Handbook of Micro and Nano Tribology P5

• Probe Geometry 5.3 Surface Forces The Derjaguin Approximation • Electrostatic Forces • Electrodynamic Forces • Electromagnetic Forces • Forces in and Due to Liquids • Overview 5.4 Adhe

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

Handbook of Micro/Nanotribology

Ed Bharat Bhushan Boca Raton: CRC Press LLC, 1999

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 ofpaper Why? What is different about the interaction of small things with a surface as opposed to bigthings? 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 — ton’s law F f = µN As the normal load N is increased, the frictional force F f also increases, with a constantproportionality factor µ, the coefficient of friction Remembering that even “atomically flat” surfaceshave 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 realarea of contact between the two bodies

Amon-In nanotribology, where one considers the interaction of a single asperity contact, the researcher hasthe 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 dimensionalanalysis 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 µm, whereasthe eraser itself may have dimensions in the range of 1 cm The surface-to-volume ratio for spheres equals

4πR2/4/3πR3= 3/R, which means that the residue will have a surface-to-volume ratio 100 times greaterthan 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 is4/3ρπR3 and itsattraction to a flat piece of paper is 2πRϖ, 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 3ϖ/2ρR2 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 Insummary, 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 arebrought 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 forcesare 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 textsand 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, becausethis is a configuration likely to occur in microelectromechanical systems, and is the most commonsituation in scanning probe microscopy studies which are used to probe materials properties withnanometer-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 bediscussed 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 bethe 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 incommon with other instruments, notably the surface force apparatus (SFA), the indentor, and thescanning acoustic microscope (SAM) The overlap extends from the materials properties desired, to howforce and displacement are controlled and measured, to calibration procedures, to the ease with whichimaging is performed It can be seen from Table 5.1 that SPMs are capable of measuring surface forcesand adhesion, of determining mechanical properties such as elasticity and hardness, and are optimizedfor 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 overscales less than a micron, for example, in new composite materials, or across a cell membrane So althoughimaging to capture the lateral variations in properties is ultimately desired, we restrict our discussion tothe SPM mode of operation most closely related to SFA and the indentor — that of force curve acquisitionusing 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 stillmust avoid the trap of measuring a property of the instrument, rather than of the sample or its interactionwith the tip One can model the AFM–sample system as two springs and two dashpots in series Thesprings symbolize energy storage or stiffness, and the dashpots symbolize energy dissipation One springand dashpot combination represents the AFM cantilever and the other the tip–sample interaction, asrepresented in Figure 5.1 Normally, a cantilever with an effective spring constant not too different fromthat of a Slinky™— approximately 1 N/m — is used The spring stiffness representing the interactionbetween 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 furtherdevelopment

TABLE 5.1 Tabular Comparison of SFA, Indentor, SAM, and SPM

Surface Forces and Adhesion Mechanical Properties Imaging Lateral Resolution

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 βc and βi 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.

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 thetip into the sample The curve has an attractive region near contact, a repulsive region when the tip isfirmly indented into the sample, and no interaction when the tip is far removed from the sample Theorigin 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, orindentation, 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 balancesthat 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 right on the graph), it is accelerated over the triple-valued region, following the upper dashed line to theright, 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 thecantilever restoring forces do not balance those of the interaction, and the tip finds its new steady-stateposition at the left-hand end of the lower dashed line

(left-to-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 toeverything below the x-axis of Figure 5.2 For a sufficiently stiff (high k c) cantilever, the shape of thecurve 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-inevents, and they are often equivalently graphically (but perhaps more confusingly) explained using thedashed lines labeled –1 N/m in Figure 5.2

Mathematically, the reason for the instabilities can be seen from the following equations The first isthe simple harmonic oscillator expression for the cantilever, set equal to the forces acting upon it by thetip–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 βc, and its spring constant by k c The distance [z – d] represents the separation betweentip and sample, or the indentation of the tip into the sample The tip–sample interaction has damping

βi, 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 theseparation [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.

m d* ˙˙+2m*βc d k d k z d z d˙+ c = i( ), [ ]− +2m*βi[ ]z d˙−˙

m d* ˙˙+[k c+k z d d k z d z i( ), ] = i( ),

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.

in contact with a rigid sample, the detection system response voltage is taken to be equal to the scannermovement 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 oftenemploy 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 untilrecently (pp 421–454 in Bhushan, 1997) a dilemma between using stiff cantilevers for materials propertiesmeasurements and compliant cantilevers for imaging surface topography

