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Crystallographic relationships in aged copper-beryllium alloys
CRYSTALLOGRAPHIC
RELATIONSHIPS J. S. BOWLESt
IN AGED and W. J. McG.
COPPER-BERYLLIUM
ALLOYS*
TEGARTS
The precipitation of CuBe from copper-beryllium (Ysolid solutions involves a transformation from a facecentered cubic (F) to a body-centered cubic (B) structure. The habit plane of the relief effects produced on polished surfaces by this transformation has been determined. It is not a simple lattice plane but is close to (110)~. The association between variants of the habit plane and variants of the orientation relationship has been found to be such that a variant of the habit plane is associated with that variant of the (1OO)z pole which lies in the same stereographic triangle. With these results the precipitation reaction has been analysed using the geometrical theory developed for martensitic transformations by Bowles and Mackenzie. 1,226The theory then predicts the orientation relationship determined by Guy.” RELATIONS
CRISTALLOGRAPHIQUES
DANS
LES
ALLIAGES
CUIVRE-GLUCINIUM
La pr&ipitation de CuBe B partir des solutions solides cuivre-glucinium comporte une transformation du systeme cubique 2 faces centrees (F) dans le systeme cubique centre (B). Le plan d’habitat des reliefs sur une surface polie produits par cette transformation a et6 determine; ce n’est pas un plan simple mais il est voisin de (110)~. Le rapport entre les variantes du plan d’habitat et les variantes de la relation d’orientation est tel qu’une variante du plan d’habitat est allike a la variante du p61e (100)~ qui appartient au meme triangle stereographique. A l’aide de ces r&ultats la precipitation est analysee par la theorie geometrique developpee pour les transformations martensitiques par Bowles et Mackenzie. r,%8 La thCorie prtvoit les orientations observbes par Guy.” EMBER DIE
KRISTALLOGRAFISCHEN ZUSAMMENHANGE KUPFER-BERYLLIUM LEGIERUNGEN
BEI
AusGELAGERTEN
Die Ausscheidung von CuBe aus (Y Mischkristallen des Systems Kupfer-Beryllium ist gleichbedeutend mit einer Umwandlung einer kubisch-flachenzentrierten (F) in eine kubisch-raumzentrierte (B) Struktur. Es wurde die Habitusebene der Reliefwirkungen bestimmt, die diese Umwandlung auf polierten Flachen hinter&St. Die Habitusebene ist keine einfache Gitterebene, sie liegt jedoch nahe bei (110)~. Der Zusammenhang zwischen Veranderungen der Habitusebene wie such der Orientierungsbeziehung ist derart, dass eine Anderung der Habitusebene von der Anderung desjenigen (lOO)B-Poles begleitet wird, der im gleichen Bestimmungsdreieck leigt. Mit diesen Ergebnissen wurden die Ausscheidungsreaktionen auf Grund der geometrischen Theorie der Die Theorie sagt dann die von Guy” Martensitumwandlung von Bowles und Mackenzie.‘s2,6 analysiert. bestimmte Orientierungsbeziehung voraus.
1. INTRODUCTION
Although the details of the atomic mechanisms of martensitic transformations have not yet been elucidated, the phenomenological theory of the over-all atomic displacements has been developed to the point where accurate predictions can be made of habit planes, orientation relationships and the homogeneous strains accompanying the transformations.1-4 For the diffusioncontrolled transformations no such quantitative theory exists. X-ray studies have disclosed a wealth of information about intermediate structural states produced during these transformations, but no quantitative theory of the crystallographic relations has yet been derived. Some time ago the suggestion was made by Bowles and Barrett5 that, during the stage of “coherence,” the geometry of the atomic processes involved in diffusion-controlled transformations may be identical with that involved in martensitic transformations. * Received
May 2,. 1955. t C.S.I.R.O., Division of Tribophysics, University of Melbourne, Carlton, N.3., Victoria! Australia; now New South Wales University of Technology,. High Street, Kensington, N.S.W., Australia. $ C.S.I.R.O., Division of Tribophysics, University of Melbourne, Carlton, N.3., Victoria, Australia. ACTA
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1955
At the time this suggestion was made, the only experimental evidence available to support it was the general similarity of the orientation relationships produced by the two kinds of transformation. However, the recent developments in the phenomenological theory of martensitic transformations enable a more satisfactory test to be made of the merits of this idea. If the geometry of the processes involved in the two kinds of transformation is identical, then the theory developed for martensitic transformations should be capable of predicting the geometrical features of diffusion-controlled transformations. The work to be described in this paper was carried out to test the validity of the predictions of the theory for the precipitation of CuBe from copperberyllium a! solid solutions. This particular precipitation process was selected for two reasons. First, the lattice transformation involved is f.c.c.-+b.c.c; the theory for this case has been worked out in detail6 and shown to yield predictions consistent with observation for a number of martensitic transformations. Second, the precipitation of CuBe is accompanied by the production of visible relief effects on the surfaces of polished specimens. These relief effects 590
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have been described as similar to those produced by martensitic transf~rmations7.8 and, in the present work, they are interpreted as having a similar origin. It is proposed that the relief effects arise as a direct consequence of the net change in shape accompanying the formation of plate-shaped crystals of CuBe. The phenomenalogical theory’sa has been developed from the proposal that the total atomic displacements produced by martensitic transformations can be described consistently by means of a homogeneous strain which is composed of an invariant plane strain and a small dilatation, followed by a shear, which is part of a twinning shear in the final lattice but which occurs inhomogeneously. It has been shown that these conditions determine the total strain, and hence the orientation relationship, in terms of a single parameter, tl. In applying the theory to a particular transformation, it is necessary to evaluate 0 by calculating the variation of the habit plane with @and determining that value, if any, for which the predicted and observed habit planes agree, The predicted habit plane variation for facecentered cubic to bay-centered cubic tmnsfo~tions has already been derived6 so it remains to determine the habit plane of copper-beryllium, i.e., the plane of the relief effects, and to evaluate 8. The experimental work is described in $52, 3. This consisted of determining the general indices of the habit plane and of &ding which variant of the habit plane is associated with a particular variant of the precipitate orientation. In $4 these results are used to evaluate 6’and tbc orientation relationship is then determined by substitution in the equations developed elsewhere.” The predicted orientation relationship is compared with that measured by Guy.‘r The notation used in this paper is the same as that used previously.“j”r6 2. EXPERIMENTAL
METHODS
The alloys used were prepared from oxygen-free high conductivity copper (> 99.98 wt. 7o copper) with traces f <0.005 wt. yo> of lead, tin, nickel and iron, and a copper-beryllium mother alloy (95.85 wt. ye copper, 4.14 wt. y0 beryllium) supplied by Brush Beryllium Corporation. Initially the melting procedure described by Guy, Barrett and Mehlr2 was followed but later the alloys were prepared by melting the mother alloy and the copper together in silica under an atmosphere of purified nitrogen” The solidified rods were swaged down from 12 mm diameter to 7-8 mm diameter with intermediate anneals at 800-850°C. The alloys used were restricted to the range 1.80-1.95 wt. $$c beryllium since it was found that relief effects were not produced in specimens containing smafler concentrations of beryllium. To avoid grain-boundary precipitation during the ageing treatments, all the experiments were performed on singIe crystals. Crystals made by the Bridgman technique were found to be badly cored, and therefore
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repeated zone melting was used to produce more homogeneous crystals. The specimens, sealed under nitrogen in a silica tube, were passed through a small coil fitted in the centre of a tube furnace. The coil was maintained above the liquidus temperature and the furnace at 800°C. After zone melting the specimens were maintained at 800°C for three days and then water-quenched. The single crystals produced in this way were not cored but only small lengths of uniform composition could be obtained. Specimens were cut from the single crystals with a jeweller’s hacksaw, lapped on Carborundum paper and then polished with diamond dust. They were then carefully electropolished in an orthophosphoric acid electrolytero for one half to three quarters of an hour to obtain a strain-free surface. ~~echanically polished specimens were found to be unsuitable for studying the relief effects because discontinuous precipitation. occurred in the deformed surface layer during ageing. Specimens were aged at temperatures ranging from 250°C to 330°C for periods of 1.5 minutes to 24 hours to determine the optimum conditions for the d~e~o~ment of relief effects. The habit plane of the relief effects was determined by plotting the zone normal to each trace direction into a single stereographic triangle and determining the common point of intersection of all zones.13 The directions of the relief effects in a given set varied slightly, but the trace measurements were reproducible to within 3 degrees. The orientations of the parent crystals were determined, both before and after ageing, from back reflection Laue and oscillating crystal photographs. In spite of the care taken in preparation of the crystals, the diffraction spots were diffuse even before ageing However, the orientations determined prior to ageing are considered to be accurate to within =tr1”. In the aged specimens the spots were much more diffuse, reflections occurring over a range of several degrees. For this reason the determination of the complete pole figure of the precipitate axes was not attempted. However, in connection with the problem of finding out which variant of the habit plane is associated with a particular variant of the orientation relationship, the locations of those (100)~” poles which occur in the neighbourhood of (lOO)~‘* poles were determined. Owing to the slight uncertainty in the matrix orientation after ageing, the positions of these (lOO}a poles were determined relative to the original positions of the (100)~ p01es. Oscillating crystal photographs taken with filtered OI monochromatic copper radiation were used for this purpose. Even in specimens aged for 20 hours the reflections from the precipitate were very weak and exposures of about 3 or 14 hours were needed for tiltered and monochromatic radiation respectively. For the correlation of habit plane and orientation relationship variants, the directions of the relief effects *The SU&~S B and P are used to denote the body-centered cubic (CuBe) and face-centered cubic f&j phases respectively.
