**5. Recent Experiments**

### *5.1. Reconstruction of Temperature Dependence of the Shear Modulus Using Calorimetric Data*

Equation (4) for the heat flow *W*(*T*) can be used in the "opposite" way, i.e., for the calculation of the shear modulus relaxation using input calorimetric data . This leads to a relation [51]

$$G(T) = \frac{G\_{rt}}{\mu\_{rt}}\mu(T) - \frac{\rho\mathcal{G}}{T} \int\_{T\_{rt}}^{T} \mathcal{W}(T)dT.\tag{7}$$

where the subscript "*rt*" refers to the room temperature. Figure 5 gives experimental temperature dependences of the shear modulus of a Zr-based glass in the initial state, after relaxation obtained by heating into the supercooled liquid region and after full crystallization. The figure also shows temperature dependences of the shear modulus in the initial and relaxed states calculated with Equation (7) using experimental calorimetric heat flow *<sup>W</sup>*(*T*), temperature dependence of the shear modulus in the crystalline state *μ*(*T*), experimental parameters and material constants entering this equation. It is seen that the calculation reproduces experimental *G*(*T*)-data quite well, including shear modulus growth due to structural relaxation below *Tg* and shear softening in the supercooled liquid region.

**Figure 5.** Experimental and calculated using Equation (7) temperature dependences of the shear modulus *G* of glassy Zr65Cu15Al10Ni10 in the initial and relaxed states. Temperature dependence of the shear modulus *μ* after full crystallization is also shown. Calorimetric *Tg* is indicated by the arrow [51]. With permission from Elsevier, 2019.

### *5.2. Heat Absorption Occurring upon Heating of Relaxed Glass*

Within the IT framework, the heat absorbed upon heating from room temperature *Trt* to a temperature *Tsql* in the supercooled liquid region can be calculated as [52]

$$Q \approx \frac{1}{\beta \rho} \left( G\_{T\_{rt}} - G\_{T\_{sq}} - \mu\_{T\_{rt}} + \mu\_{T\_{sq}} \right), \tag{8}$$

where *GTrt* , *GTsql* , *μTrt* and *μTsql* are the shear moduli of glass and maternal crystal at temperatures *Trt* and *Tsql*, respectively, other quantities are the same as above. Since the shear moduli *GTsql* and *μTsql* in the supercooled liquid state do not depend on the thermal prehistory, the only quantity in Equation (8), which varies upon structural relaxation, is the room-temperature shear modulus. The moduli *GTsql* , *μTrt* and *μTsql* are only temperature dependent. The temperature *Tsql* can be accepted as a constant and the quantities *GTsql* , *μTrt* and *μTsql* are then constants as well. Thus, the heat *Q* is dependent on a single variable *GTrt* . The latter can be changed by preliminary heat treatment. Figure 6 shows the experimental data on the heat absorbed upon warming up of *Pd*40*Ni*40 *P*20 glass into the supercooled

liquid as a function of the shear modulus measured at 330 K. The data points correspond to different preannealing treatments as indicated. It is seen that *Q*(*GT*330 )-data nicely fall onto a straight line, as implied by Equation (6). The slope of this line within a few percent error agrees with its theoretical value given by this equation as *∂Q*/*∂G*330 = 1*βρ* [52]. Similarly, one can describe [53] the widely known so-called "sub-*Tg* enthalpy relaxation" effect, which consists in the growth of the heat absorption near the glass transition temperature *Tg* in MGs subjected to prolonged preannealing well below *Tg* [54,55].

The data discussed above in Sections 3.5, 5.1 and 5.2 convincingly demonstrate a close relationship between shear modulus relaxation and heat effects in MGs, in full agreemen<sup>t</sup> with the IT predictions.

**Figure 6.** Dependence of the integral heat *Q* absorbed upon heating from 330 K to 610 K (supercooled liquid region) as a function of the shear modulus *G*330 measured at 330 K just after heating onset. The points correspond to different preannealing treatment applied for shear modulus measurements and DSC tests as indicated. The solid line gives the lest square fit [52]. With permission from Elsevier, 2019.

