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Origin and Characterization of Different Stick-Slip Friction Mechanisms† Alan D. Berman, William A. Ducker,‡ and Jacob N. Israelachvili* Department of Chemical Engineering, University of California, Santa Barbara, California 93106 Received October 17, 1995. In Final Form: January 25, 1996X Mechanical parts often move, not smoothly, but in jerks known as stick-slip. Stick-slip motion may be regular (repetitive or periodic) or irregular (erratic or intermittent). In the case of frictional sliding, stick-slip can have serious and often undesirable consequencessresulting in noise (chatter), high energy loss (friction), surface damage (wear), and component failure. We review the origins of stick-slip friction and present new experimental results on model surfaces that clarify its different origins, its dependence on experimental conditions or “system parameters”, and how stick-slip can be controlled in practical situations.
Introduction Stick-slip motion manifests itself in a variety of everyday phenomena: a creaking door, screeching brakes, the sound of tearing and grinding teeth, earthquakes, and the rich sound produced by a violin string simply by stroking it gently with a bow. In all these cases, a simple input of constant force or steady velocity V has resulted in a complex spectrum of stick-slip motions ν or sounds (Figure 1). When we try to model the stick-slip friction between two rubbing surfaces in terms of the friction force F acting between them, we soon find that the stick-slip phenomenon depends not only on F but also on other system parameters such as the inertia, stiffness (K), and mass (m) of the moving parts. Concerning the properties of the rubbing surfaces themselves, a sufficient condition for stick-slip is that the static friction force Fs (the force needed to initiate sliding) is higher than the kinetic friction force Fk (the force during sliding). The reason for this is given in the legend to Figure 1. There are several more detailed models for stick-slip friction that include the effects of molecular or asperity size, sliding velocity, various relaxation times, previous history, and other system parameters as mentioned above. The various models are based on different mechanical or molecular properties of the surfaces or interacting bodies. Each model is strictly applicable only to a particular class of real systems, and experience has shown that in this field of tribology it can be dangerous to expect that results or trends obtained on one system will apply to another. Following a brief review of the more common types of stick-slip models or theories (summarized in Table 1), we describe our experimental studies using a surface forces apparatus (SFA) in which we tried to establish which frictional model best described a wide range of data obtained on different types of surfaces (Figure 2). An additional aim was to better understand how macroscopic (whole system) properties and nanoscopic (molecular or interfacial) properties act together to determine the resulting motion and how one may be able to predict and thereby control unwanted stick-slip. † Presented at the Workshop on Physical and Chemical Mechanisms in Tribology, Bar Harbor, ME, August 27 to September 1, 1995. * To whom correspondence should be addressed. E-mail: jacob@ squid.ucsb.edu. ‡Present address: Department of Chemistry, University of Otago, Dunedin, New Zealand. X Abstract published in Advance ACS Abstracts, Sept. 15, 1996.
S0743-7463(95)00896-1 CCC: $12.00
Figure 1. Basic tribological system. Note that due to the spring coupling between the drive and the surfaces, the applied force F and drive velocity V are not necessarily the same as the friction force F and sliding velocity v of the surfaces. When starting from rest, as soon as the applied (spring) force, F, reaches the friction force F the slider will begin to move. If the static friction force, Fs, is greater than the kinetic friction force Fk during sliding, the slider will accelerate rapidly to a velocity v higher than the drive velocity V, and the spring will soon become extended beyond its free length. At this point the slider will rapidly decelerate to a stop, at which point the friction force jumps back to its high static value Fs, and the whole processes (stick-slip cycle) will be repeated so long as the drive continues to move at a steady velocity. The motion of the slider is governed by the equation of motion:
mx¨ ) K (X - x) + F
(1)
Surface Topology (Roughness) Model of StickSlip. The surface topology model (Table 1, row 1) explains stick-slip in terms of the topology or roughness of the sliding surfaces.1 As the slider climbs an asperity on the substrate, a resisting force is encountered. Once the peak is reached, the slider will slide down rapidly into the valley, resulting in a slip. The measured friction trace with time will show regular or irregular stick-slip “spikes” depending on whether the surface corrugations are themselves regular, as for a lattice plane, or irregular, as for a randomly rough surface. The controlling factors of this type of stick-slip are the topology of the surface (the 2D amplitude and periodicity of protrusions) and the elastic and inertial properties of the sliding surfaces which determine the rate of slip. In addition, a stiffer material (1) Rabinowicz, E. Friction and Wear of Materials; John Wiley & Sons: New York, 1965, and references therein.
