Noncontact Atomic Force Microscopy - Yale School of Engineering ...

P.I-30
Internal Resonances and Spatio-Temporal Instabilities in Nonlinear
Multi-mode NC-AFM Dynamics
Oded Gottlieb, Sharon Hornstein, Wei Wu and Arthur Shavit
Department of Mechanical Engineering, Technion - Israel Institute of Technology, Haifa, Israel
The use of multi-mode excitation in atomic force microscopy has been proposed and
validated in the past decade. Examples include the combined excitation of the first two
bending modes where the latter has been found to be sensitive to low surface force
variations [1], and excitation of a third bending mode in the vicinity of a torsion mode to
map nanomechanical changes of a polymer near its glass transition [2]. While small
amplitude excitation results in a single-valued frequency response, finite amplitude
dynamics can yield nonlinear coexisting bi-stable solutions and complex aperiodic
dynamics that reveal nonstationary energy transfer between multiple spatial modes. The
accuracy of force estimation from measured data crucially depends on the quality of the
mathematical model in use. Thus, as lumped mass models cannot describe multi-mode
dynamics, a continuum mechanics description is required to consistently incorporate
nonlinear atomic interaction and coupled elastic and internal damping forces that govern
noncontact operation in ultrahigh vacuum. Thus, the objectives of this paper include
theoretical derivation and analysis of a continuum spatio-temporal model for the
vibrating NC-AFM cantilever that consistently incorporates both nonlinear atomic
interaction and coupled thermo-visco-elastic damping. We derive a nonlinear initialboundary-value
problem using the extended Hamilton's principle [3] for the coupled
viscoelastic and temperature fields for torsion, transverse and out-of-plane bending. The
continuum system is then reduced via a Galerkin procedure to a multi-mode dynamical
system which is analyzed numerically. System response reveals that the dynamic jumpto-contact
bifurcation threshold is not sensitive to the damping level but includes lengthy
chaotic transients for low damping. Furthermore, both subharmonic and quasiperioic
solutions are found when combination and internal resonances are excited. The latter
reveal energy transfer between the 3rd and 2ond microbeam modes due to a strong 3:1
internal resonance [Fig.1] and complex chaotic like spatio-temporal instabilities when
this internal resonance is coupled with either its out-of-plane or torsion resonances.
Figure 1: Quasiperiodic dynamics of a 3:1 internal resonance between the 3 rd and 2 nd bending
modes in UHV: time series (left), power spectra(center), and Poincare' map (right).
[1] T.R. Rodriguez and R. Garcia, APL, 83, 449 (2004).
[2] O. Sahin et al. Nature Nanotechnology 2, 507 (2007).
[3] S. Hornstein and O. Gottlieb, Nonlinear Dynamics 54, 93 (2008).
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