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Heating rate dependence of the shear viscosity of a finemet glassy alloy
Scripta Materialia 45 (2001) 1393±1400
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Heating rate dependence of the shear viscosity of a ®nemet glassy alloy V.A. Khonika*, M. Ohtab, and K. Kitagawac a
Department of General Physics, State Pedagogical University, Lenin Str. 86, Voronezh 394043, Russia b Shimadzu Corporation, Nishinokyo-Kuwabaracho 1, Nakagyo-ku, Kyoto 604-8511, Japan c Kanazawa University, Kodatsuno 2-40-20, Kanazawa 920-8667, Japan
Received 9 July 2001; accepted 27 August 2001
Abstract Temperature dependencies of the shear viscosity of a Finemet glassy alloy under the heating rates ranging from 0.3 to 100 K/min are obtained. It is shown that isothermal cuts of these dependencies give linear viscosity increase with the inverse heating rate that accords with the ¯ow equation derived within the framework of the directional structural relaxation model. Ó 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Finemet; Metallic glasses; Viscosity; Heating rate
Introduction The Newtonian viscosity and its temperature dependence is one of the most important parameters characterizing the atomic mobility in non-crystalline materials. In particular, it concerns the case of metallic glasses (MGs), which are very far from the equilibrium state due to high quenching rates needed for glass formation. Since the beginning of the seventies, a number of MGs viscosity data obtained under various experimental conditions have been published [1±13]. These experiments were mainly carried out at constant temperatures far below or near the glass transition temperature Tg . They deal with (i) the eect of structural relaxation on the viscosity, (ii) the viscosity temperature dependence in the so-called isocon®gurational state (i.e. if structural relaxation is suppressed by preliminary heat treatment) and (iii) the viscosity temperature dependence in the metastable equilibrium state (i.e. at T > Tg ). In order to clarify the same problems several investigations [9±14] were carried out under non-isothermal
*
Corresponding author. E-mail address: [email protected] (V.A. Khonik).
1359-6462/01/$ - see front matter Ó 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 1 7 5 - 7
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conditions at the constant heating rates. However, since the viscosity is inversely proportional to the strain rate and the latter is highly dependent on the kinetics of structural relaxation, it is natural to expect that the viscosity at a given temperature will be dependent on the heating rate. Surprisingly, this question is practically unexplored in spite of three-decade-long investigations. We found in the literature few scarce data concerning the heating rate dependence of the viscosity far below or near Tg . First, the strain and strain rate during non-isothermal testing were determined to depend on the heating rate [15±19] indicating that the viscosity also depends on the latter. Second, it was found that the higher the heating rate, the lower the temperature corresponding to a given viscosity value [15,18]. To our knowledge, any other information on viscosity heating rate dependence is absent in the literature. Meanwhile, this information is evidently important for better understanding of the physics of the non-crystalline state. It can also contribute to verifying the models proposed for explanation of MGs plastic ¯ow. Amongst them the model of directional structural relaxation (DSR) has been shown to give a good quantitative explanation of a number of mechanical relaxation phenomena induced by structural relaxation of MGs (see Ref. [20] for a review and references cited therein). In particular, the DSR model predicts that the isothermal cuts of viscosity temperature dependencies g T measured under various heating rates T_ must give linear g increase with T_ 1 [19]. Having taken into account the aforementioned the purpose of the present work was twofold: (i) to perform a detailed investigation of the viscosity heating rate dependence and (ii) to compare the obtained results with the predictions of the DSR model. Experimental A typical representative of MGs, commercially available soft magnetic Finemet Fe73:5 Cu1 Nb3 Si13:5 B9 (at.%) produced by single roller melt spinning as a 22 lm thick ribbon was used for the investigation. The non-crystallinity of the ribbon was con®rmed by X-ray diraction. Mechanical polishing was employed to prepare 2 mm width samples. The viscosity temperature dependencies were calculated using the tensile creep data. Those latter were obtained by a Shimadzu TMA-50 thermomechanical analyzer (TMA) in tension arrangement with the absolute resolution of 0:01 lm. The gauge length of a sample was accepted to be 15 mm. Linear heating tests were carried out in the ¯owing nitrogen atmosphere under 18 heating rates in the range 0:3 6 T_ 6 100 K/ min. The sampling frequency increased with the heating rate so that elongation measurements were carried out every 70 s at T_ 0:3 K/min and every 0.2 s at T_ 100 K/ min. Tests at each heating rate were done twice, under tensile stresses of r0 2 and r 111 MPa. After that, the sample's tensile strain was calculated as e r ; T e r; T e r0 ; T , where e Dl=l, Dl is the elongation, l the sample's length and r r r0 . Such a method allows to remove parasitic thermal expansion of the TMA instrument [19]. The e r ; T dependence thus obtained was smoothed by 10-point adjacent averaging, numerically dierentiated and temperature dependence of the shear _ ; T . viscosity was then calculated as g T r =3e r
