Application of lever law in dual-cluster formulas model to interpret eutectic and bulk metallic glass

It is well known that the structure of bulk metallic glass has a close relationship with that of its crystalline eutectic phase, but how this is manifested on the atomic scale is not yet clear. The present paper further develops and optimizes dual-cluster formulas model to describe the eutectic and bulk metallic glass, as C1[(principal cluster)1(glue atom)1 or 3] + C2[(principal cluster)2(glue atom)1 or 3]. C1, C2 are the proportions governed by the lever law, and the two principal clusters which have the highest atomic close-packing efficiency and large degree of isolation are derived from the eutectic-related phases. Cu-Zr binary system are studied using this model and the results match well with the experiments. According to dual-cluster formulas and valence electrons, the correlation between deep eutectic and bulk metallic glass may be disclosed, and a detailed step towards composition design of bulk metallic glass is established as well.


Introduction
It is acknowledged that bulk metallic glasses (BMGs) formation is in direct competition with the growth of the eutectic [1,2].This kind of competition leads to structure similarity between glassy state and eutectic counterpart.But how this structure similarity is reflected remains unclear.
In our previous works, cluster formula model [principal cluster](glue atom) x (x = 1, 3) is proposed to explain the composition and structure of ideal metallic glasses [3][4][5], and the total valence electron number per unit formula, / e u, is close to 24 [6].This structural description unit comprises of a nearest-neighbor cluster part and a glue atoms part which are located between the clusters.Cluster formula model can be regarded as molecule like structural units, which carry information about atomic structure, chemical composition, and electronic structure.In this paper, principal cluster is proposed because within a certain system may define many different clusters, and only the cluster, so-called the 'principal cluster', appears to be associated with the best metallic glass.
Generally speaking, there are two eutectic-related phases which locating at the two sides of eutectic composition, and when participating reaction they will contribute different composition proportions.Therefore two single cluster formula and two principal clusters will be used when dealing with eutectic-related BMG.The two principal clusters which will show up in the dual-cluster formulas model are derived just from the two eutectic-related phases, respectively.In this case, dual-cluster formulas is put out to interpret the eutectic-related BMG composition, as C 1 [(principal cluster) 1 (glue atom) 1 or 3 ] + C 2 [(principal cluster) 2 (glue atom) 1 or 3 ].C 1 , C 2 are the weight proportion of two single cluster formula in the whole structure and can be determined by lever law which is widely used to determine the compositions in different reactions.
In this paper, selecting criteria of principal cluster is proposed firstly, then eutectic and eutectic-type BMG compositions of Cu-Zr binary system are interpreted via dual-cluster formulas which is modulated by lever law, and valence electron characteristics of dual-cluster formulas are also investigated.According to the dual-cluster formulas and features of valence electron, the local atomic short-range-order structure similarity of deep eutectic and BMG can be revealed, and a new approach leads to quantitatively design eutectic-type BMG is also developed.

The principal cluster
Because of liquid rapidly cooling below the glass transition temperature, the glassy state inherits the structural characteristics of its parent crystal [7,8].Recently works point out that this structure heritage is characterized by the special nearest-neighbor polyhedral cluster, called 'principal cluster' [9], which dominates local atomic short-range-order features of both glassy state and its competing parent crystal.Meanwhile, mathematical Voronoi tessellation scheme has also proved that some kinds of principal clusters, for instance the icosahedron, trigonal prism and Archimedes antiprism are the popular local atomic environment for metallic glasses [10].There are different atomic sites within unit cells, and centered by each of them may define different polyhedral clusters, so the rule of selecting the principal cluster should be given.As glassy state is rapidly quenched liquid, and the principal cluster is a short-range-order being a bridge between the liquid phase and the solid phase, it is then expected that the principal cluster should possess the highest atomic close-packing efficiency and a large degree of cluster isolation [11,12].
The radial atomic-number density is used to evaluate the atomic close-packing efficiency of a cluster, expressed as 3 .

