence of load and sliding velocity for agiven set of surface descriptors and viscoelastic data.

Figure 7 shows the simulations of the real area of contactin the range of low sliding velocities on the granite surface,calculated on the basis of the frequency-dependent complexmodulus at 3.5% strain for the various S-SBR samples. Theload is fixed at 0=12.3 kPa. The curves are simulatedwithin the velocity range of stationary sliding friction�10−6–1 m/s� for the silica- and carbon-black-filled S-SBRsin comparison with the same polymer without any filler sys-tem. The results indicate that the real area of contact is muchhigher in the case of unfilled material and lies around 0.3%of the nominal area at very low velocities. This effect isattributed to the soft character of unfilled rubber that pro-motes an improved ability for the rubber to follow the as-perities of the rough substrate. A monotone decrease of Ac

with rising velocity is observed, which can be related to anincrease of the material stiffness while approaching theglassy domain, limiting the penetration of the surface corru-gations in the rubber.

The examination of the filler effect gives an indicationabout the filler performance regarding interlocking proper-ties. The silica-filled S-SBR exhibits a significantly higherreal area of contact compared to the carbon-black-filled sys-tem �about 50% larger at 10−6 m/s�, while the differencetends to become smaller at higher velocities �but still liesaround 20% at 1 m/s�. This numerical evaluation is in ac-cordance with the lower activation energy values exhibitedby the dynamic-mechanical analysis of the silica-filled com-pounds. They indicate that silica fillers inserted with silane ascoupling agent in the polymer matrix result in dynamically

softer rubber composites, as compared to carbon black. Atypical example of this technology is the introduction ofsilica-filled tire treads, generally referred to as “green tires,”with an improvement of the dynamic contact conditions ofthe tire with the road, as indicated by better dry and wet gripproperties.

C. Hysteresis and adhesion friction contributions

Recent models have investigated the rubber friction onrough surfaces with the main objective of providing an ana-lytical tool for a better understanding of the dynamic contactof tires with road tracks. In particular, the frictional force isin this case originated via the dissipated energy resultingfrom stochastic excitations of the sliding rubber by the sur-face asperities on various length scales.2–5 Indeed, a slidingphase with a velocity v between rubber and a rigid asperityis equivalent to dynamic excitation with a frequencyf =2v /�, where � corresponds to the length of the asperity.

Fractal descriptors consider the scaling behavior exhib-ited by many rough surfaces over a typical range generallycomprised between micron and millimeter length scales.Therefore, for one defined velocity, the dynamic contact canbe expressed as the sum of elementary excitation processescontinuously distributed over many decades of frequency.This allows the evaluation of the hysteresis friction coeffi-cient on rough surfaces on the basis of rubber viscoleasticdata and the power spectrum density S��� issued from thefractal analysis of the profile,4

�H �FH

FN=

1

2�2�2

���0v

��min

�max

d � � E����S��� . �7�

Here, 0 is the load, v is the sliding velocity, and ��� is theexcited layer thickness of the rubber, which is assumed to beproportional to the mean penetration depth �zp� of the asperi-ties in the rubber, given by �zp�= F1�d / �. In the presentmodel ��� cannot be evaluated exactly and the hysteresisfriction level depends on the free parameter b= ��� / �zp�.However, we note that indentation experiments monitored byphotogrammetry have been carried out and compared to fi-nite element simulations, recently. This was done in view ofan estimation of the parameter b via a quantitative character-ization of the strain field in the vicinity of the surface asperi-ties. This experimental setup can give a better insight of thecontact conditions for different compounds, pointing out inwhich way the free parameter b is varying and how the me-chanical energy is dissipated during the frictional process.25

In the following, the friction curves predicted by Eq. �7�are compared to experimental stationary friction data per-formed on filled systems under different contact conditions.The measurements are carried out at low sliding velocitybetween 10−5 and 10−2 m/s so that no heat buildup occursand the temperature at the interface remains constant. Theobtained results emphasize the role of the hysteresis duringthe frictional process especially when the surface conditionsare modified through the addition of lubricant. Figs. 8 and 9demonstrate that the model predictions agree fairly well withexperimental friction results for the carbon-black- and silica-filled S-SBR samples on a granite surface at fixed load of

FIG. 7. Calculated ratio between real and nominal areas of contact for theunfilled and filled S-SBR composites on the granite surface. Simulations aremade for 3.5% strain amplitude at an external load of 0=12.3 kPa.

