The Effects of Molecular Weight and Temperature on Kinetic Friction

7/29/2019 The Effects of Molecular Weight and Temperature on Kinetic Friction

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Th e E ff e c t s o f M o le c u l a r We i g h t a n d Te m p e r a t u r e o n t h e

Ki n e t i c F r i c t i o n o f S i l i c o n e R u b b e r s

Kath erine Vorvolakos a nd Manoj K. Chaudhu ry*

Dep ar tm en t of Ch em ica l E n gin eeri n g, L eh igh Un iv ers it y, B eth leh em , Pen n sy lv an ia 18 01 5

R eceiv ed Decem ber 23 , 20 02 . In Fi n al For m : M ay 14 , 20 03

The frictional stresses ofp oly(dimeth ylsiloxane) elastomer s of various molecular weights were measu redagain st a supp orted m onolayer of hexadecylsiloxane a nd a th in film of polystyrene a s a fun ction of slidingvelocity and temperature. On both surfaces, friction decreases with molecular weight, but increases withsliding velocity, reaches a maximum, and thereafter it decreases or displays a plateau. While the velocitycorresponding to the maximum shear stress is nearly independent of the molecular weight of the polymer,it differs between the two substrates. These results are consistent with the models proposed earlier bySchallama ch as well as by Cher nyak a nd Leonov,a ccording to which the detachm ent force per load-bear ingchain increases with velocity while th e num ber ofcha ins supp orting th e total frictional load decreases withvelocity and molecular weight. From the temperature-dependent studies, the activation energy of frictionon both surfaces is estimated to be ∼25 kJ /mol, which is lar ger th an th e activation ener gy of viscous flowof silicone fluids, but compares well with the values obtained from r ecent studies of melt dyna mics.

I n t r o d u c t i o n

Fr ictional properties ofsoft elastomers are ofimportan cein a variety of settings, such as the shear resistance of viscoelasticadhesives,1,2 biofouling contr ol,3 road tractionof automotive tires,4 dur ability of windshield wipers,5,6

and slippery prosthetic devices,7-10 to nam e a few. Thereis,however, an incomplete understa nding ofthe molecularlevel parameters that control the frictional behavior of elastom eric surfaces. Early experim ents 11-13 on com-mercial na tur al ru bber products were performed for thesole pur pose oft abu latin g propert ies for consum ers. Suchtabu lation persisted until the ear ly 1950s, when Roth eta l.14 and Thirion 15 began experiments with the purposeof understanding the physics of rubber sliding. Quantita-tive physical analysis began with the observation thatth e classic Coulombic laws, obeyed consisten tly at int er-

faces between rigid bodies, fail at the interface betweena rigid solid and a rubber.

Papenhuyzen 16 as well as Roth et al .14 observed that

th e friction force of commer cial ru bbers on st eel increasesmonotonically with velocity. Beyond a certain velocity,however, sliding becomes unstable and t he ru bber sample“chat ter s”, or exhibits st ick -slip sliding. Thirion, 15 on th eother hand, observed th at the friction increases withnormal load, which wa s inter preted by Schallamach 17 t obe due to th e increase of contact ar ea resulting from th edeformation ofru bber asperities. Similar suggestions werema de by Bowden and Tabor. 18 Assuming th e asperities tobe hemispheres in Hertzian contact with smooth glass,Schallamach predicted that friction force would increasein a power law man ner, with a n exponent of 2/3. Indeed,th is prediction was verified over a limited ra nge ofnorm alload. However, Schallamach did not immediately addressa potentially fascinating finding that the friction force

increases with modulus. If friction force depend s on cont actarea, it is clear th at a softer mat erial would have a greatercontact area for any load, therefore exhibiting higherfriction, contrar y t o several experimenta l observations.Schallama ch’s hypothesis is th erefore incomplete. H emoved on to examine the effects of velocity and temper-a t u r e19 on rubber friction. As tem perature increases,frictional force decreases. Alternatively, at a given tem-pera tu re, th e friction force increases with slidin g velocity.Schallama ch sh owed tha t t he velocity- and temperat ure-dependent behavior of rubber friction follows Eyring’s 20

theory of reaction rates. When this theory is applied toexplain friction, inter facial sliding is presu med to proceedby the forma tion and breakage of molecular bonds at t heinterface in separate, thermally activated events.

While Schallama ch focused on th e molecula r pr ocessesat th e inter face, Greenwood and Tabor21 as well as Buecheand Flom 22 pointed out that the energy of sliding a soft

* To whom correspondence should be addressed at [email protected].

(1) Newby, B. Z.; Chaudh ury, M. K.; Brown, H. R. S cience 1995, 26 9(5229), 1407.

(2) Newby, B-m.; Chaudhury, M. K. L a n g mu i r 1998, 14 , 4865.(3) Vorvolakos, K.; Chau dhury, M. K. In Microstructure and Mi-

crotribology of Polymer S urfaces;ACS Symposium Series 741;AmericanChemical Society: Washingt on, DC, 2000; pp 83-90.

(4) Aggarwa l, S. L.; Fabris, H . J .; Hargis, I. G.; Livigni, R. A. Polym.Prepr. ( Am . Ch em . S oc., Div . Poly m . Ch em .) 1985, 26 (2), 3.

(5) Theodore, A. N.; Sam us, M. A.; Killgoar, P. C. Ind. Eng. Chem. R es. 1992, 31 (12), 2759.

(6) Extra nd, C. W.; Gent, A. N.; Kaang, S. Y. Rubber Chem. Technol.1991, 64 (1), 108-117.

(7) Dong, H.; Bell, T.; Blawert, C.; Mordike, B. L. J. Mater. Sci. Lett.2000, 19 (13), 1147.(8) Wang, J .; Stroup, E.; Wang, X.; Andrade J. D. Proc. SPIE-Int.

Soc. Opt. Eng. Int. Conf. Thin Film Phys. Appl. 1991, Pt. 2 835.(9) Murayam a, T.; McMillin, C. R. J. Appl. Polym. S ci. 1983, 28 (6),

1871.(10) Nusbaum , H. J .;Rose, R. M.;P aul, I. L.; Crugn ola, A. M.;Ra din,

E. L. J. Appl. Polym. Sci. 1979, 23 (3), 777.(11) Arian o, R. I n d i a Ru b b er J . 1930, 79 (2), 56-58.(12) Derieux, J. B. J. Elisha M itchell Sci. Soc. 1934, 50 , 53-55.(13) Dawson, T. R. Rubber : Physical and Chemical Properties;

Dawson, T. R., Porritt , B. D., Eds.; The Resear ch Association of BritishRubber Manufacturers: Croydon, U.K., 1935; pp 381-386.

(14) Roth, F. L.; Driscoll, R. L.; Holt, W. L. J. Res. Natl. Bur. Stand.1942, 28 (4), 439-462.

(15) Thirion, P . Rev. Gen. Caoutch. 1946, 23 (5), 101-106.

(16) Papenhuyzen Ingenieur 1938, 53 , V75.(17) Schallamach, A. Proc.Phys.S oc.,London, Sect. B 1952 , 65 ,657-

661.(18) Bowden, F. P .;Ta bor, D. Th e Friction and L ubrication of Solids;

Clarendon Press: Oxford, 1950.(19) Schallama ch, A. Proc.Phys. Soc.London, Sect.B 1952, 66 ,386-

392.(20) Eyring, H. J. Chem. Phys. 1936, 4, 283.(21) Greenwood, J . A.; Tabor, D. Proc. Phys. Soc., London 1958, 71 ,

989-1001.(22) Bueche, A. M.; F lom, D. G. Wear 1959, 2, 168.

6778 Langmui r 2003, 19 , 6778-6787

10.1021/la027061q CCC: $25.00 © 2003 American Chem ical SocietyPublished on Web 07/01/2003



viscoelastic material over a rigid substrate is not spent

entirely in breaking m olecular contacts at the interface,but at least partially on deforming the soft material.

