Healing of Defects in
a Two-Dimensional Granular Crystal
by
Marie C. Rice
SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN
PARTIAL FUFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
BACHELOR OF SCIENCE IN MECHANICAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
|
June 2014
MASSAC
JUL 302014
@ 2014 Marie C. Rice. All rights reserved.
LIBRARIES
The author hereby grants MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole
or in part in any medium now known or hereater created.
Signature redacted
Signature of Author:_
_
Department of Mechanical Engineering
May 27, 2014
Certified by:.
I
LI
I
-
-
Signature red 3cted
Pedro M. Reis
Assistant Professor of Mechanical Engineering and Civil & Environmental Engineering
Thesis Supervisor
Signature redacted
Accepted by:
Anette Hosoi
Professor of Mechanical Engineering
Undergraduate Officer
UV
Healing of Defects in
a Two-Dimensional Granular Crystal
by
Marie C. Rice
Submitted to the Department of Mechanical Engineering on
May 27, 2014 in partial fufillment of the requirements for the
degree of Bachelor of Science in Mechanical Engineering
Abstract
Using a macroscopic analog for a two dimensional hexagonal crystal, we perform an experimental investigation of the self-healing properties of circular grain defects with an emphasis
on defect orientation. A circular grain defect is introduced into a nearly perfect hexagonal
array of millimeter-sized spherical brass particles enclosed in a square tray. The array is
oscillated uniaxially, causing the particles to vibrate randomly with respect to each other,
which in turn induces the curved grain boundary around the misoriented defect region to
migrate toward its center of curvature. Images of the healing crystal are acquired and analyzed to determine particle locations and quantify the size of the defect at prescribed time
intervals. This procedure was repeated ninety-four times in order to collect data on a range
of misorientations. In some cases, the misorientation angle varied significantly during healing so both initial misorientation and time averaged misorientation angle were considered as
possible driving variables for healing rate. Healing times were fit to an exponential curve
dependent on misorientation angle but there was a high degree of scatter from this correlation. Despite this variation in path shape, there was some correlation between healing time
and misorientation, though there was significant scatter. In an effort to identify the source
of this scatter by differentiating between defects with different healing times but nearly the
same misorientation angle, the time dependence of healing rate was investigated. This more
detailed examination of the time evolution of defect size revealed substantial variation in
time dependence type. Both linear and nonlinear time dependence of defect area size was
observed among the healing samples. The nonlinear time dependence of defect size was not
common among defects with low healing times. However, degree of linearity did not effectively distinguish between defects with similar misorientation and widely different healing
times nor did it correlate meaningfully with misorientation. Though the self-healing behavior of grain defects has not yet been fully characterized, there is evidence that geometric
parameters influence overall healing time.
Thesis Supervisor: Pedro M. Reis
Title: Assistant Professor of Mechanical Engineering and Civil & Environmental Engineering
Acknowledgements
I would like to express my greatest thanks to my supervisor Professor Pedro Reis for his
insightful and positive support. Thank you very much for creating this phenomenal introduction to academic research that has taught me innumerable lessons this past year.
In addition, I would like to acknowledge Francisco L6pez Jimenez for his image analysis code
and thank him for his feedback during the writing process. Ryan McDermott's work on the
developement of the experimental apparatus was similarly invaluable. To all the members of
the EGSLab, thank you for providing a delightful and encouraging laboratory environment.
Contents
Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Characterizing grain boundaries . . . . . . . . . . . . . . . .
Theories of grain boundary structure . . . . . . . . . . . . . . . . .
1.2.1 Coincidence site lattices .. .. . . . ..............
1.2.2 Atomic relaxation and periodicity of special grain boundaries
Grain boundary migration . . . . . . . . . . . . . . . . . .
1.3.1 Early models of grain boundary migration . . . . .
1.3.2 Computational models and experimental observations
1.3.3 Observations from macroscopic crystal analogs
.
1.3
.
.
1.2
.
.
1.1
.
.
.
.
.
.
.
.
.
.
.
.
2.4.4
Particle locations . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Determining misorientation angle . . . . . . . . . . . . . . .
3.3
Identifying and delineating regions
.
.
3.1
.
.
. . . . . . . . . . . . . .
Correlation between healing time and misorientation angle .
Variation in healing path . . . . . . . . . . . . . . . . . . . .
.
3.5
4
Digital film parameters and image quality optimization
Analysis and Results
3.4
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2 Experiment
2.1 Electro-mechanical system . . . . . . . . . . . . . .
2.1.1 Optical table base . . . . . . . . . . . . . . .
2.1.2 Shaking particle bed . . . . . . . . . . . . .
..........
2.1.3 Particles . . . ........
2.1.4 Electro-mechanical shaker and accelerometer
2.1.5 Air-bearing and platform . . . . . . . . . . .
2.2 Construction of the primary crystal grain . . . . . .
2.3 Grain defect generation . . . . . . . . . . . . . . . .
2.3.1 Neoprene pad . . . . . . . . . . . . . . . . .
2.3.2 Semi-circular cavity tool . . . . . . . . . . .
2.4 Image acquisition . . . . . . . . . . . . . . . . . . .
2.4.1 Camera control and suspension . . . . . . .
2.4.2 Tim ing . . . . . . . . . . . . . . . . . . . . .
2.4.3 Illumination . . . . . . . . . . . . . . . . . .
3
9
9
Introduction
.
1
.
.
.
.
.
36
36
39
42
43
45
50
Conclusion
5
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
2.1
2.2
Two dimensional hexagonal lattice defined by lattice vectors a' and d where
lai = a2 l and #= 120* . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A grain boundary between two square crystals that has misorientation angle
O about an axis normal to the page. Figure adapted from [17]. . . . . . . . .
The misorientation distribution for an assembly of randomly oriented grains
with simple cubic structure. Figure adapted from [17]. . . . . . . . . . . . .
A low angle symmetrical tilt boundary is composed of an array of periodically
spaced dislocations with a dislocation density that depends upon misorientation angle. Figure adapted from [17]. . . . . . . . . . . . . . . . . . . . . . .
Relative boundary energy with respect to misorientation about the < 110 >
axis for tilt boundaries in aluminum. Figure adapted from [15]. . . . . . . .
a.) A coincident site lattice formed by two superimposed simple cubic lattices
of misorientation 36.90 about an axis normal to the paper. The points of
only one of the misoriented lattices are marked with either circles or crosses
and the coincident sites, which are part of both lattices, are marked with solid
circles. Figure adapted from [10]. b.) A coincidence site boundary for crystals
misoriented 380 from each other about an axis normal to the paper. Given
the symmetry of a hexagonal lattice, a misorientation of 380 also corresponds
to a misorientation of 22'. Figure adapted from [10]. . . . . . . . . . . . . .
The lowest boundary energy configuration is not the pure CSL arrangment
but does retain periodic structure. Figure adapted from [10]. . . . . . . . . .
a.) Boundary energies of symmetrical (110) tilt boundaries in aluminum calculated with molecular dynamics simulations. b.) Experimental values. Figures
adapted from [15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Misorientation and temperature dependence of (111) tilt boundaries in 99.999
at% aluminum. Figure adapted from [25]. . . . . . . . . . . . . . . . . . . .
a.) Triple junction in a Bragg-Nye bubble raft. Figure adapted from [6]. b.)
Symmetrical tilt boundary with misorientation angle 9 = 250. Figure adapted
from [23]. c.) Three-dimensional raft made from multiple layered sheets of
bubbles. Figure adapted from [6]. . . . . . . . . . . . . . . . . . . . . . . .
Asymmetric curved grain boundary in a dynamic ball model. Figure adapted
from [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a.) CAD model of the horizontally oscillating tray actuated by an electromechanical shaker. b.) Photograph of the electromechanical system without the
granular crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic forces between magnetized steel particles may induce clustering. . .
6
9
12
13
14
15
16
17
17
20
22
22
24
25
2.3
2.4
2.5
2.6
2.7
2.8
2.9
a.) Single vacancy b.) Double vacancy . . . . . . . . . . . . . . . . . . . . .
Low particle density lattice defect region . . . . . . . . . . . . . . . . . . . .
Subgrains created by dislocations in the crystal . . . . . . . . . . . . . . . .
A neoprene pad adhered to a compliant base was used to rotate crystallographic axes of the grain defect by translating the particles in the defect
region to grain defect lattice sites as a rigid block. . . . . . . . . . . . . . . .
PMMA disk with two thin-walled semicircular cavities used to make grain
defects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a.) Time from discrete time stamps is smoothed to reflect the true image
timing. b.) Smoothed time is linear and matches the raw recorded time from
the image time stamps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LED pad grid lines have vertical and horizontal alignment that deviates little
from im age axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
29
30
. . .
35
2.10 Cropped image of the granular crystal with a misoriented grain defect.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
a.) Original color image b.) Greyscale image c.) Binary image d.) Binary
image with reflections filtered e.) Binary image with eroded edges to ensure
particle separation f.) Greyscale image overlaid with particle centroid locations marked by crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
There was a small disparity between the particle locations obtained using
the reflection method and those from the silhouette method. Particle indices
which have a disparity of one particle diameter correspond to particles which
were detected by the projected area method but not the reflection method.
Excluding undetected particles, the disparity was 0.0172 t 0.008 particle diameters for this example frame. More than 99 % of all detected particles had
a disparity below 0.04 particle diameters. . . . . . . . . . . . . . . . . . . . .
