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Overview no. 57 Morphology, crystallography and kinetics of sympathetic nucleation
OOOI-6160/87 53.00 iO.00 Ltd
AC~Qmerail. Vol. 35. No. 3, pp. 549-563. 1987 Printed in Great Britain. All rights resewed
Copyright ‘i: 1987Pergirmon Journals
OVERVIEW NO. 57 MORPHOLOGY, CRYSTALLOGRAPHY AND OF SYMPATHETIC ~~CLEAT~O~
Department
KINETICS
E. SARATH KUMAR MENON? and H. 1. AARONSON of Metallurgical Engineering and Materials Science, Carnegie-Mellon Pittsburgh, PA 15213, U.S.A. (Received
28 August 198.5; in
reri.sedform IO
April
University,
1986)
Abstract-The phenomenon of sympathetic nucleation was investigated in detail, employing the Widmanstatten z precipitation in Ti-6.6 at% Cr and Ti-8.6 at% Mn alloys as a typical example. Two types of mo~holo~cal arrangement, resulting from edge-to-edge and edge-to-face sympathetic nucleation are clearly identified. The question of whether or not the latter arrangement is formed by branching in the style of solidification dendrities was considered. By expressing the orientation relationship of precipitates in the form of a matrix, a search was made for the relationships which permit branching with equivalent pairs of conjugate habit planes. From the results of such analyses, carried out on commonly observed orientation relationships among f.c.c., b.c.c. and h.c.p. phases, it is concluded that crystallographic constraints essentially prohibit branching in all of these systems. A comparative theoretical study of the relative nucleation kinetics of homogeneous, grain boundary and sympathetic nucleation demonstrated that the activation barrier associated with these processes become comparable when sympathetic nucleation occurs on the terraces of plate shaped precipitate and the matrix:precipitate interface is replaced by a relatively low energy precipitate:precipitate boundary. TEM examination of regions in which sympathetic nucleation occurred indicated that the interphase boundaries are invariably replaced by small-angle precipitate:precipitate boundaries essentially in accord with theory. R&urn&-Nous avons itudii en detail le phinomine de la germination sympathique en prenant comme exemple caractkristique la prkcipitation a de Widmanstiitten dans les alliages Ti-6,6at% Cr et Ti-8,6 at% Mn. Nous identifions nettement deux types d’arrangements morphologiques risultant des germinations sympathiques bord-$-bord et bord-&face. Nous nous sommes pod la question de savoir si cette derni&e disposition etait provoquk ou non par un phenomene de rami~cation analogue ii celui qu’on rencontre dans les dendrites de solidification. En exprimant la relation d’orientation des pticipitts sous la forme d’une matrice, nous avons cherchi les relations qui permettent la ramification avec des paires iquivalentes de plans d’accolement conjugui?s. A partir des rt?sultats de telles analyses, me&es sur les relations d-orientation couramment observkes pour les phases c.f.c., C.C.et h.c., nous concluons que les contraintes cristallographiques interdisent par nature la ramification dans tous ces systimes. Une btude thiorique comparative de la cidtique de ge~ination relative dans le cas d’une ge~ination homo&ne, intergranulaire ou sympathique a montrk que les barrieres d’activation associies $ ces m&canismes deviennent comparables lorsque la germination sympathique a lieu sur les terrasses de pticipit& lamellaires et que I’interface matricelpricipite est remplad par un joint pr&cipiti/prtcipitl d’knergie relativement faible. L’examen par microscopic Clectronique en transmission des r&ions ori s’est produite une germination sympathique a montrk que les joints interphases sont invariablement rempla&s par des joints pr&pit& pr&cipitC faiblement disorient&s. en accord avec la theorie. Zusummenfassung-Die Erscheinung der sympathetischen Keimbildung wird ausfiihrlich an Widmanstlttensz-Ausscheidungen in den Leaierungen Ti-6.6 At.% Cr und Ti-8.6 At,% Mn als tvbische Beispiele untersucht. Zwei mo$hologische inordningen, die von sympathetischer Keimbildung*&tntezu-Kante oder Kante-zu-Fllche herriihren, kijnnen klar unterschieden werden. Die Frage wird untersucht, ob die Anordnun~ Kante-zu-FIBche von einem Verzweigen der Erstarrung~endriten herriihrt. Mir der Beschreibung des Orientierungszusammenhanges der Ausschieidungen in Form einer Matrix wurden Orientierungszusammenh&nge gesucht. welche ein Verzweigen in iquivalente Paare konjugierter Habitebenen erlauben. Aus dieser an den iiblicherweise unter k.f.z., k.r.z. und hex. Phasen beobachteten Orientierungsbeziehungen ausgefiihrten Untersuchung IaDt sich folgern, daD die kristallografischen Einschrlnkungen Verzweigen in allen diesen Systemen im wesentlichen verbieten. Eine vergleichende theoretische Untersuchung der relativen Keimbildungskinetik von homogener. Korngrenz- und sympathetischer Keimbildung zeight, da13 die Aktivierungsschwellen dieser Prozesse vergleichbar werden, wenn die sympathetische Keimbildung an Terrassen von plattenfijrmigen Ausschieidungen auftritt und die GrenzflPche Matrix Ausscheidung durch eine relativ niederenergetische Grenze zwischen Ausscheidung und Ausscheidung ersetzt wird. Elektronenmikroskopische Untersuchungen in Durchstrahlung zeiplen. dal3 die Zwischenphasengrenzen unweigeriich durch Kleinwinkelgrenzen zwischen Ausscheidungen ersetzt urrden: diese Beobachtung stimmt im wesentlichen mit der Theorie itberein.
*Present address: !+\a1 Chemical & Metallurgical Laboratory. 549 4u 1.’1--4
Naval Dockyard. Tiger Gate. Bombay 400 023, India.
550
MENON and AARONSON:
OVERVIEW
NO. 57
1. INTRODU~ION
Sympathetic nucleation can be defined as the nucleation of a precipitate crystal, the composition of which differs from that of the matrix, at the interphase boundary of another crystal of the same phase [I, 21. This type of nucleation has been observed to occur frequently in a considerable variety of Fe-base [I-S], Cu-base [6] and Ti-base [2,7, 81 alloys. Both the sympathetically nucleated crystal and the one at whose interphase boundary this crystal is nucleated often, though not always develop in the plate morphology. Two basic configurations of the sympathetically nucleated and the “substrate” crystals may be recognized [I, 21. As illustrated schematicalIy, and with optical micrographs of structures developed during the proeutectoid 01reaction in Ti-X alloys presented in Fig. 1, these are the edge-to-edge and edge-to-face configurations. Problems arise in the identification of each of these co~gurations as the product of sympathetic nucleation rather than of some other process. In the edge-to-face, an alternative explanation is branching of the substrate plate [9-121. Particularly when the microstructure is examined with optical metallography, this is appealing because the boundary between the sympathetically nucleated and the substrate crystals is often difficult to detect and, when observed, can be construed as deriving from poly gonization or perhaps a growth fault rather than as a consequence of fresh nucleation. This problem is dealt with crystallographically in the next section, where it will be demonstrated that this explanation can often be unequivocally eliminated if the orientation relationships and lattice parameters of the precipitate and matrix crystals are known. The boundary detection and origin problem also arises in connection with edge-to-edge sympathetic nucleation. This is examined experimentally in the following section, using the proeutectoid a reaction in Ti-X alloys as a convenient model system. The result obtained on the orientation relationship between the sympathetically nucleated and substrate crystals during these studies delineates the nature and probable origin of the grain boundary between them and provides further insight into the mechanism of sympathetic nucleation. The kinetics of sympathetic nucleation are examined in the following section by means of classical heterogeneous nucleation theory in order to examine some of the circumstances in which such nucleation is favored. 2. EXPERIMENTAL
PROCEDURE
In the course of this study, two alloysTi-6.62at% Cr (contained 340 ppm of 0, and 30ppm of HZ) and Ti-8.6at% Mn (contained 590ppm of O,)---were used. However, most of the investigation was conducted on the Ti-Cr alloy. The Ti-Mn alloy was prepared at the Rockwell Inter-
v Fig. 1. Examples of morphologies resulting from sym pathetic nucleation. (A) and (C) show schematic diagrams while (B) and (D) show optical micrographs illustrating the edge-to-edge and the edge-to-face configurations respectively. (B) Ti-6.6 at% Cr, B solution treated and isothermally reacted at 973 K for 1h. (D) same as (B).
national Science Center. A 0.030 kg button of the alloy was melted 12 times and then homogenized for 7 days at 1273 K in vacuum. the Ti-Cr alloy was prepared at Titanium Metals Corporation of America by arc melting in an argon atmosphere. The alloy was obtained in the form of a bar 0.045 x 0.045 x 0.225 m (approx. 2.25 kg). The alloys were encapsulated in evacuated quartz tubes and homogenized at 1273 K for 3 days. ~n~vidua~ specimens,
MENON
and AARONSON:
0.010 x 0.010 x 0.008 m, were cut from the bulk homogenized alloy and were encapsulated in vycor tubes. Each capsule was evacuated three times to a pressure of 2-6 x 10-6Torr and flushed twice with a reduced pressure of purified He prior to sealing. The specimens were solution treated at 1273 K for 20 min, isothermally reacted in deoxidized lead baths and quenched in iced water. Specimens for TEM were prepared by ion milling.
OVERVIEW
and A is the transformation matrix. The column vectors in matrix A represent the unit vectors of the fl lattice expressed in terms of the a lattice. The matrix A is found by choosing any three non-coplanar vectors in the reciprocal lattice of a and finding the corresponding vectors in the B reciprocal lattice. Planes in the b lattice can be expressed in terms of the a lattice by the relation: h k
3. BRANCHING VS EDGE-TO-FACE SYMPATHETIC NUCLEATION OF PRECIPlTATE PLATES
551
NO. 57
h =A-’
This problem can be conveniently addressed by utilizing the clearly documented observations that precipitate plates normally have a well-defined lattice orientation relationship with respect to the matrix grain in which they nucleated and that their broad faces can be characterized by conjugate habit planes in the matrix and precipitate phases. Dub& [3] had pointed out that a proeutectoid ferrite plate obeying the Kurdjumov-Sachs orientation relationship with respect to austenite cannot have more than one { 1IO}, habit plane parallel to a { 11 I}, habit plane in the austenite matrix.t Hence, ferrite plates, which readily exhibit edge-to-face configurations [ 1,2], cannot have produced them by branching. Dube’s important result was obtained by means of stereographic analysis. His considerations will now be extended with matrix algebra to all possible pairs of the three most common metallic crystal structures, f.c.c., b.c.c. and h.c.p. (in which atomic arrangements will be assumed not to be complicated by the presence of long range order). All of the most frequently observed orientation relationships between these crystal structures will be utilized. In the case of h.c.p. crystals. a wide range of c/a ratios will also be considered. Each extension of the DubC finding thus made will mean that branching of a single precipitate crystal (in the style of solid state dendrities [9-141) is eliminated as an explanation for the edge-to-face configuration shown in Fig. I The orientation relationship between two crystals of phases r and 8. each of which has a different structure. can be expressed in a matrix form as:
(1) where /I. k. I refer to the Miller indices of the planes tThe macroscopic conjugate habit planes of individual ferrite plates normally deviate appreciably from these low index planes. but on an atomic scale, Dube’s description has been proved correct [13]. :Both the four index and the three index notations for the h.c.p. structure have been used in this paper. sh’ote that the reciprocal lattice corresponding to the h.c.p. lattice is rotated 30 degrees with respect to the realspace lattice [16].
k
(2)
01 z
01 B
where A-’ is the inverse of the matrix A. The relationship between the directions [WV] in the two lattices can be written as:
uu [I [I V
=AT
v
(3)
WE
WP
where AT represents the transpose of matrix A. Conjugate habit planes in each pair of crystal structures consistent with matrix A will next be deduced. Then the possibilities for two or more habit planes of a given form in one phase being parallel to their counterparts in the other phase must be investigated. These considerations are best explicated by an example. 3.1. h.c.p. precipitation from fc.c. The commonly observed orientation relationship, e.g. during precipitation of AgzAI from a Al-Ag [15] and of K from a Cu-Si [15] is:
(0001 )h.c.p.IIt 111Lc.c. < 1m,,,
II( 1~%,
For the following particular tation relationship1 viz.:
variant
.
