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Crystallography and solidification behaviour of nanometric Pb particles embedded in icosahedral and decagonal quasicrystalline matrix
PII:
Acta mater. Vol. 46, No. 13, pp. 4641±4656, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00137-2 1359-6454/98 $19.00 + 0.00
CRYSTALLOGRAPHY AND SOLIDIFICATION BEHAVIOUR OF NANOMETRIC Pb PARTICLES EMBEDDED IN ICOSAHEDRAL AND DECAGONAL QUASICRYSTALLINE MATRIX ALOK SINGH{ and A. P. TSAI National Research Institute for Metals, Tsukuba 305, Japan (Received 9 December 1997; accepted 29 March 1998) AbstractÐSolidi®cation behaviour of lead particles embedded in icosahedral and decagonal quasicrystalline matrix has been studied. Dierential scanning calorimetry traces for the icosahedral matrix show a sharp lead melting peak but a very ¯at solidi®cation exotherm with several peaks, while lead in the decagonal matrix shows sharp melting and solidi®cation peaks. Transmission electron microscopy shows that lead particles are faceted on major planes of the matrix. The solidi®ed lead particles show well de®ned orientation relationships with the matrix. Three relationships were observed in the case of the icosahedral matrix in all of which there is a tendency for the lead {022} planes to align with icosahedral {211111} and {221001} planes. In the decagonal matrix a single relationship is observed. The calculated contact angles for nucleation show that lead solidi®cation on quasicrystalline surfaces occur at higher undercoolings for the same contact angles, as compared to solidi®cation on crystals. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Since most liquids solidify by heterogeneous nucleation, nucleation of solids on substrates has been a subject of intense investigations. However, to achieve this, it is important to avoid the in¯uence of undesirable impurities in the melt on nucleation. To achieve this an experiment with liquid droplets embedded in a matrix of higher melting point was devised by Wang and Smith [1]. With this procedure it is possible to carry out controlled studies of heterogeneous nucleation, with solidi®cation of the particles nucleating on the surrounding higher melting point matrix and monitoring with the help of a calorimeter. This procedure has been extensively used by Chadwick and co-workers [2, 3] and more recently by Cantor and co-workers [4±10] and Goswami and Chattopadhyay [11±15] in studying heterogeneous solidi®cation using immiscible binary alloys. Solidi®cation of nanometric lead particles in a cubic matrix have been studied in many alloys such as in aluminium matrix [5], in copper matrix [9] and in zinc matrix [11], and melting studies in aluminium, copper and nickel [13]. Lead solidi®es on these cubic matrices with a de®nite orientation relationship, usually cube-on-cube. In this study nucleation and solidi®cation of crystalline lead in a quasicrystalline matrix has been studied. Lead is immiscible in the alloys chosen as matrix. A three-dimensional (icosahedral) and a {To whom all correspondence should be addressed.
two-dimensional (decagonal) quasicrystal were chosen as the matrix. It was found that the lead indeed solidi®ed with well de®ned orientation relationships with the quasicrystals and preliminary results on the orientation relationships have been reported [16]. This paper describes the results with the icosahedral Al±Cu±Fe phase [17] matrix and the decagonal Al± Cu±Co matrix. The solidi®cation behaviour and the crystallographic relationship of the lead with the matrix is studied in detail here. 2. EXPERIMENTAL PROCEDURE
Pure metals corresponding to Al65Cu20Fe15 and Al70Cu8Co22 were melted in an arc furnace under argon atmosphere. The Al±Cu±Fe alloy was melt spun with 20 wt% Pb and the Al±Cu±Co alloy with 10 wt% Pb by melting the alloy and lead in a quartz tube and ejecting onto the rotating copper wheel by argon pressure. The melt spun ribbons were thinned by ion milling for observation in transmission electron microscopes (TEM). A JEOL 2000FX-II and 2010F microscopes operated at 200 and 100 kV were used for micro and convergent beam diraction experiments and a JEOL 4000EX-II TEM operated at 400 kV was used for high resolution electron microscopy. A Perkin Elmer model DSC7 for dierential scanning calorimetry (DSC) was used for monitoring the melting and solidi®cation of the lead particles inside the matrix. About 10 mg of sample was
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scanned between 20 and 5008C at rates varying from 5 to 208C/min. The sample was held at the maximum temperature for 10 min before cooling.
3. RESULTS
3.1. Shape and size of the lead particles Figure 1(a) is a typical electron micrograph showing the lead particles in Al±Cu±Fe icosahedral
matrix. The lead particles are typically of diameter of about 70 nm in icosahedral matrix of average grain size of about 0.5±1.0 mm. There was some additional dark contrast associated with the particles which was found to be due to mechanical strain. Energy dispersive X-ray analysis con®rmed that the elements of the matrix phase and the lead were immiscible in each other. Figure 1(b) is a bright ®eld micrograph showing the distribution of the lead particles in the decago-
Fig. 1. Low magni®cation electron micrographs showing the distribution of lead nanoparticles in (a) icosahedral matrix and (b) decagonal matrix.
SINGH and TSAI: CRYSTALLOGRAPHY AND SOLIDIFICATION BEHAVIOUR
nal matrix. The particles were lens shaped ¯attened in the quasiperiodic plane with a diameter of about 150 nm and thickness 30 nm. The matrix was found to be free of any strain. No miscibility was found between the lead and matrix phase. The nucleation sites of the solid lead will depend on the facets of the matrix phase it is in contact with. Therefore, the shape of the particles in the icosahedral phase was studied. Figure 2(a) shows two particles A and B observed along a twofold axis of the matrix phase. Both the matrix and the particles are in major orientations and therefore dark in contrast. In addition, the matrix shows strain contrast. A ®vefold plane is faintly resolved in this micrograph and it is noticed that particle A has a facet along this plane of the matrix. Particle B also has a facet nearly along this plane, as marked in the micrograph. Another facet in particle A is seen perpendicular to the other ®vefold direction of the matrix. The ®vefold facets are marked ``x''. There are other facets which are parallel to twofold planes, marked ``y'' in the ®gure. The grain is then tilted nearly 598 about one of the twofold vectors, to another zone axis ``D'' (notation of Singh and Ranganathan [18]) which contains a ®vefold and a twofold direction perpendicular to each other. In this grain orientation, shown in Fig. 2(b), both the particles show facets perpendicular to ®vefold axes. Facets parallel to twofold axes are also still observed. Thus the cavities in which the lead particles solidify have facets parallel to ®vefold and twofold planes of the matrix icosahedral phase and also threefold planes. This observation is in keeping with the fact that the morphology of the Al±Cu±Fe icosahedral phase is a pentagonal dodecahedron and Audier et al. [19] have observed a tendency for faceting on ®vefold planes, followed by twofold and threefold planes, at an atomic scale. 3.2. Melting and solidi®cation behaviour of the lead particles Figure 3(a) and (b) show the DSC traces of heating and cooling cycles, respectively, for the icosahedral matrix samples. On heating, there is a sharp endothermic peak at 327.58C, corresponding to the melting of lead. Thus there is no superheating of lead. The cooling cycle, however, does not show a single clear peak. In each cooling cycle several small exothermic peaks were observed. In the curve shown in Fig. 3(b) there is a small peak corresponding to 310.68, and three well distinguished peaks at 305.0, 302.3 and 301.38C. These peaks are marked 1, 2, 3 and 4 in the inset showing details in Fig. 3, corresponding to undercoolings of 17.1, 22.7, 25.4 and 26.48C, respectively. This suggests that there are several well distinguishable sites in the matrix phase for nucleation of the solid lead of which three are prominent. Figure 4(a) and (b) show the DSC traces of heating and cooling cycles in the decagonal matrix.
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Both the melting endothermic peak while heating and the solidi®cation exothermic peak while cooling are fairly sharp and almost identical. There is no signi®cant superheating for melting. The solidi®cation shows about 58C undercooling. The areas under both the peaks are almost identical. 3.3. Orientation relationships Microdiraction of particles was performed to determine the orientation relationship between the particles and the matrix. For this, the matrix grain was oriented with a major zone axis along the electron beam and then the nano lead particle orientations were checked by microdiraction. Favourite orientations for the icosahedral matrix were the twofold axis, since in this zone occur all the major planes, the ``D'' zone axis (containing a ®vefold and a twofold plane), ®vefold zone axis and threefold axis. In this method, however, there is a possibility of selectively looking at lead particles only in prominent contrast. Therefore another approach followed to determine the orientation relationship was to check the orientation of the particles in arbitrary matrix orientation and then determine the matrix orientation by obtaining Kikuchi bands through convergent beam electron diraction. This approach was expected to give all possible orientations between matrix and particles. Figure 5 shows pairs of diraction patterns from the icosahedral matrix and the particles. Figure 5(a) and (b) show a twofold diraction pattern from the matrix and corresponding microdiraction pattern from a particle. The particle orientation is along the cubic [111] axis. A [011]* reciprocal vector is along each of the two ®vefold vectors. Another of the [011]* relvectors is along a twofold icosahedral vector. Figure 5(c) and (d) show microdiraction patterns from a matrix oriented along a ``D'' zone axis and from a particle in it. The particle orientation is along the [100] cubic direction. Again, a {022} vector is along a ®vefold and a twofold vector of the icosahedral phase. Discs due to {200} re¯ections are observed faintly. Figure 5(e) and (f) show a matrix orientation along a ®vefold icosahedral zone axis. The corresponding microdiraction from the particle shows spots in the [122] zone axis. Spots to only one side of the transmitted beam (in all observations) shows that the orientation is slightly away from the [122] axis, around the h220i* vector. It is noteworthy that the angle between two reciprocal vectors in the cubic [122] zone axis is about 728. Figure 5(g) and (h) show a matrix orientation along a threefold icosahedral zone axis and a corresponding microdiraction from a particle along the cubic [111] zone axis. The cubic {011} reciprocal vectors are along the twofold icosahedral vectors. Figure 6 shows the orientation relationship between the decagonal matrix and the particles. Figure 6(a) and (b) show a diraction from a matrix grain in twofold ``G'' orientation and a
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Fig. 2. Two lead particles A and B observed along (a) a twofold axis and (b) 598 away along the ``D'' axis of the icosahedral phase matrix. Facets on ®vefold planes are marked as ``x'' and on twofold planes as ``y''.
