Boundary Lubrication: Squeeze-out Dynamics of a Compressible 2D Liquid U. Tartaglino,1,2 B.N.J. Persson,3,4 A.I. Volokitin,3,5 and E. Tosatti2,1,4 1INFM Democritos National Simulation Center, and Unit`a INFM, Trieste 2SISSA, Via Beirut 4, I–34014 Trieste, Italy 3IFF, FZ-Ju¨lich, 52425 Ju¨lich, Germany 4ICTP, Strada Costiera 11, I–34014 Trieste, Italy 3 5Samara State Technical University, 443100 Samara, Russia 0 0 The expulsion dynamics of the last liquid monolayer of molecules confined between two surfaces 2 has been analyzed by solving the two-dimensional (2D) Navier-Stokes equation for a compressible n liquid. Wefindthatthesqueeze-outischaracterizedbytheparameterg0 ≈P0/ρc2,whereP0 isthe average perpendicular (squeezing) pressure, ρ the liquid (3D) density and c the longitudinal sound a J velocity in the monolayer film. When g0 ≪ 1 the result of the earlier incompressible treatment is recovered. Numerical results for the squeeze-out time, and for the time-dependence of the radius 9 of thesqueezed-out region, indicate that compressibility effects may benon-negligible both in time and in space. In space, they dominate at the edge of the squeeze-out region. In time, they are ] i strongest right at theonset of thesqueeze-out process, and just before its completion. c s - 81.40.Pq, 46.55.+d, 68.35.Af, 62.20.Qp l r t m I. INTRODUCTION of order 10-100 µm (much greater than atomic dimen- . t sions),it isreasonableto expectthatduring the layering a transitionthesqueeze-outcanbedescribedintheframe- m Slidingfrictionisoneoftheoldestproblemsinphysics, work of 2D continuum fluid mechanics. - andhasundoubtedlyahugepracticalimportance1,2,3. In d For the first time such squeeze-out layering transi- recent years, the ability to produce durable low-friction n tions were quite recently observed in a chain alcohol, surfaces and lubricants has become an important factor o C H OH(Ref.23,24),byimagingthelateralvariationof in the miniaturization of moving components in techno- 11 23 c the gap between the two anvil surfaces as a function of [ logically advanced devices. For such applications, the time. These experiments addressed the n=1→0 tran- interestis focusedonthe stability under pressureof thin 1 sition. More recently, in a refined experimental setup, lubricant films, since the complete squeeze-outof the lu- v Mugele et al.25 were able to image several layering tran- bricant from an interface may give rise to cold-welded 1 sitions (n→n−1, n=5,4,3) of the silicon oil OMCTS 2 junctions, resulting in high friction and catastrophically (spherical molecule, diameter ∼10 ˚A) in great detail. 1 large wear. 1 It has been shown both experimentally and theoreti- 0 cally that when simple fluids (quasi-spherical molecules 3 and linear hydrocarbons) are confined between atomi- 0 / cally flat surfaces at microscopic separations,the behav- t a ior of the lubricant is mainly determined by its inter- m action with the solids that induce layering in the per- - pendicular direction4,5,6,7,8,9,10,11. The thinning of the d lubrication film under applied pressure occurs step-wise, n by expulsion of individual layers. These layering tran- o sitions appear to be thermally activated12,13,14,16. Un- c : der strong confinement conditions, some lubricant flu- v ids become solid-like4,5,6,7,8,9,10,11. Other fluids, notably i FIG. 1: Because of the curvature of the solid walls at the X water17,18,remainliquid-likeuptothelastlayerthatcan boundaryline, theperpendicularpressure P0 will giveriseto r beremoveduponsqueezing. Thisisrelatedtotheexpan- aparallel