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Dendrite growth velocities in highly undercooled, dilute NiC melts
730
Materials Science and Engineering, A133 ( 1991 ) 730-733
Dendrite growth velocities in highly undercooled, dilute Ni-C melts K. Eckler and D. M. Herlach lnstitut fiir Raumsimulation, Deutsche Forschungs- und Versuchsanstalt fiir Lufi- und Raumfahrt, D-5000, Kdln 90 (F.R.G.)
R. G. Hamerton and A. L. Greer University of Cambridge, Department of Materials Science and Metallurgy, Pembroke Street, Cambridge CB2 3QZ (U.K.)
Abstract Dendrite growth velocities have been measured in droplets of dilute Ni-C alloys undercooled by up to 300 K using an electromagnetic levitation technique. For alloys containing 1.7 at.% C and 3 at.% C, the observed behaviour is in good agreement with the predictions of current models. Velocities in pure nickel and a 0.6 at.% C alloy at low undercooling ( < 80 K) are in excess of those expected for the pure material and suggest effects due to the presence of an impurity.
1. Introduction
Owing to its containerless nature, electromagnetic levitation melting permits high undercoolings in liquid droplets prior to crystal nucleation, giving the possibility of extremely high initial growth rates, which can be monitored by observation of the sample surface. Furthermore, nucleation can be externally induced at a well-determined temperature, enabling correlation between growth rate measurements and the droplet undercooling [1]. A number of models (e.g. [2, 3]) predict the growth rate V of a dendrite (approximated by a paraboloid of revolution with tip radius R) in an undercooled liquid. The total undercooling A T is written as [2] AT=ATt+ATc+ATr+ATk
(1)
The thermal and solutal undercoolings A T t and ATc are obtained from steady-state solutions to the relevant diffusion equations [4], while the curvature undercooling A Tr is the lowering of the equilibrium temperature at the tip due to the Gibbs-Thomson effect. Although the actual tip temperature is lower than the equilibrium value by the kinetic undercooling A Tk, the difference is small for the velocities considered here if the interracial kinetics are collision-limited [5]. However, at large V there is the additional possibility that higher than equilibrium solute concentra0921-5093/91/$3.50
tions may be trapped in the growing solid [6]. With less partitioning, conditions of essentially solute-diffusion-controlled growth at low undercooling may give way to thermal control at higher undercooling, reflected in a transition in the relative contributions of A T t and ATc to A T. An equation such as (1), however, gives only the product VR at a given A T, hence a second criterion, termed the marginal-stability hypothesis, in which the dendrite tip is assumed to have the largest possible radius while remaining morphologically stable, is used to extract a single solution for V [7]. For ease of calculation, the marginally-stable tip radius is associated with the minimum wavelength of perturbation which would grow on a planar interface subject to the same thermal and solutal gradients as exist at the dendrite tip [2, 3]. The computational procedure followed in this work is, firstly, to select a dendrite velocity V. The tip radius can then be calculated under the marginal-stability assumption using an iterative technique, and finally the tmdercooling is obtained by summing the contributions in eqn. ( 1 ). 2. Experimental
The technique of measuring dendrite growth velocities in highly undercooled, levitated metallic droplets has been described elsewhere [1].
731
Only brief details are included here. The levitation chamber was evacuated to < 1 0 - 7 Tort before being back-filled with pre-cleaned helium. In situ alloying from samples of high purity nickel rod (Johnson Matthey "Specpure", 99.998%) and graphite powder (99.97%) gave a typical sample mass of 1 g. The levitated droplet was cooled using jets of helium or a H e - H 2 mixture to achieve the desired A T (monitored using a twocolour pyrometer), whereupon nucleation was triggered with a boron nitride plate. Owing to the localized recalescence that occurs at the solid-liquid interface, the solidification front can be detected as it sweeps across the sample surface using a silicon photodiode. It is assumed that the solidification front corresponds to the locus of the tips of an array of dendrites growing from the nucleation point. From the photodiode signal and a knowledge of the size of the observed region of sample surface, the steady-state growth velocity of the dendrites can be deduced. For these experiments, several samples of each of 0.6at.%C, 1.7at.%C and 3at.%C were studied. Measurements were also made on a single sample of pure nickel, concentrating on a range of lower undercoolings not examined previously [ 1]. 3.
