Enthalpy recovery on thermal cycling within the non-equilibrium state of a glass

Journal of Non-Crystalline Solids 261 (2000) 52±66

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Enthalpy recovery on thermal cycling within the non-equilibrium state of a glass G.P. Johari *, J.G. Shim Department of Materials Science and Engineering, McMaster University, Hamilton, Ont., Canada L8S 4L7 Received 20 April 1999; received in revised form 20 September 1999

Abstract The so-called reversible structural relaxation of a Ni-based glassy metal-alloy, two organic polymers and As2 Se3 glass has been studied by di€erential scanning calorimetry (DSC) on their thermal cycling below Tg . The similarity of the relaxation in these glasses creates a diculty in reconciling the earlier conclusion that reversible relaxation in glassy metal-alloys involves changes in both chemical (i.e., a preference for unlike neighbors) and topological short range orders. Here, its origin is seen in the time- and temperature-dependent regain of the equilibrium state. Groups of atoms di€use fast enough to allow local loss of con®gurational enthalpy and entropy on annealing. On heating, they absorb energy and reach their new equilibrium con®gurations of higher enthalpy and entropy while still below Tg . Hence each mode of atomic di€usion in the structure has its own `mini glass-softening endotherm', and reversible relaxation is a re¯ection of a broad distribution of di€usion times arising from temporal and spatial variations in the atomÕs environment. The relaxation is modeled in terms of the stretched-exponential, non-linear character of glass relaxation. An experimentally testable consequence is that reversible relaxation will be minimum in a glass with a narrow distribution of di€usion times and low contribution from the sub-Tg relaxation processes. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction One of the features of a glassy material is that its enthalpy and entropy decrease spontaneously with time towards a temperature-dependent equilibrium value. Known as structural relaxation, this is investigated generally by annealing a sample isothermally and then heating it at a constant rate to a temperature, T, above the calorimetric glass-softening temperature, Tg , until its thermodynamic equilibrium state is reached * Corresponding author. Tel.: +1-905 525 9140; fax: +1-905 528 9295. E-mail address: [email protected] (G.P. Johari).

without crystallization [1±5]. In a di€erential scanning calorimetry (DSC) experiment, the heating scan of an annealed glass usually shows an endothermic peak or overshoot superposed on a sigmoid-shaped endothermic rise at T > Tg . The area under this peak is used to determine the enthalpy lost on annealing. Materials in which Tg is not discernible, observations of structural relaxation itself have been used to show that the material is vitreous. For example, the regain of the enthalpy and entropy lost on structural relaxation of hydrated proteins [6,7], DNA [8] and interpenetrating network polymers [9,10] appears as a small and broad endothermic peak above the annealing temperature.

0022-3093/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 6 1 3 - 4

G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

In addition to the above-mentioned endothermic peak observed for most materials, which Chen and co-workers [11±14] referred to as sub-Tg structural relaxation, a second, broad endothermic feature in the DSC scan was observed at T < Tg when a glass preannealed at T < …Tg ÿ 100 K† was heated in their studies. This was called the sub-subTg structural relaxation [11±14]. Chen and coworkers showed that both relaxations occur in poly(styrene) [11], B2 O3 glass [12] and metallic glasses [13,14]. Concurrently, from model calculations, Hodge and Berens [15] showed that the subTg and sub-sub-Tg relaxations are compatible with the Moynihan±Tool±Narayanaswamy treatment [16±18] of the spontaneous structural relaxation of a glass, and that both arise from the stretched-exponential and non-linear relaxation of a material in the glassy state. One may note that ChenÕs terminology has been occasionally confusing, since the peak observed after annealing at T < …Tg ÿ 100 K† is routinely referred to as a sub-Tg peak, whereas the peak resulting from annealing in the sub-Tg region is called the endothermic Tg -peak. There is however a third situation which occurs under special conditions of annealing and thermal cycling a glassy metal-alloy over a temperature range below its Tg or the crystallization temperature when Tg is not observed. To elaborate, when a metal-alloy glass sample was annealed at Ta < Tg , and then heated at a certain rate to a temperature Th , which was also below Tg , a broad endotherm was observed. As the sample was not heated to a T even close to Tg , it remained solid and, in contrast with a case when the sample was heated to a T > Tg , not all modes of atomic di€usion reached their equilibrium state. In the terminology used for glassy metal-alloys, the feature observed on thermal cycling between two temperatures both below Tg , has been called `reversible relaxation'. Similar reversible relaxation has been observed in the electrical resistivity of a variety of Cu±Ti and Ni±Ti [19], and Ti±Be±Zr [20] glassy metal-alloys. However, resistivity measurements of some glassy metals have shown no analogous relaxation. For example, Fe±B glasses showed a reversible enthalpy relaxation [21,22], but not the equivalent relaxation in the electrical resistance [23]. Further investigations of the so-called reversible structural relax-

