Fractal atomic-level percolation in metallic glasses

Freeze a liquid fast enough, and it becomes a glass, a material that is structurally similar to the liquid but incapable of flow. This concept, albeit not well understood (1, 2), is so ubiquitous that it holds even for metals (3). Vitrified metals, or metallic glasses, are a class of disordered materials with nondirectional bonding and possess a suite of lucrative mechanical properties, such as high elastic limit and strength (4). Unlike most crystalline metals and alloys, metallic glasses earn their name from a lack of long-range atomic order and the absence of typical defects, such as dislocations, rendering their microstructure challenging to conceptualize and model. Some studies suggest the existence of short-range order, for which solute-centered clusters serve as the building blocks, and medium-range order characterized by cluster packing (5–7). These short- and medium-range packing schemes inevitably break down over longer coordinates as a result of spatial incompatibility, and they do not fully describe the atomic organization within these complex glasses. The incomplete understanding of atomic-level structure in glassy materials has made it challenging to capture the physics of their response to mechanical deformation. We propose a model that describes a short-range order and encompasses the long-range structural details of metallic glasses. The model has considerable implications for understanding glass properties and the origin of the glass transition.

Diffraction experiments characterize the structure of amorphous materials by mapping the atomic neighbor–separation distances and statistical density distributions. Dissimilar glasses and liquids commonly possess distinct short- and medium-range orders due to variations in chemical bonding, but the atomic structure becomes fluid like and nearly indistinguishable among different glasses beyond the first few nearest neighbors (8). The similarity of atomic-level environments in liquids and glasses makes it difficult to understand how glasses get their rigidity. Glass rigidity may be related to the jamming of atoms as density increases (9). The marked difference between the short- and long-range configurations in glassy systems sets glasses apart from crystals. In contrast to crystals, simplifying the underlying structure in a glass is problematic, because the short- and medium-range orders do not repeat in a recognizable pattern. For this reason, no two glasses, produced under the same conditions and with similar diffraction patterns, are identical at the atomic level. The question of how repeatable long-range structures in glasses can emerge from nonrepeating atomic clusters remains unanswered.

Studies suggest fractal properties in metallic glasses (10, 11). Fractal behavior manifests in the relationship between mass (M) and volume. For crystals, this relationship, M(r) ~ r³ (where r is the radius of a region within the material), has a dimensionality () of 3. The dimensionality of metallic glasses is closer to ~2.5 (11), and any non-integer corresponds to a fractal (12). Many naturally occurring random fractals have D ~ 2.5, including crumpled balls of paper and thin sheets (13), which are fractals down to the size of nanoballs of graphene oxide (14). Fractal concepts may be useful in developing an atomic-level understanding of amorphous materials, because they imply underlying order in inherently chaotic and random arrangements. The specific nature of fractals in metallic glasses is not obvious, because most mass fractals have macroscopic pores at large (e.g., crumpled paper), and metallic glasses are monolithic materials. Metallic glasses have packing fractions close to or exceeding those of close-packed crystalline metals (15). The puzzle of how metallic glasses can simultaneously possess fractal properties and remain fully dense is unresolved (16). One possible explanation is that the diffraction experiments only probe the short-range dimensionality. In this work, we observed a fractal short-range D < 3 and a homogeneous long-range D = 3 for several metallic glasses, indicating the presence of a dimensionality crossover at an intermediate length scale.

