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Multiscale Analysis of Liquid Lubrication Trends fromIndustrial Machines to Micro-Electrical-Mechanical Systems

Donald W. Brenner, Douglas L. Irving, Angus I. Kingon, Jacqueline Krim, and Clifford W. PadgettLangmuir, 2007, 23 (18), 9253-9257• DOI: 10.1021/la701280k • Publication Date (Web): 28 July 2007

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Multiscale Analysis of Liquid Lubrication Trends from IndustrialMachines to Micro-Electrical-Mechanical Systems

Donald W. Brenner,*,† Douglas L. Irving,† Angus I. Kingon,† and Jacqueline Krim‡

Departments of Materials Science and Engineering and of Physics, North Carolina State UniVersity,Raleigh, North Carolina 27695

Clifford W. Padgett

Department of Chemistry and Physics, Armstrong Atlantic State UniVersity, SaVannah, Georgia 31419

ReceiVed May 2, 2007. In Final Form: June 22, 2007

An analytic multiscale expression is derived that yields conditions for effective liquid lubrication of oscillatingcontacts via surface flow over multiple time and length scales. The expression is a logistics function that dependson two quantities, the fraction of lubricant removed at each contact and a scaling parameter given by the logarithmof the ratio of the contact area to the product of the lubricant diffusion coefficient and the cycle time. For industrialmachines the expression confirms the need for an oil mist. For magnetic disk drives, the expression predicts thatexisting lubricants are sufficient for next-generation data storage. For micro-electrical-mechanical systems, the expressionpredicts that a bound+ mobile lubricant composed of tricresyl phosphate on an octadecyltrichlorosilane self-assembledmonolayer will be effective only for temperatures greater than∼200 K and up to∼MHz oscillation frequencies.

I. IntroductionLiquid lubrication has played a major role throughout history

in the development of new technologies involving contactinginterfaces. These technologies run the scale from the slowmovement of massive structures for building temples andpyramids1 to current challenges in overcoming stiction ofmicrometer-scale contacts in micro-electrical-mechanical systems(MEMSs).2 While specialized natural and synthetic lubricantshave been available for many years for conventional machinessuch as combustion engines, effective lubrication strategies formicrometerandsmallerscalecontactsarestillbeingdeveloped.3-11

To begin to unify these efforts, and to provide a global multiscaleperspective on liquid lubrication, we introduce below a newanalytic scaling relation from which lubrication across disparatescales can be analyzed.

Several researchers have recently developed models for thelubrication of oscillating contacts. Sawyer and Blanchet developeda model for the lubrication of combined sliding and rollingmacroscopic scale contacts that was based on the deposition of“islands” of solid lubricant from the vapor.12Unlike prior modelsthat assumed either the presence or complete absence of a thinlayer of lubricant,13 the Sawyer-Blanchet analysis was able to

account for the gradual transition of friction coefficients as afunction of the partial pressure of vapor-phase lubricants. Inrelated work, Dickrell et al. developed a closed-form expressionfor the vapor lubrication of a reciprocating contact that providesfriction coefficients that are both time and position dependent.14

Like the Sawyer-Blanchet analysis, the Dickrell et al. expressionassumes that replenishment of the lubricant is from the depositionof a vapor without a contribution from surface flow.

It was recently reported that when a mobile overlayer is presenton a metal surface, a megahertz transverse oscillation created bya combined quartz crystal microbalance-scanning tunnelingmicroscope (QCM-STM) can produce a clearer tunneling imageof the metal surface relative to the same system withoutoscillation.15 This observation was attributed to a “windshieldwiper” effect in which rapid oscillation at the tip-surface interfacecreates a region below the tip where at least part of the mobileoverlayer is wiped away. To support the feasibility of thisobservation as being due to a windshield wiper effect, numericalmodeling was carried out to determine the steady-state coverageof a mobile lubricating layer at an oscillating contact due tosurface flow. The modeling predicted that a region below the tipwith a reduced coverage of mobile overlayer can be maintainedby megahertz transverse substrate oscillations for a physicallyrealistic range of overlayer diffusion coefficients.

