Theory, Modeling and Numerical Simulation [1 ed.] 9783038132004, 9783908451563

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Theory, Modeling and Numerical Simulation of Multi-Physics Materials Behavior

Theory, Modeling and Numerical Simulation of Multi-Physics Materials Behavior

Selected, peer reviewed papers from the Symposium: Theory, Modeling and Numerical Simulation of Multi-Physics Materials Behavior organized within the MRS Fall Meeting 2007 held in Boston MA, USA, November 26-30, 2007

Edited by

Veena Tikare, Graeme E. Murch, Frédéric Soisson and Jeung Ku Kang

TRANS TECH PUBLICATIONS LTD Switzerland • UK • USA

Copyright  2008 Trans Tech Publications Ltd, Switzerland

All rights reserved. No part of the contents of this book may be reproduced or transmitted in any form or by any means without the written permission of the publisher.

Trans Tech Publications Ltd Laubisrutistr. 24 CH-8712 Stafa-Zurich Switzerland http://www.ttp.net

Volume 139 of Solid State Phenomena ISSN 1012-0394 (Pt. B of Diffusion and Defect Data - Solid State Data (ISSN 0377-6883)) Full text available online at http://www.scientific.net

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Preface No present day research and development program is complete without a robust modeling and numerical simulation component. Models and model-based numerical simulations are widely and extensively used to probe complex materials behavior or structure in order to obtain deep insight into basic materials understanding. Multi-physics models are becoming increasingly common with advances in computational sciences and are advancing our basic understanding of materials. The idea of the Symposium: “Theory, modeling and numerical simulation of multiphysics behavior” organized within the Fall Meeting of the Materials Research Society, Boston November 26-30, 2007 was to provide a focused forum to highlight the theory, models and numerical techniques that have enable multi-physics simulations to be an integral part of research and development programs. The present volume contains 25 selected and refereed papers presented at the Symposium.

Veena Tikare Graeme Murch Frédéric Soisson Jeung Ku Kang

Publication support: Sandia National Laboratories and CEA Saclay

Table of Contents Preface Atomistic Simulations of the Aluminum-Silicon Interfaces under Shear Loading A. Noreyan and V. Stoilov Shock Loading of Bone-Inspired Metallic Nanocomposites D. Sen and M.J. Buehler Hydrogen Storage in MgH2 Matrices: An Ab-Initio Study of Mg-MgH2 Interface S. Giusepponi, M. Celino, F. Cleri and A. Montone First-Principles Calculations of the Atomic and Electronic Structures in Au-Pd Slab Interfaces N. Taguchi, S. Tanaka, T. Akita, M. Kohyama and F. Hori In-Diffusion and Out-Diffusion of Oxygen from a Composite Containing Random Traps I.V. Belova and G.E. Murch Effects of Supports on Hydrogen Adsorption on Pt Clusters K. Okazaki-Maeda, Y. Morikawa, S. Tanaka and M. Kohyama First-Principles Calculations of Pd/Au(100) Interfaces with Adsorbates S. Tanaka, N. Taguchi, T. Akita, F. Hori and M. Kohyama In-Plane Rotated Crystal Structure in Continuous Growth of Bismuth Cuprate Superconducting Film S. Kaneko, K. Akiyama, T. Ito, Y. Hirabayashi, H. Funakubo and M. Yoshimoto Dynamical Interaction between Thermally Activated Glide of Screw Dislocation and SelfInterstitial Clusters in Bcc Fe X.Y. Liu and S.B. Biner The Effects of Solute Segregation on the Evolution and Strength of Dislocation Junctions Q. Chen, X.Y. Liu and S.B. Biner Physics Mechanisms Involved in the Formation and Recrystallization of Amorphous Regions in Si through Ion Irradiation I. Santos, L.A. Marqués, L. Pelaz, P. Lopez and M. Aboy Hotspot Formation in Shock-Induced Void Collapse Y.F. Shi and D.W. Brenner Molecular Dynamics Simulation of Nanocrystalline Tantalum under Uniaxial Tension Z.L. Pan, Y.L. Li and Q.M. Wei Diffusion Mechanisms near Tilt Grain Boundaries in Ni3Al Intermetallide M. Starostenkov, D.V. Sinyaev, R.Y. Rakitin and G.M. Poletaev Phase-Transformation Wave Dynamics in LiFePO4 D. Burch, G. Singh, G. Ceder and M.Z. Bazant Molecular-Dynamics Analysis of the Structural Properties of Silica during Cooling B.M. Lee, S. Munetoh, T. Motooka, Y.W. Yun and K.M. Lee Atomistic Simulations of Copper Precipitation and Radiation Induced Segregation in α-Iron F. Soisson and C.C. Fu Ab-Initio Calculation for the Study of Nano Scale Silicon Based Device Structure S. Chakraborty, G. Sashidhar, S.V. Ghaisas and V. Sundararajan Modelling of Elastic Modulus and Molecular Structure Interrelationship of an Oriented Crystalline Polymer U. Gafurov Reaction Rate as an Effective Tool for Analysis of Chemical Diffusion in Solids M. Sinder, Z. Burshtein and J. Pelleg Simulation of the Columnar-to-Equiaxed Transition in Alloy Solidification - The Effect of Nucleation Undercooling, Density of Nuclei in Bulk Liquid and Alloy Solidification Range on the Transition H.J. Dai, H.B. Dong, H.V. Atkinson and P.D. Lee Simulation of Surface-Enhanced Ordering in Smectic Films N.M. Abukhdeir and A.D. Rey

1 11 23 29 35 41 47 53 59 65 71 77 83 89 95 101 107 113 119 123

129 135

b

Theory, Modeling and Numerical Simulation

Atomic Scale Modelling of Materials: A Prerequisite for any Multi-Scale Approach to Structural and Dynamical Properties M. Matsubara, M. Celino, P.S. Salmon and C. Massobrio Morphological Evolution of Intragranular Void under the Thermal-Stress Gradient Generated by the Steady State Heat Flow in Encapsulated Metallic Films: Special Reference to Flip Chip Solder Joints T.O. Ogurtani and O. Akyildiz Effect of C on Vacancy Migration in α-Iron C.C. Fu, E. Meslin, A. Barbu, F. Willaime and V. Oison

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Solid State Phenomena Vol. 139 (2008) pp 1-10 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.1

Atomistic Simulations of the Aluminum-Silicon Interfaces Under Shear Loading Alice Noreyana, Vesselin Stoilovb Mechanical, Automotive and Materials Engineering, University of Windsor, Windsor, ON, N9B3P4, Canada a

b

[email protected], [email protected]

Keywords: Aluminum-Silicon interface, shear stress, MEAM, stick-slip, amorphisation.

Abstract. In the present study molecular dynamics simulations were carried out to investigate the deformation of pure FCC aluminum and diamond cubic silicon interfaces under shear stress. A second nearest-neighbor modified embedded atom method was used to describe the interactions between Al-Al, Si-Si and Al-Si atoms. The critical shear stress (CSS) was determined for various Al/Si and Al/Al interfaces with different alignments and orientations. Structural analyses show that the deformation is localized at approximately 10 Å thickness of the interface in Al. The critical shear stress of Al/Si interface was found to be significantly lower than the critical tensile stress due to the partial stick-slip in sliding. In addition, it has been proven that there is no explicit relationship between shear and tensile critical stresses, which is fundamentally different from isotropic materials, where the shear stress is about half of the tensile stress. The misorientation has a dramatic effect in reducing shear stress at Al/Al interfaces, but has no effect on CSS in Al/Si. As a result, it was shown that introducing Si improves the strength of the interface (and the composite material in general) for different grain orientations. Introduction The need of composite materials with increased strength has made the study of metal/ceramic interfaces a critical issue. Al matrix with silicon inclusions is of particular interest due to wide application of AlSi alloys in the automotive industry. Previous atomic level simulations on metal-ceramic interfaces were mainly focused on interface decohesion [1-5]. Gall et al. [1] and Ward et al [1] used molecular dynamics simulations to study the deformation and fracture characteristics of Al-Si interfaces under tensile stress. They found [1] that relaxed interface possesses a rippled structure, and these ripples act as local stress concentrators and initiation sites for failure at the interface. They also observed that when crack-like vacancy defects in Al and Si base material reach some critical value, it causes the shift of failure from the interface to the bulk materials. In a similar study, Ward et. al [2] found that imperfectly-matched Al-Si interfaces have high tensile strengths and fracture energies. They also compared the nature of failure for Al-Si interfaces and nanocomposits consisting of Al and Si nanograins under tensile loading, and showed that the composite failure stresses are lower than the strength of the Al-Si interfaces. Extensive studies were carried out on the sliding at metal/metal interfaces with same material (grain boundary) in Al [6, 7], Cu [8] and Ni [9] or different materials of Ni/Zr [10], Ta/Al, Cu/Ag [11]. However the sliding at a sharp heterogeneous metal/ceramic interface has not been studied, although it is the dominant deformation process. The current MEAM simulations consider the nature of deformation and force-displacement response of Al and Si materials under shear stress. Different crystallographic orientations between Al and Si materials were considered. The effect of different applied forces is investigated, and size of interface was defined, as well as comparison with all-Al material was made.

2

Theory, Modeling and Numerical Simulation

Model description All simulations were carried out using a second nearest-neighbor MEAM potential [12]. The potential not only does fit to most bulk properties for Al, Si and AlSi, but also correctly predicts FCC as the lowest energy phase for Al, Al surface energies, Si surface re-construction, and solubility of Si and Al [12, 13]. Fig. 1 presents a snapshot of a typical simulation cell and the settings for the applied shear loading. In the present simulations the workpiece was taken to have a cubic diamond structure for Si with a lattice constant of 5.4 Å and FCC Al with 4.05 Å, both Al and Si pieces were put into contact at the interface parallel to the xy plane. The separation distance between two materials at an interface planes was initially set at 2Å and was fully relaxed before applying shear stress. Periodic boundary conditions were assumed in x and y directions (parallel to the free surface and interface) while the free surface was allowed to relax in the z direction (Fig. 1). The workpiece was initially annealed at 600K, and then cooled back to 300K for equilibrium. The equilibrated workpiece for each material was divided into 3 zones: a fixed zone that contains 2 layers of Al(Si) at the free surface, a moving zone that is adjusted to the surface and contains 4 layers of Al(Si) and free zone, that includes all remaining atoms in the system near the interface (see Fig.1). The external strain rate or force was applied every time step to each atom in the moving zone in -x (Al) and +x (Si) directions. The atoms in the fixed zone were allowed to move only along x and y directions, while atoms in the free zone of the both materials were allowed to move freely. The Hoover method was applied for thermostating all the atoms in the system at a fixed temperature of T = 300K during dynamic simulations.

} Fixed zone

}

} }

Z

}

}

Y

Moving zone

-Fx

Free zone

Free zone

Fx Moving zone Fixed zone

X

Figure 1. Snapshot of simulation cell for Al(111)/Si(111) interface: gray Si atoms, light blue Al atoms. In shear loading simulations, the workpiece is divided into three zones for Al and Si, and the external forces/strains are applied to atoms in moving zone. It should be noted that since periodic boundary conditions were applied in the x and y directions, the length for both materials should be equal along these two directions. For example, in the case of Si(001)||Al(001), the x and y size of both materials is 32.4Å, which is 6 lattice parameters for Si and 8 lattice parameters for Al. Rotating Al with 45o in xy plane about , the smallest size possible to minimize the lattice mismatch is ~ 48.62Å, with 9 lattice parameters for Si and 17/2 lattice parameters for Al. In total, four Al/Si interfaces were

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studied , and the length of each type of material in x/y directions was chosen to get the minimum possible cell length in x and y directions, with smallest possible lattice mismatch. Results and discussions Critical shear stress. Threshold applied force was defined as the minimum applied force to initiate sliding at the interface. Different forces in the range of 0.0005 to 0.1 eV/Å were applied to find the threshold of force for initiating sliding at the interface or plastic deformation in the base material. If all atoms returned to their original positions, then sliding was not achieved. The threshold force was determined by the averaged force between the largest forces without sliding and the smallest force initiating sliding. These two forces also lead to an error bar of 0.0005eV/Å. The critical shear stress (CSS) was calculated based on applied force (fi) and number of atoms in moving zone (N) and the cross-sectional area of a layer (S) (Eq. 1). N

τ=

∑f i =1

S

i

⋅

(1)

The threshold of applied forces for initiating sliding at the interface was determined for various Al/Si interfaces with different alignments and orientations. In total, we have studied three alignments of Al/Si interfaces, namely: Al(001)/Si(001), Al(111)/Si(001) and Al(111)/Si(111). Within Al(001)/ Si(001) we have studied two different orientations to understand the misorientation effect. Table 1 summarizes the results for threshold forces and the corresponding critical shear stresses for initiating sliding at different interfaces. The obtained results clearly indicate that the critical shear stress necessary to induce sliding/fracture in heterogeneous Al/Si interfaces is strongly dependent on the crystallographic alignment. The critical shear stress required to initiate motion in Al(001)/Si(001), exceeds the corresponding critical shear stress in Al(111)/Si(111) more than 3.5 times (Table 1). On the other hand, the mutual orientation of the surfaces within a specific crystallographic alignment has very little impact on the critical stress. For different surface orientations within Al(001)/Si(001) crystallographic alignment, the critical shear stresses vary between 3-13%(Table 1). This result contradicts the findings for homogeneous metal/metal interfaces [9], where the mutual surface orientation within a crystallographic alignment could lead to 100 times change in the critical shear stresses. The unique orientation dependence of the critical shear stress at the Al/Si interface needs to be understood from the deformation behavior at the interface. Table 1: Threshold forces and critical shear stresses Interface

Applied threshold force for initiating

Applied threshold

Al/Si Tensile

sliding at interface eV/ Å (N)

stress (GPa)

strength (GPa) [2]

Si(001)||Al(001)

0.015 ± 0.0005 (~2.48 × 10-11)

1.172 ± 0.04

6.4

Si(001)||Al(001)

0.0145 ± 0.0005 (~2.32 × 10-11)

1.136 ± 0.039

-

Si(001)||Al(111)

0.0075 ± 0.0005 (~1.2 × 10-11)

0.657 ± 0.044

5.6

Si(111)||Al (111)

0.0035 ± 0.0005 (~5.6 × 10-12)

0.318 ± 0.045

7.2

Al(001)||Al (001)

0.045 ± 0.005 (~7.2 × 10-11)

3.52 ± 0.39

-

Al(001)||Al (001)

0.001 ± 0.0005 (~1.6 × 10-12)

0.078 ± 0.008

-

Al(001)||Al (111)

0.003 ± 0.0005 (~4.8 × 10-12)

0.263 ± 0.044

-

Al(111)||Al (111)

0.005 ± 0.0005 (~8.01 × 10-12)

0.438 ± 0.088

-

4

Theory, Modeling and Numerical Simulation

The location of the failure during sliding. To reveal the nature of the plastic deformation and fracture, we employed different statistical analysis to characterize the details of structural changes at the Al/Si interface. In the critical shear stress calculation, after sliding was initiated, the sliding speed was not well controlled. To facilitate the comparison of the evolution of atomic structures during sliding, we used constant sliding rate simulations for the following study. Both constant force and constant velocity method can be used to compute the maximum shear stress, typically the constant force simulation will slightly underestimate the threshold stress due to the inertia effect of the moving slab [14]. Quantitatively, the thickness of the interface for Al and Si was determined by using the bond angle distribution function (BADF) [15]. The BADF was defined as the bond angle count per atom per angle with cutoff bond distance about 4% larger than the bond length of the first nearest neighbors at T=0K (2.96 Å for Al and 2.46 Å for Si). For perfect Al, the angles between the first nearest neighbor atoms are 60°, 90°, and 120°, for perfect Si, the angles between the four nearest neighbors are 109.5°. The total bond count within the perfect lattice angles was determined by integrating the BADFs within the regions of these angles (±5% of the angle). In a perfect crystal, the integration of BADF will lead to the total N(N-1)/2 angles formed by the N nearest neighbors. Fig. 2 shows the average bond count per atom corresponding to Al and Si at different distances from the interface. Away from the interface the bond count coincides with the one for the perfect lattice (66 for perfect Al and 6 for perfect Si). However, close to the interface the bond count significantly drops, indicating a strong deviation from the perfect crystal structures. This distortion from perfect crystal means that the presence of the interface already introduced distortions in Si and Al. Thus the as-build sharp interface needs to be considered as an interface region with certain thickness. As a quantitative measure of the thickness of the interface, the 30% margin of decrease of the bond count from that of the perfect crystal was used. As a result the thickness of the deformation/fracture zones in Al and Si were determined to be 10 Å and 5.5Å, indicating that that more than 70 percent of the lattice distortions are localized in these regions. The atomic structures before sliding and after sliding for 5 and 10 ps also confirm that the sliding zone is about 10 Å near the Al (see Fig. 2c). Further, to precisely locate the failure region, the pair correlation function (PCF) (Fig. 3) was calculated for the deformation/fracture regions of 10 Å of the Al and Si respectively. The pair correlation function for Al and Si in a 10 Å region near the interface clearly indicates the essential changes and differences in the atomic structure after 10ps of sliding (Fig. 3). For the perfect FCC Al, the distances between the first, second, third and forth-nearest neighbors at 0K are 2.86 Å, 4.05 Å, 4.96 Å and 5.72 Å, and for the perfect Si in diamond structure the distances between the first, second, third and forth-nearest neighbors are 2.34 Å, 3.82 Å, 4.48 Å and 5.4 Å. After sliding (Fig. 3a), the magnitude of all peaks in Al decreased and the peaks widened showing a significant deviation from neighbors’ distribution characteristics for FCC, indicating a distorted and slightly disordered FCC structure in Al. Another change on the PCF for the Al curve is the decrease of the first maximum and the increase of the first minimum after sliding. After sliding the change in PCF indicates partial amorphization of Al near the interface. In comparison the PCF for Si within the 10 Å region near the interface does not show much noticeable changes after sliding for 10ps (Fig. 3b), confirming that the fracture is localized mainly within the Al. Local structural evolutions in Al due to shear stress were investigated by the method of shortest-path ring distribution [15-18]. The occurrence of structural deformation is marked with presence of non-three atom rings in FCC structure (Al) while in Si diamond structure with the presence of non-six atom rings. In perfect FCC structure the shortest path ring contains three atoms. Fig. 4 presents the shortest path ring analysis in Al for Si(100)||Al(100), when strain rate was 1 Å/ps (only atoms forming non-three atom rings are considered). Detailed analysis in Al showed partial amorphisation in the region near the interface when applied force is above the threshold force for a particular configuration. Ring analysis in Si indicated absence of any dislocations and structural changes before and after sliding. The results are consistent with pair

Solid State Phenomena Vol. 139

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correlation function analysis, implying that the deformation region is concentrated in Al near the interface, although the extent of plastic deformation varies depending on the orientation of the interface. 50

6.5

Bond-angles per atom

Bond-angles per atom

a) 40

30

0ps 5ps 10ps 20

b)

6 5.5 5 4.5

0 ps 5 ps 10 ps

4 3.5 3

10

0

10

20

30

40

50

60

70

80

2.5 0

5

10

Region (Å)

15

20

25

30

35

Region (Å)

c)

10Å

t = 0ps

t = 5ps

t = 10ps

Figure 2: Average bond angles per atom in a) Al and b) Si before and after 5 and 10 ps sliding, and (c) the corresponding atomic structures. Comparison of Al/Si and Al/Al interfaces. As the structural analyses collectively shown, the fracture in heterogeneous Al/Si interfaces under shear loading is localized in the Al within a 10 Å region near the interface (Fig. 2). Although, the overall shearing at heterogeneous-Al/Si interfaces have shown typical “stick” and “slip” characteristics, the sliding path at different locations at the interface actually varies (Fig. 5). Fig. 5a presents three displacement versus time dependence for shearing in the case of Si(001)||Al(001) for three different regions in Al (12 atoms in each region located in the first two layers near the interface) at the Al/Si interface separated by ~8.1Å along x direction under the applied force of 0.015eV/Å, which is just above the threshold value. All three regions slide in a “stick-and-slip” pattern, however they were trapped at the “stick” position at different time. Vertical dash lines in Fig. 5a illustrate the shift of “stick” points for these different regions, which means while on average atoms in region 1 had moved (“slipped”) at the highlighted time, the atoms of region 2 have no or small displacement at the same instance. Similar behavior was observed in the case of Si(111)||Al(111) (Fig. 5b). A few drops in the sliding displacement are caused by some of the Al atoms in a region moving with Si in the opposite

6

Theory, Modeling and Numerical Simulation

direction, instead of moving along with other Al atoms. These moves are associated with certain energy minima along the sliding path. Although (111) plane is a slip plane for Al, the presence of Si distorts the structure and causes significant change in the observed deformation mechanism. In order to compare with the nature of fracture in Al/Si interface, homogeneous Al/Al interface was also studied. The critical shear stresses in Al/Al exhibit primary dependence on the misorientation characteristic for homogeneous interfaces [2]. The obtained values for the critical shear stresses in Al/Al ranged between 78MPa-3520MPa (see table 1), where the highest shear stress and the lowest shear stress shared the same crystal alignments but with different mis-orientations. The highest magnitude corresponds to Al(001)||Al (001) perfect single crystal, which is close to the theoretical value of the critical shear stress estimated using Frenckel’s model. The lowest value of the critical shear stress of 78MPa was obtained for the Al(001)||Al (001) orientation. In comparison, the values of the critical shear stresses in Al/Si interfaces (all orientations - see Table 1) are between the minimum (32MPa) and the maximum (1172MPa). This is an indication that the presence Si in the Al leads to formation of inter-granular Al/Si interfaces which are “stronger” than some of the inter-granular interfaces in polycrystalline Al (Al(001)||Al (001) etc.). On the other hand the inter-granular Al/Si interfaces are “weaker” (Table 1) compared to the Al/Al perfect crystal ones. These can be visualized by the atomic structures after sliding shown in fig. 6. For both Si/Al interfaces (same alignments but different orientations), the sliding is localized on the Al side near the interface in Fig 6a and 6b. When two Al surfaces have the same alignments and same orientations, a commensurate interface is formed like a perfect crystal (Fig. 6c), where the essential damage in the form of partial amorphization happens at the interface. However, the applying shear force causes inter-granular sliding between two mis-oriented Al grains, without any distortion of the lattice at the Al/Al interface nor inside the Al. The locations of the sliding in the four samples are evidence of the strong anisotropy of critical shear stress depending on the mis-orientations found in homogenous Al/Al interfaces, but not in heterogeneous Al/Si interfaces.

0.7

0.8

a)

b)

Al 0.6

0.6

Normalized p.c.f.

Normalized p.c.f.

0.7

Before After

0.5 0.4 0.3

Si Before After

0.5 0.4 0.3 0.2

0.2 0.1

0.1

0

0 2

2.5

3

3.5

4

4.5

r (Å)

5

5.5

6

2

2.5

3

3.5

4

4.5

5

5.5

6

r (Å)

Figure 3. PCF for a) Al and b) Si in 10 Å regions near the interface for Si(001)||Al(001) before and after sliding for 10ps in the case of applied strain rate for 1 Å/ps

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Non-threeatom rings

50

40

0 ps 5 ps 10 ps 15 ps 20 ps 25 ps

30

20

10

0 3

4

5

6

7

8

9

10

# of atoms in ring

Figure 4. Number of non-three-atom rings in Al before and after sliding for 5, 10, 15, 20 and 25 ps, 10 Å regions near the interface was considered.

7

1.2

a)

b) 1

Region_1 Region_2 Region_3

5

Displacement (Å)

Displacement (Å)

6

4

3

2

Region_1 Region_2 Region_3

0.8

0.6

0.4

0.2

1

0

0 1

2

3

Time (ps)

4

5

6

1

2

3

4

5

6

Time (ps)

Figure 5. Displacement-time for three different regions (12 atoms in each region) at the Al/Si interface for a) (Si(001)||Al(001)) separated by ~8.1Å, applied force of 0.015eV/Å – threshold value b) b) (Si(111)||Al(111)) separated by ~8.5Å, applied force of 0.0035eV/Å – threshold value. Vertical dash lines indicate the difference between “stick” points for the regions.

8

(a)

Theory, Modeling and Numerical Simulation

(b)

(c)

(d)

Figure 6: Snapshots of simulations cells after shearing for a) Si(001) ||Al(001) for applied strain rate of 1 Å/ps, b) Si(001)||Al(001) applied force of 0.02 eV/Å, c) Al(001)||Al(001) for applied force of 0.05 eV/Å, d) Al(001)||Al(001) for applied force of 0.002 eV/Å. All the highlighted atoms were aligned before sliding. The existence of the critical shear stress indicates that an interface is trapped in a potential minimum. A critical force, CSS, must therefore be applied to overcome the potential barrier along the sliding path. The interfacial energy is the summation of all the inter-atomic energies on each atom sitting on a periodic lattice. If the interface is commensurate (like the prefect Al/Al), the energies of all the atoms add in phase, giving quite a large energy barrier and a large CSS. For the flat incommensurate Al/Al interface, if the ratio of the lattice constants of two surfaces is irrational, then the total interfacial energy was canceled out leading to a zero CSS. All Al/Si interfaces are incommensurate due to their lattice mismatch. However, the Al-Si covalent bonds form across the interface, which induces surface reconstruction and residual stresses in Al atoms. Therefore there are always multiple local energy minima at Al/Si interfaces, causing CSS and stick-slip motion during sliding. The observed partial stick-slip sliding behavior at Al/Si interfaces are also due to these local minima. The residual stresses on Al atoms lower the forces required to slide the Al slab, thus all CSS at Al/Si interfaces are higher than mis-oriented Al/Al interfaces, except when two Al surfaces are perfectly aligned and oriented into a single crystal. As a result, the overall strength of AlSi composites (with Al/Si interfaces under all orientations) should be less than the single crystal Al, but higher than polycrystalline Al (with mis-oriented Al/Al interfaces). Conclusions 1) The critical shear stress for various Al/Si interfaces with different alignments (normal to the interface) and orientations (parallel to the interface) were determined. It was found that the interface misorientation has insignificant effect on the critical shear stress, whereas the primary influence parameter was the general crystallographic alignment. 2) Bond angle distribution, pair correlation functions and shortest path ring analysis were used to determine the nature of plastic deformation due to the sliding. It was shown that the deformation is localized at approximately 10 Å thickness of the interface for Al. 3) The critical shear stress of Al/Si interface was found to be significantly lower than the critical tensile stress due to the partial stick-slip in sliding. In addition, it was proven that there is not explicit relationship between shear and tensile critical stresses, which is dramatically different from isotropic materials, where the shear stress is about half of the tensile stress. 4) In comparison, the mis-orientation effects exhibited significant difference in homogenous Al/Al interfaces and heterogeneous Al/Si interface: the mis-orientation can reduce the CSS at Al/Al

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9

interfaces by two orders of magnitude; while it has insignificant effect on CSS in Al/Si. The only CSS of Al/Al larger than that of the Al/Si interfaces occurs when two Al surfaces are perfectly aligned and oriented into a single crystal. As a result, the overall strength of AlSi composites (with Al/Si interfaces under all orientations) should be less than the single crystal Al, but higher than polycrystalline Al (with misorientated Al/Al interfaces).

References [1] D.K. Ward, W.A. Curtin, Yue Qi, Acta Mater. Vol. 54 (2006), p. 4441. [2] K. Gall, M.F. Horstemeyer, M. Van Schilfgaarde, M.I. Baskes, J. Mech. and Phys. of Solids Vol. 48 (2000), p. 2183 [3] D.K. Ward, W.A. Curtin, Y. Qi, Comp. Sc. Tech. Vol. 66 (2006), p. 1148 [4] R. Komanduri, N. Chandrasekaran, L.M. Raff, Int. J. Mech. Sci. Vol. 43 (2001), p. 2237 [5] D. Spearot, K.I. Jacob, D.L. McDowell, Mech. Mater. Vol. 36 (2004), p. 825. [6] Y. Qi, P.E. Krajewski, Acta Mater. Vol. 55 (2007), p. 1555 [7] V. Yamakov, D. Wolf, S. Phillpot, H. Gleiter, Acta Mater. Vol. 50 (2002), p. 5005 [8] F. Sansoz, J.F. Molinari, Acta Mater. Vol. 53 (2005), p. 1931 [9] Y. Qi, Y.T. Cheng, T. Cagin, W.A. Goddard, Phys. Rev. B Vol. 66 (2002), p. 085420 [10] F. Delogu, G. Cocco, Phys. Rev. B Vol. 72 (2005), p. 014124 [11] J.E. Hammerberg, B.L. Holian, T.C. Germann, L. Ravelo, Metalurgical and Materials Transactions Vol. 35A (2004), p. 2741 [12] B.J. Lee, J.H. Shim, M.I. Baskes, Phys. Rev. B Vol. 68 (2003), p. 144112 [13] M.I. Baskes, Phys. Rev. B Vol. 46 (1992), p. 2727 [14] Q. Zhang, Y. Qi, L.G. Hector, T. Cagin, W.A. Goddard, Phys. Rev. B Vol. 75 (2007), p. 144114 [15] I. Szlufarska, R. Kalia, A. Nakano, P. Vashishta, Appl. Phys. Lett. Vol. 85 (2004), p. 378 [16] J. Rino , I. Ebbsjo, R. Kalia, A. Nakano, P. Vashishta, Phys. Rev. B Vol. 47 (1993), p. 3053 [17] X. Yuan, L. Hobbs, Nucl. Instr. & Meth. in Phys. Res. B Vol. 191 (2002), p. 74 [18] J. Rino, I. Ebbsjo, R. Kalia, A. Nakano, F. Shimojo, P. Vashishta, Phys. Rev. B Vol. 70 (2004), p. 045207.

Solid State Phenomena Vol. 139 (2008) pp 11-22 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.11

Shock loading of bone-inspired metallic nanocomposites Dipanjan Sen1,2, Markus J. Buehler1,a 1

Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Mass. Ave. Room 1-272, Cambridge, MA, 02139, USA 2

Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Mass. Ave., Cambridge, MA, 02139, USA a

Corresponding author, e-mail: [email protected]

Keywords: Nanocomposite, nanocrystal, bone, nacre, shock loading, atomistic simulation, metal composite

Abstract. Nanostructured composites inspired by structural biomaterials such as bone and nacre form intriguing design templates for biomimetic materials. Here we use large scale molecular dynamics to study the shock response of nanocomposites with similar nanoscopic structural features as bone, to determine whether bioinspired nanostructures provide an improved shock mitigating performance. The utilization of these nanostructures is motivated by the toughness of bone under tensile load, which is far greater than its constituent phases and greater than most synthetic materials. To facilitate the computational experiments, we develop a modified version of an Embedded Atom Method (EAM) alloy multi-body interatomic potential to model the mechanical and physical properties of dissimilar phases of the biomimetic bone nanostructure. We find that the geometric arrangement and the specific length scales of design elements at nanoscale does not have a significant effect on shock dissipation, in contrast to the case of tensile loading where the nanostructural length scales strongly influence the mechanical properties. We find that interfacial sliding between the composite’s constituents is a major source of plasticity under shock loading. Based on this finding, we conclude that controlling the interfacial strength can be used to design a material with larger shock absorption. These observations provide valuable insight towards improving the design of nanostructures in shock-absorbing applications, and suggest that by tuning the interfacial properties in the nanocomposite may provide a path to design materials with enhanced shock absorbing capability.

