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Theoretical Modeling and Molecular Dynamics Simulation of Atomic Theoretical Modeling and Molecular Dynamics Simulation of Atomic Scale Wear: a Combined Study by Yuchong Shao A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy. Baltimore, Maryland August, 2016 c Yuchong Shao 2016 All rights reserved Abstract In this dissertation work, I have developed an analytical theory and performed molecular dynamics simulations of atom-by-atom wear that arises during sliding con- tact between a nanoscale asperity and a surface. The \multi-bond" theoretical wear model treats interfacial bond formation, wear-less rupture and transfer of atoms as three competing thermally activated processes. It is numerically implemented and asymptotic analysis is performed on its predictions. Numerical and asymptotic theo- retical calculations are compared to experimental atomic force microscope measure- ments in the literature and those performed by research collaborators. I have also simulated wear in passivated diamond-like carbon using molecular dynamics tech- niques and have analyzed these data in light of the \multi-bond" wear theory. First Reader: Michael L. Falk Second Reader: Mark O. Robbins ii Acknowledgments First and foremost, I wish to thank my advisor, Michael Falk, a source of in- spiration and guidance throughout the process. Although there were many difficult times during the course of my research, he always encourages me and keeps my study in the right direction. I have learned so much from his broad knowledge of physics and theory about computer simulation. His high standard of scientific research and persistent pursuit of creativity will benefit me for years to come. I would like to gratefully thank Prof. Mark Robbins, Prof. Peter Armitage, Prof. Jeffrey Gray and Prof. Jaafar El-Awady for serving on my PhD committee and providing scientific guidance along the way. I would also like to express my gratitude towards the Department of Physics and Astronomy, as well as Department of Materials Science and Engineering at Johns Hopkins. The administrative staff members are extremely helpful and friendly. I am also fortunate to have excellent collaborators: Prof. Robert Carpick, Prof. Kevin Turner and Yijie Jiang at University of Pennsylvania; Prof. Tevis Jacobs at University of Pittsburg and Dr. Jingjing Liu at Applied Materials. I am also grateful iii ACKNOWLEDGMENTS to former and current Falk research group members for their camaraderie and much fun, to name a few, Prof. Woo-Kyun Kim, Dr. Pavan Valavala, Dr. Sylvain Patinet, Prof. Pengfei Guan, Dr. Tonghu Jiang, Dr. Shengwei Deng, Dr. Peng Yi, Rongguang Xu, and one of my best friends Darius Alix-Williams. Last but not least, I would like to thank my parents for their unconditional sup- port. iv Dedication To my family. v Contents Abstract ii Acknowledgments iii List of Tables x List of Figures xi 1 Introduction 1 1.1 Tribology: a historical note . 1 1.2 Nano-tribology and AFM . 3 1.3 Introduction to wear . 5 1.3.1 Classical wear theories . 5 1.3.2 Nano-scale wear . 7 1.4 Molecular dynamics simulation . 9 1.4.1 Main achievements . 9 1.4.2 Challenges and issues . 12 vi CONTENTS 1.5 Structure of the thesis . 15 2 Experiments and theoretical models of nano-tribology: a review 17 2.1 AFM Experiments . 18 2.1.1 Atomic Force Microscopy . 18 2.1.2 Atomic stick-slip . 19 2.1.3 Smooth sliding and superlubricity . 20 2.2 Continuum contact mechanics . 21 2.2.1 Hertzian contact . 23 2.2.2 Adhesive contacts: JKR and DMT limits . 24 2.2.3 Maugis-Dugdale theory . 25 2.2.4 The breakdown of continuum contact theory . 26 2.3 The Prandtl-Tomlinson model . 27 2.4 The multi-bond friction model . 33 2.5 Introduction to atomistic wear . 41 2.5.1 Experimental investigations of wear . 43 2.5.2 A review of nano-scale wear models . 46 3 A multi-bond model of single-asperity wear at the nano-scale 55 3.1 Methodology . 57 3.2 Results and discussion . 61 vii CONTENTS 3.2.1 Low effective normal stress regime . 61 3.2.2 Intermediate effective normal stress regime . 65 3.2.3 High effective normal stress regime . 68 3.3 Conclusion . 71 3.4 Appendix . 72 3.4.1 Derivation of wear rate . 72 3.4.2 Supplementary analysis of the high stress regime . 77 3.4.3 Summary of Fitting Procedure . 82 4 Molecular dynamics investigations of atomistic wear in diamond-like carbon 84 4.1 Introduction . 84 4.2 Simulation methods . 86 4.3 Results and discussion . 90 4.3.1 The evolution of contact area . 92 4.3.2 The evolution of tip shape . 99 4.3.3 Analysis in light of the \multi-bond" wear model . 101 4.4 Conclusion . 109 5 Conclusion and future work 112 5.1 Conclusion . 112 viii CONTENTS 5.2 Future work . 