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The Thermophysical Properties of Bulk Metallic Glass-Forming Liquids

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15/12/2006

Bulk metallic glass-forming liquids are alloys with typically three to five metallic components that have a large atomic-size mismatch and a composition close to a deep eutectic. They are dense liquids with small free volumes and viscosities that are several orders of magnitude higher than in pure metals or previously known alloys. In addition, these melts are energetically closer to the crystalline state than other metallic melts due to their high packing density in conjunction with a tendency to develop short-range order. These factors lead to slow crystallization kinetics and high glass-forming ability. Crystallization kinetics is very complex, especially in the vicinity of the glass transition, due to the influence of phase separation and the decoupling of the diffusion constants of the different species.

Ralf Busch

TABLE OF CONTENTS
INTRODUCTION THERMODYNANICS OF SUPERCOOLED BMG LIQUIDS VISCOSITY OF THE UNDERCOOLED LIQUID AND STRONG LIQUID BEHAVIOR DIFFUSION, VISCOSITY, AND RELAXATION CRYSTALLIZATION ACKNOWLEDGEMENTS References


Figure 1. A time-temperature-transformation diagram for the primary crystallization of V1. Data obtained by electrostatic levitation (s)¹⁰ and processing in high-purity carbon crucibles (l)¹¹ are included. Calculated times for a crystalline volume fraction of x = 10^-4 , using D_eff m h^-1(solid line) and D_eff m exp(–Q _eff/kT) (dashed line).

INTRODUCTION

In recent years new families of multicomponent glass-forming alloys, such as La-Al-Ni,¹ Zr-Ni-Al-Cu,² Mg-Cu-Y,³ and Zr-Ti-Cu-Ni-Be,⁴ that exhibit very good glass-forming ability have been discovered. These bulk metallic glass (BMG) formers show high thermal stability of supercooled (undercooled) liquid with respect to crystallization, enabling the study of the thermophysical properties of metallic melts in the supercooled state and the exploration of their properties and possible applications.⁵

Since their discovery in 1960 by Duwez and coworkers,⁶ much research has been devoted to the study of metallic glasses. Crucial contributions were made by Turnbull, Chen, and collaborators,^7,8 who showed, for example, that rapidly quenched Au-Si alloys exhibit a well-defined calorimetric glass transition and viscosity. In the following years, thermophysical properties like viscosity, relaxation, diffusion, and thermodynamics were explored (e.g., Reference 9). However, the lack of thermal stability in the supercooled liquid of metallic systems with respect to crystallization did not allow studies deep into the supercooled liquid region, and most studies were done below or in the vicinity of the glass transition region.

The novel BMG-forming liquids can now be studied in a much broader time and temperature range. It is now even possible to measure time-temperature- transformation (TTT) diagrams, such as the diagram shown for Zr_41.2Ti_13.8Cu_10.0Ni_12.5Be_22.5 (V1) alloy in Figure 1. In this diagram, the onset times for isothermal crystallization are plotted as a function of temperature. The data were obtained by electrostatic levitation¹⁰ and crystallization in high-purity graphite crucibles.¹¹ The diagram shows the typical “C” or nose shape and a minimum crystallization time of 60 s at 895 K. For previously known glass-forming alloys, the times were of the order of milliseconds, resulting in the need for rapid quenching for vitrification. The TTT diagram of V1 reflects a very low critical cooling rate of about 1 K/s, which is 5–6 orders of magnitude lower than in earlier metallic glass-forming systems. The C shape is the result of the competition between the increasing driving force for crystallization and the slowing of kinetics (effective diffusivity) of the atoms. In simple liquids, atomic mobility is connected to viscosity via the Stokes-Einstein relation.

If steady-state nucleation is assumed, the rate of nucleation is determined by the product of a thermodynamic contribution and a kinetic contribution as

I_s = A·D_eff exp(–DG*/kT)

(1)

where, D_eff is the effective diffusivity, T is temperature, and A is a constant. For high temperatures, the diffusivity is set proportional to the inverse of the viscosity as D_eff µ 1/h. The activation barrier for nucleation, DG^*, is given as DG^* = 16ps³/DG², where s is the interfacial energy between liquid and solid, and DG is the driving force for crystallization.