53

ation in several Fe- and Ni-based metal-alloy glasses have shown no evidence that chemical short-range order (CSRO, i.e., a preference for unlike atoms [1,4]) plays a role in that relaxation. Based on the earlier interpretation of reversible relaxation in glassy metal-alloys [24±26], Bruning et al. [27] modeled the reversible structural relaxation in terms of thermal repopulation of con®gurational states occupying sites in a two-level system. A further reversible enthalpy relaxation study of the Zr± Ni±Cu glasses [28] has indicated that two reversible relaxations occur: (i) the low-annealing temperature relaxation when only CSRO changed, and (ii) the high-annealing temperature relaxation when topological short-range order (TSRO, i.e., local packing determined by the spatial constraints [1,4], and not by preference for unlike atoms) changed by co-operative atomic di€usion. To investigate whether the so-called reversible enthalpy relaxation is unique to a glassy metalalloyÕs structure, or whether it is part of a general phenomenon of glass relaxation, we have studied the enthalpy and entropy relaxation of a typical glassy Ni-based alloy, several organic polymers and an inorganic network glass, As2 Se3 . The results show that the `reversible relaxation' occurs also in these glasses, with features remarkably similar to those for the glassy metal-alloy, and that the features can be qualitatively simulated by using one of the several general models of non-exponential, non-linear kinetics of glass relaxation based on the Moynihan±Tool±Narayanaswamy postulates [16±18]. This shows that the `reversible' relaxation of a glassy metal-alloy is one aspect of the distribution of di€usion times in local regions, arising from spatial and temporal distinctions of the atomic environment in the structure of a glass. 2. Experimental methods 25 lm thick and 25 mm wide ribbons of MBF50 Ni-based glassy metal-alloy of atomic composition, Cr18:36 Si13:06 C0:34 B23:56 Ni44:68 was donated by AlliedSignals, New Jersey, USA. A DSC (PerkinElmer model DSC-4) was used for the measurements, with a data acquisition software written for the purpose [10]. The samples were accurately

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G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

weighed in 10±20 mg amounts, contained in sealed aluminum pans, and argon was used as purge gas. The DSC output divided by the glass sampleÕs mass, was checked repeatedly. It was found to remain the same for samples of di€erent mass. Thus the values were not measurably a€ected by the sampleÕs mass. A base line signal recorded with the empty pan was subtracted from the measured signal during the course of a DSC scan of a sample to remove the e€ect of the mismatch in the instrument's sample holders and the pans. The thermal drift of the instrument was minimized by stabilizing the calorimeter for at least 2 h in the temperature range of interest. The DSC scans were corrected for the thermal lag of the instrument which was 1.5 K for a heating rate of 40 K/min. The relatively high heating rate of 40 K/min was chosen so that the features observed could be discerned easily and the areas under the curves measured with sucient accuracy. The as-received sample of the glassy metal-alloy, MBF-50, was ®rst heated in the DSC instrument from 300 to 800 K, which led to its partial crystallization. The scan obtained is shown in Fig. 1(a), where the ordinate scale is given by a vertical bar in Watt per gm (W gÿ1 ) of the sample. As the scan showed no feature below 500 K (as seen in Fig. 1(b)), and as we needed to show clearly the width of the sharp exotherm at 720 K, the lower limit of T in Fig. 1(a) was 500 K. The DSC scan shows a broad, barely perceptible exotherm due to structural relaxation on ®rst heating of the as-received sample over the 550±700 K range, and thereafter a single sharp and narrow exotherm at high temperatures when the sample crystallized, but no appreciable endothermic feature attributable to Tg . (The procedure for determining Tg in glassy metal alloys given by Ram and Johari [59] was not used to determine the Tg of MBF-50.) The enlarged curves in Fig. 1(b) show the broad exotherm observed on ®rst heating of the as-received sample. From the shape of the scans in Fig. 1, an upper temperature limit, Th …ˆ 700 K†, which was below both the crystallizationÕs onset temperature and the Tg of MBF-50, was chosen for heating the sample in subsequent experiments. A new sample of MBF50 was then heated from 300 K to Th at 40 K/min, during which an exotherm identical to that in Fig.

Fig. 1. The DSC scans of the MBF-50 glassy metal-alloy, obtained during heating the as-received samples at 40 K/min. The vertical bar in this and all subsequent ®gures provides the scale.

