Bhushan B. Nanotribology and Nanomechanics: An Introduction

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474 Marina Ruths and Jacob N. Israelachvili

the phenomenaseen experimentally, including the diﬀerencesbetween adhesive and

non-adhesive systems, the issue of the dependence of the observed friction on the

real contact area (a parameter that is diﬃcult to deﬁne or measure at the nanoscale),

and the molecular origin of friction responses that follow Amontons’ law [287,302,

315–317].

9.7.4 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 Figs. 9.16

and 9.18, and sliding now proceeds smoothly with the surfaces separated by a 10–

100nm forest of wear debris (in this case, mica ﬂakes). 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 [308]. 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.

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, the material properties of the underlying substrates con-

trol the friction. In Figs. 9.16, 9.18, and 9.20a the friction for the damaged surfaces

is that of any damaged mica–mica system, μ ≈ 0.3, independent of the initial sur-

face coatings or liquid ﬁlms between the surfaces. A similar friction coeﬃcient was

found for damaged silica surfaces [41].

In order to modify the frictional behavior of such brittle materials practically, 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. [318] found that the friction of a strongly bound octadecyl phosphonic

acid monolayer on alumina surfaces was higher than for untreated, undamaged α-

alumina surfaces, but the bare surfaces easily became damaged upon sliding, result-

ing in an ultimately higher friction system with greater wear rates than the more

robust monolayer-coated surfaces.

Clearly, the mechanism and factors that determine normal friction are quite dif-

ferent from those that govern interfacial friction (Sects. 9.7.1–9.7.2). This eﬀect is

not general and may only apply to brittle materials. For example, the friction of duc-

tile surfaces is totally diﬀerent and involves the continuous plastic deformation of

contacting surface asperities during sliding, rather than the rolling of two surfaces

on hard wear particles [263]. Furthermore, in the case of ductile surfaces, water and

other surface-active components do have an eﬀect on the friction coeﬃcients under

“normal” sliding conditions.

9 Surface Forces and Nanorheology of Molecularly Thin Films 475

9.8 Liquid Lubricated Surfaces

9.8.1 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

(Fig. 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 sur-

faces of area A separatedbyadistanceD and moving laterally relative to each other

with velocity v, if the intervening liquid is Newtonian, i.e., if its viscosity η is in-

dependent of the shear rate, the frictional force experienced by the surfaces is given

F =

ηAv

, (9.44)

where the shear rate ˙γ is deﬁned by

˙γ =

. (9.45)

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 viscos-

ity may become non-Newtonian, one form given by

η ∝ ˙γ

, (9.46)

where n = 0(i.e.,η

eﬀ

= constant) for Newtonian ﬂuids, n > 0 for shear-thickening

(dilatant) ﬂuids, and n < 0 for shear-thinning (pseudoplastic) ﬂuids (the latter be-

come 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 as ˙γ increases. Since the viscosity is generally also sensitive to the pres-

sure (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 hydrodynamic forces acting on the sliding surfaces. For example,

Fig. 9.23 shows how two surfaces deform elastically when the sliding speed in-

creases to a high value. These deformations alter the hydrodynamic friction forces,

and this type of friction is often referred to as elasto-hydrodynamic lubrication

(EHD or EHL), as mentioned in Table 9.4.

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 in Sect. 9.4.3 that ﬁlms composed of simple

liquids display continuum behavior down to thicknesses of 4–10 molecular diam-

eters. Similar eﬀects have been found to apply to the dynamic properties, such as

476 Marina Ruths and Jacob N. Israelachvili

Stationary

Viscous film

Sliding

Fig. 9.23. Top: Stationary sur-

faces (one more deformable

and one rigid) separated by

a thick liquid ﬁlm. Bottom:

Elasto-hydrodynamic defor-

mation of the upper surface

during sliding (after [1], with

permission)

the viscosity, of simple liquids in thin ﬁlms. Concerning viscosity measurements,

a number of dynamic techniques were recently developed [11–13,43,51,319,320]

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.

The results show that, for simple liquids including linear chain molecules such

as alkanes, the viscosity in thin ﬁlms is the same, within 10%, as the bulk even for

ﬁlms as thin as 10 molecular diameters (or segment widths) [11–13,319,320].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 [22,313,319–321] 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 tenths of a nanometer of the solid–liquidinterfaces. 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 sur-

faces 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 [322].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 with polystyrene (PS) adsorbed onto surfaces from

solution, the far-ﬁeld viscosity is again equal to the bulk value, but with the non-

slip plane (hydrodynamic layer thickness) being located at D = 1−2R

away from

each surface [11,47], or at D = L or less for polymer brush layers of thickness L

per surface [13, 323]. In contrast, the same technique was used to show that, for

non-adsorbing polymers in solution, there is actually a depletion layer of nearly

pure solvent that exists at the surfaces that aﬀects the conﬁned solution ﬂow prop-

9 Surface Forces and Nanorheology of Molecularly Thin Films 477

erties [321]. These eﬀects are observed from near contact to surface separations in

excess of 200nm.

