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The solidification behaviour of Bi particles embedded in an Al matrix
Pergamon 0956-7151(95)00332-O
Acto makr. Vol. 44, No. 6, Pp. 2421-2429, 1996 Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
THE SOLIDIFICATION BEHAVIOUR OF Bi PARTICLES EMBEDDED IN AN Al MATRIX R. GOSWAMI Department
of Metallurgy,
Indian
and K. CHATTOPADHYAY Institute
of Science,
Bangalore,
560 012, India
(Received 23 May 1995; in revised form 8 August 1995) Abstract-This
paper reports
heterogeneous
nucleation
behaviour
of embedded
nano- and micron-scaled
Bi particles in an aluminium matrix synthesized by rapid solidification and chill casting. It is shown that only a fraction of the nanoparticles that have a truncated octahedron shape show orientation relation with the matrix. The solidification of the embedded Bi melt was found to take place over a wide temperature range. The micron sized particles do not show any solidification peak while a broad shallow peak can be observed for nano-scaled particles. The observed broad differential scanning calorimetry exotherm has been deconvoluted into two peaks, one sharp and the other broad in shape. The exotherms are also simulated using classical theory of heterogeneous nucleation using a parametric approach and matched with the experimental one. It is shown that the broad peak can be best explained by a spread of contact angles during nucleation while the sharp peak can be fitted with a single contact angle. The latter is most likely associated with particles showing orientation relation and represents nucleation of Bi on the 111 plane of aluminium. Analysis of the present result emphasizes the important role of the local microscopic structures of the liquid-solid interface in promoting the catalytic nucleation efficiency of the Al matrix.
1. INTRODUCTION
2. EXPERIMENTAL
Solidification of metals and alloys mostly occurs by heterogeneous nucleation and subsequent growth of the solid phase. In order to study heterogeneous nucleation, the nucleant on which it is solidifying should be very well characterized. The nucleation process is sensitive to the presence of extraneous particles. There exist various ways of avoiding extraneous particles [ 11. Rapid solidification of binary monotectic alloys result in the formation of a large number of nanometre size particles embedded in a matrix [2-41. These provide ideal systems to study heterogeneous nucleation. Recently, heterogeneous nucleation from melt has been studied in detail on nanometre size particles of Pb [3], In [4], Cd [5] and Sn [6] embedded in an Al matrix and Pb [7] and Bi [8] in a Zn matrix. These studies have revealed that the hemispherical cap model of classical heterogeneous nucleation works very well when the catalytic efficiency is not good. A break down of the spherical cap model is predicted [3,7] when the substrate acts as an efficient catalyst. This suggests that the catalytic sites play an important role in controlling the mechanism of nucleation. The aim of the present investigation is to synthesize and characterize nanometric dispersion of Bi in an Al matrix and use it as a model material for the study of heterogeneous nucleation during solidification designed to understand the nature and role of catalytic sites controlling the nucleation process.
Al-10 wt% Bi (1.4 at% Bi) was prepared by induction melting from 99.999% purity Al and 99.999% Bi in a chamber which was evacuated and purged with Ar. A part of the as-cast alloy was rapidly solidified on Cu wheel rotating with a velocity of 10 m/s. The microstructure was characterized using a transmission electron microscope (JEOL 2000 FX II) operated at 200 kV. The sample for electron microscopic observation was prepared in a Gatan twin gun ion mill thinner using Ar ions with a sputtering angle of 15”, voltage of 6 kV and current of 0.25 mA for each gun. The solidification and melting behaviour of Bi particles were studied using a Perkin Elmer DSC 2C differential scanning calorimeter (DSC) under dynamic Ar atmosphere.
2421
3. RESULTS
Figure l(a) shows a large number of nanometre size Bi particles embedded in an Al matrix in melt spun Al-B1 alloy. The corresponding distribution of Bi particles is shown in Fig. l(b). The size of the Bi particles ranges from 20 to 100 nm. A polarized light micrograph of as-cast sample of Al-Bi is shown in the inset of Fig. l(a). The Bi particle size in the as-cast condition ranges from 5 to 15 pm. Figure 2(a) is the diffraction pattern from an as-spun sample consisting of matrix and the second phase reflections. The corresponding indexed pattern is shown
GOSWAMI
and CHATTOPADHYAY.
SOLIDIF~ICATIOh
BEHAVIOIIK
(4
Al - Ri
1
10
20
30
40 Radius
50
60
70
83
I nm)
1.(a) A bright field electron micrograph of a rapidly solidified Al-Bi sample showing the nanometre size Bi particles embedded in an Al matrix. (A polarized light optical micrograph is shown in the inset.) (b) The histogram showing the corresponding size distribution of the rapidly solidified Al-Bi sample. Fig.
schematically in Fig. 2(b). Two types of reflections could be observed from the Bi particles. One set is oriented randomly to form broken Debye rings indicating no orientation relationship with the matrix. The other set shows well arranged spot patterns indicating the existence of orientation relationship. Only a fraction of the Bi particles shows orientation relationship with the matrix given by
101ii,, im%,
.
Figure 2(c) gives the corresponding stereogram. The Bi particles in the samples are faceted. The shape of
the Bi particles can be determined by analysing the bright field image of the particles at the symmetric zone axes of the matrix. Figures 3(a) and (b) show the morphology of Bi particles in the four- and two-fold zone axes of Al, respectively ([loo] and [l lo] zones). The morphology of the Bi particle was found to be a truncated octahedron consisting of { 1 1 1} and { 100) planes of Al (Fig. 4). Careful examination of micrographs near [ 1 lo] type of zones also reveals the existence of small facets of 110 planes. In contrast, the micron size particles in the as-cast sample are almost spherical. The Bi particles were not found to be oriented with respect to one another. The differential scanning calorimetric plots showing the melting and cooling cycles, respectively, are
GOSWAMI and CHATTOPADHYAY:
0
SOLIDIFICATION
BEHAVIOUR
2423
a
Of
oii2
2,
;;;o.” .
.
0 Iii
..
[Oli lpd
ioo
.
