MULTISCALE MODELING OF CONTACT PLASTICITY AND

NANOINDENTATION IN NANOSTRUCTURED FCC METALS

A Dissertation Presented

by

Virginie Dupont

to

The Faculty of the Graduate College

of

The University of Vermont

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Specializing in Mechanical Engineering

October, 2008

Accepted by the Faculty of the Graduate College, The University of Vermont, in partial fulfillment of the requirements for the degree of Doctor of Philosophy, specializing in Mechanical Engineering.

Dissertation Examination Committee: Advisor Frédéric Sansoz, Ph.D. Dryver Huston, Ph.D. Yves Dubief, Ph.D. Chairperson Dennis Clougherty, Ph.D. Vice President for Research Frances E. Carr, Ph.D. and Dean of Graduate Studies Date: September 5th, 2008

ABSTRACT

Nanocrystalline thin films are materials with a grain size less than 100 nm which

are commonly used to fabricate microscale electro-mechanical devices. At such small scale, nanoindentation is the only standard experimental technique to study the mechanical properties of thin films. However, it is unclear if the continuum laws commonly used in nanoindentation analysis of polycrystalline materials are still valid for nano-grained metals. It is therefore critical to better understand the behavior of nanocrystalline materials under nanoscale contact. This dissertation summarizes the results of atomistic simulations aimed at modeling the nanoindentation of nanocrystalline metal thin films for which the grain size is smaller than the indenter diameter.

The nanoindentation of aluminum thin films was first studied using the Quasicontinuum method, which is a concurrent multiscale model where regions of small gradients of deformations are represented as a continuum medium by finite elements, and regions of high gradients of deformation are fully-treated atomistically. Two embedded-atom-method potentials for aluminum were used in order to study the effect of the potential on the nanoindentation behavior. The aim is to better understand the effects of a grain boundary network on the plasticity and the underlying mechanisms from an atomistic perspective. Our results show that a grain boundary network is the primary medium of plasticity at the nanoscale, via shear banding that causes flow serration. We also show that although the dislocation mechanisms are the same, the mechanisms involving grain boundaries are different depending on the interatomic potential.

In a second part, abnormal grain growth in aluminum thin films under nanoindentation is studied using both the Quasicontinuum method and parallel molecular dynamics simulations. The effects of the potential, the nature of the indenter and of its size on the grain growth under nanoindentation are investigated. Our results show that the potential used, which can be related to the purity of the material, can reduce grain growth. We also show that the size and material used for the indenter both have significant effects on grain growth. More specifically, grain growth under the indenter is found to occur via atom diffusion if the indenter is of the same material as the thin film.

Finally, the sample size effects were studied using parallel molecular dynamics simulations on nickel thin films and nanowires. Single crystals with different sizes are modeled in order to investigate the effects of the free boundaries as well as of the thickness of the samples. It is shown that the yield point and the incipient plasticity mechanisms are similar for all simulations. However, the hardness of the nanowires is found to decrease with the nanowire size during nanoindentation, due to the interaction of prismatic loops and dislocations with the free boundaries.

This dissertation has shed light on the plastic deformation mechanisms under nanoscale contact. The results obtained will help the scientific community gain a better understanding of the behavior of nanomaterials, which will lead to the fabrication of more reliable nanodevices.

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ACKNOWLEDGMENTS

I would like to thank my advisor, Frederic Sansoz, for allowing me to work

with him and for his guidance and support during my graduate career at the University of

Vermont. The subject we worked on was very interesting, and he continued to push me to

achieve my goals and challenged me by sharing our thoughts on the projects.

I would also like to thanks Bertrand Rollin, for helping me through this

endeavor, research-wise, physically and psychologically. He helped through the most

difficult parts of the PhD.

I would like to extend my thanks to Drs. Yves Dubief, Dryver Huston and

Dennis Clougherty for taking time out of their busy schedules and agreeing to be on my

committee.

Thanks to Steve Plimpton and Ellad Tadmor for the codes they developed and

shared, and for their support to all my questions (Tadmor; Plimpton). Thanks to Jim

Lawson for his support with the VACC.

Finally, thanks to my family back in France for their support during this

difficult period of my life. Thanks to Karen Bernard and Michelle Mayette, for their help

throughout my studies and their unwavering good mood. Thanks to Ida Russin and Mike

Cook for helping me the many times I went to see them at the Grad College. Finally,

thanks to all my friends: Lucie, Sophie, Aurélie L, Aurélie G, Cécile, Amélie, Marie-

Agnès, Montse, Xabier, Fahmi, Carl, Ana, Chris, Benji, Nirav, Ben and those I am

forgetting.

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TABLE OF CONTENTS

ACKNOWLEDGMENTS................................................................................................... ii

LIST OF TABLES............................................................................................................ vii

LIST OF FIGURES..........................................................................................................viii

CHAPTER 1: INTRODUCTION........................................................................................ 1

1.1. Motivations and Objectives ................................................................................ 1

1.2. State of Knowledge............................................................................................. 8

1.2.1. Contact Plasticity in Nanocrystalline Thin Films....................................... 8

1.2.2. Grain Growth Mechanisms at Atomic Scale ............................................ 12

1.2.2.1. Thermodynamically-activated Grain Growth............................. 12

1.2.2.2. Stress-assisted Grain Growth...................................................... 13

1.2.3. Size Effects in Nanosized Structures........................................................ 15

1.3. Plan of the Dissertation..................................................................................... 17

CHAPTER 2: NUMERICAL METHODS....................................................................... 18

2.1. The Quasicontinuum Method ........................................................................... 18

2.2. Parallel Molecular Dynamics............................................................................ 20

2.3. Modeling of Spherical/Cylindrical Contact ...................................................... 23

2.4. Interatomic Potentials ....................................................................................... 24

2.4.1. Embedded Atom Method Potentials ......................................................... 24

2.4.2. Tersoff Potential ....................................................................................... 25

2.4.3. Morse Potential ......................................................................................... 27

2.5. Calculation of Stresses...................................................................................... 27

iv

2.6. Voronoi Construction........................................................................................ 28

2.7. Tools for the Visualization of Defects, Dislocations and Grain Boundaries in

Atomistic Simulations....................................................................................................... 29

2.7.1. Centro-symmetry Parameter ..................................................................... 29

2.7.2. Ackland Parameter.................................................................................... 30

2.8. Validity of the 3D Models ................................................................................ 31

CHAPTER 3: EFFECTS OF A GRAIN BOUNDARY NETWORK ON INCIPIENT PLASTICITY DURING NANOSCALE CONTACT...................................................... 34

3.1. Objectives ......................................................................................................... 34

3.2. Model ................................................................................................................ 35

3.3. Characterization of the EAM Potentials ........................................................... 38

3.4. Force and Contact Pressure Calculations.......................................................... 40

3.5. Determination of the Yield Point...................................................................... 41

3.6. Shear Localization Mechanisms ....................................................................... 44

3.7. Effects of the Interatomic Potential on Plasticity in a 7 nm-grain-size Model. 46

3.7.1. Effects of EAM Potential on GB Structure and Energy at Equilibrium... 46

3.7.2. Mechanical Response Under Nanoscale Contact ..................................... 49

3.7.3. Incipient Mechanisms of Plasticity at Yield Point (δ < 20 Å).................. 53

3.7.4. GB-mediated Plasticity Mechanisms (δ > 20 Å)...................................... 55

3.8. Discussion ......................................................................................................... 58

3.8.1. Similarities Between Simulation and Experiments in Nanocrystalline

Metal Indentation .............................................................................................................. 58

v

3.8.2. Effects of Interatomic Potential on GB-mediated Plasticity..................... 60

3.8.3. Incidence of Impurity on Flow Stress and GB-mediated Plasticity in

Nanocrystalline Metals ..................................................................................................... 61

3.9. Conclusions....................................................................................................... 63

CHAPTER 4: FUNDAMENTAL MECHANISMS OF GRAIN GROWTH DURING THIN FILM NANOINDENTATION............................................................................... 64

4.1. Objectives ......................................................................................................... 64

4.2. Models............................................................................................................... 65

4.2.1. Quasicontinuum Model ............................................................................ 65

4.2.2. Molecular Dynamics Model ..................................................................... 65

4.3. Stress-assisted Grain Growth............................................................................ 67

4.3.1. Grain Growth at 0K .................................................................................. 67

4.3.2. Grain Growth at 300K .............................................................................. 71

4.4. Grain Growth under Spherical Contact............................................................. 75

4.5. Discussion ......................................................................................................... 80

4.5.1. Stress-assisted Grain Growth.................................................................... 80

4.5.2. Thermodynamically-activated Grain Growth........................................... 81

4.6. Conclusions....................................................................................................... 85

CHAPTER 5: STUDY OF SIZE EFFECTS ON SINGLE CRYSTAL PLASTICITY IN THIN FILMS AND NANOWIRES.................................................................................. 87

5.1. Objectives ......................................................................................................... 87

