**2. The Interstitialcy Theory**

The Interstitialcy theory is essentially based on the experimental investigations of the effect of low-temperature (4 K), low-dose soft neutron irradiation on the elastic moduli of single crystal copper carried out by Granato's group in the 1970s [17,18]. The irradiation leads to the formation of isolated Frenkel pairs (vacancy+interstitial) remaining mostly immobile in the liquid helium temperature range. A careful analysis of the experimental data led to the conclusion that the formation of interstitial defects results in a strong decrease of the shear modulus (diaelastic effect) *C*44 according to Δ *C*44/*C*44 = <sup>−</sup>*Bici*, where *ci* is the interstitial concentration and *Bi* ≈ 30 is the shear sensitivity. For the vacancies, the shear sensitivity is by an order of magnitude smaller, *Bv* ≈ 2. The effect of point defects on the bulk modulus is also small, comparable to the diaelastic effect produced by the vacancies. At the same time, similar results were obtained upon electron irradiation of single crystal aluminum [19]. It was argued that a strong diaelastic effect can be observed only if the interstitials do not make up an octahedral configuration, as considered before the 1970s (red circle in Figure 1a), but form a dumbbell (split) structure, which is characterized by the two atoms trying to occupy the same lattice cite (red circles in Figure 1b). To underline the difference between these defects, the latter defect is also called an interstitialcy [20]. It is of major importance that the octahedral interistitial is a spherically symmetrical defect (similarly to the vacancy) while the dumbbell interstitial is clearly anisotropic and, thus, constitutes an elastic dipole strongly interacting with the external shear stress [19,21]. It is this feature of the defect, which leads to a strong diaelastic effect. It is now generally accepted that dumbbell intertitials exist in all main lattices and constitute the defect state with the lowest formation enthalpy [22–26].

**Figure 1.** Octahedral (**a**) and dumbbell (split) (**b**) interstitial defects in a computer model of a face-centered cubic lattice [27]. All of the dumbbell atoms (marked by red circles) are characterized by < 0, 2, 8, 0 > Voronoi indexes. With permission from JETP Letters, 2019.

An important consequence of the interstitial dumbbell structure consists in the appearance of the low-frequencies in the vibrational spectrum, which are by a few times smaller than the Debye frequency [28]. This in turn leads to the high formation entropy of the defect (for copper, *Sf* ≈ (5 − 10) *kB* according to the Granato's estimate [14]) and, respectively, to a decrease of the Gibbs formation free energy at high temperatures. The extrapolation of the elastic moduli of irradiated crystals towards high defect concentration in the experiments [17,18] showed that *C*44 → 0 at *ci* ≈ 0.02 ÷ 0.03 providing a guess that such big defect concentration should lead to the crystal →liquid transition, because the liquid is characterized by a vanishing (or very small) shear modulus [29,30]. This allowed Granato to sugges<sup>t</sup> that melting of metals should be related to the rapid thermoactivated generation of dumbbell interstitials. Another important point realized by Granato was the understanding that the defect formation enthalpy is proportional to the unrelaxed shear modulus *G*, in line with earlier investigations [31,32]. The above hypotheses led Granato to the formulation of the interstitialcy theory, which includes melting as an integral part [13,14] (see also a discussion given below). In fact, the mathematical formalism of the IT is based on the two equations,

$$\frac{\partial H\_i^F}{\partial c\_i} = \kappa G \Omega\_\prime \tag{1}$$

$$G = \mu \exp(-B\_i c\_i),\tag{2}$$

where *H<sup>F</sup> i* is the interstitial formation enthalpy, *α* is a dimensionless parameter close to unity, *μ* is the shear modulus of the defectless crystal, Ω is the volume per atom and *Bi* = *αβ* with *β* being the dimensionless shear susceptibility. This quantity was estimated by Granato as *β* = −3*C*4444/*C*<sup>44</sup> ≈ 40, where *C*44 is the shear modulus of the crystal and *C*4444 is its fourth-rank (anharmonic) shear modulus. On the other hand, the shear susceptibility is related to the internal energy *U* of the crystal as *β* = 116*μ ∂*4*U ∂ε*<sup>4</sup> , where *ε* is the shear strain. The above equations show that the shear susceptibility constitutes a fundamental parameter of the material since it is proportional to the ratio of the fourth-rank shear modulus to the "usual" shear modulus.

The general approach of the IT to the crystal →melt →glass transformation and the relationship between these states consists in the following. Crystal melting is related to a rapid increase of the concentration of dumbbell interstitials, which remain identifiable structural units in the liquid state (as confirmed by later computer modeling [33]). Rapid melt quenching freezes the melt defect structure in the solid glass and different relaxation processes occurring in it upon heat treatment can be interpreted in terms of the changes of the defect concentration by using Equations (1) and (2). For the glassy state, the quantities *G* and *μ* in these equations correspond to the shear moduli of glass and maternal crystal, respectively. Despite the simplicity of these equations, it has been found that the IT, although originally derived for simple metals, provides explanations for many experiments on multicomponent MGs, as reviewed earlier [15,16] and discussed below.

