- Email: [email protected]
Computer simulation of symmetrical high angle boundaries in aluminium
COMPUTER SIMULATION OF SYMMETRICAL ANGLE BOUNDARIES IN ALUMINIUM D. .\. SMITH.*
V. VITEIi+
HIGH
and R. C. POND*
*Department of Metallurgy and Science and Materials, University of Oxford, Parks Road. Oxford. England. and +Materials Division. Central Electricity Research Laboratories. Kelvin Avenue, Leatherhead. Surrey. England (Rrcrired
27 l~unrv
1976; in rrrisedjmn
33 Arqrsr
1976)
Abstract-The internal energy of symmetrical high angle tilt boundaries with an (001) axis has been calculated using a potential for aluminium. The computer program used permits relaxations involving rigid translations parallel and perpendicular to the boundary as well as individual atom rearrangements. subject to overall conservation of volume. The calculated energy waj found to vary by a factor of about two for coincidence boundaries with 5 $ Z C 41 but did not depend simply on the size of the boundary period. The condition for low energy configurations appears to require a translation of the adjacent grains away from the coincidence position to a state where characteristic interlocking groups of closely packed atoms occur in the boundaries. R&urn&-On a calcult l’energie interne de joints de grains de Rexion symitriques de forte d&orientation et d’axe (001). B l’aide d’un potentiel pour l’aluminium. Le programme utilisi permettait les relaxations par translations rigides parall~lement et perpendiculairement au joint. ainsi que des riarrangements atomiques individuels. a condition de conserver le volume total. On a trouvt que l’energie calculee variait d’un facteur deux environ pour des joints de coincidence avec 5 C 1 C 41, mais qu’elle ne dCpendait pas simplement de la grandeur de la periode du joint. Les configurations de faible tnergie semblaient necessiter une translation des grains adjacents i partir de la position de cdincidence vers un &at OC I’on trouve dans les joints des groups compacts CaractCristiques d’atomes. Zusammenfassung-Die innere Energie symmetrischer GroBwmkelknickgrenzen mit einer (001)-Achse wurde mit einem Potential fir Aluminium berechnet. Das benutzte Rechnerprogramm erlaubte Relaxationen. die starre Translationen parallel und senkrecht zur Grenze und die Umlagerun! einzelner Atome umfassen. entsprechend der Erhaltung des Gesamtvolumens. Die berechncte Energle schwankte urn einen Faktor von etwa zwei ftir Koinzidenzgrenzen mit 5 $ Z < At, hing aber nicht allein von der Gral3e der Grenzenperiode ab. Bedingung fiir eine niederenergetische Konfiguration scheint die Translation benachbarter KGrner weg von der Koinzidenzlage in eine Lage. in der charakteristische ineinandergreifende Gruppen dicht gepackter Atome in den Grenzen entstehen, zu erfordern.
1. INTRODUCTIOS
Recent experimental results from electron microscopy suggest that high angle grain boundary structure depends sensitively on the orientation of the adjacent grains [i]. In turn it is expected that the energy of high angle grain boundaries will also depend on these boundary parameters. However, it has proved difficult to measure even relative values of grain boundary energy (i’)for a range of boundaries in order to investigate the factors which govern ;:. McLean [Z] has shown that the free energy for 1: = 3 twins in copper depends on the boundary plane orientation and that there is a definite minimum for the coherent (111) plane. It is observed that the I: = 5 (33 boundary in lead. and this and other coincidence boundaries [4] in aluminium have lower than average energies. These results are difficult to interpret fully since whilst the grain rotations were carefully controlled, the grain boundary planes were : E is defined as the ratio of the volume of the coincidence site lattice unit cell to that of the crystal lattice unit cell.
not. In the case of the work by Hasson and Goux [4] on tilt boundaries the experimental measurements were corrected by assuming a value for d-//d+ where 4 is an azimuthal angle between the actual boundary plane and the plane of a symmetric boundary. In addition there is recent experimental evidence that a wide range of coincidence boundaries often with high values of Z, have low energies in copper and silver [j]. In view of the experimental difficulties involved, a number of attempts to study 7 have been made using computer simulation [6-lo]. Haason et a/ [6] use a LMorse potential and do not impose a condition of volume conservation which introduces an arbitrary assumption of zero volume dependent energy terms (see Section 2). Calculations in Ref. 9 are confined to a two dimensional model and in Ref. 8 the structure of periodic boundaries is not computed. The procedure used in [lo] is to minimize 7 as a function of translational relaxation alone and then separately to allow individual atomic relaxations from the structure which had the minimum energy after rigid translation. This procedure does not necessarily lead to minimum energy structures as the
476
SMITH, VITEK
AND
POND:
COMPUTER
SI.MULATION OF HIGH ANGLE BOUNDARIES
two relaxations are not fully coupled. None of these calculations has incorporated all of the following desirable features El 11: (a) use of three dimensional “crystals’.; (b) allowing the two crystals translational freedom; (c) use of a realistic potential applicable to metals and avoidance of arbitrary assumptions concerning volume dependent energy terms; (d) including explicitly more than one repeating unit in the boundary block. Conditions (a). (b) and (c) were satisfied in our calculations, but so far we have assumed that the periodicity of the unrelaxed boundary is conserved, although the relaxation algorithm does not depend fundamentally on this assumption. Computer calculations have been done For all periodic symmetrical (001) tilt boundaries with C < 50 and these results, in principle can be extended to the full range of rotation angles by adding an energy representative of an appropriate superimposed low angle type boundary calculated according to the Read-Shockley [12-J equation. Owing to the crystal symmetry, for each value of Z there are two symmetrical tilt boundaries with different periodicities and both have been investigated. These correspond respectively to the boundary plane occupying the diagonal and edge faces, containing [QOlJ, of the body centred tetragonal unit ceils of the coincidence site lattices formed by rotations about [OOl] [13]; both non-zero indices are odd for the edge type while one is even and the other odd for the diagonal type. 2. METHOD OF CALCULATION ~TE~TO_MIC FORCES
AND
The block of f.c.c. lattice containing a chosen periodic boundary was constructed in the computer and the configuration subsequently relaxed so as to obtain a minimum internal energy. The bicrystal used was two dimensional from the computational point of view as the periodicity of the lattice in the direction of the tilt axis was used. The dimension of the block in the direction perpendicular to the tilt axis and in the boundary plane. and also that in the direction perpendicular to both the boundary and the tilt axis has been chosen as at least twice the cut-off radius of the interatomic potential {see below). The relaxation was carried out using a gradient method. Displacements both parallel and perpendicular to the boundary were allowed but the total volume of the bicrystal was kept constant. This is consistent with the assumption that the interatomic interaction is described by central Forces only; otherwise the volume dependence of the energy would have to be considered. Therefore, if an expansion or contraction took place in the boundary, the corresponding volume change was automatically compensated by a compression or expansion which occurred as an elastic deformation extending Far away from the boundary. However, this long range compensating elastic field is imposed by the calculations. An alter-
