Dynamic Mechanical Relaxation in Bulk Metallic Glasses: A Review

Accepted Manuscript Dynamic Mechanical Relaxation in Bulk Metallic Glasses: A Review J.C. Qiao , J.M. Pelletier

PII:

S1005-0302(14)00093-0

DOI:

10.1016/j.jmst.2014.04.018

Reference:

JMST 355

To appear in:

Journal of Materials Science & Technology

Received Date: 14 February 2014 Accepted Date: 4 April 2014

Please cite this article as: J.C. Qiao, J.M. Pelletier, Dynamic Mechanical Relaxation in Bulk Metallic Glasses: A Review, Journal of Materials Science & Technology (2014), doi: 10.1016/j.jmst.2014.04.018. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Dynamic Mechanical Relaxation in Bulk Metallic Glasses: A Review J.C. Qiao1,2), J.M. Pelletier1,2)*

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1) Université de Lyon, CNRS, France 2) INSA-Lyon, MATEIS UMR5510, F-69621 Villeurbanne, France

[Manuscript received February 14, 2014, in revised form April 4, 2014] *

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Corresponding author. Prof.; Tel.: +33 4 72 43 83 18; Fax: +33 4 72 43 85 28; E-mail address: [email protected] (J.M. Pelletier).

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Prof. Jean-Marc Pelletier’s research concerns the mechanical response of amorphous materials, especially bulk metallic glasses. Zr-based, Ti-based, La-based metallic glasses are for instance investigated. One of the main topics is the influence of thermo-mechanical treatments on mechanical properties of these materials, both at room temperature or at elevated temperature. Mechanical relaxations are investigated in detail in order to understand the influence of chemical composition and atomic arrangement on the atomic mobility in these disordered materials. He has published more than 120 SCI journal papers and presented more than 100 contributions in International Conferences, with the citation of more than 1000 and h-index of 19. Before studying the mechanical response of amorphous materials he was interested in the phase transformation (especially precipitation and ordering), high power laser engineering and various topics. He developed international collaboration, especially with groups in China (Prof. W.H. Wang in Beijing) and in Japan (Prof. A. Inoue and Prof. H. Kato, in Sendai).

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Metallic glasses have aroused considerable interest in the past decades because they exhibit fascinating properties. First, this article briefly outlines the mechanical, thermal properties and application of the metallic glasses. In addition, we focus on the dynamic

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mechanical relaxation behaviours, i.e. main (α) and secondary (β) relaxations, in metallic glasses. The mechanical relaxation behaviors are connected to the mechanical properties and physical properties in glassy materials. The main relaxation in glassy materials is related to the glass transition phenomenon and viscous flow. On the other hand, the β relaxation is linked to many fundamental issues in metallic glasses. In these materials relaxation processes are directly related to the plastic deformation mechanism. The mechanical relaxations, particularly, the β relaxation provides an excellent opportunity to design metallic glasses with desired physical and mechanical properties. We demonstrate the universal characteristics of main relaxation in metallic glasses. The phenomenological models and the physical theories are introduced to describe the main relaxation in metallic glasses. In parallel, we show the dependence of the α and β relaxations on the thermal treatments in metallic glasses. Finally, we analyze the

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correlation between the atomic mobility and the thermo-mechanical treatments in metallic glasses. On the one hand, the atomic mobility in metallic glasses is reduced by physical aging or crystallization. On the other hand, the atomic mobility in metallic glass is enhanced by deformation (i.e. compression and cold rolling). Importantly, to analyze the atomic mobility in amorphous materials, a physical theory is introduced. This model invokes the concept of quasi-point defects, which correspond to the density fluctuations in the glassy materials.

KEY WORDS: Metallic glasses; Mechanical properties; Main relaxation; Secondary relaxation; Physical analysis

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1. Introduction

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Many materials exist in the amorphous state, such as polymers, oxide glasses, organic liquids and metallic glasses. Of particular interest is the fact that metallic glasses have attracted considerable attention over the last decades, since they combine two particular characteristics: their mechanical properties are very outstanding at room temperature (in particular their elastic properties) and they can easily be processed at a moderate temperature compared with their crystallized counterparts[1–5]. The present review addresses these different topics. Taking account of the outstanding properties of metallic glasses, the following outlines will be illustrated in the current review. (1) Fundamental concepts relative to amorphous materials and in particular to metallic glasses: - Different states of the matter: amorphous, supercooled liquid, crystalline. - Brief survey of the development of metallic glasses. - Fascinating properties of metallic glasses. - Some applications of metallic glasses (mainly related to their special mechanical properties). - Open questions in metallic glasses. (2) Thermal stability of the metallic glass (i.e. structural relaxation and crystallization). (3) Mechanical response of bulk metallic glasses (in the amorphous state) either at room temperature or at high temperature. Different components of this response, i.e. elastic, viscoelastic and viscoplastic behaviors are discussed. In parallel, mechanical relaxations in metallic glasses investigated by dynamic mechanical analysis (DMA) are analyzed. (4) It is well-known that the mechanical and physical properties are determined by the microstructure in crystalline solids. There are many works focused on the effect of the physical aging and plastic deformation on the mechanical relaxation for glassy polymers. Unfortunately, the information of the metallic glasses is limited. In this section, we will discuss the influence of the thermo-mechanical treatments on the mechanical properties of the metallic glasses. (5) Physical models developed for the description of mechanical relaxation in the amorphous materials. (6) Conclusions and outlook.

2. General Information on Amorphous Materials and Metallic Glasses 2.1. Concept of amorphous state/supercooled liquid/crystalline solid To analyze the difference between a crystalline solid and the glassy material, the simplest is to describe phenomena that occur during cooling from the liquid state. Fig. 1 displays the variation of the enthalpy (H) or volume (V) depending on the temperature. From the liquid state, when the temperature

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decreases and reaches the melting temperature Tm, a decrease in enthalpy or volume is observed, which corresponds to the formation of the crystalline solid. This is a first-order phase transition, which means that, as long as solid and liquid coexist, the equilibrium of the temperature remains constant and equal to the melting temperature. It is well-known that the crystalline metal alloys have a periodic microstructure. In other words, metallic alloys present a long range order. On the other hand, when the liquid temperature reaches the melting temperature Tm, if the cooling rate is very rapid, the melting liquid system does not have sufficient time to create crystalline germs. Then, the system is in a metastable equilibrium, which is called the supercooled liquid. The decrease in enthalpy (or volume) with the temperature is low and does not present discontinuity with the liquid state. If the temperature continues to decrease, there is a transition at a temperature called glass transition temperature (Tg). When the temperature is below this temperature, the atoms are frozen because the time required for ordering is much longer than that corresponding to the experimental one. Therefore, the system is disordered and structural state is a non-periodic arrangement. It should be noted that the glass transition phenomenon is an open question in the field of condensed matter physics[6,7].

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2.2. History- What is a metallic glass? As we introduced earlier, metallic glasses are solid metallic alloys, which have an amorphous structure with a long-distance disorder at the atomic scale. However, their atomic configuration may exhibit topological short range ordering (TSRO) or chemical short range order ordering (CSRO)[8]. Compared with “conventional” materials, such as metallic alloys, mineral glass or polymers, the metallic glass is new comer of the family of amorphous materials. The first metallic glass (Au75Si25) was obtained by Klement et al. in 1960 (Caltech, USA) with a cooling rate of the order of 106 K/s obtained by splat quenching[9]. This method allows a very high cooling rate, but provides samples only in the form of thin ribbons (typically, the thickness is of the order of tens of micrometres). Starting in 1990, many new alloys (Zr-[10–12], Ti-[13,14], Cu-[15], Pd-[16], Pt-[17]... based metallic glasses) have been developed with more traditional methods and appropriate compositions. Up to now, bulk metallic glasses can be prepared with cooling rate around 1 K/s. The thickness or diameter of amorphous samples reaches the centimetric scale, for example, the very famous composition Vit1 (Zr41.2Ti13.8Cu12.5Ni10Be22.5)[10]. The record has been obtained in the research group managed by Prof. A. Inoue (Tohoku University, Japan) with the Pd42.5Cu30Ni7.5P20 alloy. The sample has a length of 85 mm and a diameter of 80 mm (see Fig. 2)[18]. In addition, Fig. 3 shows some examples of Pd-, Zr-, Cu- and Ni-based bulk metallic glasses[4]. How to obtain bulk metallic glasses? What criteria hold for the choice of composition? Inoue et al. suggested the following empirical rules for the selection of chemical elements[3]: ● The alloy must be composed of at least three elements. ● The difference of size in the atomic elements must be as large as possible, at least 12% between the main elements. ● The enthalpy of mixing between the main elements must be negative. 2.3. Attractive properties of the metallic glass How do metallic glasses differ from other amorphous materials (polymers, organic glasses, oxides glasses…)? Fig. 4 shows a map based on the model of Ashby and represents the relationship between the mechanical resistance and toughness for different materials at room temperature[19]. Metallic glasses have a very high mechanical resistance at room temperature with interesting toughness.

