Double kink mechanisms for discrete dislocations in BCC...

Int J Fract (2012) 174:29–40DOI 10.1007/s10704-012-9681-7

ORIGINAL PAPER

Double kink mechanisms for discrete dislocations in BCCcrystals

M. P. Ariza · E. Tellechea · A. S. Menguiano ·M. Ortiz

Received: 17 October 2011 / Accepted: 23 January 2012 / Published online: 11 February 2012© Springer Science+Business Media B.V. 2012

Abstract We present an application of the discretedislocation theory to the characterization of the ener-getics of kinks in Mo, Ta and W body-centered cubic(BCC) crystals. The discrete dislocation calculationssupply detailed predictions of formation and interactionenergies for various double-kink formation and spread-ing mechanisms as a function of the geometry of thedouble kinks, including: the dependence of the forma-tion energy of a double kink on its width; the energy offormation of a double kink on a screw dislocation con-taining a pre-existing double kink; and energy of forma-tion of a double kink on a screw dislocation containinga pre-existing single kink. The computed interactionenergies are expected to facilitates the nucleation ofdouble kinks in close proximity to each other and topre-existing kinks, thus promoting clustering of dou-ble kinks on screw segments and of ‘daughter’ dou-ble kinks ahead of ‘mother’ kinks. The predictions of

M. P. Ariza (B) · E. Tellechea · A. S. MenguianoEscuela Técnica Superior de Ingeniería,Universidad de Sevilla, 41092 Sevilla, Spaine-mail: [email protected]

E. Tellecheae-mail: [email protected]

A. S. Menguianoe-mail: [email protected]

M. OrtizDivision of Engineering and Applied Science,California Institute of Technology, Pasadena, CA 91125,USAe-mail: [email protected]

the discrete dislocation theory are found to be in goodagreement with the full atomistic calculations based onempirical interatomic potentials available in the litera-ture.

Keywords Dislocations · Crystal plasticity ·Double-kinks · BCC metals

1 Introduction

The thermally activated double kink nucleation mecha-nism has attracted considerable attention over the years(Duesbery 1983; Tang et al. 1998; Cai et al. 2000;Yang and Moriarty 2001). This interest arises fromthe observation that, in body-centered cubic (BCC)metals, the motion of a

2 〈111〉 screw dislocations iscontrolled by the nucleation and spreading of doublekinks. The particular core structure of screw disloca-tions in BCC metals is accepted to be the underly-ing cause of this behavior (Vitek 1976). Some modelsof double-kink dynamics and BCC metal plasticityare based on linear-elastic dislocation theory (Hirthand Lothe 1968; Seeger 1981). Atomistic simulationsbased on empirical interatomic potentials or on latticeGreen’s functions have also been carried out in orderto study the structure and energetics of the double kinkconfigurations in BCC crystals (cf., e.g., Yang et al.2001). In addition, dislocations mobility at low temper-ature in tantalum has been studied using a mesoscop-ic approach (Tang et al. 1998). In order to overcome

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30 M. P. Ariza et al.

the sample-size limitations of dislocation dynamicsmodels, Cai et al. (2000, 2001) have modeled disloca-tion motion in silicon and molybdenum using a kineticMonte Carlo scheme.

Discrete dislocation dynamics is a widely usedcomputational tool for assessing the plasticity of sin-gle crystals. A number of approaches are commonlyadopted in constructing discrete dislocation models.These approaches may roughly be grouped into line-tracking based methods (e.g. Zbib et al. 2002; Ghoniemet al. 2000; Madec et al. 2002) and non-tracking meth-ods such as the phase-field and related models (Wanget al. 2001, 2004; Koslowski et al. 2002). Line-trackingmethods require the discretization of dislocation linesand the subsequent computation of the long-range elas-tic interactions between pairs of dislocation segments.Special rules need to be set forth in order to account fortopological transitions arising from dislocation reac-tions, annihilation, and others, and in order to regu-larize the logarithmically divergent self-interaction ofthe segments. A common additional limitation of linearelastic dislocation dynamics models arises from theirreliance on isotropic elasticity, which neglects the elas-tic anisotropy of the crystals. Furthermore, the modelsfail to account for a dislocation-core structure and aminimum separation between slip-planes. Dislocationnucleation is often accounted for by means of ad hoccriteria and mobility laws require to be prescribed apriori.