0
‘*\
lOi
i0
\
170 2
[zoii] Bi
i2io
loii. 011
(c) Fig. 2. (a) The selected area diffraction pattern from the matrix and the second phase particles. (b) The corresponding indexed schematic pattern. (c) A stereogram showing orientation relationship with the matrix.
shown in Fig. 5 for as-cast samples. The heating cycle yields a sharp endothermic peak near the eutectic temperature corresponding to the melting of the Bi particles. No sharp exothermic peak corresponding to the solidification of the Bi could be observed. This indicates that the solidification events must take place over a large temperature range, thus completely smearing the DSC peaks of the individual events. In
order to know the range of solidification temperature and the amount of Bi solidified at each temperature interval in the as-cast sample, the DSC scan during cooling is stopped at certain predetermined temperatures and reheated beyond the eutectic temperature to get the sharp endotherm of melting (Fig. 5). The areas of these endothermic peaks give us an estimate of the fraction of the particles solidified upto the
2424
GOSWAMI
Fig. 3. The bright
and CHATTOPADHYAY:
field electron
SOI_IDIFI~‘ATION
BEHAVIOR.
micrograph showing the faceted Bi particle: (b) viewed along [OI I]*,.
temperature when the cycling is reversed. The analysis of the results indicate that 42% of the Bi droplets have solidified when the sample is cooled to 470 K with a maximum at 547 K. The fraction solidified upto 420 K is still only 70% and the complete solidification of the remaining 30% of the melt requires an additional 50 K undercooling. Figure 6 shows the DSC traces of the melt-spun sample. The solidification exotherm is flat. However, a broad peak beginning at 410 K and ending at 375 K could be discerned. From the endothermic peaks of the heating experiments from different temperatures, it is seen that only 2% of the Bi droplets have solidified in the range 543-440 K. Thirty percent of the Bi droplets have solidified when the sample is cooled to 410 K, which is the onset of the detectable
(a) viewed along [OOI],, and
solidification. The observed broad exotherm corresponds to the solidification of 70% of the Bi droplets. Figure 7 shows the fraction transformed during
$
F 1 t E 2 6
4 Exo
I 390
I
I 430
I
I 470
TEMPERNURE
Fig. 4. A schematic diagram showing the morphology of the Bi particles bounded by the {OOI}, { 11 I} and (01 I} planes of the matrix.
K
I
I 510
(1
1 550
ll
590
( K )
Fig. 5. DSC traces showing melting and solidification of the as-cast sample. (a), (c), (e) and (g) are the DSC traces during heating showing the endothermic peaks due to melting of Bi particles. (b), (d) and (f) are the DSC traces during cooling showing the exotherms corresponding to the solidification of Bi particles.
GOSWAMI and CHATTOPADHYAY:
SOLIDIFICATION
360
2425
BEHAVIOUR
440
520
TEMPERATURE
600 (K)
Fig. 7. A plot showing the fraction of the particles solidified as a function of temperature for chill cast (a) and rapidly solidified alloys (b) heated above the melting point of Bi prior to cooling.
360
400
440
480
520
560
TEMPERATURE ( K)
Fig. 6. DSC traces showing melting and solidification of the as-spun sample. (a), (c), (e), (g) and (i) are the DSC traces during heating showing the endothermic peaks due to melting of Bi particles. (b), (d), (f) and (h) are the DSC traces during cooling showing the exotherms corresponding to the solidification of Bi particles. solidification as a function of temperature for both as-cast and melt-spun alloys. The melt-spun sample was heat treated at 600 K for 30 h in the solid + liquid phase field. The heat treated melt-spun sample was then heated in the DSC upto 580 K and subsequently cooled to obtain a solidification peak (Fig. 8). A very flat DSC trace with a small peak detectable at 400 K is observed.
orientation relationship. This can be rationalized in terms of the lattice disregistry (20.6%) with respect to Al in the close-packed direction of Al and Bi. The composite diffraction patterns indicate that the common symmetry element between the matrix and the second phase particle is i. The equilibrium shape of the Bi particle in Al matrix will be governed by the intersection point group symmetry in the observed relation [9] and the number of Bi particle variants will be the ratio of the order of the point group symmetry of the matrix (m3m) to that of the intersection point group symmetry (1) [lo]. Thus, there are expected to be 24 orientational Bi particle variants. The total number of orientational Bi particle variants observed is one. Further, the shape of the experimentally
1
4. DISCUSSION 4.1. Morphology The melt-spun microstructure shows nanometric Bi particles embedded in Al matrix. The distribution shows a peak at 15 nm. However, the distribution is skewed towards a higher size with 15% of the particles having a size SO-60 nm and a small fraction (2.5%) having a size between 70 and 80nm. It is important to note that the selected area electron diffraction experiments indicate that not all the Bi particles show orientation relation with the matrix. In general, it is seen that the smaller particles show the
Fig. 8. DSC traces of rapidly solidified and heat treated (600 K, 30 h) alloy showing the melting and solidification of Bi particles. (a) Melting of Bi particles. (b) Solidification of Bi particles.
2426
GOSWAMI
and CHATTOPADHYAY:
observed Bi particle is truncated octahedron (Fig. 4) which conforms to the point group symmetry of the matrix (m3m). The shape of the Bi particles and the number of experimentally observed orientational variants suggest that Bi particles retained the shape of the molten Bi droplets. Similar observation of the shape of the second phase particle is reported in Zn-Pb [7], AI-In [4], Al-Sn [6], Al-Cd [5] and Zn-Bi [8] alloys. The Bi droplets formed at higher temperature will be equilibrated rapidly. Assuming self-diffusivity of Bi in the liquid state to be approximately 10m9m2/s [l l] and a cooling rate of IO4 K/s in the temperature interval of 933-544K [12], the diffusion distance is calculated to be 10 pm. The self-diffusivity of Bi in solid state is very low and anisotropic. The upper limit of diffusivity is 3 x 10-‘* m2/s at 541 K [13]. The diffusivity in the temperature range of 410-375 K will be much lower. This is the temperature range within which 70% of the Bi particles have solidified in the as-spun condition. Using the above self-diffusivity value and cooling rate, the diffusion distance calculated over the temperature range of 540-300 K turned out to be between 1 and 2 nm. This is less than the minimum observed particle size. This suggests that the kinetic constraints of mass transfer provide a major restriction for changing the particle shape from m3m symmetry to T symmetry. 4.2. Solidl@ation
of Bi particles
It is observed from the DSC traces that the solidification of Bi in as-cast and melt-spun samples has taken place over a range of temperatures and in both cases the solidification exotherms are flat. The nature of solidification remains similar even after annealing for 30 h at 600 K. This clearly shows that nucleation of solid Bi in embedded liquid is taking place over a very wide temperature range. The heterogeneous nucleation rate per droplet is given by the following equation [ 141 I = (NKT/h).exp(-AG,/kT).exp(-AG*/kT).