5.2. Model ................................................................................................................ 87

vi

5.3. Results............................................................................................................... 91

5.3.1. Elastic Behavior and Limit of Elasticity................................................... 91

5.3.2. Plastic Behavior in Thin Films ................................................................. 93

5.3.3. Plastic Behavior in Nanowires.................................................................. 96

5.4. Discussion ....................................................................................................... 101

5.5. Conclusions..................................................................................................... 104

CHAPTER 6: CONCLUSIONS......................................................................................106

REFERENCES................................................................................................................108

vii

LIST OF TABLES

Table 3.1: Stacking fault energy (γSF), unstable stacking fault energy (γUSF), unstable twinning fault energy (γUTF) and GB energy (γGB) for three Σ tilt symmetric grain boundaries calculated from quasicontinuum method on the two EAM Al potentials investigated, and reference values from first-principles simulations for pure Al and Al with impurities. All units of energy are in mJ/m². ............................................................ 40

Table 3.2: Constitutive parameters extracted from force-displacement nanoindentation curves obtained by quasicontinuum simulation in nanocrystalline Al with a mean grain size of 7 nm. The parameters δf and δmax represent the depth of the residual impression after unloading and the maximum penetration depth, respectively. ................................. 51

Table 4.1: Molecular simulations performed on an aluminum thin film. All the tips are rigid. .................................................................................................................................. 66

Table 5.1: Young’s modulus and mechanical characteristics at yield point in Ni thin films and nanowires from molecular dynamics simulations of spherical indentation. .............. 93

Table 5.2: Number of dislocations nucleated and absorbed by free surfaces for each nanowire............................................................................................................................ 99

viii

LIST OF FIGURES

Fig. 1.1: (a) Scanning Electron Microscope (SEM) picture of a made-to-measure nanoindentation probe for the group’s Atomic Force Microscope (AFM); (b) AFM picture of nanoindentation tests performed on a thin film. Pictures courtesy of Travis Gang, Helix 2008. ............................................................................................................... 2

Fig. 1.2: In situ nanoindentation experiments of Al thin film. (a) Load as a function of displacement. (b) Load as a function of displacement for the leading portion of the loading curve. (c) Initial image of the Al grain. Note that it is free of dislocations. (d) Pictures corresponding to event 1 on the loading curve. Note the appearance of dislocations in the grain, characterized by the darker color. (e) Pictures corresponding to event 2 on the loading curve. Note the appearance of more dislocations in the grain. Pictures courtesy of Minor et al. (Minor et al., 2006)......................................................... 3

Fig. 1.3: TEM observation during an in situ nanoindentation on nanocrystalline aluminium. (a) no grains in strong diffraction condition under the tip area indicated by the white arrow; (b) a grain with size about 10 nm has rotated into strong diffraction condition; (c) a group of grains in bright contrast; (d) the size of the group has become larger with increasing load. Picture courtesy of Jin et al. (Jin et al., 2004). ....................... 4

Fig. 1.4: A comparison of (a) number of fraction and (b) volume fraction grain size distribution after 30 min of indenter dwell time for indents made at room temperature and -190°C. The arrow in (a) indicates the presence of the large grains after 30 min dwell time at the low temperature. The presence of large grains is more evident in the volume distribution. Picture courtesy of Zhang et al. (Zhang et al., 2005b). .................................. 5

Fig. 1.5: Elastic moduli of ZnO nanowires ENW and GNW obtained from nanoindentation measurement and fitted as functions of the nanowire radius. Graph courtesy of Stan et al. (Stan et al., 2007). ............................................................................................................... 6

Fig. 1.6: A SEM image of a Ni 20-µm-diameter nanopillar after application of a 4% compression strain. The arrow points at the steps left on the surface by the escaping dislocations. Picture courtesy of Uchic et al. (Uchic et al., 2004). ..................................... 7

Fig. 1.7: Thermodynamically-activated grain growth processes. (a) and (b) curvature driven grain growth; (c) and (d) atom diffusion. .............................................................. 13

Fig. 1.8: Stress-assisted grain growth. (a) Original configuration; (b) grain rotation and coalescence or (c) shear-coupled motion. Grains boundaries are represented by thick continuous lines. Thin lines represent the crystal orientation........................................... 14

Fig. 2.1: Schematics of the Voronoi construction of a 2D model..................................... 29

ix

Fig. 2.2: Evolution of the contact pressure as a function of the indentation depth for both models. The arrows indicate the yield points.................................................................... 33

Fig. 2.3: Indentation step 250 ps after the yield point (a) for the 40 nm ä 12 nm ä 40 nm model and (b) the 60 nm ä 12 nm ä 60 nm model. The atoms in perfect FCC lattice have been removed. ................................................................................................................... 33

Fig. 3.1: Quasicontinuum model of a 7 nm-grain size Al thin film indented by a 15 nm radius cylindrical indenter. (a) Full view of both finite element domain and atomistic region. (b) Close-up view of full atomistic zone near the contact region in unrelaxed configuration. .................................................................................................................... 36

Fig. 3.2: Generalized stacking and planar fault energy curves obtained by quasicontinuum method with the Mishin-Farkas EAM potential for Al..................................................... 39

Fig. 3.3: Effect of nanosized grains on the nanoindentation response of Al substrates from molecular static simulation using the Al-VC potential. The indenter radius is 15 nm. Serrated plastic flow clearly appears in the two nanocrystalline Al substrates under nanoindentation................................................................................................................. 42

Fig. 3.4: Contact pressure versus penetration depth plot for a 7-nm polycrystal with the Al VC potential, along with the corresponding theoretical fitting.................................... 43

Fig. 3.5: Thin shear band formation in 5-nm-grain-size nanocrystalline Al after 2.5-nm-deep indentation. (a) Partial view of the contact interface and location of the grain cluster associated with the shear band. (b) Enlarged view of the shear plane. A mechanical twin nucleated at the triple junction in the prolongation of the shear place is clearly visible in grain 2. (c) Magnitude and direction of atomic displacements between two loading increments represented by arrows. The shear band results from sliding of aligned grain boundaries (grains 3 and 4) and intragranular partial slip (grain 2). ................................ 45

Fig. 3.6: Statistics of misorientation angle and GB structure in simulated nanocrystalline Al films after force relaxation as a function of interatomic potential. (a), (b) Distribution of misorientation angles (ψ + ψ’) between grains. (c), (d) Degree of symmetry of the GB structure from perfectly-symmetrical tilt GB (STGB, ψ – ψ’ ~ 0) to highly-asymmetrical tilt GB (ATGB, ψ – ψ’ ~ 180°). ....................................................................................... 48

Fig. 3.7: Atomic energies (in eV) calculated after relaxation of a cluster of 6 nano-grains in the contact zone. (a) Schematic representing the GB distribution and corresponding grain number as indicated in figure 2. (b) Voter-Chen EAM potential. (c) Mishin-Farkas EAM potential................................................................................................................... 49

Fig. 3.8: Contact pressure – displacement curves predicted by quasicontinuum simulation using Al-VC and Al-MF interatomic potentials. The curve for the Al-MF potential has been shifted to the right for clarity. The dashed curves correspond to the contact response

x

of an isotropic elastic surface deformed by a perfectly-rigid, wedge-like cylinder, obtained by continuum theory. ......................................................................................... 50

Fig. 3.9: Evolution of contact pressure as a function of penetration depth for shallow indentation using Al-VC potential in a 7 nm-grain-size simulation. Close-up views represent atomic details of deformation in the contact region. (a) Nucleation of the very first dislocation. (b) Nucleation and evolution of new dislocations. (c) Drop in curve corresponding to a sudden increase in contact area, just before the yield point. (d) Nanocrystal after the yield point....................................................................................... 52

Fig. 3.10: Evolution of contact pressure as a function of penetration depth using Al-MF potential, for shallow indentation in a 7 nm-grain-size simulation. Close-up views represent atomic details of deformation in the contact region. (a) Nucleation of the very first dislocation. (b) Nucleation and evolution of the structure. (c) Nanostructure before the yield point. (d) Nanostructure after the yield point..................................................... 53

Fig. 3.11: Effects of interatomic potential on the structure of a low-angle GB (ψ + ψ’ ~ 164°) before indentation (δ = 0 Å) and at final indentation (δ = 80 Å). (a) Al-VC potential; (b) Al-MF potential. Continuous lines indicate the [110] crystal lattice direction. ........................................................................................................................... 56

Fig. 3.12 : Effects of interatomic potential on the structure of a high-angle GB (ψ + ψ’ ~ 117°) before indentation (δ = 0 Å) and at mid indentation (δ = 40 Å). (a) Al-VC potential; (b) Al-MF potential. Continuous lines indicate the [110] crystal lattice direction........... 57

Fig. 4.1: Polycrystalline 3D model of an aluminum thin film deformed by a spherical tip of aluminum. The radius of the tip is 9 nm, and the mean grain size is 7 nm. The crystallographic orientations of the tip are as indicated. The model is made of 3 million atoms. ................................................................................................................................ 66

