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Thermodynamics of undercooled liquids
Journal
of the Less-Common
THERMODYNAMICS
Metals,
145 (1988)
131
OF UNDERCOOLED
131
- 144
LIQUIDS*
A. L. GREER University Pembroke
of Cambridge, Department of Materials Street, Cambridge, CB2 392 (U.K.)
Science
and Metallurgy,
(Received May 31,1988)
Summary Undercooled alloy liquids may exhibit an excess specific heat capacity, indicating ordering. On cooling, the enthalpy and entropy of fusion decrease, and the driving force for crystallization does not rise as rapidly as would otherwise be expected. These effects are found to a large extent in systems showing deep eutectic freezing, solid state amorphization and inverse melting. There are consequences for the ease of hypercooling and for crystal growth kinetics.
1. Introduction At the melting point of a pure metal the crystal and liquid have closely similar specific heat capacities. It is noticeable, however, that at high undertoolings the liquid specific heat tends to exceed that of the solid. Such an excess specific heat is found to be particularly large for some alloy liquids, and is correlated with glass-forming ability. The thermodynamic properties of undercooled alloy liquids are of importance in understanding and modelling many solidification processes (especially rapid solidification and glass formation), and when the excess specific heat has been measured below the melting point the properties can readily be’derived. Metallic glasses have specific heats similar to those of the corresponding crystalline solids, but they are configurationally frozen liquids and have thermodynamic properties related to those of the undercooled liquid at the glass transition temperature Tg. Figure 1 shows schematically the excess of the enthalpy, entropy and Gibbs free energy (AH, AS and AG respectively) of the undercooled liquid or glass over that of the crystalline solid. The effect of an excess liquid specific heat AC, (= C,, , - C,, ,) is shown in comparison with the behaviour expected when AC, = 0. It should be noted that the assumption AC, = 0 is usual, both in phase diagram fitting and in kinetic calculations. It is suggested in this paper that this assumption may be seriously misleading, and *Paper presented at the Symposium on the Preparation and Properties of Metastable Alloys at the E-MRS Spring Meeting, Strasbourg, May 31 - June 2,1988. 0 Elsevier Sequoia/Printed
in The Netherlands
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Tg Temperature Fig. 1. The excess of the enthalpy, entropy and Gibbs free energy of the liquid (1) or glass (g) over that of the crystal between the glass transition temperature Tg and the melting point T,. The full lines are for a material with AC, = 0, the broken lines are for AC, > 0.
that it is not necessary even in the absence of AC, data in the undercooled regime. Usually such data are not available, but estimation is possible. The thermodynamics of undercooled pure metal and alloy liquids are briefly reviewed below, and the origins of the excess specific heat are considered. Some consequences of the excess specific heat in thermodynamic and kinetic modelling are illustrated both for solidification and for solid state amorphization. The latter can be considered here because of the similarity between amorphous alloys produced by solid state reaction and metallic glasses produced from the melt.
2. Undercooled Liquid pure metals For many pure metals the liquid specific heat at the melting point 7’, is known [l].It may be greater or less than that of the crystalline solid, but is usually quite similar. The specific heat of undercooled liquid metals, however, is very difficult to measure because of their tendency to crystallize. This tendency can be reduced by hindering nucleation. In the emulsion technique [2] this is achieved by dispersing the liquid metal as droplets in an organic carrier fluid, and coating the droplets with a surfactant to prevent coalescence. Using this technique undercoolings of 0.3 - 0.4 T, can be reached, and this capability has been exploited recently [ 31 to measure C,, 1 below T,. It was found that C,, 1 increases as the temperature falls, such that AC, = A + BT, where A and B are constants. For indium, for example,
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A = 16.69 J g-atom-’ K-’ and B = -0.404 J g-atom-’ KP2 [3]. For the metals studied (bismuth, mercury, indium and tin) the C!,,, value rose by 3% to 20% in the accessible undercooling range. This leads to only a small deviation of AG(T) from the linear behaviour when AC, = 0 (Fig. 1). The value of AG(T) at the maximum undercooling attained differs by just a few per cent from the value estimated using AC, = 0. Nonetheless, the positive excess liquid specific heat, increasing as the temperature is lowered, does indicate that configurational entropy is lost as the liquid is cooled. This loss of entropy, or ordering, was first pointed out by Kauzmann [4] for organic liquids, and may in such cases lead to AS < 0 at finite temperature. This apparent paradox is averted by glass formation. In metallic systems, pure liquids do not have a sufficiently large AC, value to lead to a dramatic lowering of AS and consequent glass formation, but this behaviour can be observed in alloy liquids.
3. Ordering in alloy liquids For many binary metallic systems the enthalpy of mixing the liquid components has been determined (normally well above the melting point of either component). For glass-forming systems it has always been found to be negative [5]. The entropy of mixing has not often been measured, but it seems that in glass-forming systems it may be much less than ideal. Indeed in one case, Al-Ca, it has been found to be negative [6], indicating not only a low configurational entropy but also other significant effects (vibrational or electronic) on the entropy. Both the enthalpy and entropy of glass-forming systems suggest ordering, and there is direct evidence for this from diffraction experiments. Glass-forming liquids exhibit radial distribution functions with significant correlations out to greater spacing than is usual for melts [7]. In addition, there is much evidence of substantial short-range order in metallic glasses, e.g. NisIBig [8], based on the coordination polyhedra found in the corresponding crystalline compounds [9, lo]. The most direct approach to determining the thermodynamic properties of undercooled liquid alloys is derivation from measured specific heat values. These difficult measurements have been undertaken in just a few cases : Au-Si [ll], Au-Ge-Si [12] and Mg-Cu [13]. For glass-forming liquids such as these, however, another approach is possible. If a glass is formed, by rapid quenching or otherwise, its heat of crystallization AH, can be measured. This gives the enthalpy difference between the glass and the crystal close to T,. It has been found for a wide variety of eutectic systems, which on rapid quenching are glass-forming, that at the eutectic composition -AH, is only 0.5 to 0.7 times the heat of fusion at the eutectic temperature T, [ 141. This corresponds to an average excess specific heat of the liquid of about 0.8 AS( T,) in the temperature range between Tg and T, [ 151. The excess specific heat of the liquid alloy must reflect mainly an excess specific heat of mixing in the liquid state, since the excess specific heat of the
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elemental liquid components is not expected to be large. Since the specific heat of the crystalline alloy is expected to be very similar to the specific heats of the elemental crystalline components, we find that [ 51 AHmix
-AHmix
e AH(T,) -
AH(Tg)
(1)
where AHmix and AH are the heat of mixing in the liquid and the heat of fusion respectively. This relationship has been verified using data for Aus,.,Si,s., and Zr,,Cu,, [5]. The heat of mixing referred to here is the heat of formation of the liquid alloy from its liquid components, and it is related not only to simple mixing but also to the structural state of the alloy. That AHmix becomes more negative on cooling, as is revealed by eqn. (1) and the straightforward measurements of AH, (= -AH(Z’,)) and AH(T,), indicates an increased order on cooling. There is accordingly a decreased entropy of the mixed configurations at low temperature, but it should be remembered that even at high temperature (greater than T,) the liquid can be significantly ordered. By making use of eqn. (l), the measured relationship between AH, and AH(T,) and the approximate constancy of As(T,) in metallic systems, it has been shown [5] that in glass-forming alloys there should be a linear relationship between (T, - T,)/T, and (Z’, - T,)/T,, where T, is the melting point of the main component. This relationship has been verified experimentally and has been used to predict the Tg value of a glass-forming system (Fe-B) in which the glass transition on heating is obscured by crystallization. In more stable metallic glasses the crystallization on heating may occur only at some temperature interval above Tg. In this case the specific heat measured in the interval between Tg and crystallization is that of the highly undercooled liquid alloy. The specific heat of the liquid alloy from T, down to Tg may then be obtained by interpolation rather than extrapolation. This has been done for Pd&Ni,,,P,, and AG(T) derived for this alloy is shown in Fig. 2 (from ref. 16). The other, curves in the figure show the predictions of various models. The most successful is that of Dubey and Ramachandrarao
500
600
700
800
900
Temperature (K)
Fig. 2. The excess of the Gibbs free energy of liquid Pd4$JL,,,PZ0 over that of the crystalline solid as a function of themperature (+), calculated from measured specific heat. The straight line shows the behaviour for AC, = 0, and the other curves are for various models: DR [17], TSI and TSII [16,18], BG [15] and JC [19].
