UNDERSTANDING the INTERACTION BETWEEN GRAIN BOUNDARIES and PRECIPITATES in Ni-Al USING MOLECULAR DYNAMICS

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UNDERSTANDING THE INTERACTION BETWEEN GRAIN BOUNDARIES AND PRECIPITATES IN Ni-Al USING MOLECULAR DYNAMICS

RACHEL L. MORRISON

Submitted in partial fulﬁllment of the requirements

For the degree of Master of Science

Thesis Adviser: Dr. Jennifer L.W. Carter

Department of Materials Science and Engineering

CASE WESTERN RESERVE UNIVERSITY

August, 2018 Understanding the Interaction between Grain Boundaries and

Precipitates in Ni-Al Using Molecular Dynamics

Case Western Reserve University Case School of Graduate Studies

We hereby approve the thesis1 of

RACHEL L. MORRISON

for the degree of

Master of Science

on June 26, 2018

Dr. Jennifer L.W. Carter Committee Chair, Adviser Department of Materials Science and Engineering

Dr. John J. Lewandowski Committee Member Department of Materials Science and Engineering

Dr. David H. Matthiesen Committee Member Department of Materials Science and Engineering

Dr. Saryu J. Fensin Secondary Adviser Los Alamos National Laboratory

1We certify that written approval has been obtained for any proprietary material contained therein. Dedicated to Joe Ledger, who believed in me during those moments when I did not believe in myself. Table of Contents

List of Tables vi

List of Figures vii

Acknowledgements xiv

List of Abbreviations xv

Abstract xvi

Chapter 1. Introduction1

Chapter 2. Background2

Nickel Superalloys2

Grain Boundaries7

Molecular Dynamics 11

Molecular Dynamics and Interfaces 19

Chapter 3. Methodology 22

Simulation Resources 22

Σ11 Boundary System 24

γ0 Precipitate 29

Chapter 4. Results 35

Σ11 Boundary Structure 35

System Relaxation 36

Σ11 Deformation Behavior 41

iv γ/γ0 Deformation Behavior 43

Σ11/γ0 Deformation Behavior 44

Chapter 5. Discussion 57

Σ11 Behavior 57

γ/γ0 Deformation 58

γ0 Formation 59

γ0 Strain Fields 60

γ0 Stress Fields 60

Orientation Study 61

Size Study 62

Interaction Study 63

Chapter 6. Conclusions 65

Chapter 7. Suggested Future Research 66

Grain Boundary Morphology 66

Grain Boundary Conﬁguration 66

Appendix A. 67

Appendix. References 71

v List of Tables

3.1 The elastic constants and standard Gibbs free energy as

calculated by the Mishin potential1. 23

3.2 An overview of the grain boundary creation steps. The x, y,

and z boundaries conditions are listed as periodic (p) or ﬁxed

(f). 29

vi List of Figures

2.1 SEM micrograph of the superalloy Rene´ 104 with a deposited

oxide grid and speckle pattern. Grain boundary sliding is

observed along the cracked boundary, highlighted by the

white circles. HRSEM backscatter inset image showing the two

0 primary phases, FCC Ni γ channels and L12 Ni3Al cuboidal γ

precipitates. Micrographs taken from Carter2.3

2.2 Grain boundaries are conventionally deﬁned by a rotation axis

(o = [hokolo]), a rotation angle (θ), and a boundary plane (nˆ =

3 (hnAknAlnA)). Figure recreated from Lejcek .8

2.3 A grain boundary between two grains that are described with

a Σ5 coincidence site lattice. Every 5th atom is colored white to

represent those lattice sites that overlap within the two grains3. 10

2.4 The center (gray) simulation cell is surrounded by images

of itself (white), and the equivalent atoms in each cell move

in tandem, as shown by the arrows. The atoms within the

simulation cell are allowed to interact with the atoms in the

periodic cells, and atoms can travel freely across the simulation

cell/periodic cell boundary. 15

3.1 Each simulation contains two grains (G1 and G2) separated

by a grain boundary (gray line). A γ0 precipitate with an

orientation (tilt, square, and inline) and size (small, large) is

inserted onto the boundary, coherent with G1. 31

vii 3.2 The system is bisected by a grain boundary (light gray plane),

and a γ0 is inserted in the center of the GB (orange cube). The y

boundaries have ﬁxed regions (dark gray) in order to apply a

velocity that will drive shear deformation (arrow). The system

is cut down in the x and z directions to allow the γ0 to interact

with its periodic images, as shown partially by the dotted lines. 33

4.1 The kite structure for the Σ11 boundary as modeled by (left)

MD and (right) TEM4 looking down the [110] tilt axis. 35

4.2 The temperature as a function of time during the relaxation

of an initialized MD system. The temperature increases to a

setpoint, and then reaches a steady state regime. Once the

system has reached steady state, further experiments can be

run. 36

4.3 The stress per atom averaged in rectangular regions

perpendicular to the grain boundary normal. The three

principal stress directions, σxx, σyy, and σzz are shown. 37

4.4 The Gibbs free energy of nucleation for a sphere of radius r

and cube of edge length 2r. The energy is calculated based off

values from the Mishin potential1. 38

4.5 Strain ﬁelds caused by the γ0 precipitates prior to deformation.

The strain is calculated by the displacement of atoms before

and after relaxation, normalized to one unit of slip for nickel.

viii Images are a x/z slice of the system. The position of the grain

boundary is indicated by the gray line. 39

4.6 Stress ﬁelds caused by the γ0 precipitates prior to deformation.

Images are a x/z slice of the system. 40

4.7 The stress vs. strain of a (left) Σ11 GB system and (right) γ/γ0

system without a grain boundary. Both systems experience

a stress drop after deformation (GBS or dislocation formation

respectively). 41

4.8 The centrosymmetry, excess volume, and normalized

displacement of atoms averaged in bins along the y axis of

the (left) Σ11 and (right) γ/γ0 systems before and after shear

deformation. 42

4.9 Snapshots of the Σ11 GB system during sliding. The (left)

initial structure, (center) an intermediate structure, and (right)

the disordered boundary after sliding. The atoms are colored

according to centrosymmetry; dark blue atoms have a perfect

FCC environment and all other colors deviate from an FCC

environment. 43

4.10 A snapshot of the γ/γ0 system under shear deformation.

The γ0 is colored white, and the remaining atoms are colored

according to centrosymmetry; dark blue atoms have a perfect

FCC environment and all other colors deviate from an FCC

environment. 44

ix 4.11 (left) The stress vs. strain of the Σ11 and small γ0 tilt, square,

and inline systems after shear deformation. (right) The

displacement of atoms averaged in bins along the y axis of

the Σ11 and small γ0 tilt, square, and inline systems after

deformation. 45

4.12 The maximum stress of the small tilt, small square, small

inline, and Σ11 systems prior to deformation, measured from

the stress strain curves. 46

4.13 The width of the centrosymmetry curves before and after

deformation for the small tilt, small square, small inline, and

Σ11 systems, which represents the extent to which the system

disorder is localized to the grain boundary (narrower = more

localized, wider = less localized). Centrosymmetry curves are

shown in AppendixA. 47

4.14 An x/y plane of the small tilt γ0 conﬁguration before (left

and center) and after (right) GBS. The γ0 is coherent with the

right grain. Atoms that have an FCC environment are colored

dark blue (nickel grains) or white (γ0), and all other atoms are

colored according to centrosymmetry. 47

4.15 An x/y plane of the small square γ0 conﬁguration before (left

and center) and after (right) GBS. The γ0 is coherent with the

right grain. Atoms that have an FCC environment are colored

dark blue (nickel grains) or white (γ0), and all other atoms are

colored according to centrosymmetry. 48

x 4.16 An x/y plane of the small inline γ0 conﬁguration before (left

and center) and after (right) GBS. The γ0 is coherent with the

right grain. Atoms that have an FCC environment are colored

dark blue (nickel grains) or white (γ0), and all other atoms are

colored according to centrosymmetry. 48

4.17 (left) The stress vs. strain of the small vs. large γ0 tilt, square,

and inline systems after shear deformation. (right) The

displacement of atoms averaged in bins along the y axis of the

small and large γ0 tilt, square, and inline systems after shear

deformation. 50

4.18 The maximum stress of the small and large tilt, square, inline

systems, and the Σ11 system prior to deformation, measured

from the stress strain curves. 51

4.19 The width of the centrosymmetry curves before and after

deformation for the small and large tilt, square, inline systems,

and the Σ11 system, which represents the extent to which the

system disorder is localized to the grain boundary (narrower

= more localized, wider = less localized). Centrosymmetry

curves are shown in AppendixA. 51

4.20 An x/y plane of the large tilt γ0 conﬁguration before (left

and center) and after (right) GBS. The γ0 is coherent with the

right grain. Atoms that have an FCC environment are colored

dark blue (nickel grains) or white (γ0), and all other atoms are

colored according to centrosymmetry. 52

xi 4.21 An x/y plane of the large square γ0 conﬁguration before (left

and center) and after (right) GBS. The γ0 is coherent with the

right grain. Atoms that have an FCC environment are colored

dark blue (nickel grains) or white (γ0), and all other atoms are

colored according to centrosymmetry. 52

4.22 An x/y plane of the large inline γ0 conﬁguration before (left

and center) and after (right) GBS. The γ0 is coherent with the

right grain. Atoms that have an FCC environment are colored

dark blue (nickel grains) or white (γ0), and all other atoms are

colored according to centrosymmetry. 53

4.23 The (left) stress vs. strain and (left) normalized displacement

of the 0D, 1D, and 2D systems after deformation. 54

4.24 The maximum stress of the 0D, 1D, 2D, and Σ11 systems prior

to deformation, measured from the stress strain curves. 54

4.25 The width of the centrosymmetry curves before and after

deformation for the 0D, 1D, 2D, and Σ11 systems, which

represents the extent to which the system disorder is localized

to the grain boundary (narrower = more localized, wider = less

localized). Centrosymmetry curves are shown in AppendixA. 55

4.26 An x/y plane of the 1D γ0 conﬁguration before (left and center)

and after (right) GBS. The γ0 is coherent with the right grain.

Atoms that have an FCC environment are colored dark blue

xii (nickel grains) or white (γ0), and all other atoms are colored

according to centrosymmetry. 55

4.27 An x/y plane of the 2D γ0 conﬁguration before (left and center)

and after (right) GBS. The γ0 is coherent with the right grain.

