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SOME KINETICS AND THERMODYNAMIC ISSUES IN SOLID-STATE AMORPHISATION
R. Highmore
To cite this version:
R. Highmore. SOME KINETICS AND THERMODYNAMIC ISSUES IN SOLID-STATE AMORPHI- SATION. Journal de Physique Colloques, 1990, 51 (C4), pp.C4-37-C4-47. �10.1051/jphyscol:1990404�.
�jpa-00230764�
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Colloque C4, suppl6men-k au n 0 1 4 , Tome 5 1 , 15 j u i l l e t 1990
SOME KINETICS AND THERMODYNAMIC ISSUES IN SOLID-STATE AMORPHISATION
R.J. HIGHMORE
University of Cambridge, Department of Materials Science and Metallurgy, Pembroke Street, GB-Cambridge CB2 392, Great-Britain
Rksumk
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La premikre partie de cet article discute comment on peut supprimer la germination d'un composC cristallin B l'interface entre une alliage amorphe et un Clement cristallin jusqutB que la couche amorphe aura atteint une Cpaisseur critique, laquelle est fonction de la tempkrature et du systkme. La discussion se focalise sur le systkme qui a Ct6 le plus etudiC, Ni-Zr, quoique les arguments avancCs pourraient ttre valable aussi pour l'interpretation de l'amorphisation en Ctat solide pour des autres systkmes. La deuxieme partie examine la relation entre le modkle bask sur la germination transitoire et celui basC sur la croissance competitive, tandis que la troisikme essaye une gCnCrdisation du critkre, dit"eutectique profond, pour la facilitC de fabriquer les verres mitalliques par refroidissement rapide, de fqon de comprendre aussi la formation les verres par reaction en Ctat solide.
Abstract
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The first part of this paper discusses how nucleation of a crystalline compound at an advancing interface between an amorphous alloy and a crystalline element may be suppressed until the amorphous layer has attained a temperature- and system- dependent critical thickness. The discussion concentrates upon the most-studied amorphising system, Ni-Zr, though the points made may be relevant in interpreting solid-state amorphisation in other systems. The second section examines the relationship between transient nucleation and competitive-growth models for phase suppression in diffusion couples, and the third section attempts to generalise the so-called "deep eutectic" criterion for ease of manufacturing metallic glasses to include ease of glass formation by solid-state reaction.1: Nucleation Of The Second Product Phase
Transmission electron microscopy of Ni-Zr diffusion couples which have been annealed at 300°C shows that after the amorphous layers have attained a thickness of about 100 nm, the crystalline compound NiZr forms at interfaces between amorphous alloy (a-NiXZrl.,) and crystalline Zr and grows instead of the amorphous phase [l]. Meng et al. [2] observed that, if a diffusion couple is annealed at 360°C, the crystalline NiZr can grow backward at the expense of the amorphous alloy; they deduced that formation of the second product phase may be limited by the ease with which it can nucleate at the moving interface. Two groups [2,3] have proposed what have been termed [4] "nucleation-time" models in attempts to explain why formation of the crystalline intermetallic NiZr is suppressed until the amorphous alloy has attained a temperature- and system- dependent critical thickness.
Both groups assume that NiZr is able to nucleate if the speed of the a-NixZrl-,/Zr interface is less than than a critical speed given by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990404
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critical speed
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the size of a critical cluster of NiZrthe time needed to form a critical cluster (1).
The suggestion is that the interface will advance quickly in the early stages of reaction and surround clusters of NiZr before they attain the critical size for nucleation; the clusters then dissolve as the surrounding alloy becomes more rich in Ni. The critical speed is the speed of the a-Ni,Zrl.x/Zr interface below which nuclei of NiZr can just survive. Meng et al. [2] proposed that the formation time be approximated by the reciprocal of the steady-state nucleation frequency. By contrast, Highmore et al. [3] suggested that for temperatures far below the equilibrium liquidus, and for nucleation at a moving interface, transient nucleation kinetics might predominate, and that there may be insufficient time to establish steady-state nucleation kinetics. They proposed that the formation time be approximated by the time lag for transient nucleation, 0. Transient nucleation kinetics have been reviewed by Kelton et al. [5]: the present article uses the nomenclature and the approximations employed in that work.