previously described, information regarding the equilibrium phase diagram for the Ni-C system is required as well as a value for the diffusivity D E of carbon in molten nickel. To perform the calculations here, linear equilibrium liquidus and solidus lines were assumed. The linear liquidus gives a good approximation to the available data [8]. Unfortunately, no data have been found for the solidus. Assuming non-linear equilibrium phase boundaries, incorporating a polynomial fit to the available liquidus data was found to have only a small effect on calculated dendrite velocities over the range of undercoolings of interest. Values quoted for D E vary considerably. For example, Osugi et al. have suggested 5-10 x 10 -s m 2 s ~ [9], whereas Strong and Hanneman give a
100
I
I0
o 0
"
Results
Measured velocities as a function of undercooling for the three Ni-C alloys are shown in Figs. 1-3, where data for all the samples studied at each composition are included. To calculate the growth behaviour on the basis of the model
i
1
) 10
100 Undercooling
(K)
Fig. 2. Dendrite growth velocities measured in two samples each containing 1.7at.%C ( Q o), and calculated for pure nickel ( . . . . ) and a 1.7 at.% C alloy assuming D~. = 4 x 10-" m2s i( )andDt=5xl0 Sm_,s-i( .... ).
100
100
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jo
-"'b
10
~
;;,oo
;"o
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1
CD
° o
m
10
209
1
>
_ 0
,,i
,/.
~ 2o/oO
o _
//
....
..... 10
100 Undercooling
(K)
Fig. 1. Dendrite growth velocities measured in samples containing 3.0 at.% C (o) and 3.1 at.% C (o), and calculated for pure nickel ( . . . . ) and a 3 a t . % C alloy assuming Dl=3xl0-gm2s i( )andDL=5xl0 am2 s l( .... ).
100 Undercooling
(K)
Fig. 3. Dendrite growth velocities measured in three samples each containing 0.6 at.% C (o, o D), and calculated for pure nickel ( . . . . ) and a 0.6 at.% C alloy assuming Di. = 4 × 10 '~ m2s I( .)andDt=5xl0-~m2s I( .... ).
732 100 --
. . . . . . . .
r
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~ o oO
~m 10
,
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,,
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# 10
100 Undereooling (K) Fig. 4. Dendrite growth velocities in pure nickel measured by Schleip et aL (o) and in the present work (o). Also shown are the calculated curves for pure nickel ( ) and nickel containing 0.5 at.% oxygen(. . . . ). value of 2 - 4 x 10 -9 m 2 s -l [10]. Best fits of the model to the data were obtained when D L was taken to be 3 × 1 0 - 9 m 2 s-~ in the case of the 3at.%C alloy and 4 x 10 -9 m 2 s -~ for the 1.7 at.% C alloy (see Figs. 1 and 2). If a value for DE of 5 X 10-8 m 2 s-~ is assumed, then clearly the model does not provide a good fit. However, from Fig. 3 it is apparent that the observed behaviour for the 0.6 at.% C alloy is consistent with this higher carbon diffusivity and not with the lower values. Data for the pure nickel sample are shown in Fig. 4 along with the earlier measurements of Schleip et aL [1] made using the same technique. Also shown in each of Figs. 1-4 is the calculated behaviour for pure nickel which gives a good fit to these latter measurements except for undercoolings beyond about 190 K. This abrupt departure from the growth theory is as yet unexplained, but there is evidence to suggest that it is associated with a grain refinement effect found to occur at a similar critical A T [ 1, 11 ]. 4. Discussion
Considering the 1.7 at.% C and 3 at.% C alloys first, the most interesting part of the behaviour predicted by the model occurs at intermediate undercoolings where there is a distinct kink in the otherwise smooth variation of V with AT. Although in the case of the 3 at.% C alloy the experimental scatter is too great to make such behaviour clearly identifiable, it is observed for the 1.7 at.% C alloy. At the lowest undercoolings investigated in this case, the alloy dendrites grow
more slowly than the pure metal. This is the expected result of the slower diffusion of solute in the liquid as compared with heat. However, as A T rises, V increases rapidly to become comparable with that predicted for the pure metal at the same undercooling. In the model, this behaviour is a consequence of the marginal-stability assumption: as the dendrite velocity increases, the solute diffusion field at the dendrite tip becomes steeper, which destabilizes the tip, resulting in a sharper, faster dendrite. At still higher A T and hence V, the onset of solute trapping results in a gradual transition from growth controlled by solute-diffusion to thermal-diffusion-controlled growth, as discussed earlier. The trapping reduces the solute gradient in the liquid, thus diminishing the destabilizing effect and consequently reducing the rate at which V continues to rise with A T. However, there still remains the question of the much higher value for D L which is suggested by the behaviour of the 0.6 at.