# Atomistic Origin of Rate-Dependent Serrated Plastic Flow in Metallic Glasses

- SY Jiang
^{1}, - MQ Jiang
^{1}, - LH Dai
^{1}Email author and - YG Yao
^{2}

**3**:524

**DOI: **10.1007/s11671-008-9192-7

© to the authors 2008

**Received: **7 October 2008

**Accepted: **17 October 2008

**Published: **4 November 2008

## Abstract

Nanoindentation simulations on a binary metallic glass were performed under various strain rates by using molecular dynamics. The rate-dependent serrated plastic flow was clearly observed, and the spatiotemporal behavior of its underlying irreversible atomic rearrangement was probed. Our findings clearly validate that the serration is a temporally inhomogeneous characteristic of such rearrangements and not directly dependent on the resultant shear-banding spatiality. The unique spatiotemporal distribution of shear banding during nanoindentation is highlighted in terms of the potential energy landscape (PEL) theory.

### Keywords

Metallic glasses Serrated plastic flow Strain rate Molecular dynamicsBulk metallic glasses (BMGs), because of their long-range atomic disorder, deform uniquely [1–4]: the plastic deformation is highly localized into narrow shear bands at room temperature [5–8]. Under deformation-constrained loading modes such as compression [9–11] and nanoindentation [12–19], serrated plastic flow phenomena have been widely observed and found to be rate-dependent: as strain rate decreases, the flow serrations become more distinct. Moreover, rate-dependent shear-band patterns were observed on the surfaces or inside the post-deformed specimens under compression [10] and nanoindentation [15–17]. Therefore, currently, it is well accepted that the macroscopic serrated plastic flow behavior is associated with the shear-banding operations on a nanoscale within BMGs. Considerable efforts have been made to uncover the relationship between them. Schuh et al. [13], Schuh and Nieh [14] and Zhang et al. [18] suggested that the simultaneous operations of multiple shear bands at high strain rate result in the smooth plastic flow; with increasing strain rate, the deformation mode transits from inhomogeneous to homogeneous. However, Jiang and Atzmon [15], Greer et al. [17], and Jang et al. [19] considered that the disappearance of flow serrations at high strain rate arises from the limitation of data acquisition and the instrumental response; even at extremely high strain rate, the plastic deformation is still inhomogeneous. Since these hypotheses were mainly gained from ex situ experimental observations on shear-band patterns, the physical origin of rate-dependent serrated plastic flow, even if well accepted as a temporal event or process behavior [10, 11, 15], is still not well grounded.

Very recently, infrared camera technique has been used for in situ observing dynamic shear banding operations in the study of the serrated plastic flow during compression by Jiang et al. [10]. Based on the information from experimental observations, they conjectured a spatiotemporal picture of shear-banding during serrated flow, and further explained the inhomogeneous deformation during the indentation [10]. However, as they pointed out the propagation of a shear band is very fast; hence only mature shear bands can be captured [10], not to mention the fine shear-banding events on atomic scale. In addition, during indentation, since the shear bands develop underneath indenter, direct observation in situ on them is very difficult, even impossible. The molecular dynamics (MD) simulation is generally believed to be an effective way in modeling various indentation processes [20–22], providing an in situ observation on atomic motions. In this aspect, remarkable progress has been made by Falk, Langer, Shi et al. [20–23] by developing Argon’s shear transformation zone (STZ) concept [2]. They found that the load-drop events in a load-displacement curve, i.e., serrated plastic flow, correspond closely to the bursts in deformation activity (irreversible atomic rearrangement) associated with shear bands [21]. Spatially, the suppression of wing-like shear bands of post-deformed specimen at higher strain rate leads to milder serrations [22]. However, how do the strain rates affect this inherent correlation between serrations and atomic rearrangements? How does the temporal behavior of such rearrangements produce the final shear-banding patterns after loading? These questions are still not well understood and deserve further investigation. In this letter, we rely on MD computer simulations to roundly probe how spatiotemporal distribution of shear-banding events on an atomic scale is related in situ to macroscopic rate-dependent serrated plastic flows in BMGs undergoing nanoindentation. Strain rate effect on this relationship and its underlying physics are discussed.

