ARCHNE
MASSACHUSFTTs NSTTTE
Grain Boundary Network Design
JUN 17 2015
by
LIBRARIES
Oliver Kent Johnson
B.S., Brigham Young University (2010)
Submitted to the Department of Materials Science and Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Materials Science and Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
Massachusetts Institute of Technology 2015. All rights reserved.
Signature redacted
Author ..........
Department \f Materials Science and Engineering
May 19, 2015
Signature redacted
Certified by.......................................................
Christopher A. Schuh
Department Head, Danae and Vasilis Salapatas Professor of Metallurgy
z62
Thesis Supervisor
Signature redacted
Accepted by ..................
/R
C r DDonal
on Grad
Chair, Department a;ommittee
1
Sadoway
R. Sadoway
e Students
2
Grain Boundary Network Design
by
Oliver Kent Johnson
Submitted to the Department of Materials Science and Engineering
on May 19, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Materials Science and Engineering
Abstract
Grain boundaries in polycrystals form a complex interconnected network of intercrystalline interfaces. The crystallographic character of individual grain boundaries
and the network structure of the grain boundary ensemble have been experimentally
observed to have a strong influence on many materials properties. This observation
suggests that if we could control the types of grain boundaries present in a polycrystal
and their spatial arrangement then it would be possible to dramatically improve the
properties of polycrystalline materials and tailor them to specific engineering applications. However, there are a number of major obstacles that have, until now, precluded
the realization of this opportunity: (1) methods capable of simultaneously quantifying the crystallographic and topological structure of grain boundary networks do not
exist; (2) theoretical models relating grain boundary network structure to physical
properties have not yet been developed; and, consequently, (3) there are no techniques
to quantitatively identify grain boundary network structures that would be beneficial
for a given property. In this thesis I address these obstacles by first developing a new
statistical description of grain boundary network structure called the triple junction
distribution function (TJDF), which encodes both crystallographic and topological
information. I establish new results regarding the physical symmetries of triple junctions and find a relationship between crystallographic texture and grain boundary
network structure. I then use the TJDF to develop a model for the effective diffusivity of a grain boundary network. Finally, using the relationship between texture and
grain boundary network structure that I develop, I describe a method for texturemediated grain boundary network design. This process permits the theoretical design
of grain boundary networks with properties tailored to a given engineering application and is applicable to any polycrystalline material. I demonstrate the potential
of this technique by application to a specific design problem involving competing
design objectives for mechanical and kinetic materials properties. The result is a
designed microstructure that is predicted to outperform an isotropic polycrystal by
seven orders of magnitude.
Thesis Supervisor: Christopher A. Schuh
Title: Department Head, Danae and Vasilis Salapatas Professor of Metallurgy
3
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Acknowledgments
In completing the work described in this thesis, I readily acknowledge help, support,
encouragement, and inspiration that I have received from a loving Father, frequently
in the form of amazing people with whom I have had the privilege of associating.
I would like to especially thank my advisor Professor Chris Schuh for taking a risk
on me, even before I was admitted to MIT. His encouragement has been overwhelming
and his confidence in me has always been greater than my own. I am particularly
grateful for his efforts to teach me the art of scientific communication and for giving
me just the right mix of direction and autonomy to help me gain scientific maturity
and independence as a researcher. I hope to be the kind of scientist and mentor that
Chris has been to me.
Chris' support has been second only to that of my incredible wife, Julie, whom I
love dearly and for whom I am eternally grateful. Her faith in me has been unwavering
and she has made this thesis possible through her unbounded love and encouragement.
I am grateful to her for being willing to have this adventure in Boston, and for making
it worth having.
I am grateful also to my parents and sister for their love, support and patience; for
believing in me, encouraging my curiosity, giving me every opportunity a son could
hope for, sharing their experience and wise advice, and never letting me settle for less
than my best. I would also like to thank the rest of my friends and family for their
love, support, and encouragement.
I express my gratitude, also, to the members of my thesis committee, Professor
Michael Demkowicz and Professor Carl Thompson, for their incisive comments and
suggestions regarding this research. I would also like to thank Michael for mentoring
me on an individual level as a writer and a researcher, and for valuable life and career
advice. I am grateful to Carl for teaching me the ins and outs of being an educator
and giving me opportunities to grow as a teacher.
Thank you to all the members of the Schuh group, past and present. We have
enjoyed a wonderful esprit de corps and I am grateful for many valuable scientific
discussions, non-scientific diversions, and mutual support.
There are many others in DMSE, and at MIT broadly, to whom I owe a great debt
of gratitude, including: Diane Johansen, Jazy Ma, Angelita Mireles, Elissa Haverty,
Professor Tom Mason, Professor Sam Allen, JP, Mike Tarkanian, Ike Feitler, Dr.
Shaoyan Chu, and Dr. Charlie Settens.
I also gratefully acknowledge financial support from the US Department of Energy
(DOE), Office of Basic Energy Sciences under award no. DE-SC0008926 and from the
Department of Defense (DoD) through the National Defense Science & Engineering
Graduate Fellowship (NDSEG) Program.
5
Contents
1
Introduction
1.1 Background .......
..........
....
1.1.1 The Forward Problem: Characterization
1.1.2 The Inverse Problem: Design . . . . .
1.1.3 Limitations of MSDPO . . . . . . . . .
1.1.4 Grain Boundary Engineering . . . . . .
1.1.5 Percolation Theory . . . . . . . . . . .
1.2 Structure of Thesis . . . . . . . . . . . . . . .
2 A Triple Junction Distribution Function
2.1 Introduction . . . . . . . . . . . . . . . . .
2.2 Conventions . . . . . . . . . . . . . . . . .
2.3 The Triple Junction Distribution Function
2.4 D iscussion . . . . . . . . . . . . . . . . . .
2.5 Validation . . . . . . . . . . . . . . . . . .
2.6 C onclusions . . . . . . . . . . . . . . . . .
3
4
The
3.1
3.2
3.3
3.4
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4.3.1
4.3.2
Step 1: Sample ODFs from MH.
Step 2: Map from M(1) - m .
4.3.3
4.3.4
Step 3: Compute texture dependent properties . . . . . . . . .
M
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Step 4: Map from M( )
64
64
Triple Junction Hull
Introduction . . . . . . . . . . . . . . .
Triple Junction State Space . . . . . .
Grain Boundary Network Configuration
Texture & Topology . . . . . . . . . .
C onclusions . . . . . . . . . . . . . . .
Grain Boundary Network Design
4.1 Design Strategy . . . . . . . . .
4.2 Design Problem . . . . . . . . .
4.2.1 Y ield . . . . . . . . . . .
4.2.2 Elastic Compliance . . .
4.2.3 Grain Boundary Network
4.3 Properties Closure . . . . . . .
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Diffusivity
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4.4
4.5
4.3.5 Step 5: Compute grain boundary network
4.3.6 Step 6: Define the boundary of P . . . .
Solving the Design Problem . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . .
5 Localization
5.1 Introduction . . . . . . . . . . . . .
5.2 Model .......
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5.3 Strategy . . . . . . . . . . . . . . .
5.3.1 O verview . . . . . . . . . .
5.3.2 Hom ogenization . . . . . . .
5.3.3 Microstructural Parameters
5.3.4 Localization . . . . . . . . .
5.4 D iscussion . . . . . . . . . . . . . .
5.5 Conclusions . . . . . . . . . . . . .
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Summary
95
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Directions for Future Work
97
A Definition of Special Functions
99
B Integral of the Product of Three Hyperspherical Harmonics
101
C Inversion Symmetry for Hyperspherical Harmonics
105
D Relations for the Coefficients of ODFs and TJDFs
107
D.1 The Reality Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 107
D.2 Triple Junction Symmetries . . . . . . . . . . . . . . . . . . . . . . . 108
E Mathematical Theorems
111
E.1 Addition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
E.2 Linearization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .111
F Error Scaling for Bicrystal Experiments
8
113
List of Figures
1-1
The microstructure set for cubic crystals in the first three dimensions
(left), and the corresponding microstructure hull (right), taken from [61.
The Fl' are the coefficients in the basis of the generalized spherical
harmonics, analogous to Cnm. ....
1-2
1-3
2-1
.................
21
Young's Modulus-Shear Modulus properties closures for various materials [251. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Intersection of iso-property hyper-surfaces with the microstructure hull,
for elastic compliance of a two-dimensional composite material 1131.
Microstructures defined by coefficients, F4 and F2 , whose values fall
within region C satisfy both design constraints: ((S)-1)1
> 10 GPa
and ((S)-), > 10 GPa. . . . . . . . . . . . . . . . . . . . . . . . .
23
Schematic illustration showing the gaps in the microstructure design
literature (dotted lines). Much work has focused on prediction of properties that depend on texture (solid lines). This is typically done by
using the ODF in homogenization relations. However, much less has
been done in the realm of property prediction or materials design when
the properties of interest are dependent on the misorientations of grain
boundaries and their spatial distribution in the grain boundary network. This work attempts to fill this gap by making an explicit connection between the ODF and the TJDF. . . . . . . . . . . . . . . . .
28
2-2
Schematic triple junction illustrating the nomenclature used in this
thesis for grain orientations, qi, and grain boundary misorientations, qij. 30
2-3
{111} Pole figures for some of the simulated ODFs. (a) A fiber ODF
with a half-width of 50; (b) a fiber ODF with a half-width of 750; (c)
a unimodal ODF with a half-width of 250; (d) a unimodal ODF with
a half-width of 750; (e) a multimodal ODF with 19 modal orientations
and a half-width of 120; and (f) a multimodal ODF with 17 modal
orientations and a half-width of 1780. In all cases the modal orientations were randomly selected. For the fiber and unimodal ODFs the
half-widths were chosen in 5' increments from 5' to 180'. For the mul-
timodal ODFs the half-widths were chosen randomly between 5' and
.....................
1750........................
9
37
2-4
3-1
3-2
3-3
3-4
Comparison of the TJDF coefficients computed using Eqs. (2.15) and
(2.21a) respectively. The vertical axis corresponds to the theoretical
values computed from Eq. (2.15) and the horizontal axis to the "expected" values obtained directly from the misorientations by means of
Eq. (2.21a). The real parts of the complex valued TJDF coefficients
are plotted on the left, and the imaginary parts are plotted on the
right. Each set of uniquely colored points corresponds to a separate
ODF. Note that the data points closely follow a line of slope equal to
1, indicating nearly perfect agreement between the TJDF coefficients
computed using these two independent methods. . . . . . . . . . . . .
38
(a) The grain boundary network of a sample of Inconel 690, representing a global configuration of triple junctions. Grain boundaries in (a)
are color coded by their disorientation axis (in the standard stereographic triangle) and angle (see the legend at right) [99]. (b) The local
state of a representative triple junction as indicated by its triple junction misorientations, qAB. The grains coordinating the triple junction
are labeled 1, 2, and 3, with dotted arrows defining a circuit that encloses it. Composition of the grain boundary misorientations around
such a circuit must result in the identity operation (Eq. 3.3). (c) All
six distinct paths surrounding a triple junction. These correspond to
all possible ways of choosing the two independent misorientations coordinating a triple junction, and thus, represent physically equivalent
descriptions. Colors distinguish circuits starting in different grains,
and line type (solid/dashed) distinguish circuits of different sense. . .
41
(a) The triple junction hull, M(, in the discrete basis with K = 4,
corresponding to the standard 3-simplex, or tetrahedron. The vertices are points for which the corresponding Pk = 1 and all others are
0, edges are points for which two of the Pk are non-zero, faces correspond to points for which three of the Pk are non-zero, and the interior
is populated by points for which all of the Pk are non-zero. (b) Orthogonal projection of m( in the first three non-trivial dimensions.
Two-dimensional projections are also shown as outlines for clarity. .
45
A schematic of a columnar microstructure in which every triple junction possesses the same state. The three grain orientations are distinguished by color (red, blue, and white). . . . . . . . . . . . . . . . . .
47
A microstructure with columnar grains and periodic boundary conditions, formed by "gluing" together four lamellae, each with a single
triple junction state. The triple junction state of each lamella is shown
in the lower detail views, along with the actual values of Pk that are attained. For ease of visualization the orientations of the incident grains
are colored rather than the grain boundaries themselves. The states of
triple junctions inhabiting the interfaces are shown in the upper detail
view s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
10
3-5
3-6
In (a) The green surface contains a three-dimensional orthogonal projection of mg, and the points are samples, cs, taken uniformly from the
six-dimensional orthogonal projection of m , as defined in Section 3.4.
In (b) the red surface contains a three-dimensional orthogonal projection of m(, the blue surface corresponds to in , and the points are
the t, obtained from the c, via Eq. 3.15. . . . . . . . . . . . . . . . .
51
Histogram of the ratio of the six-dimensional measures of in) and
mH. The mean and standard deviation are also plotted. . . . . . . .
52
4-1
Graphical summary of the process of constructing the properties closure. 62
4-2
Comparison of microstructural sampling techniques for Ns = 1 x 103.
Points resulting from uniform sampling of the standard simplex are
shown as red markers. Points resulting from the hierarchical simplex
sampling technique are shown as green markers. The purple shaded
region is the convex hull of 1 x 10' samples taken using the hierarchical
technique and is shown for comparison, being a higher order approximation of the true 0 (O: 1 , S 111 ), for the texture dependent properties
Oly1, and S i ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4-3
A {100} pole figure showing the orientations used to define
. .
66
4-4
Properties space representation of each of the sampled textures. Only
half of the 10' points are shown to reduce visual clutter. . . . . . . .
67
a-shapes for a point cloud with various values of the parameter a. The
figure to the far right with a = oc shows the convex hull of the point
clou d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4-5
4-6
4-7
4-8
The properties closure, 2, for the properties
.')
S.
iy1, Sill,, and D as
approximated using the procedures outlined in Section 4.3. . . . . . .
69
Intersection of the design constraint Sill
Sja1 1 with Y, for Siii=
2.5665 x 10-11 Pa-1 . This design constraint is represented as the semitransparent vertical plane. . . . . . . . . . . . . . . . . . . . . . . . .
70
The feasible region for the design problem. Contours of the objective
function, fobj, as defined in Eq. 4.25, are plotted over the feasible region.
The best performing design solution is shown (black dot) along with the
worst performing solution (white dot) and the isotropic solution (blue
dot). The isotropic solution does not satisfy the design constraints
and falls outside of the feasible region. The textures of the best and
worst solutions are also shown as {100} pole figures where the gray
discrete points are the elements of Mf. The color scale for the texture
components is in multiples of the random distribution (MRD). .....
71
11
The grain boundary networks corresponding to the best, worst, and
isotropic design solutions shown in Fig. 4-8. Grain boundaries are colored according to their misorientation according to the Patala coloring
scheme [99, 1021. The color code on the right indicates the disorientation angle of a misorientation as well as the rotation axis by its position
in the relevant standard stereographic triangle. The grain boundaries
colored in light gray correspond to disorientations of 0' and therefore
are not really grain boundaries, but are shown to illustrate the geometry of the grain boundary network. . . . . . . . . . . . . . . . . . . .
72
Schematic plot of the model microstructure showing the boundary conditions used for the diffusion simulations. . . . . . . . . . . . . . . . .
78
5-2
Two-dimensional ODFs with varying Omax according to Eq. 5.1. . . . .
79
5-3
Effective diffusivity of simulated grain boundary networks for various
fractions of type-2 (high-diffusivity) grain boundaries. The error bars
for many of the microstructures are smaller than the markers, and, as
expected, they are very large near the percolation threshold. Marker
color corresponds to the sharpness of the texture. As 0 max was varied,
I obtained networks with Ds'M ranging from ~5 x 10-1 m 2 /s to
5 x 106 m2 /s. The red solid line connects the values predicted using
4-9
5-1
DeGEM. This is shown as a solid line rather than discrete markers only
. . . . . . . . . . . . . . . . . . . . . . . . . . .
79
Comparison of the effective diffusivity predicted by the GEM, DGEM
to that obtained via simulation, DsjM. The dashed red line indicates
a 1 : 1 relationship (i.e. perfect agreement). Error bars for many of
the data points are smaller than the respective markers. Marker color
indicates the sharpness of the texture, and is referenced to the color
scale of Fig. 5-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
to im prove visibility.
5-4
5-5
Isosurfaces through the solution space of E(P), with N = 3 and P E A,
where A = Di x D2 x Qt is a grid of points defined by Di = {108, 10-7-*,... , 101} m2 /s,
2
D 2 = {10-, 10-75,...,1012} m /s, and Qt = {0', 1.5 ,..., 45'}. The
true values of the parameters are represented by the location of the
blue star. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5-6
All (51) = 20, 825 possible sets of N = 3 distinct microstructures as
represented by their respective fractions of high-angle grain bound(2 ) i3)), indicate the fraction
aries. The coordinates of a point, (,)
of high-angle boundaries in each of the three microstructures in the
set, where
(l2
<
P23). The red points represent microstructure
sets that did not result in successful recovery of the true constitutive parameters. For the sake of visual clarity, the points representing
microstructure sets that did result in successful recovery of the true
constitutive parameters are not shown, instead they are indicated by
the green semi-transparent region. . . . . . . . . . . . . . . . . . . . .
12
91
5-7
5-8
F-I
95% confidence intervals for each of the parameters as obtained by
solving the localization problem with N microstructures chosen uniformly at random from among those considered here. The process was
repeated 100 times for each N. . . . . . . . . . . . . . . . . . . . . .
The number of experiments, N, required for one to recover the values
of all of the constitutive parameters in Eq. 5.2 to within a given level
of uncertainty (%Error). The performance of the localization methods introduced in this chapter is compared with that of an exhaustive
bicrystal approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solutions of Eq. F.1 for various values of %ERR E {0.001, 0.002, ...
13
92
92
, 0.100}.114
'I
List of Tables
1.1
Design case studies in the literature. Where crystallographic texture
was the design variable, the coefficients of the ODF were optimized.
The symbols used in the table are: length (L), deflection (6), Taylor factor (M), stress concentration (K), load carrying capacity (o-o),
maximum crack extension force (Gmax), components of the elastic stiffness tensor (Cijkl), elastic crack driving force/ J-integral (J), critical
resolved shear stress (TCRSS), yield stress (oy), angular velocity (w),
4.1
5.1
5.2
and anisotropy ratio (A). The performance enhancement is the improvement in material performance for the optimal solution relative to
the worst case scenario. . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Numerical values of the independent elastic constants for Al [1111 appearing in Eq. 4.7. All values are in units of 10-12 Pa-1 . . . . . . . .
58
Summary of the applications of Eq. 5.3. In this table P is the macroscopic effective property of interest, M is the microstructure, and P
represents the relevant local constitutive equation (i.e. the local properties). The '?' symbol indicates the unknown quantity that is to be
solved for in a given application, and the '/' symbol indicates a known
or specified quantity. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical values of the coefficients appearing in Eq. 5.5a. . . . . . .
80
84
15
-,---.-:-R--------------...NNOwy
n------
x~~~~~~~~w
opw
0-
Chapter 1
Introduction
Much of materials science is focused on two intimately related tasks: characterization
and design. Characterization consists of determining a material's structure and predicting the resulting properties. Design consists of predicting and synthesizing a structure so as to satisfy some desired property. Both of these endeavors are fundamentally
rooted in the notion that material structure determines material performance. Mathematically this idea has been canonized in various structure-properties relations (see,
e.g. [1, 2] and references therein). However, within the meso-scale' microstructure
design community, rigorous theoretical developments have been largely restricted to
the forward problem of characterization. Various predictive models of material performance have been formulated and have proven extremely successful [3-5]. However,
microstructure design has mostly consisted of an empirical trial-and-error approach.
In the context of polycrystalline materials, recent developments have revolutionized
the field of microstructure design by providing mathematical tools that allow the
designer to explore the complete universe of physically realizable microstructures,
and identify those that meet various performance objectives/design constraints [6, 71.
Whereas the materials designer formerly had a finite (and rather incomplete) catalog
of observed microstructures and forward models at his/her disposal, the advent of
these new microstructure design tools has allowed for the consideration of material
structure as a continuous design variable and the rigorous solution of complex inverse
design problems. While this advancement can hardly be understated, the present
microstructure design paradigm is incapable of treating materials properties that depend upon the structure of the network of interfaces that are present in polycrystals.
In its current form many properties such as elasticity, conductivity, thermal expansion, and a limited version of plasticity can be treated [6, 8-14]. However, other
properties of scientific and engineering interest such as fracture and corrosion, which
depend upon the structure of the grain boundary network, lie outside of its scope.
It is the purpose of this thesis to address this gap in the literature by developing
'Meso-scale refers to the intermediate length scale, which is much larger than atomic constituents,
but much smaller than the macroscopic scale of an engineering component. Within this thesis I will
restrict myself to a discussion of microstructural features that exist on this scale, and characterization/design of the same. For brevity I will use the term "microstructure design" to refer to the
design of meso-scale microstructural features.
17
the mathematical apparatus needed for the design of grain boundary networks in
polycrystalline materials.
1.1
1.1.1
Background
The Forward Problem: Characterization
Before delving into the inverse problem of microstructure design, it will be useful
to briefly discuss the forward problem of properties prediction within the context of
a polycrystalline ensemble. The computation of an orientation dependent effective
macroscopic tensor property, P, requires knowledge of the orientation dependence of
P for a single crystal 2 , P(q) = P(w, 0, <), and knowledge of the distribution of crystal
orientations, f(q), referred to as the orientation distribution function (ODF). The
procedure to obtain P was provided by Bunge [15], and is briefly summarized here.
As is common in mean field theories, the effective property is expressed as a sum of
the mean value, P, and a perturbation term, AP:
P
+ A P(.)
In the first order treatment, the approximation, P e P, is made, considering
that the perturbation term is often of much smaller order than P. The effective
macroscopic property, P, is then given by the volume average of P(q):
P=
f (q) P(q) dq
(1.2)
where the ODF is seen to be the weighting term, and the domain of integration
is Q = SO (3). Equation 1.2 is a bound on the value of P. Another bound can be
obtained by following the same procedure for P1 . For elasticity these upper and lower
bounds correspond to the Voigt and Reuss averages respectively [7, 15-17]. While
higher-order homogenization relations exist [2, 18-20], the first-order approximation
is sufficient to illustrate the concept: effective macroscopic material properties of a
polycrystalline aggregate may be computed from the orientation dependence of the
property and a statistical description of the microstructure (i.e. the ODF).
1.1.2
The Inverse Problem: Design
Microstructure Sensitive Design for Performance Optimization (MSDPO) was originally proposed by Adams, et al. [61, as a rigorous theoretical framework for the
2
Bunge's treatment of the homogenization of polycrystalline properties was formulated in the
context of direction cosine matrices, g, in terms of the Bunge-Euler angles (#1, <P, #2). However, to
maintain consistency of notation, the treatment given here as well as any other formulas that are
reproduced from the literature will be expressed in terms of the quaternion parameterization of the
rotation space, with the components of the unit quaternion, q, given as functions of (w, 0, #), the
rotation angle and polar angles of the rotation axis, respectively.
18
consideration of crystallographic orientation in the design of polycrystalline materials. Since its inception, MSDPO has seen extensive use. The design case studies that
have been reported in the literature are summarized in Table 1.1.
Relying heavily upon the spectral representation of structure [21, 221 and properties [15], MSDPO provides a means to determine the theoretically optimal microstructures with respect to some set of design constraints or objectives. Once a candidate
material system has been chosen, the MSDPO formalism consists of the following
steps:
1.
2.
3.
4.
Define Homogenization relations for the properties of interest
Construct the MicrostructureHull
Construct the Properties Closure
Back-mapping to identify optimal microstructures
The essential process of homogenization was illustrated in section 1.1.1; the balance
of the MSDPO process will now be treated in detail.
