ATOMISTIC INVESTIGATION OF DEFORMATION TWINNING IN NC FCC AND BCC MULTILAYERS

by

Ashok Kumar

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Department of Mechanical and Industrial Engineering University of Toronto

© Copyright by Ashok Kumar 2018

ii

Atomistic Investigation of Deformation Twinning in NC FCC and

BCC Multilayers

Ashok Kumar

Master of Applied Science

Department of Mechanical and Industrial Engineering University of Toronto

2018

Abstract

The deformation twinning is observed to increase both ductility and strength of nanocrystalline

(NC) materials. Hence, deformation twinning in NC materials is very important and can be used

to improve structural performance of NC materials and of those materials which employ both CG

and NC layers, e.g. Multilayers (MLs). This thesis is focused on finding twinning trends in NC

FCC materials and study interface-mediated deformation twinning in BCC ML using molecular

dynamics. The first study investigates the competition between twin nucleation and twin

thickening in NC FCC materials. The second study extends the FCC ML code to BCC materials

and investigates CG-NC interface-mediated deformation twinning in BCC Ta ML material. The

results of both studies provide insight into the work hardening behavior of the NC FCC and BCC

ML due to deformation twinning.

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To my parents

iv

Acknowledgments

I would like to thank my supervisor Professor Chandra Veer Singh for his invaluable advice and

continuous support throughout the course of my graduate studies. I would also like to extend my

gratitude towards the members of the Computational Materials Engineering Laboratory, especially

Matthew Daly and Hao Sun for their mentorship and encouragement.

I would also like to thank NSERC and the Department of Mechanical and Industrial Engineering

for funding this research. I would also like to acknowledge Scinet, Calcul Quebec, and Compute

Canada for providing the computing resources for carrying out this research.

Finally, I want to thank my wife and my parents for their loving support and encouragement.

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Table of Contents

Acknowledgments.......................................................................................................................... iv

Table of Contents .............................................................................................................................v

List of Tables ................................................................................................................................ vii

List of Figures .............................................................................................................................. viii

List of Acronyms and Symbols ................................................................................................... xvi

List of Appendices ..................................................................................................................... xviii

1 Introduction .................................................................................................................................1

1.1 Background ..........................................................................................................................1

1.1.1 Deformation Twinning in FCC Materials ................................................................1

1.1.2 Deformation Behavior of Metallic Multilayers .......................................................9

1.2 Thesis Motivation ..............................................................................................................13

1.3 Thesis Objectives ...............................................................................................................15

1.4 Thesis Organization ...........................................................................................................16

2 Computational Methodology ...................................................................................................17

2.1 Molecular Dynamics ..........................................................................................................17

2.1.1 Interatomic Potential ..............................................................................................19

2.2 Nanowire Model and Calculation Methodology................................................................21

2.2.1 Nanowire Model Generation..................................................................................21

2.2.2 EAM Potential Selection........................................................................................21

2.2.3 Crystal Analysis Tool ............................................................................................22

2.2.4 Twinning Parameters .............................................................................................22

2.3 BCC Multilayer Model Generation ...................................................................................23

vi

3 Investigate Competition between Twin nucleation and Twin Thickening in NC FCC Materials ..................................................................................................................................28

3.1 Introduction ........................................................................................................................28

3.2 Kinetic Monte Carlo Model ...............................................................................................29

3.3 Investigate Competition between Twin Nucleation and Twin Thickening using MD Nanowire Simulations ........................................................................................................36

3.3.1 Computational Methodology and Simulations Detail............................................36

3.3.2 Results and Discussions .........................................................................................44

3.4 Summary ............................................................................................................................65

4 Deformation behavior of a BCC Tantalum Multilayer with a modulated grain size distribution ..............................................................................................................................66

4.1 Introduction ........................................................................................................................66

4.2 Methods..............................................................................................................................68

4.3 Results and Discussions .....................................................................................................72

4.3.1 Coarse Grain Dimension Selection ........................................................................72

4.3.2 Deformation Behavior of Ta BCC ML with Zone Axis ............................74

4.3.3 Comparison with FCC ML and NC BCC Microstructure .....................................80

4.3.4 Grain Size Effect ....................................................................................................82

4.3.5 Strain Rate Sensitivity............................................................................................83

4.3.6 Temperature Effect ................................................................................................85

4.4 Conclusion .........................................................................................................................86

5 Conclusions and Future Work .................................................................................................87

5.1 Summary and Overall Contribution ...................................................................................87

5.2 Future Work .......................................................................................................................88

References .....................................................................................................................................90

vii

List of Tables

Table 3. 1: Parameters used in KMC simulations for different materials such as GPFEs(a) and

external shear stress (𝜎 )(b) etc. [20]. ............................................................................................ 32

viii

List of Figures

Figure 1. 1: Change in grain shape above the twin boundary due to deformation twinning above

the twin boundary [1]. ..................................................................................................................... 2

Figure 1. 2: (a) The three equivalent Shockley partials on the (111) closed packed plane. (b) The

stacking sequence of {111}-type slip planes with three equivalent Shockley partials directions [2].

......................................................................................................................................................... 2

Figure 1. 3: (a) The process of four-layer monotonic twin formation by glide of Shockley partials

with same Burgers vectors. (b) The process of four-layer twin formation by the glide of Shockley

partials with different Burgers vectors [2]. ..................................................................................... 3

Figure 1. 4: Crystal geometry used by Yamakov et al. See main text for description of symbols

[3]. ................................................................................................................................................... 5

Figure 1. 5: Simulation snapshot after loading the cell to the indicated stress intensity factor(KI).

The deformation twins are shown in a, b, and c which have an orientation (𝜃, 𝜑) of (35.26o, 0o),

(54.74o, 0o), (70.53o, 0o) respectively. Full dislocation slip is shown in d, e and f which have an

orientation of (𝜃, 𝜑) of (35.26o, 30o), (54.74o, 30o), (70.53o, 30o) respectively [3]. ....................... 6

Figure 1. 6: GPFE curve for Nickel created using Molecular Dynamics. ...................................... 7

Figure 1. 7: Deformation twinning and full dislocation slip Hall-Petch relationship with decreasing

grain size in CG materials, where 𝜏 represents the shear stress and d is the grain size [2]. ........... 8

Figure 1. 8: Cu/Nb ML with two different layer thicknesses of (a) 0.8nm and (b) 20nm [15]. ... 10

Figure 1. 9: Deformation mechanism map of ML with varying layer thickness [15]. ................. 11

Figure 1. 10: Multilayer structure, where 𝑑 and 𝑑 is grain size of coarse grain and

nanocrystalline layer, 𝑡 and 𝑡 is layer thickness of coarse grain and nanocrystalline layer [20].

....................................................................................................................................................... 14

ix

Figure 2. 1: MD algorithm flowchart ............................................................................................ 19

Figure 2. 2: GPFE curve calculated for Nickel using MD. ........................................................... 22

Figure 2. 3: Image file containing only twin boundaries and black lines represents simulation cell.

....................................................................................................................................................... 23

Figure 2. 4: (a) 3d and (c) 2d Voronoi tessellations for 20nm grain size. (b) 3d and (d) 2d BCC NC

microstructure generated using custom MATLAB code. ............................................................. 24

Figure 2. 5: The schematic of BCC ML generation (where 𝑡 ≈ 𝑑 ) for MD simulations. The

NC grains generated by Voronoi tessellation are split along transecting pathway and a 100 nm CG

layer is inserted, while enforcing periodic conditions atoms are populated in to the grains and CG

layer. The purple color represents BCC atoms with a lattice constant of 0.3304 nm and black color

shows grain boundary atoms. The BCC and grain boundary atoms are identified using common

neighbor analysis algorithm in OVITO [37]. ................................................................................ 26

Figure 2. 6: ML microstructure generated by (a) Straight boundaries and (b) zig zag boundaries at

NC and slab layer interface. The purple color represents BCC atoms with lattice constant of 0.3304

nm and black color shows grain boundary atoms. The BCC and grain boundary atoms are identified

using common neighbor analysis algorithm in OVITO [37]. ....................................................... 26

Figure 3. 1: (a) KMC simulation cell which is a representative of a single grain in NC FCC

materials. 𝐸 represents the barrier for leading partial nucleation, the energy barriers (𝐸 ,...., 𝐸 )

are listed next to the appropriate faults. 𝜎 is the external applied shear stress and 𝜎 is the stress

required for partial glide (b) Typical FCC GPFE curve for deformation twinning

[20]............................................................................................................................30

Figure 3. 2: Twin evolution in Lead (Pb). Purple region represents twinned area and leading