_{46}Zr

_{54}, was used in our MD simulations. In this system, atoms interact via a modified Lennard-Jones 4–8 potential of the form [24, 25]:

where *r*
_{
ij
} is the distance between the atoms *i* and *j*, A, B, C, and D are constants whose values are available in Ref. [25], and *r*
_{
t
} is the truncation distance with the values of 5.08, 5.58, and 6.00 Å for Cu–Cu, Cu–Zr, and Zr–Zr pairs, respectively. The motion of each atom was evaluated by integrating the Newtonian equations of motion using velocity-Verlet method with a time step of 1 fs. To form an amorphous sample, an initial structure for the sample was built by placing all atoms into a face-centered cubic (fcc) crystal lattice in a random order, and the initial velocities of all atoms were set to be zero [26]. The initial structure was gradually heated to 2400 K for sufficiently melting, and then cooled to 1 K with the cooling rate of 25 K/ps. In this process, the NPT ensemble was used, and the pressure was kept at zero; periodic boundary conditions were used in all three directions. Then, for the subsequent indentation simulations, we set the top boundary free and fixed a layer of 6.0 Å in thickness at the bottom. Another 100 ps was carried out to a new equilibration. Finally, a three-dimensional sample (sample I) which contains 432,000 atoms with the size of 250 × 250 × 125 Å^{3} and a two-dimensional sample (sample II) which includes 250,000 atoms with the size of 1950 × 1050 Å^{2} were formed in this way.

where *r* is the distance from the indenter center to a sample atom, *R* is the radius of the indenter which is chosen as 20 Å for sample I and 400 Å for sample II, *θ*(*R* − *r*) is the standard step function, and *E* is a constant which is equal to 3.0 and 3.9 nN/Å^{2} for a Cu atom and a Zr atom, respectively [28]. The indenter was displaced toward the top surface of the sample at a constant strain rate by keeping an invariable displacement interval of 0.1 Å and adjusting the relaxation time for each displacement interval. In this process, the control to the temperature (1 K) was only allowed in a layer of 10.0 Å in thickness which is just above the fixed layer at the bottom. For sample I, the total indentation depth was 15 Å, and three strain rates, 10^{11}, 10^{10}, and 10^{9} s^{−1}, were executed; for sample II, the total indentation depth was 50 Å, and three strain rates, 4 × 10^{10}, 4 × 10^{9}, and 4 × 10^{8} s^{−1}, were performed. Parallel computing was used in all the simulation processes.

*D*

_{min}

^{2}(−Δ

*t*

*t*), therefore, is adopted to identify such irreversible rearrangement with the form [23, 29]:

*n*runs over the atoms within the interaction range of the reference atom (

*n*= 0) and

**r**

_{ n }(

*t*) is the position vector of the

*n*th atom at time

*t*. The parameter

*D*

_{min}

^{2}(

*t*− Δ

*t*

*t*) then denotes the local deviation between the true deformation denoted by

**r**

_{ n }(

*t*) −

**r**

_{0}(

*t*)] and the affine deformation indicated by (X Y

^{−1})

**r**

_{ n }(

*t*− Δ

*t*)−

**r**

_{0}(

*t*− Δ

*t*)] during the time interval

*t*− Δ

*t*

*t*[29]. We calculated

*D*

_{min}values of all atoms during each displacement interval (0.1 Å) to get information of in situ deformation. We selected 1.5 Å, which is about half of the average distance between a Cu atom and a Zr atom in the samples, as a cutoff of

*D*

_{min}to characterize the rearrangements that make up a plastic event at all strain rates [23, 29]. It is important to point out that the method of choosing the cutoff may be a little rough, considering its value may be affected by strain rates; nevertheless, it is efficient to judge the plastic deformation, and should not significantly change the trend. In addition, we find that choosing different cutoffs cannot change the trend of plastic flow under various strain rates. Any atom whose

