## 3 Mathematical Model and Calculation Results
It takes the following form characterised by a constant value of the diffusion coefficient in a spherical system:  
\[\frac{\partial C(l,\tau)}{\partial\tau}=D\cdot\left(\frac{\partial^{2}C(l, \tau)}{\partial l^{2}}+\frac{2}{l}\frac{\partial C(l,\tau)}{\partial l}\right), \tag{2}\]  
where \(C\) denotes the concentration of the element (mass%), \(l\) denotes the distance from the grain centre (m), and \(D\) denotes the diffusion coefficient of the element (m\({}^{2}\cdot\)s\({}^{-1}\)). The change in zinc concentration in the growing grain is shown in Figure 3. The concentration of zinc in the grain is denoted as \(C_{grain}\). The concentration of zinc at the phase boundary on the grain side is denoted as \(C_{grain/amorph}\), while on the amorphous side, it is denoted as \(C_{amorph/grain}\). The concentration of zinc in the amorphous phase is denoted as \(C_{amorf}\). The growth of the grain in the numerical simulation takes place in time steps, which are denoted as \(k\) in the diagram. After one time step (\(k\)), the grain has grown by a unit of length denoted as \(dl\) in the diagram. Meanwhile, the concentrations at the phase boundary change. The initial assumption is that the zinc concentration in the grain (\(C_{grain}\)) is 0%, and in the amorphous phase (\(C_{amorf}\)), it is 28%. The calculations performed in the simulation used a finite-volume numerical method. This method was selected for its speed and ease of implementation for the amorphous Mg\({}_{72}\)Zn\({}_{28}\) alloy. The equations of [PERSON]'s second law describing the concentration change in the grain are given by the following equation:  
\[\frac{\partial C_{grain}}{\partial\tau}=D_{grain}\cdot\left(\frac{\partial^{ 2}C_{grain}}{\partial l^{2}}+\frac{2}{l}\frac{\partial C_{grain}}{\partial l }\right), \tag{3}\]  
and in the amorphous phase,  
\[\frac{\partial C_{amorf}}{\partial\tau}=D_{amorf}\cdot\left(\frac{\partial^{ 2}C_{amorf}}{\partial l^{2}}+\frac{2}{l}\frac{\partial C_{amorf}}{\partial l }\right), \tag{4}\]  
and at the phase boundary, \(X_{k}\), the mass balance equation is introduced [28]:  
\[\left(C_{grain/amorf}^{*}-C_{amorf/grain}^{*}\right)\frac{dl}{d\tau}=D_{amorf }\left(\frac{\partial C_{amorf}}{\partial l}\right)_{X_{k}^{+}}-D_{grain} \left(\frac{\partial C_{grain}}{\partial l}\right)_{X_{k}^{-}}. \tag{5}\]The zinc diffusion coefficient in the grain of the primary magnesium phase is equal to \(D_{grain}=2.67\cdot 10^{-4}\), m\({}^{2}\cdot\)s\({}^{-1}\), and in the amorphous phase, \(D_{amorf}=2.67\cdot 10^{-3}\), m\({}^{2}\cdot\)s\({}^{-1}\)[29]. The following boundary conditions were adopted for the calculations:  
* In the grain centre, \(l=0,\rightarrow\frac{\partial C_{grain}}{dl}=0\);
* In the amorphous phase, _very far_ from the growing grain, \(l=X_{D}\), \(\rightarrow\) the constant concentration value equal to the initial concentration \(C_{0}\) (Dirichlet boundary condition);
* At the phase boundary, the concentrations were calculated using Equation (5), and \[\int_{0}^{X_{k}}C_{grain}dl+\int_{X_{k}}^{X_{D}}C_{amorf}dl=\int_{l_{0}}^{X_{d}} C_{0}dl,\] (6) where \(l_{0}\) is the initial grain size.