Hidden electronic rule in the "cluster-plus-glue-atom" model.

Electrons and their interactions are intrinsic factors to affect the structure and properties of materials. Based on the "cluster-cluster-plus-glue-atom" model, an electron counting rule for complex metallic alloys (CMAs) has been revealed in this work (i. e. the CPGAMEC rule). Our results on the cluster structure and electron concentration of CMAs with apparent cluster features, indicate that the valence electrons' number per unit cluster formula for these CMAs are specific constants of eight-multiples and twelve-multiples. It is thus termed as specific electrons cluster formula. This CPGAMEC rule has been demonstrated as a useful guidance to direct the design of CMAs with desired properties, while its practical applications and underlying mechanism have been illustrated on the basis of CMAs' cluster structural features. Our investigation provides an aggregate picture with intriguing electronic rule and atomic structural features of CMAs.

Composition-structure-properties correlations are important topics with great significance in materials science research fields12345. Crystallographic method describes simple crystalline materials’ structure by means of “atomic positions plus space lattice”, and usually knowledge of a few atoms within the unitcell is sufficient to deduce their partial properties67. For complex metallic alloys (CMAs) like some intermetallics, quasicrystals and amorphous alloys, the problem becomes complicated because their structural information is often submerged in a long list of atomic coordinates. In this case, the structural characteristics of CMAs cannot be reflected through the crystallographic method, not to mention their structure-related properties8910. Since the atomic clusters are advocated as primary units to represent materials’ structural features, to solve the above problem, various cluster-based models have been developed during the past decades11121314151617. Among these cluster-based models, Dong’s “cluster-plus-glue-atom” model15 can be used to describe the atomic structure of nearly all materials. Denoted by a uniform cluster formula of [cluster](glue atoms)x18192021, this cluster-plus-glue-atom model regards the atomic structure of any materials, no matter whether crystalline or non-crystalline, to be composed of the clusters part and the glue atoms part22232425262728. Accordingly, all atoms in a given structure belong to three kinds of the central atoms, the shell atoms and the glue atoms2930, as the red spheres, the blue spheres and the green spheres shown in Supplementary Figure S1, respectively. In this context, the cluster-plus-glue-atom model contains materials’ basic composition information and structure information in its cluster formula, thus it lays the foundation to uncover the connections among composition, structure and properties of materials, especially for those CMAs with complicated atomic configuration.

Electrons and their interactions are believed as the most intrinsic factors to dominate the structure and properties of materials3132333435. When different atoms gather into molecules, there are electrons transferring and these electrons share or overlap in the bond-making process. Consequently, different materials behave different properties, and for a long time, electron factors have attracted considerable attentions to investigate the composition-structure- properties correlations of materials3637. For example, the exclusive principle proposed by Pauli at the beginning of last century, has been successfully used to extend the formal classification of valence electrons by four quantum numbers38394041. Afterwards, the valence bond theory and molecular orbital theory have been developed successively to explore the structure and properties of materials4243. One of the most important corollaries of these electronic theories, is that the stable electronic configuration of common covalent compounds and ionic compounds follow the octet rule444546. As for alloy phases, Hume-Rothery points out that electron concentration plays an important role in stabilizing the structure of electron compounds, which is known as the Hume-Rothery rule474849. Actually, many structural distinctions of metals and alloys can be discussed directly from their electron concentration differences324750. Furthermore, the structure and structure related properties of CMAs originate from the local atomic bonding inside and between the cluster structural units via electronic interactions515253. Therefore, it becomes necessary to investigate CMAs’ composition-structure-property correlations from the electron perspective.

Depending on the cluster-plus-glue-atom model, the cluster formula of CMAs is equivalent to the molecular formula of covalent compounds and ionic compounds. Meanwhile, it is known that a majority of covalent compounds and ionic compounds follow the octet rule. Accordingly, we speculate that CMAs should follow analogous electronic rule as well. In this work, an electron counting rule hidden in the cluster formula for CMAs is revealed. Our analysis on the cluster structure and electron concentration for typical kinds of CMAs2022242654 (including intermetallic compounds, quasicrystals and metallic glasses), indicates that the valence electrons’ number per unit cluster formula for these CMAs are close to specific constants of eight-multiples and twelve-multiples. It is thus termed as CMAs’ specific electrons cluster formula. This electron counting rule has been demonstrated as a useful guidance to direct the design of CMAs with desired properties, and its practical application has been illustrated accordingly. Furthermore, an underlying mechanism behind this electron counting rule is presented on the basis of CMAs’ cluster structural features. The present work will help people to better understand the composition-structure-properties correlations of CMAs.

