Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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because some of these are connected via spontaneous — and therefore thermodynamically irreversible —

instabilities or transitions (Figure 9.11C) where energy is liberated and therefore “lost” via heat or phonon

release (Israelachvili and Berman, 1995). This is an area of much current interest and activity, especially

regarding the fundamental molecular origins of adhesion and friction, and the relationships between them.

9.9 Rheology of Molecularly Thin Films: Nanorheology

9.9.1 Different Modes of Friction: Limits of Continuum Models

Most frictional processes occur with the sliding surfaces becoming damaged in one form or another

(Bowden and Tabor, 1967). This may be referred to as “normal” friction. In the case of brittle materials,

the damaged surfaces slide past each other while separated by relatively large, micron-sized wear particles.

With more ductile surfaces, the damage remains localized to nanometer-sized, plastically deformed

asperities.

There are also situations where sliding can occur between two perfectly smooth, undamaged surfaces.

This may be referred to as “interfacial” sliding or “boundary” friction, which is the focus of the following

sections. The term boundary lubrication is more commonly used to denote the friction of surfaces that

contain a thin protective lubricating layer, such as a surfactant monolayer, but here we shall use this term

more broadly to include any molecularly thin solid, liquid, surfactant, or polymer ﬁlm.

Experiments have shown that as a liquid ﬁlm becomes progressively thinner, its physical properties

change, at ﬁrst quantitatively then qualitatively (Van Alsten and Granick, 1990a,b, 1991; Granick, 1991;

Hu et al., 1991; Hu and Granick, 1992; Luengo et al., 1997).

The quantitative changes are manifested by

an increased viscosity, non-Newtonian ﬂow behavior, and the replacement of normal melting by a glass

transition, but the ﬁlm remains recognizable as a liquid. In tribology, this regime is commonly known

as the “mixed lubrication” regime, where the rheological properties of a ﬁlm are intermediate between

the bulk and boundary properties. One may also refer to it as the “intermediate” regime (Table 9.3).

For even thinner ﬁlms, the changes in behavior are more dramatic, resulting in a qualitative change

in properties. Thus, ﬁrst-order phase transitions can now occur to solid or liquid-crystalline phases (Gee

et al., 1990; Israelachvili et al., 1990a,b; Thompson and Robbins, 1990; Yoshizawa et al., 1993; Klein and

Kumacheva, 1995), whose properties can no longer be characterized — even qualitatively — in terms of

bulk or continuum liquid properties such as viscosity. These ﬁlms now exhibit yield points (characteristic

FIGURE 9.12 Measured advancing and receding radius vs. load curves for two surfactant-coated mica surfaces of

initial, undeformed radii R ≈ 1 cm. Each surface had a monolayer of CTAB (cetyl-trimethyl-ammonium-bromide)

on it of mean area 60 Å

per molecule. The solid lines are based on ﬁtting the advancing and receding branches to

the JKR equation, Equation 9.20), from which the indicated values of γ

and γ

were determined, in units of mJ/m

or erg/cm

. The advancing/receding rates were about 1 µm/s. At the end of each unloading cycle the pull-off force,

, can be measured, from which another value for γ

can be obtained using Equation 9.21).

of fracture in solids) and their molecular diffusion and relaxation times can be ten orders of magnitude

longer than in the bulk liquid or even in ﬁlms that are just slightly thicker. The three friction regimes

are summarized in Table 9.3.

