Bhushan B. Nanotribology and Nanomechanics: An Introduction

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464 Marina Ruths and Jacob N. Israelachvili

vary between 1 and 0, it determines whether a particular friction force will be large

or close to zero. Molecular simulations oﬀer the best way to understand and predict

the magnitude of ε, but the complex multibody nature of the problem makes simple

conclusions diﬃcult to draw [299–302]. Some of the basic physics of the energy

transfer and dissipation of the molecular collisions can be drawn from simpliﬁed

models such as a 1D three-body system [268].

Finally, the above equation may also be expressed in terms of the friction

force F:

F = S

A = C

A+ C

L . (9.37)

Equations similar to (9.36) and (9.37) were previously derived by Derjaguin [303,

304] and by Briscoe and Evans [305], where the constants C

and C

were inter-

preted somewhat diﬀerently than in this model.

In the absence of any attractive interfacial force, we haveC

≈0, and the second

term in (9.36)and (9.37) should dominate(Fig. 9.15).Such situations typically arise

when surfaces repel each other across the lubricating liquid ﬁlm. In such cases, the

total frictional force should be low and should increase linearly with the external

load according to

F = C

L . (9.38)

Friction force

Load–5 30

0 5 10 15 20 25 460 500 540 10,000 12,000 14,000

120

100

3,200

2,400

1,600

800

L = 0

JKR

Hertz

Amontons

Fig. 9.15. Friction as a function of load for smooth surfaces. At low loads, the friction is dom-

inated by the C

A term of (9.37). The adhesion contribution (JKR curve) is most prominent

near zero load where the Hertzian and Amontons’ contributions to the friction are minimal.

As the load increases, the adhesion contribution becomes smaller as the JKR and Hertz curves

converge. In this range of loads, the linear C

L contribution surpasses the area contribution to

the friction. At much higher loads the explicit load dependence of the friction dominates the

interactions, and the observed behavior approaches Amontons’ law. It is interesting to note

that for smooth surfaces the pressure over the contact area does not increase as rapidly as the

load. This is because as the load is increased, the surfaces deform to increase the surface area

and thus moderate the contact pressure (after [307], with permission of Kluwer Academic

Publishers)

9 Surface Forces and Nanorheology of Molecularly Thin Films 465

An example of such lubricated sliding occurs when two mica surfaces slide in water

or in salt solution (see Fig. 9.20a), where the short-range “hydration” forces be-

tween the surfaces are repulsive. Thus, for sliding in 0.5M KCl it was found that

= 0.015 [283]. Another case where repulsive surfaces eliminate the adhesive

contribution to friction is for polymer chains attached to surfaces at one end and

swollen by a good solvent [219]. For this class of systems, C

< 0.001 for a ﬁnite

range of polymer layer compressions (normal loads, L). The low friction between

the surfaces in this regime is attributed to the entropic repulsion between the oppos-

ing brush layers with a minimum of entanglement between the two layers. However,

with higher normal loads, the brush layers become compressed and begin to entan-

gle, which results in higher friction (see [306]).

It is important to note that (9.38) has exactly the same form as Amontons’ Law

F = μL , (9.39)

where μ is the coeﬃcient of friction.

Figure 9.14c,d shows the kinetic friction force measured with both SFA and

FFM (friction force microscopy, using AFM) on a system where both surfaces were

covered with a chemically bound benzyltrichlorosilanemonolayer [286]. When im-

mersed in ethanol, the adhesion in this system is low, and very diﬀerent contact

areas and loads give a linear dependence of F on L with the same friction coeﬃ-

cients, and F → 0asL → 0. In the FFM measurements (Fig. 9.14d), the plateau in

the data at higher loads suggest a transition in the monolayers, similar to previous

observations on other monolayer systems. The pressure in the contact region in the

SFA is much lower than in the FFM, and no transitions in the friction forces or in the

thickness of the conﬁned monolayers were observed in the SFA experiments (and

no damage to the monolayers or the underlying substrates was observed during the

experiments, indicating that the friction was “wearless”). Despite the diﬀerence of

more than six orders of magnitude in the contact radii, pressure, loads, and friction

forces, the measured friction coeﬃcients are practically the same.

At the molecular level a thermodynamic analog of the Coulomb or cobblestone

models (see Sect. 9.7.1) based on the “contact value theorem” [3,283,307] can ex-

plain why F ∝ L also holds at the microscopic or molecular level. In this analysis

we consider the surface molecular groups as being momentarily compressed and

decompressed as the surfaces move along. Under irreversible conditions, which

always occur when a cycle is completed in a ﬁnite amount of time, the energy

“lost” in the compression–decompression cycle is dissipated as heat. For two non-

adhering surfaces, the stabilizing pressure P

acting locally between any two ele-

mental contact points i of the surfaces may be expressed by the contact value theo-

rem [3]:

= ρ

T = k

T/V

, (9.40)

where ρ

= V

−1

is the local number density (per unit volume) or activity of the in-

teracting entities, be they molecules, atoms, ions or the electron clouds of atoms.

