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On the significance of grain junction topology and crystallography in polycrystals
Scripta M E T A L L U R G I C A et M A T E R I A L I A
Vol. 28, pp. 889-893, 1993 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
ON THE SIGNIFICANCE OF GRAIN JUNCTION TOPOLOGY AND CRYSTALLOGRAPHY IN POLYCRYSTALS
Valerie Randle Department of Materials Engineering, University College, Swansea, SA2 8PP, U.K.
(Received January
29,
1993)
Introduction Traditionally, studies of the topological nature of grain boundary networks in polycrystals have been separate from investigations of grain boundary crystallography. Topological studies have been concerned primarily with mathematical descriptions of grain shapes, sizes and coordination in a polycrystalline aggregate, particularly as a function of grain growth (1). Crystallographic studies, on the other hand, have concentrated on individual orientation measurements, principally expressed as misorientations between neighbouring grains (2). It has been usual to classify misorientations in terms of the coincidence site lattice (CSL) and use this scheme for recognition of boundaries with potentially 'special' properties. Much of the reported misorientation data is interpreted on a statistical basis only: proportions of special boundaries are deemed to be significant rather than their spatial distribution in the microstructure. In other words, boundaries in a polycrystal are treated as isolated bicrystal pairs. Physically this is an oversimplification since the assemblage of grains in a polycrystal constitute a connected system of interfaces (3). The connectivity aspects of grain boundary crystallography are sometimes addressed in terms of the interaction between grains which share a common edge, particularly where twinning is involved. For example, it is well known that where two E = 3 grain boundaries meet a E = 9 CSL is formed. These simple multiplication rules extend to all CSLs, although they are most commonly observed for E = 3~ types because in materials of low stacking fault energy these are abundant (4). Recently, there have been attempts to introduce a more physically realistic connectivity approach to the study of grain boundaries in polycrystals. One impetus for this has been the observation in both real materials (5) and computer simulations (6) of local clustering of certain orientations, and correspondingly misorientations. Another influence on the study of grain boundary crystallography which has been recently developed is the geometrical characterisation of grain junctions based on the orientation of the component grains. The analysis yields two types of grain junction, 'I-lines' and 'U-lines', which have distinct properties (7,8). It is the purpose of this paper to discuss how the misorientation-based approach to grain boundary studies in polycrystals can be incorporated into a topological view of the boundary network, and the consequences of these factors on the overall material properties. Grain boundary_ topology Grain boundary geometry is observed and measured on a two-dimensional section through the polycrystalline aggregate. However the true connectivity of the grain assembly is only revealed by serial sectioning and stereology. In three dimensions it is apparent that grains can join in three ways: at faces, edges and at corners, commonly referred to as grain boundaries, triple/multiple lines or junctions and quadruple points or nodes respectively. Where boundaries are of equal energy, surface tension is the major factor in shaping the grain boundary network and leads to the equilibrium condition of three grain faces meeting at 120 ° and four grain edges meeting at 109.5 ° as shown in figure 1. Hence grain edges are composed of three boundaries and so referred to as triple lines. This situation is complicated by the presence of low energy boundaries, predominantly twins. For this case dihedral angles can deviate considerably from the equilibrium 120 ° and consequently more than three boundaries can co-exist at a grain edge (3). These topological features can be readily observed experimentally in two-dimensional sections.
889 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press
Ltd.
890
GRAIN JUNCTION TOPOLOGY
Vol.
28, No.
