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Correlation between kinetic strength, volumetric properties, and glass forming ability in metallic liquids
LETTER TO THE EDITOR Journal of Non-Crystalline Solids 376 (2013) 205–208
Contents lists available at SciVerse ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Letter to the Editor
Correlation between kinetic strength, volumetric properties, and glass forming ability in metallic liquids J.C. Bendert, K.F. Kelton ⁎ Washington University, Department of Physics and the Institute of Materials Science and Engineering, One Brookings Drive, Campus Box 1105, Saint Louis, MO 63130-4899, USA
a r t i c l e
i n f o
Article history: Received 1 April 2013 Received in revised form 13 May 2013 Available online xxxx Keywords: Metallic glass; Liquids; Viscosity; Fragility; Glass forming ability
a b s t r a c t A systematic study of the viscosity in the equilibrium and supercooled states is reported for Cu100 − xZrx (30 ≤ x ≤ 55) liquids. While no correlation between the magnitude of the viscosity at T = Tg/0.6 with the glass forming ability is observed, peaks in the kinetic strength are observed at the best glass forming compositions. This not only confirms the rule that strong metallic liquids are better glass formers, but also sheds light on recent correlations reported between volumetric properties and glass forming ability. © 2013 Elsevier B.V. All rights reserved.
1. Introduction
can be parameterized by fits to the Vogel–Fulcher–Tammann (VFT) equation,
Since the discovery that metallic alloys could be cooled and held below their equilibrium melting temperature (supercooled) [1,2], volume and atomic packing have played central roles in the theories used to describe these systems. From Frank's hypothesis [3], to Turnbull, Cohen and Grest's free volume models [4,5] and more recently Miracle's packing model [6], dense packing has been identified as a key reason for greater thermodynamic stability and slower kinetics in liquids. Although atomic packing in amorphous phases has been correlated with critical casting thickness [7], an empirical connection to the dynamics of the liquid and volume has not been reported previously. Here, we present a systematic study of the viscosity of Cu100 − xZrx (30 ≤ x ≤ 55) liquids, establishing relationships between the volumetric properties of the liquid, the kinetic fragility, and the glass forming ability (GFA). The unifying concept of strength/fragility, first proposed by Angell [8], has played an important role in understanding the thermodynamic and dynamical properties of glass forming liquids for many diverse materials classes (organic and inorganic compounds, metallic alloys, colloids). Liquids are kinetically “strong” when the temperature dependence of the response functions (viscosity, diffusivity, relaxation time etc.) follows an Arrhenius behavior. They are “fragile” when these quantities show a non-Arrhenius (or super Arrhenius) behavior with temperature. The degree of non-Arrhenius behavior of the viscosity, η,
η ¼ η0 exp D T 0 =ðT−T 0 Þ ;
⁎ Corresponding author. Tel.: +1 314 935 6228. E-mail addresses: [email protected] (J.C. Bendert), [email protected] (K.F. Kelton). 0022-3093/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnoncrysol.2013.05.032
ð1Þ
where η0 is the viscosity in the infinite temperature limit, T0 is the temperature at which the viscosity becomes infinite, and D∗ is a measure of the fragility of the liquid, called here the kinetic strength, to differentiate it from the commonly-used fragility index, which is defined near the glass transition temperature. As D∗ increases the VFT equation becomes more Arrhenius, consistent with a stronger (less fragile) liquid [9]. The empirical rule, for metallic liquids, that strength correlates with glass formability is reasonable, since stronger glasses have higher viscosities at high temperatures, which inhibit the nucleation and growth kinetics [10]. There are, however, exceptions to this rule. For example, Zr57Cu15.4Ni12.6Al10Nb5 (Vitreloy 106) requires a higher cooling rate to form a metallic glass than Zr58.5Cu15.6Ni12.8Al10.3Nb2.8 (Vitreloy 106a) [11,12] despite being the kinetically stronger of the two liquids [13]. This occurs because, the local order in the liquid in the former case is more similar to that of the primary crystal phase [14], which decreases the nucleation barrier and promotes crystallization. Density provides a measure of the degree of local or medium range order in a liquid because of the positive correlation between energy and volume in metallic systems. Maxima in the density of the glass [7] and the thermal expansion coefficients of the liquid [15] at the best glass forming compositions in Cu–Zr alloys (i.e., Cu50Zr50, Cu54Zr44 and Cu64Zr36 [16–20,7,21]), corresponding to presumed minima in the internal energy of the glasses, were therefore interpreted as manifestations of strong liquid behavior. However, in the absence of experimental data for viscosity and diffusivity, this could not be verified. The viscosity data
LETTER TO THE EDITOR 206
J.C. Bendert, K.F. Kelton / Journal of Non-Crystalline Solids 376 (2013) 205–208
Fig. 1. Viscous damped surface oscillations recorded for a Cu56.5Zr43.5 liquid at 1200 K.