This equation is essentially the osmotic or entropic pressure of a gas of conﬁned

466 Marina Ruths and Jacob N. Israelachvili

molecules. As onesurface movesacross the other,local regionsbecome compressed

and decompressed by a volume ΔV

. The work done per cycle can be written as

εP

ΔV

,whereε (ε ≤ 1) is the fraction of energy per cycle “lost” as heat, as deﬁned

earlier. The energy balance shows that, for each compression–decompressioncycle,

the dissipated energy is related to the friction force by

= εP

ΔV

, (9.41)

where x

is the lateral distance moved per cycle, which can be the distance between

asperities or the distance between surface lattice sites. The pressure at each con-

tact junction can be expressed in terms of the local normal load L

and local area

of contact A

as P

= L

. The volume change over a cycle can thus be expressed

as ΔV

= A

,wherez

is the vertical distance of conﬁnement. Inserting these into

(9.41), we get

= εL

) , (9.42)

which is independent of the local contact area A

. The total friction force is thus

F =



εL

) = εz





= μL , (9.43)

where it is assumed that on average the local values of L

and P

are independent

of the local slope z

. Therefore, the friction coeﬃcient μ is a function only of the

average surface topography and the sliding velocity, but is independent of the local

(real) or macroscopic (apparent) contact areas.

While this analysis explains non-adhering surfaces, there is still an additional

explicit contact area contribution for the case of adhering surfaces, as in (9.37).

The distinction between the two cases arises because the initial assumption of the

contact value theorem, (9.40), is incomplete for adhering systems. A more appro-

priate starting equation would reﬂect the full intermolecular interaction potential,

including the attractive interactions, in addition to the purely repulsive contribu-

tions of (9.40), much as the van der Waals equation of state modiﬁes the ideal gas

law.

9.7.3 Examples of Experimentally Observed Friction of Dry Surfaces

Numerous model systems have been studied with a surface forces apparatus (SFA)

modiﬁed for friction experiments (see Sect. 9.2.3).The apparatus allows for control

of load (normal force) and sliding speed, and simultaneous measurement of surface

separation, surface shape, true (molecular)area of contactbetween smooth surfaces,

and friction forces. A variety of both unlubricated and solid- and liquid-lubricated

surfaces have been studied both as smooth single-asperity contacts and after they

have been roughened by shear-induced damage.

Figure 9.16 shows the contact area, A, and friction force, F, both plotted against

the applied load, L, in an experiment in which two molecularly smooth surfaces of

mica in adhesive contact were slid past each other in an atmosphere of dry nitrogen

9 Surface Forces and Nanorheology of Molecularly Thin Films 467

Friction force

F (N)

Normal load L (N)

0 0.3

0.5

0.4

0.3

0.2

0.1

2 u 10

0.1 0.2

Dynamic contact

area A

(Pm

)

Damage

observed

Negative

load

Damaged

(F=

μL)

Friction force

Contact area

JKR

(F = S

Fig. 9.16. Friction force F and contact area A versus load L for two mica surfaces sliding

in adhesive contact in dry air. The contact area is well described by the JKR theory, (9.22),

even during sliding, and the friction force is found to be directly proportional to this area,

(9.28). The vertical dashed line and arrow show the transition from interfacial to normal

friction with the onset of wear (lower curve). The sliding velocity is 0.2 µms

−1

(after [45],

gas. This is an example of the low-load adhesion-controlled limit, which is excel-

lentlydescribed by (9.28).In a number ofdiﬀerent experiments,S

was measuredto

be 2.5×10

−2

and to be independent of the sliding velocity [45,308]. Note that

there is a friction force even at negative loads, where the surfaces are still sliding in

adhesive contact.

Figure 9.17 shows the correlation between adhesion hysteresis and friction for

two surfaces consisting of silica ﬁlms deposited on mica substrates [41]. The fric-

tion between undamaged hydrophobic silica surfaces showed stick–slip both at dry

conditions and at 100% relative humidity.Similar to the mica surfaces in Figs. 9.16,

9.18, and 9.20a, the friction of damaged silica surfaces obeyed Amontons’ law with

a friction coeﬃcient of 0.25–0.3 both at dry conditions and at 55% relative humid-

ity.