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based on the hole theory of the liquid. In this theory, developed by Hirai and Eyring [ 20, 211, it is assumed that the free volume in a liquid is distributed as discrete holes of constant volume and formation energy. The Dubey and Ramachandrarao expression for A G( 2’) is [I?]
AG(T) =
AH(T T
m
-
(2 )
where AT is the undercooling. The expression was also found to agree well with measurements on Aus,.$i,s.6 and Au,,Ge,,,Si,, [17]. The other curves in Fig. 2 show pr~ictions based on a constant AC, value. The expression of Jones and Chadwick [19], used with the value AC,(T,), works rather well. The remaining treatments do not use a measured AC, value, but instead estimate it. Hoffman f22] assumes AH = 0 at T x Tg, Thompson and Spaepen [ 181 assume AS = 0 at T 3 Tg and Battezzati and Garrone [15] assume AC, = 0.8 AS(T,). The methods for predicting AG(T) are considered in more detail in ref. 16. Here, we can note that the methods based on the measured AC,(T,) values are good, with the Dubey and Ramachandrarao expression giving a particularly impressive fit. The parameters in this expression, Tm, AH(T,) and AC,(T,), control the degree of deviation of AGfT) from the linear behaviour expected if AC, = 0. Thus the expression can be used to model liquids with widely differing glassforming abilities. The main imitation of this approach to pr~icting the thermodynamic properties of undercooled liquid alloys is that it can be applied only to those cases in which there is congruent freezing (including eutectic systems). For other systems which do not freeze congruently, prediction is more difficult. If high temperature thermodynamic mixing data are available for the liquid, the associated solution model may be useful. This model, originally proposed by Hildebrand and Eastman 1231, is based on the assumption that clusters or associates exist in the melt. Evidence for clustering has been provided by the composition dependence of viscosity and structure factor 1241, but direct thermodynamic data are used to obtain the model parameters. Only one type of associate AiBi (i, i are integers) is considered in a melt of A and B atoms. Although this must be a very crude description of the topological and chemical order in the liquid alloy, it seems to be adequate in at least some cases. With one type of associate there are five model parameters: three regular solution parameters, C, C1 and CZ and the enthalpy and entropy of formation of the associates. The regular solution parameters quantify the non-ideality of the A-B, A-AiBj and B-AiBj interactions. The simultaneous evaluation of all five parameters is not straightforward, but has been achieved, for example, by non-linear least-squares fitting to thermodynamic mixing data for In-Sb and Mg-Sn [25]. For these systems the phase diagrams were calculated without using any adjustable parameters [Z5] ; remarkable agreement with experiment was found, considering that the equilibria represented in the diagrams are at a temperature about 600 K lower than the temperature at which the mixing data were obtained.
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4. Euteetic melting andsolid state foliation It is well established that in binary alloy systems there is a correlation between glass-forming ability and deep eutectics [26]. The glass transition temperature in concentrated alloys is usually not strongly dependent on composition; this has been verified experimentally [27], and has some basis in free volume theory [28]. Accordingly it is at eutectics that T, is depressed closest to Tg and crystallization on cooling the melt is least likely. Deep eutectics reflect unusual stability of the liquid, and this is a consequence of a negative heat of mixing. The thermodynamics are most easily treated at the eutectic composition. Figure 3(a) is an enthalpy-temperature diagram (referred to the enthalpy of the crystalline phases as a standard state) for a eutectic alloy. At the eutectic point the liquid must have a higher entropy than the solid phases, i.e. AS and AH are positive. The enthalpy change on melting at the eutectic point is given by the algebraic sum of the appropriately weighted average heat of fusion of the two solid phases and the heat of mixing of the two resulting liquids. The negative AH,i, value shown on the figure must be of smaller magnitude than the average heat of fusion A Ha,,, of the solid phases. Direct measurements on simple eutectic systems, with no intermediate compounds, bear this out. For Aus,.,Si,,,, for example, AHmix = -10 kJ g-atom-l, while (assuming negligible solid solution) AH,,,, = 19.75 kJ g-atom-‘.
I
I h
i
T,
Fig. 3. The enthalpy of the liquid alloy (l.a.), liquid components (l.c.) and crystalline phases (e.p.) (referred to the last as a standard state) for (a)eutectic melting and (b) solid state amorphization. AH,, is the average enthalpy of fusion of the components.
In solid state amorphization, alternating layers of two polycrystalline elements, made by deposition or mechanical reduction, react on annealing to give a solid amorphous phase. As such a phase appears to be identical with metallic glasses formed from the melt, the process of solid state amorphization can be viewed as eutectic melting. The product is a glass, not a liquid, because the reaction occurs below Tg . The melting occurs on heating, not because a critical thermodynamic temperature (the melting point) is reached, but because sufficient atomic mobility is attained to permit the transformation. The system can be regarded throughout as being above its eutectic melting temperature.
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It is possible to conceive of a deep eutectic, with thermodynamics as illustrated in Fig. 3(a), in which T, < Tg . In such a case the two crystalline solid phases should react to form an amorphous phase at around Tg on heating. The superheated eutectic solid would melt endothermically. Yet in normal solid state amorphization experiments, the amorphization which occurs on heating is strongly exothermic. In such a case the heat of mixing in the liquid phase must outweigh the average heat of fusion of the solid phases, as shown in Fig. 3(b). The ~orphous phase which is formed, being configurationally frozen, is likely to have a specific heat close to that of the crystalline phase mixture. If not, it may exhibit an excess specific heat. In either case, the heat of amorphization will not become positive at any lower temperature, as it must if there is to be an equilibrium amorphization, or melting, reaction. Thus solid state amorphization in systems such as Ni-Zr, in which the reaction is known to be exothermic [29], although it may be structurally the same as the melting of a superheated eutectic solid, cannot be so regarded in a strict thermodynamic sense. The equilibrium eutectic temperature does not exist. The two extrapolated liquiduses, for the equilibrium between the liquid or solid amorphous phase and the two elemental crystalline phases, do not meet. When solidification is sufficiently rapid to be partitionless (and ignoring congruent freezing of compounds) the composition range between the T,, lines is one in which a glass must be formed [30]. Often glasses may be formed over a wider range due to kinetic constraints on crystal growth. Solid state amorphizing systems such as Ni-Zr are cases in which, if intermetallic compounds could be avoided, rapid quenching of the melt would give glasses over a particularly wide range. With their liquidus curves not meeting they show an even greater departure from ideal liquid behaviour than in the case in which the two T, lines for the elemental solid solutions do not meet. This arises because of the highly negative heat of mixing in the liquid phase. In practice, rapid quenching of the melt does not avoid intermetallic compounds. Solid state ~orphization is more effective in avoiding crystallization, because at the low temperature typical for the process, crystal nucleation is difficult; this is considered in more detail in the next section. If the mixing in the liquid phase is made still less ideal compared with that in the solid, not only can the heat of ~o~hization become negative, but so too can the entropy change on ~orphization. In this case an equilibrium amorphization reaction is again possible, but with the amorphous phase being the low temperature phase in the transformation. This “inverse melting” is discussed later.
