Computer simulation of triple line character distributions in FCC materials

Scripta METALLURGICA et MATERIALIA

Vol. 24, pp. 2325-2328, 1990 Printed in the U.S.A.

Pergamon Press plc All rights reserved

COMPUTER SIMULATION OF T R I P L E LINE CHARACTER DISTRIBUTIONS IN FCC MATERIALS E.G. Doni*, G. Palumbo and K.T. Aust Depamnent of Metallurgy and Materials Science, University of Toronto, Toronto, Canada M5S-1A4 (Received September

24, 1990)

Introduction

Many properties of conventional polycrystalline materials, which are of significance in their industrial applications, depend on both the specific properties of grain boundaries, and as has been recently demonstrated [1-6], the intersection of these interfaces at triple junctions. Furthermore, it has been recently shown [7], that triple fines play an important role in the properties of nanocrystalline materials (grain size < 10 nm), since they comprise a significant fraction of the bulk volume. Triple line corrosion in high purity Ni has been extensively studied by Palumbo and Aust [3-5] on the basis of Bollmann's O-lattice theory [8] and Bollmann's criteria [9-11] for the characterization of a triple line either as an I-fine or a U-line. The authors show that there exists preferential corrosion at junctions which satisfy the crystallographic criterion of Bollmann's U-lines (i.e., disclinations) [4], thus providing experimental support for Bollmann's disclination treatment of triple junctions. A theoretical study of u'iple fines in polycrystalline materials has been made [12,13] on the basis of the CSL/DSC model [14]. It has been shown that the symmetry of the CSL [15,16] can be used for the determination of the geometric characteristics of triple junctions. Attention has been paid to junctions composed of one or more high symmetry, low Y~CSL interfaces (which generally have a high symmetry rotation axis as their common intersection). When all of the adjoining interfaces are low Y~CSL types, the junction is classified as a "special triple junction". Experimental evidence for the existence of"special triple junctions" in polycrystalline Si, composed of ~=3 n type CSL grain boundaries (i.e., £3, £9, £27a) and having the <110>-axis as their common intersection axis, has been presented [17]. A Coincident Axial Direction (CAD)approach[18,19] to the structure of triple junctions in polycrystalline materials has been recently presented by Palumbo and Aust [20], outlining the applicability of the CAD approach to triple lines which do not display three dimensional periodicities in the framework of the O-lattice and CSL/DSC models. It is anticipated that, as in the case of grain boundaries, the properties of bulk materials will be affected by the distribution of triple line 'types' in polycrystalline materials. In the present work a computer simulation of triple line character distributions in the FCC system is made. Three comp~mentary models for the interpretation of triple line structure, i.e., Bollmann's disclination treatment [9-11], the "special triple junction" and CSL/DSC classifications [12,13], and the Coincident Axial Direction approach [20] have been investigated. Random crystal orientations have been considered at fn'st, and "fibre texture" in the directions <100>, <110> and <111> has also been imposed in order to study the possible relaxations at the triple junctions and associated interfaces. In total, over 200 triple junctions and 600 associated interfaces have been analyzed. The results of this preliminary investigation are presented here. Method Three sets of random crystal orientations K1, K2, K3 in the cubic system were produced, each corresponding to an adjoining lattice at the junction. "Fibre texture" was imposed by extracting, from each random set, crystal orientations lying within A0 of the axes <100>, <110>, and <111>. A0 is considered to be 10o for "strong" texture and 20 ° for "weak" texture. Crystal orientations were combined in order to produce the possible triple junctions (TJs) of the boundaries Ba, Bb and Bc. In each case the rotation matrices Ra, Rb and Rc expressing the rotation relationship between the crystal orientations K1, K2 and K3 were determined as well as the corresponding rotation angle and the rotation axis * Permanent address: Solid State Section 313-1, Department of Physics, Aristotle University of Thessaloniki, 540 06 Thessaloniki, Greece.

