Relaxation of the shear modulus of a metallic glass ...

JOURNAL OF APPLIED PHYSICS 109, 073518 (2011)

Relaxation of the shear modulus of a metallic glass near the glass transition Yu. P. Mitrofanov,1 V. A. Khonik,1,a) A. V. Granato,2 D. M. Joncich,2 and S. V. Khonik3 1

Department of General Physics, Voronezh State Pedagogical University, Lenin St. 86, Voronezh 394043, Russia 2 Department of Physics, University of Illinois at Urbana-Champaign, West Greet St. 1110, Urbana, Illinois 61801, USA 3 Technopark, Voronezh State University, Universitetskaya Sq. 1, 394006 Voronezh, Russia

(Received 10 November 2010; accepted 7 February 2011; published online 13 April 2011) Precise in situ measurements of the high-frequency (f 560 kHz) shear modulus G of Pd40 Cu30 Ni10 P20 bulk metallic glass at heating rates 0:38 T_ 7:5 K/min have been performed. It has been found that structural relaxation leads to an increase of G below the glass transition temperature Tg while decreasing it at T > Tg . A quantitative analysis of this phenomenon within the framework of the interstitialcy theory has shown that structural relaxation below Tg can be understood as a decrease of the concentration of interstitialcy-like defects frozen-in upon glass production. The relaxation turns into defect multiplication on continued heating above Tg . The beginning of defect multiplication represents the glass transition temperature. An excellent agreement between calculated and experimental Tg ’s as a function of the heating rate has been C 2011 American Institute of Physics. [doi:10.1063/1.3569749] found. V

I. INTRODUCTION

Understanding the glass transition and structural relaxation of glasses is one of the most challenging problems in the physics of condensed matter.1,2 In spite of decades-long extensive investigations and numerous articles in the field, the problem is far from an adequate interpretation. One of the basic ingredients of this problem consists in the identification of nanostructural atomic groups (“defects” or “relaxation centers”) responsible for viscous flow in the supercooled liquid state (that is, above the glass transition temperature Tg ) as well as structural relaxation and related viscoelasticity below Tg . In both cases, these defects rearrange themselves by surmounting a potential barrier, and it is important to understand the nature of this barrier. The literature on these issues was reviewed recently by Nemilov3,4 and Dyre.5 One of the most attractive approaches implies that the above-mentioned potential barrier is controlled by the elastic resistance of the surrounding material while this resistance is determined by the instantaneous macroscopic shear modulus G.3–5 This allows probing the internal potential energy landscape6,7 and relating G to the shear viscosity g of glass.8 On the other hand, the ratio G=B (B is the bulk modulus) is responsible for the ductile ! brittle transition occurring upon annealing of metallic glasses.9 Because B is insensitive to annealing,10 it is G that controls the embrittlement. The same G=B -ratio correlates with the packing density,11 which is an important structural parameter. All this underlines the role of the shear modulus as an important thermodynamic and kinetic parameter governing the properties and relaxation of the glassy state.

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected].

0021-8979/2011/109(7)/073518/4/$30.00

Meanwhile, the instantaneous shear modulus is the central physical quantity in the interstitialcy theory of condensed matter states (ITCM).12 The ITCM starts from arguing that melting of a crystal occurs through a rapid generation of interstitials in the dumbbell (split) configuration (two atoms sharing the same lattice place ¼ interstitialcies) at the melting point, and the resulting liquid state contains a few percent of these defects.13 The development of the ITCM is based on the two basic properties of interstitialcies: i) high susceptibility to the applied shear stress, which provides large anelastic strain (in addition to homogeneous elasticity), and ii) high vibrational entropy. The first property provides the melting and fluidity of the liquid state, whereas the second one explains the necessary latent heat upon melting. Within this approach, the glass is a frozen liquid with 2–3% of these defects, and structural relaxation of glass is viewed as a spontaneous decrease of their concentration c.14 The latter defines the instantaneous shear modulus via the main equation of the ITCM, G ¼ Gx expðbcÞ;

