LINEAR AND NONLINEAR VIBRATION ANALYSIS

LINEAR AND NONLINEAR VIBRATION ANALYSIS

OF TAPPING MODE ATOMIC

FORCE MICROSCOPY

by

RACHAEL VIRGINIA MCCARTY

S. NIMA MAHMOODI, COMMITTEE CHAIR

BETH A. TODD

KEITH A. WILLIAMS

W. STEVE SHEPARD

TIM A. HASKEW

A DISSERTATION

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy

in the Department of Mechanical Engineering

in the Graduate School of

The University of Alabama

TUSCALOOSA, ALABAMA

2014

Copyright Rachael Virginia McCarty 2014

ALL RIGHTS RESERVED

ii

ABSTRACT

Atomic force microscopy (AFM) uses a scanning process performed by a microcantilever

probe to create a three dimensional image of a nano-scale physical surface. The dynamics of the

AFM microcantilever motion and tip-sample force need to be understood to generate accurate

images. Most AFMs use a laser system to take readings from the microcantilever. The bulky and

expensive laser system can be replaced by piezoelectric actuators and sensors along with an

electrical circuit. However, the dynamics of the piezoelectric microcantilever probe must be

accurately modeled in order to generate accurate images and to take accurate readings when

using the microcantilever for other applications. Additionally, minimizing the effect of

nonlinearities in the dynamic response of the microcantilever allows for less calculation intensive

software packages for AFMs without sacrificing accuracy.

In this dissertation, the linear and nonlinear dynamics of a microcantilever probe in

tapping mode AFM is investigated. First, different methods of including contact force in the

linear equations of motion and boundary conditions are analyzed then compared to experimental

results, which leads to the conclusion that including the contact force in the equation of motion

and the inertial force due to the tip mass in the boundary conditions is the preferable method for

most applications.

The nonlinear vibrations of the tapping mode AFM microcantilever are investigated due

to nonlinear contact force. The outcome shows that of the methods studied, the superior method

of decreasing the nonlinearity effect is to find the optimal initial tip-sample distance and

iii

excitation force. Next, the effects of the nonlinear excitation force on the microcantilever’s

frequency and amplitude response are analytically studied. The results show a frequency shift in

the response resulting from the force nonlinearities. The results of a sensitivity analysis show

that parameters can be chosen such that the frequency shift is minimized. Additionally, a

convergence analysis is used to determine the number of modes necessary to describe the motion

of the microcantilever in tapping mode. It is determined that one mode is insufficient, and two

modes are required and, for most applications sufficient.

iv

DEDICATION

This dissertation is dedicated to my family, friends, and colleagues, who have contributed

to my success. Specifically, I would like to dedicate this dissertation to my husband, Kevin, for

being such an amazing partner and friend and for always loving and supporting me; my parents,

Terry and Kristn Click, for all of their love, patience, and guidance over the years; and my

beautiful daughters, Norah and Robin, for allowing me to know the amazing joy of being their

mommy.

v

ACKNOWLEDGMENTS

I am pleased to have this opportunity to thank the many colleagues, friends, and faculty

members who have helped me with this research project. The first person I would like to thank is

Dr. Nima Mahmoodi, the chairperson of this dissertation, for the time and energy he has invested

in sharing his research knowledge and expertise with me. Dr. Mahmoodi has been an excellent

advisor, and I would especially like to thank him for his patience and understanding during my

pregnancy, hospital stay, and the early infancy days with my twin daughters.

I would also like to thank all of my committee members, Dr. Beth Todd, Dr. Keith

Williams, Dr. Steve Shepard, and Dr. Tim Haskew for their invaluable input, inspiring questions,

and support of both the dissertation and my academic progress. I would also like to specifically

thank Dr. Todd for being my mentor of eleven years and contributing so greatly to my success. I

would like to thank Dr. Stan Jones for his invaluable help with nonlinear partial differential

equations. Also, I would like to express my appreciation to all the professors that I’ve interacted

with over my nine years at UA.

I am also grateful to the ladies in the mechanical engineering office, Lynn Hamric, Lisa

Hinton, and Betsy Singleton for their encouragement and friendship and for taking care of so

many administrative tasks above and beyond the call of duty over the years. I would like to thank

my fellow graduate students who I have worked with over the last five years: Alston Pike,

Michael Carswell, Andrew Truitt, Ehsan Omidi, Ben Carmichael, and Gary Frey.

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This research would not have been possible without the support and encouragement from

my family, friends, and fellow graduate students. Specifically, I would like to thank my husband,

Kevin McCarty, for his abundant support of my decision to come back to school five years ago

and continuing support and encouragement on both the good and the bad days.

I would also like to acknowledge and thank everyone who has taken care of my two little

girls while I’ve been working on this research, most notably: Kristn Click, Jane and Sam

Ledbetter, and Rhonda and Krista McCarty. These people rearranged their schedules and their

lives in order to lift the time and money restraints of child care. Without the comfort of knowing

that my babies were in good hands, I would not have been able to focus and finish.

This material is based upon work supported by the National Science Foundation Graduate

Research Fellowship under Grant No. 23478.

vii

CONTENTS

ABSTRACT .................................................................................................................................... ii

DEDICATION ............................................................................................................................... iv

ACKNOWLEDGMENTS ...............................................................................................................v

LIST OF TABLES ....................................................................................................................... viii

LIST OF FIGURES ....................................................................................................................... ix

CHAPTER 1. OVERALL INTRODUCTION ................................................................................1

CHAPTER 2. DYNAMIC ANALYSIS OF TAPPING ATOMIC FORCE MICROSCOPY

CONSIDERING VARIOUS BOUNDARY VALUE PROBLEMS ...............................................4

CHAPTER 3. FREQUENCY RESPONSE ANALYSIS OF NONLINEAR TAPPING-

CONTACT MODE ATOMIC FORCE MICROSCOPY ..............................................................33

CHAPTER 4. PARAMETER SENSITIVITY ANANYSIS OF NONLINEAR

PIEZOELECTRIC PROBE IN TAPPING MODE ATOMIC FORCE MICROSCOPY

FOR MEASUREMENT IMPROVEMENT ..................................................................................59

CHAPTER 5. DYNAMIC MULTIMODE ANALYSIS OF NONLINEAR

PIEZOELECTRIC MICROCANTILEVER PROBE IN BISTABLE REGION OF

TAPPING MODE ATOMIC FORCE MICROSCOPY ................................................................85

CHAPTER 6. OVERALL CONCLUSIONS ...............................................................................120

viii

LIST OF TABLES

CHAPTER 2

2.1. Microcantilever Properties [35] ............................................................................................17

2.2. First 3 Natural Frequencies for 3 Cases ................................................................................19

CHAPTER 3

3.1. Sample properties [24]. .........................................................................................................48

3.2. Geometric and material properties of the AFM microcantilever probe [23, 26]. .................48

CHAPTER 4

4.1. HOPG sample properties [22]. ..............................................................................................70

4.2. Piezoelectric microcantilever properties [23]. ......................................................................71

4.3. Values of Altered Parameters. ..............................................................................................78

CHAPTER 5

5.1. Nondimensional Quantities. ..................................................................................................90

5.2. Piezoelectric microcantilever properties [32]. ......................................................................93

5.3. HOPG sample properties [30]. ..............................................................................................94

5.4. Natural frequencies of the microcantilever. ..........................................................................96

5.5. Error when considering n modes. .......................................................................................112

ix

LIST OF FIGURES

CHAPTER 2

2.1. Microcantilever beam with a spring attached to the free end. .............................................7

2.2. Picture of the Bruker Innova Atomic Force Microscope. ..................................................15

2.3. Picture of the experimental setup. ......................................................................................16

2.4. The first mode shape, ϕ1, for Cases 1, 2, and 3. .................................................................19

2.5. Maximum amplitude of vibrations at the microcantilever tip over a range of

frequencies for all three numerical cases. ..........................................................................20

2.6. The time response function, q1, for (a) Case 1 and (b) Case 3. .........................................21

2.7. The complete response at the free end of the microcantilever, w(L,t), for (a) Case 1

and (b) Case 3. ...................................................................................................................22

2.8. Phase portraits for the free end of the microcantilever for (a) Case 1 and (b) Case 3. ......22

2.9. A zoomed in view of the complete response at the free end of the microcantilever,

w(L,t), for (a) Case 1 and (b) Case 3. .................................................................................23

2.10. Tip deflection data gathered experimentally from the AFM with an MPP-11123-10

microcantilever. .................................................................................................................23

2.11. Maximum amplitude of vibrations at the microcantilever tip over a range of

frequencies for numerical cases 1 and 3 and experimental results. ...................................24

2.12. Steady state amplitude at resonance as it varies with tip to microcantilever mass ratio,

R, for Case 1. ......................................................................................................................25

CHAPTER 3

3.1. Schematic of the AFM microcantilever probe. ..................................................................36

3.2. Nonlinear tip-sample force; solid line is the force presented in equation (14); and

circles show the force based on equation (17). ..................................................................49

x

3.3. First natural frequency of the non-dimensional equations of motion for different

values of a1. ........................................................................................................................50

3.4. Non-dimensional natural frequency for practical values of a1. .........................................50

3.5. Frequency response curve for the first mode, a) =120, b) =180, and c) =210. ...........51

3.6. Frequency response curve for the first mode (=210), a) nondimensional force is f0=

0.31, b) the dashed line represents f0=0.31, and the solid line represents f0=0.34. ............52

3.7. Force response curve for the first mode when =120 (the dashed line shows the

unstable region). .................................................................................................................53

3.8. Force response curve for the first mode when =210 (the dashed line shows the

unstable region). .................................................................................................................53

CHAPTER 4

4.1. Schematic of the piezoelectric microcantilever motion. ....................................................61

4.2. Frequency response of the piezoelectric microcantilever probe: nondimensionalized

microcantilever tip amplitude as it varies with σ. ..............................................................71

4.3. Effect of (a) microcantilever length and (b) thickness on the magnitude of the

frequency shift. ..................................................................................................................73

4.4. Effect of (a) microcantilever width and (b) tip radius on the magnitude of the

frequency shift. ..................................................................................................................74

4.5. Effect of (a) vo, and (b) piezoelectric constant, d31, on the magnitude of the frequency

shift. ...................................................................................................................................75

4.6. Effect of (a) the tip-sample distance distinguishing the contact and non-contact

regions, δ, and (b) effective modulus of elasticity (between microcantilever and

sample) on the magnitude of the frequency shift. ..............................................................76

4.7. Effect of (a) microcantilever and (b) piezoelectric modulus of elasticity on the

magnitude of the frequency shift. ......................................................................................77

4.8. Effect of z0, the distance between the tip of the AFM and the sample at equilibrium,

on the magnitude of the frequency shift. ...........................................................................77

4.9. Frequency response of the piezoelectric microcantilever probe with three altered

parameters: nondimensionalized microcantilever tip amplitude as it varies with σ. ........79

xi

CHAPTER 5

5.1. Schematic of the piezoelectric microcantilever motion. ....................................................89

5.2. First natural frequency based on changing tip-sample force 1 coefficient. ....................95

5.3. First mode shape of the piezoelectric microcantilever for two different values of 1

coefficient. .........................................................................................................................95

5.4. First 6 mode shapes of the microcantilever. ......................................................................96

5.5. 3D surface plot of amplitude response of the piezoelectric microcantilever probe

with changing input voltage, v0, and excitation frequency, σ. .........................................101

5.6. Frequency response of the piezoelectric microcantilever probe: nondimensionalized

microcantilever tip amplitude as it varies with σ. (v0 = 100 mV) ....................................102

5.7. Force response of the piezoelectric microcantilever probe: nondimensionalized

microcantilever tip amplitude as it varies with v0. (σ = 15 Hz) .......................................102

5.8. Phase portrait for one mode with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)

low amplitude responses. .................................................................................................105

5.9. The (a) zoomed in phase portrait and (b) time history of one cycle of the response for

one mode with σ = 26 Hz and v0 = 0.1 V. ........................................................................106

5.10. Power spectra for one mode with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


5.11. Phase portrait for two modes with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


5.12. Power spectra for two modes with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


5.13. Contributions to time histories of the low and high amplitude limit cycles for σ = 26

Hz and v0 = 0.1 V. ...........................................................................................................108

5.14. Phase portrait with σ = 26 Hz and v0 = 0.1 V for the high amplitude responses with n

modes. ..............................................................................................................................110

5.15. Power spectra for six modes with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


5.16. Contributions of 6 modes to the time history of the microcantilever probe

nondimensionalized tip displacement with σ = 26 and v0 = 0.1 V. ..................................111

xii

5.17. Time history of microcantilever probe nondimensionalized tip velocity with

contributions from n modes and σ = 26 Hz and v0 = 0.1 V. ...................................................... 112

1

CHAPTER 1

OVERALL INTRODUCTION

I. Problem Statement

The goal of this research is threefold. The first goal is to determine appropriate and novel

mathematical models to correctly predict the vibrations of an AFM microcantilever probe. The

second goal is to validate these models with computer simulations and experimental data.

Finally, the third goal is to analyze the response including stability, sensitivity, and convergence

analyses to provide methods of avoiding instability, decreasing the effect of nonlinearities in the

dynamic response, and simplifying the solution and analysis without neglecting significant

solution dynamics.

II. Objectives

The following tasks were needed to successfully complete the research work:

1. Linearization of the contact force in dynamic tapping mode AFM and derivation of the

linear equations of motion and boundary conditions of a microcantilever,

2. Frequency response analysis to obtain the natural frequencies and mode shapes, and

complete time response of the microcantilever beam mechanics,

3. Experimental verification of the mathematical model and computer simulations using

Bruker Innova Scanning Probe Microscope available in the Nonlinear Intelligent Structures

(NIS) Laboratory,

2

4. Derivation of the nonlinear equations of motion and boundary conditions of an AFM

microcantilever probe in dynamic tapping mode considering the nonlinear contact force

and curvature of the microcantilever,

5. Mode shape analysis of the nonlinear system,

6. Utilization of the method of multiple scales to solve the nonlinear response analysis of the

system and derive the frequency and amplitude modulation equations,

7. Stability analysis of the nonlinear response to provide methods of avoiding regions of

instability and decreasing the effect of the nonlinearities in the dynamic response of the

microcantilever,

8. Derivation of the nonlinear equations of motion and boundary conditions of an AFM

piezoelectric microcantilever probe in dynamic tapping mode considering the nonlinear

contact force,

9. Mode shape analysis of the nonlinear system,

10. Utilization of the method of multiple scales to solve the nonlinear response analysis of the

system and derive the frequency and amplitude modulation equations,

11. Sensitivity analysis to determine methods of reducing the effect of nonlinearities in the

dynamic response of the microcantilever,

12. Analysis of the frequency shift in the solution of the nonlinear system to determine the

effect of the solution lying on the monostable versus bistable region as well as lying on the

low or high amplitude branch in the bistable region, and

13. Convergence analysis to determine the number of modes necessary to describe the complex

dynamics of the nonlinear solution.

3

CHAPTER 2

DYNAMIC ANALYSIS OF TAPPING ATOMIC FORCE MICROSCOPY CONSIDERING

VARIOUS BOUNDARY VALUE PROBLEMS

An accurate understanding of the microcantilever motion and the corresponding tip-

sample force is needed to generate accurate images in atomic force microscopy (AFM). In this

paper, different methods to apply the tip-sample force to the dynamic equations of motion and

boundary conditions are derived and compared to determine which method, of those studied, is

the superior method for dynamic analysis of these systems. Hamilton’s principle and the

Galerkin method are employed to investigate the vibration of the microcantilever probe used in

tapping mode AFM. Three different methods of including contact and excitation force in the

equations of motion and boundary conditions are analyzed then compared. The first case

considers the contact force at the tip and the inertial force due to tip mass to be a part of the

boundary conditions of the microcantilever. The second case assumes that the force is a

concentrated force that is applied in the equations of motion, and the boundary conditions are the

same as for the free end of a microcantilever beam. The third case is a combination where the

contact force is included in the equation of motion, but the inertial force due to the tip mass is

included in the boundary conditions. For the three cases, the equations of motion, the modal

shape functions including the natural frequencies, and the time and frequency response functions

are obtained. The numerical results are compared to the experimental results obtained from the

Bruker Innova AFM. Results show that the first and third methods produce results that

4

accurately match the experimental outcomes. However, since including the forces in the

boundary conditions is considerably more complex mathematically, this research indicates that

including the forces in the equations of motion is preferable unless the tip mass is relatively

large.

I. Introduction

Atomic force microscopy (AFM) was originally invented and used for nano-scale

scanning to create a three dimensional image of a physical surface. The scanning process is

performed by a microcantilever that contacts or taps the surface. More recently, microcantilever

probes have been used extensively for Friction Force Microscopy (FFM), Lateral Force

Microscopy (LFM), Piezo-response Force Microscopy (PFM), biosensing, and other applications

[1-4]. Most AFMs operate by exciting the microcantilever using a piezoelectric tube actuator at

the base of the probe. However, some microcantilevers have a layer of piezoelectric material on

one side for actuation purposes. This layer is usually Zinc Oxide (ZnO) [5] or Lead Zirconate

Titanate (PZT). The application of the piezoelectric microcantilever is widespread; it has been

used for force microscopy, Scanning Near-field Optical Microscopy (SNOM), biosensing, and

chemical sensing [6-9]. An accurate understanding of the microcantilever motion and tip-sample

force is needed to generate accurate images.

The force between tip and sample consists of two main components: a van der Waals

force and a contact force [10]. Numerical and experimental studies have investigated these

nonlinear forces in some detail [11, 12]. In non-contact mode, there is only van der Waals force

between AFM tip and sample. However, in a tapping contact AFM, both forces are applied to the

5

tip. In this work, only the linear contact force is considered since it is much larger than the van

der Waals force.

Dynamics of the microcantilever have been experimentally and analytically studied in

some research works. Experimental investigations have been performed in air and liquid on

dynamic AFMs and the frequency response of the systems were obtained [13-16]. The nonlinear

dynamics of a piezoelectric microcantilever have been studied considering the nonlinearity due

to curvature and piezoelectric material [17, 18]. In other works, linear dynamic models have

been developed for contact AFM probes and numerically solved [19-23]. Some works have

conducted numerical studies to determine the number of modes necessary to fully model the

complex dynamics of the microcantilever [24, 25].

Two methods have been used in research works when including the force at the free end

of the microcantilever during dynamic analysis. One method is to consider the force at the end of

the microcantilever in the boundary conditions [13, 19, 20, 26-28]. The other method is to

consider the force to be a part of the equation of motion using some type of step function, such as

the Heaviside or Dirac delta function [17, 18, 22, 24, 25, 29, 30]. Additionally, a hybrid method

will be introduced that combines these two methods. In this third method, the contact force will

be considered in the equation of motion while the inertial force due to the tip mass will be

considered in the boundary conditions.

This research work investigates the vibration of dynamic tapping mode AFM using these

three different methods of analysis and directly compares the results. For the three different

cases, the equations of motion are derived using Hamilton’s principle, the modal shape functions

including the natural frequencies are obtained using separation of variables, and the time

response functions are obtained using the Galerkin method. The AFM microcantilever probe and

6

sample is a nonlinear system. However, in this work, the system is linearized by using a spring

on the free end to approximate the linear contact force [22, 27]. Three linear systems are

compared in order to determine the superior method.

Experimental results are obtained using the Bruker Innova AFM with an MPP-11123-10

microcantilever. The resonance frequency of the microcantilever is determined using the

NanoDrive software package. The point spectroscopy function is used to collect data at

increments of 1% of resonance across a range of 90% to 110% of resonance. This procedure is

repeated four times to decrease the effects of statistical bias. The displacement data at each

frequency are analyzed to find the maximum amplitude of the tip displacement.

Numerical results are compared to the experimental results. The second method is shown

to be simpler than the other methods derivationally, but it yields inaccurate results. The first and

third methods produce equally accurate results. Also, the results are very similar to each other.

This indicates that either method is equally reliable. However, including the forces in the

boundary conditions is considerably more complex mathematically. Therefore, this research

indicates that the preferable method is the third method – including the contact and excitation

forces in the equation of motion and the inertia force due to tip mass in the boundary conditions.

Additionally, most research works neglect the effect of tip mass completely from the

equation of motion and boundary conditions [13-30]. In this work, the tip mass is included and

its effect on the microcantilever dynamics are analyzed. For the first method, tip mass is shown

to have a rather large effect on the resulting amplitude. For the third method, the inclusion of tip

mass makes practically no difference. These results indicate that if the tip mass is large, as in for

biosensing applications [31-34], it may be necessary to use the first method, despite its more

complex derivation.

7

II. Methods

The governing equations of motion, natural frequencies, mode shapes, and time response

functions for the dynamics of a microcantilever with a spring at the free end are mathematically

derived in this section for three cases: 1) the contact force and excitation force at the tip and the

inertial force due to the tip mass are considered to be a part of the boundary conditions of the

microcantilever, 2) the contact force and excitation force at the tip and the inertial force due to

the tip mass are considered to be a concentrated force that is applied in the equations of motion,

and the boundary conditions are the same as that of a free microcantilever beam, and 3) the

contact force and excitation force are considered to be a concentrated force that is applied in the

equations of motion, and the inertial force due to the tip mass is considered to be a part of the

boundary conditions of the microcantilever.

Figure 2.1 shows a microcantilever with a spring attached to the free end. The spring

represents the elements that produce tip-sample contact force. The bending displacement of the

microcantilever in the negative z direction at position x along the microcantilever and at time t is

w(x,t). The coordinate system (x, z) is used to describe the dynamics of the microcantilever, and t

denotes time.

Figure 2.1. Microcantilever beam with a spring attached to the free end.

