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The glass transition of Pd40Ni10Cu30P20 studied by temperature-modulated calorimetry
Journal of Non-Crystalline Solids 260 (1999) 228±234
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The glass transition of Pd40Ni10Cu30P20 studied by temperature-modulated calorimetry X. Hu a, T.B. Tan a, Y. Li a, G. Wilde b,*, J.H. Perepezko b b
a Department of Materials Science, National University of Singapore, Singapore Department of Materials Science and Engineering, University of Wisconsin±Madison, 1509 University Avenue, Madison, WI 53703, USA
Received 18 June 1999
Abstract The frequency dependence of the heat capacity in the glass-transition region of Pd40 Ni10 Cu30 P20 was studied by temperature-modulated dierential scanning calorimetry (TMDSC) during slow heating and cooling. Such data for low frequencies between 0.1 and 0.01 Hz are not available, especially for metallic glasses. A crossover between mixed static/ dynamic and purely dynamic response signals was observed for the lowest frequencies between 1/80 and 1/100 sÿ1 , which allows a direct determination of the average relaxation time at a given cooling rate during the static glass transition. Further, these results were used to evaluate the experimental parameters necessary to truly separate the static and dynamic response in low-frequency modulation calorimetry experiments to obtain the moduli of the dynamic speci®c heat. Ó 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 64.70.P; 82.20.R; 02.30. F, D; 65.20
1. Introduction It is widely accepted that the decrease in the measured speci®c heat during cooling that is characteristic for the glass transition re¯ects the freezing-in of liquid-like modes and is thus related to the slowing dynamics of the system upon cooling [1,2]. It has been shown that the thermodynamic aspect of the glass transition as revealed by the anomaly of the speci®c heat as well as the
* Corresponding author. Present address: Forschungszertrum Karlsruhe GmbH, INT, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany. Tel.: +49-7247 82 6414; fax: +49-7247 82 6368. E-mail address: [email protected] (G. Wilde)
kinetics associated with a divergence of relaxation times can be connected by heat-capacity spectroscopy [1]. Moreover, this approach is related to all modes of molecular motion and thus to the thermodynamics of the system in contrast to e.g. measurements of dielectric loss spectra [3] which probe the response of polarization-sensitive modes only. However, this method is restricted to nonmetallic glasses for several reasons: ®rst, metallic glasses of sucient stability against crystallization have been discovered rather recently. Secondly, the experimental set-up used for the heat-capacity spectroscopy requires a low thermal diusivity to keep the wavelength of the temperature oscillation small compared to the linear dimension of the temperature sensor. This boundary condition is ful®lled for organic systems and minimum
0022-3093/99/$ - see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 5 7 9 - 7
X. Hu et al. / Journal of Non-Crystalline Solids 260 (1999) 228±234
frequencies of about 1 sÿ1 , but is not compatible with the rather high thermal diusivity (and the small dimensions) of metallic glasses. However, temperature-modulated dierential scanning calorimetry (TMDSC) now provides a suitable means for speci®c heat measurements at low frequencies on metallic glasses. Such measurements on the easy glass-forming alloy Pd40 Ni10 Cu30 P20 were performed here, to our knowledge for the ®rst time, to identify the range of suitable measurement parameters. The results of these TMDSC experiments are then compared to conventional DSC measurements at constant rate to evaluate the frequency dependence of the calorimetric glasstransition signal in the low-frequency domain. 2. Experimental details 2.1. Sample preparation Bulk Pd40 Ni10 Cu30 P20 ingots of nominal composition were prepared by melting the pure element Pd (purity 99.9+%) with a pre-melted Cu±Ni±P master alloy (purity better than 99.9%) in an induction furnace. Glassy ribbons (width 3 mm, thickness 40 lm) of the same composition, as con®rmed by energy dispersive X-ray analysis (EDX), were produced by a single-roll melt-spinning facility, applying a tangential wheel speed of about 20 m/s. Microstructural characterization of the samples was carried out by standard X-ray diraction (Philips XRD) techniques. A well-de®ned state with respect to relaxation was obtained by heating the amorphous samples to 610 K, which is just above the glass transition temperature, equilibrating the sample at that temperature for 5 min, followed by cooling at 5 K/min to ambient temperature. Subsequent XRD measurements con®rmed the absence of a detectable crystalline fraction in the samples. 2.2. Modulation calorimetry Temperature-modulated heat capacity measurements were performed with a dierential heat¯ow calorimeter (MDSC, TA-Instruments) at underlying rates of 5 K/min. Time periods for
229
one sinusoidal temperature oscillation ranged from 10 to 100 s; the amplitude of the temperature modulations was held constant at 0.8 K. The time dependence of the sample temperature is given as T T1 bt A sin 2pt=s
1
