# Structural Relaxation, Rejuvenation and Plasticity of Metallic Glasses: Microscopic Details from Anelastic Relaxation Spectra

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{6}°C/s or higher, which limited at least one dimension to $<$10

^{−4}m. A major breakthrough was achieved when new alloy compositions were discovered that required far lower cooling rates, resulting in bulk metallic glasses with dimensions that exceeded 10

^{−2}m [3,4,5]. As a result, new, especially structural, applications became possible [6,7]. Additional experimental techniques became accessible, e.g., calorimetry in the supercooled liquid region and macroscopic mechanical testing, contributing to enhanced scientific understanding.

- A summary of Argon’s analysis of the mechanics and thermal activation of STZs.
- Our approach, which consists of (i) quasi-static anelastic recovery experiments that span more than ten orders of magnitude of time and (ii) computational determination of relaxation-time spectra by direct spectrum analysis (DSA).
- Relaxation-time spectra were determined numerically from the strain/time data. These provided valuable information on STZ size and property distribution, revealing an atomically-quantized hierarchy of STZs.
- Analysis of anelastic relaxation in the nonlinear regime, related to that of Argon and Shi’s creep experiments [19], provided an independent determination of the STZ transformation strain. Similar to the dislocation core in crystalline solids, this strain is far larger than the macroscopic yield strain.
- STZ spectra were computed from published dynamic-mechanical data. The results provide further, consistent, confirmation of the prior results and their analysis.
- Simple calculations show that stretched exponent fits, commonly used to fit non-exponential relaxation, are of limited utility. In particular, the time constant is ambiguous, and its apparent activation energy is not expected to reflect a specific physical process.
- The systematic error is evaluated for spectrum determination based on measurements conducted at discrete temperature increments and the assumption that the evolution at each temperature is dominated by a single activation free energy.
- Characterization of the details of structural relaxation and induced rejuvenation through their effect on STZ properties shows that these processes cannot be described with the evolution of a single variable.
- Anelastic relaxation spectra were obtained for La-based metallic glasses, some of which exhibit a distinct high-frequency/low-temperature (β) relaxation. Among the results, the following was found: contrary to suggestions by many authors, the α and β relaxation correspond to the same mechanism. Both are reversible when the corresponding STZs occupy a small volume fraction. The results also suggest that different elements are involved in slow vs. fast STZs, corresponding to the α and β relaxation, respectively. Simulations of dynamic-mechanical behavior for experimentally obtained STZ spectra further support the notion that the α and β relaxation correspond to the same mechanism. That curves obtained at different temperature can be shifted into a single master curve cannot be seen as proof of a single activation energy.
- By comparing metallic glasses that exhibit different degrees of plasticity at similar composition, plasticity is explained in terms of the volume fraction occupied by kinetically active potential STZs.

## 2. Theory of Thermally-Activated Shear Transformation

_{s}to account for a spectrum of STZ types, indexed initially with m, each contributing additively to the total shear strain rate [18]:

_{m}are resolved by STZ type, m, and obtained from experiment, as shown below. It is noted that in the notation used, overlapping potential STZs are counted multiple times. Equation (1) is valid as long as only a small fraction of them undergoes shear transformations.

## 3. Experiments and Spectrum Determination

^{−3}–200 s, using a nanoindenter at fixed force to monitor the displacement of a cantilever (Figure 1a) as a function of time provided the strain evolution. For long time constants, up to ~6 × 10

^{7}s, instrumented measurements pose stability challenges. Instead, therefore, 20–40 μm thick ribbon samples were constrained for 2 × 10

^{6}s around a mandrel at a fixed radius of curvature; subsequently, their radii of curvature were monitored as a function of time in a stress-free state. Except for the early study of Al

_{86.8}Ni

_{3.7}Y

_{9.5}[18], the sample curvature determination was performed using an automated fit to its image. Based on the confirmed linearity of the relaxation process, the strain and stress at any distance from the neutral midplane were calculated as a function of time. The strain at the surface is used in all reported data.

_{1}and N

_{2}, were used as an approximation of a continuum spectrum, fewer than the number of data points in order to avoid overdetermination. Extensive tests were conducted with simulated data corresponding to assumed input spectra, with added noise, verifying that these spectra can be recovered by the fitting procedure.

## 4. An Atomically Quantized Hierarchy of STZs [18]

_{86.8}Ni

_{3.7}Y

_{9.5}metallic glass ribbons. For the mandrel experiments, the anelastic strain, ${\epsilon}_{anel}\left(t\right)$, was monitored at room temperature after constraint removal as a function of time for ~8 × 10

^{7}s, as it recovered its original shape. It is shown in Figure 2, normalized by ${\epsilon}_{el}^{0}$ for several mandrel radii used to constrain the samples. All curves coincide, indicating that the anelastic processes are in the linear regime. This directly supports the assumption that the strain profile across the sample thickness is linear. It also implies that no significant yield had taken place. Visual inspection reveals that multiple time constants govern the anelastic recovery. Figure 3 shows representative ${\epsilon}_{an}\left(t\right)$/${\epsilon}_{el}^{0}$ curves, along with corresponding computed spectra, $f\left(\tau \right)$, for the cantilever (Figure 3a) and mandrel (Figure 3b) experiments. Fits obtained with different numbers of fitting points, N

_{1}and N

_{2}, demonstrate the consistency of spectrum computation.

_{0}), representing elastic behavior, in series with a series of Voigt units, each consisting of a spring (effective modulus ${E}_{m}^{\prime}$) and dashpot (effective viscosity ${\eta}_{m}^{\prime}$) in parallel (Figure 4, top). Under zero or fixed stress, each Voigt unit relaxes exponentially with a time constant

_{0}(see Figure 4) is the sample’s high-frequency Young’s modulus. By definition,

_{s}is the net shear stress on the dashpot in Unit m. Straightforward algebra [18] yields a simple expression for c

_{m}:

_{m}, is obtained and displayed, normalized by the atomic volume of Al, in Figure 5e. The error in Ω

_{m}is small since it appears in the exponent in Equation (1). Somewhat fortuitously, the slope of this plot is within < 1% of 1. This one-atom increment in Ω

_{m}leads to the conclusion that the peaks in the spectrum represent an atomically-quantized hierarchy of STZs — the spectrum peaks correspond to STZs that consist of n = 14,…, 21 atoms. The dominance of a single element, Al, likely facilitates the resolution of this hierarchy. The activation free energies corresponding to this STZ hierarchy, ${\u2206F}_{n}$, range from 0.85 to 1.26 eV (Equation (2), Figure 5f).

^{6}s and anelastic recovery for 1.1 × 10

^{8}s) further confirmed the hierarchy, showing the signature of STZs consisting of 22 atoms. The c

_{n}obtained allowed for modeling the size-density distribution. It was assumed that a cluster needed to contain a sufficient amount of free volume, >$v$

^{*}, in order for it to be capable of a shear transformation. Using Poisson statistics for the free-volume distribution, the best fit to the data was obtained when this threshold, $v$

^{*}, varied only slightly with size, as n

^{0.22}[27]. This weak dependence is expected if free volume is shared dynamically within the STZ on a time scale required for shear transformation.