8

2.1 Case 1: Forces and Tip Mass Considered in Boundary Conditions

The first case to be examined, as stated previously, is a system including a spring at the

free end where the contact force, excitation force, and inertial force due to the tip mass are

included in the boundary conditions. The relevant equation from Hamilton’s principle is

1

0

0t

tdtWUT , (1)

where T is kinetic energy, U is potential energy, and W is the work done by external loads on the

microcantilever. To derive the equation of motion, expressions for the kinetic energy, potential

energy, and external work need to be determined. First, the expression for the kinetic energy is

derived. The kinetic energy will be the combined kinetic energy of the microcantilever (Tb) and

the tip (Ttip).

L

b dxt

wT

0

2

1½m , (2)

2

2½m

t

wT L

tip , (3)

where m1 is the mass per unit length of the microcantilever, m2 is the tip mass, and L is the length

of the microcantilever. Also, wL is the displacement of the microcantilever at the free end and is a

function of time.

The potential energy term comes from two sources. Ub is the potential energy due to the

strain energy of the microcantilever, and Us is the potential energy due to the spring.

L

b dxx

wU

0

2

2

2

½EI , (4)

2

LS ½kwU , (5)

9

where E is the is the elastic modulus of the microcantilever, I is the mass moment of inertia of

the microcantilever, and k is the spring constant.

The external work is

LwtFW sin , (6)

where F and Ω are the amplitude and frequency of the excitation force.

Substituting Equations (2) through (6) into Equation (1) and simplifying results in the

equation of motion and boundary conditions for the bending vibrations of the microcantilever of

the AFM shown in Figure 2.1 for Case 1:

0)(

1 ivEIwwm , (7)

00 w , 00 w , 0Lw , tFwmkwwEI LLL sin2

. (8)

For simplicity, primes ( ) denote the partial derivative with respect to x, and dots ( ) denote the

partial derivative with respect to time. In order to homogenize Equations (7) and (8), the

following variable transformation must be performed for analyzing the full response of the

microcantilever:

tLxxCtxvtxw sin3,, 23 . (9)

where

2232

dL

FC . (10)

The appendix contains the details of this substitution. Substituting Equation (9) into Equations

(7) and (8) yields

tLxxCmEIvvm iv sin3 232

1

)(

1 , (11)

00 v , 00 v , 0Lv , LLL vmkvvEI

2 . (12)

10

In order to derive the mode shapes and natural frequencies of the system, the force term

is removed from Equation (11) and separation of variables is implemented. The mode shapes are

LL

LLxxxxAx

nn

nnnnnnnn

coshcos

sinhsincoscoshsinhsin , (13)

where n = 1, 2,…∞ indicating the number of the mode, and λn are the roots of the following

frequency equation:

0sinhcoscoshsincoshcos13

LLLLEI

LL nnnn

n

nn

, (14)

where

1

4

4

2m

EI

Lmk n , (15)

4 1

2

EI

mnn

, (16)

and ωn are the natural frequencies. The values for An can be found using the orthogonality

condition.

Now the Galerkin method that uses the following definition is utilized:

m

i

ii tqxtxw1

, . (17)

For more realistic analysis a damping term is added to the equation of motion. After substituting

Equation (17) into Equation (11), multiplying by Φj, and integrating on the limits zero to L, the

one term Galerkin solution becomes an ordinary differential equation:

tqqq sin1

2

11 , (18)

where

11

1m

, (19)

L

iv dxEI0

1

)(

1

2 , (20)

L

dxLxxCm0

23

1

2

1 3 , (21)

and μ is damping. Equation (18) can be solved as

teCtCtetCq tt sinsincoscos 3211 , (22)

where C1, C2, C3, ε, and θ are defined in the appendix. The numerical analysis of the results

obtained in this section and the comparisons with experimental results will be shown in Section

4.

2.2 Case 2: Forces and Tip Mass Considered in Equation of Motion

The second case to be examined, as stated previously, is a system including a spring at

the free end where the contact force, excitation force, and inertial force due to the tip mass are

considered to be a concentrated force that is applied in the equations of motion. The boundary

conditions are the same as those of the free end of a cantilever. Hamilton’s principle is again

utilized to derive the equation of motion and boundary conditions for this case. The equation for

the kinetic energy of the microcantilever, Equation (2), and the equation for the potential energy

of the microcantilever, Equation (4), are the same as in the first case. Equation (3), the kinetic

energy of the tip, and Equation (5), the potential energy of the spring, will not be included in this

case. Instead, the work done by the spring force and the tip mass is included in the external work

term resulting in

L

0Next wdxLxfW , (23)

where

12

tFwmkwf LLN sin2

, (24)

and δ(ζ) is the Dirac delta function and is defined as

00

0)(

. (25)

Substituting Equations (2), (4), and (23) into Equation (1) and simplifying the results

yields the equations of motion and boundary conditions for the bending vibrations of the

microcantilever of the AFM shown in Figure 2.1 for Case 2:

LxfEIwwm N

)iv(

1 , (26)

00 w , 00 w , 0Lw , 0Lw . (27)

In order to derive the mode shapes of the system, the force term is removed from Equation (26).

The derivations will be similar to Case 1. Following the process used in Section 2.1, the resulting

mode shape equation can be solved and is the same as Equations (13) and (14). Only the

equations are the same, not the actual mode shapes since the natural frequencies are different.

The natural frequency equation becomes identical to that of a microcantilever beam without a tip

mass:

0coshcos1 LL nn . (28)

With the mode shapes and natural frequencies determined, the full response of the system

can be found using the Galerkin method. Following the same procedure as in Section 2.1, the one

term Galerkin solution becomes an ordinary differential equation of the same form as Equation

(18); however, the coefficients are different and defined as

1m

A , (29)

LkdxAEI

Liv 2

10

1

)(

1

2 , (30)

13

LAF 1 , (31)

221

1

LmA

. (32)

The numerical analysis of the results obtained in this section and the comparisons with

experimental results will be shown in Section 4.

2.3 Case 3: Forces Considered in Equation of Motion; Tip Mass Considered in Boundary

Conditions

The third case to be examined is a system with a spring at the free end where the contact

and excitation forces are considered to be a concentrated force that is applied at the tip and

reflected in the equations of motion and the boundary conditions include the effect of the tip

mass. Hamilton’s principle is, again, utilized to derive the equation of motion and boundary

conditions for this case. The equations for the kinetic energy of the microcantilever and tip mass,

Equations (2) and (3), and the equation for the potential energy of the microcantilever, Equation

(4), are the same as in the first case. Equation (5), the potential energy of the spring, will not be

included in this case. Instead, the work done by the spring force is included in the external work

term resulting in

L

0Next wdxLxfW , (33)

where

tFkwf LN sin . (34)

Substituting Equations (2) through (4), and (33) into Equation (1) and simplifying results

in the equations of motion and boundary conditions for the bending vibrations of the

microcantilever of the AFM shown in Figure 2.1 for Case 3:

LxfEIwwm N

)iv(

1 , (35)

14

00 w , 00 w , 0Lw , LL wmwEI

2 . (36)

In order to derive the mode shapes of the system, the force term is removed from Equation (35).

The derivations will be similar to Case 1. Following the process used in Section 2.1, the resulting

mode shape equation can be solved and is the same as Equations (13), (14), and (16). The only

difference is the definition of ν, which is defined for Case 1 in Equation (15), which, for Case 3,

becomes

1

4

4

2m

EI

Lm n . (37)

Again, this difference leads to different mode shapes. Even though the equation is the same, the

natural frequencies are different.

With the mode shapes and natural frequencies determined, the full response of the system

can be found using the Galerkin method. Following the same procedure as in Section 2.1, the one

term Galerkin solution becomes an ordinary differential equation of the same form as Equation

(18); however, the coefficients are different and are defined as

LkdxEI

Liv 2

10

1

)(

1

2 , (38)

LF 1 . (39)

The full results for all three cases are obtained mathematically. Section 3 will describe

the experimental setup, and Section 4 will discuss and compare the computational and

experimental results.

III. Experimental Setup

This section details the experimental setup used to generate the experimental results,

which will be discussed along with the numerical results in the next section. All measurements

15

were collected from a Bruker Innova Atomic Force Microscope, as shown in Figure 2.2, with an

MPP-11123-10 microcantilever using the proprietary NanoDrive software.

Figure 2.2. Picture of the Bruker Innova Atomic Force Microscope.

The software allowed for accurate data capture of the high oscillation frequencies of the

microcantilever from the AFM photodiode. The output system of the Innova sent this voltage

signal to an external monitor for observation. For the purposes of this experiment, the data was

collected by a digital oscilloscope, which was capable of recording the signal without aliasing.

The MPP-11123-10 microcantilever was manufactured by Bruker out of antimony doped silicon.

The sample was a surface topography reference provided with the microscope, which was chosen

for its low surface variability. The entire experiment was performed on a pneumatic passive

isolation table to eliminate external sources of vibration (ThorLabs PTP603). Figure 2.3 shows

the experimental setup.

The microcantilever was positioned over a flat region of the sample using the point

spectroscopy function to avoid any topological irregularities that could influence its response.

The resonance frequency of the microcantilever was determined using the “autotune” function in

16

the NanoDrive software package. Then, the tapping excitation frequency was systematically

varied by increments of 1% of the resonance frequency across a range of 90% to 110% of the

resonance frequency. This procedure was repeated four times to decrease the effects of statistical

bias. At each frequency, the response of the microcantilever was allowed to settle before data

were recorded. Photodiode voltage data were recorded for three time scales of increasing

duration. The constant of proportionality between the photodiode voltage and tip displacement

was determined beforehand by performing a force-ramping procedure on the sample. The slope

of the resultant graph provided the conversion factor. The displacement data for the longest time

scale at each frequency were analyzed to find the maximum amplitude of the tip displacement.

Figure 2.3. Picture of the experimental setup.

IV. Results

The dynamics of a microcantilever have been obtained mathematically for three different

boundary value problems. The first case considers the forces at the tip to be a part of the

Atomic Force

Microscope

NanoDrive

Software

AFM Processing Unit

17

boundary conditions of the microcantilever. The second case assumes that the forces are a

concentrated force that is applied in the equations of motion, and the boundary conditions are

similar to a fixed-free microcantilever beam. The third case breaks up the forces. In this section,

numerical and experimental results are presented. Table 2.1 shows the values of properties used

in the numerical analysis.

Table 2.1. Microcantilever Properties [35]

Constant Value

L (μm) 125±10

E (GPa) 200±10

b (μm) 35±5

m1 (mg/m) 34.8±2

m2 (pg) 2.19±0.5

The force between tip and sample varies between approximately -4 and 4 nN over the tip-

sample separation distance of 0 to 2 nm [12]. This represents both repulsive and attractive forces

that occur when the separation distance is very small, mainly the contact and van der Waals

forces. In this paper, the tip force is modeled as a spring. This means that the spring constant in

terms of N/m is needed instead of simply a force. Therefore, several key locations were selected

from the figure of tip-sample force vs tip-sample distance in [12], and the force was divided by

the distance to determine a range of potential k values. The range found was -15 to 40 N/m. In

this paper, we are only considering the linear contact force which is positive, narrowing the

range for the spring constant to 0 to 40 N/m. This range includes all possible tip and sample

18

materials whether hard or soft so it is necessary to use system identification to determine the

actual value of k.

In reality, the tip force is only applied over a very small range of vertical tip displacement.

For example, for a ZnO tip and Highly Ordered Pyrolytic Graphite (HOPG) sample, the contact

force is applied only when the tip-sample distance is less than 0.38 nm [37]. In this paper, the

spring is considered to always be attached to the tip of the microcantilever probe. Therefore, a

small value for the spring constant must be chosen. The k value has been obtained from

experimental results by system identification. In order to maximize the effect of the spring, the

maximum value that could still result in an analytical natural frequency that matches the

experimental natural frequency for all three cases was chosen. By this method, k was identified

as 42.24 pN/m.

First, the natural frequencies are compared. The analytical natural frequencies are found

by numerically solving Equations (14) through (16) for Case 1, Equations (16) and (28) for Case

2, and Equations (14), (16), and (37) for Case 3. The microcantilever thickness has been altered

in the offset range defined by the manufacturer for each case to achieve results for the natural

frequency that match the experimental results. The exact dimensions of the microcantilever

cannot be determined, and the manufacturer gives an acceptable range for the thickness of the

microcantilever (3 to 4.5 μm) [35]. For all three cases, the thickness used is in this range. Using

this method of altering thickness, all methods result in a first natural frequency, ω1, that matches

the experimental natural frequency, 365.965 kHz. Table 2.2 presents the first three natural

frequencies of the three different cases.

Case 1 and Case 3 have natural frequencies that are more closely related than Case 2,

which is identical to that of a free microcantilever. Case 1 has more discrepancy from the free

19

microcantilever (Case 2) than Case 1. The reason for this is that the natural frequency equation

for Case 1 takes into account both spring force and tip mass, while Case 3 only considers tip

mass.

Table 2.2. First 3 Natural Frequencies for 3 Cases

Case Number ω1 (kHz) ω2 (kHz) ω3 (kHz)

1 365.965 1 707.45 4 734.73

2 365.965 2 293.46 6 421.76

3 365.965 1 441.02 4 034.91

The mode shapes for the three different cases are compared. All three cases use Equation

(13) to find the mode shapes. Figure 2.4 is the first mode shape for Cases 1, 2, and 3. The mode

shapes for the last two cases are similar. However, the mode shape for the first case slopes up

more steeply than the mode shape for Cases 2 and 3. The maximum difference between Cases 2

and 3 is 0.0009%. The largest difference between Case 1 and the last two cases is at the end of

the microcantilever: 13.4%. Case 1 is different from the last two cases because it includes k in

derivation of the mode shapes and natural frequencies.

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x (m)

1 (

x10

6)

Case 1

Case 2

Case 3

Figure 2.4. The first mode shape, ϕ1, for Cases 1, 2, and 3.

20

In order to further explore the differences between the three cases, a frequency analysis is

performed. The time dependent functions, q1, are found. Equations (19) through (22) for Case 1,

Equations (33) and (29) through (32) for Case 2, and Equations (19), (22), (38), and (39) for

Case 3 are used to solve for q1. Then, the full response can be found using Equations (9) and (17)

for Case 1 and Equation (17) for Cases 2 and 3. Figure 2.5 shows the results of the frequency

response analysis, which is the effect of the excitation frequency on the maximum amplitude of

the steady state response.

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.10

0.1

0.2

0.3

0.4

0.5

/n

Am

plit

ude (m

)

Case 1

Case 2

Case 3

Figure 2.5. Maximum amplitude of vibrations at the microcantilever tip over a range of

frequencies for all three numerical cases.

From Figure 2.5, it can be seen that Cases 1 and 3 generate very similar results. Case 2,

however, shows a very large shift in the resonance frequency. The reason for this is related to the

equation of motion that is shown in Equations (24) and (26). The equation of motion for this

case, unlike the other two cases, contains the inertial force of the tip mass, which includes Lw .

After the Galerkin method is implemented, this tip mass term changes the coefficient of the 1q

term from Equation (18) from 1 to A-1, as defined in Equation (32). This coefficient cannot be

altered with also altering the natural frequency of the system. As a result, Case 2 is not an

accurate boundary value problem for deriving the response of the system. Even a very small tip

21

mass results in an unacceptable shift in natural frequency. Therefore, tip mass cannot be included

in the equation of motion.

Cases 1 and 3 are compared to determine which of these remaining two cases are

superior. First, the time response functions at resonance are compared. The results are shown in

Figure 2.6. The excitation frequency, Ω, is set to the natural frequency, ω1. Also, the excitation

force, F, has been set experimentally so that at resonance the analytical amplitude of vibrations

after settling matches the experimental data. The second mode was investigated, and results

showed that the time response for the second mode was much smaller than the first mode.

Therefore, only the results of the first mode are presented here.

0 10 20 30 40-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (ms)

q1 (

x10

-12)

0 10 20 30 40-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (ms)

q1 (

x10

-12)

(a) (b)

Figure 2.6. The time response function, q1, for (a) Case 1 and (b) Case 3.

Figure 2.6 shows that the results for the two different methods are very similar. The

steady state frequency is the same, and the steady state amplitude is about 11.8% different

between the two cases. Next, the complete response of Cases 1 and 3 are compared in Figure 2.7.

Figure 2.7 shows the time response of the microcantilever at the free end as derived in

Section 2. The two graphs are very similar. The amplitude after settling and frequency are the

same. The difference in amplitude in the time response, shown in Figure 2.6, is compensated for

22

by the difference in mode shape at the tip, shown in Figure 2.4. Figure 2.8 shows the phase

portraits for Case 1 and Case 3.

0 10 20 30 40-0.5

0

0.5

time (ms)

w(L

,t)

(m

)

0 10 20 30 40-0.5

0

0.5

time (ms)w

(L,t)

(m

)

(a) (b)

Figure 2.7. The complete response at the free end of the microcantilever, w(L,t), for (a) Case 1

and (b) Case 3.

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Tip Displacement (m)

Tip

Velo

city (

m/s

)

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

Tip Displacement (m)

Tip

Velo

city (

m/s

)

(a) (b)

Figure 2.8. Phase portraits for the free end of the microcantilever for (a) Case 1 and (b) Case

3.

The phase portraits show a spiral that is very slowly approaching the origin. Without the

external force and with some initial displacement or velocity, the microcantilever will oscillate

with damping slowly bringing the microcantilever to rest. For a direct comparison, Figure 2.9

23

shows the complete response over 0.1 ms, which is well past the settling time. Additionally,

Figure 2.10 shows 0.1 ms of tip deflection data gathered experimentally from the AFM.

30 30.02 30.04 30.06 30.08 30.1-0.5

0

0.5

time (ms)

w(L

,t)

(m

)

30 30.02 30.04 30.06 30.08 30.1-0.5

0

0.5

time (ms)

w(L

,t)

(m

)

(a) (b)

Figure 2.9. A zoomed in view of the complete response at the free end of the microcantilever,

w(L,t), for (a) Case 1 and (b) Case 3.

30 30.02 30.04 30.06 30.08 30.1-0.5

0

0.5

time (ms)

w(L

,t)

(m

)

Figure 2.10. Tip deflection data gathered experimentally from the AFM with an MPP-11123-10

microcantilever.

Figure 2.9 shows that the two numerical methods generate very similar results. Figure

2.10 of the experimental data shows that the frequency is the same as the numerical results, and,

although the experimental amplitude is not completely uniform, the maximum amplitude is the

same as the numerical amplitude calculated from Cases 1 and 3. Figure 2.11 shows the results of

the frequency response analysis for Cases 1 and 3 with the experimental data also plotted on the

same graph for comparison.

24

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.10

0.1

0.2

0.3

0.4

0.5

/n

Am

plit

ude (m

)

Experimental Data

Case 1

Case 3

Figure 2.11. Maximum amplitude of vibrations at the microcantilever tip over a range of

frequencies for numerical cases 1 and 3 and experimental results.

Figure 2.11 shows that numerical cases 1 and 3 present very similar results. The

numerical results are much sharper than the experimental results. However, both numerical

results are accurate at resonance, and with AFM the resonance frequency is the most used and,

therefore, the most important frequency.

All of the steps detailed so far for deriving the complete response of the system at

resonance are repeated for Cases 1 and 3 while varying the tip to microcantilever mass ratio, R,

defined as

100m

mR

1

2 . (40)

Figure 2.12 shows the steady state amplitude as it varies with R for Case 1. Figure 2.12

shows that the tip mass makes a considerable difference in the microcantilever dynamics for

Case 1. The last data point at 6.3 is the actual tip to microcantilever mass ratio for the MPP-

11123-10 probe used to generate the experimental data. The difference in amplitude between

R=0 (no tip mass considered) and R=6.3 is 0.20 μm or 47.9% error. The reason that the tip mass

makes such a significant difference in this method is related to the substitution defined in

25

Equations (9) and (10). The substitution results in the new equation of motion, Equation (11),

having m2 in the denominator of the right hand side. This direct involvement of the tip mass in

the equation of motion means that the effective force on the right hand side varies with the tip

mass, which has a direct effect on the resulting microcantilever dynamics.

0 2 4 6 80.2

0.25

0.3

0.35

0.4

0.45

0.5

R

Am

plit

ude (m

)

Figure 2.12. Steady state amplitude at resonance as it varies with tip to microcantilever mass

ratio, R, for Case 1.

For Case 3, the same analysis reveals very different results. A comparison of R=0 to

R=10, resulted in only a 0.02% change in steady state amplitude. Case 3 is the only case that did

not include the tip mass in the equation of motion. (Case 1 does not include it in the original

equation of motion, but after substitution, it is included in the force term of the new equation of

motion.) The effect of tip mass in Case 3 is limited to its role in defining the natural frequencies

and mode shapes.

V. Conclusions

Three different ways of handling the forces applied to the microcantilever of an AFM

were examined. The first case included the forces in the boundary conditions. The second case

26

included them in the equation of motion with boundary conditions like that of a free end. The

third case considered the contact and excitation force in the equation of motion and the inertial

force due to tip mass in the boundary conditions. The equations of motion were derived using

Hamilton’s principle, the natural frequencies and mode shapes were determined by analytical

solution, and the time response was found using the Galerkin method. An AFM with an MPP-

11123-10 microcantilever was used to gather the experimental data. The results show that the

second case was simple to derive but inaccurate. The results of the first and third case showed

very little to no difference between the two theoretical results.

Comparison of the numerical results with the experimental data showed that the

amplitude at resonance was accurate. However, the numerical models did not match up well

away from the resonance frequency. This was most likely due to the simple nature of the

numerical models, e.g., only including linear contact force.

