Structure and Configuration of Boundary Dislocations
on Low Angle Tilt Grain Boundaries in Alumina
Atsutomo Nakamura
1, Eita Tochigi
2, Naoya Shibata
2, Takahisa Yamamoto
2and Yuichi Ikuhara
2;31
Graduate School of Engineering, Osaka City University, Osaka 558-8585, Japan
2Institute of Engineering Innovation, The University of Tokyo, Tokyo 113-8656, Japan 3WPI, Tohoku University, Sendai 980-8577, Japan
Structure and configuration of boundary dislocations on various low angle tilt grain boundaries in alumina were considered based on the ideas that the boundary is composed of regularly arrayed edge dislocations and that the dislocations could dissociate into partial dislocations with maintaining the hcp-like oxygen sublattice. Moreover, the separation distance between the partial dislocations formed by the dissociation was evaluated by the calculations based on an elastic theory. The calculations indicated that the width of the stacking fault region between partial dislocations decreases with increasing tilt angles. As a consequence, the hypothesis and calculations used here would enable us to predict the structures of various low angle boundaries with dissociated boundary dislocations. [doi:10.2320/matertrans.MC200821]
(Received December 24, 2008; Accepted March 9, 2009; Published April 22, 2009)
Keywords: sapphire, low angle grain boundary, partial dislocation, multiple dissociation
1. Introduction
The mechanical and electrical properties of ceramic materials are closely related with the structure of crystal defects.1–8)In particular, the grain boundaries, which are the
boundaries between neighbor grains, have a strong influence on the properties of ceramic materials since ceramics is ordinarily composed of a lot of crystal grains. For the understanding of the properties, accordingly, it is important to clarify the structure of grain boundaries.
In case that a low angle grain boundary, where the misorientation angle between neighbor grains is lower than about 15, only has a tilt component, the boundary is called a ‘‘low angle tilt grain boundary’’.9–16) Low angle tilt grain
boundaries are generally considered to be composed of edge type of perfect dislocations, where Burgers vector of the perfect dislocations corresponds to a translation vector in the crystal structure.
It was found from the recent bicrystal experiments, however, that some of low angle grain boundaries in ceramics were composed of dissociated dislocations, where individual dislocation is divided into partial dislocations with very narrow separation distance, forming a stacking fault between partial dislocations.10,12,15,16) In addition, the dis-sociated dislocations often give rise to the structure transition with increasing tilt angle such as change in the separation distance, variation of the Burgers vectors and so on.10,12)
In case of alumina, which is one of the most widely used ceramic materials, it was reported that the boundary dislocation ofb¼1=3h12210idissociates into the two partial dislocations of b¼1=3h11100i andb¼1=3h01110i.12) Here,
the two partials are not edge but mixed dislocations. Figure 1 shows a schematic illustration expressing such a low angle tilt grain boundary. Left one in the figure shows the ideal boundary composed of perfect edge dislocations while right one represents the boundary with partial dislocations formed due to the dislocation dissociation. Alumina is one of the most eagerly studied ceramic materials. However, there are a lot of unclear subjects in the structure of low angle tilt grain boundaries even in alumina.
As for the slip dislocations moving in slip deformation of alumina, it is known that a basal dislocation17–23)dissociates by1=3h12210i !1=3h11100i þ1=3h01110iand a prism plane dislocation24–26) dissociates by h10110i !1=3h10110i þ
1=3h10110i þ1=3h10110i. Here note that the lattice disconti-nuity of1=3h11100i,1=3h01110ior1=3h10110iis equivalent to the minimum translation vector of oxygen ion sublattice in alumina. In the case of dissociation in these slip dislocations, therefore, the structure of oxygen ion sublattice is basically maintained in the dissociation and a stacking fault is formed only in aluminum ion sublattice.27–31)
It can be considered that oxygen ions arrange as a nearly hexagonal close-packed structure and aluminum ions regu-larly occupy 2/3 of octahedral sites according to the corundum structure of alumina.32) That is, the oxygen ions
take close-packed arrangement while aluminum ions occupy just a part of vacant sites. This is considered to be a reason that brings about a stacking fault not in oxygen ion sublattice but in aluminum ion sublattice.
Therefore, the purpose in this study is to demonstrate the structure and configuration of dissociated boundary
disloca-Rotation axis
Perfect dislocation array
θ θ θ θ
Rotation axis
Partial dislocation array
SF
SF
SF
Fig. 1 Schematics showing the low angle tilt grain boundaries which consist of (a) perfect dislocations and (b) partial dislocations with stacking fault (SF).
