observed incubation periods. By applying classical theory of homogeneous nucleation, it is found that the density of quenched-in nuclei will be very small in the ...
Acta metall, mater. Vol. 39, No. 5, pp. 925--936, 1991 Printed in Great Britain. All rights reserved
0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc
ISOTHERMAL CRYSTALLIZATION KINETICS OF Ni24Zr76 AND Ni24(Zr-X)76 AMORPHOUS ALLOYS G. GHOSHI", M. CHANDRASEKARAN:~ and L. D E L A E Y Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B 3030, Belgium (Received 26 March 1990; in revised form 31 July 90)
Abstract--Isothermal devitrification kinetics of the melt-spun Ni24Zr76 and Ni24(Zr-X)76amorphous alloys has been studied by means of DSC, optical microscopy, TEM and X-ray diffraction. These amorphous alloys exhibit eutectic crystallization. Isothermal devitrification kinetics, as obtained by DSC, has been analyzed in terms of Kolmogorov-Johnson-Mehl-Avrami (KJMA) model. The classical KJMA plots exhibit significant non-linearity from the beginning of the transformation. Such non-linearity has been attributed to a number of factors such as surface-induced crystallization, anisotropic growth of crystals, impingement effects (hard and soft) and variation of the nucleation rate during devitrification etc. In these alloys, the steady state nucleation frequency is established after a well-defined transient period. A quantitative analysis of the transient period has been performed in terms of non-steady state nucleation theory and the theoretically estimated transient times are found to be consistent with the experimentally observed incubation periods. By applying classical theory of homogeneous nucleation, it is found that the density of quenched-in nuclei will be very small in the absence of heterogeneous nucleation. Rrsamr--La cindtique de la drvitrification isotherme d'alliages emorphes Ni24Zr76 et Ni24(Zr-X)76obtenus par trempe sur roue a 6t6 6tudire par DSC, microscopic optique, MET et diffraction des rayons X. Ces alliages amorphes prrsentent une cristallisation eutectique. La cinrtique de drvitrification isotherme drterminre par DSC a 6t6 analysre ~i l'aide du modrle de Kolmogorov-Johnson-Mehl-Avrami (KJMA). Les courbes KJMA classiques pl'rsentent une non-linrarit6 importante drs le drbut de la transformation. Un tel comportement a 6td attribu6 ~i plusieurs facteurs tels que la cristallisation induite par la surface, la croissance anisotrope des cristaux, les effets d'obstacles (durs et mous), la variation de la vitesse de germination pendent la d6vitrification etc . . . . Darts ces alliages, on drtermine la frrquence de germination stationnaire aprds une prriode transitoire bien drfinie. On analyse quantitativement la prriode transitoire :i partir de la throrie de la germination non stationnaire; les temps transitoires estimrs throriquement sont compatibles avec les pdriodes d'incubation que l'on observe exprrimentalement. En appliquant la throrie classique de la germination homogrne, on trouve que la densit6 de germes provoqurs par la trempe devrait &re trrs faible en l'absence de germination h&rrogrne. Zusammenfassung--Mittels DSC, optischer Mikroskopie, Elektronenmikroskopie und Rrntgenbeugung wird die Kinetik der isothermen Entglasung der schmelzgesponnenen amorphen Legierungen Ni24ZrT6 und Ni24(Zr-X)76 untersucht. Diese amorpben Legierungen zeigen eutektische Kristallisation, Die Kinetik, ermittelt mit DSC, wird mit dem Kolgomorov-Johnson-Mehl-Avrami-Modell (KJMA) analysiert. Die klassischen KJMA-Diagramme weisen vom Anfang der Umwandlung an eine bedeutende Nichtlinearit/it auf. Diese Nichtlinearit/it wird einer Anzahl yon Faktoren, wie oberfl/icbeninduzierter Kristallisation, anisotropem Kristallwachstum, Auftreffeffekten (harte oder weiche) und Ver/inderungen in der Keimbildungsrate w/ihrend der Entglasung zugeschrieben. In diesen Legierungen wird die station/ire Keimbildungsfrequenz nach einer wohldefinierten iibergangsperiode erreicht. Diese iibergangsperiode wird quantitativ mit der Theorie der nichtstatiomiren Keimbildung analysiert; die theoretisch ermittelten iibergangszeiten find mit den experimentell beobachteten Inkubationszeiten vertr/iglich. Aus der klassischen Theorie der homogenen Keimbildung folgt, daft die Dichte der eingeschreckten Keime ohne heterogene Keimbildung sehr klein ist.