(1)
Here N is the number of nucleation site, AG, is the activation energy of diffusion and AC* is the activation energy for critical nucleus. The latter is given by AG* = (16rr/3)(rJ3/(AG,)*).f(6)
(2)
where AG,, is the volume free energy change for nucleation, 0 is the solid-liquid surface energy of Bi which is taken as 83 mJ/m2 [15]. The f(0) is a function of contact angle given by f(B)=(2-3cosO+Cos3,)/4.
(3)
In the hemispherical cap model of classical nucleation the contact angle is given by fJ, - 02 = 0 cos 0 where 0,) CS~ and strate-liquid and
(4)
c are the substrate-solid, subsolid-liquid surface energies,
SOLIDIFICATION
BEHAVIOL;R
respectively. Clearly Ihr an amsotropic solid hke I%. g, and 0 will depend upon the crystallography of the situation. Therefore. the contact angle for nucleation at different facets of Bi will be different, In the present case Al acts as the substratc for the nucleation of Bi during solidification. During cooling below the eutectic temperature, solidification of Bi takes place on the available sites of the surrounding Al. As discussed earlier, the available sites are { 1 I 1). { 110) and j 100) facets of Al. There are, however. additional heterogeneous sites such as curved faces. edges, corners and matrix dislocation ending or originating at the matrix particle interface. The broad exothermic peak in the melt-spun sample and the flat curve in the as-cast sample. together with the knowledge of the orientation of the Bi particles determined in the present investigation, suggest that Bi nucleates on various sites of the matrix. This is in contrast to the sharp solidification exotherms observed in Al-Pb [3], AI-Cd [5], Al-In [4], Zn-Pb [7] and ZnBi [8]. Definite orientation relationships could be observed in these systems for all particles. It is instructive to analyse in detail the nucleation behaviour of Bi in the melt-spun sample where a broad but distinct exotherm can be detected. Comparing the distribution of particles and the fraction solidified in the temperature interval covered by the exotherm as well as the consideration of high undercooling achieved prior to the nucleation led us to assume with some justification that the exotherm represents the solidification behaviour of smaller particles which have a mode of 15 nm in the distribution curve. We first compute the nature of the DSC curve followed by attempts to match the calculated curve with the observed one. It is assumed that the solidification is controlled by a classical heterogeneous nucleation process and once the nucleation has taken place, the particle solidifies instantaneously due to the high growth rate in the undercooled liquid [7]. The procedure followed by Kim et a/. [16] has been adopted in the present case. Briefly, the rate of solidification is given by [16-181 i = 1(1 -Z)
(5)
where Z is the fraction of the solidified droplets. The Bi droplet solidification rate is proportional to the measured excess heat flow Q and therefore, to the height of the DSC exotherm at each temperature. The measured excess heat flow Q is written as Q = (4/3)nr*‘H,M,n,,i
+f(T)
(6)
where r* is the average size of the droplets, H, is the latent heat on volume basis, M, is the mass of the specimen and nBi is the number of particles per unit mass of the specimen. The use of equations (5) and (6) allows us to calculate the solidification exotherm. In order to evaluate nucleation rate 1, we need to also know of contact angle 0 and the catalytic site density N. Figure 9 shows a series of exotherms for different
GOSWAMI and CHATTOPADHYAY:
0
380
400 TEMPERATURE
420 (K
4 4(
)
Fig. 9. The calculated solidification exotherms obtained at different values of contact angles. The number of nucleation sites per particle is assumed to be 10.
contact angles assuming a constant site per particle (N, = 10). It is observed that the contact angle has a strong influence on the undercooling of Bi droplets. The onset, peak and end change sharply with the increase in contact angle. Further, the solidification exotherm becomes slightly broader with the increase in contact angle. However, at a given contact angle the undercooling does not vary significantly with the nucleation sites per particle [Fig. 10(b)]. Figure 10(a) shows calculated solidification peak temperatures as a function of contact angle for different values of nucleation sites per particle. The results indicate that the increase in undercooling is quite considerable with the increase in contact angle, whereas undercooling decreases with the increase in sites per particle [Fig. 10(b)]. Considering the experimentally observed undercooling for the exothermic peak, the parametric study indicates that the contact angle will be in the range of 72-78” assuming a wide range of site per particle. The width of the calculated solidification exotherm (T,,,,t-T,,d) at a cut-off level of 0.01 mW is shown as a function of contact angle (Fig. 11). It is observed that in all cases, this value is below 10 K. This is less than the experimentally observed width of the exotherm and suggests that a single contact angle is not operative. To explain the DSC peak, we therefore need to consider spread of contact angles during solidification of Bi droplets. To analyse the nucleation behaviour further, we have adopted a matching procedure between the experimental and calculated exotherms. The broad exotherm obtained from the melt-spun sample shown in Fig. 7 is digitized and deconvoluted. This procedure yields two curves having a broad and a sharp peak, respectively. The broad curve can be matched with the calculated one using a range of contact angles (72-80”). A normal distribution of contact angles is assumed. The sharper and relatively smaller exotherm can easily be fitted with a single contact angle of 80” and the match is found to be reasonable. Figure 12 shows the observed peaks after deconvolution and the superimposed simulated peaks. The fitting is less perfect in the case of broad exotherm.
SOLIDIFICATION
BEHAVIOUR
2427
The good fitting of small exotherm with a single contact angle suggests that a fraction of Bi droplets have solidified on a facet and experienced highest undercooling. This confirms the results of the electron diffraction experiment which indicate the existence of an orientation relation for some of the particles. The orientation relation most likely developed when a fraction of Bi particles nucleated on closed-packed { 111) planes of Al surrounding the liquid. The variable contact angles associated with the broad peak indicate that the nucleation of the majority of the droplets takes place at different sites. This trend is also observed in melt-spun + heat-treated samples, The long time equilibration annealing at 600 K makes the particle achieve a more or less similar kind of bounding surface. Thus the behaviour is intrinsic to
_
N:, ..-
N
=,o
---~
N=,W
--
N=,GoO
-.-
N = 10000
(a) 170
W Fig. 10. (a) The undercooling calculated at the maxima of the DSC exotherms as a function of contact angle. The number of nucleation sites per particle is indicated for each curve. (b) The undercooling as a function of contact angle as a function of nucleation sites per particle. The contact angle corresponding to each curve is indicated. Nucleation sites per particle are represented in log scale.
