Computer simulation study of grain boundary and triple junction distributions in microstructures formed by multiple twinning

Ada metall.mater. Vol.43, No. 6, pp. 2317-2324, 1995

Pergamon

0956-7151(94)00422-6

Copyright 0 I995 ElsevierScienceLtd Printedin Great Britain.All rightsreserved 0956-7151/95 $9.50+ 0.00

COMPUTER SIMULATION STUDY OF GRAIN BOUNDARY AND TRIPLE JUNCTION DISTRIBUTIONS IN MICROSTRUCTURES FORMED BY MULTIPLE TWINNING V. Y. GERTSMAN’,*t

and K. TANGRI’

‘Metallurgical Sciences Laboratories, Department of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 and ‘Institute for Metals Superplasticity Problems of the Russian Academy of Sciences, 450001 Ufa, Russia (Received 6 May 1994; in revised form 16 September 1994)

Abstract-Microstructures formed as a result of multiple twinning have been simulated by means of computer modelling. Grain boundary misorientation (character) and triple junction distributions have been studied with the emphasis on the effect of initial texture and multiple twinning process. Although grain boundary distributions are similar in all the microstructures modelled, sharp initial texture leads to a somewhat enhanced amount of 23 boundaries and to a considerable increase in the number of triple junctions containing two C3 boundaries. The impact of these parameters on the material susceptibility to intergranular crack propagation has been analysed and implications for grain boundary engineering has been discussed.

1. INTRODUCTION

It is a commonplace statement that the bulk polycrystal properties primarily depend upon the microstructure. However, it is necessary to emphasize that a full description of the (one-phase) microstructure includes not only crystallite sizes but also their crystallographic orientations in respect both to some outer reference coordinate system and to each other. Relative orientation of adjacent crystallites, i.e. misorientation is believed to be the most important crystallographic parameter that determines the grain boundary character and properties (see, e.g. [l-3]). The understanding of the strong dependence of the grain boundary properties on misorientation has stimulated the idea of “grain boundary design and control” (e.g. [4-61) which means the manipulation of the polycrystal properties by means of altering the misorientation distribution and thereby the grain boundary character distribution. Although a number of studies have been performed in this field over the last decade [420], we have to admit that no significant breakthrough has yet been achieved regarding the tailoring of real materials through grain boundary design.$ In our opinion, one of the reasons for such a situation is as follows. So far the attempts of most of the researchers have been concentrated on the mean parameters of the grain boundary character distribution (for example, simply on the determi-

tTo whom all correspondence should be addressed. $An example of successful application of grain boundary engineering to the problem of intergranular degradation of a Ni-base alloy has been reported recently [21].

nation of the proportion of special boundaries without distinguishing their specific types). However, mechanical behaviour, particularly fracture-related properties could be determined not by the mean values but by the “tails” of the distribution in the first place, i.e. a relatively small amount of some microstructural features may be responsible for the beneficial or, conversely, deleterious effect on the material properties. One and the same microstructure can be described in many different ways depending on the concrete goal of a study. Therefore, it is important to find an appropriate description of a microstructure in order to understand how we can control, e.g. cracking resistance or some other material property. The present study centres on one type of microstructure, viz. the microstructure formed as a result of multiple twinning. Such a microstructure is characteristic of a large class of materials that are prone to the formation of annealing twins on recrystallization. This class includes not only a large number of f.c.c. metals and alloys with low and medium stacking fault energies but also intermetallics with an Ll, crystal lattice (e.g. Ni,Al) and an Ll, lattice (e.g. TiAl), and semiconductors with a diamond structure (Si and Ge). The most comprehensive experimental data on grain boundary distributions have been collected for this class of materials (see [22] for review). This study has been carried out by means of computer modelling. Several computer simulation studies of the grain boundary misorientation distributions have been performed recently. In [23-271 different crystallographic orientations representing either a random orientation set or some texture were

2317

2318

GERTSMAN and TANGRI:

MICROSTRUCTURES FORMED BY MULTIPLE WINNING

first ascribed to each grain forming a polycrystalline aggregate, and misorientations across every grain boundary were computed after that. Both the model textures [23,24,26] and textures corresponding to some real materials [27,28] were considered. (However, in [28] only orientation pairs were simulated, therefore the grain system did not actually form a polycrystalline aggregate but represented a set of bicrystals.) The procedure employed in these studies does not model the microstructure formed as a result of multiple twinning even though correlation between orientations of nearest-neighbour grains has been introduced and a case of the preference of 23 (twin) boundaries has been considered [27]. Another appreach was used in [29] (briefly described elsewhere [22]) intentionally for the purpose of the multiple twinning modelling. However, in 1291the simulation procedure used a single crystal as an initial state, and hence no misorientations other than those produced by multiple twinning could be found in the resulting grain boundary distribution. In the present work we have endeavored to combine the advantages of both of these simulation methods. The objectives of the present study were the following: (1) to create a more realistic model of the microstructure formed as a result of multiple twin-

ning; and (2) to try to evaluate those microstructural parameters that are the most important for the goal of grain boundary design. 2. APPROACH A planar model was used for modelling. It has been shown before [24,30] that grain boundary misorientation distribution obtained in any planar section is representative for the volume distribution. Initial configuration consisted of 19 hexagon-shaped grains (Fig. 1). No dimensional or shape factors were taken into consideration, and all the initial grains are shown in Fig. 1 as regular hexagons for the sake of simplicity only. Actually the results of modelling are independent of whether or not the initial grains are equiaxed or elongated and whether all the initial grain boundaries are of equal length. For each step of modelling ten independent variants were computed and the results were averaged. Each initial grain was randomly assigned some crystallographic orientation so that the set of 19 original orientations was either chaotic or represented some simple model texture. The procedure of multiple twinning modelling was similar to that used in [29] (briefly described elsewhere [22]). At each step, a new twin appears in a randomly

Fig. 1. An example of the modelling microstructure. Initial grain boundaries are shown by thicker lines.

GERTSMAN and TANGRI: Table

I. Percentages

Nt Zl x3 z9 x27 tAverage

MICROSTRUCTURES

of’ Z = 3” (n Q 3) boundaries in modelled initial grains 1

3.0 + 1.2 28.6 + 1.2 I .9 + 2.0 number

2 2.8 33.4 7.2 0.5

3

+ 1.4 + 2.1 _+ 1.5 +_0.5

of twins per initial

2.0 35.6 10.3 1.3

microstructures

with random

4

f 0.7 + 1.5 f 1.3 +_0.8

1.6*0.6 36.8 k I .3 13.0 i 1.9 2.5 +_0.9

orientations

5 1.3 * 0.4 37.4 * 1.4 14.0 f 2.0 3.3 +_0.8

2319

of

10 1.5 39.1 18.6 5.8

* 1.3 f 1.8 f I.8 +_ 1.9

grain.

chosen region of the microstructure. It is randomly assigned one of the four possible orientations in respect to the parent grain. A new twin boundary may simply divide a grain into two parts; or a twin may be of a plate-form, i.e. a grain is divided into three parts separated by twin (C3) boundaries; or a set of several parallel twin plates may appear within a chosen parent grain. An example of a modelled microstructure is shown in Fig. 1. Again, no dimensional or shape factors were taken into consideration. For example, the results of modelling would be the same if twin boundaries were assumed to be faceted. Although it is known that grain boundary plane is an important parameter determining the grain boundary properties, it has been assumed that a grain boundary is characterized by misorientation in the first place, and grain boundary planes were not considered in the present work. 3. RESULTS AND DISCUSSION