Further experiments with surfaces closer than a few molecular diameters (D <

2−4nm for simple liquids, or D < 2−4R

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 that, 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 conse-

quence 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.

9.8.2 Friction of Intermediate Thickness Films

For liquid ﬁlms in the thickness range between 4 and 10 molecular diameters, the

properties can be signiﬁcantly diﬀerent from those of bulk ﬁlms. Still, the ﬂuids re-

main 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 et al. [44,273,274,276,277],who used a friction attachment [43,44] to the

SFA where a sinusoidal input signal to a piezoelectric device makes the two sur-

faces slide back and forth laterally past each other at small amplitudes. This method

provides information on the real and imaginary parts (elastic and dissipative com-

ponents, respectively) of the shear modulus of thin ﬁlms at diﬀerent shear rates and

ﬁlm thickness. Granick [273] and Hu et al. [277] found that ﬁlms of simple liquids

become non-Newtonian in the 2.5–5 nm regime (about 10 molecular diameters, see

Fig. 9.24). Polymer melts become non-Newtonian at much larger ﬁlm thicknesses,

depending on their molecular weight [47].

Klein and Kumacheva [46,280,324] studied the interaction forces and friction

of small quasi-spherical liquid molecules such as cyclohexanebetween molecularly

smooth mica surfaces. They concluded that surface epitaxial eﬀects can cause the

liquid ﬁlm to solidify already at six molecular diameters, resulting in a sudden (dis-

continuous) jump to high friction at low shear rates. Such dynamic ﬁrst-order tran-

sitions, however,may depend on the shear rate.

A generalized friction map (Fig. 9.24c,d) has been proposed by Luengo et al.

[325] that illustrates the changes in η

eﬀ

from bulk Newtonian behavior (n = 0,

eﬀ

= η

bulk

) through the transition regime where n reaches a minimum of −1 with

decreasing shear rate to the solid-like creep regime at very low ˙γ where n returns

to 0. A number of results from experimental, theoretical, and computer simulation

work have shown values of n from −1/2to−1 for this transition regime for a variety

of systems and assumptions [273,274,299,326–332].

478 Marina Ruths and Jacob N. Israelachvili

Effective viscosity η

eff

(Poise)

Film thickness D (nm)

3,000

2,000

1.000

Effective viscosity η

eff

(Poise)

log

eff

–1

)

10,000

1,000

100

12345

Effective viscosity

eff

Shear rate γ

–10

Sliding velocity v

+10

–10

Friction force F=

eff

+10

eff

 γ

Slope: n = –2/3

Slope: n =–1

Dodecane film

D = 2.7nm

Boundary regime

Stick–slip regime

Discontinuous

liquid–solid transitions

eff

∝ γ

(Continuous)

Low loads, L

Bulk Newtonian η

EHD

regime

Bulk

shear

thinning

High L

Low L

EHD

regime

Boundary regime

Stick–slip

regime

High L

Small D

Low L

Solid-

(creep)

Liquid-like

bulk Newtonian flow

Amorphous

eff

∝ γ

–1

≈ Constant

De≈1

a) b)

c) d)

Theintermediateregimeappearsto extendovera narrowrange ofﬁlm thickness,

from about 4 to 10 molecular diameters or polymer radii of gyration. Thinner ﬁlms

begin to adopt boundary or interfacial friction properties (described below, see also

Table 9.5). Note that the intermediate regime is actually a very narrow one when

deﬁned in terms of ﬁlm thickness, for example, varying from about D = 2to4nm

for hexadecaneﬁlms [273].

A ﬂuid’s eﬀective viscosity η

eﬀ

in the intermediate regime is usually higher than

in the bulk, but η

eﬀ

usually decreases with increasing sliding velocity, v (known

as shear thinning). When two surfaces slide in the intermediate regime, the mo-

tion tends to thicken the ﬁlm (dilatancy). This sends the system into the bulk EHD

regime where, as indicated by (9.44),the friction force now increases with velocity.