The discrete dislocation theory of Ariza and Ortizemployed here overcomes many of the aforementionedlimitations of linear elastic dislocation dynamics mod-els. Thus, by acknowledging the discreteness of thelattice from the outset, the theory is endowed withseveral desirable features, including the elimination ofunphysical singularities and the need for core cutoffs;fully anisotropic discrete lattice elasticity; an atomis-tic dislocation core structure; and topological transi-tions that are automatically accounted for without theneed for any explicit rules of interaction. The computa-tional advantages that are derived from the discrete dis-location theory of Ariza and Ortiz are also noteworthy.Thus, an appeal to algebraic topology greatly facilitatesbook-keeping in numerical calculations. Additionally,the discrete Fourier transform (DFT) enables the con-struction of fast and efficient computational schemesfor large systems (Ramasubramaniam et al. 2007).

In this paper, we present an application of thediscrete dislocation theory of Ariza and Ortiz to

Case I

Case II

Case III

LI

LII

LIII

Fig. 1 Schematic representation of the transitions considered inthis study

the characterization of the energetics of kinks in Mo,Ta and W BCC crystals. For definiteness, we restrictattention to the following transitions (Fig. 1):

– Case I: Nucleation of a double kink at an arbitraryposition on an infinite screw dislocation and subse-quent separation of each kink.

– Case II: Nucleation of a double kink at a certainposition away from a preexisting double kink on aninfinite screw dislocation.

– Case III: Nucleation of a double kink at a certainposition away from a preexisting kink in an infinitescrew dislocation.

In all cases, the transitions are effected by the addi-tion of elementary loops, termed loopons by Shenoy etal. (1999). More generally, it is possible to transitionbetween two arbitrary dislocation densities by means ofa sequence of ‘flips’, each flip consisting on the additionor subtraction of an elementary loopon. For complete-ness, in Sect. 2 we begin by briefly summarizing thediscrete dislocation theory that provides the basis forsubsequent calculations. A study of the three configura-tions defined above is presented in Sect. 3. The discretedislocation calculations supply detailed predictions offormation and interaction energies for various double-kink formation and spreading mechanisms as a functionof the geometry of the double kinks. These predictionsare found to be in good agreement with the full atomis-tic calculations based on empirical interatomic poten-tials available in the literature.

2 The lattice complex of BCC crystals

This section summarizes the general theory of discretedislocations in crystal lattices, and its specializationto body-centered cubic (BCC) crystals, as developedin Ariza and Ortiz (2005), Ramasubramaniam et al.(2007).

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Double kink mechanisms 31

Fig. 2 CW-complexrepresentation of thebody-centered cubic latticeand complex latticeindexing scheme. a Atomsor 0-cells. b Atomic bondsor 1-cells

l

l+ 1

l+ 2

l+ 3

l+ 7

l- 1

l- 2

l- 3

l- 7

l+ 4

l- 4

l+ 6l- 6

l+ 5

l- 5

(a)

l

(l,1)

(l-4,4

)

(l,6)

(l- 6,6) (l- 5,5)

(l-1,1

)

(l,7

)(l

-7,7

)

(l,5)

(l,2)

(l-2,2)

(l,4)

(l,3)

(l-3 ,3)

(b)

2.1 The cell complex of the BCC crystal lattice

Following Ariza and Ortiz, we regard the BCC lat-tice as a cell-complex C, i.e., as a collection ofcells of different dimensions equipped with discretedifferential operators and a discrete integral. In par-ticular, the BCC complex is three-dimensional andconsists of: atoms, or 0-cells; atomic bonds, or 1-cells;atomic faces, or 2-cells; and tetrahedral cells, or 3-cells(Figs. 2, 3, 4). The BCC Bravais lattice is gener-ated by the basis {(−a/2, a/2, a/2), (a/2,−a/2, a/2),(a/2, a/2,−a/2)}. The lattice complex representationof the BCC lattice is shown in Figs. 2, 3 and 4, whereε1 = (1, 0, 0), ε2 = (0, 1, 0), ε3 = (0, 0, 1), ε4 =(1, 1, 1), ε5 = (0, 1, 1), ε6 = (1, 0, 1) and ε7 =(1, 1, 0). Each element of the three-dimensional com-plex Bravais lattice can be indexed by means of threeintegers l ≡ (l1, l2, l3) ∈ Z

