Local stress analysis in Cu 50% Zr 50% metallic glass under shear strain by means of first principle modeling
SShear softening in a metallic glass: first principle local stressanalysis
I. Lobzenko a , Y. Shiihara a , T. Iwashita b , T. Egami c,d a Toyota Technological Institute, Hisakata, Tempaku-ku, Nagoya 468-8511, Japan b Oita University, Dannoharu, Oita 870-1192, Japan c University of Tennessee, Knoxville, Tennessee 37996, USA d Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Abstract
Metallic glasses deform elastically under stress. However, the atomic-level origin of elasticproperties of metallic glasses remain unclear. In this paper using ab initio moleculardynamics simulations of the Cu Zr metallic glass under shear strain, we show thatthe heterogeneous stress relaxation results in the increased charge transfer from Zr to Cuatoms, enhancing the softening of the shear modulus. Changes in compositional short-range order and atomic position shifts due to the non-affine deformation are discussed.It is shown that the Zr subsystem exhibits a stiff behavior, whereas the displacements ofCu atoms from their initial positions, induced by the strain, provide the stress drop andsoftening.Amorphous materials formed by metal atoms, usually referred to as metallic glasses(MG) or glassy metals, have gained significant attention after the procedure of massproduction of the bulk form of such materials had been developed in the 1990s [1, 2],made possible by the discovery of glass-forming alloys requiring relatively low coolingrates of <
100 K · s − [3]. This enabled broad applications of MGs, for instance, fornanoimprinted technology, bio-implants, and coating, to name a few examples [2, 4].Along with the applications, metallic glasses have received intense scientific interest.One of the striking features of MGs is shear modulus softening (SMS), namely the shearmodulus of MGs is significantly lower than that of crystalline counterparts [5]. This phe-nomenon has been extensively studied theoretically with assumptions being supported bycomputer simulations in a number of early [6, 7] and most recent [8, 9, 10, 11] works, andthe microscopic origin has been attributed to non-affine heterogeneous atomic motionsduring deformation. However, the precise atomic mechanisms of SMS remain elusive.Particularly little quantum level understanding of SMS has been achieved [9].In the case of crystalline materials, inelastic behavior is governed by the well-definedlattice defects in the periodic structure. However, attempts of defining defects in glassesface a formidable conceptual and practical barrier because of structural disorder. Anumber of theories of deformation in MG were proposed, among which the most widelyused relies on the so-called shear transformation zone (STZ) [12, 13, 14]. It attributes thedeformation in amorphous material to the emergence and development of regions with Preprint submitted to PRL March 10, 2020 a r X i v : . [ c ond - m a t . m t r l - s c i ] M a r .01.02.03.04.0 0.0 2.0 4.0 6.0 8.0 σ xy [ G P a ] Strain [%]affinerelaxed σ xy (r) / σ xy ( a ) Strain [%] (a) (b)
Figure 1: (a) Stress - strain curves of Cu Zr glass for affine and relaxed structures. (b) Ratio of relaxedstress σ (r) xy to affine stress σ (a) xy for various shear strains. Vertical bars show errors due to averaging (seediscussion in text). higher mobility. Despite advances in the STZ theory, a number of its key features remainveiled. For example, many simulations and experiments show contradictory results forSTZ sizes, from a few atoms to several hundred, and also the precise physical pictureof the STZ is missing [15, 16, 17, 18]. Furthermore, the particular mechanism of STZemergence is yet to be understood.Advances in modern computational methods allow one to obtain insights into thedeformation behavior of individual atoms. One of the most accurate atomistic modelingmethods is based on the density functional theory (DFT), which provides the evaluationof structural parameters with accuracy up to 1% [19]. This approach is effective in defin-ing the starting point of shear transformation zone in metallic glass since relatively smallstructures (order of 20 atoms) are involved in the STZ [20]. In order to examine the localstress state around the STZ, which is highly heterogeneous at the microscopic scale, weuse the atomic-level stresses analysis [21, 22, 23], coupled with the DFT calculation [24].In the current work, we study Cu Zr glass under shear strain. The choice of thesystem is dictated by the fact that a large amount of data are available for CuZr alloysin both crystalline and glassy structures [24, 25]. It is known that the CuZr glass couldbe obtained in a wide composition range. Therefore, the influence of the stoichiometrypresented in the current work may be verified by future experimental work.This paper is organized as follows. At first, the details on the structures under studyand settings for the first principle calculations are given. Next, the main results arepresented and discussed with emphasis on the fundamental properties of the structureleading to shear modulus softening.Initially, a random atomic configuration of 96 atoms was prepared at a density of57.1 nm − [26]. The first principle molecular dynamics (FPMD) simulation at 3000 Kfor 2 ps was performed under the NVE ensemble with periodic boundary conditions tothermalize the original structure, thus obtaining an equilibrium liquid structure. Thetime step of the simulations was 2 fs. The system was quenched to a supercooled stateat 1200 K, which led to the relatively stable glass structure, then FPMD was performedfor 1 ps. Next the system was gradually cooled down to a glass state at 700 K with a2 σ [ G P a ] Strain [%]affine Cuaffine Zrrelaxed Curelaxed Zr -1.0-0.50.00.51.00.0 2.0 4.0 6.0 8.0
