Mechanism of friction across molecularly confined films of simple liquids

Lubrication at the Frontier / D. Dowson et al. (Editors) 1999 Elsevier Science B.V.

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Mechanism of friction across molecularly confined films of simple liquids Jacob Klein Weizmann Institute of Science, Rehovot 76100, Israel Motivated by the results of recent experiments on the sliding of smooth surfaces across confined monolayers of simple organic materials, we develop a model for friction in such systems based on a shear-melting picture. By analysing the stick-slip motion during sliding we conclude that the effective viscosity of the confined films during the slip regime is low, and that nearly all the frictional dissipation takes place at the instant of stopping when the confined films abruptly solidify. The extension of this model to more general situations is briefly considered. 1. INTRODUCTION When two solid surfaces separated by a lubricant layer are compressed, the liquid squeezes out until a thin film only a few monolayers thick remains, which resists further extrusion[i]. For a large class of liquids consisting of simple (quasi-spherical) or regular (short linear chains) molecules, where the compressed solid surfaces are sufficiently smooth, layering of the molecules occurs parallel to the surfaces[2]. Such layering is characterised by oscillating surface forces, with repulsive humps, f ,

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Recent experiments[3, 4] using a mica surface force balance (SFB) have probed in detail the manner in which the shear properties of such compressed liquid films vary as the liquid is progressively squeezed out. This has obvious implications for understanding friction and lubrication in the presence of interracial films. Here we consider the frictional sliding of the surfaces across the confined liquid layer, and discuss the nature of the frictional dissipation. 2. RESULTS AND DISCUSSION

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separated by adhesive wells, corresponding to discrete numbers of layers (generally up to some 6 - 10 molecular layers in all). A typical structural profile for the model liquid octamethylcyclotetrasiloxane (OMCTS) is shown on the left.

60

70

Force (F)-distance (D) profile between curved mica surfaces (plotted as F(D)/R where R is mean radius of

curvature) in a crossed cylinder configuration in OMCTS. The values n correspond to the number of confined monolayers (from ref. [3]).

The mica surfaces in the SFB used in these experiments were in the usual crossedcylinder geometry. Their relative configuration, together with the springs whose bending was measured to yield the normal and shear forces across the confined layer in the intersurface gap, are shown schematically in fig. 1. The top surface, mounted on a lens, is free to move in the x-direction (lateral sliding) only, and is acted on by a spring of constant K1, whose end is pulled in the xdirection as shown (the lens and mounting have a mass M1). The lower surface, mounted

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on a lens of mass M2, is free to direction only, and is acted on constant K2; the compression K2 results in the normal force surfaces.

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Figure 2 Figure 1 It was found that the effective shear viscosity of the confined films was characteristic of a f l u i d down to a well defined surface separation, corresponding to nc molecular layers, following which the film abruptly behaved in a solid-like fashion[3-5]. For layer thicknesses nc or fewer monolayers, the confined films were capable of sustaining a shear stress over macroscopic times, characteristic of a solid, while the effective shear viscosity increased at the transition ((nc +1)---, nc monolayers) by some 7 orders of magnitude or more. This is clearly an indication of a liquid-to-solid transition. When a shear force Fs was applied to the surfaces in the solid-like regime (n < nc) they would slide past each other whenever Fs exceeded a certain yield value Fs > Fsfv). The value of the yield stress depended bottibn the normal load between the surfaces and on the value of n (< nc), and once sliding commenced, it proceeded by a clear stick-slip motion, as indicated schematically in fig. 2. On pulling on the end of the spring in the x-direction the lateral tension Fs increases, as shown schematically in the figure. This is the 'stick' part of the stick-slip cycle. As the

trace - the shear stress across the film reaches the yield value, Fs = Fs(v) = Klx0, and the confined film yields. Soih'e rapid sliding of the top surface relative to the lower one then occurs, for example from point U to point V in the Fs trace in fig. 2 (the 'slip' part of the stick-slip cycle), the tension in the spring relaxes somewhat, and the film then resolidifies abruptly at the point V. As the end of the spring continues to be pulled (at velocity v) the shear stress builds up again in a new 'stick' part of the cycle, 'slip' at a critical value occurs once again, and so on. A better appreciation of the processes occuring in the confined film under shear may be obtained by a more detailed consideration, as illustrated in fig. 3, where a schematic magnification of the gap between the surfaces is indicated. Initially, fig. 3a, the confined liquid film, thickness D, is in a layered state, n layers say, between the confining surfaces (in fig. 3, n = 4). The mean normal pressure on the film is P = F/A, where F is the normal force due to compression of the spring K2 and A is the effective area of the film. When the surfaces are stationary with respect to each other, the normal forces between them are oscillatory due to the layering as illustrated above, and in the unsheared state, for n s no, the confined film is solid-like, as indicated in fig. 3a. The

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This gives the value of S]tYm)in terms of the structure of the confined and the normal oscillating force profile, and describes rather well the experimentally observed variation of the yield stress with applied pressure P and the number of confined layers n. In what follows we are interested in the processes taking place during the stick-slip cycle itself.

shear force Fs is applied to the top surface as indicated via stretching of the spring, fig. 1, (the corresponding mean shear stress is S Fs/A). At the yield point (e.g. U in fig. 2) the solid-like film abruptly melts. The corresponding critical value of the shear stress at the yield point is Scfy). The issue of shear induced melting was finalysed in detail in ref.[4] using a Lindemann-type criterion.