Introduction Shock compression of materials occurs in a variety of situations such as during projectile impact, armor penetration, and vehicle collisions. From an engineering design point of view, it is important to minimize the damage to underlying structures or materials through a shock-absorbing protective material. Developing new materials with advanced shock mitigation potential is thus an important technological problem. In particular, nanostructured materials may have a wide range of potential applications in such thin-film shock absorbents. This is because nanostructures may feature several shock-absorbing elements and mechanisms at different length scales, which reduce shock impact over smaller distances than possible through micro- and macro-scale shock absorbents such as viscous fluids, honeycomb structures and other approaches. Applications of these materials could be in reducing ballistic impact of body armor systems [1], shock protection of disk storage devices, or vibration protection of sensitive instrumentation. The key mechanical feature required in these applications is the ability to absorb shock-wave energy over small length scales and its conversion

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Theory, Modeling and Numerical Simulation

into extensive plastic deformation and heat, which can be transported away from the site of impact. One measure of this ability is the toughness of materials, that is, the area under the compressive stress-strain curve until failure occurs. Shock loading implies both severe strain rates (1E8 s-1 and above) and high pressures, thus requiring new constitutive laws for the materials [2]. Several experimental and computational studies have been reported in the literature that were geared towards improving the hardness and toughness under shock loading. Studies of voids in single metal crystals under shock loading have shown collapse of voids in the wave of the shock and formation of nano-grains [3,4]. Studies of nanocrystals under shock loading have shown suppression of grain-boundary sliding mechanisms and improvement in hardness and flow strength of shocked samples [5,6]. These studies have illustrated the ability to tailor microstructure at the nanoscale can affect shock-loading properties profoundly. Nanostructured composites with design inspired by nanostructure of biological structural materials have been proposed for high strength applications under laboratory strain rates [7]. Bone, nacre, dentin have been shown to possess similar nanostructures at the basic organizational level, providing strong evidence that these design principles can be transferred to design new synthetic structural materials. Studies of nanostructured composites in the bone-inspired design (see Figure 1 for a simple schematic of bone’s nanostructure) using empirical constitutive equations have shown improved strength and toughness properties over base materials under tensile loading [8-10]. Other simulation work, based on atomistically informed mesoscale modeling of collagen-mineral nanocomposites in bone, has shown that localized failure in these composites leads to manifold increase in energy dissipation and fracture strength [11].

Figure 1. Two-dimensional, simple schematic model of hard-soft phases nanocomposite based on the ultrastructure of bone. In bone, the inorganic platelets are 2-4 nm thick and up to 100 nm long embedded in a collagen-rich protein matrix. This structure is realized with metallic components using a ‘soft’ Cu metal matrix and modified ‘hard’ EAM metal platelets. A single crystal of Cu is chosen in [1 1 0] x [111] (x-y) orientation, and a regular array of rectangular voids is created in it. The platelet crystals are inserted in the voids in the same crystallographic orientation, ensuring no overlapping atoms (distance closer than Cu-Cu nearest neighbor distance). Structures are then relaxed with respect to global normal stresses.

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Figure 2. The size dependence of flow strength for a bone-like nanostructure of Ni-Al composite, as reported in [10]. This plot illustrates that the systematic reduction of the building block size leads to an increase of the strength (following a Hall-Petch curve), until a maximum is reached. Below a critical size the flow stress decreases. The size of the design geometry of the nanocomposite has been shown to influence flow stress under quasi-static tensile loading in previous simulation experiments, as depicted in Figure 2. Particular configurations closely mimicking bone nanostructure have been shown to have large flow stresses compared to far-from-bone design for Ni-Al nanocomposites [10], and there exist optimal characteristic dimensions of the nanostructure at which the flow stress reaches a maximum. In this paper we will extend similar design principles to shock loading applications, to determine whether bioinspired nanostructures provide an improved shock mitigating performance, and if yes, which structural features are most critical to improve the performance. We are motivated by the order of magnitude improvement in toughness properties under tensile loading that the bone design provides over its constituent materials [12]. Our hypothesis is that a similar performance increase as observed under tensile loading of a bioinspired nanostructure will also occur under shock loading. This hypothesis is based on the consideration of the simultaneous improvement in strength and toughness that is observed in the design of bone over the properties of its constituents. To test this hypothesis, we carry out a systematic analysis of deformation of varying nanostructured composites under shock loading, while changing the geometry and constituent mechanical properties. Atomistic and computational model We employ a large-scale atomistic representation of a nanocomposite structure. simulation is then used to simulate the effect of large mechanical deformation.

Atomistic

Interatomic potential. We use a modified version of an embedded atom method (EAM) alloy potential [13] to describe the various interactions between matrix and platelets. The key properties to be considered while choosing materials for matrix and platelets are a soft, ductile matrix, and hard, more brittle platelets. We choose copper metal for the matrix, described by the EAM Mishin

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Theory, Modeling and Numerical Simulation

potential [14], and use a modified version of this EAM potential for the platelet and interface as described below. The reason for choosing a modified potential is the ability to control the EAM potential parameters to vary the interfacial strength, and thus observe the dependency of shock response on this parameter. Our atomistic model is based on earlier EAM potentials developed for two species alloy systems such as Ni-Al, Cu, Ag alloys [15, 16]. The Baskes-Daw model of an EAM potential for facecentered cubic (FCC) metals consists of a pair interaction term, and an electron density term which contributes through an embedding energy term that charges an energy penalty for deviating from an FCC environment (Equation 1). The development of an alloy potential requires the development of a cross-pair potential, and density functions for atoms of type A in an environment of type B, and vice versa. The total potential energy is given by E tot = ∑ Fi (ρ h ,i ) + i

ρ h ,i =

∑ ρ aj (Rij ) ,

1 ∑ ∑φij (Rij ) , 2 i j ( ≠i )

(1)

j ( ≠i )

where Etot is the total energy of the system, ρ h,i is the density contribution at atom i due to remaining atoms of the system, Fi (ρ ) is the energy to embed atom i in the density ρ , φ ij (R ) is the pair-pair interaction between atoms separated by distance R. To design the modified interatomic potentials, we selectively change the pair interaction of the Cu EAM potential while keeping the density and embedding energy terms unmodified. Modification of only the pair term allows us to modify the bulk modulus, cohesive energy, and unstable and stable stacking fault energies, while maintaining the lattice parameter of the FCC phase nearly constant. We use the modified potentials for describing the platelets and interface which give the potential properties shown in Figures 3 and 4.

Figure 3. Generalized stacking fault curves for the interatomic EAM potentials for matrix (Mishin Cu potential [14]), platelets, ‘strong’ interface (interface1) and ‘weak’ interface (interface2).

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Figure 4. Theoretical shear strength against (111) plane shearing in the [11 2 ] direction, for matrix, platelet, interface1 and interface2 EAM potentials. The interfacial potential parameters are adjusted so that the interface is always weaker than the matrix and platelets. Relatively stronger and weaker interfaces are developed and named ‘interface1’ and ‘interface2’ here, respectively. Their shear and slip properties are summarized in Figures 3 and 4. Several different geometrical configurations are generated for different matrix-platelet combinations. The cell sizes are approximately 1,100Ǻ x 250Ǻ x 70Ǻ, including approximately 6 million atoms. Such large systems are required to allow sufficient dislocation network development as the uniaxial shock wave travels across the sample [17]. Molecular dynamics approach and application of shock loading. MD simulations are performed with the parallelized classical MD code IMD [18]. Structures are relaxed with respect to global normal stresses by relaxing system cell parameters while maintaining periodic boundary conditions in all directions. It is observed that the lattice registry is maintained between matrix and platelets after relaxation; as the lattice mismatch is minimal. This leads to absence of any interfacial defects in the relaxed structures. Shock loading of these nanocomposites is achieved by moving in a massive rigid piston at a constant velocity, Up from one end of the system, while maintaining a rigid boundary at the other end (see Figure 5). This generates a shock wave moving away from the piston, at a speed Us > Up. Below a critical Up, called the Hugoniot elastic limit (HEL), the resulting shock behavior is completely elastic. Above this speed a plastic front is formed, which may be underdriven, that is, moves slower than the elastic shock front or overdrive the elastic shock front at higher piston velocities. Shock loading of the un-defected bulk structures obtained from structure relaxation calculations, however, requires extremely large shock velocities (on the order of a few km/s) to induce plasticity, as the presence of a large volume fraction of hard platelet phase increase the shock speed needed to observe plasticity tremendously. This would increase the size of the system tremendously to observe a sizeable travel distance of the shock wave. We thus induce plasticity at low shock speeds (≈500 m/s) through two methods: (a) placing a single void in the sample in the path of the shock wave [3], and (b) joining a slab of pure copper to the sample at the incoming end of the shock wave. These sample setups are shown in Figure 6.

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Figure 5. Schematic showing simulation setup of shock loading of a sample. A rigid massive piston compresses the sample moving in at a speed of Up. A shock wave moves away from the piston face at a speed Us, and after a distance, a plastic front develops (moving at speed Ups). This plastic front may be underdriven (UpsUp). Placing a void in the shock path reduces piston speeds required to initiate plasticity tremendously. The void induces the nucleation of partial dislocation loops as soon as an elastic shock front of a certain strength passes across it and leads to a large dislocation density as the piston speed, and hence, the shock speed are increased. The other setup that consists of joining the nanocomposite with a slab of pure copper induces plastic behavior as copper, though un-defected, initiates a plastic front behind the elastic shock front at much smaller speeds than the nanocomposite. The plastic front follows the elastic shock front into the composite material where it evolves differently coming in contact with the composite structure. Calculation of dislocation density. The shock response is measured through the analysis of the dislocation density in the wake of the shock. The dislocation density is a measure of the energy dissipated by the shock as a higher dislocation density implies larger work done to induce permanent plastic deformation. The dislocation density is measured by finding number of core dislocation atoms in the simulation (possessing nearest neighbors between 10 and 13, leaving out perfectly coordinated atoms with 12 neighbors) and calculating length of dislocation line per unit volume of system. This method provides only an upper limit on the dislocation density, as it is possible that more defects than just dislocations are captured.

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Figure 6. Two system setups for studying plasticity under shock loading. Subplot (a) shows system with single spherical void of radius 1 nm; inset shows zoomed in view of void. Subplot (b) shows a system that combines a pure copper slab at the left side with the nanostructure. The snapshot is taken when shock wave has just moved into the nanocomposite. The cloudy green area in the middle indicates regions of plasticity; the inset shows zoomed in view of core dislocation structure in the plastic region.

For the analysis of shock deformation experiments it is important to distinguish sessile from other dislocations. The sessile component is measured by observing atoms that remain in the dislocation core over a length of time ∆ t (taken sufficiently large here, at about 12% of the total shock travel time) [19]. Visualization of the dislocation structure is performed using coordination number coloring using Atomeye [20]. Computational results The geometrical arrangement and dimensions of platelets in matrix have been shown to be important for strength properties of bone-like nanostructures under tensile loading [8-10], and our attempt here is to determine the importance of these effects under shock loading. Interfacial properties, on the other hand, determine the distribution of stresses between platelets and matrix and plastic response through interfacial deformation. The simulation studies focus here on the strength of the matrix-platelet interfaces, which directly affects their ability to deform plastically under given stress.

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Theory, Modeling and Numerical Simulation

In summary, we identify two important design aspects of nanocomposites for shock loading: (a) geometrical arrangement of platelets in matrix including dimensions, spacing and placement, and (b) mechanical properties of the matrix-platelet interfaces. Geometry effect. We begin with a study of the effect of design geometry on the shock response. We test the dependence of the shock response on the geometric arrangement in a first, simple model study in which we change the stagger from zero to the typical bone-like arrangement in which the platelets show maximum overlaps. The results of these simulations are shown in Figure 7. We do not observe substantial change in dislocation density under shock loading of these structures (Table 1). This is in contrast to results under tensile loading of the two nanostructures, where a large difference in flow stress according to the shear-tension loading model [8] is seen (results shown in Figure 8).

Figure 7. Geometry and dislocation networks after shock loading for systems with void present. Subplots (a) and (b) show schematics of nanocomposites with similar platelet size and volume fraction; (a) shows the zero-stagger arrangement and (b) shows the bone-inspired design of staggered arrangement of platelets (color scheme same as Fig. 1). Subplots (c) and (d) show actual 3-D views of these two structures. Subplots (e) and (f) show dislocation network snapshots (showing only dislocation cores) for systems (a) and (b), respectively, at a simulation time of 15.3 ps. The dislocation densities are 1.91E13 cm-2 and 1.81E13 cm-2, respectively. The figures show minimal dependence of dislocation density on arrangement geometry of nanocomposites.

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Figure 8. Tensile flow stress of nanocomposites with similar platelet size and volume fraction, but different amount of platelet stagger arrangement (from 0 to a maximum of 0.5). The insets show schematics of the nanocomposite structures corresponding to different amounts of stagger. The figure shows a five-fold increase in flow stress as the stagger is changed from 0 to 0.5 (see also Table 1).

Figure 9. Dislocation density (showing only dislocation cores) for ‘weak’ and ‘strong’ interfaces in systems with void present, for different piston speeds. Subplots (a) and (b) show entire dislocation network for ‘strong’ and ‘weak’ interfaces respectively, at a piston speed of 393 m/s at a time of 25.5 ps. Subplots (c) and (d) show entire dislocation network for ‘strong’ and ‘weak’ interfaces respectively, at a higher piston speed of 589 m/s at a time of 12.7 ps. At both speeds, we observe much higher dislocation densities for the ‘weak’ interface, about twice as much as for ‘strong’ case. The horizontal lines in subplots (b) and (d) indicate slip along matrix-platelet interfaces.

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Figure 10. Dislocation density (showing only dislocation cores) for ‘weak’ and ‘strong’ interfaces in systems with pure copper slab attached at left end for different piston speeds. (a) and (b) show entire dislocation network for ‘strong’ and ‘weak’ interfaces respectively, at a piston speed of 825 m/s at a time of 17.8 ps, there is no plastic response for the strong interface; (c) and (d) show entire dislocation network for ‘strong’ and ‘weak’ interfaces respectively, at a higher piston speed of 884 m/s at a time of 12.7 ps. At 884 m/s piston speed, the dislocation density in (d) is about 40% higher than in (c). The exactly horizontal lines in (b) and (d) indicate slip along matrix-platelet interfaces. Other studies with changing the shapes of platelets also show that the geometry does not play a significant role in shock response (results not shown here).

Interface effect. The interfacial strength is varied by changing parameters for the matrix-platelet interaction to provide a ‘weak’ and ‘strong’ interface (see Figures 3 and 4). The geometrical parameters are held constant throughout the computational experiments for studying effects of the interface. Figures 9 and 10 show the dislocation density response for ‘weak’ and ‘strong’ interfaces for the samples with void present, and with slab of pure Cu attached, respectively. These plots reveal a large difference in the shock loading response. The generated dislocation density is seen to be twice as large for the ‘weak’ interface as for the ‘strong’ one in the case of the sample with a void present. For the sample with pure Cu attached in the front, we observe extensive plasticity for the ‘weak’ interface case at piston speeds where only elastic response is seen for ‘strong’ interface. This is seen to be due to addition plasticity induced by interfacial slip at the matrix-platelet interface for the ‘weak’ interface. The sessile component of the dislocation density, arising from dislocation entanglement, contributes to strain hardening of the shocked material; a large fraction of which is retained upon relaxation of the shocked crystal over microsecond timescales. We measure the sessile fraction of dislocation density at different time steps in the movement of the shockwave for the ‘strong’ and ‘weak’ interfaces (see Figure 11). We find that a larger fraction of dislocation density is sessile for the ‘weak’ interface case. Both the larger dislocation density and large fraction of sessile dislocations are indications of larger transient and permanent plastic response for the ‘weak’ interface case.

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Figure 11. Fraction of sessile component of the dislocation density, for the ‘strong’ interface (interface1) and the ‘weak’ interface (interface2), for systems with void present. The data is plotted against the simulation time, before the shock front impacts the back surface of specimen. The graphs show the increasing sessile component with time, indicating that the dislocation generation stage is far from the equilibrium dislocation density. The ‘weak’ interface shows a higher component of sessile dislocations, at all times, indicating greater work hardening and larger dissipation of elastic shock energy.

These results strongly suggest that interfacial plasticity plays an important role in shock response, and below a critical interfacial strength, the interfacial slip is activated and increases energy dissipated tremendously. For a given volume fraction of platelet phase, a nanostructure design with large interfacial area is thus proposed as a design guideline for improvement to shock response. Discussion and conclusions In this study, we observe the shock loading response of nanostructured model materials. We focused on the biomimetic design of a nanostructure, specifically mimicking the bone and nacre nanostructure as a combination of a soft matrix phase and hard platelet phase. We developed and applied a generalized EAM model to represent interactions between platelets and matrix.

Nanocomposite with zero stagger a arrangement of platelets Dislocation density under shock loading at t=15.3 ps Flow stress under tensile loading (Figure 8)

13

1.91 · 10

cm

0.1 GPa

-2

Nanocomposite with bone-like b stagger arrangement of platelets

13

1.81 · 10

cm

-2

0.56 GPa

Table 1. Variation of dislocation density under shock loading and flow stress under tensile loading with change in geometry of platelet arrangement (stagger between platelets). See Figure 7 for schematic of geometrical arrangements of nanocomposites (note: a refers to the arrangement as in Figure 7(a), b refers to the arrangement as in Figure 7(b)). The most important result of this study is the finding that the geometrical arrangement of platelets does not play a significant role in shock loading response, unlike the behavior seen in tensile loading of the same structures (Table 1 and Figures 7 and 8). However, we observe a large effect of

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interfacial strength on shock response, with extended interfacial slip occurring for weak interfaces. A higher dislocation density and a larger fraction of sessile dislocations are observed for the weaker interface, indicating higher hardness and energy dissipated by weak-interface nanostructures. These results are critical on the path to enable the design of nanocomposite materials. Our results show that with a nanocomposite with weak interfaces between hard platelets embedded in a soft matrix that easily slip are advantageous in inducing significant permanent deformation, thereby dissipating the elastic impact of the shock loading. The increase in interfacial area that arises with very thin large aspect-ratio platelets will increase the contribution of interfacial deformation to energy absorption from shock waves. This leads to a series of interesting questions: Is there a critical strength of the interface below which it is too weak to transmit load between platelets and matrix? Too weak interfaces would also lead to a drastic decrease in tensile strength, and thus, an interesting point for investigation would be to assess the ability of the material to perform well under both tensile and shock deformation condition. These points will be addressed in future studies. References [1] Y.S. Lee, E.D. Wetzel, and N.J. Wagner: J.Mat. Sci. Vol. 38 (2003), p. 2825. [2] E.M. Bringa, J.U. Cazamias, P. Erhart, J. Stolken, N. Tanushev, B.D. Wirth, R.E. Rudd, and M.J. Caturla: J. Appl. Phys. Vol. 96 (2004), p. 3793. [3] L.P. Davila, P. Erhart, E.M. Bringa, M.A. Meyers, V.A. Lubarda, M.S. Schneider, R. Becker, and N. Kumar: Appl. Phys. Lett. Vol. 86 (2005), p. 161902. [4] P. Erhart, E.M. Bringa, M.. Kumar, and K. Albe: Phys. Rev. B Vol. 72 (2005), p. 052104. [5] E.M. Bringa, A. Caro, Y. Wang, M. Victoria, J.M. McNaney, B.A. Remington, R.F. Smith, B.R. Torravla, and H.V. Swygenhoven: Science Vol. 309 (2005), p. 1838. [6] Y.M. Wang, E.M. Bringa, J.M. McNaney, M. Victoria, A. Caro, A.M. Hodge, R. Smith, B. Torralva, B.A. Remington, C.A. Schuh, H. Jamarkani, and M.A. Meyers: Appl. Phys. Lett. Vol. 88 (2006), p. 061917. [7] Z. Tang, N.A. Kotov, S. Maganov, and B. Ozturk: Nature Mat. Vol. 2 (2003), p. 413. [8] H. Gao, B. Ji, I.L. Jager, E. Arzt, and P. Fratzl: Proc. Nat. Acad. Sci. Vol. 100 (2003), p. 5597. [9] B. Ji and H. Gao: J. Mech. Phys. Sol. Vol. 52 (2004), p. 1963. [10] N.C. Broedling, A. Hartmaier, M.J. Buehler, and H.Gao: J. Mech. Phys. Sol. (2007), in press, available online doi:10.1016/j.jmps.2007.06.006 [11] M.J. Buehler: Nanotechnology Vol. 18 (2007), p. 295102. [12] J.Y. Rho, L.K. Spearing, and P. Zioupos: Med. Engg. Phys. Vol. 20 (1998), p.92. [13] M.S. Baskes, S.M. Foiles, and M.I. Baskes: Mat. Sci. Reports (The Netherlands) Vol. 9 (1993), p. 251. [14] Y. Mishin, M.J. Mehl, D.A. Papconstantopoulos, A.F. Voter and J.D. Kress: Phys. Rev. B Vol. 63 (2001), p. 224106 [15] R.A Johnson: Phys. Rev. B Vol. 39 (1989), p. 12554. [16] S.M. Foiles, M.I. Baskes, and M.S. Daw: Phys. Rev. B Vol. 55 (1986), p. 7983. [17] B.L. Holian and P.S. Lomdahl: Science Vol. 280 (1998), p. 2085. [18] J. Stadler, R. Mikulla, and H.R. Trebin: Int. J. Mod. Phys. C Vol. 8 (1997), p. 1131. [19] E.M. Bringa, K. Rosolankova, R.E. Rudd, B.A. Remington, J.S. Wark, M. Duchauneau, D.H. Kalantar, J. Hawreliak, and J. Belak: Nature Mat. Vol. 5 (2006), p. 805. [20] J. Li: Model. Simul. Mater. Sci. Eng. Vol. 11 (2003), p. 173.

Solid State Phenomena Vol. 139 (2008) pp 23-28 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.23

Hydrogen storage in MgH2 matrices: an ab-initio study of Mg-MgH2 interface Simone Giusepponi1,a, Massimo Celino1,b, Fabrizio Cleri2,c and Amelia Montone1,d 1

ENEA, Centro Ricerche Casaccia, CP 2400, 00100 Rome, Italy

2

IEMN, Université des Sciences et Technologies de Lille (CNRS UMR 8520), F-59652 Villeneuve d’Ascq cedex, France a

[email protected], [email protected],

c

[email protected], [email protected]

Keywords: Hydrogen storage, molecular dynamics, ab-initio simulation, magnesium, interface.

Abstract. We studied the atomic-level structure of a model Mg-MgH2 interface by means of the Car-Parrinello molecular dynamics method (CPMD). The interface was characterized in terms of total energy calculations, and an estimate of the work of adhesion was given, in good agreement with experimental results on similar systems. Furthermore, the interface was studied in a range of temperatures of interest for the desorption of hydrogen. We determined the diffusivity of atomic hydrogen as a function of the temperature, and give an estimate of the desorption temperature. Introduction The remarkable hydrogen capacity of magnesium has fostered intense research efforts in the last years in view of its future applications where light and safe hydrogen-storage media are needed. Magnesium can reversibly store about 7.6 wt% hydrogen, has light weight and is a low-cost material. However, further research is needed since Mg has a high operation temperature and slow absorption kinetics that prevent for the moment the use in practical applications. Two separate reasons could explain this problem: (a) hydrogen molecules do not readily dissociate on Mg surfaces, and (b) magnesium hydride is very stable up to high temperatures. Improvements on H2 absorption and desorption kinetics can be obtained by mechanically milling MgH2 and adding catalysts to magnesium and magnesium hydrides. The addition to Mg hydride of small amounts of catalytic metals, such as Pd or Ni, has been indicated as a promoting factor in the surface dissociation of the H2 molecule (the clean Mg surface alone being unable to catalyze the dissociation). Moreover, other metallic additives, such as Al, Li, Mn, Ti etc., have been suggested to improve the hydriding properties of metallic Mg [1-5]. However a full understanding of the desorption mechanism at the interface Mg-MgH2 is still lacking. From an experimental point of view there is no clear evidence of which interfaces are involved in the hydrogen diffusion, and which is the atomic dynamics at the interfaces [6]. For these reasons a detailed study of the interface between Mg and MgH2 is needed to characterize the dynamics of hydrogen at the interface. Further insights are gained by characterizing Mg-MgH2 interfaces which are supposed to play a major role in the hydrogen diffusion during absorption and desorption cycles. By means of accurate ab-initio molecular dynamics simulations based on density-functional theory with norm-conserving pseudopotentials and plane-wave expansion (CPMD code [7,8]), in this work an interface was selected, studied and characterized. Extensive electronic structure calculations are used to determine the equilibrium properties and the behavior of the surfaces in terms of total energy considerations. Furthermore, the interface is studied at several values of temperature, thereby characterizing the hydrogen atomic displacement.

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Theory, Modeling and Numerical Simulation

In the first section the computational details are reported while in the second section energy considerations are discussed to characterize the interface. In the third section a study of the interface at different values of the temperature is performed, focussing on hydrogen diffusion. Computational details We employed for all the calculations the CPMD (Car Parrinello Molecular Dynamics) code [8] based on the DFT (Density Functional Theory) and plane wave basis pseudopotential method. Goedecker-Teter-Hutter pseudopotentials for magnesium and hydrogen, together with Padé approximant LDA exchange-correlation potentials were used [9]. The electronic wave functions are expanded in a plane-wave basis set with a kinetic energy cut-off equal to 80 Ryd. The latter value was optimized by preliminary calculations both on simple molecules (Mg2, MgH and MgH2), and on the crystalline structures of metallic Mg and magnesium hydride. All the calculations were performed in the supercell approximation, in view of the large-scale molecular dynamics simulation of the interface, with periodic boundary conditions meant to mimic an infinitely extended system. The crystal structure of Mg is hexagonal close-packed. For MgH2 we considered the β-MgH2 that is observed at atmospheric pressure and low temperatures. The atomic structure β-MgH2 is of the TiO2-rutile type [10]. The crystal structures were relaxed over a range of possible cell volumes, varying by hand the cell parameters, and minimizing at each step the total energy. The final values obtained for the cell parameters are in good agreement with experimental ones within 3%. For the Mg crystal structure we found the following parameters a = b = 3.15 Å and c/a = 1.62. For the case of MgH2 we found a = b = 4.39 Å, c = 2.96 Å and u = 0.304 (experimental values: a = b = 4.501 Å, c = 3.02 Å). We used a supercell approach to simulate the interfacial system. Each surface, selected according to periodic boundary conditions, was relaxed without taking into account any possible reconstructions upon changing the temperature. Zero-temperature total energy calculations were firstly used to evaluate interface stability. Subsequently, MD simulations at constant volume and constant temperature are performed by using Nose-Hoover thermostats [11,12]. One key quantity to predict the mechanical properties of an interface, and to verify the reliability of model interfaces, is the work of adhesion, W. The latter quantity is defined as the bond energy per unit area needed to reversibly separate an interface into two free surfaces, neglecting plastic and diffusional degrees of freedom. Formally, W can be defined in terms of either the surface and interfacial energies (relative to the respective bulk materials), or by the difference in total energy between the interface and its isolated slabs: W = σ 1 + σ 2 − γ 12 = ( E1 + E 2 − E12 ) / A

[1]

Here σi is the surface energy of slab i, γ12 is the interface energy, Ei is the total energy of the slab i, and E12 is the total energy of the interface system. The total interface area is fixed by A. Model interface In the literature there are no references describing Mg-MgH2 interfaces, neither experimentally nor theoretically. For this reason, our work can be considered as the first attempt to describe at the atomic level the energies and atomic-scale dynamics of the interface. Our main constraint is that we have to choose two commensurate surfaces (one for Mg and one for MgH2), so as to fill a common simulation cell. Such a constraint narrows the possibilities of finding two suitable free surfaces, in order to build a proper interface. Among the low Miller index surfaces of both materials, it is found that the (110) of MgH2 can be attached to either (010) or (100) of Mg, according to periodic boundary conditions in x and y directions, by minimizing the lattice mismatch. A small distortion in the Mg cell parameter is needed to obtain fully commensurable surfaces: the a and c parameters are thus scaled about 1%. The final supercell is shown in Fig. 1. It is the result of a balance between computational load

Solid State Phenomena Vol. 139

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and physical reliability. Later on we will see that the calculated work of adhesion of this interface is of the same order of a typical interface, giving a reasonable estimate in the case of Mg-MgH2. Fig. 1a depicts the Mg surface. The Mg portion is composed by 4 layers exposing on both sides the (010) surface. The external surface (on the left) is the same of the internal surface (on the right) but translated by a half lattice parameter in the z direction. In Fig. 1b the MgH2 part is composed by 4 layers and also in this case the external surface (on the right) is the same of the internal one (on the left) but translated by a half lattice parameter in the x direction. Then, the two half-systems are narrowed to shape an interface as shown in Fig. 1c. The total length of the system is Ly = 38.71 Å, while in the x and z direction the system has Lx = 6.21 Å and Lz = 15.09 Å. On both sides of the system, a void slab of length Lx is introduced to suppress the interaction, due to periodic boundary conditions, between the external surfaces of Mg and MgH2. Total energy calculations are performed to optimize the distance between the internal surfaces that minimizes the energy of the whole system. As shown in Fig. 2, at the distance d = Lx/2 the total energy of the system is minimized. Subsequently, we performed a ionic relaxation to optimize the atomic geometry. This optimization has been performed by keeping fixed 2 layers of Mg atoms on both sides of the system. This is done both to impose a bulk behavior to the external parts of the systems and to limit the influence of the external surfaces on the internal atoms near the interface. During the ionic relaxation, the internal surfaces are free to adapt to each other. This relaxation reveals a modest displacement of one surface with respect to the other, and a significant displacement of hydrogen atoms towards the interface. The final configuration is characterized by a work of adhesion, computed by using equation (1), of W= 318.28 mJ/m2. For the moment is not possible to compare this value with experimental values. However, this W value is of the same order of other adhesion energies of structurally similar interfaces [13,14].