From these considerations, it is clear that the driving force and the diffusivity and viscosity, respectively, are crucial parameters for understanding the glass-forming ability of supercooled BMG-forming liquids.

THERMODYNAMICS OF SUPERCOOLED BMG LIQUIDS

The driving force for crystallization is approximated by the Gibbs free energy difference between supercooled liquid and crystal, DG. Strictly speaking, DG is the driving force for a polymorphic transformation (without a composition change) and a lower bound for the driving force for transformations with composition change. Models for the Gibbs free energy of metallic liquids have been developed by Turnbull,¹² Thompson and Spaepen,¹³ and others.^14,15 The thermodynamics of undercooled metallic liquids with respect to the crystal have also been incorporated in calculation of phase diagrams (CALPHAD) calculations.¹⁶ An experimental assessment of the Gibbs free energy difference between liquid and solid requires the determination of fusion heat and the difference in the specific heat capacity, Dc_p(T), between supercooled liquid and crystal.

Figure 2 shows examples of the specific heat capacity, c_p, of several glass-forming alloys as a function of temperature in the supercooled liquid state. The temperature axis is normalized to the melting temperature of the respective alloy. With the exception of the c_p of Nb-Ni, which was calculated by the CALPHAD method,¹⁷ all curves are based on experimental results. The c_p of the liquid at the melting temperature is higher than that of the crystalline state and increases with undercooling even further. However, for the good BMG formers (V1,¹⁹ Zr_46.75Ti_8.25Cu_7.5Ni₁₀Be_27.5 [V2], and Mg-Cu-Y²⁰), the curves are much shallower than the other two alloys, which are materials with less favorable glass-forming ability. This is due to the strong glassy nature of BMG liquids, leading to a slower change in the configurations of the system as the glass transition is approached.



Figure 2. Specific heat capacities in the supercooled liquid for several alloys normalized to the eutectic temperature, T_eut . Data on Nb-Ni is taken from Reference 17; data on Au-Pb-Sb is from Reference 18. For the other data, see References 19 and 20. Good BMGs show a shallow c_p curve that is indicative for strong liquid behavior.		Figure 3. The difference in Gibbs free energy between the liquid and the crystalline state for glass-forming liquids. The data on Zr-Ni are taken from CALPHAD calculations.²⁴ For the other alloys, see References 19, 20, and 23. The critical cooling rates for the alloys are indicated in the plot.

The Gibbs free energy of the undercooled liquid with respect to the crystal, DG_1–x (T), can be calculated by integrating the specific heat capacity difference between supercooled liquid and crystal and taking into account the fusion heat. For example, this has been done for Pd-Ni-P,²¹ Pd-Ni-Cu-P,²² Zr-Ti-Cu-Ni-Be,¹⁹ Mg-Cu-Y,²⁰ and Cu-Ti-Zr-Ni.²³ In Figure 3, the Gibbs free energy difference for a selection of glass-forming systems is plotted as a function of undercooling. All temperatures are normalized to the melting temperature of the respective alloy. The alloys show different critical cooling rates between 1 K/s for the pentary V1 and about 10⁴ K/s for the binary Zr₆₂Ni₃₈. The glass formers with the lowest critical cooling rates have smaller Gibbs free energy differences than do the glass formers with high critical cooling rates.

The driving force for crystallization decreases with increasing BMG forming ability. This originates mainly from the smaller entropies of fusion found in the BMG system, since the fusion entropy determines the slope of the free-energy curve at the melting point. A low value of entropy indicates a small free volume and a tendency to develop short-range order at the melting point and in the supercooled liquid. In fact, it was shown for V1 that the free volume at the melting point is only one percent.^11,25 These findings are consistent with the fact that BMG formers are very viscous liquids at the melting point and upon undercooling.