Figures 5.2 through 5.5 emphasize the intractable nature of the force curve measurement process—themeasurement sensor, i.e., the cantilever—is influencing your results The instabilities and mechanicalhysteresis 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 stances should the mechanical hysteresis of the entire cantilever be confused with the possible andinherently more interesting adhesion hysteresis of the tip–sample contact (Section 5.4) Nor should thepresence of instabilities be automatically associated with the existence of water layers on surfaces underambient conditions Any attractive interaction gives rise to instabilities and hysteresis if the cantilever issufficiently compliant

circum-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 off force — the maximum adhesive force during retraction of the scanner (Observe that the most negative

pull-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 elasticand 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 tipend and a flat sample will be used in this chapter Let the reader beware that if the range of force interactionextends 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 thedistance 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 bepresented: electrostatic, electrodynamic, electromagnetic, and liquid forces Because of the breadth anddepth of this subject, this section is necessarily written in summary form; for full details, consultIsraelachvili (1992) Users new to SPM should become familiar with the functional dependencies of theforce interactions (Figures 5.6 through5.8), and their typical relative strengths, as well as be exposed tothe 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 detailhere, although they can greatly change the overall measured attractive and adhesive interactions Onelayer of a nonmetallic film can completely destroy the welding, or junction formation, that wouldnormally 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 assumptionsmay 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

F( )δ sphere–flat= π2 Rϖ δ( )planes

between them, r The proportionality factor is 1/4 πεε0 = 8.99 × 109 Nm2/C, where εo is the permittivityconstant factor which has the value 8.85 × 10–12 C2/Nm2, and ε is the relative permittivity for the mediumacross which the force acts.

(5.4)

If, as an illustration, we set q1 = q2 = 1.6 × 10–19 C, the charge of one electron, and solve for the forcewhen 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 chargesinduce surface charge on nearby surfaces that acts as an image charge buried within the material Imagecharges always carry the opposite sign of the original charge In this case, the force relationship becomes

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 polarizableatom or molecule is related to whether the dipole is fixed or free to rotate The functional dependenceupon 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 potentialover the volumes where the charges, molecules, or dipoles are located Once again, this changes thefunctional dependence of the force law For example, the interaction potential between a fixed dipole

and a polar molecule is proportional to 1/r6 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/r3 The derivationsmay 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 anelectric field, the positively charged nucleus moves slightly in the direction of the field, and the electronsagainst 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 C2m2/J Because electric field strengths andfunctional 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/r4 or 1/r6 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 voltagebetween the tip and sample, forming a capacitor between them The energy stored in such an electric

field is equal to W = – ½CV2, where C is the tip–sample capacitance and V the applied voltage The

−+

4 ε ε

ε ε

ε ε ,

capacitance of two parallel plates separated by a distance δ is ε0 A/δ, where A is the area of the plates.

Therefore, the work per unit area for two planes is ϖ(δ)planes = –½ε0V/δ, and by using the Derjaquinapproximation above (Equation 5.3), we arrive at

(5.6)

The presence of a dielectric material with dielectric constant ε of thickness b/2 upon each electrode (tip

and sample) will modify the equation to

(5.7)

Try taking the limits b → 0 and b →δ in order to check this latter result with the former one A typicalvalue for the force at contact (δ≈ 0.2 nm) with an applied voltage of 10 V and tip radius of 10 nm is–14 nN The functional dependence on δ for a capacitive interaction is plotted in Figure 5.6

5.3.2.5 Innate Electrostatic Fields

If there is no voltage applied between two conductors that form a capacitor, there may still be anelectrostatic field between them If the work functions Φi (nominally, the potential difference betweenthe Fermi level and the vacuum level) of the two materials are not equal, when electrically connectedthe Fermi levels equilibrate, and the voltage difference is Φ1 – Φ2 The value can be of the order of a volt,

giving rise to a nonnegligible force An offset voltage may be applied to compensate this contact potential

difference.

Even when the contact potential difference has been removed, the system may continue to be affected

by innate electrostatic fields due to work function anisotrophies The work function is very sensitive to

perturbations at a surface Surface preparation, uneven distribution of adsorbates, crystallographic entation, and the presence of surface steps, hillocks, pits, or defects can all influence the work functionand make its value change with position on the surface

ori-Let us now perform a short gedanken experiment We assume a metallic material, the surface of whichincorporates a patch of adsorbates having work function ΦA and the rest of the surface, bare, havingwork function ΦB We know that energy must be conserved, and that everywhere inside the metal, theelectron has a mean energy equal to the Fermi level An electron taken out of the material via the patchand put back into it through the bare surface will not conserve energy (ΦA – ΦB≠ 0) unless there existelectric fields external to the material The electric fields emanate from the adsorbate patches, which can

be modeled as dipole sheets The fields are strong enough to affect electron trajectories in field emissionmicroscopy, and they induce surface charge in the sample The resulting tip–sample forces are known as

patch charge forces.

The calculation of patch charge forces is complex, first, because of the nontrivial nature of the electricfield associated with the dipole sheet, and second because the distribution of image charges induced innearby bodies is heavily dependent on their geometry Nonetheless, the resulting force is notable notonly in that it can initially be repulsive, as in Figure 5.6, or attractive, but also because induced imagecharges are always of the opposite sign, and therefore the force always turns attractive as the distance

between tip and sample is reduced (Section 5.3.2.1) The normal component of the electric field E z alongthe central axis of a dipole disk with dipole moment per unit area µ and diameter ρ is

E

z

z z

In the limits of either an infinitely large disk or at an infinite distance from the patch, E z → 0 Themagnitude of the electric field is maximum at about ρ≈ z If we assume that there is one extra electron

on the tip, then the force felt by that electron 10 nm from a patch on the sample of diameter ρ = 10 nmand dipole moment per unit area µ = 1.6 × 10–8 C/m will be F z = qE z = ±5 nN, and its image force will

be –2 pN As the tip is lowered onto the sample, the image force grows and the charge-patch field forcewill drop There will also be image charges built up in the tip from the patch field, which is attractive tothe patch