5. Nucleation in solid state amorpbization Here we consider the effect that a strongly negative heat of mixing in the liquid (such as is found in Ni-Zr) may have on the nucleation of a
138
product phase at the interface between layers of the two elements. This is of relevance in understanding why an amorphous phase is found to grow initially at such an interface, in preference to thermodynamically more stable crystalline compounds. In some cases there may be no nucleation stage in the formation of the product phase. In deposited Ni-Zr multilayers there is structural [31] and calorimetric [32] evidence that there is an initial amorphous layer at the interface. Nevertheless, such an initial reacted layer may not always be present, and it is of interest to calculate what would be involved in its nucleation. The negative enthalpy of mixing for amorphous Ni-Zr is only slightly smaller than for the equ~ibrium interme~llic compounds, and has been estimated for the equiatomic alloy to be -43 kJ g-atom-i [ 33]. Using this value, and taking a regular solution model with only nearest-neighbour bonding, it is found that there is a decrease in enthalpy of approximately 1 J m-’ on forming a reacted amorphous monolayer at the Ni-Zr interface. This decrease is, to a good approximation, the free energy of formation of the monolayer. The entropy change is likely to be small, since the amorphous phase is chemically ordered, but even if the ideal mixing entropy were to be obtained, the contribution to the free energy change would be less than 10% of the enthalpy change. The free energy of formation calculated in this way includes the chemical contribution to the interfacial energies, but not the structural con~ibution. A typical value of the ~terfacial energy due only to a structural difference is 0.1 J m- 2 1341. Using this value, the critical nucleation radius of a circular disc, one monolayer thick, of amorphous alloy at the Ni-Zr interface is calculated to be much less than one atomic diameter. The critical radius for an intermetallic compound, which would have a somewhat greater free energy of formation, would be even smaller. Thus the strongly negative enthalpy of mixing in a system such as Ni-Zr ensures that there is no nucleation barrier for formation of either an amorphous phase or crystalline intermetallic phases at the interface. The amorphous phase may form in preference to the intermetallics because it can form over a wider composition range. This range is indicated in Fig. 4, and is wider than that given by the common tangent construction.
Composition
Fig. 4. Schematic free energy curves for two crystalline elemental phases and a liquidamorphous phase (l/a) involved in solid state amorphization. Initially the amorphous phase may form in the composition range a to b.
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Once the amorphous phase has formed, the common tangent construction would give the equilibrium compositions at the crystalline-amorphous interfaces. The solid state amorphization takes place far from equilibrium (and, as pointed out above, the equilibrium may not exist at any temperature), however, and just as in solute trapping in rapid solidification, there may be deviations from local equilibrium at the interfaces. Another reason for the amorphous phase to form first is its structure. The amorphization is structurally like melting, and as such, may not require mobility of atoms in the dissolving phase. In the case of Ni-Zr, for example, the formation of the amorphous phase may require motion of only nickel, while the formation of the crystalline compounds may require motion of both nickel (fast) and zirconium (slow) to bring about the structural ordering in the new phase. Once the amorphous phase has formed, the formation of the stable crystalline compound is made much more difficult by the removal of the Ni-Zr interface. In solid state amorphization there is a limiting thickness of amorphous phase which can be formed. The growth of the amorphous phase.is restricted by the nucleation and growth of a stable crystalline intermetallic phase at the interfaces between the amorphous phase and the elemental layers. The nucleation in this case is not affected by the negative heat of mixing; a quantitative treatment based on transient nucleation kinetics may be found elsewhere [ 351.
6. Hypercooling Small alloy droplets, produced for example by atomization, may be cooled rapidly, and in the absence of a solid container may become highly undercooled. When crystal nucleation does occur, the solidification of the droplet, at least in its early stages, is very rapid, and highly metastable microstructures may be produced. It would be desirable for such a microstructure to be uniform throughout each droplet, but this is normally prevented by the release of latent heat. The initial solidification is so rapid as to be essentially adiabatic, and the droplet warms up (recalescence). As it does so the solidification becomes less rapid, resulting in microstructural change (e.g. less solute trapping and a coarser microstructure), and the temperature may rise above the melting point of a given metastable phase so that it cannot form at all. In the final stage the solidification rate is controlled, not by the solidification processes themselves but by the rate of heat extraction. This final stage, giving microstructures close to equilibrium, can be avoided if the droplet can be “hypercooled”. In this condition the droplet is so undercooled that the enthalpy of the liquid is less than or equal to the enthalpy of the solid phase (or phase mixture) at some maximum acceptable temperature. This is shown in Fig. 5. The maximum acceptable temperature would be the equilibrium melting temperature T,,, if rapid solidification to the equilibrium phases was desired, but less than T, if
140
TH
TH
TM
Temperature
Fig. 5. An enthalpy-temperature diagram showing lines for a liquid (1) and a crystalline phase mixture (c). The broken line is for a liquid with AC, > 0. 1, isothermal freezing at T,; 2, recalescence up to T,, followed by 1; 3, adiabatic freezing of a hypercooled droplet.
metastable phases were to be produced throughout the droplet. Taking the former case, the critical temperature for hypercooling TH is such that H,(T,) =H,(T,), i.e. Till $ C,,,(T) dT = AH(Tm) ti
(3)
Clearly, if C,, 1 is enhanced, hypercooling is more readily achieved (the critical temperature is moved from TH to THl in Fig. 5). This will be true whether or not C,, ,is changed. If the temperature of the droplet at the end of complete adiabatic solidification is to be less than T,, then hypercooling will be aided when C,,, is not only large, but larger than C,, c. This is precisely the case for liquid alloys, and it suggests that for at least some alloys hypercoohng may be easier to achieve in practice than predictions based on AC,, = 0 would suggest, and easier than for pure metals. This is just as well, since hypercooling of pure metals is very difficult, having been achieved with certainty only for lead (small droplets on a substrate [ 361) and for mercury (in an emulsion [37]).
7. Crystal growth kinetics It is of importance in modelling glass formation and the effects of rapid solidification in general, to know how the kinetics of solidification are affected by the thermodynamics of the undercooled liquid. Here we consider only the rate of crystal growth in a congruently freezing system. A onedimensional model for crystal growth, described in detail elsewhere [ 381, is used. The model takes account of the release of latent heat at the crystalliquid interface and solves the heat flow equation using a finite difference
141
scheme. The spatial element typically has a dimension of 0.1 - 1.0 pm. The crystal growth rate is calculated from the instantaneous interface temperature, assuming a diffusive-type atomic jump frequency. The model is applied to solidification of a splat on a substrate, with heat extracted into the substrate. The substrate temperature is taken to be 300 K and the heat transfer coefficient to be lo7 W m-* K-l. The initial, uniform temperature of the splat may be selected. If the liquid is initially undercooled, the calculation may be relevant for droplets hitting a substrate; if superheated for melt spinning. In either case, the intention is to calculate whether any glass will be formed because of the rapid heat extraction even when nucleation on the substrate is automatic. Nucleation is assumed to occur on contact provided that the temperature of the alloy in contact is less than T,. For the purposes of illustration we model a hypothetical alloy with parameters similar to those for Fe&S,, [ 391, but which is assumed to freeze congruently. The parameters are: T, = 1448 K; Tg = 700 K; AH(T,) = 13.76 kJ g-atom-‘; thermal diffusivity, 1.2 X lo-’ m* s-l; atomic diffusivity, D,, exp{--B/(T - To)}; B = 1300 K; T, = 581 K; Do= 1.4 X lo-* m* s-l; atomic diameter, 2.5 X lo-lo m; AC,(T,) = 0 or 12 J KM1g-atom-‘. The free energy of the undercooled liquid is calculated using eqn. (2). Figures 6 - 8 show the evolution of the crystal-liquid interface temperature as the solidification proceeds on a planar front through the splat. On the distance scale, 0 pm corresponds to the top of the splat and 5 ,um corresponds to the surface in contact with the substrate, i.e. solidification proceeds from right to left. There is a competition in determining the interface temperature between the release of latent heat and the external heat extraction. Three cases are considered. In the first (Fig. 6), AC,(T,) is taken to be zero. The behaviour is shown for a selection of initial melt temperatures. For initial melt temperatures of 1140 K or less the interface temperature
21400 -
c i
0
12 Position
3 (pm]
4
5
Fig. 6. The evolution of the crystal-liquid interface temperature from right to left through a splat at the initial temperatures undercooled liquid is assumed to have no excess specific heat.
