Composition formulas of Fe-based transition metals-metalloid bulk metallic glasses derived from dual-cluster model of binary eutectics

It is known that bulk metallic glasses follow simple composition formulas [cluster](glue atom)1 or 3 with 24 valence electrons within the framework of the cluster-plus-glue-atom model. Though the relevant nearest-neighbor cluster can be readily identified from a devitrification phase, the glue atoms remains poorly defined. The present work is devoted to understanding the composition rule of Fe-(B,P,C) based multi-component bulk metallic glasses, by introducing a cluster-based eutectic liquid model. This model regards a eutectic liquid to be composed of two stable liquids formulated respectively by cluster formulas for ideal metallic glasses from the two eutectic phases. The dual cluster formulas are first established for binary Fe-(B,C,P) eutectics: [Fe-Fe14]B2Fe + [B-B2Fe8]Fe ≈ Fe83.3B16.7 for eutectic Fe83B17, [P-Fe14]P + [P-Fe9]P2Fe≈Fe82.8P17.2 for Fe83P17, and [C-Fe6]Fe3 + [C-Fe9]C2Fe ≈ Fe82.6C17.4 for Fe82.7C17.3. The second formulas in these dual-cluster formulas, being respectively relevant to devitrification phases Fe2B, Fe3P, and Fe3C, well explain the compositions of existing Fe-based transition metals-metalloid bulk metallic glasses. These formulas also satisfy the 24-electron rule. The proposition of the composition formulas for good glass formers, directly from known eutectic points, constitutes a new route towards understanding and eventual designing metallic glasses of high glass forming abilities.

electrons per unit cluster formula (e/u) is close to 24 21 . In a recent work, standard procedures towards designing BMGs were practiced in Ti-Cu-based BMGs 22 . Two major steps are: (1) analyzing a binary devitrification phase to obtain the principal cluster for use in the relevant glass-forming cluster formula, and (2) matching one or three glue atoms to the principal cluster so that e/u ≈ 24. By principal cluster here we mean the most strongly bonded nearest-neighbor order that is shared between the glassy and the crystalline states. They can be identified from the usually multiple clusters present in a given crystalline phase using high cluster isolation and atomic dense packing criteria 23 . It should be noted that special types of clusters have been clearly identified by computer simulation in Cu-Zr system 24,25 , and among them, icosahedron Cu 8 Zr 5 well interprets the Cu 64 Zr 36 BMG, formulated as [Cu-Cu 7 Zr 5 ]Cu~Cu 64.3 Zr 35.7 . However, this method still suffers from a major difficulty in the accurate determination of a cluster formula in the basic binary system, on which multi-alloying is further conducted for enhanced glass forming abilities. While the principal cluster is readily established in a devitrification following the well-established procedures, the appropriate glue atoms are poorly defined, often empirically fitted from known BMG compositions. Also the lack of precise information on cluster radius and atomic density hinders the accurate calculation of e/u. Therefore, despite of our previous success in understanding various metal-metal metallic glass systems such as Cu-(Zr, Hf, Ti), Ni-(Nb, Ta) 19,20 , the composition rule for metal-metalloid ones is still missing.
In this paper, the cluster-plus-glue-atom structural model and the relevant composition formulas will be used to understand the composition rule of Fe-(B,P,C)-based transition metal-metalloid multicomponent BMGs. The above-mentioned difficulties will be particularly addressed, by introducing a new approach towards obtaining the appropriate Fe-(B,P,C)binary cluster formulas, which is based on the dual cluster formulism for eutectic liquids 26 . After obtaining the basic Fe-(B,P,C) formulas, compositions of some typical Fe-(B,P,C) based binary, ternary and multi-component BMGs will be explored on the basis of the binary basic formulas. It will be shown that the good glass-forming compositions are well interpreted using the cluster formulas containing nearly 24 valence electrons.

Dual-cluster formulas for eutectic liquids
In this section, the cluster-plus-glue-atom approach for eutectic composition interpretation will be explored in detail, covering 1) dual-cluster model of eutectic liquids, and 2) its application in the composition interpretation of binary Fe-rich eutectic points, close to which most Fe-based BMGs are formed. Dual-cluster model of eutectic liquids. As already stated in ref. 26, binary eutectic liquids are characterized, in terms of the cluster-plus-glue-atom model for describing short-range-order local structures, by: (1) Two stable liquid subunits issued from the two eutectic phases, and (2) Each subunit formulated as [cluster](glue atoms) 1 or 3 of ideal metallic glasses .