with the initial temperature, T1 , the amplitude of the temperature modulation, A and the period of temperature modulation, s. At suciently small underlying scan rates, i.e. if the non-oscillating component can be regarded as constant during one oscillation period, the response signal consists of the superposition of two independent signals, the underlying heat ¯ow that corresponds to the conventional DSC signal and an oscillating heat ¯ow. From the underlying heat ¯ow, the apparent speci®c heat, Cb , is obtained. Under the condition that the system is close to a local equilibrium, i.e. for suciently small temperature modulations, the oscillating heat-¯ow signal can be analyzed within the framework of linear response theory. Using complex notation for the dynamic response upon a periodically varying attenuation, the real and imaginary parts of the complex heat capacity signal of the sample are obtained as [4] jCj
AHF ; Aq m0
C 0 jCj cos u;
C 00 jCj sin u
2
with the amplitude of the heat ¯ow modulations, AHF , the amplitude of the heating-rate of the modulations, Aq , the sample mass, m0 and the complex heat capacity, C. C 0 and C 00 are the real and imaginary parts of the heat capacity of the sample, respectively. u is the phase angle between the heat ¯ow signal and the temperature oscillation, indicating that the heat-¯ow modulations lag behind the temperature modulations [5]. In analogy to frequency dependent rheology, the imaginary part, C 00 , of the complex speci®c heat that is related to entropy production (or damping) is called the loss speci®c heat. The real part, C 0 , that is correlated to reversible molecular motion is denoted as the storage speci®c heat. At equilibrium, i.e. without the occurrence of any thermal event, the apparent speci®c heat, Cb , and the storage speci®c heat, C 0 , are equal to the
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thermodynamic equilibrium speci®c heat, Cp . Correspondingly, the loss speci®c heat vanishes at equilibrium. In the case of the glass transition as studied here, both irreversible and reversible contributions to the heat ¯ow that occur during heating correspond to enthalpy relaxation and the successive activation of vibrational and translational modes respectively. Besides the frequencydependent complex response, a glass transition signal will be observed in the underlying heat ¯ow that is similar to the apparent speci®c heat measured at the underlying rate. In the following, this transition signal will be referred to as the static glass transition to distinguish it from the dynamic glass transition that occurs in response to the temperature modulations.
3. Experimental results Fig. 1 shows the apparent speci®c heat capacity for the Pd40 Ni10 Cu30 P20 alloy in the glass, undercooled liquid, and the crystalline state measured at an underlying heating rate of 1 K/min. The period of temperature modulation amounted to 80 s, the amplitude of the temperature modulation was 0.8 K. The best ®t for the speci®c heat capacity differences between the undercooled liquid and the crystalline phase near the glass transition is ob-
Fig. 1. Static heat capacity of glassy, undercooled liquid and crystalline Pd40 Ni10 Cu30 P20 in the glass transition region, obtained by TMDSC at an underlying heating rate of 1 K/min with temperature-modulation amplitude of 0.8 K and modulation period of 80 s.
tained mathematically from data given by Lu et al. [6] as DCp 0:494 ÿ 4:289 10ÿ4 T (J/g K). The total integrated melting enthalpy, DHm , and the eutectic temperature are 6.82 0.12 kJ molÿ1 and 798 K respectively, as determined by dierential scanning calorimetry [6]. The width of the melting interval amounts to 60 K at a heating rate of 40 K/ min. The shape of the melting signal indicates that the major part of the sample melts near the eutectic temperature. Therefore, within the accuracy of the enthalpy measurements (3%), the alloy can be treated in a ®rst order approximation as a congruently melting system with respect to the calculation of the melting entropy, DSf . Integration of the DSC melting signal and division by the eutectic temperature yields DSf 8:55 0:15 J molÿ1 K. The value of DSf is in agreement with RichardsÕ rule. The high value for the reduced glass temperature of Trg Tg =Tm 0:71 supports the easy glass-forming tendency of this alloy. The entropy dierence between undercooled liquid and crystalline solid has been calculated as a function of temperature from the thermodynamic data obtained by continuous heating and cooling. The isentropic (Kauzmann-) temperature [7] was thus obtained by linear extrapolation as TK 497 10 K [6]. Fig. 2 shows the frequency dependent real and imaginary parts of the complex heat capacity signal as a function of temperature for modulation periods between 10 and 100 s. The underlying heating rate amounted to 5 K/min throughout the measurements. The samples have been equilibrated before each TMDSC-measurement as described previously, to ensure that the relaxational state (and thus the ¯uidity) prior to the measurements was identical. Analogously, though not shown here, TMDSC-measurements have been performed during cooling from T 610 K to T 400 K at an underlying rate of ÿ5 K/min. The frequency dependence of the real and imaginary parts of the complex speci®c heat response is observed for both sets of measurements as a shift towards higher temperatures and a slight decrease of the width of the glass-transition signal with increasing frequency. As a comparison, the apparent speci®c heat obtained by continuous heating at a constant rate of 5 K/min is included in Fig. 2. It is observed,
X. Hu et al. / Journal of Non-Crystalline Solids 260 (1999) 228±234
Fig. 2. Frequency dependence of the real and the imaginary parts of the heat capacity of amorphous Pd40 Ni10 Cu30 P20 obtained at an underlying heating rate of 5 K/min. The apparent heat capacity of the same sample obtained by conventional DSC is also shown for comparison.
that the maximum in C 00 and correspondingly the in¯ection point in C 0 are located at temperatures that are above the static glass transition observed by conventional DSC-measurements.
