*Article* **X-ray Thermo-Diffraction Study of the Aluminum-Based Multicomponent Alloy Al58Zn28Si8Mg<sup>6</sup>**

**Yoana Bilbao 1,\* , Juan José Trujillo <sup>2</sup> , Iban Vicario <sup>3</sup> , Gurutze Arruebarrena <sup>2</sup> , Iñaki Hurtado <sup>2</sup> and Teresa Guraya <sup>1</sup>**


**Abstract:** Newly designed multicomponent light alloys are giving rise to non-conventional microstructures that need to be thoroughly studied before determining their potential applications. In this study, the novel Al58Zn28Si8Mg<sup>6</sup> alloy, previously studied with CALPHAD methods, was cast and heat-treated under several conditions. An analysis of the phase evolution was carried out with in situ X-ray diffraction supported by differential scanning calorimetry and electron microscopy. A total of eight phases were identified in the alloy in the temperature range from 30 to 380 ◦C: α-Al, α'-Al, Zn, Si, Mg2Si, MgZn<sup>2</sup> , Mg2Zn11, and SrZn13. Several thermal transitions below 360 ◦C were determined, and the natural precipitation of the Zn phase was confirmed after nine months. The study showed that the thermal history can strongly affect the presence of the MgZn<sup>2</sup> and Mg2Zn<sup>11</sup> phases. The combination of X-ray thermo-diffraction with CALPHAD methods, differential scanning calorimetry, and electron microscopy offered us a satisfactory understanding of the alloy behavior at different temperatures.

**Keywords:** lightweight multicomponent alloys; X-ray thermo-diffraction; differential scanning calorimetry; Al–Zn; Zn precipitation; Mg–Zn phases; strontium modification

#### **1. Introduction**

Historically, metallic alloys have been developed by selecting one or two major components and adding several minor ones that confer specific properties, such as corrosion resistance or higher mechanical properties. The multicomponent alloy concept, however, is based on the design of alloys where there are several main components that cover the central areas of phase diagrams [1].

Initial developments in the field focused mainly on steel-like alloys for industrial applications. They were based on equiatomic and near-equiatomic compositions of Co, Cr, Cu, Fe, Mn, or Ni, sometimes adding Al, Ti, or Zr, that resulted in single or dual phase microstructures [2–7]. Yeh et al. suggested that the prevalence of solid solutions over intermetallic phases could be explained by the high mixing entropy generated by the multiple components in the alloy, hence the term "high-entropy alloys" (HEAs) [8]. The conditions that the alloys should satisfy to be considered HEAs may be found in [9].

Over the past decade, research has been extended to other alloy classifications that have evolved from the original HEA concept: "medium-entropy alloys" (MEAs) [9], "nonequiatomic HEAs", or "multi-phase HEAs" that may contain bulky secondary phases [10]. In fact, the idea of intentionally having secondary phases in this type of alloy was first suggested by Miracle et al. [11]. Other alloy families have also been explored, leading

**Citation:** Bilbao, Y.; Trujillo, J.J.; Vicario, I.; Arruebarrena, G.; Hurtado, I.; Guraya, T. X-ray Thermo-Diffraction Study of the Aluminum-Based Multicomponent Alloy Al58Zn28Si8Mg6. *Materials* **2022**, *15*, 5056. https://doi.org/10.3390/ ma15145056

Academic Editor: Lijun Zhang

Received: 21 June 2022 Accepted: 19 July 2022 Published: 20 July 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

to refractory metal HEAs for high temperature structural applications based on Cr, Hf, Mo, Nb, Ta, Ti, V, and W or lightweight multicomponent alloys in the aeronautical field involving Al, Li, Mg, or Zn. Several studies may be found in the literature as proof of this tendency [12–17].

Among the lightweight multicomponent alloys, Yang et al. explored the Al–Li–Mg– (Zn, Cu, Sn) system, obtaining structures dominated by intermetallic compounds. The aluminum face-centered cubic (FCC) structure predominated only in selected alloy compositions [18]. In fact, Sanchez et al. highlighted the difficulty of forming solid solutions in medium entropy alloys based on aluminum (65–70 at. %) with elements such as Cu, Mg, Cr, Fe, Si, Ni, Zn, or Zr. The magnitude of the negative mixing enthalpy of aluminum with transition metals gave rise to intermetallic phases [19]. The presence of intermetallics was also reported by Tun et al. [20]. Only the most recent research suggests that rapid solidification processes may enhance single phase microstructures in this type of alloy [21]. However, in a previous work by Nagase et al. on Al–Mg–Li–Ca equiatomic and non-equiatomic alloys, a single solid solution could not be obtained, even with rapid solidification [22].

Asadikiya et al. considered that the application of the entropy concept in aluminum alloys may be the answer to the challenge of developing novel Al alloys with improved properties [10]. Therefore, multicomponent lightweight alloys continue to be researched for their potential applications.

In the present study, the objective was to characterize the novel Al58Zn28Si8Mg<sup>6</sup> cast alloy. It was designed to obtain as much solid solution of aluminum and zinc as possible, reinforced with intermetallics based on Zn, Mg, and Si. On the one hand, zinc is highly soluble in aluminum, enhancing the obtention of a solid solution matrix. On the other hand, Mg–Zn phases are the usual precipitates in 7xx.x aluminum cast alloys, while Mg–Si phases are common in 3xx.x alloys. In addition, Al–Zn-based alloys have attracted the interest of researchers beyond their usual use as coatings. In fact, Al–Zn cast alloys have potential applications where tribological and damping properties are required [23–25]. In terms of entropy, our multicomponent alloy would be classified as a multi-phase MEA.

The approach was explored by the CALPHAD (calculation of phase diagrams) method. This technique requires databases that are valid in composition ranges that may not be found in conventional alloys, thus demanding further experimental verification [26]. However, it is considered the most direct method for compositional design [27] and has already been used in the design of lightweight multicomponent alloys with differing degrees of success [28–30].

The study is focused on identifying and evaluating the effect of the temperature on the phases that are generated at different initial thermal conditions. Differential scanning calorimetry (DSC), electron microscopy, and X-ray thermo-diffraction are the techniques used in this evaluation. X-ray thermo-diffraction, also known as "high temperature X-ray diffraction" (HT-XRD), enables the in situ study of the solution and precipitation phenomena in the alloys [31–33].

#### **2. Materials and Methods**

#### *2.1. Material Manufacturing*

Aluminum was melted at 750 ◦C in a resistance furnace with forced convection; silicon and zinc were subsequently added. Magnesium followed, and once all the elements were melted, strontium was added as a silicon modifier. Aluminum, magnesium, and silicon were of commercial purity, whereas zinc was incorporated by adding a Zamak Zn4Al1Cu alloy so that the final alloy composition contained some residual copper. Samples were obtained to determine the chemical composition by inductively coupled plasma mass spectrometry (ICP) (Table 1).

The metal was gravity cast into a graphite mold, and samples were obtained that were 50 mm long, 22.5 mm wide, and 4 mm thick.


 **Al Zn Mg Si Cu Fe Sr** 

**Table 1.** Chemical composition of the alloy analyzed by ICP. wt. % 41.15 48.40 4.11 5.83 0.43 0.05 0.03

**Table 1.** Chemical composition of the alloy analyzed by ICP.

*Materials* **2022**, *15*, 5056 3 of 15

#### *2.2. Selection of Sample Thermal Treatments and Study Temperatures 2.2. Selection of Sample Thermal Treatments and Study Temperatures*

In order to select the temperatures of interest, DSC tests were performed on as-cast samples (Figure 1). A Netzsch STA 449 Fe Jupiter calorimeter was used, and measurements were made under argon atmosphere in a temperature range between 25 and 675 ◦C with a heating rate of 10 ◦C/min. Samples were then cooled down back to room temperature under these same conditions. In order to select the temperatures of interest, DSC tests were performed on as-cast samples (Figure 1). A Netzsch STA 449 Fe Jupiter calorimeter was used, and measurements were made under argon atmosphere in a temperature range between 25 and 675 °C with a heating rate of 10 °C/min. Samples were then cooled down back to room temperature under these same conditions.

**Figure 1.** DSC curves for the as-cast sample. Peak temperatures during heating were considered for the thermal treatment selection of the samples. **Figure 1.** DSC curves for the as-cast sample. Peak temperatures during heating were considered for the thermal treatment selection of the samples.

We decided to subject the samples to seven different thermal conditions to try to separate and simplify the identification of the different phases appearing and disappearing during the heating process (Table 2 and Figure 2). We decided to subject the samples to seven different thermal conditions to try to separate and simplify the identification of the different phases appearing and disappearing during the heating process (Table 2 and Figure 2). *Materials* **2022**, *15*, 5056 4 of 15

along with the FTlite (2021) database. Only the four main elements in the alloy were

As-cast and Q380 samples were observed with scanning electron microscopy (SEM of Shottky field emission, JEOL JSM-7000F) and energy dispersive X-ray spectroscopy (INCA EDX detector X-sight Serie Si (Li) pentaFET Oxford) at an electron beam voltage of 5.0 kV at room temperature. Specimens had been previously cleaned, ground, and

The equipment used for the X-ray thermo-diffraction tests was a Bruker D8 Advance diffractometer that operated at 30 kV and 20 mA for reflection measurements. It was equipped with a copper anode (λ = 1.5418 Å), a Vantec-1 PSD detector, and an Anton Parr HTK2000 high temperature furnace. The sample holder used, on which the test

The seven specimens, which were 10 × 10 mm2 with a thickness between 1 and 2 mm, were subjected to a heating cycle from 30 to 360 °C and cooling again to 30 °C in the diffractometer. Diffraction tests were performed at room temperature (30 °C), at three temperatures during heating (260, 320, 360 °C), and at three temperatures during cooling (260, 180, 30 °C), based on the temperatures of interest found in the DSC curves (Figure 3). The measurements were recorded in the range 10° ≤ 2θ ≤ 100° at increments of 0.033°,

Equilibrium and Scheil non-equilibrium solidification simulations were carried out with the CALPHAD method for the cast alloy composition with FactSage 7.3 software, 3

polished to obtain a proper surface finish for the analysis.

temperature was controlled, was made of platinum.

*2.3. Thermodynamic Simulations* 

*2.4. Microstructural Observations* 

*2.5. X-ray Thermo-Diffraction Tests* 

with each stage lasting 0.8 s.

considered.

**Table 2.** Heat treatments for each sample condition. The two-step solution treatment in Q360 and Q380 was applied to prevent any partial melting during one-step solution treatment at the solution temperature.


## *2.3. Thermodynamic Simulations*

Equilibrium and Scheil non-equilibrium solidification simulations were carried out with the CALPHAD method for the cast alloy composition with FactSage 7.3 software, along with the FTlite (2021) database. Only the four main elements in the alloy were considered.

## *2.4. Microstructural Observations*

As-cast and Q380 samples were observed with scanning electron microscopy (SEM of Shottky field emission, JEOL JSM-7000F) and energy dispersive X-ray spectroscopy (INCA EDX detector X-sight Serie Si (Li) pentaFET Oxford) at an electron beam voltage of 5.0 kV at room temperature. Specimens had been previously cleaned, ground, and polished to obtain a proper surface finish for the analysis.

## *2.5. X-ray Thermo-Diffraction Tests*

The equipment used for the X-ray thermo-diffraction tests was a Bruker D8 Advance diffractometer that operated at 30 kV and 20 mA for reflection measurements. It was equipped with a copper anode (λ = 1.5418 Å), a Vantec-1 PSD detector, and an Anton Parr HTK2000 high temperature furnace. The sample holder used, on which the test temperature was controlled, was made of platinum.

The seven specimens, which were 10 <sup>×</sup> 10 mm<sup>2</sup> with a thickness between 1 and 2 mm, were subjected to a heating cycle from 30 to 360 ◦C and cooling again to 30 ◦C in the diffractometer. Diffraction tests were performed at room temperature (30 ◦C), at three temperatures during heating (260, 320, 360 ◦C), and at three temperatures during cooling (260, 180, 30 ◦C), based on the temperatures of interest found in the DSC curves (Figure 3). The measurements were recorded in the range 10◦ ≤ 2θ ≤ 100◦ at increments of 0.033◦ , with each stage lasting 0.8 s.

Thermo-diffraction tests were performed three months after the samples were prepared. Twelve months after the preparation; that is, nine months after being subjected to the thermal cycle in the thermo-diffractometer, samples were retested in the same conditions as before but only at 30 ◦C, in order to observe whether natural precipitation had taken place.

