SPECIAL GRAIN BOUNDARIES IN RHOMBOHEDRAL MATERIALS

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SPECIAL GRAIN BOUNDARIES IN RHOMBOHEDRAL MATERIALS

H. Grimmer

To cite this version:

H. Grimmer. SPECIAL GRAIN BOUNDARIES IN RHOMBOHEDRAL MATERIALS. Journal de

Physique Colloques, 1990, 51 (C1), pp.C1-155-C1-160. �10.1051/jphyscol:1990123�. �jpa-00230281�

COLLOQUE D E PHYSIQUE

Colloque Cl, suppl6ment au n o l , Tome 51, janvier 1990

SPECIAL GRAIN BOUNDARIES IN RHOMBOHEDRAL MATERIALS

H. GRIMMER

Paul Scherrer Institute, Laboratory of Materials Sciences, Wiirenlingen and Villigen, CH-5232 Villigen PSI, Switzerland

RCsumC: Une collection concise de rCsultats concernant les orientations de coyncidence de cristaux rhombokdriques est prksentke avec les rkfbrences des travaux complets. On mon- tre la relation avec l'approche de Bonnet et Durand. Cette relation est va.lable aussi pour des rkseaux hexagonaux et tktragonaux. Des applications B l'interprktation d'observa.tions en MET de joints de grains spkcia.ux dans l'alumine sont dkcrites.

Abstract: A concise collection of results on coincidence orientations of rhomboledral crysta.1~

is given with references to detailed presentations. The connection with the Bonnet-Dura.nd approach is explained. This connection is valid d s o for hexagonal and tetragonal lattices. It is shown how the results can be used to interprete TEM micrographs of special grain boundaries in corundum.

Metals and ceramics are used in polycrystalline form for most of their a.pplications. Two neighbouring grains, 1 and 2, of the same phase are in coincidence orientation if the lattices of their symmetry tran~lat~ions have vectors in common that form a 3-dimensional lattice, called the coincidence lat- tice. The volume of a primitive cell of this lattice is an integral multiple C of the corresponding value for the crystal lattice. The crystal translation lattice 1 can always be mapped onto lattice 2 by a

rotation. The inultiplicity C depends on the rotation and, in general, also on the axial ratio.

For lattices of any symmetry there is a connection between coincideilce orientations and the matrix R describing the rotation in a primitive basis i.e. a basis with axes spanning a primitive cell of the lattice. Coincidence orientations correspond to matrices R with rational elements; C is given by the

C-Theorem: The multiplicity C is the least positive integer such thak

CR

and CB.-' are integral matrices.

For rhombohedral lattices a primitive basis

G , G,

can be chosen such that the 3 axes have the same length and that the 3 pairs of axes enclose the same angle a,

leil

⁼ lZ21 =

1 4

and = = L(&,&) = a

.

(1)

Writing

7- = cosor/(l+ 2coscu)

,

(2)

a. rotation by an angle

O

= 229 about an axis UG

+

V &

+

WZ3 can be written as

where

and

m = J(1- ~ T ) ( u ~

+ vZ +

^{w Z )}

+

~ T ( V W -b

wu +

UV)/ tan ~9

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990123

Cl-156 COLLOQUE DE PHYSIQUE

If R has rational elements then it is possible to choose m, U, V, W to be integers without common divisor,

gcd(m, U, V, W) ⁼ l . (6)

Conversely, if (6) is satisfied then

R

has rational elements if in addition either T is rational or U ⁼ V = W or in = U

+

V

+

W = 0. If ^T is rational then it can be written as

T = v/p where p and v are integers satisfying gcd(p, 11) = 1

.

₍₇₎

Instead of using the C-Theorem, Doni, Fanides and Bleris [l] assumed that C is the least positive integer such that CR is an integral matrix. Also this leads in many but not all cases to the correct result. A counterexample is the rotation R obtained by choosing ( r , m, U, V, W) = (13/49,15,11,4, -10) in (3) and (5)) i.e.

