Intermolecular and Interfacial Forces:

Elucidating Molecular Mechanisms using

Chemical Force Microscopy

A thesis presented

by

Paul David Ashby

to

The Department of Chemistry and Chemical Biology

in partial fulfillment of the requirements

For the degree of

Doctor of Philosophy

In the subject of

Physical Chemistry

Harvard University

Cambridge, Massachusetts

May 2003

© 2003 by Paul David Ashby

All Rights Reserved

Intermolecular and Interfacial Forces: Elucidating Molecular Mechanisms using Chemical Force Microscopy

Professor Charles M. Lieber Paul David Ashby

May 2003

Abstract

Investigation into the origin of forces dates to the early Greeks. Yet, only in

recent decades have techniques for elucidating the molecular origin of forces been

developed. Specifically, Chemical Force Microscopy uses the high precision and

nanometer scale probe of Atomic Force Microscopy to measure molecular and interfacial

interactions. This thesis presents the development of many novel Chemical Force

Microscopy techniques for measuring equilibrium and time-dependant force profiles of

molecular interactions, which led to a greater understanding of the origin of interfacial

forces in solution.

In chapter 2, Magnetic Feedback Chemical Force Microscopy stiffens the

cantilever for measuring force profiles between self-assembled monolayer (SAM)

surfaces. Hydroxyl and carboxyl terminated SAMs produce long-range interactions that

extend one or three nanometers into the solvent, respectively. In chapter 3, an ultra low

noise AFM is produced through multiple modifications to the optical deflection detection

system and signal processing electronics. In chapter 4, Brownian Force Profile

Reconstruction is developed for accurate measurement of steep attractive interactions.

Molecular ordering is observed for OMCTS, 1-nonanol, and water near flat surfaces. The

molecular ordering of the solvent produces structural or solvation forces, providing

insight into the orientation and possible solidification of the confined solvent. Seven

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molecular layers of OMCTS are observed but the oil remains fluid to the last layer. 1-

nonanol strongly orders near the surface and becomes quasi-crystalline with four layers.

Water is oriented by the surface and symmetry requires two layers of water (3.7 Å) to be

removed simultaneously. In chapter 5, electronic control of the cantilever Q (Q-control)

is used to obtain the highest imaging sensitivity. In chapter 6, Energy Dissipation

Chemical Force Microscopy is developed to investigate the time dependence and

dissipative characteristics of SAM interfacial interactions in solution. Long-range

adhesive forces for hydroxyl and carboxyl terminated SAM surfaces arise from solvent,

not ionic, interactions. Exclusion of the solvent and contact between the SAM surfaces

leads to rearrangement of the SAM headgroups. The isolation of the chemical and

physical interfacial properties from the topography by Energy Dissipation Chemical

Force Microscopy produces a new quantitative high-sensitivity imaging mode.

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Acknowledgements

Graduate school at Harvard has been a wonderful time of learning and growth in

all aspects of life. First, I would like to thank my advisor, Charles Lieber. I greatly

appreciate his support and unique treatment of me since it best fit our personalities and

methods of doing science. He was like a father teaching a young child how to ride a bike.

At first, the father holds the seat and runs alongside giving direction and stability. As the

child becomes more skillful then the father lets go and allows the child to ride away, but

continues to shout encouragement and direction. I have also appreciated the diverse

opinions and feedback from my other committee members, Rick Heller and Sunney Xie.

I am grateful to Jeff, Linda, Natalya, and Emily for running the group smoothly and

shouldering the administrative load so that I can focus on science.

I am greatly indebted to my fellow graduate students and postdoctorates whom I

have worked with day to day. Alex Noy first introduced me to the AFM and kindled my

enthusiasm for intermolecular forces. Liwei Chen was a great partner for the magnetic

feedback work because of his curiosity and playful demeanor. Tjerk Oosterkamp was

instrumental during the noise reduction work and his friendship and encouragement were

revitalizing during the darkest and most difficult years of my Ph.D. I would also like to

thank Julia Forman for her help progressing the Energy Dissipation work. Forces have

puzzled philosophers for many years because they are “spooky action at a distance”.

Similarly, Jason Cleveland significantly shaped this work both over distance and through

time. Jinlin Huang and Steve Shepard have been of great assistance in instrument and

sample fabrication. The friendship of Adam, Andrew, Chen, Chin Li, Ernesto, Jason,

Lincoln, Mark, Sung Ik, and Teri has made my many years in the Lieber lab a very

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enjoyable experience. I also thank Angie, Scott, Jason, David, and Alec for the

exhilarating experience with Potentia.

To all the saints in Christ Jesus at the Graduate Christian Fellowship, I thank my

God every time I remember you. God has profoundly shaped who I am through you.

Dave Landhuis was one of the first people I met and our times of running and eating

together were a great support. There is no one I would rather study the Bible with other

than Lou Soiles, who was not only a profound teacher but a wonderful friend, counselor,

and companion. Dave Nancekivell’s gentleness, generosity, and loyalty are without

parallel and I will always treasure our friendship. My roommates Stephen, Bob, David,

Mark, and Danny made the apartment into a home and a place of hospitality, play, and

accountability. Memorizing scripture with Jason, Michael, and Kelly has “renewed my

mind”. My rich relationship with Walter has brought new meaning and understanding to

the stories about David and Jonathan.

My family has been a great support through the whole process. I loved “talking

shop” with Dad. Mom’s unwavering faith in my ability and encouraging words were

always refreshing. My sister, Pam, has become one of my closest friends. My wife,

Keng Boon, has consistently encouraged me with words, deeds, and notes. Her practical

nature helps keep me focused and moving forward. Her kindness, gentleness, love for

others, and love for Jesus are unique. I am truly blessed to start our journey together

here.

I dedicate this thesis to Jesus, my God, Savior, teacher, and friend. Thank you for

creating this beautiful world with all of its intricacies, giving me the opportunity to study

at Harvard, and surrounding me with a cloud of witnesses.

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Table of Contents

Chapter 1: Intermolecular Forces and Atomic Force Microscopy

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Surface Forces Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Chapter 2: Magnetic Feedback Chemical Force Microscopy

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Low Bandwidth Magnetic Feedback Theory . . . . . . . . . . . . . . . . . . 13

2.3 Low Bandwidth Magnetic Feedback Experiments . . . . . . . . . . . . . . . 15

2.3.1 Magnetic Tip Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Chemical Surface Preparation . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.3 Instrument Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.4 Tip and Feedback Calibration . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.5 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.6 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.7 Hydroxyl Terminated SAM Surfaces . . . . . . . . . . . . . . . . . . . 20

2.3.8 Hydroxyl Surfaces Approach-Separation Hysteresis . . . . . . . . . . . 22

2.3.9 Van der Waals Model Fit to Hydroxyl Data . . . . . . . . . . . . . . . 24

2.3.10 Carboxyl Terminated SAM Surfaces . . . . . . . . . . . . . . . . . . 26

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2.3.11 DLVO Model Fit to pH 7.0 Carboxyl Data . . . . . . . . . . . . . . . 26

2.3.12 Comparison of Attractive Hydroxyl and Carboxyl Interactions . . . . . 30

2.4 Effects of Limited Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 High Bandwidth Magnetic Feedback Theory . . . . . . . . . . . . . . . . . . 35

2.6 High Bandwidth Magnetic Feedback Experiments . . . . . . . . . . . . . . . 37

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 3 Noise Reduction

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Contact and Tapping Mode Noise . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Low Frequency Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.1 Low Coherence Length IR Laser . . . . . . . . . . . . . . . . . . . . . 50

3.3.2 AFM Base Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.3 Wind Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 High Frequency White Noise Reduction . . . . . . . . . . . . . . . . . . . . 54

3.4.1 Laser Beam Truncation and Diffraction . . . . . . . . . . . . . . . . . 54

3.4.2 Feedback Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 White Noise Correlation between Photodiode Segments . . . . . . . . . . . 59

3.6 Position Fluctuation Noise Reduction by Laser Beam Truncation . . . . . . . 62

3.7 Total Noise Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.8 Further increases to signal to noise . . . . . . . . . . . . . . . . . . . . . . . 63

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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3.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Chapter 4 Solvation and Structural Forces

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Model for Solvent Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Force Profile Measurement Error . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Brownian Force Profile Reconstruction (BFPR) . . . . . . . . . . . . . . . . 73

4.5 Instrument Noise Compensation for BFPR . . . . . . . . . . . . . . . . . . 80

4.6 Data Collection and analysis for BFPR . . . . . . . . . . . . . . . . . . . . 85

4.7 Octa-methyl-cyclotetrasiloxane (OMCTS) . . . . . . . . . . . . . . . . . . . 86

4.8 1-Nonanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.9 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Chapter 5 Q-control for Optimizing AFM

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Q-Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3 Feedback Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Cantilever Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5 Lock-in Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.6 Noise Power as a Function of Q . . . . . . . . . . . . . . . . . . . . . . . 119

5.7 Relationship between Amplitude and Phase Noise During Tapping . . . . . 121

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5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Chapter 6 Energy Dissipation Chemical Force Microscopy

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2 Contact Mode Force Profiles at Low Deborah Number . . . . . . . . . . . 129

6.3 Energy Dissipation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.4 Energy Dissipation Force Curves . . . . . . . . . . . . . . . . . . . . . . . 140

6.5 Tapping Mode Force Profile Reconstruction (TMFPR) . . . . . . . . . . . 150

6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.5.2 TMFPR Theory and Noiseless Simulations . . . . . . . . . . . . . . . 152

6.5.3 TMFPR Simulations with Noise and Reduced Bandwidth . . . . . . . 155

6.5.4 TMFPR of Dissipative Interactions between SAM Surfaces . . . . . . 161

6.6 Mechanism for Energy Dissipation . . . . . . . . . . . . . . . . . . . . . . 166

6.7 The Phase Signal and Energy Dissipation . . . . . . . . . . . . . . . . . . 171

6.8 Energy Dissipation Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Appendix

A.1 General Techniques and the Digital Instruments Multimode AFM . . . . . 182

A.1.1 Contact Mode Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 182

A.1.2 Contact Mode Force Curves . . . . . . . . . . . . . . . . . . . . . . 182

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A.1.3 Tapping Mode Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.1.4 Tapping Mode Force Curves . . . . . . . . . . . . . . . . . . . . . . 185

A.1.5 Digital Instruments Multimode AFM . . . . . . . . . . . . . . . . . . 186

A.2 Data Collection with National Instruments 5911 . . . . . . . . . . . . . . . 188

A.3 Cantilever Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

A.4 Cantilever Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . 194

A.5 Brownian Force Profile Reconstruction . . . . . . . . . . . . . . . . . . . 198

A.6 Energy Dissipation Force Curves . . . . . . . . . . . . . . . . . . . . . . 204

A.7 Tapping Mode Force Profile Reconstruction . . . . . . . . . . . . . . . . . 211

A.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

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List of Figures Chapter 1: Intermolecular Forces and Atomic Force Microscopy Figure 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

SFA measurements of classic DLVO forces between sapphire surfaces in 0.001 M NaCl solutions. The arrows depict the trajectory of the surface during the instability.

Figure 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 SFA measurement of water solvent ordering between mica sheets. The force required to remove subsequent layers increases exponentially with distance.

Figure 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sketch of essential AFM components. The tip is mounted on the cantilever. The interaction with the surface is measured by the deflection of the laser beam on the photodiode. A 3-dimensional piezo stage controls the motion of the sample.

Chapter 2: Magnetic Feedback Chemical Force Microscopy Figure 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Sketch of a force profile (left) and resulting force curve (right). The gray arrows indicate the trajectory of the cantilever in the region where the gradient of the force profile exceeds the fixed spring constant, k, of the cantilever leading to snap-in and snap-out.

Figure 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

A magnetic feedback (MF) schematic, where the feedback loop is comprised of the cantilever, split photodiode, low pass filter, variable gain amplifier, and solenoid.

Figure 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Force diagram where FPot is balanced by the feedback loop, Fmag, and the cantilever, Fcant. The effective stiffness is the sum of the two stiffness components.

Figure 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Sketch of a thiol Self-Assembled Monolayer on gold. The alkyl chains pack in a crystalline structure tilted ~30 degrees from normal.

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Figure 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 SEM image of the tip of the cantilever used in Magnetic Feedback Chemical Force Microscopy measurements of the carboxyl functionalized surfasce.

Figure 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Deflection trace (displayed as force) of hydroxyl-terminated SAM surfaces in solution during approach without magnetic feedback has characteristic snap-in (arrow).

Figure 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Data traces of hydroxyl-terminated SAM surfaces in solution during approach. (a) Deflection trace (displayed as force) with magnetic feedback has no instabilities and a reduced total deflection. (b) Magnetic force trace from solenoid current.

Figure 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Approach (black) and separation (gray) force profiles for the hydroxyl-terminated tip-sample interaction.

Figure 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Force profile (gray) for hydroxyl-terminated SAMs in deionized water with a van der Waals model fit (black). The Hamaker constant value is 1.0×10-19 J.

Figure 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Data traces of carboxyl-terminated SAM surfaces in solution during approach. (a) Deflection trace (displayed as force) with magnetic feedback has no instabilities. (b) Magnetic Force trace from the solenoid current reveals the major features in the force profile.

Figure 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Force profile (gray) for carboxyl terminated SAMs in 0.010 M, pH 7 phosphate buffer with fit (black) using a charge regulation DLVO model. The Hamaker constant value is 1.2×10-19 J.

Chapter 3 Noise Reduction Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Power spectra near DC showing contact mode noise (shaded regions) in a 1 kHz bandwidth for a cantilever with significant white and 1/f noise (gray) and a cantilever with reduced instrument noise. The instrument noise contribution is significantly more than the thermal noise. Inset shows the same spectra over a larger frequency range to show resonance.

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Figure 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Sidebands A and B around the reference frequency of 70 kHz are shifted down to DC by the lock-in amplifier.

Figure 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Noise power spectra of cantilevers in water (a) and in air(b). Measurements with large (gray) and small (black) contributions from instrument noise are shown in each frame. The shaded region depicts the tapping mode noise in a 1.5 kHz lock-in bandwidth. The relative instrument noise is more significant at low Q.

Figure 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (a) Sketch of laser light passing by the cantilever and scattering off the surface. The reflected beam and scattered light can cause interference. (b) Oscillations in a force curve caused by interference (gray). Using a low coherence IR laser eliminated the interference (black).

Figure 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Noise spectra of deflection signals from the AFM base (black) and breakout box using a difference instrumentation amplifier INA106 (gray). The base adds both significant low frequency periodic noise and white noise. 52

Figure 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Low frequency noise spectra of AFM instrument when uncovered (black) and covered (gray). When uncovered, wind currents can add large low frequency oscillations.

Figure 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Drawing of AFM head showing laser beam traces with (solid) and without (dotted) truncating slit. The slit increased the aspect ratio of the beam and caused a diffraction pattern that focused light on the boundaries of the photodiode segments.

Figure 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Drawing of diffraction pattern caused by the truncating slit. Distances are representative of the size of the beam at the photodiode.

Figure 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 (a) Schematic of photodiode amplifier with noise sources, en and In. (b) Bode plot of amplifier open loop gain, signal gain, and noise gain.

Figure 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Noise spectrum for photodiode amplifier output. Noise peaking is clearly seen at high frequencies but does not contribute in the working frequencies of the AFM.

xiv

Figure 3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Noise spectra for a single photodiode segment (black) and the difference between the segments (gray). The lower noise in the difference signal indicates that the noise is correlated.

Figure 3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Noise power measured at ~35 kHz as a function of photodiode voltage (laser power). (a) Noise from single segment compared to shot and amplifier noise. (b) Difference noise signal compared to shot and amplifier noise. The difference signal is almost shot noise limited.

Figure 3.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 (a) Comparison of power fluctuation (correlated) and position fluctuation (anti-correlated) noise (b). Position fluctuation noise is significantly reduced by laser beam truncation by the slit.

Figure 3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Cantilever noise spectra before instrument modification (gray) and after modification (black). The white noise was significantly reduced from 800 fm/ Hz to 36 fm/ Hz . The spectrum is thermally limited and well fit (dotted) by a damped harmonic oscillator model.

Chapter 4 Solvation and Structural Forces Figure 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Lattice model of molecular interfacial density. (a) Each molecular layer is defined by a Gaussian whose variance is a function of surface distance. (b) The total molecular density (gray) is similar to a decaying sine function (black).

Figure 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 (a) Force profile (black) with markers (gray bars) showing the span of the thermal noise of the cantilever during a force curve. The intensity of the gray bars correlates with the probability of the cantilever position and the black mark indicates the average position. (b) The resulting force profile (gray) badly misses the force profile used for the simulation (black).

Figure 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 (a) Force curve showing deflection with all thermal noise. (b) Histograms of sections of force curve. (c) Histogram converted to energy using Boltzmann's equation. Includes both spring and tip-sample interaction. (d) Energy after spring contribution is subtracted away and positioned for tip-sample distance. (e) Derivative of energy is force. (f) All force sections together. (g) Average of force sections is the Brownian Reconstruction Force Profile.

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Figure 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 (a) Brownian reconstruction force sections superimposed on the force profile used in the simulation. (b) Brownian Reconstruction Force Profile (dark gray) from force sections with ordinary force curve calculated from the same data (gray). The ordinary curve deviates significantly from the shape of the force profile (black) while the Brownian Reconstruction Force Profile is a much better approximation.

Figure 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Power spectrum of cantilever noise without instrument noise (black) and with instrument noise (gray).

Figure 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Brownian Force Profile Reconstruction with instrument noise. The Brownian reconstruction is shown in dark gray and the ordinary force profile in light gray. The ordinary curve more closely matches the force profile used in the simulation (black) when there is no noise compensation (a), but the Brownian force profile is more accurate after noise compensation (b).

Figure 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Force profile of Octa-methyl-cyclotetrasiloxane. The model (gray) fits the data (black) very well revealing that the OMCTS is liquid down to a few layers. The inset reveals the last layer is excluded near 15 mN/m.

Figure 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Force profile of Octa-methyl-cyclotetrasiloxane. The advancing (gray) and receding (black) traces overlap perfectly. The absence of hysteresis is a result of the low viscosity associated with a liquid and not a glass.

Figure 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Average of reconstructed force profiles (gray) for 1-nonanol between hydrophobic surfaces and exponentially decaying sine function (black). The force profile has a period of 4.5 Å.

Figure 4.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Brownian reconstruction force sections for two different force curves a and b of 1-nonanol between hydrophilic surfaces. The force profiles show liquid behavior at distances greater than 1.5 nm but crystalline behavior with phase transitions at distances less than 1.5 nm.

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Figure 4.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Force profile of water ordering against hydroxyl terminated SAM surfaces. (a) Force sections for Brownian Force Profile Reconstruction. (b) Brownian Reconstruction Force Profile from force sections (black) and ordinary force curve from the same data (gray). The ordinary force curve significantly misses the shape of the profile. (c) Average of many Brownian Force Profiles (gray) and an oscillatory fit (black). The oscillations have a period of 3.6 Å.

Chapter 5 Q-control for Optimizing AFM Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Average tapping force for two different Q values as a function of amplitude setpoint, S = A/A0, where A and A0 are the tapping amplitude with and without tip-sample interaction respectively.. The circles and triangles are a Q of 350 and ~2 respectively.

Figure 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A mixed polymer sample imaged with and without Q control used as an advertisement for a commercial product. The region imaged with Q-control seems to show more sensitivity to surface features.

Figure 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Q-control cantilever feedback block diagram. Cantilever deflection is shifted by π/2 and added to the tapping mode drive. The composite signal drives the cantilever motion through the transducer.

Figure 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Schematic of Q-control cantilever feedback system. The cantilever deflection is AC coupled, by a high pass filter, and amplified in a 20-turn variable gain amplifier before being phase shifted by a low pass filter. The shifted signal is summed at the power amplifier, which produces a magnetic field through the solenoid to deflect the cantilever.

Figure 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Noise Power spectra of a Q-controlled cantilever at five different Q values. The integrated noise power increases with Q because the effective temperature is changed by Q-control.

Figure 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Cantilever position plotted on the quadrature phase plane with (a) ωr=0 and (b) ωr=ω. The cantilever sweeps a circle around the origin in a. The amplitude and phase are readily interpreted graphically by the steady cantilever position in b.

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Figure 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 (a) Thermal noise plotted on the quadrature phase plane and has circular symmetry. (b) Thermal noise of a cantilever with amplitude, A, and phase, φ. (c) Amplitude, NA, and phase, Nφ, noise resulting from cantilever thermal noise.

Figure 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Amplitude (a) and phase (b) noise spectra for different amplitudes and Q values. (a) Dark curves are for lower Q values and lighter curves are for higher Q values. (b) Dark curves are for large amplitudes and light curves are for small amplitudes.

Figure 5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 (a) Phase noise for the 4 different amplitudes from figure 5.8 now overlap after being scaled by the amplitude. (b) Scaled phase and amplitude spectra overlap, which supports the model for the origin of amplitude and phase noise.

Figure 5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Effect of lock-in bandwidth. (a) Unfiltered (solid) and filtered (dashed) amplitude noise along with the filter transfer function (gray). (b) Integrated amplitude noise for three different Q values.

Figure 5.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Q-control noise power as a function of Q (gray) follows a Q0.8 power law (black). The cantilever heating contributes a Q and the bandwidth limiting

should add another Q . The discrepancy is a result of too open a bandwidth.

Figure 5.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Amplitude and phase noise as a function of amplitude setpoint. Interacting with the surface moves the amplitude noise to the phase noise. Lowering the setpoint reduces amplitude and phase noise unless Z-piezo oscillations start.

Figure 5.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Interaction with the surface causes thermal noise squeezing which lowers the amplitude noise but increases the phase noise.

Figure 5.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Amplitude (a) and phase (b) noise spectra for different proportional gain values. The setpoint>1 spectrum is included for comparison. Proportional gain reduces the noise at high frequencies. Amplitude (c) and phase (d) noise spectra for different integral gain values. Integral gain decreases the noise over all frequencies.

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Figure 5.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Amplitude and phase noise in contact (gray) and out of contact (black) with the surface for three Q values. Higher Q values cause the Z-piezo feedback loop to oscillate.

Chapter 6 Energy Dissipation Chemical Force Microscopy Figure 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Height image of an atomically flat gold surface used for experiments.

Figure 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Contact mode force profiles. The hydroxyl (a) and carboxyl terminated surfaces at low pH (b) and high pH (c) show no hysteresis or energy dissipation. The tip is coated with hydroxyl terminated SAM for all three interactions.

Figure 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 (a) Deflection time course during tapping showing significant nonsinusoidal periodic motion from tip-sample interaction. (b) Power spectrum of the deflection time course. Harmonics of the fundamental contain some of the power dissipated to the background through drag.

Figure 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Power spectrum of cantilever thermal noise. The cantilever parameters are calculated from the fit. A vertical arrow marks the tapping frequency. The transfer function is used to compute the drive force and phase offset for off-resonance tapping.

Figure 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Deflection (a) and Z-piezo (b) time courses used for energy dissipation force curves. The numerous oscillations of the deflection time course mark the envelope of oscillation or amplitude. The amplitude is reduced as the piezo brings the surface into contact with the tapping tip.

Figure 6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Tapping amplitude (a) and phase (b) calculated from numerical lock-in of time course data. Energy dissipation (c) calculated from amplitude of all harmonics, phase, and cantilever variables. (d) Energy dissipation plotted as a function of tapping amplitude shows that energy dissipation varies significantly with tapping amplitude.

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Figure 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Energy dissipation force curves using a hydroxyl terminated SAM on the tip tapping against SAM surfaces terminated with hydroxyl (black), carboxyl at high pH (dashed), and carboxyl at low pH (gray). Curves were collected with Q = 6.6 (a, b) and Q = 30 (c, d) and for a free tapping amplitude of A1 ~ 4 nm (a, c) and A1 ~ 2 nm (b, d).

Figure 6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Simulated noiseless deflection (black) and reconstructed interaction force (gray) time courses. The adhesion hysteresis is readily observed between the left (advancing) and right (receding) side of the peaks.

Figure 6.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Reconstructed advancing (light gray) and receding (dark gray) force profiles from the noiseless simulated deflection time course. The reconstructed force profiles show hysteresis and are indistinguishable from the force profiles (black) used in the simulation.

Figure 6.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Power spectra of deflection time courses. The harmonics contain the important information about the tip-sample interaction. The 4th order Savitxky-Golay smooths (gray) remove the high frequency instrument noise from the raw signal (black).

Figure 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Reconstructed interaction force (gray) time courses of the tip-sample distance (black) for simulations including cantilever thermal and instrument noise. Instrument noise considerably degrades the interaction force signal.

Figure 6.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Reconstructed force profiles (gray) from a simulation with noise for three different tapping amplitudes (a-c). They show hysteresis and match the force profiles (black) used in the simulation well.

Figure 6.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Reconstructed force profiles (gray) for three different tapping amplitudes (a-c) from a simulation including noise where attractive force profiles (black) with a very stiff contact region were used. The reconstruction does not match the original force profiles because the instrument noise obscures the information about the stiff contact region in the higher harmonics.

Figure 6.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Reconstructed force profiles from carboxyl data in Figure 6.7a at high pH for three tapping amplitudes (a-c). The equilibrium force profile (dashed) from Figure 6.2c matches the advancing (light gray) trace well. The receding (dark gray) trace shows hysteresis at reduced amplitudes.

xx

Figure 6.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Reconstructed force profiles from the carboxyl data at low pH from Figure 6.7a for three tapping amplitudes (a-c). The equilibrium force profile (dashed) from Figure 6.8b overlaps the advancing (light gray) trace well. The receding force profile (dark gray) hysteresis is localized to the contact region.

Figure 6.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Reconstructed force profile for hydroxyl surface data in Figure 6.7c for three tapping amplitudes (a-c). The advancing (light gray) and receding (dark gray) show hysteresis in the contact region and the receding trace has significantly more adhesion (c).

Figure 6.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 (a) Amplitude, (b) phase, and (c) energy dissipation images of a patterned SAM surface of hydroxyl surrounding a carboxyl square. The black square highlights the edges of the pattern. The topography is coupled into the amplitude and phase but compensated in the energy dissipation leading to significantly more contrast.

Appendix Figure A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Contact Mode force curve (a) raw photodiode signal and (b) scaled force as a function of Z-piezo displacement. The contact region is used to determine the detection sensitivity. (c) The tip-sample distance is calculated by subtracting the deflection from the Z-piezo distance to make a force profile.

Figure A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Tapping Mode force curve (a) Amplitude and (b) phase signals from the lock in amplifier. The phase change is positive when tapping in the attractive regime. The amplitude increases in value and the phase changes becomes negative at the transition to repulsive regime.

Figure A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Sketch of the experimental setup.

Figure A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Labview code for data collection scheme of NI 4911.

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xxii

List of Tables Chapter 2: Magnetic Feedback Chemical Force Microscopy Table 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chi squared values for determining the wellness of a curve fit to the hydroxyl and pH 2.2 carboxyl data. The equations used were a single exponential to model entropic disordering and a second order power law to model van der Waals forces.

Chapter 3 Noise Reduction Table 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Sensitivity values for the signal from one photodiode segment (A) or the difference between segments (A-B), with and without the truncating slit and for three different positions on the cantilever.

Table 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Noise values for the components of the amplifier noise of the AD827 at 50 A for the feedback resistor values of 10 kΩ and 200 kΩ.

Table 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Noise values for the components of the amplifier noise for AD827 and OPA655 at 50 µA and 200 kΩ feedback resistors.

Chapter 6 Energy Dissipation Chemical Force Microscopy Table 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Comparison of signal and signal to noise ratio for Phase and Energy Dissipation Imaging.

Chapter 1 Intermolecular Forces and Atomic Force Microscopy 1.1 Intermolecular Forces

The four fundamental forces mediate all interactions in nature: strong and weak

nuclear, electromagnetism, and gravity. * Although each is crucial, electromagnetism

most directly shapes everyday experience since it determines the nature of atoms,

molecules, and intermolecular forces.1 Together, these microscopic forces mediate all

aspects of life including protein folding and molecular recognition. For many proteins

the substitution of a single amino acid can change the fold and modification of the

binding pocket by only 0.2 Å is enough to change the binding affinity by orders of

magnitude.2

Other processes where forces mediate important interactions include

chemisorption, adhesion, self-assembly, lubrication, fracture, solvation, emulsions,

detergents, and tectonic fault rupture.1,3-6 For instance, adhesion is mediated by both

specific and non-specific interactions. The organization of the molecules and their

binding of solvent play an important role in shaping the strength and distance scale of

adhesive forces. Although poorly understood, adhesion has the most industrial

applications from airplanes to paints to biosensors.7 Also, the necessity of lubrication has

been known for millennia. Early Egyptian hieroglyphics show slaves applying mud to

large stones as they are shoved into place on the pyramids. Lubrication is more important

* Current high-energy physics experiments support the unification of the weak nuclear and electromagnetism.

1

today with high precision parts sliding by each other ceaselessly in factories and on the

road. Lubrication is fundamentally molecular as solvent molecules help surfaces slip past

each other under confinement.4

Understanding these vast phenomena requires intimate knowledge of the

intermolecular force from which they arise. Historically, these forces have been inferred

from macroscopic measurements and phenomena such as adsorption calorimetry, surface

tension studies, pressure induced chemical or vibrational line shifts, equilibrium constants

and elastic moduli.1,4,7 Although significant information has been gleaned from indirect

measurements, the true nature of the interaction is microscopic, accessible only through

direct measurement.

1.2 Surface Forces Apparatus

One of the most important direct force measurement tools is the Surface Forces

Apparatus (SFA) developed by Tabor8 and Israelachvili.1 The SFA probes two

atomically flat sheets of mica affixed on glass surfaces with a radius of curvature (R)

around 1 cm. A piezo moves one surface into contact with the other, which is mounted

on a spring, and the distance between the mica sheets is measured using optical

interferometry. Functionalized surfaces can be prepared but they are required to be

transparent. The great utility of the SFA lies in its precise distance resolution (1 Å) and

well-defined surfaces. The interaction is well-defined by the regularity of the surfaces

and the distance resolution allows molecular scale measurements. The SFA was used to

confirm the validity of the DLVO theory of electrostatic repulsion and van der Waals

adhesion, as shown in figure 1.1.9-13 The data (symbols) are sparse but they correlate

with the theoretical plot well. Incremental steps toward the surface were observed at

2

Forc

e/R

adiu

s (m

N/m

)

Distance (nm)

Figure 1.1 – SFA measurements of classic DLVO forces between sapphire surfaces in 0.001 M NaCl solutions. The arrows depict the trajectory of the surface during the instability.

Distance (nm)

Forc

e (µ

N)

Figure 1.2 – SFA measurement of water solvent ordering between mica sheets. The force required to remove subsequent layers increases exponentially with distance.

3

short distance scales in repulsive contact, implying solvent ordering.14-17 (figure 1.2)18

Steric repulsion between polymer brushes was also quantified, together with the adhesion

between biomembranes.1

The above examples are only a few of the diverse experiments that have been

performed with the SFA. Unfortunately, the normalized stiffness (k/R) of the spring is

very low and there is considerable loss of information resulting from instabilities in the

attractive (adhesion and van der Waals) or rapidly changing (solvent ordering)

interactions as depicted with arrows in Figure 1.1. Also, the calculation of the distance

using interferometry is time consuming. The most sophisticated SFA is automated to

produce an astounding 18 pm of baseline noise but it can only sample at 30 Hz.19 The

more recently invented Atomic Force Microscope, has a greater distance resolution than

the SFA. It consists of a probe that is six orders of magnitude smaller with a greater

normalized spring constant, which makes it ideal for direct measurement of

intermolecular forces.

1.3 Atomic Force Microscopy

Atomic Force Microscopy (AFM) is a very versatile and precise surface force

analysis technique.20 It consists of an ultrasharp tip (radius of curvature ~10 nm) that is

mounted on a spring, through which interaction forces are measured when the tip is

placed in contact with a surface. The most common implementation uses a cantilever as

the spring, and deflection of a laser beam off the back of the cantilever surface to

quantitate the interaction. The tip and sample can be precisely positioned relative to each

other using piezo electric materials. The basic setup of the AFM is drawn in Figure 1.3.

AFM is related to Scanning Tunneling Microscopy (STM), which was created earlier for

4

Laser

Split Photodiode

AFM tip and Cantilever

X, Y, and Z Piezo

Figure 1.3 – Sketch of essential AFM components. The tip is mounted on the cantilever. The interaction with the surface is measured by the deflection of the laser beam on the photodiode. A 3-dimensional piezo stage controls the motion of the sample.

ultra precise imaging of conducting surfaces.21 AFM has the distinct advantage of being

able to sense non-conducting surfaces and this in addition to its technical simplicity, has

made it immensely popular over the last decade.

The AFM is perfect for measuring the force as a function of distance (force

profile) for intermolecular and interfacial interactions since it has a small probe size and

the high position sensitivity. The small radius of curvature reduces the contact area of the

interaction to a few molecular contacts enabling single molecule force experiments.

Interactions with single proteins have been measured using the small AFM probe5,22,23

and the possibility of measuring single chemical interactions is being entertained.24,25

Position sensitivity is significantly higher for AFM than SFA. A sensitivity of 1 Å in a 1

5

kHz bandwidth is common and resolution of 0.05 Å is possible. Unfortunately, the

higher sensitivity has not been utilized and the normalized spring constants of AFM

experiments are only an order of magnitude larger than SFA experiments. As a result,

the tip continues to experience instabilities and only repulsive interactions or the total

adhesion can be measured. Hence, experiments with stiff springs, which measure the

whole force profile, are needed to understand intermolecular and interfacial forces.

The small probe size also makes AFM very useful for creating images of

nanoscale structure and interactions. Images are produced by scanning the surface

underneath the tip while holding the tip-sample interaction constant with a feedback loop.

The topography and changes in physical properties are recorded for the whole area.

Many beautiful AFM images have been circulated over the last decade revealing a

phenomenally complex nanoscale world. Unfortunately, an understanding of the tip-

sample interaction is not always known such that even though contrast is observed most

images are devoid of physical meaning. Quantitative methods of surface analysis during

imaging are needed.

Chemical Force Microscopy (CFM) is a variant of AFM, created in the laboratory

of Charles Lieber, that selectively measures chemical interactions between the tip and

sample.26 Specific and well-defined chemical interactions are created by coating the

surface and tip with Self-Assembled Monolayers (SAMs) terminated with functional

groups. Both contact and tapping mode have been used to detect specific chemical

functionality and measure their corresponding adhesion values in different solvents.26-30

CFM was further advanced in the Lieber laboratory by using nanotubes as AFM probes.

6

The small size of the probe and the capability of chemically functionalization make

nanotube tips ideal.22,31-35

1.4 Thesis

In this thesis, Chemical Force Microscopy is used to probe intermolecular forces

at surfaces. Many novel techniques are developed for imaging and force curves to

enhance accuracy and precision while using cantilevers stiff enough to observe the whole

force profile. In chapter 2, Magnetic Feedback Chemical Force Microscopy is

developed, where a magnetic feedback loop stiffens the cantilever so that whole force

profiles between SAM surfaces are measured. Hydroxyl terminated SAMs produce

short-range interactions that only extend 1 nm into the solvent while carboxyl terminated

SAMs extend up to 3 nm. The technical challenge of implementing magnetic feedback

reveals that intrinsically stiff cantilevers with a low noise instrument are better for force

profile measurement. In chapter 3, an ultra low noise AFM is produced through multiple

modifications to the optical deflection detection system and signal processing electronics.

In chapter 4, molecular ordering is observed for a silicone oil, a long chain alcohol, and

water near flat surfaces. The molecular ordering of the solvent produces oscillatory force

profiles, called structural or solvation forces, which provide insight into the orientation

and possible solidification of the solvent under confinement. Brownian Force Profile

Reconstruction is developed to aide the accurate measurement of these steep attractive

interactions. In chapter 5, electronic control of the cantilever Q is used to obtain the

highest imaging sensitivity. In chapter 6, Energy Dissipation Chemical Force

Microscopy is developed to investigate the time dependence and dissipative

characteristics of SAM interfacial interactions. The isolation of the chemical and

7

physical interfacial properties from the topography by Energy Dissipation Chemical Fore

Microscopy produces a new quantitative ultra-high sensitivity imaging mode. The

appendix contains a short description of the Digital Instruments Multimode AFM and

explanations with Igro Pro code for the techniques used in this thesis.

1.5 References

1. Isaelachvili, J. Intermolecular and Surface Forces (Academic Press, San Diego, 1992).

2. Stryer, L. Biochemistry (W.H. Freeman and Company, New York, 1995).

3. Bhushan, B. Handbook of Micro/Nanotribology (CRC Press, Boca Raton, 1995).

4. Myers, D. Surfaces, Interfaces, and Colloids (John Wiley & Sons, New York, 1999).

5. Oberhauser, A. F., Marszalek, P. E., Erikson, H. P. & Fernandez, J. M. The molecular elasticity of the extracellular matrix protein tenascin. Nature 393, 181-185 (1998).

6. Brooks, C. L., Gruebele, M., Onuchic, J. N. & Wolynes, P. G. Chemical Physics of Protein Folding. Proceedings of the National Academy of Sciences USA 95, 11037-11038 (1998).

7. Birdi, K. S. (ed.) Surface and Colloid Chemistry (CRC Press, Boca Raton, 1997).

8. Tabor, D. & Winterto.Rh. Surface Forces - Direct Measurement of Normal and Retarded Van Der Waals Forces. Nature 219, 1120-& (1968).

9. Grabbe, A. Double-layer interactions between silylated silica surfaces. Langmuir 9, 797-801 (1993).

10. Israelachvili, J. N. The calculation of van der Waals dispersion forces between macroscopic bodies. Proc. R. Soc. London A331, 39-55 (1972).

11. Sivasankar, S., Subramaniam, S. & Leckband, D. Direct molecular level measurements of the electrostatic properties of a protein surface. Proceedings of the National Academy of Sciences USA 95, 12961-12966 (1998).

12. Tabor, D. & Winterton, R. H. S. The direct measurement of normal and retarded van der Waals forces. Proc. R. Soc. London A312, 435-450 (1969).

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13. Horn, R. G., Clarke, D. R. & Clarkson, M. T. Direct Measurement of Surface Forces between Sapphire Crystals in Aqueous-Solutions. Journal of Materials Research 3, 413-416 (1988).

14. Heuberger, M., Zäch, M. & Spencer, N. D. Density Fluctuations Under Confinement: When Is a Fluid Not a Fluid? Science 292, 905-908 (2001).

15. Israelachvili, J. Solvation Forces and Liquid Structure, as Probed by Direct Force Measurements. Accounts of Chemical Research 20, 415-421 (1987).

16. Israelachvili, J. N. & Pashley, R. M. Molecular Layering of Water at Surfaces and Origin of Repulsive Hydration Forces. Nature 306, 249-250 (1983).

17. Israelachvili, J. N. & Wennerstrom, H. Hydration in electrical double layers. Nature 385, 689-690 (1997).

18. McGuiggan, P. M. & Pashley, R. M. Molecular Layering in Thin Aqueous Films. Journal of Physical Chemistry 92, 1235-1239 (1988).

19. Heuberger, M. The extended surface forces apparatus. Part I. Fast spectral correlation interferometry. Review of Scientific Instruments 72, 1700-1707 (2001).

20. Binnig, G., Quate, C. F. & Gerber, C. Atomic Force Microscope. Physical Review Letters 56, 930-933 (1986).

21. Binning, G., Rohrer, H., Gerber, C. & Weibel, E. Surface Studies by Scanning Tunneling Microscopy. Physical Review Letters 49, 57-61 (1982).

22. Wong, S. S., Joselevich, E., Woolley, A. T., Cheung, C. L. & Lieber, C. M. Covalently functionalized nanotubes as nanometre- sized probes in chemistry and biology. Nature 394, 52-55 (1998).

23. Rief, M., Gautel, M., Oesterhelt, F., Fernandez, J. M. & Gaub, H. E. Reversible Unfolding of Individual Titin Immunoglobulin Domains by AFM. Science 276, 1109-1112 (1997).

24. Skulason, H. & Frisbie, C. D. Direct detection by atomic force microscopy of single bond forces associated with the rupture of discrete charge-transfer complexes. Journal of the American Chemical Society 124, 15125-15133 (2002).

25. Skulason, H. & Frisbie, C. D. Detection of discrete interactions upon rupture of Au microcontacts to self-assembled monolayers terminated with - S(CO)CH3 or -SH. Journal of the American Chemical Society 122, 9750-9760 (2000).

26. Frisbie, C. D., Rozsnyai, L. F., Noy, A., Wrighton, M. S. & Lieber, C. M. Functional Group Imaging by Chemical Force Microscopy. Science 265, 2071-2074 (1994).

9

10

27. Noy, A., Frisbie, C. D., Rozsnyai, L. F., Wrighton, M. S. & Lieber, C. M. Chemical Force Microscopy - Exploiting chemically-modified tips to quantify adhesion, firction, and functional-group distributions in molecular assemblies. Journal of the American Chemical Society 117, 7943-7951 (1995).

28. Noy, A., Vezenov, D. V. & Lieber, C. M. Chemical Force Microscopy. Annual review of Material Science 27, 381-421 (1997).

29. Noy, A., Sanders, C. H., Vezenov, D. V., Wong, S. S. & Lieber, C. M. Chemically-Sensitive Imaging in Tapping Mode by Chemical Force Microscopy: Relationship between Phase Lag and Adhesion. Langmuir 14, 1508-1511 (1998).

30. Vezenov, D. V., Noy, A., Rozsnyai, L. F. & Lieber, C. M. Force Titrations and Ionization State Sensitive Imaging of Functional Groups in Aqueous Solutions by Chemical Force Microscopy. Journal of the American Chemical Society 119, 2006-2015 (1997).

31. Wong, S. S., Woolley, A. T., Joselevich, E., Cheung, C. L. & Lieber, C. M. Covalently-Functionalized Single-Walled Carbon Nanotube Probe Tips for Chemical Force Microscopy. Journal of the American Chemical Society 120, 8557-8558 (1998).