The n otion that fr iction m ight be a com bination of surface and bulk effects prompted Grosch 23 to performthe most systematicstu dy in the field to date. He measuredthe effects ofvelocity, tempera tu re, an d surface roughness,while noting the synthetic m akeup of the elastom ers.Grosch observed that the rubber friction increases non-linearly with velocity, m uch like the shear thinningbehavior ofh igh viscous polymers . Above a cert ain criticalvelocity, t he friction force exhibits a stick -slip behaviorwith th e maximu m friction in each pulse decrea sing withvelocity. F urt hermore, at each sliding velocity, frictiondecreases with increasing tem peratur e. All these t em -peratu re- and velocity-dependent frictional dat a can be

assembled in a single master curve with the help of thewell-known superposition principle of Williams, Landel,and Ferry.24

For rubber sliding on optically smooth glass, Groschnoted that velocity corresponding to maximum frictionand the frequencycorresponding to maximum viscoelasticloss form a rat iot ha t is near ly const ant (∼7 nm) for variousmat erials. He rationalized this observation by assert ingtha t th e int erfacial relaxat ion processes responsible forfriction are related to the segm ental relaxation of thepolym er chain. The length scale of 7 nm represents amolecular length, presumably the characteristic lengthby which the molecular jumps occur during the slidingprocess. For rough sur faces,t he relevant length scale wasfound to be the characteristic spacing between surfaceasperities. Grosch’s general observations of th e depen-dence of friction on velocity an d temperat ure were alsosupported by Extrand et al . ,6 who exam ined the m orepractical geometry of sharp rubber edges against rigidsur faces. Extr an d et al. noted th at t he coefficient offrictiondepends strongly on the local load and the results aredependent on th e surface prepara tion, i.e.,chlorination of natural rubber.

Prompted by Grosch’s observations, Schallamach 25

refined his m odel of int erfacial friction, since a predictionof a monotonic dependence of friction on velocity wasclearly insufficient. He m aintained that unlubricatedsliding on sm ooth surfaces is essentially adhesive innature, m ediated by separate bonding and debonding

events between the rubber and the rigid surface, depictedin Figure 1.Schallamach’s25 explana tion of Grosch’s23 observations

was based on the rate-dependent molecular debondingm odel of Frenkel24 a n d E y r i n g .20 I n t h i s m o d el , t h eprobability of debonding a polymer chain from a surfaceis a product oft wo functions, the first being the frequencyfactor th at in creases exponen tially with th e applied forceand the second being the nu mber of load-bearing chainstha t decreases with velocity. The solution of the kinet icrate equation resulting from such considerations leads to

an expr ession for the debondin g force th at in creases withvelocity, while t he nu mber of the load-bearing polymerchains (Σ) decreases (Figure 2). The net effect is that thetotal interfacial stress at first increases with velocity,r e a ch e s a m a x im u m , a n d t h e r ea ft e r d ecr e a se s w it hvelocity.

Recently, Chernyak and Leonov 27 refined Scha llama ch’sm od el b y u s in g a s t e a dy s t a t e s t och a s t ic m od e l f ordebonding kinetics. Within this m odel, s tretching of polymer chains occurs as a result of an external force,leading tot he detachment oflinking chains from the wall.The detached chain r elaxes before reattaching to theinterface, during which tim e it dissipates energy andrelieves the tension accumulated during stretching. Byconsidering the stochastic nature of detachment force,Chernyak and Leonov27 derived th e shear str ess in dr ysliding a s given by eq 1

In eq 1, Σo is the a real density of the load bear ing chains

at zero velocity, φ(r (t )/ δ) is th e elastic energy stored in th epolymer chain, V is the sliding velocity, ⟨t ⟩b is the m eanlifetime ofcontact, ⟨t ⟩f is the time the polymer chain sp endsin free stat e, and p(V ,t ) is th e transition probability of th epolymer chain in going from the bonded to the relaxeds t a t e. T h e n u m er a t or of t h e C h er n ya k a n d L eon ovequa tion (eq 1)is th e work done in str etching th e polymerch a i n t o t h e b r ea k i n g p o in t , w h il e t h e d en om i n a t orr e p r es en t s t h e m e a n d is t a n ce t r a v el ed b y t h e ch a i n .Multiplication oft his stochast icforcewith th e areal densityoft he linking chains gives rise to th e expression for shearstress. Using a steady-state stochastic m odel of bonddissociation, Chernyak an d Leonov showed tha t th e meanlifetime ofconta ct ⟨t ⟩b and t he tran sition probability dependon the sliding velocity as shown, respectively, in eqs 2

a n d 3 .

Here, δ( z) repr esents Dira c’s delta function correspondingto th e deter mina te pr ocess of forced break -off, and θ(z) isthe Heaviside step function. With the above definitions(23) Grosch, K. A. Proc. R. Soc. London, S er. A 1963, 27 4, 21-39.

(24) Kontorova, T.; Frenkel, Y. I. Zh. Eksp. T eor. Fiz. 1938, 8.(25) Williams, M. L.; Landel, R. E.; Ferry, F. D. J. Am. Chem. Soc.

1955, 77 , 3701-3707.(26) Schallama ch, A. Wear 1963, 6 , 375-382.(27) Chernyak, Y. B.; Leonov, A. I. Wear 1986, 108 , 105-138.

F i g u r e 1 . The classic depiction of a polymer chain in contactwith a laterally moving countersurface. The chain stretches,detaches, relaxes, and reatta ches to th e surface to repeat thecycle.

F i g u r e 2 . The left figure qualitatively depicts the behavior of

the a real den sity of contact points an d th e force per a dsorptionpoint as a function of velocity. The former d ecreas es, while thelatter increases up to a value limited byt he interaction strengthbetween the polymer chain and the coun ter sur face.Th e productofth ese twoqu ant ities yields the shea r str ess, which increasesand subsequently decreases, depicted on the right.

σ t ) Σo

∫0

∞φ(r (t )

δ ) p(V ,t ) dt

V [⟨t ⟩b + ⟨t ⟩f ](1)

⟨t b⟩ ) τ o{1 - exp(- V

V o)} (2)

p(V ,t ) ) ex p(- t

τ o){δ(t - t b) -

θ(t b - t )

τ o} (3)

Kin etic Friction of S ilicon e R u bbers L an gm uir, V ol. 19, N o. 17, 2003 6779



of th e bond su rvival time and t he tr ansition probability,eq 1 can be integrated for simple Gaussian polymer chains,the elastic energy of which is proportional to th e squar eof the extension. Shear stress can then be expressed asfollows:

where m is the fun dament al ra tio of the lifetimes of thepolymer chain in t he free and bound st ates at zero slidingvelocity, s is the ra tio of th e viscous ret ar dat ion t ime overthe lifetime at r est, and u is th e dimen sionless velocity of sliding defined by eq 5

where τ o is the l ifetim e of the bound stat e at rest an d δis the average distan ce between th e polymer body an d thewall. σ a is defined by eq 6.

R F is the Flory radius of the polymer chain. Equation 4predicts that the shear stress first increases with velocityin an S-shaped manner. After exhibiting a rather broadm axim um , σ u s u a ll y d e cr e a se s a t v er y h i gh s li di n gvelocities.

