A Molecular-Dynamics Study of the Frictional Anisotropy on the 2-Fold

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9-16-2008 A Molecular-Dynamics Study of the Frictional Anisotropy on the 2-fold Surface of a d-AlNiCo Quasicrystalline Approximant Heather McRae Harper University of South Florida

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A MolecularDynamics Study of the Frictional Anisotropy on the fold Surface of a

dAlNiCo Quasicrystalline Approximant

Heather McRae Harp er

A thesis submitted in partial fulllment

of the requirements for the degree of

Master of Science

Department of Physics

College of Arts and Sciences

University of South Florida

Ma jor Professor David Rabson PhD

William Matthews Jr PhD

Brian Space PhD

Matthias Batzill PhD

Date of Approval

Septemb er

Keywords atomistic simulation quasicrystals nanotrib ology contact mechanics

ap erio dicity

DEDICATION

I would like to dedicate this work to my mom for doing everything she can to encourage

me help me and make my life a little easier to Alan for reminding me that in the end

its the money that matters and to Doug for putting up with me every day and helping

me stay p ositive Without you guys I would not have b een able to nish this Thank you

ACKNOWLEDGMENTS

I would rst like to thank all of my committee memb ers for their time and patience

This research was the combination of suggestions insights and help from many p eople

Most imp ortantly I would like to thank my advisor Dr David Rabson for his continued

supp ort

Thank you to Dr Susan Sinnott and her research group at the University of Florida

for all of their help and insight on measuring friction through molecular dynamics and

allowing me the invaluable exp erience of sp ending a week in their lab Dr Sagar Pandit

for freely oering his exp ertise in molecular dynamics and Dr Patricia Thiel at Ames

National Lab oratory through which a p ortion of this research was funded

The author would also like to acknowledge the use of the services provided by Research

Computing University of South Florida

TABLE OF CONTENTS

LIST OF TABLES iii

LIST OF FIGURES iv

LIST OF ABBREVIATIONS vi

ABSTRACT vii

CHAPTER INTRODUCTION

CHAPTER A DISCUSSION OF CONTINUUM CONTACT MECHANICS

The Hertz Theory

The JohnsonKendallRob erts Theory

The DerjaguinMullerTop orov Theory

The Tab or Parameter and Other Metho ds of Interp olating Between The

ories

CHAPTER AN EXAMINATION OF SOME TECHNIQUES USED TO PER

FORM MOLECULARDYNAMICS SIMULATIONS AND SOME

POPULAR PACKAGES

Limitations on the Simulation of Quasicrystals Using Quasicrystalline

Approximants

WidomMoriarty Pair Potentials

Using an Adamant Tip

Averaging and Error Analysis

MolecularDynamics Packages

DL POLY

NAMD

Gromacs

LAMMPS

CHAPTER PRELIMINARY RESULTS

CHAPTER FINAL RESULTS

CHAPTER FUTURE WORK

Comparison of Dierent Approximants

Larger Simulations i

Tailoring the Pair Potentials

Creating and Using a More Realistic Tip

Monitoring Phonon Propagation

REFERENCES

APPENDICES

App endix A LAMMPS Workarounds

A Obtaining Forces on FixedRigid Atoms

A Bugs Noted With the LAMMPS Splining Routine

A Using a Triclinic Box in a Data File

App endix B Calculating an Appropriate Timestep

App endix C Required Files

C Data File

C Potential File

C Simulation Parameters File

C CIRCE Submission Script

App endix D Original Atom Unit Cell

App endix E Previous Publication ii

LIST OF TABLES

Table Atomic Co ordinates for the Atom Approximant Unit Cell

Table UnitCell Vectors for the Atom Approximant Unit Cell

Table Sliding Velo cities and Required Sliding Times

Table Table of Co ecients of Friction at Highest Compressions for the Ap e

rio dic Sliding Direction

Table Table of Co ecients of Friction at Highest Compressions for the Peri

o dic Sliding Direction

Table Table Showing the Ratio Between the Perio dic and Ap erio dic Frictional

Co ecients

Table C SimulationBox Vectors

Table D Atomic Co ordinates for the Full Atom Approximant Unit Cell

Table D UnitCell vectors for the Atom Approximant Unit Cell iii

LIST OF FIGURES

Figure Sphere on Disk Hertzian Contact Prole

Figure Sphere on Sphere Hertzian Contact Prole

Figure JKR Contact Prole

Figure fold Surface

Figure Bulk Bilayer Structure

Figure WidomMoriarty Pair Potentials

Figure Truncated WidomMoriarty Pair Potentials

Figure Mean Squared Displacement

Figure AdAd and AlAl Pairwise Interactions

Figure AdX Pairwise Interactions

Figure Delineation of the Simulation Groups

Figure Ap erio dic Lateral Force Over Time Preliminary Results

Figure Perio dic Lateral Force Over Time Preliminary Results

Figure Ap erio dic Normal Force Over Time Preliminary Results

Figure Perio dic Normal Force Over Time Preliminary Results

Figure Normal vs Lateral Force Preliminary Results

Figure Extended Normal vs Lateral Force Preliminary Results

Figure Visualization of Current Simulation Box fold Face

Figure Visualization of Current Simulation Box

Figure Ap erio dic Temp erature Over Time Final Results iv

Figure Perio dic Temp erature Over Time Final Results

Figure Ap erio dic Lateral Force Over Time Final Results

Figure Perio dic Lateral Force Over Time Final Results

Figure Ap erio dic Normal Force Over Time Final Results

Figure Perio dic Normal Force Over Time Final Results

Figure Ap erio dic Normal vs Lateral Force Final Results

Figure Perio dic Normal vs Lateral Force Final Results

Figure Perio dic and Ap erio dic Normal vs Lateral Force Final Results

Figure Sliding Velo city vs Co ecient of Friction Final Results

Figure Atom Approximant Unit Cell

Figure Ap erio dic Lateral vs Normal Force

Figure Ap erio dic Temp erature vs Lateral Force

Figure Perio dic Temp erature vs Lateral Force

Figure Ap erio dic Temp erature vs Normal Force v

LIST OF ABBREVIATIONS

QC Quasicrystal

LAMMPS Largescale AtomicMolecular Massively Parallel Simulator

VMD Visual Molecular Dynamics

NAMD Nanoscale Molecular Dynamics

AFM Atomic Force Microscop e

UHV UltraHigh Vacuum

JKR JohnsonKendallRob erts

DMT DerjaguinMullerTop orov

STM ScanningTunneling Microscop e vi

A MOLECULARDYNAMICS STUDY OF THE FRICTIONAL

ANISOTROPY ON THE FOLD SURFACE OF A DALNICO

QUASICRYSTALLINE APPROXIMANT

Heather McRae Harp er

ABSTRACT

In Park et al demonstrated that the fold surface of a dAlNiCo quasicrystal

exhibits an fold frictional anisotropy as measured by atomicforce microscopy b etween

the p erio dic and ap erio dic directions It has b een well known that quasicrystals

exhibit lower friction than their crystalline counterparts however

the discovery of the frictional anisotropy allows for a unique opp ortunity to study the eect

of p erio dicity on friction when chemical comp osition oxidation and wear are no longer

variables

The work presented herein is fo cused on obtaining an understanding of the mechanisms

of friction and the dep endence of friction on the p erio dicity of a structure at the atomic

level fo cusing on the dAlNiCo quasicrystal studied by Park et al Using the LAMMPS

package to simulate the compression and sliding of an adamant tip see x on

a dAlNiCo quasicrystalline approximant substrate we have demonstrated in preliminary

results an fold frictional anisotropy but in more careful studies the anisotropy is found to

b e much smaller The simulations were accomplished using WidomMoriarty pair p otentials

to dene the interactions b etween the atoms

The studies presented in this work have shown a clear velo city dep endence on the

measured frictional resp onse of the quasicrystalline approximants surface The nal results

show b etween a fold and fold anisotropy b etween sliding in the p erio dic and

ap erio dic directions dep ending on the sliding velo city vii

CHAPTER

INTRODUCTION

Quasicrystals were discovered in by Shechtman et al Since that rst icosahe

dral metastable AlMn quasicrystal was discovered there have b een numerous quasicrys

tals with various forbidden p oint group symmetries discovered and studied The

ma jority of known quasicrystals contain icosahedral symmetry This class of quasicrystals

are ap erio dic in all three dimensions Contrary to this the decagonal AlNiCo quasicrystal

on which this work fo cuses contains two ap erio dic directions and a third p erio dic direction

The indepth analysis in by Park et al showed that the fold face of decagonal

Al Ni Co contained b oth p erio dic and ap erio dic surface order on the atomic scale as

72 11 17

predicted by the bulk structure This showed that dAlNiCo quasicrystals p ossess the

unique prop erty of exp osing a surface containing a p erio dic arrangement of atoms along

the fold axis with a A p erio dicity and p erp endicular to it an ap erio dic arrangement

of atoms following a Fib onacci sequence This quickly prompted further study into the

eect of p erio dicity versus ap erio dicity on the co ecient of friction

It had b een suggested that the low co ecient of friction measured in previous studies

could b e explained through the eects of wear In friction exp eriments where plas

tic deformation takes place the measured frictional co ecient is dep endent on a highly

complex set of factors including but not limited to slip planes and the propagation of

defects the sloughing o of an oxide layer to create a lubricating eect the breaking of

chemical b onds etc To combat these issues the Park exp eriments were p erformed

using such low loads as to eliminate plastic deformation This was veried by scanning

tunneling microscop e STM images b oth b efore and after the friction exp eriments

The original Park exp eriments were p erformed in ultrahigh vacuum UHV by

sliding a hexadecanethiol passivated AFM tip across the fold surface in b oth the p erio dic

and ap erio dic directions Later exp eriments used an alkanethiol passivated AFM tip in

UHV conditions By measuring the torsional resp onse of the cantilever Park et al found

an fold frictional anisotropy The data were also a go o d t to the DMT mo del of contact

mechanics see x

Though the frictional anisotropy itself has b een well do cumented by Park et al there

still lacks a broader understanding of the role that atomic p erio dicity plays in the friction

co ecient of a material The aim of this work is to employ moleculardynamics techniques

and recreate the frictional anisotropy seen in the Park exp eriment In doing so we aim to

answer some very fundamental questions ab out the eect of p erio dicity on friction at the

atomic scale

The analysis b egins with an investigation into the theories b ehind contact mechanics

Beginning with Hertzs theory from which idealized mechanical contacts to

elastic homogeneous isotropic p erfectly smo oth b o dies in the absence of adhesive forces

we see that even the simplest mo del of contact mechanics is quite complicated Following

the Hertz Theory were the JKR JohnsonKendallRob erts and DMT DerjaguinMuller

Top orov theories which built on the original Hertz theory but most notably

included adhesive forces Due to the limitless variety of contacting surfaces there isnt a

one size ts all theory but rather they each work well under dierent situations As a

rule of thumb the JKR Theory works well in describing the contact b etween soft materials

with high adhesion while the DMT Theory ts well with hard materials p ossessing a low

adhesion Luckily in Tab or published the idea that surface roughness played a

role in adhesion along with an analysis of the JKR and DMT theories This led to the

Tab or parameter a criterion for determining if two contacting b o dies would fall in the

JKR regime the DMT regime or somewhere inb etween Following Tab or Maugis

presented a new mo del to interp olate b etween the JKR and DMT regimes and describ e

materials that fell in the intermediate range These theories and their corresp onding mo dels

only scratch the surface of the neverending debate over contact mechanics though they

will b e the only ones presented in x

Following the discussion of contact mechanics is an overview of the basic principles of

applying molecular dynamics to a system as complicated as a quasicrystal The ap erio dicity

of quasicrystals p oses a unique problem when trying to mo del them One way to think

of a quasicrystal is as a crystal with a unit cell of innite length When mo deling such a

system a p erio dic quasicrystalline approximant must b e used using p erio dic b oundary

conditions to mimic an innite sample Although approximants are p erio dic they still

retain some of the lo cal symmetry and b ehavior of their quasicrystalline counterparts and

are used for b oth mo deling and exp erimental research see x

The approximants were mo deled using WidomMoriarty pair p otentials

see x using the molecular dynamics simulation package LAMMPS see x The

LAMMPS package from the Sandia National Lab oratory was determined to b e b est suited

