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The growth mechanism of AuCu II plates
ocoi-6160 79~0501-0833802.00’0
Arro ,Mt~tallwqico. Vol. 27. pp. 833 10 X39 0 Pergamon Press Ltd 1979. Prmted in Great Britain
THE GROWTH
MECHANISM
OF AuCu II PLATES
J. S. BOWLES School of Metallurgy, The University of New South Wales, Kensington, New South Wales, Australia and
C. M. WAYMAN Department of Metallurgy and Mining Engineering and Materials Research Laboratory, University of Illinois, Urbana, Illinois, U.S.A. (Received 12
October
1978)
Abstract-The relationship between the growth mechanisms of AuCu II and martensite plates is considered and it is concluded that lengthening and thickening of AuCu II plates involves the gliding of transformation dislocations at a rate determined by the rate of ordering at the dislocations. This mechanism, which is compatible with the phenomenological theory of martensitic transformations, is shown to account for the reported lengthening kinetics and the observed dependence of thickening rate on interface orientation. It is also shown to be capable of generating the essential features of the morphology of the pyramidal groups of four plates characteristically produced by this transformation. This morphology is not compatible with a proposed ledge mechanism in which disordered ledges are considered to migrate a thousand times faster than the plates lengthen. RksumGNous avons 6tudi6 la relation entre les mtcanismes de croissance des plaquettes de AuCu II et de martensite, et nous avons conclu que I’allongement et l’epaississement des plaquettes de AuCu II met en jeu le glissement de dislocations de transformation g une vitesse dtterminte par la vitesse
de mise en ordre sur les dislocations. On montre que ce mecanisme, qui est compatible avec la theorie ph&nomtnologique des transformations martensitiques, rend compte des cinCtiques d’allongement publikes et de la variation expkrimentale de la vitesse d’tpaississement en fonction de I’orientation de I’interface. On montre que ce mecanisme permet Cgalement d’obtenir les traits essentiels de la morphologie des groupes pyramidaux de quatre plaquettes qui caracttrisent cette transformation. Cette morphologie n’est pas compatible avec un m&canisme de marches dans lequel des marches desordonnk migreraient mille fois plus vite que les plaquettes ne s’allongent. Zusammenfassung-Das Verhlltnis zwischen den Wachstumsmechanismen von AuCu II und Martensitplatten wird diskutiert. Die SchluDfolgerung ist, daB Llngenund Dickenwachstum von AuCu IIPlatten die Gleitung von Umwandlungsversetzungen mit einer Geschwindigkeit, die von der Geschwindigkeit der Ordnungseinstellung an der Versetzung bestimmt ist, umfal3t. Dieser Mechanismus ist vertrlglich mit der ph2inomenologischen Theorie martensitischer Umwandlungen. Er erkllrt die ermittelte LSingenwachstumskinetik und die beobachtete Abhlngigkeit der Geschwindigkeit des Lgngenwachstums von der GrenzflPchenorientierung. AuDerdem kann dieser Mechanismus die wesentlichen Merkmale der Form der Pyramidengruppen von je vier Platten, die charakteristischerweise bei dieser Umwandlung erzeugt werden, beschreiben. Diese Pyramidenform ist nicht vertrlglich mit einem vorgeschlagenen Vorsprungsmechanismus, bei dem gestijrte Vorspriinge eintausendmal schneller wandern
als die Platten sich verlgngern. 1. INTRODUCHON
with the phenomenological theory is not a sufficient condition for the identification of ‘a shear transformation mechanism’. Aaronson and Kinsman proposed two distinct mechanisms for the growth of AuCu II plates, one for lengthening and one for thickening. The proposed lengthening mechanism involves the migration of disordered ledges across a postulated coherent ‘leading face’ at 60” to the habit plane of the developing plate. The twins in the plates are considered to form behind the advancing order-disorder interface. This lengthening mechanism was proposed as an alternative to one involving alternate nucleation of twins, because lengthening rates calculated for the twin nucleation mechanism were found to be orders of magnitude slower than measured rates. This conclusion can be criticized because the calculation of incubation time for ‘twin on twin’ nucleation did not take account of the catalytic effect that the shear strain in the matrix resulting
The formation of the ordered orthorhombic phase, AuCu II, from the disordered cubic phase generates invariant plane strain relief and conforms with the phenomenological theory of martensitic transformations. These characteristics led Smith and Bowles [l] to conclude that the mechanism of this transformation is similar to that of a martensitic transformation even though place exchanges between neighbouring atoms are necessarily involved. The contrary view that the transformation ‘cannot be co-operative as expected in a martensitic transformation’ because an order-disorder transformation is involved was recently put by Pedraza and Kittl[2]. This view was subsequently elaborated by Aaronson and Kinsman [3] who claimed not only that the atomic jumps required by the ordering process are ‘inconsistent with a shear transformation’ but also that conformity 833
834
BOWLES
AND
WAYMAN:
GROWTH
from the formation of one twin would have on the next. The incubation periods calculated by Aaronson and Kinsman really refer to nucleation of plates in the disordered matrix and in fact are in fair agreement with incubation periods reported by Smith and Bowles Cl]. It should also be noted that if the twinning shear occurred behind the advancing orderdisorder interface, it would change the orientation of the c axis relative to the layers of copper and gold atoms unless it was accompanied by place exchanges of neighbouring atoms. Thus the atomic movements in the proposed lengthening mechanism would not be restricted to the disordered ledges on the leading face. The mechanism proposed for thickening involves the migration of disordered ledges across the habit plane interface. The twins are considered to extend right up to this interface which is fully coherent until subsequent coarsening of the twins produces a misfit dislocation structure. In both the proposed lengthening and thickening processes the ledges are considered to migrate at the same rate, a rate characteristic of a disordered boundary. The difference between the rates of lengthening and thickening is attributed to a postulated order of magnitude difference in the spacing of the ledges in the habit plane and the ‘leading face’ interfaces. No explanation is offered of how the proposed growth mechanisms generate the observed invariant plane strain relief, but in earlier papers [4,5] these authors have attributed relief effects produced by transformations involving diffusion, to the maintenance of partial coherence across the broad faces of the ledges. In martensitic transformations the maintenance of a partially coherent interface involves the progressive displacement of the untransformed matrix and it is these matrix displacements which generate the invariant plane strain relief. It is difficult to reconcile such matrix displacements, which certainly occur in the AuCu II transformation, with a mechanism in which the atomic displacements are restricted to disordered ledges in the interface. It seems more likely that, apart from the place exchanges involved in the ordering, the mechanism of formation of AuCu II plates is basically similar to that of a martensitic transformation. The purpose of this paper is to examine in more detail than has previously been attempted the extent of this similarity between the AuCu II transformation and martensitic transformations. In Section 2 the relationship between the phenomenological theory and the mechanisms of martensitic transformations is reviewed briefly and a mechanism for the AuCu II transformation, which involves ordering at steps in the interface which are transformation dislocations, is proposed. In Section 3 it is shown that the proposed mechanism is consistent with reported measurements of lengthening and thickening rates of AuCu II plates and in Section 4 the mechanism is shown to be capable of generating the essential features of the morphology of the pyramidal groups of
MECHANISM
OF AuCu II PLATES
four plates that are usually produced mation.