Position
(pm)
as solidification proceeds (in Kelvin) marked. The
Fig. 7. As for Fig. 6, but the effect of an excess specific heat of the liquid (ACP K-r g-atom-r) on the driving force for solidification is included.
= 12 J
142
0
1
2 Position
3 (pm)
4
5
Fig. 8. As for Fig. 7, but the effect of the excess specific freezing is also included.
on the latent heat of
drops below T, before solidification is complete, i.e. some glass will be formed. This initial temperature represents an initial undercooling of 310 K, however. Next the effect of a positive AC,(T,) value, such as might be found for a good glass-former, is calculated. In Fig. 7 the effect of this AC,( T,) value in lowering A G( T), and therefore also the crystal growth rate, is shown. Now some glass can be formed if the initial melt temperature is 1210 K or less. This is still a substantial undercooling of 240 K. However, AC,(T,) also has the effect of lowering the latent heat of fusion as the liquid is cooled. When this effect is taken into account, the behaviour in Fig. 8 is calculated. The effect is dramatic: now a glass can be obtained even for an initial melt superheat of 150 K. Figures 6 - 8 serve to illustrate the importance of the excess specific heat of undercooled alloy liquids. It should be noted, however, that for FessBZO, although Fig. 8 is the best approximation, the predicted depth of crystallization is still too great. This could be because the crystal growth rate is slower than calculated because of the need for eutectic partitioning, or because nucleation on the substrate is in some way hindered.
8. Inverse melting Blatter and von Allmen [40] reported that annealing a thin amorphous film of Ti&rsO at 1073 K caused transformation to a metastable supersaturated b.c.c. solid solution (/3 phase), and that subsequent annealing at 873 K caused reversion to the amorphous phase. This suggests a sluggish, but nonetheless equilibrium reversible transformation in which a crystal transforms to a liquid on cooling. This postulated transformation has been termed inverse melting [41], and it appears to occur in this case without solute partitioning. The Dubey and Ramachandrarao expression for AG(T) can, therefore, be used as a basis for modelling the thermodynamics in this system. The p phase can be formed by normal freezing of the melt, so if
143
there is to be an inverse melting point at lower temperature the free energy curves G(T) of the p phase and of the liquid or amorphous phase must intersect twice. At the inverse melting point the liquid-amorphous phase must have a lower entropy than the crystal. Although the Dubey and Ramachandrarao expression does lead to a reduction in AS and to a curvature in AG(T), even for a good glass-former such as Pd,,,Ni$,, neither AS nor AG becomes negative. In Ti-Cr some additional effect must be operating, and one possibility is that the liquid-amorphous phase is chemically ordered while the /3 phase is not [ 411. The ideal mixing entropy for Ti,,Cr,, would be 5.08 J K-’ g-atom-‘, and this is likely to be realized in the fl phase which has a tendency to phase separate rather than order. If the entropy of chemical mixing in the liquid-amorphous phase is taken as zero, the excess mixing entropy of the 0 phase could outweigh the entropy of partitionless fusion, reduced at substantial undercooling from its value of 6.98 J K-’ g-atom-’ at Z’,. If so, the chemically ordered undercooled liquid could have an entropy lower than that of the chemically disordered crystal, thus being a system exhibiting Kauzmann’s apparent paradox. Although the calculation based only on configurational entropy is very crude, it does at least show that inverse melting is thermodynamically possible, even without vibrational and electronic contributions to AS( 7’). Inverse melting provides an interesting example of amorphous phase formation by a route other than rapid quenching. Indeed for Ti,,Cr,, rapid quenching (except at the very highest rates obtainable using pulsed lasers) yields the fl phase because this simple crystal structure nucleates with ease at low undercooling. However, the amorphous phase can be obtained by annealing the fl phase at a temperature which is sufficiently low (e.g. 873 K) for the amorphous phase to be more stable. At the comparatively low temperature of amorphization, the stable intermetallic compounds cannot nucleate and spinodal decomposition of the p phase is suppressed by strain effects. Since the /_Iphase can be obtained in bulk on comparatively slow cooling, this amorphization may be practically significant in providing a route to bulk amorphous phase formation without the necessity for rapid heat extraction and a thin dimension.
Acknowledgments The author is pleased to acknowledge his collaborations with L. Battezzati, P. V. Evans and R. J. Highmore in the field covered by this paper.
References 1 R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly and D. D. Wagman, Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. 2 D. Turnbull, J. Chem. Phys., 20 (1952) 411.