1(b) was observed. It was kept for 5 min at Th , thereafter cooled at the nominal rate of 320 K/min (according to the instrument setting) to a preselected temperature, Ta , where it was annealed for a period ta . It was then cooled at 320 K/min similarly from Ta to 300 K, and thereafter (the annealed sample) heated at 40 K/min to Th during which its DSC scan was obtained. The MBF-50 sample was then cooled at a nominal 320 K/min rate from Th to 300 K, equilibrated for 5 min, and then reheated (without annealing) to Th at 40 K/ min. The DSC scan obtained during reheating was taken as the curve for the unannealed sample. The annealing conditions were varied by varying both ta and Ta . The DSC scans obtained on heating to Th were for both the annealed and unannealed samples. For comparison against the behaviour of the glassy metal-alloy, two non-crystallizable polymers, namely, a linear chain polymer, polystyrene± polymethylmethacrylate blend (PS±PMMA, calo-

G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

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rimetric Tg ˆ 383 K), and 60% polyurethane± polystyrene interpenetrating network polymer (IPN, its Tg is not known because its glass-softening endotherm is exceptionally broad) and an inorganic network glass, As2 Se3 (calorimetric Tg ˆ 460 K) were also studied in a similar manner. 3. Results The DSC scans of the annealed and unannealed samples of MBF-50 are shown in Fig. 2(a). For this study, Th was chosen as 700 K, ta was 1 h and Ta were: 520, 530, 540, 550, 560 and 570 K, as indicated by arrows in Fig. 2. The di€erence curves were obtained by subtracting the scan for the unannealed sample from the scan for the annealed sample using the data in Fig. 2(a), and these curves are shown in Fig. 2(b). MBF-50 was further studied after annealing for di€erent periods at a ®xed Ta of 520 K, and the DSC scans obtained are shown in Fig. 3(a), where ta is indicated, and the di€erence curves are shown in Fig. 3(b). Typical DSC scans for the IPN, the interpenetrating network polymer, annealed for di€erent periods at 353 K and heated to Th of 423 K and for the unannealed sample are shown in Fig. 4(a), and the di€erence curves in Fig. 4(b). Fig. 5(a) shows the DSC scans of the PS±PMMA annealed at 333, 353 and 363 K for 15 min then heated to 420 K …T > Tg † at 30 K/min. The di€erence curves are shown in Fig. 5(b). Fig. 6(a) shows the DSC scans of the annealed and unannealed samples of As2 Se3 during heating to 500 K …T > Tg † at 30 K/min. The di€erence curves are shown in Fig. 6(b). The di€erence between the area enclosed by the DSC scans for the annealed and the unannealed samples divided by the heating rate q is equal to the enthalpy recovered on heating, which should be the same as the enthalpy lost on structural relaxation isothermally, plus that lost on heating. The corresponding di€erence between the areas enclosed by the plots constructed against the logarithm of temperature yields the entropy lost on structural relaxation isothermally, plus that lost on heating. These were determined by measuring the area between the scans for the annealed and unannealed sample, by using an algorithm, as before

Fig. 2. (a) The DSC scans of the unannealed and annealed samples of MBF-50 obtained during heating at 40 K/min. The samples were annealed for 1 h at each temperature, 520, 530, 540, 550, 560 and 570 K, as indicated by the arrows. (b) The di€erence curves of the annealed and unannealed samples.

[8,29]. Estimated errors in these values are less than 4%. The noise in the DSC signal for the MBF-50 is higher (Figs. 2 and 3) than for the other glasses. As the 25 lm thick ®lm of MBF-50 sample was hand-folded and pressed, unlike the lumps of other glasses, its amount in the DSC pan was small. Moreover, its heat capacity is high. This decreased the signal to noise ratio.

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Fig. 3. (a) The DSC scans of the annealed and unannealed samples of MBF-50 obtained during heating at 40 K/min. The samples were annealed at 520 K for di€erent periods, as indicated. (b) The di€erence curves of the annealed and unannealed samples.

4. Discussion 4.1. Thermodynamic functions and structural relaxation The terms, reversible relaxation, rejuvenation, etc., have been used also to refer to the recovery of a physical property and, by implication, the recovery of the original structure of a glass after it had been subjected to a variety of thermal histories. In some of these studies, the glass had

Fig. 4. (a) The DSC scans of the annealed and unannealed samples of IPN obtained during heating at 30 K/min. The sample was annealed at 353 K for di€erent periods, 15 min for set 1 and 30 min, 1 h, 6 h, 16 h, 74 h, 120 h and 720 h for sets 2±8. (b) The di€erence curves of the annealed and unannealed samples.

been heated to a T above Tg so that its state could be brought to an internal equilibrium state of a liquid of viscosity < 1010 P before thermally cy-

G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

Fig. 5. (a) The DSC scans of the annealed and unannealed sample of PS-PMMA obtained during heating to 420 K …> Tg † at 30 K/min. The annealing time was 15 min and the annealing temperature was 333, 353, and 363 K for curves 1±3, respectively. (b) The di€erence curves of the annealed and unannealed samples.