microdiraction from a particle in it, which is in the [111] orientation. Figure 6(c) and (d) show a diraction from a matrix grain in the twofold ``H'' orientation and a microdiraction from a particle
in it which is in the [100] orientation. Microdiraction of particles from the tenfold matrix orientation was quite dicult, as the particles in this orientation were very thin and the con-
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Fig. 3. Dierential Scanning Calorimetry curves for the case of icosahedral matrix (a) heating and (b) cooling curves scanned at the rate of 108C/min. Details of exothermic peaks on cooling are shown as inset.
tribution from the matrix could not be eliminated. Figure 6(e) shows a microdiraction from a particle in a matrix close to the tenfold orientation. The microdiraction shows that the particle is oriented in the [110] zone axis. All these orientations correspond to one of the orientation relationships determined between the icosahedral matrix and lead particle. It will be shown that due to a one-dimensional periodicity, the ®t between the particle with the decagonal matrix becomes better. Figure 7 shows a lead particle, of size about 3444 nm across, along the twofold orientation of the icosahedral matrix. Faceting on ®vefold planes is clearly observed. A set of (022) lattice planes in the particle is clearly visible to be parallel to a set of ®vefold planes in the matrix. Since the particles were quite small, the interface between the particles and the matrix was curved and therefore dicult to see clearly. However, the visible lattice in the particles appears to be coherent with the matrix planes on the left part of the particles. Figure 8(a) shows a high resolution micrograph of a particle in a twofold ``G'' matrix orientation. In this micrograph half the particle shows a dierent fringe contrast. The coarser fringes which are rectangular are attributed to moire eect due to an overlap of the particle with a part of the matrix. These moire fringes suggest a good lattice match between the particle and the matrix. In the other
half, lattice fringes in [111] lead orientation are observed. The interface between the particle and the matrix is observed quite clearly (shown as detail in Fig. 8(b)) since the particle in this orientation is elongated parallel to the electron beam and therefore the interface is almost parallel to the direction of observation. It is observed in the regions marked ``X'' with arrowheads that there is a good registry between a set of lead (022) planes and quasiperiodic planes across the interface. 3.4. Stablility of the phases An important observation has been about the stability of the quasicrystalline matrix phase under the electron beam during observation. At the start of the observation of a region the icosahedral phase displayed sharp diraction peaks but after a certain duration the peaks became broader and fewer. Due to this, part of the observation was carried out at 100 kV. The lead particles, however, were found to be stable. In the Al±Cu±Co alloy matrix, the decagonal phase was observed to decompose into microdomains producing a pseudo tenfold symmetry which can be explained by tenfold related domains of a monoclinic phase with b = 728. Similar decagonal±rhombic monoclinic phases have been reported before and shown to be reversible [20]. It is not clear in the present case whether this trans-
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Fig. 4. DSC curves for the case of decagonal matrix (a) heating and (b) cooling, showing endothermic melting and exothermic solidi®cation peaks, respectively.
formation is reversible and needs further investigations. 4. DISCUSSION
The heating curve of DSC shows a sharp endothermic peak of melting. The cooling curve in the case of the icosahedral matrix, however, shows multiple exothermic peaks of melting, three of which are quite prominent and well distinguished. The area under these three peaks, however, accounts for only about 20% of the area of the melting peak. The solidi®cation is thus spread over a wide temperature range with widely dierent undercoolings. These several small peaks of solidi®cation suggests several nucleation sites with signi®cantly dierent nucleation energies. Some of them may correspond to nucleation on grain edges and boundaries. The three most prominent ones are likely to be from nucleation on the faces on ®vefold, twofold or threefold planes. In the case of the decagonal matrix the shape of the particles is ¯attened in the quasiperiodic plane of the matrix. The possible nucleation sites are thus narrowed. The lens shape of the particle in the high resolution micrograph of Fig. 8 shows that the faceting is not always on the quasiperiodic planes of the matrix but there are smaller facets on other planes. The DSC curves show a sharp melting peak
as well as a sharp solidi®cation peak which look identical in shape and area. This suggests that the nucleation sites for melting and solidi®cation are the same. Kim et al. [7] showed that the ideal simulated droplet solidi®cation exotherms were asymmetric, with approximately 70% of the droplets solidifying up to the peak temperature. In the exothermic peak shown in Fig. 4(b), approximately 68% of the droplets are estimated to solidify up to the peak temperature. All the nucleation sites thus maybe of the same kind. 4.1. Orientation relationships Figure 9 shows stereographic representations of various orientation relationships observed between lead particles and the matrix. Figure 9(a) shows a stereogram giving the orientation relationship called OR1. In this relationship a cubic [111] axis coincides with an icosahedral twofold axis (as shown in Fig. 5(a) and (b)) such that the two ®vefold reciprocal vectors nearly coincide with two h110i* cubic vectors. Diraction spots of the kind {211111} along ®vefold and {221001} along twofold vectors nearly coincide with {022} spots. Table 1 summarizes the important features of this orientation relationship. Three of the other twofold axes will nearly coincide with h110i axes. Three of the ®vefold axes will be close to cubic h110i and other
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Fig. 5. Diraction/nanodiraction patterns from the icosahedral matrix and particles along (a, b) twofold axis of the icosahedral matrix, (c, d) ``D'' axis of icosahedral matrix, (e, f) ®vefold axis, and (g, h) threefold axis of the matrix.
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three ®vefold axes at cubic h113i. It is important to note here that the angle between vectors 011* and 112* in this zone axis is about 728. It is also interesting to note that in this orientation relationship all the three cubic h100i axes coincide with the icosahedral ``D'' zone axes, which also have a squarelike appearence (Fig. 5(c) and (d)). This orientation
relationship gives the ``best ®t'' between the cubic and icosahedral phase studied here since more of the major icosahedral axes coincide with more of the similar cubic ones. Seven of the ``D'' axes will coincide with h221i and four with h112i. Six of the icosahedral threefold axes will coincide with h123i and one is near a h111i.
Fig. 6. (a, b) Selected area electron diraction of a decagonal matrix grain in a twofold G orientation and the microdiraction from a lead particle in it. The particle is in [111] orientation. (c, d) Electron diraction from a matrix grain in twofold H orientation and microdiraction from a particle in it. The particle is in [100] orientation. (e) Microdiraction from a lead particle in a grain oriented near its tenfold axis. The particle is in [110] orientation. Periodic (tenfold) reciprocal vectors are marked ``A''.
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Fig. 7. A high resolution electron micrograph showing a lead particle along a twofold axis of the icosahedral matrix. The arrows mark the directions of important planes in the matrix. The particle displays a set of (022) lattice fringes parallel to a set of ®vefold planes of the matrix.
Figure 9(a) shows an orientation relationship where a cubic h111i axis is parallel to an icosahedral twofold axis. If the two phases are rotated about this common axis by 908, so that another of the two icosahedral twofold directions is now coincident with a cubic h022i* direction, then another orientation relationship OR2 is obtained, shown in Fig. 9(b). Important coincident axes are summarized in Table 1. In this relationship a ®vefold axis
will lie in between a [233] and a [122] axis along a common h022i* direction. As observed in the diffraction patterns of Fig. 5(e) and (f), this h022i* direction is parallel to one of the twofold vectors in the ®vefold zone axis of the icosahedral phase. Other ®vefold axes coincide with, or are near to, two h112i axes, one h100i axis and two h315i axes. Only one each of h110i, h111i and h112i cube axes coincide with icosahedral twofold axes. Thus the
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Fig. 8. (a) (Continued opposite).
registry of the major axes between the two phases is poor in comparison to the relationship OR1. The formation of this relationship therefore must be dictated by nucleation on a ®vefold plane, since the interplanar angle in the cubic h122i are characterized by 728. In the orientation relationship OR3 shown in Fig. 9(c), a cubic threefold axis [111] coincides with an icosahedral threefold axis as observed in diraction patterns of Fig. 5(g) and (h). As shown in
Table 1, three each of twofold icosahedral axes coincide with cubic h112i and h123i axes. Both of these cubic axes can be considered to approximate the indices h1tt2i of the icosahedral twofold axes. Three of the ®vefold axes coincide with cubic h012i, which can be considered to approximate the ®vefold axis indices h01ti. Three of the icosahedral ``D'' axes are close to cubic h320i axes, which approximate the icosahedral ``D'' indices ht10i. Three other icosahedral threefold axes coincide with cubic h210i,
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Fig. 8. (a) Lattice image of a lead particle in a decagonal matrix grain in twofold G orientation. The upper part of the particle shows moire fringes, while the lower part shows lattice fringes of [111] orientation. (b) Magni®ed image of a part of the interface between particle and the matrix. Periodic direction is marked ``A''. Arrowheads mark regions showing good coherency across the interface.
