Rubber friction on smooth surfaces

Transcription

1 Eur. Phys. J. E 1, DOI 1.114/epje/i THE EUROPEAN PHYSICAL JOURNAL E Rubber friction on smooth surfaces B.N.J. Persson 1,a and A.I. Volokitin 1, 1 IFF, FZ-Jülich, 545 Jülich, Germany Samara State Technical Uniersity, 4431 Samara, Russia Receied 3 July 6 and Receied in final form 19 September 6 / Published online: 9 Noember 6 c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 6 Abstract. We study the sliding friction for iscoelastic solids, e.g., rubber, on hard flat substrate surfaces. We consider first the fluctuating shear stress inside a iscoelastic solid which results from the thermal motion of the atoms or molecules in the solid. At the nanoscale the thermal fluctuations are ery strong and gie rise to stress fluctuations in the MPa-range, which is similar to the depinning stresses which typically occur at solid-rubber interfaces, indicating the crucial importance of thermal fluctuations for rubber friction on smooth surfaces. We deelop a detailed model which takes into account the influence of thermal fluctuations on the depinning of small contact patches stress domains at the rubber-substrate interface. The theory predicts that the elocity dependence of the macroscopic shear stress has a bell-shaped form, and that the low-elocity side exhibits the same temperature dependence as the bulk iscoelastic modulus, in qualitatie agreement with experimental data. Finally, we discuss the influence of smallamplitude substrate roughness on rubber sliding friction. PACS e Polymers, elastomers, and plastics 6..Qp Tribology and hardness 6.4.+i Anelasticity, internal friction, stress relaxation, and mechanical resonances 1 Introduction The friction between rubber and smooth substrate surfaces is a topic of extreme practical importance, e.g., for wiper blades in particular on hydrophobic glass, rubber O-ring seals, and in the contact region between the tire rubber and the steel rim on a wheel [1,]. When a rubber block is sliding on a ery rough substrate, such as a tire on a road surface, the friction is almost entirely due to the energy dissipation in the bulk of the rubber as a result of the fluctuating in time and space iscoelastic deformations of the rubber by the substrate asperities [3 5]. This mechanism becomes unimportant when the substrate is ery smooth. In the limiting case of a perfectly smooth substrate, the friction is instead due to local stick-slip eents at the sliding interface. Schallamach [6] has proposed a molecular mechanism for the local stick slip, see Figure 1, a rubber polymer chain at the interface attaches to the moing countersurface, stretches, detaches, relaxes, and reattaches to the surface to repeat the cycle similar models hae been studied in Refs. [7, 8]. During each cycle, the elastic energy stored in the polymer chain is dissipated as heat during the rapid detachment and relaxation phase, and this is assumed to be the origin of the macroscopic friction. This model is probably suitable to describe situations indiida b.persson@fz-juelich.de stretches detaches, relaxes reattaches Fig. 1. The classical description of a polymer chain in contact with a lateral moing countersurface. The chain stretches, detaches, relaxes, and reattaches to the surface to repeat the cycle. ual molecules can bind to scarce but strong pinning sites presented on the substrate surface, i.e., situations the pinning potential is strongly corrugated. Howeer, the Schallamach idea cannot be applied to weak pinning in a quasi-periodic potential. First, for physisorption systems the energy barriers for ertical detachment are usually much higher than the energy barriers for lateral sliding [9], so one would not expect any detachment to occur. Secondly, with respect to the stresses rubber materials are usually exposed to, rubber is nearly incompressible and it is not easy to imagine how single molecules strongly confined at the interface are able to switch between an elongated stretched state and a relaxed curled-up state as indicated in the figure. Furthermore, the sliding friction tends to exhibit the same temperature dependence as the bulk rubber iscoelastic modulus Eω as described by the

2 7 The European Physical Journal E a b c d rubber hard flat substrate opening closing crack crack hard way substrate stress block perfect adhesie contact hard substrate, small amplitude roughness Schallamach waes opening closing crack crack hard flat substrate closing opening crack closing opening crack crack Fig.. Four different processes the adhesie rubbersubstrate interaction contributes to rubber friction. Williams-Landel-Ferry WLF [1] shift factor a T. This indicates the crucial role of the bulk rubber in the friction process. We beliee that the local stick-slip processes, which must occur at the sliding interface, inole relatie large rubber olume elements, always in adhesie contact with the substrate. That is, during sliding, small patches with a diameter of order D 1 1 nm or stress domains [11] of rubber at the sliding interface perform stick-slip motion: during stick the shear stress at the interface increases continuously with time until the local shear stress reaches a critical depinning stress σ c, after which a rapid local slip occurs, but with the rubber patch in continuous adhesie contact with the substrate. During the local slip the elastic deformation energy stored in the rubber during the loading phase will be dissipated partly inside the rubber in a olume of order D 3 and partly at the interface. The deformation field in the icinity of a stress domain of area D will extend a distance D into the rubber block; we denote this basic unit olume D 3 as a stress block see olume elements surrounded by dashed lines in Fig. a. Figure illustrates three other mechanisms of rubber friction, which all depend on the rubber-substrate adhesional interaction. Figure b illustrates a case a rubber block is sliding on a smooth way substrate. Here we assume only roughness wainess on a single length scale, i.e., the substrate bumps hae no roughness on length scales smaller