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A molecular-dynamics study of the frictional anisotropy on the 2-fold surface of a d-AlNiCo quasicrystalline approximant

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Title:
A molecular-dynamics study of the frictional anisotropy on the 2-fold surface of a d-AlNiCo quasicrystalline approximant
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Book
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English
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Harper, Heather McRae
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University of South Florida
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Tampa, Fla
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Subjects / Keywords:
Atomistic simulation
Quasicrystals
Nano-tribology
Contact mechanics
Aperiodicity
Dissertations, Academic -- Physics -- Masters -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Abstract:
ABSTRACT: In 2005, Park et al. demonstrated that the 2-fold surface of a d-AlNiCo quasicrystal exhibits an 8-fold frictional anisotropy, as measured by atomic-force microscopy, between the periodic and aperiodic directions 37, 38. It has been well known that quasicrystals exhibit lower friction than their crystalline counterparts 35, 17, 47, 28, 12, 50; however, the discovery of the frictional anisotropy allows for a unique opportunity to study the effect of periodicity on friction when chemical composition, oxidation, and wear are no longer variables. The work presented herein is focused on obtaining an understanding of the mechanisms of friction and the dependence of friction on the periodicity of a structure at the atomic level, focusing on the d-AlNiCo quasicrystal studied by Park et al. Using the LAMMPS 41 package to simulate the compression and sliding of an 'adamant' tip, see section 3.3, on a d-AlNiCo quasicrystalline approximant substrate, we have demonstrated, in preliminary results, an 8-fold frictional anisotropy, but in more careful studies the anisotropy is found to be much smaller. The simulations were accomplished using Widom-Moriarty pair potentials to define the interactions between the atoms 33, 52, 51, 9. The studies presented in this work have shown a clear velocity dependence on the measured frictional response of the quasicrystalline approximant's surface. The final results show between a 1.026-fold and 1.127-fold anisotropy between sliding in the periodic and 'aperiodic' directions, depending on the sliding velocity.
Thesis:
Thesis (M.S.)--University of South Florida, 2008.
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Includes bibliographical references.
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by Heather McRae Harper.
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A Molecular-Dynamics Study of the F rictional Anisotrop y on the 2-fold Surface of a d -AlNiCo Quasicrystalline Appro ximan t b y Heather McRae Harper A thesis submitted in partial fulllmen t of the requiremen ts for the degree of Master of Science Departmen t of Ph ysics College of Arts and Sciences Univ ersit y of South Florida Major Professor: Da vid Rabson, Ph.D. William Matthews Jr., Ph.D. Brian Space, Ph.D. Matthias Batzill, Ph.D. Date of Appro v al: Septem ber 16, 2008 Keyw ords: atomistic sim ulation, quasicrystals, nano-tribology con tact mec hanics, aperiodicit y c r Cop yrigh t 2008, Heather McRae Harper

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DEDICA TION I w ould lik e to dedicate this w ork to m y mom for doing ev erything she can to encourage me, help me, and mak e m y life a little easier, to Alan for reminding me that in the end, it's the money that matters, and to Doug for putting up with me ev ery da y and helping me sta y positiv e. Without y ou guys, I w ould not ha v e been able to nish this. Thank y ou.

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A CKNO WLEDGMENTS I w ould rst lik e to thank all of m y committee mem bers for their time and patience. This researc h w as the com bination of suggestions, insigh ts, and help from man y people. Most importan tly I w ould lik e to thank m y advisor Dr. Da vid Rabson for his con tin ued support. Thank y ou to Dr. Susan Sinnott and her researc h group at the Univ ersit y of Florida for all of their help and insigh t on measuring friction through molecular dynamics and allo wing me the in v aluable experience of spending a w eek in their lab, Dr. Sagar P andit for freely oering his expertise in molecular dynamics, and Dr. P atricia Thiel at Ames National Laboratory through whic h a portion of this researc h w as funded. The author w ould also lik e to ac kno wledge the use of the services pro vided b y Researc h Computing, Univ ersit y of South Florida.

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T ABLE OF CONTENTS LIST OF T ABLES iii LIST OF FIGURES iv LIST OF ABBREVIA TIONS vi ABSTRA CT vii CHAPTER 1 INTR ODUCTION 1 CHAPTER 2 A DISCUSSION OF CONTINUUM CONT A CT MECHANICS 5 2.1 The Hertz Theory 5 2.2 The Johnson-Kendall-Roberts Theory 7 2.3 The Derjaguin-Muller-T oporo v Theory 9 2.4 The T abor P arameter and Other Methods of In terpolating Bet w een Theories 11 CHAPTER 3 AN EXAMINA TION OF SOME TECHNIQUES USED TO PERF ORM MOLECULAR-D YNAMICS SIMULA TIONS AND SOME POPULAR P A CKA GES 15 3.1 Limitations on the Sim ulation of Quasicrystals: Using Quasicrystalline Appro ximan ts 15 3.2 Widom-Moriart y P air P oten tials 20 3.3 Using an Adaman t Tip 24 3.4 Av eraging and Error Analysis 24 3.5 Molecular-Dynamics P ac k ages 29 3.5.1 DL POL Y 30 3.5.2 NAMD 30 3.5.3 Gromacs 31 3.5.4 LAMMPS 31 CHAPTER 4 PRELIMINAR Y RESUL TS 33 CHAPTER 5 FINAL RESUL TS 42 CHAPTER 6 FUTURE W ORK 61 6.1 Comparison of Dieren t Appro ximan ts 61 6.2 Larger Sim ulations 64 i

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6.3 T ailoring the P air P oten tials 64 6.4 Creating and Using a More Realistic Tip 65 6.5 Monitoring Phonon Propagation 65 REFERENCES 69 APPENDICES 73 Appendix A LAMMPS W ork arounds 74 A.1 Obtaining F orces on Fixed-Rigid A toms 74 A.2 Bugs Noted With the LAMMPS Splining Routine 75 A.3 Using a T riclinic Bo x in a Data File 75 Appendix B Calculating an Appropriate Timestep 77 Appendix C Required Files 79 C.1 Data File 80 C.2 P oten tial File 82 C.3 Sim ulation P arameters File 83 C.4 CIR CE Submission Script 92 Appendix D Original 75-A tom Unit Cell 94 Appendix E Previous Publication 98 ii

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LIST OF T ABLES T able 3.1 A tomic Coordinates for the 25-A tom Appro ximan t Unit Cell 17 T able 3.2 Unit-Cell V ectors for the 25-A tom Appro ximan t Unit Cell 20 T able 5.1 Sliding V elocities and Required Sliding Times 48 T able 5.2 T able of Coecien ts of F riction at Highest Compressions for the Aperiodic Sliding Direction 57 T able 5.3 T able of Coecien ts of F riction at Highest Compressions for the P eriodic Sliding Direction 57 T able 5.4 T able Sho wing the Ratio Bet w een the P eriodic and Aperiodic F rictional Coecien ts 59 T able C.1 Sim ulation-Bo x V ectors 81 T able D.1 A tomic Coordinates for the F ull 75-A tom Appro ximan t Unit Cell 94 T able D.2 Unit-Cell v ectors for the 75-A tom Appro ximan t Unit Cell 97 iii

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LIST OF FIGURES Figure 2.1 Sphere on Disk Hertzian Con tact Prole 6 Figure 2.2 Sphere on Sphere Hertzian Con tact Prole 10 Figure 2.3 JKR Con tact Prole 10 Figure 3.1 `10-fold' Surface 16 Figure 3.2 `2-fold' Surface 18 Figure 3.3 Bulk Bila y er Structure 19 Figure 3.4 Widom-Moriart y P air P oten tials 21 Figure 3.5 T runcated Widom-Moriart y P air P oten tials 22 Figure 3.6 Mean Squared Displacemen t 23 Figure 3.7 Ad-Ad and Al-Al P airwise In teractions 25 Figure 3.8 Ad-X P airwise In teractions 26 Figure 4.1 Delineation of the Sim ulation Groups 34 Figure 4.2 Aperiodic Lateral F orce Ov er Time (Preliminary Results) 36 Figure 4.3 P eriodic Lateral F orce Ov er Time (Preliminary Results) 36 Figure 4.4 Aperiodic Normal F orce Ov er Time (Preliminary Results) 37 Figure 4.5 P eriodic Normal F orce Ov er Time (Preliminary Results) 37 Figure 4.6 Normal vs. Lateral F orce (Preliminary Results) 38 Figure 4.7 Extended Normal vs. Lateral F orce (Preliminary Results) 39 Figure 5.1 Visualization of Curren t Sim ulation Bo x (`10-fold' F ace) 43 Figure 5.2 Visualization of Curren t Sim ulation Bo x 44 Figure 5.3 Aperiodic T emperature Ov er Time (Final Results) 46 iv

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Figure 5.4 P eriodic T emperature Ov er Time (Final Results) 47 Figure 5.5 Aperiodic Lateral F orce Ov er Time (Final Results) 49 Figure 5.6 P eriodic Lateral F orce Ov er Time (Final Results) 50 Figure 5.7 Aperiodic Normal F orce Ov er Time (Final Results) 52 Figure 5.8 P eriodic Normal F orce Ov er Time (Final Results) 53 Figure 5.9 Aperiodic Normal vs. Lateral F orce (Final Results) 54 Figure 5.10 P eriodic Normal vs. Lateral F orce (Final Results) 55 Figure 5.11 P eriodic and Aperiodic Normal vs. Lateral F orce (Final Results) 56 Figure 5.12 Sliding V elocit y vs. Coecien t of F riction (Final Results) 58 Figure 6.1 132-A tom Appro ximan t Unit Cell 62 Figure 6.2 343-A tom Appro ximan t Unit Cell 63 Figure 6.3 Aperiodic Lateral vs. Normal F orce 66 Figure 6.4 Aperiodic T emperature vs. Lateral F orce 66 Figure 6.5 P eriodic T emperature vs. Lateral F orce 67 Figure 6.6 Aperiodic T emperature vs. Normal F orce 67 v

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LIST OF ABBREVIA TIONS QC Quasicrystal LAMMPS Large-scale A tomic/Molecular Massiv ely P arallel Sim ulator VMD Visual Molecular Dynamics NAMD Nanoscale Molecular Dynamics AFM A tomic F orce Microscope UHV Ultra-High V acuum JKR Johnson-Kendall-Roberts DMT Derjaguin-Muller-T oporo v STM Scanning-T unneling Microscope vi

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A MOLECULAR-D YNAMICS STUD Y OF THE FRICTIONAL ANISOTR OPY ON THE 2-F OLD SURF A CE OF A D -ALNICO QUASICR YST ALLINE APPR O XIMANT Heather McRae Harper ABSTRA CT In 2005, P ark et al. demonstrated that the 2-fold surface of a d -AlNiCo quasicrystal exhibits an 8-fold frictional anisotrop y as measured b y atomic-force microscop y bet w een the periodic and aperiodic directions [40, 41]. It has been w ell kno wn that quasicrystals exhibit lo w er friction than their crystalline coun terparts [38, 18, 51, 30, 12, 54]; ho w ev er, the disco v ery of the frictional anisotrop y allo ws for a unique opportunit y to study the eect of periodicit y on friction when c hemical composition, o xidation, and w ear are no longer v ariables. The w ork presen ted herein is focused on obtaining an understanding of the mec hanisms of friction and the dependence of friction on the periodicit y of a structure at the atomic lev el, focusing on the d -AlNiCo quasicrystal studied b y P ark et al. Using the LAMMPS [44] pac k age to sim ulate the compression and sliding of an `adaman t' tip, see x 3.3, on a d -AlNiCo quasicrystalline appro ximan t substrate, w e ha v e demonstrated, in preliminary results, an 8-fold frictional anisotrop y but in more careful studies the anisotrop y is found to be m uc h smaller. The sim ulations w ere accomplished using Widom-Moriart y pair poten tials to dene the in teractions bet w een the atoms [36, 56, 55, 9]. The studies presen ted in this w ork ha v e sho wn a clear v elocit y dependence on the measured frictional response of the quasicrystalline appro ximan t's surface. The nal results sho w bet w een a 1.026-fold and 1.127-fold anisotrop y bet w een sliding in the periodic and `aperiodic' directions, depending on the sliding v elocit y vii

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CHAPTER 1 INTR ODUCTION Quasicrystals w ere disco v ered in 1982 b y Shec h tman et al. [48]. Since that rst icosahedral, meta-stable AlMn quasicrystal w as disco v ered, there ha v e been n umerous quasicrystals with v arious `forbidden' poin t group symmetries disco v ered and studied [52]. The majorit y of kno wn quasicrystals con tain icosahedral symmetry This class of quasicrystals are aperiodic in all three dimensions. Con trary to this, the decagonal AlNiCo quasicrystal on whic h this w ork focuses con tains t w o aperiodic directions and a third periodic direction. The in-depth analysis in 2005 b y P ark, et al. [42] sho w ed that the 2-fold face of decagonal Al 72 Ni 11 Co 17 con tained both periodic and aperiodic surface order, on the atomic scale, as predicted b y the bulk structure. This sho w ed that d -AlNiCo quasicrystals possess the unique propert y of exposing a surface con taining a periodic arrangemen t of atoms along the 10-fold axis with a 4 A periodicit y and perpendicular to it an aperiodic arrangemen t of atoms follo wing a Fibonacci sequence. This quic kly prompted further study in to the eect of periodicit y v ersus aperiodicit y on the coecien t of friction [40, 41, 39]. It had been suggested that the lo w coecien t of friction measured in previous studies [41] could be explained through the eects of w ear. In friction experimen ts where plastic deformation tak es place, the measured frictional coecien t is dependen t on a highly complex set of factors including, but not limited to, slip planes and the propagation of defects, the sloughing o of an o xide la y er to create a lubricating eect, the breaking of c hemical bonds, etc. [41]. T o com bat these issues the P ark experimen ts w ere performed using suc h lo w loads as to eliminate plastic deformation. This w as v eried b y scanning tunneling microscope, STM, images both before and after the friction experimen ts [42]. 1

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The original P ark experimen ts [40] w ere performed in ultra-high v acuum, UHV, b y sliding a hexadecane-thiol passiv ated AFM tip across the 2-fold surface in both the periodic and aperiodic directions. Later experimen ts [41] used an alk anethiol passiv ated AFM tip in UHV conditions. By measuring the torsional response of the can tilev er, P ark et al., found an 8-fold frictional anisotrop y The data w ere also a good t to the DMT model of con tact mec hanics, see x 2. Though the frictional anisotrop y itself has been w ell documen ted b y P ark et al., there still lac ks a broader understanding of the role that atomic periodicit y pla ys in the friction coecien t of a material. The aim of this w ork is to emplo y molecular-dynamics tec hniques and recreate the frictional anisotrop y seen in the P ark experimen t. In doing so, w e aim to answ er some v ery fundamen tal questions about the eect of periodicit y on friction at the atomic scale. The analysis begins with an in v estigation in to the theories behind con tact mec hanics. Beginning with Hertz's theory from 1882 [22, 19], whic h idealized mec hanical con tacts to elastic, homogeneous, isotropic, perfectly smooth bodies in the absence of adhesiv e forces, w e see that ev en the simplest model of con tact mec hanics is quite complicated. F ollo wing the Hertz Theory w ere the JKR (Johnson-Kendall-Roberts) and DMT (Derjaguin-MullerT oporo v) theories, [29, 16], whic h built on the original Hertz theory but, most notably included adhesiv e forces. Due to the limitless v ariet y of con tacting surfaces, there isn't a `one size ts all' theory but rather they eac h w ork w ell under dieren t situations. As a rule of th um b, the JKR Theory w orks w ell in describing the con tact bet w een soft materials with high adhesion while the DMT Theory ts w ell with hard materials possessing a lo w adhesion. Luc kily in 1977 T abor [53] published the idea that surface roughness pla y ed a role in adhesion, along with an analysis of the JKR and DMT theories. This led to the T abor parameter, a criterion for determining if t w o con tacting bodies w ould fall in the JKR regime, the DMT regime, or somewhere in-bet w een. F ollo wing T abor, Maugis [33] presen ted a new model to in terpolate bet w een the JKR and DMT regimes and describe materials that fell in the in termediate range. These theories and their corresponding models 2