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2.1. Yield, liquification and dilatation At the instant of yield and liquification of the confined film, and commencement of slip between the surfaces (as at point U in fig. 2), we expect some dilation of the gap between them. It has long been known that shear of an ordered array of particles can lead to a dilation effect[6] and we believe this occurs also when the confined film undergoes shearinduced melting (when the shear force reaches the yield point Fs(y,)). Computer simulations show precisely sucn a dilation for thin films sheared between two plates[7]. The dilation may also be viewed as resulting from the density difference between the solid and the liquid (typically around 5 - 10% for a range of materials. For example, for bulk OMCTS, the material used in the experiments of ref. [3, 4], the differences are ca. 10%[8, 9]). This dilation must manifest itself in the increased separation of the surfaces, from D to D + 6 say, just as the film melts, as indicated in fig. 3. A 10% density change would

correspond to 6 of order 1 - 2/~ for D around 5 nm. Such a change in D would be difficult to observe from the motion of the optical interference fringes in the surface force balance experiments, and indeed was not observed in the experiments of ref.[4] Immediately following the yield point U the surfaces begin to move relative to each other (slip), since the film separating them is no longer rigid. Motion takes place both laterally, in the x-direction (top surface), driven by the shear force Fs, and, at the same time, normal to the surfaces, in the z-direction (lower surface), driven by the normal force F = PA (fig. 3). In our model, motion continues as long as the confined film remains liquid: when the surfaces have moved together by 6 and the separation between them returns to D, confinement-induced freezing occurs, the film solidifies and the relative motion stops abruptly (point V in fig. 2. The stick-slip cycle then begins anew). The total sliding motion Ax (fig. 3) is of order nanometers to tens of nanometers in the experiments of ref. 4, while the time 6t over which the slip occurs

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(fig. 2, inset) is around 10 -2 s. The equations describing the motion in the two directions (x and z) during slip have similar forms, consisting of an inertial term, a viscous (friction) term and a term describing the driving force.

2.2 Motion during slip The relative motion of the two surfaces while in the 'slip' regime may be described by two second order differential equations: M(d2q/dt 2) + B(dq/dt) = K(q0- q)

(1)

where q = x o r z ; M = M1 o r M 2 ; a n d K = K a or K2 respectively (see fig. 1). In these equations the term on the RHS of the equation is the force on the surface due to the respective springs. This is balanced on the LHS by the inertial term, and by the friction term B (Bx or Bz) which represents the viscous effects due to the confined liquid in the gap, and are discussed below. The two boundary conditions for this equation are clearly q = (dq/dt) = 0 at t = 0, where t = 0 defines the instant where yield and dilation have just occured (the start of the slip cycle), and q is the displacement during the slip. There are some simplifications in writing the equations in this form, but none of them are important[10]. E l s e w h e r e [ l l , 12] we consider in more detail the solutions to the two equations of motion (1), in particular the viscous term in B. This term can be written in terms of the effective viscosity rleff of the confined liquid during the slip phase, and from the solution to the equations it is possible to evaluate the time fit expected for the slip to occur between the points U and V in fig. 2. By comparing the experimentally observed time (ca. 10 -2 secs from the experimental traces in our experiments[3, 4]) to that predicted from the equations, the value of rleff may be estimated, and is found to have an upper bound of some 30 poise (P). The implications of this result are summarised below: a) The low effective viscosity of the film during the slip part of the stick-slip cycle implies that the molecular mobility within the confined liquid is high, and thus that re-