The X-ray diffraction patterns were indexed with the PDF-4+ 2021 database from the International Center for Diffraction Data (ICDD). For the search of non-indexed phases, least squares-based Rietveld refinement was carried out in selected patterns with the FullProf software (FullProf.2k Version 7.40, January 2021, J. Rodriguez-Carvajal, ILL, Grenoble, France). The shape of the Bragg peaks was represented by a Pseudo-Voigt function. Conventional R-values, corrected for background, are given in the figures as agreement of the fitting to the observed values [34,35]. The term "intensity" is used to refer to the "integrated intensity".

**Figure 3.** Thermal cycle of the samples in the thermo-diffractometer. Measurements were performed at the temperatures indicated in each step. **Figure 3.** Thermal cycle of the samples in the thermo-diffractometer. Measurements were performed at the temperatures indicated in each step.

#### Thermo-diffraction tests were performed three months after the samples were **3. Results**

#### prepared. Twelve months after the preparation; that is, nine months after being subjected *3.1. Thermodynamic Simulation Results*

to the thermal cycle in the thermo-diffractometer, samples were retested in the same conditions as before but only at 30 °C, in order to observe whether natural precipitation had taken place. The X-ray diffraction patterns were indexed with the PDF-4+ 2021 database from the International Center for Diffraction Data (ICDD). For the search of non-indexed phases, least squares-based Rietveld refinement was carried out in selected patterns with the FullProf software (FullProf.2k Version 7.40, January 2021, J. Rodriguez-Carvajal, ILL, Grenoble, France). The shape of the Bragg peaks was represented by a Pseudo-Voigt function. Conventional R-values, corrected for background, are given in the figures as agreement of the fitting to the observed values [34,35]. The term "intensity" is used to Thermodynamic simulations performed with FactSage for equilibrium cooling conditions (Figure 4a) predicted a high proportion of the FCC aluminum solid solution at temperatures between 360 and 380 ◦C, with the Si and Mg2Si phases being precipitated at these temperatures. As cooling went on, the solid solution decomposed and around 350 ◦C a second aluminum phase (Al#2) was generated but disappeared soon after. This phase would correspond to the zinc-rich α' aluminum phase of the miscibility gap in the Al–Zn system [36,37]. At about 340 ◦C, the intermetallic phase Mg2Zn<sup>11</sup> was formed, and MgZn<sup>2</sup> precipitated from Mg2Zn<sup>11</sup> at around 140 ◦C. The simulation under non-equilibrium conditions (Scheil approximation) predicted the precipitation of Mg2Zn<sup>11</sup> and MgZn<sup>2</sup> at about 370 ◦C and that of hexagonal zinc at 350 ◦C (Figure 4b). *Materials* **2022**, *15*, 5056 6 of 15

**Figure 4.** Thermodynamic simulations with FactSage for the studied alloy considering (**a**) equilibrium solidification conditions and (**b**) non-equilibrium solidification (Scheil model). **Figure 4.** Thermodynamic simulations with FactSage for the studied alloy considering (**a**) equilibrium solidification conditions and (**b**) non-equilibrium solidification (Scheil model).

The microstructure resulting from the as-cast state was heterogeneous, with different phases distributed throughout the interdendritic region depending on the

Mg–Si, and Mg–Zn phases were found by EDX measurements. The Si phase solidified in certain areas as eutectic and in other areas as primary silicon. In addition, isolated Al–Fe–

*3.2. Microstructure of the Samples Depending on the Initial Thermal Condition* 

(**a**) (**b**)

Mg–Si phases were detected.

#### *3.2. Microstructure of the Samples Depending on the Initial Thermal Condition 3.2. Microstructure of the Samples Depending on the Initial Thermal Condition*  The microstructure resulting from the as-cast state was heterogeneous, with

(**a**) (**b**)

*Materials* **2022**, *15*, 5056 6 of 15

The microstructure resulting from the as-cast state was heterogeneous, with different phases distributed throughout the interdendritic region depending on the solidification rate (Figure 5a). In the Q380 condition (Figure 5b), the globulization and reduction in the size of the phases after the solution treatment were remarkable. The Si, Mg–Si, and Mg–Zn phases were found by EDX measurements. The Si phase solidified in certain areas as eutectic and in other areas as primary silicon. In addition, isolated Al–Fe–Mg–Si phases were detected. different phases distributed throughout the interdendritic region depending on the solidification rate (Figure 5a). In the Q380 condition (Figure 5b), the globulization and reduction in the size of the phases after the solution treatment were remarkable. The Si, Mg–Si, and Mg–Zn phases were found by EDX measurements. The Si phase solidified in certain areas as eutectic and in other areas as primary silicon. In addition, isolated Al–Fe– Mg–Si phases were detected.

**Figure 4.** Thermodynamic simulations with FactSage for the studied alloy considering (**a**)

equilibrium solidification conditions and (**b**) non-equilibrium solidification (Scheil model).

(Figure 6).

**Figure 5.** SEM micrographs of the material with x1000 magnification (**a**) As-cast. (**b**) Q380. Numbers 1 to 4 refer to the EDX results provided below. 1: Al-Zn matrix, 2: Mg-Zn phases, 3: Si phases and 4: Mg-Si phases. **Figure 5.** SEM micrographs of the material with ×1000 magnification (**a**) As-cast. (**b**) Q380. Numbers 1 to 4 refer to the EDX results provided below. 1: Al-Zn matrix, 2: Mg-Zn phases, 3: Si phases and 4: Mg-Si phases.

As for the matrix, it showed a two-phase microstructure of aluminum and zinc. Precipitation of the Zn phase was observed in the as-cast material, unlike in sample Q380, where the Zn phase was not detected, indicating that it was dissolved within the matrix As for the matrix, it showed a two-phase microstructure of aluminum and zinc. Precipitation of the Zn phase was observed in the as-cast material, unlike in sample Q380,

The identification of the phases present in each initial thermal condition was performed by room temperature X-ray diffraction (before the heating cycle). As is shown in Figure 7, in addition to the Al phase (PDF: 00-004-0787) and the Pt phase from the sample holder (PDF: 04-013-4766), which are not indicated for clarity, the phases detected were Zn (PDF: 01-078-9363), Si (PDF: 00-027-1402), MgZn2 (PDF: 04-003-2083), Mg2Zn11

(**a**) (**b**) **Figure 6.** SEM micrographs of the Al–Zn matrix. (**a**) As-cast. (**b**) Q380.

(PDF: 04-007-1412), and Mg2Si (PDF: 01-083-5235).

where the Zn phase was not detected, indicating that it was dissolved within the matrix (Figure 6). Precipitation of the Zn phase was observed in the as-cast material, unlike in sample Q380, where the Zn phase was not detected, indicating that it was dissolved within the matrix (Figure 6).

As for the matrix, it showed a two-phase microstructure of aluminum and zinc.

**Figure 5.** SEM micrographs of the material with x1000 magnification (**a**) As-cast. (**b**) Q380. Numbers 1 to 4 refer to the EDX results provided below. 1: Al-Zn matrix, 2: Mg-Zn phases, 3: Si

**Figure 6.** SEM micrographs of the Al–Zn matrix. (**a**) As-cast. (**b**) Q380. **Figure 6.** SEM micrographs of the Al–Zn matrix. (**a**) As-cast. (**b**) Q380.

*Materials* **2022**, *15*, 5056 7 of 15

phases and 4: Mg-Si phases.

The identification of the phases present in each initial thermal condition was performed by room temperature X-ray diffraction (before the heating cycle). As is shown in Figure 7, in addition to the Al phase (PDF: 00-004-0787) and the Pt phase from the sample holder (PDF: 04-013-4766), which are not indicated for clarity, the phases detected were Zn (PDF: 01-078-9363), Si (PDF: 00-027-1402), MgZn2 (PDF: 04-003-2083), Mg2Zn11 (PDF: 04-007-1412), and Mg2Si (PDF: 01-083-5235). The identification of the phases present in each initial thermal condition was performed by room temperature X-ray diffraction (before the heating cycle). As is shown in Figure 7, in addition to the Al phase (PDF: 00-004-0787) and the Pt phase from the sample holder (PDF: 04-013-4766), which are not indicated for clarity, the phases detected were Zn (PDF: 01- 078-9363), Si (PDF: 00-027-1402), MgZn<sup>2</sup> (PDF: 04-003-2083), Mg2Zn<sup>11</sup> (PDF: 04-007-1412), and Mg2Si (PDF: 01-083-5235). *Materials* **2022**, *15*, 5056 8 of 15

**Figure 7.** Diffraction patterns of the samples in each thermal condition at 30 °C prior to the heating cycle in the thermo-diffractometer. Indexation is shown above the patterns of samples Eq280 (Mg2Zn11 and Mg2Si phases) and Eq360 (Si, Zn and MgZn2 phases). Peaks corresponding to Al and Pt phases (the latter from the sample holder) are omitted for clarity. (**a**) As-cast samples and those cooled slowly. (**b**) Quenched samples. However, the microstructure obtained depended on the applied treatment; that is, **Figure 7.** Diffraction patterns of the samples in each thermal condition at 30 ◦C prior to the heating cycle in the thermo-diffractometer. Indexation is shown above the patterns of samples Eq280 (Mg2Zn<sup>11</sup> and Mg2Si phases) and Eq360 (Si, Zn and MgZn<sup>2</sup> phases). Peaks corresponding to Al and Pt phases (the latter from the sample holder) are omitted for clarity. (**a**) As-cast samples and those cooled slowly. (**b**) Quenched samples.

the temperature at which cooling had started and the cooling rate. In as-cast conditions Zn precipitated, as did both MgZn2 and Mg2Zn11 to a lesser extent. When slowly cooling from 280 °C (Eq280 sample), MgZn2 was obtained again, as in the previous case, but now Mg2Zn11 precipitated preferentially, while HCP Zn was hardly detected. When the cooling began at 360 °C (Eq360 sample), on the other hand, no precipitation of Mg2Zn11 was observed and zinc was present in the HCP Zn and MgZn2 phases. MgZn2 phases However, the microstructure obtained depended on the applied treatment; that is, the temperature at which cooling had started and the cooling rate. In as-cast conditions Zn precipitated, as did both MgZn<sup>2</sup> and Mg2Zn<sup>11</sup> to a lesser extent. When slowly cooling from 280 ◦C (Eq280 sample), MgZn<sup>2</sup> was obtained again, as in the previous case, but now

were found in greater quantities than in the as-cast or Eq280 conditions. As for the

confirm them by X-ray diffraction. The most intense Bragg peak for Al8FeMg3Si6 (PDF: 03-065-5936) would overlap with the Al (111) reflection. Given its condition as a minor phase, further peaks could not be detected. Therefore, if other Cu- and Fe-bearing phases found in aluminum alloys containing Zn, Mg, Si, and/or Cu [38,39] were present in very small amounts in this alloy, specific X-ray diffraction conditions and equipment would

The appearance of the Mg2Si phase and the dissolution and precipitation of the Zn

The profiles obtained for the as-cast sample are representative of the evolution of the zinc-containing phases with temperature (Figure 8). The description is thus valid for the rest of the samples, while the matrix will be dealt with in the next section. This evolution

*3.3. Evolution of the HCP Zn and Intermetallic Phases with Temperature* 

be required to identify them.

is summarized below.

phase are discussed in the following section.

Mg2Zn<sup>11</sup> precipitated preferentially, while HCP Zn was hardly detected. When the cooling began at 360 ◦C (Eq360 sample), on the other hand, no precipitation of Mg2Zn<sup>11</sup> was observed and zinc was present in the HCP Zn and MgZn<sup>2</sup> phases. MgZn<sup>2</sup> phases were found in greater quantities than in the as-cast or Eq280 conditions. As for the quenched samples, the Mg2Zn<sup>11</sup> phase was dissolved when reaching 360 ◦C.

Regarding the Fe-bearing quaternary phases observed by SEM, it was not possible to confirm them by X-ray diffraction. The most intense Bragg peak for Al8FeMg3Si<sup>6</sup> (PDF: 03-065-5936) would overlap with the Al (111) reflection. Given its condition as a minor phase, further peaks could not be detected. Therefore, if other Cu- and Fe-bearing phases found in aluminum alloys containing Zn, Mg, Si, and/or Cu [38,39] were present in very small amounts in this alloy, specific X-ray diffraction conditions and equipment would be required to identify them.

The appearance of the Mg2Si phase and the dissolution and precipitation of the Zn phase are discussed in the following section.

#### *3.3. Evolution of the HCP Zn and Intermetallic Phases with Temperature*

The profiles obtained for the as-cast sample are representative of the evolution of the zinc-containing phases with temperature (Figure 8). The description is thus valid for the rest of the samples, while the matrix will be dealt with in the next section. This evolution is summarized below. *Materials* **2022**, *15*, 5056 9 of 15

**Figure 8.** Diffraction patterns of the as-cast sample showing the evolution of the intermetallic phases, Zn, and Si with temperature (**a**) during the heating cycle (from 30 to 360 °C) (**b**) and cooling cycle (from 360 to 30 °C) in the thermo-diffractometer. The main Bragg peaks for the SrZn13 phase are identified. **Figure 8.** Diffraction patterns of the as-cast sample showing the evolution of the intermetallic phases, Zn, and Si with temperature (**a**) during the heating cycle (from 30 to 360 ◦C) (**b**) and cooling cycle (from 360 to 30 ◦C) in the thermo-diffractometer. The main Bragg peaks for the SrZn<sup>13</sup> phase are identified.