3.1. Evolution of grain boundary comparison with experimental data

FORMED BY MULTIPLE TWINNING

distribution

and

Table 1 shows the averaged results of modelling of grain boundary character distributions resulting from multiple twinning in textureless microstructures. Parallel twin plates were considered as one twin orientation when counting the average number of twins per initial grain. Grain boundaries were ascribed to a certain type taking into account the Brandon criterion for the allowable deviation: A0 = 1Y/X”* [3 11.Some typical grain boundary misorientation distributions are shown in Fig. 2(a, b). The histograms represent disorientation (i.e. the smallest misorientation) angle distributions, and the distributions of disorientation axes within the zones of the standard stereographic triangle (Fig. 3) are given in the insets. At first glance, the grain boundary distribution is slightly dependent on the development of multiple twinning, at least when the mean number of twins per initial grain exceeds 1 (i.e. when actual multiple twinning takes place). Indeed, differences between the data in most pairs of the adjacent columns of Table 1 are within statistical scatter. These results seem to support the earlier proposition [32-361 that grain boundary distributions should be similar (quasistable) in all statically recrystallized materials that are prone to annealing twinning. However, close examination of the data shows that percentages of C3”

(n > 0) boundaries increase with the development of multiple twinning. Let us compare the results of computer modelling with the experimental data on grain boundary statistics in materials susceptible to annealing twinning (see the collection of such data in Table 1 and Fig. 3 of Ref. [22]). Both the grain boundary character distributions, particularly for N ranging from 2 to 5 (see Table 1) which is consistent with experimental observations [37], and the disorientation distributions (see Fig. 2) resemble very much the experimentally obtained distributions. This suggests that the model employed in the present work is rather realistic. It should be noted that experimental statistics are much poorer than the model statistics, therefore it is hardly possible to relate the data on any concrete material to any definite step of modelling. Because of statistical scatter, experimental data can be matched fairly well by different sets of model data (including those for textured microstructures-see below). Hence as a first approximation, the grain boundary distributions in the materials under consideration are very much alike and slightly dependent on the regimes of annealing treatment, as has been suggested earlier [32-361. However, we should keep in mind that this conclusion relates to average (and rather poor) statistics, and actually the experimental microstructures studied may represent different stages of the development of annealing twinning. In one case, a stainless steel microstructure formed at an early stage of primary recrystallization was studied [13], and it was found that the percentage of twin (C3) boundaries was lower in that state and very few C3” (n > 1) boundaries were observed. It was concluded [13] that multiple twinning had not developed at that stage of annealing, which is in agreement with the present results (compare column 1 of Table 1 with the other columns).

3.2. Eflect of texture For the initially textured microstructures, grain boundary character distributions are shown in Table 2 and some typical disorientation distributions are plotted in Fig. 2 (c-f). Model axial textures considered here are ideal, i.e. all the initial grains have the corresponding cyrstallographic axes exactly parallel to one outer direction (the normal to the plane of modelling). It should be pointed out that these are textures in the initial microstructures. In the course of the multiple twinning process, texture gradually spreads, which confirms the earlier modelling results [38, 391, although traces of initial texture remain (at

2320

GERTSMAN and TANGRI: 40

I--

FORMED BY MULTIPLE TWINNING 40

Cd

Distribution 30

MICROSTRUCTURES

-1

of disorientation

SST zone no. Fraction

axes within

the SST

Distribution

11)213)4)5161718

I%]

0.0)0.9~1.7~13.4~39.0

(b)

5.4136.7

30

2.8

-1

of disorientation

axes within

SST zone “a.

1112)31415161718

Fraction

~0.0~0.6~1.6~13.5~39.0~5.3

[%I

the SST

38.0

2.0

B h x

20

20

10

10

8 H lb

0

0 10

40

20

40

50

of disorientation

1I

Fraction

(12.3 0.0

[%]

axes within

the SST 30

3.1

30

40

50

60

(4

Distribution

12131415161718 0.0~9.0~28.1~10.2~37.3

20

10

(cl

SST zone no.

