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Statistical Physics of Grain Boundary Engineering

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STATISTICAL PHYSICS OF GRAIN BOUNDARY
ENGINEERING

By

Erin Scott McGarrity

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2005

 

_. wizu

ABSTRACT
STATISTICAL PHYSICS OF GRAIN BOUNDARY ENGINEERING
By

Erin Scott McGarrity

Polycrystalline materials are used in a wide variety of modern engineering applications.
From steam pipes in nuclear reactors to ceramic superconductors to battery terminals, they
can be found in almost every engineering application. One of the key indicators of poly-
crystalline material performance is the behavior of the grain boundary network. Experi-
mental results have shown that the nature and connectivity of this network in a material
directly affects its corrosion resistance, ductility, conductivity, and fracture properties. It is
for this reason that it is of import to study and control these boundary networks. This field
is known as grain boundary engineering (GBE).

This thesis presents an investigation of computational techniques to simulate and an-
alyze the statistics of GBE materials in two and three dimensions. First, two models for
generating realistic polycrystalline microstructures are presented. Once these polycrys-
talline microstructures have been generated, they may be examined for their percolative
behavior and the properties of their critical manifolds. From these Studies, scaling laws
are developed which help experimentalists eliminate unimportant variables. The results
described herein are also compared to real experimental data. The data generated by the

given models and techniques compare favorably to these experimental data.

© Copyright December 15, 2005 by

Erin Scott McGarrity
All Rights Reserved

For my daughter, Neysa, who teaches me something new every day.

ACKNOWLEDGMENTS

I would like to begin by giving thanks to my advisor, Phil Duxbury. His guidance and
assistance helped me overcome my lack of an undergraduate physics education. I also
appreciate the many discussions we have had, both about physics and other realms. He can
be on my debate team any time he would like! Mind you, this team almost always meets at
a bar.

Next, I extend a warm thank you to my collaborators at Sandia and Lawrence Livermore
national labs. Liz Holm and Bryan Reed are at the top of this list since they provided good
problems to be solved and the resources for me to work on them. I would also like to thank
Corbett Battaile at Sandia, and Mukul Kumar, and Roger Minich at Livermore for their
time and input through our discussions. To this list I would add my advisory committee,
Professors Mahanti, Tessmer, Danielowicz, and Bieler, for their questions and discussions
about my work. I would also like to add Professor MacCluer from the Mathematics depart-
ment for all of his advice, and for employing me in his MATLAB venture, which helped
me avoid the cramped lifestyle that normally besets a graduate Student due to lack of funds.

I would be remiss if I didn’t mention the people with whom I enjoyed my free time.
These include Kim Cooper, who helped me through troubled times; Neil Aaronson and the
choir, “The Grand Canonical Ensemble”; and Matt Goupell for introducing me to Ultimate
and playing rock-n-roll with me. These activities were essential for sanity management.
Also, I would like to thank my office mates, Jan Meinke, Radu Cojocaru, Chip Fay, Jiwu
Liu, Chris Farrow and Dan Olds for many exciting discussions as well as for putting up

with me on my not so cheery days. Extra honorable mention goes to Jan, as he was the one

who Showed me the ropes for my research.

vi

TABLE OF CONTENTS

LIST OF FIGURES
1 Introduction
2 Experimental Techniques of GBE Materials

3 Models for Grain Growth and GBE

3.1 Potts Model ....................................
3.2 Label Differencing GBE Model .........................
3.3 Crystallographic Constraints ...........................

4 Percolation

4.1 Scaling Laws of Percolating Behavior in GBE Materials ............
4.2 Percolation in Random GBE Microstructures ..................
4.3 Percolation in CSL GBE Microstructures ....................

5 Critical Manifolds

5.1 Low Concentrations c < < c5 A p .........................
5.2 The Critical Regime c z c5 A p ..........................
5.3 High concentrations, c >> C5 A p, e —> 0 .....................
5.4 Unified form of scaling theory ..........................
5.5 Critical Manifolds in CSL GBE Microstructures .................

6 Conclusion and Further Work
6.1 Conclusion ....................................
6.2 Things to Do ...................................

APPENDICES

A Glossary

B Algorithms

8.] Conventional Potts Model .............................
B.2 n-fold Potts Model ................................

B.3 CSL Microstructure Generation .........................
B.4 Dijkstra’s Shortest Path Algorithm ........................

C Derivations
C.l Percolation Threshold on a Bethe Lattice .....................

BIBLIOGRAPHY

vii

viii

17
18
21
24

32
35
38
41

46
48
5 1
60
68
69

76
76
77

80
81

82
82
83
85
86

88
88

92

1.1

2.1
2.2
2.3
2.4
2.5

3.1
3.2
3.3
3.4
3.5
3.6
3.7

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10

5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14

LIST OF FIGURES

Critical Manifold Example ............................ 4
EBSD Image of non-GBE Inconel 600 ...................... 12
EBSD Image of non-GBE Inconel 600 ...................... 13
EBSD Image of GBE Inconel 600 ........................ l4
EBSD Image of GBE Inconel 600 ........................ 15
Material Cracking Setup ............................. 16
Simulated Microstructures in Two and Three Dimensions ............ 19
GBE Microstructure ................................ 23
Grain Boundary Character Distribution ...................... 25
CSL Example ................................... 27
CSL Topology ................................... 28
Computer Generated CSL Cluster ........................ 30
Experimentally Observed CSL Cluster ...................... 31
Weak Bond Penetration in a GBE Crystal .................... 33
Spanning Strong Aggregate in a GBE Crystal .................. 34
Strong Aggregate Size .............................. 35
Cluster Correlation Length ............................ 36
Spanning Cluster Probability in Two Dimensions ................ 39
Spanning Cluster Probability in Three Dimensions ............... 40
Order Parameters in Two Dimensions ...................... 42
Order Parameters in Three Dimensions ...................... 43
Order Parameters in Two Dimensional CSL Polycrystals ............ 44
Weak Bond Penetration Depth in Two Dimensional CSL Polycrystals ..... 45
Critical Manifold Energy in Two Dimensions .................. 49
Critical Manifold Energy in Three Dimensions ................. 50
Fraction of Weak Boundaries on the CM in Two Dimensions .......... 51
Fraction of Weak Boundaries on the CM in Three Dimensions ......... 52
Number of Bonds on the CM in Two Dimensions ................ 53
Number of Bonds on the CM in Three Dimensions ............... 54
Sealed Roughness of the CM in Two Dimensions ................ 55
Sealed Roughness of the CM in Three Dimensions ............... 56
Roughness Scaling in Two Dimensions ...................... 57
Roughness Scaling in Three Dimensions ..................... 58
CM Energy for CSL Model ............................ 71
Fraction of Weak Bonds on CM for CSL Model ................. 72
Number of Bonds on CM for CSL Model .................... 73
Sealed Roughness of CM for CSL Model .................... 74

viii

5.15 CM for CSL Model ................................ 75
CI The Bethe Lattice ................................. 89

ix

Chapter 1

Introduction

Polycrystalline materials are used in a wide variety of modern engineering applications.
From steam pipes in nuclear reactors to ceramic superconductors to battery terminals, they
can be found in almost every engineering application [42]. The microstructure of these ma-
terials is comprised of Single crystals or grains whose lattices are oriented differently from
each other. The boundaries between these grains form a mesh or network. One of the key
indicators of polycrystalline material performance is the behavior of this grain boundary
network [33, 58]. Experimental results have shown that the nature and connectivity of the
grain boundaries in a material directly affects its corrosion resistance, ductility, conduc-
tivity, and fracture properties [8, 9, 22, 28,40]. For this reason, the Study and control these
boundary networks is desirable [34,41]. This field is known as grain boundary engineering

(GBE).

GBE was first introduced as “grain boundary design” by Watanabe in 1984 [58]. In this
groundbreaking work, he was able to improve the ductility and strength of a material by ap-

1

plying a thermomechanical process which changed the distribution of its grain boundaries.
The process is accomplished by deforming and annealing the material several times inside
its ductile regime. The effect of this process is to change the grain boundary character
distribution (GBCD) of the material. The GBCD describes the statistical properties of the
grain boundary network in a microstructure [34]. A comprehensive review of experimental

techniques and results for GBE can be found in [44].

The primary goal of the work presented in this dissertation is to capture the essence
of real materials and develop simulations which can be used to design engineered mate-
rials. Simulating materials on a computer can be done fairly quickly compared to real
experiments and, therefore, could be used to eliminate some of the search space required
by experimentalists when attempting to optimize material properties. The most important,
and probably the most difficult, aspect of these Simulations is determining a metric with

which to compare Simulations to experiments.

This thesis presents an investigation of computational techniques to Simulate and ana-
lyze GBE materials in two and three dimensions. First, two models for generating realistic
polycrystalline microstructures are presented. Once a polycrystalline microstructure has
been generated, its percolative behavior [35,43,47,50,52,55,57,60] is studied. At the same
time, a minimum energy manifold is calculated for the sample [16, 26, 28,29,36—38]. From
these studies, scaling laws are developed which help experimentalists eliminate unimpor-

tant variables. Each of these steps is introduced in more detail in the following passages.

Two methods for incorporating GBE into a q-label Potts model [6, 7, 25, 27, 48, 54] are
studied; one which has uncorrelated grain orientations, and one which uses a coincidence

2

site lattice (CSL) model for the grain orientations [46, 56]. The relative grain orientations
are used to determine “good” and “bad” boundaries. By controlling the fractions and rel-
ative strengths of these boundaries, the onset of emergent behaviors such as percolation
can be determined. The CSL model also has the capability to incorporate realistic values
for the GBCD into our model because it contains the actual Spatial orientations for each

grain [5,14,20,21,32,34,39,41,45,51,59].

The next step in the modeling process is to Study the percolative characteristics of the
generated microstructures. In these studies two types percolation have been identified as
important in ascertaining material properties [35]. The first is percolation of a network
of “bad” or weak grain boundaries across a given sample. The penetration depth of weak
bonds into a sample is an indicator of how far an impurity can diffuse into it. By varying the
concentration of strong boundaries in the sample this percolation can be halted at or above
a critical concentration, CW 3 p. This leads to improved corrosion resistance and crack arrest

properties in the material.

The second type of percolation is a form of Site percolation. It is found that at a given
concentration, c5 A p, an aggregate of strongly bonded grains spans the sample. This ag-
gregated cluster forms a sort of backbone in the sample. In two dimensions, the critical
threshold is the same as for weak bond percolation, so that the strong cluster serves as a
roadblock to weak bond percolation. This cluster also can be used to analyze and deduce

properties of the minimal energy surface, or critical manifold, which spans the sample.

Another important statistical property of a GBE polycrystal is its critical manifold
(CM). The CM is a surface which divides the sample into two parts, a source Side and

3

 

 

 

 

Figure 1.1: Mapping of a polycrystal to a graph. The nodes in the graph correspond to the
sites in the Potts model. The numbers indicate the Potts label. Edges with energy 6,- and 68
represent intergranular, and intragranular bonds in the model, respectively. The bold line is
a critical manifold which divides a graph into source side and the sink side. The sum of the
bond energies which the manifold crosses is minimal.

a Sink Side. This surface is highly degenerate in large samples has the minimum possible
energy, which is the sum of the costs of the bonds that it crosses. Figure 1.] depicts a criti-
cal manifold for a Simple network. For a polycrystal, the Sites of a Potts model are mapped
to nodes of a graph which is identical to the one shown in this figure. Bonds are made
between nearest neighbors and are assigned energies based on the labels of the Sites they
connect. This seemingly hard combinatorial problem can be solved in polynomial time
(0 (n2 log n)) [13]. This is accomplished by mapping the grain boundary network to a
graph and applying the max-flow/min-cut algorithm from computer science [2, 12, 13, 23].
The geometric properties of this manifold varies with the concentration of “good” bound-
aries and the energy contrast of “good” to “bad” boundaries. At a critical concentration,
the GBE polycrystal undergoes a toughening transition, where the critical manifold must
wander considerably to span the sample. This point can be used as a predictor of material

toughness and fracture character.