The Microstructure Hull
In order to identify optimal microstructures with respect to some design constraints,
it must be possible to consider the entire space of all possible microstructural configurations. This is precisely what the microstructure hull is: the complete set of all
physically realizable microstructures [1, 6, 9, 23-26]. Probability distribution functions describing the structure of a polycrystalline ensemble, such as the ODF, are
square integrable, and therefore permit a harmonic expansion of the form3 127]:
00
f (q)
n
I
c,"Zlm(q)
=
(1.3)
n=0,2,... 1=0 m=-1
where the basis functions, Z?m(q), are hyperspherical harmonics [27-311, and cm
are the coefficients of the expansion. The value of this spectral representation, and
the reason for its use as the foundation for all of MSDPO, is that it allows for the
concise parameterization of the microstructure, consideration of the entire "universe"
of physically realizable microstructures, and the construction of efficient and invertible
structure-property relationships [1, 2, 6, 14, 261. All that can possibly be known about
the distribution of orientations in the microstructure is encoded in the coefficients,
Cm. Consider a single crystal of orientation q0 . The ODF of such a microstructure
will be a Dirac delta function centered at q0 . The corresponding coefficients for this
"delta microstructure" are given by:
cI'M = E'm (q)(1.4)
3
MSDPO was originally formulated in terms of the harmonic expansions used by Bunge, namely,
with the generalized spherical harmonics as basis functions. Once again, for clarity, the theory is
expressed here using basis functions and notation that are consistent with the balance of this thesis.
19
Objective
Design Variables
Material/Sample Symmetry
Fixed-guided compliant beam
min (L) |6
Crystallographic Texture
Al, Fe, Mo, Cu, Si, W, Ni
Plate w/ circular hole
max (M/K)
Crystallographic Texture
Cu/Orthotropic
Plate w/ circular hole
Turbine rotor
max (o-) w/o yield
min (Gmax)
Crystallographic Texture
Crystallographic Texture
Cu/Orthotropic
Ni/Transversely Isotropic
Turbine rotor
min (Gmax)
Crystallographic Texture
N/A
min (C33 3 3 /Ci1 1)
Cylindrical pressure vessel
min (J)
Plate w/ circular hole
max (TCRSS)
Plate w/ circular hole
Rectangular cantilever beam
Flywheel
Geometry
Performance Enhancement
13
Source
34%
[6]
> 100%1
[8]
59%
9 6 %b
[91
110]
Ni/Triclinic
> 96%'
[10]
A, vol% Al
Al-Pb
12%
111]
Crystallographic Texture
Ni/Orthotropic
31%
[12]
Crystallographic Texture
Ni/Orthotropic
29%
[12]
max (ay/K)
max (6) w/o yield
Fiber Orientations
Crystallographic Texture
Carbon fiber-epoxy
a-Ti/ Orthorhombic
50 - 255%c
228%
[13]
max (w) w/o yield
Crystallographic Texture
a-Ti/Transversely Isotropic
156%
[14]
>
-
[14]
Table 1.1: Design case studies in the literature. Where crystallographic texture was the design variable, the coefficients of the ODF were optimized.
The symbols used in the table are: length (L), deflection (6), Taylor factor (M), stress concentration (K), load carrying capacity (oo3, maximum
crack extension force (Gmax), components of the elastic stiffness tensor (Cijkl), elastic crack driving force/J-integral (J), critical resolved shear stress
(rCRSS), yield stress (ay), angular velocity (w), and anisotropy ratio (A). The performance enhancement is the improvement in material performance
for the optimal solution relative to the worst case scenario.
aNo specific values were given; authors simply stated that the maximum ratio was "more than 100% larger than the minimum ratio."
bNo worst case scenario was given; this performance enhancement is measured relative to the isotropic case.
cNo
numbers were given; this is estimated from a plot of the properties closure for (0y)/K vs. C1111.
where the * denotes complex conjugation. The set of sets of Cm for all single crystal
orientations, is called the material set or microstructure set (see Fig. 1-1) 16, 91.
1.0
1.0
0.5
F1
F
Fe
.1.0 F
.1.0 F 'a
Fig. 1-1: The microstructure set for cubic crystals in the first three dimensions (left), and the
corresponding microstructure hull (right), taken from [6]. The Ff are the coefficients in the basis of
the generalized spherical harmonics, analogous to e"'.
Because any polycrystalline ensemble is merely a collection of single crystals, the
coefficients for any ODF must be a linear combination of the single crystal coefficients 191. This leads to the surprising result that the space of all physically realizable
microstructures is compact and convex [6, 9]. The convex hull of the microstructure set is precisely the microstructure hull (Fig. 1-1) [6, 91. The convexity of the
microstructure hull makes it amenable to efficient search algorithms during the optimization process.
The Properties Closure
Once the microstructure hull is constructed, the homogenization relations for the
properties of interest are exercised over its entirety. This is aided by the fact that,
when the orientation dependence of the properties of interest is expressed in spectral
form (as in Eq.1.5), it is often found to contain a finite number of terms 11, 13, 32]:
o
P (q) =
1
n
PlmZIm (q)
E
(1.5)
n=0,2,... 1=0 mn=-l
As a result of the orthogonality of the basis functions used to describe both the
structure and the properties, the effective macroscopic properties, as computed via
21
Eq. 1.2, are of the following form:
00
P(q) =
n
ZZc"Pi"m
(1.6)
n=0,2,... 1=0 m=-1
Equation 1.6 indicates that even though the generalized Fourier space in which the
microstructure hull resides is infinite dimensional, only a finite subspace is relevant
to a given design problem. Because all possible microstructures were considered, the
result is a region in the property space containing all theoretically possible simultaneous values of the material properties under consideration [23-26, 32, 331. An example
elastic properties closure for several materials is given in Fig. 1-2.
400-
300-
, -Molybdenum
Nickel
0 200Copper
100Aluminum
0
0
50
100
150
G13(GPa)
Fig. 1-2: Young's Modulus-Shear Modulus properties closures for various materials [25].
Back-mapping
In order to determine the set of microstructures which satisfy some design constraint
it is necessary to perform the process of back-mapping 17, 13]. For a given value of P,
Eq. 1.6 may be recognized as the equation of a hyper-plane in the same space in which
the microstructure hull resides. All points on such a hyper-surface represent distinct
microstructures, which are, nevertheless, predicted to exhibit an identical value of P.
Therefore, these hyper-surfaces are named iso-property surfaces (see Fig. 1-3). Thus,
the final step of determining microstructures that meet specified performance criteria
22
consists of simply finding the intersection of all relevant iso-property surfaces and the
microstructure hull, i.e. solving the system of homogenization equations (each of the
form of Eq. 1.6) subject to the constraint that the resulting microstructure lie within
the microstructure hull.
S
F4
-1.5
>10 GPa
,
--0.5
-1
((s)1)12 12 =
=10 GPa
U
M
1
1.5
10 GPa
-1.5
1212 > 10 GPa
-
S
F2
Fig. 1-3: Intersection of iso-property hyper-surfaces with the microstructure hull, for elastic compliance of a two-dimensional composite material [13]. Microstructures defined by coefficients, F4 and
F2 , whose values fall within region C satisfy both design constraints:
((S)-')
1.1.3
>
((SV1)
> 10 GPa and
10 GPa.
Limitations of MSDPO
The MSDPO framework relies on the use of distribution functions of the local state
(usually restricted to crystallographic orientation), and spatial correlation functions.
This places a fundamental limitation on the scope of MSDPO, as noted by its creators:
Representations in terms of the local state distributionfunctions and their
spatial correlations contain sufficient information to predict effective properties where the microstructure-propertiesmodel belongs to the class of
mean-field constitutive relations. Examples of this class include estimates
for linear properties, such as elastic and conductive properties, thermal expansion, and others. Another important example is the theory for initial
23
yielding. Other types of properties, however, are sensitive to a microstructure's internal interfacial structure, and these require a
different approach. Theory relating the character of interfaces to
the properties of material points lying on the interface is rather
limited at this time. Also, theories of homogenization, linking
the distribution of interfacial properties to effective properties,
are also very limited. ([341, emphasis added)
The lack of mathematical tools to allow MSDPO to handle properties that are sensitive to the grain boundary network is a significant gap in the literature. There are
many material properties of engineering importance that are known to be governed
by the structure of the grain boundary network. These include intergranular fracture [35], intergranular corrosion[361, electromigration [37, 38], and liquid metal embrittlement 1391. Furthermore, empirical evidence has confirmed that changes in the
structure of the grain boundary network can have a dramatic impact on macroscopic
material properties (see Section 1.1.4). Thus, it would appear that microstructure
design for properties that are sensitive to the structure of the grain boundary network
is possible, but a complete and rigorous mathematical theory for design is lacking.
1.1.4
Grain Boundary Engineering
In 1984 Tadao Watanabe suggested the possibility of what he called "grain boundary design and control" [40]. Now commonly referred to as Grain Boundary Engineering (GBE), this term refers to the use of various thermomechanical processing
(TMP) techniques in an effort to manipulate the grain boundary character distribution (GBCD) and the connectivity of weak/susceptible grain boundaries [41-43].
Various researchers have reported dramatically enhanced material properties as a result of GBE. Some examples include a projected two- to four-fold increase in the
service life of battery electrodes [36, 44], a seven-fold increase in critical current density of a high-Tc superconductor 145], a sixteen-fold decrease in creep rate 146], and a
fifty-fold increase in weldability [471. While these investigations have clearly demonstrated the potential to improve materials properties through manipulation of the
grain boundary network, it is important to note that GBE is currently limited in its
applicability. The TMP treatments used in GBE rely upon plasticity and twinning
mechanisms to enhance the population of beneficial grain boundaries. Therefore,
these methods are only effective for ductile materials that readily form annealing
twins (i.e. low-stacking fault energy fcc metals). The introduction of a texture based
approach to GBE (as will be discussed below) might be effective at extending the observed performance gains of GBE to a broader class of materials. Such an approach
might build upon the predictive power of MSDPO in order to design for the large
performance enhancements that GBE has demonstrated are possible.
1.1.5
Percolation Theory
Grain boundaries of differing structure can exhibit a broad variation in materials
properties. Because of the strong contrast observed in many grain boundary con24
trolled properties, mean field theories, such as those described in Section 1.1.3, are
often inadequate to predict homogenized macroscopic materials properties for the
grain boundary network [481. However, the observed critical phenomena (percolation threshold) exhibited by grain boundary networks can be modeled with excellent
accuracy via the scaling laws of percolation theory. While the spatial distribution
of grain boundaries within the global network is inherently non-random, it has been
noted that grain boundary networks do indeed belong to the same universality class
as standard percolation theory [49]. This implies that, while the percolation thresholds for crystallographic networks differ from those of random networks, the critical
exponents are identical and the scaling laws of standard percolation theory apply.
One formulation of percolation theory, which has proven effective at predicting properties over a wide range of contrast ratios (i.e. from low-contrast systems that might
appropriately be modeled using mean-field theories to those that require a percolation
approach), has been the Generalized Effective Medium (GEM) model [50, 51]. The
GEM model has been successfully applied to predict the macroscopic diffusivity of
two-dimensional crystallographically consistent grain boundary networks with local
diffusivity contrast ranging from 10010 [51]. Additionally, whereas traditional percolation theory is concerned with a binary classification of phases (e.g. conductor and
insulator), an adaptation of the GEM model by Chen and Schuh has also extended it
to consider an arbitrary, but discrete, spectrum of component properties [511. Thus,
the tools of percolation theory, in the form of the GEM model, show promise for
developing homogenization relations for grain boundary network sensitive properties
that might be used to extend the scope of MSDPO.
1.2
Structure of Thesis
This thesis addresses the above mentioned deficiencies in existing theory for microstructure design that have precluded the consideration of grain boundary network
sensitive properties. In this work I develop the mathematical tools necessary for the
design of grain boundary networks. This involves (1) the development of statistical
tools to characterize the structure of grain boundary networks, (2) the derivation of
a relationship between crystallographic texture and grain boundary network structure, (3) the construction of a model for the effective macroscopic properties of grain
boundary networks, and (4) the establishment of a design space for grain boundary
networks.
At the time of writing, much of the work presented in this thesis has already been
published or is currently under review and the modular structure of this document
reflects this fact, each chapter corresponding to the content of a publication. A brief
overview is presented here:
* In Chapter 2 I develop a triple junction distribution function (TJDF) as a
statistical tool to quantitatively characterize the structure of grain boundary
networks in polycrystals. The spectral formulation used in its development
provides the mathematical basis for the remainder of this thesis. In this chapter
I also derive a relationship between the orientation distribution function (ODF)
25
and the TJDF that enables a texture mediated approach to grain boundary
network design that was proposed above as a solution to the limited scope of
traditional GBE.
" In Chapter 3 I investigate the local state space of triple junction misorientations
and establish their symmetries. I then consider the global configuration space
of grain boundary networks and define a triple junction hull containing all possible triple junction distributions. I give a constructive proof that every TJDF
contained in this space is guaranteed to correspond to a physically realizable
microstructure. I also investigate how much of this space is accessible through
the control of texture alone.
" Chapter 4 is concerned with the application of the tools developed in the previous chapters to a specific design problem. I develop a model that predicts
the effective diffusivity of a grain boundary network from its TJDF. Using
this model, in connection with other previously established structure-properties
models, I define a properties closure for yield strength, elastic compliance, and
grain boundary network diffusivity. This properties closure is the design space
for the specific design problem that I consider and it contains all physically
possible combinations of these three properties that can be exhibited simultaneously by a single microstructure. I then identify an optimal microstructure
that satisfies the competing design objectives for the texture dependent and
grain boundary network sensitive properties involved in this design problem.
I show that through this process of texture-mediated grain boundary network
design it is possible to design materials with dramatically improved properties.
* Chapter 5 extends the work of the previous chapters on the design of grain
boundary networks to a related and complementary problem: grain boundary properties localization. This is a tool that allows one to measure the effective properties of a grain boundary network and infer the parameters of a
structure-properties relation for individual grain boundaries. By leveraging the
structure of the grain boundary network, this methodology is shown to be far
more efficient that a brute force attempt to measure the properties of individual
bicrystals in an attempt to establish structure-properties correlations. I give a
unifying treatment that shows how localization completes a family of complementary mathematical materials problems comprised of properties prediction,
microstructure design, and localization.
26
Chapter 2
A Triple Junction Distribution
Function*
2.1
Introduction
Grain boundary engineering (GBE) techniques have demonstrated the potential for
dramatic materials properties enhancement; in some cases, improvements of more
than an order of magnitude have been observed [44-47]. While such studies have
empirically demonstrated the enhancement of materials properties through manipulation of the grain boundary network, a theory for the design of microstructures with
optimized grain boundary networks is still lacking. Fig. 2-1 schematically illustrates
the current state of the art in microstructure characterization, materials properties
prediction and materials design. For materials properties that are sensitive to crystallographic orientation, the rigorous mathematical tools of texture analysis and homogenization theory provide the underpinning methodology to address the forward
problem of predicting effective macroscopic properties. These techniques are detailed
in the seminal work by Bunge [15]. The inverse problem of designing a microstructure with a specified effective macroscopic materials property has also been treated
for orientation dependent properties thanks to the Microstructure Sensitive Design
for Performance Optimization formalism developed by Adams, et al. [7].
However, the materials properties that are most likely to be enhanced by GBE
are dominated, not by the orientation of grains, but rather by the misorientation of
grain boundaries and their spatial arrangement within the grain boundary network.
Examples include intergranular fracture [35], intergranular corrosion [36], and liquid
metal embrittlement 139, 40]. For such network-dominated properties as these, neither
the forward problem of properties prediction, nor the inverse problem of materials
design has been solved, and a fundamental framework on which to approach a solution
does not even exist. It is this gap in the literature that the present work attempts to
address.
The improvement in properties that has been achieved through GBE has been
ostensibly linked to an increase in the population of certain "special" grain bound*The contents of this chapter were published previously in reference [52].
27
Texture Analysis
GB Network
Characterization
ODF
TJDF
\/
\/
Properties
/\/
GB Network
Design
Microstructure Design
Fig. 2-1: Schematic illustration showing the gaps in the microstructure design literature (dotted
lines). Much work has focused on prediction of properties that depend on texture (solid lines).
This is typically done by using the ODF in homogenization relations. However, much less has been
done in the realm of property prediction or materials design when the properties of interest are
dependent on the misorientations of grain boundaries and their spatial distribution in the grain
boundary network. This work attempts to fill this gap by making an explicit connection between
the ODF and the TJDF.
aries [44, 46, 47, 53-561, e.g. low-angle or coincidence site lattice (CSL) boundaries.
However, it is known that grain boundary networks in real materials are not, and
in fact never can be, randomly assembled, due to the existence of certain crystallo-
graphic constraints [42, 49, 57-75]. Such spatial arrangement issues have been studied
heavily 135, 41, 61, 62, 67, 72, 76-791, and are strongly linked to, e.g, triple junction
types [67, 72]. It has also been noted that the constraints imposed by higher order
topological features of the microstructure play a decreasingly significant role [80].
Thus, it would appear that triple junctions occupy a unique position in the hierarchy of topological descriptors: they are the simplest microstructural component that
encodes sufficient topological information to characterize the major features of real
grain boundary networks. Additionally, triple junctions play the unique role of decision points or hubs for the transmission of various forms of intergranular damage.
Depending upon the specific crystallography of a triple junction, it may either impede
or permit the propagation of an incident crack (for example). The triple junction thus
emerges as a key feature of the grain boundary network, constraining the assembly
of grain boundaries, controlling (to a large degree) the long-range connectivity of
boundaries of specific types, and regulating the flow of various damage processes.
Accordingly, I envision a materials design paradigm for GBE materials as compris28
ing tools like those described for texture control, with triple junctions at the center
(see Fig. 2-1). Just as in texture analysis, where the orientation distribution function
(ODF) is the weighting function that allows one to compute average or "effective"
properties, the triple junction distribution function (TJDF) could serve a similar
purpose for homogenization over the grain boundary network. With tools to connect
texture to triple junctions, and triple junctions to properties, one can envision an
approach to the inverse problem. It is our purpose in this chapter to present the
TJDF as the central component in this design paradigm, as illustrated in Fig. 2-1.
While various researchers have studied the theoretical aspects of grain boundary
networks (see, e.g., [68, 71, 81-83]), the concept of the TJDF has only been touched
upon in a relatively limited way. For example, Mason derived the TJDF for the
case of crystallographically consistent grain boundary networks with perfect in-plane
textures 1681, and both Mason and Frary independently derived the triple junction
fractions for the same specific case [60, 68]. Hardy and Field considered the TJDF
for the special case where triple junctions are composed of at least one coherent twin
boundary [84, 851.
In the present chapter, I derive a general formula for the uncorrelated TJDF of
materials possessing arbitrary texture. I also provide a mathematical relation that
links the TJDF to the ODF. This result connects our work to the existing body of
microstructure design literature (e.g. [7, 15, 86, 87]), and should facilitate solution
of the inverse problem through a texture-based approach to grain boundary network
design. The present chapter thus lays mathematical groundwork necessary for the
rational design of grain boundary networks.
2.2
Conventions
In this thesis I follow the active rotation convention, as in Ref. 188]. This treatment
regards a rotation operation as an active transformation of the points of configuration
space, with respect to a right-handed orthonormal set of space-fixed axes. From this
point of view a crystal orientation is defined as the rotation, which, when applied
to the coordinate system of a reference crystal, brings it into coincidence with the
coordinate system of the oriented crystal. All such rotations belong to the proper
rotation group, SO (3), and can be defined by three independent parameters. In this
thesis I use the quaternion parameterization of SO (3). Explicitly, the components of
a unit quaternion, q = [a, b, c, d], are related to elements of SO (3) by [88]:
a = cos (w/2)
(2.1)
#
b = sin (w/2) sin 0 cos
c = sin (w/2) sin 0 sin
d = sin (w/2) cos 0
where the rotation angle is given by w E [0, 27r], and the rotation axis is defined
by the polar and azimuthal angles: 0 E [0, 7r] and q E [0, 27r] respectively. This
parameterization maps elements of SO (3) to points on the unit hypersphere, S3.
29
The domain of the angular parameters reveals that this mapping yields a double
covering of SO (3), i.e antipodal points on S3, given by q, correspond to identical
rotations. While a discussion of the subtleties of this parameterization is beyond the
scope of this thesis (see, e.g., [88]), this point is important to emphasize because,
although I am interested in SO (3), mathematical convenience will compel us to work
entirely in S3 in what follows.
For the sake of brevity, I will frequently denote a crystal orientation by its respective quaternion representation, qi, with a single subscript, i E N. I will likewise
denote the misorientation between crystals i and j by its quaternion representation,
qij, this time with two subscripts, i, j E N. Additionally, I will use a counterclockwise
convention when labeling the crystals that coordinate a triple junction (see Fig. 2-2).
q12
q1
q2
q3
q2 3
q3 1
Fig. 2-2: Schematic triple junction illustrating the nomenclature used in this thesis for grain orien-
tations, qi, and grain boundary misorientations, qij.
As a clarifying point, the literature on triple junctions has used the term "triple
junction distribution" (sometimes "triple junction character distribution") in two different contexts, which deserve distinction. The first, and by far the most common
usage, refers to the population of triple junction types within a material [67, 69, 74, 79,
89-91]. These are the so-called Ji, with i E {0, 1, 2, 3} indicating the number of "special" boundaries coordinating the triple junction [61, 691. The second usage refers to
the probability density function (pdf) that describes the probability of finding a triple
junction in a given sample possessing specific values of certain geometrical variables.
In the most rigorous case, such variables would include the grain boundary misorientations, grain boundary plane inclinations and the triple line direction [84, 85].
Historically, however, most attention has been focused on only the misorientation
variables 168].
The two notions are related by the fact that the triple junction fractions, Ji, can
be obtained from the triple junction pdf by integration over particular portions of the
30
triple junction space [68]. In this thesis I use the term "triple junction distribution
function" (TJDF) to denote the pdf that gives the probability of finding a triple
junction possessing misorientations q1 2 and q2 3 in a given sample. The related notion
of the population of triple junctions composed of 0, 1, 2, or 3 "special" grain boundaries
I refer to as "triple junction fractions", in analogy to the term "special fraction" that
describes the population of grain boundaries that satisfy some criteria of interest.
2.3
The Triple Junction Distribution Function
An orientation distribution function (ODF) is a mapping f : SO (3) -+ R1, which
describes the probability of observing a crystal of a particular orientation within
a polycrystalline aggregate. In the context of the quaternion parameterization of
SO (3), mathematical convenience dictates that I instead consider the mapping f :
R 0, and constrain any ODF, so defined, to posses antipodal symmetry, so that
Sf (-q) = f (q).
Any square-integrable function on S3 may be expanded as an infinite linear combination of the complex hyperspherical harmonics [27]:
0C
f (w,,
(W
n
1
ZZECE
1 )
n=0,2,... 1=0 m=-l
Zn
W
(2.2)
1ZT'mW,,c/)
Note, that the index n takes only even integer values. The coefficients of this expansion are computed using [31]:
cnm
1 j2-
2
jj
f
(w, 0, #) Z* (w, 9, #) sin 2 (w/2) sin 0dwd~do
(2.3)
0
In Eq. (2.3), as in the remainder of this thesis, I use a superscript of * to denote
complex conjugation.
As it will be of use in what follows, and to make our phase convention explicit, I
include the definition of the hyperspherical harmonics here [27]:
Zim(W,0, #) =
(-i)'21 1!
2(n + 1) (n - l)!
[cos (w/2)] Y{" (0,q)
sin' (w/2) C'_'
(
n).
7r (n + 1 +
(2.4)
where C'j [cos (w/2)] is a Gegenbauer polynomial, and Ym (0, #) is one of the spherical harmonics. Definitions of these functions, consistent with the conventions of this
thesis, are provided in A. The fact that the hyperspherical harmonics partition into
a product of functions depending on only one of each of the rotation parameters is
one of their useful mathematical properties, which I will use in what follows.