In this study the flow model used to describe the STM-QCMdynamics is extended to an analytic multiscale expression foreffective liquid lubrication of oscillating contacts via surfaceflow where a fraction of the lubricant is removed from the contactat each cycle. The expression takes the form of a logistics functionthat depends on the fraction of lubricant removed at eachoscillation and a scaling parameter given by the logarithm of theratio of the contact area to the product of the lubricant diffusioncoefficient and the cycle time. This scaling relation is used toanalyze lubrication requirements and technology trends for three

† Department of Materials Science and Engineering.‡ Department of Physics.(1) Dowson, D.History of Tribology; Longman: London, 1979.(2) Bhushan, B. InHandbook of Micro/Nano Tribology; Bhushan, B., Ed.;

CRC Press: Boca Raton, FL, 1995; Chapter 11.(3) Eapen, K. C.; Patton, S. T.; Zabinski, J. S.Tribol. Lett. 2002, 12, 35.(4) Choi, J. H.; Kawaguchi, M.; Kato, T.Tribol. Lett. 2003, 15, 353.(5) Katano, T.; Oka, M.; Nakazawa, S.; Aramake, T.; Kusakawa, K.IEEE

Trans. Magn.2003, 39, 2489.(6) Satyanarayana, N.; Sinha, S. K.J. Phys. D: Appl. Phys.2005, 38, 3512.(7) Eapen, K. C.; Patton, S. T.; Smallwood, S. A.; Phillips, B. S.; Zabinski,

J. S.J. Microelectromech. Syst.2005, 14, 954.(8) Tambe, N. S.; Bhushan, B.Appl. Phys. Lett. 2005, 86, 061906.(9) Tayebi, N.; Polycarpou, A. A.J. Appl. Phys.2005, 98, 073528.(10) Irving, D. L.; Brenner, D. W.J. Phys. Chem. 2006, 110, 15426.(11) Smallwood, S. A.; Eapen, K. C.; Patton, S. T.; Zabinski, J. S.Wear2006,

260, 1179.(12) Sawyer, W. G.; Blanchet, T. A.J. Tribol. 2001, 123, 572.(13) Blanchet, T. A.; Lauer, J. L.; Liew, Y. -F.; Rhee, S. J.; Sawyer, W. G.

Surf. Coat. Technol.1994, 68/69, 446.

(14) Dickrell, P. L.; Sawyer, W. G.; Heimberg, J. A.; Singer, I. L.; Wahl, K.J.; Erdemir, A.J. Tribol. 2005, 127, 82.

(15) Abdelmaksoud, M.; Lee, S. M.; Padgett, C. W.; Irving, D. L.; Brenner,D. W.; Krim, J. Langmuir2006, 22, 9606.

9253Langmuir2007,23, 9253-9257

10.1021/la701280k CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 07/28/2007

systems with different size and time scales: industrialmachines lubricated via an oil mist, magnetic hard disk drives(MHDDs), and a “bound-plus-mobile” (B+ M) lubricant forMEMSs.

II. Lubrication Model

The idealized geometry of the model consists of the periodiccontact of a cylinder with radiusrt onto a flat surface in thepresence of a mobile lubricant (Figure 1). The system containsa reservoir of lubricant with a constant coverage ofC0at a distancer0 from the center of the contact. It is assumed that the coverageof the lubricant below the contacting cylinder within the distancert from the center of the contact is instantaneously and uniformlyreduced by some fraction at each contact. If there were no externalsource of lubricant to flow back under the contact, the coverageof lubricant under the contact would eventually become negligible.With a mobile lubricant, however, the contact region can fill toan extent that depends on the flow rate, the oscillation frequency,and the size of the contact. As more material is removed frombelow the contact, the coverage gradient increases and the rateof flow back under the contact increases. As periodic contactsare made some fraction of the lubricant is removed at each contact,resulting in an increase in the coverage gradient; this in turnresults in an increase in the diffusive flow back into the contact.These two processes eventually balance each other, resulting ina steady-state coverage distribution of lubricant below the contact.