3 and a label α ∈{1, . . . , N } that designates which of the N simple Brav-ais sublattices the element belongs to.

The lattice cell complex supplies the support fordefining functions, or forms, of different dimensions.Thus, a form of dimension p assigns a vector to eachcell of dimension p of the lattice. As we shall see,forms provide the vehicle for describing the behaviorof the BCC lattice, including its displacements, eig-endeformations and dislocation densities. In addition,the description of lattice defects such as dislocations,and the formulation of the attendant equilibrium equa-tions, is greatly facilitated by defining discrete differ-ential operators over the lattice complex. To this end,we note that the BCC lattice complex defined in (4) is a

simplicial complex and, hence, it can be endowed withthe standard simplicial differential structure (Munkres1984). To this end, we begin by orienting all cells(Figs. 2, 3, 4). Specifically, we recall that a p-simplexis defined as a collection σp = {v0, . . . , vp} of verticesand that a simplex is oriented by choosing an orderingσp = [v0, . . . , vp] of its vertices. Two simplices arethen said to have the same orientation if the orderingof their vertices differs by an even permutation, andto have the opposite orientation if it differs by an oddpermutation. The (p − q)-faces of a p-simplex σp aredefined by removing q vertices from σp. An orienta-tion of σp induces an orientation of its faces simplyby keeping the vertices of the faces in the same order.Suppose that ω is a 0-form defined over the atoms andlet σ1 = [va, vb] be an ordered atomic bond defined byatoms a and b (cf. Fig. 5). Then, the differential dω(σ1)

of ω at σ1 is

dω(σ1) = ω(vb) − ω(va). (1)

Suppose now that ω is a 1-form defined over theatomic bonds and let σ2 = {va, vb, vc} be a trian-gular cell whose boundary contains the atomic bondsσab = {va, vb}, σbc = {vb, vc} and σca = {vc, va}(cf. Fig. 5). Then, the differential dω(σ2) of ω at σ2 is

dω(σ2) = ±ω(σab) ± ω(σcb) ± ω(σca), (2)

where the + sign is assigned to those 1-cells on theright-hand side that are consistently oriented with σ2

and the − sign is assigned otherwise. The differen-tial of a 2-form is defined likewise. Thus, the differ-ential operator maps p-forms, defined over p-cells, to

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32

4 1

(a)

6

7

5

8

(b)

101112 9

(c)

Fig. 3 CW-complex representation of the body-centered cubic lattice and complex lattice indexing scheme. Elementary faces or 2-cells

(a) (b) (c)

(d) (e) (f)

Fig. 4 CW-complex representation of the body-centered cubic lattice and complex lattice indexing scheme. Elementary volumes or3-cells

(p +1)-forms, defined over (p +1)-cells. The discretedifferential operators thus defined may be regarded asthe discrete counterparts of the familiar grad, curl and

div of vector calculus. In particular: the differential of0-forms is the discrete counterpart of the grad operator;the differential of 1-forms is the discrete counterpart

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Fig. 5 Diagram for thedefinition of the discretedifferential operators ofBCC crystals

of the curl operator; and the differential of 2-forms isthe discrete counterpart of the div operator from vectorcalculus. It is readily verified from the definition of thediscrete differential operators that

d2 = 0, (3)

which is the discrete counterpart of the identities curl ◦grad = 0 and div ◦ curl = 0. In applications, we shalladditionally resort to the dual complex of the crystallattice for representing discrete dislocation lines.

