Relationship between local geometrical factors and mechanical properties for Cu–Zr amorphous alloys

Intermetallics 15 (2007) 139e144www.elsevier.com/locate/intermet

Relationship between local geometrical factors and mechanicalproperties for CueZr amorphous alloys

Masato Wakeda a,*, Yoji Shibutani a,b, Shigenobu Ogata a,b, Junyoung Park a

a Department of Mechanical Engineering, Graduate School of Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japanb Center of Atomic and Molecular technologies, Graduate School of Engineering, Osaka University, 2-1, Yamadaoka, Suita, Osaka 565-0871, Japan

Received 10 February 2006; received in revised form 6 April 2006; accepted 14 April 2006

Available online 22 June 2006

Abstract

CuxZr1�x (x¼ 0.30e0.85) amorphous alloy models were constructed using molecular dynamics simulations. In order to estimate the localstructures characterized by pentagonal short-range order and free volume, Voronoi polyhedra analyses were performed for the initial states andalso for the deformed states of the models. Both these geometrical factors are intimately related to each other and exhibit a large influence onmechanical properties. The elastic properties tend to increase as the free volume decreases. Moreover, flow stress drastically decreases with in-creasing free volume content. It was found that the non-pentagonal regions, which are sources for the generation of high free volume structures,preferentially undergo irreversible atomic rearrangements in the early stage of the deformation process.� 2006 Elsevier Ltd. All rights reserved.

Keywords: B. Elastic properties; B. Glasses, metallic; B. Plastic deformation mechanisms; E. Mechanical properties, theory; E. Simulations, atomistic

1. Introduction

Amorphous metals have received much attention due totheir excellent mechanical properties, including high yieldstrength, high fracture toughness and high elastic limit. Theseproperties arise from their disordered atomic structures.Unlike crystalline materials, amorphous metals have randomclosed-packed structures. Therefore, plastic deformation ofamorphous metals is not dependent on dislocation-basedmechanisms. Argon [1] first proposed the simplified plasticunits, the so-called STZ (Shear Transformation Zone). Froma microscopic viewpoint, it is hypothesized that the appliedstress generates local shear shuffling among the atoms withina STZ, and chain-reactions of STZs occur in sequence [2]. Themacroscopic shear bands observed in some previous experi-mental studies [3,4] have been recognized to form due toself-assembly of STZs. This suggests that the local structure

* Corresponding author. Tel./fax: þ81 06 6879 4121.

E-mail address: [email protected] (M. Wakeda).

0966-9795/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.intermet.2006.04.002

variation observed in amorphous alloys is crucial, and atomis-tic modeling, taking microscopic inhomogeneities intoaccount, is necessary to understand the deformation mecha-nism. The question arises as to what kind of local structurepreferentially undergoes irreversible atomic rearrangement inthe early stages of the deformation process. Some studieshave pointed out the role of the density and the free volumevariation on the deformation process. Spaepen [5] indicatedthat shear viscosity can be appreciably reduced in the locallydilated region of a high free volume, and predicted the local-ized deformation by the increase in the free volume inducedby the plastic deformation. Argon and Kuo [6] also showedthat plastic deformation in the local shear zone was accompa-nied by an increment in free volume. However, since free vol-ume can be defined as the global volume difference withrespect to the reference state, its extension to the atomic-levelspecific structure is not clearly understood. In addition, theevolution process of local short-range structure, such as thenon-periodic icosahedron in the deformation process, is stillnot clear, although such a geometry-specific factor is thoughtto play a critical role in the early stages of deformation.

140 M. Wakeda et al. / Intermetallics 15 (2007) 139e144

Our aims in the present study are, therefore, to identify thegeometrical factors characterizing the free volume structure,and to clarify the role of these factors in the deformation pro-cess. We computationally modeled the atomic CuxZr1�x amor-phous alloys, with x ranging from 0.30 to 0.85, usingmolecular dynamics simulations. These models are enforcedby uniaxial tension and simple shear loading to determine in-ternal structural evolution. Voronoi analyses were applied tolink these stress states to the mechanical properties of theamorphous alloys with the local geometrical factors.

