Handbook of Micro and Nano Tribology P5

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

The Derjaguin Approximation • Electrostatic Forces •
Electrodynamic Forces • Electromagnetic Forces • Forces in and
Due to Liquids • Overview

5.4 Adhesive Forces

Anelasticity • Adhesion Hysteresis in Elastic Continuum
Contact Mechanics • Adhesion in Nanometer-Sized Contacts •
Overview

5.5 Closing Words

Interpreting Your Data • Outlook

Acknowledgments
References

5.1 Introduction

5.1.1 Goals and Motivations

Small bits of eraser cling to your homework assignments, yet the eraser itself easily slides off a piece of
paper. Why? What is different about the interaction of small things with a surface as opposed to big
things?

The goal of this chapter is to familiarize you, on a conceptual basis, with the forces acting
between asperities, or between an asperity and a flat surface.

Asperity behavior is thought to determine the most famous relationship in macrotribology — Amon-

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

area of contact between the two bodies.
In nanotribology, where one considers the interaction of a single asperity contact, the researcher has
the luxury of studying an ideologically far simpler system. Not only is this an area of intensive research

© 1999 by CRC Press LLC

in materials science and physics, but also its major applications area — microelectromechanical systems,
where moving parts touch only at one or few point contacts — has a commercially lucrative future. One
of the fascinations with nanotribology is accurately expressed by the example of the eraser above.

The
behavior of small things is different from the behavior of big ones.

Let us perform a dimensional
analysis for the case of an eraser and its residue.
A particle of eraser residue may be roughly spherical, with a radius

R

of the order of 100

µ

m, whereas
the eraser itself may have dimensions in the range of 1 cm. The surface-to-volume ratio for spheres equals
4

π

R

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

/

3

ρπ

R

3

and its
attraction to a flat piece of paper is 2

π

R

ϖ

, where

ϖ

is the work of adhesion (Sections 5.3.1 and 5.4.2).
Therefore, the ratio between the surface forces and the weight for a spherical particle near a flat surface
is 3

ϖ

/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 (

adhesive forces

or

adhesion

). Other authors have differentiated them by using the nomenclature

advancing/receding

or

loading/unloading

. Surface forces are in general attractive, but under some conditions can be repulsive.
Adhesive forces, as the name implies, tend to hold two bodies together. If a process between two bodies
is perfectly elastic, that is, if no energy dissipates during their interaction, the adhesive and surface forces
are equal in magnitude. Normally, however, the adhesion is greater than any initial attraction, giving rise
to

adhesion hysteresis.

Why this is so is one of the subjects of Section 5.4.

5.1.3 Previous Knowledge Assumed

In this chapter, the assumption is that the reader is already familiar with first-year college physics,
chemistry, and calculus, and Chapter 2 of this book. We draw broadly from a variety of existing texts
and conference proceedings listed at the end of this chapter, wherein many detailed references are given.

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

common with other instruments, notably the surface force apparatus (SFA), the indentor, and the
scanning acoustic microscope (SAM). The overlap extends from the materials properties desired, to how
force and displacement are controlled and measured, to calibration procedures, to the ease with which
imaging is performed. It can be seen from Table 5.1 that SPMs are capable of measuring surface forces
and adhesion, of determining mechanical properties such as elasticity and hardness, and are optimized
for imaging surfaces.

Overly enthusiastic readers must be chided into remembering that an instrument
optimized for imaging is not necessarily the best for surface forces, adhesion, or mechanical properties
measurements.

Issues concerning SPM usage for all materials properties measurements are found on
pp. 421–454 in Bhushan (1997) and references therein.
Scanning probe microscopy will excel in applications where changes in materials properties vary over
scales less than a micron, for example, in new composite materials, or across a cell membrane. So although
imaging to capture the lateral variations in properties is ultimately desired, we restrict our discussion to
the SPM mode of operation most closely related to SFA and the indentor — that of force curve acquisition
using an AFM.

5.2.2 What Are You Measuring?

Care must be taken to avoid artifacts and to calibrate the instrument properly. Then the researcher still
must avoid the trap of measuring a property of the instrument, rather than of the sample or its interaction
with the tip. One can model the AFM–sample system as two springs and two dashpots in series. The
springs symbolize energy storage or stiffness, and the dashpots symbolize energy dissipation. One spring
and dashpot combination represents the AFM cantilever and the other the tip–sample interaction, as
represented in Figure 5.1. Normally, a cantilever with an effective spring constant not too different from
that of a Slinky

™

— 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

m
SAM —

√

√

~1

µ

m
SPM

√

√

√

~1 nm

Checks mean that the instrument was designed to measure surface forces and adhesion, or mechanical
properties, or is optimized for imaging.

