Bhushan B. Nanotribology and Nanomechanics: An Introduction

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494 Marina Ruths and Jacob N. Israelachvili

Examples of such liquids are some perﬂuoropolyethers and polydimethylsiloxanes

(PDMS).

Recent computer simulations [144, 151, 287, 315,372] of the structure, inter-

action forces, and tribological behavior of chain molecules between two shearing

surfaces indicate that both linear and singly or doubly branched-chain molecules

order between two ﬂat surfaces by aligning into discrete layers parallel to the sur-

faces. However, in the case of the weakly branched molecules, the expected os-

cillatory forces do not appear because of a complex cancelation of entropic and

enthalpic contributions to the interaction free energy, which results in a monotoni-

cally smooth interaction, exhibiting a weak energy minimum rather than the oscilla-

tory force proﬁle that is characteristic of linear molecules. During sliding, however,

these molecules can be induced to further align, which can result in a transition from

smooth to stick–slip sliding.

Table 9.5 shows the trends observed with some organic and polymeric liquids

between smooth mica surfaces. Also listed are the bulk viscosities of the liquids.

From the data of Table 9.5 it appears that there is a direct correlation between the

shapes of molecules and their coeﬃcient of friction or eﬀectiveness as lubricants (at

least at low shear rates). Small spherical or chain molecules have high friction with

stick–slip, because they can pack into ordered solid-like layers. In contrast, longer

chained and irregularly shaped molecules remain in an entangled, disordered, ﬂuid-

like state even in very thin ﬁlms, and these give low friction and smoother sliding. It

is probably forthis reason that irregularlyshaped branchedchain molecules are usu-

ally better lubricants. It is interesting to note that the friction coeﬃcient generally

decreases as the bulk viscosity of the liquids increases. This unexpected trend oc-

curs because the factors that are conducive to low friction are generally conducive

to high viscosity. Thus molecules with side groups such as branched alkanes and

polymer melts usually have higher bulk viscosities than their linear homologues

for obvious reasons. However, in thin ﬁlms the linear molecules have higher shear

stresses, because of their ability to become ordered. The only exception to the above

correlations is water, which has been found to exhibit both low viscosity and low

friction (see Fig. 9.20a,and Sect. 9.7.3).In addition, the presence of water can dras-

tically lower the friction and eliminate the stick–slip of hydrocarbon liquids when

the sliding surfaces are hydrophilic.

If an “eﬀective” viscosity, η

eﬀ

, were to be calculated for the liquids of Table 9.5,

the values would be many orders of magnitudehigher than those of the bulk liquids.

This can be demonstrated by the following simple calculation based on the usual

equation for Couette ﬂow (see (9.44)):

eﬀ

= F

D/Av , (9.50)

where F

is the kinetic friction force, D is the ﬁlm thickness, A the contact

area, and v the sliding velocity. Using typical values for experiments with hex-

adecane [362]: F

= 5mN,D = 1nm,A = 3×10

−9

,andv = 1µms

−1

, one gets

eﬀ

≈ 2000Nsm

−2

,or20,000Poise, which is ∼ 10

times higher than the bulk vis-

cosity, η

bulk

, of the liquid. It is instructive to consider that this very high eﬀective

9 Surface Forces and Nanorheology of Molecularly Thin Films 495

viscosity nevertheless still produces a low friction force or friction coeﬃcient μ of

about 0.25. It is interesting to speculate that, if a 1nm ﬁlm were to exhibit bulk vis-

cous behavior, the friction coeﬃcient under the same sliding conditions would be as

low as 0.000001. While such a low value has never been reported for any tribolog-

ical system, one may consider it a theoretical lower limit that could conceivably be

attained under certain experimental conditions.

9.9.2 Eﬀects of Surface Structure

Various studies [44,273,274,276,284–286] have shown that conﬁnement and load

generally increase the eﬀective viscosity and/or relaxation times of molecules, sug-

gestive of an increased glassiness or solid-like behavior (Figs. 9.32 and 9.33). This

is in marked contrast to studies of liquids in small conﬁning capillaries where the

opposite eﬀects have been observed [373,374]. The reason for this is probably be-

cause the two modes of conﬁnement are diﬀerent. In the former case (conﬁnement

of molecules between two structured solid surfaces), there is generally little oppo-

sition to any lateral or vertical displacement of the two surface lattices relative to

each other. This means that the two lattices can shift in the x–y–z planes (Fig. 9.32a)