V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
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The relative error of viscosity determination is easily shown to be Dg De_ Dr Dr0 De_ DS ; g r r0 r r0 S e_ e_ _ Dr and DS give the absolute errors of the where S is the sample's cross-section; De, corresponding quantities. The relative error DS=S was estimated to be 0.095. Therefore, the absolute error of the viscosity calculation is ! De_ 0:095 g: 1 Dg e_ _ -plots, which is The quantity De_ in Eq. (1) was accepted as the mean scatter in the e T dependent on temperature and heating rate. Results and discussion Fig. 1a shows temperature dependencies of the tensile strain under the stress of 2 MPa and heating rates of 0.3 and 100 K/min. This strain represents mainly the thermal expansion of samples. In both cases, crystallization-induced length contraction is clearly seen and, as expected, it shifts towards high temperatures with the increase of
Fig. 1. Tensile strain at indicated stresses and heating rates. Fig. 2. Temperature dependencies of the shear viscosity at indicated heating rates.
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the heating rate. At the heating rate of 0.3 K/min, the thermal expansion coecient starts to decrease near 500 K, goes to zero at T 620 K and increases again after that. This phenomenon should be attributed to irreversible structural relaxation and is practically absent at high strain rate of 100 K/min. Under the high stress of 111 MPa, the tensile strain increases by an order of magnitude and crystallization is manifested only as a change of slope of e T dependencies, as seen in Fig. 1b. Fig. 2 shows temperature dependencies of the shear viscosity in the logarithmic scale on the Y -axis calculated according to the procedure described above. The data scatter below T 500 K is large due to low strain rates, while quite smooth ln g T dependencies are observed at T > 550 K. The eect of heating rate on the viscosity temperature dependence is seen to be fairly large. The plots ln g T shift towards low viscosities almost parallel each other with an increase of the heating rate. The 330 times increase of the latter (from 0.3 to 100 K/min) results in approximately two-order viscosity decrease at any temperature within the glassy state. It is interesting to note that in all cases ln g T -dependencies are close to straight lines (except a sharp rise in the viscosity at high temperatures due to crystallization). This means that the shear viscosity decreases with temperature approximately in accordance with an exponential law. Notice that g T dependencies do not merge at high temperatures, as it could be expected near the glass transition of a non-crystalline material. This fact correlates with the absence of the endothermic glass transition reaction in dierential scanning calorimetry tests of the alloy under investigation. Within the framework of the DSR model mentioned above, the temperature dependence of the shear viscosity is conditioned by the activation energy spectrum of irreversible structural relaxation and given as [19,20] g
r 1 ; 3e_sr 3AN0 XC T_
2
where e_sr is the strain rate determined by the rate of irreversible structural relaxation, N0 is the number of relaxation centers per unit activation energy and unit volume (i.e. the initial activation energy spectrum, AES), which is dependent on the characteristic activation energy E0 (de®ned as the energy corresponding to the maximal rate of structural relaxation at a given instant) and E0 is linearly dependent on temperature as (see Refs. [20,21] for details) E0 AT ;
3
A 3:13 10 3 eV/K [21], X is the volume embracing an elementary event of irreversible structural relaxation and C is a constant accounting for the orienting in¯uence of external stress on these events. Since MGs viscosity is well known to be extremely dependent on the degree of irreversible structural relaxation [4], the strain rate measured in the experiment described above is believed to coincide with the strain rate e_sr in Eq. (2). Therefore, this equation is expected to describe the heating rate dependence of the shear viscosity in a real experiment. At a given temperature, the product
V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
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Fig. 3. The dependence of the shear viscosity on the inverse heating rate at indicated temperatures. The solid lines are drawn as the least square ®ts for the corresponding data sets. The slopes of these lines in a, b and in c, d are the same for the same temperatures.
