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Experimental determination and symmetry related analysis of orientation relationships in heterophase interfaces: a case study in the ZrB system
Vol. 44. No. 10. pp. 4169 4179. 1996 Copyright f' 1996 Acta Metallurgica Inc. Published by Elsevier Science Lid Printed in Great Britain. All rights reserved 1359-645496 $15.00 + 0.00
Acta mater.
~
Pergamon PII S 1359-6454(96)00055-9
EXPERIMENTAL DETERMINATION AND SYMMETRY RELATED ANALYSIS OF ORIENTATION RELATIONSHIPS IN HETEROPHASE INTERFACES: A CASE STUDY IN THE Zr-B SYSTEM
Y. CHAMPIONH"and S. HAGEGE~ ~ETCA-CREA, 94114 Arcueil, and 2CNRS-CECM, 94407 Vitry-sur-Seine, France (Received 14 December 1994: in revised form 3 January 1996)
A~traet--Orientation relationships between phases in the Zr-B binary system were determined using electron diffraction. Heterophase interfaces were produced during the solidification of the eutectic (Zr-ZrB2) followed by phase transformation, i.e. allotropic transformation of zirconium, formation of ZrB by peritectoid transformation and stabilisation of a metastable Zr phase, zZr. Despite the complexity of the various phases and their transformation upon cooling, the experimental analysis of all the orientation relationships (cubic-hexagonal and hexagonal-hexagonal) found in this system yields only three sets of orientations characterised by the parallelism of low index planes and directions. Using symmetry considerations and group theory, it has been demonstrated that these orientations can be understood as the lowest energy configurations which can be produced from one single orientation formed at high temperature between flZr and ZrB_,. Copyright ~ 1996 Acta Metallurgica bw.
1. INTRODUCTION It is now well established that grain and phase boundaries play a key role in material properties and research in this field is oriented towards a correlation between the structure of the boundaries and the physical behaviour of materials. Therefore, the "interface community" has been very active in experimental characterisation and structural description of boundaries, the final aim being to provide a quantification of the boundary structure which could be related to property measurements. One of the most widely used approaches has been the characterisation of model boundaries which were assumed to be in energetically favourable configurations. The accurate geometrical and structural characterisations reported in these studies have led to a better understanding of the boundary structure and have emphasised the basic ideas for the structural description and subsequent classification of boundaries. An interface is geometrically defined by nine parameters describing the orientation relationship between two neighbouring crystals (3), the boundary plane which separates the two phases (3) and a rigid body translation linking an arbitrarily defined origin within each crystal (3). Particular interest has been placed very early on the orientation relationship. Firstly, it remains the parameter which is experimentally the most easily accessible. Secondly, it has often been assumed to be related to the stability of tPresent address: CNRS-CECM, 15 Rue Georges Urbain, 94407 Vitry-sur-Seine Cedex, France
interfaces and this was supported by geometrical models based on coincidence orientations containing no information on the actual atomic structure near the plane of the interface. The orientation relationship has been described using different methods. The description in terms of a transformation matrix is a very useful mathematical tool for classification. For the case of grain boundaries, the rationality of the matrix and the index of coincidence ~ has been widely used in the literature [1-9]. In the case of phase boundaries, the matrix description suffers from the irrationality of the matrix and the volume variation between the two phases not allowing a relevant definition of a general index of classification. Duneau et al. [10] have suggested a novel approach in terms of elementary shears matrix decomposition, which is seen as an appropriate criterion but only for very restrictive cases. So far, the most common method for describing heterophase interfaces is the parallelism or the close parallelism of a set of low index planes and directions of the crystals. This description is very limited since it does not contain information on the metric of the system. Nevertheless one can correlate different orientation relationships for analogous but structurally different interfaces. As an example, the parallelism of densest planes and densest directions is frequently observed for hexagonal-hexagonal, f.c.c.hexagonal and f.c.c.-f.c.c, interfaces. In these systems, which would yield very different transformation matrices, densest planes and directions are
4169
4170
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES 2. MATERIAL AND EXPERIMENTAL
2.1. Sample preparation 300
250
P ~200C
1500
~
1855 ° .~/
1680 ° 1250 °
I 100
800
°
.L.
_
_
5°°f o Zr
io
,;,o
ATOMICPER CENT BORON
6o ZrB2
Fig. 1. Section of the Zr-B phase diagram after Glaser [11]. strictly equivalent and the comparisons of the various orientation relationships are straightforward. The present work is partly devoted to the analysis and understanding of the orientation relationship as a parameter in the interface stability. Its aim is also to provide experimental results as a support for the description of general interfaces. We have dealt with cubic-hexagonal and hexagonal-hexagonal heterophase interfaces in the Zr-B system. All these have been produced by eutectic solidification and subsequent phase transformations as in the phase diagram of Fig.1 [l 1]. Thus, interfaces were assumed to be in energetically favourable conditions and a very small number of specific orientation relationships was expected. In order to validate these assumptions experimentally, a significant number of orientation measurements were performed in many samples for the different interface systems. In this paper, the results are reported in two parts. The experimental results have provided characteristics on the orientation relationships for each case by direct comparisons between the different systems. A theoretical analysis of the formation of orientation variants during phase transformations has been carried out. This treatment was based on the group theory and elementary symmetry considerations and has allowed establishment of a correlation between experimental orientation relationships and energetic criteria.
tThe crystallographic structure of the zZr phase has been identified as slightly deformed f.c.c, by diffraction and high resolution electron microscopy [13]. It has been considered as f.c.c, in the present study.
Samples of the Zr-ZrB: eutectic composition (1.9% boron) have been prepared and brought above the melting temperature of the metal (1855°C), using a R F induction heating device. Then they were cooled, leading to the solidification of eutectic composite materials at 1680°C with the formation of an oriented microstructure of ZrB_, needles in a Zr matrix. The global morphology of the samples is characterised by a preferred orientation of ceramic needles in each grain of the polycrystalline metallic matrix, This texture is in full agreement with the microstructure of a stable oriented rod-like eutectic [12] and hence supports an energetically favourable configuration at the heterophase interfaces. Crystallographic and chemical analysis, performed using X-ray diffraction and electron energy loss spectroscopy, revealed the coexistence of four different phases in the samples. As expected, the hexagonal close-packed (h.c.p.) phase ~tZr and the simple hexagonal (s.h.) phase ZrB: were observed. The non-equilibrium ZrB with a NaCl face-centred cubic structure (f.c.c.) has been also found. In addition, an original phase with a near-f.c.c, structure was identified but only in thin samples for transmission electron microscopy. This new phase with a chemical composition very close to that of ~tZr has been denoted zZr in the following.t The details of sample preparation [13, 14] have shown thereafter that the non-equilibrium phases ZrB and zZr were related to the post-solidification cooling conditions of the samples. ZrB was obtained by slow cooling of the sample, zZr was observed in thin specimens obtained from samples quenched after solidification, from a temperature over 1250°C. Moreover, it has been proved that xZr is related to ctZr by a phase transformation in the ShojiNishiyama (f.c.c.-h.c.p.) orientation [15]. This transformation and the zZr stability are believed to be closely related to the combination of thinning processing, thin specimen effects and boron impurity content in the bulk of the aZr matrix of the eutectic samples. The zZr phase is seen as an artefact since it does not exist in a bulk specimen. Nevertheless, as it is assumed to be produced only under favourable conditions, it fully contributes to the analysis of orientation relationships in our study.
2.2. Experimental procedures The orientation relationships between phases at interfaces have been determined using transmission electron microscopy in the selected area electron diffraction mode. The thin specimens were prepared by dimpling and Ar + ion milling. Diffraction experiments were performed on a JEOL 1200FX microscope working at 120kV, using +60 ° tilt/ rotation specimen holder.
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
4171
Table 1. Experimentalresults on orientation relationships Orientation relationship R~'" R~"c
a b c d e f g
(0001):/(0001) [1120] [11'~0] Exactparallelism
R2"-h R~"
Interfaces ~Zr-ZrB: ZrB~xZr ~Zr-ZrB ZrB~ZrB ~Zr-~Zr ~Zr-ZrB~ ZrBr-zZr
Relation
Misorientation
R~"~
ZrB~ZrB
h
(ITO0)//,~(I 11) [l'~lO]//¢L~[O1l] [0001]//[01]']
R~"~
~Zr-ZrB:
i
( lOT 1)Zr//lO00 l )Z,B:
(0001)//(111) [1120] ,[O'fl] (0001y/(lll) [1120]/[0"1"1]
Few betweenplanes Exact parallelism
(ITO0)//(O001)[I 1'~0]//[II'~0] (11"00)//(11I) [11'~O]//[Oll]
Few between planes Few between planes
[OOOll//I2TI]
[f2T0]z,//[I2T0]z~B.