With these Statistical properties in mind, computer studies were performed in which the
growth of two and three-dimensional polycrystals were Simulated and the CM and percolat-
ing structures were computed. The parameters for these simulations were the weak/strong
bond energy contrast, 6, the concentration of strong bonds, C, lattice Size, L, and the av-
erage ensemble grain size, g. Each of these parameters was selected and varied to elicit a

particular behavior in our models.

The energy contrast, 6 = ew/es, was varied from 1/ 10 to 1/ 10000. Define em and
£5 to be the energies of the weak and Strong bonds in the model, respectively. In the 1 / 10
limit, the bond energy contrast is similar to that of a real material. The roughness scaling

5

looks much like that seen in real fracture. To determine the critical point and find scaling
exponents, a contrast of 1 / 10000 was used. At this point the roughening transition sharpens
markedly and the critical concentration can be computed for the model. It is at this point

that the model begins to exhibit interesting scaling behavior.

The fraction of strong bonds, c, was varied between 0 and 1 in small steps in order to
search for the critical points. In the label differencing model (see Section 3.2), the con-
centration was calculated by including larger and larger label differences until the desired
fraction was met. For the CSL model, the total concentration of strong bonds (£3, 29,
and 22% was varied but their relative fractions were left unchanged as is seen in experi-

ments [44, 49].

In simulations it is important to account for finite Size effects. These effects can cause
both Shifts and broadening in the observed critical points. To determine these effects, the
lattice sizes, L, were varied from 125 to 2000 in two dimensions and 50 to 125 in three
dimensions. The ensemble average grain size, g, also plays a role in scaling effects. In fact,
it is possible to define a lattice constant, L / g, to describe the effective lattice size in the

Simulations.

Once the data from the simulations was collected, it was analyzed and the mathemat-
ics which adequately described it were written down. The CM properties examined were
the cut energy, the cut roughness, the total number of cut bonds, and the fraction of weak
cut bonds. It was found that there are three regimes based on concentration. The general
mathematical properties of each are described herein. In addition, the percolation proper-
ties of each ensemble were examined. The two main properties were the size of the largest

6

strongly bonded aggregate and the penetration depth of the weak bond network, both as
functions of concentration. These are duals of one another in two-dimensions, so that a
percolating strong aggregate blocks bond percolation at C5 A p = ngp = c*. In three-
dimensions, however, both a percolating Strong aggregate and a percolating weak network
can coexist in what is known as an inter-penetrating phase.

The remainder of this thesis is divided into the following chapters. Chapter 2 describes
the experimental techniques used in the field of grain boundary engineering and gives some
motivation for studying the subject computationally. Chapter 3 describes the models used
for simulating the growth and grain boundary network properties of polycrystals. In Chap-
ter 4, the criteria for percolation are defined and the mathematics describing the data are
introduced. Chapter 5 describes critical manifolds, their computation, and their mathemat-
ical properties as applied to the models for grain growth. Finally, in Chapter 6, a summary

of the results is provided and further research goals are postulated.

Chapter 2

Experimental Techniques of GBE

Materials

In order to validate the computational models, it is important to understand the experimen-
tal techniques and results for GBE materials. This chapter contains a brief overview of the
processing of these materials. In short, the raw material is treated by some thermomechani-
cal process, then it is characterized by using electron back-scatter diffraction (EBSD). This
characterization gives information on the orientations of all the grains and thus the types of
grain boundaries present in the sample. Finally, the materials undergo various mechanical
failure tests, depending upon the intended application. These tests are typically designed
to determine the failure modes, such as cracking or corrosion rates. Unfortunately, these
tests usually destroy the experimental sample, and given the amount of labor that goes into

processing them, expensive.

The literature is rife with experiments involving GBE. A good overview of the major

8

experiments in the field of GBE is given by Randle [44]. There have been many materials
investigated including Inconel 600, stainless steel 304, nickel, copper, brass, and lead. Each
of these materials has one or more important engineering applications. For example, the
experiments with lead were aimed at improving the intergranular corrosion resistance of

battery terminal plates, thereby increasing battery life and performance characteristics.

Improved grain boundaries are achieved by performing several iterations of a thermo-
mechanical process upon the material. This process is known in the colloquial as “heat
and beat.” First the sample is reduced along one dimension by cold rolling. The typical
range of the reduction is between 5 and 30 percent and remains in the ductile regime. The
compression causes strain energy and recrystallization nuclei to be Stored in the crystal.
This energy is then annealed away in the form of grain growth and boundary adjustment
during the recrystallization phase. This phase is performed by either heating the sample for
a long time or applying a strain to the material. This process is repeated from one to seven
times and has been shown to cause a significant increase in low angle or “Special” CSL
boundaries in a sample. Typical concentrations of low angle CSL boundaries achieved are

from 30 to 90 percent.

The primary method used to determine the number of Special boundaries in a given
polycrystal is by examining orientation imaging microscopy (OIM) data. OIM is a method
which uses electron scattering to determine the lattice orientations of the grains within a
polycrystal. When an electron beam is incident to a crystal it diffracts off the lattice planes
and produces a scattering pattern [31]. This pattern can be used to indicate the orientation
of the lattice in three dimensional space. By examining the relative orientations of two

9

grains it is possible to determine the angle between them. This angle determines the 2

number in the CSL model.

Figures 2.1 and 2.2 Show orientation images of non-GBE polycrystalline microstruc-
tures which were acquired using electron back scatter difiraction (EBSD) imaging. The
low angle boundaries seen in these OIM images correspond to strong boundaries in the
computer model. The concentration of good boundaries can be measured from this image
so that a correlation can be made between the boundaries and the material properties which
are of interest in the experiment to be performed. Figures 2.3 and 2.4 Show EBSD data for
two GBE samples. Note that there is a Si gnificantly higher number of twin boundaries (23)
in the GBE samples. (Twins are indicated in red in these images.) Note: images in this

dissertation are presented in color.

Once the GBCD is determined for a sample, it can be tested for its engineering prop-
erties. The properties which have been examined in experiments include counting of triple
junctions and grain boundary statistics, creep, ductility, crack resistance, and intergranular
corrosion resistance. All of these tests are rather laborious in nature and most involve the

destruction of the sample.

Creep is defined as plastic deformation of a material due to strain. To test this property,
strain is applied to the GBE material which is above its elastic limit. In this regime the
material is permanently deformed. The strain is held for some time and the deformation is
measured. Usually the material is taken to its yield limit in these experiments. Ductility is
tested in a Similar fashion, except that the material is strained until it fractures. The amount
of inelastic deformation that it can undergo before it breaks is recorded as the ductility.

10

Crack resistance and roughness tests are performed by applying a transverse quaSiStatic
strain to the sample. Figure 2.5 Shows a setup for such an experiment. The crack is ob-
served as it moves through the sample, typically over a period of days or weeks. Once
the crack makes its way through the entire sample, the surface roughness exponent can be
determined by a surface measurement technique, such as atomic force microscopy (AFM).
This technique drags a probe with a single atom tip across the surface and records the fluc—
tuations in height. From the surface measurement, a roughness exponent can be extracted.

Intergranular corrosion resistance is measured by placing a sample in an environment
which has penetrants in it. The substance is allowed to diffuse into the sample for some
time. A percolation Study can then be done by sectioning the material serially and searching
for the impurities which have penetrated into the grain boundaries. The deepest penetration

can be correlated with the diffusion time to estimate the lifespan of the material.

11

 

—
300.0 um = 100 steps

Figure 2.1: EBSD orientation image of non-GBE Inconel 600 alloy sample. The bound-
aries are colored by twin-related domain: 23 (red), 29 (blue), 227 (green) and random
boundaries are black. The black bands are scratches on the sample surface.

12

 

— Boundary levels: 2°
280.0 um = 80 steps Unique Grain Color

Figure 2.2: EBSD orientation image of non-GBE Inconel 600 alloy sample. The bound-
aries are colored by twin-related domain: 2.3 (red), 29 (blue), 2127 (green) and random
boundaries are black. The black bands are scratches on the sample surface.

 

—
200.0 um = 80 steps

Figure 2.3: EBSD orientation image of GBE Inconel 600 alloy sample. The boundaries are
colored by twin-related domain: 23 (red), 29 (blue), 227 (green) and random boundaries
are black. Notice the increased number of twin boundaries as compared with Figures 2.1
and 2.2. The black bands are scratches on the sample surface.

lllllllll

 

— Boundary levels: 2°
280.0 pm = 70 steps Unique Grain Color

Figure 2.4: EBSD orientation image of GBE Inconel 600 alloy sample. The boundaries are
colored by twin-related domain: 23 (red), 29 (blue), 227 (green) and random boundaries
are black. Notice the increased number of twin boundaries as compared with Figures 2.1
and 2.2. The black bands are scratches on the sample surface.

 

 

 

 

 

Figure 2.5: Experimental setup for quasistatic cracking. A crack is started with a notch and
forces are applied according to the vectors. The crack is observed as it evolves from the
notch. Once the material fails, the surface roughness can be measured with AFM.

16

Chapter 3

Models for Grain Growth and GBE

This chapter gives the details of the computational models which were used to Simulate
GBE materials. Both models presented are based on the q-label Potts model, which has
been Shown to produce realistic growth rates and network topologies for polycrystals [6,
54]. The actual implementation is based on the n-fold way algorithm [25], which was

designed to avoid the critical slowing down typical of Monte Carlo methods [7, 25].

Once the structures have been grown, the effects of GBE are simulated by defining two
types of grain boundaries, “good,” or strong boundaries, and “bad,” or weak boundaries.
When modeling mechanical properties, the strong and weak boundaries would be associ-
ated with low and high angle boundaries found in real materials. Energies are associated
with the two types of boundaries, 6, for strong, and ew for weak, and the concentration of

strong boundaries, c is varied from 0% to 100%.

Two approaches to varying the concentration were implemented. The first used the
modulus of the difference of the Potts labels [16, 35, 37] and the second used a group

17

theoretic method to enforce constraints on the crystallographic orientations of the grains
[46,49,50]. Both methods give interesting and qualitatively realistic results when compared

to actual experiments [44].

This chapter is divided into three sections. In Section 3.1 the basic Potts model is
described and its properties are reviewed. Section 3.2 gives details of the label differenc-
ing method for simulating GBE materials. Section 3.3 contains a theory and method for

introducing crystallographically correct boundaries.

3.1 Potts Model

The q-label Potts model maps a polycrystalline microstructure onto a discrete lattice. Each
lattice Site, 5,- is assigned a label between 1 and q. These labels are used to indicate the
crystallographic orientation of the grain that they belong to. This allows for q different
orientations in the crystal, where q = 100 in two dimensions and q = 256 in three dimen-
sions in order to reduce the probability of having grain neighbors with like labels. It also
represents the larger number of orientations that occur in real materials. Examples of two
and three dimensional microstructures are shown in Figure 3.1. In this figure, grain labels

are indicated by color. The Hamiltonian for this model is given by

H=E0£ZI—5(Si,5j), (3.1)

where 50 is the site interaction energy, N is the total number of sites, z(i) is the connectivity
of Site i, and (S is the Kronecker delta function.