A binary eutectic composition is therefore expressed as [cluster α + cluster β ](glue atoms) 2 or 4 or 6 , where the two clusters in the brackets belongs to the two liquid subunits resulted from the corresponding eutectic phases α and β.
The identification of the right clusters from eutectic phases then becomes the key step toward establishing the dual-cluster model for a eutectic liquid. Here the clusters are derived from the eutectic phases that bound the eutectic point. Yet, in a given crystal structure, there are often multiple nearest-neighbor clusters. There are two criteria for the selection of the right cluster (called the principal cluster) in the cluster formula, the maximum cluster isolation and atomic dense packing 21 . A cluster of such a type satisfies ideal atomic interaction and constitutes the most strongly bonded part in the structure. This cluster is then assumed to be inherited from the liquid state, via the amorphous solid state, down to the devitrification/eutectic phase from which it is derived. The well-measured eutectic points serve a good check for the appropriate matching of glue atoms. In the next part, the Fe-(B,P,C) binary eutectic compositions relevant to Fe-based BMGs will be addressed using the dual-cluster formulism, and from which the basic formulas for glass-forming formulas will be derived.
Dual-cluster formulas of Fe-rich Fe-(B, P, C) binary eutectics. It has been well-established that BMG compositions satisfy simple composition formulas [principal cluster](glue atom) 1 or 3 with e/u = 24, where the principal cluster is derived from a devitrification phase. The binary Fe-(B, P, C) BMG-relevant eutectic points will be analyzed via the dual cluster formulism.
For the Fe-B-based BMGs, the relevant eutectic point is Fe 83 B 17 and the eutectic phases are γ-Fe (FCC, Cu type) and BFe 2 (tetragonal, Al 2 Cu type).
The γ-Fe phase is characterized by a unique cuboctahedral cluster [Fe-Fe 12 ] (Fig. 1a), typical for FCC metals that do not contain any solute (B is almost insoluble with Fe). In expressing a cluster, the center atom is placed first and is separated from the 1 st -neighbor ones by a hyphen, both square-bracketed. However, as far as the liquid structure is concerned, on which our model is really based, it is widely accepted that the nearly pure Fe liquid structure is in fact related to the high-temperature BCC δ-Fe phase 27 , characterized by a unique rhombidodecahedron [Fe-Fe 14 ] cluster (Fig. 1b). Therefore, in dealing with the Fe-B system, the BCC [Fe-Fe 14 ] cluster will be use instead.
The other crystalline phase BFe 2 is the devitrification phase for the Fe-B-based BMGs, and the cluster formula issued from this phase should be responsible for the glass formation. In the unit cell of BFe 2 (Al 2 Cu structure type), there are two non-equivalent sites, B at (0, 0, 0.25) and Fe at (0.1661, 0.6661, 0). All crystal structure data are taken from Pearson's handbook 28 . Two clusters can be defined centered by the two sites, i.e., Fe-centered [Fe-B 4 Fe 11 ] with coordination number (CN) of 15 (Fig. 1c), and a B-centered Archimedean octahedral antiprism CN10 [B-B 2 Fe 8 ] (Fig. 1d).
Scientific RepoRts | 7: 9150 | DOI:10.1038/s41598-017-09100-9 The degree of cluster isolation is measured by comparing the complete cluster size (i.e., the number of atoms in the cluster) with the one with a reduced size after considering inter-cluster overlapping. For the ideal case in which no overlap between the neighboring clusters occurs, the two sizes are equal. However, in general, clusters overlap with each other due to the periodic constraint.  (Fig. 3).
Next step is to assign the four glue atoms to each cluster, producing two individual cluster formulas. There are actually only two kinds of combinations for glue atoms,  25 33 . It will also be illustrated that this formula is responsible for most Fe-B-based BMGs. The determination of cluster formula for glass formation from dual cluster formulism of eutectic liquid therefore constitutes a simple and accurate route towards BMG formulation, which overcomes the ambiguity in glue atom definition.
The Fe-P based BMGs are related to eutectic point Fe 83 P 17 , involving two eutectic phases α-Fe and Fe 3 P. α-Fe (BCC, W type) is characterized by the same rhombidodecahedral cluster as δ-Fe (Fig. 1b). However, considering the substantial solubility of P in α-Fe and the negative enthalpy of mixing between P and Fe, it is reasonable to assume a P-centered [P-Fe 14 ] cluster, rather than [Fe-Fe 14 ] as in the Fe-B case.