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only scratc h the surface of the nev er-ending debate o v er con tact mec hanics though they will be the only ones presen ted in x 2. F ollo wing the discussion of con tact mec hanics is an o v erview of the basic principles of applying molecular dynamics to a system as complicated as a quasicrystal. The aperiodicit y of quasicrystals poses a unique problem when trying to model them. One w a y to think of a quasicrystal is as a crystal with a unit cell of innite length. When modeling suc h a system, a periodic quasicrystalline appro ximan t [25] m ust be used, using periodic boundary conditions, to mimic an innite sample. Although appro ximan ts are periodic, they still retain some of the local symmetry and beha vior of their quasicrystalline coun terparts and are used for both modeling and experimen tal researc h [25, 45, 17]; see x 3.1. The appro ximan ts w ere modeled using Widom-Moriart y pair poten tials [36, 56, 55, 9], see x 3.2, using the molecular dynamics sim ulation pac k age LAMMPS [44], see x 3.5.4. The LAMMPS pac k age, from the Sandia National Laboratory w as determined to be best suited to this w ork, although other pac k ages suc h as Gromacs [11], DL POL Y [50], and NAMD [43] w ere installed, tested and ev aluated, see x 3.5.3, 3.5.1, 3.5.2. T o exactly mimic the experimen tal set-up, an alk anethiol passiv ated TiN AFM tip w ould be required. T o simplify the process an `adaman t' tip w as created out of an F CC arrangemen t of atoms that has purely repulsiv e in teractions with the appro ximan t, to mimic the eect of the alk anethiol passiv ation, see x 3.3. Chapter 4 con tains the preliminary results obtained for measuring the friction in the periodic and aperiodic directions of the d -AlNiCo quasicrystalline appro ximan t. Sim ulating normal forces ranging from appro ximately 15-50 nN, an 8-fold anisotrop y w as found in the measured coecien t of friction. Though the anisotrop y seems to reproduce that seen in experimen t, the o v erall magnitude of the frictional forces is quite lo w. Upon further in v estigation, n umerous impro v emen ts and c hanges w ere made to the sim ulation procedure. This c hapter includes a discussion of the kno wn problems in the preliminary results and the adv ances that ha v e tak en place since they w ere rst obtained. 3

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After emplo ying the modications men tioned in x 4, the nal results of this w ork w ere obtained and discussed in x 5. Though w e do not see the 8-fold frictional anisotrop y that w as presen t in the preliminary results, the latest data are m uc h more reliable and pro vide an excellen t jumping-o poin t for a more sophisticated analysis of the d -AlNiCo system. Lastly it is importan t to note that using the simplest model possible to re-create the anisotrop y w ould giv e us great insigh t in to the basic mec hanisms of friction in this system. A more sophisticated analysis in the future could be performed b y comparing dieren t appro ximan ts, running larger sim ulations, tailoring the pair poten tials for surface eects, monitoring phonon propagation, and using a more realistic tip, as discussed in x 6. F or clarit y an appendix discussing the format and commands of the required les to perform the sim ulations w as included. This allo ws for a more detailed description of the LAMMPS commands and the procedures undertak en to perform the friction experimen ts, see Appendix C. 4

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CHAPTER 2 A DISCUSSION OF CONTINUUM CONT A CT MECHANICS 2.1 The Hertz Theory Heinric h Hertz began his study of con tact mec hanics in graduate sc hool b y studying the optical in terference patterns created b y t w o glass lenses coming in to con tact [22]. The no w-famous Hertz theory ga v e a method of calculating the con tact area bet w een t w o bodies under a rather large set of assumptions. Hertz assumed that the con tacting materials w ere elastic, homogeneous and isotropic, and that the con tacting surfaces are smooth and their shape does not c hange o v er time after the initial deformation. F or the Hertz theory to remain accurate, there m ust not be adhesion bet w een the con tacting bodies, and the radius of con tact m ust be m uc h smaller than their individual radii. Keeping with the assumption that the con tacting bodies do not deform outside of the area of con tact, as sho wn in Fig. 2.1, the Hertz Theory predicts a con tact radius, a of a sphere on a rigid plane under a normal load, P as presen ted b y Grierson et al. [23], to be a = PR K 1 3 where R is the radius of the con tacting sphere and K a measure of the elastic constan ts of the materials, is giv en b y K = 4 3 1 2 1 E 1 + 1 2 2 E 2 1 5

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where E 1 and E 2 are the Y oung's modulii and 1 and 2 are the P oisson ratios of materials 1 and 2 respectiv ely It is easy to see that in teractions bet w een the t w o materials are not tak en in to accoun t; only the individual properties of the materials are considered. D R1 Figure 2.1. This illustration sho ws the prole of a Hertzian con tact bet w een a sphere of radius R1 and and a rat plane. D represen ts the diameter of the circular con tact area. Most notably outside of the con tact area the surfaces are not deformed. One can also look at the radius of the Hertzian con tact area, a 0 bet w een t w o spheres of radii R 1 and R 2 under a normal load of P 0 as presen ted b y Johnson, et al. [29] and visualized in Fig. 2.2. a 3 0 = 3 4 ( K 1 + K 2 ) R 1 R 2 R 1 + R 2 P 0 The elastic constan ts K 1 and K 2 are giv en as K 1 ; 2 = 1 2 1 ; 2 E 1 ; 2 where, once again, 1 ; 2 and E 1 ; 2 correspond to the P oisson ratios and Y oung's modulii respectiv ely As the t w o spheres come in to con tact, the region around the area of con tact is compressed and distan t poin ts in the t w o spheres will approac h eac h other b y a distance also called the elastic displacemen t: 3 = 9 16 2 ( K 1 + K 2 ) 2 R 1 + R 2 R 1 R 2 P 2 0 6

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Johnson, et al.'s notation can be simplied in to that of Grierson, giving the radius of the con tact area, a 0 as a 0 = R 1 R 2 R 1 + R 2 P 0 K 1 3 and using the same denition for K as in the sphere on disk equations abo v e. W e can also look at the pressure distribution o v er the area of con tact giv en as p : p = 3 P 2 a 2 s 1 r 2 a 2 0 where r is the distance from the cen ter of the circular con tact area. In the 1930's Derjaguin calculated that t w o rigid bodies separated b y a distance d w ould experience an attractiv e force [33]. In the mid 1900's scien tists began to measure con tact areas that disagreed with the original Hertz model [33]. It th us became quite clear that a more detailed description of con tact mec hanics w ould be required. One of the major c hanges that began in the eld w as the addition of an adhesion term to the original Hertz equations. 2.2 The Johnson-Kendall-Roberts Theory In 1958 Johnson [28] published a short theoretical w ork in v estigating the adhesion bet w een t w o elastic spheres and concluded that, ev en with an adhesiv e force, t w o elastic spheres in con tact cannot ha v e a con tact radius greater than that of the Hertz theory and that adhesion is ph ysically impossible due to innite stresses along the edge of the con tact area. In the 1960's Dutro wski, see [33], published results con tradicting the Hertz theory and Johnson's 1958 paper b y sho wing a larger con tact area than the Hertz model predicted, and, v ery notably the con tact area w as nite under zero applied load. Because the con tact area w as nite under zero applied load, a force w as required to separate the t w o bodies, sho wing a measurable adhesion. These new results b y Dutro wski w ere in agreemen t with the JKR Theory whic h w as to be published later, in 1971 [29]. 7

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Johnson, Kendall, and Roberts [29] ac kno wledged that under lo w loads, t w o elastic bodies in con tact ha v e an equilibrium con tact area that is due, in large part, to surface forces. They also stated that mec hanical w ork is required to o v ercome the force of adhesion in order to separate the t w o bodies. Starting with the original Hertz theory the authors dev eloped a new theoretical model for con tact mec hanics and v eried it experimen tally The JKR Theory predicts a con tact radius, a JKR giv en b y a 3 JKR = R K P + 3 R + q 6 RP + (3 R ) 2 where is the surface energy per unit con tact area, R is used to represen t the radii of the con tacting bodies as R = R 1 R 2 R 1 + R 2 P represen ts the applied load, and K the elastic constan t term, as previously in x 2.1: K = 4 3 1 2 1 E 1 + 1 2 2 E 2 1 The last three terms in the JKR con tact area equation are the modications to the Hertz theory that tak e in to accoun t surface energies, One can easily see that neglecting surface energies, i.e. = 0, w e are left with the original Hertz equation. It is also importan t to note that under conditions of zero applied load, P = 0, there is a nite con tact area with radius a JKR ( P = 0) = 6 R 2 K 1 3 : This allo ws for the calculation of the required load to separate the bodies, P C ( JKR ) = 3 2 R: Johnson, et al. measured the con tact radii experimen tally for gelatin and rubber spheres under v arying normal loads, P Their results sho w good agreemen t with the modied Hertz theory presen ted abo v e, including the required `pull-o force' to separate the bodies. Ev en more than predicting a larger con tact radius, the JKR theory predicts a c hange in shape 8

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of the con tacting bodies. In the original Hertz theory the surfaces of the t w o con tacting spheres approac h the con tact area tangen tially with no deformation outside of the con tact area, as seen in Fig. 2.2. In the JKR Theory the surfaces are locally deformed, due to surface forces, and they approac h the area of con tact perpendicularly as seen in Fig. 2.3. After the JKR Theory in 1975 Derjaguin, et al. [16] proposed the DMT Theory Derjaguin-Muller-T oporo v Theory of con tact mec hanics. 2.3 The Derjaguin-Muller-T oporo v Theory Lik e the JKR Theory [29], the DMT Theory [16] starts from a Hertz perspectiv e then includes adhesiv e forces. The dierence bet w een the JKR and DMT theories is that the DMT Theory assumes the adhesiv e forces act in a ring outside of the con tact area but cannot deform the t w o surfaces outside of the area of con tact. This leads to a Hertzian con tact prole, in whic h the surfaces approac h the con tact area tangen tially as men tioned in x 2.2 and visualized in Fig. 2.2. The calculated con tact radius giv en b y Derjaguin, et al. [16], modied to the same notation as presen ted previously is giv en as a DMT a DMT = R K ( P + 2 R ) 1 3 As can be easily seen abo v e, the DMT Theory predicts a nite con tact area at zero applied load, P = 0, with radius giv en as a DMT ( P = 0) = 2 R 2 K 1 3 : The pull-o force required to separate the t w o bodies is giv en as [23] P C ( DMT ) = 2 R 9

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R2 D R1 Figure 2.2. The illustration sho ws the prole of a Hertzian con tact bet w een spheres of radii R1 and R2. D represen ts the diameter of the circular con tact area. As with the sphere on disk model, outside of the con tact area the surfaces are not deformed. R2 R1 D1D2 Figure 2.3. The illustration sho ws the prole of a con tact bet w een spheres of radii R1 and R2 where the local deformation, indicated b y the dashed lines, outside of the Hertzian con tact area can be seen. D1 represen ts the diameter of the Hertzian circular con tact area and D2 represen ts the diameter of the circular con tact area predicted b y the JKR theory 10

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whic h can be though t of as a measure of the adhesion bet w een the con tacting bodies. The critical loads, or pull-o forces, and nite con tact areas at zero load giv en b y the JKR and DMT Theories dier only b y a constan t. After the DMT Theory w as published, there w as a long and heated debate o v er whether the JKR Theory or the DMT Theory w as correct. The debate will not be elaborated upon here, but it w as ev en tually accepted that both theories w ere accurate; ho w ev er they w ere accurate under dieren t regimes [23]. The JKR Theory accurately describes soft, largeradii materials with high adhesion. The DMT Theory accurately describes hard, small-radii materials with lo w adhesion. This led to the need for a w a y to in terpolate bet w een the theories. 2.4 The T abor P arameter and Other Methods of In terpolating Bet w een Theories In 1977 D. T abor, [53], presen ted an in v estigation in to some of the theoretical problems in the eld of con tact mec hanics and analyzed both the JKR and DMT Theories. T abor concluded that not only w ere surface forces importan t in adhesion, but that surface roughness also pla ys a role. T abor used a v ery simple experimen t to sho w ho w surface roughness aects adhesion [53]. Using an optically smooth rubber ball and a rat P erspex surface, pull-o forces w ere measured as the P erspex surface w as roughened. It w as found that as the roughness increased, the pull-o force, or adhesion, bet w een the t w o bodies decreased. T abor's analysis [53] of the JKR and DMT theories, along with his in v estigations in to surface roughness and adhesion led to the calculation of a parameter, T commonly referred to as T abor's P arameter [23], used to determine if either the JKR or DMT theories w ould best describe a system. The T abor parameter as presen ted b y Muller, et al. [37] in 1980 is giv en as T = 16 R 2 9 K 2 z 3 0 1 3 11

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with z 0 represen ting the equilibrium separation of the surfaces. F or clarication, one can think of z 0 as the distance corresponding to the poten tial w ell in a Lennard-Jones st yle poten tial. When T is small, T 1, the DMT Theory is appropriate, and when T is large, T 1, the JKR Theory is appropriate. In 1992 Daniel Maugis [33] presen ted a JKR-DMT transition that not only w ork ed for the extreme cases, tting nicely in to JKR and DMT, but also applied to the materials that fell bet w een them. Using a square-w ell Dugdale poten tial to describe the in teraction bet w een the materials, Maugis calculated a parameter similar to the T abor parameter, M = 2 0 R K 2 1 3 where 0 represen ts a constan t adhesional stress whic h, when m ultiplied b y its range of in teraction, giv es the w ork of adhesion, or as previously men tioned, the surface energies per unit area. The DMT model applies when M < 0.1, and the JKR model applies when M > 5 [23]. Bet w een the t w o extremes is the transition region. Unfortunately the Maugis form ulation is dicult to implemen t, when compared to the previous theories, because of the complexit y of solving complicated sim ultaneous equations. A simpler appro ximation to Maugis' w ork w as presen ted b y Carpic k, et al. in 1999 [13]. Carpic k, et al. found an appro ximation that tted the transition regime of the MaugisDugdale equations to within 1% and w as exact for the JKR and DMT regimes. The con tact radius at zero applied load and the critical load required to separate the bodies are appro ximated as a 0 = R 2 K 1 3 1 : 54 + 0 : 279 2 : 28 r 1 : 3 1 2 : 28 r 1 : 3 + 1 !! L C = ( R ) 7 4 + 1 4 4 : 04 r 1 : 4 1 4 : 04 r 1 : 4 + 1 !! 12

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where r is called the transition parameter and is giv en as r = 0 : 924ln(1 1 : 02 ) with being a n um ber bet w een 0 and 1 inclusiv e that represen ts the DMT regime at = 0, the JKR regime at = 1, and the transition regime in bet w een. As Carpic k, et al. describe, is found experimen tally b y tting con tact radius vs. load or friction vs. load measuremen ts. Once is determined r can be easily calculated. Presen ted abo v e is a small o v erview of the eld of con tact mec hanics. A basic understanding of the history and v arious theories w as importan t in starting this researc h. In 2004 P ark, et al. [38] studied the surfaces of both clean and o xidized d -AlNiCo quasicrystals in UHV using an AFM. When comparing both the JKR and DMT models it w as found that the clean quasicrystal surface adhered so strongly to the metallic AFM tip that the JKR model w as appropriate, ev en though the quasicrystal surface and AFM tip are considered `hard'. On the other hand, the o xidized d -AlNiCo surface exhibited signican tly less adhesion, and the DMT model w as appropriate. In the 2006 Ph ysical Review B [41] paper, the AFM tip w as passiv ated with a la y er of alk anethiol molecules resulting in lo w adhesion bet w een the quasicrystal and the AFM tip, and th us the DMT model w as used to analyze their ndings. P ark et al. [41] used the DMT model to calculate the con tact area of a C 16 alk anethiol passiv ated TiN AFM tip on the 2-fold surface of a single grain Al 72 Ni 11 Co 16 decagonal quasicrystal. Using a 100 nm tip radius, the calculated con tact area w as 115 nm 2 The threshold load, the load at whic h there is a loss of the passiv ation la y er, w as 380 nN giving a maxim um pressure of 3.3 GP a. The pressure is calculated b y dividing the normal force b y the con tact area. The frictional data sho wn b y P ark et al. sho w the torsional response of the can tilev er o v er normal forces reac hing only 100 nN corresponding to pressures of 0.869 GP a. 13

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Due to the nature of the sim ulations in this w ork, the con tact area is supplied b y the researc her. There is no surface roughness on the adaman t tip, see x 3.3, or the quasicrystalline appro ximan t substrate, see x 3.1. Though there is no explicit surface roughness, the quasicrystalline appro ximan t does undergo a small amoun t of buc kling during the initial relaxation. In the nal results, the size of the adaman t tip is suc h that it is smaller than the buc kling features of the substrate. This allo ws for a simple calculation of the con tact area and resulting pressure. The con tact area bet w een the adaman t tip and appro ximan t substrate in this w ork is 24.9512 nm 2 The maxim um normal force used in the nal results is 95.67024 nN leading to a maxim um pressure of 3.83 GP a. The maxim um pressure ac hiev ed in the nal results is greater than in the P ark experimen ts, but it is of the same order of magnitude. F uture w ork, as discussed in x 6, should use larger sim ulations with larger tips to more accurately recreate the pressures and normal loads seen in the P ark experimen ts. Though our highest normal loads lead to pressures larger than the threshold pressure measured b y P ark et al. [41] the range of normal loads presen ted in x 5 begin at 0.2941 GP a and reac h our maxim um of 3.83 GP a, to co v er most of the pressures seen in the P ark experimen ts. 14

