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UNIVERSITY OF CALIFORNIA
Santa Barbara
Dynamic Friction Measurement and Modeling at the Micro/Nano Scale
A Dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Mechanical Engineering
by
Abhishek Srivastava
Committee in charge:
Professor Kimberly Turner, Chair
Professor Karl Astrom
Professor Noel MacDonald
Professor Jacob Israelachvili
September 2006
The dissertation of Abhishek Srivastava is approved.
____________________________________________ Karl Astrom
____________________________________________ Noel MacDonald
____________________________________________ Jacob Israelachvili
____________________________________________ Kimberly Turner, Committee Chair
August 2006
iii
ACKNOWLEDGEMENTS
I would like to express my heartfelt gratitude towards Professor Kimberly L.
Turner, committee chair, advisor, mentor, for her guidance and support. This project
could not have been completed without her vision, thoughtfulness and brilliance. I
shall remain forever indebted to her for shaping my thinking as required for research
and continually steering my career in the right direction. She is probably the person
with the greatest influence in my life and her guidance during the course of my
research and course-work has invariably led to many positive changes in my outlook
and personality. I feel privileged to have had the opportunity to obtain supervision
from the finest advisor and person one could hope for.
I feel honored to have worked under the expert guidance of Prof. Karl
Astrom, co-advisor, a person of high eminence and profound knowledge, a rare
privilege indeed. I am deeply indebted to him for his active involvement in spite of
limitations in the actual time we could spend together and for his constructive
criticism on many occasions.
I would like to thank Professor Jacob Israelachvili for the many discussions
and for extending his expertise to provide the right guidance in my research work. I
am also grateful to Professor Noel MacDonald for participating in my doctoral
dissertation committee and for providing important feedback from time to time. I am
thankful to Prof. Pirouz Kavehpour at UCLA for his insightful contribution and
active involvement granting both time and resources as required for collaborative
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research. I would also like to thank Professor Eckart Meiburg, chair, and Professor
Bud Homsy for supporting me in my delayed application to UCSB’s graduate
program.
Few works are the product of the efforts of one person alone. Insight,
assistance and suggestions go a long way in shaping any work to the final form and I
shall forever remain indebted to all my friends at the Turner-group including people I
spent a long time: Weibin, Wenhua, Mike North, Mike Requa, Laura, Barry, Kari,
Mark and Benedikt – for providing exactly that. Kirk Fields, Jeff Oakes and Dave
Bothman were people I could always resort to for any difficulties I faced during my
work. My deepest thanks to Kenny Rosenberg for being a great friend and for
offering his insight in guiding my research work, to Hongbo Zeng for his immense
help with my experimentation, to Seth Downs at Hysitron for patiently replying to
my long emails helping me figure things out and to Alejandro Bonilla at Asylum for
his advice and help. I would like to thank Professor Samir Mitragotri for his
friendship, advice and support. I am also thankful to Dr. Peter Hartwell and Dr. Steve
Naberhuis at HP Research Labs with whom I had much fun learning for two
consecutive summers, and to Dr. Amitava De at IIT Bombay for his guidance and
support throughout.
I shall forever remain indebted to all my friends and family listed below and
forgotten. I was fortunate to have in Professor Makarand Ratnaparkhi, Department of
Statistics at Wright State University, an experienced researcher in the family who,
through various discussions, helped me shape a well-rounded view on many things.
v
I would like to extend my heartfelt thanks to my friend, Professor Amit Kanvinde at
UC-Davis, for the various insightful discussions. I really cherish the good times we
had, the tennis games and all the great food he so enthusiastically cooked. I am
indebted to Rahul Khatod for graciously letting me spend two summers at his place
during my internship and for helping me with programming. I would like to thank
my friend Harith, for all the tennis games, discussion and cooking! I would also like
to thank my friends - Shiva, Devleena, Sameer, Anuradha, Mayur, Gauri, Kaushik,
Nilesh, Sachin, Rahul, Siddharth, Gaurav, Shaunak, Shriram, Mihir, Ashish and all
others. I would also like to thank everyone at the Vedanta Society for acting as
friends, philosophers and guides throughout.
Finally, I would like to thank my parents Arvind and Dr. Mukul Srivastava -
for everything and my sister, Kanupriya, for all the fun times we have shared. Most
of all, I would like to thank my wife, Dr. Priya Kulkarni for always having
unshakable faith in me. Her patience, understanding, and support throughout has
helped me come a long way.
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VITA OF ABHISHEK SRIVASTAVA August 2006
EDUCATION Bachelor of Technology in Mechanical Engineering, Indian Institute of Technology, Bombay, April 1999 Master of Arts in Mechanical Engineering, University of California, Santa Barbara, June 2004 Doctor of Philosophy in Mechanical Engineering, University of California, Santa Barbara, August 2006 (expected) PROFESSIONAL EMPLOYMENT 1999-2001: Systems Analyst, Deloitte Consulting, Santa Ana 2001-2005: Teaching Assistant, Department of Mechanical Engineering, University of California, Santa Barbara 2002-2006: Research Assistant, Department of Mechanical Engineering, University of California, Santa Barbara Summer 2002-2003: Summer Internship, Hewlett-Packard Research Labs, Palo Alto PUBLICATIONS “Experimental Characterization of Micro-friction on a mica surface using a nanoindenter”, Tribology Letters, (Submitted). “Quantitative measurement of sliding friction dynamics in the micro-regime using the scratch functionality of a traditional nanoindenter”, Review of Scientific
Instruments, (To be submitted). FIELDS OF STUDY Major Field: Micro/Nanoscale Systems Studies in Dynamic Friction Modeling with Prof. Karl Astrom and Prof. Kimberly L. Turner
vii
ABSTRACT
Dynamic Friction Measurement and Modeling at the Micro/Nano Scale
by
Abhishek Srivastava
An experimental characterization of friction forces between asymmetric surfaces
in the micro-regime is performed by exploring the lateral scratching functionality of
a traditional nanoindenter (Hysitron Triboindenter) to determine friction properties at
low velocities. Classical friction experiments are reproduced using the triboindenter
with high repeatability. It is observed that real-time depth measurements closely
follow the Hertzian prediction. Friction spikes with magnitude dependent upon the
desired velocity input function (higher peak values in reduced rise times) are
observed, indicating an onset of kinetic friction effects even before the motion
begins. Anisotropy is observed between surfaces of different materials with stick-slip
occurring only at specific relative orientations. Experimental estimation of the
control parameters of the triboindenter is combined with the LuGre model for
describing friction for simulation of the experiments. Data obtained from a tribo-
rheometer is least-square fitted to the LuGre model and modifications to the model
that describe the friction behavior more accurately and provide a physical basis for it
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are investigated. Comparison experiments are performed with the tribo-rheometer
and the surface force apparatus (SFA) using similar material and scratch parameters
to determine the validity of the triboindenter as a nano/micro-scale friction testing
tool. Limitations of the triboindenter as a tribological tool and methods to overcome
some of them are discussed. Advanced equipment for expanding the current range of
the triboindenter which allow experimentation from the nano to the macro scale are
tested. Novel methods for scaling friction are described along with possible
applications of such tribological investigations in MEMS and nano-scale devices.
ix
TABLE OF CONTENTS
1. Introduction ..........................................................................................................1
1.1. Motivation for the Study.................................................................................1
1.2. Objectives and Scope of the Study..................................................................4
1.3. Organization and Outline ................................................................................7
2. Theoretical Background ....................................................................................10
2.1. Dynamic Regimes in Friction .......................................................................10
2.1.1. Static Friction .....................................................................................11
2.1.2. Boundary Lubrication.........................................................................12
2.1.3. Partial Fluid Lubrication ....................................................................12
2.1.4. Full Fluid Lubrication ........................................................................13
2.2. Empirical Models..........................................................................................16
2.2.1. Bo and Pavelescu Model ....................................................................16
2.2.2. Armstrong’s Seven Parameter Model ................................................17
2.3. Theoretical Models .......................................................................................18
2.3.1. Dahl Model.........................................................................................18
2.3.2. Lugre Model .......................................................................................20
3. New Friction Measurement Instrument - The Hysitron Triboindenter® .....24
3.1. Historical Overview of Tribological Instrumentation...................................24
3.2. The Hysitron Triboindenter® - Components and Design ..............................26
3.2.1. Stage and Optics.................................................................................27
x
3.2.2. Triboscanner.......................................................................................28
3.2.3. 1-D and 2-D Transducers ...................................................................29
3.2.4. Displacement/Force Application and Measurement ..........................30
3.2.5. Limitations of the Lateral Force Transducer: Nonlinearity and
Transducer ‘Breakpoint’ ........................................................................31
3.3. The Hysitron Triboindenter® - Data Acquisition ..........................................33
3.3.1. Load Functions ...................................................................................33
3.3.2. Software-based Tilt Correction ..........................................................35
3.3.3. Drift ....................................................................................................37
4. Experiments with the Triboindenter: Design, Results and Analysis.............40
4.1. Sample Preparation and Characterization .....................................................40
4.1.1. Structure of Mica................................................................................41
4.1.2. AFM Imaging of Mica Surface ..........................................................42
4.1.3. Preparation of the Mica Sample .........................................................44
4.2. Triboindenter Tips.........................................................................................46
4.2.1. Cono-spherical Tip Design.................................................................47
4.2.2. Tip Selection ......................................................................................48
4.2.3. AFM Imaging of Cono-spherical Tips ...............................................49
4.3. Experiment Design........................................................................................51
4.4. Experimental Results and Analysis...............................................................56
4.4.1. Velocity Limits of the Triboindenter..................................................56
4.4.2. Friction Force Variation with Normal Load.......................................57
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4.4.3. Friction Force Variation with Velocity at Different Normal Loads...58
4.4.4. Repeatability of Experiments .............................................................59
4.4.5. Steady State Attainment through Repeated Scratching......................60
4.4.6. Friction Spikes and Effect of Resting Time .......................................60
4.4.7. Stick-Slip and Anisotropy ..................................................................63
4.4.8. Depth of Scratches and Orientation Effects .......................................70
4.4.9. Friction Force Variation with Tip Radius and Geometry...................74
4.4.10. SEM Imaging of Mica Cleavage Structure ........................................77
5. Further Analysis of Data and Triboindenter Control System Modeling ......80
5.1. Actuation and Sensing Mechanisms of the Transducer ................................80
5.1.1. Electrostatic Actuation .......................................................................80
5.1.2. Displacement Sensing ........................................................................81
5.2. Tilt Characterization of the Triboindenter ....................................................85
5.3. Analysis of the Transition from Static to Kinetic Friction Regime ..............89
5.3.1. Friction Force and Velocity Characteristics .......................................89
5.3.2. Lateral Displacement (Input and Output) Characteristics ..................93
5.3.3. Lateral Displacement and Friction Force Variation with Normal Load.
............................................................................................................96
5.4. Modeling and Simulation..............................................................................99
5.4.1. PID Parameters for Displacement Control .......................................100
5.4.2. Modeling of the Experiment ............................................................103
5.4.3. Matlab Simulations ..........................................................................105
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6. Tribo-rheometer: Data Fitting to the Lugre Model and Comparison
Experiments .....................................................................................................109
6.1. Rheology and Tribology..............................................................................109
6.2. The Tribo-rheometer ...................................................................................111
6.2.1. Experimental Setup ..........................................................................111
6.2.2. Alignment.........................................................................................113
6.2.3. Experimental Measurements ............................................................113
6.3. Tribo-rheometer Data Fitting to the Lugre Model ......................................114
6.3.1. Model Description............................................................................114
6.3.2. Curve Fitting ....................................................................................115
6.4. Experimental Comparison of the Nanoindenter and Tribo-rheometer .......121
6.4.1. Experiment Setup .............................................................................122
6.4.2. Nanoindenter – Experimental Results and Discussion ....................123
6.4.3. Tribo-rheometer – Experimental Results and Discussion................129
7. SFA Comparison Experiments: Friction Measurements with Polystyrene 133
7.1. The Surface Force Apparatus......................................................................133
7.1.1. Experimental Setup ..........................................................................133
7.1.2. Polystyrene .......................................................................................135
7.2. Triboindenter: Comparison Experiments and Results ................................136
7.2.1. Tip and Sample Preparation .............................................................136
7.2.2. Experimental Results and Discussion ..............................................137
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8. Conclusions, Applications, New Equipment and Future Work...................145
8.1. Conclusions.................................................................................................145
8.2. Future Work: Advanced Equipment and Techniques .................................146
8.2.1. High Velocity Experiments – The Flexure Stage.............................146
8.2.2. Low Velocity Experiments – The Closed-loop Scanner ..................150
8.2.3. Scaling of Friction............................................................................154
8.3. Applications of Nano/Microscale Friction Testing.....................................156
8.3.1. Correcting Nano- and Micro-scale Friction Measurements .............156
8.3.2. Applications in MEMS ....................................................................157
8.3.3. Bio-Inspired Applications ................................................................161
8.3.4. Nano-tribological Applications ........................................................162
Appendix A .............................................................................................................164
Appendix B .............................................................................................................165
Appendix C .............................................................................................................169
Bibliography ...........................................................................................................171
xiv
LIST OF TABLES
Table 3.1 Comparison of the SFA, SFFM and the Nanoindenter [1, 2] ....................26
Table 3.2 Specifications of the Transducer [3] ..........................................................32
Table 4.1 Design Parameters for Nanoindenter Tips of Radii 50 µm and 100 µm....49
Table 5.1 Parameters from Experimental Observations at Varying Normal Loads and
Input Velocities ................................................................................................103
Table 6.1 Comparison of the Nanoindenter [3], Tribo-rheometer [4] and Hess and
Soom Equipment [5] ........................................................................................110
Table 6.2 Coefficients obtained by least-square fitting a function of the form
31 2 4
p xp p e p xµ −= + + to data obtained from the Tribo-rheometer..................119
Table 8.1 Comparison of Capability Enhancement Add-ons to the Triboindenter [3,
6, 7] ..................................................................................................................147
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LIST OF FIGURES
Figure 1.1 Sketch of Two Surfaces with Interlocking Asperities [8]...........................2
Figure 2.1 True Contact Between Engineering Surfaces [9]......................................12
Figure 2.2 The Stribeck Curve – Friction as a Function of Velocity [9]....................12
Figure 2.3 Idealized Contact in Static Friction, Asperity Deformation and in Break-
Away Friction [9] ...............................................................................................13
Figure 2.4 Spring Force Profile During Stick-Slip Motion [9] ..................................15
Figure 2.5 Typical Friction-Speed Time Shift [5]......................................................15
Figure 2.6 Friction as a Function of Velocity; 0: Experimental -: Theoretical [5] ....15
Figure 2.7 Friction as Function of Steady State Velocity for Various Lubricants [9]19
Figure 2.8 Friction Interface Between Two Surfaces Modeled as Contact Between
Bristles [10] ........................................................................................................21
Figure 3.1 The Hysitron Triboindenter (Nanoindenter) [6] .......................................28
Figure 3.2 (a) TriboScanner and Optics [6] (b) Schematic of Piezo Scanner [3] ......29
Figure 3.3 (a) Triboindenter Transducer [6] (b) 1-D and 2-D Transducers [3] ......29
Figure 3.4 Cross Section Schematic of (a) 1-D and (b) 2-D Transducer [3]..............31
Figure 3.5 (a) Displacement Measurement (b) Force Measurement [3] ....................31
Figure 3.6 Schematic of Center of Transducer Showing Breakpoint [3] ...................34
Figure 3.7 The TriboView Positioning Window [3] ..................................................34
Figure 3.8 Typical Loading Function for Scratch Testing .........................................36
Figure 3.9 Typical Scratch Testing Results Window.................................................36
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Figure 3.10 Sample Normal Displacement before Tilt Correction (Top Right) ........38
Figure 3.11 Sample Normal Displacement after Tilt Correction (Top Right) ...........38
Figure 3.12 Lateral Force Profile resulting from a 50 µm diamond tip being held
stationary on a mica surface for 1000 s at 7000 µN normal load.......................39
Figure 4.1 Trace of Friction Forces at Increasing Sliding Velocities on Mica [11] .41
Figure 4.2 High Grade Mica Sheets (Src: Ted Pella, Inc.).........................................41
Figure 4.3 (a) Close-packing representation of a SiO4 tetrahedron (b) Fundamental
unit of all silicates (c) Formation of phyllosilicate [12].....................................43
Figure 4.4 (a) Schematic 3-D diagram of the structure and composition of mica (b)
Development of 2-D mica structure from phyllosilicate [12] ............................43
Figure 4.5 (a) AFM image of Mica cleavage (top view) (b) A-A profile cut along the
two adjacent apexes of K+ ions (c) B-B profile cut along the two seperate
apexes of K+ ions [12] .......................................................................................45
Figure 4.6 AFM Image of the Mica Sample using Asylum Corp’s Molecular Force
Probe 3D System................................................................................................45
Figure 4.7 Surface profile (Z) along the length of scan..............................................46
Figure 4.8 Cono-spherical Tip Geometry [3] .............................................................48
Figure 4.9 Cylindrical Tip Geometry (e.g. for 50 µm Sapphire Flat Punch) .............49
Figure 4.10 AFM Imaging of the 5 µm cono-spherical tip along perpendicular
directions ............................................................................................................50
Figure 4.11 AFM Image of the 50 µm cono-spherical tip (unclean)..........................52
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Figure 4.12 AFM Image of the 50 µm cono-spherical tip (cleaned with Acetone,
Isopropyl Alcohol and Blow Drying).................................................................52
Figure 4.13 AFM Image of the 50 µm cono-spherical tip showing Section along
which Trace is taken...........................................................................................53
Figure 4.14 AFM Image of the 50 µm cono-spherical tip – Trace (smooth light line)
and Derivative (more irregular dark line)...........................................................53
Figure 4.15 An Oscillating Scratch at 3000 µN Normal Load and 0.001 µm/s velocity
............................................................................................................................57
Figure 4.16 Friction Force vs. Normal Load plot for a 50 µm diamond tip sliding
against a mica surface at 13.5 µm/s....................................................................58
Figure 4.17 Friction Force vs. Velocity at Various Normal Loads for a 50 µm
diamond tip sliding against a mica surface ........................................................59
Figure 4.18 Multiple Execution of the Same Load Function for Repeatability .........60
Figure 4.19 Repeated Scratching of a 50 µm cono-spherical diamond tip on a mica
surface at 0.07 µm/s velocity at normal load (a) 500 µN (b) 5000 µN ..............61
Figure 4.20 Friction Spike behavior at increasing velocities for a 50 µm sapphire flat
punch scratching on a mica surface at 7000 µN normal load ............................62
Figure 4.21 Friction Spike behavior at increasing velocities for a cono-spherical
diamond tip of radius 50 µm scratching on a mica surface at 9810 µN load.....62
Figure 4.22 Friction Spike Magnitude Variation with Resting Time.........................64
Figure 4.23 A typical lateral force profile vs. time produced by a 1 µm diamond tip
on a mica sample at 5000 µN normal load and 0.2 µm/s velocity .....................64
xviii
Figure 4.24 Typical Stick-slip type behavior observed for a sapphire 50 µm flat
punch sliding on a mica surface at (a) 5000 µN normal load and 0.1 µm/s
velocity (b) 1000 µN normal load and 0.6 µm/s velocity...................................66
Figure 4.25 Friction force vs. time plots for the 50 µm Sapphire flat punch sliding
against a mica surface at 1000 µN normal load and 0.6 µm/s velocity at various
sample orientations.............................................................................................67
Figure 4.26 Friction force vs. Sample Rotation for the 50 µm Sapphire flat punch
sliding against a mica surface at 1000 µN normal load and 0.6 µm/s velocity ..68
Figure 4.27 Friction force vs. Sample Rotation for the 50 µm Diamond tip sliding
against a mica surface at 9810 µN normal load and 1.5 µm/s velocity..............68
Figure 4.28 Friction force vs. time plots for 50 µm Diamond tip sliding against a
mica surface at 9810 µN load and 1.5 µm/s velocity at various orientations.....69
Figure 4.29 Friction Force Vs Velocity at four different angles for a 50 µm diamond
tip sliding against a mica surface at 8500 µN Normal Load ..............................70
Figure 4.30 Friction force vs. sample rotation for a 50 µm diamond tip scratching on
a Silicon Surface at 9810 µN normal load .........................................................70
Figure 4.31 Friction plots for a velocity reversal experiment for a 50 µm diamond tip
sliding against a mica surface at 9810 µN normal load and 0.7 µm/s velocity..72
Figure 4.32 Maximum depth vs. applied normal load and its comparison with Hertz
prediction for 50 µm diamond tip sliding against a mica surface ......................72
Figure 4.33 Maximum depth vs. velocity comparison at occurrence of stick-slip and
no stick-slip data for a 50 µm diamond tip sliding against a mica surface ........73
xix
Figure 4.34 Max Depth vs. Rotation Angle for a 50 µm diamond tip sliding against a
mica surface at 8500 µN Normal Load ..............................................................73
Figure 4.35 Max Depth vs. Velocity varation for a 50 µm diamond tip sliding against
a mica surface at 8500 µN Normal Load............................................................74
Figure 4.36 Maximum depth vs. tip radius for 1µm, 5µm, 50 µm Diamond tips and
50µm Sapphire tip sliding against a mica sample at 1000 µN normal load and
3.3 µm/s velocity................................................................................................75
Figure 4.37 Friction force vs. velocity plots at 5000 µN normal load for the 1 µm, 5
µm and 50 µm diamond tips for (a) Mica Surface, and (b) Aluminum Surface 76
Figure 4.38 SEM Micrographs of Mica Cleavage Structures ....................................78
Figure 5.1 Schematic of transducer plates showing forces and displacement ...........81
Figure 5.2 Voltage versus displacement for a parallel plate capacitive actuator .......82
Figure 5.3 A Differential Capacitor – motion of the central plate leads to a change in
individual capacitances on either side of the central plate .................................82
Figure 5.4 Theoretical output curve for a differential capacitor ................................84
Figure 5.5 Experimental X Voltage Vs Displacement curve for 9810 µN
NormalLoad and 4.0 µm/s velocity for the 50 µm diamond tip sliding on mica84
Figure 5.6 Typical Experimental Result – Lateral Force Vs Time ............................86
Figure 5.7 Schematic of Tilt in the Nanoindenter ......................................................86
Figure 5.8 Calculated Tilt of Nanoindenter Tip with a Mica Sample at 9810 µN
Normal Load at Various Orientations ................................................................87
xx
Figure 5.9 Schematic of Orientation Experiment with a Wedge for Computing
Sample Tilt .........................................................................................................88
Figure 5.10 Positively oriented force during tip loading as observed in friction
experiment with a wedge....................................................................................88
Figure 5.11 Friction Force & Velocity vs. Time Graphs for 9810 µN normal load (a)
Velocity = 0.2 µm/s (b) Velocity = 4.0 µm/s (c) Velocity = 18.0 µm/s (d)
Velocity = 36.4 µm/s ..........................................................................................91
Figure 5.12 Lateral Displacement (Input & Output) & Acceleration Vs Time Graphs
for 9810 µN load. Bold line indicates moving average filter of period 4.(a)
Velocity = 0.2 µm/s (b) Velocity = 4.0 µm/s (c) Velocity = 18.0 µm/s (d)
Velocity = 36.4 µm/s ..........................................................................................93
Figure 5.13 Lateral Displacement (Input & Output) & Velocity Vs Time Graphs (a)
Velocity = 0.2 µm/s (b) Velocity = 4.0 µm/s (c) Velocity = 18.0 µm/s (d)
Velocity = 36.4 µm/s ..........................................................................................95
Figure 5.14 Schematic Representation of Lateral Displacement Vs. Time Graphs ...95
Figure 5.15 Lateral Displacement Vs. Time Graphs at Various Normal Loads (a)
Velocity = 0.2 µm/s (b) Velocity = 4.0 µm/s (c) Velocity = 18.0 µm/s (d)
Velocity = 36.4 µm/s ..........................................................................................98
Figure 5.16 Friction Force Vs. Time Comparison Graphs at Various Normal Loads
(a) Velocity = 0.2 µm/s (b) Velocity = 4.0 µm/s (c) Velocity = 18.0 µm/s (d)
Velocity = 36.4 µm/s ..........................................................................................99
xxi
Figure 5.17 Typical Normal Load Output of the Triboindenter for a 500 µN constant
Input Force Profile............................................................................................101
Figure 5.18 Typical PID Controller for the Hysitron Triboindenter ........................101
Figure 5.19 Matlab Simulation of Lateral Displacement, Friction Force, Velocity and
Friction State as a function of Time at 9810 µN Normal Load and 4 µm/s
Velocity (all units are in MKS) ........................................................................106
Figure 5.20 Matlab Simulation of Lateral Displacement Input (Dotted Line) &
Output (Dark Line) Vs Time For Normal Loads 5000 µN and 9810 µN and 18
µm/s Velocity ...................................................................................................107
Figure 6.1 AR 2000 Torsional Rheometer (Src.TA Instruments Inc.). ....................112
Figure 6.2 Schematic of Tribo-rheometer and annual text fixture with radii R1 and R2
respectively [4] .................................................................................................112
Figure 6.3 Effect of Normal force on friction coefficient plotted vs. Gumbel number.
