How the toughness in metallic glasses depends on topological and chemical heterogeneity

Significance

This article reports and explains how the cavitation in metallic glasses is controlled by topological structure as well as chemical heterogeneity. It is shown that in the tough metal–metalloid Pd-Si metallic glass, cavitation initiation is controlled by both chemical separation and particular types of low coordination number (LCN) Pd-centered polyhedra. In contrast, cavitation in the brittle metal-metal Cu-Zr metallic glass is only governed by topological factors. A high-energy barrier to form LCN polyhedra and the process of chemical separation during cavitation initiation are believed to contribute to a higher metallic glass toughness, thereby allowing a larger plastic strain to fracture.

Abstract

To gain insight into the large toughness variability observed between metallic glasses (MGs), we examine the origin of fracture toughness through bending experiments and molecular dynamics (MD) simulations for two binary MGs: Pd₈₂Si₁₈ and Cu₄₆Zr₅₄. The bending experiments show that Pd₈₂Si₁₈ is considerably tougher than Cu₄₆Zr₅₄, and the higher toughness of Pd₈₂Si₁₈ is attributed to an ability to deform plastically in the absence of crack nucleation through cavitation. The MD simulations study the initial stages of cavitation in both materials and extract the critical factors controlling cavitation. We find that for the tougher Pd₈₂Si₁₈, cavitation is governed by chemical inhomogeneity in addition to topological structures. In contrast, no such chemical correlations are observed in the more brittle Cu₄₆Zr₅₄, where topological low coordination number polyhedra are still observed around the critical cavity. As such, chemical inhomogeneity leads to more difficult cavitation initiation in Pd₈₂Si₁₈ than in Cu₄₆Zr₅₄, leading to a higher toughness. The absence of chemical separation during cavitation initiation in Cu₄₆Zr₅₄ decreases the energy barrier for a cavitation event, leading to lower toughness.

Engineering ceramics are strong, with high yield strength, but suffer from brittleness. In contrast, crystalline metals tend to have high fracture toughness because dislocation motion promotes plastic deformations that suppress cracks propagations, but concomitantly this dislocation motion reduces yield strength. Metallic glasses (MGs) tend to have high strength, and for some compositions, the high strength is accompanied by a high fracture toughness, making MGs promising engineering materials (1). The fracture toughness in MGs, which is accommodated by shear banding and limited by cavitation, is thought to arise from initiation of a crack opening at the core of an extending shear band (2–4). Then, new high-strength and high-toughness MGs may be designed by identifying compositions capable of suppressing cavitation during shear band extension. However, the complex physics of cavitation in MGs has obscured the development of models to illustrate cavitation’s origin.

For MGs, cavitation leads to the crack opening process that controls directly the fracture toughness, a fundamental property for material design and applications. Since the first amorphous alloy (Au₇₅Si₂₅) reported at the California Institute of Technology in 1960 (5), tremendous effort has been dedicated to understand why the amorphous structure leads to such excellent mechanical properties as high elastic limit, yield strength, and hardness (2, 6). However, toughness, which varies dramatically between MG compositions, ranging from values typical of brittle ceramics to those typical of engineering metals (2, 6, 7), is still poorly understood. More recently, improved alloys have been developed that demonstrate very high toughness, including a bulk Pd-rich, Si-bearing glass, Pd₇₉Ag_3.5P₆Si_9.5Ge₂ (7), and a bulk Zr-rich, Cu/Al–bearing glass, Zr₆₁Ti₂Cu₂₅Al₁₂ (8), in which shear band plasticity suppresses crack opening.

The fracture resistance of MGs is understood to arise from a competition between two processes: shear band plasticity and void nucleation. Currently, the process of shear band plasticity is widely recognized to be accommodated by the cooperative shearing of local atomic clusters [shear transformation zones (STZs)] (9, 10). However, to describe the fracture process, a condition for cavitation is needed coupled with the description of shear band plasticity to account for a crack opening along an operating shear band. Recently, Rycroft and Bouchbinder (11) coupled a continuum STZ model with a condition for cavitation to describe the fracture of MGs. The authors found that cavitation plays an essential role in the initiation of fracture, where they found a crack to evolve by successive void nucleation events along an operating shear band. In the context of molecular dynamics (MD) simulations, we and others proposed that cavitation precedes the onset of crack formation at the core of an operating shear band (12–15).

Our previous study showed that cavitation in metallic liquids is a homogeneous process that can be described with our revised classical nucleation theory (CNT) that incorporates the Tolman length correction (12). Furthermore, Falk et al. showed that our revised CNT remains valid down to picosecond time scales, whereas strain-aging effects become important at longer time scales in MGs (13). Murali et al. found that fracture in a brittle MG is governed by nanoscale void nucleation and coalescence (14), whereas Wang et al. established a correlation between shear transformations and dilatation (15). However, despite the success in numerical calculations, the atomistic origins of the cavitation process in MGs remains obscure, especially the role of topological structure and chemistry. Recent progress has been made to understand the nature of the atomic structures in MGs (16–18). Furthermore, Ding et al. examined the correlation between local atomic packing structure and quasilocalized soft modes correlated with STZ (19). The authors found that the geometrically unfavored motifs (GUMs) contribute the most to the soft modes that might lead to shear transformations (19). These GUMs prefer to concentrate in the quasilocalized soft spots that promote widespread shear transformations under external stress (20). Thus, although MGs lack long-range order, there is evidence that both short-range order (SRO) and medium-range order (MRO) play essential roles in determining such properties of MGs as glass-forming ability (GFA), elasticity, and strength/hardness (16–20). To understand the role of SRO and MRO on the fracture process of MGs, it is essential to establish the relationship between local cluster morphology and cavitation processes.