(4) of this orien-
(~1)h.c.p.ll(111)r.c.c.
ww,,.,. II[I m,c.c.
(5)
to provide a basis for deriving the matrix A, the following pairs of parallel directions in the reciprocal lattices (denoted with an asterisk) of the two structures are noted:8 [ 1w:,p,
IIP’fm,:.,
IO1OIL.,II[11wc.c.
(6)
P011,*,,II[1~11iL.’ Denoting ah.c.p. and %c.p. as the lattice constants of the h.c.p. structure and (I~.~.~. as the lattice parameter of the f.c.c. structure, the relationship between reciprocal lattice vectors in the h.c.p. and the f.c.c. lattices (or equivalently, between planes in the corresponding real-space lattices) can now be found as:
MENON and AARONSON:
552
OVERVIEW
NO. 57
however, these calculated ratios represent situations rarely, if ever. encountered ex~rimentally. Since the permit ( 1mkt. II‘i 1lmh.c.p. habit planes-which branching at all C/Q ratios-also do not appear to have been reported (insofar as we are aware), it is now clear that sidebranching of plates formed during f.c.c.-h.c.p. transformations is not to be expected, and thus the configuration (ii) in Fig. 1 is likely to result from sympathetic nucleation when developed during such a reaction. 3.2. h.c.p. precipitation from b.c.c. The Burgers orientation relationship [ 171. the commonly observed orientation relationship between the Q (h.c.p.) and the /3 (b.c.c.) phases in Zr-base [ 181and Ti-base 1191alioys, is expressed as: (OOO1)ll.C.,.ll~O~ 1fb.c.c.
Table 1 shows the planes in the h.c.p. lattice that are parallel to specific (lOO)r,,. , {110)rss. and ( 11 l)r,_. planes when the orientation relationship in equation (5) is obeyed.? It is clear that if the set of conjugate ((@ol)h.c., 11I1 1 1 )f.c.c. represents habit planes, branching is not permitted. However, as can be seen in Table I, since all the three { 1l~O)h,c,p, planes are parallel to (llO)eC.C. planes, if this pair represents the conjugate habit planes, sidebranches can form on h.c.p. plates growing into an f.c.c. matrix. In addition to this pair of multiple conjugate habit planes, Table 1 shows that there are also three { IOO), {11 I} and {1101 planes parallel to sets of three h.c.p. planes, each of the same form, at specific values of the c/a ratio. For example, it can be deduced from Table 1 that branching is permitted if c/a = -,/3/,/S, and (01 lfr.C.E. /I {IO1 l),,_. represents the conjugate habit planes. These results are listed in Table 2, indicating the precise c/a ratios at which multiple conjugate habit planes exist and at which sidebranching is thus c~stallographicaily feasible. Recalling that the ideal c/a ratio in h.c.p. is ~81~3, TThese considerations are restricted to the forms of f.c.c. planes with the lowest indices. There seems no need at the present time to investigate higher index planes in connection with f.c.c.-h.c.p. transformations.
Following the procedure described in the previous subsection, the variant expressed by: (Al),,,.
11 to 1I)b.c.c.
[loolh.c.,
11tlT1lb.c,c.
can be written in the matrix notation
wOC,,
IIWO,
E.P.
$=
1.225
~~o~h~.~.~t(toTth,,;(o~o~r.~~ i~(Ttot)~~.~;fwtfr,,,
$=
1.225
(~TI~.~,II~~ZO~~~.~.~.; (11~),,,,.11(02~~),,,.; (TtJ)~,o,c,l1(~021)h.~p
J6 = 2.449
(~~),.,,.il(~OT2),,.,;
,/6 = 2.449
(ITl)css.~~(lTOl)h.rp.; (l1Tbc.c 11(0lTk.~,;
* =0.612 2-42
(Ot ~)~.~~.II(ToII)~.~~; (110),.,, U~O~TIt~,~.,;(iOik,,
All
(OITL.
11(T2T0),4
(9) as:
This relationship between the b.c.c. and the h.c.p. lattices has been obtained previously by Notkin et al. [20]. The h.c.p. planes that are parallel to some of the low index planes in the b.c.c. lattice are calculated from equation (10) and are listed in Table 3. Unlike the f.c.c.-h.c.p. transformation, there are no forms of conjugate habit planes that permit sidebranching at al1 c/a ratios. Table 4 shows some specific cases where sideb~nching is permitted at particular c/a ratios. However, in all of these cases, the c/a ratios are not usually encountered and the habit planes in the h.c.p. lattice are either high index or irrational. Other observed orientation relationships between b.c.c. and h.c.p. structures are the Pitsch-Schrader
Tabk 2. c/a ratios and conjugate habit plane pairs that may resull in side branch formation c _ n
(8)
(11~0>,,,11(lil)~.~.~..
II(oT~~~.~.~.
(OW~,~,~ 11(T102h.e.~.:(ooI)~.,.II(oTI~x.,.~,
~To~),,.II~TT~o),,,
(TII)II(TOI
~b,,. II(ITo~~~.~.
i (T w,,,ATT~,,<,
MENON and AARONSON: Table 3. Parallel Wk.~
planesin b.c.c. and
5.26 from (1120)
(loo) (010)
-0.81.0.90,-0.09.i
(
(001) (III)
h.c.p. lakes
>
5.06 from(T100)
WIT)
(
0.81. -0.9010.09 F 'a >
(110)
(
0.41,0.30.-0.71,FY ll >
(TO])
(2TTo)
010)
(IIT)
(T3Zo)
(101)
C
WOl)
(011)
(ITI)
(TII)
(hk%,
(W,,,
WC,,,
r.
553
OVERVIEW NO. 57
( (
-
(
OIT: (I>
(
OTI: a>
1.62, 0.60.1.02, : ll>
1.62,-0.60, -1.02,~ (I >
-0.41, -0.30,0.71,~ a>
Table4. c/a ratios and conjugatehabitplanepairsthatmay result in sidebranch formation c
VW,,,
a 0.798 I.125 I.125 I.414 I.414
3.3. b.c.c. precipitation from j2.c.
<100),,, ll(oT1L (NW) orientation
3
[):‘
equations
(16)
Since for this variant, (1 1l)LC.C. )I(01 ,/2)b.c.c., the particular variant of the NW orientation relationship:
can be obtained by rotating the b.c.c. crystal about [TOO],C,,,by 9.74” (the angle between the (01 l)b,c,c, and the (01&L.). Jn matrix notation, this is conveniently done by premultiplying the matrix in equation (16) by the rotation matrix, R,. The elements of this rotation matrix is obtained from the general formula [24]. In the present case R, takes the form:
orientation
re-
(13)
(OOl),,, IL(Oll),,,
~[oii],,,
;
representing
re-
The matrices representing these orientation relationship can be derived by starting with the Bain orientation relationship for the variant:
[loo],,,
relationship
[1@4,.,.c. IIPT 1ltc.c.
(12)
~~~o~~h,,
I~000,,,.
The orientation (13) is then:
(15)
(11)
1011Lc II (1 11L
(lil),,,
W1lb.c.c. lIuo1lr.c.c..
(01 J)b.C.C. IIU 1l)f.c.c
~OO1Jh.c.c IIto1 1L
and the Kurdjumov-Sachs (KS) lationship is: ~~l~j,,,Iljlll)~,,.,,
in which:
[);;;=$;
As in the case of b.c.c.-h.c.p. systems, the commonly observed orientation relationships are interconnected by simple rotations [21]. The Bain relation can be expressed as:
uw,,,
P
(010~ il(T2T3),,,; (~l),,,,1t(l~l3),,, (IW,,, 11(0443),,,,; (TO]),,, II(W~ (Tl2),,, 11(@49),,,,: (121~ IIW~),,, (IlO),,, ll(0, 1.06.-1.06,I)h~.P;(T~l)~.cc IIW. -I.06,1.06,~)~.~.p (TIo),,J(-l.l5,0.42,0.73, l),,,,;(lOl),,, Il(l.15, -0.42, -0.73.1)
[21] and the Potter [22] relationships; both can be obtained by simple rotations from the Burgers’ orientation relationship [21]. The orientation relationship between c-carbide and ferrite described by Jack [23] is very similar to the Burgers’ relationship. It is similarly found that these orientation relationships also do not permit sidebranching except under very unusual combinations of c/a ratio and conjugate habit planes.
The Nishivama-Wassermann lationship -is expressed as:
IWW,,
(14)
R,={;
;;
-$:}.
(17)
(18)
Now, in the NW orientation relationship (11 l)LC.C. is parallel to (1, cos 6 + J2 sin 8, sin 8 - J2 cos 0). If $ represents the angle between this b.c.c. plane and (01 l)bc,c, then a rotation of the b.c.c. lattice by
MENON and AARONSON:
554
OVERVIEW NO. 57
Table 5. Parallel Dlanes in b.c.c. and f.c.c. lattices
(~0,,,c BAIN
WJ)
WJ)
@W
(Too
KS
(W (011)
@IT) (IW
mt CTW
uw (111)
01 I)
(1W (IIT)
$~~2
b.74’ (1. -Jz, 9.74” (T, 42, 9.74” (1. J2, 9.74”
I) from (TI I) 0 from (ITI)
1)
from (TTtf
1)
from (111)
(0.A x/2,
9.74” from (0, T, J2) 9.74”from L/2, Lo) 9.74” from 1-J% LO) 9.74” from
(0,0.99. - 0.17) 9.74’ from (010) (-0.7l,O.J2,0.70) 6.88’ from (TOi) (0.71, 0.12. 0.70) 6.88’ from (101) (0, 0.24, 1.39) 9.74’ from (001) 000) (-0.71, 1.11, 0.53) 17.19” from (TII) (0.71, -0.87, 0.87) 5.26” from firi) (-0.71, -0.87, 0.87) 5.26” from (JTl) (-0.71, 1.11, 0.53) 17.19” from (111) (011)
(-0.07. 0.98. -0.17) 10.53 from (010) (-0.67.0.07.0.74) 5.26 from (TO11 (0.74, 0.17. 0.65) 10.30 from (101) (0.07. 0.24. I .39) 10.30 from (001) 5.26‘ from (TOO) (-0.74. 1.06. 0.57) 14.21 from (rll) (ITI)
(0, -0.75. 1.56) 1.04” from (072) (1.41, 0.99, -0.17) 11.56” from (110) (-1.41,0.99, -0.17) 11.56” from (TIO)
(0.15, -0.74, 1.56) 5.09” from (oTz) (1.33. 1.07, -0.26) 10.53’ from (I IO) ( - 1.48,0.89, - 0.07) 14.21” from (TIO)
(oh) (110) (TIo)
(01 f)t..C.E. IIU 1~h.c.c. (19)
[I 1%ce. IIm%.c.~..
Accordingly, the NW and the KS variants represented by equations (17) and (19) respectively are expressed in matrix notation as: h NW 0I
=b.c.c.
