Amorphous physics and materials: Secondary relaxation and dynamic heterogeneity in metallic glasses: A brief review

1 / 24  Secondary relaxation and dynamic heterogeneity in metallic glasses: A brief review J.C. Qiaoa,b,c,*, Q. Wangd, D. Crespoe, Y. Yangb, J.M. Pelletierc,* aSchool of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, China bCentre for Advanced Structural Materials, Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon Tang, Kowloon, Hong Kong SAR, China cUniversité de Lyon, MATEIS, UMR CNRS5510, Bat. B. Pascal, INSA-Lyon, F-69621 Villeurbanne cedex, France dKey laboratory for structures, Institute of materials science, Shanghai University, Shanghai, China eDepartament de Física, EETAC, Universitat Politècnica de Catalunya, 08860-Castelldfels, Barcelona, Spain (Submitted to Chinese Physics B) Revised version: November 22, 2016 *Corresponding author: Dr. J.C. Qiao E-mail address: [email protected] Prof. J.M. Pelletier E-mail address: jean-m[email protected] 2 / 24  Understanding mechanical relaxation, such as primary ( and secondary (relaxation, is key to unravel the intertwined relation between the atomic dynamics and non-equilibrium thermodynamics in metallic glasses. At a fundamental level, relaxation, plastic deformation, glass transition and crystallization of metallic glasses are intimately linked to each other, which can be related to atomic packing, inter-atomic diffusion and cooperative atom movement. Conceptually,  relaxation is usually associated with structural heterogeneities intrinsic to metallic glasses. However, the details of such structural heterogeneities, being masked by the meta-stable disordered long-range structure, are yet to be understood. In this paper, we briefly review the recent experimental and simulation results that were attempted to elucidate structural heterogeneities in metallic glasses within the framework of  relaxation. In particular, we will discuss the correlation among  relaxation, structural heterogeneity and mechanical properties of metallic glasses. Keywords: Metallic glass; Structural heterogeneity; Mechanical property; Secondary relaxation; Physical model PACS: 64.70.pe, 62.40.+i,78.55.Qr            3 / 24  1.Introduction A glassy material is a matter with a disordered structure possessing no long-range translational periodicity. Metallic glasses, also known as amorphous alloys, are relatively newcomers to the family of glassy materials. Metallic glasses are usually obtained by quenching from the melts, which exhibit a unique combination of physical, chemical and mechanical properties due to their intrinsic disordered atomic structure and (or) production route.[1-5] As prepared by rapid solidification, metallic glasses are in a non-equilibrium state below the glass transition temperature (Tg), which differs from that of their crystalline counterparts. It is well documented that plastic deformation of conventional crystalline solids is closely connected to intrinsic structural defects (i.e. dislocations and grain boundaries), on the basis of which a theoretical framework of structure-property relations has been well constructed. In contrast to crystalline solids, it is quite difficult to characterize “defects”, if any, in a glassy material, such as polymer glass, oxide glass, granular and chalcogenide glass, soft colloidal glass or metallic glass. Conceptually, this is so partly because it is much easier to describe order than disorder. By definition, each construct of a disordered structure is different and the notion of a glass structure is simply an ensemble of many such different disordered structure constructs with equivalent macroscopic properties. In the metallic glass literature, one of the longstanding and central research topics is to establish the structure-property correlation. The relaxation modes in metallic glasses provide an important key to probe “defects” in their amorphous structure and to understand the corresponding structure-property correlation.[6-10] Yet the glass transition is consequence of a relaxation process, designated as primary or . The primary relaxation process is a global, structural atomic reordering process, which eventually ends in the glass transition if temperature is increased. However, the intensity of  relaxation decreases as the glass is undercooled. Well below Tg, the most important relaxation mode for glassy materials is  relaxation, which is closely connected to localized motions of atoms or molecules. Contrary to other glasses, such as molecular glass, metallic glasses are an “atomic glass” simply made up of individual atoms via isotropic metallic bonding, which constitute a conceptually simpler glass structure. The lack of an intrinsic microstructure in metallic glasses implies that their properties mainly stem from structural disorder. As a consequence, metallic glasses provide a model system to study the various intriguing phenomena in glass science, such as glass dynamics, structural relaxation, 4 / 24  crystallization and glass transition. Through the study of metallic glasses, one may not only gain insights into the phenomena of fundamental importance, such as glass transition, but also understand the physical origin of the unique mechanical properties that may drive future engineering applications, such as yield strength, fracture toughness, ductile-to-brittle transition temperature and damping. Nevertheless, it is still challenging as of today to develop a structure-property correlation for metallic glasses.