Composition formulas of binary eutectics

The present paper addresses the long-standing composition puzzle of eutectic points by introducing a new structural tool for the description of short-range-order structural unit, the cluster-plus-glue-atom model. In this model, any structure is dissociated into a 1st-neighbor cluster and a few glue atoms between the clusters, expressed by a cluster formula [cluster]gluex. This model is applied here to establish the structural model for eutectic liquids, assuming that a eutectic liquid consist of two subunits issued from the relevant eutectic phases, each being expressed by the cluster formula for ideal metallic glasses, i.e., [cluster](glue atom)1 or 3. A structural unit is then composed of two clusters from the relevant eutectic phases plus 2, 4, or 6 glue atoms. Such a dual cluster formulism is well validated in all boron-containing (except those located by the extreme phase diagram ends) and in some commonly-encountered binary eutectics, within accuracies below 1 at.%. The dual cluster formulas vary extensively and are rarely identical even for eutectics of close compositions. They are generally formed with two distinctly different cluster types, with special cluster matching rules such as cuboctahedron plus capped trigonal prism and rhombidodecahedron plus octahedral antiprism.

In a typical eutectic reaction, two solid phases transform to or from a single liquid phase at a specific composition and temperature. In spite of the obvious fundamental as well as engineering interests in such alloys, important especially for their low-melting points, the structures and compositions of eutectic liquids remain open issues. It is widely accepted that a eutectic liquid is characterized by topologic and chemical short-range-order clusters that maintain certain similarities with the eutectic phase structures [1][2][3][4][5][6][7][8][9] . These alloys in liquid state at temperatures not far from the melting point were considered as cluster solutions, where these clusters are self-associated atomic groups or chemically ordered structural units 1 . According to first-principles molecular dynamics simulations for liquid and undercooled eutectic alloys Au-(Si,Ge) at various temperatures 2 , the local structure presents a well-defined chemical short-range order that enhances dissimilar element interactions in contrast with random solid mixture and may explain the high stability of the liquid phase on the basis of preferential Au-(Si,Ge) bonds. However, the available information on the short-range-order structures in eutectic melts is far from complete and the structures and composition rules of eutectics are largely unknown.
Necessarily a short-range-order structural model is required for any understanding about the structures and compositions of the eutectic points. Actually binary eutectic compositions frequently occur near simple composition ratios 10 such as 8/1, 5/1, 3/1, 2/1 and 3/2. Based on Frank's conjecture 11 that icosahedral cluster might be responsible for liquid undercooling, these ratios were tentatively explained as chained arrangements of icosahedral clusters with the lowest solute atom neighbors 12 . These simple ratios were recently addressed 13 using the dense cluster-packing model for metallic glasses 14 . The eutectic liquid was treated as being composed of efficiently packed solute-centered atomic clusters where are present four topologically distinct atomic sites. Choosing different coordinate close-packing clusters and changing the occupation style of the interstitial solutes, simple ratios of binary eutectics could be reached. A similar model for metallic glasses was also proposed 15 where shared quasi-equivalent clusters are packed into an icosahedral-like structure, which explained some binary eutectics and metallic compositions near the solid solution side of binary phase diagrams. However, all these efforts fail in interpreting quantitatively the occurrence of eutectics at various and specific compositions. For this objective, a structural model, enabling a quantitative description of short-range orders, is required to describe precisely the eutectic liquid structure. It is reasonable to anticipate that a structural unit, reflecting the characteristic short-range ordering in the liquid, should be present for a specific eutectic composition. We have developed a so-called cluster-plus-glue-atom model that suits specifically for short-range-order structure descriptions in quasicrystals, amorphous alloys 16 , and solid solutions 17 . In this model, any structure is dissociated into a 1 st -neighbor coordination polyhedral cluster and a glue-atom part that are situated outside the cluster part at the 2 nd neighbors and beyond. Then the structure can be expressed by a cluster formula [cluster]glue x , where the cluster is the coordination polyhedron representative of the 1 st -neighbor short-range order of the structure (notice that the term cluster may mean any agglomeration of atoms in a general sense, but here we confine the cluster concept to cover only the 1 st -neighbor coordination polyhedra), and the glue atoms between the clusters mark the short-range-order feature on and beyond the 2 nd neighbors. For a bulk metallic glass, the number of glue atoms is either 1 or 3 16,18 . It was further pointed out 19,20 that the total number of valence electrons per unit cluster formula for an ideal bulk metallic glass is universally about 24, so that the cluster formula for a bulk metallic glass resembles the 'molecular' unit of a chemical substance 20 . This formula can also be understood as arising from a certain spatial averaging scheme of a complicated disordered structure into a short-range-order structural unit of a dozen of atoms, covering only the first few neighbors, generally 1 st and 2 nd . It should be emphasized that the clusters are assumed to be isolated from each others in metallic glasses, which is necessary to avoid the center-shell type of nearest-neighbor short-range orders to develop into longer-range ones. Solid-solution alloys have been treated in a similar manner, because they are also characterized by chemical short-range orders, and the formulas well explain the industrial specification composition selection, as exemplified by Cu-Zn and Cu-Ni alloys 17 .
