Scripta METALLURGICA Vol. 23, pp. 285-290, 1989 Pergamon Press plc Printed in the U.S.A. All rights reserved

REFUTATION OF "FURTHER REBUTTAL" BY AARONSON AND COWORKERS

J.W. Christian and D.V. Edmonds Department of Metallurgy and Science of Materials, Oxford University,

Parks Road, Oxford OXI 3PH, U.K.

(Received October 3, 1988) (Revised November 16, 1988)

We thank ACO for their "further rebuttal" (I) which merits yet another reply, despite the optimistic title of CE2. (To avoid needless repetition, we use both the designations ACO, CE, AR,etc. and the reference list of (I)). We first make it clear that we revised our original "final response" only because the editor sent us a preprint of HEA which we had not previously seen, but which was emphasised in AR2. A similar difficulty might now arise from the reference in (I) to unpublished work by Ramanujan et al. However, it seems likely that their conclusion is not really opposite to ours, so we have not asked to see this paper. We now consider in turn the various headings of (I).

Coherenc~ of Interfaces and of Enclosed Particles

The use of the term "partially coherent" in the first heading of (i) results in an incorrect statement of this aspect of the dispute, which arose initially from the claim of ACO that whilst the terraces are partly coherent, the ledges have a disordered (or incoherent) structure. In CE, CEI and CE2 we distinguished coherent or partly coherent interfaces which maintain a lattice correspondence during migration, and which consequentially produce a shape change, from other (incoherent) interfaces. We have never denied the possible co-existence of coherent and partly coherent interfaces, which from the description in (I) and contrary to the claim in AR2, is what RamanuJan's work probably demonstrates. An incoherent interface is characterised by a lack of continuity of atomic rows and planes, and its migration is inconceivable without appreciable atomic mobility. If, as in bainite formation, the M atoms have negligible mobility, the change in structure must entail a lattice deformation S with associated correspondence C and shape deformation B. This is why we concluded in CE that all interfaces of a growing bainitic ferrite plate must be at least partly coherent. Moreover, this theoretical conclusion is strongly supported by the experimental observation of a shape change with a shear component, since shear strains cannot be transmitted across an incoherent interface.

As briefly mentioned in CE2, the shape change has the further implication that an enclosed particle can only have one macroscopic (fully or partly) coherent interface; this follows since a finite deformation can have, at most, only one invarlant plane. All such particles should thus be plate-shaped with habit plane terraces identical with (or close to) the invariant plane of E and all other interfaces represented by small coherent steps ("transformation dislocations") in these terraces. If two macroscopic planes are invariant, there is no net shape deformation (E = I, the unit matrix). This is possible only in special cases some of which were considered by Sargent and Purdy (2).

We note that an important corollary of these results is that the formation of superledges on coherent plate-shaped particles which exhibit finite shape strains is extremely improbable; each such superledge is in effect a super transformation dislocation with a very large Burgers vector and stress field. In special cases, superledges with a partly coherent structure could form on coherent or partly coherent interfaces, but there is then no shape change and no reason for the particle to remain plate-shaped.

Misunderstandln~ by CE of the Barrier to Growth of Terraces

We believe, with respect, that any misunderstanding is not ours. Misfit dislocations are not free to migrate as ordinary dislocations; they are inherent in the structure of the interface, and it may not be correct to assign individual properties to them. However, certain geometrical rules must apply to the interface. Thus the "climb" of the misfit dislocations to which our "repeated statements" refer is not climb out of the interface but is simply the

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geometrically necessary addition or removal of atoms as the interface is displaced. High strain energies would certainly develop if individual misfit dislocations were to climb out of the interface in either direction, or (equivalently) if the interface were displaced (by whatever mechanism) without the necessary change in atomic density. Similarly, the motion of an array of glissile interface dislocations is not slip in the usual sense, but is simply a convenient description of the geometrically necessary atomic displacements. If the coherent regions of interface are completely immobile, sessile misfit dislocations are unable to climb and glissile misfit dislocations are unable to glide; the immobility of the interface is not "reinforced" by any additional barrier.