cling the sample [13,14,30±32]. But here the concern is with the loss and regain of H and S as a result of structural relaxation when a glass is thermally cycled in a temperature range substantially below its calorimetric Tg , and it is not heated to an internal equilibrium state above Tg . The structural recovery in our studies occurs when the sample is heated from one kinetically unstable, non-equilibrium state at Ta , to another kinetically unstable non-equilibrium state at Th . The decrease in the enthalpy, H, and entropy, S, during structural relaxation is a combined e€ect of two occurrences: (i) loss of the con®gurational enthalpy and entropy as a result of localized atomic di€usion in the disordered structure, and (ii) changes in the phonon frequencies and the corresponding density of states as volume decreases. The results in Figs. 2, 5 and 6 thus show generally how the regain of con®gurational enthalpy and entropy occurs by a variety of local

57

Fig. 6. (a) The DSC scans of the annealed and unannealed sample of As2 Se3 obtained during heating to 500 K …> Tg † at 30 K/min. The annealing time was 1 h and the annealing temperature was 333, 353 and 373 K for curves 1±3, respectively. (b) The di€erence curves of the annealed and unannealed sample.

atomic di€usion processes and by changes in phonon behavior. They further show how these occurrences produce an endothermic feature when a sample preannealed for a ®xed ta at di€erent Ta s was heated at a constant rate. Figs. 3 and 4 show how the same processes occur when a sample was preannealed for di€erent ta s, but at a ®xed Ta . The phenomenon of spontaneous decrease in H and S of a glass on annealing has been described in several review articles. (Citations to these may be found in Ref. [33].) Brie¯y, nine di€erent models for structural relaxation have been used for ®tting the DSC scans, as mentioned earlier [33], and although these models di€er in details, all admit to a phenomenology of structural relaxation based upon the original observations of Winter±Klein [34,35] and Tool [17] and, in most cases, the formalisms by Gardon and Narayanaswamy [36], and Narayanaswamy [18]. Here, we use an alternative approach in which the observed changes in the thermodynamic state

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functions are analysed without the use of a model. For that purpose, an irreversible decrease in enthalpy on annealing, dDHa , may be written as the sum of several contributions     oDHa oDHa dT ‡ dta dDHa ˆ oTa ta ;q;qc ota Ta ;q;qc     oDHa oDHa ‡ dqc ‡ dq; oqc Ta ;ta ;qc oq Ta ;ta ;qc …1† where dDHa is the di€erence between the enthalpy of the material before and after the annealing experiment and all other notations are as de®ned before. The pre®x D is used to maintain that the quantities determined by experiments are [H ÿ H (0 K)] and [S ÿ S(0 K)], where H[0 K] and S[0 K] are the enthalpy and entropy of a material at 0 K. It should be noted also that H(0 K) decreases on annealing as the structure densi®es, as does S(0 K) because the randomness of the zeropoint energy changes, as pointed out earlier [37]. The ®rst term on the RHS of Eq. (1) represents the enthalpy loss with respect to temperature for constant annealing time, ta , heating rate, q, and cooling rate, qc , and the second term represents this loss with respect to ta , for constant q, qc and Ta . The third term represents the enthalpy loss over a certain temperature range with respect to the cooling rate, for constant ta , Ta and q and the fourth term represents the enthalpy loss with respect to the heating rate for constant ta , Ta and qc . The plots in Figs. 2, 5 and 6 show the magnitude of the ®rst term, and those in Figs. 3 and 4 that of the second term. Because qc and q for a set of our experiments are kept constant, the magnitudes of the third and the fourth terms are zero. The corresponding decrease in the entropy is     oDSa oDSa dT ‡ dta dDSa ˆ oTa ta ;q;qc ota Ta ;q;qc     oDSa oDSa ‡ dqc ‡ dq; oqc Ta ;ta ;q oq Ta ;ta ;qc …2† where the terms have the same meaning as in Eq. (1). The two equations describe how H and S of a chemically stable disordered solid in storage, or in

use as a device, will change if its temperature during the storage period ¯uctuated at a certain rate. We discuss the DSC results as changes in the various terms in Eqs. (1) and (2) by using the data given in Figs. 2±6. The coecients of Eqs. (1) and (2) were calculated from the area con®ned by the DSC scans for the annealed and the unannealed samples, or from the area under the endothermic peak in the di€erence scans of the types shown in Figs. 3(b) and 4(b). This yields the H regained on heating, whose magnitude is equal to the H lost on isothermal annealing. (Note that if a dip or a small exothermic feature, which indicates a further enthalpy loss as a result of structural relaxation during the heating of the sample, appears in the DSC scan prior to the appearance of the broad peak, the area of the dip is included in the calculations.) The area of the broad peak plus the area of the exothermic dip is equal to the true decrease in DH and DS on annealing before scanning was begun. Hodge [38,39] has reviewed this subject recently. The quantity dDHa was determined from the di€erence in the DSC signal of the annealed and unannealed sample plotted on a linear T-scale, and dDSa was determined from the area of the di€erence curves plotted on a ln T -scale. The decrease in DH and DS on annealing MBF-50 for ta of 1 h at di€erent Ta s is plotted against Ta in Fig. 7. The set of curves in Fig. 3(a) were used to construct the plots in Fig. 8, which show the DH and DS regain on heating the sample preannealed at 520 K for di€erent periods. Since only the di€erence in the magnitudes of DH and DS is known, the curve labeled 1 for the unannealed and annealed samples of MBF-50 was calculated from the integral using an arbitrary lower limit, according to the equations   Z Th  1 oH dT dDH ˆ q T
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59