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CHAPTER 3 AN EXAMINA TION OF SOME TECHNIQUES USED TO PERF ORM MOLECULAR-D YNAMICS SIMULA TIONS AND SOME POPULAR P A CKA GES 3.1 Limitations on the Sim ulation of Quasicrystals: Using Quasicrystalline Appro ximan ts Quasicrystals are crystals lac king translational symmetry in one or more dimensions and can con tain symmetries `forbidden' b y the classic denition of a crystal [48]. This poses a unique problem when attempting to sim ulate a quasicrystal, rather than a traditional periodic crystal, because one can think of quasicrystals as ha ving innitely large unit cells due to the lac k of periodicit y F ortunately there are quasicrystalline appro ximan ts. Quasicrystalline appro ximan ts con tain local symmetries similar to their quasicrystalline coun terparts [25]. That being said, they can pro vide a good substitute for quasicrystals in the molecular-dynamics sim ulations in this w ork. An experimen tal study on an AlPd-Mn quasicrystalline appro ximan t and an i -AlPdMn quasicrystal [32] sho w ed that the appro ximan t has a frictional coecien t t wice that of its quasicrystalline coun terpart, both before and after o xidation. Unfortunately w e are restricted to the use of appro ximan ts, so naturally it w ould seem that the frictional responses seen in this w ork should be somewhat larger than the experimen tal w ork done b y P ark et al. on the d -AlNiCo quasicrystal. The larger the appro ximan t unit cell, the more closely the appro ximan t w ould resem ble the real quasicrystal. The appro ximan t structure used in this researc h w as supplied b y Mik e Widom [1]. The unit cell used for the nal results consists of one bila y er and only con tains 25 atoms. A table of the atomic coordinates of the appro ximan t structure used for the nal results 15

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is presen ted in T able 3.1 follo w ed b y the unit-cell v ectors in T able 3.2. T o see that the unit cell mimics 10-fold symmetry it is adv an tageous to visualize m ultiple unit cells as in Fig. 3.1. An appro ximan t with a larger unit cell w ould more closely mimic the d -AlNiCo being studied experimen tally b y P ark, et al.; ho w ev er, an in-depth comparison of dieren t and larger appro ximan ts will be left for future w ork. In x 6.1 an initial examination of t w o more appro ximan t structures is presen ted. Mihalk o vi c et al. [34] deriv e the quasicrystalline appro ximan t structures b y starting with the same Widom-Moriart y pair poten tials used in this researc h. Using experimen tal data, suc h as the distance bet w een planes of atoms, along with the minima in the pair poten tials, trial quasicrystal structures are constructed. These initial structures are then Mon te Carlo annealed to minimize the energy Figure 3.1. The appro ximan t's `10-fold' surface. Though this is a periodic structure, 25 unit cells seen here, it appro ximates a 10-fold surface. F or clarit y one of the appro ximately 10fold features is highligh ted in red and one unit cell is outlined in white. The green spheres are Al, white are Ni, and pink are Co. Original image rendered b y VMD [26] The `10-fold' face pictured in Fig. 3.1 is perpendicular to the `2-fold' face, whic h is the object of our study and is pictured in Fig. 3.2. In the d -AlNiCo quasicrystal studied b y P ark, et al. [40, 41, 42, 39, 38] the 2-fold face displa ys both periodic and aperiodic order 16

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Elemen t X ( A) Y ( A) Z ( A) Al 9.82553 7.16461 0.780714 Ni 10.0143 0.843104 0.0925141 Ni 8.2999 1.13609 2.1651 Ni 8.59727 5.20415 0.0925141 Ni 7.38254 3.95944 2.1651 Ni 2.20804 0.717434 0.0807941 Ni 3.89953 1.26703 2.16063 Al 6.15416 1.99961 1.55191 Co 3.83923 5.21954 0.0845941 Al 8.43596 2.74101 0.0860341 Co -0.038051 3.95974 2.06831 Al 1.55681 4.47794 0.107924 Al 1.63299 2.02915 2.08816 Al 0.179072 6.50384 2.08816 Al 7.527 6.41778 2.17358 Al 9.69997 4.6415 2.20145 Al 10.5756 1.94645 2.20145 Al -1.01477 5.08674 0.0546241 Al -0.165799 2.47387 0.0546241 Al 3.67271 2.63577 0.0740341 Al 4.52055 0.0263807 0.0740341 Al 3.72628 6.59979 2.17641 Al 4.55913 4.03651 2.17641 Al 6.2728 4.51163 0.0864041 Al 7.72668 0.0370727 0.0864041 T able 3.1. These are the coordinates for one unit cell of the quasicrystalline appro ximan t bulk bila y er used in this researc h. The structure of the appro ximan t w as supplied b y Mik e Widom [1] along directions separated b y 90 with a 4 A periodicit y in the periodic direction. Our appro ximate 2-fold face, the `2-fold' face, has a periodicit y of 4.03265902 A in the periodic z direction and a 12.22488050 A periodicit y in the `aperiodic' x direction. As men tioned previously the unit cell used for the nal results in this researc h con tains a 25-atom bila y er, whic h can clearly be seen in Fig. 3.3. The original structure supplied b y Mik e Widom [1] con tained three 25-atom bila y ers. There w ere t w o surface bila y ers sandwic hing a bulk bila y er. W e w ere giv en this specic structure because of our in terest in surface eects, and the t w o surface bila y ers w ere originally though t to be benecial. After in v estigation of the structure it w as found that the surface bila y ers w ere parallel to the 17

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Figure 3.2. The `2-fold' surface of the appro ximan t. This is the surface used for the friction experimen ts. The x direction represen ts the appro ximated aperiodic direction and has a periodicit y of around 12.225 A. The z direction represen ts the periodic direction and has a periodicit y of around 4 A. The green spheres are Al, white are Ni, and pink are Co. Original image rendered b y VMD [26] 18

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Figure 3.3. Looking at the y-z plane of the appro ximan t structure allo ws us to see a clear picture of the bila y er structure. Eac h bila y er is dened b y a line for ease of visualization. The unit cell in the z direction extends only about 4 A, making this our periodic direction, as opposed to our appro ximated aperiodic direction, x, whic h has a unit cell length of around 12.225 A. The green spheres are Al, white are Ni, and pink are Co. Original image rendered b y VMD [26] 19

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V ector X ( A) Y ( A) Z ( A) a -2.33474440 7.18560440 0.00000000 b 12.22488050 0.00000000 0.00000000 c 0.00000000 0.00000000 4.03265902 T able 3.2. These are the unit cell v ectors for the quasicrystalline appro ximan t used in this researc h. The structure of the appro ximan t w as supplied b y Mik e Widom [1] `10-fold' surface and not the `2-fold' surface, whic h is where w e are performing our studies. The surface bila y ers w ere remo v ed from the sim ulations after the preliminary results, and the bulk bila y er w as the only con tributor to the unit cell. The atomic coordinates and unit-cell v ectors for the original 75-atom unit cell can be found in Appendix D. 3.2 Widom-Moriart y P air P oten tials The Widom-Moriart y pair poten tials [36, 56, 55, 9] used in this researc h w ere supplied b y Marek Mihalk o vi c in a tabulated format [2]. The poten tials describe the pairwise in teractions bet w een the quasicrystalline appro ximan t atoms: Al-Al, Al-Co, Al-Ni, Co-Co, Co-Ni, and Ni-Ni. In their original form, the poten tials sho w F riedel Oscillations [10], and they extend as far as 17 A; see Fig. 3.4. The preliminary results w ere obtained b y using the poten tials as in Fig. 3.4. It w as later suggested b y Marek Mihalk o vi c, through personal comm unication with Da vid Rabson, that a truncated v ersion of the poten tials ma y more accurately t to the quasicrystalline appro ximan t used. As per the suggestion, the poten tials w ere truncated to 7 A and smoothed to zero at the tail, see Fig. 3.5. T o test the v alidit y of the truncated poten tials, relaxation sim ulations w ere performed on the 5 10 25 unit cell quasicrystalline appro ximan t, used in all friction experimen ts for the nal results, but without the adaman t tip. The relaxation sim ulations w ere performed using a 0.004 ps timestep for 10,000 timesteps with all atoms allo w ed to relax. The a v erage mean squared displacemen t of the appro ximan t's atoms as the temperature of the system w as brough t to 0 K, using a Langevin thermostat [3], w as measured and 20

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Figure 3.4. The original unsmoothed Widom-Moriart y pair poten tials. Only the rst 10 A are sho wn for ease of visualizing the oscillations though the poten tials extend to 17 A 21

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Figure 3.5. The truncated and smoothed Widom-Moriart y pair poten tials. 22

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graphed in Fig. 3.6 for sim ulations using both the original Widom-Moriart y poten tials and the truncated poten tials. Figure 3.6. The a v erage mean squared displacemen t o v er time of the 5 10 25 unit cell quasicrystalline appro ximan t, during relaxation to 0 K, using both the original and truncated Widom-Moriart y pair poten tials. The mean squared displacemen t is a w a y to measure the a v erage displacemen t of all atoms from their starting position, o v er time. The results of the relaxation test sho w that using the truncated poten tials allo ws the appro ximan t to relax faster, with the nal atomic positions closer to the original structure, than the un-truncated Widom-Moriart y poten23

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tials. Because the truncated poten tials more accurately describe the original structure, they will be used in all in v estigations follo wing the preliminary results. 3.3 Using an Adaman t Tip The friction experimen ts performed b y P ark, et al. [40, 41] w ere performed in UHV conditions emplo ying an alk anethiol passiv ated AFM tip to reduce adhesion with the quasicrystalline surface. This reduction in adhesion allo w ed for the in v estigation of w ear-less friction. Sim ulating this specic tip is a dicult and time consuming process, mostly due to the passiv ation la y er. T o mimic w ear-less friction, without sim ulating alk anethiols, an `adaman t' tip w as created. The goal w as to ha v e a tip that adhered strongly to itself but not to the quasicrystalline appro ximan t substrate. The adaman t, or Ad, tip w as created as F CC Alumin um, using a modied Al-Al Widom-Moriart y pair poten tial [36, 56, 55, 9], see Fig. 3.7. The Ad atoms ha v e the same mass as Alumin um, and the Ad-Ad pair poten tial w as deriv ed b y m ultiplying the Al-Al poten tial v alues b y 10 so that the hills and v alleys w ere more pronounced, making the Ad-Ad in teraction stronger than that of regular Al-Al. T o eliminate adhesion bet w een the adaman t tip and the quasicrystalline appro ximan t, a pair poten tial w as created out of a deca ying exponen tial, see Fig. 3.8. This is a purely repulsiv e poten tial and w as used for in teraction bet w een the tip atoms, Ad, and all appro ximan t atoms: Al, Ni, and Co. One of the consequences of using a purely repulsiv e poten tial for the tip is that w e cannot measure an y pull-o forces, as seen in the P ark experimen ts [40, 41, 38, 42]. This limits our range of normal forces to be positiv e. Using a more realistic tip and poten tial bet w een the tip and appro ximan t substrate is left for future w ork, as discussed in x 6.4 3.4 Av eraging and Error Analysis F or both the preliminary and nal results, the lateral and normal forces during sliding are recorded at discrete time in terv als. F or the preliminary results, the a v erage forces w ere 24

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Figure 3.7. The original Al-Al Widom-Moriart y pair poten tial is sho wn alongside the created Ad-Ad pair poten tial. 25

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Figure 3.8. The purely repulsiv e poten tial bet w een the adaman t tip and all appro ximan t atoms. 26

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calculated b y summing all of the recorded force v alues and dividing b y the n um ber of data poin ts. This method w as later found to be insucien t, and the nal results w ere calculated b y taking the in tegral o v er the force curv e and dividing b y the total time in terv al. Due to the closeness of the measured v alues this tec hnique w as used so as to minimize the n umerical uncertain ties. According to Abramo witz and Stegun [8], the extended trapezoidal rule for in tegrating a function F ( t ) is Z t m t 0 F ( t ) dt = h [ F 0 2 + F 1 + ::: + F m 1 + F m 2 ] with a corresponding error of E 1 = mh 3 12 F 00 ( ) where h is the time in terv al bet w een an y t w o poin ts F n and F n +1 m is the total n um ber of poin ts, and is an y random poin t. W e then divide this in tegral, and the corresponding error, b y the total time in terv al, mh to obtain the a v erage force and the error associated with the a v eraging the force using the trapezoidal rule. As discussed at the end of x 4, the tip needs to slide across exactly one sim ulation bo x length. Our sampling time is not commensurate with our required stopping time, and th us w e had to in terpolate bet w een t w o data poin ts to obtain the force at the required sliding distance. T o add this nal poin t in to our a v erage presen ted abo v e, it needs to be w eigh ted. The nal equation for obtaining the a v erage force o v er time is F = [ 1 2 F 0 + F 1 + :::F m 1 + 1 2 F m ] h + 1 2 [ F m + ~ F ]( ~ t t m ) ~ t t 0 where ~ F = F m + F m +1 F m h ( ~ t t m ) is the in terpolated force at the required stopping time ~ t Calculating the a v erage force in this manner pro vides an estimate for the error, E 1 due to the discrete sampling of poin ts. T o obtain the maxim um error, the maxim um curv ature, 27

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F 00 max w as needed. The curv ature of a sin w a v e is giv en as j F 00 j = d 2 dt 2 A sin (2 ft ) with amplitude A and frequency f By performing the deriv ativ e and ev aluating at a peak, 2 ft = 2 w e nd that the maxim um curv ature is F 00 max = 4 2 f 2 A giving an error, E 1 of E 1 = 1 3 h 2 2 f 2 A E 1 should be divided b y 2 to accoun t for our using only the maxim um curv ature of the sin w a v e; ho w ev er, there is appro ximately a 25 percen t discrepancy bet w een the calculated frequency and what w e kno w the frequency is supposed to be whic h leads to a factor of 2 dierence in the error. These factors then cancel, lea ving E 1 as presen ted abo v e. Because of the noise inheren t in the data, going from poin t to poin t, a rough estimate of error w as calculated as E 2 = 1 + 1 4 m q P m 1 j =2 ( F j 1 2 F j + F j +1 ) 2 m : The 1 4 m term comes from the error in the additional trapezoidal area created b y adding the in terpolated poin t. The bulk of E 2 comes from the idea that random noise, giv en an innite time, w ould cancel. Since w e are not dealing with an innite time series, it should be accoun ted for. A third method for estimating error w as gained through sim ulation. In the nal results, the tip does not co v er the en tire surface of the appro ximan t in either the periodic or `aperiodic' directions. T o accoun t for the eect of the tip position as sliding is performed, the tip w as shifted b y 1 A in the direction perpendicular to both the sliding and compression directions. This w as done at a sliding speed of 0.1 A/ps, for both the periodic and `aperiodic' 28

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sliding directions. The dierence bet w een the original frictional force and the frictional force obtained b y using the shifted tip w as calculated for sliding in both directions. This dierence w as then divided b y the original friction result to obtain the fractional error, e F or eac h sim ulation the error associated with c hoosing a dieren t in titial tip position is calculated as E 3 = e F where F is the a v erage force. The error bars used in the nal results, presen ted in x 5, are calculated for eac h sim ulation as a com bination of all three error estimates, suc h that E = q E 2 1 + E 2 2 + E 2 3 : W e do not rule out the possibilit y that other sources of error ma y need to be accoun ted for. W e ha v e ruled out the possiblilit y of statistical error associated with the Langevin thermostat [3] b y performing sim ulations using dieren t random seeds. Changing the random seed used for the thermostat does not c hange the results of the sim ulations. The equations used to calculate the error bars and a v erage force w ere pro vided b y Dr. Da vid Rabson, priv ate comm unication. 3.5 Molecular-Dynamics P ac k ages W riting a molecular-dynamics program sophisticated enough for the researc h performed in this w ork is a daun ting task. The goal of this project w as not to form ulate a new method of molecular sim ulation but to use existing tools to study a v ery complicated and unique system. With that in mind, some of the most prominen t molecular-dynamics pac k ages a v ailable w ere researc hed, installed, and tested for their capabilit y of handling the system to be studied, ease of modication, performance, and ho w `user-friendly' they turned out to be. This section con tains a list and brief description of eac h of the molecular dynamics pac k ages in v estigated. 29

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3.5.1 DL POL Y DL POL Y is a molecular-dynamics sim ulation pac k age created at Daresbury Laboratory [4, 50]. Tw o v ersions exist: DL POL Y 2 is suitable for small sim ulations of up to 30,000 atoms on 100 processors or less, and DL POL Y 3 is designed to be used with large sim ulations on the order of 1,000,000 atoms on up w ards of 1,000 processors. DL POL Y is equipped to handle constan t NVT (particle n um ber, v olume, and temperature), NVE (particle n um ber, v olume, and energy), and NPT (particle n um ber, pressure, and temperature) sim ulations and emplo ys either a V elocit y V erlet or V erlet Leapfrog integration algorithm [47]. F or parallel jobs MPI is used for in ter-processor comm unication. One of the reasons DL POL Y w as not c hosen for this researc h is that it is written in F ortran 90, whic h is not a language familiar to the researc her. Though the pac k age seemed capable of handling solid-state materials, when compared with LAMMPS, it w as found that the LAMMPS pac k age allo w ed more user con trol of the sim ulation while emplo ying simpler and more user-friendly input les. 3.5.2 NAMD NAMD w as dev eloped b y the Theoretical and Computational Bioph ysics Group in the Bec kman Institute for Adv anced Science and T ec hnology at the Univ ersit y of Illinois at Urbana-Champaign [43]. NAMD w as in tended to be run in conjunction with VMD, Visual Molecular Dynamics [26], whic h is not only an adv anced visualization program but also in terfaces with NAMD and allo ws the user to modify the sim ulation. The source code is written in C++ and uses MPI when executing parallel jobs. NAMD is also kno wn to be able to handle large sim ulations of 300,000 atoms on 1,000 processors. One of the unique features of NAMD is the standard capabilit y of reading input les from other sim ulation pac k ages suc h as X-PLOR, CHARMM, AMBER, and GR OMA CS. The main reason that NAMD w as not c hosen for this researc h is that it is specialized for biomolecular systems. This specialization mak es it dicult to sim ulate solid-state materials. 30

