Israelachvil, J. N. et al.“Surface Forces and Microrheology of ...”
Handbook of Micro/Nanotribology.
Ed. Bharat Bhushan
Boca Raton: CRC Press LLC, 1999
© 1999 by CRC Press LLC© 1999 by CRC Press LLC

9

Surface Forces and
Microrheology of
Molecularly Thin

Liquid Films

Jacob N. Israelachvili and
Alan D. Berman

9.1 Introduction
9.2 Methods for Measuring Static and Dynamic Surface
Forces

Adhesion Forces • Force Laws • The Surface Force Apparatus
and the Atomic Force Microscope

9.3 van der Waals and Electrostatic Forces between
Surfaces in Liquids

van der Waals Forces • Electrostatic Forces

Tribology

General Interfacial Friction • Boundary Friction of Surfactant
Monolayer-Coated Surfaces • Boundary Lubrication of
Molecularly Thin Liquid Films • Transition from Interfacial
to Normal Friction (with Wear)

9.11 Theories of Interfacial Friction

Theoretical Modeling of Interfacial Friction: Molecular
Tribology • Adhesion Force Contribution to Interfacial
Friction • Relation between Boundary Friction and Adhesion
Energy Hysteresis • External Load Contribution to Interfacial
Friction • Simple Molecular Model of Energy Dissipation

ε

9.12 Friction and Lubrication of Thin Liquid Films

Smooth and Stick-Slip Sliding • Role of Molecular Shape and
Liquid Structure

9.13 Stick-Slip Friction

Rough Surfaces Model • Distance-Dependent Model •
Velocity-Dependent Friction Model • Phase-Transition
Model • Critical Velocity for Stick-Slip • Dynamic Phase
Diagram Representation of Tribological Parameters

Acknowledgment

particles elastically or plastically when they come into contact for the first time.
When exposed to

vapors

(e.g., atmospheric air containing water and organic molecules), two solid
surfaces in or close to contact will generally have a surface layer of chemisorbed or physisorbed molecules,
or a capillary condensed liquid bridge between them. These effects can drastically modify their adhesion.
The adhesion usually falls, but in the case of capillary condensation the additional Laplace pressure or
attractive “capillary” force between the surfaces may make the adhesion stronger than in inert gas or
vacuum.
When totally immersed in a

liquid

, the force between two surfaces is once again completely modified
from that in vacuum or air (vapor). The van der Waals attraction is generally reduced, but other forces
can now arise which can qualitatively change both the range and even the sign of the interaction. The
overall attraction can be either stronger or weaker than in the absence of the intervening liquid medium,
for example, stronger in the case of two hydrophobic surfaces in water, but weaker for two hydrophilic
surfaces. Since a number of different forces may be operating simultaneously in solution, the overall
force law is not generally monotonically attractive, even at long range: it can be repulsive, oscillatory, or
the force can change sign at some finite surface separation. In such cases, the potential energy minimum,

© 1999 by CRC Press LLC

which determines the adhesion force or energy, occurs not at true molecular contact but at some small
distance farther out.
The forces between two surfaces in a liquid medium can be particularly complex at


Exchange interaction (v)
Strong short-ranged forces responsible for contact binding
of crystalline surfaces
Hydrophobic Attractive hydration force (s) Strong, apparently long-ranged force; origin not yet
understood
Ion–correlation van der Waals force of
polarizable ions (s)
Requires mobile charges on surfaces in a polar solvent
Solvation Oscillatory force (s)
Depletion force (s)
The oscillatory force generally alternates between attraction
and repulsion; mainly entropic in origin
Specific binding “Lock and key” binding (v & s)
Receptor–ligand interaction (s)
Antibody–antigen interaction (s)
Subtle combination of different noncovalent forces giving
rise to highly specific binding; main “recognition”
mechanism of biological systems.
Repulsive
Quantum
mechanical
Hard-core (v)
Steric repulsion (v)
Born repulsion (v)
Forces stabilizing attractive covalent and ionic binding
forces, effectively determine molecular size and shape
van der Waals van der Waals disjoining
pressure (s)
Arises only between dissimilar bodies interacting in a
medium

; s, applies only to interactions in

solution

, or to surfaces separated
by a liquid; v & s, applies to interactions occurring both in vacuum and in solution.