than the lateral size of the bumps. In this case, if the adhesional interaction and the external applied normal load is unable to bring the rubber into perfect contact with the substrate, as in the figure, during sliding opening and closing cracks will occur at the exit and the front of each asperity contact region. In general, negligible bulk iscoelastic energy dissipation occurs at the closing crack, while a huge energy dissipation may occur at the opening crack [1,13]. It has been shown that in some cases the dominant contribution to the friction force arises from the energy dissipation at the opening cracks [1, 13]. If the substrate has small-amplitude roughness, the adhesie rubber-substrate interaction may een in the absence of an external load lead to complete contact between the rubber and the substrate at the sliding interface see Fig. c. In this case, during sliding iscoelastic deformations will occur in the bulk of the rubber in the icinity of the substrate, which will contribute to the obsered friction. This contribution can be calculated using the theory deeloped in references [3, 14]. Finally, for ery soft rubber it has often been obsered that detachment waes propagate throughout the entire contact area, from its adancing to the trailing edge, see Figure d. Schallamach [15] first discoered these waes at high sliding elocities and for elastically soft rubber. Roberts and Thomas [1,16] hae shown that when such instabilities occur, the frictional stress is mainly due to the energy dissipation at the opening crack, i.e., similar to the situation in Figure b. In this paper we study the rubber friction process shown in Figure a. This is probably the most important rubber friction mechanism in most applications inoling ery smooth surfaces. In Section and App. A we consider the fluctuating shear stress inside a iscoelastic solid, which results from the thermal motion of the atoms or molecules in the solid. At the nanoscale, the thermal fluctuations are ery strong giing stress fluctuations in the MPa-range, which is similar to the depinning stresses which typically occur at solid-rubber interfaces, illustrating the crucial importance of thermal fluctuations for rubber friction on smooth surfaces. In Section 3 and Apps. B and C we deelop a detailed model which takes into account the influence of thermal fluctuations on the depinning of small contact areas stress domains at the rubber-substrate interface. The theory predicts that the elocity dependence of the macroscopic shear stress has a bell-shaped form see Sect. 4, and that the low-elocity side exhibits the same temperature dependence as the bulk

3 B.N.J. Persson and A.I. Volokitin: Rubber friction on smooth surfaces 71 ν = λ/µ + λ and Fig. 3. The thermal motion of the atoms in a solid gies rise to shape fluctuations for a small olume element in the solid schematic. iscoelastic modulus, in qualitatie agreement with experimental data Sect. 5. In Section 5 we also discuss the role of the surface roughness, which exists een on ery smooth surfaces. We show that the countersurfaces used in the study by Vorolakos and Chaudhury [17] hae such small surface roughness that the roughness contributes negligibly to the friction. Section 6 contains the summary and conclusion. Brownian motion in iscoelastic solids A small particle in a liquid performs random motion caused by the impact of the surrounding liquid molecules. Similarly, a small olume element in a iscoelastic solid performs shape fluctuations as a result of the thermal motion of the atoms or molecules in the solid, see Figure 3. Here we will estimate the magnitude of the fluctuating shear stress which acts on any internal surface in a iscoelastic solid. Assume that ux, t is the displacement ector in an infinite iscoelastic solid. The equation of motion for u is ρ u t = ˆµ u + ˆµ + ˆλ u + f, 1 ˆµ and ˆλ are time integral operators, e.g., ˆµGt = t dt µt t Gt, and fx, t is a randomly fluctuating-force density. If we define the Fourier transform f i k, ω = 1 π 4 d 3 xdt f i x, te ik x ωt and we can write f i x, t = d 3 kdω f i k, ωe ik x ωt f i k, ωf j k, ω = k BT πω π 3 Im µωk δ ij + Im µω 1 ν k ik j δk + k δω + ω, µω = dt µte iωt. Let us study the fluctuating stress in the solid. The aerage stress σ ij = but the aerage of the square of any components of σ ij will be finite. Here we consider the magnitude of the fluctuating shear stress within some plane which we can take to be the xy plane. Thus we consider σ = σ zx + σzy = σ zx. In Appendix A we show that σ σ = kb T CD 3 E E, 3 E = E and E = E are the low- and highfrequency rubber modulus both real, respectiely, and 8π4 5ν C = 451 ν. 4 In deriing 3 we hae used the relation µ = E/1 + ν, Eω is Young s iscoelastic modulus and ν the Poisson ration which is assumed to be independent of the frequency. In 3 D is a cut-off length, of order the distance between the cross-links, or larger. Assuming ν =.5 gies C 1.1. In a typical case E 1 9 Pa E and if D = 1 nm we get at room temperature σ 1 MPa, which typically is of the same order of magnitude as the depinning stress at a rubber-substrate interface. It is also interesting to estimate the fluctuation in the displacement u and the fluctuation in the strain. Using the same approach as aboe one obtains see App. A u = kb T C D 1 1 E 1 E C = 5 6ν1 + ν 31 ν., In a similar way one can calculate the aerage of the square of the strain ɛ = ɛ 1 zx + ɛ zy = kb T C D 3 1, E E C = 3π4 5ν1 + ν 451 ν With E = 1 MPa and D = 1 nm one gets at room temperature ū 1 nm and ɛ.3. The analysis aboe indicates that, on a length scale of 1 nm, ery large strain and stress fluctuations occur in normal rubber. The stress fluctuations are of similar magnitude as the typical rubber-substrate depinning stress, suggesting that thermally excited transitions oer the lateral energy barriers play a crucial role in rubber friction on smooth substrates. Howeer, the stress fluctuations 1 MPa are negligible compared to the stress necessary for detaching a rubber patch in the normal direction, since the latter is determined by the adhesional.

