• No results found

Structure and Configuration of Boundary Dislocations on Low Angle Tilt Grain Boundaries in Alumina

N/A
N/A
Protected

Academic year: 2020

Share "Structure and Configuration of Boundary Dislocations on Low Angle Tilt Grain Boundaries in Alumina"

Copied!
7
0
0

Loading.... (view fulltext now)

Full text

(1)

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

2

and Yuichi Ikuhara

2;3

1

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]
(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

(3)
[image:3.595.70.544.66.480.2]

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]
(4)

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]
(5)
[image:5.595.75.261.71.210.2]

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]
(6)

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).

REFERENCES

1) J. P. Hirth and J. Lothe:Theory of Dislocations, 2nd edition, ed. by John Wiley (McGraw-Hill, New York, 1982).

2) A. P. Sutton and R. W. Ballufi:Interfaces in Crystalline Materials, (Oxford Sci. Pub. 1995).

3) R. Hutson: Phys. Rev. Lett.46(1981) 1159–1162.

4) A. Nakamura, K. Matsunaga, J. Tohma, T. Yamamoto and Y. Ikuhara: Nature Mater.2(2003) 453–456.

5) K. Szot, W. Speier, G. Bihlmayer and R. Waser: Nature Mater.5(2005) 312–320.

6) Z. L. Zhang, W. Sigle, R. A. De Souza, W. Kurtz, J. Maier and M. Ru¨hle: Acta Mater.53(2005) 5007–5015.

7) Y. Sato, J. P. Buban, T. Mizoguchi, N. Shibata, M. Yodogawa, T. Yamamoto and Y. Ikuhara: Phys. Rev. Lett.97(2006) 106802. 8) P. B. Hirsch, Z. Zhou and D. J. H. Cockayne: Phil. Mag.87(2007)

5421–5434.

9) W. T. Read and W. Shockley: Phys. Rev.78(1950) 275–289. 10) N. Shibata, N. Morishige, T. Yamamoto, Y. Ikuhara and T. Sakuma:

Phil. Mag. Lett.82(2002) 175–181.

11) Z. Zhang, W. Sigle and M. Ru¨hle: Phys. Rev. B66(2002) 094108. 12) Y. Ikuhara, H. Nishimura, A. Nakamura, K. Matsunaga, T. Yamamoto

and K. P. D. Lagerlo¨f: J. Am. Ceram. Soc.86(2003) 595–602. 13) F. Oba, H. Ohta, Y. Sato, H. Hosono, T. Yamamoto and Y. Ikuhara:

Phys. Rev. B70(2004) 125415.

14) N. Shibata, F. Oba, T. Yamamoto and Y. Ikuhara: Phil. Mag.84(2004) 2381–2415.

15) A. Nakamura, K. Matsunaga, T. Yamamoto and Y. Ikuhara: Phil. Mag. 86(2006) 4657–4666.

16) E. Tochigi, N. Shibata, A. Nakamura, T. Yamamoto and Y. Ikuhara: Acta Mater.56(2008) 2015–2021.

17) M. L. Kronberg: Acta Metall.5(1957) 507–525.

18) B. J. Pletka, A. H. Heuer and T. E. Mitchell: Acta Metall.25(1977) 25–33.

19) K. P. D. Lagerlo¨f, A. H. Heuer, J. Castaing, J. P. Rivie`re and T. E. Mitchell: J. Am. Ceram. Soc.77(1994) 385–397.

20) J. B. Bilde-Sørensen, B. F. Lawlor, T. Geipel, P. Pirouz, A. H. Heuer and K. P. D. Lagerlo¨f: Acta Mater.44(1996) 2145–2152.

21) A. Nakamura, T. Yamamoto and Y. Ikuhara: Acta Mater.50(2002) 101–108.

22) A. Nakamura, K. P. D. Lagerlo¨f, K. Matsunaga, J. Tohma, T. Yamamoto and Y. Ikuhara: Acta Mater.53(2005) 455–462. 23) N. Shibata, M. F. Chisholm, A. Nakamura, S. J. Pennycook, T.