© 1996 American Chemical Society
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Figure 2. Various surfaces and lubricant films used in surface force (SFA) experiments: typical liquid, OMCTS (octamethylcyclotetrasiloxane, a nonpolar liquid of molecular diameter 9 Å); typical hydrocarbon, tetradecane. The surfaces themselves are mica, silica, alumina (sapphire) or metal-coated mica surfaces. Figure 4. Distance-dependent friction model (also known as the creep model) in which a characteristic distance Dc has to be moved to break adhesive junctions. The model also has a characteristic, τs, this being the time needed for the adhesion and friction forces per junction to increase after each contact is made.
Figure 3. Typical stick-slip friction traces. Analyzing the shapes of stick-slip spikes in detail can provide important additional information on the origin of the stick-slip. Table 1. Models of Stick-Slip Friction (refer also to Figure 3)
(high K) will have shorter slips because of the shorter recoil to elastic equilibrium. This would allow sticking to more of the smaller asperities, resulting in a richer stickslip spectrum. With increasing hardness of the materials,
plastic deformations during sliding are reduced and the friction pattern approaches a true “contour trace” of the surfaces. Topological stick-slip is observed in the sliding of macroscopically rough surfaces, as well as in atomic force microscopy (AFM) experiments where the intermittent motion of the slider (AFM tip) is a measure of the molecular-scale roughness or atomic-scale corrugations of the surface lattice.2 Distance-Dependent Model of Stick-Slip. A second theory of stick-slip, also observed in solid-on-solid sliding, is one that involves a characteristic distance (but also a time, τs, this being the characteristic time required for two asperities to increase their adhesion strength after coming into contact). Proposed in the 1950s,1,3 this model suggests that two rough macroscopic surfaces adhere through their microscopic asperities of characteristic length Dc. During shearing, each surface must first creep a distance Dcsthe size of the contacting junctionssafter which the surfaces continue to slide, but with a lower (kinetic) friction force than the original (static) value. The reason for the decrease in the friction force is that even though, on average, new asperity junctions should form as rapidly as the old ones break, the time dependent adhesion and friction of the new ones will be lower than the old ones. This is illustrated in Figure 4. The friction force therefore remains high during the creep stage of the slip, but once the surfaces have moved the characteristic distance Dc, the friction rapidly drops to the kinetic value. As explained in Figure 1 and Table 1, any system for which the kinetic friction force is less than the static force (or one that has a negative slope over some part of its F-v curve) will exhibit regular stick-slip sliding motion for certain values of K, m, and driving velocities, V. This type of friction has been observed in a variety of mainly dry (unlubricated) systems such as paper-on(2) Mate, C. M.; McClelland, G. M.; Erlandsson, R.; Chiang, S. Phys. Rev. Lett. 1987, 59, 1942. (3) Rabinowicz, E. Proc. Phys. Soc. 1958, 71, 668.
Different Stick-Slip Friction Mechanisms
Figure 5. Stick-slip cycle governed by a characteristic molecular length and relaxation time for surfaces composed of chain molecules. The arrows indicate the sliding velocity v of the surface.
