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The critical strain - A crossover from stochastic activation to percolation of flow units during stress relaxation in metallic glass
Scripta Materialia 134 (2017) 75–79
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular Article
The critical strain - A crossover from stochastic activation to percolation of flow units during stress relaxation in metallic glass Y.C. Wu a, B. Wang a, Y.C. Hu a, Z. Lu a, Y.Z. Li a, B.S. Shang b, W.H. Wang a, H.Y. Bai a,⁎, P.F. Guan b,⁎ a b
Institutes of Physics, Chinese Academy of Sciences, Beijing 100190, China Beijing Computational Science Research Center, Beijing 100190, China
a r t i c l e
i n f o
Article history: Received 11 February 2017 Received in revised form 27 February 2017 Accepted 28 February 2017 Available online xxxx
a b s t r a c t The stress relaxation processes of strained Zr50Cu50 metallic glass (MG) are investigated by molecular dynamics simulations. We provide the direct evidence for the strain-accelerated relaxation which can be attributed to the activation of flow units in MGs. Moreover, a crossover from stochastic activation to percolation of flow units corresponding to a critical applied strain can be observed. The temperature influence on the strain-accelerated dynamics is also discussed. Our results may be helpful for the understanding of the relaxation dynamics under applied strain field and deepen the comprehension of the nature of MG. © 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Metallic glasses (MGs), which have disordered atomic structures, are unique materials with a wide range of potential applications due to their high strength, elasticity and corrosion resistance [1–4]. It is believed that the β relaxation modes are correlated with structural heterogeneity and mechanical properties [5–7]. The intensive research has currently focused on the relaxation dynamics of MGs [8–11]. Despite extensive efforts have been made, the physical origin of relaxation modes is still not yet well understood. Due to the fact that relaxation processes in MGs are sluggish, it is difficult to get information of the relaxation dynamics directly. However, the application of a step strain can provide a straightforward method for extending the dynamic range of glasses, therefore these sluggish relaxations can be detected effectively. Combining with tensile experiments and stress relaxation spectra, some researchers proposed a flow unit perspective to microscopically describe the evolution of the dynamical heterogeneities with temperatures [12, 13]. However, the picture of flow units or deformation units is conceptual and phenomenological, the precise identification of the flow units remains elusive and ambiguous. Therefore a clear picture of evolution of the dynamic mechanisms in MGs with microscopic information is still required. Recently, some experimental results showed the crossover of relaxation modes from the stochastic activation of the localized shear transformation zones to cooperative motion of these deformation units [14–16]. Nevertheless, limited by the experiment conditions, the applied strain is usually confined within small elastic regime, which is below 2% for the conventional MGs. The processes of activation and evolution of flow units under large applied strain, even above yield strain,
⁎ Corresponding authors. E-mail addresses: [email protected] (H.Y. Bai), [email protected] (P.F. Guan).
http://dx.doi.org/10.1016/j.scriptamat.2017.02.048 1359-6462/© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
are unachievable in experiments. Thus the dynamics of metallic glasses with large imposed strain remains unclear. In this letter, we employ molecular dynamic simulations as a computational microscope to obtain detailed insight into the dynamics of Zr50Cu50 MG by means of stress relaxation. We find direct evidences of strain-accelerated relaxation dynamics in MGs. We correlate this phenomenon with the evolution of local irreversible rearrangements of atoms in glasses upon increasing applied strain, and these local irreversible rearrangements can be pictured as activated flow units. Below the critical applied strain, stress relaxation dynamics can be progressively accelerated with increasing imposed strain, which relates to the propagation of stochastic activation of the flow units. However, beyond the critical applied strain, the dynamics of the strained-glass keeps nearly unchanged due to the occurrence of avalanche of flow units and the uniform mechanical flow behavior. Furthermore, we show that the difference of dynamics at different temperatures decreases with increase of the applied strain below the critical applied strain, while this difference can be almost negligible when the imposed strain exceeds the critical applied strain. The model system is made of 27, 000 atoms, representing an alloy of Zr50Cu50 MG. Atoms interact with the embedded atom method (EAM) potential [17]. Molecular dynamics (MD) simulation was carried out using the open source LAMMPS package [18]. The periodic boundary condition was maintained throughout the simulation, and temperature was kept a constant through the Nose-Hoove thermostat [19]. A glassy state was obtained by melting and equilibrating the system at high temperature (2000 K) for 50 ps, then cooling the liquid to the designated temperatures with the cooling rate of 1012K/s, during which the cell sizes were adjusted to give zero pressure within the constant number, pressure and temperature (NPT) ensemble. The change in the energy
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Y.C. Wu et al. / Scripta Materialia 134 (2017) 75–79
at Tg(= 850 K) suggests a glass transition at the time scale of the MD simulation. Various initial tension strains εi were applied to the system with a strain rate of 109/s and the constant number, volume and temperature (NVT) ensemble was employed for stress relaxation measurement. Then the stress decay with the loading time up to 2000 ps at 300 K and 400 K (≪Tg = 850 K) was recorded. The applied strain ranges from 1% to 20%, which covers both elastic and plastic regime and no cavitation occurs in both tensile and stress relaxation process. In order to reveal the influence of initial state on the stress attenuation in the stress relaxation process, isoconfigurational ensemble introduced by Widmer-Cooper and Harrowell was employed [20,21]: 100 independent stress relaxation simulations were performed, which all started from the same particle configuration but with momenta randomly assigned from the Maxwell-Boltzman distribution at the interested temperature. Typical isothermal stress-relaxation curves are showed in Fig. 1(a) for various applied strains at T = 300 K, and the temporal-dependent stress σ(t) is normalized by the initial stress σ0. Generally, the calculated σ(t)/σ0 shows a rapid decrease from 1 at the initial time, and then obviously slows down and decays sluggishly with time. These features are consistent with experimental stress relaxation findings [13,14], confirming the validity and reliability of studying of stress relaxation dynamics of glasses with molecular dynamics simulations. It is apparent that the decay of σ(t)/σ0 accelerates as the applied strain increases until it almost collapses to a master curve with the imposed strain exceeding a certain value (εc = 8%). As shown in the inset of Fig. 1(b), the critical
