Composition, Elastic Property and Packing Efficiency Predictions for Bulk Metallic Glasses in Binary, Ternary and Quaternary Systems

Despite new metallic glass systems being continuously investigated and reported, and a broadening of the number of engineering applications for this kind of material, the discovery, development and manufacture of bulk metallic glass (BMG) systems is still a complex and long process. Materials science and engineering states that when a new material is to be produced, prior to this, some properties would be expected from that new material, from its shape, component materials, structure, etc. The other way around would be, knowing the structure, component materials and production of new materials in order to fit some specification. However, for metallic glasses, the structure is still an unsolved puzzle, even though the international community has witnessed their advantageous properties and behaviour i.e. high strength, corrosion and wear resistant, chemical inertness, high toughness, among others. In 1926, Goldschmidt1 correlated the ability to form a glass with the value of the radius ratio rA/rO for oxides AmOn. He found that for all the oxides which had been prepared in the vitreous form, the radius ratio was around 0.2 0.4. Zachariasen2, proposed a model where the SiO4 polyhedron is repeated to produce a continuous random network (CRN). Regarding the study of BMG structure, different structural models have been proposed in the past 50 years; Bernal’s “dense random packing”3 model and Miracle’s “efficient solute-centred cluster packing”4, amongst many others. The results of the latter are consistent with the high densities measured in bulk metallic glasses. Table 1 summaries some historical ideas pertaining to atomic packing and glass formation. Packing in BMG is very dense with melt viscosities that are several orders of magnitude higher than in pure metallic melts. The dense packing accomplished by structural and chemical atomic ordering below the glass transition temperature also brings the BMG-forming liquids energetically and entropically closer to the corresponding crystalline state. These factors lead to slow crystallisation kinetics and consequentially to high glass forming ability (GFA)12. Since GFA is an influential factor in studying the formation of BMG, the search for systems with sufficiently high GFA is a critical task in this field. The mechanical behaviour of metallic glasses is also one of the most important topics, attracting a huge amount of research effort. Given its large practical relevance to the implementation of BMG as structural materials in real applications, it is important to know whether a material shows plastic deformation or brittle rupture under certain loading conditions. As a tool for this, the use of Blackman diagrams Composition, Elastic Property and Packing Efficiency Predictions for Bulk Metallic Glasses in Binary, Ternary and Quaternary Systems


Introduction
Despite new metallic glass systems being continuously investigated and reported, and a broadening of the number of engineering applications for this kind of material, the discovery, development and manufacture of bulk metallic glass (BMG) systems is still a complex and long process. Materials science and engineering states that when a new material is to be produced, prior to this, some properties would be expected from that new material, from its shape, component materials, structure, etc. The other way around would be, knowing the structure, component materials and production of new materials in order to fit some specification. However, for metallic glasses, the structure is still an unsolved puzzle, even though the international community has witnessed their advantageous properties and behaviour i.e. high strength, corrosion and wear resistant, chemical inertness, high toughness, among others.
In 1926, Goldschmidt 1 correlated the ability to form a glass with the value of the radius ratio r A /r O for oxides A m O n . He found that for all the oxides which had been prepared in the vitreous form, the radius ratio was around 0.2 -0.4. Zachariasen 2 , proposed a model where the SiO 4 polyhedron is repeated to produce a continuous random network (CRN). Regarding the study of BMG structure, different structural models have been proposed in the past 50 years; Bernal's "dense random packing" 3 model and Miracle's "efficient solute-centred cluster packing" 4 , amongst many others. The results of the latter are consistent with the high densities measured in bulk metallic glasses. Table 1 summaries some historical ideas pertaining to atomic packing and glass formation.
Packing in BMG is very dense with melt viscosities that are several orders of magnitude higher than in pure metallic melts. The dense packing accomplished by structural and chemical atomic ordering below the glass transition temperature also brings the BMG-forming liquids energetically and entropically closer to the corresponding crystalline state. These factors lead to slow crystallisation kinetics and consequentially to high glass forming ability (GFA) 12 . Since GFA is an influential factor in studying the formation of BMG, the search for systems with sufficiently high GFA is a critical task in this field.
The mechanical behaviour of metallic glasses is also one of the most important topics, attracting a huge amount of research effort. Given its large practical relevance to the implementation of BMG as structural materials in real applications, it is important to know whether a material shows plastic deformation or brittle rupture under certain loading conditions. As a tool for this, the use of Blackman diagrams have been suggested and employed 13 to explain the tendency for permanent deformation that a system may exhibit.
The work presented in this manuscript is based on several theoretical models for metallic glasses in terms of glass formation and elastic properties, and is intended to offer an alternative route to design and obtain BMG by estimating the system with GFA from a structural perspective (densest atomic packing) and predict its theoretical elemental elastic properties, from which some indication of plasticity can be obtained. In addition to the above, quaternary alloys were experimentally produced in order to get bulk metallic glasses.