freezing (point V in fig. 2, and fig. 3c) induced by confinement when the dilation 8 has been eliminated due to approach of the surfaces - can take place very rapidly. The idea of a 'critical shear rate' beyond which stick-slip is suppressed because the molecules cannot rearrange rapidly enough may still be valid, but only at shear rates that exceed the molecular relaxation rate of the liquid in the gap. A rough estimate[13] based on the above results suggests that, for the conditions of the experiments in ref. [4] (for OMCTS and cyclohexane), shear rates of at least 106 s -1 would be required for this to happen. Since the shear rates used in the experiments did not exceed ca. 103 s -1, the observation that stickslip persisted at all shear velocities used is fully consistent with these estimates. b) The dissipation of energy during the stick-slip cycle may be evaluated from the value of heft. We come to the rather surprising conclusion that only a very small fraction (O(1%) at most) of the frictional dissipation is due to this viscous heating. Since there is no energy dissipation during the stick part of the cycle (where purely elastic deformation occurs), the bulk of the dissipation must occur at the point of resolidification (point V in fig. 2). At this point the moving surface stops abruptly, and the impulse imparted to the surfaces generates phonons that are lost as heat in the apparatus. It may be shown explicitly[12] that the frictional work done in sliding the top surface via a stick-slip process past the lower surface is equal to the energy lost at the re-freezing point. These ideas may form the basis of a more general model for static and sliding friction, in particular where a thin liquid layer separates the two surfaces. For the case of friction between two dry solid surfaces in contact (with no extrinsic separating layer), there is a clear analogy in that local shear melting at the interface between them (or at the points where asperities are in contact) may provide such a transient liquid-like film during the slip regime. The situation is illustrated in fig. 4:

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Figure 4: On shearing an asperity contact, the molecules at the sheared interface (o) are confined between the smooth undistorted lattice plane molecules (o), and on shearing may behave in analogy to the interfacial film indicated in fig. 3 The generality of our model may thus extend beyond the particular model system[3, 4] which motivated it. A c k n o w l e d g e m e n t s " I thank D. Tabor, S. Safran and R. Yerushalmi-Rozen for discussions. Financial support by the Levin Fund, the Israel Academy of Sciences, the Schmidt Minerva Center for Supramolecular Architecture, the US-Israel BSF and the Ministry of Arts and Sciences (Tashtit grant) is acknowledged with thanks. REFERENCES

1. S. Granick, Science, 253 (1991) 1374. 2. R. Horn and J.N. Israelachvili, J. Chem. Phys. 75 (1981) 1400; H. K. Christenson, J. Chem. Phys., 78 (1983) 6906-6913. 3. J. Klein and E. Kumacheva, J. Chem. Phys., 108 (1998) 6996- 7009. 4. E. Kumacheva and J. Klein, J. Chem. Phys., 108 (1998)

5. J. Klein and E. Kumacheva, Science, 269 (1995) 816-9. 6. O. Reynolds, Phil. Mag., 8 (1885) 2 2 53. 7. P. A. Thompson and M. O. Robbins, Science, 250 (1990) 792. 1. S. Granick, Science, 253 (1991) 1374. 2. R. Horn and J.N. Israelachvili, J. Chem. Phys. 75 (1981) 1400; H. K. Christenson, J. Chem. Phys., 78 (1983) 6906-6913. 3. J. Klein and E. Kumacheva, J. Chem. Phys., 108 (1998) 6996- 7009. 4. E. Kumacheva and J. Klein, J. Chem. Phys., 108 (1998) 5. J. Klein and E. Kumacheva, Science, 269 (1995) 816-9. 6. O. Reynolds, Phil. Mag., 8 (1885) 2 2 53. 7. P. A. Thompson and M. O. Robbins, Science, 250 (1990) 792. 1. S. Granick, Science, 253 (1991) 1374. 2. R. Horn and J.N. Israelachvili, J. Chem. Phys. 75 (1981) 1400; H. K. Christenson, J. Chem. Phys., 78 (1983) 6906-6913. 3. J. Klein and E. Kumacheva, J. Chem. Phys., 108 (1998) 6996 - 7009. 4. E. Kumacheva and J. Klein, J. Chem. Phys., 108 (1998) 5. J. Klein and E. Kumacheva, Science, 269 (1995) 816-9. 6. O. Reynolds, Phil. Mag., 8 (1885) 2 2 53. 7. P. A. Thompson and M. O. Robbins, Science, 250 (1990) 792. 8. J. Levien, J. Chem. Thermodynamics, 5 (1973) 679. 9. J. Hunter, J. Amer. Chem. Soc., 68 (1946) 669. 10. For example, the term on the RHS describing the driving force ignores, for the case of the x-motion, the velocity v at which the end of the spring K 1 is stretched; however, as long as the slip velocity (dx/dt) is much larger than v, which can be shown to be the case over most of the slip, this simplification is negligible. Also, for the z-direction motion, z 0 should strictly be (z 0 -5), but since z 0 >> ~5, this makes little difference. 11. J. Klein, J. Non-Crystalline Solids - in press, (1998) 12. J. Klein,- to be published, 13. If we make the usual assumption that the effective viscosity can be equated to the

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product of a modulus G and a relaxation time x, so that rleff = Gx, then estimating G ( k B T / m o l e c u l a r volume), where k B is Boltzmann's constant, gives x = 10 -6 sec for the upper limit Vleff = 30P. This yields the estimate x -1 = 106 s -1 for the lower limit of the relaxation rate.