Many procedures have been developed for precise determination of orientation relationships, based on either the analysis of spot diffraction patterns [I 6, 17] or the analysis of Kikuchi line patterns [18]. These methods for grain boundary analysis combine experimental measurements and computer treatment. They provide with high precision the rotation matrix, the axis and angle of misorientation, directly from a minimum of simple parameters such as tilt angles or directions in reciprocal space. For the determination of orientation relationships at heterophase boundaries, we have derived one of those methods, developed for grain boundary analysis by Karakostas et aL [19]. The aim of the method is to find the simplest planes and directions (lowest index), which are parallel or close to parallel in the two crystals. In this experimental procedure, each crystal is oriented in the spherical reference system of the microscope by tilting the specimen to successive reflections in the Laiie conditions. The tilt angle and the direction of the reciprocal lattice row measured in that condition, give directly and simultaneously the relative orientation of each crystal on a stereographic projection. The precise orientation relationship is derived from this stereographic projection. In some cases, the results were provided directly by the diffraction patterns, when both crystals were parallel or nearly parallel to a simple zone axis. 3. EXPERIMENTAL RESULTS All the possible combinations of two of the four phases identified in our samples, combinations which can be effectively obtained through the in situ formation process, have been found. Only the Z r B - z Z r interface has not been observed, since these two phases do not appear in samples at the same cooling conditions after solidification. As expected, a very small number of orientation relationships characterised by parallel low index planes and directions was found in this investigation. The structural equivalence which exists between the densest planes {0001 }, { 111 } and between the densest directions (11]0) and (110), respectively, for hexagonal and f.c.c, structure allows a classification
Few between planes Few between directions Exact parallelism Few between planes Exact parallelism
of the experimental orientation relationships into only three families denoted as R~, R,. and R~. These are detailed in Table 1. The orientations in each family are described with both the hexagonal-hexagonal and hexagonal-cubic index when they exist. They have been denoted R~'¢ for an orientation relationship belonging to the family R~ and describing an hexagonal--cubic interface. Orientation relationships of the R~ and R_, families were observed with almost identical frequency whereas R3 was observed only once. 3. I. Experimental orientation relationships Rj
The R~ orientation relationship follows the exact rule of the parallelism of the densest planes and the densest directions in both hexagonal and f.c.c. structures, except for the ZrB_,-zZr interface where a slight deviation has been found. By analogy with the so-called "cube-on-cube" orientation in interfaces with the cubic-cubic symmetry, the R~'h orientation has been called the "hexagon-on-hexagon" orientation. The orientation relationship family R~ is very common. It has been experimentally observed at numerous interphase boundaries in the literature, whatever the nature of the phases (metallic, ceramic, semiconductor) and whatever the preparation procedure (phase transformation, internal reduction or oxidation, oriented solidification, layer growth on a substrate). A non-exhaustive list of experimental results [14] shows a large proportion of interfaces belonging to R~. However, most of the experimental results concern hexagonal-f.c.c, systems (see for example Refs [20, 21]) and the orientation relationship R~h for hexagonal-hexagonal symmetry has been seldom observed so far [22,23]. Figure 2 shows the results of electron diffraction experiments of the R~ orientation relationship. Figures 2(a)-(c) are experimental diffraction patterns and Figs 2(d)-(f) are their respective indexations. Thus, Figs 2(a) and (d) and 2(b) and (e) are, respectively, the diffraction patterns of ctZr and ZrB2 phases taken for the exact same orientation of the sample. The parallelism of their respective [ I 1~0] zone axes and the strict alignment of the [0001] as well as [i100] respective reciprocal lattice rows are clearly apparent. Figures 2 (c) and (f) correspond to the
4172
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
O O
a
b
0001 •
•
d
•
•
•
[11~0]
•
c
•
0001 •
•
•
•
• "~100
•e IN
•
•
•
[ 1 ~o]
[l 1~0] //[O~l]
Fig. 2. Orientation relationship Rn. Experimental diffraction patterns of :tZr [(a), (d)], ZrB.~[(b), (e)] and the superimposition of 7Zr and ZrB: [(c), (f)].
superposition of the diffraction patterns of the zZr and the ZrB: phases, showing that the two analogous zone axes [11]0] and [0T1] are parallel. However, for this system a slight deviation from the exact alignment of the densest planes (0001) and ( l l l ) is also visible in the figure. In addition, a slight elongation of the spots of the xZr phase is observed. These two phenomena have been related to the relaxation of the strains associated with the phase transformation from ~Zr to zZr phase.
The family R] is described in Fig. 3 by the stereographic projections of an hexagonal-hexagonal interface [Fig.3(a)] and an hexagonal-cubic interface [Fig.3(b)].
3.2. Experimental orientation relationships Re In contrast to R~, the family R2 has never been reported in the literature as far as the authors are aware. By comparison with R~, the family R2 is characterised by the conservation of the densest
2 o:oI/o
,p
• c.b\
(0!01)
~
~0)
.
Fig. 3. Stereographic projections of the orientation relationships R,. R~'" hexagonal-hexagonal interface (a) and R~'~ hexagonal-cubic interface (b).
CHAMPION and HAGEGE: HETEROPHASE INTERFACES
4173
C ~100 0001
0 0 0
0001o • •
d
•
• •
[11"201
100
•
•
• "h.oo 111 •
•
•
•
•
[ 11~01
f
[6'il]
Fig. 4. Orientation relationship R2. Experimental diffraction patterns of ~Zr [(a). (d)]. ZrB., [(b), (e)] and the zZr [(c), (f)].
directions, (11'50) and (110), and the near-parallelism of the densest plane of one of the two phases with a prismatic plane { 1T00} of an hexagonal phase. The family R,, is illustrated by experimental electron diffraction patterns in Figs 4(a)-(c) and their indexing in Figs 4(d)-(f). These diffraction experiments were carried out on both sides of a ZrB2 fibre where the ~Zr-ZrB,, and zZr-ZrB2 interfaces were simultaneously present. Figures 4(b) and (e) in the middle show the diffraction pattern of the ZrB, phase where the [11]0] zone axis is exactly parallel to the electron beam. The diffraction patterns of =cZr [Figs 4(a) and (d)] and •Zr [Figs 4(c) and (f)] taken in that exact same orientation of the sample reveal a slight deviation from the exact parallelism of planes and directions for the two interfaces. The family R2 is divided into three sub-orientations denoted R~h for hexagonal-hexagonal interfaces, R~"~ and R~:~ for hexagonal--cubic interfaces. The orientation relationships R~h and R~c are illustrated by the stereographic projections in Figs 5(a) and (b), and they also correspond to the experimental results in Fig.4. In an exact orientation [as in Fig.5(b)] this corresponds for hexagonal-f.c.c, interfaces to the additional parallelism of the [0001] and [2"i"i] directions. The orientation relationship R~,', observed for ZrB2-ZrB interfaces, illustrated on the stereographic projections in Figs 5(c) and (d), is more complex. It has been classified in the R2 family orientation since
it involves, as the R~'" orientation, the close parallelism of (1]'00) and (111). However, in that case, the couple of planes (1]10), (110) which are also close to each other as in R .,h' , are not now at 90 ° from the common (1100), (111). Because the angle between (1100) and (1510) is 30 ~' and the angle between (111) and (110) is 35.26 ~, it is now impossible for both sets of planes to be simultaneously parallel. Therefore R~:¢ corresponds to a case of frustration for which the exact parallelism of planes of the respective phases can never be reached as shown in Fig.5(d). Experimentally, none of these two sets of planes are exactly parallel as there is an energetic compromise to handle the frustration of the angle. In addition, it results from this configuration that the third set of parallel planes is now (0001), (011) and not (0001), (2H) as in R~c. These points are seen as the major differences between the orientation relationships R~" and Rh,:". However, these differences are minor in comparison with the close parallelism of dense planes which has allowed us to classify R~'¢ and R~:" within the same R2 set of orientations. The analogies and differences between the orientation relationships R~'c and R~'~ are emphasised by the comparison of the stereographic projections in Figs 5(b) and (c). 3.3. Experimental orientation relationships R3 As previously mentioned, over a significant number of experimental measurements, R3 corresponds to one single observation at an ~Zr-ZrB2
4174
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
interface (hexagonal-hexagonal) and is denoted by R h'h. No experimental evidence of the equivalent R3h'¢ has been observed in this work. In comparison with the orientation relationships R~'h and R2h'h, the parallelism of the densest direction [1~10] is maintained in R~"h. However, it involves, as a second criterion, the close parallelism of the basal plane (0001) of ZrB2 with a pyramidal plane (10]'1) of ~Zr. Therefore the relative orientation for any hexagonal-hexagonal interphase boundary following R~'h and the subsequent stacking of atomic planes throughout the interface depends on the parameters of the hexagonal structures. 4. ANALYSIS OF THE ORIENTATION RELATIONSHIPS THROUGHOUT THE PHASE TRANSFORMATIONS
4. I. Phase transformation characteristics The Zr-B system seems rather complex since it involves many phases obtained by solid state transformations occurring for various experimental
conditions. However, all these phases and the subsequent interfaces are produced from the eutectic flZr-ZrB2 solidified at high temperature. The purpose of this section is to define whether the experimental orientation relationships observed at room temperature can be related to a unique primitive orientation relationship of the high temperature eutectic, which subsequently evolved through solid state transformations. According to our sample preparation conditions, two types of unrelated sequences of solid state transformations could modify the phases and the orientation present in our specimens. The first involves the formation of the zZr phase due to quenching, and the second involves the formation of the ZrB phase due to a slow cooling of the specimen.
4.1.1. Phase transformation during fast cooling. Figure 6(a) illustrates the case of quenched specimen (path I). In path I, the ZrB2 phase is not involved and the phase transformations occur in the matrix of metallic zirconium. In addition, the structural
(213)
/ • .
\
~•
<0001) .o
.
, oo,
.
(1 2o) - ' o
:
. ,
I
.
,,rooo) .
2 .xzr" : " / (off)
,26 °
• ZrB2
~
"
o Z r B ~ Z r B mZrB2 ~ . ~ . _ _ . ~ ~
d
Fig. 5. Stereographic projections of the orientation relationships R~. (a) R~h hexagonal-hexagonal orientation, (b) R~"c, (c) and (d) R2hf hexagonal-cubic orientations. Parts (b) and (c) show the two types of planes and directions {I TOO}, ( 1~10> and {111}, (011 ) which can be involved. In the case of Rh:cpart (d) emphasises the frustration and the near-orientation relationship character of the orientation Rh~ due to the angular difference in couples for the hexagonal and cubic systems.