by this transfor-
2. THE PHENOMENOLOGICAL THEORY AND ITS RELATIONSHIP TO THE MECHANISM OF MARTENSITIC TRANSFORMATION The phenomenological theory of martensitic transformations [6-83 was developed to provide a description of the overall atomic displacements involved in these transformations, consistent with the observed geometrical features. It describes the displacements of lattice points required to transform the parent lattice into the martensite lattice in its observed orientation and to generate the observed shape change. In martensitic transformations between structures whose primitive unit cells contain only one atom, the displacements of atoms and lattice points are the same; in other cases shuffling movements of some of the atoms are also required. Although the theory was developed for martensitic transformations, in which neighbouring atoms suffer only small relative displacements, it is also applicable to transformations involving place exchanges between neighbouring atoms if they exhibit the same geometrical features as martensitic transformations. Such place exchanges are redundant in the sense that they do not contribute to the shape change accompanying the transformation so that the theory still prescribes the displacements involved in producing the lattice transformation. The AuCu II transformation is an example of a transformation involving place exchanges between neighbouring atoms which produces martensitic relief effects. There are, however, no known examples of transformations involving substitutional diffusion to or from an interface, which have all the geometrical features of martensitic transformations. The phenomenological theory does not specify the order in which individual atoms are displaced during transformation and so does not prescribe a particular transformation mechanism. It simply describes a set of displacements with which the growth mechanisms must be compatible. Growth of martensite plates involves both the edgewise propagation of the plate across the parent crystal and the thickening of the plate. Current views on these growth mechanisms have been summarized by Christian [9]. In the usual type of martensitic transformation in which the shape and lattice deformations differ, a glissile interface parallel to the habit plane which can produce thickening by moving in the direction of its normal, contains areas of coherence separated either by matching dislocations or twin boundaries. Such a glissile boundary can move by homogeneous transformation of the coherent regions combined with gliding of the matching dislocations in their slip planes or extension of the twin boundaries. The movement of such boundaries is of necessity accompanied by the matrix displacements which generate the shape change.
BOWLES
AND
WAYMAN:
GROWTH
If the interface is not parallel to the habit plane, as in a lenticular or tapered plate, it must contain steps. The migration of these steps parallel to the habit plane provides an alternative mechanism for thickening as well as a mechanism for edgewise growth. Since the gliding of the steps produces the displacements of the matrix which generate the shape strain, they are transformation dislocations [lo] with Burgers vector mdh, where md is the displacement vector of the shape strain and h is the step height. For irrational habit planes, it is uncertain what value h will take. When the transformation dislocations generate the final lattice, as in the martensitic transformation in cobalt, no other defects are required in the interface. More usually, however, the lattice that would be generated by the transformation dislocations, i.e. the lattice obtained by applying the shape strain homogeneously to the parent lattice, differs from the martensite lattice. It is then necessary to propose that additional displacements occur within the transformation dislocations. An analogous situation exists in mechanical twinning where shuffling movements must accompany the gliding of twinning dislocations. The additional displacements are those described by the complementary strain of the phenomenological theory [6,7]. They are suitably small and contribute no additional change of shape if the interface contains matching dislocations or twin boundaries as well as transformation dislocations. In a martensitic transformation occurring by this mechanism, edgewise growth in the direction of the matching dislocations or along the intersection of the twinning plane and the habit plane involves only extension of existing defects. Growth at right-angles, however, requires the periodic generation of new dislocations or twin boundaries. Both the above mechanisms; propagation of a glissile habit plane interface in the direction of its normal and migration of transformation dislocations, are compatible with the phenomenological theory. In transformations involving atomic interchange, however, propagation of the interface in the direction of its normal, a mechanism involving co-ordinated displacements of whole planes of atoms, is clearly highly improbable. Migration of transformation dislocations, on the other hand, is a plausible mechanism by which both lengthening and thickening of plates can occur in such transformations. The atomic displacements involved are restricted to the vicinity of the dislocation steps and it is necessary only to propose that the dislocations migrate at a rate governed by the rate of ordering at the steps. The migration of the transformation dislocations produces the matrix displacements that generate the relief. 3. GROWTH KINETICS OF AuCu II PLATES The proposal
that growth of AuCu II plates occurs
MECHANISM
OF AuCu II PLATES
835
by glide of transformation dislocations parallel to the habit plane differs from the growth mechanisms proposed by Aaronson and Kinsman in two important respects. The first of these is that the atomic displacements are not restricted to disordered ledges in the interface; the migration of the transformation dislocations produces co-ordinated displacements of atoms in the parent phase. The second is that separate mechanisms are not required for lengthening and thickening. A lenticular plate will contain closed loops of transformation dislocation which, on expansion, lengthen the plate and produce increments of thickening equal to the step height. Continued thickening requires the nucleation of new loops of dislocation and if the rate of thickening is less than the rate of migration of the dislocations, the length-to-thickness ratio of the plate will increase during growth. In their study of the growth kinetics of AuCu II plates, Pedraza and Kittl observed precisely this effect. Pedraza and Kittl measured lengthening and thickening rates at various degrees of undercooling on pairs of plates which nucleated at grain boundaries and grew adjoined along a mid-rib plane. Near the edges of these pairs of plates the boundaries between the plates and the matrix were at an angle 0 to one another but remote from the edges; the boundaries tended to become parallel to each other and to the mid-rib. They found that not only did the length to thickness ratio increase during growth but that the rate of thickening varied with the angle 0 and became very small when the boundaries became parallel (0 = 0). The lengthening rate, on the other hand, was constant for any given pair of plates but varied from one pair to another and increased rapidly with undercooling below the critical temperature. This behaviour is consistent with the proposed dislocation mechanism for growth. In this mechanism, the rate of growth, G,, in the direction of the normal to the habit plane is
where h is the step height of the transformation, dislocations, b is the spacing between dislocations and G, is the dislocation velocity, which is equal to the lengthening rate. Periodic generation of dislocations at a point source and a constant radial rate of growth would produce an interface consisting of a stepped conic surface. If the specimen surface is not perpendicular to the habit plane, the step height, h’, and distance, b’, between dislocations in the boundary differ from h and b and the thickening rate measured in such a surface is given by G,=q=ZG,tan$ since growth is occurring at two interfaces. Thickening rates derived from values of Gr and 0 measured by Pedraza and Kittl are compared with the corresponding measured values in Table 1.