Atoms that have an FCC environment are colored dark blue

(nickel grains) or white (γ0), and all other atoms are colored

according to centrosymmetry. 56

A.1 The centrosymmetry and excess volume of atoms averaged in

bins along the y axis of the Σ11 and γ0 tilt, square, and inline

systems before (left) and after (right) shear deformation. 67

A.2 The centrosymmetry of atoms averaged in bins along the y axis

of the small and large tilt, square, and inline systems before

(left) and after (right) shear deformation. 68

A.3 The excess volume of atoms averaged in bins along the y axis

of the small and large tilt, square, and inline systems before

(left) and after (right) shear deformation. 69

A.4 The centrosymmetry (left) and excess volume (right) of atoms

averaged in bins along the y axis of the 0D, 1D, and 2D systems

before and after shear deformation. 70

xiii Acknowledgements

First and foremost I would like to thank my adviser, Dr. Jennifer Carter, for her

support and mentoring throughout my undergraduate and graduate education.

She inspired my interest in materials science and provided a balance of ﬁrmness

and understanding in her expectations. Dr. Carter served as my primary resource

for all things material science and superalloy related, and I am forever grateful for

her commitment to my education and professional growth. I would also like to

thank my secondary adviser, Dr. Saryu Fensin, who provided expertise relating to

molecular dynamics and simulations, along with valuable advice about achieve-

ment in the face of adversity. I am blessed to have such intelligent and strong role

models as a young professional.

A big thanks to Dr. Eric Hahn and Dr. Richard Hoagland for their assistance with LAMMPS and MD. I would also like to thank the remainder of my thesis committee, Dr. John Lewandowski and Dr. David Matthiesen for their time and interest in my project. A special thanks goes to Mom, Dad, and the rest of my loving family for their constant support, and an extra special thanks to Joe Ledger, who provided love, encouragement, and lots of snacks.

This work made use of the High Performance Computing Resource in the Core

Facility for Advanced Research Computing at Case Western Reserve University,

as well as high performance computing at Los Alamos National Laboratory. This work was funded by the CWRU Startup Funds and the C2 Marie Program.

xiv List of Abbreviations

CSL - Coincidence Site Lattice

DFT - Density Functional Theory

DOF - Degrees of Freedom

EAM - Embedded Atom Method

FCC - Face-Centered Cubic

GB - Grain Boundary

GBE - Grain Boundary Energy

GBS - Grain Boundary Sliding

HPC - High Performance Computing

LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator

MD - Molecular Dynamics

NIST - National Institute of Standards and Technology

NPT - Ensemble with constant mass, volume, pressure

NVE - Ensemble with constant mass, volume, and energy

NVT - Ensemble with constant mass, volume, and temperature

TEM - Transmission Electron Microscopy

xv Abstract

Understanding the Interaction between Grain Boundaries and Precipitates in Ni-Al Using Molecular Dynamics

Abstract

RACHEL L. MORRISON

0.1 Abstract

This thesis investigates the interaction between γ0 precipitates and grain boundaries in a Ni-Al system during deformation. This interaction is investigated using molecular dynamics, and γ0/boundary configurations were built to investigate how the orientation, size, and interaction of γ0 change the deformation behavior of the grain boundary. The γ0 aided in nucleating defects (i.e., dislocations) that contributed to the boundary sliding mechanism. By increasing the size of precipitates that bisect the boundary, the boundary becomes stronger, whereas increasing the size of precipitates adjacent to the boundary makes the boundary weaker. Ad- ditionally, the interaction of multiple γ0 plays a role in grain boundary sliding behavior. Low concentrations of γ0 produce sliding dominated by atomic shuffling, whereas high concentrations of γ0 produce sliding dominated by dislocation emission. More work is needed to investigate the effects of temperature, initial defects, and different grain boundary configurations on sliding behavior.

xvi 1

1 Introduction

The objective of this thesis is to use molecular dynamics (MD) simulations to

investigate how the presence of γ0 precipitates in a superalloy system inﬂuences

grain boundary deformation. Although the high temperature strength of superal-

loys is primarily attributed to the order strengthening of the γ/γ0 interface, grain

boundaries play a role in the ultimate creep life of the alloys. Serrated grain bound-

ary structures are created by heterogeneously nucleating γ0 on the boundaries, and

these serrated microstructures have a lower propensity for grain boundary sliding

(GBS) than the conventional, ”smooth” boundaries. We hypothesize that the ori-

entation of the γ0 relative to the grain boundary inﬂuences the structure in the vicinity of the grain boundary and thus the propensity for GBS. This hypothesis

is tested by inserting γ0 precipitates on grain boundaries in different conﬁgura-

tions and deforming the system. The orientation and size of the γ0 as well as the

interaction of multiple γ0 are all studied as they relate to the deformation of the

boundary. 2

2 Background

2.1 Nickel Superalloys

Nickel superalloys are a class of alloys that are characterized by exceptional high

temperature properties, including creep strength, toughness, and resistance to cor-

rosion and oxidation. Based on these properties, superalloys have found applica-

tions in extreme environments such as jet turbines. The properties of superalloys vary primarily with composition and microstructure, as discussed in the next sec-

tion.

2.1.1 Microstructure and Behavior

Nickel superalloys are composed of nickel (base), chromium (10-20 wt%), titanium

(up to 8 wt%), aluminum (up to 8 wt%), and an array of other transition metal

elements5. Superalloys consist of two main phases, the γ and γ0 phases. The γ

phase is FCC nickel that comprises the continuous matrix of the alloy. The γ phase

is stabilized by alloying element including cobalt, iron, chromium, molybdenum,

and tungsten, meaning the γ region of the Ni-Al phase diagram is expanded over

5 0 a wider range of temperatures and/or compositions . The γ phase has an L12

structure made of nickel, titanium, aluminum and/or tantalum5. Each unit cell

has the Ni3(Al, Ti, Ta) structure with nickel on the face centers and aluminum, Background 3 titanium, or tantalum on the corners. Other elements such as niobium, ruthenium, and rhenium, partition to the γ0 phase. An example of a nickel superalloy is shown in Figure 2.1.

Figure 2.1. SEM micrograph of the superalloy Rene´ 104 with a deposited oxide grid and speckle pattern. Grain boundary sliding is observed along the cracked boundary, highlighted by the white circles. HRSEM backscatter inset image showing the two primary phases, 0 FCC Ni γ channels and L12 Ni3Al cuboidal γ precipitates. Micro- graphs taken from Carter2.

The γ0 precipitates’ morphology changes at different size ranges; extremely

small precipitates form spheres to minimize surface energy, whereas larger pre-

cipitates form cubes that maintain coherency with the matrix and minimize the

interfacial energy of the precipitate and the matrix5–7. Basic calculations of the Background 4

Gibbs free energy associated with homogeneous nucleation of precipitates, shown in equation 2.1, can be used to determine the stability of spherical vs. cuboidal γ0

precipitates.

∆G = −V(Gγ − Gγ0 ) + Aγγ/γ0 (2.1)

V is the volume of the precipitate, Gγ and Gγ0 are the Gibbs free energies per unit volume of the γ and γ0 phases respectively, A is the surface area of the precipitate, and γγ/γ0 is the interfacial energy per unit area. Volume and surface area expressions can be plugged in for both spherical and cuboidal precipitates. There is a critical particle size for which the cuboidal precipitates are more energetically favorable than the spherical precipitates, which depends on the values of Gγ,Gγ0 , and γγ/γ0 . After this critical size is reached, there is an energetic driving force for coherent cuboidal morphologies to form.

In addition to the free energy of the system, the lattice misﬁt plays a role in the critical precipitate size for the spherical to cuboidal transition. The misﬁt strain between the γ and γ0 is dependent on the chemical composition of the superalloy, and it can be calculated according to equation 2.2.

aγ0 − aγ δ = 2 (2.2) aγ0 + aγ 0 δ is the misﬁt strain, aγ0 is the lattice parameter of the γ phase, and aγ is the

lattice parameter of the γ phase. If aγ0 > aγ, the two phases have a positive misﬁt strain. Under standard supersolvus heat treatments, the γ0 precipitates maintain

coherency with the γ matrix such that the {110} in the precipitate are parallel with

the {110} of the matrix8. Superalloy compositions with low misfit strain between Background 5 phases require a larger precipitate size before the misfit strain causes a precipitate shape change5. As the size of the precipitate grows, the precipitate becomes semi- coherent, and misfit dislocations are formed to relax the misfit strains caused by the difference in γ and γ0 lattice parameters. The spacing between misfit dislocations,

D, can be calculated using equation 2.39.

b D = (2.3) δ

dγ + dγ0 b = (2.4) 2 δ is the misﬁt strain, calculated by equation 2.2, and b is the burgers vector of dis-

locations, calculated by equation 2.4. dγ and dγ0 are the unstressed interplanar spacings of the γ and γ0 phases, which correspond to the 110 interplanar spac-

ings. If the precipitate size is smaller than D, misﬁt dislocations will not form and

a completely coherent interface will be observed. As the precipitate grows very

large, dislocations can no longer accommodate the total strain caused by the γ0

precipitate, and the precipitate can become incoherent with the γ matrix and is

termed overaged. Typically, however, the lattice mismatch between the γ0 and the

γ is minimal, resulting in coherent γ/γ0 boundaries that are stable, even at high

temperatures10.

2.1.2 Strengthening Mechanisms

Nickel superalloys are strengthened by a number of mechanisms. Alloying ele-

ments strengthen the alloy via solid solution strengthening, and the γ0 precipitates Background 6 strengthen the alloy via precipitate hardening. The ordered structure of the γ0 further enhances the precipitate strengthening. When a dislocation moving through the γ phase reaches a γ0 precipitate, it disrupts the order of the precipitate and creates high energy antiphase boundaries5. Because of this energy penalty, in order for a dislocation to move through the γ0, a second dislocation must follow the

ﬁrst dislocation to reorder the γ0. This coupled motion of dislocations impedes slip and thereby increases the strength of the alloy, allowing the alloys to be used at temperatures up to 70% of their melting point5,11. Additionally, the γ0 phase is ductile, so γ0 precipitates increase the strength of the alloy without lowering the fracture toughness5. These combined properties and the interaction of the two phases contribute to the high creep strength of superalloys.

2.1.3 Creep Performance

Superalloys find applications at high temperatures, and thus creep deformation is a key consideration in alloy design and processing. Grain boundaries play an important role in the ultimate creep of superalloys12, as grain boundaries can serve as preferential sites for cracking and ultimately failure13. Specifically, grain boundary sliding (GBS) is a mechanism for accommodating creep deformation, and the morphology of the boundary can influence the extent to which GBS accommodates the total deformation12,14. In order to remove the GBS mechanism, grain boundaries were removed entirely by creating single crystal alloys12. These single crystal alloys are used for turbine blades, as the loading conditions and simple axial ge- ometry allow for single crystal use. However, superalloys for disc applications do not benefit from single crystal design, as the loading conditions are much more Background 7 complex. In order to improve the performance and lifetime of polycrystalline disc alloys, γ0 are heterogeneously nucleated on the grain boundaries in order to produce a serrated morphology15. Serrations ranging in scale from 100 nm to a few microns help decrease the overall contribution of GBS to the total strain accommodation, and the creep fatigue crack growth of serrated microstructures is slower than conventional microstructures2,10,13.

