Dislocations Proceedings

27-31 August 2012 Budapest, Hungary

th4 International Conference onFundamental Properties of Dislocations

PROCEEDINGS

Proceedings

1

4th

International Conference on

Fundamental Properties of Dislocations

DISLOCATIONS 2012

PROCEEDINGS

27-31 August, 2012 Budapest, Hungary

4th

International Conference on Fundamental Properties of Dislocations


2

4th


DISLOCATIONS 2012


Proceedings

ISBN 978-615-5270-01-7

Editor-in-chief: István Groma

Technical editor: Róbert Hohol

All rights are reserved for the Experimental Physics at Eötvös University and the Department of Materials

Physics, except the right of the authors to (re)publish their materials wherever they decide. This book is a

working material for the 4th International Conference on Fundamental Properties of Dislocations.

The professional and grammatical level of the materials is the authors' responsibility.

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TABLE OF CONTENTS

Articles are edited in the order of the scientific programme

Inga VATNE Three-dimensional Crack Initiation Mechanisms BCC-Fe

under Loading Modes I, II and III

7

Enrique GALINDO NAVA A Thermostatistical Theory for Plastic Deformation 13

Aarne POHJONEN Dislocation Mechanism of Protrusion Formation in the

Presence of a High Electric Field

18

Vaclav PAIDAR Dislocations in C11b and BCC Lattices 23

Ichiro YONENAGA Dislocation Activities in Si under High-magnetic-field 29

Masood HAFEZ HAGHIGHAT In-situ Transmission Electron Microscopy of Dislocation-

defect Interaction in BCC-Fe (P-11)

33

Genichi SHIGESATO Screw Dislocation Slip Behavior in Single Crystals of Pure

Iron and FE-4.5%SI Alloy (P-29)

37

Kenta TSUJII The Effect of Deformation Twinning on the Brittle-to-

ductile Transition in Fe-Al Single Crystalline Alloys (P-31)

41

Akihiro UENISHI Crystal Plasticity Analysis of Work Hardening Behaviour

at Large Strains in Ferritic Single Crystals (P-33)

46

Tomáš ZÁLEŽÁK Interactions Between Dislocation Boundaries and

Spherical Precipitates at High Temperatures (P-40)

50

Authors’ index 55

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ARTICLES

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THREE-DIMENSIONAL CRACK INITIATION MECHANISMS UNDER

LOADING MODES I, II AND III

Inga Vatne1, Alexander Stukowski

2, Christian Thaulow

2, Jaime Marian

2

1 Department of Engineering Design and Materials, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2 Lawrence Livermore National Laboratory. Livermore, CA 94551, USA

Abstract

Simulations of fracture initiation and crack propagation under modes I and II are translationally invariant and thus typically performed in 2D to simulate a semi-infinite solid.

When out-of-plane mode III or mixed mode are considered, however, the crack responds three-dimensionally in terms of the growth mechanism and/or the plastic features observed. In addition, in real crack fronts, dislocations are emitted as loops with a characteristic lengthscale.

To capture this length scale and faithfully represent materials behavior, 3D simulations must be employed. Because crack growth is mediated by processes operating at the atomic scale, atomistic resolution is desirable near the crack tip. However, far away from it the laws of elasticity are sufficient to describe the material response and a continuum representation of the material is adequate. The Quasicontinuum method (QC) suggests itself as an ideal technique to bridge both of these limits, yet providing a seamless and consistent description between them. Here, we perform QC simulations of crack growth in static conditions for bcc Fe and compare quasi-2D to fully 3D conditions. We calculate the critical stress intensity factors under both scenarios and identify the growth mechanisms operating in each case. We analyze and categorize the dislocation structures emitted and compare the QC results to expected solutions from analytical models.

Keywords: fracture, quasicontinuum modeling, dislocations

Introduction

Crack propagation and fracture are complex phenomena operating on multiple length scales ranging from atomistic bond-breaking and dislocation emission to macroscopic failure. Under linear elasticity, stress (strain) fields are singular at the crack tip. Thus, methods capable of resolving non-linear fields are required to remove the singularity and to study incipient crack plasticity and propagation. This has led to a voluminous literature on atomistic and multiscale modeling of fracture, studying the influence of crystallographic orientation, geometry, grain boundaries, precipitates etc. (see e.g. [1, 2, 3, 4, 5, 6]). These works reveal the nucleation of dislocations at or near the crack tip that act as carriers of matter away from it. There is thus a strong link between fracture mechanics and dislocation theory, as dislocations emitted from crack tips often mediate and control crack propagation and growth. Dislocation motion and the effect of dislocations on material strength are also complex phenomena that require atomistic modeling and have attracted significant attention in the last decade [7, 8, 9, 10].

However, the vast majority of crack propagation simulations is done in a fairly narrow parametric space, typically consisting of 2D (or quasi 2D) models loaded under mode I. Modes II and III have been comparatively much less studied, even though they are equally significant for understanding fracture. The importance of the influence of modes II and III loading on cracks, how it affects crack propagation and the importance of studying it at an atomistic level are discussed in Refs. [11, 6]. Pons et al. [11] investigates a crack under mixed-mode loading using a 3D phase-field

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model. The influence of mode III loading on the fatigue behavior of a crack in single crystal iron has been studied by Uhnáková et al. [12].

Vatne et al. [13] investigated a crack in single crystal iron under mode I, II and III loading and mixed mode loading.

Here we systematically address the impact of crack depth on 3D crack configurations and compare it to analytical models and 2D simulations. In addition we extend our previous simulations in bcc Fe [1, 13] to three dimensions and mode I, II and III loading considering several crystallographic orientations.

Method and model

To achieve 3D simulations we have employed the Cluster based QC-method [14,15,16,17].

The boundary conditions employed are the isotropic version of the MBL method, as described by Vatne et al. [1,13]. The simulations are performed using the EAM potential by Mendelev [18] for iron, as it is known to predict a compact screw dislocation core in agreement with DFT calculations [19].

To emphasize the comparison and to better identify the differences between 2D and 3D configurations, here we have used two different thicknesses along the z-direction. The dimensions

of the ‘thick’ samples were 288×288×192 unit cells, whereas those of the ‘thin’ samples

were 288×288×10 unit cells. A crack is inserted with the crack tip in the center of the sample. At the crack tip atomistic resolution is used from the outset, while a coarsened description of the system is gradually imposed as one moves farther away from it. The crystallographic orientation investigated has the crack plane oriented as (010) and the crack front along [101].

Results

As mentioned above, the displacement field is imparted by varying the stress intensity factor incrementally. At the point of instability, whether via plasticity or cleavage, we measure the critical stress intensity factor Kc and take note of the mechanism observed. The critical stress intensity factor is taken at the point of which the first crack opening is observed for the crack propagation, and at the point at which the first dislocation is observed to be fully emitted from the crack tip for the dislocation emission. We chose to use Kc as a measure for both crack initiation and dislocation nucleation in order to compare the loading level at which the first dislocations emission event occurs, and to see which events occur first. The results considered here are listed in Table 1, and illustrated in Figure 1 and 2, visualized employing OVITO and DXA, described in refs. [20,21,22] To compare the results to the analytical predictors of Kc calculated as described in ref.[23,24], we use the surface and unstable stacking fault energies computed by Gordon et al. [2] and Müller et al. [25] for the empirical potential employed here.

Table 1: Critical stress intensity factor and crack tip event for both simulations and analytical calculations

The crystallographic orientation investigated is known as the ”easy” twinning orientation and often exhibits crack propagation accompanied by twinning, especially at low temperatures

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and mode I loading. As shown in refs. [1, 2, 26, 27] the mechanism for this orientation is brittle fracture for long and semi-infinite cracks and brittle fracture accompanied by twinning for shorter ones [5, 26, 28, 29, 30, 31].

Figure 1 shows an analysis of the deformation mechanisms for this case. In summary, we observe crack propagation on the (010) plane in mode I loading, edge dislocations accompanied by twinning under mode II loading, and screw dislocations on the 101 plane under mode III loading. Investigating the crack propagation under mode I loading more thoroughly in the 3D sample we observe that the crack front is kinked, and the crack tip shape is varying through the sample, see Figure 4.

Figure 1: Orientation (010)[101]. a) showing the unit cell of the bcc lattice for this orientation. b-c) showing dislocations and Burgers vector as analyzed by DXA [26]. d-f) showing the crystal structure in OVITO

[23]. Colors from common neighbor analysis, where blue is bcc structure, green is fcc, red is twinning and white is no specific crystallographic structure.

Discussion and conclusion

3D effects

The mechanisms and stress intensity factors of the simulations performed of thin samples are very similar to the those using thick samples. This is also consistent with the quasi 2D simulations carried out by Vatne et al. [1, 13], where similar mechanisms as those observed here were observed. This may suggest that 2D simulations are sufficient to capture the essential behavior of

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crack propagation and associated plastic phenomena. This may be intuitively expected because the three basic loading modes are translationally symmetric, but one might consider this to not be the case under asymmetric or mixed-mode loading.

However, the 3D simulations reveal the presence of three-dimensional features such as dislocation loops and their behavior, which evidently do not appear in the thin and semi-2D simulations. Spielmannova et al. [32] also investigated 3D effects of fracture in bcc-Fe and found that dislocations are emitted at lower stresses during the 3D simulations than in 2D simulations.

Another aspect not captured by 2D simulations is the morphology and structure of the crack front in 3D simulations, see Figure 4 and 8. Under most orientations, we see that crack fronts in the thick samples are not straight, but kinked, and that the crack tip does not always maintain the same shape along the entire thickness. For orientation 1, a kinked crack front varying between mostly cleavage crack along the 011 to crack propagation on the 112 plane and crack propagation by void coalescence and growth is observed. Orientation 2 is the one with the straightest crack front, exhibiting an almost kink-free crack front with virtually no differences between the thin and thick samples.

Analytical solutions vs. simulations

Generally speaking, the analytical formulas for Kc overestimate the vales measure in the simulations by up to a factor of 2. Our study suggests that the availability and combination of slip systems plays an important role that cannot be captured with models based on surface and stacking-fault energies. Nucleation energies for dislocations, twins, and fcc-transformed regions are a strong function of orientation and crystallography, and our results suggest that they need to be accounted for to reliably predict Kc and the governing mechanisms.

Dislocation nucleation

Dislocations in bcc materials are expected to mainly glide on 011, 112, or 123 planes with 1/2<111> Burgers vector.

The 1121/2<111> dislocations observed have been comprehensively studied in the literature using atomistic simulations. The reader is referred to published works on screw [33, 9, 34, 35] and edge [8, 7, 36] dislocations with 1/2<111> Burgers vector.

Further studies with this model will be to investigate the influence of temperature, alloying elements and grain boundaries. Iron is known to undergo a brittle-to-ductile transition with temperature [5] and dislocation motion in bcc materials is also known to change with temperature [33, 35, 10]. Thermal effects would also be an interesting topic of study, to explore how finite temperature affects the different modes of loading. The development of the finite temperature cluster-based QC [37,38] is a step toward such possible investigations in the future.

References:

[1] I. Vatne, E. Řstby, C. Thaulow, D. Farkas, Quasicontinuum simulation of crack propagation in bcc-Fe, Materials Science and Engineering: A 528 (15) (2011) 5122–5134.

[2] P. A. Gordon, T. Neeraj, M. J. Luton, D. Farkas, Crack-tip deformation mechanisms in α Fe and binary Fe alloys: An atomistic study on single crystals, Metallurgical and Materials Transactions; A; Physical Metallurgy and Materials Science 38 (13) (2007) 2191, 1073-5623.

[3] G. Beltz, A. Machová, Reconciliation of continuum and atomistic models for the ductile versus brittle response of iron, Mod Sim in Mater Sciand Eng 15 (2007) 65.

[4] A. Latapie, D. Farkas, Molecular dynamics simulations of stress-induced phase transformations and grain nucleatio at crack tips in Fe, Mod Sim in Mater Sciand Eng 11 (2003) 745–753.

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[5] Y. Guo, C. Wang, D. Zhao, Atomistic simulation of crack cleavage and blunting in bcc-fe, Materials Science and Engineering A 349 (1-2) (2003) 29–35.

[6] M. Buehler, Z. Xu, Mind the helical crack, Nature 464 (2010) 4.

[7] Y. Osetsky, D. Bacon, An atomic-level model for studying the dynamics of edge dislocations in metals, Mod Sim in Mater Sciand Eng 11 (2003) 427.

[8] G. Monnet, D. Terentyev, Structure and mobility of the edge dislocation in BCC iron studied by molecular dynamics, Acta Materialia 57 (5) (2009) 1416–1426.

[9] P. Gordon, T. Neeraj, Y. Li, J. Li, Screw dislocation mobility in BCC metals: the role of the compact core on double-kink nucleation, Mod Sim in Mater Sciand Eng 18 (2010) 085008. 13

[10] M. R. Gilbert, S. Queyreau, J. Marian, Stress and temperature dependence of screw dislocation mobility in α-fe by molecular dynamics, Phys. Rev. B 84 (2011) 174103. doi: 10.1103/PhysRevB.84.174103.

[11] A. Pons, A. Karma, Helical crack-front instability in mixed-mode fracture, Nature 464 (7285) (2010) 85–89.

[12] A. Uhnáková, J. Pokluda, A. Machová, P. Hora, 3D atomistic simulation of fatigue behaviour of cracked single crystal of bcc iron loaded in mode III, International Journal of Fatigue.

[13] I. Vatne, E. Řstby, C. Thaulow, Multiscale simulations of mixed-mode fracture in bcc-fe, Mod Sim in Mater Sciand Eng 19 (2011) 085006.

[14] E. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids, Phil Mag A 73 (6) (1996) 1529–1563.

[15] R. Miller, E. Tadmor, The quasicontinuum method: Overview, applications and current directions, Journal of computer-aided materials design 9 (3) (2002) 203–239.

[16] J. Knap, M. Ortiz, An analysis of the quasicontinuum method, Journal of the Mechanics and Physics of Solids 49 (9) (2001) 1899–1923.

[17] B. Eidel, A. Stukowski, A variational formulation of the quasicontinuum method based on energy sampling in clusters, Journal of the Mechanics and Physics of Solids 57 (1) (2009) 87–108.