BOWLES
836
Table 1. Comparison
Spec. no.
AND
WAYMAN:
GROWTH
MECHANISM
OF AuCu II PLATES
of measured thickening rates with those derived from measured lengthening rates and angles 8. Data from Reference [Z] Undercooling (“C)
Gr
R&ge of 9 (deg.)
2G, tan 012 (x 10m7cm/s)
Measured G, ( x lo-’ cm/s)
5.3-2.2 10.5-7.8 6.46.0 7.c6.0
5.8 11.2 9 9.6
(x 10d6 cm/s)
5
9.4 10.4 11.75 11.75
3.9 4.58 6.93 6.93
1.75-3.25 13-9.7 5.25-5 4.7555
8
3.8 6.0 9.0 9.8
4.14 9.62 15.4 22
3 4 3.75 4
The agreement between the calculated and measured thickening rates is convincing evidence that the order of magnitude difference between the thickening and lengthening rates arises from the proposed dislocation mechanism for growth. The measured lengthening rates are several orders of magnitude slower than the migration rates calculated for incoherent ledges by Aaronson and Kinsman. The rapid migration rates calculated for incoherent ledges arise from the plausible assumption that the activation enthalpy for diffusion at these sites is half that for volume diffusion, (AH, = 44,2OOcal/ mole). Such rapid migration rates are not expected for transformation dislocations. The arrangement of atoms within the dislocations will be more regular than in an incoherent boundary and the activation enthalpy should be between iAH, and AH,. Following Aaronson and Kinsman [3] the relationship between activation enthalpy AH, lengthening rate G, and undercooling AT may be written?, G, = 1.32 x 107AT/T*e-m’RT.
(3)
Values of AH derived by substituting the measured lengthening rates (l-22 x 10e6 cm/s) and undercoolings (3-22 K) are between 30,000 and 35,000 cal/mole. As Pedraza and Kittl have noted, the transformation kinetics are also likely to be influenced by the elastic strain field within which the plates develop and to which their growth contributes. Strain energy effects are probably the cause of the variations in lengthening rate observed at constant undercooling. There seems little doubt, however, that the order of magnitude of the observed lengthening rates implies a growth step in which the atomic arrangement is more regular than in an incoherent ledge. 4. MORPHOLOGY OF AuCu II PRECIPITATES Having demonstrated that the measured thickening and lengthening rates are consistent with the pro-
t Reference [S] gives 2.91 as the coefficient in this expression. The correct value (1.32) has, however, been used to derive the growth rates shown in Fig. 2 of this reference.
2.3 9.8 10.1 15.3
3.5 8.6 13.0 29.0
posed dislocation mechanism for growth, we must now consider how the dislocations are nucleated and how the tapered edges of the plates are generated. The morphology of the pyramidal groups of four plates [l] which characteristically form when AuCu II is nucleated within the parent crystal provides a valuable insight into these problems. Serial sectioning studies by Corderoy [ll] and Morton Cl23 have established that the pyramids have the form illustrated in Fig. 1. Double pyramids and quadruple pyramids, consisting of two double pyramids at right-angles sharing a common apex, are also observed. It seems likely that all pyramids develop from nuclei located at the initial position of the apex. Hot-stage microscopy [ 131 shows that growth occurs both on the inside and outside of the pyramid, causing the sides to extend and the apex to migrate. In multiple pyramids where nucleation presumably occurs at the common apex and the apex cannot migrate, growth occurs predominantly in the inside, relatively little thickening occurring at the ‘apex (Fig. 2). The lengthening rate of the plates is much greater than the thickening rate and the sides of the pyramid extend more rapidly than its interior transforms. Where the plates grow together, inside the pyramid, they meet along junction planes. The junction plane for plates AB and CD (Fig. 1) is (Oil) and for plates AC and BD is (100). The inner and outer surfaces of the plates are not parallel and the angles between the pyramid axis [011] and the junction lines along which the inner surfaces meet differ significantly from the angles between the axis and the corresponding habit plane intersections. For pyramids formed between 405 and 400°C Corderoy[8] found the inner junctions of plates AC and BD to be 4.5” + 1 from [01 l] compared with the value 2.3” calculated from the theoretical habit plane [l] and 3.25” + 1 measured from the outer junctions of these plates. Similarly, and more strikingly, the inner junctions of plates AB and CD are 76” f 5 from [Oil] compared with the angle 47” made by the intersection of the two (calculated) habit planes. Comparison with the outer junctions of these plates is not possible, for these edges have a marked curvature, as shown in Fig. 1. This curvature implies a curvature in the outer surfaces of the four plates
BOWLES
AND
WAYMAN:
GROWTH
MECHANISM
OF
AuCu
837
II PLATES
Fig. 2. Multiple pyramid in a specimen
transformed
at
395°C for 2 h x 125.
with velocity G,, stepped interfaces will develop. In unit time, the inner apex will move through distance Nh/cos a where N is the number of dislocations generated, h is the step height and ‘1 is the angle between the [011] axis and the normal to the habit plane. For the habit plane calculated [1] from the phenomenological theory, a = 87.7”. As shown in Fig. 4, the angle p between the inner junction lines and the pyramid axis is given by tan p =
Fig. I. Schematic representation of AuCu II pyramid morphology (after Corderoy [l I]). The inner junctions are shown as dotted lines and the form of the diamond-shaped sections is indicated at three levels. The relative orientations of plates A B C and D is indicated in the stereographic projection.
and this can be detected on some lateral (diamond shaped) sections (Fig. 3). It seems likely that the dislocation steps needed for growth on the inside of a pyramid are generated at the apex of the inner surface. The Burgers vectors of the dislocations on all four plates in a pyramid are equal in magnitude and are all near the same (110) direction, so that the dislocations can form cooperatively and glide across the plates, their ends meeting at points that generate the inner junctions of the plates. The co-operative generation of dislocations explains why all four plates grow at the same rate, an observation that would be difficult to explain if the plates grew independently of one another, This growth model also reveals why the inner surfaces and their junction lines are not parallel to the corresponding habit planes and habit plane intersections. Generation of dislocations will cause the inner apex to move along the [011] axis of the pyramid, and if the dislocations glide across the habit planes
r sin w r cos w -
1’
(4)
where o is the angle between the habit plane intersection and the pyramid axis and r = G,cosx/Nh is the ratio of the dislocation velocity to the rate of migration of the inner apex along the pyramid axis as a result of the generation of new dislocations. For the intersections, AB and CD, w = 47.1” and for intersections AC and BD, o = 2.3”. Substitution in equation (4) of these values for o and the corresponding measured values of p = 76” and p = 4.5” leads in each case to r = 2. This agreement between the values of r, derived from the two inner junction lines, is, however, fortuitous and when the ranges of values of the angles p are considered, r varies between 1.7
Fig. 3. ‘Diamond-shaped formed by transformation
sections through pyramids at 400°C for 22 h. Polarized light x 250.
