Medium-range order structure analysis of metallic glasses based on reverse Monte Carlo modeling

The elucidation of amorphous alloy characteristics presents a formidable challenge due to the absence of crystalline arrangements and enigmatic imperfections. This study aimed to tackle this challenge through a comprehensive investigation utilizing synchrotron X-ray diffraction (XRD), extended X-ray absorption fine structure spectroscopy (EXAFS), ab initio molecular dynamics (AIMD) simulations, and reverse Monte Carlo (RMC) modeling. The primary objective was to delve into the medium range order (MRO) structure of metallic glasses (MGs) containing CuXZr100-X, wherein X represents values of 50, 56, 60, and 64. The results obtained from this research demonstrate that an increase in the Cu atom concentration within Cu-Zr MGs leads to the formation of complete icosahedral clusters and high-coordination polyhedral connections, thereby bolstering the stability and heterogeneity of local structures. Examination of the stiff full icosahedral bones shows that defects exist in the network connections on the MRO, resulting in a lack of connectivity in the positive icosahedron. Conversely, the density of defects decreases with an increase in Cu content. The observation suggests that the observed low MRO defects in Cu64Zr36 MGs may be accountable for their excellent glass-forming ability and kinetic stability.


Introduction
Microscopic defects are ubiquitous in various materials and significantly impact their macroscopic mechanical behavior [1] .As a result, a considerable number of microstructure-based studies have been conducted to comprehend and characterize the intrinsic material defects and reveal their response mechanisms under external loading [2] .Recently, Voronoi analysis has been introduced in metallic glasses (MGs) to provide better insights into the short-order range microscopic arrangement of such disordered structures [2,3] .On this basis, the Voronoi index [2] and the local five-fold symmetry (LFFS) [4] parameter, among others, have been proposed to establish a relationship between MG structure and properties.Previous molecular dynamics simulations [5] have unveiled that MGs possess defective forms like voids and MRO boundaries at scales of 1-2 nm, which are closely involved in porosity and shear transformation zone events [6] .However, pure MD simulation cannot entirely reflect objective facts due to significant differences in time and space scales compared to the real world.To remove this limitation, the RMC method was introduced, combining AIMD and synchrotron XRD to remodel the three-dimensional atomic structure of the Cu-Zr series [7] .
However, the partial pair distribution function (ppdf) in AIMD used as a constraint in the RMC modeling has some limitations.While density flooding theory-based AIMD can represent some main characteristics of glass in the objective world [8] , it still differs from the real cooling rate of the glass in the time domain, making this modeling method incomplete.To overcome this challenge, the K-edge spectrum of Cu/Zr obtained by the EXAFS experiment was integrated as a supplementary constraint in the RMC simulation process.The combination of AIMD, EXAFS, and synchrotron XRD data through RMC simulations leading to the acquisition of an MGs atomic structure model with diffraction characteristics is very similar to that of the actual sample.Subsequently, we applied our developed technique, as outlined in reference [9] , to elucidate the atomic structure of MGs and provide robust evidence regarding the presence of medium-range order (MRO) defects.Additionally, a comparative analysis was conducted between the previous 3D atomic models [7] to discern the disparities and investigate their underlying origins.

Experiment
The four alloys (including Cu 50 Zr 50 , Cu 56 Zr 44, Cu 60 Zr 40 Cu 64 Zr 36 ) were synthetically prepared in the laboratory, followed by the acquisition of high-resolution synchrotron XRD datasets at an energy of 17.9 keV (λ = 0.693 Å).The complete data processing workflow was executed employing PDFgetX3 software [10] .To obtain Cu and Zr K-edge EXAFS spectra, a transmission mode setup was utilized with subsequent data processing performed using Athena software [11] .The AIMD calculation involves all 256 atoms per sample model and is performed on a supercomputing platform using CP2K [12] software.The specific preparation process, synchrotron radiation experimental, and AIMD calculation procedures are detailed in previous work [7,9] .