force component acting on the2D-lubrication film. a sionofwateruponfreezing.19,20,andshouldalsoholdfor (Schematic.) other liquids which expand upon freezing. Thephenomenologyoflayeringtransitionsin2D-solid- Thebasictheoryof2Dsqueeze-outdynamicswasout- likeboundarylubricationhasbeenstudiedinRef.20,21,22. lined in Ref.12. Initially the system is trapped in a Ithasbeenshowninaseriesofcomputersimulationsthat meta-stable state at the initial film thickness. Squeeze- for solid-like layers, the layering transitions are some- out starts by a thermally activated nucleationprocess in times initiated by a disordering transition, after which whichadensityfluctuationformsasmallhole,ofcritical the lubricantbehavesin a liquid-like manner for the rest radius R ∼ 10 ˚A. Once formed, a 2D pressure differ- c of the squeeze-out process. Since the typical lateral ex- ence ∆p develops between the boundary line separating tension in surface force apparatus (SFA) experiments is the squeezed out regionfrom the rest of the system, and 2 the outer(roughly circular)boundary line ofthe contact on the nature of the solid walls, e. g. on the amplitude area,thusdrivingouttherestofthe2Dfluid. Theorigin of the atomic corrugation and on the structure of the of ∆p is the elastic relaxation of the confining solids at solidwalls(amorphousvs.crystalline,commensuratevs. the boundary line as is illustrated in Fig. 1. incommensurate). As long as the lubricant can be con- All earlier analytical studies of squeeze out have as- sideredas a 2D fluid, as in the recentsqueeze out exper- sumed that the lubricant behaves as an incompressible iments by Mugele et al., all the details of the solid walls 2Dliquid. Whilethatassumptionisquitegoodformany areproperlytakenintoaccountbythefrictioncoefficient practical situations, recent computer simulations20 have η¯. Inprincipleη¯dependsonthenormalpressuretoo,but shown that, at least at high squeezing pressures, strong in the measurements of Mugele et al. this dependence is density fluctuations may occur in the lubrication film. slight enough that it appears to be negligible15. For example, in Ref.20 it was found that during the lay- Simple dimensional arguments (see Ref.16) show that eringtransitionn=2→1,whileislandsof(temporarily) one can usually neglect the nonlinear and the viscosity trapped bilayer (n=2) were removedby being squeezed terms in (2), and that one can also assume the velocity intothemonolayer,thedensityofthemonolayerfilmwas field to change so slowly that the time derivative term much higher in the region close to the trapped n=2 is- can be neglected too. Thus, lands than further away. This resulted in a 2D pressure gradient in the film which induced a flow of the lubri- ∇p+mnη¯v=0. (3) cantmoleculesawayfromthetrappedislands. Thiskind In what follows we will assume that the squeeze-out of situations clearly calls for a considerationof the finite nucleated in the center (r = 0) of the contact area, compressibility of the film. and spread circularly towards the periphery. The 2D- Inthispaperwestudythedynamicsoftheexpulsionof pressure p at the outer boundary r = R of the contact the last liquid monolayer of molecules confined between areatakestheconstantvaluep (thespreadingpressure) twosurfacesbysolvingthetwo-dimensional(2D)Navier- 0 while it takes a higher value p at the inner boundary Stokes equation for a compressible liquid monolayer. We 1 towards the n = 0 area (Fig. 1). In fact, if P(r) is the find that the squeeze-out is characterized by the param- eter g ≈P /ρc2, where P is the average perpendicular perpendicular pressure acting in the contact area, then 0 0 0 p (r)=p +P(r)a,whereaisthethicknessofthemono- (squeezing) pressure,ρ the liquid (3D) density and c the 1 0 layer(seeRef.12). Wemayinmostpracticalapplications longitudinalsoundvelocityinthe monolayerfilm. When g ≪ 1 the 2D liquid can be considered as incompress- assume a Hertz contact pressure 0 ible, in which case the results of the earlier treatment 3 r2 1/2 are reproduced. We present numerical results for the P(r)= P 1− (4) squeeze-outtime, andforthe time-dependence ofthe ra- 2 0(cid:18) R2(cid:19) dius of the squeezed-out region, for several values of the Finally, in order to have a complete set of equations we parameter g . The main changes due to compressibility 0 must specify the relation between the 2D pressure p and occur right at the onset of the squeeze-out process, and the 2D density n. We will assume that just before its completion. p=p +mc2(n−n ) (5) 0 0 II. THEORY wherethe compressibilityis B =1/mc2, c being the lon- gitudinal 2D sound velocity. Here n is the 2D lubricant 0 We assume the lubricant film to be in a 2D-liquid-like density at the periphery r =R of the contact area. state, and the squeeze-out to be described by the 2D Writing the velocity field as v = rˆv(t), equations (1) Navier-Stokes equations. For a compressible 2D liquid and (3) takes the form these equations take the form ∂n ∂ 1 + + (nv)=0 (6) ∂n +∇·(nv)=0 (1) ∂t (cid:18)∂r r(cid:19) ∂t ∂p ∂v 1 +mnη¯v =0 +v·∇v=− ∇p ∂r ∂t mn Using (5) the last equation takes the form +ν1∇2v+ν2∇∇·v−η¯v (2) ∂n η¯ + (nv)=0 (7) where mn and v are the local 2D mass density and the ∂r c2 velocity of the fluid, p is the 2D-pressure, ν1 and ν2 are Combining (6) and (7) gives viscosities. The last term in (2), i. e. −η¯v, describes the dragforceactingonthe fluidasitslides relativelyto the ∂n c2 ∂ 1 ∂n − + =0 (8) solidwalls.26 The magnitude of the friction alsodepends ∂t η¯ (cid:18)∂r r(cid:19) ∂r 3 The density n satisfies the boundary conditions includethefinitecompressibilityofthelubricantinorder to accuratelydescribe the squeeze-outdynamics. InSec. n(R,t)=n0 (9) III weshallpresentnumericalresultsbasedonthe above equations for a range of values of the parameter g . 0 a Let us first calculate the squeeze-out time in the limit n(r1(t),t)=n0+ mc2P(r1(t)) (10) when the compressibility B = ∞. In this limit, all the adsorbates from the region r < r (t) will be piled-up 1 where r = r (t) is the equation for the inner boundary right at the boundary line r =r . If we consider a small 1 1 line. Finally, we note that the velocity v(r1(t),t) of the angular section ∆φ then the driving force acting on the 2D-liquidattheinnerboundaryr =r (t)mustequalthe boundaryline is F =r ∆φ[p −p ]=r ∆φP(r )a. This 1 1 1 0 1 1 radialvelocityr˙1(t)oftheboundaryline. Thus,ifweput must balance the frictional drag force which is Nmη¯r˙1 r =r (t) and v(r (t),t)=r˙ (t) in (7) we get where the number of adsorbates N = πr2(∆φ/2π)n . 1 1 1 1 0 Thus we get ∂n η¯ (r (t),t)=− n(r (t),t)r˙ (11) ∂r 1 c2 1 1 2P(r)a r˙ r = (19) 1 1 n mη¯ 0 Let us at this point introduce dimensionless variables. If we measure the radius r in units of R, time t in units of Assume first that P =P is constant. Thus, (19) gives 0 τ =η¯R2/c2 and density n in units of n we get 0 R2 r˙ r = (20) 1 1 2T ∂n ∂ 1 ∂n − + =0 (12) where ∂t (cid:18)∂r r(cid:19) ∂r mn η¯R2 and the boundary conditions becomes T = 0 (21) 4P a 0 n(1,t)=1 (13) is the squeeze out time for an incompressible 2D fluid, see