whose indices approximate the icosahedral threefold axis indices ht210i. Three of the h111i axes are close to threefold axes and all of the h100i axes are near twofold axes. Thus the orientation relationship OR3 corresponds to the orientation relationship between an icosahedral phase and a cubic approximant to the icosahedral phase. Many common coincidence conditions are common between dierent orientation relationships, as observed in Table 1. Figure 9(d) is a stereogram showing the orientation relationship of the particles with the decagonal matrix. In the case of the icosahedral phase matrix it was shown that the orientation relationship OR1 is the best ®t between the particles and the matrix. The unique orientation relationship between the decagonal matrix and the lead particles corresponds to OR1 and therefore con®rms it being the best ®t. It will be shown here that in the case of the decagonal matrix this ®t is much better, leading to a single orientation relationship. 4.2. Epitaxy An interface between a quasicrystal and a crystal cannot be coherent but only semicoherent since the structure on one side of the interface is periodic and on the other side is quasiperiodic. This will also produce interfacial strain which may be considerable in many cases. Due to very dierent structures
of the two phases, it has been observed here that the solidifying lead particle chooses an orientation with the best ®tting local structure on the plane of the matrix phase acting as a substrate. There are mainly three planes of the icosahedral phase oered to the lead droplets as substrateÐthe ®vefold, twofold and threefold planes. It would appear that the lead particles solidify with their {111} planes at the interface when solidifying on the twofold or threefold planes, and a plane near to {122} when solidifying on a ®vefold plane. This could not be con®rmed from high resolution micrographs as the interface could not be observed clearly. The {111} planes of lead have lowest energy [5]. In all the orientations observed, the common factor is the registry of {211111} ®vefold and {221001} twofold planes with {022} planes. The {211111} planes have a planar spacing of about 2.1 AÊ and the {221001} planes of about 2.0 AÊ, while the {022} planes have an interplanar distance of about 1.75 AÊ in the bulk phase. There is a small angular mismatch when the icosahedral twofold and {111} planes are at the interface. The angle between two ®vefold directions is 63.48 while that between two h022i* directions is 608. Several solidi®cation peaks in the DSC cooling curves indicate several dierent nucleation sites on the matrix, with signi®cant dierences in surface
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Fig. 9. Stereograms showing orientation relationships (a) OR1, (b) OR2 and (c) OR3, observed between the icosahedral matrix and the lead nanoparticles and (d) between decagonal matrix and nano-lead particles (A is tenfold axis and G and H are twofold axes).
energy. The small peak 1 at 310.68C maybe due to nucleation on grain boundaries and edges, which will aord easier nucleation. Of the possibilities considered above, the best match between lead and the icosahedral phase at the interface will be when the threefold planes of both the phases are on the interface. The nucleation would be then easier in this case, and the solidi®cation peak 2 at 306.38C maybe due to nucleation on threefold faces of the icosahedral matrix. The next two possibilities are of nucleation on ®vefold and twofold planes. As shown, the nucleating lead has a tendency to align its h011i* direction with ®vefold or twofold directions of the icosahedral phase. This is satis®ed better if lead nucleates on a twofold plane of the matrix (with its {111} plane on the interface). The exothermic peak 4 must correspond to nucleation on ®vefold planes.
The Al±Cu±Fe icosahedral phase has a tendency to facet on ®vefold planes and therefore a greater number of ®vefold facets are expected. It is thus no surprise that this exothermic peak is the largest. It is emphasized here that further experiments are required to con®rm these arguments. It will now be shown that due to a one-dimensional periodicity, the lattice match between the lead and the decagonal quasicrystal in this orientation is better than the OR1 observed between lead and the icosahedral quasicrystal. Figure 10(a)±(c) show twofold diraction patterns from the Al±Cu± Fe icosahedral phase, an icosahedral twin and the Al±Cu±Co decagonal phase, respectively. As observed in Fig. 10(a), the angle between two ®vefold vectors (X) in the icosahedral phase is 63.48. The twofold vector is denoted Y. In the icosahedral twin, Fig. 10(b), the X and Y directions from the
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Table 1. The three orientation relationships OR1, OR2 and OR3 between the icosahedral matrix and lead nanoparticles Icosahedral axis
Twofold Fivefold ``D'' Threefold Twofold Fivefold ``D'' Threefold Twofold Fivefold ``D'' Threefold
Coincident or close cubic major axis (in parantheses are number of such matches) Orientation relationship OR1 h111i (3), h110i (3), h112i (1) h110i (3), h113i (3) h100i (3), h112i (4), h122i (7) h123i (6), near h111i (1) Orientation relationship OR2 h111i (1), h110i (1), h112i (1), h014i (2) near h122i (1), h100i (1), h112i (2), h315i (2) h110i (3), h120i (2), h140i (2), h113i (3), h122i (2), h223i (2) h112i (1), h133i (1), near h123i (3) Orientation relationship OR3 h112i (3), h123i (3) h012i (3) h122i (3), near h320i (3) h111i (1), h210i (3)
twin contributions are denoted by primes. The common ®vefold vector (X±X') has become a tenfold axis [21]. It is observed that the Y' axis of the twin is close to the X axis of the matrix and the X' axis of the twin is close to the Y axis of the matrix. In a decagonal phase the X±X' direction becomes periodic X. Introduction of a periodicity along the tenfold axis has two eects in the decagonal phase, as observed in Fig. 10(c). The closeby X and Y' (and X' and Y) directions coincide, denoted as XY. The angle between X and XY directions is 608. Thus when lead particles are oriented with the [111] axis parallel to this twofold axis, the angular match is better (as compared to the icosahedral phase in which the angle between X directions is 63.48). Due to this angular match, the decagonal axis is exactly parallel to a h011i axis, as compared to the case of the icosahedral matrix where ®vefold axes were only nearly parallel to h011i axes. The second eect of the one-dimensional periodicity is on the epitaxial relationship between the substrate and the nucleus. It was observed in the case of the icosahedral matrix that the orientation of the nucleating lead was chosen such that its h022i* directions preferred to coincide with ®vefold and twofold vectors. In the case of the decagonal phase it is seen that these two vectors coincide, into a vector pointing towards XY in Fig. 10(c). Thus a very good epitaxial relationship is expected (and is indeed found) between the nucleating solid lead and the decagonal matrix, and nucleation on either of the several facets will lead to the same orientation of the lead. This is observed in the lattice image in Fig. 8(b). In places marked ``X'' and ``Z'' with arrowheads a good lattice match across the interface is clearly observed.
Fig. 10. Comparison of the twofold patterns from (a) Al± Cu±Fe icosahedral phase (®vefold vectors marked X and a twofold vector Y), (b) icosahedral twofold pattern with contributions from a twin (marked with primes) and (c) twofold G pattern from Al±Cu±Co decagonal phase (tenfold axis marked X). The angle corresponding to 63.48 between ®vefold vectors (marked X) in the icosahedral phase becomes 608 (between X and XY) in the decagonal phase.