Fe 3 P is a commonly identified devitrification phase for BMGs 34,35 . It is of Ni 3 P structure type. There exist four non-equivalent sites in its unit cell, one being occupied by P and the other three by Fe's. From the four sites are developed four clusters, i.e., capped trigonal prism CN9 [P-Fe 9 ], CN14 [Fe-P 2 Fe 12 ], CN13 [Fe-P 3 Fe 10 ] and CN14  [Fe-P 4 Fe 10 ] (Fig. 4). The degree of isolation is the highest for the P-centered one, with the reduced cluster being [P-Fe 3 ]. The Fe-centered clusters are reduced to [Fe-PFe 2 ], but from much larger initial sizes than the P-centered one. Therefore, the P-centered capped trigonal prism [P-Fe 9 ] is taken as the principal cluster, instead of any of the Fe-centered ones.
There are two options to separate the glue atoms P 3 Fe, either P 3 -Fe or P 2 Fe-P 1 . [P-Fe 9 ], after being matched with each of the four possibilities, gives [P-Fe 9 ]Fe = Fe 90.9 P 9.1 , [P-Fe 9 ]P 1 = Fe 81.8 P 18.2 , [P-Fe 9 ]P 2 Fe = Fe 76.9 P 23.1 , and [P-Fe 9 ]P 3 = Fe 69.2 P 30.8 . Fe-P metallic glasses have been obtained by liquid quenching over a composition range of 13~24 at.% P 36,37 . Among the four formulas, [P-Fe 9 ]P 2 Fe is chosen as it is in the glass forming zone. [P-Fe 9 ]P 1 , being also in the range but quite close to the eutectic point, is eliminated because glass formation composition usually deviates from the eutectic one 38 . As will be illustrated later, [P-Fe 9 ]P 2 Fe is indeed responsible for BMG formation in multi-component Fe-P-based alloys.
Fe-C-based BMGs are related to eutectic point Fe 83 C 17 , involving two eutectic phases γ-Fe and Fe 3 C. γ-Fe (FCC, Cu type) is characterized by cuboctahedral cluster [Fe-Fe 12 ]; however, since it dissolves a substantial amount of C in its octahedral interstitial site, the more reasonable cluster should be C-centered octahedron [C-Fe 6 ] (Fig. 6a), rather than [Fe-Fe 12 ] (when a substitutional type of solute is nearly insoluble) or [C-Fe 12 ] (C is too small to be a substitutional element).
The commonly observed devitrification phase for the Fe-C-based BMGs is cementite Fe 3 C. The three non-equivalent sites, one C and two Fe, define capped trigonal prism CN9 [C-   glass formers because it is related to a devitrification phase. For Fe-C binary alloys, the glass forming ability is weak and there is no report on the best glass former in this system, though the glassy state by liquid quenching has been reported near eutectic point 35 .     9 ]P 2 Fe and [C-Fe 9 ]C 2 Fe, which are related to devitrification phases Fe 2 B, Fe 3 P, and Fe 3 C, respectively. They will be used as the basis for BMG composition analysis.

Cluster formulas for Fe-based metallic glasses
Fe-based metallic glasses usually have complex compositions but can always be regarded as being developed from M-(B,C,P) binary systems 14 .
Existing Fe-(B,C,P)-based multi-component compositions with good glass forming abilities are carefully scrutinized using the proposed binary basic formulas within the framework of the cluster-plus-glue atom model. The Fe-(B,C,P)-based multi-component compositions are shown in Table 1. As the example, BMG Fe 75 Mo 4 B 4 C 4 Si 3 P 10 of a critical size of 4 mm is chosen to illustrate the procedures of composition interpretation, as stated below.
(1) Select the alloys with good glass forming abilities; This step guarantees, to the maximum degree, that the compositions selected corresponds to high glass forming ability, as our approach only works for ideal metallic glasses stabilized at specific compositions. This is of course often inaccurate, especially in multi-component systems, as the trial-and-error experiments become quite tedious. In BMG series Fe 79  , where ρ a is the atomic density (number of atoms per unit volume), Z the total number of atoms in the cluster formula, and r 1 the cluster radius of the basic binary clusters 21 . The cluster radius r 1 of a complex alloy is an unknown parameter but can take the value of the relevant basic binary cluster. This is a reasonable simplification because at most M is mainly composed of 3d transition metals of similar atomic sizes like Cr and Mn. Even if large atoms such Mo and Ta are introduced, their amounts are minor so that their influence on r 1 can be ignored. Atomic density ρ a can be transformed from mass density ρ by multiplying ρ with Avogadro constant N 0 and divided by the average atomic weight ΣC i A i (C i and A i are respectively the atomic fraction and atomic weight of element i in the alloy). ρ a is also equal to the reciprocal of average atomic volume , where η is the atomic packing efficiency and is empirically fitted as 0.671 from the experimentally measured densities in Fe-P based metallic glasses following the method reported in ref. 29. Note that here we deal with the global packing efficiency of the whole cluster formula, including both the cluster and the glue atom parts, which is different from the atomic packing efficiency of the cluster itself 39,85 . By using this atomic packing efficiency, densities can be evaluated for all Fe-P-based compositions as shown in Table 1. e/u = 24.4 for M 75 Mo 4 B 4 C 4 Si 3 P 10 = [P-M 9 ]M 1.3 B 0.5 C 0.5 Si 0.4 P 0.3 can be calculated using a calculated mass density of 7.2 g/cm 3 .
