PRL 110, 248301 (2013)
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PHYSICAL REVIEW LETTERS
Onset of Sliding in Amorphous Films Triggered by High-Frequency Oscillatory Shear J. Le´opolde`s,1,* G. Conrad,1 and X. Jia1,2,† 1
Universite´ Paris Est, LPMDI, 5 Bd Descartes, 77454 Marne-la-Valle´e Cedex 2, France Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, 1 rue Jussieu 75005 Paris, France (Received 23 January 2013; published 11 June 2013)
2
We investigate the change of the static friction threshold of weakly adhesive amorphous interfaces in the presence of the shear ultrasonic oscillation. Prior to sliding, a softening of the shear interfacial stiffness is observed under either static or high-amplitude oscillatory shear. We find that the nonlinear shear ultrasound, regardless of its polarization, triggers the macroscopic sliding at these interfaces far below the static threshold. Such unjamming transition is due to the vibration-induced decrease of the apparent coefficient of static friction, which provides a mechanism for understanding the reduction of the yielding threshold of granular media by the acoustic fluidization. DOI: 10.1103/PhysRevLett.110.248301
PACS numbers: 83.60.La, 43.35.+d, 62.20.Qp, 68.35.Ja
The stick-slip motion between solids is associated with the transition of the amorphous interfacial layer from a static solid state to a sliding fluid state under shear load [1–3]. This transition plays an essential role in a wide range of natural processes and technological applications involving amorphous systems such as molecular thin films, glasses, granular materials, and seismic fault gouges [3–9]. The threshold rheology at the wide range of length scales could be described with the jamming phase diagram [10] and rationalized by various rate- and state-dependent constitutive laws [5,11,12]. For solid friction, the elastic dissipation occurs within the amorphous thin films on the nanometer scale via mechanical instabilities. These flips primarily affect a small cluster of a few molecular units, termed shear transformation zones (STZs) [13], and are activated by a thermal noise displaying a logarithmic rate dependence [3,14]. Moreover, a dynamical noise generated by the flip of a STZ, i.e., acoustic emission random in time and space, may act in parallel with the thermal one on the other STZs and trigger avalanches of correlated flips [3,15]. These vibrational effects can be modeled in terms of an effective temperature [16]. In athermal amorphous systems such as granular media, the experiments have clearly shown that externally applied vibrations significantly reduce the yielding stress s [17,18] and the angle of avalanche s [19]. However, it still remains unclear whether these effects are due to a collective effect [8,20], a reduction of the normal stress by the acoustic pressure so that slip of granular media can occur at low shear stress [21], or a vibrationinduced decrease of the static friction coefficient between solid grains. The frictional rheology is controlled by confined thin films, e.g., obtained by the surface grafting of nanometric organic films. Oscillatory rheological measurements performed on various thin films [4] showed a linear viscoelastic response for < s ; however, for > s the elastic modulus suddenly drops below the viscous one due to the 0031-9007=13=110(24)=248301(5)
onset of sliding. To study precisely the precursor behaviour before the macroscopic sliding, Bureau, Baumberger, and Caroli have measured the displacement response of a slider at a multicontact interface submitted to a biased oscillating shear force of low frequency f < 1 kHz as compared to the eigenfrequency f0 of the slider interface [22]. In this quasistatic regime, the oscillating force Fac is added to the static shear Fdc . When the maximum shear force F ¼ Fdc þ Fac is close below the static threshold Fs , the slider undergoes incipient creep. If Fac is such that F Fs , the slider undergoes abrupt accelerating motion, i.e., triggered sliding. These observations raise another important question. Indeed, the transition from the static state to the sliding state occurs over a typical slip distance m; for low velocity sliding ( mm=s), this corresponds to a characteristic time ms [3,8]. What would be the role, if any, of a high-frequency oscillation f f0 in the triggering of macroscopic sliding? Addressing these issues may be of fundamental interest for understanding the rheology of jammed granular media under vibration [17–19], but also the fault gouge weakening by the acoustic fluidization [21]. In this Letter, we monitor the change of the static friction threshold of weakly adhered interfacial films, in the presence of the high-frequency shear oscillation. The shear ultrasound is used here both as a nondestructive probe and a controlled pump. We observe that the shear stiffness is weakened by either static or high-amplitude oscillatory shear before sliding. Our main finding is the onset of sliding, triggered well below the static threshold by nonlinear shear ultrasound, whatever the direction of polarization. Beyond the nonlinear behavior shown in [23], this result points to the important role of an effective temperature played by the high-frequency shear oscillations in a jamming transition diagram. Experiments.—The experimental setup is shown Fig. 1(a), which combines static shear and ultrasonic measurements of interfacial films. The sphere-plane contact
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PRL 110, 248301 (2013)
PHYSICAL REVIEW LETTERS
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FIG. 1 (color online). (a) The cell for shear stiffness measurements is mounted on an inclined plane. (b) Side view of the interfacial film between the probe and the shear quartz resonator covered with electrode.