AN0 E0 T XC is a constant and Eq. (2) predicts that the viscosity should be proportional to the inverse heating rate. To check this prediction the g T_ 1 data obtained by isothermal cuts of the g T dependencies were plotted. The corresponding results are shown in Fig. 3 as separate graphs for high (Fig. 3a and c) and low (Fig. 3b and d) inverse heating rates. The error bars in these graphs are calculated according to Eq. (1). It is seen that within the experimental error linear dependences are observed in all cases in accordance with Eq. (2). The quality of data recti®cation can be characterized by the correlation coecient, which was found to be in the range of 0:986 6 R 6 0:999, depending on the data set. Since the volume spectral density of relaxation centers N0 increases with the characteristic activation energy (that is with temperature), Eq. (2) predicts a decrease of the slope of g T_ 1 -dependencies with testing temperature. Fig. 3 shows that such a situation is indeed observed. The quantities X and C in Eq. (2) are believed to be nearly temperature independent. Therefore, the product N0 XC, accurate to a constant, is the apparent AES. Eq. (2) allows to reconstruct the apparent AES by the two independent methods: (i) using the slopes of g T_ 1 dependencies at constant temperatures shown in Fig. 3, i.e. as
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Fig. 4. Apparent AES of irreversible structural relaxation independently calculated from (i) isothermal cuts of the g T_ 1 dependencies shown in Fig. 3 (large dark circles), (ii) viscosity temperature dependences measured at indicated heating rates and (iii) isothermal creep tests at indicated temperatures reported in Ref. [22].
N0 XC 3AB 1 , where B og=o T_ 1 , and (ii) using temperature dependencies of the _ T_ . creep strain rate at ®xed heating rates, i.e. as N0 XC e=rA The results of AES reconstruction are shown in Fig. 4 where the characteristic activation energy calculated according to Eq. (3) is plotted on the abscissa. Reconstruction using the ®rst method produces ®ve large dark circles, which correspond to the temperatures indicated in Fig. 3. Open symbols in Fig. 4 give the AES reconstruction using the second method for the same set of heating rates, which was applied for obtaining the viscosity data shown in Fig. 2. It is seen that this method gives quite close results nearly independent of the heating rate if the latter belongs to the range 0:3 6 T_ 6 20 K/min. The creep curve measured at T_ 100 K/min shows somewhat lower N0 XC-values. Independence of AES reconstruction on the heating rate using this method was recently demonstrated for a Fe±Ni±B metallic glass [23]. It is also worthy of note that a drop of the AES reconstructed for the heating rates of 0.3 and 1.18 K/min at E0 > 2:4 eV appears to be due to crystallization of samples. With the use of the two aforementioned methods, AES reconstruction is based on the analysis of linear heating creep tests. These results can be directly compared with the results of AES reconstruction from independent isothermal tests using the creep ¯ow equation obtained within the framework of the DSR model for a constant temperature [20]. In this case it was assumed that the volume spectral density of relaxation centers is a constant within a narrow activation energy range scanned during an isothermal experiment (the ``¯at spectrum approximation'') and, therefore, the AES can be reconstructed as a set of horizontal segments on the N0 XC f E0 plot. The corresponding calculations have been recently reported in Ref. [22]. These calculations were based on