4. Discussion The frequency dependent measurements of the real and the imaginary parts of the complex speci®c heat displayed in Fig. 2 show the similar characteristics that have been observed by socalled heat-capacity spectroscopy measurements [1] or by mechanical spectroscopy [8,9] on deeply undercooled organic and inorganic samples in the glass transition region. This result indicates the general characteristics of the glass transition that are not dependent on the type of intermolecular interactions. However, it should be noted that the frequency dependence of the speci®c heat that is
231
directly related to the spectrum of internal relaxation times has to our knowledge not been reported for a metallic glass previously. The real part C0 measured at modulation periods of 10 s and 30 s, respectively, shows the sigmoidal increase at the glass transition that is unaected by relaxation or by the enthalpy hysteresis of well-relaxed glasses. The shape of the transition signal resembles the glass-transition signal obtained by conventional DSC measurements if enthalpy relaxation does not occur, i.e. if the speci®c heat is measured during cooling. However, the C 0 -curves that were measured at lower frequencies, i.e. s 80 s and 100 s, show a weak maximum at the end of the C 0 -increase that corresponds to the dynamic glass transition. In order to interpret this result and following the analysis of Donth [10], an average frequency xr can formally be associated with a constant heating (or cooling) rate in order to interrelate the rate dependence of the static ± and the frequency dependence of the dynamic glass transition. Therefore, independence of the static and the dynamic contributions to the response signal requires that the modulation frequency x0 2p=s xr . However, the measurements at s 80 s and s 100 s indicate that the static glass transition at an underlying heating rate of 5 K/min apparently still contributes to the con®gurational readjustments in the dissipative region, thus resulting in an interdependence of the static and the dynamic response. The crossover from mixed static/dynamic to purely dynamic C 0 -curves occurs at modulation periods between 30 s and 60 s. This result indicates the bounds on the experimental parameters that can be applied for modulation calorimetry, i.e. rates lower than 5 K/min have to be used to explore the frequency dependence of the glass transition of this Pd±Cu±Ni±P alloy at modulation periods longer than 50 s. Moreover, the possibility to observe the calorimetric contributions due to the static and the dynamic glass transition during one measurement allows for an independent estimation of the average relaxation time that governs the static glass transition. Heat-capacity spectroscopy does not provide that opportunity because modulation periods for this method are limited to s < 5 s, while
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the typical relaxation time of deeply undercooled liquids near the glass transition, sr , (at a cooling rate of the order of 10 K/min) is generally estimated for various glass forming substances as 60±120 s [11], i.e. well within the interval of modulation periods that can be applied in TMDSCexperiments. The crossover from mixed to purely dynamic response indicates that the typical relaxation time for the governing molecular modes of undercooled liquid Pd40 Ni10 Cu30 P20 is about sr 50 s at 578 K. The result is further supported by the observation that the static speci®c heat response on cooling by 5 K/min shows ®rst detectable deviations from the approximately constant value corresponding to the liquid state at slightly higher temperatures, i.e. at 582 K. The dynamics of the glass transition in the context of modulation experiments are usually discussed by frequency-dependent quasi-isothermal measurements at dierent temperatures in the glass transition region [1,12,13]. The isothermal C 0 and C 00 curves given by heat-capacity spectroscopy as functions of the modulation frequency characteristically show a sigmoidal increase (C 0 ) and a broad maximum (C 00 ) of non-Debye shape that is characteristic for the distribution of intrinsic times for structural relaxation. The temperature dependence of the frequency, mmax , where C 00 is a maximum (the inverse average relaxation time) then gives a measure of the dynamics of the glass transition. The TMDSC experiments were conducted at a constant modulation frequency that is superimposed on an underlying heating-or cooling rate. Therefore, the frequency dependence of the temperature Tmax , where C 00 is a maximum, is obtained from the present measurements. Moreover, mmax (T) can be obtained from the measured Tmax (m) values under the assumption that the frequency-dependent signal contributions of the measurements are not aected by the static contributions to the glasstransition signal. This assumption is ful®lled for the measurements at modulation periods of 10 and 30 s respectively. Fig. 3 shows the modulation frequency, m, with m 1=s and the corresponding peak temperatures of C 00 measured during heating (open circles). The closed circles give the respective values that were
Fig. 3. Frequency dependence of the peak temperature of the imaginary part of the heat capacity obtained during heating (open circle) and cooling (closed circle) at a rate of 5 K/min.
