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Thermodynamics and kinetics of the Zr41.2Ti13.8Cu10.0Ni12.5Be22.5 bulk metallic glass forming liquid: glass formation from a strong liquid
Journal of Non-Crystalline Solids 250±252 (1999) 566±571
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Thermodynamics and kinetics of the Zr41:2Ti13:8Cu10:0Ni12:5Be22:5 bulk metallic glass forming liquid: glass formation from a strong liquid A. Masuhr, R. Busch *, W.L. Johnson W.M. Keck Laboratory of Engineering Materials, California Institute of Technology, Pasadena, CA 91125, USA
Abstract The ¯ow properties of Zr41:2 Ti13:8 Cu10:0 Ni12:5 Be22:5 have been measured both in the liquid and supercooled liquid state, spanning 15 orders of magnitude in viscosity. We used beam bending experiments in the vicinity of the glass transition regime and concentric cylinder rheometry near the liquidus temperature to ®nd very large viscosities for a metallic system of 2 Pa s in the equilibrium liquid. The temperature dependence of the viscosity is discussed in the framework of the Adam±Gibbs entropy model as well as the free volume model for viscous ¯ow. The thermodynamic functions on the one side and the speci®c volume on the other have been measured previously and allow for a detailed comparison of the two phenomenological descriptions. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Monatomic or binary metallic liquids at the melting point have viscosities, g, on the order of 10ÿ3 Pa s [1,2]. To our knowledge, no direct viscosity measurements have been performed in the supercooled liquid state of metallic systems near the liquidus temperature. Recently, viscosity measurements on multicomponent bulk metallic glass (BMG) forming alloys near their calorimetric glass transition have found considerable attention [3±6]. The experiments are made possible by the increased stability of these alloys with respect to crystallization. The thermodynamic functions of
* Corresponding author. Present address: Department of Mechanical Engineering, Oregon State University, 204 Rogers Hall, Corvallis, OR 97331, USA. Tel.: +1-541 737 2648; fax: +1-541 737 2600; e-mail: [email protected]
the deeply supercooled liquid can be determined by dierential scanning calorimetry (DSC) on reheating amorphous samples. Relatively few data is available for transport coecients in the equilibrium or slightly supercooled liquid of BMG forming alloys. The anity of the transition metal alloys for impurities such as oxygen pose severe challenges for experimental studies at elevated temperatures. This temperature range is, however, of importance in the process of understanding atomic transport mechanisms and decisive for the glass forming ability of the alloys [7]. In the following we will present results from viscosity measurements on the Zr41:2 Ti13:8 Cu10:0 Ni12:5 Be22:5 (Vit1) BMG forming alloy [8]. With a critical cooling rate for crystallization of 1 Ksÿ1 it is one of the best metallic glass forming systems [9] and thermodynamic data [10] and speci®c volume measurements [11] are already available.
0022-3093/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 1 3 3 - 7
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
567
2. Experimental We have designed a Couette viscometer to measure the rheological properties of Zr-based glass forming alloys in the range from 10ÿ1 to 103 Pa s. The viscometer's concentric cylinder shear cell is machined from graphite and designed to hold about 5 cm3 of the liquid alloy. The cell is mounted vertically inside a vacuum induction furnace with the outer cylinder attached to a static torque sensor (Eaton 2127-10). Ingots of ca. 30 g of Vit1 are prepared by melting the elements in an arc-melting device under argon. The ingots are placed inside the shear cell and melted at 1200 K. Subsequently, the inner rotatable cylinder was inserted into the liquid. We measure the temperature, T , with a type K thermocouple mounted inside the crucible wall of the outer cylinder. The torque and temperature signals are digitally processed and are linked to the temperature controller of the induction furnace. The liquid alloy wets the graphite and allows a straightforward calculation of the ¯ow ®eld between the cylinders [12]. A more detailed description of the experiment can be found elsewhere [13]. Isothermal measurements in the equilibrium liquid are performed by both, clockwise and counterclockwise rotation of the inner cylinder to improve the resolution of the apparatus. The static torque on the outer cylinder is proportional to the Newtonian viscosity, g, where the proportionality constant can be calculated from the angular frequency of the inner cylinder and the geometry of the shear cell [14]. We ®nd the ¯ow to be Newtonian for strain rates from 0.4 to 4 sÿ1 used in this study. Additional viscosity measurements of the supercooled liquid were performed with a cooling rate of 0.7 K sÿ1 at constant shear rate. The graphite containers do not alter the times to nucleation as compared to electrostatic levitation experiments [15,16].