POLY and NAMD to this work although other packages such as Gromacs DL

were installed tested and evaluated see x To exactly mimic the

exp erimental setup an alkanethiol passivated TiN AFM tip would b e required To simplify

the pro cess an adamant tip was created out of an FCC arrangement of atoms that has

purely repulsive interactions with the approximant to mimic the eect of the alkanethiol

passivation see x

Chapter contains the preliminary results obtained for measuring the friction in the

p erio dic and ap erio dic directions of the dAlNiCo quasicrystalline approximant Simulating

normal forces ranging from approximately nN an fold anisotropy was found in the

measured co ecient of friction Though the anisotropy seems to repro duce that seen

in exp eriment the overall magnitude of the frictional forces is quite low Up on further

investigation numerous improvements and changes were made to the simulation pro cedure

This chapter includes a discussion of the known problems in the preliminary results and

the advances that have taken place since they were rst obtained

After employing the mo dications mentioned in x the nal results of this work were

obtained and discussed in x Though we do not see the fold frictional anisotropy that

was present in the preliminary results the latest data are much more reliable and provide

an excellent jumpingo p oint for a more sophisticated analysis of the dAlNiCo system

Lastly it is imp ortant to note that using the simplest mo del p ossible to recreate the

anisotropy would give us great insight into the basic mechanisms of friction in this system

A more sophisticated analysis in the future could b e p erformed by comparing dierent

approximants running larger simulations tailoring the pair p otentials for surface eects

monitoring phonon propagation and using a more realistic tip as discussed in x

For clarity an app endix discussing the format and commands of the required les to

p erform the simulations was included This allows for a more detailed description of the

LAMMPS commands and the pro cedures undertaken to p erform the friction exp eriments

see App endix C

CHAPTER

A DISCUSSION OF CONTINUUM CONTACT MECHANICS

The Hertz Theory

Heinrich Hertz b egan his study of contact mechanics in graduate scho ol by studying

the optical interference patterns created by two glass lenses coming into contact The

nowfamous Hertz theory gave a metho d of calculating the contact area b etween two b o dies

under a rather large set of assumptions

Hertz assumed that the contacting materials were elastic homogeneous and isotropic

and that the contacting surfaces are smo oth and their shap e do es not change over time

after the initial deformation For the Hertz theory to remain accurate there must not b e

adhesion b etween the contacting b o dies and the radius of contact must b e much smaller

than their individual radii

Keeping with the assumption that the contacting b o dies do not deform outside of the

area of contact as shown in Fig the Hertz Theory predicts a contact radius a of a

sphere on a rigid plane under a normal load P as presented by Grierson et al to b e

where R is the radius of the contacting sphere and K a measure of the elastic constants

of the materials is given by

2 2

2 1

E E

1 2

where E and E are the Youngs mo dulii and and are the Poisson ratios of materials

1 2 1 2

and resp ectively It is easy to see that interactions b etween the two materials are not

taken into account only the individual prop erties of the materials are considered

Figure This illustration shows the prole of a Hertzian contact b etween a sphere of

radius R and and a at plane D represents the diameter of the circular contact area

Most notably outside of the contact area the surfaces are not deformed

One can also lo ok at the radius of the Hertzian contact area a b etween two spheres

of radii R and R under a normal load of P as presented by Johnson et al and

1 2 0

visualized in Fig

R R

1 2

K K P a

1 2 0

R R

1 2

The elastic constants K and K are given as

1 2

where once again and E corresp ond to the Poisson ratios and Youngs mo dulii

12 12

resp ectively As the two spheres come into contact the region around the area of contact

is compressed and distant p oints in the two spheres will approach each other by a distance

also called the elastic displacement

R R

1 2

2 2 2 3

P K K

1 2

R R

1 2

Johnson et als notation can b e simplied into that of Grierson giving the radius of

the contact area a as

R R P

1 2 0

R R K

1 2

and using the same denition for K as in the sphere on disk equations ab ove We can also

lo ok at the pressure distribution over the area of contact given as p

r P

a a

where r is the distance from the center of the circular contact area

In the s Derjaguin calculated that two rigid b o dies separated by a distance d would

exp erience an attractive force In the mid s scientists b egan to measure contact

areas that disagreed with the original Hertz mo del It thus b ecame quite clear that

a more detailed description of contact mechanics would b e required One of the ma jor

changes that b egan in the eld was the addition of an adhesion term to the original Hertz

equations

The JohnsonKendallRob erts Theory

In Johnson published a short theoretical work investigating the adhesion

b etween two elastic spheres and concluded that even with an adhesive force two elastic

spheres in contact cannot have a contact radius greater than that of the Hertz theory and

that adhesion is physically imp ossible due to innite stresses along the edge of the contact

area

In the s Dutrowski see published results contradicting the Hertz theory and

Johnsons pap er by showing a larger contact area than the Hertz mo del predicted

and very notably the contact area was nite under zero applied load Because the contact

area was nite under zero applied load a force was required to separate the two b o dies

showing a measurable adhesion These new results by Dutrowski were in agreement with

the JKR Theory which was to b e published later in

Johnson Kendall and Rob erts acknowledged that under low loads two elastic

b o dies in contact have an equilibrium contact area that is due in large part to surface

forces They also stated that mechanical work is required to overcome the force of adhesion

in order to separate the two b o dies Starting with the original Hertz theory the authors

develop ed a new theoretical mo del for contact mechanics and veried it exp erimentally

The JKR Theory predicts a contact radius a given by

JKR

a P R R P R

JKR

where is the surface energy p er unit contact area R is used to represent the radii of the

R R

1 2

contacting b o dies as R P represents the applied load and K the elastic constant

R +R

1 2

term as previously in x

2 2

2 1

E E

1 2

The last three terms in the JKR contact area equation are the mo dications to the

Hertz theory that take into account surface energies One can easily see that neglecting

surface energies ie we are left with the original Hertz equation It is also imp ortant

to note that under conditions of zero applied load P there is a nite contact area

with radius

2 3

a P

JKR

This allows for the calculation of the required load to separate the b o dies

R P

C (JKR)

Johnson et al measured the contact radii exp erimentally for gelatin and rubb er spheres

under varying normal loads P Their results show go o d agreement with the mo died Hertz

theory presented ab ove including the required pullo force to separate the b o dies Even

more than predicting a larger contact radius the JKR theory predicts a change in shap e

of the contacting b o dies In the original Hertz theory the surfaces of the two contacting

spheres approach the contact area tangentially with no deformation outside of the contact

area as seen in Fig In the JKR Theory the surfaces are lo cally deformed due to

surface forces and they approach the area of contact p erp endicularly as seen in Fig

After the JKR Theory in Derjaguin et al prop osed the DMT Theory

DerjaguinMullerTop orov Theory of contact mechanics

The DerjaguinMullerTop orov Theory

Like the JKR Theory the DMT Theory starts from a Hertz p ersp ective then

includes adhesive forces The dierence b etween the JKR and DMT theories is that the

DMT Theory assumes the adhesive forces act in a ring outside of the contact area but

cannot deform the two surfaces outside of the area of contact This leads to a Hertzian

contact prole in which the surfaces approach the contact area tangentially as mentioned

in x and visualized in Fig

The calculated contact radius given by Derjaguin et al mo died to the same

notation as presented previously is given as a

DMT

P R a

DMT

As can b e easily seen ab ove the DMT Theory predicts a nite contact area at zero applied

load P with radius given as

a P

DMT

The pullo force required to separate the two b o dies is given as

P R

C (DMT ) R1

Figure The illustration shows the prole of a Hertzian contact b etween spheres of radii

R and R D represents the diameter of the circular contact area As with the sphere on

disk mo del outside of the contact area the surfaces are not deformed

Figure The illustration shows the prole of a contact b etween spheres of radii R and

R where the lo cal deformation indicated by the dashed lines outside of the Hertzian

contact area can b e seen D represents the diameter of the Hertzian circular contact area

and D represents the diameter of the circular contact area predicted by the JKR theory

which can b e thought of as a measure of the adhesion b etween the contacting b o dies The

critical loads or pullo forces and nite contact areas at zero load given by the JKR and

DMT Theories dier only by a constant

After the DMT Theory was published there was a long and heated debate over whether

the JKR Theory or the DMT Theory was correct The debate will not b e elab orated up on

here but it was eventually accepted that b oth theories were accurate however they were

accurate under dierent regimes The JKR Theory accurately describ es soft large

radii materials with high adhesion The DMT Theory accurately describ es hard smallradii

materials with low adhesion This led to the need for a way to interp olate b etween the

theories

The Tab or Parameter and Other Metho ds of Interp olating Between The

ories

In D Tab or presented an investigation into some of the theoretical prob

lems in the eld of contact mechanics and analyzed b oth the JKR and DMT Theories

Tab or concluded that not only were surface forces imp ortant in adhesion but that surface

roughness also plays a role

Tab or used a very simple exp eriment to show how surface roughness aects adhesion

Using an optically smo oth rubb er ball and a at Persp ex surface pullo forces

were measured as the Persp ex surface was roughened It was found that as the roughness

increased the pullo force or adhesion b etween the two b o dies decreased

Tab ors analysis of the JKR and DMT theories along with his investigations into

surface roughness and adhesion led to the calculation of a parameter commonly re

ferred to as Tab ors Parameter used to determine if either the JKR or DMT theories

would b est describ e a system The Tab or parameter as presented by Muller et al in

is given as

2 3

K z

with z representing the equilibrium separation of the surfaces For clarication one can

think of z as the distance corresp onding to the p otential well in a LennardJones style

p otential When is small the DMT Theory is appropriate and when is

T T T

large the JKR Theory is appropriate

In Daniel Maugis presented a JKRDMT transition that not only worked for

the extreme cases tting nicely into JKR and DMT but also applied to the materials

that fell b etween them Using a squarewell Dugdale p otential to describ e the interaction

b etween the materials Maugis calculated a parameter similar to the Tab or parameter

M 0

where represents a constant adhesional stress which when multiplied by its range of

interaction gives the work of adhesion or as previously mentioned the surface energies

p er unit area The DMT mo del applies when and the JKR mo del applies when

Between the two extremes is the transition region Unfortunately the Maugis

formulation is dicult to implement when compared to the previous theories b ecause of

the complexity of solving complicated simultaneous equations

A simpler approximation to Maugis work was presented by Carpick et al in

Carpick et al found an approximation that tted the transition regime of the Maugis

Dugdale equations to within and was exact for the JKR and DMT regimes The

contact radius at zero applied load and the critical load required to separate the b o dies

are approximated as

13 2

L R

where is called the transition parameter and is given as

with b eing a numb er b etween and inclusive that represents the DMT regime at

the JKR regime at and the transition regime in b etween As Carpick et al

describ e is found exp erimentally by tting contact radius vs load or friction vs load

measurements Once is determined can b e easily calculated

Presented ab ove is a small overview of the eld of contact mechanics A basic under

standing of the history and various theories was imp ortant in starting this research In

Park et al studied the surfaces of b oth clean and oxidized dAlNiCo quasicrystals

in UHV using an AFM When comparing b oth the JKR and DMT mo dels it was found

that the clean quasicrystal surface adhered so strongly to the metallic AFM tip that the

JKR mo del was appropriate even though the quasicrystal surface and AFM tip are con

sidered hard On the other hand the oxidized dAlNiCo surface exhibited signicantly

less adhesion and the DMT mo del was appropriate In the Physical Review B

pap er the AFM tip was passivated with a layer of alkanethiol molecules resulting in low

adhesion b etween the quasicrystal and the AFM tip and thus the DMT mo del was used

to analyze their ndings

Park et al used the DMT mo del to calculate the contact area of a C alkanethiol

passivated TiN AFM tip on the fold surface of a single grain Al Ni Co decagonal

72 11 16

quasicrystal Using a nm tip radius the calculated contact area was nm The

threshold load the load at which there is a loss of the passivation layer was nN giving

a maximum pressure of GPa The pressure is calculated by dividing the normal force

by the contact area The frictional data shown by Park et al show the torsional resp onse

of the cantilever over normal forces reaching only nN corresp onding to pressures of