32. Wong, S. S., Harper, J. D., Lansbury, P. T. & Lieber, C. M. Carbon Nanotube Tips: High-Resolution Probes for Imaging Biological Systems. Journal of the American Chemical Society 120, 603-604 (1998).

33. Wong, S. S., Woolley, A. T., Joselevich, E. & Lieber, C. M. Functionalization of carbon nanotube AFM probes using tip-activated gases. Chemical Physics Letters 306, 219-225 (1999).

34. Hafner, J., Cheung, C. L. & Lieber, C. M. Growth of nanotubes for probe microscopy tips. Nature 398, 761-762 (1999).

35. Hafner, J. H., Cheung, C. L. & Lieber, C. M. Direct Growth of Single-Walled Carbon Nanotube Scanning Probe Microscopy Tips. Journal of the American Chemical Society 121, 9750-9751 (1999).

Chapter 2 Magnetic Feedback Chemical Force Microscopy 2.1 Introduction

The Atomic Force Microscope1 can probe forces between the tip and sample with

high sensitivity and spatial resolution in solution, providing critical information about

potential energy surfaces through the measurement of force profiles. A force profile is the

derivative of a one-dimensional projection of the potential energy surface, where the

reaction coordinate is defined by the pulling direction. The potential energy surface,

which determines both energetics and dynamics of the reaction, can then be reconstructed

by integration. Unfortunately, in most AFM experiments a region in the potential energy

surface exists where the derivative of the force profile (second derivative of the potential

energy surface) exceeds the stiffness of the AFM cantilever. This condition causes the tip

to snap to contact during approach and snap out during separation,2 thus precluding

measurement of the attractive portion of the potential at small separations. A sketch of an

Forc

e

k

∆x1 ∆F2=∆x2*k

Forc

e ∆F1=∆x1*k ∆x2

Z-piezo displacement Tip-sample distance

Figure 2.1 - Sketch of a force profile (left) and resulting force curve (right). The gray arrows indicate the trajectory of the cantilever in the region where the gradient of the force profile exceeds the fixed spring constant, k, of the cantilever leading to snap-in and snap-out.

11

idealized force profile and resulting force curve is shown in Figure 2.1. In

supramolecular systems such as molecular recognition and protein folding, where the

potential energy surface is very rugged, this drawback of conventional AFM leads to the

loss of crucial information. The potential energy surface can be reconstructed using

indirect methods such as measuring the anharmonicity of a tapping cantilever3-5 or the

Brownian motion of an undriven cantilever.6-8 However, the tapping methods do not

work well in solution at low Q and the Brownian motion technique still requires a

relatively stiff cantilever. To measure the whole force profile in solution, the cantilever

stiffness must be increased.

Two schemes to control the effective stiffness of the cantilever and thus eliminate

mechanical instabilities have been reported.9-13 First, interfacial force microscopy (IFM),9

which is based on a differential-capacitance sensor and a force feedback system,

electrostatically stiffens the cantilever. IFM has been used to measure force profiles

between SAM modified substrates and probes in air.10 Unfortunately, this electrostatic

technique cannot be used in high ionic strength solutions since the electrostatic screening

by ions in solution reduces the usable distance over which the feedback is effective. In

addition, polarizable samples could be perturbed by the electric fields used for controlling

cantilever stiffness. The second approach uses magnetic feedback to stiffen the

cantilever,11 and has been utilized to measure force profiles in ultrahigh vacuum and

air.12,13 These initial studies did not, however, investigate the applicability of magnetic

feedback to measure interactions between chemically well-defined surfaces in the

condensed phase. In this chapter, the development and implementation of Magnetic

Feedback Chemical Force Microscopy (MFCFM) for the study of intermolecular forces

12

and potentials is presented. Low bandwidth MFCFM was used to map effectively force

profiles between hydroxyl and carboxyl-terminated self-assembled monolayers (SAMs)

in aqueous solution. The hydroxyl surfaces have short-range attractive forces while the

carboxyl surfaces at pH 2.2 have longer-range attraction. The carboxyl surfaces at pH

7.0 showed long range repulsion, which was well fit by the DLVO model. Although

generally accurate, low bandwidth MFCFM does not control well the motion of the

cantilever, leading to uncertainty in tip-sample distance. High bandwidth MFCFM in

principle resolves this issue but instrument noise and phase lag in high power electronics

renders high bandwidth MFCFM infeasible.

2.2 Low Bandwidth Magnetic Feedback Theory

A schematic of a low bandwidth magnetic feedback loop is shown in Figure 2.2.

The set-up implements a servo loop, which includes optical detection of cantilever

Photodiode

Voltage Out ∝ Force

Solenoid

Laser

SmCo5

HH

Current Sensing Resistor

Low Pass Filter

Variable Gain Power Amplifier

Figure 2.2 - A magnetic feedback (MF) schematic, where the feedback loop is comprised of the cantilever, split photodiode, low pass filter, variable gain amplifier, and solenoid.

13

Fpot

Fcant Fmag Fcant= kspring • ∆x

Fmag= Gain • kspring • ∆x keff= kspring(Gain + 1)

Figure 2.3 - Force diagram where FPot is balanced by the feedback loop, Fmag, and the cantilever, Fcant. The effective stiffness is the sum of the two stiffness components.

deflection, a variable gain amplifier, a solenoid, and a cantilever with magnetic particle,

to balance the tip-sample interaction. A simple picture using a force diagram to gain an

intuitive understanding of magnetic feedback is shown in Figure 2.3. When the tip on the

cantilever senses the surface, the cantilever will deflect under the force. The deflection

follows Hooke’s law such that the cantilever component of the restoring force, Fcant, is the

spring constant, kspring, times the deflection of the cantilever, ∆x. The photodiode signal,

which can be expressed as ∆x, is inverted and multiplied by the loop gain, inducing a

solenoid current. The gradient of the solenoid magnetic field interacts with the magnetic

particle on the cantilever, which exerts a restoring force on the cantilever. The magnetic

force component, Fmag, can be written in a similar form to Hooke’s law such that

Fmag=Gain*kspring*∆x. The two equations can be combined to solve for the effective

spring constant, keff=kspring*(Gain+1).

The low pass filter in the feedback loop increased the stability by lowering the

bandwidth. The cutoff frequency was chosen such that the loop gain fell to unity before

phase shifts totaling 360° (180° from the inverter and 180° from the cantilever resonance

and feedback electronics) cause oscillations. The Gain used in the above expression for

effective stiffness is thus frequency dependent and the cantilever only has a higher

stiffness at low frequencies.

14

2.3 Low Bandwidth Magnetic Feedback Experiments

2.3.1 Magnetic Tip Preparation

An inverted optical microscope equipped with micromanipulators was used to

prepare the magnetic tips by gluing small SmCo magnets onto triangular Si3N4 (NP

probes, k = 0.06-0.6 N/m, Digital Instruments, Inc., Santa Barbara, CA) cantilevers with

UV curable adhesive. The magnets (5-20µm in diameter) were prepared by crushing a

larger SmCo magnet (CR54-314, Edmond Scientific, Inc., Barrington, NJ), spreading the

resulting powder on a plastic film, and stretching the film to separate small pieces. The

inverted microscope (Epiphot 200, Nikon, Excel Technologies, Enfield, CT) was

equipped with three micromanipulators (461-XYZ-M, Newport, Irvine, CA), which were

used to manipulate the Si3N4 cantilever, a sharpened tungsten wire probe, and the plastic

film with magnet fragments. The magnets were attached using the the following

sequence: (i) the tungsten wire probe, which was dipped in glue (optical adhesive #63,

Norland Products, New Brunswick, NJ), was used to deposit a small patch of glue on the

backside of the cantilever; (ii) the wire was used to pick up a single magnet fragment

from the plastic film and deposit this on the back of the cantilever; (iii) the glue was

cured for at least 1 h using an UV lamp (UVGL-25 mineralight lamp, UVP Inc., San

Gabriel, CA).

2.3.2 Chemical Surface Preparation

SAM surfaces composed of organic thiols on gold were prepared because they are

clean, flat, and chemically well-defined. Gold layers were prepared on the magnetic tips

using a thermal evaporator to slowly (1 Å/s) deposit 70 nm of Au on a 7 nm Cr adhesion

layer. Flat gold substrates 20 nm thick were deposited at 1.5 Å/s on freshly cleaved mica

15

Gold Sulfur

Hydrogen

Carbon

Oxygen

Figure 2.4 – Sketch of a thiol Self-Assembled Monolayer on gold. The alkyl chains pack in a crystalline structure tilted ~30 degrees from normal.

by electron beam evaporation at a substrate temperature of 350 oC. The mica was baked

for 6 hours before evaporation and annealed for 2 hours after evaporation. The

evaporation pressures were typically 5×10-7 and 7×10-7 torr for the thermal and electron

beam evaporators, respectively. The SAM layers were made by immersing tips and

samples in 400 µM ethanol solutions of either 11-mercaptoundecanol or 16-

mercaptohexadecanoic acid for 1-2 hours14 before rinsing with ethanol and drying under

a stream of nitrogen. A cartoon of an alkane thiol SAM is shown in Figure 2.4. The

alkane chains pack into a crystalline monolayer that is tilted by ~30 degrees to fill space

because the van der Waals radii of the chains are different than the spacing of the

threefold hollow binding sites of the underlying gold (111) lattice. The measurements on

hydroxyl-terminated SAM tip and sample surfaces were performed in deionized water,

while the experiments using carboxyl-terminated SAM tip and sample surfaces were

carried out in pH 7.0 0.010 M phosphate buffer and pH 2.2 0.010 M phosphoric acid

solutions.

16

2.3.3 Instrument Setup

A Digital Instruments Multimode AFM and Nanoscope III controller equipped

with a signal access module between the microscope and phase extender, were used

unmodified. The photodiode difference voltage was obtained from the signal access

module and passed through a single pole low pass filter before it was input into the power

operational amplifier (PA10, Apex Microtechnologies, Tucson, AZ) configured as a

variable inverting amplifier. The amplified signal drove the solenoid (400 turns of 32-

gauge wire wrapped around a hollow aluminum spindle with an inner diameter of 1.5 mm

and height of 5 mm producing an outer diameter of 15mm) and current sensing resistor

(10 ohms). The output voltage from the current sensing resistor was input into the

controller using an auxiliary data channel. The AFM sample disc and mica sample were

glued to opposite sides of the spindle so that the whole sample assembly could be

mounted securely on top of the scanner (D scanner, Digital Instruments, Inc., Santa

Barbara, CA) used for the experiments.

2.3.4 Tip and Feedback Calibration

Calibration of the cantilever was completed after force curves were acquired so

that the calibration process did not damage the sample surfaces. With magnetic feedback

off, the sensitivity (nm/Vc) of the cantilever deflection detection system was determined

from the region of the force curve dominated by cantilever compliance. The open loop

gain and sensitivity of the magnetic feedback were calibrated by input of a low frequency

square wave (~2Hz) into the low pass filter. The input wave, solenoid current, and

cantilever deflection were recorded. The force sensitivity of the system (N/Vs) could be

determined by using the solenoid voltage (Vs), cantilever deflection (Vc), sensitivity

17

200 nm

Figure 2.5 - SEM image of the tip of the cantilever used in Magnetic Feedback Chemical Force Microscopy measurements of the carboxyl functionalized surfasce.

(nm/Vc), and spring constant (N/m). Multiple input voltages were used to ensure

linearity. The thermal noise spectrum in air was used to calibrate the spring constant of

each cantilever.15 The spring constants of 0.69 N/m and 0.092 N/m were determined for

the cantilevers containing hydroxyl and carboxyl terminated SAMs, respectively.

A scanning electron microscope (SEM) image of the apex of the carboxyl

terminated SAM tip is shown in Figure 2.5. The image is representative of the tips used

for MFCFM experiments and was taken after the tip was used for collecting data. The

gold grain at the apex defines the radius of curvature of the tip. The tip radii were

calculated from SEM images using Igor Pro (Wavemetrics Inc., Lake Oswego, OR);

specifically, the tip was modeled as a circle segment defined by three points near the

apex. The radii of curvature were 75 and 70±15 nm for the hydroxyl and carboxyl

fuctionalized tips, respectively.

2.3.5 Data Collection

Force curves (force vs. z-piezo displacement) were acquired by recording both the

cantilever deflection, ∆x, and solenoid current, Vs, while the sample surface was moved

18

in and out of contact with the tip. Each curve contains 512 points over a scan range of 20

nm for the hydroxyl-hydroxyl interaction data and 75 nm for the carboxyl-carboxyl

interaction data. Piezo extension (tip-sample approach) and retraction (tip-sample

separation) were done at a rate of 0.5-1Hz. Typically, sets of fifty cycles were saved with

the feedback on and the feedback off. The Z-piezo movement should not push into the

tip compliance region too far. The magnetic feedback increases the stiffness of the

cantilever so that higher forces are applied in the contact region possibly causing damage.

Also, the current flowing through the coil during feedback can heat the solenoid. The

resulting expansion is similar to Z-piezo movement, which pushes the surface further into

the tip. This positive feedback can ruin a MFCFM experiment.

2.3.6 Data Analysis

The force profiles used for model fitting are an average of 6-17 curves. The raw

data were imported into Igor Pro and scaled for force and z-piezo displacement as

follows. The portion of the curves corresponding to large tip-sample separation (no

interaction) was used to define zero force. The sum of the force contributions from

deflection and solenoid current was plotted against a tip-sample distance scale adjusted

for the cantilever deflection. The movement of the cantilever caused the tip-sample

spacing between data points to be irregular. An interpolation algorithm was used to

reconstruct the curve with even spacing. The hydroxyl and pH 2.2 carboxyl data are

similar in shape to the Lennard-Jones potential and show a long-range attractive portion

at large tip-sample distance and a steep contact region at short distances. The pH 7.0

carboxyl data show a long-range exponential repulsion with a small dip in force

associated with van der Waals forces. To adjust for drift, the z-piezo displacement of the

19

hydroxyl and pH 2.2 carboxyl data were shifted such that all curves overlap at zero force

in the contact region. The pH 7.0 carboxyl data were similarly shifted such that the

curves overlapped at the van der Waals dip. The hydroxyl and pH 2.2 carboxyl approach

data were fit with a model for the van der Waals interaction, and the carboxyl-carboxyl

pH 7.0 force profile was fit with a model that includes attractive van der Waals and

repulsive electrostatic terms (details below).

2.3.7 Hydroxyl Terminated SAM Surfaces

Magnetic feedback effectively removes the snap-in and snap-out associated with

soft spring force profile measurements. At short distances, a shallow barrier separates the

minimum of the tip-sample potential and the spring potential at no deflection. Thermal

fluctuations are enough to overcome the barrier and the system will snap from one well to

the other or show bistability. A characteristic approach curve with magnetic feedback off

for hydroxyl-terminated SAMs is displayed in Figure 2.6. The instability in the force

profile is evident as the tip snaps to contact with the sample when it enters the steep

Z-piezo Displacement (nm)

-0.4

0.0

0.4

543210-1

Forc

e (n

N)

Figure 2.6 - Deflection trace (displayed as force) of hydroxyl-terminated SAM surfaces in solution during approach without magnetic feedback has characteristic snap-in (arrow).

20

a

b

-0.6

-0.4

-0.2

0.0

543210-1

-0.2

0.0

0.2Fo

rce

(nN

)

Tip Sample Distance (nm)

Figure 2.7 - Data traces of hydroxyl-terminated SAM surfaces in solution during approach. (a) Deflection trace (displayed as force) with magnetic feedback has no instabilities and a reduced total deflection. (b) Magnetic force trace from solenoid current. region of the potential surface. The cantilever stiffness must be greater than the steepest

gradient in the force profile to map continuously the interaction. The magnetic feedback

increases the stiffness at the observation frequencies and achieves this goal. Figures 2.7a

and 2.7b show the force contributions from the intrinsic stiffness of the cantilever and the

magnetic feedback, respectively, for the same hydroxyl-terminated tip and sample used to

record Figure 2.6. These data demonstrate several important points. First, the data clearly

show that magnetic feedback has eliminated the mechanical instability (snap-in) during

approach. Similarly, the instability during separation (snap-out) was eliminated with

magnetic feedback. Second, deflection of the cantilever was greatly reduced during

approach while magnetic feedback contributed most of the restoring force.

21

2.3.8 Hydroxyl Surfaces Approach-Separation Hysteresis

Force profiles for the hydroxyl-hydroxyl interaction during both tip-sample

approach (black) and separation (gray) are shown in Figure 2.8. These force profiles are

the average of many individual traces. Some hysteresis exists between the approach and

separation curves as evidenced by a 0.3 nm difference at the point of zero force in the

contact region. The total adhesion for the separation curve is 850 pN, which is ~300 pN

greater than the adhesion determined from the approach curve. Adhesion hysteresis can

be attributed to inherent irreversibility when bonding and unbonding the SAM surfaces

during the force curve cycle.16 Hysteresis in the loading and unloading of a SAM surface

has previously been studied using IFM.17 The contact pressure in these IFM experiments

was greater than 3.0 GPa, and was suggested to rearrange the packing of the alkane

chains in the SAM.18 For a sphere interacting with a flat surface, the maximum pressure,

Pmax, can be estimated using the JKR model:16

1

0

-1

86420

Forc

e (n

N)

Tip Sample Distance (nm)Figure 2.8 - Approach (black) and separation (gray) force profiles for the hydroxyl-terminated tip-sample interaction.

22

2/1

max 23

23

−=

aKW

RKaP

ππ, (2.1)

where a is the contact radius, R is the radius of curvature of the tip, K is the Young

modulus, and W is the surface energy. The contact radius is calculated using

( )

+++= 23 363 RWWFRWF

KRa πππ . (2.2)

A maximum pressure of 1.4 GPa is calculated for a tip with 75 nm radius of curvature,

maximum applied load of 25 nN, gold’s Young modulus of 77 GPa, and a surface energy

of 1.4 mJ/m2 from the adhesion of 0.6 nN. The surface energy was obtained using the

following relation based on the Derjaguin approximation,

23 RWF π−

= . (2.3)

This is less than the pressure expected to rearrange substantially the SAM layer. Hence,

the hysteresis is caused instead by molecular rearrangements of the SAM terminal groups

to facilitate interfacial bonding and hysteretic deformation of the underlying gold support.

This same principle is responsible for the discrepancy in advancing and receding values

for contact angle measurements.19

The minimum of the MFCFM separation force profile is a measure of the total

adhesion. A histogram of the total adhesion for the hydroxyl surfaces data is grouped

around a value of 1100 pN with a standard deviation of 400 pN. In measurements with

magnetic feedback off, a histogram of the snap-out adhesion is grouped around 800 pN

with a standard deviation of 300 pN. The higher effective spring constant of the

cantilever during magnetic feedback leads to a larger total applied load for the same piezo

movement into the contact region. This larger applied force leads to more rearrangement

23

and elastic deformation at the interface, and thus can explain the discrepancy between the

histograms. Due to the rearrangement of the surface upon loading, the approach force

profiles will provide a better measure of the intrinsic potential surface.

2.3.9 Van der Waals Model Fit to Hydroxyl Data

The approach force profile for the hydroxyl-hydroxyl interaction (gray dots) is

shown in Figure 2.9. The long-range attractive region of the force profile was fit by an

inverse square power law

20 )(6 DD

ARF−

−= (2.4)

expected for the van der Waals interaction between a sphere and a flat surface where A is

the Hamaker constant, R is the radius of curvature of the tip, D is the tip-sample distance,

and D0 is an arbitrary offset. The van der Waals interaction for point particles follows

1/r6 for energy and 1/r7 for force. When the interaction is integrated over the whole

interacting sample volume the relationship becomes a 1/r2 power law.16 The parameters

-0.6

-0.4

-0.2

0.0

0.2

86420

Forc

e (n

N)

Tip Sample Distance (nm)Figure 2.9 - Force profile (gray) for hydroxyl-terminated SAMs in deionized water with a van der Waals model fit (black). The Hamaker constant value is 1.0×10-19 J.

24

used in the fit were the Hamaker constant and a tip-sample distance offset.

The value of the Hamaker constant obtained from fit to the force profile for the

hydroxl-terminated SAM surfaces was 1.0±0.2×10-19 J. The error is a result of the

uncertainty in measurement of the tip radius. Comparison of this value to the reported

Hamaker constants for alkanes, 4×10-21 J16, and gold surfaces, 1×10-19 J20, interacting

through water shows that the MFCFM results are more similar to the gold surfaces. The

Hamaker constant is also consistent with the range of values, 0.9-3×10-19 J, computed

from the most reliable spectroscopic data for gold surfaces.21,22 These comparisons

suggest that the gold support dominates the long-range attractive interaction in the gold-

SAM system. For a symmetric system with adsorbed layers on the substrate the model

for the force is

+

++

−= 2121

2123

2232

)2()(2

6)(

TDA

TDA

DARDF , (2.5)

where R is the tip radius, T is the thickness of the adsorbed films, and D is the distance

between the adsorbed layers.16 A232 is the Hamaker constant for the adsorbed layers

interacting through the medium, A123 is the Hamaker constant for the substrate and

medium interacting through the adsorbed layer, and A121 is the Hamaker constant for the

substrates interacting through the adsorbed layers. When the distance is much smaller

than T, the equation reduces to

2232

6)(

DRADF = . (2.6)

At larger distances, the full expression can be simplified to

2131

6)(

DRADF = , (2.7)

25

where A131 is the Hamaker constant for gold layers interacting through water. In

previous experiments, A131 = 1×10-19 J, is two orders of magnitude larger than the

Hamaker constant for alkane-water-alkane, A232 = 4×10-21 J, since alkanes have such a

small polarizability. With a SAM thickness23 of 1.3 nm, the van der Waals contribution

from the SAM layer is small until the separation is 0.3 nm or less and therefore does not

affect the determination of the Hamaker constant. MFCFM allows the measurement and

modeling of the whole force profile, which provides important information about the

sources of adhesion.

2.3.10 Carboxyl Terminated SAM Surfaces

The ability of MFCFM to map the whole force profile was further demonstrated

in studies of carboxyl-terminated SAM modified tips and samples. In this experiment, a

much softer cantilever was used than for the hydroxyl-terminated SAM surfaces, 0.092

N/m vs. 0.69 N/m, respectively. Figure 2.10 shows that the magnetic feedback enables

complete control of the cantilever during approach even though the attractive part of the

carboxyl-carboxyl interaction at pH 7.0 has a stiffness of ~ 0.9 N/m—ten times that of

the cantilever alone. In these magnetic feedback experiments, the deflection of the

cantilever was undetectable compared to the instrument noise and the major force

contribution was from magnetic feedback.

2.3.11 DLVO Model Fit to pH 7.0 Carboxyl Data

The physical and chemical properties of carboxyl terminated SAM surfaces were

determined using MFCFM. The approach force profile for the pH 7.0 carboxyl-carboxyl

interaction is shown in Figure 2.11. At pH 7.0, the carboxyl end groups on both surfaces

26

0.4

0.2

0.0

0.4

0.2

0.0

543210-1

b

aFo

rce

Tip Sample Distance (nm)Figure 2.10 - Data traces of carboxyl-terminated SAM surfaces in solution during approach. (a) Deflection trace (displayed as force) with magnetic feedback has no instabilities. (b) Magnetic Force trace from the solenoid current reveals the major features in the force profile.

27

54

3

2

Forc

e (n

N)

.1

67

54

3

0 2 4 6 8Tip Sample Distance (nm)

Figure 2.11 - Force profile (gray) for carboxyl terminated SAMs in 0.010 M, pH 7 phosphate buffer with fit (black) using a charge regulation DLVO model. The Hamaker constant value is 1.2×10-19 J.

are partially deprotonated.24,25 At long range, the electrostatic repulsion dominates the

force profile. Electrostatic repulsion is a result of the energy cost associated with

reducing the entropic freedom of the counterions that keep charge neutrality between the

surfaces. This repulsive regime has been observed with the surface forces apparatus and

in previous AFM experiments, and can be analyzed using a double-layer model.16,24-26

Significantly, the MFCFM data show that at small separations the steep attractive van der

Waals contribution to the overall interaction also can be reliably mapped.

A model including a repulsive electrostatic term and an attractive van der Waals

term was used to analyze the force profile. This model, often called DLVO for the

initials of its four developers, Derjaguin, Landau, Verwey, and Overbeek, can also be

formulated in a way to compensate for charge regulation at the surface. The resulting

expression for force is,

( ) 20

/)2(

/)2(200

)(612

0

0

DDAR

aeeDF DD

DD

−−

+=

−−−

−−−Ψλ

λ

λεε

, (2.8)

where Ψ is the surface potential, ε is the dielectric constant, ε0 0 is the permitivity of

space, λ is the Debye length, is the charge regulation parameter, and A, D and Da 0 have

the same meaning as with the hydroxyl surfaces fitting.16 The fit (black line) to the data

in Figure 2.10 is excellent with good agreement through the electrostatic repulsion

regime and into the van der Waals interaction. The parameters of the fit were the

Hamaker constant, surface potential, Debye length, charge regulation parameter, and the

tip-sample distance offset. Notably, there is only a single self-consistent solution for the

parameters used to fit the data. The Hamaker constant, 1.2±0.2×10-19 J, from the van der

28

Waals term, confirms that the gold-gold interaction dominates the long range van der

Waals interaction in these Au-SAM systems.

The parameters due to electrostatic interactions were analyzed and compared to

previous studies,24,25,27-30 since this repulsive regime can be measured without magnetic

feedback. The MFCFM results for the electrostatic terms are: Debye length = 2.9 nm,

surface potential = –1.5×102 mV, and charge regulation parameter is -0.71. The Debye

length for this system is slightly longer than the 2.3 nm value calculated from the charge

and concentration of the ions in the solution ([Na+] = 0.0138, [H2PO4-] = 0.00619, [HPO4

-

2] = 0.00381). The value measured with magnetic feedback, 2.9 nm, is very similar to that

obtained in a separate experiment without magnetic feedback, 3.1 nm, using freshly

prepared 0.010 M phosphate buffer. The deviation does not correspond to a systematic

error of MFCFM. The relationship between surface charge and surface potential is

−= ∑∑ ∞i

ii

ikT ρρεεσ 002 2 and kT

ez

ii

i

e0

0

ψ

ρρ−

∞= , (2.9) and (2.10)

where σ is the surface charge density, ε is the dielectric constant, ε0 is the permittivity of

space, k is Boltzmann’s constant, T is temperature, i∞ρ is the concentration of each ionic

species in the bulk, zi is the charge of the species, e is the electronic charge, and Ψ0 is the

surface potential. Using the measured surface potential of -1.5×102 mV and the

concentrations of our buffer solution components, a surface charge of -0.12 C/m2 was

calculated. The surface potential value from these new experiments, -1.5×102 mV, is

consistent with previous work in the Lieber lab.24 The surface charge density, -0.12

C/m2, obtained from the potential31 indicates that the surface is 15.3% ionized at pH 7 and

that the surface pKa is 7.7. This elevated value compared to the typical carboxyl value of

29

4.5 is consistent with previous work24-29 and is a result of poor solvation of the anion and

close proximity of other charged carboxyl groups because of the tight packing of the

SAM. The charge regulation parameter, -0.71, strongly implies that the carboxyl surface

exhibits constant charge behavior during approach. This conclusion is also consistent

with previous experiments carried out in the Lieber lab24 and by Hu and Bard.25

2.3.12 Comparison of Attractive Hydroxyl and Carboxyl Interactions

Laslty, magnetic feedback was used to measure the force profile for carboxyl

surfaces in pH 2.2 solution, shown in Figure 2.12. The carboxyl interaction is much

longer range than the hydroxyl. Fitting with the van der Waals model yields a Hamaker

constant of 2.4±0.4×10-19 J. This falls within the acceptable values from theory but is

significantly different than the values obtained for the hydroxyl interaction and the

carboxyl interaction at higher pH. This discrepancy is especially apparent when viewing

the hydroxyl and pH 2.2 carboxyl interactions together in figure 2.13. The only

-0.6

-0.4

-0.2

0.0

20151050

Forc

e (n

N)

Tip Sample Distance (nm)

Figure 2.12 - Force profile (gray) for carboxyl terminated SAMs in 0.010 M pH 2.2 phosphate acid solution with fit (black) using a van der Waals model. The Hamaker constant value is 2.4×10-19 J.

30

1.5

1.0

0.5

0.0

-0.5

121086420-2

Hydroxyl Carboxyl

Forc

e (n

N)

Tip Sample Distance (nm)

Figure 2.13 - Force Profiles of hydroxyl surfaces (black) in water and carboxyl surfaces (gray) in pH 2.2 phosphoric acid showing significant discrepancy in distance dependence of long-range forces.

difference between the two samples is the chemical surface and a couple extra angstroms

of alkane thickness, so the van der Waals interactions should be very similar. Because of

the significant difference in long-range forces, A new chemical model for the attractive

interactions is required to explain these data.

The pH 2.2 carboxyl data contact line is softer than the hydroxyl data contact line,

which maybe a result of hydration forces. Hydration forces are repulsive forces that are

exponential in character with a shorter decay length (~2 Å) than double layer forces (1

nm-100 nm). The origin is not known exactly. In experiments with mica, the strength of

the hydration force is correlated with the hardness of the ion leading to the hypothesis

that hydration forces are a result of removing solvation from surface bound ions.32

Conversely, other experimental results found that for Al2O3 slurries the hydration forces

were anticorrelated with the hardness of the ions leading to the hypothesis that hydration

forces were a result of compression of trapped ions.33 The strong ionic character of

31

carboxyl functional groups and the decay scale of the contact line imply that hydration

forces are the cause of the repulsion instead of steric repulsion between actual SAM

surfaces.

The long-range attractive forces for the hydroxyl and pH 2.2 carboxyl data may

be a result of imperfect solvation of the SAM endgroups. The surfaces are very

hydrophilic, meaning that it is energetically favorable for water to contact the surfaces

rather than contact air. In the MFCFM measurements, the interactions are attractive

which means that it is energetically more favorable for the surfaces to be near each other

than to be exposed only to the solvent thus the interfacial solvent molecules are higher in

energy than when they are in the bulk. The increased energy for the solvent could be a

result of both enthapic and entropic effects. If the water cannot arrange to form full

hydrogen bonds then that will be an enthalpic energy cost and if the molecules are

structured compared to the bulk then they will lose entropic energy. Recent sum

frequency generation experiments have shown that water at hydrophobic surfaces is both

weakly hydrogen bonded and strongly oriented.34 Also the absolute adhesion of recent

CFM experiments shows a temperature dependence hinting at an entropic contribution to

these forces.35 The spacing of SAM functional groups (5.0 Å)31 is determined by the

three-fold hollow sites of the supporting gold (111) lattice. The center-to-center spacing

of bulk water is ~2.8 Å36 so the solvent will have a “lattice mismatch” with the SAM end

groups. This mismatch may cause the water to rearrange in order to achieve the lowest

energy configuration. This new configuration may not provide the strongest hydrogen

bonds and also requires solvent ordering. Also, the carboxyl surfaces have four or five

lone pairs per head group available for bonding while the hydroxyl surfaces only has two.

32

The forces arising from orientational effects would be expected to decay exponentially

with the decay constant near the correlation length of water.37 Fitting the hydroxyl and

pH 2.2 carboxyl data with exponentials yields mixed results. The exponential curve fit

better to the carboxyl data but the van der Waals (power law) curve fit better to the

hydroxyl data (Table 2.1).

Chi Squared Carboxyl data Hydroxyl data van der Waals fit 2.1×10-20 1.2×10-20 Exponential fit 6.0×10-21 2.9×10-20

Table 2.1 – Chi squared values for determining the wellness of a curve fit to the hydroxyl and pH 2.2 carboxyl data. The equations used were a single exponential to model entropic disordering and a second order power law to model van der Waals forces.

The long-range character of the forces between hydroxyl and carboxyl terminated

SAMs in solution shows that solvation determines the forces of these systems. The van

der Waals model inadequately describes the difference in decay rate between the two

interactions but the exponential fit does not provide a clear confirmation of the

mechanism of the adhesion. Measurement of the correlation length of interfacial water at

the SAM surface, presented in chapter 4, will be crucial for helping elucidate the

chemical nature of this adhesion. Collecting more precise data, which conclusively

supports a specific functional form of the distance dependence will also provide great

insight into the mechanism of the long-range adhesion.

2.4 Effects of Limited Bandwidth

The low pass filter in the magnetic force feedback loop produces cantilever

stiffness values that are frequency dependent. Transfer functions for a cantilever and a

low pass filter (depicting the gain of low bandwidth MFCFM) are displayed in Figure

2.14. For cantilevers with Q>2 most of the movement of the cantilever has frequency

components near the resonant frequency including the snap-in and snap-out instabilities.

33

During approach, the feedback loop does not respond to these high frequency

components and the cantilever is allowed to jump to the surface. The new position is

measured as a deflection and with delay the feedback loop applies a force to the

cantilever and pushes it back out from the surface. This process happens repeatedly

producing an oscillatory motion near the unity gain frequency of the feedback loop, ωf,

with significantly greater noise than that of the intrinsic cantilever but not the violent

oscillations associated with positive feedback. The oscillations can be seen as noise in

the attractive regime of the deflection signal of Figure 2.7a. The uncertainty in deflection

results in tip-sample distance uncertainty during the conversion from Z-piezo movement.

More importantly, the measured force is inaccurate because it is an average over the

increased range of motion of the cantilever. Since the force profile is steep, it changes

rapidly over this region so that very attractive regions are averaged with lightly attractive

6

5

4

3

2

1

0121086420

ωo

ωf

Am

plitu

de (A

.U.)

Frequency (kHz)

Figure 2.14 - Transfer functions of low pass filter (black) and cantilever (gray). Most motion of the cantilever has frequency components near the resonant frequency, ω0. When using low bandwidth magnetic feedback and interacting with the sample the tip will gently oscillate near the unity gain frequency of the feedback loop, ωf.

34

regions, which misestimates the curvature of the force profile. This situation is similar to

the cantilever experiencing bistability and will be covered more thoroughly in chapter 4

(Figure 4.2). Low bandwidth MFCFM is a significant achievement since it allows the

measurement of the whole force profile but for more accurate magnetic feddback

experiments the cantilever must be controlled at all frequencies.

2.5 High Bandwidth Magnetic Feedback Theory

Significantly increasing the bandwidth of the magnetic feedback loop removes the

noise associated with the tip-sample interaction and increases the measurement accuracy.

The high frequency components of the cantilever motion are controlled and the cantilever

stiffness is no longer gain dependent. A more rigorous derivation of magnetic feedback

includes solving for the gain dependant transfer function of the cantilever motion. The

different features of the magnetic feedback servo loop are shown in Figure 2.15. The

cantilever motion is detected by the sensor and given a phase shift, θ. The phase-shifted

signal is amplified and exerts a force through the magnetic transducer. The wave

equation for the cantilever motion under force feedback becomes,

( ) ( )χθω ω +⋅⋅+=⋅+⋅+⋅ iti exGeFxkxbxm 0&&& . (2.11)

In the equation x , , and are the displacement of the cantilever from equilibrium and

its first and second time derivatives respectively, m is the effective mass, b is the

x& x&&

35

Variable Gain

Amplifier

Phase Shifter, θ

Cantilever Sensor Transducer

Figure 2.15 - Block diagram of general cantilever feedback loop

damping, k is the spring constant, ω is the angular frequency, F0 is the thermal fluctuation

force (which is constant for all frequencies), G(ω) is the loop gain, and χ is the loop

phase shift. The transfer function,

( )( ) ( )( ) ( ) ( ) ( ) ( ) 222222

202

sin2cos2 ωωχθωωχθωωωω

bGbGmkGmk

FA+++++−−−

= , (2.12)

is calculated using the Ansatz, ( )ϕω −= tiAex , with arbitrary phase, ϕ, and solving for the

amplitude as a function of frequency. For idealized high bandwidth magnetic feedback,

the equation simplifies to

( )( )( ) 2222

202

ωωωω

bmGkFA

+−+= . (2.13)

since χ is 0, G(ω) is frequency independent, and θ is π. A comparison with the transfer

function without feedback,

( ) ( ) 2222

202

ωωω

bmkFA

+−= , (2.14)

reveals that the effective spring constant is ( ) ( ) ( )( )ωωω KkkKkGkeff +=k +=+= 1 ,

using the substitution, ( ) ( )k

GK ωω ≡ .

Experiments are not idealized and the feedback electronics cause χ to be nonzero.

The main contributors are the low pass filter produced by the RL circuit of the solenoid

and the roll off of the open loop gain of the driving operational amplifier. Assuming that

only one of these components dominates such that the loop gain has a single pole, the

stability criteria as a function of gain and damping are shown in Figure 2.16.38 The white

areas are regions of instability where snap in or oscillations will occur. The shaded

regions are for progressively lower damping or higher Q. Typical values of damping in

36

Figure 2.16 - Region of stability (dark regions) for feedback loop as a function of amplifier cut off frequency, proportional gain of amplifier, and cantilever damping. For typical damping values the area of stability is extremely small.

AFM are 2×10-8 kg/s for a Q of 150 and 2×10-6 kg/s for a Q of 1.5. Too low of an

amplifier cutoff frequency leads to the region of instability on the left of the graph. In

this region, the cantilever cannot be controlled, as seen in the low bandwidth experiments

at the beginning of this chapter. A minimum level of gain is required when the attractive

force profile is stiffer than the intrinsic cantilever. Large differences require large gains,

which is represented by the white strip at the bottom of the figure. The white region at

the top of the graph is from high gains causing oscillations. Stiff tip surface interactions

have a very small window of stability using feedback.

2.6 High Bandwidth Magnetic Feedback Experiments

Implementation of high bandwidth magnetic feedback required making

component changes in the feedback loop. ESP probes (0.1 N/m Digital Instruments,

Santa Barbara, CA) were used for their better-reflected laser spot. Their resonant

frequency was 2-3Khz in water and 9Khz in air without magnets and 1-2kHz in water

and 3kHz in air with magnets. The bandwidth limiting low pass filter was removed. The

high impedance coil was replaced with a coil of simply 3-30 turns with an inner diameter

of 0.7mm. This new coil had an inductance of ~4 µH which lead to a cutoff frequency of

37

~60 kHz at 10 Ω. To compensate for the lack of field produced by the coil, larger

magnets were glued to the cantilevers. Lastly, the slow high power operational amplifier

was replaced with a significantly faster OPA627, which is capable of providing 25mA

and whose 20db bandwidth is 1 MHz. With these new parts the solenoid would be the

dominant contributor of electronic phase shift.

The new instrument setup had sufficient bandwidth to be stable at low gains but

the stability and noise were still unsatisfactory. The thermal noise spectrum near the

resonance was dynamically observed on a spectrum analyzer. As the gain was increased

an increase in resonant frequency and Q signified a change in spring constant.

Unfortunately, both of those factors lead to instability of the system. The higher resonant

frequency demands more bandwidth for the feedback loop and the higher Q reduces the

phase margin more quickly. Replacing the proportional amplifier with a Proportional-

Integral-Differential (PID) amplifier similarly made from fast op amps increased the

stability since the differential gain reduces the Q and provides more phase margin in the

feedback loop. Yet, these efforts did not adequately increase the spring constant. Lastly,

when measuring force curves the tip sample interaction in the repulsive regime greatly

increases the cantilever stiffness shifting the resonant frequency even higher inducing

oscillations. The stability limitations made implementation quite challenging.

The implementation of high bandwidth magnetic feedback was limited also by

instrument noise. The instrument was not thermally limited at frequencies other than the

resonance, which means that the power spectrum measured was not dominated by the

intrinsic thermal motion of the cantilever but instead by electronic sources of noise. A

sketch of the measured noise from the photodiode during magnetic feedback experiments

38

MF Off MF On

d

b

Chi

p

a

c

50

100

150

0

50

100

1500

Noi

se P

ower

(A.U

.)

Can

tilev

e r

0 50 100 150 200 0 50 100 150 200

Frequency (Hz)Figure 2.17 - Power spectrum of noise from the photodiode. Laser on chip without (a) and with (b) magnetic feedback. Laser on cantilever without (c) and with (d) magnetic feedback. The dotted line represents the intrinsic cantilever noise. The instrument noise is significantly greater than the cantilever noise so that most of the feedback signal is instrument noise. Amplification of the feedback signal causes the cantilever to move as compensation for the instrument noise.

is shown in Figure 2.17. Power spectra from the photodiode are shown for the laser on

the chip (a and b) and on the cantilever (c and d). No magnetic feedback is used in

spectra a and c while magnetic feedback is used in spectra b and d. The dotted line

represents the intrinsic cantilever noise. The measured noise from the photodiode is

comprised of both the cantilever noise and the instrument noise, which in turn is split

between white noise and 1/f noise. White noise is spectrally flat and 1/f noise decays

with a first order power law as a function of frequency. When the laser is on the chip the

magnetic feedback cannot move the cantilever to compensate for noise so frames a and b

are the same. When the laser is on the cantilever, magnetic feedback is able to reduce the

noise at the photodiode. In frame c the noise is not truly the motion of the cantilever but

39

instead a measure of the motion of the electrons in the circuitry. When magnetic

feedback is used, the servo loop induces the cantilever to move to compensate the motion

of the electrons in the circuitry and not the thermal motion of the cantilever. Ironically,

contrary to the goals of magnetic feedback to control the cantilever, the system instead

contributed noise to the cantilever motion.

As a result, using an intrinsically stiff cantilever is more effective for measuring

stiff interactions than using magnetic feedback. While efforts were made to reduce the

instrument noise for better magnetic feedback performance, it was found that the signal to

noise ratio (SNR) for thermally limited systems is not dependent on the stiffness of the

cantilever. An expression for the SNR of a force signal near DC is

( )22

4 iseDetectorNokTbBk

kF

SNRB +

= ,39 (2.15)

where F is the force signal, k is the spring constant, kB is Boltzmann’s constant, T is

temperature, and b is the damping. B is the bandwidth, which represents the integral

from DC to frequency B. The transfer function must be flat within the bandwidth for this

equation to be correct. The numerator is the force signal in units of distance. The two

terms in the denominator represent the thermal noise of the cantilever and the detector

noise. For most instruments the detector noise dominates the denominator such that

reducing the spring constant will increase the signal to noise. When the system is thermal

noise limited the dependence on the spring constant cancels so that the SNR does not

change as spring constant is increased to avoid instability. Thus, it is more effective to

reduce the detector noise and use intrinsically stiff cantilevers than to use electronic

feedback with weak cantilevers.