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FIG. 1. Relief effects produced on electropolished surface of ;;;r-beryllium single crystal by ageing for 4 hours at 285°C.
and the locations of (100)~ poles were measured using specimens aged at 285°C for 20 hours, Stereographic projections were constructed showing the orientations, relative to the parent crystal, of the (100)~ axes and of those variants of the habit plane capable of producing the observed relief effects. As will be described in the next section, these projections indicate a unique association of habit plane variants and of variants of the (100)~ poles. 3. EXPERIMENTAL
VOL.
3, 1955
dices; a similar conclusion was reached by Guy, Barrett and Mehl.12 When the trace normals were plotted on the stereographic projection, it was noted that several variants of the habit plane could be associated with each trace. This multiplicity makes it difficult to determine the habit plane more accurately, for the mean traces used in the determinations may actually be derived from more than one variant. Under these conditions, it is not surprising that attempts to improve the accuracy of the determination by using traces on two surfaces14 were not successful. Although relief effects were visible in specimens aged for four hours at 285°C X-ray reflections from the precipitate were not detected until after about 20 hours. Metallographic examination of specimens aged at 285°C revealed a striated structure parallel to the relief effects (Fig. 3). After 24 hours at 285°C the discontinuous precipitation of a nodular constituent was observed (Fig. 4.)
RESULTS
It was found that the relief effects developed best in specimens containing 1.80-1.95 weight per cent beryllium at 285°C. Relief effects were not observed outside the range 260” to 330°C for ageing periods of four hours. On ageing at 285’C faint relief effects became visible after an hour and three quarters and were clearly developed after four hours. A typical photomicrograph of the relief effects is shown in Fig. 1. The results of the habit plane determination are shown in Fig. 2; the mean normals to 28 traces in six different specimens passed through the region indicated. It will be noted that the habit plane is not a plane of low in-
FIG. 2. Rest&s of habit pIane determination. Each spot represents a determination from a specimen surface; the range of orientations __ .. _within _which_ the habit plane is most likely to lie is defined by the rough circle.
FIG. 3. Striated structure in crystal aged 19 hours at 285°C. Etched with ammonia-hydrogen peroxide reagent. 100X.
FIG. 4. Discontinuous precipitation in crystal aged 100 hours at Etched with ammonia-hydrogen peroxide reagent *,.,..285°C. I IUUK.
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The data obtained for the correlation of habit plane and orientation relationship variants are shown in Figs. S(a) and S(b). The interpretation of these data involves a knowledge either of the approximate orientation relationship or of the “correspondence”2 between the initial and final lattices. Guy” has shown that the orientation relationship is such that one of the (100)~ axes is in the neighbourhood of a (100)~ axis whilst the other two (100)~ axes are near (110),~ axes. In this respect the copper-beryllium orientation relationship is similar to those observed in the various iron alloys. This means that in any specimen there are as many variants of the precipitate orientation as there are (100)~ poles near (100)~ poles and, further, that when the orientation relationship is known, a particular variant of the precipitate orientation can be specified by describing the location of that one of its (100)e poles which is near a (100)~ pole. The association of habit plane and orientation relationship variants can therefore be specified in terms of the relative positions of the habit plane and the relevant (100)~~ pole. It is necessary to find from the data of Figs. S(a) and S(b) a particular combination of a variant of the habit plane and a variant of the (100)~ pole, which, when repeated by the symmetry operations of the parent crystal, generates all the other observed (100)~ poles and habit planes. Such a combination is found by systematic trial and error. For example, it might be proposed that a (100)~ pole observed at [u v W]F is associated with an observed habit plane variant (Iz k 2)~. If another (100)~ pole is observed at [U-W VIP, then for the proposed association to be valid, another habit plane must occur at ( h-Z k)~ and so on. In the specimens examined a given trace could usually be attributed to several variants of the habit plane. Accordingly, all variants of the habit plane capable of producing the observed traces have been plotted in Figs. S(a) and S(b). This does not imply that all the variant.s plotted are really present but there is no reason to suppose that each trace is produced by a single variant of the habit plane. Another complication is the possibility that a habit plane variant could escape detection if its orientation relative to the specimen surface were such that the surface was not tilted at all or was only tilted through a very small angle. notwithstanding this uncertainty in the number of habit plane variants present, the trial and error analysis of the data in Fig. S(a) leads to the unique result that a variant of the habit plane is associated with that variant of the (100)~ pole which lies in the same stereographic triangle. If we denote the variants of the habit plane and precipitate by the indices of the form (I, 10, 121~ and (1, 1.3, ZO}, respectively, then this result states that a variant of the habit plane at (1, 12, 10)~ is associated with the variant of the precipitate at [l, 20, 1.31~. The trial and error analysis of the data in Fig. 5(b) leads to two results, one of which is in agreement with the result of Fig. S(a), and another which
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F~c. 5. Sterographic projection of the data obtained to determine the association of variants of the habit plane and variants of the precipitate orientation. The dotted great circles are the normals to the observed traces and the cross-hatched areas are the habit plane variants capable of producing these traces. The filled rectangles indicate the locations where (001)~ poles were observed.