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
/--;z-q
4175
planes and the densest directions of the h.c.p, and f.c.c, structure. It may be written as T ~'~ (0001)~//(lll)z
and
[II~0]~//[0T1]~.
YZr
TiL=
T=,X :cZr
~ xZr
Fig. 6(a). Sequence of phase transformations and phases formation for a quenched specimen. TP'" and T,.x denote, respectively, the allotropic b.c.c. (fl)-h.c.p. (ct) and the unusual h.c.p. (ct)-f.c.c. (Z) phase transformations of Zr. The ZrB., phase is not involved in this path of phase transitions.
I Path H ] ~r
-
Zrl~
Table 1 shows that six types of interfaces out of nine (a, b, e, f, g, i) may be related to path I. 4.I.2. Phase transformation during slow cooling. As revealed by Fig.6(b), path II is more complex than path I. It involves the formation of the monoboride phase (ZrB) between the ZrB2 and the Zr phase in its two allotropic structures flZr, and it theoretically disappears by the inverse peritectoid transformation with ~tZr, leading to the formation of new ctZr and ZrB2 phase [denoted as (ctZr) and (ZrB2) on Fig.6(b)]. The peritectoid transformation and its inverse which combine several structural transformations and variations of the chemical composition, are still unknown. Moreover, it is easily assumed according to the sample preparation, that the transformations are not total, especially at low temperature due to the slow diffusion of boron [we have assumed in fact that this last stage does not actually occur and has been placed between brackets in Fig.6(a)]. Accordingly, the experimental interfaces may be produced at any stage in path II, mostly at high temperature and stabilised at room temperature after aliotropic transformation of Zr. Table 1 shows that three types of interfaces out of nine may be related to the path II (c, d, h).
4.2. The primitive orientation at high temperature lYZr - Z r B
-
ZrB2
T[~'~ ecZr - Z r B - ZrB2
{ a,Zr - (ZrB2) - ZrB- ZrB2 ] Fig. 6(b). Sequence of phase transformations and phase formation for a slowly cooled specimen. TB."still denotes the allotropic b.c.c. (fl)-h.c.p. (~t) phase transition of Zr. P and P-' denote, respectively, the peritectoid and the inverse peritectoid reactions in which ZrB is involved.
The preliminary stage consists of the assumption of a primitive orientation relationship between flZr and ZrB2. It may be deduced by applying the (~t-fl) allotropic phase transition [T"~= (T~'0 -~] to the ~Zr-ZrB2 interface in the hexagon-on-hexagon orientation R~"h. This orientation has been chosen because it is the simplest orientation. It has always been observed with the exact parallelism of planes and direction. The primitive orientation relationship denoted R0a'h may be written as in the following expression
p~.h = T..,(R~.h) (llO)~//(O001)h and [T1 l]d/[1210]h. 4.3. Group theory analysis
characteristics of these phase transformations, i.e. the orientation relationship between phases throughout the transformations, are known. TP." denotes the (fl, ~t) b.c.c, to h.c.p, allotropic phase transformation of the zirconium at 860°C. It is known as the Burgers transformation [24] and it may be written as T ~'~ (110)~//(0001)~
and
[Illb//[I210L.
T ~'z denotes the (~,X) h.c.p, to f.c.c, phase transformation of zirconium. Previous investigations have shown [13] that this transformation follows the Shoji-Nishiyama orientation which keeps the densest
The interfaces between ZrB2 and r,Zr which should be produced at room temperature from a/~Zr-ZrB2 interface are derived from elementary symmetry considerations. This analysis is separated into two parts: firstly the allotropic phase transition in zirconium (/~--~) is considered and the orientation variants of ctZr in the parent crystal /~Zr are characterised. Secondly the orientation of each variant is considered with respect to ZrB2. (ZrB2 is kept unchanged during the phase transition.) The aim of this last stage is to define whether the number of orientations obtained correspond effectively to different interfaces.
4176
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
(211o)
These two stages lead to the total number of possible orientation relationships for the ~Zr-ZrB_, and subsequently 7Zr-ZrB2 interfaces.
4.3.1. Variants obtained in the allotropic transformation of Zr. The number and orientation of ~tZr
(111
phase variants obtainable within the parent phase flZr are defined using the group theory. The concept based on Curie's law? has been developed by different authors [26-28] and has been applied recently for the study of precipitation in minerals [29]. As we are dealing with an orientation relationship using electron diffraction measurements, the translation symmetry operators are not accessible. Therefore, only the point-group symmetry of crystals will be used. According to Curie's law, the symmetry of the morphology of the orientation variants is related to the symmetry group of iso-probability of nucleation and the symmetry group of the nuclei. The symmetry group of the iso-probability of nucleation Go is composed of the set of common symmetry elements of the parent phase and of the external acting parameters. During the cooling of the specimen, it is assumed that the only external acting parameter is the temperature with a spherical symmetry and then Go remains the symmetry of the parent phase. If H0~ denotes the symmetry group of the bicrystal flZr-~Zr, it constitutes a set of common symmetry elements of Go (flZr) Go = m~m (Oh) and of the symmetry group of the embryo phase G~ = 6/mmm (D6h). It is called the intersection group and it is given by the superimposed stereographic projections of flZr and ctZr (Fig.7)
(1i20)~
H01 = Go n Gt = 2/m(C2h). The number of orientation variants N0~ corresponds to the number of times H0~ is contained in Go. It is given by the ratio of group orders of Go and H0~ N0~ - order(m~m) order(2/m)
48 _ 12. 4
The 12 orientation variants are defined by the coset decomposition of the point group Go = m~m with respect to the point group H0~ = 2/m (Table 2). Each variant is associated with a pair of orthogonal directions (110), (T l l ) of the b.c.c, structure which will yield the couple of orthogonal directions (0001), (T2i0) of the hexagonal structure of ctZr. This set of 12 variants is, as a whole, invariant with respect to the symmetry operation of H0~ = 2/m. However, for the decomposition of Table 2, variants 1 and 2 have the symmetry H0~, whereas the 10 others are related two by two (3/4, 5/6, 7/8, 9/10 and 11/12) by H0j.
/"
/
(i2/0) ocl
~h01 ~oll)
Om ,o
(I11
•_
\
•
=
o,,
"
"X6 H
"
,
oo~,) •
/ /io,~o
(2]i0) Fig. 7. Orientation relationship of the fl--ctphase transition of Zr and the symmetry elements of the bicrystal. intersection group between flZr and ZrB2, in the Rg'h orientation relationship, is H0h = H0~ = 2/m. Therefore, the application of the symmetry operations of the 2/m group to the set of ctZr variants leaves unchanged the orientation relationship of each variant with respect to the ZrB2 crystal. As a result the variants 3 and 4, 5 and 6, 7 and 8, 9 and 10, 11 and 12 are in the same orientation relationship with respect to ZrB2. However, this does not hold for variants 1 and 2 which will remain two distinct orientation relationships with respect to ZrB2. The total number of the orientation relationships between ~tZr and ZrB2 is then ~+1+I=7. The seven orientation relationships are listed in Table 3 and depicted in Fig.8. They are given as parallelism or misorientation between low index planes and directions of the ~Zr and ZrB2 crystals. According to this classification they may be grouped into three families which are consistent with those found experimentally. R~ corresponds to the orientation relationships which are exact or close to the hexagon-on-hexagon type of orientation. R2 is characterised by the near alignment of densest directions and a misorientation between basal planes of 90 °. R3 is mostly characterised by a misorientation between basal planes of exactly 60 °. By a direct comparison with experimental results in Table 1, V~, (V3/V4) and (VT/V~o) have been easily identified respectively as R~", R~'h and R~'h.
4.3.2. Orientation relationships with respect to ZrBe. Each one of the 12 variants is then considered
4.4. Orientation relationships of zZr.
with respect to its own orientation with the hexagonal ZrB2 phase (point group G2 = 6/mmm (Drh)). The
The (c~-Z) phase transition is considered similarly. Firstly, the number and orientations of variants of gZr are determined from previous variants of =Zr. Secondly, each orientation variant of xZr is considered with respect to the ZrB2 phase. The
?"The symmetry of the resulting effects reflects the symmetry of the originating causes" [25].