4 7 The European Physical Journal E stress [18,19] which typically is of order 1 GPa. Thus, rubber sliding on smooth surfaces neer inoles thermally actiated detachment of rubber from the substrate surface, but only lateral sliding of rubber patches the lateral depinning stress is typically 1 1 times smaller than the adhesional stress. Detached areas may form and propagate at the interface as for Schallamach waes, see fig. d, but in these cases they are generated by the external applied stress, and depend on the shape of the bodies; the detached regions usually form at the edge of the rubber-substrate contact region, elastic instabilities e.g., buckling of the rubber may occur []. In the next section we will deelop a model of rubber sliding friction based on the picture presented aboe. a b rubber stress block hard flat substrate model iscoelastic spring 3 Theory We consider a rubber block with a smooth flat surface sliding on a perfectly smooth and flat substrate. We assume that the adhesie rubber-substrate interaction will result in perfect contact between the two solids i.e., we assume that no Schallamach waes occur at the interface. During sliding at low elocities, small nanometer-sized regions stress blocks at the sliding interface will perform stick-slip motion. We model the real physical system in Figure 4a with the block-spring model shown in Figure 4b. Howeer, the springs in the model are not elastic springs but iscoelastic springs determined by the iscoelastic modulus of the rubber see below. Furthermore, the blocks in the model experience not only the stress from the substrate and the forces from the springs, but also randomly fluctuating in time forces deried from the thermal motion of the molecules in the solid. The strength of the fluctuating forces is determined by the temperature and by the iscoelastic properties of the springs ia the fluctuation-dissipation theorem. The combination of the thermal fluctuating forces, and the forces deried from the external pulling of the upper surface of the rubber block, and the stress acting on the bottom surface of the block from the substrate, determine the motion of the stress blocks. We assume that the motion at low sliding elocity occurs by a thermally actiated process, small surface areas or patches of rubber at the interface perform stick-slip motion. We will refer to these patches as stress domains. The stress domains are pinned by the substrate potential, and we assume that some characteristic shear stress σ c must be reached before local slip can occur. Thus, the pinning force F c = σ c D, D is the characteristic linear size of a stress domain. The equation of motion for the coordinate q i of the stress domain i is assumed to be m q i = ˆk z x q i + ˆk x q i+1 q i +ˆk x q i 1 q i + f i + F i, 5 x = t and ˆk z is a time integral operator, ˆk z xt = t dt k z t t xt, 6 Fig. 4. a Rubber block in adhesie contact with a flat substrate. During sliding at low elocities, small rubber olume elements stress blocks perform stick-slip motion. b The model used in the mathematical description of the system in a. The iscoelastic springs k are determined by the rubber iscoelastic modulus and by the lateral size D of the stress blocks ia kω EωD. and similar for ˆk x. We consider N stress domains, i = 1,..., N, and assume periodic boundary conditions so that q = q N. In 5, f i t is a stochastically fluctuating force which results from the thermal motion of the rubber molecules. The force F i acts on the stress domain i from the substrate and is defined as follows: When the stress domain i slips, then F i = mη q i. In the pinned state, F i is just large enough to balance the total force exerted by the stress block on the substrate: F i = ˆk z x q i ˆk x q i+1 q i ˆk x q i 1 q i f i. 7 Slip will start when F i reaches a critical alue F c, either as a result of the external applied force or as a result of a large enough thermal fluctuation, or, in general, as a combination of both these effects. We define the Fourier transform xω = 1 dt xte iωt, 8 π xt = dω xωe iωt. 9 The fluctuating force f i t results from the thermal motion of the rubber molecules and must satisfy the fluctuationdissipation theorem. That is, if we write ˆK ij q j = ˆk z q i ˆk x q i+1 q i ˆk x q i 1 q i, 1 then from the theory of Brownian motion see App. D f i ωf j ω = k BT πω Im K ijωδω + ω, 11 K ij ω = dt K ij te iωt

