## Abstract

Tapping-mode atomic force microscopy provides a means for successful and non-intrusive characterization of soft physical and biological structures at the nanoscale. Its full potential can only be realized, provided that the response of the oscillating probe tip to the strongly nonlinear, near-field force interactions with the structure and the intermittency of contact can be accurately modelled, analysed, controlled and interpreted. To this end, this paper reviews some experimental observations of fundamentally nonlinear behaviour of the tip dynamics. It discusses the nonlinear phenomenology that explains their presence in the tapping-mode operation of the atomic force microscope. Particular emphasis is placed on the coexistence of different steady-state responses and their origin in transitions across regions of rapidly varying force characteristics. The heuristics of a recently developed method for treating such transitions are presented and insights into its implications are drawn from related micro- and nanoscale applications.

## 1. Introduction

Tapping-mode atomic force microscopy finds a wide range of application in the probing of nanoscale surface and subsurface properties of a variety of materials in a variety of environments. Tapping mode involves only intermittent contact between the microscope probe tip and the specimen surface. Consequently, it greatly reduces the effects of adhesion and friction on the dynamics of the microscope probe as compared with contact-mode atomic force microscopy. Relatively low contact velocities also imply significantly less damage to the specimen as compared with contact mode. Thus, tapping mode has become the method of choice for high-resolution topographical measurements of soft and fragile materials that are difficult to examine otherwise.

The interactions between the microscope probe tip and the specimen are inherently nonlinear, for example owing to rapid variations in the force interactions over atomic length-scales and the intermittency of contact. Nonlinear phenomena, such as the coexistence of multiple system attractors and a variety of bifurcation scenarios, affect the operation and, more importantly, the interpretation of the signal produced by the microscope. Indeed, dramatic instabilities may occur as a result of the tip–sample interactions that deteriorate the imaging quality as well as result in damage to the specimen. Advanced methods of dynamical systems analysis should therefore be employed to predict, explain and control the probe dynamics.

This paper establishes the nonlinear phenomenology that underlies challenges associated with modelling and analysis of the probe-tip dynamics and the consequent problems with nanoscale surface characterization. The discussion highlights features in the particular application that are common to a broad range of micro- and nanoscale applications, e.g. microelectromechanical systems (MEMS) and nanoelectromechanical systems. Simple representative models of such systems are used to contrast the necessary analysis techniques with those applicable to macroscopic mechanical and physical systems. Finally, recent results reported in the dynamical systems literature are shown to enable accurate prediction of probe-tip behaviour and thus to anticipate future dramatic improvements in atomic force microscope-based nanoscale characterization techniques.

## 2. Atomic force microscopy

### (a) Experimental modalities

In the scanning force microscope (Binnig *et al*. 1986)—commonly known as the atomic force microscope—direct measurements of interatomic forces are possible for a wide array of materials including semiconductors, polymers, carbon nanotubes and biological cells. Here, the microscope probe consists of a large-aspect-ratio cantilever beam with a finely engineered tip that deforms owing to interactions with the specimen surface (see García & Perez (2002), Giessibl (2003) and Jalili & Laxminarayana (2004) for recent reviews) as shown in figure 1. These deformations, as measured by directional changes of a reflected laser beam, represent local force interactions between the cantilever tip and the specimen. By scanning the probe across the specimen surface under different operating modalities, topographical mapping as well as characterization of material properties of the specimen can be obtained with atomic resolution.

As with all scanning-probe techniques, the near-field characteristics upon which the scanning force microscope rely vary dramatically over small changes in tip–surface separation. Fine-grained position control, for example using piezoceramic actuators, must be applied to allow the probe tip to carefully approach the operating separation from the specimen surface (Binnig *et al*. 1982). This is particularly important when scanning across the specimen, during which a feedback control strategy is typically employed to maintain a constant value for some operationally defined quantity, in what is known as *constant interaction mode*.