9.9.2 Viscous Forces and Friction of Thick Films: Continuum Regime

Experimentally, it is usually difﬁcult to unambiguously establish which type of sliding mode is occurring,

but an empirical criterion, based on the Stribeck curve (Figure 9.13) is often used as an indicator. This

curve shows how the friction force or the coefﬁcient of friction is expected to vary with sliding speed

depending on which type of friction regime is operating. For thick liquid lubricant ﬁlms whose behavior

can be described by bulk continuum properties, the friction forces are essentially the hydrodynamic or

viscous drag forces. For example, for two plane parallel surfaces of area A separated by a distance D and

moving laterally relative to each other with velocity v, if the intervening liquid is Newtonian, i.e., if its

viscosity η is independent of the shear rate, the frictional force experienced by the surfaces is given by

(9.24)

where the shear rate

γ is deﬁned by

(9.25)

TABLE 9.3 The Three Main Tribological Regimes Characterizing the Changing Properties of Liquids Subjected

to an Increasing Conﬁnement between Two Solid Surfaces

Regime

Conditions for Getting into

this Regime

Static/Equilibrium

Properties

Dynamic Properties

Bulk • Thick ﬁlms (>10σ, »R

)

• Low or zero loads

• High shear rates

Bulk, continuum properties:

• Bulk liquid density

• No long-range order

Bulk, continuum properties:

• Newtonian viscosity

• Fast relaxation times

• No glass temperature

• No yield point

• EHD lubrication

Mixed or

intermediate

• Intermediately thick ﬁlms

(4–10 molecular

diameters ~R

for

polymers)

• Low loads

Modiﬁed ﬂuid properties include:

• Modiﬁed positional and

orientational order

• Medium to long-range molecular

correlations

• Highly entangled states

Modiﬁed rheological properties

include:

• Non-Newtonian ﬂow

• Glassy states

• Long relaxation times

• Mixed lubrication

Boundary • Molecularly thin ﬁlms (<4

molecular diameters)

• High loads

• Low shear rates

• Smooth surfaces or

asperities

Onset of non-ﬂuidlike properties:

• Liquidlike to solidlike phase

transitions

• Appearance of new liquid-crystalline

states

• Epitaxially induced long-range

ordering

Onset of tribological properties:

• No ﬂow until yield point or critical

shear stress reached

• Solidlike ﬁlm behavior

characterized by defect diffusion,

dislocation motion, shear melting

• Boundary lubrication

Based on work by Granick (1991), Hu and Granick (1992), and others (Gee et al., 1990; Hirz et al., 1992; Yoshizawa et al.,

1993) on the dynamic properties of short-chain molecules such as alkanes and polymer melts conﬁned between surfaces.

Conﬁnement can lead to an increased or decreased order in a ﬁlm, depending both on the surface lattice structure and the

geometry of the conﬁning cavity.

In each regime both the static and dynamic properties change. The static properties include the ﬁlm density, the density

distribution function, the potential of mean force, and various positional and orientational order parameters.

Dynamic properties include viscosity, viscoelastic constants, and tribological yield points such as the friction coefﬁcient and

critical shear stress.

.γ=

At higher shear rates, two additional effects often come into play. First, certain properties of liquids

may change at high

γ values. In particular, the “effective” viscosity may become non-Newtonian, one

form given by

(9.26)

where n = 0 for Newtonian ﬂuids, n > 0 for shear thickening (dilatent) ﬂuids, and n < 0 for shear thinning

(pseudoplastic) ﬂuids (the latter become less viscous, i.e., ﬂow more easily, with increasing shear rate).

An additional effect on η can arise from the higher local stresses (pressures) experienced by the liquid

ﬁlm

γ increases. Since the viscosity is generally also sensitive to the pressure (usually increasing with

P), this effect also acts to increase η

eff

and thus the friction force.

A second effect that occurs at high shear rates is surface deformation, arising from the large hydro-

dynamic forces acting on the sliding surfaces. For example, Figure 9.13B shows how two surfaces elastically

deform when the sliding speed increases to a high value. These deformations alter the hydrodynamic

FIGURE 9.13 (A) Stribeck curve: empirical curve giving the trend generally observed in the friction forces or

friction coefﬁcients as a function of sliding velocity, the bulk viscosity of the lubricating ﬂuid, and the applied load.