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12.3 kPa. The simulated friction curves were estimated withviscoelastic data measured at 3.5% strain amplitude. The useof this strain amplitude for the complex modulus is consid-ered to be representative for the average strain originatedfrom the highly inhomogeneous strain field at the appliedmoderate load.

If the surface is lubricated with a stabilized water-detergent film, one can expect that adhesion effects are to-tally suppressed and friction is due to pure hysteresis. In-deed, the observed continuous increase of the experimentalresults with respect to the sliding velocity in Fig. 8 can bewell described by the simulated hysteresis friction curve.Both the experimental data and simulations indicate that atlow sliding velocities the hysteresis level is somewhat largerfor the S-SBR/N339 composite, even though the free frontfactor b= ��� / �zp� is found to be slightly smaller for the car-bon black system �Table IV�. With rising velocity the hyster-esis friction curve of the silica system increases more rapidlythan the carbon black system and a crossover appears atabout 10−3 m/s. For larger sliding velocities the hysteresis ofthe silica system is exceeding that of the carbon black sys-tem. This behavior can be traced back to the lower activationenergy of the silica system �Table III�, implying that the dy-namic stiffness of this composite increases more slowly withrising excitation frequency �or sliding velocity� than that ofthe corresponding carbon black composite. It implies that inthe case of silica the excited layer thickness ���=b�zp�, en-

tering the friction integral Eq. �7�, is larger and decreases lessrapidly with rising velocity. The resulting higher hysteresisfriction values of the silica composite at sliding velocitiesabove 10−3 m/s correlate with the well-known advantagesconcerning wet traction properties of silica tires, since thecharacteristic sliding velocity, e.g., for ABS braking is in therange of 1 m/s. Note, however, that at such high velocitiesthe predicted hysteresis friction isotherms become very largeand the associated thermal heating of the rubber cannot beneglected. Accordingly, the final level of hysteresis frictionin the velocity range v�1 m/s depends also on the thermalmaterial parameters of the rubber composite such as heatcapacitance or thermal conductivity, which may differ for thesilica and carbon black systems.

The influence of the contact conditions on the dry andlubricated friction data is significant for both filler systems.The friction values of filled S-SBR on the unlubricated drygranite surface are quasi-independent of the sliding velocityand with a magnitude strongly influenced by the filler type.The additional contribution under dry conditions is inter-preted in terms of adhesion effects occurring at the rubber-substrate interface, which scale with the real area of contactAc. Accordingly, the friction coefficient can be written as

� = �A + �H with �A =FA

Fn=

s

0·

Ac

A0. �8�

The adhesion force FA is determined by the interfacial trueshearing stress s, required to break the contact junctions.This quantity is treated as an open parameter that can beestimated from a fit to the experimental data.

Figure 9 shows the difference of the friction coefficients � obtained under dry and lubricated �water detergent� con-ditions on the granite surface for the two filler systems in therange of low sliding velocities. Obviously, the data can be

FIG. 8. Stationary friction curves �symbols� for thecarbon-black- �left� and silica- �right� filled S-SBRsamples on a granite surface at load 0=12.3 kPa underdry and wet conditions, as indicated. Adaptations �lines�with Eq. �7� of the lubricated friction data �water deter-gent� yield ��� / �zp�=42 and ��� / �zp�=50, respectively.

FIG. 9. Difference � �symbols� of the experimental friction data under dryand lubricated �water detergent� conditions on granite at 0=12.3 kPa andadaptations �lines� with Eqs. �6� and �8� for the carbon-black- and silica-filled S-SBRs. The fitting parameters are s=13.2 MPa and s=13.8 MPa,respectively.

TABLE IV. Physical parameters governing the hysteresis and adhesion fric-tion component for the filled S-SBR samples on a granite surface in the lowsliding velocity range.

