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Extended elastic model for flow in metallic glasses

We report that both shear and bulk moduli, not only shear modulus, are critical parameters involved in both homogeneous and inhomogeneous flows in metallic glass. The flow activation energy (\Delta F) of various glasses when scaled with average molar volume Vm, which is defined as flow activation energy density \rho_E =\Delta F/Vm, can be expressed as: D=(10G+K)/11 . The extended elastic model is suggestive for understanding the glass transition and deformation in metallic glasses.

The mysterious glass transition phenomenon, which connects the liquid and glassy state, has wide applications in daily life, industry, and organism preservation [1][2][3][4]. In the past decades, intensive efforts have been made to understand the glass transition [1,[5][6][7][8][9]. To understand the flow in supercooled liquid and glass, many models have been proposed. The well-known models are the free volume model, the Adam-Gibbs entropy model, the mode-coupling theory and elastic models [1][2][3][4][5][6]. A successful model of viscous liquids and glasses must explain why the activation energy has such strong temperature dependence and can correlate the activation energy to simple and readily measurable parameter. Among these models, the elastic models link the glass transition and elastic moduli of the glasses [1]. All the elastic models link the activation energy to the readily measurable instantaneous shear modulus G. In metallic glasses, the glass transition temperature (T g ) indeed shows clear correlation with the elastic moduli such as Young's modulus E and G [10][11][12][13][14][15][16][17][18].
In this letter, based on the scaling laws between T g and elastic moduli in metallic glasses, we demonstrate that the V m scaled flow activation energy (∆F), that is, flow activation energy density, ρ Ε =∆F/V m , is determined by both G and K in a way of ρ Ε = (10G+K)/11. The physical origin of the extended elastic model is discussed.
The temperature dependence of the viscosity of liquids approaching glass transition is [1]:  [12][13]. The linear fitting results are E ∝ (0.206±0.017)T g and G ∝ (0.0759±0.0065)T g . The relations between T g and G (and E) are key evidence for the elastic model [1], however, it can be seen that the data are rather scattered, and the adjusted R-square (the adjusted coefficients of determination [19]) are 0.755 and 0.741, respectively, which denotes that G or K should not be the only parameter that involved in glass transition.
In fact, according to shoving model, a characteristic volume V c is involved in homogeneous flow of glass-forming liquids as [1,14]: ∆F= GV c . In inhomogeneous flow of glasses, the activation energy of a flow event unit (shear transformation zone, STZ) also correlates with volume as [15][16]: , where Ω is the volume of STZ and γ C is the shear strain limit. Recently, it is found that the elastic moduli scaled with V m show better correlations with the thermal and mechanical properties for metallic glasses [17][18][20][21]. Thus, the characteristic volume could be an important parameter involved in the flow event in glass transition and glass. Figure   1 It is found that the atom number N involved in the cooperative flow events of STZ in metallic glasses is similar and around ~100 [15][16], and the volume of STZ can be expressed as inhomogeneous flow in glass is a self-organized of a large number of local shearing events (or STZ) [15][16][17], and the transition from local shearing to macroscopic shear band results from the dramatic increase of the atom mobility and softening along a shear plane motivated by the input mechanical energy [15][16][17]. Thus, the transition is akin to a process of stress driven glass-to-liquid transition or glass transition [17]. Then, the involved activation energy GV= NGV m /N A should have linear correlation with k B T g . The good linear correlation between GV m (and EV m ) and T g in Fig. 1(b) confirms that the N involved in the cooperative flow event for various metallic glasses is almost the same.
Based on above scaling laws and elastic model, we propose that it is the flow activation energy density (ρ E ), not the flow activation energy itself, correlates with the elastic moduli as: The extended elastic model means that the energy per volume needed in glass transition or in STZ in metallic glass is proportional to the elastic moduli.
Previous elastic models [1] suggest that the atoms or atomic groups go through pure shearing displacement which is independent of K, and the ρ E depends only on G.
Recent works [18,20] and the jamming model of granular systems [22] find that both shear and dilatation are involved in the flow during glass transition and deformation.
Next, we further justify the flow activation energy density ρ E relates not to G or K but both G or K. That is, the flow event relate to both shearing transformation (corresponding to volume-preserving G) and dilatation (corresponding to volume-nonpreserving K). At T g , the ∆F/T g should be the same for all glasses that is independent of Poisson's ratio or other moduli [3]. However, the statistics analysis of both GV m /T g and KV m /T g for glasses listed in Table I , where U is the atomic potential energy, V 0 is the atomic equilibrium volume, U/V 0 is the energy density and V/V 0 correlates with elastic strain. In harmonic approximation around V 0 , the U can be expressed in a parabolic form as U=U 0 (1-αV/V 0 ) 2 [1], where α is constant depending on the atomic bonding nature. This gives K= 2α 2 U 0 /V 0 , which correlates with the potential energy density at equilibrium state. Thus, both G and K are proportional to their corresponding deformation energy density. Therefore, it is reasonable for ρ E rather than the activation energy shows correlation with the combination of K and G.
Most models for flow in glasses and supercooled liquids consider the case of simple shear, which involves only shear stress and shear modulus. Our model suggests that the both homogeneous and inhomogeneous flows at one hand is shearing process and on the other hand must generate free volume which is a form of dilatation (In fact, the shear induced dilatation has been widely observed [22]), and demonstrates that both shear and free volume are important for flow in glass transition and deformation, and provides an intuitional picture of the flow of the atoms or atomic groups in glass or liquid.
Furthermore, the formation of shear bands when the BMGs deform plastically is thought to be akin to the process of glass transition [17,23]. Thus, this means that both the shear [26] and the dilatation [22] could be involved in the formation of shear bands. However, due to the critical difference between the two phenomena: the glass transition is constraint-free, while the formation of shear bands is stress-constraint, and the formation of shear bands then may involve less dilatation.  [11][12][13][14][15][16][17][18][19][20][21].