18 9 1 48 -30 -67

R =

l

_l*

(

^-8 _{-4 G} ^l

-:q ⁾

^and

^R-'

^{= -} ₉₈

⁽

_-22 ⁷⁶ _-72 ^{124 94} ₄₅

^{) .}

(Notice that R-' can be obtained by replacing 13 by -13 and therefore m by -7n.) The assumption of Doni et al. gives C = 14 whereas the true value of C is 7 X 14 = 98.

Also an explicit formula for C has been derived for rhombohedral lattices, called the C-rhomb Theorein [2]. It makes use of a 4-axes coordinate system:

+ + - ,

h = & - & ,

& = & - e ' 3 , g 3 = e 3 - e l l i 4 = Z 1 + & + z 3 . It follows from (1) that

a.nd = =

l&l.

The ratio

I&l/l&I

is called the axial ratio c/a of the lattice. It satisfies ( c / ~ ) ~ = 3/(2 - 67). Equation (7) gives for rational ^T

where p = p - 3v. It follows that p - p is a multiple of 3 and that p and (p - p)/3 have no factor in common,

gcd(p, (P - = 1

.

⁽⁹⁾

The rotation axis is written as

abbreviated as [U v.w] which are called the Weber indices of the rotation axis. The constant K

>

0 indica.tes that the rotation determines the axis only up to a constant. The plane perpendicular to

[21 v.w] has Miller-Bravais indices ( U v

.

2 / 3 ( ~ / a ) ~ w ) . Putting K = 3 it follows from (6) that m, ^{U ,} v, w are integers, that 2u

+

^v

+

^W is a multiple of 3 and that

The multiplicity can be expressed as follows in terms of p, p, m, U , v , W satisfying (9) and (11) C-rhomb Theorem: The rotation with axis [u.v.w] and angle O = 279 given by

genera.tes a coincidence lattice with multiplicity

where and

It follows that C is independent of p and p for rotations with axis parallel to the 3-fold a.xis of the rlloinbohedral lattice and for 180'-rotations with axis perpendicular to the 3-fold a.xis: If ^U = zj = 0 then C = (3m2

+

w2)/(3

f:);

if m = W = 0 then C = (u2

+

riv

+

v2)/3. Rotations with pa.ra.meters satisfying (11) and either u = v = 0 or m = w = 0 are called coinnlon (or exact) rotations. They have the same value of .E for each axial ratio c/a. If (cla)' is rational then there are also specific (or a.pproximate) r o t a t i o n s . Their parameters satisfy (11); their multiplicities depend on c/n. Common rotations are common t o rhombohedral lattices with any axial ratio; specific rota.t,ions are a.ssociated with a specific value r of the axial ratio c/a. If the experimental value r e of c / a slight,ly tleviat,es from r then the coincidence of symmetry translations in the 2 lattices is only approxima.te for a, specific rota,tion whereas one always has & coincidence for common rotations.

Low energy boundaries can be expected between grains in a common orientation with a small value of C or in a specific orientation if C is small and re = r . If re

#

r then the boundary must contain secondary dislocations in order to locally preserve the structure of a coincidence boundary. The minimum density of dislocations is related to the deformation parameters of Bonnet and Durand 131: Choose a cell Ml of lattice 1 and a cell Mz of lattice 2 that coincide if re = r and which both have volumes C times larger than the volume of a primitive cell of either lattice. The deformation D needed to restore coincidence between IIFl and J/12 for re

#

r can be written as

1+E1 0

D =

(

₀ ^{0 ~ + E Z 0 with &l}

⁵

^{E Z}

⁵

^{EZ and}

⁺

^c2

⁺

^{= 0}

for ^8.11 a.ppropriate choice of the orthogonal coordina.te system [3]. For 11exa.gonal [4], rhoinl~ol~edra.1 [S]

a.nd tetragonal lattices one has E Z = 0, i.e. ~3 = -&l = E. It can be shown [G] that

xvliere A is the relative deviation between the experimental and specific values of the axial ratio. In order to express E as a function of A and of the specific rotation R the latter is clecomposecl as a product R = RIRll of a rotation Rll with axis parallel to the principal axis of lattice 1 followed by a rotation Rl with axis perpendicular to the principal axis. The angles Q = 2$ of Rill Q = 2 9 of Rl and O = 26 of R satisfy cos+ cosp = cos6, i.e.