## 5. The Transformation Strain [28]

_{m}. They cannot be determined independently from the data above because only the product ${({{\gamma}_{0}^{T})}^{2}\Omega}_{m}$ appears in the (linearized) sinh term in Equation (1) and in Equation (2), recalling that the third term in Equation (2) is negligible. Therefore, ${\gamma}_{0}^{T}$ had been estimated from experiments conducted in colloidal glass [16] and from molecular dynamics simulations [29,30], ${\gamma}_{0}^{T}\approx 0.2$, which affects the resulting values of Ω

_{m}. An independent determination of ${\gamma}_{0}^{T}$ requires experiments in the nonlinear regime of the sinh term of Equation (1). Such analysis was carried out by Argon and Shi [19] for nonlinear creep data. To complement the linear results presented above, we conducted nonlinear anelastic relaxation experiments on Al

_{86.8}Ni

_{3.7}Y

_{9.5}metallic glass ribbons by using smaller mandrel diameters, 0.35 to 0.49 cm, resulting in bending strain values up to 0.0155, compared with 0.00303 for the prior experiments in the linear regime. For the resulting stress, the sinh term in Equation (1) is nonlinear. The volume fraction occupied by STZs is still small, ≤7.2%, so that STZ interactions are negligible and the STZs can be considered isolated. Yield was ruled out by verifying the absence of change in the sample geometry following brief constraint. The normalized, apparent, anelastic strain, determined from the stress-free curvature at t = 4 × 10

^{6}s after the release of the constraint, is shown as a function of the elastic constraining strain in Figure 6, for both the nonlinear and earlier linear data. At such a point in time, the fast STZs have relaxed, and the largest STZs activated, with n = 21, dominate the relaxation behavior.

_{m}, this small error is due to the fact that ${\gamma}_{0}^{T}$ appears in the exponent in Equation (2). The value obtained, ${\gamma}_{0}^{T}$ = 0.17, is reasonably close to the value assumed in the analysis in Ref. [18], 0.2, as summarized above. It is much greater than the universal, low-temperature, macroscopic yield strain observed in metallic glasses, 0.036 [31]. An important parallel to crystalline solids helps illustrate this difference in magnitude: the strain in a dislocation core is of the order of 1, yet the yield strain in metals is below 0.01. It is worth noting that in some studies, equating the transformation strain to the yield strain resulted in unphysically large STZ sizes being backed out from the data [32,33].

## 6. Dynamic-Mechanical Analysis [34]

^{2}, the coefficient of determination, began to increase. This tolerance value was used as the best-fit criterion when analyzing the experimental data.

_{46.8}Ti

_{13.8}Cu

_{12.5}Ni

_{10}Be

_{27.5}, were used in the analysis. The fits and corresponding spectra are shown in Figure 7. The time constants obtained from each peak are shown in Arrhenius plots in Figure 8 as a function of temperature. The goal was to obtain simultaneous fit lines for all STZ sizes, based on an atomically quantized hierarchy of STZs. For each trial STZ size n, the time constant was expressed as a function of temperature based on the theory reviewed above:

_{g}, and the shear modulus varies significantly with temperature in the latter range, its approximate linear temperature dependence above T

_{g}was included in the fits [36,37,38,39]. They were carried out simultaneously for all values of T and n. The main challenge was determining which set of data points corresponded to the same STZ size, n, within a multi-n simultaneous fit. Several plausible groupings were attempted each below and above T

_{g}. The only combination of such sets that yields continuity and the same n values across T

_{g}is that shown in Figure 8a. The resulting n values range from 25 to 33, with corresponding activation free energies of 1.75–2.3 eV. These results are consistent with those of Ref. [18], further confirming them and the model used. These higher values of n, compared with 14–22 at room temperature in Ref. [18], are expected since the spectra increase monotonically and larger STZs become active with increasing temperature.

## 7. The Stretched Exponent [45]

^{0.5}), were fitted with an unshifted stretched exponent (Figure 10). The fitting parameters depend on the range of t values and the manner in which the points are spaced on the t axis. However, as shown in Figure 10, similar results are obtained for linear (a) and logarithmic (b) spacing, where the former gives greater weight to long time values. Both yield good fits with similar fitting parameters. Remarkably, the ${\tau}_{s}$ values obtained are higher by >30% than the value of 30 used to simulate the data points. This is a result of the fact that, unlike for a simple exponent, the relative rate of change of the stretched exponent is not constant in time. The common assumption that the temperature dependence of ${\tau}_{s}$, however obtained, can yield an activation energy [53,57] is therefore not supported. For these reasons, the presently reviewed work is based on spectrum determination from the data without prior assumptions.

## 8. Systematic Error in Spectrum Determination by Temperature Stepping [61]

_{i}, is dominated by a single activation free energy given by $\u2206{F}_{i}=-k{\left.\frac{\partial ln\dot{\gamma}}{\partial \left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\right)}\right|}_{{T}_{i}}$. The assumption implicit in this approximation is that at each step, processes with lower activation free energy have equilibrated, while those with higher activation free energy are frozen. Argon and Kuo [62] proposed this method to evaluate the activation free energy spectrum for torsional creep experiments. One aspect of the spectrum they obtained was a drop at the highest value of $\u2206{F}_{i}$. In contrast, Refs. [18,27] exhibit a monotonically increasing spectrum, which is also consistent with the free-volume model [27]. In this context, it is instructive to assess the error introduced by the approximation of a dominant activation free energy at each temperature step. For this purpose, we assumed a simple, monotonic, spectrum of activation free energies, qualitatively similar to that in Figure 5f. By simulating the process of anelastic relaxation at stepwise increasing temperatures, we obtained a simulated, apparent spectrum, based on the approximation of Ref. [62], which exhibits a decrease at the highest activation free energy (Figure 11). Comparison with the assumed input spectrum illustrates that the observed decrease is an artifact of the temperature-stepping method: processes with high activation energy are not completely frozen at lower temperatures, thus reducing their apparent contribution. Their participation at lower temperature also explains the shift to lower activation energies, seen in Figure 11.

## 9. Characterization of Structural Evolution [63]

_{g}[65] or by plastic deformation, including shot peening [66]. In addition, cyclic elastic loading [67], constrained loading [68] and irradiation [69,70] have led to rejuvenation. Cycling between room and cryogenic temperature has also been reported to lead to rejuvenation [71], as determined from measurements of stored enthalpy and yield. The authors proposed a rejuvenation mechanism due to heterogeneity of the thermal expansion coefficient, leading to microscopic stresses and local yielding. This novel result holds promise for practical applications, being non-destructive, controllable and isotropic [72,73,74]. It is noted, however, that, the authors have recently reported that the effect of cryogenic rejuvenation decays over time, likening the rejuvenation process to anelastic strain accumulation [75].

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}, were investigated [63]. Figure 12 shows the anelastic strain as a function of time after constraining and releasing the samples, following the same protocol as above. Curves were obtained for samples that were allowed to age and structurally relax for several durations, 1.9 × 10

^{6}to 2.9 × 10

^{7}s, prior to constraining them. In one intermediate case, samples were also cycled between room and liquid-nitrogen temperature following the aging step. Not surprisingly, the amount of anelastic strain developed during the constraining period decreased with prior room-temperature aging. For the as-prepared La

_{70}Cu

_{15}Al

_{15}alloy, the anelastic strain was higher than the elastic strain at the end of the constraining period. That the stress-free strain was entirely anelastic was verified by annealing above room temperature, which resulted in a complete recovery of the original sample shape before it was constrained (See Figure 12).