Fig. 9.32. Schematic view of interfacial ﬁlm composed of spherical molecules under a com-

pressive pressure between two solid crystalline surfaces. (a) If the two surface lattices are free

to move in the x–y–z directions, so as to attain the lowest energy state, they could equilibrate

at values of x, y,andz, which induce the trapped molecules to become “epitaxially” ordered

into a “solid-like” ﬁlm. (b) Similar view of trapped molecules between two solid surfaces

that are not free to adjust their positions, for example, as occurs in capillary pores or in brittle

cracks. (c) Similar to (a), but with chain molecules replacing the spherical molecules in the

gap. These may not be able to order as easily as do spherical molecules even if x, y,andz can

adjust, resulting in a situation that is more akin to (b)(after [362] with permission; copyright

1993 American Chemical Society)

496 Marina Ruths and Jacob N. Israelachvili

Fig. 9.33. Schematic representation of the ﬁlm under shear. (a) The lubricant molecules are

just conﬁned, but not oriented in any particular direction. Because of the need to shear, the

ﬁlm dilates (b). The molecules disentangle (c) and get oriented in a certain direction related

to the shear direction (d). (e) Slowly evolving domains grow inside the contact region. These

macroscopic domains are responsible for the long relaxation times. (f) At the steady-state,

a continuous gradient of conﬁnement time and molecular order is established in the contact

region, which is diﬀerent for molecules adsorbed on the upper and lower surfaces. Molecules

entering into the contact are not oriented or ordered. The required sliding distance to modify

their state deﬁnes a characteristic distance. Molecules leaving the contact region need some

(short) characteristic time to regain their bulk, unconﬁned conﬁguration (after [344], with

to accommodate the trapped molecules in the most crystallographically commen-

surate or epitaxial way, which would favor an ordered, solid-like state. In contrast,

the walls of capillaries are rigid and cannot easily move or adjust to accommodate

the conﬁned molecules (Fig. 9.32b), which will thereforebe forced into a more dis-

ordered, liquid-like state (unless the capillary wall geometry and lattice are exactly

commensurate with the liquid molecules, as occurs in certain zeolites [374]).

Experiments have demonstrated the eﬀects of surface lattice mismatch on the

friction between surfaces [257,258,375]. Similar to the eﬀects of lattice mismatch

9 Surface Forces and Nanorheology of Molecularly Thin Films 497

on adhesion (Fig. 9.11), the static friction of a conﬁned liquid ﬁlm is maximum

when the lattices of the conﬁning surfaces are aligned. For OMCTS conﬁned be-

tween mica surfaces [258] the static friction was found to vary by more than a fac-

tor of 4, while for bare mica surfaces the variation was by a factor of 3.5 [375].

In contrast to the sharp variations in adhesion energy over small twist angles, the

variations in friction as a function of twist angle were much broader both in magni-

tude and angular spread. Similar variationsin friction as a function of twist or misﬁt

angles have also been observed in computer simulations [376].

Robbins and coworkers [315] computed the friction forces of two clean crys-

talline surfaces as a function of the angle between their surface lattices. They found

that, for all non-zero angles (ﬁnite“twist” angles), the friction forces fell to zero due

to incommensurability eﬀects. They further found that sub-monolayer amounts of

organic or other impurities trapped between two incommensurate surfaces can gen-

erate a ﬁnite friction force. They therefore concluded that any ﬁnite friction force

measured between incommensurate surfaces is probably due to such “third-body”

eﬀects.

The reason why surface texture (lattice structure, roughness, granularity, topog-

raphy, etc.) has a larger eﬀect on the lateral (shear or friction) forces between two

surfaces than on their normal (adhesion) forces is because friction is proportionalto

the adhesion hysteresis(Sect. 9.7.2),which can be lowevenwhen the adhesion force

is high. It is also important to recognizethat a system might be deﬁnedby more than

one lengthscale. Some systems havewell-deﬁned dimensionsor size (e.g., a perfect

lattice, monodisperse nanoparticles), while others have diﬀerent lateral and vertical

dimensions and macroscopic curvature [249]. Furthermore, the morphology or tex-

ture of many systems, such as asperities that are randomlydistributed overa surface,

aﬀects adhesion and tribological properties [244,249,287,317,377–379].

With rough surfaces, i.e., those that have random protrusions rather than being

periodically structured, we expect a smearing out of the correlated intermolecular

interactions thatare involvedin ﬁlm freezing andmelting (andin phase transitions in

general).This shouldeﬀectivelyeliminate the highly regularstick–slip and mayalso

aﬀect the location of the slipping planes [151,287,314,348].The stick–slip friction

of “real” surfaces,which aregenerally rough,may,therefore, be quite diﬀerent from

those of perfectly smooth surfaces composed of the same material. We should note,

however, that even between rough surfaces, most of the contacts occur between the

tips of microscopic asperities, which may be smooth over their microscopic contact

area [380].

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