+

5 6 and cp 5 19. Grinlmer and Bonnet [G] obtained

sin @ =

2~3?773,,/-'

r(3m2

+

w2)

+

^r-13(u2

+

^uv

+

^{V') (15)}

and

E = A s i n @

,

i.e. E is the product of a factor depending only on the specific rota.tion and a factor clepentling only

011 the relative deviation of the axial ratios.

The following lower bound for the multiplicity of specific rotations as a f~unction of 11 and p can be derived from (13):

E 2 6 1 3 if p and p are odd, C 2 6 1 3 otherwise. (17) This bound can be used to compute tables of all specific coincidence rot.ations ~ v i t h c / a in a given interval and multiplicity up to a given value C,.

A given relative orientation of 2 grains can be described by different rot,ations clue to the trigonal sym~netry of the rhombol~edral lattice. These rotations form an equivalence class. Exactly one rotation in each class, called the representative { n z u v . ~ ~ ) , satisfies the conditions:

v r u if zo=0,

v > ^U if ^U)

>

0 and m = m ( u

+

V).

Cl-158 COLLOQUE DE PHYSIQUE

Eq. (18) defines a spherical triangle bounded by 2 mirror planes of lattice 1 and the plane perpendicular to its 3-fold symmetry axis. This triangle comprises 1/12 of the full solid angle and is called the standard spherical triangle or SST. Each rotation angle occurring in an equivdellce class occurs in it also with an axis in the SST. The representative has its axis [uv.w] in the SST and (because of (19)) the smallest angle occurring in the class. If there are several such rotations then (20) malces a uniclue choice.

All classes of common rotations and many classes of specific rotations contaiil rotations by 180'. The last column of Tables 1 to 3 gives the Miller-Bravais indices of the planes perpendicular to those axes of 180"-rotations that lie in the SST. Only one among the specific classes in Tables 1 allcl 3 does not contain 180"-rotations. This class has the symbol (15,3)19b, where 15, 3 and 19 are the values of p, p and C, respectively. (Remember that p and p determine the specific value r of c / n accortlillg to (8).) A letter a,b,.

.

.after the value of C arranges classes with the same values of p , p and C in t,he order of increasing angles O of the representative; a subscript distinguishes cases with the same vahw of O.

For common rotations, the values p, p are replaced by the letter c.

Table 1: The equivalence classes of common rotations with C 5 30.

The representatives have the form (n200.w).

Table 2: The equivalence classes of specific rota.tions with C

<

20 and 7.25 5 (c/a)'

<

^7.65.

Cl is the value of the lower bound (17) rounded to the next higher integer.

Coruildum (a-Al2O3), the most important oxide ceramic material, has space group R%. Its rhombohe- dral lattice has the axial ratio re = 2.730. Haematite (Fe203) with re = 2.732 is of the same structure

r=c/a 2.699 2.711

2.717

2.739

2.755 2.763

C, 8 11

14

7

19 12

Class symbol ( p , p ) C ( 3 4 , 7 ) 8 (34, 7)15 (49,10)14 (49,10)18 (64,13)14 (64,13)15 (64,13)20 (15, 3) 71 (15, 3) 7' (15, 3)11al (15, 3)11a2 (15, 3 ) l l b (15, 3)13a (15, 3)13bl (15, 3)13b2 (15, 3)17a (15, 3)17bl (15, 3)17b2 (15, 3)19al (15, 3)19a2 (15, 3)19b (86,17)20 (56,11)12 (56,11)13 (56,11)17

sill@

0.9091 0.9697 0.7454 0.5797 0.9712 0.9065 0.6799 0.9035 0.9035 0.5750 0.5750 0.9959 0.8427 0.9730 0.9730 0.6444 0.9843 0.9843 0.8807 0.8807 0.9986 0.9012 0.9750 0.9000 0.6882 O"

86.42 93.82 75.52 68.83 94.10 86.18 72.54 85.90 85.90 68.68 68.68 95.22 57.42 94.41 94.41 71.12 96.76 96.76 65.10 65.10 86.98 85.70 94.78 85.59 72.90

Miller-Bravais illdices Representative

m U v . w 1 0 2 . 1 7 0 1 7 . 7 5 0 7 . 5 1 1 0 . 1 13 0 32 . l 3 1 0 2

.