Fig. 3. The viscosity (red △) and D∗ (■) for Cu100 − xZrx liquids at Tg/T = 0.6.
reported here for 30 Cu100 − xZrx (30 ≤ x ≤ 55) equilibrium and supercooled liquid alloys provide a direct verification of that interpretation. Peaks in kinetic strength are observed for the best glass forming compositions, confirming the inverse correlation between fragility and GFA.
The surface deformation was captured by high speed (1560 frames per seconds) video of the sample's silhouette. Intensity integration of each video frame varied in proportion to the amplitude of the second harmonic. After the perturbation was removed, the amplitude of the surface harmonic oscillations damped exponentially, with a time constant inversely proportional to the viscosity [26]. A representative intensity signal, decaying due to viscous damping, is shown in Fig. 1. The viscosity was measured as a function of temperature and liquid composition during slow cooling, Fig. 2. Since the oscillating drop technique only allows the viscosity to be measured over a limited, low viscosity range, fits to the VFT equation (Eq. (1)) are over parameterized. This results in unphysical T0 values (i.e., T0 > Tg) and artificially reduces D∗. Because Cu–Zr liquids are fragile, since T0 → Tg with increasing fragility [27], and since the measurements were made at high temperatures, an approximate form of the VFT was used to mitigate the problems with unphysical values of T0,
2. Experimental Master ingots of approximately 1 g for each composition were prepared by arc-melting high-purity Cu (99.995%) and Zr (99.9+% with nominal 3% Hf) in stoichiometric quantities. The arc-melting was performed in a high-purity Ar gas atmosphere (99.998%) after at least three cycles of evacuation (to less than 13 Pa) and Ar gas backfilling of the chamber. Samples of mass 70–90 mg were prepared from portions of the master ingots by additional arc-melting. These samples were levitated and melted in an electrostatic levitation facility under high vacuum (∼ 10−5 Pa) [22]. By levitating the sample in vacuum, heterogeneous nucleation catalyzed by contact with air and/or the container is avoided, allowing for data be to acquired in the supercooled liquid [23,22]. The temperature was measured using a Metis MQ22 two-color ratio pyrometer (Process Sensors Corp.), operating at 1.40 and 1.64 μm wavelengths. The viscosity at each temperature was measured by the oscillating drop technique [24,25]. In brief, the oscillating drop technique was used to determine the viscosity by modulating the levitation electric field near the l = 2 resonant frequency (120–140 Hz) of the liquid to induce surface vibrations.
(a)
h i as T g −T 0 =T → 0: η ∼ η0 exp D T g = T−T g
ð2Þ
Fits to Eq. (2) gave the D∗ values discussed in this paper. The modified VFT fit now has the same number of free parameters as an Arrhenius fit, but with lower residuals. Due to this approximation, however, the values of D∗ are expected to be systematically lower than those obtained from fits over a wider viscosity range,
(b)
Fig. 2. (a) The viscosity (△) of Cu56.5Zr43.5 as a function of Tg/T, Tg = 696 K and a fit to Eq. (2) (line); (b) the cooling curve during which damping time measurements were made.
Fig. 4. The kinetic strength, parametrized by D∗, from the present measurements (■, lines inserted to guide the eye) and the critical casting thickness (red △, taken from Ref. [7]) for Cu–Zr liquids.