2325 0036-9748/90 $3.00 + .00 Copyright (c) 1990 Pergamon Press plc

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pairs (0, ). The boundaries of each triple junction were classified according to the smallest angular deviation from exact CSL relationships in the range Y=I to ~=51, by using the small angle rotation criterion [21]. The CSLs which can be attributed to each rotation relationship were found by using the computer program GB3 [22], modified for an IBM compatible personal computer. In order to determine the maximum allowed deviation A0 of an exact CSL, Brandon's criterion (i.e., A0=15y.'l/2) [23] was used. The classification of triple lines as I-lines or U-lines in the framework of the O-lattice theory, was made by using the procedure described in detail by Bollmann [9-11]. In each case the corresponding general unimodular matrices (Ua, Lib and Uc) which convert the orientation relationships of the adjoining grain boundaries to the nearest-neighbour relationships (NNR) are constructed [9,10]. The classification of a triple line as either an I-line or U-line is made by the determination of the characteristic tensor T123 (i.e., T123 = UcUbUa ) of the triple junction. In the cases that T123=I (i.e., the identity matrix), the triple line is classified as an I-line, while in the cases that T123~I, the triple line is classified as a U-line. The classification of triple junctions on the basis of the CAD model was made by determining the axis which remains invariant under the sequence of rotations producing the triple line [20]. To this axis a descriptor YI (i.e., l-I = h 2 + k 2 + 12 ) can be attributed [19]. The descriptor H defines the one dimensional CAD periodicity at the triple junction, which arises through the continuity of {hid} planes across the three adjoining lattices at the junction [20]. Eigenvectors (i.e., invariant axes) of the triple junctions were characterized by minimizing the angular deviation from axes having H in the range 3s;I-I~72. R e s u l t s and Discussion

CSL/DSC Description Distribution Table 1 shows the (1) percentage of low angle (~-1) grain boundaries, (2) percentage of CSL grain boundaries in the range ~=3 to ~=51, (3) preferential symmetry of the CSL grain boundaries, (4) percentage of "special triple junctions", and (5) percentage of triple junctions composed of one or two CSL interfaces for random and textured distributions. Determined CSL distributions in a random set are in good agreement with the results given by Warrington and Boon [24], who have evaluated the probability of random rotations lying within deviations from a CSL according to Brandon's criterion. The total percentage of CSL grain boundaries in the range ~=3 to ~=51 is 12.15% in a random distribution. The majority of identified CSL's exhibit low symmetry and only 5.39% of them are high symmetry CSL grain boundaries, i.e,, 2.36% hexagonal, 1.68% rhombobedral and 1.35% tetragonal CSL grain boundaries. There are also 4.05% orthorhombic CSL grain boundaries. Moreover, 1.68% of the randomly distributed boundaries are low angle grain boundaries (~-1). As far as triple junctions are concerned, there exist only 24.24% Iriple junctions composed of one or two low ,Y-value CSL grain boundaries. The majority of triple junctions (i.e., 75.76%) are composed entirely of non-CSL grain boundaries. Moreover, there is no case of "special triple junctions". From Table 1, it is evident that the total number of CSL grain boundaries in the range Z~=I (i.e., low angle grain boundaries) to Y~=51,that is the number of low Y-CSL grain boundaries, increases where "fibre texture" has been imposed. This increase depends on the character of the texture and it is greater in the case of "strong" texture. There is also a tendency for the CSL grain boundaries to exhibit preferential symmetry. The number of triple junctions composed of one or more low • CSL grain boundaries, also increases. Only one of the 203 triple junctions examined was determined to have "special triple junction" characteristics (i.e., all adjoining interfaces described by low ~ CSL relationships). This junction was identified with the "strong" <1 l l>-type texture (i.e., 12.50%). It should be noted that such intersections are commonly observed experimentally, as a result of clustering of CSL interfaces [25]. Furthermore, their occurrence in FCC materials can be enhanced through the interaction of energetically favoured annealing twins [26], resulting in special triple junctions composed of ~,=3n interfaces. In the present work, interfacial energies have not been considered; however, the observed effects indicate that CSL/DSC relaxations occur more readily in the presence of crystallographic texture. Disclination Dbwibugon (O-Lattice) Table 2 shows that the great majority (i.e., 95.96%) of triple junctions in randomly oriented crystal distributions satisfy the crystallographic criterion of Bollmann's U-lines. As shown in Table 2, the effect of "fibre texture" is to significantly reduce the U-line frequency (i.e., increased number of I-lines); however, U-lines continue to comprise a substantial fraction of the triple line character distribution in oriented systems. These results are in general agreement with

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Bollmann's suggestions [27] that (1) U-lines can be quite numerous in randomly oriented systems, and (2) the fraction of I-lines can be enhanced in strongly oriented material.

