Introduction

Despite their similar inherent amorphous structure, which endows metallic glasses with high strength and elasticity1,2,3,4, they exhibit a broad range of damage tolerance, from ideal brittle to remarkably tough5,6,7,8. Such wide spectrum of mechanical properties is in stark contrast to oxide glasses, and understanding the origin has been the focus of the metallic glass community for the last decade6,7,8,9,10,11,12. Mechanical response of a metallic glass is further influenced by extrinsic effects, such as cooling rate13, sample size14,15 and testing conditions (temperature and strain rate)16,17. Besides these extrinsic factors, the understanding of composition-dependent plasticity in metallic glasses continues to defy proposed theories. For example, Pd-based metallic glass is significantly more brittle compared with Pt-based metallic glass prepared at a similar cooling rate18. Plasticity of Zr-Cu-Al metallic glasses is very sensitive to composition, whereas their microstructure and elastic constants are largely unaffected19,20. The atomic-level structure underlying this different mechanical response of metallic glasses is difficult to characterize. Computational modelling offers some insight about the composition–structure–property relationship; however, the lowest achievable cooling rates in simulation are still several orders of magnitude higher than experimental cooling rates11.

Following the analogy from crystalline metals, the toughness of metallic glasses has been correlated with their elastic constants6,9,21,22. Higher Poisson’s ratio or lower G/B (G: shear modulus, B: bulk modulus) ratio is predicted to result in higher toughness in metallic glasses, mechanistically by increasing the resistance for crack opening compared with shear band formation and extension. Therefore, a high Poisson’s ratio or low G/B has been often used as an indicator for designing tough metallic glasses6,9. In recent years, increasing experimental evidence has mounted about the limited applicability of G/B in predicting the mechanical response of metallic glasses18,23,24,25. In particular, a marginal change in elastic constants cannot account for the sever annealing-induced embrittlement observed in metallic glasses23. These findings suggest that a simple, elastic moduli-based approach does not capture the essential features of the multifaceted plasticity problem in amorphous metals. A comprehensive description for the mechanical behaviour of metallic glasses requires the understanding of the complex interplay between their composition, processing, structure and properties.

In polymer and oxide glasses, it has been widely observed that the properties of a glass at room temperature are reminiscent of its supercooled liquid at the fictive temperature (Tf)26,27,28. For metallic glasses, it has been shown that the room-temperature elastic constants can be directly correlated with the values of the supercooled liquid through the linear Debye-Grüneisen thermal expansion29. Therefore, knowledge of the supercooled liquid state and the fictive temperature can provide a key insight about the room-temperature properties of metallic glasses.

In this work, we study the effect of fictive temperature on the room-temperature plasticity of bulk metallic glasses (BMGs). Samples that are brittle at room temperature may deform plastically at higher temperatures as reported earlier17. However, the goal of the present study is to understand the room-temperature mechanical response and its correlation with the properties of their supercooled liquids among different glass-forming alloys. Three BMG formers are considered: Pt57.5Cu14.7Ni5.3P22.5 (Pt-BMG), Pd43Cu27Ni10P20 (Pd-BMG) and Zr44Ti11Cu10Ni10Be25 (Zr-BMG). In this study, 2.5% strain to failure is set as the ductile–brittle transition to accommodate ±0.5% measurement error, because even the most brittle metallic glasses still exhibit approximately 2% elastic strain4. Our results show that the room-temperature plasticity of BMGs decreases with lowering the fictive temperature, and falls below the ductile–brittle transition at a critical fictive temperature, Tfc, which is characteristic of the BMG former. We demonstrate that Tfc is a key parameter that defines the mechanical behaviour of BMGs and its sensitivity to cooling rate and annealing.

Results

Bending strain and elastic constants

Room-temperature strain to failure corresponding to different fictive temperatures for the considered BMG formers is shown in Fig. 1a. The temperature scale is normalized to the calorimetric glass transition temperature, Tg, listed in Table 1. A common feature of all three BMGs is that their strain to failure decreases with lowering Tf and it drops below the ductile–brittle transition for Tf<Tfc. The plastic strain appears to correlate with Tf−Tfc, which is remarkably similar to the temperature dependence of free volume (vf~T−T0, vf is free volume and T0 is Vogel–Fulcher–Tammann temperature) predicted by the Vogel–Fulcher–Tammann equation30. Besides this common trend, there are several notable distinctions among three BMGs. The strain to failure for the Pt-BMG remains significantly higher than that for the Zr-BMG and the Pd-BMG after annealing at the same normalized fictive temperatures. The Tfc for the Pd-BMG former is highest among considered BMG formers and occurs above calorimetric Tg, whereas for the Zr-BMG former and the Pt-BMG this transition is below Tg. Figure 1b shows the variation in G/B values as a function of normalized fictive temperature. The G/B values for the as-quenched BMGs are listed in Table 1. The change in G/B with fictive temperature is small even though plasticity changes significantly. For all three BMG formers, the G/B ratio corresponding to the critical fictive temperature Tfc is far below that of the previously proposed lower limit of 0.41 for brittleness9.