Fig. 4.2: Evolution of the GB structure at the interface between grains 3 and 5 for (a) Al-VC potential and (b) Al-MF. ............................................................................................ 68

Fig. 4.3: Atomic-level shear stress with respect to the orientation of the interface between grains 3 and 5 at different stages of the boundary motion (in units of GPa). The boundaries of grains 3 and 5 have been highlighted for clarity. ....................................... 69

Fig. 4.4: Evolution of the misorientation angle at the interface between grains 3 and 5. Open symbols are used when the GB migrates, while plain symbols indicate that the GB is not moving..................................................................................................................... 70

Fig. 4.5: Evolution of the microstructure of the sample indented by a 9 nm-radius Al indenter. General view (a) after relaxation and before indentation, ε = 0% and (b) at maximum indentation ε = 94%; (c) close-up view of grains 1, 2, 5, 6, 7 and 8 for ε = 0%; (d) close-up view of grains 1, 2, 5, 6, 7 and 8 for ε = 94%. The grain boundaries and

xi

lattice defects are colored in white, and the atoms in FCC lattice are colored according to the grain they belonged to at the beginning of the simulation.......................................... 72

Fig. 4.6: Evolution of the microstructure of grains 1, 5, 6, 7 and 8 from the samples indented by a 4, 9 and 15 nm-radius Al indenter under a 75% indentation strain. (a) With a 4 nm-radius Al indenter; (b) With a 9 nm-radius indenter; (c) With a 15 nm-radius indenter. The atoms belonging to grain boundaries or lattice defects are colored in white, and the other atoms are colored according to the grain they belonged to at the beginning of the simulation. .............................................................................................................. 74

Fig. 4.7: Close-up view of the region below the tip for the nanocrystalline thin films indented by (a) and (b) a 4 nm-radius Al tip, (c) and (d) a 9 nm-radius Al tip and (e) and (f) a 15 nm-radius Al tip. The left column represents the Ackland parameter, and the right column shows the same view with atoms in grain boundaries or lattice defects in white and other atoms colored according to the grain they belonged to at the beginning of the simulation.......................................................................................................................... 76

Fig. 4.8 : Microstructure of the thin films (a) at 0% indentation strain for the virtual tip; (b) the polycrystalline thin film indented by a 9 nm-radius virtual tip after a 75% indentation strain; (c) a polycrystalline thin film indented by a 9 nm-radius Al tip after a 75% indentation strain; (d) a single crystalline thin film indented by a 9 nm-radius Al tip after a 75% indentation strain. .......................................................................................... 77

Fig. 4.9: (a) Evolution of Ly and Lz for the simulation with the aluminum tip with a radius of 9 nm and (b) Evolution of the energies below the tip for the polycrystal indented by a 9 nm-radius Al and virtual tip, and the single crystal indented by a 9 nm-radius Al tip. .... 79

Fig. 4.10 : Atom diffusion processes. (a) Direct interchange; (b) Rotation; (c) By vacancy formation........................................................................................................................... 82

Fig. 4.11 : 7 nm-grain size Al thin film indented by a diamond tip under a 50% indentation strain............................................................................................................... 84

Fig. 5.1: Atomistic models of spherical indentation for thin films and nanowires. The radius of the virtual indenter is 18 nm. (a) 30 nm-thick film. (b) 30 nm-diameter nanowire. (c) 12 nm-thick film. (d) 12 nm-diameter nanowire. The crystallographic orientation of the films and nanowires is as indicated. Free surfaces appear in dark-gray, while atoms in FCC lattice are colored in light-gray........................................................ 89

Fig. 5.2: Simulated force-displacement nanoindentation curves for Ni thin films and nanowires indented by a spherical tip. Eq. (5.2) fitted to the 30 nm-thick film model is also represented................................................................................................................. 93

xii

Fig. 5.3: Evolution of the mean contact pressure as a function of penetration depth in (a) thin films and (b) nanowires. The numbers indicate the occurrence of dislocation absorption by free surfaces in the nanowires. ................................................................... 94

Fig. 5.4 : Atomic-level representation of plastic deformation in Ni thin films during spherical indentation. (a) 12 nm-thick film at yield point. (b) 12 nm-thick film at maximum indentation depth. (c) 30 nm-thick film at yield point. (d) 30 nm-thick film at maximum indentation depth. Dashed lines represent the trajectory of prismatic dislocation loops. X = ]111[ ; Y = ]211[ ; Z = ]101[ . ....................................................... 95

Fig. 5.5: Atomic-level representation of plastic deformation in a 12 nm-diameter Ni nanowire during spherical indentation. (a,b) Yield point. (c,d) Maximum indentation depth. X = ]111[ ; Y = ]211[ ; Z = ]101[ . .......................................................................... 97

Fig. 5.6: Atomic-level representation of plastic deformation in a 30 nm-diameter Ni nanowire during spherical indentation. (a,b) Yield point. (c,d) Maximum indentation depth. Dashed lines represent the trajectory of prismatic dislocation loops escaping at free surfaces. X = ]111[ ; Y = ]211[ ; Z = ]101[ . ...................................................................... 98

Fig. 5.7: Dislocation mechanisms in the nanowires. (a) Local von Mises strain at the yield point indicating a homogeneous nucleation of the first dislocation in the 12-nm nanowire; (b) Absorption of the first half-dislocation loop in the 12-nm nanowire; (c) Creation of the first four-sided prismatic loop in the 30-nm nanowire, and comparison with the size of the 12-nm nanowire. ................................................................................................... 100

1

CHAPTER 1: INTRODUCTION

1.1. Motivations and Objectives

Nanocrystalline materials are materials with a grain size less than 100 nm. They

are commonly used to fabricate microscale electro-mechanical systems (MEMS).

However, MEMS technology could be substantially improved if we could broaden our

knowledge about the mechanical behavior of nanomaterials. Indeed, it is now well-

established in FCC metals and alloys that a marked transition in plasticity mechanism,

accompanied by a continuous change in mechanical behavior, operates with a reduction

of grain size from the microcrystalline to the nanocrystalline regime (Schiotz and

Jacobsen, 2003; Trelewicz and Schuh, 2007; Chang and Chang, 2007). Several

experimental techniques (Hemker and Sharpe, 2007) have been used to study the

mechanical behavior of thin films, such as tensile testing (Bagdahn et al., 2003; Tsuchiya

et al., 1996; Sharpe et al., 1997; Chasiotis and Knauss, 2002; Chasiotis and Knauss, 2000;

Cho et al., 2005), or membrane deflection experiment (MDE) (Espinosa et al., 2003a;

Espinosa et al., 2003b). Small-scale contact experiments spanning from simple micro-

hardness testing to fully instrumented nanoindentation have been used for some time to

characterize the nature of yield phenomena and the influence of grain size on hardness

and strengthening in nanocrystalline metals (Nieman et al., 1989; Elsherik et al., 1992;

2

Fougere et al., 1995; Qin et al., 1995; Farhat et al., 1996; Sanders et al., 1997; Malow et

al., 1998). Nanoscale contact probes are particularly well-suited for the studies of the

plasticity transition in nanograined metals, because they can be highly sensitive to the

heterogeneous nature of plastic deformation in very confined volumes of materials, and

they provide quantitative insights into the mechanisms governing incipient plasticity at

reduced length scales. A picture of an indenter tip that was made-to-measure for the

group’s Atomic Force Microscope (AFM) is shown in Fig. 1.1a, and resulting indentation

tests performed on a nickel thin film are shown in Fig. 1.1b.

Fig. 1.1: (a) Scanning Electron Microscope (SEM) picture of a made-to-measure nanoindentation

probe for the group’s Atomic Force Microscope (AFM); (b) AFM picture of nanoindentation tests

performed on a thin film. Pictures courtesy of Travis Gang, Helix 2008.

3

Fig. 1.2: In situ nanoindentation experiments of Al thin film. (a) Load as a function of

displacement. (b) Load as a function of displacement for the leading portion of the loading curve.

(c) Initial image of the Al grain. Note that it is free of dislocations. (d) Pictures corresponding to

event 1 on the loading curve. Note the appearance of dislocations in the grain, characterized by

the darker color. (e) Pictures corresponding to event 2 on the loading curve. Note the appearance

of more dislocations in the grain. Pictures courtesy of Minor et al. (Minor et al., 2006).

Recent experiments on aluminum thin films have shown that the onset of

plasticity in nanomaterials is more complex than was previously thought. Minor et al.

(Minor et al., 2006) performed in situ transmission electron microscopy (TEM)

experiments of nanoindentation on a polycrystalline aluminum film. They show that

contrary to previous conclusions on the onset of plasticity in single crystals, dislocation

4

activity could occur before the yield point was attained in the film. In Fig. 1.2, they show

that the two very small events marked 1 and 2 correspond to the emission of dislocations,

as can be observed in Fig. 1.2d and e when the grain becomes darker. The nucleation of

the first dislocation in a single crystal is believed to correspond to the onset of plasticity,

but this experiment shows that polycrystalline materials behave differently. It is then

worth investigating what corresponds to the onset of plasticity for nanocrystalline

materials and what are the mechanisms associated with plasticity.