VISCOSITY OF THE UNDERCOOLED LIQUID AND STRONG LIQUID BEHAVIOR

Besides thermodynamic considerations, viscosity is the kinetic key parameter that determines the nucleation and growth of crystals in the moderately undercooled liquid (Equation 1). Viscosities of amorphous alloys have been previously measured (e.g., by Chen and Turnbull⁷ and Spaepen and coworkers^26,27). The viscosities were determined in the glass-transition region, but crystallization did not allow measurements of the equilibrium viscosity below 10⁹ Pa · s or for times long enough to eliminate relaxation effects.

Figure 4. Viscosity data of V1 (solid symbols) and V4 (open symbols) measured by various methods. Fits to the data were obtained using the VFT equation (dashed curve: V1; dotted curve: V4). The data on V1 were additionally fitted using the modified free volume model by Cohen and Grest ³²(solid curve). See References 11 and 28–30 for details.

Figure 5. An Angell plot comparing the viscosities of different types of glass-forming liquids. The plot shows that V1, V4, and Mg-Cu-Y are relatively strong liquids. See References 11, 20, 29, and 30 for discussion. The data on the nonmetallic liquids are taken from Reference 31.

Viscosity can be measured in bulk glass-forming systems in a much larger temperature and time range than before. Figure 4 shows the viscosities for V1 and V4 in an Ahrrenius plot, obtained by different methods (see References 11, 28–30 for details). The data cover 15 orders of magnitude, with the exception of the temperature range where the crystallization nose in the TTT diagram (Figure 1) is observed. All equilibrium viscosity data measured in the supercooled liquid can be described well with the Vogel-Fulcher-Tammann (VFT) relation

h = h₀ ·exp[D* ·T₀/(T – T₀)]

(2)

Equation 2 represents a formulation of the VFT equation according to Angell³¹ that includes the fragility parameter, D^*, and the VFT temperature, T₀, where the barriers with respect to flow would go to infinity. Another successful description of the data of V1¹¹ is given by the Cohen and Grest model,³² which is a modification of Turnbull’s free volume model. Surprisingly, the apparent singularity at T₀ in the VFT equation occurs far below the calorimetric glass transition in BMGs,^20,29 in contrast to what was generally expected from earlier work on metallic systems. This indicates that BMG-forming liquids behave kinetically much closer to silicate melts, which are far more robust against crystallization than previous alloys.

The increasing viscosity of the liquid as a function of undercooling reflects the decreasing mobility of atoms that occurs during supercooling. This is observed in all supercooled liquids, whether they are metallic or nonmetallic. Silicate liquids, called strong liquids, usually show high equilibrium melt viscosities and Ahrrenius behavior in the slowing mobility in the supercooled melt. The other limits are fragile liquids with low melt viscosities and a more abrupt change in the kinetics close to the glass transition. The fragility concept^31,33 is used to classify the different temperature dependencies of the viscosity. To compare the viscosities measured in different glass-forming systems, data are plotted in an Ahrrenius plot in which the inverse temperature axis is multiplied by the temperature, T_g, at which the viscosity of the respective alloy is 10¹² Pa · s.

Figure 5 shows the viscosities of the BMG-forming V1 and V4 liquids in comparison with a selection of some nonmetallic liquids as well as the Mg₆₅Cu₂₅Y₁₀ BMG. As mentioned, strong glass formers like SiO₂ are one limit. They exhibit a very small VFT temperature and a very high melt viscosity. Fragile glass formers show a VFT temperature near the glass transition temperature as well as low melt viscosities. The parameter D^* in the VFT equation (Equation 2) is a measure of the fragility of the liquid. D^* is on the order of two for the most fragile liquids and yields 100 for the strongest glass former, SiO₂. The Mg₆₅Cu₂₅Y₁₀ and Zr-Ti-Cu-Ni-Be BMGs behave closer to the strong glasses than the fragile glasses and have fragility parameters of about D^* = 20. The melt viscosity of BMGs is on the order of 2–5 Pa · s. They are about three orders of magnitude more viscous than pure metals or some binary alloys, where viscosities on the order of 5 ´ 10^–3 Pa · s are observed. It is worth noting that the relaxation behavior of BMG-forming liquids as probed by neutron scattering,^34,35 creep experiments,^29,30,36 and the calorimetric glass transition²⁹ is also consistent with the strong-liquid nature of BMGs.