Figure 1. Representation of the structures in the simulation box. a) Mg slabs with free surfaces; b) MgH2 slabs with free surfaces; c) Mg-MgH2 interface with free surfaces. The structures are infinitely extended in x and y directions; Free surfaces and Mg-MgH2 interface are perpendicular to z axis.

26

Theory, Modeling and Numerical Simulation

Figure 2. Work of adhesion for different distances for the Mg-MgH2 interface. The distance d is expressed in units of Lx (see text).

Figure 3. Snapshots of Mg-MgH2 interface at different temperatures: a) T = 300 K; b) T= 500 K; c) T = 700 K; d) T = 900 K. We represent the Mg atoms in green and H atoms in blue.

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Hydrogen diffusion To understand the atomic level dynamics of hydrogen diffusion at the interface, we performed MD simulations at constant volume and constant temperature, starting from room temperature (T= 300 K) and up to T= 900 K. In this range of temperature, a phase transition from MgH2 to Mg + H2 is experimentally observed, accompanied by the desorption of hydrogen. Figure 3 depicts four atomic configurations, from which it is evident the influence of the temperature on the displacement of hydrogen atoms. At the temperatures of T= 300 K (Fig. 1a) and T= 500 K (Fig. 1b) there is no diffusion of hydrogen atom. Diffusion of hydrogen atoms starts at T= 700 K (Fig. 1c). At this temperature there is a clear tendency of hydrogen atoms to move towards the interface. The dynamics of hydrogen at the interface is characterized by jumps from one hydrogen lattice site to the nearest lattice site. At higher temperature, T= 900 K (Fig. 1c), hydrogen atoms move forward the Mg surface, by leaving their lattice sites. Such results are in rather good agreement with the experimental ones, since hydrogen desorption of unmilled MgH2 without catalyst is observed to start at 780 K. At this temperature, diffusion of hydrogen atoms in the interface is clearly visible in the MD simulations. It is worth to underline that no H atoms diffuse into the Mg lattice. To better quantify the hydrogen diffusion at the interface, in Fig. 4 is reported the diffusion coefficient of hydrogen atoms versus temperature. As shown in Fig. 4, "bulk" hydrogen atoms have a lower diffusivity than those near the interface. This difference increases at higher temperatures.

3.5

3.0

Interface Bulk

2

Diffusion (10 cm /s)

2.5

-5

2.0

1.5

1.0

0.5

0.0 200

300

400

500

600

700

800

Temperature (K)

Figure 4. Diffusion coefficient of hydrogen atoms versus temperature.

900

28

Theory, Modeling and Numerical Simulation

Conclusions In this paper we report the first attempt to rationalize the atomic level dynamics of hydrogen atoms at a model Mg-MgH2 interface, composed by the (010) face of Mg and (110) of MgH2 under the constraint of periodic boundary conditions in a common supercell. After optimization of the atomic position and minimization of the total energy by varying the interfacial distance, the work of adhesion is computed. While it is not possible to compare such a value of work of adhesion with experimental results on the same system, this value is of the same order of magnitude of that of other interfaces computed for similar systems. Furthermore, a study of the dynamics of hydrogen atoms at the interface allowed to show that their diffusivity is largely increased with respect to bulk hydrogen atoms. At a temperature of 700 K we observe a steep increase in the diffusion of hydrogen atoms at the interface, that can be interpreted as first steps of hydrogen desorption. Such a value of temperature is in rough agreement with the experimental desorption temperature. Acknowledgment The authors thank the CPMD consortium for the use of the code. Drs. M. Vittori, A. Aurora and J. Grbovic are also acknowledged for useful discussions and suggestions. References [1] D. Sun, H. Enoki, F. Gingl and E. Akiba, J. Alloys Comp. Vol. 285 (1999) p. 279. [2] A. Zaluska, L. Zaluska and J.O. Strom-Olsen, J. Alloys Comp. Vol. 288 (1999), p.217. [3] A. Montone, J. Grbovic, Lj. Stamenkovic, A. L. Fiorini, L. Pasquini, E. Bonetti and M. Vittori Antisari: Mat. Sci. Forum Vol. 494 (2005), p.137. [4] A. Bassetti, E. Bonetti, L. Pasquini, A. Montone, Y. Grbovic and M. Vittori Antisari: Eur. J. Phys. B Vol. 43 (2005), p.19. [5] F. Cleri, M. Celino, A. Montone, E. Bonetti and L. Pasquini: Mat. Sci. Forum Vol. 555 (2007), p. 349. [6] T. Schober, Metall. Trans. A Vol. 12 (1981), p.951. [7] R. Car and M. Parrinello: Phys. Rev. Lett. Vol. 55 (1985), p. 2471. [8] CPMD, Copyright IBM Corp. 1990 – 2006, Copyright MPI für Festkörperforshung Stuttgart 1997 – 2001. [9] S. Goedecker, M. Teter and J. Hutter: Phys. Rev. B Vol. 54 (1996), p. 1703. [10] M. Bortz, B. Bertheville, G. Bøttger and K. Yvon, J. Alloys Comp. Vol. 287, (1999), p.L4. [11] S. Nosé, J. Chem. Phys. Vol. 81, (1984) p. 511; Mol. Phys. Vol. 52, (1984), p. 255. [12] W. G. Hoover, Phys. Rev. A Vol. 31, (1985), p. 1695. [13] Y. Han, Y. Dan, D. Shu, J. Wang and B. Sun: Appl. Phys. Lett. Vol. 89 (2006), p. 144107. [14] D. J. Siegel, L. G. Hector and J. B. Adams: Phys. Rev. B Vol. 67 (2003), p. 092105.

Solid State Phenomena Vol. 139 (2008) pp 29-34 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.29

First-principles calculations of the atomic and electronic structures in Au-Pd slab interfaces Noboru Taguchi1a, Shingo Tanaka2b, Tomoki Akita2c, Masanori Kohyama2d, Fuminobu Hori1e 1

Department of Materials Science, Osaka Pref. Univ., Gakuen-cho 1-2,Naka-ku, Sakai, Osaka 599-8531, Japan 2

Research Institute for Ubiquitous Energy Devices, National Institute of Advanced Industrial Science and Technology (AIST), Midorigaoka1-8-31, Ikeda, Osaka 563-8577, Japan a

[email protected], [email protected], [email protected] d

[email protected], [email protected]

Keywords: First Principles Calculation, Core-Shell Nanoparticles, Au, Pd, Interface

Abstract. We performed first-principles calculations using the projector augmented-wave (PAW) method for Au/Pd slab interface models. The calculations of relaxed configurations and energies for the thin Pd layers (3 layers) stacking on Au (111) and Au (100) slabs with an epitaxial relationship represent that Pd overlayers have a lateral expansion in both cases. This trend is in good agreement with experimental results for Pd/Au slabs and Au-Pd core-shell nanoparticles, obtained by electron microscopy, X-ray diffraction, and positron annihilation. In addition, an intermixing configuration near the Au-Pd interface was shown to be more stable than the binary separated one. Introduction Sonochemically synthesized Au-Pd nanoparticles by using sodium dodecyl sulfate (SDS) as an additive have core-shell (Au core and Pd shell) structures [1], and show higher activity for hydrogenation of 4-pentenoic acid than Au and Pd monometallic nanoparticles [2, 3]. In addition, Au-Pd core-shell nanoparticles have a much higher activity in comparison with the mixtures of monometallic nanoparticles and the Au-Pd alloy nanoparticles. These facts reveal that the atomic and electronic structure of core-shell nanoparticles should be responsible for catalytic activities. Hence, it is very important to understand the detailed atomic configurations of the Au-Pd core-shell nanoparticles. Interestingly, the lattice misfit around the Au/Pd interface was hardly observed in these particles from HRTEM observations. Moreover, Electron and X-Ray diffraction measurements for Au-Pd core-shell nanoparticles did not show the radial functions of bulk fcc Pd [1, 4]. These facts suggest that the Pd shell lattice was expanded to match the lattice of fcc Au. Positron annihilation experimental results also showed the trend of Pd lattice expansion in Au-Pd core-shell nanoparticles [5]. About stacking of Pd on Au, HRTEM observations also revealed that the Au/Pd interface was mainly fcc stacking sequence but there were some core-shell nanoparticles containing stacking faults [4]. In addition to the Au-Pd core-shell nanoparticles, epitaxial growth of Pd overlayers on Au (100) and Au (111) surfaces are also reported [6, 7]. Takahashi et al. also pointed out Pd deposition at (threefold or fourfold) hollow sites on the Au (111) surface [8]. Detailed atomic configurations around the Au/Pd interface are not understand yet in Au-Pd core-shell nanoparticles. In the discussion on thin films, formation of Au-Pd alloy near the surface is also reported [9, 10]. Moreover, the Au-Pd binary system intrinsically should be an alloy structure from the perspective of the phase diagram as shown in Fig. 1 of Ref [11]. In the case of core-shell nanoparticles we consider, however, experimental measurements did not clarify whether the intermixing layer of Au and Pd exists or not at the interface. In order to discuss detailed atomic structures, especially of Au/Pd interfaces, for the core-shell

30

Theory, Modeling and Numerical Simulation

Au-Pd nanoparticles or Pd thin overlayers on a Au surface, we have performed first principles calculation of several types of Au/Pd interface models as models for core-shell Au-Pd nanoparticles and thin Pd overlayers on a fcc Au surface. Theoretical method We performed first principles calculation based on density functional theory. The first principles calculations were performed using PAW [12-14] program code QMAS (Quantum MAterials Simulator) [15]. The exchange-correlation effects were described within the generalized gradient approximation [16]. The periodic Au-Pd slab models were used as depicted in Fig. 1 and Fig. 2; thin Pd overlayers on fcc Au (111) and (100) crystal surfaces. (1x1) supercell was used for all the models. 3 Pd layers were deposited to above and below 5 and 7 Au fcc layers for (100) and (111) respectively. A plane wave cutoff energy of 40 Ry was used. 52 k-points and 20 k-points per irreducible Brillouin zone were used for the (111) and (100) interface systems, respectively. The calculated equilibrium lattice constants for bulk Au and Pd were 4.16 Å and 3.96 Å, respectively. The lattice constants parallel to the interface were fixed to those of Au (lattice-Au), Pd (lattice-Pd) and intermediate: (Au+Pd)/2 (lattice-mid). For a vertical direction to the interface, a full atomic relaxation was carried out with symmetry conditions. Various stacking sequences of the Au/Pd interface and Pd layers were dealt with in order to clarify the detailed Pd stacking at the interface and the presence of stacking faults near the interface. In the case of binary separate Au/Pd interface, as the Pd stacking sites on fcc Au at the interface, we deal with an on-top site and a hollow site for the (100) slabs and an on-top site (T1), a threefold hollow site (H3) and fourfold hollow site (T4) for the (111) slabs. The H3 site model of the (111) slab with the stacking fault in the Pd region is also constructed. Moreover, we constructed models for the (100) slab with intermixing Au-Pd layers at the interface to discuss the presence of intermixing near the interface region in Au-Pd core-shell nanoparticles. The intermixing layers were constructed from exchanging Au and Pd atoms at the interface in each binary separate Au-Pd (100) slab. Pure Au and pure Pd slabs fixed lattice constants (lattice-Au, lattice-Pd, and lattice-mid) for parallel to the surface were also constructed to calculate adhesive energy mentioned after.

Figure 1 Calculated models for Au-Pd (100)

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Figure 2 Calculated models for Au-Pd (111) Results and Discussion We calculated an adhesive energy Ead that is an energy gain by the formation of a Au/Pd interface from two slabs given by the following equation, Ead = EAu-Pd - (EAu+EPd)

(1)

where EAu-Pd, EAu and EPd are the total energies for Au-Pd, pure Au and pure Pd slabs, respectively. Table 1 Adhesive Energy Ead for each slab models with 100 and 111 surfaces Au-Pd (111)

Au-Pd (100)

(Pd lattice expansion)

All energies are given in J/m2.

32

Theory, Modeling and Numerical Simulation

Calculated adhesive energies of all the Au-Pd slab models are listed in Table 1. Values of Ead of lattice-Au are more stable than those of lattice-Pd or lattice-mid for all stacking models, so that structures with the Pd lattice being close to fcc Au are energetically stable. Moreover, pseudomorphical thin Pd overlayers on Au (100) has large Ead than that on Au (111). In particular, in the binary separated case the Au-Pd (100) interface, 5.44 J/m2, is much larger than the Au-Pd (111) one, 1.64 J/m2. In experimental reports of epitaxial growth of Pd overlayers on Au (100) and Au (111) [6, 7], the number of Pd layers that can grow pseudomorphically on Au (100) is larger than that on Au (111). The present calculated results are consistent with this trend of experimental facts. The stability of the Au/Pd interface is important for the stability of pseudomorphic Pd structure. (The structure of the Au/Pd interface) The fcc stacking sequence is the most stable for both (100) and (111) models, and the on-top sites are less stable than the hollow sites. These results are in good agreement with experimental results on thin films. However, for the (111) models, the difference of Ead between H3 and T4 is slight, especially at lattice-Au models (H3: 1.64, T4: 1.63 J/m2). Moreover, the value of Ead of the model with stacking faults at the Pd region near the Au/Pd interface (1.63 J/m2) is almost the same as that without stacking faults. Thus, the difference in the stacking sequences of Pd does not induce a large energy difference. For all the (100) models, the intermetallic models have a large Ead compared with the binary separate ones. The formation of intermixing Au/Pd interfaces is energetically advantageous. Thus, the (100) interface can construct not only fcc stacking structure but also the intermixing or alloy structure. Summary We have performed first principles calculations for stable structure of thin Pd overlayers on Au substrate as a model of Au-Pd core-shell nanoparticles and thin Pd overlayer on fcc Au. Thin Pd lattice deposited on Au substrate with lateral expansion and being close to fcc Au lattice constant is stable. The fcc stacking sequence is the most stable structure at the interface. An intermixing Au-Pd layer at Au/Pd interface is also suggested. It is concluded that Pd lattice expansion is very important to explain the peculiar structure of the Au/Pd interface. Acknowledgement This work was supported by the Japan Society for the Promotion of Science (JSPS-Grant-in-Aid for Scientific Research (B) 17360314). The authors are grateful to A. Iwase and T. Kojima (Osaka Pref. Univ.), K. Okazaki (JST), S. Ishibashi and T. Tamura (AIST) and H. Takatani for their valuable comments and fruitful discussions. References [1] T. Akita, T. Hiroki, S. Tanaka, T. Kojima, M. Kohyama, A. Iwase and F. Hori: Mater. Res. Soc. Symp. Proc. (2007), p. 982. [2] Y. Mizukoshi, T. Fujimoto, Y. Nagata, R. Oshima, and Y. Maeda: J. Phys. Chem. B Vol. 104 (2000), p. 6028. [3] H. Takatani, H. Kago, Y. Kobayashi, F. Hori, and R. Oshima: Trans. Mater. Res. Soc. Jpn. Vol. 28 (2003), p. 871.

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[4] T. Akita, T. Hiroki, S. Tanaka, T. Kojima, M. Kohyama, A. Iwase, and F. Hori: Catal. Today. (2007) (in press) [5] F. Hori, T. Kojima, S. Tanaka, T. Akita, T. Iwai, T. Onitsuka, N. Taguchi and A. Iwase: Phys. Status Solidi C Vol. 4 (2007), p. 3895. [6] A.L.N. Pinheiro, M.S. Zei, M.F. Luo and G. Ertl: Surf. Sci. Vol. 600 (2006), p. 641. [7] L.A. Kibler, M. Kleinert, R. Randler and D. M. Kolb: Surf. Sci. Vol. 443 (1999), p. 19. [8] M. Takahashi, Y. Hayashi, J. Mizuki, K. Tamura, T. Kondo, H. Naohara and K. Uosaki: Surf. Sci. Vol. 461 (2000), p. 213. [9] L.A Kibler, M. Kleinert and D.M. Kolb: Surf. Sci. Vol. 461 (2000), p. 155. [10] B.E. Koel, A. Sellidj and M.T. Paffett: Phys. Rev. B Vol. 46 (1992), p. 7846. [11] M.H.F. Sluiter, C. Colinet and A. Pasturel: Phys. Rev. B Vol. 73 (2006), p. 174204. [12] P. Blöchl: Phys. Rev. B Vol. 50 (1994), p. 17953. [13] N.A.W. Holzwarth, G.E. Matthews, R.B. Dunning, A.R. Tackett, and Y. Zeng: Phys. Rev. B Vol. 55 (1997), p. 2005. [14] G. Kresse and D. Joubert: Phys. Rev. B Vol. 59 (1996), p. 1758. [15] S. Ishibashi, T. Tamura, S. Tanaka, M. Kohyama, and K. Terakura (unpublished) [16] J.P. Perdew, K. Burke and M. Ernzerhof: Phys. Rev. Lett. Vol. 77 (1996), p. 3865.

Solid State Phenomena Vol. 139 (2008) pp 35-40 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.35

In-diffusion and Out-diffusion of Oxygen from a Composite Containing Random Traps I.V. Belova1,a and G.E. Murch1,b 1

Centre for Mass and Thermal Transport in Engineering Materials School of Engineering The University of Newcastle Callaghan NSW 2308 Australia a [email protected], b [email protected] Keywords: composites, segregation, diffusion, Lattice Monte Carlo Abstract. In this paper, we investigate oxygen in-diffusion and out-diffusion with respect to a cermet composite where oxygen segregates at the interface between the metal matrix phase and the ceramic oxide phase. This phenomenological diffusion problem is treated by overlaying it with a fine-grained lattice that was addressed using a Lattice Monte Carlo method and a little-known exact expression for the lattice-based effective diffusivity in the presence of random traps. It is shown that there is very good agreement for the oxygen concentration depth profiles between the Monte Carlo results and the exact expression. Introduction During the initial synthesis and later in-service lifetime of a ceramic oxide-cermet, oxygen from the external surface diffuses through the metal matrix and segregates at the interfaces of the metal/ceramic oxide inclusions [1]. The presence of this oxygen leads to significant weakening of the bonding between the metal matrix and the ceramic oxide inclusions and subsequent serious deterioration of the mechanical properties of the cermet. This oxygen can generally be removed by annealing of the composite in vacuum at high temperature by an out-diffusion process that is essentially diffusion-limited evaporation [1]. Both processes of in-diffusion and out-diffusion are phenomenological mass diffusion problems that are amenable to being addressed using finitedifference and finite element methods. Recently, another method, termed the Lattice Monte Carlo (LMC) method, has been developed to address phenomenological mass as well as heat diffusion problems [2]. This method shows special promise as an alternative that is particularly well-suited for addressing diffusion problems that have complex geometries and/or which have singularities in the sources and sinks of diffusant. The LMC method has been used to describe the detailed process of in-diffusion and segregation of oxygen at the interfaces of (square) MgO inclusions in an Ag matrix [3,4]. In this case, there was very good agreement of the local oxygen concentration profiles obtained from the LMC method with those obtained by the finite element method. In the present paper we take what might be regarded as a ‘bigger picture’ model of this phenomenological diffusion problem wherein the ceramic oxide inclusions are distributed randomly in the matrix and can be represented as featureless traps for oxygen within the matrix. In this paper, we investigate the effective diffusivity of oxygen in this model and deduce concentration profiles for both in-diffusion and out-diffusion using the LMC method. We also make contact with an exact lattice-based expression for the effective long-time limit diffusivity for a diffusant in the presence of random traps.

36

Theory, Modeling and Numerical Simulation

The Model In keeping with the spirit of the LMC method [2] a fine-grained lattice is constructed to overlay the phenomenological diffusion problem. For convenience, in this problem we use a 2D lattice. Each inclusion is represented as a single trap situated at a given site of this lattice. Accordingly, the mesh size of the lattice then determines the apparent size of the inclusions. The trapping sites are distributed randomly. In adopting such a model we are assuming in effect that the oxygen mobility along the interfaces of the metal/ceramic oxide inclusions is very rapid compared with diffusion in the matrix. Thus oxygen can exit (and enter) a trap with exactly the same probability in all four directions. This seems to be quite a reasonable simplifying assumption. We define D1 as the oxygen diffusion coefficient in the matrix and D2 as the diffusivity of oxygen associated with the interface i.e. with the trap itself. The diffusivities Di are related to the jump frequencies Γi by the lattice random-walk expression in two dimensions: Di = Γi a 2/4

(1)

where a is the lattice mesh size and the factor 4 comes from the lattice being two dimensional. For a three dimensional lattice this factor would be 6. The segregation factor s is defined in the usual Henry’s Law form (but a site blocking-type expression can also be used in principle [5]): s=

ψ O2 ψ O1

(2)

where ψ O2 and ψ O1 are the equilibrium concentrations of the diffusant oxygen at the traps and in the matrix respectively. The segregation factor is also given by: s = Γ12/ Γ21

(3)

where Γ12 and Γ21 are the jump frequencies into and out of a trap from the matrix. For convenience, in the absence of more detailed information, we set Γ12 equal to Γ1 and Γ21 equal to Γ2. Thus we have a simple two-frequency trapping model. For this trapping model, there exists a little-known exact lattice-based expression for the effective long-time diffusivity in the presence of random traps [6]. In the present notation, the expression is:

Deff Γ2 = , D1 (1 − g )Γ2 + gΓ1

Deff Γ1 = D2 (1 − g )Γ2 + gΓ1

(4)

where g is the volume (area) fraction of the inclusions and Deff is the effective diffusivity of the composite. In effect, with respect to the lattice, the area of a given trap is a2. Results and Discussion The expression Eq. 4 for the effective diffusivity was first verified by making use of the EinsteinSmoluchowski Equation:

Deff = / 4t

(5)

where R is the displacement of a given particle in time t and the Dirac brackets < > indicates an average over a large number of particles. Eq. 5 is readily realized in an equilibrium simulation wherein completely non-interacting particles are permitted to diffuse in a lattice containing random traps for a fixed number of jump attempts per particle (proportional to time) [2]. We explored the

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situation where D1 is greater than D2 which is of course appropriate for the diffusion in the cer-met and, for completeness, we also investigated the converse situation. A range of values of the segregation factor was investigated. As can be seen in Fig. 1, the LMC results reveal that the latticebased expression [6] for the effective long-time limit diffusivity in the presence of random traps is indeed correct. 1 s=10, D =D 1 s = 0 .1 , D = D 2 0 .1 eff

D /D

s=100, D =D 1 s = 0 .0 1 , D = D 2 0 .0 1 s=100, D =D 1 s = 0 .0 0 1 , D = D 2 0 .0 0 1

0

0 .1

0 .2

0 .3

0 .4

0 .5

g

Figure 1. Calculated (symbols, Monte Carlo simulations) and theoretical values (solid lines, Eq. 4) for the ratios of the effective diffusivity Deff to D as a function of g – fraction of the random inclusions at several values of s.

Next, we determined the oxygen concentration depth profiles for both in-diffusion and out-diffusion (with the constant source/sink boundary condition) of oxygen in the presence of random traps for reasonable values of the ratio of the matrix and trap diffusivities and segregation factor. For simulating the phenomenological process of in-diffusion from a constant surface source of oxygen into a solid we followed the procedure described in detail in [2,3]. For simulating the process of out-diffusion the procedure described in [2] was followed in general terms. We started with an equilibrium concentration of oxygen in the composite according to a specified value of the segregation factor. At diffusion times greater than zero the surface concentration of oxygen Ψ0 was set equal to zero. This ensured a net flux of oxygen out from the model composite. In both cases of in-diffusion and out-diffusion at relatively short diffusion times the finite lattice sample can be considered infinite in the sense that the standard solutions of the Diffusion Equation can be employed. These solutions are: In-diffusion



x  4 Deff t 

ψ Oeff ( x, t ) = ψ Oeff (0, t )erfc 

   

(6)

Out-diffusion 

x  4 Deff t 

ψ Oeff ( x, t ) = ψ Oeff ( x,0)erf 

   

(7)

Figs. 2, 3 show results for the oxygen concentration profiles at different in-diffusion times and for s = 10 and 100 respectively. It is seen that there is generally very good agreement between the LMC

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Theory, Modeling and Numerical Simulation

results (symbols) and the expression (Eq. 4) for the exact long-time limit effective diffusivity in the presence of traps and Eq. 6 (shown by lines). At short diffusion times, there are small deviations because the long-time limit effective diffusivity is then not appropriate (these deviations are exacerbated at larger degrees of trapping i.e. larger segregation factors). 1.5

ψO/ψO(x=0)

1.0

1

1

t=104 0.5 t=74 t=36 0

0

5

10

15

x

Figure 2. Oxygen concentration profiles at different in-diffusion times and for s = 10. Fig. 4 shows results for the oxygen concentration profiles at different out-diffusion times for s = 100 and a constant initial composition. Again it is seen that there is generally very good agreement between the LMC results (symbols) and the expression (Eq. 4) for the exact long-time limit effective diffusivity in the presence of traps and Eq. 7 (shown by lines). Again, there are small deviations at short diffusion times for the same reasons as given above. 6

1

O

1

O

ψ /ψ (x=0)

4

t=3600

2

t=2590 t=1270 0

0

10

20

30

40

50

x

Figure 3. Oxygen concentration profiles at different in-diffusion times and for s = 100.

Solid State Phenomena Vol. 139

39

1.0

0.8

ψO/ψO(t=0)

t=825

t=1325

0.6

t=1882 t=2516

0.4

0.2

0

0

5

10

15

x

Figure 4. Oxygen concentration profiles at different out-diffusion times and for s = 100.

Summary In this paper, we investigate oxygen in-diffusion and out-diffusion with respect to a cer-met composite where oxygen segregates at the interface between the metal matrix phase and the ceramic oxide phase. This phenomenological diffusion problem is treated by overlaying it with a finegrained lattice and making use of a Lattice Monte Carlo method. A little-known exact expression for the lattice-based effective diffusivity in the presence of random traps was also employed. It was shown that there is very good agreement for the oxygen concentration depth profiles between the Monte Carlo results and the exact expression. Acknowledgments We wish to thank the Australian Research Council for its support of this work.

