Dynamics of shear localization and stress relaxation in amorphous Cu50Ti50

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Acta Materialia 57 (2009) 1437–1441 www.elsevier.com/locate/actamat

Dynamics of shear localization and stress relaxation in amorphous Cu50Ti50 M. Neudecker *, S.G. Mayr I. Physikalisches Institut, Georg-August-Universita¨t Go¨ttingen, Friedrich-Hund-Platz 1, 37077 Go¨ttingen, Germany Received 24 October 2008; received in revised form 17 November 2008; accepted 18 November 2008 Available online 30 December 2008

Abstract Dynamic heterogeneities in metallic glasses are investigated for the model glass Cu50 Ti50 with the help of classical molecular dynamics computer simulations. By rapid quenching from melt at various cooling rates (spanning five orders of magnitude), differently relaxed amorphous cells are prepared. During subsequent shearing at T ¼ 10 K, we observe a series of highly localized shear events, termed shear transformation zones (STZs). Detailed analysis focuses on the dynamics of STZ formation and the correlation with local properties. We identify a local mechanical stress bias as the physical origin of STZ formation during low-temperature shear deformation. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: MD-simulations; Metallic glasses; Deformation inhomogeneities; Low-temperature deformation

1. Introduction Metallic glasses are a promising new class of materials with attractive mechanical properties such as a high elastic limit, yield strength and hardness, when compared to conventional crystalline alloys [1,2]. However, deformation at temperatures well below the glass temperature, Tg, usually causes localization of shear (e.g. [3,4]), which successively leads to rapid failure via nanometer-sized shear bands (e.g. [5]). While the detailed atomic-scale kinetics that underlie deformation mechanics are still a subject of controversy, different phenomenological pictures for plasticity have emerged over the past 50 years, among them the concept of shear transformation zones (STZs). Based on early ideas of Adam and Gibbs [6] and analytical descriptions by Argon [3], STZs have attracted considerable interest during the past years, particularly in view of recent corroboration by computer simulations [7–9,4,10]. Basically, in the context of STZs, a localized cooperative motion of tens to hundreds of atoms is regarded as the carrier of plastic strains. *

Corresponding author. E-mail addresses: [email protected] (M. Neudecker), smayr@gwdg. de (S.G. Mayr).

Recently, Johnson and Samwer [11] have extended the STZ concept by interpreting cooperative shear events within a potential energy landscape (PEL) picture. As strongly promoted by Stillinger [12], the latter assigns a potential energy to any possible atomic configuration. Transitions within deep energetic minima of the PEL, so-called basins, can be identified with one-dimensional chain-like excitations (strings [13] or b-processes), while inter-basin transitions are ascribed to STZs (a-processes) [10,14]. Generally transitions across these energy barriers can be facilitated either by thermal activations or externally applied stress (or a combination of both), which is reflected by a time– temperature equivalence observed in the mechanical response of glasses [15]. The key open question, which we aim to address in the present work, concerns the origin and dynamics of STZ formation in amorphous solids – focussing on the low-temperature limit – and their correlation with local properties. 2. Simulation details We perform classical molecular dynamics (MD) simulations on a Cu50Ti50 model system employing the embedded

1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.11.032

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atom method (EAM) potential as parameterized by Sabochick and Lam [16]. Amorphous cells with a total number of 35152 atoms are prepared by rapidly quenching from liquid phase (6000 K) down to 10 K with different cooling rates, ranging from 50 K/ps to 5 K/ns, while periodic boundary conditions were applied in all spatial directions. Temperature and pressure controls are realized via a Berendsen thermostat and barostat [17], respectively. Volume and energy changes upon cooling indicate a glass transition at T g  610 K, while the radial distribution functions (RDFs) of the quenched systems confirm amorphicity. The subsequent shear process at the base temperature, T = 10 K, is realized by releasing the periodic boundaries in the z-direction, fixing the atoms in the top ˚ thick) and moving the top and bottom layers (each 5.5 A vs. the bottom layer corresponding to a constant shear rate of c_ yz ¼ 10%=ns in the y-direction1. Atomic configurations are analyzed at equidistant time frames of Dt ¼ 10 ps and Dt ¼ 1 ps (corresponding to strain increments of Dcyz ¼ 1  104 and Dcyz ¼ 1  105 ) with respect to local non-affine displacements (NADs) [4], d p . While determination of the corresponding global stress tensors from the virials is well established (e.g. [18]), generalizations down to atomic levels are less standardized in the literature. Although the definition of atomic-scale stresses on a regularly rastered grid by Lutsko [19] is presumably the most systematic, we presently assign a stress level to an individual atom a (of atomic mass and velocity, ma and ~ va ) by the Basinski, Duesberry and Taylor [20] expression 0 1 raij ¼