In the following, the cluster-plus-glue-atom structural model and the relevant composition formulas for eutectic liquids will first be proposed, and then boron-containing binary eutectics will be fully covered using this model. It will be shown that, at least for normal eutectics whose compositions do not fall too close to the phase diagram terminals, the model applies perfectly, which arrives at formulated eutectic compositions within accuracies below 1 at.% from experimental ones.

Cluster-plus-glue-atom model for binary eutectic liquids
A binary eutectic alloy liquid normally decomposes in a coupled growth mode into two eutectic phases with distinct composition differences. It is then reasonable to assume that the eutectic liquid would contain two subunits that evolve towards their respective eutectic phases. Thereof we propose the first assumption for modeling the eutectic liquids: 1) A eutectic liquid is comprised of two subunits issued from the two eutectic phases.
As already stated, the compositions at which metallic glass stability reaches the maximum are well expressed by cluster formulas out of eutectic/devitrification phases in accordance with the cluster-plus-glue-atom model 18 . Since metallic glasses can be regarded as frozen liquids, their formulas in fact describe some stable liquids that resist crystallization upon solidification. Also metallic glass formation is generally associated with eutectics. Thereof we introduce the second assumption: 2) Each subunit is described by a cluster formula of ideal metallic glasses, expressed as [cluster](glue atoms) 1 or 3 A eutectic liquid is then composed of two subunit liquids, each being formulated either by [cluster](glue atoms) 1 or by [cluster](glue atoms) 3 , so that the final eutectic composition is expressed by a dual cluster formula: where the two clusters in the brackets belong respectively to the two liquid subunits and are inherited from corresponding eutectic phases α and β . The eutectic composition interpretation then relies on the identification of the right clusters from the eutectic phases. In explaining a bulk metallic glass composition, the cluster is taken from a known crystalline phase, assuming local structural heritage between the two states. In a given crystal structure, however, there are often multiple nearest-neighbor clusters (centered by any non-equivalent site in unit cell is defined a cluster). Among the multiple clusters, there must be at least one cluster, termed 'the principal cluster' 21 , that represents the dominating short-range order feature of the structure. This principal cluster should be the most strongly bonded part in the structure, which leads to high cluster isolation and atomic dense packing, as well as high elastic coefficients 22 . The atomic dense packing of a cluster could be measured by the center-to-shell atomic radius ratios, because an ideally densely-packed cluster of certain coordination number (CN) shows a special such ratio 14 . Cluster isolation refers to the cluster size reduction to account for the commonly present cluster overlapping in crystals, and the principal one should show the highest cluster separation.
For instance, the BCo 3 phase (CFe 3 structure type, space group Pnma, the crystal structure data are all from Pearson's handbook 23  , which are also phase formulas expressed using the three clusters because there is no glue atoms. The cluster reduction rates are respectively 4/9, 3/16, and 1.5/15. Therefore the principal cluster should be CN9 [B-Co 9 ] for the highest cluster isolation. This cluster type is frequently encountered in explaining eutectic points.
In the above example, the phase structure is completely occupied by the clusters and the phase formulas expressed using the clusters do not contain the glue atom part. In many cases, such as in B 4  . Since the principal cluster should dominate the phase structure, the smallest number of glue atoms are desirable, so that the last one [Y-B 18 Y 5 ] is finally selected. In many cases, the principal clusters can be directly identified from the phase formulas, where the largest phase formula, the smallest cluster size reduction, and the smallest number of glue atoms serve the finger prints for the principal clusters. A phase formula can simply be arrived at by matching the atomic multiplicity of each atom in unit cell. For instance, to reach the phase formula expressed by the [B-Co 9 ] cluster, the multiplicity of the central B atom being 4, and those of the two Co non-equivalent sites bring 4 and 8, the reduced cluster becomes [B 1 -Co 1 Co 2 ] = [B-Co 3 ], which is also the phase formula for the absence of glue atoms here. The use of atomic dense packing is restricted, for atoms are not strictly spherical of constant atomic radii. In the following, the phase formula characteristics are mainly used to determine the principal cluster.