RMC simulation
Reverse Monte Carlo (RMC) computational simulation methodologies facilitate the generation of three-dimensional structural models that align with experimental data, primarily derived from diffraction techniques.Here,   from synchrotron, XRD, Cu/Zr K-edges from EXAFS, and three ppdfs from AIMD (data from reference [7] ) are employed as constraints within the RMC simulation.It is widely acknowledged that incorporating both long-range ordered structure information from synchrotron X-ray diffraction and short-range ordered structure information dataset from EXAFS in the RMC simulation can enhance the reliability of the final structure.In the RMC simulation, each sample contains 20000 atoms with dimensions of 777 nm 3 .The initial simulated boxes were randomly placed according to the material-chemical composition ratio.The whole RMC process was performed using RMC_POT software [13] .The simulation process was closely monitored utilizing an error factor, denoted as χ², which serves as a metric for quantifying the disparity between the experimental and simulated data.
In Equation ( 1),   and   represent the experimental and simulation data, respectively.The variable i pertains to the intersection point between the two datasets concerning , with m denoting the count of such intersections.Furthermore,   signifies the experimental error associated with the ith data point.It is important to highlight that during the fitting of EXAFS data, only   necessitates replacement with the corresponding K-edge spectral data.Throughout the simulation procedure, a state of dynamic equilibrium is attained as χ² progressively decreases, indicating a favorable agreement between the simulation model and the experimental data.

Reverse monte carlo model
The structural characteristics of four MGs with the composition Cu X Zr 100−X (X= 50, 56, 60, 64) were systematically analyzed by integrating six unique data sets for each constituent material.These encompass three separate partial pair distribution functions (PDFs) generated from AIMD, structure factors derived from synchrotron XRD, and two distinct sets of K-edge spectra (Cu and Zr) acquired through EXAFS experiments.Figure 1 depicts the fit outcomes for all six constituents, indicating a strong concurrence between the computed results and the experimental measurements, which provides direct evidence that the RMC program produces a dependable and trustworthy configuration.In contrast to the previous fitting outcomes [7] that did not incorporate EXAFS data, our current K-edge spectra fitting results for Cu and Zr exhibit a remarkable consistency with the experimental measurements.Despite some discrepancies observed in the fitting process to ppdfs, as illustrated by the slightly deviated fitting outcome in Figure 1(c), the inclusion of EXAFS data has significantly enhanced the tractability of our model.Furthermore, it is important to acknowledge the phase correction applied to the K-edge spectra obtained from EXAFS measurements.

Short-range order structure
We conducted Voronoi tessellation analysis [14] to assess the short-range order of the four MGs samples investigated in this study.The statistical analysis of the twelve most prevalent cluster atoms is presented in Figure 2, which demonstrates that icosahedral-like clusters are the most abundant, particularly as the Cu atom content increases.Moreover, we quantified the 3 (including<0, 0, 12, 0>, <0, 1, 10, 2>, <0, 2, 8, 2>) occurrence of cluster atoms exhibiting high levels of fifth-fold symmetry and the highest content, which account for 0.1497%, 0.18812%, 0.25137%, and 0.25941% of the composition in four alloys MGs, respectively as shown in Table 1.These findings suggest that the solid-like region with high resistance to loading [2] expands as the Cu content increases, indicating that Cu 64 Zr 36 has a more stable structure than the other three compositions.Furthermore, we compared the RMC model before and after incorporating K-edge spectra obtained from EXAFS experiments, as detailed in Table 1.The findings indicate a minor reduction in the quantity of high-content icosahedral-like clusters while preserving the overall trend after the inclusion of K-edge spectra.
Furthermore, through our analysis, we have recognized that the incorporation of K-edge spectra into the simulation leads to the development of greater numbers of icosahedral-like clusters with high coordination numbers, such as <0, 1, 10, 4>, <0, 1, 10, 5>, and so on, which could account for the decrease in statistical rate.This observation supports the idea that the RMC model improves the packing distribution of certain elements with high coordination numbers, such as Zr, by introducing new fitting parameters.In the subsequent analysis, LFFS parameters were employed to visualize the local structural patterns of the four distinct components, as depicted in Figure 3.Our observations indicate that the distribution of LFFS is uneven, with the full icosahedron, <0, 0, 12, 0>, exhibiting a red color and an LFFS value of 1.Moreover, an increase in the Cu content was found to correlate with a broader range of red coloration, indicating a higher abundance of dominant class icosahedra as depicted in Figure 1 and Table 1.