Fig. 7. The enthalpy and entropy loss of MBF-50 after annealing for 1 h is plotted against the annealing temperature. The cumulative error is within 6%. It varies with the magnitude of enthalpy and entropy.

 oH d ln T ot unann T
1 dDS ˆ q

Z

Th



Fig. 8. The enthalpy and entropy loss of MBF-50 after annealing for di€erent periods at 520 K. Curve 1 is for the unannealed sample and subsequent low lying curves up to 5 are for the annealing time, 5, 18, 167 and 500 h, respectively.

…4†

where q is the heating rate, and Th , as de®ned before, is the highest temperature to which the glass was heated. The subscripts, `ann' and `unann' refer to the states of the sample, annealed and unannealed, whose …oH =ot† had been measured. The ®rst term alone yields curve 1 in Fig. 8, on a relative scale, not absolute. The second term yields the net decrease in H and S on annealing at Ta for a certain ta (which is equal to the net regain on heating to Th ), and the third term yields the regain of H and S on heating to a temperature T < Th . The magnitudes of dDH and dDS decrease towards zero on heating and the curves for the annealed and the unannealed samples meet at high temperatures.

In Fig. 8, curve 1 is for the sample cooled from 700 to 300 K at 320 K/min, and heated to 700 K at 40 K/min. The subsequent unlabelled curves correspond to the sample cooled to 450 K at 320 K/ min, heated to 520 K thereafter, annealed for 5, 18, and 167 h, respectively, at 520 K, cooled to 300 K and ®nally heated to 700 K. Curve 5 was obtained after a similar procedure, but for a sample annealed for 500 h. Fig. 9 shows the enthalpy and entropy loss of IPN after annealing for di€erent periods at 353 K. Curve 1 is for the unannealed sample and the successively lower curves, 2±6, are for samples annealed for 6, 16, 74, 120, and 720 h, respectively, which correspond to the data in Fig. 4. These curves show the net e€ect of (a) structural relaxation during cooling, (b) during isothermal annealing at Ta and (c) during heating to Th . The

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where DHa ˆ ‰DH …ta ˆ 0† ÿ DH …ta ! 1†Š; DSa ˆ ‰DS…ta ˆ 0† ÿ DS…ta ! 1†Š at the annealing temperature Ta , b is the stretched-exponential relaxation parameter whose value is between zero and 1, and s is the characteristic structural relaxation time. The latter is written as s ˆ s0 exp …E =RTa †, where s0 is the pre-exponential factor, E the Arrhenius energy and R the gas constant. The decrease in DH and DS for MBF-50 on annealing at 520 K for di€erent periods, as determined from the areas of the broad peaks in Fig. 3, is plotted against the annealing time in Fig. 10. The data can be described by Eqs. (5) and (6), as shown by the lines in Fig. 10 calculated for DHa ˆ 10:12  6 J gÿ1 ; DSa ˆ 0:015  0:008 J gÿ1 Kÿ1 ; s ˆ 312  245 h and b ˆ 0:37. The plot indicates that both s and b remain constant with increasing ta . The corresponding plots for the IPN polymer annealed for di€erent periods at 353 K are shown in Fig. 11. Here the shape of the curves Fig. 9. The enthalpy and entropy loss of IPN after annealing for di€erent periods at 353 K. Curve 1 is for the unannealed sample and subsequent lower lying curves up to 6 are for the annealing time, 6, 16, 74, 120 and 720 h, respectively.

di€erent curves meet the curves for the unannealed sample at a T that increases with increase in ta . At these temperatures in Figs. 8 and 9, the MBF-50 and the IPN samples regained the H and S lost on their annealing at Ta , and attained the same thermodynamic state as the unannealed sample on heating at the speci®ed rates. 4.2. The e€ects of annealing time and temperature The loss of H and S on structural relaxation may be written as [8] " (  b )# ta oDHa …ta ; Ta † ˆ DHa …Ta † 1 ÿ exp ÿ ; s…Ta † …5† " oDSa …ta ; Ta † ˆ DSa …Ta † 1 ÿ exp

(

 ÿ

ta s…Ta †

b )# ; …6†

Fig. 10. The enthalpy and entropy lost during annealing of MBF-50 is plotted against the annealing time at 520 K. The cumulative error is within 6%. It varies with the magnitude of enthalpy and entropy.