5.3.3 Electrodynamic Forces

5.3.3.1 The Dispersion Force

Now we need to consider oscillating electrons The position of an electron about the nucleus of an atom

is not fixed with time; it oscillates, generating a fluctuating dipole field The field interacts with nearbyatoms, inducing the appropriate instantaneous dipole moments in them that are always attractive This

is known as the dispersion force because the frequencies correspond to those of visible and ultraviolet light, which the fluctuations disperse The London equation gives the dispersion potential W(r) between two nonpolar molecules a distance r apart:

(5.9)

The electrons orbit the nuclei at a frequency ν, h is the Planck constant, and α is the polarizability of

the atom The dispersion force acts between all materials, because all atoms have fluctuating electrons.

For two atoms with α/4πε0 = 10–30 m3, hv = 10–18 J, and interatomic distance of 2 × 10–10 m, W = –1.2

× 10–20 J This is three times more energy than the thermal energy kT at 300 K, so the dispersion force may not be ignored at room temperature The London constant 3hνα2/4 (4πε0)2 for Ar–Ar is approxi-

FIGURE 5.6 General behavior of electrostatic and electrodynamic forces as a function of tip–sample separation δ Capacitive forces approximately follow a 1/ δ force, with van der Waals forces following 1/ δ 2 Patch charge forces can first be repulsive before becoming attractive.

2 6

ναε

mately 50 × 10–79 Jm6, whereas for CCl4–CCl4 it is about 1500 × 10–79 Jm6, the difference being largelydue to the sixfold change in polarizability of the molecular pairs.

The electronic motion is correlated at visible and ultraviolet frequencies, which are of the order of

1015 Hz During one period of the fluctuation frequency, the electromagnetic field travels 300 nm.Correlated electron motion does not occur if upon leaving one atom, the field travels to another atomand returns to the original one to find that the motion is out of phase Therefore, at distances greater

than approximately 100 nm, the dispersion potential for two atoms drops off at a rate of 1/r7, instead of

1/r6 This is known as the retardation effect Integrated over the volume of a flat surface and an approaching

spherical tip, the distance dependence for nonretarded dispersion forces become 1/δ2; that for retardeddispersion forces is 1/δ3

5.3.3.2 van der Waals Forces

There is not just one van der Waals force, but rather there are van der Waals forces The terminology

van der Waals forces encompasses three forces of different origin The dominant contribution is the

dispersion, or London force, due to the nonzero instantaneous dipole moments of all atoms and molecules,

as described in the previous section The second contribution is the Keesom force, which originates from

the attraction between rotating permanent dipoles The interaction between rotating permanent dipoles

and the polarizability of all atoms and molecules generates the third contribution, the Debye force The interaction potential between atoms or molecules of each force is a function of 1/r6 The dispersion force

is the most important component of van der Waals forces because all materials are polarizable, whereasfor Keesom and Debye forces, there must be permanent dipoles present

The Hamaker constant, A, reflects the strength of the van der Waals interaction for two bodies 1 and

2 in medium 3, with permittivities εi and indexes of refraction n i The first term includes Keesom andDebye interactions, the second the London interaction

(5.10)

Inspection of this equation reveals that for identical materials across any medium (ε1 = ε2 and n1 =

n2), A is positive (attractive), whereas if ε3 and n3 are intermediate to ε1, n1 and ε2, n2, A is negative

(repulsive) If ε3 and n3 equal the permittivity and index of refraction of either of the two bodies, A

vanishes For ε3 = n3 = 1 (air or vacuum), A is always positive In other words, van der Waals forces can

be attractive, repulsive, or zero The judicious choice of the medium in which an SPM experiment is carried out helps control the van der Waals forces between tip and sample.

The form of the van der Waals interaction potential for two flat surfaces is W(δ)planes = –A/12πδ2 Using

Derjaguin’s equation from Section 5.3.1, we find that the force between a sphere and a flat is F(δ)sphere–flat =

–AR/6δ2, as seen in Figure 5.6 A typical value for A is 10–19 J; in air or vacuum a force at contact of the

order of –4 nN would be expected for a 10 nm tip radius In water, A is drastically reduced because of

the high permittivity of water (≈80)

5.3.4 Electromagnetic Forces

There are three classes of magnetism — diamagnetism, paramagnetism, and ferromagnetism In

paramag-netic and diamagparamag-netic materials, electron spins are randomly oriented due to thermal fluctuations,yielding no net permanent magnetic moment for the material However, in ferromagnetic materials, astrong quantum effect called exchange coupling causes the spins to align, giving the ferromagnet apermanent magnetic moment

Spins will respond to an external magnetic field In all materials, the change in electron orbital moment

is opposite to the external field, giving a repulsive force (This is an atomic analog of Lenz’s law, which

2 3

2 1 22 2 3

2 1 21 2 3

2 1 22 2 3