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The relationship between texture and CSL boundaries distribution in polycrystalline materials—II. Analysis of the relationship between texture and coincidence grain boundary distribution
Acta metall,mater.Vol. 41, No. 2, pp. 475-483, 1993 Printed in Great Britain.All rights reserved
0956-7151/93 $6.00+ 0.00 Copyright © 1993 PergamonPress Ltd
THE RELATIONSHIP BETWEEN TEXTURE A N D CSL BOUNDARIES DISTRIBUTION IN POLYCRYSTALLINE MATERIALS--II. ANALYSIS OF THE RELATIONSHIP BETWEEN TEXTURE A N D COINCIDENCE GRAIN B O U N D A R Y DISTRIBUTION A. G A R B A C Z t and M. W. GRABSKI
Faculty of Materials Science and Engineering, Warsaw University of Technology, 02-524 Warsaw, Narbutta 85, Poland
(Received 20 November 1991; in revisedform 1 July 1992) AMtraet--The spatial model of a polycrystal was used to prove the existence of exact (quantitative) relationship between texture and distribution of CSL boundaries for (hkl) type if textures and also to determine features of spatial distribution of CSL boundaries: number CSL boundaries per grain and number CSL boundaries in triple-line. The main conclusion is that the texture strongly affects the grain boundary misorientation distribution and distribution of CSL boundaries and each type of texture generates characteristic types of CSL boundaries.
1. INTRODUCTION
2. TEXTURE MODELS USED FOR MODELLING CGBD
The effect of the fabrication method on grain orientation distribution and simultaneously on grain boundary character distribution in polycrystal leads to the conclusion that certain direct relationship exists between texture and grain boundary misorientation distribution (GBMD) and coincidence grain boundary distribution (CGBD). The existence of that relationship was reported in many recent experimental works [l-10]. The approximate, qualitative relationship between texture and CGBD has been already established, which consists in prediction of the predominance of certain types of CSL boundaries for sharp textures: (100), ( I l l ) and (110) [7]. Generally, it was based on assumption that for sharp (hkl) textures low-angle boundaries and CSL boundaries with (hkl) misorientation axes are dominating. This is valid in the case of axial texture, for which it is accepted that crystallographic co-ordinates system associated with grains is randomly rotated in the relation to the preferred (hkl) direction. Otherwise, for example in the case of the texture formed during recrystallization of the deformed single crystal, the observed CGBDs are different from predicted. The aim of the present part of the paper is to prove the existence of exact (quantitative) relationship between texture and CGBD on the basis of simulation of spatial model of polycrystal built with Kelvin polyhedra-shaped grains (the method B2 discussed in Part I of this work). tPresent address: Institute of Technology and Organisation of Building Production, Warsaw University of Technology, A1. Armii Ludowej 16, 00-637 Warsaw, Poland. AM 41/2--K
The existence of the relationship between texture and distribution of CSL boundaries was studied for (hkl) textures, often observed in materials with cubic lattice. The spatial model of polycrystal built with Kelvin polyhedra (tetrakaidekahedra) grains was used for simulation. This model is a reasonable approximation of a real polycrystalline aggregate, which additionally allows to analyse features of spatial CSL boundary distribution. Orientations described by sets of three Euler's angles were ascribed to particular grains that form aggregate. Brandon's criterion [11] was used to determine if given boundary is of CSL type. The two different models of (hkl) texture were applied (Fig. 1). In both models deviation of (hkl) direction from the sample axis was described by the normal distribution N(0, a), where a is given in degrees, determining the sharpness of texture [Fig. l(c)]. The details of these models are given elsewhere [12-14]. The first model [Fig. l(a)] is an approximation of the axial textures, i.e. wire and fibre textures, characteristic for many engineering materials. The second model approximates textures occurring in deformed and subsequently recrystallized single-crystal (without deformation bands, etc.). For the given type of texture the second model [Fig. l(b)] leads to smaller differences of grain orientations than in the case of the first model. For both models it was assumed that grain orientation is uncorrelated with respect to its position within a polycrystalline aggregate. It means that texture was homogeneous and deviations from
475
476
GARBACZ and GRABSKI: TEXTURE IN POLYCRYSTALLINE MATERIALS--II
b) Model
a) Model T_
~
c)
A-N(O,3)
<.kl
rondom(_~~
==>
<3==
\ /2-N(0,5) ~ , ~ 3 - N (0,15)
v
]As
JAs AS
-
SYMPLE
IAS
--36
AXIS
Fig. 1. The models of (hkl) texture used in this work (a, b) and three normal distributions describing sharpness of the analysed textures (c).
ideal direction ( h k l ) were randomly ascribed to particular grains. The adopted texture models allowed to determine changes of CGBD dependent on texture, which can be regarded as the reference for analysis of more complicated grain orientation distributions. The worked-out method makes it possible to calculate hypothetical GBMD and CGBD for any grain orientation distribution and also for the case of inhomogeneous textures.
jx (1)
°>
/-/ [
.
(2)
b)~
z
Z
D
3. COMPUTER SIMULATION OF EXPERIMENTAL DISTRIBUTIONS OF CSL BOUNDARIES FOR (hkl) TEXTURES Among numerous experimental works concerning the distributions of CSL boundaries it is difficult to find a work containing both the data on CGBD and the well-described texture. The exception are reports of Watanabe and co-workers [2, 3], in which CGBDs were studied with ECP method in rapidly solidificated ribbons with sharp textures (100) and (110) in purpose to corroborate the quantitative relationship between texture and CGBD. This data was used in the present work to verify the correctness of the adopted model. The analysis of grain orientation distributions for different simulated textures showed that distribution obtained for mixed texture: 90 % (100) N(0, 3.5°) + 10%(111) N(0, 8°) described by model I was most close to the experimental grain orientation distribution obtained by Watanabe et al. [2]. Comparison of the experimental and modelled grain orientation distributions for similar number of investigated boundaries are presented in Fig. 2. The modelled CGBD exhibits 25 -<100> 20
~.~ 15 O
1
5-
\1/2
o ~
Fig. 2. The comparison of grain orientation distributions: 1--experimental distributions obtained by Watanabe and co-workers for sharp (100) texture [2]. 2---computer simulation for 90% (100) + 10%( 111) texture.
~
C,l C4 C,I
Fig. 3. The distributions of CSL boundaries for sharp (100) texture: 1--experimental by Watanabe et al. [2]. 2--computer simulation.