Medium range order structure
Extensive reports [15] have highlighted that the MRO characteristics of MGs tend to provide greater insight into their dynamics, thermal properties, and plastic deformation.In our previous investigations, we employed a customized methodology [9] to extract the core structures of complete icosahedral clusters.Subsequently, we focused on characterizing the structural defects associated with MRO in our subsequent studies [7] .Specifically, we identified a type of missing degree of full icosahedral connectivity represented by weakly connected regions, which play a critical role in glass failures [5,9] .Similarly, in our new RMC model, the full icosahedral bones were extracted using a self-developed procedure as shown in Figure 4.An evident observation from our analysis is that the Cu 50 Zr 50 sample with low Cu content contains numerous super large cluster groups.With an increasing Cu content, a super bone progressively emerges, encompassing a significant portion of the atoms.The presence of MRO defects within this structure is manifested through weakly connected regions between freely formed complete icosahedral clusters and the prominent bones of the configuration.Moreover, a specific defect structure was extracted, as illustrated in Figure 5(b).Additionally, Figure 5(a) showcases the MRO defect structure obtained from the RMC model without the incorporation of Kedge spectrum data [7] .A comparison of Figures 5(a) and (b) highlights the presence of a similar boundary layer, which represents the MRO defect in both RMC models.Furthermore, this defect was also identified in our model through molecular dynamics simulations [5] , suggesting that this MRO defect is prevalent across various three-dimensional atomic models.
Our investigation encompassed a comprehensive statistical analysis of the atomic structure characteristics for each component, as detailed in Table 2. Notably, a significant rise in the number of full icosahedra is observed as the Cu content swiftly escalates from 50% to 60%, indicating an expansion of the entire full icosahedral cluster of atoms and the gradual formation of an extensive bone network.Although no further increase in full icosahedra is observed between Cu content values of 60-64%, the solid-like regions (groups of full icosahedral clusters) tend to expand, suggesting the emergence of potential MRO defect regions.However, there is an overall increase in the maximum atomic number of bones, suggesting enhanced structural stability as Cu atoms are introduced within a specific range.(b) RMC model includes EXAFS data.

Conclusion
In summary, a novel three-dimensional atomic model was constructed through the integration of AIMD, synchrotron XRD, and EXAFS data using RMC simulations.The remarkable agreement observed between the simulated and experimental data provides substantial support for the credibility of this newly proposed atomic structure.Subsequently, we conducted a comprehensive analysis of the characteristic properties of both the previous and current models employing Voronoi tessellation, LFFS, and an assessment of MRO defects.The specific conclusions derived from this analysis are enumerated below: (1) The inclusion of EXAFS data in the RMC modeling process offers increased structural information and enhances the feasibility of the proposed atomic model.
(2) Our investigation into MRO defects revealed that these defects are a common intrinsic feature across various models.
(3) It can be found that the full icosahedral atomic bone expands with increasing Cu content over a specific range, resulting in a more stable structure.This observation provides insight into why Cu 64 Zr 36 MGs exhibit superior amorphous formation ability and strong stable kinetic properties.

Figure 3 .
Figure 3.The distribution of LFFS in distinct MGs components is illustrated, with the red indicating atoms exhibiting a high LFFS and the blue representing a low LFFS.

Figure 4 . 6 Figure 5 .
Figure 4.The cube bones of Cu-Zr MGs are depicted in Figures (a-d) correspondingly.Clusters that are interconnected have been assigned the same color for visualization purposes.

Table 1 .
Statistics of high-content icosahedral-like clusters of Cu-Zr MGs from different RMC model.

Table 2 .
Statistical analysis was conducted to examine the parameters of full icosahedral bones in Cu-Zr MGs.