FIGURE 5.1

Schematic diagram of the cantilever and its interaction with the sample.
The cantilever moves a distance of

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

i

(

z, d

) (cantilever and interaction, respectively). Note
that

k

i

(

z, d

) may vary. It depends on the position of the tip relative to the sample. For
example, when the tip is far from the sample, the stiffness is zero, but when the tip is
indented into a sample, the stiffness is high.

© 1999 by CRC Press LLC

5.2.2.1 Cantilever Instabilities and Mechanical Hysteresis

Figure 5.2 displays a typical theoretical force curve (curved line) of an elastic tip–sample interaction,
with force plotted as a function of the separation between tip and sample, or as the penetration of the
tip into the sample. The curve has an attractive region near contact, a repulsive region when the tip is
firmly indented into the sample, and no interaction when the tip is far removed from the sample. The

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

= –

k

c

d,

where

d

is the deflection of the cantilever.
The tip of the cantilever will find an equilibrium position such that the cantilever restoring force balances
that of the tip–sample interaction. Summed, the two curves in Figure 5.2 become the plot of Figure 5.3,
in which a region that is triple valued in force exists. Should the tip be approaching the sample (left-to-
right on the graph), it is accelerated over the triple-valued region, following the upper dashed line to the
right, until it finds the new equilibrium position at the same force magnitude at the right-hand termi-
nation of the upper dashed line. Similarly, if the scanner is withdrawn such that the tip moves right-to-
left on the graph, the tip follows the thick line until the triple-valued region, where once again the
cantilever restoring forces do not balance those of the interaction, and the tip finds its new steady-state
position at the left-hand end of the lower dashed line.

FIGURE 5.2

A typical force–distance curve (curved line) and the force applied by the cantilever (straight line).
The force is plotted as a function of tip–sample separation or penetration depth. At the point labeled “Contact” (the
inflection of the curve), the tip touches the sample. The dashed lines represent the path that a 1 N/m cantilever
would follow if it were used for data acquisition, i.e., it jumps over sections of the curve.

© 1999 by CRC Press LLC

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.

Mathematically, the reason for the instabilities can be seen from the following equations. The first is
the simple harmonic oscillator expression for the cantilever, set equal to the forces acting upon it by the
tip–sample interaction.
(5.1)
The cantilever is assumed to act as a spring, with an effective mass of

m

*. Its position is represented by

d,

its damping by

β

c

, and its spring constant by

k

c

. The distance [

z – d

] represents the separation between
tip and sample, or the indentation of the tip into the sample. The tip–sample interaction has damping

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

follow the thick line, but rather it follows the dashed lines over the triple-valued region — the upper one upon
loading and the lower one upon unloading.
md m d kd k zd z d m z d
cc i i
*
˙˙
*
˙
,*
˙
˙
.++=
()
−
[]
+−
[]
22ββ
md k k zd d kzd z
ci i
*
˙˙
,,.++
()
[]
=
()