© 1999 by CRC Press LLC

determined solely by its bulk properties; the size and shape of its molecules begin to play an important
role in determining the overall interaction. In addition, the surfaces themselves can no longer be treated
as inert and structureless walls (i.e., mathematically flat) — their physical and chemical properties at the
atomic scale must also now be taken into account. Thus, the force laws will now depend on whether the
surface lattices are crystallographically matched or not, whether the surfaces are amorphous or crystalline,
rough or smooth, rigid or soft (fluidlike), hydrophobic or hydrophilic.
In practice, it is also important to distinguish between

static

(i.e., equilibrium) forces and

dynamic

(i.e., nonequilibrium) forces such as viscous and friction forces. For example, certain liquid films confined
between two contacting surfaces may take a surprisingly long time to equilibrate, as may the surfaces
themselves, so that the short-range and adhesion forces appear to be time dependent, resulting in “aging”
effects.

9.2 Methods for Measuring Static and Dynamic



S

, the upper surface will jump
from

A

into contact at

A

′

(

A

for advancing). On separating the surfaces by raising the base, the two surfaces will
jump apart from

R

to

R

′

(

© 1999 by CRC Press LLC

apparatuses such as the surface forces apparatus (SFA) (Israelachvili, 1989, 1991) or the atomic force
microscope (AFM) (Ducker et al., 1991).
If

K

S

is the stiffness of the force-measuring spring and

∆

D

the distance the two surfaces jump apart
when they separate, then the adhesion force

F

S

is given by
(9.1)
where we note that in liquids the maximum or minimum in the force may occur at some nonzero surface
separation (see Figures 9.3 and 9.4 below).
From

F


(9.3)
where

γ

is in units of J/m

2

.

9.2.2 Force Law

The full force law

F

(

D

) between two surfaces, that is, the force

F

as a function of surface separation

D


4)
The force difference

∆

F

between the initial and final separations is given by
(9.5)
The above equations provide the basis for measuring the force difference between any two surface
separations. For example, if a particular force-measuring apparatus can measure

∆

D

0

,

∆

D

S

, and

K



∆

D

= (

∆

D

0

–

∆

D

S

), the full force law

F

(

D

) can be constructed over any desired

∆∆∆DDD
S
==
0
.
∆∆FK D
SS
= .

© 1999 by CRC Press LLC

In practice, it is difficult to measure the forces between two perfectly flat surfaces because of the
stringent requirement of perfect alignment for making reliable measurements at the angstrom level. It
is far easier to measure the forces between curved surfaces, for example, two spheres, a sphere and a flat,
or two crossed cylinders. As an added convenience, the force

F

(

D

) measured between two curved surfaces
can be directly related to the energy per unit area

E

(

D

S

,

∆

D

, and

K

S

, are directly or indirectly measured, from which the third displacement and resulting force
law

F

(

D

) are deduced using Equations 9.4 and 9.5. For example, in SFA experiments,

∆

D

0

g). For the typical surface radii of R ≈ 1 cm used in these experiments, γ
values can be measured to an accuracy of about ±10
–3
mJ/m
2
(±10
–3
erg/m
2
).
Various surface materials have been successfully used in SFA force measurements including mica
(Pashley, 1981, 1982, 1985), silica (Horn et al., 1989b), and sapphire (Horn et al., 1988).

It is also possible
to measure the forces between adsorbed polymer layers (Klein, 1983, 1986; Patel and Tirrell, 1989; Ploehn
and Russel, 1990), surfactant monolayers and bilayers (Israelachvili, 1987, 1991; Christenson, 1988a;
Israelachvili and McGuiggan, 1988), and metal and metal oxide layers deposited on mica (Coakley and
Tabor, 1978; Parker and Christenson, 1988; Smith et al., 1988; Homola et al., 1993; Steinberg et al., 1993).
The range of liquids and vapors that can be used is almost endless, and so far these have included aqueous
solutions, organic liquids and solvents, polymer melts, various petroleum oils and lubricant liquids, and
liquid crystals.
Recently, new friction attachments were developed suitable for use with the SFA (Homola et al., 1989;
Van Alsten and Granick, 1988, 1990b; Klein et al., 1994; Luengo et al., 1997). These attachments allow
for the two surfaces to be sheared past each other at varying sliding speeds or oscillating frequencies
while simultaneously measuring both the transverse (frictional or shear) force and the normal force or
load between them. The externally applied load, L, can be varied continuously, and both positive and
negative loads can be applied. Finally, the distance between the surfaces D, their true molecular contact
area A, their elastic (or viscoelastic or elastohydrodynamic) deformation, and their lateral motion can
all be monitored simultaneously by recording the moving interference fringe pattern using a video
camera–recorder system.

