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Investigation of specific heat and thermal expansion in the glass-transition regime of Pd-based metallic glasses
Journal of Non-Crystalline Solids 274 (2000) 294±300
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Investigation of speci®c heat and thermal expansion in the glass-transition regime of Pd-based metallic glasses I.-R. Lu a,*, G.P. G orler a, H.-J. Fecht b, R. Willnecker a b
a Institute of Space Simulation, DLR, D-51170 Cologne, Germany Faculty of Engineering, Materials Division, University Ulm, D-89081 Ulm, Germany
Abstract The alloys Pd±Ni±P and Pd±Ni±Cu±P have an outstanding glass-forming ability and stability against crystallisation in the undercooled liquid state at temperatures greater than the glass-transition temperatures. Samples of the copper containing alloys can be vitri®ed readily by cooling the liquid at rates P 0:1 K/s. This rate allows us to investigate the thermodynamic quantities of this alloy system in the liquid as well as in the glassy state during continuous cooling or heating without interference from crystallisation. In the present work, the speci®c heat and the volume expansion coecient in the glassy and in the undercooled liquid state were determined at temperatures less than and greater than the glass-transition temperature by applying dierential scanning calorimetry (DSC) and sessile drop (SD) technique, respectively. The glass temperatures of the samples resulting from the cooling rate were investigated during re-heating by both the methods. By comparing the results, a correlation between enthalpy and speci®c volume as a function of temperature could be demonstrated experimentally for these metallic alloys. We attribute the characteristic course of the enthalpy and the speci®c volume in the range of the glass-transition to structural relaxation processes and to the fraction of free volume in the glass as a function of temperature and time. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The phenomenon of the liquid-to-glass-transition is interesting because of the ergodicity breaking of the thermodynamic properties, such as enthalpy and speci®c volume. However, there is still a lack of fundamental insight into the basics of glass formation and the properties of glasses [1]. Metallic glasses have simple atomic con®gurations compared to the oxide glasses or glassy organic polymers. With the development of bulk glass* Corresponding author. Present address: Institut f ur Raumsimulation, DLR, 51170 K oln, Germany. Tel.: +49-2203 6012986; fax: +49-2203 61768. E-mail address: [email protected] (I.-R. Lu).
forming alloys [2±6], glass-transition phenomena can now be measured in metallic systems. Investigations on the change of thermal expansion and heat capacity of metallic systems at the glasstransition are meaningful to test atomistic models proposed to describe the relaxation processes in this regime [7±9]. The Pd-based metallic alloys, Pd±Cu±Si and Pd±Ni±P, are well known for their glass-forming ability [2,3]. Glassy samples of Pd±Ni±P can be prepared at cooling rates of about 0.2 K/s [10]. Especially, metallic glasses composed of Pd±Ni± Cu±P can be vitri®ed at 0.1 K/s and have the largest interval between glass-transition temperature and crystallisation temperature in comparison to other alloys [6,12]. Therefore, the investigation
0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 2 2 0 - 9
I.-R. Lu et al. / Journal of Non-Crystalline Solids 274 (2000) 294±300
of the properties of the undercooled liquid and the vitreous state during continuous cooling or heating cycles is achievable for these metallic alloys without interference from crystallisation. In the previous papers, we have reported upon dierential scanning calorimetry (DSC) measurements of the speci®c heat as a function of temperature on these alloys covering the liquid, undercooled, and glassy state. From the heat-capacity data, the melting enthalpies and the melting temperatures, the excess entropies, and the Kauzmann temperatures were obtained [11,12]. In the present work, the thermodynamic glass temperatures, Tg , de®ned in the same way as the limiting ®ctive temperatures [13], are derived independently from calorimetric and sessile drop (SD) measurements for samples, which have been vitri®ed at dierent cooling rates. The results are used to identify correlations between the investigated thermodynamic properties as a function of temperature with respect to the existing glass-transition models and to draw conclusions on the relaxation processes within the glass-transition regime. As known from earlier investigations [14], the glass temperature of a sample depends on the cooling rate during vitri®cation, i.e., on the thermal history of the sample, and it can be determined by a subsequent DSC measurement at increasing temperature. Tg can be derived from the experimentally received data of the speci®c heat, Cp (T), by integration, using an extrapolation of the speci®c heat of the liquid, Cpl , and of the glassy state, Cpg , according to Eq. (1) Z
T2 Tg
Cpl ÿ Cpg dT
Z
T2
T1
Cp ÿ Cpg dT :
1
T1 and T2 are temperatures well within the region of the stable glass and the supercooled liquid, respectively. In principle, the same method could be applied, analogously, to the data of the coecient of thermal volume expansion, b(T), to derive a glass temperature of the sample from the volumetric measurements. The correlation between these thermodynamic quantities of second order, Cp (T)
295
and b(T), is the basis of a theory developed for the glass-transition from the model of free volume [15]. Moreover, the change of the glass temperature in dependence on the cooling rate, q, is investigated to describe the relaxation kinetics at the glass-transition [14]. This dependence corresponds to that of a thermally activated process (Eq. (2)). d ln jqj Dh : RTg2 dTg
2
Here, Dh is regarded as an apparent activation energy for structural, enthalpy or volume relaxation and can be obtained from the slope of an appropriate Arrhenius diagram.