[18] M. Mendelev, S. Han, D. Srolovitz, G. Ackland, D. Sun, M. Asta, Development of new interatomic potentials appropriate for crystalline and liquid iron, Philosophical Magazine 83 (35) (2003) 3977–3994.

[19] L. Ventelon, F. Willaime, Core structure and peierls potential of screw dislocations in alpha-Fe from first principles: cluster versus dipole approaches, Journal of Computer-Aided Materials Design 14 (2007) 85–94,

[20] A. Stukowski, Ovito-the open visualization tool, Mod Sim in Mater Sci and Eng 18 (2010) 015012.

[21] A. Stukowski, Structure identification methods for atomistic simulations of crystalline materials, Arxiv preprint arXiv:1202.5005.

[22] A. Stukowski, K. Albe, Extracting dislocations and non-dislocation crystal defects from atomistic simulation data, Mod Sim in Mater Sciand Eng 18 (2010) 085001.

[23] M. Müller, P. Erhart, K. Albe, Analytic bond-order potential for bcc and fcc iron — comparison with established embedded-atom method potentials, Journal of Physics: Condensed Matter 19 (32) (2007) 326220.

[24] T. L. Anderson, Fracture Mechanics, 3rd Edition, Taylor and Francis group, 2005.

[25] J. Rice, Dislocation nucleation from a crack tip: An analysis based on the peierls concept, Journal of the Mechanics and Physics of Solids 40 (2) (1992) 239 – 271.

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[26] A. Machová, Atomistic simulation of stacking fault formation in bcc iron, Modelling Simul. Mater. Sci. Eng. 7 (1999) 949–974.

[27] A. Machová, G. Beltz, Ductile-brittle behavior of (001)[110] nano-cracks in bcc iron, Materials Science and Engineering A 387-389 (2004) 414–418.

[28] A. Spielmannová, A. Machová, P. Hora, Transonic twins in 3d bcc iron crystal, Computational Materials Science 48 (2) (2010) 296–302.

[29] P. Hora, V. Pelikán, A. Machová, A. Spielmannová, J. Prahl, M. Landa, O. Cerven, Crack induced slip processes in 3d, Eng Frac Mech 75 (12) (2008) 3612 – 3623, microstructurally Aided Fracture Mechanisms.

[30] L. Cao, C. Wang, Atomistic simulation for configuration evolution and energetic calculation of crack in body-centered-cubic iron, J. Mater. Res. 21.

[31] A. Spielmannová, A. Machová, P. Hora, Crack orientation versus ductile-brittle behavior in 3d atomistic simulations, in: Materials Science Forum, Vol. 567, Trans Tech Publ, 2008, pp. 61–64.

[32] A. Spielmannová , A. Machová, P. Hora, Crack-induced stress, dislocations and acoustic emission by 3D atomistic simulations in bcc iron, Acta Materialia 57 (14) (2009) 4065–4073.

[33] J. Marian, W. Cai, V. Bulatov, Dynamic transitions from smooth to rough to twinning in dislocation motion, Nature Materials 3 (3) (2004) 158–163.

[34] J. Chaussidon, M. Fivel, D. Rodney, The glide of screw dislocations in bcc Fe: atomistic static and dynamic simulations, Acta materialia 54 (13) (2006) 3407–3416.

[35] C. Domain, G. Monnet, Simulation of screw dislocation motion in iron by molecular dynamics simulations, Physical review letters 95 (21) (2005) 215506.

[36] S. Queyreau, J. Marian, M. R. Gilbert, B. D. Wirth, Edge dislocation mobilities in bcc fe obtained by molecular dynamics, Phys. Rev. B 84 (2011)

[37] J. Marian, G. Venturini, B. Hansen, J. Knap, M. Ortiz, G. Campbell, Finite-temperature extension of the quasicontinuum method using langevin dynamics: entropy losses and analysis of errors, Modelling and Simulation in Materials Science and Engineering 18 (2010) 015003.

[38] Y. Kulkarni, J. Knap, M. Ortiz, A variational approach to coarse graining of equilibrium and nonequilibrium atomistic description at finite temperature, Journal of the Mechanics and Physics of Solids 56 (4) (2008) 1417–1449.

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A THERMOSTATISTICAL THEORY FOR PLASTIC DEFORMATION

Enrique I. Galindo-Nava1,2

, Jilt Sietsma2, Pedro E.J. Rivera-Díaz-del-Castillo

1

1 Department of Materials Science and Metallurgy, University of Cambridge, United Kingdom

2 Department f Materials Science and Engineering, Delft University of Technology, The Netherlands

Abstract

A new theory for describing dislocation evolution in metals is presented. The novelty of the approach stems from obtaining an expression for the dynamic recovery term in the Kocks-Mecking equation. A thermodynamic analysis on an annihilating dislocation segment is performed to determine the energy barrier for dislocation annihilation. The statistical entropy associated to energy dissipation of energetically favourable dislocation paths during deformation is introduced. It is demonstrated that statistical entropy features strongly in modelling plasticity at low and high temperatures: 1) the transition between low to intermediate, and intermediate to high temperature dislocation annihilation regimes are delimited by the transitions in the number of microstates; and 2) the average dislocation cell size and misorientation angle evolution as a function of strain, strain rate and temperature are obtained by performing an energy balance between the dislocation forest and the cellular structure formation, expressing the slip energy to form the latter in terms of the statistical entropy. Employing only input parameters obtained from experiments, the new theory is able to reproduce experimental saturation stress, stress-strain relationships, and average cell size evolution at temperatures ranging from cryogenic to near–melting temperature conditions for Cu, Al and Ni at a variety of strain rates.

Keywords: Theory, modelling, stastistical mechanics, thermodynamics, plastic deformation, dislocations

Introduction

One of the most frequently employed phenomenological models to describe dislocation behaviour and work hardening is the Kocks-Mecking (KM) formulation; this accounts for the competition between dislocation generation and annihilation, describing the evolution of the

average dislocation density ρ during deformation [1]

(1)

where k1 is the dislocation storage coefficient, which has been obtained by Kocks and Mecking [1], b is the magnitude of the Burgers vector and f is the dynamic recovery coefficient.

This approach can be directly applied to obtain the flow stress during deformation. It is often input to more sophisticated techniques such as crystal plasticity, micromechanics or discrete dislocation dynamics, providing the material’s hardening behaviour via the average dislocation density considering different slip systems, temperature and strain rate effects. However, a number of parameters are fitted for each alloy composition, and they remain valid only for specific temperature ranges. Moreover, it has been demonstrated that the dislocation annihilation term (f) is the main controller of dislocation evolution during deformation [2], and more fundamental approaches are required to understand further the interplay of dislocations and other defects such as vacancies and the formation of dislocation patterns.

dρ

dγ=

k1

bρ − f ρ

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Theory

A thermodynamic analysis on a dislocation segment l undergoing annihilation is performed to determine the dynamic recovery rate resulting from dislocation slip [3]. Such analysis rests on three assumptions:

i. Dislocation annihilation is a thermally activated process which characteristic velocity v can range from zero up to the speed of sound in the material (c); when the energy barrier for annihilation vanishes, v approaches the speed of sound.

ii. Once dislocations are in close proximity to each other, their strain fields screen their

neighbours’ and undergo impingement.

iii. The energy necessary for a dislocation segment to migrate towards annihilation is proportional to the yield stress of the material (σY ), accounting for the stored mechanical work during deformation.

Assumption i) leads to an expression for the expected dislocation velocity <v> for annihilation in the form of an Arrhenius equation:

(2)

where <∆G> is the energy barrier to be overcome, and it is composed by [3,4]: 1) a dislocation formation energy term Uform, approximated by the strain energy around the segment; 2) a migration energy term Umig; 3) a vacancy energy contribution to dislocation annihilation Uvac, induced by the vacancy chemical work around the segment that is present when the temperature increases; and 4) a statistical entropy contribution accounting for the energy dissipation due to the

energetically favourable dislocation paths during deformation T∆S. <∆G> is expressed:

(3)

where the factor b/l scales the energy contributions to the number of particles along the dislocation line (of length l) participating in the annihilation process [3]. In order to obtain the

dissipation effect (∆S), a microstate is defined as the number of interatomic subunits a dislocation

segment glides during an arbitrary time step ∆t [3]. At high temperatures, where vacancy-assisted dislocation climb prevails, additional microstates are incorporated to account for their interaction [4]. A canonical formalism is employed to obtain the total number of microstates (dislocation

migration paths) Ω:

(4)

where Ωdis and Ωv-d are the number of microstates due to dislocation slip and to the vacancy

dislocation interaction respectively; is the strain rate, =bcρY is a limiting value for the strain

rate [3], a constant related to the speed of sound, ρY is the dislocation density consistent with the

yield point; ϖ=ϖDexp(-Em/RT) is the vacancy migration frequency, ϖD is the Debye frequency and Em is the vacancy migration energy.

From this result, the transition points T0 and Tf , where different annihilation mechanisms

prevail, are obtained by comparing Ωdis and Ωv-d [4]: T0=Em/Rln(ϖD/ ), when only one vacancy–

dislocation interaction microstate is available; and Tf=Em/Rln(ϖD/ ), when vacancy-dislocation

microstates equal those for slip. Below T0, no vacancy effect is present and cross–slip is the main

v = cexp −∆G

kBT

,

∆G =b

lU form +Umig +Uvac −T∆S( ),

Ω = Ωdis +Ωvac =ε0

ε+

ϖ

ε,

ε ε0

εε0

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annihilation mechanism; above Tf , vacancy–assisted dislocation climb is the predominant annihilation mechanism; between T0 and Tf, cross-slip and vacancy-assisted dislocation climb compete.

The statistical entropy then equals [3]

(5)

Combining equations (2-4) the length of the annihilating segment l is obtained [3]. The dynamic recovery term f is defined as the fraction of substance undergoing dislocation annihilation per

dislocation [3]. By defining Vsys=bl∗l as the annihilating volume per dislocation; i.e. the volume of

substance per dislocation that is not dislocated after a certain strain increment, where l∗=12.5b is

the dislocation’s distortion field length (∼98% of the total strain field induced by the dislocation

[3]), then [4]: f=NAρaVsys/wa= NAρa bl∗l /wa, where NA is the Avogadro’s number, wa is the alloy’s

atomic weight and ρa is the material’s density. A detailed formula of f as a function of the temperature, strain rate and material’s physical parameters can be found in [4].

On the other hand, the average cell size (dc) evolution is described by performing a balance between the energy to form a dislocation cell, and the addition of the dislocation forest energy in the non-cellular structure plus the dislocation–slip energy necessary to form the cellular structure

[5]; the statistical entropy features strongly on this analysis as the latter is proportional to ∆S. The average cell size behaviour as a function of temperature, strain rate and the average dislocation

density ρ is equal to [5]:

dc=κc/ρ1/2, (6)

where κc=24π(1-v)/(2+v)(1/2+T∆S/µb3), where v is the Poisson ratio. Additionally, the Young-

Laplace equation is employed to describe the evolution of the average misorientation angle θ, and the dislocation density evolution during stage IV (where dislocation generation and annihilation rates are in equilibrium and only dislocation rearrangement takes place), by expressing the pressure build-up across the cell walls in terms of the entropy of dislocations displacing towards

the walls. Details on the derivation of θ ανδ stage IV can be found in [5].

Figure 1 shows the model results at different deformation conditions for single crystal/coarse grained pure copper describing the saturation and flow stress response (a,b); the average cell size (c,d); the average misorientation angle (f); and severe deformation strains where the Kocks-Mecking equation no longer is valid and a stage IV features due to dislocation rearrengement, [3-5].

∆S= kB lnε0 +ϖ

ε

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Figure 1: Deformation phenomena at various scales. Milimetre: (a) single crystals saturation shear stress, (b) shear stress–shear strain curves. Micrometre: average

dislocation cell size variation with (c) temperature and (d) shear strain. Submicrometre: (e) Severe deformation strains and (f) the average misorientation angle [2-4].

Figure 2 shows additional results for single crystal/coarse grained pure aluminium and nickel showing the flow stress response and average cell size evolution at different temperatures [3,4].

Figure 2: Stress-strain curves of pure (a) nickel and (b) aluminium [3]; (c) average cell size-strain for nickel and aluminium [4]; and

(d) stress-strain curves for nickel incorporating stage IV at different temperatures [4].

Conclusions

Dynamic recovery and average cell size evolution is described for FCC metals employing thermostatistical concepts.

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The introduction of the statistical entropy:

• Incorporates energetically favourable dislocation paths at a given temperature and strain rate are considered.

• Features strongly in modelling plasticity at low and high temperatures.

• Allows for the transition temperatures for vacancy-assisted and cross-slip deformation mechanisms to be recovered.

• Permits the description of plasticity with a single parameter: the average dislocation density.

References:

[1] U.F. Kocks & H. Mecking. Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48 (2003) 171-273

[2] P. E. J. Rivera Diaz del Castillo, M. Huang, Dislocation annihilation in plastic deformation: II. Multiscale

irreversible thermodynamics, Acta Mater. 60 (2012) 2606-2614

[3] E. I. Galindo–Nava, J. Sietsma & P. E. J. Rivera–Díaz–del–Castillo. Dislocation Annihilation in plastic

deformation: II. Kocks-Mecking Analysis. Acta Mater. 60 (2012) 2615-2624.

[4] E. I. Galindo–Nava & P. E. J. Rivera–Díaz–del–Castillo. A thermostatistical theory of low and high temperature

deformation in metals. Mater. Sci. Eng. A. 543 (2012) 110-116

[5] E.I. Galindo-Nava, P. E. J. Rivera Diaz del Castillo, A thermodynamic theory for dislocation cell formation and

misorientation in metals, Acta Mater. 60 (2012) 4370-4378

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INTERACTION OF A SCREW DISLOCATION WITH A NEAR-

SURFACE VOID UNDER TENSILE SURFACE STRESS

Aarne S. Pohjonen1, Flyura Djurabekova

1, Antti Kuronen

2, Steven P. Fitzgerald

3

1 Helsinki Institute of Physics, Helsinki, Finland

2 University of Helsinki, Helsinki, Finland 3 EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon

Abstract

Presence of a near-surface void can affect the dynamics of a metal surface evolution held under high electric field. The tensile stress applied at the surface due to the electric field is able to activate the mobility of dislocations. We show that a screw dislocation which connects the void and the surface with 110 face, where the tensile stress is applied, can perform a cross slip. Such behavior can eventually lead to the helical motion of the surface material outwards the surface forming a protrusion.