A copper fixture is used with Pennzoil 80W-90 is the lubricant [4]................116
Figure 6.4 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve at
Normal Load 5 N..............................................................................................117
Figure 6.5 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve at
Normal Load 10 N............................................................................................117
Figure 6.6 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve at
Normal Load 15 N............................................................................................118
Figure 6.7 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve at
Normal Load 20 N............................................................................................118
xxii
Figure 6.8 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve to
sample curve using a different function form2
31 2 4
p xp p e p xµ −= + + .............121
Figure 6.9 Design of a non-standard Stainless Steel tip used for comparison
experiments with the nanoindenter ..................................................................124
Figure 6.10 Friction Coefficient vs. velocity for a spherical stainless steel sliding on
a polished stainless steel plate observed with the nanoindenter.......................124
Figure 6.11 Friction force vs. time plots showing stick-slip at 5000 µN load at low
velocities and µ vs. velocity comparison with non stick-slip type data ...........125
Figure 6.12 Typical failure curve obtained when the nanoindenter fails to produce
reliable result, observed in the 4.5 µm/s to 13.5 µm/s velocity range..............126
Figure 6.13 Typical failure curve obtained when the nanoindenter fails to produce
reliable data, observed in low normal-load experiments (~ 500 µN)...............126
Figure 6.14 Friction coefficient vs. velocity plots obtained using a nanoindenter at
5000 µN load for dry friction and heavy paraffin oil as the lubricant..............128
Figure 6.15 Friction coefficient vs. velocity plots obtained using a nanoindenter at
10000 µN load for dry friction and using heavy paraffin oil as the lubricant ..128
Figure 6.16 Friction Coefficient vs. velocity for a stainless steel fixture rotating on a
stainless steel plate observed with tribo-rheometer on two different days.......130
Figure 6.17 Repeatability of the experiments conducted with the tribo-rheometer on
two different days (both plots are plotted to the same scale for comparison)..130
Figure 7.1 Schematic of the Surface Force Apparatus (SFA) [13] ..........................134
xxiii
Figure 7.2 Interferometry based measurement in the Surface Force Apparatus [14]
..........................................................................................................................135
Figure 7.3 Schematic of Mica Surfaces used in the SFA [14] .................................135
Figure 7.4 Typical Results Window for a Zero-Load Experiment Performed with the
Nanoindenter for a PS Sphere sliding against a PS Surface (MW 280,000)....138
Figure 7.5 (a) Friction Forces Between a PS Sphere and an Untreated PS Surface
(MW 280,000) as a function of Sliding Speed measured with the Triboindenter
(b) Friction Forces Between Two PS 280,000 Surfaces before and after
Crosslinking as a function of Sliding Speed measured with the SFA [15] ......139
Figure 7.6 Friction Force Between PS (MW 280,000) Deposited on a Steel Sphere of
Radius 3 mm Sliding Against a similar PS Surface deposited on Mica as a
function of Normal Load Measured by Triboindenter at 0.5 µm/s and SFA [15]
..........................................................................................................................140
Figure 7.7 Friction Force vs. Rotation Angle Plot for a PS Sphere sliding against a
PS Surface of MW 280,000 indicating the Isotropic nature of PS...................141
Fig 7.8 (a) Friction Spike vs. Stopping Time Plot for a PS (MW 280,000) Deposited
on a Steel Sphere of Radius 3 mm Sliding Against a PS Surface (MW 280,000)
deposited on Mica at 1000 µN Normal Load
(b) Height of Stiction Spike of PS 2,000,000 in stop-start experiments as a function
of Stopping Time [15]......................................................................................142
Figure 7.9 Surface Separation between PS Sphere and Surface at Zero Load vs.
Velocity ............................................................................................................143
xxiv
Figure 8.1 Flexure Stage [6].....................................................................................147
Figure 8.2 Scratch results using the Multirange Nanoprobe and flexure stage using
50 µm cono-spherical tip at 9.81 mN normal load (a) 7 µm/s (b) 225 µm/s vel
..........................................................................................................................149
Figure 8.3 Visible damage of the mica surface in scratch experiments conducted with
the 50 µm cono-spherical tip at 300 µm/s and normal load (a) 500 mN (b) 750
mN....................................................................................................................149
Figure 8.4 Closed-loop Scanner [7] .........................................................................151
Figure 8.5 Friction Force vs. Velocity Plots Obtained Using the Closed-loop Scanner
at High Velocity ...............................................................................................152
Figure 8.6 Friction Force vs. Velocity Plots Obtained Using the Closed-loop Scanner
at Low Velocities .............................................................................................152
Figure 8.7 Friction Force vs. Velocity Comparison between the Triboindenter and the
Closed-loop Scanner ........................................................................................153
Figure 8.8 Asperity Patterns with the JDX-5D11 E-Beam Lithography System .....155
Figure 8.9 MEMS (a) Multiple and (b) Linear Rack Gear Speed Reduction Drives
(Courtesy of Sandia National Laboratories, SUMMiTTM Technologies).......158
Figure 8.10 Cross-sectional View of Polysilicon Plate and Bushing of an SDA [16]
..........................................................................................................................159
Figure 8.11 Gecko Foot and Spatulae [17] ..............................................................162
1
1. Introduction
1.1 Motivation for the Study
The science of friction, i.e., tribology, together with astronomy, is possibly one of
the oldest sciences. Unlike astronomy, the interest in tribology over the centuries has
been purely practical – to move mechanical pieces past each other as easily as
possible. Leonardo da Vinci first introduced the concept of friction coefficient in the
15th century [1]. The next tribologist was Amontons around the year 1700 who
proposed a simple model relating friction to the macroscopic structure of the surface
[8]. According to Amontons, surfaces are tilted on a microscopic scale [18].
Therefore, when two surfaces are pressed against each other and moved, a certain
lateral force is needed to lift the surfaces against the loading force. From purely
geometrical arguments it can be stated based on the above model
loadlat FF ).tan(α= …(1)
where α is the tilting angle on the microscopic scale as shown in Fig. 1.1.
This model is too simple to explain everyday friction and in spite of efforts and
progress made by scientists and engineers, tribology is still far from being a well-
understood subject. In contrast to many other fields in physics, a fundamental theory
of friction does not exist and determination of relevant tribological phenomenon
from first principles is a very complicated task. Also, it is difficult to find a simple
experimental system which would serve as a model system.
2
Fig 1.1 Sketch of Two Surfaces with Interlocking Asperities [8]
Within tribology, experimental and theoretical understanding of friction is well
developed. Wear is the central interest of tribologists and frictional dynamics was
used as a tool to explore basic phenomenon, especially in the early years of
tribological investigation [9]. In the later years, more powerful means were
developed to explore interface physics and frictional dynamics, although not
forgotten by tribologists, has become of secondary interest. In the recent years,
tribology has been most concerned with issues of wear and machine life on one hand
and of surface chemistry and physics on the other. Great progress has been made
towards understanding the physical processes of sliding machine contacts: bearings,
transmission elements, brushes, seals etc. [19]. For the controls engineer, the
implications of the modern understanding of friction are substantial. Of the several
ways that friction affects machine performance, stick-slip poses perhaps the greatest
challenge to precise control. With the last four decades of progress in surface and
lubricant physics, interest in dynamics is rising again for predictive models of the
friction mechanism [9]. Early work has found friction to be highly repeatable and
from the standpoint of control theory, repeatability, coupled with more exact
Fload
Flat
3
modeling, opens up the possibility of theoretical results that more accurately reflect
the observed phenomenon.
Friction plays a role in the simplest actions of living, such as walking, grasping
and stacking. The importance of the forces or friction are not small but when friction
is addressed, the models used are often those of Leonardo da Vinci or Charles de
Coulomb. Experimental evidence pertinent to the situation under scrutiny is rarely
sought out or presented, perhaps because of the fact that friction may vary from one
situation to another.
A model of friction is necessary for many purposes. In some cases it is desirable
to have a model which provides insight into the physical mechanisms of the friction
interface [20]. In others it is suffices with a model that can predict global, qualitative
behavior of a system with friction. The study of friction mechanism for control
applications has been driven by the desire to understand and compensate for limit
cycles observed in the mechanism motion, namely stick-slip. The limit cycles
themselves produce measurable phenomenon: slip distance, period and relative time
in stick-slip which have been the principal data guiding the choice of models for
analysis or compensation.
There are many different models for friction, ranging from purely empirical
models [19, 21] to detailed models based on material physics [22]. The
understanding of the friction mechanisms that have emerged in the past 20 years has
not yet been translated into dynamic models that are easy to use for simulation and
friction compensation. This is in part because some effects are still not well modeled.
4
Current experimental data that is based on precision servos are not accurate enough
to invalidate key hypotheses. Strong experimental support in the form of careful
studies of the behavior of friction for small displacements and slow motions is
required. The Hysitron Triboindenter (nanoindenter) is an instrument primarily
designed for indentation and it can also measure lateral force, i.e. perform scratch
testing, using a 2D transducer. It is very likely that many issues can be resolved
using experiments with the nanoindenter and it could serve as an effective tool for
performing the experiments required for improved modeling of friction at the
microscale. In this thesis, we plan to test the feasibility of Hysitron Triboindenter as
a friction measurement tool and investigate such dynamic friction models at the
micro-scale for control applications as well as for applications in MEMS and other
nano-scale devices.
1.2 Objectives and Scope of the Study
An important challenge in the development of models and methods for these
types of systems has often been the lack of accurate friction data and reliable
instruments for obtaining essential data. The Atomic Force Microscope (AFM) [23,
24], Surface Force Apparatus (SFA) [25] and the Scanning Force and Friction
Microscope (SFFM) [26] have been developed and used in recent years for
tribological studies. While the AFM and the SFFM are specifically suited to very low
load (1-100 nN and 1-500 nN respectively) and very small (nearly atomic) scale
friction experiments (radii of contact ~ 20 nm and 1-300 nm respectively), devices
5
including the SFA are less limited in this regard. The Hysitron Triboindenter is a
recently developed tool for measuring the hardness and elastic properties of thin
films and coatings and provides both normal and lateral force loading configurations.
Although the instrument is well-suited for friction experiments through accurate
control and measurement of the normal and lateral forces through feedback, it had
not been previously characterized for friction testing. Using the scratch capability of
the nanoindenter, we aim to perform experiments to increase an understanding of
friction at the micro-scale and to aid future modeling efforts. We plan to validate the
obtained experimental results by performing comparison experiments with other
friction measurement tools such as the SFA and the Tribo-rheometer [4].
A good model is an essential element of control design. A friction model should
be of moderate complexity, work for different engineering contacts and under
various operating conditions, yet have built in as much built in friction structure as
possible [20]. The aim is to derive a model that qualitatively captures the complex
nonlinear behavior of friction and is fairly simple with as few parameters as possible
to tune. There is, of course a tradeoff between model complexity and its ability to
describe intricate friction behavior.
Apart from seeking to modify existing models such as the LuGre model, we will
also attempt to develop first-principle physics-based models. Another idea is to
model the local contact forces using materials models and to get global behavior by
averaging over the ensemble of contact points. The models will have additional
states and the nonlinearity is generated by the shape of the distribution of the contact
6
surfaces. Our goal in modeling is to develop models which accurately represent the
physics phenomenon, but are simple enough to be used for control.
The main objectives of this study are
1. To perform an extensive literature review of existing friction models, both
theoretical and empirical, focusing on the LuGre model in particular.
2. To characterize the Hysitron Triboindenter for friction testing at the
nano/micro scale and understand its capabilities and current limitations.
3. To repeat classical friction experiments using the triboindenter and gather
reliable data for different material samples for accurately modeling friction
at the micro scale.
4. To perform a multitude of different kinds of friction experiments, e.g.
oscillations of increasing magnitude, variation of tip material and radius,
testing at different normal loads, altering the waiting times before
experiment commencement etc. to further understand and quantify the
behavior of friction under different conditions.
5. To estimate functions and parameters in existing models such as the
LuGre model to see how accurately they can fit the experimental data.
6. To explore new friction models that could better explain the observed
friction characteristics and to investigate a physical basis for models that
work best.
7
7. To perform simulations of the friction experiments using the new friction
model based on an estimation the parameters of the measurement tool.
8. To perform comparison experiments with other tribological measurement
systems such as the SFA and the tribo-rheometer to validate the friction
results obtained from the triboindenter.
9. To investigate potential enhancements to the triboloigcal measurement
equipment and techniques and their potential applications in micro/nano-
scale devices.
1.3 Organization and Outline
In this study, we plan to explore the capabilities of the Hysitron Triboindenter to
generate precise data for friction tests. After characterizing the equipment for friction
measurement, we plan to perform friction tests and develop dynamic models of
friction that can be used for friction compensation in MEMS systems.
In Chapter 2, we present a theoretical background of the various characteristics
exhibited by friction followed by a literature review of the currently existing
empirical and theoretical models used to describe them. We explore the advantages
and drawbacks of each of these modeling approaches. This chapter highlights the
need for a thorough experimental investigation that could help better estimate the
parameters of this model as well as open the possibility of new friction models from
first principles.
8
In Chapter 3, we present a historical overview of tribological measurement
equipment followed by a detailed description of the components, design and testing
methodology of the Hysitron Triboindenter, the scratch functionality of which we
intend to further explore for tribological measurements. Its currently known
limitations are highlighted along with techniques to overcome some of these
limitations.
Chapter 4 details our experimental results with the Hysitron Triboindenter. The
choice of tips and samples used in our experiments are described in detail along with
the experimental procedure. The experimental results are presented along with an
analysis of the results derived from a variation of various experimental parameters
and an investigation of their repeatability. The entire load and displacement range of
the triboindenter is utilized in these experiments and unique features such as the
depth measurement capability of the triboindenter are investigated in detail.
In Chapter 5, we study and model the actuation and sensing mechanisms of the
triboindenter and characterize some of its observed limitations such as the tilt. This is
followed by a detailed analysis of few particular experimental results with a view to
understand the finer details of the transition from the static to the kinetic regime. The
experimental parameters obtained from the above analysis are combined with an
estimation of the control-loop parameters of the triboindenter to develop a MATLAB
simulation of the friction experiments based on a modification to the LuGre model.
9
In Chapter 6, a tribo-rheometer setup [4], which has a larger experimental range than
the Hysitron Triboindenter in terms of both velocity range and normal load although
at a lower resolution, is described in detail. Comparison experiments with the tribo-
rheometer using similar contacting surfaces in both systems are performed and the
results are discussed. Data fitting to the LuGre model using previously obtained
results from the tribo-rheometer is performed and the insights into the modeling
obtained from the curve fitting are discussed.
In Chapter 7, friction experiments with Polystyrene (PS), which is a common testing
material used with the surface force apparatus (SFA), are performed with the
triboindenter and the equipment modifications required to perform these experiments
are described. A comparison between the results obtained from the triboindenter and
the SFA are used to determine the validity of the triboindenter as a tribological tool
in comparison to the SFA.
Chapter 8 presents a conclusion to the research and investigates new research tools
and methods to characterize friction in load and velocity ranges not possible with the
current set up. The testing results of some of these tools are presented along with an
analysis of their usability and drawbacks in the context of the current research.
Lastly, possible application areas of the measurement tool and techniques developed
in this study for tribological investigation are described.
10
2. Theoretical Background
2.1 Dynamic Regimes in Friction
Early friction models assumed that friction was intimately related to the
roughness of the opposing surfaces and the fundamental cause of friction was the
force required to lift interlocking asperities over each other in sliding motion, a view
strongly promoted by Amontons and Coulomb based on their experimental
investigations. This view had great appeal and merit since it was beautiful in its
simplicity, yet concordant with the laws of friction but many of these early models
failed because the surface topography was misunderstood to be conformal. In Fig.
2.1, part A rests on part B and the parts deform to create a nonconformal contact.
The true area of contact, an area that increases with increasing load, is much smaller
than the apparent area of contact [19]. Over a broad range of engineering materials,
the asperities have slopes ranging from 0 to 25 degrees and are concentrated in the
band from 5 to 10 degrees [27]. When asperities come into contact the local loading
will be determined by the strength of materials, which will deform as necessary to
take up the total load. As the load grows, the junction area grows; but, to first order,
the shear strength (measured per unit area) remains constant. As proposed in most
models, friction is proportional to the shear strength of the asperity junctions and in
this way friction is proportional to load.
11
Traditionally, friction has been regarded as a static function of velocity. In the
presence of a lubricant, friction as a function of velocity exhibits four dynamic
regimes – static friction, boundary lubrication, partial fluid lubrication and full fluid
lubrication. These are shown in Fig. 2.2, which is called the Stribeck curve and
shows the three moving regimes. The interesting characteristics of static friction such
as presliding and break-away friction force, described in the following subsections,
are traditionally not considered to be dependent on velocity.
2.1.1 Static Friction
The asperity junctions deform elastically giving rise to motion that appears to be a
solid connection with a stiff spring as shown in Fig. 2.3. Both the boundary film and
the asperities deform plastically under the load, giving rise to increasing static
friction as the junction spends more time at zero velocity. In the presliding regime,
i.e. motion prior to fully developed slip, the force is a linear function of the
displacement, until a critical displacement (force), as in a spring. When the applied
force exceeds the level of static friction, the junctions break and true sliding occurs.
This phenomenon is called the Dahl effect and it is observed to occur for true relative
deflections of 5 µm or more.
2.1.2 Boundary Lubrication
In the second regime, the velocity is not adequate enough to build a fluid film
between the surfaces. When lubricant is added to the contact, for low velocities, the
lubricant acts as a surface film, and its lower shear strength helps in reducing the
12
Fig 2.1 True Contact Between Engineering Surfaces [9]
Fig 2.2 The Stribeck Curve – Friction as a Function of Velocity [9]
friction. The friction in this regime is largely independent of velocity and strongly
dependent upon lubricant chemistry.
2.1.3. Partial Fluid Lubrication
In the third regime, the lubricant is brought into the load bearing region through
motion, either by sliding or rolling. Some of it is expelled by the pressure arising
13
Fig 2.3 Idealized Contact in Static Friction, Asperity Deformation and in Break-
Away Friction [9]
from the load but viscosity prevents all the lubricant from escaping and thus a film is
formed. The greater the viscosity or motion velocity, the thicker the fluid film will
be. When the film is not thicker than the height of the asperities, some solid-to-solid
contact will result and there will be partial fluid lubrication. If the static friction is
greater than kinetic friction, the friction will decrease with increasing velocity.
2.1.4. Full Fluid Lubrication
When the film is sufficiently thick, separation is complete and the load is fully
supported by the fluid. Hydrodynamic and elastohydrodymanic (EHL) are two forms
14
of full fluid lubrication. Hydrodynamic lubrication arises in conformal contacts and
EHL in nonconformal contacts. The wear is reduced by orders of magnitude and
friction is a function of velocity. The goal of lubrication engineering is often to
maintain full fluid lubrication effectively at a low cost.
Causes of stick-slip: As seen from the Stribeck curve, the relative motion between
two bodies is accompanied by a reduction in the friction force. The sliding body
accelerates until the point where the elastic restoring force and the friction force
between the sliding bodies equalize and deceleration takes place until a new period
of stick occurs [21].
The time dependent characteristics of friction are listed below –
1. Rising Static Friction with Increasing Dwell Time – The dwell time is the
time during which the surfaces are in fixed contact viz. the time intervals a-b,
c-d, e-f, g-h and i-j in Fig. 2.4 The static friction is observed to increase with
dwell time and this accounts for the larger stick-slip at lower velocities.
2. Frictional Lag – The Stribeck curve shows a dependence of friction upon
velocity. If there is a change in velocity, one might presume the
corresponding change in friction to occur simultaneously but experimental
data [5] indicates a lag in the frictional change, as shown in Fig. 2.5. This
results in hysteresis as the velocity varies as shown in Fig. 2.6. Hess and
Soom carefully measured the frictional lag and found it to range from 3 to 9
milli-seconds in a range of load and lubricant combinations for experiments
with 52100 Steel, the lag increasing with increasing lubricant viscosity and
15
Fig 2.4 Spring Force Profile During Stick-Slip Motion [9]
Fig 2.5 Typical Friction-Speed Time Shift [5]
Fig 2.6 Friction as a Function of Velocity; 0: Experimental -: Theoretical [5]
16
with increasing contact load. This leads to a pure delay in the onset of the
destabilizing drop in friction. Whether this process is better modeled by a
simple time delay or some other formulation is still an open question and
experimental data would be instrumental for investigating this further.
3. Time Dependence of the Stiction Force - Time dependence of the stiction
force means that the force required to start sliding depends upon how long
time the system has rested since the binding force between two bodies
increases with contact time. Further experimental evidence [28] is required
for investigation.
From the above discussion we can conclude that there is evidence that suggests
that friction can be repeatable. The derivation of an accurate friction model that
captures all of the above effects would permit analytical prediction of performance,
correct decoupling of multi-degree-of-freedom mechanisms and design of friction
compensation. If stick-slip cannot be eliminated by proper choice of lubricants, an
accurate friction model would aid in the determination of mechanism and control
performance required to achieve smooth action.
2.2 Empirical Models
2.2.1 Bo and Pavelescu Model
There have been several empirical models proposed for friction. Bo and Pavelescu
[21] reviewed several models from existing literature and adopted and linearized an
exponential model of the form
17
vFeFFFvF V
vv
CSC
s
s +−+=−
δ
)()( …(2)
where, F(v) is the friction as a function of velocity
FC is the kinetic friction (Coulomb Friction)
FS is the static friction
FV is the viscous friction parameter, incorporated later [9]
v is the motion velocity
vs is an empirical parameter called the Stribeck velocity
δs is an empirical parameter whose value was suggested to be between 0.5 and 1 [9]
used a Gaussian parametrization with δs = 2.
The exponential model (2) is not a strong constraint and by appropriate
choice of parameters, curves of types (a), (b) and (c) as shown in Fig. 2.7 can be
realized. More data such as that presented in [5] over a range of materials, conditions
and lubricants is required.
2.2.2 Armstrong’s Seven Parameter Model
Another empirically motivated model was Armstrong’s seven parameter model [19]
in which the pre-sliding displacement is described by
xkxF t−=)( …(3)
And sliding is described by
)sgn()/)((1
1),(),( 22 v
vtvtFvFFtvF
sL
SVC
−+++−=
τγ …(4)
18
where the rising static friction (friction level at breakaway) is given by
γγ
+−+= ∞
2
2,,,2 )(),(
t
tFFFtF aSSaSS …(5)
FS,a is the Stribeck friction at the end of the sliding friction
FS,∞ is the Stribeck friction after a long time at rest (with a slow application of force)
t2 is the dwell time
τL is the time constant of frictional memory
γ is the temporal parameter of the rising static friction
kt is the tangential stiffness of the static contact
As the name states, the model requires seven parameters – kt, FV, FC, FS,∞, vS, τL, and
γ as defined earlier. Each of the seven parameters of the model represents a different
friction phenomenon and their magnitudes depend upon the mechanism and the
lubrication. Also, since the model consists of two separate models, one for sticking
and one for sliding, a logical statement probably requiring an eighth parameter would
determine the switching [20]. The model states would have to be initialized every
time a switch occurred.
2.3 Theoretical Models
2.3.1 Dahl Model
Another approach to modeling friction is to use the simplified pictures of the
physical contact for example as the ones shown in Fig. 2.3 as the starting point. Dahl
[29] introduced two models for frictions essentially based upon a reformulation of
the stress-strain curve, thus being closely related to physics. In the first model, the
friction interface is modeled as a junction at which shearing takes place and the
19
Fig 2.7 Friction as Function of Steady State Velocity for Various Lubricants [9]
resulting friction force depends on the strain caused by the external force. This
corresponds to pre-sliding displacement. If the strain is large enough, the junction
breaks and the friction force remains constant at the level at which the rupture took
place. When the external force is removed the result is a permanent deformation.