2. Experimental 2.1. Sample preparation and calorimetric analysis The Pd40 Ni40 P20 (PNP), Pd40 Ni10 Cu30 P20 (PNPCu30 ) and Pd43 Ni10 Cu27 P20 (PNP-Cu27 ) alloys (concentrations in at.%) were prepared by induction melting of the pure elements under argon atmosphere in a quartz glass tube. Prior to cooling, the alloys investigated in the present study were heat-treated above the respective liquidus temperature in a B2 O3 -¯ux to suppress the crystallisation caused by impurities. After water quenching, rods of metallic glasses with a diameter of 5 mm were obtained. Samples of the as-quenched ingots of the dierent alloys were investigated by X-ray diffraction. The presence of crystalline phases from the measured spectra was not detected. Samples for volume expansion measurements, in shape of a droplet with a diameter of about 6 mm, were prepared from the ingot within the crucibles of a heat-¯ow DSC (Netzsch DSC 404). The DSC trace observed during cooling ensures that the sample was vitri®ed without any crystallisation. The calorimetric investigation on the glasstransition of the Pd-based metallic glasses was performed in DSC (Perkin Elmer DSC-2C). The samples were heated to the undercooled liquid state at 620 K and cooled to room temperature at dierent rates (5, 10, 20, 40, 80 K/min) passing the range of transition into the vitreous state.
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Subsequently, all measurements were performed at a heating rate of 10 K/min. 2.2. Volume expansion measurement The volume expansion experiments were performed in an SD facility as shown in Fig. 1. The metallic droplet is located on a sapphire substrate, which is heated by a resistance heater under argon atmosphere in a vacuum chamber. A platinum resistance thermometer which is calibrated at the melting points of indium and lead is used for measuring the temperature of the substrate. The temperature±time courses during heating or cooling are continuously scanned by a data acquisition unit. A high-resolution monochrome CCD digital camera (Kodak DCS 420M) with an array of 1524 1012 pixels and a grey-value depth of 12 bits in combination with an image processing software (Halcon) was applied for the recording and the analysis of the cross-section of the metallic droplet. The image of the droplet is magni®ed by a
factor of 1.5±2.0 using a macro-lens (Micro-Nikkor 4.0/200 mm) to improve the experimental resolution. Utilizing a sub-pixel algorithm for image processing, the resolution with respect to the contour of the metallic droplet could be improved up to at least 1/10 of a pixel. Within a reduced dynamic range of 8 bits, the boundary of the droplet was arbitrarily settled on the grey value of 140, approximately the mean between the dark and bright ®elds of the image. Thus, the area, A, within the boundary could be evaluated. The shape of the droplet is assumed to be symmetrical with respect to the rotation about the y-axis, which is perpendicular to the substrate and passing through the center of the droplet. Based on this assumption, the method for the numerical evaluation of the volume of the droplet, V, follows (Fig. 1): the cross-section is divided along the symmetry axis into two equal areas. The distance of the center of gravity of the half cross-section from the symmetry axis is determined as Xs . The
Fig. 1. Scheme of the SD facility. The sample is located on a sapphire substrate in a chamber ®lled with argon atmosphere. A digital camera and a commercial image analysing software are applied to evaluate the cross-section of a metallic droplet. A is the area of the cross-section of the droplet and the co-ordinates (X0 ,Y0 ) and (Xs ,Y0 ) are the centres of gravity of the whole and half cross-sections, respectively.
I.-R. Lu et al. / Journal of Non-Crystalline Solids 274 (2000) 294±300
volume of the droplet then can be derived from GuldinÕs rule as V pXs A. During heating, the cross-section of the sample was photographed and evaluated at every 5 K. To verify the reliability and the sensitivity of the optical measuring system, the thermal expansion of a nickel disk was measured. The coecient of linear expansion, a(T), for nickel was found to increase from 13 ´ 10ÿ6 Kÿ1 at 300 K to 18 ´ 10ÿ6 Kÿ1 at 620 K. Compared with the literature data given in Ref. [16], the deviation of our data was within 3%. As for the calorimetric measurements, the specimen for volume expansion measurements was heated to the metastable liquid state at 620 K and vitri®ed at various cooling rates. The thermal expansion measurements were performed subsequently in the temperature range between 370 and 620 K during isochronous heating. A rate of
297
4 K/min was applied to reduce the oset between substrate temperature and droplet temperature. In fact, the limiting ®ctive temperature, Tg , of the sample achieved by cooling does not depend on the applied heating rate. 3. Results The apparent speci®c heat data of the Pd-based metallic glasses measured at a heating rate of 10 K/ min on samples after various cooling treatment are illustrated in Fig. 2. The glass temperatures achieved at a cooling rate of 10 K/min are determined from these measurements as 564 1, 572 1 and 566 1 K for PNP-Cu30 , PNP-Cu27 , and PNP, respectively. Due to the melting temperature of the alloys containing copper, which is decreased by 80 K compared to PNP [12], sample
Fig. 2. Speci®c heat in dependence on the temperature from DSC measurements at isochronous heating of 10 K/min for the metallic glasses, PNP (a), PNP-Cu27 (b) and PNP-Cu30 (c), which were vitri®ed previously by various cooling rates (5, 10, 20, 40 and 80 K/min).