1

60 40

Distribution 30

30

of disorientation

axes within

the SST

SST zone no. 1 I 121314]5161718 Fraction

[%I

I5.310.1

0.2 10.7 34.41 7.6 40.211.6

20

30

55 2 2

20

20

10

10

1 m z L

0

0 10

20

30

40

50

60

‘t IO

40

SO

60

(e) 40

40

Distribution 30

SST zone no. Fraction

of disorientation

axes within

11121314

[%I

15

0.010.010.0

I.6

161

the SST

Distribution

7 18

11121314

30

46.714.6j37.619.5

of disorientation

axes within 15

161

the SST 7 18

E ,x 9 $

20

20

10

10

e Ll.

0

0 10

20

Disorientation

30

40

50

60

IO

angle [degrees]

20

30

40

50

Disorientation angle [degrees] Fig. 2

60

GERTSMAN and TANGRI:

MICROSTRUCTURES FORMED BY MULTIPLE TWINNING

2321

the possible misorientation axes for X3 while this is not a case for the (100) axis. These propositions are in agreement with the present work (see Table 2). 3.3. Implications for grain boundary design

100 00

150

30”

110 45”

Fig. 3. Zones of the standard stereographic triangle. least at the stages of multiple twinning studied). Comparison of Tables 1 and 2 shows that for a given step of modelling, percentage of C3 boundaries is always slightly higher if initial microstructure is textured. The difference is most profound in the case of the (110) texture. It is pertinent to mention that topologically exactly the same microstructures were modelled for the textureless state as well as for the textured states, with the sole difference in crystallographic orientations of initial grains. Therefore, “additional” X3 boundaries in the textured microstructures are not primary twin boundaries but are parts of the initial grain boundary network, whose orientations have changed due to multiple twinning. This conclusion was confirmed by examination of misorientation changes in the initial grain boundary network. This fact may be important for the idea of grain boundary design and control (see Section 3.3). The possible reason for the enhanced amount of X3 boundaries in the textured microstructures is as follows. A random misorientation may accidentally become close to X3 in the course of multiple twinning, i.e. when multiplication by different variants of C3 is repeatedly applied to it. This is likely since every random misorientation may be described as close to some C3” misorientation with a relatively small n (see Appendix), and therefore “decomposed” with a high precision into n E3 misorientations. If the set of grain orientations is not chaotic but represents some sharp texture, grain boundary misorientations represent not the whole misorientation space but only a part of it. Therefore the choice of misorientations, even if all of them are random, is limited in a textured case, and during the multiple twinning process, the probability of accidental changes of an initial misorientation to that close to X3 might be higher. This probability must be larger for the (110) axial texture than for the (100) texture because the (110) direction is one of

Let us consider designing a material resistant to intergranular fracture or stress-corrosion, etc. Apparently, it would be not only necessary to increase the proportion of “strong” boundaries, but also to prevent formation of continuous paths of “weak” boundaries, because if such a path exceeded some critical length, a sample would be prone to failure even if these weak boundaries constitute only a small fraction of all grain boundaries. The latter task relates to the problems considered in the mathematical percolation theory, but such a consideration is a challenge for future work, and here the problem will be analysed within the framework of a general physical metallurgy approach. First of all, it is necessary to know what types of grain boundaries are strong and what types are weak. Although there are quite a few speculations on the intrinsic cracking resistance of grain boundaries, only very limited reliable experimental data are available for the class of materials we are interested in this paper. Determinations of grain boundary types along the crack propagation paths were done in [14, 19,401. Unfortunately, the concrete C values are not reported in [14], and all the boundaries are merely classified into the three groups: CSL (coincident site lattice), low-angle, and general high-angle boundaries. However, it is indicated that the vast majority of the CSL boundaries (which have been reported to be more resistant to cracking) are 23” boundaries, and one may expect that most of them are E3 boundaries, as should be in a material susceptible to annealing twinning. Studies [14, 191 claim that low-angle boundaries are less prone to cracking than general boundaries. Although such a suggestion seems to be reasonable, it should be noted that experimental statistics cannot be considered quite significant: there were only a few low-angle boundaries in the microstructures studied. The present study shows that low-angle boundaries (near-xl within 15” allowable deviation) constitute only few percent of the grain boundaries in the microstructures formed by multiple twinning, and their proportion decreases while the process develops (see Tables 1 and 2). It is pertinent to mention that percentages of low-angle boundaries are higher in textured microstructures, which is in agreement with earlier results [23, 24,261. In spite of their small fraction, low-angle boundaries might be important for the material resistance to intergranular

