Garabedian Udel 0060

A DIRECT EXPERIMENTAL LINK BETWEEN ATOMIC-SCALE AND

MACROSCALE FRICTION

Nikolay T. Garabedian

A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering

Spring 2019

A DIRECT EXPERIMENTAL LINK BETWEEN ATOMIC-SCALE AND

MACROSCALE FRICTION

Nikolay T. Garabedian

Approved: ______Ajay K. Prasad, Ph.D. Chair of the Department of Mechanical Engineering

Approved: ______Levi T. Thompson, Ph.D. Dean of the College of Engineering

Approved: ______Douglas J. Doren, Ph.D. Interim Vice Provost for Graduate and Professional Education

I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.

Signed: ______David L. Burris, Ph.D. Professor in charge of dissertation

Signed: ______M. Zubaer Hossain, Ph.D. Member of dissertation committee

Signed: ______Chaoying Ni, Ph.D. Member of dissertation committee

Signed: ______Erik Thostenson, Ph.D. Member of dissertation committee

ACKNOWLEDGMENTS

I would like to sincerely acknowledge all people and institutions that were part of my PhD degree. I am first grateful to the institutions and organizations that provided financial support for my research: the National Science Foundation, the

Society of Tribologists and Lubrication Engineers, the Gore Fellowship through the College of Engineering at the University of Delaware, the University of Delaware Office of Graduate and Professional Education, and the Department of Mechanical Engineering. I would like to thank my dissertation committee members for the valuable discussions we had during my time as a student: Drs. M. Zubaer Hossain, Chaoying Ni, Erik Thostenson. My doctoral work would have not been possible without the invariable support from the Department of Mechanical Engineering: Lita Toto, Lisa

Katzmire, Ann Connor, Amy Adams, Melissa Arenz. The people that support the Keck Advanced Microscopy Facility were another vital part of my experimental work: Jen Sloppy, Thomas Barkley, Yong Zhao. I need to express my enormous gratitude to the staff at the Department of Mechanical Engineering Machine Shop: Scott Nelson and Jeff Ricketts. Their help and expertise improved my designs and taught me engineering skills that I could have never obtained otherwise. I also need to thank the community of researchers and engineers that are part of the broader tribology community for their advice whenever we have met at conferences.

iv My collaborators on projects and papers were another vital part of my work at the University of Delaware and I thank them for inspiring scientific interest and allowing me to learn about fields I never expected to venture into: Drs. Jillian Emerson, Brandon McClimon, James Hilbert, Harmandeep Khare, Eric Furst, Thomas Epps, III, Robert Carpick, Gary Doll, Kevin Dicker, Xinqiao Jia, Jing Qu, David Martin, and especially Brian Borovsky for bringing his character and originality of research to our lab for a full summer.

During my doctoral training I spent the most time with my labmates. Thank you for all your support and richness of experiences you provided me with: Dr. Jiaxin Ye, Dr. Axel Moore, Dr. Diana Haidar, Dr. Benjamin Gould, Arnab Bhattacharjee, Istiaque Alam, Aman Garodia, Viraj Sachpara, Steven Voinier, Jamie Benson. And although a PhD required my full attention and I very often was not fully available to my friends, these people were there and helped me get through the marathon which this degree represents: Erisa Saraçi, Albraa Jaber, Georgi Manolov,

Brandie Pugh, Tugce Yuksel, Mariya Petrova, Anita Toncheva, Milen Kisov, Raja Ganesh, Shyamola Athaide, Daniela Hristova, Eriselda Danaj, the Kutsilevi family, Stephen Berniker, Yulian Karapetkov, Rohit Kakodkar, Mariana Stoitseva, Swing Club, Rexhi, Adam Stager, Krasimir Dimitrov, Stoyan Yanchev, Diana Gomes.

I thank my family members for their unconditional support and understanding during my degree: Tzveta Garabedian, Takvor Garabedian, Tanya Petrova, Martina Garabedian, Mariya Petrova, Nikolay Petrov, Mitka Garabedian, Margarita Gesheva,

Stoyan Geshev, Sofia Gesheva, Krasimir Geshev, Stefan Garabedov, Toni Garabedova.

v Finally, I need to express my deepest gratitude to my mentor Dr. David L. Burris for guiding me through his insatiable interest for scientific discovery, eagerness for new lab endeavors and dedication to my progress.

v i TABLE OF CONTENTS

LIST OF TABLES ...... x LIST OF FIGURES ...... xi ABSTRACT ...... xxi

Chapter

1 INTRODUCTION ...... 1

2 PROBLEM STATEMENT AND OBJECTIVES ...... 9

3 QUANTIFYING, LOCATING, AND FOLLOWING ASPERITY SCALE WEAR WITHIN MACROSCALE CONTACT AREAS ...... 11

3.1 Abstract ...... 11 3.2 Introduction ...... 12 3.3 Methods ...... 15

3.3.1 The Topographic Difference Method ...... 15 3.3.2 Wear Volume Uncertainty Characterization ...... 16 3.3.3 Validation Experiment ...... 18 3.3.4 Quantifying and Correcting Repositioning Error ...... 20

3.4 Results ...... 22

3.4.1 Repositioning Uncertainty ...... 22 3.4.2 Validation Results ...... 23 3.4.3 Wear Mapping ...... 25

3.5 Discussion ...... 30 3.6 Conclusions ...... 32

4 MICROTRIBOMETRY ...... 34

4.1 Setup and Calibration ...... 34 4.2 Friction Force Dependence on Material and Probe Size ...... 37

vii 5 INTEGRATED QCM-MICROTRIBOMETRY: FRICTION OF SINGLE- CRYSTAL MOS2 AND GOLD FROM mm/s TO m/s ...... 45

5.1 Abstract ...... 45 5.2 Introduction ...... 46 5.3 Methods ...... 49

5.3.1 Materials ...... 49 5.3.2 Instruments ...... 51 5.3.3 Measurement and Analysis Methods ...... 55

5.3.3.1 QCM-based Force Measurements ...... 55 5.3.3.2 Spring-based Force Measurements ...... 57

5.3.4 Experimental Design ...... 57

5.4 Results ...... 59

5.4.1 Benchmarking Friction with the QCM ...... 59 5.4.2 The Effect of Integration on the QCM Measurement ...... 62 5.4.3 Comparing QCM and Spring-based Microtribometry ...... 63 5.4.4 Effects of Other Experimental Variables ...... 64

5.5 Discussion ...... 66 5.6 Conclusions ...... 70

6 HIGH-FORCE AFM PROBES ...... 72

6.1 AFM at the Macroscale: Methods to Fabricate and Calibrate Probes for Millinewton Force Measurements ...... 72

6.1.1 Abstract ...... 72 6.1.2 Introduction ...... 73 6.1.3 Methods and Materials ...... 75

6.1.3.1 Preparation of High-force AFM Cantilevers ...... 75 6.1.3.2 Quantifying Cantilever Flexural Stiffness: The Direct Calibration Method ...... 77 6.1.3.3 Experimental Validation of Direct Calibration ...... 79 6.1.3.4 Experimental Application ...... 80

6.1.4 Results ...... 81

6.1.4.1 Validation of the Direct Calibration Method ...... 81

viii 6.1.4.2 Calibration of High-force Cantilevers with Colloidal Probes ...... 83 6.1.4.3 High-force AFM ...... 87 6.1.4.4 Lateral Force Sensitivity ...... 88

6.1.5 Discussion ...... 91 6.1.6 Conclusions ...... 94

6.2 New Methods for AFM Lateral Force Calibration ...... 94

7 FIRST EVIDENCE OF CONNECTING SCALES ...... 103

8 CONCLUSIONS AND FUTURE WORK ...... 105

REFERENCES ...... 106

Appendix

PERMISSIONS ...... 121

ix LIST OF TABLES

Table 4.1 Summary of the ten probes that were assembled and calibrated for this study. The uncertainty is listed in parentheses and represents the Monte-Carlo simulated error that stems from the three independent repeats of the calibration procedure...... 36

Table 5.1 Summary of all linear regressions and associated standard errors for the measurements in this study. The fits are based on all data shown in Figure 5.5...... 64

Table 6.1 Measured properties of the seven cantilevers used in this study (No. 1- 7). Length measurements were made in an optical microscope with a resolution of 0.5 μm. The measurements were fit to Euler beam theory (Eq. 6.1) to quantify Young’s modulus, which was subsequently used to back-solve for the theoretical stiffness of each cantilever. The best- fit to modulus was 120±32 GPa; our uncertainty analysis showed that ~90% of this error is attributable to uncertainty in the thickness measurement (~5% due to uncertainty in length and ~5% due to uncertainty in the calibrated spring constant). This modulus error was subsequently propagated into the theoretical stiffness calculation...... 85

Table 6.2 List of constants used for Eqs. 6.2 and 6.3 to obtain the beam stiffness of each configuration of the calibrating device...... 98

Table 6.3 Summary of stiffness calibration of one integrated and four colloidal probes with the reference lever method. The reference was set to 5 N/m and the error is calculated through propagation of uncertainty coming from the measurement and the error in the reference...... 102

x LIST OF FIGURES

Figure 1.1: Experimental microscale and macroscale friction coefficient values summarized from reference [36]. The values which have been obtained for the different normal load ranges differ due to available tip materials, scanning speed and testing techniques’ normal load limitations. It is important to notice that the friction coefficient varies by orders of magnitude when the load range or velocity is changed...... 5

Figure 3.1 The mechanics of our topographical difference method using a representative pair of worn and unworn surface scans. (a) Topography of a steel surface (Ra = 84 nm) before a test. (b) The same surface after 100 sliding cycles against a 6.35 mm diameter steel pin at 2 N under lubricated conditions. The reference regions, shown in grey, have been used to establish a common reference between surfaces. (c) Average height (average of each 236 µm long vertical column) versus position for the worn and unworn surface. Material addition and removal are indicated with yellow and blue, respectively...... 16

Figure 3.2 Wear volume uncertainty quantification. (a) A sample was mounted to a high-resolution piezo positioning stage, placed under a SWLI objective, and scanned after each of a series of controlled displacements from 0 to 200 µm. (b) Raw topography scans are shown for 0, 2, 18, and 180 µm displacements for the roughest surface in the study (Ra = 216 nm). (c) Two-dimensional averages (the “front-view” arithmetic averages) of the surfaces before and after a 2 µm displacement. Topographic subtraction detects material removal and addition (despite the absence of both) due to the repositioning error. While the net difference in the reference regions is zero, topographic subtraction detects a 0.85 µm2 difference, which is the wear measurement uncertainty for this set of conditions...... 18

xi Figure 3.3 In-situ wear measurements experimental setup. (a) A microtribometer consisting of a cantilever-based load cell and two piezo stages used for loading and actuation. The steel surface is mounted to a Maxwell kinematic coupling, which relocates the ball relative to the wear track following interruption with a repositioning error of ~50 nm. (b) The steel counterface and the top half of the kinematic coupling were positioned beneath the SWLI using edge alignment rather than a second kinematic coupling to achieve repositioning errors on the order of 1-5 µm during profilometry...... 19

Figure 3.4 (a) The error per pixel plotted as a function of post-hoc repositioning coordinate for the 10-cycle wear experiment reported in the results section. The origin represents the original uncorrected topographical difference measurement. The repositioning error is the vector of the coordinate that minimizes the error. The vector itself is used to realign the surfaces and improve the topographical difference measurement. In this case, the repositioning error was (1, 0) µm and realignment reduced the error per pixel from 0.8 to 0.4 nm. (b) Topographical differences between the unworn and worn (10-cycles; see Figure 3.7) surface profiles before (upper) and after (lower) realignment. Actual locations of asperity scale wear emerge clearly once repositioning errors were corrected...... 21

Figure 3.5 Wear volume uncertainty as a function of repositioning error (sample displacement) and surface roughness for the three surfaces shown. The uncertainty is the magnitude of the volume difference per unit area of the scan window and represents the height uncertainty at each pixel. The error bars are the standard deviation of 5 repeat measurements at random locations of the same surface. The solid lines represent logistic function fits to the experimental data...... 23

Figure 3.6 (a) Topography of the unworn surface. (b) Wear volume as a function of sliding cycles for a lubricated steel on steel reciprocating test (blue data points). The error bars represent the uncertainty in volume due to the measured repositioning error. The inset shows the steady-state wear region (70-500 cycles). The wear measurements following post- hoc repositioning error correction are shown in red...... 25

xii Figure 3.7 Topographical subtractions characterizing wear after 10, 20, 30, and 500 sliding cycles without (a) and with (b) post-hoc correction of repositioning errors. The repositioning error is shown in the bottom right of each corrected subtraction image. Whereas worn volumes can be detected with remarkable resolution under typical repositioning errors (~1-5 µm), locating asperity scale wear requires either precise mechanical repositioning or post-hoc correction...... 27

Figure 3.8 Corrected topographical subtractions mapped onto the unworn surface topography for various distances of wear testing. The distribution of peak heights is shown on a gray scale and wear is showed in solid black; wear is defined here by loss >50 nm. Two regions and asperities of interest are highlighted for further analysis...... 28

Figure 3.9 Topographical subtractions of the regions surrounding the two asperities highlighted in Figure 3.8...... 29

Figure 3.10 Wear volume versus number of wear cycles for the orange highlighted regions in Figure 3.9. The trend at the asperity scale is generally the same as that observed for the entire observation window. The maximum wear volumes reached are 6×10-9 mm3 and 1.2×10-9 mm3 for asperities 1 and 2, respectively...... 30

Figure 4.1 Custom microtribometer designed and fabricated for use in this project. This tribometer has been designed for speeds down to 100 nm/s and has achieved a friction force noise floor as low as 200 nN in the latest iteration. The normal and lateral actuation are provided by high-precision piezo positioning stages. (a) Zoom-in of the load cell with an AFM chip adhered to the tip of the flexible cantilever; capacitance probes are used to sense the deflection of the steel beam. The single-crystal MoS2 countersurface is adhered to the one of the electrodes of a quartz crystal microbalance (QCM) to study speed effects. (b) Live camera view of the AFM chip and probe. (c) Live camera view of custom probe comprising a 27 μm diameter colloid mounted to a commercial AFM cantilever. *The AFM chip is not necessary for the studies in this thesis, and the colloid is mounted directly to the steel beam...... 35

Figure 4.2 Friction versus normal load for gold tested with colloidal probes of different material and diameter (listed in legend). Each gold-ball pair was tested at six different locations in an increasing and decreasing normal load sequence. The points of higher density between 0 and 0.5 mN were used for friction coefficient quantification, which is shown later...... 38

xiii Figure 4.3 Friction versus normal load for single-crystal MoS2 tested with colloidal probes of different material and diameter (listed in legend). Each gold-ball pair was tested at six different locations in an increasing and decreasing normal load sequence. The points of higher density between 0 and 0.5 mN were used for friction coefficient fitting, which is shown later...... 39

Figure 4.4 Friction versus normal load for single-crystal MoS2 between 0 and 0.5 mN of Normal Load...... 40

Figure 4.5 Example of wear occurring at a critical load for 22 µm Alumina probe on single-crystal MoS2. While most repeats exhibited no sudden changes in their friction (blue arrows), one of the repeats demonstrated a sudden and lasting increase in friction (red arrows) which has previously been correlated with wear of the single-crystal MoS2. Only 5 out of the 60 measurements displayed this behavior, and were excluded from the fitting of friction coefficient...... 41

Figure 4.6 Friction coefficient for 4 materials (Steel, Alumina, Silica, Glass) and different ball diameters (listed above each material) in contact with gold and single-crystal MoS2; a total of 10 different balls were tested against two different materials. The friction coefficients of gold are almost invariably an order of magnitude higher than those of MoS2. What is more interesting however is that there were not any strong evidence of the friction coefficient being related to the probe diameter. The error bars represent the standard error in the linear regression...... 42

Figure 4.7 The friction force at 0 nN according to the linear regression that was fit. Surprisingly, this property of the results not only correlates between gold and MoS2, but is also usually almost equal in magnitude. The error bars represent the standard error in the linear regression...... 43

Figure 4.8 Simulation of 1 µm2 contact areas for two different colloids’ topographies. The alumina contact (right) was measured to have three times higher friction than the steel contact (left). This is evidence for the dependence of friction on the number of asperities rather than the contact size...... 44

xiv Figure 5.1 A: The benchmark Quartz Crystal Microbalance paired with a Hysitron capacitive normal force transducer. (i) Side view of an opening in the chamber that contains the QCM. (ii) The normal load transducer positioned directly above the QCM and in the middle of the chamber. (iii) Live camera view of the 50 µm alumina colloid before contact with the gold surface of the QCM. B: The integration of the QCM with a spring-based microtribometer. The microtribometer comprises piezo positioning stages for lateral actuation of the counterface and vertical positioning of the measurement probe. The counterface is the same QCM chip used in the setup on the left. (iv) The capacitance probes are used to measure the deflection of a steel cantilever both vertically and laterally upon force application. (v) A 50 µm alumina colloid is glued to the end of the cantilever and is shown just before the contact with the single-crystal MoS2...... 53