40

2.7 Conclusion

Low bandwidth Magnetic Feedback Chemical Force Microscopy was used to map

the force profiles between chemically well-defined surfaces in solution. Magnetic

feedback removes cantilever instabilities so that important high stiffness intermolecular

interactions can be fully characterized. Complete force profiles for hydroxyl-terminated

and carboxyl-terminated SAMs were obtained. The force profiles were fit using a van

der Waals model. Hamaker constants of 1.0×10-19 J, 1.2×10-19 J, and 2.4×10-19 J for

hydroxyl surfaces and carboxyl surface in pH 7.0 and pH 2.2 respectively were

calculated. These values are consistent with the best theory for van der Waals but the

large difference in value for such similar systems implies that the long-range attractive

behavior is not physical but chemical and related to solvent stability. The measurement

of the attractive portion of the force profile is the first step to gaining an understanding of

these important interactions , and future experiments to determine the functional

dependence of the force and the correlation length of water at the surface will be crucial.

Low bandwidth MFCFM removes the snap-in and snap-out instabilities but in the

process it significantly increases the uncertainty of the measurement. High bandwidth

MFCFM could reduce the noise by using the feedback to control the motion of the

cantilever near resonance. The bandwidth of the feedback loop was increased to the

limits of inductive circuits and medium power op amps, and it was found that an

instrument with impractically high bandwidth and low noise is required to successfully

implement magnetic feedback. Using an intrinsically stiff cantilever with a low noise

instrument is easier and effective. Thus, developing a low noise instrument becomes

paramount for advancing studies of the molecular mechanism of nanoscale adhesion.

41

2.8 References

1. Binnig, G., Quate, C. F. & Gerber, C. Atomic Force Microscope. Physical Review Letters 56, 930-933 (1986).

2. Noy, A., Vezenov, D. V. & Lieber, C. M. Chemical Force Microscopy. Annual review of Material Science 27, 381-421 (1997).

3. Holscher, H., Allers, W., Schwarz, U. D., Schwarz, A. & Wiesendanger, R. Determination of tip-sample interaction potentials by dynamic force spectroscopy. Physical Review Letters 83, 4780-4783 (1999).

4. Gotsmann, B., Anczykowski, B., Seidel, C. & Fuchs, H. Determination of tip–sample interaction forces from measured dynamic force spectroscopy curves. Applied Surface Science 140, 314-319 (1999).

5. O'Shea, S. J. & Welland, M. E. Atomic Force Microscopy at Solid-Liquid Interfaces. Langmuir 14, 4186-4197 (1998).

6. Willemsen, O. H., Kuipers, L., Werf, K. O. v. d., Grooth, B. G. d. & Greve, J. Reconstruction of the Tip-Surface Interaction Potential by Analysis of the Brownian Motion of an Atomic Force Microscope Tip. Langmuir 16, 4339-4347 (2000).

7. Cleveland, J. P., Schaffer, T. E. & Hansma, P. K. Probing oscillatory hydration potentials using thermal-mechanical noise in an atomic-force microscope. Physical Review B 52, R8692-R8695 (1995).

8. Heinz, W., Antonik, M. D. & Hoh, J. H. Reconstructing Local Interaction Potentials from Perturbations to the Thermally Driven Motion of an Atomic Force Microscope Cantilever. Journal of Physical Chemistry B 104, 622-626 (2000).

9. Joyce, S. A. & Houston, J. E. A new force sensor incorporating force-feedback control for interfacial force microscopy. Review of Scientific Instruments 62, 710-715 (1991).

10. Thomas, R. C., Houston, J. E., Crooks, R. M., Kim, T. & Michalske, T. A. Probing Adhesion Forces at the Molecular Scale. Journal of the American Chemical Society 117, 3820-3834 (1995).

11. Jarvis, S. P., Dürig, U., Lant, M. A., H.Yamada & Tokumoto, H. Feedback stabilized force-sensors: a gateway to the direct measurement of interaction potentials. Applied Physics A 66, S211-S213 (1998).

12. Yamamoto, S.-i., Yamada, H. & Tokumoto, H. Precise force curve detection system with a cantilever controlled by magnetic force feedback. Review of Scientific Instruments 68, 4132-4136 (1997).

42

13. Jarvis, S. P., Yamada, H., Yamamoto, S.-I., Tokumoto, H. & Pethica, J. B. Direct mechanical measurement of interatomic potentials. Nature 384, 247-249 (1996).

14. Bain, C. D. et al. Formation of Monolayer Films by the Spontaneous Assembly of Organic Thiols from Solution onto Gold. Journal of the American Chemical Society 111, 321-335 (1989).

15. Hutter, J. L. & Bechhoefer, J. Calibration of atomic-force microscope tips. Review of Scientific Instruments 64, 1868-1873 (1993).

16. Isaelachvili, J. Intermolecular and Surface Forces (Academic Press, San Diego, 1992).

17. Joyce, S. A., Thomas, R. C., Houston, J. E., Michalske, T. A. & Crooks, R. M. Mechanical Relaxation of Organic Monolayer Films Measured by Force Microscopy. Physical Review Letters 68, 2790-2793 (1992).

18. Tupper, K. J. & Brenner, D. W. Compression-induced Structural Transition in a Self-assembled Monolayer. Langmuir 10, 2335-2338 (1994).

19. Isaelachivili, J. & Berman, A. Irreversibility, Energy Dissipation, and Time Effects in Intermolecular and Surface Interactions. Israel Journal of Chemistry 35, 85-91 (1995).

20. Kane, V. & Mulvaney, P. Double-Layer Interactions between Self-Assembled Monolayers of Mercaptoundecanoic Acid on Gold Surfaces. Langmuir 14, 3303-3311 (1998).

21. Parsegian, V. A. & Weiss, G. H. Spectroscopic Parameters for computation of van der Waals Forces. Journal of Colloid and Interface Science 81, 285-289 (1981).

22. Schrader, M. E. Wettability of Clean Metal-Surfaces. Journal of Colloid and Interface Science 100, 372-380 (1984).

23. Harder, P., Grunze, M., Dahint, R., Whitesides, G. M. & Laibinis, P. E. Molecular Conformation in Oligo(ethylene glycol)-Terminated Self-Assembled Monolayers on Gold and Silver Surfaces Determines Their Ability To Resist Protein Adsorption. Journal of Physical Chemistry B 102, 426-436 (1998).

24. Vezenov, D. V., Noy, A., Rozsnyai, L. F. & Lieber, C. M. Force Titrations and Ionization State Sensitive Imaging of Functional Groups in Aqueous Solutions by Chemical Force Microscopy. Journal of the American Chemical Society 119, 2006-2015 (1997).

25. Hu, K. & Bard, A. J. Use of atomic force microscopy for the study of surface acid- base properties of carboxylic acid-terminated self-assembled monolayers. Langmuir 13, 5114-5119 (1997).

43

26. Zhmud, B. V., Meurk, A. & Bergstrom, L. Evaluation of surface ionization parameters from AFM data. Journal of Colloid and Interface Science 207, 332-343 (1998).

27. White, H. S., Peterson, J. D., Cui, Q. & Stevenson, K. J. Voltammetric Measurement of Interfacial Acid/Base Reactions. Journal of Physical Chemistry B 102, 2930-2934 (1998).

28. Creager, S. E. & Clarke, j. Contact-angle Titrations of Mixed Omega-mercaptoalkanoic acid Alkanethiol Monolayers on Gold-reactive vs Nonreactive Spreading, and Chain-length effects on Surface pKa values. Langmuir 10, 3675-3683 (1994).

29. Smalley, J. F., Chalfant, K., Feldberg, S. W., Nahir, T. M. & Bowden, E. F. An Indirect Laser-Induced Temperature Jump Determination of the Surface pKa of 11-Mercaptoundecanoic Acid Monolayers Self-Assembled on Gold. Journal of Physical Chemistry B 103, 1676-1685 (1999).

30. Wang, J., Frostman, L. M. & Ward, M. D. Self-assembled Thiol Monolayers with Carboxylic-acid Functionality-measuring pH-dependant Phase-transitions with the Quartz Crystal Microbalance. Journal of Physical Chemistry 96, 5224-5228 (1992).

31. Strong, L. & Whitesides, G. M. Structures of Self-Assembled Monolayer Films of Organosulfur Compounds Adsorbed on Gold Single-Crystals - Electron- Diffraction Studies. Langmuir 4, 546-558 (1988).

32. Pashley, R. M. Hydration Forces between Mica Surfaces in Electrolyte-Solutions. Advances in Colloid and Interface Science 16, 57-62 (1982).

33. Colic, M., Franks, G. V., Fisher, M. L. & Lange, F. F. Effect of counterion size on short range repulsive forces at high ionic strengths. Langmuir 13, 3129-3135 (1997).

34. Scatena, L. F., Brown, M. G. & Richmond, G. L. Water at hydrophobic surfaces: Weak hydrogen bonding and strong orientation effects. Science 292, 908-912 (2001).

35. Noy, A., Zepeda, S., Orme, C. A., Yeh, Y. & Yoreo, J. J. D. Entropic Barriers in Nanoscale Adhesion Studied by Variable Temperature Chemical Force Microscopy. Journal of the American Chemical Society 125, 1356-1362 (2003).

36. Head-Gordon, T. & Hura, G. Water structure from scattering experiments and simulation. Chemical Reviews 102, 2651-2669 (2002).

37. Marcelja, S. & Radic, N. Repulsion of Interfaces Due to Boundary Water. Chemical Physics Letters 42, 129-130 (1976).

44

45

38. Kato, N., Kikuta, H., Nakano, T., Matsumoto, T. & Iwata, K. System analysis of the force-feedback method for force curve measurements. Review of Scientific Instruments 70, 2402-2407 (1999).

39. Viani, M. B. et al. Small cantilevers for force spectroscopy of single molecules. Journal of Applied Physics 86, 2258-2262 (1999).

Chapter 3 Noise Reduction

3.1 Introduction

The Atomic Force Microscope (AFM) and other scanning probe technologies

have been an important part of many of the nanoscience discoveries of the last decade.

The high sensitivity and ultra small probe size makes them ideal for measuring and

manipulating the nanoscale world. In vacuum at cryogenic temperatures, STM and AFM

have imaged individual atoms.1 The most significant advantage of AFM is its ability to

image non conducting substrates and work in solution, which makes it ideal for

characterizing chemical and biological samples. Unfortunately, features larger than

single atoms become convoluted with the tip shape reducing resolution to ~10 nm. The

Lieber group has worked extensively with nanotube probes2-6 to increase the lateral

resolution and has observed individual domains of proteins (5 nm).

Similarly, instrument and thermal noise limit force resolution. The low cantilever

Q of working in ambient conditions, causes the instrument noise to be relatively more

significant. For tapping mode imaging in air, reducing the instrument noise is necessary

to resolve subtle details, especially at small tapping amplitudes. For force curves, the

noise precludes detailed measurement of tip-surface interactions. As mentioned in the

previous chapter, very low instrument noise is required to accurately probe stiff regions

of adhesive interactions, which is important for work such as developing a theory of

molecular adhesion.

46

In this chapter, the many sources of cantilever position detection noise are

carefully analyzed and significant sources are reduced. It was found that interference

from the laser, electronics in the AFM base, air currents around the instrument, and

positional fluctuations in the laser direction were significant sources of noise. The noise

was decreased by using a low coherence length laser to reduce interference, bypassing the

noise producing circuitry in the AFM base, enclosing the AFM to protect from air

currents, and removing the fringes of the laser beam that experience the most significant

variation during fluctuations. Truncating the laser beam had two other advantages. It

allowed for increased laser power densities to be used and also caused diffraction of the

laser beam, which focused the laser intensity on the borders of the photodiode segments,

both of which considerably increased the signal to noise ratio (SNR). The high frequency

white noise was reduced from 800 fm/ Hz to 36 fm/ Hz matching the specifications of

the best AFMs reported in the literature.7,8

3.2 Contact and Tapping Mode Noise

The frequency components near DC contribute to the noise in contact mode,

while the frequency components near the cantilever resonance contribute to the noise is

tapping mode. The deflection functions as the measure of tip-sample force and is

produced by removing high frequency noise with a low pass filter. The total noise is the

sum of the major components near DC such as the white instrument noise, pink (1/f)

instrument noise, and cantilever thermal noise. Simulated power spectra for a 2 N/m

cantilever in water with two different instrument noise signatures are shown in Figure

3.1. The gray curve is representative of a stock DI multimode AFM and the black curve

is a thermally limited cantilever with some low frequency noise. The shaded regions

47

4

3

2

1

0125010007505002500

0.8

0.4

0.060200

(pm

/

)

Noi

se P

ower

(pm

/

)

40(kHz)

Frequency (Hz)Figure 3.1 – Power spectra near DC showing contact mode noise (shaded regions) in a 1kHz bandwidth for a cantilever with significant white and 1/f noise (gray) and a cantilever with reduced instrument noise. The instrument noise contribution is significantly more than the thermal noise. Inset shows the same spectra over a larger frequency range to show resonance.

represent the measurement bandwidth and the integrated noise power. The noise on the

stock system is many times greater than a thermally limited system, with the 1/f noise

being the largest contributor in the first few hertz. The instrument noise can reduce the

sensitivity by an order of magnitude compared to the thermally limited sensitivity.

Therefore, reduction of the pink and white noise is necessary for applications such as the

measurement of force curves in solution, where forces are small.

The white noise near the tapping frequency reduces the sensitivity of tapping

mode. The amplitude of oscillation is the feedback signal and the phase is a sensitive

measure of tip-sample interaction. The amplitude and phase signals are derived from the

oscillatory deflection signal using a lock-in amplifier. The cantilever noise in the

sidebands of the lock-in reference frequency is shifted to DC by the lock-in amplifier

48

49

Figure 3.2 – Sidebands A and B around the reference frequency of 70 kHz are shifted down to DC by the lock-in amplifier.

0.8

0.6

0.4

0.2

0.0420

0.4

0.3

0.2

0.1

0.07472706866

1.0

0.8 a0.6

0.4

0.2

0.06050403020100

Noi

se P

ower

(pm

2 /Hz)

Frequency (kHz)

Sideband B

0.6

0.5 Sideband A

Noi

se P

ower

(pm

2 /Hz)

Frequency (kHz)

Sideband A+

Sideband B

1.2

1.0

Resonance Lock-in output

Noi

se P

ower

(pm

/

)

1.6

1.2

0.8

0.4

0.076747270686664

Frequency (kHz)

b

Figure 3.3 – Noise power spectra of cantilevers in water (a) and in air(b). Measurements with large (gray) and small (black) contributions from instrument noise are shown in each frame. The shaded region depicts the tapping mode noise in a 1.5 kHz lock-in bandwidth. The relative instrument noise is more significant at low Q.

multiplier, Figure 3.2.* For a more complete understanding, read the exhaustive

treatment of tapping mode noise in chapter 5. For tapping mode, the reference signal is

often chosen to be the near the cantilever resonance frequency where oscillations are

easily induced, but the cantilever thermal noise is also most significant in that region.

Simulated power spectra comparing tapping mode noise contributions are shown in

Figure 3.3. Cantilevers in water and air are shown in panels a and b respectively. When

the cantilever is highly damped the Q is low and the thermal noise is spectrally broad

such that instrument noise becomes the most significant component of the total noise. In

air, the low damping leads to a high Q, which localizes the thermal noise in a narrow

bandwidth but the instrument noise is still strong enough to obscure detailed features

such as PNA labels on DNA.9 Reducing the white noise is important for imaging and

force curve measurement using tapping and contact mode. Other AFM applications, for

example Tapping Force Profile Reconstruction (chapter 6), require exact knowledge of

the motion of the cantilever at high frequencies and low instrument noise is more critical

for these applications.

3.3 Low Frequency Noise Reduction

3.3.1 Low Coherence Length IR Laser

One significant source of low frequency noise is caused by interference between

laser light scattered by the surface and the beam reflected off the cantilever. Often, some

laser power passes through or spills over the edges of the cantilever. The stray laser light

is scattered by the surface and some of it reaches the detection photodiode as shown in

*Digital Instruments phase extenders do not use a real lock in but instead an amplitude demodulator and other electronics that approximate phase for small phase shifts but the principle is similar.

50

-8

-4

0

4

2.01.51.00.50.0

8 IR Laser Red Laser

b

LaserPhotodiode

Scattered light

a D

efle

ctio

n (n

m)

Tip-Surface Distance (µm)Figure 3.4 – (a) Sketch of laser light passing by the cantilever and scattering off the surface. The reflected beam and scattered light can cause interference. (b) Oscillations in a force curve caused by interference (gray). Using a low coherence IR laser eliminated the interference (black).

Figure 3.4a. The two beams of light can interfere and produce oscillations in the

deflection signal that are dependent on the tip-surface distance as shown in Figure 3.4b.

The oscillations can lead to inaccurate force measurement and artificial deflection signals

of up to 30nm.

Using a low coherence length IR laser eliminates the interference (Figure 3.4b).

The coherence length of a laser or light source, lc, is the distance that the laser travels

before phase information is lost. The coherence length can be approximated by the

formula, scl ωλ2= , where λ is the wavelength of the light and ωs is the spectral width.

Reducing the coherence length is accomplished by increasing the spectral width. The

51

spectral width of the laser diode used in the AFM is unknown but it is estimated to be

150nm, by using a wavelength of 1000 nm and an interferometer path length of 40 µm

(based on the distance from the cantilever, down the tip to the surface, and back).

Changing the laser diode to a low coherence length IR laser diode greatly reduced the

interference.

3.3.2 AFM Base Noise

The AFM base adds a 15 Hz square wave and some white noise to the deflection

signal. The noise is possibly caused by a difference amplifier to compute the deflection

and a divider to normalize the deflection by the total laser power. The harmonic peaks

from this signal are seen clearly in the power spectrum in Figure 3.5. The divider

compensates for the variable reflectivity of different cantilevers so that they have similar

sensitivities. A broad range of sensitivity values is expected for AFM measurements for

reasons other than laser power so for accurate quantitative applications the cantilever

must be calibrated for every experiment; therefore, a divider is not necessary. A small

20

15

10

5

1500 1200900600300

Signal from Base Signal from INA106

0 0 N

oise

Pow

er (µ

V/

)

Frequency (Hz)Figure 3.5 – Noise spectra of deflection signals from the AFM base (black) and breakout box using a difference instrumentation amplifier INA106 (gray). The base adds both significant low frequency periodic noise and white noise.

52

breakout box was built to fit between the AFM base and the head, providing access to the

signals from the transimpedance amplifiers of the photodiode. An instrumentation

amplifier, INA106, was included in the breakout box to compute the ten times difference

signal, resulting in a very low noise and high gain deflection signal. Bypassing the base

electronics lead to a significant reduction in low frequency noise.*

3.3.3 Wind Shield

A cover to shield the instrument from drafts and air currents is crucial for quality

force curve collection. Typical force curves are collected with a repetition rate of 1 Hz.

As a result, the time of tip-sample interaction is about 200 ms and noise in the frequency

range of 5 Hz or below can unfortunately masquerade as tip-sample interaction. Air

currents are the most significant source of low frequency noise in this region. Low

Noi

se P

ower

(µV

/

)

uncovered covered

0 5 10 15 0

200

400

600

Frequency (Hz)

Figure 3.6 – Low frequency noise spectra of AFM instrument when uncovered (black) and covered (gray). When uncovered, wind currents can add large low frequency oscillations.

* Completely bypassing the base, by inserting the instrumentation amplifier output into the phase extender, caused the tip to crash on engagement. As a result, the transimpedance amplifier signals were allowed to pass through the base to the controller for the crucial engagement function and the low noise deflection signal was accessed through the instrumentation amplifier and sent to other input channels.

53

frequency power spectra for the instrument when uncovered and covered are shown in

Figure 3.6. The noise power at these low frequencies is enormous and can be on order of

~10 pm of total noise. Keeping the instrument covered is an integral step in collecting

reproducible and accurate force curves.

3.4 High Frequency White Noise Reduction

3.4.1 Laser Beam Truncation and Diffraction

A narrow slit in the optical path, depicted in Figure 3.7, greatly increases the

sensitivity and signal to noise ratio (SNR) by allowing the power density of the laser

beam to be increased and causing a diffraction pattern that focused laser light on the

boundary of the photodiode segments. The deflection signal results from the laser

slightly moving over the boundary between photodiode segments. Truncating the laser

beam would decrease the total laser power hitting the photodiode but leave the important

light that crosses the boundary intact. The laser power could then be increased such that

the power density at the boundary is higher, resulting in more sensitivity, without going

over the input power limit of the photodiode.

PD AFMHead

laser

slit

Figure 3.7 – Drawing of AFM head showing laser beam traces with (solid) and without (dotted) truncating slit. The slit increased the aspect ratio of the beam and caused a diffraction pattern that focused light on the boundaries of the photodiode segments.

54

without slit with slit Position A A-B A A-B

1 14.34 26.92 19.35 39.00 2 9.16 17.46 11.66 23.80

3 7 94 15 81 14 28 29 66

Table 3.1 – Sensitivity values for the signal from one photodiode segment (A) or the difference between segments (A-B), with and without the truncating slit and for three different positions on the cantilever.

Surprisingly, the truncating slit also increased the sensitivity without a laser

power increase. Detector sensitivities with and without the truncating slit, in Table 3.1,

reveal the inherent increase in sensitivity associated with inserting the slit. Also tabulated

are the measurement of the signal from only one photodiode segment (A) and the

difference signal between the two photodiode segments (A-B) for three laser positions on

the cantilever. The sensitivities were calculated using the thermal noise of the cantilever

at resonance instead of measuring the contact line of a force curve. The thermal energy

stored in the motion of the cantilever will be constant and result in the amplitude of the

thermal noise spectrum being constant, if the cantilever properties, k, Q, and f0, do not

change. But, the measured thermal noise is a function of the detector sensitivity and can

be used to calculate the sensitivity. A slit width of 280 µm maximized the sensitivity for

all laser positions on the cantilever.

The sensitivity of the difference signal was expected to be twice the sensitivity of

a single photodiode segment because the laser moves from one segment to the other.

Also, the sensitivity with and without the truncating slit was expected to be the same

because the slit only removed light intensity from regions far from the photodiode

segment boundary. The data show that without the slit, the difference signal is less than

twice the single segment signal, while the presence of the slit causes the difference signal

55

to be more than twice the single segment signal. More importantly, the sensitivity for the

signals with the slit are significantly higher than the comparable signals without the slit.

The increased sensitivity after inserting the slit results from the laser beam being

diffracted, which focuses light on the photodiode boundary. A drawing of the laser beam

profile with the dimensions it would have at the photodiode is shown in Figure 3.8. The

size of the beam is 4 mm X 625 µm and consists of the two regions separated by 125 µm.

The first region (bottom of Figure) is 250 µm wide and has uniform brightness, while the

second region consists of two areas. The first area (top of Figure) is 125 µm and has a

similar brightness to region one. The second area near the dark boundary (middle of

Figure) is brighter and is also 125 µm wide. When no slit is in the optical path the beam

is 4 mm X 2 mm with a generally gaussian intensity distribution except extra stray

intensity along the minor axis. A separation of 250 µm is between the segments of most

two-segment photodiodes, which matches the separation of the laser intensity regions

very well. The diffraction pattern essentially moved the laser power from the dead region

between the photodiode segments to the active region at the boundaries. Using the

truncating slit along with increasing the laser power significantly increased the detection

signal to noise ratio.

500

µm

Figure 3.8 – Drawing of diffraction pattern caused by the truncating slit. Distances are representative of the size of the beam at the photodiode.

56

3.4.2 Feedback Resistors

Increasing the value of the feedback resistors on the transimpedance amplifiers

slightly increases the sensitivity. A schematic of a photodiode transimpedance amplifier

with noise sources and a Bode plot (log amplitude vs. log frequency) of the gain

bandwidth is shown in Figure 3.9. The photodiode can be considered a current source, Ip,

with internal resistance, RD, and capacitance, CD. The gain is determined by the feedback

resistor, R1, and limited by the shunt capacitor, CS, for stability. The amplifier has input

current, In, and input voltage, en, noise sources. The expression for the resulting output

voltage is

nS

DnPBP e

CRJCRJRIqIRTRkRIe

1

111110 1

124ωω

++

++++= . (3.1)

The terms on the right side of the equation are the signal, resistor noise, shot noise, input

current noise, and input voltage noise, respectively. The current, IP, is limited by the

laser output because high laser powers damage the laserdiode and shorten the lifetime.

The laser power was not increased to more than 2mW, which produces 50 µA on each

ba

Figure 3.9 – (a) Schematic of photodiode amplifier with noise sources, en and In. (b) Bode plot of amplifier open loop gain, signal gain, and noise gain.

57

photodiode segment with the 280 mm slit in the optical path. The noise values for each

term in the above equation are written in Table 3.2, based on the AD827 (AFM

transimpedance amplifier) at 10 kHz with 50 µA for the feedback resistor values of 10

kΩ and 200 kΩ.

AD827 10 kΩ 200 kΩ Resistor noise 12.7 nV/ Hz 57 nV/ Hz Input voltage noise 15 nV/ Hz 15 nV/ Hz Input current noise 15 nV/ Hz 300 nV/ Hz Shot noise 40 nV/ Hz 800 nV/ Hz Total amplifier noise* 47 nV/ Hz 856 nV/ Hz

SNR42.6 46.7

Table 3.2 – Noise values for the components of the amplifier noise of the AD827 at 50 µA for the feedback resistor values of 10 kΩ and 200 kΩ.

At high laser powers, the shot noise is the largest component with other sources

contributing about 18% for a 10 kΩ feedback resistor. Baseline input voltage noise is

unaffected by changing the feedback resistor and the Johnson noise increases by 1R .

Unfortunately, both the input current noise and the shot noise increase by R1, like the

signal, and provide no increase to the signal to noise ratio. Nonetheless, the signal to

noise does rise by increasing the resistor value because the Johnson noise and input

voltage noise become insignificant.

At high frequencies, the voltage noise experiences gain peaking from the

photodiode capacitance and is no longer insignificant. The instrument after modification

is shown in the Bode plot of Figure 3.10. The gain increase starts at 250 kHz from which

a photodiode capacitance of 8 pF is calculated. The photodiode capacitance is consistent

58

* Addition of noise power follows the relation, 2222

lkji σσσσσ +++=

567

10-6

2

3

4

1kHz 10kHz 100kHz 1MHzFrequency

Noi

se P

ower

(V/

)

Figure 3.10 – Noise spectrum for photodiode amplifier output. Noise peaking is clearly seen at high frequencies but does not contribute in the working frequencies of the AFM.

with a small segmented photodiode of about 5 mm2 or 1.5 × 3 mm per segment. Gain

peaking noise would intrude on the frequency range used during experiments (0-150

kHz) if the resistor value was further increased.

Aside from increasing the signal to noise ratio, an increase in resistor value

increases the measurability of the signal. The DI Nanoscope IIIa is equipped with a 14-

bit ADC running at ±10 V giving 1.2 mV resolution or 1.2 Å resolution with typical

sensitivities. Increasing the feedback resistor increases the resolution so that crucial

position information is not lost during digitization. The AFM head from DI was

originally equipped with 10 kΩ resistors. The resistors were changed to 200 kΩ to

increase the signal to noise by 9% in the frequently used bandwidth and to increase the

resolution per bit on the ADCs.

3.5 White Noise Correlation between Photodiode Segments

The instrument noise is composed of components that are correlated,

anticorrelated, and uncorrelated between the two photodiode segments. The noise spectra

59

8

6

4

2

403020100

A Signal Noise A-B Difference Signal Noise

Noi

se P

ower

(µV

/

)

Frequency (kHz)

Figure 3.11 – Noise spectra for a single photodiode segment (black) and the difference between the segments (gray). The lower noise in the difference signal indicates that the noise is correlated.

for the individual photodiode segment signal (A) and the difference signal (A-B) are

shown in Figure 3.11. The A signal has many inductive noise peaks and stronger

baseline noise closer to DC, which do not exist in the A-B difference signal. The absence

of the inductive peaks is a result of both signal lines sensing the stray field equally such

that the noise cancels because it is correlated. Surprisingly, the baseline white noise is

reduced also by calculating the difference, which implies that some of the white noise is

also correlated. The observation is more clearly depicted in Figure 3.12 where the

baseline white noise at ~35 kHz is plotted as a function of photodiode segment voltage

(laser power) along with the shot and amplifier input current noise which sets the lower

bound. The A signal, Figure 3.12a, is plotted separately from the difference signal,

Figure 3.12b, because the shot noise of the difference between two channels is square

root two greater than the shot noise of one channel. The difference signal noise is

significantly lower than the single segment and at low laser powers it is limited only by

the shot and input current noise.

60

b

aN

oise

Pow

er (µ

V/

)

0

1

2

0

1

2

3

Shot Noise

Shot and Amplifier Input Current Noise

A-B Difference Signal Noise

A Signal Noise

2 6 8 10 4

Photodiode Voltage (V)

Figure 3.12 – Noise power measured at ~35 kHz as a function of photodiode voltage (laser power). (a) Noise from single segment compared to shot and amplifier noise. (b) Difference noise signal compared to shot and amplifier noise. The difference signal is almost shot noise limited.

The separate components of the noise are calculated by comparing the individual

and difference signal measurements. The resulting noise for two noise components

follows the formula ( )θσσσσσ cos222jiji ++= , where σi and j are the noise components

and θ is the angle of correlation between the two sources. Two uncorrelated sources have

θ equal to zero and the formula simplifies to, 22ji σσσ += , the familiar additivity

formula for noise sources. The first formula is generalizable to many noise sources by

including a correlation term for every combination of noise sources in the expression.

The noise was modeled as originating from shot and amplifier input current noise

(uncorrelated), laser position fluctuations (anti-correlated), and laser power fluctuations

61

(correlated). The magnitudes of the laser power and position fluctuations were calculated

since the shot and amplifier input current noises were known values.*

3.6 Position Fluctuation Noise Reduction by Laser Beam Truncation

The truncation of the laser beam by the slit reduced the stray light and

considerably cut the position fluctuation noise. The magnitudes of the position and

power fluctuation noise components are plotted as a function of photodiode segment

voltage (laser power) in Figure 3.13. Measurements with the laser on the supporting chip

and on the cantilever are shown for both beam truncation and without beam truncation.

The slit causes little difference in the power fluctuation noise (Figure 3.13a) because

Noi

se P

ower

(µV

/

)

Chip without Slit

Cantilever without Slit

Chip with Slit

Cantilever with Slit

b.

a.

1

2

3

0 2 4 6 8 10.0

0.5

1.0

1.5

Photodiode Voltage (V)

Figure 3.13 – (a) Comparison of power fluctuation (correlated) and position fluctuation (anti-correlated) noise (b). Position fluctuation noise is significantly reduced by laser beam truncation by the slit.

* The difference between an uncorrelated and anti-correlated noise source was not distinguishable since a sum signal was not also collected. If the source was uncorrelated instead of anti-correlated then the value would be a factor square root two larger. An anti-correlated noise source is modeled because including laser positional fluctuations as the anti-correlated noise sources fits intuition and the data well.

62

most of the laser power is in the center of the beam. The position fluctuation noise

(Figure 3.13b) is greatly reduced by inserting the truncating slit, because the slit cuts off

the laser light that would normally be along the outer edges of the photodiode. Both

position and power fluctuation noises are stronger when the laser is on the cantilever

because position fluctuations that cause the beam to shift off of the cantilever would

cause both correlated and anti-correlated modulations. Positional fluctuations can also

cause correlated noise if the shift is parallel to the segment boundary, but it is expected to

be lower intensity than the change resulting from perpendicular motion and like power

fluctuations, it will be canceled during calculation of the difference signal.

3.7 Total Noise Reduction

After completing the modifications, the instrument noise was decreased

significantly. The noise spectra for a 1.3 N/m cantilever in water before the

modifications and after are shown in Figure 3.14. Before modification, the resonance is

barely perceptible above the 800 fm/ Hz of instrument noise. After modifications, the

baseline noise is only 36 fm/ Hz , the lowest reported value of position noise for AFM

measurements.7,8 The low noise allows the resonance to be observed very clearly and fit

by a damped harmonic oscillator model to determine k, f0, and Q. Detailed motion of

cantilevers as stiff as 20 N/m can be measured accurately.

3.8 Further increases to signal to noise

Two additional changes could enhance the signal to noise of the optical cantilever

detection mechanism. Changing the transimpedance amplifier to a FET op amp can

increase the signal to noise by another 7%. The AD827 is a bipolar op amp, which has

significant current noise of 1.5pA/ Hz . FET op amps such as the OPA655 or OPA656

63

have input current noise of 1.5 fA/ and voltage noise of only 6 nV/ . A

comparison of the amplifier noise values for the AD827 and OPA655 are recorded in

Table 3.3. Unfortunately, the fast FET op amps do not have 15 V rails like the AD827,

and a new 5V power source would be required, which is too much work for a small gain.

Hz Hz

0.6

0.4

0.2

0.0100806040200

Noise Spectrum before Modifications Noise Spectrum after Modifications Fit

Hz Hz

1.0

0.8N

oise

Pow

er (p

m/

)

Frequency (kHz)

Figure 3.14 – Cantilever noise spectra before instrument modification (gray) and after modification (black). The white noise was significantly reduced from 800 fm/ to 36 fm/ . The spectrum is thermally limited and well fit (dotted) by a damped harmonic oscillator model.

Replacing the laser with a higher power laser would increase significantly the

signal to noise ratio. High power (>5mW) superluminescent LEDs are readily available

and many also have fiber optic coupled outputs. Increasing the current on the photodiode

by a factor of 2 and reducing the resistors to restore the same dynamic range provides

40% more signal to noise. The fiber optic coupling would also reduce positional

variations in the beam and remove the residual anti-correlated noise left after the beam-

truncating slit. Replacing the laser is the first step if more sensitivity is desired.

64

AD827 (bipolar Op Amp)

OPA655 (FET Op Amp)

Resistor noise 57 nV/ Hz 57 nV/ Hz Input voltage noise 15 nV/ Hz 6 nV/ Hz Input current noise 300 nV/ Hz 0.2 nV/ Hz Shot noise 800 nV/ Hz 800 nV/ Hz Total amplifier noise 856 nV/ Hz 802 nV/ Hz SNR 46.7 49.9

Table 3.3 – Noise values for the components of the amplifier noise for AD827 and OPA655 at 50 µA and 200 kΩ feedback resistors.

3.9 Conclusion

The instrument noise of the optical cantilever detection system was reduced

through a number of modifications. Significant increases in sensitivity were obtained by

using a low coherence length IR laser to reduce interference, shielding the AFM from air

currents, and bypassing the noise producing electronics in the AFM base. The most

sensitivity was gained by truncating the laser beam with a slit to increase the power

density near the photodiode segment boundaries and remove the stray light that caused

laser position fluctuation noise. The slit also caused a diffraction pattern that further

increased power density at the segment boundary. Fluctuations in the laser power add

noise but this effect is canceled by calculating the difference signal. The high frequency

white noise was reduced from 800 fm/ Hz to 36 fm/ Hz and low frequency sources

were eliminated, making the AFM as sensitive as the best reported values in the

literature.7,8 Detailed motion of cantilevers as stiff as 20 N/m with Q of 3 can be

accurately measured. With precise knowledge of the cantilever position, new techniques

to probe steep tip-sample interactions such as Brownian Force Profile Reconstruction or

Tapping Force Profile Reconstruction can be used to uncover the mechanism underlying

interfacial adhesion and other intermolecular processes.

65

66

3.10 References

1. Lantz, M. A. et al. Quantitative Measurement of Short-Range Chemical Bonding Forces. Science 291, 2580-2583 (2001).

2. Dai, H., Hafner, J. H., Rinzler, A. G., Colbert, D. T. & Smalley, R. E. Nanotubes as nanoprobes in scanning probe microscopy. Nature 384, 147-150 (1996).

3. Hafner, J. H., Cheung, C. L. & Lieber, C. M. Direct Growth of Single-Walled Carbon Nanotube Scanning Probe Microscopy Tips. Journal of the American Chemical Society 121, 9750-9751 (1999).

4. Hafner, J., Cheung, C. L. & Lieber, C. M. Growth of nanotubes for probe microscopy tips. Nature 398, 761-762 (1999).

5. Wong, S. S., Joselevich, E., Woolley, A. T., Cheung, C. L. & Lieber, C. M. Covalently functionalized nanotubes as nanometre- sized probes in chemistry and biology. Nature 394, 52-55 (1998).

6. Wong, S. S., Woolley, A. T., Joselevich, E., Cheung, C. L. & Lieber, C. M. Covalently-Functionalized Single-Walled Carbon Nanotube Probe Tips for Chemical Force Microscopy. Journal of the American Chemical Society 120, 8557-8558 (1998).

7. Manalis, S. R., Minne, S. C., Atalar, A. & Quate, C. F. Interdigital cantilevers for atomic force microscopy. Applied Physics Letters 69, 3944-3946 (1996).

8. Rugar, D., Mamin, H. J., Erlandsson, R., Stern, J. E. & Terris, B. D. Force Microscope Using a Fiber-Optic Displacement Sensor. Review of Scientific Instruments 59, 2337-2340 (1988).

9. Hahm, J. I., Personal Communication, (2002)

Chapter 4 Solvation and Structural Forces

4.1 Introduction

The behavior of liquids near a solid interface is important to many areas of

science, from physics to biology. A theoretical study in 1978 suggested that the presence

of the surface restricts the motion of the molecules and causes the liquid molecules to

order into layers near the surface.1 The force required to exclude these layers and the

increased viscosity associated with confinement are of specific interest in tribology and

wear. Also, electrochemistry and solid-phase catalysis are considerably affected by the

altered diffusion and solute adsorption resulting from solvent ordering. Lastly, interfacial

ordering of water molecules influences protein stability, ion-channel conductance, ligand

binding dynamics, and protein folding.2 The forces associated with surface-solvent

interactions are called solvation or structural forces.

Israelachvili first observed solvent ordering by measuring the layering of Octa-

methyl-cyclotetrasiloxane (OMCTS) between mica plates using the Surfaces Forces

Apparatus.3 The layering was periodic and the force required to remove each layer

increased roughly exponentially with interfacial distance. Subsequently, solvation forces

have been observed for many different liquids using both the Surface Forces Apparatus

(SFA)3-6 and Atomic Force Microscope (AFM).7-11 Also, Transmission Electron

Microscopy12 and X-ray reflectivity13,14 studies revealed that solvent ordered near single

surfaces without requiring confinement by two surfaces.

67

The initial SFA experiments were very insightful since they observed the distance

range of the interaction and also helped to initiate an understanding of the confinement-

induced solidification of the solvent. Unfortunately, the large contact area of the SFA is

unphysical since many contacts happen at small asperities. Also, the weak spring

snapped in and out from these rapidly changing force profiles and only periodicity could

be observed instead of the shape of the force profile. AFM experiments were extremely

noisy and only revealed that two large atomically flat surfaces are not required to observe

solvent ordering.

Direct observation of whole force profiles, especially attractive regions,

associated with solvent ordering could lead to a greater understanding of interfacial

phenomena. Theory predicts that an ordered liquid will have a sinusoidal force profile

while a semi-solid will be non-sinusoidal and have hysteresis from plasticity. Also, more

precise measurement of the distance scale will lead to a better understanding of the

packing of the solvent near the surface and the ability of other molecules to penetrate and

move through the solvent.

In this chapter, high precision force profiles are presented for OMCTS, 1-

nonanol, and water near smooth, flat surfaces. The force profiles for solvent ordering

have high stiffness in many regions, and Brownian Force Profile Reconstruction was

developed to accurately measure these unique interactions. The OMCTS data imply that

OMCTS is a fluid at all times. The 1-nonanol became crystalline upon confinement of

four molecular layers. Water between hydrophilic surfaces showed three oscillations

within 1 nm, placing an upper limit on the range of attractive forces originating from

68

alignment of molecules at the surface. Also, the period was 3.6 Å, which revealed that

two water molecules must be expunged simultaneously to properly solvate the surfaces.

4.2 Model for Solvent Structure

A solid surface confines the motion of the nearby solvent reducing their motion

and causing the molecules to occupy specific positions relative to the surface. As a

result, a lattice model of solvent dynamics15 is an appropriate representation of the

molecules. The layers can be modeled with regularly increasing root-mean-square (rms)

displacement from the lattice sites since each successive layer is less confined by the

surface. In the model, each lattice layer is modeled as a gaussian distribution

( )nw

npz

i enw

⋅⋅−−

⋅=

2

21

1ρ , (4.1)

where ρi is the molecular density of the nth layer, w is the initial width, z is the distance

from the surface, p is the interlayer spacing, and n is an integer. The rms displacement

amplitude, nwA ⋅= , increases with each successive layer. The sum of all the layers is

the total molecular density. The Gaussians representing each individual layer and the

total molecular density are depicted in Figure 4.1. Numerically, the resulting molecular

density distribution is equivalent to an exponentially decaying sine (black fit), which can

be easily fit to data.14

The AFM’s small probe is ideal for measuring the equilibrium molecular density

distribution. Variations of molecular density on opposite sides of the probe will lead to a

pressure difference, which can be measured as a force or deflection. Simulations have

shown that the pressure is independent of probe size even in the limit of a single atom.16-

19 The probe must be large enough that the force is perceptible but a second surface

69

a

b

1.5

1.0

0.5

0.0

1 .5

1 .0

0 .5

0 .06 05 04 03 02 01 00

FitTotal Density

Mol

ecul

ar D

ensi

ty

Surface Distance (Å)Figure 4.1 – Lattice model of molecular interfacial density. (a) Each molecular layer is defined by a Gaussian whose variance is a function of surface distance. (b) The total molecular density (gray) is similar to a decaying sine function (black).

causes it own ordering. The probe must not be too large, or else solvent molecules will

not be free to move between the surfaces in the timescale of the observation.

