A Full-Process Method from Composition Design to Glass Forming Ability Prediction in Zr-Al-Co-Cu Metallic Glasses

Abstract

In this paper, a systematic approach utilizing clusters and mixing entropy is proposed to design efficient glass-formers and evaluate glass forming ability (GFA) without relying on thermal properties’ parameters. Firstly, under the guidance of this method, glass-formers in Zr-Al-Co-Cu system were designed as mixtures of topologically packed Zr-Al, Zr-Co and Zr-Cu clusters. Among these, two novel clusters, namely a trigonal prism Co-Zr₉ and an Archimedean octahedral anti-prism Co-Co₂Zr₈ were obtained and used. The best composition is Zr_55.79Al_12.66Co_7.74Cu_23.81, expressed as Cu (Cu₇+Zr₅)+0.867Co (Co₂+Zr₈)+0.851Al (Al₄+Zr₈), which exhibits a critical diameter of up to 10 mm. Additionally, based on this method, a novel parameter (ξ), independent of thermal properties’ parameters, was proposed to predict the GFA of metallic glasses. The results indicate that the lower ξ value correlates with better GFA. Simultaneously, the average local five-fold symmetry (ALFFS) from molecular dynamics (MD) simulations was employed to provide supplementary explanations for a few outlier points, further validating the proposed method’s effectiveness. The findings not only confirm the effectiveness of the clusters and mixing entropy approach in designing high-GFA metallic glasses, but also demonstrate the new parameter’s potential in predicting GFA. This study provides important theoretical and practical guidance for optimizing the design of metallic glasses.

1. Introduction

Glass forming ability (GFA) is one of the major factors affecting application of metallic glasses [1–8]. For years, researchers have focused on creating superior glass-formers and on assessing their GFA via thermal analysis parameters efforts that have yielded impressive results [9–12]. However, until now, research in these areas has been fragmented, lacking a systematic approach to understanding and designing metallic glasses. For instance, in existing studies, although indicators such as γ and T_rg have been proposed and successfully used to predict the GFA of metallic glass, the validity of these indicators is based on the necessary thermal analysis having been performed [13, 14]. Therefore, there is an urgent need to develop a set of systematic research methods that are not limited to understanding and designing GFA, but can also evaluate GFA without relying on thermal properties’ parameters. This will bring new insights into the field of GFA research.

Zr-based bulk metallic glasses (BMGs) have been extensively studied due to their unique chemical properties, excellent corrosion resistance, and superior GFA [15–21]. In our previous research, a method based on clusters and mixing entropy was proposed to successfully design a series of Zr-based BMG alloys with high GFA, such as Zr-Al-Co-(Nb) and Zr-Al-Fe-Cu, which exhibit optimal GFA in their respective systems [22–27]. It was found through calculations that testing only a few compositions was sufficient to achieve high GFA, demonstrating the method’s effectiveness in understanding and designing superior metallic glass compositions. In Zr-based BMGs, Ni-free metallic glasses have garnered significant attention as potential biomaterials [28–30]. Numerous researchers have made outstanding contributions to the field of Zr-Al-Co-Cu based metallic glasses, which are considered highly promising for surgical instrument applications [31–33]. However, this system has not yet been analyzed and designed from the perspective of clusters.

Additionally, molecular dynamics (MD) simulations, as a powerful tool, are widely used to study and characterize the local structural features of metallic glasses at the atomic scale [34–36], particularly the average local five-fold symmetry (ALFFS) [37]. Numerous studies have shown that the local five-fold symmetry of atomic clusters plays a crucial role in GFA [37–39]. In this paper, by introducing MD simulations, we utilize the ALFFS parameter to effectively describe the overall five-fold symmetry in the local environment of atomic clusters, thereby explaining the causes of anomalous points and further validating the effectiveness of the proposed method.

This study employs the clusters and mixing entropy method to understand and design a series of Zr-Al-Co-Cu metallic glasses. Additionally, a parameter (ξ) independent of thermal properties’ parameters is proposed to evaluate the GFA of the designed compositions. This parameter effectively predicts GFA in most cases, with only a few anomalies. By incorporating the AFFLS from simulations, these anomalies can be successfully explained, enhancing the model’s overall accuracy and reliability. This study introduces a full-process approach that systematically encompasses all stages, from composition design and cluster modeling to the prediction of GFA without relying on thermal properties. This method deepens the understanding of glass formation in Zr-based systems and provides a rapid way to identify and optimize high-GFA metallic glass compositions.

2. Experimental Procedure

2.1 Compositions understanding

According to our method, glass-formers were regarded as a mixture of clusters [22]. These clusters are believed to originate from the significant enthalpies of mixing between their constituent elements.

For the basic system Zr-Al-Co-Cu, the enthalpies of mixing of Zr-Al, Zr-Co, Zr-Cu, Al-Co, Al-Cu and Co-Cu at Equi-atomic compositions are respectively ΔH_Zr-Al = −44 KJ/mol, ΔH_Zr-Co = −41 KJ/mol, ΔH_Zr-Cu = −23 KJ/mol, ΔH_Al-Co = −19 KJ/mol, ΔH_Al-Cu = −1 KJ/mol and ΔH_Co-Cu = 6 KJ/mol [40]. It can be inferred that Zr-Al, Zr-Co and Zr-Cu clusters are more favored as their enthalpies of mixing are more negative than other pairs. Hence, the glass former (C_am) in Zr-Al-Co-Cu system could be written as:

  

\begin{align} \mathit{C}_{\text{am}} &= \alpha [\text{cluster(Zr-Co)}] + \beta [\text{cluster(Zr-Al)}] \\ &\quad + \gamma [\text{cluster(Zr-Cu)}] \end{align} (1)

Here, α, β and γ are coefficients of clusters. Glass-former in Zr-Al-Co-Cu system could be understood as a mixture of α (Zr-Co) clusters, β (Zr-Al) clusters and γ (Zr-Cu) clusters.