Schallamach26 and Chernyak and Leonov27 developedtheir models envisioning pur elya dhesivesliding. However,Savkoor28 as well as Ludema an d Tabor29 suggested thateven seemingly adhesive sliding could never be purelyadhesive. Savkoor 28 proposed a hybrid model, in whichthe interface consists of discrete patches of asperities of molecular d imensions in adh esive cont act with th e rubbersurface. When a shear force is imposed, the patch storeselastic energy until it overcomes the adhesive energy,causing the propagation of a shear crack. According toSavkoor28 as well as Ludem a and Tabor,29 sliding mayproceed by an activated process, but th e extent to whichthe two surfaces come into conta ct depends on modulusand sliding velocity. Hidden in more macroscopic terms,these approaches of Savkoor 28 and Ludem a a nd Tabor 29

are similar to the model of Schallamach. 26

In ad dition to th e above molecular d escriptions ofru bberfriction, there are other viewpoints, which can be impor-tant especially when the adhesion between the surfacesis dominated by specific short-range interactions, and/orwhen one of the materials is excessively compliant. Inthese cases, the surfaces do not easily slide relative to

each other. Instead, the surfaces start peeling locally anddetachment waves propagate th roughout the entire areaofcont act star ting from its advan cing to the t railing edge.Schallamach 30 first discovered th ese waves at high slidingvelocities. Roberts and J ackson 31 suggested that whensuch instabilit ies occur, the frictional stress betweens u r fa ce s ca n b e d es cr i be d i n t e r m s of t h e a d h e si onhysteresis (∆W ), which is t he difference between theenergies involved in m aking and breaking interfacialcontacts, and the wavelength (Λ) of the Schallam achinstability, as σ ) ∆W / Λ. Recent th eories of interfacial

friction of Rice,32 Johnson 33 a n d K im 34 invoke otherdislocation models to describe the sliding of one surfaceagainst the other.

The above models, all of which offer plausible explana-tions for interfacialfriction, ha ve yet to be rigorously testedexperimenta lly. The decoupling of the myriad factorscontributing to interfacial friction requires not only acomprehensive experimental design but also the use of modelelastomericn etworks and well-char acterized, modelcountersurfaces. The elastomeric networks would have

to be chemically similar but differing in modulus, free of resins and fillers,and transparent for opticalexamination.Coun tersu rfaces would have t o be as smooth as possibleand free of secondary interactions. The model interfaceswould h ave to be r obust enough t o vary sliding velocityand tem perature without com prom ising th e ideality of the sliding materials.

Model stu dies ofth ese types have recently been initiat edby several a ut hors. For example, Brown 35 and Casoli eta l.36 examined the pulling out of polymer chains fromelastomeric networks and the associated friction. We 37

studied friction ofpoly(dimethylsiloxane)(PDMS)on somelow energy surfaces as a function of molecular weight of the polymer and the sliding velocity. Although we notedthat fr iction decreases with m olecular weight, these

stu dies were incomplet e as the sliding speeds were rat hersmall (V < 4 mm/s) an d a limited molecula r weight ra ngeofPDMS was used. In this paper, we extend these previousstudies. The current stu dies were carried out with cross-linked elast omeric networks of PDMS sliding on two lowenergy surfaces: a m ethyl functional self-assem bledmonolayer (SAM) of hexad ecylsiloxan e an d a th in film of polystyrene, both of which interact with PDMS viadispersion interactions. Usingt hesesimple modelsystems,we carried out the measurements ofadhesion and frictionto investigate h ow the latter depends on surface energy,temperat ure, velocity, a nd inter-cross-link molecularweight oft he elast omer. Roughn ess was purp osely avoidedso th at we could observe pu rely adh esive sliding as closelyas possible.

E x p e r i m e n t a l S e c t i o n

Materials. T h e P DMS elas t om ers were cros s -l in k ed b yplatinu m-cat alyzed hydrosilation ofvinyl end-capped siloxane 38

oligomers (H2CdCH(Si(CH3)2O)nSi(CH3)2CHdCH 2) with m eth-ylhydrogen siloxane cross-linker 39 (Syloff 7678: (H3C)3O-(SiHCH 3O) p(Si(CH 3)2O)qSi(CH3)3). This reaction system withoptimu m combina tion of divinylsiloxane oligomer an d the cross-linker yielded a highly cross-linked network with negligiblebyproducts. The molecular weights of the oligomers M were 1.3,1.8, 4.4, 8.9, 18.7, an d 52.2 k g/mol. The oligomers were mixedthoroughly with Pt(IV) catalyst and maleate inhibitor beforeadding t he cross-linker. The mass ratio of oligomer/cata lyst/ maleat e was 97.4:1.9:0.7 for all molecular weights . The pr opor-tional amount ofcross-linker added after thorough mixing variedwith molecular weight as 23 M -0.97, where M is in kg/mol.

(28) Savkoor, A. R. Wear 1965, 8, 222-237.(29) Ludema, K. C.; Tabor, D. Wear 1966, 9, 329-348.(30) Schallam ach, A. Wear 1971, 17 , 301.(31) Roberts, A. D.; Jackson, S. A. Na t u r e 1975, 257 (5522), 118-20.

(32) Rice, J . R.; Ben -Zion, Y. Proc. Natl. Acad. Sci. 1996, 93 , 3811.(33) Johnson, K. L. L a n g mu i r 1996, 12 (19), 4510.(34) Hurtado, A. J.; Kim, K.-S. Mater. Res. Soc. Symp. Proc. 1999,

539 , 81-92.(35) Brown, H. R. S cience 1994, 263 , 1411.(36) Casoli, A.; Brendle, M.; Schultz, J.; Philippe, A.; Reiter, G.

L an gm ui r 2001, 17 (2), 388.(37) Ghatak, A.; Vorvolakos, K.; She, H.; Malotky, D.; Chaudhury,

M. K. J. Phys. Chem. London, Sect. B 2000, 10 4, 4018-4030.(38) The PDMS oligomers were synth esized by ionic polymerization

and su pplied to us by Dow Corning Corporation. The num ber avera gedmolecular weights ofthese polymers were deter mined by gel-permeat ionchromatography and NMR. Oligomers ha ving the following M wereused: 1.3, 1.85, 2.738, 4.44, 8.88, 18.72, 52.17 kg/mol.

(39) Syloff7678 was char acterized by Jim Tonge ofDow Corn ing andfound to have M n a n d M w ) 3.5 an d 7.5 kg/mol, respectively. The SiHfunctionality makes up 70% of the molecule.

σ ) σ a(m + 1)

u (1 + s)(1 - exp(-1

u ) - ex p(-1

u ))m + 1 - exp(

-1

u

)(4)

u ) V τ o / δa (5)

σ a )k T Σoδ

(1 + m ) R F 2

(6)

6780 L an gm u ir, V ol. 19, N o. 17, 2003 V orvolak os an d Ch au d h u ry



The resulting mesh sizes ofth e networks were estimated usingthe sta ndar d method of swelling in solvent (see for example Pa telet al.,40 who studied th e properties of ideal PDMS networks). Asheet (1 mm t hick) of each net work was cured, from which smallrectan gular pieces having dimensions 2 cm × 2 cm were cut out.These were immer sed in toluen e (with which PDMS ha s a solventinteraction parameter χ of0.465) overnight t oen sure equilibriumswelling, after which their swollen dimensions were carefullymeasured. The equilibrium PDMS volume fraction φ i n t h epresent systems very closely resembled the result s ofP atel et a l.In both cases, it may be approximat ed as φ) 0.7 M -1/3, where M

is in kg/mol. The classic Flory-Rehner equation,41,42 whichassumes that the only connections between network chains arethe chemical cross-links, fails in predicting the equilibriumvolume fraction ofPDMS. As described byP atel et al., the swollen

dimensions of the present networks a lsocorr esponded t oeffectivemesh sizes smaller tha n t he oligomeric precursors. Patel et al.ascribe this phenomenon to the int erspersion ofoligomericchainsthat are not relieved and, in fact, tr apped by the formation of chemical cross-links. To calculate the effective mesh size, Pat elet al. consider the experimentally measured equilibrium elasticmodulus E and invoke the affine model described by eq 7

where F is the density of the polymer.We followed a procedure similar to that of Patel et al. To

measure the equilibrium elastic modulus of each network, themeth od of contact m echanics as developed by Johnson, Kendall,and Roberts 43 (JKR) was used. Hemispherical lenses of each

network were pr epared by depositing small drops ofth e reactionmixtur e onto perfluorinat ed glass slides and cross-linking themat 120° for 50 min. These lenses were th en brought in to and outof contact with the substrate of choice under controlled loads.The load-deforma tion data obtained from these experimentsyielded not only the elastic moduli oft he networks but also theiradhesion energies with the countersurface.