The lattice model predicts that liquid behavior near the surface will have a

decaying sinusoidal force profile. Increased confinement from the probe surface will

cause the viscosity to increase. If the viscosity and contact area are too great then the

solvent molecules will not be able to escape and hysteresis will be observed between the

advancing and receding force curves. Also, if the molecules are solidified by the

confinement then transitions from one layer to the next are cooperative such that the

removal of a portion of the molecular layer causes the removal of the whole layer. The

transitions from repulsive to attractive force occur over a smaller distance scale causing

the oscillatory force to be asymmetric. Deviations from an exponentially decaying

sinusoidal force profile are an important marker of non-fluid or glassy dynamics.

70

The force profiles expected for solvent ordering are extremely stiff (10 N/m for

R=10 nm), with the force profile increasing stiffness with each layer, since the overall

interaction decays exponentially. The many closely spaced energy minima also cause the

tip position to become bistable as it jumps from one minima to the other, which causes

errors in the deflection measurement. Brownian Force Profile Reconstruction was

developed to correct the errors in force profile measurement when the cantilever stiffness

is near the interaction stiffness.

4.3 Force Profile Measurement Error

The development of a correct model of molecular interfacial phenomena requires

the accurate collection of force profiles. The need to increase the cantilever stiffness to

avoid instabilities was thoroughly discussed in chapter 2. TO meet this need, magnetic

forces were used to increase the stiffness of the cantilever. Unfortunately, magnetic

feedback adds noise to the cantilever and is very difficult to implement because of

bandwidth limitations. Superior force profile measurement can be obtained by simply

using a stiffer cantilever and a low noise detection scheme. Methods for significantly

reducing the instrument noise were covered in chapter 3. These allow a thermally-limited

position measurement of stiff cantilevers.

These improvements make force profile measurement more accurate but thermal

noise can still cause significant error when measuring the equilibrium landscape of

interactions with stiffness near the stiffness of the cantilever because the thermal noise

simultaneously samples significantly different regions of the force profile. The results of

a simulation of an ordinary force curve are shown in Figure 4.2. The simulation used the

71

-250

-200

-150

-100

-50

0

3.02.52.01.51.0

100

0

-100

-200

a Span of thermal noiseForce Profile

b

Ordinary Force ProfileForce Profile

Forc

e (p

N)

Tip Sample Distance (nm)Figure 4.2 – (a) Force profile (black) with markers (gray bars) showing the span of the thermal noise of the cantilever during a force curve. The intensity of the gray bars correlates with the probability of the cantilever position and the black mark indicates the average position. (b) The resulting force profile (gray) badly misses the force profile used for the simulation (black).

wave equation of motion to calculate the trajectory of the cantilever. The thermal force

noise is calculated from the cantilever parameters and the temperature. The cantilever

properties used were k = 0.2 N/m, Q = 3, f0 = 25,000, and T= 300K. The motion of the

cantilever was sampled at 200 kHz although time steps for the simulation were 500 ns for

increased accuracy of the numerical algorithm. A complete explanation of the simulation

algorithm and the code are in the appendix (A.4). The force profile used in the

simulation is shown in Figure 4.2a. Gray bars indicate the range of motion of the

cantilever from thermal excitation in the potential energy well with the intensity

corresponding to position occupation probability. The black line in the center of each

72

gray bar represents the average tip-sample distance. The large span of the thermal noise

causes the tip to sense both the strongly attractive regions along with the weakly

attractive regions within a very short interval of time. Ordinary force curves simply

average the thermal noise to obtain the deflection, which averages the strongly attractive

and weakly attractive regions together producing an inaccurate result. Figure 4.2b shows

the force profile resulting from an ordinary force curve compared to the real force profile.

The simulated force profile overestimates the attractive forces at large tip-sample

separations and underestimates the attractive forces near the bottom of interfacial

attractive well. The real force profile has a maximum stiffness of 0.31 N/m while the

cantilever stiffness used in the simulation was 0.2 N/m. Thermal noise is intrinsic to the

cantilever and cannot be removed without working away from ambient or physiologic

conditions. A method that harnesses the thermal (Brownian) motion of the cantilever to

measure the force profile is a very powerful and useful tool for measuring intermolecular

and interfacial interactions.

4.4 Brownian Force Profile Reconstruction

Brownian force profile reconstruction (BFPR) is a data collection and analysis

technique that accurately reconstructs the force profile by using the thermal noise or

Brownian motion to probe the tip-sample interaction. The tip moves around in the

potential energy landscape using the energy provided by the thermal bath. At

equilibrium, the position dependence of this Brownian motion is related to the potential

energy landscape by Boltzmann’s equation,

( ) ( )( )TkdU BCedP −= , (4.2)

73

where P(d) is the probability density at distance d, C is a scaling constant, U(d) is the

potential energy, kB is Boltzmann’s constant, and T is the temperature. Recording the tip

position probability allows the energy landscape to be calculated by inverting

Boltzmann’s equation,

( ) ( )( ) CdPTkdU B +−= ln . (4.3)

Cleveland first used this technique to measure the perturbation to the cantilever harmonic

potential caused by oscillatory hydration forces near calcite and barite surfaces.20 The

whole energy landscape and force profile can be reconstructed by measuring the

perturbation to the harmonic potential by the tip-sample interaction as the force curve

progresses.

The steps involved in Brownian Force Profile Reconstruction are demonstrated in

Figure 4.3.

Step 1. The deflection during a force curve measurement is sampled very quickly to

include the thermal noise (Figure 4.3a). The whole bandwidth of the cantilever motion

must be included so it is best to sample around 4f0, which makes the Nyquist frequency

2f0. Data sets for BFPR can be very large. For example, when sampling at 100 kHz for 2

seconds with 16-bit resolution, he resulting file is 400 kB. Oversampling at frequencies

greater than 8f0 does not provide more information about cantilever position and only

increases the data set size. Sampling at less than 4f0 cuts off some of the thermal noise

bandwidth producing inaccuracies.

Step 2. The force curve is parsed into many small sections, typically 10,000 points each.

Each section is binned into a histogram as a function of cantilever deflection as seen in

Figure 2.2b. The left side of the figure shows regions of the force curve with significant

74

CdPTkdU B +−= ))(ln()(

-1.0

-0.5

0.0

For

ce (n

N)

3.02.52.01.51.00.5Tip-sample Distance (mn)

-1.0

-0.5

0.0

For

ce (n

N)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Def

lect

ion

(nm

)

3.53.02.52.01.51.0Z-Piezo Displacement (nm)

-1.0

-0.5

0.0

For

ce (n

N)

4321Tip-sample Distance (mn)

-100

-50

0

Ene

rgy

(kBT

)

-1.0

-0.5

0.0

For

ce (n

N)

4321Tip-sample Distance (mn)

-100

-50

0

Ene

rgy

(kBT

)

4

2

0

Ene

rgy

(kBT

)

-200 -100 0 100Deflection (pm)

12

10

8

6

4

2

0

Hun

dred

s of P

oint

s

4

2

0

Ene

rgy

(kBT

)

-800 -600 -400 -200Deflection (pm)

6

4

2

0

Hun

dred

s of P

oint

s b

c

d

e

g

f

b

c

d

e

a

Figure 4.3 – (a) Force curve showing deflection with all thermal noise. (b) Histograms of sections of force curve. (c) Histogram converted to energy using Boltzmann's equation. Includes both spring and tip-sample interaction. (d) Energy after spring contribution is subtracted away and positioned for tip-sample distance. (e) Derivative of energy is force. (f) All force sections together. (g) Average of force sections is the Brownian Reconstruction Force Profile.

75

tip-sample interaction and negligible tip sample interaction on the right. The histogram

on the left is bimodal showing the strong influence of the tip-sample well on the

harmonic cantilever potential.

Step 3. Scaling the histograms by the total number of points produces the probability as a

function of position, which is related to the energy by the inverse of Boltzmann’s

equation (4.3) as shown in Figure 4.3c. The energy curve on the left shows the two

minima clearly while the energy curve on the right resembles a harmonic well. The

sections are clipped on both sides to remove the points that had zero probability.

Step 4. An energy curve similar to the one on the right side of Figure 4.2c is fit with a

simple quadratic function,

( ) ( ) cddkdU +−= 202

, (4.4)

where U(d) is the energy, k is the spring constant, d0 is a deflection offset (near zero), and

c is an energy offset. The energy curve is derived from a very large number of points

from the region of the force curve with no tip-sample interaction. The resulting fit is

used to subtract off the cantilever contribution of the energy curves in Figure 4.3c, which

leaves only the energy of the tip-sample interaction, Figure 4.3d. The section is also

scaled for tip-sample distance based on the overall deflection of the cantilever and the

position of the cantilever support relative to the sample. The latter is determined by a

procedure similar to computing tip-sample distance from Z-piezo displacement for a

regular force curve. Because the arbitrary constant C in equation 4.3 is unknown the

energy sections do not overlap.

Step 5. The derivative is calculated, which removes the necessity of obtaining C but also

introduces noise as shown in Figure 4.3e. The force profile used to do the simulation for

76

Figure 4.3 is also displayed to show the quality of the reconstruction.

Step 6. All the force sections are averaged together. The force sections are plotted

together in Figure 4.3f and the average of the sections is plotted in Figure 4.3g. The

curve in 4.3g is called the Brownian Reconstruction and according to simulations it is an

excellent fit to the real force profile.

Treating the thermal noise as an informative distribution instead of meaningless

obtrusion produces a significantly more accurate force profile. The simulation used to

produce Figure 4.2 is reshown in Figure 4.4 along with the Brownian Reconstruction.

The BFPR algorithm split the wave into 200 sections each with 2×104 points. Each wave

-250

-200

-150

-100

-50

3.02.52.01.51.0

Force profile used in simulation Ordinary Force Curve Brownian Reconstruction Force Profile

a

b

300200100

0-100-200-300

Force Sections Force Profile used in simulation

Forc

e (p

N)

Tip Sample Distance (nm)Figure 4.4 – (a) Brownian reconstruction force sections superimposed on the force profile used in the simulation. (b) Brownian Reconstruction Force Profile (dark gray) from force sections with ordinary force curve calculated from the same data (gray). The ordinary curve deviates significantly from the shape of the force profile (black) while the Brownian Reconstruction Force Profile is a much better approximation.

77

section overlapped with the adjacent wave sections. The bin size on the histograms was

0.1 Å. The 200 force sections along with the force profile used in the simulation are

shown in Figure 4.4a. The force sections match the force profile well with some scatter.

The Brownian Reconstruction Force Profile and the ordinary force profile are displayed

in Figure 4.4b. Brownian reconstruction is able to accurately measure the force profile

even when the stiffness of the interaction is greater than the stiffness of the cantilever.

Calculation of the force for each section is a crucial component to the success of

Brownian Force Profile Reconstruction. As mentioned earlier, the unknown C from

equation 4.3 makes reconstruction of the energy landscape difficult. Two other methods

of solving the problem were attempted concurrently with the development of Brownian

Force Profile Reconstruction. The first method developed by Heinz et al.21 removed the

frequency components of the force curve between DC and 150 Hz thus removing the

deflection signal or force component. The energy sections were calculated and the

harmonic well of the spring was subtracted. Since the force component was previously

removed, all sections centered at zero deflection. The interaction was assumed to have

constant stiffness over the whole range of thermal motion. A harmonic well was fit to the

energy data to calculate the stiffness of the interaction at that position. A stiffness versus

tip-sample distance curve was compiled from the sections which was later integrated to

calculate force. This method was used with limited success to measure an electrostatic

double layer interaction. This method has the following limitations: (1) information is

lost during the removal of the DC frequency components, (2) the assumption that the

interaction has constant stiffness over the range is an extremely bad approximation, and

(3) their method required a user defined error factor, є, to compensate for inaccuracies in

78

the calculated stiffness because instrument noise broadened their histograms. The

limitations of Heinz’s method are severe. The second method developed by Willemsen et

al. tries to solve for the arbitrary constants by maximizing the overlap between the energy

sections after the harmonic well has been removed.22 Although great care is used to

describe the statistics of the averaging technique used to compute the average of the

sections, no description is given for the process by which the overlap is maximized.

Possibly a subjective determination of the overlap by the researcher was used.

During the development of BFPR, automated techniques were sought to remove

researcher bias. A moderately successful automated technique employed linear fits to the

overlap sections of two waves. The waves were then offset to reduce the χ2 statistic of

the two lines. A quadratic fit would be more effective but also more computationally

expensive. Unfortunately, the automated technique caused systematic errors to the

potential reconstruction. The error between the first and second section is propagated to

the next segment and its error is also added. Typically the errors were small but the

effect was amplified as the offsets for the other sections were computed. Also, the errors

changed for different reconstruction parameters (total number of points in a section, the

bin size, or the total number of sections) causing the potential curve to swing by 1×10-19 J

or 25 kBT. In contrast, calculating the force from the energy sections makes automation

easy and accurate, costing only precision, since taking the derivative increases the noise.

To achieve high accuracy, it is important to properly weight the force components

during the averaging process. Cantilever deflection values that are infrequently sampled

during the binning of the data into histograms have more error because of the finite time

of collection. The error is clearly seen in the barrier between the two potential wells in

79

the left side of Figure 4.3c. Willemsen performed a careful analysis of the weighting

factor for the force components of the segments and found that the weighting factor for

each tip-sample distance value of the wave should be the probability of the tip residing at

that distance.22 All the probability information is contained in the histogram so

multiplying the force section by the histogram gives the correct weighted force value at

each tip-sample distance.

4.5 Instrument Noise Compensation for Brownian Force Profile Reconstruction

Brownian Force Profile Reconstruction requires precise knowledge of the

cantilever position to produce accurate probability distributions. The degradation of the

deflection signal by instrument noise needs to be compensated to restore the correct

potential.

The power spectra of a 17 N/m tip with (gray) and without (black) 65 fm/ Hz of

white instrument noise is shown in Figure 4.5. The instrument noise accounts for 40% of

the total noise power. The noise figures presented here are a little worse than the noise

values obtained on the modified AFM described in chapter 3. A force curve was

simulated using a force profile expected for oscillatory solvation forces with a maximum

200

150

100

50

0100806040200

Noi

se P

ower

(fm

/

)

Frequency (kHz)Figure 4.5 – Power spectrum of cantilever noise without instrument noise (black) and with instrument noise (gray).

80

-1.5

-1.0

-0.5

0.0

0.5

543210

Force Profile for Simulation Ordinary Force Curve Brownian Reconstruction Force Profile

with Noise Compensation

-1.5

-1.0

-0.5

0.0

0.5

Force Profile for Simulation Ordinary Force Curve Brownian Reconstruction Force Profile

without Noise Compensation

b

aFo

rce

(nN

)

Tip-Sample Distance (Å)Figure 4.6 – Brownian Force Profile Reconstruction with instrument noise. The Brownian reconstruction is shown in dark gray and the ordinary force profile in light gray. The ordinary curve more closely matches the force profile used in the simulation (black) when there is no noise compensation (a), but the Brownian force profile is more accurate after noise compensation (b).

attractive stiffness of 32 N/m. The cantilever parameters were k = 13 N/m, f0 = 25 kHz,

and Q = 3. The sampling rate was 200 kHz but the time step for the simulation was 500

ns. Instrument noise was generated using a random number picker from a gaussian

distribution and added to the deflection signal. The ordinary force profile (light gray) and

the Brownian Reconstruction without noise compensation (dark gray) are compared to

the real force profile in Figure 4.6a. The ordinary force profile experiences instability

and snaps over the last oscillation. The instrument noise causes the Brownian

reconstruction to be worse, and it is not qualitatively accurate.

81

The error is caused by disruption of two important steps in BFPR. The instrument

noise makes the deflection distribution seem broader so that the spring constant is

incorrectly estimated when the fit to the harmonic well at large tip-sample separations is

performed. This causes the measured deflection to produce a weaker force signal than

what is accurate. The noise also causes each section histogram to be wider such that the

stiffness of the interaction is miscalculated after the harmonic well is removed. A

method of restoring the original width of the distributions corrects these errors.

Deconvoluting the signal from the instrument noise greatly increases the accuracy

of Brownian Force Profile Reconstruction. Gaussian distributions follow the simple

relationship that the variance of a sum of distributions is the sum of the variances of the

individual distributions,

222nsm σσσ += (4.5)

The variance is the square of the standard deviation, σ. If the instrument noise standard

deviation is known, σn, then the true standard deviation of the cantilever, σs, is calculated

by assuming that the cantilever thermal noise is gaussian and using the measured

standard deviation, σm, in equation 4.5.

The Brownian Force Profile Reconstruction algorithm was modified to include

compensation of the instrument noise. The instrument noise, σn, is estimated by

measuring the deflection when the tip is in hard contact with the surface. The standard

deviation of the measured signal is dominated by instrument noise in the contact region

because the stiffness of the tip-sample interaction is high. The average value of the

deflection for each section is recorded and removed. Removing the average deflection

protects the force signal from being scaled along with the noise. The standard deviation,

82

σm, is calculated and then a multiplicative factor, C, scales the deflection. The

multiplicative factor,

m

nmC

σ

σσ 22 −= , (4.6)

is derived from equation 4.5. The absolute value ensures that the multiplicative factor is

real in the rare event of σn > σm. Once the deflection is scaled the offset is restored and

the algorithm continues as usual by computing a histogram of the scaled deflection data.

The Brownian Reconstruction resulting from using the noise-compensating algorithm is

shown in Figure 4.6b. The Brownian Reconstruction is a very accurate measure of the

real force profile even though the tip stiffness is only 13 N/m while the interaction (32

N/m) is much stiffer. The code for the whole BFPR algorithm is recorded in the

appendix (A.5).

The noise compensation performs well in spite of being based on the bad

assumption that the signal is a Gaussian distribution. The distribution of the signal is

non-gaussian when significant tip-sample interaction is mixed with the harmonic well of

the cantilever (left side of Figure 4.3b). The tip-sample interaction increases the overall

distribution width and causes the correction factor, C, to become unity. The magnitude

of the deflection between the two peaks of the histogram determines the absolute force

and therefore the curvature. This makes the compensation of the broadening of the

individual peaks inconsequential for interactions with large positive stiffness. For

interactions with large negative stiffness the distribution will be nearly gaussian but

narrow. The noise compensation adjusts the width to produce the appropriate stiffness.

Noise compensation works well for measuring very stiff interactions even though the

83

assumptions on which it is based are incorrect for distributions broadened by tip-sample

interaction.

The interaction force must be tightly coupled to the cantilever for Brownian Force

Profile Reconstruction to be effective. Many AFM experiments, especially those that

pull biological molecules, use long linkers and non-specific interactions to attach the

molecules of interest to the tip. The long linkers are weak springs that stretch

significantly when pulled. Since force is conserved, measuring the deflection of the

cantilever still provides an accurate measure of the force experienced at the end of the

weak linker. On the other hand, BFPR measures the force profile by mixing the potential

of the cantilever with the potential of the tip-sample interaction. The position of the tip

must actually enter and leave the potential energy landscape of the interaction. A weak

linker will isolate the cantilever potential from the tip-surface interaction and remove the

information content from the cantilever motion. The forces that the chemical or

biological interactions exert must be strongly coupled to the cantilever.

Brownian Force Profile Reconstruction is a very powerful tool for accurately

measuring the force profile. Ordinary force curves are inaccurate because the thermal

noise from strongly attractive and weakly attractive regions is averaged together and

information is lost. BFPR harnesses the distribution of the thermal noise to provide

information about the shape of the force profile. Lastly, BFPR is robust when instrument

noise is abundant because the broadening caused by instrument noise can be

compensated. Brownian Force Profile Reconstruction is a nearly ideal method for

investigating the mechanism and character of adhesive forces and mapping the energy

landscapes of stiff interactions.

84

4.6 Data Collection and analysis for Brownian Force Profile Reconstruction

Data collection for Brownian Force Profile Reconstruction requires oversampling

the deflection signal. Deflection time courses were recorded using a National

Instruments 5911 ADC board and Labview. The 5911 is capable of 100 MS/s at 8 bit

resolution. Higher position resolution is achieved by use of oversampling and using

digital filters to produce more bit resolution at the cost of time resolution. At 100 kS/s

the 5911 has 20 bit resolution. The Labview code using the VIs provided with the board

is recorded in the appendix (A.2). Sampling rates of 100 and 200 kS/s were commonly

used for data collection since FESP tips (1-5 N/m) have a resonant frequency of 25-30

kHz in solution.

The instrument was assembled after thoroughly cleaning the liquid cell tip holder

and O-ring. Solutions were injected directly from a glass syringe because the plastic

tubing was found to cause contamination. The whole system was allowed to equilibrate

for an hour. After equilibration, the laser power was increased for higher sensitivity

(chapter 3) and to heat the AFM head. The heating produced a constant drift of the tip

toward the surface at 2-10 Å/s. Force curves were collected by recording the deflection

time course as the surface drifted toward the tip. Force Calibration mode was used to

move the surface away from the tip after contact. Each data file was limited to a

maximum of 8MB or ~ 40 s of data. Gaps of 4 s existed between files because the 5911

cannot “data log”.

During analysis, the photodiode drift, caused by slow expansion of the AFm parts,

was initially removed by calculating the drift in V/s from the regions of negligible tip-

sample interaction. Data with irregular or non-linear photodiode drift were not further

85

analyzed. Next, the rate of surface drift (Å/s) was calculated by measuring the time

between points of equal deflection in the contact region and dividing it into the step size.

Time courses with irregular or non-linear contact regions were not further analyzed. The

detector sensitivity calculated from the contact region of the time course was compared to

the sensitivity calculated using normal force curves at the end of the experiment for

accuracy. Next, a section of the contact region was used as the measure of pure

instrument noise for compensation in the Brownian Force Profile Reconstruction

algorithm. The linear surface drift was removed. The BFPR algorithm calculates the

spring constant of the cantilever in Step 4 outlined above by fitting a harmonic well to the

scaled logarithm of the probability distribution. The standard deviation of the pure

instrument noise reference was adjusted such that the spring constant calculated using

BFPR matched the value calculated by fitting the power spectrum (appendix A.3).

Finally, the time course was parsed into individual force curves and the Brownian Force

Profile Reconstruction performed.

Brownian Force Profile Reconstruction is an involved technique. A large

proportion of the data is unacceptable for analysis because of non-linear drift of the

surface. A common procedure included recording normal force curves with a 1 Hz

repetition rate for security.

4.7 Octa-methyl-cyclotetrasiloxane (OMCTS)

Octa-methyl-cyclotetrasiloxane (OMCTS) is a large spherical silicone oil. Its size

and inert properties make it ideal for studying a “Leonard-Jones” fluid. OMCTS has

been the object of numerous SFA studies of liquid solidification under confinement. This

phenomenon is especially applicable to the fields of tribology and lubrication.

86

Unfortunately the SFA literature is not in agreement on the properties of this ideal liquid.

Klein’s studies of the onset of shear force with the SFA suggest solidification at 7

molecular layers23-25 and Granick’s work measuring the elastic and dissipative

components of the shear force suggests OMCTS behaves as a glass under confinement.26

The AFM is ideal for probing solvent ordering. The small probe size is more

similar to the asperities that protrude from polished surfaces than the atomically flat mica

sheets of the SFA. Previous experimental efforts to measure OMCTS ordering with the

AFM required AC techniques to measure the stiffness since the noise was too high for

direct force measurements.7-9 Clear periodicity was observed and an estimate of the

stiffness at each layer could be inferred.7 Unfortunately, the noise was not low enough

for additional conclusions. Probing OMCTS ordering using a sensitive AFM is necessary

to understand the ordering and lubrication properties of this liquid for tribological

applications.

OMCTS force profiles were measured with the ultra-low noise instrument

described in chapter 3 using ordinary force curve techniques. A methyl terminated SAM

was prepared on a silicon FESP AFM tip. The tip apex was blunted by applying high

loads to a silicon wafer before coating to promote adhesion of the gold layer. A piece of

freshly cleaved HOPG (graphite) was used as the surface. OMCTS (99% pure, Fluka)

was used as received. Hydrophobic surfaces were used to avoid water contamination.7-

9,23,25,26 One hundred twenty ordinary force curves were collected at a 1 Hz repetition

rate. The curves were shifted to overlap in both the tip-sample distance and force axes to

compensate for photodiode and Z-piezo drift. The ordinary force curves are accurate

87

-81086420

Fit Average of Data-4

0

4

8N

orm

aliz

ed F

orce

(mN

/m)

Tip-Sample Distance (nm)Figure 4.7 – Force profile of Octa-methyl-cyclotetrasiloxane. The model (gray) fits the data (black) very well revealing that the OMCTS is liquid down to a few layers. The inset reveals the last layer is excluded near 15 mN/m.

because the interaction stiffness (1 N/m) is much weaker than the spring constant (2.2

N/m).

The curves were subsequently averaged and fit with an exponentially decaying

sine curve with two exponential terms as the long-range attraction and repulsive contact.

The model fits the data extremely well (Figure 4.7), revealing a period of oscillation of

9.0±0.1 Å . The force data was normalized using the tip radius of 50 nm, obtained by

TEM, for direct comparison with previous SFA data. The force data in Figure 4.7 is

attractive at long-range. The previous AFM experiments also show long-range

attraction8,9 but the SFA data tends to be repulsive. A possible explanation for the

difference between the SFA and AFM trends is that the AFM measures the true force

profile but the SFA has shear forces which couple into the normal component as

confinement reduces the ability of the solvent to escape. Fitting the force profile with a

curve that has an inverse square term modeling the attractive region (like van der Waals

forces) did not match as well. The mechanism of the attractive forces is not van der

88

Waals, and it is not clearly understood. It may result from long-range ordering of the

fluid near the surface, which confines the entropic freedom. Upon expulsion from the

surface it gains the entropic freedom which lowers the energy of the whole system.

Conversely, pulling the tip away comes at an energy cost to reorder the molecules.

The inset in Figure 4.7 reveals that the last layer is removed near 15 mN/m. The

AFM does not have the ability to measure absolute distance between the tip and surface,

like the SFA, but the absolute scale for the AFM can be obtained by comparing the

results with the previous SFA work. The peak to peak magnitude of the oscillations from

1 nm to 4 nm for the present AFM data is similar to values obtained by SFA for the last

few layers.26 The final oscillation in the AFM data requires significantly more force to

remove the solvent. This is characteristic of the last layer since the packing of the

molecules would no longer transfer normal force into lateral force through the closest

packed structure.25 The removal of the last layer would be initiated by an asperity in the

tip that causes slight lateral force. This agrees with the observation that during retrace the

tip jumps out at that same region with only one mN/m of hysteresis. Molecules are still

under a portion of the tip causing strong repulsive forces. If the tip shape was perfectly

spherical, then a “vacuum” would be expected during retrace, leading to very strong

adhesion.

Most importantly, the excellent fit by the model shows that OMCTS is still fluid

when probed with a tip of small dimensions. The sinusoidal shape matches the lattice

model of fluid confinement where the fluid has freedom to move from its lattice positions

and the tip measures the density of the fluid. If the OMCTS was not fluid then more

abrupt transitions to attractive forces would be observed and the force profile would be

89

-8

-6

-4

-2

0

2

RecedingAdvancing

Nor

mal

ized

For

ce (m

N/m

)

108642Tip-Sample Distance (nm)

Figure 4.8 – Force profile of Octa-methyl-cyclotetrasiloxane. The advancing (gray) and receding (black) traces overlap perfectly. The absence of hysteresis is a result of the low viscosity associated with a liquid and not a glass.

asymmetric. Further confirmation of the fluid character of OMCTS in the advancing and

receding traces is shown in Figure 4.8. The advancing curve is one of the many used in

the average to produce Figure 4.7. There is very little hysteresis between the curves,

which is especially important at the turn around point in the positive stiffness region

between layers at 1 nm of tip-sample distance. These results confirm the tip follows the

reversible equilibrium trajectory and not a visco-elastic response expected for

solidification.4 These results are also very consistent with Steve Granick’s work

measuring the glass transition. At a couple molecular layers he calculates that a kBT of

lateral elastic energy is contained in over 60 molecules so the glass transition requires the

cooperation of many molecules.26 The AFM tip has only ~20 molecules underneath it

which is not enough to solidify or even substantially increase the lateral viscosity. In

conclusion, for a probe the size most applicable to industrial applications of interest,

OMCTS remains a fluid down to 1 layer and the AFM measures the pressure changes

associated with molecular layering near the surface.

90

4.8 1-Nonanol

The behavior of long chain alkanes near flat surfaces is of great interest to the

field of petroleum lubrication. Alkane chains order side by side in crystalline solids but

the extent of twisting and knotting resulting from the entropy increase upon melting is

unclear and the effects of confinement on the alkane dynamics are also unknown. Early

SFA studies show that long chain alkanes lay down on the mica surface such that the

periodicity is ~5 Å (the width of the molecules) and the number of layers is correlated

with the chain length of the molecule.27 An AFM experiment using n-docecanol revealed

layering with a periodicity slightly under 4 Å.9 Like the previous OMCTS experiment,

the early work is exceptionally noisy and hard to interpret. The experiments revealed that

ordering could be measured but did little to understand the surface dynamics of alkane

ordering.

Alkane ordering between hydrophobic surfaces was probed using Brownian Force

Profile Reconstruction. The alkane used was 1-nonanol and the hydrophobic surfaces

consisted of HOPG (graphite) and a methyl-terminated SAM on the tip. The nonanol

was obtained from Aldrich (98% pure) and used without further purification. The surface

was allowed to drift toward the tip at a rate of 2.0 Å/s and the deflection was sampled at

100 kHz at 16 bit resolution. The spring constant (3.0 N/m) was independently calibrated

from fitting the thermal spectrum (appendix A.3). The spring constant, detector

sensitivity, surface drift, and measure of instrument noise were compared for self-

consistency as described above. A TEM was used to estimate the tip radius of 10 nm

resulting from a single gold grain. The overall time trace was cut into force curves that

were typically 2×106 points. The Brownian force profile reconstruction further divided

91

-40

-20

0

20

40

4.03.53.02.52.01.5

Average of Data Fit

Forc

e (m

N/m

)

Tip-Sample Distance (nm)

Figure 4.9 – Average of reconstructed force profiles (gray) for 1-nonanol between hydrophobic surfaces and exponentially decaying sine function (black). The force profile has a period of 4.5 Å.

the force curves into 200 sections of 1.5×104 points each. The histograms bin size was

0.05 Å. After reconstructing the brownian force profile for each curve they were

averaged together and fit with an exponentially decaying sine function plus an attractive

exponential. The average force profile and the fit are depicted in Figure 4.9. The

stiffness in the region between 1.5 and 1.8 nm is 5.4 N/m, which is 1.8 times greater than

the stiffness of the cantilever. The interaction stiffness for the region less than 1.3 nm is

too high such that BFPR could not measure the whole force profile with this spring. The

forces were very strong even for the small probe surface, which lead to enormous

normalized forces on the order of 100 mN/m. These forces are about an order of

magnitude larger than comparable SFA measurements but similar in magnitude to

previous AFM work. The origin of the discrepancy is not known.

Although the forces were very strong, BFPR measured the whole force profile for

distances greater than 1.4 nm and it was well fit by a decaying exponential, which implies

that the solvent is liquid in this region. The period of oscillation is 4.5±0.2 Å and the

92

error is due to uncertainty in Z-piezo drift during data collection. Simulations using a

nonbonded potential energy minimum of 4.38 Å lead to a spacing of 4.5 Å in the liquid

state in perfect agreement with the measured data.28 The value is also in excellent

agreement with the molecular width of 4.5 Å calculated from the density (0.828g/ml),

molecular weight (144 g/mol), and the length to width ratio (Length/Diameter = 3.18) of

the all trans molecule computed using Chemdraw.

The region greater than 1.4 nm is sinusoidal but at shorter distances deviations

from a sinusoidal force profile suggest the onset of non-liquid behavior. The Brownian

force sections (equivalent of Figure 4.3f) are displayed for 2 individual force profiles in

Figure 4.10. The force profiles contain discontinuities because the interaction stiffness is

too great and the cantilever snaps through to the next layer. The interlayer transition near

300200

1000

-100200

100

0

-1004.03.53.02.52.01.51.00.5

b

a

Forc

e (m

N/m

)

Tip-Sample Distance (nm)

Figure 4.10 – Brownian reconstruction force sections for two different force curves a and b of 1-nonanol between hydrophilic surfaces. The force profiles show liquid behavior at distances greater than 1.5 nm but crystalline behavior with phase transitions at distances less than 1.5 nm.

93

1.2 nm is only 3.9 Å. The transition is much smaller than measured for the fluid layers of

1-nonanol at distances greater than 1.5 nm. The smaller interlayer distance can be

interpreted as solidification of the 1-nonanol into a hexagonal structure, which would

have an interlayer spacing of 3.9 Å for cylinders of radius 4.5 Å. The transition was

likely a transition from one layer to the next of the closest packed solid. The next

transition in Figure 4.10a is 1.5 Å followed by a 2.6 Å step while Figure 4.10b has a 1.9

Å step followed by a 2.2 Å step. The sum of the two steps in both curves is 4.1 Å,

another molecular layer. A possible interpretation of the smaller step is that the solid

changes to a new phase or crystal surface. Rotating the hexagonal lattice by 90 degrees

such that the rows of atoms are perpendicular to the surface and the atoms touching the

surface are staggered every other layer allows transitions of ~2 Å. The reorganization

does cause some “vacuum” near the surface, which would only be advantageous for a

solid confined to a space of only a few molecular layers. An estimate of the exact

number of layers between the surfaces is challenging since AFM can’t measure absolute

distance but the last 1.5 Å jump at 0.4 nm in Figure 4.10a suggests that at least 1.5 layers

remains. The transition between 1.0 and 1.4 nm should be an integer multiple of layers

where the layers are in the energetically most favorable position, parallel with the

surface. The transitions between 0.6 and 1.0 nm make another layer. Since the transition

from 0.4 to 0.6 in Figure 4.10a is a half layer, another half layer would be expected

before the last full molecular layer. The onset of solid behavior is at a minimum of 4

molecular layers.

The measurement of the interlayer transitions of 1-nonanol between hydrophobic

surfaces suggest that 1-nonanol is a freely moving liquid with at least 5 molecular layers

94

laying flat against the surface and further confinement crystallizes the nonanol into a

hexagonal lattice. The crystal also reorders such that half layers can be incrementally

excluded.

4.9 Water

Water is undoubtedly the most important molecule in science. Its special

properties are what make the earth the cradle of life in the universe.29,30 The structure of

water near surfaces is of particular importance to biology and interfacial science because

biological processes are performed in aqueous solution and surfaces exposed to air

contain a 2-3 nm water layer from the ambient humidity. Understanding the behavior of

water near surfaces is extremely important for applications such as binding, cell adhesion,

interfacial friction, and lubrication.

Previous attempts to understand interfacial water behavior have used a variety of

techniques to measured water structure near flat surfaces. The first surface forces

apparatus experiments in buffer between mica revealed a large oscillatory component

superimposed on the electrostatic repulsion. The oscillations were between 2.5 and 3.0 Å

and extended up to 8 molecular layers.31,32 Oscillatory behavior has also been observed

using Boltzmann’s equation and the thermal noise of the AFM cantilever near crystalline

salt surfaces. This early precursor to Brownian Force Profile Reconstruction measured

water layers between 1.5 and 3.0 Å.20 Six waters layers spaced by 2.2 Å were also

measured by AFM using a large multiwall nanotube tip against a carboxyl terminated

SAM surface. The AC techniques used measured the dissipation between the tip and

sample and did not provide a direct measure of the force profile.10 Shear force

microscopy was also used with a large (R ~ 50 nm) flat glass probe to observe mean

95

periodicity of 2.54 Å and gradual stepwise increases in both the elastic and viscous forces

upon confinement.33 The gradual increase in the elastic and viscous forces is similar to

the glasslike transition observed for OMCTS by Granick.26 Methods that do not confine

the water between two surfaces have also observed orientation and layering of the

solvent. X-ray reflectivity experiments of hydrated mica reveal the first hydration layer

is 2.5 Å from the surface while some of the hydrating molecules occupy a site 1.3 Å from

the surface. The next layer is 2.7 Å away with the following layers being spaced by 3.7

Å. Vibrational sum frequency spectroscopy suggests strong orientation of the water

molecules at a tetrachloromethane/water interface.34 These results confirm the ordering

of the solvent without confinement. Extensive effort has been expended to understand

the behavior of water at interfaces but a great need still exists for higher precision data to

reveal subtleties of water structure and orientation at interfaces.

Brownian Force Profile Reconstruction was used to probe precisely the whole

force profile of water structure between hydroxyl terminated SAM surfaces. Atomically

flat gold surfaces were used so that instrument drift did not cause irreproducibility of the

force curves. The flat gold was prepared in an ion pumped load-locked UHV thermal

evaporator at 2×10-9 torr. The mica surfaces were annealed at 400 C overnight before

evaporation and for at least 2 hours after evaporation before slowly cooling. The

evaporation rate was 7 Å/s. The surfaces were placed in either hydroxyl terminated

alkane thiol solution immediately after removal from the evaporation chamber. Tips

were prepared by growing a thermal oxide in a tube furnace open to air at 900 C

overnight. A thermal evaporator was used to apply a chromium and gold layer of 70 and

400 Å respectively at a rate of 1 Å/s. Tips were placed in hydroxyl alkane thiol solution

96

immediately after removal from the evaporation chamber. SAMs were formed for at

least 1 hour before rinsing with ethanol and blowing dry with clean nitrogen. The

solution was prepared from Burdick and Jackson ultra high purity water and Fisher

Scientific salts, passed through a 0.2 mm filter under vacuum to remove particulates and

gas, and stored in a refrigerator until used.

Force curves were collected by sampling the deflection at 100 kHz while scanning

the surface 5 nm at a repetition rate of 0.1 Hz, which produced a surface velocity of 1

nm/s. Each force curve contained ~4×105 usable data points for performing BFPR.

Although not advantageous for BFPR experiments, the effective cantilever temperature

was 180 C as a result of an artificially lowered Q value (chapter 5) used for low noise

tapping mode force curves collected previously (data not shown). The force curves were

split into 100 sections but the number of points per section depended on the region of

analysis. In the contract region the average deflection is changing rapidly, which can

broaden the histogram producing an artificially soft contact region thus ~500 points were

analyzed per section. The rest of the curves used ~5,000 points per section. A large bin

size of 0.2 Å reduced the noise resulting from small sample size during the binning

process. A broad deflection range was used for the histograms to gather all the

information in the steep attractive regions and consequentially many of the force sections

required manual deletion of singularities resulting from histogram values of zero.

The Brownian Reconstruction Force Profiles for water confined between two

hydroxyl SAM surfaces is shown in Figure 4.11. The force sections that comprise the

brownian reconstruction of a single advancing trace are shown in part a. The average of

the force sections is displayed in black in part b and for comparison the force profile

97

-400

-300

-200

-100

0

100

2.01.51.00.50.0

-400

-200

0

200

Ordinary Force Profile Brownian Reconstruction Force Profile

600

400

200

0

-200

-400

3.71 Å

H

HH

O

OH

HO

O

c

b

a

Forc

e (p

N)

Tip-Sample Distance (nm)

Figure 4.11 – Force profile of water ordering against hydroxyl terminated SAM surfaces. (a) Force sections for Brownian Force Profile Reconstruction. (b) Brownian Reconstruction Force Profile from force sections (black) and ordinary force curve from the same data (gray). The ordinary force curve significantly misses the shape of the profile. (c) Average of many Brownian Force Profiles (gray) and an oscillatory fit (black). The oscillations have a period of 3.6 Å.

98

calculated using ordinary force profile techniques is shown in gray. The average of 16

brownian force profiles is shown in part c. The interaction is attractive with three

oscillations in force at long range. The stiffness in the attractive region is 4.2 N/m,

causing the cantilever to snap through the attractive interaction (part b) because the

cantilever spring constant is only 1.5 N/m. The average curve was fit with an

exponentially decaying sine function with two extra exponential terms for the long-range

attractive force and the repulsive contact region.

The fit matches very well and produced a period of 3.6±0.2 Å for the oscillations

in force due to solvent structure. The period is longer than the equilibrium length

between water molecules of 2.78 Å. The closest spacing between planes of oxygen

atoms is 2.27 Å, which is in good agreement with the previous measurements between

mica and salt surfaces.10,20,31-33 The spacing between two layers of water such that some

of the oxygen-hydrogen bonds are perpendicular to the water planes (inset Figure 4.11c)

is 3.71 Å, in very good agreement with the results between hydoxyl terminated SAM

surfaces. This result also implies that the solvent layers are specifically oriented when

solvating the SAM surface. Both surfaces are identical which requires identical solvation

characteristics. Solvation with hydrogens perpendicular to both surfaces requires

removing two layers at the same time as the surfaces come together. The underlying gold

lattice for the hydroxyl terminated SAM defines the unit cell with dimensions 4.99 × 8.66

Å. The interface between the SAM surface and the water structure has a lattice mismatch

because the face of the water unit cell has dimensions of 4.53 × 7.86 Å. The lattice

mismatch and the resulting poor solvation of the surface could be the origin of the strong

99

attractive forces as the last solvent layers are removed from the surfaces and the

commensurate SAM surfaces touch.

The decay of the sinusoidal force component reveals that solvent orientation

information is lost within 1 nm from the surface. Experiments with the SFA revealed

solvent structure to much larger length scales but the surfaces in the SFA confine the

fluid too strongly such that it cannot escape, which artificially induces structure. The

small correlation length in these experiments caps the length scale of solvent orientation

as a mechanism of long-range adhesion. The correlation length of 6 molecular layers or 3

molecular layers for each surface agrees well with other measurements of the structural

correlation between water molecules in clusters.15 In chapter 2 the mechanism for the

very long-range adhesion (3 nm) for carboxyl terminated SAM surfaces in low pH buffer

was unknown. A suggested mechanism was strong orientation of the solvent, which

upon displacement by the tip gains entropic freedom lowering the free energy and

making the interaction attractive. The decay of the solvent structure near the surface

reveals that water 1 nm from a hydroxyl surface is the same as bulk water causing no

attractive forces. It is possible that the solvent is more strongly oriented by the carboxyl

terminated surface because there are more hydrogen bonding sites to confine the water.