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Though NAMD w as not c hosen to perform the molecular dynamics, VMD w as used to render the snapshots sho wn in this w ork. 3.5.3 Gromacs Gromacs w as initially dev eloped to sim ulate biological materials suc h as lipids and proteins, but unlik e NAMD it can handle non-bonded solid-state materials relativ ely w ell [11]. The source code is written in C, and parallel jobs use standard MPI comm unication. Gromacs uses the leap-frog algorithm for the in tegration of Newton's La ws to update both the positions and v elocities of the atoms. As input, Gromacs uses fairly complicated topology les that are used to describe bonds, angles, and molecules. F or biological systems it is necessary to specify all of the information in the topology les, but it is unnecessarily complicated for the system researc hed in this w ork. Though Gromacs claims to be 3 to 10 times faster than most molecular-dynamics pac k ages [5], LAMMPS w as a better t to our eld of study 3.5.4 LAMMPS The LAMMPS pac k age, dev eloped at Sandia National Laboratory w as the moleculardynamics pac k age c hosen to complete this w ork [44]. LAMMPS has all of the basic functionalit y of the other molecular-dynamics pac k ages review ed, including the abilit y to w ork on a single processor or in parallel using MPI comm unication, implemen ts the v elocit y V erlet in tegration sc heme [47], and can run under n umerous ensem bles including NPT (n um ber of atoms, pressure, and temperature), NVE (n um ber of atoms, v olume and energy), and NVT (n um ber of atoms, v olume and temperature). LAMMPS is compatible with CHARMM, AMBER, and GR OMA CS force elds and can output data in to m ultiple formats, including one view able using VMD, as men tioned in x 3.5.2. The source code is written in C++ and is easily modied to include an y new features. The input les required to run a sim ulation using the LAMMPS pac k age are relativ ely simple and easy to understand. An in-depth description of the commands used to 31

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perform this researc h is giv en in Appendix C. Though LAMMPS is capable of sim ulating biomolecular systems, it is not specialized for that purpose; ho w ev er it easily sim ulates systems in the gas, liquid, or solid state. Though some w ork arounds w ere required | see Appendix A | the LAMMPS pac k age w as able to perform the required sim ulations and output the appropriate data. 32

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CHAPTER 4 PRELIMINAR Y RESUL TS Preliminary results w ere obtained for compressing and sliding a 3,960-atom adaman t slab on a 10,625-atom quasicrystalline appro ximan t slab. All sim ulations use a Langevin thermostat to con trol the temperature of the sim ulation b y k eeping it less than 1 K. All of the atoms in the sim ulation are brok en up in to groups. The top-most la y ers of the tip and the bottom-most la y ers of the appro ximan t are grouped as xed tip atoms and xed appro ximan t atoms respectiv ely The cen ter-most la y ers of the tip and appro ximan t are used for the thermostat and are called the tip thermostat atoms and appro ximan t thermostat atoms. The la y ers at the tip-appro ximan t in terface do not ha v e an y constrain ts on them and are left to act freely hence they are called the free tip atoms and free appro ximan t atoms. These la y ers are visualized in Fig. 4.1. There are four main portions to eac h sim ulation: relaxation, compression, relaxation, and sliding. Initially the en tire system is allo w ed to relax. This is ac hiev ed b y applying a Langevin thermostat to all atoms in the sim ulation, bringing the temperature to 0 K. The thermostat is applied for 4.5 ps. After relaxation, the v elocities and forces on the top-most la y ers of the adaman t tip and the bottom-most la y ers of the appro ximan t are set to zero. This forces the atoms to mo v e as rigid bodies, and they are called the `xed' atoms as seen in Fig. 4.1. The Langevin thermostat, implemen ted previously on all atoms, is remo v ed and replaced b y a Langevin thermostat acting only on the cen ter-most la y ers of the tip and appro ximan t: these are called the thermostat atoms. T o a v oid the thermostat aecting lateral or normal force data, the thermostat is allo w ed to act only in the direction perpendicular to the sliding 33

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Figure 4.1. Dieren t areas of the sim ulation are used for dieren t purposes. The outer-most la y ers of atoms are xed and held rigid. The cen ter-most la y ers of atoms are used for the thermostat, and the in terface la y ers are left to act freely The green spheres are Al, white are Ni, pink are Co, and blue are adaman t (Ad). Original image rendered b y VMD [26]. 34

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v elocit y and the compression direction. The desired temperature is 0 K. The xed atoms of the adaman t slab are then giv en a v elocit y to w ard the quasicrystalline appro ximan t surface. Multiple compressions w ere ac hiev ed b y using dieren t compression v elocities for 2 ps eac h. After the desired compression is ac hiev ed, the system is allo w ed to relax once more for 15 ps. The heigh t after compression is held constan t b y the xed rigid la y ers at the top and bottom of the sim ulation. This procedure has been used in previous molecular-dynamics studies of friction [24]. T o ac hiev e the sliding portion of the sim ulation, the rigid tip atoms are giv en a constan t sliding v elocit y of 0.05 A/ps in either the periodic or `aperiodic' direction of the appro ximan t. The sliding lasts for 50 ps and corresponds to sliding a distance of 2.5 A. During this time the forces that oppose the sliding of the tip, the lateral forces, are recorded for eac h compression in both the periodic and `aperiodic' directions. Examples of the lateral forces during sliding are presen ted in Fig. 4.2 and Fig. 4.3 for the appro ximated aperiodic and periodic directions respectiv ely Also sho wn in Fig. 4.4 and Fig. 4.5 are the corresponding normal forces o v er time during sliding. The ructuations seen in the normal force and lateral force graphs are though t to be mostly due to noise. Con trary to this, the ructuations seen in the nal results correspond to surface features of the appro ximan t and will be discussed in x 5. After all of the desired compressions ha v e been sim ulated, a plot of the a v erage normal force v ersus a v erage lateral force is made, for both sliding directions, see Fig. 4.6. According to Amon tons's La w, the coecien t of friction bet w een t w o sliding bodies is the slope of the normal vs. lateral force curv e [20]. The preliminary results sho wn in Fig. 4.6 clearly sho w a frictional anisotrop y The o v erall magnitude of the frictional forces dier b y a factor of appro ximately 4 with the periodic direction being higher. More importan tly the friction coecien ts sho w an 8-fold anisotrop y The measured coecien ts of friction are 0.008 for the periodic direction and 0.001 for the `aperiodic' direction. These results w ere v ery promising; ho w ev er, when the system 35

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Figure 4.2. An example graph of the lateral force o v er time as the tip slides across the quasicrystalline appro ximan t surface in the `aperiodic' x direction at a compression of 19.888 nN. The lateral force, a v eraged o v er time, is calculated to be 0.065699 nN Figure 4.3. An example graph of the lateral force o v er time as the tip slides across the quasicrystalline appro ximan t surface in the periodic z direction at a compression of 20.01824 nN. The lateral force, a v eraged o v er time, is calculated to be 0.2844608 nN. 36

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Figure 4.4. An example graph of the normal force o v er time as the tip slides across the quasicrystalline appro ximan t surface in the `aperiodic' x direction at an a v erage compression of 19.888 nN. Figure 4.5. An example graph of the normal force o v er time as the tip slides across the quasicrystalline appro ximan t surface in the periodic z direction at an a v erage compression of 20.01824 nN. 37

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Figure 4.6. This graph sho ws the a v erage lateral forces, or frictional forces, for eac h normalforce compression in both the periodic (+) and appro ximated aperiodic (X) directions. The frictional coecien t for the periodic direction is 0.001, while the frictional coecien t of the aperiodic direction is 0.008, as calculated b y the slope according to Amon tons's La w [20]. 38

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w as studied further, discrepancies started to appear. After the initial preliminary results, w e started studying a wider range of compressions along with dieren t initial tip positions. W e began to realize that the preliminary results required further in v estigation when the measured lateral force started getting smaller as the compressiv e force increased. A graph of the preliminary results along with the in v estigations at higher compressions can be seen in Fig. 4.7. Figure 4.7. This graph sho ws the a v erage lateral forces, or frictional forces, for eac h normal force compression in both the periodic (+) and appro ximated aperiodic (X) directions. Lines of best t to the original force data are sho wn to illustrate that the frictional forces started decreasing at increasing normal force. 39

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It w as ob vious that more sophisticated sim ulations w ere necessary The rst, and easiest, modication to the preliminary results w as to sim ulate a larger system. W e w en t from sim ulating a total of 14,585 atoms to sim ulating 34,770 atoms, an increase more than doubling the sim ulation size. The preliminary results w ere performed using a tip that co v ered 94% of the appro ximan t's surface area. Because periodic boundary conditions w ere used in the sliding directions, there w as a gap in the tip small enough that phonons could pass from one side to the other, see Fig. 4.1. It w as suggested b y Dr. Sagar P andit [6] that one of our problems migh t be due to these phonons. One of the modications curren tly used is that the tip is no w small enough so that one side does not in terfere with the other through the periodic boundary conditions. The third modication to the preliminary sim ulations w as the in troduction of the compression direction in to the Langevin thermostat, as suggested b y Dr. Susan Sinnott at the Univ ersit y of Florida [7]. As can be seen in Fig. 4.2 and Fig. 4.3, the lateral force data are noisy In troduction of the thermostat in the compression direction cleaned up the data considerably and surface features of the appro ximan t can no w be clearly seen in the force data; see x 5. The fourth modication to the preliminary results w as to a v erage the forces during sliding o v er the en tire sim ulation bo x rather than a small portion, as in the preliminary results. W e use the term `sim ulation bo x' to mean all of the atoms in the sim ulation before periodic boundary conditions are applied. The dimensions of the sim ulation bo x are the bo x boundaries specied in the data le. During man y sim ulations testing dieren t w a ys of measuring friction, it w as disco v ered that the surface of the appro ximan t experiences a small amoun t of buc kling during relaxation. The buc kling is periodic on the length scale of the sim ulation bo x. The preliminary results only slid o v er 2.5 A of the sim ulation bo x, whic h extended 60 A in the periodic x direction and 67 A in the aperiodic z direction. Whether this small amoun t of sliding occurred going `up-hill' or `do wn-hill' will c hange the results. This w ould accoun t for the decrease in the a v erage lateral force as the normal 40

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force is increased. By a v eraging the forces o v er sliding exactly one sim ulation-bo x length in eac h direction, this eect will be negated. The fth modication w as to use a modied v ersion of the Widom-Moriart y pair poten tials [36, 56, 55, 9], as discussed in x 3.2. The Widom-Moriart y pair poten tials that w ere shortened to 7 A created less deviation from the original structure during relaxation. 41

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CHAPTER 5 FINAL RESUL TS The nal results w ere ac hiev ed b y sim ulating a total of 34,770 atoms. There are 31,250 atoms in the appro ximan t and 3,520 atoms in the adaman t tip. In Fig. 5.1 and Fig. 5.2 it is clearly seen that the tip co v ers only a portion of the surface of the appro ximan t. This modication to the preliminary results is discussed in x 4. Just as in the preliminary results, there are four parts to eac h sim ulation: relaxation, compression, relaxation, and sliding. Initially the en tire system is allo w ed to relax. This is ac hiev ed b y applying a Langevin thermostat to all atoms in the sim ulation to bring the temperature to 0 K. The thermostat is applied for 25 ps. The compression procedure in the nal results is the same as in the preliminary results, see x 4. Once the `xed' atoms are constrained to mo v e as rigid bodies the thermostat is applied to the `thermostat' atoms only The delineation of the groups can be seen in Fig. 4.1. T o a v oid the thermostat aecting the lateral-force data, the thermostat is only allo w ed to act in the directions perpendicular to the sliding v elocit y including the compression direction. Allo wing the thermostat to act in the compression direction is one of the modications discussed in x 4. The desired temperature throughout the sim ulation is 0 K. The xed atoms of the adaman t slab are then giv en a v elocit y to w ard the quasicrystalline appro ximan t surface. Multiple compressions w ere ac hiev ed b y using dieren t compression v elocities for 38.5 ps eac h. 42

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Figure 5.1. A snapshot of the `10-fold' face and adaman t tip used in the sim ulations that produced the nal results. The green spheres are Al, white are Ni, pink are Co, and blue are adaman t (Ad). Original image rendered b y VMD [26]. 43

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Figure 5.2. A snapshot looking do wn the x direction of the appro ximan t along with the adaman t tip used in the sim ulations that obtained the nal results. The green spheres are Al, white are Ni, pink are Co, and blue are adaman t (Ad). Original image rendered b y VMD [26]. 44

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After the desired compression is ac hiev ed, the system is allo w ed to relax once more for 35 ps. The heigh t after compression is held constan t b y the xed rigid la y ers at the top and bottom of the sim ulation. T o ac hiev e the sliding portion of the sim ulation, the rigid tip atoms are giv en a constan t sliding v elocit y in either the periodic or appro ximated aperiodic direction of the appro ximan t. T o test the eect of sliding v elocit y on friction, m ultiple sliding v elocities w ere used. The sliding v elocities tested range from 0.04 A/ps to 0.12 A/ps in 0.02 A/ps incremen ts, giving a total of 5 dieren t sliding v elocities. Example graphs of the temperature ructuations o v er time, during sliding, can be seen in Fig. 5.3 and Fig. 5.4 for sliding in the `aperiodic' and periodic directions respectiv ely As discussed in x 4, due to a small amoun t of buc kling on the surface of the appro ximan t, the frictional forces experienced b y the tip need to be a v eraged o v er sliding the length of one sim ulation bo x to negate the eect of the `hills' and `v alleys' created b y the buc kling. Because of this, eac h v elocit y required a dieren t amoun t of sliding time to more than completely co v er the sim ulation bo x. When the sliding is initially begun, the data are noisy but quic kly relax. T o get clean data for sliding o v er one sim ulation-bo x size, the required sliding time w as increased. A table of the v elocities with the required sliding time and the actual time spen t sliding, in order to accoun t for an y transien t beha vior, is giv en in T able 5.1. As the sliding is performed, the forces opposing the motion of the xed tip atoms are recorded, as seen in Fig. 5.5 and Fig. 5.6, for sliding in the `aperiodic' and periodic directions respectiv ely This is the frictional force, and it is time a v eraged o v er sliding one sim ulation-bo x length. F or a more detailed description of the a v eraging procedure see x 3.4. The fact that the frictional forces go belo w zero is though t to partially be due to the lac k of adhesion bet w een the tip and the appro ximan t com bined with surface features of the appro ximan t. Similar features w ere sho wn b y Harrison et al. [24] for the frictional response of diamond. 45

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Figure 5.3. An example graph of the temperature o v er time as the tip slides across the appro ximan t surface in the `aperiodic' x direction at a compression of 37.18448 nN. The lateral force a v eraged o v er time is calculated to be 1.3533408x10 4 nN. The graph w as obtained from sliding at a speed of 0.1 A/ps. 46

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Figure 5.4. An example graph of temperature o v er time as the tip slides across the appro ximan t surface in the periodic z direction at a compression of 37.2816 nN. The lateral force a v eraged o v er time is calculated to be 1.1181488x10 4 nN. The graph w as obtained from sliding at a speed of 0.1 A/ps. 47

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V ( A/ps) Direction D ( A) t D (ps) t T (ps) 0.04 periodic 122.248805 3056.220125 3160 0.06 periodic 122.248805 2037.48008 2080 0.08 periodic 122.248805 1528.110063 1600 0.1 periodic 122.248805 1222.48805 1400 0.12 periodic 122.248805 1018.74 1200 0.04 aperiodic 100.8 2520 2600 0.06 aperiodic 100.8 1680 1720 0.08 aperiodic 100.8 1260 1320 0.1 aperiodic 100.8 1008 1200 0.12 aperiodic 100.8 840 1040 T able 5.1. Presented above is a breakdown of the required sliding times for the fve dierent sliding velocities tested. V is the sliding velocity D is the length of the simulation box or the required sliding distance, t D is the amount of time at the given velocity to slide the distance D and t T is the total amount of time spent sliding. The total amount of time spent sliding is larger than the time required to slide exactly one simulation box to allow for any transient behavior, when the sliding is initially begun, to not be included in the averaged force data. One interesting feature of the lateral-force data is the period of the peaks. It was calculated that the distance between two peaks corresponds to the amount of time it takes for the tip to slide over roughly one unit cell of the approximant in that direction. This is true for sliding in both the periodic and `aperiodic' directions of the approximant. The example plots in Fig. 5.5 and Fig. 5.6 show a period of approximately 122 ps for sliding in the `aperiodic' direction and a period of approximately 40 ps for sliding in the periodic direction. Both plots were obtained from sliding at a speed of 0.1 A/ps. The peaks correspond to sliding roughly 12 A in the `aperiodic' direction, which has a unit cell length of 12.2248805 A, and 4 A in the periodic direction which has a unit cell length of 4.032659 A. During the error calculations it was noticed that there is a discrepancy between the actual unit-cell lengths and the period. This discrepancy is taken into account in the error calculation. 1 1 The frequency discrepancy is noticed in a simulation sliding in the `aperiodic' direction with a sliding speed of 0.1 A/ps and a compression speed of 0.64 A/ps. The frequency should be 0.00818 ps )Tj/T3_4 0.12 Tf5.76 0 Td(1 but the calculated frequency from the data fles is 0.0098 ps )Tj/T3_4 0.12 Tf5.76 0 Td(1 Most of the other simulations have not been tested for this discrepancy The discrepancy does not appear in the 0.1 A/ps sliding speed, 0.65 A/ps compression speed, `aperiodic' simulation. 48

