Colchero, J. et al. “Friction on an Atomic Scale” Handbook of Micro/Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press LLC, 1999 © 1999 by CRC Press LLC © 1999 by CRC Press LLC 6 Friction on an Atomic Scale Jaime Colchero, Ernst Meyer, and Othmar Marti 6.1 Introduction 6.2 Instrumentation The Force-Sensing System • The Tip 6.3 Experiments Atomic-Scale Imaging of the Friction Force • Thin Films and Boundary Lubrication • Nanocontacts • Quartz Microbalance Experiments in Tribology 6.4 Modeling of an SFFM Resolution in SFFM • Deformation of Tip and Sample • Modeling of SFM and SFFM: Energy Dissipation on an Atomic Scale 6.5 Summary Acknowledgments References 6.1 Introduction The science of friction, i.e., tribology, is possibly together with astronomy one of the oldest sciences. Human interest in astronomy has many reasons, the awe experienced when observing the dark and endless sky, the fear associated with phenomena such as eclipses, meteorites, or comets, and perhaps also practical issues such as the prediction of seasons, tides, or possible floods. By contrast, the interest in tribology is purlye practical: to move mechanical pieces past each other as easily as possible. This goal has not changed essentially since tribology was born. Ultimately, the person who a few thousand years ago had the brilliant idea to pour water between two mechanical pieces was working on the same problem as the expert tribologist today, the only difference being their level of knowledge. A better understanding of friction and wear could save an enormous amount of energy and money, which would be positive for economy and ecology. On the other hand, friction is not only negative, since it is fundamental for basic technological applications: brakes as well as screws are based on friction. The first approach to tribology is due to Leonardo da Vinci at the beginning of the 15th century. In a certain sense he introduced the idea of a friction coefficient. For smooth surfaces he found that “friction corresponds to one fourth its weight”; in other words, he assumed a friction coefficient of 0.25. To appreciate these tribological studies one should bear in mind that the modern concept of force was not introduced until about 200 years later. The next tribologist was Amontons around the year 1700. Surprisingly, the © 1999 by CRC Press LLC model he proposed to explain the origin of friction is still quite modern. According to Amontons, surfaces are tilted on a microscopic scale. Therefore, when two surfaces are pressed against each other and moved, a certain lateral force is needed to lift the surfaces against the loading force. Assuming that no friction occurs between the tilted surfaces, one immediately finds from purely geometric arguments where α is the tilting angle on a microscopic scale. This model relates the friction to the microscopic structure of the surface. Today we know that this model is too simple to explain the friction on a macroscopic scale, i.e., everyday friction. In fact, it is well known that surfaces touch each other at many microasperities and that the shearing of these microasperities is responsible for friction (Bowden and Tabor, 1950). Within this model the friction coefficient is related to such parameters as shear strength and hardness of the surfaces. On an atomic scale, however, the mechanism responsible for friction is different. As will be discussed in more detail in this chapter, the model for explaining energy dissipation in a scanning force microscope (SFM) is that the tip has to overcome the potential well between adjacent atoms of the surface. For certain experimental conditions, which are in practice almost always realized, the tip jumps from one stable equilibrium position on the surface to another. This process is not reversible, leads to energy dissipation, and, therefore, on average to a friction force. The similarity between Amon- tons’ model of friction and these modern models for friction on an atomic scale is evident. In both cases asperities have to be passed, the only difference is the length scale of these asperities, in the first case assumed to be microscopic, in the second case atomic. Although tribology is an old science, and in spite of the efforts and progress made by scientists and engineers, tribology is still far from being a well-understood subject, in fact (Maugis, 1982), “It is incredible that, all properties being known (surface energy, elastic properties, loss properties), a friction coefficient cannot be found by an a priori calculation.” This is in contrast to other fields in physics, such as statistical physics, quantum mechanics, relativity, or gauge field theories, which in spite of being much younger are already well established and serve as fundamental theories for more complex problems such as solid state physics, astronomy and cosmology, or particle physics. A fundamental theory of