LETTER TO THE EDITOR J.C. Bendert, K.F. Kelton / Journal of Non-Crystalline Solids 376 (2013) 205–208
207
Table 1 Measured correlations with strong or fragile behavior in Cu–Zr liquids. Kinetic strength, D∗
GFA
Temperature range
Viscosity, η
Density, ρ
Thermal expansion, dln(ρ)/dT
High
Higha
Near Tg Above 1.6 Tg
No study Uncorrelateda
Highc Uncorrelatedb
No study Highb
a b c
This work. [15]. [7].
but trends across composition are maintained. For the fits to the Cu– Zr data, Tg(K) = 866.48 − 3.91x (linear fit to Ref. [28]), where x is the atomic percent of Zr [15]. An example of the quality of the fit is shown for the Cu56.5Zr43.5 data in Fig. 2. 3. Results The viscosities were measured within the temperature range 1100–1300 K (approx. 0.01–0.1 Pa s). Because the liquidus temperatures vary significantly across the alloy series, a meaningful comparison of the viscosities and the temperature dependence of the fit parameters as a function of composition can be made only at some reference temperature. Since within an Angell plot the relevant temperature is Tg, the viscosities are compared at a common temperature of Tg/T = 0.6 in Fig. 3. The viscosity shows a nearly monotonic increase with increasing Zr concentration, which does not correlate with the GFA. In contrast, D∗, derived from fitting Eq. (2) to the experimental data, show local maxima at 36, 44 and 50 at. % Zr, which are precisely the compositions of maximum critical thickness (best glass formation) [7], as shown in Fig. 4. The uncertainty in the absolute viscosity in Fig. 3 is dominated by uncertainty in the temperature identified at Tg/T = 0.6. The pyrometer calibration gives a tolerance in the absolute temperature of ± 5 K. Because of the exponential dependence on temperature, a 5 K shift in T can change the viscosity by 10%. This is reflected in the error bars of Fig. 3. The pyrometer calibration can also stretch or compress the temperature scale. This propagates to D∗ as an uncertainty of approximately ±1%. Combined with the confidence interval of the VFT fit, the values for D∗ reported in Fig. 3 have an uncertainty of approximately ±2%. The additional scatter observed in the data is likely an artifact from the approximation T0 = Tg in the VFT fit. Since Tg varies linearly with composition, the possibility of a variation in T0 from this trend merits some discussion. In the VFT equation (Eq. (1)) both T0 and D∗ parameterize kinetic strength; an increase in D∗ with increasing strength, causes a decrease in T0 [27]. By fitting to the approximate VFT (Eq. (2)), T0 is not permitted to decrease with increasing strength, so the fits underestimate the variation in D∗ with composition. If the VFT fits were constrained to true T0 values, it is expected that the correlation between kinetic strength and glass forming ability would be even more pronounced than is demonstrated in Fig. 4. 4. Discussion
that a stronger liquid will be more ordered than a fragile one. The additional order in the strong liquid is manifest as a larger density near the glass transition temperature and in the amorphous phase [7]. The better-established order of the stronger liquid would raise the nucleation barrier of crystal phases that may form if they have a structure different from the liquid and decrease the growth rate due to an increase in the viscosity, both favoring glass formation. This would explain the observed correlation of the density and thermal expansion coefficient with GFA in the Cu–Zr liquids. 5. Conclusions A systematic study of the viscosity of equilibrium and supercooled Cu100 − x–Zrx liquids (30 ≤ x ≤ 55) showed no correlation between the viscosity of the liquid at Tg/T = 0.6 and the glass forming ability. However, peaks in the kinetic strength were observed at the best glass forming compositions, confirming in these liquids the empirical rule that stronger metallic liquids are better glass formers [10]. The new experimental results validate earlier studies of volumetric properties [7,15] of these alloys and establish a clear connection between GFA, the volumetric properties of the liquid and glass, and the kinetic strength. Acknowledgments We thank A.K. Gangopadhyay and N.A. Mauro for useful discussions and C. Miller for help with sample preparation. This research was supported by the National Aeronautics and Space Administration (NNX07AK27G and NNX10AU19G) and the National Science Foundation (DMR-08-56199 and DMR-12-06707). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