native procedure would be to include a volume dependent energy term. In the linear approx~ation this is inadequate and would not change our results substantially. whilst a non-linear approach would involve arbitrary assumptions comparable in significance with our assumption that volume is conserved. An important aspect of the present calculation is that in the procedure for relaxations parallel to the boundary. relative shifts of one grain as well as displacement of individual atomic planes with respect to the other are allowed (for more details see Refs. 14 and 15). An infinite number of boundary structures are geometrically possible for a given crystal rotation and boundary orientation; these correspond to the infinite number of states of relative translation. Thus in order to find the minimum energy structure for a given boundary a scheme of incremental changes of translation state, covering the whole range of possible translation. was necessary. The procedure adopted was to begin with the coincidence structure and allow this to relax; the magnitude of the maximum subsequent translation in the plane of the boundary did not usually exceed 0.6~ (a is the lattice parameter). Next, the first plane of atoms parahel to the boundary was removed from the upper crystal and this crystal shifted by one inte~lanar spacing in order to keep the total volume constant. This structure was then taken as the next starting point for relaxation. In this way, up to six planes were removed successively from the upper and lower _mi.ns before relaxation was carried out. This corresponds to starting the relaxation from a number of possible translational states. As a further check the starting position was also varied by making a shift of u/7 along the tilt axis. This procedure ensures that the symmetry of the continuous plane configuration perpendicular to the rotation axis does not impose structures which are not true minima. Thus we are able to find low energy structures but it is still possible that some low energy configurations may not have been discovered since no relaxation procedure can ensure that all minima of a function have been found. The interaction between the atoms was described by a central Force potential for aluminium which was constructed by Dagens, Rasolt and Taylor [16-j on the basis of a pseudopotential theory. This potential was shown to give a value of O.lOjJm-’ for the intrinsic stacking fault energy and O.O%Jm-’ for the twin boundary energy[lQ Since this potential is long range and osciilates as cos @k&/r3 (kf is the Fermi vector) For large P, a cut-off radius r, has to be chosen. In the present work r, = 1.6~ was used, i.e. the potential was truncated between fifth and sixth nearest neighbours. The effects of the choice of the cut-off radius on the calculated grain boundary structures and energies were studied in more detail by Pond and Vitek [14]. They concluded that the structure of the boundary, i.e. the atom positions and relative translations of the grains are practically uninfluenced
SMITH. VITEK
.LVD
POND:
COMPUTER
SIMULATION
OF HIGH .L\XiLE BOUNDARIES
477
Table 1. Crystallographic data. translation, Ti and local expansion for the lowest energy configurations of the grain boundaries studied. Ti components x and y, and the expansion are in units of the lattice parameter n: areas of the repeating units are in the units of u’
I 5
13 17 25 29 37 41
0
Plane
7 (Jm-?I
53.13
310 310 510 320 410 330 710 430 730 520 750 610 910 540
0.208 0.161 0.241 0.242 0.195 0.260 0.235 0.260 0.271 0.268 0.234 0.257 0.285 0.247
36.87 22.62 67.38 28.07 61.93 16.26 73.74 46.40 43.60 71.08 18.92 12.68 11.32
Translation state T, 5 .” 0.5 0.5 0 0 0 0.5 0 0.5 0
0.112 -0.192 - 0.565 0.895 - 0.763 1.230 -0.89i -0.910 -0.196 - 0.242
i 0 0.5 0
1.iMo -0.338
by the choice of r,. However, the absolute values of the boundary energies may vary by + 40:; of the mean when different cut-off radii are used. 3. RESLXTS Table 1 contains the crystallographic data character&g the lowest energy configuration for each orientation investigated. the final translation state, T, the area of the repeating unit in the boundary and the local increase in separation at the boundary of the atom planes parallel to the boundary. A state of relative rigid body transiation can be defmed in an infinite number of ways, but descriptions, Ti, which lie in the plane of a boundary have particular advantages [14]. In this work, Ti, represents the movement of the lower grain with respect to the upper away from the coincidence position (the coincidence atoms “belonging” to the upper crystal at that position). The coordinate system used for defining Ti is orthogonal with the s-axis pointing in Figs. 1-8 into the page parallel to the tilt axis [OOl] and the positive y-axis horizontally to the right. Figures 1-8 show the lowest energy relaxed structures for the (9101, (710), (310). (5201, (7301, (210), (230) and (WI) boundaries projected onto (Wl). This sequence corresponds to inTable 2. Calculated values of grain bound-
ary energy for Z = .Sboundaries (J m-‘) i’
x
0.161 0.366 0.372
(310) 0.5 0 0
0.207 0.211 0.306
OS 0.5 OS
Ti Y -0.192 -0.316 0.559
(120) 0.112 0.427 0.671
-0.827 -0.701
Area of repeating unit
Expansion 0.16 0.16 0.21 0.20 0.10 0.23 0.13 0.15 0.24 0.21
I.118 I.%1 2.550 I.503
2.061 2.915
3.536 2.500 3.808 1.693 4.301 3.M1 4.528 3.201
0.77 0.23 0.26 0.23
creasing values of 8, where Q is the angle between (100) directions in the two crystals which are equally inclined to the boundary plane. Figure 9 is a plot of the lowest energy value. 7. for each boundary studied, against 8. Three distinct relaxed boundary structures were found for both the (210) and (310) cases: the energies and translation states of these are shown in Table 2. In addition it is noted that symmetry degenerate forms of all these (210) and (310) structures were obtained. Symmetry degenerate forms of a given boundary structure are identical structures that can be recognised as such by relating the two by some symmetry operation of the associated CSL. Moreover, their corresponding Ti are also related by that CSL symmetry axis [ 143. The general results of the present work were as folIows: (a) Ail minimum energy structures involved translations away from the coincidence position which resulted in there being no atoms at coincidence sites. Translations away from starting con~guration~ were aiways in a direction perpendicular to the tilt axis. The final translation states, T,, of minimum energy Table 3. .;i.jrta calculated in the present work and in the quoted references; * gold; t atuminium Symmetrical tilt boundary piane 310 310 410 510 610 710 910 230 350 250 540
Weins, Gleiter and Chalmers* 1.00 0.82 0.9 1 0.72
0.72 1.12 1.25 -
Hasson er al.
Present aorkf
1.00 0.95 1.09 t.03 1.03 1.08 0.90 0.94 1.06 1.09 0.90
1.00 0.78 0.94 I.16 1.24 1.13 1.37 1.19
1.16 I.25 1.19
178
SMITH. VITEK
.\SD
PO%D:
COMPL’TER SIML:LATION
configurations were very specitic and in several cases identical final states were obtained from different starting positions. (b) For some boundaries more than one low energy structure exists, e.g. (210). as shown in Table 2. The translations associated with low energy structures appear in general to lead to groups of atoms at the boundary which approach close packing. (c) The calculated energy did not depend monotonically on Z or the boundary period, e.g. ;:A1O(Z= 17) is less than ~siO(E = 13). and the primitive period in the (210) diagonal boundary is smaller than that in its edge counterpart. the (310) boundary, yet 7310
<
i’?lO.