obtained on cooling with an underlying rate of ÿ5 K/min. The data at m 1/10 sÿ1 and 1/30 sÿ1 obtained during heating and cooling show a good agreement. The residual dierence is a measure for the accuracy of the determination of Tmax from the experimental TMDSC-data. However, the peak temperatures for lower frequencies show a pronounced dependence on the sign of the underlying rate. This result and the observed maxima in C 0 further indicate that both equilibrium and nonequilibrium relaxation modes contributed to those measurements. Thus, to exclude hysteresis eects due to the static glass-transition, heating rates have to be chosen such that the modulation periods are shorter than the average time for structural relaxation at the static glass transition. Certainly, the range of accessible modulation frequencies by TMDSC alone is too narrow to distinguish between an Arrhenius-like behavior and the expected Vogel±Fulcher±Tammann± Hesse-(VFTH)-like dependency [14±16] A 3 m m0 exp ÿ T ÿ T0 with the activation barrier, A and a constant parameter, m0 . Especially, the limited range of frequencies and corresponding peak temperatures does not allow for a reliable determination of the divergence temperature, T0 . However, to check the data for consistency, ®ts for both approximations to the data obtained on cooling are included in
X. Hu et al. / Journal of Non-Crystalline Solids 260 (1999) 228±234
Fig. 4. Frequency dependence of the peak temperature of the imaginary part of the heat capacity in comparison to theoretical curves according to Arrhenius and VFT ®ts.
Fig. 4. The Arrhenius-like behavior gives log m 45:85 ÿ 27:46 1000=T . The VFTH-®t (Eq. (3)) was then calculated with a T0 in the range of TK , to estimate whether a reasonable approximation of the measured data can be achieved for T0 TK . The result for this ®tting is v 9:15 106 exp ÿ1359= T ÿ 511. That expression describes the best mathematical ®t to the experimental data. However, a reasonable ®t with respect to the accuracy of the experimentally obtained data can be obtained by a T0 that is within the con®dence interval obtained for TK . Fig. 4 shows that, as expected, the two ®ts are nearly indistinguishable within the experimentally accessed frequency range. However, the data indicate that, as observed e.g. by [17], the frequencydependent data can be described by the divergence temperature chosen as the Kauzmann-temperature of the system, TK that was obtained by static equilibrium measurements.
233
lower values compared to heat-capacity spectroscopy. The results of these measurements at low frequencies show a dissipative behavior of the heat-¯ow response of the deeply undercooled liquid state of this metallic alloy that is qualitatively similar to results obtained on covalent and ionic glasses. TMDSC at dierent modulation periods indicated that the relationship between temperature and peak-frequency of the imaginary part of the complex speci®c heat can be obtained from measurements of the frequency dependence of the peak-temperature. However, this requires that the static and dynamic contributions to the total response signal corresponding to the glass transition are independent. Therefore, the underlying average rate and the modulation periods have to be chosen such that the static glass transition occurs at temperatures below the dissipative region where the internal relaxation times of the undercooled liquid are comparable to the modulation periods. At the same time, measurements at low frequencies in the crossover-domain where the static and dynamic glass transition overlap provide the unique opportunity to directly determine the average relaxation time at a given rate for the occurrence of the static glass transition.
Acknowledgements The authors gratefully acknowledge the support by the Academic Research Fund (RP3970633) of the National University of Singapore and by the Alexander von Humboldt-Foundation via the Feodor±Lynen Program (G.W., V-2-FLF1052606) and NSF (DMR-9712523).
References 5. Conclusions Calorimetric measurements on the easy glassforming Pd40 Ni10 Cu30 P20 alloy have been conducted by applying modulated-temperature calorimetry during heating and cooling through the dynamic glass-transition regime. The range of accessible frequencies can be extended towards considerably
[1] N.O. Birge, S.R. Nagel, Phys. Rev. Lett. 54 (1985) 2674. [2] N. Menon, S.R. Nagel, Phys. Rev. Lett. 74 (1995) 1230. [3] P.K. Dixon, L. Wu, S.R. Nagel, B.D. Williams, J.P. Carini, Phys. Rev. Lett. 65 (1990) 1108. [4] J.E.K. Schawe, Thermochim. Acta 260 (1995) 1. [5] Z. Jiang, C.T. Imrie, J.M. Hutchinson, Thermochim. Acta 315 (1998) 1. [6] I.-R. Lu, G. Wilde, G.P. G orler, R. Willnecker, J. NonCryst. Solids 250±252 (1999) 577.
