Materials Science and Engineering A281 (2000) 148 – 155
www.elsevier.com/locate/msea
On the disclination-structural unit model of grain boundaries
A.A. Nazarov a, O.A. Shenderova b,*, D.W. Brenner b
a
b
Institute for Metals Superplasticity Problems, Russian Academy of Science, 39 Khalturin str., 450001 Ufa, Russia
Department of Materials Science and Engineering, North Carolina State Uni6ersity, Raleigh, NC 27695 -7907, USA
Received 4 August 1999; received in revised form 22 October 1999
Abstract
The relation between two elastic continuum approaches to grain boundary structure, the dislocation and disclination models,
is discussed. It is shown that the disclination model has two advantages: a well-behaved expression for the elastic energy of
disclination dipole walls, which describes the elastic energy over a wide interval of misorientations, and a continuous
misorientation angle dependence of the elastic energy of grain boundaries in an interval between two delimiting boundaries. The
elastic energy of the most general, faceted disclination wall is calculated. For cases in which both the energies of delimiting
boundaries and elastic constants are available from atomic simulations (Ž001 and Ž111 tilt boundaries in copper and Ž001 and
Ž011 tilt boundaries in diamond) quantitative agreement between the disclination model and simulation results is obtained.
© 2000 Elsevier Science S.A. All rights reserved.
Keywords: Dislocation; Disclination; Grain boundaries; Structural units
1. Introduction
In 1950, Read and Shockley [1] derived their classical
formula for the elastic energy of low-angle dislocation
tilt boundaries
gel =
eh
Gb 2
ln
4p(1− n)h 2pr0
(1)
where gel is the energy per unit area of the boundary, G
and n are the shear modulus and Poisson ratio, respectively, h is the period of the boundary, b is the magnitude of the Burgers vector of the dislocations, and r0 is
the core radius. Since then, numerous attempts have
been made to describe the energy of more general,
high-angle grain boundaries in terms of the continuum
theory of dislocations. The main difficulty encountered
by these attempts is that it is impossible to uniquely
define a value for the dislocation core energies that are
added to the elastic energy given by Eq. (1) and that
plays an important role in high angle boundaries. In the
1980s, insights from atomistic computer simulations
lead to the structural unit/dislocation model of grain
* Corresponding author.
E-mail address: [email protected] (O.A. Shenderova)
boundaries [2–4]. Based on this model, Wang and
Vitek [5] proposed a modification of Eq. (1) that includes the core energy of grain boundary dislocations,
and obtained energy versus misorientation angle (g(u))
curves for Ž001 and Ž111 tilt boundaries in copper
that reproduce the results of computer simulations relatively accurately. However, the entire misorientation
range for each axis had to be divided into small
subintervals in order to get meaningful elastic energies
from Eq. (1). As Sutton and Balluffi [6] pointed out,
this shortcoming is not a result of the structural unit
model, but rather is due to the logarithmic term in Eq.
(1) that imposes a severe constraint on the misorientation range covered.
Another way to describe grain boundaries in terms of
linear defects is the use of the disclination model [7,8].
A general grain boundary with a misorientation angle u
can be considered to consist of alternating segments of
two special boundaries uA and uB, which are delimited
by partial disclinations of strength 9 v= 9 (uB −uA).
Thus the elastic energy of the boundary is given by the
energy of a disclination dipole wall [8]
gel =
Gv 2h
f(l)
32p 3(1−n)
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 7 2 7 - 3
(2)
A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
where h is the period of the wall, l = pLB/h, LB is the
length of the segment of special boundary B, and
2l
6 d6
f(l)= 2 sin u du
0
0 cosh 6 −cos u
&
= − 16
&
&
l
(l− 6) ln 2 sin 6d6
(3)
0
Eq. (2) produces reasonable results for any deviation
from the special misorientation, and is better suited for
calculating the high angle grain boundary energies than
Eq. (1).
In Refs. [9,10], the disclination model has been considered as a direct consequence of the structural unit model.