1. INTRODUCTION Technological applications of metallic glasses require that such materials to be thermally stable with time and temperature during use. Scientifically, kinetics o f 1"Present address: The Department of Materials Science and Engineering, The Technological Institute, Northwestern University, Evanston, IL 60208, U.S.A. ~:Present address: Solid State Physics Laboratory, Lucknow Road, Delhi 110054, India.
crystallization is equally important in understanding the atomic processes involved in the formation of crystalline phases. The thermal stability of the amorphous alloys can be defined by the crystallization temperature, crystallization mode, activation energy and the driving force for crystallization. While the latter is dictated by the thermodynamics of the alloy concerned, the first three can be exprimentally studied by differential scanning calorimetry (DSC), electrical resistivity, transmission electron microscopy (TEM), 925
926
GHOSH et al.: CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS
hardness measurements and X-ray diffraction techniques. Generally, crystallization kinetics of amorphous alloys is studied in two ways: (i) dynamic crystallization at constant linear heating rates which can be analyzed by Kissinger method [1] or Ozawa method [2], (ii) isothermal crystallization kinetics which can be analyzed in terms of KolmogorovJohnson-Mehl-Avrami (KJMA) method [3-5]. For in-depth understanding of the crystallization process, it is necessary to employ a combination of the above experimental techniques so as to obtain activation energy for crystallization, nucleation rate, growth rate, growth dimension etc. In this investigation, the effect of micro-additions ( ~ 1 at.%) of two transition metals (Mo and Ti) and two metalloids (Si and P) on the isothermal crystallization kinetics of Ni24Zr76 amorphous alloy has been studied. The micro-mechanism of the crystallization process has been studied by DSC, optical microscopy and TEM and X-ray diffraction. The results of the dynamic crystallization behaviour of these alloys have been reported elsewhere [6]. 2. EXPERIMENTAL Ribbons of 2-3 mm wide and 25-30/zm thick were produced by "chill-block melt spinning" technique in which the melt in a quartz tube was forced through a 0.8 mm diameter nozzle onto a 150 mm diameter copper wheel rotating with a tangential velocity of 35 m/s. The entire operation was carded out in an argon atmosphere. The chemical compositions of the melt-spun ribbons, as determined by electron probe microanalysis, are listed in Table 1. Isothermal "crystallization kinetics were studied in a Du-Pont DSC 910 cell coupled with computer controlled thermal analyzer (TA 9900) having system control and data acquisition capabilities. Isothermal runs were carded out at 5 different temperatures for each alloy. During isothermal tests the DSC cell was continuously purged with high-purity Ar (3 l/h) and the temperature was controlled within + 0.15°C. Isothermal annealing of the amorphous alloys were also carded out in salt-bath after wrapping the specimens with Zr-foils and encapsulating them in quartz tubes partially filled with Ar. The temperature of the salt-bath was controlled within + 1.5°C. Partially crystallized samples were etched with a solution containing glycerin, HNO3 and HF in the ratio of 20:1 : 1 for metallographic observations. The thin foils for TEM were prepared by dual jet electropolishing using an electrolyte of methanol (80%) and Table
1. C h e m i c a l c o m p o s i t i o n s (in a t . % ) o f solidified r i b b o n s d e t e r m i n e d b y E P M A
Alloy No.
Ni
Zr
Mo
1 2 3 4 5
23.9 24.0 24.5 24.1 24. I
76.1 74.9 74.2 74.7 74.8
. 1.1 ----
Ti .