[6,10] The thermal events of glassy materials always go hand-in-hand with their dynamic features. The typical dynamical features of glassy materials may be summarized as follows: (i) the non-Arrhenius temperature dependence of viscosity [11], (i.e. since the steepness of a relaxation can be characterized by the fragility parameter m, which measures the deviation of a liquid viscosity from an Arrhenius relation,.[12] fragile liquids (with large m values) usually exhibit pronounced  relaxations in the form of distinct peaks or broad humps, while strong liquids (with small m values) display  relaxations as “excess wings” . Meanwhile, strong liquids often show an Arrhenius-like behavior whereas fragile ones do not, as shown in Fig.1(a).) (ii) the split of  and  relaxation, (i.e. although there is one relaxation behavior at a high temperature,  relaxation becomes “frozen” when the temperature is near Tg; however,  relaxation persists below Tg and is decoupled from  relaxation. It is important to note that  relaxation could be described bythe Vogel– Fulcher–Tammann (VFT) equation while  relaxation obeys an Arrhenius law (Fig.1 (b)) (iii) complex relaxation processes, (i.e. in general, there are two kinds of atomic cooperative rearrangements which correspond to two types of relaxation processes, i.e. the primary relaxation ( relaxation) and the secondary relaxation ( relaxation).[13-15] As illustrated in Fig.1 (c),  relaxation prevails within a lower frequency domain (~10-2 Hz) while slow  relaxation, as a fast process, prevails within the range from 102 to109 Hz. A boson peak can be detected when the frequency further increases to around 1012 Hz. Physically,relaxation stems from large scale rearrangements of atoms or molecules, being closely related to dynamic glass transition; while  relaxation constitutes the principal source of glass dynamics as  relaxation disappears below the glass transition temperature Tg.) (iv) non-exponential relaxation dynamics. (i.e. it is widely accepted that the relaxation dynamics in amorphous materials displays a typical non-exponential behavior, which could be described by Kohlrausch-Williams-Watts (KWW) equation (as shown in Fig.1 (d)). Many recent investigations have revealed that structural or dynamical 5 / 24  heterogeneities are intrinsic to metallic glasses.[4] The correlation between the  relaxation and structural heterogeneities is one of the challenging issues in glassy physics. In this paper, we briefly review the recent advances in the research on secondary relaxation in metallic glasses. In Section 2 and Section 3, we discuss the state-of-art research on the relaxation behaviors (i.e. the slow and fast  relaxations) of metallic glasses of metallic glasses. In Section 4, we review the various connections among mechanical properties, physical properties and relaxations in metallic glasses. Finally, we give general conclusions and an outlook for the research of mechanical relaxations in metallic glasses. 2.Slowrelaxationofmetallicglasses When a glass forming liquid is cooled fast enough below its melting point, crystallization is inhibited and the liquid undergoes supercooling. The relaxation time of the supercooled liquid increases dramatically in the supercooled temperature region until it becomes so large that the liquid fails to flow on an experimental timescale, thereby leading to glass transition. From a kinetic viewpoint, one may argue that glass transition is purely of a kinetic origin, involving no actual phase transition. Therefore, the atomic structure remains unaltered with only the timescale for atomic movements increasing with supercooling. In such a case, relaxation phenomena can be regarded as a manifestation of the kinetic behavior in a frozen liquid. The atomic movement is global in the liquid state while constrained or local in the glass state. Dynamical mechanical analysis, also called mechanical spectroscopy, is a powerful tool to investigate the mechanical relaxation behavior of metallic glasses. The complex shear modulus ( G"G'G i ) is usually obtained with the storage (G') and loss (G") shear modulus. The loss factor (also named as internal friction) tan = G"/G' can be also extracted. Early studies on the dynamic characteristics of Pd-[18,19], Fe-, and Co-based metallic glasses [20,21] varied with physical aging or crystallization. Because those prior measurements were carried out over a wide temperature range, from low temperatures to the glass transition temperature, the activation energy of internal friction can be estimated, falling in the range of 0.5-1.5 eV. The observed internal friction was then considered to result from the diffusion of constituent atoms. Morito et al. examined the effect of heat treatment on internal friction and the Curie point of the Fe-Ni-Cr-P-B amorphous alloys, which showed that these alloys underwent a reversible change among various metastable states. In addition, they proposed that the change in internal friction during an isochronal heating process stemmed from an activation-energy distribution of the relaxation process in a glass structure.