Langmuir 2001, 17, 6045-6047
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Origin of Friction Derived from Rupture Dynamics Hitoshi Suda† Department of Biological Science and Technology, Tokai University, 317 Nishino, Numazu, Shizuoka 410-0321, Japan Received April 30, 2001. In Final Form: July 24, 2001 In the field of friction, pull-off force (rupture or unbinding force) required to break a bond has been thought to be independent of the necessary time, which is termed the bond lifetime, to dissociate the bond. However, the author presents here that friction forces are equal to pull-off forces and that they are revealed as an exponential function of the bond lifetime through rupture dynamics. The result of rupture dynamics that expresses essentially stochastic behavior was employed here as a tool to connect the atomic friction theory of Tomlinson and a modified adhesion model of Israelachvili that was first developed by Bowden and Tabor.
Ff ) ScA
I. Introduction Friction between dry surfaces results from an energydissipating process at the interface of sliding contact bodies. Classical fundamental laws of friction are sometimes inconsistent with recent experimental results at clean surfaces or atomic level. For example, as the third basic law of friction, kinetic friction is independent of sliding velocity. However, according to frictional studies by using a friction force microscopy (FFM),1 a surface forces apparatus (SFA),2,3 and Bristol boards,4 friction depends on velocity. By use of FFM in an ultrahigh vacuum (UHV) chamber and a SFA in a controlled atmosphere, in the present high technology one can measure frictional characteristics while maintaining the structure of a clean surface or without wear. According to the classical second law of friction, the friction force is proportional to the normal load, N0. If two surfaces adhere to each other, then a finite friction force can arise even in the absence of any weight or externally applied compressive load. This is because the two surfaces now experience an effective additional load L0 due to the attractive or adhesion forces between them. Thus, between two adhering surfaces, the total friction force (F) can be phenomenologically expressed 5,6 as
F ) µ(L0 + N0) ) Ff + µN0
(1)
where Ff ) µL0 is a constant arising from the friction force due to adhesion and µ ) dF/dN0 is the normal friction coefficient. Bowden and Tabor 6,7 extended the laws of friction to the case where the friction is dominated by adhesive contacts. Their adhesion model is simple in principle and postulates that the friction (Ff) arises from the forces required to shear the adhesive junctions, that is †
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(1) Gnecco, E.; Bennewitz, R.; Gyalog, T.; Loppacher, Ch.; Bammerlin, M.; Meyer, E.; Gu¨ntherodt, H.-J. Phys. Rev. Lett. 2000, 84, 1172. (2) Homola, A. M.; Israelachvili, J. N.; Gee, M. L.; McGuiggan, P. M. J. Tribol. 1989, 111, 675. (3) Yoshizawa, H.; Chen, Y. L.; Israelachvili, J. N. J. Phys. Chem. 1993, 97, 4128. (4) Heslot, F.; Baumberger, T.; Perrin, B.; Caroli, B.; Caroli, C. Phys. Rev. E 1994, 49, 4973. (5) Derjaguin, B. V. Zh. Fiz. Khim. 1934, 5, 1165. (6) Tabor, D. Adhesion and Friction Surface Physics of Materials; Blakely, J. M., Ed.; Academic Press: New York, 1975; Vol. II, pp 475529. (7) Bowden, F. P.; Tabor, D. The Friction and Lubrication of Solids; Oxford University Press: New York; 1964.
(2)
where Sc is the critical shear stress at the contacting interface and A is the real molecular contact area. It is interesting to note that in adhesive sliding, the frictional force is not proportional to the load and that in the absence of any external load both the friction and contact area are finite. This is in accordance with the Johnson-KendallRoberts (JKR) theory of adhesion or mechanical contact,8 even though their theory applies only to static, not sliding, conditions. It is generally accepted that friction is related to adhesion, but Israelachvili3,9,10 concluded that the friction force rather correlates with adhesion hysteresis during loading (or approaching, γΑ) and unloading (or separating, γR) cycles whose hysteresis corresponds to contact angle hysteresis of advancing/receding liquid. Let ∆γ ) (γR - γΑ) be the adhesion energy hysteresis per unit area, as measured during a typical loading-unloading cycle. The following relationships were induced from experimental results that were obtained by using a SFA
Ff ) (A/δ) ∆γ ) (πr2/δ)(γR - γΑ)
(3)
Sc ) Ff/A ) ∆γ/δ and ∆γ > 0 where δ is the characteristic molecular length scale, r is the contact radius, and the contact area A ) πr2. To lead these results, adhesion energies (γ) per unit area were measured in two ways. First, they were measured from the adhesion force, Fad (or the pull-off force), needed to separate two surfaces from contact, from which one obtains the unloading (or receding) adhesion energy or surface energy through the relation
γR ) -Fad/(3πR)
(4)
where R is the local radius of mica surface in the SFA. Second, they were measured by the contact radius versus load dependence, and the results were analyzed in terms of the JKR theory. In these studies, the adhesion force was treated as a constant. However, is this force truly a constant during the loading-unloading cycles? The answer is “no”, because the friction process is practically in (8) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (9) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J. Phys. Chem. 1991, 95, 10736. (10) Israelachvili, J. N.; Chen, Y. L.; Yoshizawa, H. J. Adhesion Sci. Technol. 1994, 8, 1231.