© 1999 by CRC Press LLC

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

/ [

k

c

+ k

i

(

z, d

)]. But when

k

i

(

z, d

) is the negative of the
cantilever spring constant

k

c

, 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

of Figure 5.5, one of these two very close values has been subtracted from the other, and the noise becomes
significant, obscuring the nonlinearity of the data. The second limiting factor is that weak cantilevers are
often used to calibrate the detection system response of the microscope. It is assumed that a compliant
cantilever moves as much as the scanner does. If the scanner is calibrated, then by placing a weak cantilever
FIGURE 5.4 How the force curve of Figure 5.2 would appear when measured by cantilevers of 0.1, 1.0, and 10
N/m stiffnesses. The axes are labeled in terms of the data before conversion into distance and force, that is, the voltage
driving the scanner, and the voltage corresponding to the cantilever position. The thin lines indicate the cantilever
instabilities, and the arrows the motion of the tip.
© 1999 by CRC Press LLC
in contact with a rigid sample, the detection system response voltage is taken to be equal to the scanner
movement. In fact, for this example, the 0.1 N/m cantilever did not constitute 100% of the compression,
but rather 98.3%. This small error of 1.7% leads to the infinite slope in Figure 5.5.
5.2.2.3 Where’s the Beef?
As mentioned above, AFMs have been designed and optimized for imaging sample surfaces, and often
employ a compliant cantilever to enhance force resolution and to avoid compressing the sample surface,
which distorts topographic features and may permanently disfigure them. Therefore, there has been until
recently (pp. 421–454 in Bhushan, 1997) a dilemma between using stiff cantilevers for materials properties
measurements and compliant cantilevers for imaging surface topography.
Figures 5.2 through 5.5 emphasize the intractable nature of the force curve measurement process—the
measurement sensor, i.e., the cantilever—is influencing your results. The instabilities and mechanical
hysteresis caused by the weakness of the cantilever in comparison with the tip–sample interaction lead
to loss of important data and insensitivity to exactly what you would like to observe. Under no circum-
stances should the mechanical hysteresis of the entire cantilever be confused with the possible and
inherently more interesting adhesion hysteresis of the tip–sample contact (Section 5.4). Nor should the
presence of instabilities be automatically associated with the existence of water layers on surfaces under
ambient conditions. Any attractive interaction gives rise to instabilities and hysteresis if the cantilever is
sufficiently compliant.
The oft-found linearity in the raw force-scanner position curves usually implies that the only thing
you are recording is the stiffness of the cantilever itself. Indeed, the only tip–sample interaction char-
acteristic that can be readily measured with a weak cantilever using force curve acquisition is the pull-

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.

Short-ranged forces, that is, those due to chemical or metallic bonding, will not be discussed in detail
here, although they can greatly change the overall measured attractive and adhesive interactions. One
layer of a nonmetallic film can completely destroy the welding, or junction formation, that would
normally occur between clean metal tips and samples.
5.3.1 The Derjaguin Approximation
The Derjaguin approximation is a useful method by which to arrive at a force law for the sphere–flat
geometry. It states that if the interaction energy per unit area, ϖ, as a function of separation for two
semi-infinite parallel planes is known, then the force law for a sphere near a flat surface becomes
(5.3)
The assumptions used to obtain this expression are that both the range of the forces and the separation
δ are much shorter than the tip radius. In some scanning probe microscope geometries, these assumptions
may not hold.
5.3.2 Electrostatic Forces
Electrostatic forces include those due to charges, image charges, and dipoles. Electric fields polarize
molecules and atoms, so that there exist forces that act between the electric field and the polarized
object. Electric fields can purposefully be applied between tip and sample, or may exist due to differ-
ences in the work function between them. Moreover, electric fields may surround the tip and/or sample
due to variations in the work function over their surfaces.
5.3.2.1 Charges and Image Charges
The expression for the force between two charges should be well known to you. The force F is equal to
the charges multiplied together, divided by a proportionality constant and the square of the distance
FRδϖδ
()
=π
()
sphere–flat planes
2.
© 1999 by CRC Press LLC
between them, r. The proportionality factor is 1/4 πεε
0

= 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

and ε
s
representing the permittivities of the medium and sample, respectively. For metals, where
the permittivities are infinite, the term in parentheses approaches one. Because of the high permittivity
of liquids, if the system is immersed in a liquid, the force is drastically reduced and can even be repulsive
(i.e., positive), depending on the relative values of ε
m
and ε
s
.
5.3.2.2 Dipoles
Molecules can have positive and negative ends to them, due to one atom or atomic group having a
stronger electronegativity than the others. Dipoles have associated electric fields, and these electric fields
interact with other charges and dipoles. The magnitude of a dipole moment of a molecule or an atomic
bond is p = ql, where charges ±q lie a distance l apart. Ubiquitous water’s dipole moment is 6.18 ×
10
–30
Cm, which is a modestly high value, exceeded in general only by strongly ionic pairs such as NaCl.
It is interesting that the interaction potential of a dipole with a charge, another dipole, or a polarizable
atom or molecule is related to whether the dipole is fixed or free to rotate. The functional dependence
upon distance changes. Although the force between two dipoles can be quite weak, collective effects may
be large enough to be measurable using SPM techniques. Typically, one integrates the force or potential
over the volumes where the charges, molecules, or dipoles are located. Once again, this changes the
functional dependence of the force law. For example, the interaction potential between a fixed dipole
and a polar molecule is proportional to 1/r
6
. If the fixed dipole finds itself in front of a semi-infinite half-
space of polar molecules (a flat sample), the interaction potential is then a function of 1/r
3
. The derivations

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
=

π
12
0
2
4 εε
.
F
qq
r
m
sm
sm
=
−
π
−
+






12
0
2
4 εε
εε
εε
,