4.3. Contact angles and nucleation site density DSC results have been used to calculate the contact angles for nucleation and the catalytic site density using classical nucleation theory. The catalytic site density estimations are, however, often unrealistically low. The values of contact angles and catalytic site densities estimated in various systems are summarized by Goswami and Chattopadhyay in a
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table [14]. Such unrealistic values are attributed to a breakdown of the classical nucleation theory for nanometric sized particles [5, 11]. Goswami and Chattopadhyay [15], however, argue that this is not due to a breakdown of classical nucleation theory at low undercoolings but due to variations in the microscopic details of the surfaces oered by the matrix for heterogeneous nucleation. Eorts to ®t a single value to these dierent sites can lead to unrealistic results. Nevertheless, here estimations are made for the lead solidi®cation on the quasicrystalline phase for comparison with other systems. A method for estimating the contact angle of a solidifying liquid droplet on a substrate is described by Moore et al. [5]. At any temperature T below the melting point, the fraction of solid particles Z is given by dZ I 1 ÿ Z dt
1
where I is the nucleation frequency within each particle, which is given by [22, 23] I A expÿB= Tm ÿ T 2 T
2
where Tm is the particle melting point A Nc kT=hexp ÿQ=kT
3
where Nc is the number of catalytic sites per particle and k and h are Boltzmann's and Planck's constants, respectively, Q is the activation energy for transporting a particle atom across a particle solid± liquid interface. B Ks3 T 2m f y=kL2
4
where K is a shape factor equal to 16p/3 for a hemispherical shaped nucleus, s the particle solid±liquid surface energy, f y 2 ÿ 3 cos y cos3 y=4, y the contact angle at the substrate±solid/liquid particle triple point as in the classical heterogeneous nucleation theory, and L the particle latent heat of solidi®cation. From equations (1) and (2) it can be seen that ln dZ=dt= 1 ÿ Z varies linearly with 1= Tm ÿ T 2 T, with a slope and intercept of ÿB and ln A, respectively. In the icosahedral matrix case the solidi®cation is not on one single plane of the matrix and therefore there is no singular value of contact angle. For each of the three facets in the icosahedral phase that has been considered above, there is a value for y. To obtain each of the values of y, each of the three most prominent peaks in the DSC cooling curve (Fig. 3) is considered as a separate peak since each of these peaks is assumed to correspond to solidi®cation in a speci®c nucleation site. Thus ln dZ=dt= 1 ÿ Z vs 1= Tm ÿ T 2 T is plotted for each peak. Results can be aected by overlap of peaks, but since y is calculated from the slope of this plot, considerations of the tops of these peaks
is enough. The solidi®cation rate dZ/dt at each temperature is directly proportional to the height of the exothermic peak and the fraction of solid particles at any temperature is given by the partial integration of the exothermic peak. These plots are shown in Fig. 11(a)±(c). The central portion of each of these plots is linear. This linearity is eected at the peak ends perhaps due to overlaps from neighbouring peaks. These plots give contact angles y of 7.55, 10.45 and 9.158 for peaks 2, 3 and 4 (in Fig. 3), respectively. As expected, the level of undercooling will depend on the contact angle. A lower angle will make nucleation easier. Thus for the solidi®cation peak 2 the contact angle is lower (7.558) than peak 3 (10.458). The y for peak 4 (9.158), however, is slightly lower than that of peak 3. This is because of the shape of the peak, since dZ/dt and Z are measured from the height and the area of the peak. Kim et al. [7] and Goswami and Chattopadhyay [15] have shown that with lowering of undercooling the solidi®cation peaks become sharper (with the same area). It is observed in Fig. 3 that while peaks 2 and 3 do not dier much in undercooling levels, peak 3 is sharper and peak 2 is ¯atter. For the DSC cooling curve solidi®cation peak of lead in the decagonal matrix shown in Fig. 4(b), the ln dZ=dt= 1 ÿ Z vs 1= Tm ÿ T 2 T plot is shown in Fig. 11(d). The value of constant B from this ®gure gives a contact angle value of 3.18. The catalytic nucleation sites per particle Nc are calculated to be of the order of 10ÿ9 for the icosahedral matrix and 10ÿ7 for the decagonal matrix, which are unrealistically low, but are comparable to the calculated values in the case of lead solidi®cation in crystalline matrices. In case of lead solidi®cation in aluminium it is 10ÿ6 [5], in copper it is 10ÿ12 [9] and in zinc 10ÿ9 [11]. Kim et al. [7] and Goswami and Chattopadhyay [15] show that at a given undercooling the contact angle does not change signi®cantly with Nc. Kim et al. [7] show that the change in particle size does not change undercooling very signi®cantly. The size of embedded lead particles studied here was fairly uniform. An increase in the size range of particles, or a size range of y only broadens the exothermic peaks slightly [7]. Thus the strongest correlation of y is with the undercooling, and thus it is reasonable to compare the y calculated here to those reported in other systems. Goswami and Chattopadhyay [14] have tabulated values of undercooling (DT) with calculated contact angles and nucleation site densities from various works. The calculated y of solidifying lead in an aluminium matrix is 218 (DT = 228C) [5], in copper it is 48 (DT = 0.68C) [9] and in zinc it is 238 (DT = 328C) [11]. The DT vs y for various systems is plotted in Fig. 12. Since examples are taken from other particles too, like tin and bismuth, the DT is plotted as a percentage. A fairly good agreement between DT and y is observed. The y values
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Fig. 11. Plot of ln dZ=dt= 1 ÿ Z vs 1= Tm ÿ T 2 T for the DSC solidi®cation exothermic peaks numbered (a) 2, (b) 3 and (c) 4 in Fig. 3 for the icosahedral matrix and (d) for the decagonal matrix.
obtained in the case of the icosahedral matrix are lower in comparison to crystalline systems. It is observed that in the case of the icosahedral quasicrystalline matrix, for the same undercooling levels the y values are smaller as compared to those obtained for crystalline matrices. In other words, for the same y, the undercooling required for nucleation of lead on the quasicrystal will be much higher. This could be due to a poor registry expected at the interface between crystal and quasicrystal. However, other factors such as the chemical component of the interface also play an important part in determining the contact angle.
2.
3.
5. CONCLUSIONS
Solidi®cation of lead by heterogeneous nucleation on icosahedral and decagonal quasicrystalline substrate has been studied by embedding lead particles in a matrix of the icosahedral phase by rapid solidi®cation. A study by DSC and TEM leads to the following conclusions: 1. The lead particles embedded in the icosahedral matrix had a size of about 70 nm and are faceted mainly on ®vefold, twofold and threefold planes of the matrix phase. The particles embedded in the decagonal phase matrix were lens shaped,
4.
5.
¯attened in the quasiperiodic plane, with a diameter of up to 150 nm and thickness 30 nm. DSC traces for the icosahedral matrix showed a sharp lead melting peak but very ¯at solidi®cation exotherm with several small peaks, suggesting several nucleation sites for solidi®cation. DSC traces for lead in decagonal matrix exhibit a sharp melting and an identical solidi®cation peak. Three orientation relationships between the icosahedral matrix and the particles were determined, which is selected by the site on which the nucleation occurs. Of the three orientation relationships one (OR1) gives the best ®t between the particle and the matrix lattice, while another (OR3) is similar to the orientation relationship of an icosahedral phase with its cubic approximant. In all these orientations, a preference of lead {022} reciprocal spots to coincide with icosahedral {211111} spots (along the ®vefold direction) and {221001} spot (along the twofold direction) is observed. In the case of the decagonal matrix only one orientation relationship was found, which corresponds to the orientation relationship OR1 of lead in the icosahedral matrix.
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Fig. 12. Plot of calculated values of nucleation contact angles y against observed undercooling reported for various systems. The undercooling is plotted as per cent for normalization of various systems: 1, Cu±(Pb) [9]; 2, Al±(Pb) [5]; 3, Zn±(Pb) [11]; 4, Al±(In) [4]; 5, Al±(Cd) [8]; 6, Al±(Sn) [6]; 7, Zn±(Bi) [13]; 8, Al±(Bi) [15]; 9, decagonal quasicrystal±(Pb) (present work); 10±12, icosahedral quasicrystal±(Pb) (present workÐcorresponding to DSC peaks 2, 3 and 4, respectively, in Fig. 3).
6. One-dimensional periodicity of the decagonal phase leads to a better lattice match with the particles. Due to this, nucleation at many dierent possible sites in the matrix will lead to the same orientation relationship. 7. The calculated contact angles for heterogeneous nucleation for three dierent solidi®cation nucleation sites in the icosahedral matrix are 7.55, 10.45 and 9.158, corresponding to undercooling levels of 22.7, 25.4 and 26.48C, respectively. In the decagonal matrix, the calculated contact angle is 3.18 which occurs at an undercooling level of 58C. These angles are smaller in comparison to those reported for calculations for lead nucleation in cubic matrices at the same undercooling levels. AcknowledgementsÐThis work is supported by Core Research for Evolutional Science and Technology, Japan Science and Technology Cooperation.
REFERENCES 1. Wang, G. C. and Smith, C. S., Trans. Am. Inst. Min. Engrs, 1950, 188, 136. 2. Southin, R. T. and Chadwick, G. A., Acta metall., 1978, 26, 223. 3. Boswell, P. G. and Chadwick, G. A., Acta metall., 1980, 28, 209. 4. Zhang, D. L. and Cantor, B., Phil. Mag. A, 1990, 62, 557.
5. Moore, K. I., Zhang, D. L. and Cantor, B., Acta metall., 1990, 38, 1327. 6. Kim, W. T. and Cantor, B., J. Mater. Sci., 1991, 26, 1266. 7. Kim, W. T., Zhang, D. L. and Cantor, B., Metall. Trans., 1991, 22A, 2487. 8. Zhang, D. L., Chattopadhyay, K. and Cantor, B., J. Mater. Sci., 1991, 26, 1531. 9. Kim, W. T. and Cantor, B., Acta metall. mater., 1992, 40, 3339. 10. Kim, W. T. and Cantor, B., Acta metall. mater., 1994, 42, 3045. 11. Goswami, R., Chattopadhyay, K., Kim, W. T. and Cantor, B., Metall. Trans., 1992, A23, 3207. 12. Goswami, R. and Chattopadhyay, K., Acta metall. mater., 1994, 42, 383. 13. Goswami, R. and Chattopadhyay, K., Acta metall. mater., 1995, 43, 2837. 14. Goswami, R. and Chattopadhyay, K., Phil. Mag. Lett., 1995, 72, 411. 15. Goswami, R. and Chattopadhyay, K., Acta mater., 1996, 44, 2421. 16. Singh, A. and Tsai, A. P., Phil. Mag. Lett., 1998, 77, 89. 17. Tsai, A. P., Inoue, A. and Masumoto, T., Japan. J. appl. Phys., 1987, 26, L1505. 18. Singh, A. and Ranganathan, S., Scripta metall. mater., 1991, 25, 409. 19. Audier, M., Guyot, P. and Brechet, Y., Phil. Mag. Lett., 1990, 61, 55. 20. Hiraga, K., Lincoln, F. J. and Sun, W., Mater. Trans. JIM, 1991, 32, 308. 21. Singh, A. and Ranganathan, S., Acta metall. mater., 1994, 43, 3539. 22. Turubull, D., J. appl. Phys., 1950, 21, 1022. 23. Cantor, B. and Doherty, R. D., Acta metall., 1979, 27, 33.
