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Layering transition: dynamical instabilities during squeeze-out
7 July 2000
Chemical Physics Letters 324 Ž2000. 231–239 www.elsevier.nlrlocatercplett
Layering transition: dynamical instabilities during squeeze-out B.N.J. Persson ) IFF, FZ-Julich, D-52425 Julich, Germany ¨ ¨ Received 16 March 2000
Abstract A lubrication fluid confined between two approaching surfaces form, in the limit of thin interfaces, well-defined layers of molecular thickness, whose number decreases in discontinuous steps with increasing applied pressure. It will be shown that, for two-dimensional Ž2D. liquid-like layers, the squeeze-out exhibit instabilities which may result in trapped islands of lubrication molecules, as observed in recent experiments. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Sliding friction is one of the oldest problems in physics and has undoubtedly a huge practical importance w1–3x. In recent years, the ability to produce durable low-friction surfaces and lubricant fluids has become an important factor in the miniaturization of moving components in technologically advanced devices. For this application, the interest is focused on the stability under pressure of thin lubricant films, since the complete squeeze-out of the lubricant from an interface may give rise to cold-welded junctions, resulting in high friction and catastrophically large wear. In this Letter, the late stages of the approach of two elastic solids limited by two curved surfaces, wetted by an atomic lubricant film of microscopic thickness, will be investigated. Under these conditions, the behavior of the lubricant is mainly determined by its interaction with the solids, that induce a two-dimensional Ž2D. order along the surfaces, and layering in the perpendicular direction w4–11x. The )
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thinning of the lubrication film occurs step-wise, by the squeeze-out of individual layers. These layering transitions appear to be thermally activated, in agreement with the theoretical prediction of Ref. w12x. In Ref. w12x we have studied the nucleation and spreading of the n n y 1 layering transition Žsee Fig. 1.. In this Letter, we present further theoretical results for this fundamental process. First, experimental results will be briefly discussed where the layering transition has been observed directly by imaging the lateral variation of the gap between the solid surfaces as a function of time. These results are for 2D liquid-like layers, for which the theory developed in Ref. w12x is applicable. Next a detailed theoretical discussion will be presented about the dynamics of the boundary line separating the n and n y 1 regions during squeeze-out. Finally, the nature of the layering transition when the lubrication film is in a 2D solid-like state w13,14x will be discussed.
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2. Basic experimental observations The dynamics of the layering transition has been studied with the surface forces apparatus by imaging
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 0 7 - 2
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
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Fig. 1. The ns 2
™1 layering transition.
the gap width in two dimensions w15x. The experiment was performed with a chain alcohol ŽC 11 H 23OH., where the unit of liquid that is expelled in a layering transition corresponds to a bilayer of molecules, with the OH groups pointing towards each other. The mica surfaces are covered by monolayers of C 11 H 23 OH, chemically bound Žvia the OH group. to the mica surfaces, leading effectively to a CH 3terminated substrate for any additional material inside the gap w16x. The coated surfaces are very inert, and the additional alcohol does not wet the surfaces. Shear experiments showed that the static friction force vanish when the passivated mica surfaces was separated by a lubricant bilayer. Thus, the bilayer film is likely to be in a 2D liquid-like state, and the theory developed in Ref. w12x should be applicable to the present system. The experimental data presented in Ref. w15x show the expulsion of the last bilayer, i.e., the n s 1 0 layering transition. We focus on the dynamics of the layering transition n s 1 0. During squeeze-out the local curvature of the boundary between the n s 1 and n s 0 regions becomes negative in some areas. Some of these areas eventually detach from the boundary and leave behind pockets of n s 1 layer trapped material in the final n s 0 state. In some cases, pockets that are located close to the rim of the contact area eventually disappear. These pockets move towards the edge as a whole. There they form little necks through which liquid is squeezed out. A series of approximately 100 subsequent expulsion events resulted in more than 20 non-overlapping locations of pockets, indicating that the island formation is not due to pinning by defects but rather due to intrinsic instabilities of the non-linear equations of motion of the boundary line Žsee below.. The diameters of the islands range from 5 to 10 mm. The dynamics of the layering transition observed in the experiments separates into two phases. In the first phase, the system is trapped in a metastable
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state at the initial film thickness, i.e. one bilayer of alcohol molecules between the substrate-bound monolayers. Thermal fluctuations of the 2D density in the bilayer eventually lead to the formation of a hole with a radius that exceeds the critical radius R c . Once the nucleus is formed the growth phase begins, and the rest of the bilayer gets expelled quickly. Since the film seems to be in a liquid-like state Žas manifested by the vanishing of the static friction force. we can analyze the data using the theory presented in Ref. w12x.
3. Dynamics of the boundary line for 2D liquid-like lubrication films Let us discuss the evolution of the boundary of the hole-island during the layering transition n s 1 0 when the nucleation of the layering transition occurs off-center. We assume, for simplicity, that the pressure in the contact area is constant Žequal to P0 .; the qualitative picture presented below does not change if the pressure varies with r. The basic equation of motion for the lubrication film are the continuity equation and the Žgeneralized. Navier– Stokes equations for the 2D velocity field zŽ x,t . Žwe assume an incompressible 2D fluid. w1,12x:
™
=Pzs0 Ez Et
q z P =z s y
Ž 1. 1 mn a
=p q n = 2 z y h z
Ž 2.
where p is the 2D pressure and n the 2D kinematic viscosity. The last term in Ž2. describes the ‘dragforce’ from the substrate acting on the fluid. Neglecting the non-linear and the viscosity terms in Ž2., and assuming that the velocity field changes so slowly that the time derivative term can be neglected, gives =p q mn ah z s 0 .
Ž 3.
From this equation it follows that z s =f ,
Ž 4.
and the continuity Eq. Ž1. gives = 2f s 0 .
Ž 5.
Substituting Ž4. in Ž3. gives
f s yprmn ah .
Ž 6.