states that a variant of the habit plane at (1, 12, 10)~ is associated with that variant of the (100)~ pole at [%, 1.3, l]F. Such a result is not unexpected in view of the uncertainty in the number of habit plane variants present, but agreement of one of the results with the unique result of Fig. S(a) strongly suggests that this is the correct association. 4. COMPARISON
OF THEORY
AND
EXPERIMENT
In applying the theory of the transformation from face-centered cubic to body-centred cubic lattices6 to the transformation in copper-beryllium alloys, the assumptions involved are that the lattice correspond-
594 ence is described
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3, 1955 1. Values
TABLE
of 8 for Cu-Be transformation
1.019
0.988
and that the twinning plane (the plane of the slip inhomogeneity) is (112)~. The theory involves a variable parameter 8= (~~~~~)~~ where QB and a~ are the lattice parameters of the final and initial structures respectively, and 6 is a dilatation which must be within a few per cent of unity for the theory to be applicable. The predicted variation of habit plane with f?$is shown in Fig. 6. The first step in applying the theory is to determine @ by finding that value for which the predicted and observed habit planes agree. Since at this stage it is not known which variant of the experimental habit plane is to be compared with the predicted habit planes, it is assumed provisionally that the experimental habit plane lies in the stereographic triangle [OlO]r ; [Oll]~; [lll]~ On this assumption the experimental habit plane agrees well with the predicted habit planes for 82=0.59 to 0.60. This assumption will only be justified if the subsequent analysis predicts the association between habit plane an.d orientation relationship variants that has been found in the present investigation. Specifically the requirement is that the predicted orientation relationship be such that the fOOl]B axis, which is generated from the [010]~ axis [Eq. (l)] lies in the same triangle as the habit plane. Since this implies that the habit plane lies in a triangle containing [Olo]~, the assumed location of the habit plane is not an impossible one. Having derived &he value of 8 the next step is tu see whether the impkd value of the unknou?r dilatation 6 is close to unity. The values of 6= S/ (a~/aF) for 82” 0.59 and 82=0.60 are shown in Table 1. The values labelled Sactual have been calculated from the lattice parameters, as==3.S7& aB= 2.69& (uB/uF)~= 0~768,1~ whilst for those labelled &arrectedr an attempt has been made to take into account the ~ss~bi~ity that the beryllium
1.028 0.996
For this
atoms segregate prior to the transformation. purpose the relatioxP ap= (~.~~-~*~3X~)~
(2)
where C is the concentration. of beryllium in atomic per cent, has been used to calculate the value of aF for 50 at. 7o beryllium, whence (aJuF)2= 0.605, All the values of 6 are within a few per cent of unity, as required, and 6 could be exactly unity for a critical degree of segregation of beryllium atoms. The orientation relationship is defined bi the theory in terms of an invariant line strain S. This strain differs from the “total strain”” which generates the final lattice in its observed orientation, only by a dilatation I,/&which does not affect the orientation ~elationshiF. Since equations describing the strain S were derived earIier,6 the application of the theory to the copperberyllium transformation merely involves substituting the derived value of B into these equations. Some simplifications arise in the present case because the axial ratio of the fmaf body-centred lattice is unity and Eqs. (L&Y),(4.9) and (4.10) of reference c6] become: co&X= (2-3P)+/e 7
(3)
(LW =
and
r 1, - P coscr sinol/2, P sinor/2 0, ~@cos#, {B”co* cosw- 28 sinw>iG (P COSEV sinw+ 28 cost)/% j i 0, v%P sir&, (4) COW= (48+3)/2G(l+@
(9
The basis L to which the strain S is referred is related to the basis F, defining the f.c.c. unit cell, by the transformation
F (DF)
=
[
sin&Z,
- sinajti,
COS&~~
-cos&2,
l/fi
3
1/a,
CfJsa! -
sim . 0 I
(6)
The values of ty and w for 82=0.59 and 0.60 are given in Table II, Of the four possible solutions which can be obtained for (L&I,) by combining positive and negative values of a! and w, it is only necessary to consider the (a-k, w-i-1 TABLE
FIG, 6. Stereographic projection showing the predicted variation of habit plane with g0, and the experimentally determined habit plane region.
II, Values of a, w
and k for the Cu-Be transformation
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and (cy+, a--) solutions; the other two are variants of these. Further, the (a+, W-) solution involves a much greater fraction, K, of the twinning shear than the (of, a+) solution (Table II) and thus might be expected to be less favoured from an energetic point of view. In fact, calculation with the (a+, W-) solution leads to an orientation relationship which is incompatible with the experimental observations and thus we consider only the (IX+, w+) solution. The values of K have been calculated from Eq. (4.15),‘j which in the present case reduces to 1+2K= The (a+,
(2-3f?2)~(3-482)+,‘0.
w+) solution
(LSL)=;
0;
-0.143875; 1.013155; 0.133476;
:,i I
for @=0.59 is 0.230434 0.095013 0.907123
(7)
1. (8)
FIG. 8. Stereographic projection showing the orientation lationship determined by Guy. ‘1 The filled rectangles indicate locations where (001)~ poles were observed.
The orientation relationship produced by this strain is most conveniently described by the transformation matrix (E’TB) describing the relative positions of the bases F and B. It follows from Eq. (1) that the axes [lOO]e, [OlO], and [OOI]B are generated from [lOl&, [lOi$ and [010]~ respectively. The columns of the matrix (FTB) are therefore unit vectors in the directions (FTL) (LSL) (~T~)[lol] F, etc. Thus, for 02=0.59 (IFTB)=
0.694027 ; -0.145274; i 0.705139;
Similarly
for 02= 0.60
(BTB)=
0.691563; -0.151348; ( 0.706282 ;
0.715100; 0.025617; -0.698553;
0.083419 0.989060 . 0.121662 1
0.716416; 0.01~3; -0.697414;
0.092133 0.988298 . (10) 0.121570 I
(9)
The orientation of [OOI] B relative to the basis F is given by the last column of (FTB). It will be noted that
both cases [001-J* lies in the stereographic triangle [OlO&; [Oll]F; [Ill.&, thereby confirming the validity of the provisional assumption made earlier. The theory thus predicts the observed association between habit plane variants and variants of the orientation relationship. The pole figure predicted by the theory is shown in Fig. 7, the numerical values of the angles indicated being given in Table III. For comparison the pole figure determined by Guy” using a 1.75 percent beryllium alloy is reproduced in Fig. 8. It can be seen that the predicted orientation relationship reproduces all the essential features of the observed pole figure. Perhaps the most striking feature is the interpretation that is provided for the symmetry of the pole spreading. The arcs in the observed pole figure arise from the merging of groups of two (or four) poles belonging to different variants; this merging can be attributed to plastic deformation in the matrix, for the observed regions of intensity would be reproduced within experiments error if the predicted poles were spread uniformly over a range of about 1 to l+ degrees. Another possible origin for the pole spreading is variation of oz. As can be seen from Fig. 7 and Table III, variation of f12changes the angles of separation of the pairs of poles; this could also account for the lack of precision of the habit plane determination.
in
TABLEIII. Values for the angles of the predicted orientation Angle
FIG. 7. Stereographic relationship. Table III.
The
values
projection of the predicted orientation of the angles indicated are listed in
rethe
a1 Pl (Yz I32 ff3 P3 Y3
82
relationship
=0.59 8.49 3.10 8.36 0.90 1.60 1.32 2.95
ez=0&O 8.77 “8*;; 1:20 1.32 1.53 2.12
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The orientation relationship predicted for @= 0.60 has the property that [lil]Fjj[201]B. This relation was found by Neerfeld and Mathie@ in specimens of iron30 percent nickel and of 18-8 stainless steel, transformed in tension. Neerfeld and Mathieu assumed that the additional relation [iOl]Flj[OiO]B also held. For the e2=0.60 orientation, these directions are 1.32 degrees apart. It is possible that the orientation studied by Neerfeld and Mathieu was actually that corresponding to @=0.60. For this case there is no parallelism of simple rational lattice planes, but the irrational planes (l,l,fl)~ and (l+G, l-a, 2)~ are parallel. 5. DISCUSSION
The present investigation has shown that the geometrical features of the precipitation of CuBe are described consistently by the phenomenological theory that was developed for martensitic transformations. It should be noted that this theory does not impose any restrictions on the transformation mechanism other than the specification of the over-all atomic displacements involved.* It does not imply that the transformation mechanism involves homogeneous strains nor even that whole planes of atoms are moved without segregation or ordering of solute atoms, as suggested by Geis1er.l’ Recent discussionssJ7J* of the significance of double strain descriptions in relation to mechanism have been complicated by conflicting interpretations of the nature of the relief effects produced by transformations. Geislers considers that there is no change of shape inherently associated with the lattice transformation, but that the observed change of shape arises as a result of slip caused by transformation stresses. Since transformation stresses presumably arise from an inherent shape change, the view that the relief effects are due both to the inherent shape change and to slip seems much more reasonable. He proposes that the slip simulates a homogeneous strain, and that this strain is responsible for the observed tilting of polished surfaces about their intersections with the habit plane. The only homogeneous strains that can be simulated by slip on a single slip system are shears on the slip plane in the slip direction. Since a simple shear rotates planes about their intersections with the shear plane, surfaces will be rotated about their intersections with the habit plane only when the intersection of habit plane and slip plane lies in the specimen surface. For all other surfaces the martensite plate would become tipped up out of the surface. This limitation can be overcome by invoking double slip, but, since the homogeneous strains involved in the transformation have dilatational components, no combination of slip processes can simulate such strains. *Strictly, the phenomenoIogical theory describes the correspondence between lattice points in the initial and final lattices. This correspondence defines the over-all atomic displacements involved in the lattice transformation, but does not include the atomic displacements due to diffusion.