As an example, in constant interaction, *contact-mode* atomic force microscopy, the probe–specimen separation is controlled so as to maintain a constant deflection of the cantilever tip while ensuring that the tip remains in direct contact with the surface. Sustained contact ensures a large signal-to-noise ratio and a strong measurable signal. On the other hand, it raises concerns about strong adhesive forces, friction, damage to soft specimens and subsequent poor repeatability of measurements.

While the destructive features of contact-mode atomic force microscopy would be eliminated if a non-zero tip–surface separation could be maintained, this would reduce the deformation of the cantilever and, correspondingly, the signal-to-noise ratio. Moreover, at some critical tip–surface separation, the rate of growth of the attractive force from the specimen begins to exceed that of the restoring force of the cantilever beam. Under further decreases in the tip–surface separation, this results in so-called snap-through of the tip to contact with the specimen. The ability to maintain a large tip deflection in non-contact-mode static atomic force microscopy is thus compromised by the nonlinear characteristics of the near-field force interactions.

In contrast to the static measurement modalities considered in the previous paragraphs, the operating principle of dynamic modalities is based on changes in the oscillatory response of an externally driven cantilever to the surface force field (Radmacher *et al*. 1992; Hansma *et al*. 1994; García & San Paulo 2000). Here, the cantilever support is excited by a vibrating dither piezo. For a given excitation amplitude and frequency, a unique steady-state oscillatory response of the cantilever results when the probe tip is sufficiently removed from the specimen surface. As the probe–specimen separation is reduced, the effects of the local force field on the cantilever oscillations become measurable. Constant interaction mode during surface scanning is achieved, for example, by maintaining a constant oscillation amplitude.

In *non-contact-mode* dynamic atomic force microscopy, the probe tip remains separated from the specimen at all times, yet approaches it sufficiently closely to have its dynamics affected by the attractive force from the surface. As the tip only samples the attracting region, the effects on the tip motion are relatively weak, resulting in poor resolution and noise sensitivity without appropriate signal conditioning. As with the static modality, non-contact-mode dynamic microscopy also requires careful feedback control to avoid undesired dynamic snap-through, whereby the cantilever oscillation jumps to an oscillatory motion that involves intermittent contact between the tip and the specimen.

Indeed, such intermittent-contact dynamic atomic force microscopy—also known as *tapping-mode* atomic force microscopy—is an appealing measurement modality, as it samples attractive as well as repulsive force interactions with the specimen. As a result of the intermittency of physical contact with the specimen, the effects of force interactions on the oscillatory response are stronger and more distinct than those that result from the non-contact modality. Intermittency also greatly reduces the risk of surface damage and poor measurement repeatability, in particular, when it can be maintained with a minimum of peak contact velocity. Consequently, tapping mode has the potential to provide non-destructive high-resolution topographical measurements of soft and fragile materials.

### (b) Experimental evidence of nonlinear phenomenology

Repeatable and reliable operation of physical measurement devices relies on the ability to calibrate the response of the device to the phenomenon being investigated. Although some scatter in calibration data is inevitable, this is usually ascribed to stochastic effects owing to the presence of noise and not an inherent problem with the device. In contrast, in the event that multiple states of the device coexist under identical operating conditions or that identical states of the device occur under multiple operating conditions it becomes impossible, without alternative corroborating means of measurement, to quantify an observed phenomenon with any degree of certainty.

Nonlinear system characteristics are well known to result in complex dynamic behaviours. These include the coexistence of multiple system attractors under identical operating conditions and the large sensitivity to initial conditions (known as chaos) that constrains any attempt to quantitatively predict the system dynamics over long periods of time. A wealth of experimental evidence outlined in the existing literature has established the occurrence of both of these phenomena in the behaviour of the atomic force microscope probe tip under dynamic operating modalities (Gleyzes *et al*. 1991; Burnham *et al*. 1995; Lee *et al*. 2002).