3. Results Fig. 1 summarizes results obtained from the high temperature viscometer as well as from beam
Fig. 1. Viscosity of Vit1 as measured by beam bending (h) and concentric cylinder (s). The data were ®tted to entropy model, Eq. (1), and the free volume model, Eqs. (7) and (8). The dotted, dashed and solid curves represent the least squares ®ts, respectively. Also shown are the viscosities (d) of Sn, Bi, Tl, Cd, Pb, Zn, Te, Sb, Mg, Ag, Ac, Au, Cu, Mn, Be, Ni, Co, Fe, Sc, Pd, V, Ti, Pt, Zr, Cr, Rh, B, Ru, Ir, Mo, Os, Re, and W near their melting point (in order of increasing melting point) [2]. The viscosity of Zr was calculated with Andrade's formula, which can be found in [2].
bending rheometry [16]. The latter were performed by heating amorphous samples above the calorimetric glass transition temperature. Extrapolating the measured viscosity of Vit1 to higher temperatures according to Eqs. (1) and (2), discussed below, we arrive at `normal' melt viscosity of 10ÿ3 to 10ÿ2 Pa s at about the average melting point of Vit1's constituents. This extrapolation merely re¯ects the vicinity of Vit1's composition to a deep eutectic. The high viscosity of Vit1 which is about 3 orders of magnitude larger than for monatomic metallic liquids at its liquidus temperature, Tliq 1026 K, is of great importance for the alloy's glass forming ability [7]. For the following discussion of the ¯ow of the supercooled melt, however, the liquidus temperature itself has no signi®cance.
568
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
4. Discussion Adam and Gibbs [17] developed a theory on cooperative relaxation in liquids and supercooled liquids. The increase in relaxation times and viscosity with increasing supercooling is directly linked to a decrease in con®gurational entropy, Sc , of the system. Within their model, the viscosity is expressed as C ; 1 g g0 exp TSc where the constant C represents an eective free enthalpy barrier for cooperative rearrangements. The pre-factor, g0 h=vm in Eq. (1), with h as Planck's constant, is calculated according to Eyring's theory to be 4 10ÿ5 Pa s [18]. The atomic volume, vm , of Vit1 was measured by Ohsaka [10] in an electrostatic levitation experiment and is vm 1:67 10ÿ29 m3 near the liquidus temperature. Its temperature dependence is neglected in comparison to the exponential term in Eq. (1). The temperature dependence of the con®gurational entropy, Sc , can be approximated from the dierence, DS, between the total entropies of the liquid and the crystal, assuming that their vibrational parts of the entropy are equal. DS results directly from an integration of the measured heat capacity dierence DCp Cp;liq ÿ Cp;xtl between the liquid and the crystal [6,9]. Prior to the measurement of Cp;xtl , an amorphous sample was crystallized by heating to 800 K at 20 K/min. We obtain Sc by Sc
Scliq
Tliq Z
ÿ T
DCp ~ dT ; T
If Dcp is proportional to 1=T an integration of Eqs. (1) and (2) yield a Vogel±Fulcher±Tammann (VFT) relation. For the present alloy, however, a 1=T dependence clearly underestimates the temperature dependence of Dcp as shown in Fig. 2. So, a VFT relation cannot be justi®ed on the basis of the Adam±Gibbs entropy model for the present system. Instead, we use the following relations to describe the heat capacities of the liquid and the crystal: Cp;xtl 3R aT bT 2 ;
3
Cp;liq Cp;xtl bT ÿ2 cT ÿ3 :
4
The parameters are (in SI units) R 8:314; a ÿ8:021 10ÿ3 ; b 2:076 10ÿ5 ; b 2:940 106 , and c 3:934 109 . Eqs. (3) and (4) describe the experimental data (Fig. 2) and yield a ®nite high temperature limit, g0 , for the viscosity if inserted in Eqs. (1) and (2) because limT !1 Dcp 0. Kubaschewski's formula [9,20] describes the heat capacity data equally well, but fails to comply with limT !1 Dcp 0 and therefore with limT !1 g T g0 .
