On the characteristic length scales associated with

On the characteristic length scales associated with plastic deformation in metallic glasses P. Murali, Y. W. Zhang, and H. J. Gao Citation: Appl. Phys. Lett. 100, 201901 (2012); doi: 10.1063/1.4717744 View online: http://dx.doi.org/10.1063/1.4717744 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i20 Published by the American Institute of Physics.

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APPLIED PHYSICS LETTERS 100, 201901 (2012)

On the characteristic length scales associated with plastic deformation in metallic glasses P. Murali,1,a) Y. W. Zhang,1,b) and H. J. Gao2,c) 1

Institute of High Performance Computing, Singapore 138632 School of Engineering, Brown University, Rhode Island 02912, USA

2

(Received 21 December 2011; accepted 24 April 2012; published online 14 May 2012) Atomistic simulations revealed that the spatial correlations of plastic displacements in three metallic glasses, FeP, MgAl, and CuZr, follow an exponential law with a characteristic length scale ‘c that governs Poisson’s ratio , shear band thickness tSB , and fracture mode in these materials. Among the three glasses, FeP exhibits smallest ‘c , thinnest tSB , lowest , and brittle fracture; CuZr exhibits largest ‘c , thickest tSB , highest , and ductile fracture, while properties of MgAl lie in between those of FeP and CuZr. These findings corroborate well with existing experimental observations and suggest ‘c as a fundamental measure of the shear transformation zone size in C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4717744] metallic glasses. V Plastic deformation in metallic glasses (MGs)1 is understood to be mediated by shear transformation zones (STZs),2–5 which are clusters of atoms that undergo local structural rearrangements in shear. STZs are said to be the most fundamental unit carriers of inelastic strain in MGs. Since it has been argued that sufficiently small STZs may lead to brittle fracture,6–9 an interesting, often debated issue is how many atoms would comprise an STZ.10–12 Johnson and Samwer10 proposed a cooperative shear model (CSM) for STZ activation in MGs and that the number of atoms in an STZ to be of the order of one hundred. Zink et al.11 used molecular dynamics (MD) simulations to estimate the STZ size in CuTi glass to be about 1.5 nm, which appears to be consistent with Johnson and Samwer’s prediction.10 Recently, Delogu has conducted MD simulations using a semi-empirical tight-binding potential, and studied the structural instabilities associated with the perturbation of local atomic displacements and provided a scheme for identifying potential STZs from atomistic simulations and found that a STZ comprises about 20 atoms.4 Pan et al.12 characterized the sizes of STZ in various glasses experimentally, with results showing that different MG systems have different STZ sizes, and that there exists a direct correlation between STZ size and Poisson’s ratio. It should be noted that the experiments of Pan et al.12 were based on an indirect measurement of STZ size through strain rate sensitivity estimated from the hardness data and then using CSM to relate to STZ. Jiang et al.6 used similar methods to characterize STZ and showed that the ductile to brittle transition in MGs is governed by the STZ volume. Currently, it is still challenging to directly measure the size of STZs experimentally owing to the disordered atomic structure of MGs. How to define a measurable size for STZs and reliably extract its value remains an active research topic. The size of the STZ is also important to understand the size effects in metallic glasses that have been observed recently by many groups.13–17 It is believed that STZs operate in a cascade-like manner, leading to the formation of shear bands (SBs) and highly a)

Electronic mail: [email protected]. Electronic mail: [email protected]. c) Electronic mail: [email protected]. b)

0003-6951/2012/100(20)/201901/4/$30.00

heterogeneous plastic deformation in MGs.1,18–25 Generally, SBs exhibit larger length scales compared to STZs, and efforts have been made to measure the thickness of SBs in various MGs.18 Masumoto and Maddin19 used transmission electron microscopy (TEM) to study the SB thickness in PdSi glass samples. Subsequently, Lewandowski et al.21 used high resolution TEM and a Sn-coating technique to measure the SB thickness and assess the temperature rise inside SBs. Li et al.22,23 analyzed the SBs from atomistic simulations. In all these studies, the SB thickness was found in the range from 2.5 nm to 20 nm in different MGs using different measurement techniques and theoretical models.18–25 An important outstanding question is whether and how the two length scales, i.e., the STZ size and SB thickness, are interrelated.25 In spite of the above studies on characteristic length scales of plastic deformation in metallic glasses, including shear transformation zone size and shear band thickness, what exactly governs these length scales still remains an open question. In the present study, we use MD simulations26 to show that the spatial correlation functions of atomic scale fluctuations of plastic displacements in three chosen metallic glass systems, FeP, MgAl, and CuZr, follow an exponential law with a characteristic length scale ‘c that governs the shear band thickness and the fracture mode in these materials. We have previously shown that atomic scale fluctuations play a governing role in brittle and ductile fracture behaviors in MGs.27 Here, we demonstrate that the characteristic length ‘c may be adopted as a clear, measurable, and convenient definition for the STZ size in MGs. Atomistic simulations are performed on three different MG systems, Fe80P20, Mg75Al25, and Cu50Zr50, using a freely available open source code LAMMPS.26 Atomic interactions are modeled by embedded atom method (EAM) potentials with parameters given by Mendelev et al.28,29 These glass samples are produced by an initial melting-andquenching simulation of a randomly substituted solid solution of the two components.30 Periodic boundary conditions are applied in all three dimensions, and the temperature and pressure are maintained using Nose-Hover thermo- and barostats, respectively. The samples are initially equilibrated at 1