In conclusion, the mathematical models from the first and third case resulted in equally

accurate results. However, it should be noted that including the forces in the equations of motion

and multiplying by a step function is more physically accurate than including it in the boundary

conditions. For example, the microcantilever used to generate the experimental data in this

research work did not have the tip located exactly at the end of the microcantilever. Instead it

was set back 12%. A Heaviside function can be adjusted to compensate for this discrepancy.

Additionally, including the forces in the boundary conditions was considerably more complex

mathematically due to the substitution process. Therefore, including the forces in the equations

of motion is preferable for most situations.

An investigation of the effect of the tip mass on the microcantilever dynamics revealed

that the third case was practically unaffected by a large variation of the tip mass, while the first

27

case showed a significant effect from variation of the tip mass. Therefore, for applications with a

large tip mass, such as biosensing, it may be necessary to use the first case, despite its more

complex derivation.

References

[1] Salehi-Khojin, A., Thompson, G.L., Vertegel, A., Bashash, S., and Jalili, N. (2009).

Modeling piezoresponse force microscopy for low-dimensional material characterization:

theory and experiment. Journal of Dynamic Systems, Measurement, and Control, 131(6),

061107.

[2] Lekka, M. and Wiltowska-Zuber, J. (2009). Biomedical applications of AFM. Journal of

Physics: Conference Series, 146, 012023.

[3] Mahmoodi, S. N. and Jalili, N. (2008). Coupled flexural-torsional nonlinear vibrations of

piezoelectrically-actuated microcantilevers. ASME Journal of Vibration and Acoustics,

130(6), 061003.

[4] Omidi, E., Korayem, A. H., and Korayem, M. H. (2013). Sensitivity analysis of

nanoparticles pushing manipulation by AFM in a robust controlled process. Precision

Engineering, 37(3), 658-670.

[5] Shibata, T., Unno, K., Makino, E., Ito, Y., and Shimada, S. (2002). Characterization of

sputtered ZnO thin film as sensor and actuator for diamond AFM probe. Sensors and

Actuators A, 102, 106-113.

[6] Itoh, T. and Suga, T. (1993). Development of a force sensor for atomic force microscopy

using piezoelectric thin films. Nanotechnology, 4, 2l8-224.

[7] Yamada, H., Itoh, H., Watanabe, S., Kobayashi, K., and Matsushige, K. (1999). Scanning

near-field optical microscopy using piezoelectric cantilevers. Surface and Interface

Analysis, 27(56), 503-506.

[8] Rollier, A-S., Jenkins, D., Dogheche, E., Legrand, B., Faucher, M., and Buchaillot, L.

(2010). Development of a new generation of active AFM tools for applications in liquid.

Journal of Micromechanics and Microengineering, 20, 085010.

[9] Rogers, B., Manning, L., Jones, M., Sulchek, T., Murray, K., Beneschott, B., and Adams, J.

D. (2003). Mercury vapor detection with a self-sensing, resonating piezoelectric cantilever.

Review of Scientific Instruments, 74(11), 4899-4901.

http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=ASMEDL&possible1=Salehi-Khojin%2C+Amin&possible1zone=author&maxdisp=25&smode=strresults&pjournals=AMREAD%2CJAMCAV%2CJBENDY%2CJCNDDM%2CJCISB6%2CJDSMAA%2CJEPAE4%2CJERTD2%2CJETPEZ%2CJEMTA8%2CJFEGA4%2CJFCSAU%2CJHTRAO%2CJMSEFK%2CJMDEDB%2CJMDOA4%2CJMOEEX%2CJPVTAS%2CJSEEDO%2CJOTRE9%2CJOTUEI%2CJVACEK%2CJTSEBV&aqs=true

http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=ASMEDL&possible1=Thompson%2C+Gary+Lee&possible1zone=author&maxdisp=25&smode=strresults&pjournals=AMREAD%2CJAMCAV%2CJBENDY%2CJCNDDM%2CJCISB6%2CJDSMAA%2CJEPAE4%2CJERTD2%2CJETPEZ%2CJEMTA8%2CJFEGA4%2CJFCSAU%2CJHTRAO%2CJMSEFK%2CJMDEDB%2CJMDOA4%2CJMOEEX%2CJPVTAS%2CJSEEDO%2CJOTRE9%2CJOTUEI%2CJVACEK%2CJTSEBV&aqs=true

http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=ASMEDL&possible1=Vertegel%2C+Alexey&possible1zone=author&maxdisp=25&smode=strresults&pjournals=AMREAD%2CJAMCAV%2CJBENDY%2CJCNDDM%2CJCISB6%2CJDSMAA%2CJEPAE4%2CJERTD2%2CJETPEZ%2CJEMTA8%2CJFEGA4%2CJFCSAU%2CJHTRAO%2CJMSEFK%2CJMDEDB%2CJMDOA4%2CJMOEEX%2CJPVTAS%2CJSEEDO%2CJOTRE9%2CJOTUEI%2CJVACEK%2CJTSEBV&aqs=true

http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=ASMEDL&possible1=Jalili%2C+Nader&possible1zone=author&maxdisp=25&smode=strresults&pjournals=AMREAD%2CJAMCAV%2CJBENDY%2CJCNDDM%2CJCISB6%2CJDSMAA%2CJEPAE4%2CJERTD2%2CJETPEZ%2CJEMTA8%2CJFEGA4%2CJFCSAU%2CJHTRAO%2CJMSEFK%2CJMDEDB%2CJMDOA4%2CJMOEEX%2CJPVTAS%2CJSEEDO%2CJOTRE9%2CJOTUEI%2CJVACEK%2CJTSEBV&aqs=true

28

[10] Seo, Y. and Jhe, W. (2008). Atomic force microscopy and spectroscopy. Reports on

Progress in Physics, 71, 016101.

[11] Xu, D., Ravi-Chandar, K., and Liechti, K. M. (2008). On scale dependence in friction:

transition from intimate to monolayer-lubricated contact. Journal of Colloid and Interface

Science, 318, 507-519.

[12] Hölscher, H. (2006). Quantitative measurement of tip-sample interactions in amplitude

modulation atomic force microscopy. Applied Physics Letters, 89, 123109.

[13] Delnavaz, A., Mahmoodi, S. N., Jalili, N., Ahadian, M. M. and Zohoor, H. (2009).

Nonlinear vibrations of microcantilevers subjected to tip-sample interactions: theory and

experiment. Journal of Applied Physics, 106, 113510.

[14] Chu, J., Maeda, R., Itoh, T., and Suga, T. (1999). Tip-sample dynamic force microscopy

using piezoelectric cantilever for full wafer inspection. Japanese Journal of Applied

Physics, 38, 7155–7158.

[15] Asakawa, H. and Fukuma, T. (2009). Spurious-free cantilever excitation in liquid by

piezoactuator with flexure drive mechanism. Review of Scientific Instruments, 80, 103703.

[16] Stark, R. W. and Heckl, W. M. (2000). Fourier transformed atomic force microscopy:

tapping mode atomic force microscopy beyond the Hookian approximation. Surface

Science, 457, 219-228.

[17] Mahmoodi, S. N., Daqaq, M., and Jalili, N. (2009). On the nonlinear-flexural response of

piezoelectrically-driven microcantilever sensors. Sensors and Actuators A, 153(2), 171–

179.

[18] Mahmoodi, S. N., Jalili, N. and Daqaq, M. (2008). Modeling, nonlinear dynamics and

identification of a piezoelectrically-actuated microcantilever sensor. ASME/IEEE

Transaction on Mechatronics, 13(1), 58-65.

[19] Kuo, C., Huy, V., Chiu, C., and Chiu, S. (2012). Dynamic modeling and control of an

atomic force microscope probe measurement system. Journal of Vibration and Control,

18(1), 101-116.

[20] Ha, J., Fung, R., and Chen, Y. (2008). Dynamic responses of an atomic force microscope

interacting with sample. Journal of Dynamic Systems, Measurement, and Control, 127,

705-709.

[21] El Hami, K. and Gauthier-Manuel, B. (1998). Selective excitation of the vibration modes of

a cantilever spring. Sensors and Actuators A, 64, 151–155.

[22] Chang, W. and Chu, S. (2003). Analytical solution of flexural vibration responses on taped

atomic force microscope cantilevers. Physics Letters A, 309, 133-137.

29

[23] Rabe, U., Janser, K., and Arnold, W. (1996). Vibrations of free and surface-coupled atomic

force microscope cantilevers: theory and experiment. Review of Scientific Instruments, 67,

3281.

[24] Zang, Y., Zhao, H., and Zuo, L. (2012). Contact dynamics of tapping mode atomic force

microscopy. Journal of Sound and Vibration, 331, 5141-5152.

[25] Andreaus, U., Placidi, L., and Rega, G. (2013). Microcantilever dynamics in tapping mode

atomic force microscopy via higher order eigenmodes analysis. Journal of Applied Physics,

113, 224302.

[26] Bahrami, A. and Nayfeh, A. H. (2012). On the dynamics of tapping mode atomic force

microscope probes. Nonlinear Dynamics, 70, 1605-1617.

[27] Abdel-Rahman, E. M. and Nayfeh, A. H. (2005). Contact force identification using the

subharmonic resonance of a contact-mode atomic force microscopy. Nanotechnology, 16,

199-207.

[28] Bahrami, A. and Nayfeh, A. H. (2013). Nonlinear dynamics of tapping mode atomic force

microscopy in the bistable phase. Communications in Nonlinear Science and Numerical

Simulation, 18, 799-810.

[29] Jovanovic, V. (2011). A Fourier series solution for the transverse vibration response of a

beam with a viscous boundary. Journal of Sound and Vibration, 330(7), doi: 10.1016/

j.jsv.2010.10.007

[30] Hilal, M. A. and Zibdeh, H. S. (2000). Vibration analysis of beams with general boundary

conditions traversed by a moving force. Journal of Sound and Vibration, 229(2), 377-388.

[31] Sokolov, I., Dokukin, M. E., and Guz, N. V. (2013). Method for quantitative measurements

of the elastic modulus of biological cells in AFM indentation experiments. Methods, 60(2),

202-213.

[32] Darling, E. M., Topel, M., Zauscher, S., Vail, T. P., and Guilak, F. (2008). Viscoelastic

properties of human mesenchymally-derived stem cells and primary osteoblasts,

chondrocytes, and adipocytes. Journal of Biomechanics, 41, 454-464.

[33] Darling, E. M., Zauscher, S., Block, J. A., and Guilak, F. (2007). A Thin-layer model for

viscoelastic, stress-relaxation testing of cells using atomic force microscopy: do cell

properties reflect metastatic potential? Biophysical Journal, 92, 1784-1791.

[34] Darling, E. M., Zauscher, S., Block, J. A., and Guilak, F. (2006). Viscoelastic properties of

zonal articular chondrocytes measured by atomic force microscopy. OsteoArthritis and

Cartilage, 14, 571-579.

30

[35] MPP-11123-10 AFM Probe, Bruker Corporation, (n.d.), Retrieved June 13, 2013 from

https://www.brukerafmprobes.com/Product.aspx?ProductID=3324

[36] Lee, S.I., Howel, S.W., Raman, A., and Reifenberger, R. (2002). Nonlinear dynamics of

microcantilevers in tapping mode atomic force microscopy: a comparison between theory

and experiment. Physical Review B, 66, 115409.

Appendix

The details for derivation Equations (9) and (10) are described below. Initially, w(x,t) is

defined as follows:

),(,, txatxvtxw , (A1)

where

3

3

2

210),( xtaxtaxtatatxa . (A2)

To solve for a(x,t), the first three boundary conditions from Equation (8) are

implemented. This results in:

23 3),( LxxtBtxa . (A3)

To solve for B(t), the fourth boundary condition from Equation (8) is used, which results in the

second order ordinary differential equation:

t

Lm

FtBdtB sin

2 3

2

2 , (A4)

where

3

2

32

2

26

Lm

kLEId

. (A5)

Any solution of Equation (A4) will work in Equation (A3). The following solution is chosen:

tCtB sin , (A6)

where C is defined in Equation (10).

31

Definitions of terms from Equation (22) are as follows:

222221

C , (A7)

1

22

2 CC

, (A8)

213

CCC

, (A9)

½ ,

224½ . (A10)

32

CHAPTER 3

FREQUENCY RESPONSE ANALYSIS OF NONLINEAR TAPPING-CONTACT MODE

ATOMIC FORCE MICROSCOPY

The nonlinear vibrations of the tapping mode atomic force microscopy (AFM) probe are

investigated due to both the nonlinearity in the tip-sample contact force and the curvature of the

microcantilever probe. The nonlinear equations of motion for the vibrations of the probe are

obtained using Hamilton’s principle. In this work, the contact force is considered to be more

dominant while previous works only consider van der Waals force. The nonlinear contact force

is expanded using a Taylor series to provide a polynomial with coefficients that are functions of

the tip-sample distance. The outcome of this work allows the proper distance to be chosen before

scanning to avoid instability in the response. Instability regions must be avoided for accurate

imaging. The results show that the initial tip-sample distance has a major effect on the stability

of the frequency response and force response curves. For the analytical investigation, the mode

shapes of the AFM probe are derived based on the presence of the nonlinear contact force as a

boundary condition at the free-end of the probe. The frequency response curve is obtained using

the method of multiple scales. The results show that the effects of the nonlinearities due to the

probe geometry and contact force can be minimized. Minimizing the effects of nonlinearities

allows for less cumbersome and calculation intensive software packages for AFMs. This

research shows that one possible method of decreasing the nonlinearity effect is increasing the

excitation force; however, this can increase the contact region and is not the best solution for

canceling the nonlinearity effect. The superior method which is the major contribution of this

33

paper is to find the optimal initial tip-sample distance and excitation force that minimize the

nonlinearity effect. It is shown that at a certain tip-sample distance the quadratic and cubic

nonlinearities cancel each other and the system responds linearly.

I. Introduction

Tapping mode atomic force microscopy (AFM) is used for scanning the image of a

physical surface, sensing biological properties of cells, and many other applications. The

common issue in all of these applications is the nonlinear forces applied to the AFM tip due to

tapping or contact with the sample. The nonlinear contact and van der Waals forces at the AFM

tip are the main tools for the AFM to scan or measure. Also, these nonlinearities can cause

regions of instability that must be avoided for accurate imaging or sensing. Therefore, studying

the response of the AFM to these nonlinear forces is significantly important for AFMs and all

other sensing systems that use the AFM microcantilever as a sensor. If the nonlinearities of the

microcantilever system can be minimized, a linear model can be used which allows for quicker

and less bulky AFM software without sacrificing scan precision. Scans are quicker and can be

accomplished by computers with smaller processors. AFM microcantilevers have been widely

used for mechanical property measurement of cells and biological tissues [1-3]. Nonlinear

contact force between the tip and sample plays a critical role in measurement of cell elasticity [4,

5]. In addition, the nonlinear contact force is the tool for other Spectrum Probe Microscopy

Methods (SPM); therefore, studying this nonlinear force improves the sensing function [6-9].

AFM tip forces have been studied in previous research works for improving control of

the measurement systems of the AFMs. In non-contact mode, the AFM tip does not touch the

sample, but it gets close enough to sense the van der Waals force. The governing linear equations

34

of motion for a tapping mode AFM in non-contact mode are derived (only the van der Waals

force was applied and the contact force was not considered) [10]. In that work, the tip-sample

distance was estimated using the Lennard Jones model. Nonlinear behavior of non-contact

tapping mode AFM was studied in the presence of the van der Waals forces to analyze the

stability of the system [11, 12]. The van der Waals potential model was used as a force

expression, and the AFM was considered as a lumped-parameter system. The nonlinear

dynamics of the AFM were investigated for non-contact mode when there are deterministic and

random excitations applied to the tip [13]. The non-contact force was considered to be van der

Waals and was modeled using the Lennard Jones equation. The microcantilever motion was also

modeled as a lumped-parameter system. However, there are studies that considered the

microcantilever as a continuous system. In a series of works, linear and nonlinear responses of a

tapping mode AFM were analytically and experimentally studied for a non-contact system (only

the van der Waals force was considered) [14-18]. The microcantilever was considered as a

continuous system that has nonlinearities in inertia and stiffness due to large curvature of the

microcantilever. Most of the previous works investigated the AFM tip-sample nonlinear

behavior, but they only considered the non-contact, i.e., van der Waals force.

For contact mode AFM, both the contact and van der Waals forces must be included in

the analysis. The tip initially enters the non-contact area where the attractive van der Waals force

affects the tip. Then, it contacts the sample where the force is repulsive. An algorithm for the

reconstruction of the tip-sample interactions in amplitude modulation AFM was introduced [17],

which is based on the recording of amplitude and phase versus distance curves. Analysis of the

contact force showed that the moduli of elasticity of both the tip and the sample affected the

value of the contact force [18]. The nonlinear response of a tapping mode AFM microcantilever

35

probe to a combination of contact and van der Waals forces was studied [19]. Amplitude jumps

and turning points in the response were investigated. The AFM probe was considered to be a

linear continuous microcantilever excited at its base using a piezoelectric actuator. Experimental

investigations were also performed to obtain the amplitude and phase of a silicon AFM tip on a

Highly Oriented Pyrolytic Graphite (HOPG) sample [20]. In a different study, the nonlinear

response of the AFM subject only to contact force was studied for the condition of no probe

motion at the base. Instead, the sample was excited using a piezoelectric actuator [21]. The

primary and subharmonic resonances were investigated to find a method for measuring contact

stiffness of the surface. A research study was performed to model an AFM sensing oscillations of

a microcantilever [22]. The AFM microcantilever moved over an oscillating microcantilever

(without tip) to measure its vibrations. Both vibrations of the AFM probe and microcantilever

were modeled, and the model was verified by experiments.

This work will develop a new nonlinear model for the tapping mode AFM probe as a

continuous microcantilever considering both nonlinearities due to the force and geometry.

Therefore, the effect of all the nonlinear terms will be compared and the dominant nonlinear

terms will be found. The nonlinear tip-sample force includes the contact and van der Waals

forces, while the nonlinear geometry provides the nonlinear inertia and stiffness terms. A novel,

analytical, closed-form solution for the model will be obtained along with the amplitude-

frequency modulation equations, frequency response, and force response. The closed-form

solution assists in distinguishing the nonlinearity effects, and determining that the nonlinear

force has the dominant effect on the response. Results will show that the nonlinear tip-sample

contact force will generate a jump phenomenon; however, tuning the tip-sample distance and

excitation frequency can control the instability. The previous solution was high amplitude

36

excitation which could cause a flat response curve at the tapping mode frequency. The new

analytical method will provide a precise measurement without instability or flat response.

II. Dynamic Modeling of the AFM Probe

In this section, the governing equations of motion for the nonlinear bending vibrations of

an AFM probe in dynamic contact mode are derived with both the van der Waals and contact

forces applied to the tip. Figure 3.1 shows an AFM probe with length, l, in contact mode with a

sample with summation of the contact and van der Waals forces applied to the AFM tip as a

nonlinear force, fN. It is considered that for an arc length element, s, a longitudinal displacement,

u(s,t), and a flexural displacement, w(s,t), emerge in the probe due to bending. To describe the

AFM microcantilever dynamics, two coordinate systems, i.e., the inertial system (x, z) and the

local principal system (ξ, ζ) are utilized. The relationship between the inertial and the local

coordinates is described by the Euler rotation, ψ(s,t).

Figure 3.1. Schematic of the AFM microcantilever probe.

For an element of length ds, ψ can be written as:

u

w

1arctan , (1)

x

z AFM microcantilever

AFM tip

z0

Piezoelectric actuator

f(t)

fN

x

37

where the prime denotes the derivative with respect to s. The Euler-Bernoulli beam theory is

utilized to derive the strain-displacement relations. The reason for choosing this theory is that the

AFM microcantilevers have a large length to height ratio. Therefore, it can be assumed that the

angular deformation due to shear is negligible when compared to the flexural deformations due

to bending, and that the rotation of a differential element is very small relative to its translation.

Using Figure 3.1, the position vector of an arbitrary point on the neutral axis of the

microcantilever can be given by x0 er s . The position of the same point after deformation

becomes ζξ eer wus . Using these position vectors, the strain along the neutral axis for an

element, ds, can be written as

11 22

21

00

21

wu

ssss

rrrr. (2)

The angular velocity of the bending can be obtained using equation (1),

HOTwwuwuww 2 , (3)

where HOT stands for higher order terms. Similarly, the curvature of the microcantilever is

calculated as

HOTwwuwuww 2 , (4)

and the axial strain at a point using the coordinates (ξ, ζ) is expressed as

11 . (5)

Using the constitutive equations, the axial stress induced in a differential element is

expressed as 1111 mE , where Em is the modulus of elasticity of the microcantilever. Now, the

total strain energy of the microcantilever is written as

ll

A

dswwuusEAdsdAU0

422

01111 25.0)(

2

1

2

1, (6)

38

where dA is the cross sectional area of a microcantilever element, and A(s)=b.h, where b denotes

the width and h denotes the thickness. The kinetic energy of the system can be written as

dswumTl

0

22

2

1 , (7)

where m is the mass per unit length of the microcantilever. Using equations (4)-(7), the

Lagrangian of the system, L T U is expressed as

,222

25.02

1

2222

0

42222

dsuwwuwwwwIE

wwuubhEwumL

mm

l

m

(8)

where 12

3bhIm . It is considered that the length of the AFM microcantilever does not elongate at

its neutral axis, thereby the strain along the neutral axis is equal to zero and equation (2) reduces

to [23]

11 22 wu . (9)

Equation (9) is used to relate the bending and the longitudinal vibrations of the

microcantilever. To obtain the equations of motion for the bending vibrations of the AFM

microcantilever probe, Hamilton’s principle which states that 2

1

0t

tL W dt is used, where

indicates the variation, and W is the work of external forces and damping. The variation of work

on the system can be expressed as

ll

N

l

dxwwBdxwtlFdxwtfW000

,)( , (10)

where B is the damping coefficient, f(t) is the excitation force, and FN is the nonlinear tip force.