(1)

where Gx is the shear modulus of the reference crystal and b the shear susceptibility. Because b 25 30, Eq. (1) allows precise monitoring the defect concentration in glass by shear modulus measurements. This approach provides a comprehensive understanding of a number of basic thermodynamic and kinetic properties of (supercooled) liquid and glassy states within a common framework. In particular, the ITCM gives quantitative explanations of Richard rule12 (1897, the entropy of melting DS 1:2kB per particle that holds all over the Periodic Table with only a few exceptions), Lindemann melting rule13 (1910, a Tm ¼ const, where a is the thermal expansion coefficient and Tm the melting temperature), temperature

109, 073518-1

C 2011 American Institute of Physics V

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

073518-2

Mitrofanov et al.

dependence of the heat capacity of the liquid state,15 Vogel—Fulcher–Tamman viscosity law in the supercooled liquid state,16 linear increase of the glass viscosity and kinetics of the shear modulus relaxation upon structural relaxation,14,17 and some other important properties of the glassy state18 (for a review, see Refs. 17 and 19). Thus measurements of the high-frequency shear modulus are important for an adequate understanding of the physics of the glass transition, structural relaxation, and physical properties as well as the application issues of noncrystalline materials. Meanwhile, such measurements are being performed most often ex situ at room temperature after the heat treatment of glass is finished, decreasing thus the importance of the obtained results. Due to the experimental difficulties, precise in situ (that is, just in the course of annealing) high-frequency shear modulus measurements are only scarcely available (for example, Refs. 20–23). Recently, we employed a high-precision electromagnetic acoustic transformation (EMAT) technique for in situ shear modulus measurements in bulk glassy Pd40 Cu30 Ni10 P20 ,24–26 which is often used as a model glass. The purpose of the present work is twofold: i) to obtain high-precision data on the relaxation of the shear modulus near the glass transition by means of the EMAT technique and ii) analyze the obtained data within the framework of the ITCM. II. EXPERIMENTAL

Bulk glassy Pd40 Cu30 Ni10 P20 (at. %) was used for the investigation. The initial 2 5 60 mm3 bars were prepared by melt jet quenching.27 X-ray checked bars (Thermo Scientific ARL X’TRA diffractometer) were next cut into 2 5 5 mm3 samples for EMAT measurements. In this method, vibrations of the sample occur due to the Lorentz force induced by the interaction of the external magnetic field with the current excitation coil. The primary advantage of this technique consists in the absence of a direct acoustic contact between the sample and current/receiving coils. Transverse vibrations at a frequency f 560 kHz were continuously monitored at a relative precision of about 10 ppm. The absolute error of G calculations was estimated to be 5 103 . Further details are given elsewhere.26 III. RESULTS AND DISCUSSION

We measured the shear modulus of different samples in the as-cast state at different heating rates 0:38 T_ 7:5 K/ min. The results are given in Fig. 1 in terms of the normalized shear modulus change gexp ðTÞ ¼ GðTÞ=G0 1, where GðTÞ and G0 are the shear moduli at temperature T and 330 K, respectively. At any heating rate, structural relaxation starts at 360–370 K as manifested by an increase of gexp ðTÞ over the quasilinear anharmonic modulus decrease. As _ At temexpected, the degree of relaxation decreases with T. peratures 530–560 K depending on the heating rate, the slope jdgexp =dTj increases by several times that corresponds to the glass transition. The glass transition temperature Tg determined by the differential scanning calorimetry (DSC) (shown by the arrow) approximately coincides with the one obtained from the twist of gexp ðTÞ. Above Tg , the shear

J. Appl. Phys. 109, 073518 (2011)

FIG. 1. (Color online) Normalized shear modulus change at indicated heating rates. The vertical arrow gives Tg as determined by DSC at 5 K/min.