Filler system Load �kPa� ��� / �zp� s �MPa� ls �nm�

8.0 42 8.8 8.0Carbon black 12.3 42 13.2 5.3

16.5 42 19.2 3.78.0 50 11.2 7.1

Silica 12.3 50 13.8 5.816.5 50 16.0 5.0

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well adapted with the adhesion friction coefficient shown asthe dashed lines, demonstrating that a proper description ofrubber friction on dry surfaces is possible with Eq. �8� andlead to satisfying correlation in the range of low sliding ve-locities. As expected, the adhesive component decreases suc-cessively with increasing sliding velocity. It is interesting tonote that the different levels of � for the carbon black andsilica systems depicted in Fig. 9 correlate to that of the realarea of contact shown in Fig. 7. Indeed, the same differenceof about 40% is found for the silica- and carbon-black-filledsystems. This is quantified by the examination of the fittingparameters s, which is found to be almost equal for bothfiller systems. It confirms that the level of adhesion friction isgoverned by the real area of contact, which is found to besignificantly larger for the silica system.

The values of the two open parameters b and s, ob-tained from adaptations to experimental friction data underdry and lubricated conditions for three different loadinglevels, are summarized in Table IV. The front factorb= ��� / �zp� is found to be independent of load and differsslightly for the two filler systems. The obtained values arecomparable with those from similar friction studies carriedout on two corundum surfaces.8 The real shearing stress s,required to break the contact patches, increases with load forboth filler systems. The opposite behavior is observed for thecharacteristic length scale of adhesion ls, also listed in TableIV, which is defined by s as follows:

ls = �/s. �9�

Here � is the change in interfacial free energy per unit area�work of adhesion� when the two solids come into contact. Ithas been evaluated from measurements of the surface energy�S and �R of the granite surface and the rubber samples,respectively, by applying the formula

� = �S + �R − ���S − ��R�2. �10�

From contact angle measurements one obtains �S

=62.9 mN/m for the smooth polished granite surface,�R=19.8 mN/m for the carbon black and �R=25.1 mN/mfor the silica-filled S-SBR sample. This yields quite similarchanges in the interfacial free energy �=70.6 mN/m and �=79.5 mN/m for the carbon black and silica systems,respectively, when coming in contact with the granite sur-face.

By applying Eqs. �9� and �10�, the values of ls are foundin the range of a few nanometers �Table IV�, indicating thatthe adhesion energy is dissipated on atomic length scales.This seems to support the classical models of adhesion fric-tion of rubber, as proposed, e.g., by Cherniak and Leonov,26

since ls can be identified with the jumping distance of poly-mer chains while moving over the substrate. However, itmust be criticized that the use of the static work of adhesion � does not consider the strong time or velocity dependencyof adhesion effects of viscoelastic solids. This is closely re-lated to the strain energy release rate G when the crack be-tween sliding rubber and substrate opens at the edge of thecontact patches.27,28 In the case of linear viscoelastic solids,G can be identified with an effective work of adhesion �eff,which increases with rising velocity according to a power

law. When a critical velocity is exceeded �v�vc� , �eff ap-proaches a constant saturation value determined by the ratioof the elastic modulus E��� in the limit of infinite and zerofrequency,29,30

�eff = ��E�/E0� . �11�

From the viscoelastic data shown in Fig. 3 we can concludethat in the case of carbon-black-filled S-SBR at small strainamplitudes, E� is about two orders of magnitude larger thanE0. Accordingly, the effective work of adhesion at large slid-ing velocities is significantly larger than the static one� �eff / ��102�. If �eff is inserted into Eq. �9� instead of �, one obtains for the length scale of adhesionls�0.5 �m, which can be identified with the characteristicsize of the so-called process zone of the propagating crackbetween the adhering rubber surface and the substrate. Withthis approach one can now understand the observed load de-pendency of s �Table IV�, because �eff is expected to in-crease with load, due to the well-known decrease of the rub-ber elastic plateau E0 with rising strain amplitude �Payneeffect�.

We finally point out that the critical velocity vc, where �eff approaches the saturation value given by Eq. �11�,strongly depends on the glass transition temperature Tg of therubber and the amount of filler. A crude estimate of vc can beobtained with a recent approach of Persson and Brener,30

which is based on a relaxation time spectra for the glasstransition according to the Rouse model. It yields

