Bhushan B. Nanotribology and Nanomechanics: An Introduction

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424 Marina Ruths and Jacob N. Israelachvili

Friction attachments for the SFA [43–48] allow for the two surfaces to be

sheared laterally past each other at varying sliding speeds or oscillating frequen-

cies, while simultaneously measuring both the transverse (frictional or shear) force

and the normal force (load) between them. The ranges of friction forces and slid-

ing speeds that can be studied with such methods are currently 10

−7

–10

−1

Nand

−13

–10

−2

−1

, respectively [49]. The externally applied load, L, can be varied

continuously, and both positive and negative loads can be applied. The distance be-

tween the surfaces, D, their true molecular contact area, their elastic (or viscoelastic

or elasto-hydrodynamic)deformation, and their lateral motion can all be monitored

simultaneously by recording the moving interference-fringe pattern. Equipment for

dynamic measurements of normal forces has also been developed. Such measure-

ments give information on the viscosity of the medium and the location of the shear

or slip planes relative to the surfaces [11–13,50,51].

In the atomic force microscope (Fig. 9.3), the force is measured by monitoring

the deﬂection of a soft cantileversupportinga sub-microscopictip (R≈10−200nm)

as this interacts with a ﬂat, macroscopic surface [8,52,53]. The measurements can

be done in a vapor or liquid. The normal (bending) spring stiﬀness of the can-

tilever can be as small as 0.01Nm

−1

, allowing measurements of normal forces

Quadrant displacement sensor

Lateral

Normal

Base

Piezo

transducer

Cantilever

spring

Light beam

Cantilever

displacements

ΔD

Tip

Surface

ΔD

Fig. 9.3. Schematic drawing of an atomic force microscope (AFM) tip supported on a trian-

gular cantilever interacting with an arbitrary solid surface. The normal force and topology are

measured by monitoring the calibrated deﬂection of the cantilever as the tip is moved across

the surface by means of a piezoelectric transducer. Various designs have been developed that

move either the sample or the cantilever during the scan. Friction forces can be measured

from the torsion of the cantilever when the scanning is in the direction perpendicular to its

long axis (after [60] with permission)

9 Surface Forces and Nanorheology of Molecularly Thin Films 425

as small as 1 pN (10

−12

N), which corresponds to the bond strength of single

molecules [54–57]. Distances can be inferred with an accuracy of about 1nm, and

changes in distance can be measured to about 0.1 nm. Since the contact area can

be small when using sharp tips, diﬀerent interaction regimes can be resolved on

samples with a heterogeneous composition on lateral scales of a few nanometers.

Height diﬀerences and the roughness of the sample can be measured directly from

the cantilever deﬂection or, alternatively, by using a feedback system to raise or

lower the sample so that the deﬂection (the normal force) is kept constant during

a scan over the area of interest. Weak interaction forces and larger (microscopic)

interaction areas can be investigated by replacing the tip with a micrometer-sized

sphere to form a “colloidal probe” [9].

The atomic force microscopecan also be used for friction measurements(lateral

force microscopy, LFM, or friction force microscopy, FFM) by monitoring the tor-

sion of the cantilever as the sample is scanned in the direction perpendicular to the

long axis of the cantilever [10,53,58,59].Typically, the stiﬀness of the cantilever to

lateral bending is much larger than to bending in the normal direction and to torsion,

so that these signals are decoupled and height and friction can be detected simulta-

neously. The torsional spring constant can be as low as 0.1Nm

−1

, giving a lateral

(friction) force sensitivity of 10

−11

Rapid technical developmentshave facilitated the calibrations of the normal [61,

62] and lateral spring constants [59,63–65], as well as in situ measurements of the

macroscopic tip radius [66, 67]. Cantilevers of diﬀerent shapes with a large range

of spring constants, tip radii, and surface treatments (inorganic or organic coatings)

are commercially available. The ﬂat surface, and also the particle in the colloidal

probe technique, can be any material of interest. However, remaining diﬃculties

with this technique are that the distance between the tip and the substrate, D,and

the deformations of the tip and sample, are not directly measurable. Another im-

portant diﬀerence between the AFM/LFM and SFA techniques is the diﬀerent size

of the contact area, and the related observation that, even when a cantilever with

a very low spring constant is used in the AFM, the pressure in the contact zone is

typically much higher than in the SFA. Hydrodynamic eﬀects in liquids also aﬀect

the measurements of normal forces diﬀerently on certain time scales [68–71].