The model on which the calculations are based drasticallysimplifies the dynamics of the lubricant around the contact,ignoring effects such as drag forces on the lubricant, thedependence of lubricant properties on the contact scale, transientpressure under the contact, contact roughness, capillary forces,etc.16 Furthermore, although the model is idealized in Figure 2as a cylinder making normal contact with a lubricated surface,the model does not distinguish between a normal contact andcontacts at non-normal angles (including sliding), only that thereis an explicit contact area from which lubricant is removed. Asimilar assumption has been made in prior related studies.14Ratherthan attempt to explicitly address these and related effects, theyare implicitly incorporated within the model through twoparameters, the diffusion coefficient and the assumed fractionof lubricant that is removed from below the contact at the start

of each cycle. In addition, the model does not distinguish betweendifferent mechanisms of diffusion (e.g., spreading diffusion,atomic diffusion, etc.). By significantly simplifying the analysisin this way, the scaling behavior of lubricated oscillating contactsbecomes more apparent, and a single analytic expression can beused to describe multiscale lubrication based on surface flow.A further discussion of these approximations is given below insection IV.

Assuming a diffusion coefficientD that is independent ofcoverage, and a boundary condition of a constant coverage ofC0 at some distancer0 from the center of the contact, the lubricantcoverage profile due to flow back into the region below thecontact is given by a solution of the form

(16) Wu, L. J. Appl. Phys.2006, 100, 024505.

Figure 1. Illustration of the simplified contact model consisting of a cylinder in oscillating contact with a surface on which there is a liquidlubricant. (a) Cylinder above the surface at the start of the cycle. (b) Cylinder in contact with the surface. A given fractionf of the lubricantis removed in the region below the cylinder. (c) The cylinder is removed, and the lubricant flows into the contact. This cycle is repeateduntil a steady-state coverage below the contact is reached. The vertical arrow in (a) indicates the point at which the relative lubricant coverageis given by eq 2.

Figure 2. Top panel: Calculated steady-state coverage at the contactcenter as a function ofγ for 10% (dashed line) and 1% (solid line)loss of lubricant below the contact at each oscillation. The arrowtail corresponds to MHDD technology circa 1990; the arrow headcorresponds to current technology. Bottom panel: ParametersA(solid line) andB (dashed line) in eq 2 as a function of the fractionof lubricant lost at each contact.

C(r,tc) )

2

r02∑n)1

∞ J0(ânr)

ânJ1(ânr0)2e(-ân

2Dtc)∫r′)0

rt r′J0(ânr′) F(r′) dr′ + C0 (1)

9254 Langmuir, Vol. 23, No. 18, 2007 Brenner et al.

to the standard Fick’s second law continuum diffusion equation.17

In this expressionr andr′ are the distances from the center ofthe evacuated region,rt is the radius of the partially evacuatedregion,tc is the time between contacts, andân are the roots tothe Bessel functionsJ0(ânr0) ) 0. The expressionF(r′) is theinitial coverage profile for the system under the contact, definedas the deviation fromC0, so thatF(r′) ) (1- f)C0, wheref equalsthe fraction removed by the contact. To obtain the steady-statecoverage, the calculated coverage within the area impacted bythe cylinder at each contact is uniformly reduced byf, and thecoverage for the entire surface is recalculated from eq 1. Thisprocess is continued until the coverage below the tip at the endof each cycle has converged to a constant value.

In prior work steady-state coverages below an oscillatingcontact were calculated for a single contact area of 0.01µm,oscillation frequencies from kilohertz to megahertz, and differentfractions of lubricant removed at each contact.15 It was observedthat the steady-state coverage at a point in the center of thecontact at timetc is well described by a logistics function thatdepends on the logarithm of the product of the diffusion coefficientand the oscillation cycle timetc for each value of the fractionof lubricant lost at each contact. To develop this result into amultiscale analysis, the same computational procedure was carriedout for conditions that correspond to oscillation frequencies of1 Hz to 1 GHz and contact areas ranging from square micrometersto square centimeters. All of the steady-state coverages at a pointin the center of the contact at timetc calculated by this procedurecan be fit to within 1% or less by the logistics function

where the unitless scaling quantityγ is the (base 10) logarithmof the ratio of the area of the contact to the product of thelubrication diffusion coefficientD and the cycle timetc

andAandBare parameters that depend on the fraction of lubricantremoved at each cycle. To a very good approximation the valuesof A and B are related to the fractionf of lubricant removedbelow the contact at each cycle time by the relations

The coverage given by eq 2 as a function ofγ assuming that 1%and 10% of the lubricant below the contact is removed at eachcycle is plotted in the top panel of Figure 2. The calculatedparametersA andB as well as the fitting relations in eq 4 areplotted in the bottom panel of Figure 2 as a function of thefraction of lubricant removed at each cycle.