The lines at which different types of structural units meet
are disclinations. This allows the minority structural
units to be considered as disclination dipoles. To emphasize the origin from the structural unit model, the new
disclination model has been called the disclination-structural unit model of grain boundaries. This model has
recently been used to calculate the energies of Ž001
symmetrical tilt grain boundaries in diamond, and a
reasonably good fit to simulation data has been obtained
[11].
In the present paper, the relationship between the
dislocation and disclination models is discussed. It is
demonstrated that despite the more widespread application of the former, the disclination model is more
convenient, requires fewer parameters, and can be used
Fig. 1. Atomic structures of the S = 25 [001] (170) symmetric tilt
boundary with a misorientation angle u = 73.74° in diamond: (a) the
lowest energy configuration with coupled Az units and the corresponding disclination representation; (b) the higher energy zig-zag
configuration with separated Az units.
Fig. 2. A schematic rendering of the step associated with an Az unit
inserted in a (010) plane in diamond cubic crystals.
149
to obtain a reasonably good approximation of the energy
of high angle tilt boundaries. The elastic energy of the
most general, faceted disclination dipole wall is then
calculated. Energy versus misorientation angle curves are
calculated for tilt grain boundaries in copper and diamond, systems for which elastic constants and grain
boundary energies are available from computer simulations.
2. Disclinations and dislocations in the structural unit
model
Grain boundary structures in face-centered cubic
(f.c.c.) metals are usually characterized by straight arrangements of two types of structural units that can be
modeled by flat disclination walls. More complicated
structures appear, however, in covalent materials having
tetra-coordinated atoms. For example, computer simulations show that the stable configurations of some Ž001
and Ž011 tilt boundaries in silicon and diamond have
zig-zag arrangements of units [12–15]. Zig-zag tilt
boundaries have also been observed by high-resolution
electron microscopy in silicon and germanium [16,17].
Illustrated in Fig. 1 are two structures of the boundary
S= 25 [001] (170) with a misorientation angle u=73.74°
in diamond that have been obtained by simulations using
an analytic potential [14]. Both structures are composed
of structural units of two types: two units of the S=5
(130) u= 53.13° boundary (we denote them type Az
units), and four units of the (010) plane of the perfect
lattice, or S= 1/u = 90° ‘boundary’ (F% units). In the first
structure, which has the lowest energy, the Az units are
coupled into pairs. These pairs constitute the cores of
dislocations with Burgers vector b= a0[010]. In the
second structure, the Az units are split and constitute the
cores of lattice dislocations with alternating Burgers
vectors b= 12a0[110] and b= 12a0[1( 10]. It is easily seen that
the pairs of Az units in the first type of structure can be
considered as disclination dipoles whose arm and
strength are equal to 2dA = 10/2 and v=90−
53.13°=36.87°, respectively (Fig. 1(a)). In the second
structure Az units shift the planes of F% units (Fig. 1(b)),
and the corresponding disclination wall is faceted, i.e. the
disclinations lie on two different planes with spacing a.
The exact value of the distance a can be obtained using
the condition that the Burgers vector of the dislocation
calculated from the disclination model equals the Burgers vector known from the coincidence site lattice
geometry. A single Az unit inserted between units
of the ideal lattice plane (010) as well as a corresponding disclination dipole with an arm L and a strength
equal to v= uF% − uA = 36.9° are illustrated in Fig. 2.
The Burgers vector of a dislocation equivalent to this
dipole can be calculated as b= 2L sin[(uF% −uA)/2] =
2L/
10 [18]. This must be equal to the magnitude
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A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
values, the elastic energy of the disclination dipole wall
approaches that given by the dislocation model. For
example, at u“uA (l=pLB/h“0) the function f(l)
behaves as f(l): 8l 2(3/2 − ln 2l). Substituting this approximation into Eq. (2) yields
gel =
Fig. 3. The dependence of the elastic energy of Ž001 symmetric tilt
grain boundaries in an f.c.c. crystal calculated by the disclinationstructural unit (solid line) and dislocation models. The long-dash line
corresponds to Eq. (4), short-dash curve to Eq. (5).
of the lattice dislocation Burgers vector b = a0/
2;
therefore, L=
5a0/2. This is larger than the dimension of the Az unit along the facet it forms, and is equal
to the length of a complex unit that includes an additional unit of the perfect lattice. This complex unit is
shown filled in Fig. 2. Components of the Burgers
vector normal and tangential to the grain boundary
plane are both equal to a0/2, i.e. the dipole must be
inclined at an angle of 45° to the grain boundary plane.