GARBACZ and GRABSKI: TEXTURE IN POLYCRYSTALLINE MATERIALS---II high goodness of fit with the experimental distribution (Fig. 3). It confirms the existence of high fraction of low-angle boundaries and also of 2~5, 13a 17a and 25a CSL boundaries with (100) disorientation axis. Greater difference was observed only for the case of Z 17a boundaries. To confirm agreement between both modelled and experimental distributions the Kolmogorov-Smirnov test of goodness of fit was applied. The value of statistic for analysed distributions was found to be equal 2 = 0.849 and the critical value of this statistic 2 , was equal 1.358 for the level of significance ~ --0.05. It indicates that there is no basis to reject hypothesis about goodness of fit of both distributions. The conclusion is that the distribution of coincident boundaries in Watanabe's experiment was controlled by the geometrical factors with the exception of Z 17a boundaries. The reason for surplus of 2~17a boundaries is most probably related to their particular properties and/or to the specific mechanism of the microstructure formation. The similar analysis was performed for CGBD obtained for the same material but with (110) texture [3]. The computer modelling was performed with assumption that material had 1 0 0 % ( l l 0 ) N(0,3 °) texture described by model I. The comparison of experimental and modelled distributions again shows very good agreement (Fig. 4). It was also confirmed by Kolmogorov-Smirnov test. The calculated value of statistic is 2 = 0.636. It indicates that also in this case there was no basis to reject hypothesis about goodness of fit of both experimental and modelled distributions. The obtained results indicate existence of the direct quantitative relationship between texture and CGBD. This shows that the worked-out model of a polycrystal can be used to determine the reference CGBD for any grain orientation texture.
20
--
<110>
15
o
10
/ 5
o
/
I I
I I
I]-J
I ,,I I I
x Fig. 4. The distributions of CSL boundaries for (110) texture: 1---experimental by Watanabe et al. [3]. 2---computer simulation.
477
4. ANALYSIS OF CGBD FOR BASIC TYPES
OF (hkl) TEXTURE The GBMDs and CGBDs (coincidences from the range Z1-65) for textured polycrystal were computed for four types of ( h k l ) textures, namely 100%(100), 1 0 0 % ( I l l ) , 5 0 % 0 0 0 ) + 50%(111) and 100%(110). The simulations were performed for three levels of the texture sharpness equal to 3 °, 5 ° and 15° [Fig. 1(a-c)]. The changes of type, sharpness and character of the texture (Fig. 5) exert a strong influence on G B M D (Fig. 6) and CGBD (Table 1). The obtained results indicate that GBMDs and particularly CGBDs are strongly dependent on existing texture and that each type of texture is associated with characteristic set of CSL boundaries. The most probable CSL boundaries for given texture type are listed in Table 2. As an example of changes of CGBD with texture, the distributions of CSL boundaries for texture described by first model [Fig. l(a)] with sharpness equal to N(0,5 °) are shown in Fig. 7. With increasing deviation from ideal orientation and decreasing sharpness of texture the randomness of grain orientation distribution increases and the fraction of CSL boundaries decreases, mainly due to the decreasing of fraction of lowangle (271) boundaries. Fraction of2~ 1 boundaries for model II is about two times greater than for model I (Fig. 8). CGBDs for particular type of texture originate from GBMDs generated by that texture. The changes of the character of distributions of disorientation angles and of disorientation axes, which define disorientation distribution, are more distinct for texture described by second model. For a pure texture the distributions of disorientation axes in eight zones of standard stereographic triangle (SST) and particularly the distributions of disorientation angles have characteristic features, which allow to recognise their type [Fig. 6(b)]. Corresponding distributions obtained for the first model exhibits small differences in GBMDs, which practically disappears for the lowest investigated sharpness of texture, namely N(0,15°). It was also observed that for different types of texture similar disorientation distribution can be obtained. For example, very similar distribution of disorientation axes in 8 zones of SST and of distribution of disorientation angles were obtained for both models and for all sharpness in the cases of mixed 50%(100) + 50%(11 l ) and pure 100%(110) textures. In fact it is necessary to apply a more precise analysis consisting on determination of the proportion of axes deviated maximum by 5° from (100), (11 l ) and (11 l ) to find that these distributions are different. On the contrary, the distributions of CSL boundaries in mostly cases were different not only in total fraction of CSL boundaries but also in types of these boundaries. This result shows that CGBDs are often a more meaningful polycrystal characteristic than GBMDs. The distributions of disorientation angles and axes used
478
G A R B A C Z and GRABSKI:
T E X T U R E IN P O L Y C R Y S T A L L I N E MATERIALS---II
LO
.z
o~ .= z
"Ld
c:;
v
V
z
Iml
I=1
t'Xl
o
..=
tll ill
r~
"12
r~
O
t::t O
O 0"2
m
Z
o
oS.
O O
G A R B A C Z and GRABSKI:
T E X T U R E IN P O L Y C R Y S T A L L I N E M A T E R I A L S - - I I
479
(a)
10
80
:t
<100>+ <111>
<100>
<100> + <111>
60 1-..~
,3H
40
3
o
2/
20
3
v
e. = P
10
°
8
~A1 <110>
<111>
"~
80
<111>
t~
<110>
1
60 6 I
2
40 4
1
3
1
20
2 30
30 D i s o r i e n t a t i o n angle (deg)
(b) 10
3
600
•
0
60
2
/
2 3 4 5 6 7 8 l 2 3 4 5 6 7 8 Zone in SST
80
1 14.9 <100> + <111>
<100>
<100> + <111>
60 2 o ×
40
4
1~i/
3
2O
2 v
=
0
8 0 8o
~r 10 <111> 1 4 . 5 ~
[
1.
<110>
<110>
6O 3
/2
3\
40
32
2
\
20
0
30
60 0
30
60
D i s o r i e n t a t i o n angle (deg)
0
1
rhar~ 2 3 4
6 7 8
2 3 Zone in SST
4
A...In 5 6 7 8
Fig. 6. The distributions of disorientation angles and the distributions of disorientation axes in 8 zones of standard stereographic triangle (SST) for model I (a) and model II (b).
