Mechanical properties, failure mechanisms, and scaling laws of bicontinuous nanoporous metallic glasses

Abstract

Molecular dynamics simulations are employed to study the mechanical properties of nanoporous Cu_xZr_1-x metallic glasses (MGs) with five different compositions, x = 0.28, 0.36, 0.50, 0.64, and 0.72, and porosity in the range 0.1 < ϕ < 0.7. Results from tensile loading simulations indicate a strong dependence of Young's modulus, E, and Ultimate Tensile Strength (UTS) on porosity and composition. By increasing the porosity from ϕ = 0.1 to ϕ = 0.7, the topology of the nanoporous MG shifts from closed cell to open-cell bicontinuous. The change in nanoporous topology enables a brittle-to-ductile transition in deformation and failure mechanisms from a single critical shear band to necking and rupture of ligaments. Genetic Programming (GP) is employed to find scaling laws for E and UTS as a function of porosity and composition. A comparison of the GP-derived scaling laws against existing relationships shows that the GP method is able to uncover expressions that can predict accurately both the values of E and UTS in the whole range of porosity and compositions considered.

Introduction

Benefiting from a disordered atomic arrangement, metallic glasses (MGs) have many unique properties, such as high hardness, strength, and wear resistance [1], [2], [3]. By introducing pores in their structure, porous MGs (PMGs) synergize the qualities of porous structures and MGs. That enables strong and lightweight materials, which are suitable for various structural and functional applications, such as catalysis [4,5], sensing [6], dye degradation [7], membrane filters [8], and gas absorption [9]. PMGs can be obtained by various methods such as water vapor release [10], salt dissolution [11], and low-pressure infiltration of hollow carbon microspheres [12]. Some possible applications of PMGs benefit from a high surface to volume ratio, i.e., high specific area. PMGs with pore size in the nanometer scale maximize the specific area and are gaining more and more interest from the community. Tanaka et al. [13] fabricated Pd-based nanoporous MGs (NPMGs) by chemical dealloying a Pd₃₀Ni₅₀P₂₀ MG. The 30–60 nm sized nanopores were shown to enhance remarkably the catalytic activity and reusability. Jiao et al. [14] fabricated 3-dimensional bicontinuous NPMGs by selectively dealloying of spinodally decomposed MG precursors and passivation. The underlying dealloying process can be adjusted for tuning the pore size in such nanoporous structures. The stochastic nature of the spinodal decomposition process and the nano-length scale of the porous features result in a high specific surface area making such material a promising candidate for gas absorption.

While MGs have attracted considerable interest due to their mechanical properties, their significant brittleness, which can lead to catastrophic failure, notably limits their possible applications. Surprisingly, recent studies of the mechanical properties of NPMGs have suggested that the introduction of pores and nanopores is able to alleviate the brittleness inherent to MGs. Atomistic investigations, e.g., using molecular dynamics (MD) simulations, have been increasingly used to complement experiments and unveil the underlying deformation mechanisms of NPMG during tensile loading [15], [16], [17], [18], [19]. Sopu et al. [15] employed MD to study Cu₆₄Zr₃₆ NPMGs and concluded that a structure with nanopores with optimized size and distribution could achieve homogenous plastic deformation while maintaining the strength close to that of monolithic bulk MGs (BMGs). Liu et al. [16] investigated the mechanical behavior of Ta NPMGs and found that the spatial distribution of pores and pore sizes have a strong influence on the displayed ductility. Lin et al. [17] simulated the tensile loading mechanical response of Cu₅₀Zr₅₀ NPMGs with well-defined cylindrical ligaments and observed a slenderness ratio dependent failure by necking. Liu et al. [18] investigated the tensile loading behavior of stochastic Cu₆₄Zr₃₆ bicontinuous NPMGs showing an anomalous ductile behavior, displaying an extended plastic regime and a gracious failure by collective ligaments necking. They described the deformation mechanism as a synergistic combination of delocalized necking of ligaments mostly aligned with the loading direction and concurrent progressive reorientation of remaining ligaments. In a similar study, Zhang et al. [19] investigated the sensitivity of the mechanical properties of Cu₅₀Zr₅₀ bicontinuous NPMG to porosity and temperature. They also attributed the plasticity found to the bending of the ligaments. Experimentally, it is challenging to directly apply tensile loading simulations on nanoporous samples. Instead, nanoindentation experiments are often conducted to evaluate the tensile behavior of nanoporous samples. Zhang et al. [20] employed nanoindentation, tension, and compression loading to evaluate the deformation behavior of Cu_55.4Zr_35.2Al_7.5Y_1.9 NPMG prepared using selective dissolution and found bending of ligaments and collapse of ligament nodes in the early stage of deformation, followed by plastic deformation and failure of ligaments.