### *5.3. Density Changes upon Structural Relaxation and Crystallization*

The interpretation of volume changes using the IT is based on the expected change of the volume Δ*V* upon creation of a dumbbell interstitial defect, which can be represented as Δ*V*/Ω = −1 + *ri*, where *ri* is the so-called relaxation volume reflecting the relaxation of the structure after defect creation, and Ω is the volume per atom [24,56]. Then, if a defect concentration *c* is created, the volume increases by Δ*V* and the relative volume change becomes Δ*V*/*V* = (*ri* − <sup>1</sup>)*<sup>c</sup>*. Using the shear modulus given by Equation (2), one arrives at the relative change of the density upon isothermal structural relaxation at a particular temperature as

$$
\left[\frac{\Delta\rho(t)}{\rho\_0}\right]\_{rel} = \frac{(r\_i - 1)}{a\beta} \ln \frac{G(t)}{G\_0},\tag{9}
$$

where <sup>Δ</sup>*ρ*(*t*) = *ρ*(*t*) − *ρ*0, the densities *ρ*(*t*) and *ρ*0 correspond to the shear moduli *G*(*t*) and *G*0 = *G*(*t* = <sup>0</sup>), respectively, *α* and *β* are the same as in Equation (2). An example of relative density changes as a function of shear modulus change of glassy *Pd*30*Cu*30*Ni*10*P*20 measured at room temperature after isothermal annealing is given in Figure 7 [57]. It is seen that this dependence can be fitted by a straight line, in accordance with Equation (9). This equation implies the slope of this line equal to (*ri* − 1) *αβ* . With *α* = 1 (as usually assumed), the shear susceptibility for this glass *β* = 19 (Table 1) and the relaxation volume *ri* = 1.6 (as for FCC metals [24]), one arrives at the slope equal to 0.031, in a good agreemen<sup>t</sup> with the experimental slope of 0.037 (Figure 7).

Equation (9) can be then modified as [57]

$$\frac{d\ln G}{d\ln V} = \frac{a\beta}{r\_i - 1}.\tag{10}$$

With *β* and *ri* given above, one arrives at *dlnG*/*dlnV* = 32, rather close to the experimental value *dlnG*/*dlnV* = 25 reported in Ref. [58] for the same glass.

For the density change upon crystallization, the IT gives [57]

$$\left(\frac{\Delta\rho}{\rho}\right)\_{cryst} = \frac{(r\_i - 1)}{a\beta} \ln\frac{\mu}{G'}\tag{11}$$

where Δ*ρ* = *ρcrys<sup>t</sup>* − *ρ* with *ρ* and *ρcrys<sup>t</sup>* being the densities of glass and maternal crystal, respectively. It is seen that, if the relaxation volume *ri* > 1, the density change Δ*ρ*/*ρ* > 0 upon crystallization. It was shown that Equation (11) then provides a reasonable explanation of crystallization-induced density changes of Zr-based MGs [57]. However, for loosely packed crystalline structures, one can expect that the relaxation volume *ri* is less than unity. In this case, Equation (11) predicts a *decrease* of the density upon crystallization. Indeed, the literature gives a few examples of crystallization-induced density decrease of about 1% [59,60] or even more [61,62].

**Figure 7.** Dependence of the relative density change on the quantity *ln*(*G*/*G*0), where *G*0 is the initial room-temperature shear modulus and *G* is the room-temperature shear modulus after annealing of bulk glassy *Pd*40*Cu*30*Ni*10*P*20 at *T* = 533 K [57]. With permission from Elsevier, 2019.

For warming up from room to the temperature of the full crystallization, the above reasoning leads to the relationship

$$\frac{\Delta\rho(T)}{\rho\_{rt}} = \frac{r\_i - 1}{a\beta} \ln\left[\frac{\mu\_{rt}}{G\_{rt}} \frac{G(T)}{\mu(T)}\right],\tag{12}$$

where <sup>Δ</sup>*ρ*(*T*) is the density change upon heating. It was shown that this equation provides a good description of <sup>Δ</sup>*ρ*/*ρ*-changes occurring upon heating up to the temperature of the full crystallization [63]. It can be concluded that changes of the density are controlled by the shear moduli of glass and maternal crystal, which in turn reflect the evolution of the interstitial-type defect system.