[22] In the 1990s, a series of metallic glasses, such as La-based and Zr-based systems, 6 / 24  were developed following Inoue et al. and Johnson et al, which exhibited high thermal stability in the supercooled liquid state.[23-25] Okumura et al. were the first to identify a dynamic glass transition phenomenon in these metallic glasses and also reported sub-Tg relaxation.[26-28] Subsequently, Perera et al. measured the activation energy and/or relaxation time distribution for the Pt-Ni-P, Pd-Ni-P, and Zr-Ti-Cu-Ni-Be metallic glasses.[29,30] Pelletier et al. showed that the dynamic glass transition and sub-Tg relaxation observed in the Pd-Ni-Cu-P alloys were equivalent to the  and  relaxations, respectively, as observed in other glasses.[31] Fig.2 (a) shows the storage modulus G′ and the loss modulus G″ as a function of temperature in a typical La60Ni15Al25 metallic glass.[32] The features of the dynamic mechanical behaviors of the La60Ni15Al25 metallic glass can be described as follows. (i) At temperatures below 450 K, the material stays in the amorphous state and the storage modulus G′ remains nearly constant and high whereas G″ is negligibly low. This is a temperature range that corresponds a typical elastic behavior of the metallic glass. Interestingly, a secondary peak is observed in the loss modulus G″, around 360 K, being accompanied by a slight decrease in the storage modulus G′. These slight variations are typical of a secondary relaxation for amorphous materials. (ii) At intermediate temperatures, G′ decreases strongly while G″ reaches a maximum value. This corresponds to the so-called main or α relaxation. The  relaxation is connected to a dynamic glass transition process in amorphous materials. (iii) At higher temperature (above about 500 K), crystallization occurs, which causes a large increase in the storage modulus G′ and a maximum in the loss modulus G″ Fundamentally, the  and  relaxations of glassy materials are temperature and frequency (or time) dependent. As mentioned above, the  relaxation time shows a non-Arrhenius temperature dependence, while the  relaxation time shows an Arrhenius behavior over a wide range of temperatures. They approach to each other with increasing temperature and frequency. The mode coupling theory predicts that these two relaxations merge into a single relaxation termed “-process” in a crossover region. In polymer glasses, the merging phenomenon can be well observed e.g. by dielectric spectroscopy.[33] In metallic glasses, the merging of  and  relaxation is always impeded by crystallization at a high temperature. For this reason, no direct experimental observation of the merging between them is reported for metallic glasses. The prominent peak of β relaxation in La-based metallic glasses allows one to obtain the activation energy based on the Arrhenius equation. The information one can obtain in this manner the apparent activation energy of the β relaxation in amorphous materials can be calculated from the Arrhenius equation: f = f∞ exp(−Uβ/RT), where R is the gas constant, and f∞ is a pre-factor equal to the high temperature limit of the frequency. With the Arrhenius equation, one can extract the activation energy of the β relaxation of metallic glasses using the Kissinger 7 / 24  method [34]. The activation energy of β relaxation (or excess wing) determined this way for metallic glasses generally obeys~26, as for organic glass liquid formers and other glassy materials.[35-38] This result is important, which supports the proposal from the cooperative shear model that the shape of the potential landscape associated with   relaxation should be similar to that with   relaxation. As this scaling relation holds for non-metallic glasses, Ngai et al. provided an alternative explanation from the perspective of coupling model [32]. Aside from humps,  relaxation of glassy materials also manifests as an excess wing on the mechanical spectroscopy. Many previous publications demonstrated that βrelaxation and  relaxation are the universal characteristics of glassy materials [4,6,7,10]. Contrary to La-based and Pd-based metallic glasses, the Zr-based metallic glass exhibits another form of secondary relaxation according to Rösner et al.,[39] the so-called excess wing at the high-frequency side (low-temperature side) of the dynamic  relaxation peak – as shown in Fig.2(b) [11]). Thereafter, Ngai theoretically demonstrated that the  relaxation of the Zr-Al-Cu alloy can occur at the temperature and/or frequency region where an excess wing is observed, which supported the idea of Rösner et al. [40] Subsequently, Zhao et al. reported that, depending on the coupling strength (difference in activation energy) of Johari-Goldstein (JG) relaxation and  relaxation, the JG relaxation of glassy materials is either distinctly isolated and observed as slow  relaxation, or partially covered by  relaxation and observed as an excess wing. These behaviors are strongly dependent on the fragility of the supercooled liquid.