For sliding friction mea sur ement s, the lenses were allowed toslide later ally on the substra tes. It was however noticed tha t thelow modulus ( M > 4.4 kg/mol) hemispheres deform laterally,thus compromising the accuracy of the shear stress measure-ments. To avoid such complications, we transferred thin filmsof these high molecular weight PDMS onto more rigid lenses of P DMS ( M ) 3.5 kg/mol) according to a method described byChaudhury.44 Briefly, thin films (∼20 µm) of the high molecularweight PDMS elast omers were cast onto a silicon wafer, whichwas ma de nonadherend to the PDMS film by coating it with a

monolayer of hexad ecyltr ichlorosilan e. After both the PDMS filmand lens were oxidized using plasma , they were pressed togetherfor about 1 m in, during which the plasma-oxidized polymersbegan to weld toea ch oth er. When the normal load was released,the P DMS lens peeled off the t hin P DMS film from the silanizedsilicon wafer (Figure 3). These specially made lenses were notused immediately but were left in contact overnight t o ensuresecure welding between the thin film an d t he u nderlying lens.

Unreacted oligomers were extra cted from all lenses withchloroform in a Soxhlet extra ctor for 12 h. They were th en allowedto dry at room t emperatur e for 1 week under a gentle vacuum(∼0.8 atm) before being used in any m easurements.

Contact mechanics and friction measurements were performedagainst two low-energy surfaces: a self-assembled monolayer(SAM) of hexadecylsiloxane, and a thin film (∼10 0 µm ) o f polystyrene. The SAM was prepar ed by rea cting a clean polishedsilicon wafer with t he vapor ofh exadecyltrichlorosilane a ccordingto a method previously published.45 The sur face energy of theresultant surface was ∼19 mJ/m 2, as estimated by th e contactangle ofh exadecane (45°),wh ich exhibited n egligible hyster esis,indicating lack of gross sur face imperfections. The polystyrenefilm was prepa red by casting a toluene solution of the polymer

( M ) 1.5 × 10 6 g/mol) on a clean silicon wafer a nd a llowing th esolvent to evapora te slowlyin a covered Petri dish at at mosphericpressure for 1 week.

C o n t a c t M e c h a n i c s . Following the well-known method of Johnson, Kendall, and Roberts,43 a h em i sp h erical l en s wasbrought into conta ct with the substrat e ofinter est at zeroa ppliedload. The lens was then loaded extern ally in a qua si-equilibriummanner up to a maximum load of 0.2 g. Subsequently, it wasunloaded in the same mann er. During each loading and unloadingcycle, the contact area was viewed using a reflection microscope,while the load was recorded in an electrobalance interfaced toa personal comput er. These load-deformation data were ana lyzedusing th e well-known J KR equation (8) in order to estima te th eadhesion energy W of the int erface and th e elastic modulus E of the P DMS lenses

where a is the contact radius, R is the ra dius of curvatu re of thelens, and P is the normal force.

M e a su r e m e n t o f F r i c t i o n . Frictional properties betweenthe lenses and the countersurfaces were examined using twometh ods, both of which requ ire use of the setup r epresent ed byF i g u re 4 . In a m an n er rem i n i s cen t o f R o t h et al . , 14 velocityrelaxation data were combined with steady-stat e data. The formerdata were obtained using a method described by Brown 35 a n dC h au d h u ry2,46 to mea sure the friction at low sliding velocities.

(40) Patel, S. K.; Malone, S.; Cohen, C.; Gillmor, J . R.; Colby, R. H. M acrom olecu les 1992, 25 (20), 5241.

(41) Flory, P. J . Principles of Polymer Ch emistry; Cornell Un iversityPress: Ithaca, NY, 1953.

(42) Flory, P. J.; Rehner, J. J . Ch em. Ph y s. 1943, 11 , 521.(43) John son, K. L.; Kendall, K.; Roberts, A. D. Proc. R. S oc. Lond on

1971, A324, 301.(44) Chaudhur y, M. K. J . Ph y s. Ch e m. B 1999, 103 , 6562.

(45) Chaudh ury, M. K.; Whitesides, G. M. L a n g mu i r 1991, 7 (5),1013.

(46) Chaudhury, M. K.; Owen, M. J . L a n g mu i r 1993, 9, 29.

F i g u re 3 . The preparation of low-modulus samples forfrictional testing. A th in (∼20 µm) film of the desired network is cast on a hydrophobic Si wafer. The a ir-exposed surface of the film and a higher-modulus lens are both plasma oxidizedand welded together to form a composite lens which does not

deform during friction measurements.

M e ) F R T / E (7)

F i g u r e 4 . The appa rat us for frictional testing of elastomericlenses. The lens is placed on t he end of a calibrated spring, thedeflection ofwhich gives the frictional force. The frictiona l force,normalized by the contact area, yields the shear stress. Boththe deflection and the contact area are viewed using a high-speed camera . Stea dy-state and velocity relaxat ion experiment sare both performed using this setup.

a3/2

R) 9

16 E P

a3/2+ 3

4(6π W E )

1/2 (8)




In t his method, the lens was placed on one end of a calibratedspring, the other end of which was rigidly supported. After thelens was brought into contact with the substrate of interest, thelatt er was given a sudden displacement. The lens at first movedwith th e substr ate but then relaxed back to its original positionas t he spring r ecovered its neutr ality.

With the deflection of the spring monitored as a function of time, the force acting on the lens was determined as a functionofsliding velocity.Division oft his force byt he cont act ar ea yieldedthe sh ear st ress as a function ofvelocity. The velocity ra nge thu sachieved was from 10-7 to 10-3 m/s.

Measurements at higher velocities were obtained by slidingthe su bstra te relat ive to the lens at uniform velocities while thelens rested on the edge of the calibrated spring. The velocityrange ofth ese steady-state m easurements was from 10-4 t o 1 0-1

m/s. Some mea surem ents wer e tak en at even lower velocities toensure agreement with the relaxation data. Inertial forces werenegligible in all these measurements. Up to a critical velocity,slidingwa s stable, beyond which the friction force exhibited stick -slip dynam ics similar t o that r eported by Grosch.23 When theseinsta bilities occur red, friction force corr esponding to the highestd efl ect i on of t h e s p rin g was record ed , i n accord an ce wit hGrosch’s23 procedure. This force, divided by t he corr espondingconta ct area (mea sured simu ltan eously by the camera), yieldedthe shear stress. The combination of the above two methodsallowed us to measur e t he friction force in the range of 10-8-

10-1 m/s,comparable to the ran ge of velocities employed byReiter

et al.36 to study t he effect of the pu ll-out of gra fted PDMS cha insfrom PDMS networks.

S h ear s t res s es were i n v ari an t wi t h res p ect t o ch an g es i nnormal load (5.5-120 mN for stable sliding, 24-50 mN forunstable sliding) and t hus to the contact area. These findingsare consistent with ear lier findings of Homola et al.47 a n dC h au d h u ry et al .2,46 an d i n di cat e t h at t h e r at i os o f act u al t onominal contact areas do not increase with increasing load, incontr ast with t he findings of Scha llamach 17 an d B ah ad u r,48 a n dthe predictions of Ludema and Tabor.29

Measurements at various temperatures were carried out byh eat i n g t h e s u b s t rat e wi t h an i n frared l am p . T h e s u b s t rat etemperature was carefully controlled by adjusting the distanceof t h e l a m p f r om t h e s u bs t r a t e a n d m e a su r e d u s i n g a fl a tthermocouple (OMEGA SAJ-1) adhered to th e substr ate. Thetemperature range employed was 298-348 K. The substrates

and the PDMS networks are all hydrophobic, but lower tem-peratur es were not attempted as a precautionary measure, soa sto avoid condensation effects. Higher t emperatur es were n otatt empted so as to avoid morphological chan ges in the S i wafersand/or an incipient glass tr ansition in the P S th in film.