V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
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the analysis of isothermal creep experiments carried out using the same Finemet ribbon and same creep instrument that were used in the present investigation. The results of these calculations for nine testing temperatures are shown by the horizontal segments in Fig. 4. It is seen that these results generally correspond quite well to the results of AES reconstruction from non-isothermal creep tests using the two methods described above. Thus, the three independent ways of AES reconstruction give consistent results indicating validity of the plastic ¯ow law derived within the framework of the DSR model. Like the previous results of AES calculation according to the DSR model [20, 23±25], the AES reconstructed in the present investigation takes the form of an increasing function. It was pointed out earlier [20] that microscopic interpretation of the increase of the relaxation centers density with the activation energy is not clear. A possible reason can lie in the fact that the energy structure of relaxation centers in MG is rather complicated and has a number of local energy minima separated by barriers of various heights. In this case, a relaxation center in a glassy sample can transform several times into states with sequentially decreasing potential energy. This assumption correlates with the current notions of the energy landscape in glassy materials [26]. However, the ¯ow equations of the DSR model imply only a two-well energy structure of relaxation centers. Therefore, the AES reconstructed with these equations should be considered as an apparent one determined by their limited applicability. Conclusions 1. A detailed investigation of non-isothermal creep of a Finemet glassy alloy has been performed for a wide range of heating rates T_ . It is shown that the eect of the heating rate on temperature dependencies of the shear viscosity g is fairly large: an increase of the heating rate by 330 times results in viscosity decrease approximately by two orders of magnitude at any temperature from 500 K up to the crystallization onset temperature. Independently of the heating rate, the viscosity decreases with temperature approximately according to an exponential law. 2. At a ®xed temperature, the shear viscosity linearly increases with the inverse heating rate and the slopes of g T_ 1 dependencies decrease with temperature, in accordance with Eq. (2) derived earlier within the framework of the DSR model. 3. Reconstruction of the AES of irreversible structural relaxation from linear heating and isothermal creep tests by the three independent methods based on the DSR model yield consistent results, indicating validity of the corresponding ¯ow equations. Acknowledgements This work was carried out under the ®nancial support of the Ministry of Education of Russian Federation in the ®eld of natural sciences (grant E00-3.4-48). The authors are thankful to Dr. A.Yu. Vinogradov (Osaka City University, Japan) and Dr. Smith (Moscow State University, Russia) for their kind help.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