In the above Zr-Al-Co-Cu expression, both Co and Cu belong to transitional elements. Their chemical properties and atomic radii are similar, with Goldschmidt radii of 1.25 nm for Co and 1.28 nm for Cu, respectively. The difference between their atomic radii is only 2.3%. According to the literature, elements possessing similar radii and chemical properties can be regarded as similar elements, capable of substituting for each other within clusters [41]. However, in this method, Co cannot be regarded as a similar element to Cu for replacing Cu’s position in the Cu-Zr cluster. That is because the large repulsive interaction, which comes from the atomic pair Co-Cu, as the enthalpy of mixing of Co-Cu is 6 KJ/mol, would bring an unstable state if Co element replaces the location of Cu in Cu-Zr cluster’s shell. Thereby, glass-formers in Zr-Al-Co-Cu system could be regarded as a mixture of ternary clusters, namely Zr-Al, Zr-Co and Zr-Cu clusters, rather than the mixture of Zr-Al and Zr-Co/Cu clusters.

2.2 Compositions design

According to eq. (1), there are two factors affecting the final compositions in Zr-Al-Co-Cu system, namely, the specific clusters in each binary system and the coefficients of clusters.

The choice of microstructure for metallic glass is a crucial issue. Metallic glass is formed through rapid cooling of the molten liquid state. In combination with Gaskell’s view, Miracle further pointed that the microstructure of metallic glass is similar to the microstructure of the competing crystalline phase of the amorphous system, indicating that there is an inheritance of internal microstructure between the metallic glass and the competing crystalline phase [42]. Thus, specific clusters could be derived from the competitive phase. This method of selecting microstructures has been widely applied in numerous component designs.

However, not all microstructures obtained by the aforementioned methods could be used to effectively characterize the local structural properties of metallic glass. The selection of clusters is typically based on the following criteria: Firstly, the clusters should satisfy the topological dense packing criterion. Secondly, there should be a large negative mixture between the cluster components. A larger negative mixture helps to produce more chemical short-range order.

In our previous work, appropriate Zr-Al and Zr-Cu clusters, namely Al-Al₂Zr₈, Al-Al₄Zr₈, Cu-Cu₄Zr₆, Cu-Cu₅Zr₅ and Cu-Cu₇Zr₅ have been obtained and successfully used to design glass-formers. These clusters were also adopted in this paper. In Zr-Co binary system, a trigonal prism Co-Zr₉ and an Archimedean octahedral anti-prism Co-Co₂Zr₈ have been obtained, as shown in Fig. 1.

Fig. 1

Images of Zr-Co clusters ((a) Cluster Co-Zr₉ derived from phase CoZr₃, (b) Cluster Co-Co₂Zr₈ derived from phase CoZr₂).

The degree of topological dense packing of the clusters is one of the important factors influencing the formation of metallic glasses. Therefore, it is necessary to determine if the clusters mentioned above conform to the topological dense packing criterion.

Miracle has calculated the ideal radius ratio R^* for clusters with different coordination numbers (CN) [43]. The actual atomic radius ratio R_a can be calculated by dividing the cluster’s center atom’s radius by the average radius of the atoms in the cluster shell [43]. The percentage value between R_a and R^*, which is defined as Δ, is expressed as (R_a − R^*/R^*) × 100%. As for the clusters with CN of 9 and 10, the ideal ratio has been calculated [41].

Miracle’s analysis of a large amount of published data [42] shows that topological dense packing conditions are very effective in the composition design of metallic glass when the deviation of Δ is within the evaluation accuracy of ±10%. It is worth noting that Miracle used an ideal model to calculate topological dense packing, without considering the chemical interactions between components and the effects of internal strain. Therefore, there would be some degree of error in the dense packing deviation Δ between the actual cluster and the theoretical cluster.

In this paper, for the computation of actual dense packing, we employed the method utilizing the average atomic radius of the actual atoms in the cluster shell [44].

Take the calculation of Δ in cluster Co-Co₂Zr₈, for example, the Goldschmidt radii of Zr and Co are 1.60 nm and 1.25 nm respectively. The radius of the atom at the center of the cluster is 1.25 nm. The average radius of the atoms in the shell of the cluster is 1.53 nm. The actual level of dense packing in cluster Co-Co₂Zr₈ is 0.817. The ideal ratio for the CN10 cluster is 0.799. Δ in cluster Co-Co₂Zr₈ is 2.3%. Similarly, Δ in cluster Co-Zr₉ is 10.0%. Both clusters meet the requirement of the topological dense packing criterion. Thereby, the above two clusters Co-Zr₉ and Co-Co₂Zr₈ were adopted.

The second aspect is the calculation of the coefficients of clusters. Entropy plays a significant role in the formation and design of amorphous materials, particularly in the process of forming amorphous alloys, where entropy serves as a crucial thermodynamic parameter.

Wang [45, 46] proposed that low melting entropy is favorable for the formation of amorphous materials, contradicting previous theories regarding high melting entropy. This new perspective has brought a fresh outlook to our understanding of amorphous material formation. Additionally, mixing entropy also plays a vital role in amorphous formation. Mixing entropy describes the mixing state of components in a mixed system. Greer [47] once proposed the principle of disorder in metallic glasses formation, suggesting that the higher the system’s disorder, the stronger the tendency for metallic glasses formation.

Besides, according to our method, given that the microstructure characteristics of metallic glasses are satisfied, the greater the mixing entropy of the system, the lower the likelihood of crystalline phase precipitation [22]. The coefficients of clusters were taken at the critical value to make the system own the largest mixing entropy.