2
which is equal to DS except for an additive constant. This equation takes into account, that in a multicomponent system the crystal can have a considerable amount of entropy of mixing. In fact, a disordered solid solution can have a larger entropy of mixing than a supercooled liquid of the same composition with tendency to topological and chemical short range order [6,19]. Inserting Eq. (2) in Eq. (1), we treat Scliq , the con®gurational entropy at the liquidus temperature, as a ®tting parameter as details about short range ordering in the present alloy are not yet known.
Fig. 2. Speci®c heat capacities of Vit1 in the liquid (s) and crystalline (n) states. Eqs. (3) and (4) are shown as solid curves and the resulting Dcp as a dotted curve. A Dcp 1=T ®t to the spec®c heat capacity of the liquid is shown (- - -).
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
A ®t of Eq. (1) to the viscosity data of Vit1, shown in Fig. 1, yields the two ®t parameters, C and 203 kJ g atomÿ1 or 2:1 eV atomÿ1 Scliq 15:8 J Kÿ1 g atomÿ1 . We ®nd qualitative agreement between the measured viscosity and the entropy model but, e.g., the tendency of the viscosities to be ®t by an Arrhenius function (log g 1=T ) at temperatures < 700 K is not entirely accounted for by Eq. (1). Using the ®t parameters, C and Scliq , and Eq. (1) to solve for the con®gurational entropy, we obtain Fig. 3. Although the temperature dependence of the viscosity in Fig. 1 is in qualitative agreement with Adam's and Gibb's theory, it is clear from Fig. 3 that the decrease in entropy with supercooling cannot quantitatively account for the exact form of g T . According to the free volume model [21], viscous ¯ow occurs via structural rearrangements without appreciable activation energy compared to kB T . The viscosity can be expressed as bvm 5 g g0 exp vf
with vf as the average free volume per atom and bvm a critical volume for ¯ow [21,22]. In the following discussion b and vm are treated as temperature independent. Again, the pre-factor is set constant to the above g0 . In Fig. 1 the logarithm of the viscosity data is ®tted to Eq. (5) with two different expressions for the temperature dependence of the free volume. First, a linear relation vf vm af T ÿ T0 between the free volume and the temperature is used, which yields the VFT equation b : 6 g g0 exp af T ÿ T0 Here, af is a measure of the increase in free volume with temperature and according to Williams et al. [23] can be approximated by the dierence in thermal expansion coecients between the liquid and the glass af aliq ÿ aglass :
7
Often, the dimensionless parameter, D b= af T0 , is introduced, to classify a liquid according to its fragility [24]. At the temperature, T0 , treated as a ®t parameter, the free volume vanishes and viscous ¯ow is no longer possible, according to this model. For the Vit1 data, shown in Fig. 1, we ®nd D 16:5 and T0 426:3 K comparable to the relatively strong liquid property of sodium silicate glasses [25]. These parameters vary slightly from previously published results [25] due to the recent higher temperature measurements of g. In an extended version of the free volume model by Cohen and Grest [21] the free volume decreases to zero, only for T 0 and the viscosity takes the form 2 3 6 g g0 exp 4
Fig. 3. Entropy dierence DS between the liquid and the crystalline states of Vit1 (á á á) and con®gurational entropy Sc (±±±) of the liquid as obtained from a ®t of Eq. (1) to the viscosity data in Fig. 1. The viscosity measurements are also converted via Eq. (1) into Sc (s).
569
2bvm f0 =kB 7 q 5; 2 4va f0 T ÿ T0 T ÿ T0 kB T
8
resulting in the three ®t parameters: bvm f0 =kB 4933 K, T0 672 K, and va f0 =kB 40:5 K for Vit1. Similar to Sc in Fig. 3 we plot in Fig. 4 the ratio vf = bvm as obtained from the viscosity and the Cohen±Grest ®t, Eq. (8). This representation con®rms the excellent agreement between the ®t and the present viscosity data in
570
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
as the free volume model. Comparing the results from the least squares ®ts to the viscosity with thermodynamic measurements and speci®c volume measurements, we conclude that the interpretation in terms of the free volume is more consistent.
Acknowledgements This work was supported by the U.S. Department of Energy, Grant No. DEFG-03-86ER45242 and by the ALCOA Technical Center.
References Fig. 4. Ratio vf = bvm as obtained from the VFT-relation, Eq. (5) (- - -), the Cohen±Grsest formula, Eq. (8) (±±±), and from viscosity measurements (s). The free volume vf =vm as obtained from the Cohen±Grest ®t and Eq. (7) is also shown (á á á).