144 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
J. H. Perepezko and J. S. Paik, J. Non-Cryst. Solids, 61, 62 (1984) 113. W. Kauzmann, Chem. Rev., 43 (1948) 219. L. Battezzati and A. L. Greer, Znt. J. Rapid Solidification, 3 (1987) 23. F. Sommer, J. J. Lee and B. Predel, 2. Metallkd., 74 (1983) 100. E. Nassif, P. Lamparter, B. Sedelmeyer and S. Steeb, Z. Naturforsch. Teil A, 38 (1983) 1093. P. Lamparter, W. Sperl, S. Steeb and J. Bletry, Z. Naturforsch. Teil A, 37 (1982) 1223. P. H. Gaskell, J. Non-Cyst. Solids, 75 (1985) 329. J. M. Dubois, P. H. Gaskell and G. Le Caer, in S. Steeb and H. Warlimont (eds.), Rapidly Quenched Metals, North-Holland, Amsterdam, 1985, p. 567. H. S. Chen and D. Turnbull, J. Appl. Phys., 38 (1967) 3646. H. S. Chen and D. Turnbull, J. Chem. Phys., 48 (1968) 2560. F. Sommer, G. Bucher and B. Predel, J. Phys. (Paris) Colloq., 8 (41) (1980) 563. E. Garrone and L. Battezzati, Philos. Mag. B, 52 (1985) 1033. L. Battezzati and E. Garrone, Z. Metallkd., 75 (1984) 305. P. V. Evans, A. Garcia-Escorial, P. E. Donovan and A. L. Greer, MRS Symp. Proc., 57 (1987) 239. K. S. Dubey and P. Ramachandrarao, Acta Metall., 32 (1984) 91. C. V. Thompson and F. Spaepen, Acta Metall., 27 (1979) 1855. D. R. H. Jones and G. A. Chadwick, Philos. Mag., 24 (1971) 995. N. Hirai and H. Eyring, J. Appl. Phys., 29 (1958) 810. N. Hirai and H. Eyring, J. Polym. Sci., 37 (1959) 51. J. D. Hoffman, J. Chem. Phys., 29 (1958) 1192. J. H. Hildebrand and E. D. Eastman, J. Am. Chem. Sot., 37 (1915) 2452. F. Sommer, Z. Metallkd., 73 (1982) 72. P. Ramachandrarao a,nd S. Lele, Mater. Sci. Forum, 3 (1985) 449. H. A. Davies, in F. E. Luborsky (ed.), Amorphous Metallic Alloys, Butterworths, London, 1983, p. 8. H. A. Davies, in B. Cantor (ed.), Rapidly Quenched Metals ZZZ,The Metals Society, London, 1978, p. 1. D. Turnbull and M. H. Cohen, J. Chem. Phys., 34 (1961) 120. R. J. Highmore, J. E. Evetts, A. L. Greer and R. E. Somekh, Appl. Phys. Lett., 50 (1987) 566. W. J. Boettinger, in B. H. Kear, B. C. Giessen and M. Cohen (eds.), Rapidly Solidified Amorphous and Crystalline Alloys, Elsevier, New York, 1982, p. 15. B. M. Clemens, Phys. Rev. B, 33 (1986) 7615. R. J. Highmore, R. E. Somekh, A. L. Greer and J. E. Evetts, Mater. Sci. Eng., 97 (1988) 83. N. Saunders and A. P. Miodownik, J. Mater. Res., 1 (1986) 44. F. Spaepen and R. B. Meyer, Ser. Metall., 10 (1976) 257. R. J. Highmore, A. L. Greer, J. A. Leake and J. E. Evetts, Mater. Lett., 6 (1988) 401. M. J. Stowell, T. J. Law and J. Smart,Proc. R. Sot. London, Ser. A, 318 (1970) 231. J. H. Perepezko, Mater. Sci. Eng., 65 (1984) 125. P. V. Evans and A. L. Greer, Mater. Sci. Eng., 98 (1988) 357. F. Spaepen and C. J. Lin, in M. von Allmen (ed.), Amorphous Metals and NonEquilibrium Processing, Les Editions de Physique, Les Ulis, 1984, p. 65. A. Blatter and M. von Allmen, Phys. Rev. Lett., 54 (1985) 2103. A. L. Greer, J. Less-Common Met., 140 (1988) 327.
of the Less-Common
THERMODYNAMICS
Metals,
145 (1988)
131
OF UNDERCOOLED
131
- 144
LIQUIDS*
A. L. GREER University Pembroke
of Cambridge, Department of Materials Street, Cambridge, CB2 392 (U.K.)
Science
and Metallurgy,
(Received May 31,1988)
Summary Undercooled alloy liquids may exhibit an excess specific heat capacity, indicating ordering. On cooling, the enthalpy and entropy of fusion decrease, and the driving force for crystallization does not rise as rapidly as would otherwise be expected. These effects are found to a large extent in systems showing deep eutectic freezing, solid state amorphization and inverse melting. There are consequences for the ease of hypercooling and for crystal growth kinetics.
1. Introduction At the melting point of a pure metal the crystal and liquid have closely similar specific heat capacities. It is noticeable, however, that at high undertoolings the liquid specific heat tends to exceed that of the solid. Such an excess specific heat is found to be particularly large for some alloy liquids, and is correlated with glass-forming ability. The thermodynamic properties of undercooled alloy liquids are of importance in understanding and modelling many solidification processes (especially rapid solidification and glass formation), and when the excess specific heat has been measured below the melting point the properties can readily be’derived. Metallic glasses have specific heats similar to those of the corresponding crystalline solids, but they are configurationally frozen liquids and have thermodynamic properties related to those of the undercooled liquid at the glass transition temperature Tg. Figure 1 shows schematically the excess of the enthalpy, entropy and Gibbs free energy (AH, AS and AG respectively) of the undercooled liquid or glass over that of the crystalline solid. The effect of an excess liquid specific heat AC, (= C,, , - C,, ,) is shown in comparison with the behaviour expected when AC, = 0. It should be noted that the assumption AC, = 0 is usual, both in phase diagram fitting and in kinetic calculations. It is suggested in this paper that this assumption may be seriously misleading, and *Paper presented at the Symposium on the Preparation and Properties of Metastable Alloys at the E-MRS Spring Meeting, Strasbourg, May 31 - June 2,1988. 0 Elsevier Sequoia/Printed
in The Netherlands
132
Tg Temperature Fig. 1. The excess of the enthalpy, entropy and Gibbs free energy of the liquid (1) or glass (g) over that of the crystal between the glass transition temperature Tg and the melting point T,. The full lines are for a material with AC, = 0, the broken lines are for AC, > 0.
that it is not necessary even in the absence of AC, data in the undercooled regime. Usually such data are not available, but estimation is possible. The thermodynamics of undercooled pure metal and alloy liquids are briefly reviewed below, and the origins of the excess specific heat are considered. Some consequences of the excess specific heat in thermodynamic and kinetic modelling are illustrated both for solidification and for solid state amorphization. The latter can be considered here because of the similarity between amorphous alloys produced by solid state reaction and metallic glasses produced from the melt.
2. Undercooled Liquid pure metals For many pure metals the liquid specific heat at the melting point 7’, is known [l].It may be greater or less than that of the crystalline solid, but is usually quite similar. The specific heat of undercooled liquid metals, however, is very difficult to measure because of their tendency to crystallize. This tendency can be reduced by hindering nucleation. In the emulsion technique [2] this is achieved by dispersing the liquid metal as droplets in an organic carrier fluid, and coating the droplets with a surfactant to prevent coalescence. Using this technique undercoolings of 0.3 - 0.4 T, can be reached, and this capability has been exploited recently [ 31 to measure C,, 1 below T,. It was found that C,, 1 increases as the temperature falls, such that AC, = A + BT, where A and B are constants. For indium, for example,
133
A = 16.69 J g-atom-’ K-’ and B = -0.404 J g-atom-’ KP2 [3]. For the metals studied (bismuth, mercury, indium and tin) the C!,,, value rose by 3% to 20% in the accessible undercooling range. This leads to only a small deviation of AG(T) from the linear behaviour when AC, = 0 (Fig. 1). The value of AG(T) at the maximum undercooling attained differs by just a few per cent from the value estimated using AC, = 0. Nonetheless, the positive excess liquid specific heat, increasing as the temperature is lowered, does indicate that configurational entropy is lost as the liquid is cooled. This loss of entropy, or ordering, was first pointed out by Kauzmann [4] for organic liquids, and may in such cases lead to AS < 0 at finite temperature. This apparent paradox is averted by glass formation. In metallic systems, pure liquids do not have a sufficiently large AC, value to lead to a dramatic lowering of AS and consequent glass formation, but this behaviour can be observed in alloy liquids.