. -1.3 ---
the
as-
Si
P
. --1.1 --
---1.2
Fig. 1. Bright field TEM micrograph of the Ni24Zr76 amorphous alloy and the corresponding diffraction pattern. perchloric acid (20%) at 223 K. The specimens were examined in a JEOL200CX TEM operated at 200 kV. 3. RESULTS 3.1. As-solidified ribbons
The amorphous nature of the as-cast ribbons were confirmed both by X-ray diffraction and TEM. The X-ray diffraction gave broad diffuse peaks, which were typical of amorphous alloys. The scattering vector (4n sin0/2) in all cases were found to be 0.251 _+ 0.001 nm -I which is in very good agreement with that reported by Buschow et al. [7]. This corresponds to an average interatomic distance of about 0.307 nm. Figure 1 shows a typical TEM bright field micrograph and the corresponding diffraction pattern, confirming the amorphous nature of the as-cast NiuZr76 alloy. No trace of crystallinity could be detected, in any of the as-spun alloys, either by X-ray diffraction or by TEM. 3.2. Isothermal crystallization behaviour
The crystallization kinetics under isothermal conditions was followed by means of DSC. Some typical DSC thermograms obtained, after crystallization NiuZr76 amorphous alloy, at different annealing temperatures are shown in Fig. 2 along with the baseline used to integrate the peak. For all the alloys and temperature range investigated here only one peak was obtained under isothermal conditions. This is in contrast to the results of Kolb-Telieps [8] who reported two DSC peaks for an amorphous Ni:4Zr76 alloy, the first one was claimed as due to the formation of (w-Zr) and the second one was due to the formation of (,~-Zr) + NiZr2. The isothermal crystallization kinetic curves of NiuZr76 amorphous alloy, as shown in Fig. 3, are typically sigmoidal in nature. The presence of a significant incubation period or an effective time lag, z, can also be noticed. The incubation period was estimated by measuring the time from the instant the DSC cell had reached the required temperature till the onset of the crystallization. This satisfies the operational definition of z given by Christian [9].
GHOSH
CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS
et al.:
927
nuclei) or constant, and (b) isotropic growth rate is either linearly varies with time t or as t °5. Once the steady state nucleation rate is established, the equation (1) takes the form C
X = 1 - e x p [ - k (t - z)"]
(2a)
v
or
o "1-
ln[--ln(l - X)] = m ln(t - z) + In k
K
o
lb
lg
Time
(mini
g
2b
25
Fig. 2. DSC thermograms associated with the crystallization of NiuZr76 amorphous alloy at different annealing temperatures. A quantitative assessment of the isothermal crystallization kinetics can be made by applying the concepts first developed by Kolmogorov [3] and later independently derived by Johnson and Mehl [4] and Avrami [5]. The general form of crystallization kinetics in terms of transformed fraction (X) can be expressed as In(1
- X) = A o
- z)q dz
UPI~(t
(1)
where A0 is the shape factor of the growing particle, p and q are constants related to the dimensionality of growth and growth law respectively, U is the isotropic growth rate, Iv is the nucleation rate per unit volume, t is the total time of isothermal annealing and z is the incubation period for nucleation. In order to facilitate the integration of equation (1) the following usual assumptions are made: (a) nucleation rate is either zero (i.e. crystallization takes place due to the growth of pre-existing 1.0
,
I
!
--.oxe~ 0.8
(
-
i
,
; K
o !
!
where E= is the apparent activation energy describing the overall crystallization process, T is the temperature, and ko is a rate constant. The apparent activation energy, Eo, has contributions from both the activation energy, of nucleation EN and that of growth Eo. Thus, the effective activation energy for crystallization, Eo will be ( E a / m ) . As can be seen from equation (2b), by plotting l n [ - l n ( l - X ) ] vs l n ( t - z ) one would expect to obtain a straight line whose slope gives the KJMA exponent. Such a procedure has been used extensively in order to understand the crystallization behaviour of metallic glasses. The conventional KJMA plot of Ni24aZr74.TSiH alloy at different temperatures are shown in Fig. 4 over the full range of volume fraction transformed. It is obvious that deviations from linearity occur from the beginning of crystallization. Nevertheless, one can obtain an average KJMA exponents from these plots with correlation coelficients greater than 0.975 and these are listed in Table 2.