The present work combines mechanical testing experiments with MD simulations to investigate the origin of the large variability in toughness observed between MG compositions (2, 6, 7). For ease of simulation and analysis, we chose to study binary MGs. Two MG compositions were chosen that are capable of forming bulk rods (1–2 mm in diameter) such that the MGs can be mechanically tested. The two binary MGs chosen are Cu₄₆Zr₅₄, a metal-metal alloy in which metallic bonding dominates, and Pd₈₂Si₁₈, a metal-metalloid alloy in which chemical SRO is well established (21). We synthesized 1-mm-diameter rods of Cu₄₆Zr₅₄ and Pd₈₂Si₁₈ and performed bending experiments to assess the rods’ bending ductility. Then, we used MD simulations to study the cavitation mechanisms and cluster packing (chemical and topological SRO) effects in these two types of MGs at various conditions of tension and shear under negative pressure. Lastly, the difference in ductility between the two MGs is rationalized in terms of the differences in their cavitation mechanisms and cluster-packing effects.

Results and Discussion

Processing and Testing of Pd₈₂Si₁₈ and Cu₄₆Zr₅₄.

To quantitatively measure the fracture toughness of MGs in accordance with established standards, samples considerably thicker than 1–2 mm are generally needed. Therefore, it would not be possible to quantitatively assess the fracture toughness of the chosen MGs in any meaningful way, because glass formation is limited to 1- to 2-mm-diameter rods. Nonetheless, one can semiquantitatively estimate a critical mode I stress intensity factor, K_IC, from knowledge of the plastic strain to failure in bending (i.e., the bending ductility). Conner et al. (22) derived an analytical expression to evaluate K_IC in bending of unnotched MGs by relating K_IC to the bending strain at fracture. The underlying assumption in the authors’ analysis is that a stable crack developing at the core of an operating shear band is akin to a precrack, and hence when such crack develops, the bending test transitions to a fracture toughness test of a [single-edge notched bending SEN(B)] specimen.

To avoid any adverse effects on toughness from extrinsic parameters such as crystallinity, oxygen, and impurities, the alloy ingots were prepared using high-purity elements with low-oxygen content and were alloyed under an ultrapure inert atmosphere (Supporting Information). The MG test samples were carefully inspected by X-ray diffraction and differential scanning calorimetry to ensure the absence of crystallinity. Samples for the two MGs were synthesized differently because of their different melt reactivity with crucible materials. Specifically, MG rods of Cu₄₆Zr₅₄ were produced by arc-melting over a water-cooled copper hearth followed by suction casting in a copper mold, whereas MG rods of Pd₈₂Si₁₈ were produced by furnace melting in quartz capillaries, followed by rapid water quenching. Because the latter method produces a somewhat lower cooling rate (because of a slower heat removal rate across the low conductivity quartz walls), the Pd₈₂Si₁₈ MG rods are associated with a lower quenched-rate state in relation to Cu₄₆Zr₅₄. Because fracture toughness is known to depend on the processing cooling rate (23), the fracture toughness of the Pd₈₂Si₁₈ MG rods may be expected to be somewhat depressed relatively to Cu₄₆Zr₅₄ because of a lower quenched-rate state.

Three rods of each MG were tested in three-point bending (Supporting Information). A representative bending load-displacement diagram of each MG is presented in Fig. 1A. Because the loading and specimen geometries were identical for the two samples, the values for load and displacement are direct reflections of stress and strain. Within the elastic region, the load-displacement slopes are roughly equal, indicating approximately the same Young’s moduli, which is consistent with the measured Young’s moduli of the two alloys (24). On the other hand, the load at yielding is higher for Cu₄₆Zr₅₄ than Pd₈₂Si₁₈ by about 15%, indicating a higher yield strength. Within the plastic regime, the Cu₄₆Zr₅₄ sample fractures soon after it yields, demonstrating a rather limited ductility. In contrast, the Pd₈₂Si₁₈ sample continuously deforms plastically to a very large displacement without fracturing, indicating a very high plastic strain in the absence of fracture and hence a very high ductility. This picture was consistent in all three samples tested for each alloy. Specifically, the bending displacement to fracture for all three Cu₄₆Zr₅₄ samples was found to be between 0.4 and 0.6 mm, whereas all Pd₈₂Si₁₈ samples exceeded 1.2 mm in bending displacement (which is the maximum displacement applicable by the bending fixture) without fracturing. Through the concept of Conner et al. (22), one can therefore conclude that Pd₈₂Si₁₈ has a considerably higher fracture toughness compared with Cu₄₆Zr₅₄, because the plastic strain attainable before fracture is considerably larger. Moreover, considering that the Pd₈₂Si₁₈ samples are associated with a somewhat lower quench rate state in relation to Cu₄₆Zr₅₄, the difference in the intrinsic fracture toughness may be even larger than the difference in bending ductility suggests.