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61

…ta ! 1†Š, decrease and become zero as Tg is approached on heating. Thus the shape of the oDHa and oDSa plots against Ta is determined by two processes: (i) a kinetically limiting process that increases the spontaneous oDHa and oDSa as Ta is increased and (ii) a thermodynamically limiting process that decreases the terms, ‰DH …ta ˆ 0† ÿDH …ta ! 1†Š and ‰DS…ta ˆ 0† ÿ DS …ta ! 1†Š, as Ta is increased. The peaks in oDHa and oDSa appear at a Ta when e€ect (ii) begins to control the thermodynamic consequences of annealing. Thus for a ®xed ta and varying Ta , oDHa …ta ; Ta † and oDHa …ta ; Ta † reach their peak values according to ÿ

o ln DHa …Ta † btb E ˆ b a 2 oTa s RTpeak

…7†

and ÿ Fig. 11. The enthalpy and entropy lost during annealing of IPN is plotted against the annealing time at 353 K. The cumulative error is within 6%. It varies with the magnitude of enthalpy and entropy.

is similar to that of MBF-50, and the data can be described also by Eqs. (5) and (6), as shown by the lines in Fig. 11 calculated for DHa ˆ 2:45  0:4 J gÿ1 ; DSa ˆ 0:006  0:001 J gÿ1 Kÿ1 ; s ˆ 66:8  20 h and b ˆ 0:26. The magnitude of the ®rst term on the RHS of Eqs. (1) and (2), or the e€ect of Ta is indicated by the manner in which the area under the peaks seen in Figs. 2, 5 and 6 changes when Ta is changed and ta is kept constant. The e€ects of Ta on the enthalpy and entropy loss of MBF-50 are shown in Fig. 7. The feature of the plots of oDHa and oDSa against Ta are determined by several temperatureand time-dependent e€ects. When Ta is low, s is exceedingly long, which makes oDHa and oDSa in a given annealing time too small to be measurable, and as Ta is increased s decreases rapidly such that at each Ta there is a progressively larger increase in oDHa and oDSa until the point of in¯exion in the sigmoid-shaped part on the LHS of the peak is reached. At still higher Ta , the magnitudes of ‰DH …ta ˆ 0† ÿ DH …ta ! 1†Š and ‰DS…ta ˆ 0† ÿ DS

o ln DSa …Ta † btb E ˆ b a 2 : s RTpeak oTa

…8†

By combining Eqs. (5) and (7) and (6) and (8) s btab E ; …9† Tpeak ˆ ÿsb R‰o ln DHa …Ta †=oTa Š

Tpeak

s btab E : ˆ ÿsb R‰o ln DSa …Ta †=oTa Š

…10†

This gives the conditions for the appearance of the peak for both the H and S loss on annealing for a ®xed period, ta . The data here show that the second condition is not reached in the annealing experiments of MBF-50. In this respect the behavior observed on thermal cycling below Tg di€ers fundamentally from the other reversible relaxations in which a glass is thermally cycled between T  Tg and T  Tg , and where a peak is observed [13,14,33]. 4.3. The nature of structural relaxation The irreversible relaxation in glasses is a common occurrence when an unannealed glass is heated more slowly than it was cooled from the liquid state, and the state approaches that of a lower energy and entropy during heating. This is

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represented also as a decrease in the ®ctive temperature, Tf , de®ned as the temperature [17] at which a glass is kinetically stable with respect to its equilibrium liquid. But, in the production of glassy metal-alloys, the high rate of quenching produces a state with an exceptionally high Tf , where the arelaxation has not evolved from the Johari± Goldstein relaxation (the currently used term, which distinguishes it from the b-relaxation of the mode-coupling theory [40]). In addition, the manner of metal±glass ribbon production causes the cooling rate of the ribbon to di€er across its thickness, thereby producing a Tf -gradient across the thickness. There are also attendant strains in the sample, as the density and the mechanical moduli of the glassy states of di€erent Tf s di€er across the ribbonÕs thickness. Thus, on ®rst heating of the sample, not only do the enthalpy and entropy decrease, but also some of the elastic energy stored in the sample is released. For these reasons, mathematical modeling of its structural relaxation is not appropriately done in the same manner as for other types of glasses. However, once the as-received sample has been heated to Th , most of its strain energy is expected to have been released, and any further relaxations that occur on thermal cycling can be treated in the same manner as for other glasses. Despite that, the reversible relaxation in metal-alloy glasses has been seen as unique, and therefore modeled in terms of independent two-level systems [24±27], originally given by Phillips [41] and Anderson et al. [42]. In this view, an atom or a group of atoms have two local minima in con®gurational space separated by a certain energy di€erence. A given two-level system contributes according to its activation energy and the energy di€erence between the two minima. On the assumption that they are independent of each other in con®gurational space, Bruning et al. [27] considered that the rate of approach of a two-level system towards an equilibrium with time itself varies with T in an Arrhenius manner. By assuming that the activation energy has a rectangular box distribution, they deduced that the height of the apparent Cp maximum in the difference curves of the type seen in Figs. 3(b) and 4(b) is directly proportional to the total heat