geometry [Fig. 1(b)] is composed of three equidistant steel beads of radius 1 mm clamped in a thin steel disk, referred to as probe, and a shear quartz resonator coated with a self assembled monolayer of undecanethiol or mercaptoundecanoic acid, providing, respectively, the low and high adhesion to the steel beads. Before each measurement, the clamped beads are slightly polished with diamond paste, followed by water rinsing and air drying, and let sit at ambient condition (17 C, 50% RH) for 15 min. The quartz is cleaned with a ‘‘piranha’’ solution for 5 min followed by thorough rinsing with water, then immersed for 24 h in a 1 mmol solution of thiol. The samples are rinsed with milli-Q water, dried with a flow of nitrogen, and stored under vacuum until use. For making the shear experiment, the homemade cell is mounted on an inclined plane [Fig. 1(a)] whose angle is controlled by means of a rack-and-pinion at 0:5 with a protractor. The probe is gently placed on the quartz (at ¼ 0 ) such that its axis coincides with the centre of the quartz. To obtain the reproducible measurements, a high-amplitude ultrasonic oscillation is applied to shear the interface for 10 s [22]; the contact is left further ageing for 10 min. We determine the static friction coefficient s by measuring the angle of sliding s and the threshold force Fs ¼ W sins for various weights of the probe W ¼ 2:3–21 mN. The results are correctly fitted by a modified Coulomb’s law Fs ¼ F0 þ s W coss where F0 is the zero-load threshold. F0 1 mN is not significantly affected by the nature of the monolayer, while a larger value of the friction coefficient is obtained for the adhesive substrate (COOH) s 0:30, compared to 0.15 for the less adhesive one (CH3). Our data are consistent with those obtained by others [24]. The ultrasonic measurement is realized by bringing the probe in contact with the adsorbed surface of the quartz resonator, which shifts the resonance peak towards higher frequency. The increased frequency shift f is related to the shear stiffness of the contact by kT ð¼ Fac =Uac Þ ¼ f where ¼ 4ðMKÞ1=2 , M ¼ 3:5 105 kg and K ¼ 3 1010 N=m are the effective mass and stiffness of the quartz, Fac and Uac are its oscillating shear force and displacement [23]. Figure 2(a) shows the typical elastic response of the adhesive interface (COOH) as a function of the oscillatory amplitude Uac at an inclination well below the angle of sliding s 34:5 . Two distinct regimes [23] can be
FIG. 2 (color online). (a) Decrease of the shear stiffness for the COOH interface under various angles and W ¼ 6:7 mN. Inset: experimental protocol for measuring f at ¼ 19 . Each color (ten measurements) corresponds to a given oscillatory Uac where the last 5 points are averaged for a data point shown in the main panel. At Umax 5 nm the delayed slip occurs and the probe slides. (b) Schematic illustration of a Hertzian contact of radius aH and a microslip annulus of width aH c induced by shear.
identified before the macroscopic sliding: a linear viscoelastic response at low amplitude (Uac < 1 nm) and a nonlinear frictional regime at high-amplitude oscillation (Uac > 2 nm), accompanied with an important decrease of the interfacial stiffness. In the linear regime, when increasing the inclination angle or shear load, we observe a significant softening of the stiffness of about 5% [Fig. 2(a)], both with COOH and CH3 interfaces before sliding [Fig. 3(a)]. Furthermore, when the macroscopic sliding occurs at the angle of sliding s the resonance peak exhibits abrupt erratic shifts, providing a supplementary sensitive measurement of s . The main finding is obtained in the nonlinear regime: at ¼ 19 far below the static threshold s , a high-amplitude ultrasonic oscillation (U > 5 nm) provokes a discontinuous shift in f and triggers the macroscopic sliding of the probe. Before sliding, a creeplike softening is observed [inset of Fig. 2(a)], pointing to a delayed slip induced by shear ultrasound.
FIG. 3 (color online). (a) Softening of the shear stiffness for COOH (red) and CH3 interfaces (blue) as a function of , extracted from measurements similar to Fig. 2(a). The average angle of sliding is indicated by the vertical lines. Various symbols correspond to different experiments. (b) Schematic illustration of a noncohesive (Hertzian) contact and cohesive (JKR) contact.
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Discussion.—We seek to understand the combined shear and ultrasonic measurements before sliding [Fig. 2(a)]. At low shear displacement U < Uc 1 nm, the interfacial layer is pinned and responds to shear force elastically as FðUÞ ¼ kT0 U with kT0 a constant stiffness. Beyond a certain threshold U > Uc , the contact zone flows plastically due to the structure change within the nanometric amorphous film. In the sphere-plane geometry, this plastic flow may initiate by a fracture at the edge of the adhesive contact [25] [Fig. 3(b)], along with the growth of a microslip annulus towards the center [Fig. 2(b)] [26]. To account for the interplay between friction and adhesion, we use here a modified friction model F ¼ F0 ðUÞ þ FM ðUÞ where the Mindlin friction force is FM ðUÞ ¼ s W cosf1 ½1 16G aH U=ð3s W cosÞ 3=2 g with G (12 GPa) the reduced shear modulus and aH the radius of the Hertzian contact area [23,26]. F0 ðUÞ kT0 SðUÞU is an effective force associated with the pinned sites or adhesive area; SðUÞ ( 1 for U < Uc ) is a parameter depending on the adhesive area and decreases with increasing U (see below). At the macroscopic sliding Umax , FM ¼ s W cos leads to the Coulomb-like law Fs ¼ F0 ðUmax Þ þ s W cos. Let us examine the measured shear