This would lead to a longer-range interaction but many experiments were performed with

carboxyl terminated SAM surfaces and no solvent ordering was observed. Further

experimentation is required to understand the origin of the very long-range adhesion for

carboxyl terminated surfaces.

The observation of water structure between hydroxyl terminated SAM surfaces

suggests that the water is strongly oriented with bonds pointing perpendicular to the

100

surface. Two layers of water must be removed together to keep the surfaces properly

solvated. The correlation length of the solvent structure is only 0.5 nm limiting the

length scale of solvent orientation as a mechanism of long-range attractive forces.

4.10 Conclusion

The role of solvent is extremely important in physics, chemistry, and biology.

Since solvent is composed of molecules of finite size with their own chemical physical

properties the solvent can substantially affect analyte interactions. In this chapter, the

structure and properties of solvent near flat surfaces have been investigated using force

profiles. A model of fluid solvent near a flat surface, which uses progressively broadened

gaussians to describe the density distribution of each solvent layer predicts an

exponentially decaying sinusoidal force distribution. These oscillating forces can have

very stiff attractive interactions and using ordinary force curves with spring constants

near the interaction stiffness causes inaccuracies resulting from the averaging of the

thermal noise. Brownian Force Profile Reconstruction was developed to harness the

thermal noise as a probe of the steep interactions leading to much more accurate force

profile measurement. Also, techniques for compensating the instrument noise were

developed which makes the accurate measurement of extremely stiff force profiles

possible.

The structure of OMCTS, 1-nonanol, and water near well-ordered, smooth

surfaces was investigated using force profiles. OMCTS is a large waxy spherical

molecule that is has been thoroughly investigated with the surface forces apparatus. The

force profile for OMCTS between hydrophobic surfaces is an exponentially decaying

sine function that is reversible. These properties show that OMCTS is a fluid under

101

confinement down to a single molecular layer settling the dispute within the SFA

community.

Brownian Force Profiles were collected for 1-nonanol between hydrophobic

surfaces. The long chain alcohol laid down on it side and showed fluid behavior when

more than 4 molecular layers were confined but exhibited solid-like behavior at smaller

distances. The crystal also rearranged to allow half molecular diameter steps.

The structure and orientation of water between hydroxyl terminated SAM

surfaces was elucidated using Brownian Force Profile Reconstruction. The water was

strongly attractive and showed 3 oscillations in the attractive force with a period of 3.6 Å.

The period is larger than the spacing between the closest planes of oxygen atom in ice but

closely matches the spacing between pairs of oxygen atom planes that have oxygen-

hydrogen bonds perpendicular to the surface. Pairs of water molecule planes are

removed simultaneously to preserve similar solvation of both hydroxyl-terminated SAM

surfaces. The decay of the solvation forces revealed that molecular correlation does not

extend beyond three layers. The smaller correlation distance places an upper limit of the

length scale of attractive forces resulting from orienting the solvent molecules near the

surface.

Measuring force profiles is the best method to directly probe intermolecular and

interfacial interactions. Brownian Force Profile Reconstruction accurately probes stiff

interactions such as solvation and structural forces, which are important to all areas of

science.

102

4.11 References

1. Abraham, F. F. Interfacial Density Profile of a Lennard-Jones Fluid in Contact with a (100) Lennard-Jones Wall and Its Relationship to Idealized Fluid-Wall Systems - Monte-Carlo Simulation. Journal of Chemical Physics 68, 3713-3716 (1978).

2. Sorenson, J. M., Hura, G., Soper, A. K., Pertsemlidis, A. & Head-Gordon, T. Determining the role of hydration forces in protein folding. Journal of Physical Chemistry B 103, 5413-5426 (1999).

3. Horn, R. G. & Israelachvili, J. N. Direct Measurement of Structural Forces between 2 Surfaces in a Non-Polar Liquid. Journal of Chemical Physics 75, 1400-1411 (1981).

4. Heuberger, M., Zäch, M. & Spencer, N. D. Density Fluctuations Under Confinement: When Is a Fluid Not a Fluid? Science 292, 905-908 (2001).

5. Israelachvili, J. Solvation Forces and Liquid Structure, as Probed by Direct Force Measurements. Accounts of Chemical Research 20, 415-421 (1987).

6. Isaelachvili, J. Intermolecular and Surface Forces (Academic Press, San Diego, 1992).

7. Han, W. & Lindsay, S. M. Probing molecular ordering at a liquid-solid interface with a magnetically oscillated atomic force microscope. Applied Physics Letters 72, 1656-1658 (1998).

8. O'Shea, S. J., Welland, M. E. & Pethica, J. B. Atomic Force microscopy of local compliance at solid-liquid interfaces. Chemical Physics Letters 223, 336-340 (1994).

9. O'Shea, S. J. & Welland, M. E. Atomic Force Microscopy at Solid-Liquid Interfaces. Langmuir 14, 4186-4197 (1998).

10. Jarvis, S. P., Uchihashi, T., Ishida, T., Tokumoto, H. & Nakayama, Y. Local Solvation Shell Measurement in Water Using a Carbon Nanotube Probe. Journal of Physical Chemistry B 104, 6091-6094 (2000).

11. Franz, V. & Butt, H. J. Confined liquids: Solvation forces in liquid alcohols between solid surfaces. Journal of Physical Chemistry B 106, 1703-1708 (2002).

12. Donnelly, S. E. et al. Ordering in a fluid inert gas confined by flat surfaces. Science 296, 507-510 (2002).

13. Cheng, L., Fenter, P., Nagy, K. L., Schlegel, M. L. & Sturchio, N. C. Molecular-scale density oscillations in water adjacent to a mica surface. Physical Review Letters 87, 156103-1-156106-4 (2001).

103

14. Magnussen, O. M. et al. X-Ray Reflectivity Measurements of Surface Layering in Liquid Mercury. Physical Review Letters 74, 4444-4447 (1995).

15. Hansen, J. P. & McDonald, I. R. Theory of Simple Liquids (Academic Press, San Diego, 1986).

16. Ho, R., Yuan, J.-Y. & Shao, Z. Hydration Force in the Atomic Force Microscope: A Computational Study. Biophysical Journal 75, 1076-1083 (1998).

17. Gelb, L. D. & Lyndenbell, R. M. Effects of Atomic-Force-Microscope Tip Characteristics on Measurement of Solvation-Force Oscillations. Physical Review B 49, 2058-2066 (1994).

18. Gao, J. P., Luedtke, W. D. & Landman, U. Origins of solvation forces in confined films. Journal of Physical Chemistry B 101, 4013-4023 (1997).

19. Gao, J. P., Luedtke, W. D. & Landman, U. Layering transitions and dynamics of confined liquid films. Physical Review Letters 79, 705-708 (1997).

20. Cleveland, J. P., Schaffer, T. E. & Hansma, P. K. Probing oscillatory hydration potentials using thermal-mechanical noise in an atomic-force microscope. Physical Review B 52, R8692-R8695 (1995).

21. Heinz, W., Antonik, M. D. & Hoh, J. H. Reconstructing Local Interaction Potentials from Perturbations to the Thermally Driven Motion of an Atomic Force Microscope Cantilever. Journal of Physical Chemistry B 104, 622-626 (2000).

22. Willemsen, O. H., Kuipers, L., Werf, K. O. v. d., Grooth, B. G. d. & Greve, J. Reconstruction of the Tip-Surface Interaction Potential by Analysis of the Brownian Motion of an Atomic Force Microscope Tip. Langmuir 16, 4339-4347 (2000).

23. Klein, J. & Kumacheva, E. Confinement-Induced Phase-Transitions in Simple Liquids. Science 269, 816-819 (1995).

24. Klein, J. & Kumacheva, E. Simple liquids confined to molecularly thin layers. I. Confinement-induced liquid-to-solid phase transitions. Journal of Chemical Physics 108, 6996-7009 (1998).

25. Kumacheva, E. & Klein, J. Simple liquids confined to molecularly thin layers. II. Shear and frictional behavior of solidified films. Journal of Chemical Physics 108, 7010-7022 (1998).

26. Demirel, A. L. & Granick, S. Glasslike transition of a confined simple fluid. Physical Review Letters 77, 2261-2264 (1996).

104

105

27. Christenson, H. K., Gruen, D. W. R., Horn, R. G. & Israelachvili, J. N. Structuring in liquid alkanes between solid surfaces: Force measurements and mean-field theory. Journal of Chemical Physics 87, 1834-1841 (1987).

28. Jin, R. Y., Song, K. Y. & Hase, W. L. Molecular dynamics simulations of the structures of alkane/hydroxylated alpha-Al2O3(0001) interfaces. Journal of Physical Chemistry B 104, 2692-2701 (2000).

29. Eisenberg, D. S. & Kauzmann, W. The structure and properties of Water (Oxford University Press, New York, 1969).

30. Denny, M. W. Air and Water: the biology and physics of life's media (Princeton University Press, Princeton, NJ, 1993).

31. McGuiggan, P. M. & Pashley, R. M. Molecular Layering in Thin Aqueous Films. Journal of Physical Chemistry 92, 1235-1239 (1988).

32. Israelachvili, J. N. & Pashley, R. M. Molecular Layering of Water at Surfaces and Origin of Repulsive Hydration Forces. Nature 306, 249-250 (1983).

33. Antognozzi, M., Humphris, A. D. L. & Miles, M. J. Observation of molecular layering in a confined water film and study of the layers viscoelastic properties. Applied Physics Letters 78, 300-302 (2001).

34. Scatena, L. F., Brown, M. G. & Richmond, G. L. Water at hydrophobic surfaces: Weak hydrogen bonding and strong orientation effects. Science 292, 908-912 (2001).

Chapter 5 Q-control for Optimizing AFM

5.1 Introduction

The Atomic Force Microscope (AFM) is a powerful tool for investigating the

nanoscale world. In principle, the AFM can image any substance, including individual

molecules, since the sample does not need to be conducting. The versatility of AFM

makes it especially useful for imaging biological samples where physiologic conditions

require high ionic strength buffer at ~300 K. Unfortunately, biological samples are soft

and adhere poorly to the surface, which requires the AFM to sense the surface without

applying too much force to the sample.

Since the first AFMs using contact mode, a couple significant developments

reduced the interaction force during imaging for Atomic Force Microscopy. In contact

mode the tip is dragged along the surface and significant lateral forces are produced

which can easily dislodge or tear the sample. Tapping mode was invented in 1994 to

reduce the lateral forces by oscillating the tip near its resonant frequency and allowed it

tap on the surface.1 The intermittent contact alleviates some of the lateral force on the

sample since most of the lateral movement occurs when the tip is not in contact with the

sample. However, imaging forces during tapping mode can still be quite significant in

solution, dislodging the sample and causing deformation, since the force applied to the

106

Figure 5.1 – Average tapping force for two different Q values as a function of amplitude setpoint, S = A/A0, where A and A0 are the tapping amplitude with and without tip-sample interaction respectively.. The circles and triangles are a Q of 350 and ~2 respectively.

sample is inversely proportional to the quality factor,* Q and the Q is reduced in

solution.2 Recently, Q-control was developed, promising to overcome this difficulty of

low-Q imaging in solution and produce high sensitivity images by gently imaging the

surface.

Q-control uses a feedback mechanism to increase the Q electronically and

decrease the impact force.3,4 The average force between the cantilever and surface as a

function of amplitude setpoint is shown in Figure 5.1 for a Q of 2 and 340.4 The average

force is two orders of magnitude lower for the higher Q measurement at the same tapping

amplitude. Also, the transition from attractive to repulsive imaging is a function of Q

such that large phase contrast for small changes in the attractive potential can be

produced as seen in the advertisement for a commercial Q-control system reproduced in

* The impact force during tapping as a function of quality factor is not an analytic expression but is a result of the nonlinear motion of the cantilever along the potential energy surface. The inverse dependence is a good first approximation.

107

Figure 5.2 – A mixed polymer sample imaged with and without Q control used as an advertisement for a commercial product. The region imaged with Q-control seems to show more sensitivity to surface features.

Figure 5.2.*3,5,6 Similarly, it was thought that increasing the Q would significantly

increase the sensitivity during Energy Dissipation imaging since the phase signal follows

the relationship, ( ) QEdis∝θsin . These early impressions produced great excitement and

optimism for increasing the sensitivity of imaging soft biological samples with Q-control

AFM.

In this chapter, the sensitivity of Q-control AFM is thoroughly analyzed. It was

found that Q-control feedback changes the effective temperature of the cantilever. The

increased temperature coupled with the narrow spectral width of the thermal noise at high

Q led to little increase in signal to noise. Also, the increased propensity for Z-piezo

* High Q values make non-sinusoidal motion of the cantilever more difficult. The non-harmonic potential energy surface during tip-sample interaction necessarily requires non-sinusoidal tapping motion. As a result, higher Q tapping tends to probe the interaction less deeply for the same tapping amplitude. Also, the shallow interaction of high Q tapping may only consist of attractive interactions. A very distinct transition occurs when the cantilever transitions from sampling the attraction portion of the potential to the repulsive portion. The shift from attractive to repulsive tapping is associated with an increase in amplitude and the shift of the phase from leading to lagging.

108

Surface

Cantilever SensorTransducer

VariableGain

Amplifier

π/2 Phase Shifter

Tapping Drive

Figure 5.3 – Q-control cantilever feedback block diagram. Cantilever deflection is shifted by π/2 and added to the tapping mode drive. The composite signal drives the cantilever motion through the transducer.

oscillations and a slower feedback mechanism makes using higher Q for more sensitive

imaging unfeasible.

5.2 Q-Control Theory

Q-control uses a cantilever feedback mechanism to actively change the damping

of the cantilever. A block diagram of the feedback mechanism is shown in Figure 5.3.

The feedback shifts the cantilever response by ±π/2 and adds it to the tapping drive signal

in a summing amplifier, which applies a force to the cantilever through a transducer. Q-

control is very similar to Magnetic Feedback Chemical Force Microscopy (MFCFM)

except a ±π/2 phase shift is used instead of π, therefore the resulting wave equation of

motion for the Q-control feedback is,

20

πω iti exGeFxkxbxm ⋅⋅+=⋅+⋅+⋅ &&& . (2.11), (5.1)

In the equation x , , and are the displacement of the cantilever from equilibrium and

its first and second time derivatives respectively, m is the effective mass, b is the

damping, k is the spring constant, ω is the angular frequency, F

x& x&&

0 is the thermal fluctuation

force (which is constant for all frequencies), and G is the loop gain. Using the Ansatz,

( ϕω −= tiAex ) , (5.2)

109

where A is the amplitude and φ is the phase shift of the cantilever relative to the driving

force, F0eiωt, the last term, G , can be approximated by Gθiex ⋅⋅ x&⋅ since eiπ/2=i and ω is

relatively constant over the frequency range of interest. The velocity terms can be joined

together to produce

( ) tii eFxkxGbxm ω

0=⋅+⋅−+⋅ &&& . (5.3)

As a result, the effective damping is gain dependant and follows,

( MbbGbGbb i

iiie −⋅=

−⋅=−= 11 ) . (5.4)

Substituting Q for b using 02 Qfkb π= , produces the expression for the effective Q,

( )MQQ i

e −=

1. (5.5)

The transfer function provides a more thorough understanding the gain dependant

changes in damping. The general feedback transfer function developed in chapter 2 is

( )( ) ( ) 2

02

2

2

2

02

0

22

20

2

2

20

2

sin2cos121fQ

fkG

fkQGf

ff

kG

ff

kF

fA

ii

+++

−−

−

=

θθ

. (2.12), (5.6)

The cantilever parameters were changed to k, f0, and Q using ω0=2πf0, 02 Qfkb π= , and

( )202 fkm π= . When θ = π/2, the transfer function simplifies to

( )2

0

2

20

2

2

20

2

1

++

−

=

kG

fQf

ff

kF

fA

i

, (5.7)

110

but the general equation, 5.6, can be helpful for understanding behavior when the phase

shifts are not exactly π/2 due to instrument imperfections. Reorganization of the

damping term of equation 5.7 leads to

2

0

0

1

+

kfGf

Qff

i

, (5.8)

revealing the effective Q,

kfGf

QQ ie

011+= , (5.9)

is a function frequency. The frequency dependence is negligible at high Q because the

damping term only dominates the value of the denominator at frequencies very close to

f0. Assuming that , the expression for Q simplifies to 0ff =

MQ

kGQ

QQ i

i

ie +

=+

=11

, (5.10)

with the substitution, kGQM i≡ . . At lower Q, the frequency dependence of the

dampening term changes the shape of the transfer function seemingly changing k and f0

also.

5.3 Feedback Hardware

Cantilever feedback for Q-control was implemented using magnetic forces.

Preparation of the magnetic cantilevers followed the MFCFM procedure given in chapter

two with a few changes. First, larger magnets, 30-50 µm, were glued on the back of the

cantilevers. Second, FESP cantilevers (2-5 N/m, Digital Instruments, Santa Barbara,

CA.) were used for their higher stiffness, low drift, and excellent laser spot. Third,

Norland UV cure optical adhesive (#63, Norland Products, New Brunswick, NJ) was

111

Summing Amplifier

AC Couple

Gain Control

π/2 Shifter Solenoid

Tapping Mode Drive

Cantilever Deflection

+-+

-

Figure 5.4 – Schematic of Q-control cantilever feedback system. The cantilever deflection is AC coupled, by a high pass filter, and amplified in a 20-turn variable gain amplifier before being phase shifted by a low pass filter. The shifted signal is summed at the power amplifier, which produces a magnetic field through the solenoid to deflect the cantilever.

used for the initial transfer. Forth, a second layer of heat cure epoxy (#377, Epoxy

Technology, Billerica, MA) was applied and cured at 150 C for at least 4 hours because

the first glue was not strong enough for vigorous oscillations in fluid. The heat cure

epoxy enveloped the whole magnet permanently attaching it and trapping contaminates

so that the experiments were cleaner.

A schematic of the feedback loop is shown in Figure 5.4. The cantilever

deflection was AC coupled using a high pass filter with 1-2 Hz cutoff to compensate

deflection drift. The resulting signal was amplified using a variable gain amplifier

(AD711) with a 20-turn resistor for high precision adjustment of the gain. A low pass

filter with 10 kHz cutoff provided π/2 phase shift at frequencies above the knee without

significant loss of gain. The phase shifted signal was joined with the tapping mode drive

using a high power op amp (PA01, Apex Microtechnologies, Pheonix, AZ) as a summing

amplifier. This high power op amp also drove the solenoid.

112

The tolerance for phase error is much greater for Q-control than for MFCFM.

Deviations from π/2 skew the resonance shape or cause changes in spring constant but

the system is less likely to become unstable during tip-sample interactions unless the Q

has been increased by a couple orders of magnitude. Phase shifting components such as

the power op amp and high inductance solenoid were used because the application was

non-critical. The resulting useable bandwidth for doing Q-control was 8-22 kHz.

5.4 Cantilever Heating

The Q-control feedback amplifies or cancels the cantilever thermal noise, which

changes the effective cantilever temperature. The noise power, <x2>, is related to the

spring constant and temperature of the cantilever by the equipartition theorem,

kTkx B=2 (5.11)

The noise power is computed by integrating the transfer function,

( ) 2

200

0

22

2kFQf

dffAx iπ== ∫∞

, (5.12)

and the thermal force noise, F0, as a function of cantilever parameters can be solved for

by joining equations 5.11 and 5.12.

i

B

QfTkkF

0

20

2π

= . (5.13)

Q-control applies extra forces to the cantilever and drives the system away from

equilibrium. When integrating the Q-control transfer function the gain term does not

cancel with the thermal force noise so that the total noise power is gain dependant,

113

( ) ( )∫∞

+=

++

−

=0

2

0

2

20

2

2

20

2

111Mk

Tkdf

fQMf

ff

kF

x B

i

. (5.14)

For Q-control the spring stiffness is constant so the noise power changes resulting from

the gain appear to be changes in effective cantilever temperature,

MTT i

e +=

1. (5.15)

The noise power spectra for different Q values of the same cantilever in water are shown

in Figure 5.5. The equipartition theorem no longer applies when the effective

temperature is changed because the feedback system is adding or removing energy from

the cantilever.

The change in the mechanical response and temperature make Q-control useful

Frequency (kHz)

10-25

10-24

10-23

10-22

2015105

Q=130 Q=6.4

Noi

se P

ower

(m2 /H

z)

Figure 5.5 – Noise Power spectra of a Q-controlled cantilever at five different Q values. The integrated noise power increases with Q because the effective temperature is changed by Q-control.

114

for many applications. Q-control was implemented using laser power7 and acoustic

pressure8 as the modulating force to decrease the response time such that the cantilever

more quickly followed force gradient changes. Conversely, Q-control was used to

increase the Q such that a phase locked loop could be used to more precisely follow mass

changes for a cantilever biosensor.9 A mirror was cooled using feedback to remove

thermal noise in an attempt to measure gravity waves.10 Also, the force sensitivity of

Magnetic Resonance Force Microscopy was enhanced down to attonewton levels by

using Q-control cooling.11 Lastly, it has also been suggested to use Q-control cooling to

clamp the cantilever for more precise measurement of force profiles.12

5.5 Lock-in Noise

The noise of the lock-in signals determines the sensitivity and signal to noise ratio

of tapping mode and Q-control AFM as mentioned briefly in chapter 3. One approach to

understanding lock-in noise is to graph the cantilever position as a vector with quadrature

and phase axes. The oscillating cantilever motion can be described by the function

x = Aei ωt−ϕ( ) (5.2) as depicted by the rotating vector in Figure 5.6a. A lock-in amplifier

essentially rotates the frame by the reference frequency, ωr. If the ω of the cantilever

motion is equal to ωr then the cantilever will appear to be still and the amplitude and

phase are easily interpreted graphically, as shown in Figure 5.6b.

Cantilever thermal noise are positional fluctuations that have little phase or

amplitude coherence, although their spectral gain characteristics are well defined by the

cantilever parameters k, f0, and Q. The lock-in reference frequency, ωr, is typically near

the resonance frequency of the cantilever, ω0, and the thermal motion of the cantilever

with frequency components greater than ωr rotate clockwise while the components with

115

quad

ratu

re

inphase

ba

φ A qu

adra

ture

inphase

Figure 5.6 – Cantilever position plotted on the quadrature phase plane with (a) ωr=0 and (b) ωr=ω. The cantilever sweeps a circle around the origin in a. The amplitude and phase are readily interpreted graphically by the steady cantilever position in b.

frequency components less than ωr rotate counter-clockwise. The thermal noise is

depicted as a diffuse spot at the origin of the plot in Figure 5.7a, and the circular

symmetry is a result of the many frequency components and the lack of phase coherence.

The size of the diffuse spot is dependent on the noise power in the measurement

bandwidth, 2x . The limited bandwidth noise power value is smaller than the value

used to calculate k, since the bandwidth used for calculating k is infinite.

During tapping, the position fluctuations are imposed on the end of the vector

representing the tapping signal, as shown in Figure 5.7b. The positional fluctuations are

measured as amplitude and phase noise, as depicted in Figure 5.7c. The amplitude noise

value,

2xNA = , (5.16)

is simply the limited bandwidth noise power value. The phase noise power,

116

, (5.17)

a

quad

ratu

re

Aφ

quad

ratu

re

b c

Nφ NA

quad

ratu

re

inphase inphase inphase

Figure 5.7 – (a) Thermal noise plotted on the quadrature phase plane and has circular symmetry. (b) Thermal noise of a cantilever with amplitude, A, and phase, φ. (c) Amplitude, NA, and phase, Nφ, noise resulting from cantilever thermal noise.

Ax

Ax

N22

1tan ≈

= −

θ

is dependant on the amplitude since increasing the amplitude decreases the angle for the

same noise power value. The angles are typically small for common working amplitudes

so removing the arctangent is a good approximation.

The spectral character of the noise is determined by the shape of the noise in the

sidebands of the reference frequency. The frequency of the noise components is the

absolute value of the difference between the original frequency and the reference

frequency. The sidebands above and below the reference frequency will be joined

together at zero frequency. For a simple harmonic oscillator this leads to a half-

lorentzian shape at zero frequency. The amplitude and phase noise for 4 different

amplitudes and 4 different Q values are displayed in Figure 5.8. The amplitude curves,

shown in panel a, overlap well for each Q value and all curves overlap in the instrument

noise region (f>1000 Hz) revealing the independence of amplitude noise on tapping

amplitude. The phase curves in panel b overlap in the instrument noise region only when

117

10 -5

2

4

6810 -4

2 Q =350 Q =130 Q =50 Q =18

10 -5

10 -4

10 -3

10 -2

200 0150010005000F requen cy (H z)

A = 0 .37 n m A = 1 .5 nm A = 5 .6 nm A = 1 6 .8 n m

b

a A

mpl

itude

Noi

se (V

/

) Ph

ase

Noi

se (V

/

)

Figure 5.8 – Amplitude (a) and phase (b) noise spectra for different amplitudes and Q values. (a) Dark curves are for lower Q values and lighter curves are for higher Q values. (b) Dark curves are for large amplitudes and light curves are for small amplitudes.

they have the same tapping amplitude and the curves with the same Q are simply offset

vertically by the tapping amplitude, A. The greater thermal noise from cantilever heating

is clearly observed for the curves with increased Q.

The amplitude and phase noise curves can be scaled to overlap by converting

from phase to noise power using equation 5.17. Scaled phase curves for all 4 amplitudes

are shown in Figure 5.9a. The amplitude and scaled phase curves for a specific Q also

match each other well (Figure 5.9b) confirming the model of the spectral gain

characteristics of the lock-in signal noise

118

a

b

1 0 -5

2

4

681 0 -4

2

2 00 01 5 001 00 05 0 00F r e q ue n c y (H z )

S c al e d P h a s e A m p l itu d e

1 0 -5

2

4

681 0 -4

2 A = .3 7 n m A = 1 .5 n m A = 5 .6 n m A = 1 6 .8 n m

Sc

aled

Pha

se

Noi

se (V

/

) N

oise

Pow

er

(V/

)

Figure 5.9 – (a) Phase noise for the 4 different amplitudes from figure 5.8 now overlap after being scaled by the amplitude. (b) Scaled phase and amplitude spectra overlap, which supports the model for the origin of amplitude and phase noise.

5.6 Noise Power as a Function of Q

The low pass filter on the lock-in amplifier and the heating of the cantilever cause

the lock-in noise to increase roughly linearly with Q. The peak of the thermal noise and

the lock-in noise value at zero frequency increase proportionally to Q because the

spectral width is reduced. On the lock-in amplifier the low pass filters frequently limit

the bandwidth of the lock-in signal noise such that the spectral character of the noise is

determined by the lock-in time constant and filter slope. Filtered and unfiltered lock-in

noise signals are shown in Figure 5.10a. The transfer function of a 24db/octave filter

with 1ms time constant is plotted as gray dots on the right axis. The spectral character of

the signal follows the filter and as a result the limited bandwidth noise power value is

simply proportional to the noise value at zero frequency. The Q dependence of the

119

3

2

1

04003002001000

Frequency (Hz)

20

10

0

Q=28 Q=14 Q=7

1.5

1.0

0.5

0.0

1.0

0.5

0.0

Unfiltered signal Q=7 1ms time constant 24db/octave Filtered signal

a

b

Noi

se P

ower

(pm

/

)

Atte

nuat

ion

Noi

se (p

m)

Figure 5.10 – Effect of lock-in bandwidth. (a) Unfiltered (solid) and filtered (dashed) amplitude noise along with the filter transfer function (gray). (b) Integrated amplitude noise for three different Q values.

filtered signal and integrated noise is shown in Figure 5.10b. The zero frequency noise

value and integrated noise value of the bandwidth limited signal are both proportional to

Q when there is no cantilever heating.

The cantilever heating from Q-control causes the total noise power to increase by

another Q since,

x 2 =kBT

k 1+ M( ). (5.18)

When the Q is changed by Q-control the limited bandwidth and the cantilever heating

work together to cause the noise to increase the noise proportionally with Q.

Experimental Q-control noise values as a function of Q when the cantilever is not

interacting with the surface are shown in Figure5.11. The breadth of Q values, 6 to 155,

120

4

68

0.1

2

7 8 910

2 3 4 5 6 7 8 9100

Noise Power Fit C*Q0.8

Q

Q

Figure 5.11 – Q-control noise power as a function of Q (gray) follows a Q0.8 power law (black). The cantilever heating contributes a and the bandwidth limiting should add

another . The discrepancy is a result of too open a bandwidth.

Noi

se (n

m)

Q factor

is very large and the bandwidth was optimized for imaging speed and signal to noise at Q

= 6. The fit has Q0.8 dependence and the deviation from Q1 is a result of a large

bandwidth such that the value of the unfiltered signal was not flat in the filter bandwidth.

This is especially evident in the Q=155 point when the natural width of the cantilever is

extremely narrow. The noise power as a function of Q implies that increasing the Q for a

constant bandwidth would be slightly advantageous if the signal increases linearly with

Q.

5.7 Relationship between Amplitude and Phase Noise During Tapping

The tip-sample interaction and the incorporation of the Z-piezo feedback loop

significantly change the lock-in signal noise characteristics. The amplitude and phase

noise for 4 different amplitude setpoint values are shown in Figure 5.12. The free

amplitude is A0=14 nm and the setpoint, S=A/A0, represents the ratio of the tapping

amplitude with feedback, A, to the free amplitude, A0. The integral and proportional

121

S>1S=0.8 S=0.6 S=0.4

Am

plitu

de N

oise

(p

m/

)

S>1S=0.8 S=0.6 S=0.4

0.0

0.1

0.2

0.3

0.4

0.5

0

1

2

3

4

5

Phas

e N

oise

(m

rad/

)

0 50 100 150 200 250 300Frequency (Hz)

Figure 5.12 – Amplitude and phase noise as a function of amplitude setpoint. Interacting with the surface moves the amplitude noise to the phase noise. Lowering the setpoint reduces amplitude and phase noise unless Z-piezo oscillations start.

gains were set at 0.03 and 0.1 respectively, and the Q was 17.4.* Interestingly, amplitude

noise was reduced and the phase noise initially increased but then was reduced. A

possible mechanism is that the cantilever experiences phase squeezing where the Z-piezo

feedback loop compensates the changes in amplitude and limits the larger excursions of

the noise causing the energy to be redirected into the quadrature phase where the

feedback loop does not respond as well.13 A cartoon of the thermal noise squeezing is

depicted in Figure 5.13. The noise reduction in the phase as a function of amplitude

could be a result of the resonance shifting away from the lock-in reference frequency

during tip-sample interaction. Both the amplitude and phase noise are lower for lower

122

* The gain values quoted are specific to the Digital Instruments Nanoscope IIIa SPM system and should only be considered as relative values.

Nφ

NA

quadrature

inphase

Figure 5.13 – Interaction with the surface causes thermal noise squeezing which lowers the amplitude noise but increases the phase noise.

setpoints. But, lower setpoints more easily cause Z-piezo feedback oscillations,

significantly offsetting the advantages.

Higher gain reduces the noise in the amplitude channel but not in the phase

channel. The amplitude and phase noise for different values of integral or proportional

gain are shown in Figure 5.14. The amplitude and phase noise for three values of

proportional gain, 0.1, 1, and 10, and a setpoint of 0.8 are shown with the curve from

setpoint > 1 in panels a and b. Similarly, the amplitude and phase noise for four values of

integral gain, 0.001, 0.01, 0.03, and 0.08, and a setpoint of 0.8 are shown with the curve

from setpoint > 1 in panels c and d. The Q is 6 for all curves displayed. In the amplitude,

the proportional gain compensates noise at high frequencies while the integral gain

compensates all frequencies and both push the noise to the phase channel. Increasing the

gain reduces the amplitude noise until the system starts to oscillate, seen as a peak in the

proportional gain of 10 and setpoint of 0.8 curve.

Increasing the Q causes the Z-piezo feedback to oscillate more easily. The

amplitude and phase noise (gray) for different Q values at a setpoint of 0.8, integral gain

123

Proportional Gain Integral Gain

0.0

0.1

0.2

0.30

ca S>1 I=0.001 I=0.01 I=0.03 I=0.08

S>1 P=0.1 P=1 P=10

b d

Am

plitu

de N

oise

(p

m/

)

Phas

e N

oise

(m

rad/

)

3

2

1

200 100 3000 100 200Frequency (Hz)

Figure 5.14 – Amplitude (a) and phase (b) noise spectra for different proportional gain values. The setpoint>1 spectrum is included for comparison. Proportional gain reduces the noise at high frequencies. Amplitude (c) and phase (d) noise spectra for different integral gain values. Integral gain decreases the noise over all frequencies.

of 0.08, and proportional gain of 0.1 are shown in Figure 5.15 compared to the noise with

setpoint > 1 (black). At Q=6 only the transfer of noise from the amplitude to the phase is

seen. At Q=37 an oscillatory peak is developing near 200 Hz. Lastly, at Q=155 the

oscillatory noise is significant and the integral gain had to be reduced to 0.03 to collect

the spectra.

Reducing the gains and scanning rate can alleviate the oscillations but at

significant time cost. Since scanning at slower speeds also increases the signal to noise at

124

2.0

1.5

1.0

0.5

0.0

0.2

0.1

0.03002001000

10

8

6

4

2

0

1.0

0.5

0.03002001000

80

60

40

20

0

10

5

03002001000

Q=155 Q=37 Q=6

Phas

e N

oise

(m

rad/

)

A

mpl

itude

Noi

se

(pm

/

)

Frequency (Hz)

Figure 5.15 – Amplitude and phase noise in contact (gray) and out of contact (black) with the surface for three Q values. Higher Q values cause the Z-piezo feedback loop to oscillate.

lower Q there is no need to originally increase the Q. Increase of noise almost

proportional with Q and the greatly increased propensity to oscillate makes Q-control

very unattractive for increasing signal to noise. Since tapping force is roughly

proportional to kA0/Q it is more productive to lower the Q and use smaller oscillation

amplitudes with slow scanning to increase signal to noise while tapping. Adjusting the

amplitude can also adjust the transition from attractive to repulsive tapping if artificial

phase contrast from surface features is desired.

5.8 Conclusion

Atomic Force Microscopy has developed very rapidly in the 16 years since its

invention. It has become a tool for precise and sensitive measurement of all samples

including soft low adhesion biological samples. A significant advance for AFM was the

invention of tapping mode. Recently, Q-control was thought to be the next technological

125

advance for producing more sensitive images of soft materials and generally increasing

signal to noise. Q-control was found to drive the cantilever away from equilibrium and

change the effective cantilever temperature proportionally with Q. Using Q-control to

boost imaging sensitivity by raising Q is thwarted by the localization of noise in the

measurement bandwidth, the added noise from the increased effective temperature, and

the strong tendency for Z-piezo feedback oscillations. The advantages of Q-control are

that it can change the time constant of amplitude response to increase scanning speed,

cool the cantilever for position clamp experiments (if thermal noise is disrupting the

system), and heat the cantilever to probe deeper interactions using Brownian Force

Profile Reconstruction.

5.9 References

1. Hansma, P. K. et al. Tapping Mode Atomic-Force Microscopy in Liquids. Applied Physics Letters 64, 1738-1740 (1994).

2. Tamayo, J. & García, R. Deformation, Contact Time, and Phase Contrast in Tapping Mode Scanning Force Microscopy. Langmuir 12, 4430-4435 (1996).

3. Anczykowski, B., Cleveland, J. P., Kruger, D., Elings, V. & Fuchs, H. Analysis of the interaction mechanisms in dynamic mode SFM by means of experimental data and computer simulation. Applied Physics a-Materials Science & Processing 66, S885-S889 (1998).

4. Tamayo, J., Humphris, A. D. L. & Miles, M. J. Piconewton regime dynamic force microscopy in liquid. Applied Physics Letters 77, 582-584 (2000).

5. Chen, X., Roberts, C. J., Zhang, J., Davies, M. C. & Tendler, S. J. B. Phase contrast and attraction-repulsion transition in tapping mode atomic force microscopy. Surface Science 519, L593-L598 (2002).

6. Asylum_Research, http://www.asylumresearch.com/qbox.asp, (2003)

7. Mertz, J., Marti, O. & Mlynek, J. Regulation of a Microcantilever Response by Force Feedback. Applied Physics Letters 62, 2344-2346 (1993).

8. Sulchek, T. et al. High-speed tapping mode imaging with active Q control for atomic force microscopy. Applied Physics Letters 76, 1473-1475 (2000).

126

127

9. Mehta, A., Cherian, S., Hedden, D. & Thundat, T. Manipulation and controlled amplification of Brownian motion of microcantilever sensors. Applied Physics Letters 68, 1637-1639 (2001).

10. Cohadon, P. F., Heidmann, A. & Pinard, M. Cooling of a mirror by radiation pressure. Physical Review Letters 83, 3174-3177 (1999).

11. Bruland, K. J., Garbini, J. L., Dougherty, W. M. & Sidles, J. A. Optimal control of ultrasoft cantilevers for force microscopy. Journal of Applied Physics 83, 3972-3977 (1998).

12. Liang, S. et al. Thermal noise reduction of mechanical oscillators by actively controlled external dissipative forces. Ultramicroscopy 84, 119-125 (2000).

13. Rugar, D. & Grutter, P. Mechanical Parametric Amplification and Thermomechanical Noise Squeezing. Physical Review Letters 67, 699-702 (1991).

Chapter 6 Energy Dissipation Chemical Force Microscopy 6.1 Introduction

The great utility of the Atomic Force Microscope (AFM) is rooted in its sharp tip,

high sensitivity, and versatility, which make it ideal for performing molecular scale

measurements. Atomic Force Microscopy studies have typically focused on interaction

forces but switching the focus to energy and energy dissipation can yield tremendous

insight into the physical properties of the sample. Energy dissipation (ED) is associated

with irreversible and time dependant processes. A temporal understanding of interfacial

and intermolecular interactions is very important for tribology and rheology where

contact times and relaxation rates influence the friction between materials. For many

biological interactions such as cell adhesion, recognition, and motility, energy dissipation

is important for determining function, 1-4 for example, as cells shear past one another,

many molecular contacts are formed and broken. Energy dissipated through this process

regulates the migration rate of the cell. Early studies of irreversible interfacial

phenomena included measuring the discrepancy of advancing and receding contact angles

on the rate of motion and surface properties.3 More sophisticated investigations into

energy dissipation must focus on the molecular origin of interfacial and intermolecular

interactions.

In this chapter, the dissipative properties of solvent-surface and surface-surface

interactions between Self Assembled Monolayers are analyzed with a variety of novel

energy dissipation techniques. First, the equilibrium force profiles between SAM

128

surfaces in contact mode are investigated. Second, a method for calculating the energy

dissipation between the tip and sample in tapping mode is developed. This method is

used for collecting Energy Dissipation Force Curves, which measures the energy

dissipation experienced while the systematic motion of the surfaces modulates the tip-

sample interaction. Third, Tapping Mode Force Profile Reconstruction investigates the

force profiles experienced by the tip as it moves toward and away from the sample during

each oscillation of the tapping motion. These techniques were used to elucidate the

molecular mechanism of energy dissipation between hydroxyl and carboxyl terminated

Self-Assembled Monolayers (SAM) in solution. The largest contributor to the energy

dissipation is the rearrangement of the SAM headgroups during contact. Energy

dissipation also originated from ions redistributing themselves in the electrostatic double

layer of a carboxyl terminated surface at high pH. Lastly, Energy Dissipation Imaging is

new imaging technique that isolates physical, chemical, and biological interactions from

the topography making it more sensitive and informative than phase imaging.

6.2 Contact Mode Force Profiles at Low Deborah Number

Energy dissipation has been a topic of study for over a century in traditional

thermodynamics. According to thermodynamics textbooks, equilibrium processes are

reversible and require infinite time. In the laboratory with finite timescales, many

processes progress along “reversible” trajectories but most are irreversible and dissipate

energy as the entropy of the total system increases. The difference between these

reversible and dissipative processes is only time and the Deborah number is a

normalization method for understanding the timescale of dissipative interactions. It is

defined as the relaxation time of the interaction divided by the measurement time.3

129

Experiments performed near a Deborah number of unity have the highest energy

dissipation. An ideal example is a piston in a gas cylinder with a pinhole. Plunging and

withdrawing the piston quickly such that no gas is pushed out of the pinhole and no heat

escapes to the walls of the cylinder is the equivalent to an adiabatic compression and

expansion which has no energy dissipation. This is a measurement at high Deborah

number because the observational time is small. On the opposite extreme, the piston can

be moved slowly such that the pressure outside the cylinder and the pressure inside the

cylinder are always equal, since gas transfers easily through the pinhole. The

compression and expansion are reversible and no energy is dissipated for these

experiments at low Deborah number. Lastly, if the piston pushes the gas out with a

pressure differential then the energy dissipation is maximized and the Deborah number is

near unity since the relaxation time scale of the gas flowing through the pinhole matches

the timescale of the piston movement. Many processes will have diverse relaxation rates

producing different Deborah numbers. Measuring energy dissipation and distinguishing

between different sources of energy dissipation can lead to a deeper knowledge of the

mechanism of interfacial interactions.

The Deborah number is very low for force profiles measured using stiff springs in

contact mode. Stiff springs lower the barrier between the energy minima of the

interaction and spring and the lower barrier reduces the time required for thermal

crossings, effectively lowering the Deborah number. At low Deborah number the force

profile is the equilibrium energy surface with no energy dissipation. The equilibrium

surface provides insight into the mechanism and origin of intermolecular forces. In

chapter 2, Magnetic Feedback Chemical Force Microscopy was used to measure the

130

equilibrium force profile for carboxyl and hydroxyl terminated SAM surfaces. The

discrepancy in length scale for the long-range adhesive forces revealed that the origin of

the forces must be different. Similarly, in chapter 4, the profile of water structure led to

the conclusion that orientational information decays within 0.5 nm of the hydroxyl

terminated SAM surface producing a limit to the long-range effects of solvent ordering.

Although these experiments were insightful, the need remained for gaining a deeper

understanding of the molecular origin of the long-range adhesive forces.