-0.70711 0 0.98560 0.11957 af.c.c. 0.69692 ( -0.16910
=b.c,c.
0.70711 0.11957 0.69692 i
h NW x
k
m
0 1 LGE. and, h KS 01
-0.07491 =‘ abf.e
k
af.cs. b&c.
0.98316 -0.16667 (
-0.66667
0.74158
0.0749 1 0. I6667 0.74158
0.64983 1
h KS . (21) I 0 Lc.c. Table 5 lists the b.c.c. planes that are parallel to some of the low index planes in f,c.c. It can be seen from this table that for systems that obey the Bain orientation relationship, if the conjugate habit planes are represented by either (IOO),,~, /if 11Oh,,., or (01 l)r,,,, l](~i)b,,~., then a number of ~ssibilities for sidebranch formation is obtained provided that any of them can serve as the conjugate habit planes of b.c.c. or f.c.c. precipitate plates. For (lOO)r,,,c,//(I 10)b,G,,, if the NW orientation relationship xk
(011)
(01 I)
JI = 5.26” about [Ol l]bf,E.will produce the following variant of the KS orientation relationship:
k
(-0.59, -0.91. 0.91) 10.53’ from (TTI) (0.67, I .I 5, 0.48) 20.144- from (I 11)
holds, then branching is predicted only if the corresponding b.c.c. planes are irrational. However, if the KS orientation relationship is obeyed, branching is not permitted at all. All other possible cases (e.g. (1 IOf,,,. and (111 jf.c.c.)where more than one pair of conjugate habit plane is predicted (Table 5) involve irrational planes and hence side branch formation is infeasible. The results of the analyses which have been recounted make it easy to understand why in all experimentally observed examples of true sidebranching, i.e. solid state dendrite formation, so far reported, the crystal structure of the precipitate is either identical to that of the matrix or differs significantly only in that it is a long-range ordered version of the matrix crystal structure, and also the lattice orientations of the two phases are identical [l I]. Thus, precipitation of y from fl Cu-Zn [IO]. of y’ from Ni-base superalloys [9], cubic carbide in austenitic stainless steels [25], Ni,Mo precipitation in Ni-Mo [26] etc. fulfill the identical orientations and either of the crystal structure specifications. Dendrite formation during c~stallization of amorphous phases [27-29,12]-or during solidification-does not confront the type of crystallographic constraints considered here since the matrix phase is not crystalline. 4. MORPHOLOGY AND MECHANISMS SYMPATHETIC NUCLEATION
OF
The present studies were undertaken primarily for the purpose ofapplying TEM techniques to elucidate certain important aspects of sympathetic nucleation which could not be clarified by means of the optical microscoov techniaue. A Ti-6.62 at% Cr alloy was
MENON
and AARONSON:
OVERVIEW
NO. 57
555
(b)
(4
Fig. 2. Ti-6.6 at% Cr, /? solution treated and isothermally reacted at 973 K for (a) 1h, (b) and (c) 2 h. These TEM micrographs illustrate the appearance of the edge-to-edge configuration.
employed for this purpose. Unlike hypoeutectoid steels. in which sympathetic nucleation was first
reported [l], retention of the fl matrix at room temperature greatly facilitates the microstructural and crystallographic studies in Ti-base alloys. Typical examples of edge-to-edge sympathetic nucleation are presented in Fig. 2. These TEM micrographs illustrate that what appears as a single plate during optical microscopy examinations is often a contiguous array of smaller crystals. Similar observations have been reported earlier in Ti-Cr [7] and Ti-V [S. 301 alloys. Orientation analysis clearly showed that all of these r crystals examined are related to the matrix B crystal through the Burgers’ orientation relationship. Kikuchi pattern analysis indicated that the alpha crystals obey the Burgers’ orientation relationship to within -2. and that the alpha crystals are slightly misoriented with respect to each other. Figure 3 shows an example. The bright field micrograph (Fig. 3a) shows an z crystal (marked A) that has sympathetically nucleated at the edge of another r crystal designated B. The accompanying SAD patterns obtained from the matrix p phase and from r crystals A and B are shown in Fig. 3(b). (c) and (d) respectively. The zone axes as determined
from the spot patterns are [I 131, and [2TT3], and correspond to the Burgers’ orientation relation variant: (OOOl),[/(Ol l),; [2iiO],II[lTl],. Accurate orientations determined from the Kikuchi patterns indicated that the two tl crystals are misoriented by 0.94 with respect to each other. The observation of sympathetically nucleated 51crystals in Ti-V alloys that are separated by small angle boundaries has been earlier reported [S, 301. Figure 4(a) shows an example of edge-to-face sympathetic nucleation. From the diffraction contrast alone, it is clear that though crystal B may appear to be a sidebranch of crystal A, they are in fact two different crystals. Analysis of the SAD pattern shown in Fig. 4(b) indicates that these two crystals are again Burgers’ related. In fact, as can be seen from the stereographic projection in Fig. 5. these two crystals are related through a simple rotation of _ 60’ about the [1120], direction. A 57.42. rotation about the [I 1201, direction in Ti corresponds to a coincidence site lattice boundary with 2: = 13 [31]. Another frequently observed morphology of the z phase is shown in Fig. 6. The SAD pattern in Fig. 6(b) and the dark field micrographs presented in Fig. 6(c) and (d) clearly illustrate another case of
5.56
MENON and AARONSON:
OVERVIEW NO. 57
Fig. 3. Ti-6.6 at% Cr, /? solution treated and isothermally reacted at 973 K for 1 h. (a) is a bright field electron micrograph and (b) is the SAD pattern obtained from the region marked by a circle in (a), zone axes [I134 and [2rT3], (c) and (d) are SAD patterns obtained from the a crystals marked A and B respectively. Tbe bisectors of the Kikuchi pairs, (132T) and (3033). are shown.
sympathetic nucleation. This morphology differs from that shown in Fig. 4 in that the sympathetically nucleated a crystals have not grown into large plates and probably are examples of face-to-face nucleation. A stereographic analysis of this orientation is also shown in Fig. 5; the substrate and the sympathetically nucleated crystals are seen to he related by a rotation of -32” about the ~IOTO]~axis. A 35.10’ rotation
about [ lOTO],corresponds to a Z = 1 I boundary [3 I]. An important difference between the sympathetically nucleated a crystals shown in Fig. 6 and that in Fig. 4 is that the former crystals are not Burgers’ related to the grain into which they are growing. This morphology of the a phase has heen observed in a wide variety of titanium alloys [8, 19,32-351 and is referred to in the literature as Type 2a or non-
MENON and AARONSON:
557
OVERVIEW NO. 57
io13
2023 no
2111 , -I
’
--“‘_O I
I
I
I
I
?I
0523
I I I
or12
I B
Fig. 4. Ti-6.6 at% Cr, j solution treated and isothermally reacted at 973 K for 2 h. (a) is a bright field electron micrograph and (b) is the SAD pattern obtained from the central region of (a). (c) indexes the SAD pattern shown in (b). Zone axes are [JO32],and [7256],. These correspond to the Burgers’ orientation relationship variants. (OOOl),II(01I),; [I 12O],Ii[I 171, and (OOOl),~~(lOl),;[ll~O],/~[llT], respectively.
Burgers’ r [32]. Non-Burgers’ x and Burgers’ z are frequently related through { lOf2j,, twinning [32, 81. Margolin et al. [36] have observed the formation of Burgers’ z crystals on r :p interfaces and found them to be {lo71 iI twin related to the substrate z crystal, which obeys another variant of the Burgers’ relationship. They proposed that the new r crystals form by deformation twinning. The operation of sympathetic nucleation in producing this morphology has been suggested [19. 35.81, however. and the present work supports this mechanism. The question arises as to why edge-to-edge and face-to-face sympathetic nucleation leave behind a
small-angle boundary. The activation free energy for nucleation, AC*, would appear to be further reduced if the sympathetically nucleated crystal had the same orientation as its substrate crystal, thereby eliminating the interfacial energy needed to form a nucleus:substrate boundary of any type. A possible explanation for this is that, following Shewmon [37] and as is discussed further in the next section, sympathetic nucleation appears likely to take place at partially coherent facets of interphase boundaries in regions between growth ledges. While these boundaries have a relatively low interfacial energy, they are also immobile. thus preventing them from over-
558
MENON and AARONSON:
OVERVIEW
NO. 57
iii 1X +
--K _.--
/.R‘
ioio
ioio
/'
x, A hkil : sympothctically nucleated hcp
Fig. 5. (OOOl), standard stereographic projection for Ti superimposed on (01 l)b. The b.c.c. poles are shown by solid circles and the h.c.p. poles by open circles, crosses and triangles. Great circle a represents the foil plane in Fig. 4. As indicated by the dotted lines, the poles corresponding to crystal B in Fig. 4(a) are related to the crystal A by a simple rotation. Notice that the a crystals A and B in Fig. 4(a) obey the variants (OOOl),~](Oll),; [1120],]1[11T]Band (OOl),~](lOl),; [I 120~]][1IT], respectively. Great circle b shows the foil plane in Fig. 6. The sympathetically nucleated crystals are seen to be related to the substrate crystal by the 32” rotation indicated.
running embryos being formed by sympathetic nucleation and also, as suggested by Shewmon [371, permitting them to form in regions where appreciable supersaturation is available. Now, if the nucleus of a sympathetically nucleated crystal were to form in the vicinity of misfit dislocation(s) associated with the partially coherent facet, the fully coherent character now anticipated of all nucleation [38] would require that nucleation occur some distance away from the misfit dislocations. This is analogous to coherent nucleation at dislocations in single phase alloys. Precise positioning of the sympathetically formed nuclei so as to maximize the reduction in transformation strain energy achieved by nucleating in association with the dislocation, while at the same time replacing as much as possible of the fully coherent matrix:substrate boundary between the misfit dislocations with fully coherent substrate:substrate boundary, seems to require a small misorientation of the sympathetically formed nucleus with respect
to the substrate crystal in order to optimize satisfaction of both requirements.
the
5. RELATIVE NUCLEATION KINETICS In this section, we examine the kinetics of sympathetic nucleation and evaluate the conditions that will promote its occurrence. This is accomplished by comparing the kinetics of sympathetic nucleation with those of homogeneous and grain boundary nucleation. The rate of steady state nucleation, J:, is given by [39]: (22) where Z is the Zeldovich non-equilibrium factor, jI* is the rate of attachment of single atoms to the critical nucleus, N is the density of atomic nucleation sites being employed per unit volume of the matrix phase,
MENON
and AARONSON:
OVERVIEW
559
NO. 57
ii
101, ::
Fig. 6. Ti-8.6at% Mn, fi solution treated and isothermally reacted at 973 K for 10s. (a) bright field micrograph showing z plates in a p matrix. (b) SAD pattern corresponding to (a). The zone axes are [2TT3], and [OOOl],. (c) and (d) are dark field micrograph showing the substrate a and the sympathetically nucleated interface a respectively.
k is Boltzmann’s constant and T is absolute temperature. As previously discussed in detail [40], as
long as the undercooling below the equilibrium temperature is not very large, AC* is by far the most influential term in this equation; hence the following calculations may be safely restricted to this term. AG* is sensitively dependent upon the shape of the critical nucleus [38]. A reasonably rigorous calculation of AG* has only recently been achieved, on considerations of classical nucleation theory, for homogeneously formed nuclei of an f.c.c. phase in an f.c.c. matrix [41]. The problem of nucleation at grain boundaries-even when the crystal structures of the matrix and the precipitate are the same-is yet to be resolved. Hence. in this analysis, resort has been made to the frequently employed pillbox, envisioned as lying coplanar with a grain boundary. Selfconsistent results have usually been achieved with this simple approximation [42-44]. Figure 7 illustrates the pillbox, and identifies all of the parameters associated with it, including the critical radius. r*. and the critical height. h*. Subscripting all relevant parameters associated with grain bound-
ary, homogeneous and sympathetic nucleation by b, h and s, respectively, the free energy change AGO, associated with formation of embryos of pillbox shaped nuclei at each of these three sites is: AC: = nr~h,r$, + 2nr:y$
+ 27~,,h,g:~
AGO,= rt&,&
+ y’l”s- ypp]
+ nr;[&
+ 2nr,h,y :fl AG; = rrr:h,& + rrr:y,, + 27rr,h,y’lp
(23)
(24) (25)
where r$=AG,+
W
(26)
where AG, is the volume free energy change and W is the strain energy attending nucleation. The activation energy, AC*, associated with critical nucleus formation is obtained by finding the maximum in AG” in equations (23x25) with respect to r. The resulting equations for AG* are: (27)
560
MENON and AARONSON:
OVERVIEW
NO. 57
/ 00
aa
iYC aQ
Fig. 7. Schematic drawing of the critical nucleus model used in the analysis for (a) grain boundary nucleation and (b) sympathetic nucleation, identifying the interfacial energies involved [42].