strain εc(=8%) can be regarded as the yield strain of the simulated metallic glass system at 300 K, implying that the decay of stress in strained metallic glass is correlated with the applied strain and a transition of relaxation behavior may occur around the critical strain εc. Since the scaled stress attenuation degree Δσ =(σ0 −σt)/σ0 during the stress relaxation can reflect the intrinsic features of the relaxation behavior of glasses [13,22], we collect Δ σ = 1 − σ(t = 200 ps)/σ0 for various imposed strains at 300 K, and the Δ σ as a function of εi is shown in Fig. 1(b) (we assume Δσ =0 for εi = 0%). It is found that Δσ shows a linear increment with εi for εi b εc; nevertheless, Δ σ maintains constant as εi N εc, confirming that the relaxation process is sensitive to εi for εi b εc, but insensitive to εi for εi N εc from the results shown in Fig. 1(a) and (b). The change of the correlations between εi and Δ σ suggests the crossover of the relaxation behaviors near to εc. The deviation of the data point with εi = 0% may suggest another relaxation mechanism for ultra-small applied strain, which is difficult to investigate due to the limited timescale in computer simulations. To explore the underlying atomic level mechanism that accounts for the crossover behaviors of the stress relaxation distinguished by εc, we utilize the ability of MD simulations to examine individual atomic displacement processes. We calculate the mean square atomic displacement u for each atom during a time interval of Δt = 200 ps for various applied strain at T = 300 K. Fig. 1(c) shows a typical set of the displacement probability density function p(u) for a range of strain at 300 K, and , where P(u) is the distribution that p(u) is defined as pðuÞ ¼ PðuþΔuÞ−PðuÞ Δu quantifies the probability of finding atomic displacement x ≤ u. It is
Fig. 1. (a) Stress relaxation data (symbols) normalized by stress at initial time σ0 at T = 300 K, and the corresponding fitting curves (solid lines) of KWW function. (b) Stress drop data of stress relaxation Δσ at T = 300 K. The inset is a typical tensile stress-strain curve of the Zr50Cu50 metallic glass at T = 300 K, and the yield strain is about 8%. (c) Displacement probability density p(u) at T = 300 K for different ε. up representing the most probable value of u of all the atoms for different ε. (d) Relationship between 〈u2〉 and Δσ for all ε.
Y.C. Wu et al. / Scripta Materialia 134 (2017) 75–79
found that the probability distribution broadens with increasing applied strain for εi b εc, reflecting the increase of dynamic heterogeneity. The increase of up, the peak of p(u), suggests the faster dynamics of atoms under higher applied strain. More interesting, the collapse of the curves for εi N 8% implies the identical atomic dynamic properties as the applied strain is larger than the critical strain, which is consistent with the results shown in Fig. 1(a) and (b). It provides the direct evidence that the dynamic properties of atoms determine the stress relaxation behaviors in MGs. To quantify the overall atomic dynamics, we calculate the N
with mean-squared displacement u2 ðtÞ ¼ ∑i¼1 jri ðtÞ−ri ð0Þj2 t = 200 ps for all applied strains during the stress relaxation. The Δ σ as a function of 〈u2〉 is shown in Fig. 1(d). It presents a linear relationship between 〈u2〉 and Δσ for all strains, and the data points for εi N 8% converge to one point. In addition, the deviation of the data point with εi = 0% is in good accordance to the result shown in Fig. 1(b). The strong correlation between Δσ and 〈u2〉 confirms that the stress relaxation is an effective method to investigate the dynamic properties of MGs in atomic level. Furthermore, it provides an opportunity for understanding the corresponding atomic relaxation mechanisms in MG under various applied strains. The crossover of the stress relaxation behaviors around the yield strain implies the distinct atomic relaxation mechanisms controlled by the applied strain. To characterize the individual atomic relaxation during the stress relaxation, we further investigate the local irreversible rearrangement of atomic configuration evaluated by nonaffine displacement of the central atom i relative to its neighbor atoms j: D2mini ¼
h i2 1 ! ! ! ! ε r j ðt þ Δt Þ− r i ðt þ Δt Þ ; ∑ j r j ðt Þ− r i ðt Þ−! N
! where N is the number of nearest neighbor of atom i, r i ðtÞ is the position of atom i at the time t, Δ t is the time interval for the plastic rear! rangement, and ε is the maximum local elastic strain tensor [23]. Here we set Δt =200 ps as used in Fig. 1. The 2D snapshots that capture the distribution of nonaffine displacement D2min under various strains are shown in Fig. 2. Here, we chose D2c ~7 as the cutoff value of D2min for
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all strained samples (for more details, see Supplemental Material, Fig. S1), and the regions with D2min N D2c , just analogous to flow units, present inhomogeneous distribution [12,13,24,25], which confirms the intrinsic heterogeneity in MG [26–28]. As shown in Fig. 2(a), the individual flow units are activated even when the applied strain is only 1%, which is in excellent agreement with the previous simulation works of the existence of local atomic rearrangements in elastic deformation region in MG [24]. As the applied strain increases, the density of the activated flow units increases (see Fig. 2(b)). However, these activated regions are relatively independent (see Fig. 2(a) and (b)), which are corresponding to a stochastic and isolated activation of the flow units. As the imposed strain exceeds yield strain or εc = 8%, the connective percolation of activated flow units can be observed (see Fig. 1(c)). The stress relaxations in system with εi = 15% N εc have similar behaviors of atomic rearrangement (see Fig. 2(d)), showing that the plastic flow occurs as the applied strain becomes larger than the yield strain in the simulated system. The different choices of cut off value do not affect the general propagation trend of flow units and similar results are obtained. Since the composition always plays an important role in controlling the behaviors of MG, we carefully examined the distribution of D2min of each constituent (Zr and Cu, see Supplemental Material, Fig. S2), and found that they exhibit similar dynamic behaviors. It suggests that the constituent effect is weak in describing the dynamic heterogeneity behaviors during the stress relaxation, at least in our system. The crossover from stochastic activation to percolation of flow units reveals the distinct atomic level mechanisms of the stress relaxation behaviors, which are closely dependent on the critical applied strain εc. For more comprehensive understanding of the strain-dependent stress relaxation process in MG, we further investigate the related relaxation time under various applied strains, and thereby the temperature influence. According to previous studies, the stress relaxation curves shown in Fig. 1(a) can be described by Kohlrausch-Williams-Watts β
(KWW) form of σðtÞ ¼ σ 0 expð− τt Þ þ σ r, where σ0 is the initial stress, σr is the residual stress at infinite time, τ is the characteristic relaxation time and β is the non-exponential parameter related with dynamic heterogeneity. Since the stress relaxation is based on the activation of
Fig. 2. (a)–(d) A series of snapshots capturing nonaffine displacement D2min in tensiled Zr50Cu50 MG samples, and from (a) to (d), ε=1%, 5%, 8%, 15%, respectively.