Figure 1: Effect of fictive temperature on strain and elastic constants.
figure 1

(a) Room-temperature bending strain to failure as a function of normalized fictive temperature for Pt-BMG, Zr-BMG and Pd-BMG. The critical fictive temperature for ductile–brittle transition is 0.89 Tg, 0.96 Tg and 1.02 Tg for Pt-BMG, Zr-BMG and Pd-BMG, respectively. The open triangle is the strain to failure for Pt-BMG annealed at 0.89 Tg for 30 days. (b) G/B ratio shows a weak dependence on fictive temperature. The G/B ratio for the embrittled BMGs is well below the proposed critical value for ductile–brittle transition.

Full size image
Table 1 Calorimetric Tg measured at a heating rate of 20 K min−1, Tfc, Tg−Tfc and G/B ratio for the considered BMG formers.
Full size table

Thermal embrittlement of metallic glasses, as displayed in Fig. 1a, is typically attributed to reduction in free volume through structural relaxation31,32. The structural relaxation time at Tg is approximately 100 s and increases exponentially with decreasing temperature33. At temperature far below Tg, it is impractical to attain completely relaxed glassy state. This suggests that a metallic glass exhibiting Tfc far below Tg should not become brittle during experimental annealing time. To test this correlation, Pt-BMG was annealed for 30 days at its estimated Tfc of 446 K, which is 57 K below the Tg. The sample remained ductile and displayed a bending strain of 6% (open triangle in Fig. 1a). According to the relaxation kinetics, it would take about 30 years to structurally relax Pt-BMG at 446 K (ref. 34). In contrast, the Zr-BMG with a Tfc of only 25 K below its Tg, is sensitive to embrittlement during annealing near Tg (ref. 23). The Pd-BMG is even more susceptible to annealing-induced embrittlement because of its Tfc above Tg, which corresponds to a relaxation time of only 50 s. Consequently, the difference between Tg and Tfc of a metallic glass correlates with its resistance to annealing-induced embrittlement.

Effect of cooling rate

Mechanical properties of some BMGs strongly depend on the cooling rate during vitrification, whereas others show little or no variation13,18,21. The degree of cooling-rate sensitivity among BMG formers can be rationalized based on their Tfc values. If the cooling rate is sufficiently fast, the resulting Tf is higher than Tfc, and the BMG is ductile. A slower cooling rate will result in lower Tf, and the glassy state will become brittle if Tf drops below Tfc. Typical cooling rates that result in bulk metallic glass formation (thickness >1 mm) span over four orders of magnitude ranging from 0.1 to 1,000 K s−1 (ref. 35). Experimental measurements in a wide range of glass-forming liquids reveal that Tf decreases by 5–10 K for every order-of-magnitude decrease in cooling rate28,36,37. The maximum variation in Tf, as a result of cooling rate, can be approximated to 40 K for BMG formers (Fig. 2). This implies that for a BMG whose Tfc differs from its Tf by more than 40 K, its ductile or brittle behaviour will remain essentially unaffected by the cooling rate. BMG formers with Tf−Tfc>40 K will be ductile and the ones with Tf−Tfc≤40 K will be brittle for typical cooling rates used in BMG formation. In contrast, the BMG formers with Tfc in the proximity of Tf (Fig. 2) will be sensitive to the cooling rates.

Figure 2: The predicted mechanical behaviour of BMG formers under typical cooling rates.
figure 2

The maximum variation in Tf is about 40 K as a result of critical cooling rate, which spans over four orders of magnitude among BMG formers. BMG formers with Tfc out of the accessible Tf range are either always ductile (Tf−Tfc>40 K) or brittle (Tf−Tfc≤40 K). However, BMG formers with Tfc~Tf can change from ductile–brittle within practical cooling-rate variations.