CHAMPION and HAGEGE:
4177
HETEROPHASE INTERFACES
Table 2. Coset decomposition of the symmetry group m~m with respect to the 2/m subgroup m]m 1
P2im 1 xyz
2
1io, TI1 iyz
1
2 yxi
2~1
liO, ITT iii
TT0, lit
2,,0
3 iyZ
4 9iz
T
TTo, ITT
m,]<,
TTo. TII
2~r
xyi 110, TIT
m~
yxz 110. ITI
mtio
TTO,TIT
3
iyi Tlo, l i t
20]o
yiz fro, l i t
4~]
x~z ]To. TTI
mo,0
~xi i1o, TIT
~
4
x~i ]To, TIT zxy
2,~
~xz TIO, TIi xz~
4~t
iyz Tio, Ill iiy
mt~
yii TlO. tiT iiy
~
5
3~]
Oil, ITI
6 7 8 9 10
47~
]oL TiT
zi~ 0TT, tiT i~y OTl. Tll ix~ 011, TTT yzx 101. ill
3~
~zi
3~t
3~ 3fi, 3fi~
ToT, Ti]
~fi,
oTT, TiT
Y.zy T01, ill i~ ToT, iTT xiy 101, TTI zyi 0]L 111
2o.
z~x
2,0~
2oa 4?~ 4~0
oTi, liT
~,~
IOl, ITi
ix)' 011. IT1 zx~ o i l iTT zi)' Oil, 111 9i£ TOT.TT1 yix
~fi~
xiy i0T, TTT xzy 1Ol. Tll iz~ TOT, llT iyx 0Tl. TTT iyi
~T
~fi, ~ ~1
ioi, iTT
m~,~
mo,r ~;~ ]~0 m~o,
o i l TIi
11
yli 3fi~ iyx 4Go ~zx ]fi, z~i ~o lOT, IT1 o11, TIT iOl. T1T oil ]h 12 yix 3fi~ i~i 2~o~ yzi ~r zyx mro~ TOl, TIT OrT. TTI lOT. 111 llO. lit Each elementof the coset is givenby its Wyckoffposition and point symmetryelement.The application of the symmetryoperators to the pair 110, TII is also given. transition is associated with an increase of the symmetry of the crystal from the 6 / m m m symmetry group of ~Zr to m]~m (Oh) symmetry group of zZr. Furthermore, in the Shoji-Nishiyama orientation, the intersection symmetry group remains that of the parent phase ~Zr. It results in a single orientation variant of the gZr and also a single orientation relationship between xZr and ZrB2. In other words, the orientation relationships found in the previous paragraph are invariant under the T =-~ transformation.
4.5. The peritectoid transformation and the case of ZrB 4.5.1. Orientation relationships obtained from the RI orientation. The origin of the interfaces ~tZr-ZrB and ZrB-ZrB2 in the orientation relationship R~" may be found by considering the first stage of the path II. ZrB grows as a buffer at the /TZr-ZrB,, interfaces during the peritectoid transformation occurring at about 1200°C. At this temperature, /TZr and ZrB2 follows the primitive orientation relationship p~.h defined in the previous section. If we assume that the ZrB phase grows in perfect epitaxial conditions in-between /TZr and ZrB2, the orientation relationships between phases may be written as in the following expressions
(llO)Bz,//(1 i 1)Z,B/t(0001)Z,B: and IT i l]p,,t/[l ] o],,Bl/[T2Tok,~:. This condition corresponds to the parallelism of densest planes and densest directions in a f.c.c.-hex (ZrB-ZrB2) interface. This induces an epitaxial condition in a b.c.c.-f.c.c. (/TZr-ZrB) interface which is the K u r d j u m o v - S a c h s orientation relationship. Together with the Nishiyama-Wassermann orientation, this orientation is the most c o m m o n l y observed in b.c.c.-f.c.c, phase transitions. After cooling of the specimen, the allotropic (/7--~) phase transition of the zirconium occurs, leading a m o n g others, to the formation of ~tZr-ZrB and ZrB-ZrB2 interfaces in the R~" orientation relationship (c and d in Table 1). The other ~tZr-ZrB interfaces which would be able theoretically to be generated by the 11 orientation variants of ctZr embryos were not observed experimentally. 4.5.2. Another orientation. The last interface which has to be considered is ZrB-ZrB2 in the R~,c orientation relationship (h in Table 1). This orientation relationship cannot be included in the previous treatment since it is not related to a known phase transformation orientation between f.c.c, and hexagonal crystals. In addition, as indicated by the reaction path II [Fig.6(b)], this type of interface is
Table 3. Orientation relationships between aZr and ZrB2 as derived from the 12 orientation variants of ~(Zr V, after the allotropic//--~ phase transition of zirconium Variants
0001
11'I0
V~
II
11
--
P.,
10.53 T
--
V3 V.,
ti 90'
V7V~o
60 =
!i
7.8~
VaP~ VsVii
60° 60~ 60°
V6VI2
10.53= 10.5317.Y
0001z, ^ lOT Iz,B:
-
-
7.8' 12 12=
Exp.
Results
#t~ .h
R~
R~ "h
R2
R~h
R3
4178
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
> VI
V2
VT~V=o
Vt=V9
V~=V~I
V6~Vu
V3 or V4
Fig. 8. Schematical representation of the seven orientation relationships of ~tZr with respect to ZrB2 as derived from the 12 Vi orientation variants. The black and white six-fold axis symbols depict the basal plane of, respectively, ZrB, and ctZr, whereas the two-fold axis symbols represent similarly the (l~10) directions.
Fig. 9.
found experimentally that only interfaces originating from the variants V~ and ~ , V~ are observed, and with the same frequency.t Energy considerations can provide a clear explanation for this discrepancy. The orientation relationships found experimentally are all based on the (close) parallelism of low index planes or directions in the structures involved in the phase transformations. Then, they can be assumed to be in the most favourable energetic configurations. It can be stressed here that if the nucleation of all 12 variants of ~Zr within flZr is equiprobable, the fact that this 5. ENERGY CONSIDERATIONS ON THE nucleation has to occur in contact with ZrBz should ORIENTATION RELATIONSHIPS somehow induce a lifting of the degeneracy. The previous analysis has been summarised in the As noted in Tables l and 3, R~ is based on the exact flow chart (Fig.9) relating the experimentally parallelism of the (0001) planes and is found observed interfaces and the orientation variants experimentally when generated by V~ (parallelism of produced by the phase transformation fl--~. Starting the densest planes and densest directions) but not with the assumption that the 12 variants V~ are when generated by V2 (the two basal planes are equiprobable, this figure recalls the fact that the parallel but rotated by 10.53°). As for R2, the close probability of appearance of an interface originating parallelism of the densest directions of the structures from either the variants V~ or V: is 1/12, whereas the is maintained and the 90 ° between the (0001) planes probability of appearance of an interface originating implies the exact parallelism of relatively dense planes from one of the other variants V3V4... Vi2 is 1/6 (e.g. of the structures: for hexagonal interfaces, basal V3 and V4 generate the same interface). If we assume planes are parallel to prismatic planes and for that the probability of occurrence of these interfaces hexagonal-cubic interfaces, basal planes and prisis directly related to the equiprobability of the matic planes are parallel to (211) and (l l l) planes, variants, the probability of appearance of an interface respectively. These hexagonal-hexagonal and hexagin an orientation R3 (8/12) should be much larger onal-cubic interfaces are found experimentally. All than for the families R~ or R2. However, it has been the orientations of R3 are based on 60 ° between the tThe unique case of the ctZr-ZrB2 interface in the basal planes and this angle does not lead to the exact orientation relationship R~*, originating from VT, I,'10, (or even close) parallelism of dense planes or obviously has no real statistical value. direction. Moreover, as indicated above, this
produced either by the peritectoid transformation or by the inverse peritectoid transformation. In any case, the mechanism of this transformation has not been elucidated and the experimental interfaces cannot be located in the reaction path. In fact, those particular interfaces were observed only in a single sample and so they are not as significant as the other experimental interface systems. Finally, further investigations reported in Ref. [30] have revealed that the misorientation between the neighbouring crystals is related to the interface structure.
CHAMPION and HAGI~GE: orientation is related to the c/a ratio of the hexagonal structures. However, the only orientation found experimentally and belonging to this group happened to have an exact parallelism of the 11~0 directions. We can assess that the activation energy for the nucleation of an ~Zr grain in relation R~ or R2 is lower than the one for R3. The nucleation energy for an orientation generated by Vi is necessarily the lowest as it corresponds to 50% of the experiments with only a 25% chance of appearance within R~, R2. The orientation generated by 112 is not found experimentally and therefore the relatively large misorientation angle of 10 ° increases notably the strain energy and, subsequently, the total interface energy (not considering the influence of the interface plane). As the remaining 50% of the experimental results are generated by V3, 114, the small misorientation of 7 ° between the 11'~0 planes has to be considered as energetically acceptable. However, its activation energy should be higher than the one related to V~. Acknowledgements--The authors are grateful to Dr U. Dahmen from the NCEM, Lawrence Berkeley Laboratory (Berkeley), Dr D. Michel from CECM-CNRS (Vitrysur-Seine) and Prof. M. Guymont from the University of Paris XI (Orsay) for their fruitful interest in this work and for long hours of discussion. This work is supported by the DGA/DRET under the contract DRET/REP 93 # 93/ 1213.
REFERENCES 1. W. Bollmann, Crystal Lattices, Interfaces, Matrices, and an Extension of Crystallography. Published by the author (1982). 2. H. Grimmer, Acta crystallogr. A A30, 685 (1974). 3. H. Grimmer, W. Bollmann and D. H. Warrington, Acta crystallogr. A A30, 197 (1974). 4. H. Grimmer, Acta crystallogr. A A40, 108 (1984). 5. H. Grimmer and D. H. Warrington, Acta crystallogr. A A43, 232 (1987).