nonconducting liquids and solids interacting in vacuum or air (ε
2
= n
2
= 1), their Hamaker constants are
typically in the range (5 to 10) × 10
–20
J, rising to about 4 × 10
–19
J for metals, while for interactions in
a liquid medium, the Hamaker constants are usually about an order of magnitude smaller.
For inert nonpolar surfaces, e.g., of hydrocarbons or van der Waals solids and liquids, the Lifshitz
theory has been found to apply even at molecular contact, where it can predict the surface energies (or
tensions) of solids and liquids. Thus, for hydrocarbon surfaces the Hamaker constant is typically A = 5 ×
10
–20
J. Inserting this value into the appropriate equation for two flat surfaces (Table 9.2) and using a
“cut-off” distance of D = D
0
≈ 0.15 nm when the two surfaces are in contact, we obtain for the surface
energy γ (which is conventionally defined as half the interaction energy):
(9.8)
a value that is typical for hydrocarbon solids and liquids (for liquids, γ is sometimes referred to as the
surface tension and is expressed in units of mN/m).
TABLE 9.2 van der Waals Interaction Energy and Force Between Macroscopic Bodies
of Different Geometries
Geometry of Bodies With Surfaces D Apart
van der Waals Interaction
(D Ӷ R) Energy Force
Two flat surfaces (per unit area) E = A/12pD

nn
=
−
+






+
−
()
+
()
3
4
16 2
12
12
2
1
2
2
2
2
1
2
2
2

for elastic bodies that deform on coming into adhesive contact, their radius R changes during the
interaction and the measured adhesion force is 25% lower — see Equation 9.21). The above example
shows how the surface energies of solids can be directly measured with the SFA and, in principle, with
the AFM (if the geometry of the tip and surface at the contact zone can be quantified). The measured
values are generally in good agreement with calculated values based on the known surface energies γ of
the materials and, for nonpolar low-energy solids, are well accounted for by the Lifshitz theory (Israelach-
vili, 1991).
For adhesion measurements in vacuum or inert atmosphere to be meaningful, the surfaces must be
both atomically smooth and clean. This is not always easy to achieve, and for this reason only inert, low-
energy surfaces, such as hydrocarbon and certain polymeric surfaces, have had their true adhesion forces
and surface energies directly measured so far. Other smooth surfaces have also been studied, such as bare
mica, metal, metal oxide, and silica surfaces but these are high-energy surfaces, so that it is difficult to
prevent them from physisorbing a monolayer of organic matter or water from the atmosphere or from
getting an oxide monolayer chemisorbed on them, all of which affects their adhesion.
Many contaminants that physisorb onto solid surfaces from the ambient atmosphere usually dissolve
away once the surfaces are immersed in a liquid, so that the short-range forces between such surfaces
can usually be measured with great reliability. Figure 9.2 shows results of measurements of the van der
Waals forces between two crossed cylindrical mica surfaces in water and various salt solutions, showing
the good agreement obtained between experiment and theory (compare the solid curve, corresponding
to F = AR/6D
2
, where A = 2.2 × 10
–20
J is the fitted value, which is within about 15% of the theoretical
FIGURE 9.2 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 measured nonretarded Hamaker constant is A = 2.2 × 10
–20
J.
Retardation effects are apparent at distances above 5 nm, as expected theoretically. Agreement with the continuum
Lifshitz theory of van der Waals forces is generally good at all separations down to five to ten solvent molecular

dissociation of ions from the surfaces into the solution or the preferential adsorption of certain ions from
the solution. The surface charge is balanced by an equal but opposite layer of oppositely charged ions
(counterions) in the solution at some small distance away from the surface. This distance is known as
the Debye length which is purely a property of the electrolyte solution. The Debye length falls with
increasing ionic strength and valency of the ions in the solution, and for aqueous electrolyte (salt)
solutions at 25°C the Debye length is
(9.10)
where the salt concentration M is in moles. The Debye length also relates the surface charge density σ
of a surface to the electrostatic surface potentials ψ
0
via the Grahame equation:
(9.11)
where the concentrations [M
1:1
] and [M
2:2
] are again in M, ψ
0
in mV, and σ in C m
–2
(1 C m
–2
corresponds
to one electronic charge per 0.16 nm
2
or 16 Å
2
). For example, for NaCl solutions, 1/κ ≈ 10 nm at 1 mM,
and 0.3 nm at 1 M. In totally pure water at pH 7, where [M
1:1