2.2 Grain Boundaries

Most metallic materials in use today consist of many grains, or regions of the metal that have long range, periodic order. Atoms in the grain occupy lattice sites, and these sites correspond to the lowest energy position for said atoms given a certain set of thermodynamic parameters, i.e., temperature, pressure and chemical potential. A grain boundary is the region separating two grains or crystals of the same phase (composition and structure), and the atoms along this boundary are shifted from their equilibrium positions as a result of two separate crystals meet- ing3. Generally ﬁve macroscopic degrees of freedom (DOF) are used to describe grain boundaries: a rotation that aligns two adjoining crystals comprised of an axis common to both grains ([hokolo], two DOF) and an angle (θ, one DOF), and

3 a boundary plane between the crystals (nˆ = (hnAknAlnA), two DOF) , as shown in ◦ Figure 2.2. Grain boundary notation speciﬁes these parameters as such: θ [hokolo

], (hnAknAlnA). The axis of rotation, [hokolo] is identical for both grains and can

be expressed in any coordinate system. Because of the symmetry associated with

cubic systems, many boundary notations can be used to describe the same grain Background 8 boundary. Conventionally the lowest-index rotation axis is used to describe the boundary.

Figure 2.2. Grain boundaries are conventionally deﬁned by a rotation axis (o = [hokolo]), a rotation angle (θ), and a boundary plane (nˆ 3 = (hnAknAlnA)). Figure recreated from Lejcek .

In addition to the ﬁve macroscopic degrees of freedom, grain boundaries con-

tain microscopic degrees of freedom. The two grains can be translated via rigid

body motion parallel or perpendicular to the grain boundary plane3. These micro-

scopic degrees of freedom are constrained by temperature, pressure, and chemical

potential, and they are thus not fully independent degrees of freedom.

In order to simplify grain boundary classiﬁcation, boundaries are grouped by

the relationship between the rotation axis (o = [hokolo]) and the grain boundary normal (n). If the rotation axis and normal are perpendicular (o ⊥ n) the boundary is classified as tilt, whereas if the rotation axis and normal are parallel (okn) the boundary is classified as twist. All boundaries that do not fall into these two categories are classified as mixed grain boundaries. Background 9

Additionally, grain boundaries can be deﬁned as symmetric or asymmetric.

Symmetric grain boundaries have a mirror symmetry along the grain boundary plane, and thus atoms along the grain boundary plane are described with the same

Miller indices for both grains3. Asymmetric boundaries do not satisfy this condi- tion. Boundaries can thus be classiﬁed as symmetric tilt, asymmetric tilt, twist

(inherently symmetric), and mixed. Mixed grain boundaries are often referred to as random.

The rotation angle for a given grain boundary can further classify the boundary as low-angle or high-angle. For low rotation angles, grain boundary structure can be described using dislocations; tilt boundaries have an array of edge dislocations and twist boundaries an array of screw dislocations. The grain boundary structure can be described with dislocation theory for grain boundary angles up to 15◦ 3. At higher misorientation, the dislocations begin to overlap and become indistinguish- able from one another, and new models must be used to describe the structure. All grain boundaries with rotation angles greater than 15◦ are classiﬁed as high angle grain boundaries.

2.2.1 Coincidence Site Lattice

In addition to deﬁning the grain boundary via degrees of freedom, the coincidence- site lattice (CSL) model was introduced to describe special grain boundaries that exhibit lower grain boundary energy than would be predicted by overlapping dislocations theory16. When examining two crystals and the boundary between then, some lattice sites may overlap between the two crystals, and these sites are called coincidence sites. Starting with one reference atom, a CSL model describes the Background 10

common lattice points or sublattice of two grains17. The CSL model must describe

the maximum number of overlapping lattice sites, and this lattice is deﬁned using

equation 2.5.

number of coincidence sites in cell Σn = (2.5) total number of lattice sites in cell Thus, a Σ11 boundary would have a CSL site at one out of every eleven lattice sites within the crystal. Σ boundaries are also referred to as special boundaries. This

model operates under the assumption that boundaries with more coincidence sites will have lower grain boundary energies because fewer bond are broken across the

boundary. CSLs form because the coincidence sites impose a reduction in the dis-

order, or dislocation content, on the grain boundaries. This means there are fewer

dislocations and consequently less energy stored in the grain boundary, making

special boundaries more energetically favorable3. An example of a CSL lattice is

shown in Figure 2.3.

Figure 2.3. A grain boundary between two grains that are described with a Σ5 coincidence site lattice. Every 5th atom is colored white to represent those lattice sites that overlap within the two grains3. Background 11

2.2.2 Grain Boundary Migration

Grain boundaries within a material can migrate during deformation in directions

parallel or normal to the grain boundary plane. Boundary migration parallel to

the boundary plane is termed grain boundary sliding (GBS), and boundary mi-

gration perpendicular to the grain boundary plane is termed grain boundary cou-

pling. For GBS, there are two dominant mechanisms of motion: Lifshitz sliding

and Rachinger sliding. Lifshitz sliding relies on the diffusion of vacancies as a re-

sult of stress to cause boundary offsets18. This mechanism is active at high temper-

atures and lower stresses, and thus is not observed frequently in nickel superalloy

systems. Rachinger sliding, however, refers to the displacement of grains relative

to one another due to intergranular dislocation motion18. GBS typically occurs

at higher temperatures due to the motion of grain boundary dislocations19, how-

ever sliding at low temperatures can be observed in nano-scaled grains20. Equi-

librium boundaries require local grain boundary defects in order for sliding to oc-

cur21. Grain boundary migration is characterized by ”stick-slip” boundary behavior, whereby the stress increases up to a critical value, a sliding event is initiated, and the boundary transitions to another state that is an energetic minimum22–24.

2.3 Molecular Dynamics

Molecular dynamics (MD) is a computer simulation technique that uses classical

physics and interatomic potentials to simulate dynamic particle (often atomic) in-

teractions. MD is a type of N-body simulation that relies on solving Newton’s

second law25 for all the particles in a given simulation. Background 12

dv d2r F = ma = m = m (2.6) dt dt2 Newton’s second law states that the force acting on a given body is equal to its

mass (m) multiplied by its acceleration (a). The acceleration is equal to the deriv-

ative of the velocity (v) and the second derivative of the position (r) with respect

to time (t). The net force acting on a given particle is equal to the sum of the all

the forces exerted upon said particle by other bodies in the system. The interac-

tions, and thus the forces, between two bodies in the system is described by an

interatomic potential function.

Fi = −∇iU (2.7)

The force acting upon a given particle i is equal to the ﬁrst derivative of the poten-

tial energy function, U. The force is then used to calculate the acceleration of the

particle, and after a speciﬁed timestep a new position can be calculated using this

acceleration. In addition to the formulation above, Lagrangian mechanics can be

used in order to eliminate the need for constraint forces, and generalized coordi-

nates are used in equation 2.8 to calculate the potential and kinetic energy without

knowing the constraint forces.

N 2 ~ρi L = ∑ − U(~ri) (2.8) i=1 2mi L is the Lagrangian, and it is the sum of all atoms i to N in a system containing N

atoms, mi is the mass, ~ri is the position, and ρi is the momentum of atom i. The calculation of forces in a many bodied system is computationally expensive and must be done with relative frequency in order to produce accurate simulation Background 13

results. Thus, high performance computing capabilities are necessary for running

MD simulations.

2.3.1 Interatomic Potentials

The interatomic potentials describe the assumed physics of the system, and are

therefore the most important part of an MD simulation; atom trajectories and re-

sulting behavior are calculated using the potentials. Many different types of po-

tentials exists for a given atom type, and the potentials can be ﬁtted to different

physical constants. Pairwise potentials describe the interaction between two bod-

ies, and the most commonly known pairwise potential is the Lenard-Jones (L-J)

potential. The L-J potential provides accurate simulation results for inert gases

and material non-speciﬁc interactions26. Many-bodied potentials describe interac-

tions of many bodies, as the higher order terms are calculated explicitly. One such

many-bodied potential is the embedded atom method (EAM). EAM potentials are

calculated based on the equation 2.9.

1 Etot = ∑ Φij(rij) + ∑ Fiρi (2.9) 2 ij i

The total potential energy (Etot) of a collection of atoms is calculated by three em-

pirically ﬁtted sub functions: the pair interaction function (Φij) the embedding

function (Fi) and the electron cloud function (ρi). The pairwise interaction func-

tion describes the energy between atoms i and j separated by a distance rij.A pairwise interaction function for each atom type combination must be speciﬁed.

For example, in a system containing nickel and aluminum the following pairwise

interactions are necessary to create the EAM potential: ΦNiNi, ΦAlAl, and ΦNiAl. Background 14

The embedding function is the embedding energy of atom type i, and this function

describes the energy necessary to place atom i in an electron cloud. This function

must be speciﬁed for all atoms types: FNi(ρ) and FAl(ρ). Finally, the electron cloud function can be calculated according to equation 2.10.

ρi = ∑ i 6= jρj(rij) (2.10)

This function describes the contribution of atom j at location i to the electron charge density27. Like the functions above, the electron cloud function must be deﬁned for all atom types: ρNi(r) and ρAl(r). Thus, in order to fully describe the Ni-Al binary system, seven sub functions are necessary.

There are a number of other many-body potential functions available for modeling different states of matter, molecule orientations etc., but EAM potentials have shown high accuracy when modeling metallic materials systems in comparison to observed experimental results28–30. Typically EAM potentials are ﬁt to the lattice constant of the crystal, elastic moduli, vacancy formation energy, and energies of other defects.

2.3.2 Boundary Conditions

Because of the computational expense for calculating forces from potential functions, large simulations often cannot be run in a feasible time frame. In order to reduce the number of particles in a system, and therefore the number of calculations that must be performed, boundary conditions are imposed on the simulation.