2. Atomic amorphous alloy model

Our intention here is not to obtain precise physical proper-ties but rather to investigate the general principles of the rela-tionship between the locally inhomogeneous structure and thecomplex deformation mechanism. The simple pair potentials[7], which refer to short-range two-body effects, are thusthought to be sufficient to describe CueCu, CueZr and ZreZrinteractions. These potentials are based on the LennardeJones4e8 potential, and the potential parameters are determined soas to satisfy the equilibrium values of the nearest neighbor dis-tances of the FCC lattices of Cu and Zr, the elastic constants ofcrystalline Cu and Zr, and other conditions.

CuxZr1�x amorphous alloy models with values of x of 0.30,0.40, 0.50, 0.57, 0.64, 0.75, 0.80, and 0.85 were prepared asfollows. The source model was randomly packed with Cuand Zr atoms with the specific composition into the simulationcell. It was numerically heated to 2000 K, which is muchhigher than the melting point. After a sufficient relaxation of20 ps, it was cooled to 0 K at a rate of 1012 K/s. The externalpressure during this heat treatment was kept at zero by theParrinelloeRahman ensemble [8], and the temperature of thewhole cell was controlled by scaling the velocity of eachatom. Periodic boundary conditions were applied to the threedimensional directions in order to eliminate surface effects.The radial distribution functions and the density of the con-structed Cu57Zr43 model were in good agreement with the ex-perimentally obtained data [9,10], as shown in Table 1.

To understand the local structural evolution during defor-mation, we performed Voronoi analyses [11], indicating the3D atomic configuration between the centered and the sur-rounding atoms. As an example, we represent the Voronoipolygon and its atomic configuration that existed in theCu57Zr43 model in Fig. 1 and also the Voronoi index list inFig. 2. The Voronoi index shows that most of the Voronoipolygons have a specific kind of center atom. In the(0,0,12,0) case, the Cu atom accounts for more than 99% of

Table 1

Density and the first peaks of the radial distribution function (RDF) of the

Cu57Zr43 model along with the experimental values

Density (g/cm3) First peaks of RDF (A)

CueCu CueZr ZreZr

Model 7.366 2.53 2.83 3.13

Exp. 7.382 [9] 2.65 [10] 2.80 [10] 3.15 [10]

the center atom. Therefore, the contents of the Voronoi poly-gon are expected to largely depend on the global atomiccomposition.

3. Results

3.1. Structural analyses of initial states

The fraction of Voronoi triangles, tetragons, pentagons,hexagons, and heptagons in the quenching process is shownin Fig. 3. Similar to some previous computational results[12,13], only the fraction of pentagons increased with

Cu-Zr (Pentagon)

Cu-Zr (Pentagon)

Cu-Cu (Hexagon)

Cu-Cu (Tetragon)

Cu-Zr (Pentagon)

Cu-Cu (Pentagon)

Cu-Cu (Tetragon)Cu-Zr (Pentagon)

Cu-Cu (Pentagon)

Cu-Zr (Hexagon)

Cu-Zr (Heptagon)

Cu-Cu (Triangle)

Cu-Zr (Hexagon)

Cu-Cu (Triangle)

Fig. 1. Voronoi polyhedron, its center and surrounding atoms. This polyhedron

consists of one triangle, two tetragons, six pentagons, three hexagons and one

heptagon, representing as the Voronoi index of (1,2,6,3,1).

0 5 10 15

Fraction of Voronoi polyhedra (%)

(0, 0, 12, 0)

(0, 2, 8, 2)

(0, 1, 10, 2)

(0, 3, 6, 4)

(0, 1, 10, 4)

(0, 2, 8, 5)

(0, 1, 10, 5)

(0, 2, 8, 6)

(0, 2, 8, 4)

(0, 2, 8, 1)

(0, 3, 6, 3)

(0, 0, 12, 3)

(0, 1, 10, 3)

(1, 2, 6, 3, 1)

(0, 3, 6, 5)

Vor

onoi

inde

x

Cu

Zr

Fig. 2. Voronoi index list of the constructed Cu57Zr43model. These are the ma-

jor components among the more than 100 kinds of Voronoi polygons contained

in the model. Dark and light parts represent the ratio of Cu and Zr centers,

respectively.

141M. Wakeda et al. / Intermetallics 15 (2007) 139e144

decreasing temperature. Since the drastic increase in the pen-tagonal short-range order starts right before the glass transitiontemperature Tg, the pentagon can be regarded as one of theimportant geometrical features of amorphous structure.