III. Multiscale Analysis of Lubrication Technology

The efficacy of lubrication via surface flow can be establishedfor technologies across disparate time and length scales from thescaling parameterγ. To illustrate this, three technologies thatoperate at different scales are analyzed in terms ofγ, industrial-scale machines, MHDDs, and a B+ M lubricant for MEMSs.

III.i. Oil Mist Lubrication of Industrial Machines. Tradi-tional machines with moving parts such as industrial pumps andcombustion engines have typical apparent contact areas of roughly

square centimeters and contact times up to seconds. They arelubricated with either pumped oil, which is not addressed by ourmodel, or an oil mist that continuously introduces lubricant froma vapor into the contacting region.18 By inference, it can beconcluded that lubrication via surface flow would not be effectiveat this scale. To quantify this conclusion, our scaling relation canbe applied to this regime.

Industrial oils used for oil mist lubrication have kinematicviscosities that range from about 5 to 500 cSt depending on thetemperature and oil grade. Using the Einstein-Stokes relation,and assuming an effective molecular size of 1 nm, yields bulkdiffusion coefficients of approximately 10-11 to 10-13 cm2/s.Together with the cycle times and contacting areas listed above,these diffusion coefficients yield values forγ of 11-13. FromFigure 2 it is clear that these values are well above those requiredfor effectively replenishing a lubricant under an oscillating contactvia surface flow, hence the need for replenishing the lubricantvia vapor deposition.

III.ii. Lubrication of Magnetic Hard Disk Drives. MHDDsconsist of a spinning aluminum or glass disk with a magneticlayer that is covered with a hard carbon overcoat. Data are readfrom the disk by a flyer arm that passes laterally across the head.To maintain a clean environment, the combination of disk andflyer arm, together with associated electronics, is maintained ina closed container. An air bearing is used to maintain anappropriate flyer height for the arm,19 and a lubricating layersuch as a perfluoropoly(alkyl ether) (PFPE) is added to the platter.

MHDDs circa 1990 used a data track width of∼0.1 mm anda rotation speed of 5200 rpm. Because the arm traverses the headas the head rotates, the rotation speed leads to a rough lower limitof 10 ms for the cycle time. Using a surface diffusion coefficientof 10-7 cm2/s for PFPE20 leads to a value forγ of about 6.Therefore, under these conditions the scaling relation suggestspoor liquid lubrication. This has been confirmed by recentexperimental studies using picosliders with flyer heights of lessthan 10 nm, which have measured a lubricant depletion track ofradius∼0.2 mm.21,22In the case of MHDDs, however, the flyerheights circa 1990 were∼20 nm,23 and therefore, the PFPE wasonly required to mitigate an intermittent head crash.

A current leading goal of data storage technology is to engineerMHDDs with capacities greater than 300 Gb/in.2. Meeting thisgoal requires read tracks with widths of less than∼100 nm,which in turn forces flyer heights down to a few nanometers. Atthese tolerances an effective lubricant becomes crucial tocontinuous device operation, yet there appears to be little researchinto developing new lubricants for MHDDs. This research anddevelopment trend can be understood using the scaling analysis.Current MHDDs rotate at speeds of about 7200-15000 rpm,which leads to a lower bound for the cycle time of about 4 ms.Combining this cycle time and the area from a 100 nm contactwidth with the PFPE diffusion coefficient given above leads toa value forγ of about-1. Hence, the scaling of the data trackwidth necessary for meeting data storage goals, which requiresa decrease in the flyer height, moves this technology into a goodlubrication regime without redesign of the lubricant.