System boundaries can be periodic or ﬁxed. In order to mimic an inﬁnite cell, periodic boundary conditions are applied by repeating an image of the simulation Background 15

in the periodic direction. The atoms in the simulation cell are allowed to interact with the atoms in the boundary cells, and atoms can move freely into and out of

the periodic boundary cells, as shown in Figure 2.4. Consequently, given a large

enough system size and periodic boundary conditions, changing the size of the cell

should not alter obtained results, because periodic boundary conditions help avoid

ﬁnite cell size effects. Fixed boundaries are free surfaces in which the atoms have

unsatisﬁed bonds. This type of boundary typically results in higher energy atoms, which can give insight into surface behavior. However, because MD simulations

typically simulate fewer atoms than would be represented in a physical sample,

the effects of ﬁxed surfaces can be disproportionately large. Care must be taken to

ensure data is not affected by simulation size constraints.

Figure 2.4. The center (gray) simulation cell is surrounded by images of itself (white), and the equivalent atoms in each cell move in tandem, as shown by the arrows. The atoms within the simulation cell are allowed to interact with the atoms in the periodic cells, and atoms can travel freely across the simulation cell/periodic cell boundary. Background 16

2.3.3 Ensembles

Systems may have identical thermodynamic properties but different microscopic

properties. These systems with identical macroscopic states are grouped as an

ensemble. The NVE ensemble, known as the microcanonical ensemble, is a con-

stant mass (N, or number of atoms), constant volume (V), and constant energy (E)

ensemble and is calculated by integrating Newton’s equations of motion directly.

The system cannot exchange particles or energy with its surroundings, and it cor-

responds to an isolated system. However, many laboratory systems are not iso-

lated and rely on other controls, i.e., temperature and pressure controls. The NVT

ensemble, referred to as the canonical ensemble, is constant mass (N), constant vol-

ume (V), and constant temperature (T), and the NPT ensemble is a constant mass

(N), constant temperature (T), and constant pressure (P).

In order to regulate the temperature of the NVT or NPT ensembles, a thermo-

stat is used, and the system is in contact with a heat reservoir at constant temper-

ature. In this study, a Nose-Hoover´ thermostat is used to control the temperature,

and an additional variable is introduced to the Lagrangian31,32.

N mi 2~ ~ Q 2 LNose´ = ∑ s r˙i − U(r˙i) + )s˙ − ( f + 1)kBTln(s) (2.11) i=1 2 2 The additional variable, s, has a mass of Q and a velocity of ˙s. This variable, although ﬁctitious, represents a friction which is used to slow down or accelerate particles in order to adjust the set temperature, T. It is a dynamic variable that represents the extra degree of freedom of the heat reservoir. The value of s is the time-scaling parameter. The magnitude of Q determines how quickly the system adjusts to the temperature of the heat reservoir. Too large a value of Q will result Background 17

in the system reaching the set temperature slowly, whereas too small a value of Q will result in oscillations around the set temperature. f is the number of degrees of

freedom, ~r˙i is the time derivative of position, and ˙s is the time derivative of s.

2.3.4 Statistical Mechanics

Molecular dynamics simulations calculate atomistic (microscopic) quantities, how-

ever the bulk properties of the system are often of interest. Statistical mechanics is

used to understand macroscopic properties based on microscopic quantities. The

thermodynamic state of the system is deﬁned by a small number of parameters that

can then be used to derive other thermodynamic quantities. For example, the tem-

perature, pressure, and number of particles are deﬁned, and using these quantities

other properties, such as volume, can be calculated using fundamental thermody-

namic equations. The microscopic state of the system is deﬁned by the positions

(x y z) and momenta (px py pz) of the atoms within the system. Each position and

momentum quantity can be thought of as a coordinate value in a six dimensional

phase space. Each point in phase space represents a unique state an atom can oc-

cupy, and the distribution of all atoms in phase space dictates the thermodynamic,

macroscopic properties of the system. Macroscopic system properties for a given

ensemble, or the ensemble averages, represent the average quantity for all pos-

sible states that satisfy a set of thermodynamic constraints. Molecular dynamics

calculations, however, are time averages of a system as it evolves. The Ergodic hy-

pothesis, however, states that the ensemble average is equal to the time average33.

This hypothesis operates under the assumption that a system that evolves indef-

initely will pass through all available states. One of the challenges of molecular Background 18

dynamics is to ensure the system evolves long enough that the Ergodic hypothesis

is true and thus ensemble averages can be obtained from time averages.

2.3.5 Initialization and Integration

Atom positions and velocities must be initialized in order for the system to run.

The atom positions are created to mimic a given structure (i.e., crystal structure,

grain boundary structure etc.), and the velocities are initialized randomly around

a set temperature using a Maxwell-Boltzmann distribution.

1 2 m 2 mvx P(vx) = exp − (2.12) 2πkBT 2kBT

P is the probability density function for velocity component vx at temperature T.

kB is the Boltzmanns constant, and m is the mass. Equivalent expressions for the

remaining velocity components can be found by simply substituting vx with vy

or vz in the equation above. Note that the distribution of velocities for a given

temperatures is set so the momentum (ρ) of the system is zero.

N ρ = ∑ mivi = 0 (2.13) i=1 Once the positions and velocities have been initialized, the equation of motion can

be solved in order to calculate the forces acting upon each atom. Note, that due to

the complicated nature of the potential energy function, the equations of motion

must be solved numerically. A variety of algorithms exist for integrating Newtons

equations, and each of these algorithms conserve energy and momentum. Background 19

The algorithm most commonly used in molecular dynamics is the velocity Ver-

let algorithm34. The position (~r) and velocity (~v) after a given timestep are calcu-

lated as follows.

1 ~r(t + ∆t) = ~v(t)∆t + ~a(t)∆t2 (2.14) 2 ~(a)(t) +~(a)(t + ∆t) ~v(t + ∆t) =~(v)(t) + ∆t (2.15) 2 The standard implementation of the velocity Verlet algorithm is as follows.

(1) Calculate ~v(t+∆t).

(2) Calculate~r(t+∆t).

(3) Derive~a(t+∆t) using interatomic potential and~r(t+∆t).

(4) Calculate ~v(t+∆t) again.

The procedure can be further shortened by removing the ﬁrst velocity calculation, and the quantities are simply shifted by a half a timestep in the velocity.

The velocity Verlet algorithm is used to move the atoms forward in time and update their trajectories. The algorithm iterates until the total simulation time has been completed. Smaller timesteps can provide more accurate results, but at the cost of more calculations and thus longer simulation runtimes.

2.4 Molecular Dynamics and Interfaces

Nickel grain boundaries, particularly Σ boundaries, have been modeled exten- sively in MD, and the energetics, dislocation behavior, and mobility have been studied35–38. There are three typical conﬁgurations for modeling boundary mobility in MD: 3D nanocrystalline, columnar nanocrystalline, and bicrystalline39. Background 20

The bicrystal conﬁguration allows for the most control over the boundary defor-

mation, but the system does not describe the complex interactions between grain

boundaries in the material. Grain boundary sliding typically occurs at higher tem-

peratures due to the motion of grain boundary dislocations19, however sliding at

low temperatures can be observed in nano-scaled grains20, particularly in systems with high strain rates40. At low temperatures, equilibrium boundaries require lo-

cal grain boundary defects in order for sliding to occur21. Grain boundary sliding has been studied using molecular dynamics with a number of FCC materials including copper41, aluminum21,42,43, and nickel40,44. Some studies also examine the interaction of coherent precipitates and grain boundaries in Fe45 and Al-Mg systems46, focusing on how precipitates impede grain boundary motion and re- orient to maintain coherency. Simpliﬁed superalloy systems, i.e., γ/γ0 systems, have been examined with respect to dislocation networks47 and γ/γ0 interfacial energy48, but the researcher could ﬁnd no studies in the literature that focus on the interaction between γ0 and grain boundaries.

Although MD simulations allow for the examination of systems at resolved time and length scales, the simulated results should be compared to experimental results with caution. The simulated systems have nano-scaled grains and strain rates that are typically much higher than experimentally observed, and the deformation mechanisms active in such simulations may be different than those that occur in larger grains at lower strain rates. Although the grain boundary structure in nanocrystalline metals does not differ from the structure seen in coarse grain mate- rials49,50, such small grained systems favor grain boundary migration mechanisms Background 21 over intragranular lattice activities (dislocations, twins etc.), even at low tempera- tures51. High angle grain boundaries tend to slide more readily than low energy twins, as the high angle boundaries produce a lower energy barrier for the sliding event52. Although sliding is active in the small grain/high strain rate regime, the sliding is primarily stress driven and relies on local atomic shufﬂing rather than diffusion51. The local atomic shufﬂing can trigger GBS and partial dislocation emission, and it is theorized that larger grains experience a greater number of partials in order to accommodate the deformation in the grain that the GBS cannot accommodate on its own50. The onset of dislocation activities, although less sensitive to strain rate than diffusion based mechanisms53, is delayed with increasing strain rate54. 22

3 Methodology

3.1 Simulation Resources

3.1.1 LAMMPS

This project uses the MD code Large-scale Atomic/Molecular Massively Parallel

Simulator (LAMMPS)55, which is developed and maintained by Sandia National

Laboratory. LAMMPS takes advantage of parallel computing to run MD code for a variety of materials systems, including the metallic systems used in this study.

LAMMPS codes were run on high performance computing (HPC) clusters at Case

Western Reserve University and Los Alamos National Laboratory.

3.1.2 Mishin Potentials

The National Institute of Standards and Technology (NIST) has an Interatomic

Potentials Repository Project as part of the Materials Genome Initiative, and potentials developed by various researchers and are available to the public56. This project utilizes the EAM potentials developed by Mishin for modeling Ni and

1 Ni3Al . EAM potentials for both nickel and aluminum were created as support- ing functions for the Ni3Al potential, and the potentials are accurate enough to be used for pure nickel and aluminum systems. The potentials were ﬁtted to experimental values for lattice parameters, cohesive energies, elastic constants, vacancy Methodology 23

formation, and migration energy1. First principle data was incorporated into the

ﬁtting process as well, in particular with respect to the energy-volume relations

for different crystal structures. The ﬁtted properties can be found in the Table 3.1.

Each potential also speciﬁes a cutoff distance, which describes the distance over which the atomic interactions are calculated. The Mishin Ni-Al potential has a cut-

off distance of 6.72 A˚ , which includes the atomic interactions of an atom with its

ﬁrst through seventh nearest neighbors.

Table 3.1. The elastic constants and standard Gibbs free energy as calculated by the Mishin potential1.

