Physically Representative Atomistic Modeling of Atomic-scale Friction

Transcription

1 The University of Akron Mechanical Engineering Faculty Research Mechanical Engineering Department 2013 Physically Representative Atomistic Modeling of Atomic-scale Friction Yalin Dong The University of Akron, Main Campus, Please take a moment to share how this work helps you through this survey. Your feedback will be important as we plan further development of our repository. Follow this and additional works at: Part of the Mechanical Engineering Commons Recommended Citation Dong, Yalin, "Physically Representative Atomistic Modeling of Atomic-scale Friction" (2013). Mechanical Engineering Faculty Research This Dissertation is brought to you for free and open access by Mechanical Engineering Department at IdeaExchange@UAkron, the institutional repository of The University of Akron in Akron, Ohio, USA. It has been accepted for inclusion in Mechanical Engineering Faculty Research by an authorized administrator of IdeaExchange@UAkron. For more information, please contact mjon@uakron.edu, uapress@uakron.edu.

2 Graduate School ETD Form 9 (Revised 12/07) PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Yalin Dong Entitled Physically Representative Atomistic Modeling of Atomistic-scale Friction For the degree of Doctor of Philosophy Is approved by the final examining committee: Ashlie Martini Xiulin Ruan Arthur F. Voter Arvind Raman Chair To the best of my knowledge and as understood by the student in the Research Integrity and Copyright Disclaimer (Graduate School Form 20), this thesis/dissertation adheres to the provisions of Purdue University s Policy on Integrity in Research and the use of copyrighted material. Approved by Major Professor(s): Ashlie Martini Xiulin Ruan Approved by: David C. Anderson 03/11/2013 Head of the Graduate Program Date

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4 PHYSICALLY REPRESENTATIVE ATOMISTIC MODELING OF ATOMIC-SCALE FRICTION A Dissertation Submitted to the Faculty of Purdue University by Yalin Dong In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 Purdue University West Lafayette, Indiana

5 UMI Number: All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI

6 ii This dissertation is dedicated to my parents and my lovely wife, who have supported me all the way through my PhD study.

7 iii ACKNOWLEDGMENTS I would like to express my greatest gratitude to my advisor Dr. Ashlie Martini, who leads me all the way through my PhD training. Dr. Martini sets up a model for me in many aspects. Her vision and enthusiasm on research are my most important source of motivation and inspiration. I appreciate all her effort, patience, time, idea, and funding to make me stand here. I am also thankful to my committee members for their help in my PhD pursuit. Dr. Xiulin Ruan s personality and research philosophy are most admirable to me, I will definitely practice the things I learned from him in my future career. In addition, the experience of mingling in Dr. Ruan s Group meeting also significantly broadens my research horizon. Thanks to Dr. Arthur F Voter and his invention of state-of-art accelerated Molecular Dynamics simulation methods, I am able to investigate atomic friction at very low velocity regime. Although I have never gotten the chance to meet with Dr. Voter in person, his professional responsibility is reflected in many e- communication details. A significant amount of my PhD is to simulate atomic friction measured by Atomic Force Microscopy. Many thanks go to Dr. Arvind Raman, for my knowledge about AFM mainly comes from his class Fundamentals of Atomic Force Microscopy. The contributions from my collaborators can never be forgotten. We have been working closely with experimentalists in Dr. Robert W Carpick s research group at University of Pennsylvania. All the experimental data in this thesis originate from them. I am indebted to Dr. Qunyang Li, Dr. Phillips Egberts, and Xin Z. Liu, who take care of the experimental measurements. I also want to acknowledge Dr. Danny Perez in Dr. Voter s group at Los Alamos National Lab. With an admirably impressive physical intuition and programming ability, his ideas and methodologies show up in many chapters of this thesis.