Figure 5.2 (a) The quartz crystal resonates at ~5 MHz and with a track length of about 20 nm, which corresponds to an average sliding velocity of up to 0.2 m/s. The contact diameter at 1 mN is approximately 2 µm, which is orders of magnitude larger than the track length. (b) With the QCM turned off, the lateral nanopositioning stage reciprocated at 25 µm/s over a 25 µm track length. The microtribometer speeds were 5 orders of magnitude slower than those of the QCM and the tracks were 3 orders of magnitude larger. (c) Data from a single measurement of friction forces of gold at a normal load of 0.2 mN with a QCM. The slope of the energy loss vs. amplitude trend during full slip is used to determine the kinetic friction force [102]. (d) Normal force and “friction loop” raw data for i-µT of alumina on MoS2. For the 2 µm analysis window, sliding at a normal force of 15.6±1.8 µN produced 5.8±0.8 µN of friction force...... 54

xv Figure 5.3 (a,b) QCM friction force measurements for 50 m diameter alumina spheres against gold and MoS2. The error bars reflect statistical uncertainty in the fit of energy versus amplitude as illustrated in Figure 5.2c. (c,d) QCM contact area measurements for gold and MoS2; the orange subset of data points for gold represents one of the four locations that were tested and exhibited a consistent but offset trend. In fact this is one of the few cases we observed the expected non-zero contact are at zero load. (e,f) QCM shear and normal pressure measurements based on the previous graphs. The orange data points for gold align with the common shear pressure coefficient after their larger areas are taken in account together with the corresponding larger friction forces. The purple data points for MoS2 are derived from normal loads between 1 and 3 mN, which are not part of the fit and are not shown in (b) and (d), but follow the trend of the rest of the data...... 61

Figure 5.4 Comparing the benchmark QCM data (light colors) to i-QCM data (symbols with black outlines). The friction measurements were not strongly affected by the mechanical attributes of the measurement system for either material pair...... 63

Figure 5.6 Effect of colloid diameter on friction measurements. (a) The alumina- gold friction measurements turn out to be independent of testing condition, testing method, and colloid size. (b) The larger diameter alumina-MoS2 measurements, while producing the same friction coefficient, showed a slight increase in overall friction force. (c,d) Shear strength versus pressure for all measurements in the study; in every case, stress calculations were based on contact stiffness measurements made with the QCM under nominally slip-free conditions. The pressure-based analysis decreases the deviations for both materials, demonstrates shear strengthening with load, and showcases the benefits of method integration...... 65

xvi Figure 6.1 Top: Setup for assembly of custom AFM colloidal probes. A high- resolution 3-axis manual positioning stage controls an ultra-sharp (<1 μm) tungsten needle. The colloid (50 μm diameter alumina in this example) attaches to the tip of the needle with the help of electrostatic forces. The tipless AFM chip sits on a 2-axis manual stage for positioning in the field of view of the microscopes. The two microscopes provide a simultaneous top and front views of the process to assure perfect positioning. (1-7): Profile views of the seven cantilevers prepared for this study. The distance of the steel colloid from the free end of 40 N/m cantilevers decreases with increased cantilever number. These optical images were subsequently used for determining the effective length, as illustrated in (1), following mounting of steel colloids; (8) an illustrative plan-view of an unmodified cantilever used to quantify the lateral dimensions (length and width) the cantilever. All eight images have the same scale...... 76

Figure 6.2 (a) Schematic of the calibration setup; (b) close-up view of the contact between the AFM probe and the microbalance; (c) in-situ view of the approach to contact; (d) Side-view of the silicon cantilever with a steel sphere mounted and loaded against the microbalance to obtain load- displacement relationships...... 79

Figure 6.3 (a) Stiffness curves for tip-less CLFC-NOBO reference cantilevers from a representative chip. Black dots represent measured data; black lines represent fits to measured data and are labeled by the corresponding slope; red lines represent the manufacturer-reported values from thermal-tuned calibration. Each curve represents the stiffness of a different cantilever on the same chip. The inset shows an in-situ image of the long beam calibration measurement. (b) Results of direct calibration (shaded bars) with the long, medium, and short pre- calibrated cantilevers on each of four chips: nominal spring constants were 0.16, 1.3, and 10.4 N/m, respectively. Outlines (in red) represent the reporter value from pre-calibration and nominal values are denoted for each cantilever family by dashed lines. Error bars represent the uncertainties in direct calibration constants...... 82

xvii Figure 6.4 Flexural stiffness from direct calibration for each of seven cantilevers before and after mounting the steel colloid. Error bars represent the statistical standard deviation in the slope of each independent calibration curve. Four independent repeat measurements were made for beams 1 and 7 to test for other error sources (e.g. lab temperature, chip placement in the holder, user repeatability, etc.). The consistency of the repeat results indicate that the experimental uncertainty is a reasonably comprehensive predictor of overall measurement error...... 84

Figure 6.5 (a) The measured stiffness for all eight beams plotted versus effective beam length, which is defined as the distance between the fixed end and the center of the colloid as illustrated with beam 1 in the inset. The results indicate strong agreement with the theoretical k ~ length-3. (b) A comparison between measured stiffness using and stiffness calculated based on the dimensions obtained from side-view micrographs. The error bars in the measured stiffness represent the combined error in the regression and the error in the assembly’s compliance while the error bars in the calculated stiffness reflect the propagation of errors from individual measurements into Eq. 6.1 (~90% from uncertainty in the thickness measurement)...... 86

Figure 6.6 (a) Indentation curves for cantilevers 1 and 7. The approach and retract curve show substantial snap-in and snap-out due to adhesive forces. The smooth sloped part of the loading curve was used to quantify the PSD sensitivity. (b) The same indentation curves but with PSD voltages converted into quantitative forces. We achieved forces well above our 1 mN target using setpoint voltages well-below the limit (3 of 20 V) of a commercial AFM...... 88

Figure 6.7 (a) Friction versus position for set-point voltages from 0.5-8 V for a steel colloid against single-crystal MoS2 using Lever 7, the stiffest cantilever in the study (k = 10,900 N/m). (b) Lateral signal (friction loop half-width) versus normal force set-point for a stiff (Lever 7, k = 10,900 N/m) and a soft (Lever 3, k = 122 N/m) cantilever. The application of friction to these beams produced similar voltage responses despite enormous differences in normal stiffness, torsional stiffness, and applied forces...... 89

xviii Figure 6.8 Lateral force (per the extended wedge method) versus normal force for the stiffest (k = 10,900 N/m) and a soft (k = 122 N/m) cantilevers in the study using two different steel colloid-attached probes and a MoS2 sample. The relationship between friction force and normal force was obviously sub-linear at lower forces with the soft cantilever and closer to linear at higher forces with the stiff cantilever...... 91

Figure 6.9 (a) Reference lever calibration device consisting of two steel beams with variable length. The micrometer stage controls the effective length of these beams, while the high-precision locking collets provide fixed support in all directions. To expand the scope of permissible beam lengths beyond the range of the micrometer, the locking collets can be placed at 5 different reference positions using the key (Position 3). When the device is placed in an AFM, the colloidal probes are loaded against one of the silicon indentation platforms glued at the ends of the beams. Direct calibration of the two reference levers follows a two-step process involving (1) direct calibration of a capacitance-probe-based indentation device with the methods described in Chapter 4.1 and (2) indentation measurements to calibrate the reference lever calibration device (b)...... 96

Figure 6.10 Example of the thin beam’s calibration when locked in its shortest configuration (Position 1). The uncertainty (expressed as a fraction of the stiffness) is calculated through a Monte Carlo simulation to account for the compound effect of each calibration and the uncertainty in the calibrating device...... 97

Figure 6.11 Available stiffness at each configuration of the calibration device...... 99

Figure 6.12 Results of independently targeted values through the fits in Eqs. 6.3 and 6.4 and the constants in Table 6.3. The results show that the predicted error range matches the distribution of independent attempts which validates the usability of the device...... 99

Figure 6.13 Theoretical error in the calibration of an AFM cantilever as a function of the uncertainty in the calibrating device. The message of this graph is that errors in the AFM lever calibration are mitigated by the use of the softest reference beam possible...... 100

xix Figure 6.14 Use of the reference spring device to calibrate lateral forces of AFM cantilevers. The AFM beam is oscillated laterally without slipping at the interface. First, we use the existing standard practice to obtain the lateral sensitivity; i.e. the AFM is loaded against a nominally rigid surface (top left) and the lateral sensitivity (V/nm) is obtained from the slope of the corresponding data in the graph. The lateral sensitivity procedure is repeated against the calibrated lateral force reference spring (bottom left) to obtain the combined lateral sensitivity (one known reference spring and one unknown spring in series). The lateral stiffness of the unknown AFM cantilever is: 푘푐푎푛푡𝑖푙푒푣푒푟, 푙푎푡푒푟푎푙 = 푘푠푝푟𝑖푛푔, 푟푒푓푒푟푒푛푐푒 ∙ 푆푅푆푆 − 1. Using measured sensitivities (right) and the pre-calibrated reference spring stiffness (5 N/m), the calibrated AFM lateral stiffness of this cantilever is: 푘푐푎푛푡𝑖푙푒푣푒푟 = 73.6 ± 3.34푁푚 and the lateral force calibration constant is: 퐶퐿퐹 = 푘푐푎푛푡𝑖푙푒푣푒푟푆푅 = 1.58 ± 0.071 휇푁/푉...... 101

Figure 6.15 The reference spring method can be used in the same manner for normal force calibration. In this case instead of completing lateral oscillations, the AFM chip is actuated vertically, to obtain force – distance curves. The red repeats represent indenting on hard, while the blue ones represent indenting on the reference beam. The loading portions of the curves are used in the calculation and statistics of the AFM cantilever stiffness...... 102

Figure 7.1 Friction force results over six orders of magnitude coming from both AFM and microtribometry of single-crystal MoS2. Data from literature is shown in yellow squares: nanoscale friction coefficients are reproduced from [126] for 40% relative humidity, while the data point for macroscale was given by Saito et al. [135]; since the macroscale data for this material is limited, the general range of macroscale friction coefficients is given with a purple rectangle. The dark blue triangles represent AFM results from an integrated tip which show good agreement with the literature data. The rest of the triangles represent colloidal probe AFM. The circle data points represent the microtribometry data for all ten colloidal probes that is discussed previously. The coefficients used for the guiding lines on the graph are derived from the equation that is given; the adhesive force is assumed to be zero in all cases, except when shown to be nonzero for the friction coefficient of 0.01...... 104

xx ABSTRACT

This dissertation describes the development and application of new experimental tools designed to close the gap between industrially-relevant friction and wear and their molecular origins.

Industrially relevant friction and wear account for a significant fraction of global energy consumption and, as a result, have massive economic consequences. The complexities, uncertainties, and inaccessibility of these tribological contacts makes fundamental studies of friction and wear inherently difficult. One approach is to isolate and a study a single-asperity interaction under well-controlled nanotribological conditions. Although nanotribological interfaces have given important insights into the physics of friction, it remains unclear how their responses relate to those of industrially-relevant multi-asperity ensembles.

This dissertation describes the development and application of new experimental methods to address the most important experimental barriers, which include: (1) controlling load, probe material and geometry, and sliding speed; (2) quantifying interfacial forces at conditions that vary by many orders of magnitude within a single interface; (3) controlling the transition between interfacial slip and wear. These new methods were used to construct tribological maps that bridge the gaps in load, size, speed, pressure, and number of asperities in contact between the nanoscale and the macroscale. The first challenge was identifying a model interface capable of surviving a wide spectrum of frictional conditions without wear – we

xxi identified single-crystal MoS2. Using this model material system, friction was mapped for varying loads (10 nN to 10 mN), contact sizes (10 nm to 100 µm diameters), and sliding speeds (25 µm/s to 100 mm/s) to bridge the gap between the nanoscale and the macroscale. Without the confounding effects of wear, the results showed that atomic scale friction is likely connected to macroscale friction. Analysis of the contact areas suggest that friction is sensitive to both the contact area and the way in which contact forces are distributed between asperities. Future work will be needed to elaborate this connection and the apparently dominant effect of multi-asperity versus single asperity interactions. The major contribution of this thesis is twofold: (1) the discoveries that provide an elucidation of macroscale friction and wear as a function of atomic-scale phenomena and (2) the various technologies that were developed along the way that now enrich the family of available tribology, and not only, research methods.

xxii Chapter 1

INTRODUCTION

Friction is part of every mechanical contact and very often plays a key role in the performance of engineered devices – minimizing it generally results in more efficient systems through a decrease in the dissipated energy, while maximizing it in certain applications improves the functional reliability by securing the intended contacts. Friction and wear manifest themselves as energy dissipation mechanisms when two bodies are slid against each other and that is why they are directly linked to the operational limits, durability, efficiency and control of mechanical systems. In contrast to many other mechanical properties of a design, friction cannot be derived from first laws, it is not an intrinsic physical property and is strongly dependent on specific conditions such as surface features, chemical environment and sliding history [1–5]. Therefore, engineers are limited in their ability to use fundamental scientific knowledge when attempting to improve on any design that involves contacting bodies. This thesis aims at bridging the gap between the existing scientific knowledge about the molecular origins of friction and practical macrocontact size scales used by most engineers. Historically, people have harnessed friction and its effects to facilitate their lives in multiple ways: using rolling elements, like wheels and bearings, lowered friction and made it easy to transport heavy objects, while producing frictional heat resulted in sparks that started fires. These early practical tribological solutions gave rise to many developments that framed the way humankind progressed over the

1 centuries. Scientifically, friction theories developed between the 15th and 19th century were aimed at understanding the relationship between friction forces, normal load and contact area [6]. It was proven experimentally that friction force is proportional to normal load and independent of apparent area of contact [7] and it was also believed that roughness has a major effect on friction. Technological advances in the 20th century, namely the findings in the areas of surface chemistry and electrical engineering, allowed scientists to hypothesize and prove that friction force is proportional to real area of contact. By measuring electrical conductivity of contacts it was shown that the real area of contact is much smaller than the apparent area of contact and that most surfaces come into contact only at very small bumps on the surface called asperities. Deforming these asperities results in plastic deformation and increased adhesion which leads to higher friction. All these findings have motivated researchers in the field of tribology to conduct fundamental studies and show that friction has its roots at scales that are below the observable limits of optical devices

[8–11]. With the invention of the Atomic Force Microscope (AFM) tribologists found a way to investigate the contact between single asperities, which are the building blocks of any macro-scale contact. Friction Force Microscopy (FFM) is a branch of studies performed with AFMs which focuses on fundamental experimental research of friction. It allowed for isolating specific contributors to friction and opened the door for exploring the molecular origins of friction [12–16] and its relation to adhesion [10,

17–23]. Advances in computing technology let scientist conduct Molecular Dynamics (MD) simulations which have explained some of the observations made with AFM [24–28]. Friction coefficients predicted by MD simulations have previously been

2 found to be in agreement with experiments. For example, both MD simulations and experiments showed that the friction coefficient between single-walled carbon nanotubes and diamond surfaces was either 0.79 or 0.09 depending on the orientation of the nanotubes [29]. Despite these positive examples of agreement between MD simulations and experiments, there are still obstacles to making MD a driving research tool in tribology. The lack of accurate interatomic potentials is a problem that is valid for any MD simulation and also holds in the tribological simulations. Larger systems can be modeled by increasing the number of processors involved, and new technologies in computer engineering will certainly allow that in the future. However, larger computers cannot solve the issue of achieving a realistically small time scale that accurately represents a real dynamic contact [1]. Scientists continue to explore ways to overcome this issue and one of the ways that has been proposed is using multiple simulations of the same event [30]. Continuing efforts of this kind have the potential of eventually matching the time and velocity scales of experimental studies.

Although the molecular origins of friction have been investigated extensively using FFM techniques and MD simulations, the results of these studies have rarely been applied to practical size-scale engineering systems. This is partially due to the fact that a comprehensive experimental link between nano-scale and macro scale friction has not yet been established [31]. Since macro scale friction is usually guided by a complex system of phenomena, a phenomenological approach needs to be taken in order to find the effect of each nano-scale principle [32]. Scaling rules have previously been proposed but their experimental validation has been challenging due to the limitations of current testing techniques.

3 Despite being difficult to test, comprehensive rules on how friction scales have previously been proposed. Bhushan et al. proposes a theory where friction force is a function of whether the contact includes a single or multiple asperities and whether they are elastically or plastically deformed [33]. There are several critical elements to this theory. First, friction contributors are separated at the nano-scale level and analyzed one by one; these include an adhesion, plowing, ratchet and deformation components each of which is being investigated. Secondly, rough surface topography has been represented mathematically in two ways: using an empirical power-rule and a fractal model [34]. Third, to find the dependence of friction coefficient on debris formation and adhesion between them and the surfaces, a statistical model was established evaluating the resulting scaling effects [35]. To prove the rules set forth by this comprehensive model, Bhushan et al. used friction data collected through FFM [36]. The collection of data is summarized in the Figure 1.1. The experimental limitation become obvious by comparing the ranges in which the different tests were performed and the fact that the tip materials and scanning speeds are also not consistent. As a result, the variation of results reaches orders of magnitude in some cases. It is also evident that none of the ranges that have been investigated overlap in terms of normal force.

4 0.7 0.6 0.5 0.4 0.3 0.2

Friction Friction Coefficient 0.1 0 AFM NTT/AFM AFM NTT/AFM AFM Silicon Macroscale Macroscale Diamond Tip Diamond Tip Diamond Tip Diamond Tip Nitride Tip Friction Friction (0.2-15 µN) (1-15 µN) (20-35 µN) (20-50 µN) (50-150 µN) (0.1 N) (1 N) Silicon (111) Dry-oxidized SiO3 Polished natural diamond

Experimental microscale and macroscale friction coefficient values [36]. The values which have been obtained for the different normal load ranges differ due to available tip materials, scanning speed and testing techniques’ normal load limitations. It is important to notice that the friction coefficient varies by orders of magnitude when the load range or velocity is changed. Further investigations on how friction and adhesion scale have been performed by Liu et al. using an AFM and a microtriboapparatus for two distinct normal force ranges [37]. It becomes evident that there is a sharp discontinuity in the friction coefficient for identical materials when tested with different experimental setups. One explanation for that could be the different sliding speeds, but another test the authors performed refutes that since there is no systematic friction dependence on velocity.