Shockley partials are shown by blue arrows. Merging of twins is shown in (b) and thickening of

twins can be seen in (c) [20]. ................................................................................................... 32

x

Figure 3. 3: Colored regions represents twin evolution in different materials at different twinning

fractions [20]. ........................................................................................................................... 33

Figure 3. 4: (a) Twinning trends in different FCC materials shown by data points. (b) Evaluation

of average twin thickness at different twinning fractions calculated from equation 3.4 also shown

by data points. Results from eq. 3.5 and 3.4 in dash line format are overlaid with the KMC

simulation results [20].............................................................................................................. 34

Figure 3. 5: (a) GPFE curve calculated from different Ag EAM potentials, (b) Selected Ag

potential [53] GPFE curve along with Ag GPFE DFT points [48] are shown. ....................... 37

Figure 3. 6: (a) GPFE curve calculated from different Al EAM potentials, (b) Selected Al potential

[66] GPFE curve along with Al GPFE DFT points [48] are shown. ....................................... 38

Figure 3. 7: (a) GPFE curve calculated from different Cu EAM potentials, (b) Selected Cu potential

[70] GPFE curve along with Cu GPFE DFT points [48] are shown. ...................................... 39

Figure 3. 8: (a) GPFE curve calculated from different Ni EAM potentials, (b) Selected Ni potential

[78] GPFE curve along with Ni GPFE DFT points [48] are shown. ....................................... 40

Figure 3. 9: GPFE curve calculated from different Pb EAM potentials. The GPFE curves don’t

match with Pb GPFE DFT points [48] are shown. .................................................................. 41

Figure 3. 10: Nanowire loading direction orientation. ............................................................. 42

Figure 3. 11: (a) Atomistic Square Nanowire Model. (b) Square cross-section of nanowire model.

Atoms are colored according to CNA algorithm of OVITO [37]. ........................................... 43

Figure 3. 12: Twin boundaries identified by CAT [36] and visualized in OVITO [37]. Black lines

represent simulation cell boundaries. ....................................................................................... 43

Figure 3. 13: Deformation twinning evolution in nanowires at twinning fraction of (a) 0.1, (b) 0.2,

and (c) 0.3. Navy Blue color shows FCC atoms, red-colored lines show stacking faults and yellow

colored lines show twin boundaries. The FCC atoms, stacking faults and twin boundaries are

identified using CAT [36] and visualized in OVITO [37]. ...................................................... 45

xi

Figure 3. 14: Number of deformation twins evolution in nanowires as a function of twinning

fraction. Each point represents the statistical average of 5 simulations. ................................. 46

Figure 3. 15: Average twin thickness evolution in nanowires as a function of twinning fraction.

Each point represents the statistical average of 5 simulations. ................................................ 46

Figure 3. 16: TEM images for deformation twinning in Cu alloys with decreasing stacking fault

energy, (a) 61 mJ/m2 (b) 12 mJ/m2. Distribution of deformation twin thickness in Cu alloys with

decreasing stacking fault energy (c) 61 mJ/m2 (d) 12 mJ/m2 calculated from various TEM images

[49]. .......................................................................................................................................... 48

Figure 3. 17: Variation of average twin thickness with decreasing stacking fault energy in Cu alloys

[49]. .......................................................................................................................................... 48

Figure 3. 18: (a) Thick deformation twin in NC Al [51] and (b) Multiple thin twins in NC Cu [50].

.................................................................................................................................................. 49

Figure 3. 19: Number of deformation twins evolution in nanowires as a function of twinning

fraction. .................................................................................................................................... 50

Figure 3. 20: Average Twin thickness evolution in nanowires plotted as a function of twinning

fraction. .................................................................................................................................... 50

Figure 3. 21: Number of deformation twins evolution in nanowires plotted as a function of

twinning fraction. ..................................................................................................................... 51

Figure 3. 22: Average twin thickness evolution in nanowires plotted as a function of twinning

fraction. .................................................................................................................................... 52

Figure 3. 23: Number of deformation twins evolution in Ag nanowires at different lengths. . 53

Figure 3. 24: Average twin thickness evolution in Ag nanowires at different lengths. ........... 53

Figure 3. 25: Histogram showing number of deformation twins and average twin thickness at

twinning fraction of 0.3 for different lengths of Ag nanowires. .............................................. 54

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Figure 3. 26: Number of deformation twins evolution in Al nanowires at different lengths. . 54

Figure 3. 27: Average twin thickness evolution in Al nanowires at different lengths............. 55

Figure 3. 28: Histogram showing number of deformation twins and average twin thickness at

twinning fraction of 0.3 for different lengths of Al nanowires. ............................................... 55

Figure 3. 29: Number of deformation twins evolution in Cu nanowires at different lengths .. 56

Figure 3. 30: Average twin thickness evolution in Cu nanowires at different lengths. ........... 56

Figure 3. 31: Histogram showing number of deformation twins and average twin thickness at a

twinning fraction of 0.3 for different lengths of Cu nanowires. .............................................. 57

Figure 3. 32: Number of deformation twins evolution in Ni nanowires at different lengths .. 57

Figure 3. 33: Average twin thickness evolution in Ni nanowires at different lengths............. 58

Figure 3. 34: Histogram showing number of deformation twins and average twin thickness at a

twinning fraction of 0.3 for different lengths of Ni nanowires. ............................................... 58

Figure 3. 35: Number of deformation twins evolution in Ag nanowires at different temperatures.

.................................................................................................................................................. 59

Figure 3. 36: Average twin thickness evolution in Ag nanowires at different temperatures... 60

Figure 3. 37: Histogram showing number of deformation twins and average twin thickness at a

twinning fraction of 0.3 for different temperatures of Ag nanowires. ..................................... 60

Figure 3. 38: Number of deformation twins evolution in Al nanowires at different temperatures.

.................................................................................................................................................. 61

Figure 3. 39: Average twin thickness evolution in Al nanowires at different temperatures. .. 61

Figure 3. 40: Histogram showing number of deformation twins and average twin thickness at a

twinning fraction of 0.3 for different temperatures of Al nanowires. ...................................... 62

xiii

Figure 3. 41: Number of deformation twins evolution in Cu nanowires at different temperatures.

.................................................................................................................................................. 62

Figure 3. 42: Average twin thickness evolution in Cu nanowires at different temperatures. .. 63

Figure 3. 43: Histogram showing number of deformation twins and average twin thickness at

twinning fraction of 0.3 for different temperatures of Cu nanowires. ..................................... 63

Figure 3. 44: Number of deformation twins evolution in Ni nanowires at different temperatures.

.................................................................................................................................................. 64

Figure 3. 45: Average twin thickness evolution in Ni nanowires at different temperatures. .. 64

Figure 3. 46: Histogram showing number of deformation twins and average twin thickness at a

twinning fraction of 0.3 for different temperatures of Ni nanowires. ...................................... 65

Figure 4. 1: (a) Monolithic Multilayer template, (b) BCC Ta ML where 𝑡 ≈ 𝑑 [20], and (c)

simulation cell used for uniaxial tensile MD studies. The purple color represents BCC atoms and

green color shows grain boundary atoms. The atoms are identified using common neighbor

analysis (CNA) algorithm in OVITO [37]................................................................................ 69

Figure 4. 2: The schematic of BCC ML generation (where 𝑡 ≈ 𝑑 ) for MD simulations. The

NC microstructure is modified to accommodate a single large crystal. The purple color represents

BCC atoms and green color shows grain boundary atoms. The BCC and the grain boundary atoms

are identified using CNA algorithm in OVITO [37]. .............................................................. 69

Figure 4. 3: Atomic snapshot of the Ta ML after relaxation. All the grains and slab have same zone

axis (Z axis). The X and Y orientation of slab are shown, and the NC grains are randomly oriented.

The load is applied along the X direction. The purple color represents BCC atoms and green color

shows grain boundary atoms. The BCC and the grain boundary atoms are identified using CNA

algorithm in OVITO [37]. ........................................................................................................ 71

Figure 4. 4: (a) Uniaxial tensile response of Ta ML with slab dimensions ranging from 20 nm to

120 nm. The tensile response of 80 nm, 100 nm, and 120 nm is similar. (b) The yield stresses

xiv

which are calculated at 2% offset [20] are almost identical for 80 nm, 100nm, and 120nm

simulation cells. ....................................................................................................................... 73

Figure 4. 5: (a) Uniaxial tensile response of Ta ML with NC grain size of 20nm. (b) Relaxed Ta

ML with locations of deformation twins in (c), (d), (e), and (f). (c), (d), (e) and (f) shows nucleation

of deformation twins at various strain levels. (g) Ta ML at yielding (2% offset) [20]. Various twins

can be seen in CG slab and NC grains. The stress and strain levels of (c), (d), (e), (f) and (g) are

shown by annotations in (a). The purple color represents BCC atoms and green color shows non-

bcc atoms. The BCC and the non-bcc atoms are identified using CNA algorithm in OVITO [37].