1 13 16 0 . l 3

1 0 2

.

¹

3 6 0

.

3 1 1 0

.

1 3 0 3 . 3 1 0 3

.

0 2 0 3

.

0 6 0 15

.

6 2 5 0

.

2 3 2 2 . 3 1 2 1

.

1 3 3 6

.

3 3 0 5

.

1 9 15 0

.

3 2 3 3

.

0 1 0 2

.

1 11 0 28 .l1 1 0 2 . 1 11 14 0 . l 1

of twin 7 0 . 3 4 l 0

.

4 1 0 . 7 0 5

.

49 1 0

.

4 13 0

.

64 0 1

.

8 1 0

.

5 0 1 . 5 0 1

.

10 1 0

.

10

1 0 . 4 0 1

.

4 1 1

.

15 1 2

.

10 2 1 . 10

17 0

.

86 1 0 . 4 11 0 . 5 6 0 1

.

8

planes 0 1 . 2 0 7 . l 7 0 5 . 7 1 0

.

1 0 13 .32 0 1

.

2 13 0 . l 6 0 1

.

2 1 0

.

2 1 0 . 1 0 1

.

¹

1 1 . 6 1 1 . 3 0 2

.

5 2 0 . 5

1 2 . 5 . 2 1 . 5 0 1

.

2 0 11 .28 0 1 . 2 11 0 .l4

type. In both cases, the most common types of twins are the basal twin with twin plane (00.1) and the rhombohedral twin with a (01.2) plane [7,8]. The description of these twins by the coincidence rotations (c)3 in Table 1 and (15,3)T1 in Table 2 makes it possible to predict the possihle Burgers vectors of dislocations in the twin plane. They are vectors of short length of the D S c lattice. This lattice can be defined as the coarsest lattice that contains lattices 1 and 2 (whereas the coincidence lattice is the finest lattice contained in lattices 1 and 2). If V, VC and VD denote the volumes of primitive cells of crystal lattice 1 (and 2), ~oincidence lattice and DSC lattice one has

Burgers vectors of dislocations in twin boundaries have been determined experimentally in [Q] for basal twins and in [l01 for basal and rhombohedral twins. They have the predicted form. A multiplicity C = 8 instead of 7 was proposed in [8] for the rhombohedral twin because systematic tables of all possible coincidence orientations were not available [10]. For the'same reason, TEM observations on a-A1203 were interpreted in [l11 using tables for hexagonal lattices. These observatiorls have been revisited on the basis of the present results ([12], see also [5] and [13]).

Networks of dislocations are expected in a grain boundary if the rotation connecting the two adjacent grains is a specific rotation because of the mismatch due to the difference between r e and r . Adrlitional dislocations are expected if the rotation slightly deviates from a specific or coinmon rotation. This explains why Shiue and Phillips [l01 always found dislocations in the boundary of rhoinbol~eclral twins but only sometimes for basal twins.

Brandon [l41 estimated that the structure of a coincidence boundary will locally be preserved in cubic materials as long as the angle Od of the deviation froin a coinciclellce orientation sa.tisfies

It was suggested in [l21 to use this criterion also to discuss the distributioil of rela.t.ive orientations in corundum. If the grains are oriented at random then one expects 0.57% low angle l>oundaries, i.e.

boundaries satisfying (22) with C = l [15], 0.23% boundaries satisfying (22) for a. coininon rotation with 1 < C 5 36, and 1.19% boundaries satisfying (22) for a. specific rotation ~ 4 t h r = 2.739 and 1 < C < 3 6 .

S. Lartigue found [5] that the frequency of such special boundaries was compatible with the expected value for pure a-A1203 and for low angle and common boundaries in a-A1203 doped with 500 ppm of MgO. She analysed 56 boundaries in the doped a-A1203, 23 in as pressed samples and 33 in deformed samples [12]; 3+5=8 among them were high angle boundaries of the specific type definecl above, i.e.