GPa

Due to the nature of the simulations in this work the contact area is supplied by the

researcher There is no surface roughness on the adamant tip see x or the quasicrys

talline approximant substrate see x Though there is no explicit surface roughness the

quasicrystalline approximant do es undergo a small amount of buckling during the initial

relaxation In the nal results the size of the adamant tip is such that it is smaller than

the buckling features of the substrate This allows for a simple calculation of the contact

area and resulting pressure The contact area b etween the adamant tip and approximant

substrate in this work is nm The maximum normal force used in the nal results

is nN leading to a maximum pressure of GPa

The maximum pressure achieved in the nal results is greater than in the Park exp eri

ments but it is of the same order of magnitude Future work as discussed in x should use

larger simulations with larger tips to more accurately recreate the pressures and normal

loads seen in the Park exp eriments Though our highest normal loads lead to pressures

larger than the threshold pressure measured by Park et al the range of normal loads

presented in x b egin at GPa and reach our maximum of GPa to cover most

of the pressures seen in the Park exp eriments

CHAPTER

AN EXAMINATION OF SOME TECHNIQUES USED TO PERFORM

MOLECULARDYNAMICS SIMULATIONS AND SOME POPULAR

PACKAGES

Limitations on the Simulation of Quasicrystals Using Quasicrystalline

Approximants

Quasicrystals are crystals lacking translational symmetry in one or more dimensions

and can contain symmetries forbidden by the classic denition of a crystal This p oses

a unique problem when attempting to simulate a quasicrystal rather than a traditional

p erio dic crystal b ecause one can think of quasicrystals as having innitely large unit cells

due to the lack of p erio dicity Fortunately there are quasicrystalline approximants

Quasicrystalline approximants contain lo cal symmetries similar to their quasicrystalline

counterparts That b eing said they can provide a go o d substitute for quasicrystals

in the moleculardynamics simulations in this work An exp erimental study on an Al

PdMn quasicrystalline approximant and an iAlPdMn quasicrystal showed that the

approximant has a frictional co ecient twice that of its quasicrystalline counterpart b oth

b efore and after oxidation Unfortunately we are restricted to the use of approximants so

naturally it would seem that the frictional resp onses seen in this work should b e somewhat

larger than the exp erimental work done by Park et al on the dAlNiCo quasicrystal The

larger the approximant unit cell the more closely the approximant would resemble the real

quasicrystal

The approximant structure used in this research was supplied by Mike Widom The

unit cell used for the nal results consists of one bilayer and only contains atoms A

table of the atomic co ordinates of the approximant structure used for the nal results

is presented in Table followed by the unitcell vectors in Table To see that the

unit cell mimics fold symmetry it is advantageous to visualize multiple unit cells as in

Fig An approximant with a larger unit cell would more closely mimic the dAlNiCo

b eing studied exp erimentally by Park et al however an indepth comparison of dierent

and larger approximants will b e left for future work In x an initial examination of two

more approximant structures is presented

Mihalkovic et al derive the quasicrystalline approximant structures by starting

with the same WidomMoriarty pair p otentials used in this research Using exp erimental

data such as the distance b etween planes of atoms along with the minima in the pair

p otentials trial quasicrystal structures are constructed These initial structures are then

Monte Carlo annealed to minimize the energy

Figure The approximants fold surface Though this is a p erio dic structure unit

cells seen here it approximates a fold surface For clarity one of the approximately

fold features is highlighted 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 by VMD

The fold face pictured in Fig is p erp endicular to the fold face which is the

ob ject of our study and is pictured in Fig In the dAlNiCo quasicrystal studied by

Park et al the fold face displays b oth p erio dic and ap erio dic order

Element X A Y A Z A

Table These are the co ordinates for one unit cell of the quasicrystalline approximant

bulk bilayer used in this research The structure of the approximant was supplied by Mike

Widom

along directions separated by with a A p erio dicity in the p erio dic direction Our

approximate fold face the fold face has a p erio dicity of A in the p erio dic

z direction and a A p erio dicity in the ap erio dic x direction

As mentioned previously the unit cell used for the nal results in this research contains

a atom bilayer which can clearly b e seen in Fig The original structure supplied

by Mike Widom contained three atom bilayers There were two surface bilayers

sandwiching a bulk bilayer We were given this sp ecic structure b ecause of our interest in

surface eects and the two surface bilayers were originally thought to b e b enecial After

investigation of the structure it was found that the surface bilayers were parallel to the

Figure The fold surface of the approximant This is the surface used for the friction

exp eriments The x direction represents the approximated ap erio dic direction and has a

p erio dicity of around A The z direction represents the p erio dic direction and has

a p erio dicity of around A The green spheres are Al white are Ni and pink are Co

Original image rendered by VMD

Figure Lo oking at the yz plane of the approximant structure allows us to see a clear

picture of the bilayer structure Each bilayer is dened by a line for ease of visualization

The unit cell in the z direction extends only ab out A making this our p erio dic direction

as opp osed to our approximated ap erio dic direction x which has a unit cell length of

around A The green spheres are Al white are Ni and pink are Co Original image

rendered by VMD

Vector X A Y A Z A

Table These are the unit cell vectors for the quasicrystalline approximant used in this

research The structure of the approximant was supplied by Mike Widom

fold surface and not the fold surface which is where we are p erforming our studies

The surface bilayers were removed from the simulations after the preliminary results and

the bulk bilayer was the only contributor to the unit cell The atomic co ordinates and

unitcell vectors for the original atom unit cell can b e found in App endix D

WidomMoriarty Pair Potentials

The WidomMoriarty pair p otentials used in this research were supplied

by Marek Mihalkovic in a tabulated format The p otentials describ e the pairwise

interactions b etween the quasicrystalline approximant atoms AlAl AlCo AlNi CoCo

CoNi and NiNi In their original form the p otentials show Friedel Oscillations and

they extend as far as A see Fig

The preliminary results were obtained by using the p otentials as in Fig It was later

suggested by Marek Mihalkovic through p ersonal communication with David Rabson that

a truncated version of the p otentials may more accurately t to the quasicrystalline ap

proximant used As p er the suggestion the p otentials were truncated to A and smo othed

to zero at the tail see Fig

To test the validity of the truncated p otentials relaxation simulations were p erformed

on the unit cell quasicrystalline approximant used in all friction exp eriments

for the nal results but without the adamant tip The relaxation simulations were p er

formed using a ps timestep for timesteps with all atoms allowed to relax

The average mean squared displacement of the approximants atoms as the temp erature

of the system was brought to K using a Langevin thermostat was measured and

Figure The original unsmo othed WidomMoriarty pair p otentials Only the rst A

are shown for ease of visualizing the oscillations though the p otentials extend to A

Figure The truncated and smo othed WidomMoriarty pair p otentials

graphed in Fig for simulations using b oth the original WidomMoriarty p otentials and

the truncated p otentials

Figure The average mean squared displacement over time of the unit cell qua

sicrystalline approximant during relaxation to K using b oth the original and truncated

WidomMoriarty pair p otentials

The mean squared displacement is a way to measure the average displacement of all

atoms from their starting p osition over time The results of the relaxation test show that

using the truncated p otentials allows the approximant to relax faster with the nal atomic

p ositions closer to the original structure than the untruncated WidomMoriarty p oten

tials Because the truncated p otentials more accurately describ e the original structure

they will b e used in all investigations following the preliminary results

Using an Adamant Tip

The friction exp eriments p erformed by Park et al were p erformed in UHV

conditions employing an alkanethiol passivated AFM tip to reduce adhesion with the qua

sicrystalline surface This reduction in adhesion allowed for the investigation of wearless

friction Simulating this sp ecic tip is a dicult and time consuming pro cess mostly due

to the passivation layer

To mimic wearless friction without simulating alkanethiols an adamant tip was cre

ated The goal was to have a tip that adhered strongly to itself but not to the quasicrys

talline approximant substrate The adamant or Ad tip was created as FCC Aluminum

using a mo died AlAl WidomMoriarty pair p otential see Fig The

Ad atoms have the same mass as Aluminum and the AdAd pair p otential was derived

by multiplying the AlAl p otential values by so that the hills and valleys were more

pronounced making the AdAd interaction stronger than that of regular AlAl

To eliminate adhesion b etween the adamant tip and the quasicrystalline approximant

a pair p otential was created out of a decaying exp onential see Fig This is a purely

repulsive p otential and was used for interaction b etween the tip atoms Ad and all ap

proximant atoms Al Ni and Co

One of the consequences of using a purely repulsive p otential for the tip is that we

cannot measure any pullo forces as seen in the Park exp eriments This

limits our range of normal forces to b e p ositive Using a more realistic tip and p otential

b etween the tip and approximant substrate is left for future work as discussed in x

Averaging and Error Analysis

For b oth the preliminary and nal results the lateral and normal forces during sliding

are recorded at discrete time intervals For the preliminary results the average forces were

Figure The original AlAl WidomMoriarty pair p otential is shown alongside the cre

ated AdAd pair p otential

Figure The purely repulsive p otential b etween the adamant tip and all approximant

atoms

calculated by summing all of the recorded force values and dividing by the numb er of data

p oints This metho d was later found to b e insucient and the nal results were calculated

by taking the integral over the force curve and dividing by the total time interval Due

to the closeness of the measured values this technique was used so as to minimize the

numerical uncertainties

According to Abramowitz and Stegun the extended trap ezoidal rule for integrating

a function F t is

F F

m 0

F F F tdt h

1 m1

with a corresp onding error of

E F

where h is the time interval b etween any two p oints F and F m is the total numb er

n n+1

of p oints and is any random p oint We then divide this integral and the corresp onding

error by the total time interval mh to obtain the average force and the error asso ciated

with the averaging the force using the trap ezoidal rule

As discussed at the end of x the tip needs to slide across exactly one simulation b ox

length Our sampling time is not commensurate with our required stopping time and thus

we had to interp olate b etween two data p oints to obtain the force at the required sliding

distance To add this nal p oint into our average presented ab ove it needs to b e weighted

The nal equation for obtaining the average force over time is

1 1 1

t t F F F F h F F

m 0 1 m1 m m

2 2 2

t t

where

F F

m+1 m

F F t t

m m

is the interp olated force at the required stopping time t

Calculating the average force in this manner provides an estimate for the error E due

to the discrete sampling of p oints To obtain the maximum error the maximum curvature

F was needed The curvature of a sin wave is given as

max

jF j Asin f t

with amplitude A and frequency f By p erforming the derivative and evaluating at a p eak

we nd that the maximum curvature is f t

00 2 2

F f A

max

giving an error E of

2 2 2

E h f A

E should b e divided by to account for our using only the maximum curvature of the

sin wave however there is approximately a p ercent discrepancy b etween the calculated

frequency and what we know the frequency is supp osed to b e which leads to a factor of

dierence in the error These factors then cancel leaving E as presented ab ove

Because of the noise inherent in the data going from p oint to p oint a rough estimate

of error was calculated as

F F F

j 1 j j +1

j =2

m m

term comes from the error in the additional trap ezoidal area created by adding The

the interp olated p oint The bulk of E comes from the idea that random noise given an

innite time would cancel Since we are not dealing with an innite time series it should

b e accounted for

A third metho d for estimating error was gained through simulation In the nal results

the tip do es not cover the entire surface of the approximant in either the p erio dic or ap eri

o dic directions To account for the eect of the tip p osition as sliding is p erformed the tip

was shifted by A in the direction p erp endicular to b oth the sliding and compression direc

tions This was done at a sliding sp eed of Aps for b oth the p erio dic and ap erio dic

sliding directions The dierence b etween the original frictional force and the frictional

force obtained by using the shifted tip was calculated for sliding in b oth directions This

dierence was then divided by the original friction result to obtain the fractional error e

For each simulation the error asso ciated with cho osing a dierent intitial tip p osition is

calculated as

E eF

where F is the average force

The error bars used in the nal results presented in x are calculated for each simu

lation as a combination of all three error estimates such that

2 2 2

E E E E

1 2 3

We do not rule out the p ossibility that other sources of error may need to b e accounted

for We have ruled out the p ossiblility of statistical error asso ciated with the Langevin

thermostat by p erforming simulations using dierent random seeds Changing the

random seed used for the thermostat do es not change the results of the simulations

The equations used to calculate the error bars and average force were provided by Dr

David Rabson private communication

MolecularDynamics Packages

Writing a moleculardynamics program sophisticated enough for the research p erformed

in this work is a daunting task The goal of this pro ject was not to formulate a new metho d

of molecular simulation but to use existing to ols to study a very complicated and unique

system With that in mind some of the most prominent moleculardynamics packages

available were researched installed and tested for their capability of handling the system

to b e studied ease of mo dication p erformance and how userfriendly they turned out

to b e This section contains a list and brief description of each of the molecular dynamics

packages investigated

DL POLY

DL POLY is a moleculardynamics simulation package created at Daresbury Lab ora

tory Two versions exist DL POLY is suitable for small simulations of up to

POLY is designed to b e used with large atoms on pro cessors or less and DL

simulations on the order of atoms on upwards of pro cessors

DL POLY is equipp ed to handle constant NVT particle numb er volume and temp er

ature NVE particle numb er volume and energy and NPT particle numb er pressure

and temp erature simulations and employs either a Velo city Verlet or Verlet Leapfrog in

tegration algorithm For parallel jobs MPI is used for interpro cessor communication