BOWLES
838
AND
WAYMAN:
[Oll],
LOCUS
OF
GROWTH INNER
MECHANISM
OF AuCu 11 PLATES
APEX TRACE IN INNER
OF
HABIT
JUNCTION
PLANE
PLANE
JUNCTION
LINE
t
Nh
lb/
COS
w
Fig. 4. Relationship between inner junction line, habit plane intersection AuCu II pyramid. and 2.9 and the two values are no longer equal. This range of values of r is confirmed for the AC type edge by measurements on multiple pyramids where there is no migration of the outer apex. In this case, the ratio of the total length of the AC edge to the distance between the inner and outer apices is equal to r. Measured values of this ratio are in the range
1.9-2.7. Equality of the values of r would imply that the dislocations move along both types of habit plane intersection at the same rate and would suggest that the growth rate is isotropic. The serial sectioning studies indicate that growth along the AC intersection is about 1.5 times faster than along the AB intersection. These studies of AuCu II pyramid morphology provide no support for the proposal of Aaronson and Kinsman that ledge migration should occur most rapidly along the intersection of the habit plane and the twinning plane. In the dislocation growth model, accelerated dislocation glide in this direction would cause the plates to lengthen preferentially along directions approximately 35” from the AC and BD edges. This would lead to separation of these pairs of plates in transverse sections and such separation was never observed in the hundreds of sections examined by Corderoy and Morton. In the proposed growth mechanism, the rate of growth normal to the habit plane resulting from the generation of new dislocations inside the pyramid is the same as that resulting from the migration of an array of dislocations of spacing b and step height h. Thus G,
=
Nh
=
‘;.
and the axis [Oil] of an
the growth rate normal to the habit plane and an angle? of 1.1” between the inner interface and the habit plane. This pyramid morphology is not compatible with the proposal by Aaronson and Kinsman that thickening ledges migrate with a velocity about a thousand times greater than G,. With such a rapid migration rate the interface would be only a few thousandths of a degree from the habit plane. Thus, the ledge mechanism would generate almost parallelsided plates and the inner junction lines would be within a few hundredths of a degree of the intersections of the relevant variants of the habit plane. The relationship between the pyramid morphology and the pairs of plates studied by Pedraza and Kittl is uncertain. The opposite tilts produced by these pairs of plates suggest that they are AB pairs with the two habit planes and the (Oil) junction plane making nearly parallel traces in the surface of the specimen. Depending on the inclination of the intersection of the two habit planes to the surface, such paired plates could exhibit either a forked or a pointed end, as observed by Pedraza and Kittl. Growth at a forked end would involve generation of new dislocations between the plates, presumably at the fork where the junction line meets the surface. Growth at a pointed end would involve the gliding of dislocations on the outside of the plates. The generation of parallel or nearly parallel sides behind an advancing pointed end implies that the rate of formation of dislocations on the outside has initially been rapid but has then virtually ceased. The same behaviour is needed to produce the curved AB edges of the pyramids. The reason for this behaviour is not apparent.
It follows that 2_h G
CONCLUSION
cosa
GA=7 which for r = 2.0 indicates a lengthening
rate 50 times
t This assumes a plane interface. For constant N and a constant radial rate of migration of dislocations, the interface would be a sheared conic surface, the shear arising from the fact that the source of the dislocations moves along the [Oil] pyramid axis.
Although growth of AuCu II plates cannot occur by migration of a glissile interface because thermal activation is necessary for the atomic displacements involved in the ordering, the process can nevertheless be said to ‘take place by shear’ in the sense that it can occur by the gliding of transformation dislocations at a rate governed by the rate of ordering at
BOWLES
AND
WAYMAN:
GROWTH
the dislocation steps. This mechanism has been shown to be compatible with reported lengthening kinetics and accounts for the observed dependence of thickening rate on the angle between the boundary and the habit plane. It has also been shown to generate the essential features of the morphology of the pyramidal groups of plates produced by the transformation. The partially coherent interfaces and the invariant plane strain relief effects resulting from this mechanism are in contrast to the incoherent interfaces and absence of relief that are characteristic of massive transformations and there seems to be no justification for the proposal [3] that the formation of AuCu II should be considered to be a massive transformation.
Acknowledyements-The authors are indebted to Dr A. Morton for permission to reproduce Fig. 2 and to Dr D. Corderoy for permission to reproduce Fig. 3. We also wish to thank Professor J. W. Christian for his comments on the manuscript. This work was supported in part by the National Science Foundation through the Materials Research Laboratory at the University of Illinois. Urbana
MECHANISM
OF AuCu II PLATES
839
REFERENCES R. Smith and J. S. Bowles, Acta metall. 8, 405 (1960). A. J. Pedraza and J. Kittl, Acta metal 24, 835 (1976). H. 1. Aaronson and K. R. Kinsman, Acta metall. 25, 367 (1977). H. 1. Aaronson, C. Laird and K. R. Kinsman, Phase Transformations, p. 313. A.S.M., Metal Park, Ohio
(1970). 5. K. R. Kinsman, E. Eichen and H. I. Aaronson. Metall. Trans. A6, 303 (1975). 6. J. S. Bowles and J. K. Mackenzie, Acta melall. 2, 129 (1954). 7. 3. K. Mackenzie and J. S. Bowles, Acta metall. 2, 138 (1954). 8. M. S. Wehler, D. S. Lieberman and T. A. Read, Trans. Aus. Inst. Min. Metall. Engrs 197, 1503 (1954). 9. J. W. Christian, The Theory of Transformations in Metals and Alloys, p. 809. Pergamon Press, Oxford (1965). 10. Ibid, p. 274. 11. D. J. H. Corderoy, B.Sc. Thesis, The University of New South Wales (1958). 12. A. J. Morton, B.Sc. Thesis, The University of New South Wales (1959). 13. R. Smith. Ph.D. Thesis, University of Melbourne (1957).