Previous studies have focused on the principal (first) diffraction peak only (10, 11). We extended the analysis beyond the first peak, because the information contained in diffraction experiments is spread out in momentum space, and each peak contains information that represents a part of the total structure. We conducted in situ high-pressure x-ray diffraction and full-field nanoscale transmission x-ray microscopy experiments on ~40-μm-diameter cylindrical samples of Cu₄₆Zr₄₆Al₅Be₃ metallic glass (Fig. 1A). We made diffraction and sample volume measurements in situ as a function of hydrostatic pressure in a diamond anvil cell. We related the scattering vector (q) from diffraction peak positions to volume by increasing the hydrostatic pressure from ~0 to 20 GPa (Fig. 1B). Compared with previous methods, using multiple data points improved accuracy in measuring the exponent (10). Structural information was sensitive to the magnitude of the scattering vector. We found that D ~ 2.51 for q₁ (the scattering vector from the first peak position), consistent with previous experiments on other metallic glasses (10, 11). The value of ~2.64 from q₂ measurements was 5% higher than that from q₁ (Fig. 1C). To explore the repeatability of this finding, we analyzed data obtained from a La₆₂Al₁₄Cu_11.7Ag_2.3Ni₅Co₅ metallic glass (11). First-peak data for both systems had the same exponent of ~2.5, whereas the q₂ data for the La₆₂Al₁₄Cu_11.7Ag_2.3Ni₅Co₅ had an exponent of nearly 3 (Fig. 1C). This shift is greater than the one observed in the Cu₄₆Zr₄₆Al₅Be₃ system, and it supports the observation that a change in the dimensionality arises from probing different extents within the atomic structure in momentum space. Extracting structural information from momentum space measurements is difficult, because the information is spread out. Real-space radial distribution functions (RDFs) are needed, where peak positions correspond directly to atomic separations. Background noise and the limited range of q restrict the accuracy of Fourier transforms applied to experimental RDFs. Atomistic simulations allow for this type of investigation.

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Molecular dynamics (MD) simulations can replicate the glass structure, but the simulation time scales (picoseconds) are many orders of magnitude shorter than in the experiments. We ensured that the system had sufficient time to relax at each pressure increment to address this issue. We held the loading rate constant at 50 GPa/ns (5 × 10¹⁹ Pa/s), and we allowed the system to relax for ~0.1 ns to reach thermodynamic equilibrium at each pressure interval. Higher quench rates in simulations may produce less relaxed glasses, although their structures often closely match those produced in experiments (17–19). The differences in compression rates result in quantitative discrepancies, but the qualitative and phenomenological aspects of the simulations should represent a realistic physical system. We generated Cu₄₆Zr₅₄ RDFs by using two embedded-atom-method force fields, described by Cheng et al. (FF₁) (18) and Mendelev et al. (FF₂) (19). The neighbor separation–volume relationship for RDF peaks r₁ and r₂ indicated a D of ~2.54, similar to the experimental result, but it transitioned to ~3 between r₂ and r₃ (Fig. 2A). We also simulated Ni₈₀Al₂₀, which exhibited a similar crossover between r₁ and r₂ (Fig. 2B) (20).

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The percolation cluster (21) is probably the most relevant fractal model to describe the structure of metallic glasses. The cluster represents a disordered system with fractal dimension D ~ 2.52 and appears across many physical systems (22, 23). Percolation models incorporate the probability of occupied (p) and empty (1 – p) sites. At low p values, the system is not fully connected (for example, as with an electrical insulator) (Fig. 3A). The percolation threshold (p_c) is reached when a percolating network forms, allowing incipient conduction. Systems characterized by large p have many conduction paths (Fig. 3B). What sets the percolation model apart is the existence of a correlation length, ξ, which characterizes the size of the finite clusters at concentrations below and above p_c. The correlation length is defined as the average distance that spans two sites within the same cluster and has units equal to the size of the smallest constituent unit in the model. At p< p_c and p> p_c, ξ is finite, and the system is only fractal at length scales shorter than the correlation length. On length scales longer than ξ, the structure is homogeneous. This property of percolation clusters may help reconcile the notion that fractals need not exhibit self-similarity across all length scales (Fig. 3C) (23).

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We used a continuum percolation model, where is analogous to the atomic packing fraction (φ), and the percolation threshold is analogous to a critical volume fraction (φ_c), such that φ_c = φp_c (23, 24). The correlation length isξ ∝ | φ– φ_c|^–v (1)for continuum percolation, where v = 0.8764 (25).

We estimated φ_CuZr to be 0.717 to 0.728 using the chemical compositions and the atomic radii of the simulated glasses (29). A reasonable model for the packing of a binary metallic glass involves continuum packing of hard spheres, with a p_c of ~0.310 (26). We obtained a φ_c of 0.257 by averaging the hard-sphere value (φ_c ~ 0.224 = p_cφ_CuZr) with an overlapping sphere value (φc ~ 0.2896) (27), because atoms in metallic glass are not ideally rigid (28). The correlation length was ~2 for Cu₄₆Zr₅₄ (ξ_CuZr ~ 1.93 to 1.98), suggesting that the information in the first and second peaks pertains mostly to the angstrom-sized fractal clusters, whereas information in the third peak pertains to the homogeneous bulk. This result is consistent with our observations of a crossover in dimensionality between r₂ and r₃ (Fig. 2A), and it provides evidence for the presence of percolation structure in metallic glasses. The short-range considerations for high local densities favor the formation of Cu-centered clusters, giving rise to a large number of Cu-centered icosahedra in lieu of the close-packed structures in native Cu and Zr (Fig. 3D) (16). The atoms with local icosahedral order form a percolating network (Fig. 3E).