% C alloy. No composition dependence of DL is included in the model, and in fact an implicit assumption is that it is constant throughout the solute diffusion field. If D L is significantly composition-dependent, then it is possible that the value that is found to lead to the best fit of model to data is representative of an average diffusivity for solute atoms in the diffusion field. However, it seems unlikely that the dependence of this apparent diffusivity on the bulk alloy composition could be so strong as to give rise to the order of magnitude difference between the values used to fit the data for the 1.7 at.% C and 0.6 at.% C alloys. A plausible alternative explanation arises from the measurements made at low undercooling on nominally pure nickel. These data suggest that the pure nickel dendrites grow faster in the undercooling range 30-80 K than is expected from the behaviour at higher undercooling (see Fig. 4). This effect may be produced by a strongly partitioning impurity present at a concentration such that its destablizing influence on the dendrite tip is greater than the effect of A Tc. In such a case, the impure material may actually grow faster than the truly pure dendrite. As an example, the effects of dissolved oxygen were calculated and Fig. 4 shows that the observed behaviour is consistent with the presence of 0.5 at.% of oxygen in solution. This level does seem rather high considering the high purity of the starting material and the clean conditions under which the levitation melt-
733
ing was performed. Other dissolved gases are likely to be present, for instance helium or hydrogen, although the possible effects in these cases have not been assessed owing to the unavailability of information regarding partitioning between the solid and liquid phases. If the observed enhancement of velocities in the pure nickel case is due to an impurity, then the same impurity would be expected to be present in the experiments with the Ni-C alloys. It is thus possible that when the alloy is dilute, the effects of the impurity dominate, giving the appearance of a much higher diffusivity for carbon in liquid nickel. At higher concentrations, growth may become controlled by the carbon in solution. The presence of the impurity might still be detected, however, and this may explain the slightly higher diffusivity suggested by the behaviour of the 1.7 at.% C alloy as compared with the 3 at.% C alloy. 5. Conclusions Dendrite growth velocities in undercooled Ni-C alloys calculated on the basis of current models are in good agreement with experimental measurements on electromagnetically levitated droplets containing 1.7 at.% C and 3 at.% C, provided a suitable choice for the solute diffusivity is made. In particular, the effects of several characteristics of the model, notably the influence of rejected solute in the liquid upon the dendrite tip stability and the trapping of non-equilibrium solute concentrations in the solid, are apparent in the observed behaviour. It is suggested that the
anomalously high velocities observed in the 0.6 at.% C alloy are due to the presence of an additional impurity which is also responsible for the observation of enhanced dendrite velocities in a sample of nominally pure nickel.
Acknowledgments The authors thank Dr. P. V. Evans and Dr. R. F. Cochrane for their assistance in this work. R.G.H. is grateful to the Science and Engineering Research Council for a Studentship and to the Deutsche Forschnngs- und Versuchsanstalt f/Jr Luft- und Raumfahrt for financial support.
References 1 E. Schleip, R. Willnecker, D. M. Herlach and G. P. G6rler, Mater. Sci. Eng., 98 (1988) 39. 2 W. J. Boettinger, S. R. Coriell and R. Trivedi, in R. Mehrabian and P. A. Parrish (eds.), Rapid Solidification Processing, Principles and Technologies IV, Claitor's Baton Rouge, 1988, p. 13. 3 R. Trivedi, J. Lipton and W. Kurz, Acta MetulL, 35 (1987) 965. 4 G.P. lvantsov, Dokl. Akad. Nauk. SSSR, 58(1947) 567. 5 D. Turnbull, J. de Physique, 35 (1974) Colloque 4, C4.1-4.9. 6 M.J. Aziz, J. Appl. Phys., 53(1982) 1158. 7 J.S. Langer, Rev. Mod. Phys., 52 (1980) 1. 8 L.D. Lucas, MOrn. Sci. Rev. MOtall., 66(1969) 747. 9 J. Osugi, T. Arase, K. Hara and F. Amita, High Temperatures-High Pressures, 16 (1984) 191. 10 H, M. Strong and R. E. Hanneman, J. Chem. Phys., 46 (1967) 3668. 11 R. Willnecker, D. M. Herlach and B. Feuerbacher, Appl.
l'hys. Lett., 56(1990)324.