for this purpose regards more precise distribution of disorientation axes distribution than disorientation axes distribution in 8 zones of SST. 5. T E X T U R E I N F L U E N C E O N F E A T U R E S OF S P A T I A L D I S T R I B U T I O N OF CSL B O U N D A R I E S
The investigation of relationship between grain boundary properties and bulk properties of poly-
crystalline materials have revealed that in many cases not only the fraction of boundaries with special properties but also their spatial distribution in the relation to the loading system is an important factor [15]. The features of spatial distribution of C S L boundaries may provide new information about influence of this distribution on the propagation of fracture but also on plastic deformation or recrystallization
480
GARBACZ
and GRABSKI:
TEXTURE
IN POLYCRYSTALLINE
MATERIALS--II
Table 1. The coincidence grain boundary distribution for different types o f (hkl> texture. A - - N ( 0 , 3), B---NO, 5), C - - N ( 0 , 15). <100> 2~
A
<111>
B
C
22.8 -8.7 0.0 0.0
A
A
ModelI 3.9 12.3 1.4 2.7 0.8 4.0 0.8 1.9 0.8 0.0 0.5 3.0 0.8 2.3 0.5 0.0 0.4 1.2 0.6 0.5 0.7 0.5 0.4 -0.4 1.2 0.5 -0.3 0.5
C
A
B
C
10.1 2.6 2.1 1.0 0.2 1.8 1.0 0.1 1.0 0.3 0.4 0.I 1.1 0.1 0.4
2.8 1.7 0.9 0.8 0.8 0.7 0.6 0.6 0.5 0.5 0.7 0.4 0.5 0.4 0.3
14.5 13.2 --4.0 3.0 --1.5 1.5 0.0 -0.1 1.0 --
11.5 7.3 -0.1 1.8 1.3 0.2 0.1 0.8 0.7 0.3 -0.5 0.5 0.2
2.7 1.7 0.8 0.8 0.8 0.7 0.6 0.5 0.5 0.4 0.7 0.4 0.5 0.5 0.4
12.0 27.5
9.4 18.3
24.4 ~.1
13.8 30.5
9.1 17.9
7.7 1.2 1.4 0.5 0.8 0.6 1.1 0.4 0.8 0.6 0.6 0.3 0.9 0.4 0.5
54.8 -------7.1 ---0.4 ---
46.4 ----0.1 0.2 -3.2 ---2.2 -0.1
10.8 0.3 1.6 0.3 1.3 0.4 1.7 0.5 1.0 0.6 0.4 0.3 1.2 0.6 0.5
10.0 24.2
7.4 63.3
5.9 57.6
11.0 28.5
were
calculated
--
--
0.3
--
5.2 -3.7 -0.0 0.0 2.0 -1.6
2.6 0.1 1.6 0.1 0.2 0.1 0.9 0.1 0.7
1.1 0.4 0.5 0.5 0.6 0.4 0.4 0.6 0.3
3.8 --2.1 1.9 ---0.0
17.1 5.7 -3.6 0.0 0.0 1.5 -0.1 1.0 0.9 0.0 0.3 0.1 0.3
3-29 1-65
28.3 59.6
15.1 42.4
8.8 21.3
25.6 52.1
13.5 35.6
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
99.8 -----
90.2 -----
19.6 0.3 1.6 0.3 1.3
49.5 43.7 ----
41.7 23.1 ----
--
--
0.4
--
--
0.7
--
----------
0.1 -0.1 0.2 --0.6 ---
1.8 0.2 0.6 1.4 0.4 0.2 0.9 1.0 0.2
----------
0.1 -0.5 0.1 0.1 -0.4 ---
1.6 0.3 0.9 0.4 0.9 0.3 0.9 0.2 0.4
--0.8 ---1.1 ---
31.8 5.9 ---0.3 0.1 0.1 1.6 0.1 0.1 -1.8 -0.2
3-29 1-65
-99.8
1.0 91.2
10.5 37.8
43.7 94.2
24.4 71.4
11.8 28.0
12.7 53.6
10.0 47.9
for this purpose
tion of stereological describe
spatial
is o n l y
of
CSL
CGBD
should
to
show
The influence
the
The
existence
of texture was
Parameters
selec-
parameters
in the relation
model
that
such
A-C
of polycrystal
shaped
to investigated
grains.
determined
aim of this section
of
17.8 35.6
Mo~llI 9.5 37.0 2.5 10.8 1.1 -1.1 -0.5 --
the accurate
topological
be made.
boundaries
characterised
and
9.0 19.0
The
possibilities.
analysed
crystalline
by the following
example
aggregate.
account •
A--number
of CSL
boundaries
per
•
B---number
of CSL
boundaries
in triple-line---
•
C--number
of CSL
boundries
different
N~
Figure forming
9(a)
all these
of grain--Nqp.
runs
of investigation inside
ensured
distributions
of polythat
par-
was
o f Ngb, N t l a n d
were determined orientation
distribution
frequency there
was
271-29.
or triple-lines were taken into
grain
shows
and cumulative
corners
boundaries
corners
method
polycrystal
random
spatial
polyhedra-
the range
consisted
The
only once. The
Nqo f o r r a n d o m
grain--Ngb
from
and
using
Kelvin
of CSL
procedure
triple-lines
ticular surfaces, corners
parameters:
with
Distribution
calculating
of all grains,
an
build
for coincidences
on the spatial distribution on
<110>
B
27.9 -15.7 ---
phenomena
21.7 9.7 -8.0 --
C
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
kinetics. However,
6.4 0.5 1.8 0.7 0.7
<100> + <111>
B
of
FNgbo f no
for seven
distributions. Ngb ( h i s t o g r a m )
this distribution.
grain
found
with
In
more
Table 2. The characteristic types o f CSL boundaries for different types o f
1
3
<100) (111> (100> + <111> (110)
+ + + +
-+ + +
<100>
+
.
+
<100> + <111) <110>
+ +
5
.7
+ -+ --
9
-+ + --
11
---+
--+ +
13
15
17
19
21
23
+a +b ab --
Model I -----
+a -+a +b
-+b -+a
-+a ---
-. ---
+a . +b +b
-. . .
.
25
27
29
+a . +a --
-. -+a
+a ---
--
--
Model II . .
+
. .
.
. .
. .
. .
. .
. .
. .
. .
. .
. .
. . .
.
--
--
. . .
. . .
+a .
. .
. . .
+b +a
GARBACZ and GRABSKI:
TEXTURE IN POLYCRYSTALLINE MATERIALS--II
481
Fraction, [%]~'t t ~ 2.5
2.0
~3
20,F-Ill
=2.5
II1 1
=17.2
/11 ~3 111 z 7
=5.5 =3.7
%
~
9 17 2 5 3 3 4 1 4 9 5 7 6 5 ~ 1.0
0,
<100> F r a c t i o n , [%] f
E 1
E5 I3 Total
2.0 7_ 1.5 1.0
23.2 =8.8 = 2.5 = 43.5
% % % %
lid ,dlll.,llhll, , .d..
19 1611 / o
2.5
F r a c t i o n , [%]
i
Random total = 17.8 %
t
Fraction
1.0
20 i
0.
1.5
[%]
'£1 E3 Total
=11.8 =7.4 = 31.3
a % % %
0 1 9 1 7 2 5 3 3 4 1 4 9 5 7 6 5 y.I.0
0.5
0.5
0 9 17 2 5 3 3 4 1 4 9 5 7 6 5 3"
Fig. 7. The examples of distributions of CSL boundaries up to 2'65 obtained for (hkl> textures described by model I for sharpness N(0,5°).
A 100%<100>, B 50%<100> + 50%<111>, C 100%<111>
(a) 100 80 60 40 20 0
_-..
•d
o
(b)
15
A
B
C
50 40
-
0,0100 Fig. 8. The effect of sharpness of texture on the fraction of Z' 1 boundaries (low angle boundaries) for different types of (hkl> texture for model II (a) and model I (b).
than 6 CSL boundaries. The fraction of grains without CSL boundaries was 18.4_ 6.6%. Distribution of CSL boundaries in triple-lines Ntl, distribution of CSL boundaries in grain corners Nqp and cumulative frequencies of these distributions FNtl and FNqp are presented in Fig. 9(b). The fraction of grain corners without CSL boundaries showed a considerable majority amounting to 70.1 + 3.9%, whereas fraction of corners with junction of 3 CSL boundries were less numerous 0.20+0.10%. Similar percentages were obtained for N~ distributions, i.e. fraction of triplelines without CSL boundaries equals 69.6 _+ 2.9% and with 3 CSL boundaries~).16 + 0.08%. It was also established that between Ntl and Nqp exists approximate relationship: Ntl ~ 1.5Nqp. It corresponds to the ratio of the number of edges and number of corners for tetradeikahedra-shaped grain (36/24). The effect of the texture on parameters of spatial CGBD was determined for formerly investigated (hkl) textures. Figure l0 shows cumulative frequencies FNgb and FNtl for different types of texture. The validity of the relationship N~ ~ 1.5Nqp was also confirmed for the case of textured polycrystal. Influence of texture on Ngb and Ntl distributions can be characterised also by the changes of values of selected parameters that describe the fraction of CSL boundaries. For example such parameters can be a number of grains which have (A)--zero, and (B)--no more than 6, CSL boundaries for Ngb distribution and a number of triple-junctions with (C)--zero and
482
G A R B A C Z and GRABSKI:
T E X T U R E IN P O L Y C R Y S T A L L I N E M A T E R I A L ~ - I I
t I
;
_
_
.