evolved. The implication is that the shape of the apparent Cp maximum does not change with change in ta and Ta , thus eliminating the need for integration of the di€erence curves. The occurrence of a remarkably similar relaxation in polymers implies that either the same mathematical treatment in terms of a two-level system be used for glasses in general, or else the concepts and mathematical treatment used currently for the glassy state in general be extended to metal-alloy glasses. There is little doubt that localized di€usion leading to a spontaneous decrease in H and S of a glass is a re¯ection of the approach of the glass structure to a lowest con®gurational energy state either as a whole, or as only of those group of atoms and molecular segments that have been trapped in a deep energy minimum during the cooling of the glass from Th (as opposed to the cooling of the equilibrium liquid). In one model developed by Perez and co-workers [37,43±45], the high energy sites in con®gurational space are termed defects, i.e., local groups of atoms whose potential energy corresponds to a point on either the attractive side of the potential energy minimum or the repulsive side of it, and there are alternative representations referred to as `rugged energy landscapes' containing numerous minima of varying depths [46±51]. The potential energy landscapes have also been regarded as time-dependent [6,52]. In either representation, variations in the atomic environment in a glass structure lead to a distribution of di€usion coecients, which may be envisaged as a distribution of mainly the activation energy, E , in the Arrhenius equation, D ˆ D0 exp …ÿE =kT †, where D0 is the pre-exponential term and k the Boltzmann constant. On cooling at a certain rate, those atoms which di€use too slowly to achieve, in the time determined by the cooling rate, their lowest con®gurational H and S state of their equilibrium disordered structure, become kinetically frozen-in. As cooling progressively decreases the self-di€usion coecient, the majority of atoms becomes kinetically frozen-in just below Tg . A relatively small fraction, whose di€usion coecient lies at the short time extreme of the distribution, remains mobile in the otherwise rigid glassy state. Within the time scale of the experiments, only the mobility of this small

G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

fraction of the total population contributes to the con®gurational H and S. The fraction of the atom population that ceases to contribute to a thermodynamic function during the cooling of a glass at a certain rate leads to further, and gradual, development of a kinetic instability with respect to now those parts of the structure that fail to attain a thermodynamic equilibrium with decrease in kT. This appears as a gradual decrease in Cp , H and S of a glass over a temperature range extending to 0 K. On isothermal annealing, only those local con®gurations in which atomic di€usion is suciently rapid, approach an equilibrium structure within a given ta . On heating a preannealed glass, self di€usion in the local regions becomes faster according to the same Arrhenius equation and the increase in Cp , H and S of a glass contains the sum of contributions from all states that achieve the con®gurational equilibrium plus the vibrational contribution from all. For each state that has attained a con®gurational equilibrium at its Tf on heating at a certain rate, the slope of the H and S curves rapidly increases once T > Tf , goes through a point of in¯exion and the curves meet the curves for the con®gurationally equilibrated state from below. (For details, articles by Moynihan et al. [16], Hodge [38,39], Johari and Sartor [33], may be consulted.) In the usual DSC scan, when an annealed glass is heated towards its equilibrium liquid state, this regain appears as an overshoot at T > Tg , but before the equilibrium liquid state corresponding to that heating rate is reached. The area of the overshoot is a measure of the H and S lost during the annealing. In this study, even when the majority of the atoms in a glass structure at T < Th are kinetically frozen-in, the rapid approach towards an equilibrium of a small but increasing number of locally di€using atoms produces a small `overshoot' at a temperature determined by its di€usion rate and the rate of heating. In this view, the endothermic peak observed on heating an annealed glass to T < Tg or the regain of H and S as observed here is a re¯ection of the sum of a multitude of Cp overshoots, each corresponding to a `mini glass±liquid transition' of the localized groups of atoms or molecular segments.

63

4.4. Simulation for reversible relaxation A distribution of di€usion times may be represented as a sum of contributions from all in the population distribution, or in a macroscopic sense, as a stretched-exponential relaxation function. To examine whether the reversible relaxation in glassy metal-alloys may be attributed to a distribution of di€usion times in the same manner as for other glasses, we qualitatively simulate its features by using the equations for the structural relaxation of organic and inorganic polymers [38,39], hydrogenbonded [53] and molecular [16] and ionic glasses [54]. As mentioned earlier [33], there are at least nine di€erent models for mathematically describing the behaviour, and all share two common features: (i) a distribution of relaxation times represented by a stretched-exponential relaxation function and (ii) an increase in the characteristic or macroscopic relaxation time with increase in ta , or decrease of Tf , on annealing. In terms of the Narayanaswamy [18] and Moynihan et al. [16] formalisms, the distribution of relaxation time is written in the form of a relaxation function given by Douglas [55,56] b