.................................................................................................................................................. 75

Figure 4. 6: (a) The GSFE curve calculated for single crystal Ta using Ravelo et al. [92] EAM

potential. (b) Reflection twin formation by the glide of /6 dislocations on successive {112}

planes, (C) Isosceles twinning formation by the dissociation of /6 dislocations in two

/12 partial dislocations and their simultaneously glide on adjacent {112} planes. The purple

color shows BCC atoms and green color represents non-bcc atoms, identified using CNA

algorithm in OVITO [37]. ........................................................................................................ 76

Figure 4. 7: (a) Two periodic images of the simulation cell at 4.5% strain is shown to see

deformation twins more clearly. (b) Simulation cell at 20.5%. The purple color shows BCC atoms

and green color represents non-bcc atoms, identified using CNA algorithm in OVITO [37]. 77

Figure 4. 8: (a) Relaxed Ta ML microstructure. The annotations in (a) corresponds to location of

deformation events in (b), (c), (d), (e), (f), (g) and (h). (i) Twin – Twin interactions in CG layer.

Yellow arrow shows the void formation in (h). The purple color shows BCC atoms and green color

represents non-bcc atoms, identified using CNA algorithm in OVITO [37]. .......................... 78

Figure 4. 9: Atomic snapshots of Ta ML at various strain levels. Red arrows show void formation

in BCC ML at various strain levels. The purple color represents BCC atoms and green color shows

non-bcc atoms. The BCC and the non-bcc atoms are identified using CNA algorithm in OVITO

[37]. .......................................................................................................................................... 79

Figure 4. 10: Atomic snapshots of NiCo FCC ML [20], Ta BCC ML and nanocrystalline Ta BCC

at 10%, 20% and 60% strain with zone axis. In FCC ML snapshot, green color and black

xv

color shows FCC atoms and non-FCC atoms, respectively. In BCC ML and BCC NC snapshot,

purple color and black color represents BCC atoms and non-BCC atoms, respectively. Arrows

show voids in each individual snapshot. The FCC atoms, non-FCC atoms, BCC atoms, and non-

bcc atoms are identified using CNA algorithm in OVITO [37]. ............................................. 81

Figure 4. 11: Atomic snapshots of the Ta ML with grain sizes of (a) 15 nm, (b) 20 nm, (c) 25 nm,

and (d) 30 nm. Top atomic configurations are at 20% strain and bottom at 30% strain. Red arrows

show void formation in BCC ML. The purple color represents BCC atoms and green color shows

non-bcc atoms. The BCC and the non-bcc atoms are identified using common neighbor analysis

algorithm in OVITO [37] ......................................................................................................... 82

Figure 4. 12: (a) Ta ML Stress-Strain curves at different multiples of 100 MPa load increments

with their corresponding strain rates in brackets. (b) The flow stresses at 2% offset from (a) are

plotted with respect to strain rate. Both stresses and strain rates are in logarithmic format. ... 84

Figure 4. 13: Atomic configurations of the Ta ML at strain rates of (a) ≈ 6.25x107 s-1, (b) ≈ 9.50x107

s-1, and (c) ≈ 1.30x108 s-1. Top atomic configurations are at 15% strain and bottom at 20% strain.

Red arrows show void formation in BCC ML snapshots. The purple color represents BCC atoms

and green color shows non-bcc atoms. The BCC and the non-bcc atoms are identified using

common neighbor analysis algorithm in OVITO [37]. ............................................................ 84

Figure 4. 14: Ta ML deformed at (a) 200 K, (b) 400 K, and (c) 600 K. Top atomic configurations

are at 10% strain and bottom at 15% strain. The purple color represents BCC atoms and green

color shows non-bcc atoms. The BCC and the non-bcc atoms are identified using common

neighbor analysis algorithm in OVITO [37]. ........................................................................... 85

xvi

List of Acronyms and Symbols

Acronym

NC CG ML GPFE FCC BCC HCP UFG ECAP SHBP HREM GBMP MD LAMMPS EAM CAT esf isf ROM CNA GSFE GB SRS DFT KMC tf

Description

Nanocrystalline Coarse-grained Multilayer Generalized Planar Fault Energy Face centered cubic Body centered cubic Hexagonal close packed Ultra-fined grain Equal channel angular pressing Split Hopkinson pressure bar High-resolution electron microscopy Grain boundary mediated plasticity Molecular dynamics Large-scale Atomic/Molecular Massively Parallel Simulator Embedded Atom Method Crystal Analysis Tool extrinsic stacking fault intrinsic stacking fault Rule of mixture Common neighbor analysis Generalized stacking fault energy Grain boundaries Strain rate sensitivity Density functional theory Kinetic Monte Carlo Twin fault

Symbol

𝛾 𝛾 𝛾 𝜎 𝜎 𝜀 𝜎 𝛾 𝛾 F NT

Units

mj m-2 mj m-2 mj m-2 MPa MPa, GPa mm/mm MPa, GPa mj m-2 mj m-2 mm/mm

Description

Unstable stacking fault energy Stable stacking fault energy Unstable twin fault energy External applied shear stress True stress True strain Yield Strength extrinsic stacking fault energy intrinsic stacking fault energy Twin fraction Number of twins

xvii

�̅� E ao b d d111 E G h dCG dNC tCG tNC

�̇� mj m-2 nm nm nm nm GPa GPa nm, µm nm, µm nm, µm nm, µm nm, µm

Average twin spacing Activation energy barrier Lattice parameter Burgers vector Grain size Spacing between {111}-type slip planes Elastic modulus Shear Modulus Layer thickness multilayer Grain size of coarse-grained layer Grain size of nanocrystalline layer Thickness of coarse-grained layer Thickness of nanocrystalline layer

xviii

List of Appendices

Appendix A: MATLAB Code for Twinning Parameters Calculation ........................................102

1

Chapter 1

Introduction

1.1 Background

In the following sections, the literature review relevant to this thesis is presented. The literature

review is divided into two sections. In the First section basics of deformation twinning in FCC

materials are presented. The factors which cause competition between dislocation slip and

deformation twinning in FCC materials are also covered. The effect of grain size, strain rate and

temperature on deformation twinning mechanism in FCC metals is also discussed. In the second

section, what are multilayers (MLs) and what type of interfaces exists in MLs are covered. Length

scale dependent deformation behavior of MLs is also discussed. Basics of deformation twinning

mechanism in BCC materials is also presented. The goals of this literature review are bifold. The

main objective of the review is to familiarize the reader with deformation twinning mechanisms

in FCC and BCC materials which is relevant to understand upcoming results. Secondly, to provide

the reader with a better understanding of concepts related to MLs structures.

1.1.1 Deformation Twinning in FCC Materials

1.1.1.1 Basics of Deformation Twinning in FCC Materials

Traditionally, deformation twins were believed to be created by the glide of partials with same

Burgers vector on adjacent {111}-type slip planes. The partials Burgers Vector is 𝑏 = <

112 >, where is 𝑎 is lattice constant of the material. The shear deformation is homogeneous and

produces a large macroscopic strain. Figure 1.1 [1] illustrates the change in shape of the grain

above the twin boundary due to the deformation twinning which takes place above the twin

boundary. The deformation twinning produces a kink angle of 141o above the twin boundary which

is two times the angle between two {111}-type slip planes.

2

Figure 1. 1: Change in grain shape above the twin boundary due to deformation twinning above the twin

boundary [1].

These twinning partials are also known as Shockley partials. There are three equivalent Shockley

partials (e.g. b1, b2, b3) on each close-packed plane and there also exist three Shockley partials with

opposite Burgers vector signs (e.g. -b1, -b2, -b3). Figure 1.2(a) [2] illustrates these Shockley

partials. Figure 1.2(b) [2] illustrates the stacking sequence of {111}-type slip planes and the

directions of Shockley partials.

Figure 1. 2: (a) The three equivalent Shockley partials on the (111) closed packed plane. (b) The stacking

sequence of {111}-type slip planes with three equivalent Shockley partials directions [2].

3

The stacking sequence of {111}-type slip planes is ABCABCABC…... A stacking fault is

produced by the glide of first Shockley partial on {111}-type slip plane and atoms above the

stacking fault change their position. The Burgers vector b1, b2, b3 changes the stacking sequence of

{111}-type slip plane in the same way i.e.