_{86.8}Ni

_{3.7}Y

_{9.5}(Figure 3), they contain distinct peaks. Room-temperature aging leads to a decrease in the intensities of the peaks, especially that of the peak with the longest time constant, and their shift to longer time constants. Interestingly, cryogenic cycling reduces the corresponding time constants (Figure 14), restoring them to pre-aging values. However, the peak intensities are not affected by cryogenic cycling: the areas under resolvable peaks or peak sets for the cycled samples fit on the same curve, as a function of aging time, as those for the aged samples that were not cycled (Figure 15). Based on the discussion in Section 4, we conclude that structural relaxation associated with aging leads to a reduction in the number of potential STZs. The increase in time constants is likely due to an increase in the modulus of the glass, which increases the activation free energy for shear transformations (see present Equation (2) and Figure 7 in Ref. [76]). Cryogenic rejuvenation likely restores the elastic modulus. However, it does not lead to a recovery of the number density of potential STZs. The impact of structural relaxation on the number density of potential STZs, as seen in the amount of normalized anelastic strain, is mainly on those consisting of a larger number of atoms, which are the slowest. This is seen qualitatively in Ref. [63], and in further detail for La

_{55}Ni

_{20}Al

_{25}in Figure 16 [76] which shows the evolution of each c

_{m}with aging time. The decrease of c

_{m}with aging is likely a result of a decrease in free volume [12,27], as the density is known to increase with structural relaxation.

- (1)
- Although cryogenic rejuvenation does not restore the c
_{m}, plasticity is improved by this process because of the increased fraction of potential STZ with a sufficiently short time constant to participate in deformation. - (2)
- A comparison of the time scale for structural relaxation, > 10
^{6}s, with the shorter times for anelastic relaxation indicates that the mechanisms underlying the two processes cannot be assumed to be the same. The driving force for the former is thermodynamic, whereas for the latter it is mechanical. - (3)
- While a measurement of a single variable, e.g., stored enthalpy or plasticity, may give the impression that the cryogenic cycling process leads to a reversal of structural relaxation due to aging, these results clearly show that the details are more nuanced. Generally, structural relaxation and rejuvenation cannot be described with a single variable.

## 10. The Mechanism of the β Relaxation [76]

_{55}Ni

_{20}Al

_{25}, a metallic glass with significant β relaxation [76]. A plot of the STZ volume, ${\Omega}_{m}$, as a function of m, Figure 17, reveals two regimes. The atomic volume obtained from the slope is 0.161 × 10

^{−28}m

^{3}for small and fast STZs, and 0.236 × 10

^{−28}m

^{3}for large and slow STZs. As before, the straight-line fits are excellent. The former value is close to that of an Al atom, 0.166 × 10

^{−28}m

^{3}, whereas the latter is within about 12% of the mean atomic volume of the alloy. Similar results, with two slope regimes, were observed in La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}[86]. While it may be speculative to take these slope values literally, they suggest that different elements play a role in fast vs. slow STZs.

## 11. STZ Properties and Plasticity [86]

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}. Despite their similar compositions, the latter exhibits an intense β relaxation, whereas the former only exhibits a shoulder in the loss modulus, E″(ω) [90]. The same methodology was used as in the cases above to determine the STZ spectra, followed by comparison with the tensile behavior. The normalized strain evolution (Figure 18) and the corresponding spectra (Figure 19) show that La

_{70}Ni

_{15}Al

_{15}contains a higher fraction of fast STZs, which correspond to β relaxations, than La

_{70}Cu

_{15}Al

_{15}, whereas the opposite is true for slow STZs, which correspond to α relaxations. This agrees qualitatively with E″(ω) data [90]. It should be noted that loss-modulus data are typically normalized by the α peak, so the β intensity observed in E″(ω) is relative. In contrast, the present spectrum peak areas provide absolute information on the volume fraction occupied by STZs. One therefore finds that when E″(ω) is normalized, a lower α intensity for La

_{70}Ni

_{15}Al

_{15}further enhances its apparent β intensity, as compared with La

_{70}Cu

_{15}Al

_{15}.

^{−6}s

^{−1}to 10

^{−4}s

^{−1}(Figure 20). At the two lower rates, La

_{70}Cu

_{15}Al

_{15}exhibited far greater plasticity, with up to > 17% engineering strain. It is noted that this example shows a negative correlation of β intensity with plasticity, the opposite of that proposed in Ref. [83]. While 17% engineering strain is beyond the linear regime of non-interacting STZs, the STZ spectra can be used to qualitatively explain the difference in mechanical behavior between the alloys.

_{70}Cu

_{15}Al

_{15}alloy contains an overall higher volume fraction occupied by potential STZs than La

_{70}Ni

_{15}Al

_{15}does, the former reaches higher strains before it fails.

## 12. Additional Properties

_{0}is the temperature-dependent maximum size of such potential STZs, has to be below the percolation threshold. Otherwise, the matrix is rigid. The requirement for long-range diffusion is less strict: 1 − $\mathrm{e}\mathrm{x}\mathrm{p}\left(-\sum _{n=1}^{{n}_{0}}{c}_{n}\right)$, the volume fraction occupied by active potential STZs, has to exceed the percolation threshold. As noted above, overlapping potential STZs are counted multiple times, so $\sum _{n=1}^{{n}_{0}}{c}_{n}>1$ is possible [27].