HETEROPHASE INTERFACES
4179
6. H. Grimmer. Acta crystallogr. A A45, 320 (1989). 7. S. Erochine and G. Nouet, Scripta metall. 17, 1069 (1983). 8. H. Grimmer, Acta crvstallogr. A A45, 505 (1989). 9. H. Grimmer and R. Bonnet. Acta crystallogr. A A46, 510 (1990). 10. M. Duneau, J. Physique 1 2, 871 (1992). 11. F. W. Glaser and B. Post, Trans. metall. Soc. AIME 223, 1117 (1953). 12. K. A. Jackson and J. D. Hunt, Trans. metall. Soc. AIME 236, 1129 (1966). 13. Y. Champion and S. Hag~ge. J. Mater. Sci. to appear 14. Y. Champion, Ph.D. Thesis, University Paris VI, France (1994). 15. Z. Nishiyama, Martensitic Transformation. Academic Press, New York (1978). 16. R. Bonnet and E. E. Laufer, Physica status solidi A 40, 599 (1977). 17. L. Qing, H. Xiaoxu and Y. Mei, Ultramicroscopy 41, 317 (1992). 18. R. Bonnet and F. Durand, Physica status solidi A 27, 543 (1975). 19. T. Karakostas, G. Nouet, G. L. Bl&is, S. Hag~ge and P. Delavignette, Physica status solidi A 50, 703 (1978). 20. J. Du, Ph.D. Thesis, University of Paris XI. France (1992). 21. M. Loubradou, J. M. P6nisson and R. Bonnet, Ultramicroscopy 51, 270 (1993). 22. L. A. Tietz and C, B. Carter, Phil. Mag. A 67, 699 (1993). 23. L. A. Tietz and C, B. Carter, Phil. Mag. A 67, 729 (1993). 24. W. G. Burgers, Phisica 1, 561 (1934). 25. P. Curie, J. Physique (Paris) (1894). 26. D. Gratias and R. Portier, Proc. Structure et propribtbs des joints de grains, J. Physique, C-15. Les Editions de Physique, Caen, France (1982). 27. G. Kalonji and J. W. Cahn, Proc. Structure et propribtbs des joints de grains, J. Physique, C-25. Les Editions de Physique, Caen, France (1982). 28. G. Kalonji, Proc. Int. Conf. Structure and Properties of Internal Interfaces, J. Physique, p. 249. Les l~ditions de Physique, lrsee, Germany (1985). 29. M. Boudeulle, J. appl. Crystallogr. 27, 567 (1994). 30. Y. Champion and S. Hag~ge, submitted.
Acta mater.
~
Pergamon PII S 1359-6454(96)00055-9
EXPERIMENTAL DETERMINATION AND SYMMETRY RELATED ANALYSIS OF ORIENTATION RELATIONSHIPS IN HETEROPHASE INTERFACES: A CASE STUDY IN THE Zr-B SYSTEM
Y. CHAMPIONH"and S. HAGEGE~ ~ETCA-CREA, 94114 Arcueil, and 2CNRS-CECM, 94407 Vitry-sur-Seine, France (Received 14 December 1994: in revised form 3 January 1996)
A~traet--Orientation relationships between phases in the Zr-B binary system were determined using electron diffraction. Heterophase interfaces were produced during the solidification of the eutectic (Zr-ZrB2) followed by phase transformation, i.e. allotropic transformation of zirconium, formation of ZrB by peritectoid transformation and stabilisation of a metastable Zr phase, zZr. Despite the complexity of the various phases and their transformation upon cooling, the experimental analysis of all the orientation relationships (cubic-hexagonal and hexagonal-hexagonal) found in this system yields only three sets of orientations characterised by the parallelism of low index planes and directions. Using symmetry considerations and group theory, it has been demonstrated that these orientations can be understood as the lowest energy configurations which can be produced from one single orientation formed at high temperature between flZr and ZrB_,. Copyright ~ 1996 Acta Metallurgica bw.
1. INTRODUCTION It is now well established that grain and phase boundaries play a key role in material properties and research in this field is oriented towards a correlation between the structure of the boundaries and the physical behaviour of materials. Therefore, the "interface community" has been very active in experimental characterisation and structural description of boundaries, the final aim being to provide a quantification of the boundary structure which could be related to property measurements. One of the most widely used approaches has been the characterisation of model boundaries which were assumed to be in energetically favourable configurations. The accurate geometrical and structural characterisations reported in these studies have led to a better understanding of the boundary structure and have emphasised the basic ideas for the structural description and subsequent classification of boundaries. An interface is geometrically defined by nine parameters describing the orientation relationship between two neighbouring crystals (3), the boundary plane which separates the two phases (3) and a rigid body translation linking an arbitrarily defined origin within each crystal (3). Particular interest has been placed very early on the orientation relationship. Firstly, it remains the parameter which is experimentally the most easily accessible. Secondly, it has often been assumed to be related to the stability of tPresent address: CNRS-CECM, 15 Rue Georges Urbain, 94407 Vitry-sur-Seine Cedex, France
interfaces and this was supported by geometrical models based on coincidence orientations containing no information on the actual atomic structure near the plane of the interface. The orientation relationship has been described using different methods. The description in terms of a transformation matrix is a very useful mathematical tool for classification. For the case of grain boundaries, the rationality of the matrix and the index of coincidence ~ has been widely used in the literature [1-9]. In the case of phase boundaries, the matrix description suffers from the irrationality of the matrix and the volume variation between the two phases not allowing a relevant definition of a general index of classification. Duneau et al. [10] have suggested a novel approach in terms of elementary shears matrix decomposition, which is seen as an appropriate criterion but only for very restrictive cases. So far, the most common method for describing heterophase interfaces is the parallelism or the close parallelism of a set of low index planes and directions of the crystals. This description is very limited since it does not contain information on the metric of the system. Nevertheless one can correlate different orientation relationships for analogous but structurally different interfaces. As an example, the parallelism of densest planes and densest directions is frequently observed for hexagonal-hexagonal, f.c.c.hexagonal and f.c.c.-f.c.c, interfaces. In these systems, which would yield very different transformation matrices, densest planes and directions are
4169
4170
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES 2. MATERIAL AND EXPERIMENTAL
2.1. Sample preparation 300
250
P ~200C
1500
~
1855 ° .~/
1680 ° 1250 °
I 100
800
°
.L.
_
_
5°°f o Zr
io
,;,o
ATOMICPER CENT BORON
6o ZrB2
Fig. 1. Section of the Zr-B phase diagram after Glaser [11]. strictly equivalent and the comparisons of the various orientation relationships are straightforward. The present work is partly devoted to the analysis and understanding of the orientation relationship as a parameter in the interface stability. Its aim is also to provide experimental results as a support for the description of general interfaces. We have dealt with cubic-hexagonal and hexagonal-hexagonal heterophase interfaces in the Zr-B system. All these have been produced by eutectic solidification and subsequent phase transformations as in the phase diagram of Fig.1 [l 1]. Thus, interfaces were assumed to be in energetically favourable conditions and a very small number of specific orientation relationships was expected. In order to validate these assumptions experimentally, a significant number of orientation measurements were performed in many samples for the different interface systems. In this paper, the results are reported in two parts. The experimental results have provided characteristics on the orientation relationships for each case by direct comparisons between the different systems. A theoretical analysis of the formation of orientation variants during phase transformations has been carried out. This treatment was based on the group theory and elementary symmetry considerations and has allowed establishment of a correlation between experimental orientation relationships and energetic criteria.
tThe crystallographic structure of the zZr phase has been identified as slightly deformed f.c.c, by diffraction and high resolution electron microscopy [13]. It has been considered as f.c.c, in the present study.
Samples of the Zr-ZrB: eutectic composition (1.9% boron) have been prepared and brought above the melting temperature of the metal (1855°C), using a R F induction heating device. Then they were cooled, leading to the solidification of eutectic composite materials at 1680°C with the formation of an oriented microstructure of ZrB_, needles in a Zr matrix. The global morphology of the samples is characterised by a preferred orientation of ceramic needles in each grain of the polycrystalline metallic matrix, This texture is in full agreement with the microstructure of a stable oriented rod-like eutectic [12] and hence supports an energetically favourable configuration at the heterophase interfaces. Crystallographic and chemical analysis, performed using X-ray diffraction and electron energy loss spectroscopy, revealed the coexistence of four different phases in the samples. As expected, the hexagonal close-packed (h.c.p.) phase ~tZr and the simple hexagonal (s.h.) phase ZrB: were observed. The non-equilibrium ZrB with a NaCl face-centred cubic structure (f.c.c.) has been also found. In addition, an original phase with a near-f.c.c, structure was identified but only in thin samples for transmission electron microscopy. This new phase with a chemical composition very close to that of ~tZr has been denoted zZr in the following.t The details of sample preparation [13, 14] have shown thereafter that the non-equilibrium phases ZrB and zZr were related to the post-solidification cooling conditions of the samples. ZrB was obtained by slow cooling of the sample, zZr was observed in thin specimens obtained from samples quenched after solidification, from a temperature over 1250°C. Moreover, it has been proved that xZr is related to ctZr by a phase transformation in the ShojiNishiyama (f.c.c.-h.c.p.) orientation [15]. This transformation and the zZr stability are believed to be closely related to the combination of thinning processing, thin specimen effects and boron impurity content in the bulk of the aZr matrix of the eutectic samples. The zZr phase is seen as an artefact since it does not exist in a bulk specimen. Nevertheless, as it is assumed to be produced only under favourable conditions, it fully contributes to the analysis of orientation relationships in our study.
2.2. Experimental procedures The orientation relationships between phases at interfaces have been determined using transmission electron microscopy in the selected area electron diffraction mode. The thin specimens were prepared by dimpling and Ar + ion milling. Diffraction experiments were performed on a JEOL 1200FX microscope working at 120kV, using +60 ° tilt/ rotation specimen holder.
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
4171
Table 1. Experimentalresults on orientation relationships Orientation relationship R~'" R~"c
a b c d e f g
(0001):/(0001) [1120] [11'~0] Exactparallelism
R2"-h R~"
Interfaces ~Zr-ZrB: ZrB~xZr ~Zr-ZrB ZrB~ZrB ~Zr-~Zr ~Zr-ZrB~ ZrBr-zZr
Relation
Misorientation
R~"~
ZrB~ZrB
h
(ITO0)//,~(I 11) [l'~lO]//¢L~[O1l] [0001]//[01]']
R~"~
~Zr-ZrB:
i
( lOT 1)Zr//lO00 l )Z,B:
(0001)//(111) [1120] ,[O'fl] (0001y/(lll) [1120]/[0"1"1]
Few betweenplanes Exact parallelism
(ITO0)//(O001)[I 1'~0]//[II'~0] (11"00)//(11I) [11'~O]//[Oll]
Few between planes Few between planes
[OOOll//I2TI]
[f2T0]z,//[I2T0]z~B.