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relaxation units similar to those happened in creep process, a good fitting of stretched-exponential function can be obtained, and it well demonstrates that the stress relaxation process is highly heterogeneous on microscopic scale [29]. It can be obviously seen that τ decreases exponentially with εi b 8 %(=εc) from Fig. 3(a), and the strain dependence of the characteristic relaxation time τ can be fitted by the equation τ = τ∞ exp [ε1/3]. In the contrast, τ keeps almost a constant for εi N 8%, which is coincident with the tendency of stress attenuation with increasing strain as described in Fig. 1(b). In Fig. 3(b), it shows that the exponent β decreases with strain within elastic regime, evidencing that the distribution of relaxation time is broadened markedly by increasing the strain, which is consistent with the increasing width of p(u) associated with enhancing dynamic heterogeneity as shown in Fig. 1(c). The decrease of τ for lower strain region implies that the external strain reduces the apparent activation energy barrier dramatically in the local energy landscape through the self-energy of the related external stress. For the larger strain region that is over the yield strain, the platform illustrates that the activation energy barrier becomes insensitive to the external strain, implying that the activation energy barrier reaches the minimum value and the relaxation process is mainly contributed by highly cooperative movement. The conclusion is evidently supported by the forming of plastic flow in high strain region (see Fig. 2(c) and (d)). Furthermore, the temperature effect for the relaxation process was studied and the related data for 300 K and 400 K are shown in Fig. 3(c). It is found that both temperature and strain are equally important in controlling relaxation dynamics of MGs [30–32]. Either temperature or external strain (stress) increases can reduce the relaxation time
and accelerates the dynamic process in MGs, confirming the regulation of the equivalence between force and temperature. MGs consist of tightly bonded atomic regions (also called solid-like regions) and loosely bonded atomic regions (or liquid-like regions) [25,33]. The liquid-like sites in glasses were even directly imaged by aberration-corrected transmission electron microscope [33]. Recent simulations and experiments have showed that these liquid-like regions with viscoelastic behavior act as the flow units in MGs [24,25]. Therefore, we can model the MG into two parts: flow units and elastic matrix. Flow units are simplified as a distribution of dashpots within elastic matrix. These localized liquid-like deformation units or flow units acting like a liquid in a sufficient long time can accommodate external stress (strain) or elevated temperature by relaxing or flowing, and stress decay should be ascribed to the inhomogeneous activation of flow units [12,13,15,16,25]. Under a step strain loading, strain causes energy minima to disappear by a decrease of energy barriers, which renders the system mechanically unstable and leads to local structural change [16, 34,35]. During the process of stress relaxation, a fraction of atoms will undergo nonaffine deformation, and thus transform into flow units [13]. When the applied strain is largely limited, only few localized nano-scale flow units with smaller energy barriers at initial state can be activated, just as shown by the atoms experiencing larger nonaffine displacement (red balls) in Fig. 2(a). With an increase of applied strain, flow units are gradually stochastically activated corresponding to the proliferation of flow units shown in Fig. 2(b). Consequently, relaxation rate of system progressively accelerates. As imposed strain exceeds critical strain, adjacent weak-bonded regions around flow units also
Fig. 3. (a) Relaxation time τ extracted from fitting KWW function to raw stress relaxation data for all ε, and the corresponding fitting curve (solid line). (b) The fitting parameter β for all ε, and the solid line guides the eyes. (c) Comparison of relaxation time τ between 300 K and 400 K at ε=2%, 4%, 8%, 15%, respectively.
Y.C. Wu et al. / Scripta Materialia 134 (2017) 75–79
increasingly transform into a liquid-like state, owing to the applied strain lowers energy barrier between adjacent mega-basins in energy landscape [13,16]. Meanwhile the concentration of flow units reaches a critical value, the majority of flow units or liquid-like zones in the sample start to cooperate with each other, and a connective percolation of flow units occurs corresponding to the coalescence process as shown in Fig. 2(c) and (d). At this stage, critical transition from broken-ergodic to ergodic on energy landscape happens. This process is similar to the thermally driven glass to supercooled liquid transition. Therefore, the whole system can accommodate the applied strain, and the rate of stress relaxation first comes to peak and then turns to a constant. Finally, the relaxation dynamics of strained-glasses is nearly unvarying due to the forming of plastic flow. In summary, with the analysis of stress relaxation of a strained metallic glass, the evolution of relaxation dynamics and its intimate relation to the activation of flow units are investigated. We show a crossover from stochastic activation to percolation of flow units corresponding to a critical applied strain in a simulated Zr50Cu50 MG system. A conception of the critical applied strain is proposed as a watershed between stochastic activation and percolation of flow units in stress relaxation of MGs. Our results may shed a light for comprehensive understanding of the relaxation dynamics, mechanical behaviors, and especially their correlation with structural evolution in MGs. Acknowledgements Insightful discussions with P. Luo, Y. T. Sun, and L. J. Wang are highly acknowledged. We also thank D. Q. Zhao and D. W. Ding for helpful discussions. We also acknowledge the computational support from Beijing Computational Science Research Center (CSRC). B. S. and P. G. are supported by the National Natural Science Foundation of China (Grant No. 51571011, U) and the MOST 973 Program (Grant No. 2015CB856800). This work was also supported by the NSF of China (51571209 and 51461165101) and the Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-JSC017). Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.scriptamat.2017.02.048.