Acta mater. Vol. 46, No. 13, pp. 4641±4656, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00137-2 1359-6454/98 $19.00 + 0.00
CRYSTALLOGRAPHY AND SOLIDIFICATION BEHAVIOUR OF NANOMETRIC Pb PARTICLES EMBEDDED IN ICOSAHEDRAL AND DECAGONAL QUASICRYSTALLINE MATRIX ALOK SINGH{ and A. P. TSAI National Research Institute for Metals, Tsukuba 305, Japan (Received 9 December 1997; accepted 29 March 1998) AbstractÐSolidi®cation behaviour of lead particles embedded in icosahedral and decagonal quasicrystalline matrix has been studied. Dierential scanning calorimetry traces for the icosahedral matrix show a sharp lead melting peak but a very ¯at solidi®cation exotherm with several peaks, while lead in the decagonal matrix shows sharp melting and solidi®cation peaks. Transmission electron microscopy shows that lead particles are faceted on major planes of the matrix. The solidi®ed lead particles show well de®ned orientation relationships with the matrix. Three relationships were observed in the case of the icosahedral matrix in all of which there is a tendency for the lead {022} planes to align with icosahedral {211111} and {221001} planes. In the decagonal matrix a single relationship is observed. The calculated contact angles for nucleation show that lead solidi®cation on quasicrystalline surfaces occur at higher undercoolings for the same contact angles, as compared to solidi®cation on crystals. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Since most liquids solidify by heterogeneous nucleation, nucleation of solids on substrates has been a subject of intense investigations. However, to achieve this, it is important to avoid the in¯uence of undesirable impurities in the melt on nucleation. To achieve this an experiment with liquid droplets embedded in a matrix of higher melting point was devised by Wang and Smith [1]. With this procedure it is possible to carry out controlled studies of heterogeneous nucleation, with solidi®cation of the particles nucleating on the surrounding higher melting point matrix and monitoring with the help of a calorimeter. This procedure has been extensively used by Chadwick and co-workers [2, 3] and more recently by Cantor and co-workers [4±10] and Goswami and Chattopadhyay [11±15] in studying heterogeneous solidi®cation using immiscible binary alloys. Solidi®cation of nanometric lead particles in a cubic matrix have been studied in many alloys such as in aluminium matrix [5], in copper matrix [9] and in zinc matrix [11], and melting studies in aluminium, copper and nickel [13]. Lead solidi®es on these cubic matrices with a de®nite orientation relationship, usually cube-on-cube. In this study nucleation and solidi®cation of crystalline lead in a quasicrystalline matrix has been studied. Lead is immiscible in the alloys chosen as matrix. A three-dimensional (icosahedral) and a {To whom all correspondence should be addressed.
two-dimensional (decagonal) quasicrystal were chosen as the matrix. It was found that the lead indeed solidi®ed with well de®ned orientation relationships with the quasicrystals and preliminary results on the orientation relationships have been reported [16]. This paper describes the results with the icosahedral Al±Cu±Fe phase [17] matrix and the decagonal Al± Cu±Co matrix. The solidi®cation behaviour and the crystallographic relationship of the lead with the matrix is studied in detail here. 2. EXPERIMENTAL PROCEDURE
Pure metals corresponding to Al65Cu20Fe15 and Al70Cu8Co22 were melted in an arc furnace under argon atmosphere. The Al±Cu±Fe alloy was melt spun with 20 wt% Pb and the Al±Cu±Co alloy with 10 wt% Pb by melting the alloy and lead in a quartz tube and ejecting onto the rotating copper wheel by argon pressure. The melt spun ribbons were thinned by ion milling for observation in transmission electron microscopes (TEM). A JEOL 2000FX-II and 2010F microscopes operated at 200 and 100 kV were used for micro and convergent beam diraction experiments and a JEOL 4000EX-II TEM operated at 400 kV was used for high resolution electron microscopy. A Perkin Elmer model DSC7 for dierential scanning calorimetry (DSC) was used for monitoring the melting and solidi®cation of the lead particles inside the matrix. About 10 mg of sample was
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scanned between 20 and 5008C at rates varying from 5 to 208C/min. The sample was held at the maximum temperature for 10 min before cooling.
3. RESULTS
3.1. Shape and size of the lead particles Figure 1(a) is a typical electron micrograph showing the lead particles in Al±Cu±Fe icosahedral
matrix. The lead particles are typically of diameter of about 70 nm in icosahedral matrix of average grain size of about 0.5±1.0 mm. There was some additional dark contrast associated with the particles which was found to be due to mechanical strain. Energy dispersive X-ray analysis con®rmed that the elements of the matrix phase and the lead were immiscible in each other. Figure 1(b) is a bright ®eld micrograph showing the distribution of the lead particles in the decago-
Fig. 1. Low magni®cation electron micrographs showing the distribution of lead nanoparticles in (a) icosahedral matrix and (b) decagonal matrix.
SINGH and TSAI: CRYSTALLOGRAPHY AND SOLIDIFICATION BEHAVIOUR
nal matrix. The particles were lens shaped ¯attened in the quasiperiodic plane with a diameter of about 150 nm and thickness 30 nm. The matrix was found to be free of any strain. No miscibility was found between the lead and matrix phase. The nucleation sites of the solid lead will depend on the facets of the matrix phase it is in contact with. Therefore, the shape of the particles in the icosahedral phase was studied. Figure 2(a) shows two particles A and B observed along a twofold axis of the matrix phase. Both the matrix and the particles are in major orientations and therefore dark in contrast. In addition, the matrix shows strain contrast. A ®vefold plane is faintly resolved in this micrograph and it is noticed that particle A has a facet along this plane of the matrix. Particle B also has a facet nearly along this plane, as marked in the micrograph. Another facet in particle A is seen perpendicular to the other ®vefold direction of the matrix. The ®vefold facets are marked ``x''. There are other facets which are parallel to twofold planes, marked ``y'' in the ®gure. The grain is then tilted nearly 598 about one of the twofold vectors, to another zone axis ``D'' (notation of Singh and Ranganathan [18]) which contains a ®vefold and a twofold direction perpendicular to each other. In this grain orientation, shown in Fig. 2(b), both the particles show facets perpendicular to ®vefold axes. Facets parallel to twofold axes are also still observed. Thus the cavities in which the lead particles solidify have facets parallel to ®vefold and twofold planes of the matrix icosahedral phase and also threefold planes. This observation is in keeping with the fact that the morphology of the Al±Cu±Fe icosahedral phase is a pentagonal dodecahedron and Audier et al. [19] have observed a tendency for faceting on ®vefold planes, followed by twofold and threefold planes, at an atomic scale. 3.2. Melting and solidi®cation behaviour of the lead particles Figure 3(a) and (b) show the DSC traces of heating and cooling cycles, respectively, for the icosahedral matrix samples. On heating, there is a sharp endothermic peak at 327.58C, corresponding to the melting of lead. Thus there is no superheating of lead. The cooling cycle, however, does not show a single clear peak. In each cooling cycle several small exothermic peaks were observed. In the curve shown in Fig. 3(b) there is a small peak corresponding to 310.68, and three well distinguished peaks at 305.0, 302.3 and 301.38C. These peaks are marked 1, 2, 3 and 4 in the inset showing details in Fig. 3, corresponding to undercoolings of 17.1, 22.7, 25.4 and 26.48C, respectively. This suggests that there are several well distinguishable sites in the matrix phase for nucleation of the solid lead of which three are prominent. Figure 4(a) and (b) show the DSC traces of heating and cooling cycles in the decagonal matrix.
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Both the melting endothermic peak while heating and the solidi®cation exothermic peak while cooling are fairly sharp and almost identical. There is no signi®cant superheating for melting. The solidi®cation shows about 58C undercooling. The areas under both the peaks are almost identical. 3.3. Orientation relationships Microdiraction of particles was performed to determine the orientation relationship between the particles and the matrix. For this, the matrix grain was oriented with a major zone axis along the electron beam and then the nano lead particle orientations were checked by microdiraction. Favourite orientations for the icosahedral matrix were the twofold axis, since in this zone occur all the major planes, the ``D'' zone axis (containing a ®vefold and a twofold plane), ®vefold zone axis and threefold axis. In this method, however, there is a possibility of selectively looking at lead particles only in prominent contrast. Therefore another approach followed to determine the orientation relationship was to check the orientation of the particles in arbitrary matrix orientation and then determine the matrix orientation by obtaining Kikuchi bands through convergent beam electron diraction. This approach was expected to give all possible orientations between matrix and particles. Figure 5 shows pairs of diraction patterns from the icosahedral matrix and the particles. Figure 5(a) and (b) show a twofold diraction pattern from the matrix and corresponding microdiraction pattern from a particle. The particle orientation is along the cubic [111] axis. A [011]* reciprocal vector is along each of the two ®vefold vectors. Another of the [011]* relvectors is along a twofold icosahedral vector. Figure 5(c) and (d) show microdiraction patterns from a matrix oriented along a ``D'' zone axis and from a particle in it. The particle orientation is along the [100] cubic direction. Again, a {022} vector is along a ®vefold and a twofold vector of the icosahedral phase. Discs due to {200} re¯ections are observed faintly. Figure 5(e) and (f) show a matrix orientation along a ®vefold icosahedral zone axis. The corresponding microdiraction from the particle shows spots in the [122] zone axis. Spots to only one side of the transmitted beam (in all observations) shows that the orientation is slightly away from the [122] axis, around the h220i* vector. It is noteworthy that the angle between two reciprocal vectors in the cubic [122] zone axis is about 728. Figure 5(g) and (h) show a matrix orientation along a threefold icosahedral zone axis and a corresponding microdiraction from a particle along the cubic [111] zone axis. The cubic {011} reciprocal vectors are along the twofold icosahedral vectors. Figure 6 shows the orientation relationship between the decagonal matrix and the particles. Figure 6(a) and (b) show a diraction from a matrix grain in twofold ``G'' orientation and a
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Fig. 2. Two lead particles A and B observed along (a) a twofold axis and (b) 598 away along the ``D'' axis of the icosahedral phase matrix. Facets on ®vefold planes are marked as ``x'' and on twofold planes as ``y''.