friction does not exist. Moreover, and although recently considerable progress has been made, the determination of relevant tribological phenomena from first principles is right now a very complicated task, indeed (Anonymous, 1995): “What is needed … would be to calculate the results of moving a probe of known Miller surface of a perfect crystal and calculate how energy is generated in the various phonon modes of the crystal as a function of time.” From another point of view, the difficulties encountered in tribology are not so surprising taking into account the diversity of phenomena which in principle can contribute to the process of friction. In fact, for a detailed understanding of friction the precise nature of the surfaces and their mutual interaction have to be known. Adsorbed films which can serve as lubricants, surface roughness, oxide layers, and maybe even defects and surface reconstructions determine the tribological properties of surfaces. The essential complexity of friction has been described very accurately by Dowson (1979): “… If an understanding of the nature of surfaces calls for such sophisticated physical, chemical, mathematical, materials and engineering studies in both macro and molecular terms, how much more challenging is the subject of … interacting surfaces in relative motion.” An additional problem in tribology is that until recently it has not been possible to find a simple experimental system which would serve as a model system. This contrasts with other fields in physics. There, complex physical situations can usually be reduced to much simpler and basic ones where theories can be developed and tested under well-defined experimental conditions. Note that it is not enough if such a system can be thought of theoretically. For testing the theory this system has to be constructed FF lat load = () ⋅tan , α © 1999 by CRC Press LLC experimentally. The lack of such a system had slowed progress in tribology considerably. Recently, however, with the development of such techniques as the surface force apparatus, the quartz microbalance, and most recently the SFM, we consider that such simple systems can be prepared, which in turn has also triggered theoretical interest and progress. In recent years this has led to a new field, termed nanotribology, which is one of the subjects of the present book. Within this new field, the SFM and the scanning force and friction microscope (SFFM), which is essentially an SFM with the additional ability to measure lateral forces, have probably drawn the most attention, even though in some respects, namely, reproducibility and precision, the surface force apparatus as well as the quartz microbalance might at the moment be superior. Presumably the interest which has accompanied the SFFM is due to its great potential in tribology. The most dramatic manifestation of this potential is its ability to resolve the atomic periodicity of the topography and of the friction force as the tip moves over a flat sample surface. An important feature of modern tribological instruments is that wear can be excluded down to an atomic scale. Under appropriate experimental conditions this is true for the SFFM as well as for the surface force apparatus and the quartz microbalance. In general, wear can lead to friction, but it is known that wear is usually not the main process that leads to energy dissipation. Otherwise, the lifetime of mechanical devices — a car, for example — would be only a fraction of what it is in reality. In most technical applications — excluding, of course, grinding and polishing — the lifetime of devices is fun- damental; therefore, surfaces are needed where friction is not due to wear, even though in some cases wear can actually reduce friction. Research in wearless friction of a simple contact is thus of technical as well as of fundamental interest. From a fundamental point of view, wearless friction of a single contact is possibly the conceptually simple and controlled system needed for the well-established interplay between experiment and theory: development of models and theories which are then tested under well- defined experimental conditions. Four features makes the SFFM a unique instrument as compared with other tribological instruments: 1. The SFFM is capable of measuring simultaneously the three most relevant quantities in tribological processes, namely, topography, normal force, and lateral force. 2. The SFFM has a resolution which is orders of magnitude higher than that of classical tribological instruments. Topography can be determined with nanometer resolution, and forces can be mea- sured in the nanonewton or even piconewton regime. 3. Experiments with the SFFM can be performed with and without wear. However, due to its imaging capability, wear on the sample is easily controlled. Therefore, operation in the wearless regime, where tip and sample are only elastically but not plastically deformed, is possible. 