Where grain orientation measurements are collected with reference to an image of the microstructure, the connectivity of the boundary network is also revealed for this plane section through the grain assembly. Hence topological and orientational information can be co-assimilated, and such features as dihedral angles at grain junctions, boundary curvature or faceting, grain morphology and size, can be related to the crystallographic parameters. The 'macroscopic' crystallographic parameters are the grain orientations, grain misorientations, perhaps the grain boundary plane orientation and the 'misorientation' at grain junctions. The analysis for the last of these is considered in the next section. Crystallographic analysis of triple lines The analysis of triple line geometry is based on the assumption that the lattice mismatch across a grain boundary is accommodated by a correlation (in terms of a dislocation array) between nearest lattice points in the two neighbottring grains. Hence the transformations between three grains around a triple line (figure 1) can be expressed in terms of a 'nearest neighbour relationship', NNR. The mathematical basis for obtaining the NNRs from three measured grain orientations around a triple line are given elsewhere (7,8). If three grain misorientations are taken in anticlockwise sequence around the triple line, and R,, R b, R, are the three relevant transformations from grain to grain (figure 1) expressed as NNRs, then two cases may result: R~RbR. = I or RoRbR, F I = U
(1)
where I is the identity matrix. The first case is called an 'I-line' and corresponds to dislocation balance at the triple line. The second case is called a 'U-line' and indicates an imbalance of dislocations. The value of U will depend on which grain around the triple line is chosen as the starting grain. However, the I-line designation is independent of the starting grain so I-lines can always be unambiguously recognised. A triple line has the character of a disclination, that is, it is an oriented line with a rotation (or other tensor of rank 2) associated with it. A dislocation, on the other hand, is an oriented line with a tensor of rank 1, i.e. the Burgers vector. It is the fact that triple lines have disclination character and are described by matrices (tensors of rank 2) that makes a U-line tensor (but not that of an I-line) depend on the starting grain: the multiplication of matrices depends on the sequence of operations. The physical consequence of the I-line/u-line character of a triple line is that higher energies and stresses are associated with the latter. Several experiments have demonstrated I-line/U-line differences, for example triple line corrosion in high purity nickel was found to occur exclusively at U-lines (4). Experimental data The I-line/U-line distribution was measured for sample populations of pure nickel specimens which had been strained 2% in compression prior to annealing at 1000°C for 1 hour. One specimen, labelled 'fast', was raised to the holding temperature and also cooled after annealing quickly whereas the second specimen, labelled 'slow', had a heating and cooling rate of 330°/hour and 500°/hour respectively. The 'slow' specimens had a very weak fibre texture near 111 and 100 whereas the texture of the 'fast' specimen was nearly random. Table 1 summarises the parameters measured. The I-line/U-line characters can be compared with computer simulation data which predicts 4% I-lines for a randomly textured fcc material, and 8% and 6% for weak 100 and 111 fibre textures respectively (9). For the present data, texture is not the predominant influence on the proportion of I-lines since the 'fast' specimen, which was almost randomly textured, contains 30% I-lines whereas the 'slow' specimen, which was weakly textured, had 23% I-lines. Furthermore, previous work has shown that pure nickel specimens with a larger average grain size, 7501am, contained 76% I-lines (4). One reason for this large I-line proportion was that the microstructure contained many E = 3 ninteractions, and triple line configurations where the component boundaries are close to E = 3,3,9 or E = 3,9,27a are I-lines (10). Figure 2 shows examples of the typical I-line/U-line distributions in the 'fast' and 'slow' specimens. Analysis of the distributions shows that I-lines are not preferentially associated with any combination of boundary types - low angle (LA), CSL or random - at triple junctions. In other words the meeting of three random boundaries may balance in terms of dislocation content and form an I-line whereas three conjoining CSLs may not necessarily achieve dislocation balance, even though their total dislocation content may be lower than the random case. An
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exception to this is a junction between three LAs which will give an I-line because each boundary is a nearest neighbour. There is also some tendency for clustering of I-lines: in many cases I-lines do not occur singly on a particular grain. The basis for the different behaviour of I- and U- lines is the continuity of intrinsic structure from the component boundaries which results in low free energy at I-lines, and conversely the higher energy associated with U-lines because they are associated with 'extra' dislocations. The continuous dislocation network at an I-line has been observed and analysed by TEM for a E = 3, 9 and 27a triple junction (10). The high energy character of U-lines, both in terms of dislocation content and high free volume, is demonstrated from the fact that in the preferential etching experiments U-lines were attacked before random grain