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Figure 5.5. An example plot of lateral force o v er time as the tip slides across the appro ximan t surface in the `aperiodic' x direction at a compression of 37.18448 nN. The lateral force, a v eraged o v er time, is calculated to be 1.3533408x10 4 nN. The graph w as obtained from sliding at a speed of 0.1 A/ps. 49

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Figure 5.6. An example plot of lateral force o v er time as the tip slides across the appro ximan t surface in the periodic z direction at a compression of 37.2816 nN. The lateral force, a v eraged o v er time, is calculated to be 1.1181488x10 4 nN. The graph w as obtained from sliding at a speed of 0.1 A/ps. 50

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One can also see the same features in Fig. 5.7 and Fig. 5.8 when looking at the normal force during sliding. Because of the nature of our system, with the outer la y ers acting as rigid bodies k eeping a constan t distance bet w een the xed tip atoms and xed appro ximan t atoms, when the tip slides o v er a feature of the appro ximan t unit cell it will c hange the normal force. Due to the `squishing' or `relaxing' of the cen tral groups of atoms, as the tip slides o v er a feature, the temperature is also aected, as can be seen in Fig. 5.3 and Fig. 5.4. It is importan t to note that the v ariations in the normal force and temperature are small compared to that in the lateral force. As in the preliminary results in x 4, the a v eraged lateral force is plotted as a function of the corresponding a v eraged normal force. This w as done for eac h of the v e sliding v elocities men tioned in T able 5.1. If Amon tons's La w w ere to hold, w e w ould ha v e a straigh t line [20]. Fig. 5.9 sho ws the results of sliding in the appro ximated aperiodic x direction. Fig. 5.10 sho ws the results of sliding in the periodic z direction. As y ou can see from the data in Fig. 5.9 and Fig. 5.10 w e do not ha v e straigh t lines, and there is a clear v elocit y dependence. In the experimen tal w ork done b y P ark et al. [40, 41, 42, 39], the friction along the aperiodic direction is 8 times less than the friction along the periodic direction. W e came to a similar conclusion in the preliminary results, whic h w ere later found to be unreliable, but the curren t graphs sho w that the magnitude of the frictional forces along the `aperiodic' direction are sligh tly higher than along the periodic direction for the majorit y of the compressions in v estigated. A t the higher compressions, one can see that the periodic direction has a sligh tly steeper slope whic h, according to Amon tons's La w, w ould mean a higher coecien t of friction. Both sliding directions are plotted on the same graph in Fig. 5.11 The slope of a best-t line, through the last third of the normal-force vs. lateral-force data for eac h v elocit y w as calculated and is presen ted in T ables 5.2 and 5.3. Here w e can see that there is a v ery small dierence in the estimated frictional coecien ts, at high compression, according to Amon tons's La w. 51

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Figure 5.7. An example plot of the normal force o v er time as the tip slides across the appro ximan t surface in the `aperiodic' x direction at a compression of 37.18448 nN. The lateral force, a v eraged o v er time, is calculated to be 1.3533408x10 4 nN. The graph w as obtained from sliding at a speed of 0.1 A/ps. 52

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Figure 5.8. An example plot of the normal force o v er time as the tip slides across the appro ximan t surface in the periodic z direction at a compression of 37.2816 nN. The lateral force, a v eraged o v er time, is calculated to be 1.1181488x10 4 nN. The graph w as obtained from sliding at a speed of 0.1 A/ps. 53

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Figure 5.9. This graph sho ws the a v erage lateral forces, or frictional forces, for eac h normal force compression in the appro ximated aperiodic x direction for eac h of the v e examined sliding v elocities. 54

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Figure 5.10. This graph sho ws the a v erage lateral forces, or frictional forces, for eac h normal force compression in the periodic z direction for eac h of the v e examined sliding v elocities. 55

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Figure 5.11. This graph sho ws the a v erage lateral forces, or frictional forces, for eac h normal force compression in the periodic z direction (dashed lines) and the appro ximated aperiodic x direction (solid lines) for eac h of the v e examined sliding v elocities. 56

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Direction V ( A/ps) E( ) Aperiodic 0.12 1.31715 10 5 2.3913 10 7 Aperiodic 0.10 1.07258 10 5 1.7682 10 7 Aperiodic 0.08 8.82988 10 6 1.3099 10 7 Aperiodic 0.06 6.16378 10 6 1.1328 10 7 Aperiodic 0.04 4.42399 10 6 5.9571 10 8 T able 5.2. This table con tains the calculated slopes, of the highest v e compressions for the `aperiodic' sliding direction at eac h v elocit y along with the calculated error on the slope, E( ). These are the highest compressions graphed in Fig. 5.9. According to Amon tons's La w, the slopes are a measure of the coecien ts of friction [20]. Direction V ( A/ps) E( ) P eriodic 0.12 1.44777 10 5 6.6871 10 8 P eriodic 0.10 1.10013 10 5 5.8060 10 8 P eriodic 0.08 9.42449 10 6 5.8429 10 8 P eriodic 0.06 6.94604 10 6 5.7575 10 8 P eriodic 0.04 4.92425 10 6 3.5608 10 8 T able 5.3. This table con tains the calculated slopes, of the highest v e compressions for the periodic sliding direction at eac h v elocit y along with the calculated error on the slope, E( ). These are the highest compressions graphed in Fig. 5.10. According to Amon tons's La w, the slopes are a measure of the coecien ts of friction [20]. If w e graph the coecien ts of friction as functions of sliding v elocit y for sliding in both the periodic and `aperiodic' directions, as in Fig. 5.12, w e can see that they do not o v erlap. This allo ws us to come to the conclusion that the friction coecien t in the `aperiodic' direction of our d -AlNiCo quasicrystalline appro ximan t is lo w er than the friction in the periodic direction. This agrees with the P ark experimen ts. Ev en though our o v erall conclusions agree with P ark et al. [40, 41] the magnitude of our frictional forces and the ratios of the coecien ts of friction are extremely small in comparison. P ark et al. [41] sho w frictional responses ranging from 0 to 60 nN. The results presen ted in this w ork sho w frictional responses ranging from 0 to 0.0009 nN. Some of this discrepancy can be explained b y our use of a totally repulsiv e in teraction bet w een the tip and the appro ximan t. A more realistic adhesion is left for future w ork. One can also see in Fig. 5.9 and Fig. 5.10 and T ables 5.2 and 5.3 that the frictional forces and coecien ts of friction increase with increased sliding v elocit y P ark et al. [41] found no v elocit y dependence in the torsional response of the AFM can tilev er when sliding 57

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Figure 5.12. This graph sho ws the coecien ts of friction in the `aperiodic' sliding direction (solid line) and the periodic sliding direction (dotted line) as a function of sliding v elocit y 58

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in the aperiodic direction. When sliding in the periodic direction the can tilev er's torsional response increased with increasing sliding v elocit y The sliding v elocities in v estigated b y P ark et al. ranged from 2.0 10 10 A/ps to 1.2 10 8 A/ps. Our v elocities are extremely fast in comparison, and slo w er sliding v elocities should be probed in future w ork. Emplo ying suc h high sliding v elocities in this w ork w as necessary for computation time. T o slide 1000 times slo w er, it w ould mean the sim ulation w ould tak e appro ximately 1000 times as long to complete. This w as just not feasible with the resources curren tly a v ailable. W e can also look at the ratio of the coecien ts of friction as done b y P ark et al. [40, 41, 42, 39]. The results from the P ark et al. experimen ts sho w an 8-fold anisotrop y Our results sho w an anisotrop y ranging from 1.026 to 1.127, as seen in T able 5.4, depending on the sliding v elocit y V ( A/ps) periodic aperiodic periodic aperiodic 0.12 1.44777 10 5 1.31715 10 5 1.099 0.10 1.10013 10 5 1.07258 10 5 1.026 0.08 9.42449 10 6 8.82988 10 6 1.067 0.06 6.94604 10 6 6.16378 10 6 1.127 0.04 4.92425 10 6 4.42399 10 6 1.113 T able 5.4. This table con tains the calculated slopes, of the highest v e compressions for the periodic and aperiodic directions at eac h v elocit y along with the ratio. Through the majorit y of the compressions in v estigated, the magnitude of the frictional forces in the `aperiodic' direction are higher than in the periodic direction, but the coefcien ts of friction, at high compression, sho w that the `aperiodic' direction is lo w er than the periodic direction. The o v erall magnitudes of the frictional forces found in these experimen ts are on the order of 10,000 times smaller than in the experimen ts performed b y P ark et al. [40, 41]. The simple model presen ted here made man y appro ximations. With these extreme appro ximations w e w ere still able to sho w a dependence of friction on periodicit y Ev en with suc h small frictional forces, a frictional anisotrop y bet w een sliding in the `aperiodic' and periodic directions w as found, and w e agree with P ark et al. in that the coecien t of 59

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friction in the aperiodic direction of a d -AlNiCo quasicrystal is lo w er than the coecien t of friction along the periodic direction. 60

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CHAPTER 6 FUTURE W ORK The main goal for future w ork is to create more realistic sim ulations. By doing this, one w ould hope to obtain a larger frictional anisotrop y than is seen in the nal results of this w ork. Not only is the anisotrop y m uc h smaller than expected; the o v erall magnitude of the frictional forces is also quite small when compared with experimen t. T o ac hiev e more sophisticated sim ulations one w ould start with the comparison of dieren t quasicrystalline appro ximan ts, mak e larger sim ulations, tailor the pair poten tials to the appro ximan ts being studied, create more realistic poten tials, create and use a more realistic tip, and closely monitor phonon propagation through the system. 6.1 Comparison of Dieren t Appro ximan ts Quasicrystals are crystals lac king translational symmetry eectiv ely making them ha v e an innite unit cell in 2 or 3 dimensions. T o sim ulate quasicrystals, appro ximan ts are needed, see x 3.1. T o con tin ue this w ork it w ould be adv an tageous to compare dieren t AlNiCo appro ximan ts of v arying size. Because sim ulating a real quasicrystal is impossible, one w ould lik e to ha v e the largest appro ximan t possible to more closely resem ble a real quasicrystal. W e ha v e tested t w o new quasicrystalline appro ximan ts supplied b y Marek Mihalk o vi c [2] con taining 132 and 343 atoms per unit cell. Both of the appro ximan ts con tain one bila y er. Pictures of the appro ximated 10-fold face of a single unit cell for eac h are visualized in Fig. 6.1 and Fig. 6.2. A 4 4 4-unit-cell bloc k of the 343-atom unit-cell appro ximan t brok e apart during relaxation when using the original un-truncated poten tials. The use of the truncated 61

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Figure 6.1. The illustration sho ws the `10-fold' face of a 132-atom unit-cell quasicrystalline appro ximan t. The appro ximan t w as supplied b y Marek Mihalk o vi c [2]. The green spheres are Al, white are Ni, and pink are Co. Original image rendered b y VMD [26] 62

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Figure 6.2. The illustration sho ws the `10-fold' face of a 343-atom unit-cell quasicrystalline appro ximan t. The appro ximan t w as supplied b y Marek Mihalk o vi c [2]. The green spheres are Al, white are Ni, and pink are Co. Original image rendered b y VMD [26]. 63

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poten tials eliminated this problem in the 343-atom unit-cell appro ximan t, although there w as still a signican t amoun t of relaxation. In v estigation in to a more accurate poten tial ma y be required for this appro ximan t. The 132-atom appro ximan t did not sho w an y signican t problems with either the original or truncated poten tials. This being said, there w as some relaxation so mean squared displacemen t tests should be run, as done in x 3.2. 6.2 Larger Sim ulations Hand in hand with using larger appro ximan ts one w ould need larger sim ulation sizes. With a larger appro ximan t and tip one ma y be able to measure frictional v alues closer to the experimen tal v alues presen ted b y P ark, et al. [40, 41, 39] b y creating smaller pressures than presen ted here. The maxim um pressure studied b y P ark et al. [41] is 3.3 GP a. The maxim um pressure reac hed in this researc h is 3.83 GP a. A smaller pressure can be ac hiev ed using the same normal forces b y creating a larger con tact area. 6.3 T ailoring the P air P oten tials The Widom-Moriart y pair poten tials [36, 56, 55, 9] used in this researc h will ev en tually need to be replaced b y more accurate poten tials tailored to this system. These poten tials w ere optimized for bulk structures at high temperature for the purpose of studying alumin um migration. P oten tials optimized for surfaces at lo w er temperatures w ould be more appropriate. One could start obtaining better poten tials b y modifying the existing ones to more closely resem ble the original appro ximan t structure. Ev en the cut-o poten tials in Fig. 3.5 could use some impro v emen t. This w ould need to be done for eac h appro ximan t studied. Ev en tually pair poten tials w ould not be enough, and an EAM [14] or another similar poten tial will need to be dev eloped to more accurately describe the electronic in teractions. These more sophisticated poten tials con tain a pair-wise in teraction, suc h as the Widom64

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Moriart y pair poten tials, but also ha v e a term dependen t on the local electronic densit y These poten tials come closer to describing a real system than pair poten tials alone. 6.4 Creating and Using a More Realistic Tip The original P ark experimen t, [40, 41, 39], used an alk anethiol passiv ated AFM tip. The purpose of using a passiv ated tip w as to minimize adhesion. W e ha v e simplied this concept b y creating and using a non-in teracting `adaman t' tip; ho w ev er, a realistic adhesiv e force could pro vide more accurate frictional forces, see x 3.3. T o create a more realistic tip one needs not only structures but also poten tials. 6.5 Monitoring Phonon Propagation It is eviden t, b y w atc hing mo vies of the sim ulations, that phonons are propagating through the system. An analysis and in v estigation of these phonons could pro vide great insigh t in to the mec hanisms of friction and energy dissipation. Some in teresting information is seen if one plots lateral vs. normal force as in Fig. 6.3, temperature vs. normal force as in Fig. 6.6, and temperature vs. lateral forces as in Fig. 6.4 and Fig. 6.5. What w e ma y be looking at here are P oincare sections [21]. W e expect there to be a correlation bet w een the normal and lateral forces, and also the temperature and the forces; ho w ev er w e ma y be looking at 2-d projections of a higher dimensional phenomenon. This should be in v estigated further in the future. Along with a more sophisticated phonon analysis, a more accurate calculation of the vibrations in the system could lead to a more appropriate timestep. Phonons ha v e been studied in quasicrystals and quasicrystalline appro ximan ts before [46, 35, 27]. The problem in studying the lattice dynamics of an aperiodic system is the existence of an innitely large unit cell. In periodic crystals, phonons are described as perterbations to an underlying reciprocal lattice. Lik e amorphous solids, quasicrystals do 65

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Figure 6.3. This graph sho ws an example plot of the lateral force as a function of the normal force for sliding in the `aperiodic' direction. The a v erage normal force is 37.18448 nN. The a v erage lateral force is 0.000135334 nN. This graph w as obtained from sliding at a speed of 0.1 A/ps. Figure 6.4. This graph sho ws an example plot of the lateral force as a function of the temperature for sliding in the `aperiodic' direction. The a v erage normal force is 37.18448 nN. The a v erage lateral force is 0.000135334 nN. This graph w as obtained from sliding at a speed of 0.1 A/ps. 66

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Figure 6.5. This graph sho ws an example plot of the lateral force as a function of the temperature for sliding in the periodic direction. The a v erage normal force is 37.2816 nN. The a v erage lateral force is 0.00011181488 nN. This graph w as obtained from sliding at a speed of 0.1 A/ps. Figure 6.6. This graph sho ws an example plot of the normal force as a function of the temperature for sliding in the `aperiodic' direction. The a v erage normal force is 37.18448 nN. The a v erage lateral force is 0.000135334 nN. It is clear that an increase in the normal force corresponds to an increase in the temperature. This is expected due to the constan t heigh t during sliding. When the tip crosses a high feature of the appro ximan t, all of the atoms in the middle are compressed leading to a higher normal force and higher temperature. This graph w as obtained from sliding at a speed of 0.1 A/ps. 67