Special Issue on Nano-Materials Science for Atomic Scale Modification
[image:1.595.325.526.311.473.2]tions on low angle tilt grain boundaries in alumina on the basis of the formation of a stacking fault only in aluminum ion sublattice. That is, possible structures of dissociated bound-ary dislocations will be suggested and the separation dis-tances among the dislocations as a function of the misorien-tation angle will be discussed according to an elastic theory.
2. Geometry of the Dissociation of Boundary Disloca-tions
2.1 Translation vectors in crystal structure
Figure 2 shows the crystal structure of alumina. The representative directions and vectors in alumina are indicated by the arrows in the figure. There are the two minimum translation vectors of 1=3h12210i and h10110i on the (0001) basal plane for alumina of corundum structure. Here, the minimum translation vector for oxygen ion sublattice on the (0001) basal plane is1=3h10110i. On the other hand, there are the three minimum translation vectors ofh0001i,1=3h10111i
and1=3h22021i, as the translation vectors off the (0001) basal plane. h0001i is perpendicular to the basal plane while 1=3h10111i and 1=3h22021i are on the f11012g rhombohedral plane and thef10114gplane, respectively. Here, a minimum translation vector for oxygen ion sublattice perpendicular to the (0001) basal plane is 1=3h0001i. The 1=3h0001i
may have a potential to further divide by 1=3h0001i !
1=18h224223i þ1=18h24423i. But, this reaction is not consid-ered in this study, because the1=3h0001iitself is suggested here and its further reaction is too complex to consider.
[image:2.595.334.519.71.454.2]2.2 Dislocation dissociation reactions
Table 1 shows the six dissociation reactions of A, B, C, D, E and F, which were considered in this study. Figure 3A–3F also show schematic illustrations of the dislocation arrays that correspond to the six reactions. Here, five transla-tion vectors of 1=3h12210i, h10110i, h0001i, 1=3h10111i and 1=3h22021iin alumina of corundum structure were considered as the origin of potential dissociation reactions and they corresponds to the normal vectors of the boundary planes. The low index directions ofh0001i,h12210iandh10110i, which have been often reported in the studies of grain boundaries in alumina, were selected as rotation axes of the boundaries. As a result, the E and F have high index of boundary plane directions of nearlyf10114gand nearlyf11012g, respectively.
2.3 Interaction between dissociated dislocations
For the dissociated dislocations, both an elastic repulsive force and an attractive force act between partial dislocations as shown in Fig. 4.1,12,21,28) The elastic force is due to the stress field of dislocations and the attractive force is due to the stacking fault formed between partial dislocations. To balance these two forces, the separation distance between
Fig. 2 Atomic configuration of-Al2O3projected along the (a)½12210and
[image:2.595.46.550.629.785.2](b) [0001] directions. The arrows in the figure show the representative directions and vectors used in this study.
Table 1 Dislocation dissociation reactions and crystallographic orientation relationships in the low angle tilt grain boundaries in alumina.
No. Dislocation dissociation reactions Boundary plane direction
Rotation axis
Type of partial dislocations
A 1=3h12210i !1=3h11100i þ1=3h01110i f12210g [0001] edge
B 1=3h12210i !1=3h11100i þ1=3h01110i f12210g h10110i mixed
C h10110i !1=3h10110i þ1=3h10110i þ1=3h10110i f10110g h12210i, edge [0001]
D h0001i !1=3h0001i þ1=3h0001i þ1=3h0001i (0001) h12210i, edge h10110i
E 1=3h10111i !1=3h10110i þ1=3h0001i nearly h12210i edge f10114g
partial dislocations takes a constant value. In the case of low angle grain boundaries with periodically arranged disloca-tions, the total elastic repulsive force on a dislocation from all the other dislocations should be balanced with the attractive force due to the stacking fault. For example, a bright field image by transmission electron microscopy is shown in Fig. 5. This image is taken from the boundary with a pure 2 tilt component that is observed in the Ref. 15) and its orientation relationship corresponds to the B in Fig. 3 and Table 1. Open arrows in the figure indicate the position of partial dislocations. As can be seen in the image, the separation distance between the partial dislocations changes periodically due to the periodic presence of stacking faults in the boundary. The change can be estimated based on the total elastic repulsive force.
