Herein, we elucidate a hidden variable in a shear transformation zone (STZ) volume (Ω) versus Poisson’s ratio (ν) relation and clarify the correlation between STZ characteristics and the plasticity of metallic glasses (MGs). On the basis of cooperative shear model and atomic stress theories, we carefully formulate Ω as a function of molar volume (Vm) and ν. The twofold trend in Ω and ν is attributed to a relatively large variation of Vm as compared to that of ν as well as an inverse relation between Vm and ν. Indeed, the derived equation reveals that the number of atoms in an STZ instead of Ω is a microstructural characteristic which has a close relationship with plasticity since it reflects the preference of atomistic behaviors between cooperative shearing and the generation of volume strain fluctuation under stress. The results would deepen our understanding of the correlation between microscopic behaviors (STZ activation) and macroscopic properties (plasticity) in MGs and enable a quantitative approach in associating various STZ-related macroscopic behaviors with intrinsic properties of MGs.

Over the past few decades, it has been greatly anticipated that metallic glasses (MGs) will expand the base of metallic materials in the field of structural applications due to their outstanding mechanical properties such as high strength and large elastic limit in a glassy state and thermoplastic forming ability in a supercooled liquid state.1–4 However, their limited plasticity in a glassy state5 and a lack of understanding on physical processes responsible for the unique properties have attenuated the industrial values of MGs. As the first step in understanding the fundamental mechanism of mechanical behaviors, the concept of a shear transformation zone (STZ), the cluster of atoms which accommodate plastic strains by going through a cooperative shearing under external stress, was proposed.6 Subsequently, constitutive laws describing the deformation behaviors were suggested by adopting STZ-related terms such as activation energy and activation volume as intrinsic variables.6,7

Its importance has continued to mount even further in later years following the discovery of the equivalence between STZ activation and β-relaxation8 which is a universal phenomenon that glass and supercooled liquid exhibit.9,10 This is because it implies that STZ activation is closely related to a broad range of issues not only in MGs but also in glassy physics,10 such as diffusion,11–13 physical aging,14 jamming-unjamming transition,15,16 deformation mechanisms,8,17 mechanical properties,17 and forming ability.4 Based on this finding, the correlation between the activation energy of an STZ and glass transition temperature (Tg) has also been established.8 On the contrary, little is known with regard to the correlation between STZ volume (Ω) and intrinsic properties notwithstanding the recent development of experimental methods18–20 to characterize it, as these methods are applicable only for certain MG systems and there have been conflicting reports about whether MGs with a large Poisson’s ratio (ν) have larger or smaller Ω.18,19,21

In the present study, the correlation between Ω and intrinsic properties is explored by deriving a functional relationship based on cooperative shear model (CSM)7 and atomic stress (AS) theories.22 Along with systematic statics on various MGs, we find that Ω can be expressed in terms of molar volume (Vm) and ν, and that it shows twofold tendency in regard to ν due to the Vm versus ν relation. In addition, the atomistic origin of the correlation between STZ characteristics and the plasticity of MGs is carefully discussed. These findings help successfully settle the controversy over the Ω versus ν relation, allowing us to bridge microscopic behaviors and macroscopic properties as well as offering a simple way of estimating Ω in MGs.

Intensive efforts to experimentally characterize Ω18,19,21 were triggered by the establishment of the CSM theory which provides a constitutive description of plastic deformation in MGs.7 Following Frenkel’s approach to evaluate shear strengths in dislocation-free crystalline metals,23 the CSM theory describes the elastic energy of an STZ in MGs as a function of strain, and on the basis of this description, the equation for Ω in an unstressed MG is derived as follows:7 

Ω=WSTZ8π2γc2ζG,
(1)

where WSTZ is the activation energy of an STZ which is equal to the activation energy of β-relaxation (Eβ), γc ∼ 0.027 is the average elastic limit, ζ ∼ 3 is the correction factor that stems from the matrix confinement of a “dressed” STZ, and G is the shear modulus.7 Here, Ω is defined as the volume of the inelastically deformed plastic core of atom clusters7 since an STZ does not have an identifiable boundary.5 This means that WSTZ is large when atomic rearrangements require an inelastic distortion of larger atomic clusters (large Ω) and when these clusters are highly resistant to shear deformation (large G).7 Indeed, Eq. (1) enables the experimental determination of Ω without applying large stress which can introduce unwanted artifacts by complicating the shape of STZs.24 

Figure 1 shows the plot of experimentally determined Ω (Ωexp) of 25 typical MGs against ν, which are obtained by substituting Eβ(=WSTZ) and G values collected from the literature into Eq. (1). The data table of Ωexp and ν is provided in the supplementary material (Table S1). The Ωexp versus ν relation displays a generally decreasing trend of Ωexp with increasing ν. However, this trend becomes rather vague in the same MG system with a slight difference in compositions and even a seemingly opposite tendency arises among certain MG systems such as La-, Al-, and Tm-based MGs. In addition, it is fairly similar to the Vm versus ν relation which is also depicted in Fig. 1. However, the exact origin is not clearly understood yet. To explicate the twofold trend in the Ωexp versus ν relation and its resemblance to the Vm versus ν relation, we derived the functional relationship between Ω and intrinsic properties of MGs by approximating parameters in Eq. (1) using empirical and theoretical relations.

FIG. 1.

The plot of experimentally determined STZ volume (Ωexp) and molar volume (Vm) against Poisson’s ratio (ν) for a variety of metallic glasses.

FIG. 1.

The plot of experimentally determined STZ volume (Ωexp) and molar volume (Vm) against Poisson’s ratio (ν) for a variety of metallic glasses.

Close modal

The first modification was carried out acknowledging a close correlation between relaxation behaviors and thermal properties.8 Upon adopting the empirical relation between Eβ and Tg which is expressed as Eβ ≈ 26RTg (where R is the gas constant),8 Eq. (1) can be rewritten as follows:

Ω=26RTg8π2γc2ζNaG,
(2)

where Na is Avogadro’s number. Figure 2 shows correlation between Ωexp and Tg/G ratio using the data in Table S1 of the supplementary material. It should be noted that Tg values are collected from the literature regardless of testing conditions despite their dependence on thermal history. The regression of the data in Fig. 2 using a least squares fitting method demonstrates that the regression coefficient (R2) is 0.79, indicating that Ω exhibits a solid linear relationship with Tg/G. Also, the slope of 0.201 ± 0.02 is similar to the product of constants in Eq. (2) ∼ 0.203 within the experimental error. This result is consistent with the proposed relationship that STZ activation, or β-relaxation, is a unit event of glass transition.24 

FIG. 2.

Correlation between experimentally determined STZ volume (Ωexp) and Tg/G ratio. The black dotted line is the linear fitting data using the least squares fitting method. The gray solid lines are the 95% confidence lines.

FIG. 2.

Correlation between experimentally determined STZ volume (Ωexp) and Tg/G ratio. The black dotted line is the linear fitting data using the least squares fitting method. The gray solid lines are the 95% confidence lines.

Close modal

Then, the correlation between Tg/G and ν is investigated based on AS theory proposed by Egami et al.22 According to the theory, individual atoms in an alloy experience atomic level stress induced by the departure from an ideal packing state.22 The total elastic energy (Ev), i.e., the sum of local elastic energy due to atomic level stress, attains an equilibrium state with thermal energy at high temperature being expressed in terms of temperature (T) by a simple relation, Ev = RT/4.22 However, the scaling law breaks down below a certain temperature owing to topological frustration which finally leads to the kinetic freezing of metallic melt into a glassy state.22 This characteristic temperature at which the transition from a supercooled liquid to a glassy structure occurs is designated as Tg. Thus, the total elastic strain energy at Tg (EvTg) can be described as follows:

EvTg=RTg4=BVm2KαεvTg2,
(3)
Kα=3(1ν)2(12ν),
(4)

where B, Vm, and εvTg are the bulk modulus, the molar volume, and the standard deviation of volume strain at Tg which is a universal value of 0.095 valid for a variety of MGs, respectively.22 Here, Kα is the enhancement factor evaluated by the continuum approximation using the Eshelby theory25 to incorporate the effect of long-range stress field generated from atomic level stresses in neighboring sites.22 

Considering the relation among elastic moduli, B = 2G(1 + ν)/3(1 − 2ν), Eq. (3) can be modified to provide a functional description of Tg/G as follows:

TgG=8Vm(1+ν)9R(1ν)εvTg2.
(5)

Figure 3(a) shows the plot of Tg/G against Vm(1 + ν)/(1 − ν) for various MGs with data available from the literature. The relevant data of Tg, G, Vm, and ν are summarized in Table S1 of the supplementary material. The data points manifest a clear one-to-one correspondence with an R2 of 0.85 and the slope of 0.999 ± 0.048, which is close to the product of constants in Eq. (5) ∼ 0.965, supporting the validity of Eq. (5). To understand the correlation between ν and atomistic behaviors, the modified form of Eq. (5) where both sides are divided by Vm is considered. Whilst G represents the resistance to shear deformation,26,Tg/Vm reflects the resistance to the formation of local density or volume strain fluctuation as it is proportional to EvTg/Vm.22 Hence, for MG systems with larger ν and consequently larger Tg/GVm ratio, the applied external energy would preferentially be relaxed via cooperative shearing rather than the generation of volume strain fluctuation, thereby exhibiting larger plasticity with the delay of crack formation. This relation provides an atomistic explanation for the origin of the correlation between ν and the plasticity. We also note that the relation, G = E/2(1 + ν), where E is the Young’s modulus, can convert Eq. (3) into the following form:

TgE=4Vm9R(1ν)εvTg2
(6)

which specifies the rough proportionality suggested for Tg and E as Tg/E ∼ 2.5.28 The E values collected from the literature are also presented in Table S1 of the supplementary material. The inset in Fig. 3(a) illustrates the linear relationship between Tg/E and Vm/(1 − ν) with an R2 of 0.88.

FIG. 3.

(a) Tg/G versus Vm(1 + ν)/(1 − ν) relation for 72 kinds of metallic glasses including Ca-, Mg-, RE-, Zr-, Cu-, Fe-, Pt-, Pd-, and Au-based MGs. The black dotted line is the linear fitting data. The inset shows the Tg/E versus Vm/(1 − ν) relation of the corresponding metallic glasses. (b) A comparison of the Vm versus v relation of the corresponding metallic glasses except Pt- and Au-based MGs.

FIG. 3.

(a) Tg/G versus Vm(1 + ν)/(1 − ν) relation for 72 kinds of metallic glasses including Ca-, Mg-, RE-, Zr-, Cu-, Fe-, Pt-, Pd-, and Au-based MGs. The black dotted line is the linear fitting data. The inset shows the Tg/E versus Vm/(1 − ν) relation of the corresponding metallic glasses. (b) A comparison of the Vm versus v relation of the corresponding metallic glasses except Pt- and Au-based MGs.

Close modal

Upon combining Eqs. (3) and (5), Ω can be consequently evaluated by

Ω=26π2εvTg2Vm(1+ν)9γc2ζNa(1ν).
(7)

Equation (7) reveals the functional relationship among Ω, Vm, and ν, and enables a straightforward approximation of Ω. An important point to note is that unlike the decreasing trend with increasing ν observed in Figure 1,19 Eq. (7) implies that Ω would increase with increasing ν provided that Vm remains constant. This result suggests that ν cannot be a sole determinant of Ω and that Vm should also be taken into account. To explore the relative importance of each variable, Vm is plotted against ν in Fig. 3(b). Here, Au- and Pt-based MGs are excluded as these alloy systems exhibit unexpectedly small Tg/G values deviating from the proportional correlation with Vm(1 + ν)/(1 − ν) as denoted by colored zones in Fig. 3(a). As it can be expected from the atomic radius of constituting elements, Ca-, Mg-, and rare earth (RE)-based MGs exhibited a relatively large Vm, whilst MGs with metalloid elements such as Pd-based MGs with P and Fe-based MGs with B show a relatively small Vm. Furthermore, ν primarily depends on the covalency of the bond,27 thereby the electron configuration of major constituting elements.28,29 Thus, the Vm versus ν relation displays an interesting trend that Vm generally decreases with increasing ν besides Fe-based MGs with metalloid elements.30 To be more specific, Pd-based MGs with small Vm exhibit large ν due to the filled d-shells of Pd, but RE-based MGs with large Vm show small ν owing to the unfilled d-shells of REs.27 On the other hand, Fe-based MGs possess small ν as well as small Vm which are attributed to the unfilled d-shells of Fe and the small atomic radius of B, respectively. Here, an important point to note is that, Vm varies over a wide range from 6.85 cm3/mol to 20.25 cm3/mol, with the maximum value reaching almost three times the minimum value, whereas ν varies in a comparatively smaller range from 0.276 to 0.410. The relation indicates that the greater reduction of Vm can overwhelm the increase of ν resulting in smaller Ω in MGs with larger ν. On the other hand, one can see that Vm slightly increases or remains nearly constant in each colored zone [Fig. 3(b)], which suggests the origin of twofold propensity in the Ω versus ν relation.

Figure 4 shows the plot of Ω against ν with Ωcal values (open scatters) and Ωexp values (solid scatters), which are listed in Table S1 of the supplementary material. There are slight differences between Ωcal and Ωexp due to the series of approximations during the derivation of Eq. (7); however, the general trend remains unchanged. Similar to the Vm versus ν relation, Ω decreases when ν increases in broad outlines with the exception of Fe-based MGs. In contrast, each colored zone displays a modest inclination of Ω to rise with increasing ν. Moreover, unlike ν which is an intuitive indicator for an intrinsic plasticity or brittleness of MGs,26 the ranges of Ω in relatively ductile MGs (from 3.58 cm3 to 5.31 cm3) and relatively brittle MGs (from 2.58 cm3 to 7.41 cm3) largely overlap. The results demonstrate that Ω alone cannot serve as a single indicator of the plasticity of MGs. Instead, Eq. (7) suggests that the number of atoms in an STZ (n) can be expressed in terms of ν, when considering the relation Ω = nCfVm/Na (where Cf ≈ 1.1 is the free volume parameter19,31) as follows:

n1+ν1ν.
(8)

The finding implies that the genuine microstructural characteristics in relation with the plasticity of MGs is n, which reflects the preference of atomistic behaviors between cooperative shearing and volume strain fluctuation under stress, as revealed by Eq. (5).

FIG. 4.

The plot of STZ volume (Ω) against Poisson’s ratio (ν) for various metallic glasses. Open scatters display the calculated STZ volume (Ωcal), whereas solid scatters represent the experimentally determined STZ volume (Ωexp).

FIG. 4.

The plot of STZ volume (Ω) against Poisson’s ratio (ν) for various metallic glasses. Open scatters display the calculated STZ volume (Ωcal), whereas solid scatters represent the experimentally determined STZ volume (Ωexp).

Close modal

To summarize, we have investigated the correlation between Ω and intrinsic properties of MGs. On the basis of CSM and AS theories, the functional relationship among Ω, Tg, G, Vm, and ν is derived to be ΩTg/GVm(1 + ν)/(1 − ν). Despite the positive dependence of Ω on ν inferred from the expression, Ω generally decreases with increasing ν owing to the large variation of Vm among different MG systems as well as an inverse relation between Vm and ν. Contrastingly, an opposite trend going against the overall propensity is uncovered in the same MG system with a slight change in compositions where there is no significant reduction of Vm with increasing ν as well as the exceptional MG systems where both Vm and ν vary in the same direction. The findings obscure the correlation between Ω and the plasticity of MGs which was proposed based on its dependence on ν. However, they suggest that ν is closely correlated with the number of atoms in an STZ and thereby the plasticity of MGs, revealing an atomistic origin for the correlation between ν and plasticity.

See supplementary material for the summary of glass transition temperature (Tg), shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν), molar volume (Vm), calculated STZ volume (Ωcal), experimentally determined STZ volume (Ωexp), and references.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science, ICT and Future Planning) (No. 2014K1A3A1A20034841). One of the authors (E. S. Park) also benefited from the Center for Iron and Steel Research (RIAM) at Seoul National University.

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Supplementary Material