Thus, the height of the step associated with the dipole
is equal to a=L/
2 =
5a0/2
2. The facetted disclination dipole wall having this separation of planes will
be geometrically identical to the corresponding dislocation network. As one can see from a comparison of Fig.
1(b) and Fig. 2, the faceting of the intermediate
boundaries in the disclination model is apparently different from that in the structural unit representation.
However, both representations are virtually equivalent,
since they describe the same atomic structure. Thus,
even in the case of grain boundaries in which structural
dislocations have a component tangential to the
boundary plane, it is possible to construct an appropriate disclination model. The corresponding disclination
dipole wall will be faceted, with the step height determined from the condition of the geometrical equivalence of dislocation and disclination models.
3. Comparison of the disclination and dislocation
models
Shih and Li [8] showed that for the cases when the
misorientation angle deviates slightly from the special
GbA
e 3/2v
(u−uA) ln
2p(u− uA)
4p(1− n)
(4)
where bA = LBv and u− uA = bA/h. When starting
from the structural unit model, the exact value of the
Burgers vector is calculated as bA = 2dB sin (v/2) [18],
dB being equal to the dimension of the non-deformed
structural unit of the delimiting boundary B. Eq. (4) is
identical to the Read–Shockley Eq. (1) if r0 =dB/e 1/2 is
assumed. For the other end of the misorientation interval, the elastic energy in the dislocation model is also
given by Eq. (4), where u−uA is replaced by uB −u, dB
by dA and bA by bB.
Wolf [19] recently proposed an empirical relationship
for the energy of high angle grain boundaries in the
whole misorientation range derived from the dislocation model. In this relationship, for the case of
Ž001 symmetrical tilt boundaries, for example, u −uA
in Eq. (4) is replaced by 2 sin[(u − uA)/2]. By adjusting
the coefficients before and in the logarithm, the results
of computer simulations for the whole misorientation
angle could be fit fairly well. This fact, according
to Wolf, ‘suggests that the underlying physics may
be the same for low- and high-angle grain boundaries’
[19].
This physics, by the authors’ opinion, is based on the
disclination model. Wolf’s equation is also an approximation to the energy of disclination dipole walls. It
differs from Eq. (4) only by the use of the more exact
geometrical relationships b/h= 2 sin[(u − uA)/2] and L2/
h= sin[(u − uA)/2]/sin(v/2), but uses the same first order approximation to the function f(l). Thus, the
elastic energy of a disclination dipole wall in Wolf’s
approximation can be calculated from the following
equation:
gel =
n
GbA
u− uA
e 3/2 sin(v/2)
2 sin
ln
4p(1− n)
2
2psin[(u − uA)/2]
(5)
and from a similar equation for the misorientation
angles close to uB.
Plotted in Fig. 3 are the results of calculations of the
elastic energy of Ž001 tilt boundaries using Eqs. (2),
(4) and (5), taking as delimiting ‘boundaries’ the two
symmetry planes of the f.c.c. lattice, (110) and (010). In
this case dA = a0/
2, bA = a0/
2 and dB = a0/2, bB =
a0. The solid curve was calculated using the disclination
model, with disclination dipole arms equal to dB at the
left of the misorientation angle 36.87° and to dA at the
right of the misorientation angle. It should be noted
that different structural units are considered as disclina-
A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
tion dipoles at the left and right of the AB type
structure for calculational convenience. As a matter of
fact, the energy of a wall of disclinations does not
depend on the way they are coupled into dipoles. For
simplicity, the spacing of the dipoles is assumed to
change continuously, though this contradicts the structural unit model. This results in a smooth curve having
only one cusp at the misorientation angle S= 5/36.87°
that corresponds to a structural units stacking AB. The
long-dashed curve in Fig. 3 has been calculated from
Eq. (4) for the left branch starting from the boundary
A and its analogue for the right branch starting at the
boundary B. Similarly, the short-dashed curve corresponds to Eq. (5) and its right-branch counterpart.