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[7] W. Kauzmann, Chem. Rev. 43 (1948) 219. [8] D.J. Plazek, C.A. Bero, I.-C. Chay, J. Non-Cryst. Solids 172 (1994) 181. [9] L. Durene, R. Gy, J. Non-Cryst. Solids 211 (1997) 30. [10] E. Donth, Glas ubergang, Akademie, Berlin, 1981. [11] R. Zallen, The Physics of Amorphous Solids, Wiley, New York, 1983. [12] N. Menon, J. Chem. Phys. 105 (1996) 5246.
[13] K. Schr oter, G. Wilde, R. Willnecker, M. Weiss, K. Samwer, E. Donth, Eur. Phys. J. B 5 (1998) 1. [14] H. Vogel, Phys. Z. 22 (1921) 645. [15] G.S. Fulcher, J. Am. Ceram. Soc. 6 (1925) 339. [16] G. Tammann, W. Hesse, Z. Anorg. Allg. Chem. 156 (1926) 245. [17] P.K. Dixon, S.R. Nagel, Phys. Rev. Lett. 61 (1988) 341.
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The glass transition of Pd40Ni10Cu30P20 studied by temperature-modulated calorimetry X. Hu a, T.B. Tan a, Y. Li a, G. Wilde b,*, J.H. Perepezko b b
a Department of Materials Science, National University of Singapore, Singapore Department of Materials Science and Engineering, University of Wisconsin±Madison, 1509 University Avenue, Madison, WI 53703, USA
Received 18 June 1999
Abstract The frequency dependence of the heat capacity in the glass-transition region of Pd40 Ni10 Cu30 P20 was studied by temperature-modulated dierential scanning calorimetry (TMDSC) during slow heating and cooling. Such data for low frequencies between 0.1 and 0.01 Hz are not available, especially for metallic glasses. A crossover between mixed static/ dynamic and purely dynamic response signals was observed for the lowest frequencies between 1/80 and 1/100 sÿ1 , which allows a direct determination of the average relaxation time at a given cooling rate during the static glass transition. Further, these results were used to evaluate the experimental parameters necessary to truly separate the static and dynamic response in low-frequency modulation calorimetry experiments to obtain the moduli of the dynamic speci®c heat. Ó 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 64.70.P; 82.20.R; 02.30. F, D; 65.20
1. Introduction It is widely accepted that the decrease in the measured speci®c heat during cooling that is characteristic for the glass transition re¯ects the freezing-in of liquid-like modes and is thus related to the slowing dynamics of the system upon cooling [1,2]. It has been shown that the thermodynamic aspect of the glass transition as revealed by the anomaly of the speci®c heat as well as the
* Corresponding author. Present address: Forschungszertrum Karlsruhe GmbH, INT, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany. Tel.: +49-7247 82 6414; fax: +49-7247 82 6368. E-mail address: [email protected] (G. Wilde)
kinetics associated with a divergence of relaxation times can be connected by heat-capacity spectroscopy [1]. Moreover, this approach is related to all modes of molecular motion and thus to the thermodynamics of the system in contrast to e.g. measurements of dielectric loss spectra [3] which probe the response of polarization-sensitive modes only. However, this method is restricted to nonmetallic glasses for several reasons: ®rst, metallic glasses of sucient stability against crystallization have been discovered rather recently. Secondly, the experimental set-up used for the heat-capacity spectroscopy requires a low thermal diusivity to keep the wavelength of the temperature oscillation small compared to the linear dimension of the temperature sensor. This boundary condition is ful®lled for organic systems and minimum
0022-3093/99/$ - see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 5 7 9 - 7
X. Hu et al. / Journal of Non-Crystalline Solids 260 (1999) 228±234
frequencies of about 1 sÿ1 , but is not compatible with the rather high thermal diusivity (and the small dimensions) of metallic glasses. However, temperature-modulated dierential scanning calorimetry (TMDSC) now provides a suitable means for speci®c heat measurements at low frequencies on metallic glasses. Such measurements on the easy glass-forming alloy Pd40 Ni10 Cu30 P20 were performed here, to our knowledge for the ®rst time, to identify the range of suitable measurement parameters. The results of these TMDSC experiments are then compared to conventional DSC measurements at constant rate to evaluate the frequency dependence of the calorimetric glasstransition signal in the low-frequency domain. 