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Fragility, thermodynamic properties, and thermal stability of Pd-rich glass forming liquids
Materials Science and Engineering A 375–377 (2004) 417–421
Fragility, thermodynamic properties, and thermal stability of Pd-rich glass forming liquids G. Wilde a,∗ , I.R. Lu b , R. Willnecker b a
Institute of Nanotechnology, Research Center Karlsruhe, POB 3640, D-76021 Karlsruhe, Germany b DLR, Institute for Space Simulation, D-51170 Cologne, Germany
Abstract The thermal stability characteristics of three different Pd-rich bulk glass forming alloys, i.e. Pd40 Ni40 P20 , Pd77.5 Cu6 Si16.5 , and Pd43 Cu27 Ni10 P20 , have been investigated by continuous cooling experiments and by heating experiments following rapid melt quenching. Fragility parameters for these alloys are obtained from equilibrium viscosity data and, more directly if available, from the temperature-dependence of the average relaxation times obtained by modulation calorimetry. In addition, specific heat capacity data have been determined in dependence of temperature for the crystal, the undercooled liquid state and for the glass, respectively. Thus, a complete set of experimental data is available that allows a detailed comparison between the fragility characteristics and the thermal stabilities of the different alloy systems. As one result of this comparison, it is indicated that fragility and thermal stability are not correlated for these alloys, indicating further that vitrification is governed by nucleation control, i.e. by the avoidance of any nucleation. © 2003 Elsevier B.V. All rights reserved. Keywords: Bulk metallic glass; Fragility; Thermal stability; Relaxation time; Heat capacity; Modulation calorimetry
1. Introduction Among glass forming metallic alloys, Pd-rich systems take a special position since these were the first bulk metallic glasses that had been identified and since some improved alloy compositions based on Pd, still present the most stable examples for metallic glass formation [1]. Moreover, it was shown that bulk samples of the ternary Pd40 Ni40 P20 alloy can be vitrified by applying cooling rates as low as 0.16 K/s [2], which allowed the first direct determination of the specific heat of an undercooled metallic liquid throughout the entire undercooling range [3]. Thus, together with the more recently developed Zr-based systems [4], at least two classes of metallic alloys are known that allow vitrification at cooling rates less than 1 K/s. Experimental results obtained on the Zr-rich glass-forming alloys have led to the suggestion, that thermal stability and fragility are related such that the less fragile systems are more stable [5]. Additionally, it has been proposed that the fragility and the temperature dependence of the excess thermodynamic properties, i.e. the excess entropy, S [6], or the differences in specific heat between the undercooled ∗
Corresponding author. E-mail address: [email protected] (G. Wilde).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.10.218
melt and the crystalline equilibrium phases, Cp [7], are directly correlated, with the slope of S (or Cp (T)) being larger for more fragile systems. Consequently, it would be expected that the more fragile systems show the lowest stability and the largest temperature dependence of S (or Cp ), especially if chemically similar alloys are compared. In order to address the relation between fragility, thermodynamic properties and thermal stability experimentally, a series of Pd-base metallic glasses with distinctly different glass-forming ability have been investigated with respect to the temperature dependence of Cp and Cpl , the specific heat of the undercooled liquid, and concerning the thermal stability of the alloys against crystallization. In addition, temperature-modulated calorimetry (TMDSC) provided the opportunity to directly measure characteristic times of the slow ␣-relaxation modes of the deeply undercooled melt that are coupled to the occurrence of the glass transition upon continued cooling [8]. Thus, the fragility of the melt near the glass transition was measured directly.
2. Experimental details Glassy rods of the Pd-rich alloys of 3.5 mm in diameter and up to 35 mm in length were produced by either
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G. Wilde et al. / Materials Science and Engineering A 375–377 (2004) 417–421
flow casting or die casting in argon atmosphere. Small discs of about 100 mg were cut for the DSC measurements (Perkin-Elmer, Pyris 1 DSC). The enthalpy of fusion Hf and the heat capacity of the solid and liquid phases at higher temperatures, exceeding 1000 K, were measured with a differential heat flow calorimeter (Netzsch, DSC 404) applying a method of comparison with sapphire as the reference standard [3]. TMDSC measurements were performed with the Pyris 1 DSC operating in the modulation mode. Time periods for one heating rate oscillation, Γ , ranged from 30 to 1000 s with amplitudes of the corresponding temperature oscillations, A, always less than 1 K. The time dependence of the sample temperature in these measurements is given as: T(t) = T0 + R0 t + 8A/π2 ·[(sin 2πΓ −1 t) − (sin 6πΓ −1 t)/9 + (sin 10πΓ −1 t)/25]. The TMDSC measurements were performed during cooling the metastable melt at low average cooling rates, R0 , that ranged from 0.05 to 0.5 K/min from temperatures, T0 , well above the static glass transition. Thus, it was ensured that the static glass transition occurred always at temperatures that are below the temperature range where the frequency-dependent, dynamic glass transition was observed. Subsequent XRD measurements confirmed the absence of detectable crystalline fractions in the samples.