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References [1] A.L. Greer, Science 267 (5206) (1995) 1947. [2] W.H. Wang, Adv. Mater. 21 (45) (2009) 4524. [3] M.D. Demetriou, M.E. Launey, G. Garrett, J.P. Schramm, D.C. Hofmann, W.L. Johnson, R.O. Ritchie, Nat. Mater. 10 (2) (2011) 123. [4] J. Schroers, Phys. Today 66 (2) (2013) 32. [5] T. Ichitsubo, E. Matsubara, T. Yamamoto, H. Chen, N. Nishiyama, J. Saida, K. Anazawa, Phys. Rev. Lett. 95 (24) (2005) 245501. [6] H.B. Yu, X. Shen, Z. Wang, L. Gu, W.H. Wang, H.Y. Bai, Phys. Rev. Lett. 108 (1) (2012) 015504. [7] H.B. Yu, W.H. Wang, H.Y. Bai, K. Samwer, Nat. Sci. Rev. 1 (3) (2014) 429–461. [8] H.-B. Yu, W.-H. Wang, K. Samwer, Mater. Today 16 (5) (2013) 183. [9] P. Luo, Z. Lu, Y.Z. Li, H.Y. Bai, P. Wen, W.H. Wang, Phys. Rev. B 93 (10) (2016) 104204. [10] V.M. Giordano, B. Ruta, Nat. Commun. 7 (2016) 10344. [11] S. Ashtekar, G. Scott, J. Lyding, M. Gruebele, J. Chem. Phys. 1 (13) (2010) 1941. [12] Y.Z. Li, L.Z. Zhao, C. Wang, Z. Lu, H.Y. Bai, W.H. Wang, J. Chem. Phys. 143 (4) (2015) 041104. [13] Z. Wang, B.A. Sun, H.Y. Bai, W.H. Wang, Nat. Commun. 5 (2014) 5823. [14] W. Jiao, B.A. Sun, P. Wen, H.Y. Bai, Q.P. Kong, W.H. Wang, Appl. Phys. Lett. 103 (8) (2013) 081904. [15] L.Z. Zhao, Y.Z. Li, R.J. Xue, W.H. Wang, H.Y. Bai, J. Appl. Phys. 118 (15) (2015) 154904. [16] L.Z. Zhao, R.J. Xue, Y.Z. Li, W.H. Wang, H.Y. Bai, J. Appl. Phys. 118 (24) (2015) 244901. [17] Y. Cheng, E. Ma, H.W. Sheng, Phys. Rev. Lett. 102 (24) (2009) 245501. [18] S. Plimpton, J. Comput. Phys. 117 (1) (1995) 1. [19] S. Nosé, J. Chem. Phys. 81 (1) (1984) 511. [20] A. Widmer-Cooper, P. Harrowell, H. Fynewever, Phys. Rev. Lett. 93 (13) (2004) 135701. [21] A. Widmer-Cooper, P. Harrowell, Phys. Rev. Lett. 96 (18) (2006) 185701. [22] Z. Lu, W.-H. Wang, H.-Y. Bai, Sci. China Mater. 58 (2) (2015) 98. [23] M.L. Falk, J.S. Langer, Phys. Rev. E 57 (6) (1998) 7192. [24] Z. Lu, W. Jiao, W.H. Wang, H.Y. Bai, Phys. Rev. Lett. 113 (4) (2014) 045501. [25] Y.C. Hu, P.F. Guan, M.Z. Li, C.T. Liu, Y. Yang, H.Y. Bai, W.H. Wang, Phys. Rev. B 93 (21) (2016) 214202. [26] Y.H. Liu, D. Wang, K. Nakajima, W. Zhang, A. Hirata, T. Nishi, A. Inoue, M.W. Chen, Phys. Rev. Lett. 106 (12) (2011) 125504. [27] H. Wagner, D. Bedorf, S. Küchemann, M. Schwabe, B. Zhang, W. Arnold, K. Samwer, Nat. Mater. 10 (6) (2011) 439–442. [28] F. Zhu, H.K. Nguyen, S.X. Song, D.P.B. Aji, A. Hirata, H. Wang, K. Nakajima, M.W. Chen, Nat. Commun. 7 (2016) 11516. [29] M.D. Ediger, Annu. Rev. Phys. Chem. 51 (1) (2000) 99–128. [30] H.B. Yu, R. Richert, R. Maaß, K. Samwer, Phys. Rev. Lett. 115 (13) (2015) 135701. [31] P. Guan, M. Chen, T. Egami, Phys. Rev. Lett. 104 (20) (2010) 205701. [32] Z. Lu, X.N. Yang, B.A. Sun, Y.Z. Li, K. Chen, W.H. Wang, H.Y. Bai, Scr. Mater. 130 (2017) 229. [33] P.Y. Huang, S. Kurasch, J.S. Alden, A. Shekhawat, A.A. Alemi, P.L. McEuen, J.P. Sethna, U. Kaiser, D.A. Muller, Science 342 (6155) (2013) 224. [34] D.L. Malandro, D.J. Lacks, Phys. Rev. Lett. 81 (25) (1998) 5576. [35] D.J. Lacks, M.J. Osborne, Phys. Rev. Lett. 93 (25) (2004) 255501.