18

 

(b)

Figure 3.1: Examples of two and three-dimensional simulated grain structures grown with
a q-label Potts model. In (a) q = 100 and in (b) q = 256.

19

The simplest method that can be employed to evolve the microstructure is the Metropo-
lis Monte Carlo method. In this method, a Site is chosen at random and assigned a new
label between 1 and q. The energy difference, AB, is computed and the move is accepted
using the probabilities

(NS/""8T NE > 0
MAE) = (3.2)

1 A1330,

where kBT is the Boltzmann constant times the simulation temperature. The algorithm is

given in Appendix 8.].

The simplicity of this method is offset however due to the fact that most of the sites
chosen are in the bulk, i.e., all their neighboring sites have the same label. This means that
the grain edges are Slow to evolve and a large amount of computational effort is wasted,
especially as the grains get larger. To combat this, an algorithm was developed known as

the n-fold way algorithm [7].

The n-fold algorithm works by assigning activities to each Site which are based on the
number of unlike neighbors. Since Sites on the boundaries of grains have unlike neighbors,
in contrast to bulk sites, they are chosen for update more frequently. Once a site is chosen,
it is assigned either a “tame” label update, in which its new label is chosen at random
from one of its neighbors, or a “wild” update, where the new label is chosen completely
randomly from all q labels. This “wild” update accounts for nucleation of new grains. The

algorithm is given in Appendix B.2.

This model was used to grow grains on two dimensional square lattices with sides
ranging from L = 125 to L = 2000, and three dimensional cubic lattices with sides from

20

L = 50 to L = 125. The total number of sites is L“ where d is the number of dimensions.
The size limits for the algorithm are determined by the amount of memory available on the
computer and the amount of computation time available.

As the algorithm runs, snapshots of the growing crystal are saved to the disk at regular
intervals. The data begins to resemble actual polycrystals when the grain Size reaches g z 6
lattice units in two dimensions and g z 4 lattice units in three dimensions [6, 54]. These
are the lower bounds of the grain Sizes represented in the data. The data used has an upper
bound to the grain radii which is set by the effective lattice size L/ g. In two dimensions,
the bounds span from 10 to 200, and in three dimensions, they range from 6 to 20. Below
these limits, finite size effects become dominant in the statistical calculations, therefore,

the algorithm is terminated when it reaches this point.

3.2 Label Differencing GBE Model

Once a microstructure has been grown, a model for GBE can be applied to it. The simplest
model takes a fraction c of the grain boundaries to be strong and l — c to be weak. The
bonds across the weak grain boundaries are assigned an energy cw, while the bonds which
are interior to the grains, and the bonds across the strong boundaries are both assigned
energy 65. The energy contrast, 6 = ew/es, which in real materials, is typically around
1/ 10, determines the sharpness of the roughening transition. To capture the emerging
scaling behaviors computations were performed with e ranging from 1 / 10 to 1 / 10000.
The advantage to this model is its simplicity. With it, the scaling properties of GBE
polycrystals can be estimated. In fact, it was found that these properties approach that of

21

a pure hexagonal lattice in two dimensions. The main disadvantage of the model is that it
does not account for constraints on crystallographic orientations because the grain labels
are randomly distributed. In real materials boundaries emanating from the same junction
are necessarily correlated through a rotation matrix. This correlation is included in the

model presented in Section 3.3.

In the uncorrelated model, the strong grain boundary bonds are selected based on the
difference in Potts labels of the grains on either side of the boundary. That is, in the Potts
model, each Site has a label 5,- E {1, ..., q}. We define the normalized difference between
the Potts labels at the Sites at each end of a bond to be 55 2| s,~ — 51- | /q. We also define
the difference in labels modulo q, so that labels 1 and q differ by one. This definition
ensures that all label differences occur with equal probability. The concentration, c, is set
by creating a histogram of boundary label differences from the microstructure and finding
the threshold, 61, that corresponds to it. All boundaries with (55 S 6, are considered to be

strong and the rest are considered to be weak.

An example of a two dimensional system where c = 0.39 is presented in Figure 3.2. In
this figure, the weak grain boundaries are highlighted as thick grey lines. The remaining
grain boundaries are strong. Note that in many materials the distribution of grain boundary
orientations is not uniform, as is implicit in the model used here. The effect of varying
the grain boundary orientation distribution and including correlations in grain boundary
orientations [49] can be included in a straightforward manner and will be considered in the
next section.

22

 

Figure 3.2: This microstructure shows a label-differenced GBE simulation with c = 0.39.
The thick gray lines show the weak boundaries.

23

3.3 Crystallographic Constraints

While the label differencing model described in the previous section captures the essence
of GBE, it ignores real crystallographic constraints. These constraints cause correlations to
emerge in the grain boundaries, in particular, at junctions in the grain boundary network.
The total misorientation around any junction or any closed path through a series of grains

must be zero.

To implement the grain orientation constraints, each grain is assigned a 3 x 3 rotation
matrix which describes its orientation with respect to a reference coordinate system. Once
these matrices have been assigned, the misorientation between two grains is defined as the
minimum angle which must be rotated through to get from one grain’s coordinate system

to the other. Circulation around a closed path of grains must result in the identity matrix.

To simulate GBE with this scheme, an empirically determined GBCD was chosen and
the sample was annealed using a Metropolis Monte Carlo method. Two ways of approach-
ing the target distribution were implemented. The first was to swap two grain orientations
and the second was to rotate a Single grain. Both methods are unphysical, but they pro-
duce the correct statistics. The algorithm used is given in Appendix B.3 An example of an
experimentally determined GBCD is shown in Figure 3.3. The task is to minimize the dif-
ference between the desired distribution, 8", and the sample distribution, Sm . The objective

function is written as

"max
minimize X (5;? — 5:1)2, (3.3)
n=0

where the n-th entry of each histogram S is the bin representing the CSL misorientation

24

 

0.4 fir
[

0.35 -

0.37

0.25]
0.15,l
l
0.1r

0.05i‘

 

n

Figure 3.3: Example of an experimentally determined GBCD. The x axis contains the ex-
ponent of the CSL number, 23", and the y axis shows the relative amplitudes of each type
of boundary. A similar histogram can also be made using the actual measured misorienta-
tion angles, A6.

angle 2.3". (Note that if actual angles are used instead of the CSL model, then it represents
angles between nAO and (n + 1)A9.) The variable nmax is determined by the misorienta-
tion angle at which bonds begin to be considered weak. In the CSL case this is 2.81.

To avoid getting trapped in local minima, the algorithm is performed at finite tempera-
ture with a Boltzmann factor k3 T. The probability of accepting a more favorable configu-
ration is given by,

1 AHS S 0
P(AHS) = (3.4)

FAIR/"'9T AHs > 0.

Altemately, the crystallographic orientations of the grains can be represented using the

25

coincidence site lattice (CSL) model [56]. This model describes the grain orientations as
a sequence of special rotations about fixed axes within the crystal. These Special rotations
cause certain Sites on the lattice to coincide with each other and generate a new lattice of
the same type but with a different cell volume. The new cell volume is measured in units of
the original cell volume and is known as its 2 number. A simple example of a CSL can be
seen by overlaying two square lattices which are rotated 2 tan‘1 (1 / 3) degrees with respect
to each other. This angle is computed by noting which sites coincide after rotating. In this
case, the Site which is at (1, —3) with respect to the center of rotation overlaps with the site
at (1, 3) of the second lattice. Figure 3.4 shows the situation. The new unit cell covers an

area of 5 of the old unit cells, therefore it is a £5 rotation.

The CSL used in these simulations is body-centered cubic (BCC). The special rota-
tions begin with 21, which is aligned with the world coordinate system, a sequence of £3
rotations are applied. There are four such rotations from any given state one for each diag-
onal of the cube. Essentially, any rotation can be represented by concatenating an arbitrary

number of these simple rotations.

Figure 3.5 shows a topological schema for the CSL representation. This diagram can
be used to determine both the angular orientation of a grain as well as the misorientation
between two grains. To determine the rotation string for a grain, follow the path from the
£1 to the desired cell and concatenate the line labels together. For instance, the grain B has
orientation 21 and grain D has orientation 24. Both grains have 29 misorientation with the
world coordinate system. The misorientation between two sites is determined by walking
from the first site to the second and counting the number of Steps. For example. to get from

26

 

Figure 3.4: Example of a square CSL pattern. A unit cell from the generated lattice is
outlined. The total area covered by it in units of the original lattice is 5 squares. therefore
this rotation corresponds to a Z25 rotation.

A to B it takes 3 Steps.

As with the matrix method above, each grain has a rotation sequence that gives its crys-
tallographic orientation. The misorientation between two grains is found by concatenating

the two Strings, a" and b, together using the rules

“ibj at? 7E b-,

27

\ /5
/ \ /.
/ \ /....
m/a\m

Figure 3.5: Graphical representation of the CSL topology model for a BCC lattice Structure.

where i and j are the indices of the last elements of the two strings, a* is the reverse of
the string a and I represents identity. This process is repeated until all of the duplicates
are eliminated. The length, n, of the remaining string is taken to be the exponent of the

CSL, i. e., 213". In this representation, £3, £9 and 227 boundary misorientations are strong,

higher boundary misorientations are weak.

As with the matrix representation, the CSL distribution is randomly assigned and an-
nealed to a target empirically determined GBCD. The annealing is performed by a com-

bination of grain orientation swaps and rotations. Swapping is accomplished by choosing

28

two non—adjacent grains at random and switching their labels. Rotation is performed by
choosing a grain at random and concatenating a random rotation onto its orientation String.
In either case the acceptance criteria is given by Equation 3.4.

An interesting feature of the CSL model is the correlated Structures that emerge during
the annealing stage. It appears to be energetically favorable to form long anisotropic chains
of grains. These clusters are connected weakly to the rest of the polycrystal and provide the
main impetus for the wandering of the critical manifold at higher concentrations. Figure 3.6
shows the dominant cluster in a computer generated polycrystal. Each grain in the cluster is
connected by a strong boundary, i. e., X3, 29, or 227. This strong cluster is weakly bonded
to the rest of the crystal (dark blue.)

The experimental results shown in Figure 3.7 provide, at least a qualitative confirma-
tion of the formation of correlated clusters. This EBSD image shows the CSL correlation
map for a real GBE material. Notice that the cluster has an anisotropic character, which is
similar to the simulations. It is important to note, however, that there may be further corre-
lations which are happening in three dimensions which are not captured by either the model
or the EBSD measurement. Further research will be required to deepen the understanding

of how subsurface correlations play a role in the GBCD.

29

 

Figure 3.6: Computer generated cluster of strongly bonded grains. The grains in the cluster
are all related by either £3, 2.9 or 227 boundaries. This cluster is weakly bonded to the
rest of the sample (dark blue.)

30

 

Figure 3.7: Experimentally observed EBSD image of a cluster of strongly bonded grains.
As with the computer generated data. shown in Figure 3.6 the grains are related to one
another by £3, 29, or 227 boundaries. Note the visual Similarity between the real and
computer generated cluster geometries.

31

Chapter 4

Percolation

This chapter examines the applicability of percolation theory to grain boundary engineered
(GBE) materials. Percolative behaviors can be witnessed in many natural phenomena and
engineering applications, e.g., the coffee makers which made this document possible. An
excellent physical description of percolation is presented in [55], and some of the relevant
parts of the theory presented there will be given here. Both bond and site percolation can
be examined in polycrystalline materials to predict important mechanical and electrical

characteristics about them.