microdiraction from a particle in it, which is in the [111] orientation. Figure 6(c) and (d) show a diraction from a matrix grain in the twofold ``H'' orientation and a microdiraction from a particle
in it which is in the [100] orientation. Microdiraction of particles from the tenfold matrix orientation was quite dicult, as the particles in this orientation were very thin and the con-
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Fig. 3. Dierential Scanning Calorimetry curves for the case of icosahedral matrix (a) heating and (b) cooling curves scanned at the rate of 108C/min. Details of exothermic peaks on cooling are shown as inset.
tribution from the matrix could not be eliminated. Figure 6(e) shows a microdiraction from a particle in a matrix close to the tenfold orientation. The microdiraction shows that the particle is oriented in the [110] zone axis. All these orientations correspond to one of the orientation relationships determined between the icosahedral matrix and lead particle. It will be shown that due to a one-dimensional periodicity, the ®t between the particle with the decagonal matrix becomes better. Figure 7 shows a lead particle, of size about 3444 nm across, along the twofold orientation of the icosahedral matrix. Faceting on ®vefold planes is clearly observed. A set of (022) lattice planes in the particle is clearly visible to be parallel to a set of ®vefold planes in the matrix. Since the particles were quite small, the interface between the particles and the matrix was curved and therefore dicult to see clearly. However, the visible lattice in the particles appears to be coherent with the matrix planes on the left part of the particles. Figure 8(a) shows a high resolution micrograph of a particle in a twofold ``G'' matrix orientation. In this micrograph half the particle shows a dierent fringe contrast. The coarser fringes which are rectangular are attributed to moire eect due to an overlap of the particle with a part of the matrix. These moire fringes suggest a good lattice match between the particle and the matrix. In the other
half, lattice fringes in [111] lead orientation are observed. The interface between the particle and the matrix is observed quite clearly (shown as detail in Fig. 8(b)) since the particle in this orientation is elongated parallel to the electron beam and therefore the interface is almost parallel to the direction of observation. It is observed in the regions marked ``X'' with arrowheads that there is a good registry between a set of lead (022) planes and quasiperiodic planes across the interface. 3.4. Stablility of the phases An important observation has been about the stability of the quasicrystalline matrix phase under the electron beam during observation. At the start of the observation of a region the icosahedral phase displayed sharp diraction peaks but after a certain duration the peaks became broader and fewer. Due to this, part of the observation was carried out at 100 kV. The lead particles, however, were found to be stable. In the Al±Cu±Co alloy matrix, the decagonal phase was observed to decompose into microdomains producing a pseudo tenfold symmetry which can be explained by tenfold related domains of a monoclinic phase with b = 728. Similar decagonal±rhombic monoclinic phases have been reported before and shown to be reversible [20]. It is not clear in the present case whether this trans-
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Fig. 4. DSC curves for the case of decagonal matrix (a) heating and (b) cooling, showing endothermic melting and exothermic solidi®cation peaks, respectively.
formation is reversible and needs further investigations. 4. DISCUSSION
The heating curve of DSC shows a sharp endothermic peak of melting. The cooling curve in the case of the icosahedral matrix, however, shows multiple exothermic peaks of melting, three of which are quite prominent and well distinguished. The area under these three peaks, however, accounts for only about 20% of the area of the melting peak. The solidi®cation is thus spread over a wide temperature range with widely dierent undercoolings. These several small peaks of solidi®cation suggests several nucleation sites with signi®cantly dierent nucleation energies. Some of them may correspond to nucleation on grain edges and boundaries. The three most prominent ones are likely to be from nucleation on the faces on ®vefold, twofold or threefold planes. In the case of the decagonal matrix the shape of the particles is ¯attened in the quasiperiodic plane of the matrix. The possible nucleation sites are thus narrowed. The lens shape of the particle in the high resolution micrograph of Fig. 8 shows that the faceting is not always on the quasiperiodic planes of the matrix but there are smaller facets on other planes. The DSC curves show a sharp melting peak
as well as a sharp solidi®cation peak which look identical in shape and area. This suggests that the nucleation sites for melting and solidi®cation are the same. Kim et al. [7] showed that the ideal simulated droplet solidi®cation exotherms were asymmetric, with approximately 70% of the droplets solidifying up to the peak temperature. In the exothermic peak shown in Fig. 4(b), approximately 68% of the droplets are estimated to solidify up to the peak temperature. All the nucleation sites thus maybe of the same kind. 4.1. Orientation relationships Figure 9 shows stereographic representations of various orientation relationships observed between lead particles and the matrix. Figure 9(a) shows a stereogram giving the orientation relationship called OR1. In this relationship a cubic [111] axis coincides with an icosahedral twofold axis (as shown in Fig. 5(a) and (b)) such that the two ®vefold reciprocal vectors nearly coincide with two h110i* cubic vectors. Diraction spots of the kind {211111} along ®vefold and {221001} along twofold vectors nearly coincide with {022} spots. Table 1 summarizes the important features of this orientation relationship. Three of the other twofold axes will nearly coincide with h110i axes. Three of the ®vefold axes will be close to cubic h110i and other
SINGH and TSAI: CRYSTALLOGRAPHY AND SOLIDIFICATION BEHAVIOUR
Fig. 5. Diraction/nanodiraction patterns from the icosahedral matrix and particles along (a, b) twofold axis of the icosahedral matrix, (c, d) ``D'' axis of icosahedral matrix, (e, f) ®vefold axis, and (g, h) threefold axis of the matrix.
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three ®vefold axes at cubic h113i. It is important to note here that the angle between vectors 011* and 112* in this zone axis is about 728. It is also interesting to note that in this orientation relationship all the three cubic h100i axes coincide with the icosahedral ``D'' zone axes, which also have a squarelike appearence (Fig. 5(c) and (d)). This orientation
relationship gives the ``best ®t'' between the cubic and icosahedral phase studied here since more of the major icosahedral axes coincide with more of the similar cubic ones. Seven of the ``D'' axes will coincide with h221i and four with h112i. Six of the icosahedral threefold axes will coincide with h123i and one is near a h111i.
Fig. 6. (a, b) Selected area electron diraction of a decagonal matrix grain in a twofold G orientation and the microdiraction from a lead particle in it. The particle is in [111] orientation. (c, d) Electron diraction from a matrix grain in twofold H orientation and microdiraction from a particle in it. The particle is in [100] orientation. (e) Microdiraction from a lead particle in a grain oriented near its tenfold axis. The particle is in [110] orientation. Periodic (tenfold) reciprocal vectors are marked ``A''.
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Fig. 7. A high resolution electron micrograph showing a lead particle along a twofold axis of the icosahedral matrix. The arrows mark the directions of important planes in the matrix. The particle displays a set of (022) lattice fringes parallel to a set of ®vefold planes of the matrix.
Figure 9(a) shows an orientation relationship where a cubic h111i axis is parallel to an icosahedral twofold axis. If the two phases are rotated about this common axis by 908, so that another of the two icosahedral twofold directions is now coincident with a cubic h022i* direction, then another orientation relationship OR2 is obtained, shown in Fig. 9(b). Important coincident axes are summarized in Table 1. In this relationship a ®vefold axis
will lie in between a [233] and a [122] axis along a common h022i* direction. As observed in the diffraction patterns of Fig. 5(e) and (f), this h022i* direction is parallel to one of the twofold vectors in the ®vefold zone axis of the icosahedral phase. Other ®vefold axes coincide with, or are near to, two h112i axes, one h100i axis and two h315i axes. Only one each of h110i, h111i and h112i cube axes coincide with icosahedral twofold axes. Thus the
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SINGH and TSAI: CRYSTALLOGRAPHY AND SOLIDIFICATION BEHAVIOUR
Fig. 8. (a) (Continued opposite).