Material c11 (GPa) c12 (GPa) c44 (GPa) G0 (eV) Ni 241.3 150.8 127.3 -4.4833

Ni3Al 236 154 127 -4.6477

3.1.3 Boundary Conditions for Grain Boundary Simulations

Grain boundaries are inherently higher energy than the bulk crystal, and as such

are sensitive to interactions with other high energy regions of simulations, i.e.,

other interfaces. Periodic boundary conditions are used reduce said interactions

and other surface effects, however, in order to apply a strain to a system to drive

boundary motion, ﬁxed surfaces are required in the direction normal to the grain

boundary. These ﬁxed surfaces can interact with the grain boundary as dictated by

Saint-Venants principle, and the system must be sufﬁciently large to avoid inter-

actions between the two types of boundaries57. This experiment employs periodic boundary conditions in the grain boundary plane directions (x and z), and ﬁxed boundary conditions in the grain boundary normal directions (y). Methodology 24

3.2 Σ11 Boundary System

3.2.1 Boundary Creation

This project seeks to simulate a pseudo-random high angle grain boundary, how-

ever truly random boundaries are computationally difﬁcult to simulate because

they require large simulation cells. In order to create computationally efﬁcient

simulations, periodic boundaries are modeled, i.e., Σ boundaries as deﬁned by

Section 2.2.1. Σ boundaries have higher periodicity in their grain boundary struc-

ture, meaning the grain boundary has a smaller unit cell than a random boundary.

This periodicity coupled with periodic boundary conditions simulates an inﬁnite

cell for these Σ boundaries. FCC crystals can contain Σ3, Σ9, and Σ11 bound-

aries17, and in order to balance simulation convenience and the generalizability of

the boundary, this project models a Σ11 grain boundary in lieu of a truly random

grain boundary. Speciﬁcally, a Σ11h110i{332} symmetric tilt boundary is modeled, which has been shown to exhibit grain boundary sliding41.

Grain boundaries were generated by using a LAMMPS script developed by

Mark Tschopp et al58. The script takes grain orientations, assuming the grain

boundary plane is parallel to one of the principle axes of the simulation box, brings

the two grains together, deletes any overlapping atoms, and performs an energy

minimization to ﬁnd the lowest energy state for all the boundary atoms. This pro-

cess is repeated as the two grains traverse across one other in the plane of the

boundary. The system with the lowest grain boundary energy after this iterative

process is taken as the equilibrium grain boundary structure. The grain boundary

energy (GBE) is calculated by normalizing the excess energy of the system to the

grain boundary area, as shown in equation 3.1. Methodology 25

U − (U N ) GBE = total atom atoms (3.1) lxlz The GBE is calculated by subtracting the potential energy of an individual atom at equilibrium (Uatom as calculated using the potential) multiplied by the number of atoms (Natoms) from the total potential energy of the system (Utotal as calculated by LAMMPS) and dividing by the boundary area. lx and lz are the simulation box dimensions in the x and z directions.

The grain boundary structures are created at 0K. This particular Σ11 bound-

ary has a tilt axis of [110], and a grain boundary plane of (332), or a 50.47◦ [110],

(332) boundary. The grain boundary normal is parallel to the y direction of the

simulation box, and the tilt axis is parallel to the z direction of the simulation box.

3.2.2 System Relaxation

When an MD simulation is initialized, the atom are placed based on the assumed

physics of the system rather than a concrete representation of physical reality. For

a crystalline system this means the atoms occupy locations based on the lattice

of the material in question. However, during the stages of MD initialization (GB

creation, temperature initialization etc.), artiﬁcial stresses can be introduced into

the system. In order to prevent these stresses from inﬂuencing results, the system

must be relaxed, whereby the atoms occupy the lowest energy state possible. En-

ergy minimizations are performed in order to ﬁnd the atom positions that result in

the lowest system energy, which is the equilibrium state according to thermody-

namics. System relaxation ensures that any stresses that may have been present as Methodology 26

a byproduct of how the system was built do not artiﬁcially bias the results. Guide-

lines for system setup and relaxation were informed by Kuksin et al59.

Energy minimizations are performed when the system is created and after any

changes are made to the simulation size (i.e., the number of atoms) or composition

(i.e., when the γ0 is created). This minimization iteratively adjusts the atomic po-

sitions until one of the stopping conditions are met (minimum energy, minimum

force, or maximum number of energy or force calculations). This study utilizes

the conjugate gradient method to calculate the next atom position in the energy

minimization algorithm60. Although the lowest energy state is taken as the equi-

librium structure, the simulation cannot make the distinction between a global

energy minimum (equilibrium) and a local minimum. As such, MD results should

be compared to experimental results when possible to ensure the system structure

matches reality.

There are a number of different indicators that the system has been relaxed,

including computed atomic properties. For example, individual atom potential

energies can be examined, as high potential energies, such as positive values, do

not occur in the physical world and are thus indicators that the system is not re-

laxed. At low temperatures, very high energies, such as those that are greater than

30% different from the equilibrium energies, indicate overlapping atoms or some

other inappropriate system initialization. Centrosymmetry, a per-atom quantity

that describes the local crystal structure of an atom, can be calculated to track the

disorder in the system. Centrosymmetry can help identify crystal defects; atoms with a centrosymmetry of zero have a perfect FCC lattice, and atoms with nonzero

centrosymmetry values have a local crystal structure that deviates from FCC. In Methodology 27

particular, atoms with an HCP environment (i.e., a stacking fault) will have a cen-

trosymmetry around six for this nickel system.

In MD simulations various ﬁxes are applied to keep thermodynamic quantities

constant, as discussed in the ensemble section. For isothermal systems, the tem-

perature in the simulation should reach steady state before any additional experi-

ments are performed. Simulations were run with the NVT ensemble. NPT ensem-

bles are not typically used for grain boundary systems, because ﬁxing the global

pressure creates artiﬁcial stresses in order to counteract the intrinsic stress asso-

ciated with the grain boundary itself. A Nose-Hoover´ thermostat was employed with a damping coefﬁcient of 1 ps and a simulation timestep of 1 fs. The LAMMPS

documentation recommends a damping coefﬁcient value of 1000 timesteps61.

In order to examine if the grain boundary structure has been relaxed, the av-

erage stress per atom in regions perpendicular to the grain boundary normal can

be examined. The total stress in a perfect crystal should be zero, as all the atoms

occupy their lowest energy lattice sites. The grain boundary, which is essentially

a crystal defect, will have a higher stress than the bulk crystal. The ﬁxed system

boundaries, which are free surfaces, will have high stresses as well. In LAMMPS,

stress per atom for each of the components of the stress tensor, Sab in which a is the

plane the stress acts on and b is the direction of the applied stress (e.g., Sxx), can be

computed according to equation 3.261. Methodology 28

N h 1 p 1 Nb = − + ( + ) + ( + )+ Sab mvavb ∑ r1a F1b r2a F2b ∑ r1a F1b r2a F2b 2 n=1 2 n=1 1 Na 1 Nd ( + + ) + ( + + + ) (3.2) ∑ r1a F1b r2a F2b r3a F3b ∑ r1a F1b r2a F2b r3a F3b r4a F4b 3 n=1 4 n=1 N 1 Ni f i d + (r F + r F + r F + r F ) + Kspace(r F ) + r F ∑ 1a 1b 2a 2b 3a 3b 4a 4b ia , ib ∑ ia ib 4 n=1 n=1 The ﬁrst term describes the contribution of kinetic energy, and m is the mass of

the atoms and the v terms are velocities. The second term describes the contribu-

tion of the pairwise potential energy of the neighbor atoms (Np); r1a and r1b are the

atom positions and F1a and F1b are the forces that arise due to the pairwise inter-

action. The remaining terms describe the contributions of bond (Nb), angle (Na),

dihedral (Nd), and improper (Ni) interactions, as well as the long range coulombic

interactions (KSpace) and any external constraints/ﬁxes (N f ). In the metallic system modeled in this study, the stress calculation only takes into account the kinetic

energy, pairwise interaction, and external constraints.

Once the boundary structure was relaxed, it was replicated in the periodic di-

rections (x and z). This simulation size increase ensured that any γ0 inserted in the

system would not interact with their periodic images. All of the steps for boundary

creation, relaxation, and replication are outlined in Table 3.2

3.2.3 System Deformation

Fixed boundaries were created on the simulation boundaries normal to the grain

boundary (y boundaries). The atom positions in a region with a thickness of three

m times the cutoff distance were held constant. A constant velocity (v) of 0.001 s was Methodology 29

Table 3.2. An overview of the grain boundary creation steps. The x, y, and z boundaries conditions are listed as periodic (p) or ﬁxed (f).

Simulation Boundary Conditions Dimensions (A)˚ Temp (K) GB Creation p, p, p 70.046 x 231.741 x 59.736 0 GB Relaxation p, f, p 70.046 x 231.741 x 59.736 1 GB Replication p, f, p 560.378 x 231.741 x 477.891 1

applied to one of the ﬁxed boundaries in the positive x direction in order to impose

a shear strain on the system. The strain was calculated according to equation 3.3.

vt Shear Strain = (3.3) y t is the timestep of the simulation at the time of the calculation, and y is the length

m of the y direction of the simulation. With a constant velocity of 0.001 s , the strain rate is 4.31 x 105 s−1. The Σ11 grain boundary without any precipitate was de-

formed to establish a baseline for grain boundary motion.

3.3 γ0 Precipitate

0 The γ precipitate is Ni3Al with an ordered FCC lattice; aluminum atoms occupy

the corners sites and the nickel atoms occupy the face center sites. In order to build

the precipitate in LAMMPS, a custom lattice was created. Cuboidal precipitates are

modeled to mimic the superalloy microstructure of turbine discs.

3.3.1 γ/γ0 System

A γ/γ0 system was created without a grain boundary to establish a baseline for

how the two phases interact. A single crystal of nickel was created with a coherent Methodology 30

m precipitate in the center. The system was deformed at a velocity of 0.001 s to match the baseline grain boundary simulation.

3.3.2 γ0 Orientation Study

γ0 precipitates are typically coherent with their matrix, however the precipitates

can have a variety of orientations with respect to grain boundaries. In order to

examine the inﬂuence the γ0 orientation has on the grain boundary deformation

behavior, three different GB/γ0 conﬁgurations were built: tilt, square, and inline.

0 In physical superalloy samples, both the γ precipitate edges and lattice of Ni3Al atoms are aligned parallel to the matrix. Thus, the γ0 orientation relationship between the GB and the γ0 is dictated by the GB itself, i.e., how the two adjacent grains are oriented relative to one another. For the Σ11 symmetric tilt boundary modeled, the ”real” γ0 orientation results in a precipitate that is oriented with the

(111) parallel to the grain boundary normal. This conﬁguration is denoted as tilt.