The first type of percolation is known as bond percolation. Essentially, this type de-
scfibe$ connected paths of weak bonds on the grain boundary network of a microstructure.
At some critical concentration, CW 3 p, all paths of weak boundaries which span the poly-
crystal are cutoff. The maximum penetration depth, 1 p, of weak bonds from an edge into a
sample is also of interest because it helps determine a material’s resistance to intergranular
attack. Figure 4.] shows an example of the networks of weak bonds which are connected

32

 

Figure 4.1: The black boundaries represent networks of weak grain boundaries which are
connected to the edges of the sample. These boundaries are highly susceptible to intergran-
ular attack and corrosion. The concentration of strong bonds is c = 0.38 in this sample.
Note this is just below cw 3 p and there are no connected weak paths across the sample.

to the faces of a two dimensional polycrystal. The calculated value for l p gives the char-
acteristic depth to which an impurity may diffuse into a sample. Two examples of where
it is important to limit weak bond percolation in engineering materials are battery termi-
nals and steam pipes in nuclear reactors. The models presented here can provide estimates
of the amount of GBE that needs to go into a material to achieve the desired engineering
specifications for a particular application.

The second kind of percolation which is important in the study of GBE materials is
site percolation. As the concentration of strong bonds is increased in a microstructure,
aggregates of strongly bonded grains begin to emerge. At a critical concentration, C5,“).

33

 

Figure 4.2: The black sites show a spanning strong aggregate. The concentration of strong
bonds is c = 0.38 z CSAP. The Strong cluster blocks all of the weak paths across the
sample thereby limiting the intergranular damage propagation.

the lattice is dominated by an aggregate which spans it. Such a spanning cluster is shown
in Figure 4.2. The sizes of the clusters of grains connected by strong bonds is important in
determining the roughness properties of a minimal energy surface which divides the crystal.
This is detailed in Chapter 5. Studying the strongly bonded aggregates can also help to

predict the electrical properties of varistor and superconductor GBE polycrystals [50].

The remainder of this chapter goes into the details which are important to the Study
of percolating structures in GBE materials. In Section 4.1 the applicable scaling laws for
percolation on two and three dimensional lattices are given. Section 4.2 details a study
of the percolation thresholds and order parameters of the uncorrelated GBE model in two

34

 

0.9 r
0.8 r
0.7 r

Vl
l
1
_ .1 __ _J____L__H__J

0.6L
>- 0.5
04*
0.3 r
0.2 ~
0.1 r J

 

 

 

0 i i c* 1
C

Figure 4.3: Theoretical plot of for the percolation threshold (Equation 4.2) as a function of

concentration in the thermodynamic limit. In particular, this plot is for a two dimensional
system, so that '7 = (c — c*)5/36.

and three dimensions. Section 4.3 gives the results of a study of two dimensional CSL

constrained ensembles and describes these results qualitatively.

4.1 Scaling Laws of Percolating Behavior in GBE Materi-

als

First consider the spanning probability, P5, which is the probability that an extensive cluster
of connected sites exists in the sample. In the thermodynamic limit, this function takes on
two possible values, depending on the concentration of strong bonds, c. This function is

35

 

 

 

 

 

Figure 4.4: Theoretical plot of the correlation length (Equation 4.3) in two dimensions as a
function of concentration in the thermodynamic limit. Here, C =| c — c* |4/3.

written as

0 c < c,
P5: (4.1)

1 CBC!“

where c... is known as the critical concentration. The non-percolating and percolating phases

are connected by a second order transition.

It is important to note that in general, an analytic solution for the critical concentration
for percolation is mathematically intractable. The difficulty comes from the combinatorics
which arise due to path degeneracy (loops) on standard lattices. One exception to this is the
Bethe lattice. A closed form solution for the percolation threshold on this lattice is given in

36

 

 

 

 

..
HJ- .

 

Appendix CI.

The infinite cluster probability, 7, is the probability that a lattice Site is part of the

spanning cluster. In the thermodynamic limit, it behaves as,

’1’ % 8d(C — CSArJlfi C > CSAP (4.2)

where [3 is the order parameter exponent, which iS known exactly in two dimensions,
[320 = 5/ 36 and to high precision in three dimensions where ,6 = 0.41 [55]. Figure

4.3 shows a plot of this equation.

The correlation length scales as,

ézglc—CI ‘V. (4.3)

where v is the correlation length exponent. In two dimensions, the exact result is 1120 =
4/ 3, and in three dimensions, it is V31) 2 0.88. Figure 4.4 depicts this equation. This
applies at both c... = ngp and at c, = C5,“). The prefactors for these two cases are
non-universal and need to be determined from simulations or experiments. The critical
exponents ,8 and v are expected to be universal, so they apply to both weak boundary

percolation and to strong aggregate percolation.

The penetration depth, 1 p, from an edge scales as the correlation length, C. For the
simulations presented here, the largest depth to which the weak grain boundaries penetrate.
Analytically this gives

1p m if c > CWBp. (4.4)

37

Because the farthest Site from the penetration entry surface is used, this quantity has also
a logarithmic size dependent correction factor arising from rare fluctuations [15]. An al-
ternative definition of the penetration depth is to take the average depth to which weak
boundaries penetrate. This also diverges at the critical threshold, however with a different
exponent and it does not exhibit rare fluctuation effects.

The effective lattice Size of the polycrystalline aggregate is L/ g, where g is the average
grain size. All of the scaling calculations were performed using the effective lattice size
in order to determine and remove the finite size scaling effects from the computations. All
of the standard percolation scaling laws [55] are modified by this rescaling. This will be
evident from the results given in the next section, for instance, in the aggregate Sizes, which

tend toward 3 instead of zero in the low concentration limit.

4.2 Percolation in Random GBE Microstructures

Figures 4.5 and 4.6 present data for the spanning probability for weak boundary perco-
lation (solid symbols) and strong aggregate percolation(open symbols) in two and three
dimensions, respectively. First consider the two dimensional data in Figure 4.5. A sim-
ple model for this case is to assume that the polycrystalline microstructure consists of a
regular infinite lattice of hexagons each of Size proportional to the average grain Size, g.
The percolation thresholds in that case are known exactly. The weak boundary perco-
lation process is equivalent to bond percolation on a hexagonal lattice, while the strong
aggregate percolation corresponds to bond percolation on the dual to the hexagonal lattice,
which is the triangular lattice. From this, it is possible to deduce that the Strong aggregate

38

 

 

Figure 4.5: The spanning probabilities, PS, for weak boundary percolation (solid sym-
bols) and strong aggregate percolation (open symbols) as a function of the concentration
of strong grain boundaries c in two dimensions. In each case, the data is averaged over a
range of values of the effective lattice size L/ g. The open and filled squares are from aver-
ages over the range 10 g L / g S 50; The open and filled triangles are from averages over
50 S L / g S 100; The open and filled circles are from averages over 100 S L/ g S 200.
The data was calculated from a total of over 30, 000 different polycrystalline samples.

and weak bond percolation thresholds are equivalent in regular hexagonal lattices and that,
CSA p = cwgp = 25in( 7T/ 18) a: 0.347... [55]. The data in Figure 4.5 is quite consis-
tent with this result, though the percolation threshold is Shifted slightly to higher values of
c. Taking the crossing point of the curves in Figure 4.5 as an indicator of the percolation
threshold gives egg}, = 038(1) and Cal/381, = 038(1). One possible origin for the slightly
higher threshold in the polycrystalline model as compared to the regular hexagonal lattice
is the presence of some four fold grain-boundary junctions in the polycrystalline network.

39

 

 

C

Figure 4.6: The Spanning probabilities, P5, for weak boundary percolation (solid sym-
bols) and strong aggregate percolation (open symbols) as a function of the concentration
of strong grain boundaries c in three dimensions. In each case, the data is averaged over
a range of values of the effective lattice size L / g. The Open and filled squares are from
averages over the range 6 S L / g S 10; The open and filled triangles are from averages
over 10 S L/ g S 15; The open and filled circles are from averages over 15 S L/g S 20.
The data was calculated from a total of over 15, 000 different polycrystalline samples.

Figure 4.6 presents the data found for P5 in three dimensions. The most obvious
difference between the results in Figure 4.5 and those in Figure 4.6 is that, in three di-
mensions, the onset of a strong aggregate occurs long before the weak grain boundaries
stop percolating. In this broad “interpenetrating regime,” an extensive strong aggregate
co—exists with percolation of weak boundaries. In two dimensional grain structures this
cannot occur as the percolation of weak boundaries is blocked by the percolation of a
strong aggregate. From the results in Figure 4.6, it is evident that egg}, 2 0.12 :l: 0.03
and CW“, = 0.77 :1: 0.03. A grain in a 3-D polycrystalline network typically has 12 — 14

40

neighboring grains. The rhomboid dodecahedral lattice, i. e., FCC is quite Similar in topol-
ogy. The FCC strong aggregate and weak bond percolation thresholds are, CW 3 p = 0.802
and C5,“) = 0.119 [55], which are actually quite close to the values which were observed
in the Simulations.

Figures 4.7 and 4.8 present data for order parameters related to the two key percolation
processes in the GBE models. The first order parameter is related to percolation of the
strong aggregate. The order parameter in this case is the probability that a site is on the
largest strongly connected cluster, 7 (solid symbols). The order parameter for the onset of
a percolating weak boundary network, is the penetration depth of weak grain boundaries,
lp (open symbols). As expected the thresholds for these order parameters are consistent
with the results found from the spanning probability.

The scaling exponents for ’y and I p are expected to be those given in Equations 4.2
and 4.3 respectively, and we find that the Standard exponents are consistent with the data
of Figures 4.7 and 4.8. As a practical matter, the penetration of corrosive agents into a

material is suppressed strongly for C > CW 3 p due to the rapid decrease in I p for c > cw 3 p.

4.3 Percolation in CSL GBE Microstructures

This section shows the results of a percolation study on the computational CSL model. At
this point, there are many further computations which must be performed before a solid
theory can be developed. It is possible, however, to make some qualitative and quantitative
observations about these Simulations. These results may also be compared to the results of
others who are performing similar computations [40, 49].

41

 

Order Parameters

 

 

Figure 4.7: Two order parameters for the two different percolation processes occurring in
the GBE model. The order parameter for strong aggregate percolation is the infinite clus-
ter probability (open symbols) which has the scaling behavior given by Equation 4.2. The
order parameter for weak boundary percolation is the penetration depth of the weak bound-
aries (solid symbols). The scaling behavior of this order parameter is given by Equations
4.3 and 4.4. Results are presented for different sample sizes and in two dimensions. The
data is averaged in the same way as that described in the caption to Figures 4.5 and 4.6

The first observation that can be made is that the percolation threshold for the strong
aggregate, and hence weak boundaries, is significantly higher than that of the random strong
boundary case. In fact, the critical concentration appears to be nearly 10 percent higher than
what was found in the uncorrelated random boundary case. This is likely due to the way
the clusters must grow with the CSL constraints in place. The percolation probabilities for
the strong aggregate and weak boundaries for CSL annealed polycrystals with lattice Size
L = 500 are shown in Figure 4.9. The two percolation processes are dual to each other in

42

 

Order Parameters

 

 

0' I I a l I
0 0.2 0.4 0.6 0.8 1

C

Figure 4.8: Two order parameters for the two different percolation processes occurring in
the GBE model. The order parameter for strong aggregate percolation is the infinite clus-
ter probability (open symbols) which has the scaling behavior given by Equation 4.2. The
order parameter for weak boundary percolation is the penetration depth of the weak bound-
aries (solid symbols). The scaling behavior of this order parameter is given by Equations
4.3 and 4.4. Results are presented for three different sample Sizes and in three dimensions.
The data is averaged in the same way as that described in the caption to Figures 4.5 and 4.6

two dimensions as before. The percolation threshold for both is P5 A p 2 PW B p z 0.47.

