Burnham, N.A. and Kulik, A.J. “Surface Forces and Adhesion”
Handbook of Micro/Nanotribology.
Ed. Bharat Bhushan
Boca Raton: CRC Press LLC, 1999
© 1999 by CRC Press LLC© 1999 by CRC Press LLC

5

Surface Forces

and Adhesion

Nancy A. Burnham and Andrzej J. Kulik

5.1 Introduction

Goals and Motivations • Surfaces Forces vs. Adhesion • Previous
Knowledge Assumed • Carte Routière

5.2 Pertinent Instrumental Background

The Instrument Family • What Are You Measuring? • Probe
Geometry

5.3 Surface Forces

ton’s law

F

f

=

µ

N

. As the normal load

N

is increased, the frictional force

F

f

also increases, with a constant
proportionality factor

µ

, the coefficient of friction. Remembering that even “atomically flat” surfaces
have finite corrugation, and that most surfaces exhibit roughnesses well in excess of atomic dimensions,
increasing the normal load causes more asperities to touch, which as a consequence augments the real


2

/

4

/

3

π

R

3

= 3/

R

, which means that the residue will have a surface-to-volume ratio 100 times greater
than for the eraser. The properties of the surface and near-surface region are important for small particles,
as will be emphasized in Sections 5.3 and 5.4. The weight of a sphere of density

ρ

is

4

/2

ρ

R

2

. The value of this ratio for our residual particle is 10,000 times larger than that for the eraser,
and we might predict that the residue will cling to the paper if the value is greater than one. As long as

ϖ

is nonzero (the usual case), there is always an

R

at which surface forces are stronger than gravity. In
summary,

surface forces predominate at small enough scales.

5.1.2 Surface Forces vs. Adhesion

Throughout this chapter, we shall distinguish between the forces that are present when two bodies are
brought together (

surface forces

) and those that work to hold two bodies in contact (

We concentrate on the surface forces and adhesion that act between an asperity and a flat surface, because
this is a configuration likely to occur in microelectromechanical systems, and is the most common
situation in scanning probe microscopy studies which are used to probe materials properties with
nanometer-scale lateral resolution.

5.1.4 Carte Routière

To aid the reader, important concepts are emphasized by

boldface

type, and significant terminology by

italics.

This chapter is intended to be complementary to Chapter 9, “Surface forces and microrheology
of molecularly thin liquid films.” Here, we first cover some aspects of instrumentation that may not be
discussed in other parts of this textbook, then subsequent sections elaborate surface forces, adhesion,
and the interpretation of experimental data, before a final summary.

5.2 Pertinent Instrumental Background

5.2.1 The Instrument Family

The correct usage of scanning probe microscopes (SPMs) to study surface forces and adhesion shall be
the focus of this section. Chapter 2 details the instrumentation of atomic force microscopes (AFMs), one

© 1999 by CRC Press LLC

of the many varieties of SPMs. The researcher should bear in mind that SPMs have many features in


— approximately 1 N/m — is used. The spring stiffness representing the interaction
between the tip and the sample is usually significantly larger.

Thus, during force curve acquisition, as
the sample and cantilever are first brought together and then separated, the weaker spring (the can-
tilever) suffers most of the deflection, and the properties of the stiffer spring (the interaction of the
sample with the tip) are not observable.

This is an important concept, one that deserves further
development.

TABLE 5.1

Tabular Comparison of SFA, Indentor, SAM, and SPM

Surface Forces and Adhesion Mechanical Properties Imaging Lateral Resolution

SFA

√

— — —
Indentor —

√

— ~1

µ


d

in response to the movement of the sample

z

. The
dashpots

β

c

and

β

i

symbolize energy dissipation. Energy storage is represented by the
springs with stiffnesses