Materials Science and Engineering, A133 ( 1991 ) 730-733
Dendrite growth velocities in highly undercooled, dilute Ni-C melts K. Eckler and D. M. Herlach lnstitut fiir Raumsimulation, Deutsche Forschungs- und Versuchsanstalt fiir Lufi- und Raumfahrt, D-5000, Kdln 90 (F.R.G.)
R. G. Hamerton and A. L. Greer University of Cambridge, Department of Materials Science and Metallurgy, Pembroke Street, Cambridge CB2 3QZ (U.K.)
Abstract Dendrite growth velocities have been measured in droplets of dilute Ni-C alloys undercooled by up to 300 K using an electromagnetic levitation technique. For alloys containing 1.7 at.% C and 3 at.% C, the observed behaviour is in good agreement with the predictions of current models. Velocities in pure nickel and a 0.6 at.% C alloy at low undercooling ( < 80 K) are in excess of those expected for the pure material and suggest effects due to the presence of an impurity.
1. Introduction
Owing to its containerless nature, electromagnetic levitation melting permits high undercoolings in liquid droplets prior to crystal nucleation, giving the possibility of extremely high initial growth rates, which can be monitored by observation of the sample surface. Furthermore, nucleation can be externally induced at a well-determined temperature, enabling correlation between growth rate measurements and the droplet undercooling [1]. A number of models (e.g. [2, 3]) predict the growth rate V of a dendrite (approximated by a paraboloid of revolution with tip radius R) in an undercooled liquid. The total undercooling A T is written as [2] AT=ATt+ATc+ATr+ATk
(1)
The thermal and solutal undercoolings A T t and ATc are obtained from steady-state solutions to the relevant diffusion equations [4], while the curvature undercooling A Tr is the lowering of the equilibrium temperature at the tip due to the Gibbs-Thomson effect. Although the actual tip temperature is lower than the equilibrium value by the kinetic undercooling A Tk, the difference is small for the velocities considered here if the interracial kinetics are collision-limited [5]. However, at large V there is the additional possibility that higher than equilibrium solute concentra0921-5093/91/$3.50
tions may be trapped in the growing solid [6]. With less partitioning, conditions of essentially solute-diffusion-controlled growth at low undercooling may give way to thermal control at higher undercooling, reflected in a transition in the relative contributions of A T t and ATc to A T. An equation such as (1), however, gives only the product VR at a given A T, hence a second criterion, termed the marginal-stability hypothesis, in which the dendrite tip is assumed to have the largest possible radius while remaining morphologically stable, is used to extract a single solution for V [7]. For ease of calculation, the marginally-stable tip radius is associated with the minimum wavelength of perturbation which would grow on a planar interface subject to the same thermal and solutal gradients as exist at the dendrite tip [2, 3]. The computational procedure followed in this work is, firstly, to select a dendrite velocity V. The tip radius can then be calculated under the marginal-stability assumption using an iterative technique, and finally the tmdercooling is obtained by summing the contributions in eqn. ( 1 ). 2. Experimental
The technique of measuring dendrite growth velocities in highly undercooled, levitated metallic droplets has been described elsewhere [1].
731
Only brief details are included here. The levitation chamber was evacuated to < 1 0 - 7 Tort before being back-filled with pre-cleaned helium. In situ alloying from samples of high purity nickel rod (Johnson Matthey "Specpure", 99.998%) and graphite powder (99.97%) gave a typical sample mass of 1 g. The levitated droplet was cooled using jets of helium or a H e - H 2 mixture to achieve the desired A T (monitored using a twocolour pyrometer), whereupon nucleation was triggered with a boron nitride plate. Owing to the localized recalescence that occurs at the solid-liquid interface, the solidification front can be detected as it sweeps across the sample surface using a silicon photodiode. It is assumed that the solidification front corresponds to the locus of the tips of an array of dendrites growing from the nucleation point. From the photodiode signal and a knowledge of the size of the observed region of sample surface, the steady-state growth velocity of the dendrites can be deduced. For these experiments, several samples of each of 0.6at.%C, 1.7at.%C and 3at.%C were studied. Measurements were also made on a single sample of pure nickel, concentrating on a range of lower undercoolings not examined previously [ 1]. 3.