Proposal of the cluster-plus-glue-atom model electron counting (CPGAMEC) rule

Based on the cluster-plus-glue-atom model, we set the goal of revealing the electron counting scheme for CMAs (i. e. the CPGAMEC rule), by analogy with the octet rule and its extension for common covalent compounds and ionic compounds555657. As is known that atoms in covalent compounds and ionic compounds satisfy the octet rule by means of either sharing electrons with neighbor atoms, or transferring electrons from one atom to another444546. As a consequence, the valence electrons’ number per unit molecular formula (N) for a majority of covalent compounds and ionic compounds is specific constants of eight-multiples5658, like 8 for NaCl, 16 for CO2 and 24 for Al2O3 etc., as presented in Fig. 1 and Supplementary Table SI. Here in our work, this electron counting scheme of the N value being constants of eight-multiples, is regarded as an extension of octet rule. Noting that it is only a sufficient condition of the octet rule, rather than its necessary condition. Accordingly, the CPGAMEC rule is proposed to reveal the electron counting scheme for CMAs, since the cluster formula of CMAs is equivalent to the molecular formula of covalent compounds and ionic compounds1529. Analogously, the CPGAMEC rule is described by the valence electrons’ number per unit cluster formula (N), and expressed as the following form

Figure 1

Octet rule and its extension for common covalent compounds and ionic compounds, reflected by the valence electrons’ number per unit molecule formula (N) being specific constants of eight-multiples.

Chemical species related to some covalent compounds and ionic compounds are presented.

where e/a represents the electron concentration (i. e. the e/a-ratio), and Z represents the total number of atoms per unit cluster formula.

Given that the e/a-ratio has been proven to be an important concept in the theory of alloys39404559, just as the Hume-Rothery rule47 reflected that the structures of specific intermetallic phases (i. e. electron compounds) are stabilized by specific e/a-ratio (see Supplementary Figure S2). Thus, the e/a-ratio is used as an effective parameter to investigate the CPGAMEC rule in this work, and the methods for its calculation are presented in the following parts. Besides, the total number of atoms per unit cluster formula (Z) is obtained on the basis of [cluster](glue atoms)x. While the key point is determination of the principal cluster entering into the cluster formula: [cluster](glue atoms)x. To resolve this problem, an effective method of central force field model2960 has been developed by combining interatomic force constants (IFCs)6162 and atomic close packing principle1422, while its general utility has been validated by different CMAs in numerous alloy systems272829. For a given alloy phase, the central force field model shows that those atoms with the largest IFCs act as the central atom of the cluster, those atoms with the smallest IFCs act as the glue atoms of the model, while those atoms with the IFCs locating between the max. IFCs and min. IFCs act as either the shell atoms or the glue atoms29. Then the cutoff shell of principal cluster is determined by the atomic close-packing principle1422, as shown in the inset map (a) of Supplementary Figure S3, the cutoff radius (r) of cluster shell corresponds to the maximum radial atomic density (ρ). Thereof, the principal cluster and the corresponding cluster formula can be obtained conveniently, and thus the total number of atoms per unit cluster formula (Z) can be achieved. Hence, the CPGAMEC rule described by the valence electrons’ number per unit cluster formula (N), can be obtained via formula (1). In this work, the IFCs are computed by performing first-principles calculations within the framework of density functional perturbation theory6162, and the computational details are provided in Supplementary Materials.

