Glass-forming ability (GFA) is crucial for designing bulk metallic glasses. In this work, it was found that when the content of the solvent element with an atomic radius larger than 1.58 Å in one ternary alloy system is fixed, their entropy of mixing (ΔSmix) vs. the enthalpy of mixing (ΔHmix) curve display a slant arc shape. The compositions locating around the inflection point of each ΔHmix vs. ΔSmix curve usually show an optimal GFA. Its feasibility was verified in Zr-, La-, and Ca-based ternary systems. By considering both calculated inflection points and experimental results, the optimizing glass-forming compositions can be roughly estimated by a proposed formula under limited conditions. Our studies could provide a simple method for preliminarily selecting good glass-formers when the content of the solvent element in one ternary alloy system is fixed.
Metallic glasses (MGs) are of current interest worldwide in materials science due to their promising mechanical and functional properties especially for bulk metallic glasses (BMGs),1–3 which require relatively low critical cooling rates (Rc) to form fully amorphous phase. So far, it still has been an old but tough question about how to select an alloy composition with a good glass-forming ability (GFA), which severely impedes the further development of BMGs. In order to evaluate the GFA, the Rc is usually estimated based on the critical casting thickness (Dmax) for fully amorphous phase.3–5 However, it is a fussy and heavy task to measure the actual Dmax for different glass-forming compositions. Until now, extensive attention has been focused on exploring different GFA criteria such as Trg, ΔTx, β, γ, δ, and φ and so on,4–10 which are calculated based on the glass transition temperature (Tg), the onset crystallization temperature (Tx), and the liquidus temperature (TL). Although those GFA criteria can be helpful to estimate the GFA for the investigated alloy systems, all the characteristic temperatures should be measured after obtaining fully amorphous samples. It cannot be used to roughly predict glass-forming composition regions with a good GFA especially for unknown alloy systems before performing tedious trial and error experiments. A similar situation also happens to some other GFA criteria such as the liquid fragility (m) and the fragility of the superheated melt (M).11,12
In order to solve this puzzle, great efforts have been done to explore a theoretical GFA criterion in attempt to predict the glass formation from the thermodynamic, kinetic, or/and structural points of view.13–18 A popular topological instability criterion was proposed by Egami et al., and then some modified versions were also adopted such as λ×Δe and λ+(Δh)1/2.13–17 Dong et al. proposed a cluster-plus-glue-atom model based on the atomic and electronical structures of BMGs.18 All these GFA criteria succeed in predicting the GFA for specially appointed alloy systems to some content. However, MGs can be classified into three types:19,20 (1) Metal-metal MGs consist of a large amount of icosahedral clusters;2,20,21 (2) A network atomic configuration consists of some trigonal prisms which are connected with each other through “glue atoms” in metal-metalloid MGs;19 and (3) Two large clustered units consisting of a trigonal prismcaped with three half-octahedron and a tetragonal dodecahedron exist in Pd- or Pt-based MGs.19 As a result, complex structural features for different alloy systems gravely resist the scope of application for these GFA criteria. Therefore, until now, the three empirical component rules proposed by Inoue et al. have been still adopted to give guide to how to roughly design of bulk amorphous alloys together with both the near-eutectic compositions and “similar element” principles.19,22–24
Therefore, it is feasible to propose a simple principle to preliminarily find an alloy composition region with a good GFA only for an individual alloy system instead of creating a unified GFA criterion for all glass-forming compositions. In this study, according to the compositional dependences of the enthalpy of mixing(ΔHmix) and the entropy of mixing (ΔSmix), a simple GFA formula was suggested only for metal-metal type ternary alloy systems containing a large solvent element. It becomes easy to simply select glass-forming regions with a good GFA when the content of the solvent element is fixed for an individual metal-metal type ternary alloy system. Furthermore, such a GFA strategy was verified in Zr-, La-, and Ca-based ternary alloy systems and then the application limitation of such a GFA formula was also clarified.