Keywords: screw dislocation, void, cross slip

Introduction

In this article we consider a dislocation cross slip on a near surface void in copper under tensile surface stress, which can be caused by an application of high electric field. Let us first examine the conditions where such mechanism can operate. In a suitable stress field, a screw type dislocation line can move from one slip plane to another by a cross-slip process. The cross-slip mechanism has been experimentally observed to localize plastic deformation near a concentrated stress in a nanoindentation experiment [1]. Material irregularities, such as voids or inclusions, can also concentrate the stress by forming a local stress field significantly different from the surrounding uniform stress.

Even in carefully solidified metal crystals the dislocation density is 103 mm－2 and becomes as

high as 1010 mm－2 for heavily deformed metals [2]. Under stress dislocations become mobile, they can multiply and interact with other extended material defects, such as voids. It has been observed that under overall shear in bulk FCC metalsa screw dislocation interacts with voids by passing it through via a cross-slip mechanism [3].

The formation of voids in the presurface region can be due tothe different sources. For instance, if an oxide layer exists on the copper surface, voids can form near the surface due to the Kirkendall effect [4, 5] even in the absence of electric field. The electric field can further affect copper ions to migrate through the oxide layer while creating vacancies near the surface [6]. The voids could also in principle be created or enlarged on a dislocation line if the electric current causes electromigration along the dislocation core [7]. Hence in a vacuum insulated electrode, relevant conditions can exist for cross slip of a screw dislocation line on a near surface void under electric field induced stress.

We have earlier studied material transport caused by dislocation nucleation from a near-surface void [8, 9] under tensile surface stress and shown that the local stress concentration can cause the nucleation of dislocations on the void surface. In this article we consider a mechanism of interaction of a pre-existing screw dislocation with a near-surface void under tensile surface stress.

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19

We show that the dislocation can cross slip to another slip plane. We observe the cross slip to occur by the Fleischer mechanism [10, 11] instead of the more commonly accepted Friedel-Escaig mechanism [12]. Cross-slip via the Fleischer mechanism has earlier been observed inaluminum inmolecular dynamics simulations [13]. We explain thatthe cross-slip operates via the Fleischer mechanism in our simulations due to the small energy involved in creating the additional a/6<110> stair rod dislocation, sincethe dislocation line segment between the near-surface void and the surface itself is short.

Methods

The molecular dynamics (MD) method, which is capable of giving an atomic level description of the cross slip process, was applied in the simulations. We used the MD simulation program PARCAS [14]. Sabochick-Lam interatomic potential [15] was chosen to model the interactions between the copper atoms. A screw type dislocation was introduced in the simulation cell as passing through a near surface void by applying the displacement field with the subsequent relaxation of the cell as, for instance, in [3], as follows,

where x0 and y0 are the x and y coordinates of the straight full dislocation line.

When the system was relaxed during 20 ps, the dislocation split in two partials as shown in Fig. 1.After this a linearly increasing tensile force was exerted on the atoms of the two top atomic layers of the cell during the following 100 ps, when the maximum desired force was reached. After this the simulation was continued with the constant desired stress for 300 ps. The total simulation time was 420 ps. The three bottom atomic layers of the system were held fixed. All the simulations were performed at 600 K to increase the probability of the cross slip process within the simulation time. In order to increase the computational efficiency of the calculations of the cross slip process, in our simulations we have focused only on the upper hemisphere of the simulated void. The size of the void located at 4 nm (Fig. 1a) below the surface in such geometry could be as big as 22.5 nm in radius. We ensured that the stress distribution around hemispherical void is similar to the stress distribution around the full spherical void of the same radius.

Figure 1:Figure 1:Figure 1:Figure 1: The geometries of the simulation setups. A screw type

dislocation is introduced in the cell with a near surface void: a) upper

hemisphere of a spherical void, the dislocation is not pinned; b)

ellipsoidal void, the dislocation is pinned at the bottom of the cell. In

both cases after the relaxation the dislocations split in two partial

dislocations bound by a stacking fault. The atoms belonging to stacking

faults were identified by calculating the centrosymmetry parameter [18,

17]. The insets are the 90o rotatedimages around the vertical axis.

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We also performed the simulations for the full size ellipsoidal voids with a 12 nm semi major axis, located at the same depth as in the first case (Fig. 1b). Such geometry allowed for the insertion of the full screw dislocation line crossing the surface of the void at the bottom and the top. Using such voids we can investigate the interactions of the long dislocation lines, which may cross the void moving along the surface with one end pinned in the bulk.

Results

When the system was relaxed, the introduced dislocation splits in two partials according to the reaction

(2)

The dynamics of the cross slip process in case of the hemispherical void is illustrated in Fig. 2 (top view). For better illustration some stages of the process are shown with the different view in Fig. 3 and Fig. 4. Under the tensile stress 1.0 GPa applied to the surface of the cell with the hemispherical void we observed cross slip from the original slip plane to the intersecting slip plane (Fig. 2) through the Fleischer mechanism. The extended dislocation proceeded on the new slip plane until it was locked as apparently the shear stress on the atoms at the most probable point where the screw dislocation might have performed a cross slip again, was not sufficient. Instead, we observe the nucleation of a new dislocation in the slip plane (111), which is not intersecting with the surface and, hence, does not significantly affect the dynamics on it. However, the newly nucleated dislocation, then, assists the nucleation of a dislocation in the slip plane which is parallel to the originalplane where the screw dislocation was initially inserted, as it can be seen in the bottom inset in Fig.2 and Fig. 3b.The stacking fault formed by the partials of the last dislocation is confined between the surface of the void and the stacking fault formed by the dislocation in the (111) slip plane and from this point we did observe no further evolution of the system during the simulation time. The evolution of the process under a lower stress is much slower and in our simulations we observed only the first cross slip (Fig. 3a). When the surface stress was even lower, no cross slip was observed during the simulation time. However, these results cannot definitely exclude the possibility of the following cross slips, which might appear among the results with higher statistics and longer times.

We observed the formation of a stair-rod dislocation in the cross slip process which indicates that the mechanism operates through the Fleischer mechanismby splitting of the partial dislocation and creating an additional dislocation segment with burgers vector (a/6)[101]. The reaction formula can be written as [12]

(3).

However, since in our simulationsthe length scale is small, the energy involved in creating the

additional dislocation line segment is also small.This explains why the cross-slip observed in our

simulations is performed via the Fleischer mechanism.

[ ]( ) [ ]( ) ( )101 11 1 112 11 1 2 11 11 12 6 6

a a a+ →

[ ]( )101 11 12

a

( ) ( )211 111 112 111 1016 6 6

a a a+ →

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21

Figure 2: Evolution of the screw dislocation on a spherical void under the applied tensile surface stress (top view).The partials of the extended screw dislocation are moving along the void surface lifting the material

above the voidinitiating the formation of a surface protrusion. Only the atoms belonging to void surface and stacking faultsare shown. The left top inset shows the initial dislocation. The most right image shows the

situation stable from 200 ps to the end of the simulation. The initial dislocation is locked and the new dislocation is nucleated on the slip plane (111). The right bottom inset is side view of the same

image.

Figure 3:Detailed views of the first (a) and second (b) cross slips(cf Fig.2).The first cross slipoccurs via the Fleischer mechanism. (a) The initial relaxed dislocation intersecting the surface of the void is framed in the left top corner. The lower insets summarize the same dislocation at the later stages: the first (left to right)

shows the trailing partial exit the original slip plane; the second - both partials have exited the original slip plane, completing the cross-slip. (b) The second step does not show the cross-slip, instead the dislocation is

locked on the slip plane ( )111 , but a new dislocation is nucleated in the (111) slip plane.

For an ellipsoid shaped void 4 nm below the surface under 1.15 GPa surface stress, we observed the similar cross slip as for the spherical void from the original slip plane to the intersecting slip plane which is perpendicular to the surface (Figs. 4a and 4b) by the Fleischer mechanism as described in equation (3).

Figure 4: Screw dislocation moving along the ellipsoidal void under tensile surface stress (top view). a) and b) the partial dislocations have cross-slipped from the original slip plane to another slip plane.

Since the difference of the geometry for the spherical and ellipsoidal voids was in the presence of additional segment of the dislocation pinned at the bottom of the cell, we conclude that the upper segment of the screw dislocation which intersects with the surface and the top of the near-surface void behaves independently on the lower part, performing the cross slip if there is a tensile stress applied to the surface.

Conclusions

We observed a cross slip of a screw dislocation passing through a near surface void under tensile stress for a semi-spherical and ellipsoid shaped voids. The cross slip mechanism operated

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via the Fleischer mechanism, since the energy involved in creating the additional (a/6)<101> dislocation segment was small due to the length scale of the dislocation line. The cross-slip caused material transport from the void to the surface.

References:

[1] E. Carrascoet. al., Phys. Rev. B 68, 180102 (2003)

[2] Jr. W. D. Callister, Materials Science and Engineering, 3rd ed, 1993 (New York: Wiley)

[3] T. Hatano, T. Kaneko, Y. Abe and H. Matsui, Phys. Rev. B 77 (6) 064108 (2008)

[4] B. Selikson, Proceedings of the IEEE 57, 1594 (1969)

[5] H.J.Lee and J.Yu, Jour. of Elec. Mat. 37, 1102 (2008)

[6] J. P. Singh, T. M. Lu and G. C. Wang, Appl. Phys. Lett. 82, 4672 (2003)

[7] A. S. Nandedkar, MRS Proceedings 291,361 (1992)

[8] A. Pohjonen, F. Djurabekova et al., J. Appl. Phys. 110, 023509 (2011)

[9] A. Pohjonen, F. Djurabekova et.al., Phil. Mag (2012), doi:10.1080/14786435.2012.700415

[10] Puschl W, Progress in Materials Science 47,415 (2002)

[11] Fleischer R., ActaMetallurgica 7,134 (1959)

[12] E. Bitzeket. al., Phys. Rev. Lett.100 (2008)

[13] S. Pendurtiet al., Appl. Phys. Lett.88 (2006)

[14] K. Nordlund, PARCAS computer code (2007)

[15] M. J. Sabochick and N. Q. Lam, Phys. Rev. B 43, 5243 (1991)

Proceedings

23

DISLOCATIONS IN C11B AND BCC LATTICES

Vaclav Paidar1, V. Vitek

2

1 Institute of Physics, AS, Prague, Czech Republic 2 University of Pennsylvania, Philadelphia, USA

Abstract

The structure of dislocation cores is related to the crystal symmetry of the direction of the dislocation line and an important material characteristic is whether metastable stacking fault-like defects exist on planes containing the dislocation line. While no such stacking faults exist in BCC metals, metastable faults were found on 013) and 110) planes in MoSi2 with the bcc based C11b

structure by ab initio calculations of γ-surface. Using these results, we discuss the non-planar dissociations of screw 1/2<331] dislocations in MoSi2, analyse their impact on dislocations glide and compare with the analogous effect of non-planar cores of screw dislocations in BCC metals.

Keywords: dislocation cores, bcc metals, C11b intermetallics

Introduction

The tetragonal C11b structure would become the BCC lattice if there were not two different species in the unit cell. Hence, there are certain aspects of dislocations in the C11b compounds that appear to emulate the dislocation properties in BCC metals. This is so in spite of the important difference between the two structures when considering stacking faults. It is well established that in BCC metals there are no metastable stacking faults on any crystallographic plane [1]. On the other hand, metastable stacking faults can exist in alloys with C11b structure as found for example in MoSi2 [2]. Hence, no usual dislocation splitting exists in BCC metals while dislocations may dissociate into partials separated by metastable stacking faults in C11b crystals. In this paper we investigate possible dissociations of 1/2<331] dislocations that relate crystallographically to 1/2<111] dislocations in BCC crystals. We concentrate on non-planar splittings of screw dislocations and establish an analogy between splitting in C11b crystals and core spreading in BCC metals. The C11b alloy we consider is MoSi2 that was studied extensively experimentally [3] and for which stacking faults were identified in calculations employing a DFT based method [2].

In MoSi2 there are several possible slip systems [3-6]. However, the 1/2<331] dislocations that glide on 013) planes appear to control the plastic deformation of this compound. In fact in compression along the tetragonal axis <001], the plastic deformation is exclusively controlled by the mobility of 1/2<331] dislocations. This is the case in spite of the fact that for the ideal c/a ratio of C11b, the magnitude of the 1/2<331] Burgers vector is √6 larger than that of the <100] Burgers vector. Dislocations with the latter Burgers vector have also been observed to mediate the plastic deformation for some orientations of loading. Moreover, there are other dislocations with intermediate magnitudes of the Burgers vectors that can also be activated, for example 1/2<111] or <110] [6]. The 1/2<111] dislocations do not move easily as shown in [7] and this probably relates to their decomposition into the 1/2<001] and 1/2<110] dislocations involving a stacking fault on 001), which blocks their motion, as reported in [8].

As mentioned above, 1/2<331] dislocations are analogous to 1/2<111> dislocations in BCC crystals. It is generally accepted that in BCC metals the core of the 1/2<111> screw dislocation spreads into three 101 planes of the <111> zone [1, 9-12]. This non-planar core is responsible for a high Peierls stress of screw dislocations, which owing to their low mobility control then the plastic behaviour of BCC metals. In atomistic studies two types of non-planar dislocation cores have been

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reported in dependence on interatomic forces used. Both structures have three-fold symmetry but one of them is also invariant with respect to the diad around a <101> axis normal to the dislocation line while the other is not. In the latter case the core is degenerate i. e. two equivalent configurations related by the diad symmetry exist [13]. In both cases the core can be regarded as ‘splitting’ into fractional dislocations on three equivalent 101 planes. In the case of the structure invariant with respect to the <101> diad the Burgers vectors of the fractionals are 1/12<111> while in the non-invariant case 1/6<111>. The non-invariant core was found in most studies employing central-force potentials while calculations that include the non-central character of atomic interactions all give the invariant, non-degenerate core. Hence, it is likely that it is the non-degenerate core that exists in all transition BCC metals.

