**8. Concluding Remarks**

The Interstitialcy Theory (IT) of condensed matter states thermodynamically predicts that melting of simple metallic crystals is related to a rapid increase of the concentration of the interstitial defects in their most stable dumbbell form just near the melting temperature. Recently, rather convincing (although indirect) experimental arguments confirming this hypothesis were obtained. Computer simulations showed that these defects remain identifiable structural objects in the liquid state. Rapid melt quenching freezes them in the solid glass. In the liquid and glassy states, these defects do not have any clear topological pattern as in crystals (two atoms trying to occupy the same lattice site) but nonetheless display all the properties characteristic of dumbbell interstitials in crystals, as confirmed by computer modeling of mono-atomic metallic systems. These properties include strong susceptibility to the applied shear stress, specific strain fields and low-frequency modes in the terahertz vibration spectrum. It is found that rather numerous relaxation phenomena can be understood by assuming that these interstitial-type defects indeed exist in real multicomponent metallic glasses (MGs).

The mathematical formalism of the IT is quite simple and based on the two relations linking the unrelaxed shear modulus, which constitutes the basic thermodynamical parameter of the IT (being the second derivative of the Gibbs free energy with respect to the shear strain), with the formation enthalpy of interstitial-type defects and their concentration. A thermoactivated change of the defect concentration (due to structural relaxation below *Tg*, in the supercooled liquid state or upon crystallization) leads to an alteration of the formation enthalpy and results in numerous heat effects, which are intrinsically related to the relaxation of the shear modulus, as verified by specially designed experiments. On the other hand, changes of the defect concentration lead to certain volume changes, which can also be monitored by the shear modulus relaxation. Accepting reasonable values of the material parameter for the interstitial-type volume relaxation (i.e., the relaxation volume), one can explain the kinetics and final volume changes occurring upon structural relaxation and crystallization.

The IT leads to the conclusion that the excess internal energy and enthalpy of the glassy structure with respect to the maternal crystal is mostly related to the elastic energy of the interstitial-type defect structure frozen-in from the melt upon glass production. The full dissipation of this elastic energy constitutes the heat of crystallization. On the other hand, since the total amount of the defects is

determined by the melting, there arises a relationship between the latent heat absorbed upon melting and heat release occurring upon structural relaxation and crystallization, as verified experimentally.

Within the framework of the IT, the low temperature boson heat capacity peak originates from low-frequency resonance vibration modes of interstitial-type defects and, thus, its height should be proportional to the defect concentration. Specially performed experiments showed that this is indeed the case provided that the concentration is derived from measurements of the shear modulus, as assumed by the IT. The defect concentration determines the excess enthalpy of the glass and it is the origin, which defines the observed dependence of the boson peak height on the experimentally measured excess enthalpy.

In general, the obtained results convincingly demonstrate an intrinsic relationship of the shear modulus relaxation with the heat and volume effects occurring upon structural relaxation and crystallization of MGs. At that, the properties of the glass are tightly related with those of the maternal crystal, in line with the basic assumptions of the IT. It has also been shown that other models describing defects and properties of MGs are largely compatible with the IT. The ways for the development of the IT and its relation to other models on defects in MGs are discussed.

**Author Contributions:** The authors equally contributed to this work.

**Funding:** This research was supported by the Ministry of Science and Education of the Russian Federation under the gran<sup>t</sup> 3.1310.2017/4.6.

**Acknowledgments:** The authors are grateful to their colleagues, Yu.P. Mitrofanov, R.A. Konchakov, A.S. Makarov, G.V. Afonin and E.V. Goncharova, for long-term fruitful cooperation.

**Conflicts of Interest:** The authors declare no conflict of interest.