The equilibrium surface can also be used as a reference for experiments

investigating different timescales to discover the onset and spatial location of sources of

energy dissipation. To gain more information about the equilibrium interaction, high

precision force profiles of hydroxyl and carboxyl terminated SAM surfaces were

recorded in contact mode. Atomically flat gold surfaces were used so that instrument

drift did not cause irreproducibility of the force curves. The flat gold was prepared in an

ion pumped load-locked UHV thermal evaporator at 2×10-9 torr. The mica surfaces were

annealed at 400 C overnight before evaporation and for at least 2 hours after evaporation

before slowly cooling. The evaporation rate was 7 Å/s. The surfaces were placed in

either hydroxyl or carboxyl terminated alkane thiol solution immediately after removal

from the evaporation chamber. SAMs were formed for 1 hour before rinsing with

ethanol and blowing dry with clean nitrogen. A 600×600 nm height image of the

hydroxyl-terminated SAM surface in solution used for data in this section is shown in

Figure 6.1. The origin of the roughness is not known since it has peak-to-peak variations

of less than 2.9 Å (gold atom thickness). The roughness is not observed for images taken

in air so it is possibly do to ions bound to the surface. Tips were prepared by growing a

131

Figure 6.1 – Height image of an atomically flat gold surface used for experiments.

thermal oxide in a tube furnace open to air at 900 C overnight. The chromium-gold layer

adhered better to the silicon oxide than to hardened epoxy used for other magnetic tips.

Magnets were affixed with both UV and heat cure glue. A thermal evaporator was used

to apply a chromium and gold layer of 70 and 400 Å respectively at a rate of 1 Å/s. Tips

were placed in hydroxyl alkane thiol solution immediately after removal from the

chamber and let stand for at least an hour. Before use, they were rinsed with ethanol and

dried with nitrogen. Tips and surfaces were used in 0.01 M phosphate buffer at pH 2, 4,

and 7.

Ordinary force curve techniques were used since the surface drift was not

amenable to performing Brownian Force Profile Reconstruction. The resulting force

profiles are displayed in Figure 6.2. The hydroxyl surface is shown in a and the carboxyl

surface at low and high pH are shown in b and c respectively. The hydroxyl curves show

some snap-in and snap-out because the interaction was around the same stiffness as the

spring constant (2.0 N/m). The differences between the measured data and the model in

132

-1.0

-0.5

0.0

0.5

1.0

0.5

0.0

-0.5

864201.5

1.0

0.5

0.020151050

b

c

a Fo

rce

(nN

)

Tip-Sample Distance (nm)Figure 6.2 – Contact mode force profiles. The hydroxyl (a) and carboxyl terminated surfaces at low pH (b) and high pH (c) show no hysteresis or energy dissipation. The tip is coated with hydroxyl terminated SAM for all three interactions.

Figure 4.2 can be used help visualize the true interaction for the hydroxyl surfaces. The

carboxyl curves are smooth and continuous because their stiffness is a factor of 10 lower

than the spring constant so the interaction is relatively constant over the distance scale of

the thermal noise excursions (chapter 4.2).

The force profile quality is very high revealing subtle details that were not evident

in force profiles collected with Magnetic Feedback Chemical Force Microscopy. The

hydroxyl contact region is very stiff such that noise in the contact region causes errors

during interpolation. The extremely high stiffness is from direct contact between the

133

crystalline SAM surfaces. Fitting with an exponentially decaying curve, the contact

regions (<1 nm) of the carboxyl surface show a decay length of 2.2 Å and 3.5 Å at low

and high pH respectively. The similarity of the value and their agreement with previous

SFA data suggest that the soft contact region is a result of hydration of the surface ions.5,6

The attractive forces for the hydroxyl interaction are very short range. Many of the

curves did display solvent shells similar to those observed in chapter 4 but they are not

clearly seen in the average presented in Figure 6.2a. The long-range attractive forces for

the carboxyl interaction are significantly longer than those for the hydroxyl surface. An

inverse square power law fits the attractive carboxyl and hydroxyl surface data better

than an exponential. The χ2 statistic of the fits are 1.5 and 5 times lower for the hydroxyl

and attractive carboxyl surface data respectively. The coefficients of the inverse square

data are a factor of 4 different in value (8.8×10-28 and 3.4×10-27), which emphasizes the

inapplicability of the van der Waals model for these interfacial interactions. The physical

interpretation of the inverse square power law’s superior fit is still unknown, especially

since the carboxyl interaction is much longer-range than the 1 nm (two surfaces) limit set

by solvent-solvent orientation correlations discovered in chapter 4.

The advancing and receding force profiles in Figure 6.2 overlap well revealing the

experiments using contact mode are the equilibrium surfaces at low Deborah number.

Tapping mode increases the rate of interaction by many orders of magnitude compared to

contact mode force curves. Using tapping mode is an excellent method of increasing the

Deborah number, which may provide insight into the timescales of the interfacial

interactions and possibly the underlying molecular mechanisms. The following sections

discuss techniques developed to probe dissipative interactions using tapping mode.

134

6.3 Energy Dissipation Theory

The common paradigm in Atomic Force Microscopy focuses on the force applied

between the tip and sample. This focus works well for contact mode and force curves

where the tip-sample interactions only change within the observational bandwidth being

used. During tapping mode, the tip and sample interact with every oscillation and the

interaction forces change very rapidly, on the order of 10 kHz. Yet, the observational

bandwidth is still quite low (~1 kHz) because lock-in amplifiers and phase locked loops

are used to convert the high bandwidth information into time averaged low bandwidth

signals such as the amplitude, phase, tapping-mode deflection, and frequency. To

understand the physical origin of the low bandwidth signals requires a shift in thinking

from instantaneous forces (high bandwidth) to accumulated force over a trajectory (low

bandwidth). Cleveland pioneered this effort by developing a model of cantilever

dynamics based on time-averaged power balance.7

Power balance of the cantilever oscillations is maintained by the flow of power

out of the system continuously dissipating the flow of power into the system. The power

flow into the AFM cantilever, Pin, is a result of the sinusoidal tapping mode drive force.

The power flowing out of the cantilever is a result of hydrodynamic drag, Pdrag, and losses

between the tip and sample, Ptip. For stable oscillations these powers must be equal such

that,

tipdragin PPP += . (6.1)

The energy dissipation resulting from tip-sample interaction is easily isolated by

calculating the difference between the input power and the background losses,

dragintip PPP −= . (6.2)

135

The instantaneous power is calculated by multiplying the force applied to an object times

the velocity of the object. To calculate the power input into the cantilever the sinusoidal

driving force of the tapping mode is multiplied times the velocity of the cantilever,

( ) tvftFP odin ⋅+= ( )ϕπ2sin , (6.3)

where Fd is the magnitude of the driving force, f is the tapping frequency, ϕο is the phase

offset due to electronic delays, and v(t) is the velocity of the cantilever. The cantilever

position is not a perfect sinusoidal response because the tip-sample interaction makes the

equation of motion non-linear. Fortunately, the motion is periodic and the position is

conveniently expressed as a Fourier expansion of sinusoidal basis functions, such that the

position is

( ) )2sin(1

0 nn

n ftnAxtx ϕπ −+= ∑≥

, (6.4)

where x0 is the mean deflection, An is the amplitude of the nth harmonic, ϕn is the phase

change of the nth harmonic, and n is an integer.8 The velocity

( ) )1cos(21

nn

n ftnfAntv ϕππ −= ∑≥

(6.5)

of the cantilever is simply the first time derivative. The resulting time-averaged power

applied to the cantilever is the integral over one period of the force-velocity product,

( odin fAFP )ϕϕπ += 11 sin . (6.6)

Only the phase of the fundamental is important since all of the higher harmonic terms

integrate to zero.

Similarly, the power dissipated to the background can be calculated from the

dampening force times the velocity. The dampening force is simply the damping

constant b times the velocity such that

136

( ) ∑∑≥≥

==⋅=1

22

0

2

1

22222 2n

nn

ndrag AnQfkfAnfbtvbP ππ . (6.7)

Each cross harmonic term integrates to zero but the diagonal terms each integrate to one

half thus power in the harmonics can be very important. Lastly, the dampening term, b,

can be expressed in the more easily measured cantilever variables, k, f0 and Q. A time

course of the cantilever deflection and the associated power spectrum is shown in Figure

6.3. The tip-sample interaction is highly repulsive and causes the cantilever to turn

around very quickly when it is near the surface in Figure 6.3a. The non-sinusoidal

behavior is reflected in the many visible higher harmonics shown in the power spectrum

of Figure 6.3b. The power in these peaks drops off very quickly and the relative

contribution of the sixth harmonic is only ~0.7%. The background power loss contains

all harmonics including the fundamental. When an elastic interaction causes the motion

of the cantilever to become nonsinusoidal and many harmonics are present, the

background power loss will still equal the power input, Pin=Pdrag and Ptip=0 . Power is

simply transferred from the fundamental to the harmonics.

Tip-

Sam

ple

Dis

tanc

e (n

m)

10-26

10-25

10-24

10-23

10-22

10-21

12080400

Def

lect

ion

Pow

er

Spec

trum

(m2 /H

z)

a b

41.0 41.241.1Time (ms)

41.3

2

0

8

4

Frequency (kHz)

Figure 6.3 – (a) Deflection time course during tapping showing significant nonsinusoidal periodic motion from tip-sample interaction. (b) Power spectrum of the deflection time course. Harmonics of the fundamental contain some of the power dissipated to the background through drag.

137

The energy dissipated by the tip-sample interaction is the difference between the

tapping mode drive power and the hydrodynamic drag power,

( ) ∑≥

−+=1

22

0

2

11 sinn

nodtip AnQfkffAFP πϕϕπ . (6.8)

Three different types of variables are present in the energy dissipation equation: tapping

parameters with tip-sample interaction (An, ϕ1), tapping parameters without tip-sample

interaction (f, Fd, ϕo), and cantilever parameters (k, f0, and Q). The tapping parameters

with tip-sample interaction originate from the lock-in amplifier and contain the

information about variations in the sample properties. Their accuracy depends on the

instrumentation and their precision is determined by the instrument and thermal noise

along with the bandwidth (chapter 5.6).

The tapping parameters without tip-sample interaction play an important role as

the internal calibration of energy dissipation chemical force microscopy. The drive force

is not readily measurable but it can be inferred from the free tapping amplitude, A1free, the

frequency, f, and the cantilever parameters. The cantilever power spectral density is

shown in Figure 6.4. The power spectral density is not only a measure of the cantilever

parameters but also a measure of the response of the cantilever to driving forces. The

driving force is calculated from the transfer function using,9

2/1

2

2

0

0

01

1

+

−=

Qff

ff

ffkAF freed , (6.9)

where A1free is the free amplitude of oscillation for the first harmonic.* The amplitude of

oscillation is dependent on the tapping frequency that is chosen. Often, the tapping * In AFM there is an unfortunate convention of using A0 as the free amplitude of the first harmonic, A1free. Although this convention is used elsewhere in this thesis, A0 will not be used in this chapter but instead A1free.

138

1.2

0.8

0.4

0.0141312111098

-3

-2

-1

0 Noise spectrum Fit Phase

Phase shift (rad) N

oise

Pow

er

(pm

2 /Hz)

Frequency (kHz) Figure 6.4 – Power spectrum of cantilever thermal noise. The cantilever parameters are calculated from the fit. A vertical arrow marks the tapping frequency. The transfer function is used to compute the drive force and phase offset for off-resonance tapping.

frequency is chosen to be slightly less than the resonant frequency for stable tapping

mode feedback as shown by the arrow in Figure 6.4. The cantilever phase lag is also

displayed in Figure 6.4. The measured phase when there is no tip-sample interaction can

be used to calculate the offset associated with electronic delays,

freeo ffQff

1220

01

)(tan ϕϕ −

−= − . (6.10)

After substitution of the Fd and ϕo, the tip-sample energy dissipation expression becomes

−

−+−

+

−= ∑

≥

−

1

2222

0

0111

2/1

2

2

0

011

0

2 1)(

tansin1n

nfreefreetip AnQffQ

ffQf

fffAA

fkfP ϕϕ

π .(6.11)

The energy dissipation per tap is calculated by dividing equation 6.11 by f.

Equation 6.11 is a clean analytical expression for the energy dissipation of any

sample analyzed with tapping mode but unfortunately, it is a small number resulting from

the difference of two large numbers, heightening the need for precision and accuracy.

For the experiments performed in this chapter, the measurement of amplitude and phase

139

are limited by the intrinsic thermal noise of the cantilever (chapters 3 and 5). Using the

thermal noise spectrum, the cantilever parameters can be calculated with great accuracy

and precision since the shape of the power spectrum (Figure 6.4) determines Q and f0 and

the any error originates from inconsistent or inaccurate timing of the spectrum analyzer,

which is unlikely. The error of the detector sensitivity limits the accuracy of the spring

constant to about 5% (Appendix A.3). Fortunately, the spring constant is a scalar of both

the power flowing into the cantilever and the power flowing out of the cantilever so the

relative contributions of each component are preserved.

The energy dissipation between the tip and sample can be measured by

subtracting the power lost to the surroundings, Pdrag, from the power flowing into the

cantilever, Pin. Measuring Energy dissipation is an accurate and robust method for

quantitatively analyzing physical, chemical and biological interactions between the tip

and sample.

6.4 Energy Dissipation Force Curves

In the previous section, it was shown that the energy dissipation between the tip

and sample can be isolated and quantified during tapping mode. The energy dissipation

signal is a function of the chemical and physical properties of the surface but the

interpretation of this information is convoluted with other parameters such as the tip-

sample contact time, the rate of withdrawal, and contact area. Discerning the molecular

origin of the energy dissipation signal requires removing the dependence on the

convoluting parameters, by observing the characteristics that are independent of those

parameters. Force curves provide an excellent opportunity for investigating a large

parameter space since the energy dissipation for a large span of tip-sample interactions,

140

from very light tapping to very hard tapping, can be recorded in only a few seconds as the

surface is pushed into contact with the tip. The rate of withdrawal of the tip during each

tap and the contact area can be adjusted by changing the free amplitude of oscillation and

the Q of the cantilever respectively.

Energy Dissipation force curves have previously been explored by both Cleveland

and Tamayo. The experiments by Cleveland investigated the energy dissipation between

a silicon tip and silicon surface in air.7 The energy dissipation was found to be 4.0 aJ/tap

and relatively constant throughout the whole range of tapping amplitudes. Tamayo also

used a silicon tip but he tapped against HOPG and purple membrane.10 He similarly

found that the energy dissipation was relatively constant over the amplitude range used

but that HOPG had 4 times the energy dissipation as purple membrane. These

experiments were performed in air and the adhesion forces were due mostly to capillary

wetting. It is expected that the energy dissipation is constant since the water layer is only

3-4 nm thick and the tip experiences the same adhesion hysteresis from the wetting if it

travels 8 or 40 nm away from the surface during the tap cycle. Removing the capillary

forces by working in solution allows Energy Dissipation Force Curves to probe the

interactions specific to the tip and sample surfaces.

Specific chemical interactions between hydroxyl and carboxyl terminated SAM

surfaces were investigated using Energy Dissipation Force Curves in solution. The same

sample surfaces used for Figure 6.2 were used for the Energy Dissipation Force Curves.

The tip was functionalized with hydroxyl terminated SAM and the samples were

functionalized with carboxyl and hydroxyl terminated SAMs submerged in 0.01 M

phosphate buffer solutions of pH 2, 4, and 7. The contact area and rate of withdrawal

141

were modulated by using Q values of 6.6, 15, 36, and 60 and free tapping amplitudes of

1, 2, 4, and 8 nm. Force curves were collected by recording the deflection time course as

the sample was brought in and out of contact with the tip. The deflection data was AC

coupled and sampled at 1 MHz and 16 bit resolution for 4 s and saved to disk (appendix

A.2). The controller ramped the surface in and out of contact with the tip with a period of

0.5 Hz producing two full force curves per time course. A deflection time course along

with a sketch of Z-piezo motion is displayed in Figure 6.5. Only the envelope of the

cantilever oscillations is observed since the frequency of oscillations is high.

10

8

6

4

2

043210

Def

lect

ion

(nm

)Z-

Piez

o D

ispl

acem

ent (

nm)

Time (s)Figure 6.5 – Deflection (a) and Z-piezo (b) time courses used for energy dissipation force curves. The numerous oscillations of the deflection time course mark the envelope of oscillation or amplitude. The amplitude is reduced as the piezo brings the surface into contact with the tapping tip.

142

After collection the deflection time courses were converted into Energy

Dissipation Force Curves using numerical lock-in techniques. The inphase, xn, and

quadrature, yn, components for each harmonic were calculated using,

)2sin(2 δπ +⋅= ftnDeflectionxn (6.12)

)2cos(2 δπ +⋅= ftnDeflectionyn , (6.13)

where f is the tapping frequency, δ is an arbitrary phase shift, and n is the index of the

harmonic. The amplitude of each component and phase could be computed from the

quadrature components using,

22nnn yxA += (6.14)

= −

1

111 tan

xyϕ (6.15)

Because the phase was not locked the frequency, f, had to be found to an accuracy better

than 0.001 Hz to reduce the phase drift to within the noise. The phase offset, δ, was set

to ϕ0

(equation 6.10) to compensate the electronic phase shifts. The code for the scripts

used is recorded in the appendix (A.6).

The results of the numerical lock-in techniques are displayed in Figure 6.6 for the

hydroxyl surfaces at Q = 36 and A1free = 8 nm. The amplitude and phase as a function of

time are shown in Figure 6.6a and b respectively. The phase lag is greater than ϕfree for

tapping in the attractive regime.* After the transition to the repulsive regime the phase

lag is less than ϕ1free and the amplitude is slightly increased. The phase signal when the

amplitude is zero is meaningless. The energy dissipation was calculated using equation

6.11 as shown in Figure 6.6c. The energy dissipation significantly increases after the

* For a short discussion on tapping in the attractive and repulsive regime see the Appendix A.1.4.

143

144

3

2

1

0

43210

4203

2

1

0

3

2

1

0

86420

86

aTa

ppin

g A

mpl

itude

(nm

)

b

Phas

e (r

ad)

c

Ener

gy

Dis

sipa

tion

(aJ)

Time (s)

d

Ener

gy

Dis

sipa

tion

(aJ)

Tapping Amplitude

Figure 6.6 – Tapping amplitude (a) and phase (b) calculated from numerical lock-in of time course data. Energy dissipation (c) calculated from amplitude of all harmonics, phase, and cantilever variables. (d) Energy dissipation plotted as a function of tapping amplitude shows that energy dissipation varies significantly with tapping amplitude.

transition to the repulsive regime. Most of the force curve has very little tip sample

interaction such that A ~ A1free and Ptip ~ 0. A more informative plot of energy

dissipation as a function of tapping amplitude is shown in Figure 6.6d. The transition

from attractive to repulsive is at 6 nm of tapping amplitude during the trace (motion of

the surface toward the tip). The transition back to attractive at 7 nm, during the retrace

(motion of the surface away from the tip), shows hysteresis. The 4 or 5 points during the

transition are inaccurate as a result of the time constant (numerical smoothing) used on

the lock-in amplifier. The energy dissipation changes dramatically during the force curve

unlike the energy dissipation force curves of Cleveland and Tamayo. The repulsive

regime energy dissipation values were separated from the attractive regime since the trace

and retrace hysteresis and the small change in amplitude during the transition. Furthur

averaging was performed for curves from different time courses that were performed

using the same conditions (pH, Q, and A1free).

The data reveal many important characteristics about the interfacial interactions in

solution. Data for many Q and free tapping amplitude values were collected. The Q

directly influences the contact force for each tap, which affects the contact area.

Increased tapping amplitude increases the rate of pulling but the greater inertia also

affects the contact force and contact area. Unfortunately, analytical expressions do not

exist to translate the available parameters of Q, free tapping amplitude, and tip radius into

the more physically interesting parameters of contact time, contact area, and pulling rate.

However, important trends were clearly established as a function of these parameters.

Representative Energy Dissipation Force Curves are displayed in Figure 6.7. Data for the

hydroxyl surface (black) in pH 2 buffer along with the carboxyl surface in both pH 2

145

0.6

0.4

0.2

0.043210

2.0

1.5

1.0

0.5

0.0

0.2

0.1

0.0210

0.6

0.4

0.2

0.0

c d

ba

COO- surface COOH Surface OH Surface

Ene

rgy

Dis

sipa

tion

(aJ)

Tapping Amplitude Figure 6.7 – Energy dissipation force curves using a hydroxyl terminated SAM on the tip tapping against SAM surfaces terminated with hydroxyl (black), carboxyl at high pH (dashed), and carboxyl at low pH (gray). Curves were collected with Q = 6.6 (a, b) and Q = 30 (c, d) and for a free tapping amplitude of A1 ~ 4 nm (a, c) and A1 ~ 2 nm (b, d).

(gray) and 7 (dashed) are shown. At low pH the carboxyl surface was mostly protonated

(COOH) and at high pH the carboxyl surface was partially deprotonated (COO-). The

data in a and b were collected with a Q of 6.6 and the data in c and d had a

Q of 36. The data in a and c had a free tapping amplitude of 4 nm and the data in b and d

had an amplitude of 2 nm. All energy dissipation values are reported in joules per tap.

As mentioned above, the attractive and repulsive regime data were separated from each

other. The attractive regime data originate at the free amplitude and typically have little

146

energy dissipation. The repulsive regime data typically originate with significant energy

dissipation at lower tapping amplitude and terminate near zero tapping amplitude.

The Q changes the ability of the cantilever motion to be modified by external

forces. At higher Q the motion is required to be more sinusoidal and as a result the tip-

surface interaction is reduced leading to a small contact force. The data show that

increased Q reduces the energy dissipated between the tip and sample and tends to cause

the system to tap in the attractive regime for a larger range of tapping amplitudes. Also,

an increase in Q causes the energy dissipated by the hydroxyl and carboxyl surfaces to be

more similar which can possibly be interpreted as the sources of the energy dissipation

becoming more similar. Less significant changes are observed between the two different

values of free tapping amplitude. Smaller free amplitude also decreases the total energy

dissipation and causes the cantilever to tap in the attractive regime for a larger range of

tapping amplitudes.

The data also reveal many specific differences between the surface chemistries.

The hydroxyl terminated SAM surface has very little energy dissipation when tapping in

the attractive regime. Also, it tends to tap in the attractive regime until smaller

amplitudes (Figure 6.7b). This is supported by the equilibrium force profile in Figure

6.2a, where the attractive portion was much deeper than for the carboxyl surface at low

pH such that it requires more perturbation to the cantilever motion to shift to the

repulsive regime. The low energy dissipation when tapping in the attractive regime

suggests that the interaction is still at low Deborah number and the source of attractive

forces in the solvent near the surface has a very fast relaxation rate. When tapping in the

repulsive regime the interaction has much more energy dissipation than tapping in the

147

attractive regime or the repulsive regime for other surfaces. The distinct increase in

energy dissipation is a result of dissipative interactions being associated with removing

the last layer of solvent and having the two surfaces contact each other. Possible

mechanisms for the dissipation include slow resolvation of the surface after the last layer

has been removed and slow rearrangement of the SAMs upon interfacial contact. Both

mechanisms are strongly dependent on the contact area, which is a function of the contact

force. As a result, the difference between the hydroxyl and carboxyl surfaces is possibly

caused by the steeper contact region of the hydroxyl surfaces increasing the contact force

and contact area for the same cantilever inertia.

The low pH carboxyl terminated SAM data is similar to the hydroxyl terminated

SAM data yet still unique. Tapping in the attractive regime causes very little energy

dissipation, which is analogous to the hydroxyl surfaces interaction. This suggests that

the mechanism of attractive interactions may be similar or both have fast relaxation rates.

Uniquely, the shift to the repulsive regime causes a small increase in dissipation. The

contact region of the carboxyl surface is relatively soft (Figure 6.2b). This is likely a

result of hydrated ions and not the bare SAM surfaces touching each other. The distinct

surface characteristics affect the mechanism of energy dissipation. A soft contact region

greatly reduces the contact force and contact area, which reduces possible losses from

SAM rearrangement. Also, the resolvation dynamics of the ions may be different than

resolvation of the hydroxyl SAM surfaces changing the rate of dissipation.

Switching the buffer to pH 7 causes the carboxyl interaction to become repulsive,

which also increases the energy dissipation for the tip-sample interaction. The whole

interaction is repulsive and the tip initially taps against the electrostatic double layer

148

before reaching the hydration layer. Tapping against only the electrostatic double layer

perturbs the cantilever motion enough to reduce the amplitude. The data in Figure 6.7

imply that energy is lost while tapping in the electrostatic double layer far from the

surface since the graphs show energy dissipation for tapping amplitudes near the free

tapping amplitude. The results from the hydroxyl terminated SAM surface show that

long-range solvent rearrangement happens quickly and is not a source of energy

dissipation. Therefore, the long-range energy dissipation of the high pH carboxyl surface

must be involved with the ions of the electrostatic double layer and not the solvent

molecules. The dissipation arises from the relatively slower diffusion of ions hindering

the reestablishment of the double layer distribution upon withdrawal of the tip.

Energy Dissipation Force Curves are an excellent source of quantitative

information about the time-dependent characteristics of intermolecular and interfacial

interactions. Energy dissipation was observed to significantly increase when the tip starts

tapping in the repulsive regime. This energy dissipation could result from rearrangement

of the SAM surfaces upon contact, visco-elastic losses in the SAM and gold, or slow

resolvation of the SAM surface. The time-dependent information can also be used to

elucidate the equilibrium behavior. For example, the lack of energy dissipation for

tapping in the attractive regime of both the hydroxyl and low pH carboxyl surfaces

suggests that the mechanism for the attractive forces is similar.

Many of the above hypotheses are speculative and require more extensive data to

confirm. The speculation arises from the lack of spatial information in Energy

Dissipation Force Curves. A spatial technique could distinguish between the interfacial

rearrangement and resolvation models because the rearrangement would manifest itself as

149

hystereis in the contact region while resolvation would manifest itself near the surface in

the first solvation shell but not in the contact region. An energy dissipation method is

required that is both time-dependent and also has spatial information. In the following

section, a method is developed that can measure the force profiles experienced by the tip

as it advances and recedes from the surface in tapping mode providing both time-

dependent and spatial information about the interaction.

6.5 Tapping Mode Force Profile Reconstruction

6.5.1 Introduction

The direct measurement of force profiles is important for developing an

understanding of the mechanism of intermolecular and interfacial interactions. Static

measurement of force profiles has been limited by the instability experienced by weak

springs (Chapters 2 and 4) and the poor measurement of cantilever position (Chapter3).

Consequently, dynamic methods using tapping techniques have been developed to

measure force profiles. By tapping the cantilever, the energy required to overcome the

attractive forces is stored in the cantilever as it approaches the surface on every

oscillation, reducing the problems with instability. The tip-sample interaction changes

the cantilever dynamics such as the amplitude, phase, and resonant frequency, and the

force profile is reconstructed from these signals. These methods have been used with

some success to measure tip-sample interactions.11-13 The most notable use of AC

techniques for force profile measurement was the measurement of the different chemical

sites on the silicon surface (7X7) reconstruction.14 Unfortunately, these techniques

require high Q achieved only by placing the tip in vacuum, which is of little interest to

most of the AFM community. Also, many of the approximations used to develop the

150

theory of the potential reconstruction are not valid for the whole potential but only the

regime with positive stiffness. Lastly, the techniques cannot distinguish between

advancing and receding force profiles, which are important for understanding the sources

of energy dissipation.

More recently, a novel technique was developed where the force time course was

reconstructed using spectral analysis. In the frequency domain, the force is the deflection

times the inverse of the transfer function. For example at DC the inverse of the transfer

function is k, leading to the common expression, F=kx. Therefore, the force trace can be

calculated by computing the IFFT of the product of the deflection FFT and the inverse

transfer function.15 This method could distinguish between hard and soft samples for

individual taps but the noise was substantial. Also, reconstruction of the force profiles

was not attempted. It was a first step toward peering into the time dependant nature of

tip-sample interactions.

To understand the time-dependent nature of tip-sample interactions, techniques

must be developed for processing data sampled at high frequencies. In chapter 4,

Brownian Force Profile Reconstruction was introduced which uses the thermal noise of

the cantilever as a probe of the intricacies of the tip-sample force profile. Through

sampling and analyzing the data at higher frequencies than the resonance, BFPR detected

details in the force profile that were lost through normal force curve techniques, which

average all the deflection data together in a bandwidth of only 1 kHz or less. Noncontact

mode force profile techniques also suffer the same difficulty as normal force curves

because phase locked loops and lock-in amplifiers are used to demodulate the AC signal

to DC for measurement. In the process, the important time-dependant details of the

151

interaction are averaged together. Taping Mode Force Profile Reconstruction is a

powerful technique for observing important time-dependant details of the interaction and

reconstructing the advancing and receding force profiles experienced by the tip as it taps

near the surface. In this section, Tapping Mode Force Profile Reconstruction is

developed and applied to dissipative interactions between self-assembled monolayer

surfaces.

6.5.2 Tapping Mode Force Profile Reconstruction Theory and Noiseless Simulations

Taping Mode Force Profile Reconstruction uses the deflection time course and the

wave equation of motion to calculate the advancing and receding force profiles for the

tapping cantilever as it approaches and withdraws from the surface during each tap. The

wave equation for a tip tapping against a surface is

( ) ( ) ( ) ( ) ( )txFtFtxmtxbtxk id ,sin +=⋅+⋅+⋅ ω&&& , (6.16)

where k, b, and m are the cantilever parameters, ( )tx , ( )tx& , and ( )tx&& are the position of

the cantilever and its first and second time derivatives respectively, Fd is the driving

force, and Fi is the tip-sample interaction force. For simplicity, this equation does not

model higher order eigenmodes of the cantilever and the motion of the whole oscillator

toward and away from the surface during the force curve. The tip-sample interaction

force, Fi, is only a function of position for elastic interactions and a function of both

position and time if there is energy dissipation. The interaction force time course is

computed by moving the driving force term to the other side of equation 6.16.

( ) ( ) ( ) ( ) ( )tFtxbtxmtxktF di ωsin−⋅+⋅+⋅= &&& , (6.17)

A free oscillator with no tip-sample interaction causes all the RHS terms to cancel. The

potential energy term, , and the inertial term, ( )txk ⋅ ( )txm &&⋅ , are equal and opposite,

152

similar to the Simple Harmonic Oscillator (SHO) model for undamped, undriven

oscillators. The damping term, ( )tx&b ⋅ , removes energy from the system and to sustain

stable oscillation the driving force, ( )tFd ωsin applies equal force. The terms cancel

because the response of the cantilever has a π/2 phase shift and the first derivative adds

another π/2.

Tip-sample interaction applies a force to the tip, causing acceleration. Although

the acceleration is distributed among all the terms it is localized in the inertial term,

, such that the inertial and potential energy terms no longer cancel. Furthermore,

the tip-sample interaction adds a phase shift to the cantilever response and a reduction in

oscillation amplitude. These effects cause the damping and drive terms to no longer

cancel.

( )txm &&⋅

The tip-sample interaction time course can be constructed purely from the

deflection time course and the cantilever parameters, k, b, and m. The velocity and

acceleration can be calculated numerically from the deflection. The damping and mass

are easily derived from the resonant frequency and quality factor as described in the

appendix (A.3). The drive force is calculated from the cantilever parameters and the free

amplitude of oscillation, equation 6.9. The correct phase offset for the drive oscillation

can also be found by fitting the velocity time course when the tip is far from the surface.

The tip-sample interaction force and position time courses from a simulation, with a 100

ns time increment and no cantilever thermal or instrument noise, are shown in Figure 6.8.

The simulation parameters will be thoroughly described in the following section. The Z-

piezo motion was added to the deflection signal to produce the tip-sample distance time

course. The interaction is strongly repulsive for every tap of the tip against the sample.

153

1501005000

20

15

10

5

0

3

2

1

Tip-Sample D

istance (nm)

Tip-

Sam

ple

Forc

e (n

N)

Time (µs)

Figure 6.8 – Simulated noiseless deflection (black) and reconstructed interaction force (gray) time courses. The adhesion hysteresis is readily observed between the left (advancing) and right (receding) side of the peaks.

The advancing and receding attractive interactions on the left and right side of the peak

show clear hysteresis or energy dissipation.

The advancing and receding force profiles are reconstructed by matching the

interaction force time course to the tip-sample distance time course. After splitting the

time courses between those with positive velocity (receding) and negative velocity

(advancing) the data can be sorted by tip-sample distance and averaged over many

oscillations. The averaging techniques used included decimating the sorted waves to

1000 points, interpolating the results to produce an evenly spaced force profile in the tip-

sample distance axis, and smoothing the result with a 15 point sliding box algorithm.

The code for the Tapping Mode Force Profile Reconstruction algorithm is recorded in the

appendix (A.7). The reconstructed advancing and receding force profiles are shown in

Figure 6.9 along with the force profiles used in the simulation. The agreement between

the force profiles reconstructed from the timecourse and the simulation force profiles is

outstanding. Tapping Mode Force Profile Reconstruction is a robust technique for

154

-2321

-1

0

1 Advancing Reconstruction Advancing Force Profile Receding Reconstruction Receding Force Profile

Forc

e (n

N)

Tip-Sample Distance (nm)Figure 6.9 – Reconstructed advancing (light gray) and receding (dark gray) force profiles from the noiseless simulated deflection time course. The reconstructed force profiles show hysteresis and are indistinguishable from the force profiles (black) used in the simulation.

measuring the differences in the advancing and receding force profiles experienced by the

tapping cantilever unavailable to other reconstruction techniques.

6.5.3 TMFPR Simulations with Noise and Reduced Bandwidth

The idealized simulation in the previous section highlights the ability of Tapping

Mode Force Profile Reconstruction to measure the tip-sample force profiles from the

position time course. TMFPR is also a robust technique for waves that include noise

sources. Simulations with noise and reduced bandwidth were performed to reveal the

capabilities of TMFPR. The simulations start with two force profiles representing

advancing and receding curves. Repulsive double exponential force profiles were used to

simulate electrostatic repulsion with hydration forces associated with the carboxyl

terminated SAM surfaces in high pH (Figure 6.2c). The hydration forces region of the

receding force profile is steeper such that the force is reduced more quickly as the

cantilever moves away from the surface. Also, the electrostatic repulsion region has a

155

smaller magnitude and longer decay length than the advancing profile. The resulting

hysteresis between the force profiles is the mechanism of energy dissipation. For each

oscillation the receding force profile was adjusted in the tip-sample distance dimension

such that the advancing and receding force profiles have the same value at the point

where the cantilever turns around (velocity changes sign). By shifting the receding force

profile the cantilever travels away from the surface without experiencing large

discontinuities in tip-sample interaction force. The wave equation of motion was used to

calculate the cantilever trajectory. A description of the simulation algorithm and the Igor

Pro code are recorded in the Appendix (A.4)

Simulations were performed using k=1.5 N/m, f0=16,000 Hz, and Q=6 as the

cantilever parameters and f=15,700 and A1free = 3 nm, which are similar to the values of

the cantilever used in the energy dissipation force curves of Figure 6.7. A short time

increment of 100 ns reduced numerical error in the trajectory. The cantilever temperature

was 300 K. After the time course was calculated more gaussian noise was added to

simulate 50 fm/ Hz of instrument noise. The power spectrum of the simulated deflection

time course (black) is shown in Figure 6.10. The harmonics of the fundamental tapping

frequency contain information about non-sinusoidal motion of the cantilever and tip-

sample interaction. Harmonics that are buried under noise do not contribute useful

information. The instrument noise on the acceleration term increases by f2 because the

derivative is computed twice. Smoothing the deflection data suppresses the instrument

noise such that the signal is clearer. Sampling the deflection time course more slowly

such that the bandwidth only includes the harmonics above the noise helps to reduce the

total instrument noise collected. Unfortunately, the low sampling rate leads to numerical

156

10-28

300250200150100500

Raw Deflection Power Spectrum Smoothed Deflection Power Spectrum

10-26

10-24

10-22

10-20

Noi

se P

ower

(m2 /H

z)

Frequency (kHz)Figure 6.10 – Power spectra of deflection time courses. The harmonics contain the important information about the tip-sample interaction. The 4th order Savitxky-Golay smooths (gray) remove the high frequency instrument noise from the raw signal (black).

inaccuracies when derivatives are calculated for TMFPR. The best results are obtained

when deflection data is sampled at 4-8 times the frequency of the highest

harmonic perceptible above the instrument noise and the deflection data is smoothed. Of

the smoothing algorithms available on Igor Pro, the forth order Savitsky-Golay algorithm

is best since it most closely resembles a brick wall filter, cutting off the high frequency

noise but leaving the power in the harmonics intact. A method was attempted that

transformed the deflection into the frequency domain with an FFT, removed all noise in

between the harmonics, and retransformed the data back into the time domain using an

IFFT. This method significantly reduced the scatter but the noise at the frequencies of

the buried higher harmonics masqueraded as signal. This produced large ripples in the

data. The simple smoothing of the deflection data led to the best results and reduced the

possibility of the user shaping the data.

157

The tip-sample interaction force and position time courses simulating the carboxyl

terminated SAM surface interaction are shown in Figure 6.11. The position time course

is similar to the curve in figure Figure 6.8 but the interaction force time course shows

more noise. Fortunately, asymmetries between the advancing and receding portions of

the tapping motion are still perceptible. After sorting the interaction force time course by

velocity and tip-sample position, followed by averaging, the tapping mode force profiles

shown in Figure 6.12 are produced. Advancing and receding traces are shown for

tapping far from the surface (a) and near the surface for both full amplitude (b) and

reduced amplitude (c). All the traces match the simulation force profiles well. The traces

in b and c show noticeable energy dissipation revealing the ability of TMFPR to

accurately measure advancing and receding force profiles for noisy time courses.

The hydroxyl terminated SAM surface interaction in Figure 6.2a was modeled

using attractive force profiles with hard sphere repulsion. Similar to the carboxyl surface

interaction, the contact region of the receding force profile is steeper such that the force is

158

1

0

-1200150100500

1

0

4

3

2

3

2

Tip-Sample D

istance (nm)Ti

p-Sa

mpl

e Fo

rce

(nN

)

Time (µs)Figure 6.11 – Reconstructed interaction force (gray) time courses of the tip-sampledistance (black) for simulations including cantilever thermal and instrument noise. Instrument noise considerably degrades the interaction force signal.

1

0

1086420

c

2 Advancing Reconstruction Advancing Force P rofile Receding Reconstruct ion Receding Force P rofi le

2

1

0

2

1

0

b

aFo

rce

(nN

)

Tip-Sample Distance (nm)

Figure 6.12 – Reconstructed force profiles (gray) from a simulation with noise for three different tapping amplitudes (a-c). They show hysteresis and match the force profiles (black) used in the simulation well.

reduced more quickly as the cantilever moves away from the surface. The adhesive

region also is more attractive in magnitude than the advancing force profile (Figure 6.9).

The energy dissipation force curves from Figure 6.7 suggest that there is little energy

dissipation when the tip is tapping in attractive mode. This phenomenon was modeled in

the simulation by using the advancing force profile as the tip-sample interaction force for

both positive and negative velocities unless the tip-sample interaction became repulsive

during the oscillation cycle. If the oscillation was repulsive, then the transition to using

the hysteretic receding force profile was included in the loop. For each repulsive

oscillation the receding force profile was adjusted in the tip-sample distance dimension to

match the advancing profile at the cantilever turn around point. The reconstructed force

profiles along with the force profiles used in the simulation are shown in Figure 6.13. A

159

-2

0

2

6420

Advancing R econstruction A dvancing Force Profile Receding R econstruction Receding Force Profile

-2

0

b

c

2-2

0

2

aFo

rce

(nN

)

Tip-Sample Distance (nm)Figure 6.13 – Reconstructed force profiles (gray) for three different tapping amplitudes (a-c) from a simulation including noise where attractive force profiles (black) with a very stiff contact region were used. The reconstruction does not match the original force profiles because the instrument noise obscures the information about the stiff contact region in the higher harmonics.

large tapping amplitude with little tip-sample interaction is shown in a. Reduced

amplitude but still tapping in the attractive regime is shown in b. Repulsive tapping is

shown in c. The reconstructed advancing and receding force profiles in a and b match the

simulation force profiles very well. The repulsive reconstructed fore profiles show clear

hysteresis and match the simulation force profile qualitatively but the profiles are offset

from the well in tip-sample distance. The same simulation without noise in Figure 6.9

matched very well. The discrepancy is caused by the loss of information from the buried

higher harmonics. The higher harmonics cause the deflection trace to have a stiffer turn

around. The loss of that information makes the stiffness of the interaction smaller which

pushes the minimum of the well out from the surface. These errors were not apparent in

160

Figure 6.12 because the stiffness of the hydration layer is smaller causing less

information to be contained in the buried harmonics.

Strangely, the value of the spring constant used during analysis as an input

variable required adjustment so that the reconstructed force profile matched the force

profile used in the simulation. Using a stiffness value of 1.5 N/m in the TMFPR

algorithm produced a slight tilt of the advancing and receding force profiles. Adjusting

the spring constant 3% lower removed this tilt. This was observed for multiple

simulations where the spring constant was varied. The phenomenon results from the

skew of the data by the noise during interpolation.

Inclusion of noise revealed the tolerances of Tapping Mode Force Profile

Reconstruction. The instrument noise and reduced bandwidth can cause the important

information in the high frequency harmonics to be lost. Fortunately, these errors are only

perceptible for interactions of ultrahigh stiffness such as the hydroxyl terminated SAM

interaction. Measurements of other interactions with noise are quantitatively accurate.

6.5.4 TMFPR of Dissipative Interactions between SAM Surfaces

The Tapping Mode Force Profile Reconstruction force profiles were computed

from the time courses used to produce the Energy Dissipation Force Curves in Figure 6.7.