boundaries. Furthermore, some U-lines were not attacked, reflecting varying levels of energy at these defects (4). It is illustrative to consider comparisons between the geometrical character of triple lines and that of grain boundaries. Even though they are one-dimensional and two-dimensional defects respectively, analogies can be made between them. I-lines are the triple line analogue of a dislocation grain boundary where there is good atomic match across the boundary plane, for example a low angle or a symmetrical tilt boundary. All these defects are characterised by particularly low energies. At the other energy extreme we have highly disordered U-lines which are the counterpart of random high angle boundaries. Between these two extremes for grain boundaries there are those whose energies are lower than the random class, but not approaching the very low values of low angle or symmetrical tilt boundaries, for example non-symmetrical tilt CSLs may fall into this category. Similarly, some triple lines also display intermediate properties. Such lines are geometrically classified as U-lines although it is expected that their energy and free volume is less than the maximum value for a triple line. The most significant aspect of triple line geometry in polycrystals is their influence on transport phenomena. The coefficient for grain boundary diffusion is much higher than that for lattice diffusion, and a coefficient for U-line diffusion would be higher again than grain boundary diffusion. Furthermore, U-lines can only form loops or end at surfaces, which is exactly the same condition as for dislocations (7, 8) and hence U-lines form channels throughout the polycrystal. Clearly, the density and distribution of U-lines and I-lines will affect such material properties as intercrystalline cracking, corrosion attack and precipitation. However, even though transport along the U-line network will occur preferentially, there will still be diffusion along grain boundaries. To examine the relationship between triple lines and grain boundaries it is necessary to consider reactions in three dimensions, and to invoke the topological constraints outlined in section 2. There are five possible geometrical situations between a triple line and three component boundaries which will influence the preferential diffusion path: 1.
Three 'special' boundaries meeting at an I-line. Interracial transport is limited.
2.
Three random boundaries meeting at a U-line. Interracial transport occurs readily.
3.
Three 'special' boundaries meeting at a U-line. Inteffacial transport is preferentially along the U-line.
4.
Three random boundaries meeting at an I-line. Interracial transport is preferentially along grain boundaries.
5.
Combinations of boundary types for a U-line or an l-line. Here the transport path will be mixed.
Just as in a polycrystalline aggregate grain boundaries cannot be considered in isolation, so triple lines have to be viewed in terms of their connectivity throughout the microstructure. For an equilibrium situation, four triple lines oin in a node as shown in figure 1. At a node, the U-lines must balance, i.e. a circuit of the U transformations equation 1) around the four triple lines at the node must give the identity. This condition precludes the presence of a single U-line at a node. The clustering of I-lines observed in the experimental data (figure 2) is perhaps caused by nodal balancing conditions. Finally, it should be appreciated that the triple line analysis approach only applies to intrinsic dislocation structure. Extrinsic dislocations which impinge upon boundaries and triple lines after cold work are not accounted for in the analysis. There is evidence from 'dislocation spreading' experiments that extrinsic dislocations are absorbed less readily in special than random boundaries. It is reasonable to assume that this criterion also applies to I-lines and U-lines, namely that I-lines will be relatively resistant to extrinsic dislocations, and so the above discussion will still reflect the qualitative behaviour of triple line/grain boundary assemblies.
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Concluding remarks The geometry of triple lines can be analysed to yield an I-line or U-line classification. These are analogous to low energy and medium or high energy grain boundaries respectively. An analysis of triple line geometry in strain-annealed nickel has revealed that 23% to 30% of triple lines are I-lines, and that these tend to be clustered in the microstructure. The topology of I- and U-line networks implies that their relative distributions can play a significant role in determining the properties of the polycrystal. Acknowledzements The author would like to thank Professor W. Bollmann for his comments on the manuscript. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
F.N. Rhines and K.R. Craig, Met. Trans., 5,413, (1974). V. Randle, 'Measurement of grain boundary geometry in cubic polycrystals', Institute of Physics Publishing, Bristol, U.K., (1993). V. Randle and G. Palumbo, submitted to Acta Met. Mat. G. Palumbo and K.T. Aust, Mat. Sci. Eng., All3, 139, (1989). B.L. Adams, P.R. Morris, T.T. Wang, K.S. Willden and S.I. Wright, Acta Met., 35, 2935, 1987. C.S. Nichols, R.F. Cook, D.R. Clarke and D.A. Smith, Acta Met. Mat., 39, 1657 and 1167, (1991). W. Bollmann, Phil. Mag. A, 49, 73, (1984). W. Bollmann, Phil. Mag. A, 57, 637, (1988). E.G. Doni, G. Palumbo and K.T. Aust, Scripta Met. Mat., 24 2325, (1990). C.T. Forwood and L.M. Clarebrough, 'Electron microscopy of interfaces in metals and alloys', Institute of Physics Publishing, Bristol, U.K., (1992).