Perhaps, it may be useful to reiterate a point made in a previous discussion (3) of growth kinetics. Ledge and normal growth are alternative (parallel) modes each encompassing several (serial) processes, and actual growth will presumably utilise whichever mode is faster. If one unit step of the faster mode is much slower than the others, it will absorb most of the driving force and will effectively determine the actual growth rate, which is then described as "interface controlled", "diffusion controlled", etc. Strictly, all the serial processes have some influence on the growth rate, but this influence is usually negligible for all but the slowest. The ACO theory is based on the concept that the slowest process in the ledge growth of a ferritic plate is interstitial diffusion, but the (negligible) normal growth rate is attributed (confusingly, in our opinion) not to a single slow process but both to interface kinetics and to sessile interface dislocations. It is not explained how "the coherent regions could circumvent this kinetic barrier ... by shear", and how this is prevented by the sessile interface dislocations. Does this mean simply that the interface could adopt a glissile configuration but does not do so? Whatever alternative growth mechanism is envisaged, the only property which makes interface dislocations "sessile" is the geometrical necessity for addition or removal of atoms as the interface is displaced. Thus again we emphasise that the immobility of the terraces in the ACO model might be due either to interface kinetics or to the "climb" of intrinsic interface dislocations.

The mutually incompatible processes which we are thought to require also result from a misunderstanding. ACO frequently refer to growth by shear as though it were something mysterious and different from growth by atom attachment. The shear is only the net result of the atomic displacements, and the two are not separable. Clearly an atomic correspondence cannot be maintained in growth involving substitutional diffusion but all that is required for a systematic shape change is a correspondence of sites, and in our description a glissile or martensitic interface is one which maintains such a local correspondence and in addition conserves the number of atoms in the transformed region. The original prediction [(14) of (I)] was indeed that this site (or lattice) correspondence also should not survive if the change involves substitutional difffuslon, but the frequently quoted work of Liu and Aaronson on AIAg alloys [(31) of (1)] partly removes this difficulty by showing that the shape change disappears slowly. (In some circumstances, the shear component of the shape change averages to zero, as shown in the admirably detailed study by Howe et al (4)). As discussed later, the interface of this particular transformation, apart from the change in composition, has a structure identical with that of fully coherent martensite.

We take up the question of the assumptions of the ACO theory at the end of this reply.

The Structure of the Broad faces of Ferrite Plates

We had no wish to denigrate the RA papers and we apologise if our comment about lack of rigour caused offence. What we wished to emphasise was the contrast between the conclusion of RAI and RA2 that a variety of interfaces can be adequately modelled by one set of misfit dislocations and one set of structural ledges and the later conclusion in the O-lattice paper of Hall et al. [(15) of (I)], that at least two sets of dislocations are required. A single set of misfit dislocations implies that the lattice deformation S is an invariant line strain (ILS)~= |I and the Frank-Bilby and Bollmann equations show that the condition for this is that ITI S-11 = 0, or equivalently IS - II = 0. This is not the case for any of the interfaces considered; particular examples are given by Bollmann (5) and Knowles and Smith (6), and we can readily treat the general case (7). At any fixed orientation, there are at most three values of R for which the lattice deformation will be an ILS and for orientations from NW to KS these values of R are all outside the range expected for steels. An exact NW relation, for example, has invariant lines if R = 2/3/3 = 1.15, R = /6/2 = 1.2247 or R = /2 = 1.41, whereas, according to RA2, R for steels is generally between 1.25 and 1.30. There is an invariant llne when R = /6/2 at all orientations, but the other two values depend on the rotation 8 away from NW. For all values 1.2247 ~ R ~ 1.333., invariant lines only exist for e values greater than 5~ °, i.e.

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outside the experimental NW-KS range. Thus for the assumed orientation relations, single dislocation descriptions are only possible if R is very close to 1.225, which is improbable for steels.