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
Now, from Ž5., we see that the velocity potential can be interpreted as an electrostatic potential. Furthermore, since the pressure p is constant at both the Žouter. boundary r s r 0 of the contact area, as well as at the Žinner. boundary to the region n s 0, the problem of finding f is mathematically equivalent to finding the electrostatic potential between two conducting cylinders at different potentials, f 0 s yp 0rmn ah and f 1 s yp1rmn ah Žwhere p 0 is the spreading pressure, and p 1 s p 0 q P0 a, where a is the thickness of a monolayer; see Ref. w12x.. The outer cylinder has a circular shape Žradius r 0 ., and the inner cylinder an unknown Žtime dependent. shape to be determined. Now, suppose that the initial nucleation of the n s 1 region occurs some distance away from the
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center of the contact region, as indicated by the small circle in Fig. 2a. The lines between the two circular regions in this figure indicate the velocity field of the two-dimensional fluid at this moment in time, constructed by analogy to the electrostatic field lines between two cylinders at different potentials. Now, a little later in time, the velocity field will result in a larger n s 1 region as indicated in Fig. 2b. Fig. 2c,d shows the further spreading of the n s 1 region as time increases, constructed on the basis of the analogy with electrostatics. These results for the evolution of the boundary line are in relatively good agreement with the experimental observations for C 11 H 23 OH Žsee Ref. w15x.. One difference, however, is that the boundary line in the experiment has ‘roughness’ on the length-scale beyond a few mm,
Fig. 2. Snapshot pictures Žschematic. of the time evolution of the squeeze-out of a monolayer, n n-layer region.
™ n y 1. The dotted area denotes the
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B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
while we have drawn smooth boundary lines in Fig. 2. However, we will show below that the model presented above indeed predicts rough boundary lines. That is, any small perturbation of the boundary line will be amplified, so that the boundary line is unstable with respect to arbitrary small perturbations. In real systems such perturbations always exist, e.g., due to defects or thermal fluctuations. In order to show that the boundary line is unstable with respect to small perturbations, let us first consider a perfectly smooth circular boundary line centered at the center of the contact area as indicated in Fig. 3a. Assume that due to a fluctuation a small ‘bump’ is formed on the boundary line as indicated in Fig. 3b. By analogy to electrostatics, this will give rise to an enhanced ‘draining’ velocity of the fluid at the bump, so that the boundary line at the bump will move faster towards the periphery than the other regions of the boundary line. This argument is valid for ‘bumps’ of any size, and it follows that, within the model studied above, the boundary line will be rough on all length-scales. However, we will demonstrate below that when the free energy Žper unit length. G Žline tension. of the boundary line is taken into account Žit was neglected in the study above., the boundary line will be smooth on all length-scales below some critical length lc , while it will be rough on longer length-scales. In a typical case we calculate lc to be of the order of a few mm, in good agreement with experiments Žsee Ref. w15x..
We note that the instability described above is at the origin of many beautiful physical effects, such as the shape of snowflakes, electro-deposition or viscous fingering w17,18x. All these problems can be mapped on the same Žor slightly modified. problem as studied above, involving a scalar field f Že.g., the temperature field, the velocity potential or the electric potential. which satisfies the Laplace equations in the region between two boundaries, together with a set of boundary conditions, one at the outer Žfixed. boundary, and another at an inner moving boundary. 3.1. Linear stability analysis We now show why the boundary line separating the n s 1 region from the n s 0 region is rough on the length-scale ) 5 mm. A rough boundary line could, in principle, result from the influence of defects Žpinning centers., or be a result of instabilities due to the non-linear nature of the equation of motion for the boundary line. We have shown above Žsee Fig. 3. that such instabilities are indeed expected. However, we will now show that instabilities only occur for long enough length-scales, typically larger than a few mm. Let us first show that the local 2D pressure in the n s 1 region at the boundary line differs from p 1 s p 0 q P0 a by a term determined by the line tension G . The line-tension has a contribution from unsaturated bonds at the boundary line, and another much larger contribution from the energy stored in the elastic deformation field in the
Fig. 3. Instability of the boundary line separating the n Ždotted area. and n y 1 regions. Ža. A circular boundary line. Žb. A small bump on the boundary line results in locally enhanced squeeze-out, magnifying the perturbation.
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
235
Fig. 4. Ža. Elastic deformation energy is stored in the dotted region in the solids at the boundary line. Žb. Boundary line x B Ž y,t . separating the n s 0 area from the n s 1 Ždotted. area.
confining solids in the vicinity of the boundary line Žsee Fig. 4a.. If a denotes the difference in the separation between the solid walls in the n s 1 and n s 0 regions Žwhich is of the order of the thickness of a lubrication monolayer., then it follows from dimensional arguments that the elastic deformation energy per unit length of the boundary line must be of the order ; Ea2 . wA more detailed argument is as follows: the elastic energy stored in the boundary line is
where G s Ea2r2p Ž1 y n 2 .. This gives rise to a 2D pressure pX1 in the fluid layer at the boundary r s R, which can be obtained by calculating the adiabatic work to increase the radius from R to R q D R. We get
; 12 E d 3 x ´ 2 .
pX1 s p 0 q P0 a y GrR s p 1 q kG ,
H
Now, the strain ´ ; arl, where l is the lateral distance over which the the elastic displacement field varies along the solid surfaces perpendicular to the boundary line. Since the elastic displacement field satisfies a Laplazian-type equation in the solids, it follows that l is also the characteristic distance over which the displacement field extends into the solids. Thus the volume which contributes to the integral above will be of the order of 2p Rl 2 Žwhere 2p R is the length of the boundary line.. Hence we get the line energy ; EŽ2p Rl 2 .Ž arl . 2 s Ž2p R .Ž Ea2 .. We note that in the present case l f a but the analysis above does not depend on this assumption.x In fact, an exact calculation Žwithin the elastic continuum model. of the change in the elastic energy when a n s 0 hole of radius R is formed in a n s 1 system is given by Žfor R )) a. U Ž R . s yp R 2 P0 a q 2p R G ,
U Ž R . y U Ž R q D R . s 2p R Ž pX1 y p 0 . D R , which gives
Ž 7.