One of the reasons DL POLY was not chosen for this research is that it is written in

Fortran which is not a language familiar to the researcher Though the package seemed

capable of handling solidstate materials when compared with LAMMPS it was found

that the LAMMPS package allowed more user control of the simulation while employing

simpler and more userfriendly input les

NAMD

NAMD was develop ed by the Theoretical and Computational Biophysics Group in the

Beckman Institute for Advanced Science and Technology at the University of Illinois at

UrbanaChampaign NAMD was intended to b e run in conjunction with VMD Visual

Molecular Dynamics which is not only an advanced visualization program but also

interfaces with NAMD and allows the user to mo dify the simulation The source co de is

written in C and uses MPI when executing parallel jobs NAMD is also known to b e

able to handle large simulations of atoms on pro cessors

One of the unique features of NAMD is the standard capability of reading input les

from other simulation packages such as XPLOR CHARMM AMBER and GROMACS

The main reason that NAMD was not chosen for this research is that it is sp ecialized

for biomolecular systems This sp ecialization makes it dicult to simulate solidstate

materials

Though NAMD was not chosen to p erform the molecular dynamics VMD was used to

render the snapshots shown in this work

Gromacs

Gromacs was initially develop ed to simulate biological materials such as lipids and

proteins but unlike NAMD it can handle nonb onded solidstate materials relatively well

The source co de is written in C and parallel jobs use standard MPI communication

Gromacs uses the leapfrog algorithm for the integration of Newtons Laws to up date b oth

the p ositions and velo cities of the atoms

As input Gromacs uses fairly complicated top ology les that are used to describ e b onds

angles and molecules For biological systems it is necessary to sp ecify all of the information

in the top ology les but it is unnecessarily complicated for the system researched in this

work Though Gromacs claims to b e to times faster than most moleculardynamics

packages LAMMPS was a b etter t to our eld of study

LAMMPS

The LAMMPS package develop ed at Sandia National Lab oratory was the molecular

dynamics package chosen to complete this work LAMMPS has all of the basic func

tionality of the other moleculardynamics packages reviewed including the ability to work

on a single pro cessor or in parallel using MPI communication implements the velo city Ver

let integration scheme and can run under numerous ensembles including NPT numb er

of atoms pressure and temp erature NVE numb er of atoms volume and energy and

NVT numb er of atoms volume and temp erature

LAMMPS is compatible with CHARMM AMBER and GROMACS force elds and

can output data into multiple formats including one viewable using VMD as mentioned

in x The source co de is written in C and is easily mo died to include any new

features The input les required to run a simulation using the LAMMPS package are

relatively simple and easy to understand An indepth description of the commands used to

p erform this research is given in App endix C Though LAMMPS is capable of simulating

biomolecular systems it is not sp ecialized for that purp ose however it easily simulates

systems in the gas liquid or solid state

Though some workarounds were required see App endix A the LAMMPS package

was able to p erform the required simulations and output the appropriate data

CHAPTER

PRELIMINARY RESULTS

Preliminary results were obtained for compressing and sliding a atom adamant

slab on a atom quasicrystalline approximant slab All simulations use a Langevin

thermostat to control the temp erature of the simulation by keeping it less than K

All of the atoms in the simulation are broken up into groups The topmost layers of

the tip and the b ottommost layers of the approximant are group ed as xed tip atoms and

xed approximant atoms resp ectively The centermost layers of the tip and approximant

are used for the thermostat and are called the tip thermostat atoms and approximant ther

mostat atoms The layers at the tipapproximant interface do not have any constraints on

them and are left to act freely hence they are called the free tip atoms and free approximant

atoms These layers are visualized in Fig

There are four main p ortions to each simulation relaxation compression relaxation

and sliding

Initially the entire system is allowed to relax This is achieved by applying a Langevin

thermostat to all atoms in the simulation bringing the temp erature to K The thermostat

is applied for ps

After relaxation the velo cities and forces on the topmost layers of the adamant tip and

the b ottommost layers of the approximant are set to zero This forces the atoms to move

as rigid b o dies and they are called the xed atoms as seen in Fig The Langevin

thermostat implemented previously on all atoms is removed and replaced by a Langevin

thermostat acting only on the centermost layers of the tip and approximant these are

called the thermostat atoms To avoid the thermostat aecting lateral or normal force

data the thermostat is allowed to act only in the direction p erp endicular to the sliding

Figure Dierent areas of the simulation are used for dierent purp oses The outermost

layers of atoms are xed and held rigid The centermost layers of atoms are used for the

thermostat and the interface layers are left to act freely The green spheres are Al white

are Ni pink are Co and blue are adamant Ad Original image rendered by VMD

velo city and the compression direction The desired temp erature is K The xed atoms

of the adamant slab are then given a velo city toward the quasicrystalline approximant

surface Multiple compressions were achieved by using dierent compression velo cities for

ps each

After the desired compression is achieved the system is allowed to relax once more for

ps The height after compression is held constant by the xed rigid layers at the top and

b ottom of the simulation This pro cedure has b een used in previous moleculardynamics

studies of friction

To achieve the sliding p ortion of the simulation the rigid tip atoms are given a constant

sliding velo city of Aps in either the p erio dic or ap erio dic direction of the approxi

mant The sliding lasts for ps and corresp onds to sliding a distance of A During this

time the forces that opp ose the sliding of the tip the lateral forces are recorded for each

compression in b oth the p erio dic and ap erio dic directions Examples of the lateral forces

during sliding are presented in Fig and Fig for the approximated ap erio dic and

p erio dic directions resp ectively Also shown in Fig and Fig are the corresp onding

normal forces over time during sliding

The uctuations seen in the normal force and lateral force graphs are thought to b e

mostly due to noise Contrary to this the uctuations seen in the nal results corresp ond

to surface features of the approximant and will b e discussed in x

After all of the desired compressions have b een simulated a plot of the average normal

force versus average lateral force is made for b oth sliding directions see Fig According

to Amontonss Law the co ecient of friction b etween two sliding b o dies is the slop e of the

normal vs lateral force curve The preliminary results shown in Fig clearly show

a frictional anisotropy The overall magnitude of the frictional forces dier by a factor of

approximately with the p erio dic direction b eing higher More imp ortantly the friction

co ecients show an fold anisotropy

The measured co ecients of friction are for the p erio dic direction and for

the ap erio dic direction These results were very promising however when the system

Figure An example graph of the lateral force over time as the tip slides across the

quasicrystalline approximant surface in the ap erio dic x direction at a compression of

nN The lateral force averaged over time is calculated to b e nN

Figure An example graph of the lateral force over time as the tip slides across

the quasicrystalline approximant surface in the p erio dic z direction at a compression of

nN The lateral force averaged over time is calculated to b e nN

Figure An example graph of the normal force over time as the tip slides across the qua

sicrystalline approximant surface in the ap erio dic x direction at an average compression

of nN

Figure An example graph of the normal force over time as the tip slides across the

quasicrystalline approximant surface in the p erio dic z direction at an average compression

of nN

Figure This graph shows the average lateral forces or frictional forces for each normal

force compression in b oth the p erio dic and approximated ap erio dic X directions The

frictional co ecient for the p erio dic direction is while the frictional co ecient of the

ap erio dic direction is as calculated by the slop e according to Amontonss Law

was studied further discrepancies started to app ear After the initial preliminary results

we started studying a wider range of compressions along with dierent initial tip p ositions

We b egan to realize that the preliminary results required further investigation when the

measured lateral force started getting smaller as the compressive force increased A graph

of the preliminary results along with the investigations at higher compressions can b e seen

in Fig

Figure This graph shows the average lateral forces or frictional forces for each normal

force compression in b oth the p erio dic and approximated ap erio dic X directions

Lines of b est t to the original force data are shown to illustrate that the frictional forces

started decreasing at increasing normal force

It was obvious that more sophisticated simulations were necessary The rst and

easiest mo dication to the preliminary results was to simulate a larger system We went

from simulating a total of atoms to simulating atoms an increase more than

doubling the simulation size

The preliminary results were p erformed using a tip that covered of the approxi

mants surface area Because p erio dic b oundary conditions were used in the sliding direc

tions there was a gap in the tip small enough that phonons could pass from one side to

the other see Fig It was suggested by Dr Sagar Pandit that one of our problems

might b e due to these phonons One of the mo dications currently used is that the tip is

now small enough so that one side do es not interfere with the other through the p erio dic

b oundary conditions

The third mo dication to the preliminary simulations was the intro duction of the com

pression direction into the Langevin thermostat as suggested by Dr Susan Sinnott at the

University of Florida As can b e seen in Fig and Fig the lateral force data

are noisy Intro duction of the thermostat in the compression direction cleaned up the data

considerably and surface features of the approximant can now b e clearly seen in the force

data see x

The fourth mo dication to the preliminary results was to average the forces during

sliding over the entire simulation b ox rather than a small p ortion as in the preliminary

results We use the term simulation b ox to mean all of the atoms in the simulation b efore

p erio dic b oundary conditions are applied The dimensions of the simulation b ox are the

b ox b oundaries sp ecied in the data le During many simulations testing dierent ways

of measuring friction it was discovered that the surface of the approximant exp eriences a

small amount of buckling during relaxation The buckling is p erio dic on the length scale

of the simulation b ox The preliminary results only slid over A of the simulation b ox

which extended A in the p erio dic x direction and A in the ap erio dic z direction

Whether this small amount of sliding o ccurred going uphill or downhill will change

the results This would account for the decrease in the average lateral force as the normal

force is increased By averaging the forces over sliding exactly one simulationb ox length

in each direction this eect will b e negated

The fth mo dication was to use a mo died version of the WidomMoriarty pair p o

tentials as discussed in x The WidomMoriarty pair p otentials that were

shortened to A created less deviation from the original structure during relaxation

CHAPTER

FINAL RESULTS

The nal results were achieved by simulating a total of atoms There are

atoms in the approximant and atoms in the adamant tip In Fig and Fig it

is clearly seen that the tip covers only a p ortion of the surface of the approximant This

mo dication to the preliminary results is discussed in x

Just as in the preliminary results there are four parts to each simulation relaxation

compression relaxation and sliding

Initially the entire system is allowed to relax This is achieved by applying a Langevin

thermostat to all atoms in the simulation to bring the temp erature to K The thermostat

is applied for ps

The compression pro cedure in the nal results is the same as in the preliminary results

see x Once the xed atoms are constrained to move as rigid b o dies the thermostat

is applied to the thermostat atoms only The delineation of the groups can b e seen

in Fig To avoid the thermostat aecting the lateralforce data the thermostat is

only allowed to act in the directions p erp endicular to the sliding velo city including the

compression direction Allowing the thermostat to act in the compression direction is one

of the mo dications discussed in x

The desired temp erature throughout the simulation is K The xed atoms of the

adamant slab are then given a velo city toward the quasicrystalline approximant surface

Multiple compressions were achieved by using dierent compression velo cities for ps

each

Figure A snapshot of the fold face and adamant tip used in the simulations that

pro duced the nal results The green spheres are Al white are Ni pink are Co and blue

are adamant Ad Original image rendered by VMD

Figure A snapshot lo oking down the x direction of the approximant along with the

adamant tip used in the simulations that obtained the nal results The green spheres are

Al white are Ni pink are Co and blue are adamant Ad Original image rendered by

VMD

After the desired compression is achieved the system is allowed to relax once more for

ps The height after compression is held constant by the xed rigid layers at the top

and b ottom of the simulation

To achieve the sliding p ortion of the simulation the rigid tip atoms are given a constant

sliding velo city in either the p erio dic or approximated ap erio dic direction of the approxi

mant To test the eect of sliding velo city on friction multiple sliding velo cities were used

The sliding velo cities tested range from Aps to Aps in Aps increments

giving a total of dierent sliding velo cities

Example graphs of the temp erature uctuations over time during sliding can b e seen

in Fig and Fig for sliding in the ap erio dic and p erio dic directions resp ectively

As discussed in x due to a small amount of buckling on the surface of the approximant

the frictional forces exp erienced by the tip need to b e averaged over sliding the length of

one simulation b ox to negate the eect of the hills and valleys created by the buckling