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PNP-Cu27 has the largest reduced glass temperature, Tg =Tm 0:71, among the investigated alloys. In Fig. 3 the Tg s for the Pd-based samples as a function of the cooling rates are illustrated as evaluated from the measured speci®c heat data by applying Eq. (1). The eect of a variation in the cooling rate on the resulting change in Tg turns out smaller for the samples containing copper. This means that the temperature dependence of the kinetics of structural relaxation is increased by the addition of copper. Using Eq. (2), the apparent activation energies can be calculated as 854, 978 and 1081 kJ/mol for samples PNP, PNP-Cu27 and PNP-Cu30 , respectively. Fig. 4 shows the correspondence between the measured thermal expansion, V T ÿ V0 =V0 , and the enthalpy, H(T), calculated from speci®c heat data for sample PNP-Cu27 . Representative results are plotted for the thermal expansion and the enthalpy of samples, which were re-heated after quenching as well as after cooling at 10 K/min. Additionally, the eect of a dierent cooling rate of 0.5 K/min is shown in the enthalpy diagram for comparison.
Fig. 3. The correlation of the glass temperature, Tg , with the preceding cooling rate. Tg s were calculated from the measured data of speci®c heat by applying the Eq. (1). The accuracy of glass temperature determination is about 1 K. For the PNPCu27 sample (a), the dependence of Tg on cooling rates determined from thermal expansion measurements after cooling (1.5, 3, 8, 15 and 30 K/min) is shown for comparison.
The coecients of thermal volume expansion could be derived from the expansion data, which were ®tted with a polynomial of second order below the glass-transition and with a linear approximation for the range of the metastable undercooled liquid state. In the sample vitri®ed at a cooling rate of 10 K/min, a temperature dependence of the expansion coecient of the glass is found as bg 36:9 0:066 T ÿ 273 1:5 10ÿ6 Kÿ1 and a constant value of b1 90 3 10ÿ6 Kÿ1 for the undercooled liquid. For the quenched sample, structural relaxation eects are superimposed on the thermal expansion below the glass-transition. 4. Discussion For non-metallic glasses, a correlation has been noticed between glass-forming ability or critical cooling rate, respectively, and a greater reduced glass temperature Tg =Tm or a larger ratio of DCp =DSf [17]. The dierences in heat capacity between liquid and crystalline state at Tg , DCp Cpl ÿ Cpg , are as about 0.24 J/g K for the samples PNP-Cu30 and PNP-Cu27 , and 0.22 J/g K for the sample PNP. The addition of copper in the Pd-based alloys has an impact on the diminution of the melting enthalpy and the melting entropy [12], but negligible eect on DCp . Therefore, the ratios of DCp =DSf calculated from data according to Ref. [12] are obtained as 2.4 and 2.1 for the PNP-Cu and PNP alloys. In fact, the alloys PNPCu are the easiest bulk glass-formers among metallic systems today [18]. Comparing the apparent activation energies, Dh , of the investigated alloys as received from the evaluation of Fig. 3, the samples PNP-Cu27 and PNP-Cu30 are the more fragile glasses having a larger activation energy for the structural relaxation at the glass-transition than PNP. Obviously, the fragility of the alloys is systematically increased with the concentration of copper in PNP. From the curves of thermal expansion measured on the sample PNP-Cu27 , the thermodynamic Tg was determined as the temperature at the point of the intersection of the two approximations of V(T) for the glass and the liquid state.
I.-R. Lu et al. / Journal of Non-Crystalline Solids 274 (2000) 294±300
299
Fig. 4. Comparison between the thermal expansion (a) of a PNP-Cu27 sample and its enthalpy (b) as a function of temperature measured by SD and DSC techniques, respectively.