Fig. 1.

(A) Bending load-displacement diagram for 1-mm-diameter amorphous rods of Pd₈₂Si₁₈ and Cu₄₆Zr₅₄. (B) SEM image on the tension side of the fractured Cu₄₆Zr₅₄ rod showing the catastrophic crack adjacent to a few shear bands. (C) SEM image on the tension side of the deformed Pd₈₂Si₁₈ rod showing a dense network of shear bands and absence of any opening or microcrack.

To identify the dominant micromechanisms at play in these bending experiments, we examined the tension side of the rods using scanning electron microscopy (SEM). The micrograph of a fractured Cu₄₆Zr₅₄ rod (Fig. 1B) shows a catastrophic crack adjacent to a few shear bands, revealing that only a very limited number of shear bands were generated before crack opening. Careful inspection of the fracture surfaces did not reveal presence of any preexisting pores or inclusions that may have triggered premature fracture (Fig. S1). Therefore, one can conclude that the ductility and toughness of Cu₄₆Zr₅₄ is intrinsically low. In contrast, the micrograph of the deformed Pd₈₂Si₁₈ rod (Fig. 1C) reveals a dense network of shear bands and absence of any opening or microcrack, which is consistent with the high ductility and toughness revealed in the bending experiment.

Fig. S1.

Fracture surfaces of Cu₄₆Zr₅₄ did not reveal presence of any preexisting pores or inclusions that may have triggered premature fracture.

MD Simulations on the Cavitation of Pd₈₂Si₁₈ and Cu₄₆Zr₅₄ Under Hydrostatic Tension.

To investigate the origin of the different toughness of these two types of MGs, we performed the MD simulations of the initial stage of the cavitation. We started with amorphous structures for Cu₄₆Zr₅₄ and Pd₈₂Si₁₈ using periodic super cells, containing 32,000 atoms. We used two approaches to induce cavitation:

• i)Application of hydrostatic tension at a constant strain rate of 2.0 × 10⁸ s⁻¹
• ii)Shear at a constant strain rate of 1.0 × 10¹⁰ s⁻¹ applied under negative overall pressure

Twenty independent simulations for tension and seven independent simulations for shear were performed to obtain good statistics.

The stress–strain relationships extracted from the MD simulations (Fig. 2A) indicate that the stress/strain slope, which is a measure of the bulk modulus, is ∼52% larger for Pd₈₂Si₁₈ compared with Cu₄₆Zr₅₄. This difference is roughly consistent with the measured bulk moduli of the two alloys (24). Under hydrostatic tension, we find that the negative pressure (tension) increases to a maximum of −12.9 GPa for Pd₈₂Si₁₈ and to −11.0 GPa for Cu₄₆Zr₅₄, at which point, cavitation events initiate to relax the negative pressure very quickly. The higher cavitation pressure for Pd₈₂Si₁₈ indicates that it requires more mechanical work to introduce an opening process for crack formation than does Cu₄₆Zr₅₄, leading to a higher fracture toughness in Pd₈₂Si₁₈. This difference explains the observed higher fracture toughness of Pd₈₂Si₁₈ from experiments.

Fig. 2.

Tension-driven cavitation events for Pd₈₂Si₁₈ and Cu₄₆Zr₅₄. (A) Typical case showing the pressure as a function of time (strain) for a uniformly increasing hydrostatic tension. (B) The size of the largest void for Pd₈₂Si₁₈ as a function of time for the case in A. From ts to later times, the largest void remains at the same site, but cavitation starts as time tc. The time interval between ts and tc varies within 30 ps for 20 cases as shown in Table S1. (C) The chemical character near the largest void for Pd₈₂Si₁₈. (D) The chemical character near the largest void for Cu₄₆Zr₅₄.

Cavitation in MGs is controlled by the spatial heterogeneity in the glass, initiating at preferential nucleation sites (10). To locate the preferred cavitation sites, we started with the largest nanovoid within the bulk MGs just after initiating cavitation and traced this backward in time. This nanovoid evolution is shown in Fig. 2B for Pd₈₂Si₁₈. Based on the evolution of the largest nanovoid, the cavitation event involves the following steps.

• i)Firstly, the location within the MGs of largest void site changes with time until the cavity location is selected at time ts, after which, the largest void is always at the same site.
• ii)Secondly, the size of this largest void fluctuates with time between ts and time tc.
• iii)Finally, at tc, cavitation initiates with the cavity growing monotonically with time.

Thus, the time tc is the critical time for cavitation. Time ts should be important only in very local regions at the MD scale because there must be many cavitation sites at the macroscopic scale. This cavitation process is similar in Cu₄₆Zr₅₄, as discussed below.