After choosing the principal clusters from two eutectic phases, according to formula (1), one pair of such clusters are matched with two, four, and six glue atoms, giving a dual cluster formula for the eutectic liquid. In the following, B-Co eutectics are explored in detail to exemplify the composition interpretation procedures using the principal clusters from respective eutectic phases.
The B-Co system contains three eutectic points, exemplifying three major types, metal-compound, compound-compound, and compound-boron (Fig. 2). In the present paper, the phase diagrams are all readapted from ref. 24.
The eutectic B 18.5 Co 81.5 involves eutectic phases α -Co and BCo 3 . α -Co has the Mg structure, which presents a unique twinned cuboctahedron CN12 cluster, typical for the hexagonal close-packed metals. This CN12 [Co-Co 12 ] cluster together with CN9 [B-Co 9 ] from BCo 3 , plus four B glue atoms, explain the experimental eutectic as  where the clusters come from BCo 2 (Al 2 Cu) and BCo (BFe). The eutectic B 61 Co 39 involves BCo and β -B. Although the boron structures are generally analyzed as based on B 12 icosahedron, the unique local structure in terms of the nearest-neighbor coordination polyhedron is, however, the pentagonal pyramid [B-B 6 ] , and the commonly-used B 12 icosahedron is actually formed by twelve such pyramids enclosing an empty center. The B-richest eutectic is explained as The deviations between the calculated and experimental compositions are respectively (100*5/22-18.5)*√2 ≈ 0.0 (a scale of √2 should be used to reach the real composition distance), (100*10/27-37.0)*√2 ≈ 0, (100*14/23-61)*√2 ≈ − 0.1 at.% B (the negative sign means that the calculated composition is slightly B leaner than the experimental one), which are well below the normal experimental accuracy of about 1 at. %. The deviations are listed in Table 1 and are not mentioned below because they all lie within 1 at.% accuracy.
In reaching the final eutectic formulas using the two principal clusters from relevant eutectic phases, different combinations of glue atoms have been attempted and in general there is only one solution that fits the experimental eutectic point. For instance, the principal clusters [Co-Co 12 ] from eutectic phase Co and [B-Co 9 ] from eutectic phase BCo 3 , in combination with different glue atoms (B,Co) 2,4,6 , produce dual cluster formulas [Co-Co 12 + B-Co 9 ] (B,Co) 2,4,6 , with their total number of atoms per unit formula being 25, 27, and 29. These numbers times the    In the next section, well-established boron-containing binary eutectics are analyzed in the same manner, and they are all well explained by dual-cluster eutectic formulas using the principal clusters derived from relevant eutectic phases.

Dual cluster formulas of boron-containing binary eutectics
The boron-containing systems are analyzed in the sequence of element groups in the periodic  Fig. 4 and Table 1: Though the clusters are of the same types as the B-Sc case, the glue atoms are different. Actually, as will be further shown, eutectics are rarely the same, even for those of the nearly same chemistries and cluster structures. Again it is noticed that eutectic formulas are rarely identical, despite very similar outer-electron configurations and cluster types in both systems. It is also worth pointing out that the eutectic formation generally involves certain cluster matching rules for the two types of clusters of different geometries and chemistries, which constitutes another general property of eutectic formulas.
3) IVA elements Zr and Hf These systems are characterized by the presence of two terminal eutectics close to the two elemental ends. The B-Ti and all the B-rich eutectics, too close to phase diagram terminals, cannot be explained by any dual cluster formulas. The B 14 Zr 86 and B 13 Hf 87 eutectic points will be dealt with here. The former one is bounded by BCC β -Zr (W) and B 2 Zr (AlB 2 ). The β -Zr structure presents a unique CN14 rhombi dodecahedral cluster [W-W 14 (Fig. 6 and Table 1). The B 13 Hf 87 eutectic is bounded by β -Hf (W) and BHf (ClNa), the latter being characterized by two octahedral clusters [B-Hf 6 ] and [Hf-B 6 ]. Both clusters, giving phase formulas of the same size of two atoms, but the former one is more densely packed (B being a small atom in the octahedral interstice site of Hf) and is taken as the principal cluster: B 13 Hf 87 → [B-Hf 6 + Hf-Hf 14 ]B 2 = B 3 Hf 21 = B 12.5 Hf 87.5 ( Fig. 7 and Table 1). Notice that, again, despite of the extreme similarities between Zr and Hf, even when their eutectics are quite near each other, they corresponds to different eutectic phases and henceforth different dual cluster formulas.   (Fig. 10), where the clusters come from B 2 Ta (AlB 2 ) and B 4 Ta 3 .