/…t† ˆ exp ‰…ÿt=s† Š and s ˆ A exp



 xDh …1 ÿ x†DH  ‡ ; RT RTf

…11†

…12†

where /…t† is the relaxation function, Dh is the activation energy, A is a parameter equal to s when both T and Tf are formally in®nity, and x < 1, is an empirical parameter referred to as the non-linearity parameter. For x ˆ 1, Eq. (12) becomes the Arrhenius equation. Densi®cation of glass on annealing is expected to increase s, as is densi®cation on isothermal compression. Hence the non-linearity of the relaxation process parameterized in the terms x < 1 in Eq. (12) is implicit in the net increase in s with increase in ta , or decrease in Tf . The simulation is done by normalizing the DSC scans by the increase in Cp in the glass±liquid transition region, as described earlier [16,38,39,54]. The DSC scans and the di€erence curves simulated for the chosen

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G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

parameters: ln A ˆ ÿ230; x ˆ 0:23; b ˆ 0:25 and Dh ˆ 831:4 kJ/mol are shown in Fig. 12(a) and (b), respectively. In this simulation, a glass was cooled at 320 K/min from 400 K, 30 K below Tg of 430 K to 280 K and heated to 400 K at 30 K/min after its annealing at 323 K for di€erent periods. (Instead of using the 40 K/min heating rate of the DSC experiment, we chose 30 K/min to minimize the errors arising from the temperature-time steps used in the simulation, and to show that this rate has little e€ect on the features observed.) The simulated DSC scans and the di€erence curves in Fig. 12 are remarkably similar to those seen in Fig. 3(a) and (b) for MBF-50 and for other glasses [27,57], but somewhat di€erent from those in Fig. 4(a) and (b) for the network polymer for which the parameters to be used for simulation are likely to be di€erent. The same parameters were used for a second simulation of the DSC scans and di€erence curves

for which Th was chosen as 470 K, 40 K above Tg , and the ®xed annealing time of 1 h at di€erent temperatures. These curves are shown in Fig. 13(a) and (b). The simulation appears to give shapes similar to those shown in Figs. 5 and 6. The resemblance between the simulated results and the measured data for MBF-50 seems to support the premise that the reversible relaxation of a glassy metal-alloy may be described in terms of a distribution of relaxation times and Eqs. (11) and (12). It should be noted that rapid crystallization of MBF-50 at T < Tg kept us from determining the Cp rise on glass±liquid transition which is required for the normalization of the DSC scans. This prevented us from ®tting Eqs. (11) and (12) to the data for MBF-50 in Fig. 3(a) and (b). Although a ®t to the data for MBF-50 could be done with values of x, b and s deduced from the data in Fig. 10, the errors in the data made this approach unsatisfactory.

Fig. 12. (a) The simulated DSC scans of the annealed and unannealed sample on heating to 400 K (< Tg of 430 K) at 30 K/min. The annealing temperature was 323 K and the annealing time was 1, 2, 4, 8 and 24 h. (b) The di€erence curves calculated from the curves in (a).

Fig. 13. (a) The simulated DSC scans of the annealed and unannealed sample on heating to 470 K (> Tg of 430 K) at 30 K/min. The annealing time was 1 h and the annealing temperature was 300, 310, 320, 330, 340, 350 and 360 K for curves 1±7. (b) The di€erence curves calculated from the curves in (a).

G.P. Johari, J.G. Shim / Journal of Non-Crystalline Solids 261 (2000) 52±66

Finally, it must be pointed out that not all glassy metal-alloys have shown a reversible enthalpy relaxation, and that the magnitude of the relaxation varies with the glass composition in an alloy system, as for example in Fe±B glasses [57]. In our interpretation that is expected because, ®rstly the distribution parameter b for various glasses di€ers and secondly the contribution to thermodynamic functions from the Johari±Goldstein relaxation in various glasses di€er. The latter becomes signi®cant when its time scale is far removed from that of the relaxation processes that vitrify a liquid. This produces a second, broad Cp endotherm at temperatures far below Tg , as is observed for iso-propyl benzene glass [58] by adiabatic calorimetry.

5. Conclusions The so-called `reversible' relaxation and its time-, and temperature-dependence in glass metal-alloys is one aspect of a distribution of selfdi€usion times in the glassy state in general. Common to all types of glasses it can be simulated by the equations for a stretched-exponential relaxation function and non-linearity of the relaxation behaviour. The reproducible endotherm may be seen as the sum of a multitude of `overshoots' in the apparent Cp , usually observed when an annealed sample of a glass is heated at a certain rate. But here they correspond to localized motions as kinetic unfreezing allows rapid attainment of the thermodynamic equilibrium state from a lower to a higher energy of the localized group of atoms when the heating rate is faster than the rate of the attainment of the equilibrium.

Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of CanadaÕs grant to G.P.J. for general research.

65

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