Stress-tunable abilities of glass forming and mechanical amorphization

For ribbon preparation, binary CuZr (with nominal compositions of Zr₅₀Cu₅₀, Zr₅₅Cu₄₅, Zr₆₀Cu₄₀, Zr₆₅Cu₃₅), ternary CuZrAl (with nominal compositions of Zr₄₅Cu₄₅Al₁₀, Zr_47.5Cu_42.5Al₁₀, Zr₅₀Cu₄₀Al₁₀, Zr_52.5Cu_37.5Al₁₀, Zr₅₅Cu₃₅Al₁₀, Zr_57.5Cu_32.5Al₁₀, Zr₆₀Cu₃₀Al₁₀, Zr_62.5Cu_27.5Al₁₀, Zr₆₅Cu₂₅Al₁₀, Zr_67.5Cu_22.5Al₁₀, Zr₇₀Cu₂₀Al₁₀, Zr_72.5Cu_17.5Al₁₀), quaternary CuZrNiAl (with nominal compositions of Zr₄₅Cu₃₅Ni₁₀Al₁₀, Zr₅₀Cu₃₀Ni₁₀Al₁₀, Zr₅₅Cu₂₅Ni₁₀Al₁₀, Zr₆₀Cu₂₀Ni₁₀Al₁₀) alloys ingots were initially produced by arc melting mixtures of the raw metals Cu, Zr, Al, and Ni (with a purity of $\geq$ 99.9 wt. %) in an Ar atmosphere (with a purity of $\geq$ 99.9999%) purified with a Ti getter. All compositions are given in atomic percent. The ingots were flipped and remelted four times to ensure compositional homogeneity. Subsequently, glassy ribbons with a thickness of approximately 35 $\mu$m were produced by melt spinning on a single Cu roller at a wheel surface speed of 35 m s^-1 in a high-purity Ar atmosphere.

For powder preparation, the mixtures of elemental powders were mechanically alloyed. First, high purity metal powders ($\geq$ 99.99 at. %) were weighted according to the corresponding nominal composition and blended in a plastic vessel at a speed of 50 rpm for 24 h under a high-purity Ar atmosphere. The blends were then mechanically alloyed in a planetary ball mill at speeds of 450 and 350 rpm (QM-2SP20; apparatus factory of Nanjing University, Nanjing, China) under a protective atmosphere of high-purity Ar atmosphere. A WC ball with a diameter of 20 mm and a ball to powder weight ratio (BPR) of 10:1 was used. As a parallel comparison, the milling experiments of speed 450 rpm and BPR 7:1 was performed. Approximately 5 g of the as-milled powders were taken out from the WC vials in intervals of 5, 8, 11, 14, 17, 20, 23, 26, 30, 35, 40, 45, 50, 55, 60, 65, 70, and 75 h for phase evolution and thermal analysis. Generally, it is accepted that only the presence of a diffraction halo indicates the full amorphization of element powders during ball milling. From a calorimetric point of view, amorphous phase exhibits an exothermic behavior at the elevated temperatures, called, crystallization (Fig. S1 middle part). Due to the accumulation of disordered structures (amorphous phase) during ball milling, the value of crystallization enthalpy $H_{x}$ increased monotonically with milling times and reached a maximum one $H_{x}^{\text{max}}$ until full amorphization (Fig.S1 lower part). Furthermore, the process of mechanical amorphization can be characterized by the reduced parameter $H_{x}/H_{x}^{\text{max}}$, which is consistent with the evolution of XRD patterns (Fig.S1 upper part). Furthermore, high-resolution TEM images of as-milled amorphous phases presented the similar maze-like patterns with that of as-cast counterpart (Fig.S2). Hence, the MAA can be evaluated by amorphization time $t_{a}$Ge2017 , corresponding to the moment that $H_{x}/H_{x}^{\text{max}}$ is equal to 1. The related details of the remaining alloys were seen from Figs. S3,S4,S5,S6.

Each tested sample of approximately 25 mg was analyzed using a synchronous thermal analyzer in alumina crucibles under a high-purity of Ar atmosphere (STA-449 F3, NETZSCH, Germany). To ensure the reliability of the data, temperature and enthalpy were calibrated with an indium and a zinc standard specimen, giving an accuracy of $\pm$0.1 K and $\pm$0.01 mW, respectively. The scanning procedure was conducted at a heating rate of 20 K min^-1 until complete melting, and then cooled to ambient temperature at 40 K min^-1. A second run using the same procedure was used as a baseline for subtraction from the first run. The glass transition temperature ($T_{g}$), onset of crystallization temperature ($T_{x}$), and liquidus temperature ($T_{l}$) were determined as the intersection points of the tangents at the inflection points. The enthalpy of crystallization ($H_{x}$) was measured from the area between two curves. To investigate the kinetics of crystallization, the scanning procedure was carried out up to 823 K with various heating rates of 5, 10, 20, 50, 100, and 200 K min^-1 (DSC 8000, PE, USA) to determine the $T_{x}$ dependent on heating rates. A second run using the same procedure was used as a baseline for subtraction from the first run. Heat capacity ($C_{p}$) of alloys considered here was obtained by comparing with that of a sapphire standard sample. Identical measurement procedures were performed on the empty pan as a baseline to be subtracted from the sample and the sapphire. The specific heat capacity of the sample can be determined by