5 B.N.J. Persson and A.I. Volokitin: Rubber friction on smooth surfaces 73 and for t > K ij t = 1 π dω K ij ωe iωt. If we write the elastic modulus as [1] Eω = E 1 h n, 1 1 iωτ n n we get K ij ω = K ij K ij n h n 1 iωτ n, 13 Kij = kz + kx δij kx δi,j+1 + δ i,j 1. Equation 1 gies Im K ij ω = K ij Substituting 13 in 11 gies f i ωf j ω = k BT π K ij n n h n τ n ω 1 iωτ n. 14 h n τ n 1 iωτ n δω + ω. 15 In Appendices B and C we show how the equations aboe can be reformulated in a form conenient for numerical calculations. In the calculations presented below we hae assumed that a sliding block returns to the pinned state when the shear stress σ < σ c1 = λσ c, σ c = F c /D is the depinning stress and λ < 1. We assume that when the shear stress has decreased below σ c1 the block returns to the pinned state with the probability rate w, and we use random number to determine when the transition actually takes place. Thus, if r is a random number uniformly distributed in the interal [, 1], then if σ < σ c1 we assume that the stress block returns to the pinned state during the time interal δt the time integration step length if wδt > r. In the simulations below we use w 1 1 s 1 and λ =.1. We use δ 1 13 s in most of our simulations so that the condition wδt 1 is satisfied, and the results presented below do not depend on the time step δt. The linear size of the stress blocks, D, is most likely determined by the elastic modulus and the depinning shear stress σ c as follows. The stress block is the smallest unit which is able to slide as a coherent unit and can be determined as follows. The depinning force is F c = σ c D. If the stress σ acts at the bottom surface of a stress block, it will moe a distance x determined by kx = σd, k ED, E is the elastic modulus. The stress block experiences a quasi-periodic pinning potential from the substrate, characterized by a lattice constant a of order a few ångströms. Thus, the stress block will in general be able to occupy a good binding position in the corrugated substrate potential only if ka is less than the pinning potential []. The condition ka = F c gies the size D of the pinned domains. Using F c = σ c D and k ED gies D Ea/σ c. In a typical case E E E 1/ 1 MPa, σ c 1 MPa and a a few ångströms, giing D = 3 nm. Since D increases when the elastic modulus increases, the present theory indicates that the friction should increase with increasing elastic modulus. This is exactly what has been obsered in friction studies for smooth surfaces [17]. Since the elastic modulus E depends on frequency, during sliding the size D of the stress domains may depend on the sliding elocity, but we hae not taken this effect into account in this paper. 4 Numerical results We now present numerical results for styrene butadiene rubber both with and without filler using the measured [3] iscoelastic modulus Eω. In reference [1] we hae shown how the parameters h n and τ n in the representation 1 of Eω can be determined directly from σ MPa f N / N a b 1 C T= C x log m/s Fig. 5. a The frictional shear stress σ f and b the fraction of slipping surface area N/N as a function of the logarithm with 1 as basis of the sliding elocity of the rubber block. For styrene butadiene copolymer with 6% carbon black and for two different temperatures, T = 1 C and C. In the calculation we hae used the zero-temperature depinning stress σ c = 1 MPa and the stress block size D = 3 nm. The number of stress blocks was 18. The iscous friction coefficient during steady sliding η =.3 natural units and the critical stress below which steady sliding becomes metastable σ c1 =.1σ c =.1 MPa. The probability rate per unit time to return to the pinned state when σ < σ c1 is w = 1 1 s 1.