The normalization condition for an ODF defined as stated above is given by [31]:
/
f (w,0,1 ) dQ = 1
31
(2.5)
where Q
S3 . In the context of our quaternion parameterization, the invariant
measure in Eq. (2.5) is given by:
dQ =
1
sin2 (w/2) sin Odwd~do
2
(2.6)
and Eq. (2.5) can be written explicitly as:
2-
I
11, f
2
0 (w, 0, q) sin (w/2) sin Odwd0d# =1
,
(2.7)
With this definition in hand, I can now define the TJDF as [681:
T (qi, q2 , q3 )
=
f (qi) f (q 2 ) f (q 3 )
(2.8)
where I have denoted the rotation defined by the parameters (wi, 0i, #5) using its
quaternion representation, qi, for notational simplicity. This notation also makes the
effect of the composition of rotations on the functions which take them as arguments
more transparent. Eq. (2.8) describes the joint probability density of finding a triple
junction coordinated by grains with orientations q1 , q2 , and q3 (Fig. 2-2). This definition obviously precludes the consideration of spatial correlations as it assumes the
mutual independence of
f (qi), f (q
), and
2
f (q 3 ).
Eq. (2.8) can therefore be said to
give the uncorrelated TJDF.
Crystallographic consistency requires conservation of orientation around a triple
junction, i.e the composition of the grain boundary misorientations around a triple
junction must result in the identity rotation. This can be stated simply as:
ql2q23q3l
(2.9)
where qAB=
qB is the quaternion representation for the misorientation between
grains A and B. Eq. (2.9) indicates that a triple junction is uniquely specified by
only two rotations. Recognizing this fact, Eq. (2.8) can be rewritten in terms of
misorientations. Additionally, because the probability of finding a triple junction
composed of relative misorientations q12 and q 23 is invariant to the orientations of the
grains coordinating the triple junction, I integrate the probability density contribution
of all possible values of the remaining crystal orientation q2 . Applying these two
operations yields:
f (q2 q ) f (q 2 ) f (q 2q 23 ) dQ (q 2 )
T (q 12 , q 2 3 )
(2.10)
(q2)
This transformation not only defines the TJDF in terms of a minimal set of parameters, but it also enforces crystallographic consistency.
32
Expressing Eq. (2.10) in terms of the expansion given in Eq. (2.2) leads to:
q
Z," (q2q23) dQ
(q 2
ni
11
mi
.n2
12
mn2
Cli'mi1 Im
(q2q121
Cn2
(q2)
.
T (q2, q 2 3 ) -f
)
x
Zjn"
(2.11)
3
13
cn3
m3
.
Mason derived an addition theorem for the hyperspherical harmonics, which allows one to express a hyperspherical harmonic whose argument, q, consists of the
composition of two rotations (i.e q = qqi) in terms of hyperspherical harmonics
whose arguments are the constituent rotations, qj and qj [27]:
I~
AA
Z/2 mn (qjqi)
+
ri+
1jy
ip
(2Aj + 1) (2Aj + 1) C{r/
x
xZj,,
AjA Ai n/2
(2.12)
AAi
n/2 n/2
(qj) Z,\,, (qj)
The term in curly braces in Eq. (2.12) is a Wigner 6-j symbol (A).
Applying Eq. (2.12) to Eq. (2.11), factoring everything that does not depend
on q2 out of the integral and applying the inversion symmetry relation provided in
Eq. (C.3), I find:
T (q1 2, q 23 )
S(
-
w
/
1 )"A
3
A2 3
27r2
n3
fn2
(2A, + 1) (2A 1 2 + 1) (2A 3 + 1) (2A 23 + 1) Cni
M 2 c 3,M 3
c1,mic1
2
1)
+
(n
1)
(ni +
3
x CI'""
C
C
A1
/12p
Z
Lo(q2)
A3,P3,A23,123
f13
n 3 /2
A3
A23
n3/2
n3/2
(q12) Z"242 (q23)
xZ"1
x
A,
A 12
ni/2 ni/2 ni/2
1i
/1,M3
(q2) Z,
1'lM
2
(q2) Z"A34
q21d3(2
(2.13)
where the summation is over all valid values' of the various indices. The integral
'Here I state the summation limits explicitly: ni, n2, n3 E [0, oc); 11, A,, A 12 E [0, ni]; 12 E [0, n2];
A
13, 3 , A 23 E [0, n3]; m1 E [-11, 1i]; /'i 2 [-A1 , Ai]; [112 E [-A 12 , A12]; m 2 E [-12,12]; m 3 E [-13,13];
A3 E [-A3, A3]; P23 C [-A 23 , A 23 ]. Note that the ni take only even integers, while all other summation
33
in Eq. (2.13) is exactly equal to the coefficient of the Hyperspherical Harmonic Linearization Theorem, which I have derived, a partial derivation of which is given in B,
which is sufficient for the present discussion.
Applying Eq. (B.7) to Eq. (2.13) results in:
00
T (q
1 2, 9 2 3 )
ni
n3
A23
n3=0,2,...
A 2 3 =0
1 2 3 =-A 23
(-1) -2
=
n1=0,2.--X
00
A12
E
A12=0 A 12 =-A 1 2
(-)-AlA
3
V'2wr2 (2A 1 2 + 1) (2A 2 3 + 1)
(2A, + 1) (2A 3 + 1) V(n 2 + 1) (212 - 1)
11 n 2 12 13
x C11i4 +A /Cn2_
1
12 2 ,1 -P3
-
-111
(P1
AMiA12,112 A313,123
A3
12
11
[13)
P3
C13,3+P2
cl1,Pl+[42
Cn3
C3,P3+P23
A
A 12
13
A3
n3/2 n3/2
ni/2 ni/2 n1/2
A23
n3/2
ni/2 n 2/2 n 3 /2
A,
A3
12
Z'l (q12 ) Zn
(q 23
3
)
ni/2 n 2 /2 n 3 /2
x
J
(2.14)
The terms contained within the brackets in Eq. (2.14) evaluate to a constant coefficient
which takes ni, A 12, P12, n 3 , A 23 , and P23 as indices. Therefore, defining:
(-1)-
S=
2r2 (2 12 + 1) (2A 23 + 1)
A23
-l 3 (2A, + 1) (2A 3 + 1)
(-1)
x
(n 2 + 1) (212 + 1)
11 n 21213
X
X
Cni
C P1+P
1 1
n2
12 , -i-/-13
A,
i
12
\i1i
-P1 - /13
Cl,/1+A12
C n3
13,1P3+ 23 A 1 , 1 ,A 12 ,i 1
A3
13)
11
A1
013/,3+P23
2
A 3 ,1
A 12
nl/2 ni/2 ni/2
3 ,A 2 3 4 2 3
13
A3
A 23
n3/2 n3/2 n3/2
n3/2
n2/2 n3/2
ni/2 n2/2
nA1
A,
12
(2.15)
indices take integers.
34
A3
I can express our final result simply as:
2.4
ni
A12=0
A12
00
n3
A23
>
12=-A 12 n3=0,2,... A2 3 =0 P 2 3 =-A
,"3
Z"3
(q12) Z 3,
23
23
(q 2 3
(2.16)
Discussion
Eq. (2.16) gives a general expression for the TJDF of any polycrystalline material
possessing arbitrary texture. This series expansion is expressed in the basis of bipolar
hyperspherical harmonics,which may be more readily understood by analogy with the
more common bipolar spherical harmonics (in two fewer dimensions); just as bipolar
spherical harmonics present a reasonable basis for problems in the quantum theory
of angular momentum that depend on two three-dimensional vector directions [92],
bipolar hyperspherical harmonics provide a suitable basis for expressing functions that
depend on two four-dimensional quaternions, such as the present case, in which the
two independent misorientations of a triple junction are represented in the quaternion
parameterization.
The relation given in Eq. (2.15) between the coefficients of the ODF, Cnm and
holds exactly for materials in which there is no spaA
those of the TJDF,
of the grains themselves. This is, to a good
the
orientations
among
tial correlation
approximation, the case with high stacking fault energy materials. For this class of
materials the existence of Eq. (2.15) is of fundamental importance and is indicative
of a deeper result: the mapping from ODF to TJDF is unique, i.e, for a given ODF
there is only one commensurate TJDF.
I perceive the greatest value of Eq. (2.15) to be the fact that it provides a system of equations that can be used to solve the inverse problem of materials design. If the materials engineer can identify a target TJDF, as expressed by the set
}, then it remains only to invert Eq. (2.15) to find the required
{t , 4 , "'...
... }. In other words, Eq. (2.15) protexture, as expressed by the set {c80,o,
0 c,
vides the necessary tool for a texture-based approach to engineering grain boundary
network topology.
2.5
Validation
The coefficients of the expansion given in Eq. (2.16) can be determined in exactly the
same way as those of the ODF (see Eq. (2.3)). By multiplying both sides of Eq. (2.16)
by the complex conjugate of the basis function and then integrating over S 3 x S 3 , the
orthogonality of the hyperspherical harmonics results in:
tjlA1f,[1j2
sin 4 0 12 (s.1d)
s
T (q12, q2 3 ) Z
(q12) Z
x sin 2 (W12/2) sin 012 sin 2 (W23/2) sin 023dwi2dO12d#12dU)23dO23dO23
35
(q 2 3 )
2.17
)
00
T(q1 2 ,q 2 3 ) ="3'
nri=O,2,...
If T (q12, q2 3 ) is known analytically, then t13,A23 P23 can be obtained by simple integration. However, in most cases one would like to obtain tnj 3A23,12 from experimental
measurements. In this case t"'223 can be approximated using a procedure, outlined
by Bunge in a different context (i.e for ODFs and using different basis functions) [15].
Consider a microstructure that contains a single triple junction coordinated by grain
boundaries with misorientations q2, q2 3 , and q3 1 . The TJDF for such a microstructure
will be non-zero only 2 at (q , qO3 ), so that Eq. (2.17) becomes:
tn3,A23,1123
n1,A 12 4-12
X
2ir i-r
27r
27r f7r
27r
6 (q 1 2 , q
26
(q 23 , q23)
(q 1 2 )
ZZ12442
Z
,
q23
)
X
x sin 2 (W 12 /2) sin 612 sin2 (W23 /2) sin 0 23 dwi 2d61 2d#1 2 dW 2 3dO 2 3 dO 23
(2.18)
Since T (q 12 , q23 ) is a pdf, I impose the following normalization condition:
/(q2)
xQ(q 2 3 )
T (q 12 , q 23 ) dQ (q12 ) dQ
(q 23 )=
(2.19)
1
By this normalization condition and the definition of the Dirac delta function, Eq. (2.18)
reduces to:
tn3''1
nl,A12,W1 2
A[ 2
,1 12
(q 1
) Z0(q)
A23,P23
(2.20)
(2230
If, rather than being a perfect tri-crystal, the microstructure contains several triple
junctions then t "23 will be a weighted average of the values obtained from
Eq. (2.20) for each triple junction, where the weights are given by the respective
lengths, si, of each of the one-dimensional triple junctions. If one considers a twodimensional section of a three-dimensional microstructure, then each triple junction
is a zero-dimensional point and the weights will be equal to unity:
(qI2 ) Z
Z
tn3,A23,P23
n1,A12,P12
(q~i)
3
for 2-D
(2.21a)
for 3-D
(2.21b)
._
I
-
E siZ,
*n
1
(qi 2 ) Z
2
3
(qi 3 )
In reality triple junctions are geometric entities which posses inherent symmetries. If these
symmetries are enforced the TJDF of even a tri-crystal will be non-zero at all points that are
symmetrically equivalent to (qO 2 , q03 ). It should be possible to symmetrize the bipolar hyperspherical
harmonics to respect the symmetries of triple junctions, but the treatment given here uses the
unsymmetrized bipolar hyperspherical harmonics. In order to enforce triple junction symmetries in
this context, one may augment the list of actual triple junctions with all of the triple junctions that
are symmetrically equivalent to those observed in the microstructure so that if there are N triple
junctions then the augmented list will contain 6N triple junctions. Enumeration of these symmetries
and the mathematical relations that describe them in the context of the present formalism will be
addressed in a future publication.
36
si is the total length of
where N is the total number of triple junctions and S =
all triple junctions.
At this point I am equipped with two independent methods of obtaining TJDF
coefficients. Eq. (2.15) allows us to obtain t"'A23 P23 from the coefficients of the ODF,
Cr, while Eq. (2.21) provides a means to estimate
directly from experimen-
tn3A23
tal observations.
In order to validate the expression supplied by Eq. (2.15) I simulated fiber, unimodal, and multimodal ODFs of varying sharpness for triclinic crystal symmetry
(Fig. 2-3) and obtained values of tn3:A23 P23 using both Eqs. (2.15) and (2.21a). The
theoretical values provided by Eq. (2.15) were then plotted against the "expected"
values provided by Eq. (2.21a) for comparison.
In the case of Eq. (2.21a), 10,000 triple junctions were simulated independently
for each ODF by sampling 10,000 triplets of orientations, qi, q 2 , and q3 (see Fig. 2-2).
The misorientations, q 12 , q23 , and q31, were then computed from the orientations and
Eq. (2.21a) was used to obtain "expected" TJDF coefficients. In the case of Eq. (2.15),
in:Ti
-7
(C)
6
mx
WON
.01
101
1.0015
12
1.001
1.0005
1,10
099
099
0
1.1
09995
1 05
098
095
0965
0990
097
09985
097
096
09975
AN
Fig. 2-3: {111} Pole figures for some of the simulated ODFs. (a) A fiber ODF with a half-width of
50; (b) a fiber ODF with a half-width of 750; (c) a unimodal ODF with a half-width of 250; (d) a
unimodal ODF with a half-width of 750; (e) a multimodal ODF with 19 modal orientations and a
half-width of 12'; and (f) a multimodal ODF with 17 modal orientations and a half-width of 178'.
In all cases the modal orientations were randomly selected. For the fiber and unimodal ODFs the
half-widths were chosen in 50 increments from 50 to 1800. For the multimodal ODFs the half-widths
were chosen randomly between 50 and 175'.
the ODF coefficients were estimated from 10,000 simulated grain orientations using
the same procedure outlined by Bunge [151, but in the context of the hyperspherical
harmonics. The theoretical TJDF coefficients were then computed using Eq. (2.15).
37
I simulated a total of 144 distinct ODFs using the open source texture analysis toolbox, MTEX [931, for MATLAB [94]. Orientation sampling was also performed using
MTEX. For each simulated ODF 39 coefficients, up to max (ni) = max (n3)= 2, were
compared, for a total of 5616 coefficients. As evidenced by Fig. 2-4 the agreement beReal
tn3
Imaginary (tniJ:2N23)
)
0.01
0.40.005
0.35.
0.3-
00.25
-0.005
0.20.15
-0.01
0.1
-0.015-
0.05
0
-0.02
0
0.1
0.2
Expected
0.3
-0.02
0.4
-0.015
-0.01
-0.005
Expected
0
0.005
0.01
Fig. 2-4: Comparison of the TJDF coefficients computed using Eqs. (2.15) and (2.21a) respectively.
The vertical axis corresponds to the theoretical values computed from Eq. (2.15) and the horizontal
axis to the "expected" values obtained directly from the misorientations by means of Eq. (2.21a).
The real parts of the complex valued TJDF coefficients are plotted on the left, and the imaginary
parts are plotted on the right. Each set of uniquely colored points corresponds to a separate ODF.
Note that the data points closely follow a line of slope equal to 1, indicating nearly perfect agreement
between the TJDF coefficients computed using these two independent methods.
tween the coefficients computed using Eqs. (2.15) and (2.21a) is nearly perfect across
all of the textures considered.
2.6
Conclusions
I derived a general formula for the TJDF of any polycrystalline material possessing
arbitrary texture using a bipolar hyperspherical harmonic expansion. I provide a relation that gives the coefficients of the TJDF explicitly in terms of the coefficients of the
respective ODF. It is exact in the limit of spatially uncorrelated crystal orientations.
Our Eq. (2.15) provides a connection between grain boundary network topology
and crystallographic texture, which will be useful for a texture-based approach to
grain boundary engineering. This is particularly true in the case of materials that do
not twin easily, for which traditional thermomechanical processing routes are ineffective and which posses approximately spatially uncorrelated crystal orientations.
38
Chapter 3
The Triple Junction Hull*
3.1
Introduction
In the context of texture and grain structure in polycrystalline materials, microstruc-
ture design tools now allow the designer to explore a complete set of physically realizable microstructures and identify those that meet various performance objectives
and design constraints [6, 71. These tools have been used to address design problems in which elastic and plastic properties are concerned [6, 8-141. However, their
application to other properties of scientific and engineering interest, such as fracture
and corrosion, which depend upon the structure of the grain boundary network, remains to be addressed. This is an area of great opportunity because grain boundary
engineering (GBE) studies have demonstrated significant enhancements of materials
properties [36, 44-471, but there is currently no "inductive" method [96] by which to
design such successes a priori. For instance, Lin et al. observed a two-fold decrease in
the intergranular corrosion rate of Alloy 600 after GBE processing [97]. However, no
major change in texture accompanied this enhancement in properties. Characteristic
of GBE materials, the resultant texture was nearly random, but contained a high
fraction of low-energy grain boundaries (up to 55% of the boundaries were E3 type).
Contrast this with the fact that for a microstructure with uncorrelated grain orientations and random texture the expected population of such boundaries is extremely
small (~1.8%) [981. It would appear, then, that texture is insufficient to explain or
predict such materials properties enhancements and, consequently, design tools that
incorporate the character and connectivity of the grain boundary network are needed.
In the previous chapter, I introduced a general triple junction distribution function
(TJDF) as the core of a design strategy for grain boundary networks (see Chapter 2).
The TJDF provides a statistical description of grain boundary network connectivity
that facilitates adaptation of the existing design framework to grain boundary network design. Expressed in the context of the spectral framework of Chapter 2, the
TJDF provides a bridge between crystallographic texture and grain boundary network topology. These tools allow one to rigorously bound the universe of all possible
grain boundary network configurations. In this chapter, I employ our previous results
*The contents of this chapter were published previously in reference [95].
39
in order to define the triple junction hull, _W , which constitutes the first order design space for grain boundary networks. W3 bounds the entire space of physically
realizable grain boundary networks, and is pivotal for the adaptation of existing tools
to the problem of grain boundary network design. I also investigate the relative size of
the subspace of
_/
that is accessible through the control of crystallographic texture
alone.
3.2
Triple Junction State Space
Grain boundary network design is an intrinsically multi-scale endeavor, requiring the
practitioner to specify both (1) the local state and (2) the spatial location of grain
boundaries within the network. However, real grain boundary networks exhibit local
correlations that dominate at the length scale of triple junctions [61] and encode
network connectivity information. This suggests that the most convenient unit for
grain boundary network design is, in fact, the triple junction.
Figs. 3-la and 3-1b illustrate the two dominant length scales that must be considered in the design of grain boundary networks: that of the triple junction (i.e.
the characteristic length scale of network connectivity), and that of the network as a
whole. The duplicity of length scales in the physical grain boundary network is also
reflected in the design space, which encompasses both the local state space of the
triple junctions, and the global configuration space of the network. I will begin by
defining the local state space for triple junctions, and in the subsequent section I will
address the configuration space of the network as a whole.
A triple junction is the one-dimensional edge common to three adjacent grains,
or, equivalently, the one-dimensional intersection of three grain boundaries. It is geometrically defined by eleven macroscopic state parameters. Five of these define the
inclinations of the incident grain boundary planes. The other six characterize the
lattice misorientations between pairs of grains adjoining each of the three intersecting
grain boundaries, and it is on these that I will concentrate in this chapter. Generalization of our methods and results to include the contribution of grain boundary
planes is possible in principle, but constitutes a significant cost in complexity, and
will be left for future work.
The misorientation associated with a grain boundary between grains A and B is
defined by:
qAqB
1qAB=
(3-1)
where qA and qB are the lattice orientations of grains A and B, respectively. In this
chapter I use unit quaternions to represent general three-dimensional rotations. The
three independent parameters corresponding to such a rotation may be chosen as the
spherical angles of the rotation axis, and the angle of rotation about that axis. These
40
100 Pm
100
A
200
300
400
50*
60*
(a)
12
J#4
2
%
%
q 31
0
-
(b)
q2
/01I
,
/
----
(c)
Fig. 3-1: (a) The grain boundary network of a sample of Inconel 690, representing a global configuration of triple junctions. Grain boundaries in (a) are color coded by their disorientation axis (in
the standard stereographic triangle) and angle (see the legend at right) [99]. (b) The local state of
a representative triple junction as indicated by its triple junction misorientations, qAB. The grains
coordinating the triple junction are labeled 1, 2, and 3, with dotted arrows defining a circuit that
encloses it. Composition of the grain boundary misorientations around such a circuit must result
in the identity operation (Eq. 3.3). (c) All six distinct paths surrounding a triple junction. These
correspond to all possible ways of choosing the two independent misorientations coordinating a triple
junction, and thus, represent physically equivalent descriptions. Colors distinguish circuits starting
in different grains, and line type (solid/dashed) distinguish circuits of different sense.
41
are related to the components of a unit quaternion, q = [a, b, c, d], by f28, 100]:
a = cos(w/2)
b = sin(w/2) sin 0 cos
c = sin(w/2) sin 0 sin
d = sin(w/2) cos 0
#
(3.2)
with 6 describing the polar angle measured from the positive z-axis, # the azimuthal
angle measured from the positive x-axis, and w the rotation angle.
While there are a total of nine misorientation parameters for a triple junction
(three for each misorientation), crystallographic consistency requires that the composition of the misorientations around any closed circuit results in the identity [101].
For a triple junction coordinated by grains labeled 1, 2, and 3, (see Fig. 3-1b) this is
stated as:
(3.3)
q 1 2 q 2 3q 3 1
I
Therefore, only two of the three misorientations are independent, reducing the total
number of misorientation parameters to six. The local state space for triple junction
misorientations is, therefore, initially identified with the product space SO (3) x SO(3).
Further consideration of the physical symmetries of a triple junction reveals equivalence relations that both couple and reduce the full misorientation space. Consider
a circuit enclosing a triple junction, as depicted in Fig. 3-1b. One may choose a
starting grain and a sense in which to traverse the circuit in any one of 3 x 2 = 6
ways, each of which will result in an equivalent description of the triple junction (see
Fig. 3-1c). Consideration of all possible circuits, in conjunction with the constraint
of crystallographic consistency, leads to the following set of equivalence relations for
the ordered pair of independent misorientations defining a triple junction:
F = {[q 12 ,
231 ,
[q2 3 ,qq2
] , [q23
1
2
]
2[
qq
2
23] , [q12q 2 3 ,q23
23
2,
(3.4)
where [a, b]
{(qI 2 , q2 3 ) |(q12, q2 3) - (a, b)}. Therefore, for the most general case, the
local state space for triple junction misorientations is given by:
= (SO(3) x SO(3)) /F
-Q
(3.5)
and is referred to as the triple junction fundamental zone, or asymmetric region.
If the crystallographic point group, for a material of interest, contains symmetry
operations in addition to the identity element, then d(3 ) is further reduced. For the
purposes of this thesis, I will demonstrate my methods without including additional
crystal symmetry (i.e. for triclinic crystals). Generalization to other crystal symmetries is possible, but requires some care in handling the somewhat subtle relationship
between crystallographic symmetry and triple junction symmetry.
One noteworthy observation about Eq. 3.4, is the absence of what is referred
to as "grain exchange symmetry" [99, 102]. When considering grain boundaries in
isolation, the choice of grain A or grain B as the reference orientation is arbitrary,
42
thus one has that qAB~ qBA. However, this is not true when collectively considering
grain boundaries that intersect at a triple junction. This becomes readily apparent
by replacing q 12 with q2 1 in Eq. 3.3 and observing that the result is no longer the
identity. Inclusion of "grain exchange symmetry" is tantamount to allowing circuit
discontinuities, which violates the constraint of crystallographic consistency. One
may indeed reverse the sense of the circuit defining the misorientations around a
triple junction, or choose a different starting grain, but once a sense is chosen it must
perpetuate until the circuit has closed. As a concrete example, if one exchanges q21
for q 12 one must also exchange q32 for q2 3 and q13 for q3 1 . In this case Eq. 3.3 becomes,
q21 q13q 32 = I and one may verify that the identity is correctly obtained.