10.1021/la0106384 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/06/2001
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Langmuir, Vol. 17, No. 20, 2001
Letters
a thermodynamically nonequilibrium state, and the JKR theory is constructed under the thermodynamic equilibrium. Thus it is unreasonable to use the JKR theory to analyze the friction process or the loading-unloading cycles. It is described below that friction is closely related to the rupture force as a function of bond lifetime. Primarily the adhesion energy should be a constant, because its physical quantity is a thermodynamic one. On the other hand, the pull-off force is by no means an intrinsic constant of materials, it is rather time dependent. This feature is supported by various rupture experiments such as bulky metal-metal and glass-glass,11 a single receptorligand,12,13 a single protein-protein,14 single DNA molecules,15 and a covalent bond.16 Consequently, eq 4 is unsuitable to estimate the adhesion energy in a dynamic process. II. The Basics of Rupture Dynamics Zhurkov11 and Bell17 proposed the simplest model for the dissociation rate constant (k) or the bond lifetime (τ), which is the inverse of k. They postulated that the external applied constant force per bond (f) to break the bond acts directly along the reaction coordinate (x) on phase space to reach the value (d) at the transition state and that the force reduces the energy barrier in a linear fashion. The distance d is called the effective working distance. In this fashion, k is also assumed to increase exponentially with acting force
k ) k0 exp(f/f0) and f0 ) kBT/d
(5)
where k0 is the spontaneous dissociation rate constant under zero external load and kBT ()1/β) is the thermal energy. This model is an extension of transition state theory for reactions in gases17 in which it was proposed that intermolecular forces in a gas could be treated as a one-dimensional random walk in a potential energy well. The probability of escape depends on the height U0 (the intrinsic binding energy) of the barrier and the natural frequency of the bond in a vacuum, τ0-1 (∼1013 s-1 for atoms in a solid, 106-109 s-1 in a liquid). For a linear or parabolic energy barrier, it was shown that the bond lifetimes (τ, τ1) in the presence and absence of the external force are given by
τ1 ) 1/k0 ) τ0 exp(βU0)
(6)
τ ) 1/k ) τ0 exp[β(U0 - fd)] and
τ > τ1 where the exponential represents the probability that thermal fluctuations provide enough energy for the barrier to be surmounted (for the transition state to be reached). Here, τ1 also means the intrinsic bond lifetime of materials. (11) Zhurkov, S. N. Int. J. Fract. Mech. 1965, 1, 311. (12) Fritz, J.; Katopodis, A. G.; Kolbinger, F.; Anselmetti, D. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 12283. (13) Merkel, P.; Nassoy, P.; Leung, A.; Ritchie, K.; Evans, E. Nature 1999, 397, 50. (14) Nishizaka, T.; Seo, R.; Tadakuma, H.; Kinosita, K.; Ishiwata, S. Biophy. J. 2000, 79, 962. (15) Strunz, T.; Oroszlan, K.; Scha¨fer, R.; Gu¨ntherodt, H.-J. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 11277. (16) Grandbois, M.; Beyer, M.; Rief, M.; Schaumann, H. C.; Gaub, H. E. Science 1999, 283, 1727. (17) Bell, G. I. Science 1978, 200, 618.
On the other hand, if the acting force on the bond increases with a constant loading rate (df/dt ) uc), which is given by the pulling velocity u times the elasticity c of the cantilever, then the most probable rupture force (fmax) that is given by maximum in the distribution of yield forces depends logarithmically on the unloading rate18
fmax ) (kBT/d) ln(βucdτ1)
(7)
If we replace the term (βucd) with k, eq 7 corresponds to eq 5 or 6. This equation was derived from the reaction rate theory of Kramers.19 Since it was calculated by using Brownian dynamics, it should be noted that the observable rupture force essentially results in the stochastic process. Maximum in the distributions of yield forces arises because the force applied to the bonds increases progressively in time. Accordingly, from these results it is suggested that the pull-off forces measured by Israelachvili and his collaborators3,9,10 should not be independent of time. III. Results from Rupture Dynamics and Discussion Gnecco et al.1 showed that atomic friction (or the lateral force) increases logarithmically with sliding speed v by using a FFM. During the scanning process, the external load was kept constant at 0.44 or 0.65 nN using a feedback loop. Their experimental result was described by the following equation (eq 8 of ref 1) that was interpreted from a modified Tomlinson model20-22 based on reaction rate theory
FL(v) ) FL0 + FL1 ln(v/v1) ) FL0 + kBT/λ ln(v/v1) (8) λ ) kBT/FL1 and
1/v1 ) keffλβ/ν0 where according to their definitions FL is the lateral force, FL0 a constant, ν0 the characteristic frequency of the double well system, and keff the effective lateral spring constant of a cantilever. Sliding speed is generally defined by dividing a period L of atom-atom by a necessary time to progress during one association-dissociation cycle. Generally speaking, according to the Smoluchowski equation of coagulation, the diffusion-controlled rate coefficient katt for binding is given by katt ) (2πD/a2 + v/a) under constant speed, where D is the diffusion constant and a is the encounter radius. The time needed to break the bond is considerably longer than the necessary time to make a bond on a molecular scale, that is, usually k , katt. The necessary time to progress one step equals approximately to the time required to rupture the bonding. Therefore, sliding velocity is given by v ) dX/dt ) L (kkatt)/(k + katt) = kL ) L/τ, where X represents the coordinate on real space, and this definition can be satisfied with high precision under low velocity (