The experimental dynamical and volumetric correlations with GFA for Cu–Zr alloys from this study and others are summarized in Table 1. The results presented here allow the volumetric properties to now also be correlated with strength. Although the kinetic strength of the liquid, the density of glass and the thermal expansion coefficient of the liquids were all independently correlated with GFA, it is more appropriate to consider these volumetric properties and GFA as manifestations of the kinetic strength. At high temperatures stronger liquids are expected to develop structural order at a higher rate than more fragile liquids. In systems where volume and energy have a positive correlation this is manifest as a higher thermal expansion coefficient [15]. Although this correlation is expected to reverse near the glass transition [29–31], it is expected
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
C. Mendenhall, L. Ingersoll, Philos. Mag. Ser. 6 (15) (1908) 205–214. D. Turnbull, R.E. Cech, J. Appl. Phys. 21 (1950) 804–810. F.C. Frank, P. Roy. Soc. Lond. A Mat. 215 (1952) 43–46. D. Turnbull, M.H. Cohen, J. Chem. Phys. 34 (1961) 120–125. M.H. Cohen, G.S. Grest, Phys. Rev. B 20 (1979) 1077–1098. D.B. Miracle, Nat. Mater. 3 (2004) 697–702. Y. Li, Q. Guo, J.A. Kalb, C.V. Thompson, Science 322 (2008) 1816–1819. C.A. Angell, J. Phys. Chem Solids 49 (1988) 863–871. C.A. Angell, Science 267 (1995) 1924–1935. R. Busch, J. Schroers, W.H. Wang, MRS Bull. 32 (2007) 620–623. C.C. Hays, J. Schroers, W.L. Johnson, T.J. Rathz, R.W. Hyers, J.R. Rogers, M.B. Robinson, Appl. Phys. Lett. 79 (2001) 1605–1607. S. Mukherjee, J. Schroers, Z. Zhou, W.L. Johnson, W.K. Rhim, Acta Mater. 52 (2004) 3689–3695. Z. Evenson, I. Gallino, R. Busch, J. Appl. Phys. 107 (2010) 123529. Z. Evenson, S. Raedersdorf, I. Gallino, R. Busch, Scr. Mater. 63 (2010) 573–576. J.C. Bendert, A.K. Gangopadhyay, N.A. Mauro, K.F. Kelton, Phys. Rev. Lett. 109 (2012) 185901. M.B. Tang, D.Q. Zhao, M.X. Pan, W.H. Wang, Chin. Phys. Lett. 21 (2004) 901. D. Wang, Y. Li, B.B. Sun, M.L. Sui, K. Lu, E. Ma, Appl. Phys. Lett. 84 (2004) 4029–4031. P. Yu, H.Y. Bai, M.B. Tang, W.L. Wang, J. Non-Cryst. Solids 351 (2005) 1328–1332. O. Kwon, Y. Kim, K. Kim, Y. Lee, E. Fleury, Met. Mater. Int. 12 (2006) 207–212. Q. Zhang, W. Zhang, A. Inoue, Scr. Mater. 55 (2006) 711–713. W.F. Wu, Y. Li, Appl. Phys. Lett. 95 (2009) 011906. N.A. Mauro, K.F. Kelton, Rev. Sci. Instrum. 82 (2011) 035114. A.K. Gangopadhyay, G.W. Lee, K.F. Kelton, J.R. Rogers, A.I. Goldman, D.S. Robinson, T.J. Rathz, R.W. Hyers, Rev. Sci. Instrum. 76 (2005) 073901. W.K. Rhim, K. Ohsaka, P.F. Paradis, R.E. Spjut, Rev. Sci. Instrum. 70 (1999) 2796–2801.
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[25] R.W. Hyers, R.C. Bradshaw, J.R. Rogers, T.J. Rathz, G.W. Lee, A.K. Gangopadhyay, K.F. Kelton, Int. J. Thermophys. 25 (2004) 1155–1162. [26] H. Lamb, Proc. Lond. Math. Soc. 1 (1881) 51–70. [27] R. Böhmer, C.A. Angell, Phys. Rev. B 45 (1992) 10091–10094. [28] N. Mattern, A. Schöps, U. Kühn, J. Acker, O. Khvostikova, J. Eckert, J. Non-Cryst. Solids 354 (2008) 1054–1060.
[29] F.H. Stillinger, P.G. Debenedetti, J. Phys. Chem. B 103 (1999) 4052–4059. [30] P. Bordat, F. Affouard, M. Descamps, K.L. Ngai, Phys. Rev. Lett. 93 (2004) 105502. [31] J. Guo, X. Bian, Y. Zhao, S. Zhang, T. Li, C. Wang, J. Phys. Condens. Matter 19 (2007) 116103.