2428
GOSWAMI and CHATTOPADHYAY:
SOLIDlPIC’ATION
BEHAVIOUK
10 I
2 50
i
I
I
I
60
70
80
Contact
Angle
90
100
(8)
Fig. 11.The difference in onset and end of solidification temperature calculated from the solidification exothenn plotted as a function of contact angle. The corresponding nucleation sites per particle are indicated.
cr = - 10r#~AT+ RZ’(4 In r+5+ (1 - 4) In(l - 4))
the Al substrate and not due to some nonequilibrium sites frozen during rapid solidification or impurities. The origin of the variable contact angle can either be due to structure and crystallography of the interfaces or chemical inhomogeneities and absorption. It is therefore necessary to analyse the absorption behaviour of the Al-Bi system in order to distinguish between these two possibilities. The role of adsorption in modifying the interfacial energy is well understood [19]. An adsorbed layer can thus affect the process of heterogeneous nucleation by modifying the surface energy. The solid-liquid interfacial energy, assuming the adsorbed layer to be a monolayer, has been obtained recently [20,21]. The analytical expression of the solid-liquid interfacial energy of highly immiscible system is given by
+30m6(1
-$)ThlB+n{$w,+30(1-+)
x r,.4+(1
(7)
-~)u’,+3O~~kla~
where 4 is the interfacial solid fraction, TMAand ThlB are melting points of A and B, AT is the undercooling (TNIB- T), m and n are the fractional coordination numbers parallel and perpendicular to the boundary. w, and w, are the interaction parameters in the solid and in the liquid state. It is seen that the interface solid fraction (4) due to chemical adsorption will increase suddenly below a critical undercooling (AT&. The critical undercooling is obtained by the following equation [21]; Tads= O.ln(M;, - Nj) - 3n(T,,
- T,,).
1
-.
3/3
385
395 TEMPERATUREC
405
.. 415
K 1
Fig. 12. DSC trace showing the matching of the experimental and calculated exotherms. The dots are the experimental points. The broad exotherm is fitted with a range of contact angle (72-80”) and nucleation sites per particle of 10. The experimental DSC exotherm is digitized and deconvoluted.
(8)
GOSWAMI The
undercooling
required
and CHATTOPADHYAY: for critical
adsorption
case is 2716 K which is unrealistic. This suggests that the interfacial energy of Al( 11 l)/Bi(l) and Al(lOO)/Bi(l) will not be modified significantly during solidification. It is also interesting to note here that the difference in the solid and liquid interaction parameter is very high, which indicates that in addition to (111) and (100) facets, the interfacial energy of the curved interfaces will also not be affected significantly by the adsorption. Therefore, the origin of variable contact angle during nucleation is expected to be related to the variation in local microscopic structure of the interface between solid Al and liquid Bi. The microscopic details of the local structure, therefore, play an important role in promoting the heterogeneous nucleation sites during solidification.
in the present
5. CONCLUSIONS (1) The rapid solidification of Al-Bi alloys result in the formation of nanosized dispersoids in Al matrix. (2) The particle size in the as-cast Al-Bi sample ranges from 5 to 15 pm while in the melt-spun sample it ranges from 10 to 100nm. (3) The shape of the particle is more or less spherical in the as-cast sample, but in the melt-spun sample the morphology is found to be truncated octahedron containing (11 l}, { 1 lo} and { 100) planes of Al. (4) The selected area diffraction pattern obtained from the matrix and the second-phase particle reveals that a small fraction of the particles have orientation relation with the matrix. (5) The solidification of Bi particles takes place over a range of temperature in both cases. The solidification exotherm is very flat while the corresponding melting endotherm is quite sharp. This suggests that nucleation sites for solidification are different for different particles but for melting, sites for all the particles are similar. (6) The detectable exotherm in the as-spun sample consists of a broad and sharp peak. The observed DSC peak is simulated with the help of classical heterogeneous nucleation theory. The sharp peak can be fitted very well with a single contact angle of 80”. This is suggestive of the fact that 30% of Bi droplets solidify on { 11 l} planes of Al.
SOLIDIFICATION
BEHAVIOUR
2429
(7) The broad peak can be fitted assuming a spread in contact angle (72-80”). Further analysis suggests that this effect is not related to adsorption. The result indicates the important role of local microscopic structural state of the liquid-solid interface in catalyzing the solid. Acknowledgements-The authors would like to thank Professors S. Ranganathan and P. Ramachandrarao for many discussions. The work is supported by a grant from the Department of Science and Technology, Govt. of India. REFERENCES 1. J. H. Perepezko, Mufer. Sci. Engng 65, 125 (1984). 2. K. I. Moore, K. Chattopadhyay and B. Cantor, Proc. R. Sot. A414, 499 (1987). 3. K. I. Moore, D. L. Zhang and B. Cantor, Acta metall. 38, 1327 (1990). 4. D. L. Zhang and B. Cantor, Phil. Mug. A 62,557 (1990). 5. D. L. Zhang, K. Chattopadhyay and B. Cantor., J. Muter. Sci. 26. 1531 (1991). 6. W. T. Kim and’ B. Cantor, ‘J. Mater. Sci. 26, 2868 (1991). 1. R. Goswami, W. T. Kim, K. Chattopadhyay and B. Cantor, Metall. Trans. 23A, 3207 (1991). Phil. Mug. Lett. 8. R. Goswami and K. Chattopadhyay, in press. 9. K. W. Westmacott and U. Dahmen, in Interfaces., Structure and Properties (edited by S. Ranganathan,, C. S. Pande, B. B. Rath and D. A. Smith), p. 147. Oxford & IBH, New Delhi (1993). in Solid-Solid Phase 10. J. W. Cahn and G. Kalonji, Transformation (edited by H. I. Aaronson, D. E Laughlin, R. F. Sekerka and C. M. Wayman), p. 3. New York (1982). 11. Smithells Handbook (edited by E. A. Brandes). Butterworths, London (1983). Acta metall. 33, 1813 12. A. G. Gillen and B. Cantor, (1985). 13. W. P. Ellis and N. H. Nachtrieb, J. uppl. Phys. 40, 472 (1969). 14. D. Turnbull and R. E. Cech, J. uppl. Phys. 21, 804 (1950). 15. L. F. Mondolfo, N. L. Parisi and G. J. Kardys, Muter. Sci. Engng 68, 249 (1984-1985). 16. W. T. Kim, D. L. Zhang and B. Cantor, Metall. Truns. 27A, 2487 (1991). 17. W. T. Kim and B. Cantor, Acta metull. muter. 40, 3339 (1992). 18. W. T. Kim and B. Cantor, Acta metall. mater. 42, 3045 (1994). 19. J. W. Gibbs, Scientific Papers, Vol. I, Thermodynamics. Dover, New York (1906). 20. L. Coudurier, N. Eustathopoulos, P. Desre and A. Passerone, Actu mefull. 26, 465 (1978). 21. W. T. Kim and B. Cantor, Acfu metall. muter. 42, 31 IS (1994).