9.2.4 Some Other Force-Measuring Techniques

A largenumberof other techniquesare availablefor the measurementsof the normal

forces between solid or ﬂuid surfaces (see [5,60]). The techniques discussed in this

section are not used for lateral (friction) force measurements, but are commonly

used to study normal forces, particularly in biological systems.

Micropipette aspiration is used to measure the forces between cells or vesicles,

or between a cell or vesicle and another surface [72–74]. The cell or vesicle is held

by suction at the tip of a glass micropipette and deforms elastically in response to

the net interactions with another surface and to the applied suction. The shape of

the deformed surface (cell membrane) is measured and used to deduce the force

between the surfaces and the membrane tension [73]. The membrane tension, and

426 Marina Ruths and Jacob N. Israelachvili

thus the stiﬀness of the cell or vesicle, is regulated by applying diﬀerent hydro-

static pressures. Forces can be measured in the range of 0.1pN to 1 nN, and the

distance resolution is a few nanometers. The interactions between a colloidal par-

ticle and another surface can be studied by attaching the particle to the cell mem-

brane [75].

In the osmotic stress technique, pressures are measured between colloidal par-

ticles in aqueous solution, membranes or bilayers, or other ordered colloidal struc-

tures (viruses, DNA). The separation between the particle surfaces and the magni-

tude of membrane undulations are measured by X-ray or neutron scattering tech-

niques. This is combined with a measurement of the osmotic pressure of the so-

lution [76–79]. The technique has been used to measure repulsive forces, such as

Derjaguin–Landau–Verwey–Overbeek (DLVO) interactions, steric forces, and hy-

dration forces [80]. The sensitivity in pressure is 0.1mNm

−2

, and distances can be

resolved to 0.1nm.

The optical tweezers technique is based on the trapping of dielectric particles at

the center of a focused laser beam by restoring forces arising from radiation pres-

sure and light-intensity gradients [81,82]. The forces experienced by particles as

they are moved toward or away from one another can be measured with a sensitivity

in the pN range. Small biological molecules are typically attached to a larger bead

of a material with suitable refractive properties. Recent development allows deter-

minations of position with nanometer resolution [83], which makes this technique

useful for studying the forces during the extension of single molecules.

In total internal reﬂection microscopy (TIRM), the potential energy between

a micrometer-sized colloidal particle and a ﬂat surface in aqueous solution is de-

duced from the average equilibrium height of the particle above the surface, meas-

ured from the intensity of scattered light. The average height (D ≈ 10−100nm)

results from a balance of gravitational force, radiation pressure from a laser beam

focused at the particle from below, and intermolecular forces [84]. The technique

is particularly suitable for measuring weak forces (sensitivity ca. 10

−14

N), but is

more diﬃcult to use for systems with strong interactions. A related technique is

reﬂection interference contrast microscopy (RICM), where optical interference is

used to also monitor changes in the shape of the approaching colloidal particle or

vesicle [85].

An estimate of bond strengths can be obtained from the hydrodynamic shear

force exerted by a ﬂuid on particles or cells attached to a substrate [86, 87]. At

a critical force, the bonds are broken and the particle or cell will be detached and

move with the velocity of the ﬂuid. This method requires knowledge of the contact

area and the ﬂow-velocity proﬁle of the ﬂuid. Furthermore, a uniform stress distri-

bution in the contact area is generally assumed. At low bond density, this technique

can be used to determine the strength of single bonds (1pN).

9 Surface Forces and Nanorheology of Molecularly Thin Films 427

9.3 Normal Forces Between Dry (Unlubricated) Surfaces

9.3.1 Van der Waals Forces in Vacuum and Inert Vapors

Forces between macroscopic bodies (such as colloidal particles) across vacuum

arise from interactions between the constituent atoms or molecules of each body

across the gap separating them. These intermolecular interactions are electromag-

netic forces between permanent or induced dipoles (van der Waals forces), and

between ions (electrostatic forces). In this section, we describe the van der Waals

forces, which occur between all atoms and molecules and between all macroscopic

bodies (see [3]).