3. Results 3.1. Specific heat of the undercooled melt In order to characterize the temperature dependence of the excess thermodynamic properties of the glass forming alloys, the specific heat has been measured in the crystalline, the stable and undercooled liquid states and in the glass, respectively. Fig. 1 summarizes the results for Pd40 Ni40 P20 and Pd77.5 Cu6 Si16.5 . The specific heat measurements on Pd40 Ni40 P20 (Fig. 1a) cover the entire undercooling interval between the glass transition range and the melting temperature [3]. The results obtained for the quaternary PdCuNiP alloy that also covered the entire undercooling interval [1] are similar to the Cpl -values obtained for Pd40 Ni40 P20 . The PdCuSi alloy (Fig. 1b) could not be undercooled continuously at low rates from the melting to the glass transition range; crystallization occurred inevitably at about T = 270 K. Therefore, specific heat measurements on the glass and the deeply undercooled liquid have been performed on melt-cast samples. In order to determine the specific heat of the deeply undercooled liquid state of this alloy near the glass transition without interference of non-equilibrium relaxation effects, Cpl was determined during cooling after annealing the sample at T > Tg . For all three alloys, as for most metallic liquids that are unaffected by phase transformations [9], a linear dependence of Cpl on temperature was observed for the entire measurement interval that ranged from the glass transition up to the stable melt.
Fig. 1. (a) The specific heat of crystalline, vitreous, stable liquid, and undercooled Pd40 Ni40 P20 . For this alloy, the specific heat of the entire undercooling range of a metallic liquid could be determined for the first time. (b) Analogous results obtained on Pd77.5 Cu6 Si16.5 . Here, crystallization at deep undercooling prevented the Cp measurements over the entire undercooling interval. The circles mark the apparent specific heat data in the glass transition range where the measured values are time dependent.
3.2. Fragility and thermal stability In addition to measurements of the static specific heat, the dynamic specific heat, C∗ = C + iC , has been determined for Pd40 Ni40 P20 and Pd43 Cu27 Ni10 P20 by temperature-modulated calorimetry. In contrast to isochronal non-equilibrium measurements on the transition between undercooled liquid and glass, the present TMDSC measurements probe the so-called dynamic glass transition. This dynamic crossover that occurs in metastable thermodynamic equilibrium is governed by the coupling of an external attenuation frequency with the intrinsic frequencies for reversible structural relaxation of the liquid. At low attenuation frequencies, the structural rearrangements can follow the attenuation and the system reacts liquid-like. However, at higher frequencies the structural rearrangements cannot follow the external attenuation and the system shows a glass-like response. In the intermediate range of attenuation frequencies, external and internal time scales are similar and dispersion occurs. Thus, a maximum in the imaginary part of the complex specific heat occurs at
G. Wilde et al. / Materials Science and Engineering A 375–377 (2004) 417–421
the average inverse relaxation time, νp = 1/τa . Therefore, experimental observation of the frequency dependence of the dispersion in the response signal, i.e. of C∗ (ω), gives direct access to the intrinsic relaxation modes that are active within the selected frequency range. The imaginary contributions to the specific heat of Pd43 Cu27 Ni10 P20 that have been transformed into frequency space are summarized in Fig. 2a. Similar measurements have been conducted for Pd40 Ni40 P20 [8]. The experimental data that are given by the symbols have been fitted by Davidson–Cole stretched exponentials [10] in order to obtain νp (T) values from incomplete dispersion curves at a somewhat reduced accuracy. The details concerning the Davidson–Cole functional form, which is well-accepted to fit dynamic specific heat data accurately [11], has been published elsewhere [12]. In order to analyze the data with respect to fragility, the inverse peak frequencies of the C -data are summarized as an Arrhenius representation in Fig. 2b. For Pd40 Ni40 P20 a value of m = 41 is obtained for the fragility index, m = d log(η, τ)/dTg /T , by both, viscosity measurements [13] and TMDSC. The direct comparison of the average relaxation times for Pd40 Ni40 P20 (circles) and Pd43 Cu27 Ni10 P20 (stars) in Fig. 2b indicates a different temperature dependence of τ a in the