Chen, H. S., & Goldstein, M. (1972). J Appl Phys 43, 1642. Logan, J., & Ashby, M. F. (1974). Acta Met 22, 1047. Chen, H. S. (1978). J Non-Cryst Sol 27, 257. Taub, A. I., & Spaepen, F. (1979). Scripta Metall 13, 195. Taub, A. I., & Spaepen, F. (1980). Acta Metall 28, 1781. Haimovich, J., & Egami, T. (1983). Mater Sci Eng 61, 89. Taub, A. I., & Walter, J. L. (1984). Mater Sci Eng 62, 249. Tsao, S. S., & Spaepen, F. (1985). Acta Metall 33, 881. Bouali, A., & Tete, C. (1988). Mater Sci Eng 97, 493. Russew, K., & Stojanova, L. (1990). Mater Sci Eng A 123, 59. Russew, K., Stojanova, L., & Lovash, A. (1994). Int J Rapid Solidif 8, 147. Russew, K., Zappel, B. J., & Sommer, F. (1995). Scripta Metall Mater 32, 271. Zappel, B. J., & Sommer, F. (1996). J Non-Cryst Sol 205±207, 494. Russew, K., & Sommer, F. (1998). Int J Non-Equilibr Proc 11, 3. Taub, A. I. (1985). Rapid Quenching Metals. In S. Steeb & H. Warlimont (Eds.). Proc 5th Int Conf W urzburg, 3±7 September, 1984 (p. 1365). Amsterdam: Elsevier. Wang, J., Ding, B., Pang, D., Li, S., & Li, G. (1988). Mater Sci Eng 97, 483. Zheng, F.-Q. (1988). Mater Sci Eng 97, 487. Bhatti, A. R., & Cantor, B. (1988). Mater Sci Eng 97, 479. Khonik, V. A., Mikhailov, V. A., & Safonov, I. A. (1997). Scripta Mater 37, 921. Khonik, V. A. (2000). Phys Stat Sol (A) 177, 173. Khonik, V. A., Kitagawa, K., & Morii, H. (2000). J Appl Phys 87, 8440. Khonik, V. A., & Ohta, M. (2001). Phys Stat Sol (A) 184, 367. Csach, K., Filippov, Yu. A., Khonik, V. A., Kulbaka, V. A., & Ocelik, V. (2001). Philos Mag A 81, 1901. Bobrov, O. P., Khonik, V. A., & Zhelezny, V. S. (1998). J Non-Cryst Sol 223, 241. Belyavsky, V. I., Bobrov, O. P., Khonik, V. A., & Kosilov, A. T. (1996). J de Phys IV Coll 6, C8-715. Debenedetti, P. G., & Stillinger, F. H. (2001). Nature 410, 259.
www.elsevier.com/locate/scriptamat
Heating rate dependence of the shear viscosity of a ®nemet glassy alloy V.A. Khonika*, M. Ohtab, and K. Kitagawac a
Department of General Physics, State Pedagogical University, Lenin Str. 86, Voronezh 394043, Russia b Shimadzu Corporation, Nishinokyo-Kuwabaracho 1, Nakagyo-ku, Kyoto 604-8511, Japan c Kanazawa University, Kodatsuno 2-40-20, Kanazawa 920-8667, Japan
Received 9 July 2001; accepted 27 August 2001
Abstract Temperature dependencies of the shear viscosity of a Finemet glassy alloy under the heating rates ranging from 0.3 to 100 K/min are obtained. It is shown that isothermal cuts of these dependencies give linear viscosity increase with the inverse heating rate that accords with the ¯ow equation derived within the framework of the directional structural relaxation model. Ó 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Finemet; Metallic glasses; Viscosity; Heating rate
Introduction The Newtonian viscosity and its temperature dependence is one of the most important parameters characterizing the atomic mobility in non-crystalline materials. In particular, it concerns the case of metallic glasses (MGs), which are very far from the equilibrium state due to high quenching rates needed for glass formation. Since the beginning of the seventies, a number of MGs viscosity data obtained under various experimental conditions have been published [1±13]. These experiments were mainly carried out at constant temperatures far below or near the glass transition temperature Tg . They deal with (i) the eect of structural relaxation on the viscosity, (ii) the viscosity temperature dependence in the so-called isocon®gurational state (i.e. if structural relaxation is suppressed by preliminary heat treatment) and (iii) the viscosity temperature dependence in the metastable equilibrium state (i.e. at T > Tg ). In order to clarify the same problems several investigations [9±14] were carried out under non-isothermal
*
Corresponding author. E-mail address: [email protected] (V.A. Khonik).