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Appl. Phys. Lett. 100, 201901 (2012)

K for 100 ps, followed by a steady rise in temperature up to 2500 K at a rate of 6 K/ps, allowing the solid to melt. The samples are then held at 2500 K for 100 ps and cooled back to 1 K at the rate of 6 K/ps. Samples of various sizes are prepared from the melting-and-quenching simulations and used for the analysis. Elastic constants M of the glass samples are measured on specimens of 1 nm3 size by computing the energy density U of the relaxed configurations as a function of applied strain e and using the relation M ¼ @ 2 U=@e2 je¼0 . The measured values of bulk and shear modulii of the chosen FeP, MgAl, and CuZr systems are indicated in Fig. 1(a) along with their Poisson’s ratios. While both bulk and shear moduli of the MgAl system are lower than those of FeP and CuZr, its Poisson’s ratio happens to fall in between these two systems. These three systems then form a group of MG systems to study possible correlations between Poisson’s ratio and other mechanical properties12,21 using atomistic simulations. Previous experiments have shown that a rapidly quenched metallic glass may exhibit different mechanical behaviors compared to its slowly cooled counterpart, suggesting the properties of glass are closely related to its degree of relaxation.1,8,9 It is believed that a change in quenching rate may lead to changes in density fluctuation, free-volume distribution, and short-range atomic ordering, which can in turn affect the characteristic length of STZs. Note that the extraordinary cooling rates used in MD simulations make it difficult to study the dependence of the processing parameters such as the cooling rate on the microstructure. Within the parameter range that is accessible, we found no noticeable differences in the measured elastic constants of the MGs under investigation. We now attempt to define a measure for the STZ size from atomistic simulations and study its relationship with the Poisson’s ratio of these materials. The MG samples (size 12 nm 12 nm 12 nm) are deformed to a shear strain of 0.04 (approximately half of the yield strain),31 and the resulting plastic displacements di of each atom were calculated by subtracting the elastic part of the displacements from the total displacements. In order to measure the STZ size, we introduce the following discretized spatial correlation function11 Cd ðrÞ Cd ðrÞ ¼ hdi dj dðrij rÞi hdi i2 ;

(1)

where rij is the distance between atoms i and j; di and dj are the values of d for atoms i and j, respectively; hi represents the expectation value; and the function dðrij rÞ is defined as unity when r ¼ rij and zero for all other values of r. The normalized correlation function Cd ðrÞ=r2d , where r2d denotes the variance (square of the standard deviation) of d, can be analyzed to see if there exists a characteristic length scale associated with the atomic fluctuations of quantity d. The normalization of correlation functions with their corresponding variances is necessary in order to compare between different glasses. Figure 1(b) plots the normalized displacement correlation function Cd ðrÞ=r2d , where r2d is the variance and Cd ðrÞ is the spatial correlation function for the plastic displacement. Clearly, Cd ðrÞ=r2d follows an exponential decay function, EðrÞ ¼ D0 expðr=‘c Þ, where ‘c is a characteristic length. A least square fitting leads to ‘c ¼ 0.85 nm for the FeP glass, ‘c ¼ 1.11 nm for the MgAl glass, and ‘c ¼ 1.5 nm for the CuZr glass. Since the characteristic length ‘c provides a measure for the cooperative atomic movements during deformation, similar to the STZs, we propose to use ‘c as a clear, measurable, and convenient definition for the STZ size. Zink et al.11 studied similar correlation functions and attempted to extract the STZ size. However, they did not specify a measurable length scale that could be used to compare between different glasses. By examining the spatial correlation length, we see that the STZ size for different glasses need not be the same, but vary over a range: ‘c ¼ 0.85 nm for the FeP glass and ‘c ¼ 1.5 nm for the CuZr glass. The STZ size is known to be related to other mechanical properties of glasses. For example, it was shown12 that Poisson’s ratio and the STZ size are closely related: a higher Poisson’s ratio leads to a larger STZ size. The measured Poisson’s ratios of the three selected MGs studied in this work clearly obey this trend, showing a direct support of these observations from atomistic simulations.12 The thickness of shear bands in MGs has been measured through experiments18–21 and atomistic simulations.22–24 Although the average thickness of shear bands in MGs was found to be on the order of 10 nm, they generally exhibit substantial variations among different glasses. An important outstanding question is what controls the shear band thickness. As a preliminary effort to address this question, we performed atomistic simulations of the FeP, MgAl, and CuZr

FIG. 1. (a) Elastic constants of the FeP, MgAl, and CuZr metallic glasses measured from atomistic simulations. (b) Normalized spatial correlation functions of the plastic displacements after applying a finite shear strain of 0.04 to FeP, MgAl, and CuZr glass samples.