## 13. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Buckel, W.; Hilsch, R. Einfluß der Kondensation bei tiefen Temperaturen auf den elektrischen Widerstand und die Supraleitung für verschiedene Metalle. Z. Phys.
**1954**, 138, 109–120. [Google Scholar] [CrossRef] - Klement, W.; Willens, R.H.; Duwez, P.O. Non-crystalline structure in solidified gold–silicon alloys. Nature
**1960**, 187, 869–870. [Google Scholar] [CrossRef] - Drehman, A.J.; Greer, A.L. Kinetics of crystal nucleation and growth in Pd
_{40}Ni_{40}P_{20}glass. Acta Metall.**1984**, 32, 323–332. [Google Scholar] [CrossRef] - Inoue, A.; Zhang, T.; Masumoto, T. Production of amorphous cylinder and sheet of La
_{55}Al_{25}Ni_{20}alloy by a metallic mold casting method. Mater. Trans. JIM**1990**, 31, 425–4288. [Google Scholar] [CrossRef] - Peker, A.; Johnson, W.L. A highly processable metallic glass: Zr
_{41.2}Ti_{13.8}Cu_{12.5}Ni_{10.0}Be_{22.5}. Appl. Phys. Lett.**1993**, 63, 2342–2344. [Google Scholar] [CrossRef] - Ashby, M.F.; Greer, A.L. Metallic glasses as structural materials. Scr. Mater.
**2006**, 54, 321–326. [Google Scholar] [CrossRef] - Schroers, J.; Nguyen, T.; O’Keeffe, S.; Desai, A. Thermoplastic forming of bulk metallic glass—Applications for MEMS and microstructure fabrication. Mater. Sci. Eng. A
**2007**, 449, 898–902. [Google Scholar] [CrossRef] - Hull, D.; Bacon, D.J. Introduction to Dislocations, 5th ed.; Elsevier: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Bragg, W.L.; Nye, J.F. A dynamical model of a crystal structure. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1947**, 190, 474–481. [Google Scholar] - Hirsch, P.; Cockayne, D.; Spence, J.; Whelan, M. 50 Years of TEM of dislocations: Past, present and future. Philos. Mag.
**2006**, 86, 4519–4528. [Google Scholar] [CrossRef] - Orowan, E. Creep in metallic and nonmetallic materials. In Proceedings of the First U.S. National Congress of Applied Mechanics: Held at Illinois Institute of Technology, Chicago, IL, USA, 11–16 June 1951; pp. 453–472. [Google Scholar]
- Argon, A.S. Plastic deformation in metallic glasses. Acta Metall.
**1979**, 27, 47–58. [Google Scholar] [CrossRef] - Harmon, J.S.; Demetriou, M.D.; Johnson, W.L.; Samwer, K. Anelastic to plastic transition in metallic glass-forming liquids. Phys. Rev. Lett.
**2007**, 99, 135502. [Google Scholar] [CrossRef] [PubMed] - Demetriou, M.D.; Johnson, W.L.; Samwer, K. Coarse-grained description of localized inelastic deformation in amorphous metals. Appl. Phys. Lett.
**2009**, 94, 191905. [Google Scholar] [CrossRef] - Argon, A.S.; Kuo, H.Y. Plastic flow in a disordered bubble raft (an analog of a metallic glass). Mater. Sci. Eng.
**1979**, 39, 101–109. [Google Scholar] [CrossRef] - Schall, P.; Weitz, D.A.; Spaepen, F. Structural rearrangements that govern flow in colloidal glasses. Science
**2007**, 318, 1895–1899. [Google Scholar] [CrossRef] [PubMed] - Falk, M.L.; Langer, J.S. Dynamics of viscoplastic deformation in amorphous solids. Phys. Rev. E
**1998**, 57, 7192–7205. [Google Scholar] [CrossRef] - Ju, J.D.; Jang, D.; Nwankpa, A.; Atzmon, M. An atomically quantized hierarchy of shear transformation zones in a metallic glass. J. Appl. Phys.
**2011**, 109, 053522. [Google Scholar] [CrossRef] - Argon, A.S.; Shi, L.T. Development of visco-plastic deformation in metallic glasses. Acta Metall.
**1983**, 31, 499–507. [Google Scholar] [CrossRef] - Liu, S.T.; Wang, Z.; Peng, H.L.; Yu, H.B.; Wang, W.H. The activation energy and volume of flow units of metallic glasses. Scr. Mater.
**2012**, 67, 9–12. [Google Scholar] [CrossRef] - Kato, H.; Igarashi, H.; Inoue, A. Another clue to understand the yield phenomenon at the glassy state in Zr
_{55}Al_{10}Ni5Cu_{30}metallic glass. Mater. Lett.**2008**, 62, 1592–1594. [Google Scholar] - Cost, J.R. Nonlinear regression least-squares method for determining relaxation time spectra for processes with first-order kinetics. J. Appl. Phys.
**1983**, 54, 2137–2146. [Google Scholar] [CrossRef] - Available online: http://s-provencher.com/contin.shtml (accessed on 14 November 2023).
- Provencher, S.W. A constrained regularization method for inverting data represented by linear algebraic or integral equations. Comput. Phys. Commun.
**1982**, 27, 213–227. [Google Scholar] [CrossRef] - Provencher, S.W. CONTIN: A general purpose constrained regularization program for inverting noisy linear algebraic and integral equations. Comput. Phys. Commun.
**1982**, 27, 229–242. [Google Scholar] [CrossRef] - Lakes, R.S. Viscoelastic Solids; CRC Press: Boca Baton, FL, USA, 1999. [Google Scholar]
- Atzmon, M.; Ju, J.D. Microscopic description of flow defects and relaxation in metallic glasses. Phys. Rev. E
**2014**, 90, 042313. [Google Scholar] [CrossRef] [PubMed] - Lei, T.J.; Atzmon, M. Activation volume details from nonlinear anelastic deformation of a metallic glass. J. Appl. Phys.
**2019**, 126, 185104. [Google Scholar] [CrossRef] - Delogu, F. Identification and characterization of potential shear transformation zones in metallic glasses F. Phys. Rev. Lett.
**2008**, 100, 255901. [Google Scholar] [CrossRef] [PubMed] - Neudecker, M.; Mayr, S.G. Dynamics of shear localization and stress relaxation in amorphous Cu
_{50}Ti_{50}. Acta Mater.**2009**, 57, 1437–1441. [Google Scholar] [CrossRef] - Johnson, W.L.; Samwer, K. A universal criterion for plastic yielding of metallic glasses with a (T/T
_{g})^{2/3}temperature dependence. Phys. Rev. Lett.**2005**, 95, 195501. [Google Scholar] [CrossRef] - Pan, D.; Inoue, A.; Sakurai, T.; Chen, M.W. Experimental characterization of shear transformation zones for plastic flow of bulk metallic glasses. Proc. Nat. Acad. Sci. USA