4. In general, an SFFM setup can be considered a single asperity contact (see, however, Section 6.3.3). While some instruments used in tribology share some of these features with the SFFM, we believe that the combination of all these properties makes the SFFM a unique tool for tribology. Of these four features, the last might be the most important one. Of course, it is always valuable to be able to measure as many quantities with the highest possible resolution. The fact that an SFFM setup is a simple contact — which can also be achieved with the surface force apparatus — is a qualitative improvement as compared with other tribological systems, where it is well known that contact between the sliding surfaces occurs at many, usually ill-defined asperities. Classic models of friction propose that the friction is proportional to the real contact area. We will see that this seems to be also the case for single asperity contacts with nanometer dimension. It is evident that roughness is a fundamental parameter in tribological processes (see Chapter 4 by Majumdar and Bhushan). On the other hand, a simple gedanken experiment shows that the relation between roughness and friction cannot be trivial: very rough surfaces should show high friction due to locking of the asperities. As roughness decreases, friction should decrease as well. Absolutely smooth surfaces, however, will again show a very high friction, since the two surfaces can approach each other so that the very strong surface forces act between all the atoms of the surfaces. In fact, two ideally flat surfaces of the same material brought together in vacuum will join perfectly. To move these surfaces past each other, © 1999 by CRC Press LLC the material would have to be torn apart. This has been observed on a nanoscale and will be discussed in Section 6.3.1.3. In conclusion, it seems reasonable that for a better understanding of friction in macroscopic systems one should first investigate friction of a single asperity contact, a field where the surface force apparatus and, more recently, the SFFM have led to important progress. Macroscopic friction could then possibly be explained by taking all possible contacts into account, that is, by adding the interaction of the individual contacts which form due to the roughness of the surfaces. As discussed above, three instruments can be considered to be “simple” tribological systems: the surface force apparatus, the SFFM, and the quartz microbalance. All three represent single-contact instruments, the last being in a sense an “infinite single contact.” Since experiments with the surface force apparatus are discussed in detail in Chapter 9 by Berman and Israelachvili, we will limit our discussion to the last two and mainly to the SFFM. Accordingly, in the next section we will describe the main features of an SFFM, then present experiments which we feel are especially relevant to friction on an atomic scale, and finally try to explain these experiments in a more theoretical section. 6.2 Instrumentation An SFM (Binnig et al., 1986) and an SFFM (Mate et al., 1987) consist essentially of four main components: a tip which interacts with the sample, a force-sensing element which detects the force acting on the tip, a piezoelectric element which can move the tip and the sample relative to each other in all three directions of space, and control electronics including the data acquisition system as well as the feedback system which nowadays is usually realized with the help of a computer. A detailed description of the instrument can be found in this book in Chapter 2 by Marti. Therefore, we will limit the discussion of the instrument only to the first two components, the tip and the force-sensing element, which we consider especially relevant to friction on an atomic scale. For many applications a thorough understanding of how the SFFM works is essential to the understanding and correct interpretation of data. Moreover, in spite of the impressive performance of this instrument, the SFFM is unfortunately still far from being ideal and the experimentalist should be aware of its limitations and of possible artifacts. 