MODELLING OF TEXTURE INFLUENCE ON CSL BOUNDARIES DISTRIBUTION IN POLYCRYSTALS

HAL Id: jpa-00230341

https://hal.archives-ouvertes.fr/jpa-00230341

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MODELLING OF TEXTURE INFLUENCE ON CSL BOUNDARIES DISTRIBUTION IN POLYCRYSTALS

A. Garbacz, M. Grabski

To cite this version:

A. Garbacz, M. Grabski. MODELLING OF TEXTURE INFLUENCE ON CSL BOUNDARIES

DISTRIBUTION IN POLYCRYSTALS. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-469-

C1-475. �10.1051/jphyscol:1990173�. �jpa-00230341�

COLLOQUE DE PHYSIQUE

Colloque Cl, suppl6ment au n0l, Tome 51, janvier 1990

MODELLING OF TEXTURE INFLUENCE ON CSL BOUNDARIES DISTRIBUTION IN POLYCRYSTALS

A. GARBACZ and M.W. GRABSKI

Institute of Materials Science and Engineering, Warsaw University of Technology, Narbutta 85, 02-524 Warsaw, Poland

Resume - Nous avons etudie par la simulation sur I'ordinateur la distribution de desor ientat ions des joints de grains (DD&) character is@ par leur coincidence (C%) dans un modele de polycristal bati suivant les critPres purement geometriques. Nous avons analyse les variations de fraction des joints de coincidence, entrainees par la texture cristallographiqw. Le modele propose dome une description adequate de W& dens un polycristal aleatoire. II a Btc! ensuite utilise pour determiner la MUG pour les types particuli&res de texture de fibre. Les conclusions generales sont suivantes: la texture affecte fortement la DD& e t chaque type de texture engendre un type characteristique de DDJG.

Abstract

-

The computer modelling o f the Grain Boundary Misorientation Distribution (GBMD) in the spatial model o f p o l y c r y s t ~ l build accordinq t o purely geometrical criteria was used t o study the distribution o f C% boundaries in random polycrystal and t o analyse the changes in content o f CSL boundaries conditioned by the presence o f texture. The proposed model gives an adequate description o f the GWD in random polycrystal. The model was used t o determine a hypothetical GBMD f o r particular type o f fibre texture.

The main conclusion is that the texture strongly affects the GBMDs and each type of texture generates characteristic types o f CSL boundar ies.

1

-

INTRODUCTION

Properties o f polycrystals depend to a large extent on the properties of individual grain boundaries (GBs), which, in a given polycrystal are strongly diversified and form a certain grain boundary character distribution 2 1-43. GB properties are usually correlated with their crystallographic parameters: the angle and axis of the misorientation of adjacent grains, and the orientation of the boundary plane. Mmerous geometrical methods are used t o describe the dependence of the boundary structure on the misorientation of the grains forming the boundary. the Coincidence Site Lattice (CSL) is the most commonly applied 14-71 geometrical model o f grain boundary structure..

It has been established that boundar ies between grains characterised by certain coincidence misor ientat ions exhibit some special properties as compared with the so-called general boundaries, and that the special properties are maintained despite a small deviations from the exact CSL misorientations [ 1,2,8-141.

In recent years numerous attempts have been made in order t o experimentally determine the fraction of special boundaries in the polycrystalline microstructure, i.e. the coincidence grain boundary distribution (CGBO); the results obtained, and hence also the interpretations, have been divergent C2,3,10,12,151. The experimental evidence shows that CGBD is dependent on the technology o f the material's preparation 12,3,8-131. CGBD is controlled by the processes o f formation o f the polycrystalline structure, such as crystallization, phase transformations o r recrystallization. They lead t o the occurrence of certain preferred grain orientations in the microstructure, i.e. texture.

It follows that in order t o draw useful conclusions from experimental determined CGBD it is necessary t o understood the relationship existing between the texture and the grain boundary misorientation distribution [l81 and the coincidence grain boundary distribution C171.

The aim o f the present study was t o apply a spatial model o f polycrystal built on the basis of purely geometrical criteria t o simulate the effect o f fibrous texture on the grain boundary misor ientat ion distribution and the coincidence grain boundary distribution.