During the formation of metallic glasses, the presence of elements with different atomic radii is an important influencing factor [48]. From a microscopic structural perspective, different atomic radii can lead to distinct topological dense packing clusters. This paper has proposed a mixing entropy formula [49] that accounts for the effects of atomic radii and multiple constituents, as shown below:

  

\begin{equation} \Delta \text{S}^{\text{mix}} = - R\sum_{i = 1}^{n}C_{i} \ln \Phi_{i} \end{equation} (2)

  

\begin{equation} \Phi_{i} = \frac{c_{i}r_{i}^{3}}{\displaystyle\sum\nolimits_{i = 1}^{n}c_{i}r_{i}^{3}} \end{equation} (3)

where R represents the gas constant, r_i and Φ_i represents the atomic radius and the atomic volume fraction of the ith component respectively.

By increasing the chemical and structural disorder, we reduce the driving force for crystallization, thereby making it more difficult for the crystalline phase to nucleate. This approach helps to optimize the GFA, balancing both the liquid’s stability and the crystalline phase’s stability.

Clusters including Cu-Cu₇Zr₅, Co-Co₂Zr₈, and Al-Al₄Zr₈, which meet the topological dense packing criterion, are selected as examples. A detailed description of the specific calculation process is provided. The clusters are then substituted into the expression for the Cluster-based modeling group (C_BMG) as shown below:

  

\begin{equation} \mathrm{C}_{\text{BMG}} = \alpha [\text{Cu-Cu$_{7}$Zr$_{5}$}] + \beta [\text{Co-Co$_{2}$Zr$_{8}$}] + \gamma [\text{Al-Al$_{4}$Zr$_{8}$}] \end{equation} (4)

in this equation, α, β, and γ are coefficients that vary with different compositions. The equation can be further simplified to:

  

\begin{equation} \mathrm{C}_{\text{BMG}} = \text{($5\alpha + 8\beta + 8\gamma$)Zr} + \text{$5\gamma$Al} + \text{$3\beta$Co} + \text{$8\alpha$Cu} \end{equation} (5)

The composition of the Zr-Al-Co-Cu system is determined by the coefficients α, β, and γ, which are constrained based on high mixing entropy rules to obtain the theoretically optimal composition. The atomic percentages of Zr, Al, Co, and Cu elements are as follows:

  

\begin{equation} \text{Zr at.%} = \frac{5\alpha + 8\beta + 8\gamma}{13\alpha + 11\beta + 13\gamma} \times 100\% \end{equation} (6)

  

\begin{equation} \text{Al at.%} = \frac{5\gamma}{13\alpha + 11\beta + 13\gamma} \times 100\% \end{equation} (7)

  

\begin{equation} \text{Co at.%} = \frac{3\beta}{13\alpha + 11\beta + 13\gamma} \times 100\% \end{equation} (8)

  

\begin{equation} \text{Cu at.%} = \frac{8\alpha}{13\alpha + 11\beta + 13\gamma} \times 100\% \end{equation} (9)

The atomic percentages of Zr, Al, Co, and Cu elements are related to the coefficients α, β, and γ. Assuming the ratio of coefficient β to α is φ and the ratio of γ to α is ω, the formula can be further simplified to:

  

\begin{equation} \text{Zr at.%} = \frac{5 + 8\varphi + 8\omega}{13 + 11\varphi + 13\omega} \times 100\% \end{equation} (10)

  

\begin{equation} \text{Al at.%} = \frac{5\omega}{13 + 11\varphi + 13\omega} \times 100\% \end{equation} (11)

  

\begin{equation} \text{Co at.%} = \frac{3\varphi}{13 + 11\varphi + 13\omega} \times 100\% \end{equation} (12)

  

\begin{equation} \text{Cu at.%} = \frac{8}{13 + 11\varphi + 13\omega} \times 100\% \end{equation} (13)

By substituting the atomic percentages and atomic radii of Zr, Al, Co, and Cu into the mixing entropy eq. (2) and (3), calculate that the maximum mixing entropy is obtained when φ equals 0.867 and ω equals 0.851. The composition of ZCC12 is as follows Zr_55.79Al_12.66Co_7.74Cu_23.81.

According to the above analysis and calculation, glass-formers ZCC1–ZCC12 in Zr-Al-Co-Cu system expressed by clusters and the corresponding cluster coefficients were shown in Table 1.

Table 1 The designed compositions in Zr-Al-Co-Cu system, cluster expressions, calculated β_i/α_i and γ_i/α_i.

2.3 Experimental and computational methods

The above aforementioned Zr-Al-Co-Cu alloys in this paper were prepared in an argon atmosphere. All of the elements are highly purified as 99.5 mass% for Zr and 99.9 mass% for Al, Co and Cu. Bulk samples were produced by copper mold suction casting. The structure of the samples was identified by X-ray diffraction.

In this study, MD simulations were performed on a quaternary Zr-Al-Co-Cu alloy system using the open-source software LAMMPS [50]. The atoms were randomly distributed in a cubic box according to their elemental composition ratios. Interatomic interactions were described using the embedded atom method (EAM) potential developed by Zhou et al. [51], which has been extensively validated and applied to various metallic systems, including liquid and amorphous states [52, 53]. Simulations were conducted in the isothermal-isobaric ensemble, using the Verlet [54] algorithm for numerical integration and the Nose-Hoover thermostat for temperature control. The system was initialized as a simple cubic lattice composed of 4000 atoms with periodic boundary conditions, with atoms randomly distributed according to their elemental composition ratios. During sample preparation, the structure was first minimized for energy, followed by isothermal relaxation at 1800 K to achieve liquid equilibrium. Subsequently, rapid solidification was performed at a cooling rate of 1 × 10¹² K/s down to 300 K, and the system was further equilibrated for 2 million steps to obtain its microstructure.