The P DMS networks were n ot reinforced with an y resin orfiller. As such, they were easier to abrade than commercialmaterials. The full r ange of molecular weight was allowed toslide against the SAM-covered wafer. H owever, th e two lowestmolecular weights ( M ) 1.3 and 1.8 kg/mol) could not withstandthe entire velocity range and abraded easily on the surface.Against the PS-covered wafer, only the networks with M g 8.9kg/mol could withstand the sliding, even at low velocities.

R o u g h n e s s M e a s u r e m e n t . The root mean square rough-nesses oft he SAM (0.2 nm) and P S (0.5 nm) coat ed silicon wafersover an ar ea of1 µm 2 were mea sur ed by Olga Scha ffer (Emu lsion

Polymer Institute, Lehigh University)using the method of atomicforce microscopy (AFM).

The roughness values of the PDMS elastomers were generouslyp rov id ed b y Yu j ie S u n an d Gil bert W alk er (Un i vers it y of Pittsburgh). The root mean square r oughness values of all theelastomers wer e less tha n 0.5 nm except for t he PDMS of M 1.3kg/mol, for which the roughness wa s estima ted to be 1.0 nm. Theroughness values of the PDMS elastomers ar e consistent withthose found by Efimenko et al.49 using both AFM and X-rayreflectivity measurements.

R e s u l t s

C o n t a c t Me c h a n i c s . The cont act mechan ics dat a ar edisplayed in Figure 5,wh ere a3/2 / R is plott ed against P / a3/2.These plots, in conformity with eq 8, are stra ight lines,the slopes an d int ercepts of which yielded the values of E a n d W , respectively. F or PDMS of inter-cross-link molecular weights ( M ) of 1.3-18.7 kg/mol, the loading/ unloading dat a do n ot exhibit any n oticeable hysteresiseither on the SAM- or on PS-coated Si wafers.

The works of adhesion on the SAM-coated waferclustered ar ound 41-42 mJ/m 2, being independent of themolecular weight (Table 1). For PDMS on the PS-coatedwafer, these values were somewhat higher: 51-5 6 m J /

m2

. F or the highest m olecular weight PDMS (52.2 kg/ mol), th e loading cycle yielded valu es of W as 27 and 26m J/m 2 on the SAM and PS, respectively, whereas thecorresponding values were 55 and 68 mJ/m 2 d u r i n g t h eunloading experiments. The finite hysteresis observedwith t his molecula r weight resu lted from slight viscoelast icdeformation of the rubber, which is due to incompletecross-linking of th e network.50

As expected, th e Young’s modulus, a s obtained fromthe above J KR ana lysis, is found to be inversely propor-tional t o molecular weight (Figure 6).

C o n ta c t Ar e a d u r i n g S l i d in g . The combinat ionof the transparency of PDMS, the reflectivity of thecou n t er su r fa ces , a n d t h e u s e of t h e h igh -s pee dcamera allowed direct examination of the contact area

(47) Homola, A. M.; Isr aelachvili, J. N.; McGuiggan , P. M.; Gee, M.L. Wear 1991, 13 6 , 65.

(48) Bahadur, S. Wear 1974, 29 , 323-336.(49) Efimenko, K.; Wallace, W.; Genzer, J . J. Colloid I nterface Sci.

2002, 254, 306 and references therein.

Figure 5. The cont act ar ea as a function of norm al load allowscalculation of the net work modulus an d th e work of adhesionat ea ch inter face. As the slope ofthe line increases, the modulusdecreases. The symbols open circle, gray circle, black circle,open box, gray box, black box, and open triangle represent

networks with oligomeric precursors of 1.3, 1.8, 2.7, 4.4, 8.9,18.7, and 52.1 kg/mol, respectively.

T a b l e 1 . T h e A d v a n c i n g W o r k o f Ad h e s i o n f o r Al lN e t w o r k s o n t h e S A M- a n d P S - Co v e r e d S i W a fe r s a

M (kg/mol) W PDMS-SAM (mJ/m 2) W PDMS-PS (mJ/m 2)

1.3 42 531.9 41 552.7 44 564.4 42 538.9 42 52

18.7 42 4452.1 27 26

a The stren gth ofint eraction is largely independent ofm olecularweight. The low values for M ) 52.1 kg/mol are attr ibuted toviscoelasticity.




a s a fu n c t ion of v e loci t y. F i gu r e 7 s h ow s t h a t t h econtact area rem ains constant up to a sliding velocityof 1 m m /s . O n ly a b ou t 1 0% r e du ct ion of con t a ctradius occurred close t o t he tran sition from sm oothto stick -s l i p s l i d i n g , w h i c h a p p e a r s t o b e d u e t o t h etransition from JKR to Hertzian deformation resultingfrom the loss of adhesion as predicted by Savkoor andBriggs.51

F r i c t i o n : S y s t e m D y n a m i c s . The shear stress dataof PDMS networks sliding on the SAM-coated wafer(Figure 8) show that σ at first in creases with velocity an dthen either levels out or decreases. Frictional force is stablew h e n dσ /dV g 0. However, at the negative resistancebran ch (dσ /dV < 0) of the str ess velocity cycle, frictiona l

sliding is unstable and periodic(Figure 9)as was reportedpreviously by Grosch.

The spring deflection at the higher frictional stressachieved during these stick -slip l im it cycles is t he

un sta ble focus point.52

At a given imp osed velocity, frictionforce reaches a maximum value when t he lens slips by acertain distance before it is captured by the substrate andbrought back to the point of m axim um stress to repeatthe process.Such sliding instability occurs when th e springconstant is less than a critical value given by eq 9

where V is the imposed velocity, A is the contact area, andd o is known a s t he m emory length , which is typically on

(50) These advancing and receding works ofadh esion are essentiallythe stra in energy release rates (G). In the advancing mode, the energyto close the crack comes from the thermodynamics work of adhesion(W ), which is equal tothe strain energy release rate (G) plus the energyloss (Φ) due to viscoelastic deformation of the polymer. Thus themeasured str ain energy release rate is less than the work of adhesion,i.e., G ) W - Φ. Conversely, when the crack is opening, the viscousdissipation a d d s to the strain energy release rate, as the mat erial mustbe deformed before it detaches from t he su rface, thus increasing th ereceding work of adhesion (G )W +Φ). See the following reference formore details: Johnson, K. L. In Microstructure and Microtribology of Polymer Surfaces; ACS Symposium Series 741; American ChemicalSociety: Washingt on, DC, 2000; pp 24-41 .

(51) Savkoor, A. R.; Briggs, G. A. D. Proc. R. S oc. L ondon, S er. A1977, 356 , 103.

(52) Ronsin, O.; Coeuyrehourcq, K. L. Proc. R. Soc. London, Ser. A2001, 457 , 1277.

Figure 6. Young’s modulus E is linear with inverse molecularweight M -1.

F i g u r e 7 . The ratio of sliding to resting contact area as afunction of velocity for PDMS ( M ) 2.7 k g/mol, R ) 2.5 mm, E ) 4.8 MPa, W ) 42 mJ /mol) against t he SAM. As th e slidingv el oci t y i n creas es, t h e con t act area d rop s from t h e J K Rprediction (black line) to the purely Her tzian prediction (gra yline). The normal load P averaged 48 mN (see eq 7).