:
.
.
.
.
.
.
.
eq
'~
o ,¢.
- ' ~ - - . • ."4T~.._ J/ ~.~
O~O%
•
~,
~,
"., " ' ~ . ~ v
0%0
ffl".~'~.~
4k.
.~k "-.IK~.
H l "1~-.:~
,. 4rx,,_---..,_ :il\'x~
-4I
<00I>
<[II>
~,~.
°~o~o
"~"~N~,".~
,'-r.,-S.~.\~/ ,"r-.,:z_->._-rl ~ ~ /
4-I ~
IN
-4 00 "~"
,''r-.~:._~
+<001>
<0II> ~0
I~~
I I\ ~ i~l "~.."~ll "~'~"~\1
~,., ~ " ~ . \ ~ : . , . ,
-~'.z:.,.
"~.~\.-1 ,',:~-~.
r.~. ~. ~ J "-- " ~ ~'.~ I.I " / ,"'%.-,~o~.~, "~II ~""~, o~l.e. ~o' - : a "a ~ t r--.-~,.:: , ~ t ~
~=
~o,,
0
~.~
- . t ~ ' ~ . ~ % . -1 r ~ "~'~--(~-'--l. / "'u ""ro~s.._'.t"~;''"¢,1
ca
v
N~ [%]
.~ouanba.tj
o^.q~lnu~n
D
O
~
=_ = C..) ~
~o.Sg q) ,_., r', O ~
II II
.0.2. - ' , ..~
C~
%
z
%
E_~ ~ .--, " " ~
Z
%"'%0.
"'"'°-.
t',.\\\\\\\\\~
".."-. / ~-.\\\\\'~ ~ " ' - ~ ~ I
[%] ,iouanbazl o^.q~Intun~ ~¢ uo!l~nJd
I
I
I
~.~ ~'~.~ ~ ~
~.~
GARBACZ and GRABSKI: TEXTURE IN POLYCRYSTALLINE MATERIALS--II
483
Table 3. Effect of texture on spatial distribution of CSL boundaries: A--percentage of grains without CSL boundaries; l~-percentage of grains with 6 or less CSL boundaries; C--percentage of triple junctions without CSL boundaries; D---percentageof triple junctions with 3 CSL boundaries Model I Model II (100) (100) + + (100) (111) (111) (110) (100) (111) (111). (110) N(0, 3) A: 0.0 0.0 1.3 0.1 0.0 0.0 0.0 0.0 B: 22.9 43.6 89.9 72.6 0.0 0.2 37.1 19.8 C: 9.1 15.0 34.5 15.0 0.0 0.2 12.7 7.7 D: 22.1 13.9 3.4 6.7 98.7 82.7 13.4 26.4 N(0, 5) A: 0.8 1.3 3.3 2.9 0.0 0.0 0.3 0.0 B: 70.3 85.0 95.8 95.3 1.4 11.2 61.3 32.0 C: 26.6 34.9 47.1 44.0 1.1 5.0 20.6 12.6 D: 7.2 3.9 1.4 1.7 71.9 35.9 8,5 19.5 N(0, 15) A: 11.7 15.6 17.6 16.2 2.2 3.5 6,9 2.8 B: 99.6 100.0 100.0 100.0 84.0 97.5 99,6 97.2 C: 61.4 66.9 68.4 67.9 36.7 48.8 56.2 48.1 D: 0.6 0.3 0.3 0.2 4.4 1.4 0,6 1.6 Random: A--18.4 B---100.0 C--70.1 D---0.20
(D)--3, CSL boundaries for Ntt distribution. The values of A, B, C, D parameters for analysed textures are presented in Table 3. It seems that the importance of features of spatial distribution of CSL boundaries will be manifested if it will be possible to correlate this distribution with distribution of grain boundary plane orientations and distribution of geometrical features of grains. The connection of the geometrical description of polycrystal and features of grain boundary characteristics can give full description of polycrystal. Parameters that describe spatial C G B D can be used to investigate influence of the graininess on properties of polycrystal. It would require the strict selection of stereological parameters describing C G B D appropriately chosen for given object of analysis.
5. C G B D s computed for textures with small differences of grain orientation (model II) explain experimental data ascertaining high fraction of CSL boundaries and particularly high fraction of lowangle boundaries in deformed and recrystallized single-crystal. 6. Fraction of CSL boundaries with coincidences from the range Z" 1-29 in the whole population of considered CSL boundaries (S I - Z 65) increases from about 65% in random polycrystal to 100% for sharp texture. 7. The distribution of a number CSL boundaries per grain and a number of CSL boundaries in tripleline show that spatial distribution of CSL boundaries is also strongly dependent on texture.
6. CONCLUSIONS
REFERENCES
1. The presented model produces a reference grain boundary misorientation distribution ( G B M D ) and coincidence grain boundary distribution (CGBD) in textured materials which is a good approximation of experimental data. 2. Both G B M D and C G B D are strongly dependent on the existing texture and each type of texture is associated with characteristic types of CSL boundaries. 3. G B M D and C G B D obtained for axial ( h k l ) texture with N(0,15 °) sharpness, for which it can be assumed that crystallographic co-ordinates system associated with grains are randomly rotated in the relation to ( h k l ) direction, can be approved as random. It indicates that despite the presence of texture both G B M D and C G B D in such a case are close to randomness. 4. Similarity of G B M D s obtained for textures 5 0 % ( 1 0 0 ) + 5 0 % ( 1 1 1 ) and 100%(110) indicate that in many cases C G B D s are more useful for characterisation of polycrystal than G B M D s .
1. T. Watanabe, Mater. Forum I1, 284 (1988). 2. T. Watanabe, H. Fujii, H. Oikawa and K. I. Arai, Acta metall. 37, 941 (1989). 3. T. Watanabe, K. I. Arai, K. Yoshimi and H. Oikawa, Phil. Mag. Lett. 59, 47 (1989). 4. J. Zhao, B. L. Adams and P. R. Morris, Text. Microstruet. 8--9, 495 (1988). 5. T. Watanabe, Res. Mechanica 11, 47 (1984). 6. V. Randle, B. Ralph and D. Dingley, Acta metall. 36, 267 (1988). 7. S. Hanada, T. Ogura, S. Watanabe, O. Izumi and T. Masumoto, Acta metall. 34, 13 (1986). 8. D. E. Fleet, P. I. Welch and P. R. Howell, Acta metall. 26, 1479 (1978). 9. P. I. Welch, D. E. Fleet and P. R. Howell, Acta metall. 26, 1491 (1978). 10. P. R. Howell, D. E. Fleet, P. I. Welch and B. Ralph, Acta metall. 26, 1499 (1978). 11. D. G. Brandon, Acta metall. 14, 1479 (1966). 12. A. Garbacz and M. W. Grabski, Scripta metall. 23, 1369 (1989). 13. A. Garbacz and M. W. Grabski, J. Physique 51, C1-469 (1990). 14. A. Garbacz, Ph.D. thesis, Warsaw Univ. of Technology (1991). 15. T. Watanabe, J. Physique 49, C5-507 (1988).