14 times more than exwected for a random distribution. She observed a. network of dislocations in G aillong these 8 boundaries. It is concluded that certain high angle coincidence boundaries are highly favored in a-A1203 doped with small amounts of MgO.

Table 3 gives all classes of specific rotations with C

5

3 for arbitrary values of r . A table exteiicled to C

<

5 but without the value of sin@ will appear in [16]. The lattices having specific rotations with C = 1 show even cubic symmetry. The lattice with r =

fi

⁼ 2.4495 is face-centred cubic, the one with r = &/2 primitive cubic and the one with r = &/4 body-centred cubic. A rotation that generates a coincidence lattice in one of the cubic lattices generates coincidence lattices with the same multiplicity also in the other two [cf. 171. The rhombohedral symmetry rotations and the specific rotations with C = 1 of a cubic lattice correspond together to the cubic symmetry rotations. The common class (c)3 and the three specific classes 3a, 3bl and 3bz of a cubic lattice correspond together to the cubic equivalence class with C = 3. The splitting of cubic equivalence classes into rl~oml~ol~edral ones has to be considered in order t o interprete special grain boundaries in rl~ombohedml materials with re close to the value r of one of the cubic lattices.

The classes in Table 3 a.ppear in pairs with axial ratios r , r' satisfying

This is a consequence of the reciprocity theorem [18], which can be formulated as follou-S. "If a rotation of a lattice A generates a coincidence lattice Ac then the same rotation applied to lattice A', reciprocal to A, generates a D S c lattice Ab, reciprocal to Ac." Two rhombohedral lattices satisfying (23) are reciprocal to each other up to a dimensional constant. It follows then from the recil~rocity tlieorenl and (21) that the two specific classes of a pair contain the same rotations.

Cl-160 COLLOQUE DE PHYSIQUE

Table 3: The equivalence classes of specific rotations with C

<

^3.

References:

[l] E.G. Doni, Ch. Fanides and G.L. Bleris, Cryst. Res. Tecl~nol. 2 1 (1986) 1469.

[2] H. Grimmer, Acta Cryst. A45, (1989) 505.

[3] R. Bonnet and F. Durand, Philos. Mag. 32 (1975) 997.

[4] R. Bonnet, E. Cousineau and D.H. Warrington Acta Cryst. A37 (1981) 184.

[5] S. Lartigue, Doctoral Thesis, Universitk de Paris-Sud, ^(1988).

[G] H. Grimmer and R. Bonnet, submitted to Acta Cryst. A.

[7] L.A. Bursill and R.L. Withers, Pl~ilos. Mag. A40 (1979) 213.

[8] K.J. Morrissey and C.B. Carter, J. Am. Ceram. Soc. 67 (1984) 292.

[g] ICJ. Morrissey and C.B. Carter, Advances in Ceramics, Vol. 6, p. 85, Am. Ceram. Soc., Colurnbus, (1983).

[l01 Y.R. Shiue and D.S. Phillips, Philos. Mag. A50 (1984) 677.

[Ill S. Lartigue and L. Priester, Grain Boundary Structure and Related Pj~enomena, Supplement to Trans. Japan Inst. Metals (1986) 205.

[l21 S. Lartigue and L. Priester, J. Am. Ceram. Soc. 71 (1988) 430.

[l31 H. Grimmer, R. Bonnet, S. Lartigue and L. Priester, Philos. Ahg. A, to be published [l41 D.G. Brandon, Acta Metall. 1 4 (1966) 1479.

[l51 H. Grimmer, Scripta Metali. ¹³ (1979) 161.

[l61 H. Grimmer Scripta Metall., to be published.

[l71 H. Grimmer, W. Bollmann and D.H. Warrington Acta Cryst. A30 (1974) 197.

1181 ^H. Grimmer, Scripta Metall. 8 (1974) 1221.

r = c / a 0.6124 1.2247 2.4495 0.3062 0.3873 0.7746 0.8660 1.7321 1.9365 3.8730 4.8990 0.1936 0.2315 0.3062 0.3873 0.4629 0.6124 0.7746 0.9258 0.9682 1.2247 1.5492 1.6206 1.9365 2.4495 3.2404 3.8730 4.8990 6.4807 7.7460

Miller-Bravais ind.

