**1. Introduction**

The non-crystallinity of glasses necessarily leads to the heat release or heat absorption upon heat treatment. This applies in full to metallic glasses (MGs) [1]. Exothermal effects in MGs take place upon structural relaxation below the glass transition temperature *Tg* as well as upon crystallization. Strong endothermal reaction is observed in the supercooled liquid state, i.e., at temperatures *Tg* ≤ *T* < *Tx*, where *Tx* is the crystallization onset temperature [1]. Besides that, any physical impact (thermal cycling, plastic deformation, irradiation, etc.) leads to certain additional heat effects in MGs [2–4].

Current literature most often relates the heat effects in MGs to the changes of the free volume frozen-in upon glass production, as was originally suggested in the 1980s [5]. Later, the enthalpy relaxation in MGs was linked to the amount of the free volume in the simplest linear form [6] and subsequent experiments seemed to confirm this relationship [7,8]. As a matter of fact, heat effects in MGs are often considered as an indirect measure of the free volume since then [8].

However, it was repeatedly mentioned that the free volume concept has definite theoretical shortcomings [9] and the enthalpy release observed during relaxation cannot be simply interpreted as a measure of the free volume change [10]. On the other hand, exothermal reaction occurring upon crystallization is currently considered solely from a general viewpoint relating it to the appearance of crystalline phases, without discussing any specific details that might lead to the release of heat. We are unaware of any physical model, which could explain all heat effects taking place in the glassy state as well as upon crystallization within a unified physical concept, with the only exception discussed below.

Heat effects can be naturally explained if the defect system of glass is considered within the framework of the Interstitialcy theory (IT) suggested by Granato [11,12]. The IT argues that melting of metals is associated with the rapid increase of the concentration of interstitial defects in their most

stable dumbbell form, in line with recent experimental observations [13,14]. These "defects" remain identifiable structural units in the liquid state [15] while rapid melt quenching freezes them in the solid glass. They retain all basic properties of dumbbell interstitials in crystals—high sensitivity to the applied shear stress, specific shear strain fields as well as characteristic low- and high-frequency vibration modes—although do not have any characteristic geometrical image like in crystals (two atoms trying to occupy the same lattice site) [16]. These entities can be considered as interstitial-type "defects" (quotation marks are omitted hereafter). Heat effects can be then interpreted in terms of the changes of the defect concentration. This approach provides quantitative explanations for quite a few other relaxation phenomena in MGs [17].

A description of the heat effects within the framework of the IT is based on an expression for the formation enthalpy of interstitial-type defects [11,12], *H* = *α*Ω *G*, where the dimensionless *α* ≈ 1 is related to the defect strain field [18], Ω is the volume per atom and *G* is the unrelaxed shear modulus. The latter is related to the defect concentration *c* as *G* = *μ exp*(−*αβc*), where *μ* is the shear modulus of the maternal crystal (i.e., the one, which was melted and then used for glass production by melt quenching) measured at the same temperature and the dimensionless shear susceptibility *β* characterises the anharmonicity of the interatomic potential and by the definition equals the ratio of the fourth-order non-linear shear modulus to the second-order (i.e., "usual") shear modulus [17]. Typically, *β* = 15 − 20 depending on MGs' chemical composition [17].

Thus, any change of the defect concentration leads to heat release/absorption depending on the *c*-change sign. For the glassy state, this mechanism was first suggested in Ref. [19]. It was later found that the whole excess enthalpy (heat content) of glass with respect to the maternal crystal (i.e., the difference between the heat contents of the glassy and crystalline states) within the IT framework is related to the elastic energy of interstitial-type defect system and this energy is fully released as heat when the defect concentration drops down to zero as a result of crystallization [20]. Thus, all exo- and endothermal heat effects occurring upon structural relaxation and crystallization can be considered as different sides of the same process—a change of interstitial-type defect concentration. In this case, the heat flow upon structural relaxation and crystallization of glass should be described by the same kinetic law, which relates the heat effects with the defect concentration. At that, the latter can be monitored by shear modulus measurements.

The two main equations of the IT given above lead to the difference in the heat flow of the glassy and crystalline states (per unit mass) conditioned by this relaxation mechanism [18]:

$$
\Delta W(T) = \frac{1}{m} \frac{dH\_{\%}(T)}{dt} = \frac{T}{\beta \rho} \left[ \frac{G(T)}{\mu\_{\!}(T)} \frac{d\mu(T)}{dT} - \frac{dG(T)}{dT} \right], \tag{1}
$$

where *Hg* is the enthalpy of glass, *T*˙ is the heating rate, *ρ* is the density and *m* is the sample's mass. It is seen that the heat flow Δ *W* is fully controlled by the shear moduli of glass and maternal crystal, *G*(*T*) and *μ*(*T*). All other quantities are constants while any fitting parameters are absent. Equation (1) was successfully tested for the heat effects occurring in the glassy state [18] (A similar heat flow law was first derived in Ref. [19]. It differs from Equation (1) by slightly another expression in the square brackets: *Grt μrt dμ*(*T*) *dT* − *dG*(*T*) *dT* , where *Grt* and *μrt* are the shear moduli of glass and maternal crystal at room temperature, respectively. This heat flow law provides the results very close to those given by Equation (1). The latter, however, is more general [18]). However, as mentioned above, the same mechanism should be valid for the crystallization as well, as first argued in Ref. [18] and recently confirmed experimentally on a particular metallic glass [21]. In this work, we show that Equation (1) is valid both for the relaxation in the supercooled liquid state and the whole crystallization kinetics of four Zr-based MGs displaying diverse physical properties.