X ab ab C 1 B B a a a 1 r f C a @m vi vj þ X 2 b j i A

ð1Þ

b–a ab

where ~ r denotes the distance between atoms a and b, and ~ f ab describes the force on atom a due to atom b. We particularly chose to stick with the latter definition, as it proved to be a most significant atom-related measure to track down the kinetics of stress relaxations in amorphous systems, as we outline below. The corresponding atomic volumes, Xa , are evaluated by a Voronoi cell construction [21]. 3. Results and discussion As a starting point we first characterize the impact of preparation conditions (i.e. cooling rates) on global thermodynamic and mechanical properties. By monitoring the enthalphy during quenching we discern a slight reduction of T g with decreasing quench rate (Table 1), which seems to saturate at the slowest quench rate (T_ ¼ 0:005 K=ps). While the enthalpies, H, at 10 K after quench reveal only minor differences < 20 meV, the corre-

1

Here, cyz ¼ 2yz denotes the technical strain.

sponding mechanical properties are more severely affected. These results are consistent with previous findings on a CuZr system [22], which also reveals a strong configurational dependence of the mechanical properties. The physical origin of this behavior will be addressed in the rest of this work. All systems were investigated further, focusing on the slower quench rates (e.g. T_ ¼ 0:005 K=ps). Introducing a ˚ ) to eliminate vibrational contributions cutoff (dp > 0.15 A enables us to identify localized STZs from d p , which occur temporally and spatially distributed with varying sizes from tens to hundreds of atoms. Visual inspection indicates that STZs emerge from cores with relatively large displacements which are surrounded by more diffuse atomic rearrangements, as described previously [4]. Quantitatively we determine the average STZ size by an algorithm that counts the number of atoms within mobile clusters, based on a site-percolation criterion [23], requiring a minimum cluster size of 10 atoms (Fig. 1). Small shear events with tens of atoms emerge already in the quasilinear regime of the stress–strain curve and seem to trigger larger events (up to 250 atoms). The average STZ size increases towards higher strains up to the yielding point, which – macroscopically – is accompanied by growing deviations from linear elasticity. Investigations with a higher time resolution of Dt ¼ 1 ps revealed a characteristic transformation time of a STZ of the order of 1–10 ps – in good agreement with recent studies of Delogu [25] on NiZr. As STZs are single rapid shear events, it is reasonable to assume that this transformation time is largely independent of the shear rate. In the following we aim to gain more insight into the physical origins of STZs by evaluating correlations between the occurrence of STZs and local materials properties. Within this context the appropriate choice of the term ‘‘local” is a central issue, i.e. the properties have to be considered on the length scales of STZs. This is achieved by subdividing the simulation cells into N  N  N subcells with N = 8 (corresponding to 70 atoms; a different choice of N did not alter the results, but led to less significant signals) and employing Pearson’s correlation coefficient [26] to quantify correlations between NADs and key physical parameters. As the latter we chose average stresses ryz , pressure p ¼ 13 ðrxx þ ryy þ rzz Þ, potential energy Epot and atomic Voronoi volume X, which are computed for each subcell and timeframe. As these quantities themselves did

Table 1 Influence of quench rate, T_ , on T g , enthalphy H, shear modulus G and yielding point (ccyz , rcyz ) at 10 K. T_ (K/ps)

T g (K)

H (eV)

G (GPa)

ccyz ð%Þ

rcyz (GPa)

50 5 0.5 0.05 0.005

641 631 619 606 609

4.246 4.250 4.254 4.257 4.260

20.7 22.7 23.1 24.4 24.9

4.6 5.1 6.2 7.5 7.6

8.6 10.5 12.0 14.4 14.9

M. Neudecker, S.G. Mayr / Acta Materialia 57 (2009) 1437–1441

1800

Avg. cluster size

1600 Stress-Strain-Curve 1400

1600

1200

1200

1000

1000

800

800

600

600

400

400

200

200

1400

Average Cluster size

Shear stress [107 Pa]

