For glasses, the structural origin of their flow phenomena, such as elastic and plastic deformations especially the microscopic hidden flow before yield and glass-to-liquid transition (GLT), is unclear yet due to the lack of structural information. Here we investigate the evolution of the microscopic localized flow during GLT in a prototypical metallic glass combining with dynamical mechanical relaxations, temperature-dependent tensile experiments and stress relaxation spectra. We show that the unstable and high mobility nano-scale liquid-like regions acting as flow units persist in the glass and can be activated by either temperature or external stress. The activation of such flow units is initially reversible and correlated with β-relaxation. As the proportion of the flow units reaches a critical percolation value, a mechanical brittle-to-ductile transition or macroscopic GLT happens. A comprehensive picture on the hidden flow as well as its correlation with deformation maps and relaxation spectrum is proposed.
Normal liquids can flow instantaneously on mechanical agitation while the flow in a viscous or a solid glass appears to occur only when the external stress reaches a critical value of yield stress or the temperature reaches the glass transition temperature Tg (refs 1, 2, 3). Apart from the stress-induced instantaneous flow of yield, recent experiments and numerical simulations also demonstrated that a glass subject to stress, which is much smaller than its normal yield strength, can also undergo an extreme slow flowing, which is hard to be detected within a short period of time due to the slowness4,5,6,7. On the other hand, the universal nano-scaled localized β-relaxation in metallic glassy state has been observed before the large-scale α-relaxation, which has been demonstrated to be related to the nano-scaled microscopic hidden flowing phenomenon8,9,10,11,12. However, the structural origin of the flowing phenomena in metallic glasses (MGs) such as the slow flowing and instantaneous flow of yield, elastic and plastic deformations, the transition between the β-relaxation to the α-relaxation and glass to supercooled liquid transition are not clear yet due to the lack of clear structural information. Due to the glassy substances are usually used below their Tg (refs 13, 14, 15), the understanding of its flowing phenomena and the structural evolution path during glass-to-liquid transition, and especially its correlation with their mechanical behaviours, are of crucial practical importance. From the theoretical perspective, a general understanding of the structural origin and evolution of the flow behaviour in the extreme viscous glassy solids could be helpful for the clarification of the intriguing dilemmas of glass nature.
Intensive work concentrated on the liquid-to-glass transition or slow down process16,17. However, the reverse glass-to-liquid transition (GLT) process as well as its structural origin has not been paid enough attention. The characteristic changes and their intrinsic correlations with GLT process are still far from being thoroughly studied. Recent studies on the atomic-scale glassy structure have revealed the existence of liquid-like sites in glassy state4,18,19, which are presumed to be responsible for the viscoelastic flow behaviour in glasses20,21,22. Meanwhile, the studies on glasses have demonstrated that the β-relaxation is identified to play an essential role in the GLT process2,8,9,11,23, and the β-relaxation has comparable activation energy with that of the deformation unit and strongly correlated with mechanical brittle-to-ductile transition in MGs20,21,24,25,26, indicating that the β-relaxation is closely related to the initiation and evolution of the localized liquid-like deformation units or flow units in MGs27,28,29,30. Yet, the fraction and evolution of these liquid-like zones, leading to the flow phenomena such as elastic and plastic deformations and GLT, are still speculative due to the lack of abundant experimental evidence, and the intrinsic correlations between such deformation transition, relaxation mode and structural characteristics changes during GLT are still poorly understood.
In this paper, we apply stress relaxation method covering a wide temperature range and dynamical mechanical analysis, which can detect the β-relaxation to investigate the evolution process of GLT in MG, which can be simply treated as random stacking of hard spheres31,32,33. A La60Ni15Al25 MG, with excellent glass-forming ability, thermal stability and unusual pronounced β-relaxation behaviour among known MGs34,35, is chosen as a model system. Plausible correlations among relaxation behaviours, dynamic heterogeneity, microscopic hidden inhomogeneous flow to macroscopic large-scale homogeneous flow during GLT transition and mechanical behaviours of MGs are established and interpreted by the energy landscape theory.
