The Distribution of Internal Interfaces in Polycrystals

TLDR: In this article, the authors review techniques that can be used to study the mesoscopic crystallographic structure of grain boundary networks and summarize current findings, showing that grain surfaces within dense polycrystals favor the same low energy planes that are found on equilibrium crystal shapes and growth forms of crystals in contact with another phase.

Abstract: Recent advances both in experimental instrumentation and computing power have made it possible to interrogate the distribution of internal interfaces in polycrystals and the three dimensional structure of the grain boundary network with an unprecedented level of detail. The purpose of this paper is to review techniques that can be used to study the mesoscopic crystallographic structure of grain boundary networks and to summarize current findings. Recent studies have shown that grain surfaces within dense polycrystals favor the same low energy planes that are found on equilibrium crystal shapes and growth forms of crystals in contact with another phase. In the materials for which comprehensive data exists, the distribution of grain boundaries is inversely correlated to the sum of the energies of the surfaces of the grains on either side of the boundary.

Figures

Figure 13. The distribution of grain boundary planes averaged over all misorientations for (a) SrTiO3, (b) MgAl2O4, (c) TiO2, (d) Al. The data are shown in stereographic projection along [001], which is in the center of each projection. The [110] direction is marked with a square in each projection. In (c), the directions normal to the {101} surfaces are marked with white diamonds and the directions normal to the {111} surfaces are marked with white circles [79].

Figure 1. (a) Schematic shape a single grain from a dense polycrystal, drawn after ref. [2]. (b) Representation of a three grain junction within a polycrystal. The view is exploded so that the internal interfaces can be seen. The external surfaces are shaded and the internal surfaces are triangulated. The jth triangular facet on the ith grain is shaded and an enlarged view of this facet is shown in (c).

Figure 5. The distribution of grain boundary planes in MgO for boundaries with a 45° misorientation about [110], plotted in stereographic projection (a) along [001] and (b) along [110]. In (a), the misorientation axis is in the plane of the paper (at the position of the “-“) and in (b) it is perpendicular [4]. In (a) the zone of tilt boundaries is labeled with a t and the (100), (110), and (111) poles are marked with a “+”, “-“, and triangle, respectively. In (b) the tilts are in the plane and the symmetric tilts are labeled st.

Figure 18. The average value of the l for SrTiO3 plotted as a function of the sum of the two surface energies. The circle represents the average population for all boundaries within a range of 0.012 a.u.; the bars indicate one standard deviation above and below the mean [78].

Figure 4. Approximate number of distinguishable grain boundaries as a function of the resolution (D) with which the parameters are measured.

Figure 14. (a) Schematic sequence of tilt bicrystals with increasing misorientation about the [100] axis. If the black crystal maintains its interface plane on [001], then the interface plane of the gray crystal must tilt away from [001] by the misorientation angle. (b) The position of the normals of the black and gray crystal on a stereogram, where [001] is now points out of the plane and is in the center of the stereogram. (c) The populations for [100] tilt grain boundaries in MgO, plotted on a [100] stereogram, at 5° 15°, 25°, and 35°. The zone of tilt boundaries is the vertical great circle in the center of each stereogram. At 35°, two separate grain boundary normal peaks are resolved [4].

Full text

Brigham Young University Brigham Young University
BYU ScholarsArchive BYU ScholarsArchive
Faculty Publications
2003-11-30
The Distribution of Internal Interfaces in Polycrystals The Distribution of Internal Interfaces in Polycrystals
Brent L. Adams
b_l_adams@byu.edu
Bassem S. El-Dasher
Gregory S. Rohrer
Anthony D. Rollett
David M. Saylor
See next page for additional authors
Follow this and additional works at: https://scholarsarchive.byu.edu/facpub
Part of the Mechanical Engineering Commons
Original Publication Citation Original Publication Citation
Zeitschrift fur Metallkunde, 95 (24) 197-214
BYU ScholarsArchive Citation BYU ScholarsArchive Citation
Adams, Brent L.; El-Dasher, Bassem S.; Rohrer, Gregory S.; Rollett, Anthony D.; Saylor, David M.; and
Wynblatt, Paul, "The Distribution of Internal Interfaces in Polycrystals" (2003).
Faculty Publications
. 467.
https://scholarsarchive.byu.edu/facpub/467
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Authors Authors
Brent L. Adams, Bassem S. El-Dasher, Gregory S. Rohrer, Anthony D. Rollett, David M. Saylor, and Paul
Wynblatt
This peer-reviewed article is available at BYU ScholarsArchive: https://scholarsarchive.byu.edu/facpub/467

The Distribution of Internal Interfaces in Polycrystals
Gregory S. Rohrer , David M. Saylor
a
, Bassem El Dasher
b
, Brent L. Adams
c
, Anthony D.
Rollett, and Paul Wynblatt.
Department of Materials Science and Engineering, Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213-3890
a
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
b
current address: University of California, Lawrence Livermore National Laboratory,
P.O. Box 808, Livermore, CA 94511
c
Department Mechanical Engineering, Brigham Young University, Provo, UT 84602,
USA
Abstract
Recent advances both in experimental instrumentation and computing power have made
it possible to interrogate the distribution of internal interfaces in polycrystals and the
three dimensional structure of the grain boundary network with an unprecedented level of
detail. The purpose of this paper is to review techniques that can be used to study the
mesoscopic crystallographic structure of grain boundary networks and to summarize
current findings. Recent studies have shown that grain surfaces within dense polycrystals
favor the same low energy planes that are found on equilibrium crystal shapes and growth
forms of crystals in contact with another phase. In the materials for which
comprehensive data exists, the distribution of grain boundaries is inversely correlated to
the sum of the energies of the surfaces of the grains on either side of the boundary.
Keywords, Grain boundaries, Surface energy, Grain boundary energy, Polycrystals
Version 11/30/03, submitted to Zeitschrift für Metallkunde