paper4,5 and steel-on-steel.3,6-8 This model is also used extensively in geology to analyze rock-on-rock sliding.9,10 In a recent study of paper-on-paper sliding, Baumberger et al.4,5 analyzed the effects of slider mass m, friction spring constant K, and pulling velocity V on the measured friction forces. The system exhibited a critical velocity, which scaled differently with K in the creep-dominated regime (at low V, high K) than in the inertia-dominated regime (see Table 1). While originally described in terms of adhering asperity junctions, the distance-dependent model may also apply to molecularly smooth surfaces. For example, the adhesion and friction forces of surfactant or polymer surfaces usually increase with their time in contact, reaching a maximum value at some critical sliding velocity given when the Deborah Number is close to unity.11 The distance-dependent model would be expected to apply to such interfaces (Figure 5) where the characteristic length Dc would now be the chain-chain entanglement length and τs would be the characteristic molecular entanglement/ disentanglement time. Velocity-Dependent Models of Stick-Slip. In contrast to the above friction models which apply to unlubricated, solid-on-solid contacts, surfaces with thin liquid films between them have different stick-slip mechanisms. One of the simplest models is a pure velocitydependent friction law. In this case there is a high static friction, Fs, when the surfaces are at rest because the film has solidified. Once the shearing force exceeds this value, the surfaces slide with a lower kinetic friction force, Fk, because the film is now in the molten, liquid-like state. Stick-slip sliding proceeds as the film goes through successive freezing-melting cycles. (It is important to note that freezing and melting transitions at surfaces or in thin films may not be the same as the freezing-melting transitions between the bulk solid and liquid phases.) Such films may be considered to alternate between two states, characterized by two friction forces, Fs and Fk, as illustrated by the dashed line in Figure 6b, or they can have a rich F-v spectrum, as proposed by Persson.12 The tribological and other dynamic properties of such confined (4) Baumberger, T.; Heslot, F.; Perrin, B. Nature 1994, 367, 544. (5) Heslot, F.; Baumberger, T.; Perrin, B.; Caroli, B.; Caroli, C. Phys. Rev. E 1994, 49, 4973. (6) Rabinowicz, E. J. Appl. Phys. 1951, 22, 1373. (7) Sampson, J. B.; Morgan, F.; Reed, D. W.; Muskat, M. J. Appl. Phys. 1943, 14, 689. (8) Heymann, F.; Rabinowicz, E.; Rightmire, B. G. Rev. Sci. Instr. 1954, 26, 56. (9) Dieterich, J. H. Pure Appl. Geophys. 1978, 116, 790. (10) Dieterich, J. H. J. Geophys. Res. 1979, 84, 2161. (11) Yoshizawa, H.; Chen, Y. L.; Israelachvili, J. J. Phys. Chem. 1993, 97, 4128. (12) Persson, B. N. J. Phys. Rev. B 1994, 50, 4771.
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Figure 6. Two-state phase transition model of stick-slip. (A) Molecular schematic of the freezing/melting transition is shown. (B) At the onset of the slip, the intrinsic friction F drops from Fs in the solidlike state to Fk in the liquidlike state, and then may vary with sliding velocity v. This results in the familiar stick-slip trace (C).
films have been extensively studied by Robbins et al.13,14 and Landman et al.15,16 using computer simulations. The simplest velocity-dependent friction force law that predicts a critical velocity is one where Fs is constant and Fk increases linearly with velocity: dFk/dv ) constant ) b, as shown by the solid line in Figure 6b. The critical velocity Vc can be obtained from analytic solutions to the equations of motion (see eq 1 and Figure 1). For a sufficiently underdamped system (K > b2/4m), we find for this model
Vc ∝ (Fs - Fk)b1/2 log m log K
(2)
Equation 2 can be compared with Robbins and Thompson’s expression for Vc based on a computer simulation12
Vc ≈ xFsσ/m
(3)
and Yoshizawa and Israelachvili’s phenomenological expression17
Vc ≈ (Fs - Fk)/Kτ0
(4)
where σ is a molecular dimension and τ0 is the “freezing” time. Carlson and Batista18 have developed a comprehensive rate and state dependent friction force law. This model includes an analytic description of the time-dependent freezing and melting transitions of a film, resulting in a friction force that is a function of sliding velocity in a natural way. This model predicts a full range of stickslip behavior observed experimentally. The above classification of stick-slip is not exclusive, and molecular mechanisms of real systems may exhibit aspects of different models simultaneously. They do, however, provide a convenient classification of existing models and indicate which experimental parameters should be varied to test the different models. (13) Robbins, M. O.; Thompson, P. A. Science 1991, 253, 916. (14) Thompson, P. A.; Robbins, M. O. Science 1990, 250, 792. (15) Landman, U.; Luedtke, W. D.; Ribarsky, M. W. J. Vac. Sci. Technol. 1989, A7, 2829. (16) Landman, U.; Luedtke, W. D.; Ringer, E. M. Wear 1992, 153, 3. (17) Yoshizawa, H.; Israelachvili, J. J. Phys. Chem. 1993, 97, 11300. (18) Carlson, J. M.; Batista, A. A. Phys. Rev. E, submitted.