$$C_{p}^{\text{sample}}=C_{p}^{\text{Sapphire}}\times\frac{M_{\text{sapphire}}}{M_{\text{sample}}}\times\frac{Q_{\text{sample}}-Q_{\text{pan}}}{Q_{\text{Sapphire}}-Q_{\text{Pan}}}$$

(1)

where $M_{i}$ and $Q_{i}$ are the mass and heat flow of sample, empty pan, and sapphire respectively; and $C_{p}^{\text{Sapphire}}$ represents the heat capacity of the standard sapphire.

Samples were placed in a borosilicate capillary with an outer diameter of 0.5 mm and a wall thickness of 0.01 mm. High-resolution X-ray diffraction (HRXRD) measurements were performed with a rotating Ag anode at a wavelength $\lambda$ of 0.056 nm and a beam size of 8 mm $\times$ 0.4 mm at a power of 9 kW (Smart Lab, Rigaku, Japan). The sample scanning speed is 1.5$\degree$ per minute and the scanning range is from 20$\degree$ to 90$\degree$. Before amorphous samples were tested, a borosilicate capillary is firstly examined by HRXRD as a baseline.

Neutron diffraction measurements were performed on the Multiple Physics Instrument (MPI) at China Spallation Neutron Source (CSNS)Xu2021multiphyiscs . Approximately 3 g of samples were placed in a vanadium can with an inner diameter of 8.9 mm and thickness of 0.25 mm. Single Diffraction pattern was measured for 6 h at ambient condition in the high-flux mode. The scattering data were analyzed using the Mantid software, which corrected, normalized and collated the information from the 7 banksArnold2014Mantiddata . To obtain a high $Q$ resolution over a large distance range, the instrument resolution was initially assessed using a Si powder (SRM 640f, NIST) before amorphous samples were tested. The $dQ/Q$ was expected to reach 0.39% at 1 Å. The normalized shape profiles of single pixel for the Si (422) peak show that the resolution of FWHM is 0.36%.

TEM specimens were carefully ion milled with 3 keV Ar ions for about 5 h at the liquid nitrogen temperature (PIPS II-695.c, Gatan, USA). High-resolution transmission electron microscopy (HRTEM) measurements were performed using a field emission gun TEM (JEM F200, JEOL, Japan) at 200 kV.

The calculated difference in the Gibbs free energy $\Delta g$ between the liquid and crystalline states is given by

$$\Delta g\equiv\Delta H-T\Delta S=(\Delta H_{f}-\int_{T}^{T_{f}}\Delta C_{p}^{l-c}(T)dT)-T(\Delta S_{f}-\int_{T}^{T_{f}}\frac{\Delta C_{p}^{l-c}(T)}{T}dT)$$

(4)

where $\Delta H_{f}$ is the heat of fusion, $\Delta S_{f}$ is the entropy of fusion, rendering $\Delta S_{f}=\Delta H_{f}/T_{f}$. $T_{f}$, the temperature at which the Gibbs free energy of the liquid and the crystalline states are equal, was taken to be the temperature at which the endothermic peak is maximum during melting. The heat capacity of a crystal well above the Debye temperature can be described byGlade2000ThermodynamicsOC

$$C_{p}^{c}(T)=3R+aT+bT^{2}$$

(5)

The heat capacity of an undercooled liquid can be described byGallino2010KineticAT [41]

$$C_{p}^{l}(T)=3R+cT+dT^{-2}$$

(6)

where $R$ is gas constant, rendering R= 8.3145 J g atom^-1 K^-1, and $a$, $b$, $c$, and $d$ are fitting constants. The constants for both fits to the specific heat capacity data for each of alloy systems were summarized in Supplementary Table S2. The difference in the specific heat capacity of the liquid and the crystalline states, $\Delta C_{p}^{l-c}$, was shown in Fig. S27. The counterparts for binary and quaternary were depicted in Figs. S16a and S17a. Correspondingly, Figs. S16b, S17b, and S28 show the difference in the Gibbs free energy $\Delta{g}$ between the liquid and crystalline states for binary, quaternary, and ternary alloy systems respectively. In general, interfacial energy $\gamma_{c/a}$ is difficult to be experimentally measured. Turnbull pointed that a relation exists between the liquid-crystal interfacial energy and the heat of fusionTurnbull1950FormationOC . Such a relation can be described by

$$\gamma_{c/a}=K\Delta H_{f}V^{-2/3}N_{A}^{-1/3}$$

(7)

where $V$ is the gram-atomic volume of the crystalline phase, $N_{A}$ is the Avogadro constant, $K$ is the fitting parameter. Turnbull also found that while data for most of metals fit on a line with slope $K=0.45$, Ge, Sb, and Bi were best fit by a line with slope 0.32Turnbull1950FormationOC . Hence, $K=0.45$ is used in this work.