Kashchiev [6] has given an analytic expression for 6:
Here k is Boltzmann's constant, T the temperature,
X
the diffusional jump distance, v the volume of one"molecule" of orthorhombic NiZr, taken to consist of one atom of Ni and one atom of Zr, o the interfacial energy, Ag the driving force for addition of one "molecule" to a cluster of NiZr, and D the effective diffusivity of
"molecules" at the surface of a cluster (which is assumed to be determined by diffusion of slow-moving Zr atoms). Highmore et al. [3] took the relevant values of o and D to be those associated with the a-Ni,Zrl-aiZr interface. In reality, the appropriate values of o and D will be averages of those at the Ni,Zrl-JNiZr interface and those at the NiZr/Zr interface, weighted according to the areas of the interfaces. If one interfacial diffusivity is much larger than the other, transport of atoms across the interface with the greater diffusion coefficient will determine how quickly clusters of NiZr grow. Substitution of estimates (the values are listed in [31) for the quantities in equation (2) yields a time lag of the order of 104 S at 573 K if D is approximated by the self- diffusivity of a-Zr [7], and a time lag which is somewhat larger if D is equated with the interdiffusivity of Hf and Zr in an amorphous Ni-HflN-Zr multilayer [g]. Such long transient times could be a key reason for nucleation of crystalline compounds being suppressed during low temperature annealing.
Highmore et al. approximated the numerator in equation (1) by the critical radius of a spherical cluster,
Equations (l), (2) and (3) can be combined to yield an expression for the critical speed of the a-NixZrl-x/Zr interface, \k* (where W is the instantaneous width of the amorphous layer):
In the approximation employed here the critical interfacial speed is independent of the interfacial energy, o, because both the critical radius and the time lag for transient nucleation are proportional to the cube root of the number of molecules in a critical cluster, and hence proportional to o (see equations 2 and 3). Increasing the interfacial energy means that the interface must advance further in order that the amorphous phase can consume a critical cluster, but that it has more time in which to travel the necessary distance. In the transient nucleation analysis, the critical thickness, W*, of amorphous alloy which grows before NiZr can nucleate is [3]:
Here f is the fractional composition range over which the amorphous phase exists,
6
is the effective interdiffusion coefficient in the amorphous layer and Tl is the average composition of the amorphous layer.Equation (5) suggests that potential amorphising systems must have large values of
6D. 6
must be big in order that the a-Ni,Zrl.JZr interface can advance quickly and surround sub-critical clusters of NiZr, while D must be small to minimise the rate at which sub-critical clusters of N i grow. A weakness of equation (5) is that it assumes6
to be constant for the duration of the reaction. Relaxation of the amorphous alloy [10,1 l ] and development of voids at Nila-NixZrl, interfaces [l] may causeE
to decrease with increasing anneal time, and thus equation (5) is likely to overestimate the critical thickness.The model embodied in equation (1) is a crude one. It considers competition between phases only in the instance where the interface has slowed to near its critical speed, and provides no detail of a mechanism by which amorphous alloy can surround sub-critical clusters of NiZr. An improved analysis might avoid using ready-made formulae for r* and for 0, and instead derive an expression for W' directly £mm nucleation theory. Markedly heterogeneous conditions at the moving interface may result in a situation which cannot be described by classical nucleation theory; I discuss this problem below. The different aspects of competition between phases at a moving interface might best be combined in an atomistic computer simulation.
The discussion in this section has dealt only with nucleation of the second product phase at the interface between a-NixZrl, and crystalline Zr. The reader might refer to references [12], [13], or [l41 for further treatment of nucleation issues in solid-state amorphisation.