Ref.12. Integrating (20) gives n(r (t),t)=1+g(t) (14) 1 1/2 r (t) t 1 = (22) and R (cid:18)T(cid:19) ∂n so that the squeeze-out time, T∗, in the limit B = ∞ (r (t),t)=−[1+g(t)]r˙ (15) ∂r 1 1 is the same as for an incompressible 2D-fluid, T∗ = T. Thissuggeststhatthesqueeze-outtimeisindependentof where thecompressibilityofthe2D-liquid,whichournumerical a simulations presented below indeed show to be the case. g(t)= P(r (t)) (16) mc2n 1 Next, let us assume that the perpendicular pressure is 0 oftheHertzform,appropriateforafluidbetweencurved If we assume that P(r) is of the Hertz form, then surfaces. In that case (19) gives g(t)=g023 1−[r1(t)]2 1/2 (17) r˙1r1 = R23 1− r1 2 1/2 (cid:0) (cid:1) 2T 2(cid:20) (cid:16)R(cid:17) (cid:21) where or, with r2/R2 =ξ, 1 aP 0 g0 = mc2n0. (18) r12/R2 dξ 3t = Z (1−ξ)1/2 2T Thus, the theory depends only on a single parameterg . 0 0 Note that n0/a ≈ ρ, where ρ is the 3D density of the Performing the integral gives liquid, and with the typical values ρ ≈ 1000 kg/m3 and c≈700m/s we get ρc2 ≈500MPa. In the experimental r 2 1/2 3t studiesbyMugeleandSalmeron(aswellasinmostother 1− 1 =1− (cid:20) (cid:16)R(cid:17) (cid:21) 4T Surface Force Apparatus studies) the average squeezing pressure P ≪ 500 MPa which implies that the liquid Thus,thesqueeze-outtimefortheHertziancontactpres- 0 can be considered as incompressible and the theory de- sure is T∗ = 4T/3, and the time dependence of r (t) is H 1 veloped elsewhere canbe used12,14,16. However,in many given by practical situations the pressure P might be similar to 0 the yield stress of the solids which for metals is typically r1 = 3t 1/2 1− 3t 1/2 of order ∼ 1000 MPa. In these cases it is necessary to R (cid:18)2T(cid:19) (cid:18) 8T(cid:19) 4 or 11 (a) "r1.combined.corrected" u 1:2 1/2 1/2 0.08.8 r 2t t 1 = 1− r 1 / R R (cid:18)T∗(cid:19) (cid:18) 2T∗(cid:19) H H 0.06.6 2 0.5 The squeeze-out time for an incompressible fluid with 0.04.4 a Hertzian contact pressure is TH = (4T/3)(2−ln4) ≈ g = 0 0.8183 T which is a factor 2−ln4≈0.6137smaller than 0.02.2 0.2 0 for the B = ∞ limiting case. Thus, for a Hertzian con- tact pressure the squeeze-out time does depend on the 00 00 00..22 00..44 00..66 00..88 11 compressibility of the 2D lubrication film. t / t squeeze-out 55 (b) "dens03.combined" u 1:2 44 III. NUMERICAL RESULTS n / n 0 33 We consider first the case of a spatially constant squeezing pressure, P(r) = P0, appropriate for a fluid 22 2 between flat surfaces. In this case numerical calcula- 0.5 tions show that the squeeze-out time is independent of 11 thecompressibility,i.e.,independentoftheparameterg0. 0.2 g 0 = 0 However, the time dependence of the squeeze-out radius 00 r (t) does depend on g . In Fig. 2(a) we show this time 00 00..22 00..44 00..66 00..88 11 1 0 dependence for four cases, g = 0, 0.2, 0.5 and 2. In r / R 0 Fig. 2(b) we show the adsorbate density profile at the time point when the squeeze-out radius r = 0.3R, for FIG. 2: Constant squeezing pressure, P(r) = P0. (a) 1 the same four g -values as in (a). Squeeze-outradiusr1 (inunitsoftheradiusRofthecontact 0 area)versustime(inunitsofthesqueezeouttime). Weshow Let us now assume the Hertzian squeezing pressure results for four different cases, namely g0 =0 (corresponding profile, P(r) = PH(r). In this case the squeeze-out to an incompressible adsorbate layer), 0.2, 0.5 and 2.0. (b) time depends on the compressibility, increasing from Adsorbate density distribution n(r) (in units of the natural ≈ 0.8183 T to 1.3333 T as the compressibility increases density n0) during squeeze-out, for the same four cases as in from B = 0 to ∞. In Fig. 3 we show the squeeze-out (a) at a time when thesqueeze out radius r1 =0.3R. time as a function of g . In Fig. 4 we show the time de- 0 pendence of r (t) for four cases, g =0, 0.2, 0.5 and 2.0. 