Figure 8. Shea r str ess as a fun ction ofvelocity between PDMSand the SAM-covered Si wafer. Open circle, gray circle, black circle,open box, gray box, black box, an d open tr iangle repr esentnetworks with oligomeric precursors of 1.3, 1.8, 2.7, 4.4, 8.9,18.7, and 52.1 kg/mol, respectively.

F i g u r e 9 . Stick -slip sliding is characterized by a periodicfriction force fluctuation. As the countersurface moves at aconst an t velocity, the elast omeric lens is simult an eously slidingan d being deflected (solid curves). The actu ally sliding velocityincreases up to the imposed velocity, at wh ich point the inter faceslips (dashed lines). Here we ha ve shown a typical force tr aceobtained with a PDMS of M ) 4.4 kg/mol sliding at 2 cm/s onthe SAM surface.

k < k c ) -V

d o A

dσ

dV (9)




the order of th e distance between th e surface asperities.At the onset of instability, d o / V is on the order of therelaxation time (10-7 s) of the polymer a nd A(dσ /dV ) is onthe order of 0.1-1.0 N s/m. Substitu tion of these valuesin eq 8 yields the m agnitud e of the spring consta nt (∼10 7

N/m) that would be required t o avoid sliding instability.The spring constants used in our experiment are on theorder of 102 N/m, which is mu ch sma ller th an k c. Hence,sliding inst ability always occur s in our experiment s, evenwhen dσ /dV is slightly negative. When such instabilities

occur, we record the maximum shear stress just beforethe lens slips.

F r i c t i o n : E f fe c t s o f M o l e c u l a r We i g h t , V e l o c i t y ,a n d T e m p e ra t u r e . T h er e a r e a b ou t fou r i m por t a n tfeatu res ofth e kinetic friction observed in our experiments:

1. The friction decreases with the molecular weight.

2. The friction increases with velocity, r eaches am axim um , and then either decreases or plateaus out.

3 . T h e p e a k v el oci t y i s n e a r ly i n de p en d e n t of t h emolecular weight.

4. The friction peak broadens with molecular weight.It becomes independent of velocity when th e m olecularweight of PDMS reaches 18.7 kg/mol.

As is th e curr ent situ ation, there is no complete theory

ofk inetic friction tha t can account for all the observationssum m arized above in precise quantitative term s. Theobservations a re, h owever, qualitat ively consistent withth e stocha stic th eory ofru bber friction, as discussed below.

First, let us consider t he inverse relationship betweenrubber friction an d molecular weight, which ha s alreadyb ee n ob s er v e d w it h m e lt s 53,54 a n d g r a ft e d p ol ym e rchains.2,37 Tou nderst and this observation, let us considerLud ema an d Tabor’s29 suggested relationship between th eshear stress σ a n d t h e a r e a l d e n s it y (Σo) of the contactpoints as σ ) Σo f o, where f o is the force needed to detacha single polymer chain du ring sliding. This is similar tot h e p r e fa ct or i n t h e C h er n y a k -Leonov equa tion (4)corresponding to th e shear str ess in the high velocity limit,i.e., where the detachment of the polymer chain from the

surface is not controlled by stochastic processes. Withinthe simple model developed by Ch ernyak and Leonov,27

the areal density of the load-bearing chains is 1/ N a2, N being t he n umber of stat istical segments, each of size a .One thus obtains that the shear stress is proportional tot h e s h ea r m od u lu s a s σ ) Gf oa / k T . T h e C h e r n y a k -Leonov27 m odel is , however, not applicable to a realelastomer, where the areal density ofpolymer chains scalesas N -1/2. One th us anticipates a relat ionsh ip between shearstress and shear m odulus as σ ∼ G1/2. Experimentally,however, a power law exponent close to 3/4 has beenobserved (Figur e 10). While Grosch23 did not systemati-cally stu dy the effect of modulu s on friction, he noted t ha tthe sh ear str ess he obtained is considerably smaller th antha t expected of two surfaces in tru e molecular conta ct.

To account for the discrepancy, Grosch estimated theactual area of contact to be approxim ately 10% of th eapparent contact area during sliding.23 In our case, theAF M s t u die s i n d ica t e t h a t b ot h t h e P D MS a n d t h ecountersu rfaces are sm ooth to nan ometer length scales.Hence, gross m ism atch of interfacial contact is notexpected ba sed on roughn ess consider at ions. However, itis plausible that spontan eous roughening of the int erfaceoccurs as a result ofelasticinstability, which ensues fromth e competition between van der Waa ls and elastic forceswithin the first layer ofstretched PDMS chains in contact

w it h t h e s u r fa ce . I f w e con s id er t h a t t h e d om i n a n twavelength of such roughening scales with the thickness(δ) ofth e first layer of the polymer chain, then th e densityof th e load-bearing sites should scale as 1/ δ2 (or 1/ N a2).If one polymer chain r emains active in ea ch of the load-supporting junctions, one essentially recovers the result:σ ∼ G. Shear stress should decrease with the molecularweight becau se th e nu mber ofload-bearing polymer chainsdecreases with molecular weight. H owever, when themolecular weight reaches rather high values ( M g 18.7kg/mol), σ ma x becomes n early in dependent of molecularweight. At high m olecular weights, the above sim pleana lysis becomes less effective, du e to complicationsarising from th e enta nglement effects.

The dependence of shear stress on molecular weightalsoa ddresses a long-stan ding question on the relationshipbetween friction and en ergy losses due to bulk viscoelast icdeform at ion. Up to now, int erfacial friction force ha s beenlargely att ributed t o bulk dissipation,21,22,62 which arises

due to cyclic deformation and relaxation cycles of theru bber moving over rough asper ities. We pur posely chosemolecularly smooth surfaces so as to avoid such bulk dissipation. Even if we consider the effect of bulk dis-sipation in frictional loss, the observed trend is quiteopposite to th e predictions based on their bu lk rheologicaldata. Gordon et al.55 reported th e storage and loss moduliof several cross-linked PDMS networks sim ilar to thepresent ones, which show th at the viscoelastic lossmeasured in terms of the phase angle (δ) increases withmolecular weight (typically at a low frequ ency (aTω ∼ 10Hz), th e phas e angle varies with m olecular weight a s log-(tan δ)∼ 0.1 M , where M is in k g/mol). If friction is ca us edby bulk dissipation, shear stress sh ould increase withmolecular weight. We, in fact, observe just the opposite

behavior: shear stress decreasing with molecular weight,thus clearly pointing out that the frictional dissipationfor th e PDMS elastomers is not due to th e bulk viscoelast icdeformation.

It is noteworth y, as shown in Figur e 8, th at t he velocityat wh ich stick -slip transit ions occur is near ly independentof the molecular weight of the polymer for all molecularweight s of PDM S (except for M ) 52.2 k g/mol). Accord ingto Grosch, this transition should occur at a velocity V ogiven by the ratio of a molecular length scale ( λ ∼ 7 n m )an d the relaxat ion time oft he polymer chain . This criticalvelocity, according to Chern yak and Leonov, appears at

(53) Inn, Y.; Wang, S.-Q. Phys. Rev. Lett. 1996, 76 (3), 467-470.(54) Hirz, S.; Subbotin, A.; Frank, C.; Hadziioannou, G. Macromol-

ecules 1996, 29 , 3970-3974.(55) Gordon, G. V.; Owen, M. J .; Owens, M. S.; Per z, S. V.; Stasser,

J. L.; Tonge, J. S. Proc. Annu. Meet. Adhes. Soc. 1999, 424.

F i g u r e 1 0 . Peak s h ear s t ress σ ma x between PDMS and theSAM-covered Si wafer is proportional to G 3/4.