The data was collected at 1 MHz sampling frequency, which is greater than 4 times the

frequency of the last noticeable harmonic. The cantilever parameters were obtained from

a fit to the thermal noise data when the cantilever was not driven by the tapping signal

and the driving force was calculated from the free amplitude using equation 6.9. The

deflection signal was, unfortunately, AC coupled during data collection so the slight

deflection caused by the average attractive or repulsive forces during tapping were

161

removed. The absence of the DC deflection, ∆x, will cause the traces to be offset in the

force axis by a factor of k*∆x and offset in the tip-sample axis by ∆x. Since the

advancing and receding force profiles are reconstructed from the same oscillations, their

relative positioning will be accurate.

The reconstructed force profiles for carboxyl terminated SAM surface in pH 7.0

solution are shown in Figure 6.14. The tip was coated with a hydroxyl terminated SAM.

The force profiles of weak interactions with the surface are shown in a, while strong

interactions are shown in b and c. The reconstructed force profiles were shifted in the

force axis to match the contact mode force profile (black dashes) from Figure 6.2c, which

compensates the removal of the DC component by the AC coupling. The free amplitude

4

3

2

1

04

3

2

1

04

3

2

1

0

87654321

Advancing Force Profile Receding Force Profile Equilibrium Force Profile

b

c

a

Forc

e (n

N)

Tip-Sample Distance (nm)

Figure 6.14 – Reconstructed force profiles from carboxyl data in Figure 6.7a at high pH for three tapping amplitudes (a-c). The equilibrium force profile (dashed) from Figure 6.2c matches the advancing (light gray) trace well. The receding (dark gray) trace shows hysteresis at reduced amplitudes.

162

of oscillation was 3.80 nm and the Q was 6.6. The associated energy dissipation force

curve is shown with black dashes in Figure 6.7a. The force profiles in Figure 6.14a

overlap well and have no energy dissipation. The force profiles in b show significant

long-range hysteresis extending far into the electrostatic repulsion regime. The curves in

c are for a small tapping amplitude of 6 Å and they also show some energy dissipation as

the tip only moves within the first few solvation layers of the surface.

The reconstructed force profiles for the same hydroxyl terminated tip and

carboxyl terminated surface in pH 2 solution are shown in Figure 6.15. Attractive regime

force profiles are shown in a, while repulsive regime force profiles for two different

amplitudes are shown in b and c. The free amplitude was 3.84 nm, the Q was 6.6, and the

6

4

2

0

87654321

6

4

2

0

6

4

2

0

Advancing Force Profile Receding Force Profile Equilibrium Force Profile

b

c

a

Forc

e (n

N)

Tip-Sample Distance (nm)Figure 6.15 – Reconstructed force profiles from the carboxyl data at low pH from Figure 6.7a for three tapping amplitudes (a-c). The equilibrium force profile (dashed) from Figure 6.8b overlaps the advancing (light gray) trace well. The receding force profile (dark gray) hysteresis is localized to the contact region.

163

associated energy dissipation force curve is displayed in Figure 6.7a. These force

profiles match the equilibrium force profile (Figure 6.2b) very well in the attractive

regime. Hysteresis between the advancing and receding force profiles becomes apparent

when the tip is tapping in the repulsive regime. The hysteresis is localized exclusively to

the contact region and not the attractive region in the solvent.

The stiffness of the interactions with the carboxyl terminated SAM is relatively

soft providing a smooth turn around for the tip. The smooth turn around produces fewer

higher harmonics and the reconstruction is anticipated to the quantitatively accurate. The

area mapped out by the hysteresis between the traces in b totals 450 aJ, which is in

excellent agreement with the Energy Dissipation Force Curve value at 3.2 nm of tapping

amplitude or 6.4 nm of peak to peak motion as seen in figure 6.15b.

The reconstructed force profiles for the hydroxyl terminated tip tapping against a

hydroxyl terminated surface in pH 2 solution are shown in Figure 6.16. Force profiles

tapping in the attractive regime with a large amplitude and small amplitude are depicted

in a and b respectively. Force profiles from the repulsive regime are shown in c. The

free tapping amplitude was 3.92 nm and the Q was 28.9. The hydroxyl-hydroxyl

interaction is much harder so more harmonics were produced. The larger bandwidth

required for the reconstruction included more noise so the force profiles are noisier. The

force profiles match the contact mode force profile well except for the last few angstroms

near the surface. The trend to higher force of the profile on the near surface side is also

evident in Figure 6.16b and Figure 6.12a. It is the result of compounded inaccuracies

164

-2420

-2420

-2

543210

b

c Advancing Force Profile Receding Force Profile Equilibrium Force Profile

420

aFo

rce

(nN

)

Tip-Sample Distance (nm)

Figure 6.16 – Reconstructed force profile for hydroxyl surface data in Figure 6.7c for three tapping amplitudes (a-c). The advancing (light gray) and receding (dark gray) show hysteresis in the contact region and the receding trace has significantly more adhesion (c).

from the noise. Likewise, a trend to more negative force is seen on the end of the profile

far from the surface in b and c. Hysteresis is evident both in the hard sphere repulsion

and the attractive regions near the surface. The interaction is extremely stiff therefore the

fore profiles are only qualitatively correct.

The receding force profile in c exhibits an oscillatory force that does not originate

from solvent organization. The large impulse applied to the tip by contact with the

surface excited the second order mode of the cantilever motion. The period of the

oscillations between force profiles is constant in time but the distance varies since the tip

sweeps a full period in 60.6 µs but the pulling rate is dependent on the amplitude of

oscillation. A method of compensation could be envisioned which uses the interaction

165

force time course to drive a second oscillator with the stiffness, damping and mass

characteristics of the second order mode. The resulting position could be subtracted from

the deflection signal to obtain the true motion of the first order mode. Possible

difficulties with this correction include obtaining accurate stiffness, damping, mass, and

detector sensitivity values for the second order mode and over excitation of the mode by

noise that has been amplified by the two derivatives calculated to obtain the acceleration.

Tapping Mode Force Profile Reconstruction is a useful method for obtaining

spatial information about the sources of energy dissipation between the tip and sample at

different timescales. It uses the deflection time course and the cantilever parameters to

calculate the instantaneous tip-sample interaction force, which can be parsed and

averaged to obtain the advancing and receding force profiles. TMFPR is quantitative and

coupling the technique to Energy Dissipation Force Curves increases precision. TMFPR

was used to investigate the mechanism of energy dissipation between functionalized

SAM surfaces in solution. The following section discusses the results and proposes

specific mechanisms of energy dissipation.

6.6 Mechanism for Energy Dissipation

Energy Dissipation Force Curves and Tapping Mode Force Profile Reconstruction

are exceptional techniques for investigating time-dependence of interfacial and

intermolecular interactions. EDFC and TMFPR were used quantitatively and spatially to

investigate the mechanism of energy dissipation for functionalized SAM surfaces in

solution. The first important observation is the long-range hysteresis, extending far into

the solvent, observed in figure 6.14b, for tapping in the electrostatic double layer of the

high pH carboxyl SAM. From the very successful DLVO theory of electrostatic forces, it

166

is known that the electrostatic double layer originates form ordering of the ions in the

solution. The long-range hysteresis, conclusively revealed with TMFPR, is caused by the

disruption of ions in the double layer and their slow diffusion back to an equilibrium

distribution. The slow diffusion results from the size of the ions being relatively large

since they have strongly associated solvation shells and the ionic distribution extends

over nanometers, which requires more time to send information through the distribution

to reach equilibrium. The data from tapping within an electrostatic double layer establish

the expected observations of tapping against long-range ionic interactions. Figure 6.14

also reveals that the tapping tip contacted the hydration layer, which is similar for the

carboxyl surfaces at both pH values (Figure 6.2). The contact hysteresis is another source

of energy dissipation for the high pH carboxyl terminated SAM surface which will be

discussed with the low pH contact data below.

Equally important as the energy dissipation from the ionic reorganization is the

lack of energy dissipation observed for tapping in the attractive regime of the hydroxyl

terminated SAM. It was observed in chapter 4 that the long-range attractive forces (1 nm

from the contact region) near the hydrophilic SAM surfaces originate from solvent-

solvent interactions. Tapping interactions that exclusively sampled the attractive region

experienced little energy dissipation or showed no hysteresis on the tapping mode force

profiles.* Therefore, the solvent configuration involved in the long-range attractive

forces is able to rearrange quickly after being disrupted by the tip motion. The solvent

molecules are not required to diffuse long distances instead they are only required to

* The best TMFPR curves for attractive regime tapping were not shown since the excitation of the second order mode, when tapping in the repulsive regime for the same Q and free tapping amplitude, made the data uninformative. The data chosen for Figure 6.16 are a compromise between the behavior in the attractive and repulsive regimes.

167

rotate and slightly translate. This also further confirms the liquid behavior of the ordered

water molecules since the molecules are free to move quickly.

Tapping on the electrostatic double layer establishes the dissipation expected for

ionic interactions and tapping in the attractive regime of the hydroxyl interaction reveals

the lack of energy dissipation for solvent reordering. These two observations show that

the origin of the low-range attractive force for the low pH carboxyl surface is not ionic.

The equilibrium force profiles in figure 6.2 have quantitative differences between the low

pH carboxyl and hydroxyl terminated SAM surfaces. The carboxyl attractive forces are

much longer-range than the hydroxyl surface forces. Since the hydroxyl surface

attractive forces originate through the ordering of the water solvent near the surface and

the results from chapter 4 limit the distance scale of the ordering mechanism, the longer-

range attractive forces of the low pH carboxyl surface is perplexing. The contact region

for the two surfaces is distinctly different since the carboxyl surface has a soft contact

region and the hydroxyl surface contact is immeasurably stiff. The extremely high

stiffness of the hydroxyl terminated surface implies that the contact is between the bare

SAM surfaces while the soft contact region of the carboxyl terminated SAM surface

supports the mechanisms proposed by Pashley and Lange of removal of tightly bound

solvent from imbedded counterions.5,6 The ionic mechanism is further supported by the

low pKa of the carboxyl endgroups. A low pKa material will have significant ionic

character. The ionic origin of the repulsive contact forces makes it tempting to suggest

that the long-range character of the attractive forces could also be ionic in character. This

hypothesis is not supported by the energy dissipation data since little energy dissipation is

observed when tapping in the attractive regime for either the Energy Dissipation Force

168

Curves or the reconstructed tapping mode force profiles. The fast reorganization of these

long-range forces must result form solvent-solvent interactions similar to thos

experienced by the solvent near the hydroxyl-terminated surface. Unfortunately, the

explanation for the difference in length scales remains unknown.

The discussion above focused primarily on the origin of the energy dissipation for

long-range interactions. The most significant energy dissipation resulted from repulsive

short-range forces for both the hydroxyl and carboxyl terminated SAM surfaces. The

energy dissipation for the carboxyl surface at low pH increased during the transition to

the repulsive regime, which is confirmed by the localization of the energy dissipation to

the contact region for the reconstructed force profiles in Figure 6.15. A possible

mechanism for the energy dissipation is that the strongly bound surface ions reorient, due

to the pressure, and lose buffering solvent molecules between the surface and ion or other

tightly bound water molecules. The underlying SAM headgroups may also reorganize

from the shift in the ion positions in the inner helmhotlz plane. The rearrangements of

the large confined headgroups require more time leading to more energy loss. The

relatively small losses for the lowpH carboxyl surface could be caused by the soft contact

region, which does not produce as strong of repulsive forces as the stiff hydroxyl

interaction. As a result, the contact area is smaller reducing the amount of

reorganization.

The rearrangement mechanism for the carboxyl surface at low pH also applies at

high pH. As the tip comes into contact with the surface at high pH, ions are pushed into

contact with the surface. The confinement and necessity of charge neutrality causes the

bulk solution pH to become irrelevant. As a result the contact interactions, with tightly

169

bound solvated ions, become very similar. The reconstructed force profiles in Figures

6.14 and 6.15 confirm the similarities in the contact hysteresis for the carboxyl surface.

The differences in energy dissipation between the two pH values in the EDFC is the

added dissipation from tapping through the electrostatic double layer.

Energy dissipation for hydroxyl surfaces after contact is significant. The strong

adhesion of the equilibrium force profile reveals the great energetic advantage to

removing the solvent from the interfacial region. Similarly, the stiff contact region is a

result of the bare SAM surfaces contacting each other. The lattice match between the two

SAMs is better than the lattice mismatch between the water structure and the SAM

surfaces such that the surface may rearrange to facilitate interfacial bonding. The large

quantitative difference between the hydroxyl and carboxyl terminated SAMs could be

due to the stiffer contact region leading to higher impact forces and more contact area for

each tap. This is confirmed by increasing the Q of the cantilever. At higher Q, the

cantilever cannot be modulated by outside forces as easily. This will cause the repulsive

forces experienced by the tip to become more similar for the hydroxyl and carboxyl

surfaces as evidenced by the reduction of the relative difference between Figures 6.7a and

c.

Energy Dissipation Force Curves and Tapping Mode Force Profile Reconstruction

are powerful tools for elucidating the mechanism of energy dissipation. Long-range ionic

ordering interactions are slow and cause energy dissipation for the carboxyl-terminated

SAM surfaces at high pH. Conversely, the long-range attractive forces for the hydroxyl

and low pH carboxyl terminated surfaces do not have energy dissipation since the

solvent-solvent interactions are extremely quick. Lastly, contact between the surfaces

170

causes rearrangement of the SAM which is slow and is the largest source of energy

dissipation. Further experiments using EDFC and TMFPR will significantly increase

knowledge about intermolecular and interfacial interactions.

6.7 The Phase Signal and Energy Dissipation

The previous sections reveal the power and versatility of using Energy

Dissipation for detailed analysis of the chemical and physical properties of interfacial and

intermolecular interactions. Energy Dissipation Force Curves and Tapping Mode Force

Profile Reconstruction was used to investigate quantitatively the mechanism of energy

dissipation for functionalized SAM surfaces in solution. Energy Dissipation Force

Curves used the phase, ϕ1, and amplitude, A1, signals from a lock-in amplifier to compute

the energy dissipation signal in equation 6.11. Interestingly, the development of energy

dissipation originated from efforts to understand the physical origin of the phase signal.7

For constant tapping amplitude changes in the phase signal result exclusively from

changes in energy dissipation. However, the amplitude is rarely constant since the tip

encounters features and there is time delay in the integral gain of the feedback loop.

Therefore, the phase becomes a function of both the topography and the energy

dissipation in the sample. Isolating the energy dissipation information from the

complicated phase signal is of great utility to AFM as an analysis technique.

Both dissipative and non-dissipative interactions cause a change to the phase

signal. The relationship between the energy dissipation, phase and amplitude is

expressed in equation 6.11. Without energy dissipation the phase and amplitude are still

dynamic variables. Using the simple assumptions that f=f0 and zero tip-sample energy

dissipation, equation 6.11 simplifies to

171

( )free

nn

AA

A

11

1

2

1sin∑

≥=ϕ or (6.18)

( )freeA

A

1

11sin ≈ϕ (6.19)

for interactions at high Q. The phase signal changes when the tip-sample interaction

causes a change in amplitude. As phase images are collected the changes in topography

require the Z-piezo feedback loop to respond to keep the amplitude at the setpoint. The

time delay and finite gain in the feedback loop leads to residual error or the amplitude

signal. The amplitude signal is often a crisp image of the edges of topographical features.

Because the phase signal is a function of the amplitude, the error signal couples into the

phase signal and is displayed on both channels. Many think that the phase signal more

sensitive to surface features than the amplitude channel. By manipulating equations 6.12,

5.15, and 5.16, the signal to noise ratios for the phase and amplitude signals of non-

dissipative interactions is

AfreeAfree

SNRAA

NAA

NSNR

1

1

1

211 ===

ϕϕ

ϕ, (6.19)

which shows that the sensitivity of the phase signal is lower than the sensitivity of the

amplitude signal. Moreover, the transfer of noise from the amplitude channel to the

phase channel, as seen in Figure 5.12, further reduces the sensitivity to topographical

features making phase imaging less desirable for imaging topography.

Many tapping interactions do contain energy dissipation. As a result, the phase

signal is distorted and exaggerated. Most energy dissipation signals has complex

physical origins that are very poorly understood but like the surface force it is highly

dependent on the interaction area. The variations in interaction area are substantial when

172

surface roughness is on the order of the tip size. These variations can lead to phase

signals that are very challenging to interpret. Furthermore, the coupling of the error

signal to the phase makes meaningful interpretation of phase data more difficult.

Separating the components into the energy dissipation and topography (height,

amplitude) will greatly advance the usefulness of AFM for surface analysis.

6.8 Energy Dissipation Imaging

Energy Dissipation isolates physical, chemical, and biological interactions from

the topography causing Energy Dissipation Imaging to more sensitive to the interactions

of interest than phase imaging. Phase imaging has been on the forefront of AFM imaging

since its inception in 1997,16 due to its perceived ultra high sensitivity to surface

topography and physical characteristics. Phase changes are caused by energy loss and

tapping amplitude modulation through tip-sample interaction. By using energy

dissipation instead of phase, the height and amplitude channels can be reserved for

inspection of topography while the energy dissipation channel will investigate the nature

of the tip-sample interaction. Similarly, energy dissipation imaging becomes

significantly more sensitive to surface interactions since it removes the noise associated

with coupling of the topography to the phase.

The energy dissipation image can be obtained from equation 6.11 in real time

with a DSP chipset requiring little more computation power than what is already required

to perform the lock-in function. The only requirement is to first input the cantilever

parameters. Furthermore, close inspection of equation 6.11 reveals that the accuracy of

the cantilever parameters is not necessary for qualitative interpretation of the data (most

AFM imaging) since both the spring constant and quality factor are coefficients. An

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inaccurate value for the resonant frequency can cause the coupling of the topography to

the phase signal to not be completely compensated. Fortunately, an adequate estimate of

the resonant frequency is readily available using the cantilever tune.

The utility of Energy Dissipation Imaging was confirmed through the collection

of energy dissipation images of carboxyl and hydroxyl terminated patterned SAM

surfaces. Chemically functionalized surfaces were prepared by stamping hydroxyl

terminated alkane thiol on flat gold surfaces. Flat gold surfaces were prepared by

thermally evaporating gold onto mica. Freshly cleaved mica was placed in the evaporator

and pumped overnight using a cryopump to 8×10-8 torr. Three hundred angstroms of

gold was evaporated at 2 Å/s. After venting, the gold surfaces were stored in a dessicator

and later annealed in a tube furnace at 300 C for 1 hour before use. Soft lithography

techniques were used to make the µ-pattern using a PDMS stamp with 1um lines spaced

by 1 um. Hydroxyl terminated alkane thiol solution was inked onto the stamp and either

allowed to dry or blown dry with clean nitrogen. After drying, the stamp was gently but

quickly applied to the gold surface. Hesitation may cause smearing of the pattern. After

letting stand for 10 s the stamp was removed, reinked, dried and applied after rotating 90

degrees. After the two stampings, the hydroxyl terminated SAM was allowed to form for

10 minutes then it was submerged in carboxyl terminated alkane thiol solution so that the

remaining bare gold can form a carboxyl terminated SAM. The procedure lead to a

pattern of 1 µm × 1 µm squares of carboxyl terminated SAM surrounded by hydroxyl

terminated SAM.

The magnetic FESP Si probe was prepared using the procedures outlined in

Chapter 5 and functionalized with hydroxyl terminated SAMs. Breifly, 30 µm diameter

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SmCo5 chunks were glued on the back of the cantilever with both optical cure and heat

cure glue. Blunting the tip by coating the apex with glue promotes adhesion of the

chromium-gold surface by reducing the strain associated with a small radius of curvature.

Magnetic tips were used for their superior drive characteristics and stability.

Typical cantilever actuation in fluid involves acoustic excitation. The cantilever becomes

a coupled spring to the fluid and fluid cell chamber. The modified actuation leads to ill-

defined results and the common “forest of peaks”. Magnetic actuation applies a well-

defined, easily interpretable drive signal to the cantilever. A quantitative measure of the

cantilever drive force is important for energy dissipation imaging. Also, the drive

amplitude was found to be more stable over time and not dependant on ambient pressure,

quantity of fluid in the cell, and temperature, which are common difficulties of using

acoustic excitation. Lastly, magnetic actuation made Q-control readily available if tip-

sample interaction force needed to be tuned.

Energy dissipation images were collected by first recording amplitude and phase

images. The Digital Instruments (DI) software and controller maintained feedback using

the amplitude signal from the extender while scanning. A Stanford Instruments SR830

DSP lock-in amplifier more accurately computed the amplitude and phase while

maintaining a lock on the DI tapping drive signal. The amplitude and phase were

recorded with the DI software through auxiliary data channels along with the height data.

The free amplitude and phase were recorded on the same image by increasing the setpoint

until the feedback lost contact with the surface. After recording a few scan lines the

setpoint was restored and imaging continued. The phase image was calculated using Igor

Pro. Only the fundamental was used in the calculation since multiple lock-in amplifiers

175

were not available to record the amplitude of the higher harmonics. The harmonics are

expected to produce a correction of 10% to the energy dissipation values at the setpoint

and Q used.

The amplitude, phase and energy dissipation images for a patterned SAM surface

are shown in Figure 6.17. The phase and energy dissipation per tap are reported in

radians (1 radian = 57.3 degrees) and atto(10-18)Joules. The free tapping amplitude was

3.3 nm while the amplitude setpoint was 2.6 nm. The “flat gold” surface was

polycrystalline with facets in some regions but much of the surface contained atomic

steps and the total rms roughness was 9 Å/µm2. The most severe fault is 2.3 nm high,

which is located in the middle of the carboxyl patch. The stamp deposited contaminates

(white blotches in phase image) along the edge of the stamped region. A black box

delineates the boundary of the carboxyl region and also outlines the contamination.

The energy dissipation image shows considerably more contrast than the phase

image. The changes in topography are depicted in the amplitude and phase images (a and

b). The phase contrast between the carboxyl and hydroxyl terminated regions due to

dissipative interactions are barely perceptible even for a low surface roughness of 9

Å/µm2. The energy dissipation image (c) isolates the chemical interactions and

significantly increases the contrast between the chemically and physically similar

materials.

The phase and energy contrast is measured by calculating the difference between

the means of the hydroxyl- and carboxyl-terminated sections. The carboxyl Region of

Interest (ROI) included a box with edges 0.5 µm long that are parallel to the black box in

Figure 6.17. The hydroxyl ROI included the two faint white right triangles on the right

176

177

1.0

0.5

µm

1.5

1.0

0.5

µm

1.5

1.0

0.5

µm

1.51.00.50.0 µm

1.5

a 2.8

2.7

2.6

2.5

2.4

2.3

b 1.15 rad

1.10

1.05

1.00

c 0.9 aJ

0.8

0.7

0.6

0.5

Figure 6.17 – (a) Amplitude, (b) phase, and (c) energy dissipation images of a patterned SAM surface of hydroxyl surrounding a carboxyl square. The black square highlights the edges of the pattern. The topography is coupled into the amplitude and phase but compensated in the energy dissipation leading to significantly more contrast.

side of the energy image. The resulting mean and standard deviation for the ROIs are in

the following table. The signal to noise ratio for the energy dissipation is almost 3 times

higher. The difference in SNR will become greater for surfaces that are rougher, where

more topography is coupled to the phase signal by inadequate feedback response.

Hydroxyl Carboxyl Contrast SNR

Phase 1.082±0.032 rad 1.055±0.047 rad 0.027±0.039 rad 0.7

Energy Dissipation 0.714±0.061 aJ 0.592±0.063 aJ 0.122±0.062 aJ 2.0

Table 6.1- Comparison of signal and signal to noise ratio for Phase and Energy Dissipation Imaging.

Energy Dissipation Imaging increases the sensitivity to interactions of interest but

more importantly it is intuitively simple and quantitative. The convolution of the

dissipation with the topography and large phase changes associated with the

attractive/repulsive regime transition make phase imaging very challenging to interpret.

Conversely, Energy Dissipation Imaging results can be directly related to physical

processes such as adhesion hysteresis, visco-elasticity, and plasticity, which are easier to

understand and meaningful. Lastly, the results are quantitative which permit comparison

within and between images unlike most AFM imaging. Energy Dissipation Imaging is an

extremely powerful technique for sensitively imaging surface features quantitatively.

6.9 Conclusion

Colloid and Interface science has been investigating questions about interparticle

interactions for many years but numerous basic questions about the molecular mechanism

of interfacial forces have remained unanswered. The Atomic Force Microscope has been

on the forefront of intermolecular and interfacial science for the past decade resulting

from its sensitivity and molecular scale probe. In spite of the flurry of interest and effort

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molecular based models and quantitative analysis are strikingly absent. In this chapter,

Energy Dissipation Force Curves, Tapping Mode Force Profile Reconstruction, and

Energy Dissipation Imaging were developed for quantitative interfacial and

intermolecular analysis. These methods are powerful tools for investigating the

biological, chemical, and physical properties of surfaces. Specifically, Energy

Dissipation Imaging may become an important AFM procedure since it is intuitively

interpretable, quantitative, and isolates the interaction of interest.

Energy Dissipation Chemical Force Microscopy uses these newly developed

methods to probe the chemical properties of chemically well-defined surfaces.

Specifically, hydrophilic Self-Assembled Monolayer surfaces terminated with hydroxyl

or carboxyl groups were studied to understand the molecular mechanisms of interfacial

forces resulting from solvent-surface interactions. It was discovered that solvent

molecules order near the SAM surface and can reorder quickly after being perturbed.

The charged carboxyl surface at high pH, caused long-range ionic ordering that did not

respond as quickly causing energy dissipation when disturbed. Contact between the

SAM surfaces led to rearrangement of the SAM causing significant energy dissipation.

These results are of great interest to many areas of science. For example, the cytoplasm

is packed with organelles in close proximity with membranes separated by only a few

molecular layers of water. The reorganization properties of water are extremely

important for determining cell mobility, cytoplasmic traffic, and specific identification.

Also, understanding the energy dissipation between surfaces is the exclusive focus of

tribological and rheological studies. Energy Dissipation techniques are very powerful

tools for investigating intermolecular and interfacial phenomena.

179

6.10 References

1. Merkel, R., Nassoy, P., Leung, A., Ritchie, K. & Evans, E. Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy. Nature 397, 50-53 (1999).

2. Smith, B. L. et al. Molecular mechanistic origin of the toughness of natural adhesives, fibres and composites. Nature 399, 761-763 (1999).

3. Isaelachivili, J. & Berman, A. Irreversibility, Energy Dissipation, and Time Effects in Intermolecular and Surface Interactions. Israel Journal of Chemistry 35, 85-91 (1995).

4. Oberhauser, A. F., Marszalek, P. E., Erikson, H. P. & Fernandez, J. M. The molecular elasticity of the extracellular matrix protein tenascin. Nature 393, 181-185 (1998).

5. Colic, M., Franks, G. V., Fisher, M. L. & Lange, F. F. Effect of counterion size on short range repulsive forces at high ionic strengths. Langmuir 13, 3129-3135 (1997).

6. Pashley, R. M. Hydration Forces between Mica Surfaces in Electrolyte-Solutions. Advances in Colloid and Interface Science 16, 57-62 (1982).

7. Cleveland, J. P., Anczykowski, B., Schmid, A. E. & Elings, V. B. Energy dissipation in tapping-mode atomic force microscopy. Applied Physics Letters 72, 2613-2615 (1998).

8. Tamayo, J. Energy dissipation in tapping-mode scanning force microscopy with low quality factors. Applied Physics Letters 75, 3569-3571 (1999).

9. French, A. P. Vibrations and Waves (Norton, New York, 1971).

10. Tamayo, J. & Garcia, R. Relationship between phase shift and energy dissipation in tapping-mode scanning force microscopy. Applied Physics Letters 73, 2926-2928 (1998).

11. Albrecht, T. R., Grütter, P., Horne, D. & Rugar, D. Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity. Journal of Applied Physics 69, 668-673 (1991).

12. Gotsmann, B., Anczykowski, B., Seidel, C. & Fuchs, H. Determination of tip–sample interaction forces from measured dynamic force spectroscopy curves. Applied Surface Science 140, 314-319 (1999).

13. Holscher, H., Allers, W., Schwarz, U. D., Schwarz, A. & Wiesendanger, R. Determination of tip-sample interaction potentials by dynamic force spectroscopy. Physical Review Letters 83, 4780-4783 (1999).

180

181

14. Lantz, M. A. et al. Quantitative Measurement of Short-Range Chemical Bonding Forces. Science 291, 2580-2583 (2001).

15. Stark, M., Stark, R. W., Heckl, W. M. & Guckenberger, R. Inverting dynamic force microscopy: From signals to time- resolved interaction forces. Proceedings of the National Academy of Sciences of the United States of America 99, 8473-8478 (2002).

16. Magonov, S. N., Elings, V. & Whangbo, M.-H. Phase imaging and stiffness in tapping-mode atomic force microscopy. Surface Science 375, L385-L391 (1997).

Appendix A.1 General Techniques and the Digital Instruments Multimode AFM

A.1.1 Contact Mode Imaging

The first implementation of AFM used contact mode to image an aluminum oxide

surface.1 In contact mode, the tip is kept in constant contact with the surface and dragged

across for each scan line to make the image. A feedback loop is used to control the

position of the surface along the Z axis and keep the deflection at a specified setpoint. As

the surface experiences raised features on the surface the deflection increases beyond the

setpoint. The feedback loop amplifies this error signal and applies it to the Z-piezo to

cause the surface to back away from the tip and restore the cantilever to the setpoint

deflection. Proportional, Integral, Differential (PID) amplifiers are included in the

feedback loop for stability. By adjusting the gain of each amplifier stable feedback can

be achieved that minimizes the deflection error. With stable feedback, the voltage

applied to the Z-piezo is a good measure of the topography of the surface. During

contact mode no forces are applied to the cantilever other than the tip-sample interaction

and the thermal fluctuation force resulting in Brownian noise.

A.1.2 Contact Mode Force Curves

Contact mode can be used for imaging and also for making force curves, which

are the deflection as a function of the piezo motion in the Z axis. Force curves are used

to calibrate the detector and measure the force profile as outlined in figure A.1.

Displacements of the surface toward and away from the tip are named trace (gray) and

retrace (black) respecitively. In Figure A.1a, the photodiode signal is flat at large

distances from the surface, since there is little interaction. As the surface is brought

182

-0.4

-0.3

-0.2

-0.1

-0.6-0.4-0.20.0

54321

b

a

setpoint

sensitivity

Z-Piezo Displacement (nm)

Def

lect

ion

(V)

Forc

e (n

N)

-0.6-0.4-0.20.0

4321

Missing information c

traceretrace

Tip-Sample Distance (nm)

Forc

e (n

N)

Figure A.1 – Contact Mode force curve (a) raw photodiode signal and (b) scaled force as a function of Z-piezo displacement. The contact region is used to determine the detection sensitivity. (c) The tip-sample distance is calculated by subtracting the deflection from the Z-piezo distance to make a force profile.

closer the attractive forces start to deflect the cantilever until the stiffness of the

interaction is greater than the spring constant and the tip jumps to the surface (3 nm).

The surface continues to move toward the cantilever and the tip is deflected upward (1-3

nm) since the compliance of the cantilever dominates this region. The setpoint for

imaging is typically chosen in the tip-sample contact region at a deflection above the

equilibrium deflection without tip-sample interaction, as depicted by the arrow in

Figure A.1. As the piezo retracts, the tip stays in contact with the surface (3 to 4 nm)

until enough force is stored in the cantilever to overcome the adhesion. The contact

183

region is a good measure of the sensitivity of the detector since every nanometer of

surface motion leads to a nanometer of tip deflection (assuming no deformation). Using

the ratio of the Z-piezo displacement and the photodiode voltage signal, the force curves

can be converted to deflection. Multiplication by the spring constant leads to the force as

a function of Z-piezo displacement in Figure A.1b. The spring constant is calculated

using the thermal noise of the cantilever as described in the appendix (A.3). The force

profile is the interaction force as a function of tip-sample distance (figure A.1c). The tip-

sample distance is calculated by subtracting the cantilever deflection from the Z-piezo

displacement. The regions where the tip jumps to the surface and away do not contain

information about the force profile. A stiff cantilever is required to probe those

interactions.

A.1.3 Tapping Mode Imaging

Contact mode is excellent for hard samples but loosely bound soft samples can be

easily distorted which led to the development of tapping mode. Tapping mode measures

the surface topography and reduces lateral forces by tapping the tip along the surface

comparable to the use of a cane to avoid obstacles by a visually impaired person.

Cantilever oscillations are excited by a sinusoidal force mechanically applied to the

cantilever. The oscillatory deflection signal is converted to amplitude and phase signals

by a lock-in amplifier using the tapping mode drive signal as the reference. The feedback

loop uses the amplitude signal and a setpoint below the free amplitude of oscillation to

keep the tip tapping on the surface. The phase is often used as a measure of tip-sample

interaction while imaging in tapping mode. Other AC AFM techniques exist but they are

typically reserved for work in vacuum at high Q.2

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A.1.4 Tapping Mode Force Curves

Force curves in tapping mode are more complicated than those in contact mode.

Tapping mode force curves plot amplitude and phase as a function of Z-piezo

displacement as shown in figure A.2. As the surface is brought near the cantilever, the

tip-sample interaction causes the amplitude to be reduced. Attractive forces can cause the

amplitude to be reduced because of the non-linear nature of the tip-sample interaction

with the oscillating tip. When the forces are attractive the phase lag between the tip and

the drive force will be greater because the surface is holding onto the tip and not letting it

return immediately. As the surface continues to advance toward the cantilever the tip

pushes through the attractive region to experience repulsive forces. The amplitude jumps

higher and the phase lag is greatly reduced at this instability. The position of the

instability relative to the free tapping amplitude is a function of the Q of the cantilever

2.0

1.5

1.0

0.53.0

2.0

1.0

0.065432

b

a

Z-Piezo Displacement (nm)

Am

plitu

de (n

m)

Phas

e (r

ad)

Figure A.2 – Tapping Mode force curve (a) Amplitude and (b) phase signals from the lock in amplifier. The phase change is positive when tapping in the attractive regime. The amplitude increases in value and the phase changes becomes negative at the transition to repulsive regime.

185

resonance, drive force, and the shape and depth of the force profile. Because changes in

force profile occur during scanning it is important to choose the setpoint to be far from

the instability either in the attractive or repulsive regime so that the instability does not

occur during imaging.

A.1.5 Digital Instrumen

ts Multimode AFM

with a IIIa controller was used for all

experim

.

in 3

A Digital Instruments Multimode AFM

ents in this thesis. A sketch of the instrument components is shown in Figure

A.3. The multimode microscope is a relatively stable design where the head mounts

securely on the scanner body. The head contains the laser, tip holder, and photodiode

This design is less susceptible to drift and vibrational noise than other commercially

available configurations. The scanner contains the piezo tube for precise positioning

dimensions. An EV scanner was used for most experiments for its high precision yet

versatile scan area (10×10 µm). The scanner mounts on the base, which includes some

Figure A.3 – Sketch of the experimental setup.

Controller

Microscope

Computer Computer

ExtenderSAM SAM

Head Scanner

186

signal processing and engagement electronics. The extender contains signal processing

electronics and its main function is as a pseudo-lock-in amplifier for tapping mode. The

controller contains the ADC and DAC converters along with the high voltage piezo

drivers. The controller sends the digitized signals to a DSP board in the computer, which

controls the feedback in conjunction with the input variables from the software. The

software displays the AFM data and control windows. The Signal Access Modules

(SAM) are used for monitoring or modifying the information passing between the

microscope and the controller. Other signal processing equipment included an

oscilloscope, function generator, spectrum analyzer, DSP lock-in amplifier, and a high

speed ADC card (NI5911) controlled with Labview.

187

A.2 Data Collection with National Instruments 5911

Deflection time courses were recorded using a National Instruments 5911 high

speed Analog to Digital converter. The 5911 is capable of sampling 100 MS/s at 8-bit

resolution. Higher resolutions (17.5 bits at 1 Ms/s) are achieved for slower sampling

rates using over sampling and digital filtering. The following Labview data collection

routine was constructed from the NI Scope suite of Virtual Instruments.

188

Figure A.4 – Labview code for data collection scheme of NI 4911.

189

A.3 Cantilever Calibration

The cantilever properties can be reliably calculated from the transfer function by

measuring the thermal noise power. At thermal equilibrium, the energy imparted to the

cantilever by collisions with the surrounding medium equals the energy dissipated by the

cantilever to the surroundings. The applied force is spectrally flat* and dependent on the

damping of the cantilever,3,4

iBTbkF 40 =† (A.1)

where F0 is the thermal force, kB is Botlzmann’s constant, T is the temperature, and b is

the cantilever damping. The cantilever responds to the thermal forces depending on its

resonance characteristics. Assuming the simple harmonic oscillator (SHO) model of a

mass on a spring, the equation of cantilever motion is

tieFxkxbxm ω0=⋅+⋅+⋅ &&& , (A.2)

where k, b and m are the cantilever stiffness, damping, and mass respectively, x , ,

and , are the position and its respective time derivatives, and ω and t are frequency and

time. The cantilever parameters can be rewritten in the more familiar variables of spring

constant, k, resonant frequency,

x&

x&&

mkf

π21

0 = , (A.3)

and quality factor, bkmQ = . The resonant frequency determines at what frequency

the peak is centered and the Q determines the width of the peak. The gain response of the

* The flat spectral character of the force noise is an approximation. The damping is spectrally dependant for some liquids, which causes the force noise to also be spectrally dependant. The resulting anharmonicity is clearly seen in the poor fit of the SHO approximation to the transfer function at lower frequencies in water. † The equation for thermal force, F0, is derived from the equipartition theorem.

190

cantilever to the spectrally flat impulse is called the transfer function. Its shape is a

function of the three independent cantilever parameters and can be calculated by solving

the equation of motion for amplitude, A, using the ansatz, ( )ϕω −= tiAex , to get,

2

2

0

2

20

2

2

20

2

1

+

−

=

Qff

ff

kF

A . (A.4)

Driving the cantilever with a sinusoidal driving force while sweeping the frequency is

one way to measure the transfer function. Another method is to allow the thermal

fluctuations to drive the cantilever and measure the noise on a spectrum analyzer, which

is called the noise power spectrum. The cantilever parameters are determined by fitting

the curve with the transfer function equation. The value of k can also be obtained by

integrating the noise power spectrum since

kTkxdHzA B==∫

∞2

0

2 . (A.5)

This equation is a result of the equipartition theorem where the energy in one degree of

freedom, 2TkB , equals the energy of the cantilever,

2xk . The integrated value for k

is a more precise measurement method but accuracy is determined by the cantilever

sensitivity (nm/V), which is 5%. A robust Igor Pro script to calculate the cantilever

parameters is copied below.

Macro spring_analysis(PSD_wave, temperature) string PSD_wave variable temperature=300 prompt PSD_wave, "Which power spectral density wave do you want to use?", Popup, Wavelist("*",

";", "") prompt temperature, "What is the temperature of the experiment?" variable int_spring duplicate/o $PSD_wave integrator

191

deletepoints 0, numpnts($PSD_wave)/50 , integrator integrate integrator int_spring = 1.381e-23*temperature/integrator[numpnts($PSD_wave)-2] print int_spring variable width, Q make/o/n=5 parameters Wavestats/Q/R=[numpnts($psd_wave)/50, numpnts($psd_wave)] $psd_wave findlevels/q/B=(numpnts($psd_wave)/100)/R=[numpnts($psd_wave)/50, numpnts($psd_wave)]

$psd_wave, V_max/2 width=W_findlevels[1]-W_findlevels[0] if(V_flag==2) Q=sqrt(V_max/$psd_wave[100]) else Q=V_maxloc/width endif parameters[0]=temperature parameters[1]=int_spring parameters[2]=V_maxloc parameters[3]=Q parameters[4]=4e-27 variable/G V_fitmaxiters V_FitTol V_Fitmaxiters = 100; V_FitTol = .000001 //print parameters[0], parameters[1], parameters[2], parameters[3], parameters[4], int_spring FuncFit/Q/H="10000" KFQ_transfer_function_off parameters

$psd_wave[numpnts($psd_wave)/100,numpnts($psd_wave)] integrator = KFQ_transfer_function(parameters,x) Integrate integrator int_spring=1.381e-23*temperature/integrator[numpnts($psd_wave)-2] print parameters[0], parameters[1], parameters[2], parameters[3], parameters[4], int_spring print "k="+num2str(parameters[1])+", f_0="+num2str(parameters[2])+",

Q="+num2str(parameters[3])+", m="+num2str(parameters[1]/(4*pi^2*parameters[2]^2))+", and b="+num2str(parameters[1]/(2*pi*parameters[2]*parameters[3]))

duplicate/o $psd_wave $"fit_"+psd_wave $"fit_"+psd_wave = KFQ_transfer_function_off(parameters, x) display $"fit_"+psd_wave, $psd_wave ModifyGraph mode($"fit_"+psd_wave)=3,marker($"fit_"+psd_wave)=19,

rgb($"fit_"+psd_wave)=(0,15872,65280) killwaves W_findlevels, integrator, W_paramconfidenceinterval end Function KFQ_transfer_function(w,x) wave w; variable x // The Temperature is w[0] // The spring constant, k, is w[1] // The resonant frequency, F_0, is w[2] // The quality factor, Q, is w[3] return (2*1.381e-23*w[0])/(pi*w[1]*w[2]*w[3])*1/((1-(x/w[2])^2)^2+(x/(w[2]*w[3]))^2) End Function KFQ_transfer_function_off(w,x)

192

wave w; variable x // The Temperature is w[0] // The spring constant, k, is w[1] // The resonant frequency, F_0, is w[2] // The quality factor, Q, is w[3] // The offset for noise is w[4] return (2*1.381e-23*w[0])/(pi*w[1]*w[2]*w[3])*1/((1-(x/w[2])^2)^2+(x/(w[2]*w[3]))^2)+w[4] End

193

A.4 Cantilever Dynamics Simulations

Cantilever simulations are excellent for modeling hypotheses about the physical

origin of phenomena and testing the robustness of many of the data analysis methods

developed in this thesis. The core of the simulation is to use the wave equation for

cantilever motion to numerically solve for the trajectory. The equation of motion

expressed in units of force is,

( ) ( ) ( ) ( ) ( ) nid FtxFtFtxmtxbtxk ++=⋅+⋅+⋅ ,sin ω&&& (A.6)

where the three cantilever terms are on the LHS and the drive force, tip-sample

interaction, and the thermal force noise are on the RHS. The cantilever parameters are

input at the beginning of the simulation. The thermal force noise was computed by using

a psuedo-random number generator that produces gaussian noise that is scaled to have a

standard deviation of

TbBkdevs B4_ = , (A.7)

where kB is botlzmann’s constant, T is the temperature, b is the damping, and B is the

bandwidth, 1/(2dt), of the simulation. The drive force is calculated from inputting the

resonant frequency, tapping frequency, and desired amplitude of oscillation into equation

6.9. Starting with , t ,0= 0)( =tx , and 0)( =tx&

)( dttx&&

, the acceleration is calculated.