Using equations (8)-(10), performing necessary analysis, and using Hamilton’s principle the

equations of motion and the associated boundary conditions are obtained as

39

)(0

2 tfwwIEwwIEdsdswwwwmwBwm mmmm

s

l

s

, (11)

0 ww at s=0; 0 wandFwIE Nmm at s=l . (12)

Equation (11) is the nonlinear equation of motion governing the AFM microcantilever

probe. The nonlinear tip force appears in the boundary condition of the equation of motion. The

direct excitation of the microcantilever (using the piezoelectric actuator) emerges as f(t). The

excitation force is a periodic force and in general is expressed as

cceftf ti

02

1)( , (13)

where f0 is force amplitude, is the frequency of excitation, and cc stands for the complex

conjugate of the previous terms. The nonlinear tip force is presented based on the distance of the

AFM tip to sample. There will be only a van der Waals force if the distance is not close enough

to make contact. However, if the tip contacts and indents into the sample (if the tip-sample

distance is less than a constant value of h0), both contact and van der Waals forces are present.

Therefore, the tip-sample nonlinear force can be expressed as [24]

,3

4

6

6

0

23

0

*

2

0

02

hhhhREh

HR

hhh

HR

FN (14)

where H is the Hamaker constant, R represents the microcantilever tip radius, h is the position of

the tip and is defined as )(,)( 00 tdtlwzth , where z0 is the distance between the tip of the

AFM microcantilever probe and the sample at equilibrium as shown in Figure 3.1, and d0 is the

base displacement due to f(t). However, since the length of the AFM microcantilever is long

compared to the vertical excitation then )(, 0 tdtlw . Therefore,

40

tlwzth ,)( 0 . (15)

Also, E* in equation (14) is the effective elastic modulus and is given by

1

* 11

s

s

t

t

EEE , (16)

where E is the modulus of elasticity, is the Poisson’s ratio, and indices t and s indicate tip and

sample, respectively. The modulus of elasticity of the tip is assumed to be equal to the modulus

of elasticity of the microcantilever. Using a Taylor series expansion, equation (14) is expressed

as

.

24

1

4

1

3

2

4152010

0

3

0

2

00

22

0

1

0

32

1

hhh

h

h

h

h

h

h

hhhhh

FN

(17)

where 6

1

HR , and

23

0

*

2 2 hRE . In equation (17), the first argument has been expanded

around 1, and the second argument has been expanded around zero to study the contact force for

the indentation condition. Substituting equation (15) into (17) yields

0

0

1

2

0

2

3

0

3

),(),(),(a

z

tlwa

z

tlwa

z

tlwaFN

, (18)

where

l

l

whzh

whzzzz

a00

32

22

0

1

00

3

0

2

001

0

24

1

4

1

3

2

4152010

,

l

l

whz

whzzz

a00

32

2

00

2

001

1

8

1

2

1

123020

,

l

l

whz

whzz

a00

32

2

0001

2

8

1

4

1

1215

,

l

l

whz

whz

a00

32

001

3

24

4

, (19)

41

and 0

0

h

z .

III. Mode Shape Analysis

Before further analysis, the following non-dimensional quantities are introduced to

rewrite the equations of motion in non-dimensional form,

l

ss ˆ ,

0

ˆz

ww ,

mmImE

lBB

3

ˆ , 0

3

ˆzIE

laa

mm

ii ,

0

4

00ˆ

zIE

lff

mm

, 4

ˆml

IEtt mm , and

mmIE

ml4

ˆ . (20)

The non-dimensional form of equations (11) and (12) are rewritten as

)(ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ

1

ˆ

0

2iv tfwwwsdsdwwwwwwBws s

, (21)

0ˆˆ ww at 0ˆ s ; 0ˆˆˆˆˆˆˆˆˆ01

2

2

3

3 wandawawawaw at 1ˆ s . (22)

where

2

0

l

z. For analysis of the linear mode shapes, the nonlinear terms, damping terms,

and forcing functions are considered to be zero. In addition, the response is assumed to be a

harmonic motion that can be expressed as

ti

linear extswˆ

)ˆ,ˆ(ˆ , (23)

where is the frequency of the response. Substituting equation (23) into equations (21) and (22)

and considering the linear, non-forced, undamped motion results in

0ˆ4iv s , (24)

0 at 0ˆ s ; 0ˆˆ01 andaa at 1ˆ s , (25)

42

where 2 . By solving equation (24), the linear mode shape is obtained as

sssscs ˆsinhˆsin

sinhsin

coshcosˆcosˆcoshˆ , (26)

where c is a constant coefficient and is obtained by normalizing the equation (26). The

coefficient of and consequently the natural frequency are obtained by solving the following

expression

0sinhsin2

ˆsinhcossincoshˆcoshcos1 0

1

3 c

aa . (27)

The 0a coefficient represents a constant force, and it generates a constant deformation; therefore,

by changing the equilibrium point of the system this term can be removed from the force and

mode shape. Therefore, in the following sections, 0a is considered to be equal to zero.

IV. AFM Response Analysis to Primary Excitation

In this section, the equation of motion and boundary conditions presented in equations

(21) and (22) are solved. The method of multiple scales is used to find the modulation equations

for the nonlinear system and to obtain an analytical solution for the frequency response to a

harmonic base excitation at the primary resonance frequency in the presence of the van der

Waals and contact forces at the AFM microcantilever probe tip. Towards that objective and

using as a small scaling parameter, the equation of motion and boundary conditions are

rewritten as

)(ˆˆˆˆˆˆˆˆˆˆˆˆˆˆ 22ˆ

1

ˆ

0

22iv2 tfwwwsdsdwwwwwwBws s

, (28)

0ˆˆ ww at 0ˆ s ; 0ˆˆˆˆˆˆˆˆ1

2

2

32

3 wandwawawaw at 1ˆ s . (29)

43

Time and derivatives with respect to time are expanded into multiple time scales as

2

2

10ˆ TTTt

2

2

10

2

2

10ˆ

DDDdT

d

dT

d

T

d

td

d (30)

It is assumed that the solution to the equation of motion has the form of

),,,(),,,(),,,();,(ˆ2102

2

21012100 TTTswTTTswTTTswtsw (31)

Substituting equations (30) and (31) into equations (28) and (29), and equating coefficients of

like powers of ε yields

Order of (0):

0iv

00

2

0 wwD , (32)

000 ww at 0ˆ s ; 0ˆ0010 wandwaw at 1ˆ s , (33)

Order of (1):

010

iv

11

2

0 2 wDDwwD , (34)

011 ww at 0ˆ s ; 0ˆˆ1

2

02111 wandwawaw at 1ˆ s , (35)

Order of (2):

,)(ˆˆˆ

ˆ22

000

ˆ

1

ˆ

0

2

0000

001100

2

1020

iv

22

2

0

tfwwwsdsdwwww

wDBwDDwDwDDwwD

s s

(36)

022 ww at 0ˆ s ; 0ˆ2ˆˆ2102

3

03212 wandwwawawaw at 1ˆ s . (37)

The solution to the linear eigenvalue problem of equations (32) and (33) can be expressed

as

cceTTAswTi

nnn

0

210 ,)ˆ( , (38)

44

where An is a complex function which will be determined later in the analysis and n is the

number of the mode that has been excited. It should be noted that equation (38) shows the steady

state response in the presence of damping since all other modes that are not excited will vanish

[25]. Substituting equation (38) into equation (34) yields

cceTTADsiwwD

Ti

nnnn

0

211

iv

11

2

0 ,)ˆ(2 . (39)

In order to eliminate the secular term in equation (39), the An coefficient must be

independent of T1 or 0, 211 TTAD n , therefore An is only a function of T2. To obtain a solution

for w1, equation (38) is substituted into the boundary conditions in equation (35),

1ˆ)1(ˆˆ2

221110

satcceTAawaw

Ti

nnn . (40)

The solution of equation (34) with boundary conditions of (35) and (40) is obtained as

ccAAsgeTAsgaw nn

Ti

nnn

)ˆ()ˆ()1(ˆ

2

2

2

2

1

2

210 , (41)

where

,sinhcossincoshˆ2coshcos12

ˆsinˆsinhcoshcossinsinhˆcoshˆcos)ˆ(

2222122

3

2

222222221

nnnnnnn

nnnnnnnn

a

sssssg

,ˆ26

ˆ3ˆ)ˆ(

1

23

2a

sssg

.22

2 nn (42)

In order to solve equation (36), which contains the nonlinear terms, equations (13), (38),

and (41) are substituted into equation (36). The resultant equation is

ccefeAAeAeAAeA

sdsdesAiBADiwwD

TTiTi

nn

Ti

nnnn

Ti

nnn

Ti

nn

s s

nn

Ti

nnnnn

nnnnn

n

200000

0

0

23322332

ˆ

1

ˆ

0

2

2

iv

220

ˆ2

13

ˆˆ2ˆˆ2

(43)

and the boundary conditions are

022 ww at 0ˆ s ; 02 w at 1ˆ s and

45

cceAAggeAgaeAAeAawawti

nn

ti

nn

ti

nn

ti

nnnnnn

2

21

33

1

32

2

2333

3212 2ˆ23ˆˆ at 1ˆ s .(44)

Note that in equation (43) the closeness of the excitation frequency to the natural

frequency was shown using

2

n , (45)

where is a small detuning parameter. The solvability condition demands that the normal modes

n corresponding to the natural frequencies ωn are orthogonal to the right hand side of equations

(43) and (44); thus

2

06

2

5432

2

12ˆ

2

123324ˆ2

Ti

nnnnnnnnnnnnn efkAAkkkkkAiBADi

, (46)

where

sdsdsdks s

nnnnˆˆˆ

4

ˆ

1

ˆ

0

21

01

, (47a)

sdk nnnnnˆ

4

1

02

, (47b)

1

0

3

33ˆˆ1ˆ

4

1sdsak nnn , (47c)

1

01

42

24ˆˆ)1(1ˆ

2

1sdsgak nnn , (47d)

1

02

42

25ˆˆ)1(1ˆ

2

1sdsgak nnn , (47e)

1

06

ˆˆ sdsk nn . (47f)

In order to find the amplitude, An, using equation (46), the amplitude is expressed in polar

form as

2

2

1 Ti

nnneA

, (48)

46

where n represents the amplitude of the response and θn represents the phase, and both are real

time functions. Substituting equation (48) into (46) and separating the real and imaginary parts,

the modulation equations are obtained as

n

n

nnn f

kBD

sinˆ

2ˆ

2

10

62 , (49)

n

n

nnnnnnnn

n

nnn fk

kkkkkD

cosˆ2

23322

10

63

5432

2

12 . (50)

For the steady state condition, the derivatives of amplitude and phase vanish in equations

(49) and (50). Therefore the frequency response function can be obtained by combining the two

equations as

2

0

2

6

23

5432

2

1

2 ˆ23322ˆ fkkkkkkB nnnnnnnnnnnn . (51)

For a given excitation amplitude f0, equation (51) can be solved numerically for the associated

response amplitude, n. The phase of the response, θn, can then be obtained utilizing either

equation (49) or (50) considering steady state conditions, i.e., 022 nn DD .

V. Stability Analysis of the Solution

In order to investigate the stability of the resulting steady-state solutions, the eigenvalues,

Λ, of the Jacobian of the modulation equations, i.e., equations (49) and (50), are evaluated at the

roots. The eigenvalues can be obtained by calculating the determinant of the following matrix:

nnnnnnnnnn

nnn

fkkkkkk

fkBIJ

sin233232

cosˆ

06

2

5432

2

1

06 . (58)

47

The steady-state solutions are asymptotically stable if all the eigenvalues, Λ, contain negative

real parts and are unstable if at least one eigenvalue has a positive real part. The eigenvalues

corresponding to the trivial solution are

06

22

24

ˆ

2

ˆfk

BBnn

nn

, (59)

and the threshold for the unstable trivial solution can be expressed as

4

ˆ

4

2

22

2

0

2

6 Bfk

nn

n

. (60)

The interval [−σ, σ] of the above equation defines the frequencies within the trivial solution

which are unstable. For a force sweep, the stability boundaries for forcing levels are defined as

222

6

0ˆ2 nn

n

n Bk

f

, (61)

and the trivial solution in the interval [−f0, f0] is always unstable.

VI. Results

In this section, the response of the AFM microcantilever probe to nonlinear force and

nonlinear curvature is investigated. First, the expanded model for nonlinear force and the mode

shape are numerically verified to ensure the effectiveness of the analytical model. In order to

compare the expanded nonlinear force model presented in equation (18) with the model in

equation (14), properties of a known AFM microcantilever probe and a sample surface are used

to plot both models. The properties of the sample material are presented in Table 3.1, and the

properties of the AFM microcantilever probe are indicated in Table 3.2.

48

Table 3.1. Sample properties [24].

Property Value

s 0.17

H (J) 6.410-20

Es (GPa) 70

h0 (nm) 0.166

Table 3.2. Geometric and material properties of the AFM microcantilever probe [23, 26].

Property Value

t 0.28

ρb (kg/m3) 2330

Eb (GPa) 129

hb (μm) 2.75

l (μm) 225

R (nm) 8

wb (μm) 28

Figure 3.2 shows the nonlinear contact force between the tip of the AFM microcantilever

and the sample using both equations (14) and (18). The former equation is shown by the solid

line and the latter equation, which is the Taylor expansion of equation (14), is depicted using

circles.

It is observed that the estimated nonlinear force smoothly matches the curve of the

nonlinear force in the contact area (distances smaller than h0=0.16nm). The mode shape of the

49

AFM microcantilever probe presented in equation (26) is similar to the mode shape of a

microcantilever with zero boundary conditions; however, the natural frequency of the system in

equation (27) is a function of the coefficients of the forcing terms, i.e., a1, and a0. The effect of

the a0 coefficient is assumed to be compensated by initial deformation of the AFM

microcantilever. Figure 3.3 shows the first resonance frequency for different values of a1.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-4

-2

0

2

4

6

8

10

12

14

16

Contact distance (nm)

tip-s

am

ple

forc

e (

nN

)

Figure 3.2. Nonlinear tip-sample force; solid line is the force presented in equation (14); and

circles show the force based on equation (17).

The value of a1 is dependent on the variations of 0

0

h

z . The value of h0 was given in

Table 3.1, but the value of z0 is in order of 100 to 500 times more than h0 in real tapping mode

AFM. In this case, the a1 coefficient will be in the interval of a1=[-2.5, 2.5]. Figure 3.4 shows the

natural frequency for this interval.

The next part of this section deals with investigation of the AFM response to nonlinear

force using equation (51). The response of the AFM microcantilever probe to tapping mode

excitation is studied considering three different values of (defined in equation 19), i.e., =

120, 180, and 210. Figure 3.5 shows the frequency response of the AFM microcantilever when

the damping ratio (which has been experimentally measured [23]) is =0.034.

50

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

a1

1

Figure 3.3. First natural frequency of the non-dimensional equations of motion for different

values of a1.

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.51

1.5

2

2.5

3

3.5

4

4.5

5

a1

1

Figure 3.4. Non-dimensional natural frequency for practical values of a1.

Figure 3.5a shows that for certain values of , the linear force dominates the nonlinear

force, the response appears linear, and no frequency shift or jump is observed. However, as

increases in Figures 3.5b and 3.5c, the response becomes more unstable. In the other words, as

the distance between tip and sample increases (thus the larger amplitude of vibrations), the

instability of the response increases. However, it should be noted from equation (20) that force is

also non-dimensionalized using tip-sample distance (z0). Therefore, it cannot be concluded that

51

the larger z0 causes instability, and the force should be considered too. Figure 3.5 shows that the

frequency is a function of the tip-sample distance and the excitation force which agrees with the

results presented in reference [27].

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

n

(a)

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

n

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

n

(b) (c)

Figure 3.5. Frequency response curve for the first mode, a) =120, b) =180, and c) =210.

Figure 3.6 shows the force response of the AFM microcantilever for various values of

force. The tip-sample distance is kept constant at =210 for this case, and the excitation force has

been increased. The growth in the force amplitude will increase the response amplitude and

cancels the frequency shift due to nonlinearity. However, as can be seen in Figures 3.6a and

3.6b, there will be a range of frequencies in which the tip contacts the sample. Figure 3.6b shows

that this range will increase when the force has increased. Although it solves the nonlinearity

52

problem, this contact range will reduce the accuracy of measurement. For cases that the sample

is soft such as bio-samples, a large contact force can damage the sample. If contact force and tip-

sample distance can be adjusted to optimum values, then the tip will not penetrate the sample,

and the linear force will also be dominant. These results confirm the experimental results of

references [19, 20].

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

n

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

n

(a) (b)

Figure 3.6. Frequency response curve for the first mode (=210), a) nondimensional force is

f0=0.31, b) the dashed line represents f0=0.31, and the solid line represents f0=0.34.

For constant values of =120, and =0.02, the force response curves are shown in Figure

3.7. It is clearly shown in Figure 3.7a that the force response is linear before the tip reaches the

contact region. From Figure 3.7a, it can be concluded that the nonlinear term due to geometry

will have less effect on the response since the tip-sample distance is small in comparison to the

length of the microcantilever. In addition, for smaller tip-sample distances the value of

decreases and eventually vanishes. Therefore, the dominant nonlinear terms for small tip-sample

distances are nonlinear force terms and the nonlinear curvature can be ignored. Figure 3.7b

shows the amplitude in the contact region. It is shown that for some values of force, the

amplitude becomes unstable. However, as discussed before, this can be avoided by choosing

proper tip-sample distance. The dashed line inside the box shows the region in which the

53

response jumps from the dashed to solid line and causes a nonlinear response similar to Figures

3.5b and 3.5c. However, the dashed lines outside the box can jump to higher forces that cause

contact, and the response will be similar to the ones in Figure 3.6.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

f

n

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

f

n

(a) (b)

Figure 3.7. Force response curve for the first mode when =120 (the dashed line shows the

unstable region).

The force response curves for a constant =210 presented in Figure 3.8 confirm the

results of Figure 3.7. Although increasing the force generates a wider unstable region, the jump

in that region leads to contact and will not result in instability.

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

f

n

Figure 3.8. Force response curve for the first mode when =210 (the dashed line shows the

unstable region).

54

VI. Conclusions

The AFM is a highly accurate tool used for three dimensional imaging. Microcantilevers

like the ones in the AFM are widely used for a variety of applications. A reliable and accurate

model of the AFM microcantilever probe dynamics is necessary for accurate imaging or for

accurate readings in the many other applications. This paper studied a common issue among

these applications: the nonlinear forces applied to the AFM tip due to tapping or contact with the

sample. The nonlinear contact and van der Waals forces at the AFM tip are the main tools for the

AFM to scan or measure, but the nonlinearities can cause regions of instability that must be

avoided for accurate imaging or sensing. This work investigated the response of the AFM to

these nonlinear forces. This study is important for AFMs and all other sensing systems that use

the AFM microcantilever as a sensor both to avoid regions of instability in the response and to

minimize the effects of the nonlinearities.

The nonlinear equations of motion for nonlinear vibrations of the AFM microcantilever

probe due to the nonlinear curvature and contact force were derived using the energy method and

Hamilton’s Principle. The nonlinear contact force was expanded using a Taylor series to provide

a polynomial with coefficients that are functions of the probe-sample distance. The separation

method was used to derive the mode shapes of the AFM microcantilever probe based on the

presence of the nonlinear contact force as a boundary condition at the free-end of the

microcantilever. The results showed that the natural frequency of the microcantilever is

dependent on the coefficient of the linear term of the contact force. The results also showed that

the natural frequency is a function of the initial tip-sample distance, and for larger distances, the

values of the natural frequency can be much larger than the natural frequency of a

microcantilever with zero boundary conditions at the free end.

55

The frequency response curve was obtained using the method of multiple scales so that

the effect of the nonlinear terms could be analytically investigated. The nonlinear terms due to

the nonlinear tip-sample force appear in the form of quadratic and cubic nonlinearities, while the

nonlinear inertia and stiffness terms are cubic. The results showed that the effect of the nonlinear

force terms is dominant for small amplitude vibrations in comparison to the effect of the other

nonlinearities. Results showed that choosing optimal excitation force and tip-sample

displacement significantly decreased the effect of the nonlinear terms and provided a linear

response. A smaller tip-sample displacement decreased the nonlinear response. In addition,

increasing the force resulted in faster contact and made the response stable, but decreased the

accuracy of the measurement. The results showed that, in general, the best way to avoid a

nonlinear response is to set the initial tip-sample distance (before starting the tapping) to a

constant, and then increase the excitation amplitude to reach the stable region, but applying more

force is not recommended.

References

[1] Li, Q. S., Lee, G. Y. H., Ong, C. N., and Lim, C. T., (2008). AFM indentation study of

breast cancer cells. Biochemical and Biophysical Research Communications, 374, 609-613.



[3] Krimizis, D. and Logothetidis, S. (2010). Atomic force microscopy probing in the

measurement of cell mechanics. International Journal of Nanomedicine, 5, 137-145.

[4] Li, Q. S., Lee, G. Y. H., Ong, C. N., and Lim, C. T., (2009). Probing the elasticity of breast

cancer cells using AFM. ICBME 2008 Proceedings, 23, 2122-2125.