Because of this each velo city required a dierent amount of sliding time to more than

completely cover the simulation b ox When the sliding is initially b egun the data are

noisy but quickly relax To get clean data for sliding over one simulationb ox size the

required sliding time was increased A table of the velo cities with the required sliding time

and the actual time sp ent sliding in order to account for any transient b ehavior is given

in Table

As the sliding is p erformed the forces opp osing the motion of the xed tip atoms

are recorded as seen in Fig and Fig for sliding in the ap erio dic and p erio dic

directions resp ectively This is the frictional force and it is time averaged over sliding one

simulationb ox length For a more detailed description of the averaging pro cedure see x

The fact that the frictional forces go b elow zero is thought to partially b e due to the lack

of adhesion b etween the tip and the approximant combined with surface features of the

approximant Similar features were shown by Harrison et al for the frictional resp onse

of diamond

Figure An example graph of the temp erature over time as the tip slides across the

approximant surface in the ap erio dic x direction at a compression of nN The

lateral force averaged over time is calculated to b e x nN The graph was

obtained from sliding at a sp eed of Aps

Figure An example graph of temp erature over time as the tip slides across the approx

imant surface in the p erio dic z direction at a compression of nN The lateral force

averaged over time is calculated to b e x nN The graph was obtained from

sliding at a sp eed of Aps

A t ps t ps Aps Direction D V

D T

p erio dic

ap erio dic

Table Presented ab ove is a breakdown of the required sliding times for the ve dierent

sliding velocities tested V is the sliding velocity D is the length of the simulation b ox or

the required sliding distance t is the amount of time at the given velocity to slide the

distance D and t is the total amount of time sp ent sliding The total amount of time

sp ent sliding is larger than the time required to slide exactly one simulation b ox to allow

for any transient b ehavior when the sliding is initially b egun to not b e included in the

averaged force data

One interesting feature of the lateralforce data is the p erio d of the p eaks It was

calculated that the distance b etween two p eaks corresp onds 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 b oth the p erio dic and ap erio dic directions of the approximant

The example plots in Fig and Fig show a p erio d of approximately ps for

sliding in the ap erio dic direction and a p erio d of approximately ps for sliding in the

Aps The p erio dic direction Both plots were obtained from sliding at a sp eed of

A in the ap erio dic direction which has a unit cell p eaks corresp ond to sliding roughly

A in the p erio dic direction which has a unit cell length of A and length of

A During the error calculations it was noticed that there is a discrepancy b etween

the actual unitcell lengths and the p erio d This discrepancy is taken into account in the

error calculation

The frequency discrepancy is noticed in a simulation sliding in the ap erio dic direction with a sliding

sp eed of Aps and a compression sp eed of Aps The frequency should b e ps but the

calculated frequency from the data les is ps Most of the other simulations have not b een tested

for this discrepancy The discrepancy do es not app ear in the Aps sliding sp eed Aps compression

sp eed ap erio dic simulation

Figure An example plot of lateral force over time as the tip slides across the approx

imant surface in the ap erio dic x direction at a compression of nN The lateral

force averaged over time is calculated to b e x nN The graph was obtained

from sliding at a sp eed of Aps

Figure An example plot of lateral force over time as the tip slides across the approxi

mant surface in the p erio dic z direction at a compression of nN The lateral force

averaged over time is calculated to b e x nN The graph was obtained from

sliding at a sp eed of Aps

One can also see the same features in Fig and Fig when lo oking at the normal

force during sliding Because of the nature of our system with the outer layers acting as

rigid b o dies keeping a constant distance b etween the xed tip atoms and xed approximant

atoms when the tip slides over a feature of the approximant unit cell it will change the

normal force Due to the squishing or relaxing of the central groups of atoms as the

tip slides over a feature the temp erature is also aected as can b e seen in Fig and

Fig It is imp ortant to note that the variations in the normal force and temp erature

are small compared to that in the lateral force

As in the preliminary results in x the averaged lateral force is plotted as a function

of the corresp onding averaged normal force This was done for each of the ve sliding

velo cities mentioned in Table If Amontonss Law were to hold we would have a

straight line Fig shows the results of sliding in the approximated ap erio dic x

direction Fig shows the results of sliding in the p erio dic z direction

As you can see from the data in Fig and Fig we do not have straight lines

and there is a clear velo city dep endence In the exp erimental work done by Park et al

the friction along the ap erio dic direction is times less than the friction

along the p erio dic direction We came to a similar conclusion in the preliminary results

which were later found to b e unreliable but the current graphs show that the magnitude of

the frictional forces along the ap erio dic direction are slightly higher than along the p erio dic

direction for the ma jority of the compressions investigated At the higher compressions

one can see that the p erio dic direction has a slightly steep er slop e which according to

Amontonss Law would mean a higher co ecient of friction Both sliding directions are

plotted on the same graph in Fig

The slop e of a b estt line through the last third of the normalforce vs lateralforce

data for each velo city was calculated and is presented in Tables and Here we

can see that there is a very small dierence in the estimated frictional co ecients at high

compression according to Amontonss Law

Figure An example plot of the normal force over time as the tip slides across the

approximant surface in the ap erio dic x direction at a compression of nN The

lateral force averaged over time is calculated to b e x nN The graph was

obtained from sliding at a sp eed of Aps

Figure An example plot of the normal force over time as the tip slides across the

approximant surface in the p erio dic z direction at a compression of nN The lateral

force averaged over time is calculated to b e x nN The graph was obtained

from sliding at a sp eed of Aps

Figure This graph shows the average lateral forces or frictional forces for each normal

force compression in the approximated ap erio dic x direction for each of the ve examined

sliding velo cities

Figure This graph shows the average lateral forces or frictional forces for each normal

force compression in the p erio dic z direction for each of the ve examined sliding velo cities

Figure This graph shows the average lateral forces or frictional forces for each normal

force compression in the p erio dic z direction dashed lines and the approximated ap erio dic

x direction solid lines for each of the ve examined sliding velo cities

Direction V Aps E

5 7

Ap erio dic

5 7

Ap erio dic

6 7

Ap erio dic

6 7

Ap erio dic

6 8

Ap erio dic

Table This table contains the calculated slop es of the highest ve compressions

for the ap erio dic sliding direction at each velo city along with the calculated error on

the slop e E These are the highest compressions graphed in Fig According to

Amontonss Law the slop es are a measure of the co ecients of friction

Direction V Aps E

5 8

Perio dic

5 8

Perio dic

6 8

Perio dic

6 8

Perio dic

6 8

Perio dic

Table This table contains the calculated slop es of the highest ve compressions for

the p erio dic sliding direction at each velo city along with the calculated error on the slop e

E These are the highest compressions graphed in Fig According to Amontonss

Law the slop es are a measure of the co ecients of friction

If we graph the co ecients of friction as functions of sliding velo city for sliding in

b oth the p erio dic and ap erio dic directions as in Fig we can see that they do

not overlap This allows us to come to the conclusion that the friction co ecient in the

ap erio dic direction of our dAlNiCo quasicrystalline approximant is lower than the friction

in the p erio dic direction This agrees with the Park exp eriments Even though our overall

conclusions agree with Park et al the magnitude of our frictional forces and the

ratios of the co ecients of friction are extremely small in comparison Park et al

show frictional resp onses ranging from to nN The results presented in this work show

frictional resp onses ranging from to nN Some of this discrepancy can b e explained

by our use of a totally repulsive interaction b etween the tip and the approximant A more

realistic adhesion is left for future work

One can also see in Fig and Fig and Tables and that the frictional

forces and co ecients of friction increase with increased sliding velo city Park et al

found no velo city dep endence in the torsional resp onse of the AFM cantilever when sliding

Figure This graph shows the co ecients of friction in the ap erio dic sliding direction

solid line and the p erio dic sliding direction dotted line as a function of sliding velo city

in the ap erio dic direction When sliding in the p erio dic direction the cantilevers torsional

resp onse increased with increasing sliding velo city The sliding velo cities investigated by

8 10

Aps Our velo cities are extremely Aps to Park et al ranged from

fast in comparison and slower sliding velo cities should b e prob ed in future work Employ

ing such high sliding velo cities in this work was necessary for computation time To slide

times slower it would mean the simulation would take approximately times as

long to complete This was just not feasible with the resources currently available

We can also lo ok at the ratio of the co ecients of friction as done by Park et al

The results from the Park et al exp eriments show an fold anisotropy

Our results show an anisotropy ranging from to as seen in Table dep ending

on the sliding velo city

per iodic

V Aps

per iodic aper iodic

aper iodic

5 5

6 6

Table This table contains the calculated slop es of the highest ve compressions for

the p erio dic and ap erio dic directions at each velo city along with the ratio

Through the ma jority of the compressions investigated the magnitude of the frictional

forces in the ap erio dic direction are higher than in the p erio dic direction but the co ef

cients of friction at high compression show that the ap erio dic direction is lower than

the p erio dic direction The overall magnitudes of the frictional forces found in these ex

p eriments are on the order of times smaller than in the exp eriments p erformed by

Park et al

The simple mo del presented here made many approximations With these extreme

approximations we were still able to show a dep endence of friction on p erio dicity Even

with such small frictional forces a frictional anisotropy b etween sliding in the ap erio dic

and p erio dic directions was found and we agree with Park et al in that the co ecient of

friction in the ap erio dic direction of a dAlNiCo quasicrystal is lower than the co ecient

of friction along the p erio dic direction

CHAPTER

FUTURE WORK

The main goal for future work is to create more realistic simulations By doing this

one would hop e to obtain a larger frictional anisotropy than is seen in the nal results of

this work Not only is the anisotropy much smaller than exp ected the overall magnitude of

the frictional forces is also quite small when compared with exp eriment To achieve more

sophisticated simulations one would start with the comparison of dierent quasicrystalline

approximants make larger simulations tailor the pair p otentials to the approximants b eing

studied create more realistic p otentials create and use a more realistic tip and closely

monitor phonon propagation through the system

Comparison of Dierent Approximants

Quasicrystals are crystals lacking translational symmetry eectively making them have

an innite unit cell in or dimensions To simulate quasicrystals approximants are

needed see x To continue this work it would b e advantageous to compare dierent

AlNiCo approximants of varying size Because simulating a real quasicrystal is imp ossible

one would like to have the largest approximant p ossible to more closely resemble a real

quasicrystal

We have tested two new quasicrystalline approximants supplied by Marek Mihalkovic

containing and atoms p er unit cell Both of the approximants contain one bilayer

Pictures of the approximated fold face of a single unit cell for each are visualized in

Fig and Fig

A unitcell blo ck of the atom unitcell approximant broke apart during

relaxation when using the original untruncated p otentials The use of the truncated

Figure The illustration shows the fold face of a atom unitcell quasicrystalline

approximant The approximant was supplied by Marek Mihalkovic The green spheres

are Al white are Ni and pink are Co Original image rendered by VMD

Figure The illustration shows the fold face of a atom unitcell quasicrystalline

approximant The approximant was supplied by Marek Mihalkovic The green spheres

are Al white are Ni and pink are Co Original image rendered by VMD

p otentials eliminated this problem in the atom unitcell approximant although there

was still a signicant amount of relaxation Investigation into a more accurate p otential

may b e required for this approximant

The atom approximant did not show any signicant problems with either the orig

inal or truncated p otentials This b eing said there was some relaxation so mean squared

displacement tests should b e run as done in x

Larger Simulations

Hand in hand with using larger approximants one would need larger simulation sizes

With a larger approximant and tip one may b e able to measure frictional values closer to

the exp erimental values presented by Park et al by creating smaller pressures

than presented here The maximum pressure studied by Park et al is GPa The

maximum pressure reached in this research is GPa A smaller pressure can b e achieved

using the same normal forces by creating a larger contact area

Tailoring the Pair Potentials

The WidomMoriarty pair p otentials used in this research will eventually

need to b e replaced by more accurate p otentials tailored to this system These p otentials

were optimized for bulk structures at high temp erature for the purp ose of studying alu

minum migration Potentials optimized for surfaces at lower temp eratures would b e more

appropriate

One could start obtaining b etter p otentials by mo difying the existing ones to more

closely resemble the original approximant structure Even the cuto p otentials in Fig

could use some improvement This would need to b e done for each approximant studied

Eventually pair p otentials would not b e enough and an EAM or another similar

p otential will need to b e develop ed to more accurately describ e the electronic interactions

These more sophisticated p otentials contain a pairwise interaction such as the Widom