Regardless of shear stressHe2016insitu ; Hu2023amorphous or hydrostatic stressHemley1988PressureinducedAO ; Zhao2015pressure , each of stress state can drive amorphization. Both stress components (including the maximum shear stress or hydrostatic stress P) are determined by the generalized Hooke’s law, as detailed in Supplementary Note 1. Patel and CohenPatel1953CriterionFT were the first to quantitatively study the effect of stress states on the phase transformation under quasi-static loading by rationalizing the total work, $W=P\epsilon_{v}+\tau\gamma$. Here, $\epsilon_{v}$ is volume shrinkage between amorphous and crystalline phase. The atomic volume $v_{a}$ of amorphous phase is determined by the relationship proposed by Ma et al.Ma2009PowerlawSA based on experimental values of $q_{1}$ (the position of first peak in plot of $S(Q)$ versus of Q, see Figs. S8a and S9). The of crystalline phase is estimated from the equilibrium phase diagram using the lever rule. $\gamma$ is the shear strain, rendering (G is the shear modulusWang2012TheEP ). Hence, $W=P\frac{v_{a}-v_{c}}{v_{c}}+\frac{\tau^{2}_{max}}{G}$.

In this study, we selected a series of Zr-based alloys as representative materials. These alloys are renowned for their superior GFA and have been extensively examined in the prior studiesPhysRevLett.94.205501 ; PhysRevLett.115.165501 ; Li2008matching ; Zhu2016 ; PhysRevLett.106.125504 ; Luo2018UltrastableMG ; Yu2013ultrastable . Specifically, we considered binary ZrCu, ternary ZrCuAl, and quaternary ZrCuNiAl alloys with varying Zr content, denoted as B-Zr_x, T-Zr_x, and Q-Zr_x ($x$ representing the Zr atomic content), respectively. The composition variations within the binary ZrCu, ternary ZrCuAl, and quaternary ZrCuNiAl alloys are detailed in Figs. 1 a, b, and c respectively.

We commenced by characterizing the GFA of each composition using three well-established parameters: the calorimetric parameter reduced temperature $T_{rg}$ ($=T_{g}/T_{l}$)Turnbull1969UnderWC , the energetic parameter crystallization activation energy $E_{c}$Wang2012TheEP , and the structural parameter full width at half maximum (FWHM) of the first peak in $S(Q)$Li2021datadriven . Detailed experimental procedures are outlined in the Experimental section and the Supplementary Materials (Supplementary Note 2 and Fig. S8). For alloys with stronger GFA, all $T_{rg}$, $E_{c}$, and FWHM are expected to be larger. In our investigated systems, these parameters exhibit a consistent correlation with each other, as depicted in Figs.1 d, e, and f for binary, ternary, and quaternary alloys respectively. Specifically, the FWHM of T-Zr_72.5 and T-Zr₄₅ are a minimum and maximum value of 0.41504 and 0.56429 Å, respectively, which are among the reported range of $0.28\sim 0.67$ Åfor as-deposited ZrCuAl glass libraryLi2021datadriven . Obviously, the values of T_rg and for the bulk glass former T-Zr₄₅ are 0.616 and 0.408, larger than those of the marginal glass former T-Zr_72.5, respectively. From a structural and calorimetric point of view, these results indicate that GFA of ternary alloy system herein decreases with the increasing Zr content. Similarly, a relationship between GFA and Zr content of alloy materials also exists in binary and quaternary systems as shown in Figs.1 d and f respectively. The related details of binary and quaternary alloys are seen from the Fig. S9.

In our investigation of mechanical amorphization, we employed a technique known as mechanical alloying (MA). This process involves subjecting a mixture of powders to high-energy collisions by milling balls within a confined vessel. The continuous input of energy through mechanical milling imparts disorder to the powder materials, facilitating their transformation towards an amorphous phaseSuryanarayana2019MechanicalAA . There are several advantages to using MA for studying mechanical amorphization. Firstly, it offers a high degree of control compared to other methods such as shock waves or intense plastic deformationGe2017 . Secondly, it enables the attainment of a fully amorphous state in the materialEckert1997mechanical . Moreover, perhaps the most crucial advantage is that MA provides an effective means to characterize mechanical amorphization ability. Specifically, the time taken for milling to achieve a fully amorphous state can serve as a quantifiable measure of this ability.

