# Modelling Crystalline α-Mg Phase Growth in an Amorphous Alloy Mg72Zn28

Bartosz Opitek, Pawel Żak, Janusz Lelito, V. Vignal

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Bartosz Opitek, Pawel Żak, Janusz Lelito, V. Vignal. Modelling Crystalline α-Mg Phase Growth in an Amorphous Alloy Mg72Zn28. Applied Sciences, 2024, 14 (7), pp.3008. 10.3390/app14073008 hal-04736325

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Article

# Modelling Crystalline α-Mg Phase Growth in an Amorphous Alloy Mg₇₂Zn₂₈

Bartosz Opitek <fnref n="1" />,‡, ID, Paweł L. Żak <fnref n="1" />,‡,* ID, Janusz Lelito <fnref n="1" />,‡,* ID and Vincent Vignal <fnref n="2" />,‡

**Abstract:** A model of α-Mg grain growth in an amorphous Mg₇₂Zn₂₈ alloy matrix was developed together with a numerical software. Their application enables tracking of the process of growth of the α-Mg phase in an amorphous alloy. The model was based on a diffusion growth α-Mg in amorphous alloy under the appropriate boundary conditions for the isothermal temperature of annealing and taking into account the presence of a grain with an initial radius of 1 nm. The numerical model is based on the mathematical model of heat flow, described by the Fourier-Kirchhoff equation, and diffusion, described by Fick's second law. The initial boundary conditions necessary to simulate grain growth in the amorphous phase were established. The results of the numerical simulation indicate grain growth with an increase in the isothermal annealing temperature and an increase in the isothermal annealing time.

**Keywords:** Amorphous structure; Crystallisation; Mg₇₂Zn₂₈ alloy; Numerical methods; Metallic glass

## 1. Introduction

Computer simulations are used to describe various phenomena that occur during the crystallisation process. The development of knowledge and technology that has been progressing for several decades allows for continuous improvement of the tools used, especially at the level of numerical calculations. More and more modern computational methods are used, which give more satisfactory results and, more importantly, are very often consistent with the actual behaviour of the simulated materials [1].

In recent years, numerical models have been used for nucleation and grain growth processes during the crystallisation of metallic glasses [2]. Stefanescu [3] proposed a division into macroscopic models (1st generation) and micro/macro models (2nd generation). First-generation models are based on the relationship between the share of the local solidified phase and the temperature. The second generation models are characterised by the inclusion of the laws of nucleation and grain growth in the calculations, which is typical for modelling at the microlevel [4].

The first work related to modelling the nucleation and growth of grains in alloys with an amorphous structure (called metallic glasses) was carried out by Uhlmann [5]. He derived equations that combine classical theories of nucleation and growth and those taken from the Kolmogorov–Johnson–Mehl–Avrami theory. This allowed for the determination of time-temperature transition diagrams and the prediction of critical cooling rates for various metallic glasses. Morris [6] modelled crystallisation in the Ni-Si-B system. In this case, the author combined the thermodynamic model of Thompson and Spaepen [7] with the previously mentioned Uhlmann model. Gránásy [8], in turn, used the Gibbs free energy for his model. He calculated the crystallisation kinetics for the Fe₄₀Ni₄₀P₁₄B₆ metallic glass by measuring the difference in the specific heat of the liquid and crystalline phases. Ge et

**Citation:** Opitek, B.; Żak, P.L.; Lelito, F.; Vignal V. Modelling crystalline α-Mg phase growth in an amorphous alloy Mg₇₂Zn₂₈. *Appl. Sci.* **2024**, *1*, 0.
https://doi.org/

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**Copyright:** © 2024 by the authors. Submitted to *Appl. Sci.* for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ AGH University of Krakow, Faculty of Foundry Engineering, 30-059 Cracow, 23 Reymonta Street, Poland</footnote>

<footnote n="2">² ICB, UMR 6303 CNRS-Université de Bourgogne, BP 47870, 21078 Dijon Cedex, France</footnote>

<footnote>* Correspondence: pawelzak@agh.edu.pl (P.Z.); lelito@agh.edu.pl (J.L.)</footnote>

<footnote>‡ These authors contributed equally to this work.</footnote>
al. [9] used the Uhlmann and CALPHAD kinetics databases, which allowed the prediction of the possibility of obtaining metallic glasses of Cu-Zr and Cu-Zr-Ti.