The claim in (1) that there is good agreement between the RA and 0-lattice treatments seems inconsistent with the above results and with the further statement that we have failed to realise that ledges + misfit dislocations and two dislocation networks are alternative models of interfaces. The point of our previous calculations was to emphasiee (i) that the RA modelling assumes the Bain correspondence, (ii) that the initial fixing of the orientation relations generally results in habit planes in which there are large misfits of opposite signs (iii) that the corresponding interface models then require at least two sets of misfit dislocations, and (iv) that only slight modifications enable dislocation distributions which are close to those of the martensite theory to be specified. The "manoeuvering about" was intended simply to illustrate some limiting cases within the range of R values appropriate for steels. Similar variations in R are discussed in RA2, but the value 1.283 used to interpret the experimental results in RAI is not stated in the paper, and so was not used in our illustrations.

Turning now to the experimental evidence, we first distinguish the results for steels from those for Cu-Cr alloys. Our view that the interface must be inherently glissile strictly applies only when the M atoms are immobile; as stated in CE2, it is not obvious which model should be selected when long-range migration of M atoms can take place. (Our view also applies only to ~rowth interfaces; the various geometrical parameters which are used empirically in attempts to select the dislocation description which gives the interface of lowest energy are applicable, if at all, only to equilibrium conditions.) In the Cu-Cr paper of Hall and Aaronson (8), it is certainly claimed that the close-packed planes in the two lattices are very accurately parallel, but a careful search has failed to find any such statement about ferrite. The sentence at the top of page 372 of RAI indeed seems to imply that this parallelism was assumed (presumably (100) b is a misprint for (110) b ?). The computed angle between these planes for a martensitic interface is of the order of ~o and we are doubtful whether measurements of this accuracy can be made by electron diffraction techniques.

No comment is made in (1) on the feature of the experimental results of RAt which we emphasised, namely that in 6 out of 7 cases, the Burgers vector of the observed arrays corresponded to close-packed directions in both lattices. The 0-lattice predictions, however, for the three preferred interfaces are that the main dislocation array has a Burgers vector of [011~f which corresponds to [001~b, and Knowles and Smith (6) similarly state that the OB interfaces of RA2 are much more probable (i.e., have lower energy) than the OA interfaces. The agreement claimed between experiment and model in Table 3 of RAI is indeed impressive, but several features of the model calculations are difficult to understand. Thus the model predictions for areas 5 and 6 which have the same orientation relations are identical in all respects except for the direction and spacing of the dislocations. This is possibly related to a statement in the text of RA2 that with triatomic ledges "the misfit dislocations sometimes have a choice of more than one direction along which to lie " This is contrary to 0-lattice or surface dislocation theory, and Fig. I0 of RA2 seems to show an 0-point lattice which would require dislocations along both of the directions indicated. It is also not clear how the distinctions in Table 3 between positive and negative values of 8 were made, and why the model predictions for ±8 are not symmetrical.

The other experimental result which we are stated to ignore is the presence of structural ledges three planes high. Frankly, we see little significance in this. Structural ledges (or steps as we should prefer to call them) are not a feature unique to the models of type RA2; all irrational and high-index interfaces must be stepped on an atomic scale. For any high index or irrational interface, the structure will probably be so complex that there is no unique step description; moreover, the experimental conditions for the observation of steps are very difficult.

The Fate of "Overrun" Misfit Dislocations

We congratulate ACO on the first sentence of (I) under this heading; we were indeed "not entirely pleased" either with the HEA mechanisms or with our own suggested modifications. We begin with a clarification. If a process is described as conservative, there must be agreement about what is or is not being conserved. It was made explicit in CEI that our argument relates to the number of M atoms in an austenite region which is subsequently transformed to ferrite. Although resisted in ARI, it was accepted in AR2 that the ACO model requires that growth changes this number. Hence, any growth mechanism consistent with the ACO model must include a non-

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conservative process. For example, prismatic slip which ends within the ferrite region effectively transfers a disc of atoms into or out of this region, as does also the glide of a misfit dislocation off the end of a terrace in mechanism (i) of HEA. Clearly, prismatic Elide conserves the total number of sites (atoms ÷ vacancies) in the whole assembly, whereas climb does not, but this is not relevant to the present growth problem.