Consider, for example, constant interaction, dynamic atomic force microscopy surface scanning in which the control architecture aims to maintain a constant oscillation amplitude during lateral translation across the specimen. Then, variations in surface topography (or, alternatively, in the material properties of the surface governing near-field force interactions) result in changes in the cantilever response away from the targeted oscillation amplitude. Through feedback control, the probe–surface separation is then manipulated, so as to allow the cantilever oscillations to return to their original amplitude. Once the targeted amplitude has been reached, the change in the probe–surface separation imposed through the controller is then assumed to be directly related to the change in topography (of a homogeneous material or possibly in the material properties of a perfectly flat specimen).

Now suppose that a given oscillation amplitude of the cantilever can result from two given separations between the probe and the specimen. In such an instance, termed *bistability* in the literature (Gleyzes *et al*. 1991), the controller is unable to differentiate between the two oscillations and, consequently, to ascertain that the change in the probe–specimen separation accurately reflects a change in topography. For particular target oscillation amplitudes, the natural presence of noise in the atomic force microscope may suffice, even without any changes in the surface topography, to trigger transitions between these different oscillations (García & San Paulo 2000).

In addition to leading to great uncertainty in the interpretation of scanning data, the transitions between different oscillatory responses may also be responsible for damage to the specimen. Indeed, where one of the two oscillatory motions corresponds to a non-contact-mode oscillation, the other oscillatory motion typically corresponds to an intermittent-contact oscillation. As it happens, the smaller the minimum tip–surface separation for a non-contact-mode oscillation, the more susceptible it appears to be to small amounts of noise that trigger a transition to the impacting oscillation. The fact that stable intermittent-contact oscillations exhibit a non-zero minimum duration of contact puts further constraints on the ability to maintain non-destructive scanning. The connection between these observations and their origin in the destabilizing effects of the transition between free flight and contact is the emphasis of this paper and the discussion that follows.

### (c) Modelling of the near-field force interactions

In the absence of liquid interfaces, the near-field interactions between the atomic force microscope probe tip and the specimen surface are composed of attractive van der Waals-type forces that originate in constructive correlation of the charge distributions of the tip and surface atoms (Israelachvili 1991) and repulsive Pauli- and ionic-exclusion-type forces owing to overlapping electron clouds (Hoffmann *et al*. 2001). In particular, while the attractive forces dominate for large tip–surface separations, the repulsive forces are negligible for large separations, but experience a dramatic and sudden increase as the tip–surface separation falls below a critical distance, corresponding to physical ‘contact’ between the tip and the surface. In tapping-mode atomic force microscopy, the probe tip experiences the repulsive force only for a very short time as compared with the period of the oscillation. Upon the cessation of contact, the cantilever tip continues to vibrate away from the specimen under the influence of the restoring elastic force of the cantilever beam and the attractive van der Waals forces.

To understand and model the range of physical interactions that take place between the probe tip and the specimen under typical experimental conditions is a formidable task. For example, capillary forces may occur in the presence of liquid interfaces, the probe tip and the specimen surface may deform plastically and degrade over time, viscous drag may result from motion of the probe tip through a liquid medium and so on. However, in spite of its apparent complexity, many of the fundamental features of the cantilever dynamics observed experimentally during tapping-mode operation can be explained with relatively simple models that rely on just the attractive force or, more typically, on a combination of the attractive and repulsive effects considered in the previous paragraph.

Representative examples of models considered in the literature are those presented in Sebastian *et al*. (2001), Rützel *et al*. (2003) and Yagasaki (2004). In the model discussed by Yagasaki (2004) (cf. García & San Paulo 1999; Lee *et al*. 2002), the combined effects of attraction and repulsion are captured by a force field of the form(2.1)where *ζ* represents the tip–surface separation and *a* represents the characteristic intermolecular distance corresponding to the onset of contact. In this model, the interactions for tip–surface separations beyond the critical distance *a* consist entirely of long-range attractive forces. Here, these are modelled by the London–van der Waals forces on a spherical particle (representing the probe tip) in the vicinity of a flat surface (representing the specimen; see Goodman & Garcia 1991; Ciraci *et al*. 1992; Pérez *et al*. 1998). In contrast, following the DMT theory of Derjaguin *et al*. (1975), the force interactions during contact between the tip and the specimen are modelled by a combination of surface adhesion and the restoring forces generated by an elastic sphere pushed against a flat surface. In particular, the adhesive force here is taken to equal the van der Waals force at the onset of contact.