vc � v0�E�/E0�3. �12�

Here, v0=a0 / �2�0� depends on an atomic cutoff lengtha0�10−10 m and the largest relaxation time �0 of the Rousespectra, the so-called entanglement time, which is stronglyaffected by the glass transition temperature. For the S-SBRused in this study, the glass transition temperature is quitehigh �Tg�−20 °C� due to the large amount of vinyl units�50%�. The corresponding relaxation time �0 is relativelylarge. It can roughly be extracted from the G� data of theunfilled sample in Fig. 2. At room temperature �20 °C�, theincline from the rubbery plateau behavior to the glass tran-sition regime is found at about 1 Hz, implying that �0 is ofthe order �1 s. Then with E� /E0�102 for the filled S-SBRs,one finds vc�10−5 m/s. Accordingly, the measured frictiondata in Figs. 8 and 9 all fall into the saturation regime of �eff, indicating that the applied evaluation procedures arereasonable. However, with falling glass transition tempera-ture Tg the entanglement time �0 decreases rapidly, leading toa significant increase of vc. For that reason, the filled EPDMcomposites, with a lower Tg�−50 °C, are expected to showa qualitatively different friction behavior under dry condi-tions.

This is indeed observed, e.g., for the friction coefficientsof the carbon-black-filled EPDM system depicted in Fig. 10.The behavior of the lubricated friction is quite comparable tothat of the filled S-SBR shown in Fig. 8, but in contrast, forthe friction under dry conditions a pronounced increase ofthe friction coefficient with rising velocity is observed. Thisindicates that the effective work of adhesion �eff increasesover several velocity decades until the final saturation value,

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given by Eq. �11�, is reached above a critical velocityvc�10−3 m/s. Accordingly, due to Eq. �12� the entangle-ment time �0 of the EPDM is conjectured to be about twoorders of magnitude smaller than for the S-SBR. This is inagreement with the experimentally observed shift of the dy-namic glass transition by about two frequency decades. Asimilar friction behavior as in Fig. 10 on dry surfaces can beobserved for unfilled rubbers, because in this case the ratioE� /E0 is about one order of magnitude larger than for thefilled system. According to Eq. �12�, this leads to a pro-nounced shift of the critical velocity vc to larger values, im-plying that for low Tg polymers the saturation regime of �eff is not observed in the low sliding velocity regime up to10−2 m/s.

V. SUMMARY AND CONCLUSION

Dynamic-mechanical analysis has become a standardtechnique to measure the temperature- and frequency-dependent behaviors of viscoelastic materials such as elas-tomers. The presence of fillers in rubber components, justi-fied by the enhanced mechanical properties, introduces,however, strong nonlinear effects in the dynamic behaviordue to the complex interactions of the filler network with thepolymer matrix. As a consequence, the classical WLF equa-tion cannot be applied in its simple form, but needs the in-troduction of vertical shifting factors due to a superimposedArrhenius-type thermal activation. The activation energy hasbeen the estimation for different polymer and filler systemsand could be explained on a microscopic level via the tem-perature dependency of immobilized polymer nanobridgesbetween adjacent filler particles with a slowed-down dy-namic.

The lower activation energy of silica-filled composites isfound to be related to a favorable interlocking of the rubberwith a rough substrate. Indeed, the calculation of the realarea of contact in the frame of the proposed friction modelshows significantly larger values for the silica-filled systemin the range of low sliding velocity. Experimental investiga-tions of sliding friction demonstrate that the same tendency,

i.e., larger values for the silica-filled system, is also found forthe difference between dry and lubricated �water detergent�frictions. Consequently, prediction of rubber friction on dryrough surfaces has to include interfacial effects occurring atthe contact patches. The introduction of an adhesion frictioncomponent proportional to the real area of contact enables aglobal picture of the frictional behavior of filled elastomers.With this approach, the transition from lubricated to dry fric-tion can be analytically described. Thereby, an effectivework of adhesion �eff considering the viscoelastic energydissipation in the crack opening area at the edge of the con-tact patches, has been included in the studies.

The developed model of hysteresis and adhesion frictionprovides an analytical tool for a deeper understanding of thedynamic contact of tires with road tracks and the improve-ment of critical driving phases such as cornering or brakingwith antiblocking system �ABS�. A closer examination oftread deformation during ABS braking leads to the conclu-sion that the contact region near the entrance undergoes adeformation regime with sliding velocities smaller than10−2 m/s, while the region close to the exit of the contactpatch is subjected to larger sliding velocities of the order of�1 m/s.12,13 Accordingly, due to the deformation of treadelements the traction properties of tires are strongly affectedby the adhesion effects in the low sliding velocity range. Thismakes clear that the results concerning the role of the glasstransition temperature Tg and the modulus ratio E� /E0 on theeffective work of adhesion �eff are also relevant for tiretraction. In particular, the application of high Tg elastomersin tire treads, such as the vinyl S-SBR used in this study,implies strong adhesive interactions in the low velocityregime, because the maximum saturation value �eff= �E� /E0 is realized on a broad velocity scale below10−2 m/s, which exceeds the static work of adhesion � byabout two orders of magnitude. We note that this effect isalso partly present for wet traction, since, dependent on theroughness of the track, the real area of contact is reduced onwet roads, but is not vanishing totally as in the present caseof lubrication with a stabilized water surfactant film. Thiscan be concluded from the experimental friction data of vari-ous filled S-SBR composites on corundum surfaces.31