5 The authors suggest that the reason for this discrepancy is the presence of higher plastic deformation since the contact stresses approach the hardness of the tested materials at the higher load. On the other end of the spectrum are macro scale experiments like the ones conducted by Tajdari et al. who used a very high load system to test the change in friction coefficient from primarily being guided by adhesion theory to one which follows the junction growth theory [38]. The junction theory suggests that there are three distinct regimes of friction where at a certain point the whole apparent area of contact becomes the real area of contact and extremely high friction coefficients can be measured [39]. Being able to investigate phenomena on this scale is one of the benefits of having a tool that can continuously measure a wide range of loads. By doing friction force measurements at the nano, micro and macro scales, Achanta et al. concluded that surface roughness and variation of adhesive force is the biggest obstacle for extrapolating single-asperity data to a multi-asperity contact system [40]. They proposed a simple model that tries to explain the nature of localized friction in a single and multi-asperity arrangements. Their results, however, are clustered around four different normal loads: 20 nN, 0.5 mN, 20 mN and 1 N. It is evident that even though they used three different machines with various force ranges to conduct their tests, there is a four-fold jump in the orders of magnitude when going from nano to micro scales. Having a single machine that can study a wider range of forces will overcome the problem of discontinuous normal forces and having to match the test conditions in each lab equipment. Liu et al. proposed a model relating friction coefficient to penetration depth as a result of plowing (plastic deformation at the contact) [41]. They concluded that

6 plowing has a significant role in the development of friction forces on hard surfaces and is analytically related to friction coefficient. The model agreed with experimental results in the range of 0.1-10 mN. However, as they acknowledge, conducting more tests at lower normal loads are required to fully confirm their model; they predict very sharp rise in friction coefficient as the penetration depth gets below a certain value. The effect of tip radius, and subsequently contact size, at different length scales has been another issue of major focus for tribologists. It was discovered that friction coefficient shows trends to increase proportionally with probe radius until a saturated value is reached at the microscale [42]. Similar trends were observed by Bhushan et al, who added another level of complexity and explore the effect of humidity [43, 44]. Their conclusions are that at low humidities the increase in friction coefficient with tip radius is due to increased contact area, while at higher humidities the meniscus effect reinforces that effect. These studies both confirm that friction coefficient is roughly an order of magnitude higher at the microscale than the ones at the nanoscale. The lack of testing techniques is further exhibited when various contact radii need to be attained to prove a theory. A model for dislocation-assisted (microslip) sliding, its effects on friction and how it scales has been also proposed. Hurtado et al. came up with a model where microslip is considered to be the most energy-profitable mechanism when two asperities slide against each other for certain contact sizes [45, 46]. Their model analyzes single-dislocation-assisted slip in its first part, and a transition to multiple-dislocation-cooperated slip in its second part. Unfortunately, there are no available experimental results to fully confirm or refute the model. Bhushan et al. [47] combines the dislocation-assisted sliding model with strain-

7 gradient plasticity to come up with a simpler scaling rule. Their analytical results agree well with experimental data but once again the limitations for complete validation are clearly seen. There is a jump in the contact radius of several orders of magnitude which does not fully confirm the trends described by the model. To validate both models, an experimental technique which allows the variation of contact radius while the rest of the conditions stay unchanged is needed. Macroscale friction is believed to be a compound effect of several components: adhesive forces, plowing (plastic deformations) of the surface, roughness effects and debris formation [34]. Research suggests that materials such as PTFE and MoS2 [48, 49] maintain their low friction properties because the effects of roughness, plowing and debris formation are very low for these materials; as a result interfacial sliding at weak interfaces is the driving mechanism for their low friction coefficient. The molecular origins of interfacial sliding makes these materials suitable for conducting multiscale experiments to examine how fundamental theories can be applied at practical size scales. Thus far, the friction transitions between the nano and macro scales are often put in the framework of controlled normal load because it is a convenient independent variable in most experimental setups. It is evident that what often arises as a challenge for connecting the fundamental and practical regimes of friction, are the effects of contact size, velocity and wear. The following chapters will explore the interplay of these effects in a systematic and controlled way, and will push the boundaries of existing methods in tribology until the load – contact size – velocity – wear space is populated with enough observations to allow some general conclusions about the atomic origins of macro-scale friction to be made.

8 Chapter 2

PROBLEM STATEMENT AND OBJECTIVES

Financially, it has been estimated that the loss to US industries due to avoidable friction and wear is 1-2 % of the GDP [50]. Since friction and wear are inevitable results of almost any contact, expanding the knowledge base can have a cascade effect on many other fields of engineering which have previously been limited by available tribological solutions. Despite the vast amount of knowledge focused around the molecular origins of friction, it is still unknown and hasn’t been shown how atomic-scale friction scales continuously to macroscale friction. At the nano level, adhesive theory explains the origins of friction of any contact, while at the macroscale the effects of surface roughness, plowing and wear further increase the friction coefficient. So far, the transition between nanoscale and macroscale friction has not been fully investigated due to the lack of available experimental equipment in the range between a typical FFM and microtribometry test. Since nano friction coefficients are usually an order of magnitude lower than their macroscale counterparts, being able to show how different factors contribute continuously will provide engineers with the ability to design more efficient and durable systems. This dissertation will:

1. Trace the evolution of single-asperity wear within a multi-asperity steel-on-steel sliding.

9 2. Use wear mapping to visualize the precise area of contact in a multi- asperity interface.

3. Measure the real contact area of a multi-asperity contact in-situ for the first time. As a result, the independent effects of contact shear strength and size on friction are discussed.

4. Explore the speed dependence of friction on velocity for two model materials: gold and single-crystal MoS2.

5. Perform an extensive study of how friction and wear develop from 10 µN to 5 mN for 4 different materials and 4 different probe sizes.

6. Show the development of new AFM capabilities for high-force multi- asperity experiments.

7. Establish some of the first direct and traceable AFM normal and lateral force calibration techniques.

8. Present early evidence of low nanoscale friction (single-asperity) continuously transitioning to high macroscale (multi-asperity) friction.

Developing friction maps for certain materials continuously from the nanoscale to the macroscale in itself will be useful industrially for both product design and process development. The broader impact, however, is that the new testing equipment and procedures that are established will provide the basis for many scientists to conduct fundamental research. Isolating interfacial friction from other contributors to friction opens a new realm of possible for research work and investigating hypotheses that have previously been impossible to test. To support this notion, as pointed out in the introduction, a lot of the research work in tribology in the 20th century was profoundly influenced by the availability of new tools such as accurate ohmmeters and AFMs.

10 Chapter 3

QUANTIFYING, LOCATING, AND FOLLOWING ASPERITY SCALE WEAR WITHIN MACROSCALE CONTACT AREAS

3.1 Abstract Wear tests are inherently destructive and surfaces of interest are often consumed before enough wear has accumulated to be reliably quantified with standard gravimetric, geometric, or profilometric methods. Elucidating the subtler features of incipient wear requires improvements in our ability to quantify, locate, and follow wear at the asperity scale. The topographical difference method provides a possible solution but its practical limitation is the difficulty in perfectly repositioning samples and the effect of imperfect repositioning on the wear measurement. This paper aims to quantify the detection limits of the topographical difference method under the conditions of typical repositioning errors, surface topographies, and measurement approaches. With repositioning errors on the order of 2 µm, the topographical difference method reliably detected worn volumes as small as 10-8 mm3, which rivals the most sensitive macroscale measurements reported in the literature. Repositioning errors of that magnitude proved problematic for resolving locations of asperity scale wear, but they were effectively removed with a simple post-hoc image realignment routine. After post-hoc realignment, topographical difference measurements allowed us to map asperity scale wear onto features of the unworn surface profile, observe the gradual removal of individual asperities.

11 3.2 Introduction The sensitivity of wear volume measurements is critical in tribology whether the objective is to quantify a wear rate for engineering purposes or to elucidate the wear process for scientific purposes. Gravimetric measurements are among the most common. A high-quality laboratory scale can resolve worn masses as small as 50 µg, which corresponds to a volume resolution of 5×10-3 mm3 for steel. Because this volume of material is typically many orders of magnitude larger than the asperity scale, the original wear surface must be consumed before the worn volume can be detected. Additionally, quantifying low and ultra-low wear rates gravimetrically can require extraordinarily long tests; for example, an ultra-low wear PEEK-PTFE composite required about a month of continuous testing at 50 mm/s and 250 N of normal load to reliably quantify steady state wear rates [51]. The radionuclide method can resolve 0.1 µg (~10-5 mm3 of steel) of wear [52], which reduces the burden for quantifying ultra-low wear rates, particularly in machines [53]. Calibrated optical microscopy can improve volume sensitivity further; for example, a flat 100 µm diameter wear scar on a 6 mm diameter sphere corresponds to a worn volume of 2×10- 6 mm3 [54]. Optical and stylus profilometry can improve wear sensitivity further still; Novak and Polcar demonstrated the ability to reliably resolve 10-6 mm3 of wear volume per millimeter of track using relatively smooth DLC coatings [55]. However, the need to interpolate or extrapolate surfaces when quantifying wear with single scan profilometry measurements like these introduces unquantifiable uncertainty and limits potential sensitivity [55].

These challenges of macroscale measurements have motivated scanning probe methods for fundamental wear studies [56–64]. Zhao and Perry used a sharp silicon nitride AFM tip to model a single-asperity interaction with PbS (100) in vacuum under

12 well-controlled experimental conditions and to simultaneously map the wear process [65]. Jacobs and Carpick used in-situ TEM to quantify the wear of Si tips against a diamond surface; they demonstrated that the tip was worn by gradual atomic attrition at a wear rate of k ~ 10-1 mm3/N·m [66]. Liao and Marks used similar methods and found that a commercial CoCrMo alloy from a replacement hip wore in a qualitatively similar manner against a Si tip at quantitatively similar wear rates [67]. The literature on atomic scale wear has clearly demonstrated an ability to isolate and evaluate wear of single-asperity interactions under precisely controlled conditions with unprecedented resolution on the order of 25×10-18 mm3 [68]. Despite the inherent advantages of scanning probe measurements, the unusually high wear rates they often produce create doubt about how well they represent the most relevant phenomena within the multi-asperity contacts they presume to model. As a result, there remains significant interest in probing asperity scale wear within macroscale contacts. Archard and Hirst pioneered the first attempts to resolve asperity scale wear within macroscale multi-asperity contacts [69]. They used interrupted profilometry at a fixed location to study how metal surfaces evolved during mild and severe wear. Gahlin and Jacobson developed a more modern topographic difference method to map local worn volumes at the asperity scale by subtracting aligned surface profiles made before and after wear testing [70]. Using contact mode AFM to measure and subtract the surface profiles before and after a test, they demonstrated an ability to resolve wear volumes as small as 4×10-8 mm3 in an

80x80 µm scan window. These subtraction-based methods achieve impressive resolution and offer an opportunity to study asperity scale wear within macroscale tribological contacts.

13 Interrupted profilometry measurements typically involve physical removal of the specimen, remote measurement, and subsequent replacement in the wear tester. Using fiducial marks for post-hoc profile alignment, Gahlin and Jacobson [70] and Furustig et al. [71] eliminated the need to mechanically align surfaces within the profilometer. However, their inability to realign the contact area within the tribology experiment precluded interrupted measurements of wear, and thus they limited their study to single-point ‘before-and-after’ measurements. Sawyer and Wahl [72] addressed this issue by physically coupling the wear tester with a non-contact profilometer. In their experiment, the reciprocating stage intermittently placed the sample beneath the profilometer for measurement after each cycle of wear testing. In this case, repositioning errors perpendicular to the sliding direction were constrained by the linear stage and were on the asperity scale. The reposition error in the actuation direction was far larger (~1-5 µm), but given the nature of the experiment, this error has no impact on contact realignment or the tribological environment. Thus, this stage- based repositioning approach mitigated the effects of test interruption on the tribological experiment, which enabled them to track the evolving wear surface throughout the experiment. In principle, the topographical difference method provides the means and sensitivity to enable asperity scale wear measurements within multi-asperity macroscale tribological contacts. Although the in-situ method described by Wahl and Sawyer [73] mitigated the effects of repositioning errors on the wear test itself, the relatively larger repositioning errors of profilometry replacement (in the actuation direction) limits wear sensitivity to a currently unknown degree; in their case, they demonstrated the ability to resolve wear volumes as small as 10-6 mm3. Our first aim

14 here was to determine the sensitivity offered by the topographical difference method when influenced by ordinary repositioning errors (~1-5 µm). To this end, we systematically quantified how the wear volume uncertainties (i.e. resolution) of topographical difference measurements depend on well-controlled repositioning errors for representative surfaces of varying roughness. Our second aim was to determine the extent to which these methods can be used to locate asperity scale wear and follow the subtler aspects of the macroscale wear process.

3.3 Methods

3.3.1 The Topographic Difference Method Raw surface topography measurements were made with a Veeco NT 9100 scanning white light interferometer (SWLI). While each measurement is automatically bias and tilt corrected per standard practice, topographic difference measurements require bias and tilt correction between surfaces. Generally speaking, the sensitivity of the topographical difference method is limited by the extent to which a common reference can be established over the same unchanged (i.e. no wear) regions of the same surface (i.e. no lateral difference in the positions of identical reference features). We used the outer 15% of each side of the scan (reference regions) for height and tilt alignment between all compared surfaces as illustrated in Figure 3.1b; a sensitivity analysis showed that 15% provided sufficient information for alignment while maximizing the areas available for the wear track. Aligned surfaces were then subtracted directly to obtain a difference map. The topographical difference between Figure 3.1a and 3.1b are shown in Figure 3.1c. Whereas differences sum to zero within the reference regions, the results show evidence of material removal near the

15 center and potential redistribution outside the wear track. This particular example represents a worn volume of 2×10-7 mm3.

3.3.2 Wear Volume Uncertainty Characterization To quantify the effect of repositioning error on measurement uncertainty, we quantified wear after moving an unworn sample (the only possible wear standard) by known distances (from 20 nm to 200 µm) with a precision nanopositioning stage (PI P-622.1CD, resolution of 0.7 nm) as illustrated in Figure 3.2. Any positive or negative

16 wear volume detected was treated as the measurement uncertainty for that set of conditions (the worn volume was exactly zero). Wear uncertainty was quantified as a function of repositioning error and surface roughness using three unworn surfaces of

Ra = 6, 53, and 216 nm; measurements were repeated at five random locations of each surface to obtain statistics. Samples were displaced perpendicular to the orientation of the roughness and the ‘wear track’; as Figure 3.2b illustrates, the reference regions used as the basis of subtraction comprise completely different regions of the sample when displacements exceeded the size of the reference regions (35 µm in this case). The texture, its orientation, the displacement direction, and the orientation of the reference regions were all chosen to maximize the effect of displacement on the measurement error. The effect of a 2 µm displacement on the topographical subtraction of the roughest surface is illustrated in Figure 3.2c.

3.3.3 Validation Experiment Interrupted wear measurements were made on a typical surface to demonstrate sensitivity, validate the uncertainty, and illustrate the general utility of the topographical subtraction method for studying the evolution of incipient asperity scale wear with common surface and tribological characterization equipment. The custom tribometer used for this test (Figure 3.3a) comprised a vertical nanopositioning stage for load application (PI Q-545.240, 8 N load capacity), a reciprocating lateral piezo stage (PI Q-545.240), and orthogonal capacitance probes (Lion Precision CPL290

18 C3S, ~1 nm resolution) to quantify normal and friction forces applied to a pre- calibrated cantilever. We used a Maxwell kinematic coupling (3 balls in 3 radiating vees) to relocate the sample between profilometry measurements – we have found that the repositioning error of this particular kinematic coupling is ~50 nm, which is below where we expect changes to individual asperity contacts.

Figure 3.3 In-situ wear measurements experimental setup. (a) A microtribometer consisting of a cantilever-based load cell and two piezo stages used for loading and actuation. The steel surface is mounted to a Maxwell kinematic coupling, which relocates the ball relative to the wear track following interruption with a repositioning error of ~50 nm. (b) The steel counterface and the top half of the kinematic coupling were positioned beneath the SWLI using edge alignment rather than a second kinematic coupling to achieve repositioning errors on the order of 1-5 µm during profilometry.

The probe was a 440c stainless steel ball (d = 6.4 mm, Ra = 100 nm). The sample was a 304 stainless steel flat with moderate roughness (Ra = 84 nm) and textural alignment in the sliding direction. This sample was rigidly mounted to the removable top half of the kinematic coupling. Prior to measurement, the sample was

19 measured with the interferometer by locating the flats of the kinematic coupling on the walls of an alignment jig by hand (Figure 3.3b). We intentionally avoided using a kinematic coupling for profilometry measurements with the aim of achieving 1-5 µm of repositioning error; recall that the ultimate aim was to demonstrate how typical repositioning errors of this scale affect the detection of wear via the topographical differences. Following initial surface measurement, the kinematic coupling was reassembled in the tribometer and the sample lubricated with 1 ml of a SMF Scientific SlickLubeTM (100 cp) bio-derived lubricant (this is the only lubricant used throughout this chapter), which was selected because, unlike other more model lubricants we tested, it promoted the subtle asperity scale wear processes we aimed to detect. Once lubricated, the sample and counterface were brought into contact with a force of 2 N and reciprocation was initiated at 10 mm/s on a 10 mm long track. For the first 100 cycles, the wear test was interrupted every 10 cycles and surface topography was measured following cleaning with hexane to remove the lubricant. The test was stopped every 100 cycles thereafter up to 500 total cycles.