The chemical bonding is quite different in Pd₈₂Si₁₈ and Cu₄₆Zr₅₄. In Pd₈₂Si₁₈, the Si atoms tend to form strong covalent bonds with neighbor Pd atoms, whereas the Pd atoms tend to form weak metallic bonds among themselves. Thus, we expect that cavitation would initiate from Pd-rich regions making initiation of cavitation strongly dependent on chemical heterogeneity. This expectation is confirmed by our analysis of the nearest neighbor composition surrounding the largest nanovoid, as shown in Fig. 2C. In particular, the fraction of Pd neighbors at the largest nanovoid is nearly 100% before ts and then decreases quickly within a few picoseconds to 92% at ts. Then, as the largest void is growing after ts, the ratio of the Si component increases monotonically but remains much lower than the average ratio, until failure occurs (tc). These chemical fluctuations in the Pd can be explained from the Pd-Si phase diagram [American Society for Metals (ASM) Alloy Phase Diagram Database; mio.asminternational.org/apd/index.aspx], which shows that the Pd₈₂Si₁₈ amorphous phase lies between the pure Pd face-centered cubic crystal and the ordered Pd₃Si crystal. Thus, we find that the Pd composition in the largest void fluctuates between 100% and 82%.

In contrast, in the Cu-Zr phase diagram (ASM Alloy Phase Diagram Database; mio.asminternational.org/apd/index.aspx), the Cu₄₆Zr₅₄ amorphous phase lies between the Cu₁₀Zr₇ crystal and the CuZr₂ crystal (33%∼59% for Cu), leading to no obvious chemical trends. This speculation is confirmed by our chemical analysis of the largest nanovoid for Cu₄₆Zr₅₄, as shown in Fig. 2D, where the Cu ratios near the largest nanovoid fluctuate between 33% and 46%.

To determine the level of chemical heterogeneity responsible for initiating cavitation, we analyzed a minimum rectangular region (8.8 × 9.7 × 7.0 ų) containing all nearest-neighbor atoms of the cavity configuration at time tc, finding that the Pd concentration is 88.6%. As the critical nanovoid expands, this local enhanced Pd concentration decreases toward the bulk ratio, as shown in Fig. 3A. This decrease indicates that the chemical heterogeneity sufficient to initiate cavitation is localized within a very narrow region of ∼2.0 nm³. A similar analysis on the structure at ambient conditions (without tension) shows an even more localized chemical heterogeneity within ∼1.0 nm³. Thus, we propose that a criterion for predicting the location of the cavitation site is that the Pd concentration be greater than 87% in the structure at ambient conditions. To validate this criterion, we located cavitation sites for the 20 independent tension simulations and mapped them back to the original intact structure. As shown in Fig. 3B, all cavitation sites are within a Pd-rich region that satisfy this >87% criterion for the Pd concentration.

Fig. 3.

Chemical analysis and the related cavitation sites for Pd₈₂Si₁₈. The Pd and Si atoms are represented by yellow and blue balls, respectively. (A) The Pd composition distribution as the increase of the near cavity region. (B) The cavitation sites (purple balls) in the intact system.

To validate these ideas about the effects of local chemical heterogeneity on cavitation in Pd₈₂Si₁₈, we created a different initial configuration with the same composition by randomly exchanging 5% of the Pd with Si atoms, followed by 2.0 ns of equilibrium using the isothermal–isobaric (NPT) ensemble. The different structure is well in equilibrium after 2.0 ns as shown in Fig. S2. The Pd concentration distributions of two configurations in Fig. S3 A and B show a flatter distribution for the other configuration with more Pd-rich sites, indicating less SRO. The cavitation times for these two structures lead to a normal distribution (Fig. S3 C and D) in which the tc = 658.4 ps for the structure with more Pd-rich sites, compared with the previous value of tc = 719.1 ps. This difference indicates that the increased number of Pd-rich sites significantly reduces the cavitation time (strain) by ∼8.4%. The results show that the cavitation behavior of the same Pd₈₂Si₁₈ composition can differ substantially even with the same composition, depending instead on local fluctuations in chemical composition.

Fig. S2.

NPT equilibrium of the structure for 2.0 ns, with 5% exchange of Pd and Si. (A) Pressure evolution. (B) The potential energy evolution. (C) The mean squared displacement of exchanged atoms (the reference is the last snapshot of 2.0-ns NPT dynamics).

Fig. S3.

The Pd distribution in the Pd₈₂Si₁₈ system and the corresponding cavitation time fitting by a Gaussian distribution. (A and C) The original structure where the cavitation time is 719.1 ps. (B and D) The structure generated by exchange 5% Pd and Si atoms, where the cavitation time is 658.4 ps.

To characterize the topological SRO, we use a Voronoi tessellation analysis (25). The types and fractions of Si-centered and Pd-centered polyhedra for the intact Pd₈₂Si₁₈ and evolution under tension before failure are given in Figs. S4 and S5, with the details discussed in the Supporting Information. Here, we focus on the polyhedral clusters containing large free volumes. For Pd₈₂Si₁₈, we considered only the low coordination number Pd-centered polyhedra (termed as “LCN-Pd-polyhedra”), with less than 10 nearest neighbors and with only one or no Si atoms. A typical structural evolution during cavitation of these polyhedral clusters is shown in Fig. 4A, where the number of LCN-Pd-polyhedra around the cavity increases from three at time ts (701.5 ps) to seven at time tc (704.5 ps), which then breaks up at 705 ps. Note that the cavities are surrounded by a 4–11 LCN-Pd-polyhedra for all 20 independent cavitation events, as listed in Table S1. Generally, the number of LCN-Pd-polyhedra around the cavity increases from ts to tc for the sites at which cavitation later occurs. The average size of the LCN-Pd-polyhedra is ∼seven polyhedra, and the average critical volume is 68 ų. The main types of LCN-Pd-polyhedra are (0 3 6 0), (0 4 4 0), and (0 4 4 1), which belong to the Bernal holes class (26), constituting 78% of the total LCN-Pd-polyhedra. The critical volume increases as the number of LCN-Pd-polyhedra increases, as shown in Fig. 4B.