Full size image

Isothermal embrittlement diagrams

The existence of Tfc is experimentally manifested in the cooling-rate-dependent mechanical behaviour of BMGs. Several BMGs such as Fe-based and Mg-based, which are known to be typically brittle, become ductile when cooled at higher rates21,38. The cooling rate (Re) to prevent embrittlement and its relation with the Tfc can be understood from isothermal time–temperature–transformation (TTT) diagrams for embrittlement (Fig. 3). The embrittlement time is the annealing time required to decrease the bending strain below the ductile–brittle limit. The embrittlement time decreases with increasing temperature till Tfc and then increases abruptly before decreasing again with further increase in temperature. The discontinuity at Tfc is related to the change in embrittlement mechanism from structural relaxation (T≤Tfc) to crystallization (T>Tfc). The Tfc is the highest temperature at which a metallic glass can embrittle by structural relaxation. Consequently, the embrittlement of a glassy state through relaxation is fastest at Tfc that corresponds to the ‘nose’ of embrittlement curves in Fig. 3. The embrittlement time at the nose, te, determines the cooling rate Re to vitrify a ductile glassy state. The te values for Pd-BMG, Zr-BMG and Pt-BMG are 50 s, 5,000 s, and 9E–9 s, respectively. These values scale with the difference Tg−Tfc (Pt-BMG>Zr-BMG>Pd-BMG). The short te indicates that Re is highest for Pd-BMG among the considered BMG formers. The critical cooling rate (Rc) for glass formation, however, follows the opposite trend and is lowest (0.09 K s−1) for the Pd-BMG35. This suggests that the critical cooling rates for avoiding crystallization and embrittlement are not directly correlated.

Figure 3: TTT diagrams for embrittlement.
figure 3

The Tfc is the highest temperature where embrittlement can occur owing to structural relaxation. Above Tfc, the embrittlement time increases abruptly because of change in origin of embrittlement from structural relaxation to crystallization. The time (te) at Tfc defines the nose of TTT diagrams for embrittlement. The measured te is 5,000 s for Zr-BMG (a) and 50 s for Pd-BMG (b).

Full size image

Discussion

A comprehensive framework to account for the diverse mechanical behaviour of metallic glasses can be constructed based on the knowledge of Tf−Tfc. According to this criterion, all metallic glasses can be broadly classified into two groups: Tf−Tfc<0 (type I) and Tf−Tfc>0 (type II). For type I metallic glasses, the embrittlement nose time is shorter than the crystallization nose time and, consequently, the critical cooling rate to prevent embrittlement is higher than the critical cooling rate for glass formation (Fig. 4a). Here Tf is the fictive temperature of a glass vitrified at its critical cooling rate for glass formation. The magnitude of Tf−Tfc determines their mechanical behaviour and its sensitivity to cooling rate and annealing. Metallic glasses with a small negative Tf−Tfc are sensitive to preparation conditions and can change from ductile to brittle or vice versa with varying cooling rates. For metallic glasses with large negative Tf−Tfc, the embrittlement nose time is much shorter than the crystallization nose time. Consequently, their bulk states are always brittle. This explains why Fe-based and Mg-based metallic glasses require high cooling rates, unachievable in bulk form, to vitrify into a ductile state21.

Figure 4: Classification of BMG formers based on Tf− Tfc values.
figure 4

(a) Type I metallic glasses (Tf−Tfc<0) exhibit an embrittlement nose time shorter than the crystallization nose. As a result, these BMG formers are typically brittle but can be made ductile if a cooling rate higher than Re is applied. (b) For type II metallic glasses (Tf−Tfc>0), the embrittlement nose time is longer than the crystallization nose time; therefore, these BMG formers are always ductile. In both cases, variation of Tf with cooling rate is shown. For practical convenience, calorimetric Tg can be used to approximate Tf.

Full size image

Type II metallic glasses (Tf−Tfc>0) exhibit an embrittlement nose time that is longer than the crystallization nose time (Fig. 4b). Hence, their critical cooling rate for embrittlement is lower than the critical cooling rate for glass formation. These metallic glasses are always ductile in the as-cast bulk state and the magnitude of Tf−Tfc indicates their resistance to annealing-induced embrittlement. Metallic glasses with a large positive Tf−Tfc (for example, Pt-BMG) do not become brittle under practical annealing conditions, because their embrittlement time (≈relaxation time) at Tfc is extremely long. In contrast, metallic glasses with a small positive Tf−Tfc (for example, Zr-BMG) are ductile in the as-cast state but are susceptible to annealing-induced embrittlement because of their fast relaxation at Tfc.