Confirmation of the CPGAMEC rule

To confirm the existence of the CPGAMEC rule, different kinds of CMAs including Zr-/Ti-based intermetallic compounds (ICs), Al-based quasicrystals (QCs) and bulk metallic glasses (BMGs) in several glass forming systems, have been investigated by formula (1) upon analysis of their cluster structure and electron concentration information. As the results shown in Fig. 2, the N values of these Zr-/Ti-based ICs, Al-based QCs and BMGs, are close to specific constants of eight-multiples and twelve-multiples, verifying the existence of CPGAMEC rule for CMAs. In what follows, we will present the results and discussion about confirmation of this electron counting rule in detail.

Figure 2

CPGAMEC rule for complex metallic alloys (including the ICs in Zr-/Ti-based systems, Al-based QCs24 and BMGs in several glass-forming systems26), reflected by the correlations between electron concentration (e/a) and total number of atoms per unit cluster formula (Z).

Confirmation of the CPGAMEC rule in ICs

The Zr-Cu/Al ICs and Ti-Al/Cu ICs with apparent cluster features have been studied first272953, and their crystallographic information are listed in Supplementary Table SII. Based on the central force field model2963646566, the cluster structure information of these Zr-Cu/Al and Ti-Al/Cu ICs are obtained. The results are collected in Fig. 3, Supplementary Figure S3 and Table SIII, where the first atom represents the central atom of the principal cluster. Therewith, the total number of atoms per unit cluster formula (Z) for these Zr-Cu/Al ICs and Ti-Al/Cu ICs can be easily obtained. And thus the valence electrons’ number per unit cluster formula (N) is computed via formula (1). Here the e/a-ratio3247 of these Zr-Cu/Al ICs and Ti-Al/Cu ICs is calculated by weight averaging the valence electrons contribution of all constituent elements, as expressed in the following form

Figure 3

Principal clusters of the Zr-Cu/Al ICs and their interatomic force constants (IFCs).

(a) Correlation between radial distances (r) and radial atomic density (ρ), the red vertical line depicts the cutoff radius of the principal cluster. (b) Atomic clusters present in the structures of Zr-Cu/Al ICs.

where C and (e/a) denotes the atomic fraction and the valence electrons contribution of the i-th element, respectively. The determination of (e/a) in those TM-containing systems, however, is complicated because of sp-d hybridization35474859, thus it is still a great challenge to completely obtain the e/a-ratio of these ICs. Nevertheless, the extra-nuclear electronic configuration of TMs in the periodic table is definite67. Besides, we notice that (e/a) assignment for the e/a-ratio calculation in Hume-Rothery rule is adopted as the usual valences of the constituent elements324768. Therefore, the e/a-ratio of these ICs is calculated by adopting the outermost electrons and the common valences44 as the (e/a) assignments, respectively (see Supplementary Table SIV). The detailed discussion on the valence electrons contribution of these constituent elements is provided in Supplementary Materials.

By assigning the outermost electrons as the valence electrons contribution, the e/a-ratio of these ICs are calculated via formula (2). The results indicate that the e/a-ratio of these Zr-Cu/Al ICs and Ti-Al/Cu ICs lies within the range from 1.2 to 1.8, and it varies with the i-element’s content (C) in a linear manner (see Supplementary Figure S4). Accordingly, the N values for these ICs are obtained via formula (1), and the results are presented in Table 1. It is found that in this case, the N value of these ICs is close to a specific constant of eight-multiples and twelve-multiples 24, as shown in Fig. 2. Meanwhile, in the case when the e/a-ratios of these ICs are calculated by assigning the common valences as the valence electrons contribution, the N value for these ICs is again found to approach a specific constant of eight-multiples and twelve-multiples 48 (see Supplementary Table SV). The results indicate that in both cases, the N values of these ICs are close to the specific constants of eight-multiples and twelve-multiples, as shown in Supplementary Figure S5, which confirms the existence of the CPGAMEC rule in alloy compounds, just as the extension of octet rule for covalent compounds and ionic compounds. For convenience, this electron counting scheme for CMAs is termed as the specific electrons cluster formula. For some ICs’ N values deviating from the specific constants of eight-multiples and twelve-multiples (see Supplementary Figure S5), it arises from the (e/a) assignment in the e/a-ratio calculation process.