At 30 °C, Zn, MgZn2, and Mg2Zn11 phases were found. At 260 °C, the intensity of Zn peaks decreased while two additional Bragg peaks were detected around 2θ = 35.9° and 2θ = 54.0°. These peaks did not belong to any of the phases already indexed. Assuming they belonged to a new phase, it was clear that it arose at a temperature between 30 and 260 °C and likely dissolved between 260 and 280 °C, since it was absent in the Q280 sample at room temperature and in all the samples at any other temperature during the heating cycle. Indexing was performed considering minor elements present in the alloy, such as Cu, Fe, and Sr, and finally the SrZn13 phase was identified (PDF: 04-013-4885). At 320 °C, both Zn and SrZn13 were dissolved. In addition, between 30 and 320 °C At 30 ◦C, Zn, MgZn2, and Mg2Zn<sup>11</sup> phases were found. At 260 ◦C, the intensity of Zn peaks decreased while two additional Bragg peaks were detected around 2θ = 35.9◦ and 2θ = 54.0◦ . These peaks did not belong to any of the phases already indexed. Assuming they belonged to a new phase, it was clear that it arose at a temperature between 30 and 260 ◦C and likely dissolved between 260 and 280 ◦C, since it was absent in the Q280 sample at room temperature and in all the samples at any other temperature during the heating cycle. Indexing was performed considering minor elements present in the alloy, such as Cu, Fe, and Sr, and finally the SrZn<sup>13</sup> phase was identified (PDF: 04-013-4885).

the intensity of Mg2Zn11 increased and then became negligible at 360 °C. From the

Regarding the cooling cycle, the onset of the precipitation of the Mg2Zn11 phase was observed at 260 °C, while that of the Zn phase was not detected until 180 °C. At this temperature, the peaks belonging to SrZn13 showed slightly and disappeared again with further cooling. It should be noted that in the final measurement at 30 °C, the distribution of precipitated phases was different from what it had been at the beginning. The proportion of MgZn2 obtained at 360 °C remained stable during cooling and was higher

No evolution with temperature was observed for the Mg2Si phase. There were difficulties with detecting it in some of the measurements (see differences in Figure 7), but this was related to the specific sample (local segregations or inhomogeneities) and

As was previously mentioned, a two-phase Al–Zn matrix was found. However, a detailed observation of the indexed profiles led to the detection of some peaks whose intensity was higher than expected. These observations were confirmed when

had not completely dissolved in the matrix and may have become the MgZn2 phase.

than that found during the initial measurement.

*3.4. Evolution of Aluminum Phases* 

not to transformations taking place with temperature.

At 320 ◦C, both Zn and SrZn<sup>13</sup> were dissolved. In addition, between 30 and 320 ◦C the intensity of Mg2Zn<sup>11</sup> increased and then became negligible at 360 ◦C. From the increase in the intensity of the MgZn<sup>2</sup> peaks at this temperature, it followed that Mg2Zn<sup>11</sup> had not completely dissolved in the matrix and may have become the MgZn<sup>2</sup> phase.

Regarding the cooling cycle, the onset of the precipitation of the Mg2Zn<sup>11</sup> phase was observed at 260 ◦C, while that of the Zn phase was not detected until 180 ◦C. At this temperature, the peaks belonging to SrZn<sup>13</sup> showed slightly and disappeared again with further cooling. It should be noted that in the final measurement at 30 ◦C, the distribution of precipitated phases was different from what it had been at the beginning. The proportion of MgZn<sup>2</sup> obtained at 360 ◦C remained stable during cooling and was higher than that found during the initial measurement.

No evolution with temperature was observed for the Mg2Si phase. There were difficulties with detecting it in some of the measurements (see differences in Figure 7), but this was related to the specific sample (local segregations or inhomogeneities) and not to transformations taking place with temperature.

#### *3.4. Evolution of Aluminum Phases Materials* **2022**, *15*, 5056 10 of 15

As was previously mentioned, a two-phase Al–Zn matrix was found. However, a detailed observation of the indexed profiles led to the detection of some peaks whose intensity was higher than expected. These observations were confirmed when performing a Rietveld fitting on one of the profiles (Eq280 sample at 30 ◦C, before heating). It was verified that some of the peaks could not be fitted with the original model and there was a phase missing (Figure 9a). The addition of a phase with the same spatial group as aluminum (*Fm*3*m*) but a smaller lattice parameter managed to solve the structural model with satisfactory precision (Figure 9b). Due to the smaller atomic size of zinc compared to aluminum, a zinc-rich aluminum phase would show a smaller lattice parameter than α-Al and thus its Bragg peaks would shift to greater angles [32]. Therefore, the new phase observed could be the zinc-rich α' aluminum metastable phase of the miscibility gap in the Al–Zn system. The samples were retested with room temperature X-ray diffractometry nine months later with the aim of determining whether this was the case, and it was found that precipitation of the Zn phase from the α' metastable phase had taken place. performing a Rietveld fitting on one of the profiles (Eq280 sample at 30 °C, before heating). It was verified that some of the peaks could not be fitted with the original model and there was a phase missing (Figure 9a). The addition of a phase with the same spatial group as aluminum (3ത) but a smaller lattice parameter managed to solve the structural model with satisfactory precision (Figure 9b). Due to the smaller atomic size of zinc compared to aluminum, a zinc-rich aluminum phase would show a smaller lattice parameter than α-Al and thus its Bragg peaks would shift to greater angles [32]. Therefore, the new phase observed could be the zinc-rich α' aluminum metastable phase of the miscibility gap in the Al–Zn system. The samples were retested with room temperature X-ray diffractometry nine months later with the aim of determining whether this was the case, and it was found that precipitation of the Zn phase from the α' metastable phase had taken place.

**Figure 9.** Rietveld fitting of the Eq280 sample at 30 °C (before heating) with FullProf software. Red dots: experimental data. Black line: Fitting data. (**a**) The following six phases are considered Al (a = 4.0428 Å), Zn, Si, MgZn2, Mg2Zn11, and Mg2Si. The Pt phase comes from the sample holder. R-values with background correction: Rp = 36.6 %, Rwp = 38.6 %, Rexp = 10.72 %, χ2 = 12.91. (**b**) An α' phase with a lattice parameter a = 4.0089 Å is added to the previous case. Al phase in (a) is now labeled as α-Al. R-values with background correction: Rp = 20.5 %, Rwp = 18.7 %, Rexp = 10.54 %, χ2 = 3.15. The evolution of the intensity of the Bragg peak corresponding to the (101) plane of **Figure 9.** Rietveld fitting of the Eq280 sample at 30 ◦C (before heating) with FullProf software. Red dots: experimental data. Black line: Fitting data. (**a**) The following six phases are considered Al (a = 4.0428 A), ˚ Zn, Si, MgZn<sup>2</sup> , Mg2Zn11, and Mg2Si. The Pt phase comes from the sample holder. R-values with background correction: Rp = 36.6 %, Rwp = 38.6 %, Rexp = 10.72 %, χ <sup>2</sup> = 12.91. (**b**) An α' phase with a lattice parameter a = 4.0089 A˚ is added to the previous case. Al phase in (a) is now labeled as α-Al. R-values with background correction: R<sup>p</sup> = 20.5 %, Rwp = 18.7 %, Rexp = 10.54 %, χ <sup>2</sup> = 3.15.

the Zn phase was observed (Figure 10). The reason for choosing this peak is simple: it is the one with the maximum intensity of the Zn phase and it does not overlap with signals belonging to any other phase. For these reasons, this reflection is one of those taken as a

although only four of them are shown in the figure.

The evolution of the intensity of the Bragg peak corresponding to the (101) plane of the Zn phase was observed (Figure 10). The reason for choosing this peak is simple: it is the one with the maximum intensity of the Zn phase and it does not overlap with signals belonging to any other phase. For these reasons, this reflection is one of those taken as a reference in precipitation studies of Al–Zn alloys [31]. Intensity increased in all cases, although only four of them are shown in the figure. *Materials* **2022**, *15*, 5056 11 of 15

**Figure 10.** Close-up of the diffraction patterns of samples Eq280, Eq360, Q280, and Q360 showing the Bragg peaks of the (101) plane of the Zn phase. At the bottom, we show the profiles of the last measurements in the thermo-diffractometer at 30 °C at the end of the cooling cycle. At the top, we show the profiles obtained at the same temperature nine months later. **Figure 10.** Close-up of the diffraction patterns of samples Eq280, Eq360, Q280, and Q360 showing the Bragg peaks of the (101) plane of the Zn phase. At the bottom, we show the profiles of the last measurements in the thermo-diffractometer at 30 ◦C at the end of the cooling cycle. At the top, we show the profiles obtained at the same temperature nine months later.

#### **4. Discussion 4. Discussion**

#### *4.1. Phase Evolution with Temperature 4.1. Phase Evolution with Temperature*

The first transition temperature found in the DSC measurements (Figure 1) was 285 °C, and when compared with the evolution of the phases with the temperature observed in the diffraction patterns (Figure 8) it represents the dissolution of the Zn phase. Tests performed on Al–Zn samples with 24% atomic Zn estimate this reaction at 282 °C [32]. On the other hand, in the Al–Mg–Zn system the reaction would occur at 277 °C together with the partial dissolution of the Mg2Zn11 phase in the matrix [40]. However, it was not possible to determine to what extent this partial dissolution also takes place in the The first transition temperature found in the DSC measurements (Figure 1) was 285 ◦C, and when compared with the evolution of the phases with the temperature observed in the diffraction patterns (Figure 8) it represents the dissolution of the Zn phase. Tests performed on Al–Zn samples with 24% atomic Zn estimate this reaction at 282 ◦C [32]. On the other hand, in the Al–Mg–Zn system the reaction would occur at 277 ◦C together with the partial dissolution of the Mg2Zn<sup>11</sup> phase in the matrix [40]. However, it was not possible to determine to what extent this partial dissolution also takes place in the experimental samples at this temperature range.

experimental samples at this temperature range. The second temperature of interest in the DSC curve was about 350 °C. According to the analysis carried out, it corresponded to the complete dissolution of the Mg2Zn11 phase. This agrees with the precipitation temperature range expected for this phase by the equilibrium solidification simulations (Figure 4). In addition, as discussed in the analysis, at 360 °C a higher proportion of MgZn2 was observed so that it is possible that the dissolution of Mg2Zn11 enriched this phase. In fact, the reactions observed in the Al– Mg–Zn system fit this hypothesis, albeit at a lower temperature [40]. The reactions and The second temperature of interest in the DSC curve was about 350 ◦C. According to the analysis carried out, it corresponded to the complete dissolution of the Mg2Zn<sup>11</sup> phase. This agrees with the precipitation temperature range expected for this phase by the equilibrium solidification simulations (Figure 4). In addition, as discussed in the analysis, at 360 ◦C a higher proportion of MgZn<sup>2</sup> was observed so that it is possible that the dissolution of Mg2Zn<sup>11</sup> enriched this phase. In fact, the reactions observed in the Al–Mg–Zn system fit this hypothesis, albeit at a lower temperature [40]. The reactions and their experimental and theoretical temperatures are collected in Table 3.

their experimental and theoretical temperatures are collected in Table 3. **Table 3.** Experimental and theoretical temperatures for the reactions observed in the X-ray thermo-diffraction tests. **Reaction Experimental T (°C) in Al58Zn28Si8Mg6 Alloy T (°C) in Al–Mg–Zn System [40]**  (Al) + (Zn) (Al, Zn) ∼285 - During the cooling cycle in the diffractometer, zinc precipitation was observed at temperatures below 260 ◦C. However, such a transition was not easily detected in the DSC. According to the studies conducted by Skoko et al. with Al–Zn alloys [32], the precipitation transition occurs over a much larger temperature range than the dissolution one, so that the peak generated when cooling is much smaller. When looking specifically for this peak, it could be the one observed around 195 ◦C, a temperature consistent with the observations in the diffraction tests (Figure 11).

During the cooling cycle in the diffractometer, zinc precipitation was observed at temperatures below 260 °C. However, such a transition was not easily detected in the DSC. According to the studies conducted by Skoko et al. with Al–Zn alloys [32], the

(Al) + (Zn), Mg2Zn11 (Al, Zn) - 277


**Table 3.** Experimental and theoretical temperatures for the reactions observed in the X-ray thermodiffraction tests. *Materials* **2022**, *15*, 5056 12 of 15

**Figure 11.** DSC curve for the as-cast sample during cooling. Compressed graph to highlight possible peak showing Zn phase precipitation. **Figure 11.** DSC curve for the as-cast sample during cooling. Compressed graph to highlight possible peak showing Zn phase precipitation.