[41] Generally, a higher activation energy is required to activate local atomic motions with a higher peak temperatures Tβp in metallic glasses, which follows the empirical relationship, Uβ ≈ 33(±1)RTβp. It was found that the peak temperatures Tβp of the β relaxation increases almost linearly with its activation energy Uβ (inset of Fig.2 (c)).[11] Interestingly, this scaling behavior of metallic glasses is in good agreement with that of amorphous polymers, i.e. Uβ ≈31.5RTβp (at the driving frequency of 1 Hz). This correlation between Uβ and Tβp suggests that the high T limit, fmax, might be approximately the same (≈ 2 × 1014 Hz) for all metallic glasses. Moreover, many previous investigations indicated that  relaxation acts as a precursor to the  relaxation.[4,14] Based on the activation energy of relaxation U and relaxation U obtained for typical metallic glasses, the ratio of U /U is seen ranging from 3 to 8. [16,42,43] In general, relaxation is strongly compositional dependent [44], sometimes manifesting as a pronounced peak, sometimes as an “excess wing” or even a “shoulder”. Next, let us discuss the micro-alloying effect on the  relaxation in metallic glasses. The temperature dependence of the loss modulus G"/ G"max (G"max is the loss modulus at the peak of α relaxation) of the Cu46Zr47-xAl7Dyx (x=0, 0.5, 1, 2 and 4) 8 / 24  bulk metallic glasses is shown in Fig.3 (a).[45] Clearly, the relaxation of Cu46Zr47-xAl7Dyx is suppressed by increasing the content of Dy. From a mixing enthalpy point of view, large and similar negative values of the mixing enthalpy between the constituent atom pairs favour pronounced relaxation in metallic glasses [44,46,47] while positive or significant mismatch in the enthalpy of mixing suppresses relaxation. On the basis of the empirical rules for determination  Hmix, the mixing enthalpy of Cu46Zr47-xAl7Dyx metallic glasses can be estimated, as shown in Fig.3 (b).[45] A similar behaviour was verified for the Pd-based and La-based metallic glass systems in which relaxation tends to be suppressed with the increasing concentration of Ni, the element causing a large mixing enthalpy mismatch. [44,46,47] Structural relaxation like physical aging and rejuvenation is an intrinsic feature of glassy materials, which originates from local processes that causes the whole system migrating into a more energetically stable state [49-51]. In practice, glasses with enhanced thermodynamic and kinetic stability are attractive for both scientific research and engineering applications. Fig.4 shows the evolution of the loss factor tan δ with temperature during a continuous heating of cold-rolled samples. [48] It is important to note that the higher is the cold-rolling ratio the higher is the loss factor. In comparison with physical aging, cold-rolling introduces heterogeneities and leads to the rejuvenation of metallic glasses. In this scenario, one can anticipate that atomic mobility is increased by plastic deformation; in other words, local dynamic heterogeneities are enhanced during cold-rolling. Regarding the structural origin of relaxation and its relationship with the relaxation, Stevenson et al. suggested that  relaxation takes place through the activated events involving compact regions, while relaxation is governed by more ramified, string-like or percolation-like clusters of particles. [52] Indeed  relaxation and  relaxation are usually correlated. As discussed previously,  and  relaxation seen in various glass materials are closely correlated, and the origin of this correlation is of fundamental importance, which, once understood, could help elucidate the complicated glass dynamics in glassy materials. Two important theories emerged in the past to explain the origin of  relaxation. On the one hand, Williams and Watts proposed that relaxation is caused by the slight re-orientational motion of all atoms.[53] On the other hand, Johari and Goldstein proposed that it is caused by the re-orientational motion of atoms or a group of atoms in a loosely packed region.[54,55] The internal friction of metallic glass associated with  relaxation was observed even by the mechanical methods that usually do not cause the re-orientational motion of 9 / 24  atoms. Moreover, this relaxation does not result in the distinct creep of an entire sample. From these facts, Johari claimed that  relaxation is essentially due to the translational motion of atoms and is equivalent to the JG relaxation generated in an “island of mobility.”