Metallurgical and Materials Transactions A

, Volume 39, Issue 7, pp 1573–1577

Influence of Cooling Rate on the Enthalpy Relaxation and Fragility of a Metallic Glass


  • R. Raghavan
    • Department of Materials EngineeringIndian Institute of Science
  • P. Murali
    • Department of Materials EngineeringIndian Institute of Science
    • Department of Materials EngineeringIndian Institute of Science
Symposium: Materials Behavior: Far from Equilibrium

DOI: 10.1007/s11661-007-9262-y

Cite this article as:
Raghavan, R., Murali, P. & Ramamurty, U. Metall and Mat Trans A (2008) 39: 1573. doi:10.1007/s11661-007-9262-y


Structural relaxation behavior of a rapidly quenched (RQ) and a slowly cooled Pd40Cu30Ni10P20 metallic glass was investigated and compared. Differential scanning calorimetry was employed to monitor the relaxation enthalpies at the glass transition temperature, Tg, and the Kolrausch–Williams–Watts (KWW) stretched exponential function was used to describe its variation with annealing time. It was found that the rate of enthalpy recovery is higher in the ribbon, implying that the bulk is more resistant to relaxation at low temperatures of annealing. This was attributed to the possibility of cooling rate affecting the locations where the glasses get trapped within the potential energy landscape. The RQ process traps a larger amount of free volume, resulting in higher fragility, and in turn relaxes at the slightest thermal excitation (annealing). The slowly cooled bulk metallic glass (BMG), on the other hand, entraps lower free volume and has more short-range ordering, hence requiring a large amount of perturbation to access lower energy basins.

1 Introduction

Structural relaxation of a glass is a process in which short-range atomic rearrangements take place to account for the evolution of the as-cast property of the glass toward its equilibrium value, when annealed below Tg. The process of structural relaxation occurs in all classes of glasses, including oxide and polymeric glasses.[1]

The “frozen-in” free volume is considered as the primary “flow defect” that accounts for diffusion in metallic glasses.[2] It is now well accepted that a higher cooling rate will entrap larger amounts of free volume.[35] A slower cooling rate entraps lesser free volume but allows significant short-range ordering. Prior studies have shown that the free volume is sensitive to the structural relaxation anneals. Free volume controls the shear localization and plasticity at low temperatures, which influences the impact toughness and wear resistance of metallic glasses.[68] Thus, it is imperative to develop a detailed understanding of the structural relaxation kinetics in these materials. While the relaxation behavior of rapidly quenched glasses (RQGs) is widely studied and reported in the literature, the slowly cooled glasses–so-called bulk metallic glasses (BMGs)—are not that critically examined. In particular, there is a paucity of information that compares RQG and BMG relaxation glasses, which is the objective of this study. In this context, it is worth noting that some studies were reported recently comparing the deformation behaviors of a RQG and BMG of the same composition and imply subtle differences.[35,9]

2 Materials And Experiments

An alloy of nominal composition Pd40Cu30Ni10P20 (at. pct) was melted in a vacuum induction furnace with back-filled argon gas and then was cast in a water-cooled copper mold in the form of a 10 mm diameter rod. Ribbons of the same composition were obtained by melt spinning at a wheel speed of 40 m/s. The amorphous nature of the as-cast alloys was ascertained by X-ray diffraction (XRD) and transmission electron microscopy.

The bulk alloy was isothermally annealed at 523 K (note Tg = 573 K) for different times. The ribbon samples were annealed at 523 K as well as 483 K. Prior to the heat treatment, the samples were vacuum sealed in quartz tubes to a pressure of 10–6 MPa to avoid oxidation and a low-temperature resistance-heating furnace with a heating rate of approximately 0.083 K/s was used for the annealing. The annealed samples were cooled to room temperature within the furnace by switching off the furnace.

The as-cast and heat-treated samples were characterized by differential scanning calorimetry (DSC) at a heating rate of 0.333 K/s in argon atmosphere. The XRD analysis of all the samples was carried out using Cu Kα radiation at a scan rate of 0.033 deg/s. The samples mounted for each scan were usually in the form of 10 mm discs placed side by side from the BMG and short strips of the ribbon to maximize the area of scanning.