Fig. 1. To estimate the absolute amount of free volume as a function of temperature, we can read from Fig. 4 for the temperature range from 800 to 1200 K af 1 dvf 1:83 10ÿ4 Kÿ1 : b bvm dT
9
With aliq 5:32 10ÿ5 Kÿ1 and aglass 3:39 10ÿ5 Kÿ1 10, Eqs. (7) and (9) lead to b 0:105 and an average free volume shown as the dotted line in Fig. 4. At the liquidus vf 9:58 10ÿ3 vm , i.e. about 1% of the sample's volume is `free'. This small amount of free volume is in accordance with the already anticipated model [8,11] of a dense metallic liquid with sluggish kinetics. 5. Conclusions We have presented results from viscosity measurements near the glass transition as well as in the equilibrium liquid of Vit1. The temperature dependence is in qualitative agreement with the entropy model by Adam and Gibbs as well
[1] T.E. Faber, Introduction to the Theory of Liquid Metals, Cambridge University Press, Cambridge, 1972. [2] T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1988. [3] E. Bakke, R. Busch, W.L. Johnson, Appl. Phys. Lett. 67 (1995) 3260. [4] R. Rambousky, M. Moske, K. Samwer, Z. Phys. B 99 (1996) 387. [5] G. Wilde, G.P. G orler, K. Jeropoulos, R. Willnecker, H.J. Fecht, Mater. Sci. Forum 269±272 (1998) 541. [6] R. Busch, W. Liu, W.L. Johnson, J. Appl. Phys. 83 (1998) 4134. [7] H.A. Davies, Phys. and Chem. of Glasses 17 (1976) 159. [8] A. Peker, W.L. Johnson, Appl. Phys. Lett. 63 (1993) 2342. [9] Y.J. Kim, R. Busch, W.L. Johnson, A.J. Rulison, W.K. Rhim, Appl. Phys. Lett. 68 (1996) 1057. [10] R. Busch, Y.J. Kim, W.L. Johnson, J. Appl. Phys. 77 (1995) 4039. [11] K. Ohsaka, S.K. Chung, W.K. Rhim, A. Peker, D. Scruggs, W.L. Johnson, Appl. Phys. Lett. 70 (1997) 726. [12] P.K. Kundu, Fluid Mechanics, Academic Press, San Diego, 1990. [13] A. Masuhr, Ph.D. Thesis, California Institute of Technology, 1999. [14] M.M. Couette, Ann. Chim. Phys. 21 (1880) 433. [15] A. Masuhr, R. Busch, W.L. Johnson, Mater. Sci. Forum 269±272 (1998) 779. [16] T.A. Waniuk, R. Busch, A. Masuhr, W.L. Johnson, Acta Mater. 46 (1998) 5229. [17] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [18] S. Glasstone, K.J. Laidler, H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York, 1941. [19] H.Z. Yan, T. Klassen, C. Michaelsen, M. Oehring, R. Bormann, Phys. Rev. B 47 (1993) 8520.
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571 [20] O. Kubaschewski, C.B. Alcock, P.J. Spencer, Materials Thremochemistry, 6th ed., Pergamon Press, New York, 1993. [21] G.S. Grest, M.H. Cohen, Adv. Chem. Phys. 48 (1981) 455. [22] J. Jackle, Rep. Prog. Phys. 49 (1986) 171.
571
[23] M.L. William, R.F. Landel, J.D. Ferry, J. Am. Chem. Soc. 77 (1955) 3701. [24] C.A. Angell, Science 267 (1995) 1924. [25] R. Busch, A. Masuhr, E. Bakke, W.L. Johnson, Mater. Res. Soc. Symp. Proc. 455 (1997) 369.