Arro ,Mt~tallwqico. Vol. 27. pp. 833 10 X39 0 Pergamon Press Ltd 1979. Prmted in Great Britain
THE GROWTH
MECHANISM
OF AuCu II PLATES
J. S. BOWLES School of Metallurgy, The University of New South Wales, Kensington, New South Wales, Australia and
C. M. WAYMAN Department of Metallurgy and Mining Engineering and Materials Research Laboratory, University of Illinois, Urbana, Illinois, U.S.A. (Received 12
October
1978)
Abstract-The relationship between the growth mechanisms of AuCu II and martensite plates is considered and it is concluded that lengthening and thickening of AuCu II plates involves the gliding of transformation dislocations at a rate determined by the rate of ordering at the dislocations. This mechanism, which is compatible with the phenomenological theory of martensitic transformations, is shown to account for the reported lengthening kinetics and the observed dependence of thickening rate on interface orientation. It is also shown to be capable of generating the essential features of the morphology of the pyramidal groups of four plates characteristically produced by this transformation. This morphology is not compatible with a proposed ledge mechanism in which disordered ledges are considered to migrate a thousand times faster than the plates lengthen. RksumGNous avons 6tudi6 la relation entre les mtcanismes de croissance des plaquettes de AuCu II et de martensite, et nous avons conclu que I’allongement et l’epaississement des plaquettes de AuCu II met en jeu le glissement de dislocations de transformation g une vitesse dtterminte par la vitesse
de mise en ordre sur les dislocations. On montre que ce mecanisme, qui est compatible avec la theorie ph&nomtnologique des transformations martensitiques, rend compte des cinCtiques d’allongement publikes et de la variation expkrimentale de la vitesse d’tpaississement en fonction de I’orientation de I’interface. On montre que ce mecanisme permet Cgalement d’obtenir les traits essentiels de la morphologie des groupes pyramidaux de quatre plaquettes qui caracttrisent cette transformation. Cette morphologie n’est pas compatible avec un m&canisme de marches dans lequel des marches desordonnk migreraient mille fois plus vite que les plaquettes ne s’allongent. Zusammenfassung-Das Verhlltnis zwischen den Wachstumsmechanismen von AuCu II und Martensitplatten wird diskutiert. Die SchluDfolgerung ist, daB Llngenund Dickenwachstum von AuCu IIPlatten die Gleitung von Umwandlungsversetzungen mit einer Geschwindigkeit, die von der Geschwindigkeit der Ordnungseinstellung an der Versetzung bestimmt ist, umfal3t. Dieser Mechanismus ist vertrlglich mit der ph2inomenologischen Theorie martensitischer Umwandlungen. Er erkllrt die ermittelte LSingenwachstumskinetik und die beobachtete Abhlngigkeit der Geschwindigkeit des Lgngenwachstums von der GrenzflPchenorientierung. AuDerdem kann dieser Mechanismus die wesentlichen Merkmale der Form der Pyramidengruppen von je vier Platten, die charakteristischerweise bei dieser Umwandlung erzeugt werden, beschreiben. Diese Pyramidenform ist nicht vertrlglich mit einem vorgeschlagenen Vorsprungsmechanismus, bei dem gestijrte Vorspriinge eintausendmal schneller wandern
als die Platten sich verlgngern. 1. INTRODUCHON
with the phenomenological theory is not a sufficient condition for the identification of ‘a shear transformation mechanism’. Aaronson and Kinsman proposed two distinct mechanisms for the growth of AuCu II plates, one for lengthening and one for thickening. The proposed lengthening mechanism involves the migration of disordered ledges across a postulated coherent ‘leading face’ at 60” to the habit plane of the developing plate. The twins in the plates are considered to form behind the advancing order-disorder interface. This lengthening mechanism was proposed as an alternative to one involving alternate nucleation of twins, because lengthening rates calculated for the twin nucleation mechanism were found to be orders of magnitude slower than measured rates. This conclusion can be criticized because the calculation of incubation time for ‘twin on twin’ nucleation did not take account of the catalytic effect that the shear strain in the matrix resulting
The formation of the ordered orthorhombic phase, AuCu II, from the disordered cubic phase generates invariant plane strain relief and conforms with the phenomenological theory of martensitic transformations. These characteristics led Smith and Bowles [l] to conclude that the mechanism of this transformation is similar to that of a martensitic transformation even though place exchanges between neighbouring atoms are necessarily involved. The contrary view that the transformation ‘cannot be co-operative as expected in a martensitic transformation’ because an order-disorder transformation is involved was recently put by Pedraza and Kittl[2]. This view was subsequently elaborated by Aaronson and Kinsman [3] who claimed not only that the atomic jumps required by the ordering process are ‘inconsistent with a shear transformation’ but also that conformity 833
834
BOWLES
AND
WAYMAN:
GROWTH
from the formation of one twin would have on the next. The incubation periods calculated by Aaronson and Kinsman really refer to nucleation of plates in the disordered matrix and in fact are in fair agreement with incubation periods reported by Smith and Bowles Cl]. It should also be noted that if the twinning shear occurred behind the advancing orderdisorder interface, it would change the orientation of the c axis relative to the layers of copper and gold atoms unless it was accompanied by place exchanges of neighbouring atoms. Thus the atomic movements in the proposed lengthening mechanism would not be restricted to the disordered ledges on the leading face. The mechanism proposed for thickening involves the migration of disordered ledges across the habit plane interface. The twins are considered to extend right up to this interface which is fully coherent until subsequent coarsening of the twins produces a misfit dislocation structure. In both the proposed lengthening and thickening processes the ledges are considered to migrate at the same rate, a rate characteristic of a disordered boundary. The difference between the rates of lengthening and thickening is attributed to a postulated order of magnitude difference in the spacing of the ledges in the habit plane and the ‘leading face’ interfaces. No explanation is offered of how the proposed growth mechanisms generate the observed invariant plane strain relief, but in earlier papers [4,5] these authors have attributed relief effects produced by transformations involving diffusion, to the maintenance of partial coherence across the broad faces of the ledges. In martensitic transformations the maintenance of a partially coherent interface involves the progressive displacement of the untransformed matrix and it is these matrix displacements which generate the invariant plane strain relief. It is difficult to reconcile such matrix displacements, which certainly occur in the AuCu II transformation, with a mechanism in which the atomic displacements are restricted to disordered ledges in the interface. It seems more likely that, apart from the place exchanges involved in the ordering, the mechanism of formation of AuCu II plates is basically similar to that of a martensitic transformation. The purpose of this paper is to examine in more detail than has previously been attempted the extent of this similarity between the AuCu II transformation and martensitic transformations. In Section 2 the relationship between the phenomenological theory and the mechanisms of martensitic transformations is reviewed briefly and a mechanism for the AuCu II transformation, which involves ordering at steps in the interface which are transformation dislocations, is proposed. In Section 3 it is shown that the proposed mechanism is consistent with reported measurements of lengthening and thickening rates of AuCu II plates and in Section 4 the mechanism is shown to be capable of generating the essential features of the morphology of the pyramidal groups of
MECHANISM
OF AuCu II PLATES
four plates that are usually produced mation.