Theoretical chemical composition calculation
Chemical compositions were calculated based on a sphere-packing scheme (solute-centred clusters occupying an f.c.c. cluster unit cell) 4 . The Miracle's model includes the calculation of the three-dimensional coordination number N T , which is obtained for a radius ratio R * for maximum packing efficiency 14 . The efficiency packing was calculated from the chemical composition 14 and cluster unit cell length 15 .

Theoretical elastic properties calculation
The elastic modulus predictions were also carried out for different glassy alloys systems taken from the literature. The estimations were made by applying the "rule of mixtures" approach [16][17] once having estimated the composition for the alloy using the efficient cluster-packing model. In the same way as before, theoretical estimations were compared to experimental values reported in the open literature. Twenty different values for Poisson's ratio for typical BMG were taken from 17 ; for these twenty systems elastic constants c 11 , c 12 and c 44 were also predicted in order to elaborate a Blackman diagram (plotting c 12 /c 11 vs. c 44 /c 11 ). The elastic constants c ij were calculated with 1 -3 equations 17-18 .

( )
The transition between brittle and tough regimes is at ν crit = 0.31 -0.32 19 . Higher values of ν give higher fracture energy 19 . In other words, the larger the ν is, the more ductile the BMG become, and small variation of ν will significantly change the ductility 17 . Several theoretical compositions obtained from the database were compared to those reported in the literature, and twenty different alloys per system (binary, ternary and quaternary), reported in the literature, are considered here.  19.89 were prepared with pure elements (purity > 99.99%). The alloys were prepared under Ti-gettered inert atmosphere of argon (high purity > 99.9) using the arc-melting technique. All the ingots were re-melted five times in order to achieve a chemical homogeneity. Then, the molten alloys were poured into a copper mold with an internal cylindrical cavity of 2 mm diameter × 12 mm length by suction casting technique. The structure of the samples was characterized by means of X-ray diffraction with a Siemens D5000 diffractometer, using Cu K α radiation. Finally, the hardness of the samples was carried out with a HMV-G21D hardness tester, in order to calculate the Young´s modulus values.

Chemical composition
The prediction for the binary Pd-Si, referred to in Table 2  1984 Relation between glass formability and atomic size mismatch Binary metallic glassy alloys Granato 8 1993 The main roles of interstitial atoms is in destabilising the crystalline phase Frozen glassy and liquid states alloys Egami 9 1997 Topological instability applied to local atomic structure Metallic elements and alloys Senkov, Miracle 10 2001 Interstitial model for glass formation Prediction of glass formation Egami 11 2003 Identified several conditions that would favour multicomponent bulk metallic glass formation Prediction of BMG-formation Miracle 4 2004 Model to determine the alloy constituent concentration based on a topological atomistic approach Dense packing of atomic clusters for glass formation the reported and predicted composition was small. From the twenty binary alloys investigated, about five showed a significant difference between the reported and predicted compositions. Composition predictions for the twenty different binary alloys were obtained; these alloys are listed in Table 2 Table 3 shows 20 ternary metallic glass compositions reported, it also includes the compositions and packing efficiency calculated with the database. The 4, 5, 7, 11, 15, and 20 alloy systems showed good agreement between calculated and reported ternary compositions as shown in Figure 2. However, for the Cu-Zr-Al, Mg-Gd-Cu and Cu-Zr-Be ternary alloy systems a significant difference was evident. On the other hand, the packing efficiency calculated for the Fe 75.76 Y 6.06 B 18.18 alloy was exceptional, i.e. 96.41%. In addition, the difference between the calculated and reported composition was very small. Table 4 shows the quaternary compositions of BMG reported and calculated. Figure 3 shows good agreement of quaternary compositions. The chemical compositions difference of the alloys reported and calculated was less than 10% for most of the alloy systems studied.