where p 1 s p 0 q P0 a is the pressure which would occur in the fluid at a straight boundary line, and k s y1rR the curvature, which is positive if the origin of the radius of curvature is located in the 2D fluid region, and otherwise negative. Eq. Ž7. is, of course, valid for an arbitrary curved boundary, i.e., not just a circular boundary. In the study above of the spreading of the n s 0 hole we have neglected the line tension term. However, this term has a crucial influence on the stability of the boundary line: without it, as shown above, the boundary line is unstable on all length-scales. Let us prove this for the simplest case when the unperturbed boundary is a straight line, rather than a circle. ŽThe present analysis is similar to that presented in Refs. w19,20,17x in the context of solidification.. Let x s x B Ž y,t . be the equation for the boundary line, and assume that the n s 1 fluid layer initially occupies the half plane
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
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x ) x B Ž y,t . Žsee Fig. 4b.. The velocity potential f satisfies Žsee Eqs. Ž4. – Ž6..: = 2f s 0 , nˆ P =f s Õn
f Ž xB .
for
xsxB ,
s ypX1rmn ah
,
Ž 8. Ž 9. Ž 10 .
where x B s Ž x B , y . is a point on the boundary line. The unperturbed boundary line is given by x B0 s x 0 q Õ 0 t . Let us now study the stability of this solution by adding a small perturbation: x B s x B0 q j Ž t . e i k y .
Ž 11 .
The most general solution to Ž8. which is consistent with this boundary line is
f s f 0 Ž t . q Õ 0 x q z Ž t . e i k yykw xyx B Ž t .x ,
Ž 12 .
where we have used the fact that the velocity for large x must equal Õ 0 . Substituting Ž11. and Ž12. in Ž9. gives to linear order in j and z :
j˙s yk z .
Ž 13 .
The curvature is given by
ks
d2 x B d y2
s yk 2j e i k y ,
Ž 14 .
so that, to linear order in j and z , Ž10. takes the form
smooth on the length-scale l - lc . If the layering transition nucleates in the center of the contact area, the squeeze-out velocity Õ 0 f Rrt ) Žwhere t ) is the squeeze-out time. was derived in Ref. w12x: mn ah Õ 0 f 4 aP0rR . Thus we get
lc f
p Ea
ž / 2 P0 R
1r2
R.
Ž 15 .
˚ In the present case, using R s 40 mm, a s 8 A, E f 17 GPa and P0 s 4 MPa, gives l c equal to about 1r10 of the diameter of the contact area. The experimental boundary line for C 11 H 23 OH is indeed rough on this length-scale, while it is smooth on a shorter length-scale. Based on this result one may also argue that the linear size of the trapped islands should be of the order of lc Žor larger., which again agrees with observation. We note that unless a trapped island is centered in the center of the contact area, there will be a net tangential force acting on the island because of the spatial variation in the normal stress from a maximum in the center to zero at the periphery of the contact area. Nevertheless, in the experiments negligible Žundetectable. drift of the islands towards the periphery occurs in most cases, indicating that there may be some kind of pinning even for the liquid-like layers used in the experiments.
f 0 q Õ 0 x B0 q Õ 0 j e i k y q z e i k y sy
p1 mn ah
k 2G q mn ah
j eik y .
Comparing the coefficients in front of the expŽ iky . terms gives
z s yj Õ 0 Ž 1 y k 2Grmn ah Õ 0 . , so that, using Ž13.,
j˙s Õ 0 k j Ž 1 y k 2Grmn ah Õ 0 . . Hence, for k - k c , where 1rk c s Ž Grmn ah Õ 0 .
1r2
,
the boundary line is unstable, while it is stable for k ) k c . Thus, we expect that the boundary line is rough on the length-scale l ) l c s 2prk c , but
4. On the nature of the layering transitions for 2D solid-like lubrication films The theory presented above and in Ref. w12x is based on the assumption that the lubricant film is in a 2D fluid state. This seams to be the case in the experiments by Mugele et al. It may also be the case for solid lubricant films during sliding. We have recently performed a computer simulation study Žwith Xe as the lubricant atoms. for the case when the lubricant films are in a solid-like state w13,14x. We focus on the atomic processes by which the thickness of the interface decreases by discontinuous steps, corresponding to the decrease in the number n of lubricant layers. For solid surfaces that approach without lateral sliding, separated by unpinned or
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
weakly pinned Žincommensurate. lubrication layers, fast and complete layering transitions occur. Commensurate or strongly pinned incommensurate layers lead to sluggish and incomplete transitions, often leaving islands trapped in the contact region. As discussed above, trapped islands have been observed experimentally for 2D liquid-like lubrication films. In this latter case the island may result from dynamical instabilities of the boundary line caused by the non-linearity of the equations of motion Žsee above.. Similar instabilities may be the origin of the trapped island we observe in the present case. However, when the lubrication film is in a 2D solid-like state, plastic deformation must occur during squeeze-out in order to allow different parts of the lubrication film to move with different velocity relative to the solid surfaces. ŽNote: If strongly directional bonds occur between the lubrication atoms or molecules, squeezeout may also occur by brittle fracture; see Fig. 5. This may be the case for some solid lubricants, e.g., thin graphite layers, but is unlikely to be the case for typical lubrication fluids Že.g., hydrocarbons.; in the latter case the interaction between the lubrication molecules is usually of the van der Waals type, i.e., weak and un-directional, which favours local atomic rearrangements and plastic flow.. For commensurate layers, we observe that it is nearly impossible to squeeze-out the last few layers simply by increasing the perpendicular pressure. However, the squeeze-out rate is greatly enhanced by lateral sliding, since, in this case, the lubricant film can turn into a fluidized or disordered state, facilitating the ejection of one layer. We have performed simulations for the three different cases A–C. In all cases, the lubricant is Xe, but we have varied the Xe–substrate interaction potential so that a monolayer film of lubrication atoms
237
forms unpinned Žcase A. or pinned Žcase B. incommensurate layers, or a commensurate layer Žcase C..