As shown in Fig. 2, XRD patterns (Fig. 2a) and heating flows (Fig. 2b) of T-Zr_72.5 as a function of milling times depict the evolution of amorphization. Apparently, all the originals consist of peaks of the elementary metals Cu, Zr, and Al (Fig. 2a). With the increasing milling time, these diffraction peaks gradually attenuated and broadened, which is attributed to the introduction of disorder structures with collision-induced deformation. Such an interfacial reaction in a metastable state is promoted by continuous cold welding and fracturing of powder particles to create alternating layers with fresh interfaces, and by the generation of the abundant defects during ball millingLund2004MolecularSO ; Hellstern1986amorphization ; PhysRevLett.65.2019 . The composition varies negligibly with the advance of milling and keeps nearly with the equimolar ratio between Cu and Zr elements (see Fig. S22). In addition, no detectable contamination is found during MA. From a calorimetric point of view, due to the accumulation of disordered structures during the ball milling, this metastable phase exhibits an exothermic behavior at the elevated temperatures, called, crystallization (Fig. 2b). Furthermore, as shown in Fig. 2c, the value of crystallization enthalpy $H_{c}$ increases monotonically with milling times and reaches a maximum when the whole amorphous phase is formed. Remarkably, the amorphous microstructure obtained after ball milling may be different from that obtained by melt-spun. Mechanical amorphization via ball milling is indeed a thermodynamic-controlled stochastic process and $t_{a}$ is statistical index of the global process, similar to incubation timeTurnbull1969UnderWC during the crystal nucleation process in the undercooled liquid. Hence, this difference does not affect that $t_{a}$, the time required to achieve amorphous phase via ball milling, is capable to discuss the composition dependence of MAA.

Next, we characterized the MAA using the milling time $t_{a}$ required to achieve a fully amorphous state Generally, for superior MAA, the $t_{a}$ is shorter, and vice versa. Another parameter used to characterize MAA is the input energy $E_{\text{input}}$ for achieving full amorphization during milling, which exhibits consistent results with $t_{a}$ (see Supplementary Note 3). For a specific component, the MAA, that is, the time of amorphization does change with the milling speed and BPR (Fig. 2d). Intrinsically, the variation of MAA at different milling conditions is ascribed to the underlying mechanical work. Previous studies have shown that under the same milling experimental conditions, the difference in MAA of different compositions is mainly explained by the criterion of GFASharma2008EffectON ; Sharma2007criterion ; Zou2013effect ; Yang2015effect , in other words, the system with large GFA tends to have a large MAA (Fig. S26c). Phenomenally, the empirical criteria of MAA proposed by Suryanarayana (Fig. S26a) are analogous to the Ionue’s three rules (Fig. S26b), such as atomic size difference and negative mixing enthalpy. However, Fig. 2e show that two types of correlations between MAA and GFA were both found and the positive one shifts toward the negative with the elevated milling speed (Fig. 2f), underlying the effect of mechanical work. Both the GFA indicators, $T_{rg}$ (Fig. 2g) and FWHM (Fig. S29) parameters, exhibit an inverse relationship with MAA indicator, $1/t_{a}$. The similar relationship exists in plots of $E_{c}$ with $E_{\text{input}}$ (Fig. S15). These suggest that, for materials with excellent GFA, the milling time $t_{a}$ tends to be longer, indicating a poorer MAA (green line of Fig. 2e). This observation challenges the prevailing empirical impression that a higher GFA in an alloy result in a faster mechanical amorphization However, our experimental findings reveal an inverse correlation between MAA and GFA.

