Multi-modal analysis on the intermittent contact dynamics of atomic force microscope

Abstract

A multi-modal analysis on the intermittent contact between an atomic force microscope (AFM) with a soft sample is presented. The intermittent contact induces the participation of the higher modes into the motion and various subharmonic motions are shown. The AFM tip mass enhances the coupling of different modes. The AFM tip mass is modeled by the Dirac delta function and the coupling effects are analyzed via the Galerkin method. The necessity of applying multi-modal analysis to the intermittent contact problem is demonstrated. Unlike the impact oscillator model which assumes the impact/contact time is infinitesimal, the contact time can be a significant fractional portion in each cycle, especially for the soft sample case and thus results in different dynamic behavior from that of an impact oscillator.

Introduction

Atomic force microscope (AFM) has been a very useful tool in many fields since its inception in 1986 [1]. One of the AFM operatingmodes is called vibrating or tapping mode [2], [3], which can reduce shear [4] and it can thus be used for soft materials to prevent or reduce the damage. For example, AFM is used to measure the cell wall nanomechanical motion of the living Saccharomyces cerevisiae (baker's yeast) [5]. The intermittent contact between the AFM tip and sample often occurs in the tapping mode. Since Burham et al. [6] first linked the AFM intermittent contact dynamics to an impact oscillator model developed by Pippard for a pin hitting a loudspeaker [7], many theories/models have been developed [3], [4], [8], [9], [10], [11], [12], [13]. In those studies [3], [4], [8], [9], [10], [11], [12], [13], either one degree-of-freedom (DOF) model of a spring–mass system as shown in Fig. 1 or the single mode analysis is applied to the AFM cantilever of a continuous system. One DOF model or single mode analysis can be accurate in some special cases. For example, an AFM vibrates with a driving frequency lower than or around the first natural frequency of the AFM cantilever and has no contact with the sample [14]. However, the driving frequency can be much higher than the first natural frequency of AFM cantilever [6], [15] and the higher modes can thus participate in the motion. In the AFM imaging application, the phase shift of higher mode has been used to characterize the mechanical, electrical and magnetic interactions [15]. We demonstrate that the results computed by the single mode analysis can be very different from those computed by multimodes when the driving frequency is higher than the AFM cantilever first natural frequency. Even when the driving frequency is less than the AFM first natural frequency, the higher modes can still participate in the motion due to the repulsive tip–sample interaction of impact [16]. The repulsive tip–sample interaction has much shorter acting range and time than those of the attractive ones such as van der Waals (vdW) and Casimir forces. The repulsive tip–sample interaction acts as an impulse to excite the higher modes[16]. The total harmonic distortion (THD) is used to measure the fraction of power transferred from the fundamental into the higher harmonics and THD ranges from 0 to 100 percent. THD with the maximum value of 15 percent is observed when an AFM is driven at the first natural frequency [16]. The higher modes participation in the motion has been fully recognized in the vibration of scanning acoustic microscope, which is also a cantilever structure and has stronger tip–sample interaction [17]. Attard et al. [13] noticed that the tip mass can significantly contribute to the whole system inertia. The concentrated mass not only affects the system natural frequencies but also couples the (orthogonal) modes as reflected by the Dirac delta function used to model the concentrated mass even for the linear small vibration case [18], [19]. Furthermore, the geometrical nonlinearity of large vibration amplitude can also couple the modes [20]. The multi-modal analysis rather than one DOF model or single mode analysis should be the general approach for the study of the AFM intermittent contact dynamics. The reasons for this can be summarized as the following three: (1) that the AFM driving frequency can be higher than the first natural frequency; (2) that the repulsive tip–sample interaction can induce the motion of higher modes; (3) mode coupling due to the concentrated tip mass or the geometrical nonlinearity of large motion.

When the stiffness K₁ of the vibro-impact system as shown in Fig. 1a approaches infinity or a mass–spring system hits an infinitely hard substrate as shown in Fig. 1b, the system becomes an impact oscillator [21]. The impact of an impact oscillator is instantaneous and the instantaneous reflection velocity is only related to the instantaneous velocity just before the impact via the coefficient of restitution [3], [21]. The coefficient of restitution is also responsible for the major energy loss in an impact oscillator system. The challenges of applying the impact oscillator model to the AFM intermittent contact dynamics are recognized and summarized as the following two questions by van de Water and Molenaar [3]: Is the energy loss concentrated at the impact instance and is the impact instantaneous? The contact time is dependent on the driving frequency/amplitude, the bending and contact stiffnesses of AFM and surface, etc. [22]. The interaction/contact time of the tapping-mode AFM and sample surface in general is a considerable fraction of the cycle time [16]. Besides the coefficient of restitution, the hysteresis due to the adhesive forces can also cause the energy dissipation [9]. van de Water and Molenaar [3] demonstrated that for the grazing impact, the tapping-mode AFM displays the characteristic features of an impact oscillator. However, the grazing impact is a particular impact case with zero impact velocity [22], [23]. The AFM studied by van de Water and Molenaar interacts with the rigid surface through a liquid bridge, which offers a mechanism to realize the grazing impact [3]. However, the zero velocity impact is not general in the AFM intermittent contact dynamics. In this study the contact between an AFM tip and a surface is modeled as the AFM hitting a linear spring and a damper. The relatively small spring stiffness is chosen to model the soft sample and the contact time can thus be a significant fraction of the period. We demonstrate that the bifurcation diagram and phase portraits of the system are very different from those of an impact oscillator. The more general criterion of using the AFM tip displacement [8], [12] to tell whether the contact occurs is also adopted in this study.

The subharmonics and chaos are the two characteristic features in the intermittent contact dynamics of both the macroscopic structure [7], [21], [24], [25], [26], [27], [28], [29] and the microstructure (of AFM) [3], [6], [9]. Measuring the AFM frequency shift has been demonstrated as an effective method to determine the tip–sample potential [30], [31]. However, in those frequency shift theories/models there is an implicit assumption that the AFM subharmonic motion does not occur. When the AFM tip–sample distance is in the order of interatomic spacing [31], the very strong tip–sample interaction due to Lennard-Jones potential can induce not only the aforementioned higher modes participation in motion but also the subharmonic motion, which can cause great trouble in the frequency shift theories/models. For a microstructure, the contribution of the adhesion, vdW and Casimir forces [8], [14], [32] can be significant to the system dynamics and stability. For an AFM in the noncontact tapping mode, the vdW force is responsible for the system frequency shift and softening effect [14]. However, for the AFM intermittent contact in tapping mode, the impact/contact itself is of the primary importance to the AFM dynamic behavior [3], [9]. The nonlinearity of impact/contact forces is of the secondary importance. Although adhesion induces nonlinear contact behavior, the adhesion influence is rather weak as compared with an elastic one [33]. The adhesion influence only stands out when the external mechanical load is very small [33] and the elastic force due to impact is dominant in the AFM intermittent dynamics. Because the long-range attractive vdW force does not contribute to the energy loss of the tapping-mode AFM, its influence on the AFM dynamics is ignored [3], [9]. This study is to investigate the AFM intermittent dynamics with more than one mode participation and the interaction with a relatively soft sample. The secondary influences of adhesion, vdW and Casimir forces are thus not included.