### *5.4. Relation between the Enthalpies of Relaxation, Crystallization and Melting*

According to the general IT approach discussed above (Section 2), the interstitial-type defects in glass are inherited from the melt. Provided that the melt quenching rate is big enough, one can expect that the defect concentrations in the glass and melt are nearly equal, *cglass* ≈ *cmelt*. The quantity *cglass* determines the whole excess heat content (enthalpy) of the glass. Subsequent structural relaxation below *Tg* leads to a decrease of this concentration by Δ*crel* that results in a growth of the shear modulus from *G* up to *Grel* = *G exp*(*αβ* <sup>Δ</sup>*crel*) and corresponding release of the enthalpy Δ *Hrel*. Upon crystallization, the remaining defect concentration *cmelt* − Δ*crel* ≈ *ccryst* drops down to zero (the defects disappear), the shear modulus increases up to its value *μ* in the crystalline state with simultaneous release of the enthalpy Δ *Hcryst*. The whole heat release of the initial glass after crystallization is then given by Equation (5), where Δ*U* ≈ <sup>Δ</sup>*Hcryst*, as discussed above (Section 3.5). Since the defect concentration quenched-in from the melt is *cmelt* ≈ Δ*crel* + *ccryst*, the heat absorbed upon melting should be approximately equal to the total heat release upon structural relaxation and crystallization , i.e., in terms of the corresponding enthalpy changes,

$$
\Delta H\_{melt} \approx -\left(\Delta H\_{cryst} + \Delta H\_{rel}\right).\tag{13}
$$

The result of a specially designed experiment [64] aimed at the verification of this relationship is reproduced in Figure 8, which shows that the data taken on eleven Zr-, Pd- and La-based MGs can be approximated by a straight line with the unity slope. Thus, the relationship in Equation (13) is indeed valid within the experimental error (about 10%) confirming the idea on a connection between the defects occurring upon melting of the maternal crystal and those disappearing upon structural relaxation and crystallization of the glass.

**Figure 8.** The melting enthalpy vs. the absolute value of sum of the enthalpies of structural relaxation and crystallization. The numbers correspond to different Zr-, Pd- and La-based MGs [64]. With permission from Elsevier, 2019.

### *5.5. Relation of the Boson Heat Capacity Peak to the Defect Structure*

A peculiar universal feature of atomic dynamics in non-crystalline materials consists in the presence of excess (over the Debye contribution) low-frequency vibrational modes [65,66]. These low-frequency modes are detected as a peak in the low temperature (5–15 K) heat capacity *C* plotted as *C*/*T*<sup>3</sup> vs. *T*. These features are usually referred to as the boson peak, which is known for metallic glasses as well [67]. The nature of the boson peak constitutes a matter of intensive ongoing debates [68–71]. Granato argued that the boson peak originates from low-frequency resonance vibration modes of interstitial-type defects frozen-in upon glass production [72]. He showed that the boson peak height should be proportional to the defect concentration. A refined equation for the boson peak height has the form [73]

$$h\_B = \frac{\mathbb{C}^d}{T\_B^3} = \frac{234 \text{ R}}{\Theta\_D^3} \left[ 0.09 f \left( \frac{\omega\_D}{\omega\_r} \right)^3 + \frac{3}{2} \beta \right] c \equiv \Gamma c,\tag{14}$$

where *TB* is the boson peak temperature, *C<sup>d</sup>* is the heat capacity related to the interstitial-type defect system, Θ*D* and *ωD* are the Debye temperature and Debye frequency of the maternal crystal, respectively, *ωr* is the characteristic frequency of interstitial resonance vibrations, *f* is the number of resonance modes per interstitial-type defect, *R* is the universal gas constant and other quantities are specified above. The defect concentration *c* can be monitored by measurements of the shear moduli of glass and maternal crystal as implied by Equation (2). Thus, the boson peak within the framework of the IT is considered to be a "fingerprint" of the defect glass structure.

An experiment aimed to check the prediction given by Equation (14) was carried out on glassy *Zr*65*Cu*15*Ni*10*Al*10 [73]. The main result of this experiment is shown in Figure 9, which gives the measured height of the boson peak as a function of the defect concentration *c* calculated with Equation (2) using room-temperature measurements of the shear modulus after different annealing treatments. The annealing protocol was designed to perform measurements on both fully amorphous and partially crystalline samples. It is seen that independent of the state of the samples (amorphous/partially crystalline), the boson peak height linearly increases with the defect concentration, in line with Granato's prediction [72]. The derivative *dhB*/*dc* calculated from Figure 9 provides reasonable estimates for the resonant vibration frequencies of the defects assumed to be responsible for the boson peak [73].

**Figure 9.** The height of the boson peak as a function of the defect concentration *c* calculated using Equation (2). The line gives the least square fit [73]. With permission from John Wiley and Sons, 2019.