Note that this method to determine Tg is in mathematical agreement with the procedure according to Eq. (1), which is based on the secondorder quantities. Within the limits of the accuracy of the glass temperature determination (dTg 1:5 K for the volumetric measurement), Tg is coincidentally found as 570 K from the enthalpy and thermal expansion results for the sample cooled at 10 K/min. The glass temperatures as a function of the logarithm of the cooling rates obtained from volume expansion measurements for the sample PNPCu27 are additionally plotted in Fig. 3(a). From the diagram, we note that the Tg s, determined from independent DSC and SD measurements are identical within the experimental uncertainty. This identity implies that the kinetics and mechanism of the relaxation, evident from speci®c volume and enthalpy investigations are based on the same structural process. This coincidence between the calorimetric and volumetric Tg s, which is herewith con®rmed experimentally for a metallic system, has to be postulated when the free-volume model is applied to explain the glass-transition phenomena. The theory is based upon the hypothesis of a distinct fraction of free volume, Vf (T)/V, characterizing the liquid state and its ¯uidity u 1=g. For simple
liquids, the viscosity, g, and its dependence on the free volume can be described by the Doolittle equation g g0 exp f AV =Vf g:
3
For glass-forming liquids, the viscosity near Tg was found to follow the empirical Vogel±Tammann±Fulcher (VTF) relation g g0 exp f B= T ÿ T0 g:
4
The combination of both equations results in the simple expression Vf =V eq C T ÿ T0
5
for the equilibrium free volume. Thus, the increase of the speci®c heat at Tg , a general criterion of the glass-transition, can be directly referred to the energy necessary to produce additional free volume at heating. DCp Cpl ÿ Cpg dVf =dT const:
6
A glass-forming substance approaches this linear increase of Vf in the liquid state, but abandons the equilibrium during cooling through the glasstransition, where a residual fraction of free volume is frozen-in. The annihilation and generation of free volume in this regime is non-linear and
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non-exponential. This non-linearity is due to the fact that the actual free volume, dependent on the thermal history of the sample, is a crucial parameter determining the relaxation velocity. Using an appropriate kinetic model, the hysteresis phenomena observed in the glass-transition region can thus be explained. The free volume frozen-in at Tg can be calculated from the results of our measurements by using the approximation Vf Tg =V b1 ÿ bg Tg ÿ TK [19,20]. This approximation is based on the identi®cation of the divergence temperature T0 in the VTF-relation with the Kauzmann temperature TK . For sample PNP-Cu27 after cooling at 10 K/min, the fraction of free volume frozen-in at the glass-transition is 0.0024. From the combination of the volumetric and calorimetric data, it becomes possible to determine the speci®c energy of formation of free volume, DQf . Assuming a constant DCp according to Eq. (6), this energy follows from Z Tg ÿ Vf Tg DQf : DCp dT DCp Tg ÿ TK 7 V TK The data for the sample PNP-Cu27 give DQf 465 kJ/mol. This DQf , corresponding to 4.8 eV/atom, is reasonable because it is of the order of the heat of evaporation for the constituents of the alloy [21]. This agreement con®rms the simple connection between heat capacity and formation of free volume made by Eq. (6). 5. Conclusions The correlation of glass-forming ability with the reduced glass temperature was con®rmed for the investigated Pd-based samples. The volume expansion coecients are about 39±51 10ÿ6 Kÿ1 for the glassy state and about 90 10ÿ6 Kÿ1 for the undercooled liquid state, which shows a greater dependence of volume on temperature. The thermodynamic glass temperatures as a function of the applied cooling rates were determined both
from the speci®c heat data as well as from the volumetric results, and found to be independent of the experimental method. This correlation, which could be demonstrated now experimentally for metallic glass formers, can be used for the application of the free-volume model describing kinetic glass-transition phenomena. Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft, project No. Wi 1350/1± 4. References [1] M.D. Ediger, C.A. Angell, S.R. Nagel, J. Phys. Chem. 100 (1996) 13200. [2] H.S. Chen, Acta Metall. 22 (1974) 1505. [3] H.W. Kui, A.L. Greer, D. Turnbull, Appl. Phys. Lett. 45 (1984) 615. [4] A. Inoue, T. Zhang, T. Masumoto, Mater. Trans. JIM 31 (1990) 177. [5] A. Peker, W.L. Johnson, Appl. Phys. Lett. 63 (1993) 2342. [6] A. Inoue, N. Nishiyama, Mater. Trans. JIM 37 (1996) 1531. [7] S.S. Tsao, F. Spaepen, Acta Metall. 33 (1985) 881. [8] G.W. Scherer, J. Non-Cryst. Solids 123 (1990) 75. [9] I. Avramov, Thermochim. Acta 280 (1996) 363. [10] R. Willnecker, K. Wittmann, G.P. G orler, J. Non-Cryst. Solids 156±158 (1993) 450. [11] G. Wilde, G.P. G orler, R. Willnecker, G. Dietz, Appl. Phys. Lett. 65 (1994) 397. [12] I.-R. Lu, G. Wilde, G.P. G orler, R. Willnecker, J. NonCryst. Solids 250±252 (1) (1999) 577. [13] A.Q. Tool, J. Am. Ceram. Soc. 29 (1946) 240. [14] C.T. Moynihan, A.J. Easteal, J. Wilder, J. Tucker, J. Phys. Chem. 78 (1974) 2673. [15] M.H. Cohen, G.S. Grest, Phys. Rev. B 20 (1979) 1077. [16] F.C. Nix, D. MacNair, Phys. Rev. 60 (1941) 597. [17] K.S. Dubey, P. Ramachandrarao, Int. J. Rapid Solidi®cation 1 (1984±1985) 1. [18] A. Meyer, R. Busch, H. Schober, Phys. Rev. Lett. 83 (1999) 5027. [19] Z. Fu-Qian, Mater. Sci. Eng. A 134 (1991) 996. [20] G.W. Scherer, J. Am. Ceram. Soc. 75 (1992) 1060. [21] R. Hultgren et al., Selected Values of the Thermodynamic Properties of the Elements, ASM, Metals Park, OH, 1973.