Grain boundary and triple line character distribution in the case of CSL descriptions. random orientations 99

No. of T.J.'s % low angle (Y4) % total 7.3-51 % lmffercntlal bex symmeu'y rhomb te|r orth % TJ (CSL gb) % "special TJ" % TJ (non-CSL gb)

<100> A0<10o 4

<100> A0<90o 12

<110> A0<10o 10

<110> A0<90o 39

A0<10o 8

<111> A0~20o 31

1.68

16.67

5.56

10.00

3.42

8.33

3.22

12.15 2.36 1.68 1.35 4.05 24.24 0 75.76

16.67

13.90

19.99 10.00

13.62 0.85

16.67

8.34

50.00 0 50.00

33.33 0 66.67

40.00 0 60.00

5.13 35.90 0 64.10

20.83 8.33 4.17 8.33 37.50 12.50 50.00

12.89 2.15 1.07 4.30 29.03 0 70.97

TABLE 2 Triple line character distribution in the O-lattice framework.

No. of T.J.'s

random orientations 99

<100> AO<10o 4

<1(30> A0<20o 12

<110> AOA10o 10

<110> A0.~20o 39

<111> AO<10o 8

<111> A0<20o 31

% I-lines

4.04

25.00

8.33

20.00

10.26

12.50

6.45

% U-lines

95.96

75.00

91.67

80.00

89.74

87.50

93.55

CAD Distribution The percentage of the triple lines having H CAD periodicities in the range H=3 to N=72 are given in Table 3. The percentage of triple junctions in a random distribution having CAD periodicities with I'Ig8 and 1-I_q24were determined to be 11.11% and 33.32% respectively. These fractions axe much lower than the fractions given by Warrington and Boon [24] (i.e., 73% with II.q24) and Randle and Ralph [19] (i.e., 67% with H.f~24) concerning CAD periodicities associated with interfaces in a random crystal distribution. The reduced occurrence of low H CAD periodicitiy at triple junctions is a consequence of geometric restrictions imposed on matching three lattices instead of two (i.e., interfaces) [20]. The effect of texture is also shown in Table 3. The percentage of triple junctions having low H CAD periodicities (i.e., H<8 or IT_<~) increases significantly with imposed texture.

SumIl)arv A computer simulation of triple line character distributions in the FCC system has been conducted considering (1) Bollmann's disclination treatment [9-I1,27], (2) CSL/DSC periodicity [12,13], and (3) CAD periodicity [20]. Approximately 96% of the triple junctions in a randomly oriented distribution were determined to have disclination character (i.e., Bollmann's U-lines), with such distributions also displaying triple junctions with a low frequency of CSL/DSC and CAD pexindicitivs. It has been shown that the triple line character dislribution is sensitive to texture, with the application of "fibre texture" (i.e., <100>, <110>, <111> ) resulting in (1) diminished disciination content (i.e., reduced U-line frequency), (2) enhanced CSL/DSC relaxations, and (3) enhanced one-dimensional CAD periodicity. These results are in good agreement with previous theoretical considerations [20,27] concerning triple line characteristics,

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and dcmonstra~ that (1) the defect character of triple junctions is enhanced in highly random crystal distributions, and (2) the triple line character distribution can bc significantly modified through texture control. TABLE 3 Triple line character distribution on the basis of the CAD model.

No. of T.J.'s

random orientations 99

% II=3 6.06 % II--.4 1.01 % H=8 4.04 % total ~ 11.11 % total ~ 33.32 Note: Total H_<72is 100%.

<100> A0<10o 4

<100> A0<20o 12

<110> AO~10o 10

<110> A0<20o 39

<111> A0~10 o 8

<111> A0<20o 31

0 75.00 0 75.00 100.00

0 24.99 8.33 33.32 49.98

30.00 0 20.00 50.00 70.00

12.82 0 7.69 20.51 35.90

12.50 0 50.00 62.50 75.00

6.45 0 19.35 25.80 38.71

Acknowledgements

The authors would like to thank Mr. P. Fortier for assistance with this work. Financiai support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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