Shockley Partial: b1 : 𝐴 → 𝐵, 𝐵 → 𝐶, 𝐶 → 𝐴

Shockley Partial: b2 : 𝐴 → 𝐵, 𝐵 → 𝐶, 𝐶 → 𝐴

Shockley Partial: b3 : 𝐴 → 𝐵, 𝐵 → 𝐶, 𝐶 → 𝐴

The opposite Burgers -b1, -b2, -b3 changes the stacking sequence in the opposite way i.e. from

𝐵 → 𝐴, 𝐶 → 𝐵, 𝐴 → 𝐶.

Figure 1. 3: (a) The process of four-layer monotonic twin formation by glide of Shockley partials with same

Burgers vectors. (b) The process of four-layer twin formation by the glide of Shockley partials with different

Burgers vectors [2].

Figure 1.3(a) [2] illustrates the formation of a four-layer twin with same Burgers vector 𝑏 = <

121 >, The glide of first partial produces an intrinsic fault marked by bold letter C in layer 2,

which is equivalent to removing a layer of B atoms from column 1. The glide of second partial

(see bold letter B in column 3) produces an extrinsic fault which is equivalent to adding a layer of

C atoms between A and B layer of atoms in column 3. The glide of subsequent two Shockley

partials with same Burgers vector b1 produces a four-layer twin. As the crystal is sheared with

4

same Burgers vector b1, the resulting macroscopic strain is large similar to Figure 1.1 [1]. The

same four-layer twin can also be formed by mixtures of Burgers vector b1, b2, b3 as shown in Figure

1.3(b) [2], because these Burgers vector are equivalent, and they change the stacking sequence of

the {111}-type slip plane the same way as it is sheared by Burger vector b1 multiple times. The

macroscopic strain produced by this process is not large as they shear the {111}-type slip planes

in different directions, producing a net small macroscopic strain.

In CG materials the deformation twins are produced by the first process i.e. by the glide of the

same Burgers Vector b1 on adjacent {111}-type slip planes as shown in Figure 1.3(a) [2]and

deformation twins in NC materials are produced by the second process i.e. by the glide of mixtures

of Burgers vector b1, b2 and b3 on adjacent {111}-type slip planes as shown in Figure 1.3(b) [2].

1.1.1.2 Competition between Deformation Twinning and Dislocation Slip in FCC Metals

The interactions between twins and gliding dislocations at twin boundaries are responsible for

improving structural performance of NC materials. These interactions are the result of competition

between these two mechanisms. To understand competition between these two mechanisms, we

have to neglect polycrystalline material due to too many complexities and consider only single

crystal for simplicity. In a single crystal, the two main factors which affect the competition between

deformation twinning and dislocation slip are crystallographic orientation and generalized planar

fault energy (GPFE) curve.

A sharp edge crack is considered to be the best case, to study the effect of crystallographic

orientation on deformation mechanism, because the sharp edge crack due to its high stress

concentration somewhat nullifies the effect of energetic barrier on deformation mechanism. The

effect of crystallographic orientation on deformation twinning and dislocation slip in a single

crystal with a sharp edge crack has been studied in detail by Yamakov et al [3]. The crystal

geometry used in his study is illustrated in Figure 1.4 [3]. The (111) slip plane makes an angle 𝜃

with crack plane (x axis) and lies at the intersection of crack front which is along z axis and crack

plane. The angle 𝜑 is between normal to crack front in (111) slip plane and slip direction.

5

Figure 1. 4: Crystal geometry used by Yamakov et al. See main text for description of symbols [3].

Yamakov et al [3] reported that angle θ has no effect on the deformation mechanism with the mode

I loading case, whereas angle φ controls the deformation mechanism. The results of the study are

shown in Figure 1.5 [3], when angle φ is zero i.e. [112] direction is perpendicular to crack front,

deformation twinning is the prevalent deformation mechanism, whereas when angle φ is 30o i.e.

[011] direction lies perpendicular to crack front, full dislocation slip is the dominant deformation

mechanism.

Conventionally, stacking fault energy is considered as the fundamental property that affects the

twinning behavior. CG materials with low stacking fault energy deform by deformation twinning

process and the similar trends are reported by researchers in NC materials [4-6]. Recent studies,

however, indicate that stacking fault energy alone is not sufficient to describe the twinning

propensity and the GPFE curve also greatly affects the twinning propensity [7-9]. The GPFE curve

is created by shearing the crystal on successive {111}-type planes along direction with a

Burgers vector of b = 𝑏 = < 112 >, where 𝑎 is the lattice constant. The GPFE curves for

different materials are accurately calculated by ab initio approaches.

6

Figure 1. 5: Simulation snapshot after loading the cell to the indicated stress intensity factor(KI). The

deformation twins are shown in a, b, and c which have an orientation (𝜃, 𝜑) of (35.26o, 0o), (54.74o, 0o),

(70.53o, 0o) respectively. Full dislocation slip is shown in d, e and f which have an orientation of (𝜃, 𝜑) of

(35.26o, 30o), (54.74o, 30o), (70.53o, 30o) respectively [3].

Figure 1.6 illustrates the general GPFE curve, the most important points of GPFE are unstable

stacking fault energy (𝛾 ), stable stacking fault energy or stacking fault energy (𝛾 ) and unstable

twin fault energy (𝛾 ). NC nickel has very high 𝛾 but still deform by forming stacking faults

and twins which shouldn’t happen as per the conventional criteria. The GPFE curve can be used

to explain this discrepancy, when stacking fault is created by the glide of first partial and the glide

of trailing partial to return crystal to pristine condition is a function of 𝛾 − 𝛾 which is high

for nickel as compared to Al. This makes generation of stacking faults easier in NC nickel and also

7

the gap between 𝛾 and 𝛾 is not so large, therefore twin generation is possible in NC nickel

when stacking faults are created.

Figure 1. 6: GPFE curve for Nickel created using Molecular Dynamics.

The GPFE curve has been used by many researchers to explain the deformation mechanisms of

various NC materials e.g. NC Ni, Cu and Al [7].

1.1.1.3 Grain Size, Strain Rate and Temperature effect on Deformation Twinning in the NC FCC Metals

The formation of twins in NC FCC materials is also influenced by grain size, strain rate, and

temperature variation. Numerous experimental studies have reported that deformation twinning is

not favorable at lower grain size in CG materials and it doesn’t depend on the type of crystal

structure. The Hall-Petch are slopes are calculated for dislocation slip and deformation twinning

for various crystal structures (FCC, BCC, and HCP) by Meyers et al [10]. It was reported that Hall-

Petch slope for deformation twinning increases at a much higher rate than dislocation slip i.e.

dislocation slip is more favorable at lower grain size in CG materials. We can also say that from

Hall-Petch slopes, also shown in Figure 1.7 [2], that stress required to generate deformation twins

increases at a much larger rate than dislocation slip.

8

Figure 1. 7: Deformation twinning and full dislocation slip Hall-Petch relationship with decreasing grain

size in CG materials, where 𝜏 represents the shear stress and d is the grain size [2].

However, when the size is further reduced below 100nm deformation twinning is observed in many

NC materials i.e. Hall-Petch slopes are not valid when the grain size is reduced below 100nm i.e.

when we enter nanometer regime. The deformation twinning is reported to be the most dominant

deformation mechanism for NC FCC materials [11,12] e.g. NC copper and NC nickel which have

high to medium 𝛾 created deformation twins during deformation.

The temperature and strain rate effects are also studied by various MD and experimental studies

and it was observed that lower temperature and higher strain rate promote deformation twinning.

Zhao et al. [13] reported that ultra-fined-grain (UFG) copper sample prepared by equal channel

angular pressing (ECAP) at room temperature didn’t form deformation twins during tensile testing.

However, when the UFG copper sample was cryogenically extruded and rolled at 77k, a large

number of deformation twins are observed during the tensile testing. This study by Zhao et al.

confirms that low temperature promotes deformation twinning in NC FCC materials.

Wu and Zhu [14] experimentally investigated the effect of increasing strain and strain rate on NC

nickel foil which has approximately 130 grains with an average grain size of 25nm. The NC nickel

foil was deformed under three load conditions :(a) 3 x 10-3 s-1 strain rate to 5.5% strain and 1.5

GPa flow stress under quasi-static tension, (b) 2 x 10-2 s-1 strain rate to 9.8% strain under rolling,

(c) ~2.6 x 103 s-1 strain rate to 13.5% strain under split Hopkinson pressure bar (SHBP) test. High-

9

resolution electron microscopy (HREM) was used to examine ~130 grains of each sample to get

better statistics. Wu and Zhu [14] reported that fraction of grains with deformation twins soared

from 28% in tension test to 38% in rolling test to 44% in SHPB test. This experimental study

asserts that increasing strain rate will increase the density of deformation twins.