3.3
Grain Boundary Network Configuration Space
)
Having considered the design space for the local length scale, I now turn to the
design space for the grain boundary network as a whole. A grain boundary network
configuration may be characterized by a statistical description of its triple junction
states, i.e. a distribution function over W('). The probability density associated
with observing a triple junction characterized by the ordered pair of misorientations
(q12, q23) E d(3) is given by the six-parameter triple junction distributionfunction'
(TJDF), T(qi 2 , q2 3 )= T(wi 2 , 012, 12, 23, 023, 423).
Abstractly, the complete grain boundary network configuration space is simply
the set of all possible functions, T(q 1 2 , q2 3 ), that satisfy the properties of a probability
density function and the symmetries described by Eq. 3.4. I refer to this space as
the triple junction hull 2 and denote it & ) A key insight from the work of Adams
et al. is that spectral decomposition of the relevant distribution function, in this case
T(q 1 2 , q23 ), provides parameters with which one may concretely define the boundaries
of the space of all such functions [7]. In other words, the difficulty of defining /W( 3
directly in function space is greatly alleviated in "frequency" space. In order to make
This immediately suggests a
use of this insight, it is convenient to discretize a(3).
universal form for T(q 12 , q2 3 ), given by:
K
T(q 12 , q23)
ZPk(-) (q12
kq 12 )
(
(q 2 3 , kq 2 3 )
(3.6)
k=1
where Pk is the probability associated with the k-th ordered pair of misorientations
in the discretization, denoted (kq 1 2, kq 23 ), and the approximation becomes exact as
'The full eleven-parameter TJDF would include the boundary plane parameters. For the sake
of brevity, and because the context is clear, I will refer to the six-parameter TJDF, which contains
only the misorientation information, simply as the TJDF. However, it should be noted that alternative representations of the statistics of triple junction character exist, which include additional
crystallographic parameters such as those of the boundary plane normals [84, 85].
2
The notation here is chosen to maintain consistency with and generalize the existing notation in
the microstructure design literature. For example, the texture set and texture hull originally defined
in [6], are denoted using our scheme as ff(') and W(') respectively.
43
K -÷ oc. The basis functions in Eq. 3.6 should be regarded as a composite symbol,
which denotes a Dirac delta like function respecting the symmetries embodied in
Eq. 3.4. This can be expressed explicitly as:
(qi 2 , kq 12 ) 6(3)
(q23 , kq 23 ) =
1 [6(ql 2 ,kq 12 ) 6 (q
2 3, kq 2 3
)
6
+6(q12, k q 23 ) 6 (q2 3 (kq12 k q23 )1
+6 (q 12 ,
(kq 12 kq23 )1)
+(q12,
k q-)
+
kq12k q23 )
6(q 2 3 kq1 2 )
(37)
)
6(q 2 3, kq 12 kq 23
q1
(q2 3 ,kq-1)
+6 (q12, kq-1) 6(q 23 kq-1)
The Pk define a vector space, M(3), over RK, in which each point, p (PI, P2, ...,PK)i
represents a grain boundary network (i.e. a specific TJDF). Given that the Pk represent probabilities, M() C M(3 ) is defined by:
p P = (PP2,.
O
.,PK),
(3-8)
- 1
Pk, Ek
k=1
which is a standard (K - 1)-simplex, as shown in Fig. 3-2a. The change to the roman
typeface of Mg) is intended to indicate the Dirac basis representation of the triple
junction hull, whereas the script typeface of W43) refers only to the abstract concept,
without reference to a specific basis (a distinction that will become important in what
follows).
The vertices of M(3 correspond to points, kp, with only one non-zero component, kPk = 1. These points represent the coefficients of 6(3) type microstructures
centered on (kq 1 2 , kq2 3 ). I refer to this set of points as the triple junction set,
M
(3c
M3 C M(3), which may be defined mathematically by:
MN =
{P k
- (k
p .2
..
PK)
,
kPr =
6
rk,
k C
[1,
K]}
(3.9)
)
It is clear that Mg) is simply the convex hull of MP, which is the reason for this
3
nomenclature. Because of the simplicity of the geometry of this definition of M(H
it is efficient to sample from and easy to explore. However, another definition also
proves to be important.
Alternatively to Eq. 3.6, the TJDF may also be expressed as a generalized Fourier
series in the basis of bi-polar hyperspherical harmonics (see Eq. 2.16):
T(q1 2 ,q 2 3 )
Z(q12)Zn, 3(q23)
=
n1
1i
A 1 2 ,/L12
n3 A2 3 ,123
44
(3.10)
P4
0.06,
0.04,
,
0.02
01
-0.02,-
P3
-0.04,
-0.06,
P1
0.1
0.05
o0.05
P2
Re
(t ;:;1)
Re (t ;8;1)
-0.05
-0.05
(b)
(a)
(3)
4, corresponding to the
Fig. 3-2: (a) The triple junction hull, Al, in the discrete basis with K
standard 3-simplex, or tetrahedron. The vertices are points for which the corresponding Pk = 1
and all others are 0, edges are points for which two of the Pk are non-zero, faces correspond to
points for which three of the Pk are non-zero, and the interior is populated by points for which all
of the Pk are non-zero. (b) Orthogonal projection of m () in the first three non-trivial dimensions.
Two-dimensional projections are also shown as outlines for clarity.
where the coefficients,
2
tn234L23
ni ,A 241L12
1
may be computed by the inner product:
3
(q 23 )
(q12) ZnA23,1123
T(q12,q 2 3 ) Zni
A 1211112
(3.11)
In Eq. 3.11, * denotes complex conjugation, and f denotes integration over the entire
domain, S 3 x S3, with respect to the appropriate invariant measure (see Eq. 2.17),
which is omitted for brevity. Eq. 3.11 represents a generalized Fourier transform,
with Eq. 3.10 providing its inverse. I emphasize that Eq. 3.10 is completely general,
and triple junction statistics of any microstructure may be expressed in this form,
regardless of the presence or absence of spatial correlations. In this basis, the t
define an infinite-dimensional vector space, M(3), over C', in which each point, t =
(t0,
'
,00 .. .), represents a grain boundary network.
Using Eq. 3.11 I can take the Fourier transform of Eq. 3.6, which results in:
K
tn3A23,A23 ,
n1,A1 2,P12
ktn3,A
L..I:Pk k1,A
23
(3.12)
,71 2 3
1 2 ,A 12
k=1
where
kt'3
,i 23
",A
nlg A23
12 ,1.2
are the Fourier coefficients of 6(3)(q12,
kq 1 2 ) 6(3)
(q 23, kq 2 3 ).
In light of Eq. 3.12, alternate definitions of M(3 and M(3 may be given, respec45
tively, by:
(3)
S
_
kt
k
kt
0,0,0 k 0,0,0 k 0,0,0
0,0,0
23
kq 12 , kq 23 ) E
t
T(3)
(3
mH
(3),
=
t
I
kt 6,3,-
2,1,-1) -.
6(3 ) (q9l, kq
(3~) '12
k2(3)
kti ,A 12 ,/- 123
ktn3,2
2,0,0
(3)q 2 3 ,
2
\
4,2,1
2 3)
23k23)
12kq2
1
*t
(q 12 ) Z
3
(q23)
,(313)
k E [1, K]
K
kt
pkk
ZK
k(3)(E_
1rn
O<
t
k=1
pk,
Pk
(3-14)
k=1
where the lower-case m( and m are chosen to distinguish the Fourier representation
of the triple junction set and triple junction hull, respectively, from their representation in the Dirac basis, M-s and H)' Geometrically, H) represents a closed convex
polytope inhabiting infinite dimensional Fourier space, whose vertices are the set mn,
and clearly, mT(3) C rf (3) c (3 . Considering the real and imaginary components of
the tn3A23 4L23 as separate dimensions, Fig. 3-2b displays an orthogonal projection of
in the first three non-trivial Fourier dimensions.
In the preceding derivation I temporarily ignored a somewhat subtle, albeit important, issue regarding the convexity of the triple junction hull. Eqns. 3.8 and 3.14
define the triple junction hull as the convex hull of the triple junction set. It is evident that the coefficients of any TJDF, whether in the Dirac or Fourier bases, may
be expressed as a convex linear combination of those of the triple junction set. However, it is not immediately obvious that the converse is true, i.e that every convex
combination, so formed, generates a TJDF that is physically realizable. This point
is essential to establish because if there exist certain, otherwise valid, p that do not
correspond to physically realizable microstructures, then the triple junction hull is
not truly convex, and it contains non-physical regions, corresponding to non-feasible
design solutions. If such were the case it would significantly hamper the use of the
foregoing methodology, if not altogether thwart it.
In order to prove the convexity of the triple junction hull, I first give an example of
a grain boundary network composed of triple junctions that all possess the same local
state. Fig. 3-3 shows a hypothetical microstructure consisting of columnar grains, each
possessing one of three possible orientations. In such a microstructure every triple
junction is coordinated by each of the three possible orientations, and hence all triple
junctions are identical.
In order to prove the convexity of the triple junction hull, it is enough to show
that there exists at least one physically realizable microstructural instantiation for
any p E M(). One procedure to construct a microstructure corresponding to an
arbitrary p is by "gluing" together a succession of lamellae of the single triple junction state microstructures, depicted in Fig. 3-3, whose relative sizes are given by the
corresponding Pk. Fig. 3-4 shows an example of such a microstructure in which each
lamella is given a different color scheme. The target values of the Pk are, respecm(
46
Fig. 3-3: A schematic of a columnar microstructure in which every triple junction possesses the same
state. The three grain orientations are distinguished by color (red, blue, and white).
tively, p = (0.1875, 0.0625, 0.4375, 0.3125). At the interface between lamellae there
are triple junctions that do not correspond to any of the target states, as shown in
the interfacial detail views in Fig. 3-4, which are artifacts of the "gluing" operation.
Consequently, the Pk do not achieve their target values, as demonstrated by the actual values shown in Fig. 3-4. However, as the system size grows, the fraction of
triple junctions inhabiting the interfacial regions decreases as Nt7- 1 2 , where Ntj is
the total number of triple junctions in the system. Thus, in the limit as Nt- + oc the
target distribution is realized exactly, and this is true for any arbitrary p. Therefore,
I conclude that the triple junction hull is formally convex. The coefficients of any
TJDF may be decomposed into a convex combination of the coefficients of the triple
junction set, and every convex combination is physically realizable. The convexity
of ,3 implies that any path through the design space will be entirely composed of
physically realizable microstructural states; thus, during the design and optimization
process there is no danger of encountering any unphysical solutions. Furthermore,
this convexity permits efficient exploration of the design space via the Pk, for which
calculation and tests of feasibility are computationally inexpensive.
It is worth noting that, for triple junctions, this construction is completely general.
For an arbitrarily chosen state, characterized by the ordered pair of misorientations
(q 12 , q23 ), one may generate a microstructure in which all triple junctions are in the
chosen state. However, it is not possible to generate a microstructure in which all grain
boundaries possess the same, arbitrarily chosen, state (q 12 ). This fact suggests that
would be non-convex, which highlights the convenience
a grain boundary hull, /(,
of designing grain boundary networks in the context of triple junctions instead of
directly focusing on grain boundaries.
47
I
p1 = 0.1458
IA
P2
= 0.0208
I
P3 = 0.3958
p4
=
0.2708
Fig. 3-4: A microstructure with columnar grains and periodic boundary conditions, formed by
"gluing" together four lamellae, each with a single triple junction state. The triple junction state of
each lamella is shown in the lower detail views, along with the actual values of Pk that are attained.
For ease of visualization the orientations of the incident grains are colored rather than the grain
boundaries themselves. The states of triple junctions inhabiting the interfaces are shown in the
upper detail views.
48
3.4
Texture & Topology
The tools that I have developed here are intended to aid in the design of grain boundary networks. The triple junction hull defines the design space, and provides the
necessary parameters (the Pk or t"'A2P3) to explore its entirety in search of grain
boundary networks with enhanced physical properties.
While all points in the triple junction hull are feasible, it is not necessarily true
that they will all be equally straightforward to synthesize experimentally. Ultimately,
fabrication of a particular grain boundary network amounts to controlling the orientation of each grain and its relative position in the microstructure. Because most
standard processing routes provide control primarily over the distribution of grain
orientations, it is useful to consider how far one might get through the control of
crystallographic texture alone.
In the absence of spatial correlations of grain orientation, it is possible to derive
an uncorrelated TJDF, which I denote T(q1 2 , q23 ), and whose coefficients may be
obtained directly from those of the orientation distribution function (ODF) according
to 3 Eq. 2.15:
c2
3
E,,Mn3, 13,m3 Ln3,3'- 3t3,A23,A23
f1,A
1 2 ,P 1 2
12 -A
-
3--
n
3
12,m2 13,m3
-15,m
A1 ,jiiA12,L
12
A 3 ,P 3 , A 2 3 ,P 2 3
1,A1,i1;n2,12,m2
1,mi
n2,12,m2
13,m 3
A3 ,L
3
(3.15)
where the C~m are the coefficients of an ODF expressed as a generalized Fourier
m
are the coefficients
series in the basis of hyperspherical harmonics, the A "
of the hyperspherical harmonic addition theorem [271 as defined in E.1, and the
Ln3' 1 3,M3
are the coefficients of the hyperspherical harmonic linearization theo-
rem defined in E.2.
Consider all possible simultaneous values that the tn3 ,A 2 3 423 may take. This represents a subset of m(, which I denote i53, and whose size, relative to m provides
an estimate for how much of the grain boundary network design space is accessible
through the control of texture alone. In order to measure Hin, I must first introduce the concepts of the texture set and texture hull4 [6, 7, 14] and provide their
mathematical definitions in the context of the hyperspherical harmonic expansion.
As described by Mason [271, an ODF may be expressed as a linear combination
of hyperspherical harmonics:
00
(q)
=n
cmZn
m(q)
n
(3.16)
n=0,2,... 1=0 m=-l
3 The form of Eq. 3.15 is somewhat different than what is found in Eq. 2.15, though the two
are equivalent. The form given in Eq. 2.15 is more efficient computationally, while Eq. 3.15 of the
present chapter is more compact from a notational perspective.
4 These concepts were first introduced by Adams et al. [6] as a specific case of the more general
concepts of microstructure set and microstructure hull.
49
where the coefficients, cm, correspond to those appearing in Eq. 3.15. The texture
set, which I denote as m(), consists of the Fourier representation of all possible single
crystal textures. For the case of materials with triclinic crystal and sample symmetry,
mM can be expressed as:
Ms= {c
c=
=zT~m(i)
qcfm
, q E SO(3),
1ic
_1,...) ,
j
E 1 ,J]
(3.17)
The texture hull, m., which delineates the space of all possible crystallographic
textures, is simply the convex hull of m():
CE, p,p=
1(3.18)
m(
=
C
_
j=1
Jj=1
)
Having defined mQ, I use the following
procedure to quantify the size of inH3
H 7
relative to m (. First, I generate S samples, cs, s C [1, S], uniformly distributed
within ml. I then use Eq. 3.15 to compute ts for each of the ODFs so generated.
The convex hull of all of the Ts is then an approximation of in)
Due to the computational cost of defining high-dimensional convex hulls and generating uniform samples over them, I consider only terms up to max(ni , n3) = 2.
In this case there are six independent non-degenerate dimensions for m(, as can
be verified through the use of Eqs. D.4 and D.11. They are: Re(t2'8O'), Re (t2',28),
Re(t2',2,), Im(t2'o'o), Re(t2,22), Im(t2:2',). According to Eq. 3.15, they collectively
depend only on the following five ODF coefficients: c 0 , c2, 0 , c ,
c
, Because
(2r2)-/ 2 is a constant, and c' 0 is strictly real for even 1, by virtue of the reality
condition for ODFs (Eq. D.3), I am only concerned with the orthogonal projection of
c8,o
(1) defined by the following six dimensions: Re(c2,%), Re(c2,O), Re(c2, 1 ), Im(c2, 1 ),
Re (c2,2), I
2c,2)-
given in Eq. 3.18, to
I numerically converted the vertex representation of m
its equivalent half-space (or facet) representation by means of Jacobson's vert2lcon.m
code [103]. Subsequently, I used the hit-and-run algorithm [see 104, pp. 40-441,
as implemented in Benham's cprnd.m [1051, to sample S = 106 points uniformly
within the convex pol tope defined by the above-mentioned six-dimensional orthogonal projection of my . I performed all computations using MATLAB 1106]. A
three-dimensional orthogonal projection of m) and the samples that were generated
are shown in Fig. 3-5a. I then computed the t, from Eq. 3.15, which are shown, together with their convex hull, in Fig. 3-5b. Let pd/Pd -- i
/pH)(mI)
denote
the ratio of the d-dimensional measures of F() and M(. With 110 repetitions of this
process, I found 6 /P 6 = 0.0107 with a standard deviation of 0.0004 (see Fig. 3-6).
This is to say that, restricting our attention to the six lowest order dimensions, only
about 1% of all possible TJDFs are accessible in the absence of spatial correlations
in grain orientation. One should note, however, that the small relative size of i-( is
50
(b)
(a)
M,
0 06
03
U2,
0.04,
0.2
d
0 02.
01
0
0
z
,,
C4
-0.2
-004,.
0.3
_&OS
0
OD6
\1
0,04
.2
.
-0
1
2
2
0
006
0.02
01 02 03
O.G4
0
0,02
0
-002
-0.3
1_
( C22.2)
-0.3
0
Re
221
R t0:00
( C 22)
-GO4 -006
_0D6
-0.04
-0,02
Re
( t0::0
Fig. 3-5: In (a) The green surface contains a three-dimensional orthogonal projection of mO, and
the points are samples, c,, taken uniformly from the six-dimensional orthogonal projection of m,
as defined in Section 3.4. In (b) the red surface contains a three-dimensional orthogonal projection
,,the blue surface corresponds to
Eq. 3.15.
of m
n(, and the points are the t, obtained from the
c, via
largely a result of dimensionality. As is aparent from Fig. 3-5b, for a given dimension,
13 covers roughly half the range of mH ; more specifically,
i/Pi
= 0.4193 0.0444,
averaged over all six dimensions and all trials. The quantity p 1/pi may be regarded
as the average probability that a given coefficient from a randomly selected TJDF
I 1 - c for arbitrarily small
will lie within the bounds of (). Even if I had p/
e > 0, so long as the number of dimensions for which ii/pi < 1 is not finite, then
limd,,,z
d/Pd
=0. So, indeed,
3)
is smaller than M(3), however, how much smaller
depends upon how many dimensions one considers, and, since in all practical cases one
will truncate the series defining T(q 12 , q 23 ) to exclude the less important higher-order
terms, this limitation is at least partially ameliorated.
is concentrated
Another salient point that can be observed in Fig. 3-5b is that iabout the origin. In this space, the origin corresponds to the uniform TJDF, in which
all types of triple junction are equally likely, and for which the only non-zero coefficient
= (27r2)-1. This means that the TJDFs that are accessible in the absence of
is to
spatial correlations correspond to somewhat less exotic TJDFs than those that are
possible with spatial correlations.
51
30F
25-
20&15-
105-
0
0.0095
0.01
0.0105
0.011
0.0115
g6/iP6
Fig. 3-6: Histogram of the ratio of the six-dimensional measures of
standard deviation are also plotted.
3.5
(
and m(. The mean and
Conclusions
.
The character and connectivity of grain boundaries in polycrystalline materials controls a host of materials properties and phenomena. Empirical evidence from grain
boundary engineering (GBE) studies suggests the possibility of enhancing materials
properties through manipulation of the grain boundary network in strategic ways. At
present, however, general understanding of grain boundary network structure and its
effect on materials properties is limited. This has precluded the inductive design of
grain boundary networks.
In this chapter I have highlighted the intrinsically multi-scale nature of grain
boundary network design. I identified triple junctions as convenient topological features corresponding to a local length scale. I defined the local state space for triple
junctions and provided the equivalence relations that define its fundamental zone,
.3).I also defined the global configuration space for grain boundary networks,
and demonstrated its convexity. These tools allow one to explore the complete
.),
universe of grain boundary networks in search of application specific globally optimal network configurations. The importance of these tools is further highlighted by
the fact that existing GBE processing methods are only effective for materials that
twin readily. The prospect of inductive grain boundary network design suggests the
opportunity to achieve unprecedented properties enhancements in materials that are
not amenable to traditional GBE.
Additionally, I investigated the relative size of the portion of the grain boundary
network design space that is accessible in the absence of spatial correlations, m
I found Fn
to be a small fraction of the size of m(.
The implications of these
results are significant: the control and manipulation of crystallographic texture alone
is insufficient to access the vast majority of the grain boundary network design space.
Furthermore, spatial correlations of grain orientation expand this space and are re52
quired to reach the more extreme grain boundary network configurations. At the
same time, the fact that higher-order terms in the Fourier series of a TJDF are decreasingly important, indicates that Fin is of more significance than first impressions
might suggest.
53
54
Chapter 4
Grain Boundary Network Design
4.1
Design Strategy
Numerous experimental studies have demonstrated the possibility of improving the
performance of materials by manipulating the structure of the grain boundary net-
work f36, 44, 46, 47, 97]. However, this has been done largely by trial and error. I
wish to develop a theoretical framework that allows one to design the grain boundary
network structure of a polycrystal, not just to improve its performance with respect to
a single property, but that allows the optimization of the microstructure for complex
multi-objective design problems. Our approach will build upon the general spectral
methodology that has been used in the past for texture sensitive properties [71, expanding it to comprehend grain boundary networks and their properties. First, I
will define the design problem and specify the design constraints and objectives. I
will then develop relevant structure-properties models. Using these models and the
microstructure hulls described in Chapter 3, I will construct the space of all possible combinations of the properties relevant to the design problem and identify the
microstructure that optimally satisfies the design objectives.
4.2
Design Problem
Consider microstructure sensitive design [6, 7] of metallic interconnects for flexible
electronics. In this design problem I will assume that pure Aluminum has been
selected as the material. Chemistry and geometry will be held constant and I will
seek to optimize the microstructure of a polycrystalline sample in order to satisfy
certain design constraints and objectives.
Because of the cyclic loading characteristic to this application, fatigue failure is a
concern. To design against fatigue failure I will seek a microstructure that maximizes
the endurance limit of the material, ce. The endurance limit is the maximum stress
amplitude that the material can sustain indefinitely without failure [107]. In the
case of Al and some other materials, which do not exhibit a true plateau in the
stress amplitude vs. cycles to failure plot, -e is operationally defined as the stress
amplitude that the material can sustain for some large number of cycles (e.g. 107)
55
without failure 1107]. The endurance limit can be approximated as some fraction of
the tensile strength of a material, for Al a, ~ 0. 3 UUTS [1071. The OUTS and yield
strength, ay, are strongly correlated [108], therefore, I can use ay as a proxy for or
and design against fatigue failure by maximizing oy. Because o%
CUTS this will be a
conservative estimate. For this design problem I will focus on a simplified scenario in
which the candidate material is loaded uniaxially in the x-direction, thus, I desire to
maximize o0i, where the subscript 1 indicates the x-direction and the overbar denotes
an effective macroscopic property of the polycrystal.
Another mechanical property relevant to the flexibility of our circuit is elastic
compliance. I will require that the elastic constant Sill, of our candidate interconnect
material match that of the substrate to which it will be applied.