02
04
I
06
08
I.0
Fig. 8. Comparison of AG* for grain boundary sympathetic nucleation.
I2
and
y,,/y$ is sufficiently low, even when d1/4,, is small, sympathetic nucleation can occur at rates comparable to homogeneous nucleation. It may be pointed out here that nucleation at dislocations probably offers greater competition to sympathetic nucleation. Coherent nucleation at dislocations is a topic remaining in need of further theoretical treatment. However, Cahn [45] has compared the kinetics of incoherent nucleation at dislocation with homogeneous nucle-
where 6 =Y’ls+Y’zbs-Y&Q?.
(30)
In order to assess the feasibility of sympathetic nucleation as a viable alternative to grain boundary or homogeneous nucleation, the ratios, 4s/& and 4s/& of the driving force for sympathetic nucleation to that for the other two forms, are evaluated as a function of the interfacial energy ratios y& and yJy$ respectively when AC: = AC: or AC,*. The interfacial energy ratios are those for the net interfacial energy which must be acquired at the pillbox broad faces during the nucleation process; as indicated in equations (27)-(29), the same edge energy is needed in all three cases. Sympathetic nucleation is compared with grain boundary nucleation in Fig. 8. This plot is interpreted as follows: when yJt = 0.55, then AC: < AC,1 provided that br/bb> 0.75. As expected, &/4b increases with y&, becoming unity when y,Jc = 1. Physically, when y,,/~ = 0, the u:a boundary energy is zero and so AC: = 0. On the other hand, when y& = 1, the a:a boundary becomes energetically equivalent to the fl grain boundary and thus AC* for nucleation at the two types of sites becomes identical. In Fig. 9 sympathetic nucleation is compared with homogeneous nucleation. Plots for AC: = l/2 and l/3 of AC,* are also included to illustrate that when the interfacial energy ratio
I
I
12-
AG'
-:3 AG:
/
1
14 +ll 06-
4
’ ’ 04’ j 06’
02
’ s I.0 (
0.6
’
12
Y.. Tca@
Fig. 9. Comparison of AG* for homogeneous sympathetic nucleation.
and
MENON and AARONSON:
OVERVIEW
561
NO. 57
site considered: AC: < AC,* for $ = 0.1 l-O.29
AG~
AG,* < AG,* for ;
= 0.28-l .93.
(32)
d
I
OO
z
! 02
I
0’4
06 Y JF
,
/
08
3
I
IO
I 2
ci&
Fig. 10. Comparison of AG* for nucleation at dislocations and sympathetic nucleation.
ation. According to his analysis, AG~~AG~ = 0.15-0.40, where AC,* is the activation free energy associated with the formation of the critical nucleus at dislocations. Figure 10 shows a plot of ~~/#~ vs ~,,/-& when AG: = AG,*. This plot shows that under favorable conditions of interfacial energy ratio, sympathetic nucleation can also compete with nucleation at dislocations. Somewhat more detailed consideration of the feasibility of sympathetic nucleation can be made upon the basis of available data on interfacial energies. For disordered interphase boundaries [46], 0.6 < 3 < 0.9. (31) l.‘PP The interfacial energy of coherent interphase boundaries is considerably lower than that of counterpart disordered interphase boundaries. In the case of f.c.c.: b.c.c. boundaries in brasses, measurements made by Hu and Smith [47] indicate that the coherent f.c.c.:b.c.c. boundary energy is less than one-third that of incoherent f.c.c.: b.c.c. boundaries. The homophase boundaries which are created during sympathetic nucleation are low angle grain boundaries or coincidence site boundaries as shown in the previous section. Hence a conservative estimate of the ratio, ?%Ii:,, , is 0.02-0.1 (twin boundary energies in Cu and Ag are 0.02 and 0.05 of their grain boundary energies [48]). Using these limits, we obtain the following lower limits? driving force ratios such that AC,* is less than that corresponding to other types of
*The range 4, C#I~ etc. arises from the limiting values of the interfacial energies considered.
These results indicate that sympathetic nucleation is indeed feasible when suficient driving force is available. As earlier noted, it is necessary that adjacent growth ledges on immobile, partially coherent boundaries be spaced sufficiently apart relative to the extent of the diffusion fields associated with the ledges to permit AC, to approach that for homogeneous nucleation. Elaborate analyses of the diffusion fields associated with single [49,50] and multiple ledges [51,52] are now available. In the present context the variation of composition along the terrace of a ledge is important since the driving force for nucleation would increase as the composition at the terrace approaches the matrix composition. Figure 11 shows the variation of r the dimensionless supersaturation along the terrace of a ledge located at x = 0, obtained from Jones and Trivedi [49] and Atkinson [50]. Here, x is the distance along the terrace of the ledge and y is the distance along the riser of the ledge and f is defined as: f = %&Q- c= -c,
(33)
where qzJ.) is the composition at any point (x, y). c, is the matrix composition and c,,, is the composition at x = 0, y = 0. Accordingly, r = 1 denotes no supersaturation and r = 0 indicates the maximum available supersaturation. It should finally be pointed out that shear strain energy which can accompany the purely diffusional
1.
ATU INSOf ---JONES-TRIVEDI
7
Fig. 11,Variation of dimensionless supersaturation, 1”, with dimensioniess distance, X, along the terrace of a ledge with the riser at X = 0. Y is the dimensionless distance perpendicular to the interface. r calculated from Refs [49] and (501 is presented.
562
MENON and AARONSON:
transformations considered can also play a significant role in the kinetics of sympathetic nucleation. Lee and Johnson [53J and Russell er al. f54J have shown that if these strains have a component parallel to the broad faces of the plates which is oppositely directed in successive plates, then the net driving force for sympathetic nucleation can be markedly increased. 6. CONCLUSIONS
In this paper, various aspects of the phenomenon of sympathetic nucleation have been investigated. Various morphological features resulting from sympathetic nucleation have been illustrated with examples from TCX alloys. The question of whether or not supposedly sympathetically nucleated structures were actually formed by branching in the style of dendrites was first considered. By expressing the orientation relationship between two crystals in the form of a matrix, a search was made to see if precipitate crystals could branch in different directions while maintaining an equivalent pair of conjugate habit planes. Analyses of this question conducted on commonly observed orientation relationships among b.c.c., f.c.c. and h.c.p. phases indicated that crystallographic constraints do not permit branching in such systems except under very restricted conditions unlikeiy to he encountered experimentally. A comparative theoretical study of the kinetics of homogeneous, heterogeneous and sympathetic nucleation showed that the activation barrier associated with these processes becomes comparable when sympathetic nucleation causes the matrix:precipitate boundary to be replaced by a relatively low energy precipitate:precipitate interface. It is found that even when the driving force for sympathetic nucleation is less than half of that for heterogeneous or homogeneous nucleation, the process can become favorable. Experimental evidences showed that during sympathetic nucleation, the a$ boundary is replaced by a low angte or a coincidence lattice type a:a boundary. Acknowledgements-The authors express their appreciation for the support of the Air Force Office of Scientific Research through Grant No. AFOSR-742595E.
OVERVIEW
7. M. Unnikrishnan, E. S. K. Menon and S. Banerjee. J. Muter. Sci. 13, 1401 (1978). 8. E. S. K. Menon and R. Krishnan. J. .Vafer. Sri. 18,375
(1983). 9. R. A. Ricks. A. J. Porter and R. C. Ecob. dcfa rtvraf/.
31, 43 (1983). 10. R. D. Doherty. Crysral Growth (edited by 9. R. Pamplin), p. 576. New York (1974). 11. R. D. Doherty, Merals Sci. 16, I (1982). 12. R. Trivedi, Mater. Sci. Furam 3, 45 (f98.5). 13. J. M. Rigsbee and H. I. Aaronson, Acra metal/ 27, 365 (1979). 14. R. F. Sekerka and T. F. Marinis, Pruceedi)li)lgsof on in~erna~iona[ Conference on Solid-Solid Phase Trams~rmafions (edited by H. I. Aaronson. D. E. Laughlin.
R. F. Sekerka and C. M. Wayman). p. 67.-eds.. TMS-AIME. Warrendale. Pa (1983). 93, 78 IS. R. F. Mehl’and C. S. Barrett, TlifS-AIME (1931).
16. P. R. Okamoto and G. Thomas, Pkysicu Status SoiiJi (a) 25, 81 (1968). 17. W. G. Burgers, Pk,vsica 1, 561 (1934). 18. G. K. Dey and S. Banerjee, J. nucf. Mater. 125, 219 (1984). 19. J. C. Williams, T~~~niurnScience and Tecknoiog_v (edited by R. I. Jaffee and H. M. Burte). p. 1433. 20. A. B. Notkin, L. M. Uteviskii, -P.-V. Terent’eva and M. P. Usikov. Ind. Lab. 39. 1282 (1973). 21. U. Dahmen, ha metall. 3& 63 (1982)’ 22. D. I. Potter, 3. ie~s-common Merals 31, 299 (1973). 23. K. H. Jack, J. Iron Steel inst. 169, 26 (1951). 24. C. M. Wayman, Crystallography of Marrensitic Transformations. Macmillan, New York (1964). 25. J. M. S&cock, D. Raynor and G. Willoughby, Metals Sei. II, 551 (1977). 26. L. A. Nesbit and D. E. Laughlin, J. Cr_vst. Growth 51, 273 (1981). 27. K. H. Kuo, Y. K. Wu, J. 2. Liang and 2. H. Lai, Phil. Mug. 51A, 205 (1985).
28. B. G. Bagiey and D. Turnbull, Acra metall. 18, 857 (1970). 29. E.G. Baburaj, G. K. Dey, M. J. Patni and R. Krishnan, Scripta metah. 19, 305 (1985).
30. E. S. K. Menon. J. K. Chakravarttv. P. Mukhonadhvav and R. Krishna& Ti’gOr Science &d Te~knoiog;~ (edited by H. Kimura and 0. Izumi), p. 1481. TMS-AIME, New York (1980). 31. R. Bonnet, E. Cousineau and D. H. Warrington, Acra crystalogr. A37, I 84 (I 98 I ). 32. C. G. Rhodes and J. C. Williams, Metaif. Trans. 6A, 2103 (1976). 33. C. G. Rhodes and N. E. Paton, Merall. Trans. 8A, 1749 (1977). 34. J. C. Williams, Precipifarion in Solids (edited by K. C.