The tilt precipitate is built on the grain boundary so one half of the precipitate is coherent with its parent grain, and the other half of the precipitate is incoherent with the adjacent grain. Precipitates are built in the center of the grain boundary plane, and the boundary plane is sufficiently large so the precipitates do not interact with their periodic images. Similar to the tilt configuration, the square precipitate bi- sects the boundary plane, however the precipitate edges are oriented so (100) is parallel to the boundary plane. Again, only half of the precipitate is coherent with the parent grain, and the other half is incoherent with the adjacent grain. The final

Σ11/γ0 conﬁguration is inline. The inline precipitate is similar to the square precipitate, (100) k GB plane, but the precipitate is located entirely within the parent Methodology 31 grain. The inline precipitate is located directly against the grain boundary, but it does not cross the boundary plane into the adjacent grain. The tilt, square, and inline precipitates were created with dimensions of 14.28 A˚ cubed so each cube edge was 3 lattice units long. The small tilt, square, and inline conﬁgurations are pictured in Figure 3.1 (left).

Figure 3.1. Each simulation contains two grains (G1 and G2) separated by a grain boundary (gray line). A γ0 precipitate with an orientation (tilt, square, and inline) and size (small, large) is inserted onto the boundary, coherent with G1.

3.3.3 γ0 Size Study

In order to examine the inﬂuence precipitate size plays on the system behavior, the tilt, square, and inline γ0 system were recreated with larger precipitates. The Methodology 32

larger precipitates have edge lengths of 28.56 A˚ . Again, the simulations sizes are

large enough to prevent any interaction with the γ0 periodic images. The large tilt,

square, and inline conﬁgurations are pictured in Figure 3.1 (right).

3.3.4 γ0 Interaction Study

The orientation and size studies both examine a single γ0 precipitate and the effects said γ0 has on the grain boundary deformation behavior. However, nickel superalloys for disc applications contain upwards of 45 vol% γ0, and the space between precipitates is approximately equal to the precipitate size10. Thus, the interaction study examines how the presence of multiple γ0 precipitates in the system inﬂu- ences the grain boundary deformation behavior. Only one of the γ0 orientations was examined, the large tilt system. This system was selected because it simulates the serrated GB microstructure, and based on our hypothesis this simulation should replicate the lower propensity for grain boundary sliding seen experimentally. Additionally, the large tilt precipitate maintains full coherency with its parent grain, and thus is a good representation of the microstructure this study wishes to simulate.

The baseline Σ11/γ0 system for the interaction study is identical to the large tilt system described by the size study, and the x and z simulation cell dimensions are sufﬁciently large so no γ0/periodic image interaction occurs. This system has zero dimensional translational γ0 symmetry, and is referred to as the 0D system.

The 0D system has a γ0 area fraction on the GB of 0.003377. In order to produce a system with a higher γ0 concentration, the simulation dimensions in the z direction are reduced, shown in Figure 3.2 by the dotted lines. This allows the γ0 to interact Methodology 33 with its periodic images, effectively increasing the concentration of γ0 on the grain

boundary to 0.01630. This system has one dimensional translational γ0 symmetry,

and is referred to as the 1D system. To increase the γ0 concentration even further, the x and z dimensions of the original system are reduced, resulting in a γ0 area fraction of 0.09135. This system has two dimensional translational γ0 symmetry, and is referred to as the 2D system. Schematics of the 0D, 1D and 2D systems are shown in Figure 3.3.

As the systems are made smaller to allow γ0/γ0 interaction, the number of atoms in the simulations decreases. When atomic quantities are averaged over a different number of atoms, there is a risk of averaging out quantities for the larger systems in comparison to the smaller systems. In order to mitigate this effect, computed quantities were averaged over a volume equivalent to the smallest system for all simulations.

Figure 3.2. The system is bisected by a grain boundary (light gray plane), and a γ0 is inserted in the center of the GB (orange cube). The y boundaries have ﬁxed regions (dark gray) in order to apply a velocity that will drive shear deformation (arrow). The system is cut down in the x and z directions to allow the γ0 to interact with its periodic images, as shown partially by the dotted lines. Methodology 34

Figure 3.3. Each simulation contains two grains (G1 and G2) separated by a grain boundary (gray line). A large, tilt γ0 precipitate is inserted onto the boundary, coherent with G1, and the x and/or z dimensions are reduced to allow the γ0 to interact with periodic images of themselves in one or more dimensions. 35

4 Results

4.1 Σ11 Boundary Structure

A large number of potential Σ11 boundary structures were created, and the lowest

mJ grain boundary energy (equilibrium) measured was 1059 m2 , which is similar to the energy simulated for the same grain boundary with a different potential, 970

mJ 4 m2 . The kite structure of the modeled boundary matches the boundary structure seen experimentally, as shown in Figure 4.1.

Figure 4.1. The kite structure for the Σ11 boundary as modeled by (left) MD and (right) TEM4 looking down the [110] tilt axis. Results 36

4.2 System Relaxation

The initial Σ11 boundary structure was created at 0K, but in order to regulate the

system temperature, LAMMPS requires a nonzero temperature. The temperature

of the simulations was increased to 1K and allowed to reach steady state (i.e., the

ﬂat region of Figure 4.2) prior to applying the driving force for boundary deforma-

tion. Running the simulations for 10,000 timesteps was found to be an adequate

amount of time for the temperature to reach steady state. Thermal expansion at

such low temperatures is negligible, so no adjustments were made to the simula-

tion volume. The same temperature ﬁx procedure was run on all simulations, and

a representative example of this relaxation is shown in Figure 4.2.

Figure 4.2. The temperature as a function of time during the relaxation of an initialized MD system. The temperature increases to a setpoint, and then reaches a steady state regime. Once the system has reached steady state, further experiments can be run. Results 37

In order to ensure there are no unusual stresses in the GB system created as a

result of the initialization, the stress averaged in rectangular regions perpendicu-

lar to the grain boundary plane was calculated using equation 3.2 and plotted, as

shown in Figure 4.3. These regions had a width equal to the atomic cutoff distance, which is the distance over which atoms interact with their neighbors. The cutoff

distance is deﬁned by the potential. The stresses in the bulk material are zero, as

the atoms occupy the ideal lattice positions. At the grain boundary, the stresses be-

come nonzero due to the mismatch between the two lattices. The ﬁxed boundaries

also exhibit high stresses, as the boundary atoms have unsatisﬁed bonds. Mea-

surements of the simulation are taken away from these ﬁxed boundaries in order

to avoid the higher stresses created by these surfaces.

Figure 4.3. The stress per atom averaged in rectangular regions perpendicular to the grain boundary normal. The three principal stress directions, σxx, σyy, and σzz are shown.

Properties of the γ0 were calculated based on the potential. The Gibbs free

energy of nucleation for spherical and cuboidal γ0 (equation 2.1) is shown in Figure Results 38

4.4. An average interfacial energy for the (100), (110), and (111) was taken for the

the spherical precipitate, and the interfacial energy for the (100) was used for the

cuboidal calculation. The transition radius from spherical γ0 to cuboidal occurs at

0.0761 A˚ . The misﬁt between the γ and γ0 (equation 2.2) is 0.0145 and the misﬁt

dislocation spacing (equation 2.3) is 172.4 A˚ .

Figure 4.4. The Gibbs free energy of nucleation for a sphere of radius r and cube of edge length 2r. The energy is calculated based off values from the Mishin potential1.

The strain ﬁelds caused by the presence of the γ0 were calculated as the atomic

displacement of the Σ11/γ0 system before and after the relaxation. The initial sys-

tem state was taken after the γ0 was inserted, but before any energy minimizations or thermostats were added (i.e., simulation was at 0K). The final state was taken as the relaxed Σ11/γ0 after an energy minimization and the system temperature had reached steady state at 1K. The displacement of atoms was normalized to the ~ 1 burgers vector in an FCC nickel system: b = 2 h110i. The strain fields for the small and large tilt, square, and inline systems are shown in Figure 4.5. Strain fields for the interaction study (0D, 1D, and 2D) are identical to the large tilt strain field. Results 39

Figure 4.5. Strain ﬁelds caused by the γ0 precipitates prior to deformation. The strain is calculated by the displacement of atoms before and after relaxation, normalized to one unit of slip for nickel. Images are a x/z slice of the system. The position of the grain boundary is indicated by the gray line. Results 40

Similarly, the stress ﬁelds caused by the γ0 are shown in Figure 4.6. The stress is computed as the per atom stress shown in equation 3.2.

Figure 4.6. Stress ﬁelds caused by the γ0 precipitates prior to deformation. Images are a x/z slice of the system. Results 41

4.3 Σ11 Deformation Behavior

The stress/strain behavior of the Σ11 boundary exhibits ”stick/slip” behavior;

the system stresses increases linearly up to a critical value, and then stress drops

rapidly as the system deforms, as shown in Figure 4.7 (left). The maximum stress

the boundary reaches prior to deformation is 5.32 GPa. A shear modulus of 100.13

GPa was measured from the stress/strain curve.

Figure 4.7. The stress vs. strain of a (left) Σ11 GB system and (right) γ/γ0 system without a grain boundary. Both systems experience a stress drop after deformation (GBS or dislocation formation respectively).

The centrosymmetry (structure factor), excess volume, and normalized dis-

placement perpendicular to the grain boundary normal were averaged in bins

equal to the atomic cutoff distance, shown in Figure 4.8 (left). The grain boundary

is located at position zero, and there is a sharp increase in the displacement at the

boundary along with a relatively narrow and symmetric centrosymmetry distri-

bution, indicating the deformation is localized to the boundary and Σ11 boundary

experiences grain boundary sliding. Results 42

Figure 4.8. The centrosymmetry, excess volume, and normalized displacement of atoms averaged in bins along the y axis of the (left) Σ11 and (right) γ/γ0 systems before and after shear deformation. Results 43

The boundary structure remained periodic prior to sliding, though the periodic

ordering of the boundary changed with applied stress, indicating local shufﬂing of

atom positions, as shown in the changes in centrosymmetry in Figure 4.9 (middle).

After a critical stress, boundary sliding occurs and the boundary loses its periodic

structure, shown in Figure 4.9 (right). This disordering is also evidenced by the widening of the centrosymmetry and excess volume curves shown in Figure 4.8.

Sliding was achieved through local atomic shufﬂing.

Figure 4.9. Snapshots of the Σ11 GB system during sliding. The (left) initial structure, (center) an intermediate structure, and (right) the disordered boundary after sliding. The atoms are colored according to centrosymmetry; dark blue atoms have a perfect FCC environment and all other colors deviate from an FCC environment.