II. THEORETICAL ANALYSIS
On the basis of the Miedema model, the γ* and ε parameters were suggested to evaluate the GFA.25–27 These studies found that when the ΔHmix and ΔSmix of the glass-forming compositions achieve critical values, a high GFA of multicomponent BMGs could be observed. Generally, the Miedema model was used to calculate both the ΔHmix and ΔSmix.25 Since no elastic and structural effects were found in a melt state, the Miedema model can be expressed by the regular melt model:25,28
where Ωij (=) is the regular melt interaction parameter between the ith and jth elements, Ci is the atomic fraction of the ith element, Cj is the atomic fraction of the jth element, and is the enthalpy of mixing between the ith and jth elements which can be found in Ref. 28. By considering the effect of atomic mismatch, the entropy of mixing of a regular liquid is given by the following equation:28
where R is the gas constant of about 8.314 J·mol-1·K-1. Jiang et al. proposed that the lnCi should be replaced by lnϕi in BMGs due to the effect of dissimilar.29,ϕi (= 4/3πri3) is the volume fraction of the ith element, and the atomic radius ri in an amorphous state.13,30 Then the Eq. 2 can be given as follows:29
In order to clarify the compositional dependence of the ΔHmix and the ΔSmix on GFA for an individual ternary alloy system, the ΔHmix vs. ΔSmix curves for the Zr-Co-Al and La-Co-Al alloy systems were first plotted, respectively (Fig. 1). It is clearly seen that a nonlinear relationship between the ΔHmix and ΔSmix can be observed. When the content of the solvent element with a largest atomic radius is fixed for one ternary alloy system, an inflection point always exists around the maximum value of the corresponding ΔHmix vs. ΔSmix curve. It has been shown that the difference in energy between solid- and liquid-state (i.e. GL-GS) for multicomponent systems can be used to estimate the GFA of BMGs and then an assumption was proposed,31 i.e. GL-GS at a composition is proportional to the free energy of mixing (Gmix) of the liquid phase. The ΔGmix can be defined in terms of the ΔHmix and ΔSmix as expressed by ΔGmix =ΔHmix -TΔSmix when the consistent pure elements are standard states. Therefore, when the glass-forming compositions which locate around the inflection point usually possess a large ΔSmix as well as a very negative ΔHmix, leading to a lower ΔGmix, then the selected alloys trend to form a BMG easily. Both the confusion principle and the three empirical component rules19,24 also have shown that the good glass-forming compositions should simultaneously possess both a larger ΔSmix and a more negative ΔHmix. As shown in Fig. 1(a), when the Zr-Co-Al compositions are very far away from the inflection point of a ΔHmix vs. ΔSmix curve (i.e. right side or left side of each curve), a smaller ΔSmix would play an important role in the glass formation even though the ΔHmix is quite negative. Meanwhile, the values of ΔHmix also become less negative compared with those around the inflection points for the Zr-Co-Al alloy system with a fixed Zr content. Then the decreases of both ΔSmix and |ΔHmix| should strongly worsen the GFA. Furthermore, as shown in Fig. 1(b), a similar change tendency of the ΔHmix vs. ΔSmix curves of the La-Co-Al appears at the right side of all the curves. When the compositions locate at the left side of one ΔHmix vs. ΔSmix curve, both a smaller ΔSmix and a more negative ΔHmix can be observed. In this regard, even though the |ΔHmix| of the individual alloy system with a fixed content of the solvent element Zr increases, the rapid decrease of the ΔSmix also can reduce the GFA based on the confusion principle.24
On the other hand, it has often been argued that the atomic structure of BMGs is a key factor for the glass formation in metallic alloys.32–34 Turnbull suggested one general guiding principle to design BMGs with a good GFA based on the large differences in their atomic sizes.6 Based on the Miracle’s solute-centered packing model, when the size difference between atoms exceeds 12%, a denser randomly-packed atomic configuration can be observed.19,30,35 Based on previous experimental results,1,2,5,10,19,25 the atomic radius of the used metallic solvent element A in a ternary metal-metal type glass-forming compositions (e.g. A-B-C) is usually larger than 1.58 Å but less than 2.22 Å.13 Here the atomic radii of the metallic elements were taken from Ref. 30. In order to exceed a size difference of 12 %, the solvent element A with an atomic radius larger than 1.58 Å was usually chosen as the base element. Moreover, according to the periodic table of the elements,13 the atomic radii of the transition metal elements usually possess a smaller atomic radius less than 1.49 Å, which can be used as the solute element B. Its corresponding content should be less than the solvent element A but larger than the solute element C. When the gradual incorporation of solute atoms into crystalline host lattice could lead to the destabilization of the lattices, the collapse of the lattices and even amorphisation could be induced.13,36 Therefore, the appropriate addition of micro-alloying elements is beneficial for a good efficient atomic packing and the formation of icosahedral or icosahedral-like clusters related with glass formation.13,37 When an efficient atomic packing in clusters or polyhedral is achieved, the packing or correlation of these quasi-equivalent clusters gives rise to distinct medium-range orders, which further assists the formation of BMGs.19,30,35 Therefore, in order to achieve an efficient atomic packing in clusters or polyhedral and then obtain a good GFA, the difference of the atomic radii of the solvent and solute elements should be considered together with both the ΔHmix and ΔSmix of the metal-metal type ternary alloy systems, i.e. (1) The solvent element A should possess an atomic radius larger than 1.58 Å; (2) the atomic radius of the solute element C should is larger than the element A or is between the elements A and B; (3) The solute element B should a transition metal element; and (4) The selected compositions should locate around the inflection point of each ΔHmix vs. ΔSmix curve.
