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Analyzing the vibrational response of an AFM cantilever in liquid with the consideration of tip mass by comparing the hydrodynamic and contact repulsive force models in higher modes

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Abstract

This paper investigates the vibration of a tapping-mode Atomic Force Microscope (AFM) cantilever covered with two whole piezoelectric layers in a liquid medium. The authors of this article have already modeled the vibration of a cantilever immersed in liquid over rough surfaces. Five new ideas have been considered for improving the results of the previous work. Mass and damping of a cantilever probe tip have been considered. Since the probe tip of an AFM cantilever has a mass, which can itself affect the natural frequency of vibration, the significance of this mass has been explored. Also, two hydrodynamic force models for analyzing the mass and damping added to a cantilever in liquid medium have been evaluated. In modeling the vibration of a cantilever in liquid, simplifications are made to the theoretical equations used in the modeling, which may make the obtained results different from those in the real case. So, two hydrodynamic force models are introduced and compared with each other. In addition to the already introduced DMT model, the JKR model has been proposed. The forces acting on a probe tip have attractive and repulsive effects. The attractive Van der Waals force can vary depending on the surface smoothness or roughness, and the repulsive contact force, which is independent of the type of surface roughness and usually varies with the hardness or softness of a surface. When the first mode is used in the vibration of an AFM cantilever, the changes of the existing physical parameters in the simulation do not usually produce a significant difference in the response. Thus, three cantilever vibration modes have been investigated. Finally, an analytical approach for obtaining the response of equations is presented which solves the resulting motion equation by the Laplace method and, thus, a time function is obtained for cantilever deflection is determined. Also, using the COMSOL software to model a cantilever in a liquid medium, the computed natural frequencies have been compared.

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Abbreviations

L 1 :

Length of piezoelectric layer

L 2 :

Length of first section cantilever

L :

Length of cantilever

q(t):

Time function

Ø(x):

Cantilever mode shape

ν(x,t):

Deflection of cantilever

\(\check{M}\) :

Mass of the cantilever

\(\check{K}\) :

Stiffness of the cantilever

\(\check{C}\) :

Damping of the cantilever

E b :

Elasticity modulus of cantilever

ρ b :

Density of cantilever

ρ p :

Density of piezoelectric layer

E p :

Elasticity modulus of piezoelectric layer

w b :

Width of microcantilever

w p :

Width of piezoelectric layer

w t :

Width of microcantilever tip

t b :

Thickness of microcantilever

t p :

Thickness of piezoelectric layer

F ts :

Forces acting between cantilever probe tip and surface

F liq :

Hydrodynamic forces

\({{m}_{\text{added}}}\) :

Mass added to a cantilever due to the presence of the surrounding liquid

\({{c}_{\text{added}}}\) :

Damping added to a cantilever due to the presence of the surrounding liquid

\({{\rho }_{a}}\) :

Added density

c a :

Added hydrodynamic damping

w l :

Cantilever width

ω :

Natural frequency

\({{\rho }_{\text{liq}}}\) :

Density of liquid

\(\rho \) :

Viscosity of liquid

F vdw :

Attractive Van der Waals force

F DMT :

Contact repulsive force (DMT model)

F JKR :

Contact repulsive force (JKR model)

\( H\) :

Hamaker constant

d ts :

Distance between the tip and surface

R a :

Radius of surface asperity

\( R\) :

Probe tip radius

rms1 :

Distances between two tall asperity peaks

rms2 :

Distances between two short asperity peaks

λ :

Types of successive surface asperities

E*:

Effective elasticity modulus

a 0 :

Intermolecular distance

γ :

Surface energy

D 0 :

Initial cantilever tip distance

U(t):

Heavisid function

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Acknowledgements

The authors gratefully acknowledge the support of INSF under the grant contracts.

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Authors and Affiliations

  1. Robotic Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    Moharam Habibnejad Korayem

  2. Department of Mechanical Engineering, Roudehen Branch, Islamic Azad University, Tehran, Iran

    Amir Nahavandi

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  1. Moharam Habibnejad Korayem
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  2. Amir Nahavandi
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Correspondence to Moharam Habibnejad Korayem.

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Korayem, M.H., Nahavandi, A. Analyzing the vibrational response of an AFM cantilever in liquid with the consideration of tip mass by comparing the hydrodynamic and contact repulsive force models in higher modes. Appl. Phys. A 123, 265 (2017). https://doi.org/10.1007/s00339-017-0812-x

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  • DOI: https://doi.org/10.1007/s00339-017-0812-x

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