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Thermodynamics and kinetics of the Zr41:2Ti13:8Cu10:0Ni12:5Be22:5 bulk metallic glass forming liquid: glass formation from a strong liquid A. Masuhr, R. Busch *, W.L. Johnson W.M. Keck Laboratory of Engineering Materials, California Institute of Technology, Pasadena, CA 91125, USA
Abstract The ¯ow properties of Zr41:2 Ti13:8 Cu10:0 Ni12:5 Be22:5 have been measured both in the liquid and supercooled liquid state, spanning 15 orders of magnitude in viscosity. We used beam bending experiments in the vicinity of the glass transition regime and concentric cylinder rheometry near the liquidus temperature to ®nd very large viscosities for a metallic system of 2 Pa s in the equilibrium liquid. The temperature dependence of the viscosity is discussed in the framework of the Adam±Gibbs entropy model as well as the free volume model for viscous ¯ow. The thermodynamic functions on the one side and the speci®c volume on the other have been measured previously and allow for a detailed comparison of the two phenomenological descriptions. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Monatomic or binary metallic liquids at the melting point have viscosities, g, on the order of 10ÿ3 Pa s [1,2]. To our knowledge, no direct viscosity measurements have been performed in the supercooled liquid state of metallic systems near the liquidus temperature. Recently, viscosity measurements on multicomponent bulk metallic glass (BMG) forming alloys near their calorimetric glass transition have found considerable attention [3±6]. The experiments are made possible by the increased stability of these alloys with respect to crystallization. The thermodynamic functions of
* Corresponding author. Present address: Department of Mechanical Engineering, Oregon State University, 204 Rogers Hall, Corvallis, OR 97331, USA. Tel.: +1-541 737 2648; fax: +1-541 737 2600; e-mail: [email protected]
the deeply supercooled liquid can be determined by dierential scanning calorimetry (DSC) on reheating amorphous samples. Relatively few data is available for transport coecients in the equilibrium or slightly supercooled liquid of BMG forming alloys. The anity of the transition metal alloys for impurities such as oxygen pose severe challenges for experimental studies at elevated temperatures. This temperature range is, however, of importance in the process of understanding atomic transport mechanisms and decisive for the glass forming ability of the alloys [7]. In the following we will present results from viscosity measurements on the Zr41:2 Ti13:8 Cu10:0 Ni12:5 Be22:5 (Vit1) BMG forming alloy [8]. With a critical cooling rate for crystallization of 1 Ksÿ1 it is one of the best metallic glass forming systems [9] and thermodynamic data [10] and speci®c volume measurements [11] are already available.
0022-3093/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 1 3 3 - 7
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
567
2. Experimental We have designed a Couette viscometer to measure the rheological properties of Zr-based glass forming alloys in the range from 10ÿ1 to 103 Pa s. The viscometer's concentric cylinder shear cell is machined from graphite and designed to hold about 5 cm3 of the liquid alloy. The cell is mounted vertically inside a vacuum induction furnace with the outer cylinder attached to a static torque sensor (Eaton 2127-10). Ingots of ca. 30 g of Vit1 are prepared by melting the elements in an arc-melting device under argon. The ingots are placed inside the shear cell and melted at 1200 K. Subsequently, the inner rotatable cylinder was inserted into the liquid. We measure the temperature, T , with a type K thermocouple mounted inside the crucible wall of the outer cylinder. The torque and temperature signals are digitally processed and are linked to the temperature controller of the induction furnace. The liquid alloy wets the graphite and allows a straightforward calculation of the ¯ow ®eld between the cylinders [12]. A more detailed description of the experiment can be found elsewhere [13]. Isothermal measurements in the equilibrium liquid are performed by both, clockwise and counterclockwise rotation of the inner cylinder to improve the resolution of the apparatus. The static torque on the outer cylinder is proportional to the Newtonian viscosity, g, where the proportionality constant can be calculated from the angular frequency of the inner cylinder and the geometry of the shear cell [14]. We ®nd the ¯ow to be Newtonian for strain rates from 0.4 to 4 sÿ1 used in this study. Additional viscosity measurements of the supercooled liquid were performed with a cooling rate of 0.7 K sÿ1 at constant shear rate. The graphite containers do not alter the times to nucleation as compared to electrostatic levitation experiments [15,16].
3. Results Fig. 1 summarizes results obtained from the high temperature viscometer as well as from beam
Fig. 1. Viscosity of Vit1 as measured by beam bending (h) and concentric cylinder (s). The data were ®tted to entropy model, Eq. (1), and the free volume model, Eqs. (7) and (8). The dotted, dashed and solid curves represent the least squares ®ts, respectively. Also shown are the viscosities (d) of Sn, Bi, Tl, Cd, Pb, Zn, Te, Sb, Mg, Ag, Ac, Au, Cu, Mn, Be, Ni, Co, Fe, Sc, Pd, V, Ti, Pt, Zr, Cr, Rh, B, Ru, Ir, Mo, Os, Re, and W near their melting point (in order of increasing melting point) [2]. The viscosity of Zr was calculated with Andrade's formula, which can be found in [2].