Table 1

Cluster information for the Zr-Cu/Al and Ti-Cu/Al ICs, including the principal cluster with its coordination number (CN), cluster formula, total number of atoms per unit cluster formula (Z), electron concentration (e/a) and valence electrons’ number per unit cluster formula (N ), calculated via formula (1) with e/a obtained from formula (2), where the (e/a) is assigned as the outermost electrons of the i-element.

Alloy system	ICs’ composition	Principal cluster	Cluster formula	e/a	Z	Ne/u
Zr-Cu/Al	ZrCu5	CN12 Cu10Zr3 cluster	[Cu10Zr3](Cu5)	1.167	18	21
	Zr14Cu51	CN15 ZrCu15 cluster	[ZrCu15](Zr159/51)	1.215	19.12	23.24
	Zr3Cu8	CN12 Cu8Zr5 cluster	[Cu8Zr5](Cu16/3)	1.273	18.33	23.33
	Zr7Cu10	CN12 Zr5Cu8 cluster	[Zr5Cu8](Cu2Zr2)	1.412	17	24
	ZrCu	CN14 Zr7Cu8 cluster	[Zr7Cu8](Zr)	1.5	16	24
	Zr8Cu5	CN11 Zr6Cu6 cluster	[Zr6Cu6](Zr18/5)	1.615	15.6	25.2
	Zr2Cu	CN12 Zr9Cu4 cluster	[Zr9Cu4](Cu1/2)	1.667	13.5	22.5
	ZrAl3	CN12 ZrAl12 cluster	[ZrAl12](Zr3)	1.25	16	20
	ZrAl2	CN16 Zr5Al12 cluster	[Zr5Al12](Zr)	1.333	18	24
	Zr2Al3	CN13 Zr5Al9 cluster	[Zr5Al9](Zr)	1.4	15	21
	ZrAl	CN13 Zr7Al7 cluster	[Zr7Al7]	1.5	14	21
	Zr5Al4	CN11 Zr5Al7 cluster	[Zr5Al7](Zr15/4)	1.556	15.75	24.51
	Zr4Al3	CN14 Zr9Al6 cluster	[Zr9Al6](Al3/4)	1.571	15.75	24.74
	Zr3Al2	CN14 Zr9Al6 cluster	[Zr9Al6]	1.6	15	24
	P63/mmc-Zr5Al3	CN14 Zr9Al6 cluster	[Zr9Al6](Zr)	1.625	16	26
	I4/mcm-Zr5Al3	CN10 Al3Zr8 cluster	[Al3Zr8](Al9/5)	1.625	12.8	20.8
	Zr2Al	CN11 Zr7Al5 cluster	[Zr7Al5](Zr3)	1.667	15	25
	Zr3Al	CN12 Zr9Al4 cluster	[Zr9Al4](Zr3)	1.75	16	28
Ti-Al/Cu	TiAl3	CN12 TiAl12 cluster	[TiAl12](Ti3)	1.25	16	20
	Ti2Al5	CN12 Ti5Al8 cluster	[Ti5Al8](Al9/2)	1.286	17.5	22.5
	TiAl2	CN12 Ti3Al10 cluster	[Ti3Al10](Ti2)	1.333	15	20
	Ti3Al5	CN12 Ti3Al10 cluster	[Ti3Al10](Ti3)	1.375	16	22
	TiAl	CN14 Ti7Al8 cluster	[Ti7Al8](Ti)	1.5	16	24
	Ti3Al	CN12 AlTi12 cluster	[AlTi12](Al3)	1.75	16	28
	TiCu3	CN12 TiCu12 cluster	[TiCu12](Ti3)	1.25	16	20
	TiCu2	CN14 Ti5Cu10 cluster	[Ti5Cu10]	1.333	15	20
	Ti2Cu3	CN14 Ti6Cu9 cluster	[Ti6Cu9]	1.4	15	21
	TiCu	CN14 Ti9Cu6 cluster	[Ti9Cu6](Cu3)	1.5	18	27
	Ti2Cu	CN12 Ti9Cu4 cluster	[Ti9Cu4](Cu1/2)	1.667	13.5	22.5
	Ti3Cu	CN12 Ti9Cu4 cluster	[Ti9Cu4](Ti3)	1.75	16	28