In our molecular simulations, we focused on two model alloys, equimolar CuZr and CuZr₂, to investigate the phenomenon of mechanical amorphization. These two alloys are well-suited for this study due to their distinct GFA and their extensive examination in molecular dynamics (MD) simulationsHu2020physical . The polycrystalline structure comprises approximately 80% of these two phases. Notably, CuZr exhibits superior GFA compared to CuZr₂. And we ensured the construction of polycrystalline structures for both model systems by employing similar grain sizes and crystalline orientations and the corresponding grain boundary energy is comparable (for a comprehensive description of the modeling procedures, please refer to the Experimental section). Fig. 3a illustrates the evolution of the Disorder degree over loading time. When the Disorder degree reaches a plateau value, we define this time as the mechanical amorphization time, denoted as $t_{a}$. Consequently, the MAA of a system can be effectively characterized by its corresponding $t_{a}$. In the regular cases, where $t_{a}$ is shorter, the MAA is considered better. For amplitude strain $\gamma_{A}=0.5$, we observed that $t_{a}^{\text{CuZr}}<t_{a}^{\text{CuZr}_{2}}$, aligning with the conventional impression that strong glass-forming systems tend to exhibit better MAA. However, as depicted in Fig.3b, for $\gamma_{A}=0.6$, a different narrative unfolds. Here, the situation is reversed, with $t_{a}^{\text{CuZr}}<t_{a}^{\text{CuZr}_{2}}$ , consistent with the aforementioned experimental finding that stronger glass-forming systems paradoxically possess poorer MAA. This intriguing reversal indicates that the relationship between MAA and GFA is highly sensitive to external loading conditions. To delve deeper into this sensitivity, we conducted a series of systematic tests under various loading conditions, as elucidated in Fig.3c. These results unequivocally emphasize that the correlation between MAA and GFA is indeed contingent upon the subtle external loading conditions imposed on the system. Notably, for small stress, the MAA appears consistent with GFA, whereas for larger stress, the MAA inversely correlates with GFA. Furthermore, Fig.3d provides valuable insights into the spatial evolution of the Disorder degree throughout the mechanical amorphization process, hinting at a behavior reminiscent of melting.

The glass forming ability of an alloy system is theoretically attributed to two factors: the resistance to nucleation of crystallization and the resistance to growth of crystallization Turnbull1969UnderWC ; Johnson2016quantifying . The resistance to nucleation can be measured by the critical nucleation barrier ($\Delta G_{c}^{*}$). Based on classical nucleation theory Becker1935KinetischeBD , the nucleation barrier can be expressed as follows:

where $\delta g_{l-c}$ represents the Gibbs free energy difference between the liquid and crystalline states, and $\gamma_{l-c}$ denotes the interfacial energy between the amorphous and crystalline phases. Both $\gamma_{l-c}$ and $\delta g_{l-c}$ can be determined experimentally (see Methods). As illustrated in Fig. 4a, c, and e, $\Delta G_{c}$ evolves with the crystal nucleus size r in binary, ternary, and quaternary systems, respectively. The highest energy barrier $\Delta G^{*}_{c}$ in Fig. 4, is referred to as the critical nucleation barrier. For our investigated systems, we found that all the GFA parameters used in the experiment are positively correlated with the critical nucleation barrier and negatively correlated with the Gibbs free energy difference $\delta\gamma_{l-c}$. This indicates that the GFA of our investigated systems is primarily dominated by the resistance to nucleation.

The control factors and mechanisms for modulating MAA remain enigmatic. Notably, the stress state appears to play a pivotal role in reconciling the disparities between experimental and simulation outcomes. From the perspective of the potential energy landscape (see Fig. 5 a), it becomes evident that mechanical amorphization and glass forming represent two fundamentally distinct pathways toward the formation of an amorphous phase. One pathway involves the introduction of disorder structures from a crystalline state, while the other revolves around the preservation of disorder structures and the avoidance of crystallization. Preservation of disorder structures in the liquid state can be likened, thermodynamically, to the energy barrier that resists crystallization ($\Delta G_{c}$), In the context of mechanical amorphization, the process of crystalline structure dissolution bears a resemblance to amorphous embryo nucleation phenomena (Fig. 3d). At a temperature below melting temperature, the amorphization transformation must overcome a high energy barrier, making it impossible to occur under ambient condition. However, the high magnitude of the coupled hydrostatic pressure and associated deviatoric component dramatically lowers the energy barrier, rendering the amorphization transformation possible under shock compression. Following the Patel and Cohen methodology, the energetics of solid state amorphization of covalently bonded solids, such as siliconZhao2016AmorphizationAN ; PhysRevB.96.054118 , germaniumZhao2018ShockinducedAI , SiCZhao2021directional ; Levitas2012HighdensityAP , and B₄CZhao2021directional , were analyzed by calculating of the external work varied with pressure and shear. In this regard, we introduce a modified version of the classical nucleation theory, widely employed to characterize the transition from a crystalline to an amorphous stateGlade2000ThermodynamicsOC ; Zhao2016AmorphizationAN . Within this framework, the energy barrier resisting amorphization, denoted as $\Delta G_{a}$, can be expressed as:

$$\Delta G_{a}=4\pi r^{2}\gamma_{c/a}+\frac{4}{3}\pi r^{3}\Delta g_{c-a}-\frac{4}{3}\pi r^{3}W$$