The strong-liquid behavior of BMGs, as reflected by the temperature dependence of viscosity, leads to the fact that the kinetics stays sluggish in the supercooled liquid region compared to other metallic liquids. This results in a small nucleation and growth rate of crystals. The strong liquid nature is also expressed in the shallower c_p curves of BMGs close to the glass transition compared to the curves of more fragile glass formers (Figure 2), because the structural changes for strong liquids are more gradual as T_g is approached.

The high melt viscosities in multicomponent BMG-forming liquids, as well as the small entropy differences between liquid and solid, have structural origin. There are several experimental findings that shed some light on the structure of bulk metallic glasses.

First, specific volume measurements of the liquid and the crystalline state by electrostatic levitation²⁵ in conjunction with viscosity data¹¹ show that the free volume, at least for V1, is only one percent at the melting point. The large variety of atoms with different sizes leads to a more effective packing of the atoms in the liquid state. This view is supported by the fact that supercooled V4 liquids have a very small compressibility.³⁷

Second, the fusion entropy in BMG-forming liquids is small.^19,20,29 This suggests that there is pronounced chemical short-range order present in the melt. In fact, atom-probe field ion microscopy and small-angle neutron scattering experiments show that this chemical short-range ordering can result in clustering³⁸ or phase separation.^39,40

Third, the thermal and electrical conductivity of Zr-Ti-Cu-Ni-Be bulk metallic glasses is smaller than in previously known glass-forming alloys. This suggests that an increasing number of electrons become localized in bulk metallic glasses as a result of directional bonds due to short range order. This effect, together with the small amount of free volume, makes the liquid more rigid with respect to shear flow and brings it energetically closer to the crystalline ground state.

DIFFUSION, VISCOSITY, AND RELAXATION

Diffusion in the glassy state has previously been studied (e.g., Faupel and coworkers^41,42); in the last five years, progress has been made to understand diffusion in the glass-transition region and above. Particular attention has been paid to V1 and V4. Beginning with work by Geyer et al.⁴³ on beryllium, the mobilities of other species in the glass-transition region have been explored. Mehrer, Macht, and coworkers have conducted tracer-diffusion studies on a variety of elements, such as cobalt, nickel, and aluminum.^44,45 In addition, the isotope effect of cobalt in V4 has been measured, indicating collective hopping.⁴⁶

For temperatures below the glass transition, the differences in diffusivity between the smallest and the largest atoms are up to five orders of magnitude. This fact was known prior and has been studied extensively in connection with solid-state amorphization in binary metallic glass formers.^47,48 However, the glass-transition region and the supercooled liquid had not been accessible in the past due to crystallization.

For the small beryllium atoms, pronounced changes in apparent activation energies for diffusion were found in the glass transition region,^43,49 which were attributed to a crossover from solid-like hopping at low temperature to a cooperative shearing at high temperatures. The latter process is connected to the viscosity and structural relaxation time of the material. Tang et al. observed the hopping times of beryllium in the glass transition region directly by NMR experiments.^50,51

The fact that the glass transition temperature is a kinetic temperature makes the design and the interpretation of experiments that probe atomic motion difficult. The time scale on which a property is observed determines whether the material is considered a supercooled liquid or a glassy solid. This has prompted continual debates and further experiments.

One interesting fact is that the apparent activation energies for diffusion of the different species increase monotonously with atomic size and the temperature dependence of the diffusion for the large atoms such as aluminum (comparable to zirconium) approaches the temperature dependence of the equilibrium viscosity. Masuhr et al.¹¹ compared an overall (internal) relaxation time

t_h = h/G_h

with the characteristic diffusional relaxation time of the atoms given by

t_D,I = l²/(6D_i)

which is different for each of the species in the alloy. Here, l is the atomic diameter, and G_h is a constant.

