Forced oscillations with continuum models of atomic force microscopy (original) (raw)
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Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory
Nanoscale Systems: Mathematical Modeling, Theory and Applications, 2013
A matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous boundary conditions. Transients are identified from intial values of permanent responses. Eigenanalysis for determining frequencies and matrix mode shapes is done with the use of a fundamental matrix response that characterizes solutions of a damped second-order matrix differential equation. It is observed that surface effects are influential for the natural frequency at the nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantilever beam actuate...
Physical Review B, 2004
The dynamics of the microcantilever in atomic force microscopy ͑AFM͒ is represented by a multipledegrees-of-freedom state-space model and is discussed within the framework of system theory. The cantilever dynamics is modeled as a linear time-invariant system with a nonlinear output feedback due to the tip-sample interaction. This allows one to use the same model to analyze different aspects of atomic force microscopy such as the dynamics of contact-mode or the dynamics of tapping-mode AFM. The state-space approach to the dynamic response of the AFM cantilever allows for numerically efficient simulations. We show that not only the eigenfrequency but also the modal damping of a cantilever interacting with a surface strongly depends on the contact stiffness. This is important for a quantitative characterization of elastic sample properties. Additionally, our model shows the presence of higher harmonics in tapping-mode AFM. The excitation of higher eigenmodes can strongly distort the system response. The results illustrate that higher eigenmodes have to be considered in the analysis of dynamic AFM.
Multimode and multitone analysis of the dynamic mode operation of the Atomic Force Microscope
2013 American Control Conference, 2013
This article investigates the multimode model of the cantilever beam during probe based imaging. It develops a framework to quantify the effects of different material properties like dissipativity and stiffness in a near tapping mode operation of Atomic Force Microscope (AFM), which is the primary mode of imaging soft matter, when excitation consists of more than one sinusoids. Averaging theory forms an important basis and provides the theoretical foundations. Effect of dissipative and stiffness properties of the sample on the forces experienced by probe is modeled as changes in parameters of an equivalent linear time invariant model, of the cantileversample system. It is shown that this model can be extended to the case when multiple modes of the cantilever participate in the nonlinear interaction with the sample forces.
Coupled molecular and cantilever dynamics model for frequency-modulated atomic force microscopy
Beilstein journal of nanotechnology, 2016
A molecular dynamics model is presented, which adds harmonic potentials to the atomic interactions to mimic the elastic properties of an AFM cantilever. It gives new insight into the correlation between the experimentally monitored frequency shift and cantilever damping due to the interaction between tip atoms and scanned surface. Applying the model to ionic crystals with rock salt structure two damping mechanisms are investigated, which occur separately or simultaneously depending on the tip position. These mechanisms are adhesion hysteresis on the one hand and lateral excitations of the cantilever on the other. We find that the short range Lennard-Jones part of the atomic interaction alone is sufficient for changing the predominant mechanism. When the long range ionic interaction is switched off, the two damping mechanisms occur with a completely different pattern, which is explained by the energy landscape for the apex atom of the tip. In this case the adhesion hysteresis is alwa...
Nonlinear dynamics of atomic force microscopy with intermittent contact
Chaos, Solitons & Fractals, 2007
When the atomic force microscopy (AFM) in tapping mode is in intermittent contact with a soft substrate, the contact time can be a significant portion of a cycle, resulting in invalidity of the impact oscillator model, where the contact time is assumed to be infinitely small. Furthermore, we demonstrate that the AFM intermittent contact with soft substrate can induce the motion of higher modes in the AFM dynamic response. Traditional ways of modeling AFM (one degree of freedom (DOF) system or single mode analysis) are shown to have serious mistakes when applied to this kind of problem. A more reasonable displacement criterion on contact is proposed, where the contact time is a function of the mechanical properties of AFM and substrate, driving frequencies/amplitude, initial conditions, etc. Multi-modal analysis is presented and mode coupling is also shown.