(8)

where $\Delta g_{c-a}$ signifies the Gibbs free energy difference between the amorphous solid and crystalline states, while $\gamma_{c/a}$ characterizes the interfacial energy between these two phases. $W$ represents the mechanical work induced by external loading. Note that, the introduction of crystal defects into metal powders during mechanical amorphization may lead to an increase in the free energy of the crystal phase, surpassing that of the hypothetical amorphous phase. Hence, the third term on the right-hand side of Eq. 8 represents the contributed energy penalty to overcome the energy barrier of amorphization, including the grain boundaries energy contributed by the exterior stress. As a result, the most significant energy barrier, denoted as, stands as a pivotal metric for the assessment of a material’s MAA. Notably, a higher implies a longer duration required to achieve amorphization, thus signifying a poorer MAA. At lower temperatures, particularly below the glass transition temperature, and exhibit insensitivity to temperature fluctuationsTurnbull1950FormationOC ; Busch1995ThermodynamicsAK . We therefore adopt the approximations $\gamma_{c/a}\approx\gamma_{l/c}$ and $\Delta g_{c-a}\approx\Delta g_{l-c}$. The formula for mechanical work $W$ depends on the stress conditions. Here, we assume the material undergoes uniaxial compressive stress, denoted as $\sigma_{33}$. It’s important to note that different stress conditions can alter the value of $W$, but they don’t change the qualitative conclusions. The relationship between $W$ and $\sigma_{33}$ is detailed in the Experimental section.

Fig. 5c clearly demonstrates that various loading stresses notably influence MAA. Under a fixed loading stress condition, such as $\sigma_{33}=9.5$ GPa, different materials exhibit varying MAA levels (see Fig. 5d), consistent with the experimental results presented in Fig. 2 d. Moreover, the correlation between and reaffirms the inverse relationship between MAA and GFA in ternary ZrCuAl alloy systems (Fig. 5e), in line with the experimental observations (Figs. 2 f and 2 g).

Therefore, using the above calculation frameworks, we can establish the relationship between MAA and GFA as follows:

$$\Delta G_{a}^{*}\approx\Delta G_{c}^{*}\frac{1}{(W/\Delta g_{c-a}-1)^{2}}$$

(9)

Evidently, MAA is influenced by both external mechanical work and the composition properties, specifically the energy barrier $\Delta G_{c}^{*}$ and the free energy difference $\Delta g_{c-a}$. Eq. 9 can be dissected into two components concerning composition sensitivity. Firstly, $\Delta G_{c}^{*}$ characterizes the GFA of the composition which is insensitive with external loading. Secondly, the term $1/(W/\Delta g_{c-a}-1)$ delineates the interplay between external loading conditions. At low stress levels, the external work is comparable with the free energy difference $\Delta g_{c-a}$ and the term $1/(W/\Delta g_{c-a}-1)$ exhibits sensitive to composition, particularly a positive correlation with $\Delta g_{c-a}$. For stronger glass-forming systems, this value tends to be smaller, as exemplified by the notable difference between the marginal glass-former B-Zr₆₅ and the good glass-former B-Zr₅₀ (about one order, see Fig. 6). In contrast, at high stress levels, where $\Delta\ll\Delta g_{c-a}$, the composition insensitivity becomes apparent, leading to significantly reduced differences between B-Zr₅₀ and B-Zr₆₅. In this regime, all values fall within a comparable range of $0.8\sim 1.1$ (Fig. 6). Therefore, the composition sensitivity of MAA is predominantly governed by $\Delta g_{c-a}$ at low stress levels and shifts towards being dominated by $\Delta G_{c}^{*}$ at high stress levels.

We tested a series of loading stresses using Eq. 9, as shown in Fig. 7 a-c, all the experimental systems, including binary, ternary, and quaternary alloy systems, reveal a crossover correlation between MAA and GFA. Under low-stress conditions, MAA exhibits a positive correlation with GFA (Fig. S19), while under high-stress conditions, MAA shows an inverse correlation with GFA. This finding aligns with the simulation results (Fig. 3c) and helps reconcile the discrepancy between current experimental results and established impressionsSharma2008EffectON ; Sharma2007criterion ; Zou2013effect . Note that, the critical stress level responsible for this reversion varies with the alloy systems. At this stress level, it becomes evident that the MAA becomes entirely independent of GFA, highlighting the sensitivity of the correlation between MAA and GFA to external stress levels (Figs. 7a-c). It is intriguing to observe the inverse correlation between MAA and GFA under large stress conditions (Fig. 7d). This suggests that for materials with poor GFA, mechanical amorphization can be highly efficient, thus offering a novel avenue to expand the range of attainable glassy states PhysRevB.56.R11361 ; Eckert1997mechanical . This finding is particularly fascinating in the context of monatomic glassesZhong2014FormationOM ; ZHAO2021114018 , where mechanical amorphization, through processing techniques like shocking or various deformation loadings, could potentially offer significant advantages over rapid coolingZhao2018ShockinducedAI ; He2016insitu ; Zhao2016AmorphizationAN . Furthermore, this discovery opens up possibilities for generating new amorphous phases, especially in materials with poor GFA, such as amorphous iceRosuFinsen2023medium . Additionally, the tunable nature of mechanical amorphization provides valuable insights into controlling amorphization in material design. For instance, the creation of gradient amorphous-crystalline nanostructures with superior performance becomes feasible by mechanical amorphization techniques, such as ultrasonic vibrationFan2023rapid .