0956-7151/93 $6.00+ 0.00 Copyright © 1993 PergamonPress Ltd
THE RELATIONSHIP BETWEEN TEXTURE A N D CSL BOUNDARIES DISTRIBUTION IN POLYCRYSTALLINE MATERIALS--II. ANALYSIS OF THE RELATIONSHIP BETWEEN TEXTURE A N D COINCIDENCE GRAIN B O U N D A R Y DISTRIBUTION A. G A R B A C Z t and M. W. GRABSKI
Faculty of Materials Science and Engineering, Warsaw University of Technology, 02-524 Warsaw, Narbutta 85, Poland
(Received 20 November 1991; in revisedform 1 July 1992) AMtraet--The spatial model of a polycrystal was used to prove the existence of exact (quantitative) relationship between texture and distribution of CSL boundaries for (hkl) type if textures and also to determine features of spatial distribution of CSL boundaries: number CSL boundaries per grain and number CSL boundaries in triple-line. The main conclusion is that the texture strongly affects the grain boundary misorientation distribution and distribution of CSL boundaries and each type of texture generates characteristic types of CSL boundaries.
1. INTRODUCTION
2. TEXTURE MODELS USED FOR MODELLING CGBD
The effect of the fabrication method on grain orientation distribution and simultaneously on grain boundary character distribution in polycrystal leads to the conclusion that certain direct relationship exists between texture and grain boundary misorientation distribution (GBMD) and coincidence grain boundary distribution (CGBD). The existence of that relationship was reported in many recent experimental works [l-10]. The approximate, qualitative relationship between texture and CGBD has been already established, which consists in prediction of the predominance of certain types of CSL boundaries for sharp textures: (100), ( I l l ) and (110) [7]. Generally, it was based on assumption that for sharp (hkl) textures low-angle boundaries and CSL boundaries with (hkl) misorientation axes are dominating. This is valid in the case of axial texture, for which it is accepted that crystallographic co-ordinates system associated with grains is randomly rotated in the relation to the preferred (hkl) direction. Otherwise, for example in the case of the texture formed during recrystallization of the deformed single crystal, the observed CGBDs are different from predicted. The aim of the present part of the paper is to prove the existence of exact (quantitative) relationship between texture and CGBD on the basis of simulation of spatial model of polycrystal built with Kelvin polyhedra-shaped grains (the method B2 discussed in Part I of this work). tPresent address: Institute of Technology and Organisation of Building Production, Warsaw University of Technology, A1. Armii Ludowej 16, 00-637 Warsaw, Poland. AM 41/2--K
The existence of the relationship between texture and distribution of CSL boundaries was studied for (hkl) textures, often observed in materials with cubic lattice. The spatial model of polycrystal built with Kelvin polyhedra (tetrakaidekahedra) grains was used for simulation. This model is a reasonable approximation of a real polycrystalline aggregate, which additionally allows to analyse features of spatial CSL boundary distribution. Orientations described by sets of three Euler's angles were ascribed to particular grains that form aggregate. Brandon's criterion [11] was used to determine if given boundary is of CSL type. The two different models of (hkl) texture were applied (Fig. 1). In both models deviation of (hkl) direction from the sample axis was described by the normal distribution N(0, a), where a is given in degrees, determining the sharpness of texture [Fig. l(c)]. The details of these models are given elsewhere [12-14]. The first model [Fig. l(a)] is an approximation of the axial textures, i.e. wire and fibre textures, characteristic for many engineering materials. The second model approximates textures occurring in deformed and subsequently recrystallized single-crystal (without deformation bands, etc.). For the given type of texture the second model [Fig. l(b)] leads to smaller differences of grain orientations than in the case of the first model. For both models it was assumed that grain orientation is uncorrelated with respect to its position within a polycrystalline aggregate. It means that texture was homogeneous and deviations from
475
476
GARBACZ and GRABSKI: TEXTURE IN POLYCRYSTALLINE MATERIALS--II
b) Model
a) Model T_
~
c)
A-N(O,3)
<.kl
rondom(_~~
==>
<3==
\ /2-N(0,5) ~ , ~ 3 - N (0,15)
v
]As
JAs AS
-
SYMPLE
IAS
--36
AXIS
Fig. 1. The models of (hkl) texture used in this work (a, b) and three normal distributions describing sharpness of the analysed textures (c).
ideal direction ( h k l ) were randomly ascribed to particular grains. The adopted texture models allowed to determine changes of CGBD dependent on texture, which can be regarded as the reference for analysis of more complicated grain orientation distributions. The worked-out method makes it possible to calculate hypothetical GBMD and CGBD for any grain orientation distribution and also for the case of inhomogeneous textures.
jx (1)
°>
/-/ [
.
(2)
b)~
z
Z
D
3. COMPUTER SIMULATION OF EXPERIMENTAL DISTRIBUTIONS OF CSL BOUNDARIES FOR (hkl) TEXTURES Among numerous experimental works concerning the distributions of CSL boundaries it is difficult to find a work containing both the data on CGBD and the well-described texture. The exception are reports of Watanabe and co-workers [2, 3], in which CGBDs were studied with ECP method in rapidly solidificated ribbons with sharp textures (100) and (110) in purpose to corroborate the quantitative relationship between texture and CGBD. This data was used in the present work to verify the correctness of the adopted model. The analysis of grain orientation distributions for different simulated textures showed that distribution obtained for mixed texture: 90 % (100) N(0, 3.5°) + 10%(111) N(0, 8°) described by model I was most close to the experimental grain orientation distribution obtained by Watanabe et al. [2]. Comparison of the experimental and modelled grain orientation distributions for similar number of investigated boundaries are presented in Fig. 2. The modelled CGBD exhibits 25 -<100> 20
~.~ 15 O
1
5-
\1/2
o ~
Fig. 2. The comparison of grain orientation distributions: 1--experimental distributions obtained by Watanabe and co-workers for sharp (100) texture [2]. 2---computer simulation for 90% (100) + 10%( 111) texture.
~
C,l C4 C,I
Fig. 3. The distributions of CSL boundaries for sharp (100) texture: 1--experimental by Watanabe et al. [2]. 2--computer simulation.