Planar dissociations of 1/2<331] dislocations had already been discussed in our previous paper [14]. In this paper, the emphasis is on the non-planar splitting of 1/2<331] screw dislocation in MoSi2 when considering possible dissociations involving metastable stacking faults that were

found in DFT based calculations of γ-surfaces for 110) and 013) planes [2]. Only one type of the metastable stacking fault was found on the 110) plane while three different metastable stacking faults were identified on the 013) plane. The positions of local minima of these three stacking faults are schematically depicted in Fig. 1 where possible partial dislocation Burgers vectors are also defined. The energies of these three stacking faults on 013) are relatively high and very similar (1.22, 1.06 and 1.12 J/m2) while the stacking fault energy on the 110) plane is appreciably lower (0.357 J/m2).

Figure 1: Schematic positions of three stacking faults SF1, SF2, SF3 on the 013) plane in MoSi2 determined as minima on the γ-surface, together with the Burgers vectors of possible partial dislocations, marked bn; IC

marks positions corresponding to the ideal crystal.

Non-planar dissociation of 1/2<331] dislocations

In the C11b lattice, two 013) planes are equivalent but the third plane from the <331] zone, the plane 110), is geometrically different. This is in contrast with the BCC lattice where all three 101 planes from the <111> zone are equivalent. Hence, the non-planar splitting of 1/2<331] screw dislocations in the C11b lattice is intrinsically asymmetrical while the core spreading of the 1/2<111> screw dislocations in BCC crystals possesses symmetries mentioned above. Moreover, the stacking fault energy on 110) in MoSi2 is significantly lower than energies of stacking faults on 013) and so the dissociation widths are much larger on 110) than on 013).

There is a number of possible non-planar dissociations of 1/2<331] screw dislocations in the C11b but we consider only those that may affect significantly the dislocation glide. Since the energy of a dislocation split into partials with smaller Burgers vectors is always lower than the energy of the same dislocation dissociated into partials with larger Burgers vectors, the lowest energy dislocation splitting is that depicted in Fig. 2a where all four partials have small Burgers vectors.

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25

This splitting is wide on the 110) plane and the dislocation cannot move easily along any of 013) planes. However, when two partials with small Burgers vectors are located on one of the 013) planes and only one partial on the (110) plane, as in the configuration shown in Fig. 2b, the dislocation glide on 013) can be more easily activated. A more detailed analysis of all possible partial dislocation combinations following from the stacking faults of several types will be discussed in a separate paper.

a) b)

Figure 2: a) Wide splitting of the b = 1/2<331] dislocation into the (110) plane with simultaneous narrow

splitting into (013) and planes, b) Less extended splitting of the 1/2<331] dislocation into the (110)

plane with simultaneous wider splitting into the (013) plane and narrow splitting into the plane. The

lower case bn denote the partial dislocations on the 013) planes as defined in Fig. 1; the subscripts r and s

correspond to the (013) and planes, respectively. B1 = 1/4<111] and B2 = 1/2<110] are the

displacement vectors that lead to the same fault on the (110) plane and their sum is half of the total Burgers vector, i. e. 1/4<331].

Configurations shown in Figs. 2a and b are both composed of partial dislocations with small Burgers vectors and possess low energies when compared with other possible splittings. Interesting is the mechanism of transition from 2a to 2b that enables the glide of the screw dislocation on the (013) plane. During such transition the screw dislocation will pass through higher energy states that are best described by dissociations shown in Fig. 3. First the two partials with the Burgers vectors B1 and B2 on 110) contract into the partial with the Burgers vector b/2, the stacking fault ribbon on the (110) plane shortens and owing to the interaction between the partials the stacking fault ribbon on the 013) planes extends; at this stage the configuration shown in Fig. 3a is attained. The next step, shown in Fig. 3b, is further transition of the Burgers vector from the (110) plane into the (013) plane leaving the partial with the Burgers vector B1 on the (110) plane and transforming the partial on the (013) plane into that with the larger Burgers vector b2. The latter finally decreases its energy by splitting into partials with the Burgers vectors b5 and b7 as shown in Fig. 2b and the screw dislocation may start gliding on the (013) plane.

a) b)

Figure 3: Two intermediate higher energy dissociations of the 1/2<331] screw dislocation involved

in the transition from the configuration in Fig. 2a to that in Fig. 2b

(103)

(103)

(103)

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The full analysis of the energetics of the dislocation splittings has been made using the

anisotropic theory of elasticity to describe the interaction between partials.1 The widths of splittings are determined by the usual balance between the surface tensions of stacking faults and the forces arising from dislocation interaction [16]. In the dissociation shown in Fig. 2a the width of the dissociation on the (110) plane is about 80a and the segments of the stacking fault on 013) planes are less than 2a wide. When external forces push the dislocation to move into one of 013) planes, the width of the dissociation on 110) is substantially reduced and the dislocation can start to move along the plane inclined to 110).

When a half of the total Burgers vector b/2 = B1 + B2 is situated on the 110) plane, as in the configuration in Fig. 2a, the width of dissociation is r1 + r2 = 80 a. However, when the two partials on 110) contract into one partial, as in the configuration in Fig. 3a, the width of the segment of the stacking fault on 110) is reduced to about one half rb = 38 a. When about one half of the Burgers vector is already transferred into the (013) plane, as in the configuration in Fig. 3b, the width of dissociation on 110) is further reduced to about rc = 27 a.

Detailed analysis of dislocation dissociations and related energetics of splittings will be published elsewhere. In this paper we just utilize the so-called b squared criterion to assess the energetic of splittings. Within this approximation the sum, S, of the squares of the Burgers vectors of all partial dislocations constituting a particular splitting is used for a rough estimate of the energy of the dissociated dislocation. For configurations involved in the transition of the 1/2<331] screw dislocation into the (013) plane, as described above, S is plotted in Fig. 4 as a function of the ratio of the widths of splitting on the (013) and (110) planes.

Figure 4: Sum of squared Burgers vectors of the partial dislocations of four configurations involved in the transition of the 1/2<331] screw dislocation into the (013) plane as a function of the ratio of

the widths of splittings on the (013) and (110) planes. The four points at which S is calculated correspond to configurations shown in Figs. 2a, 3a, 3b, 2b, in this sequence.

1 The single-crystal elastic constants of MoSi2 at 0 K used have been given in [15]. They are (in GPa) c11=410, c33=514, c12=115, c13=87.5, c44=207 and c66=200; the c/a ratio is 2.447 (a = 0.3206 nm, c = 0.7846 nm).

0 0.06 0.12 0.18

r(013) / r(110)

1.6

2

2.4

2.8

S

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27

The dissociated dislocations depicted in Figs. 2a and b possess lower energies than the intermediate configurations, Figs. 3a and b, in which the Burgers vector transfers gradually from the (110) plane into the (013) plane. Significantly, the splitting shown in Fig. 2b has the lowest energy and it is reasonable to assume that under an applied stress the dislocation motion begins when the energy maximum, corresponding to the highest S value in Fig. 4, is overcome. At this point about one half of the Burgers vector from the equilibrium configuration originally situated mostly on the (110) plane has been transferred into the (013) plane. At finite temperature this transition occurs, presumably, via the thermally activated formation of pairs of kinks. Importantly, for the loading axis along the <001] tetragonal direction there is no shear stress on the 110) plane that can drive the dislocation. Hence, the transition to the (013) plane will require a large contribution of thermal activation and thus at low temperatures, such as the room temperature, the deformation cannot take place and brittle fracture may ensue. This explanation is consistent with the suggestion in [17].

Conclusions

Complex dislocation dissociations and/or core configurations may exist owing to a variety of possible stacking faults on 013) and 110) planes in materials with the C11b lattice. Exact displacement vectors and energies of these faults were found on the basis of DFT based calculations in MoSi2. The existence of the plethora of stacking faults is in a qualitative agreement with earlier suggestions of Mitchell et al. [17] though both the displacement vectors and energies of the faults are appreciably different. Similarly as the cores of 1/2<111> screw dislocations in BCC metals spread into several planes of the <111> zone, 1/2<331] screw dislocations in the C11b lattice may dissociate in a non-planar way into planes of the <331] zone. Hence, in both cases the Peierls stress of these screw dislocations is very high and they control the deformation properties. Their motion involves transformations from sessile to glissile configurations and in this paper we have suggested the path of such transition into the 013) glide plane. However, while the core of screw dislocations in BCC metals possesses three-fold symmetry, such symmetry is not present in the case 1/2<331] screw dislocations in the C11b lattice. This implies that an anomalously high yield stress can be expected if there is no shear stress driving the dislocation motion in the (110) plane. This is the case of the compressive/tensile axis in the tetragonal <001] direction. While for other orientations of the loading axis the glide of 1/2<331] screw dislocations may take place even at relatively low temperatures, fracture occurs at these temperatures for the loading axis close to <001] and plastic deformation may only occur at very high temperatures. This was, indeed, observed in [3] where for this orientation of the compressive axis plastic deformation took place only at temperatures higher than 1200K.

Acknowledgements

This research was supported by the Grant Agency of the Academy of Sciences of the Czech Republic, contract No. IAA100100920 (VP) and by the Department of Energy, BES Grant No. DE-PG02-98ER45702 (VV).

References:

[1] V. Vitek and V. Paidar, in "Dislocations in Solids", edited by J. P. Hirth (North-Holland, Amsterdam, 2008) vol. 14, p. 439.

[2] M. Cak, M. Sob, V. Paidar and V. Vitek, in "Advanced Intermetallic-Based Alloys for Extreme Environment and Energy Applications" (Mater. Res. Soc.,Symp .2009) vol. 1128, p. U07.

[3] K. Ito, T. Yano, T. Nakamoto, M. Moriwaki, H. Inui and M. Yamaguchi, Prog. Mater. Sci. 42 (1997) 193.

[4] Y. Umakoshi, Y. Sakagami, T. Hirano and T. Yamane, Acta Metall. Mater. 38 (1990) 909.

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[5] K. Ito, H. Inui, Y. Shirai and M. Yamaguchi, Phil. Mag. A 72 (1995) 1075.

[6] S. A. Maloy, T. E. Mitchell and A. H. Heuert, Acta Metall. Mater. 43 (1995) 657.

[7] K. Ito, T. Yano, T. Nakamoto, H. Inui and M. Yamaguchi, Acta Mater. 47 (1999) 937.

[8] S. Guder, M. Bartsch and U. Messerschmidt, Phil. Mag. A 82 (2002) 2737.

[9] V. Vitek, R. C. Perrin and D. K. Bowen, Phil. Mag. 21 (1970) 1049.

[10] V. Vitek, Crystal Lattice Defects 5 (1974) 1.

[11] M. S. Duesbery, in "Dislocations in Solids", edited by F.R.N. Nabarro (North-Holland, Amsterdam, 1989) vol. 8, p.67.

[12] M. S. Duesbery and V. Vitek, Acta Mater. 46 (1998) 1481.

[13] V. Vitek, Phil. Mag. 84 (2004) 415.

[14] V. Paidar, M. Cak, M. Sob and V. Vitek, J. Phys. Conf. Series 240 (2010) 012007.

[15] K. Tanaka, H. Onome, H. Inui, M. Yamaguchi and M. Koiwa, Mater. Sci. Eng. A 239-240 (1997) 188.

[16] J. P. Hirth and J. Lothe, Theory of Dislocations (McGraw-Hill, New York, 1968).

[17] T. E. Mitchell, M. I. Baskes, S. P. Chen, J. P. Hirth and R. G. Hoagland, Phil. Mag. A 81 (2001) 1079.

Proceedings

29

DISLOCATION ACTIVITIES IN SI UNDER HIGH-MAGNETIC-FIELD

Ichiro Yonenaga, Yutaka Ohno, Yuki Tokumoto, Kentaro Kutsukake

Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

Abstract

Dislocation-oxygen impurity interaction in oxygen impurity containing Si crystals was influenced by treatments at temperature of 650 °C under a magnetic field up to 10 T. The critical stress for dislocation generation from a surface scratch varies with the magnetic treatment at 650˚C, depending on the magnetic field intensity and duration. In the case of magnetic field application for 15 min duration, the critical stress starts to increase gradually up to at 8 T. Contrarily, in 1 h application, the critical stress shows a dramatic variation against the applied magnetic field as a first increase to a maximum critical stress at the magnetic field of 1 T, then a gradual decrease and a final approach to the stress level of a specimen without magnetic treatment. That is, the generation of dislocations was effectively suppressed under certain conditions of the magnetic treatments. Such phenomena could not be detected in oxygen-free Si crystals. The results were discussed in terms of spin-dependent solid-state reaction in atomistic binding with oxygen atoms around dislocation core, causing immobilization of dislocations in their macroscopic generation process.

Keywords: dislocation motion, magnetic field, impurity-dislocation interaction, semiconductor Si

Introduction

Recently magnetic field has a keen interest for controlling material structure as grain growth, recrystallization, phase transformation, precipitation and so forth (for example, a paper by Fujii et al. [1]). Even in semiconductors, modifications of defects under external magnetic fields have been reported [2-4], where dislocations become mobile, leading to softening crystals after exposure to a magnetic field. Such phenomena, so-called magneto-plastic phenomena, and relevant mechanisms are discussed in terms of dislocation motion and solid-state reaction, i.e., the spin-dependent release of dislocations from segregated paramagnetic impurities or their complexes and the subsequent relaxed motion of dislocations in matrix crystals, leading to enhancement of plasticity. Badylevich et al. [5] reported that oxygen (O) impurity in silicon (Si), may lose the ability of locking against dislocations due to singlet- triplet-state transition by exposure under a magnetic field of 2T at room temperature (RT), resulting in easy release of dislocations at elevated temperature. We showed that O impurity loses their locking feature against dislocations in Si by heat treatment under a magnetic field of 10T at 650–700˚C and that velocity of moving dislocations in Si crystals was free from influence by the magnetic field [6-8]. Here we review a magnetic efefct on dislocation-O impurity interaction in Si under a high magnetic field up to 10 T at elevated temperatures, based on data reported previously [6-8].