3.3.4 Quantifying and Correcting Repositioning Error We quantified the repositioning error between profilometry measurements using a simple computational algorithm to determine the movement necessary to minimize the sum of the errors between each pixel pair in the reference regions; the total difference is zero by definition but there are differences between all pixel pairs due to a combination of SWLI and repositioning errors as Figure 3.2b illustrates. The wear sample profile was repositioned computationally by single pixel steps (0.5 µm/pixel) in concentric ‘circles’ of increasing radius until achieving an optimum.

20 Figure 3.4 illustrates the application of the procedure when applied to the 10-cycle wear measurement (see Figure 3.7). In this case, the repositioning error of (1,0) µm fell into our target range of 1-5 µm. Post-hoc realignment reduced the error per pixel (square root of SSE divided by the number of pixels in the reference regions) from 0.8 to 0.4 nm and, more importantly, resolved the specific locations of wear.

21 3.4 Results

3.4.1 Repositioning Uncertainty The mean wear depth detected for unworn surfaces is plotted on a per pixel basis versus displacement for representative surfaces of low (6 nm), middle (53 nm), and high (216 nm) roughness in Figure 3.5; error bars in this case represent the standard deviation of the five repeat measurements made at different locations of the same sample. The results contain several noteworthy features. First, the uncertainty increases with roughness at all repositioning errors. The minimum uncertainty, umin, at 0.1 nm reflects the contribution from uncertainty in the SWLI measurement itself; this value depends on roughness and any other variable that reduces the repeatability of the profile measurement. Second, the uncertainty tends toward another asymptote at infinite displacement, umax. This is understandable if one considers that an area of a surface with a repeating structure (e.g. a sine wave) looks like all other areas of the same surface; rougher surfaces tend to be less regular, which increases the effect of repositioning error on the measurement. Third, for typical tribological surfaces (Ra < 100 nm), typical repositioning errors (1-5 µm) increase the uncertainty in wear volume measurements from SWLI-based topographical subtraction by about 10x (from 0.1 to 1 nm per pixel).

The uncertainty results were fit to the following logistic function of the minimum and maximum roughness-dependent uncertainties (u), the repositioning error (d), and another roughness-dependent fitting parameter (A):

푢푚푖푛 푢 = 푢 −푢 푑 Eq. 3.1 1− 푚푎푥 푚푖푛∙ 푢푚푎푥 푑+퐴

Fitting each parameter to all three datasets gave: 푢푚𝑖푛 = 푅푎⁄1,700 + 0.12 푛푚

2 2 2 2 (R = 0.98); 푢푚푎푥 = 푅푎⁄280 푛푚 (R > 0.99); 퐴 = 27 푛푚 ∙ µ푚⁄푅푎 (R > 0.99). The best-fits are shown in Figure 3.5 and Eq. 8.1 can be used to estimate wear uncertainty for arbitrary roughness and repositioning error.

3.4.2 Validation Results Wear volume results for the interrupted lubricated steel-on-steel experiment are plotted versus sliding cycles in Figure 3.6. Based on the repositioning error of each measurement, which varied between 0 and 1.4 µm, the expected uncertainty in each

23 measurement (Eq. 1) varied between 0.2 and 1.1 nm per pixel (between 13×10-9 mm3 and 83×10-9 mm3). Despite the significant repositioning errors, topographic subtraction measurements remained sensitive to wear in the 10 cycle sliding increments used throughout run-in, which was characterized by a wear rate of 4×10-6 ∆푉 1 mm3/N·m (푘 = ( ) ∙ , where F is the applied force and L is the length of the ∆푁 2퐹퐿 observation window). The results also suggest that the system transitioned abruptly to an ultra-low wear rate on the order of 10-8 mm3/N·m after 80 cycles.

In every case, the corrected wear measurement (after post-hoc realignment) differed from the uncorrected measurement by less than 12×10-9 mm3; in other words, the corrected measurements demonstrate that the raw interrupted topographical subtractions achieved wear sensitivity on the order of 10-8 mm3 even without unusual efforts to perfectly align or realign surfaces. The corrected results demonstrate that the occasional reductions in wear volume, which were within the bounds of the expected measurement uncertainty before correction (recall that our estimates represent an upper limit), were significant. Back transfer of material from the wear surface is a plausible contributor to momentary wear reversals like this.

Figure 3.6 (a) Topography of the unworn surface. (b) Wear volume as a function of sliding cycles for a lubricated steel on steel reciprocating test (blue data points). The error bars represent the uncertainty in volume due to the measured repositioning error. The inset shows the steady-state wear region (70-500 cycles). The wear measurements following post-hoc repositioning error correction are shown in red.

3.4.3 Wear Mapping Representative topographical differences are shown in Figure 3.7 to illustrate the extent to which wear can be mapped before and after post-hoc realignment. According to Figure 3.6, half the total volume of material lost in the entire experiment occurred in the first 10 cycles. Whereas topographical differences can be highly sensitive to minute changes in wear volume, even with significant repositioning errors on the order of 1 µm (Figure 3.6), Figure 3.7 demonstrates that repositioning errors of this magnitude made inferences about the locations of wear impossible without realignment. After realignment, topographical subtraction locates wear specifically near the center of the observation window (wear is shown as positive for visual clarity). Realignment was unnecessary for the 20-cycle measurement because the repositioning error happened to be below our detection limit. Comparison between 10 and 20 cycle measurements shows there is little to no practical difference between perfect mechanical repositioning (20 cycles) and imperfect mechanical repositioning

25 with post-hoc realignment (10 cycles) using these methods. Interestingly, the original wear strip observed after 10 cycles is relatively untouched by the subsequent 490 sliding cycles, which created another large strip on the right of the wear track and sporadic locations of what appears to be gradual asperity scale wear in between.

Figure 3.7 Topographical subtractions characterizing wear after 10, 20, 30, and 500 sliding cycles without (a) and with (b) post-hoc correction of repositioning errors. The repositioning error is shown in the bottom right of each corrected subtraction image. Whereas worn volumes can be detected with remarkable resolution under typical repositioning errors (~1-5 µm), locating asperity scale wear requires either precise mechanical repositioning or post-hoc correction.

The corrected topographical subtractions are mapped onto the unworn surface topography in Figure 3.8. After 10 cycles, wear (defined here as loss >100 nm) was essentially confined to a single strip of tall asperities near the center of the window. This line became the left boundary of the wear track later in the test (500 cycles). The interrupted measurements reveal a wear process characterized mostly by very subtle asperity-scale changes that are only occasionally disrupted by significant wear events. Not surprisingly, the locations of wear can be mapped directly to the tallest asperities on the unworn surface. It is interesting to note, however, that a line of tall asperities just to the left of the initial wear location (10 cycles) was mostly untouched by wear for the entire experiment; after 10 cycles, one would expect this strip to be removed.

28 The original strip was not at the apex of the sphere, as later cycle measurements show, but it was tall enough to make first contact. Figure 3.9 isolates and magnifies two regions of interest (located in Figure 3.8 for context). These results highlight the impressive degree to which topographical subtractions can be used with common methods to detect, locate, and follow asperity scale wear. Asperity 1, for example, was worn down significantly by the first 10 sliding cycles and continued accumulating wear depth and breadth for the remainder of the experiment. Asperity 2, which was only barely worn by the first 10 cycles, exhibited a similar pattern of gradual monotonic wear with continued sliding. The peaks of both asperities were worn down by about 200 nm after 500 cycles. This pattern of progressive asperity scale wear can be found at other asperities throughout both regions and at those of other regions. The results provide evidence of both abrupt wear events (Figures 3.6-8) and more gradual processes that might appropriately described as asperity scale attrition [57, 61].

Figure 3.9 Topographical subtractions of the regions surrounding the two asperities highlighted in Figure 3.8.

The wear volume is plotted for both asperities versus sliding cycles in Figure 3.10. In both cases, the asperity scale wear trend mirrors the global trend with rapid run-in followed by a smaller steady state wear rate. The total wear volumes for the regions bounded by the orange boxes were 6×10-9 mm3 and 1.2×10-9 mm3 for asperities 1 and 2, respectively. The wear curves themselves demonstrate that the topographical differences method detected worn volumes as small as 10-10 mm3.

3.5 Discussion The topographical difference method is simple, well established, and able to resolve asperity scale wear. However, its broad use in the tribology community has been limited, presumably, by an expectation that imperfect repositioning introduces significant wear measurement uncertainties that are difficult to quantify [74]. We used the only available wear standard, unworn surfaces, to systematically quantify the

30 extent to which typical repositioning errors limit the sensitivity of this potentially transformative measurement approach. Measurements of unworn surfaces subjected to known and systematically varied repositioning errors showed that that topographical difference method provides excellent sensitivity to wear volume in macroscale experiments under typical conditions (errors and surfaces). While the empirical fits to Eq. 3.1 can be used by other researchers to estimate the uncertainty in arbitrary topographical difference measurements, the results demonstrate that quantifying and correcting repositioning errors post-hoc is vastly superior to simply quantifying wear measurement errors with Eq. 3.1. Perez et al. [74] noted that their topographical subtraction measurements were ‘highly sensitive to lateral repositioning’; while true generally, this study shows that the contribution of repositioning error to their wear measurement uncertainty was likely negligible (based on volume measurements of ~10-4 mm3). Likewise, this study suggests that the methods used by Sawyer and Wahl could have resolved at least an order of magnitude less worn volume than the smallest worn volume their experiments generated (~10-6 mm3). Using typical SWLI measurements like theirs with typical repositioning errors of ~2 µm, we demonstrated the ability to resolve wear volumes as small as 2×10-8 mm3 without post-hoc correction (as supported by later post-hoc correction; see Figure 3.6). Post-hoc correction reduced the detection limit of asperity scale wear measurements to the order of 10-10 mm3 (Figure 3.10). To our knowledge, this is, by orders of magnitude, the smallest wear volume reported from a macroscale wear measurement (4×10-8 mm3 [70] is the smallest we are aware of). The most important benefit of the topographical difference method, in our view, is the ability to locate and follow the wear process. This paper demonstrates that

31 topographical differences can be used to track wear at the asperity scale throughout an experiment – whereas Gahlin and Jacobson [70] clearly resolved asperity scale wear, our measurements are the first to probe how the macroscale wear process evolves at the asperity scale. Not surprisingly, wear mapped reliably to the tallest asperities on the unworn surface. By tracking the wear process intermittently, we were able to identify both discrete large-scale wear events and gradual attrition of individual asperities. Although such processes have been documented for single-asperity AFM measurements [66, 75–78], this is the first direct observation of asperity scale attrition in a macroscale contact that we are aware of. With minimal effort, topographical subtraction creates an opportunity to more directly link fundamental AFM and TEM- based wear measurements to the more practical macroscale systems they aim to model.

3.6 Conclusions This paper quantified the uncertainties and detection limits of wear measurements based on topographical subtraction under the conditions of typical repositioning errors, surface topographies, and measurement approaches. The following conclusions can be drawn from the results:

1. With typical surfaces and with normal repositioning errors on the order of 2 microns, SWLI-based topographical subtraction reliably detected worn volumes as small as 10 µm3 or 10-8 mm3, which appears to be among the most sensitive macroscale wear measurements in the literature and several orders of magnitude more sensitive than typical measurements.

2. Before post-hoc realignment, imperfect repositioning prevented us from reliably locating wear. After post-hoc realignment, we improved wear sensitivity to 0.1 µm3 or 10-10 mm3, mapped wear to features of the unworn topography, and observed gradual attrition at the single

32 asperity scale; to our knowledge, this is the first direct demonstration of the latter in any macroscale sliding experiment.

3. Interrupted topographical subtraction is widely accessible, exceptionally sensitive to wear, able to resolve wear with asperity scale spatial resolution, and ideally suited to bridge fundamental and practical studies of incipient wear.

33 Chapter 4

MICROTRIBOMETRY

4.1 Setup and Calibration To close the measurement gap starting from the practical scale, we designed a new microtribometer which can replicate the conditions of an AFM test. Because traditional tribometry relies on inherently displacement-based measurements, the normal force sensitivity that can be achieved with a tribometer is only limited by the cantilever’s compliance. However, these are macroscale systems and probe mass must be reduced with spring rate to avoid excessive reductions in resonance frequency, which impedes detection of frictional dynamics and increases the noise floor. By systematically reducing the weight of the cantilever assembly and its compliance, we have achieved normal force sensitivities of ~1 μN and friction force sensitivities of ~200 nN, which is nearly 4 orders of magnitude below the maximum loads we have been able to achieve in the AFM.

Ten custom steel cantilevers with comparable stiffnesses were manufactured for this study. Each of them was calibrated for normal and lateral stiffness using a piezo positioning stage for vertical actuation and a high-precision microbalance for

35 force quantification. The calibrations consisted of three independent beam replacements within the system, each followed by three indentations that produced force-displacement curves. The total of nine curves were first averaged within the same three replacements; then using the uncertainty in these averages, the three independent replacements of the beam were combined through a Monte-Carlo simulation; in this way a realistic uncertainty is established for future replacement of the beams within the system. The results are shown below in Table 4.1 and point to the fact that 10 beams had stiffness within the same order of magnitude (comparable to an AFM cantilever) and their uncertainty was <5%. Following calibration, spheres of different sizes and materials were mounted directly to the tips of these cantilevers through a procedure described in the following chapters.

Table 4.1 Summary of the ten probes that were assembled and calibrated for this study. The uncertainty is listed in parentheses and represents the Monte- Carlo simulated error that stems from the three independent repeats of the calibration procedure.

Colloid Colloid Normal Lateral Diameter Material Stiffness (N/m) Stiffness (N/m) (µm) Steel 25 392.1 (15.7) 277.4 (3.9) Steel 50 687.6 (21.9) 602.8 (13.0) Steel 100 751.3 (16.7) 488.8 (13.5) Steel 1,000 630.3 (14.4) 434.5 (15.7) Alumina 20 377.7 (16.7) 238.6 (5.3) Alumina 50 784.7 (10.3) 313.5 (2.7) Alumina 100 665.7 (26.1) 591.4 (43) Alumina 1,000 683.9 (6.9) 415.0 (7.2) Silica 100 214.9 (12.9) 323.6 (8.4) Glass 100 523.3 (12.2) 471.9 (21.3)

36 4.2 Friction Force Dependence on Material and Probe Size Each of the ten probes that were produced was tested both on hardened gold

(>99.99% purity) and single-crystal MoS2. Each ball-material pair was tested at six independent locations to prevent any local features from affecting the conclusions of these results. At higher loads (>3 mN) some of the measurements exhibited irreversible increase in friction which was later correlated to wear particles being generated; an example is shown in Figure 4.5. The high-density data under 0.5 mN was fitted through a linear model for friction: 퐹푟𝑖푐푡𝑖표푛 퐹표푟푐푒 =

(퐹푟𝑖푐푡𝑖표푛 퐶표푒푓푓𝑖푐𝑖푒푛푡) ∙ (푁표푟푚푎푙 퐹표푟푐푒) + (퐴푑ℎ푒푠𝑖표푛푎푙 퐶표푚푝표푛푒푛푡 표푓 퐹푟𝑖푐푡𝑖표푛) and the resulting values are summarized in Figure 4.6 and Figure 4.7.

37 2 1.8 Glass 100 µm 1.6 SiO2 100 µm 1.4 Alumina 22 µm 1.2 Alumina 50 µm 1 Alumina 100 µm 0.8 Alumina 1000 µm 0.6 Friction Force (mN) Force Friction Steel 25 µm 0.4 Steel 50 µm 0.2 Steel 100 µm 0 0 2 4 Steel 1000 µm Normal Force (mN)

38 0.14 Glass 100 µm 0.12 SiO2 100 µm 0.1 Alumina 22 µm 0.08 Alumina 50 µm Alumina 100 µm 0.06 Alumina 1000 µm 0.04

Friction Force (mN) Force Friction Steel 25 µm 0.02 Steel 50 µm Steel 100 µm 0 0 1 2 3 4 5 Steel 1000 µm Normal Force (mN)

Figure 4.3 Friction versus normal load for single-crystal MoS2 tested with colloidal probes of different material and diameter (listed in legend). Each gold- ball pair was tested at six different locations in an increasing and decreasing normal load sequence. The points of higher density between 0 and 0.5 mN were used for friction coefficient fitting, which is shown later.

39 0.025 Glass 100 µm 0.02 SiO2 100 µm Alumina 22 µm 0.015 Alumina 50 µm Alumina 100 µm 0.01 Alumina 1000 µm

Friction Force (mN) Force Friction 0.005 Steel 25 µm Steel 50 µm 0 Steel 100 µm 0 0.2 0.4 Steel 1000 µm Normal Force (mN)

Figure 4.4 Friction versus normal load for single-crystal MoS2 between 0 and 0.5 mN of Normal Load.

40 0.9 0.8 0.7 0.6 0.5 0.4 0.3

Friction Force (mN) Force Friction 0.2 0.1 0 0 1 2 3 4 5 6 Normal Force (mN)

The results of the linear regressions (Figure 4.6 and 4.7) show a few surprising revelations. First, the friction coefficient does not appear to be dependent on the colloid diameter. Second, the gold has a friction coefficient which is almost perfectly an order of magnitude higher than single-crystal MoS2. Thirdly, the friction at 0 mN according to the linear regressions has the same magnitude for most colloids, regardless of the surface material (gold or MoS2).