We recall that the p-cells of the dual lattice complexmay be identified with the (n − p)-cells of the primallattice complex (Munkres 1984). For simplicial com-plexes, the barycentric block decomposition (Munkres1984) supplies a particular geometric representation ofthe dual lattice complex.

The block dual to each 3-cell of a cell-complex Cis its barycenter. The closed block dual to any 2-cellconsists of the two line segments joining the barycen-ter of this face with the barycenters of the two 3-cellshaving the former cell as a face (Figs. 6, 7). The blockdual to any 1-cell is in general a non-flat region withflat faces and straight edges, that intersects the formercell at its barycenter. Each of these faces has three ver-texes, the barycenter of the 1-cell, the barycenter ofone of the 2-cells having the 1-simplex as an edge andthe barycenter of the 3-cell having the 2-cell as a face(Fig. 8).

Likewise, elementary dual segments of the sametype are translations of each other, have the same com-plement of neighbors and are arranged as simple Brav-ais lattices. The dual block to any 0-cell of C consists ofthe volume surrounded by the dual to the 1-cells havingthe former 0-cell in common. In Fig. 9 we have plottedthe dual to a 0-cell whose edges are elementary dualsegments.

Fig. 6 Pair of elementary dual segments

Fig. 7 Elementary segments of the dual lattice of BCC crystals

Fig. 8 Group of elementary faces that constitute the dual cell toan atomic bond of the BCC lattice

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Fig. 9 Elementary dual segments that form the edges of the dualto a 0-cell of the BCC lattice

2.2 Eigendeformation theory of discrete latticedislocations

It is possible to fashion a theory of discrete dislocationsin crystals from the classical theory of eigendeforma-tions (cf., e.g., Mura 1987). In the present setting, thetheory rests on the fundamental property of crystalsthat certain uniform deformations leave the crystal lat-tice unchanged and, hence, should cost no energy. Theentire class of lattice-invariant deformations is charac-terized by a classical theorem of Ericksen (1979) as theset of unimodular affine mappings with integer latticecoordinates. An energy that satisfies this property byconstruction is

E(u, β) = 1

2〈B(du − β), (du − β)〉

≡ 1

2

∑

e1∈E1(C)

∑

e′1∈E1(C)

〈B(e1, e′1)(du(e1) − β(e1)), (du(e′

1) − β(e′1))〉

(4)

where the sums take place over the atomic bonds ofthe crystal lattice and: u(e0) is the atomic displace-ment of atom e0; du(e1) is the deformation of atomicbond e1;β(e1) is the eigendeformation at bond e1; andB(e1, e′

1) are bond-wise force constants. In (4), thelocal values β(e1) of the eigendeformation field areconstrained to defining lattice-invariant deformations.By this restriction and the form of the energy (4), uni-form lattice-invariant deformations du cost no energy,as desired. We note that, owing to the discrete nature ofthe set of lattice-invariant deformations, the energy (4)is strongly nonlinear. In particular, the reduce energy

E(u) = infβ

E(u, β) (5)

is piecewise quadratic with zero-energy wells at all uni-form lattice-invariant deformations.

The primary slip systems of a BCC lattice are tabu-lated in Table 1. The atomic bond (Fig. 2b) were chosenso as to contain the 1

2 〈111〉 slip directions and 110 slipplanes. We note from (4) that energy vanishes if theeigendeformations are compatible, i.e., if β = dv forsome atomic displacement field v. Indeed, in that casethe energy is minimized for u = v and the minimumenergy is zero. Hence, the energy at equilibrium, orstored energy, i.e., the energy that remains stored inthe crystal when the displacement field is equilibrated,can only depend on the degree of incompatibility of theeigendeformations. A measure of that incompatibilityis provided by the discrete dislocation density