The three friction/lubrication regimes are known as the thick-ﬁlm or EHD lubrication regime (see B below), the

intermediate or mixed lubrication regime (see Figure 9.14), and the boundary lubrication regime (see Figure 9.24).

The ﬁlm thicknesses, believed to correspond to each of these regimes, are also shown. For thick ﬁlms the friction

force is purely viscous e.g., Couette ﬂow at low shear rates, but may become complicated at higher shear rates where

EHD deformations of surfaces can occur during sliding, as shown in (B).

ηγ

eff

friction forces, and this type of friction is often referred to as elastohydrodynamic lubrication (EHD or

EHL) as mentioned in Table 9.3.

One natural question is: How thin can a liquid ﬁlm be before its dynamic e.g., viscous ﬂow, behavior

ceases to be described by bulk properties and continuum models? Concerning the static properties, we

have already seen that ﬁlms composed of simple liquids display continuum behavior down to thicknesses

of 5 to 10 molecular diameters. Similar effects have been found to apply to the dynamic properties, such

as the viscosity, of simple liquids in thin ﬁlms. Concerning viscosity measurements, a number of dynamic

techniques were recently developed (Chan and Horn, 1985; Israelachvili, 1986a; Van Alsten and Granick,

1988; Israelachvili and Kott, 1989) for directly measuring the viscosity as a function of ﬁlm thickness

and shear rate across very thin liquid ﬁlms between two surfaces. By comparing the results with theoretical

predictions of ﬂuid ﬂow in thin ﬁlms, one can determine the effective positions of the shear planes and

the onset of non-Newtonian behavior in very thin ﬁlms.

FIGURE 9.14 Typical rheological behavior of liquid ﬁlm in the mixed lubrication regime. (A) Increase in effective

viscosity of dodecane ﬁlm between two mica surfaces with decreasing ﬁlm thickness (Granick, 1991). Beyond 40 to

50 Å, the effective viscosity η

eff

approaches the bulk value η

bulk

. (B) Non-Newtonian variation of η

eff

with shear rate

of a 27-Å-thick dodecane ﬁlm (from Luengo et al., 1996). The effective viscosity decays as a power law, as in

Equation 9.26. In this example, n = 0 at the lowest

, then transitions to n = –1 at higher

. For bulk thick ﬁlms,

dodecane is a low-viscosity Newtonian ﬂuid (n = 0).

The results show that for simple liquids including linear-chain molecules such as alkanes, their viscosity

in thin ﬁlms is the same, within 10%, as the bulk even for ﬁlms as thin as ten molecular diameters (or

segment widths) (Chan and Horn, 1985; Israelachvili, 1986a; Israelachvili and Kott, 1989). This implies

that the shear plane is effectively located within one molecular diameter of the solid liquid interface, and

these conclusions were found to remain valid even at the highest shear rates studied (of ~2 × 10

–1

With water between two mica or silica surfaces (Chan and Horn, 1985; Israelachvili, 1986a; Horn et al.,

1989b; Israelachvili and Kott, 1989), this has been found to be the case (to within ±10%) down to surface

separations as small as 2 nm, implying that the shear planes must also be within a few angstrom of the

solid–liquid interfaces. These results appear to be independent of the existence of electrostatic double-

layer or hydration forces. For the case of the simple liquid toluene conﬁned between surfaces with

adsorbed layers of C

molecules, this type of viscosity measurement has shown that the traditional no-

slip assumption for ﬂow at a solid interface does not always hold (Campbell et al., 1996). For this system,

the C

layer at the mica–toluene interface results in a “full-slip” boundary, which dramatically lowers

the viscous drag or effective viscosity for regular Couette or Poiseuille ﬂow.