1359-6462/01/$ - see front matter Ó 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 6 2 ( 0 1 ) 0 1 1 7 5 - 7
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V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
conditions at the constant heating rates. However, since the viscosity is inversely proportional to the strain rate and the latter is highly dependent on the kinetics of structural relaxation, it is natural to expect that the viscosity at a given temperature will be dependent on the heating rate. Surprisingly, this question is practically unexplored in spite of three-decade-long investigations. We found in the literature few scarce data concerning the heating rate dependence of the viscosity far below or near Tg . First, the strain and strain rate during non-isothermal testing were determined to depend on the heating rate [15±19] indicating that the viscosity also depends on the latter. Second, it was found that the higher the heating rate, the lower the temperature corresponding to a given viscosity value [15,18]. To our knowledge, any other information on viscosity heating rate dependence is absent in the literature. Meanwhile, this information is evidently important for better understanding of the physics of the non-crystalline state. It can also contribute to verifying the models proposed for explanation of MGs plastic ¯ow. Amongst them the model of directional structural relaxation (DSR) has been shown to give a good quantitative explanation of a number of mechanical relaxation phenomena induced by structural relaxation of MGs (see Ref. [20] for a review and references cited therein). In particular, the DSR model predicts that the isothermal cuts of viscosity temperature dependencies g T measured under various heating rates T_ must give linear g increase with T_ 1 [19]. Having taken into account the aforementioned the purpose of the present work was twofold: (i) to perform a detailed investigation of the viscosity heating rate dependence and (ii) to compare the obtained results with the predictions of the DSR model. Experimental A typical representative of MGs, commercially available soft magnetic Finemet Fe73:5 Cu1 Nb3 Si13:5 B9 (at.%) produced by single roller melt spinning as a 22 lm thick ribbon was used for the investigation. The non-crystallinity of the ribbon was con®rmed by X-ray diraction. Mechanical polishing was employed to prepare 2 mm width samples. The viscosity temperature dependencies were calculated using the tensile creep data. Those latter were obtained by a Shimadzu TMA-50 thermomechanical analyzer (TMA) in tension arrangement with the absolute resolution of 0:01 lm. The gauge length of a sample was accepted to be 15 mm. Linear heating tests were carried out in the ¯owing nitrogen atmosphere under 18 heating rates in the range 0:3 6 T_ 6 100 K/ min. The sampling frequency increased with the heating rate so that elongation measurements were carried out every 70 s at T_ 0:3 K/min and every 0.2 s at T_ 100 K/ min. Tests at each heating rate were done twice, under tensile stresses of r0 2 and r 111 MPa. After that, the sample's tensile strain was calculated as e r ; T e r; T e r0 ; T , where e Dl=l, Dl is the elongation, l the sample's length and r r r0 . Such a method allows to remove parasitic thermal expansion of the TMA instrument [19]. The e r ; T dependence thus obtained was smoothed by 10-point adjacent averaging, numerically dierentiated and temperature dependence of the shear _ ; T . viscosity was then calculated as g T r =3e r
V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
1395
The relative error of viscosity determination is easily shown to be Dg De_ Dr Dr0 De_ DS ; g r r0 r r0 S e_ e_ _ Dr and DS give the absolute errors of the where S is the sample's cross-section; De, corresponding quantities. The relative error DS=S was estimated to be 0.095. Therefore, the absolute error of the viscosity calculation is ! De_ 0:095 g: 1 Dg e_ _ -plots, which is The quantity De_ in Eq. (1) was accepted as the mean scatter in the e T dependent on temperature and heating rate. Results and discussion Fig. 1a shows temperature dependencies of the tensile strain under the stress of 2 MPa and heating rates of 0.3 and 100 K/min. This strain represents mainly the thermal expansion of samples. In both cases, crystallization-induced length contraction is clearly seen and, as expected, it shifts towards high temperatures with the increase of
Fig. 1. Tensile strain at indicated stresses and heating rates. Fig. 2. Temperature dependencies of the shear viscosity at indicated heating rates.