The total elastic repulsive force can be calculated using the Peach-Koehler’s equation.1,12)For the supplement, appendix shows the used method to calculate the total elastic repulsive force acting on a dislocation from all the other dislocations for periodically arranged dislocations. Here, we deal with
alumina as an elasticity isotropic crystal in the calculation, since the elastic anisotropy of alumina is not so large in spite of the complex crystal structure; the anisotropic constant is about 0.9.33–36)
3. Discussions Concerning Structure and Configuration of Dissociated Boundary Dislocations
3.1 The [0001] low angle tilt grain boundary with the f12210gboundary plane, A
Considering the dislocation dissociation of the [0001] low angle tilt grain boundary with thef12210gboundary plane as shown in Fig. 3A, the two partial dislocations will have the Burgers vector with the same size and be edge type. In addition, the extra half planes are inclined at30degrees to the boundary plane alternatively. According to the Peach-Koehler’s equation, in this case, the relation between the total elastic repulsive force on a partial dislocation, fR and the
stacking fault energy,SFcan be expressed as follows.
SF¼ fR¼
jbpj2
4ð1Þ d
X1
n¼0
1
nþ 1
nþ1
; ð1Þ
where jbpj is the size of Burgers vector of the partial
dislocation withbp¼1=3h11100ior1=3h01110i(0.276 nm),
is the shear modulus (150 GPa33–36)) and is the Poisson’s
d1
d2 d
1 3[0110] A
SF
GB
d1
d2 d
B
SF
GB
d1
d2 d
1 3[1010]
d1 C
SF
GB
d1
d2 d d1
1 3[0001] 1 3[0001] 1 3[0001] D
SF
GB
d1
d2 d
1 3[1010] 1 3[0001] E
SF
GB
d1
d2 d d1
1 3[0001] F
SF
GB 1
3[1100]
1 3[0110] 1 3[1100]
1 3[1010] 1 3[1010]
1 3[1010] 1 3[1010]
SF SF
SF
Fig. 3 Schematic illustrations showing the predicted dissociation structure of boundary dislocations in the low angle tilt grain boundaries in Alumina. The crystallographic characteristics of the dissociation are described in Table 1.
f
γSF
(a)
d1
d2
(b)
SF
Elastic repulsive force Attractive force by Stacking fault energy
Fig. 4 A schematic showing (a) the elastic repulsive force due to elastic fields of neighbor partials and (b) the attractive force due to the shrinkage of stacking fault (SF).
(1010) [0001]
[1210]
Fig. 5 A bright field transmission electron microscopy image taken from the boundary with a pure 2tilt component that is observed in the Ref. 15).
[image:3.595.60.277.70.488.2] [image:3.595.355.502.73.233.2] [image:3.595.306.549.292.422.2]ratio (0.2433–36)). Here, d is the periodicity of boundary
dislocations and is the ratio of the separation distance between the two partial dislocations,d1against thed(d1=d). According to the Frank’s formula,37) the tilt angle of 2 is
given by2¼ jbj=d, wherejbjis the size of Burgers vector of ideal perfect boundary dislocation. That is, the periodicity of boundary dislocations,dis the function of2. It can be said that the eq. (1) gives the separation distance between partial dislocations as a function of tilt angle, since the stacking fault energy, SF can be treated as a constant and only thed and
are changeable. Figure 6 shows the separation distance with increasing tilt angle, which can be estimated using SF¼0:2J/m2, 0.3 J/m2, 0.4 J/m2. It can be seen that the
separation distance decrease with increasing tilt angle in all the cases. Ikuharaet al.12)observed the boundary A with the
misorientation angle of 2, 6and 8by transmission electron microscopy including the bicrystal experiment and found that the boundary dislocations dissociated into two partials as shown in Fig. 3A. It was reported that the observed structure of boundary dislocations is good agreement with the estimation usingSF¼0:3J/m2.
3.2 The h10110ilow angle tilt grain boundary with the f12210gboundary plane, B
Considering the dislocation dissociation of theh10110ilow angle tilt grain boundary with thef12210gboundary plane as shown in Fig. 3B, the two partial dislocations will be the mixed dislocations with both the1=6h12210iedge component and the 1=6h10110i screw component. Accordingly, this boundary dislocation of b¼1=3h12210i corresponds to the basal dislocation, which is one of the slip dislocations in alumina. According to the Peach-Koehler’s equation, in this case, the relation between the total elastic repulsive force on a partial dislocation, fRand the stacking fault energy,SFcan
be expressed as follows.