V o ) δ cot χ / τ o. As δ∼ N 1/2 a an d cot χ∼ f oδ / k T , V o ∼ f o N a2 / k T τ o. Here th e relaxation time of the polymer chain τ ο is

related to the segmental level relaxation time τ a s τ ο )τ N β, β being an exponent th e value of which depends onthe mode of relaxation of the polymer chain. Chernyak and Leonov proposed the value of β to be 3/2 (an un likelyscenario in dense state), which results in the molecularvelocity V o decreasing with N following a 1/2 power law.Experimental observation is that V o is nearly independentofth e molecular weight ofth e polymer, suggesting a valueof β close to un ity. Thus, V o appears to be the segm entallevel velocity of th e polym er chain on the surface asconjectured by Grosch,23 who estimated this relaxationtime from the frequency (ω) at which th e loss modulu s of the polymer exhibits a m aximum. Un fortu nat ely, such acomparison is not possible for PDMS, as its segmentalrelaxation frequen cyis so high th at it ha s not been possibleto measu re it by rh eological spectr oscopy. On t he ba sis of the fact th at th e segmental length ofPDMS is 0.6 nm, andthat the friction peaks appear at a sliding velocity of 1cm/s, the r elaxation time of PDMS segments in conta ctwith the methyl SAM coated surface is estimated to be∼10-7 s. This time is considera bly lar ger th an t he viscousrelaxation time (10-11 s) of dimeth ylsiloxan e monomer, 56

suggesting that the m obility of the polym er chain isseverely m odified by its interaction with the surface.Further support to this conjecture is given below.

Fr ic tion as an Ac tivate d R ate Pr oc e s s . Shear stressof PDMS depends on temperature, as shown in Figure 11for PDMS sliding on the SAM surface. If rubber frictionis a therm ally activated rate process, then it should be

possible to shift the shear str ess data obtained at differenttem peratures to room tem perature by m ultiplying thesliding velocities with a su itable sh ift factor. As the glasstransition tem perature (T g ∼ -130 °C) of PDMS is farlower than an y ofth e testing tempera tur es, an Arrheniusshift factor aT (eq 10) is adequa te for th e above purpose.

F r o m t h e s h i f t f a c t o r u s e d t o u n i f y t h e t e m p e r a t u r e -depend ent dat a (see Figur e 12),t he activation energy ( E a)of sliding of PDMS on th e SAM-covered wa fer was foun d

to be 25 kJ /mol. This activat ion en ergy is also found t o beindependent of molecular weight but is five to six t imes

larger tha n th e typical depth ofa van der Waals potentialwell.

Stein et al.57 ha ve stu died th e dyna mics ofP DMS cha insin the melt state by measuring the fluorescent decay of a probe chrom ophore. They noted that the therm allyactivated local dynamics follow an exponential behaviorwith activation energies in the range of 20-27 kJ/mol,which are considerably higher than the activation energy(13-16 kJ /mol) of viscous flow but close to th at observedin our dynamic friction stu dies.

On t he ba sis of the above values of activation energy(25 kJ/mol) and relaxation time (10-7 s), it is tempt ing toestimate the pre-exponent ial factor τ * of the Arrheniusequation τ ) τ * exp( E a / R T ). τ * is estimated to be on theorder of 10-12 s, which is very close to the value (h / k T )

predicted by Eyring.20 This is som ewhat a surprisingresu lt, as th e pre-exponen tial t ime scale for t he diffusionofpolymericsegmen ts on surfaces61 is usually a few ordersof m agnitude higher th an the elem entary tim e scale inEyring’s kinetics.

P e a k B r o a d e n in g w i t h Mo l e c u l ar We i g h t . Animportant observation of these friction data is that thepeak a t which ma ximum friction occur s broadens as th emolecular weight increases.To un derstan d this effect, werecall the models of Schallamach 26 a n d C h e r n y a k a n dLeonov,27 which su ggest two t ypes of processes occurr ingat t he interface dur ing frictional sliding: th e debondingforce increasing with velocity, while th e n umber of theload-bearing polymer chains (Σ) decreases wit h velocity.

The deta chmen t force, in general, is contr olled by th estochast ic process, un til very high velocities. The arealdensity of the load-bearing sites however decreases withvelocity, as the time of detachment of the polymer chaindecreases and thus approaches its free relaxation time.At an y given velocity, Σ(t ) can be expressed in terms oft hebonded and relaxed time of the polymer as follows

where ⟨t ⟩b is the time ofcontact between t he polymer chain

(56) Appel, M.; F leischer, G. Ma cromolecules 1993, 26 , 5520.(57) Stein, A. D.; Hoffman, D. A.; Marcus, A. H.; Leezenberg, P. B.;

Frank, C. W.; Fayer, M. D. J. Phys. Chem. 1992, 96 , 5255.

F i g u r e 1 1 . Shear stress as a function of velocity and tem-perat ure between PDMS a nd t he SAM-covered Si wafer for M ) 2.7 kg/mol.Open circle,gr ay circle, an d black circle represen tdata at 298, 318, and 348 K, respectively.

log aT ) E a

2.3 R [1

29 8-

1

T ] (10)

F i g u r e 1 2 . T h e t emp erat u re-d epen d en t s h ear s t ress d at ashifted to room temperature using an Arrhenius shift factor.The activation energy of sliding between PDMS a nd t he SAMis thu s estima ted t o be 25 kJ /mol. Open circle, gra y circle, andblack circle represen t dat a at 298, 318, an d 348 K, respectively.

Σ ) Σo

⟨t ⟩b

⟨t ⟩b + ⟨t ⟩f

(11)




a n d t h e s u b s t r a t e a n d ⟨t ⟩f is the relaxation tim e of thep ol ym e r ch a i n i n t h e u n b on d e d s t a t e . C h e r n y a k a n dLeonov27 argued t hat all the segments of a polymer chainha ve to be activated for it to desorb from a su rface. Whilea cata strophic desorption of the polymer chain may n otrepresent the reality, the alternative possibility thatdesorpt ion is a sequent ial process akin to peeling is equallyconsist ent wit h th e lifetime of cont act being proportiona lto molecular weight. Th e lifetime of conta ct, as sh own ineq 2, however, decreases with velocity. For higher mo-lecular weight polymers, a larger velocity mu st be reachedbefore t he chain desorbs from th e sur face. I t is th usexpected that the peak corresponding to the stick -sliptransition should broaden with molecular weight.

On t he basis of the above discussions, we note th at th eChernyak -Leonov27 model, as embodied by eq 4, takes

into account most of the results of PDMS ru bber slidingon the SAM-coated silicon wafer. To examine the fullprediction oft his model,we simulat ed the frictional shearstresses of a rubbery network on a surface using eq 4under two simplified assumptions. The first is that theterm (m + 1)σ a cot χ is replaced by the sh ear m odulus G.The second assum ption is that th e nondimensional velocityu is independent of the molecular weight. The pa ram eterm , which is the ra tio oft he relaxation time of th e polymersegm ent in the detached state to that in the adsorbedstat e, is varied from 0.1 t o 0.003 in order to observe thegenera l effect of molecula r weight on peak breadt h. Thesesimulations, as summarized in Figure 13, show that thepeak width indeed increases with molecular weight ofthepolymer. For very small values of m (i.e., at very highmolecular weight s),a plat eau is observed. Experiment ally,however, we are restricted to the plateau region of thepeak.

F r i c t i o n a l B e h a v i o r o f P D M S o n P o l y s t y r e n e . Asshown in Figure 14,th e general pattern offriction ofPDMSon PS (i.e., its depend ence on sliding speed a nd m olecularw ei gh t ) i s s im i la r t o t h a t on t h e S AM . T h e fr i ct i ondecreases with molecular weight. It increases with thesliding speed, th en reaches a critical velocity beyond whichit either decreases or exhibits a plateau. The velocity a twhich the friction reaches a maximum or a plateau is, atm ost, a n order of m agnitude lower t han that observedwith SAM. What is significantly different between thetwo sur faces is th eir behavior in the ran ge of low velocity,

where polystyrene exhibits a much larger t ail than SAM(Figure 15).