Fig. 2. Typical examples of the grain boundary disorientation distributions. The histograms represent disorientation (i.e. the smallest misorientation) angle distributions, and the inset tables give the percentages of disorientation axes in the zones of the standard stereographic triangle. (a) No texture in the initial microstructure, average number of twins per initial grain N = 4; (b) no texture, N = 10; (c) (100) texture, N = 3; (d) (100) texture, N = 10; (e) (1 IO) texture, N = 2; (f) (110) texture, N = 5.

AM 4316-M

2322

GERTSMAN

and TANGRI:

MICROSTRUCTURES

FORMED

BY MULTIPLE

TWINNING

Table 2. Percentages of Z = 3” (n Q 3) boundaries in modelled microstructures in textured starting material Nt (100) XI

23 Z9 227

I

2

Axial Texture ofInitial Grains 9.9 + 2.9 6.4 i I .O

32.7 f I .6 0.7 f 0.8

36. I + 2.0 7.3 f 1.1 0.7 ? 0.6

(I IO) Axial Texture cf Initial Grains Zl 11.9+ 3.4 8.7 f 2.1 23 37.0 f 2.8 40.0 + 2.4 Z9 6.1 + 2.4 11.8 i 2.5 227 2.4 f 0.6 2.7 k 1.3

3

4

I .6

5.4 + 1.8 38.0 + 1.7 10.8 + 0.9 1.5 f0.8

4.6 k 39.0 f 13.2 + 2.6 +

6.6 f 41.4 + 14.5 + 3.4 *

5.4 +_1.4 42.1 + 2.3 16.7 +_2.1 4.1 f 1.0

1.7 2.2 2.4 1.5

1.3 1.3 0.9

5

10

4.1 & 1.2 39.8 f I .3 14.7 f 1.4 3.4 +_0.8

2.7 +_1.5 41.3 f 1.6 20.0 * I .2 6.7 k 0.8

5.4 * 42.5 f 18.0 k 4.7 +

4.4 * 43.1 ? 21.6 k 7.5 *

1.3 2.2 3.0 1.2

3.4 2.1 2.0 1.0

tAverage number of twins per initial grain.

fracture because they appear mainly within the initial grain boundary network, thus breaking the continuous paths of random boundaries (as the previously mentioned “additional” C3 boundaries do). However, we propose that the chief tool for the grain boundary engineering in this class of materials is Z3 boundaries. The study [19] has revealed that C3 are crack-resistant while other low-x boundaries are weak. The importance of C3 boundaries for intergranular crack arrest has also been indicated in [40]. Of course some variations in crack resistance are possible for the grain boundaries other than X3, but as a first approximation we can assume that in regard to fracture, all grain boundaries may be divided into the two groups: C3 boundaries that are strong, and other (high-angle) boundaries that are weak. Intergranular cracks most probably can be arrested at triple junctions.? According to the above assumptions (only 23 boundaries are strong among highangle boundaries), the crack will propagate easily through the triple junction if at least two of the boundaries in it are non-X.3 (of course if external factors, i.e. orientation of applied stress is favourable for the crack propagation). Then crack blunting must almost surely occur at the triple junctions where two X3 boundaries meet, i.e. X3-X3-X9. An example of such a junction in an experimental microstructure is shown in Fig. 4. Low-angle boundaries are less important from this point of view since the triple junctions containing two low-angle boundaries must be extremely rare in the microstructures under investigation. If only one of the boundaries in the triple junction is a low-angle one, the other two must be of one and the same type, e.g. X1-X3-X3, Xl-C9-Z9, etc. If cracks are stopped by the X3-C3-C9 junctions, an ideal crack resistant polycrystal should contain only C3 and X9 boundaries in proportion 2: 1 (as suggested also in 1421).The presence of a boundary of another type means that there is an active intergranular path, i.e. a chain of grain boundaries that

