Growth mechanisms of AuCu II plates

GROWTH

MECHANISMS

OF AuCu II PLATES

H. 1. .iARONSON Department of Metallurgical Engineerin,.0 Michigan Technological University. Houghton. MI 39931. U.S..-\. and Ii. R. KINSXL~X

Metallurgy Department. Engineering and Research Staff. Ford Motor Company. P.O. Box 3053. Dearborn. Ml -!Sl?l. U.S.A. (Rirc Pedrazu and Klttl (.4cra Jfer. 24. Y35. 1976). These rates are 31 orders of magnitude smaller than those estimated for incoherent AuCu II: disordered matrix (interphase) boundaries. Durin g growth. the interphase boundaries are deduced to be coherent. At the edges of plates. coherency is maintained because the succession of ledges is too rapid to give the misfit dislocation structure time to develop. At the broad faces of plates. the closely spaced twins (initially. cu. 50 f, apart) in the AuCu II greatly diminish the driving force for introduction of misfit dislocations until substantial coarsening of the twinned structure has occurred. Ledge growth at the broad faces is inhibited by the requirement that such growth occur parallel to the twins; this effect appears responsible for the order of magnitude larger spacing between ledges at the broad faces than at the edges of the plates and for the growth of AuCu II crystals as plates rather than as idiomorphs. The principal features of AuCu II growth were shown to be consistent with a massive rather than a martensitic mode of transformation. R&sum&-Les mecanismes de croissance des plaques de CuAu II sont considertes 6 partir drs mesures de leur vitesses d’allongement et d’epaississment rapportees par Pedraza et Kitti (.&cl .Lfer. 24. S35. 1976). Ces vitesses sont plus basses par 3-t ordres de grandeur que celles estimies pour le AuCu II incohirent: des limites de matrice (l’interphase) d&ordonnkes. II est dkduit qu’au tours de la croissance les limites des interphases sont cohirentes. Aux bords des plaques la coherence est conservee h cause de la succession trop rapide des corniches ce qui ne permet pas 6 la structure non-adapt& de la dislocation le temps de divelopper. AU larges faces des plaques Ies macles ttroitement espactes (initialement. cu. 50 A d’tiart) dans le AuCu II diminuement fortement la force motrice pour I’introduction de dislocations non-adaptees jusqu’h ce qu’un grossissement important de la structure ma&e ait eu lieu. La croissance de comiches aux faces larges est empechee par la condition que telle diveloppement prenne lieu paralltlement aux macles: cet effet semble 2tre responsable pour un espacement dix fois plus grand entre les comiches aux faces larges qu’aux bords des plaques. et pour la croissance des cristaux de AuCu II en forme de plaques plutBt qu’en forme d’idiomorphes. II est montrt? que les traits principaux de la croissance de AuCu II s’accordent avec une mode de transformation massive plutBt que martensitique. von der gemessenen LPngen- und Dickenwachstumsgeschwindigkeiten (Pedraza und Kittl, Acto Jfet. 24. S35, 1976) werden die Wachstumsmechanismen von AuCu-II-Platten betrachtet. Diese Geschwindigkeiten sind drei bis vier GraDenordnungen kleiner als diejenigen. die Fir (Interphasen-)Grenzen zwischen inkohirenten AuCu-II und entordneter Matrix abgeschitzt werden. Man findet, dal3 die Interphasengrenzen wlhrend des Wachsens kohlrent sind. An den Kanten der Platten wird die Kohlrenz aufrechterhalten. da die Folge der Stufen zu schnell ist fiir die Entwickluno einer Struktur von Misfitversetzungen. Die eng beieinanderleigenden Zwillinge (anfinglich. ca. 50 4 Abstand) an den breiten Seiten der Platten vermindem die treibende Kraft fiir die Einf_ihrung von Misfitversetzungen im XuCu-II aeitgehend. bis eine betrlchtliche Vergrbberung der verzwillingten Struktur entstanden ist. Das Stufenwachstum ist an der breiten Seite wegen der Forderung verhindert. dal3 dieses Wachstum parallel zu den Zwillingen erfolgt. Dieser Effekt scheint daftit verantwortlich zu sein. da13 der Abstand von Stufen an der breiten Seiten urn eine GrBDenordnung griit3er ist als an den Kanten der Platten und dal3 AuCu-II-Kristalle plattenfijrmig anstatt idiomorph wachsen. Die wesentlichen Ziige des AuCu-II-Wachstums sind konsistent eher mit einer massiven anstatt einer martensitischen .Art der Umwandlung. Zusammenfassung-Ausgehend

INTRODCCTION

The growth mechanisms of AuCu II plates are of unusual interest since they appear to incorporate features of two quite different types of transformation.

martensitic and the massive. A study of the growth kinetics and morphology of AuCu II plates [l, 21. in which the present authors had the opportunity to participate. thus provides an unusually

the

367

x3

AARONSON

AS0

KINSM;\W:

GROWTH MECHANISMS OF AuCu II PLATES

interesting basis upon which to develop a better understanding nor oniy of this transformation per se but also of certain fundamental aspects of both martensitic and massive transformations. The temperature range of the AuCu II phase field is approximately 385’ to rtlO’C (3): bv quenching through the narrow disordered Au-Cu + AuCu II regions or by using an equiatomic alloy-. the formation of AuCu II from disordered Au-Cu can take place without a change in composition. AuCu II is a face centered orthorhombic long period superlattice of the following desc~ption [4.5]. Ten f.c.0. cells lie side by side parallel to the a-axis; atomic planes perpendicular to the c-axis are alternately pure Au and pure Cu. but the stacking order of the planes is reversed every five cells. The a- and b-parameters of the f.c.o. cells are slightly unequal, being 3.976 and 3.963 A as compared with 3.872 A for the lattice parameter of the disordered F.c.c. Au-Cu matrix; the c-parameter, at 3.678A, is noticeably shorter, and thus the rows of cells are slightly flattened in one direction perpendicular to their length. Presumably as a result of the eiastic strain energy associated with the t~~nsfo~ation, individual AuCu II plates are finely twinned, on a [I10jfc, plane [6-S]. The c-axes of adjacent twins are nearly normal to each other [S-10]. thereby minimizing their combined strain energy [I 1, 12-j. Initially the twins are only ca. 50 A thick (91, but subsequent coarsening [9,2] increases their thickness by up to two orders of magnitude [7]. Perhaps for the same reason. AuCu II plates nucleated at grain boundaries appear in pairs [ 1) and those. formed intragranularly usually develop in sets of four [13]; “midribs” separate plates grouped in this manner {I, 13-J. Amongst the latter groupings, eight different orientations (because each plate contains two sets of twins) are observed [13]. The orientation refationships between AuCu II and disordered Au-& are within a few degrees of cube planes and directions in each lattice being parallel to each other, and the average habit plane of the plates is within a few degrees of {l 10tdi5[13]. AuCu II plates form an invariant plane strain surface relief. Both the surface relief and the transformation crystailography are very accurately described by the phenomenonological theory of martensite (131. OF GR0~~l-H KIXETICS DATA [ I.21 Data on the lengthening and thickening kinetics of AuCu II plates growing into a disordered Au-Cu matrix were obtained by Pedraza and Kittl [I] and by Pedraza [2] by means of hot stage optical microscopy. Lengthening of an individual plate proceeds at a constant rate which increases rapidly with undercooiing (AT) beIow the critical temperature. At a given AT, however, the lengthening rate may vary from one plate to another by as much as an order of magnitude. In the range of undercoolings investigated (AT = 3 - 22 K). lengthening rates varied from SUMMARY

x 10V6cmisec. Thickening kinetics exhibited l-22 similar behavior, but were also found to be dependent upon the distance from the plate tip at which measurements are made. At distances well removed from the tips the sides of the plates are accurately parallel and thicken~g can evidently be very slow. Between the sharply curved tips and the parallel-sided region, however, the sides of the plates form an angle of j--13”, and over the range AT= 3.80 - 11.75’K. rate from 3.5 to the thickening varies 29 x IO-’ cm;sec. and hence averages about an order of magnitude lower than the lengthening rate. NOiMENCLATLRE Because of uncertainty as to whether the twins form behind the advancing order:disordcr boundary or whether they represent separate plates nucleated in the disordered titrix at this boundary. it is necessary to define two sets of growth rates. as shown in Fig. 1. G) and G, represent the lengthening and thickening rates of overall AuCu II plates, while gr and gt are the lengthening and thickening rates of individual twins. Although gi and gt are without meaning if the ex post facto mechanism of twin formation is correct. these terms are usefu1 in discussing the possibilities of the nucleated twin mechanism. In the latter connection, it should be noted that gr is the individual growth rate related, approximately by cos 60”. to G,, whereas g1 provides, on the same reIationship. and together with the kinetics of twin nucleation, the basis for understanding G *. BASIC DATA FOR ANALYSES Dz~lisiun data

Published data on Do,, the pre-exponential constant, and AH,. the activation enthalpy for volume diffusion, in disordered and ordered AuCu are summarized in Table 1. On the usual criteria [14], these data are of uncertain value, the AH, and especially the Dow vaiues being considerabiy lower than expected. Accordingfy, interdiffusion coefficients in disordered At&u alloys, determined by Badia [IS] by means of electron microprobe analysis at four temperatures in the range 733-857X. were used to compute least squares values of D,,<.and AH, at the equiatomic composition, yielding the following

Fig. 1. Definitions of lengthening (G,) and thickening (G,) rates of an AuCu II plate and of lengthening (gl) and of thickening (9,) rates of twin elements v;ithin this plate.

,L-\RONSON

.ASD

KINSM4NN:

GROWTH

MECH;\SISMS

369

OF .AuCu II PLATES

Table 1. Published Amhenius Constants for volume diffusion in disordered

Au-Cu

dlO)S

Cllmposltion

In~c~t1~2torc

Pinnel and Bennett [ 1S] Pure Cu pure Au Toumon and Kuczinski [ 161 jO:50 Khobaib and Gupta [ 171 Purs cu r\ucu

RT).