bending rheometry [16]. The latter were performed by heating amorphous samples above the calorimetric glass transition temperature. Extrapolating the measured viscosity of Vit1 to higher temperatures according to Eqs. (1) and (2), discussed below, we arrive at `normal' melt viscosity of 10ÿ3 to 10ÿ2 Pa s at about the average melting point of Vit1's constituents. This extrapolation merely re¯ects the vicinity of Vit1's composition to a deep eutectic. The high viscosity of Vit1 which is about 3 orders of magnitude larger than for monatomic metallic liquids at its liquidus temperature, Tliq 1026 K, is of great importance for the alloy's glass forming ability [7]. For the following discussion of the ¯ow of the supercooled melt, however, the liquidus temperature itself has no signi®cance.
568
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
4. Discussion Adam and Gibbs [17] developed a theory on cooperative relaxation in liquids and supercooled liquids. The increase in relaxation times and viscosity with increasing supercooling is directly linked to a decrease in con®gurational entropy, Sc , of the system. Within their model, the viscosity is expressed as C ; 1 g g0 exp TSc where the constant C represents an eective free enthalpy barrier for cooperative rearrangements. The pre-factor, g0 h=vm in Eq. (1), with h as Planck's constant, is calculated according to Eyring's theory to be 4 10ÿ5 Pa s [18]. The atomic volume, vm , of Vit1 was measured by Ohsaka [10] in an electrostatic levitation experiment and is vm 1:67 10ÿ29 m3 near the liquidus temperature. Its temperature dependence is neglected in comparison to the exponential term in Eq. (1). The temperature dependence of the con®gurational entropy, Sc , can be approximated from the dierence, DS, between the total entropies of the liquid and the crystal, assuming that their vibrational parts of the entropy are equal. DS results directly from an integration of the measured heat capacity dierence DCp Cp;liq ÿ Cp;xtl between the liquid and the crystal [6,9]. Prior to the measurement of Cp;xtl , an amorphous sample was crystallized by heating to 800 K at 20 K/min. We obtain Sc by Sc
Scliq
Tliq Z
ÿ T
DCp ~ dT ; T
If Dcp is proportional to 1=T an integration of Eqs. (1) and (2) yield a Vogel±Fulcher±Tammann (VFT) relation. For the present alloy, however, a 1=T dependence clearly underestimates the temperature dependence of Dcp as shown in Fig. 2. So, a VFT relation cannot be justi®ed on the basis of the Adam±Gibbs entropy model for the present system. Instead, we use the following relations to describe the heat capacities of the liquid and the crystal: Cp;xtl 3R aT bT 2 ;
3
Cp;liq Cp;xtl bT ÿ2 cT ÿ3 :
4
The parameters are (in SI units) R 8:314; a ÿ8:021 10ÿ3 ; b 2:076 10ÿ5 ; b 2:940 106 , and c 3:934 109 . Eqs. (3) and (4) describe the experimental data (Fig. 2) and yield a ®nite high temperature limit, g0 , for the viscosity if inserted in Eqs. (1) and (2) because limT !1 Dcp 0. Kubaschewski's formula [9,20] describes the heat capacity data equally well, but fails to comply with limT !1 Dcp 0 and therefore with limT !1 g T g0 .
2
which is equal to DS except for an additive constant. This equation takes into account, that in a multicomponent system the crystal can have a considerable amount of entropy of mixing. In fact, a disordered solid solution can have a larger entropy of mixing than a supercooled liquid of the same composition with tendency to topological and chemical short range order [6,19]. Inserting Eq. (2) in Eq. (1), we treat Scliq , the con®gurational entropy at the liquidus temperature, as a ®tting parameter as details about short range ordering in the present alloy are not yet known.
Fig. 2. Speci®c heat capacities of Vit1 in the liquid (s) and crystalline (n) states. Eqs. (3) and (4) are shown as solid curves and the resulting Dcp as a dotted curve. A Dcp 1=T ®t to the spec®c heat capacity of the liquid is shown (- - -).