TABLE SUMMARY OF MEASURED PARAMETERS FOR THE 'FAST' AND 'SLOW' SPECIMENS FAST
SLOW
Microstructure features
anomalous growth; Mainly straight X = 3 boundaries
very anomalous growth; Mainly curved X = 3 boundaries
Average grain size
244Lum
3551am
Texture
random
weak
Misorientations
4% LA, 27% X = 3
12% LA, 26% X = 3*
Number of triple junctions analysed
194
113
I-lines
30%
23%
* Not close to exact CSL misorientation
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TOPOLOGY
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Fig. 1. Grain edge (triple line) topology in a polycrystal. The sketch shows a shared corner (node) of four grains. There are six surfaces (boundaries), labelled a, b, c, d, e, f, between the four grains and four grain edges joining the following grain boundaries; abc, dea, efb and fdc. The equilibrium angle between adjoining grain boundaries is 120 ° and that between adjoining grain edges is 109.5 ° . One example of each angle is marked.
,""" Fig. 2. Grain boundary and triple line crystallographic characteristics combined with spatial information for (a) the 'fast' and (b) the 'slow' specimen. CSL (mostly Z = 3s), low angle boundaries and random boundaries are represented as dotted, dashed and full lines respectively. I-lines are represented by full circles; all other triple lines are U-lines.
O.lmm
Vol. 28, pp. 889-893, 1993 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
ON THE SIGNIFICANCE OF GRAIN JUNCTION TOPOLOGY AND CRYSTALLOGRAPHY IN POLYCRYSTALS
Valerie Randle Department of Materials Engineering, University College, Swansea, SA2 8PP, U.K.
(Received January
29,
1993)
Introduction Traditionally, studies of the topological nature of grain boundary networks in polycrystals have been separate from investigations of grain boundary crystallography. Topological studies have been concerned primarily with mathematical descriptions of grain shapes, sizes and coordination in a polycrystalline aggregate, particularly as a function of grain growth (1). Crystallographic studies, on the other hand, have concentrated on individual orientation measurements, principally expressed as misorientations between neighbouring grains (2). It has been usual to classify misorientations in terms of the coincidence site lattice (CSL) and use this scheme for recognition of boundaries with potentially 'special' properties. Much of the reported misorientation data is interpreted on a statistical basis only: proportions of special boundaries are deemed to be significant rather than their spatial distribution in the microstructure. In other words, boundaries in a polycrystal are treated as isolated bicrystal pairs. Physically this is an oversimplification since the assemblage of grains in a polycrystal constitute a connected system of interfaces (3). The connectivity aspects of grain boundary crystallography are sometimes addressed in terms of the interaction between grains which share a common edge, particularly where twinning is involved. For example, it is well known that where two E = 3 grain boundaries meet a E = 9 CSL is formed. These simple multiplication rules extend to all CSLs, although they are most commonly observed for E = 3~ types because in materials of low stacking fault energy these are abundant (4). Recently, there have been attempts to introduce a more physically realistic connectivity approach to the study of grain boundaries in polycrystals. One impetus for this has been the observation in both real materials (5) and computer simulations (6) of local clustering of certain orientations, and correspondingly misorientations. Another influence on the study of grain boundary crystallography which has been recently developed is the geometrical characterisation of grain junctions based on the orientation of the component grains. The analysis yields two types of grain junction, 'I-lines' and 'U-lines', which have distinct properties (7,8). It is the purpose of this paper to discuss how the misorientation-based approach to grain boundary studies in polycrystals can be incorporated into a topological view of the boundary network, and the consequences of these factors on the overall material properties. Grain boundary_ topology Grain boundary geometry is observed and measured on a two-dimensional section through the polycrystalline aggregate. However the true connectivity of the grain assembly is only revealed by serial sectioning and stereology. In three dimensions it is apparent that grains can join in three ways: at faces, edges and at corners, commonly referred to as grain boundaries, triple/multiple lines or junctions and quadruple points or nodes respectively. Where boundaries are of equal energy, surface tension is the major factor in shaping the grain boundary network and leads to the equilibrium condition of three grain faces meeting at 120 ° and four grain edges meeting at 109.5 ° as shown in figure 1. Hence grain edges are composed of three boundaries and so referred to as triple lines. This situation is complicated by the presence of low energy boundaries, predominantly twins. For this case dihedral angles can deviate considerably from the equilibrium 120 ° and consequently more than three boundaries can co-exist at a grain edge (3). These topological features can be readily observed experimentally in two-dimensional sections.