_W..lcal mole1

secl

23.600 39.000 13.630

1.5 x lo-’ 3 x 10-I 1.36 x lo-”

have the same elastic constants:

expression for D,. the volume diffusivity: D,. = 1.28.eup(--44.200

&,lcm-

AF =

ill

r

Eb’k:,)’ -,

I-r

These values of the constants are more appropriate: it seems possible that the other values recorded in Table 1 were affected by short-circuit diffusion. No useful data were found for D, in any of the ordered temperature regimes. Hence, to approximate the diffusivit!- at incoherent AuCu 1I:disordered matrix boundaries. Dh. the usual approximations [19] for f.c.c. metals are applied to equation (I), rather than to an average of the volume diffusivities in both phases. D, is taken to be about the same as that for volume diffusion and AH, is assumed to be approximately one-half AH,.:

where E = Young’s modulus = 1.08 x 10” erg, cm3 [21]. L.= molar \olumr of AuCu II = 5.74 cm’ mole and 1’= Poisson’s ratio % 0.3. yielding AFc = 0.1 cal mole. Once Ai- is 3 K or greater (the AT range used experimentally). AF, is sufficiently small relative to AF to be disregarded. and hence equation (3) expresses the driving force for AuCu II formation with adequate accuracy.

D,, = 1.28~esp( -22,OOO’RT).

Although incoherent boundaries may be present to a significant extent only on the non-Widmanstatten morphology sometimes found at grain boundaries in the disordered AWCU matrix [ 11. this calculation nonetheless provides a “baseline” for subsequent considerations of growth kinetics. The usual relationship for the diffusional migration kinetics of an interphase boundary in the absence of a composition change may be written [22]:

(3

Driciny force fbr r\uCu II formation

From the thermodynamic data of Orr et al. [ZO]. the free energy change accompanying the transformation from disordered Au-Cu to AuCu II may be estimated as: AF = -0.8~ATcal/mole

GRONTH

KINETICS OF INCOHERENT I\ucu II: DISORDERED M4TRIX BOUSDARIES

(3 G = civexp(AS,,R).exp(-AHb!Rr).(-~F.RT).

in the absence of strain energy. Pedraza [2] has calculated that the dilatational strain energy alone attending formation of a single coherent AuCu II plate is nearly two orders of magnitude higher than AF at AT = 10 K, and that the shear strain energy is about 15fold higher. However, elimination of the shear strain energy is readily accomplished if the transformation shear in adjacent twins is oppositely directed. as is observed [9]. The dilatational strain energy is reduced, though not eliminated, by the circumstance that the c-axes of adjacent twins are orthogonal to each other. Then, for example, the long a-axis is succeeded by the short c-axis. When viewed over a pair of adjacent twins, the average dilatational strain energy is considerably reduced, as previous investigators have recognized [5-12-J. Since the twin lamellae are initially only ca. 50 A thick [9], it seems a reasonable approximation to extend this averaging process to the third dimension. The volume change relative to the disordered matrix is only 0.00166, and on this basis the average stress-free linear transformation strain, of, = 0.00056. The molar strain energy viewed on this basis can now be calculated from Eshelb”s [‘l] relationship for the isotropic coherent stram energy when both matrix and product phases

where d = jump distance. v = vibration frequency in the jump direction. and A.Sb and AHb = activation entropy and enthalpy for transinterphase boundary diffusion. Following Shewmon [23]. this relationship may be written:

D = -_e0 6

Substituting equations d = 3.9 x lo-8.

-MbRT

(2) and

G,+= 2.91 x lo- exp

(3) and

letting

(9

Figure 2 shows that Gdis is 10-3-10-’ cm set within the range of AT’s used experimentally. The validity of equations (2) and (3) may be examined in approximate fashion by scaling the growth rates determined for the /I- z, transformation in Cu-Zn b>- Karlyn. Cahn and Cohen [25] to the driving force given bq equation (3). Gdir for 1, Cu-Zn normalized in this

370

AAROXSON

ASD

KINSMLI;\NN: GROWTH MECH.ANISXIS OF AuCu II PLATES

al. concluded that AuCu II plates lengthen by the repeated nucleation of twins. However. since the twinning dislocations will appear on alternate planes. resolution of these dislocations is unlikely. Further consideration of the possible mechanisms through which AuCu II plates may lengthen thus seems desirable. This question is first examined from the viewpoint of solid-solid nucleation theory. Because the matrix and product lattices are geometrically similar and are nearly identically oriented. it seems likely (as will be discussed later) that most of the boundaries involved in the nucleation of a twin in the matrix at a broad face of a previously formed twin will have a fully or partially coherent structure. Hence the transport mechanism operative during nucleation should be that of volume diffusion. The incubation time, r, for nucleation by this mechanism may be written [30]: t= Fig. 1. Calculated growth rate of incoherent (disordered) boundaries between AuCu II crystals and their disordered matrix vs undercooling (AT). manner at the mean undercooling of 7 K. or 672 K, is 1.8 x lo-‘cm/see. or just about twice that for AuCu II. That Gdis is higher in Cu-Zn is consistent

with the greater mobility of Zn than of Au atoms. This result suggests that equation (8) provides reasonable baseline data for the migration kinetics of incoherent AuCu II: disordered matrix (order:disorder) boundaries. ANALYSIS OF PLATE LENGTHENING KINETICS Does plate lengthening of twinned regions?

