Exploring Amorphous Alloys: Advanced Electron Microscopy and Cluster Analysis

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1. INTRODUCTION

Amorphous alloys, or metallic glasses, exhibit extraordinary mechanical, thermal, and magnetic properties owing to their disordered atomic structures. Metallic glasses, as a broader category, offer promising potential for diverse engineering applications. A fundamental prerequisite for unlocking the full potential of amorphous alloys lies in comprehensively understanding their atomic structure and degree of ordering, a focal point of this study.

This research introduces an innovative method for exploring local atomic ordering applicable to structures with any symmetry, including non-crystallographic ones. Based on the analysis of electron microscopic images of an amorphous structure, the study reveals insights into the thermal-induced changes in the density of atomic clusters within amorphous alloys like CoP, CoNiP, and NiW.

Notably, a 30% change in the density of 1–2 nm-sized atomic clusters with an ordered structure is observed under thermal action, demonstrating both an increase and decrease in the degree of order upon heating [1].

Early investigations into metallic glasses, pioneered by Turnbull and Cohen in 1970, laid the foundation for understanding the transition from liquid to amorphous solid states [2]. The subsequent evolution of experimental techniques and computational tools has significantly deepened our comprehension of the inherent atomic structure in metallic glasses. Despite numerous studies exploring the universal characteristics of amorphous alloys, the specific intricacies of NiW alloys have remained relatively unexplored.

In the realm of electron microscopy, high-resolution techniques such as Transmission Electron Microscopy (TEM) and Scanning Transmission Electron Microscopy (STEM) have become indispensable for visualizing nanoscale structures. Pioneering studies underscored the potential of advanced electron microscopy in unraveling the complexities inherent in metallic glass structures [3; 4].

Addressing a crucial gap in the existing literature, this study focuses on the comprehensive exploration of local atomic arrangements within amorphous alloys. Through a combi- nation of advanced electron microscopy and sophisticated cluster analysis methods, the aim is to provide valuable insights into the unique atomic architecture of metallic glasses. This approach not only lays the foundation for tailored material design but also opens avenues for enhanced applications in diverse technological domains. The study delves into the atomic level characterization of metallic glass alloys, utilizing state-of-the-art electron microscopy techniques and advanced cluster analysis methods. By exploring nanoscale features and employing sophisticated analytical approaches, the goal is to illuminate the nuanced atomic arrangements and level of ordering within metallic glasses.

2. MATERIALS AND METHODS

Sample preparation

To initiate the study, samples of CoP, CoNiP, and NiW amorphous alloys were meticulously prepared using the electro-chemical deposition method. This ensured the controlled formation of amorphous structures within the alloys, laying the foundation for subsequent analyses.

Microscopy Techniques

The investigation of atomic structures involved high-resolution transmission electron microscopy (HRTEM) conducted on an FEI Titan 80–300 microscope, operating at 300 and 80 kV with aberration correction. Samples, strategically placed on standard copper grids with thicknesses ranging from 2 to 10 nm, allowed for the examination of local atomic structures, revealing distinct levels of ordering within the amorphous alloys [5].

Temperature variation study

A comprehensive study unfolded with a series of HRTEM images acquired over a temperature range extending from 20 to 300 °C. This temperature-dependent analysis provided crucial insights into the dynamic changes in the atomic structure of CoP, CoNiP, and NiW amorphous alloys under varying thermal conditions.

HRTEM image processing

Post-image acquisition, the HRTEM images underwent meticulous processing using GPU software. Initial steps involved precise image calibration for accurate measurements. To enhance clarity and reduce noise, denoising techniques were implemented. Following the denoising phase, the Particle Analyzer tool was used for particle detection, and supplementary filtering procedures were employed to refine the identification of atomic particles, considering factors such as size and circularity. The subsequent generation of coordinate points for each particle resulted in a robust dataset.

The processing involved cross-correlation with a double-core function, mathematically represented as:

$H_{\phi ,r_0}\left(x,y\right)=h\left(x,y\right)\sum _{\phi }h\left(x-r_0\sin \phi ,y-r_0\cos \phi \right),$

where h(x, y) = sin c (ρ/ρ₀) – h₀. Parameters r₀ = 0,25 nm, ρ₀ = 0,15 nm, and h₀ were precisely selected to meet the condition:

$\sum _{x,y}h\left(x,y\right)\approx 0.$

This intricate processing enhanced the resolution and clarity in the obtained HRTM images. Moreover, this holistic approach, incorporating both denoising and filtering methodologies, not only bolstered the accuracy of atomic particle localization but also played a crucial role in extracting meaningful insights into the intricate structures at the atomic scale.