previously described, information regarding the equilibrium phase diagram for the Ni-C system is required as well as a value for the diffusivity D E of carbon in molten nickel. To perform the calculations here, linear equilibrium liquidus and solidus lines were assumed. The linear liquidus gives a good approximation to the available data [8]. Unfortunately, no data have been found for the solidus. Assuming non-linear equilibrium phase boundaries, incorporating a polynomial fit to the available liquidus data was found to have only a small effect on calculated dendrite velocities over the range of undercoolings of interest. Values quoted for D E vary considerably. For example, Osugi et al. have suggested 5-10 x 10 -s m 2 s ~ [9], whereas Strong and Hanneman give a
100
I
I0
o 0
"
Results
Measured velocities as a function of undercooling for the three Ni-C alloys are shown in Figs. 1-3, where data for all the samples studied at each composition are included. To calculate the growth behaviour on the basis of the model
i
1
) 10
100 Undercooling
(K)
Fig. 2. Dendrite growth velocities measured in two samples each containing 1.7at.%C ( Q o), and calculated for pure nickel ( . . . . ) and a 1.7 at.% C alloy assuming D~. = 4 x 10-" m2s i( )andDt=5xl0 Sm_,s-i( .... ).
100
100
o ,' o
jo
-"'b
10
~
;;,oo
;"o
o: f r o
1
CD
° o
m
10
209
1
>
_ 0
,,i
,/.
~ 2o/oO
o _
//
....
..... 10
100 Undercooling
(K)
Fig. 1. Dendrite growth velocities measured in samples containing 3.0 at.% C (o) and 3.1 at.% C (o), and calculated for pure nickel ( . . . . ) and a 3 a t . % C alloy assuming Dl=3xl0-gm2s i( )andDL=5xl0 am2 s l( .... ).
100 Undercooling
(K)
Fig. 3. Dendrite growth velocities measured in three samples each containing 0.6 at.% C (o, o D), and calculated for pure nickel ( . . . . ) and a 0.6 at.% C alloy assuming Di. = 4 × 10 '~ m2s I( .)andDt=5xl0-~m2s I( .... ).
732 100 --
. . . . . . . .
r
"
~ o oO
~m 10
,
°v i"
o o 9
,,
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°?4 ¢9
>
# 10
100 Undereooling (K) Fig. 4. Dendrite growth velocities in pure nickel measured by Schleip et aL (o) and in the present work (o). Also shown are the calculated curves for pure nickel ( ) and nickel containing 0.5 at.% oxygen(. . . . ). value of 2 - 4 x 10 -9 m 2 s -l [10]. Best fits of the model to the data were obtained when D L was taken to be 3 × 1 0 - 9 m 2 s-~ in the case of the 3at.%C alloy and 4 x 10 -9 m 2 s -~ for the 1.7 at.% C alloy (see Figs. 1 and 2). If a value for DE of 5 X 10-8 m 2 s-~ is assumed, then clearly the model does not provide a good fit. However, from Fig. 3 it is apparent that the observed behaviour for the 0.6 at.% C alloy is consistent with this higher carbon diffusivity and not with the lower values. Data for the pure nickel sample are shown in Fig. 4 along with the earlier measurements of Schleip et aL [1] made using the same technique. Also shown in each of Figs. 1-4 is the calculated behaviour for pure nickel which gives a good fit to these latter measurements except for undercoolings beyond about 190 K. This abrupt departure from the growth theory is as yet unexplained, but there is evidence to suggest that it is associated with a grain refinement effect found to occur at a similar critical A T [ 1, 11 ]. 4. Discussion
Considering the 1.7 at.% C and 3 at.% C alloys first, the most interesting part of the behaviour predicted by the model occurs at intermediate undercoolings where there is a distinct kink in the otherwise smooth variation of V with AT. Although in the case of the 3 at.% C alloy the experimental scatter is too great to make such behaviour clearly identifiable, it is observed for the 1.7 at.% C alloy. At the lowest undercoolings investigated in this case, the alloy dendrites grow
more slowly than the pure metal. This is the expected result of the slower diffusion of solute in the liquid as compared with heat. However, as A T rises, V increases rapidly to become comparable with that predicted for the pure metal at the same undercooling. In the model, this behaviour is a consequence of the marginal-stability assumption: as the dendrite velocity increases, the solute diffusion field at the dendrite tip becomes steeper, which destabilizes the tip, resulting in a sharper, faster dendrite. At still higher A T and hence V, the onset of solute trapping results in a gradual transition from growth controlled by solute-diffusion to thermal-diffusion-controlled growth, as discussed earlier. The trapping reduces the solute gradient in the liquid, thus diminishing the destabilizing effect and consequently reducing the rate at which V continues to rise with A T. However, there still remains the question of the much higher value for D L which is suggested by the behaviour of the 0.6 at.