The second model introduced by Dahl [29] is based on the assumption that the
change in friction force can be described by
dt
dx
dx
dF
dt
dF= …(6)
This would imply that the friction force is only position dependent. The model
exhibits hysteresis between velocity and friction force. The hysteresis depends upon
the rate of change of velocity. A good agreement between the model and true friction
was observed and Dahl also noted that the model is a generalization of ordinary
Coulomb friction. The second model is further studied by Dahl in [30] [31] where it
is used to describe frictional damping of a wire pendulum and is also applied to
20
internal friction. The position dependency of the friction force is further explored and
Dahl proposes the relation
−
−=
dt
dx
F
F
dt
dx
F
F
dx
dF
C
i
C
sgn1sgnsgn10σ …(7)
where σ0 is the stiffness and the exponent i is a model parameter. The second
factor is present to stabilize the differential equation for simulation purposes. In [32]
the model is used when experimentally studying friction in ball bearings and
parameters of the model are fitted to agree with the experiments. In one type of
fitting, i is estimated to 1.5 quite consistently and it is observed that the rest stiffness
σ0 is important to the fit. In the other type, i is fixed to 1 and 2 but it is observed that
the fitted data do not agree with the measured data over an as large friction force
range.
When referred in literature the Dahl model is often simplified using i = 1 to
dt
dx
dt
dx
F
F
dx
dF
C
−= sgn10σ …(8)
The Dahl model has been used extensively for simulation of systems with
friction. The model captures many properties of real friction phenomenon but not the
important stick-slip effect.
2.3.2 Lugre Model
The LuGre model is an extension of Dahl’s model which captures many
properties of the friction including stick-slip motion. The starting point of the LuGre
model’s derivation is force caused by solid-to-solid contact as visualized in Fig. 2.8.
21
Fig 2.8 Friction Interface Between Two Surfaces Modeled as Contact Between
Bristles [10]
Surfaces are very irregular at the microscopic level and therefore two surfaces make
contact at a number of asperities. This can be visualized as two rigid bodies that
make contact through elastic bristles. When a tangential force is applied, the bristles
deflect like springs giving rise to the friction force. If the force is sufficiently large,
some of the bristles deflect so much that they slip off each other. New contacts are
then formed as the two surfaces continue to move and the process goes on. The
contact phenomenon is highly random due to the irregular forms of the surfaces. The
average deflection of the bristles is denoted by z and is modeled by
zvg
vv
dt
dz
)(−= …(9)
where v is the relative velocity between the two surfaces.
The first term gives a deflection that is proportional to the integral of the relative
velocity. At steady state, the deflection z approaches the steady-state value
)sgn()( vvgz ss = …(10)
when v is constant.
22
The function g is positive and depends upon many factors such as material
properties, lubrication and temperature. For typical bearing friction, g(v) will
decrease monotically from g(0) when v increases, corresponding to the Stribeck
effect. A proposed parametrization of g is [20]
( )
−+=
−
2
0
1)( sv
v
CSC eFFFvgσ
…(11)
where FC is the Coulomb friction level, FS is the level of stiction force and vs is the
Stribeck velocity.
The friction force generated from the bending of the bristles is described as
vFdt
dzzF v++= 10 σσ …(12)
where σ0 is the stiffness, σ1 is a damping coefficient and Fv is the coefficient of
viscous friction. This is known as the LuGre Model which is characterized by six
parameters σ0, σ1, Fv, FC, FS and vs. Thus for steady-state motion, the relation
between friction force and velocity is given by
vFveFFvFvF v
v
v
CSCSSs +−+=
−
)sgn()()sgn()(
2
…(13)
This model reduces to the Dahl Model if g(v) = FC /σ0 and σ0 = σ1 = 0
The model is simple yet captures most friction phenomena that are of interest for
feedback control. The low velocity friction characteristics are particularly important
for high performance pointing and tracking. The model can describe arbitrary steady-
state friction characteristics. It supports hysteretic behavior due to frictional lag,
spring-like behavior in stiction and gives a varying break-away force depending on
23
the rate of change of the applied force. All these phenomena are unified into a first-
order nonlinear differential equation. The model can readily be used in simulations
of systems with friction. A drawback of the LuGre model is that it does not describe
presliding and the dependence of friction on dwell time. Some of these can be
captured by modification of nonlinearities but they require an additional state. A new
class of single-state models was defined by [33] in which presliding is elastoplastic
and under loading, frictional displacement is first purely elastic and then transitions
to plastic. In [28] the origin memory effects in plasticity and ways to deal with them
are discussed. In [34] limit cycles caused by friction are treated and necessary
conditions for limit cycle and conditions for local stability are given.
The nonlinear function in the LuGre model (equation 13), which describes the
steady state friction force at constant velocities, has been measured using servo
systems. The precision of these systems do not give accurate results for very slow
sliding speeds below 0.01 m/s. This behavior is crucial to describe stick-slip
behavior and precision tracking. Strong experimental support in the form of precise
measurements at low tracking speed is critical for the development of improved
models.
24
3. New Friction Measurement Instrument - The Hysitron
Triboindenter®
3.1 Historical Overview of Tribological Instrumentation
Tribology has been a part of Physics and Engineering for a long time since the
classic results of French physicists Amonton and Coulomb. There was significant
progress in this field in the 1970s [27] and greater insight was gained in the past 15-
20 years when tribologists began to work in the nanoscale [35].
Atomic Force Microscopy (AFM) is an early offspring of scanning tunneling
microscopy (STM). The force between a tip and the sample was used to image the
surface topography. The surface force apparatus (SFA) was developed by
Israelachvili [36] for measuring van der Waals forces between molecularly smooth
mica surfaces. Recently, new friction attachments have been developed suitable for
use with the SFA which allow for the two surfaces to be sheared past each other at
varying sliding speeds at varying sliding speeds while simultaneously measuring
both the transverse force and the normal force between them [11]. The Scanning
Force Microscope (SFM) was introduced in 1986 to measure the topography of
nonconducting surfaces and within a year its potential to measure forces was applied
successfully to measure the atomic scale variation of of the friction force as a sharp
tip scans over a surface [18]. The simple but clever idea of turning the SFM around
by 900 in order to measure the lateral force instead of the normal force led to the
birth of the Scanning Force and Friction Microscope (SFFM). The SFFM has the
25
ability to resolve the atomic periodicity of the topography and the friction force as
the tip moves over the surface. The dynamic-SFFM, provides for increased
sensitivity through lock-in technique to small difference in friction forces and offers
enhanced scanning speeds [2].
Nanoscale experiments give a deeper understanding of the basic mechanisms that
generate friction and physical explanations for dependence of friction on contact area
velocity. The AFM does not yield a very high force range, and thus is limited to very
low load and very small (nearly atomic) scale friction experiments, more suited for
tribological purposes. The Hysitron Triboindenter is a stand-alone nanomechanical
testing system which allows users to characterize materials on nanometer length
scales. Traditionally it is used to characterize materials by making a nanoscale indent
using a precisely formed tip. This indent and resulting force vs. displacement data
can be used to extract material properties of the sample [3]. Quasistatic
nanoindentation can measure properties including Young’s modulus, hardness,
viscoelasticity and fracture toughness. Scratch testing using lateral can be used to
quantify scratch resistance, critical delamination forces, friction coefficients etc. A
comparison of the performance characteristics and ranges for the SFA, SFFM and the
Hysitron Triboindenter® (nanoindenter) are given in Table 3.1. The nanoindenter
seems to be well suited for measuring motion in the micro-load scale and in the low-
velocity regime. The tip can move laterally and vertically, and can either be position
or force controlled using feedback on the transducer. Lateral motions can be used to
26
Operating Parameter
Radius of Mating
Surface/Tip
Normal Load Sliding Velocity
SFA 0.2 – 2 cm 10 - 100 mN 0.001 - 100 um/s
SFFM/dyn. SFFM 10 - 300 nm 0.1 - 500 nN 0.02 - 2 um/s
Nanoindenter 1 - 400 um 0.001 - 10 mN 0.1 - 100 um/s
Table 3.1 Comparison of the SFA, SFFM and the Nanoindenter [1, 2]
extract friction data. The instrument is well suited for our friction experiments
because it can measure displacement and normal and lateral forces simultaneously.
The normal and lateral forces can be controlled accurately using feedback and it also
has potential for experiments that bridge the gap between the macro and nano scales.
3.2 The Hysitron Triboindenter® - Components and Design
As shown in Fig. 3.1, the basic components of the nanoindenter are -
- Triboindenter base
- XYZ Staging system
- Top Down Optics
- TriboScanner
- Transducer Assembly
- Vibration Isolation Assembly
- Acoustic Enclosure
27
- Electronics for transducer, stage, piezo and optics
- Computer system with Data Acquisition Boards
A brief description of the important components is as follows –
3.2.1 Stage and Optics
Triboindenter Base
The primary function of the granite base is to support the other components
which are connected to it using a special bracket. The base is designed to minimize
drift and noise transfer while maximizing the stability of the instrument.
XYZ Staging System
The coarse control of the samples and tip positions is controlled by the XYZ
staging system. The X-axis and Y-axis stages are mounted to the bottom of the base
while the Z-axis stage is mounted on the bridge. The sample stage is connected
directly to the XY stage via a dovetail mount. The TriboScanner and optics are
mounted to the Z stage which ensures that the probe tip and the optics move
together. The step resolution of the XY encoder is 50 nm while that of the Z stage is
13 nm.
Top-Down Optics
The optics are located at the right side of the Z-stage. There is a 10X
objective on the end which sends the magnified image to the zooming optics which
magnify the image again upto 10X. The image is then sent to the CCD detector
which is then sent to a video capture board in the computer system. The maximim
field of view is 850µm X 650µm.
28
Fig 3.1 The Hysitron Triboindenter (Nanoindenter) [6]
3.2.2 Triboscanner
As shown in Fig. 3.2 (a), the TriboScanner is designed to provide fine scale
positioning of the indenter tip and for imaging of the surface before and after the
tests. The final approach and positioning of the tip is provided by a 3-axis piezo
scanner that has a precision 300 times higher than the Z stage and 10 times higher
than the XY stage. The piezo-scanner, shown in Fig. 3.2 (b), has a rigid tube
configuration – the dimensions increase in one direction and decrease in another
maintaining a constant volume. The top half of the TriboScanner tube has four
seperate quarter cylinders, each controlling motion in a different direction +X, -X,
+Y and –Y. When each separate portion of the top half of the TriboScanner tip is
energized, the ceramic of that portion lengthens along the axis of the scanner while
the walls become thinner causing the tube to bend to the side. Energizing the lower
tube lengthens it providing motion along the Z axis. Thus 3D motion is achieved by
manipulating voltages sent to all five parts of the tube.
Acoustic Enclosure
Triboscanner and Optics
Electronics Rack
29
Fig 3.2 (a) TriboScanner and Optics [6] (b) Schematic of Piezo Scanner [3]
Fig 3.3 (a) Triboindenter Transducer [6] (b) 1-D and 2-D Transducers [3]
3.2.3 1-D and 2-D Transducers
Transducer Assembly
The transducer assembly, as shown in Fig. 3.3 consists of the
force/displacement sensor, drive circuit board and the hardware used to mount the
TriboScanner. The transducer has a three-plate capacitive design as shown in Fig. 3.4
(a) that allows both electrostatic force actuation and capacitive displacement
measurement in a single device. It also provides for high sensitivity, large dynamic
range and a linear force or displacement output signal.
30
Lateral Force Transducer
The lateral (X-Axis) forces and displacements using a 2D transducer system
are applied and measured similar to the indentation (Z-Axis) systems. The X axis is
under displacement control rather than force control and there is a control loop to
apply proper forces which has been investigated in detail in section 5.4.1. Because of
the larger range of motion along X-axis, there are large displacement corrections for
the force that are taken into account. A schematic of the lateral force transducer is
shown in Fig. 3.4 (b). The transducer has two additional force-displacement sensors
to monitor and control position in the X-direction.
3.2.4 Displacement/Force Application and Measurement
The force is applied electrostatically while the displacement is measured
simultaneously by the change in capacitance. The sensor consists of two fixed outer
electrodes (drive plates) which are driven by AC signals 1800 out of phase with each
other as shown in Fig. 3.5 (a). The electric field potential at the drive plates is
maximized (equal to the applied signal) and minimized (zero) at the exact center of
the drive plates. This results in a bipolar output signal equal in magnitude to the
drive plate at the maximum deflection, zero at the center position, and varying in a
linear manner between the maximum displacement and center position. This creates
an electrostatic attraction between the center plate and the bottom plate, pulling the
center plate down. The force can be calculated from the magnitude of the voltage
applied. The maximum force available from the 1-D transducer is approximately 30
mN. The specifications of the transducer are listed in Table 3.2.
31
Fig 3.4 Cross Section Schematic of (a) 1-D and (b) 2-D Transducer [3]
Fig 3.5 (a) Displacement Measurement (b) Force Measurement [3]
3.2.5 Limitations of the Lateral Force Transducer: Nonlinearity and
Transducer ‘Breakpoint’
Nonlinearity
The spacings between the plates inside the lateral transducer are
approximately 100-150 µm. This means that the total physical travel that can be
allowed in the X-axis is limited to this amount although the linear range is much less.
The center plate of the transducer is moved by actuating one of the drive plates. The
center plate would then be electrostatically attracted towards the drive plate. When
the center plate moves, the distance between the drive plate and the center plate is
|Electric Field Potential|
1800 out of phase Linear variation
0 max
Drive Signal Plate Pulled Down Electrostatically
600 V DC Bias
32
Z-Axis X-Axis
Maximum Force 10, 30 mN 2 mN
Load Resolution 1 nN 3 µN
Load Noise Floor 100 nN 10 µN
Maximum Displacement 20 µm 15 µm
Displacement Resolution 0.04 nm 4 nm
Displacement Noise Floor 0.2 nm 10nm
Thermal Drift <0.05 nm/s <0.05 nm/s
Table 3.2 Specifications of the Transducer [3]
reduced, causing an increase in the electrostatic force.
Transducer ‘Breakpoint’
‘Breakpoint’ is another issue with the operation of the lateral transducer
which further limit its range. When plates reach a critical distance apart the force
constant gets too high causing them to snap together. This typically happens well
before the maximum physical displacement of the transducer has been reached.
Another critical point in the X-axis transducer occurs when the scratch switches from
one plate to another. When center plate passes through zero, the voltage on the drive
plate moving the center plate goes to zero and the other plate begins moving the
center plate. Since voltages on both plates goes to zero for an instant, the control
loop loses control of the force and a “breakpoint” occurs as shown in Fig. 3.6. Some
key transducer constants such as the Electrostatic Force Constant and the Plate
33
Spacing change, thus all scratches must be made completely on one side of the
breakpoint
3.3 The Hysitron Triboindenter® - Data Acquisition
All of the control of the nanoindenter is done through the software,
TriboScan® 5.0, which is designed to fully automate most of the tasks required from
the nanoindenter. The stage is controlled using the Positioning window as shown in
Fig. 3.7 which is also the main window of the software. The Video window enables
the user to view the stage and sample with the top down optics. The Method editor is
the control center for automated experiments.
3.3.1 Load Functions
Load Functions are set up in the Data Acquisition and Data Analysis section
of the software. The load function for lateral force analysis looks like the one shown
in the Fig. 3.8. There are two graphs that display the Normal Load versus Time and
the X-Position versus the time. The individual segments of each graph are linked
with respect to time. The user can change the beginning and end loads and the
beginning and end positions along with the time for each segment thus defining the
exact load and velocity pattern required. As seen in the figure, a typical loading
function consists of five segments. During the first segment, as seen from the top
window, the normal load is kept at zero and the tip is moved to +4 µm, which is half
the length of the desired scratch, over a period of 5 seconds. During the second
segment, the tip is held at that location and the normal load is increased to the
34
Fig 3.6 Schematic of Center of Transducer Showing Breakpoint [3]
Fig. 3.7 The TriboView Positioning Window [3]
35
desired value, 1000 µN in this case over a period of 3 seconds. In the third segment,
the normal load is maintained at the desired value and tip is moved from +4 µm to -4
µm over a period of 16 seconds which corresponds to a velocity of 0.5 µm/s. At
theend of the motion, the displacement is held constant at -4 µm and the normal load
is decreased to zero in the fourth segment. The tip is then moved back to the zero
displacement value in the last segment. This method ensures that the scratch made is
centered for imaging purposes. After loading the required tip and performing the
necessary calibrations, the desired load function can be executed. The resulting
Scratch Data window contains four graphs as shown in Fig. 3.9. Normal Force,
Normal Displacement, Lateral Force and Lateral Displacement are all plotted as
functions of time. The Normal Force (top-left) and the Lateral Displacement
(bottom-right) plots are the real-time results of the input load function. The Lateral
Force (bottom-left) plot yields the required friction force versus time.
3.3.2 Software-based Tilt Correction
There is always some tilt to the sample no matter how level we try to make it
and the measured Z displacement is a combination of the tip moving into the sample
and the tilt of the sample. If, during the first few seconds of a scratch, the tip is
moved to the starting position of the scratch with no normal force applied, we may
observe a slope in the result because of the tip being pushed up on account of sample
sloping up in the direction of travel. We may also observe the tip coming off the
surface completely if the sample slopes away from the tip since the sample cannot
pull the tip down with it. The tip might have to be moved in both directions first to
36
Fig 3.8 Typical Loading Function for Scratch Testing
Fig 3.9 Typical Scratch Testing Results Window
37
determine which way the sample is tilting because the data is only valid in one
direction. The Tilt Correction functionality in the software calculates the tilt and
adjusts the data to compensate for it so that only the displacement of the tip into the
sample is plotted on the Z Displacement graph. To correct for the tilt, the red cursors
in the normal displacement plot have to be positioned on an area of the plot where
the normal force is constant and the X-axis is displacing, as shown in Fig 3.10. This
is done preferably at the beginning of the first segment. Upon clicking ‘tilt
correction’, the software measures the slope of the sample, assuming all
displacement is from the sample slope, and removes this slope from the rest of the
data. The generated output file, as shown in Fig 3.11, displays the new tilt-corrected
normal displacement while the rest of the plots remain unchanged. This utility is
applicable only if the sample tilts upwards in the direction of travel so the
experiment might have to be repeated multiple times at different sample orientations
to accurately compensate for the sample tilt.
3.3.3 Drift
The Hysitron triboindenter employs the three-plate capacitive transducer
described earlier that is used as both the actuator and sensor of the instrument. The
force is applied electrostatically while the displacement is simultaneously measured
by the change in capacitance. Electrostatic actuation requires virtually no current,
which is expected to result in virtually no drift due to heating during actuation
compared to many other designs, such as electromagnetic devices, that realize
significant heating during actuation due to high current requirements. However, as
38
Fig 3.10 Sample Normal Displacement before Tilt Correction (Top Right)
Fig 3.11 Sample Normal Displacement after Tilt Correction (Top Right)
39
Fig 3.12 Lateral Force Profile resulting from a 50 µm diamond tip being held
stationary on a mica surface for 1000 s at 7000 µN normal load
seen from Fig. 3.12, a considerable amount of drift in the lateral force is observed if
the tip is held stationary on the surface for certain amount of time after the
application of a normal force. Typical magnitudes of this drift are observed to vary
between about half the value of the lateral force which would be created if the tip
were in motion, in this case between -300 and 300 µm. This may adversely effect the
friction measurements and needs to be further investigated. The new closed-loop
control for the normal displacement of the transducer combined with the
temperature-controlled stage could be aids in reducing this drift.
40
4. Experiments with the Triboindenter: Design, Results and
Analysis
Tribological studies have led to substantial understanding of the friction forces
active during the steady motion of lubricated systems. The dynamic of the friction
process around zero velocity has been investigated by Israelachvili [11] reproduced
in Fig. 22. In [5, 9, 37-39] macro-scale experiments on friction are performed which
we aim to reproduce using the precise capabilities of the nanoindenter. Since we
have good control over tip velocity, tip shape and the force profile applied, we
should be able to make accurate measurements on friction. In [33], the authors create
a model which takes into account presliding displacement (the motion prior to fully
developed slip). Initially, our aim is to repeat the classical friction experiments. By
doing so the experimental setup would be verified and the scaling issues can be
worked out. The controlled nature of the experiments would then lead us to a
physical basis for model changes.
4.1 Sample Preparation and Characterization
Mica, a widespread and common rock-forming mineral, was chosen as the
primary material for our experiments. The choice of mica as one of the contacting
surfaces was based upon that fact that it provides with atomically flat planes and is
thus a common friction element in which to compare obtained friction data to
published experimental data [26, 39]. Ruby-red mica, grade #1 from S&J trading
company with asperity height smaller than ~ 10 pm and ~ 1 µm width was used for
41
Fig 4.1 Trace of Friction Forces at Increasing Sliding Velocities on Mica [11]
Fig 4.2 High Grade Mica Sheets (Src: S&J Trading Co.)
our experiments. The choice of high grade ensures that the mica is optically flat,
resilient, incompressible, splits into thin films along its cleavage planes and which
remains tough and elastic, even at elevated temperatures.
4.1.1 Structure of Mica
Chemically, mica is a complex hydrous silicate of aluminum, containing
potassium, magnesium, iron, sodium fluoride and/or lithium and traces of other
elements. It is stable and inert to water, acids (except hydrofluoric and concentrated
42
sulfuric acids), alkalis, conventional solvents and oil. The exact chemical formula of
mica is SiO2, Al2O3, K2O, Fe2O3, MgO, CaO, Hg2O and SiO2 (45.09%), Al2O3
(34.5%) and K2O (9.51%) form its major components.
Mica, being a silicate, its structure is based on a fundamental unit of four O2- ions
at the apexes of a regular tetrahedron surrounding and coordinated by one Si4+ ion at
the centre as shown in Fig 4.3 (a) and (b). When three of these oxygen ions are
shared with adjoining tetrahedral leading to a Si:O ratio of 2:5, infinitely extending
flat sheets of unit composition Si2O5, known as phyllosilicates, shown in Fig 4.3 (c),
are formed. Furthermore, substituting aluminum for one quarter of the silicon gives a
negatively charged layer to bind univalent cations, K+ ions, in 12-coordination to
tetrahedral-octahedral-tetrahedral (T-O-T) layers as shown in Fig 4.4. This creates
the mica structure in which the K+ ions are sandwiched between the two T-O-T
layers and occupy large holes between 12 oxygen atoms so that the K-O electrostatic
bond strength is only one twelfth of the interlayer K-O bonds. These bonds are easily
broken at the position of the K+ interlayer cation, and the mica accordingly possesses
perfect cleavage parallel to the layers [12]. Thus a simple Scotch adhesive tape can
be used to remove the top layers of the mica to form the cleavage to expose fresh
layers of mica.
4.1.2 AFM Imaging of Mica Surface
Fig 4.5 shows a top view of an AFM image of the mica cleavage reported in
literature [26], with a grayscale to indicate the range of surface heights. The nearly
43
Fig. 4.3 (a) Close-packing representation of a SiO4 tetrahedron (b) Fundamental
unit of all silicates (c) Formation of phyllosilicate [12]
Fig. 4.4 (a) Schematic 3-D diagram of the structure and composition of mica (b)
Development of 2-D mica structure from phyllosilicate [12]
44
hexagonal array of light spots in the image of Fig. 4.5 (a) corresponds to hexagonal
rings of K+ ions in the cleavage planes. The two-dimensional roughness was
measured by cutting along the two adjacent apexes of K+ ions (line A–A) and the two
separate apexes of K+ ions (line B–B) on the AFM image. Two profiles with
different spatial wavelengths of 5.2 and 6.2 Å were reported for two different
directions on the surface, shown in Figs. 4.5 (b) and 4.5 (c), respectively. The
arithmetic mean deviation roughness, Ra and peak-to-valley, P-V of the A-A section
are Ra = 0.25 Å and P-V = 0.91 Å. For the B–B section, Ra = 0.29 Å and P-V = 1.05
Å.
Fig 4.6 shown an AFM image of the mica sample used in our experiments taken
with Asylum Corp’s Molecular Force Probe 3D (MFP3D) system which has a noise
floor of 20 pm, a sensitivity of 42 nm/V on the mica surface and 1 kHz bandwidth.
From Fig 4.7, we can see that the peak value along this scan is about 1.1 Å, which is
in good agreement with the reported peak values in literature in the range 0.91 Å –
1.05 Å, discussed earlier. The visible asperities are of the order of tens of picometers
in height and thus the mica surface can be considered to be essentially flat.
4.1.3 Preparation of the Mica Sample
A mica sheet was glued on to a small steel plate of diameter approximately 15
mm using superglue. The sample was placed in the center position towards the rear
end of the nanoindenter. A new cleaved surface was prepared every time by placing a
piece of scotch tape on top and lifting away, removing the uppermost layers.