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular Article
The critical strain - A crossover from stochastic activation to percolation of flow units during stress relaxation in metallic glass Y.C. Wu a, B. Wang a, Y.C. Hu a, Z. Lu a, Y.Z. Li a, B.S. Shang b, W.H. Wang a, H.Y. Bai a,⁎, P.F. Guan b,⁎ a b
Institutes of Physics, Chinese Academy of Sciences, Beijing 100190, China Beijing Computational Science Research Center, Beijing 100190, China
a r t i c l e
i n f o
Article history: Received 11 February 2017 Received in revised form 27 February 2017 Accepted 28 February 2017 Available online xxxx
a b s t r a c t The stress relaxation processes of strained Zr50Cu50 metallic glass (MG) are investigated by molecular dynamics simulations. We provide the direct evidence for the strain-accelerated relaxation which can be attributed to the activation of flow units in MGs. Moreover, a crossover from stochastic activation to percolation of flow units corresponding to a critical applied strain can be observed. The temperature influence on the strain-accelerated dynamics is also discussed. Our results may be helpful for the understanding of the relaxation dynamics under applied strain field and deepen the comprehension of the nature of MG. © 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Metallic glasses (MGs), which have disordered atomic structures, are unique materials with a wide range of potential applications due to their high strength, elasticity and corrosion resistance [1–4]. It is believed that the β relaxation modes are correlated with structural heterogeneity and mechanical properties [5–7]. The intensive research has currently focused on the relaxation dynamics of MGs [8–11]. Despite extensive efforts have been made, the physical origin of relaxation modes is still not yet well understood. Due to the fact that relaxation processes in MGs are sluggish, it is difficult to get information of the relaxation dynamics directly. However, the application of a step strain can provide a straightforward method for extending the dynamic range of glasses, therefore these sluggish relaxations can be detected effectively. Combining with tensile experiments and stress relaxation spectra, some researchers proposed a flow unit perspective to microscopically describe the evolution of the dynamical heterogeneities with temperatures [12, 13]. However, the picture of flow units or deformation units is conceptual and phenomenological, the precise identification of the flow units remains elusive and ambiguous. Therefore a clear picture of evolution of the dynamic mechanisms in MGs with microscopic information is still required. Recently, some experimental results showed the crossover of relaxation modes from the stochastic activation of the localized shear transformation zones to cooperative motion of these deformation units [14–16]. Nevertheless, limited by the experiment conditions, the applied strain is usually confined within small elastic regime, which is below 2% for the conventional MGs. The processes of activation and evolution of flow units under large applied strain, even above yield strain,
⁎ Corresponding authors. E-mail addresses: [email protected] (H.Y. Bai), [email protected] (P.F. Guan).
http://dx.doi.org/10.1016/j.scriptamat.2017.02.048 1359-6462/© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
are unachievable in experiments. Thus the dynamics of metallic glasses with large imposed strain remains unclear. In this letter, we employ molecular dynamic simulations as a computational microscope to obtain detailed insight into the dynamics of Zr50Cu50 MG by means of stress relaxation. We find direct evidences of strain-accelerated relaxation dynamics in MGs. We correlate this phenomenon with the evolution of local irreversible rearrangements of atoms in glasses upon increasing applied strain, and these local irreversible rearrangements can be pictured as activated flow units. Below the critical applied strain, stress relaxation dynamics can be progressively accelerated with increasing imposed strain, which relates to the propagation of stochastic activation of the flow units. However, beyond the critical applied strain, the dynamics of the strained-glass keeps nearly unchanged due to the occurrence of avalanche of flow units and the uniform mechanical flow behavior. Furthermore, we show that the difference of dynamics at different temperatures decreases with increase of the applied strain below the critical applied strain, while this difference can be almost negligible when the imposed strain exceeds the critical applied strain. The model system is made of 27, 000 atoms, representing an alloy of Zr50Cu50 MG. Atoms interact with the embedded atom method (EAM) potential [17]. Molecular dynamics (MD) simulation was carried out using the open source LAMMPS package [18]. The periodic boundary condition was maintained throughout the simulation, and temperature was kept a constant through the Nose-Hoove thermostat [19]. A glassy state was obtained by melting and equilibrating the system at high temperature (2000 K) for 50 ps, then cooling the liquid to the designated temperatures with the cooling rate of 1012K/s, during which the cell sizes were adjusted to give zero pressure within the constant number, pressure and temperature (NPT) ensemble. The change in the energy
76
Y.C. Wu et al. / Scripta Materialia 134 (2017) 75–79
at Tg(= 850 K) suggests a glass transition at the time scale of the MD simulation. Various initial tension strains εi were applied to the system with a strain rate of 109/s and the constant number, volume and temperature (NVT) ensemble was employed for stress relaxation measurement. Then the stress decay with the loading time up to 2000 ps at 300 K and 400 K (≪Tg = 850 K) was recorded. The applied strain ranges from 1% to 20%, which covers both elastic and plastic regime and no cavitation occurs in both tensile and stress relaxation process. In order to reveal the influence of initial state on the stress attenuation in the stress relaxation process, isoconfigurational ensemble introduced by Widmer-Cooper and Harrowell was employed [20,21]: 100 independent stress relaxation simulations were performed, which all started from the same particle configuration but with momenta randomly assigned from the Maxwell-Boltzman distribution at the interested temperature. Typical isothermal stress-relaxation curves are showed in Fig. 1(a) for various applied strains at T = 300 K, and the temporal-dependent stress σ(t) is normalized by the initial stress σ0. Generally, the calculated σ(t)/σ0 shows a rapid decrease from 1 at the initial time, and then obviously slows down and decays sluggishly with time. These features are consistent with experimental stress relaxation findings [13,14], confirming the validity and reliability of studying of stress relaxation dynamics of glasses with molecular dynamics simulations. It is apparent that the decay of σ(t)/σ0 accelerates as the applied strain increases until it almost collapses to a master curve with the imposed strain exceeding a certain value (εc = 8%). As shown in the inset of Fig. 1(b), the critical