stiffness kT [Fig. 2(a)] in terms of the contact area. Two types of approaches appear available for these weakly adhered interfaces. (i) One relates kT to a bonded contact of kT a2E G=h, where h (1 nm) and G (10 MPa) are the thickness and elastic modulus of the interfacial film. aE aJKR is the radius of the adhesive area closely predicted by the Johnson-KendallRoberts (JKR) model [Fig. 3(b)] [23,25] and can be reduced by either static or high-amplitude oscillatory shear. (ii) Alternatively, kT can be derived from a noncohesive sphere-plane contact using the Hertz-Mindlin model via the static friction s [27] kT kM ½1 Fac =ð6s WÞ (at ¼ 0 ), where kM ¼ 8G aH ð kT0 Þ is the linear shear stiffness at vanishing Fac . This model does not account for the linear oscillatory response originating from the adhesion, but describes conveniently the decrease of kT in the nonlinear regime, relating kT =kT Fac =s W to the growth of the microslip annulus of radius c ¼ aH ½1 Fac =ðs WÞ 1=3 [Fig. 2(b)], indicated by kT =kT ðaH cÞ=aH [26]. These two apparently disconnected approaches predict a similar decrease of kT induced by the high-amplitude Fac if the reduction of the adhesive area a2E is the same as that of the nonslip area inside a2H . A softening of kT of about 5% induced by static shear Fdc ¼ W sin is observed before sliding using the lowamplitude ultrasound [Fig. 3(a)]. It likely arises from the reduction of a2E initiated at the edge via the opening crack. According to the adhesion models [25], a2E may be reduced under shear from a2JKR to a2H [Fig. 3(b)] prior to failure, by developing a fractured zone similar to the microslip during the incipient stage of sliding friction. This would define an upper limit for the decrease of SðUmax Þ a2H =a2JKR 40% and for the softening kT =kT 2aE =aE 50% where
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we use aH 3:2 m for a normal load W=3 2:2 mN and aJKR 5 m estimated from the interfacial energy of the adhesive layers [23]. Such decrease of kT larger than 50% is observable at these weakly adhered brittle interfaces (aJKR aH ) before failure [Fig. 2(a)] if applying precisely ramped high-amplitude oscillatory shear [3], supporting thus the above picture. Moreover, the zeroload threshold deduced from the ultrasonic measurement F0 ðUmax Þ kT0 SðUmax ÞUmax 2 mN agrees with those obtained from the sliding experiment F0 1 mN. The complex interplay between adhesion and friction requires further study to describe more precisely the softening regime [6,25]. We now interpret the triggering of sliding by the nonlinear shear ultrasound, far below the static threshold Fs [Fig. 2(a)] using the friction approach. As mentioned above, the onset of sliding has been previously observed for a multicontact interface below the angle of sliding s , triggered by a low-frequency (f=f0 101 ) oscillating pffiffiffiffiffiffiffiffiffiffiffiffi force Fac parallel to Fdc [22]. Here f0 ¼ kT =m=2 1 kHz, kT is the interfacial stiffness and m the mass of the slider. When Fac is ramped so that F ¼ Fdc þ Fac precisely approaches the threshold Fs , a self-accelerated unlimited slip occurs reaching the averaged velocity about 0:1 mm=s. We use this range of velocity as a criterion to define the triggering of sliding. The bifurcation between the jamming creep regime (F Fs ) and the sliding regime (F Fs ) is well described by a rate- and state-dependent friction law [3,8,9]. Unlike the simple Coulomb failure law, the process of thermal activation included in the Rice-Ruina model implies a creep prior to sliding, as observed in the above experiments. In the ultrasonic measurements, the frequency of the oscillating force Fac is much higher than the eigenfrequency of the slider interface (f=f0 102 ). Unlike the above quasistatic regime, Fac shall not provoke any macroscopic sliding motion of the slider in the high-frequency limit due to the inertial effect [28]. Here we propose a new scenario of the triggering of sliding by the nonlinear shear ultrasound. As stated above, the high-amplitude oscillation Fac most significantly reduces the nonslip area s from the initial a2H to c2 via the growth of the microslip annulus. Accordingly, such high-frequency oscillation which works as a lubrication decreases the static threshold Fs ¼ s s and triggers the macroscopic sliding under a static shear Fdc below the shear threshold Fs . Figure 4(a) displays the reduced static threshold sins ¼ Fs =W as a function of the oscillatory shear Fac . We additionally notice that the triggering of sliding is independent of the polarization of shear oscillation relative to the sliding direction. Such behavior could be captured by the Mindlin friction model,
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sins = sins ¼ Fs =Fs c2 =a2H 1 ð2=3ÞFac =ðs W coss Þ:
(1)
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PHYSICAL REVIEW LETTERS
FIG. 4 (color online). (a) Reduced static threshold versus the oscillating force for CH3 (blue symbols) and COOH interfaces (red symbols). Circles and diamonds correspond to two sets of experiments for a given direction of shear oscillation ( ¼ 0 ), stars are related to those where the shear resonator is rotated by ¼ 45 for CH3 and by 90 for COOH films. The straight lines correspond to best fits with Eq. (1). Inset: angles of sliding of a steel cylinder versus direction of polarization [Fig. 1(a)], measured at low-amplitude (black circles) and high-amplitude shear ultrasound (green points). (b) Normalized vibrational energy necessary for the triggering of sliding (see the text). The solid curve delimits the jammed and flowing states. Inset: jamming phase diagram [10].