This initial decrease, followed by an increase, in the frictional forces of many lu-

bricant systems is the basis for the empirical Stribeck curve of Fig. 9.13. 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 dis-

continuous qualitative changes that result in yield stresses and non-liquidlike be-

havior.

9 Surface Forces and Nanorheology of Molecularly Thin Films 479



Fig. 9.24a–d. Typical rheological behavior of liquid ﬁlms in the mixed lubrication regime.

(a) Increase in eﬀective viscosity of dodecane ﬁlm between two mica surfaces with decreas-

ing ﬁlm thickness. At distances larger than 4–5 nm, the eﬀective viscosity η

eﬀ

approaches

the bulk value η

bulk

American Association for the Advancement of Science). (b) Non-Newtonian variation of η

eﬀ

with shear rate of a 2.7-nm-thick dodecane ﬁlm at a net normal pressure of 0.12 MPa and at

◦

C. The eﬀective viscosity decays as a power law, as in (9.46). In this example, n = 0atthe

lowest ˙γ and changes to n= −2/3and−1 at higher ˙γ. For ﬁlms of bulk thickness, dodecane is

a low-viscosity Newtonian ﬂuid(n= 0). (c) Proposed general friction map of eﬀective viscos-

ity η

eﬀ

(arbitrary units) as a function of eﬀective shear rate ˙γ (arbitrary units) on logarithmic

scales. Three main classes of behavior emerge: (i) Thick ﬁlms: elasto-hydrodynamic sliding.

At L = 0, approximating bulk conditions, η

eﬀ

is independent of shear rate except when shear

thinning might occur at suﬃciently large ˙γ. (ii) Boundary layer ﬁlms, intermediate regime.

A Newtonian regime is again observed [η

eﬀ

= constant, n = 0 in (9.46)] at low loads and low

shear rates, but η

eﬀ

is much higher than the bulk value. As the shear rate ˙γ increases beyond

˙γ

min

,theeﬀective viscosity starts to drop with a power-law dependence on the shear rate [see

panel (b)], with n in the range −1/2to−1 most commonly observed. As the shear rate ˙γ

increases still more, beyond ˙γ

max

, a second Newtonian plateau is encountered. (iii) 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. (c) Proposed friction map of friction force as

a function of sliding velocity in various tribological regimes. With increasing load, Newto-

nian ﬂow in the elasto-hydrodynamic (EHD) regimes crosses into the boundary regime of

lubrication. Note that even EHD lubrication changes, at the highest velocities, to limiting

shear stress response. At the highest loads (L) and smallest ﬁlm thickness (D), the friction

force goes through a maximum (the static friction, F

), followed by a regime where the fric-

tion coeﬃcient (μ) is roughly constant with increasing velocity (meaning that the kinetic

friction, F

, is roughly constant). Non-Newtonian shear thinning is observed at somewhat

smaller load and larger ﬁlm thickness; the friction force passes through a maximum at the

point where De = 1. De, the Deborah number, is the point at which the applied shear rate

exceeds the natural relaxation time of the boundary layer ﬁlm. The velocity axis from 10

−10

to 10

with permission from Elsevier Science)

The rest of this section is devoted to friction in the boundary regime. 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.

9.8.3 Boundary Lubrication of Molecularly Thin Films: Nanorheology

When a liquid is conﬁned between two surfaces or within any narrow space whose

dimensions are less than 4 to 10 molecular diameters, both the static (equilibrium)

and dynamic properties of the liquid, such as its compressibility and viscosity,

can no longer be described even qualitatively in terms of the bulk properties. The

molecules conﬁned within such molecularly thin ﬁlms become ordered into layers

(“out-of-plane” ordering), and within each layer they can also have lateral order

480 Marina Ruths and Jacob N. Israelachvili

(“in-plane” ordering). Such ﬁlms may be thought of as behaving more like a liquid

crystal or a solid than a liquid.

As described in Sect. 9.4.3, the measured normal forces between two solid sur-

facesacross molecularlythin ﬁlms exhibitexponentiallydecayingoscillations, vary-

ing between attraction and repulsion with a periodicity equal to some molecular di-

mension of the solvent molecules. Thus most liquid ﬁlms can sustain a ﬁnite normal

stress,and the adhesion forcebetween two surfacesacross suchﬁlms is “quantized”,

depending on the thickness (or number of liquid layers) between the surfaces. The

structuring of molecules in thin ﬁlms and the oscillatory forces it gives rise to are

now reasonably well understood, both experimentally and theoretically, at least for

simple liquids.