α = dβ, (6)

which may be regarded as the discrete curl of theeigendeformations. Thus, α provides a discrete coun-terpart of Nye’s dislocation density tensor field (Nye1953). Since the eigendeformations β are defined onthe atomic bonds, it follows that the discrete disloca-tions of BCC crystals are defined on the triangular cellsof the lattice, i.e., the discrete dislocation density α

assigns a Burgers vector to every 2-cell of the BCC lat-tice. From the fundamental property (3) of the discretedifferential operator if follows that

dα = 0, (7)

i.e., the net Burgers vector of the discrete dislocationdensity must vanish. This condition is the discrete coun-terpart of the classical divergence-free property of theNye’s dislocation density tensor field, which may inturn be regarded as a conservation of Burgers vectorproperty. A basis for the free-abelian group of the dis-crete dislocation densities generated by the eigende-formations β is shown in Fig. 10. In this figure, theelementary loops, or loopons (Shenoy et al. 1999), arerepresented as cycles of edges in the dual lattice cellcomplex (cf. Sect. 2). The loopons represent the dis-location density corresponding to unit slip on a sin-gle cartesian and diagonal primary edges. An arbitrarydiscrete dislocation density may be obtained throughan integer linear combination of dislocation densitiesassociated to elementary loops (Fig. 10). In particular,it follows that it is possible to transition between twoarbitrary dislocation densities by means of a sequence

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Table 1 BCC slip systems in Schmid and Boas’ nomenclature. The unit normal to the slip plane is denoted by m and the unit vectoralong the Burgers vector by s

s. s. A2 A3 A6 B2 B4 B5

√3s [111] [111] [111] [111] [111] [111]√2m (011) (101) (110) (011) (101) (110)

s. s. C1 C3 C5 D1 D4 D6

√3s [111] [111] [111] [111] [111] [111]√2m (011) (101) (110) (011) (101) (110)

(a) (b)

Fig. 10 Elementary dislocation loops, or ‘loopons’, in BCCcrystals. a Cartesian loopon. b Diagonal loopon

of ‘flips’, each flip consisting on the addition or sub-traction of an elementary loopon.

A theorem of Ariza and Ortiz (2005) shows that per-fect lattices, including the graphene lattice (Ariza andOrtiz 2010), possess a Helmholtz–Hodge decomposi-tion. By this discrete Helmholtz–Hodge decomposi-tion, it follows, in particular, that α = 0 if and only ifβ = dv for some displacement field v, i.e., if and onlyif the eigendeformations are compatible. Thus, the dis-crete dislocation density does indeed provide a measureof the incompatibility of the eigendeformations. It alsofollows from the discrete Helmholtz–Hodge decompo-sition that α is determined by β up to an arbitrary dis-placement field. From these properties it may be shown(Ariza and Ortiz 2005)

infu

E(u, β) = E(α), (8)

i.e., that the stored energy of a crystal may be writtenas a function of the discrete dislocation density. By thequadratic dependence of the energy (4) on the displace-ment field it follows that the stored energy must be ofthe form

E(α) = 1

2

∑

l∈Z3

∑

l ′∈Z3

〈Γ (l − l ′)α(l ′), α(l)〉

≡ 1

2〈Γ ∗ α, α〉,

(9)

where the kernel Γ (l) gives the interaction energybetween Burgers vectors (Table 1) at the origin andat position l on the simple Bravais lattice defined bythe triangular cells (Fig. 3). Explicit expressions forthe kernel Γ in terms of the force constants of the lat-tice are given in Ariza and Ortiz (2010). Again we notethat, despite is harmonic appearance, the stored energy(9) is rendered strongly nonlinear by the constraint thatthe local Burgers vectors α(l) must be integer linearcombinations of the basic Burgers vectors (Table 1).This strong nonlinearity renders the determination oflow-energy dislocation structures mathematically chal-lenging.

The DFT (cf., e.g., Babuska et al. 1960; Ariza andOrtiz 2005) provides a computationally convenientalternative representation of the stored energy (9). Inorder to define this representation, we begin by group-ing cells of the same dimension by type (Figs. 2, 3, 4).Thus, cells of the same type are translations of eachother and have the same complement of neighbors, orenvironment. According to this definition, a BCC crys-tal has one type of atoms, seven types of elementarysegments, along each of the cube directions and diago-nal directions, twelve types of elementary areas and sixtypes of elementary volumes. The fundamental prop-erty of cells of the same type is that they are arranged assimple Bravais lattices (Figs. 2, 3, 4). Thus, the atomsof BCC crystals define one simple Bravais lattices, theatomic bonds define seven simple Bravais lattices, thetriangular cells define twelve simple Bravais lattices,and the tetrahedral cells define six simple Bravais lat-tices. We recall that the DFT of a function f defined