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not ha v e a reciprocal lattice. This does not mean that the task is impossible and, as in this w ork, one can use quasicrystalline appro ximan ts. When comparing periodic crystals with amorphous solids, one m ust accept that the absence of long-range order in an amorphous material ma y localize the vibrational modes [27]. This can cause an absence of long-range w a v es propogating through the system. Con trary to this, quasicrystals posess long-range order though they lac k translational periodicit y When studying an AlNiCo quasycrystalline appro ximan t Mihalk o vi c et al. found that at lo w frequency there is localization of the phonon modes and that the locatlization rapidly increases for increasing frequency Experimen tally de Boissieu et al. [15] compare the lattice dynamics of an icosahedral quasicrystal and its corresponding 1/1 appro ximan t using inelastic neutron and x-ra y scattering. F or both the quasicrystal and appro ximan t, a w ell-dened transv erse acoustic mode is found. The calculated sound v elocities are 2,670 (+-30) m/s for the quasicrystal and 2,660 m/s for the corresponding appro ximan t. Along with a more sophisticated phonon analysis, a more accurate calculation of the vibrations in the system could lead to a more appropriate timestep. Because experimen tal studies ha v e sho wn the similarities bet w een acoustic phonon modes in quasicrystals and their appro ximan ts [15], con tin uing with this theoretical w ork on and AlNiCo quasicrystalline appro ximan t could lead to further insigh t on the d -AlNiCo quasicrystal and the correlation bet w een phonons and friction. 68

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REFERENCES [1] Mik e Widom. personal comm unication. [2] Marek Mihalk o vic. personal comm unication. [3] See the LAMMPS User's Man ual at http://lammps.sandia.gov/ [4] http://www.ccp5.ac.uk/DL POLY/ as retriev ed on 31 July 2008. [5] http://www.gromacs.org as retriev ed on 2 August 2008. [6] Sagar P andit. personal comm unication. [7] Susan Sinnott. personal comm unication. [8] Milton Abramo witz and Irene A. Stegun, Handb o ok of mathematic al functions v ol. 9, Do v er, 1970. [9] I. Al-Leh y ani, M. Widom, Y. W ang, N. Moghadam, G.M. Stoc ks, and J. Moriart y T r ansition-metal inter actions in aluminum-rich intermetallics Ph ysical Review B 64 (2001), no. 075109. [10] Neil W. Ashcroft and Da vid N. Mermin, Solid State Physics Harcourt, Inc., 1976. [11] H.J.C. Berendsen, D. v an der Spoel, and R. v an Drunen, GR OMA CS: A messagep assing p ar allel mole cular dynamics implementation Comp. Ph ys. Comm. 91 (1995), 43{56. [12] Pierre Brunet, L.-M. Zhang, Daniel J. Sordelet, Matt Besser, and Jean-Marie Dubois, Comp aritive study of micr ostructur al and trib olo gic al pr op erties of sinter e d, bulk ic osahe dr al samples Materials Science and Engineering 294-296 (2000), 74{78. [13] Robert W. Carpic k, F rank D. Ogletree, and Miquel Salmeron, A gener al e quation for tting c ontact ar e a and friction vs lo ad me asur ements Journal of Colloid and In terface Science 211 (1999), 395{400. [14] M.S. Da w and M.I. Bask es, Emb e dde d-atom metho d: Derivation and applic ation to inpurities, surfac es, and other defe cts in metals Ph ysical Review B 29 (1984), no. 12, 6443{6453. [15] M. de Boissieu, S. F rancoual, M. Mihalk o vic, K. Shibata, A.Q.R. Baron, Y. Sidis, T. Ishimasa, D. W u, T. Lograsso, L.-P Regnault, F. Gahler, S. Tsutsui, B. Hennion, 69

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P Bastie, T.J. Sato, H. T ak akura, R. Currat, and A.-P Tsai, L attic e dynamics of the zn-mg-sc ic osahe dr al quasicrystal and its zn-sc p erio dic 1/1 appr oximant. Nature Materials 6 (2007), 977{984. [16] B.V. Derjaguin, V.M. Muller, and YU.P T oporo v, Ee ct of c ontact deformations on the adhesion of p articles Journal of Colloid and In terface Science 53 (1975), no. 2, 314{326. [17] J. Dolinsek, P Jeglic, M. Komelj, S. V rtnik, A. Smon tara, I. Smiljanic, A. Bilusic, J. Ivk o v, D. Stanic, E.S. Zijlstra, B. Bauer, and P Gille, Origin of anisotr opic nonmetallic tr ansp ort in the A l 80 Cr 15 F e 5 de c agonal appr oximant Ph ysical Review B 76 (2007), 174207. [18] L.P F eng, T.M. Shao, Y.J. Jin, E. Fleury D.H. Kim, and D.R. Chen, T emp er atur e dep endenc e of the trib olo gic al pr op erties of laser r e-melte d A l-Cu-F e quasicrystalline plasma spr aye d c o atings Journal of Non-Crystalline Solids 351 (2004), 280{287. [19] An thon y C. Fisc her-Cripps, Intr o duction to Contact Me chanics Mec hanical Engineering Series, Springer-V erlag New Y ork, Inc., 175 Fifth Av en ue, New Y ork, NY, 2000. [20] J Gao, W.D. Luedtk e, D. Gourdon, M. Ruths, J.N. Israelac h vili, and U Landman, F rictional for c es and Amontons' Law: F r om the mole cular to the macr osc opic sc ale J. Ph ys. Chem. B 108 (2004), 3410{3425. [21] Nic holas J. Giordano, Computational Physics Pren tice-Hall, Inc., 1997. [22] I.G. Gory ac hev a, Contact Me chanics in T rib olo gy Solid Mec hanics and its Applications, v ol. 61, Klu w er Academic Publishers, P .O. Bo x 17, 3300 AA Dordrec h t, The Netherlands, 1998. [23] D.S. Grierson, E.E. Flater, and R.W. Carpic k, A c c ounting for the JKR-DMT tr ansition in adhesion and friction me asur ements with atomic for c e micr osc opy J. Adhesion Sci. T ec hnol. 19 (2005), no. 3-5, 291{311. [24] J.A. Harrison, C.T. White, R.J. Colton, and D.W. Brenner, Mole cular-dynamics simulations of atomic-sc ale friction of diamond surfac es Ph ysical Review B 46 (1992), no. 15, 9700{9708. [25] Christopher L. Henley Cell ge ometry for cluster-b ase d quasicrystal mo dels Ph ys. Rev. B 43 (1991), 993. [26] W. Humphrey A. Dalk e, and K. Sc h ulten, VMD Visual Mole cular Dynamics Journal of Molecular Graphics 14 (1996), 33{38. [27] C. Janot, Quasicrystals a primer v ol. 2, Clarendon Press, 1994. [28] K.L. Johnson, A note on the adhesion of elastic solids British Journal of Applied Ph ysics 9 (1958), 199{200. [29] K.L. Johnson, K. Kendall, and A.D. Roberts, Surfac e ener gy and the c ontact of elastic solids Proc. R. Soc. Lond. A. 324 (1971), 301{313. 70

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[30] J.S. Ko, A.J. Gellman, T.A. Lograsso, C.J. Jenks, and P .A. Thiel, F riction b etwe en single-gr ain A l 70 Pd 21 Mn 9 quasicrystal surfac es Surface Science 423 (1999), 243{255. [31] D.C. Lo v elady H.M. Harper, I.E. Brodsky and D.A. Rabson, Multiphase r e gion of helimagnetic sup erlattic es at low temp er atur e in an extende d six-state clo ck mo del J. Ph ys. A: Math. Gen. 39 (2006), 5681{5694. [32] C. Mancinelli, C.J. Jenks, P .A. Thiel, and A.J. Gellman, T rib olo gic al pr op erties of a B2-typ e A l-Pd-Mn quasicrystal appr oximant Journal of Metrials Researc h 18 (2003), 1447{1456. [33] Daniel Maugis, A dhesion of spher es: The JKR-DMT tr ansition using a Dugdale mo del Journal of Colloid and In terface Science 150 (1992), no. 1, 243{269. [34] M. Mihalk o vic, I. Al-Leh y ani, E. Coc k a yne, C.L. Henley N. Moghadam, J.A. Moriart y Y. W ang, and M. Widom, T otal ener gy-b ase d pr e diction of a quasicrystal structur e Ph ysical Review B 65 (2002), no. 104205. [35] M. Mihalk o vic, H. Elhor, and J.-B. Suc k, L ow-ener gy phonon excitations in the de c agonal quasicrystal Materials Science and Engineering 294-296 (2000), 654{657. [36] J.A. Moriart y and M. Widom, First-principles inter atomic p otentials for tr ansitionmetal aluminides: The ory and tr ends acr oss the 3d series Ph ysical Review B 56 (1997), no. 13, 7905{7917. [37] V.M. Muller, V.S. Y ushc henk o, and B.V. Derjaguin, On the inruenc e of mole cular for c es on the deformation of an elastic spher e and its sticking to a rigid plane Journal of Colloid and In terface Science 77 (1980), no. 1, 91{101. [38] J.Y. P ark, D.F. Ogletree, M. Salmeron, C.J. Jenks, and P .A. Thiel, F riction and adhesion pr op erties of cle an and oxidize d A l-Ni-Co de c agonal quasicrystals: a UHV atomic for c e micr osc opy/sc anning tunneling micr osc opy study T ribology Letters 17 (2004), no. 3, 629{636. [39] J.Y. P ark, D.F. Ogletree, M. Salmeron, C.J. Jenks, P .A. Thiel, J. Brenner, and J.M. Dubois, F riction anisotr opy: A unique and intrinsic pr op erty of de c agonal quasicrystals Journal of Materials Researc h 23 (2008), 1488{1493. [40] J.Y. P ark, D.F. Ogletree, M. Salmeron, R.A. Ribeiro, P .C. Caneld, C.J. Jenks, and P .A. Thiel, High frictional anisotr opy of p erio dic and ap erio dic dir e ctions on a quasicrystal surfac e Science 309 (2005), 1354{1356. [41] T rib olo gic al pr op erties of quasicrystals: Ee ct of ap erio dic versus p erio dic surfac e or der Ph ysical Review B 74 (2006), no. 024203. [42] J.Y. P ark, D.F. Ogletree, M. Salmeron, R.A. Ribiero, P .C. Caneld, C.J. Jenks, and P .A. Thiel, A tomic sc ale c o existenc e of p erio dic and quasip erio dic or der in a 2-fold A lNi-Co de c agonal quasicrystal surfac e Ph ysical Review B 72 (2005), no. 220201(R). 71

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[43] James C. Phillips, Rosemary Braun, W ei W ang, James Gum bart, Emad T ajkhorshid, Elizabeth Villa, Christophe Chipot, Robert D. sk eel, Laxmik an t Kale, and Klaus Sc h ulten, Sc alable mole cular dynamics with NAMD Journal of Computational Chemistry 26 (2005), 1781{1802. [44] S. Plimpton, F ast p ar allel algorithms for short-r ange mole cular dynamics Journal of Computational Ph ysics 117 (1995), 1{19. [45] K. Pussi, N. F erralis, M. Mihalco vic, M. Widom, S. Curtarolo, M. Gierer, C.J. Jenks, P Caneld, I.R. Fisher, and R.D. Diehl, Use of p erio dic appr oximants in a dynamic al LEED study of the quasicrystalline tenfold surfac e of de c agonal A l-Ni-Co Ph ysical Review B 73 (2006), 184203. [46] M. Quilic hini and T. Janssen, Phonon excitations in quasicrystals Rev. Mod. Ph ys. 69 (1997), no. 1, 277{314. [47] D.C. Rapaport, The Art of Mole cular Dynamics Simulation v ol. 2, Cam bridge Univ ersit y Press, 1995. [48] D. Shec h tman, I. Blec h, D. Gratias, and J.W. Cahn, Metallic phase with long-r ange orientational or der and no tr anslational symmetry Ph ysical Review Letters 53 (1984), 1951{1953. [49] W ei-Mei Sh yu and G.D. Gaspari, Sound velo city in metals Ph ysical Review 177 (1969), no. 3, 1041{1043. [50] W. Smith and T.R. F orester, DL POL Y 2.0: A gener al-purp ose p ar allel mole cular dynamics simulation p ackage Journal of Molecular Graphics 14 (1996), 136{141. [51] D.J. Sordelet, M.F. Besser, and J.L. Logsdon, A br asive we ar b ehavior of A l-Cu-F e quasicrystalline c omp osite c o atings Materials Science and Engineering A255 (1998), 54{65. [52] W alter Steurer, Twenty ye ars of structur e r ese ar ch on quasicrystals. p art I. p entagonal, o ctagonal, de c agonal, and do de c agonal quasicrystals Z. Kristallogr. 219 (2004), 391{ 446. [53] D. T abor, Surfac e for c es and surfac e inter actions Journal of Colloid and In terface Science 58 (1977), no. 1, 2{13. [54] P .A. Thiel, Quasicrystal surfac es Ann ual Review of Ph ysical Chemistry 59 (2008), 129{152. [55] M. Widom, I. Al-Leh y ani, and J.A. Moriart y First-principles inter atomic p otentials for tr ansition-metal aluminides. III. extension to ternary phase diagr ams Ph ysical Review B 62 (2000), no. 6, 3648{3657. [56] M. Widom and J.A. Moriart y First-principles inter atomic p otentials for tr ansitionmetal aluminides. II. applic ation to A l-Co and A l-Ni phase diagr ams Ph ysical Review B 58 (1998), no. 14, 8967{8979. 72

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APPENDICES 73

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Appendix A LAMMPS W ork arounds This c hapter is concerned with some of the tec hnical issues, and corresponding w ork arounds implemen ted, with the LAMMPS code. Only one bug in the code w as found, but there are n umerous features that are not w ell documen ted in the LAMMPS User's Man ual [3]. A.1 Obtaining F orces on Fixed-Rigid A toms The procedure used to perform the friction experimen ts requires xing the forces on the top-most la y ers of the tip and the bottom-most la y ers of the appro ximan t to zero in all three directions. T o obtain the normal and frictional force data, w e had to kno w what the forces acting on the xed tip atoms w ould ha v e been, had the forces not been set to zero. A t rst, w e w ere only a w are of obtaining force information using the dump command. When the forces on the xed atoms w ere prin ted using the dump command, LAMMPS prin ted the xed force, and th us w e did not kno w ho w to obtain the required information. Ho w to obtain the forces calculated for a group of atoms before a fix setforce command w as applied w as not w ell documen ted. The older v ersions of the User's Man ual did not, in the fix setforce command description, specify a w a y to do this. An email w as sen t to Stev e Plimpton, the author of most of the LAMMPS pac k age, and he informed us that the v alues calculated for a group of atoms, before a x is applied, can be prin ted using the thermo style command. The syn tax is thermo_style custom step temp f_1[1] f_1[2] f_1[3] The abo v e command c hanges the thermodynamic output to prin t timestep, temperature, the x v alue from x 1, the y v alue from x 1, and the z v alue from x 1, where x 1 w ould be the x setforce command used to set the forces initially to zero. In the April 2008 User's Man ual it is stated in the documen tation for the fix setforce command that these v alues can be accessed b y v arious output commands, and the reader is inevitably led to the thermo style command, where the syn tax abo v e is documen ted. 74

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Appendix A (Con tin ued) A.2 Bugs Noted With the LAMMPS Splining Routine The Widom-Moriart y pair poten tials came in a table format whic h listed, in columns, the distance bet w een t w o atom t ypes follo w ed b y the poten tial at that distance. LAMMPS is set up to handle this st yle of poten tial b y using v arious methods to in terpolate bet w een the poin ts giv en b y the table. The cubic spline in terpolation w as c hosen because it pro vides more accurate results than a straigh t linear in terpolation. The syn tax of the command is pair_style table spline N where N is the n um ber of v alues in the table. By implemen ting the poten tials in this manner, LAMMPS w ould ev en tually incur an error sa ying that t w o atoms w ere closer than the inner table cuto, whic h in our case is less than 1 A. This w ould mean that the forces felt b y these t w o atoms is almost innite. F or the atoms to be allo w ed to v en ture that close to eac h other w as highly unlik ely After looking through the LAMMPS source-code, the problem w as narro w ed do wn to the splining routine. Keith McLaughlin in v estigated the routine that reads the tabulated poten tials and found that not only does it fail using a spline in terpolation, but it also fails using a linear in terpolation sc heme. The problem is associated with the w a y LAMMPS stores and ev aluates the tabulated information. The curren t w ork around for this issue is to not ha v e N be the n um ber of en tries in the table, but rather a v ery large n um ber (N=5000 w as used in this researc h). This seems to w ork for both splining and linear in terpolation. A.3 Using a T riclinic Bo x in a Data File The v ersion of the LAMMPS User's Man ual a v ailable during the start of this researc h did not documen t ho w to specify a triclinic sim ulation bo x using a separate data le and the read data command. This feature w as necessary for the unit cell being used. A t rst w e though t that the code w ould need to be modied; ho w ev er, ev en tually the 75

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Appendix A (Con tin ued) appropriate command w as found in the source code, although no documen tation of it existed in the man ual. The April 2008 v ersion does include this documen tation, so it will not be elaborated upon here except to refer the reader to Appendix C. 76