Work has also recently been done on the dynamic, e.g., viscous or shear, forces

associated with molecularly thin ﬁlms. Both experiments[38,46,257,275,280,281,

335,336] and theory [254,255,326,337] indicate that, even when two surfaces are

in steady-state sliding, they still prefer to remain in one of their stable potential-

energy minima, i.e., a sheared ﬁlm of liquid can retain its basic layered structure.

Thus even during motion the ﬁlm does not become totally liquid-like. Indeed, if

there is some “in-plane” ordering within a ﬁlm, it will exhibit a yield point before

it begins to ﬂow. Such ﬁlms can therefore sustain a ﬁnite shear stress, in addition to

a ﬁnite normal stress. The value of the yield stress depends on the number of layers

comprisingthe ﬁlm and represents another“quantized”property of molecularlythin

ﬁlms [254].

The dynamic properties of a liquid ﬁlm undergoing shear are very complex.

Depending on whether the ﬁlm is more liquid-like or solid-like, the motion will

be smooth or of the stick–slip type illustrated schematically in Fig. 9.25. During

sliding, transitions can occur between n layers and (n −1) or (n + 1) layers (see

Fig. 9.27). The details of the motion depend critically on the externally applied

load, the temperature, the sliding velocity, the twist angle between the two surface

lattices, and the sliding direction relative to the lattices.

Smooth and Stick–Slip Sliding

Recent advances in friction-measuring techniques have enabled the interfacial fric-

tion of molecularly thin ﬁlms to be measured with great accuracy. Some of these

advances have involved the surface forces apparatus technique [38,44–47,274,275,

280,281,285,286,296,297,308,313,335,336,338] while others have involved the

atomic force microscope[10,58,59,284,290,339,340].In addition, computer simu-

lations [111,151,254,255,287,295,299–302,315–317,333,337,341] have become

suﬃcientlysophisticated toenable fairly complextribologicalsystems to bestudied.

All these advances are necessary if one is to probe such subtle eﬀectsassmoothor

stick–slip friction, transient and memory eﬀects, and ultralow friction mechanisms

at the molecular level.

9 Surface Forces and Nanorheology of Molecularly Thin Films 481

Potential energy (k

–10

Lattice spacing (Å)

4 8 12 16 20 24

Time

D F

A (start)

Friction

force

Fig. 9.25. Simple schematic illustration of the most common molecular mechanism leading

from smooth to stick–slip sliding in terms of the eﬃciency of the energy transfer from me-

chanical to kinetic to phonons. The potential energy of the corrugated surface lattice is shown

by the horizontal sine wave. Let the depth of each minimum be ε which is typically > k

At equilibrium, a molecule will ‘sit’ at one of these minima. When the molecule is connected

to a horizontal spring, a smooth parabolic curve must be added to the horizontal curve. If this

spring is now pushed or pulled laterally at a constant velocity v, the sine curve will move like

a wave along the parabola carrying the molecule up with it (towards point A). When the point

of inﬂection at A is reached the molecule will drop and acquire a kinetic energy greater than

ε even before it reaches the next lattice site. This energy can be “released” at the next lattice

site (i.e., on the ﬁrst collision), in which case the processes will now be repeated – each time

the molecule reaches point A it will fall to point B. This type of motion will give rise to peri-

odic changes in temperature at the interface, as predicted by computer simulations [333]. The

stick–slip here will have a magnitude of the lattice dimension and, except for AFM meas-

urements that can detect such small atomic-scale jumps [59,334], the measured macro- and

microscopic friction forces will be smooth and independent of v. If the energy dissipation (or

“transfer”) mechanism is less than 100% eﬃcient on each collision, the molecule will move

further before it stops. In this case the stick–slip amplitude can be large (point C), and the

kinetic friction F

can even be negative in the case of an overshoot (point D) (after [287],

The theoretical models presented in this section will be concerned with a situ-

ation commonly observed experimentally: stick–slip occurs between a static state

with high friction and a low-friction kinetic state, and a transition from this sliding

regime to smooth sliding can be induced by an increase in velocity. Experimen-

482 Marina Ruths and Jacob N. Israelachvili

tal data on various systems showing this behavior are shown in Figs. 9.27, 9.30b,

and 9.31a. Recent studies on adhesive systems have revealed the possibility of other

dynamic responses such as inverted stick–slip between two kinetic states of higher

and lower friction and with a transition from smooth sliding to stick–slip with in-

creasing velocity, as shown in Fig. 9.30c [342]. Similar friction responses have also

been seen in computer simulations [343].