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V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
the heating rate. At the heating rate of 0.3 K/min, the thermal expansion coecient starts to decrease near 500 K, goes to zero at T 620 K and increases again after that. This phenomenon should be attributed to irreversible structural relaxation and is practically absent at high strain rate of 100 K/min. Under the high stress of 111 MPa, the tensile strain increases by an order of magnitude and crystallization is manifested only as a change of slope of e T dependencies, as seen in Fig. 1b. Fig. 2 shows temperature dependencies of the shear viscosity in the logarithmic scale on the Y -axis calculated according to the procedure described above. The data scatter below T 500 K is large due to low strain rates, while quite smooth ln g T dependencies are observed at T > 550 K. The eect of heating rate on the viscosity temperature dependence is seen to be fairly large. The plots ln g T shift towards low viscosities almost parallel each other with an increase of the heating rate. The 330 times increase of the latter (from 0.3 to 100 K/min) results in approximately two-order viscosity decrease at any temperature within the glassy state. It is interesting to note that in all cases ln g T -dependencies are close to straight lines (except a sharp rise in the viscosity at high temperatures due to crystallization). This means that the shear viscosity decreases with temperature approximately in accordance with an exponential law. Notice that g T dependencies do not merge at high temperatures, as it could be expected near the glass transition of a non-crystalline material. This fact correlates with the absence of the endothermic glass transition reaction in dierential scanning calorimetry tests of the alloy under investigation. Within the framework of the DSR model mentioned above, the temperature dependence of the shear viscosity is conditioned by the activation energy spectrum of irreversible structural relaxation and given as [19,20] g
r 1 ; 3e_sr 3AN0 XC T_
2
where e_sr is the strain rate determined by the rate of irreversible structural relaxation, N0 is the number of relaxation centers per unit activation energy and unit volume (i.e. the initial activation energy spectrum, AES), which is dependent on the characteristic activation energy E0 (de®ned as the energy corresponding to the maximal rate of structural relaxation at a given instant) and E0 is linearly dependent on temperature as (see Refs. [20,21] for details) E0 AT ;
3
A 3:13 10 3 eV/K [21], X is the volume embracing an elementary event of irreversible structural relaxation and C is a constant accounting for the orienting in¯uence of external stress on these events. Since MGs viscosity is well known to be extremely dependent on the degree of irreversible structural relaxation [4], the strain rate measured in the experiment described above is believed to coincide with the strain rate e_sr in Eq. (2). Therefore, this equation is expected to describe the heating rate dependence of the shear viscosity in a real experiment. At a given temperature, the product
V.A. Khonik et al./Scripta Materialia 45 (2001) 1393±1400
1397
Fig. 3. The dependence of the shear viscosity on the inverse heating rate at indicated temperatures. The solid lines are drawn as the least square ®ts for the corresponding data sets. The slopes of these lines in a, b and in c, d are the same for the same temperatures.
AN0 E0 T XC is a constant and Eq. (2) predicts that the viscosity should be proportional to the inverse heating rate. To check this prediction the g T_ 1 data obtained by isothermal cuts of the g T dependencies were plotted. The corresponding results are shown in Fig. 3 as separate graphs for high (Fig. 3a and c) and low (Fig. 3b and d) inverse heating rates. The error bars in these graphs are calculated according to Eq. (1). It is seen that within the experimental error linear dependences are observed in all cases in accordance with Eq. (2). The quality of data recti®cation can be characterized by the correlation coecient, which was found to be in the range of 0:986 6 R 6 0:999, depending on the data set. Since the volume spectral density of relaxation centers N0 increases with the characteristic activation energy (that is with temperature), Eq. (2) predicts a decrease of the slope of g T_ 1 -dependencies with testing temperature. Fig. 3 shows that such a situation is indeed observed. The quantities X and C in Eq. (2) are believed to be nearly temperature independent. Therefore, the product N0 XC, accurate to a constant, is the apparent AES. Eq. (2) allows to reconstruct the apparent AES by the two independent methods: (i) using the slopes of g T_ 1 dependencies at constant temperatures shown in Fig. 3, i.e. as
1398
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Fig. 4. Apparent AES of irreversible structural relaxation independently calculated from (i) isothermal cuts of the g T_ 1 dependencies shown in Fig. 3 (large dark circles), (ii) viscosity temperature dependences measured at indicated heating rates and (iii) isothermal creep tests at indicated temperatures reported in Ref. [22].