According to the generalized Hook’s law, $\sigma_{ij}=C_{ij33}\epsilon_{33}$ , where $C_{ijkl}$ presents the elastic constants, $\epsilon_{33}$ presents the uniaxial strain. Correspondingly, the hydrostatic pressure $P$ and maximum shear stress $\tau_{\text{max}}$ can be determined by $P=\frac{1+\nu}{3(1-\nu)}\sigma_{33}$ and $\tau_{\text{max}}=\frac{1-2\nu}{2(1-\nu)}\sigma_{33}$, where $\sigma_{33}$ is the impact stress. In terms of ZrCu-based metallic glass systems, the Poisson’s ration $\nu$ is under the range of $0.3\sim 0.32$. Hence, the value of v is 0.305 for all of alloy systems herein.

The apparent activation energy $E_{c}$ of crystallization is determined under isochronal heating conditions using the following Kissinger equation

$$\ln T^{2}/B=\frac{E_{c}}{k_{B}T}+C$$

(S1)

where $k_{B}$ is the Boltzmann’s constant, B is the heating rate (K s^-1), $C$ is the constant and T is the specific temperature dependence of heating rate. Generally, T is denoted as the temperature corresponding to the beginning or to the peak of the exothermic crystallization event detected by DSC. Then, through plotting $\ln(T^{2}/B)$ as a function of $1/T$, the quantity $E_{c}$ can be determined by the slope. In this work, $T$ is taken as the onset crystallization temperature $T_{x}$ and $E_{c}$ is understood as crystallization activation energy in this case. Correspondingly, Fig. S13 shows the valid of this dependence.

In terms of a planetary milling, the geometry of one vial can be schematized in Fig. S20. It can be expressed that the absolute velocity of one peripheral point $M$ can be depicted by the following equation:

$$V_{M}=[(W_{P}R_{P})^{2}+(W_{V}R_{V})^{2}+2W_{P}R_{P}W_{V}R_{V}\cos{\theta}]^{1/2}$$

(S2)

where $W_{P}$ and $W_{V}$ (RPM, rotations per minute) present the absolute angular velocity of the milling plate and steel vial respectively; $R_{P}$ and $R_{V}$ (mm) present vectorial distances from the center of the plate to the center of the vial and radius of the vial respectively; $\theta$ is the angle formed by the vectors of $\vec{R}_{P}$ and $\vec{R}_{V}$. It is assumed that the steel ball moves without sliding at the point $M$ (Fig. S20). At a specific moment, it is launched towards the opposite inner wall. And the ball moves back to the wall after a short hit and is accelerated by the vial at the next launch cycle. Note that the previous hypotheses are realistically starting from the moment at which a thin powder layer covered the ball. Hence, the condition for the ball detached from the inner wall of vial is expressed by:

$$\cos{\theta_{b}}=\frac{W_{V}^{2}R_{V}}{W_{P}^{2}R_{P}}$$

(S3)

Combined Eqs. S2 and S3, the absolute velocity of one ball leaving the wall can be expressed by:

$$V_{b}=[(W_{P}R_{P})^{2}+W_{V}^{2}(R_{V}-d_{b}/2)^{2}+2(1-2W_{V}/W_{P})]^{1/2}$$

(S4)

where $d_{b}$ (mm) refers to the diameter of one ball. When attached solidly again with the wall, the velocity of the ball after a hit, $V_{s}$ , equal to that of the inner wall and can be given by:

$$V_{s}=[(W_{P}R_{P})^{2}+W_{V}^{2}(R_{V}-d_{b}/2)^{2}+2W_{P}R_{P}W_{V}(R_{V}-d_{b}/2)]^{1/2}$$

(S5)

When the ball is launched, the energy can be shown as:

$$E_{b}=\frac{1}{2}m_{b}V_{b}^{2}$$

(S6)

where $m_{b}$ (g) refers to the mass weight of one ball. After collision, a fraction of energy is released in ways of deformation of powder materials and instantaneous temperature rise of systems. The remaining energy of the ball becomes:

$$E_{s}=\frac{1}{2}m_{b}V_{s}^{2}.$$

(S7)

Hence, the released energy $\Delta E\equiv=E_{b}-E_{s}$ of one ball during a series of collision events can be expressed by:

$$\Delta E=-m_{b}[W_{V}^{3}(R_{V}-d_{b}/2)/W_{P}+W_{P}R_{P}W_{V}](R_{V}-d_{b}/2)$$

(S8)