.
. ,
:
:
:
M
M
M
for 1:1 electrolytes such as NaCl
for 1:2 or 2:1 electrolytes such as CaCl
for 2:2 electrolytes such as MgSO
2
4
σψ
ψ
=
()
++
()
−
0 117 51 4 2
01122
25 7
12
0
. sinh . ,
::
.
MM e
E mV J m for monovalent salts

ψ
κ
κ
© 1999 by CRC Press LLC
(9.13)
The above approximate expressions are accurate only for surface separations beyond about one Debye
length. At smaller separations one must resort to numerical solutions of the Poisson–Boltzmann equation
to obtain the exact interaction potential for which there are no simple expressions (Hunter, 1987).

In the
limit of small D, it can be shown that the interaction energy depends on whether the surfaces remain at
constant potential ψ
0
(as assumed in the above equations) or at constant charge σ (when the repulsion
exceeds that predicted by the above equations), or somewhere in between these two limits. In the “constant
charge limit,” since the total number of counterions between the two surfaces does not change as D falls,
the number density of ions is given by 2σ/eD, so that the limiting pressure P (or force per unit area, F)
in this case is the osmotic pressure of the confined ions, given by
(9.14)
that is, as D → 0 the double-layer pressure becomes infinitely repulsive and independent of the salt
concentration. However, the van der Waals attraction, which goes as 1/D
2
between two spheres or as 1/D
3
between two planar surfaces (see Table 9.2) actually wins out over the double-layer repulsion as D → 0.
At least this is the theoretical prediction, which forms the basis of the so-called Derjaguin–Landau–Ver-
wey–Overbeek (DLVO) theory, illustrated in Figure 9.3. In practice, other forces (described below) often
come in at small separations, so that the full force law between two surfaces or colloidal particles in
solution can be more complex than might be expected from the DLVO theory.
9.4 Solvation and Structural Forces: Forces Due to Liquid

−−
4 61 10 103
11 2
0
. tanh .ψ
κ
mV J for 1:1 electrolytes
F kT kT zeD D=× =
−
ion number density for2
1
σκ ,Ӷ
© 1999 by CRC Press LLC
via some form of the Lennard–Jones potentials have invariably led to an oscillatory solvation force at
surface separations below a few molecular diameters (Snook and van Megan, 1979, 1980, 1981; van
Megan and Snook, 1979, 1981; Kjellander and Marcelja, 1985a,b; Tarazona and Vincente, 1985; Hend-
erson and Lozada-Cassou, 1986; Evans and Parry, 1990).
In a first approximation the oscillatory force laws may be described by an exponentially decaying
cosine function of the form
(9.15)
where both theory and experiments show that the oscillatory period and the characteristic decay length
of the envelope are close to σ (Tarazona and Vincent, 1985).
It is important to note that once the solvation zones of two surfaces overlap, the mean liquid density
in the gap is no longer the same as that of the bulk liquid. And since the van der Waals interaction
depends on the optical properties of the liquid, which in turn depend on the density, one can see why
the van der Waals and oscillatory solvation forces are not strictly additive. Indeed, it is more correct to
think of the solvation force as the van der Waals force at small separations with the molecular properties
and density variations of the medium taken into account.
FIGURE 9.3 Classical DLVO interaction potential energy as a function of surface separation between two flat
surfaces interacting in an aqueous electrolyte (salt) solution via an attractive van der Waals (VDW) force and a