8 iv I am grateful to my dear friend and colleague Ajay Vadakkepatt. During his three years stay at Purdue, I had a wonderful time to share with him my thought on life and research. The members of the Martini s research group contribute to my work in different ways. Special thanks to Jianguo Wu, Lingqi Yang, Xiawa Wu, Anirudh Udupa, Zhenjia Gao, Jose Garcia and Hildur Gylfadottir at Purdue University and Zhijiang Ye, Hongyu Gao, Xiaoli Hu and Dr. Chun Tang at UC Merced. They together serve as a research platform where I can stand higher and dance with support and friendship. I appreciate the time I spent with my friends at Purdue. My life is enriched by friends at Christian church at Lafayette, school of mechanical engineering and Chinese soccer association. At last, the most important credit goes to my lovely, elegant, delightful, supportive, and patient wife. It s your love makes my life meaningful and enjoyable.

9 v TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ix ABSTRACT xix 1. INTRODUCTION ANALYTICAL METHODS FOR ATOMIC FRICTION Tomlinson Model and its Extended Versions Mathematical Formulation and Algorithm Thermal Effects: Temperature, Velocity and Mass Dependence Temperature Speed Mass Transitions between Friction Regimes Friction Modulation due to Surface Reconstruction Dynamic Actuation D Characteristics of Atomic Friction D Corrugation Potential Scan Line Dependence FKT Model: Superlubricity and Area Dependence D FKT D FKT Conclusion MOLECULAR DYNAMICS SIMULATION OF ATOMIC FRICTION Materials Generic Potentials Material-Specific Potentials Tip-Substrate Interactions Surface Structure Substrate Surface Orientation Model Considerations Compliance Contributions to Lateral Stiffness Model Considerations viii

10 vi 3.4 Load and Contact Area Breakdown of Continuum Mechanics MD Measurements of Load Dependence MD Measurements of Area Dependence Temperature Mechanisms of Temperature Dependence Model Considerations Velocity Accelerated MD Methods Parallel Replica Dynamics of Atomic Friction Conclusion INTERPRETATION OF AFM EXPERIMENTS Velocity Dependence of Friction Optimal Match between AFM experiment and MD simulation Parallel Replica Dynamics to Bridge Velocity Gap Attempt Frequencies Discrepancy Friction Modulation on Reconstructed Au (111) Surface AFM Measurement of the Modulated Friction MD Revealed Modulation Mechanism Friction at Graphite Step Edge AFM Measurement and MD Simulation Comparison between Simulation and Experiment Discrepancy in Peak Width Real-time Tip Wear Monitor Conclusion RATE PROCESS IN ATOMIC FRICTION Thermal Activation in Stick-Slip Friction: a Theoretical Revisiting Form of Energy Barrier Variable Attempt Frequency Velocity and Temperature Dependance Instrumental Noise Inherent in AFM Measurement Modeling Instrumental Noise: Master Equation Method Suppression of Atomic Friction under Cryogenic Conditions Towards Understanding the Attempt Frequency Discrepancy Apply Rate Theory to MD Simulation Wear as a Rate Process Adhesive Wear: Archard s Wear Model Thermal Activation Wear: Rate Process SUMMARY Conclusion Remarks Outlook

11 vii LIST OF REFERENCES VITA

12 viii Table LIST OF TABLES 2.1 Typical values or range of values for the parameters used in the PT model to describe an AFM friction experiment [16 19] Fourth-order, time-varying Runge-Kutta coefficients The parameters used in the fitting for the wear between a silicon tip and copper substrate