[5] Kuznetsova, T. G., Starodubtseva, M. N., Yegorenkov, N. I., Chizhik, S. A., and Zhdanov,

R. I. (2007). Atomic force microscopy probing of cell elasticity. Micron, 38, 824-833.

56

[6] Kalinin, S. V., Rar, A., Jesse, S. (2006). A decade of piezoresponse force microscopy:

progress, challenges, and opportunities. IEEE Transactions on Ultrasonics, Ferroelectrics,

and Frequency Control, 53, 2226- 2252.




061107.

[8] Salehi-Khojin, A., Jalili, N., and Mahmoodi, S. N. (2009). Vibration analysis of vector

piezoresponse force microscopy with coupled flexural-longitudinal and lateral-torsional

motions. Journal of Sound and Vibration, 322, 1081-1099.

[9] Morozovska, A. N., Eliseev, E. A., and Kalinin, S. V. (2007). The piezoresponse force

microscopy of surface layers and thin films: Effective response and resolution function.

Journal of Applied Physics, 102, 074105.

[10] Jalili, N., Dadfarnia, M., and Dawson, D. M. (2004). A fresh insight into the

microcantilever-sample interaction problem in non-contact atomic force microscopy.

ASME Journal of Dynamic Systems, Measurement, and Control, 126, 327-335.

[11] Couturier, G., Nony, L., Boisgard, R., and Aimé, J.-P. (2002). Stability analysis of an

oscillating tip–cantilever system in NC-AFM. Applied Surface Science, 188, 341-348.

[12] Nony, L., Boisgard, R., and Aimé, J.-P. (2001). Stability criterions of an oscillating tip-

cantilever system in dynamic force microscopy. The European Physical Journal B, 24,

221-229.

[13] Pishkenari, H. N., Behzad, M., and Meghdari, A. (2008). Nonlinear dynamic analysis of

atomic force microscopy under deterministic and random excitation. Chaos, Solutions &

Fractals, 37, 748-762.

[14] Delnavaz, A., Mahmoodi, S. N., Jalili, N., and Zohoor, H. (2010). Linear and non-linear

vibration and frequency response analyses of microcantilevers subjected to tip–sample

interaction. International Journal of Non-Linear Mechanics, 45, 176–185.




[16] Delnavaz, A., Mahmoodi, S. N., Jalili, N., and Zohoor, H. (2010). Linear and nonlinear

approaches towards amplitude modulation atomic force microscopy. Current Applied

Physics, 10, (2010) 1416-1421.







57

[18] Rabe, U., Amelio, S., Kester, E., Scherer, V., Hirsekorn, S., and Arnold, W. (2000).

Quantitative determination of contact stiffness using atomic force acoustic microscopy.

Ultrasonics, 38, 430–437.

[19] Lee, S. I., Howel, S. W., Raman, A., and Reifenberger, R. (2003). Nonlinear dynamic

perspectives on dynamic force microscopy. Ultramicroscopy, 97, 185-198.






199-207.

[22] Llic, B., Krylov, S., Bellan, L. M., and Craighead, H. G. (2007). Dynamic characterization

of nanomechanical oscillators by atomic force microscopy. Journal of Applied

Physics, 101, 044308.

[23] Mahmoodi, S. N. and Jalili, N. (2007). Non-linear vibrations and frequency response

analysis of piezoelectrically driven microcantilevers. International Journal of Non-Linear

Mechanics, 42, 577-587.

[24] Stark, R. W., Schitter, G., and Stemmer, A. (2003). Tuning the interaction forces in tapping

mode atomic force microscopy. Physical Review B, 68, 085401.

[25] Mahmoodi, S. N., Khadem, S. E., and Rezaee, M. (2004). Analysis of nonlinear mode

shapes and natural frequencies of continuous damped systems. Journal of Sound and

Vibration, 275, 283-298.

[26] FESPA Scilicon AFM Probe, Bruker Corporation, (n.d.), Retrieved January 7, 2012 from

http://www.brukerafmprobes.com/p-3734-fespa.aspx

[27] Hu, S. Q. and Raman, A. (2007). Analytical formulas and scaling laws for peak interaction

forces in dynamic atomic force microscopy. Applied Physics Letters, 91, 123106.

58

CHAPTER 4

PARAMETER SENSITIVITY ANLAYSIS OF NONLINEAR PIEZOELECTRIC PROBE IN

TAPPING MODE ATOMIC FORCE MICROSCOPY FOR MEASUREMENT

IMPROVEMENT

The equations of motion for a piezoelectric microcantilever are derived for a nonlinear

contact force. The analytical expressions for the natural frequencies and mode shapes are

obtained. Then, the method of multiple scales is used to analyze the analytical frequency

response of the piezoelectric microcantilever probe. The effects of the nonlinear contact force on

the microcantilever beam’s frequency and amplitude are analytically studied. The results show a

frequency shift in the response resulting from the force nonlinearities. This frequency shift

during contact mode is an important consideration in the modeling of AFM mechanics for

generation of more accurate imaging. Also, a sensitivity analysis of the system parameters on the

nonlinearity effect is performed. The results of the sensitivity analysis show that it is possible to

choose parameters such that the frequency shift minimizes. Certain parameters such as the tip

radius, microcantilever beam dimensions, and modulus of elasticity have more influence on the

nonlinearity of the system than other parameters. By slightly changing only two parameters – tip

radius and microcantilever length – a 20% reduction in the nonlinearity effect was achieved.

I. Introduction

AFM was initially invented and used for scanning the image of a physical surface in

nano-scale. The sensing process is performed by a microcantilever that contacts or taps the

59

sample. Recently, microcantilever probes have been used extensively for Friction Force

Microscopy (FFM), Lateral Force Microscopy (LFM), Piezo-response Force Microscopy (PFM),

biosensing, and other applications [1-3]. In most AFMs the microcantilever is excited using a

piezoelectric tube actuator at the base of the probe and a laser measurement system. However,

some microcantilevers have a layer of piezoelectric material, which is usually ZnO [4] or Lead

Zirconate Titanate (PZT), on one side of the microcantilever for actuation purpose. The

application of the piezoelectric microcantilever is widespread; it has been used for force

microscopy, biosensing, and chemical sensing [5-8]. Piezoelectric microcantilevers for surface

scanning and imaging of different samples are an improvement for sensing in AFMs [9]. The

piezoelectric actuators and sensors along with an electrical circuit can replace the bulky and

expensive laser measurement system [10, 11].

However, an accurate understanding of the microcantilever motion and tip-sample force

is needed to provide an inclusive sensing model. If the nonlinearities of the microcantilever

system can be minimized, a linear model can be used which allows for quicker and less bulky

AFM software without sacrificing scan precision. This research work investigates two major

parameters in the sensing process that should be carefully studied to guarantee the accuracy of

measurement. The first parameter is the nonlinear force between microcantilever tip and sample,

and the second parameter is the effect of the piezoelectric layer on the motion of the

microcantilever.

The nonlinear force between tip and sample may consist of two forces: a van der Waals

force and a contact force. In a series of work, the nonlinear response of the non-contact tapping

mode AFM was studied considering a van der Waals force at the tip [12-14]. In those works, the

microcantilever did not have a piezoelectric layer and the force was estimated using the Lennard

60

Jones model. The nonlinear response of a tapping contact AFM microcantilever probe to a

combination of contact and van der Waals forces was investigated [15] to study the jump

phenomena and stability of the response. The AFM microcantilever was considered to be a linear

continuous microcantilever excited at the base using a piezoelectric actuator. Based on the

recording of the amplitude and phase, an algorithm for reforming the tip-sample interactions in

the amplitude modulation was constructed and the response studied [16]. In another work, the

nonlinear response of the AFM microcantilever probe to the contact force was studied

considering that the microcantilever has no motion at the base, but the excitation is applied to the

sample from a piezoelectric actuator under the sample [17].

The dynamics of the piezoelectric microcantilever have been experimentally and

analytically studied in some research works. Experimental investigations have been performed in

air and liquid on dynamic AFMs and the frequency response of the systems were obtained [18].

The nonlinear dynamics of a piezoelectric microcantilever have been studied considering the

nonlinearity due to the curvature and piezoelectric material [19, 20]. The microcantilever model

was developed and the equations of motion were analytically solved. However, only free

vibrations were considered. In other works, the linear dynamic models have been developed for

contact piezoelectric AFM microcantilever probes and numerically solved [21]. These works did

not consider the obtained analytical model for the nonlinear response of the system. They only

considered the nonlinear contact force and piezoelectric effect on the microcantilever. This work

will develop a new nonlinear model for the tapping mode AFM microcantilever probe as a

continuous microcantilever considering the nonlinearities due to the force and the effect of the

piezoelectric layer on the motion of the microcantilever. A novel, analytical, closed-form

solution for the model will be obtained along with the frequency response. Results will show that

61

the nonlinear tip-sample contact force will generate a jump phenomenon; however, selection of

certain parameters can cause this frequency shift to be minimized.

II. Dynamics of the Piezoelectric Cantilever with Tip Force

The governing equations of motion for the nonlinear dynamics of a piezoelectric

microcantilever in dynamic contact mode are derived in this Section. The schematic of the

microcantilever motion in dynamic contact mode is shown in Figure 4.1. The microcantilever

length is l, the bending displacement is w(x,t), and fN is the nonlinear force due to tip contact with

a sample, which is the summation of the contact and van der Waals forces. The coordinate

system (x, z) describes the dynamics of piezoelectric microcantilever, and t denotes time.

Figure 4.1. Schematic of the piezoelectric microcantilever motion.

In order to utilize the energy equations to obtain the equations of motion for the

piezoelectric microcantilever, the strain energy in the system should be defined. For a

microcantilever beam, the constitutive equation is written as bbb E , where and are the

stress and strain along the x-axis, and the index b stands for the microcantilever beam. Eb is the

corrected modulus of elasticity and is defined as 2b

*bb 1EE , where *

bE is the modulus of

62

elasticity and ν is Poisson’s ratio for the microcantilever. The constitutive equation for the

piezoelectric layer is given by [2]

p

pppph

tVdEE

)(31 , (1)

where 2p

*pp 1EE , *

pE is the modulus of elasticity for the piezoelectric material, index p

stands for the piezoelectric layer, d31 is the piezoelectric constant relating charge and strain, h is

the thickness of the layer, and V(t) is the applied voltage to the piezoelectric actuator and is

defined as

ccevtV ti

0½)( , (2)

where v0 is the constant voltage amplitude, is the voltage frequency (excitation frequency for

the piezoelectric actuator), and cc is the complex conjugate of the previous terms.

To solve for the equations of motion for the system, consider an infinitesimal length of

the microcantilever. From this, the moment of the microcantilever M can be found as

pppbbpnbbbp IExlHIExlHxlHzbhIExlHxM )()()()()( 2 , (3)

where zn represents the modified neutral axis [9], I is the moment of inertia, b is the width, and

H(n) is the Heaviside function. Again considering an infinitesimal length of the microcantilever,

l0, and summing the forces in the lateral direction results in

xxMxx

wtVxK

xl

2

2

p0

, (4)

where

p

nbpp

pph

bzhhdExlHxK

4

2)()(

31 . (5)

63

The external nonlinear force, fN, is the force due to the van der Waals and contact force

between the microcantilever tip and sample and is defined as [22]

.zzRE3

4

6

RA

zz6

RA

f

l

23

l

*

2

h

l2

l

h

N

(6)

The first term is the van der Waals force that is applied to the tip when the tip-sample distance is

larger than the constant distance of , and the second term is the contact force for the tip-sample

distance of less than . In Equation (6), Ah is the Hamaker constant, R represents the

microcantilever tip radius, zl is the position of the tip and is defined as t,lwz)t(z 0l , where z0

is the distance between the tip of the AFM and the sample at equilibrium as shown in Figure 4.1,

and E* is the effective modulus of elasticity [19].

In order to be able to study the nonlinear effect of the tip force on the system, a Taylor

series is used to rewrite the forcing Equation (6) into polynomial form. The first argument is

expanded around since this is the point that contact will start, and the second argument is

expanded around zero to study the contact force for the indentation condition. Therefore, the

nonlinear force is rewritten as

,zz

24

1z

4

1z

3

2cc

zz

4z

15z

2010c

f

l

3

l

2

ll21

l

3

l

2

ll1

N

(7)

where 2

1 6/ RAc h and 3*

2 2 REc . The higher order terms are not considered since the

displacement zl is relatively small and higher order terms become negligible. Applying Newton’s

64

second law in the lateral direction, considering the moment generated in the microcantilever by

the piezoelectric layer, and using equation (4) results in

Npop flxHtVxKM

x

wtVxK

x

wxM

t

wxm )()()()()()()(

2

2

4

4

2

2

, (8)

where ppb m)xl(Hm)x(m is the linear mass density and ppbo l2hhM . The boundary

conditions for the bending vibrations of a tapping mode piezoelectric AFM are

.lxat0wwand;0xat0ww (9)

The nonlinear terms only appear in the fN expression as quadratic and cubic nonlinearities

as shown in Equation (7). In some cases, piezoelectric microcantilever sensors curvature

nonlinearities can also be considered [20]. However, these nonlinearities apply to cases where

there is no tip-force. For scanning purposes, when there is a tip-sample force, the microcantilever

deformation is relatively small compared to sensors, and, therefore, the curvature is linear. In

addition, the long length of AFM microcantilever probe causes any nonlinear curvature terms to

become negligible in comparison to the nonlinear force terms.

III. Response to Excitation at Resonance

3.1. Nondimensional Governing Equations of Motion

The governing equation of motion should be rewritten in nondimensional form for

finding a general solution for the system. The following nondimensional quantities are

introduced to be replaced in the equations of motion:

l

xx ,

zz ,

ww ,

0

ˆv

VV ,

4ˆ

lm

IEtt

b

bb , and bb

4b

IE

lmˆ . (10)

65

Applying the nondimensional quantities of Equation (10), and substituting t,lwz)t(z 0l into

the tip-sample forcing function of Equation (7) leads to

01

2

2

3

3ˆˆˆ lllN wwwf ,

(11)

where )t,l(wwl and the α-terms are defined in the Appendix. Only the contact force, i.e., when

lz , will be considered from this point forward since the van der Waals force is much smaller

than the contact force. Additionally, the constant force, 0 , can be removed from the equation by

considering a new static equilibrium point for the system that compensates for the effect of this

constant force. Therefore, in the rest of this paper the constant force will not be considered.

The non-dimensional form of Equations (8) and (9) are expressed as

,ˆwˆwˆwˆ)1x(H)t(V)x(KM

x

w)t(V)x(K

x

w)x(M

t

w)x(m

0l1

2

l2

3

l3po

2

2

p4

4

2

2

(12)

0ww at 0x ; 0ww at 1x , (13)

where

0l

i

bb

4

i

zwH

IE

lˆ . (14)

In order to derive the linear mode shapes of the system, the nonlinear and force terms are

removed from Equation (12) and the linear response is considered to be tin

nex)t,x(w . Then

considering that the nonuniform thickness has negligible effect on the mode shapes, they can be

found by solving the following equations:

0)()()( 4iv xxx n , (15)

0 at 0x ; 0,ˆ1

at 1x , (16)

66

where

l

lxfor

l

lxfor

zbhIm

mmI

x

p

n

p

n

nbbb

pbb

n

ˆ

ˆ

)(

2

2

24

, (17)

n = 1, 2,…∞ indicating the number of the mode, and ωn are the natural frequencies. Considering

the properties of the microcantilever presented in reference [23], the value of n is almost

n2n for all values of x. Therefore, one solution can be found for the entire length of the

microcantilever. Solving Equations (15) and (16) leads to the following mode shapes:

)sinh()sin(

)cos()cosh()]sinh()[sin()cos()cosh()(

nn

nnnnnnn xxxxx

, (18)

where n are the roots of the following frequency equation:

0sinhcossincoshˆ

coshcos13

1

nnnn

n

nn . (19)

For other cases, depending on boundary and continuity conditions, Equation (15) should be

solved for two conditions, i.e., x<lp/l and x>lp/l.

3.2. Closed-form Solution of the Equations of Motion

The method of multiple scales is employed to derive the modulation equations, frequency

response, and closed-form solution for the equation of motion presented in (12) with the

boundary conditions of (13). The response is obtained based on excitation of the primary

resonance frequency of the AFM piezoelectric microcantilever probe while it is scanning the

surface and facing contact forces at the tip. For more realistic analysis a damping term is added

to the equation of motion. In addition, is used as a small scaling parameter. The equation of

motion and boundary conditions of (12-13) are rewritten as

67

,wˆwˆwˆ)1x(H)t(V)x(KM

x

w)t(V)x(K

x

w)x(M

t

w

t

w)x(m

1

2

2

3

3

2

po

2

2

2

p

2

4

42

2

2

(20)

0ˆˆ ww at 0ˆ x ; 0ww at 1ˆ x , (21)

where is the damping coefficient. Also note that 1ˆˆ xm for the free-end of the

microcantilever. Time and time-derivatives are expanded into multiple time scales, tT nn . The

solution to the equation of motion is expressed in the form of

),,,(),,,(),,,();,(ˆ2102

2

21012100 TTTxwTTTxwTTTxwtxw (22)

Substituting Equation (22) into Equations (20-21), considering the expanded time-derivations,

and putting coefficients of like powers of ε equal to zero, since ε can have any arbitrary small

quantity, results in

Order of (0):

)1x(Hwˆw)x(MwD)x(m 01

iv

00

2

0 , (23)

0ww 00 at 0x ; 0ww 00 at 1x , (24)

Order of (1):

2

0211010

iv

11

2

0 wˆwˆ)1x(HwDD)x(m2w)x(MwD)x(m , (25)

0ww 11 at 0x ; 0ww 11 at 1x , (26)

Order of (2):

,wwˆ2wˆwˆ)1x(H)t(V)x(KMw)t(V)x(KwD

wDD)x(m2wD)x(mwDD)x(m2w)x(MwD)x(m

102

3

0321po0p00

1100

2

1020

iv

22

2

0

(27)

0ww 22 at 0x ; 0ww 22 at 1x . (28)

68

The eigenvalue problem of Equation (23) with the boundary condition of Equation (24)

has already been solved in Section 3.1, and its solution can be written as

cceT,TA)x(w 0nTi

21nn0 , (29)

where An is a time dependent function which will be determine later by solving equations for

higher orders of . Substituting Equation (29) into Equation (25) with some simplification,

results in

2

02

Ti

21n1nn

11

iv

11

2

0

wˆ)1x(HcceT,TAD)x()x(mi2

wˆ)1x(Hw)x(MwD)x(m

0n

. (30)

The left hand side of the Equation (30) has the same general solution as Equation (23),

but a particular solution is needed due to the presence of the terms on the right hand side. The

secular terms of the right hand side should be eliminated from the Equation (30), which leads to

the fact that An must be independent of T1. In other words, 0T,TAD 21n1 . Therefore, An is a

function of T2 only. Now, Equation (30) can be expressed as

.eTA)1(ˆTATA)1(ˆ2eTA)1(ˆ

wˆ)1x(Hw)x(MwDxm

0n0n Ti2

2

2

n

2

n22n2n

2

n2

Ti2

2

2

n

2

n2

11

iv

11

2

0

(31)

The particular solution for Equation (31) is

]ccAA)x(keTA)x(k)[1(ˆw nn2

Ti2

2

2

n1

2

n210n

, (32)

where k1 and k2 are defined in the Appendix.

The next equation that should be solved to find the response is Equation (27), which

contains all the cubic nonlinear terms. Substituting Equations (29) and (32), which are solutions

to (23-26), and Equation (2) into Equation (27), results in

69

.cceTATAkkeTA)x(kxˆ2eTATA3

eTAˆ)1x(He)x(KM½e)x(KTAx"½

exTAiTADxmi2wˆ)1x(Hw)x(MwDxm

0n0n0n

0n00n

0n

Ti

2n2

2

n21

Ti3

2

3

n1

3

n

2

2

Ti

2n2

2

n

Ti3

2

3

n

3

n3

Ti

po

T)(i

p2nn

Ti

n2nn2n2n21

iv

220

(33)

The corresponding boundary conditions have been shown in Equation (28). Using a small

detuning parameter, , the excitation function, Ω, which has a frequency close to the natural

frequency, n, can be defined as

2

n . (34)

The normal modes, n, corresponding to the natural frequencies, ωn, are orthogonal to the right

hand side of Equation (33) [24]. Using this solvability condition yields

2Ti

n4n

2

nn3n2n12nn2n2n0 eg2

1AAggg4TAiTADig2

, (35)

where the g-terms are defined in the Appendix. The term An has real and imaginary parts that are

functions of the amplitude and phase of the system. To find the modulation of the phase and

amplitude, An is expressed in polar form as

2n Ti

nn ea½A

, (36)

where an is the amplitude of the response and n is the phase. Substituting the definition of An

into Equation (35) and separating the real and imaginary parts of the resultant equation, results in

the modulation equations, which can be expressed as

nn4nnn2n0 singaaDg2 , (37)

nn4

3

nn3n2n1nn0n2nn0 cosgagggag2Dag2 . (38)

Considering the fact that for the steady state condition, the derivatives of the amplitude

and phase vanish by passing time in Equations (37) and (38), the frequency response function

can be achieved by combining the two equations, which can be expressed as

70

2

n4

23

nn3n2n1nn0

2

nn gagggag2a , (39)

Equation (39) is the frequency response equation for the piezoelectric AFM microcantilever. The

term an is the amplitude response to the piezoelectric excitation which appears in the coefficient

g4n. The coefficients for the nonlinear tip-sample force are g1n to g3n.