SF¼fR¼
jbpj2ð2þÞ
8ð1Þ d
X1
n¼0
1
nþ 1
nþ1
: ð2Þ
Figure 7 shows the separation distance with increasing tilt angle, which can be estimated using SF¼0:2J/m2,
0.3 J/m2, 0.4 J/m2. The boundary B with the misorientation angle of 2 were also observed by transmission electron microscopy including the bicrystal experiment and found that
the boundary dislocations dissociated into two partials as shown in Fig. 3B.15,16) It was reported that the observed
structure of boundary dislocations is good agreement with the estimation usingSF¼0:3J/m2.
3.3 Theh12210ior [0001] low angle tilt grain boundary with thef10110gboundary plane, C
Considering the dislocation dissociation of theh12210ior [0001] low angle tilt grain boundary with the f12210g
boundary plane as shown in Fig. 3C, the three partial dislocations will be the edge dislocations with the same Burgers vector ofbp¼1=3h10110i. According to the
Peach-Koehler’s equation, in this case, the relation between the total elastic repulsive force on a partial dislocation, fR and the
stacking fault energy,SFcan be expressed as follows.
SF¼ fR¼
jbpj2
2ð1Þ d
X
1
n¼0
1
nþ 1
nþ1þ 1
nþ2 1
n2þ1
ð3Þ
Figure 8 shows the separation distance with increasing tilt angle, which can be estimated from the eq. (3) using SF¼0:2J/m2, 0.3 J/m2, 0.4 J/m2. This boundary
disloca-tion of b¼ h10110i corresponds to the prism plane disloca-tion, which is one of the slip dislocations in alumina. In the case of slip dislocation, it is known that the prism plane dislocation can dissociate into the three partial dislocations as shown here.24,26)So far, however, it does not seem that the
structure of prism plane dislocations array that compose a low angle tilt grain boundary is observed by transmission electron microscopy.
3.4 Theh12210iorh10110ilow angle tilt grain boundary with the (0001) boundary plane, D
Considering the dislocation dissociation of theh12210ior
h10110i low angle tilt grain boundary with the (0001) boundary plane as shown in Fig. 3D, the three partial dislocations will be the edge dislocations with the same Burgers vector of bp¼1=3h0001i, where the size of bp is
0.433 nm. In this case, the relation between the total elastic repulsive force on a partial dislocation, fRand the stacking
fault energy, SF can be also expressed by the eq. (3).
Fig. 6 A graph showing the separation distance between partial disloca-tions ofd1, which is calculated based on the eq. (1) as a function of tilt
angles.
Fig. 7 A graph showing the separation distance between partial disloca-tions ofd1, which is calculated based on the eq. (2) as a function of tilt
[image:4.595.77.263.71.213.2] [image:4.595.333.519.73.216.2]Figure 9 shows the separation distance with increasing tilt angle, which can be estimated from the eq. (3) using SF¼1:0J/m2, 2.0 J/m2, 3.0 J/m2. To our knowledge, the
dislocation withb¼ h0001iis not observed by experimental. The dislocation could dissociate into partial dislocations because of the too large Burgers vector of 1.30 nm. If this dissociation shown here can be observed, the energy of a stacking fault along (0001) will be estimated by experimental although the energy has been calculated only by computer simulations.29,30)
3.5 The h12210ilow angle tilt grain boundary with the boundary plane of the normal vector alongh10111i, E
Considering the dislocation dissociation of theh12210ilow angle tilt grain boundary with the boundary plane of the normal vector alongh10111i, whose plane is near thef10114g
plane, the boundary dislocation of b¼1=3h10111i can dissociate into the two partial dislocations as shown in Fig. 3E. Here, the partials have different Burgers vector about the size and orientation. According to the Peach-Koehler’s equation, in this case, the relation between the total elastic repulsive force on a partial dislocation, fR and the
stacking fault energy,SFcan be expressed as follows.
SF¼ fR¼
jbAjjbBjsin 2’A
2ð1Þ d
X1
n¼0
1
nþ 1
nþ1
;
ð4Þ
where bA¼1=3h10110i, bB¼1=3h0001i and the ’A is the
misorientation between these two directions of h10111i and
h10110i (’A¼57:61). Figure 10 shows the separation
dis-tance with increasing tilt angle, which can be estimated from the eq. (4) using SF¼0:3J/m2, 1.0 J/m2, 3.0 J/m2. It
should be noted that the two Burgers vector are perpendicular to each other. Therefore,jbj2 ¼ jbAj2þ jbBj2. This suggests
that the dissociation of E does not give the decrease of elastic energy around the boundary dislocation. That is, the dissociation of E may not occur.
