Bhushan B. Handbook of Micro/Nano Tribology, Second Edition

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FIGURE A4.3.1 Static contact between two surfaces can be modeled as contact between an equivalent surface and

a ﬂat hard plane.

Burnham, N.A. and Kulik, A.J. “Surface Forces and Adhesion”

Handbook of Micro/Nanotribology.

Ed. Bharat Bhushan

Boca Raton: CRC Press LLC, 1999

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

. As the normal load

is increased, the frictional force

also increases, with a constant

proportionality factor

, 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

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

of the order of 100

m, whereas

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

= 3/

, 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

ρπ

and its

attraction to a ﬂat piece of paper is 2

, where

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

. 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

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

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

adhesion

). Other authors have differentiated them by using the nomenclature

advancing/receding

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

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

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

SAM —

√

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

in response to the movement of the sample

. The

dashpots

and

symbolize energy dissipation. Energy storage is represented by the

springs with stiffnesses

and

(

z, d

) (cantilever and interaction, respectively). Note

that

(

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.

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

-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

of the cantilever. In this example,

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

= –

where

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.

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

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

) 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

*. Its position is represented by

its damping by

, and its spring constant by

. 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

, and its interaction stiffness can be written as

(

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 kzd d kzdz

ci i

˙˙

,,.++

()

[]

()

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

eration term

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

that satisﬁes the requirement

(

z, d

)

/ [

+ k

(

z, d

)]. But when

(

z, d

) is the negative of the

cantilever spring constant

, the [

+ k

(z, d)]d term in Equation 5.2 equals zero, and the cantilever is

accelerated by the force k

(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.

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.

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