/e at D = σ (second minimum), E
0
/e
2
at D = 2σ, etc. Such multivalued adhesion forces
have been observed in a number of systems, including the interactions of fibers. Most interesting, the
depth of the potential energy well at contact (–E
0
at D = 0) is generally deeper but of similar magnitude
to the value expected from the continuum Lifshitz theory of van der Waals forces (at a cutoff separation
of D
0
≈ 0.15 – 0.20 nm), even though the continuum theory fails to describe the shape of the force law
at intermediate separations.
There is a rapidly growing literature on experimental measurements and other phenomena associated
with short-range oscillatory solvation forces. The simplest systems so far investigated have involved
measurements of these forces between molecularly smooth surfaces in organic liquids. Subsequent mea-
surements of oscillatory forces between different surfaces across both aqueous and nonaqueous liquids
have revealed their subtle nature and richness of properties (Christenson, 1985, 1988a; Christenson and
Horn, 1985; Israelachvili, 1987; Israelachvili and McGuiggan, 1988), for example, their great sensitivity
FIGURE 9.4 Solid curve: Forces between two mica surfaces across saturated linear-chain alkanes such as n-tetrade-
cane (Christenson et al., 1987; Horn and Israelachvili, 1988; Israelachvili and Kott, 1988; Horn et al., 1989a). The
0.4-nm periodicity of the oscillations indicates that the molecules align with their long axis preferentially parallel to
the surfaces, as shown schematically in the upper insert. The theoretical continuum van der Waals force is shown
by the dotted line. Dashed line: Smooth, nonoscillatory force law exhibited by irregularly shaped alkanes, such as
branched isoparaffins, that cannot order into well-defined layers (lower insert) (Christenson et al., 1987). Similar
nonoscillatory forces are also observed between rough surfaces, even when these interact across a saturated linear
chain liquid. This is because the irregularly shaped surfaces (rather than the liquid) now prevent the liquid molecules
from ordering in the gap.
© 1999 by CRC Press LLC

±
1° away from these peak, the energy decreases by 50%. In aqueous
salt (KCl) solution, due to potassium ion adsorption the water between the surfaces becomes ordered,
resulting in an oscillatory force profile where the adhesive minima occur at discrete separations of about
0.25 nm, corresponding to integral numbers of water layers. The whole interaction potential was now
found to depend on orientation of the surface lattices, and the effect extended at least four molecular layers.
Although oscillatory forces are predicted from Monte Carlo and molecular dynamic simulations, no theory
has yet taken into account the effect of surface structure, or atomic “corrugations,” on these forces, nor any
FIGURE 9.5 Adhesion energy for two mica surfaces
in a primary minimum contact in water as a function
of the mismatch angle θ about θ = 0° between the two
contacting surface lattices (McGuiggan and Israelach-
vili, 1990). Similar peaks are obtained at the other coin-
cidence angles: θ = ±60°, ±120°, and 180° (inset).
© 1999 by CRC Press LLC
lattice mismatching effects. As shown by the experiments, within the last 1 or 2 nm, these effects can alter
the adhesive minima at a given separation by a factor of two. The force barriers, or maxima, may also depend
on orientation. This could be even more important than the effects on the minima. A high barrier could
prevent two surfaces from coming closer together into a much deeper adhesive well. Thus, the maxima can
effectively contribute to determining not only the final separation of two surfaces, but also their final adhesion.
Such considerations should be particularly important for determining the thickness and strength of inter-
granular spaces in ceramics, the adhesion forces between colloidal particles in concentrated electrolyte solu-
tions, and the forces between two surfaces in a crack containing capillary condensed water.
The intervening medium profoundly influences how one surface interacts with the other. As experi-
mental results show (McGuiggan and Israelachvili, 1990), when two surfaces are separated by as little as
0.4 nm of an amorphous material, such as adsorbed organics from air, then the surface granularity can
be completely masked and there is no mismatch effect on the adhesion. However, with another medium,
such as pure water which is presumably well ordered when confined between two mica lattices, the atomic
granularity is apparent and alters the adhesion forces and whole interaction potential out to D > 1 nm.
Thus, it is not only the surface structure but also the liquid structure, or that of the intervening film