Calculation of the velocity, )() tx&( dttx& +⋅=+ , and position,

, follow, which completes one iteration. The loop is iterated

until the desired number of points has been calculated. The equilibrium position of the

cantilever relative to the base is determined and compared to the position of the

cantilever, , to calculate the potential term for each iteration. A force curve is

)(tx+)()( dttxdttx ⋅=+ &

)(tx

194

simulated by smoothly ramping the cantilever equilibrium position. An Igor Pro script

for cantilever dynamics simulations is copied below.

function simulate_cantilever() variable length = numvarordefault("glength", 1000000) prompt length, "Number of points in waves" doprompt "Wave properties", length variable/g glength = length print "The simulation wave length is " + num2str(length) // ******* cantilever properties ********** variable k = NumVarordefault("gk", 1.5) // k = N/m variable f_0 = NumVarordefault("gf_0", 16000) variable Q = NumVarordefault("gQ", 6) variable temperature = Numvarordefault("gtemperature", 300) variable m,b prompt k, "Spring constant" prompt f_0, "Resonant Frequency" prompt Q, "Quality Factor" prompt temperature, "temperature of experiment" doprompt "Cantilever properties", k, f_0, Q, temperature variable/g gk=k variable/g gf_0=f_0 variable/g gQ=Q variable/g gtemperature=temperature m = k/(2*pi*f_0)^2 b=sqrt(k*m)/Q print "The spring constant is "+num2str(k)+", resonant frequency is "+num2str(f_0)+", the Q is

"+num2str(Q)+", the dampening is "+num2str(b)+", and the temperature is "+num2str(temperature)

// timestep depends on the resonancefrequency of the cantilever: // dt=1/sample_pts*f_0 , where sample_pts is the # pts per period variable sample_pts =numvarordefault("gsample_pts", 500) prompt sample_pts, "Number of points per period" doprompt "Timestep properties", sample_pts variable/g gsample_pts=sample_pts variable dt = 1/(sample_pts*f_0) print "The time step is "+num2str(dt)+"and the number of points per period is "+num2str(sample_pts) variable time_length=dt*length // ****************** force field ****************** variable forcewaveprompt string forcewavestring prompt forcewaveprompt, "Use a real wave as the force profile or a default force prophile?", popup,

"Yes; No" doprompt "Force Prophile", forcewaveprompt if(forcewaveprompt ==1) prompt forcewavestring, "What wave do you want to use?", popup, Wavelist("*", ";", "") doprompt "Choose the force profile", forcewavestring duplicate/o $forcewavestring forcetrace else Make/N=5000/D/O forcefield setscale/I x, 1e-12, 1e-8, "m", forcefield

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forcefield = 100*(1e-11/x)^6-.0006*(1e-11/x)^3 //565000*(1e-11/x)^11-.00942*(1e-11/x)^5 // the original coefficients are 465000 and .00242

duplicate/o forcefield forcetrace forcewavestring="forcefield" endif print "The wave used as the force trace is "+forcewavestring display forcetrace dowindow/C graph_of_forcefield SetAxis/w=graph_of_forcefield left, -1e-9,1e-10 // ****************** equilibrium_position ****************** variable equilibrium_position_begin = numvarordefault("gequilibrium_position_begin", 5e-9) variable equilibrium_position_end = numvarordefault("gequilibrium_position_end", 1e-12) prompt equilibrium_position_begin, "Starting position of the simulation" prompt equilibrium_position_end, "Ending position of the simulation" doprompt "Bounds of simulation", equilibrium_position_begin, equilibrium_position_end variable/g gequilibrium_position_begin=equilibrium_position_begin variable/g gequilibrium_position_end=equilibrium_position_end variable equilibrium_position=equilibrium_position_begin print "The start postion is "+num2str(equilibrium_position_begin)+"The end position is

"+num2str(equilibrium_position_end) dowindow/k graph_of_forcefield // ****************** driving force ****************** variable driving_frequency = numvarordefault("gdriving_frequency", f_0) variable driving_amplitude = numvarordefault("gdriving_amplitude", 5e-9) prompt driving_frequency, "Drive Frequency" prompt driving_amplitude, "Drive Amplitude if at resonance" doprompt "Drive Characteristics", driving_frequency, driving_amplitude variable/g gdriving_frequency=driving_frequency variable/g gdriving_amplitude=driving_amplitude print "The drive frequency is "+num2str(driving_frequency)+"and the drive amplitude is

"+num2str(driving_amplitude) variable drive_force=driving_amplitude*k*(driving_frequency/f_0)*sqrt((f_0/driving_frequency-

driving_frequency/f_0)^2+(1/Q)^2) // ****************** initial position and velocity ****************** Make/N=(length)/D/O position Make/n=(length/sample_pts)/O impact //velocity variable vel,acc,position_temp vel = 0; acc = 0 position_temp = equilibrium_position // ****************** the loop ****************** variable counter, oversamploop, oversampling, counter2,F_noise,Fdrive, forcetrace_temp, vel_1,

vel_turn=0//, vel_meas=0 // the counter used in the loop counter=0; oversampling=10; oversamploop=0; counter2=0 do vel_1=vel forcetrace_temp= forcetrace(position_temp) equilibrium_position=equilibrium_position_begin+(equilibrium_position_end-

equilibrium_position_begin)*counter/(length*oversampling) F_noise=gnoise(sqrt(temperature*1.38e-23*2*k/(pi*Q*2*dt*f_0)));

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Fdrive=drive_force*sin(2*pi*counter*dt*driving_frequency) // a sinewave at the driving frequency with amplitude

acc= 1/m *(-k*(position_temp-equilibrium_position) - b*vel + Fdrive+forcetrace_temp+F_noise) vel=vel+acc*dt position_temp=position_temp+vel*dt counter+=1 if(vel*vel_1<0&&vel>0) impact[vel_turn]=forcetrace_temp vel_turn+=1 endif if(oversamploop==oversampling) position[counter2]=position_temp counter2+=1 oversamploop=0 endif oversamploop+=1 while (counter2<(length-1)); position=position-((equilibrium_position_end-equilibrium_position_begin)*p/length +

equilibrium_position_begin) duplicate/o position scaled_position SetScale/P x, 0,oversampling*dt,"", position SetScale/I x, equilibrium_position_begin,equilibrium_position_end,"", scaled_position end

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A.5 Brownian Force Profile Reconstruction

Brownian Force Profile Reconstruction (BFPR) is a very robust technique for

calculating the force profile from a deflection timecourse. The Igor Pro script for BFPR

including the compensation for instrument noise is copied below along with the called

subroutines.

function Brownian_for_noise(deflection) wave deflection print "The name of the wave being analyzed is "+nameofwave(deflection) variable bin_size = NumVarOrDefault("gbin_size",10000) variable numtrace = NumVarOrDefault("gnumtrace", 100) variable tracerange = NumVarOrDefault("gtracerange", 1.5) variable tracestep = NumVarOrDefault("gtracestep", .1) variable temp = NumVarOrDefault("gtemp", 300) prompt bin_size, "Number of points for wave sections used in reconstruction" prompt numtrace, "Number of wave sections in reconstruction" prompt tracerange, "Span of wave sections along distance axis in anstroms" prompt tracestep, "Point spacing of wave sections along distance axis in angstroms" prompt temp, "The temperature during the experiment" doprompt "Enter values for wave reconstruction", bin_size, numtrace, tracerange, tracestep, temp variable sec_portion, num_pnts_sec sec_portion=round(tracerange/tracestep) if(mod(sec_portion,2)==0) num_pnts_sec=sec_portion+1 tracerange=tracestep*sec_portion else num_pnts_sec=sec_portion tracerange=tracestep*(sec_portion-1) endif print "Number of points for wave sections used in reconstruction is "+num2str(bin_size) print "Number of wave sections in reconstruction is "+num2str(numtrace) print "Span of wave sections along distance axis in anstroms is "+num2str(tracerange) print "Point spacing of wave sections along distance axis in angstroms is "+num2str(tracestep) print "the temperature during the experiment was "+num2str(temp) variable/g gbin_size = bin_size variable/g gnumtrace = numtrace variable/g gtracerange = tracerange variable/g gtracestep = tracestep variable/g gtemp = temp tracestep=tracestep*1e-10 tracerange=tracerange*1e-10 //This section finds the standard deviation of the noise for taking off instrument noise from the

deflection data //This process works really well on the harmonic regions. There should be some error in the

anharmonic regions but still less than when instrument noise is still included. wave noisewave string noisewavestring variable noisestd

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prompt noisewavestring, "What wave to use as pure instrument noise wave?", popup, Wavelist("*",";","")

doprompt "Find instrument noise wave", noisewavestring duplicate/o $noisewavestring noisewave wavestats/Q noisewave noisewave = noisewave-V_avg wavestats/Q noisewave noisestd=V_sdev //this section makes a wave that will be the normalization wave by which then has the wave sections

subtracted from it. wave normalizationwave, ln_normalization_hist string normalizationwavestring variable normalizationwaveprompt prompt normalizationwaveprompt, "Use a separate wave as the normlaization wave?", popup, "yes;

no" doprompt "Choice of normalization wave", normalizationwaveprompt if (normalizationwaveprompt == 1) prompt normalizationwavestring, "What wave do you want to use?", popup, WaveList("*", ";", "") doprompt "Choice of normalizationwave", normalizationwavestring duplicate/o $normalizationwavestring normalizationwave else duplicate/o deflection normalizationwave endif display normalizationwave dowindow/C graph_of_normalization_wave variable normalizationwaveamount = NumVarordefault("gnormalizationwaveamount", 25) prompt normalizationwaveamount, "what percent of wave is to be used for normalization?" doprompt "Enter Normalization amount", normalizationwaveamount variable/g gnormalizationwaveamount = normalizationwaveamount dowindow/k graph_of_normalization_wave duplicate/o/r=[0,numpnts(normalizationwave)*normalizationwaveamount/100] normalizationwave

sectionnormalizationwave wavestats/Q sectionnormalizationwave sectionnormalizationwave = sectionnormalizationwave-V_avg variable normalizationstd wavestats/Q sectionnormalizationwave normalizationstd=V_sdev sectionnormalizationwave=sectionnormalizationwave*(sqrt(normalizationstd^2-

noisestd^2))/normalizationstd print "The error from the noise at a distance is "+num2str(noisestd^2*100/normalizationstd^2) make/o normalization_hist histogram/b=-tracerange/2, tracestep, num_pnts_sec sectionnormalizationwave, normalization_hist duplicate/o normalization_hist ln_normalization_hist ln_normalization_hist=-(ln(normalization_hist)-ln(numpnts(sectionnormalizationwave))) ln_normalization_hist*= 1.3806e-23*temp

////This scales the wave so that it is units of energy killwaves normalizationwave, sectionnormalizationwave variable newspan = NumVarOrDefault("gnewspan", 2) prompt newspan, "Span of normalization used after fit, i.e. greatest deflection of wave. in nm" doprompt "Enter values for normalization wave span", newspan newspan=newspan*1e-9 variable norm_portion norm_portion=round(newspan/tracestep) if(mod(norm_portion,2)==0)

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newspan=tracestep*norm_portion else norm_portion-=1 newspan=tracestep*norm_portion endif variable/g gnewspan=newspan/1e-9 Make/D/N=3/O W_coef W_coef[0] = 1.1, -5e-12,1e-20 FuncFit/q SHO_pot W_coef ln_normalization_hist /D print "an estimate of the spring constant is "+num2str(W_coef[0]) redimension/n=(norm_portion) ln_normalization_hist setscale/p x, -newspan/2, tracestep, ln_normalization_hist ln_normalization_hist = SHO_pot(W_coef,x) //This section makes the wave sections along the deflection trace variable offsetprompt prompt offsetprompt, "Do you want to offset wave to zero deflection?", popup, "yes; no" doprompt "Offset wave", offsetprompt if(offsetprompt ==1) wavestats/q/r=[0,numpnts(deflection)/15] deflection deflection=deflection-V_avg endif variable histlimitprompt= NumVarOrDefault("ghistlimitprompt", 10) prompt histlimitprompt, "What threshold for cutting off histogram? higher more included suggest

10" doprompt "Hist limit", histlimitprompt variable/g ghistlimitprompt=histlimitprompt print "The hist threshold variable is "+num2str(histlimitprompt) variable tracecounter = 1, start_p = 0, stepsize, start_x, center, alignedcenter,

hist_limit=bin_size*gtracestep/histlimitprompt, hist_peak string extension stepsize = numpnts(deflection)/(numtrace+(bin_size*numtrace/numpnts(deflection))-.9) make/o hist do extension = num2istr(tracecounter) //turns the variable into a string for making extension

names duplicate/o/r=[start_p, start_p+bin_size-1] deflection cut_deflection //make section to

be compared to standard from above variable Y_offset, cutstd wavestats/q cut_deflection y_offset=v_avg cut_deflection=cut_deflection-Y_offset wavestats/q cut_deflection cutstd=V_sdev cut_deflection= cut_deflection*sqrt(abs(cutstd^2-noisestd^2))/cutstd //print sqrt(abs(cutstd^2-noisestd^2))/cutstd cut_deflection=cut_deflection+Y_offset histogram cut_deflection hist wavestats/q hist hist_peak=round(V_maxloc/tracestep)*tracestep histogram/b=-tracerange/2+hist_peak, tracestep, num_pnts_sec cut_deflection, hist

//histogram with same properties as one from above. duplicate/o hist, ln_hist, pot_section ln_hist=-(ln(ln_hist)-ln(bin_size))

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ln_hist*= 1.3806e-23*temp //scale wave for energy duplicate/o/r=[x2pnt(ln_normalization_hist, leftx(hist)), x2pnt(ln_normalization_hist,

pnt2x(hist, numpnts(hist)-1))] ln_normalization_hist cut_normalization_hist pot_section=ln_hist-cut_normalization_hist //subtract off the standard //differenciate to get force duplicate/o pot_section force_section differentiate force_section force_section= -force_section clip_brownian_edges(hist, hist_limit, hist_peak) //setting the proper scale to fit in potential well properly center=pnt2x(deflection, start_p + bin_size/2) alignedcenter= leftx(deflection)-tracestep*round((leftx(deflection)-center)/(tracestep)) start_x=alignedcenter+leftx(pot_section) setscale/p x, start_x, tracestep, pot_section, force_section //copying traces to look at later //duplicate/o hist $"hist"+extension // duplicate/o ln_hist $"ln_hist"+extension // duplicate/o pot_section $"potential"+extension duplicate/o force_section $"force"+extension tracecounter += 1 start_p = (tracecounter-1)*stepsize while (tracecounter <= numtrace) //This section is to calculate an average variable average_start_x, average_end_x, average_numpnts average_start_x = leftx($"force"+extension) average_end_x = rightx($"force1")-tracestep average_numpnts = round((average_end_x - average_start_x)/(tracestep)) + 1 //print average_numpnts make/o/n=(average_numpnts) force_average variable ext=str2num(extension) variable i for(i=0; i<=(average_numpnts-1) ; i+=1) variable x x=average_start_x+tracestep*i force_average[i]= find_average_for_brownian("force",x, tracestep, ext) endfor setscale/I x, average_start_x, average_end_x, force_average display force_average string average_name prompt average_name, "What do you want to call the average trace?" doprompt "Average Name", average_name if(waveexists($average_name)==1) variable overwrite_prompt prompt overwrite_prompt, "Do you want to overwrite previous wave?", popup, "No;Yes" doprompt "Overwrite", overwrite_prompt if(overwrite_prompt==2) duplicate/o force_average $average_name else prompt average_name, "What do you want to call the average trace?" doprompt "The trace already exists", average_name duplicate/o force_average $average_name endif else duplicate force_average $average_name

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endif print average_name AppendToGraph $average_name RemoveFromGraph force_average string parameters sprintf parameters, "\F'Times New Roman' Brownian subtracting inst noise\r Number of points/section

%g\r %g wave sections\r Span %g angstroms\r spacing %g angstroms\r Temp %g Kelvin\r Hist limit variable %g", bin_size, numtrace, tracerange*1e10, tracestep*1e10, temp, histlimitprompt

TextBox/C/N=analysis_results parameters killwaves noisewave, normalization_hist, ln_normalization_hist, fit_ln_normalization_hist, W_coef,

W_sigma, W_paramconfidenceinterval, cut_deflection, hist, pot_section, ln_hist, force_section, cut_normalization_hist

end Function find_average_for_brownian(basename,x, span, extension) String basename Variable x, span, extension Variable xsum= 0, nsum=0 Variable i for(i=1;i<=extension;i+=1) WAVE w= $basename+num2istr(i) wave p=$"hist"+num2istr(i) Variable x0= leftx(w) Variable dx= deltax(w) Variable npts= numpnts(w) if( x >= (x0-dx*.01) && x <= (x0+dx*(npts-1+.01)) ) //print i xsum += w(x)*p(x) nsum += p(x) endif endfor return xsum/nsum end function SHO_pot(w,x) wave w; variable x return w[0]/2*(x-w[1])^2+w[2] end function gaussian(w,x) wave w; variable x return w[2]*exp(-(x-w[1])^2/(2*w[0])) //w[0] is the standard deviation squared, and w[1] is the x

offset end function clip_brownian_edges(hist, hist_limit, hist_peak) wave hist variable hist_limit, hist_peak variable limit_x findlevel/q/p/r=[x2pnt(hist, hist_peak),0] hist, hist_limit //find cutoff in point number starting from

peak and working backward toward the front. if(V_flag==0) V_levelx=ceil(V_levelx)

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if(hist[V_levelx-1]==0)//if statement to accommodate the errors caused during differentiation if there is a zero.

limit_x=pnt2x(hist,V_levelx+1) deletepoints 0,V_levelx+1, force_section, hist, prob, ln_hist, pot_section setscale/p x, limit_x, deltax(hist), force_section, hist, ln_hist, pot_section else limit_x=pnt2x(hist,V_levelx) deletepoints 0,V_levelx, force_section, hist, prob, ln_hist, pot_section setscale/p x, limit_x, deltax(hist), force_section, hist, ln_hist, pot_section endif endif findlevel/q/p/r=(hist_peak) hist, hist_limit if(V_flag==0) V_levelx=floor(V_levelx) If(hist[V_levelx+1]==0) limit_x=hist[V_levelx-1] deletepoints V_levelx, numpnts(hist)-V_levelx-2, force_section, hist, ln_hist, pot_section else limit_x=hist[V_levelx] deletepoints V_levelx+1, numpnts(hist)-V_levelx-1, force_section, hist, ln_hist, pot_section endif endif end

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A.6 Energy Dissipation Force Curves

The energy dissipation force curves from chapter six were computed from

deflection timecourses recorded on the National Instruments board. The timecourses

sampled near 1 MHz allow the numerical computation of multiple harmonics so that the

energy dissipation can be more accurately calculated. The Igor Pro scripts for computing

the energy dissipation curve as a function of tapping amplitude are copied below. First,

the energy dissipation as a function of time is calculated including higher harmonics.

Second, the attractive regime is separated from the repulsive regime because the energy

dissipation is not a single valued function. Lastly, the waves are averaged together.

#include <decimation> Function Many_Energy_for_NI(basewave, velocity, frequency, f_0, targetphase, k, Q, smoothing,

startwave, endwave) string basewave variable velocity, frequency, f_0, targetphase, k, Q, smoothing, startwave, endwave prompt basewave, "What is the wave to analyze?" prompt velocity, "What velocity is the surface moving toward the tip in nm/s?" prompt frequency, "What is the frequency of oscillation?" prompt f_0, "What is the resonant frequency of the cantilever?" prompt targetphase, "Give the phase at the working frequency without interaction" prompt k, "Spring constant of cantilever" prompt Q, "Q of the cantilever" prompt smoothing, "What smoothing factor for output?" prompt startwave, "What is the starting wavenumber of the set?" prompt endwave, "What is the ending wavenumber of the set?" variable harmonic_prompt, num_har prompt harmonic_prompt, "Work with higher harmonics?", popup, "Yes; No" doprompt "Higher harmonics", harmonic_prompt if(harmonic_prompt==1) prompt num_har, "How many harmonics would you like to include?" doprompt "Number of Harmonics", num_har print "The number of harmonics used is "+num2istr(num_har) else print "No higher harmonics are included" endif variable i=startwave do //****Calculation of offsets*** variable phaseshift, A_0, driving_force//driving_force is the force applied to the cantilever to cause the

oscillation as in drving_force*sin(2*pi*f_0*t)...

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lockin_for_NI(basewave+num2istr(i), velocity, frequency, 0, smoothing) wave amplitude, phase make/n=1000/o histamp, histphase histogram phase histphase histogram amplitude histamp histamp[0]=0; histamp[1]=0 wavestats/q histphase phaseshift=targetphase-V_maxloc wavestats/q histamp A_0=V_maxloc print "The free amplitude used was "+num2str(A_0) //****First order calculation*** lockin_for_NI(basewave+num2istr(i), velocity, frequency, phaseshift, smoothing) //*****Energy Dissipation for first order***** driving_force=A_0*k*(frequency/F_0)*sqrt((F_0/frequency-frequency/F_0)^2+(1/Q)^2)//yucky

formula for driving force but neccesary duplicate/o amplitude amp_fund duplicate/o phase energy energy=driving_force*amplitude*pi*sin(energy)-k*pi*amplitude^2*frequency/(Q*f_0) //calculation for higher harmonics if(harmonic_prompt==1) variable n for(n=2; n<=num_har; n+=1) lockin_for_NI(basewave+num2istr(i), velocity, frequency*n, phaseshift, smoothing) duplicate/o amplitude $"amplitude"+num2istr(n) energy=energy-k*pi*frequency*n^2*amplitude^2/(Q*f_0) endfor duplicate/o energy $"energy"+num2istr(i)+"H"+num2istr(num_har) duplicate/o amp_fund $"amp"+num2istr(i)+"H"+num2istr(num_har) else duplicate/o energy $"energy"+num2istr(i) duplicate/o amp_fund $"amp"+num2istr(i) endif i+=1 while(i<=endwave) end function lockin_for_NI(tappingwave, velocity, frequency, phaseshift, smoothing) string tappingwave variable velocity, frequency, phaseshift, smoothing prompt tappingwave, "What is the wave to analyze?" prompt velocity, "What velocity is the surface moving toward the tip in nm/s?" prompt frequency, "What is the frequency of oscillation?" prompt phaseshift, "Give phase offset" prompt smoothing, "What smoothing factor for output?" variable length=numpnts($tappingwave) Duplicate/o $tappingwave xtrace ytrace xtrace=xtrace*2*sin(2*pi*x*frequency*1e9/velocity+phaseshift)

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ytrace=ytrace*2*cos(2*pi*x*frequency*1e9/velocity+phaseshift) FDecimate(xtrace,"dxtrace",length/1000) FDecimate(ytrace,"dytrace",length/1000) smooth smoothing, dxtrace smooth smoothing, dytrace duplicate/o dxtrace xtrace, phase, amplitude duplicate/o dytrace ytrace amplitude=sqrt(xtrace^2+ytrace^2) phase=-atan(ytrace/xtrace)+pi/2//added a negative so that it measures phase lag killwaves xtrace, ytrace, dxtrace, dytrace end macro parse_energy_dis_waves(har,num_har,wavenumber ) variable har, wavenumber string num_har prompt har, "Use harmonics?", popup, "No; Yes" prompt num_har, "number of harmonics used" if(har==2) duplicate/o $"energy"+num2istr(wavenumber)+"H"+num_har energy_wave //dif_energy duplicate/o $"amp"+num2istr(wavenumber)+"H"+num_har amp_wave // differentiate dif_energy // make/o torepulsive, toattractive // findlevels/b=3/p/q/m=2e-9/d=Torepulsive dif_energy, 7e-10 // findlevels/b=3/p/q/m=2e-9/d=Toattractive dif_energy, -7e-10 // torepulsive[0]=floor(torepulsive[0]);torepulsive[1]= floor(torepulsive[1]);toattractive[0]=

ceil(toattractive[0]);toattractive[1]=ceil(toattractive[1]) if(torepulsive[0]<toattractive[0]) make/o/n=(numpnts(energy_wave)+torepulsive[0]+torepulsive[1]-toattractive[0]-

toattractive[1]-8) $"energy_at"+num2istr(wavenumber)+"H"+num_har, $"amp_at"+num2istr(wavenumber)+"H"+num_har

$"energy_at"+num2istr(wavenumber)+"H"+num_har[0, torepulsive[0]-2]=energy_wave[p]

$"energy_at"+num2istr(wavenumber)+"H"+num_har[torepulsive[0]-2, torepulsive[1]+torepulsive[0]-toattractive[0]-6]=energy_wave[toattractive[0]+p-torepulsive[0]+4]

$"energy_at"+num2istr(wavenumber)+"H"+num_har[torepulsive[1]+torepulsive[0]-toattractive[0]-6, numpnts(energy_wave)+torepulsive[1]+torepulsive[0]-toattractive[0]-toattractive[1]-8]=energy_wave[toattractive[0]+toattractive[1]+p-torepulsive[0]-torepulsive[1]+8]

$"amp_at"+num2istr(wavenumber)+"H"+num_har[0, torepulsive[0]-2]=amp_wave[p] $"amp_at"+num2istr(wavenumber)+"H"+num_har[torepulsive[0]-2,

torepulsive[1]+torepulsive[0]-toattractive[0]-6]=amp_wave[toattractive[0]+p-torepulsive[0]+4]

$"amp_at"+num2istr(wavenumber)+"H"+num_har[torepulsive[1]+torepulsive[0]-toattractive[0]-6, numpnts(energy_wave)+torepulsive[1]+torepulsive[0]-toattractive[0]-toattractive[1]-8]=amp_wave[toattractive[0]+toattractive[1]+p-torepulsive[0]-torepulsive[1]+8]

make/o/n=(toattractive[0]+toattractive[1]-torepulsive[0]-torepulsive[1]-16) $"energy_rep"+num2istr(wavenumber)+"H"+num_har, $"amp_rep"+num2istr(wavenumber)+"H"+num_har

$"energy_rep"+num2istr(wavenumber)+"H"+num_har[0, toattractive[0]-torepulsive[0]-8]=energy_wave[p+torepulsive[0]+4]

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$"energy_rep"+num2istr(wavenumber)+"H"+num_har[toattractive[0]-torepulsive[0]-8, toattractive[0]-torepulsive[0]+toattractive[1]-torepulsive[1]-16]=energy_wave[p+torepulsive[0]+torepulsive[1]-toattractive[0]+12]

$"amp_rep"+num2istr(wavenumber)+"H"+num_har[0, toattractive[0]-torepulsive[0]-8]=amp_wave[p+torepulsive[0]+4]

$"amp_rep"+num2istr(wavenumber)+"H"+num_har[toattractive[0]-torepulsive[0]-8,toattractive[0]-torepulsive[0]+toattractive[1]-torepulsive[1]-16]=amp_wave[p+torepulsive[0]+torepulsive[1]-toattractive[0]+12]

endif else duplicate/o $"energy"+num2istr(wavenumber) energy_wave //dif_energy duplicate/o $"amp"+num2istr(wavenumber) amp_wave // differentiate dif_energy // make/o torepulsive, toattractive // findlevels/b=3/p/q/m=2e-9/d=Torepulsive dif_energy, 7e-10 // findlevels/b=3/p/q/m=2e-9/d=Toattractive dif_energy, -7e-10 // torepulsive[0]=floor(torepulsive[0]);torepulsive[1]= floor(torepulsive[1]);toattractive[0]=

ceil(toattractive[0]);toattractive[1]=ceil(toattractive[1]) if(torepulsive[0]<toattractive[0]) make/o/n=(numpnts(energy_wave)+torepulsive[0]+torepulsive[1]-toattractive[0]-

toattractive[1]-8) $"energy_at"+num2istr(wavenumber), $"amp_at"+num2istr(wavenumber)

$"energy_at"+num2istr(wavenumber)[0, torepulsive[0]-2]=energy_wave[p] $"energy_at"+num2istr(wavenumber)[torepulsive[0]-2, torepulsive[1]+torepulsive[0]-

toattractive[0]-6]=energy_wave[toattractive[0]+p-torepulsive[0]+4] $"energy_at"+num2istr(wavenumber)[torepulsive[1]+torepulsive[0]-toattractive[0]-6,

numpnts(energy_wave)+torepulsive[1]+torepulsive[0]-toattractive[0]-toattractive[1]-8]=energy_wave[toattractive[0]+toattractive[1]+p-torepulsive[0]-torepulsive[1]+8]

$"amp_at"+num2istr(wavenumber)[0, torepulsive[0]-2]=amp_wave[p] $"amp_at"+num2istr(wavenumber)[torepulsive[0]-2, torepulsive[1]+torepulsive[0]-

toattractive[0]-6]=amp_wave[toattractive[0]+p-torepulsive[0]+4] $"amp_at"+num2istr(wavenumber)[torepulsive[1]+torepulsive[0]-toattractive[0]-6,

numpnts(energy_wave)+torepulsive[1]+torepulsive[0]-toattractive[0]-toattractive[1]-8]=amp_wave[toattractive[0]+toattractive[1]+p-torepulsive[0]-torepulsive[1]+8]

make/o/n=(toattractive[0]+toattractive[1]-torepulsive[0]-torepulsive[1]-16) $"energy_rep"+num2istr(wavenumber), $"amp_rep"+num2istr(wavenumber)

$"energy_rep"+num2istr(wavenumber)[0, toattractive[0]-torepulsive[0]-8]=energy_wave[p+torepulsive[0]+4]

$"energy_rep"+num2istr(wavenumber)[toattractive[0]-torepulsive[0]-8, toattractive[0]-torepulsive[0]+toattractive[1]-torepulsive[1]-16]=energy_wave[p+torepulsive[0]+torepulsive[1]-toattractive[0]+12]

$"amp_rep"+num2istr(wavenumber)[0, toattractive[0]-torepulsive[0]-8]=amp_wave[p+torepulsive[0]+4]

$"amp_rep"+num2istr(wavenumber)[toattractive[0]-torepulsive[0]-8,toattractive[0]-torepulsive[0]+toattractive[1]-torepulsive[1]-16]=amp_wave[p+torepulsive[0]+torepulsive[1]-toattractive[0]+12]

endif endif end Macro Average_parsed_NI_EDis_waves(har, num_har, startwave, endwave) variable har variable startwave, endwave

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string num_har prompt har, "Use harmonics?", popup, "No; Yes" prompt num_har, "number of harmonics used" variable i=startwave, tracestep=1e-11, tracestart, traceend, abs_reptracestart,

abs_reptraceend,abs_attracestart, abs_attraceend, wave_pnts abs_attracestart=1e-6;abs_reptracestart=1e-6//these are trash values to make sure they are reset abs_attraceend=0;abs_reptraceend=0 if(har==2) do wavestats/q $"amp_at"+num2istr(i)+"H"+num_har tracestart=V_min if(abs_attracestart>tracestart) abs_attracestart=tracestart endif traceend=V_max if(abs_attraceend<traceend) abs_attraceend=traceend endif wave_pnts=(traceend-tracestart)/1e-11 wave_pnts=round(wave_pnts) duplicate/o $"energy_at"+num2istr(i)+"H"+num_har temp_energy duplicate/o $"amp_at"+num2istr(i)+"H"+num_har temp_amp sort temp_amp, temp_amp, temp_energy make/o/n=(wave_pnts) $"avg_at"+num2istr(i)+"H"+num_har setscale/p x, tracestart, tracestep, "", $"avg_at"+num2istr(i)+"H"+num_har interpolate/T=1/I=3/y=$"avg_at"+num2istr(i)+"H"+num_har temp_energy /x=temp_amp wavestats/q $"amp_rep"+num2istr(i)+"H"+num_har tracestart=V_min if(abs_reptracestart>tracestart) abs_reptracestart=tracestart endif traceend=V_max if(abs_reptraceend<traceend) abs_reptraceend=traceend endif wave_pnts=(traceend-tracestart)/1e-11 wave_pnts=round(wave_pnts) duplicate/o $"energy_rep"+num2istr(i)+"H"+num_har temp_energy duplicate/o $"amp_rep"+num2istr(i)+"H"+num_har temp_amp sort temp_amp, temp_amp, temp_energy make/o/n=(wave_pnts) $"avg_rep"+num2istr(i)+"H"+num_har setscale/p x, tracestart, tracestep, "", $"avg_rep"+num2istr(i)+"H"+num_har interpolate/T=1/I=3/y=$"avg_rep"+num2istr(i)+"H"+num_har temp_energy /x=temp_amp i+=1 while(i<=endwave) wave_pnts=(abs_attraceend-abs_attracestart)/1e-11 make/o/n=(wave_pnts+1) $"avg_at"+num2istr(startwave)+"_"+num2istr(endwave)+"H"+num_har setscale/p x, abs_attracestart, tracestep, "",

$"avg_at"+num2istr(startwave)+"_"+num2istr(endwave)+"H"+num_har variable j=0, x

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do x=abs_attracestart+tracestep*j $"avg_at"+num2istr(startwave)+"_"+num2istr(endwave)+"H"+num_har[j]=

find_average_X("avg_at",x, startwave, endwave, "H"+num_har) j+=1 while(j<=wave_pnts) wave_pnts=(abs_reptraceend-abs_reptracestart)/1e-11 make/o/n=(wave_pnts+1)

$"avg_rep"+num2istr(startwave)+"_"+num2istr(endwave)+"H"+num_har setscale/p x, abs_reptracestart, tracestep, "",

$"avg_rep"+num2istr(startwave)+"_"+num2istr(endwave)+"H"+num_har j=0 do x=abs_reptracestart+tracestep*j $"avg_rep"+num2istr(startwave)+"_"+num2istr(endwave)+"H"+num_har[j]=

find_average_X("avg_rep",x, startwave, endwave,"H"+num_har) j+=1 while(j<=wave_pnts) else do wavestats/q $"amp_rep"+num2istr(i) tracestart=V_min if(abs_reptracestart>tracestart) abs_reptracestart=tracestart endif traceend=V_max if(abs_reptraceend<traceend) abs_reptraceend=traceend endif wave_pnts=(traceend-tracestart)/1e-11 wave_pnts=round(wave_pnts) duplicate/o $"energy_rep"+num2istr(i) temp_energy duplicate/o $"amp_rep"+num2istr(i) temp_amp sort temp_amp, temp_amp, temp_energy make/o/n=(wave_pnts) $"avg_rep"+num2istr(i) setscale/p x, tracestart, tracestep, "", $"avg_rep"+num2istr(i) interpolate/T=1/I=3/y=$"avg_rep"+num2istr(i) temp_energy /x=temp_amp wavestats/q $"amp_at"+num2istr(i) tracestart=V_min if(abs_attracestart>tracestart) abs_attracestart=tracestart endif traceend=V_max if(abs_attraceend<traceend) abs_attraceend=traceend endif wave_pnts=(traceend-tracestart)/1e-11 wave_pnts=round(wave_pnts) duplicate/o $"energy_at"+num2istr(i) temp_energy duplicate/o $"amp_at"+num2istr(i) temp_amp sort temp_amp, temp_amp, temp_energy

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make/o/n=(wave_pnts) $"avg_at"+num2istr(i) setscale/p x, tracestart, tracestep, "", $"avg_at"+num2istr(i) interpolate/T=1/I=3/y=$"avg_at"+num2istr(i) temp_energy /x=temp_amp i+=1 while(i<=endwave) wave_pnts=(abs_attraceend-abs_attracestart)/1e-11 make/o/n=(wave_pnts+1) $"avg_at"+num2istr(startwave)+"_"+num2istr(endwave) setscale/p x, abs_attracestart, tracestep, "",

$"avg_at"+num2istr(startwave)+"_"+num2istr(endwave) variable j=0, x do x=abs_attracestart+tracestep*j $"avg_at"+num2istr(startwave)+"_"+num2istr(endwave)[j]= find_average_X("avg_at",x,

startwave, endwave, "") j+=1 while(j<=wave_pnts) wave_pnts=(abs_reptraceend-abs_reptracestart)/1e-11 make/o/n=(wave_pnts+1) $"avg_rep"+num2istr(startwave)+"_"+num2istr(endwave) setscale/p x, abs_reptracestart, tracestep, "",

$"avg_rep"+num2istr(startwave)+"_"+num2istr(endwave) j=0 do x=abs_reptracestart+tracestep*j $"avg_rep"+num2istr(startwave)+"_"+num2istr(endwave)[j]= find_average_X("avg_rep",x,

startwave, endwave, "") j+=1 while(j<=wave_pnts) endif end Function find_average_X(basename,x, startwave, endwave, har_string) String basename, har_string Variable x, startwave, endwave Variable xsum= 0, nsum=0 Variable i for(i=startwave;i<=endwave;i+=1) WAVE w= $basename+num2istr(i)+har_string Variable x0= leftx(w) Variable dx= deltax(w) Variable npts= numpnts(w) if( x >= (x0-dx*.25) && x <= (x0+dx*(npts-1+.25)) ) if(w(x)>= -1e-6 && w(x) <= 1e-6) //print w(x) xsum += w(x) nsum += 1 endif endif endfor return xsum/nsum end

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A.7 Tapping Mode Force Profile Reconstruction

Tapping Mode Force profile Reconstruction (TMFPR) is a power technique for

measuring the advancing and receding force profiles experienced by the tapping

cantilever during tapping. TMFPR calculates the force profiles from the same tapping

timetrace from which the energy dissipation force curves including higher harmonics is

computed. The method has not been fully automated so the text from a typical TMFPR

experiment along with the called functions is copied below.

load_NIAFM_file() General binary file load from "trace37.bin" (8000000 total bytes) Data length: 4000000, waves: binarydata0 The gain factor is 6.4495e-05 The offset is 0 The deflection sensitivity is 5.7 Wavestats/q trace37; trace37-=V_avg SetScale/P x 0,1e-06,"", trace37; duplicate/o trace37 test fdrive piezo fint fposition; smooth/s=4 13,test;smooth/s=4 9, test; duplicate/o test pos velocity acc;differentiate velocity acc;differentiate acc; pos*=1.99;velocity*=6.12e-7;acc*=1.908e-10; make/n=4 parafdrive; parafdrive[0]=0; parafdrive[1]=4.0697e-9*1.99*(16552.607/16560)*sqrt((16560/16552.607-16552.607/16560)^2+(1/32.414)^2); parafdrive[2]=2*pi*16552.607; CurveFit/q/H="1110" sin kwCWave=parafdrive, velocity(0,.001); fdrive=parafdrive[0]+parafdrive[1]*sin(parafdrive[2]*x+parafdrive[3]); Fint=pos+velocity+acc-Fdrive piezo(0,.336)=2e-8*x+13.28e-9;piezo(.336,1.336)=-2e-8*x+26.72e-9;piezo(1.336,2.336)=2e-8*x-26.72e-9;piezo(2.336,3.336)=-2e-8*x+66.72e-9;piezo(3.336,4.336)=2e-8*x-66.72e-9; fposition=piezo+test; tapping_FC_converter(1012000, 1302000, "a37a") macro tapping_FC_converter(startpnt, endpnt, wavesection) variable startpnt, endpnt string wavesection variable span=endpnt-startpnt variable numsec=round((span-30000)/10000) variable i=0 do duplicate/r=[(startpnt+i*10000), (startpnt+i*10000+30000)]/o velocity velcut duplicate/r=[(startpnt+i*10000), (startpnt+i*10000+30000)]/o fint fcut duplicate/r=[(startpnt+i*10000), (startpnt+i*10000+30000)]/o fposition poscut split_tapping_trace_retrace(fcut, poscut, velcut) sort p_trace, P_trace, f_trace

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sort p_retrace, P_retrace, f_retrace FDecimateXPos(F_trace,"df_trace",50,1) FDecimateXPos(f_retrace,"df_retrace",50,1) FDecimateXPos(p_trace,"dp_trace",50,1) FDecimateXPos(p_retrace,"dp_retrace",50,1) interpolate/t=1/n=1000 df_trace/x=dp_trace duplicate/o df_trace_l $wavesection+"tr"+num2istr(i) interpolate/t=1/n=1000 df_retrace/x=dp_retrace duplicate/o df_retrace_l $wavesection+"re"+num2istr(i) i+=1 while(i<=numsec) end function split_tapping_trace_retrace(forcewave, poswave, velocitywave) wave forcewave, poswave, velocitywave variable pnts=numpnts(forcewave), i=0, retr=0, tr=0 make/o/n=(round(pnts*.4)) f_trace f_retrace p_trace p_retrace for(i=0;i<=pnts;i+=1) if(velocitywave[i]>0) f_retrace[retr]=forcewave[i] p_retrace[retr]=poswave[i] retr+=1 else f_trace[tr]=forcewave[i] p_trace[tr]=poswave[i] tr+=1 endif endfor end

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A.7 References 1. Binnig, G., Quate, C. F. & Gerber, C. Atomic Force Microscope. Physical Review

Letters 56, 930-933 (1986). 2. Lantz, M. A. et al. Quantitative Measurement of Short-Range Chemical Bonding

Forces. Science 291, 2580-2583 (2001). 3. Chon, J. W. M., Mulvaney, P. & Sader, J. E. Experimental validation of

theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids. Journal of Applied Physics 87, 3978-3988 (2000).

4. Hutter, J. L. & Bechhoefer, J. Calibration of atomic-force microscope tips. Review of Scientific Instruments 64, 1868-1873 (1993).