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For the relief effects in copper-beryllium, Geisler, Mallery and Steigertlg favour the interpretation, suggested by Guinier and JacquetgJo that the relief effects result from mechanical slip on the (110) F planes of the matrix due to the formation of Guinier-Preston zones parallel to the { 100)~ planes. They examined the pole figure of Guy and claimed it to be in excellent agreement with the Nishiyama relationship, W)Fll(lol)B; [iOl]~jj[OlO]~. The discrepancy between the measured angle of approximately 8 degrees for crl and the angle of 9.73 degrees, predicted from the Nishiyama relationship, was attributed to experimental error, while the spreading of the poles was regarded as arising from plastic deformation of the matrix, produced by transformation stresses. They proposed that blocks of the matrix become rotated about that (110)~ direction lying in the plane of the relief effects. This suggestion is incompatible with the results of the present investigation for it implies that, when the [OOl]a axis lies in the stereographic triangle [OlO]F; [Oll]~; [l_ll]F, the habit plane is in the neighbourhood of (lOl)p, instead of (011)~ as determined experimentally. It is clear that Geisler’s proposal, that the relief effects can be attributed to slip alone, is not consistent with observation. Instead, the change in shape produced by the transformation must be regarded as the resultant of the strain describing the distortion of the initial lattice into the final lattice, and the slip inhomogeneity. Such an interpretation of the relief effects is in no way incompatible with Geisler’s views on the mechanism of transformations. Geisler pointed out that since relief effects are produced both by diffusion-controlled and martensitic transformations, it is not necessary to postulate any fundamental difference in transformation mechanism to explain their origin. He therefore proposed that the classical mechanism of diffusion-controlled transformations could be extended to cover martensitic transformations. According to this mechanism, the atoms move independently of each other, one at a time, under the influence of thermal activation. To overcome the difficulty that, in some cases at least, the growth of martensite plates seems to be independent of thermal activation, Geisler visualized the atomic movements as being “well organized, one atom pulling the others.“8 Such a mechanism can be reconciled with our interpretation of the relief effects. In the early stages of the growth of the new crystal the individual atomic displacements would simply be those described by the homogeneous strain that converts the initial lattice into the final lattice. At a certain stage of growth the stresses arising from this mode of transformation could produce slip, as Geisler suggests, thus modifying the change of shape caused by the “homogeneous” mode of transformation. However, there is no evidence that the slip inhomogeneity arises from slip occurring after the generation of the new structure and it is quite possible that the slip inhomogeneity is an inherent feature of the trans-
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formation mechanism. Indeed, a mechanism that produces the slipped state directly seems more probable than one that produces a highly stressed product which then slips. In contrast to Geisler’s views, other workers4s20-n have attempted to develop growth mechanisms for martensitic transformations in which the atomic displacements are produced by the spontaneous movement of transformation dislocations. The striking similarity between martensitic transformations and mechanical twinning provides strong evidence in favour of such mechanisms. Martensitic transformations and mechanical twinning are both accompanied by homogeneous strains and in both processes the new crystals are plate-shaped and produced in time intervals of the order of a few microseconds; both processes can be induced by plastic deformation and the audible click that accompanies twinning in certain metals is also a feature of transformations. These common many martensitic characteristics suggest that the two processes occur by similar mechanisms. Further, as pointed out by Orowan,2:i the existence of lens-shaped lamellae in twinning (and martensitic transformations) implies the presence of dislocations as a simple geometrical corollary. The amount of shear across a lamella is proportional to its thickness and thus where the thickness changes, even by one lattice spacing, the amount of shear changes discontinuously by a small amount. This discontinuous change must be provided by a dislocation situated at the step. An explanation of mechanical twinning in terms of dislocations has been provided by the “pole mechanism” of Cottrell and Bilbyz4 and it seems attractive to consider similar mechanisms for martensitic transformations.21 In such dislocation mechanisms, it should be possible to include the slip inhomogeneity as an integral feature of the process. If we accept the hypothesis of a dislocation mechanism for martensitic transformation then, with a view to developing a general theory applicable both to diffusion-controlled and martensitic transformations, it becomes of interest to consider the possibility that diffusion-controlled transformations also occur by a dislocation mechanism. On the existing evidence there is no reason to reject this possibility. If a dislocation mechanism is operative in martensitic transformations in which the plates grow very rapidly, then the propagation of the transformation dislocations must occur spontaneously. However, this is not an essential feature of a dislocation mechanism and there is no reason why the propagation of transformation dislocations should not be spontaneous in some transformations and require thermal activation in others. If
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thermal activation is required the rate of growth will be a function of temperature and the new crystals will grow at an observable rate. In precipitations, where a change of composition is involved, a further possibility exists for here it is possible that the rate of propagation of the transformation dislocations is governed by the rate at which the required concentration of solute atoms can be assembled at the advancing interface. The concept of a common dislocation mechanism for diffusion-controlled and martensitic transformations leads to a ready interpretation of the results of the present investigation. If the geometry of the atomic processes involved in the two kinds of transformation is the same, then it is not surprising that the phenomenological theory developed for martensitic transformations should be capable of predicting the geometrical features of a typical diffusion-controlled transformation like the precipitation of CuBe. This speculation concerning the possible role of dislocation mechanisms in diffusion-controlled transformations is, of course, intended to apply only to the early stages of growth (the so-called stage of “coherent” growth). After a non-coherent interface has been produced, growth can occur by the accretion of individual atoms and the shape assumed by the growing crystal will be determined mainly by considerations of interfacial energy. REFERENCES J. S. Bowles and J. K. Mackenzie, Acta Met. 2, 129 (1954).
f. J. K. Mackenzie and J. S. Bowles, Acta Met. 2, 138 (1954).
3: M. S. Wechsler, D. S. Lieberman and T. A. Read, Trans. A.I.M.E. 197, 1503 (1953). 4. H. Suzuki, Sci. Rep. R.I.T.U. A6,30 (1954). 5. J. S. Bowles and C. S. Barrett, Progress in Metal Physics 3, 1 (1952). 6. J. S. Bowles and J. K. Mackenzie, Acta Met. 2, 224 (1954). 7. L. Northcott, J. Inst. Met. 59, 22.5 (1936). 8. A. H. Geisler, Acta Met. 1, 260 (1953). A. Guinier and P. A. Jacquet, Rev. Met. 41, 1 (1944). 1;: A. Guinier and P. A. Jacquet, Nature 155, 69.5 (1945). A. G. Guy, Gen. Elec. Rev. 49, 28 (Aug. 1946). :1: A. G. Guy, C. S. Barrett and R. F. Mehl, Trans. A.I.M.E. 175, 216 (1948). 13. J. S. Bowies, Trans. A.I.M.E. 191, 44 (1951). 14. C. S. Barrett, S$ucture of Melds, 2nd ed. (London, McGrawHill, 1953), p. 42. S. Miyatani, J. Phys. Sot. Japan 4, 181 (1949). H. Neerfeld and K. Mathieu, Arch. Eisenh. 20, 69 (1949). A. H. Geisler, Acta Met. 2, 639 (1954). 2. S. Basinski and J. W. Christian, Acta Met. 1, 759 (19.53). A. H. Geisler, J. H. Mallery and F. E. Steigert, Trans. A. I. M. E. 194, 307 (1952). 20. M. Cohen, E. S. Machlin and V. G. Paranjpe! Thermodynamics in Physical Metallurgy (Cleveland, Ohio, A. S. M., 1950), p. 242. B. A. Bilby, Phil. Mag. 44, 782 (1953). ;1. Z. S. Basinski and J. W. Christian, Phil. Mag. 44, 791 (1953). 23: E. Orowan, Dislocations in Mel& (New York, A.I.M.E., 1954), p. 116. 24. A. H. Cottrell and B. A. Bilby, Phil. Mag. 42, 573 (1951).
RELATIONSHIPS J. S. BOWLESt
IN AGED and W. J. McG.