*D*

_{min}value is greater than the cutoff is considered to be rearranged, and the numbers of the rearranged atoms at all intervals are displayed in Fig. 1 as histograms under p–h curves. Note that the numbers of the rearranged atoms get larger, but their distribution becomes more inhomogeneous when strain rate decreases. Moreover, when comparing the histograms with the p–h curves, a strong correlation between them was surprisingly discovered: the load-drop events (i.e., flow serrations) in the p–h curves correspond to the peak values (i.e., large numbers of the rearranged atoms) in the histograms; the more obvious the flow serration is, the larger the number of the rearranged atoms in that interval is. The phenomenon is consistent with the simulation results presented by Shi and Falk [21]. In fact, the number of the rearranged atoms can be regarded as an indication of the degree of plastic deformation in the displacement interval. Thus, we can conclude that the serrated plastic flow strongly depends on the temporal characteristic of the atomic rearrangement underpinning plastic deformation: successive low degree of plastic deformation at high strain rate leads to less pronounced serrated flow, and intermittent high degree of plastic deformation at low strain rate results in more distinct serrated flow.

*D*

_{min}(0,

*t*) values; here, the darker the color the larger the

*D*

_{min}(0,

*t*) value. They are quite similar to those observed from instrumented [15–19] and simulated [20, 22] indentations. The results in Figs. 2 and 3 were taken from sample II for the reason that larger planar size is available to display the shear-band patterns. Furthermore, the potential energy versus displacement curves, as shown in Fig. 4a, were also found to be rate-dependent. With decreasing strain rate, the potential energy grows more slowly, but fluctuates more prominently; the energy-drop events essentially correspond to the load-drop events. Since our MD processes proceed at very low temperature, the potential energy which is large compared to the thermal energy must dominate the flow [30]. Thus, the unique spatiotemporal characteristic of deformation can be understood in terms of potential energy landscape (PEL) theory [30–33].

Before loading, the system stays at a local minimum of the potential energy surface (PES), and the configuration of the system is metastable. Loading will tilt the PES, and induce the disappearance of some local minima. As a result, the system will became unstable, and move to a new energy local basin. At the same time, atomic rearrangement occurs. The disappearance of a potential energy basin induced by loading is schematically shown in Fig. 4b. When transiting from one energy basin to another, the system at high strain rate will stay at a higher energy state in the new basin in a displacement interval than at low strain rate because of the shorter relaxation time it has. The comparison of the systematic states at different strain rates is schematically shown in Fig. 4c. Thus a whole transition of the system at high strain rate usually costs more displacement intervals, and the degree of plastic deformation at a particular displacement is relatively lower. On the contrary, at low strain rate, the system experiences less displacement intervals to reach the new basin, and the degree of plastic deformation at a particular displacement is generally higher. Hence the rate-dependent temporal distribution of deformation appears. It is this temporal characteristic of atomic rearrangement that dominates the macroscopic serrated plastic flow. On the other hand, since the potential energy of the system at the same indenting depth at high strain rate is higher, there will be more atoms at high energy state in the plastic zone. These atoms do not have enough time to rearrange. Thus, the rearrangement occurs at multiple regions, but the number of the rearranged atoms is small (see Fig. 2a). When next loading step is applied, atoms preferentially rearrange at the same positions owing to their relatively higher local energy. In other words, the shear banding preferentially operates at the same locations at higher strain rate [10]. So it is hard to conjecture from the shear-band patterns how many shear bands operate in a moment. On the other hand, the system has lower potential energy at low strain rate, so the number of the atoms with high energy state will be smaller. Thus the rearrangement happens at fewer regions. Nevertheless, the rearrangement in these few regions can develop more sufficiently, finally producing coarse shear bands (see Fig. 3c). In addition, since local atoms rearrange sufficiently, leading to a lower local energy, the atomic rearrangement at the next step must occur at other regions having high energy level. In other words, the shear bands nucleate and develop sufficiently at different positions under low strain rate. If the plastic deformation zone or spatial distribution is not related to its time, such as the uniaxial compression case [10], more and finer shear bands can finally be produced at low strain rates. However, in nanoindentation, the unique spatial patterning, fewer and coarser shear bands (see Fig. 3c), can be observed owing to space–time relationship of plastic deformation.