3.6 Theh12210ilow angle tilt grain boundary with the boundary plane of the normal vector alongh22021i, F
Considering the dislocation dissociation of theh12210ilow angle tilt grain boundary with the boundary plane of the normal vector alongh22021i, whose plane is near thef11012g
plane, the boundary dislocation of b¼1=3h22021i can dissociate into the three edge type of partial dislocations as shown in Fig. 3F. The two partials in the three have the same Burgers vector and another one has the different vector perpendicular to the two. According to the Peach-Koehler’s equation, in this case, the relation between the total elastic repulsive force on a partial dislocation, fRand the stacking
fault energy,SFcan be expressed as follows.
SF¼ fR¼
2ð1Þ d
(
jbAjjbBjsin 2’B
X1
n¼0
1
nþ 1
nþ1
þ jbAj2
X1
n¼0
1
nþ2 1
n2þ1
)
; ð5Þ
where bA¼1=3h10110i, bB¼1=3h0001i and the ’B is
the misorientation between these two directions of h22021i
and h10110i (’B¼38:24). Figure 11 shows the separation
distance with increasing tilt angle, which can be estimated from the eq. (5) usingSF¼0:3J/m2, 1.0 J/m2, 3.0 J/m2.
To our knowledge, the dislocation withb¼1=3h22021iis not observed by experimental. The dislocation could disso-ciate into partial dislocations because of the too big Burgers vector if this boundary is intentionally fabricated by bicrystal experiment.
Fig. 8 A graph showing the separation distance between partial disloca-tions ofd1, which is calculated based on the eq. (3) as a function of tilt
angles.
Fig. 9 A graph showing the separation distance between partial disloca-tions ofd1, which is calculated based on the eq. (3) as a function of tilt
[image:5.595.333.520.72.212.2]angles.
Fig. 10 A graph showing the separation distance between partial disloca-tions ofd1, which is calculated based on the eq. (4) as a function of tilt
[image:5.595.74.263.265.407.2] [image:5.595.314.551.568.665.2]4. The Structure of Low Angle Grain Boundaries Depending on Stacking Fault Energy
As can be seen in Figs. 6–11, the estimated separation distance between partial dislocations depends on the stacking fault energy. This is because the separation distance is inverse proportion to the stacking fault energy as can be seen in eqs. (1)–(5). In this point, there are no difference between the dissociation of a slip dislocation and that of a boundary dislocation. It can be said that it is significant to grasp the real stacking fault energies along various crystal planes in alumina, for the demonstration of the structure of low angle tilt grain boundaries on the basis of the equations proposed in this study.
The energy of thef12210gstacking fault has been studied by experimental and theoretical calculations.27–29,31)In the case
of low angle tilt grain boundaries with thef12210gboundary plane, the structure can be estimated using the equations. On the other hand, it seems that the stacking faults off thef12210g
plane have not been studied enough, although the stacking faults alongf10110gwere investigated by experimentals with conventional transmission electron microscopy24,26,28) and
theoretical calculations.29,31)Accordingly, it can be said that
the actual structures of the stacking faults off the f12210g
plane leaves a lot of unclear points. In case that a low angle tilt grain boundary is fabricated by the bicrystal experiment, the introduced boundary dislocations can be selected inten-tionally. That is, it will be possible to fabricate the low angle tilt grain boundaries shown in Fig. 3 and to experimentally observe the structure in detail. This has a potential to experimentally clarify the structure and energy of stacking faults off thef12210gplane.
5. Summary
Structure and configuration of boundary dislocations on various low angle tilt grain boundaries in alumina, which is one of the most widely used ceramics, were considered. It is here premised that the boundary should be composed of regularly arrayed edge dislocations and that the dislocations could dissociate into partial dislocations with maintaining hcp-like oxygen sublattice. Moreover, the separation dis-tances between partial dislocations formed by the
dissocia-tion were evaluated by the calculadissocia-tions based on the elastic theory combined with the Peach-Koehler’s equation. The calculations can lead to the width of the stacking fault region between partial dislocations depending on the tilt angles. The estimated structure is good agreement with the experimental conducted in the past. Thus, it becomes possible to predict the structures of various low angle boundaries containing dissociated dislocations.