modulus is also notably dependent on T_ increasing with the latter (at a given T). This is unexpected and earlier unknown behavior, which disagrees with what is normally said in textbooks (temperature dependencies of physical properties do not depend on T_ at T > Tg ). It is customary to consider that because the Maxwellian relaxation time s m ¼ g=G is about a few tens of seconds at about Tg , the relaxation should occur very fast so that any T_ dependence should be absent. Figure 1 clearly shows that this is not the case. On the other _ data shown in Fig. 1 agree with recent hand, the gexp ðTÞ measurements of the isothermal relaxation of the shear modulus in the same glass near Tg , which showed that the underlying relaxation times are by 20 – 40 times bigger than sm .25 The measurements described in the preceding text do not take the glass from one quasiequilibrium state to another (as opposed to the shear modulus measurements presented in Ref. 20) but occur under intense structural relaxation. It is then reasonable to separate the relaxation contribution from the anharmonic elasticity. To do this, we measured the temperature dependence Gann ðTÞ of a fully annealed sample (obtained by heating up to 590 K at 1.25 K/min and cooling down at the same rate) and then calculated the relaxation contribution to the shear modulus as gðTÞ ¼ ½GðTÞ Gann ðTÞ=G0 . The temperature dependencies of gðTÞ at different T_ are given in Fig. 2. Structural relaxation-induced time-dependent features are now seen very clearly: i) at any temperature, there is a relatively strong shear modulus heating rate dependence; ii) the glass transition temperature Tg can be naturally identified as a temperature at which g reaches a maximum; iii) the relaxation at T < Tg increases the shear modulus but decreases it at T > Tg . In line with the preceding data, we connect these features with T - and T_ induced changes of the concentration of interstitialcy-like defects frozen-in during glass production. Within the ITCM approach, the relaxation kinetics can be understood as follows. Let the current concentration of interstitialcy-like defects be c ¼ c0 dc, where c0 is the initial frozen-in defect concentration and dc is its change due to


073518-3

Mitrofanov et al.

J. Appl. Phys. 109, 073518 (2011)

FIG. 2. (Color online) Experimental relaxation contribution to the shear modulus at different heating rates.

the relaxation. Because the changes of c are small, the basic Eq. (1) of the ITCM becomes G ¼ Gx exp½bðc0 dcÞ ¼ G0 expðbdcÞ G0 ð1 þ bdcÞ, where G0 ¼ Gx expðbc0 Þ is the initial shear modulus of glass. Then, the normalized shear modulus change is DG=G0 ¼ ðG G0 Þ=G0 ¼ g ¼ bdc. Using the aforementioned “elastic” hypothesis3–5 for the activation energy of defect rearrangement (E ¼ GVc , where Vc is some characteristic volume), the underlying relaxation time becomes s ¼ s0 eGVc =kB T ¼ s0 eðG0 Vc =kB TÞð1þbdcÞ ¼ s0 eðG0 Vc =kB TÞð1þgÞ ; (2) where s0 is of the order of the inverse Debye frequency. Equation (2) relates the relaxation time s with the shear modulus change g while the latter is controlled by the defect concentration c. We next assume that a first-order relaxation law is obeyed, c ceq dc ; ¼ s dt

(3)

monic component. By subtracting this component, we obtained the equilibrium shear modulus Geq ðTÞ, which was fitted linearly as shown in Fig. 3. The function Geq ðTÞ allowed then calculating the equilibrium concentration ceq ðTÞ of interstitialcy-like defects. The initial shear modulus of glass was taken from the experiment (33.6 GPa), the shear susceptibility b ¼ 30 (Ref. 24), s0 ¼ 1013 s, the characteristic volume Vc ¼ 8 1030 m3 was accepted from earlier direct comparison of the shear viscosity and shear modulus for the same glass at T > Tg .26 Figure 4 gives the relaxation contribution gcalc calculated using Eq. (4) with the preceding parameters. It is seen that, in general, the calculation captures quite well the aforementioned experimental features i–iii of the modulus relaxation (see Fig. 2) showing a clear T_ dependence both below and above Tg . The height of the g peak is reproduced accurate to 25–50 %. The experimental and calculated T_ dependencies of the glass transition temperatures (determined as exemplified by the arrow in Fig. 4) are given in Fig. 5. A linear increase of Tg with the logarithm of the heating rate is seen in line with calorimetric measurements on Zr- and Pd_ the calculated Tg ’s practically based glasses.28,29 At high T, _ a small coincide with the experimental ones while at low T, 5–7 K difference is observed. We have to underline that the preceding analysis does not contain any fitting parameters. The main disagreement with the experimental g (Fig. 2) lies at low temperatures, where the calculation significantly underestimates it. The reason for this seems to be clear. While at about or above Tg the approach of a single activation energy E is valid (a small change in E ¼ GVc due to the temperature dependence of G is actually not so important), the relaxation well below Tg requires a broad distribution of activation energies.17 This is confirmed by Arrhenius temperature dependencies of G and g at T > Tg and sharply nonArrhenius behavior below Tg .29 Thus the absence of relaxations with low E decreases g at low T as compared with the experiment. In the view of the preceding text, structural relaxation of glass below Tg can be understood as a spontaneous decrease of the amount of interstitialcy-like defects frozen-in upon