6.2.1 The Force-Sensing System The force-sensing system is the central part of an SFFM. Usually, it is made up of two distinct elements: a small cantilever which converts the force acting on the tip into a displacement and a detection system which measures this often very small displacement. The force is then given by where c is the force constant of the cantilever and ∆ the displacement which is measured. The fact that the force is not measured directly but through a displacement has important consequences. The first one is evident: for an exact determination of the force, the force constant has to be known precisely and this is quite often a problem in SFM. Another implication is that an SFM setup is not stiff. If a force acts on the tip, the cantilever bends and the tip moves to a new equilibrium position. Therefore, especially in a strongly varying force field, the tip position cannot be controlled directly. Moreover, a spring in a mechanical system subject to friction forces can modify its behavior substantially (see the Chapter 9 by Berman and Israelachvili). This is specially important in SFM: since the resolution is limited by the minimum displacement that can be measured, a force measurement gives high resolution if the force constant is low. With a low force constant, however, the tip–sample distance is less easily controlled. Finally, for a low force constant the properties of the system are increasingly determined by the force constant of the macroscopic cantilever and not by the intrinsic properties of the tip–sample contact, which is the system to be studied. Therefore, a reasonable trade-off between resolution and control of the tip–sample distance has to be found for each experiment. Although some schemes, such as feedback Fc=⋅∆ © 1999 by CRC Press LLC control of the cantilever force constant (Mertz et al., 1993) and displacement controlled SFMs (Joyce and Houston, 1991; Houston and Michalske, 1992; Jarvis et al. 1993; Kato et al. 1997), have been proposed to avoid this problem, up to now these schemes have not been commonly used. 6.2.1.1 The Cantilever — The Force Transducer The cantilever serves as a force transducer. In SFFM not only the force normal to the surface, but also forces parallel to it have to be considered; therefore, the response of the cantilever to all three force components has to be analyzed. In principle, the cantilever can be approximated by three springs characterized by the corresponding force constants. Within this model, the tip is attached to the rest of the rigid microscope through these three springs, one in each direction of space. The force acting on the tip causes a deflection of these springs. To determine the force and the exact behavior of the microscope, their spring constants have to be known. A cantilever is a complex mechanical system; therefore, calculation of these force constants can be a difficult problem (Neumeister and Drucker, 1994; Sader, 1995), in some cases requiring numerical computation. Most SFFM experiments are done with rectangular cantilevers of uniform cross section, since they have a higher sensitivity for lateral forces than triangular ones, which are commonly used in SFM. Moreover, for rectangular cantilevers the relevant force constants can be calculated analytically. We will limit the following discussion to these cantilevers. The equation describing the deflection of a cantilever is (see, for example, Feynmann, 1964) (6.1) where E is the Young’s modulus of the material, I = ∫ z 2 dA the moment of inertia of the cantilever and M ( y ) the bending moment acting on the surface which cuts the cantilever at the position z ( y ) in the direction perpendicular to the long axis of the cantilever (see Figure 6.1). For a cantilever of rectangular cross section of width w and thickness t the moment of inertia is I = w · t 3 /12. Solving Equation 6.1 with the correct boundary conditions one finds the bending line (6.2) FIGURE 6.1 Geometry and coordinate system for a typical cantilever. Its length is l, its width w , its thickness t, and the tip length l tip . The y -axis is oriented in the direction corresponding to the long axis of the cantilever. Forces act at the tip apex and not directly at the free end of the cantilever. This induces bending and twisting moments as discussed in the main text. ′′ () = () ⋅ () zy My EI, zy y l y l lF EI z () = − ⋅ ⋅⋅ 2 3 3 6 , © 1999 by CRC Press LLC where l is the length of the cantilever and F z the force acting at its end. From this bending line, the force constant is read off as (6.3) This is the “normal” force constant in a double sense: it is the force constant associated with a deflection in a direction normal to the surface, and also the force constant generally used to characterize a cantilever. However, other force constants are also relevant in an SFM and an SFFM setup. Exchanging t and w in the above equation gives the force constant corresponding to the bending due to lateral force F x (see Figure 6.1): (6.4) where c is the normal force constant (Equation 6.3). Since the lateral force acts at the end of the tip and not at the end of the cantilever directly, this force exerts a moment M = F x · l tip which twists the cantilever. This twisting angle ϑ causes an additional lateral displacement ∆ x = ϑ · l tip of the tip. The corresponding force constant is (Saada, 1974) where G is the shear modulus and K Ӎ 1 for cantilevers that are much wider than thick ( w ӷ t ), which is the usual case in SFFM. It is useful to relate this force constant to the normal force constant c . With the relation G = E /2(1 + ν ) and assuming a Poisson factor ν = ⅓ , one obtains (6.5) Both lateral bending and torsion of the cantilever contribute to the total lateral force constant which is calculated from the relation of two springs in series (see Section 4.3.1, Equation 6.22): (6.6) The last case is that of a force F y acting in the direction of the long axis of the cantilever ( y -direction). This force induces a moment M = F y · l tip on the cantilever which causes it to bend in a way similar but not equal to the bending induced by a normal force. Solving Equation 6.1 one finds the new bending line: (6.7) This bending has two effects. First, the tip is displaced an amount, δ z = ˜z ( l ) = (3/2) · ( l tip / l ) · ( F y / c ) in the z direction. Second, the tip is displaced an amount δ y = α · l tip in the y direction, where α is the c F zl EI l Ew t l z = () = ⋅⋅ =⋅⋅ 3 1 4 3 3 . cEt w l w t c x bend =⋅⋅ = ⋅ 1 4 32 , c K Gw t ll x tors tip =⋅⋅ ⋅ 3 3 2 , c Kl l c l l c x tors tip tip = + () ⋅ ⋅ 2 3 1 1 2 22 ν Ӎ . 11 11 2 2 2 cc c c t w l l xx x tot bend tors tip =+=⋅ + . ˜ .zy lF EI y y () = ⋅ ⋅ ⋅ 1 2 2 tip © 1999 by CRC Press LLC bending angle α = ˜z ′ ( l ), which follows from Equation 6.7. The corresponding force constant for bending due to the force F y is then (6.8) We note that the displacement of the tip in the z -direction due to a force F y implies that the model describing the movement of the tip by three independent springs is not completely correct. The correct description of an SFM setup is in terms of a symmetric tensor ˆ C which relates the two vectors force ∆ and displacement F : The terms c yz corresponds to the displacement δ y = ϑ · l tip of the tip in the y -direction due to bending induced by a normal force F z (Equation 6.3). If the off-diagonal terms are neglected, the relation between forces and displacements is determined by the diagonal terms, the three force constants, which can then be related to three independent springs. We finally note that usually the cantilever is tilted with respect to the sample. This directly affects the relation between the different components of the forces, and has to be taken into account if the tilting angle is significant (Grafström et al., 1993, 1994; Aimé et al., 1995). 6.2.1.2 Measuring Forces Force is a vector and therefore in our three-dimensional world it has three components. A classical SFM measures the component normal to the surface, while an SFFM measures at least one of the components parallel to the surface. Since normal force and lateral force are usually intimately related, the simultaneous measurement of both is fundamental in tribological studies. In fact, nowadays practically all commercial SFMs offer this possibility. The optimum solution is, of course, the determination of the complete force vector, that is, of all three force components, and in fact such a system has been proposed (Fujisawa et al., 1994) but is not widely used. As described in Chapter 2 by Marti, the simultaneous detection of normal force and the x-component of the lateral force is easy with the optical beam deflection technique (Meyer and Amer, 1990b; Marti et al., 1990), see Figure 6.2. Since this detection technique is most commonly used in SFFM, we will briefly recall some of its properties. A very particular feature of the optical beam deflection technique is that it is inherently two dimensional: the motion of the reflected beam in response to a variation in orientation of the reflecting surface is described by a two-dimensional vector. In the case of SFFM, if the cantilever and the optical components are aligned correctly, and if the sample is scanned perpendicular to the long axis of the cantilever (x-axis), then normal and lateral forces cause motions of the reflected beam which are perpendicular to each other (see Figure 6.3). This motion can then easily be measured with a four-segment photodiode or a two-dimensional position sensitive device (PSD). Another important feature of the optical beam deflection method is that unlike other detection techniques, angles and not displacements are measured. Moreover, due to the reflection properties, the angles that are detected on the photodiode are twice the bending or twisting angles of the cantilever. This has to be taken into account when signals are converted into forces. One consequence of measuring angles instead of displacements is that, in the case of a lateral force acting on the tip, only the displacement corresponding to the torsion of the cantilever is detected. However, the tip is also displaced due to lateral bending which does not result in a variation of the c F y l l c y y ==⋅ ⋅ δ 1 3 2 tip . ∆∆= − ˆ CF 1 o ˆ .C − = =⋅ + 1 222 2 1 200 0332 0321 ccc ccc ccc c lltw ll l l ll xx xy xz yx yy yz zx zy zz tip 2 tip 2 tip tip © 1999 by CRC Press LLC measured angle. Therefore, this motion is not detected. Depending on the calibration procedure