45
Fig. 4.5 (a) AFM image of Mica cleavage (top view) (b) A-A profile cut along the
two adjacent apexes of K+ ions (c) B-B profile cut along the two seperate apexes
of K+ ions [12]
Fig. 4.6 AFM Image of the Mica Sample using Asylum Corp’s Molecular Force
Probe 3D System
46
Fig 4.7 Surface profile (Z) along the length of scan
On prolonged exposure to laboratory air, a 0.3-0.4 nm fluid layer of water and
common organics adsorb to the mica surface from the time it is cleaved until it is
installed in the nanoindenter [1]. Over time, this layer could build up to 2 nm
depending upon the specific laboratory conditions and can be usually dissolved away
by immersion in pure water. Therefore, although all experiments are conducted on
dry mica surfaces under atmospheric conditions, presence of an up to 2 nm layer of
adsorbed water and organics cannot be completely avoided
4.2 Triboindenter Tips
There are many tips provided by Hysitron for use with the triboindenter,
including Berkovich, 90-degree (cube corner), cono-spherical, cylindrical, Vickers
47
(four sided pyramidal), Knoop (four sided pyramidal with opposite sides parallel)
and fluid cell tips. Cono-spherical and in rare cases cylindrical tips are most useful
for friction testing purposes.
4.2.1 Cono-spherical Tips
Cono-spherical tips are usually used for indenting in very soft materials and are
also good for scratching on harder materials when no plastic deformation is desired.
We used cono-spherical tips for friction measurements because by choosing a high
radius tip (~50 µm), a relatively flat surface-to-sample contact can be achieved.
A cono-spherical tip geometry is depicted in Fig. 4.8. The parameters can be
calculated as
)−= αsin1(Rhs …(14)
22 hRha −= …(15)
where,
R = probe radius
α = ½ included angle of probe
hs = transition depth from spherical to conical
h = contact depth
a = contact radius
48
Fig. 4.8 Cono-spherical Tip Geometry [3]
4.2.2 Tip Selection
Initial friction experiments were conducted with cono-spherical tips of radii 1 µm, 5
µm and 50 µm. By choosing a high radius tip (e.g. 50 µm), a relatively flat surface-
to-sample contact can be achieved and parallelism issues associated with flat tips
could be avoided. Diamond (recently proposed as an alternative material for friction-
based MEMS devices [40]) has been used as the material for the tips which act as
one of the contacting surfaces. The typical parameters of the higher radius tips can be
computed from the equations (14) and (15) and are presented in Table 4.1.
In some cases a sapphire flat punch of diameter 50 µm, shown in Fig 4.9, was
used for the experiments. Also, various non-standard tips made of Al, Stainless Steel
etc. and tips made by attaching small spheres of the desired surface to the end of a
49
Fig. 4.9 Cylindrical Tip Geometry (e.g. for 50 µm Sapphire Flat Punch)
R 50 µm 100 µm
Α 90 deg 90 deg
hs 14.64 29.28
amax (at h = hs ) 35.35 µm 70.7 µm
Table 4.1 Design Parameters for Nanoindenter Tips of Radii 50 µm and 100 µm
Blank tip-holders were also used in some experiments and their designs are detailed
in the relevant sections.
4.2.3 AFM Imaging of Cono-spherical Tips
Using Asylum’s MFP3D system, AFM images of the 5 µm tip were obtained
and they are shown in Fig 4.10. Lines were observed along the scan direction and
hence the direction was changed and the scanning was performed again to determine
if the lines were true features on the tip surface or created by the scanning
mechanism. As seen in Fig. 4.10 (b), changing the scan direction caused a
corresponding shift in the orientation of those lines indicating that they were an
R
50
Fig. 4.10 AFM Imaging of the 5 µm cono-spherical tip along perpendicular
directions – scan lines indicate direction of tip travel and not real features
51
experimental artifact. Small depressions were observed on the surface of the 5 µm tip
but overall the tip surface appeared reasonably well-defined.
An AFM image of the 50 µm 90-degree cone angle tip, which is used most
frequently in our experiments, was then taken and it is depicted in Fig. 4.11. The tip
shows a number of flaky depositions, which are most probably mica flakes from
previous experiments. The tip surface was cleaned with acetone, isopropyl alcohol
and DI water and was blow dried and then re-imaged. The new image, depicted in
Fig. 4.12, did not contain the flaky particles seen before but a number of irregular
features remained which indicate that the tip surface cannot be assumed to be well-
rounded and smooth.
As shown in Fig 4.13, a section along the indicated line was taken and the tip
profile along that line was analyzed. These results are shown in Fig 4.14; the thinner
line shows the trace of the AFM tip while the thicker line shows the derivative
calculated at each point quantifying the slope. The figure indicates that the asperities
on the surface for the given cross-section have heights varying from 5-20 nm while
the asperities themselves are separated by an order of magnitude of 1 µm. These
numbers yield a useful estimate of the profiles of the contacting surfaces and help in
explaining some of the friction results obtained.
4.3 Experiment Design
The nanoindenter is primarily designed for nanoindentation purposes and is not
very well characterized for performing scratch testing. The first step, therefore, was
52
Fig. 4.11 AFM Image of the 50 µm cono-spherical tip (unclean)
Fig. 4.12 AFM Image of the 50 µm cono-spherical tip (cleaned with Acetone,
Isopropyl Alcohol and Blow Drying)
53
Fig. 4.13 AFM Image of the 50 µm cono-spherical tip showing section along
which Trace is taken
Fig. 4.14 AFM Image of the 50 µm cono-spherical tip – Trace ( ) and Derivative
(-)
54
to determine the lower and upper limits within which friction data above the noise
threshold of the instrument could be obtained. The range within which the data is
repeatable and trustworthy was determined to be 0.1 µm/s to 100 µm/s and is
described in detail in the next section.
The experiments we did were to measure friction force as a function of time as
both velocity and the normal load were varied. This resulted in data that could be
used to determine variation of friction force with velocity at different normal loads.
The repeatability of the data was tested and other experiments including those to
determine the change in friction force with dwell time were conducted by varying the
amount of wait time after application of the normal load and before starting motion.
The depth of the tip during scratch experiments was studied as a function of the
normal loading along with the variation of friction behavior with tip radius.
The normal load of the indenter can be varied between 1–10,000 µN for scratch
testing for the 10 mN head. There is another 30 mN head available for higher load
indentation purposes but that does not support the scratch feature of the
nanoindenter. Experiments were conducted by an initial normal load of 500 µN and
increasing it in steps till the maximum possible load. Data points were taken at 1000
µN, 3000 µN, 5000 µN, 7000 µN and 9810 µN (1 mg) normal loads. The available
velocity range, 0.1 µm/s – 100 µm/s, was divided into approximately 50 steps. The
initial step size was chosen to be 0.2 µm/s and it was progressively increased as the
velocity increased. Data was recorded at each of these points at each of the normal
loads mentioned above. We planned to take enough readings to be able to model
55
friction in the micro-range up to the macro scale. At each velocity, the nanoindenter
gives the option of recording 1024 – 131072 data points for the entire experiment. If
too high data number of data points were selected, it was observed that for lower
velocities that the resulting curve was a band of data points instead of a single line.
This could be caused due to clustering of points as the nanoindenter attempted to
gather more data within the same time interval. The highest number of points that
could be gathered for a particular experiment without causing this effect was
determined by trial and error and in most cases about 5000 data points were recorded
from the beginning to the end of the experiment.
The TriboScan® software has limited analysis features for scratch data such as tilt
correction and friction coefficient determination as a function of time. The software
gives the option of converting the generated data into text files for analysis outside
their software domain. These text files were further converted into MS Excel
spreadsheets for easier graphics and for utilizing its enhanced computing features.
Visual Basic was used to automate some of the repetitive tasks that needed to be
performed for each of these files and the codes for converting the text files to MS
Excel and drawing the graphs and computing the average friction force etc. are
presented in Appendix B. MATLAB 6.5 was used for calculations and curve fitting
purposes.
56
4.4 Experimental Results and Analysis
4.4.1 Velocity Limits of the Triboindenter
The typical loading function for scratch testing and the resulting results window
for scratch data have been described earlier with reference to Fig. 3.8 and 3.9
respectively. Although the instrument is well-suited for friction experiments through
accurate control and measurement of the normal and lateral forces through feedback,
it had not been previously characterized for friction testing. The velocity of the tip is
specified by specifying both the displacement and the time and the maximum lateral
displacement is limited to about 18 µm which is determined experimentally.
Although the velocity can be increased to 1mm/s, 40 µm/s is chosen as a reasonable
upper limit on the velocity to allow the friction force to develop and achieve steady-
state.
The lateral displacement noise floor of the transducer is 10 nm [3] while the
feedback on the lateral displacement is at a rate of 10Hz. Although theoretically we
can scratch very slowly, if there is no change of displacement equal to or greater than
the lateral resolution every time that the feedback loop iterates, the system may try to
overcompensate in moving the probe. Thus the minimum speed to eliminate any
possibility of errors due to the resolution and feedback is 100 nm/s i.e. 0.1 µm/s.
These resolution limitations caused very slow scratches (below 0.1 µm/s) to oscillate
as shown in Fig 4.15. At higher normal loads, this lower limit occurred more
frequently around 0.2 µm/s and certain scratches in which such oscillations occurred
only towards the end of the scratch were observed. Thus the velocity range of the
57
Fig. 4.15 An Oscillating Scratch at 3000 µN Normal Load and 0.001 µm/s
velocity
instrument for friction measurements was determined to be 0.2 µm/s – 40 µm/s.
4.4.2 Friction Force Variation with Normal Load
The nanoindenter has a normal load range of 1-10,000 µN. The friction force
obtained using a 50 µm diamond tip sliding on a freshly cleaved mica surface is
plotted against the applied normal load in that range measured at a constant velocity
of 13.5 µm/s. A linear variation is observed, as shown in Fig. 4.16 and it is in
agreement with the Amontons’ law of friction, F = µN [41]. A linear fit yields a
friction coefficient, µ, of 0.0996 which is in the reported range [42]. For testing with
more commonly used materials, a similar curve was plotted using a 50 µm sapphire
flat punch sliding on a sapphire surface yielding a friction coefficient of 0.21 which
is in close agreement with the reported value in literature of 0.2 [43], thus indicating
that the nanoindenter is well calibrated for conducting accurate friction tests.
58
Fig 4.16 Friction Force vs. Normal Load plot for a 50 µm diamond tip sliding
against a mica surface at 13.5 µm/s
4.4.3 Friction Force Variation with Velocity at Different Normal Loads
Fig. 4.17 shows the variation of friction force as a function of velocity for the
entire velocity range of the nanoindenter. At a given velocity, the average value of
friction force is computed from the time a positive friction force is observed till the
time the nanoindenter tip is maintained at a constant velocity. At low normal loads,
we see a mostly linear behavior with a slight increase in the friction force as the
velocity increases in addition to some non-linearity at low velocities. One possible
reason for this is that at higher velocities, the atoms that exert the largest opposing
force and thus are closest to becoming unstable now have reduced time for thermal
activation resulting in a higher friction force [44]. At higher loads the trend becomes
more irregular, possibly because surface irregularities, such as those observed on the
AFM scan of the mica tip in Fig. 4.12, play a greater role in determining the friction
500
10003000
9810
7000
0
200
400
600
800
1000
1200
0 2000 4000 6000 8000 10000 12000
Normal Load (µN)
Fri
ctio
n F
orc
e (µ
N)
59
Fig 4.17 Friction Force vs. Velocity at Various Normal Loads for a 50 µm
diamond tip sliding against a mica surface
force as the load increases.
4.4.4 Repeatability of Experiments
To test the repeatability of the experiments, the 3000 µN normal load, 13.5
µm/s velocity experiment was executed five times on two different days and the
resulting scratches are shown in Fig. 4.18. For experiments conducted on the same
day, the mean friction force was 362.23 µN with a standard deviation of 4.73 µN
(1.3%). The average recorded on a different day was 373.92 µN indicating that the
standard deviation between the average values of these two days was 8.27 µN (2.2%)
which suggests high repeatability of the data obtained from the nanoindenter.
0
100
200
300
400
500
600
700
800
900
1000
0.01 0.1 1 10 100
Velocity (µm/s)
Frictio
n F
orc
e (µ
N)
1000 uN
3000 uN
5000 uN
7000 uN
8500 uN
9810 uN
Normal Load
60
3000 µN Load 13.5 µm/s Vel
300
350
400
8 8.5 9
Time (s)
Fri
cti
on
Fo
rce (µ
N)
Fig 4.18 Multiple Execution of the Same Load Function for Repeatability
4.4.5 Steady State Attainment through Repeated Scratching
Friction traces were observed to change with time and as reported in [45], it
took multiple scratches for the friction traces to reach steady-state conditions. As
shown in Fig. 4.19, for a normal load of 500 µN, it took around 6 repetitions for
friction traces to no longer change with time whereas it took only 3 repetitions for a
normal load of 5000 µN to achieve the steady-state. Thus at higher normal loads the
steady state was reached in fewer repetitions.
4.4.6 Friction Spikes and Effect of Resting Time
As shown in Fig. 4.20, in friction measurement experiments using a 50 µm
sapphire flat punch sliding on a mica surface, the friction force is observed to start
from an initial negative value owing to the inherent tilt in the nanoindenter setup. If a
61
Fig 4.19 Repeated Scratching of a 50 µm cono-spherical diamond tip on a mica
surface at 0.07 µm/s velocity at normal load (a) 500 µN (b) 5000 µN
constant velocity is applied, the friction force typically reaches an initial peak and
then drops, after which it rises again and fluctuates about a steady-state value. This
phenomenon, observed at the commencement of sliding for two surfaces at rest in
adhesive contact is commonly referred to as the ‘stiction spike’ [46]. As shown in
Fig. 4.21, the magnitude of the spike increases with increasing applied initial
velocity. This suggests that although traditionally friction force is assumed to be
static and independent of velocity before the actual slip occurs, the onset of the
dynamic friction regime has already begun.
(a) 500 µN Load (b) 5000 µN Load
62
Fig 4.20 Friction Spike behavior at increasing velocities for a 50 µm sapphire
flat punch scratching on a mica surface at 7000 µN normal load
Fig 4.21 Friction Spike behavior at increasing velocities for a cono-spherical
diamond tip of radius 50 µm scratching on a mica surface at 9810 µN load
(a) Load - 7000 µN, Velocity – 0.6 µm/s (b) Load - 7000 µN, Velocity – 3.7 µm/s
(c) Load - 7000 µN, Velocity – 9.9 µm/s (d) Load - 7000 µN, Velocity – 15.2 µm/s
-550
-350
-150
50
250
450
650
850
1050
8 9 10 11
Time (s)
Friction F
orc
e (µN
)
0.2 µm/s
4.0 µm/s
18.0 µm/s
36.4 µm/s
Initially
Sticti
Steady-
Direction of
increasing velocity
63
been observed to decrease as the velocity is increased as seen in Fig. 4.21. The
position change during this rise time had been previously reported to be roughly
constant [9] but in our experiments it is observed to be roughly linear with the
velocity. At higher loads, since the peak value itself is higher, more time is needed to
reach it at the same velocity.
As shown in Fig. 4.22, there is an increase in the observed initial peak value
of the spike the longer the system rests. One hypothesis is that the binding force
between the two surfaces increases with the contact time leading to an overall
increase in the friction force. No such spikes are observed in friction results where
stick-slip occurs, because stick-slip itself is assumed to be a continuum of such
stiction spikes [46].
4.4.7 Stick-Slip and Anisotropy
Stick-slip is an important feature of friction and in most cases it is
undesirable. When we try to move the tip across the surface at a constant velocity,
the friction force initially rises and when it reaches the break- away force, the tip
starts to slide and the friction decreases rapidly in accordance with the Stribeck
curve. The tip slows down and the friction force increases and the motion stops. The
phenomenon then repeats itself and is termed as stick-slip. On other instruments such
as the SFA, stick-slip has been reported to occur in up to half the cases at low
velocities, being very sensitive to lattice orientation with mica. caused because of
the sharp tip digging into the sample rather than true stick-slip.
64
76.275.5173.5272.45
354.80340.46334.45330.62
0
100
200
300
400
0 30 100 1000
Resting Time (s)
Initia
l P
eak F
riction F
orc
e (µ
N)
500 µN
3000 µN
Fig 4.22 Friction Spike Magnitude Variation with Resting Time
Fig 4.23 A typical lateral force profile vs. time produced by a 1 µm diamond tip
on a mica sample at 5000 µN normal load and 0.2 µm/s velocity
65
When friction experiments were performed with the 50 µm sapphire flat
punch on mica surface stick-slip behavior of the type shown in Fig 4.24 was
observed. In Fig 4.24 (a), the behavior resembles the behavior of the 1 µm tip shown
in Fig 4.23 probably caused because of the edge of the cylindrical punch contacting
the surface rather than the entire flat surface. Fig. 4.24 (b) seems to be true stick-slip
behavior and it was highly repeatable unlike in the case with the 50 µm cono-
spherical diamond tip. As shown in Fig.4.25, the occurrence of stick-slip in these
experiments exhibited dependence on the relative orientations of the two samples. As
shown in Fig. 4.26, the average friction force was observed to vary almost
sinusoidally w.r.t the sample rotation and the occurrence of stick-slip was observed
near the peak of the sinusoid since higher friction forces are observed when stick-slip
occurs.
Following the lead from these experiments with the sapphire flat punch,
similar orientation experiments were conducted with the 50 µm cono-spherical
diamond tip and the resulting average friction force in this case was also observed to
vary almost sinusoidally w.r.t the sample rotation as shown in Fig. 4.27. As shown in
Fig. 4.28, stick-slip was indeed observed in data points near the peak average friction
force in the previous figure. Stick-slip was more predominant and its nature was
more well-defined in case of experiments conducted with the 50 µm sapphire flat
punch. Apart from the difference in the material itself, one possible reason for this
irregularity could be accounted by the fact that the circular radius of Hertzian contact
66
Fig 4.24 Stick-slip type behavior for sapphire 50 µm flat punch sliding on a
mica surface at (a) 5000 µN load at 0.1 µm/s vel (b) 1000 µN load at 0.6 µm/s vel
between the 50 µm diamond cono-spherical tip and the mica surface (parameter a in
Fig.4.8) is only 2.33 µm, substantially lower than that in the former case which can
be 50 µm . Also, the irregularity in the geometry of the 50 µm diamond tip as noted
in Fig. 4.12 could play a role in the non-uniformity of the stick-slip.
The friction force was plotted against velocity for the 50 µm diamond tip
sliding on mica for the entire velocity range available with the nanoindenter and the
results are presented in Fig. 4.29. The friction behavior overall follows the trend seen
in Fig. 4.27. A peculiar behavior observed is that the friction force is observed to be
maximum at 270 degree orientation and min at 90 degree orientation, but at a
similarly diametrically opposite pair of 0 degree and 180 degree orientations, the
friction values are in good agreement. One possible reason for this could be a
possible sample tilt along the 90-270 degree plane leading to a huge variation in
friction along that direction while the 0-180 degree friction measurements would
67
Fig 4.25 Friction force vs. time plots for 50 µm sapphire flat punch sliding on a
mica surface at 1000 µN load at 0.6 µm/s velocity at various sample orientations
remain in the same horizontal direction leading to fairly similar friction force
values.
In [47], the frictional anisotropy between the mica-mica interface has been
attributed to the commensurability between the contacting lattices whereas in [48],
the authors state that in the case where the surfaces deform plastically under shear,
friction anisotropy originates with the properties of the bulk crystal lattices. In our
case since the materials in contact are different in nature, neither of the above models
0 deg 90 deg
135 deg 180 deg 225 deg
45 deg
360 deg 315 deg 270 deg
68
0
50
100
150
200
250
300
350
0 50 100 150 200 250 300 350 400
Sample Rotation (deg)
Fri
ction
Fo
rce (µN
)
Stick - Slip
Fig 4.26 Friction force vs. Sample Rotation for the 50 µm Sapphire flat punch
sliding against a mica surface at 1000 µN normal load and 0.6 µm/s velocity
Fig 4.27 Friction force vs. Sample Rotation for the 50 µm Diamond tip sliding
against a mica surface at 9810 µN normal load and 1.5 µm/s velocity
Mica Sample - 50 µm Diamond Conical - Spherical Tip
9810 µN Load 1.5 µm/s Vel
400
500
600
700
800
900
1000
1100
0 100 200 300 400
Sample Rotation (deg)
Fri
cti
on
Fo
rce (µ
N)
69
Fig 4.28 Friction force vs. time plots for 50 µm Diamond tip sliding against a
mica surface at 9810 µN load and 1.5 µm/s velocity at various orientations
can be directly applied. The magnitude of stick-slip in both cases decreases with an
increase in velocity similar to that reported for the mica to mica surface contact in
[11]. As shown in Fig. 4.30, no significant variation in friction force due to
anisotropy was observed in similar tests with single-crystal Silicon with a cubically
symmetric lattice structure, indicating that the crystal structure of the contacting
substrate rather than that of the diamond tip might be the determining factor for the
occurrence of anisotropy in these experiments.
0 deg 90 deg
135 deg 180 deg
225 deg 270 deg
45 deg
360 deg
Friction Force Vs Sample Rotation
Mica Sample - 50 µm Diamond Conical - Spherical Tip 9810 µN
Load 1.5 µm/s Vel
400
500
600
700
800
900
1000
1100
0 50 100 150 200 250 300 350 400
Sample Rotation (deg)
Fri
cti
on
Fo
rce (µ
N)
70
Fig 4.29 Friction Force Vs Velocity at four different angles for a 50 µm diamond
tip sliding against a mica surface at 8500 µN Normal Load
Fig 4.30 Friction force vs. sample rotation for a 50 µm diamond tip scratching
on a Silicon Surface at 9810 µN normal load
4.4.8 Depth of Scratches and Orientation Effects
One advantage of the nanoindenter over other friction measuring instruments
is its ability to generate real-time depth measurements as the scratches are being
performed. As shown in Fig. 4.31, during a velocity reversal experiment both the
Friction Force Vs Sample Rotation
550
600
650
700
750
800
850
0 100 200 300 400
Sample Rotation (deg)
Fri
cti
on
Fo
rce (µ
N)
400
500
600
700
800
900
1000
1100
-30 20 70 120
V elo cit y (µm/ s)
0 deg
90 deg
180 deg
270 deg
71
friction force (bottom-left) and the depth (top-right) are observed to be lower in
magnitude on the reverse path although it is traversed with the same velocity. The
reason for the depth being lower could be accounted by the debris created during the
forward scratch which could accumulate in the scratch zone, thus preventing the tip
from going as deep into the sample as during the forward path.
As the scratch begins, the maximum depth first reaches a peak value and then
typically fluctuates around that value until the tip is withdrawn from the sample. Fig.
4.32 shows a plot of the maximum depth at various normal loads for the entire
velocity range. This curve closely follows the Hertzian prediction as shown, thus
suggesting that the deformation during the scratching is purely elastic.
Another feature typical to the case of scratches where stick-slip has been
observed is that the maximum depth attained in case of stick-slip is lower than that in
the case when stick-slip is not observed as shown in Fig. 4.33. For the orientation
experiments conducted with the 50 µm tip described in Fig. 4.29, the depth analysis
shown in Fig. 4.34 suggests that the depth varies inversely with the friction force i.e.
at the 90 degree orientation, the maximum depth is observed but the friction force is
the least at that value and vice versa. This also supports the previous observation of
the depth being lower during stick-slip because typically higher friction forces are
observed during stick-slip. The velocity dependence of depth for these data points is
further explored in Fig. 4.35; at the 0 and 180-degree orientation, the maximum
depth decreases with an increase in velocity while the opposite behavior is observed
in the case of the other two orientations. One hypothesis is that these curves are a
72
-10
10
30
50
70
90
110
130
150
-1000 1000 3000 5000 7000 9000 11000
Normal Load (µN)
Ma
x D
ep
th (
nm
)
Observed Max Depth
(nm)
Hertz Prediction(nm)
Fig 4.31 Friction plots for a velocity reversal experiment for a 50 µm diamond
tip sliding against a mica surface at 9810 µN normal load and 0.7 µm/s velocity
Fig 4.32 Maximum depth vs. applied normal load and its comparison with
Hertz prediction for 50 µm diamond tip sliding against a mica surface
73
Fig 4.33 Maximum depth vs. velocity comparison at occurrence of stick-slip and
no stick-slip data for a 50 µm diamond tip sliding against a mica surface
Fig 4.34 Max Depth vs. Rotation Angle for a 50 µm diamond tip sliding against
a mica surface at 8500 µN Normal Load
Comparison of Maximum Depth for 9810 µN 50 µm
Conical-Spherical Tip
0
20
40
60
80
100
120
140
0 1 2 3 4
Velocity (µm/s)
Max D
ep
th (
nm
)
Max Depth (nm) -
9810 uN Previous
Max Depth (nm) -
9810 uN Stick Slip
Variation of Max Depth with Angle
0
90
180270
50
70
90
110
130
150
0 50 100 150 200 250 300
Rotation Angle (deg)
Max D
ep
th (
nm
)
74
Fig 4.35 Max Depth vs. Velocity varation for a 50 µm diamond tip sliding
against a mica surface at 8500 µN Normal Load
part of a larger oscillatory trend averaging around a certain constant value and out of
phase with each other.