**2008**, 105, 14769. [Google Scholar] [CrossRef] - Krausser, J.; Samwer, K.; Zaccone, A. Interatomic repulsion softness directly controls the fragility of supercooled metallic melts. Proc. Natl. Acad. Sci. USA
**2015**, 112, 13762. [Google Scholar] [CrossRef] - Ju, J.D.; Atzmon, M. A comprehensive atomistic analysis of the experimental dynamic-mechanical response of a metallic glass. Acta Mater.
**2014**, 74, 183–188. [Google Scholar] [CrossRef] - Pelletier, J.M. Dynamic mechanical properties in a Zr
_{46.8}Ti_{13.8}Cu_{12.5}Ni_{10}Be_{27.5}bulk metallic glass. J. Alloys Compd.**2005**, 393, 223–230. [Google Scholar] [CrossRef] - Keryvin, V.; Vaillant, M.-L.; Rouxel, T.; Huger, M.; Gloriant, T.; Kawamura, Y. Thermal stability and crystallisation of a Zr
_{55}Cu_{30}Al_{10}Ni_{5}bulk metallic glass studied by in situ ultrasonic echography. Intermetallics**2002**, 10, 1289–1296. [Google Scholar] [CrossRef] - Rouxel, T. Elastic properties and short-to medium-range order in glasses. J. Am. Ceram. Soc.
**2007**, 90, 3019–3039. [Google Scholar] [CrossRef] - Wang, J.Q.; Wang, W.H.; Bai, H.Y. Extended elastic model for flow in metallic glasses. J. Non-Cryst. Solids
**2011**, 357, 223–226. [Google Scholar] [CrossRef] - Wang, W.H. The elastic properties, elastic models and elastic perspectives of metallic glasses. Prog. Mater. Sci.
**2012**, 57, 487–656. [Google Scholar] [CrossRef] - Ngai, K.L.; Plazek, D.J.; Rendell, R.W. Some examples of possible descriptions of dynamic properties of polymers by means of the coupling model. Rheol. Acta
**1997**, 36, 307–319. [Google Scholar] [CrossRef] - Salmén, L. Viscoelastic properties of in situ lignin under water-saturated conditions. J. Mater. Sci.
**1984**, 19, 3090–3096. [Google Scholar] [CrossRef] - Qiao, J.C.; Pelletier, J.M. Mechanical relaxation in a Zr-based bulk metallic glass: Analysis based on physical models. J. Appl. Phys.
**2012**, 112, 033518. [Google Scholar] [CrossRef] - Jeong, H.T.; Fleury, E.; Kim, W.T.; Kim, D.H.; Hono, K. Study on the mechanical relaxations of a Zr
_{36}Ti_{24}Be_{40}amorphous alloy by time–temperature superposition principle. J. Phys. Soc. Jpn.**2004**, 11, 3192. [Google Scholar] [CrossRef] - Ju, J.D.; Atzmon, M. Atomistic interpretation of the dynamic response of glasses. MRS Comm.
**2014**, 4, 63–66. [Google Scholar] [CrossRef] - Atzmon, M. The pitfalls of empirical fitting of glass relaxation data with stretched exponents. J. Appl. Phys.
**2018**, 123, 065103. [Google Scholar] [CrossRef] - Kohlrausch, V.R. Theory of the electric residue in the Leyden jar. Ann. Phys. Chem. (Poggendorff)
**1854**, 91, 179–214. [Google Scholar] [CrossRef] - Williams, G.; Watts, D.C. Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans. Faraday Soc.
**1970**, 66, 80–85. [Google Scholar] [CrossRef] - Williams, G.; Watts, D.; Dev, S.B.; North, A.M. Further considerations of non symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans. Faraday Soc.
**1971**, 67, 1323–1335. [Google Scholar] [CrossRef] - Phillips, J.C. Stretched exponential relaxation in molecular and electronic glasses. Rep. Prog. Phys.
**1996**, 59, 1133. [Google Scholar] [CrossRef] - MacDonald, J.R. Linear relaxation: Distributions, thermal activation, structure, and ambiguity. J. Appl. Phys.
**1987**, 62, R51–R62. [Google Scholar] [CrossRef] - Svare, I.; Martin, S.W.; Borsa, F. Stretched exponentials with T-dependent exponents from fixed distributions of energy barriers for relaxation times in fast-ion conductors. Phys. Rev. B
**2000**, 61, 228–233. [Google Scholar] [CrossRef] - Hodge, I.M. Enthalpy relaxation and recovery in amorphous materials. J. Non-Cryst. Solids
**1994**, 169, 211–266, and references therein. [Google Scholar] [CrossRef] - Qiao, J.C.; Pelletier, J.M. Enthalpy relaxation in Cu
_{46}Zr_{45}Al_{7}Y_{2}and Zr_{55}Cu_{30}Ni_{5}Al_{10}bulk metallic glasses by differential scanning calorimetry (DSC). Intermetallics**2011**, 19, 9–18. [Google Scholar] [CrossRef] - Raghavan, R.; Murali, P.; Ramamurty, U. Influence of cooling rate on the enthalpy relaxation and fragility of a metallic glass. Metall. Mater. Trans. A
**2008**, 39, 1573–1577. [Google Scholar] [CrossRef] - Kawai, K.; Hagiwara, T.; Takai, R.; Suzuki, T. Comparative investigation by two analytical approaches of enthalpy relaxation for glassy glucose, sucrose, maltose, and trehalose. Pharm. Res.
**2005**, 22, 490–495, and references therein. [Google Scholar] [CrossRef] [PubMed] - Hu, L.; Yue, Y. Secondary relaxation in metallic glass formers: Its correlation with the genuine Johari− Goldstein relaxation. J. Phys. Chem. C.
**2009**, 113, 15001–15006. [Google Scholar] [CrossRef] - Qiao, J.C.; Casalini, R.; Pelletier, J.M. Main (α) relaxation and excess wing in Zr50Cu40Al10 bulk metallic glass investigated by mechanical spectroscopy. J. Non-Cryst. Solids
**2015**, 407, 106–109. [Google Scholar] [CrossRef] - Zhao, Z.F.; Wen, P.; Shek, C.H.; Wang, W.H. Measurements of slow β-relaxations in metallic glasses and supercooled liquids. Phys. Rev. B
**2007**, 75, 174201. [Google Scholar] [CrossRef] - Rösner, P.; Samwer, K.; Lunkenheimer, P. Indications for an "excess wing" in metallic glasses from the mechanical loss modulus in Zr
_{65}Al_{7.5}Cu_{27.5}. Europhys. Lett.**2004**, 68, 226. [Google Scholar] [CrossRef] - Brand, R.; Lunkenheimer, P.; Schneider, U.; Loidl, A. Is there an excess wing in the dielectric loss of plastic crystals? Phys. Rev. Lett.
**1999**, 82, 1951. [Google Scholar] [CrossRef] - Ju, J.D.; Atzmon, M. Evaluation of approximate measurements of activation-free-energy spectra of shear transformation zones in metallic glasses. J. Alloys Comp.
**2015**, 643, S8–S10. [Google Scholar] [CrossRef] - Argon, A.S.; Kuo, H.Y. Free energy spectra for inelastic deformation of five metallic glass alloys. J. Non-Cryst. Solids
**1980**, 37, 241–266. [Google Scholar] [CrossRef] - Lei, T.J.; DaCosta, L.R.; Liu, M.; Wang, W.H.; Sun, Y.H.; Greer, A.L.; Atzmon, M. Microscopic characterization of structural relaxation and cryogenic rejuvenation in metallic glasses. Acta Mater.
**2019**, 164, 165–170. [Google Scholar] [CrossRef] - Zhao, J.; Gao, M.; Ma, M.; Cao, X.; He, Y.; Wang, W.; Luo, J. Influence of annealing on the tribological properties of Zr-based bulk metallic glass. J. Non-Cryst. Solids
**2018**, 481, 94–97. [Google Scholar] [CrossRef] - Kumar, G.; Rector, D.; Conner, R.D.; Schroers, J. Embrittlement of Zr-based bulk metallic glasses. Acta Mater.