4.1. Incommensurate layer (unpinned) In the computer simulations, the block and the substrate are initially separated by about four Xemonolayers. The pressure-displacement curve exhibit ‘bumps’ corresponds to the layering transitions Žwith increasing pressure. n n y 1 Ž n s 4,3,2.. We observe that these transitions are rather abrupt, and are marked by a significant pressure drop. The latter implies that the squeeze-out occurs so rapidly that during the transition, the upper surface has moved Žvelocity Õz f 1 mrs. only a small fraction of the diameter of the Xe-monolayer. We observe that the layering transitions occur at higher pressures at low temperature, indicating that they are thermally activated. Inspection of snapshot pictures of the lubrication film during the nucleation of the squeeze-out n s 2 1 shows that immediately before the nucleation of the layering transition the lubrication film in the central region has undergone a phase transformation and now exhibits fccŽ100.-layers parallel to the solid surfaces. Since the fccŽ100.-layers have a lower concentration of Xe atoms than the hexagonal layers Žassuming the same nearest neighbor Xe–Xe distance., a fraction of the Xe–solid binding energy is lost during this transformation. On the other hand, the solid surfaces can now move closer to each other Žsince the distance between the fccŽ100.-layers is smaller than between the hexagonal layers. and in this way elastic energy is released. After the phase transformation, the layering transition n s 2 1 can occur much more easily since density fluctuations Žopening up of a ‘hole’. require less energy in the more dilute fccŽ100. layers than in the higher density hexagonal layers.
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4.2. Incommensurate layer (pinned)
Fig. 5. Squeeze-out as a result of brittle fracture.
We have studied squeeze-out both without Ža. and with Žb. lateral sliding. We find, in contrast to case ŽA., that in the present case, where the lateral atomic
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
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corrugation experienced by the lubrication atoms is much higher, the squeeze-out is more sluggish, and only very weak bumps corresponding to the n s 4 3 and n s 3 2 transitions can be detected in the pressure-displacement curve. In case Ža. it is found that at the end of squeezing, a trapped n s 2 island occurs, surrounded by a single Xe-monolayer. As discussed above, trapped islands have recently been observed experimentally for 2D liquid-like lubrication films. In this latter case, the islands may result from dynamical instabilities of the boundary line caused by the non-linearity of the equations of motion. Similar instabilities may be the origin of the trapped island we observe in the present case. However, when the 2D lubrication film is in a solid-like state, plastic deformation must occur during squeeze-out in order to allow different parts of the lubrication film to move with different velocities relative to the solid surfaces. During squeezing and sliding the n s 2 1 transition is complete, i.e., no n s 2 island remains trapped.
ties: when the free energy Žper unit length. G Žline tension. of the boundary line is taken into account, the boundary line will be smooth on all length-scales below some critical length l c , while it will be rough on longer length-scales. In a typical case we calculate lc to be of the order of a few mm. The line-tension G has a contribution from unsaturated bonds at the boundary line, and another much larger contribution from the energy stored in the elastic deformation field in the confining solids in the vicinity of the boundary line. When the lubricant films are in a 2D solid-like state, similar effects as observed for liquid-like lubrication films are expected w13,14x, but now plastic deformation must occur during squeeze-out in order to allow different parts of the lubrication film to move with different velocities relative to the solid surfaces.
4.3. Commensurate layer
I thank E. Brener, F. Mugele and M. Salmeron for useful discussions and comments on the manuscript. I also thank F. Mugele and M. Salmeron for sending me a preprint of Ref. w15x and P. Ballone for collaboration on the squeezing dynamics calculations reported on in Section 4 Žsee also Refs. w13,14x.. I thank BMBF for a grant related to the German–Israeli Project Cooperation ‘Novel Tribological Strategies from the Nano-to Meso-Scales’.
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The commensurate adsorbate layers are strongly pinned, and even though the Xe–substrate binding energy in the present case is much smaller than for case A, it is Žif no lateral sliding occurs. difficult to squeeze-out the lubrication film. Thus at the end of the squeeze-out process Žno sliding. the surfaces are still separated by four Xe-layers, just as at the beginning of squeeze-out. However, lateral sliding tends to break up the pinning Že.g., fluidization of the adsorbate layer may occur., and during sliding it is much easier to squeeze-out the lubrication layer, and at the end of squeeze-out only one Xe-layer remains between the surfaces in the high-pressure region.
5. Summary A lubrication fluid confined between two approaching surfaces forms, in the limit of thin interfaces, well defined layers of molecular thickness, whose number decreases in discontinuous steps with increasing applied pressure. I have shown that for 2D liquid-like layers, the squeeze-out exhibits instabili-
Acknowledgements
References w1x B.N.J. Persson, Sliding Friction: Physical Principles and Applications, Springer, Heidelberg, 1998. w2x B.N.J. Persson, Surf. Sci. Rep. 33 Ž1999. 83. w3x J. Krim, Sci. Am. 275 Ž1996. 74. w4x J.N. Israelachvili, Intermolecular, Surface Forces, Academic, London, 1995. w5x M.L. Gee, P.M. McGuiggan, J.N. Israelachvili, J. Chem. Phys. 93 Ž1990. 1895. w6x J. Gao et al., Phys. Rev. Lett. 79 Ž1997. 705. w7x H. Tamura et al., Langmuir 15 Ž1999. 7816. w8x L. Demirel, S. Granick, Phys. Rev. Lett. 77 Ž1996. 2261. w9x L. Demirel, S. Granick, J. Chem. Phys. 106 Ž1998. 6889. w10x J. Klein et al., Science 269 Ž1995. 816. w11x E. Kumacheva et al., J. Chem. Phys. 108 Ž1998. 7010. w12x B.N.J. Persson, E. Tosatti, Phys. Rev. B 50 Ž1994. 5590.