The interaction between two permanent dipoles with a ﬁxed relative orientation

can be attractive or repulsive. For the speciﬁc case of two freely rotating permanent

dipoles in a liquid or vapor (orientational or Keesom interaction), and for a perma-

nent dipole and an induced dipole in an atom or polar or nonpolar molecule (induc-

tion or Debye interaction), the interaction is on average always attractive. The third

type of van der Waals interaction, the ﬂuctuation or London dispersion interaction,

arises from instantaneous polarization of one nonpolar or polar molecule due to

ﬂuctuations in the charge distribution of a neighboring nonpolar or polar molecule

(Fig. 9.4a). Correlation between these ﬂuctuating induced dipole moments gives an

attraction that is present between any two molecules or surfaces across vacuum. At

very small separations, the interaction will ultimately be repulsive as the electron

clouds of atoms and molecules begin to overlap. The total interaction is thus a com-

bination of a short-range repulsion and a relatively long-range attraction.

Except for in highly polar materials such as water, London dispersion interac-

tions give the largest contribution (70–100%) to the van der Waals attraction. The

interaction energy of the van der Waals force between atoms or molecules depends

on the separation r as

E(D) =

−C

vdW

, (9.7)

where the constant C

vdW

depends on the dipole moments and polarizabilities of the

molecules. At large separations (> 10 nm), the London interaction is weakened by

Surface

diffusion

Inter-diffusion

Flow

Sparks

+++

–

––

–

b)a)

–

Fig. 9.4a–c. Schematic representationof (a) van der Waals interaction (dipole–induced dipole

interaction), (b) charge exchange, which acts to increase adhesion and friction forces, and (c)

sintering between two surfaces

428 Marina Ruths and Jacob N. Israelachvili

a randomizing eﬀect caused by the rapid ﬂuctuations. That is, the induced tempo-

rary dipole moment of one molecule may have changed during the time needed for

the transmission of the electromagnetic wave (photon) generated by its ﬂuctuating

charge density to another molecule and the return of the photon generated by the

induced ﬂuctuation in this second molecule. This phenomenon is called retardation

and causes the interaction energy to decay as r

−7

at large separations [88].

Dispersion interactions are to a ﬁrst approximation additive, and their contribu-

tion to the interaction energy between two macroscopic bodies (such as colloidal

particles) across vacuum can be found by summing the pairwise interactions [89].

The interaction is generally described in terms of the Hamaker constant, A

.An-

other approach is to treat the interacting bodies and an intervening medium as con-

tinuous phases and determine the strength of the interaction from bulk dielectric

properties of the materials [90, 91]. Unlike the pairwise summation, this method

takes into account the screening of the interactions between molecules inside the

bodies by the molecules closer to the surfaces and the eﬀects of the intervening

medium. For the interaction between material 1 and material 3 acrossmaterial 2, the

non-retarded Hamaker constant given by the Lifshitz theory is approximately [3]:

H,123

= A

H,ν=0

+ A

H,ν>0

≈



−ε

+ ε



−ε

+ ε



3hν

√



−n



−n





+ n





+ n



!



+ n





+ n





(9.8)

where the ﬁrst term (ν = 0) represents the permanent dipole and dipole–induced

dipole interactions and the second (ν>0) the London (dispersion) interaction.

and n

are the static dielectricconstants and refractiveindexesof the materials, re-

spectively. ν

is the frequencyof the lowest electron transition (around3×10

−1

Either one of the materials 1, 2, or 3 in (9.8) can be vacuum or air (ε = n = 1). A

typically 10

−20

–10

−19

J (the higher values are found for metals) for interactions be-

tween solids and liquids across vacuum or air.

The interaction energy between two macroscopic bodies is dependent on the

geometry and is always attractive between two bodies of the same material [A

positive, see (9.8)]. The van der Waals interaction energy and force laws (F =

−dE(D)/ dD) for some common geometries are given in Table 9.2. Because of the

retardation eﬀect, the equations in Table 9.2 will lead to an overestimation of the

dispersion force at large separations. It is, however, apparent that the interaction

energy between macroscopic bodies decays more slowly with separation (i.e., has

a longer range) than between two molecules.