Comparing the results of the three approaches yields
the following. First, the dislocation model, which is
considered here as a limiting case of the disclination
model, provides positive elastic energy values even for
very large misorientation angles. This occurs because of
the fact that the disclination model predicts that core
radii for grain boundary dislocations equal r0 = dB/e 1/2
or r0 =dA/e 1/2, these are less than those usually accepted [5]. The energy can assume negative values,
however, even in this case, if the boundaries are composed of structural units of very different dimensions.
In such cases, the AB structure occurs at a misorientation angle close to one end of the range between the A
and B type boundaries. When approaching this structure from one side, a negative energy may be given by
Eqs. (4) and (5), because the dislocation spacing closely
approaches the dislocation core diameter. The use of
2 sin(u/2) instead of u does not improve the general
behavior of the elastic energy. In both cases of the
151
dislocation model, different elastic energies are given
for the transition structure AB when approaching it
from the two sides. This is due to the fact that the
first-order approximation to f(l) results in the loss of
the property f(p − l)= f(l). In Ref. [19], the equality
of the energies is achieved by fitting the coefficients
multiplying the logarithm. Such fitting is, however,
somewhat arbitrary because these coefficients depend
on the elastic constants and cannot be changed
independently.
The conventional dislocation model includes the core
energy of dislocations in the logarithmic term [20]. In
this way, r0 can be fit for each branch such that the
total grain boundary energy assumes the same value at
the AB type structure. Similar to the disclination-structural unit model, this requires a single parameter, the
energy of the boundary with an AB structure. However,
the partitioning of the elastic and core energies in this
case is not as clear as it is in the disclination model.
Thus, not only can the dislocation model give physically meaningless negative energy values at certain misorientations, but also different elastic energies are
obtained for the same grain boundary structure at
which the transformation between two dislocation descriptions occurs (the AB type structure). On the other
hand, the disclination-structural unit model always
yields physically meaningful positive elastic energies
and the same elastic energy for the transition structure.
(Actually, there is no transition at all, and one changes
the disclination dipoles at the AB structure only for
convenience.) We conclude, therefore, that the disclination-structural unit model, being geometrically equivalent to the dislocation model, is more convenient for
calculations of grain boundary energies over wide misorientation ranges.
4. Elastic energy of a general disclination wall
Fig. 4. A faceted disclination wall representing the structure of
zig-zag tilt grain boundaries.
Consider now the disclination wall depicted in Fig. 4.
This wall is representative of the disclination structure
of zig-zag grain boundaries, an example of which has
been presented in Fig. 1(b). The segments l1 and l2 of
the grain boundary consist of one type of structural
units, while the other type of units between them shifts
the planes of these facets to a distance 9 a with respect
to each other. A disclination wall corresponding to a
straight arrangement of the structural units will be a
particular case of the wall under consideration when
a= 0.
To calculate the elastic energy of the wall, it is
convenient to couple the disclinations into dipoles as
illustrated in Fig. 4 so that the complex wall is represented as a pair of single dipole walls each of period h
separated by the distance a along the x-axis and the
distance y along the y-axis. The elastic energy of the
A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
152
system is a sum of self energies of the single dipole walls
g ps (p=1, 2), and their interaction energy gint. The former
are given by the equation
g ps =
Kev h
f(lp ) (p = 1, 2)
32p 3
N−1
(6)
K v 2h
gint = e 3 [g(d,z +l2) − g(d,z) − g(d,z +l2 −l1)
32p
+ g(d,z −l1)]
&
(7)
2d
x dx
0
&
2t
0
sin y dy
cosh x −cos y
f(t)−q(d,t)
(8)
Similar to the function f(t), the function q(d,t) can be
expanded by the series
q(d,t)
=4d 2 ln
!