(d) There was a general trend that as the local expansion at the boundary increased. so did the energy. It has been suggested [18] that long period coincidence boundaries might consist of mixtures of the strained ‘structural units’ of short period boundaries. However, as mentioned above, symmetrical boundaries of a given coincidence system are not characterised by a unique structural unit. Moreover, the translation states of favourable short period boundary structures are highly specific and, in general, mixtures of such units may not have low energies because their translation states are not compatible. This difficulty with the simple mixture idea is illustrated in the present work by the structures of the (20) and (730) boundaries which have 0 values intermediate between the short period (310) and (210) boundaries. According to the mixture model. the (520) boundary may be regarded as consisting of equal numbers of strained (310) and (210) units. Figure 4 shows that the lowest energy (520) boundary does have this general form and that the (310) and (210) strained units resemble the units of the lowest energy (310) and (210) structures in Figs. 3 and 6. This has been schematically indicated in Fig. 4(b). On the other hand, the (730) boundary (Figs. 5 a and b) is not made up of two (210) units for each (310) unit. In Fig. 5(b) it has been indicated that the (310) ‘half unit’ of the lower grain adjoins two (210) ‘half units’ of the upper grain, so that the boundary cannot be regarded as a mixture of any of the possible (210) or (310) units. The occurrence of cusps in a 7 vs 0 plot for grain boundaries is a consequence of the presence of dislocations when there is a slight deviation from a coincidence orientation rather than a feature of the 7 vs 8 curve resulting from interpolation between calculated values of 7 for periodic boundaries (see also Ref. 19). The nature of the energy cusp around a coincidence orientation depends on the magnitude of the Burgers vectors of the gram boundary dislocations which accommodate the deviation from the coincidence orientation. The grain boundary energy rises less steeply if primitive DSC dislocations are involved since for a fixed deviation from coincidence the extent of the dislocations long range stress field is proportional to lb/. In addition the cusps should become.
OF HIGH .ANGLE BOL’SDARIES
wider as 1 increases and /b j decreases. Figure 9 shows the energy of the boundaries investigated in this work together with the shape of cusps at (a) 6 = 0’ and 90’ and (b) near the (910) and (c) (210) boundaries. In each case the shape of the cusp was calculated on the basis of the Read-Shockley [ 171 formula but for cases (a) and (b) the dislocations were taken to be crystal lattice dislocations whilst for case (c) the dislocations were assumed to have l/l0 (310) Burgers vectors (i.e. primitive vectors of the DSC lattice) as suggested by transmission electron microscopy [l]. The core energies of all dislocations were taken to be the same as a first approximation. For the cases investigated the core energy was about lo:,; or less of the total energy. This calculation gives an upper limit on the 7 for an off coincidence boundary since any energy lowering interaction between the dislocation network and the boundary is neglected. 4. DISCLSSIOS The calculated energy values are estimated to be correct absolutely (at 0 K) to f 409, and the relative energy values to be somewhat more reliable. More than usual confidence is attached to the validity of the potential used and the relaxation algorithm since there is excellent agreement between the theoretical predictions and experimental observations of the rigid lattice translation for the particular case of the E = 3 boundary in aluminium [ 141. However no correlation has been attempted with experimental measurements of grain boundary free energy owing to the difficulty of making a reliable calculation of the entropy concerned. In all cases investigated the calculated local expansion normal to the boundary plane was substantial and varied between 0.12~ and 0.30~. There was a general trend for small expansions to be associated with low energy; again in general, the expansions were greatest for high X orientations and these had the highest calculated energies. However. it is emphasised that no volume dependent terms enter into the present calculation and that no overall volume change is permitted. As mentioned earlier, all lowest energy structures contained groups of atoms at the boundary which approached close packing. We call such clusters interlocking groups since they are made up of atoms from both grains and appear to ‘dowel’ the two grains together. Two types of interlocking group were found. The first type occurred in boundaries which might be classified as low angle boundaries, i.e. 15’ 2 0 2 75’. These groups consisted of clearly recognisable regions of strained single crystal. The second type of interlocking group occurred in all the lowest energy boundaries with intermediate values of 0 which might be designated as the high angle regime. These groups contain three atoms which were located at or very close to the vertices of an equilateral triangle which has sides comparable with the perfect
SMITH.
VITEK
.x-a
POND:
COMPUTER
SIMLL.\TION OF HIGH .ANGLE BOUNDARIES
crystal nearest neighbour spacing. Thub. thy: atom arrangement approaches the close packing of hard spheres. The occurrence of these interlocking groups is discussed below with reference to Figs. 1-S. The (910) and (540) boundaries (Figs. 1 and S) can be classified as low angle and their structure might be expected to be as discussed by Read and Shockley [12]. According to this model such boundaries would be made up of an array of equally spaced perfect lattice dislocations separated by patches of strained single crystal. The (540) boundary clearly corresponds to this situation and the position? of the a,2 (110) dislocations at the terminations of pairs of extra [X0: planes have been indicated in Fig. S(b). The interlocking groups in this case are ribbons of strained single crystal betueen the dislocations in an orientation which is intermediate between those of the adjacent crystals. In the case of the (910) boundary. the Read-Shockley model predicts a combination of a’2[110] and n/Z [lTO] edge dislocations spaced _ p:‘2apart. where p is the CSL period. i.e. ai2 [190]. However. careful inspection of Fig. l(a) will show that the lattice dislocations have coalesced so that their cores are overlapping in pairs. The spacing of the combined dislocations is p as indicated in Fig. l(b). Again. ths interlocking groups are regions of single crystal in intermediate orientation. This result is interesting because it suggests that even within the low angle regime. boundary structures may gradually change from Read-Shockley like structures to alternative structures as 0 increases. The authors note that the (910) structure may be equally well described as an array of n 2 [loo] imperfect crystal dislocations with a spacing p/2. Such an array means that the boundary is an area of alternating ribbons, approximately !p]j7 wide. of strained single crystal and a;2 [lOO] stacking faults as indicated in Fig. l(c). In view of this interpretation. it is not clear whether the (910) boundary should necessarily have an energy cTose to that predicted by the Read-Shockley equation. Moreover since this equation contains constants
I
-I
Stacking (c)
fault
I
-I
-I
179
&rich cannot be measured directly. the curves shown in Fig. 9 should only be regarded as describing the general form of the ;‘ vs 0 relationship for low angle boundaries. On the other hand. it seems reasonable to adjust the constants of this equation to fit the (540) calculated energy. The (7 10) boundary (Fig. 2) would normally- be considered to be just outside of the low angle regime. Alternating ribbons of a;? [lOO] stacking fault and single crystal can still be discerned as for (910). but these have become very narrow involving only about four atoms. It is interesting to note however that a triangle structure which is characteristic of all the high angle boundaries in this work is forming at the region labelled B adjacent to the area of strained single crystal labelled .A (Fig. 2). Thus. although the idea of high and low angle regimes is useful. the present results indicate that there is a gradual transition range of Q amounting perhaps to 5’ or more. separating structures which are clearly Read-Shockley type and high angle boundaries as characterised by the presence of triangular interlocking groups. The particularly low energy of the (310) boundary presumably results from a combination of ths high density of the triangle group (as at the region marked C in Fig. 3) and a relatively small amount of energy stored in the boundary strain field. Similar triangle structures are evident in the (520) and (730) boundaries (Figs. 4 and 5). The structure of the (210) boundary appears to consist entirely of triangle structures in both layers of the ABAB stacking sequence. However, these do not approach the equilateral triangle situation as closely as those of the (310) boundary which may explain in part why ;J,~,~, < ~,~t~)- The triangle group is still present in the (230) boundary (Fig. 7) but as 0 increases further strips of single crystal again become distinguishable and the triangle structure gradually less prominent. The existence of interlocking groups appears to be very important and probably accounts for the fact that translation constitutes the major component to
--I
Single
crystal
Fig. 1. 1910)boundary. la) Calculated structure. (b) The ‘coalesced’ a,2 (110) dislocation structure. (c) The alternating ribbon a,2 (110; stacking fault structure.