Confirmation of the CPGAMEC rule in QCs and BMGs

Based on the cluster-resonance model2069, it has been found that the valence electrons’ number per unit cluster formula (N) for typical QCs and BMGs2426, is also close to the specific constant of eight-multiples and twelve-multiples 24, as shown in Fig. 4. This coincidence implies that QCs and BMGs follow the CPGAMEC rule as well. As for those QCs and BMGs, whose structures are stabilized by the Fermi sphere-Brillouin zone interaction707172, the cluster-resonance model provides another applicable method to calculate their e/a-ratio, as expressed in the following form

Figure 4

Correlation between electron concentration (e/a) and valence electrons’ number per unit cluster formula (N) for typical Al-based QCs and BMGs in several glass-forming systems2829, reflecting the CPGAMEC rule of specific electrons cluster formula for CMAs and their N values’ deviation from the specific constant.

where r1 and ρ each represents the principal cluster radius and the atomic density1550. Accordingly, the e/a-ratio for some Al-based QCs and typical BMGs is calculated. While the Z value is acquired from the cluster formula of these QCs and BMGs, which has been successfully used to explain their experimental compositions (see Supplementary Table SVI and SVII). Hence, the N values for these Al-based QCs and BMGs2426 are obtained, and the results are presented in Fig. 4 and Supplementary Figure S6. As reflected that the N values for these QCs and BMGs are close to the specific constant of eight-multiples and twelve-multiples 24, confirming the existence of the CPGAMEC rule in QCs and BMGs. Furthermore, the N values’ standard deviation from constant 24 for BMGs is smaller than that for Al-based QCs (see insets in Fig. 4). This distinction is attributed to the structural differences of these CMAs, as will be discussed in the following part.

All of the results reveal the fact that the N values of Zr-/Ti-based ICs, Al-based QCs and typical BMGs are close to the specific constants of eight-multiples and twelve-multiples (see Fig. 2), which confirms the existence of CPGAMEC rule for CMAs. Moreover, the CPGAMEC rule signifies that CMAs’ cluster formula is superior to the customary stoichiometric formula54. Meanwhile, the eight-multiples’ characteristic of the N values for CMAs suggests that CMAs follow the extension of octet rule as well. Besides, the cluster formula of CMAs may differ from the molecular formula of covalent compounds and ionic compounds only by the linkage between the primary units44, which retains the basic features of interatomic interaction instead of inter-molecular forces. As for some CMAs deviating from the CPGAMEC rule, it is attributed to the e/a-ratios involving in the N calculated process, this can be readily understood since the e/a-ratio is only an effective one because of TMs’ hybridized effects1520. The situation is similar to the exception of octet rule, where special terminology like hyper-/hypo-valence has been developed to describe those chemical species that do not follow the octet rule4657. Furthermore, the CPGAMEC rule is fairly well followed by those CMAs with apparent cluster features, just as the octet rule is strictly followed by element atoms in the period two, while element atoms in other periods may obey this rule but not necessarily in all molecules111257. Especially, the electron counting rules followed by a large number of condensed matters (including CMAs, covalent compounds and ionic compounds), imply that the valence electrons’ number per unit molecular formula is close to specific constants, which are firmly related with materials’ atomic structure. Besides, the existence of the CPGAMEC rule in ICs, QCs and BMGs further reveals the close relationship between structure and properties of these CMAs15297374.

During the past decades, some other electron counting schemes have been developed to explore the interrelationship between the structure and properties of materials75767778798081828384858687888990919293949596979899. For example, the skeletal electron pair (SEP) rule76777879 used to describe the cluster structural features of complex polynuclear molecules with varied skeletal atoms8586, the topology electron counting (TEC)8082 theory used to estimate the electron counts of polyhedral metal clusters with varying nuclearity79818487. Both SEP and TEC theories assume that each vertex atom contributes three orbitals to the cluster bonding757788. Nevertheless, this assumption is true for the main-group elements but not necessarily true for the transition metals87. Besides, the hypervalent electron counting scheme and the Zintl-Klemm electron counting rule9293, provide a route for understanding the bonding in ICs containing heavy main group elements. While the 14 electron rule9495 indicates that the total valence electrons’ number per transition metal atom in Nowotny chimney ladder phases is 14. Compared with these electron counting schemes75767778798081828384858687888990919293949596979899, the CPGAMEC rule pay much attention on those CMAs with apparent cluster structural features, like some ICs, QCs and BMGs. All of these electron counting rules made significant progress in our better understanding of the close connections among the valence electrons number, the cluster stereochemistry and the atomic cluster geometries.