6 74 The European Physical Journal E.4 a.3 σ MPa f. 4 nm D=3 nm σ f MPa..1.3 η=.1 1 b log m/s N / N.5 x 1 Fig. 7. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For the temperature T = C and for the iscous friction coefficients η =.3 and.1 natural units. All other parameters as in Figure log m/s Fig. 6. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For the temperature T = C and for the stress block sizes D = 3 and 4 nm. All other parameters as in Figure 5. the measured Eω. The τ n, h n data used in the present study is shown in Figure B in reference [1]. Figure 5a shows the frictional shear stress σ f and Figure 5b the fraction of slipping surface area N/N as a function of the logarithm with 1 as basis of the sliding elocity of the rubber block. The results are for styrene butadiene copolymer with 6% carbon black, and for two different temperatures, T = 1 C and C. Note that the low-elocity part < 1, 1 1 m/s is the elocity at which the friction is maximal of the friction cure is shifted by 1 decade towards lower elocities when the temperature is reduced from to 1 C. This is identical to the change in the bulk iscoelastic shift factor a T, which changes by a factor of 1 during the same temperature change. Figure 5b shows that the number of moing stress blocks is basically temperature independent for sliding elocities > 1 1 m/s, i.e., for > 1 the fluctuating force arising from the finite temperature has a negligible influence on the friction force. For < 1 more stress blocks are depinned at a higher temperature because the fluctuating force is larger at a higher temperature. Note also that the maximal shear stress σ f.3 MPa is about 3% of the depinning stress σ c = 1 MPa. Figure 6a shows the frictional shear stress σ f and Figure 6b the fraction of sliding blocks, as a function of the logarithm of the sliding elocity. In the calculation we hae used T = C and the stress block sizes D = 3 and σ f MPa.3..1 σ c1=. σc log m/s Fig. 8. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For the temperature T = C and for the critical stresses below which steady sliding becomes metastable σ c1 =.1σ c =.1 MPa and. MPa. All other parameters as in Figure 5. 4 nm. When the stress block size increases, the pinning force σ c D increases, and it is necessary to go to lower sliding elocities in order for temperature effects to manifest themseles as a decrease in the frictional shear stress. Figure 7 shows the frictional shear stress σ f as a function of the logarithm of the sliding elocity. Results are shown for two different iscous friction coefficients η =.3 and.1 natural units. As expected, η is only important at relatie high sliding elocities. Figure 8 shows how σ f depends on the sliding elocity for two different alues.1 and. MPa of the critical stress σ c1 below which the sliding patch can return to the pinned state. Figure 9 shows the frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block for two different alues 1 1 and 1 1 s 1 of the probability rate per unit time to return to the pinned

7 B.N.J. Persson and A.I. Volokitin: Rubber friction on smooth surfaces 75.6 σ MPa f N / N s-1 x 1 w=x1 1 s log m/s Fig. 9. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For the temperature T = C and for the probability rates per unit time to return to the pinned state when σ < σ c1 w = 1 1 s 1 and w = 1 1 s 1. All other parameters as in Figure 5. σ / σ c f.3..1 σ = 1 MPa c.7 MPa log m/s Fig. 1. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For the temperature T = C and for the zero-temperature depinning stresses σ c = 1 MPa and σ c =.7 MPa. All other parameters as in Figure 5. state when σ < σ c1. Figure 1 shows similar results for two different alues 1 and.7 MPa of the depinning stress σ c. Figure 11 shows the frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block for an unfilled styrene butadiene SB copolymer for two different alues 3 and 4 nm of the size D of a pinned region. σ MPa f.4. 4 nm D=3 nm log m/s Fig. 11. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For unfilled styrene butadiene SB copolymer for T = C. In the calculation we hae used the zero-temperature depinning stress σ c = 1 MPa and the stress block size D = 3 nm and 4 nm. The number of stress blocks was 18. The iscous friction coefficient during steady sliding η =.3 natural units, and the critical stress below which steady sliding becomes metastable σ c1 =.1σ c =.1 MPa. The probability rate per unit time to return to the pinned state when σ < σ c1 is w = 1 1 s 1. 