registry of the major axes between the two phases is poor in comparison to the relationship OR1. The formation of this relationship therefore must be dictated by nucleation on a ®vefold plane, since the interplanar angle in the cubic h122i are characterized by 728. In the orientation relationship OR3 shown in Fig. 9(c), a cubic threefold axis [111] coincides with an icosahedral threefold axis as observed in diraction patterns of Fig. 5(g) and (h). As shown in
Table 1, three each of twofold icosahedral axes coincide with cubic h112i and h123i axes. Both of these cubic axes can be considered to approximate the indices h1tt2i of the icosahedral twofold axes. Three of the ®vefold axes coincide with cubic h012i, which can be considered to approximate the ®vefold axis indices h01ti. Three of the icosahedral ``D'' axes are close to cubic h320i axes, which approximate the icosahedral ``D'' indices ht10i. Three other icosahedral threefold axes coincide with cubic h210i,
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Fig. 8. (a) Lattice image of a lead particle in a decagonal matrix grain in twofold G orientation. The upper part of the particle shows moire fringes, while the lower part shows lattice fringes of [111] orientation. (b) Magni®ed image of a part of the interface between particle and the matrix. Periodic direction is marked ``A''. Arrowheads mark regions showing good coherency across the interface.
whose indices approximate the icosahedral threefold axis indices ht210i. Three of the h111i axes are close to threefold axes and all of the h100i axes are near twofold axes. Thus the orientation relationship OR3 corresponds to the orientation relationship between an icosahedral phase and a cubic approximant to the icosahedral phase. Many common coincidence conditions are common between dierent orientation relationships, as observed in Table 1. Figure 9(d) is a stereogram showing the orientation relationship of the particles with the decagonal matrix. In the case of the icosahedral phase matrix it was shown that the orientation relationship OR1 is the best ®t between the particles and the matrix. The unique orientation relationship between the decagonal matrix and the lead particles corresponds to OR1 and therefore con®rms it being the best ®t. It will be shown here that in the case of the decagonal matrix this ®t is much better, leading to a single orientation relationship. 4.2. Epitaxy An interface between a quasicrystal and a crystal cannot be coherent but only semicoherent since the structure on one side of the interface is periodic and on the other side is quasiperiodic. This will also produce interfacial strain which may be considerable in many cases. Due to very dierent structures
of the two phases, it has been observed here that the solidifying lead particle chooses an orientation with the best ®tting local structure on the plane of the matrix phase acting as a substrate. There are mainly three planes of the icosahedral phase oered to the lead droplets as substrateÐthe ®vefold, twofold and threefold planes. It would appear that the lead particles solidify with their {111} planes at the interface when solidifying on the twofold or threefold planes, and a plane near to {122} when solidifying on a ®vefold plane. This could not be con®rmed from high resolution micrographs as the interface could not be observed clearly. The {111} planes of lead have lowest energy [5]. In all the orientations observed, the common factor is the registry of {211111} ®vefold and {221001} twofold planes with {022} planes. The {211111} planes have a planar spacing of about 2.1 AÊ and the {221001} planes of about 2.0 AÊ, while the {022} planes have an interplanar distance of about 1.75 AÊ in the bulk phase. There is a small angular mismatch when the icosahedral twofold and {111} planes are at the interface. The angle between two ®vefold directions is 63.48 while that between two h022i* directions is 608. Several solidi®cation peaks in the DSC cooling curves indicate several dierent nucleation sites on the matrix, with signi®cant dierences in surface
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Fig. 9. Stereograms showing orientation relationships (a) OR1, (b) OR2 and (c) OR3, observed between the icosahedral matrix and the lead nanoparticles and (d) between decagonal matrix and nano-lead particles (A is tenfold axis and G and H are twofold axes).
energy. The small peak 1 at 310.68C maybe due to nucleation on grain boundaries and edges, which will aord easier nucleation. Of the possibilities considered above, the best match between lead and the icosahedral phase at the interface will be when the threefold planes of both the phases are on the interface. The nucleation would be then easier in this case, and the solidi®cation peak 2 at 306.38C maybe due to nucleation on threefold faces of the icosahedral matrix. The next two possibilities are of nucleation on ®vefold and twofold planes. As shown, the nucleating lead has a tendency to align its h011i* direction with ®vefold or twofold directions of the icosahedral phase. This is satis®ed better if lead nucleates on a twofold plane of the matrix (with its {111} plane on the interface). The exothermic peak 4 must correspond to nucleation on ®vefold planes.
The Al±Cu±Fe icosahedral phase has a tendency to facet on ®vefold planes and therefore a greater number of ®vefold facets are expected. It is thus no surprise that this exothermic peak is the largest. It is emphasized here that further experiments are required to con®rm these arguments. It will now be shown that due to a one-dimensional periodicity, the lattice match between the lead and the decagonal quasicrystal in this orientation is better than the OR1 observed between lead and the icosahedral quasicrystal. Figure 10(a)±(c) show twofold diraction patterns from the Al±Cu± Fe icosahedral phase, an icosahedral twin and the Al±Cu±Co decagonal phase, respectively. As observed in Fig. 10(a), the angle between two ®vefold vectors (X) in the icosahedral phase is 63.48. The twofold vector is denoted Y. In the icosahedral twin, Fig. 10(b), the X and Y directions from the
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Table 1. The three orientation relationships OR1, OR2 and OR3 between the icosahedral matrix and lead nanoparticles Icosahedral axis
Twofold Fivefold ``D'' Threefold Twofold Fivefold ``D'' Threefold Twofold Fivefold ``D'' Threefold
Coincident or close cubic major axis (in parantheses are number of such matches) Orientation relationship OR1 h111i (3), h110i (3), h112i (1) h110i (3), h113i (3) h100i (3), h112i (4), h122i (7) h123i (6), near h111i (1) Orientation relationship OR2 h111i (1), h110i (1), h112i (1), h014i (2) near h122i (1), h100i (1), h112i (2), h315i (2) h110i (3), h120i (2), h140i (2), h113i (3), h122i (2), h223i (2) h112i (1), h133i (1), near h123i (3) Orientation relationship OR3 h112i (3), h123i (3) h012i (3) h122i (3), near h320i (3) h111i (1), h210i (3)
twin contributions are denoted by primes. The common ®vefold vector (X±X') has become a tenfold axis [21]. It is observed that the Y' axis of the twin is close to the X axis of the matrix and the X' axis of the twin is close to the Y axis of the matrix. In a decagonal phase the X±X' direction becomes periodic X. Introduction of a periodicity along the tenfold axis has two eects in the decagonal phase, as observed in Fig. 10(c). The closeby X and Y' (and X' and Y) directions coincide, denoted as XY. The angle between X and XY directions is 608. Thus when lead particles are oriented with the [111] axis parallel to this twofold axis, the angular match is better (as compared to the icosahedral phase in which the angle between X directions is 63.48). Due to this angular match, the decagonal axis is exactly parallel to a h011i axis, as compared to the case of the icosahedral matrix where ®vefold axes were only nearly parallel to h011i axes. The second eect of the one-dimensional periodicity is on the epitaxial relationship between the substrate and the nucleus. It was observed in the case of the icosahedral matrix that the orientation of the nucleating lead was chosen such that its h022i* directions preferred to coincide with ®vefold and twofold vectors. In the case of the decagonal phase it is seen that these two vectors coincide, into a vector pointing towards XY in Fig. 10(c). Thus a very good epitaxial relationship is expected (and is indeed found) between the nucleating solid lead and the decagonal matrix, and nucleation on either of the several facets will lead to the same orientation of the lead. This is observed in the lattice image in Fig. 8(b). In places marked ``X'' and ``Z'' with arrowheads a good lattice match across the interface is clearly observed.
Fig. 10. Comparison of the twofold patterns from (a) Al± Cu±Fe icosahedral phase (®vefold vectors marked X and a twofold vector Y), (b) icosahedral twofold pattern with contributions from a twin (marked with primes) and (c) twofold G pattern from Al±Cu±Co decagonal phase (tenfold axis marked X). The angle corresponding to 63.48 between ®vefold vectors (marked X) in the icosahedral phase becomes 608 (between X and XY) in the decagonal phase.