glass transition region for these two alloys. This difference of the courses of τ a (T) corresponds to a variation of the fragility of the respective alloys with Pd43 Cu27 Ni10 P20 being the more fragile system with mτ = 51. This result is in agreement with qualitative observations from enthalpy relaxation experiments on both alloys [1]. Recent measurements of the viscosity of deeply undercooled Pd43 Cu27 Ni10 P20 [14] confirm the results obtained by TMDSC. For Pd77.5 Cu6 Si16.5 , TMDSC measurements are not feasible due to the reduced thermal stability at temperatures above Tg . Yet, the average relaxation times can be estimated from viscosity data [15] by applying the Stokes–Einstein equation: τ = (VH η)/(kB T), with the hydrodynamic volume, VH . These values are given as squares in Fig. 2b. Certainly, the absolute values for τ obtained from η have to be regarded with caution due to the size-dependence of the atomic mobility near Tg . However, concerning the temperature dependence of the average relaxation time, i.e. the fragility, this dependence just enters the considerations as a higher order term and can be neglected in a first order approximation. The comparison indicates a hierarchy of fragilities of the Pd-base metallic glasses as: m(PdNiP) < m(PdCuNiP) < m(PdCuSi). Fig. 2c summarizes the DSC heating curves for bulk amorphous samples of the three alloys that were conducted under identical conditions, i.e. at identical heating rates and after identical thermal history. Clearly, the temperature intervals, Tx = Tx − Tg , that are often taken as a measure of the thermal stability, are markedly different for the three alloys ranging between about 30 K for Pd77.5 Cu6 Si16.5 and almost 80 K for the PdCuNiP system. The succession of Tx -values correlates well with the observed critical
419
Fig. 2. (a) The imaginary contribution to the complex specific heat of Pd43 Cu27 Ni10 P20 . The isothermal C -curves in dependence of the attenuation frequency are obtained from the measured signals by Fourier transformation. (b) Average relaxation times in the undercooled liquid state of the Pd-rich alloys. (c) The normalized heat flow signals as obtained during heating at 20 K/min. Vertical bars indicate the respective temperature intervals between the glass transition and the onset of crystallization.
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cooling rates as measured by the maximum dimensions of fully vitreous samples [16]. 4. Discussion The results of the equilibrium static specific heat measurements indicate that the course of Cpl is linear in temperature throughout the entire undercooling interval for all three alloys. The slope of Cpl however, shows a marked difference between Pd77.5 Cu6 Si16.5 and the two P-containing alloys. A numerical fit to the experimental Cpl -values yields a succession of slopes, mC = |dCpl /dT | as: mC (PdNiP) = 3.43 × 10−4 > mC (PdCuNiP) = 1.02 × 10−4 > mC (PdCuSi) < 10−6 . Often, the slope of Cp is regarded as an indirect measure of the fragility, although the crystalline state is not included in the framework of liquid excitations as described, e.g. in current potential energy landscape models that couple thermodynamics to liquid dynamics [17]. Especially in the case of the PdCuSi-alloy it is obvious, that the temperature dependence of Cp is entirely dominated by the temperature dependence of the specific heat of the crystalline state. However, in the case of the present Pd-rich alloys, the succession of slopes would remain identical to the succession obtained from the Cpl courses. The comparison of the succession of fragility values with the sequence of thermal stabilities indicates that the quaternary system with the highest kinetic stability against crystallization is intermediate in fragility compared to the ternary alloys. This result is unexpected, since for Zr-rich glasses, Busch et al. [5] reported that the less fragile glass-forming alloys are kinetically more stable. However, that conclusion only applies under growth controlled vitrification conditions when a high melt viscosity prevents rapid growth after initial nucleation. During nucleation-controlled vitrification where nucleation is avoided, fragility and thermal stability are not necessarily coupled. Thus, the present result indicates clearly that in contrast to the Zr-rich bulk metallic glasses, vitrification of the most stable bulk metallic glasses PdNiP and PdCuNiP is nucleation controlled. Further comparison of the successions concerning fragility and thermal stability with the temperature dependence