4.4.9 Friction Force Variation with Tip Radius and Geometry
Fig 4.36 shows the penetration depth of various tips into the mica sample at a
constant normal load. It is observed that the 1 µm tip penetrates the deepest whereas
the 50 µm tip has the least penetration among the three diamond tips which is in
accordance with the Hertzian theory. The sapphire flat punch which is also different
in material properties from the harder diamond tips, has a higher surface area leading
to lower penetration depth. The contact depth in the
Max Depth (nm) Varation With Velocity at Various
Angles
85
95
105
115
125
135
0 1 2 3 4 5
Velocity (µm/s)
Max D
ep
th (
nm
) 90 deg
270 deg
0 deg
180 deg
75
Fig 4.36 Maximum depth vs. tip radius for 1µm, 5µm, 50 µm Diamond tips and
50µm Sapphire tip sliding against a mica sample at 1000 µN normal load and
3.3 µm/s velocity
case of the diamond tips predicted by the Hertzian theory is in close agreement for
the 50 µm tip while in the case of the other two tips the observed depth is about
double the value predicted by Hertz theory indicating a departure from elastic contact
between the surfaces.
Upon testing with softer materials such as aluminum, increased friction force
values are observed for the 5 µm tip over the 50 µm tip, other factors remaining the
same, as shown in Fig. 4.37. Comparing Fig. 4.37 (a) and (b), it can be concluded
that this effect is opposite to that observed in case of mica. The softness of the
aluminum sample could possibly lead to more penetration by the sharper 5 µm tip
thus producing a higher friction force than that produced by the 50 µm tip therefore
Mica - 4 Tips - Depth Analysis - 1000 µN Load,
3.3 µm/s velocity
0
50
100
150
200
250
0 20 40 60 80
Tip Radius (µm)
Max D
ep
th (
nm
)Sapphire 50 µm flat punch
Diamond Tips
76
Fig 4.37 Friction force vs. velocity plots at 5000 µN normal load for the 1 µm, 5
µm and 50 µm diamond tips for (a) Mica Surface, and (b) Aluminum Surface
giving results which are more suggestive of fracture than friction. On the other hand,
being harder, mica does not allow for such deep penetration of the 5 µm tip and the
contact area is much smaller, following Hertz prediction, thus leading to a reduced
friction force. In both cases, as seen in the figure, similar friction experiments
conducted with the 1 µm cono-spherical tip yield much higher values for friction
forces than that produced by the 5 µm tip or the 50 µm tip for the same normal load
100
600
1100
1600
0 10 20 30 40
Velocity (µm/s)
Fri
ctio
n F
orc
e (µN
)
1 µm tip
5 µm tip
50 µm tip
A
0
2000
4000
6000
8000
0 10 20 30 40Velocity (µm/s)
Fri
ctio
n F
orc
e (µ
N)
1 µm tip
5 µm tip
50 µm tip
B
77
and velocity profile. Depth measurements show that the sharper 1 µm tip also digs
deeper into both the samples and in the case of Aluminum it exhibits surface damage
for forces higher than 1000 µN. A typical lateral force profile produced by the 1 µm
tip in mica, for the 5000 µN normal load, shown previously in Fig. 4.23 indicates
that the tip moves across various points in the direction of motion in a stop-jump
fashion, penetrating deep into the surface at each contact. For these results, fracture
or surface damage due to tip indentation is not observed to occur as long as the
compressive normal stress at the tip is, on average, less than three times the ultimate
tensile stress of the material. Although subjective, this observation can be used with
an appropriate degree of conservatism to determine the minimum tip radius that will
ensure surface to surface contact without fracture.
4.4.10 SEM Imaging of Mica Cleavage Structure
Some SEM images of mica’s cleavage structure taken with FEI Thermal
Field Emission SEM are shown in Fig. 4.38. A clean piece of mica with no previous
scratches was chosen and a single scratch was made on the surface. Since mica
adsorbs organics quickly upon exposure to ambient atmosphere, the sample needs to
be put into the SEM immediately after the scratch is made. Imaging can be done
using the triboindenter itself using lower radii tips such as the 1 µm tip but the 50
µm tip, used in to make the scratch, is too big for the purpose of imaging. If the mica
sample is placed directly into the SEM, being non-conductive it accumulates charge
and after some time the surface becomes too bright for any scratch image to be
obtained. To resolve this issue, a 1-2 nm Au/Pd film (3:1 composition) was first
78
Fig 4.38 SEM Micrographs of Mica Cleavage Structures
79
deposited on to the mica surface after making the scratch using the Hummer 62
Sputtering system. Using copper clips to hold this sample inside the SEM helps
maintain the conductivity of the mica surface and thus the sample can be easily
imaged without any charge accumulation.
The SEM images show the cleavage of the mica surface in a repetitive pattern
along the direction of travel of the tip. The overall width of the scratch and the length
is higher than that predicted by Hertz, and one possibility is that these scratches are
not a result of the triboindenter experiments but other artifacts, since it is difficult to
obtain an absolutely clean mica surface. Another possibility for the scratch length not
matching the expected dimensions is that the scratch propagates automatically once
it is initiated.
80
5. Further Analysis of Data and Triboindenter Control System
Modeling
5.1 Actuation and Sensing Mechanisms of the Transducer
5.1.1 Electrostatic Actuation
As discussed previously in Section 3.2.3, the transducer of the triboindenter
consists of a three-plate capacitor schematically shown in Fig. 5.1 The force
characteristics for such a system can be derived as follows:
The mechanical force due to spring action is given by
mF kx= …(16)
where k is the stiffness of the spring and x is the displacement.
The force due to electrostatic attraction by the plate is given by 20
2
12 ( )e
AF V
d x
ε=
− …(17)
where,
d = Plate Separation
ε0 = Permittivity of Free Space = 8.85*10-12 C2/N-m2
A = Plate Area
For the given configuration, d = 85 µm as provided by the manufacturer, k = 100
N/m as determined experimentally and as an approximation, A = 1 cm2 = 1*10-4 m2
Equating the forces as shown in Fig 16, from equations (16) and (17), we get
3 2 2 20( 2 ) ( )2
Ax d x d x V
k
ε+ − + = …(18)
81
Fig 5.1 Schematic of transducer plates showing forces and displacement
Substituting the known values into equation (18), we get
3 4 2 9 18 21.7 10 7.225 10 4.425 10x x x V− − −− × + × = × …(19)
Plotting V versus x, as shown in Fig 5.2, we see the behavior is typical to a parallel
plate MEMS actuator, which shows snap-down behavior at x = d/3. Application of
the large DC bias, shown by points marked in red, allows the operation of the
instrument in a regime of more linear displacement per unit change in voltage.
5.1.2 Displacement Sensing
In case of lateral scratch experiments, the nanoindenter generates the X-Voltage
signal as a separate output. As seen previously in Fig. 3.5 (a), for displacement
measurement two complementary AC signals are applied to the outer plates and the
sensing output is an AC signal which measures the capacitance change ∆C. A typical
differential capacitor with its capacitances and displacements is depicted in Fig. 5.3
82
0 1 2 3 4 5 6 7 8 9
x 10-5
0
100
200
300
400
500
600
700
Voltage (
V)
Displacement (m)
Fig 5.2 Voltage versus displacement for a parallel plate capacitive actuator
Fig 5.3 A Differential Capacitor – motion of the central plate leads to a change
in individual capacitances on either side of the central plate
83
The two capacitances are in parallel and hence the differential capacitance, which is
measured by the measurement circuit, can be expressed as:
0 0 02 1 02 2 2 2
(2 )2
A A A x xC C C A
d x d x d x d x
ε ε εε∆ = − = − = =
− + − − …(22)
Assuming a circuit similar to the authors [49], this output is amplified,
synchronously demodulated and low-pass filtered to give a DC output signal. The
change in DC voltage, ∆Vsense, is proportional to the capacitance change:
0sense
f
VV C
C∆ = ∆ …(23)
where ∆Vsense is the change of output voltage, V0 is the amplitude of the applied
AC voltage and Cf is the feedback capacitor.
Combining equations (18) and (19), we get
00 2 2
( )(2 )sense
f
V xV A
C d xε∆ =
− …(24)
i.e.
2 2( .)
sense
xV Const
d x∆ =
− …(25)
For the given plates, plate separation d = 85 µm. To determine the qualitative nature
of the curve, we assume the constant of proportionality to be equal to 1 and plot
∆Vsense versus the displacement x. This is plotted in Fig. 5.4 and it shows a linear
variation around zero displacement. The experimental data is plotted in Fig. 5.5 and
it consists of a closed loop representing the complete motion of the tip from the start
to its return to the same point. The curve of interest is the lower curve during which
84
-60 -40 -20 0 20 40 60-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
∆ V
sense
Displacement µm
Fig 5.4 Theoretical output curve for a differential capacitor
X Voltage Vs Displacement
-700
-600
-500
-400
-300
-200
-100
0
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement (µm)
Vo
ltag
e (
V)
Fig 5.5 Experimental X Voltage Vs Displacement curve for 9810 µN Normal
Load and 4.0 µm/s velocity for the 50 µm diamond tip sliding on mica
85
the actual scratch takes place. We observe that a DC offset of -532V has been
applied at the point of zero displacement for the purpose of measurement using the
AC signal. The qualitative nature of the experimental curve is similar to that of the
theoretical curve, represented by equation (25), in the vicinity of its linear range.
Comparison of the two curves can be used to extract the system’s unknown
parameters.
5.2 Tilt Characterization of the Triboindenter
A small amount of lateral force (friction force) is always measured during the
loading period of the tip as shown in Fig. 1 from t = 5 s to t = 8 s. A mica sample
was rotated through 3600 and it was noted that a friction force in the same direction
as shown in Fig. 5.6 occurred in all cases although its magnitude varied at different
rotation angles. This ruled out the possibility of the friction force being caused solely
due to tilt in the sample, indicating a tilt in the nanoindenter transducer-tip system
itself. A possible tilt in the assembly had also been previously reported by other users
upon observation using the naked eye.
As shown in Fig. 5.7, let θ be the assumed tilt in the transducer-tip assembly with
respect to the horizontal. The normal force, N, will act on the tip perpendicular to the
base, assuming a perfectly flat sample. Let F be the force applied by the transducer
on the sample along the tip. Thus we have F cos θ = N, and the unbalanced F sin θ
would create a friction force which is measured as the lateral force by the indenter
during the loading. The ratio of this measured lateral force to the applied force F
yields the tilt angle of the tip.
86
Fig 5.6 Typical Experimental Result – Lateral Force Vs Time
Fig 5.7 Schematic of Tilt in the Nanoindenter
To estimate the sample tilt, data from the mica sample rotated through 3600 was
used to compute the overall tilt angle at different orientations, plotted in Fig. 5.8. The
mean value of the tilt was found to be 1.50 with a standard deviation of 0.50. Thus the
mean tilt of the indenter tip itself can be assumed approximately equal to the 1.50
θ
N
F
F
F
Sample
Base
Tip
Lateral Force
Typical example of
measurement of a
lateral force
87
Tilt of Tip w.r.t. Normal to the Surface (deg)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 50 100 150 200 250 300 350 400
Rotation Angle (deg)
Til
t A
ng
le (
deg
)
Fig 5.8 Calculated Tilt of Nanoindenter Tip with a Mica Sample at 9810 µN
Normal Load at Various Orientations
although a larger number of data points would provide a better indication of this
value. The deviation from this value, as seen in Fig. 5.8, could be attributed to the
sample tilt at various locations and orientations.
Wedge-shaped samples, with known wedge-angles could provide for predefined
tilts. In cases where the slope of the sample itself is greater than the tilt in the tip-
transducer assembly, orientation experiments with them, as shown schematically in
Fig. 5.9, would yield friction force acting in a direction opposite to that observed in
current experiments. Such experiments were conducted with a 4-deg, 6-deg, 10-deg
and 30-deg wedges and although the nanoindenter failed to produce data on most
occasions upon use of a wedge-shaped sample, detecting a false surface upon
contacting the titled surface, some successful results such as one shown in Fig. 5.10,
88
Fig 5.9 Schematic of Orientation Experiment with a Wedge for Computing
Sample Tilt
Fig 5.10 Positively oriented force during tip loading as observed in friction
experiment with a wedge
α
θ
89
were conducted which indicated a higher lateral force in one direction and positive
lateral force in the other direction, therefore suggesting a reversal in the friction force
enforced by the tilt of the sample. This measured tilt in the nanoindenter transducer-
assembly could be the result a misalignment during production or initial system setup
and might be limited to this particular instrument used in our experiments.
5.3 Analysis of the Transition from Static to Kinetic Friction Regime
To understand a typical output behavior at the transition from static to kinetic
regime, data for 9810 µN normal load for 0.2 µm/s, 4.0 µm/s, 18.0 µm/s and 36.4
µm/s applied velocities are presented graphically for analysis in the following
sections. Various characteristics of friction force, lateral displacement, lateral
velocity and acceleration are analyzed along with the variation in friction force and
lateral displacement at varying normal loads is studied. This data is useful for
estimating the control performance of the triboindenter and further modeling based
on the Lugre model using these observed characteristics is performed in the next
section.
5.3.1 Friction Force and Velocity Characteristics
Fig. 5.11 shows the friction force and the velocity attained by the tip during the
transition from static to kinetic friction at four different velocities. As seen from the
figure, there is a ‘stiction-spike’ at the beginning of the motion and the output
velocity also reaches a peak during the same time interval. Such friction spikes are
observed every time there is a transition from static to kinetic friction. A comparison
90
(a) Velocity = 0.2 µm/s
(b) Velocity = 4.0 µm/s
(c) Velocity = 18.0 µm/s
Applied Velocity = 18.0 µm/s
-600
-400
-200
0
200
400
600
800
1000
1200
7.5 7.7 7.9 8.1 8.3 8.5
Time (s)
Fri
cti
on F
orc
e (µ
N)
-10
0
10
20
30
40
50
60
Velo
cit
y (µ
m/s
)
Lateral Force (uN)
Velocity (um/s)
Applied Velocity = 4.0 µm/s
-600
-400
-200
0
200
400
600
800
1000
7.5 8 8.5 9 9.5 10 10.5 11
Time (s)
Fri
cti
on
Fo
rce (µ
N)
-4
-2
0
2
4
6
8
10
12
14
16
18
Velo
cit
y (µ
m/s
)
Lateral Force (uN)
Velocity (um/s)
Applied Velocity = 0.2 µm/s
-600
-400
-200
0
200
400
600
800
1000
7.5 8.5 9.5 10.5
Time (s)
Fri
cti
on
Fo
rce (µ
N)
-1.5
-1
-0.5
0
0.5
1
Velo
cit
y (µ
m/s
)
Lateral Force
(uN)Velocity
(um/s)
91
(d) Velocity = 36.4 µm/s
Fig 5.11 Friction Force & Velocity vs. Time Graphs for 9810 µN normal load
of only the friction forces on the same plot has been presented previously in Fig. 4.21
A comparison of four different stiction spikes at different velocities suggests that the
friction spikes are higher and occur quickly as the input velocity is increased. This is
to be expected since at slower velocities there is more time for the friction value to
reach steady state and hence the rise in the friction force is more uniform. Also, the
system is seen to achieve high instantaneous velocities, roughly three times the
magnitude of the desired input velocity, at the beginning of motion caused in part
because the control system of the triboindenter is not able to compensate fast enough
during the transition. This in turn creates high acceleration of the tip during this
transition, as seen in Fig. 5.12 – it should be noted that both velocities and
accelerations for the above analysis have been computed by successive division of
the instantaneous displacement values with the difference in time. It is observed that
the acceleration values achieve higher peaks at higher input velocities with peaks as
Applied Velocity = 36.4 µm/s
-600
-400
-200
0
200
400
600
800
1000
1200
7.5 7.7 7.9 8.1 8.3 8.5
Time (s)
Fri
cti
on
Fo
rce (µ
N)
-20
0
20
40
60
80
100
120
Velo
cit
y (µ
m/s
)
Lateral Force (uN)
Velocity (um/s)
92
(a) Velocity = 0.2 µm/s
(b) Velocity = 4.0 µm/s
(c) Velocity = 18.0 µm/s
Velocity = 18 µm/s
-8.5
-7.5
-6.5
-5.5
-4.5
-3.5
-2.5
-1.5
-0.5
0.5
1.5
7.7 7.8 7.9 8 8.1 8.2 8.3 8.4 8.5
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
-2000
-1000
0
1000
2000
3000
4000
5000
Accn
um
/s^
2
Lateral Displacement Output (µm)Lateral Displacement Input (µm)Accn (µm/s 2)4 per. Mov. Avg. (Accn (µm/s 2))
Velocity = 0.2 µm/s
-8.1
-8.05
-8
-7.95
-7.9
-7.85
-7.8
-7.75
-7.7
7.7 7.9 8.1 8.3 8.5 8.7
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
-30
-20
-10
0
10
20
30
Accn
um
/s^
2
Lateral Displacement Output (µm)Lateral Displacement Input (µm)Accn (µm/s 2)4 per. Mov. Avg. (Accn (µm/s 2))
Velocity = 4 µm/s
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
7.7 7.9 8.1 8.3 8.5 8.7
Time (s)
Late
ral
Dis
pla
cem
en
t
(µm
)
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Accn
(µ
m/s
^2
Lateral Displacement Output (µm)Lateral Displacement Input (µm)Accn (µm/s 2)4 per. Mov. Avg. (Accn (µm/s 2))
93
(d) Velocity = 36.4 µm/s
Fig 5.12 Lateral Displacement (Input & Output) & Acceleration Vs Time
Graphs for 9810 µN load. Bold line indicates moving average filter of period 4
high as 9000 µm/s2 at input velocity 36.4 µm/s as shown in the figure. The
acceleration signal is very noisy and a moving average filter with period 4 is applied
to observe the overall trend, the peak values are observed to be proportional to the
input velocity while the steady state behavior consists of a fluctuation about a mean
acceleration of zero, which is to be expected since the input velocity is constant.
These unusually high values of the observed acceleration depend up on the control
system design and this behavior could be optimized if the control parameters were
allowed to be fine tuned, which is not an available option with the current system.
5.3.2 Lateral Displacement (Input and Output) Characteristics
As seen from Fig. 5.13, the lateral displacement output closely follows the input
value, with a constant difference between the two values upon reaching steady state,
suggesting the use of PID control in the system. At the commencement of the
Velocity = 36.4 µm/s
-8.5
-6.5
-4.5
-2.5
-0.5
1.5
3.5
7.7 7.8 7.9 8 8.1 8.2 8.3
Time (s)
Late
ral
Dis
pla
cem
en
t
(µm
)-5000
-3000
-1000
1000
3000
5000
7000
9000
Accn
um
/s^
2
Lateral Displacement Output (µm)Lateral Displacement Input (µm)Accn (µm/s 2)4 per. Mov. Avg. (Accn (µm/s 2))
94
(a) Velocity = 0.2 µm/s
(b) Velocity = 4.0 µm/s
(c) Velocity = 18.0 µm/s
Velocity = 18 µm/s
-8.5
-7.5
-6.5
-5.5
-4.5
-3.5
-2.5
-1.5
-0.5
0.5
1.5
7.7 7.8 7.9 8 8.1 8.2 8.3 8.4 8.5
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
-6
4
14
24
34
44
54
Velo
cit
y (µ
m/s
)
Lateral Displacement Output (µm)Lateral Displacement Input (µm)Velocity (µm/s)
Velocity = 0.2 µm/s
-8.1
-8.05
-8
-7.95
-7.9
-7.85
-7.8
-7.75
-7.7
7.7 7.9 8.1 8.3 8.5 8.7
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Velo
cit
y (µ
m/s
)
Lateral Displacement Output (µm)Lateral Displacement Input (µm)Velocity (µm/s)
Velocity = 4 µm/s
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
7.7 7.9 8.1 8.3 8.5 8.7
Time (s)
Late
ral
Dis
pla
cem
en
t
(µm
)
-6
-1
4
9
14
Velo
cit
y (µ
m/s
)Lateral Displacement Output (µm)
Lateral Displacement Input (µm)Velocity (µm/s)
95
(d) Velocity = 36.4 µm/s
Fig 5.13 Lateral Displacement (Input & Output) & Velocity Vs Time Graphs
Fig 5.14 Schematic Representation of Lateral Displacement Vs. Time Graphs
transition from the static to the kinetic regime, the lateral displacement rises very
slowly as shown in the schematic in Fig. 5.14, by the nomenclature ts for the stretch-
Lateral Displacement
Time
Desired Input Displacement
Output Displacement
ts
ds
dl
Velocity = 36.4 µm/s
-8.5
-6.5
-4.5
-2.5
-0.5
1.5
3.5
7.7 7.8 7.9 8 8.1 8.2 8.3
Time (s)
Late
ral
Dis
pla
cem
en
t
(µm
)
-6
14
34
54
74
94
Velo
cit
y (µ
m/s
)
Lateral Displacement Output (µm)
Lateral Displacement Input (µm)Velocity (µm/s)
96
time, indicating that the tip is still in contact with the surface and some displacement
is probably caused as a result of the deflection/bending of the tip. As the velocity
begins to rise, a sudden rise is noted in the displacement of the tip suggesting an
actual sliding of the tip on the surface. This rise in the displacement value continues
till the velocity reaches its peak and as soon as it starts reducing, the rate of increase
of displacement decreases until it achieves a steady state corresponding to the desired
input displacement curve. At this point, the velocity achieves a steady state about the
desired input velocity while the lateral displacement closely tracks the desired input
value.
5.3.3 Lateral Displacement and Friction Force Variation with Normal Load
As seen in Fig. 5.15, at any given velocity the higher the normal load,
proportionally higher is the ‘stretch-time’, ts, and the ‘stretch distance’, ds,
schematically represented in Fig. 5.14. After reaching steady-state, the displacement
profiles at different normal loads are in close agreement as expected. As the input
velocity is increased, the magnitudes of both the above parameters are found to
reduce. This is to be expected since the tip, assumed to be contact with the surface
during this period, would tend to slip quicker at higher velocities. Also, as seen from
Fig. 5.16, a higher amount of time is required for the friction force to achieve its
peak value as the normal load is increased at a given velocity. Also, the rate of
change of force for a given velocity is observed to be approximately the same at all
normal loads, i.e. the magnitudes of the slope for the three curves remain
approximately the same. This suggests that the rate of rise in the friction force in the
97
(a) Velocity = 0.2 µm/s
(b) Velocity = 4.0 µm/s
(c) Velocity = 18.0 µm/s
Lateral Displacement Comparison at
Varying Normal Loads, vel = 0.2 µm/s
-8.2
-7.7
-7.2
-6.77.3 12.3
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
Lateral Displacement Comparison at
Varying Normal Loads, vel = 4.0 µm/s
-8.2
-6.2
-4.27.7 8.7
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
Lateral Displacement Comparison at
Varying Normal Loads, vel = 18.0 µm/s
-8.2
-6.2
-4.2
-2.2
-0.2
7.9 8.1 8.3 8.5
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
98
(d) Velocity = 36.4 µm/s
Fig 5.15 Lateral Displacement Vs. Time Graphs at Various Normal Loads
(a) Velocity = 0.2 µm/s
(b) Velocity = 4.0 µm/s
Lateral Displacement Comparison at
Varying Normal Loads, vel = 36.4 µm/s
-9
-7
-5
-3
-1
1
7.9 8.1 8.3
Time (s)
Late
ral
Dis
pla
cem
en
t (µ
m)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
Friction Force Comparison at Varying
Normal Loads, vel = 0.2 µm/s
-500
0
500
7.3 12.3
Time (s)
Fri
cti
on
Fo
rce (µ
N)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
Friction Force Comparison at Varying
Normal Loads, vel = 4.0 µm/s
-550
-50
450
7.7 8.2 8.7 9.2
Time (s)
Fri
cti
on
Fo
rce (µ
N)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
99
(c) Velocity = 18.0 µm/s
(d) Velocity = 36.4 µm/s
Fig 5.16 Friction Force Vs. Time Comparison Graphs at Various Normal Loads
static regime for a given applied velocity is the same at different normal loads. A
detailed modeling and simulation of this behavior is performed in the next section
5.4 Modeling and Simulation
As described earlier in Chapter 3, Hysitron's testing systems employ a three-plate
capacitor design to apply a force through electrostatic actuation and measure the
Friction Force Comparison at Varying
Normal Loads, vel = 18.0 µm/s
-500
0
500
1000
7.8 8.3 8.8
Time (s)
Fri
cti
on
Fo
rce (µ
N)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
Friction Force Comparison at Varying
Normal Loads, vel = 36.4 µm/s
-500
0
500
1000
7.8 8.3
Time (s)
Fri
cti
on
Fo
rce (µ
N)
Normal Force
1000 µN
Normal Force
5000 µN
Normal Force
10000 µN
100
displacement through the change in capacitance. This transducer design is inherently
a force-controlled device that has been traditionally run open loop. The most
significant challenge presented by open loop testing is to reach or maintain the
desired load at a steady value due to a small portion of the load being absorbed by
the springs of the transducer. As a result, at low loads the transducer is unable to
maintain the tip at a constant load as shown in Fig 5.17.