References [1] E. Pippel, J. Woltersdorf, J. Genger and R. Kirchheim, Acta Mater., Vol. 48 (2000), p. 2571. [2] I.V. Belova and G.E. Murch: Solid State Phenomena, Vol. 129 (2007), p. 1. [3] I V. Belova, N. Muthubandara, G E. Murch, M Stasiek, A Oechsner: Solid State Phenomena, Vol. 129 (2007) p.111. [4] I.V. Belova, G.E. Murch, A. Oechsner, Defect and Diffusion Forum, Vol. 266, (2007) p. 29. [5] I. Kaur, Y. Mishin and W. Gust, Fundamentals of Grain and Interphase Boundary Diffusion, Wiley, Chichester, 1995. [6] J.W. Haus and K.W. Kehr, Physics Reports, Vol. 150 (1987), p. 263

Solid State Phenomena Vol. 139 (2008) pp 41-46 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.41

Effects of Supports on Hydrogen Adsorption on Pt Clusters K. Okazaki-Maeda1,a, Y. Morikawa2,b, S. Tanaka3,c, and M. Kohyama3,d 1

CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi 332-0012, Japan 2

ISIR, Osaka University, 8-1 Mihogaoka, Ibaraki 567-0047, Japan

3

UBIQEN, National Institute of Advanced Industrial Science and Technology, 1-8-31 Midorigaoka, Ikeda 563-8577, Japan a

[email protected], [email protected], [email protected], d

[email protected]

Keywords: Fuel Cell, Platinum, Graphene, Hydrogen, Adsorption, Pseudopotential method

Abstract. Pt nano-particles are supported on carbon materials at the electrode catalysts of protonexchange menbrane fuel cells. Pt nano-particles are desirable to be strongly adsorbed on carbon materials for high dispersion, although strong Pt-C interactions may affect the catalytic activity of small clusters. Thus we have examined H-atom absorption on Pt clusters supported or unsupported on graphene sheets, using first-principles calculations. For Pt-atom/graphene systems, a H atom is more weakly adsorbed than for a free Pt atom, and the H-Pt interaction becomes weaker if the interaction between a Pt atom and graphene becomes stronger. For the Ptn-cluster/graphene systems (n=2-4), the H-Pt interactions are also substantially changed from those for free Pt clusters. In the Pt clusters on graphene, the Pt-Pt distances are substantially changed associated with the electronicstructure changes by the Pt-C interactions. These structural and electronic changes in the Pt clusters as well as the presence of graphene itself seem to cause the changes in the absorption energies and preferential sites of H-atom absorption. Introduction Proton-exchange membrane fuel cells are attractive as power sources for mobile electronic equipments and zero-emission electrically powered vehicles [1, 2]. Nano-particles of Pt or Pt-alloys supported on carbon materials are used as electrode catalysts. At present, catalysts using Pt and Ptalloys are considered to be the best for hydrogen oxidation and oxygen reduction at low temperature [3]. Akita et al. observed nano-particles of the electrode catalysts by transmission electron microscopy [4]. They reported that the average diameter of particles on carbon materials is approximately 2.8 [nm]. Okamoto reported that Pt particles with diameter of a few [nm] have the same character of the hydrogen adsorption for the surface of the bulk using first-principles calculations [6]. In order to enhance the catalytic activity and to reduce the quantity of Pt, it is important to investigate small Pt clusters. Theoretical studies have been performed on the stability of Ptn clusters [7, 8] and on the hydrogen dissociation and adsorption on such small Pt clusters [9]. However, it should be noted that the effect of support for the catalytic properties may become stronger when the particle size of catalysts reduces. There are few theoretical studies about the effect of carbon supports. Therefore, we have examined the hydrogen adsorption on Ptcluster/carbon systems, using first-principles calculations based on density functional theory (DFT) [10,11]. Carbon black used as the carbon supports consists of many graphitic sheets at an atomic scale [5]. Thus we have treated the systems of Pt clusters on a graphene sheet as the model structures. Method All the calculations were carried out using the program package STATE (Simulation Tool for Atom TEchnology), which has been successfully applied to the Pt(111)/graphene [12] and Pt13/graphene [13] systems. We adopted the generalized gradient approximation in the DFT with the Perdew,

42

Theory, Modeling and Numerical Simulation

Table 1 Interaction energies between Pt and graphene, Eint, adsorption energies of H on Pt/graphene systems, Ead, and distances between H and Pt, dH-Pt. [a Reference [9]] Eint [eV]

Ead [eV]

dH-Pt [Å]

Atom

-

1.65 (1.76a)

1.53 (1.54a)

6H

1.45

0.89

1.59

T

1.97

0.80

1.58

B

2.17

0.53

1.61

V

8.01

0.18

1.70

Burke, and Ernzerhof formula [14] as the exchange-correlation energy functional. All pseudopotentials are generated from scalar relativistic [15] all-electron atomic calculations to include relativistic effects which are important for heavy elements like Pt. We constructed pseudopotentials by Vanderbilt's ultra-soft scheme [16] and by the norm-conserving scheme [17]. Two projectors are used for ultra-soft pseudopotentials. The cutoff energy for the wave function is 25 [Ry] and that for the augmentation charge is 225 [Ry]. A repeated slab model is used to simulate the graphene sheet. Each sheet was separated by a vacuum region of about 25 [Å]. Ptn clusters are introduced on one side of each sheet. And one H atom is adsorbed on the Ptn cluster. For structural optimization, clusters and the substrate sheet are allowed to relax. In the present work, we treat the 4×2√3 cell of the graphene sheet. The surface Brillouin zone is sampled by the 3×3 uniform meshe. We examined the interaction energy between Pt and graphene defined by Eint=(n×EPt+EC)-EPtn/C,

(1)

where EPtn/C, EPt, and EC are the total energies of the Ptn/graphene, a free Pt atom, and a graphene sheet, respectively. And we examined the adsorption energy of one H atom defined by Ead=(EPtn/C+1/2EH2)-EH-Ptn/C,

(2)

where EH2 and EH-Ptn/C are the total energies of a free H2 molecule and the Ptn/graphene system with the H-atom adsorption. Results Table 1 lists Eint, Ead, and the distance between Pt and H atoms, dH-Pt, for the Pt-atom/graphene systems. We considered four adsorption sites of a Pt atom on graphene; the hollow site around six C atoms (6H site), the top site above the C atom (T site), the bridge site between the neighbor C atoms (B site), and the vacancy site of one C atom (V site). When the interaction between a Pt atom and the graphene becomes stronger, the adsorption energy of H on the Pt/graphene system decreases and the H-Pt distance becomes longer. A larger portion of d electrons of the Pt atom seem to be concerned with the Pt-C interactions in the case of the stronger Pt-C interfacetions, which should be the reason of the weaker H-Pt interactions. Fig. 1 shows the projected density of states for a Pt atom adsorbed on the each site of graphene. The d states of the Pt atom hybridize with the π states of graphene. Therefore, the d states of the Pt atom extend and decrease near the Fermi level when the interaction between the Pt atom and graphene. Especially, for the V site, the d states of the Pt atom

Solid State Phenomena Vol. 139

43

Figure 1 Projected density of states for a Pt atom adsorbed on graphene; (a) 6H, (b) T, (c) B, (d) V sites. Table 2 Adsorption energies, Ead, and H-Pt distance, dH-Pt, for the free Ptn-clusters and the Ptn/graphene systems. Bridge, Edge, and Top in parenthesis indicate the adsorption sites of the H atom. Ptn/graphene

Free cluster Ead [eV]

dH-Pt [Å]

Ead [eV]

dH-Pt [Å]

Pt2

1.04 (Bridge)

1.70

1.09 (Edge)

1.58

Pt3

0.69 (Bridge)

1.70

1.24 (Top)

1.54

Pt4

0.84 (Bridge)

1.70

0.49 (Top)

1.57

strongly interact with the dangling bonds around the C-atom vacancy. It is thought that the extension of the d states of the Pt atom is a cause that weakens the H-Pt interaction. We have examined the stability of the Ptn/graphene systems [18]. Pt atoms on graphene form the dimer for two atoms, the triangular cluster for three atoms, and the tetrahedral cluster for four atoms. Therefore, we have dealt with hydrogen adsorption on the dimer, triangular, and tetrahedral clusters unsupported and supported on graphene. Table 2 lists Ead and dH-Pt for the free Ptn clusters and Ptn/graphene systems. The properties of the adsorption, such as the adsorption energy and adsorption site, for the Ptn/graphene systems are different from those for the free Ptn-clusters. The origins of these differences are the changes of the electronic structure and confgurations of the Ptnclusters. Figs. 2(a) and (b) show the stable configurations of the triangle Pt3 clusters unsupported and supported on graphene, respectively. And Figs. 2(c) and (d) show the optimized geometry of the H-Pt3 and H-Pt3/graphene systems. The H atom is stably adsorbed on the bridge site between

44

Theory, Modeling and Numerical Simulation

Figure 2 Stable configurations of the triangular Pt3 cluster unsupported (a) and supported (b) on graphene and the optimized configurations of H-Pt3 and H-Pt3/graphene systems. Numbers in figures are distance between atoms [Å]. Black, white, and gray circles indicate the C, Pt, and H atoms, respectively. two Pt atoms for the free Pt3 cluster. For the Pt3/graphene system, however, the H atom is stably adsorbed on the top site above the Pt atom. One of reasons is that the distance between Pt atoms has expanded from 2.46 [Å] to 2.63 [Å] by interacting with graphene. This change, however, cannot explain the change in the adsorption site of the H atom. We observed a substantial structural change associated with the H-adsorption on the top site of one Pt atom as shown in Fig. 2(d). The distance between the C atom and the Pt atom adsorbed the H atom becomes long from 2.11 [Å] to 2.55 [Å] and that between the C atom and the other Pt atoms becomes short from 2.26 [Å] to 2.16 [Å]. This should be the result of complex interference among the Pt-C, Pt-Pt and Pt-H interactuions. In any case, there seems to be a tendency that, in a cluster, a Pt atom with relatively weak interaction with graphene has stronger H-Pt interactions. For the tetrahedral Pt4 cluster, the stable configurations are shown in Figs. 3(a) and (b). The Pt4 cluster on graphene has three atoms interacting with the graphene and one atom not interacting with graphene. Several Pt-Pt distances become longer similarly to the case of the Pt3 cluster. The stable absorption site is changed from the bridge site into the top site of the atom not interacting with graphene. For the free Pt4 cluster, the adsorption energy on the bridge site is 0.84 [eV] and that on the top site is 0.14 [eV]. For the free Pt4 cluster of the same configuration with thar supported on graphene as in Fig. 3(b), the adsorption energy on the bridge site is 0.76 [eV] and that on the top site is 0.76 [eV]. Thus the structural change itself decreases the difference in the adsorption energies between the bridge site and the top site, and the presence of graphene itself seems to decrease the values of the absorption energies, and leads to the present results. Discussion We have investigated the effect of the support of the carbon materials on the properties of the hydrogen adsorption, such as the adsorption site and the adsorption energy, using first-principles pseudopotential method. We treated the graphene as the model of the carbon materials. For the Pt atom, we found that the interaction with graphene causes the decrease in the adsorption energy of the H atom. For the Ptn cluster (n=2~4), we observed substantial changes in the absorption energy and stable absorption site for the H absorption from those of the free clusters. For the clusters, we found that the Pt-Pt distances in the clusters become longer by the Pt-C interactions. These structural changes should be the primary reason of the changes in the the H absorption. However, as examined in the case of the Pt4 cluster, the structural change itself cannot explain the whole results. And the Pt-H interaction weakens the Pt-graphene interaction if the H atom is adsorbed on the Pt atom interacting with graphene, as shown in Figs. 2(b) and (d). As shown in Figs. 3(b) and (c), however, the Pt-C distances become shorter if the H atom is adsorbed on the Pt atom not directly interacting the graphene. It can be said that there is the tendency that H atoms prefer to be adsorbed on Pt atoms without interacting the graphene. The direct Pt-C interaction weakens the Pt-H interaction. However, the H-Pt interaction becomes stronger as the Pt-Pt distance becomes longer

Solid State Phenomena Vol. 139

45

Figure 3 Stable configurations for the free Pt4 cluster (a) and the Pt4/grapehen system(b) and the optimized geometry of the H-Pt4/graphene system (c). Numbers in figures are distance between atoms [Å]. Black, white, and gray circles indicate the C, Pt, and H atoms, respectively. by the interaction between the Pt cluster and graphene. The competition of these two influences decides the catalytic properties of the Pt nano-particles supported on the carbon material. In real electrode catalysts, the diameter of the Pt particles is a few nm [4]. The size of the cluster calculated in this work is very small. Therefore, we have to investigate the effect of carbon materials for the larger Pt clusters, such as Pt10, Pt13, and so on. And we treated graphene without the defects or edges as carbon supports. The real carbon supports, however, have any defects and edges [4, 5]. And there is the study about the stabilization of Pt clusters by substitutional boron dopans in carbon supports [19]. Therefore, we also have to investigate the effects of defects, edges, and dopants. In addition, we have to investigate the effects of the carbon supports on the dissociation of the H2 molecule in order to understand the mechanism of the reaction at the electrode in the proton-exchange membrane fuel cells. Acknowledgment This study was supported by a grant from Core Research Evolutional Science and Technology (CREST) of Japan Science and Technology Agency (JST), Japan. All calculations were carried out using the AIST Super Cluster in National Institute of Advanced Industrial Science and Technology (AIST), Japan. And thank Dr. T. Akita and Dr. Y. Maeda, for beneficial discussions for experiments. References [1] P. Costamagna and S. Srinivasan, J. Power Sources Vol. 102 (2001), p.253. [2] F. Preli, Fuel Cells Vol. 2 (2005), p.5. [3] P. Costamagna and S. Srinivasan, J. Power Sources Vol. 102 (2001), p.242. [4] T. Akita, A. Taniguchi, J. Maekawa, Z. Shiroma, K. Tanaka, M. Kohyama, and K. Yasuda, J. Power Sources Vol. 159 (2006), p.461. [5] M. Wissler, J. Power Sources Vol. 156 (2006), p.142. [6] Y. Okamoto, Chem. Phys, Lett. Vol. 407 (2006), p.209. [7] E. Aprá and A. Fortunell, J. Phys. Chem. A Vol. 107 (2003), p.2934. [8] L. Xiao and L. Wang, J. Phys. Chem. A Vol. 108 (2004), p.8605. [9] M. N. Huda and L. Kleinman, Phys. Rev. B Vol. 74 (2006), p.195407. [10] P. Hohenberg and W. Kohn, Phys. Rev. Vol. 136 (1964), p.B864. [11] W. Kohn and L. J. Smith, Phys. Rev. Vol. 140 (1965), p.A1133. [12] Y. Okamoto, Chem. Phys. Lett. Vol. 407 (2005), p.354.

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Theory, Modeling and Numerical Simulation

[13] Y. Okamoto, Chem. Phys. Lett. Vol. 420 (2006), p.382. [14] J. P. Perder, K. Burke, M Ernzerhof, Phys. Rev. Lett. Vol. 77 (1996), p.3865. [15] D. D. Koelling and B. N. Harmon, J. Phys. C Vol. 10 (1977), p.3107. [16] D. Vanderbilt, Phys. Rev. B Vol. 41 (1990), p.7892. [17] N. Troullier and J. L. Martins, Phys Rev. B Vol. 43 (1991), p.1993. [18] K. Okazaki-Maeda, S. Yamakawa, Y. Morikawa, T. Akita, S. Tanaka, S. Hyodo, and M. Kohyama, to be published in J. Phys. Conference Series. [19] C. K. Acharya and C. H. Turner, J. Phys. Chem. B Vol. 110 (2006), p.17706.

Solid State Phenomena Vol. 139 (2008) pp 47-52 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.47

First-principles calculations of Pd/Au(100) interfaces with adsorbates Shingo Tanaka1,a, Noboru Taguchi2,b, Tomoki Akita1,c, Fuminobu Hori2,d, Masanori Kohyama1,e 1

Research Institute for Ubiquitous Energy Devices, National Institute of Advanced Industrial Science and Technology (AIST), 1-8-31 Midorigaoka, Ikeda, Osaka 563-8577, Japan 2

Department of Materials Science, Osaka Pref. Univ., 1-2 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

a

[email protected], [email protected], [email protected] d [email protected], [email protected]

Keywords: First-Principles Calculation, Interface, Reactivity, Au, Pd

Abstract. Atomic and electronic structures of H-adsorbed Pd overlayers on Au(100) substrates have been studied by first-principles calculations. The geometric strain effects change the electronic structure and local reactivity of the surface. The lattice strained Pd overlayers on Au surfaces have larger adsorption energies for atomic hydrogen than the unstrained Pd slabs. Adsorption energies for several adsorption sites on the models with different numbers of Pd overlayers have been analyzed from the viewpoints of strains and H-Pd and H-substrate interactions. Introduction The study of noble bimetallic nanoparticles has been of great interest for the catalysts. The oxide supported Au-Pd bimetallic particles show high catalytic activities [1,2] Au-Pd bimetallic nanoparticles containing the surfactant, sodium dodecyl sulfate (SDS), prepared by the sonochemical technique show apparent core-shell structures confirmed by various analyses [3-5]. The Au-Pd core-shell nanoparticles have also higher catalytic activities for hydrogenation of 4-pentenoic acid than monometallic Au or Pd nanoparticles or pure Pd black. The high catalytic activity for the hydrogenation of cyclohexene of similar Au-Pd particles on Fe2O3 support is also reported [6]. Recently, the fine structure analyses of Au-Pd core-shell nanoparticles by analytical TEM [7] show important facts: 1) the Au-core and Pd-shell structure without apparent alloy regions and with uniform thickness, ~1nm, of Pd shells confirmed by annular dark field scanning TEM (ADF-STEM) and energy dispersive X-ray spectroscopy (EDS) line analyses, 2) the coherent Pd/Au interface via the epitaxial growth of Pd shells observed by high-resolution TEM (HRTEM) and ADF-STEM, and 3) the Pd lattice is strained (about 5% expansion) and coincides with the Au lattice. Also, the expansion of Pd lattice is indicated by a positron annihilation experiment [5]. In the Pd/Au slabs, there are similar features of the epitaxial growth and the stable Pd lattice strain. Pseudomorphic Pd overlayers on Au(100) [8] and Au(111) [9] substrates under electrochemical conditions can grow up to 6-10 layers and 4 layers, respectively. On the other hand, the Au-Pd alloy is formed in the vacuum condition [10]. The Pd/Au slabs show high reactivity of hydrogen adsorption/desorption [11]. In the Au-Pd systems, the surface slabs and the core-shell

48

Theory, Modeling and Numerical Simulation

nanoparticles have a lot of physical properties in common. And the atomic structure of the core-shell interface in observed nanoparticles seems to be similar to that of the interface of Pd/Au slab. Thus, the Pd/Au slab is an interesting system not only as an advanced catalyst but also as a model structure for the atomic and electronic structure and catalytic activities of the core-shell nanoparticles. First-principles studies of the local reactivity of Pd/Au(100) and (111) overlayer systems have been performed for atomic hydrogen and carbon mono-oxide adsorbates [12,13]. The adsorption energies of Pd/Au overlayers are larger than that of a pure Au or Pd surface slab and that of a lattice strained Pd surface slab which is expanded to the Au lattice. And the adsorption energies show a maximum at two Pd overlayers on Au(100) and (111) surfaces. The authors conclude that the reactivity is determined by both the lattice expansion of the Pd layers on Au surfaces (lattice strain effects) and the interaction between the Pd layers and Au substrates (substrate interaction effects). Most of the results can be explained by the d-band model [13] but the indirect interaction of adsorbates with the Au substrate and the strong hybridization between adsorbates and Pd layers should also be considered. In this paper, we have performed first-principles calculations of Pd/Au(100)-(2x2) slabs with adsorbates by the projector augmented-wave (PAW) method. We determine the stable geometric configurations, their electronic structures and the adsorption energies as the probe of the local reactivity taking into account the lattice strain effect, the substrate interaction effect and the alloying effect. Theoretical method We use the first-principles PAW method [15-17] based on density functional theory within the generalized gradient approximation [18]. The PAW method is a superior method with both the efficiency of the pseudopotential method and the accuracy of the all-electron method. We use the program code QMAS (Quantum MAterials Simulator) [19]. A plane-wave cutoff energy of maximum 40 Ry is selected based on the tests of total energy convergence. In self-consistent calculations, we use maximum 16 sampling k-points in the irreducible Brillouine zone of (2x2) supercell as mentioned below. The present supercell contains a slab of five Au layers with up to three Pd overlayers and up to two adsorbates. The symmetric condition of the supercell is Pmm2 (#25). The Pd/Au(100) slabs are relaxed with the constraints of the Au lattice parameter along the (100) plane. One hydrogen atom per (2x2) unit adsorbs on an on-top (T1), a bridge (B2) or a hollow (H4) site of the (100) surface. After the adsorption, we again perform the full geometric optimization. Then we obtain the final atomic structures and their electronic structures and estimate the adsorption energies for respective slabs with the adsorbates. Results and Discussion Figure 1 shows the stable atomic structures for the Pd/Au interfaces with the adsorbate hydrogen. The adsorbed hydrogen atom induces the specific structures near Pd/Au interfaces. In the one Pd overlayer models, the H-Pd distance on T1 site is smaller than that on B2 or H4 site while the H-Pd interlayer distance on H4 site is smaller than that on T1 or B2 site. In the B2 site model, the strained Pd layer has almost the same distance and interlayer distance of H-Pd as those in the unstrained one, caused by the relaxation of surface Pd atoms. The distances between the surface Pd atoms interacting with the hydrogen atom are 0.279 nm in the strained layer, whereas the distance without hydrogen is 0.294 nm. In the strained layer, therefore, the Pd atoms relax toward the structure of the unstrained layer for the H adsorption. In the H4 site model, the Pd-Pd distances of the strained layer is also the same as those of the unstrained one due to the large surface Pd relaxation of the strained layer. In the two Pd overlayer models, there are some differences for the atomic structures. Upon

Solid State Phenomena Vol. 139

49

Fig. 1 Stable atomic structures of (a) on-top site, (b) bridge site and (c) hollow site for the one Pd overlayer/Au(100)-(2x2) interfaces with the adsobated hydrogen for (001) direction. Light and dark gray and black circles indicate Au, Pd and hydrogen atoms, respectively.

increasing the number of Pd overlayers from one to two, the H-Pd interlayer distance decreases from 0.0473 nm to 0.0254 nm in the H4 site model whereas the H-Pd interatomic distance at the top Pd layer is almost a similar value (0.201 nm). This decrease is strongly related to the geometric structure of the subsurface. In the H4 site model, the atom species of subsurface of the two Pd overlayer slab is Pd while that of the one Pd overlayer slab is gold, which is well known to be an inert noble metal. The adsorption of the hydrogen atom on the H4 site leads to interactions with the subsurface atom, and attractive interactions between the H and subsurface Pd atoms, in contrast to repulsive ones between the H and subsurface Au atoms, is the origin of the decrease in the H-Pd interlayer distance in the two Pd overlayer case. The H-Pd interatomic distance on the B2 site, 0.172 nm, is shorter than that on H4 site, 0.201 nm. Thus, one can expect that the adsorbed hydrogen atom on B2 site forms rather strong hybridization for the surface Pd atoms than the H4 site. The geometric structures of the three Pd overlayer models are qualitatively similar to those of the two overlayer models although some quantitative changes occur. These geometric relaxations should change the electronic structures of the adsorbates-substrate and Pd/Au interfaces. Figure 2 shows the charge density redistributions, obtained by the difference between the charge density of the slab with adsorbates and the superposition of the slab charge density and the adsorbate charge density with the same atomic positions. One can clearly see that accumulation of the charge density around the hydrogen and depletion near the surface Pd atoms interacting with hydrogen. There exists the depletion region around the subsurface Pd atom just below the adsorbed hydrogen atom in the two Pd overlayer case while no depletion region around the subsurface Au atom in the one Pd overlayer one. This is caused by the charge transfer from Pd to hydrogen. This result indicates that the subsurface interaction effects are important for the

50

Theory, Modeling and Numerical Simulation

reactivity of hydrogen adsorption.

Fig. 2 Charge density redistribution of (a) one Pd overlayer and (b) two Pd overlayer cases. The dark and light transparent region indicate isosurfaces for accumulation and depletion region of the charge density, respectively. The adsorption energy of hydrogen atom is defined by 1 E ads = E int +H − (E int + E H 2 ), (1) 2 where E ads is the adsorption energy, E int + H and E int the total energies of the H-adsorbed Pd/Au interface and no adsorbed interface, E H2 the hydrogen binding energy, respectively. From the definition, a negative adsorption energy denotes the energetically favourable condition for the hydrogen adsorption. The two Pd overlayer models have larger adsorption energies than the other models. This is consistent with other calculations [12,13]. The Au(100) surface slab, no Pd overlayer case, is inert for the atomic hydrogen, so that the value of adsorption energy should be positive, which means the atomic hydrogen atom has less possibility to adsorb. On the other hand, the unstrained Pd(100) surface slab has strong reactivity for atomic hydrogen, which leads to a large adsorption energy. The strained Pd overlayer on Au(100) systems has larger adsorption energies than the unstrained and strained Pd slab. For the one Pd overlayer models, larger adsorption energies are obtained for the B2 and H4 sites than that for the T1 site, caused by the larger number of coordination for direct interactions. The adsorption energy of the B2 site is slightly larger than that of the H4 site, which is important evidence of substrate interaction effects, consistent with

Table 1 Adsorption energy of atomic hydrogen for the Pd/Au(100)-(2x2) interfaces. The unit is eV. pure Au

1-layer

2-layer

3-layer

Pd@Au

pure Pd

on-top (T1)

0.28

0.06

0.01

0.05

0.07

0.02

Bridge(B2)

0.17

-0.48

-0.52

-0.46

-0.42

-0.40

Hollow(H4)

0.26

-0.44

-0.51

-0.45

-0.41

-0.44

the electronic structure analyses. The hydrogen atom located on the H4 site should have some direct interactions with the inert subsurface Au atom, which may cause the reduction in the adsorption energy as compared with the B2 site adsorption. Moreover, the adsorption energies for the B2 and

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H4 sites are almost degenerated in the two Pd overlayer models. This is strongly related to the competition of energy cost of relaxation for the surface Pd atoms and the interaction between hydrogen and Pd atoms. In the B2 site model, the adsorbed hydrogen atom directly and strongly interacts with the two surface Pd atoms and no indirect repulsive interaction from the subsurface Pd layer. In the H4 site model, on the other hand, the adsorbed hydrogen atom interacts with the four surface Pd atoms plus the one subsurface Pd atom. As a result, the twofold coordination on the B2 site model is comparable in the adsorption energy to the semi-fivefold on the H4 site model. In the three Pd overlayer models, the adsorption energies gradually decrease, which approach to the energy of the unstrained Pd slab. These results encourage a concept that the lattice strained effects and subsurface interaction effects are important for the reactivity for adsorbates. Summary We have performed the atomic and electronic structures of Pd/Au(100) overlayer systems with adsorbed hydrogen by first-principles PAW calculations. The geometric strain strongly affects the electronic structure and local reactivity. The charge density around the adsorbate hydrogen atom is increased while that around the interacted Pd atoms is decreased, due to the charge transfer from Pd to hydrogen. The lattice strained Pd overlayers on Au(100) have larger adsorption energies for the hydrogen than the unstrained Pd slabs. And the number of Pd overlayers is important to determine the adsorption energy. The adsorption energies show the maximum at the two Pd overlayer models, which can be explained by the lattice strain effect and the subsurface interaction effect. Acknowledgement This work was supported by the Japan Society for the Promotion of Science (JSPS-Grant-in-Aid for Scientific Research (B) 17360314). The authors are grateful to Drs. K. Okazaki (JST), A. Iwase, T. Kojima (Osaka Pref. Univ.), S. Ishibashi, T. Tamura (AIST), and Ms. H. Takatani for their valuable comments and fruitful discussions. The figures were drawing by VESTA (K. Momma and F. Izumi, Commission on Crystallogr. Comput., IUCr Newslett., Vol. 7 (2006), P. 106). References [1] B.E. Solsona, J.K. Edwards, P. Landon, A.F. Carley, A. Herzing, C.J. Kiely and G.J. Hutchings: Chem. Mater. Vol. 18 (2006), p. 2689. [2] D.I. Enache, J.K. Edwards, P. Landon, B. Solsona-Espriu, A.F. Carley, A.A. Herzing, M. Watanabe, C.J. Kiely, D.W. Knight and G.J. Hutchings: Science. Vol. 311 (2006), p. 362. [3] Y. Mizukoshi, T. Fujimoto, Y. Nagata, R. Oshima and Y. Maeda: J. Phys. Chem. B . Vol. 104 (2000), p. 6028. [4] H. Takatani, F. Hori, M. Nakanishi and R. Oshima: Mater. Sci. Forum. Vols. 445-446 (2004), p. 192. [5] F. Hori, T. Kojima, S. Tanaka, T. Akita, T. Iwai, T. Onitsuka, N. Taguchi and A. Iwase: phys. stat. sol. (c) Vol. 10 (2007), p. 3895. [6] H. Nitani, M. Yuya, T. Ono, T. Nakagawa, S. Seino, K. Okitsu, Y. Mizukoshi, S. Emura and T. A. Yamamoto: J. Nanoparticle Res. Vol. 8 (2006), p. 951. [7] T. Akita, T. Hiroki, S. Tanaka, T. Kojima, M. Kohyama, A. Iwase, F. Hori: Catal. Today (2008), in press. [8] A.L.N. Pinheiro, M.S. Zei, M.F. Luo and G. Ertl: Surf. Sci. Vol. 600 (2006), p. 641. [9] L.A. Kibler, M. Kleinert, R. Randler and D.M. Kolb: Surf. Sci. Vol. 443 (1999), p. 19.

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[10] C.J. Baddeley, R.M. Ormerod, A.W. Stephenson and R.M. Lambert: J. Phys. Chem. Vol. 99 (1995), p. 5146. [11] H. Naohara, S. Ye and K. Uosaki: J. Electroanal. Chem. Vol. 500 (2001), p. 435. [12] A. Roudgar and A. Gross: Phys. Rev. B Vol. 67 (2003), p. 033409. [13] A. Roudgar and A. Gross: J. Electroanal. Chem. Vol. 548 (2003), p. 121. [14] B. Hammer and J.K. NØrskov: Surf. Sci. Vol. 343 (1995), p. 211. [15] P. Blöchl: Phys. Rev. B Vol. 50 (1994), p. 17953. [16] N.A.W. Holzwarth, G.E. Matthews, R.B. Dunning, A.R. Tackett, and Y. Zeng: Phys. Rev. B Vol. 55 (1997), p. 2005. [17] G. Kresse and D. Joubert: Phys. Rev. B Vol. 59 (1996), p. 1758. [18] J.P. Perdew, K. Burke and M. Ernzerhof: Phys. Rev. Lett. Vol. 77 (1996), p. 3865. [19] S. Ishibashi, T. Tamura, S. Tanaka, M. Kohyama and K. Terakura: unpublished.

Solid State Phenomena Vol. 139 (2008) pp 53-58 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.53

In-Plane Rotated Crystal Structure in Continuous Growth of Bismuth Cuprate Superconducting Film Satoru Kaneko1, a, Kensuke Akiyama1, b, Takeshi Ito1, c, Yasuo Hirabayashi1, d, Hiroshi Funakubo2, e and Mamoru Yoshimoto2, f 1

Kanagawa Industrial Technology Research Institute, Kanagawa Prefectrual Government, 705-1 Shimo-Imaizumi, Ebina, Kanagawa 243-0435 JAPAN 2

Department of Innovative and Engineered Materials, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku, Yokohama, Kanagawa 226-8502, JAPAN

a

[email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

d

Keywords: oxide superconductor, epitaxial growth, structural modulation, x-ray diffraction, reciprocal space mapping, pulsed laser deposition, bismuth cuprate

Abstract. Bismuth cuprate superconductor has a unique structure called a structural modulation (supercell, SC) consisting of modulated several unit cells. Strain induced by multilayered structure increases the intensity of SC modulation, while an oxygen deficient sample shows expansion of SC size. In this study, as opposed to the multilayer strain, by preparing samples with thick film thicknesses the effect of strain on crystal structure was investigated including SC structure. Epitaxial growth was verified by x-ray diffraction, and the thicker film showed other epitaxial phase rotated 32° around the surface normal with respect to the initial epitaxial phase. The SC size estimated by x-ray reciprocal space mapping was double the size of the initial epitaxial phase. Interestingly, the initial epitaxial phase became a dominant structure after further deposition. In order to evaluate the different SC size and SC modulation, a new index related with an incline of the modulation vector was proposed. Introduction Bismuth cuprate superconducter has a long periodic structure besides its unit cell, called structural modulation or supercell (SC)[1,2,3,4]. Literally the long periodic structure consists of modulated several unit cells through the whole crystal. Such a structure was first observed by electron microscopy, which shows the SC structure with fourfold satellite peaks around the main peaks. This unique structure has been investigated to find the origin of SC structure with great efforts; intercalation carried out by aliovalent substitution [5,6], or oxygen doping by annealing or by high-pressure oxidization, showing that oxygen content (extra oxygen) and induced strain affect the SC structure. Aliovalent substitution replaces cations by different cations with different valence and radius, which affects both oxygen doping by different valence, and in strain by different radius. To emphasize the effect of strain, we investigated SC structure in multilayered structure [7]. Unlike aliovalent substitution, multilayered structure induces only strain into crystal structure; decreasing thickness of bilayer must induce strong strain in multilayers. SC structure in multilayered samples showed almost the same SC size, however distortion of unit cells forming SC structure increased with increasing strain. On the other hand, Bi2Sr2Ca1Cu2OX (Bi-2212) film annealed in oxygen deficient atmosphere showed remarkable expansion in the SC size [8,9]. Besides commonly evaluated modulation size (SC size), we evaluated modulation amplitude indicating how strongly unit cells are distorted from the original sites to form the SC structure. While extra oxygen strongly affects the size of SC structure, strain induced by multilayer mainly affects modulation amplitude; how strongly unit cells are distorted to form the SC structure.

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Multilayer induces strong strain in its crystal structure. As opposed to the multilayered structure, a single film must release such strain as the film continuously grows. In this study, in order to investigate the effect of strain on SC structure in relatively thick films (~5,600 and 12,000 Å), single Bi-2212 film continuously grew on a MgO substrate by pulsed laser deposition (PLD) and the crystal structures were investigated by using several methods with x-ray diffraction to be compared with a reference film. Another phase of Bi-2212 became the dominant structure rotated by 32° around surface normal with respect to the initial epitaxial Bi-2212, which showed double size of its structural modulation. The angle of 32° was described using two lattice planes as coincidence sites lattice structure (CSL)[10,11]. Interestingly, the SC size in the thick film was expanded almost the double the size of the reference film. In the case of SC structure with different sizes, modulation amplitude cannot describe the distortion in crystal. In order to evaluate such a SC structure, we proposed a new index, which is related to an incline of the modulation vector. Experiments Figure 1 Pulsed Laser Deposition. 256 nm from Q-switched YAG laser irradiates MgO target. While flash ramp is modulated with 10 Hz, Q-switch is synchronized with 2 Hz.