2. Experimental details 2.1. Sample preparation Bulk Pd40 Ni10 Cu30 P20 ingots of nominal composition were prepared by melting the pure element Pd (purity 99.9+%) with a pre-melted Cu±Ni±P master alloy (purity better than 99.9%) in an induction furnace. Glassy ribbons (width 3 mm, thickness 40 lm) of the same composition, as con®rmed by energy dispersive X-ray analysis (EDX), were produced by a single-roll melt-spinning facility, applying a tangential wheel speed of about 20 m/s. Microstructural characterization of the samples was carried out by standard X-ray diraction (Philips XRD) techniques. A well-de®ned state with respect to relaxation was obtained by heating the amorphous samples to 610 K, which is just above the glass transition temperature, equilibrating the sample at that temperature for 5 min, followed by cooling at 5 K/min to ambient temperature. Subsequent XRD measurements con®rmed the absence of a detectable crystalline fraction in the samples. 2.2. Modulation calorimetry Temperature-modulated heat capacity measurements were performed with a dierential heat¯ow calorimeter (MDSC, TA-Instruments) at underlying rates of 5 K/min. Time periods for
229
one sinusoidal temperature oscillation ranged from 10 to 100 s; the amplitude of the temperature modulations was held constant at 0.8 K. The time dependence of the sample temperature is given as T T1 bt A sin 2pt=s
1
with the initial temperature, T1 , the amplitude of the temperature modulation, A and the period of temperature modulation, s. At suciently small underlying scan rates, i.e. if the non-oscillating component can be regarded as constant during one oscillation period, the response signal consists of the superposition of two independent signals, the underlying heat ¯ow that corresponds to the conventional DSC signal and an oscillating heat ¯ow. From the underlying heat ¯ow, the apparent speci®c heat, Cb , is obtained. Under the condition that the system is close to a local equilibrium, i.e. for suciently small temperature modulations, the oscillating heat-¯ow signal can be analyzed within the framework of linear response theory. Using complex notation for the dynamic response upon a periodically varying attenuation, the real and imaginary parts of the complex heat capacity signal of the sample are obtained as [4] jCj
AHF ; Aq m0
C 0 jCj cos u;
C 00 jCj sin u
2
with the amplitude of the heat ¯ow modulations, AHF , the amplitude of the heating-rate of the modulations, Aq , the sample mass, m0 and the complex heat capacity, C. C 0 and C 00 are the real and imaginary parts of the heat capacity of the sample, respectively. u is the phase angle between the heat ¯ow signal and the temperature oscillation, indicating that the heat-¯ow modulations lag behind the temperature modulations [5]. In analogy to frequency dependent rheology, the imaginary part, C 00 , of the complex speci®c heat that is related to entropy production (or damping) is called the loss speci®c heat. The real part, C 0 , that is correlated to reversible molecular motion is denoted as the storage speci®c heat. At equilibrium, i.e. without the occurrence of any thermal event, the apparent speci®c heat, Cb , and the storage speci®c heat, C 0 , are equal to the
230
X. Hu et al. / Journal of Non-Crystalline Solids 260 (1999) 228±234
thermodynamic equilibrium speci®c heat, Cp . Correspondingly, the loss speci®c heat vanishes at equilibrium. In the case of the glass transition as studied here, both irreversible and reversible contributions to the heat ¯ow that occur during heating correspond to enthalpy relaxation and the successive activation of vibrational and translational modes respectively. Besides the frequencydependent complex response, a glass transition signal will be observed in the underlying heat ¯ow that is similar to the apparent speci®c heat measured at the underlying rate. In the following, this transition signal will be referred to as the static glass transition to distinguish it from the dynamic glass transition that occurs in response to the temperature modulations.
3. Experimental results Fig. 1 shows the apparent speci®c heat capacity for the Pd40 Ni10 Cu30 P20 alloy in the glass, undercooled liquid, and the crystalline state measured at an underlying heating rate of 1 K/min. The period of temperature modulation amounted to 80 s, the amplitude of the temperature modulation was 0.8 K. The best ®t for the speci®c heat capacity differences between the undercooled liquid and the crystalline phase near the glass transition is ob-
Fig. 1. Static heat capacity of glassy, undercooled liquid and crystalline Pd40 Ni10 Cu30 P20 in the glass transition region, obtained by TMDSC at an underlying heating rate of 1 K/min with temperature-modulation amplitude of 0.8 K and modulation period of 80 s.