Fig. 1.3: TEM observation during an in situ nanoindentation on nanocrystalline aluminium: (a) no

grains in strong diffraction condition under the tip area indicated by the white arrow; (b) a grain

with size about 10 nm has rotated into strong diffraction condition; (c) a group of grains in bright

contrast; (d) the size of the group has become larger with increasing load. Picture courtesy of Jin

et al. (Jin et al., 2004).

5

Fig. 1.4: A comparison of (a) number of fraction and (b) volume fraction grain size distribution

after 30 min of indenter dwell time for indents made at room temperature and -190°C. The arrow

in (a) indicates the presence of the large grains after 30 min dwell time at the low temperature.

The presence of large grains is more evident in the volume distribution. Picture courtesy of Zhang

et al. (Zhang et al., 2005b).

Another interesting phenomenon that was observed during the nanoindentation

of nanocrystalline FCC metals is abnormal grain growth. It was observed in aluminum by

Jin et al. (Jin et al., 2004) during a TEM experiment of nanoindentation. Fig. 1.3 shows

the TEM images of several steps during nanoindentation of nanocrystalline aluminum.

The brightness is indicative of the orientation of the grains, so that Fig. 1.3 shows that a

bigger grain is forming under the indenter. The final size of the bright spot is five times

the original grain size after 3 seconds of indentation. Abnormal grain growth was also

observed by Zhang et al. (Zhang et al., 2005b) in nanocrystalline copper. They show that

grain growth is faster at cryogenic temperatures than at room temperature, and that the

6

purity of the material has an influence on grain growth. Their result in Fig. 1.4 shows that

for the same time under the indenter, the sample at cryogenic temperature forms grains

much bigger than the sample at room temperature, which is particularly striking in the

volume fraction plot. It is now well established that the strength of a material depends on

its grain size (Schiotz and Jacobsen, 2003; Hall, 1951; Petch, 1953), which renders the

process of grain growth undesirable in most nanotechnological applications. It is then

important to study this phenomenon in order to learn how to control it.

Fig. 1.5: Elastic moduli of ZnO nanowires ENW and GNW obtained from nanoindentation

measurement and fitted as functions of the nanowire radius. Graph courtesy of Stan et al. (Stan et

al., 2007).

The final interest of nanosized materials lies in the nanowires and nanopillars.

Those cylindrical structures have been shown to present amazing size effects under

microcompression, corresponding to an increase of the elastic moduli as their radius

7

decreases (Fig. 1.5) (Stan et al., 2007). The free boundaries introduced by the shape of

the cylinder allow dislocations to leave the volume, as demonstrated in Fig. 1.6 with the

lines left on the surface of the wire by the escaping dislocations (Uchic et al., 2004). This

process changes the properties known to apply to bulk materials. It is therefore crucial to

understand the mechanisms involved in the plasticity of nanosized metals.

Fig. 1.6: A SEM image of a Ni 20-µm-diameter nanopillar after application of a 4% compression

strain. The arrow points at the steps left on the surface by the escaping dislocations. Picture

courtesy of Uchic et al. (Uchic et al., 2004).

The objectives of this dissertation are three fold. The first objective is to

examine the onset of plasticity and the underlying mechanisms of plasticity when a

nanocrystalline grain boundary network is involved during contact. Second, the plasticity

phenomenon of grain growth is further studied. Finally, the size effects in nanowires are

analyzed, along with the plasticity mechanisms in such structures. Those studies will be

performed using atomistic simulations of nanoindentation.

8

1.2. State of Knowledge

1.2.1. Contact Plasticity in Nanocrystalline Thin Films

In past work, nanoscale contact experiments in nanocrystalline metals have

revealed two microstructure length scales producing different modes of plastic

deformation. First, particular focus has been placed on examining how dislocations

interact with surrounding grain boundaries (GBs) by performing nanoindentations at the

center of single nanograins, that is, by forcing the contact area to be much smaller than

the grain size. Yang and Vehoff (Yang and Vehoff, 2007) have observed that the

dislocations, which nucleate below the indenter, only interact directly with the

neighboring interfaces for grain sizes below 900 nm. At such scale, the point of elastic

instability is clearly defined by a “pop-in” event whose width is strongly correlated to the

size of the indented grain. The smaller the grain size, the smaller the pop-in width and the

harder the material. For grain sizes comparable to the contact area, however, Minor et al.

(Minor et al., 2006) have revealed using in-situ transmission electron microscopy (TEM)

nanoindentation that significant dislocation activity takes place in ultrafine-grained Al

thin films before the first obvious jump in displacement in the load-depth nanoindentation

curves. This result challenged the prevailing notion that the first pop-in event

corresponding to the onset of plasticity during nanoindentation occurs in a dislocation-

free crystal.

The second microstructure length scale, at which past nanoscale experiments

have been performed, corresponds to contact areas much larger than the mean grain size.

9

In this case, it is the collective deformation of the nanocrystalline GB network that

dominates the plastic behavior. All experimental evidence shows that the pile-up of

deformation left around residual impressions varies dramatically from homogeneous at

large grain size (> 20 nm) to inhomogeneous with intense plastic deformation in highly-

localized shear bands for very small grain sizes (< 20 nm) (Malow et al., 1998; Trelewicz

and Schuh, 2007; Andrievski et al., 2000; Van Vliet et al., 2003). Since shear band

propagation is routinely observed in bulk amorphous metals under nanoindentation (Kim

et al., 2002; Jana et al., 2004; Shi and Falk, 2005; Su and Anand, 2006; Antoniou et al.,

2007; Lund and Schuh, 2004), it was suggested that the plastic deformation of

nanocrystalline FCC metals with the finest grain size is comparable to the shear

localization processes observed in bulk metallic glasses (Hartford et al., 1998). This

assumption has recently been confirmed by the nanoindentation study of Trelewicz and

Schuh (Trelewicz and Schuh, 2007) in nanocrystalline Ni-W alloys.

Contact plasticity at the former length scale has been well-documented using

atomistic modeling, because the underlying mechanisms are clearly defined by slip

events and slip-GB interactions. A predictive understanding of the atomic mechanisms

leading to shear localization under an indenter at very small grain sizes, however, has

proved elusive, primarily for two reasons:

• It is commonly acknowledged that the GB networks play a critical role in the process

of plasticity in nanocrystalline metals. But GBs can also be involved in simultaneous,

yet different plasticity mechanisms at very small grain sizes. These mechanisms

include sources and sinks for lattice dislocations (Yamakov et al., 2002; Van

10

Swygenhoven et al., 2002; Tschopp and McDowell, 2008), GB sliding (Van Vliet et

al., 2003; Schiotz et al., 1998; Schiotz et al., 1999; Hasnaoui et al., 2002), grain

rotation-induced shear band formation and grain coalescence (Shimokawa et al.,

2006; Hasnaoui et al., 2002; Wei et al., 2002; Fan et al., 2006; Joshi and Ramesh,

2008; Wang et al., 2008; Yang et al., 2008; Haslam et al., 2001), and GB migration

coupled to shear deformation (Haslam et al., 2001; Gianola et al., 2008a; Gianola et

al., 2008b; Gianola et al., 2006a; Gianola et al., 2006b; Farkas et al., 2006; Gutkin et

al., 2008). From past nanoindentation experiments, there is no clear consensus

whether one or several of these atomic mechanisms truly dominate the onset of

plasticity for the finest grain size. In particular, stress-assisted grain growth under an

indenter has been observed at the onset of plasticity of nano-grained Al and Cu with

a mean grain size of 20 nm (Jin et al., 2004; Zhang et al., 2005b; Gai et al., 2007). In

a second group of nanoindentation experiments, however, neither grain growth nor

GB migration processes have been observed in nano-Cu and NiAl with a mean grain

size of 14 nm and 10 nm, respectively. Instead, it was concluded that the onset of

plasticity was related to the nucleation of lattice dislocations from GBs (Chen et al.,

2003a; Li and Ngan, 2005). Another study in plated Cu thin films with a 10 nm-grain

size reported some evidence of void formation at GBs and triple junctions as a

consequence of GB sliding during indentation (Chang and Chang, 2007).

• Earlier attempts made to model the nanoindentation of nanocrystalline metals by

atomistic simulations have employed a spherical repulsive force to model virtual tips

varying from 30 Å to 98 Å in diameter (Feichtinger et al., 2003; Lilleodden et al.,

11

2003; Ma and Yang, 2003; Jang and Farkas, 2004; Hasnaoui et al., 2004; Saraev and

Miller, 2005; Kim et al., 2006; Jang and Farkas, 2007). As such, contact areas were,

to a large extent, smaller than the grain size, and the plastic zone produced by these

tips was only limited to one or two grains. In contrast, some recent studies

(Szlufarska et al., 2005) have shown that it is critically important to simulate

nanoindentation tips with more realistic sizes, in order to put into perspective the

collective plastic processes of the GB networks in nanocrystalline materials.