Bi-2212 single layer and mltilayered films were prepared on MgO(100) substrates by PLD using “slower Q-switched Nd:YAG laser” [12] at a substrate temperature of 750°C, with a substrate-target distance of 40 mm, as shown in fig.1. The oxygen pressure during the deposition was 200 mTorr. The deposition rate was estimated by x-ray reflectivity (XRR) on the reference film. Single layer of Bi-2212 and multilayered (Bi-2212/Bi-2201)6 (six bilayers) were prepared on MgO substrates under the same conditions. Film thicknesses were about 1,300, 5,600 and 12,000 Å for single layer films, and those of multilayered films were estimated by XRR, and the crystal structure including SC structure were compared with the reference film of 1,300 Å thickness. Epitaxial growth was verified by θ 2θ and φ scan, grazing incidence XRD and x-ray reciprocal space mapping Figure 2 Reciprocal space mapping. XRSM around (XRSM) were taken on cross section (b Bi-2212(0020) main peak shows two satellite peaks x c plane) and plan view (a x b plane) to (SC peaks). Qx and Qy are along the b* and c* axes in observe SC peaks generated by structural reciprocal lattice units (r.l.u.). modulation. The details about cross section and plan view XRSM can be found elsewhere [13]. Grazing incidence diffraction (in-plane θ 2θ) observed SC peaks as well as the in-plane main peaks of Bi-2212(200).

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Results and Discussions Epitaxial growth was verified by θ - 2θ and φ scan on each film. XRSM of single Bi-2212 with 1,300 Å of film thickness as a reference showed asymmetric peaks around the main peak of Bi-2212(0020) on b* x c* plane in reciprocal space, as shown in Fig.2. The ordinate is along the surface normal (c*-axis), and the abscissa lies along in-plane direction (b*-axis). The distance between the main and satellite peaks allows one to estimate the size of the SC structure. ∆Qx along b*-axis is the reciprocal

Figure 3 Pole Figure using Bi-2212(115) peaks. (a) Twelve peaks were observed on Bi-2212 film with 5,600 Å of thickness. The four peaks (phase A,φ = 0, 90, 180 and 270°) were the initial crystal arrangement on the reference film with 1,200 Å of thickness. The other eight peaks

observed at ±32°from the Phase A. (b) Phase A becomes dominant on the film with 12,000 Å of thickness. of real SC size along the b-axis (SCb), and ∆Qy along c-axis (SCc). On the reference film, SCb and SCc were estimated to be 4.9b and 1.1c, respectively. An asymmetric peak can be simulated on crystal model modulated by sawtooth function instead of sinus function, which produces symmetric peaks around the main peak[14]. The sawtooth function for a simple crystal model is  kr  w(r) = A  + A0 , λ where k is the wave vector defining the direction, the wave length, A, is the amplitude of wave vector, and A0 is a constant displacement. The XRD peak ratio of SC to main peaks, RSC/M, is related to the amplitude A, which was estimated to be ~ 0.4 Å by comparing XRSM simulation with experimental RSC/M on the reference film. In order to emphasize the effect of strain, multilayered samples of (Bi-22124/Bi-22013)6 and (Bi-22122/Bi-22012)6 were prepared and investigated by x-ray diffraction. While the SC size slightly changed in the multilayers, the peak ratio, RSC/M, was increased with decreasing the bilayer period. The induced strain affects the modulation amplitude rather than SC size. Annealing samples in deficient oxygen atmosphere resulted in expansion of SC size. While extra oxygen strongly affects the size of SC structure, strain induced by multilayer mainly affects the modulation amplitude. Contrary to the multilayered structure, strain will be released in thick film. Bi-2212 film continuously grew on MgO (001) substrate, and the crystal structure of single films with film

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thickness of 5,600 and 12,000 Å were compared with the reference film with thickness of 1,300 Å. With increasing film thickness, another phase of Bi-2212 grew with in-plane rotation. Figure 3 is the pole figure using Bi-2212(115) peak on film with thickness of 5,600 Å, showing 12 peaks instead of four peaks on the reference film. The four peaks were initial four peaks at φ = 0, 90, 180 and 270° with relation of MgO[100] parallel to Bi-2212[100] (phase A), and the other eight peaks (phase B) were observed at ±32° from phase A; φ = 32, 58, 122, 148° ··· , as shown in fig.3(a). Interestingly, as film continuously grew the phase A (φ = 0, 90, 180 and 270°) became dominant again the same as the

Figure 4 XRSM on Bi-2212 with 5,600 Å of thickness. (a) XRSM around the main peak of Bi-2212(0020) on phase A (φ = 0°), and (b) on phase B (φ = 32°). SC peaks on the phase A shows four SC peaks. The weak SC peaks are satellite peaks from the phase A, and the other two peaks are shadow peaks from the phase B. reference film with the thickness of ~1,300 Å (fig.3(b)). The coincidence site lattice (CSL) [10,11] defines the domain coherent strain, e, as m2 + n 2b − k 2 + l2 a e=2 , m2 + n 2b + k 2 + l2 a where a and b are lattice constants of each unit cells, and k and l defines the translation vector in its crystal as Ta = ka1 + la2 and m and n as Tb = mb1 + nb2 . Taking (k, l) and (m, n) to be (3, 3) and (1, 4), the domain coherent strain takes e = 0.023, and the angle of two translation vectors is ~31°, which was good agreement with the angle between phase A and B. XRSM was also taken around each single layer Bi-2212(2020) to estimate SC size and evaluate peak intensity ratio, RSC/M, as shown in fig. 4. Since thick Bi-2212 film showed two phases, XRSM was taken on phase A (φ = 0°) and phase B (φ = 32°). Fig.4(a) taken at φ = 0° (phase A) on the film with 5,600 Å of thickness, shows four peaks below the main peak. The pair of outside SC peaks were shadows from the phase B at φ = 32°, and the inner two weak SC peaks were real satellite peaks generated by the phase A. The SC size in the phase A was estimated to be 50 Å, which is double the SC size of the reference film. Figure 5 is a XRSM around main (0020) peak on Bi-2212 film with 12,000 Å of thickness at φ = 0°. As shown in fig.5, two SC peaks were observed on the XRSM, however, weak SC peaks still existed

Figure 5 XRSM on Bi-2212 film with 12,000 Å of thickness. Two SC peaks were observed around Bi-2212(0020) main peak.

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with the same intensity as sample with 5,600 Å thickness (fig.4(a)). As in the same manner as the reference film, peak ratio, RSC/M, was used to estimate the modulation amplitude. With the ratio of RSC/M, a modulation amplitude is estimated by the simulation of x-ray reciprocal space on a simple SC model modulated by sawtooth wave function [7, 11]. The modulation amplitude, A, indicates how strongly unit cells are distorted to form the SC structure. The RSC/M of SC peak of 0.032 indicated A of 0.4 Å, which was the same as the single layered film with the thickness of 1,300 Å. For phase A, the modulation amplitude was estimated to be 0.4 Å, which was equivalent to the reference film, however, the modulation period was double the SC size of the reference film. Modulation amplitude, A, can not describe how strongly the crystal is distorted when the SC size is varied. In order to evaluate these distortions, we propose a new index Φ, related to an incline of the modulation vector as A R Φ∝ ≈ SC / M , SC SC where |SC| is the size of SC defined as SCb + SCc . Φ can qualitatively express how strongly the crystal is distorted. Table 1 shows the summary for lattice parameters of lattice constants, SCb, SCc and peak ratio of SC to main, RSC/M together with ΦN (normalized Φ) on single film including the reference film and multilayered films. Unit cells in phase A intervening between substrate and phase B were extended and contracted by the substrate and the following phase B, dispersing the intrinsic strain in crystal, and resulting in expansion of SC size in phase A. 2

2

Table 1 Summary of structure.

single 12,000 Å single 5,600 Å

phase A phase B

reference (Bi-22124/Bi-22013)6 (Bi-22122/Bi-22012)6

b (Å) 5.42 5.43 5.40 5.43 5.43 5.41

c (Å) 30.83 30.89 30.83 30.88 30.91 31.09

SCb 26.4 50.1 27.3 26.8 26.7 26.1

SCc 34.7 34.9 34.8 34.9 35.5 36.4

RSC/M 0.026 0.032 0.022 0.035 0.11 0.18

ΦN 0.89 0.67 0.78 1.0 1.6 2.0

Releasing misfit strain in phase B contracted unit cells along both b and c axes, more than in the multilayered structure, and modulation amplitude was larger than the other samples. Taking the distortion index ΦN into account, phase B was indeed the less distorted structure. However, further growth of Bi-2212 turned out increase in its modulation. Summary In summary, the strain induced by multilayered structure increases the intense of SC modulation, while an oxygen deficient sample shows expansion of SC size. The strain affects how strongly unit cells are distorted to form SC structure, and extra oxygen dominates the SC size. The misfit strain with substrate must be released with continuous growth of single layer film. As the film thickness increased, another epitaxial phase became dominant with 32° rotated around the surface normal with respect to the initial epitaxial phase. Taking the translation vector as (3, 3) and (1, 4), domain coherent strain yields 0.023, and the angle of the two vectors was in good agreement with the rotation angle in the thick film. The new index, ΦN, was proposed to evaluate the SC structure in different SC size.

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Acknowledgement We would like to thank Y. Sato of Kanagawa Industrial Technology Center for maintaining the PLD system. This research was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science under Contract No. 19656170. References [1] Y. Matsui, H. Maeda, Y. Tanaka and S. Horiuchi: Jpn. J. Appl. Phys. Vol.27 (1988), p. L372 [2] Y. Bando, T. Kijima, Y. Kitami, J. Tanaka, F. Izumi and M. Yokoyama: Jpn. J. Appl. Phys. Vol.27 (1988), p. L358 [3] Y. Hirotsu, O. Tomioka, T. Ohkubo, N. Yamamoto, Y. Nakamura, S. Nagakura, T. Komatsu and K. Matsushita: Jpn. J. Appl. Phys. Vol. 27 (1988) p. L1869 [4] V. Petricek, Y. Gao, P. Lee and P. Coppens: Phys. Rev. B Vol.42 (1990), p. 387 [5] M. Zhiqiang, F. Chenggao, S. Lei, Y. Zhen, Y. Li, W. Yu and Z. Yuheng: Phys. Rev. B Vol. 47 (1993), p. 14467 [6] M. Zhiqiang, X. Gaojie, Z. Shuyuan, T. Shun, L. Bin, T. Mingliang, F. Chenggao, X. Cunyi and Z. Yuheng: Phys. Rev. B Vol. 55 (1997) , p. 9130 [7] S. Kaneko, K. Akiyama, H. Funakubo, M. Yoshimoto: Phys. Rev. B Vol. 74 (2006), p. 054503 [8] S. Kaneko, K. Akiyama, Y. Shimizu, Y. Hirabayashi, K. Saito, T. Kimura, H. Funakubo, M. Yoshimoto and S. Ohya: Jpn. J. Appl. Phys. Vol. 44 (2005), p. 156 [9] S. Kaneko, K. Akiyama, M. Mitsuhashi, Y. Hirabayashi, S. Ohya, K. Seo, H. Funakubo, A. Matsuda and M. Yoshimoto: Europhysics Lett. Vol. 71 (2005), p. 686 [10] R. W. Balluffi, A. Brokman and A. H. King: Acta Metall. Vol. 30 (1982), p. 1453 [11] D. M. Hwang, T. S. Ravi, R. Ramesh, S. Chan, C. Y. Chen, L. Nazar, X. D. Wu, A. Inam and Venkatesan: Appl. Phys. Lett. Vol. 57 (1990), p. 1690 [12] S. Kaneko, Y. Shimizu, S. Ohya: Jpn. J. Appl. Phys. Vol. 40 (2001), p. 4870 [13] S. Kaneko, K. Akiyama, Y. Shimizu, H. Yuasa, Y. Hirabayashi, S. Ohya, K. Saito, H. Funakubo and M. Yoshimoto: J. Appl. Phys. Vol. 97 (2005), p. 103904 [14] S. Kaneko, Y. Shimizu, K. Akiyama, T. Ito, M. Mitsuhashi, S. Ohya, K. Saito, H. Funakubo and M. Yoshimoto: Appl. Phys. Lett. Vol. 85 (2004), p. 2301

Solid State Phenomena Vol. 139 (2008) pp 59-64 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.59

Dynamical Interaction Between Thermally Activated Glide of Screw Dislocation and Self-interstitial Clusters in bcc Fe X.-Y. Liua and S. B. Binerb Ames Laboratory (USDOE), Materials and Engineering Physics, Iowa State University, Ames, IA 50011 a

[email protected], [email protected]

Keywords: Molecular dynamics, screw dislocations, self-interstitial clusters, bcc iron.

Abstract. Constant strain rate molecular dynamics simulations under the modified boundary conditions were performed to elucidate the interaction processes between the kink motion of screw dislocation and the glissile self-interstitial atom cluster loops in bcc Fe by using an EAM potential for Fe fitted to ab initio forces. The junction formation and the helical dislocation mechanisms were identified as two possible interaction processes. In the junction mechanism, the initial Burgers vector 1/2 of the cluster loop was transformed into . In the helical dislocation mechanism first the absorption, followed by the formation of the helical dislocation and the emission of the cluster loop through Hirsch mechanism was observed. Substantial hardening was seen as result of the interactions. The stress-strain plots obtained for different loop sizes, temperature and strain rates were used to estimate the strengthening factors. Introduction It has been recognized that at low irradiation temperatures, high concentration of defect clusters are formed in irradiated metals, causing pronounced hardening and embrittlement effect [1]. The change in the plastic behavior is generally attributed to the dynamical interactions between the dislocation motion and the defect clusters. At low temperatures, slip in the bcc metals such as Fe is mainly controlled by thermally activated screw dislocations. The point defect cluster hardening to screw dislocations in irradiated metals was studied using the elasticity theory of dislocations [2]. Although this early study was concentrated on fcc metals, however, it established several important concepts regarding to the short-range interactions between the dislocation and the SIA clusters. Molecular dynamics (MD) simulations provide a powerful tool to characterize the specific interaction mechanisms at the atomistic level. We used the recently developed EAM potential for Fe [3] that is fitted to ab initio forces, to study such interactions in bcc Fe. This potential has been used recently to study the thermally activated motion of pure screw dislocations via double kink mechanisms [4,5]. The major advantage of such a potential is that it correctly predicted for the first time the symmetrical relaxed core structure of the screw dislocations, which is in agreement with ab initio calculations. In bcc Fe, the SIA clusters form highly mobile prismatic loops, migrating in their crowdion axis. The Burgers vector of the loop b is parallel to the crowdion axis. The loops with b of a/2 and a are found in irradiated Fe. However, the origin of a loops, which have higher structural energies than a/2 loops, is still unclear [6]. In this study, we focused on the interaction of a single a/2 loop with a ½[111] screw dislocation. Methods The simulation cell was set up with x, y, z axis in its crystallographic directions [ 11 2], [1 1 0], and [111]. Single screw dislocation of a/2[111] was created at about 20 Å away from the center of cell

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in the x direction. The Burgers vector of the SIA loops was chosen to be a/2[1 1 1], which is inclined to the slip plane of the dislocation. Previous study using elasticity theory showed that SIA loops with this configuration have elastic interactions with the screw dislocation [7]. Three SIA cluster sizes were considered, with diameters of 1.6, 3, and 4.4 nm, all having a hexagonal shape and containing 37, 127, and 271 interstitial atoms. The SIA atoms were introduced to the crystal at about 40 Å (45 Å in SIA271 case) away from the dislocation in the slip direction. Periodic boundary conditions were applied in the [111] and free surfaces in the [ 11 2] directions. For the [1 1 0] direction, we adopted a modified free boundary. Previous study on motion of screw dislocations suggests the usage of free boundary to avoid any tension-shear coupling in MD simulations [5]. We found that the free boundary gave a surface rotation at 300 K temperature, but this rotation was almost indistinguishable at lower temperatures. So we used the following: Four adjacent layers were fixed in the lower (1 1 0) surfaces, and four adjacent layers were free (except in the [111] direction in which strain was applied) in the upper surfaces. The system size was 30 nm (35 nm in SIA271 case) along [ 11 2], 16 nm along [1 1 0], and 30 nm along [111] directions, containing 1.2 – 1.4 million atoms. The constrained layers in the upper surface were then translated in the [111] direction at a constant strain rate. Two strain rates (8x106 and 3x107 s-1) were used in the simulations. Unless specified, slower strain rate results were reported here.

2

1

3 6 4

y[110 ]

5

b = a /2[111] x[112] z[111]

(a)

(b)

Figure 1. (a) The SIA loop configuration from [1 1 0] view. (b) MD simulation box and shear direction. MD simulations were performed at two temperatures, 100 K and 300 K, where the plasticity is expected to be mainly by thermally activated motion of screw dislocations. No temperature control was applied to the system and the temperature rise was less than 2 K during the course of the simulations. Total time varied from 2 – 2.5 ns, yielding 1.6 – 2% strains during the simulations. As shear strains applied to the system, the screw dislocation glides via a kink mechanism, starting at about 0.4% strain level at 100 K. In Fig. 1, each segment of the SIA loop is marked with an index number from 1-6. Before the dislocation contacts the SIA loop, the SIA loop migrates along its crowdion axis under the stress field of the dislocation and the applied stress. This migration brings the SIA defect slightly above the glide plane at onset of the interaction. In fact, our separate MD simulation results showed that even when the SIA loop initially was placed at some distance below the glide plane of the dislocation, they attracted each other and the SIA loop migrated to the similar position.

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Results We identified two interaction mechanisms during the dynamical processes, which seem to be insensitive to the strain rates used in this study. One is the junction mechanism that was seen for the large cluster sizes (SIA127 and SIA271). In this mechanism, the screw dislocation and the SIA loop interact by formation of a junction and the junction migration/sweeping processes. This process is described with representative MD snapshots taken during the simulations of the SIA127 at 100 K in Fig. 2. We interpret the dynamical processes as the following: The dislocation first reacts with segment-2 of the SIA loop to form an energetically favorable a[010] junction under Frank’s rule. This is possible since segment-2 lies parallel to the glide plane (1 1 0). Shortly after, a triangular structure develops between the segment-3 and the dislocation. However, it took a long simulation time (about 0.9 ns) before the next additional reaction took place. This reaction involves segment-4, which is now oriented 54.7o from the slip plane and 70.5o from the dislocation line. The local junction formation becomes unfavorable because of these high angles. So a large bowing force for the dislocation arm is needed. Furthermore, the glide plane of segment-4, under SIA Burgers vector, is plane (110), which is perpendicular to screw dislocation glide plane (1 1 0). This implies that this segment is sessile under the external applied stress. During these junction formation events, the lower dislocation arm (where the junction formation took place) did not cross-slip owing to the cross-slip direction being in the anti-twinning direction. 980 ps

1861 ps

½[111]

½[111]

2 3 4

1880 ps

1

2

[010] 6 3 5

½[1 -11] 4

½[111]

1 [010] 6 5

[010]

½[1 -11]

Figure 2. MD snapshots during screw dislocation interaction with SIA127 at 100 K. Upper pictures were viewed from [1 1 0] direction and lower pictures were viewed from a tilted aspect close to [111]. A schematic interpretation of the process is given below the MD pictures. After the first high angle junction was formed, the additional junction formations were easily carried over. The resulting structure had a unique feature. The “planarization” of the new junctions along the slip plane rotated the SIA loop and left the dislocation to be connected to the SIA loop via

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an Orowan type junction. Under additional strain, this final junction configuration broke up, leaving a new SIA loop with b=a[010]. The interaction process shown here was similar for the larger SIA loops and other loading rates at different temperatures. The other interaction mechanism was identified during the dynamical processes is the helical dislocation mechanism. This mechanism takes place with the SIA loops having smaller loop size. Fig. 3 shows the helical dislocation formation from MD snapshots in SIA37 simulations. A transition to a faulted loop formation can be clearly seen before the absorption of the SIA loop atoms. This maybe the reason why the smaller size SIA loop is less resistant to the junction formation processes described earlier. From 3D Burgers circuit analysis, the resulting helical dislocation had the same Burgers vector of the original screw dislocation including two superjog segments. The helical dislocation was sessile because it maintained the part of the “original” SIA loop. One of the superjog segment and a connecting [001] segment form together a sessile lock, which possibly only can travel along the screw dislocation line direction. 900 ps

1140 ps

1146 ps

1147 ps

[100]

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Figure 3. The helical dislocation formation processes. Upper pictures were viewed 39o from [1 1 0] direction and lower pictures were viewed from a tilted aspect close to [111].

Figure 4. The superjogs were pushed towards each other under high shear stresses, emitting a new SIA loop. The formation of the helical dislocation was relatively fast. For the remaining of the shear straining process, the helical dislocation remained in motion, with both arms moving through the kink migration. Finally, the superjogs were pushed towards each other at higher stress levels, emitting a SIA loop ([111] character), which is the normal Hirsch mechanism [2] and the evolution of this process is summarized in Fig. 4.

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Earlier atomistic studies included the static calculations of the interaction between screw dislocation and 1/2 SIA defects in bcc Fe [8] and MD simulations of possible growth of SIA defects and their interaction with screw dislocations [9]. The study [8], involving the stationary case, reported the possibility of helix and junction structures, in agreement with our result. However, no interaction details were given and the process was not dynamical. Moreover, both studies were carried out using an EAM potential that predicts asymmetrical screw dislocation core structures.

Figure 5. Stress-strain plots during MD simulations at 100 K temperature with different SIA loop sizes, in comparison with screw dislocation only case. The resulting stress-strain plots during MD simulations are shown in Fig. 5. The figure includes the cases for dislocation motion in the absence of any SIA defect and for three SIA loop sizes. In these simulations, the temperature was 100 K and the rate of deformation was kept as 8x106 s-1. A substantial hardening can be clearly seen for all SIA loop sizes as result of the mentioned interactions. The sudden drops at the high stress levels correspond to the end of the interaction processes and the dislocation resumes to its normal motion. For SIA cluster loop sizes, especially for SIA271, the beginning of the elastic interaction at much lower stress levels can also be seen from the figure. Based on the stress strain curves, in the following, we estimate the strengthening factors due to these interactions. These results are summarized in Table 1. In these estimations, the hardening stress was calculated as the maximum junction breakup shear stress minus the shear stress for kink motion of the screw dislocation in the absence of the SIA defect. As can be seen from the table, for SIA37 case, the estimated hardening stress values are similar at different temperatures. This is in agreement with the Hirsch mechanism. As suggested, the emission of SIA loop from helical dislocations is an athermal process [2]. For the junction mechanism, a somewhat weaker hardening stress was observed at 300K. Based on the Orowan relationship [2], an estimation of the interaction strength factor α is also given in Table 1. Here, ∆σ =αµ0b/L, where µ0 is the shear modulus (71 GPa), b is Burgers vector length, and L=L0 – d, where L0 is the dislocation line length, and d is the SIA loop diameter. The obtained values of α were in the range of 0.4 – 0.6 for 100 K and 300 K

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and they seem to be in good agreement with what has been reported for the neutron irradiation experiments, 0.5 – 0.6 [10,11]. Table 1. The hardening stress ∆σ and the interaction strength factor α.

SIA37 SIA127 SIA271

∆σ [107 Pa] 100 K 300 K 34 32 31 25 37 26

α 100 K 0.55 0.47 0.53

300 K 0.51 0.38 0.38

Conclusion In this study, constant strain rate molecular dynamics simulations under the modified boundary conditions were performed to elucidate the interaction processes between the kink motion of screw dislocation and the glissile self-interstitial atom cluster loops in bcc Fe by using an EAM potential for Fe fitted to ab initio forces. The junction formation and the helical dislocation mechanisms were identified as two possible interaction processes. In the junction mechanism, the initial Burgers vector 1/2 of the cluster loop was transformed into . In the helical dislocation mechanism first the absorption, followed by the formation of the helical dislocation and the emission of the cluster loop through Hirsch mechanism was observed. Substantial hardening was seen as result of the interactions. The stress-strain plots obtained for different loop sizes, temperature and strain rates were used to estimate the strengthening factors, which seem to be in good agreement with what has been reported for the neutron irradiation experiments. Acknowledgement: Work at the Ames Laboratory was supported by the Department of EnergyBasic Energy Sciences under Contract No. DE-AC0207CH11358. Helpful discussions with C.Z. Wang and K.M. Ho at Ames Laboratory are acknowledged. References [1] S.J. Zinkle: Phys. Plasmas Vol. 12 (2005), p. 058101. [2] For a review, see P.B. Hirsch, in: Vacancies ’76, edited by R.E. Smallman and J.E. Harris, The Metals Society, London (1977), p. 95. [3] M.I. Mendelev, S.W. Han, D.J. Srolovitz, G.J. Ackland, D.Y. Sun, and M. Asta: Philos. Mag. Vol. 83 (2003), p. 3977. [4] C. Domain and G. Monnet: Phys. Rev. Lett. Vol. 95 (2005), p. 215506. [5] J. Chaussidon, M. Fivel, and D. Rodney: Acta Mater. Vol. 54 (2006), p. 3407. [6] D.J. Bacon, Y.N. Osetsky, and Z. Rong: Philos. Mag. Vol. 86 (2006), p. 3921. [7] F. Kroupa: Philos. Mag. Vol. 7 (1962), p. 783. [8] S. Jumel, J.-C. Van Duysen, J. Ruste, and C. Domain: J. Nucl. Mater. Vol. 346 (2005), 79. [9] J. Marian, B.D. Wirth, R. Schäublin, G.R. Odette, J.M. Perlado: J. Nucl. Mater. Vol. 323 (2003), p. 181. [10] A.C. Nicol, M.J. Jenkins, and M.A. Kirk: Mater. Res. Soc. Symp. Vol. 650 (2001), R1.3.1. [11] N. Hashimoto, S.J. Zinkle, R.L. Klueh, A.F. Rowcliffe, and K. Shiba: Mater. Res. Soc. Symp. Proc. Vol. 650 (2001), R1.10.1.

Solid State Phenomena Vol. 139 (2008) pp 65-70 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.65

The Effects of Solute Segregation on The Evolution and Strength of Dislocation Junctions Q. Chena, X.-Y. Liub and S.B. Binerc Ames Laboratory (USDOE), Materials and Engineering Physics, Iowa State University, Ames, IA 50011 a)

[email protected], b)[email protected], c)[email protected]

Keywords: Dislocation dynamics, kinetic Monte Carlo, dislocation junction, solute segregation.

Abstract. In this study, the role of solute segregation on the strength and the evolution behavior of dislocation junctions is studied by utilizing kinetic Monte Carlo and 3D dislocation dynamics simulations. The different solute concentrations and the character of the junctions are all included in the simulations in an effort to make a parametric investigation. The results indicate that solute segregation can lead to both strengthening and weakening behavior depending upon the evolution of the dislocation junctions. Introduction Under multiple slip conditions, the formation and destruction of the junctions resulting from the attractive interaction of the dislocations are the main contributors to the evolution of the flow stress. The formation and breakup strength of the junctions were studied with molecular dynamics simulations [1], dislocation dynamics simulations [2,3] and the simple line tension models [4,5]. These studies have indicated that the Lomer-Cottrell junctions have the highest strength, followed by the Glissile junctions and the Hirth locks. On the other hand, solute solution hardening is one of the effective strengthening mechanisms almost for all alloy systems, resulting from the interactions between the dislocations and solutes. Solutes act as point obstacles to moving dislocations at low temperatures or in the case of slow diffusion solutes [6,7]. However, when the solute diffusion is significant, the elastic interactions between the solutes and the dislocations lead to solute segregation around the dislocation (Cottrell atmosphere [8]) and depletion elsewhere. Such solute atmosphere can produce a large drag force to a mobile dislocation [9,11]. In this study, the role of solute segregation on the strength and evolution behavior of dislocation junctions is elucidated by utilizing kinetic Monte Carlo and 3D dislocation dynamics simulations. The description of the computational procedures In our 3D dislocation dynamics simulations, the dislocation loops are discretized into linear segments having mixed character. The forces per unit length causing the mobility of the dislocations are obtained from the Peach-Koehler formula: F = ((σ D + σ

A

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(1)

where σ D is the stress tensor from the other remote segments, which can be efficiently obtained from the numerical method developed in [12], σ A is the applied stress tensor, σ S is the stress induced by the solutes and can be treated within the framework of elasticity by modeling solutes as point sources of expansion [13], b is the Burgers vector of the dislocation, t is the line direction of the segment and FSelf is the force arising from the segment itself and immediate adjacent segments. The governing equation of motion then can be cast as [15]:

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(2) ∫Γ ( Fk − Bik Vi )δ jk ds = 0 where Bik is the inverse mobility matrix, Vi is the velocity vector, δjk is the distance vector and the line integral is carried along the arc length of the dislocation ds. Once the velocities of the nodal points on the dislocation lines are determined, the position of the dislocations are updated and then the dislocation reactions based on the constitutive rules are carried out. The essential steps of our kinetic Monte Carlo algorithm can be summarized as follows. The kinetic Monte Carlo method is in the form of the N-fold way or Bortz-Kalos-Lebowitz (BKL) method [14]. It is based on a 3D lattice. All solute migration events to nearest neighbor positions are accounted for. Efficient simulation is realized through neighbor list construction and update. The rate of migration is given by an Arrhenius expression: − (Em + δEel ) /k BT (3) r = ν0 e where ν0 is a prefactor related to the vibration frequency, Em is the energy of migration between lattices, δEel is the solute-stress field interaction energy change between neighboring lattices. At each Monte Carlo step, the probability of a given event being selected is determined by the ratio of its rate in the total rates sum and the time increment dt is computed as ln( ξ ) (4) dt = − R where ξ is a random number between (0,1) and R is the total rates sum. The initial length of the dislocations was taken as 150|b| of which was composed of linear segments in 5|b| in length. For all cases, the equilibrium configurations of the junctions were achieved under zero applied stress. The breakup stress was defined as the junction length becomes zero during the unzipping process resulting from the bowing action of the arms of the junctions. Results

In the first set of simulations, the role of solute segregation on the strength of the preformed junctions was studied. The Lomer junctions composed of 1 / 2[10 1 ]( 1 1 1 ) & 1 / 2[011](11 1 ) slip systems, the Glissile junction due to 1 / 2[10 1 ](1 1 1) & 1 / 2[011](11 1 ) slip system and Hirth lock resulting from 1 / 2[01 1 ]( 1 1 1 ) & 1 / 2[011](11 1 ) slip system were considered. First, the strength of these junction systems was evaluated in the absence of the solutes. At the beginning of the simulations, two straight dislocations in equal length, intersecting each other at their midpoints and making equal angles with the intersection line of the two slip planes, were placed together. Both dislocations were also pinned from their end points during the course of the simulations. The equilibrium configurations of these junction systems are shown in Fig.1. As can be seen, due to the interaction of their stress-fields, two straight dislocations react and form a junction along the intersection of the slip planes. For the case of the Lomer junctions, the resulting lengths of the junctions were almost inversely proportional to the initial angles and ranged from 85|b| to 20|b| with increasing initial angles. Although the initial angles of the dislocations were again 30o in the case of the Glissile junction and Hirth lock, the resulting junctions were much shorter than the corresponding Lomer junction. Starting from the equilibrium configuration, during the breakup process with increasing applied stress, the four remaining dislocation arms outside the junction increasingly bow out, leading to reduction in the junction length with the so-called unzipping process. Eventually, with the complete loss of junction, the pinned dislocations expand further and become unstable once the resolved shear stress reaches to a critical level. Of course, besides the level of the applied stress, the initial length of these four arms, the self-force originating from the line tension, and the forces originating from the interaction of the dislocation segments dictate the magnitude of the bow out.