occur by repeated

nucleation

When an AuCu II plate lengthens, the question arises as to whether its profusion of twins formed behind the advancing order:disorder interface or nucleated in the disordered matrix at the advancing order:disorder boundary. The question has been extensively debated [6.8,9, 13.26-281. The principal evidence for post-transformation twinning seems to be the observation of Hunt and Pashley [9] that twins are not observed when the transformation occurs in foils with a { Ill) plane parallel to the surface. However. the smaller volume strain energy associated with transformation in a thin foil may have been responsible for this observation. Alternatively. Tong and Wayman [29] point out that the twins are invisible in { 1111 foils unless an exceptionally large tilt is imposed. Adrianovsky et al. [28] note that u/2 (110) twinning dislocations are to be expected at the order:disorder boundaries should the twins form behind the advancing interface. Transmission electron microscopic studies having failed to reveal any dislocations

at the interphase

boundaries,

Adrianovsky

et

8 kTy,,a4

K

o;.AF;.D,sU

07

(9)

where yip = interfacial energy of disordered nucleus: matrix boundaries, a r one lattice parameter, L’,= average volume/atom in AuCu II. AF, = volume free energy change attending formation of AuCu II. D, = volume diffusivity in the disordered matrix. .xu = mole fraction of solute in the matrix = 0.5. K = ratio of the volume of a faceted nucleus to that of a sphere of the same radius [30]. L = ratio of the disordered interphase boundary area of a faceted nucleus (only this area can accept or reject atoms) to that of a sphere of the same radius [30] and kT has its usual meaning. At a typical AT of 7 K, estimating ‘ix0 as 3OQerg/cm’ and taking D, from equation (1). r = 1.85 x 106sec. Assuming that the rate-controlling process in the lengthening of AuCu II plate is nucleation of the twins and that each twin is cu. 50 A thick[9], this yields a minimum lengthening rate of 2.70 x IO-i3 cm/set, or nearly eight orders of magnitude slower than the range of experimentally measured lengthening rates. An alternate assumption is that mass transport takes place by diffusion along disordered regions of the interphase boundaries. On this mechanism [31]:

where 6 = boundary thickness 2 one lattice parameter, II/ = cos-‘(1 - ~~./j.~~), yu = energy of an AuCu II twin boundary, D,, = interphase boundary diffusivity and xz8 = mole fraction of solute in the interphase boundary (assumed to be 0.5). Again taking the time for growth to be @r, the values of r. calculated (average) growth rate and average experimental growth rate are compared in Table 2. The calculated growth rates are again seen to be too small, but now by only 2-3 orders of magnitude. The principal numerical uncertainty in this calculation is in the estimation of D,,. Girifalco [32] suggests that assum-

URONSON

.XNDKINSXQN:

GROWTH

ing an activation energy for boundary dilTusion half that for volume diffusion may well be an underestimate even for a disordered interphase boundary in this system, since the presence of long-range order in one of the crystals forming these boundaries ought to reduce somewhat the kinetics of diffusion in the boundary. In order to bring the calculated and experimental growth rates into reasonable agreement. however. the activation energy for boundary diffusion must be reduced to 16,SOOcal:mole. a still less plausible (though by no means impossible) value for this quantity. Finally, it should be recalled that the r-based G’s were calculated under the assumption of an infinitely rapid growth rate. As discussed in the nest section, however. this assumption is not a good one. and thus the discrepancies between the measured and calculated lengthening rates in Table 2 represent minimum estimates. Hence the nucleation kinetics calcuiations indicate that twinning occurs after rather than during the transformation. A further argument in favor of this view is the remarkable straightness which the twins exhibit when photographed at high resolution prior to coarsening (see. e.g. Figs. 16 and 17 of Hunt and Pashley [9]). To explain this observation on the nucleation mechanism would require remarkably planar interphase boundaries. Such boundaries are normally found only in systems in which matching between the two phases is even better than in the present system, e.g. in the f.c.c. Y g h.c.p. K transformation in Cu-Si (33.34). The serrations produced by the twins in the broad faces of AuCu II plates also suggest that the twins formed subsequent to the transformation.