COPPER-BERYLLIUM
ALLOYS*
TEGARTS
The precipitation of CuBe from copper-beryllium (Ysolid solutions involves a transformation from a facecentered cubic (F) to a body-centered cubic (B) structure. The habit plane of the relief effects produced on polished surfaces by this transformation has been determined. It is not a simple lattice plane but is close to (110)~. The association between variants of the habit plane and variants of the orientation relationship has been found to be such that a variant of the habit plane is associated with that variant of the (1OO)z pole which lies in the same stereographic triangle. With these results the precipitation reaction has been analysed using the geometrical theory developed for martensitic transformations by Bowles and Mackenzie. 1,226The theory then predicts the orientation relationship determined by Guy.” RELATIONS
CRISTALLOGRAPHIQUES
DANS
LES
ALLIAGES
CUIVRE-GLUCINIUM
La pr&ipitation de CuBe B partir des solutions solides cuivre-glucinium comporte une transformation du systeme cubique 2 faces centrees (F) dans le systeme cubique centre (B). Le plan d’habitat des reliefs sur une surface polie produits par cette transformation a et6 determine; ce n’est pas un plan simple mais il est voisin de (110)~. Le rapport entre les variantes du plan d’habitat et les variantes de la relation d’orientation est tel qu’une variante du plan d’habitat est allike a la variante du p61e (100)~ qui appartient au meme triangle stereographique. A l’aide de ces r&ultats la precipitation est analysee par la theorie geometrique developpee pour les transformations martensitiques par Bowles et Mackenzie. r,%8 La thCorie prtvoit les orientations observbes par Guy.” EMBER DIE
KRISTALLOGRAFISCHEN ZUSAMMENHANGE KUPFER-BERYLLIUM LEGIERUNGEN
BEI
AusGELAGERTEN
Die Ausscheidung von CuBe aus (Y Mischkristallen des Systems Kupfer-Beryllium ist gleichbedeutend mit einer Umwandlung einer kubisch-flachenzentrierten (F) in eine kubisch-raumzentrierte (B) Struktur. Es wurde die Habitusebene der Reliefwirkungen bestimmt, die diese Umwandlung auf polierten Flachen hinter&St. Die Habitusebene ist keine einfache Gitterebene, sie liegt jedoch nahe bei (110)~. Der Zusammenhang zwischen Veranderungen der Habitusebene wie such der Orientierungsbeziehung ist derart, dass eine Anderung der Habitusebene von der Anderung desjenigen (lOO)B-Poles begleitet wird, der im gleichen Bestimmungsdreieck leigt. Mit diesen Ergebnissen wurden die Ausscheidungsreaktionen auf Grund der geometrischen Theorie der Die Theorie sagt dann die von Guy” Martensitumwandlung von Bowles und Mackenzie.‘s2,6 analysiert. bestimmte Orientierungsbeziehung voraus.
1. INTRODUCTION
Although the details of the atomic mechanisms of martensitic transformations have not yet been elucidated, the phenomenological theory of the over-all atomic displacements has been developed to the point where accurate predictions can be made of habit planes, orientation relationships and the homogeneous strains accompanying the transformations.1-4 For the diffusioncontrolled transformations no such quantitative theory exists. X-ray studies have disclosed a wealth of information about intermediate structural states produced during these transformations, but no quantitative theory of the crystallographic relations has yet been derived. Some time ago the suggestion was made by Bowles and Barrett5 that, during the stage of “coherence,” the geometry of the atomic processes involved in diffusion-controlled transformations may be identical with that involved in martensitic transformations. * Received
May 2,. 1955. t C.S.I.R.O., Division of Tribophysics, University of Melbourne, Carlton, N.3., Victoria! Australia; now New South Wales University of Technology,. High Street, Kensington, N.S.W., Australia. $ C.S.I.R.O., Division of Tribophysics, University of Melbourne, Carlton, N.3., Victoria, Australia. ACTA
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1955
At the time this suggestion was made, the only experimental evidence available to support it was the general similarity of the orientation relationships produced by the two kinds of transformation. However, the recent developments in the phenomenological theory of martensitic transformations enable a more satisfactory test to be made of the merits of this idea. If the geometry of the processes involved in the two kinds of transformation is identical, then the theory developed for martensitic transformations should be capable of predicting the geometrical features of diffusion-controlled transformations. The work to be described in this paper was carried out to test the validity of the predictions of the theory for the precipitation of CuBe from copperberyllium a! solid solutions. This particular precipitation process was selected for two reasons. First, the lattice transformation involved is f.c.c.-+b.c.c; the theory for this case has been worked out in detail6 and shown to yield predictions consistent with observation for a number of martensitic transformations. Second, the precipitation of CuBe is accompanied by the production of visible relief effects on the surfaces of polished specimens. These relief effects 590
BOWLES
AND
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have been described as similar to those produced by martensitic transf~rmations7.8 and, in the present work, they are interpreted as having a similar origin. It is proposed that the relief effects arise as a direct consequence of the net change in shape accompanying the formation of plate-shaped crystals of CuBe. The phenomenalogical theory’sa has been developed from the proposal that the total atomic displacements produced by martensitic transformations can be described consistently by means of a homogeneous strain which is composed of an invariant plane strain and a small dilatation, followed by a shear, which is part of a twinning shear in the final lattice but which occurs inhomogeneously. It has been shown that these conditions determine the total strain, and hence the orientation relationship, in terms of a single parameter, tl. In applying the theory to a particular transformation, it is necessary to evaluate 0 by calculating the variation of the habit plane with @and determining that value, if any, for which the predicted and observed habit planes agree, The predicted habit plane variation for facecentered cubic to bay-centered cubic tmnsfo~tions has already been derived6 so it remains to determine the habit plane of copper-beryllium, i.e., the plane of the relief effects, and to evaluate 8. The experimental work is described in $52, 3. This consisted of determining the general indices of the habit plane and of &ding which variant of the habit plane is associated with a particular variant of the precipitate orientation. In $4 these results are used to evaluate 6’and tbc orientation relationship is then determined by substitution in the equations developed elsewhere.” The predicted orientation relationship is compared with that measured by Guy.‘r The notation used in this paper is the same as that used previously.“j”r6 2. EXPERIMENTAL
METHODS
The alloys used were prepared from oxygen-free high conductivity copper (> 99.98 wt. 7o copper) with traces f <0.005 wt. yo> of lead, tin, nickel and iron, and a copper-beryllium mother alloy (95.85 wt. ye copper, 4.14 wt. y0 beryllium) supplied by Brush Beryllium Corporation. Initially the melting procedure described by Guy, Barrett and Mehlr2 was followed but later the alloys were prepared by melting the mother alloy and the copper together in silica under an atmosphere of purified nitrogen” The solidified rods were swaged down from 12 mm diameter to 7-8 mm diameter with intermediate anneals at 800-850°C. The alloys used were restricted to the range 1.80-1.95 wt. $$c beryllium since it was found that relief effects were not produced in specimens containing smafler concentrations of beryllium. To avoid grain-boundary precipitation during the ageing treatments, all the experiments were performed on singIe crystals. Crystals made by the Bridgman technique were found to be badly cored, and therefore
ALLOYS
591
repeated zone melting was used to produce more homogeneous crystals. The specimens, sealed under nitrogen in a silica tube, were passed through a small coil fitted in the centre of a tube furnace. The coil was maintained above the liquidus temperature and the furnace at 800°C. After zone melting the specimens were maintained at 800°C for three days and then water-quenched. The single crystals produced in this way were not cored but only small lengths of uniform composition could be obtained. Specimens were cut from the single crystals with a jeweller’s hacksaw, lapped on Carborundum paper and then polished with diamond dust. They were then carefully electropolished in an orthophosphoric acid electrolytero for one half to three quarters of an hour to obtain a strain-free surface. ~~echanically polished specimens were found to be unsuitable for studying the relief effects because discontinuous precipitation. occurred in the deformed surface layer during ageing. Specimens were aged at temperatures ranging from 250°C to 330°C for periods of 1.5 minutes to 24 hours to determine the optimum conditions for the d~e~o~ment of relief effects. The habit plane of the relief effects was determined by plotting the zone normal to each trace direction into a single stereographic triangle and determining the common point of intersection of all zones.13 The directions of the relief effects in a given set varied slightly, but the trace measurements were reproducible to within 3 degrees. The orientations of the parent crystals were determined, both before and after ageing, from back reflection Laue and oscillating crystal photographs. In spite of the care taken in preparation of the crystals, the diffraction spots were diffuse even before ageing However, the orientations determined prior to ageing are considered to be accurate to within =tr1”. In the aged specimens the spots were much more diffuse, reflections occurring over a range of several degrees. For this reason the determination of the complete pole figure of the precipitate axes was not attempted. However, in connection with the problem of finding out which variant of the habit plane is associated with a particular variant of the orientation relationship, the locations of those (100)~” poles which occur in the neighbourhood of (lOO)~‘* poles were determined. Owing to the slight uncertainty in the matrix orientation after ageing, the positions of these (lOO}a poles were determined relative to the original positions of the (100)~ p01es. Oscillating crystal photographs taken with filtered OI monochromatic copper radiation were used for this purpose. Even in specimens aged for 20 hours the reflections from the precipitate were very weak and exposures of about 3 or 14 hours were needed for tiltered and monochromatic radiation respectively. For the correlation of habit plane and orientation relationship variants, the directions of the relief effects *The SU&~S B and P are used to denote the body-centered cubic (CuBe) and face-centered cubic f&j phases respectively.