Experimental procedure

Specimens were prepared from an undoped CZ-Si crystal ([O]: 1.1×1018 cm-3) and a high-purity float-zone grown (FZ-) Si crystal ([O]: < 1015 cm-3). Both crystals were dislocation-free. Specimens,

approximately 2×3×15mm3 in size, were finished chemically, following mechanical polishing. Dislocations generated preferentially around such scratches drawn on the surfaces when heated to elevated temperatures. Specimens were first annealed at 650 °C for 1 h and subsequently were treated at the same temperature under an application of magnetic field up to 10 T for durations of

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15 min to 2 h in a furnace installed into a cyclo-cooled superconducting magnet 11 T-CSM in the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University. Then, the specimens were stressed at 650 °C in a vacuum by means of three-point bending. The macroscopic motion of dislocations into the matrix from the scratch was observed by etch pit method. The geometry of the specimens as well as the details of the experimental procedure has been described in the previous papers [6-8].

Generation of dislocations from a surface scratch

Figure 1 shows the distances traveled by the leading dislocation in an array of dislocations plotted against the resolved shear stress in CZ-Si specimens of as-grown, annealed at 650˚C for 1 h, magnetic-treated under 3 and 8 T for 1 h and kept under no magnetic field for 1 h (noted controlled), together with those in FZ-Si, annealed for 1 h and then magnetic-treated under 8 T for 1 h at 650 °C. The dislocations were traveled into the matrix from a scratch during a 110 min stress pulse at 650˚C. It is seen that there is a critical stress for generation of dislocations in all of the CZ-Si specimens and that the travel distance in the specimens increases with stress once the stress

exceeds the critical stress. The critical stress is ≈5 and 9 MPa in as-grown and annealed CZ-Si, respectively, under no application of magnetic field. The critical stress is as high as 13 MPa in the specimen magnetic treated under 3T, while the critical stress is as low as 8 MPa in the specimen treated under 8 T, similar to that of the annealed specimen. The critical stress of dislocation generation in the controlled specimen was almost same as that of annealed specimen. The critical stresses for dislocation generation of FZ-Si specimens, annealed for 1 h and then magnetic-treated under 1 and 8 T for 1 h at 650 °C, are almost zero.

From these results, it can be understood that there is no effect of magnetic treatment in FZ-Si, which means that the observed variation in the critical stress is originating from an effect of magnetic field application on dislocation generation in only CZ-Si, i.e., relating to O impurity. In addition, under some suitable magnetic treatments dislocation generation might be suppressed most efficiently in CZ-Si.

Here, it has been known that dislocation generation from a scratch or surface flaw is suppressed under low stress in Si and other semiconductors doped with certain kinds of impurities, including O impurity [9,10]. The critical stress is understood as a stress required to activate dislocation motion from the immobilized state developed through impurity segregation along the dislocation lines through a dislocation-impurity interaction.

Figure 1: Travel distance of dislocations generated from a scratch at 650 °C in magnetic-treated Si plotted against the stress together with those in the as-grown, annealed, controlled and FZ-Si.

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Critical stress for dislocation generation

The critical stress for dislocation generation from a surface scratch in CZ-Si specimen varies with the heat-treatment at 650˚C, depending on the magnetic field intensity and duration. Figure 2 shows the variation in the critical stress for dislocation generation against the intensity of the magnetic field for the duration of 15 min, 30 min and 1 h at 650 °C. The critical stress for dislocation generation in the as-grown specimen is superimposed. In the case of magnetic field application for 15 min duration, the critical stress starts to increase gradually from 3 T and becomes 15 MPa at 8 T. Contrarily, in the 1 h duration the critical stress shows dramatic variation against the applied magnetic field: first increases to a maximum 15 MPa at the magnetic field of 1 T, then decreases gradually and finally approaches to the stress level of the non-magnetic treated specimen at 8 T. Specimens applied under a magnetic field for 30 min show an intermediate feature between them.

Spin dependent mechanism for dislocation-imuprity interaction

These results suggest a spin-dependent solid-state reaction in impurity–dislocation interaction, which seems a possibility of modification of atomistic configuration and displacement of crystalline defects in semiconductors. For the observed variation of critical stress of dislocation generation, there are some possible microscopic natures of atomistic binding states around dislocations affected by application of magnetic field: First, promotion of the diffusion of oxygen atoms under a magnetic field could enhance accumulation of impurity atoms on dislocations, resulting in an increase in locking strength against dislocations by development of locking agents. However, there were any effects of the magnetic field of 8 T on macroscopic oxygen diffusion at 650 °C up to 120 h in our preliminary experiments as seen in Figure 3.

Second, Russian groups proposed singlet- to triplet-state transition by exposure to a magnetic field up to 2T in Si containing O [3,5], i.e., the locking ability of oxygen complexes against dislocations may be lost by exposure of a magnetic field due to the singlet- to triplet-state transition of atomic binding of SiO2 formed at dislocation cores, which leads to a reduction of their binding energy, resulting in easy release of dislocations. However, their model seems to be rather hard to explain straightforwardly the experimentally observed variation of the critical stress of dislocation generation.

Third, similar to the above model of single- to triplet-state transitions of atomic bonds, in an early stage of the application of a magnetic field some Si-Si binding bonds around dislocation cores

Figure 2: Variation in the critical stress for dislocation generation of 60° dislocations against the magnetic field intensity of the treatment at 650 °C in Si. Numerals show the durations of magnetic

treatment at 650 °C. The critical stress for dislocation generation in the as-grown Si is superimposed.

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may break to make intermediate Si-O binding states temporarily due to the difference in their binding energies [4,11], which leads to the enhancement of locking stress up to a maximum in the critical stress. The change may proceed more rapidly under a higher intensity of magnetic field. In the prolonged stage, such Si-O bindings may be destroyed to easy start of dislocations from immobilized states under a stress.

Summary

Here, modification of dislocation-O impurity interaction in Si by treatments under a high magnetic field was presented. The critical stress for dislocation generation from a surface scratch in CZ-Si varied with an intensity and duration of application of magnetic field. Under certain conditions of the magnetic treatments dislocations were effectively suppressed for the generation. These results suggest the necessity of detailed research of spin-dependent dislocation-impurity interaction in semiconductors, including in-situ observations under a magnetic field.

Acknowledgment

The authors express the gratitude to Dr. K. Takahashi for his experimental assistance.

References:

[1] H. Fujii, S. Tsurekawa, T. Matsuzaki, T. Watanabe, Philos. Mag. Lett. 86 (2006) 113.

[2] V.I. Alshits, E.V. Darinskaya, M.V. Koldaeva, E.A. Petrzhik, Crystallogr. Rep. 48 (2003) 768.

[3] Yu.I. Golovin, Phys. Solid State 46 (2004) 789.

[4] O. V. Koplak, A. I. Dmitriev, T. Kakeshita, R. B. Morgunov, J. Appl. Phys. 110 (2011) 044905.

[5] M.V. Badylevich, Yu.L. Iunin, V.V. Kveder, V.I. Orlov, Yu.A. Ossipyan, Solid State Phenom. 95–96 (2004) 433.

[6] I. Yonenaga, K. Takahashi, J. Phys.: Conf. Ser. 51 (2006) 407.

[7] I. Yonenaga, K. Takahashi, J. Appl. Phys. 101 (2007) 053528.

[8] I. Yonenaga, K. Takahashi, T. Taishi, Y. Ohno, Physica B 401-402 (2007) 148.

[9] I. Yonenaga, J. Appl. Phys. 98 (2005) 023517.

[10] K. Jurkschat, S. Senkader, D. Gambaro, R.J. Falster, P.R. Wilshaw, J. Appl. Phys. 90 (2001) 3219.

[11] J. A. Kerr, D. W. Stocker, CRC Handbook of Chemistry and Physics, 83rd ed. (CRC, London, 2002), pp. 9-52.

Figure 3: Comparison of oxygen concentrations in Si annealed at 650˚C under a magnetic field of 0T and 8T.

Proceedings

33

IN-SITU TRANSMISSION ELECTRON MICROSCOPY OF

DISLOCATION-DEFECT INTERACTION IN BCC-FE

S.M. Hafez Haghighat1,2

, R. Schäublin1

1 Ecole Polytechnique Fédérale de Lausanne, Centre de Recherches en Physique des Plasmas, Villigen-PSI, Switzerland

2 Max-Planck-Institut für Eisenforschung, Düsseldorf, Germany

Abstract

Mechanical properties of crystalline materials strongly depend on the interaction between mobile dislocations and defects. Here we present our observations on the interaction mechanisms of moving dislocations with dislocation type defects in ultra high purity (UHP) bcc-Fe using transmission electron microscopy (TEM) in-situ straining test. Different types of interaction such as the dislocation bowing and screw dipole formation are observed. On the study of dislocation dipole formation we found out that a screw dipole may form by the interaction of a ½ a0[11-1] edge dislocation with an immobile dislocation with screw or near screw character elongated in [-111] direction. The mechanism of the dipole formation is related to the interaction geometry of the dislocation with the screw dislocation; it leads to the bowing of the dislocation arms attached to the defect onto two different glide planes, thus stabilizing a screw dipole, before release.

Keywords: transmission electron microscopy, dislocation-defect interaction, iron

Introduction

Interaction between dislocations and defects in the plastic deformation of materials is being investigated since years using various simulation techniques used along a multiscale approach. However, experimental validation of those is still largely missing. Experimental validation of this microstructure process at the needed atomic level, typically using in-situ TEM, is not trivial due to the limitation in spatial resolution by the electron optics and to the limitation in time resolution when performing such in-situ testing. Recent in-situ TEM studies show that in the deformation of bcc-Fe dislocation sources generate dislocations that then propagate rapidly when having an edge character, while screw character dislocation segments are left in the microstructure, which later control the deformation process of the material [1-3]. Atomistic simulations show that a gliding ½ a0<111> edge dislocation may be stopped at an obstacle and, upon bowing around it, may form a screw dislocation dipole segment [4-6]. This dipole can be stabilized depending on the local stress state and temperature, which impedes the release of the dislocation from the obstacle. Indeed, the dipole screw segments must first cross slip, and then glide towards each other, before the annihilation of the dipole, which allows the release of the dislocation from obstacle [7].

In this paper we report our observation of dislocations interaction with immobile dislocations in bcc-Fe at room temperature. Two interactions are reported here; first the interaction of dislocations avalanche with an immobile dislocation and second the interaction of a moving edge dislocation with an immobile straight screw dislocation leading to the <111> screw dipole formation and annihilation at room temperature. The mechanism leading to such dipole formation is discussed and compared to the atomistic simulation results.

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The in-situ straining specimen is a rectangle 11.5 mm long and 2.5 mm wide. The sample is prepared using three separate parts to reduce magnetism, namely one necked 3 mm disk of UHP Fe and two stainless steel rectangular parts to hold it in the centre. The thickness of the UHP Fe part is ideally cut to around 1 mm. It is then mechanically polished to reduce its thickness down to 100 µm. These parts are then joined by spot-welding. A hole is created at each side of the sample for the gripping. Finally, the sample is electro-polished at -20 °C to -10 °C in a solution of 10% perchloric acid in ethanol to create a small hole with thin edges transparent to electrons as observation area. A TENUPOL 5 unit from STRUERS® was used for electro-polishing at 20.5 V. The specimen is then mounted in a straining low temperature single tilt sample holder from GATAN®. Tests were performed in a JEOL 2010, equipped with a LaB6 gun and operated at 200 kV. Observations are conducted in bright field condition, close to a Bragg condition, and at room temperature, with a diffraction vector g = 011. Stepwise straining was applied instead of continuous straining to avoid rapid deformation and rupture of the sample. Dislocation glide, if any, is recorded in between the straining steps. Acquisition is performed with the 11 Mpixels CCD Orius camera of GATAN® at full frame, providing a frame rate of 2 images per second. The results reported here have been captured in a single grain, whose normal is close to [311].

Results and discussion

The interactions between an immobile dislocation with multiple dislocations are shown in Figure 1. This image shows the interaction of a series of dislocations in the right side of the image with the bended dislocation, shown by the arrow that is immobile during the interaction. Our observations show that when the mobile dislocations reach to the immobile one junctions are formed prohibiting their glide and therefore they start to bow around the immobile dislocation without their release from it. The dislocations curvature seen in Figure 1 is the maximum bowing we observe in our in-situ straining test. From the dislocations glide mechanism and crystallographic analysis one may conclude those dislocations to be of non-screw character. It also appears that other dislocations of the same slip system with a comparable local stress, at the left side of the image, are able to move freely in the vicinity of the immobile dislocation without a direct interaction. They are entirely mobile whereas the others are trapped by the immobile dislocation.

Figure 1: TEM bright field image, g=011, of the interaction of dislocations with a curved immobile dislocation (shown by arrow), forming dislocation junctions, and then bowing around it. The mobile

dislocations appearing in the left side of the image move without a direct interaction with the immobile one

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The local stress that the dislocations experience in this region can be obtained from the force balance between the line tension and the Peach-Koehler force resolved perpendicular to the initial

dislocation line direction of the bowing dislocation; τ =cosφ Gb/D, where G=76 GPa is the shear modulus of pure Fe, b=0.25 nm is the Burgers vector of the ½ a0<111> dislocation and D is the obstacles interspacing that is taken at ~0.8 µm for the bowed dislocations observed in Figure 1 assuming that the drag effect of the free surfaces of the thin foil applied on the two endings of the dislocation is comparable to that of an obstacle row. By taking the angle between the dislocation

line direction and perpendicular to the initial dislocation line at junction, φ, at 45° the stress that the junctions are experiencing is ~17 MPa. Since this stress is within the range to move non-screw dislocations in pure Fe [1, 8] one may conclude that the observed mobile dislocations in the left side of Figure 1 is of edge or non-screw character rather than of screw ones.