might be prone to cracking. Let a microstructure contain grain boundary XX. This boundary can be single if X = 27 or 81, and in this case the shortest chain of non-Z3 boundaries will consist of three boundaries [Fig. 5(a) shows an example for the case of X = 271. Of course longer active grain boundary paths are also possible (when there are several boundaries that are neither X3 nor C9) if there are triple junctions C3-X27-X8 1, etc. If X # 3” there should be a continuous path of such boundaries which must either form a closed circuit or end at the outer surface. Such a chain may terminate (at both ends) within a sample if and only if X = 3” [Fig. S(b)]. Therefore, it is really important to establish that any misorientation may be described as close to one (sometimes to several) X3” misorientations (see Appendix). It is also obvious that the higher is the maximum value of n in the system, the more probable is the formation of long active intergranular paths. Such arguments show the importance of “additional” X.3 boundaries in textured (especially (110)) microstructures: they appear within the initial network of random grain boundaries and may break their continuity, thus preventing the formation of long crack-susceptible paths. The above consideration also shows that not only grain boundary distribution but also distribution of triple junctions should be assessed for the goal of grain boundary design. Table 3 shows the proportions of triple junctions containing C3 boundaries in the model microstructures. One can see that the triple junction distribution, particularly the fraction

TThere have been observations [41] that in materials prone to annealing twinning not only triple junctions exist, but also four-fold, five-fold, etc. junctions may occur; although their proportions are much lower than that of triple junctions. For the sake of simplicity we will consider microstructures consisting of triple junctions only.

Fig. 4. 23-X3-29

triple junction in pure copper lized at 1220 K.

recrystal-

GERTSMAN and TANGRI:

MICROSTRUCTURES

FORMED BY MULTIPLE TWINNING

2323

(a)

____a

yxy (b)

Fig. 5. Schematic of the active grain boundary paths (see text). Table 3. Percentages of triple iunctions containing I: = 3 boundaries in modelled microstructures Nt

3

4

5

10

Random Orientations ofInitial Grains 123 15.4 + 4.9 66.7 f 4.9 2X3 2.8 + 2.8 15.0 f 4.7

62.9 k 4.6 20.8 f 4.5

60.5 f 4.8 24.0 f 4.1

59.4 + 5.5 25.9 % 4.7

55.1 f 3.3 31.0 f 3.6

Axial Texture of Initial Grains 66.5 + 7.7 60.5 5 6.2 15.9? 5.2 23.7 f 5.4

56.7 _+5.3 29.0 f 5.3

55.0 f 4.7 30.8 f 3.8

54.3 + 5.2 32.3 + 4.1

51.2 + 4.3 35.3 f 5.5

( 1IO) Axial Texture of Initial Grains 123 64.7 + 7.0 56.9 f 5.9 2x3 18.8 + 6.1 28.4 + 5.8

54.5 f 5.4 32.4 & 5.3

53. I + 4.6 34.1 f 4.3

52.7 &-5.4 35.2 + 4.8

49.7 f 3.5 38.3 f 3.8

(100) 123 223

I

2

tAverage number of twins per initial grain. 1X3-triple junctions containing one Z = 3 boundary. 2X3-triple junctions containing two X = 3 boundaries.

of triple junctions containing two X3 boundaries is a parameter that is more sensitive to texture than the grain boundary character distribution (cf. Tables 1 and 2). In order to assess whether the model figures are plausible, some earlier experimental data were re-examined. In nichrome [32] 53.8% of the junctions contain one X3 boundary and 16.6% contain two X3; the corresponding figures for AISI 304 stainless steel [35] are 62.2 and 17.8%, and for AISI 316L steel [35]-57.1 and 18.4%. One can see that experimental data are in the same range as the model ones. It is also interesting to note that nichrome has a very slight texture whereas the steels are strongly textured [27,43], and there are less Z3-containing junctions in nichrome than are in the steels. It should be pointed out that the considerations given above are only tentative. First, Z3 boundaries in the model microstructures may have various planes and may have deviations from the ideal misorientation within the Brandon limit (8.66” for X3). It is not clear at present, up to what deviation of misorientation from the ideal one and of boundary plane from the symmetrical position, X3 boundary retains its special properties. [Incidentally, it is not clear for low-angle (Xl) boundaries either.] Thorough experimental studies are necessary in this field. Second, the above discussion concerns only grain boundary