by this transfor-
2. THE PHENOMENOLOGICAL THEORY AND ITS RELATIONSHIP TO THE MECHANISM OF MARTENSITIC TRANSFORMATION The phenomenological theory of martensitic transformations [6-83 was developed to provide a description of the overall atomic displacements involved in these transformations, consistent with the observed geometrical features. It describes the displacements of lattice points required to transform the parent lattice into the martensite lattice in its observed orientation and to generate the observed shape change. In martensitic transformations between structures whose primitive unit cells contain only one atom, the displacements of atoms and lattice points are the same; in other cases shuffling movements of some of the atoms are also required. Although the theory was developed for martensitic transformations, in which neighbouring atoms suffer only small relative displacements, it is also applicable to transformations involving place exchanges between neighbouring atoms if they exhibit the same geometrical features as martensitic transformations. Such place exchanges are redundant in the sense that they do not contribute to the shape change accompanying the transformation so that the theory still prescribes the displacements involved in producing the lattice transformation. The AuCu II transformation is an example of a transformation involving place exchanges between neighbouring atoms which produces martensitic relief effects. There are, however, no known examples of transformations involving substitutional diffusion to or from an interface, which have all the geometrical features of martensitic transformations. The phenomenological theory does not specify the order in which individual atoms are displaced during transformation and so does not prescribe a particular transformation mechanism. It simply describes a set of displacements with which the growth mechanisms must be compatible. Growth of martensite plates involves both the edgewise propagation of the plate across the parent crystal and the thickening of the plate. Current views on these growth mechanisms have been summarized by Christian [9]. In the usual type of martensitic transformation in which the shape and lattice deformations differ, a glissile interface parallel to the habit plane which can produce thickening by moving in the direction of its normal, contains areas of coherence separated either by matching dislocations or twin boundaries. Such a glissile boundary can move by homogeneous transformation of the coherent regions combined with gliding of the matching dislocations in their slip planes or extension of the twin boundaries. The movement of such boundaries is of necessity accompanied by the matrix displacements which generate the shape change.
BOWLES
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GROWTH
If the interface is not parallel to the habit plane, as in a lenticular or tapered plate, it must contain steps. The migration of these steps parallel to the habit plane provides an alternative mechanism for thickening as well as a mechanism for edgewise growth. Since the gliding of the steps produces the displacements of the matrix which generate the shape strain, they are transformation dislocations [lo] with Burgers vector mdh, where md is the displacement vector of the shape strain and h is the step height. For irrational habit planes, it is uncertain what value h will take. When the transformation dislocations generate the final lattice, as in the martensitic transformation in cobalt, no other defects are required in the interface. More usually, however, the lattice that would be generated by the transformation dislocations, i.e. the lattice obtained by applying the shape strain homogeneously to the parent lattice, differs from the martensite lattice. It is then necessary to propose that additional displacements occur within the transformation dislocations. An analogous situation exists in mechanical twinning where shuffling movements must accompany the gliding of twinning dislocations. The additional displacements are those described by the complementary strain of the phenomenological theory [6,7]. They are suitably small and contribute no additional change of shape if the interface contains matching dislocations or twin boundaries as well as transformation dislocations. In a martensitic transformation occurring by this mechanism, edgewise growth in the direction of the matching dislocations or along the intersection of the twinning plane and the habit plane involves only extension of existing defects. Growth at right-angles, however, requires the periodic generation of new dislocations or twin boundaries. Both the above mechanisms; propagation of a glissile habit plane interface in the direction of its normal and migration of transformation dislocations, are compatible with the phenomenological theory. In transformations involving atomic interchange, however, propagation of the interface in the direction of its normal, a mechanism involving co-ordinated displacements of whole planes of atoms, is clearly highly improbable. Migration of transformation dislocations, on the other hand, is a plausible mechanism by which both lengthening and thickening of plates can occur in such transformations. The atomic displacements involved are restricted to the vicinity of the dislocation steps and it is necessary only to propose that the dislocations migrate at a rate governed by the rate of ordering at the steps. The migration of the transformation dislocations produces the matrix displacements that generate the relief. 3. GROWTH KINETICS OF AuCu II PLATES The proposal
that growth of AuCu II plates occurs
MECHANISM
OF AuCu II PLATES
835
by glide of transformation dislocations parallel to the habit plane differs from the growth mechanisms proposed by Aaronson and Kinsman in two important respects. The first of these is that the atomic displacements are not restricted to disordered ledges in the interface; the migration of the transformation dislocations produces co-ordinated displacements of atoms in the parent phase. The second is that separate mechanisms are not required for lengthening and thickening. A lenticular plate will contain closed loops of transformation dislocation which, on expansion, lengthen the plate and produce increments of thickening equal to the step height. Continued thickening requires the nucleation of new loops of dislocation and if the rate of thickening is less than the rate of migration of the dislocations, the length-to-thickness ratio of the plate will increase during growth. In their study of the growth kinetics of AuCu II plates, Pedraza and Kittl observed precisely this effect. Pedraza and Kittl measured lengthening and thickening rates at various degrees of undercooling on pairs of plates which nucleated at grain boundaries and grew adjoined along a mid-rib plane. Near the edges of these pairs of plates the boundaries between the plates and the matrix were at an angle 0 to one another but remote from the edges; the boundaries tended to become parallel to each other and to the mid-rib. They found that not only did the length to thickness ratio increase during growth but that the rate of thickening varied with the angle 0 and became very small when the boundaries became parallel (0 = 0). The lengthening rate, on the other hand, was constant for any given pair of plates but varied from one pair to another and increased rapidly with undercooling below the critical temperature. This behaviour is consistent with the proposed dislocation mechanism for growth. In this mechanism, the rate of growth, G,, in the direction of the normal to the habit plane is
where h is the step height of the transformation, dislocations, b is the spacing between dislocations and G, is the dislocation velocity, which is equal to the lengthening rate. Periodic generation of dislocations at a point source and a constant radial rate of growth would produce an interface consisting of a stepped conic surface. If the specimen surface is not perpendicular to the habit plane, the step height, h’, and distance, b’, between dislocations in the boundary differ from h and b and the thickening rate measured in such a surface is given by G,=q=ZG,tan$ since growth is occurring at two interfaces. Thickening rates derived from values of Gr and 0 measured by Pedraza and Kittl are compared with the corresponding measured values in Table 1.