% C alloy. No composition dependence of DL is included in the model, and in fact an implicit assumption is that it is constant throughout the solute diffusion field. If D L is significantly composition-dependent, then it is possible that the value that is found to lead to the best fit of model to data is representative of an average diffusivity for solute atoms in the diffusion field. However, it seems unlikely that the dependence of this apparent diffusivity on the bulk alloy composition could be so strong as to give rise to the order of magnitude difference between the values used to fit the data for the 1.7 at.% C and 0.6 at.% C alloys. A plausible alternative explanation arises from the measurements made at low undercooling on nominally pure nickel. These data suggest that the pure nickel dendrites grow faster in the undercooling range 30-80 K than is expected from the behaviour at higher undercooling (see Fig. 4). This effect may be produced by a strongly partitioning impurity present at a concentration such that its destablizing influence on the dendrite tip is greater than the effect of A Tc. In such a case, the impure material may actually grow faster than the truly pure dendrite. As an example, the effects of dissolved oxygen were calculated and Fig. 4 shows that the observed behaviour is consistent with the presence of 0.5 at.% of oxygen in solution. This level does seem rather high considering the high purity of the starting material and the clean conditions under which the levitation melt-
733
ing was performed. Other dissolved gases are likely to be present, for instance helium or hydrogen, although the possible effects in these cases have not been assessed owing to the unavailability of information regarding partitioning between the solid and liquid phases. If the observed enhancement of velocities in the pure nickel case is due to an impurity, then the same impurity would be expected to be present in the experiments with the Ni-C alloys. It is thus possible that when the alloy is dilute, the effects of the impurity dominate, giving the appearance of a much higher diffusivity for carbon in liquid nickel. At higher concentrations, growth may become controlled by the carbon in solution. The presence of the impurity might still be detected, however, and this may explain the slightly higher diffusivity suggested by the behaviour of the 1.7 at.% C alloy as compared with the 3 at.% C alloy. 5. Conclusions Dendrite growth velocities in undercooled Ni-C alloys calculated on the basis of current models are in good agreement with experimental measurements on electromagnetically levitated droplets containing 1.7 at.% C and 3 at.% C, provided a suitable choice for the solute diffusivity is made. In particular, the effects of several characteristics of the model, notably the influence of rejected solute in the liquid upon the dendrite tip stability and the trapping of non-equilibrium solute concentrations in the solid, are apparent in the observed behaviour. It is suggested that the
anomalously high velocities observed in the 0.6 at.% C alloy are due to the presence of an additional impurity which is also responsible for the observation of enhanced dendrite velocities in a sample of nominally pure nickel.
Acknowledgments The authors thank Dr. P. V. Evans and Dr. R. F. Cochrane for their assistance in this work. R.G.H. is grateful to the Science and Engineering Research Council for a Studentship and to the Deutsche Forschnngs- und Versuchsanstalt f/Jr Luft- und Raumfahrt for financial support.
References 1 E. Schleip, R. Willnecker, D. M. Herlach and G. P. G6rler, Mater. Sci. Eng., 98 (1988) 39. 2 W. J. Boettinger, S. R. Coriell and R. Trivedi, in R. Mehrabian and P. A. Parrish (eds.), Rapid Solidification Processing, Principles and Technologies IV, Claitor's Baton Rouge, 1988, p. 13. 3 R. Trivedi, J. Lipton and W. Kurz, Acta MetulL, 35 (1987) 965. 4 G.P. lvantsov, Dokl. Akad. Nauk. SSSR, 58(1947) 567. 5 D. Turnbull, J. de Physique, 35 (1974) Colloque 4, C4.1-4.9. 6 M.J. Aziz, J. Appl. Phys., 53(1982) 1158. 7 J.S. Langer, Rev. Mod. Phys., 52 (1980) 1. 8 L.D. Lucas, MOrn. Sci. Rev. MOtall., 66(1969) 747. 9 J. Osugi, T. Arase, K. Hara and F. Amita, High Temperatures-High Pressures, 16 (1984) 191. 10 H, M. Strong and R. E. Hanneman, J. Chem. Phys., 46 (1967) 3668. 11 R. Willnecker, D. M. Herlach and B. Feuerbacher, Appl.
l'hys. Lett., 56(1990)324.