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Investigation of speci®c heat and thermal expansion in the glass-transition regime of Pd-based metallic glasses I.-R. Lu a,*, G.P. G orler a, H.-J. Fecht b, R. Willnecker a b
a Institute of Space Simulation, DLR, D-51170 Cologne, Germany Faculty of Engineering, Materials Division, University Ulm, D-89081 Ulm, Germany
Abstract The alloys Pd±Ni±P and Pd±Ni±Cu±P have an outstanding glass-forming ability and stability against crystallisation in the undercooled liquid state at temperatures greater than the glass-transition temperatures. Samples of the copper containing alloys can be vitri®ed readily by cooling the liquid at rates P 0:1 K/s. This rate allows us to investigate the thermodynamic quantities of this alloy system in the liquid as well as in the glassy state during continuous cooling or heating without interference from crystallisation. In the present work, the speci®c heat and the volume expansion coecient in the glassy and in the undercooled liquid state were determined at temperatures less than and greater than the glass-transition temperature by applying dierential scanning calorimetry (DSC) and sessile drop (SD) technique, respectively. The glass temperatures of the samples resulting from the cooling rate were investigated during re-heating by both the methods. By comparing the results, a correlation between enthalpy and speci®c volume as a function of temperature could be demonstrated experimentally for these metallic alloys. We attribute the characteristic course of the enthalpy and the speci®c volume in the range of the glass-transition to structural relaxation processes and to the fraction of free volume in the glass as a function of temperature and time. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction The phenomenon of the liquid-to-glass-transition is interesting because of the ergodicity breaking of the thermodynamic properties, such as enthalpy and speci®c volume. However, there is still a lack of fundamental insight into the basics of glass formation and the properties of glasses [1]. Metallic glasses have simple atomic con®gurations compared to the oxide glasses or glassy organic polymers. With the development of bulk glass* Corresponding author. Present address: Institut f ur Raumsimulation, DLR, 51170 K oln, Germany. Tel.: +49-2203 6012986; fax: +49-2203 61768. E-mail address: [email protected] (I.-R. Lu).
forming alloys [2±6], glass-transition phenomena can now be measured in metallic systems. Investigations on the change of thermal expansion and heat capacity of metallic systems at the glasstransition are meaningful to test atomistic models proposed to describe the relaxation processes in this regime [7±9]. The Pd-based metallic alloys, Pd±Cu±Si and Pd±Ni±P, are well known for their glass-forming ability [2,3]. Glassy samples of Pd±Ni±P can be prepared at cooling rates of about 0.2 K/s [10]. Especially, metallic glasses composed of Pd±Ni± Cu±P can be vitri®ed at 0.1 K/s and have the largest interval between glass-transition temperature and crystallisation temperature in comparison to other alloys [6,12]. Therefore, the investigation
0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 0 ) 0 0 2 2 0 - 9
I.-R. Lu et al. / Journal of Non-Crystalline Solids 274 (2000) 294±300
of the properties of the undercooled liquid and the vitreous state during continuous cooling or heating cycles is achievable for these metallic alloys without interference from crystallisation. In the previous papers, we have reported upon dierential scanning calorimetry (DSC) measurements of the speci®c heat as a function of temperature on these alloys covering the liquid, undercooled, and glassy state. From the heat-capacity data, the melting enthalpies and the melting temperatures, the excess entropies, and the Kauzmann temperatures were obtained [11,12]. In the present work, the thermodynamic glass temperatures, Tg , de®ned in the same way as the limiting ®ctive temperatures [13], are derived independently from calorimetric and sessile drop (SD) measurements for samples, which have been vitri®ed at dierent cooling rates. The results are used to identify correlations between the investigated thermodynamic properties as a function of temperature with respect to the existing glass-transition models and to draw conclusions on the relaxation processes within the glass-transition regime. As known from earlier investigations [14], the glass temperature of a sample depends on the cooling rate during vitri®cation, i.e., on the thermal history of the sample, and it can be determined by a subsequent DSC measurement at increasing temperature. Tg can be derived from the experimentally received data of the speci®c heat, Cp (T), by integration, using an extrapolation of the speci®c heat of the liquid, Cpl , and of the glassy state, Cpg , according to Eq. (1) Z
T2 Tg
Cpl ÿ Cpg dT
Z
T2
T1
Cp ÿ Cpg dT :
1
T1 and T2 are temperatures well within the region of the stable glass and the supercooled liquid, respectively. In principle, the same method could be applied, analogously, to the data of the coecient of thermal volume expansion, b(T), to derive a glass temperature of the sample from the volumetric measurements. The correlation between these thermodynamic quantities of second order, Cp (T)
295
and b(T), is the basis of a theory developed for the glass-transition from the model of free volume [15]. Moreover, the change of the glass temperature in dependence on the cooling rate, q, is investigated to describe the relaxation kinetics at the glass-transition [14]. This dependence corresponds to that of a thermally activated process (Eq. (2)). d ln jqj Dh : RTg2 dTg
2
Here, Dh is regarded as an apparent activation energy for structural, enthalpy or volume relaxation and can be obtained from the slope of an appropriate Arrhenius diagram.