IV. Results

The numerical values in Tables 4.1 and 4.2 are used for analysis in this section for

properties of the sample and microcantilever. In addition z0=450 nm and v0=220 mV. The

parameter σ, which was defined in Equation (34), is a small detuning parameter around the

natural frequency. In other words, when σ is equal to zero, the excitation frequency is equal to

the natural frequency. Also, an, which is defined in Equation (36) is the nondimensionalized

microcantilever tip amplitude.

Table 4.1. HOPG sample properties [22].

Property Value

Ah (J) 2.9610-19

E* (GPa) 10.2

(nm) 0.38

R (nm) 10

Figure 4.2 shows the frequency response of the piezoelectric microcantilever probe for

the first mode. The results clearly show a frequency shift in the response resulting from the

nonlinear force. This frequency shift during contact mode is an important consideration in the

modeling of AFM mechanics for the generation of more accurate imaging. Without

71

consideration of the force nonlinearities, there would be no shift in frequency modeled around

the natural frequency, which would lead to inaccurate imaging.

Table 4.2. Piezoelectric microcantilever properties [23].

Property Value Property Value

ρb (kg/m3) 23305 hb (μm) 40.5

ρp (kg/m3) 63905 hp (μm) 40.5

b (μm) 2505 l (μm) 5005

Eb (GPa) 1805 lp (μm) 3755

Ep (GPa) 1305 d31 (pm/V) 111

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04200

400

600

800

1000

1200

1400

an

Figure 4.2. Frequency response of the piezoelectric microcantilever probe: nondimensionalized

microcantilever tip amplitude as it varies with σ.

Figures 4.3 through 4.8 are the results of a thorough sensitivity analysis. Two main

factors are at play in several of these results. First, any parameter that stifles the amplitude of the

microcantilever vibrations will also decrease the frequency shift. The reason for this is that the

tip of the microcantilever will not get as close to the sample when the amplitude of vibrations is

72

smaller. Since the tip is farther away, the nonlinear contact force acts over less of the tip’s range

of motion. Second, any parameter that varies the contact force will also change the natural

frequencies and mode shapes, as can be seen in Equations (18) and (19). When the contact force

is small, the mode shape resembles that of a free microcantilever beam, and as the contact force

increases, the end of the mode shape starts to curve downward toward the sample. In other

words, a small contact force results in the microcantilever tip not getting as close to the sample,

which leads to the contact force acting over a smaller range of the tip’s motion. With these two

factors interacting, many parameters will lead to an optimal point or minimum, at which the

smallest frequency shift will occur.

Figure 4.3(a) shows that increasing the length of the microcantilever first decreases then

exponentially increases the frequency shift. Increasing the piezoelectric layer length only has a

very small effect on the frequency shift: 38% maximum variance between 150 and 450 μm. First

the frequency shift slightly increases then decreases. The length of the microcantilever has a

much larger effect on the frequency shift than the piezoelectric layer length. For microcantilever

length, the two parameters interacting leading to an optimal value is observed. Also, decreasing

the length of the microcantilever stunts the amplitude of vibrations. Therefore, when selecting

the length of the microcantilever to minimize the frequency shift due to the nonlinear force, a

value close to the optimal value should be used, but in order to prevent decreasing the amplitude

of microcantilever vibrations beyond an acceptable level, the exact optimal value may not be

appropriate.

As for the piezo layer length, two factors are at play in creating the frequency shift

increase then decrease. The first of these two parameters is an increase in voltage of the piezo

layer due to an increase in the length over which the piezo voltage is applied. The second

73

parameter is the increase in the effective microcantilever width. The interplay between these two

factors explains why there is not a large difference between a short and long piezo layer. At first,

the effect of increase in voltage is more than the effect of increase in the effective

microcantilever width, which is why the frequency shift increases initially. Then, after some

critical point, the effect of the microcantilever width is more than that of the increase in voltage,

causing the frequency shift to come back down.

400 450 500 550 6002

3

4

5

6

7

8

9

10

Beam Length (m)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

3 3.5 4 4.5 52

3

4

5

6

7

8

9

10

Beam Thickness (m)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

(a) (b)

Figure 4.3. Effect of (a) microcantilever length and (b) thickness on the magnitude of the

frequency shift.

The effect of microcantilever thickness has been studied in Figure 4.3(b). Increasing the

thickness of the microcantilever decreases the frequency shift to a certain point then increases it

again. It should be noted that increasing microcantilever thickness decreases the amplitude of

vibrations of the microcantilever. As mentioned previously, stifling the amplitude of vibrations

also decreases frequency shift. The thicker the microcantilever, the smaller the amplitude of

vibrations will be. However, as the microcantilever thickness becomes larger, the moment arm

distance for the piezo increases causing the nonlinearity to also increase. The data collected stops

at 5 μm because after this point, the tip no longer makes contact with the sample. The effect of

piezo length is more straightforward, the piezo thickness causes a near linear effect over 2.5 to 5

μm where it loses contact with the sample, changing the frequency shift from 8.6 to 2.9.

74

Figure 4.4(a) shows that increasing the width of the microcantilever decreases the

frequency shift. The reasoning is the result of an optimal value as described earlier in the section.

A wider microcantilever decreases the amplitude of vibrations. Additionally, it should be noted

that once a microcantilever reaches a certain width, the dynamics of the system may no longer

resemble a microcantilever and may have to be considered as a plate. Since the width of the

piezoelectric layer is the same as the microcantilever, its effect has not been studied separately

here. Figure 4.4(b) shows that increasing the tip radius also leads to an optimal point. Selection

of this optimal point is desirable.

0 100 200 300 400 5002

3

4

5

6

7

8

9

10

Beam Width (m)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

0 50 100 150 200 250 3002

3

4

5

6

7

8

9

10

Tip Radius (nm)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

(a) (b)

Figure 4.4. Effect of (a) microcantilever width and (b) tip radius on the magnitude of the

frequency shift.

Figure 4.5(a) shows that increasing the voltage amplitude applied to the piezoelectric

layer, v0, increases the frequency shift exponentially. Voltage amplitude also significantly effects

the amplitude of microcantilever vibrations. Adjusting other values to allow voltage to stay

below the point of steep increase, in this case about 200 mV, would be advisable. Figure 4.5(b)

shows that increasing the piezoelectric constant, d31, increases the frequency shift. This is a

material property and changing this property means changing the piezoelectric layer. The

microcantilever probe studied in this paper is ZnO. Increasing the voltage amplitude or the

piezoelectric constant is similar to increasing the piezo layer thickness or length except that there

75

is no increase in the effective width of the microcantilever to counteract the effect. This explains

why these two parameters have such a large effect on the response amplitude and the frequency

shift.

150 200 250 300 3502

3

4

5

6

7

8

9

10

v0 (mV)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

9 10 11 12 13 14 15 16 172

3

4

5

6

7

8

9

10

d31

(pm/V)N

ondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

(a) (b)

Figure 4.5. Effect of (a) vo, and (b) piezoelectric constant, d31, on the magnitude of the frequency

shift.

Figure 4.6(a) shows that increasing the tip-sample distance distinguishing the contact and

non-contact regions, δ, decreases then increases the frequency shift, i.e., an optimal point. The

only way to change δ is to change the tip material or the sample so the effect of this parameter

cannot stand alone. However, the effect should be considered for different samples or different

microcantilever materials. Increasing δ decreases the nonlinear contact force in Equation (7).

However, it also increases the distance over which the force acts. The combination of these two

factors explains why the frequency shift decreases then increases again.

The effect of the effective modulus of elasticity between the microcantilever and sample

is studied in Figure 4.6(b), which shows that increasing the effective modulus slightly decreases

then increases the frequency shift. The effective modulus increases the nonlinear force similar to

tip radius or δ, again leading to an optimal value.

Figure 4.7(a) shows that increasing the microcantilever modulus of elasticity

exponentially decreases the frequency shift; however, increasing the piezoelectric modulus of

76

elasticity slightly increases the frequency shift as shown in Figure 4.7(b). Increasing the

microcantilever modulus means that the microcantilever is stiffer. A stiffer microcantilever

deflects less and, therefore, has a smaller amplitude of vibrations and also a smaller frequency

shift. Increasing the piezo modulus, while increasing the overall modulus of the microcantilever,

also increases the force applied by the piezo as shown in Equation (5). These two factors work in

conjunction so that the total effect on the microcantilever dynamics is small and takes much

more change in the modulus to create much effect. The most helpful of these three modulus

properties is the microcantilever modulus of elasticity because it can be changed by changing the

microcantilever material without having a large effect on other properties. Also, it makes a large

difference in the frequency shift without significantly decreasing the amplitude of vibrations of

the microcantilever.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

3

4

5

6

7

8

9

10

(nm)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

10 20 30 40 50 60 702

3

4

5

6

7

8

9

10

E* (GPa)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

(a) (b)

Figure 4.6. Effect of (a) the tip-sample distance distinguishing the contact and non-contact

regions, δ, and (b) effective modulus of elasticity (between microcantilever and sample) on the

magnitude of the frequency shift.

Figure 4.8 shows that increasing z0, the distance between the tip of the AFM and the

sample at equilibrium, causes the frequency shift to decrease. This is because the microcantilever

tip moves farther away from the sample and, therefore, the nonlinear contact force acts over a

77

small portion of the tip’s motion. The microcantilever and piezoelectric density were also

examined in the sensitivity analysis, but their effect on the frequency shift was minimal.

20 40 60 80 100 1202

3

4

5

6

7

8

9

10

Eb (GPa)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

0 100 200 300 400 5002

3

4

5

6

7

8

9

10

Ep (GPa)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

(a) (b)

Figure 4.7. Effect of (a) microcantilever and (b) piezoelectric modulus of elasticity on the

magnitude of the frequency shift.

350 400 450 500 5502

3

4

5

6

7

8

9

10

z0 (nm)

Nondim

ensio

naliz

ed

Fre

quency S

hift (x

10-3

)

Figure 4.8. Effect of z0, the distance between the tip of the AFM and the sample at equilibrium,

on the magnitude of the frequency shift.

The point of the sensitivity analysis is to determine methods of minimizing the frequency

shift due to the force nonlinearities. Some of the parameters shown are easily changed and others

are much more difficult or even impossible to change in some situations. For example, the

microcantilever width could be easily changed by using a different microcantilever probe.

However, if a particular sample is being investigated, the modulus of elasticity of the sample,

which effects E*, could not be changed. Also, certain parameters have a much larger impact on

the frequency shift than others. For example, the microcantilever length makes a much bigger

78

difference than piezoelectric layer length. By selecting parameters and changing them in such a

way as to decrease the frequency shift, the nonlinearities can be minimized. As a case study, if

tip radius is increased to its optimal point and the microcantilever length is slightly decreased

toward its optimal point, the nonlinearities can be decreased without sacrificing amplitude

beyond an acceptable limit. Altered parameters in Table 4.3 are used to plot the frequency

response of the system as shown in Figure 4.9. Many of the parameters used for the generation of

Figure 4.2 are already very near the optimal points, so a very large decrease in frequency shift

was not possible. However, the two parameters can be altered very easily. They can be changed

by using a different microcantilever. The tip radius is actually increased, which is easier to

manufacture. Altering the two parameters listed in Table 4.3 decreased the frequency shift by

19.6% while only decreasing the amplitude by 10.6%.

Table 4.3. Values of Altered Parameters.

Property Original Value Altered Value

R (nm) 10 12

l (μm) 500 480

V. Conclusions

The mechanics of a piezoelectric microcantilever beam subject to a nonlinear contact

force were examined. The equations of motion for a microcantilever were derived using the

energy method. The analytical expressions for natural frequencies and mode shapes were

obtained. The method of multiple scales was used to investigate the analytical frequency

response of the piezoelectric microcantilever probe. The effects of nonlinear excitation force on

the microcantilever beam’s frequency and amplitude were analytically studied. The results show

79

a frequency shift in the response around the natural frequency resulting from the force

nonlinearities. In addition, the results of a sensitivity analysis of the system parameters on the

nonlinearity effect were investigated. The effect of slightly changing two parameters – tip radius

and length of the microcantilever – was shown to reduce the frequency shift by 19.6% while only

decreasing the amplitude by 10.6%. By reducing the nonlinearity effect, it may be possible to use

a linear model to analyze the microcantilever mechanics, which would make the AFM software

package less cumbersome and calculation intensive.

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04200

400

600

800

1000

1200

1400

an

Figure 4.9. Frequency response of the piezoelectric microcantilever probe with three altered

parameters: nondimensionalized microcantilever tip amplitude as it varies with σ.

References




061107.



130(6), 061003.





80



Engineering, 37(3), 658-670.



Actuators A, 102, 106-113.









[8] Mahmoodi, S. N. and Jalili, N. (2007). Nonlinear vibrations and frequency response of

piezoelectrically-driven microcantilevers. International Journal of Nonlinear Mechanics,

42(4), 577-587.

[9] Rogers, B., Manning, L., Sulchek, T., and Adams, J.D. (2004). Improving tapping mode

atomic force microscopy with piezoelectric cantilevers. Ultramicroscopy, 100, 267–276.

[10] Manning, L., Rogers, B., Jones, M., Adams, J. D., Fuste, J. L., and Minne, S. C. (2003).

Self-oscillating tapping mode atomic force microscopy. Review of Scientific Instruments,

74(9), 4220-4222.

[11] Shin, C., Jeon, I., Khim, Z.G., Hong, J.W., and Nam, H. (2010). Study of sensitivity and

noise in the piezoelectric self-sensing and self-actuating cantilever with an integrated

Wheatstone bridge circuit. Review of Scientific Instruments, 81, 035109.









Physics, 10, 1416-1421.

81







199-207.



Physics, 38, 7155–7158.



179.






705-709.




[23] Mahmoodi, S.N. and Jalili, N. (2009). Piezoelectrically-actuated microcantilevers: an

experimental nonlinear vibration analysis. Sensors and Actuators A, 150(1), 131-136.



Vibration, 275, 283-298.

82

Appendix: Coefficients Calculation

The definitions of the α-terms from equation (11) are

3

0

2

0010

2

0011

012

13

4152010

123020

1215

4

zzzc

zzc

zc

c

zfor l

.z

24

1z

4

1z

3

2cc

z

8

1z

2

11c

z

8

1

4

1c

24

c

zfor

3

0

2

00210

2

0021

022

23

l

(A1)

The definitions of k1 and k2 from Equation (32) are given as

)xcos(1C)xcosh()xcos(C)xsinh()xsin(C)x(k n13n1n12n1n111 , (A2)

234

42ˆ12ˆ8ˆ2)ˆ( xxxCxk ,

nn

nnn CCC

11

131121

coshcos

sinsinhsin

, (A3)

nn

nnnnCC

11

111132

coshcos12

coshcossinhsin1

, (A4)

2

1

3ˆˆ4ˆ

1

nwxmC

, 1

4ˆˆˆ83

1

xMC ,

xM

xm nn

ˆˆ

ˆˆ4 24

1

. (A5)

83

The g-terms from Equation (35) are defined as

pl

n

b

pxdx

m

mg

0

2

0ˆˆ1 , 1ˆ

4

3 4

31 nng , )1(1ˆ2

11

42

22 kg nn , (A6)

)1(1ˆ2

12

42

23 kg nn , 1

04

ˆˆ)ˆ(ˆ xdxxKMg npon . (A7)

84

CHAPTER 5

DYNAMIC MULTIMODE ANALYSIS OF NONLINEAR PIEZOELECTRIC

MICROCANTILEVER PROBE IN BISTABLE REGION OF TAPPING MODE ATOMIC

FORCE MICROSCOPY

Atomic force microscopy (AFM) uses a scanning process performed by a microcantilever

probe to create a three dimensional image of a nano-scale physical surface. The dynamics of the

AFM microcantilever motion and tip-sample force are needed to generate accurate images. In

this paper, the nonlinear dynamics of a piezoelectric microcantilever probe in tapping AFM are

investigated. The equations of motion are derived for a nonlinear contact force, the analytical

expressions for the natural frequencies and mode shapes are obtained, and the method of

multiple scales is used to find the analytical frequency response of the microcantilever. Results

show that the nonlinear excitation force creates a shift in the frequency response curve during

contact mode. This frequency shift is an important consideration in the modeling of the AFM

dynamics for the generation of accurate images as well as for accurate readings when using the

AFM microcantilever for other applications. The frequency shift also leads to a bistable region,

in which a high and a low amplitude solution coexist. The response of the microcantilever at a

single input frequency and voltage are analyzed for both the high and low amplitude solutions

with the main difference being that the high amplitude solution makes contact with the sample

while the low amplitude solution does not. This contact results in higher harmonics of the

microcantilever being excited. The response in the bistable region is compared to the response in

the monostable region. Additionally, a convergence analysis is used to determine the number of

85

modes necessary to describe the motion of the microcantilever in tapping mode. It is determined

that one mode is insufficient, two modes are sufficient for most applications, and it is unlikely

that more than five modes would be necessary even for applications that require very precise

readings.

I. Introduction

The Atomic Force Microscope (AFM) is a very powerful tool used for nano-scale three

dimensional imaging as well as evaluating local mechanical and chemical properties. Initially,

the AFM was invented for the purpose of scanning the image of a physical surface in nano-scale

and generating a three dimensional image of the surface. A microcantilever probe performs the

sensing process by contacting or tapping the sample. In tapping mode, the microcantilever is

excited at or near one of its natural frequencies, usually the first. The response amplitude of the

microcantilever is influenced by intermolecular interaction forces between the tip and sample

and is observed to create the image or determine properties. The analysis of the microcantilever

can be widely used for a variety of applications, such as Piezo-response Force Microscopy

(PFM), Friction Force Microscopy (FFM), Lateral Force Microscopy (LFM), biosensing, and

other applications [1-5].

In most AFMs the microcantilever is excited using a piezoelectric tube actuator at the

base of the microcantilever and a laser measurement system is used to determine the motion of

the microcantilever. This system is cumbersome and expensive and is also difficult and time

intensive to set up and use. If piezoelectric actuators and sensors are added to the

microcantilever, the need for this laser system is eliminated. Some microcantilever probes use a

piezoelectric layer on one side of the microcantilever for the purpose of actuation. This layer is

86

usually ZnO [6] or Lead Zirconate Titanate (PZT). The piezoelectric microcantilever has

widespread applications. For example, it has been used for force microscopy, biosensing,

chemical sensing, and Scanning Near-field Optical Microscopy (SNOM) [7-11]. Using

piezoelectric microcantilevers for surface scanning and imaging of different samples is an

improvement for sensing in AFMs over the laser system [12]. The bulky and expensive laser

system can be entirely replaced by the piezoelectric actuators and sensors along with an electrical

circuit [13, 14].

However, the dynamics of the piezoelectric microcantilever probe are complicated. To

ensure accurate imaging and for accurate readings for the many other applications, a reliable and

accurate model of the piezoelectric microcantilever probe dynamics is required. An inclusive

sensing model must incorporate an accurate understanding of the microcantilever motion and tip-

sample force. This research work includes two major parameters in the sensing process that must

be carefully analyzed to guarantee the accuracy of measurement. The first parameter is the effect

of the piezoelectric layer on the motion of the microcantilever, and the second parameter is the

nonlinear force between microcantilever tip and sample.

The effects of the piezoelectric layer on the microcantilever beam dynamics have been

experimentally and analytically studied in some research works. The nonlinearity due to the

curvature and piezoelectric material has been considered when studying the nonlinear dynamics

of a piezoelectric microcantilever [15, 16]. The microcantilever model was developed, and the

equations of motion were analytically solved. However, neither of these references considered

contact force at the tip. Experiments have been implemented in air and liquid on dynamic AFMs

and the frequency response of the systems were determined [17, 18]. In other works, linear

dynamic models for contact piezoelectric AFM microcantilevers have been developed and

87

numerically solved [19, 20]. However, these works did not include the obtained analytical model

in the nonlinear response of the system. They only included the nonlinear contact force and the

piezoelectric effect on the microcantilever. Additionally, almost all research works only include

the first mode when analyzing the motion of the microcantilever probe [21-23].

Besides the effect of the piezoelectric layer, the nonlinear force is also a major

contributor to the microcantilever probe dynamics. The van der Waals force and contact force

are the main components of the nonlinear force between the tip and sample. In non-contact

mode, only the van der Waals force is present between the AFM tip and sample. While in a

tapping mode, both forces are applied to the tip. In one work, the nonlinear response of the AFM

microcantilever probe to contact force was analyzed considering that the excitation is applied to

the sample from a piezoelectric actuator under the sample and the microcantilever has no motion

at the base [24]. Nonlinear response of a tapping contact AFM microcantilever probe to a

combination of the contact and van der Waals forces was examined [25] in order to analyze the

jump phenomena and stability of the response. In a series of works, the nonlinear response of the

non-contact tapping mode AFM was analyzed considering the van der Waals force at the tip [26-

28]. In those works, the Lennard Jones model was used to approximate the force and the

microcantilever did not have a piezoelectric layer. The AFM microcantilever probe was

approximated as a linear continuous microcantilever excited at the base using a piezoelectric

actuator. Based on the readings of amplitude and phase, an algorithm for improving the tip-

sample interactions in amplitude modulation was constructed and the response studied [29].

The nonlinear dynamics of the microcantilever probes used in AFM have been studied

extensively as indicated in the previous paragraph. However, almost all research works to this

point investigate the behavior of the microcantilever probe in the monostable region with a single

88

mode analysis. In this study, the bistable region is investigated. In these bistable regions, there

are two stable branches that coexist: a high amplitude branch and a low amplitude branch. The

differences in the behavior in the bistable region versus the monostable region are investigated.