Till now, we have discussed the basics of deformation twinning in FCC materials and what are the

factors that lead to competition between deformation twinning and dislocation slip. Grain size,

temperature and strain rate effect on deformation twinning is also discussed. Despite of all this

research in deformation twinning field, there is not a single analytical model which can predict the

competition between twin nucleation and twin thickening in NC FCC materials. An analytical

model is necessary because it saves time, otherwise, we must run simulations or conduct

experiments to know material tendency towards deformation twinning, which has a significant

impact on the work hardening behavior of the material.

1.1.2 Deformation Behavior of Metallic Multilayers

Metallic Multilayers (MLs) consists of two or three alternating material structures. As the

mechanical behavior of the MLs is controlled by the interface between the layers, the interface

engineering can be used to improve the structural performance of MLs.

The interfaces in MLs are of three types: coherent, semi-coherent and incoherent interfaces. MLs

with FCC/FCC alternating layers commonly have coherent interfaces. They have same lattice

structure and comparable lattice constants across the interface. The slip planes and directions are

also almost uninterrupted incoherent interfaces. However, the small difference in lattice constants

across interface leads to very high stresses, which acts as a barrier for dislocation transmission.

MLs with FCC/HCP alternating layers commonly form semi-coherent interfaces. They have same

lattice structure but a large difference in lattice constant values across the interface, this results in

a low shear strength interface. Interactions between glide and misfit dislocations make this type of

interface as a barrier to dislocation glide. MLs with FCC/BCC alternating layers usually creates

incoherent interfaces. They have different crystallographic structure and large lattice mismatch,

due to this slip planes and directions are interrupted at the interface which leads to less stable

10

interface than semi-coherent. The interface shear strength is low and interface itself is acting as a

barrier to dislocation glide. Figure 1.8 [15] shows Cu/Nb ML with different layer thicknesses.

Figure 1. 8: Cu/Nb ML with two different layer thicknesses of (a) 0.8nm and (b) 20nm [15].

Similar to polycrystalline materials, MLs also show length scale dependent deformation

mechanisms. The deformation mechanisms of MLs vary with a layer thickness (h), when the layer

thickness is in micrometer regime, the Hall-Petch relationship (𝜎 ∝ ℎ / ) can be used to

describe the yield strength of the MLs with varying layer thickness [15]. The Hall-Petch

relationship is based on the argument that yielding of ML is occurring because of dislocation pile

up mechanism. When layer thickness is reduced to nanometer scale, the Hall-Petch relationship is

11

no longer valid, even though yield strength is still increasing with decreasing layer thickness which

even surpass Hall-Petch estimates. This strengthening is because of the confined layer slip of single

dislocation with in individual layers [15]. When layer thickness is further reduced to 1~2 nm, the

decrease in yield strength is observed which is similar to inverse Hall-Petch phenomena. At this

length scale, the barrier strength of interface is reduced which promotes transmission of dislocation

between layers [15]. Figure 1.9 [15] illustrates length scale dependent deformation map of MLs.

Figure 1. 9: Deformation mechanism map of ML with varying layer thickness [15].

At nanometer length scales, the yield strength of the MLs has shown to increase by a large amount

but at the same time their ductility has significantly reduced. For example, when Ag/Cu ML is

loaded under uniaxial tension whose grain size is approximately equal to the individual layer

thickness, a large decrease in ductility is observed with a reduced layer thickness in nanometer

regime [16].

Till now, different types of interfaces in MLs as well as their deformation mechanisms at different

length scales are discussed. However, most of the studies are conducted on MLs which have

dissimilar metallic components. There are only a few studies on monolithic MLs [17-19], but as

pointed out by Kurmanaeva et al. [17], these MLs contain both FCC and BCC phases which leads

to lattice mismatch across the interface, therefore convoluting monolithic study of ML. Daly [20]

12

studied NiCo monolithic multilayer which had FCC unit cell and no improvement in ductility was

observed. To my knowledge, there is not a single study on monolithic BCC ML and grain size

variation study in monolithic BCC multilayer is also not performed. As we know deformation

twinning can increase both strength and ductility of the material, therefore in monolithic BCC ML

our main focus is on deformation twinning mechanism.

1.1.2.1 Deformation Twinning in BCC Materials

The deformation twinning is also observed in BCC materials under high strain rate and low

temperature. The deformation twinning occurs on {112} systems in BCC metals, the

stacking sequence of {112} plane is ABCDEFABCDEFABCDEF……... The Burgers vector

required for deformation twinning in direction on successive {112} planes in BCC metals

is ao/6 with a magnitude of ao/√12, where ao is lattice parameter.

The deformation mechanism in NC BCC materials is very less studied compared to NC FCC

materials. Twin band formation was observed from crack tips in MD simulations of Mo by Tang

and Wang [21]. NC BCC Mo was studied by Frederiksen et al. [22] using MD dynamics and

deformation twinning was observed. Using MD single crystal BCC Fe was simulated by Marian

et al. [23] and observed that with rising strain rate, movement of screw dislocation becomes rough

from smooth and deformation twinning emerges as the dominant deformation mechanism. Wang

et al. [24] investigated NC BCC Ta using nanoindentation. Later HREM showed substantial

deformation twinning in grains and numerous twins with different orientation were observed in

some grains, signifying in NC Ta deformation twinning is a dominant deformation mechanism.

Zhang et al. [25] deformed NC Mo with zone axis using MD and found that deformation

twinning may be the prevalent deformation mechanism for this type of orientation. Zhang et al.

[26] in another MD study on NC Mo observed that deformation twinning is a major deformation

mechanism in NC Mo with zone axis irrespective of the grain size, and as grain size is

reduced from 34.4 nm to 8.5 nm ductility of NC Mo is increased due to grain boundary mediated

plasticity (GBMP). These findings in NC BCC materials are in line with the findings in NC FCC

materials except for an increase in ductility at lower grain size.

13

It is clear from above section that zone axis promotes deformation twinning in BCC

materials, therefore the MD study of BCC ML will be performed with same zone axis.

1.2 Thesis Motivation

The interaction between gliding dislocations and twins leads to increase in strength and ductility

of the material [27, 28]. The strength of the material is increased due to large strain hardening, this

happens because twin boundaries act as obstacles to gliding dislocations, but unlike grain

boundaries, twin boundaries preserve the ductility of the material [27]. During deformation

multiple twins nucleates in a low stacking fault energy structure which causes dynamic grain

refinement of the microstructure. The continuous structural segmentation basically impedes

dislocation motion at every step of the refinement of the structure by twin boundaries and causes

very high strain hardening of the material. The effect of deformation twinning on strain hardening

has been studied in TWIP steels and also included into many mathematical models. Bouaziz with

the help of other researchers has developed one such model [29-32], where the increase to the

twinned material fraction is assumed to be caused by the nucleation of new twins with identical

average twin thickness. Which is contradictory to the true nature of twinning where nucleation as

well as thickening of existing twins can take place in response to deformation. This flaw in these

models raises an intriguing question how can we predict the true nature twinning in materials

during plastic deformation i.e. under which conditions existing twins will thicken and under which

circumstances new twins will nucleate. Matthew Daly from Computational Materials Engineering

Lab has developed a Kinetic Monte Carlo model [20] which can predict whether new twins will

nucleate, or existing twins will thicken in NC FCC materials. One of the global objectives of this

thesis is to investigate the competition between twin nucleation and twin thickening in NC FCC

materials and support KMC model [20].

MLs with heterogeneous construction due to lattice mismatch suffers from low ductility [16]. MLs

with monolithic construction are very less studied. Monolithic MLs are expected to have no lattice

mismatch and may lead to increase in ductility. One such attempt is made by Matthew Daly [20]

from Computational Materials Engineering Lab to study single chemistry NiCo ML which has

only FCC phase. Figure 1.10 [20] illustrates the ML template used by Mathew Daly for his

14

experimental studies. In this ML there are alternating layers of NC and CG NiCo material. The

NC layer is used to increase the flow strength of the material and CG layers to increase ductility.

Mathew Daly [20] reported that when the thickness of CG layer (𝑡 ) is reduced to grain size of

CG layer (𝐷 ) i.e. when 𝑡 ≅ 𝐷 , the rule of mixture calculations under predicts the stress

strain behavior of the multilayer. To understand this anomaly, Mathew Daly [20] wrote a

MATLAB code to generate the required structure (and it took him over 1 year to write this code)

and performed targeted MD studies. The cause of this behavior was found to be continuous

refinement of the CG layer due to deformation twinning. To my knowledge, there is not a single

study monolithic study of BCC material, therefore another global objective of my thesis is to

extend this code to BCC material.