Comparison to the data infers s 2 for COOH and s 1 for CH3 interfaces, larger than those obtained from the angle of sliding but consistent with the previous data [23]. Furthermore, we have triggered a similar sliding at a multicontact interface between a steel cylinder of diameter 5 mm and the quartz. As shown in Fig. 4(a) (inset), we observe the decrease of s at the highest Uac > 5 nm polarized in the different azimutal angle . This result suggests that when the local static threshold is reached [22], the nonslip contact area of the interface P ¼ N a 2 (N is the number of asperities and a the s average contact radius) would fluidize by progressive sliding of asperities, reducing Fs . Considering the scalar nature of the fluidization effect insensitive to the oscillation polarization, we examine the decrease of the static threshold versus the vibrational 2 . Here A is the ratio of the vibratenergy Ev ðA=2ÞKUac ing surface area to the area of contact. In Fig. 4(b), the solid curve described by Ev =B ½1 ðsins = sins Þ 2 with B ¼ 9AK2s W 2 =8k2 1010 kT (with kT the thermal energy) delimits roughly the jammed and sliding states of the frictional system, as expected in a jamming diagram [10] where Ev would play the role of an effective temperature Teff [inset of Fig 4(b)]. It is important to notice that the oscillation period 0:2 s used in this work is much smaller than the characteristic relaxation time >1 ms in the thin interfacial layer [29]. This high-frequency shear ( 5 MHz) allows maintaining the interface in the fluidized state within the microslip annulus (or randomly distributed patches [16]) and prevents from healing, unlike the low-frequency shear (< 0:1 kHz) [22]. Likewise, at a given amplitude, oscillations of normal force Wac of lower
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frequency (< 5 kHz) [29] whose effect is further reduced by Wac ¼ Fac =s 5Fac , would be less efficient than the present shear oscillations for fluidizing the nonslip contact area. More work is needed to account for high-frequency oscillation or Teff in the rate and state models [3]. Finally, the proposed scenario for the onset of sliding triggered by nonlinear shear ultrasound is helpful for understanding the reduction of the yielding threshold of granular media by the acoustic fluidization [17–19,21]. The applied vibration could generate high-frequency acoustic emission by the rearrangement of grains and the rupture of asperities [30]. As stated above, the highfrequency oscillatory shear reduces the apparent coefficient of static friction between grains and consequently lowers the static friction coefficient of granular layers [31] which can slip at lower shear stress. The necessary energy for the rearrangement of grains by sliding is 2 orders of magnitude smaller than the energy barrier by jumping, Ej Wh0 1012 kT where h0 1 m is the height of a surface asperity. In terms of acoustic fluidization, the mechanism discussed here offers an alternative to that relying on the balance of the overburden via the acoustic pressure, which needs an unusually high acoustic energy [21]. In conclusion, the shear stiffness of weakly adhered interfaces is significantly weakened before sliding, either by static shear or by high-amplitude oscillatory shear, due to the opening crack initiated at the edge and the development of microslip zones. The present measurements would enable us to bridge two distinct approaches for describing the failure, namely, fracture and microslip propagation. The onset of sliding triggered far below the static threshold by the nonlinear shear ultrasound is due to the fluidization of the nonslip contact area, reducing the apparent coefficient of friction. The intensity rather than the polarisation of the shear ultrasound matters in such unjamming transition from static to sliding friction, suggesting its role as an effective temperature. This work would provide an alternative mechanism to understand the granular fault weakening by the acoustic fluidization via high-frequency scattered shear waves [21,27] and the effect of the dynamic noise in flowing systems [15,19]. We thank T. Baumberger and L. Bureau for helpful discussions and the reviewers for the relevant comments.
*[email protected] † [email protected] [1] P. A. Thompson and M. Robbins, Science 250, 792 (1990). [2] B. N. J. Persson, Sliding Friction: Physical Principles and Applications (Springer, New York, 1998). [3] T. Baumberger and C. Caroli, Adv. Phys. 55, 279 (2006). [4] G. Reiter, A. L. Demirel, and S. Granick, Science 263, 1741 (1994). [5] F. A. Spaepen, Acta Metall. 23, 615 (1975).
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[6] O. Ben-David, S. M. Rubinstein, and J. Fineberg, Nature (London) 463, 76 (2010). [7] S. Nasuno, A. Kudrolli, and J. Gollub, Phys. Rev. Lett. 79, 949 (1997). [8] C. Marone, Annu. Rev. Earth Planet Sci. 26, 643 (1998). [9] C. H. Scholz, Mechanics of Earthquake and Faulting (Cambridge University Press, Cambridge, England, 2002), 2nd ed. [10] A. Liu and S. Nagel, Nature (London) 396, 21 (1998). [11] J. R. Rice and A. L. Ruina, J. Appl. Mech. 50, 343 (1983). [12] P. Sollich, Phys. Rev. E 58, 738 (1998); P. Coussot, Q. D. Ngyyen, H. T. Huynh, and D. Bonn, Phys. Rev. Lett. 88, 175501 (2002). [13] M. L. Falk and J. S. Langer, Phys. Rev. E 57, 7192 (1998). [14] B. J. Briscoe and D. C. B. Evans, Proc. R. Soc. A 380, 389 (1982). [15] A. Lemaıˆtre and C. Caroli, Phys. Rev. Lett. 103, 065501 (2009). [16] B. N. J. Persson, Phys. Rev. Lett. 71, 1212 (1993); T. Haxton and A. J. Liu, Phys. Rev. Lett. 99, 195701 (2007). [17] J. A. Dijksman, G. H. Wortel, L. T. H. van Dellen, O. Dauchot, and M. van Hecke, Phys. Rev. Lett. 107, 108303 (2011). [18] P. Johnson, H. Savage, M. Knuth, J. Gomberg, and C. Marone, Nature (London) 451, 57 (2008); P. Johnson and X. Jia, Nature (London) 437, 871 (2005). [19] H. M. Jaeger, C. H. Liu, and S. R. Nagel, Phys. Rev. Lett. 62, 40 (1989).