Acto makr. Vol. 44, No. 6, Pp. 2421-2429, 1996 Elsevier Science Ltd Copyright 0 1996 Acta Metallurgica Inc. Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
THE SOLIDIFICATION BEHAVIOUR OF Bi PARTICLES EMBEDDED IN AN Al MATRIX R. GOSWAMI Department
of Metallurgy,
Indian
and K. CHATTOPADHYAY Institute
of Science,
Bangalore,
560 012, India
(Received 23 May 1995; in revised form 8 August 1995) Abstract-This
paper reports
heterogeneous
nucleation
behaviour
of embedded
nano- and micron-scaled
Bi particles in an aluminium matrix synthesized by rapid solidification and chill casting. It is shown that only a fraction of the nanoparticles that have a truncated octahedron shape show orientation relation with the matrix. The solidification of the embedded Bi melt was found to take place over a wide temperature range. The micron sized particles do not show any solidification peak while a broad shallow peak can be observed for nano-scaled particles. The observed broad differential scanning calorimetry exotherm has been deconvoluted into two peaks, one sharp and the other broad in shape. The exotherms are also simulated using classical theory of heterogeneous nucleation using a parametric approach and matched with the experimental one. It is shown that the broad peak can be best explained by a spread of contact angles during nucleation while the sharp peak can be fitted with a single contact angle. The latter is most likely associated with particles showing orientation relation and represents nucleation of Bi on the 111 plane of aluminium. Analysis of the present result emphasizes the important role of the local microscopic structures of the liquid-solid interface in promoting the catalytic nucleation efficiency of the Al matrix.
1. INTRODUCTION
2. EXPERIMENTAL
Solidification of metals and alloys mostly occurs by heterogeneous nucleation and subsequent growth of the solid phase. In order to study heterogeneous nucleation, the nucleant on which it is solidifying should be very well characterized. The nucleation process is sensitive to the presence of extraneous particles. There exist various ways of avoiding extraneous particles [ 11. Rapid solidification of binary monotectic alloys result in the formation of a large number of nanometre size particles embedded in a matrix [2-41. These provide ideal systems to study heterogeneous nucleation. Recently, heterogeneous nucleation from melt has been studied in detail on nanometre size particles of Pb [3], In [4], Cd [5] and Sn [6] embedded in an Al matrix and Pb [7] and Bi [8] in a Zn matrix. These studies have revealed that the hemispherical cap model of classical heterogeneous nucleation works very well when the catalytic efficiency is not good. A break down of the spherical cap model is predicted [3,7] when the substrate acts as an efficient catalyst. This suggests that the catalytic sites play an important role in controlling the mechanism of nucleation. The aim of the present investigation is to synthesize and characterize nanometric dispersion of Bi in an Al matrix and use it as a model material for the study of heterogeneous nucleation during solidification designed to understand the nature and role of catalytic sites controlling the nucleation process.
Al-10 wt% Bi (1.4 at% Bi) was prepared by induction melting from 99.999% purity Al and 99.999% Bi in a chamber which was evacuated and purged with Ar. A part of the as-cast alloy was rapidly solidified on Cu wheel rotating with a velocity of 10 m/s. The microstructure was characterized using a transmission electron microscope (JEOL 2000 FX II) operated at 200 kV. The sample for electron microscopic observation was prepared in a Gatan twin gun ion mill thinner using Ar ions with a sputtering angle of 15”, voltage of 6 kV and current of 0.25 mA for each gun. The solidification and melting behaviour of Bi particles were studied using a Perkin Elmer DSC 2C differential scanning calorimeter (DSC) under dynamic Ar atmosphere.
2421
3. RESULTS
Figure l(a) shows a large number of nanometre size Bi particles embedded in an Al matrix in melt spun Al-B1 alloy. The corresponding distribution of Bi particles is shown in Fig. l(b). The size of the Bi particles ranges from 20 to 100 nm. A polarized light micrograph of as-cast sample of Al-Bi is shown in the inset of Fig. l(a). The Bi particle size in the as-cast condition ranges from 5 to 15 pm. Figure 2(a) is the diffraction pattern from an as-spun sample consisting of matrix and the second phase reflections. The corresponding indexed pattern is shown
GOSWAMI
and CHATTOPADHYAY.
SOLIDIF~ICATIOh
BEHAVIOIIK
(4
Al - Ri
1
10
20
30
40 Radius
50
60
70
83
I nm)
1.(a) A bright field electron micrograph of a rapidly solidified Al-Bi sample showing the nanometre size Bi particles embedded in an Al matrix. (A polarized light optical micrograph is shown in the inset.) (b) The histogram showing the corresponding size distribution of the rapidly solidified Al-Bi sample. Fig.
schematically in Fig. 2(b). Two types of reflections could be observed from the Bi particles. One set is oriented randomly to form broken Debye rings indicating no orientation relationship with the matrix. The other set shows well arranged spot patterns indicating the existence of orientation relationship. Only a fraction of the Bi particles shows orientation relationship with the matrix given by
101ii,, im%,
.
Figure 2(c) gives the corresponding stereogram. The Bi particles in the samples are faceted. The shape of
the Bi particles can be determined by analysing the bright field image of the particles at the symmetric zone axes of the matrix. Figures 3(a) and (b) show the morphology of Bi particles in the four- and two-fold zone axes of Al, respectively ([loo] and [l lo] zones). The morphology of the Bi particle was found to be a truncated octahedron consisting of { 1 1 1} and { 100) planes of Al (Fig. 4). Careful examination of micrographs near [ 1 lo] type of zones also reveals the existence of small facets of 110 planes. In contrast, the micron size particles in the as-cast sample are almost spherical. The Bi particles were not found to be oriented with respect to one another. The differential scanning calorimetric plots showing the melting and cooling cycles, respectively, are
GOSWAMI and CHATTOPADHYAY:
0
SOLIDIFICATION
BEHAVIOUR
2423
a
Of
oii2
2,
;;;o.” .
.
0 Iii
..
[Oli lpd
ioo
.