This work will develop a new nonlinear model for the tapping mode AFM

microcantilever probe as a continuous microcantilever considering nonlinearities due to the force

and the effect of the piezoelectric layer on the motion of the microcantilever. A novel, analytical,

closed-form solution for the model will be obtained along with frequency response. The results

will show that the nonlinear tip-sample contact force will generate a jump phenomenon. This

jump phenomenon leads to a bistable region and hysteresis. It will be investigated via a

bifurcation analysis. A convergence analysis will be performed to determine how many modes

are necessary to accurately predict the complicated AFM microcantilever probe dynamics. As a

part of this study, phase portraits, power spectra, and time response of the microcantilever tip’s

displacement and velocity will be analyzed.

II. Mathematical Modeling

In this section, the mathematical modeling for the nonlinear dynamics of a piezoelectric

microcantilever in dynamic contact mode is examined. The governing equations of motion are

derived, the natural frequencies and mode shapes are presented, and the frequency response

equation is derived. Figure 5.1 is a schematic of the piezoelectric microcantilever probe motion

in dynamic contact mode. The coordinate system (x, z) is used to describe the dynamics of the

piezoelectric microcantilever probe motion. The coordinate x is in the longitudinal direction of

the microcantilever. The coordinate z is perpendicular to the x-axis. The deflection of the

microcantilever is in the z direction and is a function of x and t, i.e. w(x,t), where t indicates time.

89

The microcantilever length is l, and fN is the nonlinear force due to the tip interaction with the

sample, which is a combination of the van der Waals and contact forces.

Figure 5.1. Schematic of the piezoelectric microcantilever motion.

2.1. The Forcing Equation

In this section, the forcing equation is examined, and the most useful form for the

analysis in later sections is presented. The external nonlinear force, fN, is the summation of van

der Waals and contact forces. It is defined as [30]

l

23

l

*

2

h

l2

l

h

N

zzRE3

4

6

RA

zz6

RA

f

(1)

When the tip-sample distance is larger than , the first term is applied to the tip. This

term is the van der Waals force. When the tip-sample distance is less than , the second term is

applied to the tip. This term is the contact force. The term is a constant distance for a given

AFM tip material and sample material. Also, E* represents the effective modulus of elasticity, R

represents the microcantilever tip radius, Ah is the Hamaker constant, and zl is the tip position

fN

Piezoelectric layer

z

x

w(x,t) AFM tip

Sample

90

and is defined as tlwztz l ,)( 0 , where z0 is the distance separating the tip and the sample at

equilibrium as shown in Figure 5.1.

The first step in altering the forcing equation into a usable form is to expand it into a

Taylor series. The forcing function of Equation (1) can be rewritten into polynomial form. The

higher order terms are not considered since they become negligibly small. Next, the forcing

equation is nondimensionalized. In the rest of the paper, a hat over a symbol will simply indicate

the nondimensionalized version of that variable, i.e, t is nondimensionalized time. Table 5.1

contains a complete list of nondimensional quantities used in this paper.

Table 5.1. Nondimensional Quantities.

Quantity Variable Name Nondimensionalization

Axis parallel to microcantilever x l

x

Axis transverse to microcantilever z

z

Bending displacement in the z-

direction w

w

Applied voltage to the

piezoelectric actuator V

0v

V

Time t 4lm

IEt

b

bb

Voltage frequency (excitation

frequency for the piezoelectric

actuator)

bb

b

IE

lm 4

91

Also, only the contact force, i.e., when lz , will be considered from this point forward

since the van der Waals force is much smaller than the contact force. Additionally, the constant

force is removed from the equation by considering a new static equilibrium point for the system

that compensates for the effect of this constant force. Therefore, in the rest of this paper the

constant force will not be considered. Therefore, the dynamic nonlinear force is rewritten as

lllN wwwf ˆˆˆ1

2

2

3

3 ,

(2)

where

,123020

1215

4

2

0011

012

13

zzc

zc

c

zfor l (3)

21

6

RAc h , ),(ˆˆ tlwwl . (4)

2.2. Governing Equations of Motion

In this section, the governing equations of motion are presented. Using the energy

method, the equations of motion are developed which include the effect of the piezoelectric layer

on the microcantilever dynamics. The equations of motion and boundary conditions for the

system shown in Figure 5.1 are [31]

l1

2

l2

3

l3po

2

2

p4

4

2

2

wˆwˆwˆ)1x(H)t(V)x(KM

x

w)t(V)x(K

x

w)x(M

t

w)x(m

, (5)

0ˆˆ ww at 0ˆ x ; 0ˆˆ ww at 1ˆ x , (6)

where

92

bb

pppp

b

nbp

IE

IEx

l

lHx

l

lHxH

I

zbhx

l

lHxM

ˆˆ)ˆ1(1ˆ)ˆ(ˆ

2

, (7)

x

l

lHvzhhd

E

E

I

blxK

p

nbp

b

p

b

pˆ2

4)ˆ(ˆ

031

2

, (8)

2

2

p

pb

o

l

l2

hhM

, (9)

0

3

ˆzw

HIE

l li

bb

i , (10)

)ˆ(ˆ xm is the nondimensionalized linear mass distribution, l is the length of the microcantilever, lp

is length of the piezo layer, b is the width of both the microcantilever and piezo layer, zn is the

neutral axis, v0 is the constant voltage amplitude, d31 is the piezoelectric constant relating charge

and strain, h is the height, I is the moment of inertia, E is the modulus of elasticity, and the

indices b and p represent the microcantilever and piezoelectric layer, respectively. Also, H(n) is

the Heaviside function and is defined as

01

00)(

n

nnH . (11)

2.3. Natural Frequencies and Mode Shapes

In this section, the equations for the natural frequencies and linear mode shapes are

derived then analytically solved. The homogeneous linear solution of Equation (5) is assumed to

be tin

nex)t,x(w , and ωn are the natural frequencies. Now, the linear mode shapes can be

determined by solving the following equations:

0)()()( 4iv xxx n , (12)

0 at 0ˆ x ; 0,ˆ1

at 1ˆ x , (13)

93

where n indicates the number of the mode. Considering the properties of the microcantilever

presented in reference [32] and shown in Table 5.2, the value of n is almost equal for all values

of x. Therefore, n2n is a reasonable assumption, and one solution can be found for the entire

length of the microcantilever.

Table 5.2. Piezoelectric microcantilever properties [32].

Property Value Property Value

ρb (kg/m3) 23305 hb (μm) 40.5

ρp (kg/m3) 63905 hp (μm) 40.5

b (μm) 2505 l (μm) 5005

Eb (GPa) 1805 lp (μm) 3755

Ep (GPa) 1305 d31 (pm/V) 111

The following mode shapes can be found by solving Equations (12) and (13):

)sinh()sin(

)cos()cosh()]sinh()[sin()cos()cosh()(

nn

nnnnnnn xxxxx

, (14)

where n are the roots of the following frequency equation:

0sinhcossincoshˆ

coshcos13

1

nnnn

n

nn . (15)

It is likely that for some other cases, depending on material properties and boundary conditions,

Equation (12) would need to be solved for two conditions, i.e., x<lp/l and x>lp/l.

In order to numerically investigate the effect of the linear tip-sample force on the natural

frequency of the microcantilever, a sample material must be selected for the purpose of

determining the values of the properties in the equations. The case of scanning a sample made of

94

Highly Oriented Pyrolytic Graphite (HOPG) is selected. Table 5.3 gives the values of the

properties for the surface of the HOPG material.

Table 5.3. HOPG sample properties [30].

Property Value

Ah (J) 2.9610-19

E* (GPa) 10.2

(nm) 0.38

R (nm) 10

A numerical study is performed to examine the effect of the coefficient 1 on the first

natural frequency and the mode shape of the microcantilever. Properties listed in Tables 5.2 and

5.3 are used for the numerical study. As presented in Equation (15), the natural frequency of the

microcantilever is a function of 1 , which is the coefficient of the linear term of the tip-sample

force, as presented in Equation (2). Figure 5.2 presents the change in the first natural frequency

for different values of 1 .

Figure 5.3 shows the first mode shape of the microcantilever, n(x), for two different

values of 1 and, subsequently, two different values of natural frequencies. When

1 is small, it

does not affect the microcantilever mode shape much. Therefore, as demonstrated in Figure

5.3(a), for 1ˆ1 , the mode shape is very similar to the mode shape of a microcantilever with no

force on the free-end. However, by increasing the value of 1 , i.e., causing the tip-sample force

to become larger, the force has more effect on the motion of the microcantilever free-end. Figure

5.3(b) shows that the end of the microcantilever starts to bend downward as the tip-sample force

increases.

95

0 50 100 150 200 250 3002

4

6

8

10

12

14

16

1

n

Figure 5.2. First natural frequency based on changing tip-sample force

1 coefficient.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

1=1 and

n=4

x

n (

x)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

2.5

1=160 and

n=14

x

n (

x)

(a) (b)

Figure 5.3. First mode shape of the piezoelectric microcantilever for two different values of 1

coefficient.

Next, in preparation for the modal analysis, the first six mode shapes and natural

frequencies are numerically investigated. Calculation of 1 using Equations (3) and (10) along

with the values in Tables 5.2 and 3 results in 1 = 204.747. Table 5.4 lists the first six natural

frequencies of the microcantilever, and Figure 5.4 shows the first 6 mode shapes of the

microcantilever.

ˆ

ˆ ˆ

96

Table 5.4. Natural frequencies of the microcantilever.

Mode

Number

Natural Frequency

(kHz)

Nondimensionalized

Natural Frequency

1 65.0894 15.1110

2 199.258 46.2594

3 374.291 86.8946

4 589.451 136.846

5 899.190 208.754

6 1309.82 304.085

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

x

Mode 1

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

x

Mode 2

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

x

Mode 3

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

x

Mode 4

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

x

Mode 5

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

x

Mode 6

Figure 5.4. First 6 mode shapes of the microcantilever.

97

2.4. Closed-form Solution of the Equations of Motion

In this section, a closed-form solution for the equation of motion presented in Equation

(12) with boundary conditions of Equation (13) along with the frequency response equation is

derived using the method of multiple scales [33]. The equation of motion is rewritten to include a

damping term for realistic analysis and , which is used as a small scaling parameter. The

equation of motion and boundary conditions of (12-13) become

wˆwˆwˆ)1x(H)t(V)x(KM

x

w)t(V)x(K

x

w)x(M

t

w

t

w)x(m

1

2

2

3

3

2

po

2

2

2

p

2

4

42

2

2

, (16)

0ˆˆ ww at 0ˆ x ; 0ˆˆ ww at 1ˆ x , (17)

where is the damping coefficient. Also, note that for the free-end of the microcantilever,

1xm . The AFM piezoelectric microcantilever is assumed to be operating near the first

natural frequency while scanning the surface, which imposes contact forces at the tip. The

method of multiple scales involves using multiple time scales for time and time-derivatives as in

2

2

10ˆ TTTt (18)

2

2

10

2

2

10ˆ

DDDdT

d

dT

d

dT

d

td

d (19)

Also, the solution to the equation of motion is expressed in the form of

),,,(),,,(),,,();,(ˆ2102

2

21012100 TTTxwTTTxwTTTxwtxw (20)

Substituting Equations (19-20) into Equations (16-17) and putting coefficients of like

powers of ε equal to zero, since ε can have any arbitrary small quantity, results in three equations

and three sets of boundary conditions. (Appendix I contains the complete ε equations.) These

98

three equations, the ε0, ε1, and ε2 equations, have three unknowns, w0, w1, and w2. By solving the

ε0 equation, w0 can be written as

cceTTAxwTi

nnn

0

210 ,)ˆ( , (21)

where An is a time dependent function which will be determine later by solving equations for

higher orders of and cc is the complex conjugate of the previous terms. The definition of w0 in

Equation (21) is substituted into the ε1 equation and the elimination of the secular terms leads to

the particular solution for w1

])ˆ()ˆ()[1(ˆ2

2

2

2

1

2

210 ccAAxkeTAxkw nn

Ti

nnn

, (22)

where

)ˆcos(1)ˆcosh()ˆcos()ˆsinh()ˆsin()ˆ( 131121111 xCxxCxxCxk nnnnn , (23)

234

42ˆ12ˆ8ˆ2)ˆ( xxxCxk , (24)

(Appendix II contains the definitions of C1, C2, C3, and C4.) This process also leads to the fact

that An must be independent of T1. In other words, 0T,TAD 21n1 . Therefore, An is a function of

T2 only.

The next equation that should be solved to find the response is the ε2 equation, which

contains all the cubic nonlinear terms. Equations (21) and (22), which are solutions to the ε0 and

ε1 equations, are substituted into the ε2 equation, and the following definitions are used

2

n , (25)

2n Ti

nn ea2

1A

, (26)

99

where σ is a small detuning parameter around the excitation frequency, an is the amplitude of the

response, and θn is the phase of the response. Using these definitions and the orthogonal

solvability condition [34] results in the modulation equations, which can be expressed as

nn4nnn2n0 singaaDg2 , (27)

nn4

3

nn3n2n1nn0n2nn0 cosgagggag2Dag2 . (28)

where

pl

n

b

pxdx

m

mg

0

2

0ˆˆ1 , (29)

1ˆ4

3 4

31 nng , (30)

)1(1ˆ2

11

42

22 kg nn , (31)

)1(1ˆ2

12

42

23 kg nn , (32)

1

0npon4 xdx)x(KMg , (33)

and mb and mp are the linear mass density of the microcantilever and the piezoelectric layer,

respectively. For the steady state condition, the derivatives of the amplitude and phase disappear,

and the frequency response function can be found by combining Equations (27) and (28), which

can be expressed as

2

4

23

3210

22 nnnnnnnnn gagggaga , (34)

Equation (34) is the frequency response equation for the piezoelectric AFM microcantilever. The

term an is the amplitude response to the piezoelectric excitation which appears in the coefficient

g4n. The coefficients for the nonlinear tip-sample force are g1n to g3n.

100

III. Numerical Results

The equations of motion for a piezoelectric microcantilever probe have been presented

for a nonlinear contact force. The natural frequencies and mode shapes have been examined, and

the closed-form solution along with the frequency response function have been derived. In this

section, numerical analysis is used to investigate the dynamics of the piezoelectric

microcantilever probe in the bistable region. The effect of the number of modes on the dynamics

of the piezoelectric microcantilever beam is studied. Equations (21-24) and (27-34) are solved

for each mode. Specifically, Equations (27-28) are used to find the time response of each mode,

and Equation (34) is used to find the stead state solutions. The numerical values in Tables 5.2

and 5.3 are used for the analysis in this section for the properties of the microcantilever and

sample; additionally, z0 = 190 nm.

3.1. Single Mode Analysis

This section presents the numerical results for single mode analysis. Both frequency

response curves (setting the excitation voltage to a constant, i.e., v0 = 100 mV) and force

response curves (setting the excitation frequency to a constant, i.e., σ = 15 Hz) are generated.

Figure 5.5 shows a 3 dimensional surface plot of the amplitude response of the piezoelectric

microcantilever for the first mode found by Equation (34) with changing input voltage, v0, and

excitation frequency, σ. The parameter σ is a small detuning parameter around the natural

frequency, as defined in Equation (25). When σ is equal to zero, the excitation frequency is equal

to the natural frequency. Also, an is the nondimensionalized microcantilever tip amplitude, as

defined in Equation (26).

The results in Figure 5.5 clearly show a shift in the response resulting from the nonlinear

force. This frequency shift during contact mode is an important consideration in the modeling of

101

AFM mechanics for accurate imaging. Essentially, the top portion of the graph is shifted forward

and to the right. For better visualization, Figures 5.6 and 5.7 show section views of Figure 5.5.

Figure 5.6 is the frequency response curve with v0 set to a constant 100 mV, and Figure 5.7

shows the force response curve with σ set to a constant 15 Hz.

Figure 5.5. 3D surface plot of amplitude response of the piezoelectric microcantilever probe with

changing input voltage, v0, and excitation frequency, σ.

Figure 5.6 shows that the frequency shift results in a bistable region above the natural

frequency where two stable solutions and one unstable solution (represented by the dashed line)

are present. The two stable solutions are a high amplitude branch and a low amplitude branch.

When the system is excited in the bistable frequency region, the initial conditions determine on

which branch the solution lies. However, when conducting a frequency sweep, the solution

102

follows a different path when increasing the amplitude than when decreasing the amplitude. In

other words, the solution contains hysteresis.

Figure 5.6. Frequency response of the piezoelectric microcantilever probe: nondimensionalized

microcantilever tip amplitude as it varies with σ. (v0 = 100 mV)

Figure 5.7. Force response of the piezoelectric microcantilever probe: nondimensionalized

microcantilever tip amplitude as it varies with v0. (σ = 15 Hz)

103

In Figure 5.6, whether sweeping the frequency up or down, the solution is the same

outside the bistable region of σ = 22.61 to 33.39 Hz. This region is marked by L1 and L2. The

behavior of the system to the right of the bistable region is straightforward. For brevity, only

increasing to the left of the bistable region will be described. When starting on the left side of

Figure 5.6 and sweeping the frequency up, the amplitude increases along the curve smoothly.

From σ = -22.61 to -11.83 Hz, the increase in the amplitude slows dramatically. Past this point,

the rate of the amplitude growth increases again, and the amplitude reaches a maximum of

633.85 at σ = 17.39 Hz. The amplitude then begins decreasing until it enters the bistable region.

This is the exact process that happens in reverse when sweeping down, i.e., no hysteresis.

In the bistable region, the solution is more complex. When starting on the left side of the

region, σ = 22.61 Hz, and sweeping the frequency up, the tip amplitude follows smoothly along

the curve as it enters the bistable region and continues decreasing in amplitude following along

the high amplitude branch. At σ = 33.39 Hz, the solution loses stability due to a cyclic-fold

bifurcation [35] such that the amplitude abruptly jumps down to the low amplitude branch

following L1. However, when starting on the right side of the bistable region and sweeping down,

the tip amplitude increases along the low amplitude branch until σ = 22.61 Hz, where the

solution becomes unstable, again, due to a cyclic-fold bifurcation. The amplitude suddenly jumps

up to the high amplitude branch following L2.

Similarly, Figure 5.7 shows a bistable region in the force response curve. The bistable

region, like in the frequency response curve, results in two stable solutions including a high

amplitude branch and a low amplitude branch, which leads to hysteresis in the solution. The

bistable region, again, is marked by L1 and L2. As was the case for the frequency response, the

behavior outside the bistable region is straightforward. Therefore, for brevity, only the behavior

104

inside the bistable region will be discussed. The bistable region falls between v0 = 79.56 and

88.72 mV. When starting on the left side of the region while sweeping the voltage up, the

amplitude follows smoothly up the curve along the low amplitude branch until it reaches L1. At

this point, the solution loses stability due to a cyclic-fold bifurcation, which causes the tip

amplitude to jump up to the high amplitude branch following L1. When starting on the right side

of the bistable region and sweeping the voltage down, the amplitude follows smoothly down the

high amplitude branch until it reaches L2. Due to a cyclic-fold bifurcation, the amplitude jumps

down to the low amplitude branch following L2.

To obtain more insight into the complex response of the piezoelectric microcantilever

beam, Figure 5.8 shows the phase portraits of the high and low amplitude (contact and

noncontact) responses with σ = 26 Hz and v0 = 0.1 V. The initial conditions are used to change

the solution from the high branch to the low branch. For both branches, the initial velocity is set

to zero. The initial displacement is set to 0.076 and 0.078 nondimensionalized units, for the

noncontact and contact branches, respectively. When the initial displacement is less than or equal

to 0.076 or greater than or equal to 0.078, the solution smoothly approaches the noncontact or

contact branch, respectively. However, when the initial displacement is inside this region, the

solution amplitude oscillates back and forth before approaching the steady state solution. The

phase portraits for the contact and noncontact solutions differ in that the noncontact branch of

Figure 5.8(b) has smaller displacement and velocity. Also, the contact solution in Figure 5.8(a) is

asymmetric about the vertical axis, i.e., zero tip displacement, due to the contact force.

In order to more clearly demonstrate the effects of the nonlinear contact force on the

response of the system, Figure 5.9(a) shows a zoomed in view of a single cycle of the phase

portrait in Figure 5.8(a). Figure 5.9(b) is the time history of a single cycle of the

105

nondimensionalized tip displacement. Following the trajectory in Figure 5.9(a), the response

enters at the top left corner following a typical elliptical pattern that is expected of a standard

vibrating or oscillating system. However, the tip then enters the contact region with the sample

which leads to the velocity decreasing rapidly while the displacement actually decreases slightly.

Essentially, the tip “bounces” then reenters the contact zone at a reduced velocity. The reduced

elliptical region on the right side of the figure is indentation of the tip into the contact region.

Then the process is repeated in reverse on the way back out of the contact region. This mirroring

of the entry process upon exiting the contact region is an effect of the way in which the contact

force is modeled.

-60 -40 -20 0 20 40 60-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Nondimensionalized Tip Displacement

Nondim

ensio

naliz

ed T

ip V

elo

city

-60 -40 -20 0 20 40 60-1000

-800

-600

-400

-200

0

200

400

600

800

1000


No

nd

ime

nsio

na

lize

d T

ip V

elo

city

(a) (b)

Figure 5.8. Phase portrait for one mode with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)

low amplitude responses.

Additionally, Figure 5.10 shows the power spectra of the high and low amplitude

responses. From these graphs, two main differences are observed. First, the contact force excites

higher harmonics; whereas, the noncontact solution is nearly a single harmonic. Secondly, the

high amplitude solution is noisier. Some noise is to be expected in numerical analysis. However,

106

the noise in the high amplitude solution is more significant because of the discontinuities in the

data due to the contact as shown in Figure 5.9(b).