3. Results and Discussions

As shown in Fig. 2, alloys with the highest GFA were achieved at the compositions of Zr_55.79Al_12.66Co_7.74Cu_23.81 (ZCC12, expressed as Cu (Cu₇+Zr₅)+0.867Co (Co₂+Zr₈)+0.851Al (Al₄+Zr₈)). The critical diameter of the aforementioned composition can reach up to 10 mm. It is known that, for the certain composition, GFA is different due to various laboratory condition. Under the same laboratory condition, the GFA of well-known Ni-free glass-former Zr₆₀Cu₂₅Fe₅Al₁₀ has been detected which is less than 5 mm. Thereby, the glass formers designed in this study demonstrate superior GFA compared to the well-known Ni-free glass formers with the Zr₆₀Cu₂₅Fe₅Al₁₀ composition and the Zr_53.6Al_18.6Co_27.8 composition reported in the previous literature [23, 24].

Fig. 2

XRD patterns of as-cast ZCC1–ZCC12 alloys.

The source of the high GFA of the designed glass-formers in Zr-Al-Co-Cu systems could be understood via clusters and mixing entropy. From a microstructural perspective, the clusters used in this paper, such as Co (Co₂+Zr₈), Cu (Cu₇+Zr₅) and Al (Al₄+Zr₈), are topologically dense packed. From a thermodynamic perspective, on the premise of satisfying the structural characteristics, the designed composition possesses the largest mixing entropy, which can decrease the possibility of precipitation of crystalline phases.

As shown in Fig. 3, GFA varies with different compositions. As analyzed above, the GFA is quite related to clusters and mixing entropy. Theoretically, for the certain glass-former, the higher its mixing entropy and the degree of topologically dense packing in clusters, the greater the GFA would be.

Fig. 3

ZCC12 glass-former’s GFA understood via clusters and mixing entropy.

For the glass-former C_am, it could be understood as the mixture of various clusters as shown below:

  

\begin{equation} C_{\text{am}} = \sum_{i = 1}^{n}k_{i} \cdot \textit{Cluster}_{i} \end{equation} (14)

where Cluster_i is the ith cluster, and k_i is the coefficient of ith cluster. Correspondingly, Δ_i was defined as the deviation between the actual and ideal level of dense packing in the ith cluster.

Considering that the level of dense packing in a certain cluster can be either positive or negative, the level of dense packing Δ_am in glass-former C_am could be calculated by using the following equation:

  

\begin{equation} \Delta_{\text{am}} = \frac{\displaystyle\sum\limits_{i = 1}^{n}k_{i} \cdot | \Delta_{i} |}{\displaystyle\sum\limits_{i = 1}^{n}k_{i}} \times 100\% \end{equation} (15)

The level of atomic dense packing in clusters is closely related to system volume, kinetics of nucleation and the growth of the competing crystalline. The parameter Δ_am, which stands for the level of dense packing, was chosen as the dimension to represent the microstructure characteristics in a certain glass-former C_am. ΔS^mix could be easily calculated by eq. (2)–(3), and it could describe the difficulty of atomic rearrangement and crystallization from the aspect of thermodynamics. According to our cluster- and mixing entropy-related method, GFA is closely related to microstructure and mixing entropy. The higher the mixing entropy and the closer the degree of deviation between the actual and ideal levels of dense packing in a certain glass-former, the greater the GFA would be. Based on the above analysis, a novel parameter ξ was proposed, as shown in eq. (16), to evaluate the GFA of our designed compositions. In this theory, smaller ξ means higher mixing entropy and degree of dense packing in clusters, which would bring a higher GFA.

  

\begin{equation} \xi = \frac{\Delta_{\text{am}}}{\Delta \text{S}^{\text{mix}}} \end{equation} (16)

To further characterize the local atomic structures in the simulation results, we employed the Voronoi tessellation method [55]. The constructed polyhedra were identified using the Voronoi index ⟨n₃, n₄, n₅, n₆⟩, where n_i (i = 3, 4, 5, 6, …) denotes the number of i-edged faces of the polyhedron. Based on Voronoi analysis, the LLFS in atomic clusters is defined as the ratio of the number of pentagonal faces to the total number of faces [39, 56], given by the following formula:

  

\begin{equation} f_{i}^{5} = n_{i}^{5}\biggm/\sum_{k = 3,4,5,6,}n_{i}^{k} \end{equation} (17)

where n_i^k (k = 3, 4, 5, 6) denotes the number of k-edged polygons in the ith type of Voronoi polyhedron. The LFFS is quantified by the ratio of the number of pentagonal faces to the total number of faces in the polyhedron, reflecting the significance of LFFS.

The introduction of LFFS provides a deeper structural insight, particularly regarding the formation mechanisms of metallic glasses. Five-fold symmetry effectively inhibits crystal nucleation, facilitating the formation of an amorphous structure and enhancing GFA. The advantage of LFFS lies in its ability to reflect the stability of local atomic arrangements, as high five-fold symmetry is typically associated with a more stable microstructure. Moreover, LFFS can reveal subtle variations in the local structure, offering a more comprehensive understanding of the relationship between the microstructure and the material’s properties.

To further quantify the impact of LFFS, we employ its average degree [57], defined as follows:

  

\begin{equation} W = \sum_{i}f_{i}^{k} \times p_{i} \end{equation} (18)

where f_i represents the fraction of the ith type of polyhedron in the sample [57]. This method allows us to comprehensively consider the contributions of different types of polyhedra, thereby providing an overall assessment of the LFFS. According to the above analysis and calculation results, the ALFFS (W) for ZCC1–ZCC12 is listed in Table 2.

Table 2 The critical diameter, calculated W and ξ of the designed glass-formers ZCC1–ZCC12 in Zr-Al-Co-Cu system.