Local lattice orientation may be determined by decomposing the edge angles
from the delaunay triangulation into multiples of 7r/3 and the additional offset
from the image horizontal axis due to lattice orientation. ...........
Histogram of local lattice vectors obtained from the delaunay triangulation
edge angles of particles belonging to hexagonal lattices. Maximum angle occurrence denotes the orientation of the primary crystal and the next highest
peak occurs at the orientation of the grain defect. . . . . . . . . . . . . . . .
Misorientation angles drifted slightly from initial values during the healing
period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. . .
a.) Each particle was assigned to the grain defect, primary lattice, or neutral
group according to the orientations of its delaunay triangulation edges. Blue
solid circles mark particles belonging to the main grain, red triangles mark
particles in the defect, and green squares mark particles which cannot be assigned to either grain. b.) The particles that were engaged in grain boundary
migration are those in proximity to the opposite grain or neutral particles. .
a.) Total healing time loosely follows an exponential curve dependent on
32
32
34
35
38
39
40
41
41
42
misorientation angle Theal = To exp(a/#). Defining misorientation angle as
the intial misorientation angle and performing a least squares regression gives
Theal = To exp(a/#) where To,i = 0.90320 s and /3 = 0.069+0*007 rad. b.)
Defining misorientation angle as the mean misorientation for the complete
trial yields fitting constants To,m = 1.270.40 s and m = 0.0720_0 rad.
7
.
.
44
3.8
Healing time and defect size time dependence varies dramatically from trial
to trial, even for nearly equivalent misorientation angles. The color of the
trace corresponds to the relative misorientation angle: bright red for relatively
high mean misorientation angles near ir/6 and bright green for relatively low
misorientation angles, near 7r/18. Each plot covers a different time domain: 0
to 4000 s, 0 to 400 s, and 0 to 40 s in order that the time dependence data for
both rapidly healing and slowly healing trials are visible. Each plot contains
data from all ninety-four trials. . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Total healing time for six selected samples with mean defect angles between
57r/36 rad and 77r/48 rad. The time dependence of defect size does not have
a consistent shape across all samples. The area of some defects decreases
linearly with time but other defects heal in rapid bursts between long periods
of constant area.
..
.............................
3.10 The coefficient of determination, R2 , for the linear regression of defect size and
time for each trial was plotted with respect to misorientation angle. Though
the lowest R2 value is also near the maximum misorientation angle, the rest of
the data does not show a correlation between the shape of the healing profile
and the misorientation angle. Defects which follow a linear path are marked
with blue, defects which have a stepped path are colored magenta, and defects
which fit into neither of these categories are marked in green. . . . . . . . . .
3.11 The healing time for each defect is plotted with respect to coefficient of determination, R2 , for the linear regression of defect size and time. The W2 value
for a the healing profile of a defect serves as an approximate measurement of
linearity. High R2 are found among defects which healed in a short time and
a long time but low R2 values are not common among defects which healed
quickly. Defects which follow a linear path are marked with blue, defects
which have a stepped path are colored magenta, and defects which fit into
neither of these categories are marked in green. . . . . . . . . . . . . . . . .
8
46
47
48
49
Chapter 1
Introduction
1.1
Crystal structure
Crystalline solids exhibit a periodic atomic structure that is described by a characteristic
repeating unit cell. A corresponding lattice describes the spatial and geometric relationships
amongst atom locations within a given crystal structure. Simple cubic, body centered tetragonal, face centered orthorhombic are examples of three-dimensional lattices and rectangular,
rhombic, hexagonal are examples of two-dimensional lattices. This study examines the behavior of a two-dimensional hexagonal crystal which follows the lattice shown in Figure 1.1.
*
*
00
0
*
00
0
00*0
0
*
000
*
00
0
0
0
0
*000
00
00
00
00
000
00
0000
0
0
a2
0
Figure 1.1: Two dimensional hexagonal lattice defined by lattice vectors a-, and d-~ where jail
1200
9
la2 I and
Since two-dimensional lattices describe planes of three-dimensional lattices, two-dimensional
crystals can act as simplified versions of three-dimensional crystals. For example, the hexagonal lattice is analogous to the close-packed plane of a face-centered cubic lattice [40].
Unlike monocrystalline materials like diamond or quartz, polycrystalline metals are composed of multiple adjacent grains, which are crystalline regions that are oriented differently
from one another. The interfaces between these neighboring grains are called grain boundaries. The effects of grain boundary migration and orientation on the mechanical properties
of a material have been studied analytically [26], computationally [47], and experimentally
[44]. Given that concepts derived from circle and sphere packing are useful to describe unit
cells and lattices, studies of grain boundaries sometimes involve the use of a macroscopic
crystal analog made from small spherical particles like the bubble raft model and the dynamic ball model [18]. These systems can have vacancies, dislocations, and grain boundaries
[6] which behave similarly to those in real crystals. These systems are also useful because
they can be controlled in ways that real crystals cannot [29]. Thus, the robust vocabulary
of crystallography may be applied to macroscopic analogs in order to characterize their behavior and the properties of these analogs may be used to gain insight about microscopic
crystalline materials in return.
The crystal analog model used in this study is a ball model, which is composed of a shaking
bed filled with a single layer of small spherical particles arranged into a two dimensional
hexagonal lattice. The model is applied to the study of self-healing grain defects in this
kind of crystal. Previous research has revealed the existence of grain boundaries in twodimensional crystals and computationally or experimentally determined their impact in a
number of material properties, such as electron transport [41] and intrinsic strength [20].
Grain size and grain boundary type also affect the properties of three-dimensional crystalline materials and have been the subject of many preceding studies [17, 8]. In this thesis,
the geometric aspects of grain defect self-healing are studied with a granular analog. In
particular, an understanding of the relationship between healing rate and relative difference
10
in lattice orientation between the primary grain and the defect grain is sought. Establishing
this kind of relationship would enable the identification of defects as self-correcting, easily
removed, or resistant to removal methods.
1.1.1
Characterizing grain boundaries
Grain boundaries are most often described by their geometry. A three dimensional flat grain
boundary has five degrees of freedom and is defined by the axis of rotation, the misalignment angle about that axis, and the inclinationplane of the grain boundary itself. However,
the inclination plane is difficult to characterize experimentally and is not part of the common axis/angle notation used to describe grain boundaries in most studies [17]. The axis
of rotation is expressed with respect to the symmetry of the crystal structure as a family of
directions such as (110). The misorientation angle is the rotation about this axis that describes the difference in alignment for the two crystals. In two dimensions, the misalignment
angle and inclination are sufficient to describe the overall geometry of a straight boundary
since there is only one possible misalignment axis and all grain boundaries are tilt boundaries. Figure 1.2 shows an example of a tilt boundary between two two-dimensional crystals
misoriented at an angle 0 about an axis normal to the page.
The relative prevalence of different misorientations may be obtained using electron backscatter diffraction and used to construct Makenzie plots like the one in Figure 1.3. A Mackenzie
plot shows the relative frequency of occurence for the range of grain boundary misalignment
angles within a material sample. Figure 1.3 gives the occurence frequency of misalignment
angles about all axes of rotation in a simple cubic sample of randomly oriented grains. For
a material sample of this kind, the mean misorientation angle is 400.
It is common to categorize grain boundaries according their misorientation angle. Grain
boundaries with misorientation angles above a certain threshold, of the order 0 = 100 but
varying depending upon the type of lattice, are considered high angle grain boundaries and
grain boundaries with misorientation angles below this threshold are considered low angle
11
Figure 1.2: A grain boundary between two square crystals that has misorientation angle
normal to the page. Figure adapted from [17].
6 about an axis
grain boundaries. This demarcation reflects the observation that the properties of high angle
boundaries tend to follow different trends than those of low angle boundaries.
Low angle grain boundaries, like the one shown in Figure 1.4, exist between slightly misoriented grains and are the result of an array of periodically spaced edge dislocations. An edge
dislocation is a crystal defect in which a row or plane of atoms begins within the main body
of a crystal, resulting in a disruption of the lattice. Accommodating this disruption results
in a slight difference in axis orientation between the two sides of the added lattice plane.
The Read-Shockley model for low angle boundaries proposes that the density of dislocations
depends upon the misalignment angle and can be used to calculate the grain boundary energy [34]. Experimental results support this theory in the case of low angle grain boundaries
but the Read-Shockley model cannot explain the behavior of all high angle boundaries.
Therefore, it has been suggested that low angle boundaries are distinguishable from high
angle boundaries because they have properties which depend upon misalignment, whereas
the majority of high angle boundaries have properties which are independent of misalignment [17]. Nevertheless, some high angle grain boundaries with particular misorientation
12
4
3
2
WI
0
0
10
30
20
Woftdng
40
60
60
70
bn (*)
Figure 1.3: The misorientation distribution for an assembly of randomly oriented grains with simple cubic
structure. Figure adapted from [17].
angles have characteristic structure and corresponding properties so this delineation is not
exact. These "special" high angle grain boundaries are frequently considered separately from
"random" grain boundaries, which do not have noticeable orientation-dependent properties.
The concept of grain boundary engineering rests upon the advantages of having a high ratio
of special boundaries to random boundaries and seeks to maximize this ratio in order to
achieve more favorable material properties.