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
A ®t of Eq. (1) to the viscosity data of Vit1, shown in Fig. 1, yields the two ®t parameters, C and 203 kJ g atomÿ1 or 2:1 eV atomÿ1 Scliq 15:8 J Kÿ1 g atomÿ1 . We ®nd qualitative agreement between the measured viscosity and the entropy model but, e.g., the tendency of the viscosities to be ®t by an Arrhenius function (log g 1=T ) at temperatures < 700 K is not entirely accounted for by Eq. (1). Using the ®t parameters, C and Scliq , and Eq. (1) to solve for the con®gurational entropy, we obtain Fig. 3. Although the temperature dependence of the viscosity in Fig. 1 is in qualitative agreement with Adam's and Gibb's theory, it is clear from Fig. 3 that the decrease in entropy with supercooling cannot quantitatively account for the exact form of g T . According to the free volume model [21], viscous ¯ow occurs via structural rearrangements without appreciable activation energy compared to kB T . The viscosity can be expressed as bvm 5 g g0 exp vf
with vf as the average free volume per atom and bvm a critical volume for ¯ow [21,22]. In the following discussion b and vm are treated as temperature independent. Again, the pre-factor is set constant to the above g0 . In Fig. 1 the logarithm of the viscosity data is ®tted to Eq. (5) with two different expressions for the temperature dependence of the free volume. First, a linear relation vf vm af T ÿ T0 between the free volume and the temperature is used, which yields the VFT equation b : 6 g g0 exp af T ÿ T0 Here, af is a measure of the increase in free volume with temperature and according to Williams et al. [23] can be approximated by the dierence in thermal expansion coecients between the liquid and the glass af aliq ÿ aglass :
7
Often, the dimensionless parameter, D b= af T0 , is introduced, to classify a liquid according to its fragility [24]. At the temperature, T0 , treated as a ®t parameter, the free volume vanishes and viscous ¯ow is no longer possible, according to this model. For the Vit1 data, shown in Fig. 1, we ®nd D 16:5 and T0 426:3 K comparable to the relatively strong liquid property of sodium silicate glasses [25]. These parameters vary slightly from previously published results [25] due to the recent higher temperature measurements of g. In an extended version of the free volume model by Cohen and Grest [21] the free volume decreases to zero, only for T 0 and the viscosity takes the form 2 3 6 g g0 exp 4
Fig. 3. Entropy dierence DS between the liquid and the crystalline states of Vit1 (á á á) and con®gurational entropy Sc (±±±) of the liquid as obtained from a ®t of Eq. (1) to the viscosity data in Fig. 1. The viscosity measurements are also converted via Eq. (1) into Sc (s).
569
2bvm f0 =kB 7 q 5; 2 4va f0 T ÿ T0 T ÿ T0 kB T
8
resulting in the three ®t parameters: bvm f0 =kB 4933 K, T0 672 K, and va f0 =kB 40:5 K for Vit1. Similar to Sc in Fig. 3 we plot in Fig. 4 the ratio vf = bvm as obtained from the viscosity and the Cohen±Grest ®t, Eq. (8). This representation con®rms the excellent agreement between the ®t and the present viscosity data in
570
A. Masuhr et al. / Journal of Non-Crystalline Solids 250±252 (1999) 566±571
as the free volume model. Comparing the results from the least squares ®ts to the viscosity with thermodynamic measurements and speci®c volume measurements, we conclude that the interpretation in terms of the free volume is more consistent.
Acknowledgements This work was supported by the U.S. Department of Energy, Grant No. DEFG-03-86ER45242 and by the ALCOA Technical Center.
References Fig. 4. Ratio vf = bvm as obtained from the VFT-relation, Eq. (5) (- - -), the Cohen±Grsest formula, Eq. (8) (±±±), and from viscosity measurements (s). The free volume vf =vm as obtained from the Cohen±Grest ®t and Eq. (7) is also shown (á á á).
Fig. 1. To estimate the absolute amount of free volume as a function of temperature, we can read from Fig. 4 for the temperature range from 800 to 1200 K af 1 dvf 1:83 10ÿ4 Kÿ1 : b bvm dT
9
With aliq 5:32 10ÿ5 Kÿ1 and aglass 3:39 10ÿ5 Kÿ1 10, Eqs. (7) and (9) lead to b 0:105 and an average free volume shown as the dotted line in Fig. 4. At the liquidus vf 9:58 10ÿ3 vm , i.e. about 1% of the sample's volume is `free'. This small amount of free volume is in accordance with the already anticipated model [8,11] of a dense metallic liquid with sluggish kinetics. 5. Conclusions We have presented results from viscosity measurements near the glass transition as well as in the equilibrium liquid of Vit1. The temperature dependence is in qualitative agreement with the entropy model by Adam and Gibbs as well
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