It should be noted that thermodynamic simulations did not agree with the experimental results regarding the MgZn2 and Zn phases. It should be noted that thermodynamic simulations did not agree with the experimental results regarding the MgZn<sup>2</sup> and Zn phases.

#### *4.2. Aluminum Phases and Ageing*

*4.2. Aluminum Phases and Ageing*  According to the equilibrium solidification simulations for the alloy under study, phase Al#2, considered the zinc-rich aluminum α' phase, should have arisen around 350 °C and decomposed shortly thereafter. The simulation with Scheil's approach did not even foresee its appearance. Therefore, the observation of this phase at room temperature According to the equilibrium solidification simulations for the alloy under study, phase Al#2, considered the zinc-rich aluminum α' phase, should have arisen around 350 ◦C and decomposed shortly thereafter. The simulation with Scheil's approach did not even foresee its appearance. Therefore, the observation of this phase at room temperature was not expected.

was not expected. However, an additional phase was observed at 30 °C in all samples subjected to the different thermal conditions (Figure 9). After the heating and cooling cycles in the diffractometer, it was still detected. Since this phase would have the same crystal structure as the α-Al phase but a smaller lattice parameter, our first hypothesis was that it However, an additional phase was observed at 30 ◦C in all samples subjected to the different thermal conditions (Figure 9). After the heating and cooling cycles in the diffractometer, it was still detected. Since this phase would have the same crystal structure as the α-Al phase but a smaller lattice parameter, our first hypothesis was that it was the α'-Al phase.

was the α'-Al phase. Analyses carried out nine months later showed that the microstructure of the samples after the diffractometer cycle was metastable, with the Zn phase precipitating during this time (Figure 10). This fact confirmed that the precipitation process usually observed in Al–Zn alloys [31,32,37,41] had occurred, a phenomenon in which the α' phase intervenes and that has been studied in detail with X-ray diffraction [31,32]. Still, it was not possible to determine the evolution of the α' phase with temperature, so the Analyses carried out nine months later showed that the microstructure of the samples after the diffractometer cycle was metastable, with the Zn phase precipitating during this time (Figure 10). This fact confirmed that the precipitation process usually observed in Al–Zn alloys [31,32,37,41] had occurred, a phenomenon in which the α' phase intervenes and that has been studied in detail with X-ray diffraction [31,32]. Still, it was not possible to determine the evolution of the α' phase with temperature, so the onset of the solid solution was not detected by diffraction.

#### onset of the solid solution was not detected by diffraction. *4.3. Effect of Strontium*

*4.3. Effect of Strontium*  As has been mentioned, the Si phase is observed both as a primary crystal and as an As has been mentioned, the Si phase is observed both as a primary crystal and as an Al-Si eutectic. It can be seen in the literature that, in comparison with Al–Si alloys,

Al-Si eutectic. It can be seen in the literature that, in comparison with Al–Si alloys, the nucleation of primary silicon crystals in Zn–Al–Si alloys is promoted by the presence of

Thus, in our microstructure we find both. The presence of strontium in the alloy is

the nucleation of primary silicon crystals in Zn–Al–Si alloys is promoted by the presence of zinc. Nevertheless, the addition of strontium promotes the generation of eutectic silicon. Thus, in our microstructure we find both. The presence of strontium in the alloy is expected to modify both primary and eutectic silicon [42,43]. Since strontium is present in all samples in equal proportions, the modification effect could not be evaluated.

However, the test results showed that when heat treating the material between 30 and 280 ◦C, the strontium ceased to be just a modifier and interacted with zinc to generate SrZn13. This is a factor that must be considered since, on the one hand, it can affect the final properties of the alloy, reducing the availability of zinc for the Mg–Zn phases; on the other hand, it can influence subsequent transformation processes that take place in its stability range. Given the interactions that occur, it should be considered whether an alternative modifier, such as sodium or rare earth elements [44], should be used.

#### **5. Conclusions**

The design and manufacturing of multicomponent light alloys give rise to nonconventional microstructures. In order to determine their potential applications, it is necessary to analyze their evolution with temperature. In this work, the multicomponent Al58Zn28Si8Mg<sup>6</sup> alloy was studied with CALPHAD methods and then cast and heat-treated under several conditions. Characterization was carried out by X-ray thermo-diffraction, differential scanning calorimetry, and electron microscopy.

As a result, a total of eight phases were identified in the alloy in the 30–380 ◦C temperature range: α-Al, α'-Al, Zn, Si, Mg2Si, MgZn2, Mg2Zn11, and SrZn13. The microstructures obtained at room temperature were metastable and the precipitation of Zn from the α' phase occurred over the course of months.

Moreover, the thermal transitions below 360 ◦C could be determined; that is, the dissolution and precipitation of Zn and dissolution of Mg2Zn11. Since the MgZn<sup>2</sup> and Mg2Si phases dissolved above 360 ◦C, where partial melting may occur, those precipitates are not expected to harden the matrix, as they do in conventional 3xx.x and 7xx.x aluminum alloys with solution and precipitation treatments. However, two remarks should be made. First, a room temperature X-ray diffraction test performed immediately after the Q380 quenching treatment could offer valuable information about the solid solution capability of the alloy and enable a more direct comparison with the simulation results. Second, it was observed that the proportion of MgZn<sup>2</sup> and Mg2Zn<sup>11</sup> phases was highly dependent on the thermal history, so the microstructure of the alloy is still susceptible to adaptation by heat treatment.

It should be noted that strontium was added to modify silicon phases. Although it was a minor element, as well as Fe and Cu, it interacted with zinc between 30 and 280 ◦C. This fact should be considered during the postprocessing (thermal and/or thermomechanical treatments) in that temperature range, as it could have unexpected effects.

Finally, it is clear from the experimental results that the used database is not designed to account for high percentages of alloying elements and thus is not able to accurately predict the actual phase evolution in the material; the Al–Zn system turned out to be tricky due to the metastable α' phase, and the precipitation of MgZn<sup>2</sup> in equilibrium conditions was predicted at a temperature that was too low. Nevertheless, the general guidelines given by CALPHAD methods, combined with SEM, DSC, and X-ray thermodiffraction results, were able to give us a satisfactory understanding of the alloy's behavior at different temperatures.

**Author Contributions:** Conceptualization, Y.B., J.J.T., I.H. and T.G.; methodology, Y.B., J.J.T., G.A. and T.G.; software, J.J.T.; validation, G.A., I.H. and T.G.; formal analysis, Y.B.; investigation, Y.B. and J.J.T.; resources, J.J.T., I.H. and T.G.; writing—original draft preparation, Y.B.; writing—review and editing, Y.B., J.J.T., I.V., G.A., I.H. and T.G.; visualization, Y.B.; supervision, G.A., I.H. and T.G.; project administration, I.V., I.H. and T.G.; funding acquisition, I.V., I.H. and T.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the BASQUE GOVERNMENT through the Elkartek project KK-2020/00047.

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

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The authors would like to acknowledge the support from A. Larrañaga (SGIker Advanced Research Facilities, University of the Basque Country (UPV/EHU)) with the X-ray thermodiffraction analysis.

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

## **References**


**Shaozhi He <sup>1</sup> , Jiong Wang 1,\*, Donglan Zhang <sup>1</sup> , Qing Wu <sup>2</sup> , Yi Kong <sup>1</sup> and Yong Du <sup>1</sup>**


**\*** Correspondence: wangjionga@csu.edu.cn

**Abstract:** The nanostructured β 00 precipitates are critical for the strength of Al-Mg-Si-(Cu) aluminum alloys. However, there are still controversial reports about the composition of Cu-containing β 00 phases. In this work, first-principles calculations based on density functional theory were used to investigate the composition, mechanical properties, and electronic structure of Cu-containing β 00 phases. The results predict that the Cu-containing β <sup>00</sup> precipitates with a stoichiometry of Mg4+*x*Al2−*x*CuSi<sup>4</sup> (*x* = 0, 1) are energetically favorable. As the concentration of Cu atoms increases, Cu-containing β 00 phases with different compositions will appear, such as Mg4AlCu2Si<sup>4</sup> and Mg4Cu3Si<sup>4</sup> . The replacement order of Cu atoms in β <sup>00</sup> phases can be summarized as one Si3/Al site → two Si3/Al sites → two Si3/Al sites and one Mg1 site. The calculated elastic constants of the considered β 00 phases suggest that they are all mechanically stable, and all β 00 phases are ductile. When Cu atoms replace Al atoms at Si3/Al sites in β 00 phases, the values of bulk modulus (*B*), shear modulus (*G*), and Young's modulus (*E*) all increase. The calculation of the phonon spectrum shows that Mg4+*x*Al2−*x*CuSi<sup>4</sup> (*x* = 0, 1) are also dynamically stable. The electronic structure analysis shows that the bond between the Si atom and the Cu atom has a covalent like property. The incorporation of the Cu atom enhances the electron interaction between the Mg2 and the Si3 atom so that the Mg2 atom also joins the Si network, which may be one of the reasons why Cu atoms increase the structure stability of the β 00 phases.

**Keywords:** Al-Mg-Si-Cu alloys; Cu-containing β 00; atomic configuration; mechanical properties; electronic structure

#### **1. Introduction**

Heat treatable Al-Mg-Si(-Cu) alloys in the 6xxx series are a common category of structural materials used in the construction and transportation industries. These alloys can be customized to have a desirable combination of properties, such as good formability, high specific strength, and corrosion resistance [1–3]. After proper aging treatment, the strength of the alloy can be greatly improved. This is mainly due to the precipitates that can contribute to the strengthening mechanisms by hindering the dislocation movement [4,5], particle strengthening σ<sup>p</sup> [6], and coherency of the particles [7]. The mechanical properties of these alloys can be greatly influenced by the composition, morphology, scale, and distribution of these solute atom nanostructures [8]. The precipitation sequence for Al-Mg-Si alloys is generally considered to be [9,10]:

```
SSSS → solute clusters → GP − zones → β00 → β
                                                  0
                                                  , U1, U2, B0 → β, Si
```
The supersaturated solid solution is denoted by the abbreviation SSSS. The Guinier-Preston zones (GP-zones) were first discovered in the Al-Cu system by Guinier [11] and Preston [12]. The GP-zones mainly refer to the nanoprecipitate phases formed in the early stage of aging, which is characterized by a certain ordered structure and completely coherent with the matrix.

**Citation:** He, S.; Wang, J.; Zhang, D.; Wu, Q.; Kong, Y.; Du, Y. A First-Principles Study of the Cu-Containing β00 Precipitates in Al-Mg-Si-Cu Alloy. *Materials* **2021**, *14*, 7879. https://doi.org/10.3390/ ma14247879

Academic Editors: Elena Pereloma, Antonino Squillace and Lijun Zhang

Received: 14 November 2021 Accepted: 14 December 2021 Published: 19 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Among the precipitates [4,13–15] formed in the aged Al-Mg-Si alloys, needle-like β 00 precipitate is the most effective strengthening phase [14] responsible for the peak-hardening effect [16]. The β 00 phase is a metastable precipitate phase, which is semi-coherent with the Al matrix in the needle cross-section, the space group is *C*2/*m*, *a* = 15.16 Å, *b* = 4.05 Å, *c* = 6.74 Å, and *β* = 105.3◦ [17,18]. The monoclinic β 00 phase was originally proposed to have the composition of Mg5Si<sup>6</sup> [17]. However, according to recent experimental and theoretical studies, the composition of β <sup>00</sup> would fluctuate around Mg5Al2Si<sup>4</sup> [19–22]. Furthermore, the most recent density functional theory (DFT) calculations inferred very minor formation enthalpy differences for β <sup>00</sup>-Mg5+*x*Al2−*x*Si<sup>4</sup> (−1 < *x* < 1) [21]. These results indicate that the composition of β 00 phase in Al matrix may change under certain conditions. For example, the dispersed nano-precipitates can be affected by the addition of Mg and/or Si, as well as other elements like Cu [5,23–27]. The addition of Cu is demonstrated to increase the age-hardening response, and it promotes the generation of higher number density and smaller size precipitates [14,28–32]. Therefore, a certain amount of Cu is usually added into Al-Mg-Si alloys. The addition of Cu increases the complexity of the precipitation sequence [32,33]. The precipitation sequence of Al-Mg-Si-Cu alloys is reported as [34]:

$$\text{RSS} \rightarrow \text{ solute clusters} \rightarrow \text{GP}-\text{ zones} \rightarrow \text{\ $}''/\text{L}/\text{S}/\text{C}, \text{QP}, \text{QC} \rightarrow \text{\$ }', \text{Q}' \rightarrow \text{Q}. \text{\"S}$$

Previous work used various experimental and theoretical methods to study the incorporation of Cu in β 00, and analyzed the Cu atoms as foreign solute atoms in the phases [20]. Cu addition could further enhance the positive effect of pre-aging on bake hardening for Al-Mg-Si alloys [35]. It has been demonstrated by high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) that Cu is mainly confined to the Si3/Al sites (Si or Al atoms completely occupy) of the β 00 structure [26,35–37], which as mentioned was also supported from DFT-based calculations [38]. The β 00 precipitates in Al-Mg-Si-Cu alloy were detected with an average composition of 28.6Al-38.7Mg-26.5Si-5.17Cu (at. %) using atom probe tomography (APT) and high-resolution energy-dispersive X-ray (EDX) mapping [36]. Furthermore, the addition of Cu has no effect on the type of β 00 precipitate, Cu atoms incorporate in β 00 and some of Mg, Si and Al in β 00 unit cell are substituted by Cu atoms [39].