3
( ) N r is the total number of atoms within the cluster, and r is the spherical radial distance from the central atom to the shell [13].The degree of cluster isolation can be obtained by the cluster-size reduction rate because of clusters overlapping with the same neighbouring clusters in crystals.After sharing atoms, the cluster is reduced to a smaller one, called the effective cluster.In our previous works, cluster formula [effective cluster](glue atom) x is proposed to describe crystalline phases [4].Effective cluster is the true composition because of clusters sharing atoms with their neighboring, therefore it is different from the isolated cluster.Taking Cu 8 Zr 3 (Cu 8 Hf 3 -type) [14] phase as example, after sharing, Cu(0.3693,1/4,0.7655)-centeredinitial icosahedron Cu 10 Zr 3 cluster is reduced to an effective one Cu 7 Zr 3 , as shown in figure 1 (Here Cu(0.3693,1/4,0.7655)means coordinate of Cu atom in a unit cell: x = 0.3693, y = 1/4 and z = 0.7655).In order to match the effective cluster composition with the phase composition, one extra Cu(0.2855,1/4,0.2735)atom is needed.So this phase can be expressed with a cluster formula as [Cu 7 Zr 3 ](Cu) 1 , where Cu 7 Zr 3 is the effective cluster and what remains outside the cluster part is the glue atom one Cu atom (figure 2).The effective cluster plus certain number of glue atoms constitute the building block, in which short-range-order structure and composition of phase are both demonstrated.The composition ratio of atoms in the effective cluster over that in the isolated cluster can be used to quantitatively measure the degree of cluster overlapping.The larger of this value means the more isolated the cluster tends to be existing in crystal.For example, after sharing Cu(0.3693,1/ 4,0.7655)-centeredCu 10 Zr 3 cluster is reduced to the effective one Cu 7 Zr 3 , therefore the degree of isolation is 10/ 13 = 0.77.

Dual-cluster formulas model and lever law
BMG with high glass-forming ability (GFA) has been related with eutectic, and recently a new method is put out to measure the similarity between BMG structure and eutectic effectively [15].Moreover, many experiments prove that eutectic liquids exist short-range-order structure [16][17][18].Therefore it is reasonable to use cluster formula model [principal cluster](glue atom) 1 or 3 to decipher eutectic.A eutectic zone is always bounded by two eutectic phases α e and β e (figure 3(a)), and each phase may develop a principal cluster.Hence, dual-cluster formulas model is conceived, written as: C 1 [(principal cluster) 1 (glue atom) 1 or 3 ] + C 2 [(principal cluster) 2 (glue atom) 1 or 3 ] to interpret eutectic composition.Here C 1 , C 2 are the weight proportion of the two phases in the eutectic reaction, and can be determined by lever law.
Figure 3(a) is an illustration of eutectic reaction for binary system.L e is eutectic, α e , β e are the eutecticrelated phases which are located on two sides of eutectic point, respectively, and horizontal line is the atomic percentage (at.%) of element B in mixture.According to the lever law, the weight proportion of α e , β e in eutectic reaction: L e ↔ α e + β e can be obtained by    ] can be formulated as two sub-units, and each one is made up of single cluster formula [principal cluster](glue atom) 1 or 3 .Each single cluster formula has 6 kinds of combinations of glue atoms, that is Cu, Zr, Cu 3 , Zr 3 , Cu 2 Zr and CuZr 2 .Therefore it can produce 36 kinds of combinations of glue atoms for dual-cluster formulas, as shown in table 1.
According to our previous works, eutectic and eutectic-type BMG have close relationship and it is reflected by short-range-order microstructural similarity [20].Ideal metallic glasses are Hume-Rothery phase stabilized, the number of valence electrons per unit cluster formula / e u is 24 [6].Therefore, it is reasonable to recognize that the / e u of eutectic should also be 24.The / e u value of single cluster formula can be calculated according to the equation: where Z is the number of atom in unit cluster formula, r 1 is the cluster radius calculated by arithmetic mean of all the radial distances, and r a is the atomic density (atomic number per unit volume) [21].
Following is taking cluster formula [Cu 5 Zr 10 ]CuZr 2 as an example to illustrate the calculation procedure of / e u.Average radius of Cu 5 Zr 10 cluster is = r 0.3118 nm. 1 The atomic volume can be calculated by So atomic volume of Cu and Zr are: In a similar way and considering weight proportion, the / e u of dual-cluster formulas for all 36 combinations of Cu 45.7 Zr 54.3 eutectic composition are calculated, as listed in table 1.The deviation of both composition and  Therefore, the steps for establishing the dual-cluster formulas model to interpret eutectic can be concluded as: firstly determining two principal clusters which are derived from the eutectic-related phases according to the selecting rule; then calculating weight proportion of C 1 , C 2 by lever law; and finally the value of / e u are checked in order to obtain the glue atoms.Using the same method, other three eutectic compositions of Cu-Zr system are also analyzed and can be expressed by dual-cluster formulas as