The positive parameters *α* and *β* model the strength of the attractive and repulsive interactions, respectively. As per the discussion above, they depend on the elastic properties of the tip and the specimen, the radius of the probe tip and the adhesive surface energy. A graphical illustration of dependence of the interaction force on the tip–surface separation using representative parameter values is given in figure 2*a*.

Although the force field is a continuous function of tip–surface separation in the model considered by Yagasaki, a discontinuous jump in force gradient occurs at *ζ*=*a*. In contrast, the model considered by Rützel *et al*. (2003) presents a perfectly smooth force field based on a Lennard-Jones potential and is given by(2.2)where *α* and *β* again model the strength of the attractive and repulsive interactions, respectively (see also Sarid *et al*. 1996; Ashhab *et al*. 1999). Here, no claim is made as to the ability of the particular expression to accurately capture the detailed interactions between the probe tip and the specimen surface. Instead, it is argued that in spite of its simplicity, the interaction model captures generic properties present in the near-field interactions.

Although there are no discontinuities present in the Lennard-Jones model, a graphical representation of the dependence of the interaction force on the tip–surface separation shown in figure 2*b* indicates a rapid change in the force gradient near the point of maximum net attraction at *ζ*=*ζ*^{*}. Indeed, whereas the magnitude of the force gradient is bounded for *ζ*>*ζ*^{*}, it grows rapidly without bounds as *ζ* decreases below *ζ*^{*}. In terms of representative numerical values obtained from Rützel *et al*. (2003), a decrease in the tip–surface separation from *ζ*^{*} by 25% results in a 30-fold increase in the force gradient relative to its maximum absolute value for *ζ*>*ζ*^{*}.

As a final example, the model of Sebastian *et al*. (2001) (cf. Sebastian *et al*. 2003; Zhao & Dankowicz 2006*a*) reduces the considerations of the attractive and repulsive forces to a piecewise linear, but continuous interaction model of the form(2.3)where *α* and *β* again model the strength of the attractive and repulsive interactions, respectively. Here, in addition to the critical distance *a* corresponding to the onset of contact, an upper cutoff for the attractive forces has been introduced through the parameter *b*>*a*. The complete model considered by Sebastian *et al*. (2001) also includes a damping element in the case of *ζ*<*a* that models energy dissipation during contact. In contrast to the more physically correct model discussed by Yagasaki, here, variations in the attractive and repulsive forces are replaced by an average force gradient yielding a piecewise linear interaction description. Nevertheless, as seen in figure 2*c*, the dependence of the interaction force on the tip–surface separation contains the fundamental qualitative features found in both of the previous models.

### (d) Theoretical analysis

Provided that the cantilever tip's motion is confined to a region of approximately constant force gradient, a linear analysis may be applied to relate the system response to the probe–specimen separation. Here, small changes in the average tip–surface separation result in unique shifts in the effective natural frequency of the cantilever beam and, consequently, in the oscillation amplitude and phase shift relative to the excitation. However, for larger excursions of the cantilever tip, large variations in the force gradient undermine the efficacy of this linear analysis and are the immediate source of the bistability mentioned previously.

In spite of the presence of strong nonlinearities, successful predictions regarding the coexistence of multiple steady-state responses for a given nominal probe–specimen separation can be obtained by assuming a harmonic approximant for the cantilever motion (Aimé *et al*. 1999; García & San Paulo 1999; Yagasaki 2004). Such methods have been employed to establish the presence of a dynamically unstable intermittent-contact steady-state oscillation in addition to the two coexisting stable steady-state oscillations associated with the bistability seen in experiments. Indeed, the heightened sensitivity of the non-contact-mode oscillations to noise for particularly small minimum tip–surface separations can be directly associated with the influence of the unstable oscillatory motion.