On the other hand, since the interstitial-type defect structure determines the excess enthalpy Δ*H* of glass, as discussed above in Section 5.3, one can expect that the boson peak height *hB* should also be related to Δ*H*. The calculation gives the relation between the boson peak height and excess enthalpy of glass Δ*H* as [74]

$$
\hbar \mathbf{h}\_B = \frac{\Gamma}{\alpha \beta} \ln \frac{\mu}{\mu - \rho \beta \Delta H'} \tag{15}
$$

where the quantity Γ is defined by Equation (14), *ρ* is the density, and *μ*, *α* and *β* are same as in Equation (2). The relationship in Equation (15) directly connects the boson peak height with the excess enthalpy of the glass. The latter may be considered as an independent variable, which can be changed by the annealing leading to either structural relaxation within the glassy state or partial crystallization. Using differential scanning calorimetry, the excess enthalpy can be determined as Δ*H* = *T* ˙ −1 *Tf Ti <sup>W</sup>*(*T*)*dT*, where *W* is the heat flow measured by the calorimeter, *T*˙ is the heating rate, *Ti* can be accepted equal to room temperature and *Tf* is the temperature leading to the full crystallization. Figure 10 shows the dependence of the boson peak height *hB* calculated using Equation (15) together with the experimental *hB*-data as function of the variable *ln* [*μ*/(*μ* − *ρβ*Δ*H*)] [74]. It is seen, first, that the calculated and experimental *hB*-points are quite close. The experimental dependence *hB*(Δ*H*) nicely falls onto a straight line and the slope of this dependence equals to (5.14 ± 0.29) × 10−<sup>4</sup> J/(mol × <sup>K</sup><sup>4</sup>). This agrees with this slope given by Equation (15) as Γ*αβ* = 4.9 × 10−<sup>4</sup> J/(mol × <sup>K</sup><sup>4</sup>). Thus, the IT-based approach reproduces the boson peak height and provides a good description of its height on the excess enthalpy of the glass.

**Figure 10.** Experimental and calculated height of the boson peak *hB* as a function of the excess enthalpy Δ*H* plotted according to Equation (15). The numbers near the data points indicate the corresponding preannealing temperatures in Kelvins [74]. With permission from John Wiley and Sons, 2019.

An interesting study of the boson peak was recently reported by Brink et al. [75]. They performed a molecular dynamic simulation of an equiatomic CuNiCoFe alloy in the crystalline and amorphous states alternatively with chemical disorder (high-entropy state), structural disorder and reduced density. They found that the density reduction and fluctuations of the elastic constants cannot be responsible for the boson peak. However, they revealed that the boson peak in the crystal increases with the concentration of dumbbell interstitials while other defects (e.g., dislocations) do not contribute to it. Interstitial atoms even at a small concentration lead to a boson peak, which is close to that in the glass of the same composition. At that, the vibrational modes of interstitial defects in the crystal resemble those of glass. Finally, the authors of [75] concluded that the softened regions provided by interstitials resemble the "soft spots" discussed in the literature on MGs [70,76] and the boson peak is due to quasi-localized defect-related modes.

### *5.6. Relation between the Properties of Glass and Maternal Crystal*

In general, one can expect that the physical properties of MGs should be somehow related with the properties of their maternal crystalline states, which were used for the production of these glasses by melt quenching. Indeed, for instance, the properties of PdNiCuP glasses and their relaxation upon annealing should necessarily be related to the properties of intermetallic and/or metal-phosphide crystalline phases. To our knowledge, however, this issue was not raised in the literature.

Meanwhile, the IT is intrinsically based on the crystal→glass relationship. It starts from the main IT equation for the shear modulus of the glass *G*, which is scaled by the shear modulus of the maternal crystal *μ* (defectless state) as *G* = *μ exp*(−*αβc*). That is why the shear moduli of glass and maternal crystal determine major thermodynamic parameters of the glass, including its excess internal energy (≈enthalpy) as given by Equation (5). As a result, the shear moduli *G* and *μ* explicitly enter all relations for the heat effects (Equations (4), (6), (8) and (13)), volume relaxations (Equations (9)–(12)) and low temperature excess heat capacity (boson peak) (Equations (14) and (15)). By that, in most cases, one must know not only *G* and *μ* at particular temperatures but also their exact temperature dependences, otherwise any quantitative agreemen<sup>t</sup> with the experiment cannot be achieved. Temperature dependences of the shear moduli of glass and maternal crystal reflect the interstitial-type defect structure of glass and the physical origin of crystal→glass relationship within the framework of the IT is intrinsically related to this defect structure, which controls the fundamental properties of the glass [77]. In a certain sense (not structural), one can accept Granato's statement that ". . . glasses and dense liquids are crystals containing a few percent of interstitials" [78].