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[20] R. R. Hartley and R. P. Behringer, Nature (London) 421, 928 (2003). [21] H. J. Melosh, Nature (London) 379, 601 (1996); D. Sornette and A. Sornette, Bull. Seismol. Soc. Am. 90, 781 (2000). [22] L. Bureau, T. Baumberger, and C. Caroli, Phys. Rev. E 64, 031502 (2001). [23] J. Le´opolde`s and X. Jia, Phys. Rev. Lett. 105, 2666101 (2010). [24] B. D. Booth, S. Vilt, C. McCabe, and G. K. Jennings, Langmuir 25, 9995 (2009). [25] A. R. Savkoor and G. A. D. Briggs, Proc. R. Soc. A 356, 103 (1977); K. L. Johnson, Proc. R. Soc. A 453, 163 (1997). [26] K. L Johnson, Contact Mechanics (Cambridge University Press, New York, 1985). [27] X. Jia, T. Brunet, and J. Laurent, Phys. Rev. E 84, 020301 (R) (2011). [28] T. Baumberger, L. Bureau, M. Busson, E. Falcon, and B. Perrin, Rev. Sci. Instrum. 69, 2416 (1998). [29] M. Heubeerger, C. Drummond, and J. Israelachivilli, J. Phys. Chem. B 102, 5038 (1998); M. Urbakh, J. Klafter, D. Gourdon, and J. Israelachvili, Nature (London) 430, 525 (2004). [30] G. C. McLaskey and S. D. Glaser, Geophys. Res. Lett. 38, L12302 (2011). [31] H. M. Jaeger, C. H. Liu, S. R. Nagel, and T. A. Witten, Europhys. Lett. 11, 619 (1990).
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PHYSICAL REVIEW LETTERS
Onset of Sliding in Amorphous Films Triggered by High-Frequency Oscillatory Shear J. Le´opolde`s,1,* G. Conrad,1 and X. Jia1,2,† 1
Universite´ Paris Est, LPMDI, 5 Bd Descartes, 77454 Marne-la-Valle´e Cedex 2, France Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, 1 rue Jussieu 75005 Paris, France (Received 23 January 2013; published 11 June 2013)
2
We investigate the change of the static friction threshold of weakly adhesive amorphous interfaces in the presence of the shear ultrasonic oscillation. Prior to sliding, a softening of the shear interfacial stiffness is observed under either static or high-amplitude oscillatory shear. We find that the nonlinear shear ultrasound, regardless of its polarization, triggers the macroscopic sliding at these interfaces far below the static threshold. Such unjamming transition is due to the vibration-induced decrease of the apparent coefficient of static friction, which provides a mechanism for understanding the reduction of the yielding threshold of granular media by the acoustic fluidization. DOI: 10.1103/PhysRevLett.110.248301
PACS numbers: 83.60.La, 43.35.+d, 62.20.Qp, 68.35.Ja
The stick-slip motion between solids is associated with the transition of the amorphous interfacial layer from a static solid state to a sliding fluid state under shear load [1–3]. This transition plays an essential role in a wide range of natural processes and technological applications involving amorphous systems such as molecular thin films, glasses, granular materials, and seismic fault gouges [3–9]. The threshold rheology at the wide range of length scales could be described with the jamming phase diagram [10] and rationalized by various rate- and state-dependent constitutive laws [5,11,12]. For solid friction, the elastic dissipation occurs within the amorphous thin films on the nanometer scale via mechanical instabilities. These flips primarily affect a small cluster of a few molecular units, termed shear transformation zones (STZs) [13], and are activated by a thermal noise displaying a logarithmic rate dependence [3,14]. Moreover, a dynamical noise generated by the flip of a STZ, i.e., acoustic emission random in time and space, may act in parallel with the thermal one on the other STZs and trigger avalanches of correlated flips [3,15]. These vibrational effects can be modeled in terms of an effective temperature [16]. In athermal amorphous systems such as granular media, the experiments have clearly shown that externally applied vibrations significantly reduce the yielding stress s [17,18] and the angle of avalanche s [19]. However, it still remains unclear whether these effects are due to a collective effect [8,20], a reduction of the normal stress by the acoustic pressure so that slip of granular media can occur at low shear stress [21], or a vibrationinduced decrease of the static friction coefficient between solid grains. The frictional rheology is controlled by confined thin films, e.g., obtained by the surface grafting of nanometric organic films. Oscillatory rheological measurements performed on various thin films [4] showed a linear viscoelastic response for < s ; however, for > s the elastic modulus suddenly drops below the viscous one due to the 0031-9007=13=110(24)=248301(5)
onset of sliding. To study precisely the precursor behaviour before the macroscopic sliding, Bureau, Baumberger, and Caroli have measured the displacement response of a slider at a multicontact interface submitted to a biased oscillating shear force of low frequency f < 1 kHz as compared to the eigenfrequency f0 of the slider interface [22]. In this quasistatic regime, the oscillating force Fac is added to the static shear Fdc . When the maximum shear force F ¼ Fdc þ Fac is close below the static threshold Fs , the slider undergoes incipient creep. If Fac is such that F Fs , the slider undergoes abrupt accelerating motion, i.e., triggered sliding. These observations raise another important question. Indeed, the transition from the static state to the sliding state occurs over a typical slip distance m; for low velocity sliding ( mm=s), this corresponds to a characteristic time ms [3,8]. What would be the role, if any, of a high-frequency oscillation f f0 in the triggering of macroscopic sliding? Addressing these issues may be of fundamental interest for understanding the rheology of jammed granular media under vibration [17–19], but also the fault gouge weakening by the acoustic fluidization [21]. In this Letter, we monitor the change of the static friction threshold of weakly adhered interfacial films, in the presence of the high-frequency shear oscillation. The shear ultrasound is used here both as a nondestructive probe and a controlled pump. We observe that the shear stiffness is weakened by either static or high-amplitude oscillatory shear before sliding. Our main finding is the onset of sliding, triggered well below the static threshold by nonlinear shear ultrasound, whatever the direction of polarization. Beyond the nonlinear behavior shown in [23], this result points to the important role of an effective temperature played by the high-frequency shear oscillations in a jamming transition diagram. Experiments.—The experimental setup is shown Fig. 1(a), which combines static shear and ultrasonic measurements of interfacial films. The sphere-plane contact
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FIG. 1 (color online). (a) The cell for shear stiffness measurements is mounted on an inclined plane. (b) Side view of the interfacial film between the probe and the shear quartz resonator covered with electrode.