strain εc(=8%) can be regarded as the yield strain of the simulated metallic glass system at 300 K, implying that the decay of stress in strained metallic glass is correlated with the applied strain and a transition of relaxation behavior may occur around the critical strain εc. Since the scaled stress attenuation degree Δσ =(σ0 −σt)/σ0 during the stress relaxation can reflect the intrinsic features of the relaxation behavior of glasses [13,22], we collect Δ σ = 1 − σ(t = 200 ps)/σ0 for various imposed strains at 300 K, and the Δ σ as a function of εi is shown in Fig. 1(b) (we assume Δσ =0 for εi = 0%). It is found that Δσ shows a linear increment with εi for εi b εc; nevertheless, Δ σ maintains constant as εi N εc, confirming that the relaxation process is sensitive to εi for εi b εc, but insensitive to εi for εi N εc from the results shown in Fig. 1(a) and (b). The change of the correlations between εi and Δ σ suggests the crossover of the relaxation behaviors near to εc. The deviation of the data point with εi = 0% may suggest another relaxation mechanism for ultra-small applied strain, which is difficult to investigate due to the limited timescale in computer simulations. To explore the underlying atomic level mechanism that accounts for the crossover behaviors of the stress relaxation distinguished by εc, we utilize the ability of MD simulations to examine individual atomic displacement processes. We calculate the mean square atomic displacement u for each atom during a time interval of Δt = 200 ps for various applied strain at T = 300 K. Fig. 1(c) shows a typical set of the displacement probability density function p(u) for a range of strain at 300 K, and , where P(u) is the distribution that p(u) is defined as pðuÞ ¼ PðuþΔuÞ−PðuÞ Δu quantifies the probability of finding atomic displacement x ≤ u. It is
Fig. 1. (a) Stress relaxation data (symbols) normalized by stress at initial time σ0 at T = 300 K, and the corresponding fitting curves (solid lines) of KWW function. (b) Stress drop data of stress relaxation Δσ at T = 300 K. The inset is a typical tensile stress-strain curve of the Zr50Cu50 metallic glass at T = 300 K, and the yield strain is about 8%. (c) Displacement probability density p(u) at T = 300 K for different ε. up representing the most probable value of u of all the atoms for different ε. (d) Relationship between 〈u2〉 and Δσ for all ε.
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found that the probability distribution broadens with increasing applied strain for εi b εc, reflecting the increase of dynamic heterogeneity. The increase of up, the peak of p(u), suggests the faster dynamics of atoms under higher applied strain. More interesting, the collapse of the curves for εi N 8% implies the identical atomic dynamic properties as the applied strain is larger than the critical strain, which is consistent with the results shown in Fig. 1(a) and (b). It provides the direct evidence that the dynamic properties of atoms determine the stress relaxation behaviors in MGs. To quantify the overall atomic dynamics, we calculate the N
with mean-squared displacement u2 ðtÞ ¼ ∑i¼1 jri ðtÞ−ri ð0Þj2 t = 200 ps for all applied strains during the stress relaxation. The Δ σ as a function of 〈u2〉 is shown in Fig. 1(d). It presents a linear relationship between 〈u2〉 and Δσ for all strains, and the data points for εi N 8% converge to one point. In addition, the deviation of the data point with εi = 0% is in good accordance to the result shown in Fig. 1(b). The strong correlation between Δσ and 〈u2〉 confirms that the stress relaxation is an effective method to investigate the dynamic properties of MGs in atomic level. Furthermore, it provides an opportunity for understanding the corresponding atomic relaxation mechanisms in MG under various applied strains. The crossover of the stress relaxation behaviors around the yield strain implies the distinct atomic relaxation mechanisms controlled by the applied strain. To characterize the individual atomic relaxation during the stress relaxation, we further investigate the local irreversible rearrangement of atomic configuration evaluated by nonaffine displacement of the central atom i relative to its neighbor atoms j: D2mini ¼
h i2 1 ! ! ! ! ε r j ðt þ Δt Þ− r i ðt þ Δt Þ ; ∑ j r j ðt Þ− r i ðt Þ−! N
! where N is the number of nearest neighbor of atom i, r i ðtÞ is the position of atom i at the time t, Δ t is the time interval for the plastic rear! rangement, and ε is the maximum local elastic strain tensor [23]. Here we set Δt =200 ps as used in Fig. 1. The 2D snapshots that capture the distribution of nonaffine displacement D2min under various strains are shown in Fig. 2. Here, we chose D2c ~7 as the cutoff value of D2min for
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all strained samples (for more details, see Supplemental Material, Fig. S1), and the regions with D2min N D2c , just analogous to flow units, present inhomogeneous distribution [12,13,24,25], which confirms the intrinsic heterogeneity in MG [26–28]. As shown in Fig. 2(a), the individual flow units are activated even when the applied strain is only 1%, which is in excellent agreement with the previous simulation works of the existence of local atomic rearrangements in elastic deformation region in MG [24]. As the applied strain increases, the density of the activated flow units increases (see Fig. 2(b)). However, these activated regions are relatively independent (see Fig. 2(a) and (b)), which are corresponding to a stochastic and isolated activation of the flow units. As the imposed strain exceeds yield strain or εc = 8%, the connective percolation of activated flow units can be observed (see Fig. 1(c)). The stress relaxations in system with εi = 15% N εc have similar behaviors of atomic rearrangement (see Fig. 2(d)), showing that the plastic flow occurs as the applied strain becomes larger than the yield strain in the simulated system. The different choices of cut off value do not affect the general propagation trend of flow units and similar results are obtained. Since the composition always plays an important role in controlling the behaviors of MG, we carefully examined the distribution of D2min of each constituent (Zr and Cu, see Supplemental Material, Fig. S2), and found that they exhibit similar dynamic behaviors. It suggests that the constituent effect is weak in describing the dynamic heterogeneity behaviors during the stress relaxation, at least in our system. The crossover from stochastic activation to percolation of flow units reveals the distinct atomic level mechanisms of the stress relaxation behaviors, which are closely dependent on the critical applied strain εc. For more comprehensive understanding of the strain-dependent stress relaxation process in MG, we further investigate the related relaxation time under various applied strains, and thereby the temperature influence. According to previous studies, the stress relaxation curves shown in Fig. 1(a) can be described by Kohlrausch-Williams-Watts β
(KWW) form of σðtÞ ¼ σ 0 expð− τt Þ þ σ r, where σ0 is the initial stress, σr is the residual stress at infinite time, τ is the characteristic relaxation time and β is the non-exponential parameter related with dynamic heterogeneity. Since the stress relaxation is based on the activation of
Fig. 2. (a)–(d) A series of snapshots capturing nonaffine displacement D2min in tensiled Zr50Cu50 MG samples, and from (a) to (d), ε=1%, 5%, 8%, 15%, respectively.