13 ix Figure LIST OF FIGURES 2.1 Illustrations of the one-dimensional PT, FK and FKT models. Large blue spheres represent tip atoms and rectangular slabs represent the sliding support. Inset with the PT model is a schematic of the relationship between an AFM tip/cantilever and the simple mass-spring model. Variables are defined in the text Atomic friction versus support displacement at T =300K solved by the fourth-order Runga-Kutta algorithm (red) and Ermak s method (blue). For reference, the friction at T =0K which is the same for both methods is also presented (black). Other model parameters: m = kg, k = 1N/m, U = 0.6eV, v = nm/s, µ = 2 k/m, a = 0.288nm An illustration of slip between two adjacent energy minima. p i is the probability of the tip residing in the current potential well, i, where the energy barrier is V i j. p j is the probability of the tip residing at the next minima, j, where V j i is the corresponding energy barrier Illustration of the temperature dependence of friction. The two regimes identified on the plot, thermal activation and thermal drift, are described in the text. Other model parameters: m = kg, U = 0.6eV, v = nm/s, µ = 2 k/m, a = 0.288nm Representative force traces in the thermal activation (top) and thermal drift (bottom) regimes identified on Figure 2.4. Several characteristic forward and backward slips are identified by blue dashed lines on the friction trace in the thermal drift regime Speed dependence of friction illustrating two different regimes. In the thermal regime there is a logarithmic scaling of friction with speed, and in the athermal regime the friction is governed by the damping term such that F v. The friction plateau (F c =0.39nN) predicted by thermal activation is identified by the dashed line. Other model parameters: m = kg, U = 0.6eV, T = 300K, v = nm/s, µ = 2 k/m, a = 0.288nm The effect of tip mass (inertia) on the temperature and speed dependence of friction. Other model parameters: U = 0.6eV, µ = 2 k/m, a = 0.288nm

14 x Figure 2.8 Illustration of potential energy with varying number of energy minima for different values of the transition parameter η The effect of damping on transitions between slip regimes where µ c = 2 k/m is the critical damping coefficient. Single, double and triple slip occur at µ = µ c, 0.6µ c and 0.3µ c, respectively. The abscissa has units of the lattice spacing a to facilitate identification of the transitions between single, double and triple regimes. Other model parameters: U = 0.6eV, T = 0K, v = 1µm/s, m = kg, k = 1N/m, a = 0.288nm The effect of sliding speed on transitions between slip regimes. Single, double and triple slip occur at v = 100µm/s, 1µm/s and 0.01µm/s, respectively. Other model parameters: U = 0.6eV, T = 0K, µ = 0.8µ c, m = kg, k = 1N/m, a = 0.288nm The effect of temperature on transitions between slip regimes. Other model parameters: U = 0.6eV, v = 1µm/s, µ = 0.6µ c, m = kg, k = 1N/m, a = 0.288nm Amplitude (friction loop width) modulation induced by multiplying the sinusoidal corrugation potential by a long-range modulation term (top) and the resulting force trace (black) and retrace (blue) predicted by the PT model (bottom); T =0K Centerline (friction loop offset) modulation induced by adding the sinusoidal corrugation potential to a long-range modulation term (top) and the resulting force trace (black) and retrace (blue) predicted by the PT model (bottom); T =0K Characteristic vibrational modes of the micro-cantilever. The normal mode that can be excited by a mechanical modulation of the normal force, the torsional mode that can be excited by a mechanical modulation of torsion. F n is the normal force and F ts is the interaction force between tip and substrate Friction dependence on α, the parameter that describes actuation of the normal mode of an AFM system. Inset is the variation of friction with excitation frequency (units ) for different α. Other model parameters: k m m = kg, U 0 = 1.2eV, v = nm/s, µ = 2 k/m, a = 0.288nm, k = 4N/m and T = 0K