at any instant, even though they may be perfectly smooth on a time average. The types of surfaces that
fall into this category are fluidlike amphiphilic surfaces of micelles, bilayers, emulsions, soap films, etc.,
© 1999 by CRC Press LLC
but also solid colloidal particle surfaces that are coated with surfactant monolayers, as occur in lubricating
oils, paints, toners, etc.
Thermal fluctuation forces are usually of short range and repulsive, and are very effective at stabilizing
the attractive van der Waals forces at some small but finite separation which can reduce the adhesion
energy or force by up to three orders of magnitude. It is mainly for this reason that fluidlike micelles
and bilayers, biological membranes, emulsion droplets (in salad dressings), or gas bubbles (in beer)
adhere to each other only very weakly (Figure 9.6).
Because of their short range, it was, and still is, commonly believed that these forces arise from water
ordering or “structuring” effects at surfaces, and that they reflect some unique or characteristic property
of water (see Section 9.6). However, it is now known that these repulsive forces also exist in other liquids.
Moreover, they appear to become stronger with increasing temperature, which is unlikely for a force that
originates from molecular ordering effects at surfaces. Recent experiments, theory, and computer simu-
lations (Israelachvili and Wennerström, 1990, 1996; Granfeldt and Miklavic, 1991)

have shown that these
repulsive forces have an entropic origin — arising from the osmotic repulsion between exposed thermally
mobile surface groups once these overlap in a liquid.
9.6 Hydration Forces: Special Forces in Water
and Aqueous Solutions
9.6.1 Repulsive Hydration Forces
The forces occurring in water and electrolyte solutions are more complex than those occurring in
nonpolar liquids. According to continuum theories, the attractive van der Waals force is always expected
FIGURE 9.6 The four most common types of thermal fluctuation forces (also referred to as steric or entropic forces)
between fluid-like, usually amphiphilic, surfaces and membranes in liquids.
© 1999 by CRC Press LLC
to win over the repulsive electrostatic double-layer force at small surface separations (Figure 9.3). How-
ever, certain surfaces (usually oxide or hydroxide surfaces such as clays and silica) swell spontaneously

In a series of experiments to identify the factors that regulate hydration forces, Pashley (1981, 1982,
1985) found that the interaction between molecularly smooth mica surfaces in dilute electrolyte solutions
obeys the DLVO theory. However, at higher salt concentrations, specific to each electrolyte, hydrated
cations bind to the negatively charged surfaces and give rise to a repulsive hydration force (Figure 9.7).
This is believed to be due to the energy needed to dehydrate the bound cations, which presumably retain
some of their water of hydration on binding. This conclusion was arrived at after noting that the strength
and range of the hydration forces increase with the known hydration numbers of the electrolyte cations
in the order: Mg
2+
> Ca
2+
> Li
+
~ Na
+
> K
+
> Cs
+
. Similar trends are observed with other negatively
charged colloidal surfaces.
While the hydration force between two mica surfaces is overall repulsive below about 4 nm, it is not
always monotonic below about 1.5 nm but exhibits oscillations of mean periodicity 0.25 ± 0.03 nm,
roughly equal to the diameter of the water molecule. This is shown in Figure 9.7, where we may note
that the first three minima at D ≈ 0, 0.28, and 0.56 nm occur at negative energies, a result that rationalizes
observations on certain colloidal systems. For example, clay platelets such as motmorillonite often repel
each other increasingly strongly as they come closer together, but they are also known to stack into stable
aggregates with water interlayers of typical thickness 0.25 and 0.55 nm between them (Del Pennino et al.,
1981; Viani et al., 1984), suggestive of a turnaround in the force law from a monotonic repulsion to
discretized attraction. In chemistry we would refer to such structures as stable hydrates of fixed stochi-

illustrated in Figure 9.8.
In recent years there has been a steady accumulation of experimental data on the force laws between
various hydrophobic surfaces in aqueous solutions. These surfaces include mica surfaces coated with
surfactant monolayers exposing hydrocarbon or fluorocarbon groups, or silica and mica surfaces that
had been rendered hydrophobic by chemical methylation or plasma etching (Israelachvili and Pashley,
1982; Pashley et al., 1985; Claesson et al., 1986; Claesson and Christenson, 1988; Rabinowich and Der-
jaguin, 1988; Parker et al., 1989; Christenson et al., 1990; Kurihara et al., 1990). These studies have found
that the hydrophobic force law between two macroscopic surfaces is of surprisingly long range, decaying
exponentially with a characteristic decay length of 1 to 2 nm in the range 0 to 10 nm, and then more
FIGURE 9.7 Measured forces between charged mica surfaces in various dilute and concentrated KCl solutions. In
dilute solutions (10
–5
and 10
–4
M) the repulsion reaches a maximum and the surfaces jump into molecular contact
from the tops of the force barriers (see also Figure 9.3). In dilute solutions the measured forces are excellently described
by the DLVO theory, based on exact numerical solutions to the nonlinear Poisson–Boltzmann equation for the
electrostatic forces and the Lifshitz theory for the van der Waals forces (using a Hamaker constant of A = 2.2 ×
10
–20
J). At higher electrolyte concentrations, as more hydrated K
+
cations adsorb onto the negatively charged surfaces,
an additional hydration force appears superimposed on the DLVO interaction. This has both an oscillatory and a
monotonic component. Inset: Short-range hydration forces between mica surfaces plotted as pressure against distance.
Lower curve: force measured in dilute 1 mM KCl solution where there is one K
+
ion adsorbed per 1.0 nm
2
(surfaces