geometry [Fig. 1(b)] is composed of three equidistant steel beads of radius 1 mm clamped in a thin steel disk, referred to as probe, and a shear quartz resonator coated with a self assembled monolayer of undecanethiol or mercaptoundecanoic acid, providing, respectively, the low and high adhesion to the steel beads. Before each measurement, the clamped beads are slightly polished with diamond paste, followed by water rinsing and air drying, and let sit at ambient condition (17 C, 50% RH) for 15 min. The quartz is cleaned with a ‘‘piranha’’ solution for 5 min followed by thorough rinsing with water, then immersed for 24 h in a 1 mmol solution of thiol. The samples are rinsed with milli-Q water, dried with a flow of nitrogen, and stored under vacuum until use. For making the shear experiment, the homemade cell is mounted on an inclined plane [Fig. 1(a)] whose angle is controlled by means of a rack-and-pinion at 0:5 with a protractor. The probe is gently placed on the quartz (at ¼ 0 ) such that its axis coincides with the centre of the quartz. To obtain the reproducible measurements, a high-amplitude ultrasonic oscillation is applied to shear the interface for 10 s [22]; the contact is left further ageing for 10 min. We determine the static friction coefficient s by measuring the angle of sliding s and the threshold force Fs ¼ W sins for various weights of the probe W ¼ 2:3–21 mN. The results are correctly fitted by a modified Coulomb’s law Fs ¼ F0 þ s W coss where F0 is the zero-load threshold. F0 1 mN is not significantly affected by the nature of the monolayer, while a larger value of the friction coefficient is obtained for the adhesive substrate (COOH) s 0:30, compared to 0.15 for the less adhesive one (CH3). Our data are consistent with those obtained by others [24]. The ultrasonic measurement is realized by bringing the probe in contact with the adsorbed surface of the quartz resonator, which shifts the resonance peak towards higher frequency. The increased frequency shift f is related to the shear stiffness of the contact by kT ð¼ Fac =Uac Þ ¼ f where ¼ 4ðMKÞ1=2 , M ¼ 3:5 105 kg and K ¼ 3 1010 N=m are the effective mass and stiffness of the quartz, Fac and Uac are its oscillating shear force and displacement [23]. Figure 2(a) shows the typical elastic response of the adhesive interface (COOH) as a function of the oscillatory amplitude Uac at an inclination well below the angle of sliding s 34:5 . Two distinct regimes [23] can be
FIG. 2 (color online). (a) Decrease of the shear stiffness for the COOH interface under various angles and W ¼ 6:7 mN. Inset: experimental protocol for measuring f at ¼ 19 . Each color (ten measurements) corresponds to a given oscillatory Uac where the last 5 points are averaged for a data point shown in the main panel. At Umax 5 nm the delayed slip occurs and the probe slides. (b) Schematic illustration of a Hertzian contact of radius aH and a microslip annulus of width aH c induced by shear.
identified before the macroscopic sliding: a linear viscoelastic response at low amplitude (Uac < 1 nm) and a nonlinear frictional regime at high-amplitude oscillation (Uac > 2 nm), accompanied with an important decrease of the interfacial stiffness. In the linear regime, when increasing the inclination angle or shear load, we observe a significant softening of the stiffness of about 5% [Fig. 2(a)], both with COOH and CH3 interfaces before sliding [Fig. 3(a)]. Furthermore, when the macroscopic sliding occurs at the angle of sliding s the resonance peak exhibits abrupt erratic shifts, providing a supplementary sensitive measurement of s . The main finding is obtained in the nonlinear regime: at ¼ 19 far below the static threshold s , a high-amplitude ultrasonic oscillation (U > 5 nm) provokes a discontinuous shift in f and triggers the macroscopic sliding of the probe. Before sliding, a creeplike softening is observed [inset of Fig. 2(a)], pointing to a delayed slip induced by shear ultrasound.