41 0.4 0.04 0.35 Gold 0.035 0.3 0.03 0.25 0.025 0.2 0.02

0.15 0.015 Friction Coefficient Friction

0.1 0.01 2

Gold Friction Coefficient FrictionGold 0.05 0.005 MoS 0 0 25 50 100 1000 22 50 100 1000 100 100 Steel Al2O3 SiO2 Glass

42 0.015

0.01

0.005

0 Gold

Friction(mN) MoS2

-0.005 Adhesional Component of AdhesionalComponent -0.01 25 50 100 1000 22 50 100 1000 100 100 Steel Al2O3 SiO2 Glass

This prompted us to explore the possibility that the roughness of the colloids has a larger effect on friction than their diameter or material. Fortunately, each ball’s topography was scanned in tapping mode AFM before the tests were conducted. This let us conduct a first-order Matlab simulation, where each ball was lowered against a flat surface until the contact area reached 1 µm2, which is a representative estimate of the contact area. The sole aim of this simulation was to show the distribution of contacting areas, and thus to achieve that it was sufficient to exclude the elastic and plastic deformation effects since the topographic features of the balls were distinct enough. The two examples shown in Figure 4.8 show the strong effect of the number of individual contacts at a rough interface on friction; it appears that area is not the primary source of high friction. This is consistent with the established single-asperity

43 low-friction AFM measurements from the literature. Measuring contact areas in-situ is the subject of the next chapter which reveals, not only increasing areas with load, but also increasing shear strength with load.

44 Chapter 5

INTEGRATED QCM-MICROTRIBOMETRY: FRICTION OF SINGLE- CRYSTAL MOS2 AND GOLD FROM mm/s TO m/s

5.1 Abstract While much is known about the fundamental mechanisms of friction, little is understood about how those mechanisms contribute to more practical macroscale interfacial forces and phenomena. Two fundamentally distinct microtribometry approaches have been developed over the last decade to help connect the dots between fundamental and practical tribology measurements. Spring-based approaches use quasi-static (low speed, low inertia, long relative slip length) displacement measurements of a sensing spring to quantify the magnitude of the friction force on a probe. Quartz crystal microbalance (QCM)-based approaches use changes in the QCM resonance response (high speed, high stiffness, short relative slip length) to quantify frictional dissipation, static friction, and contact area at the slip interface. Unfortunately, making comparisons between these approaches is impossible given their differences and the likely impacts of those differences on the tribological response. This paper integrates spring and QCM-based measurements to elucidate these unresolved effects using two model materials, gold and single-crystal MoS2, against alumina microspheres at loads between 0.01 and 1 mN. The alumina-gold tribosystem produced kinetic friction coefficients of 0.28 ± 0.03 and 0.25 ± 0.01 during QCM (~0.2 m/s) and spring-based (~25 µm/s) measurements, respectively. The alumina-MoS2 tribosystem produced kinetic friction coefficients of 0.086 ± 0.009 and

45 0.041 ± 0.001 during QCM (~0.2 m/s) and spring-based (~25 µm/s) measurements, respectively. This surprising quantitative agreement suggests that frictional processes in these fundamentally distinct regimes are more closely related than current understanding suggests. Perhaps more importantly, this paper describes and validates one method to help close the ‘tribology gap’ while demonstrating how integration creates new opportunities for fundamental studies of practical friction. It has to be acknowledged that this chapter is based on a collaborative effort with Dr. Brian Borovsky and that we share an equal position of first authors in the publication that will result from this work. My specific contributions are associated with the microtribometry methods and experiments, integration with QCM, part of the data analysis and part of the writing.

5.2 Introduction Tribological interfaces comprise many discrete areas of asperity level contact, each with a unique mechanical environment and each with the potential to contribute distinctly to the collective response [7, 79–81]. The inability to quantify the conditions at the asperity level or the variations in conditions across asperities makes fundamental studies in tribology a difficult proposition. Nanotribology addresses this challenge by isolating one single-asperity contact and probing the mechanics of the interaction under well-controlled conditions; in this case, a well-characterized probe serves as the asperity [82–84]. Although the now large body of nanotribology research has taught us a great deal about the physics of friction and wear [84–91], it has also revealed stark qualitative and quantitative differences between nanoscale and macroscale tribology [80, 82, 87]. Clarifying when, how, and why these differences

46 emerge requires improvements in our ability to quantify interfacial friction at loads, areas, and speeds that vary by orders of magnitude. Microtribometry, which we loosely define by forces between 1 μN and 10 mN, has emerged as a viable way to bridge the experimental gap between fundamental and practical tribological measurements [31, 92]. The forces involved are well below the measurement threshold for standard load cells and well above the limits of more fundamental scanning probe measurements. Nonetheless, a number of strategies have proven useful for measurements in this difficult loading regime. The most common methods use light [31, 93]. capacitive [94], interferometric [95], and other displacement sensors [96] to quantify forces based on the deflections of a transduction spring or cantilever; this method of force transduction is analogous to that used in the atomic force microscope. In this case, the model asperity or probe is mounted to the spring and dragged across a frictional counterbody. During contact, the spring extends while the probe first sticks and then relaxes when the restoring force of the spring exceeds the static friction force at the interface; such systems are often described by the Prandtl-Tomlinson friction model [97]. Because the measurement assumes negligible masses and accelerations, sliding speeds are typically limited to values well below those found in practice.

The quartz crystal microbalance (QCM) provides a fundamentally distinct technique for quantifying friction [98–101]. Its most unique attribute is the ability to probe nano and microscale friction at high slip speeds consistent with those found in many industrial applications. Relative to spring-based methods, it is a less common, complementary, and equally powerful tool for quantifying friction in nanoscale and microscale tribology measurements [102–107]. In a QCM-based measurement,

47 changes in the resonance frequency and quality factor of the QCM are used to quantify friction forces, frictional energy dissipated, slip lengths, contact areas [108], contact pressures, and shear strengths within static, partial slip, and full slip contact interfaces [100–102, 109–116]. Because the QCM resonates at extremely high frequencies (often exceeding 1 MHz), even the smallest slip lengths create slip speeds that approach 1 m/s [117, 118]; as a result, there is an enormous speed gap between friction measurements made with spring-based instruments and those based on QCM resonance. Inertial effects provide another important distinction between these methods. While inertia is negligible in spring-based measurement, it dominates the QCM measurement; in fact, the inertia of even the smallest colloids effectively confines motions to the intended slip interface [115, 116, 118]. Unlike spring-based methods where friction is fundamentally related to mechanical instabilities [119, 120] and thermal activation past the barriers to motion [87, 91], QCM-based friction measurements are related to athermal dissipation characteristics [106] that are more specific to the attributes of the contact interface and less sensitive to the mechanical details of the measurement system. In theory, integrating these complementary methods provides a unique opportunity to clarify microscale friction and its relationship to friction in more fundamental and more practical contacts. Unfortunately, it is plausible, if not probable, that extreme differences in sliding speed, slip length, and dissipation mode between QCM and spring-based methods create irreconcilable differences in the magnitudes of the measured forces. For example, friction coefficients above 1 are rare with spring- based measurements but common with QCM-based measurements [100, 102, 115]. We initiated this collaborative effort and transported the QCM-based system halfway

48 across the United States to: 1) integrate QCM and spring-based microtribometry; 2) to resolve uncertainty about how these differences manifest themselves in the respective measurements; 3) to determine if those manifestations impede integration toward complementary measurements of microscale or even nanoscale friction. Integrating these approaches into a single system was necessary to control variables like probe, sample location, load, machine compliance, temperature, and environment, each of which would have been impossible to control without integration. Following integration, we quantified friction of alumina against single-crystal MoS2, a model 2D solid lubricant, and Au, a model metallic control, to assess the degree to which friction depended on speed, slip length, and frictional dissipation within these model microscale contacts.

5.3 Methods

5.3.1 Materials The QCM crystals used in this study were obtained from Q-Sense (model number QSX 301). The crystals, which were coated with gold by the manufacturer, had an average roughness of <1 nm, were AT-cut to obtain transverse shear oscillations, and had a 5 MHz resonant frequency in their fundamental mode. The gold stock electrodes were used for the friction measurements throughout the paper.

Depositing single-crystal MoS2 onto a portion of the gold surface provided a convenient sample set for studying the effects of different materials with minimal influence over the experiment. Mineralogical single-crystal MoS2 was obtained from Structure Probe, Inc. (SPI Supplies).

49 Preparing quality single-crystal MoS2 samples, we discovered, required significant trial and error due to the lamellar structure of the sample and sensitivity of the QCM to interfacial slip and dissipation. Preserving the quality of the QCM resonance required extremely thin and well-adhered samples with minimal viscoelasticity from the adhesive. To begin, we pressed tape onto the stock MoS2 agglomeration to remove a fragment of the surface. A second piece of tape was pressed onto this fragment and then peeled away to create two new surfaces. This process was repeated until we achieved a thin sample of a convenient size and shape; a typical procedure used 5-12 such exfoliations before we obtained a sample we could begin working with. Next, we adhered the chosen set of MoS2 flakes to the crystal using a small drop of liquid adhesive. For these experiments, we used a fast-drying polyurethane (Minwax Clear Gloss), which is thin and oil based. We used a custom micropipette to deposit a 500 m diameter droplet of the adhesive to the center of the crystal. The MoS2 sample was then pressed onto the adhesive with a custom vice with two 13 mm diameter natural rubber spheres to apply and distribute load across the crystal. The adhesive was given 24 hours to cure before the vice was opened and the tape removed to reveal the sample. These freshly prepared QCM samples were usually far too large to allow the crystal to resonate. We used more tape to systematically remove layers. Following each removal, we imaged the crystal in an optical microscope (for lateral size and thickness, which we assessed qualitatively based on color) and tested resonance performance of the crystal (based on measured impedance); only samples with an impedance below 100  were used (crystals were 10-30  as received). Non-oscillating QCMs were stripped of the polyurethane and

MoS2 flakes with acetone for later use.

50 The counterfaces for these experiments were 50 or 100 µm diameter spheres of polycrystalline α-Al2O3 with a typical grain size of 5 µm (purchased from microspheresnanospheres.com); AFM measurements showed that these colloids had a roughness of Ra = ~120 nm. These spheres were attached to the native measurement probe of the instrument (see Figure 5.1) using a custom micromanipulation station comprising a XYZ micromanipulation stage (±1 μm), a XY micropositioning stage (±1 μm), one tungsten needle and one glass needle (<1 μm tip) to manipulate the colloids and adhesive, respectively, and two long-working-distance microscopes for isometric and top-down views of the assembly procedure. To begin, the measurement probe was positioned within the view field of the cameras using the 2-axis stage. The 3-axis stage was used in combination with a microscope to dip the glass needle into a drop of slow-curing epoxy, which was subsequently applied to the tip of the measurement probe. Next, the tungsten needle was brought close enough to an ensemble of colloids to grab one or more electrostatically. After a few trials, a colloid was chosen, positioned above the epoxy-coated measurement probe, and slowly lowered until it contacted the adhesive. After 24 hours of curing, the colloid was imaged to confirm quality (e.g. no adhesive, protrusions, or other unintended features on the colloid near the intended contact area) and then used for measurements.

5.3.2 Instruments We use the quartz crystal microbalance based tribometer described in detail previously [102] to establish non-integrated benchmark measurements. The instrument consists of a QCM, which creates and measures slip conditions at the contact interface, a piezoelectric positioning stage to establish contact between the probe and the surface, and a Hysitron TriboScope® transducer for applying and measuring normal

51 loads ranging from 0.01 to 1 mN (Figure 5.1A). The 5 MHz QCM sensor is powered by an Agilent E5100A network analyzer. Because of the transverse shear resonance of the quartz disk, its surface reciprocates laterally with a sinusoidal motion profile. The amplitude of this motion can be varied from 0.5 to 50 nm, depending on the drive power, which corresponds to maximum surface speeds from 0.016 to 1.6 m/s. By measuring changes in the resonance of the QCM due to contact with the probe tip, the network analyzer quantifies the total energy dissipated at the contact (per cycle of oscillation) and the lateral contact stiffness (Figure 5.2c). The observed trends in these two physical quantities as functions of the QCM’s amplitude of motion reveal a transition from partial to full slip at a critical amplitude. We determine the average kinetic friction during sliding from the rate at which the energy dissipation increases with reciprocation distance in the full-slip regime. In addition, the contact area is inferred from the lateral stiffness during stuck conditions, at the lowest shear amplitudes. The detailed analysis procedure and the underlying theory was discussed by Borovsky et al. [102]. It is important to note that the interaction forces between the tip and surface are closely confined to the interface due to the ‘near-field’ acoustic geometry of the probe- QCM setup: the wavelength of 5 MHz shear waves in quartz is over 600 µm, much larger than the contact radii obtained, near 1 µm. The waves decay rapidly as they enter the colloidal tip through the small aperture. Thus, the probe assembly holding the tip are treated as infinitely stiff and massive; [103] this is the perspective from which probe-QCM friction measurements have been understood to be insensitive to the mechanical details of the overall system. In contrast, the spring-based microtribometer

52 operates in a quasi-static regime where the finite stiffness of its cantilever is integral to force measurements.

Figure 5.1 A: The benchmark Quartz Crystal Microbalance paired with a Hysitron capacitive normal force transducer. (i) Side view of an opening in the chamber that contains the QCM. (ii) The normal load transducer positioned directly above the QCM and in the middle of the chamber. (iii) Live camera view of the 50 µm alumina colloid before contact with the gold surface of the QCM. B: The integration of the QCM with a spring-based microtribometer. The microtribometer comprises piezo positioning stages for lateral actuation of the counterface and vertical positioning of the measurement probe. The counterface is the same QCM chip used in the setup on the left. (iv) The capacitance probes are used to measure the deflection of a steel cantilever both vertically and laterally upon force application. (v) A 50 µm alumina colloid is glued to the end of the cantilever and is shown just before the contact with the single- crystal MoS2.

The spring-based microtribometer (Figure 5.1B) comprises a high resolution nanopositioning stage (PI P-628.1CD) with a range of 800 m and a resolution of <2

53 nm for lateral reciprocation, a vertical nanopositioning stage (PI Q-545.240) with a range of 26 mm and a resolution of 6 nm for loading, and a two-axis load cell for friction measurements and normal force feedback to the load control system; the load cell comprises a single cantilever (k = 785 N/m) with orthogonal capacitive displacement sensors (Lion Precision CPL290, C3S).

Figure 5.2 (a) The quartz crystal resonates at ~5 MHz and with a track length of about 20 nm, which corresponds to an average sliding velocity of up to 0.2 m/s. The contact diameter at 1 mN is approximately 2 µm, which is orders of magnitude larger than the track length. (b) With the QCM turned off, the lateral nanopositioning stage reciprocated at 25 µm/s over a 25 µm track length. The microtribometer speeds were 5 orders of magnitude slower than those of the QCM and the tracks were 3 orders of magnitude larger. (c) Data from a single measurement of friction forces of gold at a normal load of 0.2 mN with a QCM. The slope of the energy loss vs. amplitude trend during full slip is used to determine the kinetic friction force [102]. (d) Normal force and “friction loop” raw data for i- µT of alumina on MoS2. For the 2 µm analysis window, sliding at a normal force of 15.6±1.8 µN produced 5.8±0.8 µN of friction force.

54 5.3.3 Measurement and Analysis Methods

5.3.3.1 QCM-based Force Measurements For a single QCM measurement, load was fixed and the quartz disk was subjected to a sweep of decreasing drive voltage, which decreased the interfacial slip from a maximum value to zero. The downward sweep allows the interface to ‘run in’ at high amplitudes, which facilitates a consistent friction behavior during the sweep and a transition to stuck conditions at the lowest amplitudes [102, 115]. Each amplitude sweep consists of ~50 measurements of varying drive voltage and requires about 10 min of testing time; this corresponds to about 3 billion reciprocation cycles for each load condition. The resonance frequency and inverse quality factor (also called “dissipation factor”) were measured directly at each of the ~50 drive voltages, analyzed, and compared to the baseline values obtained prior to contact [102]. The lateral contact stiffness is proportional to the frequency shift, as given by 1/2 2 kMKf4   res , where M (M = 5.0 mg) and K (퐾 = 푀(2휋푓푟푒푠) ) are the effective mass and stiffness of the quartz disk, respectively; ∆푓푟푒푠is the change in resonant frequency of the crystal after load has been applied and 푓푟푒푠 is the resonant frequency prior to loading [102]. The oscillation amplitude is computed from the empirical formula: U0 = (1.4 pm/V) ⋅ Q ⋅ Vp, where Vp is the peak applied drive voltage [121]. The elastic force amplitude, which is the product of the measured stiffness and the oscillation amplitude, is shown for gold at a normal load (FN) of 0.2 mN (Figure 5.2c) to illustrate the transition to full slip at the static friction force limit.