Y
-2
h.J
--4
--6
qO Time
Isothermal
(3)
E~/RT)
6q6.25K '
,, 0.2
3.
k (T) = k o exp( -
2
,- 0.4 ._o
Fig.
where m is called the KJMA exponent which characterizes the time dependence of the nucleation rate, the time dependence of the one-dimensional growth velocity and the dimensions of growth; k is a thermally activated rate constant representing both nucleation and growth rates. Accordingly, k may be expected to exhibit an Arrhenius law over a finite temperature range
•
~c 0.6
5
(2b)
q5
20
25
50
I
3,
4
I
i~I
_
8
tn E{t-r }Csl~]
(min}
transformation
-82
632 K 636.5 K 641.5 K 646.5 K 650.75 K I I 6 7
curves
of
amorphous alloy at different temperatures.
Ni24ZrT~
Fig. 4. KIMA plot of Ni24.,Zr74.vSi,.l amorphous alloy at different isothermal test temperatures.
928
GHOSH et al.: CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS Table 2. Average KJMA exponents and incubation periods for different alloys at different isothermal test temperatures Test Average ICIMA Incubation Calculated transient Alloy No. temperature (K) exponents, m period (s), • time (s), ro 1
2
3
4
5
Alloy No. 1 2 3 4 5
616.25 622 627 632 637 617 622 627 632 637.25 634 639 641.5 644 649.5 632 636.5 641.5 646.5 650.75 629.5 634.25 639.25 644.25 649.5
2.91 3.20 2.81 3.21 3.32 2.58 2.56 2.71 2.40 2.60 2.31 2.47 2.41 2.10 2.60 2.69 2.64 2.35 2.68 2.43 2.20 2.69 2.63 2.66 2.42
670 351 245 95 78 966 519 350 156 86 1610 905 710 608 315 840 501 241 154 89 1620 940 581 343 222
451 241 142 85 51
Table 3. Activation energies of crystallization for different alloys Activation energy E, kJ/tool For overall crystallization For viscous flow From to5 in From kinetic Kissinger Ozawa From incubation isothermal test rate constant k(T) method method period, T 256 254 259 257 356 276 242 282 279 395 286 279 281 277 341 302 266 303 299 404 296 255 299 295 339
3.3. Activation energy o f crystallization T h e time for the onset o f crystallization for all the alloys c a n be described by a n A r r h e n i u s - t y p e o f e q u a t i o n for a thermally activated process = Zk e x p ( E , / R T)
(4)
where xk is a c o n s t a n t a n d E, is the activatio.n energy for n o n - s t e a d y state time. Figure 5 shows the straight line relationship between In(z) a n d the reciprocal o f the absolute temperature. The activation energies, E,, derived from this plot are listed in T a b l e 3. In this investigation, the activation energy for crystallization has been evaluated by several methods. F o r example, the activation energy for crystallization c a n be evaluated f r o m the d a t a o f isothermal crystallization kinetics. This was d o n e b y applying A r r h e n i u s - t y p e of relation to the time required for a fixed a m o u n t o f t r a n s f o r m a t i o n (say 5 0 % ) at different temperatures, i.e.
to.5 = to exp(Ec/RT)
(5)
where t0.s is the time required for 5 0 % t r a n s f o r m ation. T h e plots o f In t0.5 vs 1/T for different alloys are s h o w n in Fig. 6, the slopes o f which give the
activation energies. Similarly, Fig. 7 shows the plots o f In k ( T ) vs I / T [after e q u a t i o n (3)] for all the alloys, a n d the activation energies c a n also be o b t a i n e d f r o m the slopes o f these plots. Similarly, the activation energy for crystallization were also determined by Kissinger a n d O z a w a methods, f r o m the c o n t i n u o u s heating experiments [6]. T h e activation energies o b t a i n e d by different m e t h o d s are listed in T a b l e 3.
