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Inherent structure length in metallic glasses: simplicity behind complexity

One of the central themes in materials science is the structure-property relationship. In conventional crystalline metals, their mechanical behaviour is often dictated by well-defined structural defects such as dislocations, impurities, and twins. However, the structure-property relationship in amorphous alloys is far from being understood, due to great difficulties in characterizing and describing the disordered atomic-level structure. Herein, we report a universal, yet simple, correlation between the macroscopic mechanical properties (i.e., yield strength and shear modulus) and a unique characteristic structural length in metallic glasses (MGs). Our analysis indicates that this characteristic length can incorporate effects of both the inter-atomic distance and valence electron density in MGs, and result in the observed universal correlation. The current findings shed lights on the basic understanding of mechanical properties of MGs from their disordered atomic structures.

Modulus and strength of metallic materials are key properties for their engineering applications 1,2 , and the knowledge of structural information is prerequisite for understanding the deformation behaviour 3,4 . In crystalline materials with long-range periodic atomic packing, the relationship between their structure and mechanical properties has been theoretically developed and relatively understood [5][6][7][8] . However, in glassy materials, especially metallic glasses (MGs), such a structure-property correlation is still missing due to the difficulties in decoding the atomic packing in these disordered solids. Several efforts have been attempted to reveal a scaling behaviour between the glass transition temperature and strength [9][10][11][12][13] , and the strength has also been found to be proportional to the elastic modulus, corresponding to a nearly-constant yield strain [11][12][13] . However, a direct correlation between the characteristic length in the "structure" and these mechanical properties of MGs has not yet been established.
For amorphous solids, the total structure factor, S (q), and the resolved pair distribution function (PDF), g(r), are particularly useful in characterizing the non-crystalline structure (q is the wave vector in reciprocal space and r is the nearest neighbour distance between two atoms). From these statistical data, the atomic distribution within a given volume can be well described [14][15][16][17][18] . In principle, S (q) and g (r) was related by Fourier transformation 19 , and the first peak position of S (q) plots, usually denoted as q 1 , specifies the wavelength (λ = 2π /q 1 ) of g (r) at the medium-range length scale or above 20,21 . Particularly, q 1 was found to be a key structural quantity to reflect critical characteristics in atomic network of amorphous matters, such as intermolecular bonding in amorphous ice, interconnected layers of some borates, and the weak chain in chalcogenide glasses [22][23][24] . It was found that different metallic glasses may have different q 1 values and thereby different mechanical properties. For example, Zr-based BMGs generally have a smaller q 1 and lower yield strength, as compared with Fe-based BMGs (a typical example is shown in Fig. 1). To find out whether there exists a relationship between q 1 and mechanical properties of MGs, most of available data in literature regarding the q 1 values and elastic properties of MGs was complied in Table 1   . By analyzing the structural length and mechanical properties of these MG alloy systems, we discovered a universal relationship between the elastic properties (i.e., the yield strength and moduli) and the wavelength λ (i.e., 2π /q 1 ). Figure 2a indicates a strong dependence of the yield strength on the specific length scale λ within the experimental error of measurements, and the fitted scaling relationship is: Where σ y is the yield strength (in unit of GPa) and λ is the wavelength (in unit of Å). Note that the above relationship is generally applicable to different types of MGs, independent of their chemical compositions. With increasing λ , the yield strength of MGs rapidly decreases. For example, Co-and Fe-based MGs containing metalloid constituents have relatively small λ , but high yield strength, whereas Ca-and RE-(rare earth) based MGs exhibit large λ but low yield strength. Elastic constant, especially the shear modulus, has been shown to be the dominant parameter controlling atomic shear and, consequently, the yielding of metallic glasses [10][11][12][13] . Using the current data, an intriguing correlation between the shear modulus, G, and the specific length scale λ is also established as shown in Fig. 2b:   where G is the shear modulus (in GPa). Similar to the yield strength, the shear modulus is also strongly dependent upon the specific length scale, i.e., the wavelength λ , with a similar power index. This observation further suggests that yield strength and shear modulus of MGs, in fact, may have the same atomic structural origins [11][12][13] . Combine Eqs. (1 and 2), we have the shear strain limit of MGs, 0 0266 y 2G y ε = = .