2

-

RAMMUl DlSTRlsUTlON OF GRAIN W ~ A R Y MISORIENTATION IN A FOLYCRYSTAL

In the classical work on the CGBD in a random polycrystal, Warringtoon and Boon C183 estlmated analytically that the fraction o f CSL boundaries from the range

P

3 - 25 equals 9%. They obtained a similar result in a computer simulation by calculating random rotation matrices.according t o the method presented by Mackenzie and Thompson Ct91. Their data show that the method is burdened with a systematic error consisting in overestimating the number of low angle boundaries (LA). This can stem from the fact that calculating misorientations from randomly generated rotation matrices is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990173

Cl-470

COLLOQUE DE PHYSIQUE

not equivalent t o determining the misorientat ion o f randomly oriented grains forming a spatial aggregate.

To determine the statistical distribution o f boundary misorientations in polycrystal it is necessary t o unify the description t o pick out a unique rotation among all the equivalent ones. For that purpose the notion of disorientation introduced by Grimmer 1201 is used in this work. In order t o determine the random distribution o f grain boundary disorientation a model o f polycrystal wss worked out based on the assumptions that grains have cubic lattice; grains a m identical Kelvin polyhedra arranged in spatial aggregate, and each grain has a orientation determined in relation t o the global basis. The details o f the model are given elsewhere C21.221

The simulation was performed f o r a polycrystal consisting o f 10 layers o f 20x20 grains (i.e. 4000 grains), which made it possible t o test 24 290 boundaries. Brandon's criterion C231 was used t o decide whether a given boundary disorientation was special. CSL boundaries were determined within the coincidence range 1 E 1 6 65. The obtained distribution o f disorientation angles and axes (Fig. 1,2) shows very good agreement with the distributions determined analytically by Mackenzie C24,251. It is worth noting that the present study avoided overestimating the number o f LA boundaries (X1 +I41 att57atZ61 a+.X65a), which occurred in the work o f Warrington and Boon 11 81. The simulations performed indicate, that in random polycrystal the fraction of special boundaries from the range 1 C 2 6 65 is 17,8 X; i f one separates from the population boundaries from the range 3 f X 25, then the fraction w i l l be equal t o 8.7%, a value close t o 9% determined analytically by Warrington and Boon [ l 8 1 f o r the same range of coincidence.

3

-

ANALYSIS OF THE EFFECT OF TEXTURE ON THE DISTRIBUTION OF CSL BWN)ARIES

The presence o f a texture in a polycrystalline material is manifested by the f a c t that the distribution of grain orientations is not random and grains assume certain preferred orientations in relation t o the global coordinate system associated with the sample. Within the context o f the adopted model o f polycrystal this means that Euler's angles ascribed t o particular grains w i l l have definite values.

The simulation was performed f o r fibrous textures. Fibrous texture is characterized by the fact that a given direction (hkl> lies along the axis of the sample or forms a certain angle cp with it, and the axes o f the crystallographic coordinate system associated with the grains are rotated in relation to <hkl> direction, forming a certain distribution dependent on the material preparation.

The study was based on the two different models o f the fibrous texture. In the f i r s t model it was assumed that direction <hkl> can deviate from the sample axis and the axes o f the crystallographic coordinate system are randomly oriented in relation t o the direct ion <hkl> ( f ig.3a).

In the second case differentiation o f grains orientations in a given type o f texture resulted from the deviation o f <hkl> direction from the sample's axis in grains that had the same basic orientation (f ig.3b). It was also assumed that the grains orientations are uncorrelated with respect to the positions within the polycrystal and the deviation o f direction <hkl> can be described by normal distribution.

The adopted models o f fibrous texture were worked out by ascribing Euler's angles t o particular grains, according t o the reference system. Deviations o f the <hkl> direction from the sample's axis were realized in the calcutations by approximating the values o f deviaticns o f two out of three Euler's angles, describing a given type o f texture by a normal distribution with parameters N (0.u); the values of deviations Aq, and AqE satisfied the condition:

The value o f standard deviation of the normal distribution U determines sharpness of texture: f o r greater U the differentiation o f grain orientations in polycrystal is greater.

