![Figure 1. Correlation between the logarithm of the grain boundary population, measured in MRD units, and the grain boundary energy. (a) experimental results from measurements of polycrystalline MgO [5]. At each energy, the square is the mean population and the error bars show the standard deviation. (b) Simulated results from Grain 3D [10].](/figures/figure-1-correlation-between-the-logarithm-of-the-grain-30xvvlid.webp)
A MODEL FOR THE ORIGIN OF ANISOTROPIC GRAIN BOUNDARY CHARACTER
DISTRIBUTIONS IN POLYCRYSTALLINE MATERIALS
Gregory S. Rohrer, Jason Gruber, and Anthony D. Rollett.
Department of Materials Science and Engineering,
Carnegie Mellon University
Pittsburgh PA, 15213
ABSTRACT
A model is described for the development of anisotropic grain boundary character
distributions from initially random distributions. The model is based on biased topological
changes in the grain boundary network that eliminate and create boundaries during grain growth.
The grain boundary energy influences the rates of these topological changes by altering the
relative areas of the interfaces. The model predicts grain boundary character distributions that
are inversely related to the grain boundary energy and are consistent with experimental
observations.
INTRODUCTION
The grain boundary character distribution (GBCD) is defined as the relative areas of
grain boundaries as a function of lattice misorientation and grain boundary orientation. It can be
considered as an expansion to higher dimension space of the misorientation distribution function
(MDF) and is typically normalized to give units of multiples of a random distribution (MRD). It
has recently been observed in experiments and in simulations that the GBCD, even in an
otherwise untextured polycrystal, is anisotropic [1]. The results indicate that the most common
boundaries in anisotropic distributions have greater average areas than the less common
boundaries and that there is a higher incidence of these boundaries [2]. Peaks in anisotropic
distributions commonly reach values of 5 to 10 MRD and, even in a relatively isotropic material
(Al), peaks in excess of 3 MRD are commonly observed [3]. Furthermore, based on the results
of experiments [4,5] and computer simulations in two and three dimensions [6-10], the GBCD is
inversely correlated to the grain boundary energy. The only available comprehensive
experimental data indicates that the logarithm of the population is approximately linear with the
energy, which is consistent with the results of a three-dimensional computer simulation [6, 10,
11]. These results are compared in Fig. 1.
Holm et al. [6] were the first to propose a mechanism for the enhanced areas of low
energy grain boundaries. Assuming the grain boundaries in Fig. 2a have the same lengths
(L
1
=L
2
) and energies (γ
1
and γ
2
), then the dihedral angle, Ψ, is 2π/3. If the energies change so
that γ
1
< γ
2
, then the dihedral angle will increase, L
1
will increase by the amount ∆L
1
, and L
2
will
decrease. This lengthening and shortening of boundaries enhances the relative areas of low
energy grain boundaries. The alteration of grain boundary lengths in a two-dimensional network
was also recently used as the basis for a model to study the spatial correlation among high energy
grain boundaries [12]. However, this mechanism does not, by itself, explain why low energy
grain boundaries occur in greater numbers.
-3
-2
-1
0
1
2
3
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
ln(λ)
γ
gb
(a.u.)
(a)
-3
-2
-1
0
1
1 1.05 1.1 1.15 1.2
1.25
ln(λ)
γ
gb
(a.u.)
(b)
Figure 1. Correlation between the logarithm of the grain boundary population, measured in MRD units,
and the grain boundary energy. (a) experimental results from measurements of polycrystalline MgO [5].
At each energy, the square is the mean population and the error bars show the standard deviation. (b)
Simulated results from Grain 3D [10].
L
2
L
2
L
1
L
2
L
2
L
1
∆L
1
Ψ
(a)
(b)
Figure 2. Triple junctions for the case of (a) three equal energy grain boundaries and (b) when
the horizontal grain boundary has a lower energy. It is assumed that the grain boundary line
segments are fixed at the circles at edges of the box.