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Figure 2. Correlation of breakup stress values of junctions with junction lengths for cases with solutes (solid line) and without solutes (dash line). Left, for Lomer junctions formed with 30º, 45º and 60º initial angles. Right, comparison for Lomer, Glissile junction and Hirth lock. For these junction types, the magnitude of the breakup stress is summarized in Fig.2. In agreement with the earlier studies [2,4], we also observed a linear increase in the breakup stress level with increasing junction lengths. As can be seen, decreasing initial orientation angle of the dislocations resulted in stronger junctions in the case of Lomer junctions. Also, Hirth lock was the weakest among the three types of junctions with the same initial orientation of the dislocations. In the next set of simulations the role of solute segregation on the strength of these preformed junctions was studied. A total of 500 solute atoms (0.015% global concentration) were randomly

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distributed into a 150|b|3 simulation cell containing the preformed junction configurations in its center. For these three junction types, the kinetics of segregation to a region within a 5|b| radius along the dislocation lines is shown in Fig.3. In the figure, the total number of solutes within this 5|b| region along the dislocations are normalized by the overall number of solutes in the simulation cell. As can be seen, the rate of segregation is initially high, as a result of the very large number of empty sites near the dislocation lines. The rate of segregation decreases with time, due to the decrease in the number of available empty sites, and also due to the exhaustion of the highly mobile nearby solutes. As a result, the initial segregation behavior for all three junctions is similar. However, with increasing time, more segregation takes place for those Lomer junctions composed of dislocations having smaller initial angles. When a comparison is made between the Lomer junction, Glissile junction and Hirth lock, it can be seen that the solutes segregate more heavily to the Lomer junctions. Although it is not shown in here, this behavior is associated with the character of dislocation segments in the junctions. The solute segregation (Cottrell atmosphere) takes place in the tensile stress field region of a dislocation having an edge component. Therefore, in a junction configuration, more solutes would be segregated toward those segments approaching to the edge character. 60

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Figure 3. Left, Solute segregation kinetics to Lomer junctions formed with 30º, 45º and 60º initial angles. Right, Comparison of solute segregation kinetics for Lomer junction, Glissile junction and Hirth lock . The spatial distribution of the solutes for these three junction configurations after 0.156 µs of segregation time is also shown in Fig.1. During the breakup processes, it is assumed that all the solutes remain immobile in their final configuration. The resulting breakup stress values of the junctions, with the shown solute atmospheres, are again summarized in Fig.2. As can be seen, the increase in the breakup strengths is not the same for all junctions, even though the solute segregation is not significantly different as shown in Fig.1. In this section, the effect of the solute segregation on the evolution of the junction configurations and the resulting junction strengths are elucidated. These simulations were performed for the Lomer junction with 30º initial angles, and it is assumed that the 1 / 2[10 1 ]( 1 1 1 ) dislocation is the forest dislocation. Initially, a certain local concentration of solutes was allowed to segregate to this forest arm with the kMC simulations. Then, the mobile arm of the junction, the 1 / 2[011](11 1 ) dislocation, was introduced to interact with the forest dislocation having segregated solutes. It is further assumed that, after initial segregation, the mobility of the solutes relative to the mobility of the dislocations is considerably slow; so that the solutes remain effectively immobile during the junction formation and breakup processes. The equilibrium junction configurations for different local solute concentrations for the same Lomer junction with 30º initial angles are shown in Fig.4.

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For these cases, the resulting breakup stress values are correlated with the junction lengths as a function of the local solute concentrations in Fig.5. There was an initial increase in the breakup stress even though the junction lengths were decreasing with increasing local solute concentration. This is mainly due to the considerable reduction in the arm lengths of the forest dislocation (Fig.4). At high local solute concentration levels, where the junction configurations were at or near crossstate, there was a large drop in the breakup stress levels.

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Concluding Remarks In this study, the role of solute segregation on the strength and the evolution behavior of dislocation junctions was studied by utilizing kinetic Monte Carlo and 3D dislocation dynamics simulations. The different concentration and the character of junctions were all included in the simulations in an effort to make a parametric investigation. The results indicate that the solute segregation can lead to both strengthening and weakening behavior depending upon the evolution of the dislocation junctions. The local solute concentration seems to be the more relevant parameter to characterizing the solute and dislocation interactions, due to the short-range stress fields of solutes.

Acknowledgement: Work at the Ames Laboratory was supported by the Department of EnergyBasic Energy Sciences under Contract No. DE-AC0207CH11358. References [1] D. Rodney and R. Philips, Phys. Rev. Lett. Vol. 82 (1999), p.1704. [2] V.B. Shenoy, R.V. Kukta and R. Philips, Phys. Rev. Lett. Vol. 84 (2000), p.1491. [3] R. Madec, B. Devincre and L.P. Kubin, Phys. Rev. Lett. Vol. 89 (2002), p.255508. [4] L. Depuy and M.C. Fivel, Acta Mater. Vol. 50 (2002), p. 4873. [5] R.C. Picu, 2004, Acta Mater. Vol.52 (2004), p.3447. [6] R.Labusch, Phys. Stat. Solidi, Vol. 41 (1970), p. 659. [7] T. Mohri and T. Suzuki, “Impurities in engineering materials” (1999), p.259. [8] A.H. Cottrell, “Dislocations and plastic flow in crystals” (1953). [9] J.P Hirth and J. Lothe, “Theory of dislocations” (1968). [10] Yoshinaga H.and S. Morozumi, Philos. Mag. Vol. 23 (1971), 1367. [11] S.Y. Hu, Y.L. Li, Y. X. Zheng.and L.Q. Chen, Int. J. Plasticity, Vol.20 (2004),403. [12] N.M. Ghoniem, J. Huang and Z. Wang, Philos. Mag. Lett.Vol. 82 (2002), 55. [13] C.S. Deo, D.J. Srolovitz, W. Cai and V.V. Bulatov, J. Mech. Phys. Solids, Vol.53 (2005), p.1223. [14] A. Bortz, M. Kalos, and J. Lebowitz, J. Comput. Phys. Vol.17 (1975), p.10. [15] N.M. Ghoniem, S.H. Tong, and L.Z. Sun, Phys. Rev. B Vol.61 (2000), p. 913

Solid State Phenomena Vol. 139 (2008) pp 71-76 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.71

Physics Mechanisms Involved in the Formation and Recrystallization of Amorphous Regions in Si through Ion Irradiation. Iván Santosa, Luis Alberto Marqués, Lourdes Pelaz, Pedro Lopez, and María Aboy. Departamento de Electricidad y Electrónica, Universidad de Valladolid, E.T.S.I. Telecomunicación, Campus Miguel Delibes s/n, 47011 Valladolid, Spain. [email protected] Keywords: Multiscale modeling, Molecular Dynamics, Binary Collisions, Silicon, Amorphization.

Abstract. We focus this work on multi-scale modeling of the ion-beam-induced amorphization and recrystallization in Si, although our scheme can be applied to other materials. We use molecular dynamics to study the formation mechanisms of amorphous regions. We have observed that along with energetic ballistic collisions that generate Frenkel pairs, low energy interactions can produce damage through the melting and quenching of target regions. By quantifying these results, we have developed an improved binary collision approximation model which gives a damage description similar to molecular dynamics. We have successfully applied our model to ion and cluster implantations. In order to define the energetic of defects in a more computationally efficient Kinetic Monter Carlo code, we have used molecular dynamics results related to the recrystallization behavior of local amorphous regions. The combination of all these simulation tools, molecular dynamics (fundamental studies of damage formation and recrystallization), improved binary collisions (including ballistic and melting-related damage) and Kinetic Monte Carlo (for efficient defect kinetics modeling during the implantation and the subsequent annealing), allows us to model the effect of ion mass, beam current and implant temperature on the amount and morphology of residual defects in Si. Introduction The introduction of energetic ions during the implantation step needed to fabricate shallow junctions in Si generates a large number of defects in the lattice. These defects diffuse, interact among them and with dopants, annihilate at interfaces, etc. resulting in a final dopant and defect distribution that affect the device performance. Consequently, it is necessary to have a good understanding of all these mechanisms in order to control the device characteristics. Multi-scale modeling is required to capture the underlying physics and to access space and time scales directly comparable with experiments. The tool routinely used for the simulation of ion implantation is based on the so called binary collision approximation (BCA). This simulation technique is adequate to properly reproduce the damage generated by light ions and gives reasonably good depth profiles and projected ranges of implanted species with low computational cost [1]. However, it is not able to adequately describe the amorphous regions present in the as-implanted damage of heavy ions, which have been observed experimentally [2] and obtained from molecular dynamics (MD) simulations [3-5]. Their origin has been attributed to processes involving low energy interactions [6]. It would be desirable to develop improved BCA models able to provide the same damage structures predicted by MD but without its computational overhead. BCA Cascades In BCA, implantations are simulated by considering collisions between the ion and its closest target atom. Target atoms are displaced from their lattice positions only if they receive in a collision an

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amount of energy higher than the displacement threshold, Ed [7] (conventionally Ed is 15 eV for silicon [1, 8]). BCA gives an atomistic description of damage based on the interstitial-vacancy pairs generated in these collisions, also called Frenkel Pairs (FP). When cascades are simulated with BCA, different ions produce the same number of FP for the same nuclear deposited energy [9]. However, in MD simulations the amount of damage and its complexity increases with ion mass for the same deposited energy [5]. In a more detailed analysis of energy transfers in BCA during 1 keV B, Si and Ge cascades into silicon, only 23 % of nuclear deposited energy is used to generate FP. The remaining percentage, 77 %, is employed in energy transfers to atoms below the displacement threshold. When grouping these atoms within a second neighbor distance, the mean group size and the mean group energy increase with ion mass. Then, it seems that these low energy transfers usually ignored in BCA simulations could establish the difference in damage morphology among the three ions. Damage generation below Ed We have used MD simulations to study damage generation mechanisms at energy transfers below Ed. We use the Tersoff 3 potential [10] to describe silicon interactions. We carry out our simulations in the NVE ensemble, using cubical cells and applying periodic boundary conditions in all directions. We give a certain amount of kinetic energy to a number of atoms located in a sphere in the center of the cell with velocities in random directions. Initial energy density is chosen between 0 and 20 eV/atom. At the end of the simulation the final number of atoms displaced from perfect lattice positions is evaluated. The initial temperature of the cells is 0 K to avoid damage migration and annihilation. We choose system size so that temperature increase after thermalization is low enough to avoid dynamic annealing. We run 100 simulations for each set of initial conditions to have good statistics. For each simulation we evaluate the number of final displaced atoms per initial moving atom. We refer to this quantity as the ‘efficiency on damage generation’. In Fig. 1 we represent this efficiency as a function of the initial energy of moving atoms for different numbers of initial moving atoms. In BCA simulations, the efficiency is zero below the displacement threshold and one over it. However, in MD the efficiency can be quite high even below the threshold. Furthermore, we have that for each number of initial moving atoms, the efficiency depends linearly with the initial energy density. The crossing point of straight lines with the horizontal axis represents a threshold energy density for damage production. On the other hand, the slope of the straight lines is a kind of displacement energy. It is worth noting that below 1 eV/atom there is no final damage even for total deposited energies as high as 5 keV. 4

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The physical meaning of the 1 eV/atom threshold is understood in the context of a competition between the melting of the excited region and the out-diffusion of the deposited energy [11]. In this competition, atoms must have a certain amount of energy during enough time to be displaced from their lattice positions. This time rapidly increases with decreasing energy density with a low energy threshold at 1 eV/atom below which there is no damage production. Improved damage BCA model With the previous MD results, we have developed an improved damage model for BCA codes. This model allows to obtain a damage description similar to MD but with a much lower computational cost. First, there is a collisional phase which is still simulated with a conventional BCA code. We store the generated FP with the remaining energy of interstitials at the end of their trajectories, and the position and energy of all the target atoms that receive any amount of energy during the cascade. We call them ‘hot particles’. In the next step, it is decided whether these hot particles are locally displaced from their lattice site or not. For that purpose, and with the help of Fig. 1, we evaluate their efficiencies by taking into account their local environment. This local environment is formed by all the surrounding particles within a second neighbor distance having energies above 1 eV. For efficiencies below zero, the hot particle is not displaced. If it is between zero and one, the hot particle is displaced with a random probability given by its efficiency. If it is one or above, the hot particle is displaced and, taking into account the remaining efficiency, a first random ‘no-displaced’ neighbor is also displaced or not. Then, at the end of the cascade, we have a set of FP and locally displaced atoms from the hot particles. When we apply our improved damage model to the BCA implantations of 1 keV of B, Si and Ge into silicon, we have that the final number of displaced atoms and the distribution of cascades as a function of the generated damage are similar to MD results [5] as can be seen in Fig. 2.

MD Boron

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Figure 2. Percentage of cascades generating a certain number of displaced atoms obtained from simulations of cascades of 1 keV B, Si and Ge using MD (from [5]) and the improved BCA model. Average numbers of displaced atoms for the three ions are also shown. We also have successfully applied our improved BCA model for the description of damage generated by cluster implantations. As an example we studied the implantation of B18H22 into silicon, which has been proposed as an alternative to the use of monatomic B beams in order to improve production throughput in the fabrication of shallow p-type junctions. We have carried out 100 independent cascades of 9 keV B18H22 into Si with normal incidence (500 eV per B atom). For the sake of simplicity, only the boron atoms of the molecule are considered. We have simulated a sequential implantation of 18 B ions taking into account damage accumulation. The impact point of each B ion was randomly chosen within an implantation window whose dimensions are determined by the cluster molecule geometry. If we compare with MD results [12], we obtain a good agreement for the damage region as can be seen in Fig. 3.

Theory, Modeling and Numerical Simulation

Lateral distance to impact point (nm)

74

8 I and I2 V and V2 V3 and V4 V>4

6

Figure 3. Projection of generated damage along implant direction from MD simulations [12] and from the improved BCA damage model.

4 2 0 -2 -4

Displaced Hot Particles Interstitials

-6 -8 0

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4

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Depth (nm)

8

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Combined amorphization-recrystallization model It has been shown using MD techniques that amorphous zones can be described in terms of a Si defect called bond defect or IV pair [13]. This defect consists of a local rearrangement of bonds in Si where two atoms are displaced from their lattice positions [14, 15]. Its activation energy for recombination, and therefore the recrystallization of amorphous regions, depends on the number of neighboring IV pairs: the higher the number of surrounding IV pairs, the higher the activation energy for IV pair recombination [16]. This information about recrystallization, along with the kinetics and interactions of dopants and defects, have been introduced in the non-lattice Kinetic Monte Carlo (KMC) diffusion code DADOS [17]. This KMC model based on the IV pair description of damage has been successfully applied to reproduce and explain many experimental results [18]. In order to test the combined amorphization-recrystallization model, we have obtained the transition temperature, TC, for amorphization by ion implantation as a function of dose rate for 80 keV Si and Ge implants into silicon with a dose of 1015 cm-2. This temperature marks two different behaviors: below TC the target is amorphized, and above TC damage is recombined as it is generated. To evaluate TC it is necessary to analyze the final damage at different temperatures for each dose rate. Consequently, these simulation experiments comprise a wide range of amorphization and recrystallization conditions. We proceed as described in the following. Damage zones generated by the cascades are obtained with our improved BCA model. Then, particles that form these damage zones are transferred to DADOS and translated into IV pairs taking into account that, as mentioned before, an IV pair is formed by two Si atoms displaced from their lattice positions. Annealing at the implant temperature is carried out after each implantation cascade during a time defined by the implant dose rate. During this annealing, particles interact with the probability rates defined in DADOS. Then, a new cascade is added to the remaining damage until the specified final dose is reached. Fig. 4 shows the evolution of TC as a function of dose rate. As can be seen, there is a good agreement between our simulation results and the experimental measurements [19].

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3.5

1000/Tc (K -1)

Si Ge

Figure 4. Transition temperatures TC as a function of ion dose rate for 80 keV Si and Ge implants to a total dose of 1015 cm-2. Open symbols correspond to simulation results. Solid symbols are from the experiments of ref. [19].

3

2.5

2 11.5

12

12.5

13

13.5

14

2

Log [ Dose Rate (ions/cm /s) ]

Conclusions In conclusion, we have presented an improved BCA model that provides an atomistic description of damage similar to the one given by MD but with a much lower computational cost. This model combines ballistic and melting-related damage mechanisms. We have successfully applied our model to ion and cluster implantations and the resulting damage regions are in very good agreement with MD simulations. We have linked the improved BCA code into the non lattice kinetic Monte Carlo diffusion code DADOS, which includes a recrystallization model that describes amorphous regions as agglomerates of the silicon defect called IV pair. We have tested the combined amorphization-recrystallization model for a wide range of implantation conditions, showing good agreement with experimental results. Acknowledgments This work has been supported by the Spanish DGI under project TEC2005-05101 and the JCyL Consejería de Educación y Cultura under project VA070A05. References [1] J.M. Hernández-Mangas, J. Arias, L. Bailón, M. Jaraíz and J. Barbolla: J. Appl. Phys. Vol. 91 (2002), p. 658. [2] M.S.E. Donnelly, R.C. Birtcher, V.M. Visnyakov and G. Carter: Appl. Phys. Lett. Vol. 82 (2003), p. 1860. [3] M.J. Caturla, T.D. de la Rubia, L.A. Marqués and G.H. Gilmer: Phys. Rev. B Vol. 54 (1996), p. 16683. [4] K. Nordlund, M. Ghaly, R. S. Averback, M.J. Catrula, T.D. de la Rubia and J. Tarus: Phys. Rev. B Vol. 57 (1998), p. 7556. [5] I. Santos, L.A. Marqués, L. Pelaz, P. López, M. Aboy and J. Barbolla: Mater. Sci. Eng. B Vol. 124-125 (2005), p. 372. [6] T.D de la Rubia and G.H. Gilmer: Phys. Rev. Lett. Vol. 74 (1995), p. 2507. [7] M.T. Robinson and I.M. Torrens: Phys. Rev. B Vol. 9 (1974), p. 5008.

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[8] SRIM documentation, www.srim.org. [9] I. Santos I, L.A. Marques, L. Pelaz and P. Lopez: Nucl. Instrum. Methods B Vol. 255 (2007), p. 110. [10] J. Tersoff: Phys. Rev. B Vol. 38 (1988), p. 9902. [11] I. Santos, L.A. Marqués and L. Pelaz: Phys. Rev. B Vol. 74 (2006), p. 174115. [12] L.A. Marqués, L. Pelaz, I. Santos and V.C. Venezia: Phys. Rev. B Vol. 74 (2006), p. 201201. [13] L.A. Marqués, L. Pelaz, M. Aboy, L. Enríquez and J. Barbolla: Phys. Rev. Lett. Vol. 91 (2003), p. 131504. [14] M. Tang, L. Colombo, J Zhu and T. Diaz de la Rubia: Phys. Rev. B Vol. 55 (1997), p. 14279. [15] L.A. Marqués, L. Pelaz, J. Hernández, J. Barbolla and G.H. Gilmer: Phys. Rev. B Vol. 64 (2001), p. 045214. [16] L. Pelaz, L.A. Marques, M. Aboy, J. Barbolla and G.H. Gilmer: Appl. Phys. Lett. Vol. 82 (2003), p. 2038. [17] M. Jaraiz, L. Pelaz, J.E. Rubio, J. Barbolla, G.H. Gilmer, D.J. Eaglesham, H.J. Gossman and J. M. Poate: Mater. Res. Soc. Symp. Proc. Vol. 532 (1998), p. 43. [18] L. Pelaz, L.A. Marqués and J. Barbolla: J. Appl. Phys. Vol. 96 (2004), p. 5947. [19] R. D. Goldberg, J. S. Williams and R. G. Elliman: Nucl. Instrum. Methods. B Vol. 106 (1995), p. 242.

Solid State Phenomena Vol. 139 (2008) pp 77-82 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.77

Hotspot formation in shock-induced void collapse Yunfeng Shi1, a and Donald W. Brenner1, b 1

Department of Materials Science and Engineering

North Carolina State University, Raleigh, NC 27695-7907, USA a

[email protected], [email protected]

Keywords: energetic materials, shock, detonation, void defect, hotspot, jetting.

Abstract. We present results from molecular dynamics simulations of shock-induced hydrodynamic void collapse in a model energetic crystal. During void collapse, hotspot formation is observed that leads to subsequent detonation. The hotspot formation mechanism is identified as shock energy focusing via jetting. There is another initiation mechanism that arises from the interaction of reflected shock waves with the rigid piston, which is considered to be an artifact. Such artifact can be eliminated by altering the location of the void. The detonation threshold as a function of the velocity of the driven piston is determined for various void geometries. It is found that a system containing a void has a lower detonation threshold than that of a perfect energetic crystal. The amount of reduction of the detonation threshold depends on the geometry of the void. For square voids, there exists a minimum size above which reduction of the detonation threshold occurs. Among voids that have an equal volume, the void that is elongated along the shock direction gives the lowest detonation threshold. Introduction It is well established that detonation in solid explosives can be facilitated by void collapse and that initial chemical reactions occur at so-called “hotspots” [1-3]. However, the hotspot formation mechanism, i.e. why chemistry is localized, has not been fully established at the atomic level. Jetting occurs in many high speed phenomena and serves as an important means to concentrate energy that causes, for example, the Munroe effect [4] and cavitation erosion [5, 6]. The Munroe effect in shaped charges can be quantitatively described by liner collapsing models [4, 7]. However, the application of these models to general void collapse is not obvious because the assumption that the jetting material is an inviscid/incompressible fluid may not hold. Unlike shaped charges, void collapse in water is closely related to void collapse in energetic material. Thus it is reasonable to link jetting to hotspot formation, as supported by both numerical and experimental results. Mader studied shock collapse of a spherical void using reactive hydrodynamics [8]. It was found that a high temperature liquid jet emerges during void closure due to energy convergence. More recently, jetting has been observed in studying the shock collapse mm-size voids in gel, in which case high speed jet impact on the downstream wall is the main mechanism for ignition [2, 9, 10]. However, it is not clear whether those macroscopic observations can be extended to nanometer-scale voids. At the atomic scale, where the transient nature of these processes limit experimental studies, molecular dynamics simulations have provided many useful insights into detonation as well as hotspot formation [11-15]. Hatano has reported an elevated temperature and an increased number of energetic collisions during collapsing of a rectangular void. Such shock chemistry enhancement is due to momentum/energy focusing although no jetting was observed [15]. A planar gap defect, which precludes any convergence of the spallated material, has also been shown to reduce the detonation threshold by as much as 20% in model system [13]. A recompaction model has been used to explain the hotspot formation mechanism for such defects; this model considers heating from recompression of the ejected particles at the down-stream wall [14]. This model predicts that size-dependent sub-threshold detonation with chemical reaction initiates in the compressed gas instead of the down-stream wall. Molecular dynamics simulations by Germann et al. investigated

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two-dimensional energetic materials containing circular or elliptical voids. They observed jet formation and void shape-dependent detonation behavior [16]. Nomura et al. used molecular dynamics simulations and the ReaxFF many-body reactive potential energy expression to model the collapse of a spherical void in an RDX crystal [17]. They observed the formation of a jet that was accelerated through the void due to focusing from the void walls. We have simulated shock collapsing of rectangular voids [18] and identified the detonation mechanism at both the mesoscale and the atomic scale. At the mesocale, jetting is the primary feature that appears during void collapsing that leads to hotspot formation. At the atomic scale, vibrational up-pumping which is followed by bi-molecular reaction triggers the onset of detonation. In this work, we address two important issues that have not been sufficiently covered in our previous report. The first one is the effect of the location of the void on the detonation threshold from which we identified an undesirable detonation mechanism that arises from the finite system size. The second issue is to investigate the effect of the void shape on detonation sensitivity in terms of size and aspect ratio. These results provide useful insights in understanding shape-dependent shock chemistry during void collapsing. Simulation System The reactive force field. The energetic material used in this study is a model nitrogen cubane crystal. A recently developed reactive state summation (RSS) potential is used to model nitrogennitrogen interactions [19]. This potential is designed to model the exothermic dissociation of nitrogen cubane molecules into diatomic nitrogen molecules. This potential has been tested in thermal decomposition and mechanical shock simulations in three-dimensional systems [19]. Steady detonation has been observed to propagate at an intrinsic speed. In addition, this energetic material model has been used to simulate a plastic bonded explosive (PBX) [20] and shock collapsing of rectangular voids [18]. Model explosive containing voids. The initial configuration of the system is a thin slab geometry. Nitrogen cubane molecules are packed in simple cubic structure in a simulation box of 46×52×0.9 nm3. The system is periodic in Y and Z-directions, and shocks propagate along positive X-direction. Rectangular voids are created by removing nitrogen cubane molecules from a perfect crystal. The relevant characteristics of the void are a longitudinal length (in the X-direction) and a transverse length (in the Y-direction). The void dimension in the Z-direction is infinite due to the periodic boundary condition. Three types of voids with different locations along the X-direction were created: (L, left aligned) the left boundary of the void is fixed so that the thickness of the upstream wall (the material between the piston and the void) is 10 nm; (R, right aligned) the right boundary of the void is fixed so that the thickness of the downstream wall (the material between the far-end of the explosive and the void) is 10 nm; (C, center aligned) the center of the void is fixed at the center of the system. This is to investigate how the location of the void effects detonation behavior in a finite system. Three series of voids were investigated: (I) varying the transverse length while keeping a constant longitudinal length; (II) varying the size of the void while keeping a square shape; (III) varying the aspect ratio of the void while keeping the volume (especially the project area in the X-Y plane) constant. A shock is generated by a moving rigid piston. The detonation threshold was obtained by systematically varying the piston velocity to determine when the system detonates. Results and Discussion The effect of void locations. Plotted in Figure 1 are the detonation thresholds for series I. Three types of voids (L, R and C void) are included. The detonation thresholds of R voids and C voids almost overlap. For voids with a small longitudinal dimension, there is no reduction of detonation threshold. Above a critical length, the detonation threshold decreases monotonically with increasing longitudinal length. For L voids, the detonation thresholds deviate from R voids and C voids at small longitudinal lengths. In those cases, it was found that the initiation occurs not in the vicinity

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of the void, but near the piston. As the ejected energetic molecules hit the downstream wall, a compressive reshock wave is reflected back. When this reshock wave reaches the rigid piston, detonation occurs. This type of sub-threshold detonation is a result of the finite thickness of the upstream-wall and is thus considered an artifact. Fortunately, it is straight forward to exclude such artifacts by examining the location of initial chemical reactions. Figure 2 illustrates the two types of detonation initiation for a L void. The initiation due to the artifact discussed above is located adjacent to the rigid piston. After removing these initiations from the data, the detonation threshold curve for L voids overlaps very well with the other two curves. Therefore, the location of the voids along the X-direction, or the thickness of the upstream/downstream wall, does not affect the detonation behavior. Simulations shown below are based on C voids only.

Threshold Piston Velocity (km/s)

1.6

1.5

1.4

I 1.3 0

L (apparent) L (exclude reshock) C R

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10 15 Longitudinal Length (nm)

20

Figure 1 Detonation thresholds are shown for voids (series I) with different longitudinal lengths but keeping the transverse length constant (14 nm).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 The sequence of void collapse for an L void (12 nm by 14 nm). The piston speed is above the true detonation threshold so that the hotspots from void collapsing as well as from the artifact appear. Gray represents the potential energy of nitrogen in the perfect nitrogen cubane. Black represents low potential energy of nitrogen in detonation products including diatomic and small amount of oligomeric nitrogen molecules. Therefore, the growing black region is the detonation region. Note that there are two detonation initiation locations that are marked by the arrows in pane (f). The left one is an artifact from the interaction of the reflected shock wave and the rigid piston. The right one is from void collapsing.

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Shape evolution of the jet. The observed sub-threshold detonation behavior is due to jet formation as illustrated in Figure 2. As the shock wave reaches the upstream wall of the void, material is ejected and begins to fly toward the downstream wall. This causes the material of the upstream wall to be compressed vertically by adjacent regions above and below. This can be seen by the shape change of the upstream wall and also the curving side walls. Consequently, the material in the upstream wall expands in the shock direction and a jet is formed. Illustrated in Figure 2(a) is the shape evolution of the jet. First, the flying molecules maintain an almost flat front. At this stage, the detonation threshold remains the same as the perfect crystal. Then due to the compression from the side walls, two corner-jets emerge near the side walls and move towards the centerline, which resembles experimental observations [9]. Those corner-jets are energetic enough to reduce the detonation threshold. Therefore, it is the formation of the corner-jets that sets the critical length in the longitudinal direction as shown in Figure 1. Finally, the two corner-jets combine to form an even more energetic single jet. The jet does not travel at a constant speed. Instead, its front is accelerating as evidence by the increased spacing between each profile in Figure 3.

Figure 3 Shape evolution of a jet with equal time intervals. Only the edge of the jet and the boundary of the voids are shown. The downstream wall is removed. Size-dependent shock sensitivity of squared void. Figure 4 shows the detonation thresholds for square voids with different sizes (series II). Similar to Figure 1, there exists a critical void size to initiate sub-threshold detonation. The threshold piston speed decreases with increasing void size that is larger than 8 nm.

Threshold Piston Velocity (km/s)

1.6 1.55 1.5 1.45 1.4 1.35 0

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10 Void Length (nm)

15

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Figure 4 Detonation thresholds for square voids (series II) with different sizes.

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Aspect ratio-dependent shock sensitivity. To further illustrate the effect of the void shape, plotted in Figure 5 are the detonation thresholds of series III. It is clear that elongated voids along the shock direction are the most effective to sensitize explosives.

Threshold Piston Velocity (km/s)

1.6

1.5

1.4

1.3

1.2 0

0.5

1

1.5 X/Y

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2.5

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Figure 5 Detonation thresholds for a void (series III) with a constant volume (a constant projected area in X-Y plane of 200 nm2) but different aspect ratio. Hotspot formation mechanisms. As illustrated in Figure 3, the shock energy localizes mainly via jetting. The degree of localization, which determines the intensity of the hotspot formation, depends sensitively on the shape of the voids. Upon shock, the materials in the up-stream wall are ejected and subject to continuous accelerating from behind and by the side wall. Therefore, the impact velocity of the ejected material on the down-stream wall increases as the void length in the shock direction increases. This explains the main feature in Figure 1, 4 and 5 that larger voids lead to a lower detonation thresholds. The hotspot formation mechanism in this study agrees with the momentum and energy focusing [15]. The shape dependency observed here is similar to a temperature representation reported in Ref. [14]. Contrary to a previous report [13], we observe no reduction in detonation threshold for systems with planar gaps (up to 20 nm in width) for C and R voids. This is because jetting is impossible for a planar gap. However, systems with gaps (L voids) show reduced detonation thresholds. This is again due to the artifact that the reshock wave reaches the piston. Summary We observed void collapsing in a model energetic crystal that leads to hotspot formation through jetting. This causes a reduction of the detonation threshold of an energetic crystal containing a void. The strength of jetting depends on the geometry of the void. An artifact of detonation initiation, which is caused by the reflection of a shock wave, is identified. Such an artifact can be avoided by changing the position of the void. A critical length of the void in the shock direction exists for subthreshold detonation. Acknowledgements We thank D. Thompson, T. Sewell, Y. Hu and B. Broom for stimulating discussions. The North Carolina State University High-Performance Computing Facility is also thanked for providing computational resources. The simulations were performed using LAMMPS molecular dynamics software (http://lammps.sandia.gov) with a customized force field. This work was supported by a Multi-University Research Initiative from the U.S. Army Research Office.

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References [1] F. P. Bowden and A. D. Yoffe, Fast Reactions in Solids (Butterworths Scientific Publications, London, 1958). [2] J. P. Dear, J. E. Field, and A. J. Walton, Nature Vol. 332, (1988), p.505. [3] J. E. Field, Accounts Chem Res Vol. 25, (1992), p.489. [4] W. P. Walters and J. A. Zukas, Fundamentals of shaped charges, 1989). [5] M. Kornfeld and L. Suvorov, J Appl Phys Vol. 15, (1944), p.495. [6] J. I. Katz, P Roy Soc Lond a Mat Vol. 455, (1999), p.323. [7] G. Birkhoff, D. P. Macdougall, E. M. Pugh, et al., J Appl Phys Vol. 19, (1948), p.563. [8] C. L. Mader, Phys Fluids Vol. 8, (1965), p.1811. [9] N. K. Bourne and J. E. Field, J Fluid Mech Vol. 244, (1992), p.225. [10] N. K. Bourne and A. M. Milne, P Roy Soc Lond a Mat Vol. 459, (2003), p.1851. [11] D. W. Brenner, D. H. Robertson, M. L. Elert, et al., Phys Rev Lett Vol. 70, (1993), p.2174. [12] D. W. Brenner, in Shock compression of condensed matter, edited by S. C. Schmidt, R. D. Dick, J. W. Forbes and D. G. Tasker, 1992), p. 115. [13] C. T. White, D. R. Swanson, and D. H. Robertson, in Chemical Dynamics in Extreme Environments, edited by R. A. Dressler (World Scientific, Singapore, 2001), p. 547. [14] B. L. Holian, T. C. Germann, J. B. Maillet, et al., Phys Rev Lett Vol. 89, (2002), p.285501. [15] T. Hatano, Phys Rev Lett Vol. 92, (2004), p.015503. [16] T. C. Germann, B. L. Holian, P. S. Lomdahl, et al., in 12th International Detonation Symposium, San Diego, (2002). [17] K. I. Nomura, R. K. Kalia, A. Nakano, et al., Appl Phys Lett Vol. 91 (2007). [18] Y. F. Shi and D. W. Brenner, Journal of Physical Chemistry C, submitted (2008). [19] Y. F. Shi and D. W. Brenner, J Chem Phys Vol. 127, (2007), p.134503. [20] Y. F. Shi and D. W. Brenner, Phys Rev Lett, submitted (2007).

Solid State Phenomena Vol. 139 (2008) pp 83-88 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.83

Molecular dynamics simulation of nanocrystalline tantalum under uniaxial tension Zhiliang Pan1,a, Yulong Li2,b and Qiuming Wei1 1

Department of Mechanical Engineering, University of North Carolina at Charlotte, Charlotte, North Carolina, 28223-0001, USA 2

Department of Aircraft Structural Engineering, School of Aeronautics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, P.R.China a

[email protected], [email protected]

Keywords: Molecular dynamics; nanocrystalline tantalum; inter-grannular fracture; deformation twinning; phase transition

Abstract. Using molecular dynamics (MD) simulation, we have investigated the mechanical properties and the microstructural evolution of nanocrystalline tantalum (NC-Ta, grain size from 3.25 nm to ~13.0 nm) under uniaxial tension. The results show the flow stress at a given offset strain decreases as the grain size is decreased within the grain size regime studied, implying an inverse Hall-Petch effect. A strain rate sensitivity of ~0.14, more than triple that of coarse-grain Ta, is derived from the simulation results. Twinning is regarded to be a secondary deformation mechanism based on the simulations. Similar to nanocrystalline iron, stress-induced phase transitions from body-centered cubic (BCC) to face-centered cubic (FCC) and hexagonal close-packed (HCP) structures take place locally during the deformation process, The maximum fraction of FCC atoms varies linearly with the tensile strength. We can thus conclude that a critical stress exists for the phase transition to occur. It is also observed that the higher the imposed strain rate, the further delayed is the phase transition. Such phase transitions are found to occur only at relatively low simulation temperatures, and are reversible with respect to stress. Introduction Over the past years, studies have shown that due to the differences of deformation mechanism, nanocrystalline (NC) metals have many extraordinary properties in comparison to the coarse-grained (CG) counterparts. In coarse-grained metals and at moderate to low homologous temperatures (e.g., T 0 . The boundary conditions are: C A (0, t ) = C A0 , C B (0, t ) = C B0 for t > 0 . The starting equations are: ∂CA ∂ 2C A = DA −q ∂t ∂x 2

∂CB ∂ 2C B = DB +q ∂t ∂x 2

Here q ≡ k A C A − k BC B is the rate of the reversible reaction A ↔ B. Quasi-stationary solution (obtained after long times) is given by

(1)

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k A C(A0 ) − k BC(B0 ) ≈ 0

(2a)

∂ C (A0 ) + C (B0 )  ∂ 2 C (A0 ) + C(B0 )   =D   eff 2 ∂t ∂x

(2b)

Here D eff = (D A k B + D B k A ) (k A + k B ) . By substituting the concentration profiles A or B, C(A0 ) , or C(B0 ) into the starting equation (1a) or (1b), respectively, one obtains formulas for the reaction rate ∂ 2C(A0 ) ∂ C(A0 ) ∂ 2 C(B0 ) ∂ C(B0 ) ( 1 ) q (x , t ) ≈ D − = −D + A

B

∂t

∂x2

∂x 2

(3)

∂t

Note: if reactant B is immobile ( D B = 0 ), Eq. 3 is simplified and is valid at all times (not only under quasi-stationary conditions). It may serve in the general case as the basis for calculation and direct measurement of the reaction rate from the immobile component concentration.

1.2

q (1)

20 q (1) / (CA0 / t)

Concentration / CAO

For long specimen t 1/3).

Figure 2 A comparison between the partial pair distribution functions of amorphous GeSe2 (dotted curves) and amorphous GeSe4 (full curves). Both systems comprise 120 atoms (80 Ge and 40 Se for GeSe2, 96 Ge and 24 Se for GeSe4). Statistics have been collected after a quench from the liquid state and structural relaxation at T = 300 K lasting 20 ps.

Amorphous GeSe4 is highly chemically ordered, with partial coordination numbers relative to Ge of n GeSe = 3.9, n GeGe = 0 (i.e. essentially no homopolar bonds are detected), and partial coordination numbers relative to Se of n SeSe = 1, n SeGe = 1. The picture for GeSe4 that emerges from the results on the liquid is therefore confirmed by the more recent calculations for the glassy state. Indeed, 90 % of Ge atoms are found in Ge(Se1/2)4 units and 70 % of Se atoms are involved in Se-Se homopolar bonds. However, the large number of two-fold bonded Se atoms (95 %) calls for a deeper analysis of the bonding configurations, since three different two-fold bonding configurations can be found. In the first case Se has two Ge nearest-neighbors (hereafter termed GG configurations), in the second case Se has two Se nearest-neighbours (SS configurations) and in the third case Se is linked to both a Ge atom and a Se atom (GS configurations). Two radically different situations could therefore be encountered. On the one hand, GeSe4 could comprise a phase separated structure made of Ge(Se1/2)4 tetrahedra and Se chains. This case corresponds to a distribution of GG, SS and GS configurations equal to 50 %, 50 %, 0 %, respectively. On the other

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hand, the Se atoms atoms could be positioned in between two Ge(Se1/2)4 tetrahedra to give a structure where tetrahedra are connected via Se-Se homopolar bonds. This case corresponds to a distribution of GG, SS and GS configurations equal to 0 %, 0 %, 100 %, respectively. In our simulation the distribution of GG, SS and GS configurations is 25 %, 47 %, 23 %, respectively, showing that Ge(Se1/2)4 tetrahedra are connected both by shared Se atoms (GG configurations) and by Se chains (GS with also some SS configurations). In Fig. 2 we compare the pair distribution functions gαβ(r) obtained for amorphous GeSe4 and amorphous GeSe2 [20]. The close resemblance between the two gGeSe(r) functions is due to the presence of Ge centred tetrahedra which dominate the nearest-neighbor Ge-Se environment. The behavior of the two gGeGe(r) functions at distances larger than 4 Å differs by the presence of a fairly broad peak at r = 5 Å in GeSe4 followed by an oscillation in the range ≈ 5.5-7 Å. A single peak is found in the case of GeSe2. At r < 4 Å, both gGeGe(r) functions feature a double-peak profile, indicating the presence of both corner- and edge-sharing tetrahedra. The distances associated with these two sets of configurations are more sharply separated in the case of GeSe4. Homopolar GeGe bonds involve only a few percent of Ge atoms in amorphous GeSe2 and they are even less frequent (i.e. essentially absent) in GeSe4. The most revealing indication of the structural differences between these two glasses is provided by the gSeSe(r) pair distribution functions. In the case of amorphous GeSe2, the peak at r ≈ 2.4 Å is due to a departure from chemical order, as extensively detailed in our previous work on the topology of GeSe2 disordered networks [17-20]. The significance of this peak is, however, very different in amorphous GeSe4 since the composition ensures that Se-Se homopolar bonds are compatible with a high degree of chemical order and are a mere consequence of the presence of Se-Se connections between Ge(Se1/2)4 tetrahedra. Our results are in very good agreement with neutron diffraction work performed by the group of P. S. Salmon [14-15]. This group has studied the change in topology of bulk quenched GexSe1-x glasses over a wide composition range (x = 0, 0.2, 0.25, 0.33 and 0.4) and the Bhatia-Thornton number-number partial structure factor SNN(k) and pair distribution function gNN(r) were measured [21]. It should be recalled that for Ge-Se systems these quantities are very close to the measured total structure factor and pair distribution function, respectively, due to comparable values for the coherent neutron scattering lengths of Ge and Se of natural isotopic abundance (bGe = 8.185 fm and bSe = 7.970 fm). In this experimental work, the transition from amorphous Se to amorphous GeSe4 was viewed as a break-up of the Se chain structure caused by the addition of Ge atoms which are responsible for the appearance of fourfold coordinated units. The formation of Ge(Se1/2)4 tetrahedra leads to nearest-neighbors Ge-Se bonding distances of rGeSe = 2.36 Å, the same as the value found for the main peak in the calculated gGeSe(r). The first peak in gNN(r) gives a total coordination number n = 2.44, consistent with the CON model in which n GeSe = 4, n SeGe = 1, n GeGe = 0 and n SeSe = 1 and close to the calculated value n cal = 2.38. Si-doped heterofullerenes. The present analysis of the stability of Si-doped heterofullerenes is focused on the comparative behavior of neutral and charged systems. As a first step (part (a)), we shall recall our main results on the structural and dynamical properties of neutral Si-doped heterofullerenes. Then (part (b)), we shall briefly summarize some conclusions drawn on the basis of a preliminary study devoted to the effect of charge on heterofullerenes. The systems carry either a positive or a negative charge but thermal effects are not taken into account. So as a final step (part (c)), a comparison of temperature-induced instability effects will be presented for neutral and charged systems. (a) In a recent series of papers, we were able to propose a criterion for the stability of neutral Sidoped heterofullerenes [10-12]. First, an extended isomer search showed that the most stable configurations are those in which the Si atoms are spatially separated from the C atoms, the different species forming two distinct homogeneous sub-networks. The resultant distorted cage structure features the typical conjugated pattern of C-C bonds, while Si atoms cap the fullerene in a protruding fashion. Changes in the electronic properties of fullerenes caused by doping have the

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effect of creating new reactive sites. Bonding in Si-doped heterofullerenes is characterized by charge transfer from the Si atoms to the more electronegative C atoms. As a general trend, it can be shown that, for a given Si (C) atom, the amount of positive (negative) charge increases with an increasing number of C (Si) nearest neighbors. In the case of C atoms, we found significant negative charge (up to -0.2e) for those C atoms bonded to Si atoms. In turn, neighboring Si atoms have the largest positive values (0.2e -0.3e). Away from the border of the Si-C interface and in the Si-regions, Si atoms take vanishingly small positive or negative values (up to –0.1e) when surrounded by three neighbors of the same kind. Therefore, the observed structural segregation results in an overall electronic charge transfer from Si-populated to C-populated regions. To better understand the results obtained from the first-principles molecular dynamics simulations on C36Si24 and C30Si30 it is necessary to introduce a classification scheme for the Si atoms. Si atoms at the border with C atoms are called outer Si atoms, Siout, while Si atoms within the Si-regions are called inner Si atoms, Siin. Inner atoms can be further labeled as the first (Sifi), second (Sise) or third (Sith) neighbors of outer Si atoms. The outer Si atoms bear a substantial positive charge (average value of 0.27e) corresponding to highly coulombic Si-C bonds. This interaction stabilizes the Si-C interface by compensating for the unfavorable sp2 environment. Figure 3 Temporal evolution of the coordinates of Si and C atoms in neutral C30Si30. The displacements are calculated with respect to time t = 0 and, in the case of Sith atoms (full black line) (see text), the curve is moved upwards on the ordinate by 1 Å.

An analysis of the time-dependent displacements from the initial atomic positions of the Siout and Siin sets of atoms in C36Si24 and C30Si30 revealed that a dynamical instability occurred, as demonstrated by the drastic change observed in the atomic displacements at large times (see Fig. 3). Based on these pieces of evidence, we were able to propose a criterion for the largest number of Si atoms that can replace C atoms within a cage without causing its disruption. Positive charges on the Siout atoms are common to all C60-mSim systems for increasing m, due to charge transfer towards the neighboring C atoms. Therefore, the propensity of Siout atoms to withstand the sp2 environment does not change with increased doping. Moving from C40Si20 to C36Si24 and eventually to C30Si30, the number of Siout does not exceed 12 and it appears that structural instability is due to an increased number of Siin atoms. This rationale shows that C40Si20, in which the number of Siin atoms equals that of Siout (10) is the best candidate for the upper limit of structural instability in C60-mSim. This argument is fully consistent with our FPMD simulations, where fragmentation becomes apparent only for m > 20 due to the relatively short length of our temporal trajectories. (b) In view of these results, the question arises about the extension of these ideas to the case of charged systems. Is the dynamical stability significantly modified by the introduction of a charge, either positive or negative? The answer to this question requires some knowledge of the charge distribution after removal or addition of an electron. To address this issue we have optimized the

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structure of the most stable isomer of C30Si30 for both positively and negatively charged systems. Hereafter we shall denote the bond between a pentagon and a hexagon as ph and the bond between two hexagons as hh. We found that C-C bonds are not affected by the charge state. In the negatively charged case, Si-Si bonds involving inner Si atoms have ph bonds that are slightly shrunken and hh bonds that are slightly extended. It appears that the double-bonded nature of hh couples with an additional repulsion between Si centers which is compensated by an opposite effect for the single ph bonds. Overall, the effect of charge on Si-C bonds is very limited and cannot be rationalized in a systematic way, being no larger than 1%. By performing a population analysis to obtain the Mulliken charges for each atom, non-negligible negative charges are found on the C atoms when they have nearest neighbor Si atoms. Conversely, when C atoms have no Si neighbors, the charges are vanishingly small. In the case of Si atoms, substantial positive charges appear when Si atoms are the nearest neighbors to C atoms (outer Si atoms). This general tendency can be found in all three systems regardless of their charge state. The situation changes for inner Si atoms where the values of the Mulliken charges are found to be strongly dependent on the charge state. For negative C30Si30, 12 of the 20 inner Si atoms have negative values. This number decreases to five for the neutral case and to one for positive C30Si30. Therefore, the addition of a negative charge results in a localization of small negative charges in the inner Si region. Overall, and on the basis of these findings, it appears that negatively charged heterofullerenes are better suited to play the role of chemically reactive nanosytems than positive ones. Along these lines, two types of behavior can be clearly identified. The extraction of one electron has little effect on the structure and amounts to a redistribution of charge at the Si-C border. However, the addition of an electron is reflected by a topological modification of the inner Si network which bears an overall higher negative charge. (c) To achieve a complete description of the stability of Si-doped heterofullerenes in the presence of a charge, the positively and negatively charged isomers of C30Si30, obtained via the structural optimization described above, were studied at finite temperatures by first-principles molecular dynamics. The heating schedule followed the one used in our work on the neutral system [12]. It consists of a stepwise increase of the thermostat temperature from T = 1000 K up to T = 1500 K, T = 2000 K, T = 3000 K, T = 4000 K and finally T = 5000 K during a total temporal trajectory of 24 ps.

Figure 4 The same as in Fig. 3 but for the case of negatively charged C30Si30.

The behavior shown in Fig. 3 (neutral system) and in Fig. 4 (negatively charged system) is similar, with larger displacements characterizing the inner Sith atoms of the Si atoms at the Si-C border. In addition, the interval of stability is comparable, extending up to 16 ps in both cases. Interestingly, the interval of stability is shorter in the case of positively charged C30Si30, which undergoes fragmentation after only 12 ps (Fig. 5). By focusing on the actual fragmentation

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mechanism, one notices from the two series of snapshots presented in Fig. 6 that the C and Si atoms show drastically different temperature dependence. The C portion of the cage does not experience any bond-breaking event. As a consequence, the increase in displacement of the C atoms is due to those C atoms involved in Si-C bonds. The Si part of the heterofullerene falls apart when the repulsion between Si atoms bearing an equal sign is amplified by thermal motion. This happens for those Si atoms furthest from the Si-C border.

Figure 5 The same as in Fig. 3 but for the case of positively charged C30Si30.

Figure 6 Snapshots of the dynamical evolution of negatively charged C30Si30 (left two columns) and positively charged C30Si30 (right two columns) during the first stages of fragmentation. Si atoms are black, C atoms are grey (yellow in color).

These results confirm the atomic-scale picture already collected for the case of neutral systems and ensure that the description of heterofullerene stability and fragmentation proposed in Ref. 12 can be safely applied to charged systems. FPMD calculations extended to other C30Si30 isomers do not feature any systematic effect of the charge on the dynamical stability. These conclusions are consistent with the viewpoint that dynamical effects cannot be predicted on the basis of structural

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optimizations that are performed when temperature is not taken into account. In a multi-scale approach, in which simplified recipes are constructed by extracting information on the interactions from more accurate schemes, this observation is of fundamental importance for reproducing the observed large-scale behavior. References [1] R. Car and M. Parrinello, Phys. Rev. Lett. Vol. 55 (1985), p2471. [2] D. Marx and J. Hutter, in Modern Methods and Algorithms of Quantum Chemistry, edited by J. Grotendorst, John von Neumann Institute for Computing, Jülich, NIC Series, Vol.1 ISBN 3-00005618-1, pp. 301-449, 2000. [3] Rigidity Theory and Applications, edited by M.F. Thorpe, P.M. Duxbury, Plenum Press/Kluwer Academic, 1999. [4] X. Feng, W.J. Bresser, P. Boolchand, Phys. Rev. Lett. Vol. 78 (1997), p4422. [5] M.J. Haye, C. Massobrio, A. Pasquarello, A de Vita, S.W. de Leeuw, R. Car, Phys. Rev. B Vol. 58 (1998), p. R14661. [6] T. Guo, C. Jin, R.E. Smalley, J. Phys. Chem. Vol. 95 (1991), p.4948. [7] M. Pellarin, C. Ray, P. Mélinon, J. Lermé, J.L. Vialle, P. Kéghélian, A. Perez, M. Broyer, Chem. Phys. Lett. Vol. 277 (1997), p.96. [8] C. Ray, M. Pellarin, J. Lermé, J.L. Vialle, M. Broyer, X. Blase, P. Mélinon, P. Kéghélian, A. Perez, Phys. Rev. Lett. Vol. 80 (1998), p.5365. [9] M. Pellarin, C. Ray, J. Lermé, J.L. Vialle, M. Broyer, X. Blase, P. Kéghélian, P. Mélinon, A. Perez, J. Chem. Phys. Vol. 110 (1999), p.6927. [10] M. Matsubara, C. Massobrio, J. Chem. Phys. Vol. 122 (2005), p. 084304. [11] M. Matsubara, C. Massobrio, J. Phys. Chem. A Vol. 109 (2005), p,4415. [12] M. Matsubara, J. Kortus, J.C. Parlebas, C. Massobrio, Phys. Rev. Lett. Vol. 96 (2006), p.155502. [13] M. Matsubara, C. Massobrio, Applied Physics A Vol. 86 (2007), p.289. [14] I. Petri, P.S. Salmon, Phys. Chem. Glasses Vol. 43C (2002), p. 185. [15] P.S. Salmon, J. Non-Cryst. Solids Vol. 353 (2007), p.2959. [16] S.R. Elliott, Nature (London) 354 (1991), p. 455. [17] C. Massobrio, A. Pasquarello, R. Car, Phys. Rev. Lett. Vol. 80 (1998), p. 2342. [18] C. Massobrio, A. Pasquarello, Phys. Rev. B Vol. 68 (2003), p.020201(R). [19] C. Massobrio, A. Pasquarello, Phys. Rev. B Vol. 75 (2007), p.014206. [20] C. Massobrio, A. Pasquarello, J. Phys. Condens. Matter Vol. 19 (2007), p. 415111. [21] The expressions for all the partial structure factors in the Bhatia-Thornton form can be found in H.E. Fischer, A.C. Barnes and P.S. Salmon, Rep. Prog. Phys. Vol. 69 (2006) p. 233.

Solid State Phenomena Vol. 139 (2008) pp 151-156 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.151

Morphological evolution of intragranular void under the thermal-stress gradient generated by the steady state heat flow in encapsulated metallic films: Special reference to flip chip solder joints Tarik Omer Ogurtani1, a and Oncu Akyildiz1, b 1

Department of Metallurgical and Materials Engineering, Middle East Technical University, 06531, Ankara, Turkey a

[email protected], [email protected]

Keywords: Computer simulations; Models of non-equilibrium phenomena; Electromigration; Temperature and stress gradient.

Abstract. The morphological evolution of intragranular voids induced by the surface drift-diffusion under the action of capillary forces, electromigration (EM) forces, and thermal stress gradients (TSG) associated with steady state heat flow is investigated in passivated metallic thin films via computer simulation using the front-tracking method. As far as the device reliability is concerned, the most critical configuration for interconnect failure occurs even when thermal stresses are low if the normalized ratio of interconnect width to void radius is less than certain range of values (which indicates the onset of heat flux crowding). This regime manifests itself by the formation of two symmetrically disposed finger shape extrusions (pitchfork shape slits) on the upper and lower shoulders of the void surface on the windward side. The void growth (associated with supersaturated vacancy condensation) on the other hand inhibits anode displacement but enhances cathode and shoulder slit velocities drastically, which causes lateral spreading. Introduction Thermal stresses generated by differential thermal expansion in the encapsulated metal interconnects in ultra large scale integration (ULSI) chips have long been recognized as an important subject. The importance arises from the fact that the stresses can be large enough to cause immediate or, even worse, delayed failure of the metal lines [1]. The system of interconnects in an ULSI device includes flip-chip solder joints. EM in flip-chip solder joints has become a serious reliability concern when the current density reaches to 108 A/m2 levels, which is about two orders of magnitude smaller than that in Al and Cu interconnects. The solder joint experiences not only the EM but also thermomigration (diffusion due to a thermal gradient), which has been shown to be present by Ogurtani [2] in the unified linear instability analysis (ULISA) of surfaces and interfaces in strongly anisotropic systems. The key feature of EM and thermomigration in a flip-chip solder joint is the current crowding at the cathode and anode contacts on the chip side. The major heat source for the solder joints is the Al (or Cu) line on the silicon die side. As a result, an extremely high temperature gradient of approximately 1200 ºC/cm (1.2×105 ºK/m) is observed (reported by Roush et al. [3] in Pb-In alloy across flip-chip solder joints [4]). Similarly, Ye et al. [5] found in Pb-Sn solder joints a temperature gradient of 1500 ºC/cm (1.5×105 ºK/m) under very high current stressing (estimated at about 1.3×108 A/m2 due to current crowding). The most interesting and thorough study on the failure modes of flip-chip eutectic Pb37/Sn63 solder joints is reported recently by Basaran et al. [6]. They observed that when the direction of the EM is the same as thermomigration the damage is very severe. When the direction of the EM and thermomigration are opposite, thermomigration forces dominate and the total damage is smaller. Physical and Mathematical Modeling In the presence of the thermal stress system, induced by the applied uniform temperature gradient (TG), the evolution dynamics of surfaces or interfacial layers may be described in terms of

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surface normal displacement velocities Vord by following the enlarged and well-posed free-moving boundary value problem in 2-D space for ordinary nodal points, using normalized and scaled parameters and variables [7]. The individual contributions arise from the TSG and the uniform bias field associated with elastic strain energy density (ESED) and elastic dipole tensor intensity (EDTI) are introduced separately by considering the quasi-static temperature variations over the surface layer {∂T / ∂t ≅ 0} during the motion of a pre-existing void:

V

ord

=

∂  ∂   g   ∆g o + Ξ σ g +Σ σ h b v g r d h g r d  ∂l ∂l    2   σ g    ∆g o + κ + Σ −Μ   bv  bv grd  h    

  

2

 + χϑ + κ    

(1)

In the governing equation, Eq. 1, the superscript ‘g’ over the normalized hoop stress, σ hg , indicates that this quantity is associated with the TSG. The dimensionless parameters Σgrd and Ξgrd correspond, respectively, to the intensities of the ESED and the EDTI contributions on the stress driven surface drift diffusion, and associated with the TSG. The second group of terms in Eq. 1 is related to the growth (condensation or evaporation) kinetics. κ is the local curvature and is taken to be positive for a convex void or a concave solid surface (troughs). In the above expression l is the curvilinear coordinate along the surface (arc length) in 2-D scaled space with respect to lo . lo is an arbitrary length scale, and the metric radius of void ro is chosen as a natural scaling length: l o = ro . χ is the electron wind intensity (EWI) parameter, ϑ is the normalized electrostatic potential generated at the surface layer due to the applied electric field intensity.

(

o ∆gbv (T ) = gvo − gbo

)

represents the thermal part [2] of the Gibbs free energy of transformation

(GFEOT) for a flat interface, and is normalized with respect to the minimum value of the specific surface Gibbs free energy of the interfacial layer denoted by gσo . gvo and gbo are the volumetric Gibbs free energy densities, respectively, for the realistic void and bulk phases. In the present o context, ∆gbv (T ) >0 indicates condensation of the super saturated vacancies (void shrinkage), and o ∆gbv (T ) < 0 points to evaporation (void growth) on a flat surface. Μ bv is the normalized temperature and stress dependent generalized transformation mobility [2]. In the present enlarged formulation of the problem, the bar sign over the letters indicate the following scaled and normalized quantities:

t = t /τ o ,

ϑ=

ϑ Eol o

Ξ grd =

l = l / lo ,

,

κ = κ lo ,

wo = wo / l o ,

) e Z Eol 2o χ= , Ω go

σ hg =

σ σ

(1 + ν ) l2o 3gσo

(

)

V  − β grad T − T  , Tr λσ o  

L = L / lo ,

σh

(

)

l o  − β grad T − To   

Σ grd

,

(2)

1 − ν 2 ) l3o ( 2  − β grad (T − T )  = 2 E gσo



o 

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Similarly, EDTI energy may be given by u EDTI = −Ωσ λ ⊗ σ as suggested by Kröner [8]. Where λ is the elastic dipole tensor of mobile defects and it is given by λ = (1/ 3) Tr λ Iˆ in the case of spherical (isotropic) point defects. Iˆ is the idempotent dyadic. Since in general, we assume that the traction is equal to zero for the free surfaces in the plain strain condition in 3-D space, the trace may be given by Trσ ≡ (1 + ν )σ h , where ν is the Poisson’s ratio. The normalization here is done with respect to the applied remote TSG given by Fσ = − β  grad (T − To )  . This quantity in the present studies is associated with the constant and uniform TG generated by the steady state heat flow in the void free interconnect line (particular solution). β is called thermal modulus, and it is given by

β = α E / (1 −ν ) . The constant α is the thermal coefficient of linear expansion, ν is Poisson’s ratio, and E is the modulus of elasticity, or Young’s modulus. To is the reference temperature of the stress-free state, above and below of which thermal stresses are hydrostatic compression and triaxial tension, respectively, according to the Duhamel-Neumann law [9]. Results and Discussions

To see the effects of the TSG on the eutectic solder joint failure mechanism, we performed a few simulations using a set of parameters, specially chosen to duplicate conditions of the test modulus of Ye et al. [5], and Basaran et al. [6]: Ξgrd = -0.3 → -1.4×105 ºK/m and χ = [6:20:25] → [1.3:4.3:5.4] ×108 A/m2. Where the last stated high current densities are in the range of one used by Basaran et al. [6] during the pulsed direct current stressing. In these simulation experiments a very large aspect ratio  w = 50; L = 50  is utilized since the solder joint module employed in the above cited works has a diameter about 2w=150 µm and height of 2L=100 µm. A preexisting void of 5 µm radius is inserted in close proximity to the hot cathode region (five µm away in real space). This configuration corresponds to a void of radius 1 µm localized at x = 8 having an aspect ratio of L = 10 in normalized space. This proximity is very important to simulate the heat flux crowding, taking place between the cathode electrode (dummy silicon die with only Al conduct trace on it) and the void surface, that results extremely unusual behavior, which cannot be predicted otherwise.