Figure 4 illustrates GFA of ZCC1–ZCC12 alloy compositions. In this figure, ● indicates compositions that cannot form a metallic c glass structure with a critical diameter of 3 mm, ★ indicates compositions that can form a metallic glass structure with a critical diameter of 3 mm, ▲ indicates compositions that can form a metallic glass structure with a critical diameter of 5 mm, and ■ indicates compositions that can form a metallic glass structure with a critical diameter of 10 mm. The blue ellipse in the upper left contains most compositions that cannot form a metallic glass structure with a critical diameter of 3 mm, while the pink ellipse in the lower right contains most compositions that can form a metallic glass structure with a critical diameter of 5 mm. The green dashed line represents a critical ξ value (= 0.641), dividing the data into two main regions. Most compositions that cannot form a metallic glass structure with a critical diameter of 3 mm are concentrated above the dashed line with higher ξ values, while compositions that can form a metallic glass structure with a critical diameter of 5 mm are found below the dashed line with lower ξ values. This figure clearly demonstrates the relationship between ξ value and the GFA of ZCC1–ZCC12 compositions: the lower the ξ value, the better the GFA of the designed alloy compositions. The results are consistent with the predictive trend of the theoretical model proposed in this study.

Fig. 4

Critical diameter and parameters ξ of the designed glass-formers for ZCC1–ZCC12.

However, two data points marked with ★ in the figure deviate from this trend. ZCC3 has a ξ value greater than 0.641, suggesting its critical diameter should be less than 3 mm according to the analysis, but it actually reaches 3 mm. Conversely, ZCC10 has a ξ value lower than 0.641, theoretically indicating a critical diameter greater than 3 mm, but it only achieves 3 mm. To further investigate these anomalies, the ALFFS from MD simulations provides deeper insight into this phenomenon.

The radial distribution function (RDF) at 300 K for the simulated ZCC1–ZCC12 alloys is shown in Fig. 5. The splitting of the second peak indicates that all 12 simulated compositions are in an amorphous state. Figure 6 displays the final simulated model of ZCC12. The ALFFS (W) for ZCC1–ZCC12 is listed in Table 2. MD simulations reveal a high density of local five-fold symmetry structures in these designed compositions. In alloy compositions with lower ξ values (below the dashed line), the W is higher, whereas in compositions with higher ξ values (above the dashed line), the W value is slightly lower. This overall trend aligns with the predictions of the clusters and mixing entropy method, validating its effectiveness in designing high-GFA compositions.

Fig. 5

Simulated radial distribution function (RDF) of the designed glass-formers ZCC1–ZCC12 in Zr-Al-Co-Cu system at 300 K.

Fig. 6

The final simulated model of the ZCC12 at 300 K.

Generally, a higher W value indicates greater five-fold symmetry among atoms in the material, leading to a more stable microstructure. High five-fold symmetry helps inhibit crystal nucleation, thereby promoting the formation of the glassy phase. Consequently, amorphous alloys with higher W values typically exhibit better GFA.

By incorporating the W parameter, the anomalies of ZCC3 and ZCC10 can be further understood. The W analysis indicates that ZCC3 has the highest W among the compositions above the dashed line, resulting in a more stable microstructure, which explains its superior forming ability and the attainment of a 3 mm critical diameter. Conversely, ZCC10 has the lowest W value among the compositions below the dashed line, indicating a less stable microstructure. Thus, despite its low ξ value, ZCC10 fails to achieve a 5 mm critical diameter for its amorphous structure.

This study demonstrates that the parameter ξ plays a significant role in predicting the GFA of Zr-based metallic glasses designed using the clusters and mixing entropy method. The research reveals a strong correlation between ξ values and GFA, showing that lower ξ values generally correspond to higher GFA, validating ξ as a key predictive parameter. However, anomalies were observed with ZCC3 and ZCC10. Despite their respective high and low ξ values, their GFA deviations were reasonably explained by incorporating the W parameter. These findings further confirm the central role of ξ in predicting GFA and highlight the auxiliary value of W in explaining and correcting anomalies. This study provides a solid theoretical foundation for optimizing the design of Zr-based metallic glasses.

4. Conclusions

In this paper, a systematic method, based on clusters and thermodynamics has been proposed to design glass-formers and evaluate GFA without relying on thermal properties’ parameters in the Zr-Al-Co-Cu system.

1. (1)    The alloy composition Zr_55.79Al_12.66Co_7.74Cu_23.81 (ZCC12), expressed as Cu (Cu₇+Zr₅)+0.867Co (Co₂+Zr₈)+0.851Al (Al₄+Zr₈), designed using the clusters and mixing entropy method, possesses the best GFA. This composition has tight topological packing and the highest mixing entropy within the system, achieving a critical diameter of 10 mm. Under the same laboratory conditions, this alloy composition demonstrates superior GFA compared to the well-known Ni-free Zr-Cu-Fe-Al and Zr-Al-Co system metallic glasses.
2. (2)    A new parameter ξ (ξ = Δ_am/ΔS^mix), independent of thermal properties’ parameters, has been proposed to evaluate the GFA of the designed metallic glasses. By studying the Zr-Al-Co-Cu metallic glass system, a significant correlation between this parameter and GFA was observed. Specifically, smaller ξ values correspond to higher GFA. Additionally, the incorporation of ALFFS (W) analysis reasonably explained the deviations in the forming ability of a few anomalous points. These findings further validate the effectiveness of the new parameter, providing essential theoretical foundations and guidance for future metallic glass design.

Acknowledgements

The research work was supported by the Applied Basic Research Project of Liaoning Province, P. R. China (Grant No. 2023JH2/101300211), Dalian Science and Technology Innovation Fund Project (Grant No. 2022JJ12GX028) and the Dalian Outstanding Youth Project (Grant No. 2023RY014). Funded by the Special Funds for Basic Scientific Research Operations of Provincial Undergraduate Universities in Liaoning (LJBKY2024003).

