Molecular dynamics (MD) simulations were performed with the LAMMPS code (Plimpton, 1995) for a well-studied Cu₅₀Zr₅₀ metallic glass. The glass sample containing 19,652 atoms was obtained by fast quenching the equilibrated liquid from 2000 to 0 K with a cooling rate of 10¹⁰ K/s. The interatomic interactions are described via a many-body Finnis–Sinclair–type embedded atom potential proposed by Mendelev et al. (2009). Period boundary conditions (PBCs) were used on the simulation box with 3D dimensions of ∼ 70 , × , 70 , × , 70 ų. The temperature was controlled through the Nosé-Hoover thermostat (Nosé, 1984) while the external pressure in each direction is fully relaxed via the Parrinello–Rahman barostat (Parrinello and Rahman, 1981). The MD time step was set as 0.002 ps to numerically integrate Newton’s equation of motion.

Before we extract the possible activation barriers in the complex 3N (N is the total number of atoms) dimensional potential energy landscape (PEL), the quenched glass sample was first fully relaxed to a local potential energy minimum via the conjugate gradient algorithm. Then, the widely adopted activation-relaxation technique nouveau (ARTn) (Barkema and Mousseau, 1996; Malek and Mousseau, 2000) was applied to sample possible local hopping pathways of the structural excitations from an initial energy sub-basin to a new neighboring minimum. The specific workflow of ARTn is described in the following sections. First, a small random perturbation was imposed on a central atom and its neighbors. In this study, the magnitude of the perturbation displacement was fixed as 0.1 Å, while the perturbation direction was chosen randomly. Second, the system was pulled toward the saddle point (with a high energy level) along the direction of the weakest Hessian matrix, following the Lanczos algorithm (Cancès et al., 2009). Finally, after convergence to the connected saddle state, the glass sample was eventually allowed to relax to a nearby sub-basin. In this connection, the activation energy for a specific structural excitation is defined as the energy difference between the saddle and the initial energy minimum states. ARTn searches were applied to all of the atoms, each one as the central triggered atom. After removing the failed tries, we used all 20 successful activation events for each atom. For statistical purposes, the average activation energy of the explored 20 events was then used as the single-particle activation energy for each atom.

To quantify the local short-time thermodynamic property of metallic glass, we define the intensity of the boson peak at a single-particle level, which has been documented in Yang et al. (2022). First of all, the vibrational density of states (VDOS) of an inherent structure is obtained by direct diagonalizing its Hessian matrix. The single-particle VDOS for the ith atom is then defined as the sum contribution of ith atom over all vibrational modes. It is formulated as (Togo and Tanaka, 2015) g i ω = 1 3 N ∑ j δ ω − ω j e j i 2 , (1) where g i ω denotes the single-particle VDOS for the ith atom. ω _j and e _j are the normal model frequency and the polarization vector of the jth vibrational mode, respectively. The phenomenon of the boson peak, that is, the excess vibrational modes over Debye squared law is then exhibited by the reduced VDOS (VDOS values divided by ω ²). Thereafter, the peak value of the reduced VDOS for the ith atom, that is, I BP = max g i ω / ω 2 , is used to quantify the single-particle intensity of the boson peak.

To establish the correlation between the local structure and dynamics of metallic glasses, a measure of the structural fluctuation that quantitatively predicts the long-time structural excitation and the short-time vibrational anomaly at a single-particle level is necessary. To settle this issue, we propose a time-relevant structural predictor, which is based on the diversity of the Voronoi motifs that an atom experiences during thermodynamic vibration and possible thermal activation.