**2009**, 57, 3572–3583. [Google Scholar] [CrossRef] - Concustell, A.; Méar, F.O.; Surinach, S.; Baró, M.D.; Greer, A.L. Structural relaxation and rejuvenation in a metallic glass induced by shot-peening. Phil. Mag. Lett.
**2009**, 89, 831. [Google Scholar] [CrossRef] - Louzguine-Luzgin, D.V.; Zadorozhnyy, V.Y.; Ketov, S.V.; Wang, Z.; Tsarkov, A.A.; Greer, A.L. On room-temperature quasi-elastic mechanical behaviour of bulk metallic glasses. Acta Mater.
**2017**, 129, 343–351. [Google Scholar] [CrossRef] - Pan, J.; Wang, Y.X.; Guo, Q.; Zhang, D.; Greer, A.L.; Li, Y. Extreme rejuvenation and softening in a bulk metallic glass. Nat. Commun.
**2018**, 9, 560. [Google Scholar] [CrossRef] [PubMed] - Magagnosc, D.J.; Kumar, G.; Schroers, J.; Felfer, P.; Cairney, J.M.; Gianola, D.S. Effect of ion irradiation on tensile ductility, strength and fictive temperature in metallic glass nanowires. Acta Mater.
**2014**, 74, 165–182. [Google Scholar] [CrossRef] - Heo, J.; Kim, S.; Ryu, S.; Jang, D. Delocalized Plastic Flow in Proton-Irradiated Monolithic Metallic Glasses. Sci. Rep.
**2016**, 6, 23244. [Google Scholar] [CrossRef] - Ketov, S.V.; Sun, Y.H.; Nachum, S.; Lu, Z.; Checchi, A.; Beraldin, A.R.; Bai, H.Y.; Wang, W.H.; Louzguine-Luzgin, D.V.; Carpenter, M.A.; et al. Rejuvenation of metallic glasses by non-affine thermal strain. Nature
**2015**, 524, 200–203. [Google Scholar] [CrossRef] - Miyazaki, N.; Wakeda, M.; Wang, Y.-J.; Ogata, S. Prediction of pressure-promoted thermal rejuvenation in metallic glasses. npj Comput. Mater.
**2016**, 2, 1–9. [Google Scholar] [CrossRef] - Madge, S.V.; Louzguine-Luzgin, D.V.; Kawashima, A.; Greer, A.L.; Inoue, A. Compressive plasticity of a La-based glass-crystal composite at cryogenic temperatures. Mater. Des.
**2016**, 101, 146–151. [Google Scholar] [CrossRef] - Grell, D.; Dabrock, F.; Kerscher, E. Cyclic cryogenic pretreatments influencing the mechanical properties of a bulk glassy Zr-based alloy. Fatigue Fract. Eng. Mater. Struct.
**2018**, 41, 1330–1343. [Google Scholar] [CrossRef] - Costa, M.B.; Londoño, J.J.; Blatter, A.; Hariharan, A.; Gebert, A.; Carpenter, M.A.; Greer, A.L. Anelastic-like nature of the rejuvenation of metallic glasses by cryogenic thermal cycling. Acta Mater.
**2023**, 244, 118551. [Google Scholar] [CrossRef] - Lei, T.J.; Liu, M.; Wang, W.H.; Sun, Y.H.; Greer, A.L.; Atzmon, M. Shear transformation zone analysis of anelastic relaxation of a metallic glass reveals distinct properties of α and β relaxations. Phys. Rev. E
**2019**, 100, 033001. [Google Scholar] [CrossRef] - Gallino, I.; Cangialosi, D.; Evenson, Z.; Schmitt, L.; Hechler, S.; Stolpe, M.; Beatrice Ruta, B. Hierarchical aging pathways and reversible fragile-to-strong transition upon annealing of a metallic glass former. Acta Mater.
**2018**, 144, 400–410. [Google Scholar] [CrossRef] - Monnier, X.; Cangialosi, D.; Ruta, B.; Busch, R.; Gallino, I. Vitrification decoupling from α-relaxation in a metallic glass. Sci. Adv.
**2020**, 6, eaay1454. [Google Scholar] [CrossRef] [PubMed] - Johari, G.P.; Goldstein, M. Molecular mobility in simple glasses. J. Phys. Chem.
**1970**, 74, 2034–2035. [Google Scholar] [CrossRef] - Schneider, U.; Brand, R.; Lunkenheimer, P.; Loidl, A. Excess wing in the dielectric loss of glass formers: A Johari-Goldstein β relaxation? Phys. Rev. Lett.
**2000**, 84, 5560. [Google Scholar] [CrossRef] - Cohen, Y.; Karmakar, S.; Procaccia, I.; Samwer, K. The nature of the β-peak in the loss modulus of amorphous solids. Europhys. Lett.
**2012**, 100, 36003. [Google Scholar] [CrossRef] - Johari, G.P.; Goldstein, M. Viscous liquids and the glass transition. II. Secondary relaxations in glasses of rigid molecules. J. Chem. Phys.
**1970**, 53, 2372–2388. [Google Scholar] [CrossRef] - Yu, H.B.; Shen, X.; Wang, Z.; Gu, L.; Wang, W.H.; Bai, H.Y. Tensile Plasticity in Metallic Glasses with Pronounced β relaxations. Phys. Rev. Lett.
**2012**, 108, 015504. [Google Scholar] [CrossRef] - Yu, H.B.; Wang, W.H.; Bai, H.Y.; Wu, Y.; Chen, M.W. Relating activation of shear transformation zones to β relaxations in metallic glasses. Phys. Rev. B
**2010**, 81, 220201. [Google Scholar] [CrossRef] - Küchemann, S.; Maaß, R. Gamma relaxation in bulk metallic glasses. Scr. Mater.
**2017**, 137, 5–8. [Google Scholar] [CrossRef] - Lei, T.J.; DaCosta, L.R.; Liu, M.; Shen, J.; Sun, Y.H.; Wang, W.H.; Atzmon, M. Composition dependence of metallic glass plasticity and its prediction from anelastic relaxation–A shear transformation zone analysis. Acta Mater.
**2020**, 195, 81–86. [Google Scholar] [CrossRef] - Qiao, J.C.; Pelletier, J.M. Dynamic Mechanical Relaxation in Bulk Metallic Glasses: A Review. J. Mater. Sci. Tech.
**2014**, 30, 523–545. [Google Scholar] [CrossRef] - Hao, Q.; Pineda, E.; Wang, Y.-J.; Yang, Y.; Qiao, J.C. Reversible anelastic deformation mediated by β relaxation and resulting two-step deformation in a La
_{60}Ni_{15}Al_{25}metallic glass. Phys. Rev. B**2023**, 108, 024101. [Google Scholar] [CrossRef] - Zhao, R.; Jiang, H.Y.; Luo, P.; Shen, L.Q.; Wen, P.; Sun, Y.H.; Bai, H.Y.; Wang, W.H. Reversible and irreversible β-relaxations in metallic glasses. Phys. Rev. B
**2020**, 101, 094203. [Google Scholar] [CrossRef] - Wang, X.D.; Ruta, B.; Xiong, L.H.; Zhang, D.W.; Chushkin, Y.; Sheng, H.W.; Lou, H.B.; Cao, Q.P.; Jiang, J.Z. Free-volume dependent atomic dynamics in beta relaxation pronounced La-based metallic glasses. Acta Mater.
**2015**, 99, 290–296. [Google Scholar] [CrossRef] - Atzmon, M.; Spaepen, F. Study of Interdiffusion in Amorphous Compositionally Modulated Ni-Zr Thin Films. In Science and Technology of Rapidly Quenched Alloys; Tenhover, M., Johnson, W.L., Tanner, L.E., Eds.; Materials Research Society: Pittsburgh, PA, USA, 1987; pp. 55–59. [Google Scholar]
- Fan, Y.; Iwashita, T.; Egami, T. Energy landscape-driven non-equilibrium evolution of inherent structure in disordered material. Nat. Comm.
**2017**, 8, 15417. [Google Scholar] [CrossRef]