As mentioned above, the β 00 precipitation behavior in Al-Mg-Si-Cu alloys has been investigated using various characterization methods. However, the detailed structures and stabilities are still unclear of Cu-containing β 00 phases in these alloys, and these structural refinements could be supported by first-principles results [40]. In addition, we predict energy-lowering site occupations and stoichiometries of the β 00 phases, where experimental information is incomplete. Understanding the structure of Cu-containing β 00 precipitates is essential to elucidate the precipitation sequence in heat-treatable Al-Mg-Si (-Cu) alloys.

In the present work, first-principles calculations based on density functional theory (DFT) [41] were used to study the Cu-containing β 00 phases. Based on the structural information obtained by experimental methods, first-principles atomistic calculations can provide structural, chemical, and energetic information [40]. A large number of Cucontaining β 00 structures were constructed searching for possible stable configurations and structural stability, kinetic stability, and mechanical stability were also considered. Finally, the characteristics of Cu atoms occupying sites were analyzed through the electronic structure.

#### **2. Materials and Methods**

#### *2.1. Atomic Model*

For the β <sup>00</sup> phases, the formation enthalpies and lattice parameters of Mg4Al3Si4, Mg5Al2Si4, Mg6AlSi4, and Mg5Si<sup>6</sup> were computed for each of the models of the crystal structures available in the literature [17,18,21], allowing a critical assessment of the validity of the models. Figure 1 shows four atomic models of the β 00 without Cu. The Wyckoff site information of the energetically most favorable β 00-Mg5Al2Si<sup>4</sup> is shown in Table 1 [18,19].

**Figure 1.** Four atomic models of the β″ available in the literature [17,21]. (**a**) Mg5Si6 from Zandbergen [17]; (**b**) Mg4Al3Si4 and (**c**) the Mg5Al2Si4 from Hasting [19]; (**d**) Mg6AlSi4 from Ehlers [21]. The relative location of each site is marked in (**c**). **Figure 1.** Four atomic models of the β 00 available in the literature [17,21]. (**a**) Mg5Si<sup>6</sup> from Zandbergen [17]; (**b**) Mg4Al3Si<sup>4</sup> and (**c**) the Mg5Al2Si<sup>4</sup> from Hasting [19]; (**d**) Mg6AlSi<sup>4</sup> from Ehlers [21]. The relative location of each site is marked in (**c**).

**Table 1.** Wyckoff site information (*x*, *y*, *z*) in the β″-Mg5Al2Si4 phase [18,19]; atomic configuration is shown schematically in Figure 1c. **Table 1.** Wyckoff site information (*x*, *y*, *z*) in the β 00-Mg5Al2Si<sup>4</sup> phase [18,19]; atomic configuration is shown schematically in Figure 1c.

For the β″ phases, the formation enthalpies and lattice parameters of Mg4Al3Si4, Mg5Al2Si4, Mg6AlSi4, and Mg5Si6 were computed for each of the models of the crystal structures available in the literature [17,18,21], allowing a critical assessment of the validity of the models. Figure 1 shows four atomic models of the β″ without Cu. The Wyckoff site information of the energetically most favorable β″-Mg5Al2Si4 is shown in Table 1 [18,19].


#### *2.2. Computational Details 2.2. Computational Details*

**2. Materials and Methods** 

*2.1. Atomic Model* 

The first-principles calculations were performed utilize the plane wave pseudopotential method, as implemented in the highly efficient Vienna ab initio simulation package (VASP) [42,43], The electron-ion interactions were described through projector augmented wave (PAW) [44,45]. The exchange-correlation function were constructed by the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [46]. All structures were fully relaxed with respect to atomic positions as well as all lattice parameters in order to find the lowest-energy structure. The electron wave function was expanded in plane waves up to a cutoff energy of 450 eV. The β′′phase was represented by a conventional cell with 22 atoms according to the experimental results, and 3 × 12 × 8 Γcentered k-point meshes were employed in the Brilluion zone sampling and generated automatically by following the Monkhorst-Pack sampling scheme [47], while the 3 × 3 × 8 Γ-centered k-point meshes and 1 × 4 × 1 supercells were employed for calculation of "replacement energy" (the detailed definition is explained below). Atoms were relaxed until The first-principles calculations were performed utilize the plane wave pseudopotential method, as implemented in the highly efficient Vienna ab initio simulation package (VASP) [42,43], The electron-ion interactions were described through projector augmented wave (PAW) [44,45]. The exchange-correlation function were constructed by the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [46]. All structures were fully relaxed with respect to atomic positions as well as all lattice parameters in order to find the lowest-energy structure. The electron wave function was expanded in plane waves up to a cutoff energy of 450 eV. The β 00phase was represented by a conventional cell with 22 atoms according to the experimental results, and 3 × 12 × 8 Γ-centered k-point meshes were employed in the Brilluion zone sampling and generated automatically by following the Monkhorst-Pack sampling scheme [47], while the 3 × 3 × 8 Γ-centered k-point meshes and 1 × 4 × 1 supercells were employed for calculation of "replacement energy" (the detailed definition is explained below). Atoms were relaxed until their residual forces converged to 0.01 eV/Å. The phonon spectra were obtained using the Phonopy package [48].

The four-parameter Birch–Murnaghan equation of state with its linear form [49] is employed to estimate the equilibrium total energy (*E*0), volume (*V*0),

$$E(V) = a + bV^{-2/3} + cV^{-4/3} + dV^{-2} \tag{1}$$

where *a*, *b*, *c*, and *d* are fitting parameters. More details can be found in our previous work [50].

Compared with the energy of solid solution containing a Cu atom, the energy gain of the Cu atoms incorporated in β 00 is referred to as "replacement energy". In order to construct the Cu-containing β 00 phases, it is necessary to determine the possible occupation sites of Cu in β 00 phases. Additionally, computing the replacement energy (see Ref. [51]) can be used as a criterion for the possible occupation sites of solute atoms. There have been previous studies addressing the first-principles calculations for describing replacement energies of different sides. Since the replacement energy of Cu atoms at Mg2 and Mg3 sites were not shown in Saito's work [38], one Cu atom was introduced into a 1 × 4 × 1 supercell and the preference of Cu atoms for each non-equivalent site in β 00 was evaluated using the method described by Saito et al. [51], but with higher calculation precision.

To solve the compositional uncertainty preliminarily, the reported *C*2/*m* symmetries [18] were deliberately reduced to the level where only pairs of atoms (e.g., the two Cu atoms) were regarded as equivalent. This implies that space group *P*2/*m* was used throughout and there are 11 different sites within the unit cell. Besides, no partial occupancies were considered and vacancies were ignored. The replacement energy for Cu incorporation in β 00 can be described as follows:

$$
\Delta H(\mathfrak{F}\_0'' \colon \mathbb{X} \to \mathbb{\Sigma}) = \begin{array}{c} H(\mathfrak{F}\_0'' \colon \mathbb{3} \times \{\operatorname{Al} \to \Sigma\} ; \mathbb{1} \times \{\mathbb{X} \to \Sigma\} ) \\ + H(\operatorname{fcc} \operatorname{Al}) - H(\mathfrak{F}\_0'' \colon \mathbb{4} \times \{\operatorname{Al} \to \Sigma\} ) \\ -H(\operatorname{fcc} \operatorname{Al} \colon \mathbb{1} \times \{\mathbb{X} \to \mathbb{S}\} ) \end{array} \tag{2}
$$

where *H* are the calculated enthalpy of the system, β 00 0 are the Cu-free structure, Ξ are the sides in β 00 0 , X are the solute atoms incorporated in the precipitates, and S are substitutional sites in the Al matrix. A certain atom X incorporates on site Ξ is referred to as "{X → Ξ}". The formation enthalpy of solid solution (SS), ∆*H*form *SS* , was used to find out the most energetically favorable configurations in the atomic models. Since there is no stable fcc structure for Mg and Si, their formation energies in relation to SS were determined as follows:

$$\begin{array}{rcl}\Delta H\_{\text{SS}}^{\text{form}} \left(\text{Mg}\_{a}\text{Al}\_{b}\text{Cu}\_{c}\text{Si}\_{d}\right) &=& E(\text{Mg}\_{a}\text{Al}\_{b}\text{Cu}\_{c}\text{Si}\_{d}) - aE^{\text{sub}}(\text{Mg})\\&-bE(\text{Al}) - cE(\text{Cu}) - dE^{\text{sub}}(\text{Si})\end{array} \tag{3}$$

where *E* sub (Mg) and *E* sub (Si) are the enthalpies of substituting Al atoms by Mg and Si atoms, respectively. *E* sub (Mg) and *E* sub (Si) were calculated in a 3 <sup>×</sup> <sup>3</sup> <sup>×</sup> 3 Al supercell with one Mg/Si atom and 107 Al atoms with a k-point meshes of 5 × 5 × 5. The enthalpy of substituting a Mg atom was defined as:

$$E^{\rm sub}(\rm Mg) = E(Al\_{107}Mg) - 107/108E(Al) \tag{4}$$

where *E* (Al) is the enthalpy of a 3 × 3 × 3 Al supercell. The definition of *E* sub (Mg) was also feasible for *E* sub (Si).

Finally, in order to compare the structures with different Al content, the formation enthalpy can also be expressed in kJ/mol of solute atoms, instead of kJ/mol [52]. This transformation is achieved as follows: ∆*H*SS [kJ/mol solute] = ∆*H*SS [kJ/mol]/(*x*Mg + *x*Si + *x*Cu), where *x*Mg and *x*Si and *x*Cu are the atomic fractions of Mg and Si and Cu in the β 00 phases Mg*a*Al*b*Cu*c*Si*<sup>d</sup>* (*a* = *x*Mg, *b* = *x*Al, *c* = *x*Cu, *d* = *x*Si). This is a common definition of formation enthalpy in the literature [9,21,52].