At the same normal load of 0.2 mN, gold and MoS2 transitioned to full slip at about 5.8 nm and 0.6 nm, respectively, due to the greater frictional shear strength of the alumina-gold interface. As a result of this and the fixed QCM resonance frequency (5

55 MHz), slip speeds varied between materials (180 and 19 mm/s, respectively, in this case).

Kinetic friction forces (FF) were quantified after converting the measured inverse quality factor shifts to energy dissipation using the equation:

21 WKUQloss  0 [102, 115]. As shown in Figure 5.2c for gold at 0.2 mN, the energy dissipation increases linearly with amplitude in the full slip state when the friction force is constant with varied track length. Thus, the kinetic friction force is equal to one quarter of the slope of a linear fit to the energy dissipation versus amplitude; the amplitude is half the track length and the width of the loop is twice the friction force (Fig. 5.2d). The uncertainty in the kinetic friction force measurement is one-fourth the standard deviation in the slope of this linear fit. Contact area measurements use k0, the slip-free contact stiffness measurement (recall that this is based on the frequency shift) taken nearest the limit of zero displacement (< 0.5 nm displacement). The contact radius, a, was determined from the equation: k0 = 8G*a, and contact areas and stresses were obtained assuming circular contact areas. To obtain numerical values, we used Johnson’s approach to estimate the combined shear modulus G* for aluminum oxide on gold: G* ~ 14 GPa. [122] While these estimates contain significant quantitative uncertainties, they are sensitive to qualitative differences that can, as we show later, provide valuable insights into measurable interfacial phenomena. This single load voltage sweep measurement procedure was repeated for each load within a load ramp using a single interface (same probe, same material, same test location). The load ramp began with a low load experiment followed by experiments of increasing load up to a maximum test load. Despite our initial intent to test from

56 0.01 to 10 mN, we found that the crystal was unable to reliably produce full-slip conditions against gold at loads much beyond 1 mN due to its high shear strength. As a result, we limited our analyses to loads below 1 mN; data up to 3 mN for MoS2 were used only to match the range of normal pressures developed in the relatively stiffer contact with gold.

5.3.3.2 Spring-based Force Measurements To begin a spring-based friction force measurement, the vertical stage was used to make contact between the colloid and sample at the minimum target normal force. The lateral stage was then used to induce interfacial slip while the force sensors measured the normal and friction forces developed in response as a function of wear track position. Normal and friction force ‘loops’ collected at a target load of 15 N are shown in Figure 5.2d to illustrate the method and the detection limits of the force and displacement measurements. The friction force is half the difference between the forward and reverse friction forces. It is worth noting that this calculation is identical to that used for the QCM method – by this definition, the friction force is one quarter the total work bound by the loop divided by the amplitude (half the track length). The main differences lie in (1) how the work is quantified (dissipation versus force) and (2) the number of track amplitudes used in the analysis (many versus 1).

5.3.4 Experimental Design The first objective of this study was to determine the extent to which QCM measurements are affected by the compliance, natural frequency, and other mechanical details of the measurement system – this is important to confirm before attempting to compare QCM-based friction measurements between systems and

57 research groups. To test this effect, QCM-based measurements were used to quantify friction of alumina against gold and single-crystal MoS2 at loads from 0.01 to 1 mN

(or 10 mN for MoS2) first using the benchmark system (Figure 5.1A) to apply contact forces and then using the spring-based microtribometer (Figure 5.1B) to apply the contact forces; we refer to each method as QCM and i-QCM, respectively. For benchmark QCM testing, two different but nominally identical 50 µm diameter alumina colloids were used (colloid A and colloid B). Each of these was used at different locations on the QCM electrode, with one load ramp test conducted at each location. In the sequence of locations, the tested material was alternated to capture potential effects from material transfer. The tests for colloid A were in the order: 1)

MoS2, 2) MoS2, 3) Au, 4) MoS2, 5) Au; the tests for colloid B were: 1) Au, 2) MoS2,

3) MoS2, 4) Au. Unloading measurements followed loading measurements for colloid

A experiments at two of the three MoS2 testing locations. In these cases, loads were ramped up to 10 mN and down to zero. Because the effect of loading direction on friction was no larger than the effect of location, MoS2 loading and unloading results are reported without distinction. A different 50 m colloid C was used at different testing locations for i-QCM measurements. Like QCM measurements, i-QCM measurements were conducted from lowest loads to highest loads.

Our second objective was to test whether QCM and spring-based friction measurements differ significantly when controlling for important variables like probe size, roughness, material properties, testing location, normal force, contact area, machine compliance, and testing environment. Integrated spring-based microtribometry measurements (i-T) were conducted immediately following i-QCM measurements with wear track centers aligned precisely with the location of

58 corresponding i-QCM measurements. Two sets of experiments, one with the 50 m colloid C and one with a 100 m colloid D, were conducted and compared to determine how probe size and contact area affected the results. Load was applied, controlled, and ramped in an identical way for both i-QCM and i-T measurements. However, the measurements themselves represent either of two experimental extremes; whereas i-QCM used slip lengths of ~20 nm, slip speeds of ~0.1 m/s, and changes in QCM resonance to quantify friction, i-T used a slip length of 25 µm, a slip speed 25 µm/s, and deflections of a spring to quantify friction forces. Thus, it is not obvious that the results of these matched experiments should be comparable, much less identical. The i-T measurements were based entirely on the central 2 m of the wear track to isolate the region probed by the matching i-QCM measurement. All QCM, i-QCM, and i-T measurements were conducted in ambient laboratory conditions (~20-60 %RH, 25ºC).

5.4 Results

5.4.1 Benchmarking Friction with the QCM Kinetic friction forces from benchmark QCM measurements are shown for 50

m diameter alumina spheres against gold and single-crystal MoS2 as functions of applied load in Figure 5.3. In both cases, friction increased monotonically with normal force. In both cases, the data reflect testing results from two different probes in at least four independent locations of the sample (4 for gold and 5 for MoS2). Interestingly, the variation of the data about the trend line varied just as strongly for a single probe and location as it did across probes and locations. Additionally, we saw no effect on friction from ordering of the samples tested. It seems most likely to us that transfer

59 occurred and affected friction, but that the residence time of transferred material at the contact interface was short relative to the measurement (each data point represents billions of sliding cycles). Our observations suggest that any effects from variations in the materials and potential transfer from sample to probe on friction were small compared to frictional variations that occurred within a single interface due to time dependent deformation, debris formation, and material removal. More interesting still, our analysis of contact area revealed two distinct populations within the gold data.

One location tested with colloid (A) produced significantly larger areas (orange data points in Figure 5.3) than all other measurements including those from the other test location with colloid (A). Not surprisingly, friction measurements from the population of larger contact areas were typically larger than those from the population of smaller contact areas. Nonetheless, friction measurements appeared to belong to a single population and linear fits to the gold and MoS2 measurements with the standard friction model, FF = µ·FN, gave friction coefficients of µGold = 0.28 ± 0.03 (standard error) and µMoS2 = 0.086 ± 0.009.

Figure 5.3 (a,b) QCM friction force measurements for 50 m diameter alumina spheres against gold and MoS2. The error bars reflect statistical uncertainty in the fit of energy versus amplitude as illustrated in Figure 5.2c. (c,d) QCM contact area measurements for gold and MoS2; the orange subset of data points for gold represents one of the four locations that were tested and exhibited a consistent but offset trend. In fact this is one of the few cases we observed the expected non-zero contact are at zero load. (e,f) QCM shear and normal pressure measurements based on the previous graphs. The orange data points for gold align with the common shear pressure coefficient after their larger areas are taken in account together with the corresponding larger friction forces. The purple data points for MoS2 are derived from normal loads between 1 and 3 mN, which are not part of the fit and are not shown in (b) and (d), but follow the trend of the rest of the data.

61 This standard model neglects the fact that normal force mediates friction through the independent effects of contact area and shear strength; the ubiquity of the standard model reflects the ease of controlling and measuring normal forces and the relative difficulty in controlling or measuring contact areas and stresses. The general friction model, FF = 0A + FN, in which friction depends on an unloaded shear strength, 0, and a shear pressure coefficient, , rather than a friction coefficient [102, 123–125], can be used to distinguish between the effects of normal force on area and shear strength when contact area measurements become possible; this is a distinct advantage of the QCM method [102]. Contact area measurements were used to convert friction and normal forces into shear stress and normal stresses, which are plotted against one another for gold and MoS2 in Figures 5.3e and f, respectively. Linear fits to  = 0 + P give  = 0.19 ±

0.03 and 0 = 104 ± 53 MPa for gold and  = 0.059 ± 0.026 and 0 = 66 ± 26 MPa for

MoS2. For both material systems, changes in both the contact area and shear strength significantly affected the proportionality between friction force and normal force.

5.4.2 The Effect of Integration on the QCM Measurement The first hypothesis we tested in this study was that QCM-based friction measurements were sensitive to the mechanical attributes of the measurement system. We tested this hypothesis by repeating the benchmark QCM measurements from Figure 5.3 with the i-QCM assay. As Figure 5.4 demonstrates, the i-QCM measurement results were consistent with the results of baseline QCM measurements using equivalent materials systems (different probes, sample locations, days, etc.). Thus, we found no positive evidence to reject the null hypothesis that QCM-based friction measurements are insensitive to the mechanical attributes of the measurement

62 system; this makes intuitive sense given the slip speeds involved and the important role of inertia in the measurement. This outcome is important because it suggests (1) that QCM-based friction measurements can be compared across systems and (2) that QCM methods can be integrated into more traditional systems without fundamentally changing the nature of the measurement.

5.4.3 Comparing QCM and Spring-based Microtribometry The second hypothesis we tested was that friction measurements from spring- based microtribometry differ fundamentally from those of QCM-based microtribometry. Integrating QCM-based measurements with a more traditional spring-based microtribometer enabled us to test this hypothesis in complete isolation from otherwise uncontrollable variables. The friction results of i-T and i-QCM measurements with identical testing conditions (same probe, same sample location, same loading system, same load ranges) are shown in Figure 5.5. Given the enormous differences in the slip conditions (high speed and small relative slip versus slow speed

63 and large relative slip) and the measurement approaches (QCM dissipation versus spring deflection), we observed remarkable agreement between QCM and spring- based friction measurements. The results demonstrate the surprising result that friction at a single contact interfaces can be compared quantitatively (at least for these two very different material systems) despite enormous differences between the experimental conditions and the friction measurement approaches. The aggregate results from the entire study are given in Table 5.1 for both gold and single-crystal

MoS2.

Table 5.1 Summary of all linear regressions and associated standard errors for the measurements in this study. The fits are based on all data shown in Figure 5.5.

Au MoS2 µ 0.27 (0.03) 0.087 (0.007)  0.19 (0.03) 0.057 (0.009) 0 (MPa) 104 (53) 58 (14)

5.4.4 Effects of Other Experimental Variables We repeated the measurements with a larger 100 m diameter colloid (D) to determine if probe size disturbed the agreement we observed between measurement approaches. The friction results are shown as a function of normal force for all experiments in Figure 5.6. Doubling the probe diameter and repeating the experiment against gold had no noticeable effect on the i-T or i-QCM friction force measurements (Fig. 5.6a). Against MoS2, the larger colloid tended to produce higher friction (Fig. 5.6b) but the differences were not inconsistent with the variability observed for the field. Repeat QCM measurements with a different 100 m alumina

64 sphere against MoS2 at St. Olaf College revealed reduced friction, but it should be noted that part or all of this reduction may be attributable to the reduced humidity of the repeat experiments (~20% versus ~45% RH) [96, 126–130].

We observed similar comparability between measurements on the basis of stresses (Fig. 5.6c and 5.6d). To make this comparison, we used the i-QCM contact area measurements were used to quantify the stresses for the i-µT data; given the immense differences in the mechanical conditions of i-QCM and i-µT measurements,

65 it is reasonable to question the validity of this practice. However, we justify this synergistic benefit of integration with two important facts: (1) i-µT measurements were taken from a 2 m wide analysis window that approximates the exact interface, contact location, and load conditions of the corresponding i-QCM measurement; (2) contact area is based on contact stiffness measurements made at a slip-free interface. Thus, the validity of our contact area measurements to friction measurements with i- QCM and i-µT is limited more by the effects of interfacial stability and wear than by other attributes of the sliding system (speed, track length, mechanical stiffness, inertia, etc.). The fact that the system endured billions of wear cycles between each load- dependent measurement and still exhibited the expected systematic relationship between load and area without any evidence of history effects suggests that contact area measurements were not biased by wear between loads or between measurement approaches. Finally, we increased and decreased track length and sliding speed of i-T experiments with the 50 m alumina colloid by an order of magnitude to determine if any points of divergence between approaches emerged outside our initial design space. We observed no significant systematic changes in friction coefficient with changes in either variable over two orders of magnitude. Contrary to our initial hypothesis, we found no significant evidence that friction measurements from QCM and spring-based microtribometry are fundamentally incomparable.

5.5 Discussion It is reasonable to assume, given the rates and inertial forces involved, that QCM-based friction measurements are insensitive to the mechanical details of the measurement system. However, so few of these measurements exist in the literature

66 that there has been virtually no experimental support for this theoretically supportable hypothesis. This is important since a variety of mechanical systems have been used to implement QCM-based friction measurements of solid-on-solid contacts, including atomic force microscopes [106, 107, 111, 112], scanning tunneling microscopes [110, 117], nanoindentation [102, 113], surface forces apparatus [114], and dedicated normal force or displacement transducers to engage a sphere with the QCM surface [103, 105, 115, 118]. Another approach is to attach three spheres to a plate and set the assembly onto the QCM, placing weights onto the plate as needed to vary the normal load [100, 101, 109, 116]. We note that most of these prior experiments have studied the partial slip regime, although several different groups have observed the transition from partial to full slip [100–102, 113, 116]. Only two previous studies have reported quantitative measurements of kinetic friction in full slip conditions using the energy dissipation technique summarized here [102, 116]. The results of this study experimentally support the hypothesis that QCM- based friction measurements are insensitive to the mechanical attributes of the measurement system. This outcome suggests that the QCM-based approach can be easily adopted within any existing spring-based instrument (e.g. AFM, microtribometer) and that the results can be safely compared to other QCM-based friction measurements from the widely varying mechanical systems already in the literature. Likewise, it is reasonable to assume, given the significant differences in the measurement approaches, that QCM-based friction measurements might be inherently different from spring-based measurements. Integrating QCM-based measurements with a more traditional spring-based microtribometer enabled us to test this hypothesis

67 in complete isolation from other variables for the first time; had we simply exchanged samples and made independent measurements in our respective laboratories, it would have been impossible to know if or how differences in testing locations, roughness between probes, the mechanical systems, ambient humidity, or a range of other uncontrollable variables would have affected the results. Contrary to our hypothesis, we observed surprising consistency in friction results between approaches for alumina on gold (=0.27) and alumina on single-crystal MoS2 ( =0.084). Our observation that friction is more sensitive to material than sliding speed or measurement approach is sensible given Amonton’s third friction law, that friction is independent of speed. The result only becomes surprising after considering the physics of friction in the two measurements. Within spring-based measurements, friction is the consequence of mechanical instabilities of the spring-mounted probe as it attempts to traverse differing potential energy landscapes of the two material systems; this appears to be true of both nanoscale [91, 131, 132] and macroscale [119] friction measurements when machine compliance is a defining characteristic of the tribological system. The QCM measurement, being made with an infinitely stiff system at a well-defined slip interface [103], is presumed to access other modes of energy dissipation without regard to the details of the mechanical system. Our results suggest surprising quantitative agreement between these two fundamentally distinct measurement systems and friction regimes for reasons that are currently unclear; once clarified, these reasons may help answer broader questions about the physics of friction and its application toward practical friction and wear control. Contact area is both a fundamental driver of friction and extremely difficult to measure. As a result, there have been significant efforts to quantify contact area. The

68 most successful methods, which include electrical contact resistance [7], spring-based lateral stiffness [133], and optical methods [80] are limited to conductive materials, nanoscale contacts, and transparent bodies, respectively. QCM-based contact area measurements can be made without regard to material properties or the relative stiffness of the system and the contact. In this study, contact areas were quantified despite the opaque and insulating character of the materials and despite the contact stiffness (100,000 N/m) exceeding the cantilever stiffness (785 N/m) by orders of magnitude. QCM integration of the kind described here is the most direct, accessible, and experimentally flexible means to quantify contact area in common spring-based AFMs and microtribometers that we are aware of. The current understanding of friction, at least in a general sense, is that friction is proportional to contact area, which increases proportionally with load. Many previous studies involving measurements or estimates of contact area support this; previous measurements on cleaved mica [133] and sputtered MoS2 [123] suggest more or less constant shear stress while friction and contact area grow proportionally with load. Our contact area measurements provide significant evidence that shear strengthening was a significant contributor to the proportionality between friction and normal forces for both material systems; the effect explains why friction increased linear while area increased sub-linearly with load (Fig. 5.3). While this sublinear trend might be a measurement artifact, it is nonetheless compelling that similar effects were observed in two distinct systems.

In addition to self-consistency between approaches within a single contact interface, our microscale QCM and spring-based friction measurements, which cover a wide spectrum of system compliance, mutual overlap (track length per unit contact

69 radius), speed, and load, are consistent with prior results in other testing spaces. This study gives the following results for the basic friction parameters of gold and MoS2:

µGold = 0.27 and µMoS2 = 0.087, which are consistent with previous microscale multi- asperity measurements. For the same range of normal loads, Argibay et al. showed that gold produced a friction coefficient between 0.15 and 0.35 [134]. Although reported less often, the shear strength of gold we measured (0,Gold = 104 MPa) also agrees well with results reported by Ko et al. for chemically immiscible pairs between gold and other materials: 0 = 68 MPa [125]. The friction coefficient of single-crystal

MoS2 was found to be between 0.05 to 0.1 by Saito et al. in ambient air [135]; this is perfectly consistent with the range of values we observed with both approaches.

Finally, the shear strength of single-crystal MoS2 that we obtained (0,MoS2 = 58 MPa) is significantly larger than but comparable to the 25 MPa reported by Singer et al. for a sputter-deposited film [123]. The agreement with other studies suggests that the friction characteristics we observe are robust for these material systems and apply across a surprisingly large range of experimental conditions.