Cluster analysis

In the pursuit of cluster analysis, a multi-step approach was undertaken to unravel the complex spatial relationships within the dataset.

Clusterization based on Atomic Distance

To initiate clusterization, the Euclidean distance between pairs of points (P_i and P_j) [6] within the dataset was computed using the formula:

${\text{distance}}_{ij}=\sqrt{{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2}.$

This computation resulted in a distance matrix (D), from which edges connecting points within defined threshold criteria (min_R and max_R) were formed. Consequently, an undirected graph G = (V, E) emerged, with V representing points and E representing edges within the specified distance range.

Cluster visualization

The subsequent phase involved the visualization of clusters within the undirected graph (G). Utilizing the networkx library, each cluster was represented as a subgraph (G_k) within G.

Cluster particles distribution

Quantifying the distribution of particles within each cluster became a crucial aspect of the analysis. The Radial Distribution Function (RDF) for each cluster (C_k) was calculated, considering the distances between particles (a and b) within the cluster. Statistical analysis, including mean (RDF_µ), median (Med_RDF), and standard deviation (RDF_σ), provided insights into the spatial arrangement of particles within each cluster.

Angle calculation and distribution

The calculation of angles within clusters involved both closed and non-closed loop angle calculations. In the closed loop, the angle (angle_closed) was computed as the absolute difference between two arctangents. For the non-closed loop, a similar calculation was performed.

The probability distribution of angles within each cluster was visually represented through histograms. Statistical analysis, including mean (angle_µ), median (Med_angle), and standard deviation (angle_σ), further enhanced the understanding of angle distribution within clusters.

Clustering process and visualizing clusters

The clustering process, rooted in Euclidean distances, led to the creation of an undirected graph (G) where each connected component represented a cluster. The visualization of clusters was achieved through various means:

• Graph Visualization: Subgraphs (G_k) were created for each cluster, offering a clear representation of cluster connectivity.
• Scatterplot Visualization: A scatterplot of the entire dataset, with distinct colors assigned to nodes within each cluster, provides a n overview of the spatial arrangement of clusters.
• Individual Cluster Scatterplots: Separate scatterplots were generated for each cluster, facilitating a detailed analysis of specific points constituting the cluster.

These visualization techniques not only enhanced the interpretability of clustering results but also facilitated a comprehensive exploration of the spatial relationships among particles within the dataset.

Radial distribution function (RDF) computation

The RDF, denoted as g(r), played a pivotal role in characterizing the probability of finding particles at varying distances from a reference particle compared to a random distribution. The computation was executed considering distances between particles binned into radial shells. The formula governing RDF computation is expressed as:

$g\left(r\right)=\frac{N\left(r\right)}{2\pi r\Delta rn}.$

Here, N(r) represents the number of particles in a shell, r signifies the radial distance, Δr is the radial bin size, and n stands for the number density. This meticulous methodology offered a rigorous analysis of the spatial distribution of particles within the system, providing valuable insights into its structural characteristics.

3. RESULTS AND DISCUSSION

In the course of our investigation, an exhaustive examination of the sample electron microscopy images was conducted. Following meticulous image processing, exemplified in the scatterplot below, a total of 6881 atoms or particles were identified within the dataset. These particles were organized into 1659 distinct clusters, each exhibiting a varied number of atoms or particles.

The distribution of atoms within these clusters is summarized in the table below:

 

Table 1

Distribution of atoms within clusters

Cluster size	Number of clusters
2	981
3	327
4	179
5	94
6	37
7	15
8	12
9	6
10	7
11	4
12	3
13	1
18	2

 

The variety in cluster sizes, ranging from small assemblies to more extensive conglomerates, attests to the nuanced atomic arrangements within the examined sample. This detailed breakdown provides a comprehensive overview of the clustering patterns, emphasizing the diversity in the number of atoms or particles within each cluster.

The variety in cluster sizes, ranging from small assemblies to more extensive conglomerates, attests to the nuanced atomic arrangements within the examined sample. This detailed breakdown provides a comprehensive overview of the clustering patterns, emphasizing the diversity in the number of atoms or particles within each cluster.

The total number of identified clusters, amounting to 1659, underscores the complexity and richness of the sample’s atomic organization. This abundance of clusters further signifies the heterogeneity present in the atomic arrangements, contributing to the intricate nature of the material under investigation.

Beyond the quantitative analysis of cluster sizes, the individual graph visualizations offer a nuanced perspective on the internal organization of clusters.

The visualization reveals distinct levels of orderedness within clusters of similar sizes, shedding light on the intricate spatial arrangements of atoms or particles.