Moriarty pair p otentials but also have a term dep endent on the lo cal electronic density

These p otentials come closer to describing a real system than pair p otentials alone

Creating and Using a More Realistic Tip

The original Park exp eriment used an alkanethiol passivated AFM tip

The purp ose of using a passivated tip was to minimize adhesion We have simplied this

concept by creating and using a noninteracting adamant tip however a realistic adhesive

force could provide more accurate frictional forces see x To create a more realistic tip

one needs not only structures but also p otentials

Monitoring Phonon Propagation

It is evident by watching movies of the simulations that phonons are propagating

through the system An analysis and investigation of these phonons could provide great

insight into the mechanisms of friction and energy dissipation

Some interesting information is seen if one plots lateral vs normal force as in Fig

temp erature vs normal force as in Fig and temp erature vs lateral forces as in Fig

and Fig

What we may b e lo oking at here are Poincare sections We exp ect there to b e a

correlation b etween the normal and lateral forces and also the temp erature and the forces

however we may b e lo oking at d pro jections of a higher dimensional phenomenon This

should b e investigated 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 have b een studied in quasicrystals and quasicrystalline approximants b efore

The problem in studying the lattice dynamics of an ap erio dic system is the

existence of an innitely large unit cell In p erio dic crystals phonons are describ ed as

p erterbations to an underlying recipro cal lattice Like amorphous solids quasicrystals do

Figure This graph shows an example plot of the lateral force as a function of the normal

force for sliding in the ap erio dic direction The average normal force is nN The

average lateral force is nN This graph was obtained from sliding at a sp eed

of Aps

Figure This graph shows an example plot of the lateral force as a function of the tem

p erature for sliding in the ap erio dic direction The average normal force is nN

The average lateral force is nN This graph was obtained from sliding at a

sp eed of Aps

Figure This graph shows an example plot of the lateral force as a function of the

temp erature for sliding in the p erio dic direction The average normal force is nN

The average lateral force is nN This graph was obtained from sliding at a

sp eed of Aps

Figure This graph shows an example plot of the normal force as a function of the tem

p erature for sliding in the ap erio dic direction The average normal force is nN

The average lateral force is nN It is clear that an increase in the normal force

corresp onds to an increase in the temp erature This is exp ected due to the constant height

during sliding When the tip crosses a high feature of the approximant all of the atoms in

the middle are compressed leading to a higher normal force and higher temp erature This

graph was obtained from sliding at a sp eed of Aps

not have a recipro cal lattice This do es not mean that the task is imp ossible and as in

this work one can use quasicrystalline approximants

When comparing p erio dic crystals with amorphous solids one must accept that the ab

sence of longrange order in an amorphous material may lo calize the vibrational mo des

This can cause an absence of longrange waves prop ogating through the system Contrary

to this quasicrystals p osess longrange order though they lack translational p erio dicity

When studying an AlNiCo quasycrystalline approximant Mihalkovic et al found that

at low frequency there is lo calization of the phonon mo des and that the lo catlization

rapidly increases for increasing frequency Exp erimentally de Boissieu et al compare

the lattice dynamics of an icosahedral quasicrystal and its corresp onding approximant

using inelastic neutron and xray scattering For b oth the quasicrystal and approximant

a welldened transverse acoustic mo de is found The calculated sound velo cities are

ms for the quasicrystal and ms for the corresp onding approximant

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 exp erimental

studies have shown the similarities b etween acoustic phonon mo des in quasicrystals and

their approximants continuing with this theoretical work on and AlNiCo quasicrys

talline approximant could lead to further insight on the dAlNiCo quasicrystal and the

correlation b etween phonons and friction

REFERENCES

Mike Widom p ersonal communication

Marek Mihalkovic p ersonal communication

See the LAMMPS Users Manual at httplammpssandiagov

POLY as retrieved on July httpwwwccpacukDL

httpwwwgromacsorg as retrieved on August

Sagar Pandit p ersonal communication

Susan Sinnott p ersonal communication

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Dover

I AlLehyani M Widom Y Wang N Moghadam GM Sto cks and J Moriarty

Transitionmetal interactions in aluminumrich intermetal lics Physical Review B

Neil W Ashcroft and David N Mermin Solid State Physics Harcourt Inc

HJC Berendsen D van der Sp o el and R van Drunen GROMACS A message

passing paral lel molecular dynamics implementation Comp Phys Comm

Pierre Brunet LM Zhang Daniel J Sordelet Matt Besser and JeanMarie Dub ois

Comparitive study of microstructural and tribological properties of sintered bulk icosa

hedral samples Materials Science and Engineering

Rob ert W Carpick Frank D Ogletree and Miquel Salmeron A general equation for

tting contact area and friction vs load measurements Journal of Colloid and Interface

Science

MS Daw and MI Baskes Embeddedatom method Derivation and application to

inpurities surfaces and other defects in metals Physical Review B no

M de Boissieu S Francoual M Mihalkovic K Shibata AQR Baron Y Sidis

T Ishimasa D Wu T Lograsso LP Regnault F Gahler S Tsutsui B Hennion

P Bastie TJ Sato H Takakura R Currat and AP Tsai Lattice dynamics of

the znmgsc icosahedral quasicrystal and its znsc periodic approximant Nature

Materials

BV Derjaguin VM Muller and YUP Top orov Eect of contact deformations on

the adhesion of particles Journal of Colloid and Interface Science no

J Dolinsek P Jeglic M Komelj S Vrtnik A Smontara I Smiljanic A Bilusic

J Ivkov D Stanic ES Zijlstra B Bauer and P Gille Origin of anisotropic non

metal lic transport in the Al Cr Fe decagonal approximant Physical Review B

80 15 5

LP Feng TM Shao YJ Jin E Fleury DH Kim and DR Chen Temperature

dependence of the tribological properties of laser remelted AlCuFe quasicrystal line

plasma sprayed coatings Journal of NonCrystalline Solids

Anthony C FischerCripps Introduction to Contact Mechanics Mechanical Engineer

ing Series SpringerVerlag New York Inc Fifth Avenue New York NY

J Gao WD Luedtke D Gourdon M Ruths JN Israelachvili and U Landman

Frictional forces and Amontons Law From the molecular to the macroscopic scale

J Phys Chem B

Nicholas J Giordano Computational Physics PrenticeHall Inc

IG Goryacheva Contact Mechanics in Tribology Solid Mechanics and its Applica

tions vol Kluwer Academic Publishers PO Box AA Dordrecht The

Netherlands

DS Grierson EE Flater and RW Carpick Accounting for the JKRDMT transi

tion in adhesion and friction measurements with atomic force microscopy J Adhesion

Sci Technol no

JA Harrison CT White RJ Colton and DW Brenner Moleculardynamics sim

ulations of atomicscale friction of diamond surfaces Physical Review B

Christopher L Henley Cel l geometry for clusterbased quasicrystal models Phys Rev

W Humphrey A Dalke and K Schulten VMD Visual Molecular Dynamics Journal

of Molecular Graphics

C Janot Quasicrystals a primer vol Clarendon Press

KL Johnson A note on the adhesion of elastic solids British Journal of Applied

Physics

KL Johnson K Kendall and AD Rob erts Surface energy and the contact of elastic

solids Pro c R So c Lond A

JS Ko AJ Gellman TA Lograsso CJ Jenks and PA Thiel Friction between

singlegrain Al Pd Mn quasicrystal surfaces Surface Science

70 21 9

DC Lovelady HM Harp er IE Bro dsky and DA Rabson Multiphase region of

helimagnetic superlattices at low temperature in an extended sixstate clock model J

Phys A Math Gen

C Mancinelli CJ Jenks PA Thiel and AJ Gellman Tribological properties of a

Btype AlPdMn quasicrystal approximant Journal of Metrials Research

Daniel Maugis Adhesion of spheres The JKRDMT transition using a Dugdale model

Journal of Colloid and Interface Science no

M Mihalkovic I AlLehyani E Co ckayne CL Henley N Moghadam JA Moriarty

Y Wang and M Widom Total energybased prediction of a quasicrystal structure

Physical Review B no

M Mihalkovic H Elhor and JB Suck Lowenergy phonon excitations in the decago

nal quasicrystal Materials Science and Engineering

JA Moriarty and M Widom Firstprinciples interatomic potentials for transition

metal aluminides Theory and trends across the d series Physical Review B

VM Muller VS Yushchenko and BV Derjaguin On the inuence of molecular

forces on the deformation of an elastic sphere and its sticking to a rigid plane Journal

of Colloid and Interface Science no

JY Park DF Ogletree M Salmeron CJ Jenks and PA Thiel Friction and

adhesion properties of clean and oxidized AlNiCo decagonal quasicrystals a UHV

atomic force microscopyscanning tunneling microscopy study Trib ology Letters

JY Park DF Ogletree M Salmeron CJ Jenks PA Thiel J Brenner and JM

Dub ois Friction anisotropy A unique and intrinsic property of decagonal quasicrys

tals Journal of Materials Research

JY Park DF Ogletree M Salmeron RA Rib eiro PC Caneld CJ Jenks and

PA Thiel High frictional anisotropy of periodic and aperiodic directions on a qua

sicrystal surface Science

Tribological properties of quasicrystals Eect of aperiodic versus periodic

surface order Physical Review B no

JY Park DF Ogletree M Salmeron RA Ribiero PC Caneld CJ Jenks and

PA Thiel Atomic scale coexistence of periodic and quasiperiodic order in a fold Al

NiCo decagonal quasicrystal surface Physical Review B no R

James C Phillips Rosemary Braun Wei Wang James Gumbart Emad Ta jkhorshid

Elizab eth Villa Christophe Chip ot Rob ert D skeel Laxmikant Kale and Klaus

Schulten Scalable molecular dynamics with NAMD Journal of Computational Chem

istry

S Plimpton Fast paral lel algorithms for shortrange molecular dynamics Journal of

Computational Physics

K Pussi N Ferralis M Mihalcovic M Widom S Curtarolo M Gierer CJ Jenks

P Caneld IR Fisher and RD Diehl Use of periodic approximants in a dynamical

LEED study of the quasicrystal line tenfold surface of decagonal AlNiCo Physical

Review B

M Quilichini and T Janssen Phonon excitations in quasicrystals Rev Mo d Phys

DC Rapap ort The Art of Molecular Dynamics Simulation vol Cambridge Uni

versity Press

D Shechtman I Blech D Gratias and JW Cahn Metal lic phase with longrange

orientational order and no translational symmetry Physical Review Letters

WeiMei Shyu and GD Gaspari Sound velocity in metals Physical Review

W Smith and TR Forester DL POLY A generalpurpose paral lel molecular

dynamics simulation package Journal of Molecular Graphics

DJ Sordelet MF Besser and JL Logsdon Abrasive wear behavior of AlCuFe

quasicrystal line composite coatings Materials Science and Engineering A

Walter Steurer Twenty years of structure research on quasicrystals part I pentagonal

octagonal decagonal and dodecagonal quasicrystals Z Kristallogr

D Tab or Surface forces and surface interactions Journal of Colloid and Interface

Science no

PA Thiel Quasicrystal surfaces Annual Review of Physical Chemistry

M Widom I AlLehyani and JA Moriarty Firstprinciples interatomic potentials

for transitionmetal aluminides III extension to ternary phase diagrams Physical

Review B no

M Widom and JA Moriarty Firstprinciples interatomic potentials for transition

metal aluminides II application to AlCo and AlNi phase diagrams Physical Review

B no

APPENDICES

App endix A LAMMPS Workarounds

This chapter is concerned with some of the technical issues and corresp onding workarounds

implemented with the LAMMPS co de Only one bug in the co de was found but there are

numerous features that are not well do cumented in the LAMMPS Users Manual

A Obtaining Forces on FixedRigid Atoms

The pro cedure used to p erform the friction exp eriments requires xing the forces on

the topmost layers of the tip and the b ottommost layers of the approximant to zero in all

three directions To obtain the normal and frictional force data we had to know what the

forces acting on the xed tip atoms would have b een had the forces not b een set to zero

At rst we were only aware of obtaining force information using the dump command

When the forces on the xed atoms were printed using the dump command LAMMPS

printed the xed force and thus we did not know how to obtain the required information