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B.N.J. Persson, P. Ballone, J. Chem. Phys. 112 Ž2000. 9524. B.N.J. Persson, P. Ballone, submitted. F. Mugele, M. Salmeron, submitted. F. Mugele, S. Baldelli, G.A. Somorjai, M. Salmeron, J. Phys. Chem. B, in press.
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w17x D. Kessler, J. Koplik, H. Levine, Adv. Phys. 37 Ž1988. 255. w18x A. Bunde, S. Havlin ŽEds.., Fractals and Disordered Systems, Springer, Berlin, 1991. w19x J.S. Langer, Phys. Rev. A 36 Ž1987. 3350. w20x E. Brener, Phys. Rev. Lett. 71 Ž1993. 3653.
Chemical Physics Letters 324 Ž2000. 231–239 www.elsevier.nlrlocatercplett
Layering transition: dynamical instabilities during squeeze-out B.N.J. Persson ) IFF, FZ-Julich, D-52425 Julich, Germany ¨ ¨ Received 16 March 2000
Abstract A lubrication fluid confined between two approaching surfaces form, in the limit of thin interfaces, well-defined layers of molecular thickness, whose number decreases in discontinuous steps with increasing applied pressure. It will be shown that, for two-dimensional Ž2D. liquid-like layers, the squeeze-out exhibit instabilities which may result in trapped islands of lubrication molecules, as observed in recent experiments. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Sliding friction is one of the oldest problems in physics and has undoubtedly a huge practical importance w1–3x. In recent years, the ability to produce durable low-friction surfaces and lubricant fluids has become an important factor in the miniaturization of moving components in technologically advanced devices. For this application, the interest is focused on the stability under pressure of thin lubricant films, since the complete squeeze-out of the lubricant from an interface may give rise to cold-welded junctions, resulting in high friction and catastrophically large wear. In this Letter, the late stages of the approach of two elastic solids limited by two curved surfaces, wetted by an atomic lubricant film of microscopic thickness, will be investigated. Under these conditions, the behavior of the lubricant is mainly determined by its interaction with the solids, that induce a two-dimensional Ž2D. order along the surfaces, and layering in the perpendicular direction w4–11x. The )
Fax: q49-2461-612850; e-mail: [email protected]
thinning of the lubrication film occurs step-wise, by the squeeze-out of individual layers. These layering transitions appear to be thermally activated, in agreement with the theoretical prediction of Ref. w12x. In Ref. w12x we have studied the nucleation and spreading of the n n y 1 layering transition Žsee Fig. 1.. In this Letter, we present further theoretical results for this fundamental process. First, experimental results will be briefly discussed where the layering transition has been observed directly by imaging the lateral variation of the gap between the solid surfaces as a function of time. These results are for 2D liquid-like layers, for which the theory developed in Ref. w12x is applicable. Next a detailed theoretical discussion will be presented about the dynamics of the boundary line separating the n and n y 1 regions during squeeze-out. Finally, the nature of the layering transition when the lubrication film is in a 2D solid-like state w13,14x will be discussed.
™
2. Basic experimental observations The dynamics of the layering transition has been studied with the surface forces apparatus by imaging
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 0 7 - 2
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
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Fig. 1. The ns 2
™1 layering transition.
the gap width in two dimensions w15x. The experiment was performed with a chain alcohol ŽC 11 H 23OH., where the unit of liquid that is expelled in a layering transition corresponds to a bilayer of molecules, with the OH groups pointing towards each other. The mica surfaces are covered by monolayers of C 11 H 23 OH, chemically bound Žvia the OH group. to the mica surfaces, leading effectively to a CH 3terminated substrate for any additional material inside the gap w16x. The coated surfaces are very inert, and the additional alcohol does not wet the surfaces. Shear experiments showed that the static friction force vanish when the passivated mica surfaces was separated by a lubricant bilayer. Thus, the bilayer film is likely to be in a 2D liquid-like state, and the theory developed in Ref. w12x should be applicable to the present system. The experimental data presented in Ref. w15x show the expulsion of the last bilayer, i.e., the n s 1 0 layering transition. We focus on the dynamics of the layering transition n s 1 0. During squeeze-out the local curvature of the boundary between the n s 1 and n s 0 regions becomes negative in some areas. Some of these areas eventually detach from the boundary and leave behind pockets of n s 1 layer trapped material in the final n s 0 state. In some cases, pockets that are located close to the rim of the contact area eventually disappear. These pockets move towards the edge as a whole. There they form little necks through which liquid is squeezed out. A series of approximately 100 subsequent expulsion events resulted in more than 20 non-overlapping locations of pockets, indicating that the island formation is not due to pinning by defects but rather due to intrinsic instabilities of the non-linear equations of motion of the boundary line Žsee below.. The diameters of the islands range from 5 to 10 mm. The dynamics of the layering transition observed in the experiments separates into two phases. In the first phase, the system is trapped in a metastable
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state at the initial film thickness, i.e. one bilayer of alcohol molecules between the substrate-bound monolayers. Thermal fluctuations of the 2D density in the bilayer eventually lead to the formation of a hole with a radius that exceeds the critical radius R c . Once the nucleus is formed the growth phase begins, and the rest of the bilayer gets expelled quickly. Since the film seems to be in a liquid-like state Žas manifested by the vanishing of the static friction force. we can analyze the data using the theory presented in Ref. w12x.