used, this might lead to errors in the estimation of the lateral force when the cantilever is displaced more due to bending than due to torsion. From Equations 6.4 and 6.5 we see that this is the case for cantilevers with t/w ӷ l tip /l. The technique for measuring friction forces with the optical beam deflection method just described assumes scanning in a direction perpendicular to the long axis of the cantilever (x-axis). However, friction forces can also be measured in the other direction parallel to the surface (Radmacher et al., 1992; Ruan and Bhushan, 1994a). In this different mode for measuring friction, the sample is scanned back and forth in a direction parallel to the long axis of the cantilever (y-axis). As discussed previously, the friction force acting at the end of the cantilever then bends it in a similar way as when induced by a normal force. From Equations 6.2 and 6.7 the bending line corresponding to the back-and-forth scan can be calculated. One obtains Note that the sign of F y depends on the scan direction. A technique is needed to discriminate between bending due to a normal force and bending due to a lateral force. The friction force changes sign when the scanning direction is reversed, while the normal force remains unchanged; therefore the difference signal corresponds to the effect caused by friction and the mean signal is due to the normal force. It should be noted, however, that usually the microscope is operated in the so-called constant-force mode. In the present case, this mode is better called the constant-deflection mode, since the deflection (more precisely, the bending angle) and not the (normal) force is kept constant. To maintain a constant FIGURE 6.2 Schematic setup of the optical beam deflection method. With a four-segment photodiode, the two- dimensional motion of the reflected beam is measured. Therefore, normal and lateral forces can be detected simul- taneously. Bending of the cantilever due to a normal force causes a vertical motion of the reflected beam. Torsion of the cantilever due to a lateral force causes a horizontal motion. FIGURE 6.3 If the cantilever and the optical setup are aligned correctly, the motions n α and n β induced by normal and lateral forces cause perpendicular movements r α and r β of the reflected laser spot on the photodiode. This is not the case for arbitrary alignment of the optical axes. zy c y l y l F l l F zytot tip () = − ⋅+ ⋅ 1 2 33 2 . © 1999 by CRC Press LLC deflection, the feedback adjusts the height of the sample to correct for the difference in bending due to friction while scanning back and forth; that is, the feedback adjusts the height so that where z ′ tot+ (l)is the angle of the free end of the cantilever during the forward scan, z ′ tot– (l)the angle of the cantilever during the backward scan and ˜z'(l, F y ) the angle at the free end of the cantilever induced by a force F y according to Equation 6.7. The friction is related to the difference in height of the topographic images corresponding to the back-and-forth scan. Solving the above equation for the friction force F fric = F y as a function of the height difference ∆ z between back-and-forth scan, one finally finds with ∆ z = z tot+ (l) – z tot– (l). 6.2.2 The Tip One of the great merits of the SFFM, the nanometric size of the contact, is on the other hand a serious experimental problem, since it is almost impossible to characterize the tip and thus the contact down to an atomic scale. Different schemes have been proposed to solve this problem. One possibility is to use electron microscopy not only to image, but also to grow a well-defined tip (Schwarz et al., 1997). If the electron beam is focused on the tip, molecules from the residual gas are ionized and accelerated towards its end, where they spread out due to their charge. The result is a well-defined spherical tip end. A similar procedure is to heat a very sharp metallic tip in high vacuum (Binh and Vzan, 1987; Binh and García, 1992). Surface diffusion will induce migration of atoms from regions of high curvature to regions of lower curvature. Again, to control the process, an electron microscope is needed. As in the previous case, this process will form a well-defined and smooth tip (see Figure 6.4). These preparation methods are very effective but also have disadvantages, the first one being the immense effort needed to fabricate just one single tip. Moreover, modification of the tip during transfer, and, even more critical, during the SFFM experiment due to wear cannot be excluded. To control possible wear, the tips should be imaged before and after the measurement. FIGURE 6.4 Well-defined spherical tip ends of tungsten cantilevers produced by heating the cantilever as described in the text. The formation of these tips is controlled by the balance between surface diffusion and surface energy. By carefully tuning the experimental conditions, tip ends of different shapes can be obtained. (Courtesy of Augustina Asenjo Barahona, Universidad Autónoma de Madrid.) ′ () − ′ () = ′ () zlzl zlF ytot+ tot- 2 ˜ ,, F cl l z fric tip =⋅ ⋅ 4 ∆ , [...]