N0 XC 3AB 1 , where B og=o T_ 1 , and (ii) using temperature dependencies of the _ T_ . creep strain rate at ®xed heating rates, i.e. as N0 XC e=rA The results of AES reconstruction are shown in Fig. 4 where the characteristic activation energy calculated according to Eq. (3) is plotted on the abscissa. Reconstruction using the ®rst method produces ®ve large dark circles, which correspond to the temperatures indicated in Fig. 3. Open symbols in Fig. 4 give the AES reconstruction using the second method for the same set of heating rates, which was applied for obtaining the viscosity data shown in Fig. 2. It is seen that this method gives quite close results nearly independent of the heating rate if the latter belongs to the range 0:3 6 T_ 6 20 K/min. The creep curve measured at T_ 100 K/min shows somewhat lower N0 XC-values. Independence of AES reconstruction on the heating rate using this method was recently demonstrated for a Fe±Ni±B metallic glass [23]. It is also worthy of note that a drop of the AES reconstructed for the heating rates of 0.3 and 1.18 K/min at E0 > 2:4 eV appears to be due to crystallization of samples. With the use of the two aforementioned methods, AES reconstruction is based on the analysis of linear heating creep tests. These results can be directly compared with the results of AES reconstruction from independent isothermal tests using the creep ¯ow equation obtained within the framework of the DSR model for a constant temperature [20]. In this case it was assumed that the volume spectral density of relaxation centers is a constant within a narrow activation energy range scanned during an isothermal experiment (the ``¯at spectrum approximation'') and, therefore, the AES can be reconstructed as a set of horizontal segments on the N0 XC f E0 plot. The corresponding calculations have been recently reported in Ref. [22]. These calculations were based on
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the analysis of isothermal creep experiments carried out using the same Finemet ribbon and same creep instrument that were used in the present investigation. The results of these calculations for nine testing temperatures are shown by the horizontal segments in Fig. 4. It is seen that these results generally correspond quite well to the results of AES reconstruction from non-isothermal creep tests using the two methods described above. Thus, the three independent ways of AES reconstruction give consistent results indicating validity of the plastic ¯ow law derived within the framework of the DSR model. Like the previous results of AES calculation according to the DSR model [20, 23±25], the AES reconstructed in the present investigation takes the form of an increasing function. It was pointed out earlier [20] that microscopic interpretation of the increase of the relaxation centers density with the activation energy is not clear. A possible reason can lie in the fact that the energy structure of relaxation centers in MG is rather complicated and has a number of local energy minima separated by barriers of various heights. In this case, a relaxation center in a glassy sample can transform several times into states with sequentially decreasing potential energy. This assumption correlates with the current notions of the energy landscape in glassy materials [26]. However, the ¯ow equations of the DSR model imply only a two-well energy structure of relaxation centers. Therefore, the AES reconstructed with these equations should be considered as an apparent one determined by their limited applicability. Conclusions 1. A detailed investigation of non-isothermal creep of a Finemet glassy alloy has been performed for a wide range of heating rates T_ . It is shown that the eect of the heating rate on temperature dependencies of the shear viscosity g is fairly large: an increase of the heating rate by 330 times results in viscosity decrease approximately by two orders of magnitude at any temperature from 500 K up to the crystallization onset temperature. Independently of the heating rate, the viscosity decreases with temperature approximately according to an exponential law. 2. At a ®xed temperature, the shear viscosity linearly increases with the inverse heating rate and the slopes of g T_ 1 dependencies decrease with temperature, in accordance with Eq. (2) derived earlier within the framework of the DSR model. 3. Reconstruction of the AES of irreversible structural relaxation from linear heating and isothermal creep tests by the three independent methods based on the DSR model yield consistent results, indicating validity of the corresponding ¯ow equations. Acknowledgements This work was carried out under the ®nancial support of the Ministry of Education of Russian Federation in the ®eld of natural sciences (grant E00-3.4-48). The authors are thankful to Dr. A.Yu. Vinogradov (Osaka City University, Japan) and Dr. Smith (Moscow State University, Russia) for their kind help.
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