BOWLES
836
Table 1. Comparison
Spec. no.
AND
WAYMAN:
GROWTH
MECHANISM
OF AuCu II PLATES
of measured thickening rates with those derived from measured lengthening rates and angles 8. Data from Reference [Z] Undercooling (“C)
Gr
R&ge of 9 (deg.)
2G, tan 012 (x 10m7cm/s)
Measured G, ( x lo-’ cm/s)
5.3-2.2 10.5-7.8 6.46.0 7.c6.0
5.8 11.2 9 9.6
(x 10d6 cm/s)
5
9.4 10.4 11.75 11.75
3.9 4.58 6.93 6.93
1.75-3.25 13-9.7 5.25-5 4.7555
8
3.8 6.0 9.0 9.8
4.14 9.62 15.4 22
3 4 3.75 4
The agreement between the calculated and measured thickening rates is convincing evidence that the order of magnitude difference between the thickening and lengthening rates arises from the proposed dislocation mechanism for growth. The measured lengthening rates are several orders of magnitude slower than the migration rates calculated for incoherent ledges by Aaronson and Kinsman. The rapid migration rates calculated for incoherent ledges arise from the plausible assumption that the activation enthalpy for diffusion at these sites is half that for volume diffusion, (AH, = 44,2OOcal/ mole). Such rapid migration rates are not expected for transformation dislocations. The arrangement of atoms within the dislocations will be more regular than in an incoherent boundary and the activation enthalpy should be between iAH, and AH,. Following Aaronson and Kinsman [3] the relationship between activation enthalpy AH, lengthening rate G, and undercooling AT may be written?, G, = 1.32 x 107AT/T*e-m’RT.
(3)
Values of AH derived by substituting the measured lengthening rates (l-22 x 10e6 cm/s) and undercoolings (3-22 K) are between 30,000 and 35,000 cal/mole. As Pedraza and Kittl have noted, the transformation kinetics are also likely to be influenced by the elastic strain field within which the plates develop and to which their growth contributes. Strain energy effects are probably the cause of the variations in lengthening rate observed at constant undercooling. There seems little doubt, however, that the order of magnitude of the observed lengthening rates implies a growth step in which the atomic arrangement is more regular than in an incoherent ledge. 4. MORPHOLOGY OF AuCu II PRECIPITATES Having demonstrated that the measured thickening and lengthening rates are consistent with the pro-
t Reference [S] gives 2.91 as the coefficient in this expression. The correct value (1.32) has, however, been used to derive the growth rates shown in Fig. 2 of this reference.
2.3 9.8 10.1 15.3
3.5 8.6 13.0 29.0
posed dislocation mechanism for growth, we must now consider how the dislocations are nucleated and how the tapered edges of the plates are generated. The morphology of the pyramidal groups of four plates [l] which characteristically form when AuCu II is nucleated within the parent crystal provides a valuable insight into these problems. Serial sectioning studies by Corderoy [ll] and Morton Cl23 have established that the pyramids have the form illustrated in Fig. 1. Double pyramids and quadruple pyramids, consisting of two double pyramids at right-angles sharing a common apex, are also observed. It seems likely that all pyramids develop from nuclei located at the initial position of the apex. Hot-stage microscopy [ 131 shows that growth occurs both on the inside and outside of the pyramid, causing the sides to extend and the apex to migrate. In multiple pyramids where nucleation presumably occurs at the common apex and the apex cannot migrate, growth occurs predominantly in the inside, relatively little thickening occurring at the ‘apex (Fig. 2). The lengthening rate of the plates is much greater than the thickening rate and the sides of the pyramid extend more rapidly than its interior transforms. Where the plates grow together, inside the pyramid, they meet along junction planes. The junction plane for plates AB and CD (Fig. 1) is (Oil) and for plates AC and BD is (100). The inner and outer surfaces of the plates are not parallel and the angles between the pyramid axis [011] and the junction lines along which the inner surfaces meet differ significantly from the angles between the axis and the corresponding habit plane intersections. For pyramids formed between 405 and 400°C Corderoy[8] found the inner junctions of plates AC and BD to be 4.5” + 1 from [01 l] compared with the value 2.3” calculated from the theoretical habit plane [l] and 3.25” + 1 measured from the outer junctions of these plates. Similarly, and more strikingly, the inner junctions of plates AB and CD are 76” f 5 from [Oil] compared with the angle 47” made by the intersection of the two (calculated) habit planes. Comparison with the outer junctions of these plates is not possible, for these edges have a marked curvature, as shown in Fig. 1. This curvature implies a curvature in the outer surfaces of the four plates
BOWLES
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GROWTH
MECHANISM
OF
AuCu
837
II PLATES
Fig. 2. Multiple pyramid in a specimen
transformed
at
395°C for 2 h x 125.
with velocity G,, stepped interfaces will develop. In unit time, the inner apex will move through distance Nh/cos a where N is the number of dislocations generated, h is the step height and ‘1 is the angle between the [011] axis and the normal to the habit plane. For the habit plane calculated [1] from the phenomenological theory, a = 87.7”. As shown in Fig. 4, the angle p between the inner junction lines and the pyramid axis is given by tan p =
Fig. I. Schematic representation of AuCu II pyramid morphology (after Corderoy [l I]). The inner junctions are shown as dotted lines and the form of the diamond-shaped sections is indicated at three levels. The relative orientations of plates A B C and D is indicated in the stereographic projection.
and this can be detected on some lateral (diamond shaped) sections (Fig. 3). It seems likely that the dislocation steps needed for growth on the inside of a pyramid are generated at the apex of the inner surface. The Burgers vectors of the dislocations on all four plates in a pyramid are equal in magnitude and are all near the same (110) direction, so that the dislocations can form cooperatively and glide across the plates, their ends meeting at points that generate the inner junctions of the plates. The co-operative generation of dislocations explains why all four plates grow at the same rate, an observation that would be difficult to explain if the plates grew independently of one another, This growth model also reveals why the inner surfaces and their junction lines are not parallel to the corresponding habit planes and habit plane intersections. Generation of dislocations will cause the inner apex to move along the [011] axis of the pyramid, and if the dislocations glide across the habit planes
r sin w r cos w -
1’
(4)
where o is the angle between the habit plane intersection and the pyramid axis and r = G,cosx/Nh is the ratio of the dislocation velocity to the rate of migration of the inner apex along the pyramid axis as a result of the generation of new dislocations. For the intersections, AB and CD, w = 47.1” and for intersections AC and BD, o = 2.3”. Substitution in equation (4) of these values for o and the corresponding measured values of p = 76” and p = 4.5” leads in each case to r = 2. This agreement between the values of r, derived from the two inner junction lines, is, however, fortuitous and when the ranges of values of the angles p are considered, r varies between 1.7
Fig. 3. ‘Diamond-shaped formed by transformation
sections through pyramids at 400°C for 22 h. Polarized light x 250.