While monolithic MGs have well defined mechanical properties, those of NPMGs depend on their porous structure topology. Efforts have been spent in developing scaling laws for NPMGs based on experimental results and theoretical calculations [19], [20], [21], [22], [23], [24], [25], [26]. However, current scaling laws are developed empirically or based on simplified topology models that consider only porosity as a variable. Functional relationships depending on multiple variables, such as porosity and composition, can be developed using Artificial Intelligence (AI) based methods. The emergence of AI methods is bringing a new dawn to the development of materials science [27,28]. Machine Learning (ML) and Genetic Programming (GP) algorithms have been experiencing resurgence in recent years within the materials science community [29,30]. AI technologies can provide researchers with tools to overcome the barriers between designing, synthesizing, and processing materials by accelerating simulations [31,32], prediction of properties [26,[33], [34], [35], [36]], design of synthetic routes [37,38], optimization of experimental parameters [39,40], and enhancement of characterization methods [41,42]. As for nanoporous materials, it is challenging to predict the mechanical properties due to their complex geometry. A suitable AI method, such as GP, which mimics natural selection, can be employed to describe the functional relationship between bicontinuous NPMGs mechanical properties and system variables.

Motivated by Darwin's theory of natural selection, GP is a powerful evolutionary technique commonly used to automatically generate programs suitable for user-defined tasks [43]. AI technologies are currently enabling the uncovering of physical laws. Among the AI methods, GP is a promising technique to acomplish symbolic regression, allowing one to find suitable mathematical models describing data, when little knowledge of the data structure or distribution is available [44,45]. Within the GP-based symbolic regression (GPSR), expressions are randomly generated and evolve in successive generations, which improve the description of the relationships of interest [46]. GPSR has already been applied to numerous problems [47]. In contrast to other AI methods [48], [49], [50], one can derive physical insights from the expressions obtained with the GP method. Chopra et al. [51] developed GP models to predict the compressive strength of concrete based on in situ data from literature. Cai et al. [52] used GPSR to derived heat transfer correlations, including the equation functional and its parameters, from experimental data on heat transfer measurements, which were used to predict the performance of thermal components. Langdon and Barrett [52] applied GP in drug discovery by evolving simple, biologically interpretable, in silico models of human oral bioavailability. Barmpalexis et al. [52] found a function mapping levels of four polymers to three different properties of a pharmaceutical release tablet with help of GPSR. Recently, Im et al. [53] applied the GP methodology to identify governing equations in non-linear multi-physics systems.

In the present work, we use MD simulations to study the mechanical properties of CuZr bicontinuous NPMGs. We then apply GPSR to derive scaling laws describing the mechanical behavior as a function of system variables and compare them against existing scaling relationships. The results indicate that the GPSR-derived models are able to predict accurately both the Young's modulus, E, and ultimate tensile strength (UTS) as a function of relative density and alloy composition. This research demonstrates that the GPSR is able to uncover expressions that can predict accurately the mechanical properties and also provide physical insights into complex systems.