1.2
1.2.1
Theories of grain boundary structure
Coincidence site lattices
The existence of special grain boundaries was identified from experimental observations of
cusps in grain boundary energies at certain misorientation axes and angles. Figure 1.5 shows
the measured grain boundary energies for a set of tilt boundaries in aluminum. The minima
in boundary energy roughly correspond with misorientation angles which produce boundaries
with good atomic fit. The concept of a coincidence site lattice (CSL) was proposed to explain
observations of non-monotonic variation in boundary energy and mobility with crystal lattice
theory. A coincident site lattice is the lattice of points which is produced when two differently
oriented grains are superimposed upon one another and translated such that there is at least
13
Figure 1.4: A low angle symmetrical tilt boundary is composed of an array of periodically spaced dislocations
with a dislocation density that depends upon misorientation angle. Figure adapted from [17].
one overlapping lattice site [21] . An example of a coincident site lattice constructed from
two square lattices misoriented by 36.9* is shown in Figure 1.6a.
The ratio of coincident lattice sites to total lattice sites may be obtained and if the reciprocal
of this ratio is a small whole number, the boundary between grains of these orientations may
be characterized by that parameter. For example, two lattices which have one in five lattice
sites coincident with each other, has E5 coincidence and is called a E5 boundary.
The
explanatory power of CSL theory is that those boundaries with low E values have good
atomic fit and most exhibit special or unique properties [21].
Within a real material, exact agreement with the geometric relationship required for a particular CSL is statistically unlikely. However, the special properties associated with CSL
boundaries persist despite small deviations from the ideal misorientation angle. Brandon
[7] proposed that these deviations from the ideal may be explained by a low density array
of dislocations superimposed onto a perfect CSL boundary. The Brandon criterion gives
a relationship for the acceptable angular deviation limit and the accompanying reasoning
suggests that real grain boundaries will have a stepped structure. This stepped structure
will have facets of high coincidence site density alternating with regions of disorder but the
details of this structure will depend upon the boundary plane and the misalignment angle [7].
14
9 113
3. =T
3119
2 2.5
2.0
oL1.5
ma,
201.0
-
0.5
110>
0
40
80
120 160
Misorientation ( 0)
Figure 1.5: Relative boundary energy with respect to misorientation about the < 110 > axis for tilt boundaries in aluminum. Figure adapted from [15].
Thus, the CSL with superimposed dislocation array model advanced by Brandon represents
a synthesis of the Read-Shockley model for low angle grain boundaries and the concept of a
CSL.
1.2.2
Atomic relaxation and periodicity of special grain boundaries
The coincident lattice site theory and concept of boundary coincidence are useful for describing geometric arrangements that result from intersecting lattices of discrete particles, but do
not take into account the boundary energy concerns of real crystal systems. To account for
boundary energy effects, molecular dynamics analyses have been used to evaluate and construct models of grain boundaries [10, 43, 42, 46]. The early results of this kind of study show
that CSL configurations and proposed accommodating mechanisms do not necessarily reflect
the true atomic arrangement at special grain boundaries. Molecular dynamics simulations
show that the boundary energy can be reduced by translating one side of a strict CSL grain
boundary with respect to the other. Additionally, further relaxation of atoms moving away
from their rigid lattice location may result in a different and more energetically favorable
structure. For these types of translations and other structural arrangements derived from
molecular dynamics, it has been suggested that special boundaries are characterized as those
15
o
+
6
+
0 +
+
+
+
+ + *+a0+
0
+
+
+
e
+ +o
0
0
+
+
+
+
+4 ++ +4
0
0
g +
+
0
0
+ 0+
++
+00a0
+0+ +
+
4.
O0
+ 0+0
0
0
4+
+ +
+
0
0
+ 0+
*
+-
0
+0+
0+
~0
0
o+
0
0
+ +0+0+
++ + + + +
+0
0
+
+
+
++
+
0T
0
+
+
+
+
+
+
(b)
(a)
Figure 1.6: a.) A coincident site lattice formed by two superimposed simple cubic lattices of misorientation
36.9* about an axis normal to the paper. The points of only one of the misoriented lattices are marked
with either circles or crosses and the coincident sites, which are part of both lattices, are marked with solid
circles. Figure adapted from [10]. b.) A coincidence site boundary for crystals misoriented 380 from each
other about an axis normal to the paper. Given the symmetry of a hexagonal lattice, a misorientation of
380 also corresponds to a misorientation of 220. Figure adapted from [10].
with small structural units [10].
Experimental studies have largely confirmed the results of molecular dynamics analyses
of grain boundary energy, which emphasize periodic structure in special grain boundaries
[13, 30, 32, 44, 33, 46]. In addition, experimental determinations of boundary energies and
those calculated using molecular dynamics simulations show similar trends with respect
to misorientation [15, 24, 27]. Grain boundary energies may be obtained from the angles
subtended by each grain at a triple junction in a tri-crystal, which are observable with high
resolution electron microscopy [12]. Based on the work of Smith [35] and Wolf et al. [46],
it is also likely that the free volume intrinsic to almost all grain boundaries plays a role in
boundary energy at the microscopic level.
Though the atomic arrangement corresponding to a geometrically pure and populated CSL
does not minimize boundary energy, the orientational relationships highlighted by this model
are still valid and the E notation developed for coincident site lattices remains the primary
16
Figure 1.7: The lowest boundary energy configuration is not the pure CSL arrangment but does retain
periodic structure. Figure adapted from [10].
= 9 11 3
3 11 9
3.0
600
CD
500
CD-
400
300
0
3 11 9
2.5
2.0
1.5
DC
- w 1.0
0.5
M (D
200
0
E
I = 9 11 3
100
<110>
I
ri
0
40
I
'80
Il
I
I
120
-
0
160
40
80 120 160
Misorientation ( 0
)
)
Misorientation ( 0
<110>
(b)
(a)
Figure 1.8: a.) Boundary energies of symmetrical (110) tilt boundaries in aluminum calculated with molecular dynamics simulations. b.) Experimental values. Figures adapted from [15].
method for describing boundaries with special properties. Gleiter [10] suggested that the
orientational relationships that underlie coincident site lattice theory still influence which
boundaries possess special properties by governing the lengths of periodic structural units.
For grain misorientations deviating from ideal coincident orientations, the structure may
be built from the structural units of neighboring ideal orientations.
Similarly, for non-
symmetrical boundaries on different planes, the boundary will contain multiple types of
structural units, resulting in a faceted structure with an average orientation that is in agreement with the overall misalignment.
17
1.3
Grain boundary migration
Grain boundaries migrate as an intrinsic result of the growth of one grain and the corresponding shrinking of the adjacent grain. At the atomic level, grain growth occurs by atoms
moving from the lattice sites of the shrinking grain to those of the growing grain. The exact
mechanism by which atoms migrate to adjacent lattices depends upon the boundary structure. The microstructure of a boundary depends on the misorientation, boundary plane,
and point defects like vacancies and solutes [17]. Single grain boundaries in bicrystals have
been studied computationally and experimentally in order to characterize the dependence
of individual grain boundary mobility on structural factors and thermodynamic parameters
such as temperature [10]. Other studies of grain boundary migration seek to describe the
overall effects of grain growth and consider statistical distributions of grain sizes resulting
from recrystallization. However, given that the subject of the current project is a single
grain boundary, the review of the literature presented here will focus on isolated boundaries
in bicrystals.
1.3.1
Early models of grain boundary migration
Early studies by [4] identified that grain growth among recrystallizing grains were related to
misorientation dependence. More specifically, grain boundaries migrate toward their centers
of curvature during annealing, resulting in the shrinkage and eventual disappearance of some
grains and the complementary growth of others [3]. For curved boundaries, the pressure
acting on the boundaries which causes them to move follows p = iw- where o is the free energy
of the grain boundary and K is the radius of curvature [1]. Using these observations, Mullins
[26] proposed a mathematical model for the motion of grain boundaries shaped similar to
plane curves that obeyed this relationship, with particular attention to curves that maintain
similarity as they contract. The concept of mobility was introduced in a model proposed by
Turnbull [38] to define how the speed of a grain boundary varied with the pressure applied to
18
it. In this model, the boundary migration speed followed the relationship S = pM, where p is
the pressure on the boundary and M is its mobility [38]. The mobility in this model depends
upon several other parameters, including grain boundary orientation, lattice misorientation
angle, and alloy content of the material [10], which makes building a complete theory of
mobility challenging. The influence of these variables on migration speed in both curved and
tilt boundaries and the development of a unified theory has been the focus of subsequent
research in this area.
1.3.2
Computational models and experimental observations
Molecular dynamics models [39, 47, 14] and experiments with high purity metal samples
[36, 12, 21, 28, 11, 2, 39] have been used extensively to characterize the dependence of grain
boundary migration on temperature and misorientation. In general, these studies tend to
rely upon capillary driven migration of curved boundaries in bicrystals [17].
Overall, the temperature dependence of mobility follows an Arrhenius type relationship
[2, 36]. Molecular dynamics modeling by Wolf [45] gives evidence that special boundaries at
high temperatures transition to a more disordered state. This conclusion agrees with observations that the difference in mobility rates between special and random grain boundaries is
attenuated at high temperatures [31]. Increases in temperature also cause a discontinuous
increase in boundary mobility among certain boundaries [19, 37, 12].