The relationship between the fraction of each polygon andthe Cu concentration is shown in Fig. 4. The set of modelsplotted in Fig. 4 was all constructed by the same melt-quenchprocess. Thus, the observed differences in the internal struc-tures arise purely from their atomic compositions. It can beobserved that the fraction of pentagons increases with theCu concentration and reaches a maximum of 75%, whilethe other polygons decrease in number. It has been shownthat the rich icosahedra enhance glass-forming ability, since

0

20

40

60

Pentagon

Hexagon

Tetragon

Triangle

Heptagon

Temperature (K)

Frac

tion

of f

aces

(

)

100 1900700 1300

Volume

1.18

0.98

1.06

1.02

1.10

1.14

Vol

ume

V/V

0

Tg

Fig. 3. Change in the fractions of triangles, tetragons, pentagons, hexagons,

and heptagons during the quenching process. The cell volume normalized

by the 100 K value is also represented.

20 30 40 50 60 70 80 900

20

40

60

Pentagon

Hexagon

Tetragon

Triangle Heptagon

Concentration (at. Cu)

Frac

tion

of V

oron

oi p

olyg

ons

( )

Fig. 4. Change in the fractions of polygons as a function of the Cu

concentration.

they prevent crystallization leading to a stable amorphousstructure in the quenching process [14]. The number of icosa-hedra is proportional to the number of pentagons in the pres-ent computational studies. Therefore, the most desirablecomposition of the current binary system is around x¼ 0.75under the glass-forming criterion specified only by the num-ber of icosahedra.

The definition of free volume sometimes refers to the frac-tion of matter having a lower atomic coordination than that ofa reference material that has dense packing. We calculated thenormalized free volume, which is obtained by the volume in-crement from the FCC crystal structure, as follows.

0.025

0.030

0.035

0.040

Nor

mal

ized

fre

e vo

lum

e, V

free

Fraction of pentagons ( )

Cu57Zr43

Cu50Zr50

Cu40Zr60

Cu64Zr36

Cu75Zr25

Cu80Zr20

Cu85Zr15

Cu30Zr70

50 6256 58 6052 54

Fig. 5. Relationship between the normalized free volume and the fraction of

pentagons for eight different compositions. The solid line represents the least

squares fit.

0

0.5

1

1.5

2

Shear strain ( )

Shea

r st

ress

(G

Pa)

Cu30Zr70

Cu40Zr60

Cu50Zr50

Cu57Zr43

Cu64Zr36

Cu75Zr25

Cu80Zr20

Cu85Zr15

Lower free volume

10 20

Fig. 6. Shear stress and shear strain curves for the different compositions.

142 M. Wakeda et al. / Intermetallics 15 (2007) 139e144

Vfree ¼Vamo �Vcry

Vcry

; ð1Þ

where Vamo and Vcry represent the volume of the amorphousand crystal models, respectively. Vcry was estimated basedon the atomic volumes of one component single crystallinestates of Cu and Zr, taking the atomic composition into ac-count. The relationship between the above-defined free vol-ume Vfree and the fraction of pentagons is shown in Fig. 5for eight different compositions. It is apparently found thatthe pentagonal structure makes the matter densely packedand results in a lower free volume.

l

Δl'

Fig. 7. Inhomogeneous shear banding under 30% shear deformation. Global

homogenous shear strain g (¼Dl’/l ) is applied by increasing Dl’. Atoms

are colored according to the degree of the atomic displacement. The lighter

color represents a large displacement, while the darker color represents a small

displacement.

3.2. Structural analyses of deformation states

We performed structural analyses of the deformed atomicmodels under uniaxial tension and simple shear loading. Thesimulation cell was applied with a small strain increment ateach step under periodic boundary conditions. Subsequently,all the atoms in the cell were allowed to fall into the localequilibrium states by the conjugate gradient method [15],which means that the circumferential temperature should be0 K. The applied stress for the whole system was estimated us-ing the atomic stress definition based on the virial theorem.