35. REFERENCES I. H. I. Aaronson and C. Wells, TMS-AIME 206. 1216 (I 956). 2. H. 1. Aaronson, Decomposition of Ausfenire by ~i~~~~al Processes (edited bv V. F. Zackav and H. 1. A~onson), p. 387. fnterscien&, New York-(1962). 3. Dub&, D. SC. thesis, Carnegie Institute of Technology (1948). 4. R. W. Heckel and H. W. Paxton. Trans. Am. Sot. Metals 53, 539 (1961). 5. R. A. Ricks. P. R. Howell and G. S. Barritte. J. Mater.
Sci. 17, 732. (1982). 6. J. A. Street, H. 1..___ Aaronsan and C. W. Spencer, J. Ittsr. ~~ __ Mefals 88, 3XI (I 95Y-60).
NO. 57
36. 37. 38.
Russell and H. I. Aaronson), p. 191. AIME, New York (1978). J. C. Williams, Ti and Ti Alloys: Scientific and Technological Aspects (edited by J.-C. Williams and A. F. Belov). D. 1477. Plenum Press. New York (1982). H. M~r~olin, E. Levine and M. Young, Mita#. ?rans. 8A, 373 (1977). P. G. Shewmon, TMS-AIME 233, 736 (1965). H. 1. Aaronson and K. C. Russell, Proceedings of an Inrernational Conference on Solid-Solid Phase Trans~ormari~s, (edited by H. I. Aaronson, D. E. Laughlin,
R. F. Sekerka and C. M. Wayman), p. 371. TMS-AIME, Warrendale, Pa (1983). 39. H. I. Aaronson and J. K. Lee. Lectures in the Theory of Phuse Transformations (edited by H. I. Aaronson). p. 83. TMS-AIME, New York (1975). 40. W. C. Johnson. C. L. White, P. E. Marth, P. K. Ruf, S. M. Tuominen, K. D. Wade, K. C. Russell and H. I. Aaronson, Mefall. Trans. 6A, 91 I (1975).
MENON
and AARONSON:
41. F. K. LeGoues, H. I. Aaronson, Y. W. Lee and G. J. Fix, Proceedings of an International Conference on Solid-Solid Phase Transformations (edited by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman), p. 427. TMSAIME, Warrendale, Pa (1983). 42. W. F. Lange III, M. Enumoto and H. I. Aarunson, Metall. Trans. In press. 43. M. Enomoto and H. I. Aarunson, Metall. Trans. 17A, 1385 (1986), 44. E. S. K. Menon and H. I. Aaronson, Metall. Trans. In press. 45. J. W. Cahn, Acta metall. 5, 169 (1957). 46. C. S. Smith, Imperfections in Nearly Perfect Crystals (edited by W. Shockley, J. H. Hollomon, R. Maurer and F. Seitz), p. 377. Wiley, New York (1952).
OVERVIEW
NO. 57
563
47. H. Hu and C. S. Smith, Acta metall. 4, 638 (1956). 48. L. E. Murr, Interfacial Phenomena in Metals and Alloys. Addison-Wesley, Massachusetts (1975). 49. G. J. Jones and T. Trivedi, J. appl. Phys. 42, 4299 (1971). 50. C. Atkinson, Proc. R. Sot. Lond. A378, 351 (1981). 51. G. J. Jones and R. Trivedi, J. tryst. Growth 29, 155 (1975). 52. C. Atkinson, Proc. R. Sot. Lund. A384, 197 (1982). 53. J. K. Lee and W. C. Johnson, Scripta metall. 11, 477 (1977). 54. K. C. Russell, D. M. Barnett, C. J. Altstetter, H. I. Aaronson and J. K. Lee, Scripta metall. 11, 485 (1977).
AC~Qmerail. Vol. 35. No. 3, pp. 549-563. 1987 Printed in Great Britain. All rights resewed
Copyright ‘i: 1987Pergirmon Journals
OVERVIEW NO. 57 MORPHOLOGY, CRYSTALLOGRAPHY AND OF SYMPATHETIC ~~CLEAT~O~
Department
KINETICS
E. SARATH KUMAR MENON? and H. 1. AARONSON of Metallurgical Engineering and Materials Science, Carnegie-Mellon Pittsburgh, PA 15213, U.S.A. (Received
28 August 198.5; in
reri.sedform IO
April
University,
1986)
Abstract-The phenomenon of sympathetic nucleation was investigated in detail, employing the Widmanstatten z precipitation in Ti-6.6 at% Cr and Ti-8.6 at% Mn alloys as a typical example. Two types of mo~holo~cal arrangement, resulting from edge-to-edge and edge-to-face sympathetic nucleation are clearly identified. The question of whether or not the latter arrangement is formed by branching in the style of solidification dendrities was considered. By expressing the orientation relationship of precipitates in the form of a matrix, a search was made for the relationships which permit branching with equivalent pairs of conjugate habit planes. From the results of such analyses, carried out on commonly observed orientation relationships among f.c.c., b.c.c. and h.c.p. phases, it is concluded that crystallographic constraints essentially prohibit branching in all of these systems. A comparative theoretical study of the relative nucleation kinetics of homogeneous, grain boundary and sympathetic nucleation demonstrated that the activation barrier associated with these processes become comparable when sympathetic nucleation occurs on the terraces of plate shaped precipitate and the matrix:precipitate interface is replaced by a relatively low energy precipitate:precipitate boundary. TEM examination of regions in which sympathetic nucleation occurred indicated that the interphase boundaries are invariably replaced by small-angle precipitate:precipitate boundaries essentially in accord with theory. R&urn&-Nous avons itudii en detail le phinomine de la germination sympathique en prenant comme exemple caractkristique la prkcipitation a de Widmanstiitten dans les alliages Ti-6,6at% Cr et Ti-8,6 at% Mn. Nous identifions nettement deux types d’arrangements morphologiques risultant des germinations sympathiques bord-$-bord et bord-&face. Nous nous sommes pod la question de savoir si cette derni&e disposition etait provoquk ou non par un phenomene de rami~cation analogue ii celui qu’on rencontre dans les dendrites de solidification. En exprimant la relation d’orientation des pticipitts sous la forme d’une matrice, nous avons cherchi les relations qui permettent la ramification avec des paires iquivalentes de plans d’accolement conjugui?s. A partir des rt?sultats de telles analyses, me&es sur les relations d-orientation couramment observkes pour les phases c.f.c., C.C.et h.c., nous concluons que les contraintes cristallographiques interdisent par nature la ramification dans tous ces systimes. Une btude thiorique comparative de la cidtique de ge~ination relative dans le cas d’une ge~ination homo&ne, intergranulaire ou sympathique a montrk que les barrieres d’activation associies $ ces m&canismes deviennent comparables lorsque la germination sympathique a lieu sur les terrasses de pticipit& lamellaires et que I’interface matricelpricipite est remplad par un joint pr&cipiti/prtcipitl d’knergie relativement faible. L’examen par microscopic Clectronique en transmission des r&ions ori s’est produite une germination sympathique a montrk que les joints interphases sont invariablement rempla&s par des joints pr&pit& pr&cipitC faiblement disorient&s. en accord avec la theorie. Zusummenfassung-Die Erscheinung der sympathetischen Keimbildung wird ausfiihrlich an Widmanstlttensz-Ausscheidungen in den Leaierungen Ti-6.6 At.% Cr und Ti-8.6 At,% Mn als tvbische Beispiele untersucht. Zwei mo$hologische inordningen, die von sympathetischer Keimbildung*&tntezu-Kante oder Kante-zu-Fllche herriihren, kijnnen klar unterschieden werden. Die Frage wird untersucht, ob die Anordnun~ Kante-zu-FIBche von einem Verzweigen der Erstarrung~endriten herriihrt. Mir der Beschreibung des Orientierungszusammenhanges der Ausschieidungen in Form einer Matrix wurden Orientierungszusammenh&nge gesucht. welche ein Verzweigen in iquivalente Paare konjugierter Habitebenen erlauben. Aus dieser an den iiblicherweise unter k.f.z., k.r.z. und hex. Phasen beobachteten Orientierungsbeziehungen ausgefiihrten Untersuchung IaDt sich folgern, daD die kristallografischen Einschrlnkungen Verzweigen in allen diesen Systemen im wesentlichen verbieten. Eine vergleichende theoretische Untersuchung der relativen Keimbildungskinetik von homogener. Korngrenz- und sympathetischer Keimbildung zeight, da13 die Aktivierungsschwellen dieser Prozesse vergleichbar werden, wenn die sympathetische Keimbildung an Terrassen von plattenfijrmigen Ausschieidungen auftritt und die GrenzflPche Matrix Ausscheidung durch eine relativ niederenergetische Grenze zwischen Ausscheidung und Ausscheidung ersetzt wird. Elektronenmikroskopische Untersuchungen in Durchstrahlung zeiplen. dal3 die Zwischenphasengrenzen unweigeriich durch Kleinwinkelgrenzen zwischen Ausscheidungen ersetzt urrden: diese Beobachtung stimmt im wesentlichen mit der Theorie itberein.
*Present address: !+\a1 Chemical & Metallurgical Laboratory. 549 4u 1.’1--4
Naval Dockyard. Tiger Gate. Bombay 400 023, India.
550
MENON and AARONSON:
OVERVIEW
NO. 57
1. INTRODU~ION
Sympathetic nucleation can be defined as the nucleation of a precipitate crystal, the composition of which differs from that of the matrix, at the interphase boundary of another crystal of the same phase [I, 21. This type of nucleation has been observed to occur frequently in a considerable variety of Fe-base [I-S], Cu-base [6] and Ti-base [2,7, 81 alloys. Both the sympathetically nucleated crystal and the one at whose interphase boundary this crystal is nucleated often, though not always develop in the plate morphology. Two basic configurations of the sympathetically nucleated and the “substrate” crystals may be recognized [I, 21. As illustrated schematicalIy, and with optical micrographs of structures developed during the proeutectoid 01reaction in Ti-X alloys presented in Fig. 1, these are the edge-to-edge and edge-to-face configurations. Problems arise in the identification of each of these co~gurations as the product of sympathetic nucleation rather than of some other process. In the edge-to-face, an alternative explanation is branching of the substrate plate [9-121. Particularly when the microstructure is examined with optical metallography, this is appealing because the boundary between the sympathetically nucleated and the substrate crystals is often difficult to detect and, when observed, can be construed as deriving from poly gonization or perhaps a growth fault rather than as a consequence of fresh nucleation. This problem is dealt with crystallographically in the next section, where it will be demonstrated that this explanation can often be unequivocally eliminated if the orientation relationships and lattice parameters of the precipitate and matrix crystals are known. The boundary detection and origin problem also arises in connection with edge-to-edge sympathetic nucleation. This is examined experimentally in the following section, using the proeutectoid a reaction in Ti-X alloys as a convenient model system. The result obtained on the orientation relationship between the sympathetically nucleated and substrate crystals during these studies delineates the nature and probable origin of the grain boundary between them and provides further insight into the mechanism of sympathetic nucleation. The kinetics of sympathetic nucleation are examined in the following section by means of classical heterogeneous nucleation theory in order to examine some of the circumstances in which such nucleation is favored. 2. EXPERIMENTAL
PROCEDURE
In the course of this study, two alloysTi-6.62at% Cr (contained 340 ppm of 0, and 30ppm of HZ) and Ti-8.6at% Mn (contained 590ppm of O,)---were used. However, most of the investigation was conducted on the Ti-Cr alloy. The Ti-Mn alloy was prepared at the Rockwell Inter-
v Fig. 1. Examples of morphologies resulting from sym pathetic nucleation. (A) and (C) show schematic diagrams while (B) and (D) show optical micrographs illustrating the edge-to-edge and the edge-to-face configurations respectively. (B) Ti-6.6 at% Cr, B solution treated and isothermally reacted at 973 K for 1h. (D) same as (B).
national Science Center. A 0.030 kg button of the alloy was melted 12 times and then homogenized for 7 days at 1273 K in vacuum. the Ti-Cr alloy was prepared at Titanium Metals Corporation of America by arc melting in an argon atmosphere. The alloy was obtained in the form of a bar 0.045 x 0.045 x 0.225 m (approx. 2.25 kg). The alloys were encapsulated in evacuated quartz tubes and homogenized at 1273 K for 3 days. ~n~vidua~ specimens,
MENON
and AARONSON:
0.010 x 0.010 x 0.008 m, were cut from the bulk homogenized alloy and were encapsulated in vycor tubes. Each capsule was evacuated three times to a pressure of 2-6 x 10-6Torr and flushed twice with a reduced pressure of purified He prior to sealing. The specimens were solution treated at 1273 K for 20 min, isothermally reacted in deoxidized lead baths and quenched in iced water. Specimens for TEM were prepared by ion milling.