Fig. S4.

The Voronoi tessellation analysis for the intact Pd₈₂Si₁₈. (A) The Si-centered cluster. (B) The Pd-centered cluster.

Fig. S5.

The Voronoi tessellation analysis for the Pd₈₂Si₁₈ at tensile strain of 0.12 before cavitation events. (A) The Si-centered cluster. (B) The Pd-centered cluster.

Fig. 4.

The LCN-Pd-polyhedra cluster surrounding the critical cavities. (A) The evolution of LCN-Pd-polyhedra cluster surrounding the cavity center. (B) The relation between the number of LCN-Pd-polyhedra and the critical cavitation volume at tc. (C) Shape-related information about the critical cavity at tc, indicating that the critical cavity is spheroidal shape. The cavity center, Pd atoms, Si atoms, and Pd-center atoms are represented by black, yellow, blue, and pink balls, respectively.

Table S1.

Statistics of 20 independent cavitation events from tensile deformation

Event	Time 1, ps	Time 2, ps	N_t1*	N_t2†	Vol,‡ Å3	poly_1§	poly_2¶	poly_3#
1	660	663	3	6	50.3	4	2	0
2	669.5	670	3	8	58.8	4	1	3
3	645.5	646	3	6	64.0	2	3	1
4	647.5	651	3	6	48.7	2	4	0
5	625	656	3	11	102.4	4	5	2
6	676.5	677.5	7	7	59.4	3	2	2
7	634.5	639	4	5	51.8	0	5	0
8	660.5	665.5	2	9	77.7	4	3	2
9	644	648.5	5	9	80.2	1	3	5
10	701.5	705	4	7	88.2	0	4	2
11	684.5	691.5	4	7	48.2	3	0	4
12	642	652	5	5	39.7	2	3	0
13	654	666	2	7	73.7	3	2	2
14	696	698	4	7	47.9	3	2	2
15	670	670	5	5	39.2	1	1	3
16	646	672	3	9	139.0	2	2	5
17	670	670	5	5	56.6	2	1	2
18	692	700	3	4	61.8	1	3	0
19	704	710	3	7	77.9	3	3	1
20	700	704	7	7	86.8	1	2	4

^*The number of LCN-Pd-polyhedra at time 1.

^†The number of LCN-Pd-polyhedra at time 2.

^‡The critical cavitation volume from grid analysis.

^§The number of polyhedra with type (0 3 6 0).

^¶The number of polyhedra with types (0 4 4 0) and (0 4 4 1).

^#The number of polyhedral with others types of (0 5 2 0), (0 5 2 2), (1, 3, 3, 1), etc.

To characterize the cavity shape, we measured the distance between cavity center and the polyhedral center, as shown in Fig. 4C. The largest distance is 5.54 Å, whereas the smallest distance is only 2.64 Å, indicating that the cavity shape is spheroidal. We found that cavitation preferentially breaks bonds along one direction rather than isotropic 3D dilation. As the cavity grows to a larger size, it becomes more spherical because of the surface tension.

Shear Induced Cavitation in Pd₈₂Si₁₈.

To mimic shear induced cavitation in Pd₈₂Si₁₈, we applied shear deformation on the pretension system with a tensile strain of 0.12. Here, we pretension the system to observe cavitation within the MD timescale. Fig. 5A shows the pressure and the shear stress in the shear-driven cavitation event. We observe that the pressure decreases from −11.3 to −9.9 GPa within the first 20 ps, indicating a structural relaxation process. The shear stress also increases to −1.0 GPa within the first 10 ps. The negative pressure decreases suddenly at 225 ps, indicating a cavitation event. The shear process accelerates the initiation of cavitation. We confirmed this acceleration by another MD simulation in which we kept a fixed pressure of −9.9 GPa. Here, no cavitation was observed within 2 ns.

Fig. 5.

Shear-induced cavitation for Pd₈₂Si₁₈. (A) Pressure and shear stress evolution. (B) The size evolution of the largest void. (C) The chemistry near the largest void. (D) The structures of LCN-Pd-polyhedra clusters surrounding the cavity center. The cavity center is black, the Pd atoms are yellow, the Si atoms are blue, and the Pd-center atoms are pink.

For the shear-driven cavitation event, the size and positions of the largest void displays a similar character to that for tension, where the local cavitation is selected at time ts and cavitation initiates at time tc, as shown in Fig. 5B. The chemistry is also similar showing a preferred cavitation site within the Pd-rich region, as shown in Fig. 5C. The structural evolution of LCN-Pd-polyhedra surrounding the cavity center is shown in Fig. 5D. As the cavity size increases continuously from 214.5 to 221 ps, the number of LCN-Pd-polyhedra around cavity increases from two to seven. We examined seven independent shear-driven cavitation events with the results listed in Table S2 and shown in Fig. 4 B and C.

Table S2.