The elastic constant can be represented by a 6 × 6 matrix. Based on the symmetry of the crystal structure, the independent elastic constants of the monoclinic crystal are reduced to 13, as shown in Formula (5):

$$\mathbf{C}\_{ij} = \begin{pmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} & 0 & \mathbf{C}\_{15} & 0\\ & \mathbf{C}\_{22} & \mathbf{C}\_{23} & 0 & \mathbf{C}\_{25} & 0\\ & & \mathbf{C}\_{33} & 0 & \mathbf{C}\_{35} & 0\\ & & & \mathbf{C}\_{44} & 0 & \mathbf{C}\_{46}\\ & & & & \mathbf{C}\_{55} & 0\\ & & & & & \mathbf{C}\_{66} \end{pmatrix} \tag{5}$$

The stress-strain method based on the generalized Hooke's theorem is used to calculate the elastic constants of each crystal [53]. For more detailed stress-strain method description, please refer to [54]. The relationship between elastic constant *Cijkl*, stress tensor *δkl*, and strain tensor *δkl* can be expressed as:

$$
\sigma\_{\text{ij}} = \mathbb{C}\_{\text{ijkl}} \delta\_{\text{kl}} \tag{6}
$$

The Hill model [55] is used to further obtain the bulk modulus (*B*), shear modulus (*G*), and Youngs modulus (*E*) of the crystal through the elastic constant. The Hill model takes into account that the calculation results of the Voigt model and the Reuss model will be high and low, respectively, and take the arithmetic mean of the values of the Voigt model and the Reuss model. For monoclinic crystal structure, the formula for calculating the bulk modulus (*B*) and shear modulus (*G*) of monoclinic crystals using Voigt model and Reuss model are [56]:

$$B\_{\rm V} = \frac{1}{9} [\mathbb{C}\_{11} + \mathbb{C}\_{22} + \mathbb{C}\_{33} + 2(\mathbb{C}\_{12} + \mathbb{C}\_{13} + \mathbb{C}\_{23})] \tag{7}$$

$$\begin{array}{ll} \mathcal{B}\_{\mathsf{R}} = & \Omega \big[ a \big( \mathcal{C}\_{11} + \mathcal{C}\_{22} - 2 \mathcal{C}\_{12} \big) + b \big( 2 \mathcal{C}\_{12} - 2 \mathcal{C}\_{11} - \mathcal{C}\_{23} \big) + c \big( \mathcal{C}\_{15} - 2 \mathcal{C}\_{25} \big) + \\\ d \big( 2 \mathcal{C}\_{12} + 2 \mathcal{C}\_{23} - \mathcal{C}\_{13} - 2 \mathcal{C}\_{22} \big) + 2e \big( \mathcal{C}\_{25} - \mathcal{C}\_{15} \big) + f \big]^{-1} \end{array} \tag{8}$$

$$\mathbf{G}\_{\rm V} = (1/15)[\mathbf{C}\_{11} + \mathbf{C}\_{22} + \mathbf{C}\_{33} + 3(\mathbf{C}\_{44} + \mathbf{C}\_{55} + \mathbf{C}\_{66}) - (\mathbf{C}\_{12} + \mathbf{C}\_{13} + \mathbf{C}\_{23})] \tag{9}$$

$$\begin{array}{ll} \mathbf{G\_{R}} = & 15 \{ 4[a(\mathbf{C\_{11}} + \mathbf{C\_{22}} + \mathbf{C\_{12}}) + b(\mathbf{C\_{11}} - \mathbf{C\_{12}} - \mathbf{C\_{23}}) + \\ & c(\mathbf{C\_{15}} + \mathbf{C\_{25}}) + d(\mathbf{C\_{22}} - \mathbf{C\_{12}} - \mathbf{C\_{23}} - \mathbf{C\_{13}}) + e(\mathbf{C\_{15}} - \mathbf{C\_{25}}) + f \}/\Omega + \\ & 3 \{ g/\Omega + (\mathbf{C\_{44}} + \mathbf{C\_{66}}) / \left( \mathbf{C\_{44}} \mathbf{C\_{66}} - \mathbf{C\_{46}} \right) \}^{-1} \end{array} \tag{10}$$

wherein:

$$a = \mathbb{C}\_{33}\mathbb{C}\_{55} - \mathbb{C}\_{35}^{2} \tag{11}$$

$$b = \mathbb{C}\_{23}\mathbb{C}\_{55} - \mathbb{C}\_{25}\mathbb{C}\_{35} \tag{12}$$

$$\mathcal{L} = \mathsf{C}\_{13}\mathsf{C}\_{35} - \mathsf{C}\_{15}\mathsf{C}\_{33} \tag{13}$$

$$d = \mathbb{C}\_{13}\mathbb{C}\_{55} - \mathbb{C}\_{15}\mathbb{C}\_{35} \tag{14}$$

$$e = \mathbb{C}\_{13}\mathbb{C}\_{25} - \mathbb{C}\_{15}\mathbb{C}\_{23} \tag{15}$$

$$f = \begin{array}{c} \mathbb{C}\_{11} \left( \mathbb{C}\_{22} \mathbb{C}\_{55} - \mathbb{C}\_{25}^{2} \right) - \mathbb{C}\_{12} \left( \mathbb{C}\_{12} \mathbb{C}\_{55} - \mathbb{C}\_{15} \mathbb{C}\_{25} \right) + \\\ \mathbb{C}\_{15} \left( \mathbb{C}\_{12} \mathbb{C}\_{25} - \mathbb{C}\_{15} \mathbb{C}\_{22} \right) + \mathbb{C}\_{25} \left( \mathbb{C}\_{23} \mathbb{C}\_{35} - \mathbb{C}\_{25} \mathbb{C}\_{33} \right) \end{array} \tag{16}$$

$$g = \mathbb{C}\_{11}\mathbb{C}\_{22}\mathbb{C}\_{33} - \mathbb{C}\_{11}\mathbb{C}\_{23}^2 - \mathbb{C}\_{22}\mathbb{C}\_{13}^2 - \mathbb{C}\_{33}\mathbb{C}\_{12}^2 + 2\mathbb{C}\_{12}\mathbb{C}\_{13}\mathbb{C}\_{23} \tag{17}$$

$$\begin{array}{rcl} \Omega = & 2\left[\mathbf{C}\_{15}\mathbf{C}\_{25}\left(\mathbf{C}\_{33}\mathbf{C}\_{12} - \mathbf{C}\_{13}\mathbf{C}\_{23}\right) + \mathbf{C}\_{15}\mathbf{C}\_{35}\left(\mathbf{C}\_{22}\mathbf{C}\_{13} - \mathbf{C}\_{12}\mathbf{C}\_{23}\right) + \\ & \mathbf{C}\_{25}\mathbf{C}\_{35}\left(\mathbf{C}\_{11}\mathbf{C}\_{23} - \mathbf{C}\_{12}\mathbf{C}\_{13}\right) - \left[\mathbf{C}\_{15}^{2}\left(\mathbf{C}\_{22}\mathbf{C}\_{33} - \mathbf{C}\_{23}^{2}\right) + \mathbf{C}\_{25}^{2}\left(\mathbf{C}\_{11}\mathbf{C}\_{33} - \mathbf{C}\_{13}^{2}\right) + \\ & \mathbf{C}\_{35}^{2}\left(\mathbf{C}\_{11}\mathbf{C}\_{22} - \mathbf{C}\_{12}^{2}\right) \right] + \mathbf{g}\mathbf{C}\_{55} \end{array} \tag{18}$$

The formula for calculating the bulk modulus (*B*), shear modulus (*G*), and elastic modulus (*E*) of monoclinic crystal by Hill model [55] is:

$$B\_{\rm H} = \frac{1}{2}(B\_{\rm V} + B\_{\rm R}) \tag{19}$$

$$\mathbf{G\_H} = \frac{1}{2}(\mathbf{G\_V} + \mathbf{G\_R}) \tag{20}$$

$$E = \Re \mathbf{G} / (\Im \mathbf{B} + \mathbf{G}) \tag{21}$$

#### **3. Results and Discussion**

#### *3.1. Structure Stability*

The replacement energy is shown in Figure 2 and alternative solute atoms Mg/Si were incorporated for comparison with Cu at different sites. In order to more intuitively express the competitive occupation sites of Cu atoms, the variable ∆ is introduced and the

∆ values of Cu atoms at different sites in different β 00 configurations are shown in Figure 3. The ∆ represents the "competitiveness" between Cu atoms and other solute atoms at each site, it is the difference between the lowest replacement energy of Mg/Si solute atoms and the replacement energy of Cu atoms. The larger the value of ∆, the more likely the Cu atom will occupy the site. Consequently, due to the low Cu occupancy in β 00, only three designated Cu sites (Si1, Si3, Mg1, see Figure 1c) were allowed to host Cu atoms according to the relative value of replacement energy (refer to Figure 2). This conclusion is consistent with previous research [38]. of Cu atoms at different sites in different β″ configurations are shown in Figure 3. The Δ represents the "competitiveness" between Cu atoms and other solute atoms at each site, it is the difference between the lowest replacement energy of Mg/Si solute atoms and the replacement energy of Cu atoms. The larger the value of Δ, the more likely the Cu atom will occupy the site. Consequently, due to the low Cu occupancy in β″, only three designated Cu sites (Si1, Si3, Mg1, see Figure 1c) were allowed to host Cu atoms according to the relative value of replacement energy (refer to Figure 2). This conclusion is consistent with previous research [38].

The replacement energy is shown in Figure 2 and alternative solute atoms Mg/Si were incorporated for comparison with Cu at different sites. In order to more intuitively express the competitive occupation sites of Cu atoms, the variable Δ is introduced and the Δ values

( ) ( ) ( )( )

−+ −

2 22

( ) ( )( )

− − −+ −+

11 22 33 11 23 22 13 33 12 12 13 23 *g* = −−−+ *CCC CC CC CC CCC* 2 (17)

2 22 2

*B BB* = + (19)

*G GG* = + (20)

*E BG B G* = + 9 / (3 ) (21)

(16)

(18)

11 22 55 25 12 12 55 15 25 15 12 25 15 22 25 23 35 25 33

*C CC CC C CC CC* = −− − +

( )( )

*CC CC CC C CC C C CC C*

25 35 11 23 12 13 15 22 33 23 25 11 33 13

The formula for calculating the bulk modulus (*B*), shear modulus (*G*), and elastic

H VR ( ) <sup>1</sup> 2

H VR ( ) <sup>1</sup> 2

15 25 33 12 13 23 15 35 22 13 12 23

2 *CC CC CC CC CC CC*

Ω= − + − +

2

*f C CC C C CC CC*

*Materials* **2021**, *14*, x FOR PEER REVIEW 6 of 18

( )

*C C C C gC*

− + 

2 2 35 11 22 12 55

modulus (*E*) of monoclinic crystal by Hill model [55] is:

**3. Results and Discussions** 

*3.1. Structure Stability* 

**Figure 2.** Calculated replacement energies for Cu and alternative solute atoms Mg/Si on the different sites of three different β″ configurations. 1: Mg4Al3Si4, 2: Mg5Al2Si4, and 3: Mg6AlSi4. The position of **Figure 2.** Calculated replacement energies for Cu and alternative solute atoms Mg/Si on the different sites of three different β 00 configurations. 1: Mg4Al3Si<sup>4</sup> , 2: Mg5Al2Si<sup>4</sup> , and 3: Mg6AlSi<sup>4</sup> . The position of each column represents a different position in a different configuration. Cu, Mg, and Si replacement energies are labelled with black, shaded, and white bars, respectively. each column represents a different position in a different configuration. Cu, Mg, and Si replacement energies are labelled with black, shaded, and white bars, respectively.

**Figure 3.** The competitiveness (Δ values) of Cu atoms at different sites in different β″ configurations. The black square, red circle, and blue triangle represent Cu atoms in the Mg4Al3Si4, Mg5Al2Si4, and Mg6AlSi4 configurations, respectively. **Figure 3.** The competitiveness (∆ values) of Cu atoms at different sites in different β 00 configurations. The black square, red circle, and blue triangle represent Cu atoms in the Mg4Al3Si<sup>4</sup> , Mg5Al2Si<sup>4</sup> , and Mg6AlSi<sup>4</sup> configurations, respectively.

For checking the reliability of the calculations, Table 2 displays the structural parameters for selected β″ configuration without Cu atom, along with the results of earlier theoretical and experimental studies of β″. Available calculation results of formation enthalpies are shown in Table 3. The formation enthalpies of the 33 possible unit cells have been plotted in Figure 4, including the configuration without Cu atom. Since the given formation enthalpy of per solute atom (eV/solute atom) essentially presents the solute chemical potentials, the zero-temperature convex hull can be constructed to deduce the For checking the reliability of the calculations, Table 2 displays the structural parameters for selected β 00 configuration without Cu atom, along with the results of earlier theoretical and experimental studies of β 00. Available calculation results of formation enthalpies are shown in Table 3. The formation enthalpies of the 33 possible unit cells have been plotted in Figure 4, including the configuration without Cu atom. Since the given formation enthalpy of per solute atom (eV/solute atom) essentially presents the solute chemical potentials, the zero-temperature convex hull can be constructed to de-

precipitation order of the system [57]. It can be seen that Cu occupying one column of each

Mg4Al2CuSi4 compositions. While the formation enthalpy of Mg4Al2CuSi4 is −0.337 eV/solute atom, the formation enthalpy of Mg5AlCuSi4 is −0.335 eV/solute atom, which is similar to that of Mg4Al2CuSi4. This is consistent with the observed in previous experiments that Cu atoms mainly occupy Si3 sites [36]. The energy gained when replacing Mg/Si/Al with at the Wyckoff sites is clearly varying with *x*. When Cu atoms occupy two sites (that is, *x* = 2), Mg4AlCu2Si4 is the energetically most favorable phase, and Cu atoms occupy two Si3 columns. When Cu atoms occupy three sites, Mg4Cu3Si4 is the most stable structure, in which Cu atoms occupy one Mg1 site and two Si3 sites, which is consistent with experimental observations [36]. The results show that stoichiometry of Cu-containing β″ phase is suggested as Mg4Al3−*x*Cu*x*Si4 (1 ≤ *x* ≤ 3). Since the formation enthalpy of Mg5AlCuSi4 is very close to that of Mg4Al2CuSi4, it can also be taken into account. This result emphasizes the possibility of fluctuations between various compositions as a function of the local alloying element concentration for the physical system during precipitated phases growth. Then the structural parameters of low energy configurations from Figure 4 also have been displayed in Table 2. As discussed above, sole minimization of the β″ phase formation enthalpy supports the well-defined Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) unit cell shown in Figure 5.