First of all, the quenched glass sample is relaxed at 700 K (just below the glass transition temperature) for 1 ns; such time for thermal fluctuation is right below the α relaxation time (Yang et al., 2021), making sure that there is only a secondary β-relaxation process (Yu et al., 2014; Yu et al., 2017), and the structural transformations take place only between adjacent sub-basins (Fan et al., 2014; Fan et al., 2015). Then, the evolution of Voronoi polyhedra around each centered atom is recorded. Figure 1 shows the fluctuation history of possible short-range structures centered at two representative atoms. It is intuitively seen that the difference in atomic packing symmetry would lead to remarkably different thermal responses during relaxation at a thermal bath right below the glass transition temperature. Furthermore, it is evident that the local structural diversity is nonhomogeneous in metallic glasses since atoms are capable of retaining their initial short-range structure as shown in Figure 1A. Also, there are atoms experiencing completely distinct Voronoi motifs, causing an extremely high level of structural diversity, as evidenced by Figure 1B.

Next, we perform statistics on the distribution of Voronoi polyhedra around each atom. Thus, the probability that each Voronoi motif appears in the fluctuation history of the ith atom is recorded and calculated as P i j t , where j denotes the jth existing Voronoi motif and t is the time for the thermal bath. By using the concept of Shannon information entropy (Shannon, 1948), the multiplicity of local structural variation is then quantified by the value of atomic Shannon entropy that is formulated as S i = − ∑ j P i j t ln P i j t . (2)

Because the distribution is, in principle, time-dependent, it is necessary to monitor the temporal evolution of S _i to understand its physical meaning and extract a time-invariant entity. The ensemble average time-relevant Shannon entropy is shown in Figure 1C as a function of time. The curve of S _i versus relaxation time has a tendency to level off. It indicates that atomic Shannon entropy displays a growth trend at very early stages, which is followed by the steady state for a longer time. The critical time appears at a timescale less than 0.1 ns. This is evidence for the saturation of the distribution of the local coordinated polyhedral motif. Thus, the long-time thermal fluctuation with an annealing time of 1 ns is sufficiently long for the local glass state to experience all the possible configurational space. In Figure 1D, the atomic Shannon entropy as a function of annealing time is further displayed for different atoms with distinct initial Voronoi motifs. It shows that all types of Voronoi clusters at the initial state become saturated with the statistical S _i at a relaxation time less than 0.1 ns. The full icosahedra (with the Voronoi index as ⟨0, 0, 12, 0⟩) has the lowest value of entropy, while the “geometrically unfavored motifs” (GUMs) exhibits the highest level of structural diversity. This is in line with the existing paradigm (Ma, 2015) that atomic packing with high symmetry such as full icosahedra (⟨0, 0, 12, 0⟩) constitutes the most inflexible local environments, while those GUMs contribute preferentially to the soft sites, acting as liquid-like regions (Ding et al., 2014). It should be noted that the critical time for the saturation of Shannon entropy is roughly estimated in the present work. The main scope of the critical time is to confirm the time-invariance of our proposed Shannon entropy. It is interesting to figure out the definition of such critical times. Consideration of the quantitative one-to-one correlation between the critical time and Shannon entropy or other dynamical properties is meaningful. However, it is beyond the scope of this article and will probably be discussed in future research studies.

Figure 2 displays the probability distribution of S _i for the Cu atoms (red line), Zr atoms (blue line), and all atoms (black line), respectively. It is intuitive that S _i is distributed over a very broad range, indicating a strong structural heterogeneity (Zhang et al., 2015) as commonly demonstrated by other structural predictors. This also resembles the frequently reported nonhomogeneous distribution of the short- or long-time dynamic properties in metallic glasses (Tanaka et al., 2010; Tong and Tanaka, 2018; Wang et al., 2018). Furthermore, it shows that the peak position of the Cu atoms shifts to the left when compared with that of Zr atoms. This is ascribed to the higher coordination number of Zr atoms than that of Cu atoms, resulting in a greater diversity in the local atomic packing of Zr atoms.