( −1)kB2k 22k + 2
d 2 +t 2
d 2 +t 2
+ 4t 2 ln
+ %
2
2
d
t
k(2k)!
k=1
2t 2k + 2 − (t+ id)2k + 2 −(t −id)2k + 2
2k + 2
(t+ id)2k + 1 +(t −id)2k + 1 −2t 2k + 1
2k + 1
+t
−2(−1)k
"
d 2k + 2
2k + 2
N
% [g(x̃p − x̃k,ỹp − ỹk + lp )
k=1p=k+1
"
− g(x̃p − x̃k,ỹp − ỹk )− g(x̃p − x̃k,ỹp − ỹk +lp −lk )
+ g(x̃p − x̃k,ỹp − ỹk − lk )]
(10)
with x̃k = pxk /h and ỹk = pyk /h. Note that in Eq. (10),
x̃p − x̃k is equal to either 0 or d, since the p-th and i-th
dipoles can share the same plane or be displaced by a.
The total energy of the zig-zag grain boundary which
is represented by such a disclination dipole wall is a sum
of gel, the additive surface energy of structural units,
mdAgA + ndBgB
h
(11)
and the disclination core energies [9–11]
where z=py/h, d =pa/h, and
g(d,t)= f(t)−2
!
Kev 2h N
% f(lk )
32p 3 k − 1
+ %
2
where lp =plp/h. Note that the factor G/(1 −n) determining the energy of the disclination dipole walls in the
isotropic theory of elasticity (Eq. (2)) has been replaced
in Eq. (6) by the energetic factor Ke for edge dislocations
that can be calculated in the anisotropic theory of
dislocations [20]. The interaction energy is calculated as
the work done by interaction forces when moving one
wall to infinity in the shear stress field sxy of the other.
The use of the dislocation model of disclination dipoles
[21] and conventional calculations yield the following
expression for the interaction energy:
×
gel =
(9)
where i= − 1. The series in Eq. (9) converges relatively quickly in the range 0Bt B p/2 for d 5 2.6. This
maximum value of d is well above those that appear in
the structures of Ž001 and Ž011 grain boundaries
[14,15]. Taking account of the fact that q(d, p −t)=q(d,
t), Eqs. (8) and (9) allow easy computation of the
function g(d, t) for all relevant values of d and t.
This result can be generalized to a more complex
structure of zig-zag boundaries which is represented by
a set of more than two single dipole walls. Let the grain
boundary period be composed of N disclination dipoles
with the arms lp, ordinates yp and x-coordinates xp; the
latter assume values of 0 or a. The elastic energy of this
wall is given by a sum of the self energies of N single
dipole walls, Eq. (6), and pair interaction energies, Eq.
(7), added up over N(N −1) pairs. Hence,
Kea0v 2N
32p 3h
a
(12)
In Eqs. (11) and (12), m and n are the numbers of the
A and B type of structural units in a disclination dipole
wall, gA and gB are the energies of the delimiting grain
boundaries composed of these units, and a is a parameter determining the disclination core energy
contribution.
An analysis of Eq. (10) shows that the elastic energy
of a complex disclination dipole wall with given ordinates of the dipoles yi (i= 1,2,…N) is minimum, when
all dipoles share the same plane, i.e. a=0. The fact that
in the diamond cubic lattice, some grain boundaries have
a zig-zag structure is associated with geometric restrictions resulting from the tetra-coordinated bonding of
atoms. Due to these restrictions, a flat disclination dipole
can be formed from Az type units only by coupling them
into pairs, that is increasing the arms of disclination
dipoles, as for example, in the case of S= 25/u =73.74°
boundary (Fig. 1). The elastic energy of a faceted wall
containing single Az units is less than the elastic energy
of a straight wall which has a larger disclination dipole
arm. Nevertheless, as seen from the results of simulations, the boundary structure with a straight arrangement of units is energetically more favored. This is
explained by a lower number of disclinations per period
and consequently a lower contribution of the disclination core energy (Eq. (12)) for the straight boundary
than for the zig-zag one.
5. Energies of tilt grain boundaries in copper and
diamond
The disclination-structural unit model requires the
energies of delimiting boundaries and bulk elastic prop-
A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
Fig. 5. Misorientation angle dependence of the energies of Ž001
symmetric tilt boundaries in copper. The dashed line has been obtained by using two delimiting ‘boundaries’, S = 1/0° and 90°. The
solid curve is obtained with two additional delimiting boundaries,
S = 5/36.9° and 53.1°. The dots represent the data from computer
simulations [5].