III. RESULTS AND DISCUSSION
A. Verification in Zr-Co-Al and La-Co-Al alloy system
Since the glass formation strongly depends on the thermal stability of the supercooled liquid and the resistance to crystallization, the width of the supercooled liquid (i.e. ΔTx) can reflect the thermal stability of the supercooled liquid.4,5,23 Besides, it was also found that a glass-forming liquid with a Trg larger than 2/3 would practically crystallize only within a narrow temperature range so that they easily could be undercooled to a glass state.6 Therefore, previous results have proven that the glass-forming compositions can possess a large GFA when the values of the Trg and ΔTx are very large.4–7 Fig. 2 exhibits the correlation between the ΔHmix, ΔSmix, Trg, and ΔTx of the ZrxAlCo (x = 50, 55, and 60 at.%) alloy systems. Their values of Trg and ΔTx were collected from Refs. 38–45. It is obviously seen that both the larger Trg (i.e. Trg > 0.60) and ΔTx (i.e. ΔTx > 50 K) of the individual ZrxCoAl (x = 50, 55, and 60 at.%) alloy systems always locate around the inflection point of each corresponding ΔHmix vs. ΔSmix curve. For the individual ZrxAlCo (x = 50, 55, and 60 at.%) alloy systems (Fig. 2), both the Trg and ΔTx gradually decrease with increasing Co or Al content after the glass-forming compositions are far away from their corresponding inflection points.
Since the Dmax for the formation of fully amorphous rods can directly reflect the GFA of the glass-forming compositions,31 it would be necessary to clarify the correlation between the ΔHmix, ΔSmix, and Dmax. As shown in Fig. 3, both the ΔHmix and ΔSmix of the LaxAlCo (x = 60, 65, and 67 at.%) alloy systems were collected together with their Dmax.46,47 As shown in Fig. 3, such a phenomenon mentioned above also can be observed in the La-Co-Al alloy system. The maxima of the Dmax for the individual LaxAlCo (x = 60, 65, and 67 at.%) alloy systems locates around the inflection point of each corresponding ΔHmix vs. ΔSmix curve. When the selected compositions are far away from the inflection points, the Dmax gradually decreases with increasing Co or Al content.
B. Verification in other ternary alloy systems containing a large solvent atom
In order to further confirm the feasibility of such a “inflection point” GFA principle, the ΔHmix, ΔSmix, and Dmax of the ZrxNiAl (x = 54, 56, 58, 60, and 62 at.%) and CaxMgZn (x = 55, 60, and 65 at.%) alloy systems were collected, respectively (Figs. 4 and 5).23,40,43,45,48–54 As shown in Fig. 4, with increasing Zr content from 54 at.% to 62 at.%, the maxima of the corresponding Dmax are found to be 10 mm, 12 mm, 15 mm, 15 mm, and 12 mm, respectively. Obviously, all these glass-forming compositions do not exactly locate at but do still around the inflection points of different ΔHmix vs. ΔSmix curves for each individual ZrxNiAl alloy system, respectively. When the selected compositions are far away from the inflection points, the Dmax gradually decreases with increasing Ni or Al content. As shown in Fig. 5, the maxima of Dmax of Ca55Mg20Zn25, Ca60Mg17.5Zn22.5, and Ca65Mg15Zn20 alloys are estimated to be 2 mm, 10 mm, and 6 mm for the CaxMgZn (x = 55, 60, and 65 at.%) alloy systems, respectively.52–54 When the selected compositions are far away from the inflection points, the Dmax also gradually decreases with increasing Mg or Zn content. All these experimental results are well consistent with our calculation prediction.