k

c

and

k

origin has been placed such that the area below (above) the

x

-axis represents a net attractive (repulsive)
tip–sample force, and the negative (positive) values of separation imply that the tip is separated from
(indented into) the sample. Hence, the lower-left quadrant corresponds roughly to the forces and sepa-
rations investigated with a surface forces apparatus, and the upper-right quadrant to penetration, or
indentation, experiments. Note that in this example, contact, indicated by the appropriately labeled point
in the lower-left quadrant, commences at negative values of separation. As in human relationships,
attraction causes two bodies to reach out and touch each other.
The straight line in Figure 5.2 represents the spring constant

k

c

of the cantilever. In this example,

k

c

=
1 N/m. The force is determined using Hooke’s law

F

= –


These are examples of

cantilever instabilities,

giving rise to

mechanical hysteresis

in the force curve.

The
weaker the cantilever, the larger is the region of triple-valued force and the greater is the mechanical
hysteresis.

Indeed, for a hypothetical cantilever of zero stiffness, the triple-valued region corresponds to
everything below the

x

-axis of Figure 5.2. For a sufficiently stiff (high

k

c

) cantilever, the shape of the
curve in Figure 5.3 would become single valued everywhere, and no instabilities due to the compliance
of the cantilever could occur. Cantilever instabilities are also referred to as jump-to-contact or snap-in
events, and they are often equivalently graphically (but perhaps more confusingly) explained using the
dashed lines labeled –1 N/m in Figure 5.2.


β

i

, and its interaction stiffness can be written as

k

i

(

z, d

). It should be emphasized that the stiffness,
graphically the slope at a given point on a properly presented force–distance curve, is a function of the
separation [

z – d

], and may be positive, zero, or negative. If we for the moment suppose that damping
is negligible, the above can be rewritten as
(5.2)

FIGURE 5.3

The two curves in Figure 5.2 are summed to obtain the total force for the tip–sample system. As the
separation or penetration depth is changed, the force increases or decreases in a nonmonotonic fashion, such that
there exist three points with the same force value in the region between the dashed lines. The tip does not always

During quasi-static (slow enough such that equilibrium conditions apply) data acquisition, the accel-
eration term

m*
¨
d

will be equal to zero most of the time because the cantilever can find some position

d

that satisfies the requirement

d

=

k

i

(

z, d

)

z

/ [

, the [

k

c

+ k

i
(z, d)]d term in Equation 5.2 equals zero, and the cantilever is
accelerated by the force k
i
(z, d) z.
5.2.2.2 Measured and Processed Force Curve Data
The raw data as recorded by an AFM with different cantilever stiffnesses appear as in Figure 5.4. The
voltage corresponding to the cantilever deflection is plotted as a function of the scanner position voltage.
We use the terminology force curve to embrace both force-separation or penetration depth curves (the
theoretical or processed data), as in Figure 5.2, and force-scanner position curves (the measured data), as
in Figure 5.4. The weaker the cantilever, the greater the mechanical hysteresis, and the more linear the
cantilever response upon contact with the sample. The raw data reflect neither the actual tip–sample
separation, nor the penetration of the tip into the sample. For this one must subtract the cantilever
position from the x-axis of the raw data, in order to obtain curves such as those shown in Figure 5.5.
There are two striking features of Figure 5.5. One is that much of the 0.1 N/m curve has an infinite
slope and appears linear. The other is that many data points do not exist for the 0.1 N/m curve, fewer
for the 1.0 N/m curve, and none for the 10 N/m curve. The missing data correspond to those points
omitted because of cantilever instabilities. The infinite slope of the linear curve indicates that the
microscope was operated outside of its detection limits.
Two factors limit detectability for the case of the 0.1 N/m cantilever. The first is the noise of the system.
Figure 5.4 has ±10 pm noise added to both x- and y-coordinates — an amount hardly discernible in that
figure. The compliant cantilever deflects almost as much as the scanner moves. For the processed data