1800

0

0 0

1

2

3

4

5

6

7

8

9 10

Strain γyz [%] Fig. 1. Typical stress–strain curve for amorphous Cu50Ti50 quenched at T_ ¼ 0:05 K=ps as green (light grey) points – in comparison to the average size of mobile clusters as red (dark grey) impulses. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

not reveal significant correlations with the NADs (note that in particular the absence of correlations with the average atomic volume is in accordance with other recent studies [24] but conflicts with ‘‘free volume” models [27]), we focus on changes in these quantities within a time/stress increment due to shear events and their correlation with the corresponding NADs (Fig. 2). We first note that NADs are anticorrelated with shear stress changes, Dryz , which basically constitutes a proof of concept of STZs. A strong increase in anticorrelation is evident at higher strains (beyond linear elasticity), which can be understood from a strongly increasing STZ size (Fig. 1). In contrast, changes of key structure-related properties (atomic volume, pressure and potential energy) are barely correlated with NADs (as evident from j R j< 0:1), thus indicating that – while relieving stresses – STZs hardly affect the average structure of glass. We would like to note, though, that in the course

0.2 0.1 0

R

-0.1 -0.2 -0.3

R (Δσyz, dp) R (Δp, dp) R ( ΔΩ, dp)

-0.4 -0.5 -0.6 0

1

2

3

4

5

6

7

8

9

10

Strain γyz [%] Fig. 2. Linear correlations, R, between non-affine displacements, d P , and Dryz as red (dark grey) boxes, Dp as green (light grey) stars, and DX as blue (black) crosses, respectively. (T_ ¼ 0:005 K=ps; N ¼ 8). (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

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of a shear event additional correlations are expected – particularly between NADs and energy due to the finite activation energy required to initiate a shear event. While the latter features are accessible to MD by choosing a strongly reduced time increment, Dt, they basically constitute a determination of activation energies of STZs, which has already been performed [10] for the present system. To address the dynamics of STZs in more detail we defi˚ within ne”mobile atoms” as those which reveal d p > 0:3 A a time interval of 10 ps somewhere between c ¼ 1:0% and the yielding point. For all mobile atoms at each strain mob mob and pmob increment, the average values rmob yz ; E pot ; X are computed and compared to the corresponding average values of the total cell. This approach is particularly suitable for observing changes of properties of mobile atoms during a shear event. The result is consistent with the correlation function in Fig. 2: localized plastic deformation coincides with decreasing shear stress, whereas significant changes in other properties are not observed. Fig. 3 shows a representative stress evolution rmob yz ðcyz Þ for mobile atoms between 5.3% and 5.4% strain (where a localized shear increases approxievent occurred). In the beginning rmob yz mately linearly with applied strain, but then drops abruptly c;mob at the shear event, which defines the local yield stress ryz (one strain increment before the corresponding shear event) and the stress drop Drmob yz . The onset, structure and stress dynamics of plastic events (so-called ‘‘elastic breakdown”) were also studied in detail by Maloney and Lemaıˆtre on a two-dimensional system [28], and their findings qualitatively confirm our results. A distribution of stress drops for mobile atoms at 40 consecutive strain increments is presented in Fig. 4, with a broad distribution ranging from +10 to 30 GPa and  8 GPa. Single events an average stress drop of Drmob yz occurred at strains cyz < 3% and with positive Drmob yz involved fewer atoms than average STZs. For events with the average magnitude of local yield stresses negative Drmob yz and stress drops is independent of cell strain cyz . It is remarkable that the average initial shear stresses rmob yz ð1:0%Þ of atoms which become involved in shear events during shearing (see Table 2) are substantially higher than the overall average initial stresses rav yz ð1:0%Þ. This indicates that unstable regions are characterized by a high initial stress, i.e. STZs are regions with a high initial stress bias, which are – consequently – more prone to fail under additional loading. Slower quenching causes higher stress bias and a rise – albeit less pronounced – of the local yielding point. In particular, quenching at T_ ¼ 50 K=ps causes immediate onset of plastic flow under shearing, which is reflected in significantly lower values of rmob yz ð1:0%Þ and c;mob . Average stress drops increase from  2 GPa (T_ ¼ ryz 50 K=ps) to  9 GPa (T_ ¼ 5 K=ns). Thus the large differences in macroscopic mechanical properties (Table 1) could be attributed to a different grade of stress localization, i.e. a different density of STZs. This is strongly supported by results from Shi and Falk [29] on a two-dimensional Lennard–Jones system, who showed that slow cooling rates

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Shear stress σyzmob [GPa]

25 20 15 10 5 0 -5

0

1

2

3

4

5

6

7

8

9

10

Strain γyz [%] Fig. 3. Dramatic change of shear stress rmob of mobile atoms during a shear event (T_ ¼ 0:005 K=ps) as red (dark grey) boxes. The small box to the right yz ˚ dark d p P 1:0 A). ˚ (For interpretation of the references in shows the corresponding atomic-scale relaxation with NAD color coding (bright: d p P 0:3 A, color in this figure legend, the reader is referred to the web version of this article.)