Figure 2 shows the formation and annihilation of a [11-1] screw dislocation dipole due to the interaction of a moving edge dislocation, oriented close to the [112] direction, with an immobile screw dislocation, laid along [-111] direction. Figure 2a shows the dislocation impinging on the immobile dislocation. Since the dislocation speed is high for the available time resolution of the camera, the moving dislocation appears blurred in this image. Dislocations in this region move at a speed of about 100 nm·s-1. The moving dislocation then interacts with the immobile one, bows out onto it with further straining, and forms a [11-1] screw dipole as it appears in Figure 2b. At a length of about 120 nm, the dipole disappears, presumably because of cross-slip and mutual annihilation of the dipole screw segments and the dislocation is released, leaving the immobile dislocation behind (Figure 2c). These interaction steps take place within about 2 seconds. This is consistent with the atomistic simulation results showing the <111> screw dipole formation in the interaction of a moving edge dislocation with a nanometric void and dislocation loop [4, 5]. Curiously, while atomistic simulations indicate that the screw dipole should be annihilated in less than 1 ns for dislocation velocities of about 60 m·s-1, experimental results show that it resists for about a second at dislocation speeds of about 10-7 m·s-1 at room temperature.

(a) (b) (c)

Figure 2: Screw dislocation dipole formation and annihilation in [11-1] direction due to the interaction of a moving edge dislocation with an immobile screw dislocation laid along [-111] direction.

The reaction of an edge dislocation with a screw dislocation results in the creation of a jog on each of the dislocations [9]. The mechanism of this jog formation in the interaction of the two dislocations is as followings. In the glide of an edge dislocation towards a screw dislocation crossing the glide plane of the moving dislocation the two arms of the edge dislocation, which are divided by the screw dislocation, are led to two different glide planes by a planar interspacing of one Burgers vector of the screw dislocation due to spiral crystallographic arrangement around the screw dislocation. When the edge dislocation bows out around the immobile screw dislocation the two arms are at different slip plane. In order to allow the release of the edge dislocation from the immobile dislocation one of the screw arms needs to cross-slip to be annihilated with the second

0.1

µµµµm

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one. This later reaction results in the release of the edge dislocation from the immobile screw dislocation leaving a jog that is one Burgers vector long. It can be concluded from this observation that the interaction of the two dislocations leads to a dipole of two screw segment arms residing on two different glide plane, which is the reason for the stability of the observed [11-1] screw dipole. For annihilation of the screw dipole the cross-slip of at least one of the arms is necessary, which is dependent on the local stress state and temperature since it is a thermally activated phenomenon.

Conclusion

TEM in-situ straining experiments were used to study the dislocation-defect interaction in ultra high pure Fe. The observations show that defects of dislocation character can hinder a moving dislocation in bcc-Fe, leading to a dislocation dipole. The observations in this work show that moving dislocation can interact with an immobile dislocation in the similar way they do with nanometric obstacles. This indeed contributes to materials strengthening due to the forest hardening in crystalline materials. Due to rapid occurrence of these interactions and limitation in the spatial resolution of the in-situ TEM observations a detailed elaboration of the interaction mechanism is not trivial. Atomistic simulation techniques are recommended to be used for the detailed study of the observed interactions.

References:

[1] D. Caillard, Acta Materialia, 58 (2010) 3493-3503.



[4] D. Terentyev, Y.N. Osetsky, D.J. Bacon, Scripta Materialia, 62 (2010) 697-700.

[5] Y.N. Osetsky, D.J. Bacon, Philosophical Magazine, 90 (2010) 945-961.

[6] S.M. Hafez Haghighat, G. Lucas, R. Schaeublin, IOP Conf. Series: Materials Science and Engineering, 3 (2009) 012013.

[7] S.M. Hafez Haghighat, R. Schäublin, Philosophical Magazine, 90 (2010) 1075-1100.

[8] S.M. Hafez Haghighat, D. Terentyev, R. Schaublin, Journal of Nuclear Materials, 417 (2011) 1094-1097.

[9] D. Hull, D.J. Bacon, Introduction to dislocations, Fouth edition ed., Butterworth-Heinemann, Oxford, 2001.

Proceedings

37

SCREW DISLOCATION SLIP BEHAVIOR IN SINGLE CRYSTALS OF

HIGH PURITY IRON AND FE-4.7%SI ALLOY

Genichi Shigesato1, Ken Kimura

2

1 Advanced Technology Research Laboratories, Nippon Steel Corporation, Futtsu, Japan

2 Steel Research Laboratories, Nippon Steel Corporation, Futtsu, Japan

Abstract

Mechanical properties, dislocation slip behaviors and screw dislocation core structures of high purity iron and 4.7 mass% Si steel single crystals have been studied by performing simple shear tests at room temperature. In 4.7% Si steel, anisotropic plastic behavior was seen. The critical resolved shear stress (CRSS) of the anti-twining sense was higher than that of the twining sense. Meanwhile high purity iron showed less anisotropy in CRSS at room temperature.

High resolution STEM observation implied a difference in the dislocation core structure of these materials. The STEM images of screw dislocation cores in 4.7% Si steel clearly showed the threefold strain contrast. The difference of the core structures is considered to relate to the difference of the mechanical properties.

Keywords: screw dislocation, iron, Fe-Si alloy, core structure

Introduction

Flow stress of high purity iron and Fe-Si alloy single crystals at a low temperature has been reported to strongly depend on its orientation [1-4]. Specifically when the maximum resolved shear stress plane (MRSSP) was around 112, the flow stress of shearing in the twining sense was much lower than that of shearing in the anti-twining sense as reported in other bcc metals. At room temperature, however, high purity iron exhibited much less orientation dependence while Fe-Si alloy showed as much orientation dependence as at a low temperature.

Slip line observations indicated that the slip planes were 110 at a very low temperature (< 80K) [4-5] and 112 at a relatively high temperature [1-2] for both materials when MRSSP was around 112. The slip lines observed in high purity iron deformed at around room temperature, however, were wavy while those observed in Fe-Si alloy deformed at room temperature were remained straight. Thus the slip planes might be different in microscopic resolution in these materials.

The differences in the flow stress and the slip systems of both materials were correlated to the core structures of screw dislocations. In the current study the screw dislocation core structures in high purity iron and Fe-Si alloy were observed by using aberration corrected STEM and then compared with each other.

Experiments

Two types of steel were prepared by a vacuum melt furnace; Fe-0.0003 mass% C (high purity iron) and Fe-4.7 mass% Si (4.7% Si steel). These materials were annealed in an Ar atmosphere to enlarge the grain size. High purity iron was annealed at 880°C for one week and Fe-Si alloy was annealed at 1400°C for two days. Single crystal strips of a 25 by 18 by 1mm dimension and a plane normal to <110> and a longitudinal direction of <111> were prepared by cutting out from large grains. Simple shear tests were performed with those strips (Figure 1).

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Dislocations in samples sheared by 2% in the twining and anti-twining senses were observed. Thin foil specimens were picked out and thinned by FIB. Two types of specimens were prepared; ones for longitudinal section observations and ones for transverse section observations. The foil plane of longitudinal sections was (11-2), which was parallel to the shear direction, and that of transverse sections was (111), which was perpendicular to the shear direction. The distributions and core structures of screw dislocations were observed by aberration corrected STEM (FEI Titan3 80-300) operated at 300kV.

Mechanical property

The orientation dependence of the flow stress showed different behaviors between the two types of steel. In high purity iron, the yield stresses were identical for the twining and anti-twining senses (Figure 2). Meanwhile, the work hardening behaviors were different due to the shearing sense. Slight work hardening was found in shearing in the anti-twining sense (Figure 3(a)). By contrast, the work hardening rate of the twining sense was much smaller than that of the anti-twining sense. In 4.7% Si steel, on the other hand, the yield stress of the anti-twining sense was much higher than that of the twining sense. No remarkable difference was found in the work hardening rate due to the shearing sense (Figure 3(b)).

Figure 2: Stress-strain curves of (a) high purity iron and (b) 4.7% Si steel obtained by simple shear tests.

Figure Figure Figure Figure 1: Schematic figure

of the samples for simple shear

tests.

Figure 3: Work hardening rate of high purity iron and 4.7% Si steel

sheared in twining and anti-twining senses.

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Dislocations slip behavior: observations of longitudinal sections

In 4.7% Si steel sheared in the twining sense, no dislocations were observed in the (11-2) foil

specimens. Two specimens of about 10 by 10µm of observed area were examined. By contrast, in the specimen sheared in the anti-twining sense, tangled dislocations of [111] Burgers vector were observed in some areas (Figure 4). Dislocations could easily pass through test pieces during simple shear tests because there was neither a grain boundary nor precipitate. Assuming rare occurrences of cross slips dislocations will not accumulate. On the other hand, if cross slips occurred, tangled dislocations would be observed. Although the observations imply that cross slips would happen more easily in shearing in the anti-twining sense, it is difficult to conclude this in the current investigation because of the small sample size.

In the case of high purity iron, numerous dislocations of non-[111] Burgers vectors were observed. Those dislocations were not considered to be introduced by shearing but were considered to be inherently existent in the annealed material. Dislocations of [111] Burgers vector, which tangled with dislocations of other Burgers vectors, were observed in the specimens sheared in both senses.

Dislocation core structure: observations of transverse sections

Dislocations of [111] Burgers vector were observed from the [111] direction in 4.7% Si steel sheared in the anti-twining sense (Figure 5). High resolution STEM observation indicated a threefold strain around the screw dislocation core. A threefold diffraction contrast was observed at every screw dislocation in the current investigation.

In high purity iron, images of threefold strain at screw dislocation cores have not been observed. Since the image quality was compromised due to the disturbance of other dislocations and the specimen damages introduced by FIB and Ar ion milling, conclusive observations for the screw dislocation core structure in high purity iron have not been performed yet. The strain field around screw dislocation cores in high purity iron, however, seemed to be different from that of 4.7% Si steel.

Molecular dynamics calculations have predicted the existence of threefold strain around screw dislocation cores [6]. The attempt to prove this has been conducted by precise measurements of the atomic displacement with high resolution TEM images of dislocation cores in Mo [7,8], in which significant diffraction contrast such as observed in Figure 5 has not been reported. The current results imply that the strain at dislocation cores in 4.7% Si steel can be larger than that of high purity iron and Mo. To confirm this, the precise atomic positions in the STEM images are necessary, which are being analyzed currently. The difference in the screw dislocation core structures in the materials is considered to be linked to their slip behaviors and consequently the mechanical properties.

Figure 4: Annular dark field STEM images of a longitudinal section of 4.7% Si steel sheared in anti-

twining sense by 2%. The dislocations of b=[111]were invisible with g=1-10.

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Figure 5: (a) A high resolution STEM image of a screw dislocation observed from [111] in 4.7% Si steel sheared in the anti-twining sense by 2% and (b) the dislocation observed at a tilt of 40 degree.

Summary

Mechanical properties, dislocation distributions and the screw dislocation core structure of high purity iron and 4.7 mass% Si steel were studied. The results are summarized below.

• The yield stress of 4.7% Si steel was strongly dependent on the shearing sense while that of high purity iron showed negligible dependency.

• High resolution STEM observation implied that a threefold strain field existed around screw dislocation cores in 4.7% Si steel.

References:

[1] W. A. Spitzig and A. S. Keh, Metal. Trans., 1(1970), 2751–2757

[2] S. Takeuchi, E. Furubayashi, and T. Taoka, Acta Metal., 15( 1967), 1179–1191

[3] K. Kitajima, Y. Aono, H. Abe, and E. Kuramoto, Scr. Metal., 13(1979), 1033–1037

[4] E. Kuramoto, Y. Aono, and K. Kitajima, Scripta Metal., 13(1979), 1039–1042

[5] K. Edagawa, T. Suzuki, and S. Takeuchi, Mater. Sci. Eng. A, 234–236(1997), 1103–1105

[6] M. Duesbery and V. Vitek, Acta Mater., 46(1998), 1481–1492

[7] W. Sigle, Phil. Mag. A, 79(1999), 1009–1020

[8] B. G. Mendis, Y. Mishin, C. S. Hartley, and K. J. Hemker, Mater. Res. Soc. Symp. Proc., 839(2005), 73–78

Proceedings

41

EFFECT OF DEFORMATION TWINNING ON THE BRITTLE-TO-

DUCTILE TRANSITION IN FE-AL SINGLE CRYSTALLINE ALLOYS

Kenta Tsujii1,a

, Masaki Tanaka1,b

, Kenji Higashida1,c

, Masahiro Fujikura2,d

and

Kohsaku Ushioda3,e

1 Department of Materials Science and Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

2 Yawata R&D Lab., Technical Development Bureau, Nippon Steel Corporation, 1-1 Tobihata, Tobata, Kitakyushu 804-8501, Japan

3 Technical Development Bureau, Nippon Steel Corporation, 20-1 Shintomi, Futtsu, Chiba 293-8511, Japan

a [email protected], b [email protected], c [email protected], [email protected], e [email protected]

Abstract

The effect of aluminum content on a brittle-to-ductile transition (BDT) has been investigated in Fe–Al single crystalline alloys. Absorbed impact energy was measured using an instrumental falling weight impact tester, indicating that the BDT temperature in Fe-8%Al is higher than that in Fe-4%Al. Twin-twin intersections were seen on fracture surfaces in both Fe-4%Al and Fe-8%Al tested at low temperatures. In order to highlight the effect of deformation twinning on the BDT, the temperature dependences of yield stress due to slip deformation and that of fracture stress due to twinning intersection were measured, respectively, using tensile tests in Fe-4%Al and Fe-8%Al. The increase in the plateau stress over 300K is much larger than that in fracture stress due to deformation twining. It brings the facilitation of deformation twinning at higher temperture in Fe-Al alloys, which increases the BDT temperature in Fe-Al crystals.

Keywords: deformation twin, dislocations, Fe-Al single crystal

Introduction

Materials with body-centered cubic (b.c.c.) structure is generally hard to be deformed at low temperatures, which leads to a brittle fracture while they become easy to be deformed plastically with increasing temperature, resulting in ductile failure. The change in a fracture mode from a brittle manner to a ductile one is called a brittle-to-ductile transition (BDT). Pioneering work to elucidate the fundamental mechanism behind the BDT was performed using silicon single crystals as a model substance. It was found that increasing the amount of phosphorus or arsenic in silicon increases dislocation velocity and decreases the BDT temperature [1, 2]. Since the primitive process of dislocation gliding silicon crystals and bcc materials is the same [1, 2-6], it suggests that the BDT temperature in steel is also influenced by some solute elements via change in the dislocation velocity.