Yamamoto and Y. Ikuhara: Science316(2007) 82–85.

24) J. B. Bilde-Sørensen, A. R. Tho¨len, D. J. Gooch and G. W. Groves: Phil. Mag.33(1976) 877–889.

25) J. Castaing, J. Cadoz and S. H. Kirby: J. Am. Ceram. Soc.64(1981) 504–511.

26) J. Cadoz, J. Castaing, D. S. Phillips, A. H. Heuer and T. E. Mitchell: Acta Metall.30(1982) 2205–2218.

[image:6.595.77.261.71.210.2]

27) T. E. Mitchell, B. J. Pletka, D. S. Phillips and A. H. Heuer: Phil. Mag. Fig. 11 A graph showing the separation distance between partial

disloca-tions ofd1, which is calculated based on the eq. (5) as a function of tilt

(7)

34(1976) 441–451.

28) K. P. D. Lagerlo¨f, T. E. Mitchell, A. H. Heuer, J. P. Rivie`re, J. Cadoz, J. Castaing and D. S. Phillips: Acta Metall.32(1984) 97–105. 29) P. R. Kenway: Phil. Mag. B68(1993) 171–183.

30) A. G. Marinopoulos and C. Elsa¨sser: Phil. Mag. Lett.81(2001) 329– 338.

31) M. H. Jhon, A. M. Glaeser and D. C. Chrzan: Phys. Rev. B71(2005) 214101.

32) W. E. Lee and K. P. D. Lagerlo¨f: J. Electron Microsc. Tech.2(1985) 247–258.

33) J. B. Wachtman Jr. and D. G. Lam Jr.: J. Am. Ceram. Soc.42(1959) 254–260.

34) J. B. Wachtman Jr., W. E. Tefft, D. G. Lam Jr. and R. P. Stinchfield: J. Res. Nat. Bur. Standards64A(1960) 213–228.

35) J. H. Gieske and G. R. Barsch: Phys. Stat. Sol.29(1968) 121–131. 36) T. Goto, O. L. Anderson, I. Ohno and S. Yamamoto: J. Geophys. Res.

94(1989) 7588–7602.

37) F. C. Frank: Philos. Mag. VII/Ser.42(1951) 809–819.

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

[image:7.595.306.544.73.351.2]

Figure

Fig. 1Schematics showing the low angle tilt grain boundaries whichconsist of (a) perfect dislocations and (b) partial dislocations with stackingfault (SF).
Table 1 shows the six dissociation reactions of A, B, C, D,
Fig. 5A bright field transmission electron microscopy image taken fromthe boundary with a pure 2� tilt component that is observed in the Ref
Fig. 7A graph showing the separation distance between partial disloca-tions of d1, which is calculated based on the eq
+4

References

Related documents

CAFs: Cancer-associated fibroblasts (CAFs); CRC: Colorectal carcinoma; CSCLCs: Cancer stem-like cells; CSCs: Cancer stem cells; CT: Centre tumor; DFS: Disease-free survival;

The important point of departure is that the modi- fied Dutch mechanism we consider is dynamic rather than static, establishing that a dynamic mechanism can present the seller

The aim of this case report was confirmation and treatment of ABS using a standardised carbohydrate challenge test followed by upper and lower endoscopy to obtain

Through detailed SAR and computer modeling studies, we identified quinazoline BPR1K871 as a potent dual FLT3/AURKA inhibitor with anti-proliferative activities in MOLM-13

CRM especially utilizing internet are emerging trends.CRM on the Internet, otherwise called facilitated Customer Relationship Management, CRM online or on-interest CRM,

the combination of low tidal volume (VT) and low-level PEEP (Positive End Expiratory Pressure) will effectively reduce airway pressure, prevented lung injury and improve the effect

The difficulties inherent in the sense-tagging task in- clude the order in which words are presented to tag, a word’s degree of polysemy and part of speech, vagueness of the

In conclusion, the author demonstrated unique tumors of the skin and anus composed of mucins pools and atypical glandular epithelium that appeared primary anorectal extremely well