OVERVIEW
and A is the transformation matrix. The column vectors in matrix A represent the unit vectors of the fl lattice expressed in terms of the a lattice. The matrix A is found by choosing any three non-coplanar vectors in the reciprocal lattice of a and finding the corresponding vectors in the B reciprocal lattice. Planes in the b lattice can be expressed in terms of the a lattice by the relation: h k
3. BRANCHING VS EDGE-TO-FACE SYMPATHETIC NUCLEATION OF PRECIPlTATE PLATES
551
NO. 57
h =A-’
This problem can be conveniently addressed by utilizing the clearly documented observations that precipitate plates normally have a well-defined lattice orientation relationship with respect to the matrix grain in which they nucleated and that their broad faces can be characterized by conjugate habit planes in the matrix and precipitate phases. Dub& [3] had pointed out that a proeutectoid ferrite plate obeying the Kurdjumov-Sachs orientation relationship with respect to austenite cannot have more than one { 1IO}, habit plane parallel to a { 11 I}, habit plane in the austenite matrix.t Hence, ferrite plates, which readily exhibit edge-to-face configurations [ 1,2], cannot have produced them by branching. Dube’s important result was obtained by means of stereographic analysis. His considerations will now be extended with matrix algebra to all possible pairs of the three most common metallic crystal structures, f.c.c., b.c.c. and h.c.p. (in which atomic arrangements will be assumed not to be complicated by the presence of long range order). All of the most frequently observed orientation relationships between these crystal structures will be utilized. In the case of h.c.p. crystals. a wide range of c/a ratios will also be considered. Each extension of the DubC finding thus made will mean that branching of a single precipitate crystal (in the style of solid state dendrities [9-141) is eliminated as an explanation for the edge-to-face configuration shown in Fig. I The orientation relationship between two crystals of phases r and 8. each of which has a different structure. can be expressed in a matrix form as:
(1) where /I. k. I refer to the Miller indices of the planes tThe macroscopic conjugate habit planes of individual ferrite plates normally deviate appreciably from these low index planes. but on an atomic scale, Dube’s description has been proved correct [13]. :Both the four index and the three index notations for the h.c.p. structure have been used in this paper. sh’ote that the reciprocal lattice corresponding to the h.c.p. lattice is rotated 30 degrees with respect to the realspace lattice [16].
k
(2)
01 z
01 B
where A-’ is the inverse of the matrix A. The relationship between the directions [WV] in the two lattices can be written as:
uu [I [I V
=AT
v
(3)
WE
WP
where AT represents the transpose of matrix A. Conjugate habit planes in each pair of crystal structures consistent with matrix A will next be deduced. Then the possibilities for two or more habit planes of a given form in one phase being parallel to their counterparts in the other phase must be investigated. These considerations are best explicated by an example. 3.1. h.c.p. precipitation from fc.c. The commonly observed orientation relationship, e.g. during precipitation of AgzAI from a Al-Ag [15] and of K from a Cu-Si [15] is:
(0001 )h.c.p.IIt 111Lc.c. < 1m,,,
II( 1~%,
For the following particular tation relationship1 viz.:
variant
.
(4) of this orien-
(~1)h.c.p.ll(111)r.c.c.
ww,,.,. II[I m,c.c.
(5)
to provide a basis for deriving the matrix A, the following pairs of parallel directions in the reciprocal lattices (denoted with an asterisk) of the two structures are noted:8 [ 1w:,p,
IIP’fm,:.,
IO1OIL.,II[11wc.c.
(6)
P011,*,,II[1~11iL.’ Denoting ah.c.p. and %c.p. as the lattice constants of the h.c.p. structure and (I~.~.~. as the lattice parameter of the f.c.c. structure, the relationship between reciprocal lattice vectors in the h.c.p. and the f.c.c. lattices (or equivalently, between planes in the corresponding real-space lattices) can now be found as:
MENON and AARONSON:
552
OVERVIEW
NO. 57
however, these calculated ratios represent situations rarely, if ever. encountered ex~rimentally. Since the permit ( 1mkt. II‘i 1lmh.c.p. habit planes-which branching at all C/Q ratios-also do not appear to have been reported (insofar as we are aware), it is now clear that sidebranching of plates formed during f.c.c.-h.c.p. transformations is not to be expected, and thus the configuration (ii) in Fig. 1 is likely to result from sympathetic nucleation when developed during such a reaction. 3.2. h.c.p. precipitation from b.c.c. The Burgers orientation relationship [ 171. the commonly observed orientation relationship between the Q (h.c.p.) and the /3 (b.c.c.) phases in Zr-base [ 181and Ti-base 1191alioys, is expressed as: (OOO1)ll.C.,.ll~O~ 1fb.c.c.
Table 1 shows the planes in the h.c.p. lattice that are parallel to specific (lOO)r,,. , {110)rss. and ( 11 l)r,_. planes when the orientation relationship in equation (5) is obeyed.? It is clear that if the set of conjugate ((@ol)h.c., 11I1 1 1 )f.c.c. represents habit planes, branching is not permitted. However, as can be seen in Table I, since all the three { 1l~O)h,c,p, planes are parallel to (llO)eC.C. planes, if this pair represents the conjugate habit planes, sidebranches can form on h.c.p. plates growing into an f.c.c. matrix. In addition to this pair of multiple conjugate habit planes, Table 1 shows that there are also three { IOO), {11 I} and {1101 planes parallel to sets of three h.c.p. planes, each of the same form, at specific values of the c/a ratio. For example, it can be deduced from Table 1 that branching is permitted if c/a = -,/3/,/S, and (01 lfr.C.E. /I {IO1 l),,_. represents the conjugate habit planes. These results are listed in Table 2, indicating the precise c/a ratios at which multiple conjugate habit planes exist and at which sidebranching is thus c~stallographicaily feasible. Recalling that the ideal c/a ratio in h.c.p. is ~81~3, TThese considerations are restricted to the forms of f.c.c. planes with the lowest indices. There seems no need at the present time to investigate higher index planes in connection with f.c.c.-h.c.p. transformations.
Following the procedure described in the previous subsection, the variant expressed by: (Al),,,.
11 to 1I)b.c.c.
[loolh.c.,
11tlT1lb.c,c.
can be written in the matrix notation
wOC,,
IIWO,
E.P.
$=
1.225
~~o~h~.~.~t(toTth,,;(o~o~r.~~ i~(Ttot)~~.~;fwtfr,,,
$=
1.225
(~TI~.~,II~~ZO~~~.~.~.; (11~),,,,.11(02~~),,,.; (TtJ)~,o,c,l1(~021)h.~p
J6 = 2.449
(~~),.,,.il(~OT2),,.,;
,/6 = 2.449
(ITl)css.~~(lTOl)h.rp.; (l1Tbc.c 11(0lTk.~,;
* =0.612 2-42
(Ot ~)~.~~.II(ToII)~.~~; (110),.,, U~O~TIt~,~.,;(iOik,,
All
(OITL.
11(T2T0),4
(9) as:
This relationship between the b.c.c. and the h.c.p. lattices has been obtained previously by Notkin et al. [20]. The h.c.p. planes that are parallel to some of the low index planes in the b.c.c. lattice are calculated from equation (10) and are listed in Table 3. Unlike the f.c.c.-h.c.p. transformation, there are no forms of conjugate habit planes that permit sidebranching at al1 c/a ratios. Table 4 shows some specific cases where sideb~nching is permitted at particular c/a ratios. However, in all of these cases, the c/a ratios are not usually encountered and the habit planes in the h.c.p. lattice are either high index or irrational. Other observed orientation relationships between b.c.c. and h.c.p. structures are the Pitsch-Schrader
Tabk 2. c/a ratios and conjugate habit plane pairs that may resull in side branch formation c _ n
(8)
(11~0>,,,11(lil)~.~.~..
II(oT~~~.~.~.
(OW~,~,~ 11(T102h.e.~.:(ooI)~.,.II(oTI~x.,.~,
~To~),,.II~TT~o),,,
(TII)II(TOI
~b,,. II(ITo~~~.~.
i (T w,,,ATT~,,<,
MENON and AARONSON: Table 3. Parallel Wk.~
planesin b.c.c. and
5.26 from (1120)
(loo) (010)
-0.81.0.90,-0.09.i
(
(001) (III)
h.c.p. lakes
>
5.06 from(T100)
WIT)
(
0.81. -0.9010.09 F 'a >
(110)
(
0.41,0.30.-0.71,FY ll >
(TO])
(2TTo)
010)
(IIT)
(T3Zo)
(101)
C
WOl)
(011)
(ITI)
(TII)
(hk%,
(W,,,
WC,,,
r.
553
OVERVIEW NO. 57
( (
-
(
OIT: (I>
(
OTI: a>
1.62, 0.60.1.02, : ll>
1.62,-0.60, -1.02,~ (I >
-0.41, -0.30,0.71,~ a>
Table4. c/a ratios and conjugatehabitplanepairsthatmay result in sidebranch formation c
VW,,,
a 0.798 I.125 I.125 I.414 I.414
3.3. b.c.c. precipitation from j2.c.
<100),,, ll(oT1L (NW) orientation
3
[):‘
equations
(16)
Since for this variant, (1 1l)LC.C. )I(01 ,/2)b.c.c., the particular variant of the NW orientation relationship:
can be obtained by rotating the b.c.c. crystal about [TOO],C,,,by 9.74” (the angle between the (01 l)b,c,c, and the (01&L.). Jn matrix notation, this is conveniently done by premultiplying the matrix in equation (16) by the rotation matrix, R,. The elements of this rotation matrix is obtained from the general formula [24]. In the present case R, takes the form:
orientation
re-
(13)
(OOl),,, IL(Oll),,,
~[oii],,,
;
representing
re-
The matrices representing these orientation relationship can be derived by starting with the Bain orientation relationship for the variant:
[loo],,,
relationship
[1@4,.,.c. IIPT 1ltc.c.
(12)
~~~o~~h,,
I~000,,,.
The orientation (13) is then:
(15)
(11)
1011Lc II (1 11L
(lil),,,
W1lb.c.c. lIuo1lr.c.c..