The hidden flow and temperature-dependent tensile experiments
It is known that the flow phenomenon in MG is the percolation of the localized flow units12,14,21. To detect the extreme slow hidden flow before yielding or microscopic structural evolution during GLT, we applied Dynamical Mechanical Spectrum (DMS) measurement, which is a sensitive method to detect the viscoelastic properties of MGs34,35,36. The temperature dependence of the loss modulus E′′ measured at 1 Hz is shown in Fig. 1a. Besides dominant α-relaxation peak at Tα=481 K, a β-relaxation peak at Tβ=363 K can be clearly seen. We note that the definition of Tβ and Tα is frequency dependent. Slowest boundary frequency of 0.01 Hz for both β- and α-relaxation peaks corresponds to an experimental observing time of ~100 s, which is defined as the separation time between glass and liquid2,9, and definition is helpful for comparison among different glassy systems. The blue and the red regions in Fig. 1a, respectively, represent the fit of α- and β-relaxations by Kohlrausch–Williams–Watts (KWW) equation21 (The fitting parameters and corresponding storage modulus fitting results can be seen in Supplementary Table 1 and Supplementary Fig. 1). The β-relaxation correlates with localized flow unit regions, and its peak intensity in DMS indicates the fraction of flow units in glasses21. The unusual pronounced β-relaxation peak in DMS for the La-based MG indicates that the nano-scale localized flow behavior occurs even much below the Tg, and the gap area between the α and β peaks implied that cooperative interaction occurs8,37.
Figure 1b shows temperature-dependent macroscopic flow phenomena induced by tensile stress. Near room temperature, the MG shows large apparent elastic deformation (~2%) and abruptly breaks when the stress reaches the yield strength value; when the temperature increases to 333 K the MG still shows the pure elastic behaviour, but the fracture strength obviously decreases. By further increasing the temperature, a conspicuous transition from brittle to plastic deformation occurs. Such transition is ascribed to the gradual appearance of multiply shear bands14,38,39,40, which stem from unstable localized regions39,41. A homogeneous flowing deformation with superplasticity over 100% deformation is achieved around Tg=461 K under a normal strain rate of 1 × 10−4 s−1. When temperature is higher than Tg, the apparent macroscopic flow is dominated by viscosity and undergoes the viscous non-Newtonian flow to dilute Newtonian flow transformation14,42. However, the macroscopic tensile experiments cannot reflect the microstructural changes or microscopic slow flow before the yielding or Tg.
Dynamical mechanical relaxation measurements
Figure 2 presents the DMS isothermally measured in frequency domain at various temperatures of MG in a three-dimensional form. In frequency domain, at each temperature, we can get DMS for La-based MG to distinguish the α- and β-relaxations within our measuring frequency range (0.01–16 Hz), and the DMS can be obtained from cross-section along the temperature coordinate (indicated as the dark green section in Fig. 2). The change of the β-relaxation peak on temperature reflects the evolution of the flow units approaching Tg. Both the frequency or temperature-dependent DMS of the MG can be fit by KWW, and the KWW fitting distribution coefficient β is 0.5, which represents the dynamic and structural heterogeneities and can be obtained for either α- or β-relaxations peaks from isochronal and isothermal DMS in Fig. 1a and Fig. 2 (fitting parameters and curves see Supplementary Table 1 and Supplementary Figs 1 and 2). The strength ratio between the two relaxation peaks is ~7% from either DMS method.
The β-relaxation peaks change and shift in comparison to α-relaxation peaks are indicated in Fig. 2. The β-relaxation peaks in frequency domain at different temperatures reflect the characteristics and fraction of exact portion suffering GLT transition. Initially, only a part of the flow units is activated at room temperature and no obvious relaxation peak is observed. When the temperature exceeds, more flow units with high activation energy are gradually activated, and the β-relaxation peak appears at ~323 K and then the flow units are fully activated above Tβ. With increasing temperature, the α- and β-relaxations gradually merge together, the activated flow units percolate and the GLT occurs. It is clear that those flow units reflected by β-relaxation, mainly contribute to the hidden flow at low temperature.