of Cp indicate that the most fragile system with the lowest thermal stability, PdCuSi, has the lowest value of dCp /dT. In fact, the specific heat of the undercooled melt of this alloy is characterized by a rather temperatureindependent value. Moreover, the ternary PdNiP alloy that is intermediate in thermal stability and that has the lowest fragility shows the largest slope of Cpl and Cp . These results indicate that the thermal stability of a given glass-forming system is governed by the kinetics of nucleation that depends on the complexity of the crystalline phases and on the topology of the phase diagram rather than on the dynamics of molecular relaxation in the undercooled melt as addressed by the fragility. It is obvious that high values of Cp or Cpl promote an enhanced decrease of the
excess entropy of the undercooled liquid that leads to values for the isentropic Kauzmann temperature [18] that are close to the calorimetrically observable glass transition, as characteristic for fragile systems. However, such considerations need to account for the melting entropy, Sf , besides the specific heat. It has been shown [1] that the quaternary PdCuNiP alloy is characterized by an extremely low value of Sf that amounts to only about 80% of the value measured for the ternary PdNiP alloy. Therefore, the excess entropy of the quaternary alloy approaches zero faster with increasing undercooling compared to the ternary alloy. For the PdCuSi alloy, an identical result was obtained. If the relative temperature difference between the observable glass transition at Tg (at a standard cooling rate of −10 K/min) and the Kauzmann temperature, ∆TK = [(Tg − TK )/TK ] is considered as a measure of the degree by which the undercooled liquid approaches the isentropic limit, then following sequence is obtained: TK (PdCuSi) = 0.130 < TK (PdCuNiP) = 0.134 < TK (PdNiP) = 0.138. Thus, the fragility characteristics as measured directly by modulation calorimetry or viscosity measurements and determined indirectly from thermodynamic measurements agree for the Pd-base alloys if the excess entropy instead of Cp is taken into account. 5. Summary Experimental data from static and dynamic calorimetry, thermal analysis, and viscosity measurements obtained on three Pd-based bulk glass forming alloys have been compared with respect to correlations between fragility, thermal stability, and thermodynamic properties. The results indicate that even for these closely related systems, the thermal stability is not correlated with the fragility or the excess thermodynamic functions. In this respect, long-standing models [19] that relate the thermal stability (or glass-forming ability) to reductions of the melting temperature and the melting entropy seem to be adequate to describe the relative thermal stability of the respective alloys. Concerning the relation between fragility and thermodynamics, the results indicate that a meaningful comparison needs to entail the temperature dependence of the excess entropy including its value at the melting temperature rather than the temperature dependence of Cp . Acknowledgements The authors gratefully acknowledge support by the DFG (G.W., Emmy Noether program, WI 1899/1-2; I.-R.L. and R.W., WI 1350/1–4). References [1] I.R. Lu, G. Wilde, G.P. Görler, R. Willnecker, J. Non-Cryst. Solids 250–252 (1999) 577.
G. Wilde et al. / Materials Science and Engineering A 375–377 (2004) 417–421 [2] R. Willnecker, K. Wittmann, G.P. Görler, J. Non-Cryst. Solids 156–158 (1993) 450. [3] G. Wilde, G.P. Görler, R. Willnecker, G. Dietz, Appl. Phys. Lett. 65 (1994) 397. [4] A. Peker, W.L. Johnson, Appl. Phys. Lett. 63 (1993) 2342. [5] R. Busch, E. Bakke, W.L. Johnson, Acta Mater. 46 (1998) 4725. [6] C.A. Angell, Science 267 (1995) 1924. [7] R. Busch, W. Liu, W.L. Johnson, J. Appl. Phys. 83 (1998) 4134. [8] G. Wilde, Appl. Phys. Lett. 79 (2001) 1986. [9] G. Wilde, J. Non-Cryst. Solids 307–310 (2002) 853. [10] D.W. Davidson, R.H. Cole, J. Chem. Phys. 18 (1950) 1417.
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[11] P. Lunkenheimer, U. Schneider, R. Brand, A. Loidl, Contemp. Phys. 41 (2000) 15. [12] G. Wilde, J. Non-Cryst. Solids 312–314 (2002) 537. [13] G. Wilde, G.P. Görler, R. Willnecker, H.J. Fecht, J. Appl. Phys. 87 (2000) 1141. [14] I.-R. Lu, Thesis, University of Ulm, Germany, 2002. [15] A.I. Taub, F. Spaepen, Acta Met. 28 (1980) 1781. [16] G. Wilde, I.-R. Lu, unpublished results. [17] F.H. Stillinger, Science 267 (1995) 1935. [18] W. Kauzmann, Chem. Rev. 43 (1948) 219. [19] D. Turnbull, Contemp. Phys. 10 (1969) 473.