4.0
SMITW, VITEK AYD POND:
COIMPUTER SIMULATION
OF HIGH ANGLE BOl_KXDARIES
Fig. 2. (710) boundary.
Fig. 3. (310) boundary.
+ A A+ +A (a)
+A + A++AA+ A +A
+A
“_+A *
f A
+A
“,t + “++A +‘+ “,b4 “,6 + "+hb4
CA 4A +’ + ch A CA +A ’
+A
“,=9‘+ +h
+*
A
+
A
+
A+
A
A
+
A
“,
+“c +
b
CA A
A +
CA
A,6 +A
+AA+ A++AA+ ‘+ ‘,‘: &
“, .j. A A+‘,%
+
“+
A
+
.tA+
(f-4
Fig, 4. (520) boundary. (a) Calculated structure. (b) Schematic representation of the boundary structure as a mixture of (210) and (310) structural units.
Fig. 5. (7301 boundary. !a) Calcuiated structure. (b) Schematic repressmrarion of the boundary structure as a mixture of (210) and (310) structural units.
Fig, 6. (210) boundary.
Fig. 7. (230) boundary..
482
SMITH, VI-I-JzKAND
0
POND:
I
I
10
COMPUTER SIMULATION OF HIGH ANGLE BOUNDARIES
I 20
I 30
I 40
e
I 50
I 60
I 70
I 60
I
so
Fig. 9. Plot of the minimum calculated energy for [OOI] symmetrical tilt boundaries in coincidence orientations with .E < 41. The cusps at t) = 0 and 90’ and at the Z = 41 orientation (12.65’) were calculated for the purposes of illustration on the basis that lattice dislocations accommodated the misor~entation from the exact coincidence orientation whilst for the E = 5. 36.87” cusp primitive DSC lattice dislocations with b = l/i0 (3 10) were considered.
boundary relaxation. Furthermore. the specific translations associated with the various low energy configurations for any one boundary can be understood as translations which lead to interlocking groups at different positions in the boundary period. The absence of a monotonic relationship between E: or the boundary period and p, is consistent with other calculations and experiments [5] although at variance with earlier formulations of the CSL theory [ZO]. This result is not surprising if the occurrence of interlocking groups is an important factor in low energy structures. Figure 4 illustrates that the number of triangle groups in fully relaxed boundary is not simply related to the unrelaxed period. However neither is there a simple relationship between the number of interlocking groups and 7 which is consistent with the fact that local relaxation of the atoms near the boundary, i.e. the boundary strain field is also an important energy factor. Table 3 contains relative values of the calculated energy for symmetrical (001) tilt boundaries normalised to ~czio, according to the calculations of Hasson et al. [6] and of Weins et at. [7] and the present workers. There are substantial differences between the three sets of results but all show a notable lack of agreement with the predictions of purely geometrical models. The differences between the calculated energies and structures reflect the differing relaxation algorithms and potentials used. The relaxed (210) structure calculated in Ref. 6 is very similar to that obtained in the present work. (compare Figs. 6 and 10 from Ref. 5) but the (310) structures obtained in the two studies are si~i~cantIy different because Hasson et al. [6] predict a structure
which has mirror symmetry about their boundary plane (compare Figs. 3 and S from Ref. 6). In the present work the loss of mirror symmetry is apparently the consequence of the relative displacement of the two grains parallel to the boundary. An interesting feature of all the relaxed structures calculated in the present work is that the pr~om~ant relaxation process seems to be translation as was proposed by Weins et a[. [Tf and local expansion. The atoms at the boundary adopt positions characteristic of either grain, giving a very narrow boundary, rather than compromise positions such as are postulated by the transition lattice theory [21] and sometimes found in the results of Ref. 6).
The calculated energy of symmetrical (001) tilt boundaries in aluminium depends on the strain field of the boundary and extent to which the .perfect crystal coordination %nd near neighbour spacing are conserved in interlocking groups of atoms in the boundary rather than geometrical factors describing the boundary periodicity. Some but not all long period boundaries intermediate in orientation between short period boundaries can be described in terms of mixtures of strained units characteristic of the short period boundaries. The calculated energy rises only slowly as Z increases. In many cases more than one low energy structure exists for a given grain rotation and boundary orientation. AcknowteJgenrenrs-Tfiis work was performed in part at Oxford University and in part at Central Electricity
SMITH. VITEK
,350
POND:
COMPL’TER SIMULATIOS
Research Laboratones. and IS publtshed with the permission of the Central Electricity Generating Board. The authors are grateful to the Armourers gi Brasiers Company (DAS) and the Science Research Council IRCP) for financial support, to Professors J. W. Christian. FRS and M. F. Ashby for useful discussions and to Professor Sir Peter Hirsch FRS. for encouragement and provision of facilities. The authors also wish to thank Dr. Roger Taylor for his help wtth the use of the potenttal for alummtum. REFERENCES 1. R. W. Ball&i, Y. Komem and T. Schober. Surf: Sci. 31, 68 (1972). 7 MM.McLean. J. mater. Sci. 8, 571 (1973). ;: G. Dimou and K. T. Aust. .-lcrcr 41rt. 22. 27 (1971). 4. G. Hasson and C. Goux. Scripra .ller. 5. 889 (1971). 5. G. Herrmann, H. Gleiter and G. BHro. .dcta Ma. 24. 353 (1976). G. Hasson. J. Y. Boos. I. Herbeuval. M. Biscondi and C. Gout. Surf: Sci. 31, 115 (1973). M. Weins. H. Gleiter and B. Chalmers. J. appl. PIiys. 42. 2639 (1971). R. E. Dahl. J. R. Beeler and R. 0. Bourquin. Barrrlle Colloquitrm on Interaromic Potentials and Simuhrion of
OF HIGH ANGLE BOUNDARIES
4%
L~rricr Drfecrs (edited b! P. C. Gehlen. J. R. Beeler and R. I. Jaffee). p. 673. Plenum Press. Neu York j 19-21. 9. R. .LI. J. Cotterill. T. Leffers and H. Lilholt. PM. .Lfq. 30. 265 1197-t). 10. ht. Weins. Surf: Sci. 31. 138 (1973). 11. R. J. Harrison. G. A. Bruggeman and CT. H. Bishop. in Grbn Boundary Srrucrure and Proprrries (edited by Chadwick and Smith). Academic Press. London I 1976). I’. W. T. Read and W. Shocklev. Phvs. Rrr. 78. 275 (1950). 13. R. C. Pond. Con. met&. 0. 13. 33 (1971). L-1. R. C. Pond and V. Vitek. submitted to Proc. R. Sot. (1977). 15. V. Vitek. D. A. Smith and R. C. Pond. Comprtr
V. VITEIi+
HIGH
and R. C. POND*
*Department of Metallurgy and Science and Materials, University of Oxford, Parks Road. Oxford. England. and +Materials Division. Central Electricity Research Laboratories. Kelvin Avenue, Leatherhead. Surrey. England (Rrcrired
27 l~unrv
1976; in rrrisedjmn
33 Arqrsr
1976)