With regard to grain misorientation, the prevailing consensus is that the behavior of random grain boundaries does not depend on misorientation but the behavior of special grain
boundaries does. Grain boundaries in different materials tend to be studied individually.
For example, experimental studies have identified that the maximum migration rate for aluminum grain boundaries occurs in those which are misoriented 400 about the (111) axis
[22, 28, 11]. Other structures with special properties can be summarized in tables by E
value as well as axis/angle notation [17].
Recent molecular dynamics simulations have made substantial progress in matching experi19
W1IUIA 360C
0 410'C
ftf
10?2
34
36
38
40
Misorientation
42
44
(*)
Figure 1.9: Misorientation and temperature dependence of (111) tilt boundaries in 99.999 at% aluminum.
Figure adapted from [25].
mental results. In particular, the systematic investigations into tilt boundaries in a variety of
metals show roughly the same dependence on temperature, curvature, and misorientation as
is seen in experimental studies [47]. One example of this collection was the analysis of curved
E5 asymmetric tilt boundaries rotated about the (010) axis in nickel [39]. In this report, it
was shown that misorientation angles corresponding to low index boundary planes, particularly E7, have high mobility rates and low boundary enthalpy compared with other grain
boundaries. Though migration rates and enthalpies from the simulation differed slightly
in magnitude from experimental results from real nickel bicrystals, the study supports the
long-held assertion that boundary properties may be correlated with boundary structure
[39].
1.3.3
Observations from macroscopic crystal analogs
The development of crystal structure theory has also had a substantial experimental component. Macroscopic analogs like the Bragg-Nye soap bubble raft model [6] and dynamic ball
model [29] are well suited for building intuition about the mechanical behavior of crystalline
materials and in some cases provide quantitative information about their behavior. In these
20
kinds of experimental models, spherical particles are analogs for atoms in crystalline materials and when bubbles or balls arrange themselves in a close packed lattice, parallels between
the model's structure and crystalline structure emerge [6, 40, 29].
Bubble rafts are sheets made of small, uniformly sized bubbles arranged into a pattern similar
to the close packed plane of an fcc crystal. One advantage of this model was the ability of the
rafts to sustain tension and compression. The attractive forces between bubbles are capillary
forces due to surface tension and the repulsive forces come from the pressure inside each
bubble. The low friction between bubbles sliding past each other during plastic deformation
also distinguished the bubble raft model from its predecessors. The early work using bubble
rafts focused on their elastic and plastic mechanical properties [6, 5] and the nature of
grain boundaries between crystals was examined later [23]. Examples of grain boundaries
in bubble rafts are presented in Figure 1.10. Subsequent work focused on comparisons with
geometric and other theoretical models [16, 18]. In some cases, the bubble rafts were vibrated
as to simulate temperature effects. In one such study, Ishida [18] identified multiple order
structures for each coincidence orientation, which varied based on vibration and bubble size.
This observation complements theoretical and computational investigations of the energy
reducing effects of realigning the sides of strict CSL boundaries in certain ways. More
recently, this model has also found applications in visualizing and quantifying indentation
mechanics [9, 40].
Similar to the bubble raft model, the ball model derives from an analogy between spherical particles and atoms. The original ball models involve two-dimensional arrays of ferromagnetic steel balls immersed in an alternating magnetic field, which engage in random
interatomic vibrations that mimic thermal motion. One advantage of this kind of model is
that it directly incorporates the concept of an energy minimizing distance between adjacent
particles, which is important for characterizing atomic interactions in real crystal systems
[29]. An example of a grain boundary in a ball model is presented in Figure 1.11.
A similar model may be constructed from an array of non-ferromagnetic metal spherical
21
(a)
(b)
(c)
Figure 1.10: a.) Triple junction in a Bragg-Nye bubble raft. Figure adapted from [6]. b.) Symmetrical tilt
boundary with misorientation angle 0 = 250. Figure adapted from [23]. c.) Three-dimensional raft made
from multiple layered sheets of bubbles. Figure adapted from [6].
particles in a vibrating tray. The particles engage with the tray through frictional interactions, causing the particles to vibrate with respect to each other while maintaining roughly
even distribution. The details of the ball model used in this study are presented in the next
chapter.
d
d
r
b
Figure 1.11: Asymmetric curved grain boundary in a dynamic ball model. Figure adapted from [29].
22
Chapter 2
Experiment
In our experiment, we constructed a two-dimensional crystal analog from a granular material
composed of millimeter-sized spherical particles. Then, we. rotated a circular section of the
crystal with respect to the original lattice to create a grain defect. The primary parameters
that characterize this grain defect are the misalignment angle and its size. The misalignment
angle is defined as the difference in orientation between the lattice vectors of the main crystal
grain and those of the new defect grain. Size is measured as the area occupied by particles
in the misaligned grain. The tray of particles that comprise the crystal was subjected to
a uniaxial sinusoidal force, causing the particles to vibrate with respect to each other and
behave analogously to a thermalized crystal. Each trial consisted of creating a single defect
at the center of the crystal array, setting the shaking bed into motion, and observing the
evolution of the defect until it had completely disappeared. To monitor the behavior of the
primary crystal and misoriented region, a camera was suspended above the tray to capture
images of the arrangement of particles in the shaking bed at discrete time points during
oscillation. The subsystems that comprise the experimental apparatus are described below.
23
2.1
Electro-mechanical system
A horizontally oscillating tray and the electromagnetic shaker (2075E, Modal Shop) that actuates it were the primary components of the electro-mechanical system presented in Figure
2.1. As the shaker applied a sinusoidally varying force, the tray oscillated and stick and slip
interactions between the spherical particles and the surface of the moving tray caused the
particles to vibrate with respect to each other. The peak to peak acceleration, a, and frequency,
f,
of the oscillations characterize the motion of the tray and governed the movement
of the particles.
(a)
(b)
Figure 2.1: a.) CAD model of the horizontally oscillating tray actuated by an electromechanical shaker. b.)
Photograph of the electromechanical system without the granular crystal.
24
2.1.1
Optical table base
The complete electromechanical system was fastened to an optical table resting on quickinstall vibration-damping mounts (60915K74, supplier: McMaster) above a granite table.
The vibration-damping mounts were placed beneath the corners of the optical table in order
to isolate the system from external vibrations, damp the propagation of vibration from the
shaker to the camera support structure, and enable precise leveling of the particle tray. The
height of each mount was adjusted by varying the engaged thread length of the rod that
connected the upper and lower damping pads. To precision level the tray, several thousand
particles were placed into the shaking bed and the heights of the mounts were adjusted until
the particles showed negligible bias toward particular sides or corners of the tray.
2.1.2
Shaking particle bed
The granular material was placed in a square tray with a glass surface and PMMA walls.
The tray interior was 12.2 mm deep and had edges of length 146.3 mm. The sides of the tray
were made from a square PMMA frame that fastened to the outside edge of the tray base.
This base was made from a PMMA sheet, an LED light pad (290406060120, Rosco), and a
thin sheet of transparent and scratch-resistant Gorilla glass (8410T45, supplier: McMaster)
layered on top of each other.
nan
A
A"--w
gA
IM
mW
AK0
am
Figure 2.2: Magnetic forces between magnetized steel particles may induce clustering.
25
2.1.3
Particles
Steel particles were used in the initial phase of this study but were replaced by brass particles
during data collection. The first set of particles that were used in the development of the
experimental design were spheres of 440 C stainless steel with diameter d = 1.5000
0.0025 mm (1598K17, supplier: McMaster). However, the steel particles became magnetized
due to their proximity to the electromagnetic shaker when they were placed in the tray.
These magnetized particles exhibited bias and irregularity in their behavior that caused some
particles to remain stationary with respect to each other in rigid blocks. Figure 2.2 shows
examples of the strands and other groupings that indicated that the magnetized steel particles
had preferential affinity for other magnetized particles. Due to this irregularity caused by
magnetization, 9380 Grade 200 brass particles of diameter d = 1.5875
0.0254 mm and
sphericity 0.0051 mm (B30006250200, Abbott Ball) were used in the final data acquisition
phase and all results presented in this report were obtained with them.
2.1.4
Electro-mechanical shaker and accelerometer
An electromagnetic shaker was used to drive the oscillation of the tray. This shaker was
mounted horizontally on a trunion base that was fastened to the optical table that held the
,
complete experimental setup. The shaker was paired with a power amplifier (2100E21-400
Modal Shop) that was cooled by natural convection and had a maximum sine force output
amplitude of 178 N and stroke length of 25.4 mm peak-to peak. The primary input signal
was generated by a function generator (ATF20B, Atten Instruments) and displayed as the
first trace on an oscilloscope. A accelerometer (353B34, PCB Piezoelectronics), was fastened
to the tray using dual magnet mounting on the side farthest from the shaker. The signal
from this accelerometer was displayed as the second trace on the oscilloscope (GDS-820C,
GW Instek) and conformed closely to a sine wave. The peak-to-peak acceleration amplitude
and frequency of this sine wave fully characterized the motion of the tray and were used
to calculate the displacement amplitude, A. The dimensionless acceleration is defined as
26
I'
= 47r2Afg2a where A is the displacement amplitude of vibration,
f
is the frequency of
oscillation, and g is the standard acceleration due to free fall. Not all vibration from the
shaker was fully damped by the leveling mounts, which resulted in vibration propagating
through the optical table into the granite slab. The camera used to collect images of the
particles in the tray was suspended by a cantilevered support structure that was anchored
to the granite slab. Vibrations in the slab traveled through the aluminum support structure
and resulted in the camera experiencing transmitted vibration from the shaker. The camera
experienced vibrations of the greatest magnitude when the shaker operated near 16 Hz.