All the works cited and others [10–13] assume polymorphic conditions, which means an equal chemical composition of the particle and matrix phases. Crystallisation then depends on the rate of attachment of atoms at the phase boundary, while the coupling related to grain growth and transport through diffusion to the grain-matrix phase boundary is omitted. Suryanarayana et al. [14] and Köster et al. [15] showed that polymorphic crystallisation is quite rare in metallic glasses.

A model that fully shows the kinetics of the crystallisation of metallic glasses should describe both the nucleation of crystals and the growth of grains controlled by the diffusion and dissolution of particles. Such a model should show the phase transition under both isothermal and non-isothermal conditions. It is possible to design a numerical simulation model that, using appropriate models of the nucleation and growth division in metallic glasses, shows the evolution of the crystal size distribution over time.

The modelling of the nucleation, growth, and dissolution of crystals for the amorphous Al-Cu-Zr alloy was shown in their work by Ericsson et al. [2]. In this model, the authors use two theories: the classical nucleation theory and the diffusion controlled multicomponent growth developed by Chen [16], which in turn is implemented by the numerical scheme of Kampmann and Wagner [17] and the latter was designed by Myhr and Grong [18]. In a recent publication, it was noticed that the concentration gradient causes the movement of atoms, which are excited by high temperature, causing a largely diffusive phase transformation [19,20]. Due to the increase in temperature, secretory reactions occur. If they are formed from a supersaturated solid solution, three important processes occur [17,21,22]: nucleation, growth, and thickening. The first two processes are driven by the change in the free energy of the transformation and end when the equilibrium volume fraction of the resulting phase is reached. Grain coarsening, in turn, is driven only by a reduction in surface energy that continues until a particle emerges. The presented model uses the works of Kampmann et al. [17,22] and Langer and Schwartz [23], where the three processes mentioned above occur in numerical simulation. An appropriate model (i.e. one that can cope with rapid temperature fluctuations) should include a control volume-based flux balance formulation [24].

The model of Myhr and Grong [18] contains three important elements: (i) the nucleation model predicts the number of already stable nuclei at each computational time step, (ii) the rate law, which calculates the rate of dissolution or growth for each discrete particle size class, and (iii) the continuity equation recording the amount of solute in precipitates. In the model of Ericsson et al. [2], the model of Myhr and Gong [18] was coupled with the CALPHAD database of the Al-Cu-Zr system [24], and the following assumptions were made: (i) spherical shape of grains with a sharp boundary between the grains and the matrix, (ii) cross-diffusion of elements is neglected, while the diffusion of alloying elements in the matrix is equal, (iii) nucleation takes place at a fixed rate, (iv) a local equilibrium corrected by the Gibbs-Thomson effect is maintained at the grain-matrix boundary, (v) molar volumes of the grain and matrix are assumed to be equal, (vi) grain growth is fully controlled by diffusion in the matrix. All these assumptions made it possible to assess the thermodynamic properties dependent on the phase-transition temperature and chemical composition of the amorphous glass. In addition, they take into account the imperfect mixing present in all such materials. The numerical model was built on the basis of a comparison of time-temperature diagrams of the transformation and continuous heating/cooling transformation. The formation of intermetallic phases was simulated under isothermal and nonisothermal conditions. The results obtained were consistent with the experimental data, which confirmed the correctness of the selected model.

Numerical modelling shows great possibilities to predict crystallisation in the case of amorphous alloys. The great advantage of all simulations of this type is low cost, and other advantages include the ability to perform them under almost any condition, revealing many previously unforeseen defects in the material, or even avoiding unnecessary attempts
at costly production of a previously designed material. Some amorphous alloys crystallise after a very long time, and numerical models will now allow you to check many different properties of them [25].