With polymeric liquids (polymer melts) such as polydimethylsiloxanes (PDMS) and polybutadienes

(PBD), or polystyrene (PS) adsorbed onto surfaces from solution, the far-ﬁeld viscosity is again equal

to the bulk value, but with the no-slip plane (hydrodynamic layer thickness) being located at D = 1 to

2 R

away from each surface (Israelachvili, 1986b; Luengo et al., 1997), or at D = L for polymer brush

layers of thickness L per surface (Klein et al., 1993). In contrast, the same technique was used to show

that for nonadsorbing polymers in solution, there is actually a depletion layer of nearly pure solvent that

exists at the surfaces that affects the conﬁned solution ﬂow properties (Kuhl et al., 1998). These effects

are observed from near contact to surface separations in excess of 200 nm.

Further experiments with surfaces closer than a few molecular diameters (D < 20 to 40 Å for simple

liquids, or D < 2 to 4 R

for polymer ﬂuids) indicate that large deviations occur for thinner ﬁlms, described

below. One important conclusion from these studies is therefore that the dynamic properties of simple

liquids, including water, near an isolated surface are similar to those of the bulk liquid already within the

ﬁrst layer of molecules adjacent to the surface, only changing when another surface approaches the ﬁrst.

In other words, the viscosity and position of the shear plane near a surface are not simply a property of

that surface, but of how far that surface is from another surface. The reason for this is because when two

surfaces are close together, the constraining effects on the liquid molecules between them are much more

severe than when there is only one surface. Another obvious consequence of the above is that one should

not make measurements on a single, isolated solid–liquid interface and then draw conclusions about the

state of the liquid or its interactions in a thin ﬁlm between two surfaces.

9.9.3 Friction of Intermediate Thickness Films

For liquid ﬁlms in the thickness range between 6 and 10 molecular diameters, their properties can be

signiﬁcantly different from those of bulk ﬁlms. But the ﬂuids remain recognizable as ﬂuids; in other

words, they do not undergo a phase transition into a solid or liquid-crystalline phase. This regime has

recently been studied by Granick and co-workers (Van Alsten and Granick, 1990a,b, 1991; Granick, 1991;

Hu et al., 1991; Hu and Granick, 1992; Klein and Kumacheva, 1995) who used a different type of friction

attachment (Van Alsten and Granick, 1988, 1990b) to the SFA where the two surfaces are made to vibrate

laterally past each other at small amplitudes. This method provides information on the real and imaginery

parts (elastic and dissipative components, respectively) of the shear modulus of thin ﬁlms at different

shear rates and ﬁlm thickness. Granick (1991) and Hu et al. (1991) found that ﬁlms of simple liquids

become non-Newtonian in the 25 to 50 Å regime (about ten molecular diameters), whereas polymer

melts become non-Newtonian at much thicker ﬁlms, depending on their molecular weight (Luengo et al.,

1997).

A generalized friction map (Figure 9.16) has been proposed by Luengo et al. (1996) that illustrates the

changes in η

eff

from bulk Newtonian behavior (n = 0, η

eff

= η

bulk

) through the transition regime where

n reaches a minimum of –1 with decreasing shear rate, to the solidlike creep regime at very low

γ, where

n returns to 0. The data in Figure 9.15 show the transition for thicker polymer ﬁlms from bulk behavior

to the tribological regime where n reaches –1 (Luengo et al., 1997). With further decreasing shear rates

the exponent n increases from –1 to 0, as illustrated in Figure 9.14B for a dodecane system. A number

of results from experimental, theoretical, and computer simulation work have shown values of n from

–½ to –1 for this transition regime for a variety of systems and assumptions (Hu and Granick, 1992;

Granick, 1991; Thompson et al., 1992, 1995; Urbakh et al., 1995; Rabin and Hersht, 1993).

FIGURE 9.15 Effective viscosity plotted against

effective shear rate on log–log scales for polybutadi-

ene (MW = 7000) at four different separations, D

(adapted from Luengo et al., 1997). Open data

points were obtained from sinusoidally applied shear

at zero load (L = 0) at the indicated separations.