Excessive vibration of the camera resulted in poor quality images so this frequency domain
was avoided during data acquisition.
2.1.5
Air-bearing and platform
The tray was fastened to the shaft of a linear air bearing (Atlas 1B-100, Nelson Air) in
order to reduce friction and confine the motion of the tray to pure translation along the axis
parallel to the shaker force. A coupling made of a 1.61 mm diameter nitinol rod of length
63.5 mm connected the air bearing shaft to the shaker and was held in place on each end
by a pair of set screws. This sort of coupling was chosen because it accommodated a small
axial misalignment between the air bearing shaft and the shaker.
2.2
Construction of the primary crystal grain
In order to prevent balls bouncing out of the tray, a small funnel was used to collect and
subsequently disperse particles throughout the area of the tray. Once the particles were
thermalized by tray vibration, the movement of individual particles was not directly measurable, so the frequency and acceleration of the tray were the primary dynamic variables in
this study. Given that the tray was modeled as a forced harmonic oscillator, peak-to-peak
velocity and displacement were calculated from the acceleration amplitude and input force
frequency. In order to produce measurable healing times, between ten and ten thousand
27
seconds, the dimensionless acceleration needed to be optimized.
Once an acceptable fre-
quency and tray acceleration amplitude pair were identified, the dynamic parameters were
held constant in order to focus on the influence of misalignment angle.
Because the number of particles in the tray affected the behavior of the crystal, determining
an appropriate number of particles to place in the tray was also necessary. If the tray was
overfilled, excess particles began to form a second layer on top of the original layer. If the
tray was underfilled, voids developed and the crystal lost its integrity. Therefore, the number
of particles placed in the tray was empirically optimized for the dynamic parameters selected
for the trial. Low particle counts were suitable for high dimensionless acceleration and high
particle counts were best for low dimensionless acceleration. In practice, optimization was
accomplished by adding a minimum number of particles to fill the tray at a high dimensionless acceleration and then adding more particles to maintain complete coverage as the
dimensionless acceleration was reduced. For the dimensionless acceleration that was selected
for this experiment, the ideal number of particles was approximately 9380 particles.
Merely placing the appropriate number of particles in the tray and initiating tray oscillation
were not sufficient to create a defect-free crystal that extended to the edges of the shaking
bed. Though particle agitation resulted in a general trend of increased grain size, vacancies
and dislocations persisted. Figure 2.3 shows examples of vacancy imperfections that disrupted the crystal structure. In addition, larger regions of low density were often trapped
(a)
(b)
Figure 2.3: a.) Single vacancy b.) Double vacancy
28
Figure 2.4: Low particle density lattice defect region
within a grain. These low density regions intrinsically resulted in structural defects which
are evident in Figure 2.4. Both point defects and larger defects caused by low density regions
were addressed by individually removing particles from the disordered region at the edges
of the crystal and placing them into the crystal manually. Removing dislocations was more
challenging and required a combination of actively adjusting the overall distribution of particles and varying the dynamic parameters. The adjustment of the overall distribution was
accomplished by increasing the amplitude of the input force so that the dimensionless acceleration would increase and gently applying pressure to the edges and corners of the lattice
in order to force the particle array into the defect-free highest-density packing arrangement.
When the pressure at the edge of the crystal was released, the high degree of order remained
even though the particles spread out to fill the tray once more. The effect of compressing
the lattice and releasing this pressure caused the void space from the lower density side of
the dislocation to be redistributed to the disordered edge of the crystal.
29
Figure 2.5: Subgrains created by dislocations in the crystal
2.3
Grain defect generation
Defects were created by selecting a circular region in the center of the lattice and translating
the particles as a rigid block such that the internal relative positions remained constant
while the crystallographic axes of the region rotated through a finite angle. Thus, creating
a defect was a transformation of a preexisting defect-free crystal. The original crystal was
packed at nearly maximum density but the introduction of a defect causes the evolution
of a grain boundary.
Given that grain boundaries intrinsically included some void space
due to geometric constraints imposed by the rigidity of the spherical particles, the region
containing a grain boundary had fewer particles in it than the undisrupted crystal region
that existed before it. The generation of a grain defect resulted in excess particles being
pushed on top of the original crystal layer because there was not enough space for all the
particles to remain near the grain boundary. Excess particles were removed from the region
near the grain boundary and placed at the edges of the crystal.
The defects analyzed in this experiment were created by a circular neoprene pad which
moved particles by static friction contact on their top surface.
30
Another method, which
involved a tool with two semicircular cavities that trapped and pushed particles into place,
was developed but ultimately discarded.
2.3.1
Neoprene pad
Particles in the defect region were rolled into new lattice locations by placing a compliant
neoprene pad on top of them, applying slight pressure, and rotating the neoprene pad. The
contact between the rotating neoprene pad and a given particle caused the particle to roll or
slip across the tray surface. When a large group of adjacent particles was rotated together,
the particles translated with respect to the primary grain but not with respect to each
other, resulting in a grain defect that retained the original crystal structure but had a new
orientation. The internal integrity of the defect's crystal structure was preserved best when
the neoprene pad was adhered to a compliant base made from a 13 mm thick layer of foam
attached to a PMMA disk. Neoprene pads without the foam layer made poorer contact
with the complete circular area of the defect region. In the absence of the foam layer, the
neoprene pad needed to be nearly perfectly flat and horizontal in order to evenly distribute
a high enough normal force to maintain a static friction contact with each of the balls. The
addition of a foam base resulted in a more even distribution of load at the foam-neoprene
interface and the compliance of the foam enabled the neoprene pad to tilt to conform to the
exact plane defined by the tops of the balls.
Though effective at maintaining order within the defect, this method was susceptible to
displacing too many particles into a second layer. To prevent this, a transparent PMMA
sheet with a circular cutout was placed over the particle layer to constrain the vertical
movement of the particles while the defect was being created. The sheet was dimensioned
such that it fit into the tray frame with a small clearance so that the circular cutout could
be used to place the defect at the center of the tray. Once the defect had been introduced,
this tray was removed so it did not interfere with the horizontal movement of the particles
during oscillation and healing.
31
Figure 2.6: A neoprene pad adhered to a compliant base was used to rotate crystallographic axes of the
grain defect by translating the particles in the defect region to grain defect lattice sites as a rigid block.
Figure 2.7: PMMA disk with two thin-walled semicircular cavities used to make grain defects.
2.3.2
Semi-circular cavity tool
Though the neoprene pad was the primary tool used in this study, an alternative tool was
also designed. This tool moved particles by trapping them within two semicircular cavities
and rotating the cavity filled with trapped particles. First, the cavity side of this tool was
placed on top of the crystal vibrating at low amplitude to separate the particles in the
defect region from the main crystal. The semicircular shape of the cavity walls determined
the shape of the defect and the low height of the cavity constrained particles from moving
vertically and creating a second layer. The tool disk was then rotated, causing the particles
32
in contact with the diameter divider and the cavity edges to be pushed to new locations.
The translation of these outer particles caused adjacent particles to translate in a similar
way. The main drawback to this approach was that the created defect was often composed
of two grains that were not coherent with each other.
2.4
2.4.1
Image acquisition
Camera control and suspension
The software Camera Control Pro 2 was used to automate and control image acquisition by
a camera (D90, Nikon) with a standard macro lens (50mm F2.8 EX DG Macro, Sigma). The
camera was suspended above the tray on a tripod capable of 3-axis rotation at the end of a
cantilevered beam. The cantilevered beam was fastened to an aluminum support structure
that was anchored to the granite table by screws in five locations distributed along its base.
2.4.2
Timing
In order to obtain photos at regular intervals, the interval timer shooting option in the camera
control software was set to 1 Hz, the maximum possible frequency. During the setup phase,
prior to data collection, it was observed that the true capture frequency matched the target
frequency of 5 Hz. However, analysis of photo time stamps among images obtained later
showed that the average time interval between photographs was approximately 1.4 seconds,
rather than the expected target value of 1 second. Because the resolution of the time stamps
was also one second, the approximate time for each image was obtained from its time stamp
and recalculated using the smooth function in Matlab which smooths response data using a
5-point moving average. The original time stamps and smoothed time values assigned to a
set of frames are presented in Figure 2.8.
33
80
22
60
20
v40
E
F
018
E
F
16
10
o timesta
14
12
Frame index
-smoothedtme
0 0
16
'
14
20
flime stamp
-smoothed
tme
10
(a)
30
20
Frame index
40
50
(b)
Figure 2.8: a.) Time from discrete time stamps is smoothed to reflect the true image timing. b.) Smoothed
time is linear and matches the raw recorded time from the image time stamps.
2.4.3
Illumination
The particles were illuminated from above and below. Light from the LED pad placed below
the transparent tray surface backlit the particles and created contrast between the particles
and the background such that the particles appear nearly black and the empty space appears
nearly white. The particles were also illuminated by an annular lamp fastened around the
outside of the camera lens. Other lights in the vicinity were dimmed so that light from the
annular lamp reflected off the curved surfaces of the particles and produced a single, small,
and well-defined white dot at the center of each particle's projected area.