As shown in figure 3, the relationship between the non-contact-mode oscillation and the stable and unstable intermittent-contact oscillations is even more intimate than suggested thus far. As seen here, the family of non-contact-mode oscillations connects with that of the unstable intermittent-contact oscillations in an apparently continuous fashion. Similarly, the family of stable intermittent-contact oscillations connects with the branch of unstable intermittent-contact oscillations. Although the methods employed in the references mentioned earlier are able to predict the existence of these branches of steady-state responses, they require effectively smooth force fields or, at least, dynamics that is dominated by the smooth parts of the force field. However, as in figure 3, the non-contact-mode oscillations connect to the unstable intermittent-contact oscillations in the immediate vicinity of the discontinuity associated with the onset of contact. If the excitation amplitude is increased above the critical amplitude associated with this so-called *saddle*–*node bifurcation*, a rapid transition occurs to the stable intermittent-contact oscillations as shown in the figure. Clearly, a theory that can address the implications of the discontinuity is required to resolve these observations.

## 3. Discontinuity-induced bifurcations

### (a) General phenomenology

It is not ultimately the force of gravity that is the demise of interplanetary boulders passing in the immediate vicinity of the Earth. Instead, their break-up is owing to the gradient of the force of gravity, the tidal force. Similarly, the stability characteristics of a steady-state response of the cantilever are not a function of the force field encountered by the cantilever tip during the oscillation. Instead, it is the accumulated effect of the force gradients in the vicinity of separations visited during the oscillation that establishes whether small perturbations away from this oscillation grow or decay with time.

It should, therefore, come as no surprise that rapid changes in the gradient of the tip–surface force interactions may induce the sort of changes in the system dynamics seen in the vicinity of the onset of contact. As shown earlier, such rapid variations in the force gradient are a common feature of nonlinear models of the near-field force interactions documented in the recent literature. For example, both the models discussed by Sebastian *et al*. (2001) and Yagasaki (2004) include a discontinuous change in the force gradient at the critical tip–surface separation *a* corresponding to the point of maximum attraction. While the model considered in Rützel *et al*. (2003) is perfectly smooth, it also exhibits a dramatic change in the force gradient over a small interval containing the point of maximum attraction.

Little can be ascertained of the effect of such rapid variations of the force gradient on the system dynamics by studying non-contact-mode oscillations far away from contact. As shown later, any changes in system behaviour that occur following the onset of low-velocity contact with the surface can, instead, be directly related to the effectively discontinuous nature of the contact. Such changes are thus referred to as *discontinuity-induced bifurcations*.

### (b) Macroscopic models

Discontinuity-induced bifurcations of the type considered here have been observed in a number of macroscopic experiments on vibro-impacting systems (Shaw 1985; Fredriksson *et al*. 1999). To illustrate their essential features, consider the following simplified discussion of the dynamics of a vehicle suspension system (cf. figure 4). Here, the strut and the bump stop control the relative motion of the body and the wheels. When the relative displacement reaches a critical value, the bump stop makes contact. This results in a discontinuous and abrupt change in the forces acting on the system. Since the stiffness and damping of the bump stop typically greatly exceed those of the strut, the interaction with the bump stop can be modelled as an elastic impact with a rigid stop with some characteristic coefficient of restitution. When contact is established with the bump stop with non-zero relative velocity, there is an instantaneous change in the direction and magnitude of the velocity. At a *grazing contact* with the bump stop, the relative velocity is zero, and there is no change in the velocity at impact. As the wheels traverse an uneven terrain, oscillations are induced in the suspension system. Ride comfort is a function of the character of the induced oscillations, their robustness under additional perturbations, and the frequency and nature of interactions with the bump stop.