off force — the maximum adhesive force during retraction of the scanner. (Observe that the most negative
values of the curves in Figure 5.5 are almost the same.) With a well-calibrated instrument and a sufficiently
FIGURE 5.5 How the data of Figure 5.4 would appear after processing. The cantilever voltage is converted to force,
and the cantilever position is subtracted from the scanner position in order to arrive at the separation or penetration
depth. The solid line represents the data as taken with a 10 N/m cantilever, the small squares with a 1 N/m cantilever,
and the dots a 0.1 N/m cantilever. The dashed lines show the paths taken by weaker cantilevers. Comparing with the
original force interaction of Figure 5.2, one can see that the stiffer cantilevers can reproduce the original data well.
The shape of the force curve is lost with the compliant cantilever. In general, if the slope of the measured force curve
is linear, the cantilever is too weak for all but a pull-off force measurement.
© 1999 by CRC Press LLC
stiff cantilever, it is possible to determine the operative surface forces and adhesion, as well as the elastic
and plastic response of the sample. With a weak cantilever, there’s hardly any beef.
5.2.3 Probe Geometry
Cantilever tips come in a variety of shapes and sizes. (See Chapter 2.) Because the magnitude and
functional dependence of surface forces often depend on the shape of the tip, it is important to calibrate
the tip. Although surface forces can be calculated numerically for any given tip–sample geometry,
analytical modeling calls for tips and samples of easily defined form: spherical, hyperbolic, parabolic,
cone-shaped, or a flat-ended punch against a flat sample. The assumption of a spherically shaped tip
end and a flat sample will be used in this chapter. Let the reader beware that if the range of force interaction
extends beyond the spherical part of the tip, or if the sample is very rough, this assumption will no longer
be valid. Another important assumption that may or may not hold in a given experiment is that the
distance over which the forces act is much less than the tip radius. Nevertheless, the sphere–flat geometry
is illustrative.
5.3 Surface Forces
After stating the Derjaguin approximation, four broad classes of long-ranged surface forces will be
presented: electrostatic, electrodynamic, electromagnetic, and liquid forces. Because of the breadth and
depth of this subject, this section is necessarily written in summary form; for full details, consult
Israelachvili (1992). Users new to SPM should become familiar with the functional dependencies of the
force interactions (Figures 5.6 through5.8), and their typical relative strengths, as well as be exposed to
the wide variety of possible sources of the forces.

= 8.99 × 10
9
Nm
2
/C, where ε
o
is the permittivity
constant factor which has the value 8.85 × 10
–12
C
2
/Nm
2
, and ε is the relative permittivity for the medium
across which the force acts.
(5.4)
If, as an illustration, we set q
1
= q
2
= 1.6 × 10
–19
C, the charge of one electron, and solve for the force
when two electrons are an atomic distance 0.2 nm apart, we find that the resulting force is about 6 nN.
This is a magnitude that can be detected with most AFMs. One must also remember that free charges
induce surface charge on nearby surfaces that acts as an image charge buried within the material. Image
charges always carry the opposite sign of the original charge. In this case, the force relationship becomes
(5.5)
with ε
m

may be found in Israelachvili (1992).
5.3.2.3 Polarizability
All atoms and molecules are polarizable. The effect originates from the charged nature of atoms. In an
electric field, the positively charged nucleus moves slightly in the direction of the field, and the electrons
against it, until the force exerted by the electric field is balanced by the internal restoring forces of the atom
or molecule. This is similar to the dipole moment p = ql, but it is an induced, rather than permanent,
dipole moment. The relation among the induced dipole moment µ, the electric field E, and the polarizability
α is simply µ = αE. Polarizabilities are of the order of 10
–40
C
2
m
2
/J. Because electric field strengths and
functional dependencies on distance depend on whether the source of the field is a dipole or charge, the
interaction potentials between two individual atoms or molecules exhibit either 1/r
4
or 1/r
6
proportionalities.
5.3.2.4 Applied Electrostatic Fields
An easy way in which the experimentalist can actively control an SPM measurement is to apply a voltage
between the tip and sample, forming a capacitor between them. The energy stored in such an electric
field is equal to W = –½CV
2
, where C is the tip–sample capacitance and V the applied voltage. The
F
qq
r
=