According to this critical fictive temperature viewpoint, the knowledge of Tfc and Tf is sufficient to understand a BMG’s resistance to annealing-induced embrittlement, cooling rate sensitivity and critical cooling rate for plasticity. In practice, Tf measured by method of integration from differential scanning calorimeter (DSC) heating curves is typically few degrees lower than the Tg (ref. 39). Therefore, calorimetric Tg values can be used as a good approximation of Tf for the practical application of plasticity criteria outlined here. Our experimental results validate the use of calorimetric Tg for prediction of mechanical response of BMG formers. Pt-BMG with Tg−Tfc~57 K is ductile at any cooling rate and exhibits higher resistance to annealing embrittlement. Zr-BMG with Tg−Tfc~25 K is ductile in the as-cast state, but becomes brittle during sub-Tg annealing23. Pd-BMG with Tg−Tfc~−12 K changes from ductile to brittle when the cooling rate is decreased below 50 K s−1 (ref. 18).

The absolute values of Tfc and hence Tg−Tfc are a function of sample size because of the size-dependent plasticity of BMGs13,15. Figure 5 shows Tg−Tfc for Zr-BMG samples of different thicknesses. Tg−Tfc increases with decreasing temperature. However, this size dependence is much smaller than the effect of composition on Tg−Tfc. For Pt-BMG of 0.6 mm thickness, the Tg−Tfc is larger than that of the 0.3-mm thick Zr-BMG. Similarly, the Tg−Tfc of 0.8-mm thick Zr-BMG is larger than that of the 0.6-mm thick Pd-BMG. Thus, Tg−Tfc values measured for any sample size allow to predict the mechanical behaviour of different glass-forming alloys.

Figure 5: Effect of sample size on the critical fictive temperature.
figure 5

Tg−Tfc values for Zr-BMG of different thicknesses are compared with that of 0.6-mm thick Pt-BMG and Pd-BMG samples. The results demonstrate that the effect of alloy composition is more pronounced than sample size. Therefore, Tfc measured for any sample size can be used in the Tg−Tfc criterion for plasticity.

Full size image

In summary, we propose Tg−Tfc criterion to predict the room-temperature mechanical behaviour and its sensitivity to cooling rate and annealing-induced embrittlement for metallic glasses. We envision that application of structural models40,41 can reveal the microscopic origin of such a critical fictive temperature to design tough metallic glasses.

Methods

Sample preparation

Pt57.5Cu14.7Ni5.3P22.5 and Pd43Cu27Ni10P20 alloys were prepared by induction melting the constituents in vacuum-sealed quartz tube. The alloys were subsequently fluxed with B2O3 at 1,000 °C for 600 s. The fluxed alloys were re-melted and water-quenched in 3-mm diameter quartz tubes to prepare the bulk amorphous samples. Amorphous Zr44Ti11Cu10Ni10Be25 was acquired from Liquidmetal Technologies. Rectangular beams of 0.6±0.05 mm thickness were machined from the bulk amorphous samples. To achieve the desired Tf, the beams were annealed at various temperatures (annealing temperature=Tf). Annealing times were chosen two times longer than the relaxation time, to ensure that the equilibrium had been reached at Tf. Crystallization can be ruled out, because the annealing times are at least an order of magnitude shorter than the crystallization times. After annealing, the samples were water-quenched to obtain the glassy state corresponding to different fictive temperatures.

Characterization

The samples were characterized thermally by DSC and structurally by X-ray diffraction. The glass transition temperature, Tg, was measured from DSC heating curves recorded at a heating with 20 K min−1. Elastic constants were calculated from the shear and longitudinal sound velocities measured at room temperature by an ultrasonic technique. The bending strain to failure was measured by bending the beams around mandrels of different radii at room temperature14. The samples were mirror polished before the bending tests. The bending strain corresponds to t/2r (t is the thickness of the samples and r is the radius of the mandrel). The average bending strain and error bars were calculated by testing five samples for each fictive temperature.

Construction of TTT diagrams for embrittlement

The samples were isothermally annealed at various temperatures followed by water quenching to room temperature. The embrittlement time corresponds to the annealing time for which the room-temperature bending strain to failure reduces to 2.5% or lower. TTT diagrams for embrittlement were constructed by plotting the log of isothermal embrittlement times as a function of temperature.

Additional information

How to cite this article: Kumar, G. et al. Critical fictive temperature for plasticity in metallic glasses. Nat. Commun. 4:1536 doi: 10.1038/ncomms2546 (2013).