'///
B
t
5 5
1.5:3
1.55
1.57 I/T,
X l O -3
1.59
1.61
1.63
K -1
Fig. 5. Arrhenius plot of In ~ vs I/T for different alloys. The numbers indicate the alloy number.
GHOSH et a/.: CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS
reflected in the isothermal crystallization behaviour as will be seen in Section 4. As can be seen in Table 3, the activation energies for non-steady state time is much higher than those derived from the overall crystallization kinetics. This indicates that two different mechanisms are involved during non-steady state and crystallization periods. Higher activation energy suggests that co-operative movement of all the atomic species non-steady state period, which is probably necessary for the nucleation of the crystalline phases. It is worth mentioning that the present activation energies for the non-steady state time are similar in magnitude as that predicted by Chen [13] for stable glasses.
5
5
• 0 [] A 0
4
152 15.
Ni24 Zr76 Niz4(Zr-Si)76 Ni24(Zr-P) 76 Niz4(Zr _Ti )76 Ni24(Zr- M0)76
1; 8 l/T,
1;6o 1' 2
3.4. Microstructural evolution
x l 0 -s K -1
Fig. 6. Arrbenius plot of t0 5 vs
]/T
for different alloys.
The activation energy for crystallization, of Ni~Zr76 amorphous alloy, obtained in the present investigation (258.8 kJ/mol) is in good agreement with the previous reports of Toloui et al. [10] (255 _ 5 kJ/mol) and Altounian et al. [11] (251 kJ/mol). But it is somewhat higher than that reported by Frahm [12] (234.4 kJ/mol), which can be accounted for due to slight variation in composition or difference in processing conditions or both. As can be seen in Table 3, the activation energies determined from equation (5) agree well with those obtained from the dynamic crystallization experiments. However, the activation energies obtained from equation (3) are less than those obtained from other methods. Also, it can be seen in Table 3, that the activation energy of crystallization due to the addition of the two transition metals (Mo and Ti) are almost the same. Similar is the case for the addition of the two metalloids (Si and P). The increase in activation energy of crystallization due to the addition of a third element can be due to the change in the temperature dependences of nucleation and growth rates and also the transient effect. All these effects are clearly
-14.5
929
3.4.1. Optical microscopy. Microstructures of some samples partially crystallized in the DSC were examined by optical microscope. Figure 8(a) shows an optical micrograph of Ni24Zr76 amorphous alloy, partially crystallized at 627 K for 7 min in DSC which corresponds to about 15% of crystallization. The presence of surface-induced crystallization along with the randomly distributed bulk crystallized regions can be seen. In other words, initial stages of crystallization takes place by the simultaneous surfaceinduced and bulk crystallization processes. It may be noticed in the micrograph that the surface-induced crystallization accounts for most of the crystallized fraction compared to that of the bulk crystallization. This indicates that the rate of surface crystallization is faster than that of bulk crystallization. With the progress of devitrification, the trend is reversed and bulk crystallization is found to account for the increasing amount of the crystallized fraction as compared to the surface-induced crystallization. It is worth mentioning that even after removal of about 5/zm from both the surfaces by electropolishing, the samples exhibited similar crystallization kinetics as
-2
-16.5
-18.5
j=
-20.5 , , , I ~ , , I 1.53 1.55 1,57 l/T,
, , 1.59
X10 -3
1.61
1.63
K -1
Fig. 7. Arrhenius plot of In K(T) vs I/T for all the alloys. The numbers indicate the alloy number.
Fig. 8. Optical micrographs of partially crystallized Ni~ZrT~ amorphous alloy, (a) annealed at 627 K for 7 rains in DSC, and (b) annealed at 623 K for 16 rain in evacuated and sealed quartz tube partially filled with Ar.