Grain boundary disor ientat im distribution (GBW) and coincidence grain boundary distribution in textured polycrystal was computed f o r three types o f textures:

<loo>,

<l 1 1

>

and

<l 10> with two values o f standard deviation: U = 3 and 15.

GBDD's and CGBD's are computed for each type o f texture and both values U are shown in Tables 1

-

^3. The obtained CG8D1s indicate that the effect o f texture on CGBO is very strong and that particular texture can be associated with characteristic types o f CSL boundaries.

h

order to distinguish the boundaries, the intensity coefficient (I/I,) was calculated; the coefficient determines the r a t i o o f the obtained fraction o f the given CSL boundary I t o the fraction o f this boundary in the random polycrystals I,. For texture

<loo>

these are boundaries: LA,

I

S, 13a, 17a, 25a. 378, f o r texture <Ill> : LA,

I

3, 7, 13b, 19b, 21a and f o r texture (110): LA, 2 3, 9, 11, 17b, 19a, 27a, 33a, 33c (Table la; 2a; 3a).

With increasing deviation from the ideal texture

<loo>,

<l 1 l> o r <110> and with decreasing sharpness o f the normal distribution describing deviation o f <hkl> direct ion from the sample's axis, the randomness o f grain orientations increases and the fraction o f special

boundaries decreases, mainly due t o decreasing fraction o f low-angle boundaries.

The CGBD's for particular texture are result o f the differences o f GBW's describing these texture (Table lb-c; 2b-c; 36-c).

4

-

CONCLUSIMUS

1. The presented spatial model o f polycrystal adequately describes the distribution o f grain boundary disorientation ( G m ) and the distribution o f CSL boundaries (CGBD) in random

(non-textured) polycrystal.

2. The model can be used f o r the determination o f a hypothetical GB00 and CGBD in textured materials in the cases when the texture of the material is well described (for example by the orientation distribution function).

3. Both GBW and CGBD are strongly depended on the existing texture, and each type o f texture is associated with certain characteristic types of CSL boundaries.

4. Distribution of disorientation angles and axes should be used t o characterize polycrystals alongside with such parameters as grain size, boundary diffusivity etc.

Acknowle&emenk This work was supported by the State Office of Science, Technological Development and Implementat ion (Poland) under contract CPBR 2.4

REFERENCES

Grabski M.W.: J.Fhysique, 46 C-4-567 (1 986)

Wyrzykowsk i J.W., Grabski M.W.: Fhilos.Mag.A, 53, 505 (1 986) Watanabe T.: Res Mechanics, 11, 47 (1 984)

Sutton A.P. and Balluffi R.W.: Acta Metall., 35, 2177 (1987) Ranganathan S.A.: Acta Crystallogr., 21, 1 97 (1 966) Fortes M.A.: Fhys.Stat.Solidi Cb), 54, 31 1 (1 972)

Grimmer H., Bollmann H. and Warrington D.D.: Acta Crystallogr., A30, 197 (1 974)

Rybin V.V., Titovets Yu.F. Tepilitskiy D.M. and Zolotorevskiy N.Yu.: Fhys. Met. Metall. (in russian), 53, 544 (1 982).

Rybin V.V., Titovets Yu.F. and Kozlov A.L.: fbvierchnost (in russion), 10, 107 (1 984)

Watanabe T., Yoshikawa N., Karashima S.: Textures o f Materials, ICOTOM-6, Japan Iron Steel Inst. 1, 609 (1981)

Belluz R.V., Aust K.T: MetalL Trans.A.6,219 (1 975) Lim L.C., Raj R.: Acta Metall. 32, 1177 (1984)

hmphrey P.H. in Grain Boundary Structure and Properties (ed. Chadwick G.A, Smith D.A.) p. 139, Academic Press, New York (1 976)

Shvindlerman L.S. and Straumal B.B.: Acta Metell., 33, 1735 ( l 985)

Zhao J., Adams B.L. and Morris P.R.: Textures end Microstructures, 0-9 495 (1 988) El MIRabat and Priester L.: Mat.Sci.Eng Al01, 1 1 7 (1 988)