Each of the three panels in figure 5 sketch a segment of a reference trajectory (solid), representing the relative motion of the body and the wheels, and a discontinuity line (dashed), corresponding to the presence of the bump stop. The robustness properties of the reference motion are determined by changes in form and extent of the small neighbourhood of initial conditions shown originally as a shaded circle on the right of each figure. In figure 5*a*, the amplitude of the relative motion of the body and the wheels is insufficient to establish contact with the bump stop. It follows that nearby initial conditions do not interact with the discontinuity, as evidenced by the reappearance of the circle on the left side of the figure. In figure 5*b*, the trajectory makes tangential contact with the discontinuity, corresponding to a grazing contact of the wheels with the bump stop. Again, the lower half of the neighbourhood is unperturbed by the presence of discontinuity. In stark contrast, the upper half of the neighbourhood gets dramatically stretched. As suggested in figure 5*c*, the deformation of the circle gets further pronounced as contact is established with the bump stop at increasing relative velocity.

The observed stretching signifies a loss of stability of the original motion and a resulting discontinuity-induced bifurcation (also known as a *grazing bifurcation*) to a different oscillatory behaviour, possibly involving repeated high-velocity impacts with the bump stop. Contrary to a smooth system, there is no advance warning of this impending instability. Instead, the similarity between panels (*b*) and (*c*) suggests that the onset of the instability can be traced to the singular parameter value for which grazing contact is established.

### (c) A MEMS impact actuator

An example of a microelectromechanical device that relies on repeated and controlled impacts for its very function is the impact microactuator developed by Fujita *et al*. (2004; see also Dankowicz & Zhao 2005; Zhao & Dankowicz 2006*b*) and is illustrated in figure 6. Here, the position of the actuator along a substrate is controlled by the presence of friction between the actuator frame and the substrate and episodes of relative sliding that originate in impacts between the actuator frame and an internal oscillating element.

A motion involving repeated transitions between the *stick phase*, during which the frame is stationary relative to the underlying substrate, and the *slip phase*, during which the frame is sliding relative to the substrate, is referred to as an *impacting motion*. The actuator exhibits a *non-impacting motion* if its motion remains in the stick phase. As in the macroscopic model considered earlier, we model the transition between the stick phase and the slip phase, at which time the oscillating element collides with the frame, as an elastic impact with an accompanying instantaneous change in the velocities of the oscillating element and of the frame relative to the substrate.

The application of a periodically varying voltage of sufficient amplitude excites the motion of the oscillating element and results in repeated impacts with the frame. Subsequent brief sliding episodes allow for the positioning of the frame, relative to the substrate, to an experimentally observed accuracy of the order of tens of nanometres. The resolution of the impact actuator is directly related to the sliding distance per sliding episode, which, in turn, is directly related to the impact velocity. It follows that improvements in resolution are associated with the possibility of ensuring sustained impacting motion with reduced impact velocity.

For a given excitation frequency, there exists a critical excitation amplitude *A*^{*}, such that the actuator exhibits a non-impacting periodic motion during which the oscillating element achieves zero relative velocity, grazing contact with the frame. Given such a critical amplitude *A*^{*}, we refer to a stable system response (an attractor) for nearby amplitudes *A*≈*A*^{*} as *local*, if its deviation from the grazing periodic motion goes to zero as *A*→*A*^{*} and *non-local* otherwise. If a local attractor can be found for values of *A* in a neighbourhood of *A*^{*}, the discontinuity-induced bifurcation scenario is said to be *continuous* and *discontinuous* otherwise. Figure 6 shows two distinct grazing bifurcation scenarios. In figure 6*b*, a discontinuous transition of the asymptotic motion from a non-impacting to an impacting periodic trajectory occurs as *A* is increased above the grazing parameter value *A*^{*}; and from an impacting to a non-impacting periodic trajectory as *A* is decreased below *A*^{sn}<*A*^{*}, corresponding to a saddle–node bifurcation. For *A*∈(*A*^{sn}, *A*^{*}), three distinct periodic trajectories (one non-impacting and two impacting) coexist. The coexistence of multiple attractors implies the possibility of parameter hysteresis in the long-term response of the impact actuator. As there is no local attractor for *A*>*A*^{*}, this is an example of a discontinuous grazing bifurcation.