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METALLURGICA,
FIG. 1. Relief effects produced on electropolished surface of ;;;r-beryllium single crystal by ageing for 4 hours at 285°C.
and the locations of (100)~ poles were measured using specimens aged at 285°C for 20 hours, Stereographic projections were constructed showing the orientations, relative to the parent crystal, of the (100)~ axes and of those variants of the habit plane capable of producing the observed relief effects. As will be described in the next section, these projections indicate a unique association of habit plane variants and of variants of the (100)~ poles. 3. EXPERIMENTAL
VOL.
3, 1955
dices; a similar conclusion was reached by Guy, Barrett and Mehl.12 When the trace normals were plotted on the stereographic projection, it was noted that several variants of the habit plane could be associated with each trace. This multiplicity makes it difficult to determine the habit plane more accurately, for the mean traces used in the determinations may actually be derived from more than one variant. Under these conditions, it is not surprising that attempts to improve the accuracy of the determination by using traces on two surfaces14 were not successful. Although relief effects were visible in specimens aged for four hours at 285°C X-ray reflections from the precipitate were not detected until after about 20 hours. Metallographic examination of specimens aged at 285°C revealed a striated structure parallel to the relief effects (Fig. 3). After 24 hours at 285°C the discontinuous precipitation of a nodular constituent was observed (Fig. 4.)
RESULTS
It was found that the relief effects developed best in specimens containing 1.80-1.95 weight per cent beryllium at 285°C. Relief effects were not observed outside the range 260” to 330°C for ageing periods of four hours. On ageing at 285’C faint relief effects became visible after an hour and three quarters and were clearly developed after four hours. A typical photomicrograph of the relief effects is shown in Fig. 1. The results of the habit plane determination are shown in Fig. 2; the mean normals to 28 traces in six different specimens passed through the region indicated. It will be noted that the habit plane is not a plane of low in-
FIG. 2. Rest&s of habit pIane determination. Each spot represents a determination from a specimen surface; the range of orientations __ .. _within _which_ the habit plane is most likely to lie is defined by the rough circle.
FIG. 3. Striated structure in crystal aged 19 hours at 285°C. Etched with ammonia-hydrogen peroxide reagent. 100X.
FIG. 4. Discontinuous precipitation in crystal aged 100 hours at Etched with ammonia-hydrogen peroxide reagent *,.,..285°C. I IUUK.
BOWLES
AND
TEGART:
COPPER-BERYLLIUM
The data obtained for the correlation of habit plane and orientation relationship variants are shown in Figs. S(a) and S(b). The interpretation of these data involves a knowledge either of the approximate orientation relationship or of the “correspondence”2 between the initial and final lattices. Guy” has shown that the orientation relationship is such that one of the (100)~ axes is in the neighbourhood of a (100)~ axis whilst the other two (100)~ axes are near (110),~ axes. In this respect the copper-beryllium orientation relationship is similar to those observed in the various iron alloys. This means that in any specimen there are as many variants of the precipitate orientation as there are (100)~ poles near (100)~ poles and, further, that when the orientation relationship is known, a particular variant of the precipitate orientation can be specified by describing the location of that one of its (100)e poles which is near a (100)~ pole. The association of habit plane and orientation relationship variants can therefore be specified in terms of the relative positions of the habit plane and the relevant (100)~~ pole. It is necessary to find from the data of Figs. S(a) and S(b) a particular combination of a variant of the habit plane and a variant of the (100)~ pole, which, when repeated by the symmetry operations of the parent crystal, generates all the other observed (100)~ poles and habit planes. Such a combination is found by systematic trial and error. For example, it might be proposed that a (100)~ pole observed at [u v W]F is associated with an observed habit plane variant (Iz k 2)~. If another (100)~ pole is observed at [U-W VIP, then for the proposed association to be valid, another habit plane must occur at ( h-Z k)~ and so on. In the specimens examined a given trace could usually be attributed to several variants of the habit plane. Accordingly, all variants of the habit plane capable of producing the observed traces have been plotted in Figs. S(a) and S(b). This does not imply that all the variant.s plotted are really present but there is no reason to suppose that each trace is produced by a single variant of the habit plane. Another complication is the possibility that a habit plane variant could escape detection if its orientation relative to the specimen surface were such that the surface was not tilted at all or was only tilted through a very small angle. notwithstanding this uncertainty in the number of habit plane variants present, the trial and error analysis of the data in Fig. S(a) leads to the unique result that a variant of the habit plane is associated with that variant of the (100)~ pole which lies in the same stereographic triangle. If we denote the variants of the habit plane and precipitate by the indices of the form (I, 10, 121~ and (1, 1.3, ZO}, respectively, then this result states that a variant of the habit plane at (1, 12, 10)~ is associated with the variant of the precipitate at [l, 20, 1.31~. The trial and error analysis of the data in Fig. 5(b) leads to two results, one of which is in agreement with the result of Fig. S(a), and another which
593
ALLOYS
F~c. 5. Sterographic projection of the data obtained to determine the association of variants of the habit plane and variants of the precipitate orientation. The dotted great circles are the normals to the observed traces and the cross-hatched areas are the habit plane variants capable of producing these traces. The filled rectangles indicate the locations where (001)~ poles were observed.