5 Discussion Vorolakos and Chaudhury [17] hae performed a ery detailed experimental study of sliding friction for silicon rubber on hard flat passiated substrates see also Refs. [4 8] for other studies of elastomer friction. They used silicon rubbers with many different low-frequency elastic moduli E. In Figure 1 we show the elocity dependence of the shear stress as measured at different temperatures T = 98 open circles, 318 gray circles and 348 K black circles for a silicon rubber with the low-frequency elastic modulus E 5 MPa. The experimental data on the lowelocity side can be shifted to a single cure when plotted as a function of a T, see reference [17]. This is in accordance with our model calculations see Sect. 4 and shows the direct inolement of the rubber bulk in the sliding dynamics. Vorolakos and Chaudhury [17] hae shown that the frictional shear stress for all the studied rubbers increases with increasing elastic modulus. This is in qualitatie agreement with our theory since, as explained in Section 3, as E increases we expect the linear size D of the stress domains to increase, which will increase the sliding friction see Figs. 6 and 11. In the experiments by Grosch [4] it was obsered that the friction on smooth surfaces has a bell-like shape with a maximum at some characteristic elocity = 1. We obsere the same general behaior, but with some important differences. Thus, the experimental data of Grosch was obsered to obey the WLF transform. That is, the full friction cure could be constructed by performing measurements in a ery limited elocity range and at different temperatures, and then use the WLF transform to shift

8 76 The European Physical Journal E 1 Fig. 1. Shear stress as a function of elocity and temperature for a silicon elastomer low-frequency elastic modulus E 5 MPa sliding on a Si wafer coered by an inert self-assembled monolayers film. Open circles, gray circles and black circles represent data at 98, 318 and 348 K, respectiely. Adapted from reference [17]. the measured data to a single temperature. We also find that our calculated friction obeys the WLF transform for < 1, but not for > 1. Thus, the decrease in the friction which we obsere for > 1 is nearly temperature independent. The difference between our prediction and the Grosch results can be understood as follows: Most likely, the decrease of the friction for > 1 in the Grosch experiment is related to a decrease in the area of real contact with increasing sliding elocity. The Grosch experiments performed on smooth but way glass, and the area of real contact depends on the effectie elastic modulus of the rubber. Thus, as the sliding elocity increases or, equialently, the temperature decreases, the rubber becomes stiffer and the area of real contact decreases. Hence, if the shear stress remains constant at high sliding elocity as we indeed obsere in our calculations if the calculations are performed at low elocities and different temperatures, and shifted to higher elocity using the WLF equation, then the friction force will decrease with increasing sliding elocity because of the decrease in the area of real contact. The experiments by Vorolakos and Chaudhury [17] was performed on passiated silicon wafers with the root-mean-square roughness of at most.5 nm. For such smooth surfaces the adhesie rubber-substrate interaction will, een in the absence of a squeezing pressure, gie rise to complete contact between the rubber and the substrate within the nominal contact area. For this case we hae applied the rubber friction theory deeloped in reference [3] to obtain the contribution to the frictional shear stress from the roughness-induced bulk iscoelastic deformations in the rubber Fig. c. We hae assumed that the substrate is self-affine fractal with the fractal dimension D f =.3 and the long-distance roll-off waelength = 1 µm. We hae included roughness components λ shear stress Pa 8 4 nm λ = 1 nm log m/s Fig. 13. The frictional shear stress σ f as a function of the logarithm of the sliding elocity of the rubber block. For styrene butadiene copolymer with 6% carbon black and for T = 4 C. The friction is entirely due to the surface roughness of the substrate which is assumed to be self-affine fractal with the fractal dimension D f =.3 and the root-mean-square roughness.5 nm. The long-waelength and short-waelength roll-off wae ectors q = π/λ and q 1 = π/λ 1, λ = 1 µm and λ 1 = nm upper cure and 1 nm lower cure. down to the short-distance cut-off waelength λ 1. In Figure 13 we show the resulting frictional shear stress assuming perfect contact at the sliding interface using λ 1 = 1 nm and nm. The smallest possible λ 1 is determined by an atomic distance but a more likely cut-off length in the present case is the mean distance between rubber cross-links, which is of order a few nanometers. The calculated maximal shear stress is of order 1 Pa, which is a factor 1 4 smaller than the shear stress obsered in reference [17]. Hence we conclude that the surface roughness in the measurements of Vorolakos and Chaudhury has a negligible influence on the obsered frictional shear stress. Howeer, the surfaces used in reference [17] are exceptionally smooth, and surface roughness can be ery important for rubber friction for surfaces of more common use in rubber applications, e.g., for rubber sealing. 6 Summary and conclusion We hae studied the sliding friction for iscoelastic solids, e.g., rubber, on hard flat substrate surfaces. We hae shown that the fluctuating shear stress, which result from the thermal motion of the atoms or molecules in a iscoelastic solid, gies rise to ery strong stress fluctuations, which at the nanoscale are in the MPa-range. This is similar to the depinning stresses which typically occur at solid-rubber interfaces, indicating the crucial importance of thermal fluctuations for rubber friction on smooth surfaces. We hae deeloped a detailed model which takes into account the influence of thermal fluctuations on the depinning of small contact patches stress domains at the rubber-substrate interface. The theory predicts that the elocity dependence of the macroscopic shear stress has a bell-shaped form, and that the low-elocity side exhibits a similar temperature dependence as the bulk