Many procedures have been developed for precise determination of orientation relationships, based on either the analysis of spot diffraction patterns [I 6, 17] or the analysis of Kikuchi line patterns [18]. These methods for grain boundary analysis combine experimental measurements and computer treatment. They provide with high precision the rotation matrix, the axis and angle of misorientation, directly from a minimum of simple parameters such as tilt angles or directions in reciprocal space. For the determination of orientation relationships at heterophase boundaries, we have derived one of those methods, developed for grain boundary analysis by Karakostas et aL [19]. The aim of the method is to find the simplest planes and directions (lowest index), which are parallel or close to parallel in the two crystals. In this experimental procedure, each crystal is oriented in the spherical reference system of the microscope by tilting the specimen to successive reflections in the Laiie conditions. The tilt angle and the direction of the reciprocal lattice row measured in that condition, give directly and simultaneously the relative orientation of each crystal on a stereographic projection. The precise orientation relationship is derived from this stereographic projection. In some cases, the results were provided directly by the diffraction patterns, when both crystals were parallel or nearly parallel to a simple zone axis. 3. EXPERIMENTAL RESULTS All the possible combinations of two of the four phases identified in our samples, combinations which can be effectively obtained through the in situ formation process, have been found. Only the Z r B - z Z r interface has not been observed, since these two phases do not appear in samples at the same cooling conditions after solidification. As expected, a very small number of orientation relationships characterised by parallel low index planes and directions was found in this investigation. The structural equivalence which exists between the densest planes {0001 }, { 111 } and between the densest directions (11]0) and (110), respectively, for hexagonal and f.c.c, structure allows a classification
Few between planes Few between directions Exact parallelism Few between planes Exact parallelism
of the experimental orientation relationships into only three families denoted as R~, R,. and R~. These are detailed in Table 1. The orientations in each family are described with both the hexagonal-hexagonal and hexagonal-cubic index when they exist. They have been denoted R~'¢ for an orientation relationship belonging to the family R~ and describing an hexagonal--cubic interface. Orientation relationships of the R~ and R_, families were observed with almost identical frequency whereas R3 was observed only once. 3. I. Experimental orientation relationships Rj
The R~ orientation relationship follows the exact rule of the parallelism of the densest planes and the densest directions in both hexagonal and f.c.c. structures, except for the ZrB_,-zZr interface where a slight deviation has been found. By analogy with the so-called "cube-on-cube" orientation in interfaces with the cubic-cubic symmetry, the R~'h orientation has been called the "hexagon-on-hexagon" orientation. The orientation relationship family R~ is very common. It has been experimentally observed at numerous interphase boundaries in the literature, whatever the nature of the phases (metallic, ceramic, semiconductor) and whatever the preparation procedure (phase transformation, internal reduction or oxidation, oriented solidification, layer growth on a substrate). A non-exhaustive list of experimental results [14] shows a large proportion of interfaces belonging to R~. However, most of the experimental results concern hexagonal-f.c.c, systems (see for example Refs [20, 21]) and the orientation relationship R~h for hexagonal-hexagonal symmetry has been seldom observed so far [22,23]. Figure 2 shows the results of electron diffraction experiments of the R~ orientation relationship. Figures 2(a)-(c) are experimental diffraction patterns and Figs 2(d)-(f) are their respective indexations. Thus, Figs 2(a) and (d) and 2(b) and (e) are, respectively, the diffraction patterns of ctZr and ZrB2 phases taken for the exact same orientation of the sample. The parallelism of their respective [ I 1~0] zone axes and the strict alignment of the [0001] as well as [i100] respective reciprocal lattice rows are clearly apparent. Figures 2 (c) and (f) correspond to the
4172
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
O O
a
b
0001 •
•
d
•
•
•
[11~0]
•
c
•
0001 •
•
•
•
• "~100
•e IN
•
•
•
[ 1 ~o]
[l 1~0] //[O~l]
Fig. 2. Orientation relationship Rn. Experimental diffraction patterns of :tZr [(a), (d)], ZrB.~[(b), (e)] and the superimposition of 7Zr and ZrB: [(c), (f)].
superposition of the diffraction patterns of the zZr and the ZrB: phases, showing that the two analogous zone axes [11]0] and [0T1] are parallel. However, for this system a slight deviation from the exact alignment of the densest planes (0001) and ( l l l ) is also visible in the figure. In addition, a slight elongation of the spots of the xZr phase is observed. These two phenomena have been related to the relaxation of the strains associated with the phase transformation from ~Zr to zZr phase.
The family R] is described in Fig. 3 by the stereographic projections of an hexagonal-hexagonal interface [Fig.3(a)] and an hexagonal-cubic interface [Fig.3(b)].
3.2. Experimental orientation relationships Re In contrast to R~, the family R2 has never been reported in the literature as far as the authors are aware. By comparison with R~, the family R2 is characterised by the conservation of the densest
2 o:oI/o
,p
• c.b\
(0!01)
~
~0)
.
Fig. 3. Stereographic projections of the orientation relationships R,. R~'" hexagonal-hexagonal interface (a) and R~'~ hexagonal-cubic interface (b).
CHAMPION and HAGEGE: HETEROPHASE INTERFACES
4173
C ~100 0001
0 0 0
0001o • •
d
•
• •
[11"201
100
•
•
• "h.oo 111 •
•
•
•
•
[ 11~01
f
[6'il]
Fig. 4. Orientation relationship R2. Experimental diffraction patterns of ~Zr [(a). (d)]. ZrB., [(b), (e)] and the zZr [(c), (f)].
directions, (11'50) and (110), and the near-parallelism of the densest plane of one of the two phases with a prismatic plane { 1T00} of an hexagonal phase. The family R,, is illustrated by experimental electron diffraction patterns in Figs 4(a)-(c) and their indexing in Figs 4(d)-(f). These diffraction experiments were carried out on both sides of a ZrB2 fibre where the ~Zr-ZrB,, and zZr-ZrB2 interfaces were simultaneously present. Figures 4(b) and (e) in the middle show the diffraction pattern of the ZrB, phase where the [11]0] zone axis is exactly parallel to the electron beam. The diffraction patterns of =cZr [Figs 4(a) and (d)] and •Zr [Figs 4(c) and (f)] taken in that exact same orientation of the sample reveal a slight deviation from the exact parallelism of planes and directions for the two interfaces. The family R2 is divided into three sub-orientations denoted R~h for hexagonal-hexagonal interfaces, R~"~ and R~:~ for hexagonal--cubic interfaces. The orientation relationships R~h and R~c are illustrated by the stereographic projections in Figs 5(a) and (b), and they also correspond to the experimental results in Fig.4. In an exact orientation [as in Fig.5(b)] this corresponds for hexagonal-f.c.c, interfaces to the additional parallelism of the [0001] and [2"i"i] directions. The orientation relationship R~,', observed for ZrB2-ZrB interfaces, illustrated on the stereographic projections in Figs 5(c) and (d), is more complex. It has been classified in the R2 family orientation since
it involves, as the R~'" orientation, the close parallelism of (1]'00) and (111). However, in that case, the couple of planes (1]10), (110) which are also close to each other as in R .,h' , are not now at 90 ° from the common (1100), (111). Because the angle between (1100) and (1510) is 30 ~' and the angle between (111) and (110) is 35.26 ~, it is now impossible for both sets of planes to be simultaneously parallel. Therefore R~:¢ corresponds to a case of frustration for which the exact parallelism of planes of the respective phases can never be reached as shown in Fig.5(d). Experimentally, none of these two sets of planes are exactly parallel as there is an energetic compromise to handle the frustration of the angle. In addition, it results from this configuration that the third set of parallel planes is now (0001), (011) and not (0001), (2H) as in R~c. These points are seen as the major differences between the orientation relationships R~" and Rh,:". However, these differences are minor in comparison with the close parallelism of dense planes which has allowed us to classify R~'¢ and R~:" within the same R2 set of orientations. The analogies and differences between the orientation relationships R~'c and R~'~ are emphasised by the comparison of the stereographic projections in Figs 5(b) and (c). 3.3. Experimental orientation relationships R3 As previously mentioned, over a significant number of experimental measurements, R3 corresponds to one single observation at an ~Zr-ZrB2
4174
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
interface (hexagonal-hexagonal) and is denoted by R h'h. No experimental evidence of the equivalent R3h'¢ has been observed in this work. In comparison with the orientation relationships R~'h and R2h'h, the parallelism of the densest direction [1~10] is maintained in R~"h. However, it involves, as a second criterion, the close parallelism of the basal plane (0001) of ZrB2 with a pyramidal plane (10]'1) of ~Zr. Therefore the relative orientation for any hexagonal-hexagonal interphase boundary following R~'h and the subsequent stacking of atomic planes throughout the interface depends on the parameters of the hexagonal structures. 4. ANALYSIS OF THE ORIENTATION RELATIONSHIPS THROUGHOUT THE PHASE TRANSFORMATIONS
4. I. Phase transformation characteristics The Zr-B system seems rather complex since it involves many phases obtained by solid state transformations occurring for various experimental
conditions. However, all these phases and the subsequent interfaces are produced from the eutectic flZr-ZrB2 solidified at high temperature. The purpose of this section is to define whether the experimental orientation relationships observed at room temperature can be related to a unique primitive orientation relationship of the high temperature eutectic, which subsequently evolved through solid state transformations. According to our sample preparation conditions, two types of unrelated sequences of solid state transformations could modify the phases and the orientation present in our specimens. The first involves the formation of the zZr phase due to quenching, and the second involves the formation of the ZrB phase due to a slow cooling of the specimen.