Abstract-The internal energy of symmetrical high angle tilt boundaries with an (001) axis has been calculated using a potential for aluminium. The computer program used permits relaxations involving rigid translations parallel and perpendicular to the boundary as well as individual atom rearrangements. subject to overall conservation of volume. The calculated energy waj found to vary by a factor of about two for coincidence boundaries with 5 $ Z C 41 but did not depend simply on the size of the boundary period. The condition for low energy configurations appears to require a translation of the adjacent grains away from the coincidence position to a state where characteristic interlocking groups of closely packed atoms occur in the boundaries. R&urn&-On a calcult l’energie interne de joints de grains de Rexion symitriques de forte d&orientation et d’axe (001). B l’aide d’un potentiel pour l’aluminium. Le programme utilisi permettait les relaxations par translations rigides parall~lement et perpendiculairement au joint. ainsi que des riarrangements atomiques individuels. a condition de conserver le volume total. On a trouvt que l’energie calculee variait d’un facteur deux environ pour des joints de coincidence avec 5 C 1 C 41, mais qu’elle ne dCpendait pas simplement de la grandeur de la periode du joint. Les configurations de faible tnergie semblaient necessiter une translation des grains adjacents i partir de la position de cdincidence vers un &at OC I’on trouve dans les joints des groups compacts CaractCristiques d’atomes. Zusammenfassung-Die innere Energie symmetrischer GroBwmkelknickgrenzen mit einer (001)-Achse wurde mit einem Potential fir Aluminium berechnet. Das benutzte Rechnerprogramm erlaubte Relaxationen. die starre Translationen parallel und senkrecht zur Grenze und die Umlagerun! einzelner Atome umfassen. entsprechend der Erhaltung des Gesamtvolumens. Die berechncte Energle schwankte urn einen Faktor von etwa zwei ftir Koinzidenzgrenzen mit 5 $ Z < At, hing aber nicht allein von der Gral3e der Grenzenperiode ab. Bedingung fiir eine niederenergetische Konfiguration scheint die Translation benachbarter KGrner weg von der Koinzidenzlage in eine Lage. in der charakteristische ineinandergreifende Gruppen dicht gepackter Atome in den Grenzen entstehen, zu erfordern.
1. INTRODUCTIOS
Recent experimental results from electron microscopy suggest that high angle grain boundary structure depends sensitively on the orientation of the adjacent grains [i]. In turn it is expected that the energy of high angle grain boundaries will also depend on these boundary parameters. However, it has proved difficult to measure even relative values of grain boundary energy (i’)for a range of boundaries in order to investigate the factors which govern ;:. McLean [Z] has shown that the free energy for 1: = 3 twins in copper depends on the boundary plane orientation and that there is a definite minimum for the coherent (111) plane. It is observed that the I: = 5 (33 boundary in lead. and this and other coincidence boundaries [4] in aluminium have lower than average energies. These results are difficult to interpret fully since whilst the grain rotations were carefully controlled, the grain boundary planes were : E is defined as the ratio of the volume of the coincidence site lattice unit cell to that of the crystal lattice unit cell.
not. In the case of the work by Hasson and Goux [4] on tilt boundaries the experimental measurements were corrected by assuming a value for d-//d+ where 4 is an azimuthal angle between the actual boundary plane and the plane of a symmetric boundary. In addition there is recent experimental evidence that a wide range of coincidence boundaries often with high values of Z, have low energies in copper and silver [j]. In view of the experimental difficulties involved, a number of attempts to study 7 have been made using computer simulation [6-lo]. Haason et a/ [6] use a LMorse potential and do not impose a condition of volume conservation which introduces an arbitrary assumption of zero volume dependent energy terms (see Section 2). Calculations in Ref. 9 are confined to a two dimensional model and in Ref. 8 the structure of periodic boundaries is not computed. The procedure used in [lo] is to minimize 7 as a function of translational relaxation alone and then separately to allow individual atomic relaxations from the structure which had the minimum energy after rigid translation. This procedure does not necessarily lead to minimum energy structures as the
476
SMITH, VITEK
AND
POND:
COMPUTER
SI.MULATION OF HIGH ANGLE BOUNDARIES
two relaxations are not fully coupled. None of these calculations has incorporated all of the following desirable features El 11: (a) use of three dimensional “crystals’.; (b) allowing the two crystals translational freedom; (c) use of a realistic potential applicable to metals and avoidance of arbitrary assumptions concerning volume dependent energy terms; (d) including explicitly more than one repeating unit in the boundary block. Conditions (a). (b) and (c) were satisfied in our calculations, but so far we have assumed that the periodicity of the unrelaxed boundary is conserved, although the relaxation algorithm does not depend fundamentally on this assumption. Computer calculations have been done For all periodic symmetrical (001) tilt boundaries with C < 50 and these results, in principle can be extended to the full range of rotation angles by adding an energy representative of an appropriate superimposed low angle type boundary calculated according to the Read-Shockley [12-J equation. Owing to the crystal symmetry, for each value of Z there are two symmetrical tilt boundaries with different periodicities and both have been investigated. These correspond respectively to the boundary plane occupying the diagonal and edge faces, containing [QOlJ, of the body centred tetragonal unit ceils of the coincidence site lattices formed by rotations about [OOl] [13]; both non-zero indices are odd for the edge type while one is even and the other odd for the diagonal type. 2. METHOD OF CALCULATION ~TE~TO_MIC FORCES
AND
The block of f.c.c. lattice containing a chosen periodic boundary was constructed in the computer and the configuration subsequently relaxed so as to obtain a minimum internal energy. The bicrystal used was two dimensional from the computational point of view as the periodicity of the lattice in the direction of the tilt axis was used. The dimension of the block in the direction perpendicular to the tilt axis and in the boundary plane. and also that in the direction perpendicular to both the boundary and the tilt axis has been chosen as at least twice the cut-off radius of the interatomic potential {see below). The relaxation was carried out using a gradient method. Displacements both parallel and perpendicular to the boundary were allowed but the total volume of the bicrystal was kept constant. This is consistent with the assumption that the interatomic interaction is described by central Forces only; otherwise the volume dependence of the energy would have to be considered. Therefore, if an expansion or contraction took place in the boundary, the corresponding volume change was automatically compensated by a compression or expansion which occurred as an elastic deformation extending Far away from the boundary. However, this long range compensating elastic field is imposed by the calculations. An alter-