states that a variant of the habit plane at (1, 12, 10)~ is associated with that variant of the (100)~ pole at [%, 1.3, l]F. Such a result is not unexpected in view of the uncertainty in the number of habit plane variants present, but agreement of one of the results with the unique result of Fig. S(a) strongly suggests that this is the correct association. 4. COMPARISON
OF THEORY
AND
EXPERIMENT
In applying the theory of the transformation from face-centered cubic to body-centred cubic lattices6 to the transformation in copper-beryllium alloys, the assumptions involved are that the lattice correspond-
594 ence is described
ACTA
METALLURGICA,
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by the relation
3, 1955 1. Values
TABLE
of 8 for Cu-Be transformation
1.019
0.988
and that the twinning plane (the plane of the slip inhomogeneity) is (112)~. The theory involves a variable parameter 8= (~~~~~)~~ where QB and a~ are the lattice parameters of the final and initial structures respectively, and 6 is a dilatation which must be within a few per cent of unity for the theory to be applicable. The predicted variation of habit plane with f?$is shown in Fig. 6. The first step in applying the theory is to determine @ by finding that value for which the predicted and observed habit planes agree. Since at this stage it is not known which variant of the experimental habit plane is to be compared with the predicted habit planes, it is assumed provisionally that the experimental habit plane lies in the stereographic triangle [OlO]r ; [Oll]~; [lll]~ On this assumption the experimental habit plane agrees well with the predicted habit planes for 82=0.59 to 0.60. This assumption will only be justified if the subsequent analysis predicts the association between habit plane an.d orientation relationship variants that has been found in the present investigation. Specifically the requirement is that the predicted orientation relationship be such that the fOOl]B axis, which is generated from the [010]~ axis [Eq. (l)] lies in the same triangle as the habit plane. Since this implies that the habit plane lies in a triangle containing [Olo]~, the assumed location of the habit plane is not an impossible one. Having derived &he value of 8 the next step is tu see whether the impkd value of the unknou?r dilatation 6 is close to unity. The values of 6= S/ (a~/aF) for 82” 0.59 and 82=0.60 are shown in Table 1. The values labelled Sactual have been calculated from the lattice parameters, as==3.S7& aB= 2.69& (uB/uF)~= 0~768,1~ whilst for those labelled &arrectedr an attempt has been made to take into account the ~ss~bi~ity that the beryllium
1.028 0.996
For this
atoms segregate prior to the transformation. purpose the relatioxP ap= (~.~~-~*~3X~)~
(2)
where C is the concentration. of beryllium in atomic per cent, has been used to calculate the value of aF for 50 at. 7o beryllium, whence (aJuF)2= 0.605, All the values of 6 are within a few per cent of unity, as required, and 6 could be exactly unity for a critical degree of segregation of beryllium atoms. The orientation relationship is defined bi the theory in terms of an invariant line strain S. This strain differs from the “total strain”” which generates the final lattice in its observed orientation, only by a dilatation I,/&which does not affect the orientation ~elationshiF. Since equations describing the strain S were derived earIier,6 the application of the theory to the copperberyllium transformation merely involves substituting the derived value of B into these equations. Some simplifications arise in the present case because the axial ratio of the fmaf body-centred lattice is unity and Eqs. (L&Y),(4.9) and (4.10) of reference c6] become: co&X= (2-3P)+/e 7
(3)
(LW =
and
r 1, - P coscr sinol/2, P sinor/2 0, ~@cos#, {B”co* cosw- 28 sinw>iG (P COSEV sinw+ 28 cost)/% j i 0, v%P sir&, (4) COW= (48+3)/2G(l+@
(9
The basis L to which the strain S is referred is related to the basis F, defining the f.c.c. unit cell, by the transformation
F (DF)
=
[
sin&Z,
- sinajti,
COS&~~
-cos&2,
l/fi
3
1/a,
CfJsa! -
sim . 0 I
(6)
The values of ty and w for 82=0.59 and 0.60 are given in Table II, Of the four possible solutions which can be obtained for (L&I,) by combining positive and negative values of a! and w, it is only necessary to consider the (a-k, w-i-1 TABLE
FIG, 6. Stereographic projection showing the predicted variation of habit plane with g0, and the experimentally determined habit plane region.
II, Values of a, w
and k for the Cu-Be transformation
BOWLES
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ALLOYS
59.5
and (cy+, a--) solutions; the other two are variants of these. Further, the (a+, W-) solution involves a much greater fraction, K, of the twinning shear than the (of, a+) solution (Table II) and thus might be expected to be less favoured from an energetic point of view. In fact, calculation with the (a+, W-) solution leads to an orientation relationship which is incompatible with the experimental observations and thus we consider only the (IX+, w+) solution. The values of K have been calculated from Eq. (4.15),‘j which in the present case reduces to 1+2K= The (a+,
(2-3f?2)~(3-482)+,‘0.
w+) solution
(LSL)=;
0;
-0.143875; 1.013155; 0.133476;
:,i I
for @=0.59 is 0.230434 0.095013 0.907123
(7)
1. (8)
FIG. 8. Stereographic projection showing the orientation lationship determined by Guy. ‘1 The filled rectangles indicate locations where (001)~ poles were observed.
The orientation relationship produced by this strain is most conveniently described by the transformation matrix (E’TB) describing the relative positions of the bases F and B. It follows from Eq. (1) that the axes [lOO]e, [OlO], and [OOI]B are generated from [lOl&, [lOi$ and [010]~ respectively. The columns of the matrix (FTB) are therefore unit vectors in the directions (FTL) (LSL) (~T~)[lol] F, etc. Thus, for 02=0.59 (IFTB)=
0.694027 ; -0.145274; i 0.705139;
Similarly
for 02= 0.60
(BTB)=
0.691563; -0.151348; ( 0.706282 ;
0.715100; 0.025617; -0.698553;
0.083419 0.989060 . 0.121662 1
0.716416; 0.01~3; -0.697414;
0.092133 0.988298 . (10) 0.121570 I
(9)
The orientation of [OOI] B relative to the basis F is given by the last column of (FTB). It will be noted that
both cases [001-J* lies in the stereographic triangle [OlO&; [Oll]F; [Ill.&, thereby confirming the validity of the provisional assumption made earlier. The theory thus predicts the observed association between habit plane variants and variants of the orientation relationship. The pole figure predicted by the theory is shown in Fig. 7, the numerical values of the angles indicated being given in Table III. For comparison the pole figure determined by Guy” using a 1.75 percent beryllium alloy is reproduced in Fig. 8. It can be seen that the predicted orientation relationship reproduces all the essential features of the observed pole figure. Perhaps the most striking feature is the interpretation that is provided for the symmetry of the pole spreading. The arcs in the observed pole figure arise from the merging of groups of two (or four) poles belonging to different variants; this merging can be attributed to plastic deformation in the matrix, for the observed regions of intensity would be reproduced within experiments error if the predicted poles were spread uniformly over a range of about 1 to l+ degrees. Another possible origin for the pole spreading is variation of oz. As can be seen from Fig. 7 and Table III, variation of f12changes the angles of separation of the pairs of poles; this could also account for the lack of precision of the habit plane determination.
in
TABLEIII. Values for the angles of the predicted orientation Angle
FIG. 7. Stereographic relationship. Table III.
The
values
projection of the predicted orientation of the angles indicated are listed in
rethe
a1 Pl (Yz I32 ff3 P3 Y3
82
relationship
=0.59 8.49 3.10 8.36 0.90 1.60 1.32 2.95
ez=0&O 8.77 “8*;; 1:20 1.32 1.53 2.12
596
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The orientation relationship predicted for @= 0.60 has the property that [lil]Fjj[201]B. This relation was found by Neerfeld and Mathie@ in specimens of iron30 percent nickel and of 18-8 stainless steel, transformed in tension. Neerfeld and Mathieu assumed that the additional relation [iOl]Flj[OiO]B also held. For the e2=0.60 orientation, these directions are 1.32 degrees apart. It is possible that the orientation studied by Neerfeld and Mathieu was actually that corresponding to @=0.60. For this case there is no parallelism of simple rational lattice planes, but the irrational planes (l,l,fl)~ and (l+G, l-a, 2)~ are parallel. 5. DISCUSSION
The present investigation has shown that the geometrical features of the precipitation of CuBe are described consistently by the phenomenological theory that was developed for martensitic transformations. It should be noted that this theory does not impose any restrictions on the transformation mechanism other than the specification of the over-all atomic displacements involved.* It does not imply that the transformation mechanism involves homogeneous strains nor even that whole planes of atoms are moved without segregation or ordering of solute atoms, as suggested by Geis1er.l’ Recent discussionssJ7J* of the significance of double strain descriptions in relation to mechanism have been complicated by conflicting interpretations of the nature of the relief effects produced by transformations. Geislers considers that there is no change of shape inherently associated with the lattice transformation, but that the observed change of shape arises as a result of slip caused by transformation stresses. Since transformation stresses presumably arise from an inherent shape change, the view that the relief effects are due both to the inherent shape change and to slip seems much more reasonable. He proposes that the slip simulates a homogeneous strain, and that this strain is responsible for the observed tilting of polished surfaces about their intersections with the habit plane. The only homogeneous strains that can be simulated by slip on a single slip system are shears on the slip plane in the slip direction. Since a simple shear rotates planes about their intersections with the shear plane, surfaces will be rotated about their intersections with the habit plane only when the intersection of habit plane and slip plane lies in the specimen surface. For all other surfaces the martensite plate would become tipped up out of the surface. This limitation can be overcome by invoking double slip, but, since the homogeneous strains involved in the transformation have dilatational components, no combination of slip processes can simulate such strains. *Strictly, the phenomenoIogical theory describes the correspondence between lattice points in the initial and final lattices. This correspondence defines the over-all atomic displacements involved in the lattice transformation, but does not include the atomic displacements due to diffusion.