, where the higher value corresponds to the
interfacial energy of a pure hydrocarbon–water interface.
At a separation below 10 nm the hydrophobic force appears to be insensitive or only weakly sensitive
to changes in the type and concentration of electrolyte ions in the solution. The absence of a “screening”
effect by ions attests to the nonelectrostatic origin of this interaction. In contrast, some experiments have
shown that at separations greater than 10 nm the attraction does depend on the intervening electrolyte,
and that in dilute solutions, or solutions containing divalent ions, it can continue to exceed the van der
Waals attraction out to separations of 80 nm (Christenson et al., 1989, 1990).
The long-range nature of the hydrophobic interaction has a number of important consequences. It
accounts for the rapid coagulation of hydrophobic particles in water, and may also account for the rapid
folding of proteins. It also explains the ease with which water films rupture on hydrophobic surfaces. In
this, the van der Waals force across the water film is repulsive and therefore favors wetting, but this is
more than offset by the attractive hydrophobic interaction acting between the two hydrophobic phases
across water. Finally, hydrophobic forces are increasingly being implicated in the adhesion and fusion of
biological membranes and cells. It is known that both osmotic and electric field stresses enhance mem-
brane fusion, an effect that may be due to the increase in the hydrophobic area exposed between two
adjacent surfaces.
FIGURE 9.8 Examples of attractive hydrophobic interactions in
aqueous solutions. (a) Low solubility/immiscibility of water and oil
molecules; (b) micellization; (c) dimerization and association of
hydrocarbon chains in water; (d) protein folding; (e) strong adhe-
sion of hydrophobic surfaces; (f) nonwetting of water on hydro-
phobic surfaces; (g) rapid coagulation of hydrophobic or
surfactant-coated surfaces; (h) hydrophobic particle attachment to
rising air bubbles (basic mechanism of “froth flotation” process
used to separate hydrophobic and hydrophilic minerals).
Ee
i
D
=−

as now seems more likely — steric and entropic forces (Israelachvili and Wennerström, 1996; Marcelja, 1997).
Like the repulsive hydration force, the origin of the hydrophobic force is still unknown. Luzar et al.
(1987) carried out a Monte Carlo simulation of the interaction between two hydrophobic surfaces across
water at separations below 1.5 nm. They obtained a decaying oscillatory force superimposed on a mono-
tonically attractive curve, i.e., similar to Figure 9.9.
It is questionable whether the hydration or hydrophobic force should be viewed as an ordinary type
of solvation or structural force — simply reflecting the packing of the water molecules. It is important
FIGURE 9.9 Typical short-range solvation (hydration)
forces in water as a function of distance, D, normalized by the
diameter of the water molecule, σ (about 0.25 nm). The hydra-
tion forces in water differ from those in other liquids in that
there is a monotonic component in addition to the normal
purely oscillatory component. For hydrophilic surfaces the
monotonic component is repulsive (upper dashed curve),
whereas for hydrophobic surfaces it is attractive (lower dashed
curve). For simpler liquids there are no such monotonic com-
ponents, and both theory and experiments show that the oscil-
lations decay with distance with the maxima and minima,
respectively, above and below the baseline of the van der Waals
force (middle dashed curve) or superimposed on the contin-
uum DLVO interaction.
© 1999 by CRC Press LLC
to note that for any given positional arrangement of water molecules, whether in the liquid or solid state,
there is an almost infinite variety of ways the H-bonds can be interconnected over three-dimensional
space while satisfying the Bernal–Fowler rules requiring two donors and two acceptors per water molecule.
In other words, the H-bonding structure is actually quite distinct from the molecular structure. The
energy (or entropy) associated with the H-bonding network, which extends over a much larger region
of space than the molecular correlations, is probably at the root of the long-range solvation interactions
of water. But whatever the answer, it is clear that the situation in water is governed by much more than
the simple molecular-packing effects that seem to dominate the interactions in simpler liquids.