Solid points were obtained from friction experi-

ments at constant-sliding velocities. These tribolog-

ical results extrapolate, at high shear rate, to the bulk

viscosity.

FIGURE 9.16 Proposed generalized friction map of effective viscosity plotted against effective shear rate on a log–log

scale. (From Luengo, G. et al., 1996, Wear 200, 328–335. With permission.) Three main classes of behavior emerge:

(1) Thick ﬁlms; EHD sliding. At zero load (L = 0), approximating bulk conditions, η

eff

is independent of shear rate

except when shear thinning may occur at sufﬁciently large

. (2) Boundary layer ﬁlms, intermediate regime. A

Newtonian regime is again observed (η

eff

= constant, n = 0 in Equation 9.26) at low loads and low shear rates, but

eff

is much higher than the bulk value. As the shear rate

increases beyond

min

, the effective viscosity starts to drop

with a power law dependence on the shear rate (see Figure 9.14B), with n in the range of – ½ to –1 most commonly

observed. As the shear rate

increases still more, beyond

max

, a second Newtonian plateau is again encountered.

(3) Boundary layer ﬁlms, high load. The η

eff

continues to grow with load and to be Newtonian provided that the

shear rate is sufﬁciently low. Transition to sliding at high velocity is discontinuous (n < –1) and usually of the stick-

slip variety.

The intermediate regime appears to extend over a narrow range of ﬁlm thickness, from about four to

ten molecular diameters or polymer radii of gyration. Thinner ﬁlms begin to adopt “boundary” or

“interfacial” friction properties (described below, see also Table 9.3). Note that the intermediate regime

is actually a very narrow one when deﬁned in terms of ﬁlm thickness, for example, varying from about

D = 20 to 40 Å for hexadecane ﬁlms (Granick, 1991).

The effective viscosity η

eff

of a ﬂuid in the intermediate regime is usually higher than in the bulk, but

eff

usually decreases with increasing sliding velocity, v (known as shear thinning). When two surfaces

slide in the intermediate regime, the motion tends to thicken the ﬁlm (dilatency). This sends the system

into the bulk EHL regime where, as indicated by Equation 9.24, the friction force now increases with velocity.

This initial decrease, followed by an increase, in the frictional forces of many lubricant systems is the basis

for the empirical Stribeck curve of Figure 9.13A. In the transition from bulk to boundary behavior there is

ﬁrst a quantitative change in the material properties (viscosity and elasticity) which can be continuous, to

discontinuous qualitative changes which result in yield stresses and non-liquidlike behavior.

The rest of this chapter is devoted to friction in the interfacial and boundary regimes. The former

(interfacial friction) may be thought of as applying to the sliding of two dry, unlubricated surfaces in

true molecular contact. The latter (boundary friction) may be thought of as applying to the case where

a lubricant ﬁlm is present, but where this ﬁlm is of molecular dimensions — a few molecular layers or less.

9.10 Interfacial and Boundary Friction: Molecular Tribology

9.10.1 General Interfacial Friction

When a lateral force, or shear stress, is applied to two surfaces in adhesive contact, the surfaces initially

remain “pinned” to each other until some critical shear force is reached. At this point, the surfaces begin

to slide past each other either smoothly or in jerks. The frictional force needed to initiate sliding from

rest is known as the static friction force, denoted by F

, while the force needed to maintain smooth sliding

is referred to as the kinetic or dynamic friction force, denoted by F

. In general, F

> F

. Two sliding

surfaces may also move in regular jerks, known as “stick-slip” sliding, which is discussed in more detail

in Section 9.13. Note that such friction forces cannot be described by equations, such as Equation 9.26,

used for thick ﬁlms that are viscous and therefore shear as soon as the smallest shear force is applied.

Experimentally, it has been found that during both smooth and stick-slip sliding the local geometry

of the contact zone remains largely unchanged from the static geometry, and that the contact area vs.

load is still well described by the JKR equation, Equation 9.20.