tained mathematically from data given by Lu et al. [6] as DCp 0:494 ÿ 4:289 10ÿ4 T (J/g K). The total integrated melting enthalpy, DHm , and the eutectic temperature are 6.82 0.12 kJ molÿ1 and 798 K respectively, as determined by dierential scanning calorimetry [6]. The width of the melting interval amounts to 60 K at a heating rate of 40 K/ min. The shape of the melting signal indicates that the major part of the sample melts near the eutectic temperature. Therefore, within the accuracy of the enthalpy measurements (3%), the alloy can be treated in a ®rst order approximation as a congruently melting system with respect to the calculation of the melting entropy, DSf . Integration of the DSC melting signal and division by the eutectic temperature yields DSf 8:55 0:15 J molÿ1 K. The value of DSf is in agreement with RichardsÕ rule. The high value for the reduced glass temperature of Trg Tg =Tm 0:71 supports the easy glass-forming tendency of this alloy. The entropy dierence between undercooled liquid and crystalline solid has been calculated as a function of temperature from the thermodynamic data obtained by continuous heating and cooling. The isentropic (Kauzmann-) temperature [7] was thus obtained by linear extrapolation as TK 497 10 K [6]. Fig. 2 shows the frequency dependent real and imaginary parts of the complex heat capacity signal as a function of temperature for modulation periods between 10 and 100 s. The underlying heating rate amounted to 5 K/min throughout the measurements. The samples have been equilibrated before each TMDSC-measurement as described previously, to ensure that the relaxational state (and thus the ¯uidity) prior to the measurements was identical. Analogously, though not shown here, TMDSC-measurements have been performed during cooling from T 610 K to T 400 K at an underlying rate of ÿ5 K/min. The frequency dependence of the real and imaginary parts of the complex speci®c heat response is observed for both sets of measurements as a shift towards higher temperatures and a slight decrease of the width of the glass-transition signal with increasing frequency. As a comparison, the apparent speci®c heat obtained by continuous heating at a constant rate of 5 K/min is included in Fig. 2. It is observed,
X. Hu et al. / Journal of Non-Crystalline Solids 260 (1999) 228±234
Fig. 2. Frequency dependence of the real and the imaginary parts of the heat capacity of amorphous Pd40 Ni10 Cu30 P20 obtained at an underlying heating rate of 5 K/min. The apparent heat capacity of the same sample obtained by conventional DSC is also shown for comparison.
that the maximum in C 00 and correspondingly the in¯ection point in C 0 are located at temperatures that are above the static glass transition observed by conventional DSC-measurements.
4. Discussion The frequency dependent measurements of the real and the imaginary parts of the complex speci®c heat displayed in Fig. 2 show the similar characteristics that have been observed by socalled heat-capacity spectroscopy measurements [1] or by mechanical spectroscopy [8,9] on deeply undercooled organic and inorganic samples in the glass transition region. This result indicates the general characteristics of the glass transition that are not dependent on the type of intermolecular interactions. However, it should be noted that the frequency dependence of the speci®c heat that is
231
directly related to the spectrum of internal relaxation times has to our knowledge not been reported for a metallic glass previously. The real part C0 measured at modulation periods of 10 s and 30 s, respectively, shows the sigmoidal increase at the glass transition that is unaected by relaxation or by the enthalpy hysteresis of well-relaxed glasses. The shape of the transition signal resembles the glass-transition signal obtained by conventional DSC measurements if enthalpy relaxation does not occur, i.e. if the speci®c heat is measured during cooling. However, the C 0 -curves that were measured at lower frequencies, i.e. s 80 s and 100 s, show a weak maximum at the end of the C 0 -increase that corresponds to the dynamic glass transition. In order to interpret this result and following the analysis of Donth [10], an average frequency xr can formally be associated with a constant heating (or cooling) rate in order to interrelate the rate dependence of the static ± and the frequency dependence of the dynamic glass transition. Therefore, independence of the static and the dynamic contributions to the response signal requires that the modulation frequency x0 2p=s xr . However, the measurements at s 80 s and s 100 s indicate that the static glass transition at an underlying heating rate of 5 K/min apparently still contributes to the con®gurational readjustments in the dissipative region, thus resulting in an interdependence of the static and the dynamic response. The crossover from mixed static/dynamic to purely dynamic C 0 -curves occurs at modulation periods between 30 s and 60 s. This result indicates the bounds on the experimental parameters that can be applied for modulation calorimetry, i.e. rates lower than 5 K/min have to be used to explore the frequency dependence of the glass transition of this Pd±Cu±Ni±P alloy at modulation periods longer than 50 s. Moreover, the possibility to observe the calorimetric contributions due to the static and the dynamic glass transition during one measurement allows for an independent estimation of the average relaxation time that governs the static glass transition. Heat-capacity spectroscopy does not provide that opportunity because modulation periods for this method are limited to s < 5 s, while
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the typical relaxation time of deeply undercooled liquids near the glass transition, sr , (at a cooling rate of the order of 10 K/min) is generally estimated for various glass forming substances as 60±120 s [11], i.e. well within the interval of modulation periods that can be applied in TMDSCexperiments. The crossover from mixed to purely dynamic response indicates that the typical relaxation time for the governing molecular modes of undercooled liquid Pd40 Ni10 Cu30 P20 is about sr 50 s at 578 K. The result is further supported by the observation that the static speci®c heat response on cooling by 5 K/min shows ®rst detectable deviations from the approximately constant value corresponding to the liquid state at slightly higher temperatures, i.e. at 582 K. The dynamics of the glass transition in the context of modulation experiments are usually discussed by frequency-dependent quasi-isothermal measurements at dierent temperatures in the glass transition region [1,12,13]. The isothermal C 0 and C 00 curves given by heat-capacity spectroscopy as functions of the modulation frequency characteristically show a sigmoidal increase (C 0 ) and a broad maximum (C 00 ) of non-Debye shape that is characteristic for the distribution of intrinsic times for structural relaxation. The temperature dependence of the frequency, mmax , where C 00 is a maximum (the inverse average relaxation time) then gives a measure of the dynamics of the glass transition. The TMDSC experiments were conducted at a constant modulation frequency that is superimposed on an underlying heating-or cooling rate. Therefore, the frequency dependence of the temperature Tmax , where C 00 is a maximum, is obtained from the present measurements. Moreover, mmax (T) can be obtained from the measured Tmax (m) values under the assumption that the frequency-dependent signal contributions of the measurements are not aected by the static contributions to the glasstransition signal. This assumption is ful®lled for the measurements at modulation periods of 10 and 30 s respectively. Fig. 3 shows the modulation frequency, m, with m 1=s and the corresponding peak temperatures of C 00 measured during heating (open circles). The closed circles give the respective values that were
Fig. 3. Frequency dependence of the peak temperature of the imaginary part of the heat capacity obtained during heating (open circle) and cooling (closed circle) at a rate of 5 K/min.