Application of the CPGAMEC rule

The CPGAMEC rule can be applied to guide the composition design of CMAs with desired properties. Based on the cluster-plus-glue-atom model and the CPGAMEC rule, it is clear that this electron counting scheme endows the cluster formula of CMAs with apparent molecular features, just like the molecular formula of common covalent and ionic compounds. In this context, the cluster formula corresponding to cluster-plus-glue-atom model brings with itself the basic information on CMAs’ composition, atomic structure and electronic unit. Accordingly, the composition-structure-property correlations of CMAs can be investigated further. In the present work, we take BMGs in the ZrCu-based system as an example to explain the correlations reflected by the CPGAMEC rule, and to illustrate its practical applications in CMAs’ composition designing process.

As shown in Fig. 5, based on the Cu8Zr5 icosahedral cluster derived from Zr3Cu8 ICs29, the BMGs compositions can be designed via the known cluster formula of [cluster](glue atoms)1or3 for ideal glassy formers151921. Then the possible cluster formulas are denoted as [Cu8Zr5]Cu, [Cu8Zr5]Zr, [Cu8Zr5]Zr2Cu, [Cu8Zr5]Cu3, [Cu8Zr5]Cu2Zr and [Cu8Zr5]Zr3. Under the theoretical guidance of CPGAMEC rule, it has been verified that among these cluster formulas, the specific electrons cluster formula [Cu8Zr5]Cu = Cu64.3Zr35.7 with its N = 23.7 close to the specific constant of eight-multiple and twelve multiple 24, is in good agreement with the experimentally synthesized Cu64Zr36 BMGs. Likewise, the specific electrons cluster formula [Zr7Cu8]Zr and [Ti9Cu6]Cu3, with N = 24.2 and 23.6 close to ideal value 24, can be used to explain the composition of Cu50Zr50 and Cu50Ti50 BMGs, where the Zr7Cu8 and Ti9Cu6 principal clusters are derived from ZrCu ICs and TiCu ICs, respectively2965. By combination with the micro-alloying mechanism, multi-components BMGs’ compositions can be achieved via element substitution method15. For instance, when one of the shell atoms Zr in binary cluster formula [Cu8Zr5]Cu is substituted by one Ti atom with comparable size21, the experimental composition for ternary Cu64Zr28.5Ti7.5 BMGs can be designed via the specific electrons cluster formula [Cu8Zr4Ti]Cu = Cu64.3Zr28.6Ti7.1, with N = 23.4 close to the specific constant of eight-multiple and twelve-multiple 24. Likewise, the composition of quaternary Ti40Cu46.95Zr10Sn3.05 BMGs can be designed on the basis of binary specific electrons cluster formula: [Ti9Cu6]Cu3 via elements’ substitution method, and its resultant cluster formula is [TiCu5.45Sn0.55Ti6.2Zr1.8]Cu3 with N = 24.5 close to the specific constant of eight-multiple and twelve-multiple 24. Relevant experimental studies indicate this quaternary BMGs have good glass forming ability and high strength65. Therefore, the CPGAMEC rule provides an innovative theoretical guidance to direct the design of CMAs with desired properties.

Figure 5

Illustration for the application of the CPGAMEC rule: Specific electrons cluster formula of CuZr-based bulk metallic glasses, and the principal clusters derived from relevant eutectic phases in Cu-Zr alloy system, reflected in ternary Cu-Zr-Ti phase diagram.

(In this and subsequent figures, the orange spheres and olive green spheres denote the copper atoms and zirconium atoms, respectively).