5

mobile atoms all atoms

Frequency

4 3 2 1 0 -30

-20

-10

0

10

20

30

Δσ mob yz Fig. 4. Stress drops Drmob of mobile atoms during 40 strain increments yz from 3.0% to 7.0% (as red/dark grey impulses) compared to the overall stress change Dryz (T_ ¼ 0:005 K=ps) as green (light grey) steps. The histogram of mobile atoms is clearly shifted towards lower shear stress levels. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

Table 2 Average shear stresses for all atoms (rav yz ð1:0%Þ) and mobile atoms c;mob (rmob yz ð1:0%Þ); mobile atoms have been identified for c 2 ð3:0; 7:0Þ%. ryz denotes the average local yield stress of mobile atoms. T_ (K/ps)

rav yz ð1:0%Þ

rmob yz ð1:0%Þ

rc;mob yz

50 5 0.5 0.05 0.005

2.2 2.4 2.5 3.0 2.6

4.5 8.1 9.6 8.5 10.2

9 14 17 16 16

(0.2) (0.2) (0.2) (0.2) (0.2)

(1.5) (1.3) (1.4) (1.7) (2.4)

(2) (2) (2) (2) (3)

increase the degree of localized deformation, while quickly cooled samples deformed more homogeneously. Up to now we have only investigated shear deformation in amorphous Cu50Ti50 at temperatures as low as 10 K, i.e. in the low-temperature limit. At elevated temperatures (below Tg), modifications of the shear mode might arise due to thermally activated processes. To investigate this aspect further, we performed several shear simulations

for temperatures between 100 K and 550 K and monitored mobility with a NAD cutoff adapted to the amplitude of thermal vibrations at the respective temperature. We find that at elevated temperatures NADs delocalize into a dynamic network-like structure [4] of low-dimensional string-like entities [13]. Whereas highly localized successive shear events occur at low temperatures, these events delocalize spatially and temporally at elevated temperatures into a number of thermally activated string-like entities. That is, macroscopically the stress–strain curve reaches the steady-state-flow regime far below the theoretical yielding point, as locally enduring thermal activation of strings prevents localization of large stresses (which are responsible for shear events at T = 10 K). This is reflected by insignificant correlations between NADs and change in shear stress or atomic volume. We speculate that string-like fluctuations play a key role in facilitating the occurrence of STZs by creating stress inhomogeneities in the amorphous matrix, part of which can be activated as STZs by external strain. In fact, coupling of strings and STZs has been an underlying assumption particularly in PEL pictures [14,30,31]. 4. Summary To summarize, our simulation results indicate that plastic deformation under shear strain at low temperatures is mediated by localized shear events, termed STZs, which relieve local shear stresses. STZs in Cu50Ti50 have a typical lifetime of 1–10 ps, and consist of cores with high mobility surrounded by weaker, diffuse rearrangements. Their size increases from tens to hundreds of atoms during the shear process, until they presumably percolate to a shear band at the yielding point. We were able to show that atoms involved in STZ dynamics are characterized by a high initial shear stress bias, and the stress evolution could be quantified in terms of the quench rate. In contrast to the dynamic generation of delocalized inhomogeneities in the high-temperature regime, low-temperature deformation is dominated by spatiotemporally localized, quenched-in

M. Neudecker, S.G. Mayr / Acta Materialia 57 (2009) 1437–1441

STZs. We would like to note that this picture can also explain other important phenomena, such as the onset of creep or the glass transition itself at elevated temperatures. Acknowledgements

[10] [11] [12] [13] [14] [15]

We thank Prof. Dr. K. Samwer and the glass seminar group for valuable discussions, T. Edler and C. Vree are acknowledged for proofreading the manuscript. A grant of computing time by the Gesellschaft fu¨r wissenschaftliche Datenverarbeitung Go¨ttingen as well as funding by the German DFG–PAK 36 are gratefully acknowledged.

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