Fragility, thermodynamic properties, and thermal stability of Pd-rich glass forming liquids G. Wilde a,∗ , I.R. Lu b , R. Willnecker b a
Institute of Nanotechnology, Research Center Karlsruhe, POB 3640, D-76021 Karlsruhe, Germany b DLR, Institute for Space Simulation, D-51170 Cologne, Germany
Abstract The thermal stability characteristics of three different Pd-rich bulk glass forming alloys, i.e. Pd40 Ni40 P20 , Pd77.5 Cu6 Si16.5 , and Pd43 Cu27 Ni10 P20 , have been investigated by continuous cooling experiments and by heating experiments following rapid melt quenching. Fragility parameters for these alloys are obtained from equilibrium viscosity data and, more directly if available, from the temperature-dependence of the average relaxation times obtained by modulation calorimetry. In addition, specific heat capacity data have been determined in dependence of temperature for the crystal, the undercooled liquid state and for the glass, respectively. Thus, a complete set of experimental data is available that allows a detailed comparison between the fragility characteristics and the thermal stabilities of the different alloy systems. As one result of this comparison, it is indicated that fragility and thermal stability are not correlated for these alloys, indicating further that vitrification is governed by nucleation control, i.e. by the avoidance of any nucleation. © 2003 Elsevier B.V. All rights reserved. Keywords: Bulk metallic glass; Fragility; Thermal stability; Relaxation time; Heat capacity; Modulation calorimetry
1. Introduction Among glass forming metallic alloys, Pd-rich systems take a special position since these were the first bulk metallic glasses that had been identified and since some improved alloy compositions based on Pd, still present the most stable examples for metallic glass formation [1]. Moreover, it was shown that bulk samples of the ternary Pd40 Ni40 P20 alloy can be vitrified by applying cooling rates as low as 0.16 K/s [2], which allowed the first direct determination of the specific heat of an undercooled metallic liquid throughout the entire undercooling range [3]. Thus, together with the more recently developed Zr-based systems [4], at least two classes of metallic alloys are known that allow vitrification at cooling rates less than 1 K/s. Experimental results obtained on the Zr-rich glass-forming alloys have led to the suggestion, that thermal stability and fragility are related such that the less fragile systems are more stable [5]. Additionally, it has been proposed that the fragility and the temperature dependence of the excess thermodynamic properties, i.e. the excess entropy, S [6], or the differences in specific heat between the undercooled ∗
Corresponding author. E-mail address: [email protected] (G. Wilde).
0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.10.218
melt and the crystalline equilibrium phases, Cp [7], are directly correlated, with the slope of S (or Cp (T)) being larger for more fragile systems. Consequently, it would be expected that the more fragile systems show the lowest stability and the largest temperature dependence of S (or Cp ), especially if chemically similar alloys are compared. In order to address the relation between fragility, thermodynamic properties and thermal stability experimentally, a series of Pd-base metallic glasses with distinctly different glass-forming ability have been investigated with respect to the temperature dependence of Cp and Cpl , the specific heat of the undercooled liquid, and concerning the thermal stability of the alloys against crystallization. In addition, temperature-modulated calorimetry (TMDSC) provided the opportunity to directly measure characteristic times of the slow ␣-relaxation modes of the deeply undercooled melt that are coupled to the occurrence of the glass transition upon continued cooling [8]. Thus, the fragility of the melt near the glass transition was measured directly.
2. Experimental details Glassy rods of the Pd-rich alloys of 3.5 mm in diameter and up to 35 mm in length were produced by either
418
G. Wilde et al. / Materials Science and Engineering A 375–377 (2004) 417–421
flow casting or die casting in argon atmosphere. Small discs of about 100 mg were cut for the DSC measurements (Perkin-Elmer, Pyris 1 DSC). The enthalpy of fusion Hf and the heat capacity of the solid and liquid phases at higher temperatures, exceeding 1000 K, were measured with a differential heat flow calorimeter (Netzsch, DSC 404) applying a method of comparison with sapphire as the reference standard [3]. TMDSC measurements were performed with the Pyris 1 DSC operating in the modulation mode. Time periods for one heating rate oscillation, Γ , ranged from 30 to 1000 s with amplitudes of the corresponding temperature oscillations, A, always less than 1 K. The time dependence of the sample temperature in these measurements is given as: T(t) = T0 + R0 t + 8A/π2 ·[(sin 2πΓ −1 t) − (sin 6πΓ −1 t)/9 + (sin 10πΓ −1 t)/25]. The TMDSC measurements were performed during cooling the metastable melt at low average cooling rates, R0 , that ranged from 0.05 to 0.5 K/min from temperatures, T0 , well above the static glass transition. Thus, it was ensured that the static glass transition occurred always at temperatures that are below the temperature range where the frequency-dependent, dynamic glass transition was observed. Subsequent XRD measurements confirmed the absence of detectable crystalline fractions in the samples.