Plurr lengrhening

mechnnism

We now consider the lengthening process on the basis of the post-transformation twinning mechanism, and discuss it as a solely growth-controlled process. These considerations are necessarily begun with deduction of the structure of the boundary which evidently provides the “leading edge” of an AuCu II plate. Assume initially that this boundary is planar and is precisely described by { 110j.ord,i/[110: mat(where “ord” represents AuCu II and “mat” indicates the disordered matrix), rather than lying a few degrees away from this orientation [13], and that development of the equilibrium interfacial structure is kinetically feasible. Both of these assumptions will shortly be

XtECH\NIS\lS

OF AuCu II PLATES

relaxed. The assumption is also made that. within the present context. any alterations which the long period superlattice of AuCu II may effect in the equilibrium interfacial structure other than those associated with the lattice parameters, can be safely ignored. On this basis, two orthogonal arrays of misfit dislocations with Burgers vector u 2 (110) appear to be the appropriate interfacial structure. L’sing the lattice parameter data summarized in the Introduction and Brooks’ [3_‘] relationship for interdislocation spacing. at equilibrium the spacing between parallel dislocations in these arrays would be 79. 91 or 40 A. depending upon whether the I llO1,,, plane facing its disordered counterpart incorporates an N. h or c component of the f.c.0. lattice in addition to the usual diagonal. Three types of interface are thus possible and each contains two of these three arrays. When the requirement that : 110: planes in the two lattices be precisely parallel is relaxed. the resulting additional misorientation can be accomodated in either (or both) of two ways. One is simply the addition of a third array. orthogonal to the other two. whose Burgers vector lies normal to the boundary plane. A second is the introduction of structural lsdges in the interphase boundary. either one [36] or a few [37] atom layers high. The latter approach has been shown to be capable of explaining (apparent) habit planes deviating appreciably from a parallel pair of rational planes C36.373. Assuming that all twinning takes place behind the advancing order:disorder boundary, it appears at first glance that a misfit dislocation structure. probably not deviating too far from the equilibrium one. should be present at this boundary effectively throughout the growth process. AuCu II plates are often tens of micrometers wide and CCLX-fold longer [I. 131. On the considerations of van der Merwe [3S. 391. these dimensions seem far too large to permit maintenance of full coherency between the ordered and disordered lattices. When the details of the growth process are taken into account, however. a different picture emerges. It is now firmly established that partially or fully coherent interphase boundaries between crystals with different structures migrate by means of the ledge mechanism [JO]. As Faulkner and Ralph [-ll. 401 have shown. even when the crystal structures and orientations of the matrix and precipitate phases are the same but one phase is ordered and the other is

Table 1. Comparison of nucleation-controlled with experimental lengthening rates twin thickness (cm)

G,,,;cos 60. (cm.set)

Avg. G,,, (cm set)

j x lo-’

45 x 10m9

3 x lo-”

5 x lo-5 x lo-‘

4.5 x 10e3 1.6 x lo-‘

I X lo-’ 2 x lo-’

Assumed

7-(K) 615

670 665

AT(K) 4

9 14

T (set) 56

5.5 1.6

Average values of Gcrp[I. 21 are used. 7 calculated from equation (10) (boundary

3-1

diffusion-control

assumed).

372

AARONSON

ALLO

KINSMANN:

GROWTH

not this generalization still applies.* Each time that a ledge or ledges grow across the planar order:disorder boundary which serves as the leading face of a lengthening AuCu II plate. whatever misfit dislocation structure was present will be annihilated and a new structure will have to develop along the broad faces of the newly formed ledges. If the average rate at which successive ledges cross a given area of the boundary is high relative to the rate at which misfit dislocations are acquired. then a misfit dislocation structure will not appear until lengthening of the plate has been slowed by reduction of the rate of ledge formation. The average misfit across this boundary can be characterized as moderate, ranging from 0.023 to 0.051; acquisition of dislocations from the matrix [43] is a plausible but by no means the only reasonable mechanism [44] : the details of the dislocation acquisition process tend to be system-sensitive and their kinetics vary widely [Jo, 441. An estimate of the time available for the acquisition of misfit dislocations on the leading face of a lengthening AuCu II plate can be obtained from the measured lengthening kinetics if the assumption is made that the edges of the ledges are disordered. This is accomplished through the relationship [42] :

G, = hv.cos60a/b,

(11)

where h = ledge height, c = lateral velocity of the ledge calculated from equation (8), b = interledge spacing and cos6O” is the trigonometric correction factor for the angle between the gr and GI vectors (Fig. I). Table 3 gives the b/h ratio at representative undercoolings computed in this manner. Taking the minimum value of h to be one lattice parameter or ‘z 4 A. the values of the minimum interledge spacing, bm,n.are also shown in this table. In the final entry, the time required for a ledge to traverse the distance bminat rate Gdir is given. These times range from hundredths to thousandths of a second. Experimental observations of ledge heights on precipitate plates so far reported usually fall in the range 5-loO A ([40]) (with the exception of the quite irregular proeutectoid ferrite plates, where ledge heights in the micrometer range are found [45]). Even if a ledge heighht of 100 A *Presumably this concept will begin to fail when the difference in long-range order between the two phases falls below some critical level, whose exact value is likely to depend upon both system characteristics and the driving force for growth. Cahn et al. [42] have pretiously made a somewhat similar suggestion about a different type of interface.