However, being different with the La-Co-Al and Ca-Mg-Zn alloy systems, the best glass-forming composition of Zr-based alloys slightly deviates from the inflection point and shifts to a neighboring side with a larger |dΔSmix/dΔHmix| (Fig. 4). Such a slight deviation strongly depends on the symmetry of each ΔHmix vs. ΔSmix curve (Figs. 1–4). For an alloy system with an asymmetrical ΔHmix vs. ΔSmix curve, the ΔHmix plays a more important role in the glass formation. As a result, the best glass-forming composition is not exactly at the inflection point but still around it, which trends to locate at the position around the composition with a most negative ΔHmix (Fig. 4).
C. Determination of a compositional formula with good GFA
Based on the observations above, it can be seen that the glass-forming compositions with a good GFA usually locate around the inflection points for the corresponding ΔHmix vs. ΔSmix curve of one individual alloy system with a fixed content of the solvent element. Even though the selected compositions do not perfectly locate at the inflection point of each ΔHmix vs. ΔSmix curve of a certain alloy system, the good glass formers always locate around or near such an inflection point. The inflection points for the selected individual alloy systems were collected and their corresponding contents of the elements A and C were plotted (Fig. 6(a)). It can be seen that all the inflection points of an individual alloy system exhibit a linear relationship, whose slope is around -0.5. Then it can be inferred that the content of the solute element C with a moderate radius locates between -0.5x + 46 and -0.5x + 48, i.e. 47 ± 0.5x. Here x is the content of the solvent element A. Then the inflection points can be roughly calculated by the following formula:
where the Di is the estimated compositions of the inflection points, and x, 53∓1-0.5x, and 47±1-0.5x are the content of the elements A, B, and C, respectively. In order to further estimate better glass-forming composition regions for an individual alloy system, the content of the solvent element A (i.e. x) should be roughly fixed before GFA optimization. Then the correlations between the ΔTx, Trg, or Dmax and the content of the solvent elements were shown in Figs. 6(b–d) for the Zr-Al-Co/Ni, Ca-Mg-Zn/Cu, La-Al-Co/Cu/Ni, and Mg-based alloy systems.23,38–62 As shown in Fig. 6(b), the content of the solvent element locates between 45 at.% and 71 at.% when the ΔTx is larger than 50 K. Meanwhile, the content of the solvent element locates between 50 at.% and 80 at.% when the Trg is larger than 0.4 (Fig. 6(c)). However, as shown in Fig. 6(d), the content of the solvent element locates between 50 at.% and 69 at.% when the Dmax is usually larger than 6 mm. Obviously, the glass-forming composition regions are enlarged according to the ΔTx or Trg and the GFA evaluated by the ΔTx or Trg is not well consistent with the Dmax. Even though both the ΔTx or Trg were also usually adopted to evaluate the GFA,2–6 the ΔTx or Trg GFA criteria cannot, in many cases, properly reflect the GFA of MGs since the GFA is related with not only the thermal stability of supercooled liquids but also the resistance to crystallization.4,5,57 Therefore, the values of Dmax are still the best GFA criteria and then were adopted to testify the feasibility of the proposed formula. It can be seen that the maxima of the Dmax for different base alloy systems are different with each other so that it is impossible to clearly clarify the GFA of different base alloy systems. Here, the maxima of the Dmax for different base alloy systems were used to normalize all the Dmax for all alloy systems, respectively, which is shown in Fig. 6(c). It is obviously seen that the normalized Dmax locates between 0 and 1 for all the investigated alloys. All the glass formers for a certain base alloy system exhibit a good GFA when the normalized Dmax is larger than 0.6. As a result, the content of the solvent element A is determined to be between 50 at.% and 69 at.%. According to Eq. 4, the content of the solute element B is determined to be between 12.5 at.% and 22 at.% while the content of the solute element C is roughly between 18.5 at.% and 28 at.%. Therefore, according to the Eq. 4 and the values of the solvent element A (i.e. 50 at.% - 69 at.%), it is easy to determine the glass-forming composition regions. On the other hand, after fixing the content of the solvent element and then calculating the compositions related to the inflection points, it is only necessary to finely tune the contents of the solute elements around the inflection points in order to find best glass formers for an individual alloy system. Nevertheless, the present glass formation strategy itself does not precisely predict the GFA but can be helpful to preliminarily select a good glass-forming composition region for a certain ternary alloy system containing a solvent element whose atomic radius is larger than 1.58 Å. Then, such an “inflection point” principle can effectively reduce tedious trial and error experiments to some content.