The electronic nature of this application presents another failure mode to design
against: electromigration. Metallic interconnects are known to gradually erode due
to current induced atomic diffusion, leading to open circuits and device failure. Electromigration often occurs via diffusion along grain boundaries, which can act as fast
diffusion pathways. In order to combat this failure mechanism, I will seek to minimize
the effective diffusivity, D, of the grain boundary network of our candidate material.
The design objectives for this study are summarized below:
max o-
(4. 1a)
S111= S
(4. 1b)
(4.1c)
minD
Microstructure sensitive models for Sill, and o-i are readily available. Below, I explain the application of each of these properties models to the present design problem.
I also develop a predictive model for D that depends on the types and populations
of grain boundaries and triple junctions present in the polycrystal.
4.2.1
Yield
The effective macroscopic yield strength of our polycrystalline material can be approximated using the model developed by Sachs [26, 1091, according to:
Gryl r
TCRSS
(
Ib-() c'q
max~bi(q)nl(q))
where TCRSS is the critically resolved shear stress (0.79 MPa for Al 11101), bo(q) and
nc(q) are the x-components of the slip direction and slip plane normal, respectively, of
the a-th slip system-both of which are functions of the crystallographic orientation, q,
in a given grain. The overbar indicates a volume average over all crystal orientations
in the polycrystal. This can be expressed explicitly as
oyl ~TTCRSS sfI
0"l
(q)( (
56
xbqri())(4.3)
x
c'q
maxbq
nq|
4-()
where the quaternion q is an orientation, f(q) is an orientation distribution function
(ODF), and f again represents integration over the entire domain of the independent
variable q, which, in this case is S 3 , with respect to the appropriate invariant measure:
Isin 2 (w/2) sin OdwdOd.
Substituting the hyperspherical harmonic representation of an ODF, Eq. 1.3, into
Eq.4.3 and integrating results in
*,l0,m
n
where
[0y1Jm
=CRSSJZjlrn(q) (m xIbq)r"(q)
max b~)n~)
(4.5)
Eq. 4.4 is in the form of Eq. 1.6, which is significant. In this expression the microstructural information, as encoded in the ODF coefficients Cn, has been decoupled from
the details of the physics, as encoded in the properties coefficients [0y1]"m. The use
of this spectral framework permits such a decoupling and facilitates the fast computation of the effective properties, 0 y- in this case. This is because the properties
coefficients [a yI] 7 m only have to be computed once, and then Fyl may be evaluated for
any microstructure (Cm) using the computationally efficient inner-product operation
of Eq. 4.4.
4.2.2
Elastic Compliance
Woldemar Voigt proposed a model [15, 161 for the elastic behavior of crystalline
materials, which assumes that upon elastically loading a polycrystalline aggregate, all
grains experience the same elastic strain. This assumption does not satisfy equilibrium
constraints, but permits the derivation of a simple model that provides an estimate
for the relevant constants of the elastic compliance tensor. This estimate is an upperbound and, for the specific case at hand, can be expressed as:
Si1
=
f(q) Su1
(q)
(4.6)
where the orientation dependent function Siiii(q) is defined by
S1i11(q)= Qia(q) Qb(q(q)c(q) Qic(q) Sabcd
(4.7)
with Qj 3(q) being one of the components of a direction cosine matrix and the Einstein summation convention implied for repeated indices. Eq. 4.7 may be evaluated
in closed form, but it is lengthy and will be omitted here. Its evaluation requires
the values of the independent single crystal elastic constants, which are provided in
Table 4.1 for Al.
Again expressing the ODF appearing in Eq. 4.6 in spectral form, and integrating
I arrive at a spectral expression for the Voigt estimate of the elastic compliance of a
57
S1111
S1122
S1212
15.851
-5.7285
35.286
Table 4.1: Numerical values of the independent elastic constants for Al [1111 appearing in Eq. 4.7.
All values are in units of 10-12 Pa-'.
polycrystal:
Sn
(
c"n
[un "
(4.8)
with
[
4.2.3
1n,
=
nZi(q)
S,I(q)
(4.9)
Grain Boundary Network Diffusivity
The diffusivities of individual grain boundaries can differ by orders of magnitude. This
strong contrast of grain boundary diffusivity precludes the use of effective medium
theory (EMT) models to predict the effective diffusivity of grain boundary networks
in polycrystals. McLachlan proposed a phenomenological model for the effective
electrical conductivity of binary mixtures of insulating and conducting phases [50,
112], which was adapted to the analogous case of grain boundary network diffusivity
by Chen and Schuh 151]:
Pi
J0D - (215)1/s
D+ (p-- - 1) (2D5)'s
P2
D
- (2D) I/t .
D'It + (p-' - 1) (25)1t
0
(4.10)
In this expression pi and P2 are the fraction of low- and high-angle grain boundaries,
respectively, and D < D 2 are the corresponding diffusivities. The exponents are
constants that depend only on the dimensionality of the problem and in this work
are taken as s = 1.09 and t = 1.13. The spatial distribution of low- and highangle grain boundaries in real materials is manifestly non-random and the percolation
threshold for high-angle grain boundaries, Pc,2, is sensitive to these correlations in
the grain boundary network, which result from crystallographic constraints, texture,
etc. These correlations are known to be short-range [49] and can be quantified by the
triple junction fractions, {Ji I Z E [0, 3]}, which measure the fraction of triple junctions
coordinated by i "special" grain boundaries (low-angle in the present context). Frary
and Schuh found an empirical relation that predicts the percolation threshold1 as a
function of the Ji [621, the details of which are given in Section 5.3.3. This relation
allows us to write Pc,2 = pc,2(JO, J1, J2 , J3).
The triple junction fractions can be interpreted as the average probability of observing a triple junction coordinated by i "special" grain boundaries in a given mi'The expressions given in [62] were for pc,1, but can be used to obtain
Eq. 7b of [62] with its reciprocal.
58
Pc,2
by simply replacing
crostructure. Considering low-angle grain boundaries as "special", this interpretation
allows us to compute the Ji by integration of the triple junction distribution function
(TJDF) (see Chapters 2-3) according to:
j
Ji
T(q 1 2 ,
q2 3 )
dQi
(4.11)
In Eq. 4.11, Qi is the appropriate integration region as defined by:
Qo = {(0 12 , W 2 3 ) |
Q1
=
> Wt,
{(w 12 , W23 ) | '12
Wt,
U {(W1 2 ,W 23 )
> Wt, W23
U
Q2
3
=
> Wt,
12
U)12
{(1 2 , W23 ) | J 12 >
W23
Wt)-23 >Wj,
Wt, C23
W3 1
> W}
W
31 >Wt}
Wt,
>
-t,
'31
> W}
W1 2
Wt, U
Wt, L)2 3
(4.12b)
LJ31 < Wt}
{(W1 2 ,W 23 ) I w12 > Wt, 23 < Wt, Ch.31 Lot}
Wt, W23 > wt, W3 1 < Lot}
U {(w 1 2 ,W 23 ) I C 12
U {(W 12 ,W 23 ) I C1 2 < Wt, C;23 < Wt, &C31 >Wt}
{(W12, U23) I
(4.12a)
31
< ot}
(4.12c)
(4.12d)
where wt is the angular threshold between low- and high-angle grain boundaries,
which, based on diffusivity data for Al [113, pg. 1221, I estimate to be wt = 20'. In
Eq. 4.12, Co2 is the rotation angle of the disorientation 4ij (6Zi, ij, qii ) corresponding
to the misorientation qij (ij , Oij, ij). The disorientation, 4ij, is the unique misorientation from among all of those symmetrically equivalent to qij that (1) has the smallest
rotation angle, W12 , and (2) has a rotation axis lying in the standard stereographic
triangle. Uniqueness is guaranteed by enforcing the supplementary conditions for
"reduced rotations" given by Grimmer [114].
Substituting the spectral form of the TJDF (Eq. 3.10) into Eq. 4.11 I obtain
J.
tn3'AJ]3'
R1Al2,/L2
n
(4.13)
,",
[J ln2 ,A12,/-12
n 1 ,A 1 2,4L 2
3 ,A 2 3, 1 2 3
J" ' 12 may be
the
nr pre-computed,
t
and is defined by
[J
n]3 'A23
=(q32) Z37 3 (q23)dQi
(4.14)
Using the relationship between the Fourier and Dirac coefficients (Eq. 3.12), Eq. 4.13
can alternatively be expressed as
K
J = ZPkJik
k=1
59
(4.15)
with
['J I]
_'l~
-
ktn3,A23,123
niA12, P12
1i
n423623
(4.16)
n1,A12,112
n1,A12,1112
n 3 ,A 2 3 ,23
Comparison of Eq. 4.16 with Eq. 4.13 reveals that the coefficient [Jilk in the Dirac
basis represents the triple junction fraction of the k-th element of the triple junction
set, M(3). This observation is significant and dramatically reduces the computational
effort required to compute the Ji. Because each element,
(kq
12
, kq 2 3 ),
of M( 3 ) cor-
responds to a microstructure in which all of the triple junctions are identical, [Jilk
can be determined simply by characterizing the number of "special" boundaries coordinating (kq 1 2 , kq2 3 ). To illustrate this idea let (kq 1 2 , kq 2 3 ) be the k-th element of
M . If* 1 2 < Wt,
<wt, and kJ3 1 < wt then (kq1 2 , q2 3 ) has all low-angle grain
boundaries and is a J3 -type triple junction. I, therefore, have [Jo]k = 0, [J1]k = 0,
[J2lk
0, and [J3lk = 1. Thus, the [Ji]k take the values 0 or 1 and may be computed
directly, without the aid of Eqs. 4.14 and 4.16 and their computationally expensive
integration, simply by characterizing the triple junction types of M , which need
only be performed once. The J, for any microstructure can then be computed using
Eq. 4.15, which simply represents a weighted average of the [Ji]k where the TJDF
coefficients in the Dirac basis, Pk, are the weights and represent the probability of
observing the triple junction (kq 12 , kq 23 ) in that microstructure.
While Eq. 4.10 has no closed form analytic solution, it can be solved numerically to give D(p 2 ,pc,2 , Di, D 2 ,). Using Frary's Pc,2 = pc,2 (Jo, Ji, J2 , J 3 ) together with
Eq. 4.15, and recalling that P2
1 - IJi J2 - J3 (Eqs. 5.7-5.8), I can now express
the effective grain boundary network diffusivity in a similar fashion to what I did for
yield strength (Eq. 4.4) and compliance (Eq. 4.8): as a function of the local properties
(Di and D 2 ) and the microstructure (through the TJDF coefficients Pk), according
to
D = D(pk, D 1 , D2)
(4.17)
For pure Al, I use D 2 = 10-12 m 2 /s, which I estimate from the maximum calculated
grain boundary diffusivity for (001) tilt grain boundaries at 250'C with diffusion
perpendicular to (001) [1151. For the low-angle diffusivity I extrapolated the data
in 1115] to a tilt angle of 0 = 0' to find D, 4.5812 x 10-2 m 2 /s.
4.3
Properties Closure
With constitutive models for the materials properties relevant to our design problem,
expressed in spectral form, I am now in a position to define the properties closure [7].
Consider a space in which a microstructure is represented by a point, whose coordinates, (ug1 , S1111, ), indicate its materials properties. The properties closure is the
closed region in this space containing all physically possible combinations of these
60
three properties. It can be defined mathematically by
{ (Uyi),s
D
1 ,D)
-yl
=
S
C
o{yi])m,
Si,
E
C1Mr[Siill'm,
n~l~mn,l,m
D = D(pk,
D, D 2 ), {CHm
E mfk, {p} E
(4.18)
, {C-m
{PkJf
where the last expression, {cm} -+ {p}, indicates a relationship between the coeffi-+ M} , which will be defined
cients that is a result of a mapping from mH <below. As the microstructure hull contains all possible microstructural configurations
(see Sections 3.3-3.4), the properties closure is constructed by exercising our structure properties models (Eqs. 4.4, 4.8, and 4.17) over the entirety of the appropriate
microstructure hull. While this is conceptually straightforward-if you measure the
properties over the space of all possible microstructures you will obtain the space of
all possible properties-in practice it involves a number of steps:
Step 1: Sample ODFs from M2l)
Step 2: Map from M H M MH
Step 3: Compute texture dependent properties: cry and Sill
Step 4: Map from Ml - M- to obtain the TJDF corresponding to each ODF
Step 5: Compute grain boundary network sensitive properties: D
Step 6: Define the boundary of _
This process is graphically summarized in Fig. 4-1. In what follows, each of these
steps will be explained in detail.
4.3.1
Step 1: Sample ODFs from M'H
To construct the properties closure, I take a stochastic approach and sample many
microstructures from the appropriate microstructure hull, compute the relevant materials properties of each, and define O as the region in properties space that bounds
all of the resulting points. I wish to sample microstructures in such a way that the
resulting properties closure is maximal. In other words, I want to choose microstructures that will explore as much as possible of the true properties closure.
There are two microstructure hulls relevant to the present design problem:
&(l)
and _W . In light of Eq. 3.15, relating the texture hull to the triple junction hull, I
and subsequently map these points to _ 3)
choose to sample from /f
Because our texture dependent constitutive relations are formulated in the Fourier
basis it seems natural to sample ODFs from m(). The texture hull, m
, is defined
as the convex hull of the texture set, ml. For all practical purposes m() will be
finite, consequently, mH will be a convex polytope, which can be sampled uniformly
using a hit-and-run algorithm as discussed in Section 3.4. However, there is a major
drawback to this method. To sample from m() you must know its boundary, which
61
Step 1
-----------------------------
Step 4
MN .M'
~
H~
/
(3
Step 2
M)1
Step 3
Step 5
UY
S 111 1
Step 6.Fig. 4-1: Graphical summary of the process of constructing the properties closure.
requires the computation of a convex hull. Constitutive equations for all but the
simplest properties can require a large number of coefficients, cm,, and consequently
mM will exist in a high-dimensional space. The computational cost of constructing a
convex hull scales exponentially with the dimension 11161, making this approach too
computationally expensive for practical use.
In the Dirac basis the texture hull, M(, has a far simpler geometry: it is the
standard (J - 1)-simplex, where J is the cardinality of M ). This fact obviates the
need to explicitly compute a convex hull, and also makes the procedure for sampling
far simpler. Sampling uniformly from a standard (J - 1)-simplex can be accomplished
by the following procedure [117]:
1. Generate J uniformly distributed random variables X 1 , X 2 ,..., X 1
2. Transform them to exponentially distributed random variables via Y = In Xi
J=l Y
3. Perform the transformation P' = Yi/
The vector (Pj, Ps,...,Pj) will then be uniformly distributed over the standard
(J - 1)-simplex. While this sampling method is computationally inexpensive, it does
not result in points that adequately explore the properties closure (Fig. 4-2). The reason for this is that the boundary of the simplex (faces, edges, etc.) has measure 0 in J
dimensions, and, consequently, is never sampled. Because all of the points come from
the interior of the simplex they will be a mixture of all J elements of M , and thus
do not differ sufficiently from the uniform ODF, which is located at the barycenter
of the simplex. However, this observation suggests an alternative approach.
62
(see
set
texture
element
j-th
the
of
coefficients
Fourier
are
E
PI
{C}
Recognizing that the faces, edges, etc. of a (J - 1)-simplex are just lower-dimensional
simplices themselves, the simplex sampling approach may be performed in a hierarchical fashion, starting with the vertices (0-faces), then the edges (1-faces), faces
(2-faces), and continuing with the hyper-faces of increasing dimension. Let NS be the
desired number of samples. Our hierarchical simplex sampling algorithm is summarized in the following steps:
1. Take all J of the vertices (0-faces)
2. Take max(1, LNs x 10-']) points sampled uniformly from the (J - 1)-face (the
simplex interior)
3. Beginning with j 2 and proceeding to increasing j, take [N,(J - j)/ ZJ _]
points sampled uniformly from the standard j-simplex and assign them uniformly across all j-faces
In this procedure, Lx] is the rounding operation, and N, is the remaining number
of samples, yet to be taken. In Step 3 I use the heuristic of taking a number of
samples equal to N,(J - j)/ EJ- ] for the simplices of level j. This heuristic was
optimized experimentally. This method of hierarchical sampling retains much of the
computational efficiency of the uniform simplex sampling approach, but probes far
more of the properties closure (see Fig. 4-2). For the present design problem I use
this hierarchical sampling algorithm to sample Ns
107 textures from M(1).
will
denote this set of sampled textures
S = p'
p" = {p },, j E [1, J], s E [1, Ns]}
(4.19)
where p' is the s-th sample, and can be interpreted as a vector of coefficients
(pip2 - ,)T. In this way S, which is a set of sets, can be interpreted as a matrix
where Sj, is the j-th coefficient of the s-th sample.
4.3.2
i
Step 2: Map from M(1)
MH
TrH
Because I sampled texture coefficients, p', in the Dirac basis and our models for ay,
( in
'(1)
and S1111 are in the Fourier basis, I must perform a transformation
order to obtain the Fourier texture coefficients c' that our models require. This
transformation is accomplished by means of the following formula:
J
=pic
Z[
j=1
(4.20)
which is the texture analog to Eq. 3.12, originally developed for the TJDF coefficients.
Its derivation is identical to that of Eq. 3.12 and, therefore, omitted. In Eq. 4.20,
1
Eq. 3.17).
63
Xo1
1
4.5 -Ai
4,5
4
3.5
z 3
2.5
2
1.7
1.8
1.9
*V
Standard Simplex Sampling
*
Hierarchical Simplex Sampling
2
o'---
2.1
2.2
(Pa)
2.3
2.4
xlo6
1 x 10 3 . Points resulting
Fig. 4-2: Comparison of microstructural sampling techniques for Ns
from uniform sampling of the standard simplex are shown as red markers. Points resulting from the
hierarchical simplex sampling technique are shown as green markers. The purple shaded region is the
convex hull of 1 x 10' samples taken using the hierarchical technique and is shown for comparison,
) for the texture dependent properties
being a higher order approximation of the true Y(o'i,
.
i-y-, and S 111
4.3.3
Step 3: Compute texture dependent properties
With Dirac ODF coefficients sampled and transformed to the Fourier basis, the texture dependent properties can now be directly evaluated for each sampled texture
by means of Eqs. 4.4 and 4.8. For each texture I now have the two coordinates
(-(7, S1 1 11 ) in properties space that are shown in Fig. 4-2.
4.3.4
Step 4: Map from M(1) --+ M
In the present design problem I am using texture dependent constitutive equations
for 0 'y- and S 11 11 , and a grain boundary network sensitive model for D. To construct
a properties closure that is self-consistent, I must, therefore, be able to identify the
grain boundary network configuration corresponding to a given texture.
In general the mapping between texture and grain boundary network configura- /& , is many-to-many. For a given texture, different spatial arrangeH4j
tion,
64
ments of the grains will result in distinct grain boundary network configurations
(TJDFs). Likewise, a given grain boundary network configuration may be obtained
from many different textures. For example, all textures that differ only by a rotation,
r, e.g. f(q) and f(qr), will yield the same TJDF (this also implies that, given a TJDF,
it is only possible to recover the corresponding ODF modulo a rotation operation).
However, under the assumption that grain orientations are spatially uncorrelated I
obtain the mapping &(1) -+ _ , for which every ODF maps to only one TJDF.
Taking advantage of this fact, I can take the texture coefficients, p', sampled from
M(1) and map them to M( to obtain the corresponding TJDF coefficients, Pk. A
formula for this mapping, developed in the context of a Fourier basis, was given in
Eq. 3.15. In the Dirac basis, this mapping takes a particularly simple form and is
given by:
(4.21)
PaPbPc
k
(a,b,c)CEEk
While simple in form, Eq. 4.21 deserves some explanation. The triple junction set,
M , effectively discretizes the triple junction space into bins, where the k-th bin is
centered on the triple junction (kq 1 2 , kq2 3 ). The probability of observing a triple junction belonging to the k-th bin in a given microstructure is given by Pk. Eq. 4.21 says
that, in the absence of spatial correlations in grain orientation, this probability is given
by the joint probability of observing the triplet of grain orientations (qa, qb, qc) summed
over all grain orientation triplets belonging to the equivalence class of (kq 1 2 , kq2 3 ), denoted by Ek. The equivalence class Ek is defined by the equivalence relations induced
by both crystallographic symmetry and triple junction symmetry (Eq. 3.4).
While, strictly speaking, ///') and W(3) may be defined independently of one
another, the use of Eq. 4.21 is computationally facilitated by constructing them in
such a way that they are related. Specifically, I begin by defining
1) by the
1
crystal orientations { q,
... I q} where J is the cardinality of ,Wil) and defines its
resolution. For the present design problem I defined -Ws1) using a multilevel pseudogrid over the cubic fundamental zone composed of 10 equispaced primary orientations
each with 6 secondary orientations being defined as rotations of 100 along the six
(100)-type axes relative to the respective primary orientation. Thus, for the present
design problem I have J = 70. A pole figure of the orientations defining d 1 is
shown in Fig. 4-3.
A triple junction may be defined by the ordered triplet of orientations coordinating
and the subscripts
(aqi, bq 2 , cq 3 ), where the superscripts are an index into a
denote the grain to which the orientation belongs (see Fig. 2-2). I can define the set
of all possible triple junctions that can be constructed using the elements of A'):
it:
{ (1qi,
'q 2
,'
, (1 qq
2
2
Q
( q, q 2 ,q
3
) , - - , ( qi,
q2 , q 3 )}
(4.22)
which has cardinality J3. This can then be expressed in terms of misorientations
65
Fig. 4-3: A {100} pole figure showing the orientations used to define
.
{100}
according to
' ' 1) 12, (1,2)q 3 I .(1,2 )ql1 2,3')q23) , 1 -- ,
q12, (Ji923)
((,)
121,1q 3
1
(4.23)
Let W denote a crystallographic point group, and ci E V one of the rotational symmetry operators belonging to W.
Crystallographic symmetry induces equivalence
relations on orientations according to q
gciq V ci Ce. I denote the set of all
such equivalence relations by C. In addition to crystallographic symmetry there are
distinct triple junction symmetries as described in Section 3.2, which induce triple
junction equivalence relations. The set of all triple junction equivalence relations is
given explicitly by Eq. 3.4 and is denoted by F. I next find the K equivalence classes,
{Ek I k c [1, K]}, of Eq. 4.23 that are induced by the combined application of C and
F. From each Ek I select one unique representative and denote it (kq 1 2 , kq 2 3 ) and
. In this way 40 is
this becomes the k-th element of the triple junction set,
composed of the unique triple junctions that may be constructed exclusively from the
orientations in a
0. Having constructed -///3) in this way, I already have the Ek nec-
essary to evaluate Eq. 4.21, and thereby compute the TJDF coefficients corresponding
to a given ODF in the Dirac basis.
4.3.5
Step 5: Compute grain boundary network properties
Using Eq. 4.21 I can now find the TJDF coefficients in the Dirac basis for each of the
textures sampled from M(1. Using Eq. 4.17 the grain boundary network sensitive
property D can now be determined for each sampled point and I have the triplet of
properties coordinates (aS, S 111 , D). The properties space coordinates of all of the
sampled textures can now be plotted as shown in Fig. 4-4.
66
-13\
-14
-15
-19
10
4
X 10
511U-1)
2
17
18
19
2.1
2
dy
2.2
2.3
2.4
(Pa)
7
Fig. 4-4: Properties space representation of each of the sampled textures. Only half of the 10 points
are shown to reduce visual clutter.
4.3.6
Step 6: Define the boundary of 1?
I wish to define -, which is an envelope in this space containing all physically possible
properties combinations under the assumptions of the constitutive models that I used
and the condition of uncorrelated grain orientations. The property of convexity, which
and ji)'(, does not extend to
is guaranteed for microstructure hulls, such as
properties closures [7]. This is particularly true for properties closures involving nonlinear constitutive relations.