REFERENCES

• 1)   Y.  Zhang,  Z.  Lu,  L.  Wang,  M.  Yang,  Y.  Huang,  J.  Dong,  X.  Wei,  W.  Song and  D.  Ma: Eutectic growth kinetics as an indicator for glass-forming ability, Intermetallics 162 (2023) 108025. doi:10.1016/j.intermet.2023.108025
• 2)   D.  Lahiri,  K.V.M.  Krishna,  A.K.  Verma,  P.  Modak,  B.  Vishwanadh,  S.  Chattopadhyay,  T.  Shibata,  S.K.  Sharma,  S.K.  Sarkar,  P.H.  Clifton,  A.  Biswas,  N.  Garg and  G.K.  Dey: Comprehensive characterization of the structure of Zr-based metallic glasses, Sci. Rep. 14 (2024) 4911. doi:10.1038/s41598-024-53509-y
• 3)   M.X.  Li,  Y.T.  Sun,  C.  Wang,  L.W.  Hu,  S.  Sohn,  J.  Schroers,  W.H.  Wang and  Y.H.  Liu: Data-driven discovery of a universal indicator for metallic glass forming ability, Nat. Mater. 21 (2022) 165–172. doi:10.1038/s41563-021-01129-6
• 4)   X.C.  Li,  S.Z.  Kou,  C.Y.  Li,  Y.C.  Zhao and  Y.T.  Ding: A criterion of glass-forming ability and stability derived from pseudo-four characteristic temperatures, Intermetallics 134 (2021) 107201. doi:10.1016/j.intermet.2021.107201
• 5)   X.K.  Li,  G.H.  Li,  J.  Ma,  Y.  Cao,  Y.P.  Xu and  W.Y.  Ming: Progress in the preparation, forming and machining of metallic glasses, J. Manuf. Process. 117 (2024) 244–277. doi:10.1016/j.jmapro.2024.03.022
• 6)   S.  Sohrabi,  J.N.  Fu,  L.Y.  Li,  Y.  Zhang,  X.  Li,  F.  Sun,  J.  Ma and  W.H.  Wang: Manufacturing of metallic glass components: Processes, structures and properties, Prog. Mater. Sci. 144 (2024) 101283. doi:10.1016/j.pmatsci.2024.101283
• 7)   Y.  Yao,  T.  Sullivan,  F.  Yan,  J.Q.  Gong and  L.  Li: Machine Learning-Guided Exploration of Glass-Forming Ability in Multicomponent Alloys, JOM 74 (2022) 4853–4863. doi:10.1007/s11837-022-05549-w
• 8)   T.  Nagase,  T.  Terai,  M.  Matsumuro and  M.  Takemura: Alloy Design and Solidification Microstructure Analysis in FePCAg Immiscible Metallic Glass, Mater. Trans. 64 (2023) 1982–1990. doi:10.2320/matertrans.MT-Y2023001
• 9)   P.  Błyskun,  P.  Maj,  M.  Kowalczyk,  J.  Latuch and  T.  Kulik: Relation of various GFA indicators to the critical diameter of Zr-based BMGs, J. Alloy. Compd. 625 (2015) 13–17. doi:10.1016/j.jallcom.2014.11.112
• 10)   C.  Chattopadhyay,  K.  Idury,  J.  Bhatt,  K.  Mondal and  B.S.  Murty: Critical evaluation of glass forming ability criteria, Mater. Sci. Technol. 32 (2016) 380–400. doi:10.1179/1743284715Y.0000000104
• 11)   H.J.  Fecht and  W.L.  Johnson: Thermodynamic properties and metastability of bulk metallic glasses, Mater. Sci. Eng. A 375–377 (2004) 2–8. doi:10.1016/j.msea.2003.10.254
• 12)   L.  Battezzati,  A.  Castellero and  P.  Rizzi: On the glass transition in metallic melts, J. Non-Cryst. Solids 353 (2007) 3318–3326. doi:10.1016/j.jnoncrysol.2007.05.121
• 13)   Z.P.  Lu,  H.  Tan,  S.C.  Ng and  Y.  Li: The correlation between reduced glass transition temperature and glass forming ability of bulk metallic glasses, Scr. Mater. 42 (2000) 667–673. doi:10.1016/S1359-6462(99)00417-0
• 14)   Z.P.  Lu and  C.T.  Liu: A new glass-forming ability criterion for bulk metallic glasses, Acta Mater. 50 (2002) 3501–3512. doi:10.1016/S1359-6454(02)00166-0
• 15)   A.H.  Cai,  G.  Zhou,  P.W.  Li,  D.W.  Ding,  Q.  An,  G.J.  Zhou,  Q.  Yang,  Y.P.  Lin and  H.  Mao: Mechanical properties for a series of Zr-based bulk metallic glasses, J. Alloy. Compd. 938 (2023) 168579. doi:10.1016/j.jallcom.2022.168579
• 16)   W.  Chen,  K.C.  Chan,  P.  Yu and  G.  Wang: Encapsulated Zr-based bulk metallic glass with large plasticity, Mater. Sci. Eng. A 528 (2011) 2988–2994. doi:10.1016/j.msea.2010.12.077
• 17)   S.T.  Deng,  H.  Diao,  Y.L.  Chen,  C.  Yan,  H.F.  Zhang,  A.M.  Wang and  Z.Q.  Hu: Metallic glass fiber-reinforced Zr-based bulk metallic glass, Scr. Mater. 64 (2011) 85–88. doi:10.1016/j.scriptamat.2010.09.014
• 18)   Z.J.  Fan,  Z.Y.  Zheng and  Z.B.  Jiao: Compressive fracture characteristics of Zr-based bulk metallic glass, Sci. China Phys. Mech. 53 (2010) 823–827. doi:10.1007/s11433-010-0154-6