Compare this to the data in Figure 4.5.

In addition to the percolation threshold, the weak boundary network penetration depth,
lp, can be examined. A plot of this for the L = 500 CSL data is shown in Figure 4.10.
When comparing this data to the uncorrelated case (see Figure 4.7) there are two important
features to note. The first is that the onset of blockage of weak paths occurs approximately
at the percolation threshold as it did in the random case. In contrast to the random bound-

43

 

 

\fi/
0.9~ \\
0.8» l
l
‘L’ 0.7»- \
.3 i.
0.6 ‘1
E i l.
9 05~ \
Ii“ 1
a; 04» q. \
E l
O 0.3» \
02- » l.
01" _ \
__,., _ l f ill I"; l \rm m m r1 r) F] I_l J
82‘ 03 0.4" 05 0‘6 ”0.? ”0.73 “02$ 1

Figure 4.9: Percolation probabilities in two dimensional CSL GBE polycrystals. The
squares represent weak boundary percolation and the circles represent strong aggregate per-
colation. The curves are dual and the threshold for both processes is P5 A p = PWBP z 0.47.
In these simulations the lattices were L = 500 and the average grain size was g = 6, giving

an effective lattice size of approximately L/g z 80. This data was averaged over 25 GBE
annealing simulations.

ary case, however, the rate at which the curve falls off is considerably slower. The most
reasonable explanation at this point is that the clusters tend to grow anisotropically. If
one imagines the clusters as rods or chains of grains, with random orientations, which are
weakly bonded to the rest of the polycrystal, then the penetration depth would be smallest

on average when the rod was aligned parallel to the penetrant surface and largest when it

was perpendicular.

500,.— ..'.+--._ .t--4;H-.T... . ..t
i ‘T “ ' ’ ‘
I

450.

TL_

400 i-
350»

300*

200»
150~
100»

50*

 

 

852 (13 (14 (15 (16 (17 (18 (19 1

Figure 4.10: Weak bond penetration depth in two dimensional CSL GBE polycrystals. In
these simulations the lattices were L = 500 and the average grain size was 3 = 6, giving
an effective lattice size of approximately L/g a: 80. This data was averaged over 25 GBE
annealing Simulations.

45

Chapter 5

Critical Manifolds

Random manifolds arise in the Study of domain walls in random bond Ising magnets. The
scaling behavior of these domain walls has been elucidated in refs. [18, 29]. Two key
properties in this analysis are the energy of the domain walls, E and the roughness of the
domain walls, to. The scaling laws obeyed by continuum models for random bond Ising

models are,

E = alLd‘1 + £1ng and w = a3L‘I (5.1)

where a1, a2, (13 are non-universal constants, d is the spatial dimension, L is the sample
Size and 0 and g are universal exponents which are related by the equation 9 = 2; + d — 3
[4, 17,18, 29]. The value of C is known to be exactly 2 / 3 in two dimensions [29] and to be
approximately 5 2 041(1) [18, 38] in three dimensions. The scaling behavior of critical
manifolds in GBE materials are described by similar scaling laws in the limit c < C‘s/[p as
will be demonstrated in Section 5.1.

In the < 100 > orientation domain walls of random bond Ising magnets on cubic lat-

46

tices, the lattice Structure imposes a periodic potential on the random manifold. Random
manifolds in a periodic potential arise in a variety of other contexts and the term “peri-
odic elastic media” has been coined to describe models of these phenomena. The scaling
laws for periodic elastic media(PEM) are quite Simple generalizations of the scaling laws
presented in Equation 5.1. Random manifolds in PEM exhibit a competition between the
tendency of the periodic potential to pin the manifold and to make it flat, and the ten-
dency of the disorder to make the manifold rough. Scaling theories of periodic elastic
media [10,17,53] indicate that in hypercubic lattices in the < 10 > or < 100 > directions,

the critical length, LC, beyond which manifolds roughening scales as
LC z p/(l — p) and LC 3 exp[a0p/(1— p)] (5.2)

in two and three dimensions respectively [53]. In these equations p is the probability that a
bond is present in a hypercubic lattice and a0 is a constant. The scaling laws for the energy

and roughness of PEM’S are then, modified to,

§
E = a5Ld_1+a6L9 and w 2 517(5) . (5.3)

C

The new feature in comparison with Equation 5.1 is the appearance of LC in the scaling
of the roughness. Physically, LC is the typical Size of flat regions on the critical manifold
or domain wall. The critical length LC is important in modeling critical manifolds in GBE
materials in the regime c > CSAP, and, the analysis of LC is extended to the case of GBE
materials and Show that the scaling behavior of all of the properties calculated numerically

47

can be related to the scaling behavior of this critical length.

Four properties of CMS have been calculated as a function of c, the concentration of
strong bonds, and the bond contrast 6. Note that both the energy of the strong boundaries
and that of the grain interiors to be unity, so that e iS the ratio between the strength of
the weak boundary bonds and either of these bonds. The properties calculated are: (i) the
energy of the CM, E; (ii) the fraction of the CM which lies on weak grain boundaries, fw;
(iii) the total number of bonds which lie on the CM, N, and; (iv) the roughness of the CM,
w. The results found for these four quantities in two and three dimensions are presented
in Figures 5.1-5.8. In order to understand the results presented in Figures 5.1—5.8, scaling

laws were developed in three regimes: c z CSAp, c << CSAp and c >> CSAP.

5.1 Low Concentrations c < < c5 A p

At low concentrations of Strong grain boundaries, and provided the energy ratio, 6, is small
the CM lies almost entirely on the weak grain boundaries, as found in models of polycrys-
talline materials [36]. This is evident in Figures 5.3—5.4 which give the fraction of the CM
bonds which lie on weak grain boundaries. For c << c5 A p, fw z 1 and this leads to

Simple behaviors for all of the measured quantities. Firstly, the energy of the CM is simply,

E z (5qu z eN. (5.4)

That is, Since all of the CM bonds lie on weak grain boundaries, the CM energy is the
number of bonds on the CM times the energy contrast. A plot of the scaled energy E / L"-1

48

E/L

 

 

 

Figure 5.1: The average interface energy as a function of strong bond concentration for
three values of the bond strength ratio, 6 = 0.0001 (filled squares), e = 0.01(filled circles),
e = 0.1(filled triangles) in two dimensions. The data is averaged over a range of values of
L/g, which is the effective lattice size. The calculations were restricted to effective lattice
sizes in the range 100 < L/ g < 200. These effective lattice sizes are quite small, so finite
size effects are Significant. The data comes from averaging over a total of 3480 samples.

as a function of c is presented in Figures 5.1—5.2. It is evident that the scaling behavior of
Equation 5.4 extends all the way to cs A p as is expected based on the fact that fw remains

near unity for c < CSAp.

The roughness of the CM in the low concentration limit is the same as that of random

manifolds in polycrystalline materials in the limit of weak boundaries, 6 < < 1, so that [36],
L C
w m g (—) , (5.5)

where C is the roughness exponent (see Equation 5.1 and the discussion following it).

49

 

 

 

Figure 5.2: The average interface energy as a function of strong bond concentration for
three values of the bond strength ratio, 6 = 0.0001 (filled squares), e = 0.01(filled circles),
e = 0.1(filled triangles) in three dimensions. The data is averaged over a range of values
of L/g, which is the effective lattice size. The calculations are restricted to effective lattice
sizes in the range 15 5 L / g g 20. These effective lattice sizes are quite small, so finite
size effects are significant. The total number of samples averaged over in the selected
window was 3422.

The roughness of the CM as a function of grain size, at fixed sample size L, is then predicted
to be w o: gl‘é. Tests of this relation in two and three dimensions are presented in Figure
5.9—5.10. The exponents found are nicely consistent with the expected universal values
{20 = 2/ 3 and C30 2 041(1) [3, 38]. This lends strong support to the idea that L / 3 acts
as an effective lattice constant for CM’s in polycrystalline materials.

50

 

 

C

Figure 5.3: The fraction of the critical manifold which consists of weak grain boundaries as
a function of strong bond concentration for CM’S in two dimensions for e = 0.0001(filled
squares), e = 0.01(filled circles) and e = 0.1(filled triangles). The data are averaged in
the same way as described in the caption to Figure 5.1. The heavy lines are fits of the
e = 0.0001 data to the scaling predictions of Equation 5.28 in the concentration range
0.7 < c < 1.0. The parameter values a1 = 1.58, a2 = 0.56, (13 = 0.19 were used to
obtain the fit. The curve of best fit is presented on the interval c > 0.5 for illustration

purposes.

5.2 The Critical Regime c 2: c5 A p

Percolative effects dominate in the critical regime, and the scaling behavior may be under-
stood based on critical scaling and finite size scaling near second-order phase transitions.
First consider the case 6 —+ 0, in which case the finite-size scaling behaviors at c5 A p are
given by,

Easl; sz; NzLD (5.6)

 

 

C

Figure 5.4: The fraction of the critical manifold which consists of weak grain boundaries as
a function of strong bond concentration for CM’S in three dimensions for e = 0.0001(filled
squares), e = 0.01(filled circles) and e = 0.1(filled triangles). The data are averaged in
the same way as described in the caption to Figure 5.2. The heavy lines are fits of the
e = 0.0001 data to the scaling predictions of Equation 5.29 in the concentration range
0.7 < c < 1.0. The parameter values a1 = 0.46, a2 = 3.22, (13 = 0.50, b0 = 0.53 were
used for the fit. The curve of best fit is presented on the interval c > 0.5 for illustration

purposes.

where D is the external perimeter dimension in percolation. In two dimensions, D = 5/ 4
[55], while in three dimensions D = D f where D f = 2.53 is the fractal dimension of the
infinite cluster [55]. At the critical point, the number of bonds on the CM is proportional
to the number of bonds on the external perimeter of the largest strong-aggregate cluster, as
the CM avoids cutting any strong bonds in the small 6 limit. The energy result in Equation
5.6 is evident from the fact that on the infinite cluster there are singly connected bonds
which may be cut to disconnect the cluster. The roughness is of order the sample size due
to the fact that the infinite cluster is an isotropic fractal with holes on all length scales. The

52

 

 

C

Figure 5.5: The total number of bonds which lie on the critical manifold, N, in two dimen-
sions as a function of the strong bond concentration, c, for three values of the bond strength
ratio, 6 = 0.0001(filled squares), e = 0.01(filled circles), and e = 0.1(filled triangles).
The data is averaged in the same way as described in the caption to Figure 5.1. The heavy
line is a fit of the e = 0.0001 data to the scaling theory predictions of Equation 5.32 on the
interval 0.7 < c < 1.0. The parameter values a1 = 1.69, it; = 0.59 were used to obtain
the fit. The curve of best fit is presented on the interval c > 0.5 for illustration purposes.

behavior on approach to pc is found by using the nodes, links and blobs picture [55] to

rewrite Equation 5.6 as,

E 1 N _
__%_. ___z(:Dd-l-1; wzg(

C
Ld—l éd—l’ Ld—I ) ’ (5'7)

5
C

where g m g | c — cs A p I” is the correlation length. These equations lead to,

E .—
71-] m (C ‘CSAPW 1”, C > CSAP, (5.8)

53

 

 

C

Figure 5.6: The total number of bonds which lie on the critical manifold, N, in three
dimensions as a function of the strong bond concentration, c, for three values of the bond
strength ratio, 6 = 0.0001(filled squares), e = 0.01(filled circles), and e = 0.1(filled
triangles). The data is averaged in the same way as described in the caption to Figure 5.2.
The heavy line is a fit of the e = 0.0001 data to the scaling theory predictions of Equation
5.33 on the interval 0.7 < c < 1.0. The parameter values a1 = 0.22, a2 = 4.36, ha = 0.68
were used for the fit. The curve of best fit is presented on the interval c > 0.5 for illustration

purposes.

w z g1“€L‘-I |c — CSAp I’m—C) L >> 5, (5.9)

and

N
W m (C — CSAp)—(D—d+1)/V L >> g. (5°10)

However, if the energy contrast 6 is finite, the divergence in the roughness at c5 A p is
rounded. This is due to the fact that in that case, large excursions of the critical manifold
cost an energy proportional to the number of bonds in the excursion times 6. If the energy

54

 

 

 

C

Figure 5.7: The scaled roughness as a function of the strong bond concentration for three
values of the bond strength ratio, 6 = 0.0001 (filled squares), e = 0.01(filled circles), and
0.1(filled triangles) in two dimensions. The data is averaged in the same way as described
in the caption to Figure 5.1. The heavy line is a fit of the e = 0.0001 data to the scaling
theory predictions of Equation 5.25 on the interval 0.7 < c < 1.0. The parameter values
a1 = 0, a2 = 0.36 and C = 0.59 were used to obtain the fit. The curve of best fit is
presented on the interval c > 0.5 for illustration purposes.

of a large excursion around strongly connected grains is larger than the cost of breaking the
grain, then cleavage of the grain occurs. Define r to be the length scale of the excursion,

then the energy cost of an excursion is proportional to egd‘er .