Equation 1 suggests that higher packing fractions bring about shorter correlation lengths. We estimated that the Ni₈₀Al₂₀ has a high packing fraction, φ_NiAl, of ~0.793, although this could be an overestimation due to the covalent nature of the Al bonding. This estimate gives ξ_NiAl ~ 1.73, which is much less than 2. The result shows a crossover in dimensionality from ~2.54 to ~3 that occurs between r₁ and r₂ (Fig. 2B). We were also able to induce a shift in the Cu₄₆Zr₅₄ crossover from between r₂ and r₃ to between r₁ and r₂ at a pressure of >15 GPa by increasing the packing fraction and bringing ξ_CuZr below ~1.7 (fig. S1) (29). Some of the atoms in amorphous materials undergo local nonaffine displacements, even in response to purely hydrostatic loads. The fraction of such nonaffine atoms is low (~21.7%), and they do not appear to have any effect on the scaling behavior and crossover (figs. S2 and S3) (29).

We related the current model to the glass transition by examining the dimensionality as a function of temperature. We did not observe fractal behavior of Cu₄₆Zr₅₄ until 400 K, well below the glass transition temperature (T_g ) of 763 K (Fig. 4A). The dimensionality gradually decreased from ~3 to ~2.54 over this temperature range as the temperature decreased. This behavior suggests an intermediary process such as jamming (9, 30, 31), where the percolating cluster begins to jam at the glass transition. Complete jamming occurs at lower temperatures, along with the emergence of fractal properties, correlating with a loss of ergodicity and consistent with the characteristic kink in the volume-temperature curve during supercooling (Fig. 4B). Despite structural similarities, liquids are amenable to rearrangements in local atomic configurations, whereas in rigid solids, these configurations are preserved. Pressure elicits a mostly nonaffine response from the liquid and a comparatively affine response from the glass. Applied hydrostatic forces inevitably alter the structure and induce structural relaxation in a liquid, which is unavailable in a glass. This difference is probably the reason for the emergence of fractal properties below T_g in a glass and the lack thereof above T_g in a liquid. Metallic liquids possess packing fractions in excess of our estimated percolation threshold, which implies that their atomic structures are also percolating clusters that have not yet frozen or jammed.

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A fractal model might be useful in explaining the dynamics of metallic glasses, as concepts from percolation have been applied successfully to other glass formers (32). The dynamic heterogeneities that emerge in supercool liquids may be related to the spatial distribution of nonpercolating clusters. Estimating the average number of particles in these clusters using ~N_avg ≈ ξ³, where ξ ~ 2, we got a value (~8) that is close to experimentally observed values in colloidal glasses(~3 to 7) (33). From the perspective of packing, percolation, and jamming, a correlation between density and T_g (34) is intuitive. If metallic glasses are created from the jamming of a percolating cluster, then glass formation is simplified: Liquid metal only needs to reach the jamming packing fraction, φ_j, before nucleation occurs. This could be accomplished by a combination of hydrostatic pressure and fast cooling rates. The strong correlation of metallic-glass yield strength with T_g implies that collective atomic motions dictate both yielding and glass formation (35). Because denser metallic glasses tend to be better glass formers with higher T_g (34), the strength enhancement observed in glasses with higher T_g may originate from the size of the percolating clusters, which increases with packing fraction. Higher packing leads to larger jammed clusters, which present more substantial barriers to the initiation of collective atomic motions that lead to catastrophic shear banding. The movement of finite nonpercolating clusters may also be related to shear transformation zones, which are collective rearrangements of atoms during the deformation of metallic glasses (36). This concept is supported by the observation that typical zone sizes (~10 to 20 atoms) (37–39) are close to cluster sizes (~8 atoms). The continuum percolation model illustrates how structure and rigidity may organize in the absence of ordering; atoms percolate in the liquid, and the percolating cluster “freezes” (or jams) into a glass.