How to obtain the forces calculated for a group of atoms b efore a fix setforce com

mand was applied was not well do cumented The older versions of the Users Manual did

not in the fix setforce command description sp ecify a way to do this An email was

sent to Steve Plimpton the author of most of the LAMMPS package and he informed us

that the values calculated for a group of atoms b efore a x is applied can b e printed using

style command The syntax is the thermo

thermostyle custom step temp f f f

The ab ove command changes the thermo dynamic output to print timestep temp era

ture the x value from x the y value from x and the z value from x where x

would b e the x setforce command used to set the forces initially to zero

In the April Users Manual it is stated in the do cumentation for the fix setforce

command that these values can b e accessed by various output commands and the reader

style command where the syntax ab ove is do cumented is inevitably led to the thermo

App endix A Continued

A Bugs Noted With the LAMMPS Splining Routine

The WidomMoriarty pair p otentials came in a table format which listed in columns

the distance b etween two atom typ es followed by the p otential at that distance LAMMPS

is set up to handle this style of p otential by using various metho ds to interp olate b etween

the p oints given by the table The cubic spline interp olation was chosen b ecause it provides

more accurate results than a straight linear interp olation The syntax of the command is

pairstyle table spline N

where N is the numb er of values in the table By implementing the p otentials in this

manner LAMMPS would eventually incur an error saying that two atoms were closer than

the inner table cuto which in our case is less than A This would mean that the forces

felt by these two atoms is almost innite For the atoms to b e allowed to venture that

close to each other was highly unlikely

After lo oking through the LAMMPS sourceco de the problem was narrowed down to

the splining routine Keith McLaughlin investigated the routine that reads the tabulated

p otentials and found that not only do es it fail using a spline interp olation but it also fails

using a linear interp olation scheme The problem is asso ciated with the way LAMMPS

stores and evaluates the tabulated information The current workaround for this issue is to

not have N b e the numb er of entries in the table but rather a very large numb er N

was used in this research This seems to work for b oth splining and linear interp olation

A Using a Triclinic Box in a Data File

The version of the LAMMPS Users Manual available during the start of this research

did not do cument how to sp ecify a triclinic simulation b ox using a separate data le

data command This feature was necessary for the unit cell b eing used and the read

At rst we thought that the co de would need to b e mo died however eventually the

App endix A Continued

appropriate command was found in the source co de although no do cumentation of it

existed in the manual The April version do es include this do cumentation so it will

not b e elab orated up on here except to refer the reader to App endix C

App endix B Calculating an Appropriate Timestep

The timestep chosen for a moleculardynamics simulation is imp ortant The timestep

needs to b e small enough to capture the highestfrequency vibrations in the system but as

large as p ossible to save on computation time From watching videos of the simulations

in this work one can clearly see phonons propagating in the adamant tip From the data

les the sp eed of these phonons was calculated to b e Aps Using the sp eed of these

phonons along with the length of one unit cell of the adamant tip a A we were able

to roughly estimate or the frequency of the phonons in recipro cal space

According to Ashcroft and Mermin phonons propagating along a chain of atoms

have a velo city c calculated as

where is the frequency and k the wave vector Using k a we calculate a frequency

1 1

of ps giving a time of ps This calculation was not intended to b e

accurate or precise it served only as a starting p oint for testing various timesteps through

simulation

We can compare this to the sp eed of sound in Al which is given as Aps in the

direction Aps in the direction and Aps in the direction

by Shyu et al Taking the highest value of Aps would give a frequency of

1 1

ps giving a time of ps

Al Al

Now that we have a starting p oint we do not want our timestep to exceed one one

hundredth of or roughly ps This also keeps our timestep small enough to catch the

acoustic phonon vibrations in pure aluminum keeping in mind that our approximant is

at aluminum We use the ps estimate as our maximum timestep b ecause only the

phonons in the adamant tip were taken into account during the calculation Phonons in

the approximant are not visible and would require a much more sophisticated analysis as

discussed in x The shortest unitcell length in the approximant is just over A which

App endix B Continued

is shorter than the adamant unit cell length A shorter unit cell vector would lead to a

higher frequency and thus a smaller time

To actually cho ose the appropriate timestep friction simulations using timesteps in

cluding ps ps ps and ps were p erformed After each simulation

was run the average lateral and frictional forces were calculated and compared It was

found that increasing the timestep from to ps did not change the results At

ps the results b egan to change slightly so the ps timestep was chosen to p erform

all of the nal simulations

App endix C Required Files

The goal of running these simulations is to study any frictional anisotropy b etween

sliding in the p erio dic vs ap erio dic directions on the fold face of a decagonal qua

sicrystal Due to the inability to simulate the innite unit cell of a real quasicrystal a

quasicrystalline approximant must b e used see x We have fashioned a simulated AFM

tip using adamant a ctitious material that b ehaves as a very hard aluminum and has

only repulsive interactions with the approximant see x

To p erform the exp eriments the tip is placed ab ove the approximant and the tip

and approximant are allowed to relax to a temp erature of less than K After the initial

relaxation the b ottommost layers of the quasicrystal are held xed in space as the tip

is given a downward velo city to come in contact with the quasicrystal This allows us to

achieve a compression Once the system is compressed the top few layers of the tip are

also xed in space allowing us to keep a constant height during sliding Once we have

xed our height the system is allowed to relax once more b efore the xed tip atoms are

given a constant sliding velo city parallel to the tipquasicrystal interface

The sliding p ortion of the simulation is where the friction data are gathered For each

of the p erio dic and approximated ap erio dic sliding directions one needs to obtain normal

force and lateralforce data for multiple compressions This allows us to plot a graph of

normal force vs lateral force and following Amontons Law the slop e of this line is

the friction co ecient

A simulation is needed for each compression and in each of the two sliding directions

Every simulation b egins with a data le containing the simulationb ox b oundaries masses

and initial p ositions of all atoms a p otential le containing all of the pairp otential infor

mation an inputparameters le that has all of the information for running the simulation

and nally if running in parallel a submission script for CIRCE the computing cluster

maintained by Research Computing at the University of South Florida

App endix C Continued

C Data File

All LAMMPS data les are ASCI I text les read by the LAMMPS program when told

to do so in the parameters le The rst line is ignored so it is a free comment line Any

other comment lines have to b egin with A small p ortion of a data le used in this

research is shown b elow

Atomic coordinate file for a qc

atoms

atom types

xlo xhi

ylo yhi

zlo zhi

xy xz yz

Al Co Ni Ad

Masses

Atoms

The blank lines as seen ab ove are required The numb er of atoms in the le is sp ecied

rst by atoms If this do es not match the numb er of atom entries there will b e an

error message Next the numb er of dierent typ es of atoms is given as atom typ es

App endix C Continued

The atom typ e is not restricted by element it is used as a lab el only Dierent atom typ es

can have the same mass

The next few lines after the numb er of atom typ es give the simulation b ox b ound

aries These are the b oundary conditions Whether or not one is using p erio dic b oundary

conditions one has to sp ecify the b ounds of the simulation b ox It is p ossible to use an

innite simulation b ox essentially creating a sample in an innite vacuum by typing INF

but this is inappropriate for our purp oses Due to the unique shap e of our approximant

unit cell the p erio dic b oundary conditions are integer multiples of the approximant unit

cell vectors rather than the tip unit cell which is FCC see x

The syntax b egins with two numb ers which sp ecify the minimum and maximum co

ordinate values for that direction The last line gives the skewing There are skewing

parameters that can b e sp ecied xy will shift the upp er y face the xz plane that is

highest on the yaxis in the x direction xz will shift the upp er z face the xy plane that

is highest on the zaxis in the x direction and yz will shift the upp er z face in the y

direction all in A The LAMMPS do cumentation for this is highly insucient When one

initially inputs the b ox b oundaries LAMMPS assumes a rectilinear b ox which will then

need to b e skewed to obtain other shap es see xA The origin for the simulation b ox is

at For the example given ab ove the unit cell vectors are shown in Table C

Vector X A Y A Z A

Table C These are the vectors describing the simulation b ox b oundaries for the example

data le shown in App endix C

The Masses keyword lists the masses for all typ es of atoms in atomic mass units Each

individual atom is listed as b eing of a certain typ e rather than a certain mass not only to

save typing but for another way to group atoms in the parameters le There will b e an

App endix C Continued

error if a mass is not sp ecied for each typ e but more than one typ e may have the same

mass Also one should note that LAMMPS fail with an will error if there is a typ e atom

without a typ e this inconvenient feature will thus not allow atoms to b e given typ es

matching their atomic numb ers

The bulk of the data le comes after the Atoms keyword At a minimum the initial

p osition and typ e of every atom in the simulation must b e sp ecied here Each atom is also

given a unique integer identier this feature is utilized in the parameters le in App endix

C for grouping atoms together The format used b egins with the unique atom ID the

atom typ e xp osition yp osition and ends with zp osition The typ e of data included in

the data le is determined in the parameters le by cho osing the atom style This work

was all done using the most basic style atomic

All of the numerical values in the data le will have the same units that are sp ecied

in the parameters le by the units command

C Potential File

All of the simulations in this research used the tabulated WidomMoriarty pair p oten

tials LAMMPS is built to handle tabulated pair p otentials in the following format Note

the problem in using the LAMMPS splining routine as do cumented in xA

AlAl

potslong which is used There can b e multiple tables in one le as we have done in all

for the preliminary results and can b e found in homestudentsharperqcresearchLAMMPS

App endix C Continued

on Physics The nal results use the cuto smo othed p otentials which can b e found in

homestudentsharperqcresearchLAMMPSall potssmoothed on Physics Each ta

ble in the le b egins with a unique name This is the name used in the parameters le to

sp ecify the p otential b etween two typ es of atoms After the name such as AlAl ab ove

the keyword N is followed by a numb er tells LAMMPS how many entries are in the table

The LAMMPS format demands that there b e a blank line b etween the header and the bulk

of the table Each entry in the table has a unique integer ID in the rst column followed by

the distance b etween the atoms in A the p otential at that distance in eV and the force at

that distance in eVA The original tabulated p otentials did not include the force values so

forces script found in homestudentsharperbin they were calculated using the get

on Physics The script p erforms a rough weighted derivative

C Simulation Parameters File

The ordering of commands in a parameters le is very imp ortant 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 shown b elow I will go through each command and briey describ e its

function and purp ose for this research

an Ad tip compressing and sliding on a QC surface

log log

dimension

boundary p p p

units metal

atomstyle atomic

readdata data

LJ potentials

pairstyle table spline

paircoeff allpotslong AlAl

App endix C Continued

paircoeff allpotslong AlCo

paircoeff allpotslong AlNi

paircoeff allpotslong CoCo

paircoeff allpotslong CoNi

paircoeff allpotslong NiNi

paircoeff allpotslong AdAd

paircoeff allpotslong AdX

group qcfix id

group qctemp id

group qcfree id

group adfree id

group adtemp id

group adfix id

group temps union qctemp adtemp

fix all nve

fix all langevin

timestep

thermo

dump all custom out tag type x y z

run

fix adfix setforce

fix qcfix setforce

velocity qcfix set units box sum no

unfix

fix temps langevin axes

thermostyle custom step temp f f f

velocity adfix set units box sum no

run

velocity adfix set units box sum no

run

velocity adfix set units box sum no

run

App endix C Continued

All lines b eginning with are comments and ignored by LAMMPS For the param

eters le there are no blankline requirements they are just ignored Every line b egins

with a command on the left followed by the parameters used for that command For a

more detailed description of the commands see the LAMMPS Users Manual The

tabb ed spaces b etween the commands and the command parameters are strictly for ease

of visualization

log log The log command is where you sp ecify the name of the le to which the

standard output also called the thermo dynamic output will b e written The syntax is

command then lename The data gathered from the friction exp eriments are taken from

these log les so each simulation needs to have a unique log le

dimension The dimension command allows you to set the dimensionality of the

simulation it must match the dimensionality implied in the data le

boundary p p p LAMMPS can handle many dierent forms of b oundary conditions

In our research we use p erio dic b oundary conditions in all dimensions Perio dic b oundary

conditions are also required when skewing a simulation b ox The syntax is command x

dir b oundary condition ydir b oundary condition and zdir b oundary condition where p

stands for p erio dic

units metal This sp ecies the units used in all les aliated with the simulation

The simulations run for this research used what LAMMPS calls metal units This style of

units sp ecies all quantities in A ps eV and Kelvin The forces are given in by LAMMPS

eVA but are later converted by the author to nN

atomstyle atomic This sp ecies the format in which the atoms are entered in the

data le Atomic is the most basic style which stores atom IDs typ es co ordinates and

initial velo cities if needed For a more detailed description of this le App endix C

