Nonlinear Vibrations in Contact Atomic Force Microscopy

2003 ◽  
Author(s):  
Joseph A. Turner
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Contact Stiffness ◽  
Vibration Response ◽  
Hertzian Contact ◽  
Force Microscopy ◽  
Softening Behavior ◽  
Atomic Force ◽  
Frequency Relation ◽  
Frequency Curves

The nonlinear vibration response of an atomic force microscope cantilever in contact with a vibrating sample is investigated. The tip-sample contact is modeled using Hertzian contact mechanics. The method of multiple scales is used to analyze this problem in which it is assumed that the beam remains in contact with the moving surface at all times. The primary result from this analysis is the amplitude-frequency relation for the various flexural modes. The amplitude-frequency curves exhibit softening behavior as expected. The amount of softening is shown to depend on the linear contact stiffness as well as the specific mode. The modal sensitivity to nonlinearity is the result of the nonlinearity being restricted to a single position. The mode shape greatly affects the degree to which the nonlinearity influences the frequency response. The Hertzian restriction is then loosened slightly such that variations in nonlinear contact stiffness are examined. These results depend on the linear contact stiffness and mode number as well. The nonlinear vibration response is expected to provide new insight on the nonlinear tip mechanics present in these systems.

Contact stiffness of layered materials for ultrasonic atomic force microscopy

2000 ◽  
Vol 87 (10) ◽  
pp. 7491-7496 ◽  
Author(s):  
G. G. Yaralioglu ◽  
F. L. Degertekin ◽  
K. B. Crozier ◽  
C. F. Quate
Keyword(s):  
Atomic Force Microscopy ◽  
Contact Stiffness ◽  
Layered Materials ◽  
Force Microscopy ◽  
Atomic Force

scholarly journals Subsurface imaging of flexible circuits via contact resonance atomic force microscopy

2019 ◽  
Vol 10 ◽  
pp. 1636-1647 ◽  
Author(s):  
Wenting Wang ◽  
Chengfu Ma ◽  
Yuhang Chen ◽  
Lei Zheng ◽  
Huarong Liu ◽  
...  
Keyword(s):  
Atomic Force Microscopy ◽  
Theoretical Calculations ◽  
Contact Stiffness ◽  
Metal Layer ◽  
Experimental Results ◽  
Subsurface Imaging ◽  
Force Microscopy ◽  
Atomic Force ◽  
Cover Thickness ◽  
Contact Resonance

Subsurface imaging of Au circuit structures embedded in poly(methyl methacrylate) (PMMA) thin films with a cover thickness ranging from 52 to 653 nm was carried out by using contact resonance atomic force microscopy (CR-AFM). The mechanical difference of the embedded metal layer leads to an obvious CR-AFM frequency shift and therefore its unambiguous differentiation from the polymer matrix. The contact stiffness contrast, determined from the tracked frequency images, was employed for quantitative evaluation. The influence of various parameter settings and sample properties was systematically investigated by combining experimental results with theoretical analysis from finite element simulations. The results show that imaging with a softer cantilever and a lower eigenmode will improve the subsurface contrast. The experimental results and theoretical calculations provide a guide to optimizing parameter settings for the nondestructive diagnosis of flexible circuits. Defect detection of the embedded circuit pattern was also carried out, which indicates the capability of imaging tiny subsurface structures smaller than 100 nm by using CR-AFM.