The purpose of this paper is to describe a model for the formation of anisotropic GBCDs
from initially random GBCDs during normal grain growth. Using the observation that a grain
boundary's energy is inversely related to its area, the model for the evolution of the distribution
assumes that the rates at which grain boundaries are eliminated are inversely proportional to the
grain boundary areas, and, therefore, directly proportional to the grain boundary energies. It is
shown that the model reproduces the main characteristics of the experimental results, including
the observation that low energy boundaries occur in greater numbers than expected in a random
distribution; based on these results, it is concluded that the assumed mechanisms are a plausible
explanation for the development of anisotropic grain boundary character distributions.
THE MODEL
Overview
We begin by considering how the grain boundary energy influences the grain boundary
area. With reference to the triple junctions illustrated in Fig. 2, we begin by assuming that the
three grain boundaries are fixed at their endpoints and the triple junction geometry obeys
Young's law for interfacial equilibrium. Under these conditions, the additional length is given by
the following equation:
∆L
1
=
L
1
2
1−
3
tan cos
−1
γ
1
2
γ
2
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
(1)
Note that for the case of γ
1
= γ
2
, ∆L
1
=0. As γ
1
<< γ
2
, ∆L
1
approaches L
1
/2.
Using Eq. 1, we can estimate the influence of the boundary lengthening mechanism on
the GBCD. First, assume there are just a few boundaries (say 2%) with γ
1
<< γ
2
so that we can
assume that the low energy boundary is always attached to two higher energy boundaries. All
the low energy boundaries will have a length of 3L
1
/2, the high energy boundaries attached to it
(4% of the total) will have lengths √3L
2
/2, and the remaining 94% of boundaries in the system
will have length L
1
. Using these estimates, and assuming that boundary area is equal to the
square of the length, the population of the low energy grain boundaries is 2.22 MRD and the
population of high energy boundaries is 0.98 MRD.
Note that the estimates above are maximal, assuming a vanishingly small grain boundary
energy and configurations in which the low energy boundary is always connected to two high
energy boundaries. Therefore, this mechanism does not provide a plausible explanation for
peaks in the GBCD that commonly exceed 3 MRD and it provides no explanation for the higher
incidence of low energy boundaries. It can be concluded that boundary area changes associated
with adjustments of the triple junction positions do not have a large enough effect on the areas to
explain the observed anisotropies. Furthermore, this mechanism can not account for the
observed anisotropic number fractions [2]. However, the lengthening mechanism should be
viewed as an essential factor that contributes to the anisotropy of the GBCD. In fact, in what
follows, we assume that boundary area changes are the mechanism that biases topological
changes and alters the number fractions of grain boundary types.
There are several physical processes that occur during grain growth that can alter the
GBCD. One process involves incremental changes in area as boundaries move. Another process
is the critical events that change the topology of the network. Grain faces lose edges (and area)
until they are triangular and eventually collapse. Two grains can also join and create a new
triangular face. Faces are also destroyed when four-sided tetrahedral grains collapse. Note that
these topological changes represent the end points of incremental motion and in what follows,
we will take the topological events as proxies for positive and negative incremental area changes.
Assumptions
The model for the development of the anisotropic GBCD is based on the following
assumptions:
1. The topological structure during grain growth is scale invariant. In other words, the
average number of faces per grain is independent of the mean grain size. This assumption allows
us to focus on the changes in the distribution of grain boundary types, without considering the
dissipative loss of interfacial area. In the case of isotropic materials, three dimensional computer
simulations have been used to demonstrate the scale invariance of the network topology [13].
These simulations compare favorably to results from Al, where boundary properties are naturally
anisotropic [14]. Recent three dimensional grain growth simulations have verified that
microstructures with anisotropic grain boundary properties remain scale invariant during grain
growth [15].
2. All incremental changes in grain boundary area are represented by the appearance or
disappearance of grain faces. Note that in a real situation, all grain faces are constrained to the
same sequence of events: they are created, they grow, they shrink, and then they are eliminated.