(01 J)b.C.C. IIU 1l)f.c.c
~OO1Jh.c.c IIto1 1L
and the Kurdjumov-Sachs (KS) lationship is: ~~l~j,,,Iljlll)~,,.,,
in which:
[);;;=$;
As in the case of b.c.c.-h.c.p. systems, the commonly observed orientation relationships are interconnected by simple rotations [21]. The Bain relation can be expressed as:
uw,,,
P
(010~ il(T2T3),,,; (~l),,,,1t(l~l3),,, (IW,,, 11(0443),,,,; (TO]),,, II(W~ (Tl2),,, 11(@49),,,,: (121~ IIW~),,, (IlO),,, ll(0, 1.06.-1.06,I)h~.P;(T~l)~.cc IIW. -I.06,1.06,~)~.~.p (TIo),,J(-l.l5,0.42,0.73, l),,,,;(lOl),,, Il(l.15, -0.42, -0.73.1)
[21] and the Potter [22] relationships; both can be obtained by simple rotations from the Burgers’ orientation relationship [21]. The orientation relationship between c-carbide and ferrite described by Jack [23] is very similar to the Burgers’ relationship. It is similarly found that these orientation relationships also do not permit sidebranching except under very unusual combinations of c/a ratio and conjugate habit planes.
The Nishivama-Wassermann lationship -is expressed as:
IWW,,
(14)
R,={;
;;
-$:}.
(17)
(18)
Now, in the NW orientation relationship (11 l)LC.C. is parallel to (1, cos 6 + J2 sin 8, sin 8 - J2 cos 0). If $ represents the angle between this b.c.c. plane and (01 l)bc,c, then a rotation of the b.c.c. lattice by
MENON and AARONSON:
554
OVERVIEW NO. 57
Table 5. Parallel Dlanes in b.c.c. and f.c.c. lattices
(~0,,,c BAIN
WJ)
WJ)
@W
(Too
KS
(W (011)
@IT) (IW
mt CTW
uw (111)
01 I)
(1W (IIT)
$~~2
b.74’ (1. -Jz, 9.74” (T, 42, 9.74” (1. J2, 9.74”
I) from (TI I) 0 from (ITI)
1)
from (TTtf
1)
from (111)
(0.A x/2,
9.74” from (0, T, J2) 9.74”from L/2, Lo) 9.74” from 1-J% LO) 9.74” from
(0,0.99. - 0.17) 9.74’ from (010) (-0.7l,O.J2,0.70) 6.88’ from (TOi) (0.71, 0.12. 0.70) 6.88’ from (101) (0, 0.24, 1.39) 9.74’ from (001) 000) (-0.71, 1.11, 0.53) 17.19” from (TII) (0.71, -0.87, 0.87) 5.26” from firi) (-0.71, -0.87, 0.87) 5.26” from (JTl) (-0.71, 1.11, 0.53) 17.19” from (111) (011)
(-0.07. 0.98. -0.17) 10.53 from (010) (-0.67.0.07.0.74) 5.26 from (TO11 (0.74, 0.17. 0.65) 10.30 from (101) (0.07. 0.24. I .39) 10.30 from (001) 5.26‘ from (TOO) (-0.74. 1.06. 0.57) 14.21 from (rll) (ITI)
(0, -0.75. 1.56) 1.04” from (072) (1.41, 0.99, -0.17) 11.56” from (110) (-1.41,0.99, -0.17) 11.56” from (TIO)
(0.15, -0.74, 1.56) 5.09” from (oTz) (1.33. 1.07, -0.26) 10.53’ from (I IO) ( - 1.48,0.89, - 0.07) 14.21” from (TIO)
(oh) (110) (TIo)
(01 f)t..C.E. IIU 1~h.c.c. (19)
[I 1%ce. IIm%.c.~..
Accordingly, the NW and the KS variants represented by equations (17) and (19) respectively are expressed in matrix notation as: h NW 0I
=b.c.c.
-0.70711 0 0.98560 0.11957 af.c.c. 0.69692 ( -0.16910
=b.c,c.
0.70711 0.11957 0.69692 i
h NW x
k
m
0 1 LGE. and, h KS 01
-0.07491 =‘ abf.e
k
af.cs. b&c.
0.98316 -0.16667 (
-0.66667
0.74158
0.0749 1 0. I6667 0.74158
0.64983 1
h KS . (21) I 0 Lc.c. Table 5 lists the b.c.c. planes that are parallel to some of the low index planes in f,c.c. It can be seen from this table that for systems that obey the Bain orientation relationship, if the conjugate habit planes are represented by either (IOO),,~, /if 11Oh,,., or (01 l)r,,,, l](~i)b,,~., then a number of ~ssibilities for sidebranch formation is obtained provided that any of them can serve as the conjugate habit planes of b.c.c. or f.c.c. precipitate plates. For (lOO)r,,,c,//(I 10)b,G,,, if the NW orientation relationship xk
(011)
(01 I)
JI = 5.26” about [Ol l]bf,E.will produce the following variant of the KS orientation relationship:
k
(-0.59, -0.91. 0.91) 10.53’ from (TTI) (0.67, I .I 5, 0.48) 20.144- from (I 11)
holds, then branching is predicted only if the corresponding b.c.c. planes are irrational. However, if the KS orientation relationship is obeyed, branching is not permitted at all. All other possible cases (e.g. (1 IOf,,,. and (111 jf.c.c.)where more than one pair of conjugate habit plane is predicted (Table 5) involve irrational planes and hence side branch formation is infeasible. The results of the analyses which have been recounted make it easy to understand why in all experimentally observed examples of true sidebranching, i.e. solid state dendrite formation, so far reported, the crystal structure of the precipitate is either identical to that of the matrix or differs significantly only in that it is a long-range ordered version of the matrix crystal structure, and also the lattice orientations of the two phases are identical [l I]. Thus, precipitation of y from fl Cu-Zn [IO]. of y’ from Ni-base superalloys [9], cubic carbide in austenitic stainless steels [25], Ni,Mo precipitation in Ni-Mo [26] etc. fulfill the identical orientations and either of the crystal structure specifications. Dendrite formation during c~stallization of amorphous phases [27-29,12]-or during solidification-does not confront the type of crystallographic constraints considered here since the matrix phase is not crystalline. 4. MORPHOLOGY AND MECHANISMS SYMPATHETIC NUCLEATION
OF
The present studies were undertaken primarily for the purpose ofapplying TEM techniques to elucidate certain important aspects of sympathetic nucleation which could not be clarified by means of the optical microscoov techniaue. A Ti-6.62 at% Cr alloy was
MENON
and AARONSON:
OVERVIEW
NO. 57
555
(b)
(4
Fig. 2. Ti-6.6 at% Cr, /? solution treated and isothermally reacted at 973 K for (a) 1h, (b) and (c) 2 h. These TEM micrographs illustrate the appearance of the edge-to-edge configuration.
employed for this purpose. Unlike hypoeutectoid steels. in which sympathetic nucleation was first
reported [l], retention of the fl matrix at room temperature greatly facilitates the microstructural and crystallographic studies in Ti-base alloys. Typical examples of edge-to-edge sympathetic nucleation are presented in Fig. 2. These TEM micrographs illustrate that what appears as a single plate during optical microscopy examinations is often a contiguous array of smaller crystals. Similar observations have been reported earlier in Ti-Cr [7] and Ti-V [S. 301 alloys. Orientation analysis clearly showed that all of these r crystals examined are related to the matrix B crystal through the Burgers’ orientation relationship. Kikuchi pattern analysis indicated that the alpha crystals obey the Burgers’ orientation relationship to within -2. and that the alpha crystals are slightly misoriented with respect to each other. Figure 3 shows an example. The bright field micrograph (Fig. 3a) shows an z crystal (marked A) that has sympathetically nucleated at the edge of another r crystal designated B. The accompanying SAD patterns obtained from the matrix p phase and from r crystals A and B are shown in Fig. 3(b). (c) and (d) respectively. The zone axes as determined
from the spot patterns are [I 131, and [2TT3], and correspond to the Burgers’ orientation relation variant: (OOOl),[/(Ol l),; [2iiO],II[lTl],. Accurate orientations determined from the Kikuchi patterns indicated that the two tl crystals are misoriented by 0.94 with respect to each other. The observation of sympathetically nucleated 51crystals in Ti-V alloys that are separated by small angle boundaries has been earlier reported [S, 301. Figure 4(a) shows an example of edge-to-face sympathetic nucleation. From the diffraction contrast alone, it is clear that though crystal B may appear to be a sidebranch of crystal A, they are in fact two different crystals. Analysis of the SAD pattern shown in Fig. 4(b) indicates that these two crystals are again Burgers’ related. In fact, as can be seen from the stereographic projection in Fig. 5. these two crystals are related through a simple rotation of _ 60’ about the [1120], direction. A 57.42. rotation about the [I 1201, direction in Ti corresponds to a coincidence site lattice boundary with 2: = 13 [31]. Another frequently observed morphology of the z phase is shown in Fig. 6. The SAD pattern in Fig. 6(b) and the dark field micrographs presented in Fig. 6(c) and (d) clearly illustrate another case of
5.56
MENON and AARONSON:
OVERVIEW NO. 57
Fig. 3. Ti-6.6 at% Cr, /? solution treated and isothermally reacted at 973 K for 1 h. (a) is a bright field electron micrograph and (b) is the SAD pattern obtained from the region marked by a circle in (a), zone axes [I134 and [2rT3], (c) and (d) are SAD patterns obtained from the a crystals marked A and B respectively. Tbe bisectors of the Kikuchi pairs, (132T) and (3033). are shown.
sympathetic nucleation. This morphology differs from that shown in Fig. 4 in that the sympathetically nucleated a crystals have not grown into large plates and probably are examples of face-to-face nucleation. A stereographic analysis of this orientation is also shown in Fig. 5; the substrate and the sympathetically nucleated crystals are seen to he related by a rotation of -32” about the ~IOTO]~axis. A 35.10’ rotation
about [ lOTO],corresponds to a Z = 1 I boundary [3 I]. An important difference between the sympathetically nucleated a crystals shown in Fig. 6 and that in Fig. 4 is that the former crystals are not Burgers’ related to the grain into which they are growing. This morphology of the a phase has heen observed in a wide variety of titanium alloys [8, 19,32-351 and is referred to in the literature as Type 2a or non-
MENON and AARONSON:
557
OVERVIEW NO. 57
io13
2023 no
2111 , -I
’
--“‘_O I
I
I
I
I
?I
0523
I I I
or12
I B
Fig. 4. Ti-6.6 at% Cr, j solution treated and isothermally reacted at 973 K for 2 h. (a) is a bright field electron micrograph and (b) is the SAD pattern obtained from the central region of (a). (c) indexes the SAD pattern shown in (b). Zone axes are [JO32],and [7256],. These correspond to the Burgers’ orientation relationship variants. (OOOl),II(01I),; [I 12O],Ii[I 171, and (OOOl),~~(lOl),;[ll~O],/~[llT], respectively.