Equating this to the energy cost of cleaving a grain, gd‘l, gives a critical length for
cleavage, rc,

_1_

1- (5.11)
60

reg

If the percolative correlation length is much greater than the length scale cutoff given by

55

 

 

 

C

Figure 5.8: The scaled roughness as a function of the Strong bond concentration for three
values of the bond strength ratio, 6 = 0.0001 (filled squares), e = 0.01(filled circles), and
0.1(filled triangles) in three dimensions. The data is averaged in the same way as described
in the caption to Figure 5.2. The heavy line is a fit of the e = 0.0001 data to the scaling
theory predictions of Equation 5.26 on the interval 0.7 < c < 1.0. The parameter values
a1 = 0.12, a2 = 0.67, 170 = 1.48 and C = 0.41 were used in the fit. The curve of best fit
is presented on the interval c > 0.5 for illustration purposes.

Equation 5.11, i. e., 6 > > rc, then the energy of the CM is given by,

d—l
E 5:: gd-l i o< 6(d‘l)/DLd“1. (5.12)
8"c

AS seen in Figures 5.5—5.8, at finite energy contrasts, the number of bonds on the CM
and the roughness of the CM are reduced dramatically. This is understood as follows. The
largest possible excursion of the CM at finite e is grc. The roughness of the CM then
scales as gr,- instead of with g. In a similar way, the number of bonds on the interface also

56

 

 

 

P 0 K
a i
2— _ L 5* 5 3‘ 3» saw
1 o . ‘
. u _ a, p a E a,
. - . . a sacs:
r T Q l '5 :“L ‘ 1'7 ‘2'“;
151% g a t: 3% i ’7 LDQ' “60
NA "3 3 go Him , 7: 5338.51; E
<0 I; s ‘ i ' X .
O “ L, D
d 1 is 1:87;, 5 'flt‘omj .. 7,
a it a ,, 1 a team
V ‘ g a. C a A
E i (<9; 3 ‘ go? ,8: , a U E‘gflf
0,5 17 § ‘ 1g 00 g: D r f
. , ,. ,4,
of U as gan :1 D E
ii 52’ D c
H Cl
4.5: , .. 8 , . - . . I
1.8 2 2.2 24 2.8 3 3.2

2.6
Inml
Figure 5.9: Scaling of the roughness as a function of grain size in two dimensions for
c = 0.3 and c = 0.5. The open circles are the raw data at c = 0.3. The open squares
are the raw data at c = 0.5. The filled circles and filled squares are the data averaged over
a narrow bin (of order one) in grain size. The lines of best fit to the data are indicated.
The data agrees well with the scaling prediction of Equation 5.13. The line of best fit has

slope 030(2) for c = 0.3 and 032(2) for c = 0.5, which should be compared with the
prediction g1"§, where 1 — C = 1/ 3.

57

06* ‘ g a). .0; .4
FA ': I: 7 . :1"— fi‘ 7"
V. Ii 0 V: 5“, I
O 0.4, .1 I 75 g s L“ _
g i h -~ :3‘ C ' }_ . U 7
v r I G . .3 1‘ V
5 0.2L ‘ 6 ‘
g‘ I 3‘0 VJ. ; C
. * 1'? - so"
qr.» ! i “i j
i; Y‘ i. «1:7 2:; J”?
‘l’ 1 E .

b
M
U
1:1
D
Q

1:9

4

\l

.xl
a’ril

'72, 2.1 2.2 2.3 2.4
ln(g)

Figure 5.10: Scaling of the roughness as a function of grain size in three dimensions for

= 0.1 and c = 0.35. The open circles are the raw data at c = 0.1. The open squares
are the raw data at c = 0.35. The filled circles and filled squares are the data averaged
over a narrow bin (of order one) in grain size. The lines of best fit to the data are indicated.
The data agrees well with the scaling prediction of Equation 5.13. The line of best fit has a
slope of 059(5) for c = 0.1 and a slope of 065(5) at c = 0.35, which should be compared
with the prediction gl‘é, where 1 — C = 0.59.

58

scales with grc. In particular for the roughness, it is expected that the roughness is that of

a random manifold with renormalized lattice size L/grc, so that,

L i .
10%ng <—) =€(g_1)/081—CL€' (5.13)
8 r c
The roughness thus diverges as the energy contrast goes to zero as w re 60’267 in two

dimensions where C = 2/ 3, D = 5/ 4, while to a: 60233 in three dimensions where
C 2 041(1), D 2 2.53. The exponents describing the scaling of roughness with the
energy ratio, 6, are quite small, which indicates that a large energy contrast is required in
order to observe pronounced peaks in the roughness and in the number of bonds on the CM
near c5 A p. This is evident from Figures 5.5—5.8, where it is seen that even for a contrast
of e = 0.01, the critical behavior near c5 A p is rounded significantly. The energy and the
fraction of weak boundaries on the CM are more weakly dependent on the energy contrast
(see Figures 5.1—5.4), though as seen in Equation 5.12 there are still non-trivial scaling
laws at c5 A p.

Another feature of the behavior of w and N as a function of strong bond concentration is
that the peak value of these quantities moves to higher c as the energy contrast, 6 increases.
This is due to the fact that at finite e, the maximum manifold wandering occurs when
c > c5 A p and rc = C. These conditions maximize the number of strong bonds while still
providing weak paths up to length rc. Using these conditions, the peak in the roughness
and in the number of CM bonds occurs at c p found from,

_1_

i5 = (Q) — CSAP)-v- (5-14)
6

59

The location of the peak Shift thus scales as,

CP_CSAP sev/D. (5.15)

5.3 High concentrations, c >> CSAp, e —> 0

The limit c = 1 is trivial, as in that limit the CM is a cleavage (flat) surface, so that the
number of bonds on the CM is just the surface area and the energy is N times the energy
of the strong bonds, which are taken to be one. The roughness of a flat surface is zero and

the fraction of the CM that is on weak boundaries is clearly zero. For c = 1,

__ _—_ .7 z 1; and fw = 0 = w. (5.16)

In the regime c < 1 with 1 — c small, there is a relatively small number of weak grain
boundaries. This is the regime in which scaling concepts used in the analysis of periodic
elastic media apply. The primary area of interest is the limit of small values of 6, so that a
weak boundary is favorable at almost all angles to the average CM plane. For finite 1 — c
an Imry-Ma argument [10, 30] provides a surprisingly good theory to describe the large

scale behavior of the critical manifold.

In the analysis below, the renorrnalized length scale I = L / g is used. In the Imry-
Ma argument [10, 53], consider a fluctuation of size I from a flat surface. This fluctuation
consists of 1"-1 grain boundaries. Such a fluctuation is advantageous if it contains a larger
than average number of weak grain boundaries. The large scale roughening of the critical

60

manifold is driven by the non-perturbative or “co-operative”, Imry-Ma fluctuations which
in the context of GBE materials are derived and analyzed below. In addition to these large
scale excitations, there are small scale fluctuations which lead to terms proportional to
1 — c. These terms can be considered to be perturbative terms while the Imry-Ma terms are

non-perturbative.

According to the central limit theorem, the probability that a fluctuation of size S has

energy E5 is given by,

P(E5) cc V2 exp (—(E5 — ASP/[2023]), (5.17)

1
(27w25)

where S = l"‘1 is the number of grains in the fluctuation. Here E5 is in units of energy.
To recover the energy of the fluctuation in standard units, E5 is multiplied by gd—l. ego is
the average energy (per grain) of a grain boundary while 02 is the standard deviation (per

grain) in the energy of a grain boundary. For the given GBE model.
em = c; 0'2 2 C(1 — c). (5.18)

From Equation 5.17 the typical size of the largest energetically favorable fluctuation can be
estimated by finding the solution to,
exp (452 /(2(725)) z 1. (5.19)

gain

(27TUZS)1/2

61

The maximum energy gain achieved by these fluctuations is then proportional to,

5E8‘”" o< [0'25 111(2m725)]1/2. (5.20)

The energy cost of such a fluctuation scales as.

5£,,,, o< eooId_2. (5.21)

From Equations 5.20 and 5.21 it is evident that at large enough length scales, the energy
gain is larger than the energy cost in both two and three dimensions. The energy gain favors
a fluctuation which roughens the critical manifold, so at long length scales, the critical
manifold “wanders” to take advantage of regions of the material where there is a larger
than average number of weak boundaries. The most important quantity in the theory is the
critical length, LC, which is the length scale at which the wandering sets in. The critical
manifold is flat on length scales L < L; and “rough” on longer length scales, i.e., it is
rough for L > LC. Equating 5.20 and 5.21 and dropping the logarithmic term leads to the

following result for critical manifolds in two dimensional systems, such as thin films,

 

025 mg, so that, L, z 3C . (5.22)
(1 — C)

In three dimensions, i.e., for bulk materials, the logarithmic term in Equation 5.20 is dom-

62

inant and must be kept, which results in the expression,

9XP(b0'1{7)

0211197sz) z e3, so, L, z ng _ c)]l/Z' (5.23)

where 170 is a non-universal constant. The critical length LC diverges much more rapidly as
c ——> 1 in three dimensions than in two dimensions. This means that critical manifolds are
more prone to cleavage in three dimensions than in two dimensions, for the same degree
of grain boundary engineering. The theory leading to the algebraic prefactor of the three-
dimensional result may have two other logarithmic corrections. The first is due to the
number of ways in which a CM fluctuation may be placed in the material. This leads to
an additional factor of l multiplying the LHS of Equation 5.19. In addition the energy
cost Equation (5.21) may be reduced by a factor of In L. These logarithmic factors do not
affect the result in two dimensions or the exponential term in the three dimensional result.
However they do affect the algebraic prefactor in the three-dimensional result. The way in
which they modify the three dimensional result is incorporated into the more general form
L; = g(c(1 — c))‘yExp[b0c/(1 — c)], where the new exponent y depends on the details
of the model, but is less than or equal to one. In comparing with numerical data, several
values of y were evaluated. It was found that y = 1 / 2 provides an adequate description of

the data. This is the case stated in Equation 5.23 and in the results which follow from it.