0
‘*\
lOi
i0
\
170 2
[zoii] Bi
i2io
loii. 011
(c) Fig. 2. (a) The selected area diffraction pattern from the matrix and the second phase particles. (b) The corresponding indexed schematic pattern. (c) A stereogram showing orientation relationship with the matrix.
shown in Fig. 5 for as-cast samples. The heating cycle yields a sharp endothermic peak near the eutectic temperature corresponding to the melting of the Bi particles. No sharp exothermic peak corresponding to the solidification of the Bi could be observed. This indicates that the solidification events must take place over a large temperature range, thus completely smearing the DSC peaks of the individual events. In
order to know the range of solidification temperature and the amount of Bi solidified at each temperature interval in the as-cast sample, the DSC scan during cooling is stopped at certain predetermined temperatures and reheated beyond the eutectic temperature to get the sharp endotherm of melting (Fig. 5). The areas of these endothermic peaks give us an estimate of the fraction of the particles solidified upto the
2424
GOSWAMI
Fig. 3. The bright
and CHATTOPADHYAY:
field electron
SOI_IDIFI~‘ATION
BEHAVIOR.
micrograph showing the faceted Bi particle: (b) viewed along [OI I]*,.
temperature when the cycling is reversed. The analysis of the results indicate that 42% of the Bi droplets have solidified when the sample is cooled to 470 K with a maximum at 547 K. The fraction solidified upto 420 K is still only 70% and the complete solidification of the remaining 30% of the melt requires an additional 50 K undercooling. Figure 6 shows the DSC traces of the melt-spun sample. The solidification exotherm is flat. However, a broad peak beginning at 410 K and ending at 375 K could be discerned. From the endothermic peaks of the heating experiments from different temperatures, it is seen that only 2% of the Bi droplets have solidified in the range 543-440 K. Thirty percent of the Bi droplets have solidified when the sample is cooled to 410 K, which is the onset of the detectable
(a) viewed along [OOI],, and
solidification. The observed broad exotherm corresponds to the solidification of 70% of the Bi droplets. Figure 7 shows the fraction transformed during
$
F 1 t E 2 6
4 Exo
I 390
I
I 430
I
I 470
TEMPERNURE
Fig. 4. A schematic diagram showing the morphology of the Bi particles bounded by the {OOI}, { 11 I} and (01 I} planes of the matrix.
K
I
I 510
(1
1 550
ll
590
( K )
Fig. 5. DSC traces showing melting and solidification of the as-cast sample. (a), (c), (e) and (g) are the DSC traces during heating showing the endothermic peaks due to melting of Bi particles. (b), (d) and (f) are the DSC traces during cooling showing the exotherms corresponding to the solidification of Bi particles.
GOSWAMI and CHATTOPADHYAY:
SOLIDIFICATION
360
2425
BEHAVIOUR
440
520
TEMPERATURE
600 (K)
Fig. 7. A plot showing the fraction of the particles solidified as a function of temperature for chill cast (a) and rapidly solidified alloys (b) heated above the melting point of Bi prior to cooling.
360
400
440
480
520
560
TEMPERATURE ( K)
Fig. 6. DSC traces showing melting and solidification of the as-spun sample. (a), (c), (e), (g) and (i) are the DSC traces during heating showing the endothermic peaks due to melting of Bi particles. (b), (d), (f) and (h) are the DSC traces during cooling showing the exotherms corresponding to the solidification of Bi particles. solidification as a function of temperature for both as-cast and melt-spun alloys. The melt-spun sample was heat treated at 600 K for 30 h in the solid + liquid phase field. The heat treated melt-spun sample was then heated in the DSC upto 580 K and subsequently cooled to obtain a solidification peak (Fig. 8). A very flat DSC trace with a small peak detectable at 400 K is observed.
orientation relationship. This can be rationalized in terms of the lattice disregistry (20.6%) with respect to Al in the close-packed direction of Al and Bi. The composite diffraction patterns indicate that the common symmetry element between the matrix and the second phase particle is i. The equilibrium shape of the Bi particle in Al matrix will be governed by the intersection point group symmetry in the observed relation [9] and the number of Bi particle variants will be the ratio of the order of the point group symmetry of the matrix (m3m) to that of the intersection point group symmetry (1) [lo]. Thus, there are expected to be 24 orientational Bi particle variants. The total number of orientational Bi particle variants observed is one. Further, the shape of the experimentally
1
4. DISCUSSION 4.1. Morphology The melt-spun microstructure shows nanometric Bi particles embedded in Al matrix. The distribution shows a peak at 15 nm. However, the distribution is skewed towards a higher size with 15% of the particles having a size SO-60 nm and a small fraction (2.5%) having a size between 70 and 80nm. It is important to note that the selected area electron diffraction experiments indicate that not all the Bi particles show orientation relation with the matrix. In general, it is seen that the smaller particles show the
Fig. 8. DSC traces of rapidly solidified and heat treated (600 K, 30 h) alloy showing the melting and solidification of Bi particles. (a) Melting of Bi particles. (b) Solidification of Bi particles.
2426
GOSWAMI
and CHATTOPADHYAY:
observed Bi particle is truncated octahedron (Fig. 4) which conforms to the point group symmetry of the matrix (m3m). The shape of the Bi particles and the number of experimentally observed orientational variants suggest that Bi particles retained the shape of the molten Bi droplets. Similar observation of the shape of the second phase particle is reported in Zn-Pb [7], AI-In [4], Al-Sn [6], Al-Cd [5] and Zn-Bi [8] alloys. The Bi droplets formed at higher temperature will be equilibrated rapidly. Assuming self-diffusivity of Bi in the liquid state to be approximately 10m9m2/s [l l] and a cooling rate of IO4 K/s in the temperature interval of 933-544K [12], the diffusion distance is calculated to be 10 pm. The self-diffusivity of Bi in solid state is very low and anisotropic. The upper limit of diffusivity is 3 x 10-‘* m2/s at 541 K [13]. The diffusivity in the temperature range of 410-375 K will be much lower. This is the temperature range within which 70% of the Bi particles have solidified in the as-spun condition. Using the above self-diffusivity value and cooling rate, the diffusion distance calculated over the temperature range of 540-300 K turned out to be between 1 and 2 nm. This is less than the minimum observed particle size. This suggests that the kinetic constraints of mass transfer provide a major restriction for changing the particle shape from m3m symmetry to T symmetry. 4.2. Solidl@ation
of Bi particles
It is observed from the DSC traces that the solidification of Bi in as-cast and melt-spun samples has taken place over a range of temperatures and in both cases the solidification exotherms are flat. The nature of solidification remains similar even after annealing for 30 h at 600 K. This clearly shows that nucleation of solid Bi in embedded liquid is taking place over a very wide temperature range. The heterogeneous nucleation rate per droplet is given by the following equation [ 141 I = (NKT/h).exp(-AG,/kT).exp(-AG*/kT).
(1)
Here N is the number of nucleation site, AG, is the activation energy of diffusion and AC* is the activation energy for critical nucleus. The latter is given by AG* = (16rr/3)(rJ3/(AG,)*).f(6)
(2)
where AG,, is the volume free energy change for nucleation, 0 is the solid-liquid surface energy of Bi which is taken as 83 mJ/m2 [15]. The f(0) is a function of contact angle given by f(B)=(2-3cosO+Cos3,)/4.