5.6 Conclusions The results of this study allow us to draw the following conclusions:

1. QCM-based friction measurements were quantitatively indistinguishable whether executed within a stiff or a compliant mechanical system. This outcome suggests that QCM friction measurements can be performed with dissimilar mechanical systems and that QCM can be integrated with other tribometry approaches without risk of affecting the measurement.

2. Integrated QCM friction measurements were quantitatively comparable to spring-based friction measurements. Given the fundamental differences between measurement approaches and the tribological conditions of the experiments (i.e. speed, track length, effective mechanical compliance), the quantitative agreement we observed

70 suggests that frictional processes in these fundamentally distinct regimes are more closely related than current understanding suggests.

Quantitative comparability between approaches provides an opportunity to exploit the complementary aspects of integration. We show how in-situ contact area measurements from the QCM allow us to isolate and discriminate between the distinct effects of normal force on contact area and interfacial shear strength. Our analysis shows that area and pressure contribute comparably to the proportionality between friction and load for both material systems under the conditions of our experiments.

71 Chapter 6

HIGH-FORCE AFM PROBES

6.1 AFM at the Macroscale: Methods to Fabricate and Calibrate Probes for Millinewton Force Measurements

6.1.1 Abstract The difficulty in detecting and controlling forces in the gap between the nanoscale and macroscale tribometry regimes has so far limited the application of fundamental atomic-scale insights to practical friction and wear control. This paper describes methods to achieve and quantify millinewton forces measured by atomic force microscopy (AFM) using existing experimental tools. We mounted colloidal microspheres at different points along the span of commercial AFM cantilevers to reduce their effective flexural length from 125 µm to between 21 and 107 µm. The resulting spring constants, based on direct calibration, varied from 100 to 10,000 N/m. Within a commercial AFM (Dimension 3100), these cantilevers produced normal force calibration constants between 0.006 and 0.430 mN/V; i.e. increasing the spring constant by 100x caused a corresponding increase in the calibration constant but only a negligible increase in V/m sensitivity. We demonstrate these new capabilities by measuring friction between the colloids and single-crystal MoS2 at applied normal forces up to 3.4 mN, which is in the range of existing tribometers and well above the forces typically used in AFM-based measurements. These methods, which make use of well-established procedures and only require a modified AFM cantilever, are

72 intended for use by other researchers as a platform for bridging the gap between nanoscale and macroscale tribometry.

6.1.2 Introduction Macroscale tribological contacts typically comprise many discrete nanoscale contact areas of unknown; location-dependent; and time-varying size, shape, and pressure [7]. In addition to effects from deformation, wear, and tribofilm growth, these features make controlled studies of macroscale tribological phenomena especially difficult [136]. By contrast, nanoscale friction measurements, which typically use atomic force microscopy (AFM), can be conducted within a single-asperity contact of well-defined size and location to promote more fundamental studies of tribological phenomena [82, 83, 137]. In one of the pioneering examples [84], the spatial periodicity of stick-slip friction variations matched the lattice structure of the graphite substrate as predicted by the Prandtl-Tomlinson model of atomic-scale friction [97, 138]. Other investigators have since used scanning probe microscopy (SPM) to study lattice periodicity [139–141], frictional anisotropy [95, 142], contaminant effects, superlubricity [88, 95, 143], and thermally activated slip [20, 27, 140, 144–147] among other fundamental aspects of atomic-scale friction. More recently, large-scale molecular dynamics (MD) simulations have been used to model these nanoscale tribological interactions and strengthen the link between experimentally observable phenomena [97, 148] and their underlying mechanisms [90]. It remains unclear how these fundamental nanotribological phenomena manifest themselves at the larger loads and length scales of more typical tribological contacts. Yoon et al. studied frictional scaling by comparing the frictional response of Si and DLC to varying probe radius using low loads in the AFM and high loads in a

73 microtribometer [92]. Interestingly, AFM-based friction coefficients of both materials increased with probe radius toward the radius-independent values obtained by the microtribometer. Nonetheless, the friction coefficients, probe radii, and loads differed by 10x, 100x and 10,000x, respectively, between instruments. Bhushan and Kulkarni varied load at constant probe radius and attributed a transition from lower friction coefficients at lower loads to higher friction coefficients at higher loads to the onset of plastic deformation and wear [36]. Tambe and Bhushan [149] and Bhushan et al. [37] conducted similar experiments for a range of materials using AFM and microtribometry and showed that microscale friction coefficients were typically an order of magnitude greater than their nanoscale counterparts. They proposed that increased plastic deformation, reduced hardness, third bodies (wear) and roughness contributed to the relatively larger friction coefficients of the microscale contacts. While these results demonstrate clear differences between nano- and microscale friction, they fail to elucidate where these differences first emerge; why they occur; if they are dominated by differences in contact size, load, contact stresses, or sliding speed [132]; and if the trend of increased friction with size scale is an inherent feature of all or most tribological systems. Understanding how fundamental atomic-scale interactions ultimately contribute to macroscale friction and wear requires controlled studies of friction across the relevant length and load scales. Experimental limitations, primarily in load control and sensing, have precluded such studies to date. In this paper, we develop general methods to help close this experimental gap. Because all commercial AFM’s infer forces based on beam deflection measurements, the range of loads that can be applied and measured is primarily limited by the stiffness of the cantilever. Herein, we

74 describe: 1) how well-established probe mounting methods can be used to increase the load range of commercial AFMs beyond 1 mN, which is within the range of existing microtribometers; 2) a direct method to calibrate beam stiffness and quantify AFM forces in the millinewton testing regime; and 3) the validation of a high-force AFM measurement approach using a commercial instrument and a model tribological system.

6.1.3 Methods and Materials

6.1.3.1 Preparation of High-force AFM Cantilevers Silicon tapping mode cantilevers (with a nominal stiffness of 40 N/m) were used and customized in this study. According to Euler beam theory, cantilever beam stiffness is inversely proportional to the cube of distance between the fixed end and the loading point; the “effective length” is hereafter defined as the distance between the fixed end and the center of the colloid as illustrated in Figure 6.1. Using methods described in our recent paper [75], we mounted ~30 µm steel microspheres (NanoSteel Co., Providence RI) at controlled distances along the span of the cantilevers to systematically vary the effective length, spring constant, force constant, and load capacity of otherwise stock AFM cantilevers.

Figure 6.1 Top: Setup for assembly of custom AFM colloidal probes. A high- resolution 3-axis manual positioning stage controls an ultra-sharp (<1 μm) tungsten needle. The colloid (50 μm diameter alumina in this example) attaches to the tip of the needle with the help of electrostatic forces. The tipless AFM chip sits on a 2-axis manual stage for positioning in the field of view of the microscopes. The two microscopes provide a simultaneous top and front views of the process to assure perfect positioning. (1-7): Profile views of the seven cantilevers prepared for this study. The distance of the steel colloid from the free end of 40 N/m cantilevers decreases with increased cantilever number. These optical images were subsequently used for determining the effective length, as illustrated in (1), following mounting of steel colloids; (8) an illustrative plan-view of an unmodified cantilever used to quantify the lateral dimensions (length and width) the cantilever. All eight images have the same scale.

76 The colloidal spheres were mounted at varying locations along the cantilever using a two-part epoxy (JB Weld, Sulphur Springs TX) with the aid of an optical microscope and a custom micromanipulator, based on previously established methods [150]. Optical images of the seven mounted cantilevers prepared for this study are shown in Figure 6.1. The measured dimensions and properties of these beams are provided in Table 6.1.

6.1.3.2 Quantifying Cantilever Flexural Stiffness: The Direct Calibration Method We adapted the nano-force calibrator (NFC) methods described by Kim et al. as a direct and traceable means for calibrating flexural stiffness [151]. The custom system we constructed is functionally analogous to the NFC and is shown schematically in Figure 6.2. Briefly, the calibration system used a high resolution (±6 nm) nanopositioning stage (Physik Instrumente Q-545) to actuate the cantilever and a high sensitivity (±100 nN) analytical microbalance (Mettler Toledo XP105DR®) to quantify the force response to cantilever deformations. A digital optical microscope with a 5 MP CMOS image sensor (Dino-Lite Edge AM7915MZTL) was used to guide the approach to contact (Figure 6.2c). To perform a flexural stiffness calibration measurement, the probe was loaded and unloaded against a silicon flat, which was glued to a platform and mated with the microbalance tray at three contact points; three load-unload curves were used for each measurement. Linear regression of force versus actuation depth was used to quantify the spring constant and its statistical uncertainty.

Three independent repeat measurements were performed with one soft (Lever 1 – 78 N/m) and one stiff cantilever (Lever 7 – 10,900 N/m) to quantify repeatability error from user–system interaction (see Figure 6.1 for reference). For repeat measurements,

77 the holder was removed from the instrument and the chip was removed from the holder before each repeat. Because these analytical microbalances are self-compensating and extremely stiff, Kim et al. neglected any compliance from their NFC calibration system [151]. However, compliance of even stiff systems can become significant for stiff cantilevers like those of interest here. To correct for system compliance, we first quantified system stiffness, ksystem, by repeating the calibration measurement with a Si blank in place of the cantilever. Five repeat measurements with independent blanks on independent days revealed that ksystem = 10,600±600 N/m. The corrected stiffness of -1 -1 -1 each cantilever was determined using the expression: kcantilever = (ktotal – ksystem ) .

The total combined uncertainty in the cantilever spring constant was then quantified according to the Law of Propagation of Uncertainty as outlined by the ISO Guide to Uncertainty in Measurement [152].

6.1.3.3 Experimental Validation of Direct Calibration Four Bruker CLFC-NOBO chips, each comprising three reference cantilevers, were used to experimentally validate the direct calibration method. Each tip-less reference cantilever was pre-calibrated by the manufacturer using the thermal tune method [153, 154] and had a nominal (prescribed) spring constant of 10.4, 1.3 or 0.16

79 N/m; specific pre-calibrated values were reported for each reference cantilever but experimental uncertainties in those values were not provided. Each cantilever chip was loaded into the custom holder and calibrated independently using the direct calibration method. Each direct calibration constant was compared against the manufacturer reported value to test agreement with a widely-used industry standard.

6.1.3.4 Experimental Application To quantify how substantially increased cantilever stiffness affects real-world normal and friction force sensing, we tested the colloid-stiffened cantilevers with a commercial AFM (Bruker Dimension 3100). The AFM normal force calibration constant (force per volts) depends on the cantilever-specific flexural stiffness (force per displacement) and the cantilever deflection sensitivity constant (volts per displacement), which can vary with differences in chip and laser positioning; for this reason, calibrations and measurements were done without changing chip or laser position in between. Cantilever deflection sensitivity calibration involved the regression of a single force-displacement curve of the test cantilever against a rigid Si substrate. The normal force calibration constant was determined as the product of the spring constant and the cantilever deflection sensitivity constant. Application experiments were performed with one soft (Lever 3 – 122 N/m) and one stiff (Lever 7 – 10,900 N/m) cantilever. We used single-crystal MoS2 as a model solid lubricant due to its apparent ability to maintain low friction coefficients on the order of 0.001-0.01 [126, 155] over a range of experimental conditions. Friction loops (friction voltage versus position) were collected for varying set-point voltages from 0.5 to 8 V over a 5 μm by 5 μm scan window at 5 μm/s scan speed. The corresponding friction voltage and its uncertainty were determined by computing the

80 mean and standard deviation of the half-width, respectively, over the middle 50% of the wear track; this ‘reversal method’ of friction analysis eliminates cross-talk effects (e.g. sensor misalignment, surface tilt, colloid offset) and artifacts from the reversal region (e.g. acceleration affects, static to kinetic transition, ringing, etc.) [156, 157]. We used the extended wedge method [155] immediately following testing to quantify lateral force calibration constants (per unit normal force calibration constant).

6.1.4 Results

6.1.4.1 Validation of the Direct Calibration Method Direct calibration curves for the three CLFC-NOBO reference cantilevers from a representative chip are shown in Figure 6.3(a). Measured flexural stiffness fits and the corresponding reported spring constants (red lines) are shown for reference. Mean values of flexural stiffness and uncertainty for all tested reference cantilevers (3 cantilevers each on 4 chips) using direct calibration are shown in Figure 6.3(b), together with corresponding manufacturer-reported values of flexural stiffness. In general, the results demonstrate excellent quantitative agreement between the direct calibration method and the reported value. For the stiffest (11.3 N/m) and most compliant (0.19 N/m) cantilevers, differences between measured and reported values were less than 6% on average and can be attributed to the experimental uncertainty in the direct calibration measurement. Interestingly, we observed the worst agreement between measured and reported mean values (17% on average) for the intermediate beams (1.66 N/m). The differences, in this case, cannot reasonably be attributed to direct calibration uncertainty. There is circumstantial evidence that the differences are related to uncertainty in the reported value: 1) direct calibration constants were always

81 closer to nominal constants than reported constants; 2) direct calibration constants exhibited less lever-to-lever variation than the reported constants; 3) the greatest disagreements between direct and reported constants were observed when reported constants also deviated most from nominal constants. Despite these small differences, the general quantitative agreement between methods provides a degree of independent validation for both.

Figure 6.3 (a) Stiffness curves for tip-less CLFC-NOBO reference cantilevers from a representative chip. Black dots represent measured data; black lines represent fits to measured data and are labeled by the corresponding slope; red lines represent the manufacturer-reported values from thermal- tuned calibration. Each curve represents the stiffness of a different cantilever on the same chip. The inset shows an in-situ image of the long beam calibration measurement. (b) Results of direct calibration (shaded bars) with the long, medium, and short pre-calibrated cantilevers on each of four chips: nominal spring constants were 0.16, 1.3, and 10.4 N/m, respectively. Outlines (in red) represent the reporter value from pre- calibration and nominal values are denoted for each cantilever family by dashed lines. Error bars represent the uncertainties in direct calibration constants.

82 6.1.4.2 Calibration of High-force Cantilevers with Colloidal Probes Direct calibration spring constants are shown for each mounted and unmounted cantilever in Figure 6.4. The measured spring constants of unmodified cantilevers were between 30 and 50 N/m (40 N/m nominal). Mounting a colloid near the free end increased the spring constant by ~2x (cantilever 1), while mounting a colloid near the fixed end (cantilever 7) increased the spring constant by ~300x. Experimental uncertainty as a portion of each measurement increased with cantilever stiffness; calibration uncertainty was on the order of 1% the measured value for the softest beam and approached 10% the measured value for the stiffest beam (Table 6.1). The variabilities in repeat measurements were consistent with the corresponding experimental uncertainty. Overall, these results demonstrate that: 1) colloid placement provides a controllable means for increasing cantilever stiffness by several orders of magnitude; 2) direct calibration, which is traceable to force and displacement standards, is applicable to cantilevers between 0.1 and 10,000 N/m of flexural stiffness.

Figure 6.4 Flexural stiffness from direct calibration for each of seven cantilevers before and after mounting the steel colloid. Error bars represent the statistical standard deviation in the slope of each independent calibration curve. Four independent repeat measurements were made for beams 1 and 7 to test for other error sources (e.g. lab temperature, chip placement in the holder, user repeatability, etc.). The consistency of the repeat results indicate that the experimental uncertainty is a reasonably comprehensive predictor of overall measurement error.

84 Table 6.1 Measured properties of the seven cantilevers used in this study (No. 1-7). Length measurements were made in an optical microscope with a resolution of 0.5 μm. The measurements were fit to Euler beam theory (Eq. 6.1) to quantify Young’s modulus, which was subsequently used to back-solve for the theoretical stiffness of each cantilever. The best-fit to modulus was 120±32 GPa; our uncertainty analysis showed that ~90% of this error is attributable to uncertainty in the thickness measurement (~5% due to uncertainty in length and ~5% due to uncertainty in the calibrated spring constant). This modulus error was subsequently propagated into the theoretical stiffness calculation.

Cantilever

No. 1 No.2 No. 3 No.4 No.5 No.6 No.7

Beam Width (long base) (µm) 37.3 36.4 35.9 37.4 36.6 36.1 37.8

Beam Width (short base) (µm) 17.2 17.2 17.0 17.4 17.7 17.3 17.8

Thickness (µm) 5.0 5.0 5.0 5.0 5.0 5.0 5.0

Effective Beam Length (µm) 106.9 98.1 88.5 55.9 31.3 30.2 21.1

Measured Stiffness, kcantilever 78 85 122 310 1,930 3,710 10,900

(N/m) ±1.0 ±1.4 ±2 ±6 ±60 ±140 ±700

Theoretical Stiffness, ktheory 80 102 137 562 3,190 3,500 10,600

(N/m) ±21 ±27 ±37 ±151 ±850 ±940 ±2,800

Euler beam theory predicts that the stiffness is inversely proportional to the cube of the effective length per Eq. 6.1:

3퐸퐼 푘 = (Eq. 6.1) 푡ℎ푒표푟푦 퐿3 The results in Figure 6.5 agree well with the theory. Based on the measured

variation in beam length, the expected variation in spring stiffness is (107/21)3 = 132x; the observed variation was 136x (=10,900/80 per Table 6.1). Plotting measured stiffness versus effective beam length gives a best fit with a power-law exponent of -

85 3.03 and a coefficient of determination of 0.98 (Figure 6.5a). Fitting the experimental measurements to beam theory yielded a Young’s modulus of E = 120 ± 32 GPa with R2 = 0.98 (Figure 6.5b); although the resulting modulus of E = 120 GPa is lower than the reported value of E = 169 GPa (for the <110> direction) [158], the difference is only ~30% larger than the uncertainty in the fit, ~90% of which is attributable to uncertainty in the thickness measurements (0.5 µm). In the absence of direct calibration, estimates of flexural stiffness, which we expect to be within 50% of the true value based on optical measurements, can be significantly improved with dimensional measurements of reduced uncertainties (e.g. SEM).