889 0956-716X/93 $6.00 + .00 Copyright (c) 1993 Pergamon Press
Ltd.
890
GRAIN JUNCTION TOPOLOGY
Vol.
28, No.
Where grain orientation measurements are collected with reference to an image of the microstructure, the connectivity of the boundary network is also revealed for this plane section through the grain assembly. Hence topological and orientational information can be co-assimilated, and such features as dihedral angles at grain junctions, boundary curvature or faceting, grain morphology and size, can be related to the crystallographic parameters. The 'macroscopic' crystallographic parameters are the grain orientations, grain misorientations, perhaps the grain boundary plane orientation and the 'misorientation' at grain junctions. The analysis for the last of these is considered in the next section. Crystallographic analysis of triple lines The analysis of triple line geometry is based on the assumption that the lattice mismatch across a grain boundary is accommodated by a correlation (in terms of a dislocation array) between nearest lattice points in the two neighbottring grains. Hence the transformations between three grains around a triple line (figure 1) can be expressed in terms of a 'nearest neighbour relationship', NNR. The mathematical basis for obtaining the NNRs from three measured grain orientations around a triple line are given elsewhere (7,8). If three grain misorientations are taken in anticlockwise sequence around the triple line, and R,, R b, R, are the three relevant transformations from grain to grain (figure 1) expressed as NNRs, then two cases may result: R~RbR. = I or RoRbR, F I = U
(1)
where I is the identity matrix. The first case is called an 'I-line' and corresponds to dislocation balance at the triple line. The second case is called a 'U-line' and indicates an imbalance of dislocations. The value of U will depend on which grain around the triple line is chosen as the starting grain. However, the I-line designation is independent of the starting grain so I-lines can always be unambiguously recognised. A triple line has the character of a disclination, that is, it is an oriented line with a rotation (or other tensor of rank 2) associated with it. A dislocation, on the other hand, is an oriented line with a tensor of rank 1, i.e. the Burgers vector. It is the fact that triple lines have disclination character and are described by matrices (tensors of rank 2) that makes a U-line tensor (but not that of an I-line) depend on the starting grain: the multiplication of matrices depends on the sequence of operations. The physical consequence of the I-line/u-line character of a triple line is that higher energies and stresses are associated with the latter. Several experiments have demonstrated I-line/U-line differences, for example triple line corrosion in high purity nickel was found to occur exclusively at U-lines (4). Experimental data The I-line/U-line distribution was measured for sample populations of pure nickel specimens which had been strained 2% in compression prior to annealing at 1000°C for 1 hour. One specimen, labelled 'fast', was raised to the holding temperature and also cooled after annealing quickly whereas the second specimen, labelled 'slow', had a heating and cooling rate of 330°/hour and 500°/hour respectively. The 'slow' specimens had a very weak fibre texture near 111 and 100 whereas the texture of the 'fast' specimen was nearly random. Table 1 summarises the parameters measured. The I-line/U-line characters can be compared with computer simulation data which predicts 4% I-lines for a randomly textured fcc material, and 8% and 6% for weak 100 and 111 fibre textures respectively (9). For the present data, texture is not the predominant influence on the proportion of I-lines since the 'fast' specimen, which was almost randomly textured, contains 30% I-lines whereas the 'slow' specimen, which was weakly textured, had 23% I-lines. Furthermore, previous work has shown that pure nickel specimens with a larger average grain size, 7501am, contained 76% I-lines (4). One reason for this large I-line proportion was that the microstructure contained many E = 3 ninteractions, and triple line configurations where the component boundaries are close to E = 3,3,9 or E = 3,9,27a are I-lines (10). Figure 2 shows examples of the typical I-line/U-line distributions in the 'fast' and 'slow' specimens. Analysis of the distributions shows that I-lines are not preferentially associated with any combination of boundary types - low angle (LA), CSL or random - at triple junctions. In other words the meeting of three random boundaries may balance in terms of dislocation content and form an I-line whereas three conjoining CSLs may not necessarily achieve dislocation balance, even though their total dislocation content may be lower than the random case. An