Stress relaxation spectra measurements
Stress relaxation measurement, which is another powerful tool to reveal the time-dependent liquid-like nature of flow units in MGs43,44, is applied to study the structural evolution during GLT process. Figure 3a is a stress relaxation spectra containing 65 isothermal stress relaxation curves measured from room temperature beyond Tg at a temperature interval of 3 K (for details see the methods section). A typical stress relaxation curve measured at 363 K is shown as inset, where the stress needed to maintain constant strain gradually relaxes with increasing time. Initial stress σonset and terminal stress σend can reflect the viscoelastic properties of sample43. The lower σonset value or the larger stress drop Δσ=σonset−σend indicates the larger fraction of liquid-like activated flow units or the smaller viscosity of MG43. The shape of stress relaxation curve showing non-exponential behaviour is related with dynamic heterogeneity. The σonset and σend of each relaxation curve are enlarged as orange and green open circles, respectively. Decay of the σend accelerates above Tβ, whereas the change of σonset is not prominent until close to Tg. The σend drops to almost zero near Tg, while σonset vanishes at temperature higher than Tα. When temperature is higher than Tα, the rebirth of stress is due to the disturbance of the occurrence of the crystallization.
Figure 3b highlights several representative stress relaxation curves at different temperatures to investigate characteristic change with temperature. Each curve is rescaled to its measurement time with σonset at t=0. A log–log plot is adopted here for better identification approaching glass transition. At room temperature, the stress drop is feeble but occurs within our experimental time, this demonstrates that there exist unstable flow unit regions in the MG, even though it looks like an elastic body at low temperature by conventional detection. When the temperature exceeds, the drop of stress starts from the long experimental time tail and gradually accelerates. Above 459 K (close to Tg=461 K), the σend drops to <1 MPa as shown in Fig. 3a, indicating that the whole sample enters into macroscopic flow within an experimental time of ~103 s. The stress change with time of all curves can be fitted by KWW-type formula
where σ0 is the initial stress normally equal to σonset, σr is the residual stress at infinite time, τc is the critical stress relaxation time and βKWW is a non-exponential parameter related with dynamic heterogeneity. As glass can be regarded as a frozen liquid with considerable long relaxation time10, no pronounced residual stress should be abided at infinite time. Therefore, σr=0 is taken below 459 K for accurate comparison of fitting parameters (see Fig. 4a and Supplementary Fig. 3). All fitting curves are drawn in the figure and some typical values of τc, over 107 orders change during the process, are also shown.
We note that the definition of liquid-like or elastic-like strongly depends on observing time9. Here the experimental time of each stress relaxation test is elaborately selected at an order of 103 s, which corresponds to a critical viscosity value of about 1011 Pa s at Tg for this MG system45,46. Therefore, portions that perform viscoelastic properties within our experimental time will be treated as liquid-like, otherwise as elastic-like, as illustrated in the inset of Fig. 4a.
Based on the KWW fitting of stress relaxation curves, the changes of distribution coefficient βKWW can be obtained and are shown in Fig. 4a. As βKWW is related with the dynamic heterogeneity, smaller βKWW normally indicates wider dynamic heterogeneity distribution of system, and vice versa45,46,47,48. Three stages during the evolution can be observed below Tg from the temperature-dependent curve of the βKWW. Below 323 K (termed as TT1), the βKWW is ~0.2 and remains almost constant. It rapidly increases to 0.55 at Tβ. After this, its value only slightly changes around 0.5 between Tβ and Tg. As the system relaxation time distribution β=0.5, the value of βKWW should be 0.5 when the sample can be treated as an integral acting like a liquid in a sufficient long time and contributes to the stress decay. However, the viscoelastic property is dominated by those flow units at low temperature as shown in Fig. 2, and the smaller βKWW below Tβ should be ascribed to the inhomogeneous activation of flow units. Only isolated, stochastic and reversible flow units can be activated below TT1. This conclusion can be further verified by fitting of the stress relaxation curve based on the nature of flow units (for details see Supplementary Methods and Supplementary Fig. 4). Above Tβ, an almost constant βKWW around 0.5 indicates that the majority of flow units in the sample start to cooperate with each other and involve in a stress relaxation process at a long but finite time scale. The rapid increase of βKWW above Tg is attributed to the exceeding of experimental time when compared with the time needed to attain equilibrium, as represented in Fig. 3b, rather than signifying the homogeneity of the supercooled liquid.