2. Experimental 2.1. Sample preparation and calorimetric analysis The Pd40 Ni40 P20 (PNP), Pd40 Ni10 Cu30 P20 (PNPCu30 ) and Pd43 Ni10 Cu27 P20 (PNP-Cu27 ) alloys (concentrations in at.%) were prepared by induction melting of the pure elements under argon atmosphere in a quartz glass tube. Prior to cooling, the alloys investigated in the present study were heat-treated above the respective liquidus temperature in a B2 O3 -¯ux to suppress the crystallisation caused by impurities. After water quenching, rods of metallic glasses with a diameter of 5 mm were obtained. Samples of the as-quenched ingots of the dierent alloys were investigated by X-ray diffraction. The presence of crystalline phases from the measured spectra was not detected. Samples for volume expansion measurements, in shape of a droplet with a diameter of about 6 mm, were prepared from the ingot within the crucibles of a heat-¯ow DSC (Netzsch DSC 404). The DSC trace observed during cooling ensures that the sample was vitri®ed without any crystallisation. The calorimetric investigation on the glasstransition of the Pd-based metallic glasses was performed in DSC (Perkin Elmer DSC-2C). The samples were heated to the undercooled liquid state at 620 K and cooled to room temperature at dierent rates (5, 10, 20, 40, 80 K/min) passing the range of transition into the vitreous state.
296
I.-R. Lu et al. / Journal of Non-Crystalline Solids 274 (2000) 294±300
Subsequently, all measurements were performed at a heating rate of 10 K/min. 2.2. Volume expansion measurement The volume expansion experiments were performed in an SD facility as shown in Fig. 1. The metallic droplet is located on a sapphire substrate, which is heated by a resistance heater under argon atmosphere in a vacuum chamber. A platinum resistance thermometer which is calibrated at the melting points of indium and lead is used for measuring the temperature of the substrate. The temperature±time courses during heating or cooling are continuously scanned by a data acquisition unit. A high-resolution monochrome CCD digital camera (Kodak DCS 420M) with an array of 1524 1012 pixels and a grey-value depth of 12 bits in combination with an image processing software (Halcon) was applied for the recording and the analysis of the cross-section of the metallic droplet. The image of the droplet is magni®ed by a
factor of 1.5±2.0 using a macro-lens (Micro-Nikkor 4.0/200 mm) to improve the experimental resolution. Utilizing a sub-pixel algorithm for image processing, the resolution with respect to the contour of the metallic droplet could be improved up to at least 1/10 of a pixel. Within a reduced dynamic range of 8 bits, the boundary of the droplet was arbitrarily settled on the grey value of 140, approximately the mean between the dark and bright ®elds of the image. Thus, the area, A, within the boundary could be evaluated. The shape of the droplet is assumed to be symmetrical with respect to the rotation about the y-axis, which is perpendicular to the substrate and passing through the center of the droplet. Based on this assumption, the method for the numerical evaluation of the volume of the droplet, V, follows (Fig. 1): the cross-section is divided along the symmetry axis into two equal areas. The distance of the center of gravity of the half cross-section from the symmetry axis is determined as Xs . The
Fig. 1. Scheme of the SD facility. The sample is located on a sapphire substrate in a chamber ®lled with argon atmosphere. A digital camera and a commercial image analysing software are applied to evaluate the cross-section of a metallic droplet. A is the area of the cross-section of the droplet and the co-ordinates (X0 ,Y0 ) and (Xs ,Y0 ) are the centres of gravity of the whole and half cross-sections, respectively.
I.-R. Lu et al. / Journal of Non-Crystalline Solids 274 (2000) 294±300
volume of the droplet then can be derived from GuldinÕs rule as V pXs A. During heating, the cross-section of the sample was photographed and evaluated at every 5 K. To verify the reliability and the sensitivity of the optical measuring system, the thermal expansion of a nickel disk was measured. The coecient of linear expansion, a(T), for nickel was found to increase from 13 ´ 10ÿ6 Kÿ1 at 300 K to 18 ´ 10ÿ6 Kÿ1 at 620 K. Compared with the literature data given in Ref. [16], the deviation of our data was within 3%. As for the calorimetric measurements, the specimen for volume expansion measurements was heated to the metastable liquid state at 620 K and vitri®ed at various cooling rates. The thermal expansion measurements were performed subsequently in the temperature range between 370 and 620 K during isochronous heating. A rate of
297
4 K/min was applied to reduce the oset between substrate temperature and droplet temperature. In fact, the limiting ®ctive temperature, Tg , of the sample achieved by cooling does not depend on the applied heating rate. 3. Results The apparent speci®c heat data of the Pd-based metallic glasses measured at a heating rate of 10 K/ min on samples after various cooling treatment are illustrated in Fig. 2. The glass temperatures achieved at a cooling rate of 10 K/min are determined from these measurements as 564 1, 572 1 and 566 1 K for PNP-Cu30 , PNP-Cu27 , and PNP, respectively. Due to the melting temperature of the alloys containing copper, which is decreased by 80 K compared to PNP [12], sample
Fig. 2. Speci®c heat in dependence on the temperature from DSC measurements at isochronous heating of 10 K/min for the metallic glasses, PNP (a), PNP-Cu27 (b) and PNP-Cu30 (c), which were vitri®ed previously by various cooling rates (5, 10, 20, 40 and 80 K/min).