Figure 1. 10: Multilayer structure, where 𝑑 and 𝑑 is grain size of coarse grain and nanocrystalline

layer, 𝑡 and 𝑡 is layer thickness of coarse grain and nanocrystalline layer [20].

15

1.3 Thesis Objectives

For real-world applications, engineers choose either strong material or ductile material but not both

because materials are rarely strong and ductile at the same. NC materials are strong, but they lack

ductility, but we also know deformation twinning can help in increasing both strength and ductility

of NC materials. Therefore, deformation twinning is studied in both NC FCC materials and BCC

ML with the following global objectives:

1. Investigate Competition between Twin Nucleation and Twin Thickening in NC FCC

materials:

Select proper potentials for MD simulations, by calculating their respective GPFE

curves for deformation twinning.

Perform nanowire simulations for FCC materials and support KMC model.

Investigate the effect of strain rate, nanowire length, and temperature on twin

nucleation and twin thickening.

2. Extend FCC ML code to BCC ML Structures:

Investigate CG–NC interface mediated deformation twinning mechanism in

Tantalum ML using MD.

Investigate parametric aspects of BCC ML to improve its ductility.

Investigate the effect of simulation cell thickness, and strain rate on ML

mechanical properties.

16

1.4 Thesis Organization

The organization of this thesis as follows: chapter 1 covers theory of deformation twinning in NC

FCC and ML BCC materials. The effect of GPFE, orientation, temperature and strain rate on

deformation twinning in NC materials is also explained. It also covers literature review of MLs

related to this thesis. Chapter 2 describes the MD concepts used to simulate nanowire and the

generation of BCC ML. Chapter 3 investigate competition between twin nucleation and twin

thickening using nanowire simulations and shows plots for number of twins and average twin

thicknesses vs twinning fraction for NC FCC materials. In Chapter 4 BCC ML model is

constructed using MATLAB and simulated in MD to investigate the NC-CG interface mediated

deformation twinning mechanism in BCC ML. Chapter 5 shines light on conclusions of this work

and recommendations for future work

17

Chapter 2

Computational Methodology

2.1 Molecular Dynamics

Molecular dynamics (MD) is a simulation method which is governed by the principles of classical

mechanics. MD in simpler terms is an integration of Newtonian equations of motion to study the

dynamic evolution of the system, system here represents an ensemble of interacting particles

(atoms) in a liquid, solid, or gaseous state. In MD, atoms are considered as a particle with point

mass and the presence of nuclei and electrons are neglected. Since the presence of nuclei and

electrons are neglected, MD can simulate up to billions of atoms today, but the interatomic

potential which describes interactions between atoms have to be generated empirically.

The force acting on each atom due to its interactions with other atoms is described by the following

equation:

𝐹 = 𝑚 𝑎⃗ (2.1)

Where, 𝑚 is the atomic mass and 𝐹 is the force vector acting on the ith atom, and 𝑎 is the

acceleration vector of the ith atom.

The main ingredient of the MD simulation is choosing the right interatomic potential for the

system, which is a function of atomic positions 𝑟 and computes the potential energy of the system

when atoms are lying in a particular arrangement. The interatomic potential can be presented as

𝑉( 𝑟 ). The force vector 𝐹 can also be calculated by the following expression:

𝐹 = −( ⃗ )

⃗ (2.2)

18

We also know, the acceleration vector can also be written as:

𝑎⃗ =⃗ (2.3)

From equation 2.1 and 2.3, equation 2.2 can be rewritten as:

𝑚⃗

= −( ⃗ )

⃗ (2.4)

Therefore, in MD a system with given initial conditions, atomic positions and interatomic potential

is initialized and the new atomic positions and velocities are calculated by integrating equation 2.4

over a time interval (time step) 𝛿𝑡. The new atomic positions are used as inputs and are fed again

into equation 2.4 and by integrating this equation again over timestep 𝛿𝑡 we get new atomic

positions after 2 timesteps. This process can be repeated over the required number of timesteps

until we get the system properties of interest. Using this process, atom trajectories are calculated

in 6N-diemnsional space, where 6N represents 3N momenta and 3N positions [33]. To observe the

mechanism of interest in MD, the timestep should be correctly defined. If large timestep is defined

then simulation will take less time, but we may the miss the phenomena of interest and if time step

is too small, simulation will take forever to complete. In MD, the properties of interest such as

mechanical and structural properties are calculated via statistical analysis of raw data (forces,

momenta and atomic positions etc.) generated at each timestep. The MD algorithm is presented by

the following flowchart:

19

Figure 2. 1: MD algorithm flowchart

Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [34] is used to conduct

the molecular dynamics simulations in this thesis. LAMMPS is a free open source code and it is

optimized to run on parallel computers. LAMMPS can simulate up to billions of atoms, this due

to the spatial decomposition methods which are used to divide the simulation domain into multiple

tiny domains which are assigned to each processor on parallel computers.

2.1.1 Interatomic Potential

Today, MD can simulate atomic systems which consist of billions of atoms, because MD ignores

the contributions from nuclei and electrons. Atoms are basically treated as spheres with point mass

and to describe the interactions between these spheres we need interatomic potentials. The

interatomic potentials determine the energy balance between repulsion and attraction, when atoms

are close enough to interact with other. In MD, there is not a single universal potential which can

be used for every simulation, because every material is different. Materials are different from each

other due to different nature of interactions between their atoms, which determines their material

20

properties. MD simulation results are completely dependent on the selection of interatomic

potential; therefore, we have to give special attention while choosing interatomic potential. Even

for same material system, we have to be careful in selecting potential because potentials are

empirically generated to describe specific properties of the system. Data from experiments as well

as from ab-initio methods are used to generate interatomic potentials.

In this thesis, since we are studying metallic systems, Embedded Atom Method (EAM) interatomic

potentials are used for MD simulations. Traditionally, Pair potentials were used in MD

simulations, because they were simple to generate and captured pair wise interactions between

atoms. The pair potentials are successfully implemented in inert gases but cannot be used for

metallic system because they couldn’t capture multi-atom effect. The metallic systems have long-

range columbic interactions, extending up to 8-12 atoms, which pair potentials couldn’t capture.

Pair potentials assume that bonds between atoms are independent of each other, in reality it is not

the case. In metallic systems, as the coordination number increases strength of the bond decreases.

Therefore, pair potentials couldn’t capture many-body effect and coordination number dependent

strength of the bond between atoms. Hence, researchers started looking for a potential which can

be used for metallic systems (low-symmetry systems) and also includes coordination dependent

aspects of bonding. Keeping these factors in mind, Daw and Baskes [35] came up with embedded

atom method. This method assumes the metallic system energy as the energy value we get by

embedding an atom into the electron cloud of the neighboring atoms. The EAM potential captured

the physical aspects of bonding in metallic system, by assuming each atom is embedded into the

electron densities of the local atoms and describes the interactions between atoms which is more

complicated than pair potential. The following mathematical form of the EAM potential is given

by Daw and Baskes [35];

𝑉 = 𝑃𝑎𝑖𝑟 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝐸𝑚𝑏𝑒𝑑𝑑𝑖𝑛𝑔 𝐸𝑛𝑒𝑟𝑔𝑦

𝑉 =1

2 𝑉 (𝑟 )

, ( )

+ 𝐹 ( 𝜌 (𝑟 ))

Here, 𝑉 is two atom electrostatic interaction, 𝐹 is embedding energy, 𝑟 is separation between

two atoms, and 𝜌 is the spherically averaged electron density.

21

2.2 Nanowire Model and Calculation Methodology

2.2.1 Nanowire Model Generation

The nanowire models were generated for Silver (Ag), Aluminium (Al), Copper (Cu) and Nickel

(Ni) materials. The nanowire models were created from perfect FCC single crystals using the

lattice constants of their respective materials and then by deleting all atoms which fall outside of

the square region of length d. Further simulation details like number of atoms used in nanowire

simulations, relaxation and loading durations are given in chapter 3 of this thesis.

2.2.2 EAM Potential Selection

The most important input to MD which determines the accuracy of simulation results mainly

depends on the selection of interatomic potential selection. As mentioned earlier, the EAM

potentials are good enough to describe the properties of metallic systems, but they are not unique

e.g. for a single material system there exits many EAM potentials. Therefore, main question arises

here is, how do we select EAM potential to investigate competition between twin nucleation and

twin thickening in NC FCC materials? The answer to this question is GPFE curve, as we already

know from chapter 1 that GPFE curve determines the twinning behavior of the material.