CuZr δ [%] (a) (b) Strain
Figure 2: (Color online) (a) Shear stress in the affine and relaxed structures for Cu Zr under shearstrain. Red filled circles (triangles) show Cu (Zr) contribution to the stress in the affine structure,while blue empty circles (triangles) show Cu (Zr) contribution to stress in the relaxed structure. (b)Deviation ( δ = ( σ relaxed − σ affine ) / | σ affine | ) of the xy component of stress in the relaxed system from the xy component of stress in the affine system. Red circles (blue triangles) represent Cu (Zr) atoms. cooling rate of 0.5 K/fs, and in addition, another thermalization at 300 K was performedfor 1 ps to stabilize the glass structure. Finally, the glass structure was relaxed to 0K by applying energy minimization using the conjugate gradient method under zeromacroscopic stress, allowing the box size and shape to vary during the iterations. Thedescribed procedure provides us with an equilibrium structure of the metallic glass readyfor subsequent analyses.In our study, the quantum mechanical approach in the framework of density func-tional theory was used, which is implemented in the Vienna Ab initio Simulation Pack-age (VASP) [27]. The generalized gradient approximation was used for the exchange-correlation energy, which is essential for the achievement of high accuracy. The energycutoff regulating the number of basis functions was set to 410 eV. Because of the rela-tively large size of the system, only the Γ-point was used in the reciprocal space. Theconvergence of the self-consistent calculation was enhanced by the Methfessel-Paxtonmethod [28] with the smearing of 0 . .
01 eV/˚A.To model the response of the CuZr system to strain, the athermal quasi-static shear(AQS) simulations, which are usually used in the framework of classical molecular dy-namics, was performed at the quantum mechanics level using the FPMD. Firstly theglass structure at 0 K obtained by cooling is uniformly deformed with a simple shearstrain ε xy (affine deformation). Next, atomic positions optimization is performed withthe box geometry fixed. That results in the stress relaxation from the affine state duringthe energy minimization. To improve statistics, four different original structures, eachof which is deformed independently at the same strain in six different directions, wereprepared and the results were averaged over 24 samples (= 4 ×
6) in total.In the process of analyzing the mechanical properties of materials under strain, lo-cal stress calculation is a standard tool to unveil atomic-level correlations. Neverthe-less, until recently that tool was available only in the classical approximation. Lately,3everal approaches were proposed allowing the calculation of atomic level stress in aquantum-mechanical framework (see [24] for the detailed discussion of differences be-tween those approaches). In particular, in this technique, the local stress is obtainedas the strain derivative of local energy assigned to a single atom, as it is done in [29]and [24]. The calculations were performed using the Open source package for MaterialeXplorer (OpenMX), which utilizes the orbital-based energy decomposition scheme [30].Such approach allows direct calculation of the derivatives of atomic energy with respectto the strain tensor.Figure 1 (a) shows stress as a function of strain for several shear strains (0 . , . , . , . , . affine one (obtained directly from the originalstructure by applying affine strain) and the relaxed one (obtained after performing theoptimization of atomic positions in the deformed structure). We see that stress is lin-early proportional to applied strain for both cases of deformation. The estimated shearmodulus is 44 GPa for the affine deformation and 21 GPa for the relaxed deformation.That is to say, the stress decreases during the relaxation process from the affine stateby about 50%, as shown in Fig 1 (b), which is consistent with previous results based onclassical AQS simulations (see [31] as an example).In order to study the origin of the stress drop, we decomposed the macroscopic stressinto the contributions of individual atoms and investigated the responses of the atomic-level stress under the shear for Cu and Zr. As shown in Fig. 2 (a), the local stressanalysis revealed a fascinating behavior. The local stress of Zr increases positively, asexpected, with the strain for affine and relaxed structures, whereas the local stress ofCu becomes more negative with the strain, which is a quite unusual opposite behavior.Also, it can be seen that the relaxation decreases the magnitude of the stress on eachatom, and the deviation, δ = ( σ relaxed − σ affine ) / | σ affine | of the stress from the affinestress is plotted in Fig. 2 (b). Interestingly Zr atoms show a negative contribution tothe stress drop, while Cu atoms have a positive contribution. The origin of this behaviorwill be discussed later. At the highest strain of 8%, the change in the stress correspondsto 1.38 GPa for Cu atoms and -3.31 GPa for Zr atoms, thus leading to the stress dropof -1.93 GP in total. Therefore, we see that the change in the local environment aroundZr atoms should be mainly contributing to the total stress