The cluster [B-B 3 Ta 6 ] from B 2 Ta (AlB 2 ) cannot give any solutions to the last eutectic. This is also the rare example where the dual cluster formula is constructed from two identical clusters, which indicates that the glue atom matching can also be important. It is noted that the glue atoms Ta 3 B 3 cannot be evenly assigned to the identical cluster [Ta-B 12 Ta 6 ], so that the resultant cluster formulas for the two liquid subunits are never identical, which is also a common feature for eutectics. 5) VIA elements Cr, Mo, and W No new structure type appears in the B-Cr system. The eutectics are all well explained ( Fig. 11 and Table 1 Except the B-richest one, all the B-W eutectics are explained as shown in Fig. 13 and Notice that in explaining eutectic B 27 W 73 , the cluster is made B-centered on the basis of the pure metal cluster [W-W 14 ] in order to give a satisfactory explanation. This is different from all other B-metals systems, where the clusters retain the pure metal forms. At present, we are unable to give a sound explanation for this special case. 6) VIIA element Mn No new structure types are encountered in this system. The eutectics are explained as (Fig. 14     The B-Ni eutectics are explained as ( Fig. 16 6 ]. It should be noticed that in describing borides, we stick to the principal of nearest-neighbor coordination polyhedron, which leads to the pentagonal pyramids, rather than the B 12 icosahedron without central atom as normally used in the literature. The phase formula expressed by these  This B-C system exemplifies the eutectic systems possessing covalent bonding, signifying the universality of the present formulism for binary eutectics.    23 and B 27 (Ta,Mn) 17 .
2) A eutectic formula is not constructed from two identical subunits. Even for eutectic compositions expressed with identical clusters like B 27 (Ta,Mn) 17 = [(Ta,Mn)-B 12 (Ta,Mn) 6 + (Ta,Mn)-B 12 (Ta,Mn) 6 ]B 3 Mn 3 , the two subunits cannot be made identical, because the glue atoms B 3 M 3 cannot be evenly assigned to each cluster.
3) Eutectic formulas are generally formed from two distinct types of clusters. Many cluster types are involved. Common ones are CN9 capped trigonal prisms, CN10 octahedral antiprisms, CN12 octahedra or twinned octahedra, and CN14 rhombi dodecahedra. This fact indicates that cluster matching   plays a dominating and yet unknown role in stabilizing a eutectic liquid, and the distinctly different clusters lead to two eutectic phases of large composition differences. The exceptions are found only for [(Ta,Mn)-B 12 (Ta,Mn) 6 + (Ta,Mn)-B 12 (Ta,Mn) 6 ](Ta,Mn) 3 B 3 , where the two clusters are identical so that the two subunits should be stabilized by different but unknown attributions of glue atoms to each cluster.
Regarding the cluster matching rule, the CN12 clusters from FCC and HCP metals are generally associated with CN9 capped trigonal prisms such as [B-(Be,Co,Ni,Pd) 9 ] and [B-B 3 (Sc,Y) 6 ]. The CN14 clusters from the BCC W structure are matched to more varieties of clusters, but the occurrence of the CN10 octahedral antiprism clusters is the most frequent. 4) Simple integer ratios In confirmation of Kokandale's eutectic puzzle that eutectics occur near simple composition ratios 10 , the present B-containing binary eutectics are especially abundant by composition ratios of 1:6 (the first number representing B number), 1:5, 1:3, 3:5, 3:2, and 5:2 (Table 1), though more eutectics should be treated in order to give a better statistic account of the eutectic distribution.
By unveiling the cluster-based formulism, we have actually developed a new tool for the description of liquid structures, focusing only on their characteristic short-range-order units. Experimental validation of the cluster-based eutectic structures can now be envisaged, which has long been hindered by the lack of a suitable short-range-order structural model. Also, our approach and the formulism thereof may provide a practical composition design method for multi-element eutectic alloys via substitutions of binary eutectic formulas by similar elements.
In conclusion, eutectic structure and composition rule have been addressed using the cluster-plus-glue-atom model for the description of short-range-order structures. It is assumed that a eutectic liquid consist of two different subunits issued from the relevant eutectic phases, each being expressed by the cluster formula for ideal metallic glasses, i.e., [cluster](glue atom) 1 or 3 . Such a dual cluster formulism is well validated in B-containing eutectics (except those located by the extreme phase diagram ends). The dual cluster formulas vary extensively and are always composed of different subunits. They are generally formed with two distinctly different cluster types, with special cluster matching rules such as cuboctahedron with capped trigonal prism and rhombi-dodecahedron with octahedral antiprism.