With the added insights provided by computer simulations, a number of distinct

molecular processes have been identiﬁed during smooth and stick–slip sliding in

model systems for the more familiar static-to-kinetic stick–slip and transition from

stick–slip to smooth sliding. These are shown schematically in Fig. 9.26 for the case

of spherical liquid molecules between two solid crystalline surfaces. The following

regimes may be identiﬁed:

Surfaces at rest (Fig. 9.26a): even with no externally applied load, solvent–

surface epitaxial interactions can cause the liquid molecules in the ﬁlm to attain

a solid-like state. Thus at rest the surfaces are stuck to each other through the ﬁlm.

Sticking regime (frozen, solid-like ﬁlm) (Fig. 9.26b): a progressively increasing

lateral shear stress is applied. The solid-like ﬁlm responds elastically with a small

lateral displacement and a small increase or dilatancy in ﬁlm thickness (less than

Fig. 9.26. Idealized schematic illustration of molecular rearrangements occurring in a molec-

ularly thin ﬁlm of spherical or simple chain molecules between two solid surfaces during

shear. Depending on the system, a number of diﬀerent molecular conﬁgurations within the

ﬁlm are possible during slipping and sliding, shown here as stages (c): total disorder as the

whole ﬁlm melts; (c



): partial disorder; and (c



): order persists even during sliding with slip

occurring at a single slip plane either within the ﬁlm or at the walls. A dilation is predicted in

the direction normal to the surfaces (after [278], with permission)

9 Surface Forces and Nanorheology of Molecularly Thin Films 483

a lattice spacing or molecular dimension, σ). In this regime the ﬁlm retains its

frozen, solid-like state: all the strains are elastic and reversible, and the surfaces

remain eﬀectively stuck to each other. However, slow creep may occur over long

time periods.

Slipping and sliding regimes (molten, liquid-like ﬁlm) (Fig. 9.26c,c





): when

the applied shear stress or force has reacheda certain critical value,the static friction

force, F

, the ﬁlm suddenly melts (known as “shear melting”) or rearranges to allow

for wall slip or slip within the ﬁlm to occur at which point the two surfaces begin

to slip rapidly past each other. If the applied stress is kept at a high value, the upper

surface will continue to slide indeﬁnitely.

Refreezing regime (resolidiﬁcation of ﬁlm) (Fig. 9.26d): In many practical cases,

the rapid slip of the upper surface relieves some of the applied force, which even-

tually falls below another critical value, the kinetic friction force F

, at which point

the ﬁlm resolidiﬁes and the whole stick–slip cycle is repeated. On the other hand, if

the slip rate is smaller than the rate at which the external stress is applied, the sur-

faces will continue to slide smoothly in the kinetic state and there will be no more

stick–slip. The critical velocity at which stick–slip disappears is discussed in more

detail in Sect. 9.8.3.

Experiments with linear chain (alkane) molecules show that the ﬁlm thickness

remainsquantized duringsliding, sothat the structureof such ﬁlmsis probablymore

like that of a nematic liquid crystal where the liquid molecules have become shear-

aligned in some direction, enabling shear motion to occur while retaining some

order within the ﬁlm [344]. Experiments on the friction of two molecularly smooth

mica surfaces separated by three molecular layers of the liquid octamethylcyclote-

trasiloxane(OMCTS, see Fig. 9.27)showhow the friction increasesto higher values

in a quantized way when the number of layers falls from n = 3ton = 2 and then to

n = 1.

Computer simulations for simple spherical molecules [255] further indicate that

during slip the ﬁlm thickness is roughly 15% higher than at rest (i.e., the ﬁlm den-

sity falls), and that the order parameter within the ﬁlm drops from 0.85 to about

0.25. Such dilatancy has been investigated both experimentally [296] and in further

computer simulations [295]. The changes in thickness and in the order parameter

are consistent with a disorganized state for the whole ﬁlm during the slip [337], as

illustrated schematically in Fig. 9.26c. At this stage, we can only speculate on other

possible conﬁgurations of molecules in the slipping and sliding regimes. This prob-

ably depends on the shapes of the molecules (e.g., whether they are spherical or

linear or branched), on the atomic structure of the surfaces, on the sliding velocity,

etc. [345]. Figure 9.26c,c





shows three possible sliding modes wherein the shear-

ing ﬁlm either totally melts, or where the molecules retain their layered structure

and where slip occurs between two or more layers. Other sliding modes, for exam-

ple, involvingthe movementof dislocations or disclinations are also possible, and it

is unlikely that one single mechanism applies in all cases.

Both friction and adhesion hysteresis vary nonlinearly with temperature, often

peaking at some particular temperature, T

. The temperature dependence of these