σ , which is in excellent agreement with the value of 0.0267 obtained by Johnson and Samwer 11 , further validates the above correlations. Noted that systematic errors associated with the experimental measurements of the yield strength and the diffraction patterns collected from different groups might have contributed to the data scattering shown in Fig. 2.
As shown above, the elastic properties (elastic moduli and yield strength) can be directly assessed by the simple parameter λ , in despite of the complex chemistry. In structural materials, the forces that affect elastic stiffness are predominantly electrodynamic, that is, a combination of electrostatics and dynamic exchange forces associated with various quasi-particles, including phonons, photons, electrons, and protons 93,94 . For metallic materials, the force usually comes from electrostatic interactions resulted from the distribution of electron charge, and the elastic modulus is determined primarily by atomic spacing and valence electron density 93,94 . It is, therefore, pertinent to establish a correlation between the λ value and the atomic spacing and valence charge in BMGs.
It is necessary to point out that the physical meaning of q 1 or λ (2π /q 1 ) has been extensively studied in amorphous matters, for example, inorganic glasses bonded by directionally covalent bonding. In such materials, it was proposed that 2π /q 1 could be interpreted as spatial repeat distance of structural unit 15 . For MGs, which exhibit long-range disordered packing with atoms bonded by nondirectional metallic bonding, the physical significance of q 1 has also been discussed in recent years 12,14,19 . As mentioned before, q 1 identifies the wavelength (λ = 2π /q 1 ) of the correlation function g(r) above the first nearest neighbour (i.e., the portion at the high r region of the PDF curves). In terms of atomic packing, this wavelength actually quantifies the distance between the two adjacent atomic neighbors and therefore represents the average inter-atomic distance in MGs 14,95 .
Based on the spherical-periodic theory, q 1 (i.e., 2π λ ) is essentially the Fermi-sphere diameter 2k F for MGs, and λ is the Friedel wavelength [96][97][98] . The spherical Friedel oscillations in the effective pair potential due to the electronic interaction are similar to those in the structural PDF curves, namely, the electronic wave vector 2k F equals to the structural wave vector q 1 (2k q [97][98][99] . In other words, when the Fermi surfaces of the electrons coincide with the quasi-Brillouin boundaries, a pseudo gap is expected occur in the energy bands for MGs and a corresponding minimal of density of states at the Fermi level, which makes MGs in a relative stable state 98,100 . Therefore, q 1 (or λ ) can be treated as an indicator of the electronic interaction in MGs. As a matter of fact, the universal binding-energy relation with interatomic separation has been found in a wide range of crystalline materials 101,102 , which was believed to be resulted from the scaling between electron density parameter and interatomic distances 101,102 . However, a universal relation between elastic modulus which is the second derivative of binding energy and interatomic distances was not observed due to crystallographic lattice anisotropy. In MGs, as discussed earlier, q 1 (or λ ) can represent relative electron density, indicating the proposed universal binding-energy relation also holds well. Due to the atomic isotropic disordering structure of MGs, the universal correlation between the elastic modulus and interatomic distance can also be derived.
It should be noted that there is no distinct relationship between the yield strength and the nearest neighbour distance (r 1 ), as shown in Fig. 3a. The data points are much more scattered, as compared with those in Fig. 2a. For example, even at the same r 1 value of ~2.6 angstrom, a large scattering in the yield strength from 1 to 5 GPa was observed. From the atomic packing point of view, r 1 represents the first nearest neighbour which can be more easily affected by local atomic environment. Although λ is at the scale of 2-3 angstrom, nevertheless, it actually specifies the wavelength (λ = 2π /q 1 ) of g(r) at the large r range (as shown in Fig. 3b) 20,21 and reflects the average atomic distance of MGs. Therefore, it can be seen that the average packing characteristic is more determinative to the global elastic properties, rather than the nearest neighbour distance or the localized atomic packing features which may be more contributable to the plastic deformation behavior.
