Full length articleCorrelating defects density in metallic glasses with the distribution of inherent structures in potential energy landscape
Graphical abstract
Introduction
Understanding the structures of materials and their responses to external stimuli are important, because they determine many critical mechanical properties such as strength, ductility, and failure. In crystalline materials the featured structures are naturally defined by topological defects, e.g. dislocations, interstitials, vacancies, etc. As a result the structural evolution under different environments can be directly observed and quantified, such as the well-known Taylor equation that correlates the dislocation density to the strength of the system. In amorphous materials, however, due to the disordered atomic nature and disappearance of lattice periodicity, the structure-property relationships can hardly be explicitly defined as in their crystalline counterparts. Yet given many similarities between the two systems [1,2] it is widely believed that there exist some universal deformation units in amorphous materials, although details are subject to a large variety of concepts, such as free volume [3], shear transformation zone (STZ) [4], flow unit [5], etc. Recent experiments [[5], [6], [7], [8]] and modeling [9,10] also support such a picture that the properties of individual deformation units are invariant, while the densities and distributions of such units are closely related to the processing histories or relaxation levels of the glasses. However, quantifications of the structural variations in response to surrounding environments have been hindered because of the aforementioned difficulties in analyzing the disordered atomic packings in glasses.
Ever since the pioneering work by Stillinger et al. [[11], [12], [13]], the potential energy landscape (PEL) has provided a convenient perspective for interpreting complex phenomena in amorphous materials [4,[13], [14], [15],[17], [18], [19], [20], [21]]. This school of thought stems from the very fundamental fact that the properties of a condensed matter system are ultimately governed by the interactions between constituent atoms. The inter-atomic potentials are usually functions of the relative positions between atoms, and the entire system's potential energy would yield a complex surface in the high dimensional configurational space, which is known as the PEL. As illustrated in Fig. 1 , in disordered materials there are many local energy minima in the PEL, known as inherent structures (IS), representing some metastable states of the system. From the PEL perspective the emergent deformation units in amorphous solids correspond to hopping between contiguous IS. It has been conjectured [13] and qualitatively demonstrated [9,10] that the distributions of IS in the PEL are non-uniform: in low energy region the local minima are sparse and widely separated, while in high energy region the reverse is true. Such a picture can seem to naturally explain why a better aged glassy system is less ductile comparing with a less relaxed system [7]. However, to enable a fundamental and predictive understanding, it is critical to quantify the density of IS in the PEL, which is the primary focus of this study.
We consider a Cu56Zr44 modeling sample here as the Cu-Zr system is a well-known metallic glass former. The sample is first equilibrated at high temperature (2000 K) liquid state in the molecular dynamics (MD) simulation and then quenched to 0 K at the fixed volume with six different applied cooling rates, from 1013 K/s down to 108 K/s, respectively. The purpose of varying thermal treatment is to freeze the sample at different energy levels of the PEL. Then the local PEL structures are probed by activation relaxation technique (ART), which is known to be capable of providing the key connections between a given IS and its surrounding saddle states and neighboring local minima, such as the activation energy, atomic displacement, etc [[14], [15], [16]]. In the following sections, we will investigate how such key features are correlated with the sample's thermal stability indicator (e.g. fictive temperatures). Computational details of MD and ART can be found in Appendix. A1.
Section snippets
Results
We first show the variations of inherent structure energy (EIS) of the system under different cooling rates in Fig. 2 . In the high temperature regime, despite significant fluctuations, the average MD results are almost flat. This is also known as the free diffusion regime [11] because the system has enough kinetic energy to explore the entire PEL and can easily get equilibrated. Upon cooling to below 1000 K, the IS energy becomes very sensitive to the temperature, and the system enters the
Discussion
The present study investigated the structural variation of PEL in a Cu-Zr metallic glass modeling system under different thermal processing histories. It is observed that the local minima in the PEL are spatially more separated from each other in slowly quenched samples than in fast quenched samples. In particular, the density of local minima in the PEL shows an Arrhenius dependence on the samples' fictive temperature, which is remarkably similar to the formulation of STZ density hypothesized
Acknowledgements
The authors are grateful to T. Egami, T. Iwashita, and M. Atzmon for thoughtful discussion. Y.F. would also like to thank the support by start-up funding from University of Michigan, Ann Arbor. P.F.G is supported by the NSAF of China (Grant No. U1530401).
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