The feedback control for displacement control for the 2-D transducer used in our
experiments operates in closed-loop. The PID parameters are not tunable for the
scratch experiments and thus the optimal feedback response for friction testing
cannot be generated. To gain a better understanding of the feedback control for
displacement application and measurement, the control system of the triboindenter is
modeled and simulated in this section.
5.4.1 PID Parameters for Displacement Control
A typical PID control system for the triboindenter is shown in Fig 5.18. As shown
previously in Fig. 5.14, the various parameters for a typical friction force
measurement experiment are as follows:
Fs = Stiction Force
ts = Stretch Time
ds = Stretch Distance
dl = Lag Distance
101
Fig 5.17 Typical Normal Load Output of the Triboindenter for a 500 µN
constant Input Force Profile
Fig 5.18 Typical PID Controller for the Hysitron Triboindenter
Based on the above values, the average stiffness can be calculated as:
S
s
F
dσ = …(26)
The force generated by the controller is
20
0
0 0
( ) ( )2
s st t
i si i
mk v tF t mk r d mk v dτ τ τ τ= = =∫ ∫ …(27)
r imk
s Σ
1ms
1s
dmk mk s+
Σ
1−
X mx
102
Hence,
20
2s
i
s
Fmk
v t= …(28)
The ramp error, neglecting friction force is:
2
1[ ( ) ( )]i
d
mkmk mk s R
ms sΧ = − + Χ + − Χ …(29)
3 2[ ]d i i
s k s ks k k R∴ + + + Χ = …(30)
Thus,
3 2
3 2
3 2
3 2
3 2
(1 )
i
d i
d
d i
d
d i
kR
s k s ks k
s k s ksR
s k s ks k
s k s ksR R
s k s ks k
Χ =+ + +
+ +∴Χ = −
+ + +
+ +⇒ Ε = Χ − = −
+ + +
…(31)
Using the Final Value Theorem, the steady state value of the error for 0( )r t v t= is:
0 0l
i i
kv kvd
k kΕ = − ⇒ = …(32)
Hence,
0 0
( )l i l i
d k d mkk mk
v v= ⇒ = …(33)
Thus, substituting known values from experimental measurements and using the
equations derived above, the values for the parameters mk and mki for the system can
be determined. For the experiments described in the previous section, these
parameters have been evaluated and are listed in Table 5.1.
103
Normal Load Velocity ts ds Fs mki dl mk
(µN) (µm/s) (s) (µm) (µN) (1/s) (µm) (1/s)
0.2 1.46 0.13 414 1942 0.054 524
4 0.23 0.23 487 4603 0.385 443
18 0.114 0.31 617 5275 1.272 373
5000
36 0.09 0.30 656 4499 2.21 276
0.2 2.26 0.27 742 1453 0.27 1961
4 0.3 0.35 1273 7074 0.657 1162
18 0.147 0.41 1394 7167 1.53 609
9810
36 0.109 0.46 1520 7108 2.353 465
Table 5.1 Parameters from Experimental Observations
5.4.2 Modeling of the Experiment
Modeling of the friction data is based on the Lugre model which was
described previously in section 2.3. The experiment can be modeled by:
0 1
0
( )( )
( ) ( )
( )
( )( )
,
s
d
d d i
v
v
c s c
dxv
dt
dvm F F
dt
F mkx mk v mk I
vdzv z v h v z
dt g v
g v l l l e
dzF z f v
dt
vh v
g v
dI drx r v
dt dt
σ σ
−
=
= −
= − − −
= − = −
= + −
= + +
=
= − =
…(34)
104
Introducing the state variables
1
2
3
4
5
x x
x v
x z
x I
x r
=
=
=
=
=
the equations can be rewritten as
12
0 3 1 2 2 3 2 221 2 4
32 2 3
41 5
50
( ( ) )
( )
d i
dxx
dt
x x h x x xdxkx k x k x
dt m
dxx h x x
dt
dxx x
dt
dxv
dt
σ σ σ
=
+ − += − − − −
= −
= −
=
…(35)
where
2
( ) ( )
( )sgn( )'( )
( )( )
'( )sgn( )'( )
( ) ( )
s
s
x
v
c s c
x
vs c
s
g x l l l e
l l xg x e
v
xh x
g x
x g xxh x
g x g x
−
−
= + −
−=
=
= −
…(36)
105
The Jacobian of (35) is
1 2 1 2 3 0 1 2
2 3 2
0 1 0 0 0
'( ) ( )0
0 1 '( ) ( ) 0 0
1 0 0 0 1
0 0 0 0 0
d i
h x x h xk k k
m mJ
h x x h x
σ σ σ σ σ
+ − − − − − − −
= − −
−
…(37)
5.4.3 Matlab Simulations
Using the equations described in the previous section and the parameters of
PID control-loop determined from experimental data, we can perform simulations of
the friction experiments and compare them with real experimental results. An
iterative process would help in the understanding of the control system implemented
by Hysitron and help optimize its parameters for closed-loop control at the same time
help in improving the friction model itself.
The Matlab programs used for the simulation are described in Appendix C. In
these simulations, the lateral displacement, friction force, velocity and friction state
are computed as a function of time. As seen from Fig. 5.19, the actual position of the
tip follows the desired input with a steady-state error as observed in the experimental
results. The friction force achieves a peak and then settles to its steady state value
according to experimental observations. The velocity first rises and then drops and
attains its steady-state value as seen in the analysis done in the previous section. The
velocity signal is noisy in nature and needs to be further modified to correctly match
experimental observations.
106
Fig 5.19 Matlab Simulation of Lateral Displacement, Friction Force, Velocity
and Friction State as a function of Time at 9810 µN Normal Load and 4 µm/s
Velocity (all units are in MKS)
As shown in Fig. 5.20, if the lateral displacements of two different
experiments are plotted on the same graph, a ‘kink’ in the lateral displacement
profile is observed and similar to the behavior observed in Fig. 5.15; the higher the
load, the more the duration for which the tip seems to stay in contact with the sample
before slipping. Most of the other observations noted in the previous section can also
be similarly verified using this modeling approach.
0 0.5 10
1
2
3
4x 10
-6
Positio
n
0 0.5 10
0.5
1
1.5x 10
-7
Friction S
tate
0 0.5 1-2
0
2
4
6x 10
-6
Velo
city
t
0 0.5 10
0.5
1
1.5x 10
-3
Friction F
orc
e
t
107
0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.5
0
0.5
1
1.5
2
2.5
3x 10
-5
Late
ral D
ispla
cem
ent
(N)
Time (s)
5000 uN Output
5000 uN Input
9810 uN Output
9810 uN Input
Fig 5.20 Matlab Simulation of Lateral Displacement Input (Dotted Line) &
Output (Dark Line) Vs Time For Normal Loads 5000 µN and 9810 µN and 18
µm/s Velocity
As seen from table 5.1, the parameters mk and mki vary as the normal load or the
velocity are changed. Such large variations are not desirable since these are the
system parameters and are expected to be constant or at least in a similar range for all
experiments. The real values for these parameters need to be obtained from Hysitron
and then these simulations need to be performed again to be able to iteratively
determine the parameters in the Lugre model, although the current approximations of
108
system parameters combined with the modified version of the LuGre model seem to
reconstruct the current experimental data to with high accuracy.
109
6. Tribo-rheometer: Data Fitting to the Lugre Model and
Comparison Experiments
6.1 Rheology and Tribology
The field of rheology is concerned with the description of flow behavior of all
types of matter, especially those with properties intermediate between those of ideal
solids and liquids e.g. non-Newtonian fluids such as mayonnaise, paint, molten
plastics and foams. Although tribology and rheology are typically regarded as distinct
scientific disciplines, in many modern technological applications, for example in
those involving micromachined structures, microfluidic flow channels or
microstructured fluids, both the tribological and rheological properties become
dominant in controlling the frictional dynamics of the system. A self-centering and
self-leveling tribo-rheometry test fixture designed by H. Pirouz Kavehpour (UCLA)
and G.H. McKinley [4] can be used with a standard torsional rheometer to generate
both tribological and rheological data.
As can be seen from table 6.1, the tribo-rheometer offers a higher range of
measurement for both velocity and normal load and than that available from
nano/micro-scale friction measurement devices, although it has a higher noise-floor
than them as well. It can vary the normal stress by changing not only the normal
force but also the area of contact and thus the friction force can be modeled based on
normal stress rather than normal load. It can be used to test almost any kind of
material (metallic or nonmetallic) both for dry friction and the lubricated case.
110
Min
Normal
Force
Max
Normal
Force
Z Force
Noise
Floor
Max Lateral
Displacement
(Scratch
Length)
Min
Velocity
Max
Velocity
Hysitron
Nanoindenter 10 µN 10 mN 0.1 µN 16 µm 0.1 µm/s 50 µm/s
Tribo-rheometer 0.5 N 50 N 100 mN ∞ 0.5 µm/s 1.5 m/s
Hess and Soom
Equipment 46 N 360 N ? ∞ 0.01 m/s 1 m/s
Table 6.1 Comparison of the Nanoindenter [3], Tribo-rheometer [4] and Hess
and Soom Equipment [5]
When a lubricant is used, Stribeck curves similar to those observed by Hess and
Soom [5] are obtained and it would be useful to fit the LuGre model [10] to their
data. With that aim in mind, an active collaboration was pursued with them and not
only curve-fitting but also a comparison between the tribo-rheometer and the
nanoindenter measurements was conducted by making the experiments as similar as
possible in terms of material and overlapping velocity ranges. Apart from helping
improve the LuGre model and providing a physical basis for it, such an extended
measurement range would also help bridge the gap in quantifying friction from the
micro to the macro scale.
111
6.2 The Tribo-rheometer
6.2.1 Experimental Setup
Rheometers are viscometers which are able to measure visco-elastic properties of
materials other than only viscosity. The common types of rheometers are either
rotational, capillary or extensional, rotational being most common. The AR 2000
torsional rheometer shown in Fig. 6.1 has been used for tribological measurements in
combination with a tribo-rheometry fixture [4]. It can make controlled rate and stress
relaxation measurements over torque range of 0.1 micro N-m to 200 mN-m, with
strain detection of 0.04 micro rad, and a normal force of 0.01 to 50 N with
resolution 0.1 N. It has a stainless steel parallel-plate configuration which offers the
advantage of varying the gap size, H, which is not possible in conical systems which
have a fixed cone angle for a specified conical fixture. Decreasing the gap H enables
higher shear rates to be attained for the same fluids, which is not possible with a cone
and plate system. The lower bounding surface is a Peltier plate assembly which is
used not only to control the temperature of the fluid sample, but also to measure the
normal force acting on the surface using a force transducer that is mounted below the
surface. To obtain tribological properties using a torsional rheometer, a new tribo-
rheometry fixture, shown in Fig 6.2 is designed and installed on the surface of the
Peltier plate. The enables the user to interchange different hard or soft materials as
the lower fixture in place of using the surface of the Peltier plate, thus eliminating
the possibility of damaging the sensitive Peltier plate through wear.
112
Fig 6.1 AR 2000 Torsional Rheometer (Src.TA Instruments Inc.)
Fig 6.2 Schematic of Tribo-rheometer and annual text fixture with radii R1 and
R2 respectively [4]
113
6.2.2 Alignment
To ensure parallelism of the lower fixture with the upper rotating plate, a self-
aligning method is employed. A small amount of wax is deposited on the Peltier
element and the temperature of the Peltier plate is increased to the melting point of
the wax. The circular disk that forms the lower fixture is placed on the top of the wax
layer. The top plate is brought down and a small normal force is applied to the
fixture. The fixture surface is then forced to be in full contact with the top plate. By
decreasing the temperature of the Peltier plate back to room temperature, the wax is
solidified and the lower fixture remains rigidly attached to the substrate and in
alignment with the top plate. The top plate is then retracted using the rheometer drive
system and a small amount of the test fluid (typically between 10 and 50 µL) is then
deposited on the lower fixture and the top plate is lowered to the desired gap setting.
The gap size is measured using an optical encoder built into the AR-2000 and disk–
disk contact (corresponding to a gap separation of zero) is detected automatically by
the appearance of a finite torque signal. Alignment and concentricity of the new test
fixture can be confirmed by calibration tests in the hydrodynamic regime using
standard fluids of known viscosity.
6.2.3 Experimental Measurements
When a torque, T, is applied to the top plate, the fixture reaches a certain
constant angular velocity, X, which is measured by an optical encoder on the shaft.
Through a feedback system available in the AR-2000 rheometer, one can design a
test procedure that then varies the angular velocity over several orders of magnitude
114
at either constant gap or constant normal force and measures the corresponding
torque required for a specified velocity. The range of velocity is from 0.0001-300
rad/sec. Since it is a rotational system, the linear velocity can be correspondingly
varied by changing the radius of the fixture.
6.3 Tribo-rheometer Data Fitting to the Lugre Model
6.3.1 Model Description
As discussed previously in Chapter 2, the LuGre model is a dynamic friction
model which captures many phenomena such as stick-slip motion. It has been used
successfully to model ball-bearing friction and to make friction compensation for
mechanical systems. The model is described by
zvg
vv
dt
dz
)(−= …(38)
0 1 ( )dz
F z f vdt
σ σ= + + …(39)
where the f represents viscid friction. The function g(v) can have different forms, one
possibility, which is a slight variation of the previously discussed form of the
equation and offers more flexibility is
( )0
1( ) s
v
v
C S Cg v F F F eσ
−
= + −
…(40)
The state variable z has dimension length and can be interpreted physically as the
average deflection of asperities.
115
At steady state,
( ) ( )sgn( )v
z g v g v vv
= = …(41)
which gives a Stribeck function of the form
( ) sgn( ) ( ) sgn( ) ( )s
v
v
SS C S CF v F v F F e v f v
− = + − + …(42)
The function f is given by viscosity. Kavehpour’s data [4] measured friction
coefficient as a function of Gumbel number which is defined as:
2
N
RGu
F
η η π
σ
Ω Ω= = …(43)
where,
η = Shear Viscosity
σ = FN/πR2 = Average or nominal stress acting on the rotating plate
Ω = angular velocity
6.3.2 Curve Fitting
Typical plot of friction coefficients vs. the Gumbel number at varying normal
loads is presented in Fig 6.3. In terms of the friction coefficient obtained by dividing
throughout by the steady state friction force, equation (42) can be re-written in the
following form:
31 2 4
p xp p e p xµ −= + + …(44)
116
Fig 6.3 Effect of Normal force on friction coefficient plotted vs. Gumbel
number. A copper fixture is used with Pennzoil 80W-90 is the lubricant [4]
Since the Gumbel number is proportional to velocity, we simply fit a function of
the form in equation (44) to the measured friction coefficients. The results of a mean
square fit to logarithmic data for a sample data are shown in figures 6.4-6.7.
The simulation includes the Matlab regression routine "lsqcurvefit" which is a
nonlinear curve fitting algorithm that minimizes mean square errors. The routine
needs to be supplied with an initial set of model parameters, which it refines using an
iterative performance to maximize fit. The code for the Matlab simulation used for
the curve fitting can be found in Appendix A. Table 6.2 lists the values for the
parameters in the equation (44) obtained using the least square curve fitting
algorithm. Good fits were obtained upon removing data points at high and low
117
Fig 6.4 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve
at Normal Load 5 N
Fig 6.5 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve
at Normal Load 10 N
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Actual and Fitted Curves 10N
Gumbel Number
Friction C
oeff
ecie
nt
(u)
Data Plot
Fitted Curve
y = 0.004765 + 0.07637*exp(-(4.9E+5*x)) + 455.7*x
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Actual and Fitted Curves 5N
Gumbel Number
Friction C
oeff
ecie
nt
(u)
Data Plot
Fitted Curve
y = 0.0159 + (0.0937)*exp(-(3.067E+5*x)) + 388.91*x
118
Fig 6.6 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted
Curve at Normal Load 15 N
Fig 6.7 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted
Curve at Normal Load 20 N
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Actual and Fitted Curves 20N
Gumbel Number
Friction C
oeff
ecie
nt
(u)
Data Plot
Fitted Curve
y = 0.0061 + 0.0572*exp(-(3.15E+5*x)) + 482.75*x
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
Actual and Fitted Curves 15N
Gumbel Number
Friction C
oeff
ecie
nt
(u)
Data Plot
Fitted Curve
y = 0.00668 + 0.04765*exp(-(1.73E+6*x)) + 391.4*x
119
p1 p2 p3 p4 p1+p2
5N 0.0159 0.0937 3.07E+5 388.9 0.1096
10N 0.0048 0.0764 4.90E+5 455.7 0.0812
15N 0.0067 0.0477 1.73E+6 391.4 0.0544
20N 0.0061 0.0572 3.15E+5 482.8 0.0633
Table 6.2 Coefficients obtained by least-square fitting a function of the form
31 2 4
p xp p e p xµ −= + + to data obtained from the Tribo-rheometer
velocities. Substantial variation is observed in the friction data because of the
inherent randomness in the system, along with substantial fluctuations in the low
frequency measurements. At lower speeds/frequencies, the oscillations are due to the
lack of averaging since at such low speeds (e.g. 10-4 rad/s) it takes hours for a full
rotation of the disk. Increasing the averaging time would help get rid of these
‘random’ oscillations. At higher velocities, nonlinear changes in the nature of the
curve are observed due to non-Newtonian effects. Since a liquid lubricant is used in
the system, in the high speed regime the fluid is pushed out by centrifugal force
causing large variations in the friction coefficient. The last few data points are not
considered for the fitting to avoid giving too much emphasis to viscid friction.
120
The final parameters of the curve fitting are highly sensitive to the initially
supplied parameters. In other words, the final fit is often based on parameters that
converge to a local minimum. As a result, achieving a good fit often involves trial
and error with respect to the initial parameter estimates. Another issue is the
‘unbiasedness’ of the least-squares fit with respect to region of curve where fit is
critical. Certain points in the regions of the curves towards the ends have to be
omitted to improve the fit in the dip region. An ideal curve would weigh the dip data
differently as compared to data in the flat region, and might offer more flexibility of
fit in the dip region. There are also some difficulties with the fits caused due to
occurrence of local minima.
Other function forms were also tested to investigate if there are functions that
fit the data even better. For example, a function of the form
23
1 2 4p x
p p e p xµ −= + + …(45)
similar to those previously discussed in equations (2) and (13) was used for curve
fitting and the resulting fit is shown in Fig. 6.8. This form of the function is more
strictly limited because of the e2 term. In context of the current fitting, the function
form in equation (44) which gives a reasonably good fit, is preserved.
The obtained parameters p1 and p2 show a decreasing trend while p4 shows a
slight increase with the normal load but stays fairly constant around 400. Also, the
variation of these three parameters is roughly linear with normal load for the first
121
Fig 6.8 Friction Coefficient vs. Gumbel Number Plot of Data and Fitted Curve
to sample curve using a different function form 2
31 2 4
p xp p e p xµ −= + +
three data points. The parameter ‘p1+p2’, which represents the coefficient of friction
at zero velocity, is in good agreement with the static friction values observed
between the two surfaces. As a first approximation, the parameter p4 corresponds to
R/delta where R is the radius of the disc (~2 cm) and delta is the gap size (< 300
µm). Varying the experimental conditions and performing further curve fitting could
help relate the function g to physics.
6.4 Experimental Comparison of the Nanoindenter and Tribo-rheometer
The range of load in the tribo-rheometer is from 0-50 +/-0.1 N and at the lower
end, reliable data can be obtained for a normal load of 0.5 N and higher. Its radial
122
velocity range is 0.0001-300 rad/sec. Since it is a rotational system, the linear
velocity depends on the radius of the fixture used. For a 2 cm mean radius fixture,
this translates into a linear velocity range of 2 µm/s - 6 m/s which overlaps with the
nanoindenter’s velocity range of 0.1 µm/s - 40 µm/s. Although the maximum normal
load capacity of the nanoindenter is 10 mN which is substantially lower than the
lowest normal load at which the tribo-rheometer generates reliable data, it would still
be interesting to compare the results of tribo-rheometer measurements with those of
the nanoindenter in terms of the dimensionless friction coefficient, especially in the
overlapping velocity range by making the experiments as similar as possible in terms
of substrate material. A comparative study of the friction coefficients between the
two surfaces at substantially different normal load and stress would also help
determine if the nanoindenter is an effective tool for bridging the gap in quantifying
friction from the micro to the macro scale.
6.4.1 Experimental Setup
Most of the experiments with the nanoindenter so far have been conducted with a
standard diamond tip sliding on a mica surface. Since it is difficult to use diamond as
one of the substrates in the tribo-rheometer on account of the system’s design, other
materials need to be investigated. Stainless steel is a typical engineering material
which is also a common test material for the tribo-rheometer was chosen as the
material to be used in these experiments. Experiments on the tribo-rheometer were
conducted with an annular stainless steel fixture of inner diameter 19.26 mm and
outer diameter 23.86 mm in contact with a bottom flat plate made of the same
123
material. Since Hysitron, the manufacturer of the nanoindenter, does not manufacture
standard tips made with stainless steel, we designed and fabricated a few tips in-
house for the testing purpose. The stainless steel tip design is as shown in Fig. 6.9
while the contacting bottom surface consists of a polished stainless steel surface of
radius 1 cm. Dry experiments were conducted at first followed by lubricated
experiments employing heavy paraffin oil as the lubricant.
6.4.2 Nanoindenter – Experimental Results and Discussion
Fig 6.10 shows the results of friction coefficient vs. velocity for the entire load-
velocity range of the nanoindenter using the stainless steel tip sliding on a polished
stainless steel surface. The friction coefficients obtained from the nanoindenter lie in
the range 0.07-0.14, with an inverse relationship between the applied normal load
and the observed coefficient of friction. As shown in Fig 6.11, stick-slip was
observed at low velocities for the 5000 µN normal loading case, although it could
not be repeated on subsequent runs of the same experiment thus indicating that the
occurrence of stick-slip is highly sensitive to the experimental conditions. A
comparison between the friction results for these two cases in Fig 6.11 shows that the
coefficient of friction is higher in the case where stick-slip is observed, consistent
with the results obtained for the mica-diamond interface.