2.4.4
Digital film parameters and image quality optimization
A preliminary autofocused image with aperture constrained to f/5 was obtained before prior
to tray oscillation to ensure high resolution. After confirming that the preliminary image was
focused properly, relevant parameters were fixed and held constant throughout the duration
of the trial by using the interval timer shooting feature. Due to the short shutter speed, the
translation of the tray did not produce streaks or other image defects. Figure 2.9 confirms
that the captured image of grid on the surface of the LED pad was not warped significantly
and Figure 2.10 shows a cropped but unprocessed photo.
34
The shutter speed, ISO, and
aperture were optimized for a full tray of particles; removing particles from the tray allowed
more light from the LED pad to shine through and resulted in an overexposed image if this
increase in exposure was not accounted for. Images were 4288 pixels by 2848 pixels; the
larger dimension ran parallel to the axis of tray oscillation. After each trial was completed,
images were cropped and analyzed with Matlab.
Figure 2.9: LED pad grid lines have vertical and horizontal alignment that deviates little from image axes.
Figure 2.10: Cropped image of the granular crystal with a misoriented grain defect.
35
Chapter 3
Analysis and Results
The size of the defect, misorientation angle, and locations of individual particles were
recorded at evenly spaced time intervals from the onset of oscillation until the time when
the defect had fully disappeared. It was observed that particles migrate from the defect
lattice to the main crystal lattice. This migration causes the defect to contract over time.
Defects nearly always purely decreased in size and in cases where defect growth did occur,
the growth occurred slowly and the defect eventually collapsed and completed healing. In
order to identify the controlling variables in this system, defect healing trials were examined
individually and collectively. Total healing time was somewhat correlated with misorientation angle. However, there was a wide variation in the healing times even amongst defects
with similar initial conditions, resulting in a high degree of scatter for misorientation dependent models. Additionally, the size of the defect area does not follow the same type of time
dependence path for all samples.
3.1
Particle locations
Data for each trial was assembled from the particle location recovered from frame at equally
spaced time intervals. To obtain particle locations, photos were first cropped to exclude the
disordered edges of the primary grain which did not interact with the defect. Then images
36
were converted to greyscale and binary.
There were two methods for locating the particle centers that were considered. When illuminated, he convex surface of the particles reflects light and a single small white dot appears
at the center of the projected area of each particle. One method exploited this effect and
defines the location of the particle as the centroid of this reflection. However, particles with
reflections that were dimmer than a threshold brightness or smaller than a threshold size
were often not detected by this method. This was problematic because failure to detect a
particle had the potential to distort the results of subsequent analysis phases.
The second method, which used the whole projected area of the particle, was more reliable
alternative than the first. In this method, the reflections used by the first method were filtered
out and each particle appeared as a larger black silhouette that approximated its projected
area. In order to ensure that each particle was fully separated from its neighbors, the edges
of the particle silhouettes were converted to white space and the centroid of the remaining
black region was considered the particle's location. Examples of each step of this processing
method are offered in Figure 3.1. Although this method was more reliable than the previous
method, it was less precise. The reduction in precision was due to distortions in the apparent
shape of each particle caused by other reflections off the particle surface and nearby particles.
For example, a particle surrounded by a ring of six neighbors, the arrangement found within
the uniform crystalline lattice, results in particles appearing as regular hexagons, not circles.
Figure 3.2 presents the shortest distance between the particle locations determined using
the approximate particle area and those obtained from the reflection method. For particles
detected by both methods, the differences between the results of each method was on the
order of 0.0172 t 0.008 particle diameters.
37
'4
'4
.4
II4 U
I
I
* 040
6
A
4 P',
U
I
I
wwu
AL
qI* Ill
p
'.1
PAq
I
ii
'4
-
'4
I
I I Ii
P4
.&A
vto
4*4
(b)
(a)
10.
0 0*dU
10-o
00
04~
000 0000
0000O
000
00
00000Do
0
(d)
(C)
U
'4
'4
'4
A
.4
I
A
At
V
V
i
0
Ib
0
lip 4P
9
AL
(f
)
(e)
Figure 3.1: a.) Original color image b.) Greyscale image c.) Binary image d.) Binary image with reflections
filtered e.) Binary image with eroded edges to ensure particle separation f.) Greyscale image overlaid with
particle centroid locations marked by crosses.
38
10
'
.
0
100
E
0
10 -1
(0~~~
T
Je
*
10
6-W
9%0 0
0
0
;
10
0
500
1000
Particle Index
1500
2000
Figure 3.2: There was a small disparity between the particle locations obtained using the reflection method
and those from the silhouette method. Particle indices which have a disparity of one particle diameter
correspond to particles which were detected by the projected area method but not the reflection method.
Excluding undetected particles, the disparity was 0.0172 0.008 particle diameters for this example frame.
More than 99 % of all detected particles had a disparity below 0.04 particle diameters.
3.2
Determining misorientation angle
Once the particle locations were obtained, a delaunay triangulation was used to identify its
neighbors. Each particle was considered a vertex and the angles with respect to the image
horizontal axis were calculated.
If more than one of the edges emanating from the given
vertex point was offset by a multiple of 7r/3 from each other, the vertex was considered part
of one of the hexagonal crystal grains. Particles which were part of the the primary crystal or
the grain defect had delaunay triangulation edges that were offset by multiples of ir/3 from
each other as shown in Figure 3.3. Each absolute edge angle was decomposed into the nearest
multiple of 7r/3 and an offset from that multiple. This offset was the modulus when the angle
was divided by ir/3 and was recorded as the local lattice orientation for that edge. Once
all vertices were processed, the highest peak in the histogram of lattice orientation angles
39
indicated the orientation of the primary grain and the second highest peak corresponded to
the orientation of the defect grain. The misorientation angle for the frame was calculated
by taking the difference of these two angles. Figure 3.4 presents an example of a lattice
orientation histogram obtained by this method.
It was observed that the misorientation angle drifted significantly between frames during
some trials due to the defect rotating slightly with respect to the main crystal lattice. Therefore, both the mean misorientation angle and initial misorientation angle for each sample was
calculated and recorded. Figure 3.5 shows the variation in misorientation angle with time
for six samples with mean misorientations between 57r/36 rad (250) and 77r/48 rad (280)
.
e
2T//3
*/3
@*
-
. *
7
0 =T412
Figure 3.3: Local lattice orientation may be determined by decomposing the edge angles from the delaunay
triangulation into multiples of 7r/3 and the additional offset from the image horizontal axis due to lattice
orientation.
40
0.4
Primary crystal
orientation
0.35a 0.30
o 0.250.2S0.15
-
Defect grain
orientation
LL
~0.10.050
0
6
9
18
8-
9
6
Orientation angle [rad]
Figure 3.4: Histogram of local lattice vectors obtained from the delaunay triangulation edge angles of particles
belonging to hexagonal lattices. Maximum angle occurrence denotes the orientation of the primary crystal
and the next highest peak occurs at the orientation of the grain defect.
&
C
136
0)
iL
15
.
6 = 48 red
....
0.45 rad
am=
um =O46
...
m =045 rad
3000
3500
90a
90
9 0
.5
500
= 0 46 red
1
1000
1500
2000
2500
Time [s]
4000
Figure 3.5: Misorientation angles drifted slightly from initial values during the healing period.
41
3.3
Identifying and delineating regions
Using their connectivity edge angles, each particle was assigned to one of three groups:
the primary grain, the defect grain, or the neutral group.
Particles which had multiple
connectivity edges aligned with the primary lattice orientation were considered part of the
primary crystal.
Similarly, particles with edges that were aligned with the defect lattice
orientation were considered part of the defect. The remaining particles were considered part
of neither grain. Figure 3.6a is a section of an original greyscale image in which particles
have been labeled by their lattice group assignment.
Lattice group assignment was supplemented by the identification of particles which were
engaged with grain boundary migration because the they were in proximity to the interface
between the two grains. This region was comprised of defect particles which were next to
neutral particles or primary grain particles and primary grain particles which were adjacent
to defect particles. These grain boundary region particles are designated with asterisk signs
in Figure 3.6b.
4( -*4 -
' -4'
A'V A'V & V A
&
14 z
'40101
4 4*
.,VA:AAAAA
11
VA
4001 A
A
'0040,
401z
Y
I
(a)
(b)
Figure 3.6: a.) Each particle was assigned to the grain defect, primary lattice, or neutral group according to
the orientations of its delaunay triangulation edges. Blue solid circles mark particles belonging to the main
grain, red triangles mark particles in the defect, and green squares mark particles which cannot be assigned
to either grain. b.) The particles that were engaged in grain boundary migration are those in proximity to
the opposite grain or neutral particles.
42
3.4
Correlation between healing time and misorientation angle
The total healing time was defined as the number of seconds between the start of oscillation
and the complete disappearance of a well-defined grain defect. Due to the lower density near
the grain boundary during the healing period, few samples fully returned to a perfect crystal
even after healing was considered complete. Typically, a small low-density region without
distinct edges or orientation, like the arrangement in Figure 2.4, remained behind.