2: The Relationship Between The Transient Nucleation Model And Competitive-Growth Models
This section explores the reIationship between two apparently different types of model for phase suppression in diffusion couples: the transient nucleation model described above, and competitive-growth models of the kind proposed by, e.g., Gosele and Tu [15]. The two models are similar in the sense that they both envisage a new phase, whose growth kinetics are limited by the speed with which atoms can react at interfaces, attempting to fonn ahead of an existing phase whose growth is diffusion-limited. The two models are also analogous insofar as they both predict a critical amorphous layer thickness which is independent of the interfacial energy, o. This
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independence from o arises in competitive-growth analyses because there is assumed to be no barrier to nucleation of the new phase. It arises in the transient nucleation approximation because, as explained in the previous section, both the critical radius and the time lag for transient nucleation are proportional to o. There do, however, exist important differences between the two models. One difference is morphological, since the transient nucleation approach assumes a spherical or near-spherical critical cluster (see figure l(a)), while the competitive-growth model assumes that the new phase attempts to form as a layer with the a-NixZrl-,/NB and NiZr/Zr interfaces parallel to the existing a-NixZrl-x/Zr interface (see figure l (d)). A second difference lies in the magnitude of the thermodynamic banier to nucleation of the phase; in the transient nucleation approximation it is 16m2o313Ag2, whereas in the competitive-growth analysis it is zero. An attempt to reconcile or to unite the two models must account for these disparities in the morphology and in the thermodynamic barrier to nucleation.
I suggest that the transient nucleation and competitive-growth models represent opposite limits of behaviour as 9, the contact angle for formation of clusters of new phase at the advancing interface, is varied. Figure 1 depicts the situation at the advancing interface as 9 is changed, here 9 is being used as a parameter to characterise the extent to which the presence of an heterogeneity, in the form of the existing interface, reduces the barrier for the nucleation of the new phase. The situation will not be exactly as depicted in figure 1, because the NiZr/Zr interface must be curved in order to balance forces where the three interfaces meet, but the true (qualitative) state of affairs is unlikely to be very different from that drawn. It is improbable that the most simple form of the spherical cap model for heterogeneous nucleation [l61 will describe the situation well, since that model takes the curvature of the curved interface to be independent of 0. In the present case, the curvature of the a-NixZrl,/NiZr interface must evolve from a finite value at 9 = 180' to being zero at 9 = 0.
For large 9 there exists an appreciable barrier to nucleation and clusters of new phase are spherical; in this instance one must consider the nucleation aspects of phase suppression. For smaller 9 there is a lesser thermodynamic barrier to nucleation and clusters of new phase assume the shape of a cap. For values of 9 less than about 30' the clusters acquire a layer-like morphology and classical nucleation theory will fail to describe the situation in an adequate manner, because the predicted size of the critical cluster in the direction perpendicular to the interface becomes comparable with an atomic radius. (Evidence for the failure of classical nucleation theory to describe nucleation for spherical caps with 0 less than about 30" has come from Cantor [l71 and from Greer et al.
[B]) When classical nucleation theory does fail in the limit of small predicted critical clusters, it underestimates the difficulty of nucleating a new phase. I therefore suggest that, in the limit of small 8 (i.e., in the limit of markedly heterogeneous nucleation), phase suppression at the moving interface is best described by a competitive-growth analysis.
Figure 1 (d) suggests that the critical condition for phase suppression in a competitive-growth analysis arises when the speed of the a-NiXZrl,/Zr interface is equal to the rate of thickening of the NiZr layer. If the a-Ni,Zrl, /Zr interface moves more slowly than this, it will be unable to kinetically suppress the crystalline compound.
Using this condition and the same general approach and nomenclature as were used to derive equation (4), I obtain the expression
Increasingly Heterogeneous Nucleation
Figure 1: Orthorhombic NiZr attempting to form at the interface between amorphous NixZrl-, (whose growth is diffusion-limited) and a-Zr. The diagram depicts the situation at the advancing interface as 9, the contact angle for heterogeneous nucleation, is varied. (a) 9 = 180'; this is the limit in which a transient nucleation analysis is most appropriate. (b) 90' < 9 c 180'. (c) 0< 9 < 90°. (d) 9 = 0; this is the limit in which a competitive-growth analysis is most applicable.