Statistics of 7 independent cavitation events from shear deformation

Event	Time 1, ps	Time 2, ps	N_t1*	N_t2†	Vol,‡ Å3	poly_1§	poly_2¶	poly_3#
1	209	222	1	6	99.9	2	4	0
2	55	58.5	4	7	69.0	1	4	2
3	<50	74.5	<2	8	149.8	2	3	3
4	<110	137.5	<2	5	99.4	1	2	4
5	458	458	10	10	158.0	2	5	5
6	213	233	3	8	88.1	3	0	6
7	66	86	2	6	71.6	2	1	7

^*The number of LCN-Pd-polyhedra at time 1.

^†The number of LCN-Pd-polyhedra at time 2.

^‡The critical cavitation volume.

^§The number of polyhedra with type (0 3 6 0).

^¶The number of polyhedra with types (0 4 4 0) and (0 4 4 1).

^#The number of polyhedral with others types of (0 5 2 0), (0 5 2 2), (1, 3, 3, 1), etc.

Our shear-driven cavitation simulations show character comparable with our tension simulations. However, the critical cavitation volume for shear is 56% larger than for tension, with the volume of LCN-Pd-polyhedra increasing from 68 to 106 ų, as shown in Fig. 4B. This increase is because the minimum distance between cavity center and the polyhedral cluster center increases by 14% or 3.2 Šunder shear but only 2.8 Šin tension and the maximum distances of 5.4 and 5.5 Šare almost the same for shear and tension, as shown in Fig. 4C.

Cavitation-Rate Estimations From Transition State Theory and MD Simulations.

Our simulations show that the cavitation process can be described in terms of droplet formation with the cavitation rates estimated using our modified CNT (10, 12). However, the accumulation of LCN-Pd-polyhedra near the critical cavity leads to broken bonds preferentially along a particular direction leading to an anisotropic cavity shape, making it hard to estimate the surface energy for use in the formula for the cavitation rates (12). Alternatively, we can consider the cavitation events as rate-dependent events that can be treated by activation theory.

We used the critical cavity in Fig. 4A as an example. Through the calculations of the potential energy, virial pressure, and Voronoi volume of these LCN-Pd-polyhedra, we obtain an activation enthalpy of 0.31 eV per atom. Here, we ignore the entropy effects and estimate the prefactor value as 5.0 × 10¹¹ ps⁻¹. Then we obtain a cavitation rate of 6.0 × 10⁴² s⁻¹⋅m⁻³. From the MD simulations, the average cavitation time is estimated to be ∼10 ps and the possible cavitation sites (centers of Pd-rich polyhedral clusters where Pd content is above 87%) are estimated to be 6,000. Thus, the direct measurement of the cavitation rate from MD simulations is 3.3 × 10⁴³ s⁻¹⋅m⁻³, which is consistent with the estimation from transition state theory. The details of this calculation are in the Supporting Information.

Topological Analysis of Cavitation in Cu₄₆Zr₅₄.

Because chemical factors are less obvious for Cu₄₆Zr₅₄, we focus on the topological analysis of cavitation. As shown in Fig. 6, the number of LCN-polyhedra increases from three at time ts to nine at time tc, similar to Pd₈₂Si₁₈. However, the polyhedral type are all Cu-centered polyhedral clusters with the main types being (0 3 6 0), (0 4 4 0), and (0 4 4 1), constituting 56% of the total for this particular case. These low coordination Cu-centered polyhedra (LCN-Cu-polyhedra) have been developed from the competing motifs (27) in the intact structures under hydrostatic tension, with the fluctuation of the composition in the range of 33–46 at% Cu. This lack of chemical factors playing a role in Cu₄₆Zr₅₄ should decrease the energy requirement for a cavitation event from that of Pd₈₂Si₁₈, leading to an earlier onset of crack formation and a lower fracture toughness in Cu₄₆Zr₅₄ than Pd₈₂Si₁₈, as observed experimentally.

Fig. 6.

Tension-driven cavitation events for Cu₄₆Zr₅₄. (A) The size evolution of the largest void. (B) The structures of LCN-Cu-polyhedra cluster surrounding the cavity center. The cavity center, Cu, Zr, and Cu-center atoms are represented by black, purple, green, and pink balls, respectively.

Cavitation Nucleation in Pd₈₂Si₁₈ and Cu₄₆Zr₅₄.

In Cu₄₆Zr₅₄, the LCN-Cu-polyhedra with coordination numbers (CNs) of 8 and 9 develop faster because of the absence of chemical heterogeneity. As the polyhedra aggregate in a local region above a critical concentration, cavitation initiates from these weak sites.

In contrast to Cu₄₆Zr₅₄, the cavitation nucleation in Pd₈₂Si₁₈ is delayed by two factors. First, deformation-induced LCN-Pd-polyhedra in Pd₈₂Si₁₈ is more difficult than that of LCN-Cu-polyhedra in Cu₄₆Zr₅₄. In this Pd-rich composition, local packing is dominated by Si-centered tricapped trigonal prism and Pd-centered polyhedera with CNs of 12 or 13. For Si-centered polyhedra, the covalent character of Si makes them rather rigid and highly resistant to deformation. For Pd, it is hard to decrease the CN around Pd to form LCN polyhedra because the local environment change makes only subtle difference in CN within this Pd-rich region. So the rate of forming low CN polyhedra is lower than the case for Cu₄₆Zr₅₄. Secondly, chemical partitioning is required to form the high Pd concentration region. This phase separation is facilitated by stress-assisted “diffusion” via nonaffine displacements in shear transformations. This diffusion-like “relocation of atoms to observable segregation” process has to be accrued via many shear transformations in prolonged deformation. This process further delays the cavitation initiation, which is triggered by the weak spots.

