Nonlinear dynamics of atomic force microscopy

Atomic Force Microscopes (AFMs) are powerful devices used for surface analysis in nano-electronics, mechanics of materials and biotechnology, as they permit to topologically characterize surfaces up to micro and nano resolution levels. In a typical AFM, the topography is imaged by scanning a sharp tip, fixed to the free end of a microcantilever vertically bending over the sample surface, and by measuring the tip deflection through a laser technology. The tip-sample interaction modifies the beam dynamics and allows not only to image surfaces, but also to measure some physical properties of the sample. The most common operation modes of AFMs are the noncontact mode, in which there is absence of contact between the tip and the sample, and their interaction is governed by a solely attractive potential, and the tapping mode, in which the tip operates in both attractive and repulsive force regions and touches the surface only for short time intervals. In this lecture, both operation modes are addressed via a continuous beam model, with the interaction force being accounted for as a localized nonlinear field force. However, different continuous and Galerkin-reduced models are considered, and various aspects of nonlinear dynamics, typical of the two different AFM systems, are investigated and discussed in a somehow complementary way. In the noncontact AFM, whose cantilever tip has to maintain a design gap from the sample to ensure that the beam elastic restoring force is stronger than the atomic attraction, interest is mostly towards investigating conditions for possible occurrence of the unwanted phenomenon of jump to contact (escape), in phenomenological (dynamical) terms. To this aim, attention is systematically devoted to the strongly nonlinear dynamics of the system under beam vertical or scan horizontal excitation, which realize conditions of external or parametric resonance, respectively, by considering the relevant primary (fundamental) and subharmonic (principal) resonances of the dominant beam mode. Given the width of the bifurcation analyses, a minimal-order (single-mode) model of beam is considered, yet with the underlying continuous model consistently incorporating geometric nonlinearities, nonlinear atomic interaction, and generalized forces describing motion control of the microcantilever. In this way, a general platform to possibly conduct successive, more refined, multimodal investigations also accounting for the nonlinear coupling effects is realized. The nonlinear dynamic behavior of the single-mode model is analyzed in terms of attractors robustness and basins integrity. Local bifurcation analyses are carried out to identify the overall stability boundary in the excitation parameter space as the envelope of system local escapes. The dynamic integrity of periodic bounded solutions is studied, and basin erosion is evaluated by means of different integrity measures. The ensuing erosion profiles allow us to dwell on the possible lack of homogeneous safety of the stability boundary in terms of attractors robustness, and to identify practical escape thresholds ensuring an a priori design safety target. In contrast, in the tapping mode AFM, the interest is devoted to highlighting the effect of also higher order eigenmodes, with the relevant damping ratios, on the overall system dynamics, which is indeed addressed via a multimode approximation allowing to consider external excitation at primary or secondary resonance of different modes. In this case, a simple linear beam model is considered, with the dynamic response being investigated via numerical simulations of up to a three-mode reduced model, which appears indeed sufficient to catch the main nonlinear dynamic phenomena. Different bifurcation parameters are considered, namely the excitation frequency and the approach/retract separation between cantilever and sample. Typical features of tapping mode AFM response as nonlinear hysteresis, bistability, higher harmonics contribution, impact velocity and contact force are addressed. The analysis is conducted by evaluating damping of higher modes according to the Rayleigh criterion, which basically accounts for structural damping representative of the behavior of AFMs in air. However, nominal damping situations more typical of AFMs in liquids are also investigated, by considering sets of modal Q-factors with different patterns and ranges of values, and comparing the relevant responses. Variable attractive-repulsive effects are highlighted, along with the possible presence of a coexisting multi-periodic orbit when the system is excited at second resonance. The importance of considering excitation of also the second mode to the aim of evaluating possibly harmful tapping effects on the sample is discussed. References [1] G. Rega, V. Settimi: Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy. Nonlinear Dynamics. (2013) DOI: 10.1007/s11071-013-0771-5. [2] U. Andreaus, L. Placidi, G Rega: Microcantilever dynamics in tapping mode atomic force microscopy via higher eigenmodes analysis. (2013) DOI: 10.1063/1.4808446.