The friction force of two molecularly smooth surfaces sliding while in adhesive contact with each other

is not simply proportional to the applied load, L, as might be expected from Amontons’ law. There is an

additional adhesion contribution that is proportional to the area of contact, A, which is described later.

Thus, in general, the interfacial friction force of dry, unlubricated surfaces sliding smoothly past each

other is given by

(9.27)

where S

is the critical shear stress (assumed to be constant), A = πr

is the contact area of radius r given

by Equation 9.20, and µ is the coefﬁcient of friction. For low loads we have

(9.28)

while for high loads, Equation 9.27 reduces to Amontons’ law:

(9.29)

FF SA L

== +µ.

FSASr S

LR RL R

cc c

==π=π +π+π+π

()

























6126γγγ,

FL=µ .

Depending on whether the friction force, F, in Equation 9.27 is dominated by the ﬁrst or second terms,

one may refer to the friction as adhesion controlled or load controlled, respectively.

Figure 9.17 shows a plot of contact area, A, and friction force, F, both plotted against the applied load,

L, in an experiment where two molecularly smooth surfaces of mica in adhesive contact were slid past

each other in an atmosphere of dry nitrogen gas. This is an example of the low-load, adhesion-controlled

limit, which is excellently described by Equation 9.28. In a number of different experiments, S

was

measured to be

2.5 × 10

N/m

, and to be independent of the sliding velocity. Note that there is a friction

force even at negative loads, where the surfaces are still sliding in adhesive contact.

9.10.2 Boundary Friction of Surfactant Monolayer–Coated Surfaces

The high friction force of unlubricated sliding can often be reduced by treating the solid surface with a

boundary layer of some other solid material that exhibits lower friction, such as a surfactant monolayer,

or by ensuring that during sliding, a thin liquid ﬁlm remains between the surfaces. The effectiveness of

a solid boundary lubricant layer on reducing the forces of friction is illustrated in Figure 9.18. Comparing

this with the friction of the unlubricated/untreated surfaces (Figure 9.17) shows that the critical shear

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

to 3.5 × 10

N/m

. At much higher applied

loads or pressures, the friction force is proportional to the load rather than the area of contact (Briscoe

et al., 1977), as expected from Equation 9.27.

9.10.3 Boundary Lubrication of Molecularly Thin Liquid Films

A liquid lubricant ﬁlm 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 a high afﬁnity for the surfaces so that the liquid molecules do not 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 tribological 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. Both of these effects are illustrated in Figure 9.19

for two untreated mica surfaces sliding with a thin layer of water between them. We may note that a

monomolecular ﬁlm of water (of thickness 2.5 Å per surface) has reduced S

by a factor of more than 30,

FIGURE 9.17 Friction force F and contact area A vs. load L for two mica surfaces sliding in adhesive contact in

dry air. The contact area is well described by the JKR theory, Equation 9.20), even during sliding, and the frictional

force is found to be directly proportional to this area, Equation 9.28. The vertical dashed line and arrow show the

transition from interfacial to normal friction with the onset of wear (lower curve).

which may be compared with the factor of 10 attained with the boundary lubricant layer (of thickness

25 Å per surface).

The effectiveness of a water ﬁlm only 2.5 Å thick to lower the friction force by more than an order of

magnitude is attributed to the “hydrophilicity” of the mica surface (mica is “wetted” by water) and to

the existence of a strongly repulsive short-range hydration force between such surfaces in aqueous

solutions (see Section 9.6) which effectively removes the adhesion-controlled contribution to the friction

force (Berman et al., 1998a). It is also interesting that a 2.5-Å-thick water ﬁlm between two mica surfaces

is sufﬁcient to bring the coefﬁcient of friction down to 0.01 to 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 below.

FIGURE 9.18 Sliding of mica surfaces each coated with a 25-Å-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). (Homola et al., 1989, 1990).