There are more misfit dislocations than required on terraces being shrunk only if these dislocations glide along the terraces; moreover, if one terrace is shrinking, an adjacent terrace must be expanding. The number of misfit dislocations required in any fixed length of interface is independent of how many ledges it contains. That the force acting as a ledge approaches is a climb force can readily be seen by anchoring the dislocation in position, allowing the ledge to overrun it and then relaxing the strained structure. ACO seem frequently to invoke dislocation processes without examining the implications; for example, the difficulties in nucleating a sessile misfit dislocation without climb are ignored. They profess not to understand the formation of a pair of opposite prismatic loops in our modification of their own proposal, but an exactly equivalent process (with a larger loop) is involved in the nucleation of new sessile misfit dislocations, which they invoke so readily. Because we consider the nucleation of large loops improbable, we do not reproduce diagrams here, but we have sent sketch figures to the authors of (I) and will supply them to anyone on request.

Our difficulty with mechanism (ii) of HEA is confirmed by their own conclusion that the final rotation of the dislocation back to its original orientation is not conservative. [It is nevertheless described as conservative in (i)]. To see why it is not conservative, we must consider what happens at the ends of the dislocation segment which glides in the terrace plane. As noted in HEA, the interface dislocations should be considered as loops girdling the ferrite particle. Fig. 1 shows part of such a loop ABCD where BC is in the lower terrace and corresponds to orientation A in Fig. 2a of HEA. Rotation into the screw orientation PU (A' in HEA) means that dislocation segments BP and CU must be added to the side faces, and finally after cross-sllpping on to the upper terrace and re-rotating to RS, the whole loop is now ABPQRSTUCD instead of the required ARSD. The segments PQ and UT are unable to glide on the side faces to their required final positions BR and CS, and an atom flux is required to enable them to climb to these positions. Thus mechanism (il) involves both glide and climb of the interface dislocation; since the initial rotation would he opposed by the field of the other misfit dislocations, direct climb from BC to RS, which requires the same atom flux, is much more likely.

Our statement which "dazzled" ACO relates to the impossibility of a non screw interface dislocation which forms part of a particular interface array assuming a screw orientation. This would cause a severe local distortion and increase in energy. Similarly, and in reply to another comment in (I), the conversion of a lattice dislocation into a misfit dislocation results in a large decrease in both the self energy of the dislocation and the energy of the elastically strained coherent interface. The reverse process is thus energetically very improbable.

We do not frequently construe dislocations into screw configuration; indeed we emphasised in CE2 that the martensitic model of the fcc-bcc interface uses a single array of dislocations which are pure screw only at one limit of the lattice parameter ratio. However, the reason we are able to use screws, and ACO are not, goes to the core of the whole dispute. An array of screw dislocations is just a special case of the gllssile (i.e., conservative in the sense defined above) boundary which we advocate, and is not a sessile boundary in the sense which ACO insist is a central feature of their model.

Shape Chan~es Associated with Growth

We have shown above that the dilemma of atom fluxes or emission of numerous prismatic loops can not be avoided in the ACO model by mechanisms (ii) and (iii) of HEA, nor indeed by any conceivable alternative. The observations of surface tilt whether single or double (roof top) confirm that the atomic displacements produce a shear; no other explanation of surface tilt has been suggested. To our knowledge, no account given by ACO shows how their model relates to surface observations, although they are very ready to point out discrepancies between experimental observations and the very detailed predictions of martenslte crystallography. An example is provided by the discussion of shape changes in (I). We believe almost all the statements made here are incorrect, but even if correct, they would provide no rationale for the observed tilts.