lMECHANISMS OF AuCu II PLATES

is assumed, a given area of these boundaries will remain in contact with the matrix for less than 2 sec. For misfits in the range encountered in the AuCu II transformation, this time, experience suggests [43,44]. is quite insufficient for the acquisition of a misfit dislocation structure. The conclusion that the leading edge of AuCu II plates will be fully coherent during growth is consistent wiilth the electron microscopic observations of Adrianorsky et al. [Xj. However, a more detailed study, directed specifically at such boundaries, is necessary to test this conclusion properly. The values of bminin Table 3 fall in the range experimentally observed on 0’ Al-Cu plates [46]. Screw dislocations extending through both disordered and ordered phases and nucleation at the more highly stressed edges of these interfaces are plausible sites for the formation of the ledges. Experience now makes clear [lo, 461, however, that in a particular system the actual source(s) of ledges should be sought experimentally by means of transmission electron microscopy; the number of verified sources available is now too large to permit reliable deduction by indirect means unless a special crystallographic situation obtains which “automatically” provides the ledges, as do the misfit dislocations in f.c.c.- h.c.p. transformations [47,48]. ANALYSIS OF PLATE THICKENING KINETICS Unlike the lengthening of AuCu II plates, there need be little concern about driving force losses to volume strain energy during thickening of these plates. The stack of parallel, closely >Taced twins of which each is composed forms a continuously and almost exactly self-compensating system, as previously indicated. Even if the twins must be individually nucleated, the kinetics of this process plays no role in plate thickening (Fig. 1). However, the observation that G1 > 10 G, despite the fact that, if the twin aspect ratios can be translated directly into growth rate ratios, g1 = 10-100 gl, provides a useful nucleation the argument against additional mechanism of twin formation. Because the initial width of the twins. ca. 50A (9), is little more than the smallest equilibrium spacing between misfit dislocations, 40 A, maintenance of full coherency along the broad faces of AuCu II plates is to be expected until coarsening of the twinned structure has advanced to the point where the com-

Table 3. Calculation of ledge structure from lengthening kinetics data Z

Traverse

G&S (cm set)

G,(exeW (cm/se@

b,‘h

3 X 10-6 8 x 1O-6 2 x 10-s

1 x 103 8 X 10’ 4 x 10’

675

4

5.9 x 10d3

670 665

9 14

1.2 x 1o-2 1.6 x IO-’

cm

time (set)

4 x 10-j 3 x 1o-5 2 x 10-5

7 x 10-S 2 x 1o-3 1 x lo-”

bmi,.

.A.‘,RONSOS

ASD

KISSM;\NS:

GROWTH

Fig. 3. Ledges on a twinned broad face of an AuCu II plate. produced by a screw dislocation passing through the boundary. The possibility that ledges do not lengthen at the same rate along adjacent twins has been incorporated in this sketch.

bination of the buildup of coherency stresses and the kinetics of the operative dislocation acquisition mechanism(s) cause emplacement of a misfit dislocation structure. (The prediction of an initially fully coherent structure parallels that sometimes made for internally twinned martensite plates [-l9].) This prediction ought to be subject to experimental test. since the twinned structure should interfere less with transmission electron microscopic observations as it coarsens. The twins have been predicted and experimentally shown to produce a serrated interphase boundary morphology [9]. Until the broad faces of a plate become approximately parallel to each other. these serrations might be expected to provide a source of ledges. all supposedly directed toward the edges of the plates. However, while the serrations produced by the twins are likely to assist the nucleation of ledges. the twins themselves interfere seriously with the edgewise growth of the ledges. A ledge attempting to grow parallel to the length of a plate would be required to twin each time it crossed a twin boundary in its AuCu II “substrate” in order to maintain the transformation strain energy at a minimum lev-el. A mechanism which would induce such profuse and precisely timed twinning in a lengthening ledge, particularly in one multiple atomic planes high, is not apparent. It thus appears that ledges which can lengthen parallel to the twins enjoy a special kinetic ad\-antage over those which cannot. Fulfillment of this condition requires that the ledges nucleate at or evolve from an extended source which crosses the twin boundaries at a reasonably large angle. The screw dislocation passing through both phases as shown in Fig. 3 provides a ledge generation mechanism of this type. Another source of viable ledges on the broad faces of AuCu II plates may be the fault interfaces generated when twins parallel to another habit plane of