It has been long recognized that theoretical strength and modulus of a material depend on their bonding nature, and specifically the bulk modulus is the second derivative of the binding energy with respect to the interatomic distance 1,3,4 . For instance, molecular crystals are bonded by dipole-dipole forces (similar to that in solid inert gases) and their bulk moduli scale with inter-particle distances raised to the -3 power 94 . For metallic materials whose atomic bonding is mainly from electrostatic interactions, on the other hand, both valence charge and atomic spacing can contribute to the strength and modulus, and the Keyes parameter K= e 2 /r 4 (e-electron charge, r-atomic distance) is the key scaling parameter 94 . Limited metallic materials with the same electronic structure show the atomic spacing to be the only decisive parameter for elastic modulus, for example, alkali metals, some ionic crystals, and oxide crystals. In these crystals, the bulk modulus was observed to scale with interatomic spacing raised to the -4 power 94 . However, when the d-shell electrons become effective in atomic bonding (e.g., in transition metals) or the bonding is more coherent (such as in compounds containing metalloids), the scaling can be complex 94 .
Scientific RepoRts | 5:12137 | DOi: 10.1038/srep12137 In contrast to these crystals, MGs can be envisioned as materials which are randomly densely-packed with isotropic elastic properties, and the elastic constants can be related to each other through equations: E/G= 2(1 + ν ) and B/G = 2(1 + ν )/3(1-2ν ), where E, B, ν are Young's modulus, bulk modulus, and Poisson's ratio, respectively. Consequently, elastic constants of any MG can be derived as a function of λ using the universal correlation (Eq. 2) combined with the isotropic relationship. In other words, λ can be treated as an inherent length parameter conveniently incorporating the influence of both the average atomic distance and effective valence electron charge. It is like the "structure fingerprint" providing the information of elastic properties of MGs.
We want to particularly point out that the universal correlation in MGs (Eqs. 1 and 2) is, in fact, even simpler than that in the elemental metals. Figure 4 is the dependence of shear modulus and bulk modulus on the atomic radii for elemental metals [the original data were taken from refs 103,104]. In contrast to Fig. 2, a more scattered scaling relationship is observed even in metals with the same crystalline structure (for example, in FCC-Cu, Ni, Al, Ag, Au, and Pt, indicated by the asterisks). It is noted that the universal scaling relationship in MGs (Eqs. 1 and 2) is based on the fact that λ actually represents the effect of both atomic spacing and valence electron density. In addition, the correlation between the electronic vector 2k F and the structural vector q 1 , namely, 2k q F 1 2 = = π λ must be valid. This latter correlation is, however, not necessarily true in crystalline metals due to their lattice anisotropy at the atomic level. The atomic bonding directionality as well as interatomic distance in various directions leads to the various slip systems for deformation. A non-spherical Brillouin zone subsequently affects the yield strength and elastic modulus for crystalline metals. Metallic glasses, on the other hand, can be taken as metallic materials having isotropic electron cloud. Furthermore, their atomic structures are lack of long-range periodicity and virtually isotropic (i.e., a better spherical symmetry) even down to the atomic level, leading to the universal scaling relationship observed in Fig. 2. It should be noted that λ is obtained from the statistic structural information of MGs, and there are still many uncertainties and challenges in decoding localized/atomic structure of MGs 105,106 . Prolific motifs about structure-properties relationship still merit in-depth investigation 107 . For example, atomic density fluctuation and structural heterogeneity frozen in during rapid quenching is expected to result in the fluctuation of inter-atomic distance and valence electron density around the average value [108][109][110] , also contributes to the error bar of the universal relationship in Eqs. 1 and 2.
In summary, there exists in metallic glasses a unique characteristic length λ (or q 1 ) that exhibits a universal, yet simple, correlation with the mechanical properties such as the modulus and yield strength. The universal relationship appears to be resulted from the fact that metallic glasses are essentially structurally/chemically isotropic as compared to crystalline metals which have certain structural and electronic directionality. Our result demonstrates that, in nature, "more complex" amorphous alloys actually can have a simpler mechanical behavior.