Fig. 6. Energy versus misorientation angle curves for Ž111 symmetric tilt boundaries in copper. The dashed line is obtained with no
intermediate delimiting boundary, and the solid line corresponds to
one intermediate delimiting boundary, S= 7/38.2°. The curves fit the
data from Ref. [5].
erties as input. It is convenient to use the energies
calculated by atomistic simulations for the former, and
either experimental or calculated properties for the
latter. In this case the set of atoms described by a given
potential can be formally considered as a model material for which the validity of the disclination approach
can be tested.
153
Vitek and co-authors obtained a suitable set of simulation data for Ž001 and Ž111 symmetrical tilt
boundaries in copper [4,5]. The energy factor Ke for the
tilt axis Ž001 calculated from the elastic constants
reported in Ref. [5] is Ke = 6.81× 1010 Pa. The energy
versus misorientation angle curves for this axis are
plotted in Fig. 5. The dashed line was derived using the
minimum number of two delimiting ‘boundaries’, S=
1/0° and 90°, with the core energy parameter a=24.1.
The solid curve was obtained using two additional
delimiting boundaries, S= 5/36.9° and 53.1°, and corresponding core energy parameter values of a1 =12,
a2 = 4, and a3 = 48.
Note that there are cusps in the g(u) curves not only
at the misorientation angles of delimiting boundaries,
but also at some intermediate misorientations. These
cusps correspond to grain boundaries which have relatively short periods (for instance, structural unit stacking of type AB, AAB, ABB, etc.). Actually, the energy
curve consists of an infinite number of cusps the most
of which are very small. These local minima occur due
to the fact that the elastic energy of a disclination
dipole wall increases when slightly deviating from the
misorientation angle of any short period boundary,
since this deviation is associated with a largeperiod perturbation of the arrangement of disclination
dipoles.
A rough estimate of grain boundary energies over the
entire range of misorientation angle is given even
without intermediate delimiting boundaries. Introducing two intermediate delimiting boundaries
results in reasonably good agreement with the simulation data.
Plotted in Fig. 6 are the results for the Ž111 tilt axis
in copper, for which Ke = 8.24× 1010 Pa [5]. Again, a
good approximation is obtained with one intermediate
delimiting boundary, S= 7/38.21°. The corresponding
a values are a1 = 14 and a2 = 40.
The g(u) curves for Ž011 tilt boundaries in diamond
are plotted in Fig. 7. These curves fit simulation data
from Ref. [22], where the Tersoff analytic potential and
tight binding calculations were used. The solid line is
the fit to the Tersoff potential calculations, for which
Ke : G/(1−n)= 6.06× 1011 Pa and a1 = 22.9. The
other parameter values, a2 = 18, a3 = 12 are taken only
as an example. The dashed line fits the tight binding
calculations, for which Ke : G/(1−n)= 4.86×1011 Pa,
a1 = 22, and again sample values of a2 = 18 and a3 =12
are used for the other intervals.
Figs. 8 and 9 present the results of the disclinationstructural unit model calculations of the energies of
Ž001 and Ž011 tilt boundaries in diamond using as
input, results from atomic simulations using Brenner’s
many-body bond-order potential [14,15]. For the Ž001
axis intermediate delimiting boundaries are S=5/
36.87° and 5/53.13°. Left of the latter, the stable grain
154
A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
boundary structures have a straight arrangement of
structural units. In the range 53.13 Bu B 90° the grain
boundaries contain coupled Az units (Fig. 1(b)). The
energy factor for the Ž001 axis calculated from the
elastic constants [11] is Ke =5.83 ×1011 Pa. A reasonable fit is obtained with parameter values a1 = 21 for
0 B u B 36.9°, a2 =36 for 36.9B u B 53.1° and a3 = 27
for the range 53.1B u B90°.
For Ž011 tilt boundaries, the following delimiting
boundaries have been used [15]: S =1/0°, 19/26.53°,
9/38.94°, and 3/70.53°. In the range 0 Bu B26.53° a
straight arrangement of structural units occurs, while in
other sub-intervals the zig-zag structures yield faceted
Fig. 9. Energies of symmetric Ž011 tilt boundaries in diamond with
input from the simulation results reported in Ref. [15].