• 19)   A.H.  Cai,  D.W.  Ding,  Y.  Liu,  H.  Wu,  W.K.  An,  G.J.  Zhou,  Y.  Luo and  Y.Y.  Peng: A Series of Zr-Based Bulk Metallic Glasses with Room Temperature Plasticity, Materials 9 (2016) 408. doi:10.3390/ma9060408
• 20)   A.  Inoue,  F.L.  Kong,  X.G.  Zhu,  J.X.  Chen,  H.  Men and  W.J.  Botta: Development and industrialization of Zr- and Fe-based bulk metallic glasses and light metal-based metastable alloys, J. Alloy. Compd. 979 (2024) 173546. doi:10.1016/j.jallcom.2024.173546
• 21)   S.  Korkmaz and  İ.A.  Kariper: Glass formation, production and superior properties of Zr-based thin film metallic glasses (TFMGs): A status review, J. Non-Cryst. Solids 527 (2020) 119753. doi:10.1016/j.jnoncrysol.2019.119753
• 22)   D.  Yu,  Y.  Geng,  Z.  Li,  D.  Liu,  H.  Fu,  Z.  Zhu,  Y.  Qi and  H.  Zhang: A new method locating good glass-forming compositions, J. Alloy. Compd. 646 (2015) 620–625. doi:10.1016/j.jallcom.2015.05.184
• 23)   D.  Yu,  X.  Li,  X.  Wu and  S.  Li: Understanding and designing good glass formers in Zr-Al-Co-(Nb) system combining clusters and mixing entropy, Mater. Lett. 234 (2019) 291–293. doi:10.1016/j.matlet.2018.09.124
• 24)   D.  Yu,  X.  Shi,  H.  Fu,  Y.  Geng,  Z.  Zhu,  Y.  Qi and  H.  Zhang: Glass formation in Zr–Al–Fe–Cu system, Mater. Lett. 157 (2015) 299–302. doi:10.1016/j.matlet.2015.05.142
• 25)   D.  Yu,  Z.  Qiu,  Q.  Song,  Y.  Zhang and  G.  Yuan: Formation of a Ni- and Be-free Zr-based glassy alloy in centimeter scale, Mater. Lett. 288 (2021) 129383. doi:10.1016/j.matlet.2021.129383
• 26)   Q.  Luo,  X.  Wang,  K.  Yin,  D.  Yu and  H.  Lin: A method locating Cu-Ti-based glass formers using Ni-Zr binary clusters, J. Non-Cryst. Solids 621 (2023) 122607. doi:10.1016/j.jnoncrysol.2023.122607
• 27)   Q.  Luo,  X.  Jiang,  X.  Wang,  Y.  Wang,  B.  Chen,  D.  Yu and  H.  Lin: A method of designing multi-component Y-Zr-Al-Co glass formers, Mater. Today Commun. 40 (2024) 110213. doi:10.1016/j.mtcomm.2024.110213
• 28)   N.B.  Hua,  L.  Huang,  W.Z.  Chen,  W.  He and  T.  Zhang: Biocompatible Ni-free Zr-based bulk metallic glasses with high-Zr-content: Compositional optimization for potential biomedical applications, Mater. Sci. Eng. C 44 (2014) 400–410. doi:10.1016/j.msec.2014.08.049
• 29)   H.F.  Li and  Y.F.  Zheng: Recent advances in bulk metallic glasses for biomedical applications, Acta Biomater. 36 (2016) 1–20. doi:10.1016/j.actbio.2016.03.047
• 30)   A.A.A.  Aliyu,  C.  Panwisawas,  J.  Shinjo,  C.  Puncreobutr,  R.C.  Reed,  K.  Poungsiri and  B.  Lohwongwatana: Laser-based additive manufacturing of bulk metallic glasses: recent advances and future perspectives for biomedical applications, J. Mater. Res. Technol. 23 (2023) 2956–2990. doi:10.1016/j.jmrt.2023.01.184
• 31)   S.  Mukherjee,  H.G.  Kang,  W.L.  Johnson and  W.K.  Rhim: Noncontact measurement of crystallization behavior, specific volume, and viscosity of bulk glass-forming Zr-Al-Co-(Cu) alloys, Phys. Rev. B 70 (2004) 174205. doi:10.1103/PhysRevB.70.174205
• 32)   K.  Han,  J.  Qiang,  Y.  Wang and  P.  Häussler: Zr-Al-Co-Cu bulk metallic glasses for biomedical devices applications, J. Alloy. Compd. 729 (2017) 144–149. doi:10.1016/j.jallcom.2017.09.144
• 33)   T.  Zhang and  A.  Inoue: Formation, thermal and mechanical properties of bulk glassy alloys in Zr–Al–Co and Zr–Al–Co–Cu systems, Mater. Sci. Eng. A 375–377 (2004) 432–435. doi:10.1016/j.msea.2003.10.233
• 34)   M.  Bauchy: Deciphering the atomic genome of glasses by topological constraint theory and molecular dynamics: A review, Comput. Mater. Sci. 159 (2019) 95–102. doi:10.1016/j.commatsci.2018.12.004
• 35)   W.B.  He,  D.K.  Wang,  W.Y.  Ming,  J.  Ma,  K.  Liu and  J.G.  Du: Research progress on cutting machining simulation technology of metallic glasses, Int. J. Adv. Manuf. Technol. 122 (2022) 1167–1193. doi:10.1007/s00170-022-09968-6
• 36)   M.  Shimono and  H.  Onodera: Dynamics and Geometry of Icosahedral Order in Liquid and Glassy Phases of Metallic Glasses, Metals 5 (2015) 1163–1187. doi:10.3390/met5031163