native procedure would be to include a volume dependent energy term. In the linear approx~ation this is inadequate and would not change our results substantially. whilst a non-linear approach would involve arbitrary assumptions comparable in significance with our assumption that volume is conserved. An important aspect of the present calculation is that in the procedure for relaxations parallel to the boundary. relative shifts of one grain as well as displacement of individual atomic planes with respect to the other are allowed (for more details see Refs. 14 and 15). An infinite number of boundary structures are geometrically possible for a given crystal rotation and boundary orientation; these correspond to the infinite number of states of relative translation. Thus in order to find the minimum energy structure for a given boundary a scheme of incremental changes of translation state, covering the whole range of possible translation. was necessary. The procedure adopted was to begin with the coincidence structure and allow this to relax; the magnitude of the maximum subsequent translation in the plane of the boundary did not usually exceed 0.6~ (a is the lattice parameter). Next, the first plane of atoms parahel to the boundary was removed from the upper crystal and this crystal shifted by one inte~lanar spacing in order to keep the total volume constant. This structure was then taken as the next starting point for relaxation. In this way, up to six planes were removed successively from the upper and lower _mi.ns before relaxation was carried out. This corresponds to starting the relaxation from a number of possible translational states. As a further check the starting position was also varied by making a shift of u/7 along the tilt axis. This procedure ensures that the symmetry of the continuous plane configuration perpendicular to the rotation axis does not impose structures which are not true minima. Thus we are able to find low energy structures but it is still possible that some low energy configurations may not have been discovered since no relaxation procedure can ensure that all minima of a function have been found. The interaction between the atoms was described by a central Force potential for aluminium which was constructed by Dagens, Rasolt and Taylor [16-j on the basis of a pseudopotential theory. This potential was shown to give a value of O.lOjJm-’ for the intrinsic stacking fault energy and O.O%Jm-’ for the twin boundary energy[lQ Since this potential is long range and osciilates as cos @k&/r3 (kf is the Fermi vector) For large P, a cut-off radius r, has to be chosen. In the present work r, = 1.6~ was used, i.e. the potential was truncated between fifth and sixth nearest neighbours. The effects of the choice of the cut-off radius on the calculated grain boundary structures and energies were studied in more detail by Pond and Vitek [14]. They concluded that the structure of the boundary, i.e. the atom positions and relative translations of the grains are practically uninfluenced
SMITH. VITEK
.LVD
POND:
COMPUTER
SIMULATION
OF HIGH .L\XiLE BOUNDARIES
477
Table 1. Crystallographic data. translation, Ti and local expansion for the lowest energy configurations of the grain boundaries studied. Ti components x and y, and the expansion are in units of the lattice parameter n: areas of the repeating units are in the units of u’
I 5
13 17 25 29 37 41
0
Plane
7 (Jm-?I
53.13
310 310 510 320 410 330 710 430 730 520 750 610 910 540
0.208 0.161 0.241 0.242 0.195 0.260 0.235 0.260 0.271 0.268 0.234 0.257 0.285 0.247
36.87 22.62 67.38 28.07 61.93 16.26 73.74 46.40 43.60 71.08 18.92 12.68 11.32
Translation state T, 5 .” 0.5 0.5 0 0 0 0.5 0 0.5 0
0.112 -0.192 - 0.565 0.895 - 0.763 1.230 -0.89i -0.910 -0.196 - 0.242
i 0 0.5 0
1.iMo -0.338
by the choice of r,. However, the absolute values of the boundary energies may vary by + 40:; of the mean when different cut-off radii are used. 3. RESLXTS Table 1 contains the crystallographic data character&g the lowest energy configuration for each orientation investigated. the final translation state, T, the area of the repeating unit in the boundary and the local increase in separation at the boundary of the atom planes parallel to the boundary. A state of relative rigid body transiation can be defmed in an infinite number of ways, but descriptions, Ti, which lie in the plane of a boundary have particular advantages [14]. In this work, Ti, represents the movement of the lower grain with respect to the upper away from the coincidence position (the coincidence atoms “belonging” to the upper crystal at that position). The coordinate system used for defining Ti is orthogonal with the s-axis pointing in Figs. 1-8 into the page parallel to the tilt axis [OOl] and the positive y-axis horizontally to the right. Figures 1-8 show the lowest energy relaxed structures for the (9101, (710), (310). (5201, (7301, (210), (230) and (WI) boundaries projected onto (Wl). This sequence corresponds to inTable 2. Calculated values of grain bound-
ary energy for Z = .Sboundaries (J m-‘) i’
x
0.161 0.366 0.372
(310) 0.5 0 0
0.207 0.211 0.306
OS 0.5 OS
Ti Y -0.192 -0.316 0.559
(120) 0.112 0.427 0.671
-0.827 -0.701
Area of repeating unit
Expansion 0.16 0.16 0.21 0.20 0.10 0.23 0.13 0.15 0.24 0.21
I.118 I.%1 2.550 I.503
2.061 2.915
3.536 2.500 3.808 1.693 4.301 3.M1 4.528 3.201
0.77 0.23 0.26 0.23
creasing values of 8, where Q is the angle between (100) directions in the two crystals which are equally inclined to the boundary plane. Figure 9 is a plot of the lowest energy value. 7. for each boundary studied, against 8. Three distinct relaxed boundary structures were found for both the (210) and (310) cases: the energies and translation states of these are shown in Table 2. In addition it is noted that symmetry degenerate forms of all these (210) and (310) structures were obtained. Symmetry degenerate forms of a given boundary structure are identical structures that can be recognised as such by relating the two by some symmetry operation of the associated CSL. Moreover, their corresponding Ti are also related by that CSL symmetry axis [ 143. The general results of the present work were as folIows: (a) Ail minimum energy structures involved translations away from the coincidence position which resulted in there being no atoms at coincidence sites. Translations away from starting con~guration~ were aiways in a direction perpendicular to the tilt axis. The final translation states, T,, of minimum energy Table 3. .;i.jrta calculated in the present work and in the quoted references; * gold; t atuminium Symmetrical tilt boundary piane 310 310 410 510 610 710 910 230 350 250 540
Weins, Gleiter and Chalmers* 1.00 0.82 0.9 1 0.72
0.72 1.12 1.25 -
Hasson er al.
Present aorkf
1.00 0.95 1.09 t.03 1.03 1.08 0.90 0.94 1.06 1.09 0.90
1.00 0.78 0.94 I.16 1.24 1.13 1.37 1.19
1.16 I.25 1.19
178
SMITH. VITEK
.\SD
PO%D:
COMPL’TER SIML:LATION
configurations were very specitic and in several cases identical final states were obtained from different starting positions. (b) For some boundaries more than one low energy structure exists, e.g. (210). as shown in Table 2. The translations associated with low energy structures appear in general to lead to groups of atoms at the boundary which approach close packing. (c) The calculated energy did not depend monotonically on Z or the boundary period, e.g. ;:A1O(Z= 17) is less than ~siO(E = 13). and the primitive period in the (210) diagonal boundary is smaller than that in its edge counterpart. the (310) boundary, yet 7310
<
i’?lO.