8
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exception to this is a junction between three LAs which will give an I-line because each boundary is a nearest neighbour. There is also some tendency for clustering of I-lines: in many cases I-lines do not occur singly on a particular grain. The basis for the different behaviour of I- and U- lines is the continuity of intrinsic structure from the component boundaries which results in low free energy at I-lines, and conversely the higher energy associated with U-lines because they are associated with 'extra' dislocations. The continuous dislocation network at an I-line has been observed and analysed by TEM for a E = 3, 9 and 27a triple junction (10). The high energy character of U-lines, both in terms of dislocation content and high free volume, is demonstrated from the fact that in the preferential etching experiments U-lines were attacked before random grain boundaries. Furthermore, some U-lines were not attacked, reflecting varying levels of energy at these defects (4). It is illustrative to consider comparisons between the geometrical character of triple lines and that of grain boundaries. Even though they are one-dimensional and two-dimensional defects respectively, analogies can be made between them. I-lines are the triple line analogue of a dislocation grain boundary where there is good atomic match across the boundary plane, for example a low angle or a symmetrical tilt boundary. All these defects are characterised by particularly low energies. At the other energy extreme we have highly disordered U-lines which are the counterpart of random high angle boundaries. Between these two extremes for grain boundaries there are those whose energies are lower than the random class, but not approaching the very low values of low angle or symmetrical tilt boundaries, for example non-symmetrical tilt CSLs may fall into this category. Similarly, some triple lines also display intermediate properties. Such lines are geometrically classified as U-lines although it is expected that their energy and free volume is less than the maximum value for a triple line. The most significant aspect of triple line geometry in polycrystals is their influence on transport phenomena. The coefficient for grain boundary diffusion is much higher than that for lattice diffusion, and a coefficient for U-line diffusion would be higher again than grain boundary diffusion. Furthermore, U-lines can only form loops or end at surfaces, which is exactly the same condition as for dislocations (7, 8) and hence U-lines form channels throughout the polycrystal. Clearly, the density and distribution of U-lines and I-lines will affect such material properties as intercrystalline cracking, corrosion attack and precipitation. However, even though transport along the U-line network will occur preferentially, there will still be diffusion along grain boundaries. To examine the relationship between triple lines and grain boundaries it is necessary to consider reactions in three dimensions, and to invoke the topological constraints outlined in section 2. There are five possible geometrical situations between a triple line and three component boundaries which will influence the preferential diffusion path: 1.
Three 'special' boundaries meeting at an I-line. Interracial transport is limited.
2.
Three random boundaries meeting at a U-line. Interracial transport occurs readily.
3.
Three 'special' boundaries meeting at a U-line. Inteffacial transport is preferentially along the U-line.
4.
Three random boundaries meeting at an I-line. Interracial transport is preferentially along grain boundaries.
5.
Combinations of boundary types for a U-line or an l-line. Here the transport path will be mixed.
