Chaos: The speed limiting phenomenon in dynamic atomic force microscopy

Abstract

This paper investigates the closed-loop dynamics of the Tapping Mode Atomic Force Microscopy using a new mathematical model based on the averaging method in Cartesian coordinates. Experimental and numerical observations show that the emergence of chaos in conventional tapping mode AFM strictly limits the imaging speed. We show that, if the controller of AFM is tuned to be faster than a certain threshold, the closed-loop system exhibits a chaotic behavior. The presence of chaos in the closed-loop dynamics is confirmed via bifurcation diagrams, Poincar e sections, and Lyapunov exponents. Unlike the previously detected chaos due to attractive forces in the AFM, which can be circumvented via simple changes in operation parameters, this newly identified chaos is seemingly inevitable and imposes an upper limit for the closed-loop bandwidth of the AFM.

A comprehensive model for transient behavior of tapping mode atomic force microscope

“…The first subsection explains the effect of cantilever dynamics on image quality based on closed-loop nonlinear simulation. The second subsection graphically explains the reason behind the chaotic behavior of TM-AFM as a consequence of high control gains as previously reported in [18].…”

“…The results show the practical implications of the transient behavior of the TM-AFM on the final image quality. Also the origin for previously reported chaotic behavior of the TM-AFM [18] can be explained.…”

“…It has been reported that if the controller is tuned to be faster than a certain threshold, the closed-loop system shows a chaotic behavior [18]. Although the presence of chaos was confirmed by studying the Poincaré sections and Lyapunov exponents, the origin of the chaos was not explained.…”

“…Note that, the transient response of the cantilever is not taken into account in control design for AFM. Hence, such a transient behavior of the cantilever may cause a closed-loop chaotic behavior in presences of nonlinear tip-sample interactions [18]. It has been previously reported that this chaotic behavior strictly limits the speed of the TM-AFM and can only be avoided by reducing the control gains and imaging speed [18].…”

Robustness of attractors in tapping mode atomic force microscopy

“…Moreover, period-doubling cascades are present in both stable period-2 solutions arising from Su 1 HP 1 and HP 1 branches. This period-doubling cascade can lead to chaos in a similar fashion as encountered in standard TM-AFM systems [26].…”