... constant-deflection mode, on the other hand, should show no contrast in the normal force image, all information being in the topographic image Since feedback systems are never ideal, in this second case usually a small amount of structure is visible also in the normal force image However, in the case of the normal force image shown in Figure 6.10 the magnitude of the variation as well as its shape cannot be explained... the topographic image and the normal force image are complementary: if the feedback system does not appropriately correct the height of the piezo (topographic signal), then the cantilever will be deflected (normal force signal) Images taken in the constant-height mode show no contrast in the topographic image, in this case all information is in the normal force image Images taken in the constant-deflection... CRC Press LLC FIGURE 6.9 Set of atomically resolved images taken on HOPG with an Si3N4 cantilever and a microfabricated tip The left images are raw data and the right images have been Fourier filtered to show the different positions of the maxima in the topographic and the friction images The bottom image shows the relative positions of these maxima, which are shifted with respect to each other (From... alkanethiol monolayer at low load (about 30 nN, left images), high load (about 300 nN, middle images), and again low load (right images) The top images show a schematic setup of the tip–sample system and the images in the middle the acquired topographic signals The lower images are twodimensional Fourier transforms of the corresponding topographic images showing the relation between the alkanethiol monolayer... MoS2 surface, which is a layered material with a lattice periodicity of 0.274 nm The upper images (Figure 6.11a) correspond to the usual setup in SFFM where the fast scan is along the x-axis The left image shows the torsion of the cantilever — this is the image which is usually acquired as the friction image — and the right image the bending of the cantilever The lower images (Figure 6.11b) were taken... left image shows the torsion and the right image the bending of the cantilever The fast scan in all images is from left to right The first and the last images are easily understood: they reflect the typical one-dimensional stick-slip behavior While the first image is measured © 1999 by CRC Press LLC in the usual torsion mode, the last image is measured by scanning along the cantilever The other two images... estimated from scanning © 1999 by CRC Press LLC FIGURE 6.16 (a) Schematic view of the diamond tip, seen as if looking through the tip along the surface normal This view is aligned correctly with respect to the two lateral force images shown below (b) Lateral force image of a hydrogen-terminated diamond (100) surface (c) Lateral force image of a hydrogen-terminated diamond (111) surface For both images,... 7 × 7 surface (Landman et al., 1989a,b) These simulations also predict nanoneck formation for other tip–sample systems, in particular for metal–metal contacts (Landmann et al., 1990) If nanoneck formation occurs, atomic-scale friction is not only limited to the tip–sample interface, which is not well defined any more, but to atomic rearrangement which can happen within the whole nanoneck These processes... (a) and (c), and friction (b) and (d), images of a double bilayer of Cd–arachidate Scan size is 2 × 2 µm2, the loading force is 4 nN for the upper images and 16 nN for the lower ones The total vertical scale in the topographic image is 24 nm and in the friction image 14 nN No damage is observed in the upper images at low loading forces At higher loading, however, small islands and flakes are sheared away... distance, normal and lateral forces are calculated as ( ) Fz = − ∂Vsurf = kz ⋅ V0 cos k x x ⋅ e − kz z , ∂z Fx = − ∂Vsurf = k x ⋅ V0 sin k x x ⋅ e − kz z ∂x ( ) Therefore, minima and maxima of the normal and lateral forces are shifted by one quarter lattice spacing with respect to each other In the constant-force mode, topographic image is obtained by adjusting the position of the sample to maintain a . be deflected (normal force signal). Images taken in the constant-height mode show no contrast in the topographic image, in this case all information is in the normal force image. Images taken in. the other, the decrease of the normal force for increasing tip–sample distance, normal and lateral forces are calculated as Therefore, minima and maxima of the normal and lateral forces are shifted. constant-force mode, topographic image is obtained by adjusting the position of the sample to maintain a constant force. Hence, minima and maxima of the normal force and of the topography coincide