The healing times plotted with respect to both mean misorientation angle and initial misorientation angle in Figure 3.7 revealed that high misorientation tended to result in longer
healing times than low misorientation did. However, there was a great deal of scatter among
these results. For example, over the domain 47r/15 rad (250) and 137r/90 rad (280), the
range of healing times varied by a factor of twenty (159 s to 3359 s). A linear regression was
performed using the natural logarithm of healing time as the dependent variable and the
initial misorientation angle as the independent variable. This yielded a fitted exponential
curve, Thea. = To exp(a/#) where TO,j = 0.90O-3 s and /i = 0.069ii.00 rad. The same curve
fitting procedure was repeated with the mean misorientation angle instead and produced fitting constants To,m = 1.27+0.40 s and #m = 0.072+0-00 rad. Though there was a correlation
between misorientation angle and healing time, there was a great deal of scatter among the
results. The correlation obtained with mean misorientation was better than that of initial
misorientation angle so subsequent analysis incorporated only the mean defect angle.
43
10
-
*
**
10
.
-
10 2
Ca
U
-
E
**.
-
.
U.
101
heal
Exponential fit
0
a
1
A
36
18
12
a
36
6
Initial misorientation angle, c, [radians]
(a)
10
---
.
e
.
*
E
-
-e
U
-
*
10 3
102
0I
a
*
heal
OPOP.Om-
101
*
Exponential fit
10
0
I
A
A
36
18
a
Mean misorientation angle,
A
5R
9
36
A
6
m [radians]
(b)
Figure 3.7: a.) Total healing time loosely follows an exponential curve dependent on misorientation angle
Theal = To exp(a/3). Defining misorientation angle as the intial misorientation angle and performing a
least squares regression gives Theal = To exp(a/3) where TO,j = 0.90j0.3 s and /3 = 0.069+tg-00 rad.
b.) Defining misorientation angle as the mean misorientation for the complete trial yields fitting constants
TO,m = 1.27+04 s and Om = 0.072+0:005 rad.
44
3.5
Variation in healing path
A closer investigation into the healing rates of individual defects reveals that defect size
does not vary with time in the same manner among all samples. In order to evaluate the
time-dependence of defect size, the delaunay triangulation was used to construct a Voronoi
diagram. The Voronoi diagram constructed from the delaunay triangulation of the particle
locations divides space into regions corresponding to individual particles. The region associated with each seeding particle is the set of points which are closer to the location of that
particle than to any other particle location. Therefore, the area of a Voronoi cell for a given
particle may be considered the territory belonging to that particle. The sum of the Voronoi
cell areas for each grouping of particles was considered the total area of the region. Thus,
the size of the defect at a given time was defined as the sum of the areas of the Voronoi
cells belonging to all defect particles identified from the edge angle analysis. This total area
was normalized by the size of a standard particle territory, which is the expected size of a
Voronoi cell when the lattice has no defects and is fully packed. In Figure 3.8, defect size is
plotted with respect to time for all ninety-four trials. The color of the trace for each trial
denotes relative mean misorientation angle. Bright red traces correspond to defects with
relatively high misorientation angles and green traces belong to those with relatively low
misorientation angles.
The defect size over time for six selected samples with mean misorientations between 51r/36 rad
(250) and 77r/48 rad (25*) is displayed in Figure 3.9. Even within this small angle domain,
there was variety in the paths followed by the defect size over time. The cyan and green
traces are present defects which healed in rapid bursts between long periods of constant area.
In contrast, the red and purple traces both show nearly linear dependence with time, but
have healing speeds that differ by more than a factor of six. The blue trace is neither linear
nor stepped and includes regimes of fast healing and slower healing. The black trace belongs
to one of the few defects that increased in size before reversing its growth and contracting.
45
. 6.00
20
0
o0
400
800 1200 1600 2000 2400 2800 3200 3600 4000
0
40
80
120
160
200 240
Time [s]
280
320
360
400
0
4
8
12
16
24
20
Time [s]
28
32
36
40
Time [si
GO200
00
Figure 3.8: Healing time and defect size time dependence varies dramatically from trial to trial, even for
nearly equivalent misorientation angles. The color of the trace corresponds to the relative misorientation
angle: bright red for relatively high mean misorientation angles near ir/6 and bright green for relatively low
misorientation angles, near ir/18. Each plot covers a different time domain: 0 to 4000 s, 0 to 400 s, and 0
to 40 s in order that the time dependence data for both rapidly healing and slowly healing trials are visible.
Each plot contains data from all ninety-four trials.
46
1
600
=
0.44 rad
=0.46 rad
a = 0.49 rad aM
500
0.48 rad
a = 0.4 rad a =0.45 rad
a
400
-
= 0.45 rad
=0.45
a
=
0.46 rad
rad q, =0.46 rad
a =049 rad a =0.45 rad
t300-200-
-
100
0
0
500
1000
1500
2000
Time [s]
2500
3000
3500
4000
Figure 3.9: Total healing time for six selected samples with mean defect angles between 57r/36 rad and
77r/48 rad. The time dependence of defect size does not have a consistent shape across all samples. The
area of some defects decreases linearly with time but other defects heal in rapid bursts between long periods
of constant area.
A qualitative survey of the time dependence profile for all trials was conducted and each
defect was placed in either the "linear" dependence category, the "step function" category,
or the "neutral" category.
The defects with size that depended linearly on time had a
constant healing rate. Defects with profiles marked by long periods of nearly constant size
and short periods of rapid decrease in size comprised the "step function" defect size category.
The defects which had curved traces, either concave up and concave down, were considered
neutral since they were linear in some regions but also exhibited significant variation in
shrinking rate.
In an effort to quantify healing path type, least squares linear regressions were performed
,
on the defect size versus time curve for each sample. The coefficient of determination, R2
for each regression was recorded and matched with its originating defect.
47
This R 2 value
derived from this linear regression can be considered a rough assessment of linearity that
agrees with qualitative observations of profile shape. Mean misorientation angle and this
coefficient of determination, plotted together in Figure 3.10, do not appear to be correlated.
Figure 3.11 presents the healing time for each trial and the corresponding coefficients of
determination and shows that the quadrant containing low healing times and low linearity
is relatively unpopulated. This evaluation of path type suggests that nonlinearity in path is
more characteristic of high healing time than it is of low healing time.
".
S.
.
0.8 k
.
.
.
-
-. a'.
.
s.
S.
*
*
*
(N
e
a a.
.
I
**
..g
0
0.6F
-o
4.-
0
0.4 F
" linear
0
0
0.2
0
" step
neutral
01.
36
A
5
18
36
n
Mean misorientation angle, am [radians]
Figure 3.10: The coefficient of determination, R , for the linear regression of defect size and time for each
trial was plotted with respect to misorientation angle. Though the lowest R 2 value is also near the maximum
misorientation angle, the rest of the data does not show a correlation between the shape of the healing profile
and the misorientation angle. Defects which follow a linear path are marked with blue, defects which have a
stepped path are colored magenta, and defects which fit into neither of these categories are marked in green.
48
*
10
10 3
se
E
U.*
2
10
*.
E
10
*
*
.
S
SE"
*
linear..
.
'
e
*
estep
*neutral
10
0
0.2
0.4
0.6
0.8
1
Coefficient of determination, R 2
Figure 3.11: The healing time for each defect is plotted with respect to coefficient of determination, R 2 , for
the linear regression of defect size and time. The R 2 value for a the healing profile of a defect serves as an
approximate measurement of linearity. High R 2 are found among defects which healed in a short time and
a long time but low R2 values are not common among defects which healed quickly. Defects which follow a
linear path are marked with blue, defects which have a stepped path are colored magenta, and defects which
fit into neither of these categories are marked in green.
49
Chapter 4
Conclusion
In this thesis, we developed an experimental granular analog for a two-dimensional hexagonal crystal and investigated the time evolution of self-healing grain defects. Images of the
shrinking defects were taken at multiple time intervals and used to track evolution in size
and to collect information on the defect misorientation angle. This granular system reflects
the atomistic nature of a real grain boundary and the use of discrete particles arranged in a
hexagonal crystal structure incorporates the misalignment which intrinsically defines a grain
defect. In a manner similar to that of the theoretical grain boundaries analyzed by Mullins
[26], the macroscopic grain boundaries in this experiment migrated toward their centers of
curvature and generally retained their circular shape. Therefore, this system can be used to
visuallize the progression of curved grain boundaries.
Given that the ball model in this experiment is a good analog for two-dimensional hexagonal
crystals, the dynamic behavior of the grain boundaries inside it are worthy of investigation.
Though Bragg and Nye [6] observed that grain growth among a polycrystalline region of
bubble rafts occurred over time, this study takes a systematic and quantitative approach
to analyzing the grain boundary migration that causes this phenomenon. This analysis of
curved grain boundaries that enclosed circular grains of high misorientation yielded information about the misorientation dependence of total healing time and the various ways that
50
total defect size depended on time for individual defects.
The range of healing times for similarly sized defects stretched over several orders of magnitude and was somewhat correlated with misorientation angle. To describe this relationship,
linear regressions were performed to fit the healing time results to exponential functions of
initial and mean misorientation angle. The results of this analysis possessed substantial scatter but show some correlation between misorientation angle and healing time. Despite high
scatter, this existence of misorientation-dependent trends demonstrates that self healing is
not a purely stochastic phenomenon. Deeper examination of the exact paths taken by defect
size during healing revealed substantial variation in the time dependence of healing rate. The
shape of healing profiles ranged from nearly linear trends to paths characterized by irregular
healing speed. Defects with irregular healing speeds had healing curves marked by plateaus
of constant size interspersed with short periods of rapid contraction. This kind of stepped
profile was seen to be more common among defects with high misorientation angles than
those with low misorientation angles. Overall, this variation in healing path shape suggests
that there are additional parameters which control the overall mechanism for self healing
that have yet to be determined.