The general problem of defining the non-convex envelope of a set of points has no
unique solution. A number of methods have been developed (see the review provided
by [118] and references contained therein), each making certain assumptions that
induce additional constraints necessary to find a solution. One method that is often
employed, and which I will use, is that of finding an a-shape 1119, 120]. Let S be a
set of points in R3. Consider a ball, B, of radius a E [0, oc). When a = 0, B is a
point, and as a -* oc it becomes a half-space. The a-hull is defined as the region of
R 3 through which B cannot be translated without enclosing any of the points in S.
To give a more intuitive picture, consider a region R C R'. Initially let R = R3 and
imagine that B is translated throughout the entirety of R 3 . Anywhere that B can be
placed without enclosing any points in S is removed from R. What remains of R after
this exercise will be a closed, but not-necessarily convex or connected, region called
the a-hull of S. By replacing the curved boundaries of the a-hull with flat planes,
.&/4I)
67
I recover the polytope referred to as the a-shape. The a shape is a generalization
of the convex hull, and when a = oc the convex hull is recovered. Fig. 4-5 shows
a-shapes with various values of a for a two-dimensional point cloud.
I.~iiia"
a=10
a=5
4=.,-EW
- NG
Wl
4qWW
a=o
4..
MON A551
0I
Fig. 4-5: a-shapes for a point cloud with various values of the parameter a. The figure to the far
right with a = oc shows the convex hull of the point cloud.
Applying the a-shape algorithm to the point cloud shown in Fig. 4-4 yields our
desired result: _, which is shown in Fig. 4-6. Because of the values of ayi, Sili,, and
5 differ
significantly, choosing an appropriate value of a is difficult. To address this
the coordinates were normalized according to the following transformations:
_
oy1 - (max (-1) + min (-yj 1 ))
max (5-1) - min (-y-1)
(max (S71) + min (SR11))
max (S111) - min (S7111
-
(4.24b)
)
S 111
(4.24a)
In
5
(max (in D) + min (in
max (in D) - min (in 5)
-
1
))
(4.24c)
where max (gi) denotes the maximum observed value of 7-, among all of the samples,
etc. In the space of (x, y, z) a parameter value of a = 0.75 was found to be satisfactory.
68
-13,
-15
-- 175
-16s
-190
4
91111
(pa-1)
108
xx
3
X 10
2
8
.
2.1
92
2Z2
23
2.4
u,1 (Pa)
Fig. 4-6: The properties closure, 2, for the properties o&i, S1 1 1I, and D as approximated using the
procedures outlined in Section 4.3.
4.4
Solving the Design Problem
The objectives of the present design problem are given in Eq. 4.1. The constraint
S1 11
= SIjI given in Eq. 4.1b may be represented in the properties space as a plane,
The feasible
as illustrated in Fig. 4-7, where I take S,, = 2.5665 x 10-11 Pa'.
region of our design problem lies on the intersection of this plane and O, and is
shown in Fig. 4-8. In order to satisfy Eqs. 4.1a and 4.1c, I can define a composite
objective function to minimize:
fo
x +
2 Zin
2 |Xmin|
(4.25)
where x and z are the normalized coordinates of - and D, respectively, as defined in Eq. 4.24. The theoretical minimum values of --i and D are, respectively,
D 1 /2. The terms xmin and zmin, appearing in Eq. 4.25,
(Ui)min = 2 TCRSs and (
are the normalized coordinates of these values.
In the context of texture sensitive microstructure design, several methods have
been used to search 0 for an optimal design solution including both gradient-based 181
and stochastic methods [1211. Since I have already sampled many microstructures
to generate Y I leverage this expended effort and take the solution to the design
69
-13s
-14
-16
---
-17\
-18 1
_
"
-19
>(
10*1
4
1
111(Pa- )
X 10(
I
1.
2
2.1
2
1.9
18
Y
2.2
2.3
2.4
(Pa)
I = S1 1 with Y, for S11 = 2.5665 x 10-" Pa-.
This design constraint is represented as the semi-transparent vertical plane.
Fig. 4-7: Intersection of the design constraint S
problem to be
(4.26)
min fobj
where
M
(
, SiI,
D)
S
(s1111)
H l
-
(s1
<
10-6
(4.27)
)m
.
is the region of properties space within some tolerance of the plane S 111 2.5665 x
10-11 Pa- 1
In Fig. 4-8, the contours of f0 j are plotted over the feasible region. The optimal
microstructure (black dot) lies at the bottom tip of the feasible region. The corresponding texture is composed of roughly a 60-40 mixture of two texture components,
which share a common rotation axis that lies in the plane of the material and is
orthogonal to the loading direction. The worst performing microstructure has a texture that is reminiscent of a weak fiber texture with the shared rotation axis parallel
to the loading direction. An isotropic microstructure possessing no texture (i.e. a
uniform ODF) would not satisfy the design constraint on Sil, but it is also plotted
in Fig. 4-8 for comparison as it exhibits the properties that might be expected for an
"undesigned" microstructure. The TJDF coefficients corresponding to the textures
shown in Fig. 4-8 were also obtained; however, since the TJDF is a six-dimensional
function its direct visualization is difficult. Instead, I have reconstructed simulated
70
0.4
0.6
0.8
1
fobj
1.2
1.4
1.6
1.8
2
-100
2.2
70
}
-13-
-14-
* */
50
40
C-1-17IQ
1 100}
30
0
-17-7
20
10
-19__
0
1.7
1.8
1.9
2
2.1
dy (Pa)
2.2
2.3
2.4
x 10
Fig. 4-8: The feasible region for the design problem. Contours of the objective function, fobs, as
defined in Eq. 4.25, are plotted over the feasible region. The best performing design solution is
shown (black dot) along with the worst performing solution (white dot) and the isotropic solution
(blue dot). The isotropic solution does not satisfy the design constraints and falls outside of the
feasible region. The textures of the best and worst solutions are also shown as { 100} pole figures
where the gray discrete points are the elements of M . The color scale for the texture components
is in multiples of the random distribution (MRD).
microstructures with hexagonal grains and assigned grain orientations in a spatially
-
uncorrelated fashion from each of the respective ODFs. In Fig. 4-9 the grain boundary
networks of each of these microstructures are shown with grain boundaries colored
according to their disorientation using the Patala coloring scheme [99, 1021, which
encodes both the rotation angle and axis of the disorientation. The crystallographic
structure of the grain boundary networks is dramatically different. Most notably, the
grain boundary networks of the isotropic and worst performing microstructures both
contain a large diversity of grain boundary types, including many high-angle (D 2
type) grain boundaries. In contrast, the best performing microstructure is composed
of only one type of grain boundary: a 20' rotation about (100), which is a low-angle
grain boundary. The light gray boundaries shown in Fig. 4-9 correspond to a disorientation of 00, which are not really grain boundaries, but are shown to illustrate the
simulated grain geometry. Thus, while the best performing microstructure is poly71
crystalline, and must be so in order to achieve the compliance and yield strength that
it does, it effectively behaves like a single-crystal with respect to diffusivity.
200
30
400
0 Best
0
4A
I--cL~U
%00
o0
0
06
C:'600
Fig. 4-9: The grain boundary networks corresponding to the best, worst, and isotropic design solutions shown in Fig. 4-8. Grain boundaries are colored according to their misorientation according to
the Patala coloring scheme [99, 1021. The color code on the right indicates the disorientation angle
of a misorientation as well as the rotation axis by its position in the relevant standard stereographic
triangle. The grain boundaries colored in light gray correspond to disorientations of 0' and therefore are not really grain boundaries, but are shown to illustrate the geometry of the grain boundary
network.
The optimal microstructure outperforms the worst performing microstructure and
the isotropic solution by seven orders of magnitude in terms of D. Median time
to electromigration failure (MTF) is inversely proportional to an Arrhenius term
involving the activation energy for atomic migration 1122]. Since diffusivity is directly
proportional to this same Arrhenius term, I have that MTF cx D-. Therefore, in
terms of resistance to electromigration the optimal solution is expected to have a MTF
that is roughly 10' times longer than that of an undesigned isotropic material. While
the isotropic solution is predicted to exhibit a yield strength that is slightly higher
than the optimal solution (by 3%), it has a much higher diffusivity and therefore
72
is significantly more vulnerable to electromigration failure. The optimal solution
outperforms the yield strength of the worst solution by 8.3%.
4.5
Conclusions
In this chapter I have considered the design of an Al microstructure intended to be
used in a flexible electronics application. In this context I sought to identify a crystallographic texture (as quantified by the ODF) and commensurate grain boundary
network structure (as quantified by the TJDF) that would simultaneously maximize
the effective yield strength of the polycrystalline material, minimize the grain boundary network diffusivity, and exhibit a specified elastic compliance. I developed a model
capable of predicting the effective diffusivity of a grain boundary network from the coefficients of its TJDF. Using established texture dependent models for yield strength
and elastic compliance, together with our newly developed model for grain boundary
network diffusivity, and the relationships between texture and grain boundary network structure developed in Chapters 2-3, I computed a properties closure, 2, for
Uy-, S 1111 , and D. This is the first properties closure that has ever been computed for
grain boundary network sensitive properties-or any defect sensitive property for that
matter-and represents the space of all properties combinations of - , SI , and D
that are predicted to be physically realizable under the assumptions of our constitutive models and the absence of spatially correlated grain orientations. The shape of ;
indicates that the properties of interest vary in a complex way with one another, similar to what has been observed in other contexts [8]. For the present design problem,
this resulted in competing design objectives. Using these tools I identified an optimal
texture and its corresponding grain boundary network and found that it significantly
outperformed both the worst case scenario as well as an isotropic microstructure. This
improvement confirms the enormous potential for materials properties enhancement
that is possible through grain boundary network design.
73
74
Chapter 5
Localization*
5.1
Introduction
The effective properties of polycrystalline materials are influenced, and in some cases
dominated, by the character and connectivity of grain boundaries. For example hydrogen embrittlement 11231, corrosion [441, creep 1461, weldability [471, and superconductivity [451 all depend strongly on the structure and spatial arrangement of grain
boundaries in the microstructure. Great effort has been invested by the scientific community to correlate grain boundary structure and properties, but simple constitutive
relations have been elusive.
One fundamental difficulty stems from the complex structure of grain boundaries,
whose description requires, at a minimum, the specification of five crystallographic
parameters 1124]. Three of these define the lattice misorientation between adjacent
grains and the other two specify the interfacial plane. In addition, there are microscopic degrees of freedom related to in-plane translations and associated structural
relaxations [125-1271, though these will be neglected in the present study.
Another obstacle to correlating grain boundary structure with materials properties
is the difficulty of controlled grain boundary synthesis. Controlled production of
grain boundaries in metals has predominantly been performed via seeded growth of
bicrystals from the melt [128-1301. Single crystal seeds are first synthesized, then
cut and oriented appropriately, after which bicrystals can be grown from the melt by,
e.g. the vertical Bridgman method [128, 129]. Other techniques include sintering of
metalic thin films [131, 1321, and the direct bonding technique for thin films of brittle
materials 1133-135]. In these techniques, single crystals are first produced and aligned
relative to one another, and then bonded either by hot pressing, in the case of metals,
or through evaporation of an intermediate water layer (after a number of meticulous
surface preparation steps). Regardless of the method, controlled preparation of even
a single grain boundary involves numerous steps, sophisticated equipment, and a
significant time investment.
*The contents of this chapter have been submitted for publication and are currently under review.
Although I use the first-person, "I", in describing the work performed in this chapter-to maintain
consistency with the rest of this thesis-this work was performed in collaboration with Dr. Lin Li,
Dr. Michael Demkowicz, and my advisor, Dr. Christopher Schuh.
75
Though still requiring significant time, atomistic simulations can be employed
to ameliorate some of the experimental difficulties and sources of error, as well as
increase throughput 1136-1381. However, such simulations are not without their own
complications. Accurate interatomic potentials are not available for all materials and
the development and validation of new models is a significant undertaking. Moreover,
the computational cost of atomistic simulations restricts one to the investigation of
small material volumes. This places a fundamental limitation on the number and
types of grain boundaries that can be considered.
The largest catalog of grain boundary property data to date consists of calculated
excess energies and mobilities for 388 grain boundaries in 4 different metals 1136138]. While this constitutes a significant milestone, a sample of 0(102) points in
a five-dimensional space is still relatively sparse coverage. A brute force attempt
to explore the space adequately, whether by experimental or computational means,
remains impractical due to the resources that would be required.
In contrast to the difficulties of synthesizing and studying individual grain boundaries, it is relatively easy to produce polycrystalline samples. Commercial solutions
now permit crystallographic characterization of sample areas large enough to capture
millions of grain boundaries. Automated electron backscatter diffraction (EBSD)
techniques allow for the rapid characterization of four of the five grain boundary parameters for every boundary that terminates at the sample surface. For microstructures with through-thickness grains, and where out of plane curvature can be neglected, the final parameter can be obtained by correlating EBSD scans on opposite
faces of the sample 11391, or by comparing intensities of overlapping Kikuchi patterns
with interaction volume models [1401. For general microstructures, serial sectioning
methods or high energy diffraction techniques [141] may be used.
The relative ease of synthesizing and characterizing polycrystals suggests an alternative route to obtain grain boundary structure-property correlations, beyond the
one-by-one study of individual bicrystals. For a grain boundary sensitive property
in a polycrystal, the global effective response of the grain boundary network is the
result of a homogenization of the local properties of the constituent grain boundaries.
This is analogous to an electrical resistor network where the effective resistance of the
circuit depends upon the component resistances and the topology of the network. If
one measures the effective resistance of such an electrical network, then, under certain
circumstances, it may be possible to deduce the resistance values of the components.
If one can, similarly, infer the local properties of grain boundaries in a polycrystalline
aggregate from measurements of the macroscopic effective response of the network
it would facilitate a more efficient approach to searching the five-dimensional grain
boundary structure space for structure-properties correlations.
The process of deconvoluting local quantities of a heterogeneous system from homogenized global measurements is broadly referred to as localization. While this
concept is not new, the use of localization techniques in materials science has, until now, been limited to the calculation of local field quantities, e.g. stress, in the
presence of an applied macroscopic field [2, 142-148J, or single crystal elastic constants [149-1511. In this chapter I address a new kind of localization problem, in
which the goal is to determine the local materials properties of grain boundaries from
76
measured effective (macroscopic) properties of a polycrystal. I also establish a general unifying framework for materials properties localization in the context of any
materials property and any relevant microstructural feature. For the case of grain
boundary properties localization, I consider a highly simplified two-dimensional polycrystalline model system and attempt to infer the diffusivity of low- and high-angle
grain boundaries from the effective diffusivity of the grain boundary network. While
this model problem does not represent a general microstructure, its simplicity permits
me to identify the important qualitative features that I would expect to observe in
other microstructures.
5.2
Model
Our model microstructure was similar to that of Chen [51], consisting of a twodimensional honeycomb lattice (as shown schematically in Fig. 5-1) containing 6,889
hexagonal grains and 20,212 grain boundaries. Crystal orientations were sampled
from two-dimensional orientation distribution functions (ODFs) of the following form
(see also Fig. 5-2):
f (9)
for 0 E [-Omax, Omax]
f(20max)<
0
=(5.1)
otherwise
and assigned in a spatially uncorrelated manner. In Eq. 5.1, 0 max E [0, 7/s] is a
parameter that defines the sharpness of the texture, and s represents the order of
the appropriate symmetry point group. In the present study, I considered triclinic
crystals having a common (001) rotation axis parallel to the sample normal direction,
for which s = 1.
Grain boundary diffusivities were assigned according to the disorientation angle,
W, between neighboring grains with:
D w)~{D1
for w < wt
D2
for w > Wt
(5.2)
where w is the disorientation angle, wt is the threshold angle between low- and highangle grain boundaries, and D < D 2 are the low- and high-angle diffusivities, respectively. While the physical properties of grain boundaries are known to depend, in
some cases strongly, on the grain boundary plane inclination [1521, our model constitutive equation neglects this dependence. This simplifying assumption, as well as the
binary classification of grain boundaries into low- and high-angle classes, facilitate
tractability and clarity of exposition and can be improved upon in future work.
The boundary conditions, assumptions, and procedure for calculating the effective
diffusivity of the grain boundary network were the same as those used by Chen [511.
For our simulations I assumed that diffusion occurred only along the grain boundaries.
This simplification is a good approximation in situations for which the microstructure,
experimental conditions, and kinetic regime is such that grain boundary diffusion is
77
c=0
ay
-0
(9c
0
ay
Fig. 5-1: Schematic plot of the model microstructure showing the boundary conditions used for the
diffusion simulations.
the dominant pathway and all others (e.g. bulk, surface, dislocation, etc.) can safely
be ignored. Alternatively, as mentioned by Chen, these results can be interpreted
as simply providing the contribution of grain boundary diffusion to the overall diffusion 1511. The concentration of the diffusing species was fixed at c = co and c = 0 at
left and at the right side of the network, respectively, and zero-flux conditions were
imposed at the top and bottom (Fig. 5-1). Steady-state diffusion was considered, and
conservation conditions were enforced at triple junctions (i.e. no material accumulation at triple junctions was permitted). This resulted in a linear system of equations
that was solved to obtain the concentrations at each triple junction and, thus, the
concentration gradient and local flux along each grain boundary. By averaging the
microscopic grain boundary fluxes over any vertical section of the network, I obtained
the effective macroscopic diffusion flux, Jff, of the network [51]. Finally, the effective
diffusivity, Deff, was calculated by dividing Jeff by the macroscopic concentration gradient of the network [51]. By varying the sharpness of the textures (Omax) I obtained
grain boundary networks with different grain boundary character distributions, described by the fraction of high-diffusivity grain boundaries, P2, and, consequently,
different values of Deff. I performed simulations using 51 different textures with 0 max
varying between 7.5' and 180' (see Fig. 5-2). The reported values of Deff are the
median of 1,000 network realizations for each texture. Fig. 5-3 shows the effective
diffusivity, DsM that was obtained via simulation, as a function of the fraction of
78
2
0
max = 150
1.5
1
-max
0.5
nmax
0'
-18 0
I
90
Fig. 5-2: Two-dimensional ODFs with varying
high-diffusivity boundaries, P2, with D 1
o
106
=
1800
I
0
6(deg.)
-90
600
=
Oi..ax
180
according to Eq. 5.1.
100 m 2 /s, D 2 = 107 m 2 /s, and wt = 15'.
SIM
150
GEM
105
104
100
101
102
50
10'
100
0
0.2
0.4
0.6
0.8
P2
Fig. 5-3: Effective diffusivity of simulated grain boundary networks for various fractions of type-2
(high-diffusivity) grain boundaries. The error bars for many of the microstructures are smaller than
the markers, and, as expected, they are very large near the percolation threshold. Marker color
corresponds to the sharpness of the texture. As Omax was varied, I obtained networks with DS1M
2
2
ranging from ~5 x 10-1 m /s to ~5 x 106 m /s. The red solid line connects the values predicted
using D GEM. This is shown as a solid line rather than discrete markers only to improve visibility.
Based on the values I obtained for Deff, and a knowledge of the crystallographic details of the microstructure-including the crystal orientations, and the resulting grain
boundary misorientations-I would like to infer the parameters of our constitutive
equation (DI, D 2 , and wt). This inverse problem of grain boundary properties localization is tantamount to determining the local diffusivity of each type of boundary
from measured values of the effective network diffusivity.
79
5.3
Strategy
5.3.1
Overview
The relationship between the properties of a material and its microstructure can be
expressed as a functional of the following form:
P = H(M, P)
(5.3)
where P is the macroscopic effective property of interest, M contains microstructural
information, P is a representation of the relevant local constitutive equation, and H
is a homogenization relation. Eq. 5.3 can be used in several ways:
Materials Properties Prediction
In materials properties prediction the microstructure (M) is known, as is the constitutive equation (P), and one evaluates Eq. 5.3 to obtain the unknown effective
property (P). A common example of this is the prediction of the effective elastic
constants of a polycrystal.
Materials Design
In materials design, a target effective property (P) is specified, the constitutive
equation (P) is assumed to be known, and Eq. 5.3 is solved for the unknown microstructure (M) that is commensurate with the target effective property.
Materials Properties Localization
Materials properties localization completes this set of materials problems by asking
the following question: "Having characterized a heterogeneous microstructure (M),
and measured its effective properties (P), is it possible to infer the unknown local
properties of its constituents (P)?"
These applications and the known and unknown variables in each case are summarized
in Table 5.1.
M
Prediction
Design
?
/
V/
?
/
Localization /
Table 5.1: Summary of the applications of Eq. 5.3.
property of interest, M is the microstructure, and
equation (i.e. the local properties). The '?' symbol
solved for in a given application, and the '/' symbol
P
/
P
/
Application
?
In this table P is the macroscopic effective
P represents the relevant local constitutive
indicates the unknown quantity that is to be
indicates a known or specified quantity.
In the context of the present model problem, P = Deff. For real microstructures, it
could be obtained experimentally using, e.g., accumulation methods {153, 1541. From
Eq. 5.2, the parameters of the proposed constitutive relation are P = [D 1 , D 2 , Wt]T.
80
As will be explained subsequently, the relevant microstructural parameters for the
grain boundary diffusion problem are M = [Ji, j2, j3],
where {Ji I i E [0, 3]}, are
called triple junction fractions and denote the population of triple junctions coordinated by i "special" grain boundaries 161, 67, 68, 72, 79, 90, 155, 156], which, for
our purposes, correspond to low-angle grain boundaries with diffusivity D 1 . The elements of M are easily measured experimentally using various destructive [157-159] or
non-destructive 1141, 160-163] crystallographic characterization techniques. Because
EZ Ji = 1, only three of the four Ji appear in M.
5.3.2
Homogenization
The specific form of H(M, P) is of critical importance for accurate predictions of P
in the forward problem of homogenization, and consequently is also crucial for solving
the inverse problem of localization. One common form of H(M, P) is the standard
effective medium theory (EMT) [51, 164]. EMT works well in systems for which the
variation in the local material property values (property contrast) is small [48, 51].
However, in situations where the local values of a material property differ significantly
from point to point, the detailed structure of the medium dominates the effective
response 148, 511. In such cases, the true effective property values exhibit a sharp
transition at the percolation threshold that EMT models fail to capture.
Experimental measurements of grain boundary diffusivity in bicrystals as a function of disorientation angle have demonstrated that the difference between the minimum and maximum diffusivities can span many orders of magnitude 11131. Because of this strong contrast in the spectrum of grain boundary diffusivities, standard EMT predictions for Deff are inadequate [481. Chen and Schuh demonstrated
that the phenomenological Generalized Effective Medium (GEM) equation, proposed
by McLachlan [50, 112], provides quantitatively accurate predictions of Deff across
a broad range of contrast ratios (from 5 to 108) when compared to simulation results [511. The GEM is an implicit relation that combines aspects of effective medium
theory and percolation theory, and may be expressed as1 :
D /S - (2Def)l/s
i /s+ (-
-
D
1) (2De)/
+
(-
-
(2De) 1
-
1) (2De)/t
0
(5.4)
where pi is the fraction of grain boundaries exhibiting diffusivity Di, and pc,i is the
percolation threshold for the i-th grain boundary type. The values of the critical
exponents s and t are generally considered to be universal, depending only on the
dimensionality of the problem. For the present case I use s = 1.09 and t = 1.13. These
values differ slightly from those given in [51], however, I found that small deviations
in the numerical values of s and t did not affect the results of the GEM significantly
'There are two distinct concepts of effective diffusivity (or conductivity) in the literature. One
is the diffusivity of the actual heterogeneous network as a whole, and the other is the diffusivity of
a single boundary in a hypothetical homogeneous sample whose network diffusivity matches that
of the real sample. The two notions are related and I take the former interpretation, which is the
reason that Eq. 5.4 differs from the form appearing in [50, 51, 112].
81
,
and the value of Pc,2 proved to be of much greater moment. Eq. 5.4 can be solved
numerically, using, e.g. bisection methods 1511 to obtain Deff as a function of D1 , D 2
and the microstructural parameters pi and Pc,2.