1 0 InFig. 4(b) weshowthe adsorbatedensityprofileatthe time point when the squeeze-out radius r = 0.3R, for 1 the same four g0-values as in (a). The results of Fig. 2 1.4 "Hertz.g0.squeeze.time" u 1:2 and Fig. 4 show that a compressibility parameter g as 0 small as 0.2 should produce a measurable difference in 1.12.2 the squeeze-out evolution at all times. The main effect T 1 of compressibility appears at the beginning and at the / ut end of the squeeze-out process. Initially, compressibil- e-o 0.08.8 z ity favors piling up of fluid at the squeeze-out bound- e e u 0.6 ary, which can as a result expand more rapidly, com- sq 00..62 t pared with the case of an incompressible fluid. On the 0.04.4 other hand, when the hole approaches the boundary of the contact region, the squeezing-out speed of the com- 0.2 pressed fluid is smaller, due to its increaseddensity and, 00 consequently, friction. In the case of uniform squeezing 00 55 1100 1155 2200 pressurethese two effects compensateexactly, leading to g0 atotalsqueeze-outtimeindependentofthecompressibil- ity. ForaHertzianpressuredistribution,thesqueeze-out FIG. 3: Dependence of the squeeze-out time on the com- time instead increases with increasing compressibility: pressibility parameter g0 for a Hertzian squeezing pressure. the initial speed up is overcompensatedby the enhanced friction at the periphery of the contact area. 5 11 IV. SUMMARY (a) "r1.combined.Hertz-0.2.0.5.2.0" u 1:2 0.08.8 r 1 / R 0.06.6 0.5 2 The continuum mechanics theory of squeeze-out has beensolvednumericallyfora2Dcompressibleliquid. We 0.04.4 consideredbothaconstantnormalstress,andaHertzian g = 0 0 normalstress, and assumed a centro-symmetricsqueeze- 0.02.2 0.2 out. We found that the squeeze-out is completely char- acterized by the compressibility parameter g ≈ P /ρc2, 0 0 0 0 whereP istheaverageperpendicularsqueezingpressure, 00 00..22 00..44 00..66 00..88 11 0 ρthe(3D)liquiddensityandcthelongitudinalsoundve- t / tsqueeze-out locity inthe monolayerfilm. When g ≪1the 2Dliquid 0 55 can be considered incompressible, and the earlier results (b) "Combined.dens03.dat.Hertz" u 1:2 are reproduced. We presented numerical results for the 44 squeeze-out time, and for the time dependence of the n / n 0 squeeze-out radius, for a grid of values of the parameter 33 2 g0. For a constant squeezing pressure, the squeeze-out time was found to be independent of the compressibility parameterg ,whileforaHertziancontactpressureitin- 22 0 creasedslightlywithincreasingg . Itishopedthatthese 0.5 0 theoreticalresultswillsoonbesubmittedtoexperimental 11 0.2 g = 0 check. 0 Acknowledgments 00 00 00..22 00..44 00..66 00..88 11 r / R FIG. 4: Hertzian squeezing pressure, P(r) given by Eq. (4). (a)Squeeze-outradiusr1 (inunitsoftheradiusRofthecon- B.P.thanks BMBFforagrantrelatedto the German- tact area) versustime(in units of thesqueezeout time). We Israeli Project Cooperation “Novel Tribological Strate- show results for four different cases, namely g0 = 0 (corre- gies from the Nano-to Meso-Scales”, and ICTP/SISSA sponding to an incompressible adsorbate layer), 0.2, 0.5 and in Trieste where the main part of this work was car- 2.0. (b) Adsorbate density distribution n(r) (in units of the ried out. 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