4.1.1. Phase transformation during fast cooling. Figure 6(a) illustrates the case of quenched specimen (path I). In path I, the ZrB2 phase is not involved and the phase transformations occur in the matrix of metallic zirconium. In addition, the structural
(213)
/ • .
\
~•
<0001) .o
.
, oo,
.
(1 2o) - ' o
:
. ,
I
.
,,rooo) .
2 .xzr" : " / (off)
,26 °
• ZrB2
~
"
o Z r B ~ Z r B mZrB2 ~ . ~ . _ _ . ~ ~
d
Fig. 5. Stereographic projections of the orientation relationships R~. (a) R~h hexagonal-hexagonal orientation, (b) R~"c, (c) and (d) R2hf hexagonal-cubic orientations. Parts (b) and (c) show the two types of planes and directions {I TOO}, ( 1~10> and {111}, (011 ) which can be involved. In the case of Rh:cpart (d) emphasises the frustration and the near-orientation relationship character of the orientation Rh~ due to the angular difference in couples for the hexagonal and cubic systems.
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
/--;z-q
4175
planes and the densest directions of the h.c.p, and f.c.c, structure. It may be written as T ~'~ (0001)~//(lll)z
and
[II~0]~//[0T1]~.
YZr
TiL=
T=,X :cZr
~ xZr
Fig. 6(a). Sequence of phase transformations and phases formation for a quenched specimen. TP'" and T,.x denote, respectively, the allotropic b.c.c. (fl)-h.c.p. (ct) and the unusual h.c.p. (ct)-f.c.c. (Z) phase transformations of Zr. The ZrB., phase is not involved in this path of phase transitions.
I Path H ] ~r
-
Zrl~
Table 1 shows that six types of interfaces out of nine (a, b, e, f, g, i) may be related to path I. 4.I.2. Phase transformation during slow cooling. As revealed by Fig.6(b), path II is more complex than path I. It involves the formation of the monoboride phase (ZrB) between the ZrB2 and the Zr phase in its two allotropic structures flZr, and it theoretically disappears by the inverse peritectoid transformation with ~tZr, leading to the formation of new ctZr and ZrB2 phase [denoted as (ctZr) and (ZrB2) on Fig.6(b)]. The peritectoid transformation and its inverse which combine several structural transformations and variations of the chemical composition, are still unknown. Moreover, it is easily assumed according to the sample preparation, that the transformations are not total, especially at low temperature due to the slow diffusion of boron [we have assumed in fact that this last stage does not actually occur and has been placed between brackets in Fig.6(a)]. Accordingly, the experimental interfaces may be produced at any stage in path II, mostly at high temperature and stabilised at room temperature after aliotropic transformation of Zr. Table 1 shows that three types of interfaces out of nine may be related to the path II (c, d, h).
4.2. The primitive orientation at high temperature lYZr - Z r B
-
ZrB2
T[~'~ ecZr - Z r B - ZrB2
{ a,Zr - (ZrB2) - ZrB- ZrB2 ] Fig. 6(b). Sequence of phase transformations and phase formation for a slowly cooled specimen. TB."still denotes the allotropic b.c.c. (fl)-h.c.p. (~t) phase transition of Zr. P and P-' denote, respectively, the peritectoid and the inverse peritectoid reactions in which ZrB is involved.
The preliminary stage consists of the assumption of a primitive orientation relationship between flZr and ZrB2. It may be deduced by applying the (~t-fl) allotropic phase transition [T"~= (T~'0 -~] to the ~Zr-ZrB2 interface in the hexagon-on-hexagon orientation R~"h. This orientation has been chosen because it is the simplest orientation. It has always been observed with the exact parallelism of planes and direction. The primitive orientation relationship denoted R0a'h may be written as in the following expression
p~.h = T..,(R~.h) (llO)~//(O001)h and [T1 l]d/[1210]h. 4.3. Group theory analysis
characteristics of these phase transformations, i.e. the orientation relationship between phases throughout the transformations, are known. TP." denotes the (fl, ~t) b.c.c, to h.c.p, allotropic phase transformation of the zirconium at 860°C. It is known as the Burgers transformation [24] and it may be written as T ~'~ (110)~//(0001)~
and
[Illb//[I210L.
T ~'z denotes the (~,X) h.c.p, to f.c.c, phase transformation of zirconium. Previous investigations have shown [13] that this transformation follows the Shoji-Nishiyama orientation which keeps the densest
The interfaces between ZrB2 and r,Zr which should be produced at room temperature from a/~Zr-ZrB2 interface are derived from elementary symmetry considerations. This analysis is separated into two parts: firstly the allotropic phase transition in zirconium (/~--~) is considered and the orientation variants of ctZr in the parent crystal /~Zr are characterised. Secondly the orientation of each variant is considered with respect to ZrB2. (ZrB2 is kept unchanged during the phase transition.) The aim of this last stage is to define whether the number of orientations obtained correspond effectively to different interfaces.
4176
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
(211o)
These two stages lead to the total number of possible orientation relationships for the ~Zr-ZrB_, and subsequently 7Zr-ZrB2 interfaces.
4.3.1. Variants obtained in the allotropic transformation of Zr. The number and orientation of ~tZr
(111
phase variants obtainable within the parent phase flZr are defined using the group theory. The concept based on Curie's law? has been developed by different authors [26-28] and has been applied recently for the study of precipitation in minerals [29]. As we are dealing with an orientation relationship using electron diffraction measurements, the translation symmetry operators are not accessible. Therefore, only the point-group symmetry of crystals will be used. According to Curie's law, the symmetry of the morphology of the orientation variants is related to the symmetry group of iso-probability of nucleation and the symmetry group of the nuclei. The symmetry group of the iso-probability of nucleation Go is composed of the set of common symmetry elements of the parent phase and of the external acting parameters. During the cooling of the specimen, it is assumed that the only external acting parameter is the temperature with a spherical symmetry and then Go remains the symmetry of the parent phase. If H0~ denotes the symmetry group of the bicrystal flZr-~Zr, it constitutes a set of common symmetry elements of Go (flZr) Go = m~m (Oh) and of the symmetry group of the embryo phase G~ = 6/mmm (D6h). It is called the intersection group and it is given by the superimposed stereographic projections of flZr and ctZr (Fig.7)
(1i20)~
H01 = Go n Gt = 2/m(C2h). The number of orientation variants N0~ corresponds to the number of times H0~ is contained in Go. It is given by the ratio of group orders of Go and H0~ N0~ - order(m~m) order(2/m)
48 _ 12. 4
The 12 orientation variants are defined by the coset decomposition of the point group Go = m~m with respect to the point group H0~ = 2/m (Table 2). Each variant is associated with a pair of orthogonal directions (110), (T l l ) of the b.c.c, structure which will yield the couple of orthogonal directions (0001), (T2i0) of the hexagonal structure of ctZr. This set of 12 variants is, as a whole, invariant with respect to the symmetry operation of H0~ = 2/m. However, for the decomposition of Table 2, variants 1 and 2 have the symmetry H0~, whereas the 10 others are related two by two (3/4, 5/6, 7/8, 9/10 and 11/12) by H0j.
/"
/
(i2/0) ocl
~h01 ~oll)
Om ,o
(I11
•_
\
•
=
o,,
"
"X6 H
"
,
oo~,) •
/ /io,~o
(2]i0) Fig. 7. Orientation relationship of the fl--ctphase transition of Zr and the symmetry elements of the bicrystal. intersection group between flZr and ZrB2, in the Rg'h orientation relationship, is H0h = H0~ = 2/m. Therefore, the application of the symmetry operations of the 2/m group to the set of ctZr variants leaves unchanged the orientation relationship of each variant with respect to the ZrB2 crystal. As a result the variants 3 and 4, 5 and 6, 7 and 8, 9 and 10, 11 and 12 are in the same orientation relationship with respect to ZrB2. However, this does not hold for variants 1 and 2 which will remain two distinct orientation relationships with respect to ZrB2. The total number of the orientation relationships between ~tZr and ZrB2 is then ~+1+I=7. The seven orientation relationships are listed in Table 3 and depicted in Fig.8. They are given as parallelism or misorientation between low index planes and directions of the ~Zr and ZrB2 crystals. According to this classification they may be grouped into three families which are consistent with those found experimentally. R~ corresponds to the orientation relationships which are exact or close to the hexagon-on-hexagon type of orientation. R2 is characterised by the near alignment of densest directions and a misorientation between basal planes of 90 °. R3 is mostly characterised by a misorientation between basal planes of exactly 60 °. By a direct comparison with experimental results in Table 1, V~, (V3/V4) and (VT/V~o) have been easily identified respectively as R~", R~'h and R~'h.
4.3.2. Orientation relationships with respect to ZrBe. Each one of the 12 variants is then considered
4.4. Orientation relationships of zZr.
with respect to its own orientation with the hexagonal ZrB2 phase (point group G2 = 6/mmm (Drh)). The
The (c~-Z) phase transition is considered similarly. Firstly, the number and orientations of variants of gZr are determined from previous variants of =Zr. Secondly, each orientation variant of xZr is considered with respect to the ZrB2 phase. The
?"The symmetry of the resulting effects reflects the symmetry of the originating causes" [25].