Szulfarska et al. (Szlufarska et al., 2005) have simulated the nanoindentation of

normally-brittle nanocrystalline ceramics with a four to one ratio between tip

diameter and grain size, which revealed unusual GB-mediated plastic behavior.

Furthermore, past simulations have used different embedded-atom-method (EAM)

potentials, from which predictions of stacking fault energies can lead to strong

differences within the same metal (Zimmerman et al., 2000). The impact of the

interatomic potential on collective plastic processes, however, has never been fully

characterized.

12

1.2.2. Grain Growth Mechanisms at Atomic Scale

Intense grain refinement can promote drastic changes of plasticity mechanism in

bulk materials and thin films (Schiotz et al., 1998; Lu et al., 2000; Yamakov et al., 2002;

Chen et al., 2003b; Schiotz and Jacobsen, 2003; Kumar et al., 2003; Hasnaoui et al.,

2003; Huang et al., 2006). In metals containing nanosized grains (

13

thermodynamically driven mechanism is GB atom diffusion (Smigelskas and Kirkendall,

1947; Huntington and Seitz, 1942), during which the atoms jump in the crystal into point

vacancies, creating a new vacancy in the process (Fig. 1.7 (c) and (d)). The availability of

point vacancies follows an Arrhenius equation, so that the rate of atom diffusion

increases with temperature.

Fig. 1.7: Thermodynamically-activated grain growth processes. (a) and (b) curvature driven grain

growth; (c) and (d) atom diffusion.

1.2.2.2. Stress-assisted Grain Growth

The last two mechanisms are stress-assisted, and explain why grain growth is

still possible at cryogenic temperatures (Zhang et al., 2005b). The first one is rotation-

14

induced grain coalescence (Haslam et al., 2001), during which one grain rotates and its

orientation comes to match the orientation of a neighbor grain, thus forming a single

bigger grain (Fig. 1.8b). This process is often associated with GB sliding (Cahn et al.,

2006). The final mechanism of grain growth is called shear-coupled motion (Fig. 1.8c).

In this mechanism, the normal motion of grain boundaries results from a shear stress

applied tangentially to them and causing tangential motion, or coupled motion (Cahn and

Taylor, 2004; Suzuki and Mishin, 2005; Sansoz and Molinari, 2005). It was shown that

the GB structure greatly influences the behavior of the GB between the mechanisms of

grain sliding or shear-coupled motion (Sansoz and Molinari, 2005).

Fig. 1.8: Stress-assisted grain growth. (a) Original configuration; (b) grain rotation and

coalescence or (c) shear-coupled motion. Grains boundaries are represented by thick continuous

lines. Thin lines represent the crystal orientation.

15

The complexity and the variety of mechanisms of grain growth make it a

difficult phenomenon to study and to observe experimentally. Atomistic simulations offer

the advantage of investigating an atomistically the phenomenon given the atomic

potential as a single input. This method was shown to be very successful in characterizing

plastic deformation mechanisms in nanocrystalline metals as shown in Haslam et al.

(Haslam et al., 2001) and in Chapter 4 of this dissertation.

1.2.3. Size Effects in Nanosized Structures

One-dimensional metal nanowires (Tian et al., 2003) are the building blocks for

nanoscale research in a vast variety of disciplines that range from biology, to electro-

mechanics and photonics (Mock et al., 2002; Husain et al., 2003; Bauer et al., 2004;

Barrelet et al., 2004). These nanomaterials have recently stimulated the interest of the

mechanics community, because all experimental evidence shows a strong influence of the

sample dimension on the mechanical properties of metals at nanometer scale (Dimiduk et

al., 2005; Uchic et al., 2004; Greer and Nix, 2006; Greer et al., 2005; Wu et al., 2005;

Volkert and Lilleodden, 2006). While the strength and ductility of metals in macroscopic

samples are predominantly determined by the relevant microstructure length scale (e.g.

grain size), which is often small relative to the sample size, a distinctive behavior of

crystal plasticity emerges in metal nanostructures, where the material strength

significantly increases as the deformation length scale (diameter or volume) decreases. A

micro-plasticity mechanism has been proposed to account for the size scale dependence

of small metallic samples based on dislocation starvation, in which the density of mobile

16

dislocations created from pre-existing dislocation sources is counter-balanced by the

density of dislocations escaping the crystal at free surfaces (Greer et al., 2005; Greer and

Nix, 2006; Tang et al., 2007). The in-situ TEM compression experiments of Shan et al.

(Shan et al., 2008) have recently confirmed this mode of deformation in Ni nanopillars as

small as 150 nm in diameter. Nevertheless, it remains crucial to characterize the influence

of sample size on dislocation activity at even smaller length-scale (< 100 nm) in order to

achieve meaningful results in the crystal plasticity of metal nanowires.

Nanoindentation technique via pillar compression method (Dimiduk et al., 2005;

Uchic et al., 2004; Greer and Nix, 2006; Greer et al., 2005; Volkert and Lilleodden, 2006;

Shan et al., 2008) has enabled rapid progress in the experimental investigation of

nanomechanical properties and their size dependence in metals at the micron and

submicron scales. However, the nanopillar compression method has never been applied

to samples less than 100 nm in diameter due to complications in preparation and

mechanical testing at such a small scale. By contrast, the use of nanoindentation tips to

probe the radial elastic modulus and hardness of sub-100 nm nanowires has been found

very successful in the past (Stan et al., 2007; Lucas et al., 2008; Feng et al., 2006; Lee et

al., 2006; Li et al., 2003; Tao and Li, 2008; Zhang et al., 2008; Liang et al., 2005; Bansal

et al., 2005; Fang and Chang, 2004). While a fundamental understanding of dislocation

activity during metal nanopillar compression has already been supplemented by atomistic

simulations (Rabkin et al., 2007; Rabkin and Srolovitz, 2007; Afanasyev and Sansoz,

2007; Zhu et al., 2008; Cao and Ma, 2008), the atomic mechanisms of plasticity and

related size effects for metal nanowires deformed by nanoindentation remain elusive.

17

1.3. Plan of the Dissertation

The numerical methods used for this research, including both quasicontinuum

method and molecular dynamics as well as a description of the interatomic potentials, are

presented in Chapter 2. Chapter 3 shows the study on the effects of a grain boundary

network on the incipient plasticity of nanocrystalline Al deformed by a cylindrical

contact. The effects of different factors on grain growth under an indenter are

investigated in Chapter 4. Chapter 5 presents the study on the size effects on contact-

induced plasticity in Ni nanowires and thin films. Finally, the major conclusions of this

dissertation are summarized in Chapter 6.

18

CHAPTER 2: NUMERICAL METHODS

Two different molecular simulation techniques have been used for the

calculations: The Quasicontinuum Method (Miller and Tadmor, 2002), which is

multiscale atomistic/finite element simulation technique, and parallel three-dimensional

molecular dynamics simulations. Both methods and additional numerical tools used for

this study are presented in the following.

2.1. The Quasicontinuum Method

A complete description of the method can be found in the article written by the

developers of the Quasicontinuum (QC) method, Ronald E. Miller and E.B. Tadmor

(Miller and Tadmor, 2002).

The Quasicontinuum Method is a multiscale atomistic/finite element simulation

technique. It combines the advantages of continuum finite element simulation methods

with those of molecular dynamics. At the atomistic scale, finite element methods do not

represent accurately the behavior of a material because they are based on the hypothesis

that the material is a continuum, whereas it is actually made up of discrete particles. On

the other hand, molecular dynamics simulations allow a very accurate representation of

the material by representing all the atoms. But even with today’s supercomputers’

19

capacities, the number of atoms is limited and the simulated sample has small

dimensions. This is usually taken care of with periodic boundary conditions, but in the

case of nanoindentation, this implies that an infinite number of indenters indent the

surface at the same time. The Quasicontinuum Method combines finite elements where

deformations are small with an atomistic representation in high deformation areas. This

allows the user to model larger models with an accurate representation of the material’s

behavior where needed.

A typical mesh is constituted of atomistic zones (non local) and finite element

zones (local). The regions that sustain plastic deformations are modeled atomically,

whereas the rest of the mesh is modeled by finite elements. Each node in the model is

called “repatom” for representative atom. Each repatom can represent just itself (non

local zone as well as some atoms of the interface), or more than itself (local zone). The

total energy of the system is computed as follows:

exact

N

tot EEnErep

≈= ∑=1α

αα (2.1)

where Nrep is the total number of repatoms in the system, nα is the number of real atoms

the repatom is representing (nα = 1 for non local atoms), and Eα is the energy of each

repatom. This formulation allows having the same calculation on both local and non-local

regions, so that there is no discontinuity at the interface. The minimum energy is

calculated at each step using a conjugate gradient method then a new set of forces is

applied, the minimum energy found again and so on. The conjugate gradient method does

20

not take into account the effects of the temperature, so all calculations performed with

QC are performed at 0K.