3. Results 3.1. Specific heat of the undercooled melt In order to characterize the temperature dependence of the excess thermodynamic properties of the glass forming alloys, the specific heat has been measured in the crystalline, the stable and undercooled liquid states and in the glass, respectively. Fig. 1 summarizes the results for Pd40 Ni40 P20 and Pd77.5 Cu6 Si16.5 . The specific heat measurements on Pd40 Ni40 P20 (Fig. 1a) cover the entire undercooling interval between the glass transition range and the melting temperature [3]. The results obtained for the quaternary PdCuNiP alloy that also covered the entire undercooling interval [1] are similar to the Cpl -values obtained for Pd40 Ni40 P20 . The PdCuSi alloy (Fig. 1b) could not be undercooled continuously at low rates from the melting to the glass transition range; crystallization occurred inevitably at about T = 270 K. Therefore, specific heat measurements on the glass and the deeply undercooled liquid have been performed on melt-cast samples. In order to determine the specific heat of the deeply undercooled liquid state of this alloy near the glass transition without interference of non-equilibrium relaxation effects, Cpl was determined during cooling after annealing the sample at T > Tg . For all three alloys, as for most metallic liquids that are unaffected by phase transformations [9], a linear dependence of Cpl on temperature was observed for the entire measurement interval that ranged from the glass transition up to the stable melt.
Fig. 1. (a) The specific heat of crystalline, vitreous, stable liquid, and undercooled Pd40 Ni40 P20 . For this alloy, the specific heat of the entire undercooling range of a metallic liquid could be determined for the first time. (b) Analogous results obtained on Pd77.5 Cu6 Si16.5 . Here, crystallization at deep undercooling prevented the Cp measurements over the entire undercooling interval. The circles mark the apparent specific heat data in the glass transition range where the measured values are time dependent.
3.2. Fragility and thermal stability In addition to measurements of the static specific heat, the dynamic specific heat, C∗ = C + iC , has been determined for Pd40 Ni40 P20 and Pd43 Cu27 Ni10 P20 by temperature-modulated calorimetry. In contrast to isochronal non-equilibrium measurements on the transition between undercooled liquid and glass, the present TMDSC measurements probe the so-called dynamic glass transition. This dynamic crossover that occurs in metastable thermodynamic equilibrium is governed by the coupling of an external attenuation frequency with the intrinsic frequencies for reversible structural relaxation of the liquid. At low attenuation frequencies, the structural rearrangements can follow the attenuation and the system reacts liquid-like. However, at higher frequencies the structural rearrangements cannot follow the external attenuation and the system shows a glass-like response. In the intermediate range of attenuation frequencies, external and internal time scales are similar and dispersion occurs. Thus, a maximum in the imaginary part of the complex specific heat occurs at
G. Wilde et al. / Materials Science and Engineering A 375–377 (2004) 417–421
the average inverse relaxation time, νp = 1/τa . Therefore, experimental observation of the frequency dependence of the dispersion in the response signal, i.e. of C∗ (ω), gives direct access to the intrinsic relaxation modes that are active within the selected frequency range. The imaginary contributions to the specific heat of Pd43 Cu27 Ni10 P20 that have been transformed into frequency space are summarized in Fig. 2a. Similar measurements have been conducted for Pd40 Ni40 P20 [8]. The experimental data that are given by the symbols have been fitted by Davidson–Cole stretched exponentials [10] in order to obtain νp (T) values from incomplete dispersion curves at a somewhat reduced accuracy. The details concerning the Davidson–Cole functional form, which is well-accepted to fit dynamic specific heat data accurately [11], has been published elsewhere [12]. In order to analyze the data with respect to fragility, the inverse peak frequencies of the C -data are summarized as an Arrhenius representation in Fig. 2b. For Pd40 Ni40 P20 a value of m = 41 is obtained for the fragility index, m = d log(η, τ)/dTg /T , by both, viscosity measurements [13] and TMDSC. The direct comparison of the average relaxation times for Pd40 Ni40 P20 (circles) and Pd43 Cu27 Ni10 P20 (stars) in Fig. 2b indicates a different temperature dependence of τ a in the