III.iii. A Bound + Mobile Lubricant for MEMS. Stictionof moving interfaces in MEMS devices can be significantly

(17) Ozisik, M. N.Heat Conduction; John Wiley and Sons: New York, 1993.

(18) Shamim, A.; Kettleborough, C. F.J. Energy Resour. Technol.1994, 116,224.

(19) McFadyen, I. R.; Fullerton, E. E.; Carey, M. J.MRS Bull.2006, 31, 379.(20) Ma, X.; Gui, J.; Smoliar, L.; Grannen, K.; Marchon, B.; Jhon, M. S.;

Bauer, C. L.J. Chem. Phys.1999, 110, 3129.(21) Ma, X.; Tang, H.; Stirniman, M.; Gui, J.IEEE Trans. Magn.2002, 38,

112.(22) Deoras, S. K.; Talke, F. E.IEEE Trans. Magn.2003, 39, 2471.(23) Lee, K. M.; Polycarpou, A. A.Mech. Syst. Signal Process.2006, 20,

1322.

C(γ) ) 1

1 + e(Aγ+B)(2)

γ ) log[πr t2

Dtc] (3)

A ) 0.3434 log(f) + 3.0446

B ) 2.8601 log(f) + 4.8051(4)

Multiscale Analysis of Liquid Lubrication Trends Langmuir, Vol. 23, No. 18, 20079255

reduced by using a boundary lubricant, typically a self-assembledmonolayer (SAM).2-5,7 However, for devices that require longlifetimes, for example, in satellite technology, failure due to lossof the boundary lubricant can be a major problem. One strategybeing explored to increase the lifetime of these interfaces is toincorporate a mobile phase into the SAM that can diffuse to andprotect a region of a surface from which the SAM is lost.3-5,7

The SAM therefore acts both as a protective interface and as asource of mobile lubricant.

One B + M system that has been recently explored forincreasing the lifetime of silicon MEMSs is an octadecyltrichlo-rosilane (ODTS) SAM to which tricresyl phosphate (TCP) isadded.3,9The ODTS is composed of alkane chains with 18 carbonatoms and a trichlorosilane head group that is covalently bondedto the silicon substrate. TCP is commonly used as a wear-reducingagent in high-temperature industrial lubricants and as a flame-resistant plasticizer.

The diffusion coefficient for TCP on an ODTS SAM has notbeen experimentally measured. Recent molecular dynamicssimulations, however, yield an estimated diffusion barrier of0.0937 eV and an Arrhenius prefactor of 26.47× 10-4 cm2/s forsingle-molecule diffusion.10 Using these diffusion data and thescaling relation, estimates can be made for the conditions underwhich this B+ M lubricant combination will be effective forMEMSs. Shown in Figure 3 are contour plots of the predictedrelative coverage of lubricant at the center of contact as a functionof temperature and the logarithm of the contact area in squaremicrometers. The center coverage is calculated from eq 2 usingvalues ofA andB from eq 4 assuming that 1% of the lubricantis lost at each cycle (e.g.,f ) 0.01). For a kilohertz oscillationfrequency (top panel), lubricant is able to diffuse to the centerof the contact except for conditions where the temperature is lessthan∼150 K and the contact area is greater than∼100µm2. Formegahertz oscillations (middle panel), lubricant is able to diffuseto the center of the contact except for conditions where the contactarea is greater than∼1 µm2 and the temperature is less than∼200 K. At gigahertz frequencies (bottom panel), which maybe applicable to future nanometer-scale devices, this B+ Mcombination is predicted to be effective only at temperaturesgreater than about 300 K and very small contact areas. Thediffusion data are approximate at best, and hence, the resultsshould be used with appropriate caution. In addition, as discussedbelow the lubricant loss per cycle could vary for different contactareas and contact frequencies changing from kilohertz togigahertz. Consequently, the conclusion drawn for the efficiencyof the B+ M lubricant for MEMSs under different conditionsneeds to be verified with a more realistic treatment of the valuesof f. Nonetheless, this analysis clearly illustrates the utility of ourmultiscale analysis for succinctly exploring parameter space foreffective lubrication for existing and future technologies.