The behavior of the four measured CM properties can be deduced from the critical
length, LC. Firstly, the co-operative or non-perturbative contribution to the roughness of

63

CM’S is given by,

L C to 1
20 ’53 g (f) 50 that, W (X 17;, (5.24)
C

C

where I, = Lc/g. At fixed 3 and L. in two dimensions, (using Equation 5.22) for c >>

C5A p, w behaves as

 

w 1—c é
g—1_€L§~a1(1—c)+a2( c ) . (5.25)

The first term in this expression is the perturbative term due to the small deviations of
the CM from the flat manifold. The second term is the co-operative term, and is due to large

Imry-Ma deviations from the flat CM. In three dimensions a Similar discussion yields,

57%;? 8 “1(1- C) + a216(1- c)]é/Z EXP (—boCc/(1 — c)). (5.26)

In both two and three dimensions a2 is expected to be independent of g and L except for
finite size scaling corrections. However (11 is expected to be size dependent due to the nor-
malization of w. The perturbative correction should scale as w z a(g(1 — c), with a’1 a
Size independent constant. At large sample sizes, a1 Should thus go to zero. However for
finite grain sizes and relatively small sample sizes, there is a Significant linear term. The
cooperative contribution to the roughness of CM’s approaches zero algebraically in two
dimensions and exponentially in three dimensions. This is typical of the behavior of peri-
odic elastic media in the weak disorder limit, where roughening of manifolds only occurs
at exponentially large length scales [10]. The forms 5.25 and 5.26 are compared to the data
of Figures 5.7 and 5.8 from which it is evident that they provide a good representation of

the behavior in the large c limit.

Now consider the fraction of the weak grain boundaries which lie on the CM. The
average number of weak boundaries on a cleavage plane is fw 2 a1 (1 — c). In addition,
when the critical manifold wanders, it wanders to regions of the material where there is
an excess of weak boundaries 6fw. This is the origin of the co-operative term discussed
above. The typical value of the co-operative term 5f“, is described by the same statistics as

the energy fluctuations, so that

(wa z a[1£"“”/2/(15-2 +154”, (5.27)

where [C = Lc/ g. The term in square brackets is the total number of grain boundaries in
the favorable fluctuation. Using Equation 5.22 the behavior of fw in the large c limit for

two dimensional systems (thin films) is predicted to be,

1/2
c

1+1,

02C
za l—c +a’ ma 1—c +—————, (5.28)
fw 1( ) 2 1( ) 1+a31—:E

 

where an, a2,a3 are non-universal constants. The first term in Equation 5.28 is the pertur-
bative term, while the second term is the co-operative term. In three dimensions, a similar

argument yields,

021611 —C)11/2
1+ A ’

 

fw z a1 (1 — c) + (5.29)

where

exp (boc/l — c)

”“3 [cu—c111” ’

 

(5.30)

and £11,112, (13, b0 are non-universal constants and the co-operative term is proportional

65

to (7/ (1 + 1,), from Equation 5.27. These forms are compared to the numerical data in
Figures 5.3 and 5.4 from which it is seen that they provide a good representation of the

data.

The co-operative contribution to the number of bonds on the CM is approximately,

d—l
Nz (gL‘C’_2+L‘C’-1) (LL) . (5.31)

This leads to N / Ld “1 = 1 + 1/ 1,. Adding the linear term to this co-operative term, the

result for two dimensions is found to be,

biz

cc 1 + a1(1 — c) + a2(1 — c)/c, (5.32)

where a1, a2 are non-universal constants. Again the first term is the perturbative term, while
the second term is the co-operative term. In this case both terms have a linear dependence
on 1 — c at small 1 — c. This linear behavior is valid over a surprisingly broad regime, as
is evident from Figure 5.5. In three dimensions,

% 2 1+ “1(1 - C) + [12160 — c)]vzexp (—boC/(1 — c)). (5.33)

where a1, a2, b0 are non-universal constants. The close fit of this form to the data is evident

in Figure 5.6.

Finally, the energy is related to the number of bonds on the manifold and the weak

66

fraction through the relation,

E = wae + (1—fw)N. (5.34)

This is an exact equation valid for all c and e. In fact, this equation is a check on the
consistency of the numerical procedures for calculating, fw, N and E. Analytic expressions
for the energy as a function of c at small 1 — c are found by combining Equation 5.34, with
Equation 5.29 and Equation 5.32 for two dimensions or with Equation 5.29 and Equation
5.33 for three dimensions. As a final empirical remark, it is interesting to note that the two

dimensional energy data is well described by the Simple relation,

A. 0.97 — 1.74(1 — c), (5.35)

Mm

for a broad range of values of 1 — c. At present, there is no explanation for why the
rather complex expression found from Equation 5.34 with Equations 5.28 and 5.32 should
reduce to this simple behavior over such a broad range of 1 — c, though of course a term

proportional to 1 — c is expected from Equation 5.34.

Although the unbiased fits of the scaling laws to the numerical data are very good, the
parameter values found are not precisely determined. That is, the fits are not markedly
worse if the parameter values are varied significantly. This is more severe in three dimen-
sions where the data is restricted to small effective lattice Sizes. If the grain size is large
and the lattice size is large, the finite size effects will be less severe and the correspondence
between the scaling laws and the data to improve. It is thus not surprising that the theory is

67

quite precise in two dimensions. It is also important to note that the unbiased fits of the 3-D
data lead to different values of the key parameter be in the exponential of the critical length
(see Equation 5.23). The unbiased fits lead to the results b0 2 1.48 from the roughness data
(see the caption to Fig. 9; 170 = 0.53 from the weak fraction data (see the caption to Figure
5.3) and; b0 2 0.68 from the data for the number of bonds on the CM (see the caption to
Figure 5.5). The values for 170 Should be the same, so fits were carried out to the data where
the value of b0 is chosen to find a best fit which is most consistent with all of the data. The
fits are not significantly worse than those presented in Figures 5.3-5.8 for a fixed value of
120 = 1.0 :t 0.2 with the other parameters free. This suggests that multiple data sets are
required to extract reliable fitting parameters from the three dimensional numerical data on

the sample sizes currently available.

5.4 Unified form of scaling theory

In the statistical physics analysis of manifolds, the roughness scaling plays a central role.
In this context, it is worth noting that it is possible to state all of the above roughness results

in a unified form. To do this, define a scaling length,
R, = min [C, grc], (5.36)

which is the minimum of the percolative correlation length C = g | c — c5 A p |"’ and en-

1/ D . The smaller of these two lengths cuts off the percolative

ergy cutoff length gr, :2 g / e
fluctuations. Using R; as the lattice spacing in 3 nodes, links and blobs picture leads to the

68

roughness scaling,

w o< R,(L/R,)€ c < c5“). (5.37)

In the regime C > C5 A p the effects of the periodic potential must be accounted for, which

leads to,

ZUOCRC(L/L:.)C C > CSAp, (5.38)

where L; = Rclc and I, z LC/g is given by Equations 5.22 and 5.23. These scaling forms

apply for all values of c, provided L > > RC, LC.

5.5 Critical Manifolds in CSL GBE Microstructures

In Section 4.3 it was remarked that the properties of the critical manifold are strongly
influenced by the correlated cluster sizes found in the sample. We can see this effect qual-
itatively from the data collected on the critical manifolds calculated for the CSL model
presented earlier. The main effect of the correlations was to move the percolation threshold
for the giant cluster up and this can be seen in the critical manifold data presented in this
section.

The data shown in this section is comprised of 25 runs on the CSL model presented
in Section 3.3. The lattice Size was two-dimensional with size L = 500 x 500 and the
concentration was varied from c = 0.2 to c = 0.9 in steps of 0.05. Because the data
was annealed the concentrations given are averages over all samples since the annealing
was terminated when the sample was within 5 percent of the required CSL concentration
profile. As with the data in the previous sections, Z3, Z9, and 227 bonds are considered to

69

be strong, and hence, their fractional sum is c. For all calculations presented in this section

the ratio of bond strengths is e = 0.00001.

At each concentration step, a critical manifold was calculated using Dijkstra’s Shortest
path algorithm. This algorithm is given in Appendix 34 This is equivalent to the max-
imum flow algorithm in two dimensions. The CM was then analyzed in the same way
as described earlier to find the four features. These are energy, fraction of weak bonds,
number of bonds on the interface and scaled roughness of the interface. The results of
these calculations are Shown in Figures 5.11—5.14. The data points shown are the average
values over the ensembles at each concentration step. The lines are meant only to aid in

interpolation.

The main effect of the correlations seems to be that the percolation threshold is in-
creased. A secondary effect is that the critical manifold properties are changed in two
dimensions. Most of this effect seems to be due to the apparent anisotropic cluster growth
which occurs to satisfy the global constraints. “With the label differencing method, the
strong clusters are not constrained by the loop closure, and therefore, tend to grow isotrop-
ically (randomly). The CSL constraint forces the clusters to become worm-like. These
clusters are connected to the rest of the sample with high angle boundaries. These bound-
aries serve as paths for the critical manifold. It follows them as far as possible, then jumps

straight across to the next cluster.

The first effect that the correlations have is to cause the energy of the manifold to in-
crease in a non-linear fashion with concentration, c, in contrast to the uncorrelated model
which increases linearly with c. The correlated clusters provide long avenues of weak

70

°-7r

0.5“

0.4 —

E/L

0.2 r

0.1 r (f

 

 

8.2” " "0.3' 0.3": 0.5 0.6 0.7 0.8 0.9 1

Figure 5.11: The average interface energy as a function of concentration of strong bonds.
The data is averaged in the same way as described in Figure 5.1.

bonds across the sample. We can see this from Figure 5.11. The total manifold energy
increases much more slowly as a function of concentration. This qualitative theory is sup-
ported by the data shown in Figure 5.12 which shows the fraction of weak bonds as a
function of concentration. Even at 90 percent concentration (c = 0.9), the weak fraction
is nearly fw z 0.6. This means that the manifold is finding the weak bonds which are

connecting the strong clusters to the medium.

The second effect of the correlations is that they cause the number of bonds on the
CM to decrease. This can be seen in Figure 5.13, where the average number of bonds per
unit length on the CM is less than 2.5 at the peak. There is also a decrease in the average
roughness of the manifold with increasing c. The peak average scaled roughness for the

71

 

.,_B 0.5»
0.4 ~
0.3
0.2 L

0.1r

 

C

Figure 5.12: The fraction of weak bonds on the critical manifold as a function of concen-
tration of strong bonds. The data is averaged in the same way as described in Figure 5.1.

samples is around 0.46. This is about two-thirds of the value of the peak of the uncorrelated

data suggesting that the correlations smooth out the manifolds. Figure 5.14 shows this.

Another feature that is interesting in the CSL case is the behavior of the manifold with
respect to the clusters of strongly bonded grains. It appears that the manifold is trying
to include the weak boundaries which hold the clusters to the rest of the crystal as much
as possible. An example of a manifold following the clusters is shown in Figure 5.15.
Long snake-like clusters which are aligned along the manifolds would cause a rise in the
concentration at which roughness peaks since they provide smooth low energy paths for
the manifold. The lower concentration peak, then, would be due to the clusters aligned
against the cut. At this point the cut would have to wander half of the length of the cluster,

72

2.5“-

ML

1.5r

 

C

Figure 5.13: The number of bonds on the critical manifold as a function of concentration
of strong bonds. The data is averaged in the same way as described in Figure 5.1.

on average. Another possible explanation is the small sample Sizes L = 500, which would
cause finite Size effects. A detailed cluster moment analysis on larger ensembles will be

needed to explain what is Shown.