crystallography. However, it is evident that grain boundary properties depend upon chemistry as well. This may open additional ways for grain boundary engineering (see, e.g. [44]). Acknowledgement-This work was supported by the Natural Sciences and Engineering Research Council of Canada.

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p. 139. Academic Press, London (1976). 2. P. J. Goodhew, in Grain Boundary Structure and Kinetics (edited by R. W. Ball@, p. 115. ASM, Metals Park, Ohio (1980). 3. R. Z. Valiev, V. Y. Gertsman and 0. A. Kaibyshev, Physica stnrus solidi (a) 97, 1I (1986). 4. T. Watanabe, Res. Mechanica 11, 47 (1984). 5. T. Watanabe, Muter. Forum 11, 284 (1988). 6. T. Watanabe, CON.de Phys. 51, Suppl. No. 1,73 (1990). 7. T. Watanabe, H. Fujii, H. Oikawa and K. I. Arai, Acta metall. 37, 941 (1989).

8. V. Randle and A. Brown, Phil. Mug. A 59,107s (1989). 9. L. C. Lim and T. Watanabe, Acta metall. mater. 38, 2507 (1990).

10. V. Y. Gertsman, 0. V. Mishin, 0. K. Korotkova, S. V. Averin and V. A. Safonov, Physics Metals Metallogr. 12, 80 (1991). In Russian.

2324

GERTSMAN and TANGRI:

MICROSTRUCTURES

11. K. T. Aust and G. Palumbo, in Structure and Property Relationships for Interfaces (edited by J. L. Walter, A. H. King and K. Tangri), p. 3. ASM, Metals Park, Ohio (1991). 12. G. Palumbo, P. J. King, K. T. Aust, U. Erb and P. C. Lichtenberger, Scripta metall. mater. 25, 1775 (1991). 13. V. M. Alyabyev, V. Y. Gertsman, V. N. Kuznetsov, S. S. Lapin, 0. V. Mishin, E. G. Ponomareva, V. V. Sagaradze and V. I. Shalayev, Physics Metals Metallogr. 5, 101 (1992). In Russian. 14. D. C. Crawford and G. S. Was, MetaN. Trans. A23,

FORMED BY MULTIPLE TWINNING

42. G. Palumbo, K. T. Aust, U. Erb, P. J. King, A. M. Brennenstuhl and P. C. Lichtenberger, Physica status solidi (a) 131, 425 (1992).

43. A. P. Zhilyaev, A. I. Pshenichnyuk, V. Y. Gertsman and R. Z. Valiev, Mater. Sci. Forum. 126128, 277 (1993). 44. G. Palumbo and K. T. Aust, in Recrystallyzation ‘90 (edited by T. Chandra), TMS-AIME, p. 101 (1990). 45. A. V. Andreeva and A. A. Firsova, Poverkhnost 6, 149 (1987). In Russian. 46. J. K. Mackenzie, Biometrika 45, 229 (1958). 47. J. K. Mackenzie, Acta metall. 12, 223 (1964).

1195 (1992).

15. D. P. Field and B. L. Adams, Metall. Trans. A23, 2515

APPENDIX

(1992).

16. D. P. Field and B. L. Adams, Acta metall. mater. 40, 1145 (1992). 17. T. Watanabe, Scripta metall. mater. 27, 1497 (1992). 18. T. Watanabe. Mater. Sci. Forum 126-128. 295 (1993). 19. H. Lin and D. P. Pope, Acta metall. mater. 41, 553 (1993).