Elastic properties
The elastic modulus (E) and shear modulus (G) estimations for twenty different alloys were also calculated and compared with those reported in the literature. The corresponding alloys are listed in Tables 5 and 6, respectively and plotted in Figure 4. (It was possible to estimate values for bulk modulus (K), too, but these are not included in this work).
Regarding the Young's modulus, E, the Mg 65 Cu 25 Tb 10 system reports 51.3 GPa while its predicted value was 54.8 GPa; the Hf 55 Ni 25 Al 20 reports 117.63 GPa 13 and the predicted value is 113.8 GPa (Hf 60 Ni 30 Al 10 ). In general, the similarity between the reported and predicted elastic  moduli is rather good. However, it can be noticed that some of the composition predictions, even though they did show a slight deviation from those reported in literature, kept a narrow correlation with regards to the elastic modulus values. The elastic property predictions also showed good correlation with the corresponding experimentally reported data (Tables 5, 6 and Figure 4) Table 7; it also includes the elastic constants ratios (c 44 /c 11 and c 12 /c 11 ) calculated by the database. The absolute difference percentage   of the Poisson's ratio in most alloys compared in this work was between 3.8% -12%. Compositions 3, 7, and 8, show smaller values than 3%, due to chemical composition similarity. Compositions 10 and 17, show a greater value than 13%, due to greater discrepancy between reported and calculated compositions. Figure 5 shows a Blackman diagram constructed using the data from twenty different predicted alloys (listed in Table 7), with the Poisson's ratio. Concerning to the intrinsic toughness estimated for the twenty different predicted alloys, the Poisson's ratio was consistent with the ductile behaviour in most of the alloys. These results could provide an insight into the tough-brittle behaviour that the system might present.   With the database, it was possible to make other predictions based on elastic properties, such as the kinetic fragility index, m, which was calculated with equation (4) The kinetic fragility index is recently used to know whether an alloy will present high or low GFA 63 . Since elastic moduli can be potentially correlated to a wide range of physical, mechanical and thermal properties in BMGs, the way the database has been used here may be considered a starting point for how it can be implemented in future studies. Additionally, the database performs three different composition predictions for the ternary systems, the one that considers the gamma sites to be empty, another in which gamma sites are half occupied by β γ anti-site defects and Table 6. Predicted and reported E for several alloys; alloys 1-12 17 and 13-20 13 .   the last one that assumes all gamma sites are occupied by the presence of β γ anti-site defects. Thus, the best prediction approach was chosen to be compared with the reported data in the ternary systems.

Experimental results
In order to test the database, some glass alloys in the form of bulk were experimentally obtained. Figure 6 shows the XRD pattern of the Zr 57 19.89 alloys. The XRD patterns are constituted by a single broad peak (located between 2θ ~ 35 and 55 o ) typical of a metallic glass. These results confirm the usefulness of the work presented in this manuscript, where it was possible to design, model and produce bulk metallic glasses a-priori, reducing the experimental work time associated with standard experimental processes. Table 8 shows packing efficiency values and kinetic fragility index m of alloys produced. The efficiency packing values were between 45% and 50%. According to  , glass-forming liquids can be classified into strong and fragile liquids, depending on their fragility. The upper and lower limits of parameter are theoretically estimated between 16 for 'strong' systems and 200 for 'fragile' systems 63 . The alloys obtained are strong systems with high glass forming ability. Table 9 shows compositions calculated by "mixing rules", elastic properties, and elastic constants ratios (c 44 /c 11 and c 12 /c 11 ). The typical BMGs have Young's modulus E ~ 25 GPa -250 GPa, shear modulus G ~ 9 GPa -88 GPa, and bulk modulus K ~ 20 GPa -243 GPa 17 .
A good correlation between microhardness, H v , and Young's modulus, E, has been reported elsewhere 17  The elastic constants ratios were used to plot a Blackman diagram. Figure 7 shows the Blackman diagram of alloys mentioned above with their corresponding Poisson's ratio.

Conclusions
The results of the database presented here for estimating composition and elemental elastic properties in metallic glasses have been able to closely approach those experimentally determined and reported in the literature. They additionally provide valuable information regarding the mechanical behaviour that these alloys might present. This illustrates the usefulness of these theoretical models. In relation to the packing efficiency calculations, particularly those obtained for the quaternary systems, it can be noticed that some other factors such as chemical affinity, bonding and/or the enthalpies of mixing, must play a key role in the BMG-forming systems. The database enclosing the methods used in this work has been developed; incorporates the elemental elastic constants for the most common alloying elements as well as their atomic radii information. The quaternary alloys experimentally obtained are strong systems with high glass forming ability, which is consistent with the Miracle´s model and kinetic fragility index m. The Young's modulus estimated with microhardness values are closed to those calculated by the "mixing rules". The Poisson´s ratio and Blackman diagram, suggest ductile behaviour of the BMGs obtained.