86 6.1.4.3 High-force AFM The cantilever deflection sensitivity calibration curves for a stiff and a soft cantilever are shown in Figure 6.6(a). Both cantilevers were subject to significant snap-in forces, pull-off forces, and hysteresis between loading and unloading. The

PSD sensitivities (Cz), which were determined using the linear portion of the loading curves, were 25.2 ± 0.1 and 21.9 ± 0.1 V/µm for the stiff and soft cantilevers, respectively. The ~20% difference between these values is consistent with differences we have observed for repeat measurements of the same cantilever and indicates that spring constants can be modified radically without radically changing deflection sensitivity. For the stiffest cantilever, the resulting normal force calibration constant (ratio of stiffness and deflection sensitivity) of 433 μN/V reflects a possible measurement range of more than 8 mN (± 10V PSD range), which is well within the measurement range of existing microtribometers [159, 160]. For these calibration experiments, a set-point of only 3 V produced normal forces of 1.3 and 0.017 mN for stiff and soft beams, respectively (Figure 6.6b). These results demonstrate that millinewton forces can be achieved in commercial AFM’s using only existing materials and methods.

6.1.4.4 Lateral Force Sensitivity Representative lateral voltage loops are shown for varying set-point voltages for a stiff cantilever (Lever 7) in Figure 6.7a. Low friction between a steel colloid on single-crystal MoS2 produced clean friction loops of statistically significant half-width at set-points down to 0.5 V. The tilt of the friction loop due to sample curvature increased with load, which suggests load-dependent deformation of the adhesive underlying the MoS2 flake. We observed no evidence of load-induced curvature during follow-up testing against Si. It is also worth noting that loops have been centered about zero for visual clarity; friction loop offsets arise in such measurements due to unavoidable misalignment between the sphere and the shear center of the cantilever in addition to other sources of cross-talk [155, 157].

88 The mean lateral signal, which is proportional to the friction force, is plotted versus normal set-point in Figure 6.7b. Friction forces from nominally identical steel colloids sliding against the same MoS2 sample at set-points between 0.5 and 8 V were well within the detectable range of the instrument for a stiff (Lever 7) and a soft (Lever 3) cantilever. Friction from the stiff cantilever was a linear function of load and passed approximately through the origin of the graph; both features are typical of macroscale contacts. Friction from the softer cantilever showed evidence of non- linearity and adhesive friction near zero load; these features are more typical of AFM- based friction measurements, especially when using colloidal spheres. These results suggest that the load regime has a significant effect on frictional behavior and demonstrate that beam-length modification can be used to study these effects in a controlled manner (e.g. same materials, geometries, speeds, etc.).

Figure 6.7 (a) Friction versus position for set-point voltages from 0.5-8 V for a steel colloid against single-crystal MoS2 using Lever 7, the stiffest cantilever in the study (k = 10,900 N/m). (b) Lateral signal (friction loop half- width) versus normal force set-point for a stiff (Lever 7, k = 10,900 N/m) and a soft (Lever 3, k = 122 N/m) cantilever. The application of friction to these beams produced similar voltage responses despite enormous differences in normal stiffness, torsional stiffness, and applied forces.

89 Calibrated friction forces are plotted versus normal force in Figure 6.8 with linear fits for estimates of the corresponding friction coefficients. At low loads (~10 µN) and high loads (~1 mN), mean friction coefficients were ~0.009 and ~0.002, respectively, and consistent with previous measurements of single-crystal MoS2 friction [126, 155]. Although both cantilevers produced comparable minimum friction forces, only the soft beam showed obvious evidence of adhesive friction approaching zero load (400 nN). In fact, the fit to friction results from the stiff cantilever gives a significant negative intercept (-400 nN); it is worth noting that the pull-off force observed during PSD sensitivity calibration of this cantilever was ~100 µN. Together, the results are consistent with significant adhesive friction near zero load, decreasing friction coefficients with increasing load (sub-linear), and a transition toward linear behavior at the millinewton load regime. We are currently using these methods to better understand this transition behavior and how it depends on experimental variables like load, probe radius, pressure, and cantilever stiffness.

Figure 6.8 Lateral force (per the extended wedge method) versus normal force for the stiffest (k = 10,900 N/m) and a soft (k = 122 N/m) cantilevers in the study using two different steel colloid-attached probes and a MoS2 sample. The relationship between friction force and normal force was obviously sub-linear at lower forces with the soft cantilever and closer to linear at higher forces with the stiff cantilever.

6.1.5 Discussion It can be argued that the most important grand challenge in tribology today is the gap in our understanding of macroscale friction phenomena and its underlying atomic-scale processes. Previous studies attempting to link macroscale and nanoscale friction results have done so by comparing them [36, 37, 47, 92, 161] rather than by joining them. To our knowledge, there have been no successful attempts to bridge this measurement gap directly within a single instrument or to match the experimental conditions between nanoscale and macroscale instruments (e.g. an AFM and a tribometer). This paper addresses existing experimental limitations by substantially

91 increasing the accessible load range of commercial AFMs to the millinewton regime where it is possible for more traditional tribometers to operate [159, 162]. Mounting colloidal spheres near the fixed end of commercial cantilevers increased the effective spring constant by as much as 100x. Because these custom high-stiffness cantilevers are difficult to calibrate with confidence using more typical methods, we applied a direct method based on the Nano-Force Calibrator (NFC) [151], which proved reproducible, flexible (from 0.1 and 10,000 N/m), and reliable

(traceable to force and displacement standards). While direct calibration is best for quantitative studies, our results show that beam theory can be used to obtain reasonable estimates of cantilever stiffness, especially if dimensional uncertainties can be reduced to below 100 nm. We demonstrate that the cantilevers fabricated for this study can be used to study friction of model systems like single-crystal MoS2 continuously and with significant overlap from 1 μN to 5 mN; such measurements will help elucidate how friction depends on both load and cantilever stiffness, the latter of which is likely both important and under-appreciated as an experimental variable [132]. Additionally, the load ranges of these AFM-based measurements overlap significantly with those available with more traditional tribometers, several of which have demonstrated the ability to resolve friction forces from comparably low friction material systems at normal forces well below 1 mN [159, 160]. Measurements from a tribometer are typically easier to trace to calibration standards, serve as independent validation, and provide a direct path from mN forces to true macroscale tribological systems. We are now beginning to use the methods developed here in combination with a custom

92 microtribometer (100 nN resolution) in an effort to bridge nanoscale and macroscale friction. An unintended benefit of increased beam stiffness in the AFM is increased sensitivity to super-low friction coefficients. Friction forces for these low friction material systems can reach detections limits of multi-axial load cells whose orthogonal force sensitivities are comparable, which is often the case [163]. The transduction mechanisms of normal and friction forces in the AFM use different deformation modes, which effectively decouples orthogonal force sensitivities. Because torsional stiffness scales with length and normal stiffness with length cubed, sensitivity to low friction coefficients increases with increased stiffness. The ratios between lateral and normal force calibration constants for soft (Lever 3) and stiff (Lever 7) cantilevers were 1.1/5.6 = 0.2 and 9.3/433 = 0.02, respectively, which effectively represent an ideal friction coefficient for each measurement system. It should be noted that the corresponding drawback is reduced range. Using a set-point of 8 V with the stiff Lever

7 would saturate the PSD at any friction coefficient greater than 0.025. Thus, larger friction coefficients will tend to the limit the available load range when using this approach. Finally, we would be remiss to neglect any comment on potential limitations from the AFM chip holder, the compliance of which reduces the normal force calibration constant by an unknowable amount without prior mechanical characterization. Separate stiffness measurements of our holder revealed no detectable indication of compliance (k >> 30,000 N/m) until forces exceeded the clip spring preload limit of ~10 mN where the spring constant fell to ~5,000 N/m; it was a fortunate accident that our holder was designed for our target load range. While we

93 assume that most AFM holders are of similar design and support comparable forces without significant confounding effects, it is advisable to characterize the compliance profile of the AFM holder before attempting AFM measurements above 1 mN.

6.1.6 Conclusions The data presented and analyzed here allow us to draw the following conclusions:

1. Colloid placement provides a controllable means for increasing the stiffness of commercial AFM cantilevers by more than two orders of magnitude.

2. Direct calibration, based on the nano-force calibrator (NFC) design described in a previous paper [151], is traceable to force and displacement standards, easily reproducible, and applicable to cantilevers between 0.1 and 10,000 N/m of flexural stiffness.

Mounting a colloid 21 μm from the fixed end of a 125 μm long cantilever increased the stiffness by ~300x. In a Dimension 3100 AFM, the stiffened cantilever achieved normal forces of 3.4 mN while detecting friction forces from an ultra-low friction system (μ ~ 0.002). Stiffening the cantilever increased normal and lateral force calibration constants while simultaneously improving sensitivity to ultra-low friction coefficients (i.e., as low as 0.001).

6.2 New Methods for AFM Lateral Force Calibration In section 6.1 we showed and validated methods for the production and normal stiffness calibration of high-force AFM probes. However, in achieving my goal of measuring friction across scales, lateral force calibration looms as another major challenge. Absolute lateral forces are exceedingly difficult to quantify in the AFM and this is particularly true when applied to a wide range of cantilever stiffnesses and

94 probe radii. To address the well-recognized problems of existing lateral force calibration methods, we adapted the well-subscribed reference lever method of normal force calibration to the problem of lateral force calibration. The method of retaining traceability is simple conceptually: (1) create a reference lever (a beam of fixed but unknown stiffness); (2) calibrate the lever with the direct calibration method, which is traceable to absolute measurement standards; (3) use the standard reference lever method in lateral mode to quantify the target cantilever spring constant and its uncertainty. The device, the methods we use to calibrate the device, and the methods we use to apply the calibrated device toward AFM lateral force calibration are illustrated in the following figures.

To calibrate the device shown in Figure 6.9 we conducted three independent (each including reconfiguring the position of the beam away and back to the same

96 position) indents at five different micrometer positions, for each stopper configuration and each of the two beams. The results of these three indents were used to run a Monte Carlo simulation in order to obtain the uncertainty of the average fit. After that, the stiffness at each micrometer position for a single stopper position and beam was plotted to obtain a continuous function across all micrometer positions. The fit uses a beam theory equation, while the uncertainty is quantified as a fraction of the stiffness; a straight line is fit through the highest level of uncertainty at each of the Monte Carlo simulated points as shown in Figure 6.10. This process was repeated for each stopper position and each of the two beams.

6000 0.045 0.04 Beam Stiffness 5000

0.035 (N/m) 4000 0.03

Ref Beam Stiffness k 0.025 Fit 3000 0.02

Fraction Beam Stiffness 2000 0.015 Uncertainty 0.01 1000

Beam Beam Stiffness, Maximum 0.005 Uncertainty Beam Stiffness 0 0 Beam Stiffness 0 0.5 1 Uncertainty Fit Micrometer Position, d (in)

97 The following equations are the fitting functions used to obtain the spring constants of the beams at constant positions of the fixed collets.

−3 푘푅푒푓 = 퐴 ∙ (푑푚𝑖푐푟표푚푒푡푒푟 + 퐵) (Eq. 6.2)

푘푅푒푓,푢푛푐푒푟푡푎𝑖푛푡푦 = 푃 ∙ 푑푚𝑖푐푟표푚푒푡푒푟 + 푄 (Eq. 6.3) Table 6.2 lists all constants needed to obtain the beam stiffness at every possible configuration of the calibration device.

Table 6.2 List of constants used for Eqs. 6.2 and 6.3 to obtain the beam stiffness of each configuration of the calibrating device.

Position A B P Q 1 921 0.56 -0.010 0.040 2 661 1.30 -0.003 0.039 Thin 3 723 2.34 0.0051 0.031 Beam 4 719 3.32 0.0037 0.036 5 559 3.98 0.0112 0.037 1 619,000 2.97 -0.014 0.078 2 110,500 2.20 -0.011 0.042 Thick 3 48,000 2.41 0.00807 0.037 Beam 4 69,100 3.68 0.00534 0.037 5 60,300 4.48 -0.00055 0.037

The large range of the different configurations that can be achieved by the reference spring device are all summarized in Figure 6.11. To validate the uncertainty prediction of our calibration method, we conducted calibrations of randomly selected stiffnesses that included one micrometer position for each stopper and beam position.

The targeted values and results of the new validatory calibrations are displayed in Figure 6.12.

98 5 4 3 2 Thin Beam 1 0 Thick Beam

Stopper Position 1 10 100 1,000 10,000 100,000 Range of Available Stiffnesses (N/m)

Figure 6.11 Available stiffness at each configuration of the calibration device.

Based on the spring in series equation discussed previously, we conducted an uncertainty propagation calculation to evaluate the effect of stiffness mismatch between the calibration beam and the AFM beam of interest. Interestingly, it was discovered that, as long there is an appreciable signal coming from the AFM probe, the calibration beam should be as soft as possible. This effect is summarized in Figure

99 6.13 and it determined that we are going to aim very low-stiffness calibration beams when conducting our experiments in an AFM. An example of such lateral stiffness calibration against a soft calibration beam is given in Figure 6.14.

10000

1000 Measurement

Error Percent Error in AFM in Error 100 20 10

5 Lever Stiffness Lever 10 2

1 Resulting Percent Resulting 1 0.01 0.1 1 10 100 k Reference/k Lever Figure 6.13 Theoretical error in the calibration of an AFM cantilever as a function of the uncertainty in the calibrating device. The message of this graph is that errors in the AFM lever calibration are mitigated by the use of the softest reference beam possible.

100

Figure 6.14 Use of the reference spring device to calibrate lateral forces of AFM cantilevers. The AFM beam is oscillated laterally without slipping at the interface. First, we use the existing standard practice to obtain the lateral sensitivity; i.e. the AFM is loaded against a nominally rigid surface (top left) and the lateral sensitivity (V/nm) is obtained from the slope of the corresponding data in the graph. The lateral sensitivity procedure is repeated against the calibrated lateral force reference spring (bottom left) to obtain the combined lateral sensitivity (one known reference spring and one unknown spring in series). The lateral stiffness of the unknown 푆푅 AFM cantilever is: 푘푐푎푛푡𝑖푙푒푣푒푟,푙푎푡푒푟푎푙 = 푘푠푝푟𝑖푛푔,푟푒푓푒푟푒푛푐푒 ∙ ( − 1). 푆푆 Using measured sensitivities (right) and the pre-calibrated reference spring stiffness (5 N/m), the calibrated AFM lateral stiffness of this 푁 cantilever is: 푘 = 73.6 ± 3.34 and the lateral force calibration 푐푎푛푡𝑖푙푒푣푒푟 푚 푘푐푎푛푡푖푙푒푣푒푟 constant is: 퐶퐿퐹 = = 1.58 ± 0.071 휇푁/푉. 푆푅

Interestingly, this same method can serve as an alternative for normal force calibration of AFM probes when conducted in the normal direction. One such example is given in Figure 6.15. The feasibility and reliability of using the reference lever method for normal and lateral force calibration was tested for 4 different AFM probes. The results from this study are summarized in Table 6.3.

101 1.2 1 0.8 0.6 0.4 0.2

Voltage (V) Voltage 0 -0.2 -0.4 -0.6 -200 -100 0 100 200 300 400 Vertical Actuation (nm)

kNormal ,prescribed kNormal kLateral CNormal CLateral Cantilever (N/m) (N/m) (N/m) (µN/V) (µN/V) 38.8 316 1.921 7.94 Integrated 40 (1.77) (37.0) (0.080) (0.34) 12.7 51.4 0.767 1.00 Colloidal 1 (1.03) (3.16) (0.032) (0.06) 78.4 135.9 3.398 2.71 Colloidal 2 (11.6) (6.0) (0.141) (0.12) 32.9 73.6 1.574 1.575 Colloidal 3 (1.79) (3.34) (0.066) (0.071) 79.0 361 5.645 8.80 Colloidal 4 (4.92) (20.4) (0.235) (0.37)

102 Chapter 7

FIRST EVIDENCE OF CONNECTING SCALES

Preliminary results based on the methods presented in this dissertation were collected in order to test our hypothesis that nanoscale friction is related to macroscale friction. In Figure 7.1 we show data from literature, our preliminary AFM measurements of friction for single and multi-asperity contacts, and our microtribometry data. The data shows a good agreement with existing data and a trend approaching a common friction coefficient, after adhesion effects are controlled for. This dataset will serve as the beginning of our further efforts of fully describing the space of variables that define the nano to macro scale transition of friction.

103

104 Chapter 8

CONCLUSIONS AND FUTURE WORK

This dissertation presented methods for conducting studies that elucidate the connection between practical-scale friction and its molecular origins. This was done in the framework of studying the contribution of each asperity to multi-asperity friction. We validated ways of studying the gradual wear of single asperities in a multi-asperity contact, which showed us the precise areas of contact. We also developed ways to measure the contact area in contacts that are wear free. Using traceable calibration methods, we conducting experiments that span continuously from single-asperity AFM to multi-asperity macro scale. These last preliminary observations have inspired us to explore avenues that can establish a model describing our results. And although single-crystal MoS2 has served well in conducting fundamental macroscale experiments, the effect of wear has to be explored further. One way of doing that is to extend our efforts to fully describe the wear of gold and other metals that experience transition in wear modes.

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