15 xi Figure 2.16 Friction dependence on γ, the parameter that captures actuation of the torsional mode of an AFM system. Inset is the variation of friction with excitation frequency (units ) for different γ. Other model parameters: k m m = kg, U = 1.2eV, v 0 = nm/s, µ = 2 k/m, a = 0.288nm, k = 4N/m and T = 0K Illustration of atom arrangements and energy landscapes on FCC(100) and FCC(111) surfaces. The energy unit is U which is the amplitude in Equations 2.39 and Potential energy landscape of an FCC(111) surface (top) illustrating the position of the point tip (solid lines) traveling along different scan lines (dashed lines) and corresponding force traces (bottom). Results shown at T = 0K to clearly illustrate the motion paths Lateral force images of an FCC(111) surface for different system stiffness, k. Blue spheres identify the positions of atoms and show that the larger stiffness yields better the atomic resolution. Other model parameters: m = kg, U = 0.6eV, v = nm/s, µ = µ c = 2 k/m, lattice spacing in the [110] direction a = 0.288nm Potential energy landscape of an FCC(111) surface (top) illustrating the position of the point tip (solid lines) as the support is moved in different directions from the same starting point (dashed lines), and corresponding force traces (bottom). Results shown at T = 0K to clearly illustrate the motion paths Friction variation with the tip size N with different lattice mismatch b a. k = 5N/m and k t =50N/m are used to obtain these results Two dimensional FKT model where the tip atoms (large blue spheres) are connected to each other and to the support by harmonic springs The misfit angle dependence of friction with different tip sizes; k t = 50N/m and k = 10N/m. The N = 7 curve corresponds to the model illustrated in Figure

16 xii Figure 3.1 (a) Schematic of an AFM experiment, and a nm 2 topographic AFM image of the Au(111) surface showing large terraces separated by monatomic steps. Inset above: TEM (Transmission Electron Microscopy) image of the Pt-coated probe. (b) Snapshot of an atomistic tip/substrate model. (c) Lateral force image on Au(111). Inset: Fourier low-pass filtered image. (d) Top view of the model Au(111) substrate. White arrows in (c) and (d) denote the fast scanning direction. Scale bars are 1 nm. (e,f) The experimental (e) and simulated (f) lateral force along the black horizontal line shown in (c) and (d) respectively. The simulation and experimental results are obtained under optimally matched conditions: materials P- t/au(111), incommensurate orientation, effective stiffness 6 N/s, contact area 7.3 nm 2, normal load 0.6 nn, and temperature 293 K. Only sliding speeds differ significantly: 149 nm/s in experiment and 1 m/s in simulation Friction from a Pt tip sliding on the (a) Au(100) and (b) Au(111) surfaces from MD simulation. In (a) the Pt tip is worn off almost immediately due to junction formation and no stable friction pattern observed while in (b) a stable bimetallic interface forms and regular stick-slip friction arises. Note that the scales of the y-axes differ in the two plots. Simulation parameters: EAM potential, load 0 nn, temperature 10 K, speed 1 m/s, contact area 1.2 nm 2 (aligned) MD simulation of mean friction as a function of rotation angle between Pt(111) tip and Au(111) substrate. Due to the similar lattice constants (a P t = nm and a Au = nm) and FCC structure of both materials, high friction arises at aligned contact and low friction at misaligned contact. Inset: representative friction traces at aligned and misaligned contact where S is the position of the support. Simulation parameters: EAM potential, load 0 nn, temperature 10 K, speed 1 m/s, contact area 7.3 nm Upper images: Atomic distribution in the buried interface (top) where the silver dots are substrate atoms and the red dots are tip atoms. The relative rotation of the surfaces forms superstructures called Moiré patterns. Lower images: Shear stress (color scale in units of bar Å 3 ) distribution of the atoms in the lowermost layer of the tip. Atom and stress distributions shown at relative rotation angles of (a) 0, (b) 15, and (c) 30. Simulation parameters: EAM potential, load 0 nn, temperature 10 K, speed 1 m/s, contact area 7.3 nm A representative plot of lateral force (friction) vs. displacement; the linear slope of the curve is the measured total effective stiffness of the system. 62

