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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
iii
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.
iv
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
v
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.
vi
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
vii
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
viii
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
ix
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
x
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
xi
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.
xii
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.
xiii
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.
xv
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
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-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).