IV. Discussion

The scaling analysis presented above provides an analyticexpression with which effective liquid lubrication requirementscan be efficiently evaluated and understood using only twoparameters, the scaling parameterγ and the fractionf of lubricantlost below the contact at each cycle. This expression is not intendedto replace detailed modeling and experiment, but rather it providesa complementary analysis that can be used in conjunction withmore detailed studies.

From the examples above it is clear that this multiscale analysiscan capture much of the essential behavior associated withlubrication requirements over disparate time and length scales.However, the analysis is based on significant simplifications

that should be understood so that the expression can be effectivelyused and modified as appropriate. Foremost, the model ignoresthe dynamics associated with the flow of lubricant during contact,instead assuming that hydrodynamic effects can be collectedinto the parameterf independent of the value ofγ. We note thatlarge contact areas, small diffusion coefficients, and short cycletimes (i.e., high frequencies) all lead to conditions where due tohydrodynamic effects the total outflow of the lubricant frombelow the contact will be less than that for an equilibrium pressureand temperature; i.e., the value forf will be less than some limitingequilibrium value. Each of these conditions is associated withan increase inγ, and therefore, for a given surface and lubricantcombination in general larger values ofγ will correspond to adecreasing value forf. While deriving a quantitative relationbetweenγ andf for a given system would require results fromdetailed hydrodynamic modeling, the analytic scaling relationsderived above can provide some information about the limitsand behavior of such as expression. For values off greater thanabout 0.25, the scaling expression is effectively independent off. Therefore, as long as hydrodynamic effects do not lower thepercent of lubricant lost to below 25%, neglect of the dynamicsof outflow and the associated decoupling off from γ appears tobe a reasonable approximation. On the other hand, iff is loweredby hydrodynamic effects to a value as small as 0.001 (e.g., 0.1%

Figure 3. Contour plots of the coverage at the center of a contactas a function of temperature and contact area for the ODTS/TCP B+ M lubricant combination. The panels from top to bottom correspondto kilohertz, megahertz, and gigahertz oscillations, respectively.

9256 Langmuir, Vol. 23, No. 18, 2007 Brenner et al.

of fluid lost at each contact),γ would still have to be less thanabout 3.4 for there to be more than 5% of the lubricant left atthe center of the contact after multiple contacts. For the exampleof industrial machines discussed above, this would require atime between contacts of less than 1µs, which is clearly unrealisticfor this scale.

It is clear that the dimensionless scaling parameter is key tothe successful modeling of liquid lubrication. The appropriatechoice of an effective analytic parameter that combines multipleeffects, and is relevant over different length scales, requiressignificant insight into the systems under study. This aspect isimportant, as it implies that the modeling cannot substitute fora sound physicochemical understanding of the system, ratherthat the modeling provides a complement to this understandingto achieve a quantitative global understanding of the problem.

V. Conclusions

Using a simplified flow model, a unitless scaling quantityconsisting of the logarithm of the ratio of the contact area to theproduct of the lubricant diffusion coefficient and the cycle timehas been identified that provides a multiscale measure of theeffectiveness of a liquid lubrication that is replenished via surface

flow. Using this scaling parameter, lubrication of industrialmachines via oil mist, the lubrication of MHDDs with high datadensities, and the conditions for which a B+ M lubricantcombination for MEMSs would be effective were analyzed. Forthe former, the scaling analysis confirms that surface flow cannotprovide effective lubrication, hence the need for an oil mist. ForMHDDs, the analysis demonstrates in a straightforward mannerthat the dimensions needed to advance data storage capabilitiesare consistent with an existing lubricant. For MEMSs, the analysissuggests that for megahertz and slower oscillation frequenciesthe ODTS/TCP combination will be effective for all but thelowest temperatures (less than∼100 K), but for gigahertzoscillations a higher mobility mobile phase is needed to effectivelyincrease device lifetimes.

Acknowledgment. K. Wahl and I. Singer of the U.S. NavalResearch Laboratory are thanked for valuable discussions. Thiswork was supported by the Office of Naval Research and theExtreme Friction Multi-University Research Initiative sponsoredby the Air Force Office of Scientific Research.

LA701280K

Multiscale Analysis of Liquid Lubrication Trends Langmuir, Vol. 23, No. 18, 20079257