## 2. Materials and Methods

First, a mathematical model of grain growth was developed on the basis of the diffusion model described in the next section. On the basis of this model, several numerical calculations were performed. The results of these calculations (grain size) were compared with the results obtained from the experiment. For this purpose, a two-component eutectic alloy Mg₇₂Zn₂₈ was produced by casting in a resistance furnace under the protection of argon as an inert gas. Both magnesium and zinc were 99.9% pure. Liquid Mg₇₂Zn₂₈ alloy was poured into the steel mould cavity. In this way, a cylindrical sample with a diameter of 20 mm and a height of 50 mm was obtained. Then, this sample was rolled to a diameter of 10 mm. The sample prepared in this way was melted using a melt spinning device to cast a ribbon with an amorphous structure. The thickness of the obtained ribbon was 150 µm and the peripheral speed of the wheel was 40 ms⁻¹. Both processes, i.e. melting and casting, took place under the protection of argon as an inert gas. The obtained ribbons were subjected to X-ray diffraction tests to confirm the amorphous structure. The X-ray diffraction (XRD) results were collected using a Panalytical Empyrean diffractometer (Almelo, Nederland) equipped with a Cu Kα X-ray source. The next step of the research was to verify the simulation results with the experiment. For this purpose, the amorphous ribbons were subjected to isothermal annealing heat treatment at selected temperatures. Heat treatment was carried out with high purity argon using a TA DSC Q20 differential scanning calorimeter (Eschborn, Germany). The rate of heating the amorphous ribbon to the isothermal annealing temperature was 80 K/min. The ribbon was held at an isothermal annealing temperature for 300 seconds and then cooled to ambient temperature at a rate of 80 Kmin⁻¹. Grain size tests were performed using a JEOL JSM-7600F field emission scanning electron microscope (FE-SEM).

## 3. Mathematic model and calculation results

The model presented concerns the crystallisation of the two-component amorphous alloy Mg₇₂Zn₂₈. The study analysed the increase in the primary phase of α-Mg. In this work, the presented model concerns the analysis of the grain growth rate in the amorphous phase as a function of the isothermal annealing temperature. Therefore, certain assumptions are necessary to perform the correct numerical calculations.

The binary Mg₇₂Zn₂₈ alloy has difficulty obtaining an amorphous structure. This is due to the small amount of alloying elements. As is already widely known, multicomponent alloys in which the atomic sizes of individual alloy components differ by approximately 20% have a better tendency to glass transition (containing an amorphous structure). Therefore, attempts were made to select a chemical composition as close as possible to the eutectic one, as shown in the phase equilibrium system of the Mg-Zn alloy (blue line) in Figure 1. This chemical composition facilitates the production of an amorphous alloy as a result of the small range between the liquidus temperature and the solidus. Line number 1 is an extension of the liquidus line, and it was determined using the formula $y = -0.0077x + 5.0323$. Line number 2 is an extension of the solidus line and was determined by the formula $y = -0.0002x² + 0.0637x + 24.462$. The red line indicates the isothermal temperature for which numerical calculations were carried out.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Mg-Zn phase equilibrium system.</figcaption></figure>

The crystallisation kinetics simulation will show grain growth in the amorphous phase. Therefore, it is assumed that the seed exists from the very beginning as a ball and grows. The initial grain radius assumed for the calculations is 1 nm, while the stop of the calculations is ruled by the simulation time. The simulation takes place in a fixed domain with a radius of 200 nm. The scheme used for this calculation model is shown in Fig. 2, which additionally indicates how the zinc (Zn) concentration will change.

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<figure><img src="image_3.png" /><figcaption>Figure 2. Mg-Zn phase equilibrium system.</figcaption></figure>

The numerical model used in the simulation uses the Fourier-Kirchhoff equation [26], which is one of the simplest solutions for this type of simulation. It describes the heat flow in a solidifying casting, but it can also be used for research on amorphous alloys, where the Fourier-Kirchhoff equation will allow you to calculate how heat flows in the material. This is macrolevel modelling that ignores more detailed microlevel assumptions. The model used in this work has certain simplifications (this allowed us to largely simplify the complexity of calculations and, consequently, the time of their execution), where the internal heat source was omitted, and the thermophysical parameters were assumed as constants. The calculations were performed in a one-dimensional system. The Fourier-Kirchhoff equation used in this work to calculate the numerical model has the form:

$$c_v \cdot \frac{\partial T}{\partial \tau} + c_v \cdot u \cdot \text{grad}T = \text{div}(\lambda \cdot \text{grad}T) + q_v, \quad (1)$$

where:

$c_v$ - is the volumetric specific heat, $J \cdot m^{-3} \cdot K^{-1}$,

$T$ - is temperature, $K$,

$\tau$ - time, $s$,

$\lambda$ - is the thermal conductivity, $W \cdot m^{-1} \cdot K^{-1}$,

$u$ - is the velocity vector, $m \cdot s^{-1}$,

$q_v$ - is the solidification-related heat source function, $W \cdot m^{-3}$.