To un derst an d whether th is difference in cha in mobilityis reflected in the activation energy of kinetic friction,frictional stresses were measured at different tempera-tures. These kinetic friction data, when shifted to roomtemperat ure using the Arrhenius tr ansform as done with

the SAM data, yield an activation energy of ∼27 kJ/mol,which is nearly same as that observed on a SAM. Thusth e difference of friction on t he t wo surfaces could n ot beexplained on the basis of their energetics. Differences of surface roughness cannot clearly explain this differenceeither, as the countersurfaces used for these experimentsare sm ooth down to nan ometer length scales (0.2 nm ona SAM and 0.5 nm on PS). The slight difference in sur faceroughness, should they play a role in th e sense that energydissipation increases in the bulk, ought to shift 23 t h efriction peak on th e PS to a slight ly higher velocity. Thusthe low velocity frictional behavior on the two surfacesmust originate from other mechanistic effects not con-sidered in th e presen t theories sofar. One such possibilitywould be t o consider th e coupled dyna mics of th e poly-

Figure 13. Shear stress as a function of velocity as pr edictedbythe Chernyak -Leonov model for adhesive friction. The valuem is the r atio of the lifetimes of the polymer chain in t he freeand bound state.

Figure 14. Shear str ess as a function ofvelocity between PDMS( M g8.9 kg/mol) an d the PS -covered Si wafer. The onset velocityof stick-slip sliding (∼10-3 m/s) is an order of magnitude lowerth an on th e SAM-covered wafer. Open box, gray box, black box,and open t riangle represent networks with oligomeric precur-sors of 4.4, 8.8, 18.7, and 52.1 kg/mol, respectively. Networksofsm aller precursors abr aded against polystyrene, not allowingshear stress to be measured across the entire velocity range.

Figure 15. Acomparison oft he shear str ess exhibited by PDMS( M ) 4.4 kg/mol) on PS (black box) an d th e SAM (gra y box). Theblack tren dline is th e prediction ofeq 10. The ma ximum frictionon each surface is attained at only slightly different velocities,but the low-velocity behavior differs drastically.




styr ene and PDMS cha ins at the inter face. The possibilityof chain interdigitation is not intuitive considering thefact that polystyrene is glassy and PDMS is rubbery.However, several recent stud ies ha ve raised th e possibilityth at t he sur face of polystyrene could be in th e more relaxedstate on the surface than in the bulk at room tem pera-t u r e .58,59 In our exper iment s, ther e is no clear evidence of interdigitation between PDMS and P S as no remark ableadhesion hysteresis between the two is evident in thecont act mechan ics experiment s (Figur e 5). It can however

be argued tha t an adhesion hysteresis as low as 1 mJ /m2,which is beyond th e limit of the measur ement accur acy,could translate to an interfacial shear stress of 200 kPaby assum ing the characteristic distance of segm entalhopping to be ∼5 n m .60 Thus, while th e possibility of avery small degree of interdigitation between P DMS andPS cannot be altogether eliminated, it is also plausiblethat the m olecular tortuousity of the PS surface couldplay an important role, especially by affecting the pre-exponent ial t ime of the surface diffusion. It is plau siblethen that the kinetic friction of PDMS on polystyrenecomprises two phenomena. In the low velocity limit, theeffects of molecular ru gosity an d/or su rface diffusion couldcont ribut e to friction, wher eas a t h igh velocities, frictionis controlled by stochastic processes of adsorption and

desorption as envisaged by Schallamach 26 and Chernyak and Leonov.27 The velocity (10-3 m /s ) a t w h ich t h emaximu m friction occurs yields a cha ra cteristic time scaleof the pr ocess as τ ∼ 10-6 s, which is n ot very differentfrom that observed with the PDMS on SAM. However,th e low velocity behavior ofPDMS on PS sh ould be tr eateddifferent ly. Assuming th at the low velocity frictionalbehavior is controlled by a surface diffusion, the kineticfriction could follow an equation of the type

where σ ο is a constant and V * is a cha ra cteristic velocity.Equa tion 12 fits t he low velocity (V e 10-3 m/s) data

rather well for all the molecular weights of PDMS, fromwhich the estimate of the characteristic velocity V * isaveraged to be ∼6 µm/s. Figure 16 compares the shearstress for M ) 4.4 kg/m ol on the SAM and the PS andshows how well eq 12 fits the low velocity behavior on thelatt er surface.With th is value of V * and the char acteristiclength as the segm ental length (0.6 n m ) of PDMS, the

characteristic t im e scale of the frictional process isestimated to be ∼10-4 s, which is 2 orders of magnitudel a r g e r t h a n t h a t (∼10-6 s) corresponding to maximumshea r str ess. The pre-exponent ial factors (10-9 and 10-11,r e sp e ct i ve ly ) of t h e s e t w o l a t t er p r oce ss es a r e a l sosignifican t. The latt er time scale corr esponds to a classicalArrhenius process, whereas the form er is typical of diffusiona l processes, th e ran ge observed in polymer chainfolding kinetics.61

S u m m a r yThis study reveals th e richness and complexity ofru bberfr i ct i on on i n t er fa ce s d om i n a t ed b y v a n d e r W a a lsinteractions. The dependence of rubber friction on mo-lecular weight, temperatu re, normal load, sliding velocity,and the nature of the countersurface can be understoodqualitat ively using the original ideas of Grosch,23 Schal-lamach,26 and Chernyak and Leonov.27 A main factorcontributing to rubber friction is the molecular weight of the polymer, which determines the areal density of theload-bearing junctions.

The overall behavior of ru bber friction is consistent withthe stochastic kinetics of adsorption and desorption of polymericchains tosurfaces for which twotime scales arerelevant: the relaxation tim e in t he free state, and th e

time the polymeric chain spends in the adsorbed state.The latter time increases with the molecular weight of the polymer, leading to the broadening of the friction peak.Alth ough th ese frictional chara cteristics can be describedby the t heory of absolute r eaction r ates, th ey are largelyindependent of th e work of adhesion. Interestingly, whatvaries significant ly among different sur faces is not so muchthe activation energy, but the pre-exponential factor inthe Arrhenius equation,which indicates the contributionsofother m echan isticpr ocesses not considered in th e simplestochast ic models of Schallama ch and th at of Chernyak and Leonov.

A c k n o w l e d g m e n t . We benefited from some valua blecomm ent s received from A. N. Gent an d M. Tirrell dur ingthe early stages of this study. We thank A. Leonov forbringing ref 27 to our attention. We thank J. Tonge andG. Gordon of Dow Corning for their help with the PDMSsamples and t heir characterizations with t he NMR, GPC,and rheological m easurem ents. We are indebted to G.Walker and his students for sharing with us the AFMchara cterizations of th e PDMS elastomer s. This work wassupported by th e Office of Naval Research.

LA027061Q(58) Meyers, G. F.; Dekoven, B. M.; Seitz, J. T. L a n g mu i r 1992, 8,2230 (and references th erein.)

(59) Wallace, W. E.;F ischer, D. A.; Efimenko, K.;Wu, W.-L.;Genzer,J . Ma cromolecules 2001 , 34 15, 5081.

(60) Yoshizawa,H .;Chen, Y.L.;Israelachvili,J . J . Ph y s . Ch e m. 1993 ,97 (16), 4128.

(61) Smith, J.; Cusa ck, S.; Tidor, B.; Karplus, M. J . Ch em. Ph y s.1990, 93 (5), 2974.

(62) Persson, B. N. G. Sliding Friction : Physical Properties and Ap pl ica ti ons , 2nd ed.; Springer: Heidelberg, 2000.

σ ) σ o sinh-1(V / V *) (12)


The Effects of Molecular Weight and Temperature on Kinetic Friction

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