113 Bibliography 117 Vita 146 ix List of Tables 3.1 Table for calculated values of effective Hertzian load (FH ), contact radius (a) and average effective normal stress (¯σ) corresponding to all the external loads (N) that are surveyed in Ref. [159]. The activation volume Vact is assumed to be constant. Other relevant parameters are: the radius of the tip R = 4 nm; the Poisson's ratio of the DLC γ = 0:234 and the effective modulus E∗ = 599:8 GPa. 82 x List of Figures 1.1 Schematic of Rabinowicz's abrasive wear model [42]. 6 2.1 The average friction force measured in Ref. [75] between graphite sur- faces, plotted against rotation angle φ of the graphite sample with re- spect to the normal direction of the sample surface. Two sharp peaks of high friction are observed at the angles of 0 and π=3 amongst a wide range of ultra-low friction. The curve through the data points shows results from a calculation according to the Tomlinson model for a symmetric 96-atom graphite flake sliding over the graphite surface. Reprinted with permission from [75]. 22 2.2 Schematic of the one-dimensional Prandtl-Tomlinson model in the sys- tems with compliant springs where stick-slip take place. The total po- tential energy V is plotted as a function of the displacement of the tip, which is a superposition of the elastic strain energy of the cantilever (parabolic shape) and the periodical sinusoidal curve that represents the lattice energy (dotted line). These three solid curves represent the evolution of the energy landscape as the cantilever is displaced from its initial position A to B by dAB, and from A to C by dAC . The tip, represented by the black dot, is shown to be confined within the local minima of A and B, but becomes unstable and ready to slip over one atomic position at the configuration of C. 29 2.3 Schematic of a simplified potential energy landscape with two wells. 30 2.4 The effect of temperature on velocity dependence of friction. The dependence is logarithmic below 150 K and constant above 150 K. Reprinted with permission from [34]. 34 2.5 Schematic of the original multi-bond model [33]. Rates of bond forma- tion and breaking processes are governed by the corresponding energy barriers ∆Eon and ∆Eoff. ........................ 36 xi LIST OF FIGURES 2.6 Numerical simulation results of the original multi-bond model. Here the first three figures represent time evolution of frictional forces and bond formation ratios at three different temperatures. A transition from the characteristic stick-slip behavior at relatively low temper- ature (a) to the stochastic behavior of simultaneous multiple bonds formation and breaking processes at increased temperature (b), and fi- nally to the almost continuous sliding at even higher temperature (c). (d) Illustration of the change of characteristic time scales of the system with temperature. 37 2.7 Comparison of friction simulations by the multi-bond model and ex- periments conducted by Jansen and Schirmeisen on Si wafer [33]. (a) Experimentally measured friction-versus-temperature curves for three different sliding speeds. (b) Numerically simulated mean friction as a function of temperature for the same scan velocities. (c) Experimen- tally measured friction as a function of scan velocity at three different temperatures, showing characteristic negative and positive slopes. (d) Simulated friction-velocity curves for nearly the same temperatures with qualitatively same signature. The parameters used in simulations 0 0 10 are: !on = !off = 10 Hz, ∆Eon = 0:05 eV, ∆Eoff = 0:15 eV, κ = 1 −6 N/m, fc = 0:16 nN, N = 20, η = 5 × 10 kg/s, k = 6 N/m, and M = 5 × 10−11 kg. Reprinted with permission from Ref. [33]. 40 2.8 (a) Wear volume Nlost demonstrates a linear relationship with respect to sliding distance d; (b) Wear volume per unit distance @Nlost=@d shows a roughly linear dependence upon the externally applied load. Reprinted with permission from Ref. [159]. 43 2.9 Several AFM-based experimental investigations [44,45,55,57] of single- asperity wear. (a). The growth rate of wear pits versus normal contact force on a calcite surface during repeated linear scanning of a Si3N4 tip in an aqueous solution [55]. (b). Wear rate of a NaCl substrate versus the shear stress during the sliding of a Si3N4 tip at two different temperatures in the dry nitrogen environment [57]. (c). The change in tip radius with the sliding distance where the solid line represents a fit to a model in which the wear rate is allowed to vary exponentially with shear stress and the dashed line represents the fit in accordance with the Archard equation [44]. (d). The change in tip radius and pull-off force versus sliding distance for one of the three Si-DLC tips used in [45].
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