GARBACZ and GRABSKI: TEXTURE IN POLYCRYSTALLINE MATERIALS---II high goodness of fit with the experimental distribution (Fig. 3). It confirms the existence of high fraction of low-angle boundaries and also of 2~5, 13a 17a and 25a CSL boundaries with (100) disorientation axis. Greater difference was observed only for the case of Z 17a boundaries. To confirm agreement between both modelled and experimental distributions the Kolmogorov-Smirnov test of goodness of fit was applied. The value of statistic for analysed distributions was found to be equal 2 = 0.849 and the critical value of this statistic 2 , was equal 1.358 for the level of significance ~ --0.05. It indicates that there is no basis to reject hypothesis about goodness of fit of both distributions. The conclusion is that the distribution of coincident boundaries in Watanabe's experiment was controlled by the geometrical factors with the exception of Z 17a boundaries. The reason for surplus of 2~17a boundaries is most probably related to their particular properties and/or to the specific mechanism of the microstructure formation. The similar analysis was performed for CGBD obtained for the same material but with (110) texture [3]. The computer modelling was performed with assumption that material had 1 0 0 % ( l l 0 ) N(0,3 °) texture described by model I. The comparison of experimental and modelled distributions again shows very good agreement (Fig. 4). It was also confirmed by Kolmogorov-Smirnov test. The calculated value of statistic is 2 = 0.636. It indicates that also in this case there was no basis to reject hypothesis about goodness of fit of both experimental and modelled distributions. The obtained results indicate existence of the direct quantitative relationship between texture and CGBD. This shows that the worked-out model of a polycrystal can be used to determine the reference CGBD for any grain orientation texture.
20
--
<110>
15
o
10
/ 5
o
/
I I
I I
I]-J
I ,,I I I
x Fig. 4. The distributions of CSL boundaries for (110) texture: 1---experimental by Watanabe et al. [3]. 2---computer simulation.
477
4. ANALYSIS OF CGBD FOR BASIC TYPES
OF (hkl) TEXTURE The GBMDs and CGBDs (coincidences from the range Z1-65) for textured polycrystal were computed for four types of ( h k l ) textures, namely 100%(100), 1 0 0 % ( I l l ) , 5 0 % 0 0 0 ) + 50%(111) and 100%(110). The simulations were performed for three levels of the texture sharpness equal to 3 °, 5 ° and 15° [Fig. 1(a-c)]. The changes of type, sharpness and character of the texture (Fig. 5) exert a strong influence on G B M D (Fig. 6) and CGBD (Table 1). The obtained results indicate that GBMDs and particularly CGBDs are strongly dependent on existing texture and that each type of texture is associated with characteristic set of CSL boundaries. The most probable CSL boundaries for given texture type are listed in Table 2. As an example of changes of CGBD with texture, the distributions of CSL boundaries for texture described by first model [Fig. l(a)] with sharpness equal to N(0,5 °) are shown in Fig. 7. With increasing deviation from ideal orientation and decreasing sharpness of texture the randomness of grain orientation distribution increases and the fraction of CSL boundaries decreases, mainly due to the decreasing of fraction of lowangle (271) boundaries. Fraction of2~ 1 boundaries for model II is about two times greater than for model I (Fig. 8). CGBDs for particular type of texture originate from GBMDs generated by that texture. The changes of the character of distributions of disorientation angles and of disorientation axes, which define disorientation distribution, are more distinct for texture described by second model. For a pure texture the distributions of disorientation axes in eight zones of standard stereographic triangle (SST) and particularly the distributions of disorientation angles have characteristic features, which allow to recognise their type [Fig. 6(b)]. Corresponding distributions obtained for the first model exhibits small differences in GBMDs, which practically disappears for the lowest investigated sharpness of texture, namely N(0,15°). It was also observed that for different types of texture similar disorientation distribution can be obtained. For example, very similar distribution of disorientation axes in 8 zones of SST and of distribution of disorientation angles were obtained for both models and for all sharpness in the cases of mixed 50%(100) + 50%(11 l ) and pure 100%(110) textures. In fact it is necessary to apply a more precise analysis consisting on determination of the proportion of axes deviated maximum by 5° from (100), (11 l ) and (11 l ) to find that these distributions are different. On the contrary, the distributions of CSL boundaries in mostly cases were different not only in total fraction of CSL boundaries but also in types of these boundaries. This result shows that CGBDs are often a more meaningful polycrystal characteristic than GBMDs. The distributions of disorientation angles and axes used
478
G A R B A C Z and GRABSKI:
T E X T U R E IN P O L Y C R Y S T A L L I N E MATERIALS---II
LO
.z
o~ .= z
"Ld
c:;
v
V
z
Iml
I=1
t'Xl
o
..=
tll ill
r~
"12
r~
O
t::t O
O 0"2
m
Z
o
oS.
O O
G A R B A C Z and GRABSKI:
T E X T U R E IN P O L Y C R Y S T A L L I N E M A T E R I A L S - - I I
479
(a)
10
80
:t
<100>+ <111>
<100>
<100> + <111>
60 1-..~
,3H
40
3
o
2/
20
3
v
e. = P
10
°
8
~A1 <110>
<111>
"~
80
<111>
t~
<110>
1
60 6 I
2
40 4
1
3
1
20
2 30
30 D i s o r i e n t a t i o n angle (deg)
(b) 10
3
600
•
0
60
2
/
2 3 4 5 6 7 8 l 2 3 4 5 6 7 8 Zone in SST
80
1 14.9 <100> + <111>
<100>
<100> + <111>
60 2 o ×
40
4
1~i/
3
2O
2 v
=
0
8 0 8o
~r 10 <111> 1 4 . 5 ~
[
1.
<110>
<110>
6O 3
/2
3\
40
32
2
\
20
0
30
60 0
30
60
D i s o r i e n t a t i o n angle (deg)
0
1
rhar~ 2 3 4
6 7 8
2 3 Zone in SST
4
A...In 5 6 7 8
Fig. 6. The distributions of disorientation angles and the distributions of disorientation axes in 8 zones of standard stereographic triangle (SST) for model I (a) and model II (b).