(3)
In the hemispherical cap model of classical nucleation the contact angle is given by fJ, - 02 = 0 cos 0 where 0,) CS~ and strate-liquid and
(4)
c are the substrate-solid, subsolid-liquid surface energies,
SOLIDIFICATION
BEHAVIOL;R
respectively. Clearly Ihr an amsotropic solid hke I%. g, and 0 will depend upon the crystallography of the situation. Therefore. the contact angle for nucleation at different facets of Bi will be different, In the present case Al acts as the substratc for the nucleation of Bi during solidification. During cooling below the eutectic temperature, solidification of Bi takes place on the available sites of the surrounding Al. As discussed earlier, the available sites are { 1 I 1). { 110) and j 100) facets of Al. There are, however. additional heterogeneous sites such as curved faces. edges, corners and matrix dislocation ending or originating at the matrix particle interface. The broad exothermic peak in the melt-spun sample and the flat curve in the as-cast sample. together with the knowledge of the orientation of the Bi particles determined in the present investigation, suggest that Bi nucleates on various sites of the matrix. This is in contrast to the sharp solidification exotherms observed in Al-Pb [3], AI-Cd [5], Al-In [4], Zn-Pb [7] and ZnBi [8]. Definite orientation relationships could be observed in these systems for all particles. It is instructive to analyse in detail the nucleation behaviour of Bi in the melt-spun sample where a broad but distinct exotherm can be detected. Comparing the distribution of particles and the fraction solidified in the temperature interval covered by the exotherm as well as the consideration of high undercooling achieved prior to the nucleation led us to assume with some justification that the exotherm represents the solidification behaviour of smaller particles which have a mode of 15 nm in the distribution curve. We first compute the nature of the DSC curve followed by attempts to match the calculated curve with the observed one. It is assumed that the solidification is controlled by a classical heterogeneous nucleation process and once the nucleation has taken place, the particle solidifies instantaneously due to the high growth rate in the undercooled liquid [7]. The procedure followed by Kim et a/. [16] has been adopted in the present case. Briefly, the rate of solidification is given by [16-181 i = 1(1 -Z)
(5)
where Z is the fraction of the solidified droplets. The Bi droplet solidification rate is proportional to the measured excess heat flow Q and therefore, to the height of the DSC exotherm at each temperature. The measured excess heat flow Q is written as Q = (4/3)nr*‘H,M,n,,i
+f(T)
(6)
where r* is the average size of the droplets, H, is the latent heat on volume basis, M, is the mass of the specimen and nBi is the number of particles per unit mass of the specimen. The use of equations (5) and (6) allows us to calculate the solidification exotherm. In order to evaluate nucleation rate 1, we need to also know of contact angle 0 and the catalytic site density N. Figure 9 shows a series of exotherms for different
GOSWAMI and CHATTOPADHYAY:
0
380
400 TEMPERATURE
420 (K
4 4(
)
Fig. 9. The calculated solidification exotherms obtained at different values of contact angles. The number of nucleation sites per particle is assumed to be 10.
contact angles assuming a constant site per particle (N, = 10). It is observed that the contact angle has a strong influence on the undercooling of Bi droplets. The onset, peak and end change sharply with the increase in contact angle. Further, the solidification exotherm becomes slightly broader with the increase in contact angle. However, at a given contact angle the undercooling does not vary significantly with the nucleation sites per particle [Fig. 10(b)]. Figure 10(a) shows calculated solidification peak temperatures as a function of contact angle for different values of nucleation sites per particle. The results indicate that the increase in undercooling is quite considerable with the increase in contact angle, whereas undercooling decreases with the increase in sites per particle [Fig. 10(b)]. Considering the experimentally observed undercooling for the exothermic peak, the parametric study indicates that the contact angle will be in the range of 72-78” assuming a wide range of site per particle. The width of the calculated solidification exotherm (T,,,,t-T,,d) at a cut-off level of 0.01 mW is shown as a function of contact angle (Fig. 11). It is observed that in all cases, this value is below 10 K. This is less than the experimentally observed width of the exotherm and suggests that a single contact angle is not operative. To explain the DSC peak, we therefore need to consider spread of contact angles during solidification of Bi droplets. To analyse the nucleation behaviour further, we have adopted a matching procedure between the experimental and calculated exotherms. The broad exotherm obtained from the melt-spun sample shown in Fig. 7 is digitized and deconvoluted. This procedure yields two curves having a broad and a sharp peak, respectively. The broad curve can be matched with the calculated one using a range of contact angles (72-80”). A normal distribution of contact angles is assumed. The sharper and relatively smaller exotherm can easily be fitted with a single contact angle of 80” and the match is found to be reasonable. Figure 12 shows the observed peaks after deconvolution and the superimposed simulated peaks. The fitting is less perfect in the case of broad exotherm.
SOLIDIFICATION
BEHAVIOUR
2427
The good fitting of small exotherm with a single contact angle suggests that a fraction of Bi droplets have solidified on a facet and experienced highest undercooling. This confirms the results of the electron diffraction experiment which indicate the existence of an orientation relation for some of the particles. The orientation relation most likely developed when a fraction of Bi particles nucleated on closed-packed { 111) planes of Al surrounding the liquid. The variable contact angles associated with the broad peak indicate that the nucleation of the majority of the droplets takes place at different sites. This trend is also observed in melt-spun + heat-treated samples, The long time equilibration annealing at 600 K makes the particle achieve a more or less similar kind of bounding surface. Thus the behaviour is intrinsic to
_
N:, ..-
N
=,o
---~
N=,W
--
N=,GoO
-.-
N = 10000
(a) 170
W Fig. 10. (a) The undercooling calculated at the maxima of the DSC exotherms as a function of contact angle. The number of nucleation sites per particle is indicated for each curve. (b) The undercooling as a function of contact angle as a function of nucleation sites per particle. The contact angle corresponding to each curve is indicated. Nucleation sites per particle are represented in log scale.