### **3. Verification of the Main Starting Hypotheses of the Interstitialcy Theory**

### *3.1. Shear and Dilatation Contributions into the Defect Elastic Energy*

The IT is actually built on the hypothesis that the shear component of the elastic energy created by interstitial defects is predominant while the dilatation contribution can be neglected. This agrees with later calculations by Dyre [34] who showed that it is the shear strain component, which produces the main contribution into the elastic energy far from a point defect in a solid. He derived a simple relation for the ratio of the dilatation *Udil* and shear *Ushear* components of this energy and concluded that the former is much smaller, i.e.,

$$\frac{\mathcal{U}\_{dil}}{\mathcal{U}\_{shear}} = \frac{2B/G}{\left[9\left(\frac{B}{G}\right)^2 + 8\frac{B}{G} + 4\right]} \le 0.1,\tag{3}$$

where *B* is the bulk modulus. This equation, however, was derived within the linear elasticity approach for a spherically symmetric defect and does not account for the energy of the defect nucleus. For further verification of the above Granato's hypothesis, a molecular-static modeling of interstitial defects in four FCC metals was performed [27]. To compare the contributions *Udil* and *Ushear* into the total elastic energy, the local relative change of the Voronoi polyhedra volume *Vi* for each atom with respect to the Voronoi polyhedra volume of an atom in the ideal lattice *V*0 was accepted as a measure of the

volume change upon defect formation, i.e., *Udil* = *B*2*V*0 (Δ*V*/*V*)<sup>2</sup>*dV* ≈ *B*2 ∑*i* (*<sup>V</sup>*0 − *Vi*)<sup>2</sup> *V*20 . The shear component of the elastic energy was taken as a difference *Ushear* = *Hf* − *Udil* with *Hf* being the interstitial formation enthalpy. The calculations showed that the ratio *Udil*/*Ushear* for the interstitial dumbbell is nearly twice as that given by Equation (3) due to the accounting of the elastic energy of the defect nucleus. This correction allows using this equation for 63 elemental metals. The result shown in Figure 2 implies that this ratio does not exceed 0.15 for more than 90% of metals. Assuming that MGs contain similar interstitial-type defects in line with the IT, the same calculation procedure was applied to 189 metallic glass composition and nearly the same result was obtained (see Figure 2). Thus, in both cases, the contribution of the dilatation energy is indeed much smaller than that given by the shear energy, just as supposed by Granato [13,14].

**Figure 2.** Histogram illustrating the distribution of the ratio of dilatation *Ubulk* and shear *Ushear* components of the elastic energy for dumbbell interstitials in 63 polycrystalline elemental metals. The same data for interstitial-type defects in 189 metallic glasses are also shown [27]. With permission from JETP Letters, 2019.

### *3.2. Increase of the Interstitial Concentration before Melting*

As mentioned above, the IT argues that melting of metals is related to a rapid increase of the concentration of dumbbell interstitials. This is a crucial statement of the theory. Since dumbbell interstitials produce tenfold bigger shear softening as compared with vacancies, this effect can be detected experimentally provided that there are no contributions coming from other defects (e.g., dislocations) in the crystal. Such experiments were recently performed on single crystal aluminum and coarse-crystalline indium [35,36]. The main results of these measurements are shown in Figure 3. Despite the usual opinion that the equilibrium interstitial concentration *ci* is negligible at any temperature [37], it is seen that *ci* rapidly increases for both metals upon approaching the melting temperature *Tm*. In aluminum, this concentration remains smaller than the vacancy concentration *cv* while for indium *ci* becomes even bigger than *cv* near *Tm*. Besides that, the data on Al clearly demonstrate an increasing tendency of *ci*-growth at high temperatures *T* ≥ 926 K (0.99 *Tm*) while the formation Gibbs free energy start to rapidly decrease in this region [35]. These features agree with the predictions of the IT. It is worth noting that increasing concentration of dumbbell interstitials can explain the non-linear growth of the heat capacity of simple metals near *Tm*, whose nature remains hitherto unclear [38]. It should also be mentioned that temperature dependence of the interstitial concentration obtained for aluminum [35] allowed an estimate of their formation entropy *SFi* , which was found to be ≈7 *kB* [39], in full agreemen<sup>t</sup> with Granato's value [14].

**Figure 3.** Estimates of interstitial and vacancy concentrations in crystalline aluminum (**a**) and indium (**b**) derived from the diaelastic effect measurements. The melting temperatures are indicated [35,36]. With permission from Pleiades Publishing, LTD, 2019.

### *3.3. Identification of Interstitial-Type Defects in the Glassy State*

The topological pattern of dumbbell interstitials in crystals is very clear—two atoms, trying to occupy the same lattice cite (Figure 1b). However, any similar topological picture in the glassy state is absent. In this case, one can try to identify these defects by searching structural regions, which display the properties similar to those of dumbbell interstitials. Thus far, such attempts have been performed using computer models of glassy copper and aluminum [40,41]. It was found that certain nano-sized regions reveal large non-affine displacements and can be characterized by a strong sensitivity to the applied shear stress and distinctive local shear strain fields, which are described by the local shear susceptibility as well as by the diaelastic compliance and diaelastic polarizability tensors. Another feature consists in the characteristic low- and high-frequency modes (far below and above the Debye frequency, respectively) in the vibration spectra of atoms belonging to these regions. All these peculiarities are quite similar to those of dumbbell interstitials in crystals. Thus, interstitial-type defects indeed exist in non-crystalline simple metals. Two-component metallic structures should be analyzed in a similar way. Besides that, numerous simulations of MGs show the presence of the atoms characterized by < 0, 2, 8, 0 > Voronoi indexes (or close to them) [7], which constitute a characteristic feature of dumbbell interstitials in crystals (Figure 1b). One the other hand, an interstitial defect is characterized by the two atoms with these indexes. We are unaware of any attempts for searching two adjacent atoms with < 0, 2, 8, 0 > or close indexes in computer models of MGs.