The GPFE curves are calculated accurately using the ab initio method (e.g. density functional

theory). Hence, EAM potential in MD should be selected such that GPFE curve from EAM

potential matches closely with DFT GPFE curve points [48]. The GPFE curve in MD is calculated

by applying homogenous shear with Burgers vector of < 112 > along successive {111} slip

planes in direction of FCC single crystals. See figure 2.2 for an example of GPFE curve

calculated using MD.

22

Figure 2. 2: GPFE curve calculated for Nickel using MD.

2.2.3 Crystal Analysis Tool

Crystal Analysis Tool (CAT) developed by Alexander Stukowski [36] is used to identify twin

boundaries in FCC nanowires during strain loading. CAT works by post processing dump files

written by MD and recognizes the defect and lattice structures created by atoms in these files. The

list of all structures recognized by CAT are not hardcoded into the program but are loaded from

an external pattern catalog file. The CAT doesn’t identify all the structures loaded in pattern

catalog file automatically, as it is time consuming to search for all patterns and may lead to

incorrect results when searched on inappropriate systems. The CAT only identifies user specified

patterns e.g. stacking faults and twin boundaries etc., which is more efficient and less time

consuming.

2.2.4 Twinning Parameters

The twinning parameters such as number of twins, average twin thickness and twinning fraction

are calculated using custom made MATLAB code. The output files generated by CAT are loaded

into OVITO [37] (visualization and analysis software), this tool is also developed Alexander

Stukowski. In OVITO, only twin boundaries atoms are kept, and rest of the atoms are deleted, then

23

these files are saved as images with pixel settings of 1024 x 768. Figure 2.3 shows ones such image

which contains only twin boundaries. These images are further post processed in MATLAB to

calculate number of twins and average twin thicknesses of FCC materials until twinning fraction

reaches a value of 0.3. See Appendix A for MATLAB code details.

Figure 2. 3: Image file containing only twin boundaries and black lines represents simulation cell.

2.3 BCC Multilayer Model Generation

In this section, methodologies which are used to generate NC and BCC ML microstructure are

presented. Firstly, a methodology which is used to create NC microstructure for MD simulation is

discussed, which will be modified later to create BCC ML microstructure. The are many

approaches which can be used to generate NC microstructure. The key criteria for selecting the

scheme for NC microstructure creation is that the grains created by this scheme should fully

partition the domain. One of the most famous approach which is used to generate NC

microstructure is Voronoi tessellation and it has been used by many researchers for nanocrystalline

studies in MD [38,39]. Voronoi tessellation works by randomly assigning points within a domain

and then using those points to partition the domain into cells and these cells represent grains within

a microstructure from MD analysis point of view. Another important feature of Voronoi

tessellation is that by controlling the number of seeds within a partitioned volume, the grain size

of the microstructure can be specified [20]. In this work, Voronoi++ library created by Rycroft

[40] is used for Voronoi tessellations. Figure 2.4 shows 3d and 2d BCC NC microstructures with

grain size of 20nm generated using Voronoi tessellation and custom MATLAB code. In Figure 2.4

(a) and (c), the partitioning of the volume generated by Voronoi tessellation is not completely

dense, but by enforcing periodic conditions in Voronoi tessellation, complete partitioning of the

domain is accomplished (see Figure 2.4 (b) and (d)).

24

Figure 2. 4: (a) 3d and (c) 2d Voronoi tessellations for 20nm grain size. (b) 3d and (d) 2d BCC NC

microstructure generated using custom MATLAB code.

Using Voronoi tessellation, NC microstructure of any grain size can be generated, but due to the

limitations of present-day computational hardware, MD simulation cell size is restricted to ≤ 10

million atoms [20]. For a 3d NC microstructure, these limitations restrict the size of the simulation

cell to 20 x 20 x 20 nm [20]. Restrictions on size of the simulation cell due present-day

computational hardware are troublesome for ML analysis using MD. However, for modeling ML

microstructure quasi-3d microstructure approach can be used, where one dimension of the

simulation cell is sacrificed to have larger lateral dimensions. The sacrificial dimension can be

decreased to approx. 2nm in size, without introducing any spurious size effects on the deformation

mechanisms [20]. The slip system of the active deformation mechanism should be perpendicular

to sacrificed dimension to prevent any periodic image artifacts on the simulation cell [20].

The quasi 3d approach can only be used in NC metals, if all the NC grains have same z axis (zone

axis), to avoid any dislocations going through the periodic images of the simulation cell [20]. In

this work, quasi-3d microstructures are used instead of full 3d microstructures due to the length

scale issues of MD. The empty grains created by Voronoi tessellation are filled with atoms by

atoms generation scheme which is implemented in MATLAB. The atoms generation scheme

creates large single crystals of BCC lattice with a lattice constant of 0.3304 nm (Tantalum lattice

constant) with assigned axis. These single crystals are larger in size in comparison to grain size of

the NC microstructure. The atoms which fall outside of the grain boundary are deleted using the

same scheme. In this way, required quasi 3d NC BCC microstructure is generated. Figure 2.4 (d)

also shows one such example, the intragranular region of grains are clearly filled with BCC atoms

(Blue colored using centrosymmetry algorithm). As shown in Figure 2.4 (d), The lateral

25

dimensions of the microstructure are in the plane of the paper and the sacrificed dimension is

orthogonal to the plane of the paper. In Figure 2.4 (d), the NC BCC microstructure is fully

partitioned and most of the nodes in microstructure are triple junction nodes. Nonetheless, a small

proportion of quaternary nodes are also present in NC BCC microstructure, the number of

quaternary nodes is larger here in comparison to full 3d structure due to the sacrificial dimension

which places limitations on the distribution of Voronoi seeds. Periodicity is also maintained in the

generated NC BCC microstructure.

The BCC ML microstructure is generated by modifying the BCC NC microstructure generation

algorithm. The BCC ML is created by inserting a large slab into the BCC NC microstructure as

shown in Figure 2.5. Here, slab represents the CG layer in ML structure where 𝑡 ≈ 𝑑 , in

actual MLs the CG layers are of the order of 1 𝜇𝑚. The current hardware limitations don’t allow

us to model CG layers of that dimensions. Therefore, big enough CG layer (slab) is inserted in to

the NC microstructure which can be analyzed in MD. Figure 2.5, shows the schematic of the BCC

ML generated using modified BCC NC algorithm. In the schematic, NC grains are split along the

transecting pathway to accommodate CG layer. The common zone axis of the slab and NC grains

(NC layer) is kept same, as it is a requirement of quasi 3d approach. The periodic boundary

conditions are also maintained using the modified BCC NC algorithm.

As shown in Figure 2.5, there is overlap between slab and NC grains. Now, two different methods

can be used to generate BCC ML microstructure. Firstly, overlapping atoms from NC grains can

be deleted keeping slab atoms, which generates a straight interface between CG and NC layer in

BCC ML. Contrarily, overlapping slab atoms can be deleted keeping NC grains atoms, which

generates zig zag interface between CG and NC layer in BCC ML. Figure 2.6, shows the BCC

MLs generated from both approaches. Other than interface change between CG and NC layer, both

MLs are exactly same with identical zone axis.

26

Figure 2. 5: The schematic of BCC ML generation (where 𝑡 ≈ 𝑑 ) for MD simulations. The NC grains

generated by Voronoi tessellation are split along transecting pathway and a 100 nm CG layer is inserted,

while enforcing periodic conditions atoms are populated in to the grains and CG layer. The purple color

represents BCC atoms with a lattice constant of 0.3304 nm and black color shows grain boundary atoms.

The BCC and grain boundary atoms are identified using common neighbor analysis algorithm in OVITO

[37].

Figure 2. 6: ML microstructure generated by (a) Straight boundaries and (b) zig zag boundaries at NC and

slab layer interface. The purple color represents BCC atoms with a lattice constant of 0.3304 nm and black

color shows grain boundary atoms. The BCC and grain boundary atoms are identified using common

neighbor analysis algorithm in OVITO [37].

27

For rest of the thesis, BCC ML generated by second approach is used i.e. BCC ML having zig zag

interface between CG and NC layer. The reason behind selecting second approach is that the first

approach (straight interface between CG and NC layer) artificially decreases the grain size of the

grains which are at the boundary and introduces new grain boundaries. The BCC ML

microstructure already has only a few grains due to the length scale limitations of the MD and if

grain size is altered this might lead to wrong results of tensile simulation studies. The slab which

is inserted into the NC microstructure is pristine i.e. free from defects and dislocations sources,

therefore permits the targeted studies of NC-CG interface mediated deformation mechanisms.