FIG. 3 (color online). (a) Softening of the shear stiffness for COOH (red) and CH3 interfaces (blue) as a function of , extracted from measurements similar to Fig. 2(a). The average angle of sliding is indicated by the vertical lines. Various symbols correspond to different experiments. (b) Schematic illustration of a noncohesive (Hertzian) contact and cohesive (JKR) contact.
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Discussion.—We seek to understand the combined shear and ultrasonic measurements before sliding [Fig. 2(a)]. At low shear displacement U < Uc 1 nm, the interfacial layer is pinned and responds to shear force elastically as FðUÞ ¼ kT0 U with kT0 a constant stiffness. Beyond a certain threshold U > Uc , the contact zone flows plastically due to the structure change within the nanometric amorphous film. In the sphere-plane geometry, this plastic flow may initiate by a fracture at the edge of the adhesive contact [25] [Fig. 3(b)], along with the growth of a microslip annulus towards the center [Fig. 2(b)] [26]. To account for the interplay between friction and adhesion, we use here a modified friction model F ¼ F0 ðUÞ þ FM ðUÞ where the Mindlin friction force is FM ðUÞ ¼ s W cosf1 ½1 16G aH U=ð3s W cosÞ 3=2 g with G (12 GPa) the reduced shear modulus and aH the radius of the Hertzian contact area [23,26]. F0 ðUÞ kT0 SðUÞU is an effective force associated with the pinned sites or adhesive area; SðUÞ ( 1 for U < Uc ) is a parameter depending on the adhesive area and decreases with increasing U (see below). At the macroscopic sliding Umax , FM ¼ s W cos leads to the Coulomb-like law Fs ¼ F0 ðUmax Þ þ s W cos. Let us examine the measured shear stiffness kT [Fig. 2(a)] in terms of the contact area. Two types of approaches appear available for these weakly adhered interfaces. (i) One relates kT to a bonded contact of kT a2E G=h, where h (1 nm) and G (10 MPa) are the thickness and elastic modulus of the interfacial film. aE aJKR is the radius of the adhesive area closely predicted by the Johnson-KendallRoberts (JKR) model [Fig. 3(b)] [23,25] and can be reduced by either static or high-amplitude oscillatory shear. (ii) Alternatively, kT can be derived from a noncohesive sphere-plane contact using the Hertz-Mindlin model via the static friction s [27] kT kM ½1 Fac =ð6s WÞ (at ¼ 0 ), where kM ¼ 8G aH ð kT0 Þ is the linear shear stiffness at vanishing Fac . This model does not account for the linear oscillatory response originating from the adhesion, but describes conveniently the decrease of kT in the nonlinear regime, relating kT =kT Fac =s W to the growth of the microslip annulus of radius c ¼ aH ½1 Fac =ðs WÞ 1=3 [Fig. 2(b)], indicated by kT =kT ðaH cÞ=aH [26]. These two apparently disconnected approaches predict a similar decrease of kT induced by the high-amplitude Fac if the reduction of the adhesive area a2E is the same as that of the nonslip area inside a2H . A softening of kT of about 5% induced by static shear Fdc ¼ W sin is observed before sliding using the lowamplitude ultrasound [Fig. 3(a)]. It likely arises from the reduction of a2E initiated at the edge via the opening crack. According to the adhesion models [25], a2E may be reduced under shear from a2JKR to a2H [Fig. 3(b)] prior to failure, by developing a fractured zone similar to the microslip during the incipient stage of sliding friction. This would define an upper limit for the decrease of SðUmax Þ a2H =a2JKR 40% and for the softening kT =kT 2aE =aE 50% where
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we use aH 3:2 m for a normal load W=3 2:2 mN and aJKR 5 m estimated from the interfacial energy of the adhesive layers [23]. Such decrease of kT larger than 50% is observable at these weakly adhered brittle interfaces (aJKR aH ) before failure [Fig. 2(a)] if applying precisely ramped high-amplitude oscillatory shear [3], supporting thus the above picture. Moreover, the zeroload threshold deduced from the ultrasonic measurement F0 ðUmax Þ kT0 SðUmax ÞUmax 2 mN agrees with those obtained from the sliding experiment F0 1 mN. The complex interplay between adhesion and friction requires further study to describe more precisely the softening regime [6,25]. We now interpret the triggering of sliding by the nonlinear shear ultrasound, far below the static threshold Fs [Fig. 2(a)] using the friction approach. As mentioned above, the onset of sliding has been previously observed for a multicontact interface below the angle of sliding s , triggered by a low-frequency (f=f0 101 ) oscillating pffiffiffiffiffiffiffiffiffiffiffiffi force Fac parallel to Fdc [22]. Here f0 ¼ kT =m=2 1 kHz, kT is the interfacial stiffness and m the mass of the slider. When Fac is ramped so that F ¼ Fdc þ Fac precisely approaches the threshold Fs , a self-accelerated unlimited slip occurs reaching the averaged velocity about 0:1 mm=s. We use this range of velocity as a criterion to define the triggering of sliding. The bifurcation between the jamming creep regime (F Fs ) and the sliding regime (F Fs ) is well described by a rate- and state-dependent friction law [3,8,9]. Unlike the simple Coulomb failure law, the process of thermal activation included in the Rice-Ruina model implies a creep prior to sliding, as observed in the above experiments. In the ultrasonic measurements, the frequency of the oscillating force Fac is much higher than the eigenfrequency of the slider interface (f=f0 102 ). Unlike the above quasistatic regime, Fac shall not provoke any macroscopic sliding motion of the slider in the high-frequency limit due to the inertial effect [28]. Here we propose a new scenario of the triggering of sliding by the nonlinear shear ultrasound. As stated above, the high-amplitude oscillation Fac most significantly reduces the nonslip area s from the initial a2H to c2 via the growth of the microslip annulus. Accordingly, such high-frequency oscillation which works as a lubrication decreases the static threshold Fs ¼ s s and triggers the macroscopic sliding under a static shear Fdc below the shear threshold Fs . Figure 4(a) displays the reduced static threshold sins ¼ Fs =W as a function of the oscillatory shear Fac . We additionally notice that the triggering of sliding is independent of the polarization of shear oscillation relative to the sliding direction. Such behavior could be captured by the Mindlin friction model,