Typical Fe-metalloid-based metallic glass alloys are collected (mostly BMGs but some ternary alloys with weak glass forming abilities are also included) and explained using the procedures stated above, with their cluster formulas, critical sizes, calculated and experimental densities, and e/u's illustrated in Table 1. The estimated mass densities ρ cal. are calculated by using atomic packing efficiency of 0.7 for Fe-B-based alloys, 0.671 for Fe-P-based, and 0.77 for Fe-C-based. It is noted that RE is not commonly used in Fe-B-based BMGs, except one example Fe 71.2 Y 4.8 B 24 which is understood as a partial substitution of the glue atom Fe by Y. RE elements are practically missing in Fe-P-based BMGs. However, RE's are often present in Fe-C-based BMGs. Al and Ga, practically missing in Fe-(B, C)-based BMGs, are frequently present in Fe-P-based BMGs, with their amounts taking up to 1 atom in the formula.
As a further exploration of Table 1, the glass forming ability (critical size) is again shown in Fig. 8 as a function of e/u ratio. It is clear that e/u is not well satisfied for Fe-B and Fe-C based BMG's. This discrepancy can be understood as arising from the uncertain density and cluster radius values. As has been stated, in alloy systems containing covalent bonds, the estimation of atomic densities cannot be made accurate. What is more important is the basic binary cluster radius r 1 , which is related to e/u following = × . π ρ ⋅ a e u 1 25 3 Z r 3 1 3 . In general, the cluster radius r 1 is calculated by averaging all the nearest neighbour distance. However, such an averaging is complicated by multi-alloying, and to simplify the calculation, here we always use those of binary systems and inevitably multiple alloying introduces cluster radius variations. Since e/u is inversely proportional to the cube of r 1 , tiny variations induces quite large change in e/u. It is noted that most of the added constituent elements in Fe-(B, C)-based multicomponent BMGs have larger atomic radii, so the alloying should result in increased cluster radii and henceforth decreased e/u values.
For instance, in Fe-C-based BMGs, the composition with maximum glass diameter thickness, Fe 39 Cr 15 Mo 14 Co 9 B 6 C 15 Y 2 , is interpreted by the cluster formula [C-M 9 ]Y 0.3 (B 0.8 C 0.9 )M 1 . Here Co, Cr and Mo all show negative enthalpies of mixing with C so that they should prefer the cluster shell sites, just like Fe's. In Fe-C based binary cluster [C-Fe 9 ], 1 st neighbour distances are 2.0065 Å (one Fe), 2.0111 Å (one), 2.0197 Å (two), 2.0212 Å (two), 2.3734 (two) and 2.8064 Å (one) and their average value is 2.18362 Å, which is used as the cluster radius r 1 . The change in radius Δr as caused by adding other alloying elements is assessed by considering the Goldschmidt radii difference of the new alloying elements with respect to those that they replace in the shell sites (the center atom remains unchanged, being C). The atomic radii of Fe, Co, Cr, and Mo are respectively 1. 0.023 Å, where the three radius differences correspond those of Co, Cr, and Mo, respectively. The new cluster radius becomes: r 1 + Δr = 2.184 + 0.023 = 2.207 Å. With this new radius, e/u is calculated, i.e., 25.5, which is much less than previously calculated using the binary cluster radius.

Conclusions
The present paper solves an important issue regarding the establishment of the composition formulas for BMGs, via eutectic composition interpretation in Fe-rich metalloid-containing BMGs using the cluster-plus-glue-atom model. The dual-cluster formulas for Fe-rich Fe-(B,P,C) binary eutectics are first proposed: [Fe-Fe 14 82.7 C 17.3 . The latter formulas in each, being respectively issued from BMG devitrification phases Fe 2 B, Fe 3 P, and Fe 3 C, are then clearly identified, which constitutes a new route towards understanding the composition rule of BMGs. The compositions of existing Fe-based transition metals-metalloid bulk metallic glasses are well interpreted using these basic binary formulas. The 24-electron rule is also verified.