930
GHOSH et al.: CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS
well as surface-induced crystallization behaviour. Furthermore, as shown in Fig. 8(b), the microstructure of a partially crystallized Niz4Zr7~ sample, which was sealed in an evacuated quartz tube partially filled with Ar, and annealed at 623 K for 16 min, shows mostly bulk crystallization with very little surface-induced crystallization. One should bear in mind that in Fig. 8(a) and (b), the crystallized regions do not represent individual crystallite as the size of these crystallites is well beyond the resolution limit of optical microscopes. The presence of surfaceinduced crystallization (those performed in DSC) could be taken to mean that surface oxidation results in a creation of nucleation sites on both the surfaces. Also, the above results suggest that the surface crystallization does not arise from the melt-spinning process itself. 3.4.2. Transmission electon microscopy. The results of TEM examination of Ni~4Zr76 amorphous alloy partially crystallized at 623 K for 8, 12, 16 and 20 min are shown in Fig. 9(a)-(d) respectively. Sparsely distributed non-spherical crystallites in the amorphous matrix can be observed in Fig. 9(a). During subsequent crystallization, nucleation was found to take place both away and near to the existing crystallites with the later mechanism dominating. This is possibly due to the higher driving force for nucleation, because of higher supersaturation, at the crystallite/amorphous interface. The resulting irregularly shaped crystallite regions can be observed in Fig. 9(b), which also shows a wide distribution of crystallite size. With further transformation, the crystallite regions appeared spherical shaped [marked A in Fig. 9(d)], not due to any isotropic growth of the crystals but due to repeated nucleation at the existing crystallite/amorphous interface. The consequence of such a nucleation process can have two-fold effects: (a) the crystallite can interfere each others growth due to the overlapping of the diffusion fields (or soft inpingement), and (b) physical or hard impingement as well. As mentioned in the previous section, the samples annealed in evacuated quartz tubes, from which the TEM specimens were made, showed very little surface crystallization. Since the TEM specimens were prepared by dual jet technique, the thin areas can be considered to be almost from the middle, in the through-thickness direction, of the ribbons. So the crystallite regions, as shown by the TEM micrographs, are bulk-nucleated.
phase transformations. As assumed in the classical nucleation theory, during this transient period a steady state embryo distribution (characteristic of the annealing temperature) is built up. Recently, transient nucleation effects in condensed systems has
4. DISCUSSIONS
4.1. Quantitative analysis of transient period The presence of a well-defined transient period is noticed during isothermal devitrification process of the alloys investigated and they are listed in Table 2 as a function of temperature. Such transient period or effective time lag is a common feature in isothermal
Fig. 9. Bright field TEM micrographs of partially crystallized Ni24Zr76amorphous alloy after annealing at 623 K for (a) 8rain, (b) 12min, (c) 16min and (d) 20min.
GHOSH et al.: CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS been discussed by several authors [14L16] on a quantitative basis. The kinetics of formation of sub-critical clusters, for a one component system, has been numerically analyzed by Kelton et al. [14]. However, because of computational hardships, the analytical expressions provided by Kelton et aL have been used here for theoretical estimation of transient time. Similar strategy was also adopted by Thompson et aL [17] for the estimation of transient times in Au-based amorphous alloys. In absence of the quenched-in clusters, Kelton et al. showed that the transient nucleation rate, I(t), can be described by the analytical expression due to Kashchiev [18], i.e.
The molecular transport rate ~.. across the liquid/crystal interface is given by
~., = vA. = 4vn'2/3 = 256v72 9Ag2
.=x ~ (-1)"exp(-n:t/dp)]
8k Ty % = 3rAg 2.
(7)
where k is the Boltzmann constant, T is the absolute temperature, AG. is the Gibbs energy of formation of a cluster containing n molecules, n* is the number of molecules in a critical size cluster, and ~.. is the rate at which molecules add to the cluster. As shown by Kashchiev [18], the effective time lag % can be expressed in terms of the characteristic time ~b, and is given by n2~b ~o = - -
(8)
6
In Kashchiev's formalism the Gibbs energy of formation AG. of a cluster as a function of number of atoms, n, in the cluster can be expressed as [17]
AG, = nAg + A.y
(9)
where Ag is the Gibbs energy difference per atom between liquid and crystalline phases and can be written as AGv 17 Ag = - -
(10)
NA
where AGv is the Gibbs energy difference per unit volume between the liquid and crystalline phases, N A is the Avogadro's number and 17is the molar volume. A, is the number of atoms on the surface of cluster of n atoms and y is the interfacial energy per surface atom and is given by [17] / 3 ]7 ~2/3 =l--/ nl/sa (11) )' \4NA/ where o is the molar interfacial energy. Setting ~(AG,)/dn = 0, we obtain n* = ( -
3Ag~]-3 -~---y
The atomic jump rate v can be expressed as
9Ag + 51273.