Numerical calculations were performed on the amorphous Mg₇₂Zn₂₈ alloy, and it was assumed that the growth control process would be diffusion. The grain, which exists from the very beginning, has a surface that separates it from the remaining volume in the form of an amorphous phase. The grain surface changes with the progressive crystallisation of Mg₇₂Zn₂₈, therefore this change is described by Fick's second law [27] related to diffusion. The following form is characterised by a constant value of the diffusion coefficient in a spherical system:
$$ \frac{\partial C(l, \tau)}{\partial \tau} = D \cdot \left( \frac{\partial^2 C(l, \tau)}{\partial l^2} + \frac{2}{l} \frac{\partial C(l, \tau)}{\partial l} \right), \qquad (2) $$

where:

C - denotes the concentration of the element, mass%,

l - the distance from the grain centre, m,

D - the diffusion coefficient of the element, m²·s⁻¹.

The change in zinc concentration in the growing grain is shown in Figure 3. The concentration of zinc in the grain is marked as Cgrain. The concentration of zinc at the phase boundary on the grain side is Cgrain/amorph, while on the amorphous side it is marked with the symbol Camorph/grain. The concentration of zinc in the amorphous phase is the symbol Camorf. The growth of the grain in the numerical simulation takes place in time steps, which are marked with the symbol k in the diagram. After one time step (k), the grain has grown by a unit of length marked dl in the diagram. Meanwhile, the concentrations at the phase boundary change. The initial assumption is that the zinc concentration in the grain (Cgrain) is 0% and in the amorphous phase (Camorf) is 28%.

<figure><img src="image_4.png" /><figcaption>Figure 3. Scheme showing how the concentration changes in the grain/amorphous matrix system.</figcaption></figure>

The calculations carried out in the simulation were performed using a finite volume numerical method. This method was selected taking into account the speed and ease of calculations for the amorphous Mg72Zn28 alloy. The equations of Fick's second law describing the concentration change in the grain are shown by the following equation:

$$ \frac{\partial C_{\text{grain}}}{\partial \tau} = D_{\text{grain}} \cdot \left( \frac{\partial^2 C_{\text{grain}}}{\partial l^2} + \frac{2}{l} \frac{\partial C_{\text{grain}}}{\partial l} \right), \qquad (3) $$

and in the amorphous phase:

$$ \frac{\partial C_{\text{amorf}}}{\partial \tau} = D_{\text{amorf}} \cdot \left( \frac{\partial^2 C_{\text{amorf}}}{\partial l^2} + \frac{2}{l} \frac{\partial C_{\text{amorf}}}{\partial l} \right), \qquad (4) $$

at the phase boundary, Xk, the mass balance equation is introduced:
$$ (C_{\text{grain/amorf}}^* - C_{\text{amorf/grain}}^*) \frac{dl}{d\tau} = D_{\text{amorf}} \left(\frac{\partial C_{\text{amorf}}}{\partial l}\right)_{X_k^+} - D_{\text{grain}} \left(\frac{\partial C_{\text{grain}}}{\partial l}\right)_{X_k^-}. \quad (5) $$

The zinc diffusion coefficient in the grain of the primary magnesium phase is equal to $D_{\text{grain}} = 2.67 \cdot 10^{-4}, m^2 \cdot s^{-1}$, and in the amorphous phase $D_{\text{amorf}} = 2.67 \cdot 10^{-3}, m^2 \cdot s^{-1}$ [28].

The following boundary conditions were adopted for the calculations:

- in the grain centre: $l = 0, \rightarrow \frac{\partial C_{\text{grain}}}{\partial l} = 0$;

- in the amorphous phase, very far from the growing grain: $l = X_D, \rightarrow$ constant concentration value equal to the initial concentration $C_0$ (Dirichlet boundary condition);

- at the phase boundary: concentrations calculated on the basis of eq. 5, and

$$ \int_{0}^{X_k} C_{\text{grain}} dl + \int_{X_k}^{X_D} C_{\text{amorf}} dl = \int_{0}^{X_d} C_0 dl, \quad (6) $$

where $l_0$ is the initial grain size.