[56] In light of this, Ichitsubo et al. [57] demonstrated that, by activating  relaxation through an ultrasonic heat treatment, a Pd-Ni-Cu-P metallic glass can be transformed into a nano-scale composite structure, which consisted of a network of metastable crystalline phases. As this phenomenon could not be observed in a simple heat treatment, the formation of the crystalline network is thus suggestive of nanoscale heterogeneities, which were intrinsically frozen-in in the quenched Pd-Ni-Cu-P metallic glass.  relaxation thereby takes place in the weakly bonded regions, which behave like islands of mobility. Additionally, partial crystallization proceeds owing to the atomic translational motion in  relaxation. General speaking, these experimental findings support the description of Johari. The dynamics of supercooled liquids is thought to be spatially heterogeneous, characterized by a dynamic correlation length.[58] Many investigations [59-61] suggested that such a length scale would increase with decreasing temperature (or increasing density), causing the dynamic slow-down and eventually the glass transition. Nevertheless, Wei et al. reported an interesting observation that the correlation length of the Fe50Co50 metallic glass decreases around the glass transition temperature Tg, which is seemingly contradictory to the previous findings and worth further investigation.[62] To determine experimentally the number of units within a correlating region, different metrics have been proposed. In the previous investigations, the number of correlating units Nc (i.e., atoms in a metallic glass, segments in a polymer, molecules in a simple liquid) in metallic glass-forming liquids was found significantly larger than that in other glass-forming materials.[63,64] A causal link between dynamical slow-down and structural heterogeneity was recently suggested on the basis of simulations. For instance, by extensive computer simulations, Tanaka et al. found that the drastic slow-down and the heterogeneous dynamics are the results of critical-like fluctuations of static structural orders.[65] Their results link the structural heterogeneity and the heterogeneous dynamics, which strongly suggests that the  relaxation has a structural origin. The structural order in their work can be interpreted as the medium-range bond-orientational order, which can be linked to the geometry of an equilibrium crystal. 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There are distinct dynamic relaxation modes from the left to the right (from low frequency domain to high frequency domain): primary () relaxation, slow  relaxation, fast process and boson peak;[13] (Courtesy of P. Lunkenheimer, University of Augsburg, Germany). (d) Non-exponential relaxation dynamics of Zr-based metallic glass.[17] Reprinted from Ref. 17, with the permission of AIP publishing.  (a) (b)  (c) (d) 20 / 24  Fig.2 (a) Normalized storage modulus G′ and loss modulus G″ vs temperature in La60Ni15Al25 bulk metallic glass, Gu is the unrelaxed modulus, assumed to be equal to G′ at room temperature.[32] Reprinted from Ref. 32, with the permission of AIP publishing. (b) Temperature dependence of the normalized loss modulus 󰆒󰆒  󰆒󰆒 ⁄ ( 󰆒󰆒 is the peak value of  relaxation in the loss modulus) in various amorphous alloys. Reprinted with permission from ref.11. Copyright (2013) American Chemical Society; (c) Temperature dependence of the normalized loss modulus G″/Gu in typical La-based amorphous alloys. Reprinted with permission from ref.11. Copyright (2013) American Chemical Society (a) (b) (c) 21 / 24   Fig.3 (a) Temperature dependence of the loss modulus G"/ G"max in Cu46Zr47-xAl7Dyx (x=0, 0.5, 1, 2 and 4) metallic glass-forming liquids. (b) Mixing enthalpy of Cu46Zr47-xAl7Dyx(x=0, 0.5, 1, 2 and 4) metallic glasses. The inset exhibits the mixing enthalpy of the constituent atoms.[45] Reprinted from Ref.45, Page No.62-75. Copyright (2016), with permission from Elsevier. (a) (b) 22 / 24  Fig.4 Influence of the cold-rolling ratio (ɛ=45% and ɛ=75%) on the loss factor tan δ in Ti40Zr25Ni8Cu9Be18 bulk metallic glass during continuous heating. In comparison with as-cast state.[51] Reprinted from Ref.51. Copyright (2012), with permission from Elsevier.   23 / 24  Fig.5 (a) The emergence of two secondary relaxations in La56.16Ce14.04Ni19.8Al10 metallic glass on the isochronal spectrum of mechancial spectrosopy;[86] (b) Temperature dependence of the slow β-relaxation and the fast β’-relaxation measured at frequencies of 1, 2, 4, 8 and 16 Hz for Er55Co20Al25 metallic glass;[87] (c) The schematic illustration of mechanisms of mechanical relaxations in amorphous alloy. The red atoms represent the active atoms, and atoms surrounded by green dash line represent flow units. [87] Reprinted from Ref. 87, with the permission of AIP publishing.  (a)  (b) (c)   24 / 24  Fig.6 The intensity of β relaxation during annealing for the ordinary glass. (a) The solid lines are dielectric loss spectra of ordinary glasses annealed at T=110 K, and the annealing time is 0, 1, 4, 15, 63, and 210 h from top to bottom. The dashed line is for the ultrastable glass.[95] Reprinted figure with permission from (Ref.95), Copyright (2015) by the American Physical Society.             