In ferritic steels, Fe-Ni alloys and Fe-Si alloys show higher yield stress than that of pure iron at room temperature, which is well known as solid solution hardening, while those alloys show lower yield stress than that of pure iron at low temperatures, which is called solid solution softening [7]. It suggests

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that the dislocation mobility in those alloys should be higher than that in pure iron at low temperatures, and then the BDT temperatures in those alloys are lower than that in pure iron. Actually, the BDT temperature in Fe-Ni alloys is lower than that in pure iron, however, that in Fe-Si alloys is higher than that in pure iron. Although adding nickel or silicon in iron induces solid solution softeningin deformation at low temperatures, the effects of nickel and silicon on the BDT are contrary [8].

In this study, the contradicted effects of Ni and Si on the BDT were investigated employing Fe-Al single crystals which show nearly the same mechanical properties with those of Fe-Si single crystals. Especially, competitive operations of deformation twinning and slip deformation were focused.


Single crystalline Fe-mass4%Al and Fe-mass8%Al were employed, the carbon contents of which were 0.004 and 0.0012 mass%, respectively. Miniature size specimens of 1×1×24mm3 were cut out for subsequent impact tests. The beam distance in impact tests was set to be 18 mm. The applied tensile direction in bending was along [100]. The absorbed energy for fracture was measured from each load-displacement curve obtained during the impact tests. In addition, tensile tests were conducted, the tensile direction of which was along [100]. The test temperature was controlled by cold nitrogen gas between 77K and room temperature to investigate the temperature dependences of yield stress with an initial strain rate of 1.9×10-5 /s-1. The length of parallel portion was set to be 4 mm with a cross-section of 2×0.9 mm2. A strain gage was stuck on the parallel portion of each specimen. Side surfaces and fracture surfaces of each specimen were observed by scanning electron microscopy (SEM) after the tensile tests.

Result and discussion

Figure 1 shows temperature dependences of the absorbed energy from Fe-4%Al and Fe-8%Al measured using impact tests [9]. The average absorbed energy at a lower-shelf is 6.6 kJ/m2in both Fe-4%Al and Fe-8%Al. The BDT temperatures in each specimen were defined as the temperature at which the absorbed energy shows the middle value between those of the lower-shelf and the upper-shelf energies. The BDT temperature in Fe-4%Al was determined to be approximately 180 K while that in Fe-8%Al was approximately 320 K. The increase in aluminum content increases the BDT temperature as well as Fe-Si alloys, which is contrary to the effect of nickel on the BDT temperature [8]. It was reported that the onset of deformation twinning leads to the cleavage fracture in Fe-Si single crystalline alloys [10, 11], which suggests that the increase in the BDT temperature is enhanced by deformation twinning in Fe–Si alloys. It also suggests that the increase in the BDT temperature in Fe–Al is due to the change in twinning behavior with increasing aluminum content. Therefore, it is essential to investigate the twinning behaviour in aluminum added ferritic steels. Tensile tests were performed at 77K next.

Figure 2 shows stress-strain curves from tensile tests at 77K in Fe-4%Al and Fe-8%Al. Both specimens tested at 77K fractured in a brittle manner in the elastic regime. Fracture stress was 410MPa and 470MPa in Fe-4%Al and Fe-8%Al, respectively. The load drop in Fe-4%Al is due to the onset of deformation twinning during the tensile test. Figure 3 (a) and (b) show SEM micrographs after the tensile tests, reconstructing the specimen geometry with fracture surfaces and side surfaces of the specimens from Fe-4%Al and Fe-8%Al, respectively. Figure 3(c) shows a schematic drawing of the geometry of twins and their traces on the specimen surfaces. There were a number of straight markings of deformation twins along the [120] directions and [120] directions in both side surfaces of the specimens. The directions of intersection line of those twins were parallel to <110>. The intersected lines are found to be the initiation site of fracture as reported in Fe-Si single crystals [11]. It is to be noted here that the fracture stress in Fe-8%Al is higher than that in Fe-4%Al as seen in Fig.2. This indicates that the onset of deformation twinning becomes more difficult with increasing aluminum content. This suggests the decrease in the BDT temperature with increasing aluminum content. However, the actual tendency of the BDT temperature obtained using the impact test was contrary, i.e., the BDT temperature increases with increasing Al content as seen in Fig.1. The contradiction is due to

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the lack of the consideration of the change in the facilitation of slip deformation with aluminum content.

Figure 3: Fracture surfaces and side surfaces of the specimen tested at 77K in Fe-4%Al and Fe-8%Al. The arrows on fracture surfaces and side surfaces show the directions of the straight marks caused by twinning.

A schematic drawing of the geometry of twins and their traces on the specimen surface for Fe-8%Al are shown in Fig. 3(c)

Therefore, the competitive operations of twinning deformation and slip deformation must be discussed to understand the BDT behavior. Therefore, temperature dependence of the stress for fracture due to twinning intersection and yield stress were obtained from 77K to room temperature using tensile tests.

Figure 4 shows the temperature dependences of fracture stress due to twinning intersection and

Figure 1: Temperature dependences of the energy absorbed during fracture in Fe-4%Al and Fe-

8%Al from the impact test from Figs. 3 in Ref.

Figure 2: Stress-strain curves from tensile tests at 77k in Fe-4%Al and Fe-8%Al

(a) (b)

(c)

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yield stress in Fe-4%Al and Fe-8%Al. The fracture stress due to twinning intersection is nearly temperature independent while yield stress shows strong temperature dependence. The average of fracture stress due to twinning intersection is 420MPa and 470MPa in Fe-4%Al and in Fe-8%Al, respectively, indicating fracture stress due to twinning intersection in Fe-8%Al is 50MPa higher than that in Fe-4%Al. Yield stress has stronger temperature dependence than that of deformation twinning. Yield stress consists of two components of stress: effective stress and athermal stress. The temperature dependence of yield stress is due to that of effective stress. The yield stress decreases with increasing temperature due to temperature dependence of the effective stress. The yield stress saturates at some level of stress (stress plateau) higher than 300K. The plateau stress in Fe-8%Al is 185MPa higher than that in Fe-4%Al. The amount of increase in plateau stress in Fe-8%Al is 135MPa larger than that of the fracture stress due to twinning intersection. Since the increase in the plateau stress, which increases the BDT temperature, is larger than that of stress for fracture due to twinning intersection, which decreases the BDT temperature, the upper limit of the temperature for the onset of twinning intersection to fracture is increased. Such large increase in the plateau stress should not be seen in Fe-Ni alloys.

It can be thus concluded here that the reason why the BDT temperature increases with aluminum or silicon content is that the amount of increase in plateau stress is larger than that in stress for fracture due to twinning intersection with the increase in the aluminum content. The increase in the plateau stress in Fe-Al alloys brings the facilitation of deformation twinning at higher temperatures, which leads to the contrary effect of Al(Si) and Ni on the BDT temperature.

Figure 4: The temperature dependences of yield stress due to slip and fracture stress due to twinning intersection in Fe-4%Al and Fe-8%Al.

Conclusion

The effect of aluminum on the brittle-to-ductile transition was investigated. The temperature dependences of absorbed energy and yield stress and fracture stress due to deformation twinning were measured in Fe-mass4%Al and Fe-mass8% Al. The following results were obtained;

1. The increase in aluminum content in ferritic steels increases the BDT temperature.

2. Fracture stress due to twinning intersection is increased with aluminum content.

3. The BDT temperature increased with aluminum content whereas the increase in aluminum induces solid solution softening at low temperatures, which can be explained by the amount of increase in the plateau stress is larger than that in fracture stress due to twinning intersections with the aluminum content, inducing the difficulty of slip deformation and facilitation of twinning deformation to increase the BDT temperature.

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References:

[1] M. Brede and P. Haasen: Acta Metall. Vol. 36 (1988), p. 2003.

[2] Y.-J. Hong, M. Tanaka and K. Higashida: Materials Transactions Vol. 50 (2009), p. 2177.

[3] C. St. John: Philos. Mag. Vol. 32 (1975), p. 1193.

[4] P.B. Hirsch, S. G. Roberts and J. Samuels: Proc. R. Soc. Lond. A Vol. 421 (1989), p. 25.

[5] G. Michot, M.A.L. de Oliveira and G. Champier: Mater. Sci. Eng. A Vol. 272 (1999), p. 83.

[6] P.B. Hirsch and S. G. Roberts: Acta Mater. Vol. 44(1996), p. 2361.

[7] K. Okazaki: J. Mater. Sci. Vol. 31 (1996), p. 1087.

[8] W.W. Gerberich, Y.T. Chen, D.G. Atteridge and T. Johnson: ActaMetallurgica. Vol. 29 (1981), p. 1187.

[9] ISIJ International, Vol. 51 (2011), No. 6, pp. 999–1004

[10] R. Honda: J. Phys. Soc. Jpn. Vol. 16 (1961), p. 1309.

[11] N. Narita, K. Higashida: Journal of the Society of Materials Science Vol. 36 (1987), p. 854.

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CRYSTAL PLASTICITY ANALYSIS OF WORK HARDENING

BEHAVIOUR AT LARGE STRAINS IN FERRITIC SINGLE CRYSTALS

Akihiro Uenishi1, Satoshi Hirose

1, Yusuke Tsunemi

1, Cristian Teodosiu

2, Eiji Isogai

1,

Natsuko Sugiura1, Masaaki Sugiyama

1, Masayoshi Suehiro

1

1 Nippon Steel Corporation, Futtsu, Japan 2 ADESCO Services SRL, Bucharest, Romania

Abstract

In this study, we focus on the relationship between the work hardening behaviour of bcc single crystals at large strains and the evolution of microstructure during deformation, by both experimental and numerical methods. The work hardening behaviour of ferritic single crystals with different orientations has been characterized by planar simple shear experiments. The work hardening behaviour depends largely on the initial crystal orientation. The microstructures observed by TEM may be classified into three types, while the work hardening behaviour could be correlated to these types of microstructure. An advanced model of crystal plasticity that takes the dislocation densities on the slip systems as internal state variables have been used to analyze the results and to discuss the connection between the plastic behaviour of single crystals and the evolution of their microstructure.

Keywords: ferritic single crystal, simple shear test, crystal plasticity analysis, work hardening behaviour

Introduction

The work hardening behaviour of crystalline materials has been an important subject of materials science. Recently, several studies have shown that the anisotropic work hardening occurring at large strains depends on the deformation history, and its correlation with characteristic deformation microstructures has been discussed. A special emphasis should be placed on the connection between macroscopic and microscopic deformation properties for the development of new materials and their optimum applications. Simple shear tests on planar specimens allow attaining large amounts of almost homogeneous deformation, significantly larger than uniaxial tensile specimens, because they are free from plastic instabilities such as localization and/or necking typical of uniaxial tensile tests. The planar simple shear test was first developed by Miyauchi, in a setting where the specimen has two symmetrical shear zones. A second version of the simple shear device was proposed by G’Sell for polymers and subsequently adapted by Rauch and others for metallic materials (for details see e.g. [1]). The specimen has a simple rectangular shape with a single shear zone.

In this study, we focus on the relationship between the work hardening behaviour at large strains and the evolution of microstructure during deformation of steel sheets, with attention to the crystal orientation. The work hardening behaviour of bcc Fe-Cr single crystals with different initial orientations has been characterized by simple shear experiments. An advanced model of crystal plasticity that takes the dislocation densities on the slip systems as internal state variables has been used to analyse the results and to discuss the coupling of the plastic behaviour of single crystals with the evolution of their microstructure.

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Experimental Procedure

The material studied is a ferritic Fe-Cr alloy (mass %: 0.0017 C, 16.45 Cr and 0.10 Ni). The material was annealed at 1350 C for 72 hrs in Ar atmosphere, in order to grow grains around 20 to

30 mm diameter. Then, blocks of 30 mm × 25 mm × 25 mm were cut out from the ingot and the crystal orientation of each grain was evaluated by X-ray Laue method. Finally, single crystal planar specimens with plane normal of 111, 100, and 110 were obtained.

A simple shear apparatus was adopted for testing the work hardening behaviour of bcc single

crystals at large strains [2]. The geometry of the simple shear specimen was 20 mm × 18 mm × 1

mm with a shear zone of 20 mm × 2 mm. The shear direction of each test is shown in Table 1. Monotonic simple shear tests up to 60% amount of shear were carried out at room temperature.

Crystal # Test labels

Normal

to the planar specimen

Sheari

ng direction

Normal to

the maximum shear plane

Euler angles (in degrees)

φ1 Φ φ2

1 111<112>

( )111 [ ]211　 ( )011 90 54.74 45

2 100<0

01> ( )001 [ ]010　 ( )100 45 0 45

3 100<0

11> ( )001 [ ]011　

( )110 0 0 45

4 110<0

01> ( )110

[ ]001 ( )011 90 90 45

5 110<1

12> ( )101 [ ]211

( )111 305.26 90 315

6 110<5

57> ( )101 [ ]755

( )7710 315.30 90 315

Table 1: Crystallographic orientation in single crystal specimens

Experimental Results

Fig. 1 shows the evolution of the shear stress with respect to the shear strain for single crystal specimens determined by planar simple shear tests. Inspection of this figure reveals that the crystals # 2 (100<001>), # 3 (100<011>) and # 4 (110<001>) have hardening rates which decrease monotonously until the end of the test, and which are much larger than the hardening rates of the other three crystals. This behaviour corresponds to the activation of 2 or 4 slip systems from the very beginning of the deformation, and this has generally a double effect: increasing the hardening and stabilizing the lattice orientation. Crystals # 5 (110<112>) and # 6 (110<557>), which have close initial lattice orientations, present a low and almost identical initial hardening rate, characteristic of single slip. Therefore, the lattice rotation evolves significantly, thus leading to the activation of supplementary slip systems and then to a sharp increase of the hardening rate.

Clearly, the delay of this latter event is larger for crystal # 6 than for # 5. The stress-strain curve of crystal # 1 (111<112>), suggests a single slip throughout the test. However, this is not really the case. Actually, one of the slip systems has a strongly dominant slip activity, which hides the contributions of the other, less active, slip systems.