30 35 40 45 50 55 60 65

-800

-600

-400

-200

0

200

400

600

800


No

nd

ime

nsio

na

lize

d T

ip V

elo

city

0 0.1 0.2 0.3 0.4 0.5 0.6

-80

-60

-40

-20

0

20

40

60

80

Nondimensionalized Time

No

nd

ime

nsio

na

lize

d T

ip D

isp

lace

me

nt

(a) (b)

Figure 5.9. The (a) zoomed in phase portrait and (b) time history of one cycle of the response for

one mode with σ = 26 Hz and v0 = 0.1 V.

0 20 40 60 80 10010

-8

10-6

10-4

10-2

100

102

104

Nondimensionalized Frequency

Pow

er

Spectr

um

0 20 40 60 80 10010

-8

10-6

10-4

10-2

100

102

104


Pow

er

Spectr

um

(a) (b)

Figure 5.10. Power spectra for one mode with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


107

3.2. Two-mode Analysis

In this section, numerical results based on two modes are presented. Figure 5.11 is the

phase portrait for the contact and noncontact branch. Similar to the first mode phase portrait,

Figure 5.11(b), noncontact with two modes, is symmetric about the vertical axis, while Figure

5.11(a), the contact solution, is not. The right side of Figure 5.11(a) shows the contact region.

These are obviously more complicated than the first mode phase portraits and show more of the

intricacies of the tip dynamics.

-50 0 50-2000

-1500

-1000

-500

0

500

1000

1500

2000


Nondim

ensio

naliz

ed T

ip V

elo

city

-50 0 50-2000

-1500

-1000

-500

0

500

1000

1500

2000


No

nd

ime

nsio

na

lize

d T

ip V

elo

city

(a) (b)

Figure 5.11. Phase portrait for two modes with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


Figure 5.12 is the power spectra for the high and low contact solutions. Both solutions

show a spike at the first and second natural frequencies, and similar to one mode, Figure 5.12(a)

shows higher harmonics while Figure 5.12(b) does not.

The contributions of the first two modes to the time response of the microcantilever tip

dynamics are shown in Figure 5.13. The second mode contributes 23.45% and 26.77% to the two

mode steady state solution for the contact and noncontact branch, respectively. Considering this

and the large differences in the phase portraits between first and second mode indicates that one

108

mode alone is not sufficient for the analysis of the dynamics of the AFM microcantilever probe

dynamics.

0 20 40 60 80 10010

-8

10-6

10-4

10-2

100

102

104


Pow

er

Spectr

um

0 20 40 60 80 10010

-8

10-6

10-4

10-2

100

102

104


Pow

er

Spectr

um

(a) (b)

Figure 5.12. Power spectra for two modes with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


0 500 1000 1500-80

-60

-40

-20

0

20

40

60

80


Nondim

ensio

naliz

ed

Tip

Dis

pla

cem

ent

Mode 1

Mode 2

0 500 1000 1500-80

-60

-40

-20

0

20

40

60

80


Nondim

ensio

naliz

ed

Tip

Dis

pla

cem

ent

Mode 1

Mode 2

(a) (b)

Figure 5.13. Contributions to time histories of the low and high amplitude limit cycles for σ = 26

Hz and v0 = 0.1 V.

3.3. Higher-mode Analysis

In this section, higher-modes are investigated to determine how many modes are

necessary to accurately describe the intricacies and complexities of the piezoelectric

microcantilever probe. Figure 5.14 is the phase portrait for three, four, five, and six modes for

109

the contact solution at σ = 26 Hz and v0 = 0.1 V. Comparing Figures 5.8(a), 5.11(a), and 5.14

allow some conclusions to be made about the number of modes needed to depict the dynamics of

the system. Two modes dramatically improve the results over one mode. While additional modes

do not generate as much improvement as the second mode, modes three, four, and five do give

some improvement, which may be beneficial if very precise measurements are needed. However,

the sixth mode phase portrait is practically indistinguishable from the fifth.

The power spectra for six modes is presented in Figure 5.15 for the contact and

noncontact solutions. Again, the biggest difference between these two power spectra is the

excitation of harmonics by the contact force on the high amplitude branch. Figure 5.15 also

supports the conclusions drawn from the phase portraits that the first two modes are both

significant in the dynamics of the system as can be observed by the fact that their spikes are in

the 102 to 104 range on the power spectrum. Modes three, four, and five are in the 100 to 102

range and make some contribution, and the sixth mode practically disappears among the

harmonics in Figure 5.15(a).

Figure 5.16 shows the contribution of each mode to the time response of the

microcantilever nondimensionalized tip displacement. The time response plot of microcantilever

nondimensionalized tip velocity with contributions from one mode up to six modes are shown in

Figure 5.17. Table 5.5 lists the percentage of error for the tip displacement and velocity for each

mode when compared to six modes.

The results in Figures 5.13 and 5.14 and Table 5.5 are in agreement with the results of the

power spectrum analysis and the phase portraits for each mode. Figure 5.13 and 5.14 clearly

show that the first and second modes are the most important, while the third, fourth, and fifth

modes are much smaller, and the sixth mode is barely visible. The error drops dramatically from

110

the first to second mode as can be seen in Table 5.5. The decrease in error after two modes is not

as substantial, and the difference between the fifth and sixth mode is less than 1% for both

displacement and velocity.

-100 -50 0 50 100

-3000

-2000

-1000

0

1000

2000

3000


Nondim

ensio

naliz

ed T

ip V

elo

city

3 modes

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-3000

-2000

-1000

0

1000

2000

3000


No

nd

ime

nsio

na

lize

d T

ip V

elo

city

4 modes

-100 -50 0 50 100

-3000

-2000

-1000

0

1000

2000

3000


Nondim

ensio

naliz

ed T

ip V

elo

city

5 modes

-100 -50 0 50 100

-3000

-2000

-1000

0

1000

2000

3000


Nondim

ensio

naliz

ed T

ip V

elo

city

6 modes

Figure 5.14. Phase portrait with σ = 26 Hz and v0 = 0.1 V for the high amplitude responses with n

modes.

111

0 50 100 150 200 250 300 35010

-8

10-6

10-4

10-2

100

102

104


Pow

er

Spectr

um

0 50 100 150 200 250 300 35010

-8

10-6

10-4

10-2

100

102

104


Pow

er

Spectr

um

(a) (b)

Figure 5.15. Power spectra for six modes with σ = 26 Hz and v0 = 0.1 V for the (a) high and (b)


Figure 5.16. Contributions of 6 modes to the time history of the microcantilever probe

nondimensionalized tip displacement with σ = 26 and v0 = 0.1 V.

Additionally, multiple frequencies were compared to the results found above for phase

portraits, power spectra, and time histories of displacement and velocity. For brevity, the figures

are not included here. However, the results are that the behavior in the bistable region is similar

to the behavior outside the stable region. Initial conditions in the bistable region result in a

112

possibility of two different solutions. However, despite the fact that some extra settling time is

needed, once the initial conditions determine whether the microcantilever dynamics will result in

contact or noncontact, the solution is very similar to the solution in the monostable region.

0 500 1000 1500

-3000

-2000

-1000

0

1000

2000

3000


No

nd

ime

nsio

na

lize

d T

ip V

elo

city

1 Mode

2 Modes

3 Modes

4 Modes

5 Modes

6 Modes

Figure 5.17. Time history of microcantilever probe nondimensionalized tip velocity with

contributions from n modes and σ = 26 Hz and v0 = 0.1 V.

Table 5.5. Error when considering n modes.

Modes

Considered

Error in

Displacement (%)

Error in

Velocity (%)

1 29.51 68.27

2 7.935 38.57

3 5.357 31.66

4 2.059 15.36

5 0.06327 0.7911

6 0 0

113

IV. Conclusions

The mechanics of an AFM piezoelectric microcantilever were examined. Two major

parameters in the sensing process that were carefully studied to guarantee the accuracy of

measurement were investigated. The first parameter was the nonlinear force between the

microcantilever tip and sample, and the second parameter was the effect of the piezoelectric

layer on the motion of the microcantilever. The equations of motion for the microcantilever were

derived using the energy method. The analytical expressions for the natural frequencies and

mode shapes were obtained. The method of multiple scales was used to investigate the analytical

frequency response of the piezoelectric microcantilever probe. The effects of nonlinear excitation

force on the microcantilever beam’s frequency and amplitude were analytically studied. The

results show a frequency shift in the response around the natural frequency resulting from the

force nonlinearities. Accurate imaging depends on correct modeling of the microcantilever probe

dynamics so this frequency shift during contact mode is an important consideration in the

modeling of the AFM mechanics for accuracy when creating images or using the microcantilever

for other applications. This frequency shift or jump phenomenon led to hysteresis and was

investigated via a bifurcation analysis. The bistable region resulting from the frequency shift was

studied by comparing the phase portraits, power spectra, and time response solutions for the high

and low amplitude solutions at the same input voltage and frequency from the piezoelectric

layer. The main differences were that the high amplitude solution made contact with the sample,

while the low amplitude solution did not. As a result of this nonlinear contact force acting on the

tip, not only were the natural frequencies excited, but also the harmonics of the input frequency

were excited. A convergence analysis was performed to determine how many modes are

necessary to accurately predict the complicated AFM microcantilever probe dynamics. As a part

114

of this study, phase portraits, power spectra, and time response solutions of both tip displacement

and velocity were analyzed up to six modes. This analysis led to the conclusion that one mode is

insufficient for modeling the microcantilever dynamics. For accuracy of the AFM, at least two

modes are necessary. Also, for most uses of the AFM two modes is sufficient. However, if

highly accurate measurements are needed or if it is necessary to model the complex intricacies of

the time response, then up to five modes would be appropriate. It is unlikely that more than five

modes would be necessary even for applications that require very precise readings. Comparisons

of the behavior of the microcantilever dynamics inside the bistable region to the behavior in the

monostable region resulted in the conclusion that, while initial conditions determine whether the

solution will fall on the high or low amplitude branch, the behavior of that solution is very

similar to the behavior of the solution in the monostable region.

References




061107.





130(6), 061003.

[4] Holscher, H., Schwarz, U.D., and Wiesendanger, R. (1997). Modeling of the scan process

in lateral force microscopy. Surface Science, 375, 395-402.



Engineering, 37(3), 658-670.





115



Actuators A, 102, 106-113.



[8] Yamada, H., Itoh, H., Watanabe, S., Kobayashi, K., and Matsushige, K. (1999). Scanning

near-field optical microscopy using piezoelectric cantilevers. Surface and Interface

Analysis, 27(56), 503-506.







[11] Mahmoodi, S. N. and Jalili, N. (2007). Nonlinear vibrations and frequency response of

piezoelectrically-driven microcantilevers. International Journal of Nonlinear Mechanics,

42(4), 577-587.

[12] Rogers, B., Manning, L., Sulchek, T., and Adams, J.D. (2004). Improving tapping mode

atomic force microscopy with piezoelectric cantilevers. Ultramicroscopy, 100, 267–276.

[13] Manning, L., Rogers, B., Jones, M., Adams, J. D., Fuste, J. L., and Minne, S. C. (2003).

Self-oscillating tapping mode atomic force microscopy. Review of Scientific Instruments,

74(9), 4220-4222.

[14] Shin, C., Jeon, I., Khim, Z.G., Hong, J.W., and Nam, H. (2010). Study of sensitivity and

noise in the piezoelectric self-sensing and self-actuating cantilever with an integrated

Wheatstone bridge circuit. Review of Scientific Instruments, 81, 035109.



179.






Physics, 38, 7155–7158.

116

[18] Asakawa, H. and Fukuma, T. (2009). Spurious-free cantilever excitation in liquid by

piezoactuator with flexure drive mechanism. Review of Scientific Instruments, 80, 103703.



705-709.

[20] El Hami, K. and Gauthier-Manuel, B. (1998). Selective excitation of the vibration modes of

a cantilever spring. Sensors and Actuators A, 64, 151–155.

[21] Cantrell, S. A. and Cantrell, J. H. (2011). Renormalization, resonance bifurcation, and

phase contrast in dynamic atomic force microscopy. Journal of Applied Physics, 110,

094314.

[22] Korayem, M. H. and Ebrahimi, N. (2011). Nonlinear dynamics of tapping-mode atomic

force microscopy in liquid. Journal of Applied Physics, 109, 084301.

[23] Korayem, M. H. and Sharahi, H. J. (2011). Analysis of the effect of mechanical properties

of liquid and geometrical parameters of cantilever on the frequency response function of

AFM. International Journal of Advanced Manufacturing Technology, 57, 477-489.



199-207.











Physics, 10, (2010) 1416-1421.






117

[31] McCarty, R. and Mahmoodi, S. N. (2014). Parameter sensitivity analysis of nonlinear

piezoelectric probe in tapping mode atomic force microscopy for measurement

improvement. Journal of Applied Physics, 115, 074501.

[32] Mahmoodi, S.N. and Jalili, N. (2009). Piezoelectrically-actuated microcantilevers: an

experimental nonlinear vibration analysis. Sensors and Actuators A, 150(1), 131-136.

[33] Nayfeh, A. H. (1973). Perturbation Methods. New York: John Wiley and Sons.



Vibration, 275, 283-298.

[35] Nayfeh, A. H. and Balachandran, B. (1995). Applied Nonlinear Dynamics: Analytical,

Computational, and Experimental Methods. New York: John Wiley and Sons.

Appendix I: Complete ε Equations

The equations in this appendix are the complete ε equations used in the method of multiple

scales referred to in Section 2.4.

Order of (0):

)1ˆ(ˆ)ˆ(ˆ)ˆ(ˆ01

iv

00

2

0 xHwwxMwDxm , (A1)

000 ww at 0ˆ x ; 000 ww at 1ˆ x , (A2)

Order of (1):

2

0211010

iv

11

2

0ˆˆ)1ˆ()ˆ(ˆ2)ˆ(ˆ)ˆ(ˆ wwxHwDDxmwxMwDxm , (A3)

011 ww at 0ˆ x ; 011 ww at 1ˆ x , (A4)

Order of (2):

,ˆ2ˆˆ)1ˆ()ˆ(ˆ)ˆ(ˆ)ˆ(ˆ)ˆ(ˆ

)ˆ(ˆ2)ˆ(ˆ)ˆ(ˆ2)ˆ(ˆ)ˆ(ˆ

102

3

0321000

1100

2

1020

iv

22

2

0

wwwwxHtVxKMwtVxKwD

wDDxmwDxmwDDxmwxMwDxm

pop

(A5)

022 ww at 0ˆ x ; 022 ww at 1ˆ x . (A6)

118

Appendix II: Definitions of C1, C2, C3, and C4.

The equations in this Appendix are the definitions of C1, C2, C3, and C4 referred to in Equations

(23-24).

nn

nnn CCC

11

131121

coshcos

sinsinhsin

, (A7)

nn

nnnnCC

11

111132

coshcos12

coshcossinhsin1

, (A8)

2

1

3ˆˆ4ˆ

1

nwxmC

, (A9)

1

4ˆˆˆ83

1

xMC , (A10)

xM

xm nn

ˆˆ

ˆˆ4 24

1

. (A11)

119

CHAPTER 6

OVERALL CONCLUSIONS

The AFM is a highly accurate tool used for three dimensional imaging. Microcantilevers

like the ones in the AFM are widely used for a variety of applications. A reliable and accurate

model of the AFM microcantilever probe dynamics is necessary for accurate imaging or for

accurate readings in the many other applications. In this dissertation, novel mathematic models

for describing the dynamics of the AFM microcantilever probe were derived with the purpose of

eliminating the need for a bulky and expensive laser system for taking readings from the AFM

and with the purpose of allowing AFM software packages to be simplified. In the first chapter,

the problem statement and objectives were clearly defined.

In the second chapter, three different ways of handling the forces applied to the

microcantilever of an AFM were examined. The first case included the forces in the boundary

conditions. The second case included them in the equation of motion with boundary conditions

like that of a free end. The third case considered the contact and excitation forces in the equation

of motion and the inertial force due to the tip mass in the boundary conditions. The equations of

motion were derived, the natural frequencies and mode shapes were determined, and the time

response was found. Comparing the experimental data to numerical data showed that the second

case is simple to derive but inaccurate.

Comparisons of the experimental data with the mathematical models from the first and

third case result in equally accurate results. However, it should be noted that including the forces

in the equations of motion and multiplying by a step function is more physically accurate than

120

including it in the boundary conditions. Therefore, including the forces in the equations of

motion is preferable for most situations. However, an investigation of the effect of tip mass on

the microcantilever dynamics revealed that the third case was practically unaffected by a large

variation of tip mass, while the first case showed a significant effect from variation of tip mass.

Therefore, for applications with a large tip mass, such as biosensing, it may be necessary to use

the first case, despite its more complex derivation.

The third chapter studied a common issue among AFM microcantilever applications: the

nonlinear forces applied to the AFM tip due to tapping or contact with the sample. The nonlinear

contact and van der Waals forces at the AFM tip are the main tools for the AFM to scan or

measure, but the nonlinearities can cause regions of instability that must be avoided for accurate

imaging or sensing. The nonlinear equations of motion for the nonlinear vibrations of the AFM

microcantilever probe due to the nonlinear curvature and contact force were derived. The mode

shapes were derived based on the presence of the nonlinear contact force as a boundary condition

at the free-end of the microcantilever. The results showed that the natural frequency of the

microcantilever is dependent on the coefficient of the linear term of the contact force. The results

also showed that the natural frequency is a function of the initial tip-sample distance, and for

larger distances, the values of the natural frequency can be much larger than the natural

frequency of a microcantilever with zero boundary conditions at the free end.

The frequency response curve was obtained so that the effect of the nonlinear terms could

be analytically investigated. Results showed that choosing the optimal excitation force and tip-

sample displacement significantly decreased the effect of the nonlinear terms and provided a

linear response. A smaller tip-sample displacement decreased the nonlinear response. In addition,

increasing the force resulted in faster contact and made the response stable, but decreased the

121

accuracy of the measurement. The results showed that, in general, the best way to avoid a

nonlinear response is to set the initial tip-sample distance (before starting the tapping) to a

constant, and then increase the excitation amplitude to reach the stable region, but applying more

force is not recommended.

Piezoelectric actuators and sensors along with an electrical circuit can replace the bulky

and expensive laser measurement system. However, the dynamics of the piezoelectric

microcantilever probe must be accurately modeled in order to generate accurate images as well

as for accurate readings when using the AFM microcantilever for other applications. Therefore,

in the fourth chapter, the mechanics of a piezoelectric microcantilever beam subject to a

nonlinear contact force were examined. The equations of motion for a microcantilever were

derived. The analytical expressions for the natural frequencies and mode shapes were obtained.

The method of multiple scales was used to investigate the analytical frequency response of the

piezoelectric microcantilever probe. The effects of the nonlinear excitation force on the

microcantilever beam’s frequency and amplitude were analytically studied. The results showed a

frequency shift in the response around the natural frequency resulting from the force

nonlinearities. In addition, the results of a sensitivity analysis of the system parameters on the

nonlinearity effect were investigated. The effect of slightly changing two parameters – tip radius

and length of the microcantilever – was shown to reduce the frequency shift by 19.6% while only

decreasing the amplitude by 10.6%. By reducing the nonlinearity effect, it may be possible to use

a linear model to analyze the microcantilever mechanics, which would make the AFM software

package less cumbersome and calculation intensive.

In the fifth chapter, the equations of motion and frequency response equation found in the

fourth chapter were further investigated. The effects of the nonlinear excitation force on the

122

microcantilever beam’s frequency and amplitude were analytically studied. The results showed a

frequency shift in the response around the natural frequency resulting from the force

nonlinearities. Accurate imaging depends on correct modeling of the microcantilever probe

dynamics so this frequency shift during contact mode is an important consideration in the

modeling of the AFM mechanics for accuracy when creating images or using the microcantilever

for other applications. This frequency shift or jump phenomenon led to hysteresis and was

investigated via a bifurcation analysis. The bistable region resulting from the frequency shift was

studied by comparing phase portraits, power spectra, and time response solutions for the high

and low amplitude solutions at the same input voltage and frequency from the piezoelectric

layer. The main differences were that the high amplitude solution made contact with the sample,

while the low amplitude solution did not. As a result of this nonlinear contact force acting on the

tip, not only were the natural frequencies excited, but also the harmonics of the input frequency

were excited.

A convergence analysis was performed to determine how many modes are necessary to

accurately predict the complicated AFM microcantilever probe dynamics. As a part of this study,

phase portraits, power spectra, and time response solutions of both the tip displacement and

velocity were analyzed up to six modes. This analysis led to the conclusion that one mode is

insufficient for modeling the microcantilever dynamics. For accuracy of the AFM, at least two

modes are necessary. Also, for most uses of the AFM two modes is sufficient. However, if

highly accurate measurements are needed or if it is necessary to model the complex intricacies of

the time response, then up to five modes would be appropriate. It is unlikely that more than five

modes would be necessary even for applications that require very precise readings. Comparisons

of the behavior of the microcantilever dynamics inside the bistable region to the behavior in the

123

monostable region resulted in the conclusion that, while the initial conditions determine whether

the solution will fall on the high or low amplitude branch, the behavior of that solution is very

similar to the behavior of the solution in the monostable region.

Possible future works based on this research is to verify the nonlinear analytical results

from Chapters 3, 4, and 5 with experimental results. Additionally, based on the work presented

in Chapter 3, a software routine for the AFM could be developed to automatically optimize the

tip-sample distance and excitation force to minimize the effect of the nonlinearities. For Chapters

4 and 5, an AFM capable of using a piezoelectric microcantilever probe would be necessary for

experimental validation. Also, the equations derived could be expanded to include the effects of

tapping in a liquid medium.

LINEAR AND NONLINEAR VIBRATION ANALYSIS

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