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on a simple Bravais lattice Zn is

f (θ) =∑

l∈Zn

f (l)e−iθ ·l , (10)

where the angle variables θ range over [−π, π ]n . TheDFT admits an inverse given by

f (l) = 1

(2π)n

∫

[−π,π ]n

f (θ)eiθ ·l dθ, (11)

and has properties similar to those of the Fourier trans-form, including a discrete Parseval identity and and adiscrete convolution theorem. In the DFT representa-tion, the stored energy (9) takes the form

E(α) = 1

(2π)3

∫

[−π,π ]3

〈Γ (θ)α(θ), α∗(θ)〉 dθ, (12)

where α(θ) is the DFT of the discrete dislocation den-sity α(l) and Γ (θ) is the DFT of the discrete dislocationdensity Γ (l).

3 Discrete dislocation dynamics

In this section, we apply the discrete dislocation modelpresented in the foregoing to the evaluation of kinknucleation energies and kink mobility on 1

2 a〈111〉screw dislocations in BCC Ta, W and Mo crystals. Dis-location kink mobility on screw segments is knownto be the key rate-limiting mechanism for dislocationmotion in BCC crystals. The nucleation and motionof kinks is thermally activated and, therefore, primar-ily controlled by the energy barriers corresponding totransitions between states. All calculations are basedon force constants obtained from (Dai et al. 2006)extension of the Finnis–Sinclair potential (Finnis andSinclair 1984). Expressions for the force constantscorresponding to Embedded-Atom-Method potentialshas been given in Ramasubramaniam et al. (2007),where detailed validation comparisons of dislocationcore energies and structures may also be found.

3.1 Computation of incremental energies

Consider a sequence of dislocation densities αk at dis-crete times tk, k = 0, 1, . . . resulting from Monte Car-lo dynamics. The evolution of the dislocation densityis then governed by the energy differences ΔE =

Ek+1 − Ek between two possible successive configura-tions. In addition, the condition for quasistatic config-urational equilibrium of a discrete dislocation densityis that ΔE ≥ 0 for all elementary flips (Ramasubrama-niam et al. 2007). Incremental energies due to flips are,therefore, of particular interest for purposes of charac-terizing discrete dislocation equilibrium configurationsand evolution.

Starting from (12), a straightforward calculationgives

ΔE = Ek+1 − Ek

= 1

(2π)3

∫

[−π,π ]3

1

2〈Γ (θ)αk+1(θ), α∗

k+1(θ)〉dθ

− 1

(2π)3

∫

[−π,π ]3

1

2〈Γ (θ)αk(θ), α∗

k (θ)〉dθ

= 1

(2π)3

∫

[−π,π ]3

〈Γ (θ)αk(θ),Δα∗(θ)〉dθ

+ 1

(2π)3

∫

[−π,π ]3

1

2〈Γ (θ)Δα(θ),Δα∗(θ)〉dθ

(13)

where we write

Δα = αk+1 − αk (14)

for the active loopon defining the transition underconsideration. Thus, the incremental energy may beexpressed in the form

ΔE = Eint + Eloopon (15)

where the first term, Eint represents the interactionenergy between the initial dislocation distribution αk

and the active loopon Δα and Eloopon is the formationenergy of Δα.

3.2 Case I

Double kink formation and subsequent spreading con-stitutes a fundamental rate-limiting mechanism of plas-tic deformation in BCC crystals. The process starts withthe nucleation of a double kink on a screw dislocationsegment. Once nucleated, the kink separates within itsslip planes by ‘zipping’ along the dislocation segment,in the process furthering the plastic deformation of thecrystal. The geometry of the nucleation of a discretedouble kink by the addition of an elementary loopon

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Fig. 11 Nucleation of a discrete double kink by the addition ofan elementary loopon to an infinite screw discrete dislocation ina BCC lattice.

to an infinite screw discrete dislocation is shown inFig. 11.