4.4 γ/γ0 Deformation Behavior

A γ/γ0 system was created without a grain boundary. This system reached a max-

imum stress of 20.4 GPa, Figure 4.7 (right), prior to deformation. A shear modulus

of 87.89 GPa was measured from the stress/strain curve prior to deformation. The Results 44

centrosymmetry, excess volume, and normalized displacement along the length

of the sample were averaged in bins equal to the atomic cutoff distance, shown

in Figure 4.8 (right). After deformation, centrosymmetry and excess volume were

nonzero in the bulk, and there was no clear localization of these quantities along

the length of the sample. The displacement of atoms was linear along the length

of the sample, indicating that no sliding or localized deformation occurred. In

contrast to the sliding seen in the GB system, dislocations are formed to accommo-

date the plastic deformation, which are indicated by the light green lines of atoms

(stacking faults) created by dislocations in Figure 4.10.

Figure 4.10. A snapshot of the γ/γ0 system under shear deformation. The γ0 is colored white, and the remaining atoms are colored according to centrosymmetry; dark blue atoms have a perfect FCC environment and all other colors deviate from an FCC environment.

4.5 Σ11/γ0 Deformation Behavior

Three different Σ11/γ0 studies were conducted to examine how γ0 inﬂuences the grain boundary deformation behavior. The orientation, size, and interaction of γ0

along the boundary were examined. Results 45

4.5.1 γ0 Orientation Study

The presence of the γ0 along the boundary prompted deformation at a lower stress

than was observed in the reference Σ11 system, as shown in Figure 4.11 (left) and

4.12. The small tilt system reached a maximum stress of 3.77 GPa prior to slid-

ing, followed by the small square system at a stress of 3.85 GPa, and ﬁnally the

small inline system at a stress of 4.06 GPa. All systems experienced grain bound-

ary sliding, as most of the strain in the system is accommodated by a sharp jump

in displacement at the grain boundary, as shown in Figure 4.11 (right). The slope

of the displacement curve in the leftmost grains (-90 to 0) for all γ0 systems dif-

fers slightly from that of Σ11 system, which indicates a different amount of elastic

strain accommodated at the time of measurement.

Figure 4.11. (left) The stress vs. strain of the Σ11 and small γ0 tilt, square, and inline systems after shear deformation. (right) The displacement of atoms averaged in bins along the y axis of the Σ11 and small γ0 tilt, square, and inline systems after deformation. Results 46

Figure 4.12. The maximum stress of the small tilt, small square, small inline, and Σ11 systems prior to deformation, measured from the stress strain curves.

The centrosymmetry of atoms in regions perpendicular to the grain boundary

normal were averaged in bins equal to the atomic cutoff distance for each sys-

tem before and after deformation. Similarly, the excess volume of the system was

measured along the grain boundary normal direction and these curves are shown

in AppendixA. The width of the centrosymmetry curves was measured to help

identify how much of the disorder/deformation in the system was localized to the

grain boundary plane. As seen in Figure 4.13, the changes in the centrosymme-

try curve widths are similar for all simulations. The centrosymmetry curve width

increases after deformation, but the overall disorder in the system remains rela-

tively localized to the grain boundary. The deformation of the system can be seen visually in Figures 4.14- 4.16, and dislocation emission is evidenced by the for-

mation of stacking faults in the material as dislocations are emitted from the grain

boundary. Results 47

Figure 4.13. The width of the centrosymmetry curves before and after deformation for the small tilt, small square, small inline, and Σ11 systems, which represents the extent to which the system disorder is localized to the grain boundary (narrower = more localized, wider = less localized). Centrosymmetry curves are shown in AppendixA.

Figure 4.14. An x/y plane of the small tilt γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry. Results 48

Figure 4.15. An x/y plane of the small square γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry.

Figure 4.16. An x/y plane of the small inline γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry. Results 49

4.5.2 γ0 Size Study

The strength behavior of the orientation study simulations was initially unexpected,

so an investigation on the size effect of the γ0 was conducted. The larger pre-

cipitates produce higher initial stresses and strains, shown in Figures 4.5 and 4.6

(right). The large tilt experienced a maximum stress of 3.84 GPa prior to defor-

mation, the large square system experienced 4.03 GPa, and the large inline sys-

tem experienced 3.70 GPa, as measured from the stress in Figure 4.17 (left). The

large tilt and square systems experienced an increase in strength in comparison to

their small counterparts, whereas the large inline system experienced a decrease in

strength in comparison to its small counterpart, as shown in Figure 4.18. Bound-

ary sliding occurs in the large tilt, square, and inline systems, as evidenced by the

sharp increase in the displacement curve shown in Figure 4.17 (right). The cen-

trosymmetry curve width before and after deformation is shown in Figure 4.19,

and the width of the large tilt system does not appear to change before and after

deformation due to the initial topography of the grain boundary, shown in Figure

4.20 (left). The centrosymmetry curve widths of the large square system before

and after deformation are similar to those of the small square system. The visual

deformation is shown in Figure 4.21. The centrosymmetry curve width of the large

inline system before deformation is similar that of the small inline system, and the

centrosymmetry curve width of the large inline system after deformation is larger

than the small inline system. Visualization of the large inline system’s deforma-

tion is shown in Figure 4.22. The initial conﬁgurations of both the large square and

inline systems have defects on the γ/γ0 interface within the γ0’s coherent grain because the γ0 cube faces are not aligned with the matrix. Results 50

Figure 4.17. (left) The stress vs. strain of the small vs. large γ0 tilt, square, and inline systems after shear deformation. (right) The displacement of atoms averaged in bins along the y axis of the small and large γ0 tilt, square, and inline systems after shear deformation. Results 51

Figure 4.18. The maximum stress of the small and large tilt, square, inline systems, and the Σ11 system prior to deformation, measured from the stress strain curves.

Figure 4.19. The width of the centrosymmetry curves before and after deformation for the small and large tilt, square, inline systems, and the Σ11 system, which represents the extent to which the system disorder is localized to the grain boundary (narrower = more localized, wider = less localized). Centrosymmetry curves are shown in AppendixA. Results 52

Figure 4.20. An x/y plane of the large tilt γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry.

Figure 4.21. An x/y plane of the large square γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry. Results 53

Figure 4.22. An x/y plane of the large inline γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry.

4.5.3 γ0 Interaction Study

In order to study how the interaction of multiple γ0 changes the deformation behavior, as multiple γ0 would be observed in experimental superalloys, systems that have 0D, 1D and 2D translational γ0 symmetry were built by reducing the simulation bounds to allow the γ0 to interact with their periodic images.

The 0D system, which is identical to the large tilt system, experienced maximum stress of 3.84 GPa prior to deformation. The 1D system experienced a similar maximum stress of 3.83 GPa, however the stress strain curve had multiple ”stickslip” events, shown in Figure 4.23. The 2D system had a much lower maximum stress of 3.21 GPa, however only a single stress drop occurred, shown in Figure

4.24. The strain in the system is localized to the grain boundary for all simulations, shown in Figure 4.23. Results 54

Figure 4.23. The (left) stress vs. strain and (left) normalized displacement of the 0D, 1D, and 2D systems after deformation.

Figure 4.24. The maximum stress of the 0D, 1D, 2D, and Σ11 systems prior to deformation, measured from the stress strain curves.

The centrosymmetry curve widths before and after deformation for the 0D and

1D systems are very similar, shown in Figure 4.25, however the centrosymmetry curve width after deformation for the 2D system is very wide in comparison to the others. This indicates that the deformation is no longer localized to the grain Results 55 boundary. Visualizations of the systems during deformation are shown in Figures

4.26 and 4.27.

Figure 4.25. The width of the centrosymmetry curves before and after deformation for the 0D, 1D, 2D, and Σ11 systems, which represents the extent to which the system disorder is localized to the grain boundary (narrower = more localized, wider = less localized). Cen- trosymmetry curves are shown in AppendixA.

Figure 4.26. An x/y plane of the 1D γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry. Results 56

Figure 4.27. An x/y plane of the 2D γ0 conﬁguration before (left and center) and after (right) GBS. The γ0 is coherent with the right grain. Atoms that have an FCC environment are colored dark blue (nickel grains) or white (γ0), and all other atoms are colored according to centrosymmetry. 57

5 Discussion

5.1 Σ11 Behavior

The maximum stress the system reached prior to deformation is 5.32 GPa, which compares well with the theoretical shear strength of nickel calculated by Liu et al. of 5.1 - 15.8 GPa62. The 5.1 GPa theoretical strength assumes slip occurs along the

{111}[112] or the ”easy” direction, and the 15.8 GPa assumes the slip occurs along

the {111}[112] or the ”hard” direction. If other slip systems are active, such as the

{111}h110i the theoretical strength is 6.2 GPa. Similar DFT calculations found a theoretical shear strength of 5.05 GPa63. The Σ11 boundary strength ﬁts within the reasonable range for strength. The shear modulus measured from the Σ11 simulation is 100.13 GPa, which is higher than the moduli calculated in Section

5.2.

The Σ11 boundary experiences grain boundary sliding under shear deformation, as one grain boundary is displaced relative to the other without signiﬁcant strain accommodation in the grains themselves. Prior to sliding, the periodic structure of the boundary (i.e., kite structure) changes but remains regular. After the sliding event, the boundary structure loses its periodicity, and the boundary width, as measured by the centrosymmetry and excess volume curve widths, expands Discussion 58

slightly. Dislocations form on the boundary, however, these dislocations are not visibly emitted into the grain boundary bulk. Superalloy systems typically experi-

ence Rachinger grain boundary sliding, which requires the emission of geometri-

cally necessary dislocations for the grains to move. However, because the system is

run at low temperature and with perfect bulk grains, there are no defects present

to help initiate dislocations, which Kurtz et al. found is necessary for Rachinger

sliding to occur21. The lack of thermal energy and intrinsic or extrinsic defects

allows the bulk grains to achieve very high strengths. Thus, the deformation is

localized to the grain boundary itself and sliding is accomplished through uncor-

related atomic shufﬂing, as the boundary is the only ”defect” present in the system,

and traditional sliding mechanisms are not observed.

5.2 γ/γ0 Deformation

The γ/γ0 system deforms via homogeneous nucleation of dislocations in the bulk,

as there are no defects, such as a grain boundary, to provide heterogeneous nu-

cleation sites. This homogeneous distribution of dislocations is shown visually

in Figure 4.10, and there is no localization of centrosymmetry or excess volume

in Figure 4.8. The shear modulus of the γ/γ0 system, 87.89 GPa, compares with

the shear modulus of nickel in the shear direction [332] calculated from the elastic

constants listed by the potential, 88.37 GPa1. The same modulus in the shear direction was calculated using experimental values, providing a shear modulus of 87.11

GPa64. All three moduli are very similar, indicating that the the potential and the

γ/γ0 system in this study are in good agreement with experimental results. The Discussion 59

γ/γ0 system is similar to a pure nickel system, as the γ0 is very small and does not

signiﬁcantly alter the bulk properties of the system.