where ceq is the “equilibrium” defect concentration, which according to Eq. (1) has the form ceq ¼ b1 lnðGeq =Gx Þ with Geq being the shear modulus in the metastable quasiequilibrium state. Using Eq. (2), substituting g instead of _ the relaxation law (3) can be dc for heating at a rate T, rewritten as dg cg ; ¼ _ 0 exp½G0 Vc ð1 þ gÞ dT Ts kB T

(4)

where c ¼ bc0 ð1 ceq =c0 Þ. Equation (4) describes the shear _ modulus relaxation upon heating at a given rate T. To solve it, the Geq ðTÞ dependence must be known. The equilibrium shear modulus data for the glass under investigation near Tg were earlier determined by isothermal measurements.25 It was found that G at about and above calorimetric Tg decreases with time to an equilibrium value Geq . This “apparent” Geq ðTÞ dependence, however, contains an anhar-

FIG. 3. (Color online) Equilibrium shear modulus as determined in Ref. 25 from isothermal measurements. The solid line gives the linear approximation.


073518-4

Mitrofanov et al.

J. Appl. Phys. 109, 073518 (2011)

IV. CONCLUSIONS

FIG. 4. (Color online) Relaxation contribution to the shear modulus at different heating rates calculated using Eq. (4).

glass production at a concentration c0 ¼ lnðGx =G0 Þ= b 0:011 (this is comparable with c0 0:02 found for a Zrbased glass17). Due to structural relaxation, this concentration decreases by the amount dc ¼ g=b 0:03=30 ¼ 0:001 leading to an increase of the shear modulus and appearance of an exothermal effect on DSC thermograms. On continued heating, the current defect concentration becomes lower than the equilibrium one at the same temperature that results in the beginning of the relaxation of the reverse sign—an increase of the defect concentration leading to a sharp decrease of the shear modulus. The onset of this process manifests the glass transition. The thermal-induced defect multiplication requires energy, and this seems to be the origin of an endothermal effect connected with the glass transition on DSC thermograms. In our opinion, the results presented above constitute new evidence of the interstitialcylike defects involved in structural relaxation of glass, in addition to earlier analysis presented in Refs. 14, 17, 24, and 26.

FIG. 5. (Color online) Experimental and calculated glass transition temperatures determined as the peak temperatures of the shear modulus relaxation (Figs. 2 and 4) as a function of the heating rate.

We performed precise measurements of high-frequency shear modulus G in the course of heating of bulk glassy Pd40 Cu30 Ni10 P20 from room temperature through the glass transition region. Structural relaxation occurring upon heating increases G below the glass transition temperature Tg and decreases it at T > Tg , depending on the heating rate. The observed relaxation behavior was analyzed using the interstitialcy theory according to which structural relaxation represents either decrease or increase (depending on current conditions) of the concentration of defects similar to dumbbell (split) interstitials (¼ interstitialcies) in crystalline metals. The shear modulus exponentially depends on this concentration. Assuming a first-order relaxation law and using the basic Eq. (1) of the interstitialcy theory, we derived a differential equation, which captures all significant features of the observed shear modulus relaxation. We consider these results as further evidence of interstitialcy-like defects involved in structural relaxation of glass. 1