An interesting feature observed from these plots is the absence of any data for the
mid-range of velocities, typically from 4.5 µm/s to 13.5 µm/s. In this velocity range,
the nanoindenter either applies a very small normal load and measures a negative
lateral force as shown in Fig. 6.12, or at low normal loads (~ 500 µN) it applies a
124
Fig 6.9 Design of a non-standard Stainless Steel tip used for comparison
experiments with the nanoindenter
Fig 6.10 Friction Coefficient vs. velocity for a spherical stainless steel sliding on
a polished stainless steel plate observed with the nanoindenter
Nanoindenter - Dry Friction Coeff Variation with
Velocity, SS, LogScale
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.01 0.1 1 10 100
Velocity (µm/s)
Fri
cti
on
Co
eff
ecie
nt
(µ)
5000 µN Dry
7000 µN Dry
8500 µN dry
10000 µN Dry
1.75 Φ2
Φ1 Thread Type: 000-120
(1/16th-1/8th) length variable
All units in millimeters
Material: Stainless Steel
2
3
Φ1
Φ1
125
Fig 6.11 Friction force vs. time plots showing stick-slip at 5000 µN load at low
velocities and µ vs. velocity comparison with non stick-slip type data
negative normal load as shown in Fig 6.13. The former failure mechanism seems to
occur more frequently, also causing the nanoindenter to intermittently skip through
certain methods, thus generating no data. Both these failure mechanisms seem to
stem from the use of a non-standard tip, which weighs 65.2 mg compared to the
standard diamond tip which weighs 16.9 mg. Although Hysitron suggests the use of
tips which weigh lower than ~80 mg, failure in our case seems to set in as we
approach that limit. Another issue could be the threading of the non-standard tip,
which, being non-standard cannot be secured firmly in place using the tip-mounting
Stick-slip and No stick-slip comparison, 5000
µN Load, SS
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5
Velocity (µm/s)
Fri
cti
on
Co
eff
ecie
nt
(µ)
5000 µN - Dry Stick-
Slip
5000 µN Dry
126
Fig 6.12 Typical failure curve obtained when the nanoindenter fails to produce
reliable result, observed in the 4.5 µm/s to 13.5 µm/s velocity range
Fig 6.13 Typical failure curve obtained when the nanoindenter fails to produce
reliable data, observed in low normal-load experiments (~ 500 µN)
127
tool provided by Hysitron, possibly allowing for a play or a slip. A possible solution
to this issue could be to either get a standard tip custom-made from Hysitron or
lower the weight of the tip and redesign it so as to accurately fit the tip-mounting
tool. For example, by combining the stainless steel spherical contacting part with
another material for the top part, such as aluminum, the tip weight could be reduced
by a factor of three. Another solution would be to use an Al sphere glued to the end
of a standard blank tip holder.
Lubrication experiments with heavy paraffin oil, a common lubrication element used
with the tribo-rheometer, were conducted with the nanoindenter and the results are
shown in Fig 6.14 and Fig 6.15. The friction results in the lubricated case are in close
range with those observed for the dry-friction case as seen in the case of both the
5000 µN and 10000 µN normal load cases, the coefficient of friction in the
lubricated case for the 10000 µN and at higher velocities for the 5000 µN case being
slightly higher than that in their respective dry case experiments. Possible causes for
this could be because the lubricant itself causes enhanced adhesion or because the tip
remains fully submerged in the fluid without contacting with the surface, detecting a
false surface. 'Full-fluid lubrication' zone in which the coefficient of friction depends
on the viscosity effects is ruled out because the entire experimentation is conducted
at very low velocities. For the 10000 µN normal load case with lubrication, the
friction coefficient is seen to achieve higher value for data points which are in close
vicinity to the velocity range where the failure occurs.
128
Fig 6.14 Friction coefficient vs. velocity plots obtained using a nanoindenter at
5000 µN load for dry friction and using heavy paraffin oil as the lubricant
Fig 6.15 Friction coefficient vs. velocity plots obtained using a nanoindenter at
10000 µN load for dry friction and heavy paraffin oil as the lubricant
Nanoindenter - Dry and Lubricated Friction Comparison,
5000 µN Normal Load, SS, Loglog Scale
-1.15
-1.1
-1.05
-1
-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-1.5 -1 -0.5 0 0.5 1 1.5 2
Log[Velocity (µm/s)]
Lo
g[F
ricti
on
Co
eff
icie
nt
(µN
)]
Nanoindenter - Dry and Lubricated Friction Comparison,
10000 µN Normal Load, SS, Loglog Scale
-1.1
-1.05
-1
-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-1.5 -1 -0.5 0 0.5 1 1.5 2
Log[Velocity (µm/s)]
Lo
g[F
ricti
on
Co
eff
icie
nt
(µN
)]
5000 µN Dry 5000 µN Lubricated
10000 µN Dry 10000 µN Lub
129
6.4.3 Tribo-rheometer – Experimental Results and Discussion
The friction coefficient vs. velocity plots obtained from the tribo-rheometer
for an annular stainless steel fixture sliding on a stainless steel plate are depicted in
Fig. 6.16, the velocity being plotted in µm/s for easier comparison with the results
obtained from the nanoindenter. The coefficient of friction, computed by averaging
the shear stress with at least 1 revolution of data sampling rate of ~10 Hz, is found to
vary between 0.24-1.56 for the entire data-set and follows an inverse relationship
with the normal load, similar to that observed in case of the nanoindenter. At higher
rotation rates, the system is not able to compensate fast enough to be able to maintain
a constant normal force and hence an increase in the coefficient of friction is
observed. Stick-slip is observed for normal loads higher than 2 N and for radial
velocities lower than 0.01 rad/s.
As seen from Fig 6.17, the repeatability for experiments conducted on the
same day is higher than those conducted on different days. One possible reason for
this is the alignment of the top plate with the bottom plate. As described earlier, a
wax-based alignment method is followed which ensures a good initial contact area,
but since there is an indeterminable inherent tilt in the system, the area of contact
could be substantially different upon a 180-degree rotation. This could substantially
alter the friction force, computed from the contact area-dependent shear stress, thus
causing the observed variability in the friction coefficient. In the actual
experimentation, very little ring contact has been observed between the two surfaces
130
Fig 6.16 Friction Coefficient vs. velocity for a stainless steel fixture rotating on a
stainless steel plate observed with tribo-rheometer on two different days
Fig 6.17 Repeatability of the experiments conducted with the tribo-rheometer
on two different days (both plots are plotted to the same scale for comparison)
Tribo-Rheometer - Same Day
Experiment Repeatibility
0.2
0.4
0.6
0.8
1
1.2
1.4
0 500 1000 1500 2000
Velocity (µm/s)
Fri
cti
on
Co
eff
(µ
)
F=2N
F=2N (Repeat
Same Day)
Tribo-Rheometer - Different Day
Experiment Repeatibility
0.2
0.4
0.6
0.8
1
1.2
1.4
0 500 1000 1500 2000
Velocity (µm/s)
Fri
ctio
n C
oe
ff (µ
)
F = 4N
F = 4N (RepeatDifferent Day)
Tribo-Rheometer Friction Coeff Vs
Velocity, SS, LogScale
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
10 100 1000 10000 100000
Velocity (µm/s)
Fri
c C
oeff
(µ
)F= 1N (Day 1)
F= 1N (Day 2 -
Repeat)
F=2N (Day 1)
F=2N (Day 1 -Repeat)
F = 4N (Day 1)
F = 4N (Day 2 -Repeat)
131
even for a full revolution. Also, there are multiple circular scratches on the bottom
plate used in these experiments from over a year's usage which could affect the
friction measurements. In general, the older plates have been observed to have higher
friction coefficient than the newer ones.
The friction coefficients obtained from the nanoindenter lie in the range 0.07-
0.14, which does not overlap with the range observed in the tribo-rheometer (0.24-
1.56), although at low velocities they agree more closely, especially for the data
observed on the second day of experimentation which exhibits a higher overall
coefficient of friction possibly due to greater misalignment on that day. This
variation is to be expected since one is a linear displacement based system whereas
the other is a rotational system and there is a three orders of magnitude difference in
the applied normal loads and contact areas. Also, the normal stress in the case of the
tribo-rheometer is 20 KPa at 4 N normal load, which is substantially lower than that
for the nanoindenter, which is 0.15 GPa for the 5000 µN normal load. If the 1mm
spherical stainless steel tip used for the experiments with the nanoindenter can be
redesigned into a 1mm cylindrical flat surface, a normal stress similar to that of the
tribo-rheometer could be achieved. This could possibly lead into alignment accuracy
issues similar to those of the tribo-rheometer since there is an inherent tilt in the
nanoindenter setup which could cause the edge of the flat tip to contact the bottom
surface rather than the surface of the tip.
Small particles produced during the experimentation with the tribo-rheometer
also have a direct effect on the friction coefficient. This effect would be reduced if
132
experiments are conducted with a lubricant such as heavy paraffin oil and they would
also serve as a good comparison with the friction results reported earlier with the
nanoindenter.
133
7. SFA Comparison Experiments: Friction Measurements with
Polystyrene
7.1 The Surface Force Apparatus
7.1.1 Experimental Setup
The surface force apparatus (SFA) was developed by Israelachvili [36] for
measuring van der Waals forces between molecularly smooth mica surfaces.
Recently, new friction attachments have been developed suitable for use with the
SFA which allow for the two surfaces to be sheared past each other at varying sliding
speeds while simultaneously measuring both the transverse force and the normal
force between them [11]. The basic instrument, shown in Fig. 7.1, has a simple
single-cantilever spring to which the lower silica disk is attached. This lower mica
sheet is brought near the upper mica sheet by a piezoelectric device. If there is some
interaction between them, the distance between the mica sheets will not be the same
as that given by the piezoelectric device. Therefore, the force is measured by the gap
distance difference given by the piezoelectric device and that measured directly by
interferometry, attractive forces make the mica surfaces come closer while repulsive
forces try to move the surfaces away. Interferometry, which allows measurement of
distances down to 1/1000 of a wavelength, is used in the surface forces apparatus to
measure the distance between the two surfaces with high accuracy. Multiple beam
interferometry uses intense white light which is sent normally through the surfaces in
the surface forces apparatus. Each mica surface has a highly reflecting silver coating
134
Fig 7.1 Schematic of the Surface Force Apparatus (SFA) [13]
on one side; therefore, both surfaces form an optical cavity. The white light is
reflected multiple times on these mirrors before it leaves the interferometer, each
time interfering with the previously reflected beams. Some particular wavelengths fit
exactly an entire number of times inside the interferometer and lead to constructive
interference.
Fig. 7.2 shows a schematic of the multiple beam interferometer at work in the
instrument. A source of white light is focused between two silver-coated surfaces.
The light emerging from the interferometer is sent to a spectrograph, and it consists
of well defined wavelengths which fit exactly an entire number of times into the
optical resonator in the form of curved fringes. The fringes are called Fringes of
Equal Chromatic Order (FECO).
As shown in Fig. 7.3, the crossed cylinder geometry of mica sheets used in SFA
135
Fig 7.2 Interferometry based measurement in the Surface Force Apparatus [14]
Fig 7.3 Schematic of Mica Surfaces used in the SFA [14]
is mathematically equivalent to sphere on flat surface contact. Measurement of
adhesion forces and interfacial energy can further be analyzed by JKR (Johnson,
Kendal, Roberts) theory for large soft objects, or DMT (Derjaguin, Muller, Toporov)
for small hard objects.
7.1.2 Polystyrene
Polystyrene (PS) is a nonpolar polymer which is glassy at room temperature. It is
highly adhesive and jumps into contact immediately even upon the application of
136
zero load. It is a relatively softer material, with tensile modulus ~ 3 GPa. It is an
isotropic material and shows no orientations dependence in friction experiments.
Friction measurements using PS of various molecular weights (MW) were performed
using the SFA [15]. In the following sections, we present materials and methods used
to reproduce some of these experiments using the nanoindenter for purposes of
comparison.
7.2 Triboindenter: Comparison Experiments and Results
7.2.1 Tip and Sample Preparation
1/8” diameter polystyrene (PS) balls were purchased from Polysciences, Inc.
They were glued on to a triboindenter tip holder obtained from Hysitron using
superglue. Higher viscosity superglue is preferred since it does not slide down the
sphere causing contamination of the surface which would be contacting in the
friction testing. Adhesion accelerators were also used for quick adhesion and a
constant pressure was maintained between the tip and the sphere during the process
of adhesion.
PS of average molecular weight 280,000 was purchased from Aldrich Chemical
Co. About 5 g of a dilute 5 % solution of toluene was added to 0.25 g of PS blocks
and filtered through a 200 nm mesh size PTFE filter. A larger force on the syringe
has to be applied for filtering solutions containing higher molecular weight of PS.
The sample was produced by the ‘casting method’ in which a droplet of the
toluene solution is cast on the top of a mica surface using a syringe or is spin coated.
137
Subsequent drying leads to the deposition of a 30-200 nm of PS coating on the top of
the mica surface.
7.2.2 Experimental Results and Discussion
Most of the friction force experiments are conducted with the SFA at zero load
given the highly adhesive nature of SFA. The normal loading functionality of the
nanoindenter, being under open-loop control, is unable to apply a constant zero
normal load as shown in Fig. 7.4. It applies an increasing load which averages
around zero instead and hence the experimental results in this section depict the
average friction force measured during the entire time interval for which the zero
load is applied.
The friction force vs. velocity results obtained from the nanoindenter are plotted
in Fig. 7.5 (a) and they are and compared to the results from the SFA in Fig. 7.5 (b).
The qualitative nature of the curves shows high similarity although the friction force
at zero load between the two systems is different by an order of magnitude. The
primary reason for this behavior is that the adhesive forces in the latter case are much
higher because of a higher contact area between the mica surfaces in the SFA as
compared to that between the tip and the sample in the former case. Since the MW of
the PS sphere used in the above experiments is unknown, another set of experiments
was conducted using PS of MW 280,000 deposited on a 3 mm steel sphere attached
to a tip holder and sliding against PS of MW 280,000 deposited on a mica surface.
The friction force for this system is plotted as a function of normal load at constant
velocity in Fig. 7.6 (a) along with the values obtained for the same material using the
138
Fig 7.4 Typical Results Window for a Zero-Load Experiment Performed with
the Nanoindenter for a PS Sphere sliding against a PS Surface (MW 280,000)
SFA. A linear fit indicates a friction coefficient of 0.22 from the triboindenter which
is in good agreement with the 0.15 value obtained from the SFA. The difference
between the measured values between the two systems is higher at low loads and the
values are in better agreement as the load is increased. Possible reasons for this
behavior could be attributed to the contact area difference. The friction force between
two highly adhesive surfaces such as these has been reported to vary as [35]:
k cF S A Lµ= + …(46)
where Fk is the kinetic friction, Sc is the critical shear stress, A is the contact area
139
Friction Force Vs Velocity for Polystyrene Ball
Sliding on Polystyrene Sample (MW 280,000)
0
500
1000
1500
2000
2500
3000
0.01 0.1 1 10 100
Velocity (µm/s)
Fri
cti
on
Fo
rce (µ
N)
0 µN
1000 µN
3000 µN
5000 µN
8500 µN
7.5 (a) Friction Forces Between a PS Sphere and an Untreated PS Surface (MW
280,000) as a function of Sliding Speed measured with the Triboindenter
Fig 7.5 (b) Friction Forces Between Two PS 280,000 Surfaces before and after
Crosslinking as a function of Sliding Speed measured with the SFA [15]
140
Fig 7.6 Friction Force Between PS (MW 280,000) Deposited on a Steel Sphere of
Radius 3 mm Sliding Against a similar PS Surface deposited on Mica as a
function of Normal Load Measured by Triboindenter at 0.5 µm/s and SFA [15]
and L is the normal load. The adhesive force is proportional to the area of contact, a
typical value for which is 1980 µm2 at L=0 for SFA in experiments using PS
(MW=280,000) whereas that for the triboindenter varies between 0.2 µm2 at L=10
µN to 1110 µm2 at L=10,000 µN using Hertzian approximation. The contact areas
for the two systems at higher normal loads are of the same order or magnitude and
hence there is a better agreement in the observed friction values as expected.
Polystyrene has been reported to exhibit isotropic behavior and in Fig. 7.7
this is verified by rotating the PS sample by 3600 and measuring friction force for a
particular loading function at 450 rotation increment. Unlike the anisotropic behavior
Friction Force Vs Normal Load at v = 0.5 µm/s for PS
(MW 280,000)
µ = 0.15
µ = 0.22
0
500
1000
1500
2000
2500
3000
0 5000 10000 15000
Normal Load uN
Fri
cti
on
Fo
rce u
N
Triboindenter
SFA
Linear (SFA)
Linear
(Triboindenter)
141
Orientation Dependence of Friction Force
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400
Rotation (Deg)
Fri
cti
on
Fo
rce
(µ
N)
Fig 7.7 Friction Force vs. Rotation Angle Plot for a PS Sphere sliding against a
PS Surface of MW 280,000 indicating the Isotropic nature of PS
reported earlier between the mica-diamond and mica-sapphire experiments, no
anisotropy is observed in this case as expected.
Results of stopping time experiments at zero load are presented in Fig. 7.8 (a)
and are compared with those reported earlier with the SFA in Fig. 7.8 (b). Since the
nanoindenter cannot apply a constant zero normal load as described earlier, the
experiments with the triboindenter are conducted at a higher normal load of 1000 µN
and the friction spike value is plotted as a function of waiting time. The results are
similar to that for the untreated PS shown in Fig. 7.8 (b) with both showing an
increase in friction spike as the waiting time is increased although qualitatively the
triboindenter data suggests an approach towards a steady state value as the waiting
period is increased whereas the SFA data indicates a steady increase in friction spike
142
Height of Friction Spike in Stop Start Experiments at
1000 µN Normal Load vs. Stopping Time
320
330
340
350
360
370
380
390
400
1 10 100 1000 10000
Stopping Time (s)
Fri
cti
on
Sp
ike
(µ
N)
Fig 7.8 (a) Friction Spike vs. Stopping Time Plot for a PS (MW 280,000)
Deposited on a Steel Sphere of Radius 3 mm Sliding Against a PS Surface (MW
280,000) deposited on Mica at 1000 µN Normal Load
Fig 7.8 (b) Height of Stiction Spike of PS 2,000,000 in stop-start experiments as
a function of Stopping Time [15]
143
Surface Seperation between two PS Surfaces at
Zero Load vs. Velocity
0
50
100
150
200
0 20 40 60 80
Velocity (µm/s)
Su
rfa
ce
Se
pe
rati
on
(n
m)
Fig 7.9 Surface Separation between PS Sphere and Surface at Zero Load vs.
Velocity
value as the waiting period is increased. This difference could be caused possibly
because the experiments with the SFA are conducted at L=0 whereas those with the
triboindenter are conducted at L=1,000 µN and the contact areas between the two
surfaces are at these loads are quite different, ~1980 µm2 at L=0 for SFA and ~240
µm2 at L=1,000 µN for the triboindenter.
The nanoindenter is capable of measuring the separation between the two PS
surfaces and the maximum separation at zero load is plotted as a function of velocity
in Fig. 7.9. An overall decreasing trend is observed as the velocity is increased
indicating that the adhesive force in the case of higher velocities occurs at lower
separation between the two surfaces since lesser amount of time is available to reach
steady state.
144
Overall, the friction measurements with the nanoindenter and the SFA for PS
show high correspondence in their qualitative nature and agree well on the measured
friction coefficient value in the two cases. As discussed previously, experiments at
zero load could be performed with better accuracy if the nanoindenter is able to
implement a close-loop control during the normal loading. The nanoindenter offers
the additional capability of accurately measuring real-time depth of the tip in the
surface during friction measurements. The nanoindenter can add imaging capability
to the experiments if tips of lower radii e.g. 1 µm are used. The SFA can perform
friction measurements on the PS samples up to 50-600 C whereas using the available
heating stage, the triboindenter can extend this capacity to temperatures as high as
1000 C. Further comparison results between the SFA and the triboindenter, similar to
that previously published for the SFA and the AFM [23], can be obtained by
performing more experiments using similar material; sapphire is an example of
another material which can be easily used for experimentation with both SFA and the
nanoindenter.
145
8. Conclusions, Applications, New Equipment and Future
Work
8.1 Conclusions
The nanoindenter is determined to be a reliable device for performing friction
experiments at the micro-scale for small displacements and at low velocities (when a
tip of significant radius is utilized). The velocity limits of the nanoindenter within
which reliable data can be obtained was determined and new ways to increase this
limit using the traditional triboindenter were tested and are described in the next
section. Many issues with the device, such as the inherent tilt in the nanoindenter and
software based methods to compensate for it etc. were sorted out. High repeatability
is observed in reproduction of the classical experiments and high precision friction
data is obtained using the nanoindenter. Friction spikes achieve higher peak values in
reduced rise times as the velocity is increased. Their magnitude is also found to
increase in proportion to the time the system rests in contact before the
commencement of the experiement. Anisotropy is observed between asymmetric
surfaces with stick-slip occurring only at specific relative orientations. Depth
measurements closely correspond to values predicted by Hertzian contact theory.
Comparison experiments with other tribological measurement systems, such as the
tribo-rheometer and the SFA, have proved the validity of the triboindenter’s scratch
utility for micro-scale friction measurements within reasonable limits. The X axis is
displacement controlled and not force controlled so simple experiments which
require force control such as the amount of force needed to “break-away” cannot be
146
currently conducted. On the other hand, experiments involving friction lag and
hysteresis, which require a larger displacement range, can only be performed with the
aid of new instrumentation attachments to the triboindenter and are described in the
following section. These observed effects and associated insights are important in
using the triboindenter for modeling friction at the micro-scale and also in extending
its range of measurement from the nano to the macro scale.
8.2 Future Work: Advanced Equipment and Techniques
Although the maximum displacement range of nanoindenter is limited to 16
µm, new instrumentation, in the form of extensions to the current setup that can
apply normal loads up to 5 N and provide a extended velocity range of 4 Å/s – 1
mm/s, have been explored. Such an extended measurement range could bridge the
gap in quantifying friction from the atomic to the macro scale. It would also help
perform experiments that measure friction lag and hysteresis which require a larger
displacement range than that currently available. A comparison of the load and
velocity ranges possible with the new equipment that were tested at Hysitron is given
in Table 8.1.
8.2.1 High Velocity Experiments – The Flexure Stage
A 2D flexure stage, shown in Fig. 8.1, can be added to the XYZ staging
system of the triboindenter and combined with the high load head to give 3D force
sensing for scratch testing. In this technique, the stage is used for lateral translation,
while the indenter head applies a controlled normal load either under load or
147
Normal Load Velocity Lateral Displace-ment
Tribological Expts 5-35 mN 0.001-10 µm/s NA
Triboindenter with Closed Loop Scanner
30 mN Few nm/s 200 µm
Tribo-indenter (Current Configuration)
0.001 – 10 mN 0.1-100 µm/s 15 µm
Triboindenter with 2-D Flexure Stage
30 mN 200 µm/s 25 mm
Triboindenter with 3-D Omniprobe
5 N 30 mm/s 150 mm
Classical Experiments
40-400 N 0.01 - 1 m/s NA
Table 8.1 Comparison of Capability Enhancement Add-ons to the
Triboindenter [3, 6, 7]
Fig 8.1 Flexure Stage [6]
displacement control. The flexure stage allows for measurement of longer scratches,
up to 25 mm. There is also another similar option offered by Hysitron in which the
flexure is a part of the normal force head, which allows scratches up to 150 mm. The
maximum normal load possible is limited by the nanoprobe used – Hysitron
148
currently provides one which can apply normal load up to 5 N whereas the one
currently available to us in UCSB has 1.2 N maximum load with 7 µN resolution.
Design
The design of the flexure stage consists of springs of measured stiffness. As the
scratching is performed, the lateral force causes a deflection of the springs which is
measured with displacement sensors. From the measured displacement and the
known lateral stiffness, the lateral force can be computed. The design also allows the
flexure stage to simultaneously monitor the forces in both X and Y directions. Since
it is not under the same feedback for motion as the triboindenter, the lateral force and
displacement signal are also expected be cleaner.
Experimental Investigation and Results
The flexure stage is fully controlled by software although additional hardware is
required for its operation. The range of velocity it can provide is from 0.05 µm/s up
to potentially 1000 µm/s and high velocity experiments ranging from 1 µm/s to 300
µm/s were performed using the flexure stage. A typical scratch result is shown in
Fig. 8.2. As seen from the figure, large fluctuations in the scratch results are
observed for both slow and fast motions. At loads typically higher than 250 mN, as
shown in Fig. 8.3, the 50 µm cono-spherical tip seems to dig into the sample causing
visible scratches due to surface deformation, the higher the load the greater being the
deformation. To avoid this issue, spherical surfaces with higher radii that do not dig
into the surface at those normal loads need to be fabricated. The tips that incorporate
149
Fig 8.2 Scratch results using the Multirange Nanoprobe and flexure stage using
50 µm cono-spherical tip at 9.81 mN normal load (a) 7 µm/s (b) 225 µm/s vel
(a) 500 mN (b) 750 mN
Fig 8.3 Visible damage of the mica surface in scratch experiments conducted
with the 50 µm cono-spherical tip at 300 µm/s and normal load
9.81 mN Normal load, 7 µm/s velocity
0
100
200
300
1 3 5 7 9Time (s)
Fri
cti
on
Fo
rce (µ
N)
9.81 mN Normal Load, 225 µm/s velocity
700
900
1100
1300
2.8 3.3 3.8 4.3 4.8
Time (s)
Fri
cti
on
Fo
rce
(µN
)
150
these surfaces also would need to conform to the weight limitation for the transducer
or the nanoprobe being used, which in the case of the transducer is 100 mg. Also, for
load and velocity ranges overlapping with those of the triboindenter, the preliminary
friction values obtained using the flexure stage are roughly half of that obtained
using the triboindenter. Thus, although the flexure stage helps expand the load and
velocity range on the higher side, drawbacks such as high fluctuations in lateral force
data and consistency in friction results with those of the nanoindenter need to be
resolved.