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Experimental Section Friction studies were done using Surface Forces Apparatuses (SFA Mk 2 and Mk 3) modified with friction attachments.19,20 The pure model liquids n-tetradecane (Fisher Scientific) and octamethylcyclotetrasiloxane (OMCTS, a quasi-spherical nonpolar molecule, from Aldrich) were confined between molecularly smooth curved mica surfaces as shown schematically in Figure 2. The experimental setup and procedures were as described previously.17 On applying a large load on the surfaces, they flatten elastically resulting in a circular contact zone of radius r ≈ 50 µm (Figure 2). The lower surface was moved laterally at a uniform velocity V by means of a piezoelectric bimorph slider and the resulting friction was measured from the deflection of the friction force measuring spring system supporting the upper surface. The equivalent mechanical circuit of the apparatus is shown in Figure 1. When the surfaces are stuck together, the deflection of the friction spring increases linearly with time at the same speed V as the drive moving the lower surface. Once the threshold friction force Fs is reached, the surfaces slide or slip with respect to each other, and the friction spring begins to relax back to its equilibrium state. The measured spring deflection, when plotted as a function of time, gives the friction trace, that is, the measured friction force, F, as a function of time, t (Figures 3 and 6). Both a paper chart recorder and a storage oscilloscope were used to record the friction traces; the latter was needed to faithfully record rapidly changing friction forces as they occur at the peaks and valleys of stick-slip spikes (cf. Figure 3). Detailed analysis of the shapes of stick-slip spikes provided additional important insights into their underlying mechanisms. To enable stick-slip theories to be tested, the stiffness K of the friction spring and mass m of the slider have to be changeable during experiments. The stiffness of the slider was varied using a new attachment consisting of a wire spring that could be brought into contact with the slider. The stiffness K′ of this spring could be adjusted from outside the apparatus, so that the effective stiffness of the friction spring, when it is contacted by the adjustable wire spring, is given by Keff ) K + K′. Weights could be added to the slider to change its inertial mass, m.
Results We first describe our results with tetradecane films between mica surfaces at 21 °C. The applied loads and drive speeds were in the range L ) 5-35 mN, V ) 0.0110 µm/s, and the film thickness was ∼12 Å, corresponding to three molecular layers of tetradecane between the surfaces. As previously found in the case of distancedependent stick-slip friction (see above and refs 4 and 5), we also found two distinct types of stick-slip behavior, corresponding to underdamped (inertia-dominated) and overdamped (friction-dominated) slipping. These two regimes could be easily distinguished from the magnitude and shape of the stick-slip spikes. Underdamped Conditions. At high K and low m, one expects the slip to be controlled primarily by the inertia of the slider; indeed, at high K, the drop time from Fs to Fk was found to be approximately half the cycle time 2π(m/K)1/2 for free harmonic oscillations of the slider (Figure 7). Analysis of the equations of motion further show that in the underdamped regime the true kinetic friction, Fk, assuming it to be single valued, is half way between the top (Fs) and the bottom (Fk) of the slip. In other words, for underdamped conditions, we may write:
Fs ) Fs
(5)
Fk ≈ (Fs + Fk)/2
(6)
Even though eq 6 is intuitive, it is nevertheless in contrast to the conventional assignment of the bottom of (19) Homola, A. M.; Israelachvili, J. N.; Gee, M. L.; McGuiggan, P. M. J. Tribol. 1989, 111, 675. (20) Luengo, G.; Schmitt, F.-J.; Israelachvili, J. Macromolecules, submitted.
Figure 7. Measured friction F plotted against time for a characteristic underdamped slip. The solid line is the recorded data for tetradecane between mica surfaces. The dashed line is a cosine fit to the data, which is the predicted motion of the slider using its inertial properties and assuming a constant intrinsic kinetic friction Fk.
Figure 8. Measured friction Fs and Fk for increasing drive velocity V with OMCTS confined between mica surfaces. Fs (b) decreases with increasing velocity because at higher drive speeds the sticking time is shorter, resulting in less complete freezing of the lubricant layer. The observed Fk (O) increases as Fs decreases, resulting in a nearly constant Fk (2) from eq 6. The critical velocity is reached when Fs decreases to the intrinsic friction Fk.