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For the relief effects in copper-beryllium, Geisler, Mallery and Steigertlg favour the interpretation, suggested by Guinier and JacquetgJo that the relief effects result from mechanical slip on the (110) F planes of the matrix due to the formation of Guinier-Preston zones parallel to the { 100)~ planes. They examined the pole figure of Guy and claimed it to be in excellent agreement with the Nishiyama relationship, W)Fll(lol)B; [iOl]~jj[OlO]~. The discrepancy between the measured angle of approximately 8 degrees for crl and the angle of 9.73 degrees, predicted from the Nishiyama relationship, was attributed to experimental error, while the spreading of the poles was regarded as arising from plastic deformation of the matrix, produced by transformation stresses. They proposed that blocks of the matrix become rotated about that (110)~ direction lying in the plane of the relief effects. This suggestion is incompatible with the results of the present investigation for it implies that, when the [OOl]a axis lies in the stereographic triangle [OlO]F; [Oll]~; [l_ll]F, the habit plane is in the neighbourhood of (lOl)p, instead of (011)~ as determined experimentally. It is clear that Geisler’s proposal, that the relief effects can be attributed to slip alone, is not consistent with observation. Instead, the change in shape produced by the transformation must be regarded as the resultant of the strain describing the distortion of the initial lattice into the final lattice, and the slip inhomogeneity. Such an interpretation of the relief effects is in no way incompatible with Geisler’s views on the mechanism of transformations. Geisler pointed out that since relief effects are produced both by diffusion-controlled and martensitic transformations, it is not necessary to postulate any fundamental difference in transformation mechanism to explain their origin. He therefore proposed that the classical mechanism of diffusion-controlled transformations could be extended to cover martensitic transformations. According to this mechanism, the atoms move independently of each other, one at a time, under the influence of thermal activation. To overcome the difficulty that, in some cases at least, the growth of martensite plates seems to be independent of thermal activation, Geisler visualized the atomic movements as being “well organized, one atom pulling the others.“8 Such a mechanism can be reconciled with our interpretation of the relief effects. In the early stages of the growth of the new crystal the individual atomic displacements would simply be those described by the homogeneous strain that converts the initial lattice into the final lattice. At a certain stage of growth the stresses arising from this mode of transformation could produce slip, as Geisler suggests, thus modifying the change of shape caused by the “homogeneous” mode of transformation. However, there is no evidence that the slip inhomogeneity arises from slip occurring after the generation of the new structure and it is quite possible that the slip inhomogeneity is an inherent feature of the trans-
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formation mechanism. Indeed, a mechanism that produces the slipped state directly seems more probable than one that produces a highly stressed product which then slips. In contrast to Geisler’s views, other workers4s20-n have attempted to develop growth mechanisms for martensitic transformations in which the atomic displacements are produced by the spontaneous movement of transformation dislocations. The striking similarity between martensitic transformations and mechanical twinning provides strong evidence in favour of such mechanisms. Martensitic transformations and mechanical twinning are both accompanied by homogeneous strains and in both processes the new crystals are plate-shaped and produced in time intervals of the order of a few microseconds; both processes can be induced by plastic deformation and the audible click that accompanies twinning in certain metals is also a feature of transformations. These common many martensitic characteristics suggest that the two processes occur by similar mechanisms. Further, as pointed out by Orowan,2:i the existence of lens-shaped lamellae in twinning (and martensitic transformations) implies the presence of dislocations as a simple geometrical corollary. The amount of shear across a lamella is proportional to its thickness and thus where the thickness changes, even by one lattice spacing, the amount of shear changes discontinuously by a small amount. This discontinuous change must be provided by a dislocation situated at the step. An explanation of mechanical twinning in terms of dislocations has been provided by the “pole mechanism” of Cottrell and Bilbyz4 and it seems attractive to consider similar mechanisms for martensitic transformations.21 In such dislocation mechanisms, it should be possible to include the slip inhomogeneity as an integral feature of the process. If we accept the hypothesis of a dislocation mechanism for martensitic transformation then, with a view to developing a general theory applicable both to diffusion-controlled and martensitic transformations, it becomes of interest to consider the possibility that diffusion-controlled transformations also occur by a dislocation mechanism. On the existing evidence there is no reason to reject this possibility. If a dislocation mechanism is operative in martensitic transformations in which the plates grow very rapidly, then the propagation of the transformation dislocations must occur spontaneously. However, this is not an essential feature of a dislocation mechanism and there is no reason why the propagation of transformation dislocations should not be spontaneous in some transformations and require thermal activation in others. If
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thermal activation is required the rate of growth will be a function of temperature and the new crystals will grow at an observable rate. In precipitations, where a change of composition is involved, a further possibility exists for here it is possible that the rate of propagation of the transformation dislocations is governed by the rate at which the required concentration of solute atoms can be assembled at the advancing interface. The concept of a common dislocation mechanism for diffusion-controlled and martensitic transformations leads to a ready interpretation of the results of the present investigation. If the geometry of the atomic processes involved in the two kinds of transformation is the same, then it is not surprising that the phenomenological theory developed for martensitic transformations should be capable of predicting the geometrical features of a typical diffusion-controlled transformation like the precipitation of CuBe. This speculation concerning the possible role of dislocation mechanisms in diffusion-controlled transformations is, of course, intended to apply only to the early stages of growth (the so-called stage of “coherent” growth). After a non-coherent interface has been produced, growth can occur by the accretion of individual atoms and the shape assumed by the growing crystal will be determined mainly by considerations of interfacial energy. REFERENCES J. S. Bowles and J. K. Mackenzie, Acta Met. 2, 129 (1954).
f. J. K. Mackenzie and J. S. Bowles, Acta Met. 2, 138 (1954).
3: M. S. Wechsler, D. S. Lieberman and T. A. Read, Trans. A.I.M.E. 197, 1503 (1953). 4. H. Suzuki, Sci. Rep. R.I.T.U. A6,30 (1954). 5. J. S. Bowles and C. S. Barrett, Progress in Metal Physics 3, 1 (1952). 6. J. S. Bowles and J. K. Mackenzie, Acta Met. 2, 224 (1954). 7. L. Northcott, J. Inst. Met. 59, 22.5 (1936). 8. A. H. Geisler, Acta Met. 1, 260 (1953). A. Guinier and P. A. Jacquet, Rev. Met. 41, 1 (1944). 1;: A. Guinier and P. A. Jacquet, Nature 155, 69.5 (1945). A. G. Guy, Gen. Elec. Rev. 49, 28 (Aug. 1946). :1: A. G. Guy, C. S. Barrett and R. F. Mehl, Trans. A.I.M.E. 175, 216 (1948). 13. J. S. Bowies, Trans. A.I.M.E. 191, 44 (1951). 14. C. S. Barrett, S$ucture of Melds, 2nd ed. (London, McGrawHill, 1953), p. 42. S. Miyatani, J. Phys. Sot. Japan 4, 181 (1949). H. Neerfeld and K. Mathieu, Arch. Eisenh. 20, 69 (1949). A. H. Geisler, Acta Met. 2, 639 (1954). 2. S. Basinski and J. W. Christian, Acta Met. 1, 759 (19.53). A. H. Geisler, J. H. Mallery and F. E. Steigert, Trans. A. I. M. E. 194, 307 (1952). 20. M. Cohen, E. S. Machlin and V. G. Paranjpe! Thermodynamics in Physical Metallurgy (Cleveland, Ohio, A. S. M., 1950), p. 242. B. A. Bilby, Phil. Mag. 44, 782 (1953). ;1. Z. S. Basinski and J. W. Christian, Phil. Mag. 44, 791 (1953). 23: E. Orowan, Dislocations in Mel& (New York, A.I.M.E., 1954), p. 116. 24. A. H. Cottrell and B. A. Bilby, Phil. Mag. 42, 573 (1951).