(d) There was a general trend that as the local expansion at the boundary increased. so did the energy. It has been suggested [18] that long period coincidence boundaries might consist of mixtures of the strained ‘structural units’ of short period boundaries. However, as mentioned above, symmetrical boundaries of a given coincidence system are not characterised by a unique structural unit. Moreover, the translation states of favourable short period boundary structures are highly specific and, in general, mixtures of such units may not have low energies because their translation states are not compatible. This difficulty with the simple mixture idea is illustrated in the present work by the structures of the (20) and (730) boundaries which have 0 values intermediate between the short period (310) and (210) boundaries. According to the mixture model. the (520) boundary may be regarded as consisting of equal numbers of strained (310) and (210) units. Figure 4 shows that the lowest energy (520) boundary does have this general form and that the (310) and (210) strained units resemble the units of the lowest energy (310) and (210) structures in Figs. 3 and 6. This has been schematically indicated in Fig. 4(b). On the other hand, the (730) boundary (Figs. 5 a and b) is not made up of two (210) units for each (310) unit. In Fig. 5(b) it has been indicated that the (310) ‘half unit’ of the lower grain adjoins two (210) ‘half units’ of the upper grain, so that the boundary cannot be regarded as a mixture of any of the possible (210) or (310) units. The occurrence of cusps in a 7 vs 0 plot for grain boundaries is a consequence of the presence of dislocations when there is a slight deviation from a coincidence orientation rather than a feature of the 7 vs 8 curve resulting from interpolation between calculated values of 7 for periodic boundaries (see also Ref. 19). The nature of the energy cusp around a coincidence orientation depends on the magnitude of the Burgers vectors of the gram boundary dislocations which accommodate the deviation from the coincidence orientation. The grain boundary energy rises less steeply if primitive DSC dislocations are involved since for a fixed deviation from coincidence the extent of the dislocations long range stress field is proportional to lb/. In addition the cusps should become.
OF HIGH .ANGLE BOL’SDARIES
wider as 1 increases and /b j decreases. Figure 9 shows the energy of the boundaries investigated in this work together with the shape of cusps at (a) 6 = 0’ and 90’ and (b) near the (910) and (c) (210) boundaries. In each case the shape of the cusp was calculated on the basis of the Read-Shockley [ 171 formula but for cases (a) and (b) the dislocations were taken to be crystal lattice dislocations whilst for case (c) the dislocations were assumed to have l/l0 (310) Burgers vectors (i.e. primitive vectors of the DSC lattice) as suggested by transmission electron microscopy [l]. The core energies of all dislocations were taken to be the same as a first approximation. For the cases investigated the core energy was about lo:,; or less of the total energy. This calculation gives an upper limit on the 7 for an off coincidence boundary since any energy lowering interaction between the dislocation network and the boundary is neglected. 4. DISCLSSIOS The calculated energy values are estimated to be correct absolutely (at 0 K) to f 409, and the relative energy values to be somewhat more reliable. More than usual confidence is attached to the validity of the potential used and the relaxation algorithm since there is excellent agreement between the theoretical predictions and experimental observations of the rigid lattice translation for the particular case of the E = 3 boundary in aluminium [ 141. However no correlation has been attempted with experimental measurements of grain boundary free energy owing to the difficulty of making a reliable calculation of the entropy concerned. In all cases investigated the calculated local expansion normal to the boundary plane was substantial and varied between 0.12~ and 0.30~. There was a general trend for small expansions to be associated with low energy; again in general, the expansions were greatest for high X orientations and these had the highest calculated energies. However. it is emphasised that no volume dependent terms enter into the present calculation and that no overall volume change is permitted. As mentioned earlier, all lowest energy structures contained groups of atoms at the boundary which approached close packing. We call such clusters interlocking groups since they are made up of atoms from both grains and appear to ‘dowel’ the two grains together. Two types of interlocking group were found. The first type occurred in boundaries which might be classified as low angle boundaries, i.e. 15’ 2 0 2 75’. These groups consisted of clearly recognisable regions of strained single crystal. The second type of interlocking group occurred in all the lowest energy boundaries with intermediate values of 0 which might be designated as the high angle regime. These groups contain three atoms which were located at or very close to the vertices of an equilateral triangle which has sides comparable with the perfect
SMITH.
VITEK
.x-a
POND:
COMPUTER
SIMLL.\TION OF HIGH .ANGLE BOUNDARIES
crystal nearest neighbour spacing. Thub. thy: atom arrangement approaches the close packing of hard spheres. The occurrence of these interlocking groups is discussed below with reference to Figs. 1-S. The (910) and (540) boundaries (Figs. 1 and S) can be classified as low angle and their structure might be expected to be as discussed by Read and Shockley [12]. According to this model such boundaries would be made up of an array of equally spaced perfect lattice dislocations separated by patches of strained single crystal. The (540) boundary clearly corresponds to this situation and the position? of the a,2 (110) dislocations at the terminations of pairs of extra [X0: planes have been indicated in Fig. S(b). The interlocking groups in this case are ribbons of strained single crystal betueen the dislocations in an orientation which is intermediate between those of the adjacent crystals. In the case of the (910) boundary. the Read-Shockley model predicts a combination of a’2[110] and n/Z [lTO] edge dislocations spaced _ p:‘2apart. where p is the CSL period. i.e. ai2 [190]. However. careful inspection of Fig. l(a) will show that the lattice dislocations have coalesced so that their cores are overlapping in pairs. The spacing of the combined dislocations is p as indicated in Fig. l(b). Again. ths interlocking groups are regions of single crystal in intermediate orientation. This result is interesting because it suggests that even within the low angle regime. boundary structures may gradually change from Read-Shockley like structures to alternative structures as 0 increases. The authors note that the (910) structure may be equally well described as an array of n 2 [loo] imperfect crystal dislocations with a spacing p/2. Such an array means that the boundary is an area of alternating ribbons, approximately !p]j7 wide. of strained single crystal and a;2 [lOO] stacking faults as indicated in Fig. l(c). In view of this interpretation. it is not clear whether the (910) boundary should necessarily have an energy cTose to that predicted by the Read-Shockley equation. Moreover since this equation contains constants
I
-I
Stacking (c)
fault
I
-I
-I
179
&rich cannot be measured directly. the curves shown in Fig. 9 should only be regarded as describing the general form of the ;‘ vs 0 relationship for low angle boundaries. On the other hand. it seems reasonable to adjust the constants of this equation to fit the (540) calculated energy. The (7 10) boundary (Fig. 2) would normally- be considered to be just outside of the low angle regime. Alternating ribbons of a;? [lOO] stacking fault and single crystal can still be discerned as for (910). but these have become very narrow involving only about four atoms. It is interesting to note however that a triangle structure which is characteristic of all the high angle boundaries in this work is forming at the region labelled B adjacent to the area of strained single crystal labelled .A (Fig. 2). Thus. although the idea of high and low angle regimes is useful. the present results indicate that there is a gradual transition range of Q amounting perhaps to 5’ or more. separating structures which are clearly Read-Shockley type and high angle boundaries as characterised by the presence of triangular interlocking groups. The particularly low energy of the (310) boundary presumably results from a combination of ths high density of the triangle group (as at the region marked C in Fig. 3) and a relatively small amount of energy stored in the boundary strain field. Similar triangle structures are evident in the (520) and (730) boundaries (Figs. 4 and 5). The structure of the (210) boundary appears to consist entirely of triangle structures in both layers of the ABAB stacking sequence. However, these do not approach the equilateral triangle situation as closely as those of the (310) boundary which may explain in part why ;J,~,~, < ~,~t~)- The triangle group is still present in the (230) boundary (Fig. 7) but as 0 increases further strips of single crystal again become distinguishable and the triangle structure gradually less prominent. The existence of interlocking groups appears to be very important and probably accounts for the fact that translation constitutes the major component to
--I
Single
crystal
Fig. 1. 1910)boundary. la) Calculated structure. (b) The ‘coalesced’ a,2 (110) dislocation structure. (c) The alternating ribbon a,2 (110; stacking fault structure.