In contrast, in figure 6*c*, the asymptotic dynamics exhibits a continuous transition between a non-impacting periodic trajectory and an impacting chaotic attractor. As *A* is increased above *A*^{*}, the size of the chaotic attractor grows continuously from the grazing contact point, from which an unstable impacting periodic trajectory also emanates. Under further increases in *A*, the unstable impacting periodic trajectory becomes stable in a period-doubling bifurcation at *A*^{pd}. As a local attractor exists for all *A* on some neighbourhood of *A*^{*}, this is an example of a continuous grazing bifurcation.

From a practical point of view, understanding the nature of the transition between non-impacting and impacting dynamics is an important ingredient in the proper design of an impact microactuator. For example, when the grazing bifurcation results in a discontinuous jump to a periodic high-impact-velocity impacting attractor, it is possible to initiate fast displacement manipulation with a small increase of the voltage amplitude. On the other hand, as this type of discontinuous grazing bifurcation is associated with parameter hysteresis, a larger decrease of the voltage amplitude is necessary to return to non-impacting motions. Moreover, the impacting attractor is associated with a characteristic, non-zero sliding distance per period thus limiting the displacement resolution. In contrast, a local impacting attractor, emanating from a continuous grazing bifurcation, exhibits a very high resolution since the impact velocity reduces to zero at the grazing contact.

### (d) Grazing in tapping-mode atomic force microscopy

The sudden, unanticipated disappearance of a local attractor near the grazing trajectory in favour of a high-impact-velocity motion is the characteristic of rigid contact models. Instead, if a stiff, yet compliant response results from the collisional interaction, the effects of the discontinuity on the persistence and stability of low-impact-velocity motions are delayed, but still significant.

In contrast with the rigid contact model employed in describing the bump stop and MEMS actuator dynamics, the models of the tip–surface interactions considered earlier do not exclude the possibility of the cantilever tip penetrating into the repelling region. The very stiff response that results from contact, however, results in a rapid reversal of the direction of motion and subsequent exiting of the tip from the repelling region and cessation of contact. Indeed, entry into the repelling region effectively corresponds to interactions with a nonlinear hardening spring. It follows that tapping-mode oscillations that achieve contact with sufficiently high contact velocity will experience the response of the structure in much the same way as a rigid impact. Discontinuity-induced bifurcations associated with rigid contact are thus also expected to occur in the case of compliant contact, albeit only after a sufficiently large contact velocity has been attained.

Consider, for example, the model of the tip–surface force interactions given by equation (2.3). For a given excitation frequency of the dither piezo, there exists a critical excitation amplitude *A*^{*}, such that the cantilever tip exhibits a non-contact grazing oscillation, for which the minimum tip–surface separation corresponds to *ζ*=*a*. We again refer to a system attractor for nearby amplitudes *A*≈*A*^{*} as local, if its deviation from the grazing oscillation for nearby amplitudes goes to zero as *A*→*A*^{*} and non-local otherwise. In light of the above discussion, we consider the discontinuity-induced bifurcation scenario that follows changes in excitation amplitude away from *A*^{*} to be continuous, if a local attractor can be found for values of *A* in a (1) neighbourhood of *A*^{*} and discontinuous otherwise. The discontinuity-induced saddle–node bifurcations shown in each of the panels of figure 3 are thus examples of discontinuous bifurcation scenarios.

### (e) Theoretical stability analysis

The stability characteristics of periodic oscillations in periodically forced finite-dimensional *smooth* dynamical systems may be characterized by a finite sequence of (possibly complex) numbers known as *Floquet multipliers*, corresponding to the eigenvalues of a suitably constructed sensitivity matrix. Indeed, as long as all Floquet multipliers lie within the unit circle in the complex plane, sufficiently small deviations away from the reference oscillation are guaranteed to decay to zero after some initial transient. In contrast, in the case that at least one multiplier is found outside the unit circle, the reference oscillation is unstable and sensitive to the growth of perturbations.