Interpretation of the CPGAMEC rule

To make further progress, a possible interpretation for understanding the CPGAMEC rule has been presented on the basis of CMAs’ cluster structural characteristics. As mentioned above, the N values for these CMAs are close to the specific constants of twelve-multiples. Meanwhile, the local atomic structures of CMAs are characterized by numerous polyhedral clusters4131415, and most of these clusters are the convex polyhedron with coordination number (CN) of twelve100101102103. In particular, the short-range-ordering features induced by the CN12 icosahedral clusters in the structure of BMGs and IQCs, have been verified by many theoretical and experimental investigations104105106107108109110111112113. On this ground, we assume the specific electrons cluster formula as an entire CN12 convex polyhedron, while this polyhedron contains the basic information on composition, structure and electrons of CMAs. According to the charge distribution of Gauss’s law114115116 and under the above assumption, it is readily to understand that the N values for these CMAs are close to the specific constants of twelve-multiples. Meanwhile, the principal cluster in the cluster formula represents CMAs’ main structural features, while the glue atoms is only a small part and can be averaged into the cluster part1520. For instance, the CN12 Cu8Zr5 icosahedral cluster represents the primary structural features of Zr3Cu8 phase1529. Figure 6 presents the atomic cluster structures and the charge density distribution of Zr3Cu8 phase, it shows that the electrons mainly distribute on the twelve vertex of Cu8Zr5 clusters, which further demonstrates the rationality of this interpretation for the CPGAMEC rule. Our understanding on the CPGAMEC rule further implies that the electron counting schemes of materials are closely related to their microscopic atomic structures.

Figure 6

General view of the interpretation for the CPGAMEC rule, based on the charge distribution of Gauss’s law and the cluster structural features of CMAs: (a) Atomic clusters existed in the structure of Zr3Cu8 ICs, (b) Charge density distribution of Zr3Cu8 ICs, (c) Charge density of the CN12 Cu8Zr5 icosahedral cluster presented in the structure of Zr3Cu8 ICs (the isosurface is 0.02 e/Bohr3), (d) Color-contour maps on the Cu-Zr-Cu triangle plane of CN12 Cu8Zr5 cluster in Zr3Cu8 ICs.

From the viewpoint of CMAs’ atomic cluster structures, the above interpretation for the specific electrons cluster formula provides an underlying mechanism behind the CPGAMEC rule. Accordingly, the N values’ deviation from the specific constants of twelve-multiples for some CMAs (see the inset in Fig. 2), can be understood as distortions of the CN12 convex polyhedron474. Furthermore, the fact that the N values for some covalent compounds and ionic compounds are specific constants of eight-multiples, is regarded as an extension of the octet rule. This can be understood as the valence electrons distribute in successive shells at the corners of a cube3857. Similarly, the fact that the N values for these CMAs are specific constants of twelve-multiples, can be understood as the valence electrons distribute in successive shells at the vertexes of the CN12 convex polyhedron. Therefore, the CPGAMEC rule of the specific electrons cluster formula, provides an aggregate picture with intriguing electronic rule and structural features of CMAs. It is worthwhile to mention that there are other underlying mechanisms behind this CPGAMEC rule, and our studies along this direction are still underway.

In conclusion, an electron counting rule for CMAs (i. e. CPGAMEC rule) has been presented in this work, by analogy with the extension of octet rule for common covalent compounds and ionic compounds. It has been found that the valence electrons’ number per unit cluster formula (N) for different kinds of CMAs, are close to specific constants of eight-multiples and twelve-multiples, as exemplified by Zr-/Ti-based ICs, Al-based QCs and BMGs in several glass-forming systems. Thus we termed it as CMAs’ specific electrons cluster formula. It has been demonstrated that the CPGAMEC rule is a useful guidance to direct the design of CMAs with desired properties. Meanwhile, the cluster formula can be regarded as not only CMAs’ composition unit and structural unit, but also their electronic unit and molecular formula. Furthermore, the CPGAMEC rule for CMAs imply that the electron counting schemes of materials are closely related to their atomic structure features. The present work provides an aggregate picture with intriguing electronic rule and structural features of CMAs, and hence offers a significant theoretical guidance for researchers to further investigate the composition-structure-properties correlations of CMAs.

Additional Information

How to cite this article: Du, J. et al. Hidden electronic rule in the “cluster-plus-glue-atom” model. Sci. Rep. 6, 33672; doi: 10.1038/srep33672 (2016).