Contents lists available at SciVerse ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/ locate/ jnoncrysol
Letter to the Editor
Correlation between kinetic strength, volumetric properties, and glass forming ability in metallic liquids J.C. Bendert, K.F. Kelton ⁎ Washington University, Department of Physics and the Institute of Materials Science and Engineering, One Brookings Drive, Campus Box 1105, Saint Louis, MO 63130-4899, USA
a r t i c l e
i n f o
Article history: Received 1 April 2013 Received in revised form 13 May 2013 Available online xxxx Keywords: Metallic glass; Liquids; Viscosity; Fragility; Glass forming ability
a b s t r a c t A systematic study of the viscosity in the equilibrium and supercooled states is reported for Cu100 − xZrx (30 ≤ x ≤ 55) liquids. While no correlation between the magnitude of the viscosity at T = Tg/0.6 with the glass forming ability is observed, peaks in the kinetic strength are observed at the best glass forming compositions. This not only confirms the rule that strong metallic liquids are better glass formers, but also sheds light on recent correlations reported between volumetric properties and glass forming ability. © 2013 Elsevier B.V. All rights reserved.
1. Introduction
can be parameterized by fits to the Vogel–Fulcher–Tammann (VFT) equation,
Since the discovery that metallic alloys could be cooled and held below their equilibrium melting temperature (supercooled) [1,2], volume and atomic packing have played central roles in the theories used to describe these systems. From Frank's hypothesis [3], to Turnbull, Cohen and Grest's free volume models [4,5] and more recently Miracle's packing model [6], dense packing has been identified as a key reason for greater thermodynamic stability and slower kinetics in liquids. Although atomic packing in amorphous phases has been correlated with critical casting thickness [7], an empirical connection to the dynamics of the liquid and volume has not been reported previously. Here, we present a systematic study of the viscosity of Cu100 − xZrx (30 ≤ x ≤ 55) liquids, establishing relationships between the volumetric properties of the liquid, the kinetic fragility, and the glass forming ability (GFA). The unifying concept of strength/fragility, first proposed by Angell [8], has played an important role in understanding the thermodynamic and dynamical properties of glass forming liquids for many diverse materials classes (organic and inorganic compounds, metallic alloys, colloids). Liquids are kinetically “strong” when the temperature dependence of the response functions (viscosity, diffusivity, relaxation time etc.) follows an Arrhenius behavior. They are “fragile” when these quantities show a non-Arrhenius (or super Arrhenius) behavior with temperature. The degree of non-Arrhenius behavior of the viscosity, η,
η ¼ η0 exp D T 0 =ðT−T 0 Þ ;
⁎ Corresponding author. Tel.: +1 314 935 6228. E-mail addresses: [email protected] (J.C. Bendert), [email protected] (K.F. Kelton). 0022-3093/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnoncrysol.2013.05.032
ð1Þ
where η0 is the viscosity in the infinite temperature limit, T0 is the temperature at which the viscosity becomes infinite, and D∗ is a measure of the fragility of the liquid, called here the kinetic strength, to differentiate it from the commonly-used fragility index, which is defined near the glass transition temperature. As D∗ increases the VFT equation becomes more Arrhenius, consistent with a stronger (less fragile) liquid [9]. The empirical rule, for metallic liquids, that strength correlates with glass formability is reasonable, since stronger glasses have higher viscosities at high temperatures, which inhibit the nucleation and growth kinetics [10]. There are, however, exceptions to this rule. For example, Zr57Cu15.4Ni12.6Al10Nb5 (Vitreloy 106) requires a higher cooling rate to form a metallic glass than Zr58.5Cu15.6Ni12.8Al10.3Nb2.8 (Vitreloy 106a) [11,12] despite being the kinetically stronger of the two liquids [13]. This occurs because, the local order in the liquid in the former case is more similar to that of the primary crystal phase [14], which decreases the nucleation barrier and promotes crystallization. Density provides a measure of the degree of local or medium range order in a liquid because of the positive correlation between energy and volume in metallic systems. Maxima in the density of the glass [7] and the thermal expansion coefficients of the liquid [15] at the best glass forming compositions in Cu–Zr alloys (i.e., Cu50Zr50, Cu54Zr44 and Cu64Zr36 [16–20,7,21]), corresponding to presumed minima in the internal energy of the glasses, were therefore interpreted as manifestations of strong liquid behavior. However, in the absence of experimental data for viscosity and diffusivity, this could not be verified. The viscosity data
LETTER TO THE EDITOR 206
J.C. Bendert, K.F. Kelton / Journal of Non-Crystalline Solids 376 (2013) 205–208
Fig. 1. Viscous damped surface oscillations recorded for a Cu56.5Zr43.5 liquid at 1200 K.