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for the critical speed of the advancing interface in the competitive-growth approximation. The derivation of this equation assumes that the rate-limiting Zr atoms can cross both the a-Ni,Zrl-JZr interface and the NiZrEr interface, in each case with interfacial diffusivity 6 ~ / A 2 , in order to join the NiZr. The expression (6) for the critical interfacial speed is reduced by a factor of two if atoms can be transported across only one of these interfaces. The expression for the critical amorphous layer thickness in the competitive-growth approximation is
As was the case in deriving equation (5) from equation (41, the derivation of (7) from (6) assumes that
6
is independent of time. In reality,6
is likely to decrease as reaction progresses, causing equation (7) to overestimate the critical thickness.For Ag << 2kT, equation (6) reduces to
Dividing equation (6) by equation (4):
which for Ag << 2kT reduces to
i.e., for small driving forces the ratio tends to a factor of order one, but for large driving forces it grows exponentially. It is reasonable that the critical interfacial speed required for suppression of the second phase should be larger in the competitive-growth model than in the transient nucleation approach, since for large 8 the positive free energy associated with an additional interface makes formation of the new phase relatively difficult and hence renders its suppression comparatively easy, while for small 6 the free energy bamer is reduced, nucleation of new phase is made easier and the interface must therefore advance more quickly in order to suppress the new phase.
The argument which is outlined here provides approximate analytical expressions for critical interfacial speeds in the limits 8 = 0 and 8 = 180'. Numerical studies might provide both more accurate evaluations of these limiting
critical speeds, and determinations of critical interfacial speeds for intermediate contact angles. Possible techniques include molecular dynamics or simulations based upon the kinetics of transient heterogeneous nucleation and of competitive growth, though this latter approach might require some means of allowing interfacial curvature to vary with contact angle.
3: Eutectics And Glass Formation
In this section I study the notion that the ease of manufacturing metallic glasses by liquid quenching is enhanced by deep eutectics and by a large value of T o e (where Tg is the glass transition temperature and Te is the eutectic temperature), and that these features of a system are in turn engendered by an attractive interaction between the components of the liquid alloy. I suggest that, by using values of T o e which are greater than one, or perhaps even negative, these concepts can be generalised to include ease of glass formation by solid-state reaction. The present discussion considers only the case where the entropy of the liquid alloy is always greater than the entropy of the mixture of elemental solid solutions. The reader should refer to [l91 for an analysis of what happens if the entropy of the liquid becomes less than the entropy of the mixture of elemental solid solutions.
Deep eutectics in an equilibrium phase diagram are desirable for ease of manufacturing metallic glasses by liquid quenching. Increasing the ratio Tg/Te causes the temperature interval between Te, above which the liquid is thermodynamically stable, and Tg, below which the liquid attains a measure of kinetic stability, to be reduced.
Thus the temperature interval in which the liquid is vulnerable to crystallisation is diminished and the ease of glass formation is enhanced. Experiments suggest that there is no equilibrium eutectic for which T f l e > l; if such a system did exist, the vitrified liquid would be an equilibrium phase. Thus methods of producing metallic glasses which depend on liquid quenching might best be described by values of TglTe < 1.
Compositions at which metallic glasses can be formed by interdiffusion are centred upon deep eutectics in appropriate metastable phase diagrams. Spaepen [20] has shown how, if there exists a mechanism for
suppressing crystalline compounds, one can superimpose a metastable eutectic on an equilibrium phase diagram by extending the equilibrium liquidus and solidus lines. The minimum annealing temperature for glass formation is the metastable Te, in order that there is a thermodynamic driving force for reaction. The maximum annealing temperature for amorphisation is close to Tg, because crystalline compounds will be fast to nucleate and grow at higher temperatures. Thus manufacture of metallic glasses by solid-state reaction might best be described by values of Tg/Te > 1.