5.3 γ0 Formation

The superalloy microstructure this study seeks to replicate requires cuboidal γ0

precipitates. Based on the Gibbs free energy of cuboidal vs. spherical precipitates

seen in Figure 4.4, cuboidal precipitates are thermodynamically favored at pre-

cipitate sizes greater than 0.0761 A˚ . The small transition size is due to the high

misﬁt of this potential, and given that the transition occurs at a length smaller than

one unit cell of γ0 (3.57 A˚ ), cuboidal precipitates will always be favored with this

potential. This lower transition bound of 0.0761 A˚ corresponds to the onset of co-

herent cuboidal precipitates, but as the precipitate grows larger the strain caused

by the γ/γ0 lattice misﬁt grows and semi-coherent precipitates arise. Misﬁt dislo-

cations are formed to relax the misﬁt strains, and the equilibrium spacing between

said misﬁt dislocations for this potential is 172.4 A˚ . The precipitates used in this

study are either 14.28 A˚ and 28.56 A˚ , which fall below the misﬁt dislocation spac-

ing. Thus, the cuboidal precipitates will be fully coherent, as the misﬁt strains are

not large enough to warrant misﬁt dislocation formation.

However, this assumes the γ0 precipitates’ cube faces align with the parent ma-

trix, which is not true for the square the inline configurations. Because the γ0 interface was artificially modified in order to impose certain orientation relationships between the γ0 and the grain boundary, small incoherences may form along the

γ/γ0 interface. This is shown in the starting conﬁgurations of the large square and inline systems, shown in Figures 4.21 and 4.22. Discussion 60

5.4 γ0 Strain Fields

This study uses ﬁxed, free surfaces in the grain boundary normal direction (y),

however there is a concern that the γ0 may interact with these high energy bound-

aries. In order to determine if the simulation is sufﬁciently large, the strain ﬁelds

of the γ0 were calculated, shown in Figure 4.5. It should be noted that the strain

ﬁelds are a calculation of the displacement of the atoms before and after the γ0 have

been inserted into the system, so some of the data may be due to atoms shufﬂing

during minimization and not directly related to the amount of residual strain in

the system. The small precipitates produce minimal strain ﬁelds, as the total strain

caused by substituting a small number of nickel unit cells with Ni3Al is small and

does not cause noticeable strain in the bulk of the grains. The larger precipitates

have a greater impact on the strain experienced in the bulk. Visually, the large

tilt precipitate has the greatest atomic displacements, followed by the large inline,

and ﬁnally by the large square precipitate. However, there is no evidence of the

strain ﬁeld interacting with periodic images of the γ0, as the strain ﬁelds do not

loop around the periodic boundaries. Thus, the simulation size is sufﬁciently large

to prevent the γ0 from interacting with interfaces other than the grain boundary

itself.

5.5 γ0 Stress Fields

In addition to the strain ﬁelds calculated for the precipitates, stress ﬁelds are calcu-

lated from the precipitates after relaxation. The stress is calculated per atom using

equation 3.261, and atoms are colored accordingly in Figure 4.6. The presence of Discussion 61

both small and large precipitates causes non-zero stress in the system, particularly

in the center of the precipitates, because the γ0 atoms are constrained. The larger

γ0 cause higher maximum stresses and larger stress ﬁelds than the smaller γ0. The

small square γ0 has a less severe stress field than the small tilt and inline γ0. The stress fields in the larger γ0 systems are more similar to one another, as all simulations have a high region of stress within or close to the γ0 that tapers off away from the γ0. The stresses caused by the presence of γ0 does not appear to affect the fixed simulation boundaries. It should be noted that the leftmost simulation boundary has a higher stresses than the rightmost boundary, which may influence the propensity for dislocations to form in that grain, which is shown in the orientation study in Section 4.5.1.

5.6 Orientation Study

All of the γ0 systems (tilt, square, and inline) experienced sliding at a lower stress than the grain boundary system on its own. The boundaries became disordered after sliding, and uncorrelated atomic shufﬂing occurred on the boundary. The systems all emitted dislocations during the sliding event, evidenced by the stacking faults left behind in the bulk after dislocation emission. The γ0, which can also be considered a ”defect”, helped nucleate dislocations and provide an energetic driving force to propagate the dislocations through the grain.

The tilt system was expected to be the strongest, as the orientation of the γ0

should result in a more serrated boundary, and these serrated boundaries are less

susceptible to sliding10. However, the inline γ0, which does not cross the boundary

and remains coherent with its parent grain, had the highest strength of the small Discussion 62

γ0 systems. One potential explanation for this behavior is a critical size criteria.

The γ0 are too small to strengthen the boundary and simply act as defect nuclei.

The inline precipitate disrupts the overall system the least because it does not cross

the boundary, so there is less of a driving force for defect nucleation, and this sys-

tem would be the strongest. The tilted precipitate disrupts the system the most

and thus would provide more of a driving force for defect nucleation and be the weakest. Additionally, the initial stress state of the system provides a driving force

for sliding activity, as the small tilt conﬁguration experienced sliding at a lower

applied stress than the square conﬁguration due to higher initial stress ﬁelds, as

shown in Figure 4.6. However, this stress explanation cannot fully account for the

trends in behavior, as the inline precipitate slid at a higher stress than both the tilt

and square precipitates, despite having a stress ﬁelds that rival the tilt precipitate.

These trends indicate that both initial stress and γ0 incoherency accelerate GBS.

5.7 Size Study

Larger γ0 precipitate systems were built, and the same deformation experiments were performed. The overall sliding behavior remained unchanged; dislocations were emitted as the sliding occurred and the boundary became more disordered via atomic shufﬂing. However, the trends in strength changed; the tilt and square

simulations strengthened with size, whereas the inline simulation weakened. As

the γ0 size increased for the tilt and square simulations, the γ0, which previously

provided a diving force for defect nucleation and increased susceptibility for slid-

ing, helped prevent sliding from occurring by forcing the boundary to cut through

the precipitate in order to straighten out and slide. The large inline γ0, however, Discussion 63

provides the driving force for defect nucleation, but does not provide the beneﬁt of

strengthening the boundary because the precipitate does not cross the boundary.

5.8 Interaction Study

The 0D system exhibits GBS and the γ0 on the boundary aids in dislocation nucle-

ation, as discussed before. The 0D system experiences a single stress drop as the

boundary slides, however the stress strain behavior of the 1D system is different.

Multiple, gradual stress drops occur, and this behavior is due to the γ0 being cut

and uncut as dislocations travel through the precipitate. Each stress drop is accom-

panied by a a dislocation cutting through the precipitate, and the stacking fault left

behind is reordered as a second dislocation travels through the material behind the

ﬁrst one. The stress in the system is gradually relieved through this process. No

dislocations are observed in the bulk, unlike the 0D system. The 2D system has

a single stress drop, similar to the 0D system in shape, however, the maximum

stress accommodated prior to GBS is much lower than the 0D and 1D cases. This

is likely due to the increased concentration of γ0 on the boundary, which aid in

nucleating defects. A higher γ0 concentration means a higher defect nuclei concentration, and the system is likely to nucleate defects and relieve system stresses sooner than would be observed in the systems with lower γ0 concentrations.

The 0D and 1D systems also have similar centrosymmetry curve widths, and

the majority of the disorder in the system is localized to the grain boundary plane.

Minimal dislocations are visible in the bulk of the 0D system. The 2D system, how-

ever, experiences a dramatic increase in centrosymmetry width, and a much higher Discussion 64 concentration of dislocations and stacking faults are visible in the bulk. This indicates that the GBS in the 2D system is accommodated by geometrically necessary dislocations to a greater extent that seen in the 0D and 1D systems, whose GBS behavior is dominated by uncorrelated atomic shufﬂing on the grain boundary. The expected Rachinger GBS is more clearly evident in the 2D system, suggesting that there is a critical concentration of γ0 or other defect nucleating features necessary to reproduce the experimentally observed sliding behavior. 65

6 Conclusions

This study examined the inﬂuence γ0 precipitates had on the boundary deformation of a Σ11 boundary. Speciﬁcally, the orientation (tilt, square, and inline), size (small, large), and interaction between precipitates (0D, 1D, 2D) were modi-

ﬁed. Overall, this study concludes:

• A perfect Σ11 boundary exhibited GBS via uncorrelated atomic shufﬂing.

• The presence of γ0 in an otherwise perfect grain boundary accelerates slid-

ing by nucleating defects into the bulk.

• The strength of the tilt and square γ0 systems increased with increased size,

and the strength of the inline γ0 system decreased with increased size. The

strengthening is a result of the γ0 being cut as the boundary straightens.

• The interaction of multiple γ0 changes the stress strain behavior, and there

is a critical concentration of γ0 necessary to produce GBS behavior that

is dominated by geometrically necessary dislocation emission rather than

uncorrelated atomic shufﬂing.

Although these simulations are not a perfect representation of the superalloy system, they do provide insights to how the physics of the γ/γ0 phases (i.e., the potential) inﬂuence the deformation. 66

7 Suggested Future Research

7.1 Grain Boundary Morphology

These simulations assume the morphology of the grain boundary due to the presence of the γ0. Future work would include building a γ/γ0 system and driving the

grain boundary towards the precipitate in order to observe the resulting boundary

morphology. A Σ9 boundary has been identiﬁed that produces shear coupling65,

so this boundary can be pushed towards γ0 precipitates with different orientations

relative the boundary in order to observe the resulting morphology and how the

precipitate impedes the motion of the grain boundary.

7.2 Grain Boundary Conﬁguration

This study focused on bicrystals and inﬁnite, periodic boundaries, however this

conﬁguration does not represent the complex interactions between multiple, non-

linear grain boundaries in real materials. A study by Thomas et al. developed

a grain boundary conﬁguration that models mobile boundaries that experience

coupling, as well as stationary boundaries66. This conﬁguration could be used to

understand how the γ0 inﬂuences boundary morphology with a more ”realistic”

grain boundary setup. Appendix 67

Appendix A

Figure A.1. The centrosymmetry and excess volume of atoms averaged in bins along the y axis of the γ0 tilt, square, and inline systems before (left) and after (right) shear deformation. Appendix 68

Figure A.2. The centrosymmetry of atoms averaged in bins along the y axis of the small and large tilt, square, and inline systems before (left) and after (right) shear deformation. Appendix 69

Figure A.3. The excess volume of atoms averaged in bins along the y axis of the small and large tilt, square, and inline systems before (left) and after (right) shear deformation. Appendix 70

Figure A.4. The centrosymmetry (left) and excess volume (right) of atoms averaged in bins along the y axis of the 0D, 1D, and 2D systems before and after shear deformation. References 71

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