20. K. J. Kurzydlowski, B. Ralph and A. Garbacz, S’cripta metall. mater. 29, 1365 (1993).

21. K. T. Aust, U. Erb and G. Palumbo, Mater. Sci. Engng A176, 329 (1994).

22. V. Y. Gertsman,

K. Tangri and R. Z. Valiev, Acta

metall. mater. 42, 1785 (1994).

n

1 2

Number of different misorientations 23”

1 1 2 4

3 4

_

tt was shown earlier (see, e.g. [22,45]) that any low-Z misorientation can be approximated (within the Brandon allowable deviation limits) by a 23” misorientation with n <: 9. In [41] it was also shown that there is a large number of low-angle misorientations among 23” (n < 11). We have performed a special study to see whether 23” misorientations really cover the entire space of misorientations, and therefore any (random) misorientation may be represented as 23”. Because of the computer capacity, only n < 12 were investigated. There are the following numbers of 23” misorientations:

5

6

7

8

9

10

25

70

196

574

23. A. Garbacz and M. W. Grabski, Scripta metall. mater. 23, 1369 (1989). 24. V. Y. Gertsman, A. P. Zhilyaev, A. I. Pshenichnyuk and R. Z. Valiev, Acta metall. mater. 40, 1433 (1992). 25. A. Garbacz and M. W. Grabski, Acta metall. mater. 41, 469 (1993). 26. A. Garbacz and M. W. Grabski, Acta metall. mater. 41, 475 (1993). 27. A. P. Zhilyaev, V. Y. Gertsman, 0. V. Mishin, A. I. Pshenichnyuk, I. V. Aleksandrov and R. Z. Valiev, Acta metall. mater. 41, 2657 (1993). 28. A. Morawiec, J. A. Szpunar and D. C. Hinz, Acta metall. mater. 41, 469 (1993). 29. V. Y. Gertsman and 0. V. Mishin, Metallofizika 11,26 (1989). In Russian.

10

11

12

1681 5002 14,884

Figure 6 shows the distribution of all 23” (1
Distribution

of disorientation

ar.es within

30. V. Y. Gertsman, A. P. Zhilyaev and A. I. Pshenichnyuk, Mater. Sci. Forum 62-64, Part II, 669 (1990). 31. D. G. Brandon, Acta metall. 14, 1479 (1966). 32. V. Y. Gertsman, V. N. Danilenko and R. Z. Valiev,

the SST

t

Physics Metals Metallogr. 68, 136 (1989). 33. V. Y. Gertsman, V. N. Danilenko and R. Z. Valiev, Metallojzika 12, 112 (1990). In Russian.

34. V. Y. Gertsman, R. Z. Valiev, V. N. Danilenko and 0. V. Mishin, COIL Phys. 51, Suppl. No. 1, 151 (1990). 35. V. Y. Gertsman and K. Tangri, Phil. Mag. A 64, 1319 (1991). 36. V. Y. Gertsman, Mater. Sci. Forum. 126128, 435 (1993). 37. J. Bystrzycki, W. Przetakiewicz and K. Kurzydlowski, Acra metall. mater. 41, 2639 (1993). 38. G. Gottstein, Acta metall. mater. 32, 1117 (1984).

39. 0. V. Mishin, I. V. Aleksandrov, V. Y. Gertsman and R. Z. Valiev, in The First Proceedings of the UTAN Association, p. 71, Moscow (1990). In Russian. 40. G. Palumbo, P. J. King, P. C. Lichtenberger, K. T. Aust and U. Erb, Mater. Res. Sot. Proc. 238, 311 (1992). 41. C. V. Kopezky, A. V. Andreeva and G. D. Sukhomlin, Acta metall. mater. 39, 1603 (1991).

i

0 0

I

I

I

I

I

10

20

30

40

50

Disorientation

I+ 60

angle [degrees]

Fig. 6. Disorientation distribution for 23” (n Q 12) misorientations (disorientation angles distribution is shown by crosses). Random disorientation distribution [46,47] is shown for comparison (solid line).