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relaxation units similar to those happened in creep process, a good fitting of stretched-exponential function can be obtained, and it well demonstrates that the stress relaxation process is highly heterogeneous on microscopic scale [29]. It can be obviously seen that τ decreases exponentially with εi b 8 %(=εc) from Fig. 3(a), and the strain dependence of the characteristic relaxation time τ can be fitted by the equation τ = τ∞ exp [ε1/3]. In the contrast, τ keeps almost a constant for εi N 8%, which is coincident with the tendency of stress attenuation with increasing strain as described in Fig. 1(b). In Fig. 3(b), it shows that the exponent β decreases with strain within elastic regime, evidencing that the distribution of relaxation time is broadened markedly by increasing the strain, which is consistent with the increasing width of p(u) associated with enhancing dynamic heterogeneity as shown in Fig. 1(c). The decrease of τ for lower strain region implies that the external strain reduces the apparent activation energy barrier dramatically in the local energy landscape through the self-energy of the related external stress. For the larger strain region that is over the yield strain, the platform illustrates that the activation energy barrier becomes insensitive to the external strain, implying that the activation energy barrier reaches the minimum value and the relaxation process is mainly contributed by highly cooperative movement. The conclusion is evidently supported by the forming of plastic flow in high strain region (see Fig. 2(c) and (d)). Furthermore, the temperature effect for the relaxation process was studied and the related data for 300 K and 400 K are shown in Fig. 3(c). It is found that both temperature and strain are equally important in controlling relaxation dynamics of MGs [30–32]. Either temperature or external strain (stress) increases can reduce the relaxation time
and accelerates the dynamic process in MGs, confirming the regulation of the equivalence between force and temperature. MGs consist of tightly bonded atomic regions (also called solid-like regions) and loosely bonded atomic regions (or liquid-like regions) [25,33]. The liquid-like sites in glasses were even directly imaged by aberration-corrected transmission electron microscope [33]. Recent simulations and experiments have showed that these liquid-like regions with viscoelastic behavior act as the flow units in MGs [24,25]. Therefore, we can model the MG into two parts: flow units and elastic matrix. Flow units are simplified as a distribution of dashpots within elastic matrix. These localized liquid-like deformation units or flow units acting like a liquid in a sufficient long time can accommodate external stress (strain) or elevated temperature by relaxing or flowing, and stress decay should be ascribed to the inhomogeneous activation of flow units [12,13,15,16,25]. Under a step strain loading, strain causes energy minima to disappear by a decrease of energy barriers, which renders the system mechanically unstable and leads to local structural change [16, 34,35]. During the process of stress relaxation, a fraction of atoms will undergo nonaffine deformation, and thus transform into flow units [13]. When the applied strain is largely limited, only few localized nano-scale flow units with smaller energy barriers at initial state can be activated, just as shown by the atoms experiencing larger nonaffine displacement (red balls) in Fig. 2(a). With an increase of applied strain, flow units are gradually stochastically activated corresponding to the proliferation of flow units shown in Fig. 2(b). Consequently, relaxation rate of system progressively accelerates. As imposed strain exceeds critical strain, adjacent weak-bonded regions around flow units also
Fig. 3. (a) Relaxation time τ extracted from fitting KWW function to raw stress relaxation data for all ε, and the corresponding fitting curve (solid line). (b) The fitting parameter β for all ε, and the solid line guides the eyes. (c) Comparison of relaxation time τ between 300 K and 400 K at ε=2%, 4%, 8%, 15%, respectively.
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increasingly transform into a liquid-like state, owing to the applied strain lowers energy barrier between adjacent mega-basins in energy landscape [13,16]. Meanwhile the concentration of flow units reaches a critical value, the majority of flow units or liquid-like zones in the sample start to cooperate with each other, and a connective percolation of flow units occurs corresponding to the coalescence process as shown in Fig. 2(c) and (d). At this stage, critical transition from broken-ergodic to ergodic on energy landscape happens. This process is similar to the thermally driven glass to supercooled liquid transition. Therefore, the whole system can accommodate the applied strain, and the rate of stress relaxation first comes to peak and then turns to a constant. Finally, the relaxation dynamics of strained-glasses is nearly unvarying due to the forming of plastic flow. In summary, with the analysis of stress relaxation of a strained metallic glass, the evolution of relaxation dynamics and its intimate relation to the activation of flow units are investigated. We show a crossover from stochastic activation to percolation of flow units corresponding to a critical applied strain in a simulated Zr50Cu50 MG system. A conception of the critical applied strain is proposed as a watershed between stochastic activation and percolation of flow units in stress relaxation of MGs. Our results may shed a light for comprehensive understanding of the relaxation dynamics, mechanical behaviors, and especially their correlation with structural evolution in MGs. Acknowledgements Insightful discussions with P. Luo, Y. T. Sun, and L. J. Wang are highly acknowledged. We also thank D. Q. Zhao and D. W. Ding for helpful discussions. We also acknowledge the computational support from Beijing Computational Science Research Center (CSRC). B. S. and P. G. are supported by the National Natural Science Foundation of China (Grant No. 51571011, U) and the MOST 973 Program (Grant No. 2015CB856800). This work was also supported by the NSF of China (51571209 and 51461165101) and the Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-JSC017). Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.scriptamat.2017.02.048.