**Table 2.** First-principles (VASP-GGA) and experimental lattice parameters of β″ phases of Al-Mg-Si-(Cu) system. For the Cu-containing β″ phases, only the most stable crystal structures under different Cu

**Configurations** *a* **(Å)** *b* **(Å)** *c* **(Å)** *β* **(°) Ref.** 

Mg5Si6 (exp.) 15.16 ± 0.02 4.05 6.74 ± 0.02 105.3 ± 0.5 [17] Mg5Si6 (GGA) 15.11 4.080 6.932 110.4 [21]

Mg5Si6 15.12 4.04 6.99 110.6

concentrations are listed.

duce the precipitation order of the system [57]. It can be seen that Cu occupying one column of each Si3 column pair is found to be the energetically most favorable option for the set of Mg4Al2CuSi<sup>4</sup> compositions. While the formation enthalpy of Mg4Al2CuSi<sup>4</sup> is −0.337 eV/solute atom, the formation enthalpy of Mg5AlCuSi<sup>4</sup> is −0.335 eV/solute atom, which is similar to that of Mg4Al2CuSi4. This is consistent with the observed in previous experiments that Cu atoms mainly occupy Si3 sites [36]. The energy gained when replacing Mg/Si/Al with at the Wyckoff sites is clearly varying with *x*. When Cu atoms occupy two sites (that is, *x* = 2), Mg4AlCu2Si<sup>4</sup> is the energetically most favorable phase, and Cu atoms occupy two Si3 columns. When Cu atoms occupy three sites, Mg4Cu3Si<sup>4</sup> is the most stable structure, in which Cu atoms occupy one Mg1 site and two Si3 sites, which is consistent with experimental observations [36]. The results show that stoichiometry of Cu-containing β <sup>00</sup> phase is suggested as Mg4Al3−*x*Cu*x*Si<sup>4</sup> (1 ≤ *x* ≤ 3). Since the formation enthalpy of Mg5AlCuSi<sup>4</sup> is very close to that of Mg4Al2CuSi4, it can also be taken into account. This result emphasizes the possibility of fluctuations between various compositions as a function of the local alloying element concentration for the physical system during precipitated phases growth. Then the structural parameters of low energy configurations from Figure 4 also have been displayed in Table 2. As discussed above, sole minimization of the β 00 phase formation enthalpy supports the well-defined Mg4+*x*Al2−*x*CuSi<sup>4</sup> (*x* = 0, 1) unit cell shown in Figure 5. *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361 *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361 *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361 *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361 *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361 *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 *Materials* **2021**, *14*, x FOR PEER REVIEW 9 of 18 Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696

Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361

Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361

**Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. **Figure 4.** Formation enthalpies of calculated β 00 phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si<sup>6</sup> is also marked as a reference. **Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. represents the Mg5Si<sup>6</sup> , **Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. represents the Mg5Al2Si<sup>4</sup> , **Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. represents the Mg5AlCuSi<sup>4</sup> where Cu atoms occupy a Si3 site, **Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. represents the Mg4Al2CuSi<sup>4</sup> where Cu atoms occupy a Si3 site, **Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. represents the Mg4AlCu2Si<sup>4</sup> where Cu atoms occupy two Si3 sites, and **Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two Si3 sites. represents the Mg4Cu3Si<sup>4</sup> where Cu atoms occupy a Mg1 site and two Si3 sites.

**Table 2.** First-principles (VASP-GGA) and experimental lattice parameters of β 00 phases of Al-Mg-Si- (Cu) system. For the Cu-containing β 00 phases, only the most stable crystal structures under different Cu concentrations are listed.


Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus

**Figure 5.** The well-defined Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) unit cell that sole minimization of the β″ phase formation enthalpy

**Figure 5.** The well-defined Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) unit cell that sole minimization of the β″ phase formation enthalpy

**Figure 5.** The well-defined Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) unit cell that sole minimization of the β″ phase formation enthalpy

**Figure 5.** The well-defined Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) unit cell that sole minimization of the β″ phase formation enthalpy

supports. (**a**) Mg4Al2CuSi4; (**b**) Mg5AlCuSi4. The relative location of each site is marked in (**a**).

*3.2. Elastic Properties* 

*3.2. Elastic Properties* 

*3.2. Elastic Properties* 

*3.2. Elastic Properties* 

*3.2. Elastic Properties* 

*3.2. Elastic Properties* 

*3.2. Elastic Properties* 


**Table 2.** *Cont.*

**Table 3.** Formation enthalpies of β 00 phases with different configurations in this work. The Cu occupied sites and its number are also listed in detail.


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Mg5AlCu3Si2 2 Si1 and 1 Mg1 0.71 0.27 −0.1981 Mg6Cu3Si2 2 Si1 and 1 Si3 0.75 0.27 −0.1696 Mg5Cu3Si3 1 Si1 and 1 Si3 and 1 Mg1 0.63 0.27 −0.2361

**Figure 5.** The well-defined Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) unit cell that sole minimization of the β″ phase formation enthalpy supports. (**a**) Mg4Al2CuSi4; (**b**) Mg5AlCuSi4. The relative location of each site is marked in (**a**). **Figure 5.** The well-defined Mg4+*x*Al2−*x*CuSi<sup>4</sup> (*x* = 0, 1) unit cell that sole minimization of the β 00 phase formation enthalpy supports. (**a**) Mg4Al2CuSi<sup>4</sup> ; (**b**) Mg5AlCuSi<sup>4</sup> . The relative location of each site is marked in (**a**). **Table 4.** Calculated single crystal elastic stiffness constants (*Cij*′s) of the reported β″ phases and energy favorable Cucontaining β″ phases.

**Configuration** *C***<sup>11</sup>** *C***<sup>12</sup>** *C***<sup>13</sup>** *C***<sup>15</sup>** *C***<sup>22</sup>** *C***<sup>23</sup>** *C***<sup>25</sup>** *C***<sup>33</sup>** *C***<sup>35</sup>** *C***<sup>44</sup>** *C***<sup>46</sup>** *C***<sup>55</sup>** *C***<sup>66</sup>**

#### *3.2. Elastic Properties 3.2. Elastic Properties* Mg5Si6 110 42 42 −3 103 49 4 94 11 19 5 17 25

Si3 sites.

Here, we compare the mechanical properties of β″ with or without Cu atoms. The elastic constants of key β″ phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi4 and Mg5Al-CuSi4 all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi4 (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*33 are much greater than the other elastic constants in all calculated β″ phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus Here, we compare the mechanical properties of β 00 with or without Cu atoms. The elastic constants of key β 00 phases that are most likely to precipitate during aging were calculated by using fully relaxed crystal structures, and the results are listed in Table 4. According to the Born stability criterion [61], the elastic constants of Mg4Al2CuSi<sup>4</sup> and Mg5AlCuSi<sup>4</sup> all meet the stability criteria of monoclinic crystals. This further supports the stability of Mg4+*x*Al2−*x*CuSi<sup>4</sup> (*x* = 0, 1) obtained from the formation enthalpy. The elastic constants *C*11, *C*22, and *C*<sup>33</sup> are much greater than the other elastic constants in all calculated β 00 phases, resulting in an obvious elastic anisotropy. In order to understand the anisotropic characteristics of these precipitation phases, the Young's modulus anisotropies are evaluated by three-dimensional map as shown in Figure 6. Comparing Figure 6a and c, it can be seen that after Cu atoms substituted Al atoms on the Si3/Al sites, the Young's modulus (*E*) anisotropy increases significantly; similar results are also shown in Figure 6b,d. This phenomenon indicates that the growth rate of the Cu-containing β 00 phases may be faster than that of the β 00 without Cu. It is consistent with the previous study that Cu can accelerates the age-hardening response [14,28,30]. Mg5Si6 [62] 106 49 50 −11 90 46 6 88 9 17 1 33 30 Mg5Si6 [63] 98 50 48 8 84 46 6 88 5.4 22 −10 29 51 Mg4Al3Si4 119 52 35 −3 99 47 3 122 10 19 −1 29 20 Mg4Al3Si4 [62] 114 46 48 −4 104 49 6 104 7 21 0 34 23 Mg4Al3Si4 [63] 107 47 48 9 97 48 6 97 9 26 6 36 46 Mg5Al2Si4 111 38 44 −4 102 46 3 106 7 25 4 31 25 Mg5Al2Si4 [62] 108 42 48 −3 95 46 5 100 3 23 4 33 27 Mg5Al2Si4 [63] 107 40 46 −13 95 43 4 99 12 27 5 36 49 Mg6AlSi4 121 28 40 −5 125 28 2 117 6 28 4 35 21 Mg4Al2CuSi4 136 44 48 −13 133 43 9 130 14 25 3 35 23 Mg5AlCuSi4 127 41 46 −9 128 32 5 131 6 31 4 38 22 Mg4AlCu2Si4 128 44 70 −3 136 53 6 103 10 28 7 32 22 Mg4Cu3Si4 128 47 75 7 153 43 3 115 −6 20 3 51 26

**Figure 4.** Formation enthalpies of calculated β″ phases. When Cu atoms occupy *x* sites the number of Cu atoms in unit cell is corresponding to 2*x*. The lower energy configuration is marked with a specific shape, and Mg5Si6 is also marked as a reference. represents the Mg5Si6, represents the Mg5Al2Si4, represents the Mg5AlCuSi4 where Cu atoms occupy a Si3 site, represents the Mg4Al2CuSi4 where Cu atoms occupy a Si3 site, represents the Mg4AlCu2Si4 where Cu atoms occupy two Si3 sites, and represents the Mg4Cu3Si4 where Cu atoms occupy a Mg1 site and two

**Figure 6.** The Young's modulus anisotropies three-dimensional map of β″ phases. (**a**) Mg4Al3Si4; (**b**) Mg5Al2Si4; (**c**) **Figure 6.** The Young's modulus anisotropies three-dimensional map of β <sup>00</sup> phases. (**a**) Mg4Al3Si<sup>4</sup> ; (**b**) Mg5Al2Si<sup>4</sup> ; (**c**) Mg4Al2CuSi<sup>4</sup> ; (**d**) Mg5AlCuSi<sup>4</sup> .

Mg4Al2CuSi4; (**d**) Mg5AlCuSi4.

and Young's modulus (*E*) of polycrystalline are calculated by the Hill model [55], and the results are listed in Table 5. Comparing the values of *E*, *G*, and *B* of Mg4Al3Si4 and Mg4Al2CuSi4, it can be seen that the values of *E*, *G*, and *B* of β″ with Cu atoms are higher


**Table 4.** Calculated single crystal elastic stiffness constants (*Cij* 0 s) of the reported β 00 phases and energy favorable Cucontaining β 00 phases.

Based on the elastic constants in Table 4, the bulk modulus (*B*), shear modulus (*G*), and Young's modulus (*E*) of polycrystalline are calculated by the Hill model [55], and the results are listed in Table 5. Comparing the values of *E*, *G*, and *B* of Mg4Al3Si<sup>4</sup> and Mg4Al2CuSi4, it can be seen that the values of *E*, *G*, and *B* of β 00 with Cu atoms are higher than that of β 00 without Cu atoms. This relationship is also shown between Mg5Al2Si<sup>4</sup> and Mg5AlCuSi4. In general, the Young's modulus (*E*) can be used to measure the stiffness of the material. The stiffness of the material is greater with the increasing of Young's modulus (*E*) [64]. It is obvious that the stiffness is enhanced after Cu incorporate into Si3 sites. Pugh [65] proposes using the ratio of the bulk and shear modulus, *B*/*G*, to predict brittle or ductile behavior of materials. According to the Pugh criterion, if *B*/*G* is more than 1.75, ductile behavior is expected; otherwise, the material would be brittle. From Table 4, the *B*/*G* values of calculated β 00 phases are all larger than 1.75, therefore, all the compounds of β 00 phase are ductile with or without Cu atoms and the ductility decreases after Cu atoms incorporate into β 00. In addition, Poisson's ratio *v* has been used to measure the shear stability of the lattice, which usually ranges from −1 to 0.5. The smaller the value, the stronger the ability of the crystal to maintain stability during shear deformation [66]. The value of Poisson's ratio *v* > 0.26 means the ductility of the materials, and the Poisson's ratio of metals is usually 0.25< *v* < 0.35 [67]. As one can see, all β 00 configurations show ductility with minor differences. It is consistent with the conclusion based on Pugh criterion.

**Table 5.** Calculated mechanic properties of the reported β 00 phases and energy favorable Cucontaining β 00 phases.


#### *3.3. Phonon Spectra 3.3. Phonon Spectra*

taining β″ phases.