3 Results

The bulk alloy shows a distinct glass transition at 573 K followed by a ΔTx spanning over 80 K (Figure 1). The glass crystallizes in two events with a sharp exothermic peak at 653 K followed by a minor second exothermic peak. Prior to melting that starts at 793 K, a minor endotherm is also observed. The value of the reduced glass transition temperature, Trg (given by the ratio of Tg/Tm), calculated for this alloy is 0.72, which is in agreement with the published data.[10] Large values for ΔTx and Trg for this alloy are indicators of its high glass forming ability. The DSC scan of the melt-spun ribbon sample (Figure 1) shows that the Tg and ΔTx are identical to those measured for the BMG, with the only difference being an ∼10 J/g in the enthalpy of primary crystallization. Such differences in crystallization enthalpies have also been observed by Jiang et al.,[11] who compare the mechanical response of a RQG and BMG of the same composition. However, Jiang et al.[11] also observe an increased exothermic hump just before Tg in the RQG, which is not observed in the present study.
Fig. 1

Comparative DSC scans of the RQG and BMG at a scan rate of 20 K/min

When annealed at Ta(below Tg), the DSC scans of both the RQG and BMG showed increased enthalpy recovery, as compared to the as-cast state at Tg, for increasing ta at a given Ta (Figure 2). The data for the RQG were fit with the Kolrausch–Williams–Watts (KWW) function (explained later) and corresponding fitting parameters were obtained (Figure 3). To observe the effect of Ta, the enthalpy recovery response of the RQG for two Ta was fit with the KWW function, and corresponding fitting parameters are compared (Figure 4).
Fig. 2

DSC scans of the RQG in the as-cast and annealed state (Ta = 523 K)
Fig. 3

KWW fits for the enthalpy recovered in the RQG and BMG annealed at 523 K
Fig. 4

KWW fits for the enthalpy recovered in the RQG annealed at 483 and 523 K

The glass transition can be thought of as the structural relaxation of the glass into the supercooled liquid. Thus, the expected difference in kinetics of structural relaxation between the RQG and BMG can be captured by examining the scan rate dependence of Tg. The scan rate dependence of Tg can be fit with the Vogel–Fulcher–Tamman (VFT) equation, and the fitting parameters obtained can give an insight into the fragility. In this study, the scan rate dependence of Tg of the RQG and BMG in both their as-cast and annealed states was studied (Figures 5(a) and (b)).
Fig. 5

(a) The scan rate dependence of the RQG in the as-cast and annealed state and (b) the scan rate dependence of the BMG in the as-cast and annealed state

4 Discussion

Structural relaxation in glasses can involve the simultaneous operation of many processes, each associated with a characteristic time scale. Thus, relaxation is a nonexponential process. The KWW equation captures the deviation from exponential behavior, which describes the response of any property of a glass to thermal, electrical, and mechanical perturbation:[12]
$$ M_{p} = \exp ( - t/\tau )^{\beta } $$
where Mp is the relaxation function of the property P given by
$$ P = \frac{{p(t) - p(\infty )}} {{p(0) - p(\infty )}} $$
where p(t), p(0), and p() are the values of P at times t, 0, and ∞, respectively; and τ is the average relaxation time. The temperature dependence of τ gives an idea of the activation energy of the processes involved.

The parameter of primary interest is β, often referred to as the stretched exponent. It is assumed to be independent of Ta and can assume values between 0 and 1. While a value close to 1 implies that the system is a strong glass former, a value less than 0.5 implies that the glass is fragile.[13] Note that a strong glass is one that exhibits an Arrhenius dependence of viscosity with temperature, while a fragile glass exhibits a non-Arrhenian or VFT fit.[14] In the context of metallic glasses, the former exhibits high glass forming ability, whereas the latter is typically a poor glass former requiring high cooling rates for preventing crystal nucleation.[15]

Because we have resorted to calorimetry for characterizing the structural relaxation of Pd40Cu30Ni10P20 metallic glass, the response of the enthalpy recovery at Tg is fit with the KWW equation. Good fits are obtained for ribbon samples, but the KWW equation does not adequately capture the trend exhibited by the bulk alloy. The curve fits for the data of the ribbon and bulk samples yield β values of 0.33 and 0.11, when annealed at 523 K (Table I). The discrepancy observed in the BMG is due to high sensitivity of the KWW function to the data.
Table I

Parameters Obtained by Fitting the KWW Equation to the Enthalpy at Tg

Fitting Parameters



483 K (R = 0.96)

523 K (R = 0.97)

523 K (R = 0.97)





τ (s)



1.6 × 105

Fan et al.[16] obtained a β value of 0.75 for a Pd43Ni10Cu27P20 BMG, which is evidently a strong glass former, as the BMG could be obtained at a cooling rate of 0.167 K/s. The β value of 0.33 obtained for the Pd-based ribbon in the present study and the value obtained by Fan et al.[16] for a Pd-based BMG are in accordance with the interpretations presented previously. However, a survey of the values for Zr-based metallic glasses, another good glass former, exhibits that the values of β obtained are contrary to expectations. Table II provides a comprehensive summary of values of β and τ available literature for different metallic glasses. This is because the values of β and τ obtained by fitting the KWW function to the data of any property are very sensitive to the accuracy of the data. In other words, small variations in the data will cause significant variations in fitting parameters of β and τ.
Table II