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Figure 1: Evolution of shear stress with respect to shear strain measured by simple shear experiments and

calculated by crystal plasticity model proposed by Hoc and Forest [3] (=40, 0=30 MPa, G=5 nm, n = 50, 0=0.001 s− 1, ρ0=64×109 m−2).

(a) #1, 111<112> 60% (b)#2, 100<001> 15% (c)#3, 100<011> 60%

Figure 2: Dislocation structure in crystals #1, #2 and #3.

The results of TEM observations are shown in Fig. 2. One set of parallel dislocation boundaries, which is parallel to (110) plane, are observed in crystal #1 (Fig. 2a). Equiaxed cells are found in the microstructure of crystal #2 (Fig. 2b). The microstructure of crystal #3 can be characterized by two sets of parallel planar dislocation boundaries (Fig. 2c).

Crystal Plasticity Analysis

In order to discuss the coupling of the work hardening behavior of crystalline materials to the evolution of their microstructure, a crystal plasticity analysis that takes the dislocation densities on the slip systems as internal state variables has been applied. The model proposed by Hoc and Forest [3] has been used here. The results of the best fit to the experimental results are also shown in Fig. 1. As a whole, the calculated results show a rather good agreement with the experimental ones. However, a careful inspection reveals some discrepancies, such as the difference of flow stress level in crystal #1. In the present model, the anisotropy of dislocation interactions comes from the consideration of the 24 potential slip systems belonging to the two families <111>110 and <111>112, as well as from the difference between their interaction coefficients. More precisely, the model assumes that the values of these interaction coefficients depend on whether

0

100

200

300

400

500

0 0.2 0.4 0.6

ExperimentCalculation

Sh

ear

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ess

(MP

a)

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300

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500

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500

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500

0 0.2 0.4 0.6


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ss (

MP

a)

Shear strain

Crystal #6

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the systems are collinear and on whether they belong to the same family. In the present calculation, the interaction coefficients identified by Hoc and Forest for IF-Ti steel [3] by semi-inverse methods based on polycrystalline tests, have been used.

In crystal #1, two non-collinear slip systems of the <111>110 family that have same slip plane are activated. In contrast, four non-collinear slip systems of the <111>110 family that belong to two slip planes are activated in crystal #3. According to the model, the dislocation interactions are quite similar among active slip systems in crystals #1 and #3, except for the fact that there are more active slip systems in crystal #3. However, Fig. 2 indicates essential differences of dislocation interactions between crystals #1 and #3. The simulated flow stress of crystal #1 is larger than the experimental one, which reflects interactions of the non-collinear slip systems in the model, whereas the simulation of crystal #3 shows a relatively good agreement with the experiment. It is, therefore, apparent that the geometrical configuration of slip planes plays an important role for the determination of the anisotropy of dislocation interactions. By further refining this analysis, it is expected that simulations will show a better agreement with experiments.

Conclusion

The work hardening behaviour at large stains of bcc Fe-Cr single crystals was investigated by using experimental and numerical methods. By taking into account the anisotropic dislocation interactions, the dependence of the complex work hardening behaviour on the crystallographic orientation could be discussed.

References:

[1] S. Bouvier, H. Haddadi, P. Levee and C. Teodosiu: J. Mat. Proc. Tech., 172 (2006), 96.

[2] Uenishi, A., Sugiura, N., Ikematsu, Y., Sugiyama,M., Isogai, E., Hiwatashi, S., In: Proc. of The 2nd Int. Symp. on Steel Science, 2009, 57.

[3] T. Hoc, S. Forest: Int. J. Plasticity 17(2001), 65.

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INTERACTIONS BETWEEN DISLOCATION BOUNDARIES AND

SPHERICAL PRECIPITATES AT HIGH TEMPERATURES

Tomáš Záležák1,2

, Antonín Dlouhý1

1 Institute of Physics of Materials, Academy of Sciences, Žižkova 22, 616 62 Brno, Czech Republic

2 Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

Abstract

We present a 3D discrete dislocation dynamics model (DDD) which addresses interactions between spherical particles and flexible dislocation lines subjected to conditions of high temperature creep. The piecewise-smooth and continuous dislocation lines are represented by a discrete set of short straight line segments. The segment motion is controlled by Peach-Koehler forces (PKFs). PKFs stem from a linear combination of segment-to-segment interactions and an externally applied stress. The model further exploits a symmetry of the simulated structure, allowing for a relevant speed-up of the calculations.

The results provide a detailed insight into how the low-angle dislocation boundaries may overcome precipitate fields in 3D crystals at high temperatures. Three regimes have been encountered: (1) the low-angle dislocation boundaries may pass through the precipitate field leaving behind new dislocation loops at the precipitate surfaces, (2) alternatively, the low angle boundaries may dissolve and form several new boundaries with lower dislocation density and lower misorientation angle and, finally, (3) the low-angle dislocation boundaries may be also trapped by the precipitate field and remain in equilibrium with particles, unless a higher load is applied.

Keywords: 3D discrete dislocation dynamics, low angle dislocation boundary, subgrains, particle strengthening, high temperature creep.

The motivations and the model

3D dislocation processes can be addressed by discrete dislocation dynamics (DDD). Currently, many of the 3D DDD simulations have focused only on low temperature plasticity, where the diffusion does not significantly contribute to plastic deformation [1, 2]. Only few models extend the 3D DDD to high temperatures, e.g. [3]. Our 3D DDD model, which does include diffusion processes, may contribute to understanding of some fundamental mechanisms related to creep in metallic materials. The presented article discusses a migration of low-angle tilt boundaries in a field of rigid spherical precipitates.

The dislocation structure composed of smooth dislocation curves is approximated by continuous broken lines consisting of short straight segments. The linear theory of elasticity [4], which is the 3D DDD model based upon, allows to sum stress fields associated with all individual dislocation segments and also the externally applied stress in order to calculate local driving forces. The model takes advantage of simple calculation of the stress field for the straight dislocation segments [4]. A regularization is necessary to eliminate divergences [5]. The local stress field acquired by the summation is thus used to compute Peach-Koehler (PK) force for each

dislocation segment [4]. The velocity of each segment is controlled by a mobility function b :

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bD s

b2k T

, D s D0 expQ

R T.

Here Ω is a volume per atom, b is the length of the Burgers vector, k is the Boltzmann constant, T is the temperature, Ds is the self-diffusion coefficient, Q is an activation energy of self-diffusion

and R is a gas constant. This thermodynamic relation is further discussed in [3, 4]. The actual segment velocity v is a sum of three components:

v vs ve ,G ve ,C ,

where vs = As β(b) Fs is a component driven by a screw component Fs of PK force. The

velocities ve,G = Ae,G β(b) Fe,G and ve,C = Ae,C β(b) Fe,C come from an edge component of PK

force decomposed into a slip plane Fe,G and into its normal Fe,C. The climb prefactor Ae,C is

always set to 1. If the slip plane of the particular segment corresponds to a crystallographic plane, we set Ae,G = AS = 10, otherwise we set Ae,G = AS = 2. This choice of scaling factors makes the

glide movement along crystallographic planes one order of magnitude faster than diffusion-controlled climb. The non-compact glide is much slower, so we choose a prefactor of two as a first order approximation.

The model incorporates also rigid spherical precipitates, which are impenetrable for the dislocation lines. For further details of the model, see [6, 7].

Low angle tilt boundaries

Figure 1: A geometry of a low-angle tilt boundary in a single simulation cell.

We investigate a single low angle tilt boundary subjected to an externally applied stress σxz.

The low angle dislocation boundary is shown at Fig. 1. The particle diameter is d = 100 nm and the interparticle spacing is λ = 200 nm. The edge dislocations, which form the boundary, are parallel to the Y axis and their Burgers vector points in a positive X direction. The system is

symmetric with respect to the planes Y = 0, Z = 0. The elementary cell with dimensions aY = aZ

= 200 nm is replicated in a 3×3 pattern to approximate a large-scale periodic dislocation structure.

A range of applied stresses σxz from 20 to 100 MPa has been considered. Other input

parameters are listed in Table 1. Low-angle tilt boundaries remain planar until first dislocation lines reach the surface of the rigid precipitates. The impenetrability constraints the shapes of the dislocation lines and thus the behaviour of the boundary. The low-angle dislocation boundaries exhibit three different regimes of behaviour depending on the initial dislocation spacing h as well as on the applied loading σxz [7]. For low applied stresses and smaller dislocation spacings (denser

dislocation walls), the boundary propagation stops at the rigid precipitates (mode STOP). Even the driving forces acting upon dislocation lines, which are not in contact with the precipitates, approach zero in such situations. On the other hand, the boundaries pass through the precipitate

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field under the action of higher applied stresses (mode PASS). These results indicate that the edge dislocation boundary is a stable structure and that there is a critical stress, which delimits the boundary migration regimes depending on the line spacing σcrit.(h). However, if the dislocation

density is low enough (i. e. high values of h), the dislocation boundary may decompose into two or more boundaries with higher h. This eases the relaxation of the dislocations, allowing at least some of the boundaries to pass through the precipitate field (mode PASS & SPLIT). The map of the tree modes for variable h and σxz is plotted in Fig. 2.

μ 0.8 · 1011 Nm-

2

shear modulus N 32 # of segments per line

ν 0.3 Poisson ratio n 7, … , 17 # of lines in the cell

D0 2 · 10-4 m2s-1 diffusion factor aY;

aZ

200 nm size of the simulation

cell Q 240 kJ mol-1 activation energy h aZ / (n - 1) initial line distance

T 873 K temperature c [-50, 0, 0] nm particle center

Ω (0.35 nm)3 atomic volume d 100 nm particle diameter

b (0.2, 0, 0) nm Burgers vector λ 200 nm particle distance

Δt 30 mn time step l ⟨3, 8⟩ nm segment length

Table 1: Parameters of the simulation.

Figure 2: Modes of a dislocation boundary migration for varying σxz and h.

Discussion

Numerous experiments have documented a formation and migration of low angle dislocation boundaries as a generic process that is characteristic for many materials including the precipitation hardened alloys subjected to loading at high temperatures [8, 9]. Our results show that even fully flexible tilt boundaries can be pinned by the precipitates, as long as the applied stress is below the critical limit σcrit.(h) (mode STOP at Fig. 2). This means that the boundary migration ceases. Thus

the interaction between the dislocation lines and the precipitates may cause a non-neglible threshold stress even at high temperatures, where the dislocation climb is important. This corresponds to an observation of threshold stresses in many creep experiments, e. g. [8]. The range of critical stresses between 20 and 60 MPa corresponds to dislocation spacing h equivalent to misorientation angles from 0.3° to 0.9°, which agrees very well with typical misorientation angles

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53

observed by TEM after high temperature deformation. We conclude that interactions between the low-angle dislocation boundaries and rigid precipitates seem to be adequately described by our 3D DDD model.

A comparison with the former simulations neglecting the crystallography [7] shows that the threshold stresses do not seem to be strongly dependent either on the crystallography model or the scaling factors. On the other hand, the stability of the dislocation boundary is very sensitive to the details related to the mobility of dislocation segments. These effects should be investigated by further parametric studies.

Summary

A recently developed 3D discrete dislocation dynamics (DDD) model has been extended using mobility functions that account for the crystallography of the material. This step improved the model and allowed a numerical study addressing a system consisting of a migrating low angle dislocation boundary within a field of rigid precipitates (diameter d = 100 nm). The numerical results show that the dislocation boundaries may be considerably restricted due to the interactions with the particles. The boundary migration may be completely stopped for critical stresses below σcrit.(h) (see Fig. 2). This critical stress is inversely proportional to the initial spacing h of

dislocations composing the dislocation wall. The function σcrit.(h) is not sensitive to details that

describe the mobility of the individual dislocation segments.

Acknowledgement

The financial support was obtained from the Czech Science Foundation under contracts No. 202/09/2073 and 106/09/H035.

References:

[1] B. Devincre, L. P. Kubin, Mat. Sci. Eng. A234-236 8 (1997).

[2] D. Weygand, L. H. Friedman, E. van der Giessen, A. Needleman, Mat. Sci. Eng., A309- 310 420 (2001).

[3] D. Mordehai, E. Clouet, M. Fivel, M. Verdier, Phil. Mag. 88 899 (2008).

[4] P. Hirth, J. Lothe, Theory of Dislocations, 2nd ed., Krieger Publishing Company, Malabar 1992.

[5] W. Cai, A. Arsenlis, Ch. R. Weinberger, V. V. Bulatov, J. Mech. Phys. Sol. 54 561 (2006).

[6] T. Záležák, A. Dlouhý, Key Engineering Materials 465 115 (2011).

[7] T. Záležák, A. Dlouhý, Acta Physica Polonica A (2012), in print.

[8] J.H. Hausselt, W.D. Nix: Acta Metall. 25 595 (1977)

[9] W. Blum, in: Materials Science and Technology, Vol. 6, Ed. H. Mughrabi, VCH Verlagsgesellschaft, Weinheim 1993

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AUTHORS’ INDEX

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56

Proceedings

57

Djurabekova, F. 18

Dlouhý, A. 50

Fitzgerald, S. P. 18

Fujikura, M. 41

Galindo-Nava, E. I. 13

Hafez Haghighat, S. M. 33

Higashida, K. 41

Hirose, S. 46

Isogai, E. 46

Kimura, K. 37

Kuronen, A. 18

Kutsukake, K. 29

Marian, J. 7

Ohno, Y. 29

Paidar, V. 23

Pohjonen, A. S. 18

Rivera-Díaz-del-Castillo, P. E. J. 13

Schäublin, R. 33

Shigesato, G. 37

Sietsma, J. 13

Stukowski, A. 7

Suehiro, M. 46

Sugiura, N. 46

Sugiyama, M. 46

Tanaka, M. 41

Teodosiu, C. 46

Thaulow, C. 7

Tokumoto, Y. 29

Tsujii, K. 41

Tsunemi, Y. 46

Uenishi, A. 46

Ushioda, K. 41

Vatne, I. 7

Vitek, V. 23

Yonenaga, I. 29

Záležák, T. 50

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