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However, most metallic glasses have a catastrophic failure without significant plastic deformation[1,20]. The low plastic deformation is heterogeneous and associated with the formation of localized shear bands followed by the propagation of these shear bands[21–23]. On the other hand, homogeneous deformation takes place at higher temperature in metallic glasses (typically for T>0.8Tg, where Tg is the glass transition temperature). In those conditions, the metallic glass always exhibits pronounced plasticity[24–29]. Brittleness increases by increasing the strain rate during the deformation. In the case of the homogenous deformation, it is now generally accepted that a transition from a Newtonian viscous flow to a non-Newtonian viscous flow occurs and this transition shows a strong dependence on temperature and strain rate[24–27]. Generally speaking, the non-Newtonian viscous flow in amorphous alloys is accompanied by the existence of an overshoot phenomenon in the stress-strain curve, during the homogenous deformation[26,30]. For example, the La–Al–Ni metallic glass can be deformed up to more than 20,000% in tensile mode at a temperature near the glass transition temperature Tg (Fig. 5)[31].

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2.4. Some applications of bulk metallic glasses Having regard to the mechanical, thermal and physical properties of bulk metallic glasses, uses of metallic glasses emerged in different areas: ● The sport industry: Compared with conventional alloys, metallic glasses have a high elastic limit and an excellent energy transfer. So, they have been applied in the industry of sport, for example for tennis rackets, golf clubs, baseball bats[32]. ● The aeronautical field: Metallic glasses have a high yield strength, so they have been used in space[33]. ● Medical applications: Metallic glasses possess excellent resistance to corrosion and abrasion, so these are very good candidates for biomedical applications[4]. ● Magnetic devices: Metallic glasses based on iron (or cobalt) possess an exceptionally soft ferromagnetism, which allows applications in a sector such as the hearts of transformers or anti-theft devices[34–38]. ● Microsystems[39–42]: Metallic glasses are excellent candidates for implementing micro-systems because they present excellent process ability at a moderate temperature. Fig. 6 shows micro-gears and micro-tools manufactured by the metal glass. The interesting properties of the metallic glasses that are used for these applications are their high mechanical properties and the ability to the formatting at moderate temperature. Therefore, this paper mainly addresses the mechanical properties of these materials. So, we are focusing on their mechanical response, especially the temperature and the strain rate dependence. First, to conclude this introduction, we indicate some open questions relative to the metallic glasses. 2.5. Open questions As a result of these various bibliographic recalls, it seems interesting in this study to clarify various points, including: ● What is the thermal stability of the metallic glass? How do structural relaxation and crystallization occur? Which laws govern their kinetics? How to physically describe these evolutions? ● What are the roles of temperature and structural state of the material on the mechanical properties, either at room temperature or at high temperature? ● To interpret the results which models can be used to describe the amorphous materials and especially the metallic glasses?

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3. Thermal Stability of Bulk Metallic Glasses

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As we have discussed in section 2, metallic glasses are in a non-equilibrium state, like all other amorphous materials. Therefore, the thermal stability of glassy materials is a general problem. Here, we propose to clarify some terms concerning the amorphous materials, e.g. structural relaxation and crystallization. Above the melting temperature Tm, the liquid state corresponds to the thermodynamic equilibrium. When the temperature decreases, the supercooled liquid state corresponds to a metastable state equilibrium. Compared with the liquid state, atomic or molecular mobility is lower. When the temperature is below the glass transition temperature Tg, the system is in an unstable equilibrium. It therefore tends to shift a metastable state (the supercooled liquid) or even to the stable equilibrium state (the crystalline state). The first phenomenon corresponds to the structural relaxation (also called physical aging); the second one is crystallization phenomenon. Now let us discuss and review the results for these different evolutions in detail.

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3.1. Enthalpy relaxation by differential scanning calorimetry In glassy materials, when a heat treatment is performed below the glass transition temperature Tg, the system tends to equilibrium or at least to a more stable state. Differential scanning calorimetry (DSC) is a powerful and effective thermal analysis technique to investigate the structural relaxation in metallic glasses as well as amorphous polymers, especially the kinetics of this evolution[43–49]. Specifically, it should be noted that the structural relaxation in metallic glasses is usually linked to modification of the mechanical properties[50–54], thermodynamics characteristics[55–61] and densities[62– 64] . As a consequence, the mechanical behavior could be tuned by physical aging in metallic glasses, which induces an increase of hardness, storage modulus and gives rises to an increase in brittleness[1]. Fig. 7(a) shows the DSC curves obtained in a La55Al25Ni10Cu10 bulk metallic glass after different annealing time at 435 K. An “overshoot” is always observed around the glass transition temperature Tg. Its amplitude is sensitive to annealing time (ta) and annealing temperature (Ta). The magnitude of the overshoot decreases with annealing temperature (Ta) but increases with annealing time (ta). This is simply due to the fact that the kinetics is thermally activated and that the total amplitude of the phenomenon decreases when the temperature increases (see Fig. 1). Fig. 7(b) shows an example of the kinetics of recovery enthalpy of La55Al25Ni10Cu10 metallic glass[65].

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3.2. Reversibility of the enthalpy relaxation by physical aging As mentioned in the previous section, physical aging below Tg in metallic glasses leads to the movement of atoms, which induces a decrease of the Gibbs energy during the physical aging process. These atomic movements can induce a topological short range ordering (TSRO) as well as a chemical short range ordering (CSRO). These phenomena have been for instance studied in the Vit1 metallic glass with the help of thermoelectric power measurements[66]. In addition, influence of heating treatment on reversibility of the enthalpy relaxation in Zr-based and Cu-based bulk metallic glasses was investigated by DSC[46]. It found that the initial state before physical aging in metallic glass has little influence on the enthalpy relaxation. Thus, it is reasonable to conclude that the enthalpy relaxation occurring during physical aging below the glass transition temperature Tg is mainly reversible. In parallel, they reported that the recovery enthalpies of the metallic glasses are independent of the cooling rates and consequently independent of the initial state before physical aging.

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3.3. Analysis of the kinetics of structural relaxation For glassy materials, enthalpy change ( ∆H ) during structural relaxation can be analyzed by[44]:

∆H = ξ ⋅ ∆ν

(1)

where ξ is a constant, ∆ν is free volume change during structural relaxation. In previous

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literature[47,65,67], the Kohlrausch-Williams-Watts (KWW) model was successfully used to analyze the kinetics of the enthalpy relaxation in glassy materials[47,65–67]. The KWW equation is defined as:

  t βage  φ ( t ) = exp  −  a    τ   

(2)

where ta is the annealing time, τ is the average enthalpy relaxation time, and β age is the Kohlrausch

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exponent whose value is between 0 and 1.

In the framework of the DSC experimental results, the enthalpy relaxation is characterized by the

( )

as

[43]

:

{

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relaxed (or recovered) enthalpy (see Fig. 8(b)). Thus, the recovered enthalpy ( ∆H Ta ) is expressed

}

β ∆H (Ta ) = ∆H eq 1 − exp  − ( ta τ ) age   

(3)

where ∆H eq is the equilibrium value of ∆H eq as ta → ∞ at different temperatures of physical aging. On the average enthalpy relaxation time τ in glassy materials, many models have been proposed. The simplest one uses a classical Arrhenius equation:

( RT )

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τ = τ0 ⋅ exp  Ea

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where τ 0 is the pre-exponential factor, and Ea is the apparent activation energy. It is well documented that the temperature dependence of τ was best fitted by the Vogel-Fulcher-Tamman (VFT) type equation[68–70]:



B    T − T0 

(5)

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τ = τ 0 ⋅ exp

where B and T0 are temperature independent constants, and τ 0 is a fitting constant.

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In addition, the Tool-Narayanaswamy-Moynihan (TNM) model was also used to analyze the enthalpy relaxation or structural relaxation in glassy materials[71,72]:

 ∆h∗ 1 − x ) ∆h∗  ( τ = A ⋅ exp  x +  RTf  RT 

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where A is a pre-exponential factor, x is a nonlinear parameter, ∆h∗ is the activation energy of structural relaxation in amorphous materials, R is gas constant and Tf is the fictive temperature. Thus, the question of temperature dependence of the characteristic time is still an open problem. 3.3. Crystallization The crystallization kinetics of metallic glasses is the basic requirements of applications as structural and functional materials. Therefore, it is interesting to elevate the relevant mechanical properties. In

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the framework of the DSC curve, after the supercooled liquid region, an exothermic peak is observed, which corresponds to the crystallization behavior. Fig. 8(a) shows that the temperature corresponding to the onset of crystallization (Tx) and the temperatures of peaks of crystallization (Tp1 and Tp2) increase with the heating rate in a Zr56Co28Al16 bulk metallic glass[73]. This evolution allows to deduce the activation energy of the phenomenon from the analysis proposed by Kissinger[74],

Rh 2 θ

T

) = −

Ea +C RTθ

(7)

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ln(

where Rh is the heating rate, Ea is the activation energy for the characteristic temperature Tθ (Tg, Tp or Tx), and C is a constant. Fig. 8(b) illustrates this analysis and provides the values for the apparent activation energy.

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4. Mechanical Properties of Bulk Metallic Glasses 4.1. Different components of the mechanical response (influence of temperature) The mechanical response of materials depends on temperature, type and level of applied stress.

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Consider, for example, a constant applied load, corresponding to a creep experiment (see Fig. 9). The mechanical response of materials is the addition of three different contributions: elastic, visco-elastic and viscoplastic. These contributions depend on temperature and strain rate during the deformation. At a given strain rate, the temperature dependence is as follows: ● At low temperature: the amorphous alloys present an elastic behavior at small strain. And, this deformation process is instantaneous and reversible. ● When the temperatures close to the glass transition temperature Tg, it has already been noted that the elastic component decreases dramatically, the visco-elastic component reaches a maximum value [this corresponds to the main (α) relaxation detected by dynamic mechanical analysis, DMA] and the viscoplastic component increases. The visco-elastic behavior is governed by the strain rate. ● When the temperature is above the Tg, the viscoplastic component is the major contribution to the deformation process, at least when no parasite crystallization phenomenon occurs. In addition, at a given temperature, an increase in the strain rate is equivalent to a temperature decrease. The mechanical spectroscopy, also called dynamic mechanical analysis (DMA), is widely used for investigating these different components[75–82]. When a periodic stress σ=σ0cos(ωt) is applied ( ω is the angular frequency, and ω=2πf, where f is the frequency) to the materials, the resulting deformation

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is recorded: ε=ε0cos(ωt+δ), where δ is the phase lag. The energy loss induced during a cycle is directly related to the phase lag δ. Therefore, the atomic mobility in the model materials may be approached. In the case of a shear stress, the complex shear modulus of the materials can be expressed: G = σ/ε = G' +iG". G' is the storage modulus and represents the elastic response. G" is the loss modulus and corresponds to the visco-elastic deformation and also to the visco-plastic deformation when the magnitude or the temperature is too high. The phase lag δ is connected to the components of the modulus by the relation tan δ = G"/G'. As we discussed earlier, similar phenomena were observed in amorphous polymers, oxide glasses or other glasses. There is a debate about the interpretation of the microscopic deformation mechanisms in amorphous materials. These mechanisms have sensitivity to the nature of the material. For example, polymers consist of key chains with side groups and local deformation can occur via movements of

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these entities. On the other hand, metallic glasses are made with atoms simply related with a metallic bonding. As a result, local movements seem more difficult. However, for metallic glasses, structural heterogeneity was observed in recent experiments or numerical simulations and this heterogeneity may provide a key for an understanding of local movements and mechanical relaxations[83–85].

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4.2. Dynamic mechanical relaxations in bulk metallic glasses For crystalline metals, when a stimulus (i.e. mechanical stress) is applied, motion of atoms is accompanied by crystal defects (i.e. vacancies, dislocations or grains boundaries). However, up to now, the situation in glassy materials is not so clear. It is found that the movement of atoms can occur and mechanical relaxations can be observed whatever the chemical architecture of the glassy materials, i.e. whatever the nature of the chemical bonds involved. Fig. 10 presents the general outline of the dielectric relaxation of Poly(methyl methacrylate) (PMMA) (a similar diagram can be plotted for mechanical relaxations)[86]. A low-frequency (around 10–3 Hz) main relaxation (α relaxation), corresponding to the dynamic glass transition phenomenon, is always observed[87]. At a higher frequency, the β relaxation [or relaxation of Johari-Goldstein (JG)] or “excess wing” is observed in polymers, oxide glasses, organic materials and bulk metallic glasses (see below). This relaxation is related to local movements. It should be noted that metallic glasses display excellent conductivity as opposed to the isolate property of polymer and organic glasses. Thus, only mechanical spectroscopy can be used to study relaxation phenomena in these metallic glasses[88–91].

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5. Main (or a) Relaxation in Metallic Glasses: Experimental Results 5.1. Temperature dependence at a given frequency Fig. 11 shows the dynamic mechanical behavior in a Cu36Zr46Ag8Al8 bulk metallic glass, i.e. the temperature dependence of the normalized storage modulus (G'/Gu) and loss modulus (G"/Gu)[92]. It is worth mentioning that three distinct temperature regions are detected, which is consistent with data reported in other metallic glasses[75–81]. Region (I): The material is in the amorphous state at low temperature. G' is high and nearly constant and the viscoelastic component G" is very low (closes to zero GPa), so the behaviour is mainly elastic in this domain. Region (II): maximum of G" occurs at a

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temperature called Tα, which corresponds to α relaxation (or main relaxation) in amorphous materials. This relaxation is related to a dynamic glass transition. A decrease of G' occurs as the temperature increases. This temperature range is associated with the supercooled liquid region (SLR) for metallic glasses. Region (III): both storage modulus G' and loss modulus G" increase drastically as temperature increases, as a consequence of crystallization. Let us mention that the maximum of the loss modulus is not located at the same temperature as the loss factor, due to the rapid decrease in G' in this region (Inset of Fig. 12). Thereafter we will denote by Tα the temperature corresponding to the maximum of G". 5.2. Effect of the testing frequency in bulk metallic glasses In an effort to well understand the mechanical relaxations in metallic glass, it is necessary to illustrate the influence of the testing frequency on the evolution of the dynamic mechanical properties. Fig. 12 shows the frequency dependence for the storage modulus G' and the loss modulus G" in

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Cu38Zr46Ag8Al8 bulk metallic glass[92]. It is found that the peak of G" (i.e. Tα) and the onset of the G' decrease tend to higher temperature by increasing the frequency. Similarly, at a given temperature, the loss modulus is shifted to lower frequencies. These evolutions are simply due to an increase in atomic mobility when the temperature increases or when the frequency decreases. The master curves can be determined by the time temperature superposition (TTS) principle. Fig. 13 shows the master curves in typical metallic glasses (in the frequency domain). Obviously, some differences are observed in the high frequency domain, which are sensitive to the metallic glass composition. Particularly, a pronounced secondary maximum is observed in some metallic glasses (i.e. the Johari-Goldstein β relaxation in La-base metallic glass). For other materials, only a “shoulder” or

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“excess wing” is observed. On the contrary, it was found that the α relaxation is a common characteristic for all the metallic glasses. Actually, it is a common nature for all the amorphous materials (amorphous polymers and oxide glasses)[93]. In fact, the main relaxation in amorphous materials, corresponding to the dynamic glass transition process, is always observed, regardless the nature of the chemical bonds. As shown in Fig. 13, it should be noticed that there is no remarkable

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discrepancy on the α relaxation characteristic for all the metallic glasses in a large frequency region. 5.3. Main (or α) relaxation in metallic glasses: theoretical analysis 5.3.1. Classical models-- Phenomenological models. Many models or theories were used to describe the main (α) relaxation in amorphous materials. One of the simplest models was proposed by Debye to investigate the atomic movements in solid materials[94]. It is well known that Debye model assumes that there is only a single relaxation time. The time decaying function can be expressed as:

 t Φ(t ) = exp −   τ

(8)

G * (ω ) =

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Then, the elastic modulus is obtained: 1 1 + iωτ

(9)

It can be found that when temperature is given, the relaxation time τ has a given value, which can be

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obtained from the frequency at which a maximum (peak) of the loss modulus G" is observed (ωτ =1). As shown in Fig. 14, clearly, the main relaxation in metallic glasses exhibits non-Debye relaxation

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feature[95]. Logically, the important fact is that the curve G" (ω) of the glassy materials is broad and asymmetric. In the previous investigations, several results confirmed that the Debye model fails to describe the main relaxation of amorphous materials[95,96]. Then, some models were proposed to describe the asymmetry of curve and the non-exponential process of the relaxation behavior in glassy materials. A distribution of relaxation times is introduced. For instance, the Cole-Davidson (CD) model[97,98] leads to:

G * (ω ) =

∆G

(10)

[1 + iωτ CD ]

γ

where γ is the power-law exponent in the high frequency domain, ranging from 0 to 1. ∆G (= Gu–Gr, Gu the unrelaxed modulus and Gr the relaxed modulus) is the relaxation strength. This equation reduces to Debye one when γ =1. The empirical stretched exponential or Kohlrausch-Williams-Watts (KWW) equation[99] is often used to describe variations vs time:

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  t  β KWW  G ( t ) = ∆G exp  −     τ    *

(11)

where β KWW is the Kohlrausch exponent which value is between 0 and 1. Low values of β KWW indicate a large deviation from a pure exponential decay. It is clearly seen that Eq. (11) reduces to a

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simple Debye expression, for β KWW = 1. In order to describe the experimental results obtained by DMA or dielectric spectroscopy, Bergman has proposed a convenient representation of Eq. (11) in the frequency domain, as follows[100]:

G" =

" Gpeak

1- β KWW +

β β KWW  β ω ω ω ω + ( ) ( ) KWW peak peak 1 + β KWW 

KWW

 

(12)

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" where Gpeak is the peak maximum of the loss modulus and ωpeak is the peak frequency of the loss

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modulus. Fig. 15 shows the loss modulus G" of Zr55Cu30Ni5Al10 bulk metallic glass as a function of frequency when the temperature ranges from 697 to 715 K. The solid lines are the best fit by KWW model (Eq. (11))[95]. It can be seen that a reasonable fit for G" is obtained in the experimental frequency. Interestingly, the Kohlrausch exponent β KWW is independent of the temperature ( β KWW =0.5±0.02) (when the temperature around the glass transition temperature Tg). The experimental results are accordance with typical β KWW value for bulk metallic glasses is about 0.5[101,102]. The CD and KWW models can be used to describe the broad distribution of relaxation time in amorphous materials. However, since the CD model introduces only a single parameter, i.e. γ , the

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asymmetric broadness cannot be properly described. In contrast, KWW model can be used to analyze relaxation phenomena of glassy formers. It should be noted that only one single value for the Kohlrausch exponent βKWW =0.5 is sufficient to describe the main relaxation in all the metallic glasses based on the master curves (see Fig. 13)[92].

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Actually, the Kohlrausch exponent βKWW, has been proved to be not sensitive to the isothermal temperature in the supercooled liquid region (SLR)[103]. Their investigations are in good agreement with the Wang’s investigation based on mechanical relaxations in metallic glass-forming liquids[101,102].

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Moreover, the Kohlrausch exponent βKWW in Cu50Ti50 metallic glass is around 0.5 within the framework of massively parallel classical molecular dynamics simulations[104]. Many methods were proposed to investigate the Kohlrausch exponent βKWW, for example, viscoelastic modulus measurements in the shear and tensile modes[105], DSC[46,49,64], dielectric and mechanical spectroscopy[101–103,106–108], mechanical testing[109], X-ray photon correlation spectroscopy[110] as well as simulation methods[104]. It is obvious that the β KWW keeps nearly constant (~0.5), implying the distribution of the relaxation time remains a constant based dynamic main relaxation for various metallic glasses. Böhmer et al.[105] reviewed non-exponential relaxations in about 70 strong and fragile glass formers. Many experimental methods, i.e. dielectric and specific heat spectroscopy, viscoelastic modulus measurements in the shear and tensile modes as well as shear compliance investigations, quasi-elastic light scattering experiments and others have been employed to measure the coefficient β KWW . Lots of theoretical methods have been developed to describe a physical meaning of the Kohlrausch

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exponent β KWW . More recently, it was reported that stretching exponent β aging is independent of the driving frequency at a given aging temperature in La-based metallic glass by DMA technique[108]. Unfortunately, this is still an open question and the microscopic interpretation of the stretching parameter β KWW varies from one theory to another. Havriliak and Negami (HN) proposed a new

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expression with two parameters, which can fit very well the experimental data[111]. Unfortunately, the physical meaning of the HN model is not clear. 5.3.2. Physical aging below the glass transition temperature—Investigation by DMA. When DMA experiments are preformed, the structural relaxation can be followed using the evolution of the

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loss factor (tan δ))[103,108,112]. Fig. 16 displays the loss factor tan δ evolution versus annealing time in Cu46Zr45Al7Dy2 bulk metallic glass at different annealing temperatures[103]. Structural relaxation induces a strong decrease in tan δ. The amplitude of tan δ is sensitive to the annealing temperature. In such a scenario, the parameter ∆ varied with annealing time ta by dynamic mechanical analysis

∆=

tan δ (t ) − tan δ (t → ∞ ) tan δ (t = 0 ) − tan δ (t → ∞ )

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experiments[103,108,112]:

(13)

A similar equation to that proposed for the enthalpy recovery has been given:

{

}

β tan δ ( t ) − tan δ ( t = 0 ) = A 1 − exp  − ( ta τ )   

(14)

where A is the maximum magnitude of the relaxation.

Fig. 17 gives the correlation between ln (ln (− ∆ )) and ln ( ta ) for Cu46Zr45Al7Dy2 bulk metallic

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glass (annealing temperature is 675 K). A linear relationship is effectively observed. As shown in Fig. 18, after an incubation time (about 450 s) structural relaxation leads to an increase of the storage

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modulus G' (~5%) and a decrease of the loss factor tan δ. Namely, Physical aging in metallic glasses reduces the viscoelastic component and increases the elastic component, which is in agreement with a previous work[113]. There are many models or theories in amorphous materials have been proposed taking account of the absence of long range order. Initially, free volume theory corresponds to a global description of the materials and was initially proposed for the liquid state[114,115]. Other models correspond to a more local description and especially those based on the concept of “defects”. They are associated with fluctuations of density, entropy, energy, etc. One of the most detailed theories is that proposed by Perez et al., initially applied in polymers as well as used in metallic glasses[116–118]. More recently, investigations have consolidated this model, either by computer simulation or by experiments at a nanoscale[80–82]. In any case local movements are possible in weak regions with a given activation energy. Perez et al. called these regions “quasi-point defects (QPDs)”[116–118]. The potential energy landscape (PEL) initially proposed by Debenedetti and Stillinger[119,120] was developed by Johnson et al.[5,121,122], enabling a possible description of both secondary and main relaxations. When a mechanical stress is applied the response is linked to these weak zones. Argon et al.[123,124] introduced the concept of shear transformation zones (STZ). In addition, Egami et al.[125–127] proposed liquid-like sites (defects) to analyze the structural relaxation process in metallic glasses. Perez et al.[116–118] introduced a very similar concept of “sheared micro-domains (MSD)”. Using these concepts these authors elaborated a detailed model of the mechanical response and they proposed expressions for the three

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components of the deformation: elastic, viscoelastic and viscoplastic. Then an expression of the complex shear modulus is given which includes both temperature and frequency. Only a little information is available on the description of the mechanical relaxation in bulk metallic glasses in the framework of these different models, especially on QPD theory.

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Let’s us focus on Fig. 18, annealing time and temperature dependence of the loss factor tan δ can be discussed in the framework of the quasi-point defects theory. Structural relaxation in bulk metallic glasses reduces the atomic mobility, due to a decrease in the defect concentration towards the equilibrium value at the annealing temperature. The value of tan δ increases by increasing the temperature of physical aging in metallic glasses, which ascribes to an increase in defect mobility with temperature. In this model, density fluctuations in amorphous materials (polymers, metallic glasses as well as other non-crystalline solids) are considered as quasi-point defects[116–118]. Fig. 19(a) shows the “defects” in an elementary volume of amorphous material[128]. When an external mechanical stress is imposed on the structure, the stress resulting in the maximum shear plane causes first the activation of these defects, which becomes polarized, giving rise to the basic movements (Fig. 19(b)). The nucleation and growth from these sheared micro-domains (SMD) correspond to the onset of the anelastic response of the material (Fig. 19(c)). When this stress is applied during a longer time, new sheared micro-domains are nucleated (Fig. 19(d)). The plastic deformation is the result of a coalescence of these SMD. The defect concentration Cd strongly depends on temperature and could be derived by Boltzmann statistics. Based on the quasi-point defects theory, physical aging and crystallization reduce the concentration of defects and then atomic mobility in glassy materials[112,129–131]. On the other hand, plastic deformation (i.e. cold-rolling and compression) increases the concentration of defects and increases the atomic or molecular mobility in amorphous materials[129,131,132]. In order to predict the mechanical deformation in amorphous materials, here we should recall the concept of “shear micro-domains (SMDs)”. In the framework of the quasi-point defects theory, nucleation and percolation of shear micro-domain occur around these defects when the applied stress was maintained during a long enough time. Growth and expanding of the SMDs induce the coalescence of SMDs and then the onset of the visco-plastic deformation. Therefore, consistent with the model of shear transformation zones (STZ), formation of the SMDs is linked to the plasticity in a macroscopic picture[112,113]. Movements of atoms (or molecules) are assisted by quasi-point defects and a hierarchical

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correlation exists, as proposed initially by Palmer et al.[133]. The global characteristic time τmol, is given by[93,134,135]:

τ τ mol = t0 ( β )1/ χ

(15)

t0

τmol corresponds to the mean duration of the movement of a structural unit over a distance equal to its dimension; t0 is a time scale parameter and τβ is the time which corresponds to the thermo-mechanical activation of the elementary or flow unit. Generally, it can be described by the following Arrhenius equation[86,134,135]:

τ β = τ 0 exp (

Uβ kT

)

(16)

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where τ 0 is a pre-exponential time. And, χ is a correlation factor, which ranges from 0 (full order) to 1

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(full disorder). χ is linked to the quasi-point defect concentration, reflecting the degree of the hierarchically constrained correlated movement among the atoms or molecules during the thermomechanical process. χ =0: maximum order (fully constrained situation), corresponds to a perfect crystal, any movement of a structural unit requires the motion of all other units. χ=1: maximum disorder (constraint-free state), corresponds to a perfect gas, all the movements are independent of each other. Then τmol = τβ.

Uβ is the activation energy for the structural unit movement, which corresponds to the height of the

Gu

1 + λ ( iωτ mol )

−χ

+ ( iωτ mol )

−1

(17)

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G ∗ ( iω ) =

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energy barrier between two metastable configuration states. Uβ can be determined by the DMA experimental data. In reality, in the framework of the quasi-point defects theory, the dynamic modulus complex at the frequency domain can be expressed by[135]

where λ is a numerical factor near unity. In this expression, the three contributions of the mechanical response are included: the elastic one (through Gu), the viscoelastic one (through (iωτmol)–χ) and the viscoplastic one (through (iωτmol)–1)). In this expression, both the frequency dependence and the temperature dependence are taken into account. However, a general expression is not easy to obtain. However, since χ = 0.3–0.4 and since we can assume ωτmol >>1 in the temperature range below the glass transition temperature Tg and for not too low frequencies, then equation for the loss factor can be simplified. Indeed:

πχ G " Im(G / Gu ) = = − λ sin(− )(ωτ mol ) − χ = A(ωτ mol ) − χ G ' Re(G / Gu ) 2

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tan δ =

(18)

where A is a constant. Then the value of τmol given by Eq. (15) can be introduced and the two effects of temperature and frequency can be separated and clearly expressed. The final expression is[128,136]:

Uβ

− χ ln ω + constant

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ln ( tan δ ) = −

kT

(19)

As we demonstrated above, when the temperature is below the glass transition temperature Tg, the amorphous materials remain in a frozen or iso-configurational state. Therefore, the correlation factor χ

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remains constant and the mechanical behaviour can be described by an Arrhenius law. When T > Tg , on the other hand, it is important to note that the system of amorphous materials shifts to a metastable thermodynamic equilibrium state, and the correlation factor χ increases by increasing the temperature. Thus, according to this model, temperature dependence of χ is given by

χ (T ) = χ (Tg )

for T < Tg

(20a)

χ (T ) = χ (Tg ) + f (T )

for T > Tg

(20b)

A plot of ln( tan δ) versus the reciprocal of the temperature should give a straight line below Tg , with a slope which is not equal to Uβ, but to Uβ /k.

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It is well known that below the glass transition temperature Tg, χ is typically 0.3–0.4 in bulk metallic glasses[137]. Fig. 20 shows the variations of ln(tanδ) vs the frequency at various temperatures. All the data can be easily fitted by Eq. (19) (The solid lines in Fig. 20)[95]. In agreement with the

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predictions of the model, the correlation factor χ remains constant below the glass transition temperature. When the temperature exceeds Tg the state is no longer isoconfigurational and the temperature dependence is more complex. The VFT equation is sometimes introduced; however in the present investigation a parabolic increase allows a good fitting. Expressions for G'(ω) and G"(ω) can be derived from Eq. (17). There is only one fitting parameter. Indeed, for the description of a master curve corresponding to a given temperature (for instance Tg),

τmol is easily calculated from the relation ω·τmol= 1 at the maximum value of G". As seen in Fig. 21, the QPD model leads to a very good fit of the experimental results.

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The parameter χ can be determined by the slope at different temperatures (Fig. 22)[92]. The current experimental results are in good agreement with the predication of the quasi-point defects theory. In other words, when the system of the metallic liquids keep in an iso-structural state, then the parameter

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χ remains nearly constant when the temperature below the glass transition temperature. This behavior is in conformity with the previous experimental phenomena in many amorphous materials, including polymers and glassy oxides. On the other hand, the correlation factor χ increases by increasing the disorder degree when the temperature exceeds the glass transition temperature since concentration of quasi-point defects increases and the microstructure evolves with the change of the temperature towards a more disordered state. 6. β Ρelaxation in Metallic Glasses

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From the loss dielectric spectrum point of view, the introduction of one or more secondary relaxations is an important feature to understand molecular mobility in amorphous polymers[93,138]. Contrary to the case of amorphous polymers, only several families of metallic glasses present a

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pronounced β relaxation based on the DMA measurement, such as Pd-based[80,102,135,139,140], La-based [108,141–149] and Nd-based[150] metallic glasses. For the other metallic glasses, i.e. Zr-based[86,95,151–155], Cu-based[92,103] and Ti-based[129,156], Au-based[157], Ce-based[74,154,158], Mg-based[75,159] metallic glasses, a well resolved β relaxations is not detected. In reality, it is present only as a “shoulder” at lower

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temperature (or high frequency) domain compared with the α relaxation spectra. This relaxation process is usually called the “excess wing”. Fig. 23 shows the loss modulus spectra for typical metallic glasses. 6.1. Universal nature of β relaxation in metallic glasses It is recognized that the β relaxation is viewed as a common feature, with the temperature dependence of its peak frequency described below the glass transition temperature Tg by an Arrheniustype law for amorphous polymers[93]. The mechanisms of the β relaxation in polymers are ascribed to local non-cooperative motions and local mobility of the polymer segments for β relaxations. While secondary relaxation associated with the side group’s motion, are generally considered non JG relaxations[160]. In particular, the β relaxation may be responsible for the plasticity in amorphous polymers. As a matter of fact, many polymers can be deformed between Tβp (temperature corresponding to the β relaxation) and Tα (temperature corresponding to α relaxation)[128,132]. An

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empirical correalation between the activation energy of the β relaxation Uβ and glass transition temperature Tg in glass formers has been established as Uβ = (26 ± 2) RTg[140]. In amorphous materials, as the JG relaxation occurs before the α process it is usually regarded as its non-cooperative precursor[161,162].

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Based on the mechanical relaxation results, it was found by several authors that the β relaxation is connected to the physical and mechanical properties in metallic glasses. ● The β relaxation, when it exists, is correlated with the plastic deformation in metallic glass [141]. Importantly, Yu et al. investigated the correlation between the apparent activation energy of the β relaxation Uβ and relative energy in the transition between modes of ductility and fragility [141]. It is interesting to note that the apparent activation energies for secondary relaxation β and the transition

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between brittle-to-ductile modes have the same value (Fig. 24(a)). A number of discoveries of β relaxation play an important role in the properties in metallic glasses. Wang recently presents an overview perspective on the relationship between the β relaxation and deformation in metallic glasses [163] .

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● The activation energy Uβ of the β relaxation in metallic glasses is related to the potential energy barriers of the shear transformation zones (STZs) WSTZ: Uβ ≈ 26 (±2) RTg ≈ WSTZ[164] (Fig. 24(b)), which is in good agreement with the proposition of the Ngai et al.[161]. ● The β relaxation is linked to the diffusion motion of the smallest constituent atom in metallic glasses (i.e. Zr- and Pd-based metallic glasses)[165]. ● More recently, Yu et al. found that the β-relaxations in metallic glasses are sensitive to the chemical interactions among all the constituent atoms. In other words, β-relaxations in metallic glasses are related to a large mixing enthalpy for the atoms[149].

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In an effort to more fully understand the β relaxation in metallic glasses, it is important to note that Hu et al. examined the sub-Tg relaxation, and in particular the features of the β relaxation in glassy materials by calorimetry approach[166–169]. It has been reported the following relationship for the

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activation energy of the sub-Tg relaxations in glassy materials Uβ~26.1 RTg[166,167], which is in accord with the results obtained from DMA. 6.2. Effect of the chemical composition on the β relaxation in metallic glasses

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Of particular interest is the observation of the β relaxation in metallic glasses is related to the chemical interactions among all the constituent atoms[149]. They reported that β-relaxations in metallic glasses are linked to a large mixing enthalpy for the atoms. The enthalpy of mixing among all the constituent atoms in Pd-Ni-Cu-Al is reported in Fig. 25. Based on the investigation of Yu et al.[149], it was proposed that the β relaxation is enhanced by replacing Ni by Cu atoms. Fig. 26 shows the temperature dependence of the loss modulus as a function of temperature for four different Pd-based bulk metallic glasses. It can be seen that by increasing the content of Ni atoms, the β relaxation becomes less pronounced. Take Pd30Ni50P20 metallic glass for example, the β relaxation corresponds only to as a weak shoulder. In parallel, in the work of Yu et al.[149] the enthalpies of mixing in Pd-based metallic glasses are given (see Fig. 25). In the framework of enthalpy of mixing, it can be seen that more prominent β relaxations in Pd40Ni10Cu30P20 and Pd42.5Ni7.5Cu30P20 are directly related to large negative values of the enthalpy of mixing among the constituting atoms, in very good

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agreement with the proposition of Yu et al.[149]. More interestingly, Yu et al. systematically investigated the β relaxation in typical metallic glasses (La-, Pd- and Zr-based) based on enthalpy of mixing among the constituting atoms, they found that the pronounced β relaxation is connected to large negative values of the enthalpy of mixing of the all the constituent atoms[149]. Their findings

6.3. Reversibility and Stability of the β Relaxation in metallic glasses

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provide an important pathway to design and control the β relaxation in metallic glasses.

Although we understand the β relaxation can be occurred at lower temperature domain in

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amorphous materials, the evolution of the β relaxation with the temperature is not clear. The β relaxation is the results microscopically from fast and localized elementary process (in section 6.4). As pointed out previously, glassy materials are in an out equilibrium state after quenching. It is apparent that annealing below the glass transition temperature Tg, which induces physical aging. Indeed, structural relaxation leads to the glassy materials towards the thermodynamic equilibrium state. To

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achieve a comprehensive picture of the β relaxation in metallic glasses, Fig. 27 shows the temperature dependence of α and β relaxations for La60Ni15Al25 metallic glass during successive heating processes[108]. Physical aging leads also to a drop in the viscoelastic part and then to a decrease in the atomic mobility in metallic glasses. In this mechanism, the evolution can be ascribed to a progressive reduction of the concentration of “defect” in metallic glass. It is evident that a temperature increase up to 430 K does not induce any modification, benefitting from the temperature is still below the glass transition (the glass transition temperature of La60Ni15Al25 metallic glass is ~ 461 K). As expected, when the physical aging temperature is below the glass transition temperature, the relaxed state is metastable and then no structural evolution is induced by heating in glassy materials. Intuitively, the structural relaxation time is much longer than the time of measurement. Thus, it is reasonable to

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conclude that the characteristics of the β relaxation are stable, which suggests that β relaxation in metallic glasses corresponds to a reversible phenomenon. There has been a general consensus that crystallization can modify the mechanical and physical properties in metallic glasses. Fig. 28(a and b) present the storage and loss modulus in La60Ni15Al25 metallic glass with different annealing times at a given temperature (510 K) above the glass transition temperature Tg. In general, formation of crystalline particles, on the one hand, can lead to a progressive increase in the shear modulus (G'), and, on the other hand, to a progressive decrease in the

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viscoelastic component (G"). Particularly, it is important to note a reduction in the magnitude of the β relaxation by introducing the crystalline particles. In this instance, formation of particles at nano- or micro-scale by annealed above the glass transition temperature, which restricts the atomic mobility as well as decreases the degree of inhomogeneity in the metallic glass, that is why magnitude of the mechanical relaxations is decreased by introduction the crystalline particles. 6.4. Physical mechanisms of the β relaxation in metallic glasses As we discussed earlier, β relaxation could be related to heterogeneous structure or microstructural fluctuation at the nanoscale for the metallic glass, which is linked to local movements of “weak spots” or “soft zones” (i.e. free volume[114,115], flow defects or quasi-point defects[116–118,171], liquid-like sites[125–127], weakly bonded zones[172,173], loose packing regions[174]). The presence of structural heterogeneity in nano- or micro-scale domain has been verified in metallic glasses by transmission electron microscopy (TEM) and simulation method[83–85].

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In the framework of the soft-hard model[84], the existence of “soft” and “hard” regions in metallic glasses is suggested. It seems likely that β relaxation is associated with the motion in soft area, i.e. of

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weakly bonded zones[175]. On the contrary, the α relaxation in the metallic glass is connected to dense packing regions. It should be pointed out that on the architecture of nanoscale medium-range order in the metallic glass, it exhibits the competition between crystal-like and icosahedral ordering[174,176–179]. Specifically, in order to well describe glass transition of glassy materials, Tanaka has proposed the two-order-parameter (TOP) model based on the isotropic and disordered features of liquids[175,176,180– 182] . With the help of this model, Tanaka has ascribed a relaxation of glassy materials to “creation and annihilation of metastable islands”. On the other hand, the β relaxation process is related to the “local rotational jump motion in a cage”[175]. As we discussed previously, consider for example the model originally proposed by Debenedetti and Stillinger[119,120] (the potential energy landscape (PEL) model), and then developed by Johnson et

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al.[5,121]. It provides a plausible picture to analyze the α and β relaxations in glassy materials (Fig. 29).

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The α relaxation corresponds to jumps from one megabasin to another one, while the β relaxation corresponds to jumps between neighbouring potential energy minima. These two different jumps require very different energies.

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Unfortunately, the precise physical nature of the β relaxation is still far from being understood. Recently, Yang et al. verified the presence of heterogeneities in a metallic glass after annealing with the help of high resolution atomic force microscopy[183]. A narrow energy spectrum corresponds to a more uniform structure whereas the enlargement of the energy spectrum indicates a growing degree of heterogeneity. Liu et al. have confirmed the existence of the nano-scale heterogeneities in metallic glasses by means of dynamic atomic force microscopy[184]. More theoretical modelling in this research region and experimental results are desired and will be helpful to provide clearer picture to understand the structural origin in metallic glasses. 7. Relation between α and β Relaxation in Metallic Glasses Despite many investigations focus on the β relaxation (Johari-Goldstein-type secondary process) in metallic glasses, the correlation between α and β relaxation remains unclear. Based on the previous

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investigations, it should be noted that the Debye model unable to describe the main (α) relaxation in glassy materials. From a theoretical viewpoint, it has been shown that the loss modulus spectra in glassy materials can be analyzed by the superposition of the KWW relaxation function for the α relaxation and the Havriliak-Negami (HN) equation for the β relaxation process[185]

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∆Gβ  dφ (t , τ a )  G ∗ (ω) = G∞ + ∆Gα Liω  − α + b  dt   1 + (iωτβ ) a    where φα (t , τ a ) = exp - ( t τ α )



β KWW

(21)

 , ∆Gα and ∆Gβ are the relaxation strength of the α relaxation and 

β relaxation, respectively. Liω is the Laplace transform. The parameters a and b (0
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glass. Taking La60Ni15Al25 metallic glass for example, the Kohlrausch exponent βKWW is equal to 0.44, which is in good agreement with the other bulk metallic glasses determined by DMA[102]. With the help of Eq. (21) (with ∆Gβ=0) to fit the isothermal spectra of the α relaxation in La60Ni15Al25 metallic

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glass, the Kohlrausch exponent βKWW=0.48–0.49[108]. It can be noticed that these values are in accordance with the other metallic glasses either in isothermal mode[95,101–103] or in isochronal route[186]. It is well known that another physical model usually employed to describe the mechanical relaxation behavior in glass forming materials is the Coupling Model (CM), which is proposed by Ngai et al. It should be noted that the coupling model is concentrated on the many-body nature of the α relaxation in glassy materials[93]. As a consequence of the heterogeneous dynamics of amorphous materials, the cooperative α relaxation is interpreted by a Kohlrausch stretched exponent equation as 1− n ϕ (t ) = exp  − ( t τ α ) 



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τ α = tc − nτ 0 

(22)



1 1− n

(23)

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where n is the coupling parameter of the coupling model and n ranges from 0 to 1. τ0 is the primitive relaxation time and tc (~ 1 ps) is the temperature insensitive crossover time from exponential to stretched exponential relaxation. The larger the coupling parameter n, the higher the degree of cooperativity of the dynamics in glassy materials. Based on the Coupling Model (Eq. (23)), if it is assumed that the β relaxation time is about the primitive relaxation time, τβ~τ0, the Coupling Model can be used to predict the separation between the relaxation time τα and τβ as:

(24)

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log τ α − log τ β = n ( log τ α − log tc )

where the coupling parameter n=1–βKWW.. The peak frequencies can be obtained by Eq. (24) (fpeak=1/2πτmax) with the best fit parameters to the master curves are reported in Fig. 30 as arrow. In the case of La-based metallic glass, it can be seen that the arrow is very close to the peak frequency of the

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β process. However, it needs to be considered the limit of the validity of using the master curves for this analysis. As a matter of fact, the master curve of La-based metallic glass is constructed based on the assumption that the shape of the spectra is independent of temperature. When a spectrum consists of the superposition of two relaxation behaviors, then even if the shape and intensity of each process do not change with temperature but the two relaxation processes have very different temperature dependences, then it is inevitable that the spectra given by the superposition of these two relaxation processes will have a different shape at various temperatures. In the framework of QPD’s theory, for amorphous materials, below the glass transition temperature, amorphous materials remain in a frozen or iso-configurational state and the correlated parameter χ is constant. Therefore an Arrhenius behavior is expected. In such a sense, the static structural relaxation behavior presents Arrhenius-type relaxation time. By contrast, one can see that when the temperature surpasses the glass transition temperature, the system remains in metastabe state. Evidently, it is reasonable to get the conclusion that the α relaxation of metallic glasses exhibit VFT-type relaxation time. Note once again that the α and β relaxations both appear to be universal features of amorphous materials. A relationship exists between the two processes and in particular whether the β functions as the precursor to structural relaxation[93]. Importantly, Ngai et al. found the correlation between the α

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relaxation and β relaxation in metallic glasses based on the Coupling Model. Their founding is helpful

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to understand the connection between the β relaxation and macroscopic properties in metallic glasses[187]. More recently, Ngai et al. established the link between the Poisson’s ratio and ductility in metallic glasses, which provides an interesting image to realise the correlation between the dynamic mechanical relaxations and mechanical relaxations in metallic glasses[188]. Additionally, the linear relation between the peak temperatures Tβp of the β relaxation and its activation energy Uβ was established, which is consistent with a linear increase in Uβ with increasing Tβp. In other words, higher activation energy is required to activate local atomic motions in alloys with a higher Tβp in metallic

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glasses. In particular they found: Uβ ≈ 33(±1)RTβp[108]. As noted above, the β relaxation is closely related to the mechanical properties in glassy materials. The activation volume for deformation process in La60Ni15Al25 metallic glass is about two times larger than that reported in other metallic glasses based on high temperature compression testing. The existence of the β relaxation links to high degree of heterogeneity, may be assumed to favorite the existence of local movements[189].

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8. Influence of the Thermo-mechanical History on the Atomic Mobility in Metallic Glasses

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It is noted that an important factor to understand the physical properties of metallic glasses is the atomic mobility. Introduction of physical aging (annealing below the glass transition temperature), crystallization or plastic deformation, for example, leads to structural changes in metallic glasses. It is well-known that amorphous materials are in out-of-equilibrium state below the glass transition temperature Tg. Modifications of enthalpy and entropy are induced by various treatments. The important relation shows that mechanical properties are related to physical properties in amorphous materials. Atomic mobility could be modified during the physical aging or plastic deformation in amorphous state. Many metallic glasses exhibit a very limited plasticity at ambient temperature. Plastic deformation is heterogeneous and occurs through the formation of shear bands[190]. Fortunately, a plastic deformation can be observed, either during compression tests or during cold rolling in some bulk metallic glasses[64,129,191–200]. This plastic deformation can enhance the mechanical properties of bulk metallic glasses[191,192]. It has been suggested that plastic deformation can increase the free volume, and then enhance the atomic mobility[201]. In experiments, in contrast to physical aging, it can be seen that the amount of stored enthalpy was released in cold-rolling metallic glasses[64,196,98]. An expanded view of the impact of the structural state on the atomic mobility in metallic glasses was reported[129,200]. The experimental results based on the DMA technique evidence that the higher the cold rolling ratio, the higher the loss factor. In this manner, one can anticipate that the atomic mobility introduced by cold-rolling is increased. There are many experimental phenomena to suggest that nano-crystals could be embedded in an amorphous matrix in metallic glasses during the crystallization process[1]. In other words, local quasi-point defects (density fluctuation) promote the formation of the crystalline clusters in metallic glasses. When the crystalline particles are introduced in metallic glasses, the viscoelastic component becomes nearly negligible compared with the elastic one. As a direct consequence of the current research results present an excellent agreement with the QPD prediction. The application of the correlation factor χ provides an indicator regarding the atomic mobility. Namely, χ reflects the degree of short-range order as well as concentration of quasi-point defects in amorphous materials, indicative of the atomic mobility of the condense matter.

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It is of interest to compare the atomic mobility of metallic glasses at different structural states, and the nanoindentation techinique can outline another scenario on the issue. Fig. 31 shows the loaddisplacement curves with maximum indentation depth 2000 nm at a constant loading rate in different

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9. Summary and Outlook

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states for Zr50.7Cu28Ni9Al12.3 metallic glass: as-cast, cold-rolled ε = 58% , relaxed one and crystalline (heating to 973 K), respectively[131]. It can be seen that compared with the as-cast state, the metallic glass after cold-rolling induces a lower applied load. The samples with crystalline particles, in contrast, needs a higher applied force due to the presence of the nano-crystals in the glassy matrix limits the movement of the atomic mobility in metallic glasses. As noted in the previous section, the heterogeneity at a nanoscale in glassy materials corresponds to fluctuations of density, entropy. The QPD theory provides clear predictions for the concentration of defects is decreased by the physical aging process while increased by the plastic deformation (i.e. cold rolling, tensile and compressive testes). It is reasonable to conclude that a rejuvenation process is induced by an irreversible deformation (i.e. cold rolling) in metallic glasses, which is in a similar manner to what is detected in polymers[132].

In this review, we have addressed the dynamic mechanical relaxations and atomic mobility in metallic glasses. The mechanical relaxation behaviors are connected to the mechanical properties and physical properties in glassy materials. The main (α) relaxation in glassy materials is connected to the

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glass transition phenomenon. In addition, the β relaxation is associated with many fundamental issues in metallic glasses[202,203]. In metallic glasses these relaxation processes are directly related to the plastic deformation mechanism[1]. The microscopic origin of secondary relaxations and its dependence on the glass structure remain unclear. Despite recent advances, the origin and control of ductility in metallic glasses is a main scientific and technological problem, being still the focus of intensive

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research. In addition, the mechanical relaxations, particularly the β relaxation, provide an excellent opportunity to design metallic glasses with desired physical and mechanical properties. In our mind, there are several issues in the mechanical relaxation that are not clear: ● How to describe the structural unit mobility in metallic glasses? In amorphous polymers, the secondary relaxation is generally attributed to movements in side groups and is often considered as a precursor of α relaxation. Unfortunately, the relationship between the microstructure and the β

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relaxation is not clear in metallic glasses. More recently, Liu et al. have reported that the β relaxation in metallic glass stems from short-range collective rearrangements of the large solvent atoms[204]. It is suggested that shear transformation zone (STZ) is the unit of movement of the deformation in metallic glasses, which then leads to the formation of shear bands[190]. The β relaxation and STZs are related to “soft zone” in metallic glass, however, the nature and size of these “soft zones” remains unclear. ● How to describe the β relaxation and α relaxation in metallic glasses? It is well known that the main relaxation can be modeled by many theories (Please see the section 5). As we discussed, the β relaxation is sensitive to the thermal treatment as well as plastic deformation. However, the information on the effect of the deformation on the α relaxation and β relaxation is very limited. Fortunately, the Coupling Model provides one effective method to analyze the connection between these two relaxation behaviors in glassy materials[93].

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● It is well known that the polymers offer the possibility of realizing a large deformation (compression or tension) at a temperature between the β relaxation and the α relaxation. What’s about the metallic glasses? As we know, most metallic glasses can be deformed around 0.8Tg, is it possible that the β relaxation in metallic glasses is activated around this temperature?

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Acknowledgment

One of the authors, J.C. Qiao thanks the Centre National de la Recherche Scientifique (CNRS) for

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providing the post-doctoral financial support.

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Fig. 1 Variation of enthalpy (H) or volume (V) as a function of temperature. Fig. 2 The outer appearance and surface morphology of the resulting bulk alloy in a cylindrical form of 80 mm in diameter and 85 mm in length[18]. Reprinted from Ref. [18] with permission from Elsevier. Copyright 2012, Elsevier. Fig. 3 Pd-, Ni-, Cu- and Zr-based bulk metallic glasses[4]. Reprinted from Ref. [4] with permission from Elsevier. Copyright 2011, Elsevier. Fig. 4 Ashby map of the damage tolerance (toughness versus strength) of materials[19]. Adapted by permission from Macmillan Published Ltd: [Nature Materials] (Ref. [19]), copyright (2011). Fig. 5 A La55Al25Ni20 alloy deformed to 20,000% in the supercooled liquid region[31]. Reprinted from Ref. [31] with permission from Elsevier. Copyright 2006, Elsevier. Fig. 6 Bulk metallic glass micro-gears and micro-tools fabricated by micro-moulding and thermoprocessing techniques using silicon molds[40]. Adapted by permission from Macmillan Published Ltd: [Nature] (Ref. [40]), copyright (2009). Fig. 7 (a) Enthalpy relaxation in a La55Al25Ni10Cu10 bulk metallic glass for different annealing times. (b) Recovery enthalpy obtained based on DSC experimental results[65]. Reprinted from Zhang et al. (Ref. [65]) with kind permission from Springer Science+Business Media: Copyright © 2008, Springer Science and Business Media. Fig. 8 (a) DSC curves with different heating rates in a Zr56Co28Al16 bulk metallic glass; (b) Kissinger plots to calculate the activation energies relative to Tg, Tx and the peaks of crystallizations (Tp1 and Tp2), respectively[73]. Reprinted from Ref. [73] with permission from Elsevier. Copyright 2011, Elsevier. Fig. 9 Schematic representation of the three components of the deformation observed during a creep experiments: elastic εel , viscoelastic (or anelastic) εan and viscoplastic εvp. Fig. 10 Dielectric constant (top) and loss (bottom) of PMMA at 351.5 K[86]. Reprinted from Ref. [86] with permission from Elsevier. Copyright 2011, Elsevier. Fig. 11 The storage modulus G' and loss modulus G" vs temperature in Cu38Zr46Ag8Al8 bulk metallic glass, Gu is the unrelaxed modulus, assumed to be equal to G' at room temperature[92]. Reprinted from

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Ref. [92] with permission from Elsevier. Copyright 2014, Elsevier. Fig. 12 DMA curves in Cu38Zr46Ag8Al8 bulk metallic glass as a function of frequency at different temperatures (674-677-680-683…725 K): (a) the normalized storage modulus G'/Gu and (b) the normalized loss modulus G"/Gu, respectively[92]. Reprinted from Ref. [92] with permission from Elsevier. Copyright 2014, Elsevier. Fig. 13 Dependence of the normalized loss modulus vs the normalized frequency in typical metallic glasses. The solid line is fitted by the KWW model with the Kohlrausch exponent βKWW = 0.5[92]. Reprinted from Ref. [92] with permission from Elsevier. Copyright 2014, Elsevier. Fig. 14 Comparison of the fitting curves by a single Debye relaxation time and the experimental results for Zr55Cu30Ni5Al10 bulk metallic glass: the loss modulus G"/Gu[95]. Reprinted with permission from Ref. [95]. Copyright 2012, AIP Publishing LLC. Fig. 15 Loss modulus G"/Gu versus frequency for the temperature ranges from 697 K to 715 K. Solid lines are the fits by the KWW model. Inset is the master curve of the loss modulus G"/Gu (experimental points) fitted by the KWW equation (solid line), and the Kohlrausch exponent β KWW =0.4981[95]. Reprinted with permission from Ref. [95]. Copyright 2012, AIP Publishing LLC.

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Fig. 16 Time dependence of the loss factor tan δ = G"/G' in Cu46Zr45Al7Dy2 bulk metallic glass at various annealing temperatures[103]. Reprinted from Ref. [103] with permission from Elsevier. Copyright 2012, Elsevier. Fig. 17 Double logarithmic plot of variation of the loss factor tan δ vs logarithm of the annealing time in Cu46Zr45Al7Dy2 bulk metallic glass[103]. Reprinted from Ref. [103] with permission from Elsevier. Copyright 2012, Elsevier.

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Fig. 18 Storage modulus G' and loss factor tan δ in Cu46Zr45Al7Dy2 bulk metallic glass vs annealing time[103]. Reprinted from Ref. [103] with permission from Elsevier. Copyright 2012, Elsevier. Fig. 19 Schemes corresponding to the QPD model (a) Quasi-point defects in amorphous materials (b), (c) et (d) activation of these DQP and growth of sheared micro-domains (SMD) induced by application of a stress[128]. Reprinted from Ref. [128] with permission from Elsevier. Copyright 2011, Elsevier. Fig. 20 Influence of the driving frequency on the logarithm of the loss factor at various temperatures in the Zr55Cu30Ni5Al10 bulk metallic glass. Solid lines are fits by Eq. (19)[95]. Reprinted with permission from Ref. [95]. Copyright 2012, AIP Publishing LLC. Fig. 21 Loss modulus G"/Gu versus frequency for the temperature range from 697 K to 715 K. Solid lines are the fits by the QPD model. Inset is the master curve of the loss modulus G"/Gu (experimental points) fitted by the QPD equation (solid line), and the correlation factor χ =0.38[95]. Reprinted with permission from Ref. [95]. Copyright 2012, AIP Publishing LLC. Fig. 22 Evolution of the correlation factor χ with the temperature in various bulk metallic glasses[92]. Reprinted from Ref. [92] with permission from Elsevier. Copyright 2014, Elsevier. Fig. 23 Temperature dependence of the loss modulus G"/G"max (G"max is the peak of α relaxation in the loss modulus) in various bulk metallic glasses[108]. Adapted with permission from Ref. [108]. Copyright 2013, American Chemical Society. Fig. 24 (a) Variation of the characteristic frequency versus 1000/T, for determining the apparent activation energy for secondary relaxation and the transition between ductile and fragile modes[142], Reprinted with permission from Ref. [142]. Copyright 2012 by the American Physical Society. (b) Correlation between the energy of activation Eβ of the β relaxation and the energy WSTZ barriers of STZ[164]. Reprinted with permission from Ref. 164. Copyright 2010 by the American Physical Society. Fig. 25 Schematics of enthalpy of the mixing elements for Pd–Ni–Cu–P system metallic glasses. The values of the enthalpy of mixing come from literature[170]. Adapted with permission from Ref. [170]. Copyright 2013, American Chemical Society. Fig. 26 Temperature dependence of the loss modulus G"/G"max (G"max corresponds to the peak of α relaxation in the loss modulus) in Pd-based metallic glass-forming liquids[170]. Adapted with permission from Ref. [170]. Copyright 2013, American Chemical Society. Fig. 27 Evolution of the β and α relaxations during successive continuous heating processes in La60Ni15Al25 metallic glass The inset (a) is a schematic illustration of the DMA experiment and (b) presents the zoom of the β relaxation at lower temperature range[108]. Adapted with permission from Ref. [108]. Copyright 2013, American Chemical Society. Fig. 28 Influence of partial crystallization on the β relaxation in La60Ni15Al25 bulk metallic glass: (a) Storage modulus G'/Gu and (b) loss modulus G"/Gu (annealing temperature Ta = 510 K). The inset of (b) is a schematic illustration of the DMA experiment at various annealing time (1-as cast, 2annealing time is 5 min, 3-annealing time is 15 min…)[108]. Adapted with permission from Ref. [108]. Copyright 2013, American Chemical Society.

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Fig. 29 Schematic diagram of α and β relaxations in the framework of the potential energy landscape[122]. Reprinted with permission from Ref. [122]. Copyright 2009, AIP Publishing LLC. Fig. 30 Best fit of the master curve (red solid line) of the loss modulus of La60Ni15Al25 with Eq. (21), the dotted lines are the individual contributions of the two relaxations[108]. Adapted with permission from Ref. [108]. Copyright 2013, American Chemical Society. Fig. 31 Load-displacement curves with maximum indentation depth 2000 nm at a loading rate 0.05 s-1

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in different states: (1) as-cast, (2) cold-rolled ε=58% (3) relaxed sample (annealed at 670 K and annealing time 16 h) and (4) crystalline (heating to 973 K), respectively[131]. Reprinted from Ref. [131] with permission from Elsevier. Copyright 2013, Elsevier.

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