Cu-Zr system eutectic-type BMGs interpretation via dual-cluster formulas model
It is well known that microstructure of metallic glasses are generally characterized by nearest-neighbor polyhedral clusters [20,[22][23][24], and many studies show that BMGs with high GFAs have been related with eutectic [25,26].Based on above analysis, eutectic liquid is comprised of two sub-units issued from the two eutectic phases, and can be interpreted by dual-cluster formulas model.Therefore, eutectic-type BMGs may also be deciphered by dual-cluster formulas model.Taking Cu 45.7 Zr 54.3 eutectic-related BMG for instance, the composition of this BMG can be supposed as Cu 100−x Zr x , and using dual-cluster formulas model can be written as Cu  So optimal BMG composition Cu 33.6 Zr 66.4 can be obtained from solving above equation, and when forming the BMG the weight proportion of CuZr and CuZr 2 phases are C 1 = (66.7−x)/(66.7−50)= 0.02 and C 2 = 1−C 1 = 0.98, respectively.The very important thing is that the composition Cu 33.6 Zr 66.4 which is developed from dual-cluster formulas and lever law is very close to the experimental composition Cu 35 Zr 65 [27,28].The / e u of dual-cluster formulas 0.02[(Cu 7 Zr 8 )Cu] + 0.98[(Cu 5 Zr 10 )CuZr 2 ] is 23.59, close to the ideal value 24.
Other eutectic-type BMGs of Cu-Zr system obtained from dual-cluster formulas method are listed in table 2, and / e u is also calculated.The results show that the BMG compositions designed by dual-cluster formulas combined with lever law are very close to the experimental results [29][30][31][32][33] and with / e u are all close to 24.This proves the rationality of the dual-cluster formulas model which is modulated by the lever principle.Table 2 also reveals the local atomic short-range-order structure similarity between eutectic and eutectic-type BMG, that is, they are characterized by the same polyhedral principal clusters.Through the above analysis, a detailed procedure for composition design of binary BMG can be obtained, summarized as: (1) Finding deep eutectic and two eutectic-related phases which are located at the two sides of eutectic composition; (2) Determining the principal cluster for each eutectic-related phase based on atomic close-packing and isolated degree of cluster [20]; (3) Dual-cluster formulas model C 1 [(principal cluster) 1 (glue atom) 1 or 3 ] + C 2 [(principal cluster) 2 (glue atom) 1 or 3 ] is developed to quantitatively design the optimum BMGs, and weight proportion coefficient C 1 , C 2 can be obtained by lever law; (4) Valence electrons / e u of dual-cluster formulas should be checked because the ideal BMG should lead to the / e u close to 24.Although the analysis of the work only focuses on the Cu-Zr binary system, the method should be applicable to other systems.Taking Zr-Pd system as an example, the optimum BMG Zr 70 Pd 30 forming is related with Zr 75.5 Pd 24.5 deep eutectic [34,35], and this eutectic zone is bounded by eutectic phases Zr(W-type) and PdZr 2 (NiZr 2 -type).The principal clusters of eutectic-related phases are Zr 15 and Pd 3 Zr 8 , respectively.The eutectic composition Zr 75.5 Pd 24.5 may be understood by the dual-cluster formulas as: 0. e u = 24.20.Because close-packing and isolated for cluster are the general local atomic short-range-orders property of metallic alloys, it can be expected that this model is also applicable in other systems.Other binary systems which are easy to form metallic glasses will be analyzed in our further works.Moreover, the current work only considers geometric close-packing efficiency, our future work will consider using first principles to calculate cluster energy in order to determine the principal cluster in a more reasonable way.