Just as in a polycrystalline aggregate grain boundaries cannot be considered in isolation, so triple lines have to be viewed in terms of their connectivity throughout the microstructure. For an equilibrium situation, four triple lines oin in a node as shown in figure 1. At a node, the U-lines must balance, i.e. a circuit of the U transformations equation 1) around the four triple lines at the node must give the identity. This condition precludes the presence of a single U-line at a node. The clustering of I-lines observed in the experimental data (figure 2) is perhaps caused by nodal balancing conditions. Finally, it should be appreciated that the triple line analysis approach only applies to intrinsic dislocation structure. Extrinsic dislocations which impinge upon boundaries and triple lines after cold work are not accounted for in the analysis. There is evidence from 'dislocation spreading' experiments that extrinsic dislocations are absorbed less readily in special than random boundaries. It is reasonable to assume that this criterion also applies to I-lines and U-lines, namely that I-lines will be relatively resistant to extrinsic dislocations, and so the above discussion will still reflect the qualitative behaviour of triple line/grain boundary assemblies.
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GRAIN JUNCTION T O P O L O G Y
Vol.
28, No.
Concluding remarks The geometry of triple lines can be analysed to yield an I-line or U-line classification. These are analogous to low energy and medium or high energy grain boundaries respectively. An analysis of triple line geometry in strain-annealed nickel has revealed that 23% to 30% of triple lines are I-lines, and that these tend to be clustered in the microstructure. The topology of I- and U-line networks implies that their relative distributions can play a significant role in determining the properties of the polycrystal. Acknowledzements The author would like to thank Professor W. Bollmann for his comments on the manuscript. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
F.N. Rhines and K.R. Craig, Met. Trans., 5,413, (1974). V. Randle, 'Measurement of grain boundary geometry in cubic polycrystals', Institute of Physics Publishing, Bristol, U.K., (1993). V. Randle and G. Palumbo, submitted to Acta Met. Mat. G. Palumbo and K.T. Aust, Mat. Sci. Eng., All3, 139, (1989). B.L. Adams, P.R. Morris, T.T. Wang, K.S. Willden and S.I. Wright, Acta Met., 35, 2935, 1987. C.S. Nichols, R.F. Cook, D.R. Clarke and D.A. Smith, Acta Met. Mat., 39, 1657 and 1167, (1991). W. Bollmann, Phil. Mag. A, 49, 73, (1984). W. Bollmann, Phil. Mag. A, 57, 637, (1988). E.G. Doni, G. Palumbo and K.T. Aust, Scripta Met. Mat., 24 2325, (1990). C.T. Forwood and L.M. Clarebrough, 'Electron microscopy of interfaces in metals and alloys', Institute of Physics Publishing, Bristol, U.K., (1992).
TABLE SUMMARY OF MEASURED PARAMETERS FOR THE 'FAST' AND 'SLOW' SPECIMENS FAST
SLOW
Microstructure features
anomalous growth; Mainly straight X = 3 boundaries
very anomalous growth; Mainly curved X = 3 boundaries
Average grain size
244Lum
3551am
Texture
random
weak
Misorientations
4% LA, 27% X = 3
12% LA, 26% X = 3*
Number of triple junctions analysed
194
113
I-lines
30%
23%
* Not close to exact CSL misorientation
8
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GRAIN JUNCTION
TOPOLOGY
893
Fig. 1. Grain edge (triple line) topology in a polycrystal. The sketch shows a shared corner (node) of four grains. There are six surfaces (boundaries), labelled a, b, c, d, e, f, between the four grains and four grain edges joining the following grain boundaries; abc, dea, efb and fdc. The equilibrium angle between adjoining grain boundaries is 120 ° and that between adjoining grain edges is 109.5 ° . One example of each angle is marked.
,""" Fig. 2. Grain boundary and triple line crystallographic characteristics combined with spatial information for (a) the 'fast' and (b) the 'slow' specimen. CSL (mostly Z = 3s), low angle boundaries and random boundaries are represented as dotted, dashed and full lines respectively. I-lines are represented by full circles; all other triple lines are U-lines.
O.lmm