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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(j)

(k)

(l)

X2

(i)

FIG. 2. Snapshots of the atomic configurations of (a)(d) FeP, (e)-(h) MgAl, and (i)-(l) CuZr glasses during shear deformation at various shear strains. (a), (e) and (i) c12 ¼ 0.0; (b), (f) and (j) c12 ¼ 0.08; (c), (g) and (k) c12 ¼ 0.4; (d), (h) and (l) c12 ¼ 0.4. The color of the atoms for (a)-(c), (e)-(g), and (i)-(k) is based on the initial positions of the atoms. The color of the atoms in (d), (h) and (l) is based on the local atomic strain c12.

X1

12 nm

FIG. 3. Local shear strain, c12 variation across the samples of (a) FeP (b) MgAl, and (c) CuZr corresponding to Figures 2(d), 2(h), and 2(l) in the direction perpendicular to the shear band. The width of region with c12 exceeding half of its peak value is taken as half of the shear band thickness, tSB.

glass samples of size 12 nm 12 nm 12 nm subjected to shear deformation. Figure 2 shows some snapshot configurations of the three glasses at various levels of the shear strain c. The top row ((a)-(d)) shows the sequence of deformation patterns in the FeP glass while the middle ((e)-(h)) and bottom rows ((i)-(l)) show the corresponding sequences in MgAl and CuZr glasses, respectively. In order to clearly visualize the shear banding process, atoms are colored based on the x-coordinate of their original positions in the undeformed configuration (2(a), 2(e), and 2(i)). All three glasses undergo elastic deformation up to the yield strain. During this process, the vertical lines shown in Figures 2(b), 2(f), and 2(j) are straight, indicating nearly homogeneous deformation of the samples. Beyond this point, shear localization takes place and the grid lines are seen distorted as shown in Figures 2(c), 2(g), and 2(k). Note that the atomic positions have been displaced such that the shear bands appear in the middle of the periodic simulation box. The local shear strain can also be used to visualize the shear bands from atomistic simulations. Figures 2(d), 2(h), 2(l) display the distribution of local strain of the same configurations shown in Figures 2(c), 2(g), 2(k), respectively. Figure 3 plots the local shear strain c12 versus vertical position x2 of each. The local strain distribution shows a peak value at approximately the middle of the shear band. The width of region with c12 exceeding half of its peak value is taken as half of the shear band thickness, tSB . Based on this definition, the shear band thickness is measured to be tSB 6.4 nm for the FeP glass, tSB 8 nm for the MgAl glass, and tSB 11.2 nm for the CuZr glass. Interestingly, we find that the ratio of the shear band thickness to the characteristic

length scale ‘c of STZ remains approximately identical for the three glasses at tSB =‘c 8. This is also consistent with the observation that the shear band thickness tends to be roughly 10 times the STZ size in granular materials.25 In summary, we have performed atomistic simulations to study the spatial correlation functions of atomic-scale fluctuations in local plastic displacements in metallic glasses and discovered a characteristic length scale ‘c which can be used to define the size of STZs. Further simulations revealed that this characteristic length scale also controls the shear band thickness. Our work suggests that a larger ‘c leads to proportionally larger shear band thickness, higher fracture toughness and higher Poisson’s ratio, all of which are in agreement with existing experimental observations. Financial support for this work is provided by A-Star Singapore through VIP project on “Size effects in small scale materials.” 1

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Y. Shi and M. L. Falk, Phys. Rev. Lett. 95, 095502 (2005). M. Q. Jiang, W. H. Wang, and L. H. Dai, Scr. Mater. 60, 1004 (2009). LAMMPS: MD simulation code available from Sandi National Laboratories, USA. 27 P. Murali, T. F. Guo, Y. W. Zhang, R. Narasimhan, Y. Li, and H. J. Gao, Phys. Rev. Lett. 107, 215501 (2011). 28 M. I. Mendelev, D. J. Sordelet, and M. J. Kramer, J. Appl. Phys. 102, 043501 (2007). 29 G. J. Ackland, M. I. Mendelev, D. J. Srolovitz, S. Hans, and A. V. Barashev, J. Phys. Condens. Mater. 16, S2629 (2004). 30 P. Murali, U. Ramamurty, and V. B. Shenoy, J. Chem. Phys. 128, 104508 (2008). 31 Note that a relatively small strain has been imposed here in measuring the displacement correlation functions, as local bursts of inelastic displacements occur way before global plastic deformation is evident. The latter requires activation of a large number of STZs leading to the formation of a shear band. 25 26