17 xiii Figure 3.6 Schematic illustration of the elastic contact between spherical bodies used to derive an expression for stiffness in the framework of contact mechanics. Variables are defined in the text Illustration of an MD simulation with compliance introduced through three harmonic springs (k x, k y, and k z ) connected to the top layers of the tip apex. Along the x-direction, the support moves at contact speed to drag the tip apex to slide against the substrate Illustration of different views of the interaction between an AFM tip and substrate. Continuum mechanics assumes the tip and substrate are homogeneous elastic materials so that contact area can be determined from the geometry and elastic properties. On the other hand, the atomic view takes individual atoms into consideration and the definition of contact becomes more complicated and depends on many more factors MD simulation of mean friction as a function of normal load. Inset are illustrations of atomistic configurations during sliding at N =10nN (left) and N=17.5nN (right) Friction as a function of contact area at different orientation angles between Pt(111)/Au(111) measured by MD simulation; 0 corresponds to perfectly aligned contact Illustrations of (a) an AFM cantilever pulling the tip to slide over a substrate and (b) the corresponding Prandtl-Tomlinson (PT) model in which the AFM system is reduced to a mass-spring system. In the PT model, the model tip is confined within the potential well and hops over the energy barrier with the assistance of thermal activation (a) Temperature of the contact interface varying with sliding distance from an MD simulation with Pt(tip)/Au(substrate) system. The initial temperature of the simulation is 300 K. Once sliding begins, the average temperature increases with distance if there is no thermostat to remove thermal energy from the system (solid line) but fluctuates about the prescribed temperature if the Nosé Hoover thermostat is applied (dashed line). (b) Mean friction as a function of temperature from an MD simulation of atomic friction between a Pt(tip) and Au(substrate). Inset is a schematic illustrating a thermostat being applied to atoms away from the contact in a simulation of atomic friction. Simulation parameters: EAM potential, load 0 nn, speed 2 m/s, contact area 1.3 nm 2 (aligned)

18 xiv Figure 3.13 (a) Schematic of hyperdynamics. The potential is modified by adding a positive bias δv (r) to increase the transition rate. (b) Schematic of parallel replica dynamics. The original system is replicated to help find the transition path (a) Relative friction variation (variation divided by mean friction) predicted using parallel replica dynamics when run on 128 replicas vs. 64 replicas (circles and insets) and the expected relative variation due to MD simulation noise (solid line). (b) Mean friction as a function of sliding velocity measured between Cu(111)/Cu(111) interface: squares represent ParRep simulation and the solid line is a logarithmic fit to that data. Simulation parameters: EAM potential, load 0 nn, temperature 300 K, contact area 1.3 nm 2 (aligned) Heuristic representation of the parallel replica dynamics (ParRep) method in which the atomic stick-slip simulation is run parallel in time across multiple processors. Circles represent atomic positions: red - substrate, solid blue - tip at current time, dashed line - tip at previous time. Note that this illustration does not include the critical ParRep phases of minimization, dephasing, and decorrelation which are described in the text Mean friction measured by AFM (squares) for speeds between 1 and 1000 nm/s, and predicted via ParRep MD simulation (triangles) for speeds between m/s to 2 m/s between Pt/Au(111). The dashed curve and dotted curve are fitted with thermal activation model (Eq. 3.8) for experimental and simulation data respectively. All other parameters from materials, orientation, effective stiffness, contact area, normal load and temperature between AFM and MD are optimally matched based on the methods introduced in previous section AFM lateral force images of a reconstructed Au (111) surface at (a) a 50 nm scan size, (b) a 20 nm scan size. (c) A schematic of the Au(111) herringbone reconstruction showing two domains with different atomic stacking (FCC and HCP). Scan directions are from left to right in (a) and (b). The dotted lines are drawn to highlight the pattern of the friction variation. The atomic lattice is resolved in (b)