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The Broad Faces of Y A1-A~ and 0 ' A1-Cu

Our views on AI-Ag are broadly similar to those of Dahmen (9) and of Howe et al (4). The interface is essentially martensitic in the sense defined above; its displacement is conservative and, although its velocity is diffusion-controlled, it produces a shape change which is substantially the same as that of the corresponding martensitic transformation from f.c.c, to h.c.p. Except for trivial differences in terminology, we are also largely in agreement with the description of Y AI-Ag precipitation in a major ACO review [(13) of (I), pp. 388-90]. Unfortunately, this sensible description seems to have been abandoned; in no sense is this interface "epitaxially coherent = sessile, partially coherent" as stated in AR2. The quotation from ref (33) of (I) is agreed, but does not apply to this particular case.

The interface between e' precipitates and the AI-Cu matrix is different from the AI-Ag case since there is no matching plane of the two lattices and sessile edge misfit dislocations with Burgers vectors of [I00] and [010] produce a macroscopic match in the (001) interface. In agreement with this, there is no evidence of a macroscopic shear produced by the growth of such particles, and indeed if the misfit normal to the plane of the disc is compensated by edge misfit dislocations with [001] Burgers vectors, as often observed, there should be no net shape change.

Summary

We are challenged in (I) to state what we consider to be the assumptions of the ACO model and why we think they are mutually incompatible and we now respond to this invitation.

The original model, presented in a valuable paper [(18) of (I)], assumed plate morphology to be a growth phenomenon resulting from the formation of virtually immobile, partly coherent, planar interfaces. Growth parallel to the plane of the plate was considered to be rapid because the interfaces are disordered, and thickening was attributed to the migration of ledges, with disordered riser interfaces. The immobility of the terrace (habit plane) interface was attributed to its dislocation structure and the necessity for climb. Strain energy was considered not to influence the morphology. In a second important review paper [(13) of (i)] the rate of normal growth was stated to be restricted by structural considerations in the regions of forced coherence. Sessile dislocations were still postulated in the interface structure, but it was implied that the ledge model obviated in some way the necessity for them to climb as the plate thickened.

The inconsistencies to which we have drawn attention are:

i) Disordered riser interfaces cannot co-exist with partly coherent habit plane interfaces, and could not produce a shape change with an appreciable shear component. [The riser interfaces were later considered to be partly coherent, but as shown above and in more detail in (7) this would carry the implication that the average shape deformation is zero.]

2) Immobility of the M atoms is not reconcilable with an interface which includes large linear misfits in particular directions in its plane, or indeed with any interface containing sessile misfit dislocations. [Several attempts have been made by ACO to escape this dilemma. In ARI, it was dismissed under paragraph 9CE as making no sense. The problem was admitted in AR2 but glossed over by invoking mechanism (i) of HEA. When confronted with the implications of mechanism (i), they now write in (i) about "fully conservative mechanisms such as (ii) which permit transfer of misfit dislocations from terraces..." Mechanism (ii) in fact requires climb, as shown in Fig. I. It is obvious that there is no way of avoiding this problem.]

3) If there is a shape deformation, steps on a planar interface must be fully coherent. Superledges of this type would have impossibly high energies.

4) It is implicit in our discussion that the role of strain energy cannot be ignored in any particle in which the stress free strains of a shape change are constrained by a surrounding matrix. Unless the particle is plate-shaped with a habit parallel to the invariant plane of the macroscopic shape deformation, its energy will be very large.

References

i. H.I. Aaronson etal. Scripta Metall., this issue. 2. C.M. Sargent and A.E. Purdy, Phil.Mag.,32, 27 (1975). 3. J.W. Christian, Phase Transformations, Institution Metall. Conf. Series 3, No. II, vol. 2,

p.II-29 (1979).

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4. J.M. Howe, U. Dahmen and A. Gronsky, Phil.~ag., 56, 31 (1987). 5- W. Bollmann, phys.stat.sol.(a), 21, 543 (1974). 6. K.M. Knowles and D.A. Smith, Acta Cryst., A38, 34 (1982). 7. J.W. Christian, Bainite Conf., Chicago, (1988) to be published. 8. M.G. Hall and H.I. Aaronson, Acta Metall., 34, 1409 (1986). 9. U. Dahmen, Scripta Metall. 21, 1027 (1988).

T S

u / Figure i.

To illustrate why mechanism (ii) of HEA involves climb.