obtained on cooling with an underlying rate of ÿ5 K/min. The data at m 1/10 sÿ1 and 1/30 sÿ1 obtained during heating and cooling show a good agreement. The residual dierence is a measure for the accuracy of the determination of Tmax from the experimental TMDSC-data. However, the peak temperatures for lower frequencies show a pronounced dependence on the sign of the underlying rate. This result and the observed maxima in C 0 further indicate that both equilibrium and nonequilibrium relaxation modes contributed to those measurements. Thus, to exclude hysteresis eects due to the static glass-transition, heating rates have to be chosen such that the modulation periods are shorter than the average time for structural relaxation at the static glass transition. Certainly, the range of accessible modulation frequencies by TMDSC alone is too narrow to distinguish between an Arrhenius-like behavior and the expected Vogel±Fulcher±Tammann± Hesse-(VFTH)-like dependency [14±16] A 3 m m0 exp ÿ T ÿ T0 with the activation barrier, A and a constant parameter, m0 . Especially, the limited range of frequencies and corresponding peak temperatures does not allow for a reliable determination of the divergence temperature, T0 . However, to check the data for consistency, ®ts for both approximations to the data obtained on cooling are included in
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Fig. 4. Frequency dependence of the peak temperature of the imaginary part of the heat capacity in comparison to theoretical curves according to Arrhenius and VFT ®ts.
Fig. 4. The Arrhenius-like behavior gives log m 45:85 ÿ 27:46 1000=T . The VFTH-®t (Eq. (3)) was then calculated with a T0 in the range of TK , to estimate whether a reasonable approximation of the measured data can be achieved for T0 TK . The result for this ®tting is v 9:15 106 exp ÿ1359= T ÿ 511. That expression describes the best mathematical ®t to the experimental data. However, a reasonable ®t with respect to the accuracy of the experimentally obtained data can be obtained by a T0 that is within the con®dence interval obtained for TK . Fig. 4 shows that, as expected, the two ®ts are nearly indistinguishable within the experimentally accessed frequency range. However, the data indicate that, as observed e.g. by [17], the frequencydependent data can be described by the divergence temperature chosen as the Kauzmann-temperature of the system, TK that was obtained by static equilibrium measurements.
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lower values compared to heat-capacity spectroscopy. The results of these measurements at low frequencies show a dissipative behavior of the heat-¯ow response of the deeply undercooled liquid state of this metallic alloy that is qualitatively similar to results obtained on covalent and ionic glasses. TMDSC at dierent modulation periods indicated that the relationship between temperature and peak-frequency of the imaginary part of the complex speci®c heat can be obtained from measurements of the frequency dependence of the peak-temperature. However, this requires that the static and dynamic contributions to the total response signal corresponding to the glass transition are independent. Therefore, the underlying average rate and the modulation periods have to be chosen such that the static glass transition occurs at temperatures below the dissipative region where the internal relaxation times of the undercooled liquid are comparable to the modulation periods. At the same time, measurements at low frequencies in the crossover-domain where the static and dynamic glass transition overlap provide the unique opportunity to directly determine the average relaxation time at a given rate for the occurrence of the static glass transition.
Acknowledgements The authors gratefully acknowledge the support by the Academic Research Fund (RP3970633) of the National University of Singapore and by the Alexander von Humboldt-Foundation via the Feodor±Lynen Program (G.W., V-2-FLF1052606) and NSF (DMR-9712523).
References 5. Conclusions Calorimetric measurements on the easy glassforming Pd40 Ni10 Cu30 P20 alloy have been conducted by applying modulated-temperature calorimetry during heating and cooling through the dynamic glass-transition regime. The range of accessible frequencies can be extended towards considerably
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