4.3. Contact angles and nucleation site density DSC results have been used to calculate the contact angles for nucleation and the catalytic site density using classical nucleation theory. The catalytic site density estimations are, however, often unrealistically low. The values of contact angles and catalytic site densities estimated in various systems are summarized by Goswami and Chattopadhyay in a
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table [14]. Such unrealistic values are attributed to a breakdown of the classical nucleation theory for nanometric sized particles [5, 11]. Goswami and Chattopadhyay [15], however, argue that this is not due to a breakdown of classical nucleation theory at low undercoolings but due to variations in the microscopic details of the surfaces oered by the matrix for heterogeneous nucleation. Eorts to ®t a single value to these dierent sites can lead to unrealistic results. Nevertheless, here estimations are made for the lead solidi®cation on the quasicrystalline phase for comparison with other systems. A method for estimating the contact angle of a solidifying liquid droplet on a substrate is described by Moore et al. [5]. At any temperature T below the melting point, the fraction of solid particles Z is given by dZ I 1 ÿ Z dt
1
where I is the nucleation frequency within each particle, which is given by [22, 23] I A expÿB= Tm ÿ T 2 T
2
where Tm is the particle melting point A Nc kT=hexp ÿQ=kT
3
where Nc is the number of catalytic sites per particle and k and h are Boltzmann's and Planck's constants, respectively, Q is the activation energy for transporting a particle atom across a particle solid± liquid interface. B Ks3 T 2m f y=kL2
4
where K is a shape factor equal to 16p/3 for a hemispherical shaped nucleus, s the particle solid±liquid surface energy, f y 2 ÿ 3 cos y cos3 y=4, y the contact angle at the substrate±solid/liquid particle triple point as in the classical heterogeneous nucleation theory, and L the particle latent heat of solidi®cation. From equations (1) and (2) it can be seen that ln dZ=dt= 1 ÿ Z varies linearly with 1= Tm ÿ T 2 T, with a slope and intercept of ÿB and ln A, respectively. In the icosahedral matrix case the solidi®cation is not on one single plane of the matrix and therefore there is no singular value of contact angle. For each of the three facets in the icosahedral phase that has been considered above, there is a value for y. To obtain each of the values of y, each of the three most prominent peaks in the DSC cooling curve (Fig. 3) is considered as a separate peak since each of these peaks is assumed to correspond to solidi®cation in a speci®c nucleation site. Thus ln dZ=dt= 1 ÿ Z vs 1= Tm ÿ T 2 T is plotted for each peak. Results can be aected by overlap of peaks, but since y is calculated from the slope of this plot, considerations of the tops of these peaks
is enough. The solidi®cation rate dZ/dt at each temperature is directly proportional to the height of the exothermic peak and the fraction of solid particles at any temperature is given by the partial integration of the exothermic peak. These plots are shown in Fig. 11(a)±(c). The central portion of each of these plots is linear. This linearity is eected at the peak ends perhaps due to overlaps from neighbouring peaks. These plots give contact angles y of 7.55, 10.45 and 9.158 for peaks 2, 3 and 4 (in Fig. 3), respectively. As expected, the level of undercooling will depend on the contact angle. A lower angle will make nucleation easier. Thus for the solidi®cation peak 2 the contact angle is lower (7.558) than peak 3 (10.458). The y for peak 4 (9.158), however, is slightly lower than that of peak 3. This is because of the shape of the peak, since dZ/dt and Z are measured from the height and the area of the peak. Kim et al. [7] and Goswami and Chattopadhyay [15] have shown that with lowering of undercooling the solidi®cation peaks become sharper (with the same area). It is observed in Fig. 3 that while peaks 2 and 3 do not dier much in undercooling levels, peak 3 is sharper and peak 2 is ¯atter. For the DSC cooling curve solidi®cation peak of lead in the decagonal matrix shown in Fig. 4(b), the ln dZ=dt= 1 ÿ Z vs 1= Tm ÿ T 2 T plot is shown in Fig. 11(d). The value of constant B from this ®gure gives a contact angle value of 3.18. The catalytic nucleation sites per particle Nc are calculated to be of the order of 10ÿ9 for the icosahedral matrix and 10ÿ7 for the decagonal matrix, which are unrealistically low, but are comparable to the calculated values in the case of lead solidi®cation in crystalline matrices. In case of lead solidi®cation in aluminium it is 10ÿ6 [5], in copper it is 10ÿ12 [9] and in zinc 10ÿ9 [11]. Kim et al. [7] and Goswami and Chattopadhyay [15] show that at a given undercooling the contact angle does not change signi®cantly with Nc. Kim et al. [7] show that the change in particle size does not change undercooling very signi®cantly. The size of embedded lead particles studied here was fairly uniform. An increase in the size range of particles, or a size range of y only broadens the exothermic peaks slightly [7]. Thus the strongest correlation of y is with the undercooling, and thus it is reasonable to compare the y calculated here to those reported in other systems. Goswami and Chattopadhyay [14] have tabulated values of undercooling (DT) with calculated contact angles and nucleation site densities from various works. The calculated y of solidifying lead in an aluminium matrix is 218 (DT = 228C) [5], in copper it is 48 (DT = 0.68C) [9] and in zinc it is 238 (DT = 328C) [11]. The DT vs y for various systems is plotted in Fig. 12. Since examples are taken from other particles too, like tin and bismuth, the DT is plotted as a percentage. A fairly good agreement between DT and y is observed. The y values
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Fig. 11. Plot of ln dZ=dt= 1 ÿ Z vs 1= Tm ÿ T 2 T for the DSC solidi®cation exothermic peaks numbered (a) 2, (b) 3 and (c) 4 in Fig. 3 for the icosahedral matrix and (d) for the decagonal matrix.
obtained in the case of the icosahedral matrix are lower in comparison to crystalline systems. It is observed that in the case of the icosahedral quasicrystalline matrix, for the same undercooling levels the y values are smaller as compared to those obtained for crystalline matrices. In other words, for the same y, the undercooling required for nucleation of lead on the quasicrystal will be much higher. This could be due to a poor registry expected at the interface between crystal and quasicrystal. However, other factors such as the chemical component of the interface also play an important part in determining the contact angle.
2.
3.
5. CONCLUSIONS
Solidi®cation of lead by heterogeneous nucleation on icosahedral and decagonal quasicrystalline substrate has been studied by embedding lead particles in a matrix of the icosahedral phase by rapid solidi®cation. A study by DSC and TEM leads to the following conclusions: 1. The lead particles embedded in the icosahedral matrix had a size of about 70 nm and are faceted mainly on ®vefold, twofold and threefold planes of the matrix phase. The particles embedded in the decagonal phase matrix were lens shaped,
4.
5.
¯attened in the quasiperiodic plane, with a diameter of up to 150 nm and thickness 30 nm. DSC traces for the icosahedral matrix showed a sharp lead melting peak but very ¯at solidi®cation exotherm with several small peaks, suggesting several nucleation sites for solidi®cation. DSC traces for lead in decagonal matrix exhibit a sharp melting and an identical solidi®cation peak. Three orientation relationships between the icosahedral matrix and the particles were determined, which is selected by the site on which the nucleation occurs. Of the three orientation relationships one (OR1) gives the best ®t between the particle and the matrix lattice, while another (OR3) is similar to the orientation relationship of an icosahedral phase with its cubic approximant. In all these orientations, a preference of lead {022} reciprocal spots to coincide with icosahedral {211111} spots (along the ®vefold direction) and {221001} spot (along the twofold direction) is observed. In the case of the decagonal matrix only one orientation relationship was found, which corresponds to the orientation relationship OR1 of lead in the icosahedral matrix.
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Fig. 12. Plot of calculated values of nucleation contact angles y against observed undercooling reported for various systems. The undercooling is plotted as per cent for normalization of various systems: 1, Cu±(Pb) [9]; 2, Al±(Pb) [5]; 3, Zn±(Pb) [11]; 4, Al±(In) [4]; 5, Al±(Cd) [8]; 6, Al±(Sn) [6]; 7, Zn±(Bi) [13]; 8, Al±(Bi) [15]; 9, decagonal quasicrystal±(Pb) (present work); 10±12, icosahedral quasicrystal±(Pb) (present workÐcorresponding to DSC peaks 2, 3 and 4, respectively, in Fig. 3).
6. One-dimensional periodicity of the decagonal phase leads to a better lattice match with the particles. Due to this, nucleation at many dierent possible sites in the matrix will lead to the same orientation relationship. 7. The calculated contact angles for heterogeneous nucleation for three dierent solidi®cation nucleation sites in the icosahedral matrix are 7.55, 10.45 and 9.158, corresponding to undercooling levels of 22.7, 25.4 and 26.48C, respectively. In the decagonal matrix, the calculated contact angle is 3.18 which occurs at an undercooling level of 58C. These angles are smaller in comparison to those reported for calculations for lead nucleation in cubic matrices at the same undercooling levels. AcknowledgementsÐThis work is supported by Core Research for Evolutional Science and Technology, Japan Science and Technology Cooperation.
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5. Moore, K. I., Zhang, D. L. and Cantor, B., Acta metall., 1990, 38, 1327. 6. Kim, W. T. and Cantor, B., J. Mater. Sci., 1991, 26, 1266. 7. Kim, W. T., Zhang, D. L. and Cantor, B., Metall. Trans., 1991, 22A, 2487. 8. Zhang, D. L., Chattopadhyay, K. and Cantor, B., J. Mater. Sci., 1991, 26, 1531. 9. Kim, W. T. and Cantor, B., Acta metall. mater., 1992, 40, 3339. 10. Kim, W. T. and Cantor, B., Acta metall. mater., 1994, 42, 3045. 11. Goswami, R., Chattopadhyay, K., Kim, W. T. and Cantor, B., Metall. Trans., 1992, A23, 3207. 12. Goswami, R. and Chattopadhyay, K., Acta metall. mater., 1994, 42, 383. 13. Goswami, R. and Chattopadhyay, K., Acta metall. mater., 1995, 43, 2837. 14. Goswami, R. and Chattopadhyay, K., Phil. Mag. Lett., 1995, 72, 411. 15. Goswami, R. and Chattopadhyay, K., Acta mater., 1996, 44, 2421. 16. Singh, A. and Tsai, A. P., Phil. Mag. Lett., 1998, 77, 89. 17. Tsai, A. P., Inoue, A. and Masumoto, T., Japan. J. appl. Phys., 1987, 26, L1505. 18. Singh, A. and Ranganathan, S., Scripta metall. mater., 1991, 25, 409. 19. Audier, M., Guyot, P. and Brechet, Y., Phil. Mag. Lett., 1990, 61, 55. 20. Hiraga, K., Lincoln, F. J. and Sun, W., Mater. Trans. JIM, 1991, 32, 308. 21. Singh, A. and Ranganathan, S., Acta metall. mater., 1994, 43, 3539. 22. Turubull, D., J. appl. Phys., 1950, 21, 1022. 23. Cantor, B. and Doherty, R. D., Acta metall., 1979, 27, 33.