4.0
SMITW, VITEK AYD POND:
COIMPUTER SIMULATION
OF HIGH ANGLE BOl_KXDARIES
Fig. 2. (710) boundary.
Fig. 3. (310) boundary.
+ A A+ +A (a)
+A + A++AA+ A +A
+A
“_+A *
f A
+A
“,t + “++A +‘+ “,b4 “,6 + "+hb4
CA 4A +’ + ch A CA +A ’
+A
“,=9‘+ +h
+*
A
+
A
+
A+
A
A
+
A
“,
+“c +
b
CA A
A +
CA
A,6 +A
+AA+ A++AA+ ‘+ ‘,‘: &
“, .j. A A+‘,%
+
“+
A
+
.tA+
(f-4
Fig, 4. (520) boundary. (a) Calculated structure. (b) Schematic representation of the boundary structure as a mixture of (210) and (310) structural units.
Fig. 5. (7301 boundary. !a) Calcuiated structure. (b) Schematic repressmrarion of the boundary structure as a mixture of (210) and (310) structural units.
Fig, 6. (210) boundary.
Fig. 7. (230) boundary..
482
SMITH, VI-I-JzKAND
0
POND:
I
I
10
COMPUTER SIMULATION OF HIGH ANGLE BOUNDARIES
I 20
I 30
I 40
e
I 50
I 60
I 70
I 60
I
so
Fig. 9. Plot of the minimum calculated energy for [OOI] symmetrical tilt boundaries in coincidence orientations with .E < 41. The cusps at t) = 0 and 90’ and at the Z = 41 orientation (12.65’) were calculated for the purposes of illustration on the basis that lattice dislocations accommodated the misor~entation from the exact coincidence orientation whilst for the E = 5. 36.87” cusp primitive DSC lattice dislocations with b = l/i0 (3 10) were considered.
boundary relaxation. Furthermore. the specific translations associated with the various low energy configurations for any one boundary can be understood as translations which lead to interlocking groups at different positions in the boundary period. The absence of a monotonic relationship between E: or the boundary period and p, is consistent with other calculations and experiments [5] although at variance with earlier formulations of the CSL theory [ZO]. This result is not surprising if the occurrence of interlocking groups is an important factor in low energy structures. Figure 4 illustrates that the number of triangle groups in fully relaxed boundary is not simply related to the unrelaxed period. However neither is there a simple relationship between the number of interlocking groups and 7 which is consistent with the fact that local relaxation of the atoms near the boundary, i.e. the boundary strain field is also an important energy factor. Table 3 contains relative values of the calculated energy for symmetrical (001) tilt boundaries normalised to ~czio, according to the calculations of Hasson et al. [6] and of Weins et at. [7] and the present workers. There are substantial differences between the three sets of results but all show a notable lack of agreement with the predictions of purely geometrical models. The differences between the calculated energies and structures reflect the differing relaxation algorithms and potentials used. The relaxed (210) structure calculated in Ref. 6 is very similar to that obtained in the present work. (compare Figs. 6 and 10 from Ref. 5) but the (310) structures obtained in the two studies are si~i~cantIy different because Hasson et al. [6] predict a structure
which has mirror symmetry about their boundary plane (compare Figs. 3 and S from Ref. 6). In the present work the loss of mirror symmetry is apparently the consequence of the relative displacement of the two grains parallel to the boundary. An interesting feature of all the relaxed structures calculated in the present work is that the pr~om~ant relaxation process seems to be translation as was proposed by Weins et a[. [Tf and local expansion. The atoms at the boundary adopt positions characteristic of either grain, giving a very narrow boundary, rather than compromise positions such as are postulated by the transition lattice theory [21] and sometimes found in the results of Ref. 6).
The calculated energy of symmetrical (001) tilt boundaries in aluminium depends on the strain field of the boundary and extent to which the .perfect crystal coordination %nd near neighbour spacing are conserved in interlocking groups of atoms in the boundary rather than geometrical factors describing the boundary periodicity. Some but not all long period boundaries intermediate in orientation between short period boundaries can be described in terms of mixtures of strained units characteristic of the short period boundaries. The calculated energy rises only slowly as Z increases. In many cases more than one low energy structure exists for a given grain rotation and boundary orientation. AcknowteJgenrenrs-Tfiis work was performed in part at Oxford University and in part at Central Electricity
SMITH. VITEK
,350
POND:
COMPL’TER SIMULATIOS
Research Laboratones. and IS publtshed with the permission of the Central Electricity Generating Board. The authors are grateful to the Armourers gi Brasiers Company (DAS) and the Science Research Council IRCP) for financial support, to Professors J. W. Christian. FRS and M. F. Ashby for useful discussions and to Professor Sir Peter Hirsch FRS. for encouragement and provision of facilities. The authors also wish to thank Dr. Roger Taylor for his help wtth the use of the potenttal for alummtum. REFERENCES 1. R. W. Ball&i, Y. Komem and T. Schober. Surf: Sci. 31, 68 (1972). 7 MM.McLean. J. mater. Sci. 8, 571 (1973). ;: G. Dimou and K. T. Aust. .-lcrcr 41rt. 22. 27 (1971). 4. G. Hasson and C. Goux. Scripra .ller. 5. 889 (1971). 5. G. Herrmann, H. Gleiter and G. BHro. .dcta Ma. 24. 353 (1976). G. Hasson. J. Y. Boos. I. Herbeuval. M. Biscondi and C. Gout. Surf: Sci. 31, 115 (1973). M. Weins. H. Gleiter and B. Chalmers. J. appl. PIiys. 42. 2639 (1971). R. E. Dahl. J. R. Beeler and R. 0. Bourquin. Barrrlle Colloquitrm on Interaromic Potentials and Simuhrion of
OF HIGH ANGLE BOUNDARIES
4%
L~rricr Drfecrs (edited b! P. C. Gehlen. J. R. Beeler and R. I. Jaffee). p. 673. Plenum Press. Neu York j 19-21. 9. R. .LI. J. Cotterill. T. Leffers and H. Lilholt. PM. .Lfq. 30. 265 1197-t). 10. ht. Weins. Surf: Sci. 31. 138 (1973). 11. R. J. Harrison. G. A. Bruggeman and CT. H. Bishop. in Grbn Boundary Srrucrure and Proprrries (edited by Chadwick and Smith). Academic Press. London I 1976). I’. W. T. Read and W. Shocklev. Phvs. Rrr. 78. 275 (1950). 13. R. C. Pond. Con. met&. 0. 13. 33 (1971). L-1. R. C. Pond and V. Vitek. submitted to Proc. R. Sot. (1977). 15. V. Vitek. D. A. Smith and R. C. Pond. Comprtr