β Values Observed for Different Systems from Various Studies Compared with the Present Study







average positron lifetime

0.7 to 0.8


Zr46.7Ti8.3Cu7.5Ni10Be27.5 BMG





Pd43Ni10Cu27P20 BMG


anelastic strain

0.5 to 0.7

Zr65Al7.5Cu27.5 RQG


stress relaxation

0.5 > Tg, 0.9 < Tg

Zr41.25Ti13.75Ni10Cu12.5Be22.5 BMG





Zr41.2Ti13.75Ni10Cu12.5Be22.5 BMG

Present work




Pd40Cu30Ni10P20 BMG



Pd40Cu30Ni10P20 RQG

Furthermore, β is expected to be independent of Ta, for a given system. However, the ribbon exhibits a significant change in the value of β: 0.33 at Ta = 523 K and 0.8 at Ta = 483 K on changing Ta (Figure 4). A similar study could not be conducted on the bulk samples due to sample limitation. This implies that β has a functional dependence on temperature, though more datum points are required to obtain the actual form.

Bobrov et al.[17] have compared the stress relaxation behaviors of a RQG and BMG of the same composition (Pd40Cu30Ni10P20). Both in the as-cast and annealed states, the stress relaxation kinetics of the RQG is faster than that of the BMG. This suggests that free-volume annihilation is the mechanism by which the RQG undergoes relaxation. The BMG, which has a low free volume, can relax by restructuring of short-range ordering. In the present study, it has been shown that not only the enthalpy relaxation kinetics of the RQG is faster than the BMG, but also there is a crossover at high annealing times. Quantitative differences between the relaxation kinetics of the RQG and BMG may be captured by the scan rate dependence of Tg. The scan rate dependence of Tg can be fit with VFT[14] as follows:
$$ q = c \cdot \exp {\left[ {\frac{{m \cdot T^{0}_{g} }} {{{\left( {T^{0}_{g} - T_{g} } \right)}}}} \right]} $$
where q is the heating rate, c is a constant, m is the fragility parameter, and \( T^{0}_{g} \) is the ideal glass transition temperature.
The fragility index, D, which is directly calculated from the VFT fit to the viscosity dependence on temperature, is a quantitative estimate of the kinetics of the glass forming ability of the system. In other words, D can be thought of as the kinetic fragility of the glass forming system. Angell[14] has shown that the fragility can be correlated to the isothermal relaxation kinetics, which is the objective of this part of the study. The value of D can be calculated from the fitting parameters m and \( T^{0}_{g} \) obtained from the scan rate dependence of Tg (Eq. [3]) using the following equation:
$$ D = \frac{{m \cdot T^{0}_{g} \cdot T_{{g,\eta }} }} {{{\left( {T_{{g,\eta }} - T^{0}_{g} } \right)}^{2} \cdot \ln 10}} $$
where Tg,η is the temperature at which the viscosity of supercooled liquids reaches 1012 Pa s. Using 568 K as an estimate of Tg,η, the values of D obtained for the as-cast and annealed states of both the RQG and BMG are summarized in Table III. The lower is the value of D, the stronger is the glass. The results indicate that the as-cast BMG is a stronger glass than the as-cast RQG; however, on annealing, the same values of D are obtained. This implies that annealing induces both the RQG and BMG to fall into megabasins of similar depth in the potential energy landscape, though the density of minima might be different in the RQG and BMG. Hence, the fragility of a glass forming system in the as-cast state might be a function of the cooling rate at which it has been processed, which has also been suggested by Fan et al.[16]
Table III

Parameters Obtained by Fitting VFT Equation to the Scan Rate Dependence of Tg (R = 0.99 for All Fits)



\( T^{0}_{g} \) (K)


















5 Summary

This work compares the enthalpy relaxation and fragility of a RQG and BMG of the same composition. The annihilation of the excess free volume trapped in the RQG and restructuring of the short-range ordering of the BMG are thought to be the mechanisms of enthalpy relaxation in the RQG and BMG, respectively. The relaxation study indicates that indeed there is a difference in kinetics, though the relaxed state achieved may be different in the RQG and BMG. The difference in kinetics, which has been captured by the fragility index, indicates that, though the potential energy landscape of the RQG and BMG might be different, the RQG and BMG fall to megabasins of similar depths on annealing. It has also been suggested that the fragility of a glass forming system might be a function of the cooling rate.


The authors thank Professor S. Ranganathan for his valuable input and Dr. N. Nishiyama for providing the metallic glass samples examined in this study. The authors acknowledge the assistance rendered by Mr. P. Padaikathan in conducting the DSC experiments. This research work was funded by a grant from the Defense Research and Development Organization, Government of India.

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