Fig. 3. The viscosity (red △) and D∗ (■) for Cu100 − xZrx liquids at Tg/T = 0.6.
reported here for 30 Cu100 − xZrx (30 ≤ x ≤ 55) equilibrium and supercooled liquid alloys provide a direct verification of that interpretation. Peaks in kinetic strength are observed for the best glass forming compositions, confirming the inverse correlation between fragility and GFA.
The surface deformation was captured by high speed (1560 frames per seconds) video of the sample's silhouette. Intensity integration of each video frame varied in proportion to the amplitude of the second harmonic. After the perturbation was removed, the amplitude of the surface harmonic oscillations damped exponentially, with a time constant inversely proportional to the viscosity [26]. A representative intensity signal, decaying due to viscous damping, is shown in Fig. 1. The viscosity was measured as a function of temperature and liquid composition during slow cooling, Fig. 2. Since the oscillating drop technique only allows the viscosity to be measured over a limited, low viscosity range, fits to the VFT equation (Eq. (1)) are over parameterized. This results in unphysical T0 values (i.e., T0 > Tg) and artificially reduces D∗. Because Cu–Zr liquids are fragile, since T0 → Tg with increasing fragility [27], and since the measurements were made at high temperatures, an approximate form of the VFT was used to mitigate the problems with unphysical values of T0,
2. Experimental Master ingots of approximately 1 g for each composition were prepared by arc-melting high-purity Cu (99.995%) and Zr (99.9+% with nominal 3% Hf) in stoichiometric quantities. The arc-melting was performed in a high-purity Ar gas atmosphere (99.998%) after at least three cycles of evacuation (to less than 13 Pa) and Ar gas backfilling of the chamber. Samples of mass 70–90 mg were prepared from portions of the master ingots by additional arc-melting. These samples were levitated and melted in an electrostatic levitation facility under high vacuum (∼ 10−5 Pa) [22]. By levitating the sample in vacuum, heterogeneous nucleation catalyzed by contact with air and/or the container is avoided, allowing for data be to acquired in the supercooled liquid [23,22]. The temperature was measured using a Metis MQ22 two-color ratio pyrometer (Process Sensors Corp.), operating at 1.40 and 1.64 μm wavelengths. The viscosity at each temperature was measured by the oscillating drop technique [24,25]. In brief, the oscillating drop technique was used to determine the viscosity by modulating the levitation electric field near the l = 2 resonant frequency (120–140 Hz) of the liquid to induce surface vibrations.
(a)
h i as T g −T 0 =T → 0: η ∼ η0 exp D T g = T−T g
ð2Þ
Fits to Eq. (2) gave the D∗ values discussed in this paper. The modified VFT fit now has the same number of free parameters as an Arrhenius fit, but with lower residuals. Due to this approximation, however, the values of D∗ are expected to be systematically lower than those obtained from fits over a wider viscosity range,
(b)
Fig. 2. (a) The viscosity (△) of Cu56.5Zr43.5 as a function of Tg/T, Tg = 696 K and a fit to Eq. (2) (line); (b) the cooling curve during which damping time measurements were made.
Fig. 4. The kinetic strength, parametrized by D∗, from the present measurements (■, lines inserted to guide the eye) and the critical casting thickness (red △, taken from Ref. [7]) for Cu–Zr liquids.