Further work with this system should target indentifying the complete set of variables which
influence healing path type and evaluating their impacts on healing speed. In particular,
the dependence of healing speed on nondimensional acceleration has yet to be determined
and the density of the grain boundary shows a promising correlation with misorientation
angle. More broadly, other microstructural parameters that characterize particle interaction
are good candidates for investigation.
51
Bibliography
[1] N.K. Adam. The Physics and Chemistry of Surfaces. Oxford University Press, London,
third edition, 1941.
[2] K.T. Aust and J.W. Rutter. Temperature dependence of grain migration in high-purity
lead containing small additions of tin. Transactions of the Metallurgical Society of
A.I.M.E., 215:820-831, 1960.
[3] P.A. Beck. Metal Interfaces, page 208. American Society for Testing Materials, Cleveland, 1952.
[4] P.A. Beck, P.R. Sperry, and H. Hu. The orientation dependence of the rate of grain
boundary migration. Journal of Applied Physics, 21(5):420-424, 1950.
[5] L. Bragg and W.M. Lomer. A dynamical model of a crystal structure. ii. Proceedings of
the Royal Society of London. Series A, Mathematical and Physical Sciences, (1045):171181, 1949.
[6] L. Bragg and J.F. Nye. A dynamical model of a crystal structure. Proceedings of the
Royal Society of London. Series A, Mathematical and Physical Sciences, (1023):474-481,
1947.
[7] D.G. Brandon.
The structure of high-angle grain boundaries.
Acta Metallurgica,
14(11):1479 - 1484, 1966.
[8] J.P. Escobedo, D. Dennis-Koller, E.K. Cerreta, B.M. Patterson, C.A. Bronkhorst, B.L.
Hansen, D. Tonks, and R.A. Lebensohn. Effects of grain size and boundary structure
on the dynamic tensile response of copper. Journal of Applied Physics, 110(3):033513,
2011.
[9] J.M. Georges, G. Meille, J.L. Loubet, and A.M. Tolen. Bubble raft model for indentation
with adhesion. Nature, 320(6060):342, 1986.
[10] H. Gleiter. The structure and properties of high-angle grain boundaries in metals.
Physica Status Solidi (b), 45(1):9-38, 1971.
[11] H. Gleiter and Chalmers B. High-angle grain boundaries, volume 16 of Progress in
Materials Science. Pergamon, London, 1972.
[12] G. Gottstein and L.S. Shvindlerman. Grain boundary migration in metals: Thermodynamics, kinetics, applications. CRC Press, Boca Raton, 1999.
[13] R. Gronkski. Grain Boundary Structure and Kinetics, page 45. ASM, Ohio, 1980.
52
[14] A.J. Haslam, S.R. Phillpot, D. Wolf, D. Moldovan, and H. Gleiter. Mechanisms of
grain growth in nanocrystalline fcc metals by molecular-dynamics simulation. Materials
Science and Engineering: A, (1-2 ; Nov. 2001), 2001.
[15] G.C. Hasson and C. Goux. Interfacial energies of tilt boundaries in aluminium. experimental and theoretical determination. Scripta Metallurgica, 5(10):889 - 894, 1971.
[16] P.R. Howell, I.T. Kilvington, A. Willoughby, and B. Ralph. Simulations of grainboundary structure. Journal of Materials Science, 9(11):1823 - 1828, 1974.
[17] F.J. Humphreys and M. Hatherly. Recrystallization and Related Annealing Phenomena.
Elsevier, Oxford, second edition, 2004.
[18] Y. Ishida. Order structures and dislocations in bubble raft grain boundary. Journal of
Materials Science, 7(1):75 - 83, 1972.
[19] C.V. Kopetski, V.G. Sursaeva, and L.S. Shvindlerman. Sov. Phys. Solid State, 21(2):238,
1979.
[20] J. Kotakoski and J.C. Meyer. Mechanical properties of polycrystalline graphene based
on a realistic atomistic model. Physical Review B, 85:195447, 2012.
[21] M.L. Kronberg and F.H. Wilson. Secondary recrystallization in copper.
Metals, 2:1056, 1950.
Journal of
[22] B. Liebman, K. Lficke, and G. Masing. Untersuchung iber die orientierungsabhiingigkeit
der wachstumsgeschwindigkeit bei der primiren rekristallisation von aluminium.
Zeitschrift Zeitschrift Zeitschrift Zeitschrift fir Metallkunde, 47(2):57-63, 1956.
[23] W.M. Lomer and J.F. Nye. A dynamical model of a crystal structure. iv. grain boundaries. Proceedings of the Royal Society of London. Series A, Mathematical and Physical
Sciences, (1111):576-584, 1952.
[24] H. Miura, M. Kato, and T. Mori. Correlation between the boundary energy and precipitation in copper-[011] symmetric tilt boundaries. Colloque de Physique.
[25] D.A. Molodov, U. Czubayko, G. Gottstein, and L.S. Shvindlerman. Mobility of 111)
tilt grain boundaries in the vicinity of the special misorientation [sigma] = 7 in bicrystals
of pure aluminum. Scripta Metallurgica et Materialia, page 529, 1995.
[26] W.W. Mullins. Two-dimensional motion of idealized grain boundaries. Journal of Applied Physics, 27(2):900-904, August 1956.
[27] G. Palumbo and K. Aust. Recrystallization 90', page 101. TMS, Warrendale, 1990.
[28] M.N. Parthasarathi and P.A. Beck. The oriented grwoth mechanism of the formation
of recrystallization textures in aluminum. Transactions of the Metallurgical Society of
A.I.M.E., 221:831-838, August 1961.
[29] R. Pino Plasencia, D.L. Beke, G. Erdelyi, and F.J. Kedves. Study of the structure
and dynamic behaviour of grain boundaries using dynamic ball-model. In Colloque de
Physique C, pages 317 - 322, 1990.
[30] R.C. Pond. Grain Boundary Structure and Kinetics, page 13. ASM, 1980.
53
[31] J.W. Rutter and K.T. Aust. Migration of (100) tilt grain boundaries in high purity
lead. Ata Metallurgica, 13:181-186, March 1965.
[32] S.L. Sass and P.D. Bristowe. Grain Boundary Structure and Kinetics, page 71. ASM,
1980.
[33] D.N. Seidman. Materials Interfaces, page 58. Chapman and Hall, London, 1992.
[34] W. Shockley and W.T. Read. Quantitative predictions from dislocation models of crystal
grain boundaries. Physical Review, 75:692, 1949.
[35] D.A Smith, C.M.F. Rae, and C.R.M. Grovenor. Grain Boundary Structure and Kinetics,
page 337. ASM Metals Park, Ohio, 1980.
[36] R.C. Sun and C.L. Bauer. Tilt boundary migration in NaCl bicrystals. Acta Metallurgica, 18(6):639 - 647, 1970.
[37] A.P. Sutton and R.W. Balluffi. Interfaces in crystalline materials, volume 51 of Monographs on the physics and chemistry of materials. Oxford : Clarendon Press, 1995.
[38] D. Turnbull. Theory of grain boundary migration rates. Transactionsof the Metallurgical
Society of A.I.M.E., 191:1-7, 1951.
[39] M. Upmanyu, D.J. Srolovitz, L.S. Shvindlerman, and G. Gottstein. Misorientation
dependence of intrinsic grain boundary mobility: simulation and experiment. Acta
Materialia, 47(14):3901-3914, 1999.
[40] K.J. Van Vliet, S. Tsikata, and S. Suresh. Model experiments for direct visualization of grain boundary deformation in nanocrystalline metals. Applied Physics Letters,
83(7):1441 - 1443, 2003.
[41] P. Vancs6, G. I. Mark, P. Lambin, A. Mayer, Y. Kim, C. Hwang, and L.P. Bir6. Electronic transport through ordered and disordered graphene grain boundaries. Carbon,
64:101 - 110, 2013.
[42] V. Vitek, A.P. Sutton, G.J. Wang, and D. Schwartz. On the multiplicity of structures
and grain boundaries. Scripta Metallurgica, 17(2):183 - 189, 1983.
[43] M.J. Weins. Computer simulation of the structure of high angle grain boundaries. In
Pierre C. Gehlen, Jr. Beeler, Joe R., and Robert I. Jaffee, editors, Interatomic Potentials
and Simulation of Lattice Defects, pages 695-712. Springer US, 1972.
[44] K. William and D.A Smith. A high-resolution electron microscopy investigation of some
low-angle and twin boundary structures. Ultramicroscopy, 22:47 - 55, 1987.
[45] D. Wolf. High temperature structure and properties of grain boundaries: long-range vs.
short-range structural effects. Current Opinion in Solid State and Materials Science,
5(6):435-443, October 2001.
[46] D. Wolf and K.L. Merkle. Materials Interfaces, pages 87-150.
London,1992.
Chapman and Hall,
[47] H. Zhang, M. Upmanyu, and D.J. Srolovitz. Curvature driven grain boundary migration
in aluminum: molecular dynamics simulations. Acta Materialia, 53:79-86, 2005.
54