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PNP-Cu27 has the largest reduced glass temperature, Tg =Tm 0:71, among the investigated alloys. In Fig. 3 the Tg s for the Pd-based samples as a function of the cooling rates are illustrated as evaluated from the measured speci®c heat data by applying Eq. (1). The eect of a variation in the cooling rate on the resulting change in Tg turns out smaller for the samples containing copper. This means that the temperature dependence of the kinetics of structural relaxation is increased by the addition of copper. Using Eq. (2), the apparent activation energies can be calculated as 854, 978 and 1081 kJ/mol for samples PNP, PNP-Cu27 and PNP-Cu30 , respectively. Fig. 4 shows the correspondence between the measured thermal expansion, V T ÿ V0 =V0 , and the enthalpy, H(T), calculated from speci®c heat data for sample PNP-Cu27 . Representative results are plotted for the thermal expansion and the enthalpy of samples, which were re-heated after quenching as well as after cooling at 10 K/min. Additionally, the eect of a dierent cooling rate of 0.5 K/min is shown in the enthalpy diagram for comparison.
Fig. 3. The correlation of the glass temperature, Tg , with the preceding cooling rate. Tg s were calculated from the measured data of speci®c heat by applying the Eq. (1). The accuracy of glass temperature determination is about 1 K. For the PNPCu27 sample (a), the dependence of Tg on cooling rates determined from thermal expansion measurements after cooling (1.5, 3, 8, 15 and 30 K/min) is shown for comparison.
The coecients of thermal volume expansion could be derived from the expansion data, which were ®tted with a polynomial of second order below the glass-transition and with a linear approximation for the range of the metastable undercooled liquid state. In the sample vitri®ed at a cooling rate of 10 K/min, a temperature dependence of the expansion coecient of the glass is found as bg 36:9 0:066 T ÿ 273 1:5 10ÿ6 Kÿ1 and a constant value of b1 90 3 10ÿ6 Kÿ1 for the undercooled liquid. For the quenched sample, structural relaxation eects are superimposed on the thermal expansion below the glass-transition. 4. Discussion For non-metallic glasses, a correlation has been noticed between glass-forming ability or critical cooling rate, respectively, and a greater reduced glass temperature Tg =Tm or a larger ratio of DCp =DSf [17]. The dierences in heat capacity between liquid and crystalline state at Tg , DCp Cpl ÿ Cpg , are as about 0.24 J/g K for the samples PNP-Cu30 and PNP-Cu27 , and 0.22 J/g K for the sample PNP. The addition of copper in the Pd-based alloys has an impact on the diminution of the melting enthalpy and the melting entropy [12], but negligible eect on DCp . Therefore, the ratios of DCp =DSf calculated from data according to Ref. [12] are obtained as 2.4 and 2.1 for the PNP-Cu and PNP alloys. In fact, the alloys PNPCu are the easiest bulk glass-formers among metallic systems today [18]. Comparing the apparent activation energies, Dh , of the investigated alloys as received from the evaluation of Fig. 3, the samples PNP-Cu27 and PNP-Cu30 are the more fragile glasses having a larger activation energy for the structural relaxation at the glass-transition than PNP. Obviously, the fragility of the alloys is systematically increased with the concentration of copper in PNP. From the curves of thermal expansion measured on the sample PNP-Cu27 , the thermodynamic Tg was determined as the temperature at the point of the intersection of the two approximations of V(T) for the glass and the liquid state.
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Fig. 4. Comparison between the thermal expansion (a) of a PNP-Cu27 sample and its enthalpy (b) as a function of temperature measured by SD and DSC techniques, respectively.