glass transition region for these two alloys. This difference of the courses of τ a (T) corresponds to a variation of the fragility of the respective alloys with Pd43 Cu27 Ni10 P20 being the more fragile system with mτ = 51. This result is in agreement with qualitative observations from enthalpy relaxation experiments on both alloys [1]. Recent measurements of the viscosity of deeply undercooled Pd43 Cu27 Ni10 P20 [14] confirm the results obtained by TMDSC. For Pd77.5 Cu6 Si16.5 , TMDSC measurements are not feasible due to the reduced thermal stability at temperatures above Tg . Yet, the average relaxation times can be estimated from viscosity data [15] by applying the Stokes–Einstein equation: τ = (VH η)/(kB T), with the hydrodynamic volume, VH . These values are given as squares in Fig. 2b. Certainly, the absolute values for τ obtained from η have to be regarded with caution due to the size-dependence of the atomic mobility near Tg . However, concerning the temperature dependence of the average relaxation time, i.e. the fragility, this dependence just enters the considerations as a higher order term and can be neglected in a first order approximation. The comparison indicates a hierarchy of fragilities of the Pd-base metallic glasses as: m(PdNiP) < m(PdCuNiP) < m(PdCuSi). Fig. 2c summarizes the DSC heating curves for bulk amorphous samples of the three alloys that were conducted under identical conditions, i.e. at identical heating rates and after identical thermal history. Clearly, the temperature intervals, Tx = Tx − Tg , that are often taken as a measure of the thermal stability, are markedly different for the three alloys ranging between about 30 K for Pd77.5 Cu6 Si16.5 and almost 80 K for the PdCuNiP system. The succession of Tx -values correlates well with the observed critical
419
Fig. 2. (a) The imaginary contribution to the complex specific heat of Pd43 Cu27 Ni10 P20 . The isothermal C -curves in dependence of the attenuation frequency are obtained from the measured signals by Fourier transformation. (b) Average relaxation times in the undercooled liquid state of the Pd-rich alloys. (c) The normalized heat flow signals as obtained during heating at 20 K/min. Vertical bars indicate the respective temperature intervals between the glass transition and the onset of crystallization.
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cooling rates as measured by the maximum dimensions of fully vitreous samples [16]. 4. Discussion The results of the equilibrium static specific heat measurements indicate that the course of Cpl is linear in temperature throughout the entire undercooling interval for all three alloys. The slope of Cpl however, shows a marked difference between Pd77.5 Cu6 Si16.5 and the two P-containing alloys. A numerical fit to the experimental Cpl -values yields a succession of slopes, mC = |dCpl /dT | as: mC (PdNiP) = 3.43 × 10−4 > mC (PdCuNiP) = 1.02 × 10−4 > mC (PdCuSi) < 10−6 . Often, the slope of Cp is regarded as an indirect measure of the fragility, although the crystalline state is not included in the framework of liquid excitations as described, e.g. in current potential energy landscape models that couple thermodynamics to liquid dynamics [17]. Especially in the case of the PdCuSi-alloy it is obvious, that the temperature dependence of Cp is entirely dominated by the temperature dependence of the specific heat of the crystalline state. However, in the case of the present Pd-rich alloys, the succession of slopes would remain identical to the succession obtained from the Cpl courses. The comparison of the succession of fragility values with the sequence of thermal stabilities indicates that the quaternary system with the highest kinetic stability against crystallization is intermediate in fragility compared to the ternary alloys. This result is unexpected, since for Zr-rich glasses, Busch et al. [5] reported that the less fragile glass-forming alloys are kinetically more stable. However, that conclusion only applies under growth controlled vitrification conditions when a high melt viscosity prevents rapid growth after initial nucleation. During nucleation-controlled vitrification where nucleation is avoided, fragility and thermal stability are not necessarily coupled. Thus, the present result indicates clearly that in contrast to the Zr-rich bulk metallic glasses, vitrification of the most stable bulk metallic glasses PdNiP and PdCuNiP is nucleation controlled. Further comparison of the successions concerning fragility and thermal stability with the temperature dependence