3. Dynamics of the boundary line for 2D liquid-like lubrication films Let us discuss the evolution of the boundary of the hole-island during the layering transition n s 1 0 when the nucleation of the layering transition occurs off-center. We assume, for simplicity, that the pressure in the contact area is constant Žequal to P0 .; the qualitative picture presented below does not change if the pressure varies with r. The basic equation of motion for the lubrication film are the continuity equation and the Žgeneralized. Navier– Stokes equations for the 2D velocity field zŽ x,t . Žwe assume an incompressible 2D fluid. w1,12x:
™
=Pzs0 Ez Et
q z P =z s y
Ž 1. 1 mn a
=p q n = 2 z y h z
Ž 2.
where p is the 2D pressure and n the 2D kinematic viscosity. The last term in Ž2. describes the ‘dragforce’ from the substrate acting on the fluid. Neglecting the non-linear and the viscosity terms in Ž2., and assuming that the velocity field changes so slowly that the time derivative term can be neglected, gives =p q mn ah z s 0 .
Ž 3.
From this equation it follows that z s =f ,
Ž 4.
and the continuity Eq. Ž1. gives = 2f s 0 .
Ž 5.
Substituting Ž4. in Ž3. gives
f s yprmn ah .
Ž 6.
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Now, from Ž5., we see that the velocity potential can be interpreted as an electrostatic potential. Furthermore, since the pressure p is constant at both the Žouter. boundary r s r 0 of the contact area, as well as at the Žinner. boundary to the region n s 0, the problem of finding f is mathematically equivalent to finding the electrostatic potential between two conducting cylinders at different potentials, f 0 s yp 0rmn ah and f 1 s yp1rmn ah Žwhere p 0 is the spreading pressure, and p 1 s p 0 q P0 a, where a is the thickness of a monolayer; see Ref. w12x.. The outer cylinder has a circular shape Žradius r 0 ., and the inner cylinder an unknown Žtime dependent. shape to be determined. Now, suppose that the initial nucleation of the n s 1 region occurs some distance away from the
233
center of the contact region, as indicated by the small circle in Fig. 2a. The lines between the two circular regions in this figure indicate the velocity field of the two-dimensional fluid at this moment in time, constructed by analogy to the electrostatic field lines between two cylinders at different potentials. Now, a little later in time, the velocity field will result in a larger n s 1 region as indicated in Fig. 2b. Fig. 2c,d shows the further spreading of the n s 1 region as time increases, constructed on the basis of the analogy with electrostatics. These results for the evolution of the boundary line are in relatively good agreement with the experimental observations for C 11 H 23 OH Žsee Ref. w15x.. One difference, however, is that the boundary line in the experiment has ‘roughness’ on the length-scale beyond a few mm,
Fig. 2. Snapshot pictures Žschematic. of the time evolution of the squeeze-out of a monolayer, n n-layer region.
™ n y 1. The dotted area denotes the
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B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
while we have drawn smooth boundary lines in Fig. 2. However, we will show below that the model presented above indeed predicts rough boundary lines. That is, any small perturbation of the boundary line will be amplified, so that the boundary line is unstable with respect to arbitrary small perturbations. In real systems such perturbations always exist, e.g., due to defects or thermal fluctuations. In order to show that the boundary line is unstable with respect to small perturbations, let us first consider a perfectly smooth circular boundary line centered at the center of the contact area as indicated in Fig. 3a. Assume that due to a fluctuation a small ‘bump’ is formed on the boundary line as indicated in Fig. 3b. By analogy to electrostatics, this will give rise to an enhanced ‘draining’ velocity of the fluid at the bump, so that the boundary line at the bump will move faster towards the periphery than the other regions of the boundary line. This argument is valid for ‘bumps’ of any size, and it follows that, within the model studied above, the boundary line will be rough on all length-scales. However, we will demonstrate below that when the free energy Žper unit length. G Žline tension. of the boundary line is taken into account Žit was neglected in the study above., the boundary line will be smooth on all length-scales below some critical length lc , while it will be rough on longer length-scales. In a typical case we calculate lc to be of the order of a few mm, in good agreement with experiments Žsee Ref. w15x..
We note that the instability described above is at the origin of many beautiful physical effects, such as the shape of snowflakes, electro-deposition or viscous fingering w17,18x. All these problems can be mapped on the same Žor slightly modified. problem as studied above, involving a scalar field f Že.g., the temperature field, the velocity potential or the electric potential. which satisfies the Laplace equations in the region between two boundaries, together with a set of boundary conditions, one at the outer Žfixed. boundary, and another at an inner moving boundary. 3.1. Linear stability analysis We now show why the boundary line separating the n s 1 region from the n s 0 region is rough on the length-scale ) 5 mm. A rough boundary line could, in principle, result from the influence of defects Žpinning centers., or be a result of instabilities due to the non-linear nature of the equation of motion for the boundary line. We have shown above Žsee Fig. 3. that such instabilities are indeed expected. However, we will now show that instabilities only occur for long enough length-scales, typically larger than a few mm. Let us first show that the local 2D pressure in the n s 1 region at the boundary line differs from p 1 s p 0 q P0 a by a term determined by the line tension G . The line-tension has a contribution from unsaturated bonds at the boundary line, and another much larger contribution from the energy stored in the elastic deformation field in the
Fig. 3. Instability of the boundary line separating the n Ždotted area. and n y 1 regions. Ža. A circular boundary line. Žb. A small bump on the boundary line results in locally enhanced squeeze-out, magnifying the perturbation.