Randle V., Ralph B. and Dingley D.: Acta Metall., 36, 267 (1988) Warrington D.H., Boon M.: Acte Metall, 23, 599 (1975)

Mackenzie J.K.: Biometrika 45, 229 (1 958) Grimmer H.: Acta Crystallogr., A30, 685 (1 974) Garbacz A and Grabski M.W.: Scripta Metall. 23 (1989)

Garbacz A and Grabski M.W.: Archives o f Metallurgy (submitted f o r publication) Brandon D.G.:Acta Metall.14.1479 (1 966)

Mackenzie J.K., Thomson M.J.: Biometrika, 44, 205 (1 957) Mackenzie J.K.: Acte Metall., 12, 223 (1964)

COLLOQUE DE PHYSIQUE

disorientation angle,

deg z o n e i n S S T 1 - Method used by Warrington and Boon; 2

-

This work;

3

-

Analytical result of Mackenqie

Fig.1. The distribution of disorientation angles

Fig. 2. The distribution of disorientation axes in the Stereograf i c Standard triangle CS=

Model I Model g

9 ^!

r :

"

:

$

3 -<h&

2 1

I I I

a i ^AS i ^{AS b) i AS}

AS - sample axis

Fig. 3. Schema of the models of the fibre texture using in this work

Table 1. Coincidence Grain Boundary D i s t r i b u t i o n and Grain Boundary Misorientation D i s t r i b u t i o n for

<loo>

texture.

a. b p o r n r . tne m o s t comnon type of CSL ooundarv

' ^a)

E

L A 3 5 7 9 1 1 13 15 17 19 21 23 25 27 29 3-29 1-65

b ) 4j E

d Z

3 *

f z

2

F p 2

nw z

Y P

a!

l- Cl

c >

z o n e i n S S 1

m 1

60

S

:

⁴⁰

- ..

U 0 20

- .

0

I o n e i n SS1

* m r E L I 1

1-N(0,3) F [ % l ! / I , 29.3 12. 6

- -

15.8 14.6

- -

- -

-

-

5. 2" 8. 9

- -

3. 5- 7. 8

-

-

- -

-

-

2.2" 4.7

- -

1 . 7 4.9

28.4 3.0

60.0 3.4

14N(0, 3) F [%l I / I , 99. 8 43. 4

- -

-

-

- -

-

-

- -

- - -

- -

- -

- -

- -

- -

- -

- -

0 0

99.8 5.6

2-NCO, 15) F [%l I / I , 6. 5 2. 8 0. 6 0. 4

1.6 1.5

0. 8 1.0 0. 7 0. 8 0. 3 0. 5

1.0 1.7

0. 5 0. 7 0. 7 2.0

0.5 1.0

0.6 0. 9 0. 4 1.1 0. 4 0. 9 0. 5 1.0 0. 4 1.1 8. 9 0. 9

21.7 1.2

2-N(0,15) F [%l 1 1 1 21. 2 9. 2 0. 1 0. 1 3. 1 2. 9

0 0

0. 8 0. 9 0. 1 0.2 2.3" 3. 9 0. 3 0.4 1.3" 2.9 l. l" 2.4 0. 1 0. 2

0.2 0.6

1.3" 2.8

1.0 2.0

0.3 0. 9

11.9 1.3

38.8 2. 2

2

-

^{!Y t 15U}

.,

⁵

-

0 _Y

0 30 60

disorientation angle,deg

- $$"--J

^{.2 0 5}

- /*

!!

30 60 0

disorientation angle.deg

80

Cl-474 COLLOQUE DE PHYSIQUE

Table 2. Coincidence Grain Boundary Distribution and Grain Boundary Misorientation Distribution for < I l l > texture.

a. o ooint the most comnon t y ~ e of CSL boundary

1 ^a)

X

LA 3 5 7 9 1 l 13 15 17 19 21 23

MODEL I 1 1 -N(0, 3)

F [XI

1 / 1 ,

49.4 21.5

42.9 21.8

-

^-

- -

- -

: - -

- -

-

.-

- - -

- -

- -

- -

-

^-

- -

42. 9 4.5

94.8 5. 4

MOML I

0

!J z f a '5 g m E 2

'=W

z

E

F 8

+ E

c)