The QC method can apply a “nonlocality criterion” to the model in order to

verify whether atoms should be local or non-local. A cutoff rnl is defined, and the applied

criterion will be:

ελλ

21

( ) ( )

ii

j kkji

jji

ii

vdtrd

rrrFrrFdtvd

m

rr

rrrrrr

=

++= ∑∑∑ ...,,, 32 (2.3)

where mi is the mass of atom i, irr and iv

r are its position and velocity vectors, F2 is a

force function describing pairwise interactions between atoms, F3 describes three-body

interactions, and many-body interactions can be added. The Verlet integration algorithm

was chosen in order to calculate the atoms’ positions.

The Verlet integration algorithm calculates the position of the atoms at the next

time step from the positions at the previous and current time steps, without using the

velocity. It is derived by writing two Taylor expansions of the position vector in different

time directions:

( )433

32

2

2)(

61)(

21)()()( tOtt

dt

rdtt

dt

rdtt

dtrd

trttr iiiii ∆+∆+∆+∆+=∆+rrr

rr (2.4)

( )4333

22

2)(

61)(

21)()()( tOtt

dtrdtt

dtrdtt

dtrdtrttr iiiii ∆+∆−∆+∆−=∆−

rrrrr (2.5)

When adding the two expansions, we have:

( )422

2)()(2)()( tOtt

dt

rdtrttrttr iiii ∆+∆+=∆−+∆+

rrrr (2.6)

or

2)()()(2)( ttattrtrttr iii ∆+∆−−=∆+rrrr (2.7)

with a(t) the acceleration.

22

This offers the advantage that the first and third-order terms from the Taylor expansion

cancel out, thus making the Verlet integrator more accurate than integrations by simple

Taylor expansion alone.

By subtracting (2.4) and (2.5), and reducing the precision, we have:

( )2)(2)()( tOttdtrd

ttrttr iii ∆+∆=∆−−∆+r

rr (2.8)

or, if we reorganize:

t

ttrttrt

dtrd iii

∆∆−−∆+

=2

)()()(

rrr

(2.9)

The first step has slightly different equations, as we cannot use )( tri ∆−r . Instead,

we use the initial conditions:

( )32

2)0(

21)0()0()( tO

dt

rddtrd

rtr iiii ∆+++≈∆rr

rr (2.10)

The calculations were performed using an NVT integration (constant number of

atoms, constant volume of the simulation box, and constant temperature). This was

achieved by using a Nose/Hoover temperature thermostat (Hoover, 1985). In this model,

the equations are given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=′

⋅−=′

′=′

1

)(

0

2TT

pFpmp

tr

T

ii

ii

νς

ς rrr

rr

(2.11)

23

with T0 the desired constant temperature of the simulation, νT the thermostating rate, and

ζ the thermodynamic friction coefficient. This constant dynamically modifies the

velocities of each atom such that the temperature of the model tends to T0.

The actual algorithm used in LAMMPS for solving the equations is described in

the paper written by the developer of the code, S. Plimpton (Plimpton, 1995).

2.3. Modeling of Spherical/Cylindrical Contact

Different types of contacts were investigated during this study. The tips used

with the Quasicontinuum Method were cylindrical, with their axis perpendicular to the

indentation direction, and rigid. They were made of aluminum, using the same potential

as for the thin film.

In molecular dynamics, the indenters used were spherical, with a radius R. They

were modeled following three distinct methods. The first method, similar to the one used

in the Quasicontinuum Method, was to model the indenter with the same material as the

film and to keep it rigid. The second method consisted in modeling the indenter with

carbon atoms in the diamond structure, and keeping it fixed. The last method was to use a

virtual indenter. Similar to past atomistic works (Lilleodden et al., 2003), a repulsive

force is applied such that:

2)()( RrkrF −−= (2.12)

24

with k a specified force constant (k = 10 N/m²), R the radius of the indenter, and r the

distance between the atom and the center of the indenter. This type of indenter removes

the adhesion and friction forces that are applied by the real indenters.

2.4. Interatomic Potentials

2.4.1. Embedded Atom Method Potentials

Interactions amongst atoms for numerical applications are represented using an

interatomic potential. The Embedded Atom Method (EAM) potentials (Daw and Baskes,

1983; Daw and Baskes, 1984) accurately represent defects, surfaces and impurities in

FCC metals.

In this method, the total energy of a monoatomic system is represented as

(Mishin et al., 1999):

( ) ( )∑ ∑+=ij i

iijtot FrVE ρ21

(2.13)

where ( )ijrV is a pair potential as a function of the distance ijr between atoms i and j ,

and F is the “embedding energy” as a function of the host “density” iρ induced at site i

by all other atoms in the system. The latter is given by:

( )∑≠

=ij

iji rρρ (2.14)

25

( )ijrρ being the “atomic density” function. The second term in equation (2.13) is volume

dependent and represents, in an approximate manner, many-body interactions in the

system.

When creating a new potential, one needs to make sure it will represent very

closely the real interactions between atoms. EAM potentials can be fitted to experimental

data for the values of equilibrium lattice parameter 0a , the cohesive energy 0E , three

elastic constants and the vacancy formation energy fvE . This basic set of properties can

often be complemented by other data such as planar fault energy or phonon frequencies.

But those parameters are usually not enough in order to create a reliable potential. The

created potential is thus fitted to ab-initio calculations for the given material. Ab-initio

calculations can represent the electronic structure of the atoms, and use laws of quantum

physics instead of empirical potentials. The aim of the creator of a potential is thus to fit

the created curve very closely with the ab-initio curve.

2.4.2. Tersoff Potential

A Tersoff potential was used in order to represent the interactions between

carbon atoms for a diamond indenter. This potential is a 3-body potential, and the energy

of the system of atoms is computed as:

∑ ∑≠

=i ij

ijVE 21 (2.15)

( ) ( ) ( )[ ]ijAijijRijCij rfbrfrfV += (2.16)

26

( )

DRr

DRrDR

DRr

DRrrfC

+>

+

27

2.4.3. Morse Potential

A Morse potential was used in order to represent the interactions between carbon

atoms from the indenter and aluminum atoms from the film. The Morse potential

computes pairwise interactions with the formula:

( ) ( )[ ] crrrr rreeDE

28

where α and β are the Cartesian coordinates, ωi0 is the undeformed atomic volume of the

ith atom, Det[Fiαβ] is the determinant of the deformation gradient, j is the interatomic

potential, and rij is the distance between ith and jth atoms. Note that the kinetics terms have

been eliminated in (2.24) as compared to (Lilleodden et al., 2003). In this equation, the

use of the determinant of the deformation gradient has been shown to provide improved

accuracy for the calculation of deformed atomic volumes. Furthermore, the principal

shear stress, τ, was calculated for each atom using the components of the atomic-level

stress tensor iσ~ using the formula in (Johnson et al., 1971).

2.6. Voronoi Construction

In order to model polycrystalline thin films, grains have to be created. One

method often used to create those grains is called the Voronoi method (Voronoi, 1908).

The Voronoi construction enables the user to create a 2D or 3D grain boundary network

that is considered to be representative of a natural grain boundary (GB) network.

The user must start by randomly placing reference points at a specified mean

distance from each other in the surface or volume studied. Each reference point will be

the center of a grain, for which a random orientation can be assigned. Each grain will be

composed of atoms that are closer to the reference point of the grain than to the other

reference points in the surface/volume. This creates grain boundaries that are orthogonal

to the lines joining the reference point to neighbor reference points. Fig. 2.1 below is an

illustration of the result of the procedure for a 2D case.

29

Fig. 2.1: Schematics of the Voronoi construction of a 2D model

2.7. Tools for the Visualization of Defects, Dislocations and

Grain Boundaries in Atomistic Simulations

2.7.1. Centro-symmetry Parameter

In solid-state systems, the centro-symmetry parameter is a useful measure of the

local lattice disorder around an atom and can be used to characterize whether the atom is

part of a perfect lattice, a local defect (e.g. a dislocation or stacking fault) or at a surface.

This parameter is computed using the following formula (Kelchner et al., 1998):

∑=

++=6,1

26

iii RRPrr

(2.25)

where the 12 nearest neighbors are found and iR and 6+iR are the vectors from the

central atom to the opposite pair of nearest neighbors. This formula was projected in the

30

plane for use with the QC method. An atom in perfect FCC lattice will then have a

centro-symmetry parameter of zero. The values for other configurations depend on the

material chosen. For aluminum, those values are 32.8 Å2 for a surface atom, 8.2 Å2 for

atoms in an intrinsic stacking fault, and 2.05 Å2 for atoms halfway between fcc and hcp

sites (in a partial dislocation). The values were 16.4 Å2, 4.1 Å2 and 1.025 Å2 respectively

for the QC method.