The calculated shear stress and applied strain relation underthe simple shear simulation are shown in Fig. 6 for differentcompositions. After an initial linear response, each curve hasa maximum at 8e10% shear strain. In the uniaxial tensile sim-ulations, the stressestrain relations, which are not shown here,also show the elasticeplastic behaviors with maximum at5e7% true strain. We observed in both simulations that theflow stress is sensitive to the free volume; the flow stress con-siderably decreases with increasing the free volume content. Inthe shear simulation, shear banding, which has been observedin some experimental studies [3,4], appears parallel to theloading direction shortly after the shear stress reaches themaximum (see Fig. 7). Then, subsequent global deformationlocalizes intense shear across this narrow shear layer, andthe shear resistance drastically decreases [16]. However, underuniaxial tension, plastic deformations appear discretely, and

Table 2

Average bond lengths between the center and the neighboring atoms of

Voronoi polygons (in A)

CueCu CueZr ZreZr

Triangle 3.78 3.94 4.09

Tetragon 3.29 3.55 3.73

Pentagon 2.61 2.98 3.33

Hexagon 2.52 2.83 3.15

Heptagon 2.45 2.80 3.12

0.025 0.03 0.035 0.0470

75

80

85

Normalized free volume, Vfree

You

ng's

mod

ulus

, E (

GPa

) Cu57Zr43

Cu40Zr60

Cu30Zr70

Cu50Zr50

Cu64Zr36

Cu75Zr25

Cu85Zr15

Cu80Zr20

0.025 0.03 0.035 0.0425

26

27

28

29

30

Normalized free volume, Vfree

Shea

r m

odul

us, G

(G

Pa)

Cu57Zr43

Cu40Zr60

Cu30Zr70

Cu50Zr50

Cu64Zr36

Cu75Zr25

Cu85Zr15

Cu80Zr20

a b

Fig. 8. Relationship between elastic properties and normalized free volume. (a) Young’s modulus and (b) shear modulus. Both are obtained from uniaxial tensile

and shear simulations, respectively.

143M. Wakeda et al. / Intermetallics 15 (2007) 139e144

a b c

Fig. 9. Density distributions of (a) tetragons, (b) pentagons, and (c) hexagons in the same cross-section. Circles correspond to the pentagonal region.

the localized deformation like a shear banding couldn’t be ob-served within the applied strain range.

Focusing on the elastic properties, the Young’s modulusE and shear modulus G as a function of the free volume areshown in Fig. 8. Both moduli linearly decrease with increasingfree volume content. This free volume dependence of the elas-tic properties may include two factors; substantial differencesin the free volume due different local structures and mixingeffects due to the different elastic moduli of Cu and Zr. Thelatter is estimated to be 3% at most from Cu75Zr25 toCu30Zr70 according to the law of mixture using the Young’smodulus of Cu and Zr for a one component single crystallinestate using the same potential. Therefore, we conclude that thisfree volume dependence should be predominantly caused bythe internal structure change.

Structural stability also depends on the local atomic config-uration, especially on the bond strength. We calculated aver-age bond lengths between the center and the neighboringatoms in each polyhedron observed in the Cu57Zr43 case, asshown in Table 2. The bond length is classified not only bythe type of atom, but also by the kind of face penetrated bythe bond (with reference to Fig. 1). We found that the bondspenetrating a pentagon agree well with the equilibrium dis-tance of the present pair potentials (CueCu: 2.66 A, CueZr:2.98 A, and ZreZr: 3.29 A). Therefore, the pentagon shouldhave low energy configuration that contributes to the localstructural stability.

a b

Fig. 10. (a) Density distribution of pentagons and (b) atomic displacements

from the initial configuration under 3% shear deformation. The arrows above

and below the figures show the shear direction.

As seen in Fig. 5, the pentagon symbolizes the dense-packed states and, therefore, leads to a lower free volume.We therefore estimated the density distribution of each Voro-noi polygon before and after deformation in order to representthe structural and density variations. The number density dis-tribution was calculated as follows. First, we extracted an ar-bitrary 1/10th slice from the model and divided it into 400small cells. The number of polygons was counted for eachcell and normalized by the total number of atoms within thecell. We effectively visualized the density distributions of tet-ragons, pentagons, and hexagons in the same cross-section, asshown in Fig. 9(a), (b), and (c), respectively. The magnitude ofthe density is denoted by the size of each polygon and thelower regions are colored darker. In these figures, the distribu-tion of each polygon is not homogeneous, but has a large den-sity fluctuation. It can be noticed that the distribution oftetragons is similar to that of hexagons, by comparingFig. 9(a) and (c), while the distribution of pentagons is differ-ent in comparison. The pentagon has 5-fold symmetry and of-ten exhibits quasi-crystalline order. In contrast, the tetragonand the hexagon have translational symmetry and often form

