Mathematical model and atomic force microscopy measurements of adhesion between graphite particles and rough walls

In qPlus atomic force microscopy the tip length can in principle approach the length of the cantilever. We present a detailed mathematical model of the effects this has on the dynamic properties of the qPlus sensor. The resulting, experimentally confirmed motion of the tip apex is shown to have a large lateral component, raising interesting questions for both calibration and force-spectroscopy measurements.

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Feedback-induced instability in tapping mode atomic force microscopy: theory and experiment

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2010.0451 ◽

2010 ◽

Vol 467 (2130) ◽

pp. 1801-1822 ◽

Cited By ~ 13

Author(s):

O. Payton ◽

A. R. Champneys ◽

M. E. Homer ◽

L. Picco ◽

M. J. Miles

Keyword(s):

Mathematical Model ◽

Atomic Force Microscopy ◽

Driving Frequency ◽

Integral Control ◽

Tapping Mode ◽

Force Microscopy ◽

Atomic Force ◽

Flat Sample ◽

Realistic Representation ◽

Mode Atomic Force Microscopy

We investigate a mathematical model of tapping mode atomic force microscopy (AFM), which includes surface interaction via both van der Waals and meniscus forces. We also take particular care to include a realistic representation of the integral control inherent to the real microscope. Varying driving amplitude, amplitude setpoint and driving frequency independently shows that the model can capture the qualitative features observed in AFM experiments on a flat sample and a calibration grid. In particular, the model predicts the onset of an instability, even on a flat sample, in which a large-amplitude beating-type motion is observed. Experimental results confirm this onset and also confirm the qualitative features of the dynamics suggested by the simulations. The simulations also suggest the mechanism behind the beating effect; that the control loop over-compensates for sufficiently high gains. The mathematical model is also used to offer recommendations on the effective use of AFMs in order to avoid unwanted artefacts.

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Numerical Exploratory Analysis of Dynamics and Control of an Atomic Force Microscopy in Tapping Mode with Fractional Order

Shock and Vibration ◽

10.1155/2020/4048307 ◽

2020 ◽

Vol 2020 ◽

pp. 1-18

Author(s):

Mauricio A. Ribeiro ◽

Jose M. Balthazar ◽

Wagner B. Lenz ◽

Rodrigo T. Rocha ◽

Angelo M. Tusset

Keyword(s):

Mathematical Model ◽

Atomic Force Microscopy ◽

Feedback Control ◽

Fractional Order ◽

Delayed Feedback Control ◽

Linear Feedback ◽

Tapping Mode ◽

Force Microscopy ◽

Control Techniques ◽

Atomic Force

In this paper, we investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) forces. The dynamic behavior of the system is analyzed through a nonlinear dimensionless mathematical model. Numerical tools as Poincaré maps, Lyapunov exponents, and bifurcation diagrams are accounted for the analysis of the system. With that, the regions in which the system presents chaotic and periodic behaviors are obtained and investigated. Moreover, the fractional calculus is introduced into the mathematical model, employing the Riemann-Liouville kernel discretization in the viscoelastic term of the system. The 0-1 test is implemented to analyze the new dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional (integer derivative) and fractional models reveal the need for the application of control techniques such as Optimum Linear Feedback Control (OLFC), State-Dependent Riccati Equations (SDRE) by using feedback control, and the Time-Delayed Feedback Control. The results of the control techniques are efficient with and without the fractional-order derivative.

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Error Sources in Atomic Force Microscopy for Dimensional Measurements: Taxonomy and Modeling

Journal of Manufacturing Science and Engineering ◽

10.1115/1.4001242 ◽

2010 ◽

Vol 132 (3) ◽

Cited By ~ 16

Author(s):

F. Marinello ◽

S. Carmignato ◽

A. Voltan ◽

E. Savio ◽

L. De Chiffre

Keyword(s):

Mathematical Model ◽

Atomic Force Microscopy ◽

Scanning System ◽

Thermal Drift ◽

Scaling Effects ◽

Error Sources ◽

Force Microscopy ◽

Atomic Force ◽

Dimensional Measurements ◽

Squareness Errors

This paper aimed at identifying the error sources that occur in dimensional measurements performed using atomic force microscopy. In particular, a set of characterization techniques for errors quantification is presented. The discussion on error sources is organized in four main categories: scanning system, tip-surface interaction, environment, and data processing. The discussed errors include scaling effects, squareness errors, hysteresis, creep, tip convolution, and thermal drift. A mathematical model of the measurement system is eventually described, as a reference basis for errors characterization, with an applicative example on a reference silicon grating.

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