The centro-symmetry parameter is well adapted for calculations with no

temperature involved, because at higher temperatures, the distance between atoms is

constantly changing. An average of the centro-symmetry parameter is then needed in

order to facilitate the visualization.

The color scheme used is this study is the following, unless otherwise indicated:

atoms in a perfect FCC lattice are colored in grey or are not shown at all, those with a

HCP structure or representing a stacking fault are in blue, and all other non-coordinated

atoms are in green or red.

2.7.2. Ackland Parameter

In contrast to the centro-symmetry parameter, the method using the formulation

by Ackland (Ackland and Jones, 2006) is stable against temperature boost, because it is

based not on the distance between the atoms, but the angles. Therefore, statistical

fluctuations are averaged out a little more. This parameter classifies atoms depending on

the closest crystallographic structure it belongs to (BCC, FCC, HCP or unknown).

31

The procedure (Ackland and Jones, 2006) first calculates the mean squared

separation ∑=

=6,1

220 6

jijrr for the nearest six particles to atom i . It then finds the closest

neighbors that verify 202 55.1 rrij < . For each of the neighbor pairs found, it calculates the

bond angle cosines jikθcos . The procedure then relies on a table given by the author, that

separates the [-1;1] range of possible values for a cosine into 8 ranges. Depending on the

crystallographic structure around the atom, there are a certain number of bonds within the

( ) 2100 −NN closest neighbors that should fall into each range of cosine. The number of

cosines that should fall into the first range (from -1.0 to -0.945) is called 0χ , and the

number of cosines that should fall into the last range (from 0.795 to 1.0) is called 7χ . By

comparing the values obtained for each iχ with the given values of the table, one is able

to determine to which category atom i can be assigned. The color scheme used in the rest

of the study is the same than for the centro-symmetry parameter.

2.8. Validity of the 3D Models

Periodic boundary conditions are needed for the molecular dynamics 3D models.

In order to verify that the periodicity of the model does not influence the results, two

nickel thin films of thickness 12 nm were modeled. The first one has dimensions of 40

nm ä 12 nm ä 40 nm, and the second one has dimensions of 60 nm ä 12 nm ä 60 nm. A

virtual indenter of radius 9 nm applies a repulsive force on the surface. The EAM

interatomic potential for Ni from Mishin et al. (Mishin et al., 1999) is used. Both thin

32

films have the same orientation, which is ]211[ in the indentation direction, ]111[ and

]101[ in the directions normal to indentation. The bottom two layers of the films were

constrained in the indentation direction. Both simulations were performed at 300K, with a

time step of 5 fs. The indenter was displaced at a rate of 1 m/s, and the atomic positions

were recorded at 50 ps intervals up to 1250 ps (250,000 time steps).

The evolution of the contact pressure is presented in Fig. 2.2, and the snapshots

at the step 250 ps after the yield point are presented in Fig. 2.3. Fig. 2.2 shows that the

elastic regime for both samples is very similar. The yield point is attained for a lower

value of indentation depth for the larger sample, but the values of the yield point are very

close: 29.3 GPa for the small sample, and 30.3 GPa for the larger sample, or a difference

of only 3%. The mechanisms of plasticity are also similar, with dislocation loops and a

prismatic loop (Li et al., 2002) (Fig. 2.3). We can then say that our smaller system (40 nm

ä 12 nm ä 40 nm) not affected by the periodic boundary conditions. This will save some

computation time, as the larger system has over 4 million atoms, and the smaller one only

1.7 million.

33

Fig. 2.2: Evolution of the contact pressure as a function of the indentation depth for both models.

The arrows indicate the yield points.

Fig. 2.3: Indentation step 250 ps after the yield point (a) for the 40 nm ä 12 nm ä 40 nm model

and (b) the 60 nm ä 12 nm ä 60 nm model. The atoms in perfect FCC lattice have been removed.

34

CHAPTER 3: EFFECTS OF A GRAIN BOUNDARY

NETWORK ON INCIPIENT PLASTICITY DURING

NANOSCALE CONTACT1

3.1. Objectives

In this chapter, the effects of a nanocrystalline grain boundary network on the

plasticity of thin films are studied in aluminum, along with the mechanisms associated

with plasticity in polycrystalline thin films. Single crystalline and polycrystalline

simulations with different grain sizes and different potentials are compared. We first

investigate how to define the yield point for a polycrystalline simulation. We then study

the mechanism of shear localization in polycrystals, and finally compare the grain

boundary mechanisms of plasticity for different interatomic potentials for Al.

1 The results presented in this chapter also appeared in the following journal articles: - V. Dupont and F. Sansoz, Quasicontinuum Study of Incipient Plasticity under Nanoscale Contact in Nanocrystalline Aluminum, Acta Materialia, in press, doi:10.1016/j.actamat.2008.08.014. - F. Sansoz and V. Dupont, Atomic Mechanism of Shear Localization during Indentation of a Nanostructured Metal, Materials Science and Engineering C 27 (2007) 1509.

35

3.2. Model

Multiscale computer models of wedge-like cylindrical nanoindentations in 200

nm-thick Al films were created using the quasicontinuum method. In this study, the

region subjected to small deformation gradients outside the plastic zone was treated by

finite elements with an atomistically-informed elastic behavior, while the contact region

at the interface between the indenter and film surface was fully represented by individual

atoms. The film dimensions were 400 nm × 200 nm × 0.286 nm, and the size of the full

atomistic zone was 50 nm × 25 nm × 0.286 nm, as indicated in Fig. 3.1a for a

nanocrystalline thin film with a grain size of 7 nm. Plane-strain contact was modeled by

displacing a single crystal Al cylinder with a radius of 15 nm along the direction normal

to the top surface of the film. The indenter was oriented along the crystallographic

directions shown in Fig. 3.1b and kept completely rigid during the simulation. The single

crystalline model had a ]011[ out-of-plane orientation and a [111] surface.

The polycrystalline structure of the film was constructed as follows. Reference

atoms were placed randomly in the sample at an average distance equal to a pre-defined

grain size. GBs were created by a Voronoi construction, which was based on a

constrained-Delaunay connectivity scheme. Starting from the reference atom, all atoms in

the grains were added using the Bravais lattice vectors. The mean grain sizes studied

were 5 and 7 nm. Each grain was assigned a common tilt axis along the ]011[ direction,

and random in-plane orientation. To avoid discontinuities in the energy state during force

minimization, the continuum/atomistic frontier was modeled as a single crystal interface

36

whose crystallographic orientations are shown in Fig. 3.1a. We note that no significant

atomistic activity was found near this interface, indicating that the plastic deformation

was limited to the polycrystalline region during the simulations.

Fig. 3.1: Quasicontinuum model of a 7 nm-grain size Al thin film indented by a 15 nm radius

cylindrical indenter. (a) Full view of both finite element domain and atomistic region. (b) Close-

up view of full atomistic zone near the contact region in unrelaxed configuration.

The bottom of the film was fixed along each direction, while both sides of the

model were left free. A spacing of 10 Å was initially imposed between the tip apex and

the film surface. Periodic boundary conditions were imposed along the out-of-plane

direction in the entire model. Therefore, this study focuses on a randomly oriented 2D

columnar microstructure. A caveat here is that the plastic deformation processes in this

type of microstructure may differ from those found in fully 3D polycrystalline structures.

37

The total energy was minimized by conjugate gradient method until the addition

of out-of-balance forces over the entire system was found less than 10-3 eV/Å. The

sample was first relaxed under zero pressure condition in order to obtain the lowest state

of energy. After relaxation, the atoms of the indenter were displaced by increments of 0.9

Å until a total displacement of 90 Å, corresponding to an indentation depth of 80 Å

(increments of 0.4 Å with a final indentation depth of 30 Å for the single crystal). Energy

minimization was performed between each loading step. The centro-symmetry parameter

was calculated after each relaxation step to analyze the presence of planar defects in the

lattice and the structure of the GB network during deformation.

The semi-empirical EAM potentials for Al by Voter and Chen (Voter and Chen,

1987) and Mishin and Farkas (Mishin et al., 1999) were used. For brevity in the

following, these two potentials are referred to as Al-VC and Al-MF potentials,

respectively. For each potential, we adopted the quasicontinuum procedures used

previously by Sansoz and Molinari (Sansoz and Molinari, 2005; Sansoz and Molinari,

2004) to calculate the generalized planar and stacking fault energy curves and the GB

energy of three Σ tilt bicrystals, including Σ3(112), Σ9(221) and Σ11(113)

symmetric tilt GBs.