LETTER TO THE EDITOR J.C. Bendert, K.F. Kelton / Journal of Non-Crystalline Solids 376 (2013) 205–208
207
Table 1 Measured correlations with strong or fragile behavior in Cu–Zr liquids. Kinetic strength, D∗
GFA
Temperature range
Viscosity, η
Density, ρ
Thermal expansion, dln(ρ)/dT
High
Higha
Near Tg Above 1.6 Tg
No study Uncorrelateda
Highc Uncorrelatedb
No study Highb
a b c
This work. [15]. [7].
but trends across composition are maintained. For the fits to the Cu– Zr data, Tg(K) = 866.48 − 3.91x (linear fit to Ref. [28]), where x is the atomic percent of Zr [15]. An example of the quality of the fit is shown for the Cu56.5Zr43.5 data in Fig. 2. 3. Results The viscosities were measured within the temperature range 1100–1300 K (approx. 0.01–0.1 Pa s). Because the liquidus temperatures vary significantly across the alloy series, a meaningful comparison of the viscosities and the temperature dependence of the fit parameters as a function of composition can be made only at some reference temperature. Since within an Angell plot the relevant temperature is Tg, the viscosities are compared at a common temperature of Tg/T = 0.6 in Fig. 3. The viscosity shows a nearly monotonic increase with increasing Zr concentration, which does not correlate with the GFA. In contrast, D∗, derived from fitting Eq. (2) to the experimental data, show local maxima at 36, 44 and 50 at. % Zr, which are precisely the compositions of maximum critical thickness (best glass formation) [7], as shown in Fig. 4. The uncertainty in the absolute viscosity in Fig. 3 is dominated by uncertainty in the temperature identified at Tg/T = 0.6. The pyrometer calibration gives a tolerance in the absolute temperature of ± 5 K. Because of the exponential dependence on temperature, a 5 K shift in T can change the viscosity by 10%. This is reflected in the error bars of Fig. 3. The pyrometer calibration can also stretch or compress the temperature scale. This propagates to D∗ as an uncertainty of approximately ±1%. Combined with the confidence interval of the VFT fit, the values for D∗ reported in Fig. 3 have an uncertainty of approximately ±2%. The additional scatter observed in the data is likely an artifact from the approximation T0 = Tg in the VFT fit. Since Tg varies linearly with composition, the possibility of a variation in T0 from this trend merits some discussion. In the VFT equation (Eq. (1)) both T0 and D∗ parameterize kinetic strength; an increase in D∗ with increasing strength, causes a decrease in T0 [27]. By fitting to the approximate VFT (Eq. (2)), T0 is not permitted to decrease with increasing strength, so the fits underestimate the variation in D∗ with composition. If the VFT fits were constrained to true T0 values, it is expected that the correlation between kinetic strength and glass forming ability would be even more pronounced than is demonstrated in Fig. 4. 4. Discussion
that a stronger liquid will be more ordered than a fragile one. The additional order in the strong liquid is manifest as a larger density near the glass transition temperature and in the amorphous phase [7]. The better-established order of the stronger liquid would raise the nucleation barrier of crystal phases that may form if they have a structure different from the liquid and decrease the growth rate due to an increase in the viscosity, both favoring glass formation. This would explain the observed correlation of the density and thermal expansion coefficient with GFA in the Cu–Zr liquids. 5. Conclusions A systematic study of the viscosity of equilibrium and supercooled Cu100 − x–Zrx liquids (30 ≤ x ≤ 55) showed no correlation between the viscosity of the liquid at Tg/T = 0.6 and the glass forming ability. However, peaks in the kinetic strength were observed at the best glass forming compositions, confirming in these liquids the empirical rule that stronger metallic liquids are better glass formers [10]. The new experimental results validate earlier studies of volumetric properties [7,15] of these alloys and establish a clear connection between GFA, the volumetric properties of the liquid and glass, and the kinetic strength. Acknowledgments We thank A.K. Gangopadhyay and N.A. Mauro for useful discussions and C. Miller for help with sample preparation. This research was supported by the National Aeronautics and Space Administration (NNX07AK27G and NNX10AU19G) and the National Science Foundation (DMR-08-56199 and DMR-12-06707). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
The experimental dynamical and volumetric correlations with GFA for Cu–Zr alloys from this study and others are summarized in Table 1. The results presented here allow the volumetric properties to now also be correlated with strength. Although the kinetic strength of the liquid, the density of glass and the thermal expansion coefficient of the liquids were all independently correlated with GFA, it is more appropriate to consider these volumetric properties and GFA as manifestations of the kinetic strength. At high temperatures stronger liquids are expected to develop structural order at a higher rate than more fragile liquids. In systems where volume and energy have a positive correlation this is manifest as a higher thermal expansion coefficient [15]. Although this correlation is expected to reverse near the glass transition [29–31], it is expected
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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