Conclusions
Dual-cluster formulas model C 1 [(principal cluster) 1 (glue atom) 1 or 3 ] + C 2 [(principal cluster) 2 (glue atom) 1 or 3 ] is proposed to interpret eutectic and eutectic-related BMG, and C 1 , C 2 are determined by lever law.The principal clusters, having the highest atomic packing efficiency and larger degree of isolation, are derived from eutectic-related phases.Dual-cluster formulas governed by lever law gives a new method to establish the structural model, assuming that eutectic and eutectic-related BMG consist of two sub-units which are issued from the relevant eutectic phases.The rationality of this model is verified in Cu-Zr binary system, and the compositions developed from dual-cluster formulas match the experimental value very well.This description builds the link between eutectic and BMG, and reveals the local atomic short-range-order structure similarity between eutectic and eutectic-type BMG in a clear way.The work also confirms that eutectic and good BMG always have their / e u values close to 24.Moreover, a practical method to quantitatively design BMG is proposed.

Figure 1 .
Figure 1.One icosahedron Cu 10 Zr 3 cluster sharing atoms with two neighboring ones in Cu 8 Zr 3 (Cu 8 Hf 3 -type) phase.It show clearly that six Cu atoms double-shared on edge and other four Cu atoms (including central atom) and three Zr atoms unshared.Here, the big spheres represent the Zr atoms and the small ones Cu atoms (Two colors for Zr means two different Zr atomic sites and six colors for Cu means six different Cu atomic sites).

4 .
Cu-Zr system eutectic interpretation via dual-cluster formulas model 4.1.Establishment of dual-cluster formulas model There are four deep eutectic compositions in Cu-Zr binary system, and Cu 45.7 Zr 54.3 eutectic composition is taken as the example to illustrate the detail steps of dual-cluster formulas model establishment.As shown in figure 3(b), a coupled zone for eutectic Cu 45.7 Zr 54.3 formation is bounded by CuZr(CsCl-type) and CuZr 2 (MoSi 2 -type) phases, and the information of their crystalline structure may be reached in Pearson's handbook [14].Eutectic reaction can be written as: Cu 45.7 Zr 54.3 ↔ CuZr + CuZr 2 .According to the selecting rule for the principal cluster, Cu 7 Zr 8 , Cu 5 Zr 10 are the principal cluster of CuZr and CuZr 2 phases, respectively (configurations of the two principal clusters shown in figure 3(b)).So dual-cluster formulas model can be established to describe Cu 45.7 Zr 54.3 eutectic composition as C 1 [(Cu 7 Zr 8 )G 1 ] + C 2 [(Cu 5 Zr 10 )G 2 ], here G denotes glue atoms and the total number is 1 or 3. C 1 and C 2 can be reached by lever law, as C 1 = 0.74, C 2 = 0.26.

Figure 2 .
Figure 2. Cu 8 Zr 3 (Cu 8 Hf 3 -type) phase structure analyzed using Cu 10 Zr 3 cluster within 3 ´3 ´3 unit cells, and located between clusters are glue atoms.Here unit cells are projected along the a axis.

Figure 3 .
Figure 3. (a) Lever law of eutectic reaction for binary system.(b) Deep eutectic composition Cu 45.7 Zr 54.3 interpreted by dual-cluster formulas.Cu 7 Zr 8 and Cu 5 Zr 10 are the principal clusters of eutectic-related CuZr and CuZr 2 phases, respectively.

5
The composition of cluster formula [Cu 5 Zr 10 ]CuZr 2 is expressed by atomic percent as Cu 33.3 Zr 66.7 , therefore atomic average volume of Cu 33.3 Zr 66.7 alloy is = Zr 10 ]CuZr 2 can be obtained as: a a 3 So the / e u value of single cluster formula [Cu

Table 1 .
(Continued.)are given in order to determine the best glue atoms combination and further develop the optimum dualcluster formulas model.Table 1 obviously shows that eutectic composition Cu 45.7 Zr 54.3 can be perfectly interpreted by dual-cluster formulas 0.74[(Cu 7 Zr 8 )Cu 2 Zr] + 0.26[(Cu 5 Zr 10 )CuZr 2 ], meanwhile the / e u of dual-cluster formulas is 23.64, close to 24.