An important point is the strong influence of the strainamplitude on dynamic properties of filled compounds. Aspresented in this work, increasing the deformation leads to acompletely different picture regarding the mechanical dissi-pated energy in the rubber during cyclic deformation. Anincrease of the strain amplitude modifies the level of hyster-esis friction and leads to a less pronounced increase of thefriction coefficient with rising velocity.9 Accordingly, thestrain amplitude can be considered to be an additional freeparameter to optimize the prediction of friction models onrough surfaces under lubricated conditions. In the presentwork we found that a strain amplitude of 3.5% was wellsuited for modeling the friction behavior of filled rubbers inthe moderate load regime around 10 kPa. However, we pointout that considerations of higher strains are necessary to de-scribe the frictional behavior of tire treads on road surfaces,since the load is about two orders of magnitude larger in thiscase.

FIG. 10. Stationary friction curves for the carbon-black-filled EPDM sampleon a granite surface at load 0=12.3 kPa under dry and wet conditions. Thecritical velocity vc, where �eff approaches the saturation regime isindicated.

014704-10 Le Gal, Yang, and Klüppel J. Chem. Phys. 123, 014704 �2005�

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ACKNOWLEDGMENT

Support by the Deutsche Forschungsgemeinschaft �FOR492� is gratefully acknowledged.

1 Z. Radó, Ph.D. thesis, Pennsylvania State University, 1994.2 G. Heinrich, Rubber Chem. Technol. 70, 1 �1997�.3 G. Heinrich, M. Klüppel, and T. A. Vilgis, Comput. Theor. Polym. Sci.

10, 53 �2000�.4 M. Klüppel and G. Heinrich, Rubber Chem. Technol. 73, 578 �2000�;ACS Rubber Division Meeting, Chicago, 13–16 May 1999 �unpublished�,Paper No. 1.

5 B. N. J. Persson, J. Chem. Phys. 115, 3840 �2001�; , ACS Rubber Divi-sion Meeting, Cleveland, Ohio, 16–19 October 2001 �unpublished�, PaperNo. 24.

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8 M. Klüppel, A. Müller, A. Le Gal, and G. Heinrich, Dynamic Contact ofTires with Road Tracks, ACS Meeting, San Francisco, 28–30 April 2003�unpublished�, Paper No. 49.

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11 J. Schramm, Ph.D. thesis, University of Regensburg, 2002.12 J. Roth, Ph.D. thesis, University of Darmstadt, 1993.

13 G. Heinrich, L. Grave, and M. Stanzel, VDI-Report No. 1188, 49, 1995.14 G. Heinrich and M. Klüppel, Elastomer Friction and Adhesion on Self-

Affine Interfaces: Theory, Experiment and Applications in Tire Industry,Proceedings of the IPF-Colloquium Theory and Experiment, Dresden,FRG, 14–15 November 2002 �unpublished�.

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Stockholm, Sweden, 27–29 June 2005 �unpublished�.25 A. Le Gal, X. Yang, and M. Klüppel, Kautsch. Gummi Kunstst. �submit-

ted�.26 Y. B. Cherniak and A. I. Leonov, Wear 108, 105 �1986�.27 D. Maugis and M. Barquins, J. Phys. D 11, 1989 �1978�.28 M. Barquins, Wear 158, 87 �1992�.29 P. G. de Gennes, Langmuir 12, 4497 �1996�.30 B. N. J. Persson and E. A. Brener, Phys. Rev. E 71, 036123 �2005�.31 M. Klüppel, A. Müller, A. Le Gal, and G. Heinrich �unpublished�.

014704-11 Friction and contact mechanics of elastomers J. Chem. Phys. 123, 014704 �2005�

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