(16)
where D. is the diffusivity which can be expressed in terms of viscosity, ~/, via Stokes-Einstein relationship i.e.
kT 3naoq
D. = - -
(17)
ao is the molecular diameter and 2 is the interatomic distance. It is usual practice [19] to take the molecular diameter, ao, as the ionic diameter. In order to estimate % from equation (15), the following parameters must be known: (a) the difference in Gibbs energy per unit volume AGv between the liquid and crystalline phases; (b) the interracial energy, tr, between the amorphous and crystalline phases; and (c) viscosity, of the amorphous phase, which controls the atomic transport kinetics. Among the presently used alloys, the thermodynamic parameters are known only for the phases of the Ni-Zr binary system. So, we have calculated the transient times at different temperatures for the Ni•Zr76 amorphous alloy only. The Gibbs energies of the liquid and crystalline phases have been published by Saunders [20]. However, using his thermodynamic data the Ni-Zr phase diagram could not be reproduced. Consequently, a complete assessment of the Ni-Zr system has been done and the thermodynamic parameters of all the phases have been derived [21]. In this assessment [21], the excess Gibbs energies of the solution phases have been expressed by Redlich-Kister polynomial. Liquid-crystal interracial energy is calculated following the model proposed by Spaepen and coworkers [22, 23]. They have shown that the crystalmelt surface energies can be described by
oloASfT o" = 11'''''''''~ (N^ 172)
(18)
(12)
where ASf is the molar entropy of fusion, and ~0 = 0.86 for f.c.c, and h.e.p, crystals [22, 23]. The composition dependence of o results from those of 17 and ASr which can be assumed to vary linearly with the molar volumes and entropies of fusion of pure components respectively i.e.
(13)
17 -----XA 17A+ (1 -- XA) 17a
and [O:(AG.)/0n 2]. = .. =
(15)
(6)
where I~ is the steady state nucleation rate and (p is the characteristic time given by [18]
8kT (a = - n:~..[d2AG./dn2]....
(14)
where v is the jump frequency of the atoms across the liquid/crystal interface. Substituting equations (7), (13) and (14) in equation (8), we obtain
v = 6D./22
I(t)=I~[l+2
931
(19a)
932
GHOSH et al.: CRYSTALLIZATION KINETICS OF AMORPHOUS ALLOYS
and
15
ASc = X^AS~ + (1 - XA)AS~.
By analyzing a large number of published data, recently Battezzatti and Greer [24] concluded that the temperature-dependence of the liquid viscosity of the glass forming systems can be expressed in terms of Vogel-Fulcher-Tammann form ~/(T) = F exp ( T - - - ~ )
(20)
where F and 0 are constants, and T ' is the temperature at which the excess cortfignrational entropy of the free volume is zero. In this study F, 0 and T' are determined from the following set of conditions: (i) that the viscosity at the liquidus temperature is 5.904 x 10-3Nsm -2, a value determined from the procedure described by Battezzatti et al. [24]. So we have F exp(-123:~_ T ; ) = 5.904 x 10-3
(21)
(ii) Battezzatti et aL [24] have proposed that at the measured Tg, as determined from the specific heat capacity on heating, a reasonable value of viscosity is 101°Nsm -2. This gives us F exp(-6350---~) = 10 '°
(22)
(iii) The activation energy of the viscosity, E,, at Ts is 355.5 kJ/mol. The apparent activation energy of viscosity is given by [17]
(@),.=ro