Simulations of switching vibrating cantilever in atomic force microscopy
Applied Surface Science, 2003
We analyze the steady state tip sample interaction in atomic force microscopy by using an electrical circuit simulator. The phase shift between the cantilever excitation and tip, and the amplitude versus distance curves are obtained with sample stiffness as a parameter. The height shifts and hysteresis in amplitude and phase curves are observed as a result of the in¯uence of the force between the tip and the sample. The damping and switching mechanisms are explained using the force traces obtained from simulations. The oscillation amplitude dependence of operating mode is inspected. We ®nd that improper selection of the free tip oscillation amplitude is the cause of operating state transitions. #
Microsystem Technologies-micro-and Nanosystems-information Storage and Processing Systems, 2018
The aim of this investigation is to study the motion of an elastically restrained beam used in tapping mode atomic force microscopy (TM-AFM), which is to be utilized in manufacturing at nano-scale. TM-AFM uses high frequency oscillations to remove material or shape nano structures. Euler-Bernoulli theory and Eringen's theory of non-local continuum are used to model the nano machining structure composed of a nanobeam and a single degree of freedom, spring-mass system. The system is modelled as a beam with a torsional spring boundary condition that is rigidly restrained in the transverse direction at one end; and at the free end is a transverse linear spring attached to the tip. The other end of the spring is attached to a mass, resulting in a single degree of freedom spring-mass system. When the linear spring constant is infinite, the free end behaves as a beam with a concentrated tip mass. When the mass is infinite, the boundary condition is that of a linear spring. When the tip mass is zero, the configuration is that of a torsionally restrained-free beam. The motion of the tip of the beam and tip mass can be investigated to observe the tip frequency response, displacement and force. The tip displacement frequency contains information about the maximum displacement amplitude and therefore the sample penetration depth.
A new model for investigating the flexural vibration of an atomic force microscope cantilever
Ultramicroscopy, 2010
A new model for the flexural vibration of an atomic force microscope cantilever is proposed, and a closed-form expression is derived. The effects of angle, damping and tip moment of inertia on the resonant frequency were analysed. Because the tip is not exactly located at one end of the cantilever, the cantilever is modelled as two beams. The results show that the frequency first increases with increase in angle and then decreases to a constant value for high values of the angle. Moreover, the damping is increased at lower contact positions. The tip moment of inertia is also sensitive to the resonant frequency at small values for the odd modes and large values for the even modes.
Frequency Analysis and Identification in Atomic Force Microscopy
The class of models for tip-sample interaction dynamics in atomic force microscopy (AFM) via impact dynamics are proposed. For the analysis of oscillatory behaviors authors use a hysteresis law combined with harmonic balance (HB, see the monograph [H. K. Khalil, Nonlinear systems. NJ: Prentice Hall (2002; Zbl 1003.34002)]) techniques. In particular, the presence of jump phenomena discovered in many experiments is well-predicted and explained. Taking into account the fact that the linear part of the considered Luré system has a significant filtering effect, the authors have designed the special method based on a first order harmonic approximation which provided good quantitative results. In such a situation, the HB technique has advantages over standard numerical approaches because it requires a computational effort much smaller than the one required by a pure simulation of the same model. The paper is organized as follows. In Section 2 the AFM model class is described; in Section 3 ...
Enhancing capabilities of Atomic Force Microscopy by tip motion harmonics analysis
Bulletin of the Polish Academy of Sciences: Technical Sciences, 2013
Motion of a tip used in an atomic force microscope can be described by the Lennard-Jones potential, approximated by the van der Waals force in a long-range interaction. Here we present a general framework of approximation of the tip motion by adding three terms of Taylor series what results in non-zero harmonics in an output signal. We have worked out a measurement system which allows recording of an excitation tip signal and its non-linear response. The first studies of spectrum showed that presence of the second and the third harmonics in cantilever vibrations may be observed and used as a new method of the investigated samples characterization.