Rohrer et al. Internal Crystal Interfaces p. 2
1. Introduction
Dense polycrystalline materials consist of irregularly shaped, approximately
polygonal, single crystals joined at internal interfaces referred to as grain boundaries.
The internal structure of a polycrystal is most frequently characterized by observing
planar sections in which the grain boundary planes appear as lines. Therefore,
knowledge of the three-dimensional shapes of grains and the connectivity of the
interfacial network is limited. During the last decade, however, there have been
significant advances both in experimental instrumentation and computing power.
Because it is now possible to collect and analyze large quantities of observations in an
automated fashion and to model structures of increasing complexity, studies of the three-
dimensional structure of interfacial networks have begun to flourish. The purpose of this
paper is to review what is known about the structure of internal interfaces in polycrystals
and to put recent results in the context of more historical studies. The scope will be
confined to the macroscopic and crystallographic structure of interfaces, and will
concentrate on single phase materials. In the next section, the observable macroscopic
characteristics of grain boundaries in polycrystals are defined. Methods used to study the
structure of three dimensional interfacial networks are then outlined in Section 3, and this
is followed by a summary of recent findings in Section 4. In the final section, the most
important points are reviewed and several interesting directions for future research are
identified.

Rohrer et al. Internal Crystal Interfaces p. 3
2. Observable Characteristics of the Interfacial Network
2.1 Grain boundary degrees of freedom
To distinguish one type of grain boundary from another, the values of five
independent parameters must be specified [1]. Throughout the majority of this review,
three parameters will be used to describe the lattice misorientation (Dg) across the
boundary and two parameters will be used to describe the interface normal (n). The
parameters needed to specify the distribution of internal grain surfaces are described are
described in Section 2.2. This distribution is a function of the two parameters associated
with n. In section 2.3, the distribution of internal grain surfaces will be considered as a
function of the lattice misorientation. This is a five parameter function that is referred to
as the grain boundary character distribution.
2.2 Distribution of internal grain surfaces,
l
(n)
To define the internal distribution of internal grain surfaces, we can begin by
considering a volume of material with a constant orientation, enclosed by curved surfaces
that meet along lines where there is an abrupt change in the surface normal, as illustrated
in Fig. 1a [2]. If the curvature of the surfaces between grains is approximated by a set of
triangular tangent planes with a fixed surface orientation, then a grain can be defined as a
polyhedral volume of material with constant orientation. Referring to Fig. 1b and c, the
vertices of these triangular tangent planes have coordinates (x
k
,y
k
,z
k
). These coordinates
are measured in the sample reference frame, which is relative to the external surfaces of
the polycrystal or, more generally, the microscope stage. For the j
th
facet on the i
th
grain,

Frequently asked questions

Q1. What have the authors contributed in "The distribution of internal interfaces in polycrystals" ?

The purpose of this paper is to review techniques that can be used to study the mesoscopic crystallographic structure of grain boundary networks and to summarize current findings. 

Q2. What is the way to make the shape meet the surface tension constraints?

To make this shape meet the surface tension constraints, some curvature must be introduced in the faces with six edges and this will lead to curvature driven boundary motion. 

Q3. How many general equivalent grain boundaries are there for every observed triangular segment?

In the tetragonal system, there are 256 (=2•2•82) general equivalent grain boundaries for every observed triangular segment and 4 (=256/64) of these are in the sub-domain. 

Q4. How can the authors separate the grains of astainless steel from the polycrystal?

The grains of astainless steel have been separated by selective grain boundary corrosion [24] and the grains in aluminum can be separated from the polycrystal by infiltration of liquid gallium [25]. 

Q5. Why do some asymmetric grain boundaries in ceramics prefer low index planes?

It has been hypothesized that certain boundary planes are preferred because these geometries place a relatively high fraction of the sites on the surface of one crystal in coincidence with those on the surface of the adjacent crystal. 

Q6. What is the interesting analog of a polycrystal?

One other interesting polycrystal analog is lead shot, which has been compressed to a minimum volume to form polyhedral bodies that can then be separated and analyzed [17]. 

Q7. What is the preference for grain boundaries in ceramics?

The preference for certain grain boundary planes in ceramics is especially clear in cases where the grains grow abnormally large and/or there is an intergranular liquid phase present at the processing temperature. 

Q8. What is the correlation between the surface energies and the grain boundary?

because the binding energy changes systematically with d-spacing and is relatively constant for high index surfaces, the surface energies provide a good measure of the density of unsatisfied bonds in the interface and, therefore, the grain boundary energy. 

Q9. What is the way to visualize grain boundaries in transmission x-ray microscopy?

In the simplest mode, grain boundaries in transmission x-ray microradiographs can be visualized if they are marked with a denser material that absorbs x-rays more strongly than the grains.