3. Ordering in alloy liquids For many binary metallic systems the enthalpy of mixing the liquid components has been determined (normally well above the melting point of either component). For glass-forming systems it has always been found to be negative [5]. The entropy of mixing has not often been measured, but it seems that in glass-forming systems it may be much less than ideal. Indeed in one case, Al-Ca, it has been found to be negative [6], indicating not only a low configurational entropy but also other significant effects (vibrational or electronic) on the entropy. Both the enthalpy and entropy of glass-forming systems suggest ordering, and there is direct evidence for this from diffraction experiments. Glass-forming liquids exhibit radial distribution functions with significant correlations out to greater spacing than is usual for melts [7]. In addition, there is much evidence of substantial short-range order in metallic glasses, e.g. NisIBig [8], based on the coordination polyhedra found in the corresponding crystalline compounds [9, lo]. The most direct approach to determining the thermodynamic properties of undercooled liquid alloys is derivation from measured specific heat values. These difficult measurements have been undertaken in just a few cases : Au-Si [ll], Au-Ge-Si [12] and Mg-Cu [13]. For glass-forming liquids such as these, however, another approach is possible. If a glass is formed, by rapid quenching or otherwise, its heat of crystallization AH, can be measured. This gives the enthalpy difference between the glass and the crystal close to T,. It has been found for a wide variety of eutectic systems, which on rapid quenching are glass-forming, that at the eutectic composition -AH, is only 0.5 to 0.7 times the heat of fusion at the eutectic temperature T, [ 141. This corresponds to an average excess specific heat of the liquid of about 0.8 AS( T,) in the temperature range between Tg and T, [ 151. The excess specific heat of the liquid alloy must reflect mainly an excess specific heat of mixing in the liquid state, since the excess specific heat of the
134
elemental liquid components is not expected to be large. Since the specific heat of the crystalline alloy is expected to be very similar to the specific heats of the elemental crystalline components, we find that [ 51 AHmix
-AHmix
e AH(T,) -
AH(Tg)
(1)
where AHmix and AH are the heat of mixing in the liquid and the heat of fusion respectively. This relationship has been verified using data for Aus,.,Si,s., and Zr,,Cu,, [5]. The heat of mixing referred to here is the heat of formation of the liquid alloy from its liquid components, and it is related not only to simple mixing but also to the structural state of the alloy. That AHmix becomes more negative on cooling, as is revealed by eqn. (1) and the straightforward measurements of AH, (= -AH(Z’,)) and AH(T,), indicates an increased order on cooling. There is accordingly a decreased entropy of the mixed configurations at low temperature, but it should be remembered that even at high temperature (greater than T,) the liquid can be significantly ordered. By making use of eqn. (l), the measured relationship between AH, and AH(T,) and the approximate constancy of As(T,) in metallic systems, it has been shown [5] that in glass-forming alloys there should be a linear relationship between (T, - T,)/T, and (Z’, - T,)/T,, where T, is the melting point of the main component. This relationship has been verified experimentally and has been used to predict the Tg value of a glass-forming system (Fe-B) in which the glass transition on heating is obscured by crystallization. In more stable metallic glasses the crystallization on heating may occur only at some temperature interval above Tg. In this case the specific heat measured in the interval between Tg and crystallization is that of the highly undercooled liquid alloy. The specific heat of the liquid alloy from T, down to Tg may then be obtained by interpolation rather than extrapolation. This has been done for Pd&Ni,,,P,, and AG(T) derived for this alloy is shown in Fig. 2 (from ref. 16). The other, curves in the figure show the predictions of various models. The most successful is that of Dubey and Ramachandrarao
500
600
700
800
900
Temperature (K)
Fig. 2. The excess of the Gibbs free energy of liquid Pd4$JL,,,PZ0 over that of the crystalline solid as a function of themperature (+), calculated from measured specific heat. The straight line shows the behaviour for AC, = 0, and the other curves are for various models: DR [17], TSI and TSII [16,18], BG [15] and JC [19].
135
based on the hole theory of the liquid. In this theory, developed by Hirai and Eyring [ 20, 211, it is assumed that the free volume in a liquid is distributed as discrete holes of constant volume and formation energy. The Dubey and Ramachandrarao expression for A G( 2’) is [I?]
AG(T) =
AH(T T
m
-
(2 )
where AT is the undercooling. The expression was also found to agree well with measurements on Aus,.$i,s.6 and Au,,Ge,,,Si,, [17]. The other curves in Fig. 2 show pr~ictions based on a constant AC, value. The expression of Jones and Chadwick [19], used with the value AC,(T,), works rather well. The remaining treatments do not use a measured AC, value, but instead estimate it. Hoffman f22] assumes AH = 0 at T x Tg, Thompson and Spaepen [ 181 assume AS = 0 at T 3 Tg and Battezzati and Garrone [15] assume AC, = 0.8 AS(T,). The methods for predicting AG(T) are considered in more detail in ref. 16. Here, we can note that the methods based on the measured AC,(T,) values are good, with the Dubey and Ramachandrarao expression giving a particularly impressive fit. The parameters in this expression, Tm, AH(T,) and AC,(T,), control the degree of deviation of AGfT) from the linear behaviour expected if AC, = 0. Thus the expression can be used to model liquids with widely differing glassforming abilities. The main imitation of this approach to pr~icting the thermodynamic properties of undercooled liquid alloys is that it can be applied only to those cases in which there is congruent freezing (including eutectic systems). For other systems which do not freeze congruently, prediction is more difficult. If high temperature thermodynamic mixing data are available for the liquid, the associated solution model may be useful. This model, originally proposed by Hildebrand and Eastman 1231, is based on the assumption that clusters or associates exist in the melt. Evidence for clustering has been provided by the composition dependence of viscosity and structure factor 1241, but direct thermodynamic data are used to obtain the model parameters. Only one type of associate AiBi (i, i are integers) is considered in a melt of A and B atoms. Although this must be a very crude description of the topological and chemical order in the liquid alloy, it seems to be adequate in at least some cases. With one type of associate there are five model parameters: three regular solution parameters, C, C1 and CZ and the enthalpy and entropy of formation of the associates. The regular solution parameters quantify the non-ideality of the A-B, A-AiBj and B-AiBj interactions. The simultaneous evaluation of all five parameters is not straightforward, but has been achieved, for example, by non-linear least-squares fitting to thermodynamic mixing data for In-Sb and Mg-Sn [25]. For these systems the phase diagrams were calculated without using any adjustable parameters [Z5] ; remarkable agreement with experiment was found, considering that the equilibria represented in the diagrams are at a temperature about 600 K lower than the temperature at which the mixing data were obtained.
136
4. Euteetic melting andsolid state foliation It is well established that in binary alloy systems there is a correlation between glass-forming ability and deep eutectics [26]. The glass transition temperature in concentrated alloys is usually not strongly dependent on composition; this has been verified experimentally [27], and has some basis in free volume theory [28]. Accordingly it is at eutectics that T, is depressed closest to Tg and crystallization on cooling the melt is least likely. Deep eutectics reflect unusual stability of the liquid, and this is a consequence of a negative heat of mixing. The thermodynamics are most easily treated at the eutectic composition. Figure 3(a) is an enthalpy-temperature diagram (referred to the enthalpy of the crystalline phases as a standard state) for a eutectic alloy. At the eutectic point the liquid must have a higher entropy than the solid phases, i.e. AS and AH are positive. The enthalpy change on melting at the eutectic point is given by the algebraic sum of the appropriately weighted average heat of fusion of the two solid phases and the heat of mixing of the two resulting liquids. The negative AH,i, value shown on the figure must be of smaller magnitude than the average heat of fusion A Ha,,, of the solid phases. Direct measurements on simple eutectic systems, with no intermediate compounds, bear this out. For Aus,.,Si,,,, for example, AHmix = -10 kJ g-atom-l, while (assuming negligible solid solution) AH,,,, = 19.75 kJ g-atom-‘.
I
I h
i
T,
Fig. 3. The enthalpy of the liquid alloy (l.a.), liquid components (l.c.) and crystalline phases (e.p.) (referred to the last as a standard state) for (a)eutectic melting and (b) solid state amorphization. AH,, is the average enthalpy of fusion of the components.
In solid state amorphization, alternating layers of two polycrystalline elements, made by deposition or mechanical reduction, react on annealing to give a solid amorphous phase. As such a phase appears to be identical with metallic glasses formed from the melt, the process of solid state amorphization can be viewed as eutectic melting. The product is a glass, not a liquid, because the reaction occurs below Tg . The melting occurs on heating, not because a critical thermodynamic temperature (the melting point) is reached, but because sufficient atomic mobility is attained to permit the transformation. The system can be regarded throughout as being above its eutectic melting temperature.