The measured stress drop Δσ=σonset−σend reflects the fraction of liquid-like zones43,49. From Fig. 3a,b, the increase of temperature leads to an increment of Δσ, indicating that more fractions of atoms in the MG undergo inelastic deformation and are transformed into liquid-like state50,51. Therefore, we model the MG into two parts: liquid-like zones and elastic matrix. Liquid-like zones are simplified as a distribution of dashpots following the Newtonian law, while elastic matrix are treated as springs with ideal elastic modulus E following Hooke’s law. The fraction of liquid-like zones Fliquid and elastic matrix Felastic can be deduced from the σonset and σend of each stress relaxation test by
where Fliquid+Felastic=1, σelastic=Eε and . The σliquid can be estimated based on viscosity relation10,52 and E, ε and are temperature-independent constant values (see Supplementary Methods and Supplementary Fig. 5).
Figure 4b presents the change of Fliquid values with temperature, which shows the similar changing tendency of the distribution coefficient βKWW. Two transition temperatures TT1 and Tg are also observed and separate the evolution of Fliquid into three regions. From room temperature to TT1, the Fliquid values remain almost constant, indicating that in this region the activation process is reversible and no notable elongation is observed after each stress relaxation test (see Supplementary Fig. 6). When temperature is higher than TT1, more flow units are transformed into liquid state and the MG shows viscoelastic behavior. The Fliquid growth rate change or the flow phenomenon becomes noticeable around Tβ, related with the previous assumption of the isolated-to-cooperative deformations transition. The MG transfers into macroscopic flowing state at Tg with fully activated of liquid-like zones and Fliquid approaches 1. The changing tendency of Fliquid, showing a nonlinear relationship with temperature, further demonstrates a crucial role of the activation of the liquid-like zones in flow and properties of MGs.
The equation based on topological fluctuations in the bonding network53 can also fit our results. The simulation details can be found in Supplementary Methods. The atoms with local volume strain exceeding 0.11 are defined as liquid-like in the theory of ref. 53. Obviously, our definition from time distribution is broader than the atomic level volume strain distribution. With a latent flow unit fraction of 8%, the theoretical curve fits the evolutionary trend of liquid-like zones well at the range from room temperature to Tβ, as shown by black line in Fig. 4b. This indicates that the Fliquid obtained from a simple mechanical model, can also reflect intrinsic microscopic structural properties of MG. Furthermore, the Fliquid reaches a classical percolation threshold between 0.2–0.3 (refs 53, 54) at Tβ, suggesting that the activated liquid-like zones are no longer isolated above Tβ. The appearance of these extensive and cooperative flowing spots trigger the brittle-to-ductile mechanical transition.
Based on the above results, intrinsic correlations between evolution of flow units, deformation and relaxation maps can be established and can be interpreted by the energy landscape theory. Below TT1, only reversible flow units can be activated, as illustrated by the high mobility atoms (red balls) embedded in the elastic shell (blue balls) in Fig. 5. The flow units with high concentration region of free volume may be related with high density inhomogeneity55. The fraction of the flow units is ~8% for this specific MG system. Such isolated reversible motions within matrix can be regarded as hopping events across inherent structures and contribute to the β-relaxation26,56. These few easy flowing spots do not change the brittle nature of MGs. When the temperature continuous to exceed, adjacent weak-bonded regions around flow unit regions are also gradually transformed into liquid state, as the applied strain lowers the energy barrier between adjacent megabasins in energy landscape51,57,58 and increase the fraction of the flow units as illustrated in Fig. 5. At this stage, the liquid-like zones lead to localized plastic events and lower the yielding stress, but the MG shows no macroscopic plasticity.