**Figure 1.**Measurement techniques. (

**a**) Cantilever method. The displacement h is monitored as a function of time at a fixed load, P. The instantaneous displacement is the elastic component; (

**b**) Mandrel method. The sample was constrained for 2 × 10

^{6}s at varying radii, after which the radius of curvature was monitored as a function of time in a stress-free condition. Reproduced from Ju, J.D.; Jang, D., Nwankpa, A; Atzmon. M. An atomically quantized hierarchy of shear transformation zones in a metallic glass. J. Appl. Phys.

**2011**, 109. with permission of AIP Publishing [18].

**Figure 2.**Al

_{86.8}Ni

_{3.7}Y

_{9.5}: Anelastic strain evolution following equilibration at different mandrel radii. The strain is normalized by the elastic strain at equilibrium, prior to removal of the constraint. Reproduced from Ju, J.D.; Jang, D., Nwankpa, A; Atzmon. M. An atomically quantized hierarchy of shear transformation zones in a metallic glass. J. Appl. Phys.

**2011**, 109. with permission of AIP Publishing [18].

**Figure 3.**Al

_{86.8}Ni

_{3.7}Y

_{9.5}: Sample relaxation curves and corresponding relaxation-time spectra. (

**a**) Cantilever measurement, performed at fixed load, P = 0.2 mN, i.e., fixed stress. (

**b**) Mandrel measurement, performed in a stress-free condition after equilibration under constraint. For each case, two spectra, f(τ), are shown, obtained from fits with different numbers of fitting parameters. Reproduced from Ju, J.D.; Jang, D., Nwankpa, A; Atzmon. M. An atomically quantized hierarchy of shear transformation zones in a metallic glass. J. Appl. Phys.

**2011**, 109. with permission of AIP Publishing [18].

**Figure 4.**Top: linear solid model: n anelastic processes act in series, each represented by a Voigt unit. m-type sites are associated with Young’s modulus of ${E}_{m}^{\prime}$ and viscosity ${\eta}_{m}^{\prime}$, both effective quantities that are inversely proportional to the volume fraction of these sites. E

_{0}is the high-frequency Young’s modulus. Bottom: illustration of the contribution of each m-type STZ to Voigt Unit m. Reproduced from Ju, J.D.; Jang, D., Nwankpa, A; Atzmon. M. An atomically quantized hierarchy of shear transformation zones in a metallic glass. J. Appl. Phys.

**2011**, 109. with permission of AIP Publishing [18].

**Figure 5.**Al

_{86.8}Ni

_{3.7}Y

_{9.5}: Calculated properties of the respective anelastic processes m = 1–8. (

**a**) Time constants. (

**b**) Volume fraction of potential STZs. (

**c**) Effective macroscopic Young’s modulus. (

**d**) Effective macroscopic viscosity. (

**e**) STZ volume in units of atomic volume of Al, V

_{Al}= 16.6 × 10

^{−30}m

^{3}. Values for m = 4 were obtained by interpolation. (

**f**) Volume fraction of potential STZ as a function of ΔF/kT. The error bars are the standard deviation of the mean, obtained by averaging over multiple measurements. Reproduced from Ju, J.D.; Jang, D., Nwankpa, A; Atzmon. M. An atomically quantized hierarchy of shear transformation zones in a metallic glass. J. Appl. Phys.

**2011**, 109. with permission of AIP Publishing [18].

**Figure 6.**Al

_{86.8}Ni

_{3.7}Y

_{9.5}: Apparent anelastic strain after unconstrained relaxation for t = 4 × 10

^{6}s as a function of the apparent elastic strain at the end of the constraining period for varying constraining radii. Both are computed for the sample surface from the curvature. Each symbol represents one sample. Linear data are from Ref. [18]. Deviation from linearity occurs at high strain. Comparison between the two-parameter fit (dotted line) and two-step fit (dashed line) (see Ref. [28]): the latter yields a better fit for the small-strain data than the former. Reproduced from Lei, T.J.; Atzmon, M. Activation volume details from nonlinear anelastic deformation of a metallic glass. J. Appl. Phys.

**2019**, 126, 185104, with permission of AIP Publishing.

**Figure 7.**Zr

_{46.8}Ti

_{13.8}Cu

_{12.5}Ni

_{10}Be

_{27.5}: (

**a**) Digitized loss moduli [35] with DSA fits. (

**b**,

**c**) Spectra obtained from these fits above and below T

_{g}. Reprinted from Ju, J.D.; Atzmon, M. A comprehensive atomistic analysis of the experimental dynamic-mechanical response of a metallic glass. Acta Mater.

**2014**, 74, 183–188, Copyright (2014), with permission from Elsevier [34].

**Figure 8.**Zr

_{46.8}Ti

_{13.8}Cu

_{12.5}Ni

_{10}Be

_{27.5}: lnτ, determined from the median of the respective peak in the relaxation-time spectra of Figure 7, plotted as a function of 1/kT for three tentative groupings of τ

_{n}(

**a**–

**c**) below (gray circles) and above T

_{g}(black circles). Simultaneous fit performed using Equation (9) is shown with dashed lines for each n. Above T

_{g}, linear temperature dependence of the modulus was used. Out of nine possible combinations, continuity of the fits and n at T

_{g}is obtained only for the combination displayed in (a). Reprinted from Ju, J.D.; Atzmon, M. A comprehensive atomistic analysis of the experimental dynamic-mechanical response of a metallic glass. Acta Mater.

**2014**, 74, 183–188, Copyright (2014), with permission from Elsevier [34].

**Figure 9.**(

**a**) ${E}_{T}^{\u2033}\left(\omega \right)$, loss modulus, calculated at T

_{i}steps of 5 K from the spectrum obtained in Ref. [18]. (

**b**) Master curve obtained by shifting sets of points obtained at each T

_{i}to coincide with the curve at ${T}_{ref}$ = 388 K. A Cauchy function, corresponding to a single activated process, and a KWW fit to the main part of the curve, are included. Inset: Arrhenius plot of the shift factor as a function of reciprocal temperature. Reproduced from Ju J.D.; Atzmon, M. Atomistic interpretation of the dynamic response of glasses. MRS Comm.

**2014**, 4, 63–66 with permission from SNCSC [44].

**Figure 10.**A hypothetical time-dependent quantity, described as a stretched exponent with a small shift (y(t), open circles) is fitted with an unshifted stretched exponent (x(t), Equation (1), line). (

**a**) Linearly spaced time points; (

**b**) Logarithmically spaced time points. Significantly different τ and β are obtained. Reproduced from Atzmon, M. The pitfalls of empirical fitting of glass relaxation data with stretched exponents. J. Appl. Phys. 2018, 123, 065103, with the permission of AIP Publishing [45].

**Figure 11.**Assumed spectrum compared with that calculated from it using the temperature stepping approximation. Both the activation-energy values and spectrum shape are affected. The latter spectrum is similar to those in Ref. [62]. Reprinted from Ju, J.D.; Atzmon, M. Evaluation of approximate measurements of activation-free-energy spectra of shear transformation zones in metallic glasses, J. Alloys Comp.

**2015**, 643, S8–S10, Copyright (2014), with permission from Elsevier [61].

**Figure 12.**Normalized anelastic strain of La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}as a function of time for different aging times prior to bending, as indicated. Open circles and filled squares correspond, respectively, to measurements without and with cryogenic cycling after aging, prior to bending. Curves are not shifted. The dashed lines are all drawn with the same slope. Note that the entire strain is anelastic, as verified by annealing above room temperature (bold arrow). Reprinted from Lei, T.J.; DaCosta, L.R.; Liu, M.; Wang, W.H.; Sun Y.H.; Greer, A.L.; M. Atzmon. Microscopic characterization of structural relaxation and cryogenic rejuvenation in metallic glasses. Acta Mater.