drop.We investigate the role of charge on the stress drop under deformation. Figure 3shows the strain dependence on the charge transfer, δQ , between the affine and therelaxed states for Cu and Zr atoms. Units of δQ are number of electrons per atom. Atsmall strains ( ε ≤ δQ becomes opposite, which may be due to the plastic flow far from the elastic regionwhich tends to increase the effective temperature of the system and rejuvenate it [32].Finally, we show a clear correlation between the local shear stress deviation ( δ =( σ relaxed − σ affine ) / | σ affine | ) and charge difference, δQ for small strains in Fig. 4 (a).The data for the highest 8% strain beyond the elastic region was not included in thefigure. From Fig. 4 (b,c), it can be seen how the relaxation affects in the shrinking ofthe charge clouds around the atom denoted ’Cu0’. We can conclude that, during the4 CuZr
Strain [ e l . ] d Q Figure 3: (Color online)
Averaged total charge difference ( δQ , relaxed to affine) expressed in number ofelectrons for Cu (red circles) and Zr (blue triangles) atoms with shear strain. relaxation process, Cu gains charge, which positively increases the stress, whereas Zrloses charge, which makes the stress more negative. It is worth mentioning that there isno direct correlation between the atomic-level pressure deviation and δQ , although thedata are not shown here.To show the difference in the behavior of the Cu and Zr subsystems under strain, letus discuss the 3-atoms angle change induced by the relaxation of atom positions fromthe affine structure. For each atom, we found closest neighbors of the same type. Next,for that group of atoms, the change of all unique 3-atoms angles was calculated with thetargeted atom being the vertex. All angle change values were averaged and associatedwith each atom type(see Fig. 5a). The optimization process for the system under the 8%shear strain changes angles of the Cu subsystem by the value of 7.06 degrees, while forthe Zr subsystem that parameter equals only to 2.24 degrees. The Cu d -electron statesare full and do not participate in bonding, whereas the Zr d -states are only partiallyfilled and form covalent bonds [33]. The significant difference in the changes in angleinduced by relaxation between Cu and Zr is consistent with this difference in the d -statesoccupation.Taking only the neighbors of the same type, we can analyze the Cu subsystem andthe Zr subsystem separately. However, the interplay between the Cu and Zr subsystemscan only be understood if we take into account the types of neighboring atoms (seeSupplemental Material for more data). The Cu subsystem is rearranged significantlyduring the relaxation process, resulting in the stress of an opposite sign, whereas theZr-Zr angles are affected only slightly. At the same time, the Zr/Cu compositionalshort-range order (CSRO) is changed by deformation (see Fig. 5b). The fraction of theZr-Cu pairs is increased at the expense of the Zr-Zr pairs. The increased CSRO inducesmore charge transfer, and the lowering of the potential energy, resulting in softening. Anexample of actions of bond-breaking and bond-forming is given in Supplemental Material.However, the CSRO of the system before deformation depends on the condition of the5 CuZr Q [el.] Figure 4: (Color online) (a) Correlation between the local stress deviation ( δ = ( σ relaxed − σ affine ) / | σ affine | ) and the change in atomic charge ( δQ = Q relaxed − Q affine ) for Cu Zr under elasticdeformation. Dashed line is guide for eyes. (b) and (c) show the 2D charge density map for one particularstructure; shared plane was chosen for atoms Cu0, Cu1, Cu2 to make a section of the 3D charge density. δ θ Strain [ % ] -4-2 0 2 4 6 δ N b Strain [ % ] (a) (b) Figure 5: (Color online)
Relaxation induced changes of Cu and Zr subsystems for studied shear strainvalues. (a) Change in same type angles (see discussion in text). (b) Change in the number of chemicalbonds. Zr system, not only a geometrical rearrangement of atoms but also a change in aCSRO contributes to the softening of shear modulus.Summarizing, the effect of elastic deformation on the electronic states in metallicglass is studied under shear strain by means of ab initio calculations. The simulations ofCu Zr glassy alloy under shear strain show that atoms undergo non-affine deformationeven in the elastic regime, accompanied by substantial charge transfer. In particular theCu subsystem is severely rearranged under the shear strain, surprisingly resulting in theshear stress of the opposite sign on Cu atoms. On the other hand the Zr subsystemdeforms in a nearly affine manner. Deformation produces increased Zr/Cu CSRO, de-creasing Zr-Zr bonds, which leads to increased charge transfer. This work demonstratesthat the change in the CSRO contributes to the softening of the shear modulus throughincreased charge transfer, along with the geometrical transformation. The evidence fromour results points toward the need of considering the effect of the deformation on theCSRO in metallic glasses.
1. Acknowledgement
YS is grateful to M. Kohyama and T. Ozaki for discussion and comments. This workwas partly supported by Grant-in-Aid for Scientific Research in Innovative Areas to YS(26109705 and 19H05177). TE was supported by the U.S. Department of Energy, Officeof Science, Basic Energy Sciences, Materials Science and Engineering Division. TI wassupported by JPSJ KAKENHI Grant Number JP19K03771.
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