of Cp indicate that the most fragile system with the lowest thermal stability, PdCuSi, has the lowest value of dCp /dT. In fact, the specific heat of the undercooled melt of this alloy is characterized by a rather temperatureindependent value. Moreover, the ternary PdNiP alloy that is intermediate in thermal stability and that has the lowest fragility shows the largest slope of Cpl and Cp . These results indicate that the thermal stability of a given glass-forming system is governed by the kinetics of nucleation that depends on the complexity of the crystalline phases and on the topology of the phase diagram rather than on the dynamics of molecular relaxation in the undercooled melt as addressed by the fragility. It is obvious that high values of Cp or Cpl promote an enhanced decrease of the
excess entropy of the undercooled liquid that leads to values for the isentropic Kauzmann temperature [18] that are close to the calorimetrically observable glass transition, as characteristic for fragile systems. However, such considerations need to account for the melting entropy, Sf , besides the specific heat. It has been shown [1] that the quaternary PdCuNiP alloy is characterized by an extremely low value of Sf that amounts to only about 80% of the value measured for the ternary PdNiP alloy. Therefore, the excess entropy of the quaternary alloy approaches zero faster with increasing undercooling compared to the ternary alloy. For the PdCuSi alloy, an identical result was obtained. If the relative temperature difference between the observable glass transition at Tg (at a standard cooling rate of −10 K/min) and the Kauzmann temperature, ∆TK = [(Tg − TK )/TK ] is considered as a measure of the degree by which the undercooled liquid approaches the isentropic limit, then following sequence is obtained: TK (PdCuSi) = 0.130 < TK (PdCuNiP) = 0.134 < TK (PdNiP) = 0.138. Thus, the fragility characteristics as measured directly by modulation calorimetry or viscosity measurements and determined indirectly from thermodynamic measurements agree for the Pd-base alloys if the excess entropy instead of Cp is taken into account. 5. Summary Experimental data from static and dynamic calorimetry, thermal analysis, and viscosity measurements obtained on three Pd-based bulk glass forming alloys have been compared with respect to correlations between fragility, thermal stability, and thermodynamic properties. The results indicate that even for these closely related systems, the thermal stability is not correlated with the fragility or the excess thermodynamic functions. In this respect, long-standing models [19] that relate the thermal stability (or glass-forming ability) to reductions of the melting temperature and the melting entropy seem to be adequate to describe the relative thermal stability of the respective alloys. Concerning the relation between fragility and thermodynamics, the results indicate that a meaningful comparison needs to entail the temperature dependence of the excess entropy including its value at the melting temperature rather than the temperature dependence of Cp . Acknowledgements The authors gratefully acknowledge support by the DFG (G.W., Emmy Noether program, WI 1899/1-2; I.-R.L. and R.W., WI 1350/1–4). References [1] I.R. Lu, G. Wilde, G.P. Görler, R. Willnecker, J. Non-Cryst. Solids 250–252 (1999) 577.
G. Wilde et al. / Materials Science and Engineering A 375–377 (2004) 417–421 [2] R. Willnecker, K. Wittmann, G.P. Görler, J. Non-Cryst. Solids 156–158 (1993) 450. [3] G. Wilde, G.P. Görler, R. Willnecker, G. Dietz, Appl. Phys. Lett. 65 (1994) 397. [4] A. Peker, W.L. Johnson, Appl. Phys. Lett. 63 (1993) 2342. [5] R. Busch, E. Bakke, W.L. Johnson, Acta Mater. 46 (1998) 4725. [6] C.A. Angell, Science 267 (1995) 1924. [7] R. Busch, W. Liu, W.L. Johnson, J. Appl. Phys. 83 (1998) 4134. [8] G. Wilde, Appl. Phys. Lett. 79 (2001) 1986. [9] G. Wilde, J. Non-Cryst. Solids 307–310 (2002) 853. [10] D.W. Davidson, R.H. Cole, J. Chem. Phys. 18 (1950) 1417.
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[11] P. Lunkenheimer, U. Schneider, R. Brand, A. Loidl, Contemp. Phys. 41 (2000) 15. [12] G. Wilde, J. Non-Cryst. Solids 312–314 (2002) 537. [13] G. Wilde, G.P. Görler, R. Willnecker, H.J. Fecht, J. Appl. Phys. 87 (2000) 1141. [14] I.-R. Lu, Thesis, University of Ulm, Germany, 2002. [15] A.I. Taub, F. Spaepen, Acta Met. 28 (1980) 1781. [16] G. Wilde, I.-R. Lu, unpublished results. [17] F.H. Stillinger, Science 267 (1995) 1935. [18] W. Kauzmann, Chem. Rev. 43 (1948) 219. [19] D. Turnbull, Contemp. Phys. 10 (1969) 473.