 

Fig. 1. Examples of HRTEM images and scatterplot graph: a – HRTEM image CoP; b – HRTEM image-NiW; c – Scatterplot graph of the NiW sample

 

The fig. 2 below showcases cluster visualizations for clusters of sizes 4, 5, and 6.

 

Fig. 2. Cluster visualizations for clusters of sizes 4, 5, and 6

 

The visual representation underscores the variability in atomic configurations even within clusters of the same size. Each cluster exhibits a unique spatial arrangement, contributing to the diverse structural patterns observed in the material. The differing levels of orderedness within clusters highlight the complexity of the atomic architecture and emphasize the need for detailed investigations at the cluster level.

These visual insights complement the quantitative data, providing a holistic understanding of the material’s structural diversity. The combination of quantitative analysis and visual representation enhances the depth of our exploration, offering a comprehensive characterization of the local atomic arrangements within the investigated sample.

A notable observation from our study is the correlation between cluster size and the degree of orderedness within the clusters. Analyzing the cluster point distribution reveals a trend where larger clusters tend to exhibit higher levels of orderedness. Additionally, examining outcomes from both simulated and experimental images verifies that the vertex and bond distributions of the graphs adhere to the Pareto distribution [7; 8]. The distribution for the dataset, featuring different level ordered clusters in the image, is illustrated in Fig. 3. To quantify the dissimilarity between distributions, we employed the Kullback–Leibler divergence in its linearized form.

 

Fig. 3. Cluster points distribution for multiple samples, illustrating the relationship between cluster size and orderedness

 

The figure below illustrates the cluster points distribution for multiple samples, highlighting the relationship between cluster size and orderedness.

As depicted in the figure, clusters with larger sizes demonstrate a more pronounced and structured arrangement of points, indicative of enhanced atomic ordering. This finding aligns with the expectation that larger clusters may possess greater internal cohesion and organization.

The quantitative analysis of cluster point distribution further supports this observation, revealing a positive correlation between cluster size and the degree of orderedness. This correlation has significant implications for understanding the material’s behavior, suggesting that the size of clusters plays a pivotal role in influencing the local atomic structure.

Fig. 4 illustrates the linearized Kullback–Leibler (K-L) divergence for two metrics, Div (SP(B/V)) and Div (Mu(B/V)), which pertain to the ratio of bonds to vertices in atomic cluster graphs [9]. The figure demonstrates how these metrics vary with the degree of ordering in the images of atomic structures. Upon analyzing the trends, it becomes apparent that with an increase in ordering, both Div (SP(B/V)) and Div (Mu(B/V)) decrease. This decrease signifies a di- minished divergence between the cumulative distribution function (CDF) and Lebesgue measure for the ratio of bonds to vertices, respectively. These metrics serve as indicators of cluster ordering and facilitate the assessment and comparison of different images or samples of amorphous alloys in terms of their ordering levels.

 

Fig. 4. Linearized Kullback–Leibler divergence for bonds to vertices ratio CDF (Div (SP(B/V) on left axis) and for Lebesgue measure for bonds to vertices ratio (Div (Mu(B/V) on right axis) depending on level of ordering in the images

 

Furthermore, the K-L divergence emerges as a valuable parameter for gauging cluster ordering. Supplementary calculations, including the S-K test, confirm the equality of distributions across various levels of ordering, except for the last one featuring manually constructed clusters with perfect symmetry.

 

Fig. 5. Probability distributions of angles for six samples, with a reference sample characterized by a high degree of orderedness

 

In-depth exploration of our dataset extends to the analysis of angle distributions within clusters, shedding light on the nature of atomic arrangements. The examination of angles between particles, considering angles from 0 to 180 degrees, provides valuable information about the degree of orderedness within the clusters.

The figure above showcases the probability distri-butions of angles for six samples, alongside a sample characterized by a high degree of orderedness. Each curve represents the likelihood of finding a specific angle within the corresponding cluster. Notably, angles reaching 180 degrees signify a more ordered and structured configuration of atoms.

The observed trend indicates that clusters with a higher probability of angles approaching 180 degrees exhibit a greater level of orderedness. This aligns with the expectation that well-ordered atomic structures often manifest as angles close to 180 degrees.

The data sample characterized by enhanced order serves as a reference point for comparison, showcasing a distinctively sharper peak around 180 degrees. This reinforces the correlation between angle distribution and orderliness within clusters.

Our findings underscore the utility of angle distribution analysis as a powerful tool for discerning the structural characteristics of amorphous alloys.