SLECHANISXIS OF AuCu II PLATES

the same form develop within a single plate. Both Smith and Bowles [13] and Hunt and Pashley [9] show examples of such behavior. Where the fault interface intercepts the broad faces a row of steps capable of propagating on both sides of the fault line will be produced. This fault line will normally lie at a substantial angle to the twins on either side of it and hence ledge growth can take place simultaneously along the lengths of these twins. Possibly the reduction in thickening kinetics which develops when the broad faces of AuCu II plates become parallel is associated with annealing out of the fault planes during the twin coarsening process vvhich develops extensively during the later stages of growth [Z]. .A third source of useful ledges. of a similar nature. are the midrib between plates formed in pairs and other impingement boundaries between plates. Such interfaces have been shown to be important sources of ledges during the thickening of 0’ AlXu plates [46.50]. None of these three proposed sources of ledges on the broad faces of AuCu II plates is expected to occur frequently. This deduction is supported by the results of applying equation (11) to the experimental data on plate thickening. as summarized in Table 4. The values of h h and hence of b,i, are about an order of magnitude larger than those which obtain during lengthening. Ledges more than a micrometer apart clearly do not derive from the finely spaced twins. The restriction on ledge growth transverse to twin boundaries on the broad faces of AuCu II plates is suggested to be the basic reason why these crystals, which can form a partially or fully coherent interphase boundary at a sufficient number of orientations to enclose themselves completely. develop as plates rather than as idiomorphs. The structure of the interphase boundaries. the marked difference in long-range order across them and the relative scarcity of ledges explain why both lengthening and thickening kinetics of AuCu II plates are several orders of magnitude lower than the rates estimated for a disordered interphase boundary in this system. These considerations are fully consistent with a general theory of precipitate morphology [44].

THE ROLE OF SHEAR IN THE GROWTH OF AuCu II PLATES Smith and Bowles [ 131 have shown that the crystallography of this transformation and the invariant plane strain surface reliefs which it produces are in remarkably accurate agreement with the predictions of the phenomenological theory of martensite. This

Table 1. Calculation of ledge structure from thickening kinetics data

Gdi3 Icm set)

\-77>

Traverse

G, (exptl) (cm,sec)

b,,, (cm)

time lsec)

675

4

5.9 x 10m3

3 x lo-’

1 x 10’

1 x 1o-4

7 x lo-’

670

9

1.3 x 10-J

I x 1o-6

6 x 10’

3 x 1o-1

2 x lo-’

37-1

AARONSON

AND

KINSMANX:

GROWTH tilECHANIStilS OF AuCu II PLATES

result has been interpreted to mean that AuCu II plates grow by a shear mechanism [13,51.52]. However, the ordering process requires diffusion. Even though single atomic jumps may suffice. the randomal (though biased) nature of the diffusion process is inconsistent with a shear transformation, in which each atom in the matrix is required to take up a predestined site within the product phase, to within the radius of action of a shuffle. The finding that the phenomonological theory of martensite accurately describes the crystallography and surface relief effect of a transformation to which it cannot be applicable provides further support for the authors’ view [44] that such agreement is a necessary but not a sufficient condition for the identification of a shear transformation mechanism. The proposal that the AuCu II transformation involves conservation of lattice points rather than of atoms[51.53] seems a misuse of the phenomenological theory, for despite the disavowals of mechanistic significance [5 1,521, this theory has been shown to provide a good accounting for such physical aspects of the martensite transformation as the lattice invariant deformation and certain aspects of interfacial structure [52]. That the phenomenological theory is able to predict in some cases the crystallography of diffusional phase transformations has been ascribed to its apparent ability to select conjugate habit planes of minimum misfit strain energy [54] ; these planes are expected to appear as facets on the critical nuclei of diffusional transformations [30] and thereby directly determine orientation relationships and somewhat less directly the habit planes. AuCu II PLATE FORXATION AS A MASSIVE TRANSFORMATION A phase transformation which does not involve a change in composition and yet cannot take place by shear must, by elimination, be a massive transformation. The beautifully shaped AuCu II plates and the exact orientation relationships which they maintain with respect to their matrix crystals are, of course, just the characteristics whose absence is currently considered to be an important attribute of the massive transformation (see, particularly. a comprehensive review by Massalski [U]). However, Aaronson et nl. [56] have pointed out that the nature of the diffusional nucleation process, whether it occurs with or without a composition change, requires that specific orientation relationships develop if the lattices of the matrix and product phases are at all crystallographitally compatible. Plichta er a/. [57] have recently made a parametric analysis of nucleation kinetics in the p-x,,, transformation in Cu-38 A,‘0 Zn and have concluded that faceted nuclei, i.e. those with an orientation relationship which allows closely matching conjugate habit planes to develop at least at one boundary orientation, enjoy an overwhelming kinetic advantage over nuclei whose interphase boundaries have a disordered structure at all boundary orien-