as 40% of saturation (Fisher and Israelachvili, 1981; Christenson, 1988b). With water condensing from
vapor or from oil it appears that the bulk value of
γ
LV
is also applicable for meniscus radii as small as 2 nm.
The capillary condensation of liquids, especially water, from vapor can have additional effects on the
whole physical state of the contact zone. For example, if the surfaces contain ions, these will diffuse and
build up within the liquid bridge, thereby changing the chemical composition of the contact zone as well
as influencing the adhesion. More dramatic effects can occur with amphiphilic surfaces, i.e., those
containing surfactant or polymer molecules. In dry air, such surfaces are usually nonpolar — exposing
hydrophobic groups such as hydrocarbon chains. On exposure to humid air, the molecules can overturn
so that the surface nonpolar groups become replaced by polar groups, which renders the surfaces
hydrophilic. When two such surfaces come into contact, water will condense around the contact zone
and the adhesion force will also be affected — generally increasing well above the value expected for inert
hydrophobic surfaces.
It is clear that the adhesion of two surfaces in vapor or a solvent can often be largely determined by
capillary forces arising from the condensation of liquid that may be present only in very small quantities
e.g., 10 to 20% of saturation in the vapor, or 20 ppm in the solvent.
9.7.1 Adhesion Mechanics
Modern theories of the adhesion mechanics of two contacting solid surfaces are based on the
Johnson–Kendall–Roberts (JKR) theory (Johnson et al., 1971, Pollock et al., 1978; Barquins and Maugis,
1982). In the JKR theory two spheres of radii R
1
and R
2
, bulk elastic moduli K, and surface energy γ per
unit area will flatten when in contact. The contact area will increase under an external load or force F,
such that at mechanical equilibrium the contact radius r is given by
(9.20)
where R = R

612 6=+π+π+π
()








γγγ,
FR
SS
=− π3 γ ,
© 1999 by CRC Press LLC
curve, has no effect on the adhesion force — an interesting and unexpected result that has nevertheless
been verified experimentally (Johnson et al., 1971; Israelachvili, 1991).
Equations 9.20 and 9.21 are the basic equations of the JKR theory and provide the framework for
analyzing the results of adhesion measurements of contacting solids, known as contact mechanics (Pollock
et al., 1978; Barquins and Maugis, 1982), and for studying the effects of surface conditions and time on
adhesion energy hysteresis (see next section).
9.8 Nonequilibrium Interactions: Adhesion Hysteresis
Under ideal conditions the adhesion energy is a well-defined thermodynamic quantity. It is normally
denoted by E or W (the work of adhesion) or γ (the surface tension, where W = 2γ), and it gives the
reversible work done on bringing two surfaces together or the work needed to separate two surfaces from
contact. Under ideal, equilibrium conditions these two quantities are the same, but under most realistic
conditions they are not: the work needed to separate two surfaces is always greater than that originally
gained on bringing them together. An understanding of the molecular mechanisms underlying this
phenomenon is essential for understanding many adhesion phenomena, energy dissipation during load-
ing–unloading cycles, contact angle hysteresis, and the molecular mechanisms associated with many

surface tension, γ, and the solid–liquid adhesion energy, W, by the Dupré equation:
(9.23)
we see that wetting hysteresis or contact angle hysteresis (θ
A
> θ
R
) actually implies adhesion hysteresis,
W
R
> W
A
, as given by Equation 9.22.
WW
RA
Receding Advancing
>
∆WWW
RA
=−
()
> 0,
1+
()
=cos ,θγ
L
W
© 1999 by CRC Press LLC
Energy dissipating processes such as adhesion and contact angle hysteresis arise because of practical
constraints of the finite time of measurements and the finite elasticity of materials which prevent many
loading–unloading or approach–separation cycles to be thermodynamically irreversible, even though if

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
2