The high friction force of unlubricated sliding can often be reduced by treat-

ing the solid surface with a boundary layer of some other solid material that ex-

hibits lower friction, such as a surfactant monolayer, or by ensuring that during

sliding a thin liquid ﬁlm remains between the surfaces (as will be discussed in

Sect. 9.8). The eﬀectiveness of a solid boundary lubricant layer on reducing the

forces of friction is illustrated in Fig. 9.18. Comparing this with the friction of the

unlubricated/untreated surfaces (Fig. 9.16) shows that the critical shear stress has

been reduced by a factor of about ten: from 2.5×10

to 3.5×10

−2

.Atmuch

higher applied loads or pressures, the friction force is proportionalto the load, rather

than the area of contact [298], as expected from (9.27).

468 Marina Ruths and Jacob N. Israelachvili

Contact radius r

(cm

u 10

)

Load L (mN)

0 120

3.0

2.5

2.0

1.5

1.0

0.5

–20 20 40 60 80 100

Sliding velocity v (Pm/s)

0 1.0

0.5

Unloading

100% RH

γ =71± 4 mJ/m

=15 mJ/m

=5 mJ/m

0% RH

Friction

force

(mN)

L = 5.5 mN

L = 8.3 mN

L = 0 mN

L = 2.8 mN

0% RH

100% RH

L = 5.5 mN

L = 2.8 mN

L = 0

Fig. 9.17. (a) Contact radius r versus externally applied load L for loading and unloading

of two hydrophilic silica surfaces exposed to dry and humid atmospheres. Note that, while

the adhesion is higher in humid air, the hysteresis in the adhesion is higher in dry air. (b)

Eﬀect of velocity on the static friction force F

for hydrophobic (heat-treated electron-beam-

evaporated) silica in dry and humid air. The eﬀects of humidity, load, and sliding velocity on

the friction forces, as well as the stick–slip friction of the hydrophobic surfaces, are quali-

tatively consistent with a “friction” phase diagram representation as in Fig. 9.28 (after [41];

9 Surface Forces and Nanorheology of Molecularly Thin Films 469

5 u 10

4 u 10

3 u 10

2 u 10

Dynamic contact

area A (Pm

)

Friction force

F (N)

Normal load L (N)

0 0.6

0.20

0.15

0.10

0.05

0.1 0.2 0.3 0.4 0.5

Friction force

Contact area

JKR (F = S

Damaged (F = μL)

Water (μ = 0.02)Hertz

Fig. 9.18. Sliding of mica surfaces, each coated with a 2.5 nm thick monolayer of calcium

stearate surfactant, in the absence of damage (obeying JKR-type boundary friction) and in

the presence of damage (obeying Amontons-type normal friction). Note that both for this

system and for the bare mica in Figs. 9.16 and 9.20a, the friction force obeys Amontons’

law with a friction coeﬃcient of μ ≈ 0.3 after damage occurs. At much higher applied loads,

the undamaged surfaces also follow Amontons-type sliding, but for a diﬀerent reason: the

dependence on adhesion becomes smaller. Lower line: interfacial sliding with a monolayer

of water between the mica surfaces (load-controlled friction, cf. Fig. 9.20a), shown for com-

Yamada and Israelachvili [309] studied the adhesion and friction of ﬂuorocar-

bon surfaces(surfactant-coatedboundary lubricantlayers), which were compared to

those of hydrocarbon surfaces. They concluded that well-ordered ﬂuorocarbon sur-

faces have high friction, in spite of their lower adhesion energy (in agreement with

previous ﬁndings). The low friction coeﬃcient of Teﬂon (polytetraﬂuoroethylene,

PTFE) must, therefore, be due to some eﬀect other than low adhesion. For exam-

ple, the softness of PTFE, which allows material to ﬂow at the interface, which thus

behaves like a ﬂuid lubricant. On a related issue, Luengo et al. [310] found that

surfaces also exhibited low adhesion but high friction. In both cases the high

friction appears to arise from the bulky surface groups – ﬂuorocarbon compared to

hydrocarbon groups in the former, large fullerene spheres in the latter. Apparently,

the fact that C

molecules rotate in their lattice does not make them a good lubri-

cant: the molecules of the opposing surface must still climb over them in order to

slide, and this requires energy that is independent of whether the surface molecules

are ﬁxed or freely rotating. Larger particles such as ∼ 25nm sized nanoparticles

(also known as “inorganic fullerenes”) do appear to produce low friction by behav-

ing like molecular ball bearings, but the potential of this promising new class of

solid lubricant has still to be explored [311].