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Appendix B Calculating an Appropriate Timestep The timestep c hosen for a molecular-dynamics sim ulation is importan t. The timestep needs to be small enough to capture the highest-frequency vibrations in the system but as large as possible to sa v e on computation time. F rom w atc hing videos of the sim ulations in this w ork one can clearly see phonons propagating in the adaman t tip. F rom the data les, the speed of these phonons w as calculated to be 1.754 A/ps. Using the speed of these phonons along with the length of one unit cell of the adaman t tip, a =6.3 A, w e w ere able to roughly estimate or the frequency of the phonons in reciprocal space. According to Ashcroft and Mermin [10] phonons propagating along a c hain of atoms ha v e a v elocit y c calculated as c = k where is the frequency and k the w a v e v ector. Using k = / a w e calculate a frequency of =0.87 ps 1 giving a time of 1 =1.14 ps. This calculation w as not in tended to be accurate or precise; it serv ed only as a starting poin t for testing v arious timesteps through sim ulation. W e can compare this to the speed of sound in Al whic h is giv en as 53.35 A/ps in the [100] direction, 58.00 A/ps in the [110] direction, and 59.43 A/ps in the [111] direction b y Sh yu et al. [49]. T aking the highest v alue of 59.43 A/ps w ould giv e a frequency of Al =29.6 ps 1 giving a time of Al 1 =0.034 ps. No w that w e ha v e a starting poin t, w e do not w an t our timestep to exceed one oneh undredth of or roughly 0.01 ps. This also k eeps our timestep small enough to catc h the acoustic phonon vibrations in pure alumin um, k eeping in mind that our appro ximan t is 68 at.% alumin um. W e use the 0.01 ps estimate as our maxim um timestep because only the phonons in the adaman t tip w ere tak en in to accoun t during the calculation. Phonons in the appro ximan t are not visible and w ould require a m uc h more sophisticated analysis, as discussed in x 6. The shortest unit-cell length in the appro ximan t is just o v er 4 A, whic h 77

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Appendix B (Con tin ued) is shorter than the adaman t unit cell length. A shorter unit cell v ector w ould lead to a higher frequency and th us a smaller time. T o actually c hoose the appropriate timestep, friction sim ulations using timesteps including 0.001 ps, 0.002 ps, 0.004 ps, and 0.008 ps w ere performed. After eac h sim ulation w as run, the a v erage lateral and frictional forces w ere calculated and compared. It w as found that increasing the timestep from 0.001 to 0.004 ps did not c hange the results. A t 0.008 ps the results began to c hange sligh tly so the 0.004 ps timestep w as c hosen to perform all of the nal sim ulations. 78

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Appendix C Required Files The goal of running these sim ulations is to study an y frictional anisotrop y bet w een sliding in the periodic vs. aperiodic directions on the 10-fold face of a decagonal quasicrystal. Due to the inabilit y to sim ulate the innite unit cell of a real quasicrystal, a quasicrystalline appro ximan t m ust be used, see x 3.1. W e ha v e fashioned a sim ulated AFM tip using `adaman t,' a ctitious material that beha v es as a v ery hard alumin um and has only repulsiv e in teractions with the appro ximan t, see x 3.3. T o perform the experimen ts, the tip is placed abo v e the appro ximan t, and the tip and appro ximan t are allo w ed to relax to a temperature of less than 1 K. After the initial relaxation, the bottom-most la y ers of the quasicrystal are held xed in space as the tip is giv en a do wn w ard v elocit y to come in con tact with the quasicrystal. This allo ws us to ac hiev e a compression. Once the system is compressed, the top few la y ers of the tip are also xed in space, allo wing us to k eep a constan t heigh t during sliding. Once w e ha v e xed our heigh t, the system is allo w ed to relax once more before the xed tip atoms are giv en a constan t sliding v elocit y parallel to the tip-quasicrystal in terface. The sliding portion of the sim ulation is where the friction data are gathered. F or eac h of the periodic and appro ximated aperiodic sliding directions, one needs to obtain normalforce and lateral-force data for m ultiple compressions. This allo ws us to plot a graph of normal force vs. lateral force, and follo wing Amon ton's La w [20], the slope of this line is the friction coecien t. A sim ulation is needed for eac h compression and in eac h of the t w o sliding directions. Ev ery sim ulation begins with a data le con taining the sim ulation-bo x boundaries, masses, and initial positions of all atoms, a poten tial le con taining all of the pair-poten tial information, an input-parameters le that has all of the information for running the sim ulation, and nally if running in parallel, a submission script for CIR CE, the computing cluster main tained b y Researc h Computing at the Univ ersit y of South Florida. 79

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Appendix C (Con tin ued) C.1 Data File All LAMMPS data les are ASCII text les read b y the LAMMPS program when told to do so in the parameters le. The rst line is ignored, so it is a free commen t line. An y other commen t lines ha v e to begin with `#'. A small portion of a data le used in this researc h is sho wn belo w. Atomic coordinate file for a qc 44626 atoms 4 atom types 0.0 122.248805 xlo xhi 0.0 158.0832968 ylo yhi 0.0 100.8 zlo zhi -51.3643768 0.0 0.0 xy xz yz # 1=Al 2=Co 3=Ni 4=Ad Masses 1 26.982 2 58.933 3 58.693 4 26.982 Atoms 1 1 9.82553 7.16461 0.780714 2 3 10.0143 0.843104 0.0925141 3 3 8.2999 1.13609 2.1651 . The blank lines, as seen abo v e, are required. The n um ber of atoms in the le is specied rst b y `44626 atoms'. If this does not matc h the n um ber of atom en tries, there will be an error message. Next, the n um ber of dieren t `t ypes' of atoms is giv en as `4 atom t ypes'. 80

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Appendix C (Con tin ued) The atom t ype is not restricted b y elemen t; it is used as a label only Dieren t atom t ypes can ha v e the same mass. The next few lines, after the n um ber of atom t ypes, giv e the sim ulation bo x boundaries. These are the boundary conditions. Whether or not one is using periodic boundary conditions, one has to specify the bounds of the sim ulation bo x. It is possible to use an innite sim ulation bo x, essen tially creating a sample in an innite v acuum, b y t yping INF, but this is inappropriate for our purposes. Due to the unique shape of our appro ximan t unit cell, the periodic boundary conditions are in teger m ultiples of the appro ximan t unit cell v ectors, rather than the tip unit cell, whic h is F CC; see x 3.1, 3.3. The syn tax begins with t w o n um bers whic h specify the minim um and maxim um coordinate v alues for that direction. The last line giv es the sk ewing. There are 3 sk ewing parameters that can be specied: xy will shift the upper y face (the x-z plane that is highest on the y-axis) in the x direction, xz will shift the upper z face (the x-y plane that is highest on the z-axis) in the x direction, and yz will shift the upper z face in the y direction, all in A. The LAMMPS documen tation for this is highly insucien t. When one initially inputs the bo x boundaries, LAMMPS assumes a rectilinear bo x whic h will then need to be sk ew ed to obtain other shapes, see x A.3. The origin for the sim ulation bo x is at (0, 0, 0). F or the example giv en abo v e, the unit cell v ectors are sho wn in T able C.1. V ector X ( A) Y ( A) Z ( A) a -51.3643768 158.0832968 0.00000000 b 122.2488050 0.00000000 0.00000000 c 0.00000000 0.00000000 100.8 T able C.1. These are the v ectors describing the sim ulation bo x boundaries for the example data le sho wn in Appendix C.1 The Masses k eyw ord lists the masses for all t ypes of atoms, in atomic mass units. Eac h individual atom is listed as being of a certain t ype rather than a certain mass, not only to sa v e t yping, but for another w a y to group atoms in the parameters le. There will be an 81

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Appendix C (Con tin ued) error if a mass is not specied for eac h t ype, but more than one t ype ma y ha v e the same mass. Also, one should note that LAMMPS fail with an will error if there is a t ype-2 atom without a t ype 1; this incon v enien t feature will th us not allo w atoms to be giv en t ypes matc hing their atomic n um bers. The bulk of the data le comes after the Atoms k eyw ord. A t a minim um, the initial position and t ype of ev ery atom in the sim ulation m ust be specied here. Eac h atom is also giv en a unique in teger iden tier; this feature is utilized in the parameters le, in Appendix C.3, for grouping atoms together. The format used begins with the unique atom ID, the atom t ype, x-position, y-position, and ends with z-position. The t ype of data included in the data le is determined in the parameters le b y c hoosing the atom st yle. This w ork w as all done using the most basic st yle, `atomic'. All of the n umerical v alues in the data le will ha v e the same units that are specied in the parameters le b y the units command. C.2 P oten tial File All of the sim ulations in this researc h used the tabulated Widom-Moriart y pair potentials. LAMMPS is built to handle tabulated pair poten tials in the follo wing format. Note the problem in using the LAMMPS splining routine as documen ted in x A.2. AlAl N 303 1 0.95251 54.9408 131.907 2 1.00543 48.2862 119.59 3 1.05835 42.2834 107.273 . There can be m ultiple tables in one le, as w e ha v e done in all pots.long whic h is used for the preliminaryresults and can be found in /home/students/harper/qcr ese arch /LAM MPS 82

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Appendix C (Con tin ued) on Ph ysics. The nal results use the cut-o smoothed poten tials, whic h can be found in /home/students/harper/qc rese arch /LA MMPS /all pots.smoothed on Ph ysics. Eac h table in the le begins with a unique name. This is the name used in the parameters le to specify the poten tial bet w een t w o t ypes of atoms. After the name, suc h as `AlAl' abo v e, the k eyw ord N is follo w ed b y a n um ber tells LAMMPS ho w man y en tries are in the table. The LAMMPS format demands that there be a blank line bet w een the header and the bulk of the table. Eac h en try in the table has a unique in teger ID in the rst column follo w ed b y the distance bet w een the atoms in A, the poten tial at that distance in eV, and the force at that distance in eV/ A. The original tabulated poten tials did not include the force v alues, so they w ere calculated using the get forces script found in /home/students/harper/bi n on Ph ysics. The script performs a rough w eigh ted deriv ativ e. C.3 Sim ulation P arameters File The ordering of commands in a parameters le is v ery importan t. LAMMPS reads the le one line at a time and executes the command on that line as it is read. An example parameters le is sho wn belo w. I will go through eac h command and briery describe its function and purpose for this researc h. # an Ad tip compressing and sliding on a QC surface log log0.1 dimension 3 boundary p p p units metal atom_style atomic read_data data.4 # LJ potentials pair_style table spline 5000 pair_coeff 1 1 all_pots.long AlAl 83

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Appendix C (Con tin ued) pair_coeff 1 2 all_pots.long AlCo pair_coeff 1 3 all_pots.long AlNi pair_coeff 2 2 all_pots.long CoCo pair_coeff 2 3 all_pots.long CoNi pair_coeff 3 3 all_pots.long NiNi pair_coeff 4 4 all_pots.long AdAd pair_coeff 1 4 all_pots.long AdX pair_coeff 3 4 all_pots.long AdX pair_coeff 2 4 all_pots.long AdX group qcfix id <> 1 6250 group qctemp id <> 6251 18750 group qcfree id <> 18751 31250 group adfree id <> 31251 36114 group adtemp id <> 36115 40978 group adfix id <> 40979 44626 group temps union qctemp adtemp fix 1 all nve fix 2 all langevin 0 0 0.1 239482 timestep 0.0005 thermo 100 dump 1 all custom 1000 out0.1 tag type x y z run 50000 fix 3 adfix setforce 0.0 0.0 0.0 fix 4 qcfix setforce 0.0 0.0 0.0 velocity qcfix set 0.0 0.0 0.0 units box sum no unfix 2 fix 5 temps langevin 0 0 0.01 239482 axes 0 0 1 thermo_style custom step temp f_3[1] f_3[2] f_3[3] velocity adfix set 0.0 -0.75 0.0 units box sum no run 77000 velocity adfix set 0.0 0.0 0.0 units box sum no run 70000 velocity adfix set -0.05 0.0 0.0 units box sum no run 200000 84

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Appendix C (Con tin ued) All lines beginning with `#' are commen ts and ignored b y LAMMPS. F or the parameters le, there are no blank-line requiremen ts; they are just ignored. Ev ery line begins with a command on the left follo w ed b y the parameters used for that command. F or a more detailed description of the commands, see the LAMMPS User's Man ual [3]. The tabbed spaces bet w een the commands and the command parameters are strictly for ease of visualization. log log0.1 : The log command is where y ou specify the name of the le to whic h the standard output, also called the thermodynamic output, will be written. The syn tax is command then le-name. The data gathered from the friction experimen ts are tak en from these log les, so eac h sim ulation needs to ha v e a unique log le. dimension 3 : The dimension command allo ws y ou to set the dimensionalit y of the sim ulation: it m ust matc h the dimensionalit y implied in the data le. boundary p p p : LAMMPS can handle man y dieren t forms of boundary conditions. In our researc h w e use periodic boundary conditions in all 3 dimensions. P eriodic boundary conditions are also required when sk ewing a sim ulation bo x. The syn tax is command, xdir boundary condition, y-dir boundary condition, and z-dir boundary condition where `p' stands for periodic. units metal : This species the units used in all les aliated with the sim ulation. The sim ulations run for this researc h used what LAMMPS calls metal units. This st yle of units species all quan tities in A, ps, eV, and Kelvin. The forces are giv en in b y LAMMPS eV/ A but are later con v erted b y the author to nN. atom_style atomic : This species the format in whic h the atoms are en tered in the data le. Atomic is the most basic st yle whic h stores atom IDs, t ypes, coordinates, and initial v elocities if needed. F or a more detailed description of this le Appendix C. 85

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Appendix C (Con tin ued) read_data data.4 : This is when y ou tell LAMMPS to read y our data le and what the name of the data le is. LAMMPS goes through data the le line-b y-line and stores the information for later use. The syn tax is the command follo w ed b y the le-name. pair_style table spline 5000 : This command tells LAMMPS what kind of pair poten tials are being used. The syn tax is command, format, st yle, and ho w man y look-up v alues to use. Because our poten tials are tabulated it is best to use a splining routine to calculate a v alue bet w een t w o poin ts, but LAMMPS has a bug in the in terpolation. This is discussed in Appendix A.2 and is the reason for using 5000 poin ts. Keith McLaughlin is responsible for this w ork around. pair_coeff 1 1 all_pots.long AlAl : F or eac h pair of atom t ypes y ou m ust specify where to nd the poten tial table. The syn tax is command, t ype of one atom, t ype of the other atom, le con taining the tabled poten tial, and the name of the table. The example sho wn sa ys that the in teraction bet w een t w o t ype 1 atoms is the table titled AlAl in le all pots.long The order of the atom t ypes does not matter here; for example pair coeff 1 2 is equiv alen t to pair coeff 2 1 group qcfix id <> 1 6250 : Group commands are v ery useful, they allo w y ou to con trol a large group of atoms using the group name rather than t yping out eac h atom individually Because of the manner of the data les used in this w ork I ha v e specied most groups b y their atom IDs. The syn tax is command, group name, ho w y ou are iden tifying the atoms to be put in to the group, and the atoms y ou c hoose. The ` < and ` > are inclusiv e, so in the example abo v e atom 1, atom 2, atom 3, ... atom 6249, and atom 6250 are all in the group called qcfix See the LAMMPS User's Man ual [3] for more w a ys to specify atoms b y ID. group temps union qctemp adtemp : Sometimes it is con v enien t to specify a group as a union bet w een t w o existing groups, as sho wn here. The syn tax is command, group name, ho w y ou are iden tifying the atoms (in this case, a union bet w een t w o groups), follo w ed b y 86

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Appendix C (Con tin ued) the names of the groups. There are n umerous w a ys for specifying groups and they can be found in the LAMMPS User's Man ual [3]. fix 1 all nve : A x is just that, it xes something in the sim ulation. Here w e ha v e xed the sim ulation to run at constan t NVE (n um ber of atoms, v olume, and energy). Eac h x m ust also ha v e a unique name or n um ber associated with it. The syn tax is command, name, what atoms, st yle of x. The st yle of the x will determine if there are an y extra argumen ts that need to be set. In this example there are no extra argumen ts. The group that this x modies is all This is a predened group that includes all atoms in the sim ulation. fix 2 all langevin 0 0 0.1 239482 : This sets the Langevin thermostat [3] to bring all atoms in the sim ulation from a starting temperature of 0 K to a nal temperature of 0 K using a 0.1 ps damping parameter and beginning with a random seed of 239482. The syn tax is command, name, what atoms, st yle of x, starting temperature, ending temperature, damping parameter, and a random seed. Because the starting and ending temperatures are specied here, this command can be used to heat up or cool do wn a sim ulation, rather than trying to k eep it at a constan t temperature, as is done here. timestep 0.0005 : Since w e are using metallic units, time is specied in picoseconds. This command tells LAMMPS to use a 0.0005 ps timestep. This w as the timestep used in the preliminary results; ho w ev er, it w as found later that a m uc h larger timestep (0.004 ps) w as sucien t. Using a smaller timestep will increase the run time of the sim ulation, so one should use the largest timestep possible. The method for nding an appropriate timestep is discussed in Appendix B. thermo 100 : This command tells LAMMPS to dump the thermodynamic output ev ery 100 timesteps. It is written to standard out and recorded in the log le. Un til the thermo st yle command is used it will output the default data: timestep, temperature, a v erage energy per pair, a v erage energy per mol, the total energy and the pressure. 87