In piecewise smooth dynamical systems where the force field is continuous but has a discontinuous gradient, it is again possible to characterize the stability of periodic oscillations using the Floquet multipliers, even when the periodic oscillation crosses the discontinuity. Indeed, the analysis originally developed by Dankowicz & Nordmark (2000) and recently revisited in the context of atomic force microscopy by Zhao & Dankowicz (2006*a*) yields series expansions for the corresponding sensitivity matrix in terms of the deviation in system parameters away from the critical value corresponding to grazing.

Consider again the model of the tip–surface force interactions given by equation (2.3). Denote by *P*^{*} the sensitivity matrix for the grazing oscillation obtained using only the attracting force field. Suppose that as *A* increases past *A*^{*}, the minimum tip–surface separation decreases past *a*. Then, for Δ*A*=*A*−*A*^{*}≳0, to the lowest non-trivial order, the sensitivity matrix for the corresponding intermittent-contact oscillations takes the form(3.1)where *G*^{*} is a matrix that can be computed entirely in terms of properties of the force field at *ζ*=*a* and the excitation at the moment of grazing. The square-root dependence on the deviation in excitation amplitude shows that the rate of change of the sensitivity matrix and correspondingly of the Floquet multipliers in the limit *A*→*A*^{*} is infinite. Although not infinite, rapid variations in the Floquet multipliers would also be expected for a smooth force field near a point of rapid variation in the force gradient. Figure 7 shows the variations in the real part of the largest-in-magnitude Floquet multiplier, *λ*_{max}, of the periodic oscillations found along the branch of non-contact-mode oscillation that connects with that of the unstable intermittent-contact oscillations at the saddle–node bifurcation shown in figure 3 for each of the three models of the near-field force interactions. Although the multipliers vary continuously with location along these two branches, their rate of change clearly becomes very large on a small neighbourhood of the point of maximum net attraction.

## 4. Future directions

In this paper, the presence of discontinuities or rapid variations in the gradient of the force interactions between an atomic force microscope cantilever tip and a nanoscale surface have been shown to govern the transition between non-contact- and intermittent-contact-mode oscillations. Rather than eliminating system discontinuities by suitable smoothing as suggested by the model in Rützel *et al*. (2003), the above results advocate retaining such discontinuities as they provide an organizing centre for analysing the resultant transition. In particular, the discontinuous abstraction enables one to formulate and numerically compute quantitative descriptors of the system response and its stability characteristics near the critical excitation amplitude. In contrast, a smooth analysis of a smoothed, but rapidly varying, force field would at best be able to predict slow variations in the stability characteristics and thus, necessarily, only have limited range of applicability.

The discontinuous bifurcation scenarios shown in figure 3 are associated with saddle–node bifurcations as one Floquet multiplier exits the unit circle through 1. In contrast, a continuous scenario could result if the Floquet multipliers remained within the unit circle or if they exited the unit circle through other points on the unit circle. As the sensitivity matrix is given in closed form above, the determination of the type of bifurcation scenario that will result from grazing can be achieved without the need to locate the near-grazing intermittent-contact oscillations.

In all the three examples of discontinuity-induced bifurcations, discontinuous bifurcation scenarios were associated with potentially detrimental changes in system response. In the case of the vehicle suspension and the microelectromechanical actuator, the onset of high-impact-velocity oscillations implied reduced ride comfort or reduced positioning precision. Similarly, in the case of dynamic atomic force microscopy, the transition from non-contact mode to intermittent-contact oscillations with non-zero duration of contact increases the risk for damage to the specimen. Efforts to suppress such transitions, for example by designing the device to ensure continuous bifurcation scenarios or through the deliberate imposition of additional control, would thus appear to be highly desirable. The techniques discussed here provide a dynamical framework for formulating such control strategies.

## Acknowledgments

This material is based upon work supported by the National Science Foundation under grant nos. 0237370 and 0510044. The author gratefully acknowledges Mark Paul and Craig Prater for valuable insights into nanoscale characterization and AFM modelling, and Xiaopeng Zhao and Sambit Misra for their contributions.

## Footnotes

One contribution of 23 to a Triennial Issue ‘Mathematics and physics’.

- © 2006 The Royal Society