Shear strain ( )

Frac

tion

of V

oron

oi p

olyh

edra

(

)

( 0, 0, 12, 0)

( 0, 2, 8, 2)

( 0, 1, 10, 2)

( 0, 1, 10, 4)

( 0, 3, 6, 4)

0 10 203

14

10

11

12

13

9

8

7

6

5

4

30

Icosahedron

Fig. 11. Fraction of Voronoi polyhedra vs. shear strain in the simple shear sim-

ulation. (0,0,12,0), which corresponds to the icosahedral atomic configuration,

is one of the smallest clusters among all polyhedra.

144 M. Wakeda et al. / Intermetallics 15 (2007) 139e144

an ordinarily crystalline structure. From this viewpoint, it isunderstandable that the pentagon and the other polygons spon-taneously separate and assemble in the cooling process, andthe density variation is induced by the deviation of these geo-metrical factors. Since Fig. 5 indicates the explicit correlationbetween the free volume and the fraction of pentagons, wenext examine the relationship between the distribution of thepentagonal structure and inhomogeneities of local deforma-tion, and also the transition of each number of Voronoi poly-hedra during deformation.

Fig. 10(a) and (b) shows the density distribution of pentagonsand the atomic displacements from the initial positions in thesame cross-section, respectively. These are snapshots at the rel-atively low shear strain of 3%, at which the model mainly de-formed elastically, as shown in Fig. 6. In Fig. 10(b), we dyedthe section according to the atomic displacement calculatedby subtracting uniform cell deformation from actual deforma-tion, and thus the light-colored part corresponds to large non-affine displacement. In both the figures, circles are added torepresent the substantial deformation zones. We found that irre-versible deformation tends to occur in the non-pentagonalregion which possesses a higher free volume and a smaller shearresistance.

Fig. 11 shows the change in the fraction of Voronoi polyhe-dra under shear deformation. The icosahedral clusters contain-ing the 12 pentagons were first occupied in the ratio of 12%among all the polyhedra for Cu57Zr43. They, however, de-crease with the applied strain increment and a larger decreasecan be observed compared to the other polyhedra. In our ob-servation, both icosahedra and other polygons transform intoother structures under applied global strain. However, the ratioof newly born icosahedra during deformation is too smallcompared to the amount of annihilation. Since the maximumstress peak can be observed at around 10% in Fig. 6, a numberof collapses of the dense-packed polygons typified by the ico-sahedra, which have lower free volume and higher shear resis-tance, start before the maximum stress and result in thecontinuation of the loss in ductility of the amorphous alloys.

4. Concluding remarks

Computational observations of the internal structures of theatomic CuxZr1�x amorphous alloys indicate that the distributionof the pentagons has the large variation, which is spontaneouslygenerated during the quenching process due to pentagonalshort-range order. Thus, the density variation denoted by thefree volume can arise from a variety of geometrical aspects;

the pentagonal regions correspond to the densely packed struc-ture and non-pentagonal regions correspond to the free volumestructure. Shear simulations show that the local deformationalso exhibits characteristic variation on a macroscopic level.The pentagon-rich region tends to undergo elastic deformation,while the pentagon-poor, i.e. non-pentagonal region is easily de-formed plastically. This result indicates that the pentagonalshort-range order has structural stability, which can be attributednot only to the bond strength of the pentagon itself, but also to thelower free volume due to the dense packing. Some computa-tional studies estimating the free energy from a theoretical view-point have already indicated the decrement of shear resistancegenerated by the increment in free volume [1,5]. Similarly,our shear simulations show that the non-pentagonal region hasa high free volume and a small shear resistance, therefore pref-erentially undergoes plastic deformation.

In addition, these geometrical factors, such as the pentagonor free volume, control not only the early stage of the deforma-tion but also the flow stress, which is closely related to yieldand fracture behavior in amorphous alloys.

Acknowledgements

The author (YS) gratefully acknowledges support from theMinistry of Education, Culture, Sports, Science and Technol-ogy in Japan, Grant-in-Aid for Scientific Research on PriorityAreas (15074214).

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