19 xv Figure 4.4 (a) Line profiles showing both the raw experimental and the low-pass filtered lateral force signal (trace direction). (b,c) The low-pass filtered data for: (b) trace and retrace signals, and (c) centerline and half difference (friction) of the trace and retrace signals. (d,e) Low-pass filtered AFM lateral force images highlighting the centerline modulation effect: ( d) trace (left to right), and (e) retrace (right to left). The contrast is clearly in phase, showing that the variations are primarily due to local centerline modulation, not varying friction (a) The simulation model of a Pt tip sliding over a gold substrate. (b) A schematic showing the atomic configuration of the reconstructed Au(111) surface: yellow spheres represent the atoms from the top layer and blue spheres represent those from the second layer. Yellow spheres on the lefthand-side sit on FCC sites (ABC stacking) and they gradually shift to HCP sites (ABA stacking) in the middle region, and then they gradually shift back to FCC sites on the right-hand-side region. The upper line profile shows the height variation (rumpling) of the reconstructed surface (a) Lateral force vs. lateral displacement curves obtained from simulation on the reconstructed Au(111) surface with relatively compliant loading springs: black curve for raw data, red curve for low-passed filtered data. (b) The low-pass filtered lateral force signal for trace (red dashed curve) and retrace (blue dash-dot curve). (c) The filtered centerline modulation (green dashed curve) and friction (orange dash-dot curve) curves are also plotted side-by-side with the surface height profile (black solid curve). (d) The potential energy relative to its mean value (blue circles) is plotted side-by-side with the surface height profile (black solid curve) (a) Lateral force vs. lateral displacement curves obtained from simulation on a modified reconstructed surface that is atomically flat, i.e., where the height rumpling is deliberately suppressed with compliant loading springs: black curve for raw data, red curve for low-passed filtered data. (b) The low-pass filtered lateral force signal for trace (red dashed curve) and retrace (blue dash-dot curve). (c) The filtered centerline modulation (green dashed curve) and friction (orange dash-dot curve) curves are also plotted side-by-side with the surface height profile (black dash-dot curve) Snapshot of the molecular dynamics simulation in which a model tip slides from left to right, or in the step-up direction, over a graphite substrate. The stiffness of the cantilever is mimicked by a harmonic spring with lateral stiffness K l through which the support, moving with constant speed v, pulls the tip along the substrate. A close-up of the zigzag step edge is shown in the inset

20 xvi Figure 4.9 (a) A typical lateral force profile obtained from MD simulation with a hemispherical tip radius of 4 nm at an applied load of 0 nn. A peak in the lateral force whose width is identified by w is observed at a sliding distance of approximately 3.5 nm. The inset illustrates the tip s trajectory over the surface with a geometric drawing showing the relationship between the tip, the peak width w, and the height h of the graphite step. (b) A representative lateral force profile from an AFM experiment where the tip slides in the step up direction acquired at an applied load of 2 nn. Similar to the result from MD simulation, the width of the lateral force peak at the step is identified by w Simulation-predicted width of the lateral force peak as a function of tip radius. A snapshot of a hemispherical tip with a radius of R = 4 nm is shown in the inset. Measurements of the peak width shown here are acquired at an applied load of 0 nn and using free boundary conditions in the y-direction Experimentally-measured friction peak width as a function of tip radius. Post-mortem TEM images of the tips used in each width measurement are shown in the insets of the figure. All tips were used in the UHV-AFM system except for the smallest tip, which was used in the environmental system. Measurements were acquired in contact mode at a constant applied load of 2.0±0.5 nn Results from MD simulations showing lateral force peak width for a 2 nm radius tip versus an increasing degree of truncation. The truncation is illustrated geometrically by the schematic in the upper-left inset. In the bottom right inset, three representative lateral force traces are given to graphically show the change in lateral force profile with increasing truncation. All measurements shown in this figure were acquired at an applied load of 0 nn with fixed boundary conditions Measurement of step width variation with sliding distance measured in the environmental AFM. The initial tip radius was 19 nm based on blind tip reconstruction (BTR) conducted on a Nioprobe surface using the BTR function of SPIP (Image Metrology Inc. Denmark). The normal load was 0.9 nn throughout the tip wear measurement Snapshot of a model truncated tip (R = 3 nm, T = 0.4 nm) sliding up the graphite step with free boundary conditions in the y-direction. We observed significant compression of the step before the tip slips forward over the step edge Illustration of the local minima x a and saddle point x b at support position S, as well as the critical point (x c, S c ) where the energy barrier vanishes. 137