Acknowledgement
This work was supported by the Grant-in-Aid for Scientific Research on Priority Areas ‘‘Nano Materials Science for Atomic-scale Modification’’ (No. 19053001) from the Min-istry of Education, Culture, Sports, Science and Technology (MEXT).
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Appendix: Total Elastic Repulsive Force Acting on a Dislocation from All the Periodically Arranged Dislocations
When we estimate elastic repulsive force acting a dislocation on the low angle tilt grain boundary with periodically arranged dislocations, it is needed to consider the total elastic repulsive force from all the other boundary dislocations. Figure A·1 schematically shows the elastic repulsive forces on a partial dislocation from all the other partial dislocations on the boundary shown in Fig. 3A. Here, two groups of the partial dislocations indicated by A and B have different Burgers vector. In defining An(n¼1;2;3;. . .)
as shown in the figure, the An on the boundary are
symmetrically arranged to the partial of A0. This means that the elastic force acting on the A0 from An can be cancelled
and is not needed to be taken into account. On the other hand, In defining Bn (n¼1;2;3;. . .) as shown in the figure, the
elastic force on the A0from Bncan not be cancelled because
of the nonsymmetrical distances between A0 and Bn.
According to the Peach-Koehler’s equation, in the case of Fig. 3A, the elastic repulsive force between the two partial dislocations of A0 withb¼1=3½10110 and Bn with b¼1=3½01110, fycan be expressed as follows.
fy¼ jbpj
2
4ð1Þ
1
y; ðA:1Þ
where jbpj is the size of Burgers vector of the partial
dislocations andyis the separation distance between the A0 and the Bn. Thus, the total elastic repulsive force acting on
the A0 from all the partial dislocations indicated by Bn, fR,
which can be calculated using the eq. (A·1), is given by
fR¼
jbpj2
4ð1Þ
1
D1
þ 1
D1
þ 1
D2
þ 1
D2
þ 1
D3
þ 1
D3
þ
¼ jbpj
2
4ð1Þ
X1
n¼1
1
Dn
X
1
n¼1
1
Dn
( )
; ðA:2Þ
whereDn represents the separation distances between the A0and the partial dislocations of Bn.
That is, if n>0,Dn¼d1þ ðn1Þðd1þd2Þ ¼d ðn1þÞ ðA:3Þ
and if n<0,Dn¼d2þ ðn1Þðd1þd2Þ ¼d ð1Þ ðnþÞ; ðA:4Þ
wheredis the periodicity of boundary dislocations,d1is the separation distance between neither partial dislocations forming a stacking fault in between and¼d1=d. Therefore, the eq. (A·2) can be transformed to
fR¼
jbpj2
4ð1Þd
X1
n¼1
1
n1þ
X
1
n¼1
1
ð1Þ ðnþÞ
( )
¼ jbpj
2
4ð1Þd
X1
n¼0
1
nþ 1
nþ1
: ð1Þ
This is the way to obtain the eq. (1) in the main text. In the cases of the boundaries shown in Fig. 3B–3F, the total elastic repulsive forces can be given by the similar way.
D1 d1 d2 A0 A1 A-1 A2 A-2 B1 B-1 B2 B-2 B-3 B3
FSF(=γSF)
FA0B1
FA0B3
FA0B-3
FA0A-1
D-1
D2
D-2
D3
D-3
FA0A-2
FA0B-1
FA0A1
FA0B-2
FA0B2
FA0A2 ( )− = = n P D b F F F n n 1 1 4 2 B A A A A A 0 -n 0 0 ν πµ SF d1 d2 d1 d1 d2 d1 d2 d2 SF SF SF SF
Fig. A1 A schematic illustration showing the repulsive and attractive forces acting on partial dislocations. In order to evaluate the actual force on the particular partial dislocation described as A0, the contribution from
all the other partial dislocations on the boundary must be taken into account. As shown in Fig. 4,d1is the width of Stacking fault (SF) between
two partials while d2¼dd1 is that of coherent region. Since the
periodicity of the boundary is given byd, the distance between similar partials indicated by Anor Bnwas determined by this periodicity, i.e.,nd, wherenis an integer. The repulsive force on A0for symmetrically located
Anis completely cancelled, and one only needs to consider the effects due to dissimilar partials of Bn. Each repulsive force to A0 by Bn,FA0Bn
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