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Appendix C (Con tin ued) dump 1 all custom 1000 out0.1 tag type x y z : The dump command is curren tly being used to generate mo vies. All of the frictional-force data are tak en from the thermodynamic output in the log les. There is about a 20% decrease in computing time when the dump command is not used, but it can sho w some useful data. The syn tax is command, name, what atoms, custom st yle, dump once after ev ery 1,000 timesteps, dump to what le, and then the list of information that one w an ts in the dump le. In the example sho wn, ev ery 1000 timesteps the atom ID, t ype, x, y and z coordinates for ev ery atom are prin ted. The dump les made for the most recen t sim ulations, whic h run m uc h longer than the preliminary sim ulations, can quic kly exceed 1 gigab yte in size. Due to memory limitations only a few compressions w ere visualized. The LAMMPS visualization program xmo vie or VMD can visualize these dump les. No w the initial preparation is o v er and the sim ulation can start running. The procedure begins with an initial relaxation period in whic h the system is brough t do wn to 0 K temperature using the Langevin thermostat on all atoms. Up un til this poin t, LAMMPS is just storing information. The run command tells LAMMPS ho w man y timesteps to in tegrate using the conditions specied so far. run 50000 : Run the sim ulation at the specied conditions for 50000 timesteps. The sim ulation has no w been run for 50000 timesteps for the sole purpose of relaxation. No w w e ha v e to begin the compression stage. T o ac hiev e a compression, w e need to x the top-most la y ers of the tip and the bottom-most la y ers of the quasicrystal so that they are rigid. This is ac hiev ed b y articially requiring the forces on these atoms to be 0. fix 3 adfix setforce 0.0 0.0 0.0 : This sets the forces on the adfix group of atoms to 0.0 eV/ A in the x, y and z directions. The adx atoms are the top-most la y ers of the tip, and at the end of ev ery timestep the forces on them will be replaced with 0.0 eV/ A so that they are rigid. 88

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Appendix C (Con tin ued) fix 4 qcfix setforce 0.0 0.0 0.0 : This sets the forces on the qcx la y ers of atoms to 0.0 eV/ A in all directions as w ell. These are the bottom-most la y ers of the appro ximan t, and they are no w rigid. Just because the forces are set to zero, if the no w xed atoms had a v elocit y they will k eep it, because there are no forces acting on them to alter that v elocit y This means that for the xed quasicrystal atoms, whic h w e do not w an t to mo v e in space, their v elocities m ust be set to 0 eV/ A. The xed tip atoms are the atoms that w e will giv e a v elocit y to w ard the quasicrystal, in the negativ e y direction, to create our compression. velocity qcfix set 0.0 0.0 0.0 units box sum no : This sets the v elocit y of the qcx atoms to 0.0 A/ps in the x, y and z directions. If the `optional' argumen ts of units box sum no are not included, LAMMPS will fail, sa ying that w e ha v en't used the lattice command, so it cannot set the v elocities. The argumen t units box means that the v elocit y w e are specifying will be in A/ps; if w e w ere to c hoose units lattice a v alue of 0.5 w ould mean that the atoms w ould ha v e a v elocit y of 0.5 unit cells per picosecond. The command sum no means that the v elocit y will not be added to the v elocit y from the previous timestep; it will replace it. This k eeps the v elocit y constan t. unfix 2 : Since w e no longer w an t the thermostat acting on the xed atoms or the free in terface atoms, w e ha v e to tak e a w a y our previous x that put the thermostat on all atoms. The syn tax is unfix then the name of the x y ou w an t to get rid of. No w that w e ha v e set our rigid atoms and gotten rid of the original thermostat, w e ha v e to implemen t the thermostat on the thermostat atoms only These are the atoms in the cen ter-most la y ers of both the tip and the appro ximan t. Implemen ting the thermostat tak es a little bit of though t, because w e do not w an t the thermostat in terfering with the force data. W e are compressing in the y direction and sliding in either the x or z directions. This means that for sliding in the x direction, w e only w an t to allo w the thermostat to modify the v elocities of the thermostat atoms in the z direction. Subsequen tly when 89

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Appendix C (Con tin ued) sliding in the z direction w e will only allo w the thermostat to modify the v elocities of the thermostat atoms in the x direction. Originally the thermostat w as not allo w ed to aect the compression direction, but the resulting data w ere v ery noisy This is easily seen in the preliminary results in x 4. Allo wing the thermostat to aect the compression direction negated a lot of the noise. This is the same procedure used b y Dr. Sinnott's group at the Univ ersit y of Florida and is being used in the most curren t results as seen in x 5 fix 5 temps langevin 0 0 0.01 239482 axes 0 0 1 : This command xes the temps group to ha v e a Langevin thermostat just as before. The axes argumen t allo ws one to specify whic h axes will be utilized b y the thermostat. A 1 means that axis will be utilized and a 0 means that it will be ignored. 1 Up un til this poin t w e ha v e not prin ted out an y of the force v alues used for the data analysis. As the tip is sliding across the quasicrystal, the normal and frictional forces need to be measured. Due to LAMMPS limitations one cannot, for example, ask LAMMPS to dump the forces on the free tip atoms that are due only to the quasicrystal atoms. Because of this w e ha v e to retriev e the forces that w ould ha v e acted on the xed tip atoms if the x had not been there. During sliding at a constan t v elocit y w e record the forces that oppose sliding as the frictional forces. Because the forces w ere set to 0.0 eV/ A, the thermodynamic output information can be modied to prin t the forces that w ould ha v e aected those atoms had the force not been set to 0.0 eV/ A. This is described further in Appendix A.1. thermo_style custom step temp f_3[1] f_3[2] f_3[3] : The thermo st yle command c hanges the st yle of the thermodynamic output that is prin ted to standard out. By using the custom command, w e specify eac h column of the output individually Here w e prin t the timestep, temperature, and then the v alues from x 3 before the x is applied. 1 The axes option in the x langevin command is no longer a v ailable in the 21 Ma y 2008 LAMMPS distribution. The 22 Jan uary 2008 v ersion w as used for this researc h. The option of including or deleting an axis for consideration is no w performed through a compute command. F or a more detailed description of this command see the curren t LAMMPS User's Man ual [3] 90

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Appendix C (Con tin ued) Fix 3 is where w e set the forces on the tip atoms to be zero in all 3 dimensions, so the thermodynamic output prin ts the forces that w ould ha v e acted on those atoms in all 3 dimensions before the x is applied. This is the data that w e use for the analysis. No w that w e are getting the output that w e w an t, w e begin the compression stage b y giving the xed tip atoms a v elocit y in the negativ e y direction, to w ard the quasicrystal. This is the rst time that w e set the v elocit y of the rigid tip atoms. T o c hange the normal force, only the compression v elocit y needs to be c hanged. The run-time could be impro v ed b y increasing the v elocit y and decreasing the n um ber of timesteps, but this leads to a need for a longer relaxation time. The induced v elocit y of the xed tip atoms creates phonons in the tip as the compression is occurring. Before sliding begins, w e w an t the vibrations in the compression direction to be as small as possible. By compressing slo wly w e minimize the eect of the compression phonons so that a shorter relaxation time, bet w een compression and sliding, is required. The syn tax of the commands belo w is the same as the previous v elocit y and run commands. velocity adfix set 0.0 -0.75 0.0 units box sum no run 77000 After compression, the tip is giv en a 0.0 A/ps v elocit y in all directions to halt the compression and the system is allo w ed to relax again. velocity adfix set 0.0 0.0 0.0 units box sum no run 70000 A t this poin t w e ha v e our desired compression, and the sliding can begin. This section of the log le is where all of the data analysis is done. velocity adfix set -0.05 0.0 0.0 units box sum no run 200000 91

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Appendix C (Con tin ued) C.4 CIR CE Submission Script All of the preliminary results w ere run on the CIR CE cluster, main tained b y Researc h Computing, Univ ersit y of South Florida. Eac h sim ulation needs a submission script that requests the n um ber of processors, an estimated run time, and the actual command to run the sim ulation. A sample CIR CE submission script is sho wn belo w. #!/bin/sh # # # start in the current directory #$ -cwd # do not merge stderr into stdout #$ -j n #$ -M harper@physics.cas.usf.edu #$ -notify # name of the job #$ -N comp0.x.short #$ -m abe # use the Bourne shell #$ -S /bin/sh # parallel environment and number of processors #$ -pe ompi* 8 #$ -l h_rt=62:00:00 PATH=.:$PATH sge_mpirun lmp_crockett < input_parameter_file The template for the submission script w as giv en b y Dr. Da vid Rabson. The v ariable parts of the script are as follo ws. -M harper@physics.cas.usf.ed u : This tells CIR CE where to send emails when a job is started, nished, and aborted. -N comp0.x.short : The string follo wing -N is the name giv en to the sim ulation, used when y ou c hec k the status of the sim ulation and when CIR CE sends email updates on the job. 92

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Appendix C (Con tin ued) -pe ompi* 8 : This is the processor request. All of the sim ulations for the preliminary results w ere run on 8 processors per sim ulation; subsequen t w ork w as mostly run using 16 processors. -l h_rt=62:00:00 : As of April 2008, this is a new addition to the CIR CE submission scripts. It is an estimate of the w all-cloc k time that the job will tak e. If this statemen t is not in the script, CIR CE will kill the job after 10 min utes. The job will be aborted if it is not nished in the allotted time. In more curren t w ork, w e ha v e decreased the run time limit signican tly sge_mpirun lmp_crockett < input_parameter_file : This is the command that starts the sim ulation. When LAMMPS w as installed on CIR CE, w e installed the crockett MPI v ersion. The lename of the input parameters le described abo v e should replace input_parameter_file 93

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Appendix D Original 75-A tom Unit Cell Presen ted in T able D.1 are the atomic coordinates for the original 75-atom unit cell supplied b y Marek Mihalk o vi c [2]. The unit cell consists of t w o surface bila y ers sandwic hing a bulk bila y er. The preliminary results w ere performed with the surface bila y ers on either side of a stac k of 15 bulk bila y ers. Unfortunately the surface bila y ers are parallel to the `10-fold' surface and not the `2-fold' surface of in terest. In the nal results the surface bila y ers w ere remo v ed from the edges, and only the bulk bila y er w as repeated. T able D.1: These are the coordinates for one unit cell of the quasicrystalline appro ximan t originally supplied b y Marek Mihalk o vi c [2]. Elemen t X ( A) Y ( A) Z ( A) Al 9.79744 7.15549 0.615919 Ni 9.96585 0.867474 0.0804434 Ni 8.33196 1.11839 2.18229 Ni 8.57243 5.15598 0.0804433 Ni 7.39807 3.99261 2.18229 Ni 2.17267 0.705943 0.0878319 Ni 3.90064 1.26739 2.18392 Ni 6.17814 2.0074 2.75835 Ni 3.85917 5.22602 0.117395 Ni 8.35191 2.7137 0.0938453 Co -0.0672782 3.95024 2.28389 Al 1.91473 4.59424 1.25497 Al 1.60871 1.97881 2.23178 Al 0.129845 6.53028 2.23178 c ontinue d on next p age 94

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Appendix D (Con tin ued) A tomic coordinates, con tin ued Al 7.59446 6.43969 2.19768 Al 9.71004 4.65116 2.11771 Al 10.5895 1.94455 2.11771 Al -0.716724 5.148 0.0916514 Al 0.111335 2.5995 0.0916514 Al 3.56991 2.62732 0.116111 Al 2.09767 7.1584 0.116111 Al 3.76315 6.61683 2.15484 Al 4.59898 4.0444 2.15484 Al 6.26141 4.48668 0.0643969 Al 7.70279 0.0505632 0.064397 Al 9.82553 7.16461 0.780714 Ni 10.0143 0.843104 0.0925141 Ni 8.2999 1.13609 2.1651 Ni 8.59727 5.20415 0.0925141 Ni 7.38254 3.95944 2.1651 Ni 2.20804 0.717434 0.0807941 Ni 3.89953 1.26703 2.16063 Al 6.15416 1.99961 1.55191 Co 3.83923 5.21954 0.0845941 Al 8.43596 2.74101 0.0860341 Co -0.038051 3.95974 2.06831 Al 1.55681 4.47794 0.107924 c ontinue d on next p age 95

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Appendix D (Con tin ued) A tomic coordinates, con tin ued Al 1.63299 2.02915 2.08816 Al 0.179072 6.50384 2.08816 Al 7.527 6.41778 2.17358 Al 9.69997 4.6415 2.20145 Al 10.5756 1.94645 2.20145 Al -1.01477 5.08674 0.0546241 Al -0.165799 2.47387 0.0546241 Al 3.67271 2.63577 0.0740341 Al 4.52055 0.0263807 0.0740341 Al 3.72628 6.59979 2.17641 Al 4.55913 4.03651 2.17641 Al 6.2728 4.51163 0.0864041 Al 7.72668 0.0370727 0.0864041 Al 9.79743 7.15548 -0.368935 Ni 9.96074 0.838618 0.140225 Ni 8.37438 1.12631 2.22357 Ni 8.55133 5.17632 0.140225 Ni 7.43705 4.01113 2.22357 Ni 2.18531 0.710049 0.126995 Ni 3.90655 1.26931 2.24497 Al 6.18826 2.01069 1.75277 Co 3.84015 5.21984 0.0992046 Al 8.38696 2.72509 0.135765 c ontinue d on next p age 96

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Appendix D (Con tin ued) A tomic coordinates, con tin ued Co -0.107542 3.93716 2.21337 Al 1.87111 4.58006 1.15588 Al 1.63891 1.9924 2.22877 Al 0.162265 6.53705 2.22877 Al 7.64131 6.45492 2.23377 Al 9.75372 4.70317 2.21657 Al 10.6554 1.92814 2.21657 Al -0.725446 5.15357 0.121585 Al 0.10755 2.58987 0.121585 Al 3.60202 2.62668 0.128415 Al 4.45802 -0.00782064 0.128415 Al 3.64571 6.57196 2.25077 Al 4.47759 4.01167 2.25077 Al 6.24132 4.48008 0.171385 Al 7.68266 0.04409 0.171385 V ector X ( A) Y ( A) Z ( A) a -2.33474440 7.18560440 0.00000000 b 12.22488050 0.00000000 0.00000000 c 0.00000000 0.00000000 24.48 T able D.2. These are the unit cell v ectors for the original quasicrystalline appro ximan t used in the preliminary results. The structure of the appro ximan t w as supplied b y Marek Mihalk o vi c [2] 97

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Appendix E Previous Publication Before the researc h presen ted here w as begun, the author con tributed to w ork resulting in a paper titled, \Multiphase region of helimagnetic superlattices at lo w temperature in an extended six-state cloc k model" [31]. The w ork in said paper w as largely based on Douglas Lo v elady's Master's Thesis; ho w ev er, the author of this w ork made con tributions signican t enough to be recorded as second author. The details of the w ork will not be elaborated upon here; only a short sk eleton outline of the specic con tributions made will be men tioned. The original w ork focused on four or more magnetic la y ers sandwic hed bet w een nonmagnetic spacers where the spins in neigh boring planes w ere rotated b y 60 This author's task w as to v erify all calculations previously performed and to in v estigate one, t w o, and three magnetic la y ers bet w een non-magnetic spacers. It w as found that one and t w o magnetic la y ers w ere trivial. The in v estigation of three magnetic la y ers sho w ed it to be a special case that required the calculation of new matrices and equations. It also added three new possible excitation congurations to the lo w-temperature expansion. 98


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Harper, Heather McRae.
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A molecular-dynamics study of the frictional anisotropy on the 2-fold surface of a d-AlNiCo quasicrystalline approximant
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by Heather McRae Harper.
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[Tampa, Fla] :
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2008.
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Thesis (M.S.)--University of South Florida, 2008.
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ABSTRACT: In 2005, Park et al. demonstrated that the 2-fold surface of a d-AlNiCo quasicrystal exhibits an 8-fold frictional anisotropy, as measured by atomic-force microscopy, between the periodic and aperiodic directions [37, 38]. It has been well known that quasicrystals exhibit lower friction than their crystalline counterparts [35, 17, 47, 28, 12, 50]; however, the discovery of the frictional anisotropy allows for a unique opportunity to study the effect of periodicity on friction when chemical composition, oxidation, and wear are no longer variables. The work presented herein is focused on obtaining an understanding of the mechanisms of friction and the dependence of friction on the periodicity of a structure at the atomic level, focusing on the d-AlNiCo quasicrystal studied by Park et al. Using the LAMMPS [41] package to simulate the compression and sliding of an 'adamant' tip, see section 3.3, on a d-AlNiCo quasicrystalline approximant substrate, we have demonstrated, in preliminary results, an 8-fold frictional anisotropy, but in more careful studies the anisotropy is found to be much smaller. The simulations were accomplished using Widom-Moriarty pair potentials to define the interactions between the atoms [33, 52, 51, 9]. The studies presented in this work have shown a clear velocity dependence on the measured frictional response of the quasicrystalline approximant's surface. The final results show between a 1.026-fold and 1.127-fold anisotropy between sliding in the periodic and 'aperiodic' directions, depending on the sliding velocity.
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Atomistic simulation
Quasicrystals
Nano-tribology
Contact mechanics
Aperiodicity
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