• 37)   Y.C.  Hu,  F.X.  Li,  M.Z.  Li,  H.Y.  Bai and  W.H.  Wang: Five-fold symmetry as indicator of dynamic arrest in metallic glass-forming liquids, Nat. Commun. 6 (2015) 8310. doi:10.1038/ncomms9310
• 38)   M.Z.  Li,  H.L.  Peng,  Y.C.  Hu,  F.X.  Li,  H.P.  Zhang and  W.H.  Wang: Five-fold local symmetry in metallic liquids and glasses, Chin. Phys. B 26 (2017) 016104. doi:10.1088/1674-1056/26/1/016104
• 39)   M.Z.  Li: Correlation Between Local Atomic Symmetry and Mechanical Properties in Metallic Glasses, J. Mater. Sci. Technol. 30 (2014) 551–559. doi:10.1016/j.jmst.2014.05.001
• 40)   A.  Inoue and  A.  Takeuchi: Recent progress in bulk glassy, nanoquasicrystalline and nanocrystalline alloys, Mater. Sci. Eng. A 375–377 (2004) 16–30. doi:10.1016/j.msea.2003.10.159
• 41)   D.B.  Miracle,  W.S.  Sanders and  O.N.  Senkov: The influence of efficient atomic packing on the constitution of metallic glasses, Philos. Mag. 83 (2003) 2409–2428. doi:10.1080/1478643031000098828
• 42)   D.B.  Miracle: Efficient local packing in metallic glasses, J. Non-Cryst. Solids 342 (2004) 89–96. doi:10.1016/j.jnoncrysol.2004.05.017
• 43)   D.B.  Miracle: The efficient cluster packing model – An atomic structural model for metallic glasses, Acta Mater. 54 (2006) 4317–4336. doi:10.1016/j.actamat.2006.06.002
• 44)   G.J.  Naz,  D.  Dong,  Y.  Geng,  Y.  Wang and  C.  Dong: Composition formulas of Fe-based transition metals-metalloid bulk metallic glasses derived from dual-cluster model of binary eutectics, Sci. Rep. 7 (2017) 9150. doi:10.1038/s41598-017-09100-9
• 45)   L.M.  Wang,  Y.J.  Tian and  R.P.  Liu: Dependence of glass forming ability on liquid fragility: Thermodynamics versus kinetics, Appl. Phys. Lett. 97 (2010) 181901. doi:10.1063/1.3506900
• 46)   J.  Wang,  Z.  Cai,  H.  Kang,  B.  Huo,  Y.  Zhang,  Y.  Gao,  Z.  Li,  S.  Feng and  L.-m.  Wang: Basic thermodynamic and dynamic characteristics of the glass forming intermetallics, Mater. Des. 238 (2024) 112665. doi:10.1016/j.matdes.2024.112665
• 47)   A.L.  Greer: Confusion by design, Nature 366 (1993) 303–304. doi:10.1038/366303a0
• 48)   Y.  Zhu,  Y.  Zhou,  A.  Wang,  H.  Li,  H.  Fu,  H.  Zhang,  H.  Zhang and  Z.  Zhu: Atomic-scale icosahedral short-range ordering in a rejuvenated Zr-based bulk metallic glass upon deep cryogenic treatment, Mater. Sci. Eng. A 850 (2022) 143565. doi:10.1016/j.msea.2022.143565
• 49)   P.  Ramachandrarao,  B.  Cantor and  R.W.  Cahn: Free volume theories of the glass transition and the special case of metallic glasses, J. Mater. Sci. 12 (1977) 2488–2502. doi:10.1007/BF00553936
• 50)   S.  Plimpton: Fast Parallel Algorithms for Short-Range Molecular Dynamics, J. Comput. Phys. 117 (1995) 1–19. doi:10.1006/jcph.1995.1039
• 51)   X.W.  Zhou,  R.A.  Johnson and  H.N.G.  Wadley: Misfit-energy-increasing dislocations in vapor-deposited CoFe/NiFe multilayers, Phys. Rev. B 69 (2004) 144113. doi:10.1103/PhysRevB.69.144113
• 52)   P.  Gupta,  K.C.  Katakam,  G.  Katakareddi and  N.  Yedla: Crack and its interaction with defects in Al coated with Cu₅₀Zr₅₀ metallic glass thin film: an MD simulation study, J. Mol. Model. 26 (2020) 82. doi:10.1007/s00894-020-4335-6
• 53)   L.  Ma,  X.D.  Yang,  F.  Yang,  X.J.  Zhou and  Z.W.  Wu: Unveiling the early stage evolution of local atomic structures in the crystallization process of a metallic glass, Chin. Phys. B 33 (2024) 036402. doi:10.1088/1674-1056/ad24d6
• 54)   L.  Verlet: Computer “Experiments” on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules, Phys. Rev. 159 (1967) 98–103. doi:10.1103/PhysRev.159.98
• 55)   X.k.  Xi,  L.l.  Li,  B.  Zhang,  W.h.  Wang and  Y.  Wu: Correlation of Atomic Cluster Symmetry and Glass-Forming Ability of Metallic Glass, Phys. Rev. Lett. 99 (2007) 095501. doi:10.1103/PhysRevLett.99.095501
• 56)   H.L.  Peng,  M.Z.  Li and  W.H.  Wang: Structural Signature of Plastic Deformation in Metallic Glasses, Phys. Rev. Lett. 106 (2011) 135503. doi:10.1103/PhysRevLett.106.135503
• 57)   Y.C.  Hu,  F.X.  Li,  M.Z.  Li,  H.Y.  Bai and  W.H.  Wang: Five-fold symmetry as indicator of dynamic arrest in metallic glass-forming liquids, Nat. Commun. 6 (2015) 8310. doi:10.1038/ncomms9310