the stick-slip spike as the kinetic friction force17 (this assignment is, however, correct for the overdamped case, as described below). With increasing drive velocity, V, the stick-slip friction amplitude (Fs - Fk) is found to decrease steadily to zero (Figure 8, solid lines), indicating that the system exhibits a critical velocity. As shown in Figure 8, both Fs and ∆F are largest at low V, and both decrease with increasing V. However, Fk increases with V in such a way that the true kinetic friction Fk, as calculated in eq 6, remains relatively constant (Figure 8, dashed line). Thus, we may conclude that in this regime, the simple two-state phase transition model appears to apply. The freezing of the lubricant in the stick stage of the cycle was found to be the slowest, rate-limiting part of the process. Shorter sticking times and higher drive velocities produce lower static friction forces, Fs, because the lubricant has less time to fully solidify between the surfaces. When the static friction force is overcome and
Different Stick-Slip Friction Mechanisms
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Figure 9. Overdamped stick-slip friction. In the slip, the friction drops rapidly at first and then more slowly. On a semilog plot of the scaled friction (F - Fk) vs time, the slip data is fit by a double exponential decay before the surfaces restick. Note that during the stick the friction increases linearly with time. Inset: The measured Fs and Fk are nearly independent of drive velocity below the critical velocity.17 At drive velocities V > Vc, the smooth sliding friction is equal to Fk in the stick-slip regime. This indicates that in the stick-slip, the friction does drop to the steady state kinetic friction before sticking again in the cycle.
the lubricant melts (shear-induced melting), the friction quickly drops to a nearly constant kinetic value Fk, and the surfaces begin to accelerate (slip). The melting process during a slip also takes a finite time but appears to be much faster than the freezing process in the stick regime. Overdamped Conditions. In contrast to the underdamped regime where the slip time is limited by the system inertia, in the overdamped regime the slip times are much longer than the mechanical system response time. In this regime, either K is low or the interfacial friction damping coefficient b (Figure 6b) is large. Experimentally, b can be modified in a particular experiment by changing the temperature, the relative humidity, or some other surfacesensitive condition. In this regime, slip begins with a rapid accelerationsvery much as in the underdamped regimesfollowed by a lengthy deceleration before resticking occurs. Total slip times can be longer than 10 s (compared to milliseconds in the underdamped regime), and more than one decay time can be associated with each slip. Figure 9 shows a sample stick-slip trace that exhibits two time scales. Another salient feature of overdamped stick-slip is that the stick-slip amplitude is independent of the driving velocity V almost all the way up to the critical velocity Vc (Figure 9, inset). The transition from stick-slip to smooth sliding with increasing V goes from regular, periodic stickslip at V < Vc, through a nonperiodic stick-slip regime as V approaches Vc, to smooth sliding at V > Vc. Particularly noteworthy is the finding that in the stickslip regime the measured kinetic friction force, Fk, is the same as the intrinsic force, Fk. More generally, in the overdamped regime we find
Fs ) Fs
(7)
Fk ) Fk
(8)
This is in contrast to the underdamped regime, where Fk ) (Fs + Fk)/2 as given by eq 6. The reason for this can
Figure 10. Critical velocity as a function of slider spring constant K for OMCTS between mica surfaces. Load between surfaces is held constant at 24.5 mN. The negative correlation is consistent with previous experimental observations and theoretical predictions. The power law fit is from eq 417 and the logarithmic fit is according to eq 2.
be understood from the phase transition model: in the overdamped regime, the freezing process is relatively rapid compared to the stick times, so the static friction measured is close to the saturation (equilibrium) value. In contrast, the slow molecular processes associated with the slip (slower than the mechanical response time) allow the measured friction F to equal the intrinsic friction F. Effects of System Inertia. An experimental study of the effects of the slider spring constant on the critical velocity was done in the underdamped regime with a 20 Å layer of OMCTS (Figure 2) between two mica surfaces. Preliminary results (Figure 10) indicate the same negative correlation between spring constant and critical velocity as previously observed and theoretically predicted (refs 3-5; see also Table 1). Conclusion Recent experimental and theoretical progress is finally allowing the important but highly complex phenomenon of stick-slip sliding to be understood. The experimental results presented of sliding with liquid lubricants in a number of systems involving smooth surfaces are consistent with velocity and time dependent phase transition models for stick-slip friction (referred to as “rate and state” dependent models). For these idealized systems there was no need to invoke a distance-dependent model, nor the classical model based on a negative friction forcevelocity profile. Future experiments and theory on a larger variety of model and real systems will give a more complete molecular picture of friction mechanisms and will ultimately allow control of stick-slip in both laboratory and practical situations. Acknowledgment. The authors thank Jean Carlson for enlightening discussions. This research was supported by the Department of Energy (DOE) under Grant DEFG03-87ER45331, although this support does not constitute an endorsement by DOE of the views expressed in this article. LA950896Z