In summary, rate-dependent serrated plastic flow was observed in our MD simulations. The temporally inhomogeneous characteristic of the plastic deformation was revealed as the main determining factor of serrated flow behavior. It is not proper to directly relate the numbers of shear bands with the flow serrations. For example, although there is no serration in some Fe-based or Ce-based BMGs during nanoindentation, a number of fine shear bands are observed under the indents [34]. The unique rate-dependent spatiotemporal distributions of shear banding can be understood in terms of PEL theory. We believe that these findings can shed light on the relationship between microstructure and inhomogeneous plastic flow in BMGs.

## Declarations

### Acknowledgements

The work was supported by the Natural Science Foundation of China (Grants Nos. 10725211, 10721202, 10534030, 10674163), the Ministry of Science and Technology of China (2006CB921300, 2007CB925000), and the Knowledge Innovation Project & Key Project of Chinese Academy of Sciences (Nos. KJCX2-YW-M04 and KJCX-SW-L08). All computation of this work were carried out by Supercomputer DeepComp 6800, and we thank Dr. Yangde Feng of Super Computing Center of Chinese Academy of Science for his help in the computations.

## Authors’ Affiliations

## References

- Spaepen F:
*Acta Met.*. 1977,**25:**407. COI number [1:CAS:528:DyaE2sXltVGntrg%3D] COI number [1:CAS:528:DyaE2sXltVGntrg%3D] 10.1016/0001-6160(77)90232-2View ArticleGoogle Scholar - Argon AS:
*Acta Metal.*. 1979,**27:**47. COI number [1:CAS:528:DyaE1MXhvFSqtrg%3D] COI number [1:CAS:528:DyaE1MXhvFSqtrg%3D] 10.1016/0001-6160(79)90055-5View ArticleGoogle Scholar - Schun CA, Hufnagel TC, Ramamurty U:
*Acta Mater.*. 2007,**55:**4067. 10.1016/j.actamat.2007.01.052View ArticleGoogle Scholar - Falk ML:
*Science*. 2007,**318:**1880. COI number [1:CAS:528:DC%2BD2sXhsVygtb%2FO] COI number [1:CAS:528:DC%2BD2sXhsVygtb%2FO] 10.1126/science.1150919View ArticleGoogle Scholar - Hufnagel TC, EI-Deiry P, Vinci RP:
*Scr. Mater.*. 2000,**43:**1071. COI number [1:CAS:528:DC%2BD3cXoslSls7w%3D] COI number [1:CAS:528:DC%2BD3cXoslSls7w%3D] 10.1016/S1359-6462(00)00527-3View ArticleGoogle Scholar - Li J, Spaepen F, Hufnagel TC:
*Philos. Mag.*. 2002,**82:**2623. COI number [1:CAS:528:DC%2BD38XntFGlsL0%3D] COI number [1:CAS:528:DC%2BD38XntFGlsL0%3D]View ArticleGoogle Scholar - Liu LF, Dai LH, Bai YL, Wei BC:
*J. Non-Cryst. Solids*. 2005,**351:**3259. COI number [1:CAS:528:DC%2BD2MXhtVKjtrrN] COI number [1:CAS:528:DC%2BD2MXhtVKjtrrN] 10.1016/j.jnoncrysol.2005.07.030View ArticleGoogle Scholar - Zhang Y, Greer AL:
*Appl. Phys. Lett.*. 2006,**89:**071907. 10.1063/1.2336598View ArticleGoogle Scholar - Liu LF, Dai LH, Bai YL, Wei BC, Eckert J:
*Mater. Chem. Phys.*. 2005,**93:**174. COI number [1:CAS:528:DC%2BD2MXlt1Gisb4%3D] COI number [1:CAS:528:DC%2BD2MXlt1Gisb4%3D] 10.1016/j.matchemphys.2005.03.011View ArticleGoogle Scholar - Jiang WH, Fan GJ, Liu FX, Wang GY, Choo H, Liaw PK:
*Int. J. Plast.*. 2007,**24:**1. COI number [1:CAS:528:DC%2BD1cXitlOltrs%3D] COI number [1:CAS:528:DC%2BD1cXitlOltrs%3D] 10.1016/j.ijplas.2007.01.015View ArticleGoogle Scholar - Wright WJ, Saha R, Nix WD:
*Mater. Trans JIM*. 2001,**42:**642. COI number [1:CAS:528:DC%2BD3MXktlKjsLc%3D] COI number [1:CAS:528:DC%2BD3MXktlKjsLc%3D] 10.2320/matertrans.42.642View ArticleGoogle Scholar - Yang B, Nieh TG:
*Acta Mater.*. 2007,**55:**295. COI number [1:CAS:528:DC%2BD28Xht1KhsrnE] COI number [1:CAS:528:DC%2BD28Xht1KhsrnE] 10.1016/j.actamat.2006.08.028View ArticleGoogle Scholar - Schuh CA, Argon AS, Nieh TG, Wadsworth J:
*Philos. Mag.*. 2003,**83:**2585. COI number [1:CAS:528:DC%2BD3sXmtVGktbk%3D] COI number [1:CAS:528:DC%2BD3sXmtVGktbk%3D] 10.1080/1478643031000118012View ArticleGoogle Scholar - Schuh CA, Nieh TG:
*Acta Mater.*. 2003,**51:**87. COI number [1:CAS:528:DC%2BD38Xps1OgtLk%3D] COI number [1:CAS:528:DC%2BD38Xps1OgtLk%3D] 10.1016/S1359-6454(02)00303-8View ArticleGoogle Scholar - Jiang WH, Atzmon M:
*J. Mater. Res.*. 2003,**18:**755. COI number [1:CAS:528:DC%2BD3sXivVeltr0%3D] COI number [1:CAS:528:DC%2BD3sXivVeltr0%3D] 10.1557/JMR.2003.0103View ArticleGoogle Scholar - Dai LH, Liu LF, Yan M, Wei BC, Eckert J:
*Chin. Phys. Lett.*. 2004,**21:**1593. COI number [1:CAS:528:DC%2BD2MXltlKlsbw%3D] COI number [1:CAS:528:DC%2BD2MXltlKlsbw%3D] 10.1088/0256-307X/21/8/051View ArticleGoogle Scholar - Greer AL, Castellero A, Madge SV, Walker IT, Wilde JR:
*Mater. Sci. Eng. A*. 2004,**375:**1182. 10.1016/j.msea.2003.10.032View ArticleGoogle Scholar - Zhang GP, Wang W, Zhang B, Tan J, Liu CS:
*Scr. Mater.*. 2005,**52:**1147. COI number [1:CAS:528:DC%2BD2MXjtV2nsrs%3D] COI number [1:CAS:528:DC%2BD2MXjtV2nsrs%3D] 10.1016/j.scriptamat.2005.01.045View ArticleGoogle Scholar - Jang J-I, Yoo BG, Kim JY:
*Appl. Phys. Lett.*. 2007,**90:**211906. 10.1063/1.2742286View ArticleGoogle Scholar - Shi Y, Falk ML:
*Appl. Phys. Lett.*. 2005,**86:**011914. 10.1063/1.1844593View ArticleGoogle Scholar - Shi Y, Falk ML:
*Thin Solid Films*. 2007,**515:**3181.Google Scholar - Shi Y, Falk ML:
*Acta Mater.*. 2007,**55:**4317. COI number [1:CAS:528:DC%2BD2sXntFensbg%3D] COI number [1:CAS:528:DC%2BD2sXntFensbg%3D] 10.1016/j.actamat.2007.03.029View ArticleGoogle Scholar - Falk ML, Langer JS:
*Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics*. 1998,**57:**7192. COI number [1:CAS:528:DyaK1cXjvVKmsL4%3D] COI number [1:CAS:528:DyaK1cXjvVKmsL4%3D]Google Scholar - Kobayashi S, Maeda K, Takeuchi S:
*J. Phys. Soc. Jpn.*. 1980,**48:**1147. COI number [1:CAS:528:DyaL3cXit1CntLg%3D] COI number [1:CAS:528:DyaL3cXit1CntLg%3D] 10.1143/JPSJ.48.1147View ArticleGoogle Scholar - Lund AC, Schuh CA:
*Acta Mater.*. 2003,**51:**5399. COI number [1:CAS:528:DC%2BD3sXnsl2rt7s%3D] COI number [1:CAS:528:DC%2BD3sXnsl2rt7s%3D] 10.1016/S1359-6454(03)00396-3View ArticleGoogle Scholar - Bailey NP:
*Phys. Rev. B*. 2004,**69:**144205. 10.1103/PhysRevB.69.144205View ArticleGoogle Scholar - Kelchner CL, Plimpton SJ, Hamilton JC:
*Phys. Rev. B*. 1998,**58:**11085. COI number [1:CAS:528:DyaK1cXntVaqsbg%3D] COI number [1:CAS:528:DyaK1cXntVaqsbg%3D] 10.1103/PhysRevB.58.11085View ArticleGoogle Scholar - Ma XL, Yang W:
*Nanotechnology*. 2003,**14:**1208. COI number [1:CAS:528:DC%2BD2cXhvFGktQ%3D%3D] COI number [1:CAS:528:DC%2BD2cXhvFGktQ%3D%3D] 10.1088/0957-4484/14/11/009View ArticleGoogle Scholar - Gannepalli A, Mallapragada SK:
*Nanotechnology*. 2001,**12:**250. COI number [1:CAS:528:DC%2BD3MXotlGls7Y%3D] COI number [1:CAS:528:DC%2BD3MXotlGls7Y%3D] 10.1088/0957-4484/12/3/309View ArticleGoogle Scholar - Goldstein M:
*J. Chem. Phys.*. 1969,**51:**3728. COI number [1:CAS:528:DyaE3cXmsVSl] COI number [1:CAS:528:DyaE3cXmsVSl] 10.1063/1.1672587View ArticleGoogle Scholar - Stillinger FH, Weber TA:
*Science*. 1984,**225:**983. COI number [1:CAS:528:DyaL2cXlsFyisbs%3D] COI number [1:CAS:528:DyaL2cXlsFyisbs%3D] 10.1126/science.225.4666.983View ArticleGoogle Scholar - Osborne MJ, Lacks DJ:
*J. Phys. Chem. B*. 2004,**108:**19619. COI number [1:CAS:528:DC%2BD2cXmtlOmtrg%3D] COI number [1:CAS:528:DC%2BD2cXmtlOmtrg%3D] 10.1021/jp047715zView ArticleGoogle Scholar - Mayr SG:
*Phys. Rev. Lett.*. 2006,**97:**195501. COI number [1:STN:280:DC%2BD28jitlCnsw%3D%3D] COI number [1:STN:280:DC%2BD28jitlCnsw%3D%3D] 10.1103/PhysRevLett.97.195501View ArticleGoogle Scholar - Li WH, Wei BC, Zhang TH, Xing DM, Zhang LC, Wang YR:
*Intermetallics*. 2007,**15:**706. 10.1016/j.intermet.2006.10.010View ArticleGoogle Scholar