8.2.2 Low Velocity Experiments – The Closed-loop Scanner
The closed-loop scanner tested at Hysitron was nPoint’s XY200Z25A
nanopositioner [7] shown in Fig. 8.4. Its scan range in the X and Y directions is 200
µm while that in the Z direction is 25 µm. It has subnanometer resolution and closed-
loop capacitive feedback for position control. Unlike the flexure stage, the scanner
itself does not measure force and hence the force sensing is limited to that of the
triboindenter. The velocity range was from few nm/s up to 200 µm/s.
Since the closed-loop scanner is not integrated with Hysitron’s software,
scratches had to be performed by feeding a function generator signal into the stage
controller. By changing the voltage and frequency of the triangle wave created by
the function generator the length and velocity of the scratch were controlled.
Inputting a sinusoidal function into stage controller would allow testing at varying
velocities during a single scratch, although this was not thoroughly investigated.
151
Fig 8.4 Closed-loop Scanner [7]
Experimental Investigation and Results
Results of a high-velocity scratch are shown in Fig. 8.5. Substantial fluctuations
in the friction force similar to those observed in the case of the flexure stage are
consistently observed in experiments with the closed-loop scanner as well. Scratch
experiments in which the tip was held stationary on the mica surface exhibited a
considerable amount of drift. Low velocity results with the closed-loop scanner are
shown in Fig. 8.6 - the closed loop scanner was able to generate data accurately for
velocities as low as 4 Angstroms/s. The scanner is currently limited by the
capabilities of the function generator being used and thus the instrument has the
potential for scratching at even lower velocities. A comparison of the results
obtained from the closed-loop scanner and the triboindenter is presented in Fig. 8.7.
The scanner certainly increases the velocity range of the triboindenter at both the
lower and higher ends and friction force curves obtained using both instruments
exhibit more or less similar trend although the magnitudes differ by about a factor of
152
Fig 8.5 Friction Force vs. Velocity Plots Obtained Using the Closed-loop
Scanner at High Velocity
Fig 8.6 Friction Force vs. Velocity Plots Obtained Using the Closed-loop
Scanner at Low Velocities
3 mN Normal Load, 100 µm/s Velocity
325
425
525
625
725
25.75 26.25 26.75 27.25 27.75
Time (s)
Fri
cti
on
Fo
rce
(µN
)
3 mN Normal Load, 0.012 µm/s Velocity
250
300
350
400
45 55 65 75Time (s)
Fri
cti
on
Fo
rce
(µN
)
3 mN Normal Load, 4 Ang/s Velocity
200
220
240
260
280
300
175 225 275 325
Time (s)
Fri
cti
on
Fo
rce
(µN
)
153
Comparison of Closed Loop Scanner and Triboindenter
0
100
200
300
400
500
600
700
0.0001 0.001 0.01 0.1 1 10 100 1000
Velocity (µm/s)
Fri
cti
on
Fo
rce (µ
N)
Closed-loop Scanner
Triboindenter
Fig 8.7 Friction Force vs. Velocity Comparison between the Triboindenter and
the Closed-loop Scanner
2. Since different triboindenters were used for experimentation in the two cases, that
could be in part responsible for the observed difference.
Although the closed-loop scanner increases the capacity of scratch length to be as
high as 200 µm as compared to current limitation of 16 µm, it suffers from the same
limitation as that of the triboindenter in the reduction of scratch time as the velocity
is increased. Since function generators were being used for actuation, the exact time
of the start of the experiment cannot be accurately recorded hence friction lag
experiments cannot be performed unless the closed loop scanner is integrated with
the triboindenter’s software. This, in turn, would also permit Hysteresis experiments
at low velocities to be conducted by allowing the desired velocity profile to be input
into the software.
154
The results obtained by flexure stage, scanner and triboindenter for the exact
same scratches do not match well in numbers and pattern. One reason for this could
be that a 50 µm tip 60 deg cone angle tip was used for testing with the closed loop
scanner and flexure stage whereas the 50 µm tip 90 deg cone angle tip was used with
the triboindenter. Although this difference in cone angle does not alter the surface
curvature of the tip in contact with the surface, the other properties of the tip such as
surface roughness could be substantially different resulting in this observed
difference.
While the closed-loop scanner seems to be a good choice for extending the range
of velocity in the lower end and slightly on the higher end, the flexure stage could
prove effective in extending it substantially on the higher end. Further
experimentation could help resolve the current inconsistency in the results obtained
from the new instruments with the existing results from the triboindenter in
overlapping range of normal load and velocities.
8.2.3 Scaling of Friction
Studies in scale-dependence of friction have indicated that adhesion and
friction are scale-dependent and material behavior on one scale cannot be assumed to
hold on another scale [50]. Using E-Beam lithography technique, surfaces with
known asperity heights and asperity spacings can be produced on the nanometer
scale. If the spacing distance is reduced and the surfaces are overexposed after
spinning on negative resist, the neighboring asperities would merge providing a
surface of known corrugation with even finer features. As shown in Fig. 8.8, the
155
Fig 8.8 Asperity Patterns with the JDX-5D11 E-Beam Lithography System
currently available E-Beam lithography system using the JEOL JBX-5D11 machine
can create circular 80 nm features and a corrugated surface with minimum feature
size 50 nm. A new system, 6300 FS, is being currently installed which extends the
minimum feature size capability to as low as 8 nm. The choice of the deposition
substrate would depend on its adhesion behavior with the sample material. Polished
Silicon can help provide a truly flat sample material while an organic photoresist
such as Man2403 could be a possible deposition material although its shear modulus
is currently unknown. This method could help create controlled surfaces for friction
measurements. For surfaces with micro-level features i.e. feature sizes larger than
500 nm diameter and 100 nm height, optical lithography can be used. The capability
to design surfaces with known roughness using E-beam lithography and optical
80 nm 80 nm
50
nm
50
nm
100 nm
100 nm
80 nm
Negative Resist
Sample Surface
Sample
Top View
Side View
Created Asperities
156
lithography could serve to be instrumental in studying the effects of surface scaling
on friction behavior.
8.3 Applications of Nano/Microscale Friction Testing
8.3.1 Correcting Nano- and Micro-scale Friction Measurements
Friction measurements made at the micro- and nano-scale can differ
substantially due to changes in applied load. Some of these measured differences
have been attributed to the unintended scratching of the surface by the sharp tips
used in making the measurements themselves [51]. Researchers at NIST used a
specially designed friction tester developed jointly by NIST and Hysitron. A
carefully calibrated force was applied to diamond tips having a range of sizes and
friction forces were measured as each tip was slid across a very smooth surface of
silicon. Images of the test surface made with an atomic force microscope indicated
unintentional scratching to be the cause of the extra friction. To correct for this
effect, NIST researchers developed a way to measure precisely the size, shape and
orientation of the diamond tips so that friction forces caused by "plowing" can be
subtracted to produce a more accurate final measurement. This approach could help
develop a better method for correcting nano- and microscale friction measurements
should help designers produce more durable micro- and nanodevices with moving
parts, such as tiny motors, positioning devices, or encoders.
157
8.3.2 Applications in MEMS
To date most of the microelectromechanical system (MEMS) devices have been
based on Silicon. This is due to the technological knowledge accumulated on
manipulating, machining, manufacturing of Silicon. The MEMS field has developed
sufficiently to produce micromechanisms as complex as gear boxes capable of two
and three levels of speed reduction [52] as shown in Fig. 8.9, but tribological issues
are holding back their development and that of a myriad of other MEMS devices -
such as microturbines and relay switches - from entering commercial applications. At
the micro scale, friction and stiction can be catastrophic to MEMS rotary motors [53]
and flexible cantilevers [54]. Due to the brittle nature of Silicon, one of the core
MEMS materials, friction causes significant wear leading to the failure of most
rotating MEMS devices. Traditional lubricants such as oil cannot be used in these
devices because these microfabricated structures succumb rapidly to capillary forces
in the presence of liquids.
The Scratch Drive Actuator
Stiction forces in MEMS devices often degrade or prevent system performance.
Sliding frictional contacts contribute to the force transfer between actuators and
objects and influence their speed [55]. However, friction can also be useful for some
MEMS devices [56, 57]. Friction drives or ‘scratch-type’ drives utilize friction to
enable free-standing pieces to be moved with micro actuators [16] and their design is
described next.
158
Fig 8.9 MEMS (a) Multiple and (b) Linear Rack Gear Speed Reduction Drives
(Courtesy of Sandia National Laboratories, SUMMiTTM
Technologies)
Scratch drive actuators (SDAs) use electrostatic attraction to snap down a
polysilicon plate to the substrate. A bushing structure is incorporated in the front of
the plate that causes the actuator to step forward when the electrostatic attraction is
released. These devices can then be used to pull a structure, such a reflective surface,
out of the substrate plane.
An SDA consists of three parts - a plate, bushing and substrate electrode [58].
The plate rests at a small angle to the substrate and an AC signal (~100 V, 100 Hz-
100 kHz) powers the device such that the driver plate is alternately attracted to and
released from the substrate. The result is step-wise motion of the SDA with each step
providing nanometer scale displacement.
Fig. 8.10 illustrates a model of the step motion of a polysilicon microstructure
designed, fabricated and tested by Akiyama and Shono [16]. In Fig. 8.10 (a) a cross-
sectional view of the polysilicon plate and bushing on an insulator film (Si3N4) of an
Si wafer is shown. At the rise of an applied pulse, the plate is pulled down. Since one
a b
159
Fig 8.10 Cross-sectional View of Polysilicon Plate and Bushing of an SDA [16]
end of the plate, supported by the bushing, cannot move, the other part is pulled
down to come in contact with the surface of the insulator. The warp of the plate
causes the bushing to shift, as shown in Fig. 8.10 (b). Distortion energy is
temporarily stored in the plate. At the fall of the pulse, the distortion is released and
the plate snaps back to its original shape, as shown in Fig. 8.10 (c).
The SDAs offer many advantages for actuation purposes. They can move in
extremely fine steps ~ 10 nm and high velocities ~ 4.4 mm/s [59]. Their motion can
be accurately controlled by tuning the number of applied pulses and thus their
velocity is linearly related to electrical drive frequency. There is no fundamental
limit of travel and displacements upto 1 mm are possible. They occupy very small
area 100 Х 100 µm2 and are capable of producing over 100 µN of force. One huge
advantage is that they do not require stand-by power and they do not operate in a
resonant mode. They have the capability of providing untethered locomotion at the
MEMS scale [56]. Also, they can be easily integrated with other elements (e.g.
microoptical elements) through the same fabrication process.
However, some of the drawbacks that currently exist are that the exact details of
the actuation mechanism have not been exactly understood or modeled and that these
devices require high actuation voltages up to 200 V. CAD Modeling has been done
160
using IntelliSuiteTM [60]. They achieved a step size of 0.1 µm for their device
geometry which is in good agreement with experimental results. Their modeling was
done completely based on elastic deformation whereas plastic deformation is
assumed to be actually occurring. More work needs to be done to enable plastic
deformation and modeling friction for better visualization of the scratch drive.
During the first phase, the bushing end must move more forward than the plate
end moves backward while during the relaxation phase, the bushing end must slide
backward less than the other end slides forward. As a result, it is possible that most
of the energy being delivered to the device could be cancelled internally because of
the friction forces acting against each other [61]. Analyzing the role of friction
would give us an insight into optimizing the device geometry. The triboindenter can
accurately mimic the normal forces that are experienced in some of the current
devices and hence it can serve to be a useful tool in further studying the behavior of
the SDA.
Sandia’s “Inchworm” actuator based on an SDA was used as a test structure to
model friction at the microscale [62]. Existing theories predict a slip distance of 2
nm but in reality a slip as large as 200 nm – 100 times the expected value - was
observed. This indicates a gross slip mechanism occurring before the static friction
limit is reached. This gross sliding would be very important in MEMS, where
positioning of objects to nanometer accuracy is required for optical applications.
Other friction instruments have not yet indicated such behavior and a further study
would enable predicting friction-related motion in MEMS to a higher degree of
161
accuracy. This could help open a whole new market for micromachines such as
motors, pumps, actuators, communications devices and miniature mechanical
combination locks with sliding parts, which has not been possible so far since no
currently existing theory of friction is directly applicable to MEMS devices.
8.3.3 Bio-Inspired Applications
Biological surfaces represent the interface between living organisms and the
environment. Because of the broad diversity of functions, biological surfaces are
multifunctional and being a part of the physical world, the rules of mechanics also
apply to the living world [63]. Living creatures move on land, in air, and in water
and one always needs friction to generate force to move on a substrate or to
overcome the drag caused by friction elsewhere. A living motion system becomes
optimized when it is capable of minimizing friction at one end of the system while
maximizing it at the other end. Because of their structural and chemical complexity,
exact working mechanisms have been clarified for only a few systems. However,
biological surfaces hide a virtually endless potential of technological ideas for the
development of new materials and systems.
Gecko – Biological Solutions to Sticking to Surfaces
Gecko lizards do not have little suction cups on their feet but are able to climb up
walls and stick to ceilings. The feet of these animals, shown in Fig. 8.11, have toe
pads consisting of tiny hair-like structures called setae, made of keratin. The setae are
arranged in lamellar patterns and each seta has 400 to 1000 microhair structures,
162
Fig 8.11 Gecko Foot and Spatulae [17]
called spatulae. These tiny structures allow geckos to climb vertical walls across
ceilings. Using massively parallel MEMS processing technology, 20-150 µm
platforms, supported by single slender pillars and coated with ~2 µm long, ~200 nm
diameter, organic-looking polymer nanorods ‘organorods’ were batch fabricated to
mimic the gecko foot [64]. Adhesion testing was performed using the triboindenter
and it was concluded that in the real world, friction forces cannot be decoupled from
adhesion and testing methods developed in this work could be utilized for the
purpose of lateral testing of the ‘organorods’.
8.3.4 Nano-tribological Applications
Nano-scale devices based on moving molecular components have the potential to
radically alter technologies such as energy storage, drug delivery, computing,
communications and chemical manufacture. These nano-scale structures need to be
163
made mechanically and chemically resistant enough to withstand the extreme
conditions that can exist inside the human body or in any of the other hostile
environments where nanomachines might be expected to operate [65]. The shearing-
off or melting of even a single layer of atoms can easily spell death for a
nanomachine, The chemical and mechanical stability of moving nanostructures
underlie the field of nanotribology and the tribological considerations of these
systems have to be an integral aspect of the system design.
164
Appendix A
Matlab Programs
Least Square Curve Fitting:
polyj.m
function aa=polyj(x,xdata) [m,n]=size(xdata); c1=x(1); c2=x(2); c3=x(3); c4=x(4); for i=1:m aa(i) = (c1)+ (c2)*exp(((-1)*c3)*(xdata(i)^2))+(c4)*(xdata(i)); end aa=aa'; return regre.m % This is the main program calling the regression subroutine. load xdata.txt; load ydata.txt; x0=[0.0105;0.09;5.2E+10;350]; % Set initial parameter vectors % Run the regression routine x=lsqcurvefit('polyj',x0,xdata,ydata); yplot=polyj(x,xdata); loglog(xdata,yplot) % Plot The Fitted Curve hold on % Plot the Original Data [a,b]=xlsread('Kavehpour2.xls'); loglog(a(:,1),a(:,2),'bo'); title('Actual and Fitted Curves '); xlabel('Gumbel Number'); ylabel('Friction Coeffecient (u)'); legend('Data Plot','Fitted Curve');
165
Appendix B
Visual Basic Programs
Iteratively takes txt files generated by the Triboindenter and converts them to xls, draws graphs and computes average friction force:
Sub OpenAllFoldersAndReadTextFile() mypath = "C:\50 um tip\Data 4 3000 uN Load\" ' Set the path. ChDir mypath MyName = Dir(mypath, vbDirectory) ' Retrieve the first entry. Do While MyName <> "" ' Start the loop. If MyName <> "." And MyName <> ".." Then ChDir CurDir & "\" & MyName 'Read text file Call OpenAndReadTextFile 'Change back to root Workbooks.Open (CurDir & "\Results 0000.xls") 'Change back to root Call FindAverage ActiveWorkbook.Save ActiveWorkbook.Close ChDir mypath End If MyName = Dir Loop End Sub Sub OpenAndReadTextFile() Dim spath As String spath = CurDir Workbooks.OpenText Filename:= _ "Results 0000.txt" _ , Origin:=xlWindows, StartRow:=1, DataType:=xlDelimited, TextQualifier
_ :=xlDoubleQuote, ConsecutiveDelimiter:=False, Tab:=True, Semicolon:= _ False, Comma:=False, Space:=False, Other:=False, FieldInfo:=Array(Array
_ (1, 1), Array(2, 1), Array(3, 1), Array(4, 1), Array(5, 1), Array(6, 1),
Array(7, 1)) Range("D:D,F:F").Select
166
Range("F1").Activate Selection.Copy Sheets.Add ActiveSheet.Paste Columns("A:A").Select Application.CutCopyMode = False Selection.Cut Columns("C:C").Select ActiveSheet.Paste Columns("A:A").Select Selection.Delete Shift:=xlToLeft Rows("1:4").Select Selection.Delete Shift:=xlUp Columns("A:B").Select Charts.Add ActiveChart.ChartType = xlXYScatterSmoothNoMarkers ActiveChart.SetSourceData Source:=Sheets("Sheet1").Range("A1:B8010"),
PlotBy _ :=xlColumns ActiveChart.Location Where:=xlLocationAsObject, Name:="Sheet1" With ActiveChart .HasTitle = True .ChartTitle.Characters.Text = "Friction Force Vs Time" .Axes(xlCategory, xlPrimary).HasTitle = True .Axes(xlCategory, xlPrimary).AxisTitle.Characters.Text = "Time (s)" .Axes(xlValue, xlPrimary).HasTitle = True .Axes(xlValue, xlPrimary).AxisTitle.Characters.Text = "Friction Force
(uN)" End With ActiveSheet.Shapes("Chart 1").ScaleWidth 1.09, msoFalse,
msoScaleFromTopLeft ActiveSheet.Shapes("Chart 1").ScaleHeight 1.25, msoFalse,
msoScaleFromTopLeft ActiveChart.Legend.Select Selection.Delete ActiveSheet.ChartObjects("Chart 1").Activate ActiveWorkbook.SaveAs Filename:= _ "Results 0000.xls" _ , FileFormat:=xlNormal, Password:="", WriteResPassword:="", _ ReadOnlyRecommended:=False, CreateBackup:=False ActiveSheet.ChartObjects("Chart 1").Activate ActiveWorkbook.Close End Sub
167
Public Function GetArrayFromRange(rngARange As Excel.Range, intRows As Integer, intCols As Integer) As Variant
If intRows < 1 Or intCols < 1 Then Call Err.Raise(1, , "Negative arguments passed to GetArrayFromRange.
Contact ZS Developer.") ElseIf intRows = 1 And intCols = 1 Then Dim vntReturn(1 To 1, 1 To 1) As Variant vntReturn(1, 1) = rngARange.Value GetArrayFromRange = vntReturn Else GetArrayFromRange = Range(rngARange, rngARange.Offset(intRows - 1,
intCols - 1)).Value End If End Function Sub FindAverage() Dim vntData As Variant Dim intRow As Integer Dim dblSum1 As Double, dblSum2 As Double Dim dblAvg1 As Double, dblAvg2 As Double Dim intCount1 As Integer, intCount2 As Integer vntData = GetArrayFromRange(Workbooks("Results
0000.xls").Worksheets("Sheet1").Range("A1"), 9000, 2) For intRow = 1 To 9000 If vntData(intRow, 1) > 8 Then 'Found the section having values greater than 8 intCount1 = intCount1 + 1 dblSum1 = dblSum1 + vntData(intRow, 2) If vntData(intRow, 2) > 0 Then intCount2 = intCount2 + 1 dblSum2 = dblSum2 + vntData(intRow, 2) End If 'if > 0 If vntData(intRow + 1, 1) = "" Then Exit For End If End If 'if > 8 Next intRow 'Find averages dblAvg1 = dblSum1 / intCount1 dblAvg2 = dblSum2 / intCount2 Workbooks("Results 0000.xls").Worksheets("Sheet1").Range("D3").Value =
dblAvg1
168
Workbooks("Results 0000.xls").Worksheets("Sheet1").Range("E3").Value = dblAvg2
End Sub
Iteratively reads average friction force from different files and copies them into a single spreadsheet:
Sub OpenAllFoldersAndReadTextFile() Dim intFilesSoFar As Integer mypath = "C:\ Dry Friction\9 10000 uN Load\" ' Set the path. ChDir mypath MyName = Dir(mypath, vbDirectory) ' Retrieve the first entry. Do While MyName <> "" ' Start the loop. If MyName <> "." And MyName <> ".." Then ChDir CurDir & "\" & MyName intFilesSoFar = intFilesSoFar + 1 Call OpenAndReadTextFile_New(intFilesSoFar) ChDir mypath End If MyName = Dir Loop ThisWorkbook.Save End Sub Sub OpenAndReadTextFile_New(intFilesSoFar As Integer) 'Reads the average and pastes it in this workbook. Dim spath As String Dim wkbResults As Workbook spath = CurDir Set wkbResults = Workbooks.Open("Results 0000.xls") ThisWorkbook.Worksheets("Sheet1").Range("E3").Offset(intFilesSoFar,
0).Value = wkbResults.Worksheets("Sheet1").Range("E3").Value wkbResults.Close False Set wkbResults = Nothing End Sub
169
Appendix C
Matlab Programs for Hysitron Experiment Simulation
hyssim04.m
%Simulation or stick slip experiment with the Hysitron %Define initial simulation paramters tmax=1;str='b'; %Initial conditions x0=[0;0;0;0;0]; % x1 - position % x2 - velocity % x3 - state in friction model % x4 - integrator % x5 - ramp generator % Fd - driving force [t,x]=ode45('hysitronmod04',tmax,x0); %--------------------------- subplot(221) pl=plot(t,x(:,1),str,t,x(:,5),'r--'); set(pl,'LineWidth',2); axis([0 tmax 0 8e-6]); ylabel('Lateral Displacement');xlabel('Time');hold on %--------------------------- subplot(224) pl=plot(t,x(:,3),str); set(pl,'LineWidth',2); axis([0 tmax 0 15e-8]); ylabel('Friction State');xlabel('Time');hold on %--------------------------- subplot(223) pl=plot(t,x(:,2),str); set(pl,'LineWidth',2); %axis([0 tmax -1e-6 10e-6]); ylabel('Velocity');xlabel('Time');hold on %--------------------------- subplot(222) %Computing friction force [F,dzdt,m]=hysfricfcn(x(:,2),x(:,3)); pl=plot(t,F,str);%,t,F1,'r',t,Fss,'g--'); set(pl,'LineWidth',2);
170
axis([0 tmax 0 15e-4]); ylabel('Friction Force'); xlabel('Time');hold on
hysitronmod04.m
function dxdt=hysitronmod04(t,x) %LuGre friction model %Controller parameters %mk=1260;mki=7200;mkd=50; mk=1394;mki=7167;mkd=50; Fd=(x(4)-mk*x(1)-mkd*x(2));vpull=18e-6; [F,dzdt,m]=hysfricfcn04(x(2),x(3)); dxdt=[x(2);(Fd-F)/m;dzdt;mki*(x(5)-x(1));vpull]; hysfricfcn04.m
function [F,dzdt,m]=hysfricfcn(v,z) %Parameters %This is a round about way to estimate s0 and s1 can be improved %m=1e-3;z0=1e-7;zs=3*z0;s0=1e4;s1=2*sqrt(m*s0);s2=0.001;vs=1.4e-6; m=1e-3; zs=0.41e-6; z0=zs/3.1; % height of friction peak Fs=1400e-6; s0=Fs/zs; s1=3*sqrt(m*s0);s2=0.001; vs=1.2e-6; %kink factor %Function in friction model g=z0+(zs-z0).*exp(-abs(v)/vs);%+s2*v; %State equation for friction dzdt=v-abs(v).*z./g; F=s0*z+s1*dzdt+s2*v;
171
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