Consider, for definiteness, activity in the slip sys-tem C5 (Table 1). Consider, in addition an infinitescrew dislocation defined by the sequence of 2-cells{e2(4, l), e2(8, l), e2(10, l)}. The corresponding Fou-rier representation is

α(8, θ) = 2πδ(θ)eiθ2

α(4, θ) = −2πδ(θ)e−iθ1 (16)

α(10, θ) = 2πδ(θ)

whereas Δα(θ) for the active loopon is

α(3, θ) = e−i(θ1+θ3) − e−i(θ1+2θ3)

α(4, θ) = e−i(2θ1+θ2+2θ3) − e−i(θ1+θ3)

α(5, θ) = e−iθ1 − e−i(θ1+θ3) (17)

α(8, θ) = e−i(θ1+θ3) − eiθ2

α(10, θ) = e−i(θ1+θ2+θ3) − e−iθ3

The incremental energy due to the addition of a dou-ble-kink to a straight dislocation, or kink-pair forma-tion energy, is shown in Fig. 12 as a function of kinkseparation. Following Hirth and Lothe (1968), the dou-ble-kink energy may be expressed as

Etot = 2Efor + Eint (18)

where Efor is the energy of formation associated with anisolated kink, and Eint represents the interaction energybetween the kinks. By recourse to atomistic calcula-tions, Yang et al. (2001) have obtained kink-formationenergies ranging from 0.67 to 1.84 eV for 16 possibledifferent kink configurations in tantalum. By way ofcomparison, the computed discrete-dislocation kink-formation energy in tantalum is 0.9 eV, which is in

Fig. 12 Double-kink formation energy as a function of kinkseparation in BCC Mo, Ta and W

the ballpark of the atomistic simulations of Yang et al.(2001). Similarly, using empirical many-body poten-tials (Xu and Moriarty 1998) calculated the kink-pairformation energy in molybdenum to be 2 eV. Again, thecorresponding discrete dislocation kink-pair nucleationenergy is in good agreement with that value (Fig. 12).

As is evident from Fig. 12, the dependence of thecomputed double-kink formation energy on kink sep-aration agrees remarkably well with the prediction ofelasticity theory (Hirth and Lothe 1968). In particular,the interaction energy is attractive but decays in inverseproportion to the kink separation L I and, therefore, itis negligible for well-separated kinks. It follows fromthese decay properties that the mobility of the individ-ual kinks may be expected to be high once the double-kink width becomes sufficiently large, e.g., of the orderof ten lattice constants. This finding is in agreementwith previous atomistic calculations using an embed-ded atom method empirical potential (Cai et al. 2001).

3.3 Case II

Next, we wish to ascertain the nucleation energy of anelementary double-kink on a screw dislocation contain-ing a pre-existing elementary double kink (Fig. 13). Inparticular, we wish to determine the separation of theincipient double kinks at which their interaction energyis negligible and, consequently, the thermally activatednucleation of the double kinks are statistically indepen-dent events.

The nucleation energy of the second double kink isshown in Fig. 14 as a function of the distance to the

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LI

Fig. 13 Geometry of nucleation of a second elementary doublekink in an infinite screw dislocation containing a pre-existingelementary double kink

Fig. 14 Second double-kink nucleation energy as a function ofdistance to pre-existing elementary double kink in BCC Mo, Taand W

pre-existing elementary kink. It is seen from the fig-ure that the interaction between the two double kinksis attractive and decays rapidly with double-kink sep-aration, becoming ostensibly negligible at distancesgreater than ten lattice distances. Interestingly, theattractive nature of the interaction energy facilitates thenucleation of double kinks in close proximity to eachother, which may in turn promote clustering of doublekinks on screw segments.

Fig. 15 Nucleation energy of the double kink as a function ofthe distance to the pre-existing kink

3.4 Case III

Similarly to case II, next we wish to ascertain the nucle-ation energy of an elementary double-kink on a screwdislocation containing a pre-existing kink. In partic-ular, we wish to determine the distance between thepre-existing kink and the incipient kink at which theirinteraction energy is negligible and, consequently, thethermally activated nucleation of the kink is not influ-enced by the preexisting kink.