Class symbol

( p , p ) C ( 1, 4)l

1, l l 4, l l 1,16)2 1,10)2 ( 2, 5 2 ( 3, 6 2 ( 6, 3)2 11:;

(16, 1)2 1,40 3 1,2833 ( 1,16)3 ( 1,10)3 ( 1, 7)3 ( 1. 4)3a

1. 4 3bl 1. 4 3bz ( 2. 5)3

4. 7 3 5. 8 3 ( 1. 1)3a

1. 1 3bl 1. 1 3b2 ( 8. 5)3

7. 4 3 5. 2 3 ( 4. 1 3a ( 4. 1 3bl ( 4. 1)3bz ( 7. 1 3 ( l 0 113 16. 1)3 128. 1)3 (40. 1)3

of twin 1 0

.

1 0 1.. 2 1 0

.

4 0 2 . 1 1 0 . 1 0 1 . 2 1 0

.

1 1 0

.

2 0 1

.

5 1 0 .l 0 0 1

.

8 4 0 . 1 0 2 . 1 1 0

.

1 0 2 . 1 1 0

.

1 2 1

.

¹

0 1

.

2 1 0

.

1 0 1 . 2 0 4

.

5 1 2

.

2 1 0

.

4 0 1 . 2 0 2 . 7 2 0 . 5 2 1

.

⁴

0 1 . 8 1 0 . 7 0 1

.

5 1 0 .l6 0 1 . l 4 1 0 .l 0 sin

0.9428 0.9428 0.9428 0.9428 0.7454 0.7454 1.0000 1.0000 0.7454 0.7454 0.9428 0.9938 0.8315 0.6285 0.9938 0.8315 0.8315 0.9428 0.6285 0.9938 0.8315 0.9938 0.8315 0.9428 0.6285 0.9938 0.8315 0.9938 0.8315 0.9428 0.6285 0.8315 0.9938 0.6285 0.8315 0.9938 0'

90.00 90.00 90.00 90.00 75.52 75.52 104.48 104.48 75.52 75.52 90.00 99.59 80.41 70.53 99.59 80.41 60.00 70.53 70.53 99.59 80.41 99.59 60.00 70.53 70.53 99.59 80.41 99.59 60.00 70.53 70.53 80.41 99.59 70.53 80.41 99.59

planes 0 2

.

¹

1 0

.

1 0 1

.

2 4 0 . 1 0 5 . 1 5 0

.

2 0 1

.

1 0 1

.

2 1 0

.

1 0 1 . 2 1 0

.

4 0 5

.

1 7 0 . 1 0 8

.

1 5 0 . 2 0 7 . 2

4 0

.

¹

0 5

.

4 7 0

.

4 1 0

.

1

0 2 . 1 5 0

.

⁸

1 0 . 1 0 1

.

²

1 0 . 1 0 1 . 2 1 0

.

4 0 1

.

2 1 0 . 4 0 1

.

8 Representative

m U v

.

^W

2 0 1

.

2 1 1 0

.

1 l 0 2

.

¹

4 1 0

.

4 5 0 1

.

5 5 2 0

.

.5 3 0 3

.

3 1 0 2 . l 1 1 0 . 1 1 0 2 . 1 1 4 0

.

1 5 0 1

.

5 7 1 0 . 7 8 0 1

.

8 5 2 0 . 5 7 0 2 . 7 3 1 0 . 1 4 1 1 . 0 4 1 0

.

4 5 0 4 . 5 7 4 0

.

7 1 1 0

.

1 3 0 2

.

¹

2 1 1

.

⁰

2 0 1

.

2 5 8 0

.

5 1 1 0

.

1 1 0 2 . 1 3 4 0 . 1 1 1 1 . 0 1 . 1 0 . 1 1 0 2 . 1 1 4 0 . 1 1 0 2 . 1 1 4 0 . 1 1 0 8 . 1