."g

F

z

E 2 3:

roz a

U

a 5 8

l-

a

2-N(0, 15)

F C%]

I / I ,

10.9 4. 5

5. 3 3. 6

0 0

0. 6 0.8

1.9 2. 1

l. 6 2. 5

0.2 0. 3

0. 1 0. 1 1.5" 3 . 3 1.2' 2 . 6 0. 3 0 . 5 0. 1 0. 3

0.8 1.7

0.6 1.2

0. 4 1 . 1 1 4. 3 1 . 5 32. 4 l. 8 1 -N (0.3)

F [%l I / 1 ,

22.1 9. 6

9. 6 6. 6

- -

7.9 9. 9

- -

-

^-

3. 9- 6. 6

-

^-

-

2.0- 4. 3 l . 7- 2.5

- -

2-N(0, 15) F [%l I / 1 ,

4. 0 1.7

l . 4 1.0 0. 8 0. 7 0. 9 1 . 1

0.9 1.0

1.0 1.5

0.6 1.0

0.5 0. 7

0. 5 1 . 1 0. 5 1 . 1

0. 4 0. 6

0. 4 1.2

0.5 1.0

0.5 l . 0

0.3 0. 9

8. 9 0. 9 25. 1 1 . 4

t

29

: : l f

-

l0

a l,

r 5

.I

⁵

-

_E

- -

t

:

3-29 1-65

30 60 disorientation angle,deg

80

60

S

c 0 40

.- ..

U

-

20 0

1 2 3 4 5 6 7 8 z o n e i n S S 1

25.1 2.6

52.0 2. 9

b)

LL V)

30 60 disorientation ongle.deg

80

60 S

4 "

.- ..

U 5 20

-

0

1 2 3 4 5 6 7 8 z 5 n e i n S S 1 20

z o n e i n S S 1

Table 3. Coincidence G r a i n Boundary D i s t r i b u t i o n and G r a i n Boundary

z o n e i n SST

a)

Z

LA 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3-29 1-65

b)

b g 9 Z a

5 ^z

&?g

E: P z Q w

rB

E I- 0

c ) -

t e x t u r e . CSL boundarv

MOlXL

1 1 l-NCO, 3 )

F [%l

I / I r

55.0 23.9

- -

-

^-

-

-

- -

- -

- -

- -

7. Ob 15.6

-

-

- -

- -

0. 4 0. 7

- -

- -

7. 4 0 . 8 6 2 . 8 3. 5 M i s o r i e n t a t i o n D i s t r i b u t i o n f o r <110>

a. o o o i n t t h e most c o m n t v o e o f MXXL I

2-N(0, 15) F

[%l

! / I 11.7 5. 1

0. 7 0. 4 2 . 2 2 . 0

0. 3 0. 4

0. 8 1 . 0

0. 4 0.6

1.5 2. 6

0. 4 0.6

1.0 2. 1

0. 6 1 . 4

0. 5 0. 7

0. 3 0 . 6 1 . 2 2. 5 0. 7 1 . 3 0. 3 1 . 0

10.7 l. 1

28. 3 1. 6

l -N (O,3) F C % ] I / I r

14.6 6. 4

13. 7 6. 0

-

-

- -

4.0 4. 5

3. 1 4. 8

0 0

-

-

l . S b 3 . 6 1. 6" 3. 4

0 0

- -

0 0

1.0" 1 . 9

- -

25.0 2. 7

44.3 2. 5

!a ^-

^{m 10 5 0 1} _{30 60}

disorientation ongle,deg 2-N(0, 15)

F CXI

I / I , 3.0 1 . 3

1.8 1.2

0. 8 0. 7 0. 9 1 . 0

0. 9 1.0

0. 7 1.0

0. 6 1 . 0 0. 6 0. 8

0. 4 1.0

0. 4 0. 9 0 . 8 1.2

0. 4 1.0

0. 5 1.0

0 . 5 1.0

0. 4 1.1

9. 4 1.0

18. 6 1.0

20

k Z g 10

-

1.

.o

5

.- -

_{0 2\}

-

v

30 60 disorientation angle , deg