In addition, the dynamic stability is also taken into account. The phonon spectra of Mg4Al2CuSi<sup>4</sup> and Mg5AlCuSi<sup>4</sup> are shown in Figure 7. From Figure 7, one can see that there is no virtual frequency of configuration Mg4Al2CuSi<sup>4</sup> and Mg5AlCuSi4, which is generally considered to be dynamically stable. In addition, the dynamic stability is also taken into account. The phonon spectra of Mg4Al2CuSi4 and Mg5AlCuSi4 are shown in Figure 7. From Figure 7, one can see that there is no virtual frequency of configuration Mg4Al2CuSi4 and Mg5AlCuSi4, which is generally considered to be dynamically stable.

*Materials* **2021**, *14*, x FOR PEER REVIEW 11 of 18

than that of β″ without Cu atoms. This relationship is also shown between Mg5Al2Si4 and Mg5AlCuSi4. In general, the Young's modulus (*E*) can be used to measure the stiffness of the material. The stiffness of the material is greater with the increasing of Young's modulus (*E*) [64]. It is obvious that the stiffness is enhanced after Cu incorporate into Si3 sites. Pugh [65] proposes using the ratio of the bulk and shear modulus, *B*/*G*, to predict brittle or ductile behavior of materials. According to the Pugh criterion, if *B*/*G* is more than 1.75, ductile behavior is expected; otherwise, the material would be brittle. From Table 4, the *B*/*G* values of calculated β″ phases are all larger than 1.75, therefore, all the compounds of β″ phase are ductile with or without Cu atoms and the ductility decreases after Cu atoms incorporate into β″. In addition, Poisson's ratio *v* has been used to measure the shear stability of the lattice, which usually ranges from −1 to 0.5. The smaller the value, the stronger the ability of the crystal to maintain stability during shear deformation [66]. The value of Poisson's ratio *v* > 0.26 means the ductility of the materials, and the Poisson's ratio of metals is usually 0.25< *v* < 0.35 [67]. As one can see, all β″ configurations show ductility with

minor differences. It is consistent with the conclusion based on Pugh criterion.

**Table 5.** Calculated mechanic properties of the reported β″ phases and energy favorable Cu-con-

Mg5Si6 62 22 60 2.77 0.34 Mg5Si6 [63] 62 - - - - Mg4Al3Si4 67 26 69 2.57 0.33 Mg4Al3Si4 [63] 64 - - - - Mg5Al2Si4 63 28 74 2.23 0.30 Mg5Al2Si4 [63] 61 - - - - Mg6AlSi4 62 33 84 1.87 0.27 Mg4Al2CuSi4 72 32 84 2.26 0.31 Mg5AlCuSi4 69 34 88 2.01 0.29 Mg4AlCu2Si4 76 28 74 2.73 0.34 Mg4Cu3Si4 81 32 84 2.54 0.33

**Configurations** *B* **(GPa)** *G* **(GPa)** *E* **(GPa)** *B***/***G ν*

**Figure 7.** The phonon spectrum along a highly symmetric K-points path of (**a**) Mg4Al2CuSi4; (**b**) Mg5AlCuSi4. **Figure 7.** The phonon spectrum along a highly symmetric K-points path of (**a**) Mg4Al2CuSi<sup>4</sup> ; (**b**) Mg5AlCuSi<sup>4</sup> .

#### *3.4. Electronic Structure*

The total and partial electronic density of states (TDOSs and PDOSs) for four types of β 00 configurations are calculated to explore the influence mechanism of electronic interaction on structural stability and mechanical properties, as shown in Figure 8, with the Fermi level set to zero. It is evident that incorporating Cu does not change the metallic characteristic of the β 00 phase due to the finite DOS at the Fermi level. At the Fermi level, the TDOS for four types of β 00 configurations at the Fermi level varies. The greatest *n* (*Ef* ) is 7.41 states/eV/cell in Mg5Al2Si4, followed by 6.40 states/eV/cell in Mg4Al3Si4, 5.27 states/eV/cell in Mg4Al2CuSi4, and 4.18 states/eV/cell in Mg5AlCuSi4. This indicates that the Cu-containing β <sup>00</sup> phases have a smaller *n* (*E<sup>f</sup>* ). In general, a smaller pseudo gap value *n* (*E<sup>f</sup>* ) corresponds to a more stable structure [68]. This indicates that Mg4+*x*Al2−*x*CuSi<sup>4</sup> (*x* = 0, 1) are more stable than the β 00 phases without Cu. The Si-s (range from around 11 eV to 7 eV) and Si-p states (from around 7 eV to the Fermi level) dominate the TDOS of Mg4Al3Si<sup>4</sup> and Mg5Al2Si<sup>4</sup> below the Fermi level. In between (ranging from about −7 eV up to −4 eV) regimes, a mixture of s and p character exists, indicating strong hybridization. Especially from −7 eV to −5 eV, the shapes of Si-s and Si-p are very similar, indicating that there is a strong interaction between Si atoms. This may be the origin for the formation of the Si-network; the Si-network acts as a stable skeleton of these phases [32,69]. One can see that Mg-s/Al-s and Si-p in the range from −7 to −4 eV, originating mainly from the s-p hybridization of Si atoms and Mg/Al atoms. The s-states and p-states of Al, Mg, and Si are strongly hybridized above the Fermi level. From Figure 8, it should be noted that, below the Fermi level, the Cu-d state is formed. The s/p orbitals of Mg, Si, and Al all interact with the Cu-d state, and there is obvious electron transfer. The Si-p orbital and the Cu-d orbital are hybridized to form a covalent like bonding, and more electrons are transferred to the new orbital formed by the p-d hybridization.

**Figure 8.** The total and partial electronic density of states (PDOSs and TDOSs) for four β″ type compounds. (**a**) Mg4Al3Si4; (**b**) Mg5Al2Si4; (**c**) Mg4Al2CuSi4; (**d**) Mg5AlCuSi4. **Figure 8.** The total and partial electronic density of states (PDOSs and TDOSs) for four β 00 type compounds. (**a**) Mg4Al3Si<sup>4</sup> ; (**b**) Mg5Al2Si<sup>4</sup> ; (**c**) Mg4Al2CuSi<sup>4</sup> ; (**d**) Mg5AlCuSi<sup>4</sup> .

In order to gain a better understanding of the electronic structure of the studied system, the charge density distributions were used as an additional method. The charge-density difference between the (DFT) converged charge density and the isolated atomic charge densities were employed. Figure 9 shows the charge density difference contour plot for the (010) plane to analyze the interaction between Al, Mg, Si, and Cu atoms for the β″ phases. Here we clearly see that there has indeed been a transfer of charge to all the In order to gain a better understanding of the electronic structure of the studied system, the charge density distributions were used as an additional method. The charge-density difference between the (DFT) converged charge density and the isolated atomic charge densities were employed. Figure 9 shows the charge density difference contour plot for the (010) plane to analyze the interaction between Al, Mg, Si, and Cu atoms for the β 00 phases. Here we clearly see that there has indeed been a transfer of charge to all the Si–Si bond regions, it is consistent with the analysis by Derlet et al. [69]. A dominant feature of Figure 9a,b is the concentration of charge between the Si1-S3/Al-Si2-Mg1-Si1 nearest neighbors, and to a lesser extent, between the Si3 and Mg1 nearest neighbors,

indicating that covalency plays a role in this system, which was also reported in previous research [70]. Meanwhile, the charge distribution looks like a "charge loop", which can lead to the formation of an "Si network". Strong covalent bonds network can significantly increase the structural stability of β 00 phases. Such a charge transfer to the bonding regions originates from the core regions of both atoms on the Mg and Si sites, in addition to the homogeneous interstitial region between the Mg atoms. The depletion of charge from the Mg3 sites indicates that for this system both metallicity and covalency are present in the bonding. Moreover, the charge transfer density between the Si3 and Si2 sites is slightly decreased, and the charge transfer density between the Mg2 and Si2 sites is increased, indicating the bonds between atoms on Mg2 and Si2 sites are covalent. As shown in Figure 9, the charge ionization of all Mg3 sites is strong, and when the Cu atom on the Si3 site charge ionization becomes stronger, it means both Mg and Cu valence electron are delocalized. The difference is that Mg uniformly provides charges to the surroundings to form a metallic environment [69], while the charges of Cu atoms are delocalized toward Si atoms in unit cells, forming a directional covalent like bond. ing that covalency plays a role in this system, which was also reported in previous research [70]. Meanwhile, the charge distribution looks like a "charge loop", which can lead to the formation of an "Si network". Strong covalent bonds network can significantly increase the structural stability of β″ phases. Such a charge transfer to the bonding regions originates from the core regions of both atoms on the Mg and Si sites, in addition to the homogeneous interstitial region between the Mg atoms. The depletion of charge from the Mg3 sites indicates that for this system both metallicity and covalency are present in the bonding. Moreover, the charge transfer density between the Si3 and Si2 sites is slightly decreased, and the charge transfer density between the Mg2 and Si2 sites is increased, indicating the bonds between atoms on Mg2 and Si2 sites are covalent. As shown in Figure 9, the charge ionization of all Mg3 sites is strong, and when the Cu atom on the Si3 site charge ionization becomes stronger, it means both Mg and Cu valence electron are delocalized. The difference is that Mg uniformly provides charges to the surroundings to form a metallic environment [69], while the charges of Cu atoms are delocalized toward Si atoms in unit cells, forming a directional covalent like bond.

Si–Si bond regions, it is consistent with the analysis by Derlet et al. [69]. A dominant feature of Figure 9a,b is the concentration of charge between the Si1-S3/Al-Si2-Mg1-Si1 nearest neighbors, and to a lesser extent, between the Si3 and Mg1 nearest neighbors, indicat-

*Materials* **2021**, *14*, x FOR PEER REVIEW 14 of 18

**Figure 9.** Charge density difference of four calculated β″ phases. (**a**) Mg4Al3Si4; (**b**) Mg5Al2Si4; (**c**) Mg4Al2CuSi4; (**d**) Mg5AlCuSi4. **Figure 9.** Charge density difference of four calculated β <sup>00</sup> phases. (**a**) Mg4Al3Si<sup>4</sup> ; (**b**) Mg5Al2Si4; (**c**) Mg4Al2CuSi<sup>4</sup> ; (**d**) Mg5AlCuSi<sup>4</sup> .

formation enthalpies in Section 3.1 "Structure stability", the stoichiometry of Cu-containing β″ phases in the precipitation sequence and the sequence of Cu atoms substituting sites in the β″ phases can be inferred. For the stable phases determined from the thermodynamics, the elastic properties of β″ phases with and without Cu were calculated in Section 3.2 "Elastic properties", further supporting the proposed stoichiometry. Besides, the "Phonon spectra" study in Section 3.3 shows that they are also dynamically stable. In summary, the proposed compositions Mg4Al3−*x*CuxSi4 (1 ≤ *x* ≤ 3) are reasonable, which is consistent with the results observed in the experiment [36]. In the Section 3.4 "Electronic structure", the origin for the stability of the Cu-containing β″ phases is analyzed from the perspective of electron interaction. According to the analysis of the thermodynamic results of replacement energies and formation enthalpies in Section 3.1 "Structure stability", the stoichiometry of Cu-containing β 00 phases in the precipitation sequence and the sequence of Cu atoms substituting sites in the β 00 phases can be inferred. For the stable phases determined from the thermodynamics, the elastic properties of β 00 phases with and without Cu were calculated in Section 3.2 "Elastic properties", further supporting the proposed stoichiometry. Besides, the "Phonon spectra" study in Section 3.3 shows that they are also dynamically stable. In summary, the proposed compositions Mg4Al3−*x*CuxSi<sup>4</sup> (1 ≤ *x* ≤ 3) are reasonable, which is consistent with the results observed in the experiment [36]. In the Section 3.4 "Electronic structure", the origin for the stability of the Cu-containing β 00 phases is analyzed from the perspective of electron interaction.

According to the analysis of the thermodynamic results of replacement energies and

#### **4. Conclusions 4. Conclusions**


**Author Contributions:** Conceptualization, S.H. and J.W.; methodology, S.H. and J.W.; investigation, S.H., J.W.; resources, J.W., Q.W. and Y.D.; data curation, S.H. and J.W.; writing—original draft preparation, S.H.; writing—review and editing, J.W., D.Z., Y.K., Q.W. and Y.D.; visualization, S.H. and D.Z.; supervision, J.W. and Y.D.; project administration, J.W. and Y.D.; funding acquisition, J.W. and Y.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Key Research and Development Program of China, grant numbers 2019YFB2006500, and the National Natural Science Foundation of China, grant numbers 52171024, 51771234 and 51601228.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are contained within the article.

**Acknowledgments:** This work was financially supported by National Key Research and Development Program of China (grant numbers: 2019YFB2006500), the National Natural Science Foundation of China (grant numbers 52171024, 51771234 and 51601228) are greatly acknowledged. First-principles calculations were partially carried out at the High Performance Computing of Central South University.

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