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 , [...]... (1997), Micro/ Nanotribology and Its Applications, Vol E330, Kluwer Academic, Dordrecht Bowden, F P., and Tabor D (1950, 1964), The Friction and Lubrication of Solids, Parts I & II Clarendon Press, Oxford Briggs, G A D (1995), Advances in Acoustic Microscopy, Plenum, New York Burnham N A and Colton, R J (1993), “Force Microscopy,” in Scanning Tunneling Microscopy and Spectroscopy: Theory, Techniques and. .. ride, and the more energy is used for travel Modern experiments show that even on atomically flat surfaces the molecular packing and tilt of long-chained hydrocarbon monolayers affect the frictional response Bowden and Tabor (1950, 1964) understood the friction and wear of metal–metal sliding in terms of the plastic deformation of asperity contacts 5.5.2 Outlook Part of the excitement in nanotribology... maximum negative force upon separation of the tip and sample is often referred to as the pull-off force Unfortunately, measuring the pull-off force does not directly give the work of adhesion The pull-off force is a function of the local curvature radius, the equilibrium interatomic distance, the reduced elastic modulus of the tip and sample, as well as the work of adhesion Details are forthcoming in... when the tip is over part of the sample where the field has a gradient, for example, at the edge of a magnetic domain If both sample and tip are ferromagnets, then a magnetic force of some magnitude and direction will be detected over all of the sample The interpretation of magnetic force microscope images is not simple, and requires considerable theoretical effort to understand the data They depend... in scanning probe microscopy? Do existing theories quantitatively account for all of the features of force curves? To what extent do traditional continuum mechanics explain nanometer-scale tribological processes, and how well do they agree with atomistic simulations? • Can we develop fundamental understanding of friction, lubrication, and wear? The interpretation of surface forces and adhesion measurements... measurements is challenging Even if you understand your instrumental artifacts, and are confident about the interpretation of your data, you must still put your work in the context of the research and current thinking of other groups The future prospects for continued intriguing research in nanotribology are excellent Acknowledgments The timely work of F Oulevey on some of the graphics is very much appreciated... spectrum of materials possibilities As can be seen, large λ is for the more compliant, adhesive combinations, and vice versa, for small λ In the limit of infinite λ, the Maugis equation approaches those of JKRS theory, and the pull-off force goes to –1.5πRϖ For the limit of λ equals zero, the system responds like the DMT mechanics, and the pull-off force approaches –2πRϖ Measuring the pull-off force... between the tip of an SPM and a sample surface In this chapter, we have tried to introduce to readers those ideas, equations, and concepts that should help speed them on the way to a good grasp of the surface forces and adhesion interactions relevant to this growing field Of course, the gap between laboratory experiments that study one single contact and the real world where thousands or millions of asperities... if they had the density of water? Of plutonium? 5.4 Adhesive Forces One of the first questions that comes to mind upon viewing a hysteretic force curve is “What is the origin of the hysteresis?” Or, in other words, “Why is adhesion greater than attraction?” We shall see in this section that the chemical and mechanical properties of the surface and near-surface region of the tip and sample determine the... important role in determining the work of adhesion for the asperity and flat The yield stress of the near surface, relative to the work of adhesion and the local curvature radius, fixes the dissipation process, i.e., elastic or plastic The elastic modulus of the near surface, relative to the work of adhesion and the local curvature radius, determines the amount of energy dissipated for the elastic case . Burnham, N.A. and Kulik, A.J. “Surface Forces and Adhesion” Handbook of Micro/ Nanotribology. Ed. Bharat Bhushan Boca Raton:. 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