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References [1] A.L. Greer, Science 267 (5206) (1995) 1947. [2] W.H. Wang, Adv. Mater. 21 (45) (2009) 4524. [3] M.D. Demetriou, M.E. Launey, G. Garrett, J.P. Schramm, D.C. Hofmann, W.L. Johnson, R.O. Ritchie, Nat. Mater. 10 (2) (2011) 123. [4] J. Schroers, Phys. Today 66 (2) (2013) 32. [5] T. Ichitsubo, E. Matsubara, T. Yamamoto, H. Chen, N. Nishiyama, J. Saida, K. Anazawa, Phys. Rev. Lett. 95 (24) (2005) 245501. [6] H.B. Yu, X. Shen, Z. Wang, L. Gu, W.H. Wang, H.Y. Bai, Phys. Rev. Lett. 108 (1) (2012) 015504. [7] H.B. Yu, W.H. Wang, H.Y. Bai, K. Samwer, Nat. Sci. Rev. 1 (3) (2014) 429–461. [8] H.-B. Yu, W.-H. Wang, K. Samwer, Mater. Today 16 (5) (2013) 183. [9] P. Luo, Z. Lu, Y.Z. Li, H.Y. Bai, P. Wen, W.H. Wang, Phys. Rev. B 93 (10) (2016) 104204. [10] V.M. Giordano, B. Ruta, Nat. Commun. 7 (2016) 10344. [11] S. Ashtekar, G. Scott, J. Lyding, M. Gruebele, J. Chem. Phys. 1 (13) (2010) 1941. [12] Y.Z. Li, L.Z. Zhao, C. Wang, Z. Lu, H.Y. Bai, W.H. Wang, J. Chem. Phys. 143 (4) (2015) 041104. [13] Z. Wang, B.A. Sun, H.Y. Bai, W.H. Wang, Nat. Commun. 5 (2014) 5823. [14] W. Jiao, B.A. Sun, P. Wen, H.Y. Bai, Q.P. Kong, W.H. Wang, Appl. Phys. Lett. 103 (8) (2013) 081904. [15] L.Z. Zhao, Y.Z. Li, R.J. Xue, W.H. Wang, H.Y. Bai, J. Appl. Phys. 118 (15) (2015) 154904. [16] L.Z. Zhao, R.J. Xue, Y.Z. Li, W.H. Wang, H.Y. Bai, J. Appl. Phys. 118 (24) (2015) 244901. [17] Y. Cheng, E. Ma, H.W. Sheng, Phys. Rev. Lett. 102 (24) (2009) 245501. [18] S. Plimpton, J. Comput. Phys. 117 (1) (1995) 1. [19] S. Nosé, J. Chem. Phys. 81 (1) (1984) 511. [20] A. Widmer-Cooper, P. Harrowell, H. Fynewever, Phys. Rev. Lett. 93 (13) (2004) 135701. [21] A. Widmer-Cooper, P. Harrowell, Phys. Rev. Lett. 96 (18) (2006) 185701. [22] Z. Lu, W.-H. Wang, H.-Y. Bai, Sci. China Mater. 58 (2) (2015) 98. [23] M.L. Falk, J.S. Langer, Phys. Rev. E 57 (6) (1998) 7192. [24] Z. Lu, W. Jiao, W.H. Wang, H.Y. Bai, Phys. Rev. Lett. 113 (4) (2014) 045501. [25] Y.C. Hu, P.F. Guan, M.Z. Li, C.T. Liu, Y. Yang, H.Y. Bai, W.H. Wang, Phys. Rev. B 93 (21) (2016) 214202. [26] Y.H. Liu, D. Wang, K. Nakajima, W. Zhang, A. Hirata, T. Nishi, A. Inoue, M.W. Chen, Phys. Rev. Lett. 106 (12) (2011) 125504. [27] H. Wagner, D. Bedorf, S. Küchemann, M. Schwabe, B. Zhang, W. Arnold, K. Samwer, Nat. Mater. 10 (6) (2011) 439–442. [28] F. Zhu, H.K. Nguyen, S.X. Song, D.P.B. Aji, A. Hirata, H. Wang, K. Nakajima, M.W. Chen, Nat. Commun. 7 (2016) 11516. [29] M.D. Ediger, Annu. Rev. Phys. Chem. 51 (1) (2000) 99–128. [30] H.B. Yu, R. Richert, R. Maaß, K. Samwer, Phys. Rev. Lett. 115 (13) (2015) 135701. [31] P. Guan, M. Chen, T. Egami, Phys. Rev. Lett. 104 (20) (2010) 205701. [32] Z. Lu, X.N. Yang, B.A. Sun, Y.Z. Li, K. Chen, W.H. Wang, H.Y. Bai, Scr. Mater. 130 (2017) 229. [33] P.Y. Huang, S. Kurasch, J.S. Alden, A. Shekhawat, A.A. Alemi, P.L. McEuen, J.P. Sethna, U. Kaiser, D.A. Muller, Science 342 (6155) (2013) 224. [34] D.L. Malandro, D.J. Lacks, Phys. Rev. Lett. 81 (25) (1998) 5576. [35] D.J. Lacks, M.J. Osborne, Phys. Rev. Lett. 93 (25) (2004) 255501.