Burgers’ r [32]. Non-Burgers’ x and Burgers’ z are frequently related through { lOf2j,, twinning [32, 81. Margolin et al. [36] have observed the formation of Burgers’ z crystals on r :p interfaces and found them to be {lo71 iI twin related to the substrate z crystal, which obeys another variant of the Burgers’ relationship. They proposed that the new r crystals form by deformation twinning. The operation of sympathetic nucleation in producing this morphology has been suggested [19. 35.81, however. and the present work supports this mechanism. The question arises as to why edge-to-edge and face-to-face sympathetic nucleation leave behind a
small-angle boundary. The activation free energy for nucleation, AC*, would appear to be further reduced if the sympathetically nucleated crystal had the same orientation as its substrate crystal, thereby eliminating the interfacial energy needed to form a nucleus:substrate boundary of any type. A possible explanation for this is that, following Shewmon [37] and as is discussed further in the next section, sympathetic nucleation appears likely to take place at partially coherent facets of interphase boundaries in regions between growth ledges. While these boundaries have a relatively low interfacial energy, they are also immobile. thus preventing them from over-
558
MENON and AARONSON:
OVERVIEW
NO. 57
iii 1X +
--K _.--
/.R‘
ioio
ioio
/'
x, A hkil : sympothctically nucleated hcp
Fig. 5. (OOOl), standard stereographic projection for Ti superimposed on (01 l)b. The b.c.c. poles are shown by solid circles and the h.c.p. poles by open circles, crosses and triangles. Great circle a represents the foil plane in Fig. 4. As indicated by the dotted lines, the poles corresponding to crystal B in Fig. 4(a) are related to the crystal A by a simple rotation. Notice that the a crystals A and B in Fig. 4(a) obey the variants (OOOl),~](Oll),; [1120],]1[11T]Band (OOl),~](lOl),; [I 120~]][1IT], respectively. Great circle b shows the foil plane in Fig. 6. The sympathetically nucleated crystals are seen to be related to the substrate crystal by the 32” rotation indicated.
running embryos being formed by sympathetic nucleation and also, as suggested by Shewmon [371, permitting them to form in regions where appreciable supersaturation is available. Now, if the nucleus of a sympathetically nucleated crystal were to form in the vicinity of misfit dislocation(s) associated with the partially coherent facet, the fully coherent character now anticipated of all nucleation [38] would require that nucleation occur some distance away from the misfit dislocations. This is analogous to coherent nucleation at dislocations in single phase alloys. Precise positioning of the sympathetically formed nuclei so as to maximize the reduction in transformation strain energy achieved by nucleating in association with the dislocation, while at the same time replacing as much as possible of the fully coherent matrix:substrate boundary between the misfit dislocations with fully coherent substrate:substrate boundary, seems to require a small misorientation of the sympathetically formed nucleus with respect
to the substrate crystal in order to optimize satisfaction of both requirements.
the
5. RELATIVE NUCLEATION KINETICS In this section, we examine the kinetics of sympathetic nucleation and evaluate the conditions that will promote its occurrence. This is accomplished by comparing the kinetics of sympathetic nucleation with those of homogeneous and grain boundary nucleation. The rate of steady state nucleation, J:, is given by [39]: (22) where Z is the Zeldovich non-equilibrium factor, jI* is the rate of attachment of single atoms to the critical nucleus, N is the density of atomic nucleation sites being employed per unit volume of the matrix phase,
MENON
and AARONSON:
OVERVIEW
559
NO. 57
ii
101, ::
Fig. 6. Ti-8.6at% Mn, fi solution treated and isothermally reacted at 973 K for 10s. (a) bright field micrograph showing z plates in a p matrix. (b) SAD pattern corresponding to (a). The zone axes are [2TT3], and [OOOl],. (c) and (d) are dark field micrograph showing the substrate a and the sympathetically nucleated interface a respectively.
k is Boltzmann’s constant and T is absolute temperature. As previously discussed in detail [40], as
long as the undercooling below the equilibrium temperature is not very large, AC* is by far the most influential term in this equation; hence the following calculations may be safely restricted to this term. AG* is sensitively dependent upon the shape of the critical nucleus [38]. A reasonably rigorous calculation of AG* has only recently been achieved, on considerations of classical nucleation theory, for homogeneously formed nuclei of an f.c.c. phase in an f.c.c. matrix [41]. The problem of nucleation at grain boundaries-even when the crystal structures of the matrix and the precipitate are the same-is yet to be resolved. Hence. in this analysis, resort has been made to the frequently employed pillbox, envisioned as lying coplanar with a grain boundary. Selfconsistent results have usually been achieved with this simple approximation [42-44]. Figure 7 illustrates the pillbox, and identifies all of the parameters associated with it, including the critical radius. r*. and the critical height. h*. Subscripting all relevant parameters associated with grain bound-
ary, homogeneous and sympathetic nucleation by b, h and s, respectively, the free energy change AGO, associated with formation of embryos of pillbox shaped nuclei at each of these three sites is: AC: = nr~h,r$, + 2nr:y$
+ 27~,,h,g:~
AGO,= rt&,&
+ y’l”s- ypp]
+ nr;[&
+ 2nr,h,y :fl AG; = rrr:h,& + rrr:y,, + 27rr,h,y’lp
(23)
(24) (25)
where r$=AG,+
W
(26)
where AG, is the volume free energy change and W is the strain energy attending nucleation. The activation energy, AC*, associated with critical nucleus formation is obtained by finding the maximum in AG” in equations (23x25) with respect to r. The resulting equations for AG* are: (27)
560
MENON and AARONSON:
OVERVIEW
NO. 57
/ 00
aa
iYC aQ
Fig. 7. Schematic drawing of the critical nucleus model used in the analysis for (a) grain boundary nucleation and (b) sympathetic nucleation, identifying the interfacial energies involved [42].
02
04
I
06
08
I.0
Fig. 8. Comparison of AG* for grain boundary sympathetic nucleation.
I2
and
y,,/y$ is sufficiently low, even when d1/4,, is small, sympathetic nucleation can occur at rates comparable to homogeneous nucleation. It may be pointed out here that nucleation at dislocations probably offers greater competition to sympathetic nucleation. Coherent nucleation at dislocations is a topic remaining in need of further theoretical treatment. However, Cahn [45] has compared the kinetics of incoherent nucleation at dislocation with homogeneous nucle-
where 6 =Y’ls+Y’zbs-Y&Q?.
(30)
In order to assess the feasibility of sympathetic nucleation as a viable alternative to grain boundary or homogeneous nucleation, the ratios, 4s/& and 4s/& of the driving force for sympathetic nucleation to that for the other two forms, are evaluated as a function of the interfacial energy ratios y& and yJy$ respectively when AC: = AC: or AC,*. The interfacial energy ratios are those for the net interfacial energy which must be acquired at the pillbox broad faces during the nucleation process; as indicated in equations (27)-(29), the same edge energy is needed in all three cases. Sympathetic nucleation is compared with grain boundary nucleation in Fig. 8. This plot is interpreted as follows: when yJt = 0.55, then AC: < AC,1 provided that br/bb> 0.75. As expected, &/4b increases with y&, becoming unity when y,Jc = 1. Physically, when y,,/~ = 0, the u:a boundary energy is zero and so AC: = 0. On the other hand, when y& = 1, the a:a boundary becomes energetically equivalent to the fl grain boundary and thus AC* for nucleation at the two types of sites becomes identical. In Fig. 9 sympathetic nucleation is compared with homogeneous nucleation. Plots for AC: = l/2 and l/3 of AC,* are also included to illustrate that when the interfacial energy ratio
I
I
12-
AG'
-:3 AG:
/
1
14 +ll 06-
4
’ ’ 04’ j 06’
02
’ s I.0 (
0.6
’
12
Y.. Tca@
Fig. 9. Comparison of AG* for homogeneous sympathetic nucleation.
and
MENON and AARONSON:
OVERVIEW
561
NO. 57
site considered: AC: < AC,* for $ = 0.1 l-O.29
AG~
AG,* < AG,* for ;
= 0.28-l .93.
(32)
d
I
OO
z
! 02
I
0’4
06 Y JF
,
/
08
3
I
IO
I 2
ci&
Fig. 10. Comparison of AG* for nucleation at dislocations and sympathetic nucleation.
ation. According to his analysis, AG~~AG~ = 0.15-0.40, where AC,* is the activation free energy associated with the formation of the critical nucleus at dislocations. Figure 10 shows a plot of ~~/#~ vs ~,,/-& when AG: = AG,*. This plot shows that under favorable conditions of interfacial energy ratio, sympathetic nucleation can also compete with nucleation at dislocations. Somewhat more detailed consideration of the feasibility of sympathetic nucleation can be made upon the basis of available data on interfacial energies. For disordered interphase boundaries [46], 0.6 < 3 < 0.9. (31) l.‘PP The interfacial energy of coherent interphase boundaries is considerably lower than that of counterpart disordered interphase boundaries. In the case of f.c.c.: b.c.c. boundaries in brasses, measurements made by Hu and Smith [47] indicate that the coherent f.c.c.:b.c.c. boundary energy is less than one-third that of incoherent f.c.c.: b.c.c. boundaries. The homophase boundaries which are created during sympathetic nucleation are low angle grain boundaries or coincidence site boundaries as shown in the previous section. Hence a conservative estimate of the ratio, ?%Ii:,, , is 0.02-0.1 (twin boundary energies in Cu and Ag are 0.02 and 0.05 of their grain boundary energies [48]). Using these limits, we obtain the following lower limits? driving force ratios such that AC,* is less than that corresponding to other types of
*The range 4, C#I~ etc. arises from the limiting values of the interfacial energies considered.
These results indicate that sympathetic nucleation is indeed feasible when suficient driving force is available. As earlier noted, it is necessary that adjacent growth ledges on immobile, partially coherent boundaries be spaced sufficiently apart relative to the extent of the diffusion fields associated with the ledges to permit AC, to approach that for homogeneous nucleation. Elaborate analyses of the diffusion fields associated with single [49,50] and multiple ledges [51,52] are now available. In the present context the variation of composition along the terrace of a ledge is important since the driving force for nucleation would increase as the composition at the terrace approaches the matrix composition. Figure 11 shows the variation of r the dimensionless supersaturation along the terrace of a ledge located at x = 0, obtained from Jones and Trivedi [49] and Atkinson [50]. Here, x is the distance along the terrace of the ledge and y is the distance along the riser of the ledge and f is defined as: f = %&Q- c= -c,
(33)
where qzJ.) is the composition at any point (x, y). c, is the matrix composition and c,,, is the composition at x = 0, y = 0. Accordingly, r = 1 denotes no supersaturation and r = 0 indicates the maximum available supersaturation. It should finally be pointed out that shear strain energy which can accompany the purely diffusional
1.
ATU INSOf ---JONES-TRIVEDI
7
Fig. 11,Variation of dimensionless supersaturation, 1”, with dimensioniess distance, X, along the terrace of a ledge with the riser at X = 0. Y is the dimensionless distance perpendicular to the interface. r calculated from Refs [49] and (501 is presented.
562
MENON and AARONSON:
transformations considered can also play a significant role in the kinetics of sympathetic nucleation. Lee and Johnson [53J and Russell er al. f54J have shown that if these strains have a component parallel to the broad faces of the plates which is oppositely directed in successive plates, then the net driving force for sympathetic nucleation can be markedly increased. 6. CONCLUSIONS
In this paper, various aspects of the phenomenon of sympathetic nucleation have been investigated. Various morphological features resulting from sympathetic nucleation have been illustrated with examples from TCX alloys. The question of whether or not supposedly sympathetically nucleated structures were actually formed by branching in the style of dendrites was first considered. By expressing the orientation relationship between two crystals in the form of a matrix, a search was made to see if precipitate crystals could branch in different directions while maintaining an equivalent pair of conjugate habit planes. Analyses of this question conducted on commonly observed orientation relationships among b.c.c., f.c.c. and h.c.p. phases indicated that crystallographic constraints do not permit branching in such systems except under very restricted conditions unlikeiy to he encountered experimentally. A comparative theoretical study of the kinetics of homogeneous, heterogeneous and sympathetic nucleation showed that the activation barrier associated with these processes becomes comparable when sympathetic nucleation causes the matrix:precipitate boundary to be replaced by a relatively low energy precipitate:precipitate interface. It is found that even when the driving force for sympathetic nucleation is less than half of that for heterogeneous or homogeneous nucleation, the process can become favorable. Experimental evidences showed that during sympathetic nucleation, the a$ boundary is replaced by a low angte or a coincidence lattice type a:a boundary. Acknowledgements-The authors express their appreciation for the support of the Air Force Office of Scientific Research through Grant No. AFOSR-742595E.
OVERVIEW
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