**2019**, 164, 165–170. Copyright (2018), with permission from Elsevier [63].

**Figure 13.**Relaxation-time spectra for La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}with different aging times, as indicated. For each condition, representative data for two independent samples are shown. Open circles and crosses, vs. filled squares and pluses, correspond to samples without, vs. with, cryogenic cycling, respectively. The curves are shifted vertically for clarity. Reprinted from Lei, T.J.; DaCosta, L.R.; Liu, M.; Wang, W.H.; Sun Y.H.; Greer, A.L.; M. Atzmon. Microscopic characterization of structural relaxation and cryogenic rejuvenation in metallic glasses. Acta Mater.

**2019**, 164, 165–170. Copyright (2018), with permission from Elsevier [63].

**Figure 14.**The evolution of time constants of different STZ types, m, with aging time for La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}metallic glasses. Downwards arrows indicate the effect of cryogenic cycling following aging. Reprinted from Lei, T.J.; DaCosta, L.R.; Liu, M.; Wang, W.H.; Sun Y.H.; Greer, A.L.; M. Atzmon. Microscopic characterization of structural relaxation and cryogenic rejuvenation in metallic glasses. Acta Mater.

**2019**, 164, 165–170. Copyright (2018), with permission from Elsevier [63].

**Figure 15.**c

_{∞}= the additive term in the spectrum fit, c

_{5,6}= the integrated area of the last two peaks, and c

_{total}= the integrated area of the entire spectrum plus c

_{∞}vs. aging time for La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}MGs. Blue: cycled after aging. Lines: guide to the eye. Reprinted from Lei, T.J.; DaCosta, L.R.; Liu, M.; Wang, W.H.; Sun Y.H.; Greer, A.L.; M. Atzmon. Microscopic characterization of structural relaxation and cryogenic rejuvenation in metallic glasses. Acta Mater.

**2019**, 164, 165–170. Copyright (2018), with permission from Elsevier [63].

**Figure 16.**Volume fraction occupied by m-type potential STZs for La

_{55}Ni

_{20}Al

_{25}metallic glass, Equation (6), as a function of activation free energy ΔF

_{m}, Equation (2), divided by kT, for different room-temperature aging times. Each symbol corresponds to one aging-time value. Arrows show the direction of evolution with room-temperature aging time for each m. m = 6–8 and beyond (not active at room temperature within the time range used) correspond to the α relaxation, and m ≤ 5 correspond to the β relaxation. The last two data points for m = 8 STZs represent an underestimate due to lack of mechanical equilibration at the end of the constraining period for samples with long aging time and associated long τ

_{8}values (see discussion). Reproduced from Lei, T.J.; Liu, M.; Wang, W.H.; Sun, Y. H.; Greer, A. L.; Atzmon, M. Shear transformation zone analysis of anelastic relaxation of a metallic glass reveals distinct properties of α and β relaxations. Phys Rev. E

**2019**, 100, 033001 [76].

**Figure 17.**La

_{55}Ni

_{20}Al

_{25}: STZ volume (Ω

_{m}) as a function STZ type (m) for samples aged 2.9 × 10

^{7}s. The error bars, <0.7%, are smaller than the symbols. The slopes correspond to the volume increment between two adjacent Ω

_{m}values. The random error in these slopes is 2–3%. Reproduced from Lei, T.J.; Liu, M.; Wang, W.H.; Sun, Y. H.; Greer, A. L.; Atzmon, M. Shear transformation zone analysis of anelastic relaxation of a metallic glass reveals distinct properties of α and β relaxations. Phys Rev. E

**2019**, 100, 033001 [76].

**Figure 18.**Anelastic strain, normalized by the corresponding equilibrium elastic strain, vs. time for (

**a**) cantilever bending and (

**b**) mandrel measurements for La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}with a prior room-temperature aging time of 1.9 × 10

^{6}s. For cantilever bending, an average of all tests for the same composition is displayed, and each point is an average of 500 experimental data points. For the mandrel measurements, data corresponding to all samples are shown. Reprinted from Lei, T.J.; DaCosta, L. R.; Liu, M.; Shen, J.; Sun, Y. H.; Wang, W.H.; Atzmon, M. Composition dependence of metallic glass plasticity and its prediction from anelastic relaxation–A shear transformation zone analysis. Acta Mater.

**2020**, 195, 81–86, Copyright (2020), with permission from Elsevier [86].

**Figure 19.**Relaxation-time spectra for (

**a**) cantilever bending and (

**b**) mandrel measurements, computed from the normalized anelastic strain vs. time data of Figure 18 for La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}aged at room temperature for 1.9 × 10

^{6}s. For cantilever bending, an average of all spectra is shown for each alloy, and the standard deviation of the mean is smaller than the symbols. All spectra are included for the mandrel measurements. Peaks are numbered m = 1,…,8, corresponding to different STZ types. Reprinted from Lei, T.J.; DaCosta, L. R.; Liu, M.; Shen, J.; Sun, Y. H.; Wang, W.H.; Atzmon, M. Composition dependence of metallic glass plasticity and its prediction from anelastic relaxation–A shear transformation zone analysis. Acta Mater.

**2020**, 195, 81–86, Copyright (2020), with permission from Elsevier [86].

**Figure 20.**Engineering stress vs. engineering strain for La

_{70}Cu

_{15}Al

_{15}and La

_{70}Ni

_{15}Al

_{15}obtained from room-temperature tensile tests at strain rates of 1.6 $\times $ 10

^{−6}s

^{−1}, 10

^{−5}s

^{−1}, and 10

^{−4}s

^{−1}. Curve thickness decreases with increasing strain rates. Each curve consists of 200–20,000 data points, depending on rate. Reprinted from Lei, T.J.; DaCosta, L. R.; Liu, M.; Shen, J.; Sun, Y. H.; Wang, W.H.; Atzmon, M. Composition dependence of metallic glass plasticity and its prediction from anelastic relaxation–A shear transformation zone analysis. Acta Mater.

**2020**, 195, 81–86, Copyright (2020), with permission from Elsevier [86].

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Atzmon, M.; Ju, J.D.; Lei, T.
Structural Relaxation, Rejuvenation and Plasticity of Metallic Glasses: Microscopic Details from Anelastic Relaxation Spectra. *Materials* **2023**, *16*, 7444.
https://doi.org/10.3390/ma16237444

**AMA Style**

Atzmon M, Ju JD, Lei T.
Structural Relaxation, Rejuvenation and Plasticity of Metallic Glasses: Microscopic Details from Anelastic Relaxation Spectra. *Materials*. 2023; 16(23):7444.
https://doi.org/10.3390/ma16237444

**Chicago/Turabian Style**

Atzmon, Michael, Jong Doo Ju, and Tianjiao Lei.
2023. "Structural Relaxation, Rejuvenation and Plasticity of Metallic Glasses: Microscopic Details from Anelastic Relaxation Spectra" *Materials* 16, no. 23: 7444.
https://doi.org/10.3390/ma16237444