App endix C Continued

readdata data This is when you tell LAMMPS to read your data le and what

the name of the data le is LAMMPS go es through data the le linebyline and stores

the information for later use The syntax is the command followed by the lename

pairstyle table spline This command tells LAMMPS what kind of pair

p otentials are b eing used The syntax is command format style and how many lo okup

values to use Because our p otentials are tabulated it is b est to use a splining routine to

calculate a value b etween two p oints but LAMMPS has a bug in the interp olation This

is discussed in App endix A and is the reason for using p oints Keith McLaughlin is

resp onsible for this workaround

paircoeff allpotslong AlAl For each pair of atom typ es you must sp ecify

where to nd the p otential table The syntax is command typ e of one atom typ e of the

other atom le containing the tabled p otential and the name of the table The example

shown says that the interaction b etween two typ e atoms is the table titled AlAl in le

potslong The order of the atom typ es do es not matter here for example pair coeff all

is equivalent to pair coeff

group qcfix id Group commands are very useful they allow you to

control a large group of atoms using the group name rather than typing out each atom

individually Because of the manner of the data les used in this work I have sp ecied most

groups by their atom IDs The syntax is command group name how you are identifying

the atoms to b e put into the group and the atoms you cho ose The and are

inclusive so in the example ab ove atom atom atom atom and atom

are all in the group called qcfix See the LAMMPS Users Manual for more ways to

sp ecify atoms by ID

group temps union qctemp adtemp Sometimes it is convenient to sp ecify a group as

a union b etween two existing groups as shown here The syntax is command group name

how you are identifying the atoms in this case a union b etween two groups followed by

App endix C Continued

the names of the groups There are numerous ways for sp ecifying groups and they can b e

found in the LAMMPS Users Manual

fix all nve A x is just that it xes something in the simulation Here we have

xed the simulation to run at constant NVE numb er of atoms volume and energy Each

x must also have a unique name or numb er asso ciated with it The syntax is command

name what atoms style of x The style of the x will determine if there are any extra

arguments that need to b e set In this example there are no extra arguments The group

that this x mo dies is all This is a predened group that includes all atoms in the

simulation

fix all langevin This sets the Langevin thermostat to

bring all atoms in the simulation from a starting temp erature of K to a nal tem

p erature of K using a ps damping parameter and b eginning with a random seed of

The syntax is command name what atoms style of x starting temp erature

ending temp erature damping parameter and a random seed Because the starting and

ending temp eratures are sp ecied here this command can b e used to heat up or co ol down

a simulation rather than trying to keep it at a constant temp erature as is done here

timestep Since we are using metallic units time is sp ecied in picoseconds

This command tells LAMMPS to use a ps timestep This was the timestep used in

the preliminary results however it was found later that a much larger timestep ps

was sucient Using a smaller timestep will increase the runtime of the simulation so one

should use the largest timestep p ossible The metho d for nding an appropriate timestep

is discussed in App endix B

thermo This command tells LAMMPS to dump the thermo dynamic output

every timesteps It is written to standard out and recorded in the log le Until

style command is used it will output the default data timestep temp erature the thermo

average energy p er pair average energy p er mol the total energy and the pressure

App endix C Continued

dump all custom out tag type x y z The dump command is cur

rently b eing used to generate movies All of the frictionalforce data are taken from the

thermo dynamic output in the log les There is ab out a decrease in computing time

when the dump command is not used but it can show some useful data The syntax is

command name what atoms custom style dump once after every timesteps dump

to what le and then the list of information that one wants in the dump le In the exam

ple shown every timesteps the atom ID typ e x y and z co ordinates for every atom

are printed The dump les made for the most recent simulations which run much longer

than the preliminary simulations can quickly exceed gigabyte in size Due to memory

limitations only a few compressions were visualized The LAMMPS visualization program

xmovie or VMD can visualize these dump les

Now the initial preparation is over and the simulation can start running The pro ce

dure b egins with an initial relaxation p erio d in which the system is brought down to K

temp erature using the Langevin thermostat on all atoms Up until this p oint LAMMPS

is just storing information The run command tells LAMMPS how many timesteps to

integrate using the conditions sp ecied so far

run Run the simulation at the sp ecied conditions for timesteps

The simulation has now b een run for timesteps for the sole purp ose of relaxation

Now we have to b egin the compression stage To achieve a compression we need to x the

topmost layers of the tip and the b ottommost layers of the quasicrystal so that they are

rigid This is achieved by articially requiring the forces on these atoms to b e

fix adfix setforce This sets the forces on the adfix group of

atoms to eVA in the x y and z directions The adx atoms are the topmost layers of

the tip and at the end of every timestep the forces on them will b e replaced with eVA

so that they are rigid

App endix C Continued

fix qcfix setforce This sets the forces on the qcx layers of atoms

to eVA in all directions as well These are the b ottommost layers of the approximant

and they are now rigid

Just b ecause the forces are set to zero if the now xed atoms had a velo city they will

keep it b ecause there are no forces acting on them to alter that velo city This means that

for the xed quasicrystal atoms which we do not want to move in space their velo cities

must b e set to eVA The xed tip atoms are the atoms that we will give a velo city

toward the quasicrystal in the negative y direction to create our compression

velocity qcfix set units box sum no This sets the velo city of the

qcx atoms to Aps in the x y and z directions If the optional arguments of units

box sum no are not included LAMMPS will fail saying that we havent used the lattice

command so it cannot set the velo cities The argument units box means that the velo city

we are sp ecifying will b e in Aps if we were to cho ose units lattice a value of would

mean that the atoms would have a velo city of unit cells p er picosecond The command

sum no means that the velo city will not b e added to the velo city from the previous timestep

it will replace it This keeps the velo city constant

unfix Since we no longer want the thermostat acting on the xed atoms or the

free interface atoms we have to take away our previous x that put the thermostat on all

atoms The syntax is unfix then the name of the x you want to get rid of

Now that we have set our rigid atoms and gotten rid of the original thermostat we

have to implement the thermostat on the thermostat atoms only These are the atoms in

the centermost layers of b oth the tip and the approximant Implementing the thermostat

takes a little bit of thought b ecause we do not want the thermostat interfering with the

force data We are compressing in the y direction and sliding in either the x or z directions

This means that for sliding in the x direction we only want to allow the thermostat to

mo dify the velo cities of the thermostat atoms in the z direction Subsequently when

App endix C Continued

sliding in the z direction we will only allow the thermostat to mo dify the velo cities of the

thermostat atoms in the x direction Originally the thermostat was not allowed to aect

the compression direction but the resulting data were very noisy This is easily seen in

the preliminary results in x Allowing the thermostat to aect the compression direction

negated a lot of the noise This is the same pro cedure used by Dr Sinnotts group at the

University of Florida and is b eing used in the most current results as seen in x

fix temps langevin axes This command xes the temps

group to have a Langevin thermostat just as b efore The axes argument allows one to sp ec

ify which axes will b e utilized by the thermostat A means that axis will b e utilized and

a means that it will b e ignored

Up until this p oint we have not printed out any of the force values used for the data

analysis As the tip is sliding across the quasicrystal the normal and frictional forces need

to b e 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 we have to retrieve the forces that would have acted on the xed tip atoms

if the x had not b een there During sliding at a constant velo city we record the forces

that opp ose sliding as the frictional forces Because the forces were set to eVA the

thermo dynamic output information can b e mo died to print the forces that would have

aected those atoms had the force not b een set to eVA This is describ ed further in

App endix A

style com thermostyle custom step temp f f f The thermo

mand changes the style of the thermo dynamic output that is printed to standard out By

using the custom command we sp ecify each column of the output individually Here we

print the timestep temp erature and then the values from x b efore the x is applied

The axes option in the x langevin command is no longer available in the May LAMMPS

distribution The January version was used for this research The option of including or deleting

an axis for consideration is now p erformed through a compute command For a more detailed description

of this command see the current LAMMPS Users Manual

App endix C Continued

Fix is where we set the forces on the tip atoms to b e zero in all dimensions so the

thermo dynamic output prints the forces that would have acted on those atoms in all

dimensions b efore the x is applied This is the data that we use for the analysis

Now that we are getting the output that we want we b egin the compression stage by

giving the xed tip atoms a velo city in the negative y direction toward the quasicrystal

This is the rst time that we set the velo city of the rigid tip atoms To change the normal

force only the compression velo city needs to b e changed The runtime could b e improved

by increasing the velo city and decreasing the numb er of timesteps but this leads to a need

for a longer relaxation time The induced velo city of the xed tip atoms creates phonons in

the tip as the compression is o ccurring Before sliding b egins we want the vibrations in the

compression direction to b e as small as p ossible By compressing slowly we minimize the

eect of the compression phonons so that a shorter relaxation time b etween compression

and sliding is required The syntax of the commands b elow is the same as the previous

velo city and run commands

velocity adfix set units box sum no

run

After compression the tip is given a Aps velo city in all directions to halt the

compression and the system is allowed to relax again

velocity adfix set units box sum no

run

At this p oint we have our desired compression and the sliding can b egin This section

of the log le is where all of the data analysis is done

velocity adfix set units box sum no

run

App endix C Continued

C CIRCE Submission Script

All of the preliminary results were run on the CIRCE cluster maintained by Research

Computing University of South Florida Each simulation needs a submission script that

requests the numb er of pro cessors an estimated run time and the actual command to run

the simulation A sample CIRCE submission script is shown b elow

binsh

start in the current directory

cwd

do not merge stderr into stdout

j n

M harperphysicscasusfedu

notify

name of the job

N compxshort

m abe

use the Bourne shell

S binsh

parallel environment and number of processors

pe ompi

l hrt

PATHPATH

sgempirun lmpcrockett inputparameterfile

The template for the submission script was given by Dr David Rabson The variable

parts of the script are as follows

M harperphysicscasusfedu This tells CIRCE where to send emails when a

job is started nished and ab orted

N compxshort The string following N is the name given to the simulation used

when you check the status of the simulation and when CIRCE sends email up dates on the

job

App endix C Continued

pe ompi This is the pro cessor request All of the simulations for the preliminary

results were run on pro cessors p er simulation subsequent work was mostly run using

pro cessors

l hrt As of April this is a new addition to the CIRCE submission

scripts It is an estimate of the wallclo ck time that the job will take If this statement is

not in the script CIRCE will kill the job after minutes The job will b e ab orted if it

is not nished in the allotted time In more current work we have decreased the runtime

limit signicantly

sgempirun lmpcrockett inputparameterfile This is the command that

starts the simulation When LAMMPS was installed on CIRCE we installed the crockett

MPI version The lename of the input parameters le describ ed ab ove should replace

inputparameterfile

App endix D Original Atom Unit Cell

Presented in Table D are the atomic co ordinates for the original atom unit cell

supplied by Marek Mihalkovic The unit cell consists of two surface bilayers sandwiching

a bulk bilayer The preliminary results were p erformed with the surface bilayers on either

side of a stack of bulk bilayers Unfortunately the surface bilayers are parallel to the

fold surface and not the fold surface of interest In the nal results the surface

bilayers were removed from the edges and only the bulk bilayer was rep eated

Table D These are the co ordinates for one unit cell of the

quasicrystalline approximant originally supplied by Marek

Mihalkovic

Element X A Y A Z A

continued on next page

App endix D Continued

Atomic co ordinates continued

continued on next page

App endix D Continued

Atomic co ordinates continued

continued on next page

App endix D Continued

Atomic co ordinates continued

Vector X A Y A Z A

Table D These are the unit cell vectors for the original quasicrystalline approximant

used in the preliminary results The structure of the approximant was supplied by Marek

Mihalkovic

App endix E Previous Publication

Before the research presented here was b egun the author contributed to work resulting

in a pap er titled Multiphase region of helimagnetic sup erlattices at low temp erature in

an extended sixstate clo ck mo del The work in said pap er was largely based on

Douglas Loveladys Masters Thesis however the author of this work made contributions

signicant enough to b e recorded as second author The details of the work will not b e

elab orated up on here only a short skeleton outline of the sp ecic contributions made will

b e mentioned

The original work fo cused on four or more magnetic layers sandwiched b etween non

magnetic spacers where the spins in neighb oring planes were rotated by This authors

task was to verify all calculations previously p erformed and to investigate one two and

three magnetic layers b etween nonmagnetic spacers It was found that one and two mag

netic layers were trivial The investigation of three magnetic layers showed it to b e a sp ecial

case that required the calculation of new matrices and equations It also added three new

p ossible excitation congurations to the lowtemp erature expansion