Figure 1. The slit formation on the shoulders of a preexisting void induced by the TSG Ξgrd = 0.3 → -1.4×105 ºK/m associated with intense current crowding at the entrance into the solder bump

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in a eutectic 37Pb63Sn solder joint (ro = 5 µm, wo = 50 µm). The direction of the TSG is parallel to the direction of the electron wind (χ = 20 → 4 ×108 A/m2). In Fig. 1, the simulation results are presented in a concise fashion, where only results of the Ξgrd = -0.3 (hot cathode) and χ = 20 experiment is reported in detail. This EWI parameter corresponds to a current density of about J ≈ 4 ×108 A/m2, which is also used by Lin et al. [10] for the in situ observation of void formation and propagation in solder joints under current stressing. That means TSG and the EM driving forces are acting along the same direction, and they are reinforcing each others. The first effect of the void proximity to the cathode edge, which is assumed to be rigid Dirichlet BC in the stress calculations, may be anticipated by observing the fact that there is about a factor of eight enhancement on the magnitude of the normalized total and body stresses (associated with the unit nominal temperature gradient) compared to the void situated at the center of the film. Here the nominal thermal stress is identically equal to zero, since the bias stress is assumed to be zero (the mean temperature of sample is at the reference temperature). For the present case, the cathode and anode regions experience about ±28 MPa hydrostatic stress. This is calculated using a value of βPbSn ≈ 3.7 MPa/ºK for the thermal modulus of eutectic 37Pb63Sn, which is in turn obtained from a 15 ºC temperature drop across the sample (147 to 132 ºC). The mean temperature is TM = 139 ºC according to Fig. 8. of Ref. [5]. This results a temperature gradient given by ΞPbSn = 0.3 → 1.4×105 ºK/m in solder joint, under the zero bias stress condition.

Figure 2. Effects of void growth associated with supersaturated vacancy condensation on the morphological evolution and displacement kinetics. Extreme heat flux crowding at the cathode edge is observed due to the proximity effect. The growth inhibits anode displacement while showing drastic enhancement in the cathode and shoulder slits velocities, which causes lateral spreading. The centroid displacement does not show any influence from the growth process. The first effects of growth are seen in the anode and cathode displacement kinetics. The anode velocity drops down steadily from 24 to 13.5 in normalized time and length scale as may be seen, respectively from Fig. (1b) and Fig. (2b). The cathode velocity shows rather drastic enhancement

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with flattened front compared to the previous case, where there was almost no growth. Actually, there is a strong resemblance in the 2D-morphological evolution between the computer simulation presented in Fig. (2a), and the secondary SEM pictures of solder joint modulus tested by Basaran et al. [6], (See, Fig. 11a-b of Ref. [6]), in regards to the void growth and spreading behavior along the Ni UBM-solder interface. These SEM pictures illustrate clearly the initial as well as the final stages of the preexisting void produced during the manufacturing, where complete failure takes place after 61 hours. Conclusions As it has been demonstrated with the two consecutive computer simulation experiments that, the role played by the properly formulated growth [2, 7] term on the morphological evolution kinetics of preexisting void becomes an indispensable factor for the true understanding of the failure mechanisms of flip chip solder joints. It seems that it is also the primary term in the governing equation, Eq. 1, which causes the speedy lateral extension or spreading of the damage region, along the contact area of the flip chip solder joint by consuming EM induced supersaturated athermal vacancies in the presence of the combined effects of the heat and current crowding, which are enhanced drastically by the proximity of the void to the hot zone. Acknowledgements The senior author (T.O) has the great pleasure to dedicate this article to his teacher and close friend Professor William D. Nix of Stanford University on his 70th birthday, who inspired him to go deep into the theory of dislocation dynamics in the beginning of the sixties. This work is partially supported by the Turkish Scientific and Technical Research Council, TUBITAK, through a research Grant No. 107M011. References [1] Z. Suo, Adv. Appl. Mech. Vol. 33 (1997) p.193. [2] T. O. Ogurtani, Phys. Rev. B Vol. 74 (2006) p.155422. [3] W. Roush and J. Jaspal, Proceedings of Electron. Compon. 32nd Conference, San Diego, CA, 1982, p.342. [4] A. T. Huang and K. N. Tu, J. Appl. Phys. Vol. 100 (2006) p.03351264. [5] H. Ye, C. Basaran, and D.C. Hopkins, Appl. Phys. Lett. Vol. 82 (2003) p.1045. [6] C. Basaran, H. Ye, D.C. Hopkins, D. Fear and J. K. Lin, Journal of Electronic Packaging, Vol. 127 (2005) p.157. [7] T. O. Ogurtani, O. Akyildiz, J. Appl.Phys., (in revision_ 2007). [8] E. Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer, Berlin, 1958). [9] Y. C. Fung, Foundation of solid mechanics (Prentice-Hall, Inc. New Jersey, 1965) p.385. [10]Y. H. Lin, Y. C. Hu, C. M. Tsai. C.R. Kao and K. N. Tu, Acta Mater. Vol. 53 (2005) p.2029.

Solid State Phenomena Vol. 139 (2008) pp 157-164 © (2008) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/SSP.139.157

Effect of C on Vacancy Migration in α-Iron Chu-Chun Fua, E. Meslinb, A. Barbu, F. Willaime and V. Oison* Service des Recherches de Métallurgie Physique, CEA/Saclay, 91191 Gif-sur-Yvette Cedex France a

[email protected], [email protected]

Keywords: diffusion, impurity, carbon, iron, ab initio, thermodynamic, kinetic, equilibrium, irradiation.

Abstract. Carbon atoms are always present in Fe-based materials, either as impurities even in high purity samples or as an intrinsic constituent in steels. Density Functional Theory calculations have been performed to study the interaction between C atoms and vacancies (V) in α-Fe. We find that the formation of VCn complexes is energetically favourable for n ≤ 3, with VC2 being the most stable one. The energy gain corresponding to the clustering reaction VCn-1 + C → VCn depends mainly on the strength of C-C covalent bonds. The vacancy diffusivity is shown to be significantly modified by the formation of vacancy-carbon complexes, exhibiting non-Arrhenius behaviour. Effective vacancy diffusion coefficients in α-Fe are calculated as a function of temperature and carbon content using a simplified thermodynamic model. The results are discussed in detail in the limiting case of excess of C with respect to vacancies. Introduction Carbon is one of the essential components in steels, it is an interstitial solute in bcc iron. Although when carbon is present only in very small amounts, it is expected to have a crucial effect on the energetic and kinetic properties of point defects (vacancies and self-interstitial atoms) as suggested by electrical resistivity recovery experiments [1]. In particular, carbon interacts strongly with vacancies in α-iron whereas its interaction with self-interstitial atoms has been suggested to be weaker but also attractive [1,2]. As indicated by many experimental TEM (transmission electron microscopy) measurements giving the growth speed of interstitial loops under electron irradiation [3,4], the vacancy migration energy is indeed sensitive to the impurity content in the sample (e.g. C). It is significantly higher in carbon-doped iron (0.72 to 1.5 eV) than in high purity iron (~ 0.55 eV) [1]. In recent work [5], we have also reported the experimental fitting of the effective vacancy diffusion coefficient in iron containing C on the number and size distribution of interstitial loops under ion irradiation. An effective vacancy migration energy of around 0.83 eV has been obtained, in very good agreement with 0.83 ± 0.09 eV found by fitting thermal He desorption experimental data [6], which is indeed higher than the value for high purity iron from experiments (0.55 eV) and from DFT calculations (0.67 eV)[7]. As indicated by electrical resistivity recovery and positron annihilation experiments [1,2], it is most likely that the decrease of vacancy mobility may be due to the trapping of vacancies in small vacancy-carbon complexes. In order to understand the vacancy mobility and the evolution of microstructures in steels in non-equilibrium conditions e.g. under irradiation, a quantitative description of these complexes is required. As a first step, we report here ab initio results on structures and stabilities of the most relevant complexes containing one vacancy interacting with n C atoms, as well as the effective diffusivity of vacancies in C-doped iron as a function of carbon content in thermal equilibrium or under irradiation when thermal vacancies dominate over irradiation induced vacancies [8].

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Method of calculation The present study has been performed using the Density Functional Theory (DFT) based SIESTA code [9]. The calculations are spin polarized and use the generalized gradient approximation (GGA). Core electrons are replaced by non-local norm-conserving pseudo-potentials. Valence electrons are described by linear combinations of numerical pseudo-atomic orbitals. The basis set for Fe is the same as in [7]. The basis set for C consists in two localized functions for the 2s states and six for the 2p states, the respective cutoff radii are 2.2 Å and 3.3 Å. Also five functions for the 3d states are included as polarized orbitals. The charge density is represented on a regular 0.078 Å width grid in the real space. Supercell calculations are performed to study the defect properties. Results have been obtained on 54 and 128 atom cells using a 4x4x4 and 3x3x3 k-point grids respectively. The Methfessel-Paxton scheme with a 0.3 eV broadening is used. Unless otherwise mentioned all the results correspond to constant pressure calculations (P=cte) i.e. the structures are optimized by relaxing both the atomic positions and the shape and volume of the supercell. The binding energy between various defects is defined as the energy difference between the situations where they are infinitely separated from each other and where they are close and interact with each other. Positive binding energies (Eb > 0) mean attractive interactions.

Results and discussions Properties of C in iron, The lowest energy solution site for carbon in bcc Fe is found to be the octahedral site with a solution energy of 0.42 eV with respect to C in graphite, which is consistent with the experimental low solubility of C in ferritic steels. We have found that the tetrahedral site is less stable by 0.86 eV in good agreement with previous DFT results [10,11]. When sitting at an octahedral site, rather strong bonds are formed between C with its two first nearest neighbors and four second nearest neighbors, the respective bond lengths are 1.79 Å and 2.00 Å respectively. The hybridization of the 2p band of C and the 3d of the first Fe neighbor makes the C atom slightly antiferromagnetic (-0.3 µB) with respect to the other Fe atoms in the bulk. The solution volume of C is rather large, about 1.0 atomic volume of Fe. Significant tetragonal lattice distortion (compressive strain field) is therefore created around a C atom [11]. A C octahedral migrates to an equivalent neighboring position via a tetrahedral site with an energy barrier of 0.87 eV, this value is in excellent agreement with previous DFT results of 0.86 and 0.90 eV [10,11], and with 0.87 eV from resistivity recovery experiments [1]. The interaction between C and a vacancy is significantly attractive. First, in order to determine the optimal number of C atoms bound to a vacancy, we have studied the structures and stabilities of small VCn complexes. Their lowest energy configurations are shown in Fig. 1, which are In good agreement with previous DFT plane-wave calculations [12]. The configurations of all these complexes are indeed consistent with existing positron life time measurements showing that C atoms occupy off-centered positions around a vacancy [2]. In particular, we find the best configuration for the VC complex to be one with the C atom being at a first neighbor octahedral site of the vacancy (Fig. 1 (a)). The corresponding V-C binding energy is 0.41 eV, where the reference states are isolated vacancy and C octahedral in bcc iron. The C in the VC complex is closer to the vacancy than its original octahedral site, the V-C distance is 0.38 a0 (iron lattice parameter) instead of 0.5 a0 . The configuration where the C is a second neighbor of the V has almost zero binding energy and that with the C inside the vacancy is unstable. Concerning the VC2 complex, the best configuration found consists of two C atoms sitting in neighboring octahedral sites around a vacancy (Fig. 1 (b)). The two C atoms approach the vacancy in order to form a covalent bond. The resulting C-C bond length is 1.45 Å, which is similar to the first neighbor distance in graphite (1.45 Å) according to the present approach. This strong covalent

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bond induces a strong VC-C binding energy of 0.77 eV. When adding a third carbon to a VC2 complex, two 1.45 Å C-C bonds are built, forming a 107° angle. The clustering reaction is still exothermic, but the resulting energy gain is significantly lower (0.12 eV). Finally, the VC4 complex consists in 4 C atoms decorating a vacancy, where the four 1.57 Å C-C bonds form a square-shaped ring. The formation of VC4 from C and VC3 induces however an energy loss of 0.62 eV as shown in Fig. 2.

Figure 1. Schematic representation of the lowest energy configurations for: VCn ≤ 4 complexes, where cubes, gray and black spheres symbolize vacancies, Fe and C atoms respectively. All the C atoms are at their optimized positions, and only Fe-C within the represented cubic unit cells are shown. The energetic balance of above described reactions can be better understood by splitting the binding energies into two contributions namely ‘Chemical’ and ‘Mechanical’ in a similar way as in ref. [13], where the ‘Mechanical’ contribution accounts for the energy gain associated to the atomic relaxation when moving a C from its original octahedral site to a near-vacancy site, and the ‘Chemical’ contribution is the remaining part representing the variation of system energy related to the removing of C from its initial site and the insertion to its final without further lattice relaxation. As shown in Fig. 2, the ‘Mechanical’ term is always positive since the recovery of lattice distortion when a C is removed from its octahedral site induces a stronger energy gain (0.98 eV) than the energy loss related to the relaxation due to the insertion of carbon near a VCn-1 complex. This energy loss increases with n, and has the largest absolute value (-0.75 eV) for the addition of the fourth carbon.

Figure 2, ‘Chemical’ and ‘Mechanical’ contributions and resulting binding energies of VCn ≤ 4 complexes. Values from constant pressure 128-atom cell calculations are shown. Positive values indicate gain of energy. On the other hand, the ‘Chemical’ contribution mainly results from the energetic competition of C-C bond formation and Fe-C bond breaking during each reaction. In order to better understand the ‘Chemical’ contribution, it is essential to analyze the strength of C-C and C-Fe interactions. It is worth noticing that not only C-C but also C-Fe bonds are rather directional in view of the charge density plots [10]. We have therefore calculated the respective C-C and C-Fe bond orders for the

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VCn complexes. They are estimated by considering the off-diagonal elements of the Mulliken atomic overlap population matrix, which are proportional to the bond strength between different atoms [14]. Note that we only focus on the relative values of bond orders in the following analysis. From Table 1 we see that C-C bonds are indeed stronger than C-Fe bonds. Also, the strength of C-C bonds decreases from VC2 to VC4. The C-C bond orders in VC2 are similar to that in graphite (0.51) which has a sp2 hybridization, and the bond orders in VC4 are very close to that in diamond (0.37) with a sp3 hybridization. The corresponding values in VC3 are in between. This bond strength analysis shows consistent results with the decrease of ‘Chemical’ contribution of the binding energies. This contribution has a maximum value for VC2 due to the creation of the strongest C-C bond. We wish to point out that the binding energies follow closely their ‘Chemical’ energies, which are mainly governed by the strength of the respective C-C bonds.

Table 1. Bond orders of C octahedral and VCn complexes in Fe C–C

C – Fe

Cocta

-

0.19 (4), 0.29 (2)

VC

-

0.25 (4), 0.21 (1)

VC2

0.50 (1)

0.25 (4), 0.15 (2)

VC3

0.43 (2)

0.12 (4), 0.18 (1) 0.13 (4), 0.23 (4), 0.24 (2)

VC4

0.32 (4)

0.15 (16), 0.25 (4)

C-C and C-Fe bond orders for C octahedral and VCn ≤ 4 complexes. Values in parentheses indicate the number of bonds corresponding to each bond order. In the VC3 case (Fig. 1 (c)), Fe-C bond orders of the central and the side C atoms are shown in two consecutive lines.

To evaluate the error bar of the binding energies, we compare the present SIESTA results with those from other DFT-GGA plane-wave codes using 54-atom cells [10,15]. Binding energies of clustering reactions to form VC and VC2 complexes are listed in Table 2. We note that the discrepancies between values from different approaches i.e. different pseudo-potentials and basis sets range from 0.1 to 0.2 eV. Table 2. Comparison of complex binding energies from various DFT-GGA calculations

V + C → VC VC + C → VC2

SIESTA (this work) 0.52 0.89

PWSCF (this work) [15] 0.65 1.13

VASP [10] 0.44 0.80

Clustering reactions and corresponding binding energies (energy gain in eV) of VC and VC2 complexes. Values from constant volume 54-atom cell calculations are shown, where positive binding energies indicate attractive interactions.

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Effective vacancy diffusion coefficient in presence of C, The high stability of these VCn ≤ 3 complexes suggests that they may significantly modify the properties of vacancies e.g. their concentration and effective diffusivity in and out of thermal equilibrium. We therefore try to estimate the effective vacancy diffusion coefficients with three carbon concentrations: 1.2, 12, and 120 appm within the following assumptions: (i)

The concentration of isolated vacancies is determined by their formation energy, i.e. [V] = exp(-EVf / kBT),

(1)

where EVf = 2.12 eV according to our DFT calculations, kB is the Boltzmann constant, T is the absolute temperature, and the entropy has been neglected for simplicity, i.e. SVf = 0.0. This condition may correspond to either the system being in thermal equilibrium or under high temperature irradiation when the concentration of vacancies is controlled by sinks of point defects e.g. dislocations, surfaces and grain boundaries [8]. (ii)

All the C atoms are in solution, either as isolated impurities or forming part of a complex. The formation of carbides is neglected, which is reasonable given the amount of C considered in this work. The total concentration of carbon ([CT]) is therefore a constant and [CT] = [C] + 2[C2] + 3[C3] + [VC] + 2[VC2]+ 3[VC3]

(iii)

(1)

The concentration ratios between different species are assumed to be their respective ratios in equilibrium. The complex concentrations can therefore be defined by the mass-action law: [VmCn] = [V]m[C]n exp(EbT / kBT) with 0 ≤ m ≤ 1 and 1 ≤ n ≤ 3

(2)

where EbT is the complex total binding energy with respect to its constituents, i.e. one (or zero) vacancy and n isolated C octahedrals. The values for all the species considered are listed in Table 3. (iv)

In view of experimental evidence on the trapping of vacancies in vacancy-carbon complexes and the reduction of vacancy mobility, we adopt, by simplicity, the extreme case where all the VCn complexes are immobile.

Under these assumptions, the effective vacancy diffusion coefficient DVeff in the presence of carbon can be given by the following expression: DVeff = DV[V] / [VT]

(3)

where DV is the diffusion coefficients of isolated vacancies calculated with a migration energy of 0.67 eV and a standard pre-exponential factor of 8.6x10-3 cm2/s and [VT] is the total concentration of vacancies: [VT] = [V] + [VC] + [VC2] + [VC3]

(4)

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Table 3. Total binding energies of small vacancy-carbon complexes V2 0.30

C2 0.09

C3 0.20

VC 0.41

VC2 1.18

VC3 1.30

Total binding energies EbT (in eV) corresponding to the formation of VCn complexes from one vacancy and n C atoms infinitely separated from each other, with 0 ≤ n ≤ 3. Values from fully relaxed 128-atom cell calculations are shown. The resulting DVeff and [VT] are shown in Fig. 3. We note that, besides the C-free case, all the curves exhibit strong deviation from an Arrhenius behaviour even when the C content is as small as 1.2 appm. Three temperature regimes can be roughly identified where the diffusion coefficients show significantly different slopes, i.e. different effective migration energies can be determined assuming a local Arrhenius behaviour. These regimes are indicated in Fig. 3 for iron containing 12 appm of C. At high temperatures, all the curves coincide with that of carbon-free case, whereas at intermediate and low temperatures, the effective diffusion coefficients decrease with increasing carbon content confirming the strong reduction of vacancy mobility in presence of C. Also, as a direct consequence of Eq. 3, the total vacancy concentration increases with C content, which means a decrease of the corresponding effective vacancy formation energy assuming an Arrhenius law.

Figure 3, Variation of effective vacancy diffusion coefficients (DVeff) as a function of temperature are plotted (upper graph) for iron with four C concentrations ([CT]) from 0.0 to 120 appm. The respective total vacancy concentrations [VT] are also shown (lower graph).

To gain insight into the effects of various vacancy-carbon clusters on the vacancy mobility, we analyze in detail the case of iron with 12 appm of C (Fig.4). At first place, the best Arrhenius fit of the vacancy diffusion coefficient is obtained for each of the three temperature regimes. An effective migration energy of 0.67 eV is found as in the carbon-free case at high temperatures (T > 772 K), the corresponding pre-exponential factor is 0.008 cm2/s. As shown in Fig. 4, most complexes are dissociated at such high temperatures and isolated vacancies (V) are indeed the dominant species. On the opposite side, all the vacancies are trapped in immobile complexes at low temperatures (T < 450K), the vacancies can only migrate through dissociation. The effective migration energy found (1.86 eV) is consistent with 1.85 eV, the total dissociation energy of VC2,

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which is the dominant complex in this temperature regime. This dissociation energy is assumed to be the sum of free vacancy migration energy (0.67 eV) and the total binding energy of VC2 (1.18 eV see Table 3). Due to a co-existence of various species with similar concentrations, the diffusion coefficient is clearly not Arrhenius at intermediate temperatures ( 450 K < T < 772 K). In particular, there are practically the same amounts of V, VC and VC2 around 660 K. The best fit of DVeff gives a migration energy of 1.25 eV. It is worth stressing that there is always an excess of C atoms with respect to vacancy by many orders of magnitudes in all the cases in this study, thus, the above described results concerning DVeff and dominant species as functions of temperature depend essentially on the species binding energies but not on the specific C content. Only a shift of the temperature regimes frontiers towards high temperatures and a narrowing of the intermediate temperature window are expected at increasing C concentrations. Also, as stated in assumption (1), the present results may also be valid to predict vacancy diffusivity in presence of C under high temperature irradiation.

Figure 4, Effective vacancy diffusion coefficient (DVeff) versus reciprocal of temperature for iron containing 12 appm of C (upper graph), where dashed lines with different migration activation energies are also ploted. Only the fractional concentrations of VCn complexes are shown in the lower graph.

Summary We have performed Density Functional Theory calculations to investigate the lowest energy structures and stabilities of VCn complexes in bcc iron. A vacancy can be decorated by up to 3 C atoms, with VC2 being the energetically most favorable configuration, thanks to the formation of a strong covalent C-C bond as in graphite. A simple thermodynamic model has been applied to evaluate the effective vacancy diffusivity in systems with large excess of C with respect to vacancies. The resulting effective diffusion coefficients show non-Arrhenius behaviour even with only 1.2 appm of C. The present results reveal the important role of small vacancy-carbon complexes, in particular the VC2, to modify vacancy mobility in presence of C.

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References [*] present address: Université Paul Cézanne Aix–Marseille, Faculté des Sciences et Techniques, Case 152, Campus St Jérôme Avenue Escadrille Normandie–Niemen, 13397 Marseille Cedex 20, France. [1] S. Takaki, J. Fuss, H. Kugler, U. Dedek H. Schultz, Rad. Eff. Vol. 79 (1983) p.87. [2] A. Vehanen, P. Autojarvi, J. Johansson, J. Yii-Kauppila, Phys. Rev. B Vol. 25 (1982) p.762. [3] M. Kiritani, Ultramicroscopy Vol. 39 (1991) p.135. [4] A. Hardouin Duparc, C. Moingeon, N. Smetniansky-de-Grande, A. Barbu, J. Nucl. Mater. Vol. 302 (2002) p.143. [5] E. Meslin, A. Barbu, L. Boulanger, K. Arakawa, B. Radiguet, P. Pareige, C.C.Fu, J. Nucl. Mater., in press. [6] C.J. Ortiz, M.J. Caturla, C.C. Fu, F. Willaime, Phys. Rev. B Vol. 75 (2007) p.100102(R). [7] C.C. Fu, J. Dalla Torre, F. Willaime, J.L. Bocquet, A. Barbu, Nature Mater. Vol. 4 (2005) p.68. [8] J. Verdone, A. Bourret, P. Moser, Rad. Eff. Vol. 61 (1982) p.99. [9] J.M. Soler, E. Artacho, J.D. Gale, A. Garcia, J. Junquera, P. Ordejon, and D. Sanchez-Portal, J. Phys. Cond. Matter Vol. 14 (2002) p.2745.

[10] C. Domain, C.S. Becquart, J. Foct, Phys. Rev. B 69, 144112 (2004). [11] D.E. Jiang, E.A. Carter, Phys. Rev. B 67, 214103 (2003). [12] C. Domain, E. Vincent, C.S. Becquart, proceedings of the third international conference on ‘Computational Modeling and Simulation of Materials’, Acireale, Italie Vol. 441 (2004). [13] W.T. Geng, A.J. Freeman, R. Wu, C.B. Geller, J.E. Raynolds, Phys. Rev. B Vol. 60 (1999) p.7149. [14] C.C. Fu, M. Weissmann, M. Machado, P. Ordejon, Phys. Rev. B Vol. 63 (2001) p.085411 and reference therein. [15] Calculations are performed with the PWSCF package (S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi, http://www.pwscf.org) using an ultrasoft pseudo-potential and an energy cutoff of 30 Ry.

Keywords Index A Ab Initio Ab Initio Calculation Ab Initio Simulation Adsorption Aluminum-Silicon Interface Amorphisation Atomic Structure Atomistic Simulation Au

157 107 23 41 1 1, 71 141 11 29, 47

B BCC Iron Binary Collisions Bismuth Cuprate Bone

59 71 53 11

C Carbon Chain of Displaced Atoms Chemical Diffusion Columnar Equiaxed Transition (CET) Composite Computer Simulation Continuum Modeling Core-Shell Nanoparticle Crystal Crystalline Polymer

157 89 123 129 35 129, 151 95 29 135 119

D Defect Deformation Twinning Detonation Diffusion Diffusion Mechanism Directional Solidification Dislocation Dynamics Dislocation Junction Dislocation Step

77 53 157

F First-Principle Calculations Fuel Cell (FC) Fullerenes

29 41 141

G Grain Boundary Grain Boundary Dislocation Graphen

89 89 41

H Hotspot Hydrogen Hydrogen Storage

77 41 23

I Impurity Interface Intergranular Fracture Iron Irradiation

157 23, 29, 47 83 157 157

J Jetting

77

K 95 83 77 35, 157 89 129 65 65 89

E Elastic Modulus Electromigration

Energetic Material Epitaxial Growth Equilibrium

Kinetic Kinetic Monte Carlo

157 65

L Lamellar Lattice Monte Carlo Layer LiFePO4/C Liquid

135 35 135 95 135

M 119 151

Magnesium

23

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Theory, Modeling and Numerical Simulation

MEAM Mesophase Metal Composites Modeling Models of Non-Equilibrium Phenomena Molecular Dynamic (MD) Molecular Structure Monte-Carlo Simulation Multi-Scale Modelling

1 135 11 119 151 23, 59, 71, 83, 89, 101, 141 119 107 71

11 11 11 83 141

135 129 65 1 151 53 101 135

T Temperature Gradient Thermodynamic

N Nacre Nanocomposite Nanocrystal Nanocrystalline Tantalum Network Glasses

Smectic A Solidification Structure Solute Segregation Stick-Slip Stress Gradient Structural Modulation Structural Property Surface

151 157

V Void Defect

77

W Wave Dynamics

95

O Ordering Oxide Superconductor Oxygen Vacancy

135 53 123

P Pd Phase Transformation Phase Transition Platinum Precipitation Kinetics Pseudopotential Method Pulsed Laser Deposition (PLD)

29, 47 95 83 41 107 41 53

R Radiation Induced Segregation Reaction Front Reactivity Reciprocal Space Mapping (RSM)

107 123 47 53

S Screw Dislocation Segregation Self-Interstitial Clusters Shear Stress Shock Shock Loading Silica Silicon

59 35 59 1 77 11 101 71

X X-Ray Diffraction (XRD)

53

Authors Index A Aboy , M. Abukhdeir, N.M. Akita, T. Akiyama, K. Akyildiz, O. Atkinson, H.V.

71 135 29, 47 53 151 129

B Barbu, A. Bazant, M.Z. Belova, I.V. Biner, S.B. Brenner, D.W. Buehler, M.J. Burch, D. Burshtein, Z.

95 23, 141 113 65 23

D Dai, H.J. Dong, H.B.

Ito, T.

53

K Kaneko, S. Kohyama, M.

53 29, 41, 47

L 157 95 35 59, 65 77 11 95 123

C Ceder, G. Celino, M. Chakraborty, S. Chen, Q. Cleri, F.

I

129 129

Lee, B.M. Lee, K.M. Lee, P.D. Li, Y.L. Liu, X.Y. Lopez, P.

101 101 129 83 59, 65 71

M Marqués, L.A. Massobrio, C. Matsubara, M. Meslin, E. Montone, A. Morikawa, Y. Motooka, T. Munetoh, S. Murch, G.E.

71 141 141 157 23 41 101 101 35

N Noreyan, A.

1

F Fu, C.C. Funakubo, H.

107, 157 53

G Gafurov, U. Ghaisas, S.V. Giusepponi, S.

119 113 23

H Hirabayashi, Y. Hori, F.

53 29, 47

O Ogurtani, T.O. Oison, V. Okazaki-Maeda, K.

151 157 41

P Pan, Z.L. Pelaz, L. Pelleg, J. Poletaev, G.M.

83 71 123 89

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Theory, Modeling and Numerical Simulation

R Rakitin, R.Y. Rey, A.D.

89 135

S Salmon, P.S. Santos, I. Sashidhar, G. Sen, D. Shi, Y.F. Sinder, M. Singh, G. Sinyaev, D.V. Soisson, F. Starostenkov, M. Stoilov, V. Sundararajan, V.

141 71 113 11 77 123 95 89 107 89 1 113

T Taguchi, N. Tanaka, S.

29, 47 29, 41, 47

W Wei, Q.M. Willaime, F.

83 157

Y Yoshimoto, M. Yun, Y.W.

53 101