137
It is possible to conceive of a deep eutectic, with thermodynamics as illustrated in Fig. 3(a), in which T, < Tg . In such a case the two crystalline solid phases should react to form an amorphous phase at around Tg on heating. The superheated eutectic solid would melt endothermically. Yet in normal solid state amorphization experiments, the amorphization which occurs on heating is strongly exothermic. In such a case the heat of mixing in the liquid phase must outweigh the average heat of fusion of the solid phases, as shown in Fig. 3(b). The ~orphous phase which is formed, being configurationally frozen, is likely to have a specific heat close to that of the crystalline phase mixture. If not, it may exhibit an excess specific heat. In either case, the heat of amorphization will not become positive at any lower temperature, as it must if there is to be an equilibrium amorphization, or melting, reaction. Thus solid state amorphization in systems such as Ni-Zr, in which the reaction is known to be exothermic [29], although it may be structurally the same as the melting of a superheated eutectic solid, cannot be so regarded in a strict thermodynamic sense. The equilibrium eutectic temperature does not exist. The two extrapolated liquiduses, for the equilibrium between the liquid or solid amorphous phase and the two elemental crystalline phases, do not meet. When solidification is sufficiently rapid to be partitionless (and ignoring congruent freezing of compounds) the composition range between the T,, lines is one in which a glass must be formed [30]. Often glasses may be formed over a wider range due to kinetic constraints on crystal growth. Solid state amorphizing systems such as Ni-Zr are cases in which, if intermetallic compounds could be avoided, rapid quenching of the melt would give glasses over a particularly wide range. With their liquidus curves not meeting they show an even greater departure from ideal liquid behaviour than in the case in which the two T, lines for the elemental solid solutions do not meet. This arises because of the highly negative heat of mixing in the liquid phase. In practice, rapid quenching of the melt does not avoid intermetallic compounds. Solid state ~orphization is more effective in avoiding crystallization, because at the low temperature typical for the process, crystal nucleation is difficult; this is considered in more detail in the next section. If the mixing in the liquid phase is made still less ideal compared with that in the solid, not only can the heat of ~o~hization become negative, but so too can the entropy change on ~orphization. In this case an equilibrium amorphization reaction is again possible, but with the amorphous phase being the low temperature phase in the transformation. This “inverse melting” is discussed later.
5. Nucleation in solid state amorpbization Here we consider the effect that a strongly negative heat of mixing in the liquid (such as is found in Ni-Zr) may have on the nucleation of a
138
product phase at the interface between layers of the two elements. This is of relevance in understanding why an amorphous phase is found to grow initially at such an interface, in preference to thermodynamically more stable crystalline compounds. In some cases there may be no nucleation stage in the formation of the product phase. In deposited Ni-Zr multilayers there is structural [31] and calorimetric [32] evidence that there is an initial amorphous layer at the interface. Nevertheless, such an initial reacted layer may not always be present, and it is of interest to calculate what would be involved in its nucleation. The negative enthalpy of mixing for amorphous Ni-Zr is only slightly smaller than for the equ~ibrium interme~llic compounds, and has been estimated for the equiatomic alloy to be -43 kJ g-atom-i [ 33]. Using this value, and taking a regular solution model with only nearest-neighbour bonding, it is found that there is a decrease in enthalpy of approximately 1 J m-’ on forming a reacted amorphous monolayer at the Ni-Zr interface. This decrease is, to a good approximation, the free energy of formation of the monolayer. The entropy change is likely to be small, since the amorphous phase is chemically ordered, but even if the ideal mixing entropy were to be obtained, the contribution to the free energy change would be less than 10% of the enthalpy change. The free energy of formation calculated in this way includes the chemical contribution to the interfacial energies, but not the structural con~ibution. A typical value of the ~terfacial energy due only to a structural difference is 0.1 J m- 2 1341. Using this value, the critical nucleation radius of a circular disc, one monolayer thick, of amorphous alloy at the Ni-Zr interface is calculated to be much less than one atomic diameter. The critical radius for an intermetallic compound, which would have a somewhat greater free energy of formation, would be even smaller. Thus the strongly negative enthalpy of mixing in a system such as Ni-Zr ensures that there is no nucleation barrier for formation of either an amorphous phase or crystalline intermetallic phases at the interface. The amorphous phase may form in preference to the intermetallics because it can form over a wider composition range. This range is indicated in Fig. 4, and is wider than that given by the common tangent construction.
Composition
Fig. 4. Schematic free energy curves for two crystalline elemental phases and a liquidamorphous phase (l/a) involved in solid state amorphization. Initially the amorphous phase may form in the composition range a to b.
139
Once the amorphous phase has formed, the common tangent construction would give the equilibrium compositions at the crystalline-amorphous interfaces. The solid state amorphization takes place far from equilibrium (and, as pointed out above, the equilibrium may not exist at any temperature), however, and just as in solute trapping in rapid solidification, there may be deviations from local equilibrium at the interfaces. Another reason for the amorphous phase to form first is its structure. The amorphization is structurally like melting, and as such, may not require mobility of atoms in the dissolving phase. In the case of Ni-Zr, for example, the formation of the amorphous phase may require motion of only nickel, while the formation of the crystalline compounds may require motion of both nickel (fast) and zirconium (slow) to bring about the structural ordering in the new phase. Once the amorphous phase has formed, the formation of the stable crystalline compound is made much more difficult by the removal of the Ni-Zr interface. In solid state amorphization there is a limiting thickness of amorphous phase which can be formed. The growth of the amorphous phase.is restricted by the nucleation and growth of a stable crystalline intermetallic phase at the interfaces between the amorphous phase and the elemental layers. The nucleation in this case is not affected by the negative heat of mixing; a quantitative treatment based on transient nucleation kinetics may be found elsewhere [ 351.
6. Hypercooling Small alloy droplets, produced for example by atomization, may be cooled rapidly, and in the absence of a solid container may become highly undercooled. When crystal nucleation does occur, the solidification of the droplet, at least in its early stages, is very rapid, and highly metastable microstructures may be produced. It would be desirable for such a microstructure to be uniform throughout each droplet, but this is normally prevented by the release of latent heat. The initial solidification is so rapid as to be essentially adiabatic, and the droplet warms up (recalescence). As it does so the solidification becomes less rapid, resulting in microstructural change (e.g. less solute trapping and a coarser microstructure), and the temperature may rise above the melting point of a given metastable phase so that it cannot form at all. In the final stage the solidification rate is controlled, not by the solidification processes themselves but by the rate of heat extraction. This final stage, giving microstructures close to equilibrium, can be avoided if the droplet can be “hypercooled”. In this condition the droplet is so undercooled that the enthalpy of the liquid is less than or equal to the enthalpy of the solid phase (or phase mixture) at some maximum acceptable temperature. This is shown in Fig. 5. The maximum acceptable temperature would be the equilibrium melting temperature T,,, if rapid solidification to the equilibrium phases was desired, but less than T, if
140
TH
TH
TM
Temperature
Fig. 5. An enthalpy-temperature diagram showing lines for a liquid (1) and a crystalline phase mixture (c). The broken line is for a liquid with AC, > 0. 1, isothermal freezing at T,; 2, recalescence up to T,, followed by 1; 3, adiabatic freezing of a hypercooled droplet.
metastable phases were to be produced throughout the droplet. Taking the former case, the critical temperature for hypercooling TH is such that H,(T,) =H,(T,), i.e. Till $ C,,,(T) dT = AH(Tm) ti
(3)
Clearly, if C,, 1 is enhanced, hypercooling is more readily achieved (the critical temperature is moved from TH to THl in Fig. 5). This will be true whether or not C,, ,is changed. If the temperature of the droplet at the end of complete adiabatic solidification is to be less than T,, then hypercooling will be aided when C,,, is not only large, but larger than C,, c. This is precisely the case for liquid alloys, and it suggests that for at least some alloys hypercoohng may be easier to achieve in practice than predictions based on AC,, = 0 would suggest, and easier than for pure metals. This is just as well, since hypercooling of pure metals is very difficult, having been achieved with certainty only for lead (small droplets on a substrate [ 361) and for mercury (in an emulsion [37]).