9 B.N.J. Persson and A.I. Volokitin: Rubber friction on smooth surfaces 77 iscoelastic modulus, in qualitatie agreement with experimental data. Finally, we hae discussed the influence of small-amplitude substrate roughness on rubber sliding friction and shown that it gies a negligible contribution to the friction in the experiments of Vorolakos and Chaudhury [17]. A.I.V. and B.N.J.P. thank DFG for support, and the EU for support within the Natribo network of the European Science Foundation. Appendix A. Stress and strain fluctuations in iscoelastic solids Let us study the fluctuating stress in the solid. The aerage stress σ ij = but the aerage of the square of any components of σ ij will be finite. Here we consider the magnitude of the fluctuating shear stress within some plane which we can take to be the xy plane. Thus, we consider σ = σ zx + σzy = σ zx. The stress tensor Thus we get and σ ij = ˆµu i,j + u j,i + ˆλu k,k δ ij. σ zx = ˆµu z,x + u x,z σ = 4 ˆµuz,x + ˆµu z,x ˆµu x,z = 4 d 3 kd 3 k dωdω µωµω k x k x u z k, ωu z k, ω +k x k z u z k, ωu x k, ω. A.1 Using 1 and neglecting inertia effects gies 1 uk, ω = µωk fk, ω = + µω + λωkk 1 1 µωk 1 1 ν kk k fk, ω. A. Using A. and and assuming that ν is frequency independent gies σ = kb T C3π 5 d 3 kdω 1 Im Eω, ω A.3 C = 8π4 5ν 451 ν. In deriing A.3 we hae used the relation µ = E/1+ν, Eω is Young s iscoelastic modulus. The shear stress when we only include wae ectors up to k = π/d D is a cut-off length, of order the distance between the cross-links, or larger is gien by σ = kb T CD 3 π Using the sum rule [1,13] π dω 1 Im Eω. ω dω 1 ω Im Eω = E E, E = E and E = E are the low- and highfrequency rubber modulus both real, respectiely, we get σ σ = kb T CD 3 E E. Assuming ν =.5 gies C 1.1. It is also interesting to estimate the fluctuation in the displacement u and the fluctuation in the strain. Using the same approach as aboe one obtains u = u x + u y = kb T C D 1 π C = If we use the sum rule [1,13] we get π 5 6ν1 + ν 31 ν dω 1 1 ω Im, Eω dω 1 1 ω Im = 1 1, Eω E E u 1 = kb T C D 1 1. E E In a similar way one can calculate the aerage of the square of the strain ɛ = ɛ 1 zx + ɛ zy = kb T C D 3 1, E E C = 3π4 5ν1 + ν 451 ν Appendix B. Reformulation of the basic equations of motion Using 13 we can write K ij ωq j ω + k z ωxω = K ij q j ω + k z xω n u ni ω =.. h n u ni ω, B.1 1 K 1 ij q j ω + kz xω B. 1 iωτ n 1 iωτ n

10 78 The European Physical Journal E or u ni t + τ n u ni t = K ij q j t + k z x + g ni t. B.3 Here we hae added a stochastically fluctuating force on the right-hand side of B.3 which we can choose so as to reproduce the fluctuating force f i t in 5. That is, if we choose g ni t appropriately, we can remoe the force f i t in 5. To this end we must choose with with the k-sum oer Note that f i ω = n g nj ω = N 1/ Re k k = π D M kn = kb T τ n K k πh n h n g nj ω 1 iωτ n B.4 M kn e ikxj ξ kn, B.5 1/ B.6 r, r =, 1,,..., N 1. N K k = k z + k x [1 coskd] B.7 is the discrete Fourier transform of K ij. In B.5, ξ kn are complex Gaussian random ariables: ξ kn = ζ kn + iη kn, B.8 ζ kn ωζ k n ω = δ nn δ kk δω + ω, B.9 η kn ωη k n ω = δ nn δ kk δω + ω, B.1 ζ kn ωη k n ω =. B.11 Using B.5 and B.8 gies g nj ω = N 1/ k Using B.9 B.11 gies g nl ωg n l ω = N 1 k M kn [ζ kn coskx j η kn sinkx j ]. M kn cos[kx l x l ]δ nn δω + ω = N 1 Mkne ikx l x l δ nn δω + ω = k k B T τ n Kk e ikx l x l δ nn δω + ω = πh n N k B.1 k B T τ n πh n K ll δ nn δω + ω. B.13 Using B.4 and B.13 gies f i ωf j ω = k BT π K ij n h n τ n 1 iωτ n δω+ω B.14 which agrees with 15. Let us summarize the basic equations m q i = k z x q i + k x q i+1 + q i 1 q i n h n u ni + F i, B.15 u ni t + τ n u ni t = k z x q i + k x q i+1 + q i 1 q i + g ni t. B.16 The random force g nj ω = N 1/ Re k M kn = kb T τ n K k πh n M kn e ikxj ξ kn ω, B.17 1/. B.18 Appendix C. Numerical implementation If D is the lateral size of a stress block, then the mass of a stress block m = ρd 3. We introduce the spring constants k z = α z k and k x = α x k, k = DE, and α x and α z are of order unity. We measure time in units of τ = m/k 1/ and distance in units of l = F c /k, F c = σ c D is the stress block pinning force. We also measure u in units of F c and x in units of l. In these units B.15 gies q i = α z x q i + α x q i+1 + q i 1 q i n from B.16 h n u ni + F i /F c, C.1 u ni t + τ n /τ u ni t = α z x q i + α x q i+1 + q i 1 q i + g ni t. C. The random force g nj ω = N 1/ Re k M kne ikxj ξ kn ω, Mkn kb T τ n K 1/ k = M kn /F c = π Ek, h n E = k l /. Note also that gies ζ kn ωζ k n ω = δ nn δ kk δω + ω ζ kn tζ k n t = πδ nn δ kk δt t. In order to numerically integrate equations C.1 and C. we discretize time with the step length δt.