21 xvii Figure 5.2 Comparison of the energy barrier obtained by numerical solution (triangles) to predictions of the analytical model with β given by Riedo et al. [30] (dotted line), reported by Furlong et al. [10] (dashed line), and fitted to the analytical equation using the numerical data (solid line) Comparison of the analytical expression for the variation of attempt frequency with friction (Eq. 5.17) to numerical data for a sinusoidal corrugation potential Average friction as a function of temperature at different sliding velocities. F v (T ) is relation with variable attempt frequency and F c (T ) is the relation with constant attempt frequency (a). schematic for the potentials wells available in atomic friction system. (b).illustration for an Atomic Force Microscopy system where a nano tip mounted to a micro cantilever slides against the substrate Illustration of two-spring, two-mass Tomlinson model. The support moves at a constant velocity v to drag the cantilever and tip slides on the substrate. k t and k c are the tip and cantilever stiffness, respectively Prefactor as a function of support displacement (S) demonstrating the effect of cantilever (a) and tip apex (b) masses. Other parameters used in the calculations are the magnitude of the corrugation U 0 = 1.7eV, the cantilever stiffness k c = 10N/m, and the tip apex stiffness k t = 1N/m. the cantilever mass m c = kg is applied in (b) and the tip apex mass m t = kg is used in (a) Variation of average friction with velocity and temperature with consideration of both thermal and instrument noise. Inset depicts the average friction as a function of temperature at different velocities (a). Numerical result of maximum friction as a function of sliding velocity and temperature with only thermal noise considered. (b). Numerical result of maximum friction as a function of velocity and temperature with both thermal noise and instrumental noise considered Velocity dependence of atomic maximum friction with both thermal and instrumental noise at different temperatures. A transition point exists at the velocity dependence curve as indicated by an arrow barrier (red continuous line) and attempt frequency for slip (blue dotted line) between two adjacent local minima as a function of support position. Inset: Geometric average of the vibration frequencies (red continuous line) and deflection of the tip (blue dotted line) for one minima state as a function of support position

22 xviii Figure 5.12 Average friction force as a function of velocity and temperature. Red surface: kinetic model; blue symbols and dotted line: atomistic parallel replica dynamic simulations Experimental data of tip radius as a function of sliding distance obtained from AFM measurement where a silicon tip with oxidized tip end slides against a copper substrate at normal load F n = 100nN. The fitting equation 5.43 is from Archard s wear model. The initial tip radius R 0 = 33nm and the constant k = m 2/3 N 1/3. Data from Prof. Sriram Sundararajan s group at IOWA state university Right side is illustration of the conical tip. The big circle with radius R is the tip end after wear and R 0 is the initial tip end. On the left side is the schematics of top view of contact area. a is the contact area. The total number of atoms at contact N c = πa2 A s, where A s is the area occupied by a single atom The experimental data is fitted to Eq The only free parameter is ξv a. ξ = 0.2 and V a = V primitivecell / The value E k B as a function of sliding distance, where the energy barrier T E = E τv a