for this purpose regards more precise distribution of disorientation axes distribution than disorientation axes distribution in 8 zones of SST. 5. T E X T U R E I N F L U E N C E O N F E A T U R E S OF S P A T I A L D I S T R I B U T I O N OF CSL B O U N D A R I E S
The investigation of relationship between grain boundary properties and bulk properties of poly-
crystalline materials have revealed that in many cases not only the fraction of boundaries with special properties but also their spatial distribution in the relation to the loading system is an important factor [15]. The features of spatial distribution of C S L boundaries may provide new information about influence of this distribution on the propagation of fracture but also on plastic deformation or recrystallization
480
GARBACZ
and GRABSKI:
TEXTURE
IN POLYCRYSTALLINE
MATERIALS--II
Table 1. The coincidence grain boundary distribution for different types o f (hkl> texture. A - - N ( 0 , 3), B---NO, 5), C - - N ( 0 , 15). <100> 2~
A
<111>
B
C
22.8 -8.7 0.0 0.0
A
A
ModelI 3.9 12.3 1.4 2.7 0.8 4.0 0.8 1.9 0.8 0.0 0.5 3.0 0.8 2.3 0.5 0.0 0.4 1.2 0.6 0.5 0.7 0.5 0.4 -0.4 1.2 0.5 -0.3 0.5
C
A
B
C
10.1 2.6 2.1 1.0 0.2 1.8 1.0 0.1 1.0 0.3 0.4 0.I 1.1 0.1 0.4
2.8 1.7 0.9 0.8 0.8 0.7 0.6 0.6 0.5 0.5 0.7 0.4 0.5 0.4 0.3
14.5 13.2 --4.0 3.0 --1.5 1.5 0.0 -0.1 1.0 --
11.5 7.3 -0.1 1.8 1.3 0.2 0.1 0.8 0.7 0.3 -0.5 0.5 0.2
2.7 1.7 0.8 0.8 0.8 0.7 0.6 0.5 0.5 0.4 0.7 0.4 0.5 0.5 0.4
12.0 27.5
9.4 18.3
24.4 ~.1
13.8 30.5
9.1 17.9
7.7 1.2 1.4 0.5 0.8 0.6 1.1 0.4 0.8 0.6 0.6 0.3 0.9 0.4 0.5
54.8 -------7.1 ---0.4 ---
46.4 ----0.1 0.2 -3.2 ---2.2 -0.1
10.8 0.3 1.6 0.3 1.3 0.4 1.7 0.5 1.0 0.6 0.4 0.3 1.2 0.6 0.5
10.0 24.2
7.4 63.3
5.9 57.6
11.0 28.5
were
calculated
--
--
0.3
--
5.2 -3.7 -0.0 0.0 2.0 -1.6
2.6 0.1 1.6 0.1 0.2 0.1 0.9 0.1 0.7
1.1 0.4 0.5 0.5 0.6 0.4 0.4 0.6 0.3
3.8 --2.1 1.9 ---0.0
17.1 5.7 -3.6 0.0 0.0 1.5 -0.1 1.0 0.9 0.0 0.3 0.1 0.3
3-29 1-65
28.3 59.6
15.1 42.4
8.8 21.3
25.6 52.1
13.5 35.6
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
99.8 -----
90.2 -----
19.6 0.3 1.6 0.3 1.3
49.5 43.7 ----
41.7 23.1 ----
--
--
0.4
--
--
0.7
--
----------
0.1 -0.1 0.2 --0.6 ---
1.8 0.2 0.6 1.4 0.4 0.2 0.9 1.0 0.2
----------
0.1 -0.5 0.1 0.1 -0.4 ---
1.6 0.3 0.9 0.4 0.9 0.3 0.9 0.2 0.4
--0.8 ---1.1 ---
31.8 5.9 ---0.3 0.1 0.1 1.6 0.1 0.1 -1.8 -0.2
3-29 1-65
-99.8
1.0 91.2
10.5 37.8
43.7 94.2
24.4 71.4
11.8 28.0
12.7 53.6
10.0 47.9
for this purpose
tion of stereological describe
spatial
is o n l y
of
CSL
CGBD
should
to
show
The influence
the
The
existence
of texture was
Parameters
selec-
parameters
in the relation
model
that
such
A-C
of polycrystal
shaped
to investigated
grains.
determined
aim of this section
of
17.8 35.6
Mo~llI 9.5 37.0 2.5 10.8 1.1 -1.1 -0.5 --
the accurate
topological
be made.
boundaries
characterised
and
9.0 19.0
The
possibilities.
analysed
crystalline
by the following
example
aggregate.
account •
A--number
of CSL
boundaries
per
•
B---number
of CSL
boundaries
in triple-line---
•
C--number
of CSL
boundries
different
N~
Figure forming
9(a)
all these
of grain--Nqp.
runs
of investigation inside
ensured
distributions
of polythat
par-
was
o f Ngb, N t l a n d
were determined orientation
distribution
frequency there
was
271-29.
or triple-lines were taken into
grain
shows
and cumulative
corners
boundaries
corners
method
polycrystal
random
spatial
polyhedra-
the range
consisted
The
only once. The
Nqo f o r r a n d o m
grain--Ngb
from
and
using
Kelvin
of CSL
procedure
triple-lines
ticular surfaces, corners
parameters:
with
Distribution
calculating
of all grains,
an
build
for coincidences
on the spatial distribution on
<110>
B
27.9 -15.7 ---
phenomena
21.7 9.7 -8.0 --
C
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
kinetics. However,
6.4 0.5 1.8 0.7 0.7
<100> + <111>
B
of
FNgbo f no
for seven
distributions. Ngb ( h i s t o g r a m )
this distribution.
grain
found
with
In
more
Table 2. The characteristic types o f CSL boundaries for different types o f
1
3
<100) (111> (100> + <111> (110)
+ + + +
-+ + +
<100>
+
.
+
<100> + <111) <110>
+ +
5
.7
+ -+ --
9
-+ + --
11
---+
--+ +
13
15
17
19
21
23
+a +b ab --
Model I -----
+a -+a +b
-+b -+a
-+a ---
-. ---
+a . +b +b
-. . .
.
25
27
29
+a . +a --
-. -+a
+a ---
--
--
Model II . .
+
. .
.
. .
. .
. .
. .
. .
. .
. .
. .
. .
. . .
.
--
--
. . .
. . .
+a .
. .
. . .
+b +a
GARBACZ and GRABSKI:
TEXTURE IN POLYCRYSTALLINE MATERIALS--II
481
Fraction, [%]~'t t ~ 2.5
2.0
~3
20,F-Ill
=2.5
II1 1
=17.2
/11 ~3 111 z 7
=5.5 =3.7
%
~
9 17 2 5 3 3 4 1 4 9 5 7 6 5 ~ 1.0
0,
<100> F r a c t i o n , [%] f
E 1
E5 I3 Total
2.0 7_ 1.5 1.0
23.2 =8.8 = 2.5 = 43.5
% % % %
lid ,dlll.,llhll, , .d..
19 1611 / o
2.5
F r a c t i o n , [%]
i
Random total = 17.8 %
t
Fraction
1.0
20 i
0.
1.5
[%]
'£1 E3 Total
=11.8 =7.4 = 31.3
a % % %
0 1 9 1 7 2 5 3 3 4 1 4 9 5 7 6 5 y.I.0
0.5
0.5
0 9 17 2 5 3 3 4 1 4 9 5 7 6 5 3"
Fig. 7. The examples of distributions of CSL boundaries up to 2'65 obtained for (hkl> textures described by model I for sharpness N(0,5°).
A 100%<100>, B 50%<100> + 50%<111>, C 100%<111>
(a) 100 80 60 40 20 0
_-..
•d
o
(b)
15
A
B
C
50 40
-
0,0100 Fig. 8. The effect of sharpness of texture on the fraction of Z' 1 boundaries (low angle boundaries) for different types of (hkl> texture for model II (a) and model I (b).
than 6 CSL boundaries. The fraction of grains without CSL boundaries was 18.4_ 6.6%. Distribution of CSL boundaries in triple-lines Ntl, distribution of CSL boundaries in grain corners Nqp and cumulative frequencies of these distributions FNtl and FNqp are presented in Fig. 9(b). The fraction of grain corners without CSL boundaries showed a considerable majority amounting to 70.1 + 3.9%, whereas fraction of corners with junction of 3 CSL boundries were less numerous 0.20+0.10%. Similar percentages were obtained for N~ distributions, i.e. fraction of triplelines without CSL boundaries equals 69.6 _+ 2.9% and with 3 CSL boundaries~).16 + 0.08%. It was also established that between Ntl and Nqp exists approximate relationship: Ntl ~ 1.5Nqp. It corresponds to the ratio of the number of edges and number of corners for tetradeikahedra-shaped grain (36/24). The effect of the texture on parameters of spatial CGBD was determined for formerly investigated (hkl) textures. Figure l0 shows cumulative frequencies FNgb and FNtl for different types of texture. The validity of the relationship N~ ~ 1.5Nqp was also confirmed for the case of textured polycrystal. Influence of texture on Ngb and Ntl distributions can be characterised also by the changes of values of selected parameters that describe the fraction of CSL boundaries. For example such parameters can be a number of grains which have (A)--zero, and (B)--no more than 6, CSL boundaries for Ngb distribution and a number of triple-junctions with (C)--zero and
482
G A R B A C Z and GRABSKI:
T E X T U R E IN P O L Y C R Y S T A L L I N E M A T E R I A L ~ - I I
t I
;
_
_
.