73

 

 

0 0.2 0.4 0.6 0.8 1
C

Figure 5.14: Scaled roughness of the critical manifold as a function of concentration of
strong bonds. The data is averaged in the same way as described in Figure 5.1.

74

 

Figure 5.15: Critical manifold on a CSL microstructure. Notice how the path follows the
strong clusters where they are weakly bonded to the rest of the crystal.

75

Chapter 6

Conclusion and Further Work

6.1 Conclusion

This document summarizes two methods for simulating GBE polycrystalline materials.
The first method assigns a fixed concentration of strong bonds randomly to the grain bound-
aries based on the difference between their Potts labels. The results from this method can
be used to estimate the statistical properties of real materials. The second method enforces
CSL constraints which force the grain orientations to be correlated. The results from this

method appear to be similar, at least qualitatively, to experimentally observed polycrystals.

These polycrystalline models were analyzed using percolation theory and found to give
reasonable numbers for two and three dimensional systems. In particular, the weak bond
percolation threshold was found to be at concentration c = 0.77 in three dimensions, which
is what has been seen in real experiments for GBE materials [44]. This result is encourag-
ing since experimental percolation studies on GBE materials are laborious and require the

76

destruction of the sample. If the model captures the essence of the actual material behav-
ior, then it will be possible to narrow the amount of experiments that need to be done to
optimize the design of a material.

The models were also examined using critical manifold theory. This theory can provide
an estimate of the roughness exponents for fracture surfaces. These exponents play a role
in the design of materials for toughness or brittleness. Again, by developing good scaling
theories, it is possible to help reduce the search space for experimentalists. As with perco-
lation studies, surface roughness experiments are time consuming and they are destructive
to the sample.

It is the hope that computational modeling of polycrystalline materials can someday
reduce the amount of time and energy that goes into the experimental design and study of
GBE materials. It is important to note, also, that this work is not intended as a replacement
for real experiments. On the contrary, the computational models are put forth as tools to
help experimentalists reduce their costs and labor, and provide means to more effectively
interpret the meaning of experimental data. In order for this to become a reality, it will be
necessary to continue this work in parallel with experimentalists and keep comparing the

results.

6.2 Things to Do

One of the first things that needs to be completed is an analysis of the geometric properties
of the strongly bonded clusters. It is apparent from visual inspection of these clusters that
they are anisotropic but without some quantitative measure of this anisotropy, it will not be

77

possible to work out a theory for the scaling laws. The cluster analysis can be performed

using principal axis theory [24] upon the supergrains in the sample.

Currently there are two other methods in the literature for simulating CSL GBE mi-
crostructures. The first is that of Schuh, et al., [49] which uses a hexagonal network and as-
signs constraints at each junction so that the loops around each are consistent. This method
does not take longer loops into account, which must also satisfy the CSL constraint. It
also assumes the polycrystal is a hexagonal lattice, which to a first approximation is correct

since the average connectivity of a planar graph is six.

The second algorithm is described in Reed, et al., [46]. This algorithm also uses a
Sparse representation, but the CSL constraints are held globally. This means that the total
rotation through any loop of grains is the identity. The main difficulty this model encounters
is that the motion of the grain boundaries is unphysical. In a personal correspondence with
this author, it was determined that the model could be hybridized with the model described
herein to allow for grain boundary motion and possibly nucleation of new grains. This

would allow for a more physically based annealing Step.

One of the most exciting directions that needs to be taken is the implementation of the
CSL constraints in three dimensions. It is hoped that this study will elucidate the physics
of grain orientation correlations. The third dimension may reveal relationships between
clusters below the observation plane and show further correlations and scaling laws. This
is exciting because the EBSD technique is limited to planar studies and it is not possible to
see the three dimensional connectivity. It would be good to provide some predictions for
the up and coming serial sectioning techniques.

78

Another interesting direction that three dimensional models may take is toward ex-
plaining grain nucleation and recrystallization. The current model does not allow for new
grain formation during the GBE process. During the deformation stage, it is thought that
nucleation sites for new grains are formed. A physically-based model for rrricrostructure
generation which is based on the current CSL model has been discussed [1] and needs to
be implemented.

Another mystery that may be solved with the 3D CSL model is the formation of an-
nealing twins. Thus far there has been no sufficient explanation of the emergence of these
boundaries during the GBE process. Modeling in three dimensions may provide some in-
sight to their origin. The current state of the art models add annealing twins after grain
growth is complete using a statistical model. While this is useful for determining material

properties, it doesn’t illuminate the physics of the formation of these boundaries.

79

APPENDICES

80

Appendix A

Glossary

The following is a list of abbreviations used in this document.

CM
CSL
EBSD
GBCD
GBE
OIM
PEM

Critical Manifold

Coincidence Site Lattice

Electron Back Scatter Diffraction
Grain Boundary Character Distribution
Grain Boundary Engineering
Orientation Imaging Microscopy
Periodic Elastic Manifold

81

Appendix B

Algorithms

B.l Conventional Potts Model

For a single Monte step of N iterations. This algorithm comes from [25].
MC-POTTS()
l whilet g MaxTime

2 doforj4—2toN

3 do E,- 4— 0

4 Ef «— 0

5 choose a site i : i E {1,2,...N}

6 choose anew label sf :sf‘ 6 {1,2, ...,q}
7 for each k in 2(1)

8 do ifsk ¢ 5,-

9 then B,- <— E,- + E0

82

10

ll

12

l3

14

B.2

N-FOLD()

l

2

10

ll

12

IfSk 329 S?
then Ef 4— Ef ‘1' E0
AE <— Ef — E,’

flip site i to Spin 5;.“ with probability p(AE)

n-fold Potts Model

whilet S MaxTime

do pick an activity randomly a E {0, ..., A}

find a Site i : 211:1 7n, < a 3 [2:] 7n,
pick a random flip activity p 6 {0, ..., 7T1}
find 1* : 231p.) < p S 21917.1
ifi* = 0
then s,- (—— RANDOM E {1,...,q,\{sk,k E Z(i)}}
else 5,- <— 5,.
UPDATE(i)
for each k E z(i)
do UPDATE(k)

increment time by At

83

UPDATE(i)
1 7T: *— 71',"

2 for eachk E z(i)

3 do if number[sk] = 0

4 then flipk <— S),

5 q* <— q’ + 1

6 number[sk] <— number[sk] + 1
7 AE 2 E0 x number[s,-]

8 171,0 1—(q —q* - 1) >< MAE)

9 771 *— [71,0

10 fork=1toz

1

#

do ifflip[k] yé 0

12 then AE (—- 50 x (number[S,-] — number[sk])
13 Pu: *— MAE)
14 7T1 = 7T1 + Put

15 A4—A—7tf‘1-7T,‘

84

B.3 CSL Microstructure Generation

CSL-ANNEAL(f5~p,€)
I generate a potts microstructure with appropriate grain size statistics
2 define a target CSL histogram HT with fSp special boundaries
3 assign a random rotation string to each grain

4 compute the CSL histogram for the ensemble H E

 

5 A0 4— \/(H5 — HT)2
6 while A0 2 E

7 do a «— RANDOM e {0,...,1}

 

8 if a < swapthreshold
9 then swap two grain strings
10 else concatenate a rotation onto a single grain
ll recompute H E
12 A1<— \/(HE--HT)2
13 p +— RANDOM E {0,...,1}
14 ifp g eAl‘Ao/T
15 then keep change to microstructure
16 A0 4— A1
17 else reject change to microstructure

85

B.4 Dijkstra’s Shortest Path Algorithm

Dijkstra’s Shortest path algorithm and helper functions. The version implemented for the
computations in this thesis used a min-heap data structure for maximum efficiency. In the
algorithms below, G is a graph, 5 is the source node, u and v are arbitrary nodes in the
graph, w is the list of edge weights, 7r is the path and H is a min-heap. The functions on
the graph are V[G] which extracts all the vertices and Adj [it] which is the adjacency list of
the neighbors of u in G. Further details on this algorithm can be found in [13].
INITIALIZE-SINGLE-SOURCE(G,s)
1 for v E V[G]
2 do d [v] +— co

3 7T[v] <—0

RELAX(u, v, w)
1 ifd[v] > d[u] + w(u,v)
2 then d[v] +— d[u] + w(u,v)

3 7T[v] +— u

DIIKSTRA(G,w,S)

l INITIALIZE-SINGLE-SOURCE(G,s)
2 S <— 0

3 H <— MAKE-MIN-HEAP(V[G],w)
4 while HEAP-NOT—EMPTY(H)

86

do u +— EXTRACT-MIN(H)
S +— S U u
for v e Adj[u]

do RELAx(u, v, w)

87

Appendix C

Derivations

C.1 Percolation Threshold on a Bethe Lattice

The Bethe lattice or Cayley tree is a key model for understanding the underpinnings of
percolation theory [11,55]. It is the simplest model which has a phase transition that is not
trivial, and is still analytically tractable. Other structures commonly used in simulations
such as lattices in two or more dimensions also have non-trivial percolation thresholds,
but their connectivity makes an analytic solution impossible to carry out. Instead their

thresholds must be computed numerically.

The reason the Bethe lattice is tractable is because it is a tree structure. By this it is
meant that there is a central site, known as the root, from which all of the other sites branch
off. The number of branches, 2, which leave this root node is known as the coordination
number of the lattice. A Bethe lattice with z = 3 is depicted in Figure C]. There are
two important features of this kind of lattice which make it a useful tool in the physics of

88

 

 

 

Figure C]: A Bethe lattice of degree 2 = 3.

critical phenomena. Both features come from the tree structure.

The first feature is that there is only one path from any site in the lattice to any other
Site in the lattice. This is because there are no loops. This lack of path degeneracy is what
makes the model solvable. It will also allow the solution to be written in the form of a

recurrence relation.

The second feature is related to the dimensionality of the lattice. The number of sites

on the outer surface of a tree, 5 ,1 with z = 71 branches at generation n is given by

5,, = (z — 1)". ((2.1)

89

The volume V" of the same tree is

v, z (z — 1)"+1 — 1. (C2)

Flory used these properties to describe the physics of gelation, which is a percolation pro-

cess [19].

Suppose that each connection from one Site to its neighbors is open (unblocked) with
probability p. Let each bond be open or closed as drawn independently from an identical
distribution. To hop from one Site to another one must go down one of the (on average)
(2 — 1) p paths available. This means that to arrive at the boundary from a Site which is N
levels away through open sites the condition ((2 — 1);?)N Z 1 must hold. This gives rise

to the critical probability, pc

_ 1
pC—(Z_1)I

 

(C3)

the probability threshold at which an open path emerges from the origin node to the bound-

ary.

To find the critical exponent E for the rate of divergence for p > pc it is necessary to
write down a recurrence relation for the probability a site is connected to the boundary. As
an example, let 2 = 3 (the first non-trivial case.) Define q as the probability that a Site
is not connected to the boundary through one of its branches. This means the site is not
connected via both branches with probability qz. The probability that this site is a leaf node
is thus pqz. Using this, it is possible to write the total probability that a site is not connected

90

to the boundary as

i=1-P+pf, (Co

which has two solutions, q = 1, and q = (1 — p) / p. Moving up the tree to the root node,

the probability of a connected path from it to the boundary, 6, is given by
9 = p(1— ([3), (C5)

Since the root has 3 branches toward the boundary, instead of 2.

Plugging in the expression for q, the expression becomes
9 1— 3
—=1—(—Jg, (Gm
P P

for the non-trivial case p > 1),.
Expanding Equation C6 in a Taylor series about pc the exponent E is found to be 1,
i.e.,

9
Eat (p — pc)1. (C7)

91

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