BOWLES
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[Oll],
LOCUS
OF
GROWTH INNER
MECHANISM
OF AuCu 11 PLATES
APEX TRACE IN INNER
OF
HABIT
JUNCTION
PLANE
PLANE
JUNCTION
LINE
t
Nh
lb/
COS
w
Fig. 4. Relationship between inner junction line, habit plane intersection AuCu II pyramid. and 2.9 and the two values are no longer equal. This range of values of r is confirmed for the AC type edge by measurements on multiple pyramids where there is no migration of the outer apex. In this case, the ratio of the total length of the AC edge to the distance between the inner and outer apices is equal to r. Measured values of this ratio are in the range
1.9-2.7. Equality of the values of r would imply that the dislocations move along both types of habit plane intersection at the same rate and would suggest that the growth rate is isotropic. The serial sectioning studies indicate that growth along the AC intersection is about 1.5 times faster than along the AB intersection. These studies of AuCu II pyramid morphology provide no support for the proposal of Aaronson and Kinsman that ledge migration should occur most rapidly along the intersection of the habit plane and the twinning plane. In the dislocation growth model, accelerated dislocation glide in this direction would cause the plates to lengthen preferentially along directions approximately 35” from the AC and BD edges. This would lead to separation of these pairs of plates in transverse sections and such separation was never observed in the hundreds of sections examined by Corderoy and Morton. In the proposed growth mechanism, the rate of growth normal to the habit plane resulting from the generation of new dislocations inside the pyramid is the same as that resulting from the migration of an array of dislocations of spacing b and step height h. Thus G,
=
Nh
=
‘;.
and the axis [Oil] of an
the growth rate normal to the habit plane and an angle? of 1.1” between the inner interface and the habit plane. This pyramid morphology is not compatible with the proposal by Aaronson and Kinsman that thickening ledges migrate with a velocity about a thousand times greater than G,. With such a rapid migration rate the interface would be only a few thousandths of a degree from the habit plane. Thus, the ledge mechanism would generate almost parallelsided plates and the inner junction lines would be within a few hundredths of a degree of the intersections of the relevant variants of the habit plane. The relationship between the pyramid morphology and the pairs of plates studied by Pedraza and Kittl is uncertain. The opposite tilts produced by these pairs of plates suggest that they are AB pairs with the two habit planes and the (Oil) junction plane making nearly parallel traces in the surface of the specimen. Depending on the inclination of the intersection of the two habit planes to the surface, such paired plates could exhibit either a forked or a pointed end, as observed by Pedraza and Kittl. Growth at a forked end would involve generation of new dislocations between the plates, presumably at the fork where the junction line meets the surface. Growth at a pointed end would involve the gliding of dislocations on the outside of the plates. The generation of parallel or nearly parallel sides behind an advancing pointed end implies that the rate of formation of dislocations on the outside has initially been rapid but has then virtually ceased. The same behaviour is needed to produce the curved AB edges of the pyramids. The reason for this behaviour is not apparent.
It follows that 2_h G
CONCLUSION
cosa
GA=7 which for r = 2.0 indicates a lengthening
rate 50 times
t This assumes a plane interface. For constant N and a constant radial rate of migration of dislocations, the interface would be a sheared conic surface, the shear arising from the fact that the source of the dislocations moves along the [Oil] pyramid axis.
Although growth of AuCu II plates cannot occur by migration of a glissile interface because thermal activation is necessary for the atomic displacements involved in the ordering, the process can nevertheless be said to ‘take place by shear’ in the sense that it can occur by the gliding of transformation dislocations at a rate governed by the rate of ordering at
BOWLES
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GROWTH
the dislocation steps. This mechanism has been shown to be compatible with reported lengthening kinetics and accounts for the observed dependence of thickening rate on the angle between the boundary and the habit plane. It has also been shown to generate the essential features of the morphology of the pyramidal groups of plates produced by the transformation. The partially coherent interfaces and the invariant plane strain relief effects resulting from this mechanism are in contrast to the incoherent interfaces and absence of relief that are characteristic of massive transformations and there seems to be no justification for the proposal [3] that the formation of AuCu II should be considered to be a massive transformation.
Acknowledyements-The authors are indebted to Dr A. Morton for permission to reproduce Fig. 2 and to Dr D. Corderoy for permission to reproduce Fig. 3. We also wish to thank Professor J. W. Christian for his comments on the manuscript. This work was supported in part by the National Science Foundation through the Materials Research Laboratory at the University of Illinois. Urbana
MECHANISM
OF AuCu II PLATES
839
REFERENCES R. Smith and J. S. Bowles, Acta metall. 8, 405 (1960). A. J. Pedraza and J. Kittl, Acta metal 24, 835 (1976). H. 1. Aaronson and K. R. Kinsman, Acta metall. 25, 367 (1977). H. 1. Aaronson, C. Laird and K. R. Kinsman, Phase Transformations, p. 313. A.S.M., Metal Park, Ohio
(1970). 5. K. R. Kinsman, E. Eichen and H. I. Aaronson. Metall. Trans. A6, 303 (1975). 6. J. S. Bowles and J. K. Mackenzie, Acta melall. 2, 129 (1954). 7. 3. K. Mackenzie and J. S. Bowles, Acta metall. 2, 138 (1954). 8. M. S. Wehler, D. S. Lieberman and T. A. Read, Trans. Aus. Inst. Min. Metall. Engrs 197, 1503 (1954). 9. J. W. Christian, The Theory of Transformations in Metals and Alloys, p. 809. Pergamon Press, Oxford (1965). 10. Ibid, p. 274. 11. D. J. H. Corderoy, B.Sc. Thesis, The University of New South Wales (1958). 12. A. J. Morton, B.Sc. Thesis, The University of New South Wales (1959). 13. R. Smith. Ph.D. Thesis, University of Melbourne (1957).