A previous study showed that the Fe₈₀P₂₀ MG is very brittle compared with the relatively more ductile Cu₅₀Zr₅₀ MG (14). The brittle behavior in the Fe₈₀P₂₀ glass is associated with forming multiple nanoscale cavities, which arise from atomic scale spatial fluctuations of the local density (14). In the more ductile Cu₅₀Zr₅₀ glass, extensive shear banding tends to suppress crack propagation. In the current study, we showed that the Pd₈₂Si₁₈ glass is even more ductile than Cu₄₆Zr₅₄ glass, owing to local chemical heterogeneities and providing an additional barrier for cavitation. The common conclusion from both studies is that the intrinsic cavitation mechanism effectively controls fracture toughness.

In summary, our bending experiments revealed that Pd₈₂Si₁₈ is far more ductile and tougher than Cu₄₆Zr₅₄. The higher toughness of Pd₈₂Si₁₈ arises from an ability to deform plastically in the absence of crack nucleation through cavitation. Our MD simulations revealed that in Cu₄₆Zr₅₄, cavitation is mainly governed by the types of similar Cu-centered polyhedron. However, in Pd₈₂Si₁₈, cavitation is controlled by both this topological structure of particular types of Pd-centered polyhedron plus local chemical heterogeneity. Together, these two factors lead to the higher observed toughness. Thus, we find that cavitation initiation in Pd₈₂Si₁₈ requires formation of very Pd-rich regions. However, the bonding to Si tends to distribute the Pd inhomogeneously, thereby forming far fewer Pd-rich clusters than expected statistically. Consequently, to initiate cavitation in Pd₈₂Si₁₈, it is necessary for Pd to diffuse to form these Pd-rich clusters. The slow kinetics of the Pd-diffusion process raises the overall energy barrier for a cavitation event, which enables more extensive plastic strains before fracture, leading to a higher fracture toughness for Pd₈₂Si₁₈. This conclusion suggests that if one attempts to optimize the roles of chemical inhomogeneity and topology in the process of cavitation and crack opening, new tougher MG compositions may be developed. Chemical factors might also play essential roles in the nucleation process of crystallization in MGs, thus controlling their GFA.

Supporting Information.

The Supporting Information includes (i) details regarding processing and testing, (ii) simulation details and analysis methods, (iii) the Voronoi tessellation analysis for Pd₈₂Si₁₈, (iv) details regarding cavitation-rate estimations from transition state theory and MD simulations, (v) fractography of Cu₄₆Zr₅₄ specimens, (vi) Figs. S1−S5, and (vi) Tables S1 and S2.

1. Processing and Testing Details

The Pd₈₂Si₁₈ alloy was prepared by inductively melting high-purity elements (Pd, 99.95%; Si, 99.9999%) in quartz tubes under a high-purity argon atmosphere. The alloy ingot was fluxed with B₂O₃ in a quartz tube under a high-purity argon atmosphere at 1,200 °C for 16 h. Amorphous rods 1 mm in diameter were prepared by furnace-melting the fluxed ingots in round quartz capillaries with 100-μm-thick walls under a high-purity argon atmosphere, followed by rapid water-quenching.

The Cu₄₆Zr₅₄ alloy was prepared by arc-melting high-purity elements [Zr, 99.9% (crystal bar; Hf, 0.028% wt; O, 110 ppm wt); Cu, 99.995%] over a water-cooled copper hearth under a titanium-gettered argon atmosphere. Amorphous rods 1 mm in diameter were prepared by arc-melting the alloy ingots under a titanium-gettered argon atmosphere, followed by suction-casing into a copper mold.

Three-point bending tests of 1-mm amorphous rods of Pd₈₂Si₁₈ and Cu₄₆Zr₅₄ were performed at room temperature on a screw-driven Instron testing machine (Instron). A bending fixture with a span distance of 6.35 mm was used. Strain was recorded using a linear variable displacement transducer.

2. Simulation Details and Analysis Methods

We use the embedded-atom-method potential developed to describe the interactions for Zr-Cu and Pd-Si (28). MD simulations were performed on Cu₄₆Zr₅₄ and Pd₈₂Si₁₈ MGs using large-scale atomic/molecular massively parallel simulator (LAMMPS) (29), with periodic boundary condition applied for all three dimensions. Simulation cell sizes at ambient conditions are 83.27 and 78.21 Å for Cu₄₆Zr₅₄ and Pd₈₂Si₁₈, respectively. The samples were melted and equilibrated at 2,000 K for 1 ns with a time step 1.0 fs and quenched into the glassy state (300 K) at cooling rates of 1.0 ×10¹² K s⁻¹. Isothermal–isobaric (NPT) and canonical (NVT) ensembles are applied in the simulations using the Nose–Hoover thermostat (damping constant 100 fs) and barostat (damping constant 1,000 fs) to adjust temperature and pressures. At room temperature, the system is hydrostatically expanded along three directions with a constant rate of 2.0 ×10⁸/s by scaling the atomic coordination of all atoms at each MD step. In shear deformation, the system is sheared under a constant engineering strain rate of 1.0 ×10¹⁰ s⁻¹.