Study of the sensitivity and resonant frequency of the flexural modes of an atomic force microscopy microcantilever modeled by strain gradient elasticity theory

2013 ◽  
Vol 228 (8) ◽  
pp. 1299-1310 ◽  
Author(s):  
Mohammad Abbasi ◽  
Ardeshir Karami Mohammadi
Keyword(s):  
Atomic Force Microscopy ◽  
Resonant Frequency ◽  
Strain Gradient ◽  
Contact Stiffness ◽  
Current Model ◽  
Gradient Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Force Microscopy ◽  
Atomic Force

In this study, the resonant frequency and sensitivity of an atomic force microscopy microcantilever are analyzed utilizing the strain gradient theory, and then the governing equation and boundary conditions are derived by a combination of the basic equations of the modified strain gradient theory and the Hamilton principle. Afterward, the resonant frequency and sensitivity of the proposed atomic force microscopy microcantilever are obtained numerically. The results of the current model are compared to those evaluated by both modified couple stress and classic beam theories. Results show that utilizing the strain gradient theory in the analysis of atomic force microscopy microcantilever dynamic behavior is necessary especially when the contact stiffness is high and the thickness of the microcantilever approaches the internal material length scale parameter.

scholarly journals Generalized Hertz model for bimodal nanomechanical mapping

2016 ◽  
Vol 7 ◽  
pp. 970-982 ◽  
Author(s):  
Aleksander Labuda ◽  
Marta Kocuń ◽  
Waiman Meinhold ◽  
Deron Walters ◽  
Roger Proksch
Keyword(s):  
Contact Mechanics ◽  
Small Amplitude ◽  
Sample Surface ◽  
Hertzian Contact ◽  
Modes Of Operation ◽  
Hertz Model ◽  
Force Microscopy ◽  
Atomic Force ◽  
Resonant Modes ◽  
Shape And Size

Bimodal atomic force microscopy uses a cantilever that is simultaneously driven at two of its eigenmodes (resonant modes). Parameters associated with both resonances can be measured and used to extract quantitative nanomechanical information about the sample surface. Driving the first eigenmode at a large amplitude and a higher eigenmode at a small amplitude simultaneously provides four independent observables that are sensitive to the tip–sample nanomechanical interaction parameters. To demonstrate this, a generalized theoretical framework for extracting nanomechanical sample properties from bimodal experiments is presented based on Hertzian contact mechanics. Three modes of operation for measuring cantilever parameters are considered: amplitude, phase, and frequency modulation. The experimental equivalence of all three modes is demonstrated on measurements of the second eigenmode parameters. The contact mechanics theory is then extended to power-law tip shape geometries, which is applied to analyze the experimental data and extract a shape and size of the tip interacting with a polystyrene surface.

scholarly journals Nonlinear vibration of crowned gear pairs considering the effect of Hertzian contact stiffness

2019 ◽  
Vol 1 (5) ◽  
Author(s):  
Moslem Molaie ◽  
Farhad S. Samani ◽  
Habibollah Motahar
Keyword(s):  
Nonlinear Vibration ◽  
Contact Stiffness ◽  
Hertzian Contact

scholarly journals Indentation-Induced Mechanical Deformation Behaviors of AlN Thin Films Deposited onc-Plane Sapphire

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Sheng-Rui Jian ◽  
Jenh-Yih Juang
Keyword(s):  
Thin Films ◽  
Contact Stiffness ◽  
Maximum Load ◽  
Cross Sectional ◽  
Continuous Contact ◽  
Force Microscopy ◽  
Atomic Force ◽  
Slip Bands ◽  
Transmission Electron ◽  
Deformation Behaviors

The mechanical properties and deformation behaviors of AlN thin films deposited onc-plane sapphire substrates by helicon sputtering method were determined using the Berkovich nanoindentation and cross-sectional transmission electron microscopy (XTEM). The load-displacement curves show the “pop-ins” phenomena during nanoindentation loading, indicative of the formation of slip bands caused by the propagation of dislocations. No evidence of nanoindentation-induced phase transformation or cracking patterns was observed up to the maximum load of 80 mN, from either XTEM or atomic force microscopy (AFM) of the mechanically deformed regions. Instead, XTEM revealed that the primary deformation mechanism in AlN thin films is via propagation of dislocations on both basal and pyramidal planes. Furthermore, the hardness and Young’s modulus of AlN thin films estimated using the continuous contact stiffness measurements (CSMs) mode provided with the nanoindenter are 16.2 GPa and 243.5 GPa, respectively.