## 4. Results of numerical simulation

Numerical simulation showing grain growth in the amorphous phase of the Mg$_{72}$Zn$_{28}$ alloy required additional assumptions. The appropriate isothermal annealing temperature values were necessary. Based on the research carried out in the work of Opitek et al. [29] for the amorphous Mg$_{72}$Zn$_{28}$ alloy, the following values were selected: 350 K, 352 K, 353 K, 353.5 K, 354 K. The total crystallisation time for each of these temperature values was, respectively: 6001 s, 4067 s, 2916 s, 2644 s and 2447 s. The simulation results are shown in Figure 6. Each graph shows the increase in grain size in the amorphous phase over time for a given isothermal annealing temperature. At the same time, the diffusion of zinc into the grain continues. Therefore, the concentration of this element in the grain increases. It can also be seen how the crystallisation front shifts with the progressing phase transformation. Zinc concentrates at the phase boundary and is pushed away by the crystallisation front moving over time. This is mainly due to the inability to diffuse so much zinc into the growing grain.

At an isothermal annealing temperature of 354 K, the grain size increased from 20 nm after 50 seconds to 70 nm after 300 seconds of isothermal annealing.
<figure><img src="image_5.png" /><figcaption>**Figure 4.** The X-ray diffraction patterns of Mg<sub>72</sub>Zn<sub>28</sub> cast ribbon.</figcaption></figure>

The ribbon produced made of the binary Mg<sub>72</sub>Zn<sub>28</sub> alloy was examined by XRD. The results of these tests are presented in Figure 4. The XRD results of the ribbon showed an amorphous character. An amorphous ribbon of the binary Mg<sub>72</sub>Zn<sub>28</sub> alloy, after heat treatment and isothermal annealing at a temperature of 354 K for 300 seconds, was subjected to SEM examination. The results of these tests are presented in Figure 5.

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<figure><img src="image_6.png" /><figcaption>**Figure 5.** Results of SEM examination of the microstructure of an amorphous ribbon after heat treatment of isothermal annealing at a temperature of 354 K for 300 seconds.</figcaption></figure>

The average grain size was determined using the secant method. The average grain size was approximately 71 nm.

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<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 6.** Simulation results of grain growth over annealing time for chosen isothermal annealing temperature values: T = 350 K, T = 352 K, T = 353 K, T = 353.5 K, T = 354 K.</figcaption></figure>

The results of the numerical simulation in Figure 7 show how much grain grew in the amorphous phase at various temperature values for the given time values, changing every 50 s in the range from 50 s to 300 s. The grain grows the fastest for the highest value of the isothermal annealing temperature of 354 K, and the slowest for the lowest value of the isothermal annealing temperature of 350 K. Therefore, it is noted that as the value of the isothermal annealing temperature decreases, the grain growth rate also decreases.
This phenomenon is related to the fact that the diffusion process, which depends on the temperature value, plays a decisive role during grain growth.

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

**Figure 7.** Simulation results of grain growth at various isothermal heating temperatures after selected annealing times: 50 s, 100 s, 150 s, 200 s, 250 s, 300 s.

**5. Summary and Conclusions**

The prepared numerical simulation for the amorphous Mg₇₂Zn₂₈ alloy was based on the Fourier-Kirchhoff equations of heat conduction and Fick's second law. Zinc diffusion is
a process that controls grain growth. The mathematical model predicted the crystallisation process of the α-Mg primary phase. Assuming some simplifications, the calculations focused on the finite-volume method for the grain/matrix interface and the finite difference method for the remaining area. This allowed efficient calculations without omitting any important assumptions.

The simulation results indicate a correctly selected numerical calculation model for the amorphous Mg₇₂Zn₂₈ alloy. For a given isothermal annealing temperature, the grain size increases as the isothermal annealing time increases. For the isothermal annealing temperature of 354 K, the grain size changed from 20 to 70 nm after 300 seconds. By comparing the grain size for the isothermal annealing temperature of 354 K and the annealing time of 300 seconds, a very good agreement between the simulation results (70 nm) and the experimental results (71 nm) is visible.

The grain growth kinetics is greater with higher values of isothermal annealing temperature. This phenomenon is related to the fact that, on the one hand, the process controlling grain growth is the process of diffusion, and, on the other hand, diffusion depends on temperature. It should be remembered that the diffusion process is a very slow process and the nucleation process may still progress, which may lead to a situation in which the grain size will be very small, e.g. about 50 nm.

**Author Contributions:** Conceptualisation, J.L.; methodology, J.L.,P.Z., V.V.; software, P.Z.; investigation, B.O., V.V; formal analysis, J.L., P.Z., B.O., V.V.; data curation, B.O., P.Z., J.L., V.V.; writing - original draft, B.O., J.L.; writing - review and editing, P.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by "The Excellent Initiative—Research University" (IDUB) programme, grant number 501.696.7996 (action 4, 1658).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the
Authors upon reasonable request.

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

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