### **6. Development of the Interstitialcy Theory**

Despite the amazing matching of the IT predictions with a number of relaxation phenomena in MGs, there exist a few phenomena, which cannot be easily interpreted within the framework of the original Granato's theory. First, an increase of the apparent defect concentration above *Tg* due to the thermal activation, which follows from the original IT version (e.g., see Figure 10 in Ref. [16]), faces certain difficulties related to the high formation enthalpies necessary for this process. The same issue applies to the understanding of so-called rejuvenation of relaxation properties of MGs by quenching (or even relatively slow cooling) from the supercooled liquid region (i.e., from above *Tg*) [79,80]. However, this difficulty can be avoided by assuming that an interstitial-type defect in the glass has a few energy states and the transitions between them are accessible by thermal activation. In the original IT version, this possibility was not assumed. One can sugges<sup>t</sup> that the occurrence of a spectrum of energy states is related to the high defect density in the glass and melt, which is estimated to be a few percent [13,16]. Such high density of the defects will result in their strong interaction and this must be taken into account. In Granato's original approach, the defect interaction was considered only qualitatively.

It is long known from the physics of crystals that this interaction can lead to the formation of interstitial clusters consisting of *N* individual interstitials, from *N* = 2 up to *N* = 7 and even more [23,28,81]. Clustering is energy profitable since the formation enthalpy per interstitial decreases with the number of interstitials [23,81]. The interstitial cluster consisting of *N* = 7 split interstitials in the FCC lattice represents a *perfect* icosahedron with < 0, 0, 12, 0 > Voronoi indexes [81], as illustrated by Figure 11 for the FCC cell. Thus, if melting of metallic crystals is related to an increase of the concentration of split interstitials up to a few percent, as considered by the IT, then one can expect the formation of clusters consisting of *N* = 2 to *N* = 7 interstitial-type defects. It is to be emphasized in this relation the commonly accepted notion that icosahedral clusters constitute a major structural feature of metallic melts and their concentration increases upon supercooling defining thus the dynamic slowdown of the internal movements and eventual glass formation [7,82].

**Figure 11.** Formation of a perfect icosahedron by the creation of dumbbell interstitials on the opposite faces of the FCC cell: (**Left**) elementary FCC cell where the arrows show how two atoms are inserted instead of one atom; and (**Right**) perfect icosahedron with < 0, 0, 12, 0 > Voronoi indexes formed by six dumbbell interstitials on the faces of the cell and one interstitial in the octahedral position (i.e., in center of the cell, see Figure 1a) [81].

One can expect, therefore, that the solid glass will contain, first, individual interstitial-type defects (adjacent atoms with < 0, 2, 8, 0 > Voronoi indexes and/or close to them) and their small clusters (*N* = 2, 3), which correspond to the defect part of the structure. Larger clusters define the icosahedral-type structural backbone. Thus, the metallic glass constitutes a heterogeneous structure. The properties of the clusters should be evidently dependent on the number *N*. Specifically, the shear sensitivity *Bi* should decrease with *N* defining a reduction of the diaelastic effect produced by the clusters. By that, the vibrational entropy of interstitial-type defects in the clusters should decrease due to their interaction leading to the mutual damping of the low-frequency vibration modes, as was qualitatively noted in the original Granato's model [14]. The quasi-equilibrium balance between all these clusters will be dependent on temperature and thermal prehistory defining the evolution of glass properties [83,84]. Some qualitative estimates of clustering kinetics are given elsewhere [85,86]. Further detailed work in this direction is challenging.

### **7. Comparison with Other Models**

Quite a few models are suggested in the literature for a description of defects in metallic glasses. A common shortage in most of them is the lack of understanding of their nature. However, it turns out that many of these models are quite consistent with the IT, as sketched below.

Egami suggested that the main properties of MGs are determined by the defects, which create shear, hydrostatic compression and/or tension (so-called *τ*-, *p*- and *n*-defects) [8]. Meanwhile, the dumbbell interstitials are featured by the same combination of stresses. Figure 12 shows the changes of the Voronoi polyhedra volume (indicated by the numbers) for a dumbbell interstitial in a Cu crystal with respect to the ideal lattice. It is seen that the interstitial produces both positive

and negative changes of the Voronoi polyhedra volume. In other words, the defect creates both hydrostatically compressed and hydrostatically tensioned regions. One can expect that the same should be applicable for an interstitial-type defect in the glass. Taking into account that shear stress field is one of its main characteristics, one can conclude that the defect demonstrates the properties similar to those of Egami's *τ*-, *p*- and *n*-defects.