23 xix ABSTRACT Yalin, Dong Ph.D., Purdue University, August Physically Representative Atomistic Modeling of Atomic-scale Friction. Major Professor: Ashlie Martini, School of Mechanical Engineering. Nanotribology is a research field to study friction, adhesion, wear and lubrication occurred between two sliding interfaces at nano scale. This study is motivated by the demanding need of miniaturization mechanical components in Micro Electro Mechanical Systems (MEMS), improvement of durability in magnetic storage system, and other industrial applications. Overcoming tribological failure and finding ways to control friction at small scale have become keys to commercialize MEMS with sliding components as well as to stimulate the technological innovation associated with the development of MEMS. In addition to the industrial applications, such research is also scientifically fascinating because it opens a door to understand macroscopic friction from the most bottom atomic level, and therefore serves as a bridge between science and engineering. This thesis focuses on solid/solid atomic friction and its associated energy dissipation through theoretical analysis, atomistic simulation, transition state theory, and close collaboration with experimentalists. Reduced-order models have many advantages for its simplification and capacity to simulating long-time event. We will apply Prandtl-Tomlinson models and their extensions to interpret dry atomic-scale friction. We begin with the fundamental equations and build on them step-by-step from the simple quasistatic one-spring, onemass model for predicting transitions between friction regimes to the two-dimensional and multi-atom models for describing the effect of contact area. Theoretical analysis, numerical implementation, and predicted physical phenomena are all discussed. In the process, we demonstrate the significant potential for this approach to yield new fundamental understanding of atomic-scale friction.

24 xx Atomistic modeling can never be overemphasized in the investigation of atomic friction, in which each single atom could play a significant role, but is hard to be captured experimentally. In atomic friction, the interesting physical process is buried between the two contact interfaces, thus makes a direct measurement more difficult. Atomistic simulation is able to simulate the process with the dynamic information of each single atom, and therefore provides valuable interpretations for experiments. In this, we will systematically to apply Molecular Dynamics (MD) simulation to optimally model the Atomic Force Microscopy (AFM) measurement of atomic friction. Furthermore, we also employed molecular dynamics simulation to correlate the atomic dynamics with the friction behavior observed in experiments. For instance, ParRep dynamics (an accelerated molecular dynamic technique) is introduced to investigate velocity dependence of atomic friction; we also employ MD simulation to see how the reconstruction of gold surface modulates the friction, and the friction enhancement mechanism at a graphite step edge. Atomic stick-slip friction can be treated as a rate process. Instead of running a direction simulation of the process, we can apply transition state theory to predict its property. We will have a rigorous derivation of velocity and temperature dependence of friction based on the Prandtl-Tomlinson model as well as transition theory. A more accurate relation to prediction velocity and temperature dependence is obtained. Furthermore, we have included instrumental noise inherent in AFM measurement to interpret two discoveries in experiments, suppression of friction at low temperature and the attempt frequency discrepancy between AFM measurement and theoretical prediction. We also discuss the possibility to treat wear as a rate process.

25 1 1. INTRODUCTION The Amontons law developed 300 years ago describes the friction force at the macroscopic level in an empirical way, F = µl, in which the friction force F is linearly dependent on the load L, but independent of the contact area A m. With the development of new technology and surface science, people have begun to realize that surfaces at the macroscopic level and even at microscopic level are rough. With the help of STM (Scanning Tunnel Microscope) and AFM (Atomic Force Microscope), the topography of surfaces can be visualized. Now it is well known that the real contact area A real is much smaller and the apparent contact area is constituted by many microscale or nanoscale junctions which are called asperities. It has been proposed that fundamental insight into frictional phenomena might be gained by studying single asperity friction, where a single asperity is considered to be the basic element of friction on any length scale. The small size scale of a single asperity means that individual atoms may play a role in resisting motion, so single asperity friction is often called atomic-scale friction [1]. In single asperity friction, the friction coefficient and wear rate which are the focus of macroscopic tribology are no longer enough to describe the friction property an nanoscale; many more parameters, like crystallography, orientation, sliding direction, the chemical environment, temperature, sliding speed and so on get involved to collectively determine the friction behavior [2]. From the perspective of engineering, with the development of nanotechology and the demand of miniaturizing devices in MEMS/NEMS systems, there is an urgency to understand and control the friction at nanoscale. Due to the strong adhesion and wear at the nano and micro scales, there is still no commercialized sliding mechanical component in MEMS and NEMS systems. People are expecting to find ways to control friction at the nano scale by gaining fundamental understanding of single-asperity friction.