To validate the GEM for use as our homogenization relation, I computed Deff using
Eq. 5.4 for each of the simulated textures. I denote the effective diffusivity predicted
by Eq. 5.4 as DeEM, to distinguish it from the values that were obtained directly
from our simulations, Ds M . The agreement between the predicted and simulated
diffusivities is excellent (see Figs. 5-3 and 5-4). Strictly speaking, Eq. 5.4 holds only
for infinite systems, and a scaling factor should be used for finite microstructures.
However, the value of the scaling factor for our simulations is very close to unity
and
(see [51]), particularly after considering 1000 replications of each microstructure,
I therefore omit it.
106
101
i04
~
10-
102
101
100
100
10W
102
DsM
106
(rn2
Fig. 5-4: Comparison of the effective diffusivity predicted by the GEM, DeM to that obtained via
simulation, DYM. The dashed red line indicates a 1 : 1 relationship (i.e. perfect agreement). Error
bars for many of the data points are smaller than the respective markers. Marker color indicates
the sharpness of the texture, and is referenced to the color scale of Fig. 5-3.
5.3.3
Microstructural Parameters
An important quantity that appears in the GEM is the percolation threshold, Pc,2. For
a randomly assembled two dimensional honeycomb network the percolation threshold
is known analytically to be pc =1 - sin (7r/18) ~ 0.6527 11651. However, the spatial
distribution of grain boundary types in real polycrystals is markedly non-random
as a result of crystallographic constraints [49]. The resulting local correlations lead
to a shift in the percolation threshold 1491 and, consequently, must be accounted
for in order to obtain accurate predictions of Deff. These short range correlations
can be quantified using the triple junction fractions, Ji. Frary obtained an empirical
relation that predicts the percolation threshold for two-dimensional networks from the
Ji 1621. As mentioned earlier, I am interested in the percolation threshold for type-2
grain boundaries, Pc,2, but the common convention defines the Ji as the fraction of
82
triple junctions coordinated by i type-1 (low-angle) grain boundaries. In this case
the percolation threshold can be approximated by a second order polynomial of two
variables 2 :
Pc,2
=d, + d 2 x + d 3 x 2
+
d 4 c + d 5 yx
+ d6cr2
(5.5a)
with the values of the coefficients, dj given in Table 5.2. The variables a and x are
topological order parameters, the prior measuring the tendency of grain boundaries
of differing types to segregate or form ordered arrangements, and the latter indicating
the tendency for grain boundaries of like type to be found in elongated or compact
clusters. These are defined in terms of certain ratios of the Ji according to:
1 -j,
-
for j, <1
for j, > 1
(5.5b)
(
1
for j <
for JX >1
f1 -jx
x
J1 + J2 )(JOr
+Jr)
Jir + J2r
JO + J3
.
-Ji
(5.5c)
(5.5d)
J 2r
J2 Jir
where Eq. 5.5e is the reciprocal of the expression given in [62] because I am considering
Pc,2. In Eq. 5.5d and 5.5e, the Jir are the triple junction fractions for a randomly
assembled network (i.e. in the absence of crystallographic constraints), and are given
by a binomial distribution:
Jr
(1
P2)3-
P
(5.6)
Consideration of Eq. 5.5 reveals that if a grain boundary network were to be totally
uncorrelated then Ji = Jir and a = X = 0 resulting in Pc,2 = dl, which is the percolation threshold for the random network. Correlations in the grain boundary network
result from the physical requirements of crystallographic constraints as well as other
influences such as, e.g, crystallographic texture. Our simulations are crystallographically consistent and, therefore, include both of these effects. The strength and type
of these correlations is reflected in the non-random values of the Ji, which, in turn,
lead to non-zero values of the order parameters o- and X and a commensurate shift in
the percolation threshold due to the additional terms in Eq. 5.5a.
The other microstructural parameters in the GEM (Eq. 5.4) are the pi. These
2
There is an error in this formula as printed in [62]. If the values of the dj presented in Table II
of [62] are to be used, then the signs of all terms in Eq. 8 of [62] should be positive. This is corrected
in Eq. 5.5a of the present work.
83
j
1
2
3
4
5
6
dj
0.6527
0.0651
0.0117
-0.0961
0.0233
-0.0715
Table 5.2: Numerical values of the coefficients appearing in Eq. 5.5a.
may also be expressed in terms of the Ji according to 168]:
1
2
pi ~ - J + -J 2 + J3
3
3
(5.7)
and
P1 + P2
1
(5.8)
where the approximation becomes exact in the limit of infinite systems (or for finite
systems with periodic boundary conditions). Thus, for the present model system
the relevant details of the microstructure are fully specified by the Ji and I have
M = [J1 , J2 , j3]T
It is important to note, however, that the Ji also have a subtle dependence on the
parameters of the constitutive equation. As mentioned earlier, the Ji are defined as
the fraction of triple junctions coordinated by i low-angle grain boundaries (type-1 in
the present model); however, the distinction between type-1 and type-2 boundaries
is governed by the constitutive parameter wt. As a result, not only do the Ji depend
upon the value of wt, but so do the pi, and Pc,2. This interaction between the constitutive parameters, P, and the microstructural parameters, M, complicates the solution
of this particular localization problem. For each trial value of wt the microstructure
must be re-analyzed: the Ji must be determined anew, and the pi and Pc,2 must be
recomputed before H(M, P) can be evaluated. This does not preclude solution of
the localization problem, but care must be taken not to exclude this feedback loop in
the solution process.
5.3.4
Localization
The solution of the localization problem is accomplished by solving Eq. 5.3 for the
unknown constitutive parameters of P. However, if one considers P and M of only
a single microstructure then the system will be underdetermined if P contains more
than one parameter, as in the present case. Even if the number of microstructures, N,
is equal to the number of parameters, nP, errors arising from imperfect measurements
or inexact homogenization models (H) could still preclude the solution of the resulting
system of equations. A more flexible approach is to minimize the total error, in some
sense, between the measured values of P and the predicted values of H(M, P) for a
large number, N, of microstructures, over the domain of P. For the present problem
84
this can be stated as:
minimize
P
E(P)
subject to 0 < D 1 , D 2
0
(5.9)
< Wt < 7r/S
where E(P) is a function that provides some measure of the total error. One choice
of objective function is the sum of squared errors over all N measurements; however,
given that the values of Deff vary over many orders of magnitude, such a procedure
would give too much weight to errors at the high end of the Deff spectrum. For
instance, a difference of 10- m 2 /s is small when Deff = 0(100) m 2 /s, and a difference
of 10 m 2 /s is small when Deff = 0(106) m 2 /s, but the magnitude of the error in
the latter case would completely dominate so that measurements for small Doff would
have negligible influence on the solution. To correct for this effect, I defined the
following objective function:
E(P)
[log (()DsM
=
()DsIM
1
N
where (j)DS'M and ()D
GEM
-
i=1f
D
log ()DGM
2
2
(5.10)
eff
(iog(D
M(P))
are the effective diffusivities of the i-th microstructure (i.e.
one of the 51 possible textures) as calculated via simulation and as predicted from
the GEM homogenization relation respectively. Eq. 5.10 represents a least-squares
type measure of the error between the logarithms of the effective diffusivities obtained
via simulation and those predicted by the GEM for a fixed set of parameter values,
P, when considering N different microstructures. Using Eq. 5.10 as the objective
function, I solved Eq. 5.9 in order to infer P from the observed values of (i)DsM.
Physically, this corresponds to a situation in which I have characterized the Ji of
N different microstructures, and measured the effective diffusivity of their respective
grain boundary networks ((i) DsM for i E [1, N]). I then want to know what values
of P = [D 1 , D 2 , Wt] are commensurate with our observations. In this way, I can
infer the local grain boundary properties (D 1 and D 2 ), from measurements of the
macroscopic effective properties (()D1M).
In an effort to visualize the solution space, and to avoid uncertainty resulting from
questions of convergence, I employed a brute force method by evaluating Eq. 5.10
over a grid of points, A = D, x D2 x Qt, with Ei = {10-8, 10- 7 .5, ... , 108} m 2 /s,
2
D2 = {10-8,10-7'5, ... 1012} m /s, and Qt = {00, 1.50, ... , 450}. The resolution of
this grid defines the precision of our solutions. For each N E {3, 5, 10, 20, 50}, I selected N textures uniformly at random from those that were simulated and evaluated
Eq. 5.10 over A. This procedure was performed 100 times for each N. Figure 5-5 depicts isosurfaces through the solution space for constant values of E(P), with N = 3,
and illustrates that E(P) decreases monotonically to its global minimum near the
true values of D, = 100 m 2 /s, D 2 = 10 7 m 2 /s, and wt = 150.
85
100
50
40,
30
25
20,
10
10,
3
0
0
1
100
-5
10
-100
0 5
D 2 (m2/s)
1010
101
D 1 (m2/s)
Isosurfaces through the solution space of E(P), with N 3 and P E A,
Fig. 5-5:
{10-8, 10-7 5 , ... , 108} m 2 /s,
D, x D2 x Qt is a grid of points defined by D
where A
2
D 2 = f10-8, 10-7.5,.... 1012} m /s, and Qt = {0, 1.5,.. ., 450}. The true values of the parameters are represented by the location of the blue star.
5.4
Discussion
In the context of a general localization problem in which the true parameter values
of a proposed constitutive relation are not known a priori, one seeks to infer them
while simultaneously minimizing experimental/computational expense and effort. In
order to accomplish this, one would naturally ask (1) "What is the minimum number
of polycrystalline microstructures that I need to consider?" and (2) "What characteristics should those microstructures possess to facilitate the solution of the localization
problem?". General answers to these questions are not yet known for arbitrary microstructures and properties and they are subjects worthy of future inquiry. However,
some insight into these issues can be gained by considering the simplified and specific
case at hand.
With respect to the number of microstructures required, the theoretical minimum, Nmin, is equal to np, the number of parameters in P, if the form of the constitutive equation is known. However, multiple copies of an identical microstructure
would be redundant and would not permit the solution of the localization problem.
Distinct microstructures whose structures are statistically equivalent would also ex86
)
hibit this redundancy. Consequently, there must be at least as many non-redundant
microstructures--i.e. microstructures that are, in some sense, different enough-as
there are parameters in P. As illustrated in Fig. 5-5, it was, indeed, possible to
recover the true parameter values to within the resolution of A using Nmin = nP = 3
microstructures. However, not all sets of 3 microstructures resulted in the correct
solution.
To investigate what microstructural characteristics were required to solve the localization problem, I evaluated Eq. 5.9 for all () = 20,825 possible sets of N = 3
distinct microstructures out of the 51 that I simulated. Because of the special class of
microstructures considered here (honeycomb network with ODFs specified by Eq. 5.1,
and spatially uncorrelated grain orientations), the structure of the grain boundary network can be fully specified by the fraction of high-angle grain boundaries, P2. Thus,
(1)
(2)
(3)
an N = 3 microstructure set can be described by the ordered triplet
P2 1P2 1P2
an
where P2
p2
p2 are the fractions of high-angle grain boundaries in the first,
second, and third microstructures in the set, respectively. Fig. 5-6 reveals that microstructure sets that returned the true solution (to within the precision of A) and
those that did not were separated into, more or less, well defined regions. The planar
boundaries separating satisfactory microstructure sets (green polyhedral region) from
unsatisfactory ones (red points) define the microstructural characteristics that permit
solution of the localization problem:
C.1 P(
< Pc,2
C.2 p2 > Pc,2
These two conditions simply mean that both sides of the percolation threshold must
be sampled. While necessary, conditions C.1 and C.2 are apparently insufficient to
guarantee the correct solution as there are a small number of microstructure sets
falling in this region that still do not produce the correct results. These, however,
invariably fall near other facets of this region and can be eliminated by additionally
enforcing the following conditions:
0.0756
C.3 p(2) - (
C.4 P() > 0.0092
C.5 pf
Pc,2
or p2
C.6 p 3 )
Pc2
> 0.0202
> Pc,2
+ 0.0202
The necessity of C.3 confirms that solution of the localization problem is complicated
when the first and second microstructure in the set are too similar and quantifies how
different is "different enough" to avoid microstructural redundancy. Condition C.4 is
required only to eliminate the single anomalous point at (0, 0.3487,0.9168), and could
equally have been expressed as an upper-bound on p2. In either case, the surprising
implication is that pOM and/or p( should not be too far from Pc,2. C.5 and C.6
indicate that the second and third microstructures in the set, respectively, should
posses high-angle boundary fractions that are sufficiently far from the percolation
threshold. Extrapolating these specific results to the general localization problem, I
87
postulate that satisfactory microstructure sets should: (1) be sufficiently diverse so as
to avoid microstructural redundancy; (2) sample all regimes of the effective property
of interest; and (3) avoid critical points (e.g. percolation thresholds).
Having performed many simulations spanning the domain of P2, I had the luxury
of knowing the location of the percolation threshold, Pc,2. Consequently, it would be
possible for us to deliberately choose a set of N = 3 microstructures satisfying the
conditions C.1 to C.6. In general one may not know the salient features of a candidate
structure-properties model. In such cases one might attempt to uniformly sample the
structure space. For sets of randomly selected microstructures the accuracy and
precision of the solution to the localization problem increases with N. Using N = 3
randomly chosen microstructures (repeated 100 times), I succeeded in recovering the
true parameter values to within the precision of A in 67% of the trials. For N =
5, 10, 20, 50 the success rates were 87%, 98%, 100%, and 100%, respectively. Fig. 5-7
shows 95% confidence intervals for each of the parameters, as a function of N. The
convergence of the confidence interval for log(D 2 ) was slower than for the other two
parameters, and, expressed in terms of % error, it scaled according to
%ERRiog(D2) = 0.13 exp (-0.2449N)
(5.11)
with a coefficient of determination of 0.9983. This scaling relationship serves as an
upper bound' on the error in the solution of the localization problem and is useful as
a means of estimating the marginal value of performing an additional experiment. For
example, assume one has characterized the microstructures of N
=
7 randomly chosen
samples and measured their effective diffusivity. How much more accurate would the
estimates of D 1 , D 2 , and wt be if N = 8 microstructures were used instead? Using
the scaling relationship I conclude that, with 95% confidence, all of the parameter
estimates will be correct to within 2.34%. By adding an additional microstructure,
the maximum error in the inferred parameter values would be reduced to 1.83%.
Alternatively, if one simply wants to know the minimum number of microstructures
needed to ensure that the error is less than, e.g, 5%, use of the scaling relationship
would reveal that N = 4 microstructures are required.
For the sake of comparison, a scaling law for bicrystal experiments can be derived.
Consider a set of N bicrystals with respective misorientations, {w 1i, w 2 , ... ,WN }, which
are sampled from U(0, 180'). One could perform diffusion experiments on each of
the N bicrystals in an attempt to infer the parameters, P = [D 1 , D 2 , Wt]T, of the
constitutive equation (Eq. 5.2). In the case of these bicrystal experiments the 95%
confidence interval for wt will converge more slowly than for the other two parameters
and thus the % error on wt will be the upper bound for the bicrystal experiments.
3
The percent error for log(D 2 ) is strictly greater than or equal to that of wt. However, since
log(Di) = 0 the concept of percent error is not well defined for this parameter. Alternatively,
considering the percent error in D1 instead of its logarithm would be inconsistent with the rest of
our analysis. While the absolute error of log(D1 ) is very large for N = 3, it is zero (or at least
smaller than the resolution of A) for all other values of N that were tested, and consequently its
percent error can be reasonably considered to be zero as well. Therefore, using the percent error of
log(D 2 ) as a bound is strictly only valid for N > 5.
88
As demonstrated in F, the % error in the parameter estimates resulting from these
bicrystal measurements scales according to
%ERR, ~ 0.0227N-
(5.12)
Let us set a target value for a tolerable % error in our recovered parameter values,
e.g. %ERR < 5%. Solving Eqs. 5.11 and 5.12 for N I find that the number of
samples, NB, required to achieve a given level of error with bicrystal experiments
scales according to
1
NBoc
1
(5.13)
%ERR
In comparison, the required number of experiments using localization, NL, scales as
NL oc In
%RR
(5.14)
Figure 5-8 shows a graphical comparison of NB and NL as a function of %ERR, which
confirms that localization dramatically reduces the experimental effort required to infer P. For our chosen level of accuracy, %ERR < 5%, I would need to perform at
least 882 bicrystal experiments on average, to infer the values of the constitutive parameters. In contrast, localization would necessitate an average of only 4 experiments
for the same level of accuracy.
As just mentioned, one of the great advantages of the localization approach to
deducing the properties of microstructural constituents and structure-property correlations is efficiency. I expect this experimental economy to be a general feature of the
localization method, extending beyond the model system considered here. To obtain
good coverage of a d-dimensional space (d = 5 for grain boundaries) and make meaningful inferences about structure-property correlations would require 0 (10d) experiments (starting from a point of complete ignorance). In contrast, given a pre-specified
constitutive relation with n, undetermined parameters, the localization methods described here allow inference of structure-property relations with as few as 0(np)
experiments and there is evidence that even for more complex constitutive relations
than what I have here considered, np may be extremely small. For example, Bulatov
et al. recently developed a function for grain boundary energy in FCC metals that
requires only 2 material specific parameters [166]. Thus, the key to experimental determination of grain boundary structure-property relations may be the development
of physics-based constitutive models with clearly stated parameters. The present results suggest that systematic experimental calibration of such models may be possible
and would require far less experimental effort than an exhaustive bicrystal approach.
Furthermore, the use of the localization approach presented here, in conjunction with
model selection techniques, could allow not only for the calibration of model parameters, but also the deduction of the form of an appropriate model.
In this chapter, the localization approach has been demonstrated in the specific
context of a grain boundary structure-properties model. However, the localization
approach is not limited to a particular material, property, or microstructural feature.
89
Rather, the localization methodology is completely general and may be applied to
any material, for the inference of any property, as long as a suitable homogenization
relation (H) and an appropriate ansatz for the relevant constitutive equation (P)
exist.
90
0.8
0.6
0.4
0.2
0.8
0.6
-
0.8
.-
0.4
0..2
0
(2)
P2
Fig.
5-6: All
(5)
0..4
4
-1
0
20, 825 possible sets of N = 3 distinct microstructures as represented by their
2
,(3)
( )
respective fractions of high-angle grain boundaries. The coordinates of a point, (p
where
set,
in
the
microstructures
the
three
indicate the fraction of high-angle boundaries in each of
() < (2) < 3). The red points represent microstructure sets that did not result in successful
recovery of the true constitutive parameters. For the sake of visual clarity, the points representing
microstructure sets that did result in successful recovery of the true constitutive parameters are not
shown, instead they are indicated by the green semi-transparent region.
91
0
E -0.2
-0.4
/
-
S
7
6.5
0
-7o
6
15.5
15
50
20
10
5
3
N
Fig. 5-7: 95% confidence intervals for each of the parameters as obtained by solving the localization
problem with N microstructures chosen uniformly at random from among those considered here.
The process was repeated 100 times for each N.
105
Localization
--Bicrystal
-
104
101
102
101
1 0
0
1
2
4
3
5
6
7
8
%Error
Fig. 5-8: The number of experiments, N, required for one to recover the values of all of the constitutive parameters in Eq. 5.2 to within a given level of uncertainty (%Error). The performance of
the localization methods introduced in this chapter is compared with that of an exhaustive bicrystal
approach.
92
5.5
Conclusions
In this chapter I defined the problem of grain boundary properties localization, which
permits one to infer the parameters of a grain boundary structure-properties model
from macroscopic measurements of effective materials properties. Stated another way,
the localization approach allows one to infer local physics information from measurements of global structure and the effective response of the material. I applied this
technique to a simple model system of grain boundary diffusivity in a two-dimensional
microstructure and inferred the properties of low- and high-angle grain boundaries
from the effective diffusivity of the entire grain boundary network. Below I summarize
the main conclusions of this investigation:
1. Under appropriate conditions the localization problem has a unique solution
and the objective function converges monotonically towards it. In our model
problem I found that this unique solution coincided with the known true solution.
2. Solution of the localization problem requires at least N = nP non-redundant
microstructures.
3. General microstructural characteristics required to recover the correct solution
to the model localization problem considered here were that the microstructures
should be sufficiently different, should sample all unique effective properties
regimes (i.e. both sides of the percolation threshold), and should avoid regions
where abrupt transitions occur in the effective properties (pc,2). For a general
localization problem the details will be problem specific.
4. If appropriate selection criteria are unknown and selection of microstructures
is performed uniformly at random, the accuracy and precision of the resulting
inferences increase with N, the number of microstructures considered. At the
95% confidence level, an upper-bound to the % error in the inferred parameter
values was shown to decay exponentially with N. This scaling relationship can
assist in determining the marginal value of performing additional experiments
or to identify the minimum number of experiments one should perform to obtain
a specified accuracy in the solution.
5. Materials properties localization provides an efficient means to infer structureproperties models for grain boundaries. In the present case I found that, using
brute force bicrystal experiments, the number of experiments required to infer constitutive model parameters scales inversely with the desired accuracy
%R). In
contrast, the number(NBoc
of experiments' that localization re-
quires is much smaller and scales logarithmically with the inverse of the desired
accuracy (NL aC in (%ERR)). For more general structure-properties models I ex-
pect that, for a specified accuracy (%ERR), the effort required for localization
will scale with nP, the number of parameters in the model, which should generally be small. In contrast, the experimental effort required for a brute force
bicrystal approach should scale exponentially with d, the number of degrees of
freedom required to describe the microstructural feature of interest (for grain
boundaries d = 5).
93
Chapter 6
Summary
The characterization and control of the structure of materials at various length scales
are central objectives in the field of materials science. For engineering applications
in which the properties of interest are governed by crystallographic texture, rigorous
methods exist for the design of textures that are tailored to particular design applications. However, many important materials properties in polycrystalline materials are
governed, not by texture, but by the character and connectivity of grain boundaries.
In this thesis, I have developed the theoretical tools required for the design of grain
boundary networks in polycrystalline materials. The major contributions of this work
are summarized below.
In Chapter 2 I derived a triple junction distribution function (TJDF) as a statistical tool to characterize the structure of grain boundary networks. The TJDF
measures the distribution of triple junction misorientations in any polycrystalline
material possessing arbitrary crystal symmetry and texture. I also derived a formula
relating the spectral coefficients of an orientation distribution function (ODF) to those
of the TJDF that is exact in the limit of spatially uncorrelated crystal orientations.
This result provides a general connection between crystallographic texture and grain
boundary network structure.
In Chapter 3 I identified the crystallographic symmetries of triple junctions and
the triple junction equivalence relations that they induce. Using these relations I
defined the triple junction fundamental zone which is the region of triple junction
misorientation state space containing all physically distinct triple junctions. Having
established the symmetries of triple junctions, I considered the configuration space
for connected grain boundaries in polycrystals, as quantified by the spectral coefficients of the TJDF. In this context, I defined the triple junction hull, 'W , which
circumscribes all physically possible TJDFs allowing one to explore this universe of
grain boundary network structures in search of application specific globally optimal
network configurations. This is the first microstructural design space to consider the
H is convex and that every
influence of grain boundaries. I demonstrated that
TJDF contained therein is physically realizable. Employing the relationship between
ODF and TJDF developed in Chapter 2, I investigated how much of this space is
accessible through the control of texture alone.
In Chapter 4 I demonstrated the ability to solve a grain boundary network sen95
sitive engineering design problem using the tools developed in Chapters 2-3. I developed a hierarchical simplex sampling algorithm to sample microstructures from
a general microstructure hull in Dirac space. I also derived a model to predict the
effective macroscopic diffusivity of a grain boundary network from the coefficients of
the TJDF. Using this, and other existing structure-properties models, in conjunction with our hierarchical simplex sampling approach, I used a stochastic method
to define a properties closure, Y, for yield strength, elastic compliance, and grain
boundary network diffusivity in Al. Using these tools I found an optimal texture and
its commensurate grain boundary network that satisfied the competing objectives of
the design problem. This process of texture-mediated grain boundary network design
resulted in an optimal microstructure that is predicted to outperform an isotropic
material by seven orders of magnitude.