Having characterized the level of structural diversity by S _i , we can now establish the long-sought structure-property relationship in metallic glasses. First of all, we focus on the correlation between ergodic Shannon entropy and the activation energy of local structure excitation, the latter of which is the energy barrier of the long-time transition from one local energy minimum to a neighboring one which is usually used as a universal indicator of the difficulty of structural excitations under external mechanical or thermal loadings. To address this issue, the particle-level activation energy ΔQ is calculated for each triggered atom. The statistical correlation between S _i and ΔQ is indeed seen in Figure 3 for Cu atoms (Figure 3A) and Zr atoms (Figure 3B), respectively. It shows a clear trend that decreasing values of ΔQ corresponds to growth in the magnitude of S _i . It further suggests that particles centered around motifs with a higher degree of structural diversity tend to have multiple basins with a lower activation barrier. To demonstrate this relation more explicitly, all Cu atoms (or Zr atoms) are sorted in terms of the value of S _i into groups, each containing 50 atoms. Then, the averaged ΔQ for each group is calculated. As shown in Figure 3C, an inverse scaling law between S _i and ΔQ is even more remarkable after such a numerical coarse-graining procedure. This intimate correlation applies to both Cu and Zr atoms. Furthermore, it is evident that Zr atoms are linked with deeper valleys in PEL, which correspond to higher activation barriers compared with Cu atoms. This is because Zr atoms are heavier than Cu atoms, causing the latter more easily to be activated under external stimuli.

The next task at hand is to investigate how the structural diversity parameter S _i correlates with the short-time vibrational feature. For this purpose, the boson peak vibrational anomaly which is one of the most mysterious phenomena of metallic glasses and other disordered materials is utilized to benchmark the relation between S _i and thermodynamics. Here, the boson peak is the measure of excess vibrational density of states with respect to the Debye-squared law in 3D. It can be represented by the peak value of the reduced VDOS, that is, VDOS divided by ω ². It has been extensively discussed that there is an intimate correlation between the activation energy and intensity of the boson peak. Thus, it is expected that there should be a strong correlation between S _i and I _BP. Figure 4 shows the semi-logarithmic plot of the single-particle boson peak intensity as a function of ergodic Shannon entropy for Cu (Figure 4A) and Zr atoms (Figure 4B), respectively. It shows that atoms with different values of S _i display different intensities of boson peaks. Mostly, atoms with a higher level of structural diversity will make more contributions to the boson peak. As shown in Figure 4C, this correlation is more apparent after numerical coarse graining with a proper bin size. Here, each bin contains 0.25% of all atoms that have been sorted based on the increasing value of S _i . It points out that atoms exhibiting a high value of ergodic Shannon entropy would indeed have an extraordinary vibrational anomaly, and large S _i does necessarily mean large intensity of boson peaks. This is indicative of the exponential susceptibility of the boson peak intensity to the diversity of structural environments and complexity of the PEL since S _i is proved linearly correlated with the PEL’s topology as evidenced by Figure 3.

The strong correlation among S _i − ΔQ − I _BP can also be verified in terms of their spatial nature. The contour maps of S _i , ΔQ, and the I _BP field are shown in Figure 5. It is noticeable that all of these parameters are distributed in nonhomogeneous manners with atoms having high/low values of S _i (or ΔQ or I _BP) tend to aggregate spatially into clusters, which cover regions spanning nearly nanometers in diameter or ∼ 5 Å in radius, which is corresponding to the second valley of the radial distribution function (RDF) of Cu₅₀Zr₅₀ MG. This is a direct piece of evidence that local structural diversity, local structural excitation, and vibrational anomalies share the same physical origin embedded in both short- and medium-range order of glass. This characteristic length scale is also commensurate with the critical size of soft regions that are responsible for triggering plastic shear transformations under thermal and/or mechanical stimuli. This is expected since the robust correlation between ΔQ and shear transformation has been well reported in the literature Kosiba et al. (2020), Han et al. (2020), and Wei et al. (2019). In this connection, local regions with a high level of S _i can be identified as potential glassy defects. Furthermore, it is evident in Figure 5 that high-S _i clusters are in striking overlap with the low-barrier and high-I _BP regions. It underpins the concept that ergodic Shannon entropy that conceives the information about the flexibility of the local structural packing can quantitatively correlate with both short-time and long-time dynamic features at a particle-level resolution.