P. W. Anderson, Science 267, 1615 (1995). J. Langer, Phys. Today 60(2), 8 (2007). 3 S. V. Nemilov, J. Non-Cryst. Sol. 352, 2715 (2006). 4 S. V. Nemilov, J. Non-Cryst. Sol. 353, 4613 (2007). 5 J. C. Dyre, Rev. Mod. Phys. 78, 953 (2006). 6 A. Kahl, T. Koeppe, D. Bedorf, R. Richert, M. L. Lind, M. D. Demetriou, W. L. Johnson, W. Arnold, and K. Samwer, Appl. Phys. Lett. 95, 201903 (2009). 7 B. Zhang, H. Y. Bai, R. J. Wang, Y. Wu, and W. H. Wang, Phys. Rev. B 76, 012201 (2007). 8 W. L. Johnson, M. D. Demetriou, J. S. Harmon, M. L. Lind, and K. Samwer, MRS Bull. 32, 644 (2007). 9 J. J. Lewandowski, W. H. Wang, and A. L. Greer, Phil. Mag. Lett. 85, 77 (2005). 10 U. Harms, O. Jin, and R. B. Schwarz, J. Non-Cryst. Sol. 317, 200 (2003). 11 T. Rouxel, H. Ji, T. Hammouda, and A. Moreac, Phys. Rev. Lett. 100, 255501 (2008). 12 A. V. Granato, Phys. Rev. Lett. 68, 974 (1992). 13 A. V. Granato, D. M. Joncich, and V. A. Khonik, Appl. Phys. Lett. 97, 171911 (2010). 14 A. V. Granato and V. A. Khonik, Phys. Rev. Lett. 93, 155502 (2004). 15 A. V. Granato, J. Non-Cryst. Sol. 307–310, 376 (2002). 16 A. V. Granato, J. Non-Cryst. Sol. 357, 334 (2011). 17 S. V. Khonik, A. V. Granato, D. M. Joncich, A. Pompe, and V. A. Khonik, Phys. Rev. Lett. 100, 065501 (2008). 18 A. N. Vasiliev, T. N. Voloshok, A. V. Granato, D. M. Joncich, Yu P. Mitrofanov, and V. A. Khonik, Phys. Rev. B 80, 172102 (2009). 19 A. V. Granato, J. Non-Cryst. Sol. 352, 4821 (2006). 20 N. B. Olsen, J. C. Dyre, and T. Christensen, Phys. Rev. Lett. 81, 1031 (1998). 21 V. Keryvin, M.-L. Vaillanta, T. Rouxel, M. Huger, T. Gloriant, and Y. Kawamura, Intermetallics 10, 1289 (2002). 22 K. Tanaka, T. Ichitsubo, and E. Matsubara, Mater. Sci. Eng. A 442, 278 (2006). 23 S. Bossuyt, S. Gimnez, and J. Schroers, J. Mater. Res. 22, 533 (2007). 24 V. A. Khonik, Yu. P. Mitrofanov, S. A. Lyakhov, D. A. Khoviv, and R. A. Konchakov, J. Appl. Phys. 105, 123521 (2009). 25 V. A. Khonik, Yu. P. Mitrofanov, S. V. Khonik, and S. N. Saltykov, J. Non-Cryst. Sol. 356, 1191 (2010). 26 V. A. Khonik, Yu. P. Mitrofanov, S. A. Lyakhov, A. N. Vasiliev, S. V. Khonik, and D. A. Khoviv, Phys. Rev. B. 79, 132204 (2009). 27 S. V. Khonik, L. D. Kaverin, N. P. Kobelev, N. T. N. Nguyen, A. V. Lysenko, M. Yu Yazvitsky, and V. A. Khonik, J. Non-Cryst. Sol. 354, 3896 (2008). 28 R. Busch, Y. Kim, and W. L. Johnson, J. Appl. Phys. 77, 4039 (1995). 29 O. P. Bobrov, V. A. Khonik, S. A. Lyakhov, K. Csach, K. Kitagawa, and H. Neuha¨user, J. Appl. Phys. 100, 033518 (2006). 2