**Figure 12.** Dumbbell [001]-oriented interstitial (two red circles) and the relative changes of the volume of the Voronoi polyhedra Δ*V*/*V* (indicated by the numbers) as compared with the ideal lattice for the atoms (shaded circles) in a copper crystal. Δ*V*/*V*-changes are shown along the dumbbell axis, perpendicular to it and in the [111] direction. It is seen that the defect creates both positive and negative changes of the Voronoi polyhedra volume (i.e., changes of the local density) [27]. With permission from JETP Letters, 2019.

It is often mentioned that supercooled liquids contain "strings" ("string-like" solitons), which become frozen in the solid glass [12,87–91]. It is also sometimes said that these defects resemble the signatures of dumbbell interstitials in crystals [33,88]. Meanwhile, the "string" idea is compatible with the IT since a split interstitial in the original Granato's model was considered as a string segmen<sup>t</sup> [13,14]. Similar arguments can be applied for the defects viewed as "shear transformation zones" [3,92], "soft spots" [70,76], "soft zones" [92], "flow units" [10,93], "liquid-like regions" [11], "geometricallly unfavored motifs" [76,94] and "regions with large non-affine displacements" [3,9,76]. Since the region of an interstitial-type defect is characterized by the big shear susceptibility, the shear deformation of the surrounding material upon action of the applied stress is bigger that of the matrix and contains a large non-affine component [41]. Naturally, the above terms ("soft zones", "liquid-like regions", etc.) can be applied to this region. It should be noted that these defects also contribute to the low-frequency part of the vibration spectrum of a glassy structure [76,93] similar to what is assumed by the IT.

Another approach to the understanding of defects in MGs is the "free volume" model, which was suggested long ago [1,95,96] and subjected to numerous modifications since then (e.g., Ref. [97]). In this approach, the defects are considered as regions of the reduced density (vacancy-like "free volume"), which affect relaxation and deformation phenomena in MGs. The premise for its popularity consists, on the one hand, in a decreased density of MGs (frozen-in "free volume") with respect to their maternal crystals and, on the other hand, in the existence of numerous correlations of MGs' properties with the amount of the "free volume" [98]. Although this model was repeatedly criticized in different directions [5,99], it nonetheless constitutes perhaps the most popular approach. Meanwhile,

it is quite compatible with the IT. Indeed, one should recall that the volume change Δ *Vv* upon vacancy formation is Δ *Vv*/Ω = *rv* + 1, where *rv* is the corresponding relaxation volume and Ω is the volume per atom [56]. Taking into account the volume change Δ *Vi* occurring upon interstitial formation (see Section 5.2 above) and accepting, e.g., for Al, *ri* = 1.9 and *rv* = −0.38 [24], one can easily arrive to the ratio of the relative volume changes produced by interstitials and vacancies, (Δ *<sup>V</sup>*/*V*)*i*/(<sup>Δ</sup> *<sup>V</sup>*/*V*)*v* = (*ri* − <sup>1</sup>)/(*rv* + 1) = 1.45. This means that the volume changes for interstitials and vacancies have the same sign and are quite comparable in the magnitude [100]. Thus, a decrease of the defect concentration within both the free volume model and the IT should lead to the densification of the glass by about the same amount. Moreover, since the free volume in both models is proportional to the number of the defects, one can naturally expect a correlation of material properties with the amount of the free volume. However, there are major differences between these approaches. In the former one, it is the free volume that constitutes the principal source for the property changes. The IT considers the features of the interstitial-type defect (large shear susceptibility, high formation entropy, and specific strain fields) to be of major importance while defect-related volume changes are of secondary relevance. It is also very important that vacancy-like free volume appears to have the spherical symmetry, which is why it *should not* interact with the external shear stress. Conversely, an interstitial defect is strongly asymmetric (Figure 1b) and constitutes an elastic dipole displaying strong sensitivity to the external shear load. Moreover, if a metallic glass is formed by melting of a loosely packed crystalline structure (related to the generation of interstitial-type defects), the creation of the free volume is not necessary at all. In this case, the density of glass can be even bigger than that of the maternal crystal (see Section 5.2 above) that cannot be understood within the free volume approach.