Nonlinear vibration of rectangular atomic force microscope cantilevers by considering the Hertzian contact theory

2010 ◽  
Vol 88 (5) ◽  
pp. 333-348 ◽  
Author(s):  
Ali Sadeghi ◽  
Hassan Zohoor
Keyword(s):  
Atomic Force Microscope ◽  
Timoshenko Beam ◽  
Frequency Ratio ◽  
Contact Stiffness ◽  
Sample Surface ◽  
Hertzian Contact ◽  
Nonlinear Frequency ◽  
Lateral Contact ◽  
Atomic Force ◽  
Contact Position

The nonlinear flexural vibration for a rectangular atomic force microscope cantilever is investigated by using Timoshenko beam theory. In this paper, the normal and tangential tip–sample interaction forces are found from a Hertzian contact model and the effects of the contact position, normal and lateral contact stiffness, tip height, thickness of the beam, and the angle between the cantilever and the sample surface on the nonlinear frequency to linear frequency ratio are studied. The differential quadrature method is employed to solve the nonlinear differential equations of motion. The results show that softening behavior is seen for most cases and by increasing the normal contact stiffness, the frequency ratio increases for the first mode, but for the second mode, the situation is reversed. The nonlinear-frequency to linear-frequency ratio increases by increasing the Timoshenko beam parameter, but decreases by increasing the contact position for constant amplitude for the first and second modes. For the first mode, the frequency ratio decreases by increasing both of the lateral contact stiffness and the tip height, but increases by increasing the angle α between the cantilever and sample surface.

On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation

1992 ◽  
Vol 7 (3) ◽  
pp. 613-617 ◽  
Author(s):  
G.M. Pharr ◽  
W.C. Oliver ◽  
F.R. Brotzen
Keyword(s):  
Elastic Modulus ◽  
Contact Area ◽  
Contact Stiffness ◽  
Elastic Half Space ◽  
Elastic Contact ◽  
Simple Relationship ◽  
Depth Sensing ◽  
Force Microscopy ◽  
Atomic Force ◽  
The Relationship

Results of Sneddon's analysis for the elastic contact between a rigid, axisymmetric punch and an elastic half space are used to show that a simple relationship exists among the contact stiffness, the contact area, and the elastic modulus that is not dependent on the geometry of the punch. The generality of the relationship has important implications for the measurement of mechanical properties using load and depth sensing indentation techniques and in the measurement of small contact areas such as those encountered in atomic force microscopy.

A fresh insight into the non-linear vibration of double-tapered atomic force microscope cantilevers by considering the Hertzian contact theory

2010 ◽  
Vol 225 (1) ◽  
pp. 233-247 ◽  
Author(s):  
A Sadeghi ◽  
H Zohoor
Keyword(s):  
Atomic Force Microscope ◽  
Timoshenko Beam ◽  
Frequency Ratio ◽  
Contact Stiffness ◽  
Sample Surface ◽  
Hertzian Contact ◽  
Linear Frequency ◽  
Atomic Force ◽  
Non Linear ◽  
Contact Position

The non-linear flexural vibration for a double-tapered atomic force microscope cantilever has been investigated by using the Timoshenko beam theory. In this article, the normal and tangential tip—sample interaction forces are found from the Hertzian contact model, and the effects of the contact position, normal and lateral contact stiffness, height of the tip, thickness of the beam, angle between the cantilever and the sample surface, and breadth and height taper ratios on the non-linear frequency to linear frequency ratio have been studied. The differential quadrature method is employed to solve the non-linear differential equations of motion. The results show that the softening behaviour is seen for all cases. The non-linear frequency to linear frequency ratio increases by increasing the Timoshenko beam parameter and breadth and height taper ratios, but decreases by increasing the contact position for the first and second modes. For the first vibrational mode, the non-linear frequency to linear frequency ratio increases by increasing the height of the tip and the angle α between the cantilever and sample surface. By increasing the normal contact stiffness, the frequency ratio increases for the first mode.

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