Around Tβ, a connectivity percolation of liquid-like zones occurs. The percolation of the flow units forms the multiply shear bands59, leading to ductile deformation21,25. For most MGs, Tβ is located at about 0.8–0.9 Tg (refs 35, 60), which is close to homogeneous deformation boundary at 0.9 Tg in deformation map14. Higher strain rate should correspond to a faster boundary frequency and a higher fully activated β-relaxation temperature Tβ′, and vice versa. Another arresting phenomenon around Tβ is the sub-Tg endothermic peak or termed shadow glass transition in glasses61,62. This endothermic peak is thought to be related with β-relaxation but the intuitive physical picture is still not clear61. From our scenario, cooperative translational movements35,63 take place above Tβ and can be regarded as a kind of confined GLT, leading to an endothermic peak. Such hidden flowing movements need external energy to facilitate. This also explains the reason that the endothermic peak is always slightly higher than the Tβ measured by dynamical method, because additional mechanical disturbance is needed to motivate the cooperative flow events. Meanwhile, the slower growth rate of Fliquid also originates from the cooperative movements, as the activation energy of cooperative α-relaxation is three times larger than isolated β-process (see Supplementary Fig. 7).
When temperature approaches Tg, the sample behaves macroscopic flowing under our observing time. A heat capacity jump of about 3R/2 related to translational freedom addition64 can be observed and a critical transition from broken-ergodic to ergodic on energy landscape occurs. The sample transforms into liquid-like above Tg, the viscosity change becomes the dominating factor for the deformation behaviours. As viscosity rapidly decreases and approaches Tα, the sample transitions from non-Newtonian to Newtonian flow as marked in the deformation map of Fig. 5.
In summary, a panorama picture revealing the evolution of localized flow from low temperature glassy state to supercooled liquid state is offered to understand the deformation and relaxation maps and the GLT. Unstable flow units are demonstrated to persist in glassy state and their reversible activation contributes to the viscoelasticity. These liquid-like zones show a nonlinear increase with temperature and their properties play a crucial role in determining diverse flow phenomena in glasses. A connective percolation state is achieved above Tβ and leads to the ductile deformation and a cooperative GLT process. Our results might shed light on the mechanism of flowing phenomena in glassy state and provide a practical guideline in controlling mechanical behaviour in MGs.
Ribbons and bulk La60Ni15Al25 MG were prepared by melt-spinning and Cu-mold casting methods in argon atmosphere, respectively. The thickness of the ribbons are about 40 μm and the diameter of the MG rods are 2 mm. The glass transition temperature of the La60Ni15Al25 samples is located at 461 K. The tensile mechanical tests and stress relaxation measurements were performed with ribbons, while DMS results are obtained with rods. The conclusion obtained in the tension and the evolution trends is the same for both as thin ribbons and bulk sample form.
Structural and dynamic characterization
The glassy nature of the samples was ascertained by X-ray diffraction (a MAC Mo3 XHF diffractometer with Cu Kα radiation). The differential scanning calometer (DSC) measurements were conducted by using a Perkin-Elmer DSC-7 at a heating rate of 20 K min−1 (see Supplementary Fig. 8). Dynamical mechanical behaviour was measured using temperature ramp mode and frequency sweep mode on a TA DMA Q800. Under temperature ramp mode, the sample was measured at a heating rate of 3 K min−1 and only 1 Hz results were shown here. Meanwhile, frequencies covering 0.01–16 Hz were used with a temperature step of 10 K in frequency sweep mode. Uniaxial tensile tests of MG ribbons were carried out over a temperature range (303–483 K) under 1 × 10−4 s−1 in the same dynamical mechanical analyzer machine.
Stress relaxation measurements
Measurements were performed on TA DMA Q800 and ribbon samples were mounted in tension clamp. A constant strain of 0.5% was rapidly applied on the sample and held for 30 min for each test to record the stress decay. The stress relaxation measurements for the same sample were tested from 303 to 495 K (>Tg) with a step of 3 K, and a total of 65 stress relaxation tests were conducted. The interval between the adjacent two tests is 5 min, including a 3-min isothermal holding time at the next measured temperature to avoid thermal fluctuations. Besides, to assure the accuracy of the strain value, the length of sample was measured before and after each stress relaxation test.
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We would like to thank Professor K.L. Ngai and Professor A. Lemaitre for fruitful discussions. Experimental assistance and insightful discussions with D.Q. Zhao, D.W. Ding, M.X. Pan and P. Wen are also appreciated. This work was supported by the NSF of China (51271195) and MOST 973 Programme (No. 2015CB856800).
The authors declare no competing financial interests.
Supplementary Figures 1-8, Supplementary Table 1, Supplementary Methods and Supplementary References (PDF 357 kb)
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