For inert nonpolar surfaces, e.g., consisting of hydrocarbons or van der Waals

solids and liquids, the Lifshitz theory has been found to apply even at molecular

contact, where it can be used to predict the surface energies (surface tensions) of

such solids and liquids. For example, for hydrocarbon surfaces, A

= 5×10

−20

9 Surface Forces and Nanorheology of Molecularly Thin Films 429

Table 9.2. Van der Waals interaction energy and force between macroscopic bodies of diﬀer-

ent geometries

Geometry of bodies van der Waals interaction

with surfaces D apart (D  R)Energy,E Force, F

Two atoms or

small molecules

r  σ

−C

vdW

−6C

vdW

Two ﬂat surfaces

(per unit area)

r  D

Area = Sr

−A

12πD

−A

6πD

Two spheres or

macromolecules

of radii R

and R

 D

−A



+ R



−A



+ R



Sphere or

macromolecule

of radius R

near a ﬂat surface

R  D

−A

Two parallel

cylinders or rods

of radii R

and R

(per unit length)

 D

Length

−A

√

3/2



+ R



1/2

−A

√

5/2



+ R



1/2

Cylinder of

radius R

near a ﬂat surface

(per unit length)

R  D

−A

√

3/2

−A

√

5/2

Two cylinders or

ﬁlaments of radii

and R

crossed

at 90

◦

 D

−A

√

−A

√

A negative force (A

positive) implies attraction, a positive force means repulsion

negative) (after [60], with permission)

430 Marina Ruths and Jacob N. Israelachvili

Inserting this value into the equation for two ﬂat surfaces (Table 9.2) and using

a “cut-oﬀ” distance of D

≈ 0.165nm as an eﬀective separation when the surfaces

are in contact [3], we obtain for the surface energy γ (which is deﬁned as half the

interaction energy)

γ =

24πD

≈ 24 mJm

−2

, (9.9)

a value that is typical for hydrocarbon solids and liquids [92].

If the adhesion force is measured between two crossed-cylindrical surfaces of

R = 1cm (a geometryequivalentto a sphere with R = 1 cm interacting witha ﬂat sur-

face, cf. Table 9.2) using an SFA, we expect the adhesion force to be (see Table 9.2)

F = A

R/(6D

) = 4πRγ ≈ 3.0 mN. Using a spring constant of k

= 100Nm

−1

,such

an adhesive force will cause the two surfaces to jump apart by ΔD = F/k

= 30 µm,

which can be accurately measured. (For elastic bodies that deform in adhesive con-

tact, R changesduring theinteractionand the measuredadhesionforce is 25%lower,

see Sect. 9.5.2). Surface energies of solids can thus be directly measured with the

SFA and, in principle, with the AFM if the contact geometry can be quantiﬁed. The

measured values are in good agreement with calculated values based on the known

surface energies γ of the materials, and for nonpolar low-energysolids they are well

accounted for by the Lifshitz theory [3].

9.3.2 Charge-Exchange Interactions

Electrostatic interactions are present between ions (Coulomb interactions), between

ions and permanent dipoles, and between ions and nonpolar molecules in which

a charge induces a dipole moment. The interaction energy between ions or between

a charge and a ﬁxed permanent dipole can be attractive or repulsive. For an induced

dipole or a freely rotating permanent dipole in vacuum or air, the interaction energy

with a charge is always attractive.

