## 5 Recent Experiments
### Density Changes upon Structural Relaxation and Crystallization  
The interpretation of volume changes using the IT is based on the expected change of the volume \(\Delta V\) upon creation of a dumbbell interstitial defect, which can be represented as \(\Delta V/\Omega=-1+r_{i}\), where \(r_{i}\) is the so-called relaxation volume reflecting the relaxation of the structure after defect creation, and \(\Omega\) is the volume per atom [24; 56]. Then, if a defect concentration \(c\) is created, the volume increases by \(\Delta V\) and the relative volume change becomes \(\Delta V/V=(r_{i}-1)c\). 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  
\[\left[\dfrac{\Delta\rho(t)}{\rho_{0}}\right]_{rel}=\dfrac{(r_{i}-1)}{\alpha \beta}\ln\dfrac{G(t)}{G_{0}}, \tag{9}\]  
where \(\Delta\rho(t)=\rho(t)-\rho_{0}\), the densities \(\rho(t)\) and \(\rho_{0}\) correspond to the shear moduli \(G(t)\) and \(G_{0}=G(t=0)\), respectively, \(\alpha\) and \(\beta\) 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 \(\dfrac{(r_{i}-1)}{\alpha\beta}\). With \(\alpha=1\) (as usually assumed), the shear susceptibility for this glass \(\beta=19\) (Table 1) and the relaxation volume \(r_{i}=1.6\) (as for FCC metals [24]), one arrives at the slope equal to 0.031, in a good agreement with the experimental slope of 0.037 (Figure 7).  
Equation (9) can be then modified as [57]  
\[\dfrac{dlnG}{dlnV}=\dfrac{\alpha\beta}{r_{i}-1}. \tag{10}\]  
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 best square fit [52]. With permission from Elsevier, 2019.  
With \(\beta\) and \(r_{i}\) 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]  
\[\left(\frac{\Delta\rho}{\rho}\right)_{cryst}=\frac{(r_{i}-1)}{\alpha\beta}\,ln \frac{\mu}{G}, \tag{11}\]  
where \(\Delta\rho=\rho_{cryst}-\rho\) with \(\rho\) and \(\rho_{cryst}\) being the densities of glass and maternal crystal, respectively. It is seen that, if the relaxation volume \(r_{i}>1\), the density change \(\Delta\rho/\rho>0\) 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 \(r_{i}\) 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].  
For warming up from room to the temperature of the full crystallization, the above reasoning leads to the relationship  
\[\frac{\Delta\rho(T)}{\rho_{rt}}=\frac{r_{i}-1}{\alpha\beta}\,ln\left[\frac{ \mu_{rt}}{G_{rt}}\frac{G(T)}{\mu(T)}\right], \tag{12}\]  
where \(\Delta\rho(T)\) is the density change upon heating. It was shown that this equation provides a good description of \(\Delta\rho/\rho\)-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.