470 Marina Ruths and Jacob N. Israelachvili

Inert air

0 100

1.0

0.5

F (mN)

Decane vapor

0 100

1.0

0.5

F (mN)

–10 40

(μm)

(x 10

–4

)

10 20 300

Load L (mN)

–10 40

(μm)

(x 10

–4

)

10 20 300

Load L (mN)

c) d)

Inert air Decane vapor

= 0.04

Time, t (s)

= 0.003

Time, t (s)

= 40 mJ/m

= 28 mJ/m

= γ

= 21– 24 mJ/m

Fig. 9.19. Top: friction traces for two ﬂuid-like calcium alkylbenzene sulfonate monolayer-

coated surfaces at 25

◦

C showing that the friction force is much higher between dry mono-

layers (a) than between monolayers whose ﬂuidity has been enhanced by hydrocarbon pen-

etration from vapor (b). Bottom: Contact radius vs. load (r

vs. L) data measured for the

same two surfaces as above and ﬁtted to the JKR equation (9.22), shown by the solid curves.

For dry monolayers (c) the adhesion energy on unloading (γ

= 40 mJm

−2

) is greater than

that on loading (γ

= 28 mJm

−2

), which is indicative of an adhesion energy hysteresis of

Δγ = γ

−γ

= 12 mJm

−2

. For monolayers exposed to saturated decane vapor (d) their adhe-

sion hysteresis is zero (γ

= γ

), and both the loading and unloading data are well ﬁtted by

the thermodynamic value of the surface energy of ﬂuid hydrocarbon chains, γ = 24 mJm

−2

Figure 9.19 illustrates the relationship between adhesion hysteresis and friction

for surfactant-coated surfaces under diﬀerent conditions. This eﬀect, however, is

much more general and has been shown to hold for other surfaces as well [41,262,

292,312].

Direct comparisonsbetween absoluteadhesion energiesand friction forcesshow

little correlation. In some cases, higher adhesion energies for the same system un-

der diﬀerent conditions correspond to lower friction forces. For example, for hy-

drophilic silica surfaces (Fig. 9.17) it was found that with increasing relative humid-

ity the adhesion energy increases, but the adhesion energy hysteresis measured in

a loading–unloading cycle decreases, as does the friction force [41]. For hydropho-

bic silica surfaces under dry conditions, the friction at load L = 5.5mN was F =

75mN. For the same sample, the adhesion energy hysteresis was Δγ = 10mJm

−2

with a contact area of A ≈ 10

−8

at the same load. Assuming a value for the char-

9 Surface Forces and Nanorheology of Molecularly Thin Films 471

acteristic distance σ on the order of one lattice spacing, σ ≈ 1nm, and inserting

these values into (9.32), the friction force is predicted to be F ≈ 100mN for the ki-

netic friction force, which is close to the measured value of 75mN. Alternatively, we

may conclude that the dissipation factor is ε = 0.75, i.e., that almost all the energy

is dissipated as heat at each molecular collision.

A liquid lubricant ﬁlm (Sect. 9.8.3) is usually much more eﬀective at lowering

the friction of two surfaces than a solid boundary lubricant layer. However, to use

a liquid lubricant successfully, it must “wet” the surfaces, that is, it should have

ahighaﬃnity for the surfaces, so that not all the liquid molecules become squeezed

out when the surfaces come close together, even under a large compressive load.

Another important requirement is that the liquid ﬁlm remains a liquid under tribo-

logical conditions, i.e., that it does not epitaxially solidify between the surfaces.

Eﬀective lubrication usually requires that the lubricant be injected between the

surfaces, but in some cases the liquid can be made to condense from the vapor. This

is illustrated in Fig. 9.20a for two untreated mica surfaces sliding with a thin layer

of water between them. A monomolecular ﬁlm of water (of thickness 0.25nm per

surface)has reducedS

fromits valuefor drysurfaces(Fig. 9.16)by a factorof more

than 30, which may be compared with the factor of ten attained with the boundary

lubricant layer (of thickness 2.5nm per surface) in Fig. 9.18. Water appears to have

unusual lubricating properties and usually gives wearless friction with no stick–

slip [313].

The eﬀectiveness of a water ﬁlm only 0.25nm thick to lower the friction force

by more than an order of magnitude is attributed to the “hydrophilicity” of the mica

surface(micais “wetted”by water)and tothe existenceof a stronglyrepulsiveshort-

range hydration force between such surfaces in aqueous solutions, which eﬀectively

removes the adhesion-controlled contribution to the friction force [283]. It is also

interesting that a 0.25nm thick water ﬁlm between two mica surfaces is suﬃcient

to bring the coeﬃcient of friction down to 0.01–0.02, a value that corresponds to

the unusually low friction of ice. Clearly, a single monolayer of water can be a very

good lubricant – much better than most other monomolecular liquid ﬁlms – for

reasons that will be discussed in Sect. 9.9. A linear dependence of F on L has also

been observedfor mica surfacesseparated bycertain hydrocarbonliquids[275,285].