At much higher applied loads the undamaged surfaces also follow

Amontons-type sliding,

but for different reasons (see next section). Lower line: interfacial sliding with a monolayer

of water between the mica surfaces, shown for comparison. Note that in both cases, after damage occurs, the friction

force obeys Amontons’ law with the same coefﬁcient of friction of µ ≈ 0.3.

FIGURE 9.19 Two mica surfaces sliding past each other while immersed in a 0.01

M KCl salt solution. In both cases

the water ﬁlm is molecularly thin: 2.5 to 5.0 Å thick, and the interfacial friction force is very low: S

≈ 5 × 10

N/m

µ ≈ 0.015 (before damage occurs).

9.10.4 Transition from Interfacial to Normal Friction (with Wear)

Frictional damage can have many causes such as adhesive tearing at high loads or overheating at high

sliding speeds. Once damage occurs, there is a transition from “interfacial” to “normal” or load-controlled

friction as the surfaces become forced apart by the torn-out asperities (wear particles). For low loads,

the friction changes from obeying F = S

A to obeying Amontons’ law: F = µL, as shown in Figures 9.17

through 9.20, and sliding now proceeds smoothly with the surfaces separated by a 100 to 1,000 Å forest

of wear debris (mica or alumina ﬂakes in this case). The wear particles keep the surfaces apart over an

area that is much greater than their size, so that even one submicroscopic particle or asperity can cause

a signiﬁcant reduction in the area of contact and therefore in the friction (Homola et al., 1990). For this

type of frictional sliding, one can no longer talk of the molecular contact area of the two surfaces, although

the macroscopic or “apparent” area is still a useful parameter. A further discussion on the impact of

normal load and contact area is found in Section 9.11.4.

One remarkable feature of the transition from interfacial to normal friction of brittle surfaces is that

while the strength of interfacial friction, as reﬂected in the values of S

, is very dependent on the type of

surface and on the liquid ﬁlm between the surfaces, this is not the case once the transition to normal

friction has occurred. At the onset of damage, it is the material properties of the underlying substrates

that control the friction. In Figures 9.17 through 9.19 the friction for the damaged surfaces is that of any

damaged mica–mica system, while in Figure 9.20 the damaged surfaces friction is that of general alu-

mina–alumina sliding (with a friction coefﬁcient that agrees with literature values for the bulk materials),

independent of the initial surface coatings or liquid ﬁlms between the surfaces.

In order to modify practically the frictional behavior of such brittle materials, it is important to use

coatings that will both alter the interfacial tribological character and remain intact and protect the surfaces

from damage during sliding (Berman et al., 1998b). An example of the friction behavior of a strongly

FIGURE 9.20 Friction forces as a function of load for smooth (undamaged) and rough (damaged) surfaces (Berman

et al., 1998b). Untreated alumina surfaces (curve a) exhibit the lowest friction due to a thin physisorbed layer of

lubricating contaminants, but these are easily damaged upon sliding, resulting in rough surface (curve b) with a

higher friction. Monolayer coated surfaces (curve c) slide with higher friction at lower loads than untreated alumina,

but remain undamaged even after prolonged sliding, keeping the friction and wear substantially lower than rough

surfaces at high loads. Smooth sliding was “adhesion controlled,” i.e., the contact area A is well described by the JKR

equation (Equation 9.20), and F/A = constant (Equation 9.28). In addition, F is ﬁnite at L = 0. Rough, damaged

sliding was “load controlled”; i.e., the real contact area is undeﬁned, and F ∝ L (Equation 9.29) with F ≈ 0 at L = 0.

Experimental conditions: sliding velocity V = 0.05 to 0.5 µm/s; undeformed radius of curved surfaces, R ≈ 1 cm;

temperature T = 25°C; contact pressure range, P = 0 to 10 MPa; relative humidity, RH = 0% (curves a and c), RH =

0% and 100% (curve b).