7. Crystal growth kinetics It is of importance in modelling glass formation and the effects of rapid solidification in general, to know how the kinetics of solidification are affected by the thermodynamics of the undercooled liquid. Here we consider only the rate of crystal growth in a congruently freezing system. A onedimensional model for crystal growth, described in detail elsewhere [ 381, is used. The model takes account of the release of latent heat at the crystalliquid interface and solves the heat flow equation using a finite difference
141
scheme. The spatial element typically has a dimension of 0.1 - 1.0 pm. The crystal growth rate is calculated from the instantaneous interface temperature, assuming a diffusive-type atomic jump frequency. The model is applied to solidification of a splat on a substrate, with heat extracted into the substrate. The substrate temperature is taken to be 300 K and the heat transfer coefficient to be lo7 W m-* K-l. The initial, uniform temperature of the splat may be selected. If the liquid is initially undercooled, the calculation may be relevant for droplets hitting a substrate; if superheated for melt spinning. In either case, the intention is to calculate whether any glass will be formed because of the rapid heat extraction even when nucleation on the substrate is automatic. Nucleation is assumed to occur on contact provided that the temperature of the alloy in contact is less than T,. For the purposes of illustration we model a hypothetical alloy with parameters similar to those for Fe&S,, [ 391, but which is assumed to freeze congruently. The parameters are: T, = 1448 K; Tg = 700 K; AH(T,) = 13.76 kJ g-atom-‘; thermal diffusivity, 1.2 X lo-’ m* s-l; atomic diffusivity, D,, exp{--B/(T - To)}; B = 1300 K; T, = 581 K; Do= 1.4 X lo-* m* s-l; atomic diameter, 2.5 X lo-lo m; AC,(T,) = 0 or 12 J KM1g-atom-‘. The free energy of the undercooled liquid is calculated using eqn. (2). Figures 6 - 8 show the evolution of the crystal-liquid interface temperature as the solidification proceeds on a planar front through the splat. On the distance scale, 0 pm corresponds to the top of the splat and 5 ,um corresponds to the surface in contact with the substrate, i.e. solidification proceeds from right to left. There is a competition in determining the interface temperature between the release of latent heat and the external heat extraction. Three cases are considered. In the first (Fig. 6), AC,(T,) is taken to be zero. The behaviour is shown for a selection of initial melt temperatures. For initial melt temperatures of 1140 K or less the interface temperature
21400 -
c i
0
12 Position
3 (pm]
4
5
Fig. 6. The evolution of the crystal-liquid interface temperature from right to left through a splat at the initial temperatures undercooled liquid is assumed to have no excess specific heat.
Position
(pm)
as solidification proceeds (in Kelvin) marked. The
Fig. 7. As for Fig. 6, but the effect of an excess specific heat of the liquid (ACP K-r g-atom-r) on the driving force for solidification is included.
= 12 J
142
0
1
2 Position
3 (pm)
4
5
Fig. 8. As for Fig. 7, but the effect of the excess specific freezing is also included.
on the latent heat of
drops below T, before solidification is complete, i.e. some glass will be formed. This initial temperature represents an initial undercooling of 310 K, however. Next the effect of a positive AC,(T,) value, such as might be found for a good glass-former, is calculated. In Fig. 7 the effect of this AC,( T,) value in lowering A G( T), and therefore also the crystal growth rate, is shown. Now some glass can be formed if the initial melt temperature is 1210 K or less. This is still a substantial undercooling of 240 K. However, AC,(T,) also has the effect of lowering the latent heat of fusion as the liquid is cooled. When this effect is taken into account, the behaviour in Fig. 8 is calculated. The effect is dramatic: now a glass can be obtained even for an initial melt superheat of 150 K. Figures 6 - 8 serve to illustrate the importance of the excess specific heat of undercooled alloy liquids. It should be noted, however, that for FessBZO, although Fig. 8 is the best approximation, the predicted depth of crystallization is still too great. This could be because the crystal growth rate is slower than calculated because of the need for eutectic partitioning, or because nucleation on the substrate is in some way hindered.
8. Inverse melting Blatter and von Allmen [40] reported that annealing a thin amorphous film of Ti&rsO at 1073 K caused transformation to a metastable supersaturated b.c.c. solid solution (/3 phase), and that subsequent annealing at 873 K caused reversion to the amorphous phase. This suggests a sluggish, but nonetheless equilibrium reversible transformation in which a crystal transforms to a liquid on cooling. This postulated transformation has been termed inverse melting [41], and it appears to occur in this case without solute partitioning. The Dubey and Ramachandrarao expression for AG(T) can, therefore, be used as a basis for modelling the thermodynamics in this system. The p phase can be formed by normal freezing of the melt, so if
143
there is to be an inverse melting point at lower temperature the free energy curves G(T) of the p phase and of the liquid or amorphous phase must intersect twice. At the inverse melting point the liquid-amorphous phase must have a lower entropy than the crystal. Although the Dubey and Ramachandrarao expression does lead to a reduction in AS and to a curvature in AG(T), even for a good glass-former such as Pd,,,Ni$,, neither AS nor AG becomes negative. In Ti-Cr some additional effect must be operating, and one possibility is that the liquid-amorphous phase is chemically ordered while the /3 phase is not [ 411. The ideal mixing entropy for Ti,,Cr,, would be 5.08 J K-’ g-atom-‘, and this is likely to be realized in the fl phase which has a tendency to phase separate rather than order. If the entropy of chemical mixing in the liquid-amorphous phase is taken as zero, the excess mixing entropy of the 0 phase could outweigh the entropy of partitionless fusion, reduced at substantial undercooling from its value of 6.98 J K-’ g-atom-’ at Z’,. If so, the chemically ordered undercooled liquid could have an entropy lower than that of the chemically disordered crystal, thus being a system exhibiting Kauzmann’s apparent paradox. Although the calculation based only on configurational entropy is very crude, it does at least show that inverse melting is thermodynamically possible, even without vibrational and electronic contributions to AS( 7’). Inverse melting provides an interesting example of amorphous phase formation by a route other than rapid quenching. Indeed for Ti,,Cr,, rapid quenching (except at the very highest rates obtainable using pulsed lasers) yields the fl phase because this simple crystal structure nucleates with ease at low undercooling. However, the amorphous phase can be obtained by annealing the fl phase at a temperature which is sufficiently low (e.g. 873 K) for the amorphous phase to be more stable. At the comparatively low temperature of amorphization, the stable intermetallic compounds cannot nucleate and spinodal decomposition of the p phase is suppressed by strain effects. Since the /_Iphase can be obtained in bulk on comparatively slow cooling, this amorphization may be practically significant in providing a route to bulk amorphous phase formation without the necessity for rapid heat extraction and a thin dimension.
Acknowledgments The author is pleased to acknowledge his collaborations with L. Battezzati, P. V. Evans and R. J. Highmore in the field covered by this paper.
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