CHAMPION and HAGEGE:
4177
HETEROPHASE INTERFACES
Table 2. Coset decomposition of the symmetry group m~m with respect to the 2/m subgroup m]m 1
P2im 1 xyz
2
1io, TI1 iyz
1
2 yxi
2~1
liO, ITT iii
TT0, lit
2,,0
3 iyZ
4 9iz
T
TTo, ITT
m,]<,
TTo. TII
2~r
xyi 110, TIT
m~
yxz 110. ITI
mtio
TTO,TIT
3
iyi Tlo, l i t
20]o
yiz fro, l i t
4~]
x~z ]To. TTI
mo,0
~xi i1o, TIT
~
4
x~i ]To, TIT zxy
2,~
~xz TIO, TIi xz~
4~t
iyz Tio, Ill iiy
mt~
yii TlO. tiT iiy
~
5
3~]
Oil, ITI
6 7 8 9 10
47~
]oL TiT
zi~ 0TT, tiT i~y OTl. Tll ix~ 011, TTT yzx 101. ill
3~
~zi
3~t
3~ 3fi, 3fi~
ToT, Ti]
~fi,
oTT, TiT
Y.zy T01, ill i~ ToT, iTT xiy 101, TTI zyi 0]L 111
2o.
z~x
2,0~
2oa 4?~ 4~0
oTi, liT
~,~
IOl, ITi
ix)' 011. IT1 zx~ o i l iTT zi)' Oil, 111 9i£ TOT.TT1 yix
~fi~
xiy i0T, TTT xzy 1Ol. Tll iz~ TOT, llT iyx 0Tl. TTT iyi
~T
~fi, ~ ~1
ioi, iTT
m~,~
mo,r ~;~ ]~0 m~o,
o i l TIi
11
yli 3fi~ iyx 4Go ~zx ]fi, z~i ~o lOT, IT1 o11, TIT iOl. T1T oil ]h 12 yix 3fi~ i~i 2~o~ yzi ~r zyx mro~ TOl, TIT OrT. TTI lOT. 111 llO. lit Each elementof the coset is givenby its Wyckoffposition and point symmetryelement.The application of the symmetryoperators to the pair 110, TII is also given. transition is associated with an increase of the symmetry of the crystal from the 6 / m m m symmetry group of ~Zr to m]~m (Oh) symmetry group of zZr. Furthermore, in the Shoji-Nishiyama orientation, the intersection symmetry group remains that of the parent phase ~Zr. It results in a single orientation variant of the gZr and also a single orientation relationship between xZr and ZrB2. In other words, the orientation relationships found in the previous paragraph are invariant under the T =-~ transformation.
4.5. The peritectoid transformation and the case of ZrB 4.5.1. Orientation relationships obtained from the RI orientation. The origin of the interfaces ~tZr-ZrB and ZrB-ZrB2 in the orientation relationship R~" may be found by considering the first stage of the path II. ZrB grows as a buffer at the /TZr-ZrB,, interfaces during the peritectoid transformation occurring at about 1200°C. At this temperature, /TZr and ZrB2 follows the primitive orientation relationship p~.h defined in the previous section. If we assume that the ZrB phase grows in perfect epitaxial conditions in-between /TZr and ZrB2, the orientation relationships between phases may be written as in the following expressions
(llO)Bz,//(1 i 1)Z,B/t(0001)Z,B: and IT i l]p,,t/[l ] o],,Bl/[T2Tok,~:. This condition corresponds to the parallelism of densest planes and densest directions in a f.c.c.-hex (ZrB-ZrB2) interface. This induces an epitaxial condition in a b.c.c.-f.c.c. (/TZr-ZrB) interface which is the K u r d j u m o v - S a c h s orientation relationship. Together with the Nishiyama-Wassermann orientation, this orientation is the most c o m m o n l y observed in b.c.c.-f.c.c, phase transitions. After cooling of the specimen, the allotropic (/7--~) phase transition of the zirconium occurs, leading a m o n g others, to the formation of ~tZr-ZrB and ZrB-ZrB2 interfaces in the R~" orientation relationship (c and d in Table 1). The other ~tZr-ZrB interfaces which would be able theoretically to be generated by the 11 orientation variants of ctZr embryos were not observed experimentally. 4.5.2. Another orientation. The last interface which has to be considered is ZrB-ZrB2 in the R~,c orientation relationship (h in Table 1). This orientation relationship cannot be included in the previous treatment since it is not related to a known phase transformation orientation between f.c.c, and hexagonal crystals. In addition, as indicated by the reaction path II [Fig.6(b)], this type of interface is
Table 3. Orientation relationships between aZr and ZrB2 as derived from the 12 orientation variants of ~(Zr V, after the allotropic//--~ phase transition of zirconium Variants
0001
11'I0
V~
II
11
--
P.,
10.53 T
--
V3 V.,
ti 90'
V7V~o
60 =
!i
7.8~
VaP~ VsVii
60° 60~ 60°
V6VI2
10.53= 10.5317.Y
0001z, ^ lOT Iz,B:
-
-
7.8' 12 12=
Exp.
Results
#t~ .h
R~
R~ "h
R2
R~h
R3
4178
CHAMPION and HAGI~GE: HETEROPHASE INTERFACES
> VI
V2
VT~V=o
Vt=V9
V~=V~I
V6~Vu
V3 or V4
Fig. 8. Schematical representation of the seven orientation relationships of ~tZr with respect to ZrB2 as derived from the 12 Vi orientation variants. The black and white six-fold axis symbols depict the basal plane of, respectively, ZrB, and ctZr, whereas the two-fold axis symbols represent similarly the (l~10) directions.
Fig. 9.
found experimentally that only interfaces originating from the variants V~ and ~ , V~ are observed, and with the same frequency.t Energy considerations can provide a clear explanation for this discrepancy. The orientation relationships found experimentally are all based on the (close) parallelism of low index planes or directions in the structures involved in the phase transformations. Then, they can be assumed to be in the most favourable energetic configurations. It can be stressed here that if the nucleation of all 12 variants of ~Zr within flZr is equiprobable, the fact that this 5. ENERGY CONSIDERATIONS ON THE nucleation has to occur in contact with ZrBz should ORIENTATION RELATIONSHIPS somehow induce a lifting of the degeneracy. The previous analysis has been summarised in the As noted in Tables l and 3, R~ is based on the exact flow chart (Fig.9) relating the experimentally parallelism of the (0001) planes and is found observed interfaces and the orientation variants experimentally when generated by V~ (parallelism of produced by the phase transformation fl--~. Starting the densest planes and densest directions) but not with the assumption that the 12 variants V~ are when generated by V2 (the two basal planes are equiprobable, this figure recalls the fact that the parallel but rotated by 10.53°). As for R2, the close probability of appearance of an interface originating parallelism of the densest directions of the structures from either the variants V~ or V: is 1/12, whereas the is maintained and the 90 ° between the (0001) planes probability of appearance of an interface originating implies the exact parallelism of relatively dense planes from one of the other variants V3V4... Vi2 is 1/6 (e.g. of the structures: for hexagonal interfaces, basal V3 and V4 generate the same interface). If we assume planes are parallel to prismatic planes and for that the probability of occurrence of these interfaces hexagonal-cubic interfaces, basal planes and prisis directly related to the equiprobability of the matic planes are parallel to (211) and (l l l) planes, variants, the probability of appearance of an interface respectively. These hexagonal-hexagonal and hexagin an orientation R3 (8/12) should be much larger onal-cubic interfaces are found experimentally. All than for the families R~ or R2. However, it has been the orientations of R3 are based on 60 ° between the tThe unique case of the ctZr-ZrB2 interface in the basal planes and this angle does not lead to the exact orientation relationship R~*, originating from VT, I,'10, (or even close) parallelism of dense planes or obviously has no real statistical value. direction. Moreover, as indicated above, this
produced either by the peritectoid transformation or by the inverse peritectoid transformation. In any case, the mechanism of this transformation has not been elucidated and the experimental interfaces cannot be located in the reaction path. In fact, those particular interfaces were observed only in a single sample and so they are not as significant as the other experimental interface systems. Finally, further investigations reported in Ref. [30] have revealed that the misorientation between the neighbouring crystals is related to the interface structure.
CHAMPION and HAGI~GE: orientation is related to the c/a ratio of the hexagonal structures. However, the only orientation found experimentally and belonging to this group happened to have an exact parallelism of the 11~0 directions. We can assess that the activation energy for the nucleation of an ~Zr grain in relation R~ or R2 is lower than the one for R3. The nucleation energy for an orientation generated by Vi is necessarily the lowest as it corresponds to 50% of the experiments with only a 25% chance of appearance within R~, R2. The orientation generated by 112 is not found experimentally and therefore the relatively large misorientation angle of 10 ° increases notably the strain energy and, subsequently, the total interface energy (not considering the influence of the interface plane). As the remaining 50% of the experimental results are generated by V3, 114, the small misorientation of 7 ° between the 11'~0 planes has to be considered as energetically acceptable. However, its activation energy should be higher than the one related to V~. Acknowledgements--The authors are grateful to Dr U. Dahmen from the NCEM, Lawrence Berkeley Laboratory (Berkeley), Dr D. Michel from CECM-CNRS (Vitrysur-Seine) and Prof. M. Guymont from the University of Paris XI (Orsay) for their fruitful interest in this work and for long hours of discussion. This work is supported by the DGA/DRET under the contract DRET/REP 93 # 93/ 1213.
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HETEROPHASE INTERFACES
4179
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