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sins = sins ¼ Fs =Fs c2 =a2H 1 ð2=3ÞFac =ðs W coss Þ:
(1)
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FIG. 4 (color online). (a) Reduced static threshold versus the oscillating force for CH3 (blue symbols) and COOH interfaces (red symbols). Circles and diamonds correspond to two sets of experiments for a given direction of shear oscillation ( ¼ 0 ), stars are related to those where the shear resonator is rotated by ¼ 45 for CH3 and by 90 for COOH films. The straight lines correspond to best fits with Eq. (1). Inset: angles of sliding of a steel cylinder versus direction of polarization [Fig. 1(a)], measured at low-amplitude (black circles) and high-amplitude shear ultrasound (green points). (b) Normalized vibrational energy necessary for the triggering of sliding (see the text). The solid curve delimits the jammed and flowing states. Inset: jamming phase diagram [10].
Comparison to the data infers s 2 for COOH and s 1 for CH3 interfaces, larger than those obtained from the angle of sliding but consistent with the previous data [23]. Furthermore, we have triggered a similar sliding at a multicontact interface between a steel cylinder of diameter 5 mm and the quartz. As shown in Fig. 4(a) (inset), we observe the decrease of s at the highest Uac > 5 nm polarized in the different azimutal angle . This result suggests that when the local static threshold is reached [22], the nonslip contact area of the interface P ¼ N a 2 (N is the number of asperities and a the s average contact radius) would fluidize by progressive sliding of asperities, reducing Fs . Considering the scalar nature of the fluidization effect insensitive to the oscillation polarization, we examine the decrease of the static threshold versus the vibrational 2 . Here A is the ratio of the vibratenergy Ev ðA=2ÞKUac ing surface area to the area of contact. In Fig. 4(b), the solid curve described by Ev =B ½1 ðsins = sins Þ 2 with B ¼ 9AK2s W 2 =8k2 1010 kT (with kT the thermal energy) delimits roughly the jammed and sliding states of the frictional system, as expected in a jamming diagram [10] where Ev would play the role of an effective temperature Teff [inset of Fig 4(b)]. It is important to notice that the oscillation period 0:2 s used in this work is much smaller than the characteristic relaxation time >1 ms in the thin interfacial layer [29]. This high-frequency shear ( 5 MHz) allows maintaining the interface in the fluidized state within the microslip annulus (or randomly distributed patches [16]) and prevents from healing, unlike the low-frequency shear (< 0:1 kHz) [22]. Likewise, at a given amplitude, oscillations of normal force Wac of lower
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frequency (< 5 kHz) [29] whose effect is further reduced by Wac ¼ Fac =s 5Fac , would be less efficient than the present shear oscillations for fluidizing the nonslip contact area. More work is needed to account for high-frequency oscillation or Teff in the rate and state models [3]. Finally, the proposed scenario for the onset of sliding triggered by nonlinear shear ultrasound is helpful for understanding the reduction of the yielding threshold of granular media by the acoustic fluidization [17–19,21]. The applied vibration could generate high-frequency acoustic emission by the rearrangement of grains and the rupture of asperities [30]. As stated above, the highfrequency oscillatory shear reduces the apparent coefficient of static friction between grains and consequently lowers the static friction coefficient of granular layers [31] which can slip at lower shear stress. The necessary energy for the rearrangement of grains by sliding is 2 orders of magnitude smaller than the energy barrier by jumping, Ej Wh0 1012 kT where h0 1 m is the height of a surface asperity. In terms of acoustic fluidization, the mechanism discussed here offers an alternative to that relying on the balance of the overburden via the acoustic pressure, which needs an unusually high acoustic energy [21]. In conclusion, the shear stiffness of weakly adhered interfaces is significantly weakened before sliding, either by static shear or by high-amplitude oscillatory shear, due to the opening crack initiated at the edge and the development of microslip zones. The present measurements would enable us to bridge two distinct approaches for describing the failure, namely, fracture and microslip propagation. The onset of sliding triggered far below the static threshold by the nonlinear shear ultrasound is due to the fluidization of the nonslip contact area, reducing the apparent coefficient of friction. The intensity rather than the polarisation of the shear ultrasound matters in such unjamming transition from static to sliding friction, suggesting its role as an effective temperature. This work would provide an alternative mechanism to understand the granular fault weakening by the acoustic fluidization via high-frequency scattered shear waves [21,27] and the effect of the dynamic noise in flowing systems [15,19]. We thank T. Baumberger and L. Bureau for helpful discussions and the reviewers for the relevant comments.
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