Figure 6. Structural relaxation times from viscosity are compared with the relaxation times for atomic diffusion, which have been determined from intrinsic diffusion constants of nickel,⁴⁴ aluminum,⁴⁵ and cobalt.⁴⁵

These times are calculated from the measured viscosities and diffusivities, respectively (Figure 6). Around 600 K, the diffusivities of, for example, aluminum and nickel differ by three orders of magnitude, while they show a tendency to merge at higher temperatures. The temperature dependence of aluminum diffusion is similar to that of viscosity t_D,Al @ t_h/14 (dashed curve). The proportionality factor implies that the mean displacement of an aluminum atom in the supercooled liquid during a typical relaxation time is roughly four interatomic diameters.

The smaller atoms show a significantly smaller absolute value of t_D, indicating that the mobility of the small atoms at low temperature is decoupled from the relaxation kinetics given by viscosity. However, at high temperatures a crossover is expected. At low temperatures, the atoms predominantly move by atomic hopping because the motion due to cooperative shearing is several orders of magnitude slower, as indicated by the structural relaxation time (Figure 6). Close to the crossover, the time scale for shear becomes comparable to the hopping time, and the liquid-like motion is predominant. When a melt is undercooled, each species experiences a different (glass) transition range below which it moves in a solid-like environment.

CRYSTALLIZATION

The decoupling of the diffusivity from the structural relaxation upon undercooling strongly affects the crystallization process. The effective diffusivities of the smaller atoms stay much larger than would be expected if they would follow the viscosity. This means that at low temperatures the diffusion is not governed by the VFT law, but by Ahrrenius behavior. In Reference 11 the nucleation and growth process was modeled according to Ullmann and Davis. For simplification purposes, classical nucleation theory was applied. From this approach,

(3)

is developed for the time to crystallize a small volume fraction, x. In this equation, I_s is the steady-state nucleation rate (Equation 1), and u is the growth rate. With the commonly used relation D_eff µ h^–1, the minimum in the nucleation time, t_x, at 895 K requires an interfacial energy of s = 0.040 J/m².

The corresponding temperature dependence of t_x is plotted in Figure 1. While the low-temperature crystallization data cannot be described with the assumption D_eff µ h^–1, satisfactory agreement between the experimental findings and classical nucleation and growth theory is found (solid curve) for temperatures above 850 K. In contrast, an Arrhenius-like effective diffusivity D_eff µµ exp(–Q_eff/kT) with Q_eff = 1.2 eV describes very well the crystallization times in the vicinity of the glass transition as shown in Figure 1 (dashed curve). It is interesting to note that the qualitative temperature dependence of D_eff is very similar to that of the tracer diffusion of medium-sized atoms. This indicates that, in fact, the decoupling of the diffusion constants at lower temperatures plays an important role in the devitrification process.

It must be noted that the crystallization process (at least at temperatures below the crystallization nose) is likely to be much more complex, involving phase separation and complicated diffusion fluxes that are affected by the diffusional asymmetries between the different species. There are indications that many BMG-forming liquids become thermodynamically unstable, especially in the deeply supercooled liquid when the glass transition is approached. This leads to phase separation into two or more supercooled liquids and can trigger primary crystallization. Phase separation^52–55 and crystallization^56,57 have been studied to a large extent on V1. Alternative mechanisms for the devitrification have been proposed (e.g., by Kelton⁵⁸). Much work still needs to be done to understand the devitrification of BMGs and the structure of deeply supercooled BMG-forming liquids.

ACKNOWLEDGEMENTS

I thank W.L. Johnson for inspiration and continuing support and all his former and present group members at Caltech. My special thanks go to A. Masuhr, E. Bakke, T.A. Waniuk, W. Lui, and J. Schroers. My memories are with my late friend and fellow researcher at Caltech and JPL, Y.J. Kim. This work was supported by the U.S. Department of Energy (grant no. DEFG-03-86ER45242), the Alexander von Humboldt Foundation via the Feodor Lynen Program.

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