The present model is based on average probabilities for the events at the endpoints, without
consideration of individual grain faces or the intermediate incremental changes.
3. The distribution of grain boundary types that arises from grains growing into one
another is determined by the grain orientation distribution. This assumes that the pair-wise
spatial configuration of orientations is random. It has been shown previously that for a fixed
orientation distribution, the crystals can be positioned to exhibit non-random misorientation
distributions [16]. So, while the assumption is certainly a plausible condition, there may be cases
where it does not hold.
4. The probability that a grain face is eliminated is inversely proportional to its area. In
other words, the smallest faces are eliminated with the highest probability. This assumption is
based on observations of soap froths reported by Smith [17].
5. There is a functional relationship between grain boundary energy and grain boundary
area. Recent three dimensional grain growth simulations with anisotropic properties has shown
that this is the case [15]. As an approximate model for this relationship, we modify Eq. 1 and
assume that the length, L
i,
of the i
th
GB, when connected to two boundaries of average length, L ,
depends on the ratio between its own energy, γ
i
, and the average energy,
γ
, in the following way:
L
i
L
=1+
1
2
1−
3
tan cos
−1
γ
i
2
γ
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
(2)
This expression for L
i
has the property that for γ
i
=
γ
, L
i
= L and as γ
i
goes to 0, L
i
goes to
3 L /2. The area of the face is then given by the square of its length. Here, the average length is
fixed at 1.
It should be noted that an alternate approach to determining the relationship between the
energy and the area is to extract it from the results of three dimensional grain growth
simulations. This approach will be discussed briefly in the results section.
With these assumptions, we can quantify the probability that, for a single critical event
(e), a GB of the i
th
type is created or destroyed. The probability that it is created is equal to
fractional number of boundaries of the i
th
type expected from the grain orientation texture. For
this analysis, we assume a random grain orientation distribution. Therefore, the probability that
a boundary is created is equal to its expected population in a random distribution, (ρ
i
). The
probability that it is destroyed is equal to the normalized product of the fractional number of
boundaries of the i
th
type and the inverse of their area (n
i
/L
i
2
= α
i
n
i
). Therefore, the probability
that during a critical event the population of the i
th
type of GB is changed is,
∆n
i
∆e
=
ρ
i
−
α
i
n
i
α
i
n
i
i=1
N
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
. (3)
Note that when summed over all grain boundaries types, the creation and annihilation terms are
both equal to 1. Therefore, as critical events occur, the total number of grain boundaries under
consideration is unchanged. This reflects the assumption of scale invariance.
Details of the calculations
To determine how the GBCD changes as critical events occur, we begin by assuming an
initial random distribution and a functional form for the anisotropy of the grain boundary energy.
Here, we consider only energy anisotropies that vary with a single crystallographic parameter.
We initially consider energy functions that have a Read-Shockley like variation in the energy as
a function of misorientation.
γ
i
=
θ
i
θ
c
1−ln
θ
i
θ
c
⎛
⎝
⎜
⎞
⎠
⎟
for
θ
i
<
θ
c
γ
i
=1 for
θ
i
>1
(4)
In Eq. 4, θ
i
is the angle that characterizes the disorientation, and θ
c
is the cut-off angle beyond
which the energy is constant. The energy and GBCD are discretized in 0.5° intervals and, thus,
there are 125 discrete boundary types.
To test a situation that is more characteristic of variations that occur as a function of the
grain boundary plane orientation at a fixed lattice misorientation, we assume the following
energy anisotropy:
γ
i
=1+
ε
(sin2
ω
i
)
4
(5)
where ω parameterizes the inclination of the grain boundary in the bicrystal reference frame. For
calculations with this energy function, the misorientation is not accounted for and it is assumed
that all boundaries have this energy dependence in the domain of possible inclinations.
To begin the calculation, the initial populations of grain boundaries are defined so that
each occurs with a population of 1 MRD. The average energy (weighted by the population) is
determined according to the following equation: