*3.4. Shear Susceptibility*

The large magnitude of the shear susceptibility *β* (see Equation (2)), which determines the influence of defects on the shear modulus (diaelastic effect), constitutes a salient ingredient of the IT. This quantity is controlled by the non-linearity of the solid, specifically by the magnitude of the non-linear shear modulus *C*4444 or, in an isotropic approximation, by the quantity *γ*4 = 116 *∂*<sup>4</sup>*Uel ∂ε*<sup>4</sup> , where *Uel* is the elastic energy and *ε* is the shear strain. To determine the value of the shear susceptibility ˙

*β*, the effect of elastic loading on the ultrasound velocity was studied on two (Zr- and Pd-based) MGs [42,43]. The shear susceptibility derived in these experiments is smaller than that originally estimated by Granato (*β* ≈ 40) but has nonetheless the same order of magnitude. Moreover, this quantity is close to the estimates derived by other methods [44] (see Section 3.5 and Table 1).

### *3.5. Relation between the Shear Modulus and Heat Effects*

A simple analysis of Equations (1) and (2) shows that the IT implies an intrinsic relation between shear modulus changes and heat effects in MGs. Indeed, integration of Equation (1) taking account of Equation (2) leads to an expression for the enthalpy increment Δ*H* upon insertion of interstitial-type defects, which can be expressed through the shear moduli of glass and maternal crystal, Δ*H* ≈ (*μ* − *<sup>G</sup>*)/*βρ*, where *β* is the shear susceptibility and *ρ* is the density. Then, one can arrive at an expression relating the heat flow with the change of the shear modulus,

$$\mathcal{W} = \frac{\dot{T}}{\rho \beta} \left( \frac{G}{\mu} \frac{\partial \mu}{\partial T} - \frac{\partial G}{\partial T} \right),$$

where *T* is the rate of temperature change. This relationship was repeatedly tested and found to give an excellent description of the heat flow on the basis on shear modulus relaxation data not only upon structural relaxation below the glass transition temperature *Tg* and in the supercooled liquid state but upon crystallization as well [45,46]. The latter fact is really surprising and actually suggests that the whole excess internal energy Δ*U* (with respect to the crystalline maternal state) is mostly related to the interstitial-type defect system frozen-in upon glass production. An analysis gives a simple expression for this quantity [47],

$$
\Delta II \approx \Delta H \approx \frac{\mu}{\beta \rho} \left( 1 - \frac{G}{\mu} \right). \tag{5}
$$

It is seen that the excess internal energy is simply controlled by the shear moduli of glass and maternal crystal. Upon crystallization, the defect system disappears and this energy is released as heat, i.e., Δ*U* ≈ *Qcryst*, where *Qcryst* is the heat of crystallization. A specially designed experiment confirmed this idea [48]. From a physical viewpoint, this result means that the whole heat content of glass (i.e., the excess enthalpy with respect to the crystalline maternal state) is mostly determined by the interstitial-type defect system frozen-in upon glass production.

On the other hand, considering the initial and relaxed states of MGs and calculating the corresponding differences for the heat flow and shear modulus, Δ*W* = *Wrel* − *W* and Δ*G* = *Grel* − *G*, Equation (4) after simplification can be rewritten as [49]

$$
\Delta W = -\frac{\dot{T}}{\rho \beta} \frac{\partial \Delta G}{\partial T}.\tag{6}
$$

This relationship shows that the quantities Δ*W* and *∂*Δ*G*/*∂T* should be proportional to each other. Figure 4 shows these quantities derived from calorimetric and shear modulus data for bulk glassy *Zr*65*Al*10*Ni*10*Cu*15. It is clearly seen that they are indeed proportional. This allows determination of proportionality constant and calculation of the shear susceptibility *β*. Table 1 shows *β*-values thus derived for a few MGs. It is seen that these values belong to a relatively narrow range 15 ≤ *β* ≤ 22 that, in general, agrees with the IT. Other methods used for the determination of *β* give close results [49]. Since the shear susceptibility determines the shear softening effect (via Equation (2)), the heat effects (according to Equations (4) and (5)) and also related to the anharmonicity of the interatomic potential, it appears to be a major integral parameter of the glassy structure.

**Figure 4.** Temperature dependences of the quantities Δ*W* and *∂*Δ*G*/*∂T* entering Equation (6) derived from calorimetric and shear modulus measurements [49]. The data correspond to structural relaxation below the glass transition. With permission from Elsevier, 2019.

**Table 1.** Determination of the shear susceptiblity *β* on the basis of calorimetric and shear modulus data taken on MGs in the initial and relaxed state using Equation (6) [49]. With permission from Elsevier, 2019.


### **4. Refinement of the Parameters of the Interstitialcy Theory**

As reviewed earlier [15,16] and discussed in the present work, the IT provides a good description of different aspects of MGs relaxation behavior. At the same time, some model parameters of the IT were introduced in a phenomenological way. It is therefore desirable to clarify their physical meaning and relationship with material parameters. In the initial model, an interstitial defect was considered by Granato as an elastic string. At the same time, a split interstitial can be treated as an elastic dipole [47,50]. The corresponding "dipole" approach is based on the expansion of the energy into a series in powers of the elastic strain created by the dipoles [50]. It was found that that the Granato and "dipole" approaches give practically identical expressions for the elastic energy and shear modulus [47]. A comparison of these approaches leads to an expression for the parameter *α* introduced in the original version of the IT (see Equations (1) and (2)) as *α* = (*<sup>ε</sup>ij<sup>ε</sup>ji* − 13 *<sup>ε</sup>*2*ii*)*dV*/Ω, where *εij* is the elastic strain field created by the interstitial. Thus, the parameter *α* characterizes the "strength" of the defect. The shear susceptibility within the "dipole" approach was calculated as *β* = <sup>−</sup>*γ*4Ω*t*/*μ*, where *μ* is the shear modulus of the maternal crystal and Ω*t* = 1.38 is a parameter characterizing the elastic anisotropy of the interstitial [50]. This estimate is about two times smaller than that given by Granato. It should be noted that the "dipole" approach was found very useful upon further IT development, especially in the part that accounts for the effect of the concentration of interstitials on their interaction (see Section 6).