Spontaneouscharge transfermay occur betweentwo dissimilar materialsin con-

tact [93–97]. The phenomenon, called contact electriﬁcation, is especially promi-

nent in contact between a metal and a material with low conductivity (including

organic liquids) [95,97], but is also observed, for example, between two diﬀerent

polymer layers. It is believed that when two diﬀerent materials are in static con-

tact, charge transfer might occur due to quantum tunneling of electrons or, in some

cases, transfer of surface ions. The charging is generally seen to be stronger with

increasing diﬀerence in work function (or electron aﬃnity) between the two materi-

als [95, 97]. During separation, rolling or sliding of one body over the other, the

surfaces experience both charge transition from one surface to the other and charge

transfer (conductance) along each surface (Fig. 9.4b). The latter process is typi-

cally slower, and, as a result, charges remain on the surfaces as they are separated

in vacuum or dry nitrogen gas. The charging gives rise to a strong adhesion with

adhesion energies of over 1000mJm

−2

, similar to fracture or cohesion energies of

the solid bodies themselves [93,94,98]. Upon separating the surfaces further apart,

9 Surface Forces and Nanorheology of Molecularly Thin Films 431

a strong, long-range electrostatic attraction is observed. The charging can be de-

creased through discharges across the gap between the surfaces (which requires

a high charging) or through conduction in the solids. The discharge may give rise

to triboluminescence [99], but can also cause sparks that may ignite combustible

materials [100]. It has been suggested that charge-exchange interactions are parti-

cularly important in rolling friction between dry surfaces (which can simplistically

be thought of as an adhesion–separation process), where the distance dependence

of forces acting normally to the surfaces plays a larger role than in sliding friction.

In the case of sliding friction, charge transfer is also observed between identical

materials [98, 101]. Mechanisms such as bond formation and breakage (polymer

scission), slip at the wall between a ﬂowing liquid and a solid [102], or material

transfer and the creation of wear particles have been suggested. However, friction

electriﬁcation or triboelectriﬁcation also occurs during wearless sliding, i.e., when

the surfaces are not damaged. Other explanations such as the creation or transla-

tion of defects on or near the surface have been put forward [98]. Attempts have

been made to correlate the amount of charging with the normal force or load and

with the polarizability of the sliding materials [96,103]. Recent experiments on the

sliding friction between metal–insulator surfaces indicate that stick–slip would be

accompanied by charge-transfer events [104,105].

Photoinduced charge transfer, or harpooning, involves the transfer of an elec-

tron between an atom in a molecular beam or at a solid surface (typically an alkali

or transition metal) to an atom or molecule in a gas (typically a halide) to form

a negatively charged molecular ion in a highly excited vibrational state. This trans-

fer processcan occur at atomic distancesof 0.5–0.7 nm,which is far from molecular

contact. The formed molecular ion is attracted to the surface and chemisorbs onto

it. Photoinduced charge-transferprocesses also occur in the photosynthesis in green

plants and in photo-electrochemical cells (solar cells) at the junction between two

semiconductors or between a semiconductor and an electrolyte solution [106].

9.3.3 Sintering and Cold Welding

When macroscopic particles in a powder or in a suspension come into molecular

contact, they can bond together to form a network or solid body with very diﬀerent

density and shear strength compared to the powder (a typical example is porcelain).

The rate of bonding is dependent on the surface energy (causing a stress at the edge

of the contact) and the atomic mobility (diﬀusion rate) of the contacting materials.

To increase the diﬀusion rate, objects formed from powders are heated to about one

half of the melting temperature of the components in a process called sintering,

which can be done in various atmospheres or in a liquid.

In the sintering process, the surface energy of the system is lowered due to the

reduction of total surface area (Fig. 9.4c). In metal and ceramic systems, the most

important mechanism is solid-state diﬀusion, initially surface diﬀusion. As the sur-

face area decreasesand the grain boundariesincrease at the contacts, grainboundary

diﬀusionand diﬀusion throughthe crystallattice becomemore important[107]. The

432 Marina Ruths and Jacob N. Israelachvili

grain boundaries will eventually migrate, so that larger particles are formed (coars-

ening). Mass can also be transferred through evaporation and condensation, and

through viscous and plastic ﬂow. In liquid-phase sintering, the materials can melt,

which increases the mass transport. Amorphousmaterials like polymers and glasses

do not have real grain boundaries and sinter by viscous ﬂow [108].