Fig. 7. Energies of symmetric Ž011 tilt boundaries in diamond fitting
the data of simulations [22] with Tersoff potential (solid line) and in
tight binding approach (dashed line).
disclination walls. It can be shown [23] that in all these
cases separation of the disclination dipole planes is
a= 3a0/4.
A calculation of the anisotropic energy factor for
edge dislocations, the lines of which are parallel to
Ž011, yields K xe = 5.96× 1011 Pa for the Burgers vector
component along the Ž100 direction and K ye =6.50×
1011 Pa for the Burgers vector parallel to Ž011. The
former value is used for lattice dislocations composing
low-angle grain boundaries with median plane (100)
that correspond here to misorientation angles near
180°. The latter is used for low-angle boundaries with
(01( 1) median plane, i.e. for misorientation angles near
0°. To take into account the change of the energy factor
with grain boundary misorientation angle u, an effective factor is calculated as Kx = [(K ye cos u/2)2 +
(K ye sin u/2)2]1/2. The following values of the core
parameter have been chosen to obtain a good fit to the
simulation data for the minimum energy configurations:
a1 = 40 for 0B uB26.53°, a2 = 36 for 26.53BuB
38.94°, and a3 = 22 for 38.94B uB70.53°.
6. Discussion
Fig. 8. Misorientation angle dependence of the energies of symmetric
Ž001 tilt boundaries in diamond, to fit the data of simulations with
the bond-order analytic potential [14].
Two models describing grain boundary structure in
terms of elastic continuum theory, the dislocation and
disclination-structural unit models, have been compared. Both of these models are based on the same
structural unit model and are geometrically equivalent.
However, the elastic energy of grain boundaries and,
therefore, the partitioning of the elastic and core energies are expressed quite differently. The dislocation
model traditionally uses a cut-off, or core radius r0.
Due to the logarithmic term including this parameter,
the model can yield physically meaningless, negative
energies for large misorientation angles of the dislocation wall. If the dimensions of the two types of struc-
A.A. Nazaro6 et al. / Materials Science and Engineering A281 (2000) 148–155
tural units composing intermediate boundaries in a
given misorientation interval are similar, the elastic
energy given by Eq. (4) is positive in this interval.
If the dimensions differ greatly, the core diameter of
dislocations equivalent to large sized units can
closely approach the dislocation spacing, thus leading
to negative values of the elastic energy. This is not the
only shortcoming of Eqs. (1), (4) and (5). The
dislocation model gives two branches of the energy
versus misorientation angle curve that start from the
two delimiting boundaries A and B (Fig. 3).
These branches yield different elastic energy values for
the same misorientation angle at which a transformation from one dislocation representation to the
other occurs (that is, at which the grain boundary
has a structural unit stacking of the type AB). In
contrast, the disclination-structural unit model always
gives meaningful elastic energies, and the elastic
energy versus misorientation angle curve is continuous
at the AB-type structure. Therefore, the disclination
model allows for a more straightforward division of the
grain boundary energy into elastic and core contributions.
Due to the increase of allowed misorientation
subintervals, the disclination-structural unit model enables a reduction in the number of key structures
(delimiting boundaries) whose energies are used as input to calculate the grain boundary energies in the
whole misorientation range compare to the dislocation
approach. This feature is crucial for a multiscale
modeling approach recently introduced [11]. The main
idea of this approach is to use first-principles density
functional methods to calculate energies of the delimiting structures, and use these energies (and elastic properties) as input into the disclination-structural unit
model to yield accurate energies for intermediate
grain boundaries. Sufficiently large misorientation intervals covered by the disclination-structural unit
model enables the proposed multiscale scheme. A comparison of the results for model materials described by
analytic potentials made in the present paper demonstrates the efficiency and accuracy of this multiscale
approach.
.
155
Acknowledgements
A.A. Nazarov has been supported by NCSU through
a Subcontract No. 95-0012-01 as a part of the Prime
Grant No. N00014-95-1-0270 from the Office of Naval
Research through which O.A. Shenderova and D.W.
Brenner were supported.
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