 

Fig. 6. Radial distribution function for amorphous alloys

 

Furthermore, our investigation focuses on a tho-rough examination of the Radial Distribution Function (RDF) to elucidate the local structural characteristics of amorphous alloys. The g(r) function, representing the RDF, plays a pivotal role in providing essential insights into the probability of finding a particle at a specific distance from a reference particle, unveiling the local short-range order within the material.

In Figure 6, the radial distribution function for amorphous alloys is presented, incorporating both actual data and manually generated symmetric data. The chart visually illustrates variations in particle density at different distances, revealing peaks and troughs in the curve as a distinct fingerprint of the local structural order within the material.

Our observations confirm the existence of local short-range order within amorphous alloys. Peaks in the RDF chart indicate preferential distances between particles, reflecting specific atomic arrangements that significantly con- tribute to the distinctive mechanical, thermal, and magnetic properties observed in metallic glass alloys.

The identified peaks in the RDF chart act as markers for characteristic interatomic distances, providing valuable insights into the atomic arrangement within the material. Understanding these local structural features is crucial for tailoring material properties and optimizing performance for various technological applications. Despite our efforts, as demonstrated by the manually created symmetric simulation in Fig. 6, RDF alone may not offer significant input for distinguishing the orderedness of atomic structure.

In summary, RDF analysis reveals the presence of local short-range order in amorphous alloys, enhancing our understanding of the material’s atomic structure. While RDF contributes a crucial layer to our understanding, additional considerations may be necessary for a comprehensive assessment of the ordered nature of atomic structures in these alloys.

4. CONCLUSION

In conclusion, our research has provided valuable insights into the atomic structure of amorphous alloys, employing state-of-the-art electron microscopy techniques and rigorous analytical methodologies. The observed correlation between a higher probability of angles approaching 180 degrees and increased ordered- ness within clusters underscores the reliability of angle distribution analysis as a powerful tool for characterizing structural features.

The trends identified in Div (SP(B/V)) and Div (Mu(B/V)) metrics with respect to ordering levels offer robust indicators for assessing and comparing different amorphous alloy samples. The diminished divergence between cumulative distribution functions and Lebesgue measures highlights the effectiveness of these metrics in quantifying cluster ordering, providing researchers and engineers with essential tools for materials characterization.

Furthermore, the significance of the K-L divergence as a parameter for gauging cluster ordering is reinforced by supplementary calculations, confirming its reliability across various levels of ordering. The application of the S-K test further validates the equality of distributions, except in cases featuring manually constructed clusters with perfect symmetry.

Our investigation into RDF analysis has unveiled the existence of local short- range order in amorphous alloys, providing an extra dimension to our comprehension of the material’s atomic structure. However, our findings indicate that the utility of RDF in discerning orderliness is insignificant. Despite this, our discovery deepens our understanding of the distribution and arrangement of atoms within the alloy, supplying crucial information for advancing our knowledge in materials science.

In essence, our comprehensive investigation advances the current under- standing of amorphous alloys, offering valuable data and analytical tools that can inform future research and engineering applications in diverse fields. The nuanced insights gained from this study contribute to the broader discourse on material science and provide a solid foundation for further exploration into the unique properties and potential applications of amorphous alloys.

About the authors

Dilla Dagim Sileshi

Far Eastern Federal University

Author for correspondence.
Email: dilla.d@dvfu.ru
ORCID iD: 0000-0002-9100-1257

PhD student, Institute of Mathematics and Computer Technologies, research engineer, Electron Microscopy and Imaging Laboratory

Russian Federation, Vladivostok

Evgeniy V. Pustovalov

Far Eastern Federal University

Email: pustovalov.ev@dvfu.ru
ORCID iD: 0000-0003-1036-3975

Dr. Sci. (Phys.-Math.), Professor, Department of Information and Computer Systems, Institute of Mathematics and Computer Technologies, Head of the educational program 09.03.02 “Information systems and technologies”, profile “Programming of robotic systems”

Russian Federation, Vladivostok

Alexander N. Fedorets

Far Eastern Federal University

Email: fedorec.an@dvfu.ru
ORCID iD: 0000-0001-9007-3171

senior lecturer, Department of Information and Computer Systems, Institute of Mathematics and Computer Technologies

Russian Federation, Vladivostok

Anatoliy M. Frolov

Far Eastern Federal University

Email: frolov.am@dvfu.ru
ORCID iD: 0000-0002-5368-5694

Dr. Sci. (Phys.-Math.), associate professor, Department of Information and Computer Systems, Institute of Mathematics and Computer Technologies

Russian Federation, Vladivostok

References

Supplementary files