The nucleation energy of the double kink is shownin Fig. 15 as a function of the distance to the pre-exist-ing kink. As in Case II, it is evident from the figurethat the interaction between the double kink and thepre-existing kink is attractive and decays rapidly withseparation, becoming ostensibly negligible at distancesgreater than ten lattice distances. As in Case II, it isalso remarkable that the attractive nature of the inter-action energy facilitates the nucleation of double kinksin close proximity to pre-existing kinks, which mayin turn promote clustering of ‘daughter’ double kinksahead of ‘mother’ kinks.

4 Summary and conclusions

We present an application of the discrete-dislocationtheory of Ariza and Ortiz to the characterization of theenergetics of kinks in Mo, Ta and W BCC crystals. Fordefiniteness, we have restricted our attention to threekink and double-kink geometries. In all cases, the tran-sitions are effected by the addition of elementary loops,

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termed loopons by Shenoy et al. (1999). The discretedislocation calculations supply detailed predictions offormation and interaction energies for various double-kink formation and spreading mechanisms as a func-tion of the geometry of the double kinks, including:the dependence of the formation energy of a doublekink on its width; the energy of formation of a doublekink on a screw dislocation containing a pre-existingdouble kink; and energy of formation of a double kinkon a screw dislocation containing a pre-existing sin-gle kink. Within a thermally-activated framework, theattractive nature of the computed interaction energiesbetween kinks and double kinks is expected to facil-itates the nucleation of double kinks in close prox-imity to each other and to pre-existing kinks, thuspromoting clustering of double kinks on screw seg-ments and of ‘daughter’ double kinks ahead of ‘mother’kinks.

The predictions of the discrete dislocation theory arefound to be in good agreement with the full atomisticcalculations based on empirical interatomic potentialsavailable in the literature, including: the calculationsof Yang et al. (2001) of kink-formation energies rang-ing from 0.67 to 1.84 eV for 16 possible differentkink configurations in tantalum; the calculations of (Xuand Moriarty 1998) of the kink-pair formation energyin molybdenum; and previous atomistic calculationsusing an embedded atom method empirical potential(Cai et al. 2001). The great advantage of the discretedislocation theory over direct atomistic simulations isthat the former leads to analytical forms of the inter-action energies of lattice defects that are explicit upto simple quadratures, thus conferring discrete dis-location theory a considerable computational advan-tage. In addition, discrete dislocation theory overcomesmany of the essential difficulties that have traditionallyafflicted linear-elastic dislocation theory. Thus, discretedislocation theory may be regarded as an effective com-promise between linear-elastic dislocation theory andfull atomistic simulations based on empirical poten-tials.

Acknowledgments We gratefully acknowledge the support ofthe Ministerio de Ciencia e Innovación of Spain (DPI2009-14305-C02-01/02) and the support of the Consejería de Inno-vación of Junta de Andalucía (P09-TEP-4493). Support for thisstudy was also provided by the Department of Energy NationalNuclear Security Administration under Award Number DE-FC52-08NA28613 through Caltech’s ASC/PSAAP Center forthe Predictive Modeling and Simulation of High Energy DensityDynamic Response of Materials.

A The discrete Fourier transform

A.1 Definition and fundamental properties

The DFT of f : Zn → R is

f (θ) =∑

l∈Zn

f (l)e−iθ ·l (19)

In addition we have

f (l) = 1

(2π)n

∫

[−π,π ]n

f (θ)e−iθ ·ldnθ (20)

which is the inversion formula for the DFT. It followsfrom this expression that f (−l) is the Fourier-seriescoefficient of f (θ). We have the identity

f ∗ g = f g (21)

which is often referred to as the convolution theorem.Suppose in addition that f, g are square-summable.Then∑

l∈Zn

f (l)g∗(l) = 1

(2π)n

∫

[−π,π ]n

f (θ)g∗(θ)dnθ (22)

which is the Parserval identity for the DFT.

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