tations. Plichta er (I/. noted that there is some crystallographic evidence [j&59] that orientation relationships are apparently absent in some massive transformations. They suggested, however. that such results may be a consequence of the inherent difficulty of “proving a negative” and that interfacial structure studies will be required to test properly their orientation relationship prediction, which is basically an essentially straightforward consequence of Gibbs’ [60] nucleation theory. When a phase transformation between two not very closely matching crystal lattices such as f.c.c. and b.c.c. takes place a relatively low undercooling. the ratio of the migration rates of disordered interphase boundaries to those of dislocation boundaries is relatively low [44.61] and in the interiors of matrix gains idiomorphs are the predominant morphology [61]. In a massive transformation between two such lattices, growth rates are customarily very high [55] and thus a very small number of nuclei can transform completely a typical metallographic specimen. Hence idiomorphs may be the predominant. though not necessarily the only [62,63] morphology formed prior to termination of transformation. Even when plate or needle morphologies develop more extensively as a consequence of particularly rapid cooling through the transformation region. the general impingement which attends essentially complete transformation of a specimen and the subsequent rapid rearrangements of the impingement’ boundaries thus formed into lower energy arrangements can largely or entirely eliminate the traces of the Widmanstatten morphologies originally present [64]. In the AuCu II transformation, however, the lattices match sufficiently well so that the product crystals can be completely enclosed by partially or fully coherent boundaries. The resulting requirement of boundary migration by the ledge mechanism, and the additional kinetic difficulty of producing viable ledges at boundary orientations where dense stacks of twins in AuCu II are in contact with the boundary results. particularly when combined with the small driving force available, in low. boundary orientation-dependent growth rates. Hence plate-shaped crystals are readily formed. and are easily preserved in an otherwise untransformed matrix even during cooling to room temperature at quite slow rates. Hence the massive transformation of disordered 50 Au: 50 Cu to AuCu II differs from “the usual” massive transformation only in that the driving force is somewhat lower and the boundary mobility is both very much smaller and boundary orientation-dependent, thereby resulting in an unaccustomed product morphology. However, the unit atomic process is the usual one in a massive transformation. i.e. diffusional jumps across disordered interphase boundaries. During the formation of AuCu II, however. the disordered boundaries are restricted to the edges of ledges rather than occupying a large proportion of the total interphase boundary area.

,\.-\RONSON

.&NDKIX?.l;\XX:

GROWTH

MECHA%lS~lS

OF -\uCu II PL.\TES

3-j

AuCu II crystals debc’lop as plates despirt thz fact that they can be entir:l> enclosed b> partialI> or fuli> Analysis of data on the lengthening and thickening coherent interphase boundaries. kinetics of AuCu II plates in an equiatomic Au-Cu (6) The presence of a partially or fully coherent alloy reported by Kitti and Pedraza [l. 21 has led to *structure at the interphase boundaries of .\uCu II the following conclusions: crystals and the relative scarcity of ledges at these (1) The volume strain energy retarding the transformation is sufticientlv small. as the result of the hza- boundaries are the basic reasons why the growth rates vity twinned structure of AuCu II plates and the for- of AuCu II crystals are so much slower than those mation of adjacent twins with c-axes orthogonat to estimated for incoherent ordrr:disorder boundaries. (71 It was emphasized that Aufu II formation caneach other. so that it causes a negligible reduction not take place by shear. despite the excellent agreein the chemical free energy change (due to ordering) ment reported by Smith and Boivles [ 131 bst\veen the which drives the transformation. (2) Using the experimental data of Badia (IS) on measured crystallography and surface reliefs of AuCu II plates and the predictions of the phenomenological volume diffusivities in disordered equiatomic Au-Cu theory of martensite. because of the diffusion required to make a reasonable estimate of the interphase for ordering. boundary diffusivity. growth rates of disordered (8) The characteristics of AuCu II plate formation matrix:AuCu II boundaries are calculated to be in were shown to be consistent with an alternate view the range lo-‘-IO-’ cm.sec within the temperature region investigated by Pedraza and Kittl. These rates of the massive transformation currently being advoare 3-4 orders of magnitude higher than those experi- cated [56. 571. mentally observed. (3) The question of whether the twins in AuCu II .-lci;nowi~~~~~itgemmrs-Both authors wish to acknouledge the kindness of the Comision National ds Encrgia Atomica plates form by repeated nucleation or by twinning of Argentina in supporting visits by them to the Comision’s behind the advancing interphase boundary was examBuenos Aires research laboratories, in part to participate ined. The incubation time for nucleation by volume in a joint investigation of the growth kinetics and difftfusion was found to be far too long and that for mechanisms of AuCu II plates. the hospitalit! and exceinucleation by boundary diffusion was shown to be lent arrangements for these stimulating visits provided by Drs. Jorge Sabato and Jorge Kittl. and the courtesy of somewhat too long to explain the measured IengthenDrs. Kittl and Antonio Psdraza in making available to ing rates. The straightness of the twins. demonstrated them the esperimental data from this investi_eation for the by Hunt and Pashleq [9] with high resolution eiec- preparation of the interpretation presented m this paper. tron microscopy, is difficult to explain on the nucleaThe contribution of H.U.. begun while on the staff of the Ford Scientific Laboratory. was completed at bfichigan tion mechanism. And “growth kinetics” of individually nucleated twins appear to be inconsistent with Technological University with the support of the Army Research Office (Durham) under Grant No. D.+AROthose of the overall plates. Hence it was concluded D-3 I- 124-73-G l-!-i. that twinning occurs subsequent to transformation. (4) At equilibrium. the interface plane in contact REFERENCES with the disordered matrix which serves as the leading 1. A. J. Pedraza and J. 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376

AARONSON

XT0

KINSMANN:

GROWTH

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I

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.MECH.ANISMS OF AuCu II PLATES

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