In Chapter 5 I defined the problem of grain boundary properties localization,
which permits one to infer the parameters of a grain boundary structure-properties
model from macroscopic measurements of effective materials properties. I showed
that localization is the third and final member of a family of mathematical materials problems including properties prediction and design. In the context of a simple
model constitutive equation for grain boundary diffusivity, I showed that solution of
the localization problem is feasible, unique, and far more efficient at inferring the
parameters of a structure-properties model than the traditional brute force bicrystal
approach. I quantified the uncertainty in the inferred parameter values when a given
number of microstructures is considered, and commented on the characteristics of
microstructures that should facilitate localization in a more general context.
In this thesis I have established theoretical tools to characterize the structure
of grain boundary networks and to design them for specific engineering applications. These developments should advance our understanding of the influence of
grain boundaries and their collective network structure on the properties of materials, and facilitate the development of both structural and functional materials with
enhanced properties.
96
Chapter 7
Directions for Future Work
During the course of the work described in this thesis I discovered gaps in the current
understanding of grain boundaries and their associated networks. Building upon the
present work I have identified a number of research directions that would expand the
knowledge of the materials science community and advance the current state of the
art. Listed below are some of those that appear most promising:
" In the present work the TJDF and our constitutive model for grain boundary
properties contemplated only the misorientational degrees of freedom of a grain
boundary. While the misorientation certainly influences the properties of grain
boundaries [167], the interfacial plane of the boundary is also important [431.
Developing an enhanced version of the TJDF that quantifies not only the distribution of triple junction misorientations, but also their plane inclinations will
be a valuable enrichment of current capabilities.
" In Chapter 3 I proved, by construction, that any TJDF contained in the triple
junction hull, //(), corresponds to a physically realizable microstructure. This
constructive proof established a particular microstructure-a regular array of
hexagonal columnar grains-that could be used to construct any TJDF. While
portions of
_W(3)
can also be accessed with more general microstructural tem-
plates, it is not obvious how much. To expand the application of the methods
developed here to a more general class of microstructures, it will be important to
establish the correspondence between geometric grain boundary network topologies and feasible TJDFs.
" The texture-mediated grain boundary network design approach leveraged the
relationship between ODF and TJDF that can be derived in the absence of
grain orientation correlations. While there are realistic situations in which this
approximation is nearly true, there are also many systems for which it is not.
Consequently, future work should also take into account the existence of orientation correlations. Based on the results of Chapter 3, I expect this refinement
to significantly expand the design space.
" Physical models for the properties of individual grain boundaries have been
elusive and are a very active area of research. Models for the macroscopic
97
properties of grain boundary networks are even more rare. In fact, apart from
the model developed in the present thesis I am unaware of any such model.
This model has two major limitations: (1) it considers only two types of grain
boundaries; and (2) it is only applicable to honeycomb grain boundary network
topologies. While a formula does exist that can handle an arbitrary number of
discrete grain boundary types, it is intractable for more than a small number.
The reason for the second limitation is that the dependence of the percolation
threshold on the triple junction fractions has only been characterized for twodimensional honeycomb networks. Future work that addresses these limitations
will make it possible to model grain boundary phenomena with increased fidelity.
* The ability to design optimally performing microstructures is an important
step to the development of advanced materials. To fully realize the potential of
these tools, we will also need complementary processing methods that expand
current capabilities of microstructural control in order to synthesize the optimal
materials that we design.
98
Appendix A
Definition of Special Functions
Spherical Harmonics [92]:
(21 + 1) (1 - in)!
47r 1 + n)! ptm[Cos 0] eim7
Y1 " (0,$) =
i""
Associated Legendre Polynomials [28]:
P/m ()
=
(1 -
()
x2)m/d
2 _
1)2
Gegenbauer Polynomials [28, 168]:
C"'(z)n
(-2)" F (v + n) F (2v + n) (1
n!
F (v) F (2v + 2n)
-Z2)1/2_v
dn
dzn [(I -
z2)n+--1/2I
A-Symbol [92]:
A (abc) =
(a + b - c)! (a - b + c)! (-a + b + c)!
(a +b+ c+ 1)!
Clebsch-Gordan Coefficient [92]:
= 6-ya+ A (abc) \/(a + a)! (a - a)! (b + 0)! (b -
C/,a
z
z! (a + b - c - z)! (a - a
-
z)! (b +
-z)!
(c
-
/)!
(c + -y)! (c - 'y)! (2c + 1
b + a + z)! (c
Wigner 3-j Coefficient [92]:
i
Tml
j2
j
m2 m3
(_13+Tn3+21
J2j)
1
CiM'"
+ 1
99
3
J1,m1,32,M2
-
a-
/
+ z)!
Wigner 6-j Coefficient 1921:
ce
fJ
= A (abc) A (cde) A (aef) A (bdf)
(--1)n (n + 1)!
(n - a - b - c)! (n - c - d - e)! (n - a 1
x
(a+b+d+e-n)!(a+c+d+f
-n)!(b+c+e+f
e f)!
-
a b
d e
(n - b - d - f)!
-nf
Wigner 9-j Coefficient 192]:
a b C
=_
(-1)2 (2x
+ 1)
.
a b C
d e f1
g h j
100
f j xJ
d e f
b x h
4
g
h
x
a d
Appendix B
Integral of the Product of Three
Hyperspherical Harmonics
The integral of the product of three arbitrary hyperspherical harmonics over S3,
as encountered in Eq. (2.13), can be evaluated by first expanding the hyperspherical
harmonics in terms of the spherical harmonics and simplifying. The integral becomes:
I
fo
2ir
ir
ir
Z
Z
2
(_.)L
L!
X
3
23
-2,7r
j
(nI + 1) (n, - l1)! (n 2 + 1) (n 2 - 12)! (n 3 + 1) (n3
(ni + 1i + 1)! (n 2 + 12 + 1)! (n3 + 13 + 1)!
12+
11
Cn+t1 [cos O] C 2+2
[j2 7
sin2 a sin dadOdo
3m
r
[
x
3
Z
Jool0i12m
13
[cosa]C+_
ozsin
1CO
a)L+2
[cosa](s
-
13)!
1a
)L+da
7r
Y"l
YY"2
sin OdOd 5
(B.1)
where L = E 1j, and a = w/2. For compactness of notation I omit the arguments to
Z 1 2m (a, 0, #) and Y17 (0, #).
The integral in the second bracket is known to be 192, 169]:
7r
/27r
j Y"1Y1 "Y72"3 sin OdOd#
j0
4r
11
0
12
13
0
0) (mi
11
12
13
m2
m3
(B.2)
)
/(21i + 1) (212 +1) (213 + 1)
V
The integral in the first bracket is related to the isoscalar factor (reduction coefficient) [170, 1711 for the chain SO (4) D SO (3). The general expression for the
isoscalar factor of SO (n) D SO (n - 1) was given by Alisauskas [1721, which can be
101
specialized to our case, giving:
j/
3
7r
(
C + [cos a]
ni
n2
n3
(43)(4)(
11
12
13
(4:3
(sin a)L+2 da
n1
n2
11
2
(0
0
n3
.3
2L+11 !2! 13!(
13
0)(3
(3)
(ni + 1) (n2 +1) (n3 + 1) (ni + i + 1)! (n2 +12 +1)! (n3 + 1 3 + 1)!
(21 + 1) (2/2 + 1) (213 + 1)
where
2
(ni
- lI,)!
(n2
-
l2 )!
(n3
-
13)!
is the isoscalar factor for the chain SO (n) D SO (n - 1), and
c
is the Wigner 3-j coefficient for SO (n). In the case of SO (4) D SO (3)
(0 3 Y (n)
the isoscalar factor can be obtained by combining Eq. (17) of Ref. [173] with Eq.
(4.6b) of Ref. [172], resulting in:
ni
n2
n3
1(4:3)
n1/2
n2/2
n1/2
n2/2 n3/2
n3/2
_)l
(1)
V(21i+ 1) (2/2 + 1) (2/3 +
1)
11
12
1
(B.4)
where the term in curly braces is the usual Wigner 9-j coefficient of SO (3). I note
that this expression differs from what one would obtain by direct substitution of
Biedenharn's result [1731 into that of Alisauskas 1172] by an, as yet undetermined,
phase. It is unclear what the origin of this discrepancy is, but this modified version
is consistent with the symmetry of the problem and I have extensively validated
the final result by testing the expression given in Eq. (B.6) for 1067 distinct cases.
Additionally, the specific Wigner 3-j coefficient for SO (4) that appears in Eq. (B.3)
is given explicitly by [172, 1741:
where
Vb
ni
n2
n3
0
0
0/)
-1*
/(n1I+ 1) (n21)(13+1)
is also an undetermined phase. I choose 0a +
102
b
= L/2.
(B.5)
Substituting Eqs. (B.4) and (B.5) into Eq. (B.3) and simplifying yields:
7r
j (i
3
[cos o]
C
(sin a)L+2 da
(-1)L/27
2
L+11 1 !12! 3!
F(ni + 11 + 1)! (n2 + 12 + 1)! (ns + l +-i)!(i
(ni - li)!
(n2 - 12)!
(n3 - 13)!
0
2
1n
0
022/2
n3/2
11
12
13
(B.6)
where I have dropped the subscripts on the 3n-j coefficients since all terms involved
in this expression correspond to the usual definitions for SO (3). I note that this
expression holds as long as the reciprocal of the Wigner 3-j coefficient appearing in
Eq. (B.6) is finite. This imposes the requirement that L is even, and that the li satisfy
triangular conditions.
Substituting Eqs. (B.2) and (B.6) into Eq. (B.1) I arrive at the final result:
27
j
fo
-rr
'
j
Zj 1 i Z12
Zn 3
sin 2 a sin OdadOd$
0
(n + 1) (2
V
+1) (n
~
+1) (21 +1) (212 +1) (23
1)(/1
12ni/2
l2/
M3
22-(lM
n 2 /2
ni/2 n2/2 n3/2
11
12
13
(B.7)
This result is in exact agreement with independent derivations of the coefficients
of the Hyperspherical Harmonic Linearization Theorem (alternatively referred to as
Gaunt coefficients, 0 (4) Clebsch-Gordan coefficients, etc.) provided by Hicks [175],
and Meremianin 130], though no mention is made of its relation to the above integral.
The two quantities are identical except for a phase difference.
103
n3/2
72/272/
Appendix C
Inversion Symmetry for
Hyperspherical Harmonics
A useful identity for the hyperspherical harmonics is that of inversion symmetry.
This result was given by Talman [176], but using distinct notation and unnormalized
versions of the basis functions. For clarity, I derive it here using notation that is
consistent with the conventions of this paper.
Given a rotation, q, defined by a rotation of w about an axis defined by i = (0,#),
its inverse, q-1, can be accomplished by a rotation of -w about ii. I can determine
the value of Z,m (q- 1 ) explicitly from the definition given in Eq. (2.4):
Zjn
)
=1(-i)!
21!
(n - Y' 1)!
1) (-w/2)]
+ [cos
(n Cn+_
V2
sin (-w/2)
(0, q)
7r (n + 1 +1)
(C. 1)
Since sin' (-w/2)
(-1)' sin' (w/2) and cos (-w/2) = cos (w/2), I have:
(n + 1) (n -1)!
sin' (w/2) Cn+_1 [cos (w/2)] Yrl (0,
7r (n + I +1)
(C.2)
which, upon comparing with Eq. (2.4), yields:
(-1)' (-i)' 2'l!
2
Zjnm (_W,10,#
)
Zm (-w,Zjn2
0, #)
= (-1Zj",(W, 6,)
105
(C.-3)
Appendix D
Relations for the Coefficients of ODFs
and TJDFs
Constraints imposed upon a function that is to be expressed in the form of a generalized Fourier series result in constraints upon the values of and relationships between
the coefficients. In this section I give several of the most useful of these relations for
ODFs and TJDFs. The existence of such relations allows one to identify which coefficients are independent, and therefore reduces the computational cost of their evaluation. Furthermore, these relations reduce the dimensionality of m, thus greatly
reducing the computational cost of exploring the design space.
D.1
The Reality Condition
Because an ODF is a probability density function it must be strictly real valued, i.e.
f * (q) = f(q). Employing the expansion of Eq. 3.16 for both sides of this expression
results in:
1,*Z,,
"Z
=
I'
n,l,m
Tn)
(q)
(D. 1)
n,l,m
Making use of the complex conjugation identity for the hyperspherical harmonics
one has:
S
(
=
n,I,m
cmZ["(q)
[27]
(D.2)
n,l,m
By changing indices on the left-hand side (m + -m), setting the coefficients equal
term by term, and taking the complex conjugate of both sides, one arrives at:
*
I
as a reality condition for the coefficients of an ODF.
The same procedure may be used to obtain the reality condition for a TJDF, with
the result being:
tn3,A23,423
n1,AXl2,/-12
_
)Al2-l12+A23~1
107
23tf
3,A23,--23
n1,A12,-/112
(D4)
Triple Junction Symmetries
D.2
,
Consider a circuit enclosing a triple junction coordinated by grains labeled 1, 2, and 3
(see Fig. 3-1b). Without loss of generality, let the circuit begin in grain 1, proceeding
next to grain 2, on to grain 3 and finally returning to grain 1. 1 take, as a convention,
the misorientations of the first two grain boundaries that are crossed, q1 2 and q23
to be the two independent misorientations defining the triple junction. Traversing
the circuit in the opposite direction results in an equivalent description using the
misorientations q13 and q3 2 . Thus the TJDF must be invariant with respect to either
choice, and I have:
T(q1 2,q 2 3 ) =T(q 3 , q 32
)
(D.5)
)
T(q1 2q 23, q231
where the second line is a result of Eq. 3.3 and the definition of a misorientation.
Expanding both sides using Eq. 3.10 one has:
E1Z2,1
nl,Al2 ,,2
1A2X1
A2
(qi2) Zn3 ,3
A3A2
(q23 )
tn3A23 ,/I23 Znl
X124,
nlA 1 2 ,/112
E
(q-1)
(q2
(qi223) Z3
(A
12 q)
ZA232 3
3
)
,2'3 Z nl
n
-
t
ni,A12,tl2
f3,A 23 ,P23
nl3 ,A2 3 ,IL23
(D.6)
For the sake of typographic economy, and because all subsequent manipulations will
be performed exclusively on the right-hand side, I will henceforth omit the left-hand
side. The first hyperspherical harmonic on the right-hand side may be expanded
using the addition theorem (Eq. E.1), and the second hyperspherical harmonic may
be expressed as a function of q2 3 by means of the inversion symmetry relation for
hyperspherical harmonics (Eq. C.3), resulting in:
(q 12 ) Znm (q 23 ) Z" 33
"
Z
Z1,,1,#1lpl1A2
n1 A12,A12
(D.7)
(q 23
)
A11
(-1)fl"t3
2
1l,p1
n3,A23,4123
The product of the two hyperspherical harmonics taking a common argument is then
simplified via the linearization theorem (Eq. E.3):
i A1 ,p
nl2,12 M~2
(l)
An
A2 32023
1,[" Ln2, 1 2'"n2
n1,A 1 2 , 1 1 2 Aiq1,,1,mi
nl
11 ,ml;n 3 ,A 23 ,P23
S(q 12 )
2,M2 (q 2 3
)
A2
n3,Z23,P23
A12 ,[L12
L1,m1
(D.8)
Finally, relabeling the indices (A 1 2 + Al,
/112
1, n3 + n2, A2 3
+4 12, /123
M2),
and setting the coefficients of Eq. D.8 equal to those of the left-hand side of Eq. D.6,
108
term by term, results in the following:
tn3,A127/12
1)323423
tn21n
,12,m2 AnA11,/-1
l ,1l A12,I12,l1,ml
-
n 3 ,.\ 2 3 ,4t2 3 , ,m
niiim1;n
2 2
2
(D.9)
n2,12,m2
Al 7P1
1imi
as the circuit sense symmetry relation for the coefficients of a TJDF.
Following the same procedure for the case in which the sense is not reversed, but
the starting grain is changed results in the following:
)_
tn3,A23,A23
n1,A12412
tE
R
iA,
n212,M2
1
,,#1
A1 21,/ 12 ,
1,mi Ln3,A23,23
rl,11,ml;n2,12,m2
(D.10)
2 ,12,m2
11 ,mI
as the circuit start symmetry relation.
Eqs. D.9-D.10, are somewhat complicated in form. However, if one follows the
same procedure, but considers the simultaneous operations of a change in starting
grain and change of circuit sense, one arrives at the following simple expression:
tn,(12,/12
_
)Al2+A23tn3,A23,P23
n 1,A12,1112
n3,A23,J123
(D.11)
As only two of the three triple junction symmetry relations are independent, the use
of Eq. D.11 obviates the need for one of Eqs. D.9-D.10. Furthermore, Eq. D.11 is
clearly preferable for most applications, such as the identification of the independent
TJDF coefficients, which is its primary use in the present work (see Section 3.4).
109
110
Appendix E
Mathematical Theorems
Here I provide several mathematical results that are both necessary for the main
results of this paper and of great use generally for the mathematical treatment of
microstructures.
E.1
Addition Theorem
Mason [271 provided an addition theorem for the hyperspherical harmonics, which
I reproduce here in order to make our notation, which differs from that of Mason,
explicit. A hyperspherical harmonic, whose argument is the result of the composition
of two rotations, may be expanded according to:
Zj'(q2 qi) =
A(q2)
(E.1)
Z',m, (qi)
12,m2
11 ,rn
with the coefficients of the addition theorem provided by:
- (-1)
',"
A 12,M2,1i,1-
1
272 (212 + 1) (21, + 1) C'm
n
+1 12,2, ,Ml n/2
12
11
n/2
n/2
(E.2)
E.2
Linearization Theorem
Orthogonal polynomials come equipped with a linearization theorem, which allows
one to express the product of two polynomials of like argument, but differing degrees (and possibly different parameters), as a linear combination of other polynomials [1771. The linearization theorem for the hyperspherical harmonics takes the
following form:
n3I+n2
n3
3
L n13M3
12
Z 1,M Z
1 Z12,m2n3=jnj-n2| 13=0 M3=-13
111
Z"
(E.3)
In Eq. E.3 all of the hyperspherical harmonics take identical arguments, which are
1
omitted for brevity. The linearization coefficient, Ln3,
3,M3
issfud
found byyml
mulnl,l1,ml;n2,12,M
2'
tiplying both sides by the complex conjugate of another hyperspherical harmonic of
the same argument and integrating over the entire domain:
nl+n 2
Z
Z2
,
=
13
n3
(E.4)
1 L" . Z3
n3 fL
n3=Jn1 -n2| 13=0 M3=-13
The integral on the right-hand side is equal to fl3,n' 613, 1'mm
of the hyperspherical harmonics, which results in:
6
Ln3,I3,M
n1 11,ml;n2,12,M2 =
i
3
by the orthonormality
Z1
Zn2
Zn3*
1,7i
'12,M2
13,M3
(E.5)
In Eq. B.7, I provided the solution to integrals of the form that appears in Eq. E.5.
Substituting these results into Eq. E.5 and simplifying yields:
(ni + 1) (n 2 + 1) (n3 -1 (21 1 -- 1) (212 +
27r2ilim
L n3,13,m3
ni,1i,m1;n2,12,M2
)
n1/2 n 2 /2
13,M3
2
n1/2
n2/2 n3/2
11
12
(E.6)
Meremianin [30j derived a similar result for what he referred to as the sphericaltype hypersphericalharmonics or C-harmonics, Cn,i,m, which are related to the hyperspherical harmonics used in this work by a normalization factor1 : Znm =
7Cn,,m.
Meremianin refers to his result as the C-type Clebsch-Gordan Coefficients, Cn33"3
ni1imi;n212M21
1
and they are related to our Ln3,
3,3
by the following:
nl,I1,ml;n2,12,m2bytefloig
L fli
11ii,13,n23
i ;n2,12)M2
(ni + 1) (n2 + 1)Cn33M3
nilimi;n22m2
272 (n3 + 1)
(E.7)
Because Meremianin did not explicitly state the phase convention of what he calls "... the usual
3D-spherical harmonics ... " [301, it is possible that there is also a missing factor of (-1)'.
112
n3/2
13
1
Appendix F
Error Scaling for Bicrystal
Experiments
In attempting to
tal experiments,
that I test has a
value of D1 that
infer the parameters of the constitutive equation (Eq. 5.2) via bicrysthe parameter D 1 will be recovered exactly if at least one bicrystal
misorientation w < wt, assuming that there is no uncertainty in the
I measure. Likewise, D2 will be recovered exactly with at least one
bicrystal having w > wt. To recover the final parameter, wt, to within some specified
accuracy (%ERR), there must be at least one bicrystal with misorientation falling in
the range [wt - 6, wt] and another bicrystal with misorientation falling in the range
[Wt, Wt + 6], where %ERR = 6/t. For small 6, the error in wt will be larger than that
of both D 1 and D 2 . Specifically, this will be the case for 6 < wt. I wish to identify
the number of bicrystal experiments, N, required to recover wt to within 6 of its true
value at a 95% confidence level.
Consider a set of N bicrystals with respective misorientations, {W1, w2 ,..., WN},
U(0, Wmax), where Wma, = 180' for the crystal system
which are sampled from Q
considered in this study . Let' A be the event that at least one of these bicrystals falls
in the interval [wt - 6, wt], and let B be the event that at least one of these bicrystals
falls in the interval [Pt, Wt + 6]. I am interested in finding N such that:
IP(A n B) = 0.95
Thus, I seek an expression for the joint probability P(A
plished by considering the complement:
P(A n B) =
(F.1)
n B).
This may be accom-
- IP(Ac U BC)
1 - [P(AC) + IP(BC) - P(Ac n BC)
'The authors wish to acknowledge David K [178], of the Math Stack Exchange community, for
suggesting the derivation provided in Eqs. F.1-F.6, which I subsequently validated both numerically
and analytically.
113
The probability that none of the N samples fall within
- 6, wt] is given by
)N
6
1
P(Ac)
[t
(F.3)
Wmax)
Likewise, for the interval [ot, wt + 6] I have
(I -
P(B0 )
ia)N
Wmax
(F.4)
The probability that none of the samples fall in the interval [wt - 6, wt + 6] is given
by
26
N
(F.5)
)
(1
P(Ac n B")
Wmax
Substituting these results into Eq. F.2 I find
N
P(A n B) = 1 - 2 1
ml
-
)N
-
(F.6)
Wmax
Substituting 6 = wt (%ERR), setting Eq. F.6 equal to 0.95, and solving for N numerically for 100 values of %ERR E {0.001. 0.002,... , 0.100} I observe the N(%ERR)
dependence shown in Fig. F-1. These results suggest a power-law dependence, and a
10.5
10
S
0
9.5 L
S
9k
S
0
0
8.5-
S
S
0
S
87.5
6.5
-6.5
-5.5
-4.5
-3.5
In(%Error)
-2.5
Fig. F-1: Solutions of Eq. F.1 for various values of %ERR E {0.001, 0.002,.
.,0.100}.
fit to the data results in the following:
N = 44.0337(%ERR)114
1 .0003
(F.7)
with a coefficient of determination equal to R2 = 1.0000 and an RMS error of 1.0407.
The 95% confidence intervals for the parameters in Eq. F.7 are [44.0216, 44.0459]
and [-1.0002, -1.0003], respectively. For comparison with the error scaling of the
localization method (See Eq. 5.11), I can rearrange Eq. F.7 to get the error scaling
law for a bicrystal approach:
%ERR = 0.0227N-0. 99 97
115
(F.8)
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