11 B.N.J. Persson and A.I. Volokitin: Rubber friction on smooth surfaces 79 The fluctuating force δg ni to be used for each time step in C. can be written as But if we get δg nj = N 1/ Re k δζ = t+δt Mkne ikxj dt ξ kn t. ζtζt = πδt t t+δt t dt dt ζt ζt = πδt. Thus we can write δζ = πδt 1/ G, G is a Gaussian random number with G = 1. Thus, we take δg nj = N 1/ Re k G 1 kn can also write and G kn δg nj = N 1/ Re k M kn = t M kne ikxj πδt 1/ G 1 kn + ig kn, are Gaussian random numbers. We kb T E M kn e ikxj G 1 kn + ig kn, C.3 τ n δt τ K k h n k 1/. The calculation of δg nj Eq. C.3 is coneniently performed using the fast-fourier-transform method. Appendix D. Memory friction and fluctuating force Equation 11 is a standard result in the general theory of Brownian motion which can be deried in arious ways. The most general proof is based on the memory function formalism as described, e.g., in the book by D. Forster [9]. A simpler but less general deriation of equation 11 inoles the study of a particle coordinate qt coupled to an infinite set of harmonic oscillators the heat bath coordinates x µ. For the readers conenience, we briefly reiew this deriation here. The particle and the heat bath coordinates satisfy the equations of motion m q + µ α µ x µ =, m µ ẍ µ + m µ ω µx µ + α µ q + m µ η µ ẋ µ = f µ, D.1 D. f µ tf ν t = m µ η µ k B T δt t δ µν. D.3 If we define x µ t = dω x µ ωe iωt, x µ ω = 1 dt x µ te iωt π and similar for q and f µ, we get from D. and from D.3, x µ ω = f µω α µ qω m µ ω µ ω iωη µ D.4 f µ ωf ν ω = m µ η µ k B T δω + ω δ µν /π. D.5 From D.4 we get and α µ x µ ω = γωqω fω, µ γω = µ fω = µ Using D.1 and D.6 gies α µ m µ ω µ ω iωη µ α µ f µ ω m µ ω µ ω iωη µ mω qω + γωqω = fω. Using D.8, D.7 and D.5 it is easy to show that fωfω = k BT πω Im γωδω + ω. Note also that D.9 is equialent to t m q + dt γt t qt = ft. D.6 D.7 D.8 D.9 D.1 In Section 3 we studied a system of many coupled dynamical ariables m q i + t dt γ ij t t q j t = f i t but this problem can be reduced to the problem studied aboe by forming new dynamical ariables, as linear combination of the old dynamical ariables q i, chosen so that γ ij becomes diagonal. References 1. D.F. Moore, The Friction and Lubrication of Elastomers Pergamon, Oxford, B.N.J. Persson, Sliding Friction: Physical Principles and Application, nd ed. Springer, Heidelberg,. 3. B.N.J. Persson, J. Chem. Phys. 115, B.N.J. Persson, A.I. Volokitin, Phys. Re. B 65, M. Klüppel, G. Heinrich, Rubber Chem. Technol. 73, A. Schallamach, Wear 6, Y.B. Chernyak, A.I. Leono, Wear 18,

12 8 The European Physical Journal E 8. A.E. Filippo, J. Klafter, M. Urbakh, Phys. Re. Lett. 9, Weak absorption systems e.g., saturated hydrocarbon molecules or noble-gas atoms on metal surfaces exhibit extremely low pinning barriers for parallel motion. The barriers can be estimated from the measured frequencies of parallel adsorbate ibrations, see. e.g., B.N.J. Persson, Surf. Sci. Rep. 15, M.L. Williams, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77, B.N.J. Persson, Phys. Re. B 51, B.N.J. Persson, O. Albohr, G. Heinrich, H. Ueba, J. Phys. Condens. Matter 17, R B.N.J. Persson, E. Brener, Phys. Re. E 71, B.N.J. Persson, Surf. Sci. 41, A. Schallamach, Wear 17, A.D. Roberts, A.G. Thomas, Wear 33, K. Vorolakos, M.K. Chaudhury, Langmuir 19, U. Tartaglino, V.N. Samoilo, B.N.J. Persson, J. Phys. Condens. Matter 18, B.N.J. Persson, Surf. Sci. Rep. 61, B.N.J. Persson, Phys. Re. B 63, B.N.J. Persson, J. Phys. Condens. Matter. 18, The force acting on a stress domain should depend quasiperiodically on the lateral displacement x i.e., F = F c sinπx/a giing for small x: F F cπx/a. The pinning stress F c = σ cd and the elastic force F = kx with k = αed, α is of order 6. Thus we get ED σ cd /a. 3. A. Le Gal, M. Klüppel, J. Chem. Phys. 13, K.A. Grosch, Proc. R. Soc. London, Ser. A 74, ; K.A. Grosch, in The Physics of Tire Traction: Theory and Experiment, edited by D.F. Hays, A.L. Browne Plenum Press, New York, London, 1974 p A. Casoli, M. Brendle, J. Schultz, A. Philippe, G. Reiter, Langmuir 17, T. Baumberger, C. Caroli, Ad. Phys. 55, 79 6; T. Baumberger, C. Caroli, O. Ronsin, Eur. Phys. J. E 11, 85 3; O. Ronsin, K.L. Coeyrehourcq, Proc. R. Soc. London, Series A 457, M.R. Mofidi, E. Kassfeldt, B. Prakash, to be published in Wear. 8. J. Molter, Elastomerreibung und Kontactmechanik, DIK fortbildungsseminar, Hannoer D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions Benjamin, London, 1975.