Note that this method to determine Tg is in mathematical agreement with the procedure according to Eq. (1), which is based on the secondorder quantities. Within the limits of the accuracy of the glass temperature determination (dTg 1:5 K for the volumetric measurement), Tg is coincidentally found as 570 K from the enthalpy and thermal expansion results for the sample cooled at 10 K/min. The glass temperatures as a function of the logarithm of the cooling rates obtained from volume expansion measurements for the sample PNPCu27 are additionally plotted in Fig. 3(a). From the diagram, we note that the Tg s, determined from independent DSC and SD measurements are identical within the experimental uncertainty. This identity implies that the kinetics and mechanism of the relaxation, evident from speci®c volume and enthalpy investigations are based on the same structural process. This coincidence between the calorimetric and volumetric Tg s, which is herewith con®rmed experimentally for a metallic system, has to be postulated when the free-volume model is applied to explain the glass-transition phenomena. The theory is based upon the hypothesis of a distinct fraction of free volume, Vf (T)/V, characterizing the liquid state and its ¯uidity u 1=g. For simple
liquids, the viscosity, g, and its dependence on the free volume can be described by the Doolittle equation g g0 exp f AV =Vf g:
3
For glass-forming liquids, the viscosity near Tg was found to follow the empirical Vogel±Tammann±Fulcher (VTF) relation g g0 exp f B= T ÿ T0 g:
4
The combination of both equations results in the simple expression Vf =V eq C T ÿ T0
5
for the equilibrium free volume. Thus, the increase of the speci®c heat at Tg , a general criterion of the glass-transition, can be directly referred to the energy necessary to produce additional free volume at heating. DCp Cpl ÿ Cpg dVf =dT const:
6
A glass-forming substance approaches this linear increase of Vf in the liquid state, but abandons the equilibrium during cooling through the glasstransition, where a residual fraction of free volume is frozen-in. The annihilation and generation of free volume in this regime is non-linear and
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non-exponential. This non-linearity is due to the fact that the actual free volume, dependent on the thermal history of the sample, is a crucial parameter determining the relaxation velocity. Using an appropriate kinetic model, the hysteresis phenomena observed in the glass-transition region can thus be explained. The free volume frozen-in at Tg can be calculated from the results of our measurements by using the approximation Vf Tg =V b1 ÿ bg Tg ÿ TK [19,20]. This approximation is based on the identi®cation of the divergence temperature T0 in the VTF-relation with the Kauzmann temperature TK . For sample PNP-Cu27 after cooling at 10 K/min, the fraction of free volume frozen-in at the glass-transition is 0.0024. From the combination of the volumetric and calorimetric data, it becomes possible to determine the speci®c energy of formation of free volume, DQf . Assuming a constant DCp according to Eq. (6), this energy follows from Z Tg ÿ Vf Tg DQf : DCp dT DCp Tg ÿ TK 7 V TK The data for the sample PNP-Cu27 give DQf 465 kJ/mol. This DQf , corresponding to 4.8 eV/atom, is reasonable because it is of the order of the heat of evaporation for the constituents of the alloy [21]. This agreement con®rms the simple connection between heat capacity and formation of free volume made by Eq. (6). 5. Conclusions The correlation of glass-forming ability with the reduced glass temperature was con®rmed for the investigated Pd-based samples. The volume expansion coecients are about 39±51 10ÿ6 Kÿ1 for the glassy state and about 90 10ÿ6 Kÿ1 for the undercooled liquid state, which shows a greater dependence of volume on temperature. The thermodynamic glass temperatures as a function of the applied cooling rates were determined both
from the speci®c heat data as well as from the volumetric results, and found to be independent of the experimental method. This correlation, which could be demonstrated now experimentally for metallic glass formers, can be used for the application of the free-volume model describing kinetic glass-transition phenomena. Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft, project No. Wi 1350/1± 4. References [1] M.D. Ediger, C.A. Angell, S.R. Nagel, J. Phys. Chem. 100 (1996) 13200. [2] H.S. Chen, Acta Metall. 22 (1974) 1505. [3] H.W. Kui, A.L. Greer, D. Turnbull, Appl. Phys. Lett. 45 (1984) 615. [4] A. Inoue, T. Zhang, T. Masumoto, Mater. Trans. JIM 31 (1990) 177. [5] A. Peker, W.L. Johnson, Appl. Phys. Lett. 63 (1993) 2342. [6] A. Inoue, N. Nishiyama, Mater. Trans. JIM 37 (1996) 1531. [7] S.S. Tsao, F. Spaepen, Acta Metall. 33 (1985) 881. [8] G.W. Scherer, J. Non-Cryst. Solids 123 (1990) 75. [9] I. Avramov, Thermochim. Acta 280 (1996) 363. [10] R. Willnecker, K. Wittmann, G.P. G orler, J. Non-Cryst. Solids 156±158 (1993) 450. [11] G. Wilde, G.P. G orler, R. Willnecker, G. Dietz, Appl. Phys. Lett. 65 (1994) 397. [12] I.-R. Lu, G. Wilde, G.P. G orler, R. Willnecker, J. NonCryst. Solids 250±252 (1) (1999) 577. [13] A.Q. Tool, J. Am. Ceram. Soc. 29 (1946) 240. [14] C.T. Moynihan, A.J. Easteal, J. Wilder, J. Tucker, J. Phys. Chem. 78 (1974) 2673. [15] M.H. Cohen, G.S. Grest, Phys. Rev. B 20 (1979) 1077. [16] F.C. Nix, D. MacNair, Phys. Rev. 60 (1941) 597. [17] K.S. Dubey, P. Ramachandrarao, Int. J. Rapid Solidi®cation 1 (1984±1985) 1. [18] A. Meyer, R. Busch, H. Schober, Phys. Rev. Lett. 83 (1999) 5027. [19] Z. Fu-Qian, Mater. Sci. Eng. A 134 (1991) 996. [20] G.W. Scherer, J. Am. Ceram. Soc. 75 (1992) 1060. [21] R. Hultgren et al., Selected Values of the Thermodynamic Properties of the Elements, ASM, Metals Park, OH, 1973.