38

3.3. Characterization of the EAM Potentials

The generalized stacking and planar fault energy curves for the Al-MF potential

are shown in Fig. 3.2. The unstable stacking fault energy (γUSF), stacking fault energy

(γSF) and unstable twinning fault energy (γUTF) are also indicated in this figure. As pointed

out by Van Swygenhoven et al. (Van Swygenhoven et al., 2004), the ratios γSF/γUSF, and

γUTF/γUSF are most important in assessing the activation energy required to predict the

nucleation of stacking faults, full dislocations or twins in the material. More specifically,

if γSF/ γUSF is close to 1, the activation energy to create a trailing partial is low, which

favors the nucleation of full dislocations. In contrast, if this ratio is closer to 0, the

activation energy is too high, which decreases the propensity to nucleate trailing partials,

leaving only stacking faults after propagation of the leading partial dislocations. The

same reasoning can be made for twinning, which is more likely to occur if the ratio

γUTF/ γUSF is close to 1. The energy values obtained for both potentials are summarized in

Table 3.1, along with reference values from first principles simulations for pure Al (Murr,

1975; Bernstein and Tadmor, 2004) and Al with either H or Ge solute impurities (Lu et

al., 2002; Qi and Mishra, 2007). In this table, we find that the calculated energy values

are significantly smaller for the Al-VC potential than the Al-MF potential, which is

consistent with the predicted values in the literature (Mishin et al., 1999). Similarly, the

first-principle values for γSF and γUSF are found to be lower when adding solute impurities

to Al. Therefore, our finding is the predicted tendency that the stacking and planar fault

energy values calculated from the Al-MF potential are consistent with the ab-initio values

39

for pure Al, while the Al-VC results seem to be in better agreement with the energy

values for Al with impurities. In addition, we find that all the ratios γSF/ γUSF and

γUTF/ γUSF are similar and equal to 0.81-0.86 and 1.30-1.32, respectively, which suggests

the same slip and twinning behavior regardless of the interatomic potential for Al.

Fig. 3.2: Generalized stacking and planar fault energy curves obtained by quasicontinuum method

with the Mishin-Farkas EAM potential for Al.

40

Table 3.1: Stacking fault energy (γSF), unstable stacking fault energy (γUSF), unstable twinning

fault energy (γUTF) and GB energy (γGB) for three Σ tilt symmetric grain boundaries calculated

from quasicontinuum method on the two EAM Al potentials investigated, and reference values

from first-principles simulations for pure Al and Al with impurities. All units of energy are in

mJ/m².

Present atomistic study First-principles simulations Energy

EAM Al-VCa EAM Al-MFb Pure Al Al with impurities

γUSF 93.04 166.71 175-224c 97d

γSF 75.41 144.22 120-166c,e 73d, 82f

γUTF 120.99 220.72 207g -

γSF/γUSF 0.81 0.86 0.73-0.9 0.75 γUTF/γUSF 1.30 1.32 0.92-1.18 -

γGB − Σ9(221) 302 454 408h -

γGB − Σ11(113) 96 151 190-206i,j 96j

γGB − Σ3(112) 318 355 426i -

a(Voter and Chen, 1987) ; b(Mishin et al., 1999) ; c(Ogata et al., 2002; Hartford et al., 1998; Lu et al., 2000) ; dAl + 14.3 at. % solute H impurities (Lu et al., 2002) ; e(Murr, 1975; Rautioaho, 1982; Westmacott and Peck, 1971); fAl + 3.3 at. % solute Ge impurities (Qi and Mishra, 2007); g(Bernstein and Tadmor, 2004); h(Inoue et al., 2007); i(Wright and Atlas, 1994); jAl + 9 at. % substitution Ga impurities (Thomson et al., 2000).

3.4. Force and Contact Pressure Calculations

The force applied by the indenter was calculated using the formula:

∑∈

=Zi

iPP , (3.1)

where Z represents all atoms of the film belonging to the contact zone and Pi is the out-

of-balance force on atom i in this zone, projected along the direction of indentation. The

contact zone was computed after each loading step by only including atoms at the

indenter-film interface within a separation distance from the tip equal to the potential

41

cutoff radius. The cutoff radii were 5.555 Å and 6.287 Å for the Al-VC and Al-MF

potentials, respectively. The mean contact pressure was determined at each step as:

perioza

PH×

=2

, (3.2)

where a is the contact length, defined as half the width of the projected contact area, and

zperio is the thickness of our sample in the out-of-plane direction.

3.5. Determination of the Yield Point

The onset of plasticity in single crystals is characterized by a pop-in event

corresponding to the homogeneous nucleation of dislocations in the crystal. This is

visible on the force – displacement plot because it corresponds to a sudden drop in the

force (Fig. 3.3), which corresponds to the yield point. In a nanocrystalline thin film,

however, there is no visible drop on the plot, and rather, we find flow serration leading to

a significant softening effect. The flow serration has been found to correspond to plastic

shear localization through the formation of shear bands (Trelewicz and Schuh, 2007;

Lund and Schuh, 2004). As can be seen on Fig. 3.3, the yield point for a polycrystalline

material cannot be found from the force – displacement curve directly.

42

Fig. 3.3: Effect of nanosized grains on the nanoindentation response of Al substrates from

molecular static simulation using the Al-VC potential. The indenter radius is 15 nm. Serrated

plastic flow clearly appears in the two nanocrystalline Al substrates under nanoindentation.

The following criterion was used to quantitatively assess the onset of plasticity as

obtained by the simulations. We computed the theoretical pressure – displacement curve

corresponding to an isotropic elastic surface in contact with a rigid cylinder by using the

following equation from continuum theory (Johnson, 1985):

( )[ ])1()2ln(21

2

ννπ

νδ −−−= atE

F , (3.3)

where F is the total linear force in N/m ( = P/zperio), δ is the indentation depth, t is the film

thickness along the direction of indentation, E and ν are the Young’s modulus and

43

Poisson’s ratio of the film, respectively. Substituting this into (3.3) gives the mean

contact pressure for an elastic surface He:

( )δ

ννν

π×

⎥⎦

⎤⎢⎣

⎡−

−⎟⎠⎞

⎜⎝⎛−

=

12ln212 2ata

EH e ,(3.4)

Fig. 3.4: Contact pressure versus penetration depth plot for a 7-nm polycrystal with the Al VC

potential, along with the corresponding theoretical fitting.

Since the above equation depends on the contact length a, which changes with

the penetration δ, the parameter He was re-evaluated at each loading step. Assuming ν =

0.345 for polycrystalline Al (Meyers and Chawla, 1999), the isotropic Young’s modulus

was determined by fitting the elastic contact pressure He from (3.4) to the first portion of

the curve obtained by quasicontinuum simulation (Fig. 3.4). Using the Al-VC potential,

44

we see that the yield point is around 2.3 GPa, and that the curve shows significant

softening effects.

3.6. Shear Localization Mechanisms

Shear banding was found to occur in all polycrystalline simulations, and is best

illustrated by a 5 nm-grain-size simulation using the Al-VC potential (Fig. 3.5). At the

onset of plasticity, significant grain boundary sliding is found to occur. This behavior

results in significant rotational deformation of the grains with limited intragranular slip.

During this process, the grain boundary structure is significantly changed and, in some

cases, several grain boundaries tend to be aligned (Fig. 3.5a and b). The bands are formed

by the sliding of aligned interfaces separating the grains (grains 3 and 4 in Fig. 3.5c).

When the shear plane encounters a triple junction and is stopped by a grain that is not in

its alignment, the shear band follows its path by intragranular slip in the prolongation of

the shear plane. For example, a stacking fault left behind a partial dislocation can be seen

in grain 2 in the prolongation of the shear plane in Fig. 3.5c. Subsequently, the newly

created stacking faults are found to nucleate mechanical twins, which grow under the

applied shear stress. Mechanical twinning has also been observed in nanocrystalline Al

under indentation by Chen et al. (Chen et al., 2003b). This result suggests therefore that

our simulation is in excellent agreement with the experimental data.

45

Fig. 3.5: Thin shear band formation in 5-nm-grain-size nanocrystalline Al after 2.5-nm-deep

indentation. (a) Partial view of the contact interface and location of the grain cluster associated

with the shear band. (b) Enlarged view of the shear plane. A mechanical twin nucleated at the

triple junction in the prolongation of the shear place is clearly visible in grain 2. (c) Magnitude

and direction of atomic displacements between two loading increments represented by arrows.

The shear band results from sliding of aligned grain boundaries (grains 3 and 4) and intragranular

partial slip (grain 2).

Shear localization in nanocrystalline metals has been found to occur through

collective grain activity initiated by grain boundary sliding. It has been shown by

Hasnaoui and co-workers (Hasnaoui et al., 2002a) on uniaxial compression that three

mechanisms contribute to the formation of local shear planes by cooperative grain

activity in nanocrystalline metals: (i) GB sliding to form a single shear plane consisting

of a number of collinear GBs, (ii) continuity of the shear plane by intragranular slip and

(iii) the coalescence via reorientation of neighboring grains that have an initially low

46

angle GB. The present study clearly indicates that during indentation the first two

mechanisms also co-exist to form shear bands. We believe however that the mechanism

of grain coalescence via reorientation should occur at larger depth of indentation after