2428
GOSWAMI and CHATTOPADHYAY:
SOLIDlPIC’ATION
BEHAVIOUK
10 I
2 50
i
I
I
I
60
70
80
Contact
Angle
90
100
(8)
Fig. 11.The difference in onset and end of solidification temperature calculated from the solidification exothenn plotted as a function of contact angle. The corresponding nucleation sites per particle are indicated.
cr = - 10r#~AT+ RZ’(4 In r+5+ (1 - 4) In(l - 4))
the Al substrate and not due to some nonequilibrium sites frozen during rapid solidification or impurities. The origin of the variable contact angle can either be due to structure and crystallography of the interfaces or chemical inhomogeneities and absorption. It is therefore necessary to analyse the absorption behaviour of the Al-Bi system in order to distinguish between these two possibilities. The role of adsorption in modifying the interfacial energy is well understood [19]. An adsorbed layer can thus affect the process of heterogeneous nucleation by modifying the surface energy. The solid-liquid interfacial energy, assuming the adsorbed layer to be a monolayer, has been obtained recently [20,21]. The analytical expression of the solid-liquid interfacial energy of highly immiscible system is given by
+30m6(1
-$)ThlB+n{$w,+30(1-+)
x r,.4+(1
(7)
-~)u’,+3O~~kla~
where 4 is the interfacial solid fraction, TMAand ThlB are melting points of A and B, AT is the undercooling (TNIB- T), m and n are the fractional coordination numbers parallel and perpendicular to the boundary. w, and w, are the interaction parameters in the solid and in the liquid state. It is seen that the interface solid fraction (4) due to chemical adsorption will increase suddenly below a critical undercooling (AT&. The critical undercooling is obtained by the following equation [21]; Tads= O.ln(M;, - Nj) - 3n(T,,
- T,,).
1
-.
3/3
385
395 TEMPERATUREC
405
.. 415
K 1
Fig. 12. DSC trace showing the matching of the experimental and calculated exotherms. The dots are the experimental points. The broad exotherm is fitted with a range of contact angle (72-80”) and nucleation sites per particle of 10. The experimental DSC exotherm is digitized and deconvoluted.
(8)
GOSWAMI The
undercooling
required
and CHATTOPADHYAY: for critical
adsorption
case is 2716 K which is unrealistic. This suggests that the interfacial energy of Al( 11 l)/Bi(l) and Al(lOO)/Bi(l) will not be modified significantly during solidification. It is also interesting to note here that the difference in the solid and liquid interaction parameter is very high, which indicates that in addition to (111) and (100) facets, the interfacial energy of the curved interfaces will also not be affected significantly by the adsorption. Therefore, the origin of variable contact angle during nucleation is expected to be related to the variation in local microscopic structure of the interface between solid Al and liquid Bi. The microscopic details of the local structure, therefore, play an important role in promoting the heterogeneous nucleation sites during solidification.
in the present
5. CONCLUSIONS (1) The rapid solidification of Al-Bi alloys result in the formation of nanosized dispersoids in Al matrix. (2) The particle size in the as-cast Al-Bi sample ranges from 5 to 15 pm while in the melt-spun sample it ranges from 10 to 100nm. (3) The shape of the particle is more or less spherical in the as-cast sample, but in the melt-spun sample the morphology is found to be truncated octahedron containing (11 l}, { 1 lo} and { 100) planes of Al. (4) The selected area diffraction pattern obtained from the matrix and the second-phase particle reveals that a small fraction of the particles have orientation relation with the matrix. (5) The solidification of Bi particles takes place over a range of temperature in both cases. The solidification exotherm is very flat while the corresponding melting endotherm is quite sharp. This suggests that nucleation sites for solidification are different for different particles but for melting, sites for all the particles are similar. (6) The detectable exotherm in the as-spun sample consists of a broad and sharp peak. The observed DSC peak is simulated with the help of classical heterogeneous nucleation theory. The sharp peak can be fitted very well with a single contact angle of 80”. This is suggestive of the fact that 30% of Bi droplets solidify on { 11 l} planes of Al.
SOLIDIFICATION
BEHAVIOUR
2429
(7) The broad peak can be fitted assuming a spread in contact angle (72-80”). Further analysis suggests that this effect is not related to adsorption. The result indicates the important role of local microscopic structural state of the liquid-solid interface in catalyzing the solid. Acknowledgements-The authors would like to thank Professors S. Ranganathan and P. Ramachandrarao for many discussions. The work is supported by a grant from the Department of Science and Technology, Govt. of India. REFERENCES 1. J. H. Perepezko, Mufer. Sci. Engng 65, 125 (1984). 2. K. I. Moore, K. Chattopadhyay and B. Cantor, Proc. R. Sot. A414, 499 (1987). 3. K. I. Moore, D. L. Zhang and B. Cantor, Acta metall. 38, 1327 (1990). 4. D. L. Zhang and B. Cantor, Phil. Mug. A 62,557 (1990). 5. D. L. Zhang, K. Chattopadhyay and B. Cantor., J. Muter. Sci. 26. 1531 (1991). 6. W. T. Kim and’ B. Cantor, ‘J. Mater. Sci. 26, 2868 (1991). 1. R. Goswami, W. T. Kim, K. Chattopadhyay and B. Cantor, Metall. Trans. 23A, 3207 (1991). Phil. Mug. Lett. 8. R. Goswami and K. Chattopadhyay, in press. 9. K. W. Westmacott and U. Dahmen, in Interfaces., Structure and Properties (edited by S. Ranganathan,, C. S. Pande, B. B. Rath and D. A. Smith), p. 147. Oxford & IBH, New Delhi (1993). in Solid-Solid Phase 10. J. W. Cahn and G. Kalonji, Transformation (edited by H. I. Aaronson, D. E Laughlin, R. F. Sekerka and C. M. Wayman), p. 3. New York (1982). 11. Smithells Handbook (edited by E. A. Brandes). Butterworths, London (1983). Acta metall. 33, 1813 12. A. G. Gillen and B. Cantor, (1985). 13. W. P. Ellis and N. H. Nachtrieb, J. uppl. Phys. 40, 472 (1969). 14. D. Turnbull and R. E. Cech, J. uppl. Phys. 21, 804 (1950). 15. L. F. Mondolfo, N. L. Parisi and G. J. Kardys, Muter. Sci. Engng 68, 249 (1984-1985). 16. W. T. Kim, D. L. Zhang and B. Cantor, Metall. Truns. 27A, 2487 (1991). 17. W. T. Kim and B. Cantor, Acta metull. muter. 40, 3339 (1992). 18. W. T. Kim and B. Cantor, Acta metall. mater. 42, 3045 (1994). 19. J. W. Gibbs, Scientific Papers, Vol. I, Thermodynamics. Dover, New York (1906). 20. L. Coudurier, N. Eustathopoulos, P. Desre and A. Passerone, Actu mefull. 26, 465 (1978). 21. W. T. Kim and B. Cantor, Acfu metall. muter. 42, 31 IS (1994).