28

Chapter 3

Investigate Competition between Twin nucleation and Twin Thickening in NC FCC Materials

3.1 Introduction

In today’s world, the need for materials which are both strong and ductile is increasing, but

materials are rarely simultaneously strong and ductile. Many methods have been proposed by

researchers to improve strength and ductility of materials such as inserting hard material particles

into the softer material matrix and grain boundary refinement, but they almost all suffer from low

ductility. The basic principle behind grain boundary refinement is that by introducing high density

of incoherent grain boundaries which will act as barriers for dislocations motion, the strength of

the material can be increased i.e. by impeding the motion of dislocations, but due to the incoherent

nature of grain boundaries the ductility of the material is compromised. The boundaries can also

be created in the crystal by introducing twins. There are many ways by which twins can be

introduced in a crystal such as during deformation, during processing of material and

recrystallization of the material. Twin boundaries have low energy compared to grain boundaries,

therefore twin boundaries are expected to be more mechanically and thermally stable. Recent

studies have shown that the interactions between gliding dislocations and twin boundaries have

shown to increase both strength and ductility of the material [27,28]. The high density of twin

boundaries strengthens the material comparable to that of grain boundaries. Contradictory to grain

boundary refinement with decreasing twin boundary spacing the ductility of the material increases

[44]. This is because of interactions between gliding dislocations and twin boundaries aides in loss

of coherency in twin boundaries which leads to improvements in ductility and hardening of the

material [27,41-43].

The effect of twinning especially deformation twinning on strain hardening has been studied in

TWIP steels and also included into many mathematical models. Bouaziz with the help of other

researchers has developed one such model [29-32], where the increase to the twinned material

29

fraction is assumed to be caused by the nucleation of new twins with identical average twin

thickness. Which is contradictory to the true nature of twinning where nucleation, as well as

thickening of existing twins, can take place in response to deformation. The nucleation of new

twins will increase the density of twin boundaries in a crystal which lead to very large strain

hardening of the material, whereas thickening of existing twins will lead to a very small decrease

in grain size and therefore almost no improvement in strain hardening. This flaw in these models

raises an intriguing question, can the true nature of deformation twinning in materials can be

predicted i.e. under which conditions existing twins will thicken and under which circumstances

new twins will nucleate. My work is to investigate competition between twin nucleation and twin

thickening in NC FCC materials using nanowire simulations and support KMC model [20]. The

details of the KMC model are addressed in next section.

3.2 Kinetic Monte Carlo Model

The Kinetic Monte Carlo Model proposed by Mathew Daly is as follows [20]:

To evaluate the competition between twin nucleation and twin thickening during plastic

deformation on FCC materials, five typical FCC materials were selected which were Silver (Ag),

Aluminium (Al), Copper (Cu), Nickel (Ni) and Lead (Pb). These materials cover the extremes of

GPFEs, 2D KMC simulations were performed on these materials using Bortz et al algorithm [45].

Figure 3.1(a) [20] shows the schematic of KMC simulation cell considered for all FCC materials

with x and y oriented in and directions. The length and width of the KMC simulation

cell are 500b and 500d111, where b is the magnitude of Burgers vector for Shockley partials and

d111 is the spacing between {111}-type slip planes. The KMC simulation cell is representative of

a single grain in NC materials, therefore deformation twins are assumed to be formed by grain

boundary nucleation mechanisms. The deformation twins in KMC simulations are formed by the

glide of leading Shockley partials on successive {111} type planes as explained in chapter 1. KMC

simulations consider only deformation twinning process to evaluate the competition between twin

nucleation and twin thickening, therefore neglects dislocation slip processes such as nucleation of

trailing partials and dislocation cross slips etc.

30

Figure 3.1(b) [20] shows the typical GPFE curve for FCC materials with their stable and unstable

fault energies and activation energy barriers specified. The activation energy barriers (𝐸 ,...., 𝐸 )

is defined as the difference of the unstable energy of the subsequent fault and stable energy of the

existing fault. 𝛾 …....𝛾 are the unstable fault energies, where subscript denotes the number

of leading Shockley partials required to make the required fault. The activation barrier energy

required to make an embryonic twin is defined 𝐸 = 𝛾 − 𝛾 , but we know GPFE curve of

FCC material stabilizes after the formation of an extrinsic stacking fault (esf) [46], therefore 𝐸 ≈

𝛾 − 𝛾 . The energy of stable embryonic twin is ≈ 2𝛾 , where 𝛾 represents the energy of

a single twin boundary. 𝐸 which approx. equal to 𝐸 represents the activation barrier energy of a

thickened twin.

Figure 3. 1: (a) KMC simulation cell which is a representative of a single grain in NC FCC materials. 𝐸

represents the barrier for leading partial nucleation, the energy barriers (𝐸 ,...., 𝐸 ) are listed next to the

appropriate faults. 𝜎 is the external applied shear stress and 𝜎 is the stress required for partial glide (b)

Typical FCC GPFE curve for deformation twinning [20].

31

In KMC model, the rates (𝑅 ) of nucleation and glide of leading Shockley partials are expressed

by the following formula [20]:

𝑅 = 𝑅 exp {−( )

} (3.1)

Where i represents ith {111}-type plane of simulation cell, 𝑅 is the Debye frequency, V is the

activation volume (assumed 10b3), 𝑘 is the Boltzmann constant, T is temperature (all simulations

are performed at 300 K), 𝜎 and 𝜎 are elastic shear and activation stresses. 𝜎 is calculated after

each step in KMC model, if no Shockley partial is present on the ith plane, then 𝜎 represents the

activation barrier for leading Shockley partial nucleation and if Shockley partial is already present

then it represents a barrier for leading Shockley partial glide. Therefore, 𝜎 can be described by the

following relation [20]:

𝜎 =, 𝑓𝑜𝑟 𝑡𝑤𝑖𝑛 𝑛𝑢𝑐𝑙𝑒𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑡ℎ𝑖𝑐𝑘𝑒𝑛𝑖𝑛𝑔

( )exp − , 𝑓𝑜𝑟 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑔𝑙𝑖𝑑𝑒

(3.2)

Where 𝐸 is the activation barrier energy for the fault structure under consideration, 𝜗 is the

Poisson ratio, 𝐺 is the shear modulus and 𝜏 =( )

is the half width of dislocation core.

The elastic shear stress (𝜎 ) is the sum of the external applied shear stress and stress field of the

leading Shockley partial dislocation [20].

𝜎 = 𝜎 + ∑( )

[∆ ∆ ∆

∆ ∆] (3.3)

𝜎 is applied to make sure leading partial glide happens with in an acceptable timeframe and it is

also represents the stress required to stabilize twin embryo [47].

Equation 3.1 to 3.3 are executed in KMC model, to see the deformation twinning process.

Additional inputs which are required for KMC simulations are shown in Table 3.1 [20]. The initial

KMC simulation cell contains no dislocations or defects and simulations are stopped when the

twinning fraction (F) of simulation cell reaches 0.3. Number of twins (𝑁 ) are calculated after each

simulation step and average twin thickness (�̅�) is calculated by the following relation [20]:

32

�̅� = (3.4)

Where N is the length of the simulation cell (number of -type slip planes).

Table 3. 1: Parameters used in KMC simulations for different materials such as GPFEs(a) and external shear

stress (𝜎 )(b) etc. [20].

a Ref. [48] b Ref. [47]

c Shear modulus for different materials in < 112 > direction.

Hundred simulations per material are performed to get average values. Figure 3.2 [20] and 3.3 [20]

shows the twin evaluation in different materials at twinning fractions of 0.1, 0.2 and 0.3. Figure

3.4 (a) [20] shows the average number of twins calculated for different materials after each

simulation step and plotted with respect to twinning fraction and Figure 3.4 (b) [20] shows the

average twin thickness evolution for different materials calculated by equation 3.4 and plotted with

respect to twinning fraction. Non-integer values can be seen in figures 3.2 [20] and 3.3 [20], this

is due to the statistical average of the data.

Figure 3. 2: Twin evolution in Lead (Pb). Purple region represents twinned area and leading Shockley

partials are shown by blue arrows. Merging of twins is shown in (b) and thickening of twins can be seen in

(c) [20].

33

Figure 3. 3: Colored regions represents twin evolution in different materials at different twinning fractions

[20].

Figure 3.3 [20] shows, the largest number of twins occurs in Ag and Cu, and the smallest number

of twins occurs in Ni and Al. 𝐸 = 𝐸 − 𝐸 which is the difference between activation energy

barrier for nucleation and twin thickening, shows an inverse correlation with numbers of twins.

Ag and Cu have 𝐸 values of 6 mJ/m2 and 15 mJ/m2, and possess the highest number of twins,

where Al (55 mJ/m2) and Ni (72 mJ/m2) have lowest numbe