D. Limitation and expansion of the “inflection point” principle
It should be pointed out that there are some prerequisites for the applicability of the “inflection point” principle proposed in this work. When the atomic size of the solvent element is smaller than 1.58 Å such as transition metal elements Ti, Cu, Fe, Ni, Co, and Ta, such a GFA principle would be not simply applicable for some alloy systems. These alloys systems can be classified into three types: (1) Alloy systems such as TiCu-, TiZr-, NiNb-, and CuZr-based alloys,63–73 whose base elements Ti, Ni or Cu belong to the transition metal elements and their atomic radii are usually less than 1.58 Å.30 Here taking Cu47Zr-Al as an example, the maximum of Dmax does not locate around the inflection point of the ΔHmix and the ΔSmix whose corresponding composition would be Cu47Zr45Al8.71 In fact, besides the solvent element Cu, the element Zr also should be considered to be another solvent element. (2) Alloy systems such as Fe- and Co-based alloys, whose solvent elements also belong to the transition metal elements and their atomic radii are usually less than 1.58 Å.30 In such a case, their maximum of Dmax also does not locate around the inflection point of the ΔHmix and the ΔSmix when the content of the solvent element is fixed. Usually, a large amount of nonmetallic elements such as Si, B, C, or P should be introduced into the alloy systems.2,19,74–76 (3) Alloys systems such as Al-based alloys such as Al-Ni/Co-Rare earth metals, whose atomic radii of the solvent elements are less than 1.58 Å.30 Generally, the glass-forming compositions also locate around the inflection point of the ΔHmix and the ΔSmix, but the GFA of Al-based metallic glasses is very small.77 Even though one can use the “inflection point” principle to optimize the glass-forming composition regions, it is difficult to fabricate a bulk amorphous sample.
On the other hand, even though the glass-forming ternary compositions with GFA can be roughly determined, it is necessary to further enhance the GFA by introducing other micro-alloying elements. Recently, a “similar element” GFA principle has been proposed to improve the GFA of BMGs.22 In the past, a large amount of glass-forming compositions with excellent GFA have been developed,22,78–82 which actually happens to obey such a strategy. For example, the known Zr-Ti-Ni-Cu-Be, Zr-Ti-Hf-Ni-Be, La-Ce-Pr-Al-Co, La-Ce-Al-Co-Cu, Pd-Cu-Ni-P, and Zr-Co-Ni-Al alloy systems can be treated to originate from (Zr,Ti)-(Ni,Cu)-Be, (Zr,Ti,Hf)-Ni-Be, (La,Ce,Pr)-Al-Co, (La-Ce)-(Co-Cu)-Al, (Pt,Pd)-(Cu,Ni)-P, and Zr-(Co,Ni)-Al ternary alloy systems,22,78–82 respectively, by considering the effect of similar element on the GFA. Therefore, one should adopt both the “inflection point” principle and the “similar element” principle in order to design glass-forming compositions with good GFA.
Correlations between the ΔHmix, the ΔSmix and the GFA were established via the plotted enthalpy of mixing-entropy of mixing curves. Compositions with optimal GFA were found to usually locate around the inflection point of each ΔHmix vs. ΔSmix curve when the content of the solvent element is fixed in one ternary alloy system containing a solvent element whose atomic size is larger than 1.58 Å. Based on the previous experimental results and the “inflection point” principle, a compositional formula which is defined as (50 at.% ≤ x ≤ 69 at.%) for predicting BMGs of a ternary alloy system containing a large-sized solvent element, has been proposed. The solute element B should a transition metal element, while the atomic radius of the solute element C should be larger than the solvent element A or be between the solvent element A and the solute element B. Furthermore, the ΔHmix of constituents should be negative with each other. Such a GFA evaluation formula can be verified to be effective in preliminarily predicting BMGs of such La-, Zr-, and Ca-based BMGs. Although such a formula cannot precisely select a certain glass-forming composition with a large GFA, it indeed can help us narrow a glass-forming compositional region with large GFA before performing tedious trial and error experiments.
The authors are grateful to the financial support from the National Natural Science Foundation of China (51871132, 51761135125, 51501103, and 51501104), Young Scholars Program of Shandong University at Weihai, and Supercomputing Center, Shandong University, Weihai.