In general, a number of balls, $N_{b}$ , widely used to accelerate the process of amorphization, hinder each other so that Eq. S8 must be considered the effect of filling degree of vial by introducing a related empirical parameter, $\phi_{b}$ . To simplification, two boundary cases is: i) $\phi_{b}=1$ for only one or few balls and ii) $\phi_{b}=0$ when the vial is completely filled by the balls, no movement occurs. Correspondingly, Eq. S8 is modified as in a realistic experiment:

$$\Delta E^{*}=\phi_{b}\Delta E$$

(S9)

where $\Delta E^{*}$ refers to the kinetic energy of one ball in the system including $N_{b}$ balls. Hence, the total released energy $P$ can be given by:

$$P=\Delta E^{*}\phi_{b}N_{b}f_{b}$$

(S10)

where $f_{b}$ refers to the frequency of launching for ball, which is proportions the relative rotation speed, $f_{b}=K(W_{P}-W_{V})/2\pi$. Therefore, Eq. S10 is modified as normalizing by powder weight $m_{p}$ and multiplying by milling time t denoted as $E_{\text{input}}\equiv P_{t}/Km_{P}$:

$$E_{\text{input}}=-\phi_{b}N_{b}m_{b}t(W_{P}-W_{V})[\frac{W_{V}^{3}(R_{V}-d_{b}/2)}{W_{P}}+W_{P}R_{P}W_{V}](R_{V}-d_{b}/2)/2\pi m_{p}$$

(S11)

Note that, the above model can be used to determine the released energy by collision between balls and vial walls, while the focused energy transferred to powder materials is apparently only a portion of the derived released energy $E_{\text{input}}$. In spite of the restrictions and hypotheses, the aforementioned model was used to rationalize the previously reported experimental results.

As far as $\phi_{b}$ is concerned, it is found convenient to express it as a function of two parameters $n_{v}$ and $n_{s}$:

$$n_{v}=N_{b}/N_{b,v}$$

(S12)

Where $N_{b,v}$ refers to the number of balls that can be contained in a simple cubic arrangement in the vial and is given by the following estimation

$$N_{b,v}=\pi D_{v}^{2}H_{v}/4d_{b}^{3}$$

(S13)

Where $H_{v}$ and $D_{v}$ is the height and diameter of the vial;

$$n_{s}=N_{b}/N_{b,s}$$

(S14)

Where $N_{b,s}$ refers to the number of balls needed to cover one third of the inner surface in a simple cubic arrangement and is shown as

$$N_{b,s}=\pi(D_{v}-d_{b})H_{v}/3d_{b}^{2}$$

(S15)

In order to derive a simple analytical expression for $\phi_{b}$ , it is assumed that i) $\phi_{b}=1$ for $n_{v}=0$ (vial is completely empty); ii) $\phi_{b}=0$ for $n_{v}=1$ (vial is completely filled up); iii) $\phi_{b}$ is close to 1 (e.g. 0.95) for $n_{s}=1$ (this assumed that the reciprocal hindering of the ball is negligible until one third of the inner surface wall is not covered). According to above assumptions, a simple formulation of $\phi_{b}$ can be estimated by

$$\phi_{b}=1-n_{v}^{\epsilon}$$

(S16)

Where $\epsilon$ refers to a parameter dependent of ball diameter that can be evaluated by the assumption (iii):

$$0.95=1-(N_{b,s}/N_{b,v})^{\epsilon}$$

(S17)

In this work, the evolution of XRD patterns corresponded to a feature of type II that a decrease of elemental peaks accompanied with an increase of the amorphous broad peak for all of compositions herein (see the upper part of Figs. S1, S2, S3, S4, S5, S6). We supplemented the eds mapping to further detect the evolution of structures with the increase of milling time for the binary system. As shown in Fig. S21, each elements distributed incompatibly and no compounds formed at early milling of Zr and Cu elemental powders, corresponding to the peak intensity of pure elements decreased with the simultaneously increased amorphous halo peak. The identical amorphization type of here-considered alloy systems provided a prerequisite for discussion about relationship between MAA and GFA.

As shown in Fig. S23, $E_{c}$ is well correlated with GFA indicators FWHM and $T_{rg}$, which is consistent with the reported studies[6,7]. In terms of ball milling, a large portion of the input work $E_{\text{input}}$ can be dissipated in the form of heat, a small portion of which is stored in the crystalline powder material, achieving transformation from order to disorder.

To an extent, $E_{\text{input}}$ varied with compositions may be considered as a parameter reflecting the difficulty of amorphization transformation. That means that the poor MMA systems need more $E_{\text{input}}$ for amorphization during milling. Hence, the negative plot of $E_{c}$ with $E_{\text{input}}$ (Fig. S15) can be considered an energetic expression for the inverse relationship between GFA and MAA.