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
235
Fig. 4. Ža. Elastic deformation energy is stored in the dotted region in the solids at the boundary line. Žb. Boundary line x B Ž y,t . separating the n s 0 area from the n s 1 Ždotted. area.
confining solids in the vicinity of the boundary line Žsee Fig. 4a.. If a denotes the difference in the separation between the solid walls in the n s 1 and n s 0 regions Žwhich is of the order of the thickness of a lubrication monolayer., then it follows from dimensional arguments that the elastic deformation energy per unit length of the boundary line must be of the order ; Ea2 . wA more detailed argument is as follows: the elastic energy stored in the boundary line is
where G s Ea2r2p Ž1 y n 2 .. This gives rise to a 2D pressure pX1 in the fluid layer at the boundary r s R, which can be obtained by calculating the adiabatic work to increase the radius from R to R q D R. We get
; 12 E d 3 x ´ 2 .
pX1 s p 0 q P0 a y GrR s p 1 q kG ,
H
Now, the strain ´ ; arl, where l is the lateral distance over which the the elastic displacement field varies along the solid surfaces perpendicular to the boundary line. Since the elastic displacement field satisfies a Laplazian-type equation in the solids, it follows that l is also the characteristic distance over which the displacement field extends into the solids. Thus the volume which contributes to the integral above will be of the order of 2p Rl 2 Žwhere 2p R is the length of the boundary line.. Hence we get the line energy ; EŽ2p Rl 2 .Ž arl . 2 s Ž2p R .Ž Ea2 .. We note that in the present case l f a but the analysis above does not depend on this assumption.x In fact, an exact calculation Žwithin the elastic continuum model. of the change in the elastic energy when a n s 0 hole of radius R is formed in a n s 1 system is given by Žfor R )) a. U Ž R . s yp R 2 P0 a q 2p R G ,
U Ž R . y U Ž R q D R . s 2p R Ž pX1 y p 0 . D R , which gives
Ž 7.
where p 1 s p 0 q P0 a is the pressure which would occur in the fluid at a straight boundary line, and k s y1rR the curvature, which is positive if the origin of the radius of curvature is located in the 2D fluid region, and otherwise negative. Eq. Ž7. is, of course, valid for an arbitrary curved boundary, i.e., not just a circular boundary. In the study above of the spreading of the n s 0 hole we have neglected the line tension term. However, this term has a crucial influence on the stability of the boundary line: without it, as shown above, the boundary line is unstable on all length-scales. Let us prove this for the simplest case when the unperturbed boundary is a straight line, rather than a circle. ŽThe present analysis is similar to that presented in Refs. w19,20,17x in the context of solidification.. Let x s x B Ž y,t . be the equation for the boundary line, and assume that the n s 1 fluid layer initially occupies the half plane
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
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x ) x B Ž y,t . Žsee Fig. 4b.. The velocity potential f satisfies Žsee Eqs. Ž4. – Ž6..: = 2f s 0 , nˆ P =f s Õn
f Ž xB .
for
xsxB ,
s ypX1rmn ah
,
Ž 8. Ž 9. Ž 10 .
where x B s Ž x B , y . is a point on the boundary line. The unperturbed boundary line is given by x B0 s x 0 q Õ 0 t . Let us now study the stability of this solution by adding a small perturbation: x B s x B0 q j Ž t . e i k y .
Ž 11 .
The most general solution to Ž8. which is consistent with this boundary line is
f s f 0 Ž t . q Õ 0 x q z Ž t . e i k yykw xyx B Ž t .x ,
Ž 12 .
where we have used the fact that the velocity for large x must equal Õ 0 . Substituting Ž11. and Ž12. in Ž9. gives to linear order in j and z :
j˙s yk z .
Ž 13 .
The curvature is given by
ks
d2 x B d y2
s yk 2j e i k y ,
Ž 14 .
so that, to linear order in j and z , Ž10. takes the form
smooth on the length-scale l - lc . If the layering transition nucleates in the center of the contact area, the squeeze-out velocity Õ 0 f Rrt ) Žwhere t ) is the squeeze-out time. was derived in Ref. w12x: mn ah Õ 0 f 4 aP0rR . Thus we get
lc f
p Ea
ž / 2 P0 R
1r2
R.
Ž 15 .
˚ In the present case, using R s 40 mm, a s 8 A, E f 17 GPa and P0 s 4 MPa, gives l c equal to about 1r10 of the diameter of the contact area. The experimental boundary line for C 11 H 23 OH is indeed rough on this length-scale, while it is smooth on a shorter length-scale. Based on this result one may also argue that the linear size of the trapped islands should be of the order of lc Žor larger., which again agrees with observation. We note that unless a trapped island is centered in the center of the contact area, there will be a net tangential force acting on the island because of the spatial variation in the normal stress from a maximum in the center to zero at the periphery of the contact area. Nevertheless, in the experiments negligible Žundetectable. drift of the islands towards the periphery occurs in most cases, indicating that there may be some kind of pinning even for the liquid-like layers used in the experiments.
f 0 q Õ 0 x B0 q Õ 0 j e i k y q z e i k y sy
p1 mn ah
k 2G q mn ah
j eik y .
Comparing the coefficients in front of the expŽ iky . terms gives
z s yj Õ 0 Ž 1 y k 2Grmn ah Õ 0 . , so that, using Ž13.,
j˙s Õ 0 k j Ž 1 y k 2Grmn ah Õ 0 . . Hence, for k - k c , where 1rk c s Ž Grmn ah Õ 0 .