Figure 9.20b shows the kinetic friction forces measured at a high velocity across

thin ﬁlms of squalane, a branched hydrocarbon liquid (C

), which is a model

for lubricating oils. Very low adhesive forces are measured between mica surfaces

across this liquid [285] and the ﬁlm thickness decreased monotonically with load.

The friction force at a given load was found to be velocity-dependent, whereas the

contact area was not [285].

Dry polymer layers (Fig. 9.21) typically show a high initial static friction (“stic-

tion”) as sliding commences from rest in adhesive contact. The development of the

friction force after a change in sliding direction, a gradual transition from stick–

slip to smooth sliding, is shown in Fig. 9.21. A correlation between adhesion hys-

teresis and friction similar to that observed for silica surfaces in Fig. 9.17 can be

seen for dry polymer layers below their glass-transition temperature. As shown in

472 Marina Ruths and Jacob N. Israelachvili

Friction force F (N)

Normal load L (N)

0.4

0.12

0.10

0.08

0.06

0.04

0.02

0.1 0.2 0.3

μ = 0.02

μ = 0.33

Surface

damage

Friction force F (mN)

Load L (mN)

2.0

1.5

1.0

0.5

0.0

2500

2000

1500

1000

500

Contact area A (Pm

)

246810

Hertz

Friction force

Contact area

F = PL

Fig. 9.20. Load-controlled friction. (a) Two mica surfaces sliding past each other while im-

mersed in a 0.01 M KCl salt solution (nonadhesive conditions). The water ﬁlm is molecularly

thin, 0.25 to 0.5 nm thick, and the interfacial friction force is very low: S

≈ 5×10

−2

μ ≈ 0.02 (before damage occurs). After the surfaces have become damaged, the friction co-

(b) Steady-state friction force and contact area measured on a conﬁned squalane ﬁlm between

two undamaged mica surfaces at v = 0.6 µm/s in the smooth sliding regime (no stick–slip).

Open circles show F obtained on loading (increasing L), solid circles show unloading. Both

data sets are straight lines passing through the origin, as shown by the brown line (μ = 0.12).

The black curve is a ﬁt of the Hertz equation (cf. Sect. 9.5.2 and [3]) to the A vs. L data (open

squares)usingK = 10

N/m

, R= 2 cm. The thickness D varies monotonically from D = 2.5

to D = 1.7 nm as the load increases from L = 0toL = 10 mN (adapted from [285]; copyright

2003 American Physical Society)

9 Surface Forces and Nanorheology of Molecularly Thin Films 473

Friction force

TimeTime

Friction force

a) b)

Fig. 9.21. Typical friction traces showing how the friction force varies with the sliding time

for two symmetric, glassy polymer ﬁlms under dry conditions. Qualitative features that are

common to both polystyrene and polyvinyl benzyl chloride: (a) Decaying stick–slip motion

is observed until smooth sliding is attained if the motion continues for a suﬃciently long

distance. (b) Smooth sliding observed at suﬃciently high speeds. Similar observations have

been made by Berthoud et al. [314] in measurements on polymethyl methacrylate (after [262],

Fig. 9.12b,c, the adhesion hysteresis for polystyrene surfaces can be increased by

irradiation to induce scission of chains, and it has been found that the steady-state

friction force (kinetic friction) shows a similar increase with irradiation time [262].

Figure 9.22 shows an example of a computer simulation of the sliding of two

unlubricated silicon surfaces (modeled as a tip sliding over a planar surface) [112].

The sliding proceeds through a series of stick–slip events, and information on the

friction force and the local order of the initially crystalline surfaces can be obtained.

Similar studies for cold-welding systems [112] have demonstrated the occurrence

of shear or friction damage within the sliding surface (tip) as the lowest layer of it

adheres to thebottom surface.Recent computer simulationshaveaddressed manyof

(111)

–

21)(1

–

21)

(111)

a) b)

Fig. 9.22. Computer simulation of the sliding of two contacting Si surfaces (a tip and

a ﬂat surface). Shown are particle trajectories in a constant-force simulation, F

z,external

−2.15×10

−8

N, viewed along the (101) direction just before (a)andafter(b) a stick–slip

event for a large, initally ordered, dynamic tip (after [112] with permission of Kluwer Aca-

demic Publishers)