Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy
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Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy Giuseppe Rega · Valeria Settimi
Received: 31 August 2012 / Accepted: 9 January 2013 / Published online: 31 January 2013 © Springer Science+Business Media Dordrecht 2013
Abstract The nonlinear dynamical behavior of a single-mode model of noncontact AFM is analyzed in terms of attractors robustness and basins integrity. The model considered for the analyses, proposed in (Hornstein and Gottlieb in Nonlinear Dyn. 54:93–122, 2008), consistently includes the nonlinear atomic interaction and is studied under scan excitation (which appears as parametric excitation) and vertical excitation (which is prevalently external). Local bifurcation analyses are carried out to identify the overall stability boundary in the excitation parameter space as the envelope of system local escapes, to be compared with the one obtained via numerical simulations. The dynamical integrity of periodic bounded solutions is studied, and basin erosion is evaluated by means of two different integrity measures. The obtained erosion profiles allow us to dwell on the possible lack of homogeneous safety of the stability boundary in terms of robustness of the attractors, and to identify practical escape thresholds ensuring an a priori design safety target. Keywords Noncontact AFM · Bifurcation diagrams · Response charts · Basin erosion · Escape · Dynamical integrity · Design safety target G. Rega () · V. Settimi Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza”, 00197 Rome, Italy e-mail: [email protected]
1 Introduction 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 [1]. 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 [2–4]. Thus, referring to a proper mechanical model for the analysis of strongly nonlinear dynamics of these devices, in different operational modalities, is crucial at both the design and service phase. The most common mathematical models used in the literature are the lumped-mass spring system and the continuous beam [5]. The first one reduces the microcantilever to an equivalent linear spring and incorporates the nonlinear force derived from an effective interaction potential between the tip and the sample [6, 7], the other describes the continuum problem by accounting for the interaction force as a nonlinear boundary condition [8, 9] or a localized nonlinear field force [10]. As far as the AFM dynamics is concerned, the most common operation modes are th
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