The foregoing discussion suggests that T o e = 1 represents a diffuse, approximate boundary between a regime which is accessible by rapid quenching and a situation which can be realised by solid-state reaction. This diffuse boundary may also separate a region in which steady-state kinetics for nucleation of crystalline compounds predominate (TglTe < 1) from one in which transient nucleation kinetics are most important (T@e > 1).
Nucleation of crystalline compounds during amorphisation is likely to show significant transient effects, because reaction occurs below the glass transition temperature of the amorphous alloy. Transient effects are also present in melt quenching 1211 but, because melts are momentarily subjected to high temperatures during the early part of a quench, steady-state nucleation kinetics are more important in determining whether an amorphous phase is
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Increasing interaction in liquid
Fi~ure 2: A possible evolution of the hypothetical eutectic phase diagram as the interaction in the Iiquid alloy becomes more attractive. (a) T f l e < 1. (b) T f l e > 1. (c) T o e infinite. (d) T f l e < 0. The diagrams are drawn on the assumption that the entropy change on amorphisation remains positive.
formed.
Figure 2 illustrates how a hypothetical eutectic phase diagram might evolve as the stability of the liquid alloy relative to the mixture of crystalline elements is varied, while figure 3 shows the corresponding locus of T o e . The figures are drawn on the assumption that the entropy of the liquid alloy is always greater than the entropy of the mixture of crystalline elements. For moderately attractive interaction in the liquid alloy, the phase diagram might be qualitatively similar to the equilibrium phase diagram of a system such as Au-Si. As the interaction is increased the eutectic becomes broader and deeper, Tfle grows larger and the likelihood of glass being formed by liquid quenching is enhanced. If there exists a kinetic mechanism for suppressing the nucleation and growth of crystalline compounds, the eutectic can deepen into the metastable regime, T o e > 1, and it becomes possible to manufacture glass by solid-state reaction. As the interaction is further increased, so Te descends further below Tg, TgTe grows larger and the ranges of composition and temperature over which an amorphous alloy can be made by solid-state reaction will be enhanced. Given sufficiently strong interaction between atoms in the liquid alloy, the liquidus lines need not meet for any positive temperature, in which event Te and T o e might be thought of as negative (see figure 2(d)). Metastable phase diagrams for strong amorphising systems such as Ni-Zr might fall in this category, since it appears that in these systems the liquidus lines do not join at a positive temperature.
-RQ-- SSA- - -
lncreaslna interaction in liquid
Figure 3: A possible evolution of the quantity Tfle as the interaction in the liquid alloy becomes more attractive. The annotation shows the regions T o e < 1, which might be most relevant for production of metallic glasses by rapid quenching of the melt (RQ), and T o e > I, or negative, which may best characterise
manufacture of metallic glasses by solid-state amorphisation (SSA). The dashed vertical line marks the discontinuity in TgTe at the interaction corresponding to Te = 0.
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The third law of thermodynamics may, however, forbid a situation such as that drawn in figure 2(d), since if the liquid is in metastable equilibrium with the mixture of elements at absolute zero, the two must have equal entropies. This would force the liquidus lines to be vertical at absolute zero and so preclude a scenario such as that sketched in figure 2(d).
Figures 2 and 3 are drawn on the assumption that the entropy of the liquid alloy is always greater than the entropy of the mixture of crystalline elements. Highmore and Greer [l91 have discussed what may happen if the entropy of the liquid can become less than the entropy of the mixture of elemental solid solutions. Their analysis predicts the emergence of a so-called "inverse eutectic" and the presence of a pseudocritical point at which the eutectic and the inverse eutectic merge as the attractive interaction in the liquid alloy is increased. An inverse eutectic may have been observed in the Nb-A1 system [22]; the inverse eutectic temperature was determined to be approximately 500°C.
Acknowledgements
I am pleased to acknowledge collaborations with J.E. Evetts, A.L. Greer, J.A. Leake, S.B. Newcomb and R.E.
Somekh in the field covered by this paper. I thank B. Cantor for constructive criticism of some of this work, and R.W. Cahn for providing a translation of the abstract.
References
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