1r2
,
the boundary line is unstable, while it is stable for k ) k c . Thus, we expect that the boundary line is rough on the length-scale l ) l c s 2prk c , but
4. On the nature of the layering transitions for 2D solid-like lubrication films The theory presented above and in Ref. w12x is based on the assumption that the lubricant film is in a 2D fluid state. This seams to be the case in the experiments by Mugele et al. It may also be the case for solid lubricant films during sliding. We have recently performed a computer simulation study Žwith Xe as the lubricant atoms. for the case when the lubricant films are in a solid-like state w13,14x. We focus on the atomic processes by which the thickness of the interface decreases by discontinuous steps, corresponding to the decrease in the number n of lubricant layers. For solid surfaces that approach without lateral sliding, separated by unpinned or
B.N.J. Perssonr Chemical Physics Letters 324 (2000) 231–239
weakly pinned Žincommensurate. lubrication layers, fast and complete layering transitions occur. Commensurate or strongly pinned incommensurate layers lead to sluggish and incomplete transitions, often leaving islands trapped in the contact region. As discussed above, trapped islands have been observed experimentally for 2D liquid-like lubrication films. In this latter case the island may result from dynamical instabilities of the boundary line caused by the non-linearity of the equations of motion Žsee above.. Similar instabilities may be the origin of the trapped island we observe in the present case. However, when the lubrication film is in a 2D solid-like state, plastic deformation must occur during squeeze-out in order to allow different parts of the lubrication film to move with different velocity relative to the solid surfaces. ŽNote: If strongly directional bonds occur between the lubrication atoms or molecules, squeezeout may also occur by brittle fracture; see Fig. 5. This may be the case for some solid lubricants, e.g., thin graphite layers, but is unlikely to be the case for typical lubrication fluids Že.g., hydrocarbons.; in the latter case the interaction between the lubrication molecules is usually of the van der Waals type, i.e., weak and un-directional, which favours local atomic rearrangements and plastic flow.. For commensurate layers, we observe that it is nearly impossible to squeeze-out the last few layers simply by increasing the perpendicular pressure. However, the squeeze-out rate is greatly enhanced by lateral sliding, since, in this case, the lubricant film can turn into a fluidized or disordered state, facilitating the ejection of one layer. We have performed simulations for the three different cases A–C. In all cases, the lubricant is Xe, but we have varied the Xe–substrate interaction potential so that a monolayer film of lubrication atoms
237
forms unpinned Žcase A. or pinned Žcase B. incommensurate layers, or a commensurate layer Žcase C..
4.1. Incommensurate layer (unpinned) In the computer simulations, the block and the substrate are initially separated by about four Xemonolayers. The pressure-displacement curve exhibit ‘bumps’ corresponds to the layering transitions Žwith increasing pressure. n n y 1 Ž n s 4,3,2.. We observe that these transitions are rather abrupt, and are marked by a significant pressure drop. The latter implies that the squeeze-out occurs so rapidly that during the transition, the upper surface has moved Žvelocity Õz f 1 mrs. only a small fraction of the diameter of the Xe-monolayer. We observe that the layering transitions occur at higher pressures at low temperature, indicating that they are thermally activated. Inspection of snapshot pictures of the lubrication film during the nucleation of the squeeze-out n s 2 1 shows that immediately before the nucleation of the layering transition the lubrication film in the central region has undergone a phase transformation and now exhibits fccŽ100.-layers parallel to the solid surfaces. Since the fccŽ100.-layers have a lower concentration of Xe atoms than the hexagonal layers Žassuming the same nearest neighbor Xe–Xe distance., a fraction of the Xe–solid binding energy is lost during this transformation. On the other hand, the solid surfaces can now move closer to each other Žsince the distance between the fccŽ100.-layers is smaller than between the hexagonal layers. and in this way elastic energy is released. After the phase transformation, the layering transition n s 2 1 can occur much more easily since density fluctuations Žopening up of a ‘hole’. require less energy in the more dilute fccŽ100. layers than in the higher density hexagonal layers.
™
™
™
4.2. Incommensurate layer (pinned)
Fig. 5. Squeeze-out as a result of brittle fracture.
We have studied squeeze-out both without Ža. and with Žb. lateral sliding. We find, in contrast to case ŽA., that in the present case, where the lateral atomic
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corrugation experienced by the lubrication atoms is much higher, the squeeze-out is more sluggish, and only very weak bumps corresponding to the n s 4 3 and n s 3 2 transitions can be detected in the pressure-displacement curve. In case Ža. it is found that at the end of squeezing, a trapped n s 2 island occurs, surrounded by a single Xe-monolayer. As discussed above, trapped islands have recently been observed experimentally for 2D liquid-like lubrication films. In this latter case, the islands may result from dynamical instabilities of the boundary line caused by the non-linearity of the equations of motion. Similar instabilities may be the origin of the trapped island we observe in the present case. However, when the 2D lubrication film is in a solid-like state, plastic deformation must occur during squeeze-out in order to allow different parts of the lubrication film to move with different velocities relative to the solid surfaces. During squeezing and sliding the n s 2 1 transition is complete, i.e., no n s 2 island remains trapped.
ties: when the free energy Žper unit length. G Žline tension. of the boundary line is taken into account, the boundary line will be smooth on all length-scales below some critical length l c , while it will be rough on longer length-scales. In a typical case we calculate lc to be of the order of a few mm. The line-tension G has a contribution from unsaturated bonds at the boundary line, and another much larger contribution from the energy stored in the elastic deformation field in the confining solids in the vicinity of the boundary line. When the lubricant films are in a 2D solid-like state, similar effects as observed for liquid-like lubrication films are expected w13,14x, but now plastic deformation must occur during squeeze-out in order to allow different parts of the lubrication film to move with different velocities relative to the solid surfaces.
4.3. Commensurate layer
I thank E. Brener, F. Mugele and M. Salmeron for useful discussions and comments on the manuscript. I also thank F. Mugele and M. Salmeron for sending me a preprint of Ref. w15x and P. Ballone for collaboration on the squeezing dynamics calculations reported on in Section 4 Žsee also Refs. w13,14x.. I thank BMBF for a grant related to the German–Israeli Project Cooperation ‘Novel Tribological Strategies from the Nano-to Meso-Scales’.
™
™
™
The commensurate adsorbate layers are strongly pinned, and even though the Xe–substrate binding energy in the present case is much smaller than for case A, it is Žif no lateral sliding occurs. difficult to squeeze-out the lubrication film. Thus at the end of the squeeze-out process Žno sliding. the surfaces are still separated by four Xe-layers, just as at the beginning of squeeze-out. However, lateral sliding tends to break up the pinning Že.g., fluidization of the adsorbate layer may occur., and during sliding it is much easier to squeeze-out the lubrication layer, and at the end of squeeze-out only one Xe-layer remains between the surfaces in the high-pressure region.
5. Summary A lubrication fluid confined between two approaching surfaces forms, in the limit of thin interfaces, well defined layers of molecular thickness, whose number decreases in discontinuous steps with increasing applied pressure. I have shown that for 2D liquid-like layers, the squeeze-out exhibits instabili-
Acknowledgements
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