:
.
.
.
.
.
.
.
eq
'~
o ,¢.
- ' ~ - - . • ."4T~.._ J/ ~.~
O~O%
•
~,
~,
"., " ' ~ . ~ v
0%0
ffl".~'~.~
4k.
.~k "-.IK~.
H l "1~-.:~
,. 4rx,,_---..,_ :il\'x~
-4I
<00I>
<[II>
~,~.
°~o~o
"~"~N~,".~
,'-r.,-S.~.\~/ ,"r-.,:z_->._-rl ~ ~ /
4-I ~
IN
-4 00 "~"
,''r-.~:._~
+<001>
<0II> ~0
I~~
I I\ ~ i~l "~.."~ll "~'~"~\1
~,., ~ " ~ . \ ~ : . , . ,
-~'.z:.,.
"~.~\.-1 ,',:~-~.
r.~. ~. ~ J "-- " ~ ~'.~ I.I " / ,"'%.-,~o~.~, "~II ~""~, o~l.e. ~o' - : a "a ~ t r--.-~,.:: , ~ t ~
~=
~o,,
0
~.~
- . t ~ ' ~ . ~ % . -1 r ~ "~'~--(~-'--l. / "'u ""ro~s.._'.t"~;''"¢,1
ca
v
N~ [%]
.~ouanba.tj
o^.q~lnu~n
D
O
~
=_ = C..) ~
~o.Sg q) ,_., r', O ~
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GARBACZ and GRABSKI: TEXTURE IN POLYCRYSTALLINE MATERIALS--II
483
Table 3. Effect of texture on spatial distribution of CSL boundaries: A--percentage of grains without CSL boundaries; l~-percentage of grains with 6 or less CSL boundaries; C--percentage of triple junctions without CSL boundaries; D---percentageof triple junctions with 3 CSL boundaries Model I Model II (100) (100) + + (100) (111) (111) (110) (100) (111) (111). (110) N(0, 3) A: 0.0 0.0 1.3 0.1 0.0 0.0 0.0 0.0 B: 22.9 43.6 89.9 72.6 0.0 0.2 37.1 19.8 C: 9.1 15.0 34.5 15.0 0.0 0.2 12.7 7.7 D: 22.1 13.9 3.4 6.7 98.7 82.7 13.4 26.4 N(0, 5) A: 0.8 1.3 3.3 2.9 0.0 0.0 0.3 0.0 B: 70.3 85.0 95.8 95.3 1.4 11.2 61.3 32.0 C: 26.6 34.9 47.1 44.0 1.1 5.0 20.6 12.6 D: 7.2 3.9 1.4 1.7 71.9 35.9 8,5 19.5 N(0, 15) A: 11.7 15.6 17.6 16.2 2.2 3.5 6,9 2.8 B: 99.6 100.0 100.0 100.0 84.0 97.5 99,6 97.2 C: 61.4 66.9 68.4 67.9 36.7 48.8 56.2 48.1 D: 0.6 0.3 0.3 0.2 4.4 1.4 0,6 1.6 Random: A--18.4 B---100.0 C--70.1 D---0.20
(D)--3, CSL boundaries for Ntt distribution. The values of A, B, C, D parameters for analysed textures are presented in Table 3. It seems that the importance of features of spatial distribution of CSL boundaries will be manifested if it will be possible to correlate this distribution with distribution of grain boundary plane orientations and distribution of geometrical features of grains. The connection of the geometrical description of polycrystal and features of grain boundary characteristics can give full description of polycrystal. Parameters that describe spatial C G B D can be used to investigate influence of the graininess on properties of polycrystal. It would require the strict selection of stereological parameters describing C G B D appropriately chosen for given object of analysis.
5. C G B D s computed for textures with small differences of grain orientation (model II) explain experimental data ascertaining high fraction of CSL boundaries and particularly high fraction of lowangle boundaries in deformed and recrystallized single-crystal. 6. Fraction of CSL boundaries with coincidences from the range Z" 1-29 in the whole population of considered CSL boundaries (S I - Z 65) increases from about 65% in random polycrystal to 100% for sharp texture. 7. The distribution of a number CSL boundaries per grain and a number of CSL boundaries in tripleline show that spatial distribution of CSL boundaries is also strongly dependent on texture.
6. CONCLUSIONS
REFERENCES
1. The presented model produces a reference grain boundary misorientation distribution ( G B M D ) and coincidence grain boundary distribution (CGBD) in textured materials which is a good approximation of experimental data. 2. Both G B M D and C G B D are strongly dependent on the existing texture and each type of texture is associated with characteristic types of CSL boundaries. 3. G B M D and C G B D obtained for axial ( h k l ) texture with N(0,15 °) sharpness, for which it can be assumed that crystallographic co-ordinates system associated with grains are randomly rotated in the relation to ( h k l ) direction, can be approved as random. It indicates that despite the presence of texture both G B M D and C G B D in such a case are close to randomness. 4. Similarity of G B M D s obtained for textures 5 0 % ( 1 0 0 ) + 5 0 % ( 1 1 1 ) and 100%(110) indicate that in many cases C G B D s are more useful for characterisation of polycrystal than G B M D s .
1. T. Watanabe, Mater. Forum I1, 284 (1988). 2. T. Watanabe, H. Fujii, H. Oikawa and K. I. Arai, Acta metall. 37, 941 (1989). 3. T. Watanabe, K. I. Arai, K. Yoshimi and H. Oikawa, Phil. Mag. Lett. 59, 47 (1989). 4. J. Zhao, B. L. Adams and P. R. Morris, Text. Microstruet. 8--9, 495 (1988). 5. T. Watanabe, Res. Mechanica 11, 47 (1984). 6. V. Randle, B. Ralph and D. Dingley, Acta metall. 36, 267 (1988). 7. S. Hanada, T. Ogura, S. Watanabe, O. Izumi and T. Masumoto, Acta metall. 34, 13 (1986). 8. D. E. Fleet, P. I. Welch and P. R. Howell, Acta metall. 26, 1479 (1978). 9. P. I. Welch, D. E. Fleet and P. R. Howell, Acta metall. 26, 1491 (1978). 10. P. R. Howell, D. E. Fleet, P. I. Welch and B. Ralph, Acta metall. 26, 1499 (1978). 11. D. G. Brandon, Acta metall. 14, 1479 (1966). 12. A. Garbacz and M. W. Grabski, Scripta metall. 23, 1369 (1989). 13. A. Garbacz and M. W. Grabski, J. Physique 51, C1-469 (1990). 14. A. Garbacz, Ph.D. thesis, Warsaw Univ. of Technology (1991). 15. T. Watanabe, J. Physique 49, C5-507 (1988).