We use a grid-based method (25) to characterize the nanovoid in the deformation simulations. From a radial distribution function, we get nearest-neighbor distances (R_nn) equal to 3.85 and 3.53 Å for Pd₈₂Si₁₈ and Cu₄₆Zr₅₄, respectively. Therefore, we make very fine grids with grid spacing (R_g = 0.76 Å) about five times smaller than R_nn. Then, we fill the grid points with atom radius (R_at) (1.96 Å for Cu, 2.02 Å for Zr, 1.83 Å for Pd, and 1.70 Å for Si). We also fill the empty sites that are close (R < R_at + R_g/2.0) to more than one atom region, which means filling the empty sites that contain more than one filled nearest-neighbor atoms. This filling procedure is done to ensure no voids are found in the initial structure. Cluster analysis is performed to determine the cluster size based on the grid size.

3. The Voronoi Tessellation Analysis for Pd₈₂Si₁₈

To characterize the topological SRO, we use a Voronoi tessellation analysis (25). Here, the indices (n₃, n₄, n₅, n₆) represent the number of i-edge faces of the polyhedron, where the sum of n_i gives the CN of the center atom. The most common Si-centered polyhedra are the tricapped trigonal prism (fraction of 39.6%), corresponding to the index of (0 3 6 0); the deformed icosahedrons (fraction of 26.0%), having the indices of (0 2 8 0); and the standard square dodecahedron (fraction of 3.2%), which has the index of (0 4 4 0).

However, different from Si-centered polyhedral, Pd atoms exhibit over 18 packing-type clusters, where the most abundant ones are (0 3 6 4) (fraction of 11.5%), (0 2 8 2) (fraction of 11.6%), and (0 1 10 2) (fraction of 15.8%).

Although the amount of perfect icosahedra [with index of (0 0 12 0)] is only 5.6%, a noticeable amount of defective or deformed icosahedra form with the indices of (0 2 8 1), (0 2 8 2), and (0 1 10 2). There is also a large amount of crystal-like clusters with (0 3 6 3) and (0 4 4 4) and distorted or defective ones with (0 3 6 4), (0 4 4 3), and (0 4 4 5) that are associated with face-centered cubic- and hexagonal close packed-type structures. The Voronoi analysis on intact Pd₈₂Si₁₈ is consistent with a previous study (30).

During the tension process, the main polyhedra remain similar but the ratio changes, as shown in Figs. S2 and S3. Thus, for the Si-centered polyhedra, the (0 4 4 0) type cluster increases abruptly when the tensile strain increased to 0.12.

For the Pd-centered polyhedra, the LCN polyhedra (CN = 8–11) increase. In particular, we find Pd-centered (0 3 6 0) trigonal and (0 4 4 0) dodecahedral clusters that belong to the Bernal holes (26), which will carry large free volume if the center Pd atom is removed.

4. Cavitation-Rate Estimations from Transition-State Theory and MD Simulations

The potential energy (PE) and enthalpy (H) of these LCN-Pd-polyhedra cluster (45 Pd atoms and 5 Si atoms) are −4.03 and −5.35 eV per atom, respectively, whereas the average PE and H for these atoms in the whole system are −4.25 and −5.66 eV per atom, respectively. Because our cavitation events happened at room temperature, we ignore the entropy effects here. Thus, we can estimate the activation energy for this cavitation event is 0.31 eV.

We estimate the surface energy of the critical cavity in Fig. 4A by the slab calculation to extract the flat surface energy of 0.973 eV/Ų. Because the cavitation volume is ∼86 ų, the surface area is estimated to be ∼100 Ų if we assume a spherical surface. Thus, the surface energy of the critical cavity is estimated to be ∼10.0 eV, which is consistent with the potential energy difference between the clusters and the average of 11.0 eV.

We estimate the prefactors using the previous approach (13), where it is on the order of c_l/L, where c_l is the longitudinal wave speed and L is a cell length of the system. Here, we estimated the c_l/L = 0.56 ps⁻¹ by calculating the bulk modulus and shear modulus. Thus, the prefactor is estimated to be 5.0 × 10¹¹ s⁻¹.

To estimate the cavitation time, we need to choose a configuration before ts. Because the time interval between ts and tc is several picoseconds, we estimate the cavitation time is ∼10 ps for the direct MD waiting-time simulation.

5. Fractography of Cu₄₆Zr₅₄ Specimens

The fracture surfaces of Cu₄₆Zr₅₄ samples was investigated using SEM. The fractograph of a representative Cu₄₆Zr₅₄ sample is presented in Fig. S1. As shown, no preexisting pores or inclusions are evident in the fracture surface that could have triggered premature fracture, indicating that fracture was triggered by nucleation of a crack at the core of an operating shear band through cavitation. The left-half region in the fractogram shows evidence of plastic flow consistent with shear band extension, whereas the right-half region shows evidence of smooth facets consistent with crack formation following plastic deformation. This picture supports that shear band extension of ∼0.5 mm (i.e., halfway through the 1-mm sample), followed by crack nucleation through cavitation at the core of an extending shear band.