Some of these mechanisms (surface diﬀusion and evaporation–condensation)

reduce the surface area and increase the grain size (coarsening) without densiﬁca-

tion, in contrast to bulk transport mechanisms such as grain-boundary diﬀusion and

plastic and viscous ﬂow. As the material becomes denser, elongated pores collapse

to form smaller, spherical pores with a lower surface energy. Models for sinter-

ing typically consider the size and growth rate of the grain boundary (the “neck”)

formed between two spherical particles. At a high stage of densiﬁcation, the sinter-

ing stress σ at the curved neck between two particles is given by [108]

σ =

2γ

, (9.10)

where γ

is the solid–solid grain boundary energy, γ

is the solid–vapor surface

energy, G is the grain size, and r

is the radius of the pore.

A related phenomenon is cold welding, which is the spontaneous formation of

strong junctions between clean (unoxidized) metal surfaces with a mutual solubil-

ity when they are brought into contact, with or without an applied pressure. The

plastic deformations accompanying the formation and breaking of such contacts

on a molecular scale during motion of one surface normally (see Fig. 9.10c,d) or

laterally (shearing) with respect to the other have been studied both experimen-

tally [109,110] and theoretically [111–116]. The breaking of a cold-welded contact

is generally associated with damage or deformation of the surface structure.

9.4 Normal Forces Between Surfaces in Liquids

9.4.1 Van der Waals Forces in Liquids

The dispersion interaction in a medium will be signiﬁcantly lower than in vacuum,

since the attractive interaction between two solute molecules in a medium (solvent)

involves displacement and reorientation of the nearest-neighbor solvent molecules.

Even though the surrounding medium may change the dipole moment and polariz-

ability from that invacuum, theinteraction betweentwo identical moleculesremains

attractive in a binary mixture. The extension of the interactions to the case of two

macroscopic bodies is the same as described in Sect. 9.3.1. Typically, the Hamaker

constants for interactions in a medium are an order of magnitude lower than in

vacuum. Between macroscopic surfaces in liquids, van der Waals forces become

importantat distances below 10–15nm and may at these distances start to dominate

interactions of diﬀerent origin that have been observed at larger separations.

Figure 9.5 shows the measured van der Waals forces between two crossed-

cylindrical mica surfaces in water and various salt solutions. Good agreement is

9 Surface Forces and Nanorheology of Molecularly Thin Films 433

Force/Radius F/R (mN/m)

Distance D (nm)

–0.1

–0.2

–0.3

–0.4

–0.5

510

Non-

retarded

regime

Retarded regime

F/R = – A

/6D

= 2.2 u 10

–20

Fig. 9.5. Attractive van der Waals force F between two curved mica surfaces of radius R ≈

1 cm measured in water and various aqueous electrolyte solutions. The electrostatic interac-

tion has been subtracted from the total measured force. The measured non-retarded Hamaker

constant is A

= 2.2×10

−20

J. Retardation eﬀects are apparent at distances larger than 5 nm,

obtained between experiment and theory. At larger surface separations, above about

5 nm, the measuredforces fall oﬀ more rapidlythan D

−2

. This retardation eﬀect (see

Sect. 9.3.1) is also predicted by the Lifshitz theory and is due to the time needed for

propagation of the induced dipole moments over large distances.

From Fig. 9.5, we may conclude that, at separations above about 2nm, or 8

molecular diameters of water, the continuum Lifshitz theory is valid. This would

mean that water ﬁlms as thin as 2nm may be expected to have bulk-like properties,

at least as far as their interaction forces are concerned. Similar results have been

obtained with other liquids, where in general continuum properties are manifested,

both as regards their interactions and other properties such as viscosity, at a ﬁlm

thickness larger than ﬁve or ten molecular diameters. In the absence of a solvent (in

vacuum), the agreement of measured van der Waals forces with the continuum Lif-

shitz theoryis generally goodat all separationsdown tomolecular contact(D = D

Van der Waals interactions in a system of three or more diﬀerent materials

(see (9.8)) can be attractive or repulsive, depending on their dielectric properties.

Numerous experimental studies show the attractive van der Waals forces in var-

ious systems [3], and repulsive van der Waals forces have also been measured

directly [117, 118]. A practical consequence of the repulsive interaction obtained

across a medium with intermediate dielectric properties is that the van der Waals

forces will give rise to preferential, nonspeciﬁc adsorption of molecules with an in-

termediate dielectric constant. This is commonly seen as adsorption of vapors or

solutes to a solid surface. It is also possible to diminish the attractive interaction